From 60a29973ed7923c451878944861da29d17b29fdc Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Niccol=C3=B2=20Laurenti?= Date: Wed, 16 Mar 2022 11:46:02 +0100 Subject: [PATCH 01/71] Add notebooks --- notebook/QCDxQED_DGLAP.ipynb | 470 +++++++++++++++++++++++++++++++++++ notebook/QCDxQED_DGLAP.pdf | Bin 0 -> 93582 bytes notebook/QCDxQED_DGLAP.tex | 67 +++++ notebook/uni-dglap.ipynb | 259 +++++++++++++++++++ 4 files changed, 796 insertions(+) create mode 100644 notebook/QCDxQED_DGLAP.ipynb create mode 100644 notebook/QCDxQED_DGLAP.pdf create mode 100644 notebook/QCDxQED_DGLAP.tex create mode 100644 notebook/uni-dglap.ipynb diff --git a/notebook/QCDxQED_DGLAP.ipynb b/notebook/QCDxQED_DGLAP.ipynb new file mode 100644 index 000000000..9a945cf3b --- /dev/null +++ b/notebook/QCDxQED_DGLAP.ipynb @@ -0,0 +1,470 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": 2, + "id": "e9589ad4", + "metadata": {}, + "outputs": [], + "source": [ + "import sympy" + ] + }, + { + "cell_type": "markdown", + "id": "83c8a5aa", + "metadata": {}, + "source": [ + "## Flavor basis :\n", + "### [u+, c+, t+, d+, s+, b+, g, \\gamma, b-, s-, d-, t-, c-, u-]\n", + "## Singlet basis :\n", + "### [\\Sigma_u, \\Sigma_d, g, \\gamma, V_u, V_d, T_3^d, V_3^d, T_3^u / c+, V_3^u / c-, T_8^d / b+, V_8^d / b-, T_8^u / t+, V_8^u / t-]\n", + "## Intrinsic evolution basis :\n", + "### [g, \\gamma, \\Sigma, \\Delta_\\Sigma, V, \\Delta_V, T_3^d, V_3^d, T_3^u / c+, V_3^u / c-, T_8^d / b+, V_8^d / b-, T_8^u / t+, V_8^u / t-]" + ] + }, + { + "cell_type": "markdown", + "id": "34f7316a", + "metadata": {}, + "source": [ + "### [T_3^u, V_3^u, T_8^d, V_8^d, T_8^u, V_8^u] is: \n", + "### [c+, c-, b+, b-, t+, t-] in nf=3, \n", + "### [T_3^u, V_3^u, b+, b-, t+, t-] in nf=4, \n", + "### [T_3^u, V_3^u, T_8^d, V_8^d, t+, t-] in nf=5, \n", + "### [T_3^u, V_3^u, T_8^d, V_8^d, T_8^u, V_8^u] in nf=6" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "id": "d17e49bd", + "metadata": {}, + "outputs": [], + "source": [ + "#QCD\n", + "Pv, Pp, Pm, Pqq, Pqg, Pgq, Pgg = sympy.symbols(\"P_V P_+ P_- P_qq P_qg P_gq P_gg\")\n", + "nf = sympy.symbols(\"n_f\")" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "id": "48aa792c", + "metadata": {}, + "outputs": [], + "source": [ + "# QED\n", + "Pxv, Pxp, Pxm, Pxqq, Pxqg, Pxgq, Pxgg = sympy.symbols(\"P^x_V P^x_+ P^x_- P^x_qq P^x_qg P^x_gq P^x_gg\")\n", + "Pxqy, Pxyq, Pxyg, Pxgy, Pxyy = sympy.symbols(\"P^x_q\\gamma P^x_\\gamma\\ q P^x_\\gamma\\ g P^x_g\\gamma P^x_\\gamma\\gamma\")\n", + "eu2, ed2, es2 = sympy.symbols(\"e_u^2 e_d^2 e_\\Sigma^2\") # charges" + ] + }, + { + "cell_type": "markdown", + "id": "ea0536a9", + "metadata": {}, + "source": [ + "# Unified Evolution for generic n_f" + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "id": "b48ccd4b", + "metadata": {}, + "outputs": [], + "source": [ + "def theta(x):\n", + " if x>=0 :\n", + " return 1\n", + " else:\n", + " return 0" + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "id": "6c5d10c1", + "metadata": {}, + "outputs": [], + "source": [ + "#QCD\n", + "\n", + "def P_qcd(nf): \n", + " return sympy.Matrix([[Pp , 0, 0, 0, 0, 0, 2 * Pqg, 0, 0, 0, 0, 0, 0, 0], #u+\n", + " [0, theta(nf-4)*Pp, 0, 0, 0, 0, theta(nf-4)*2*Pqg, 0, 0, 0, 0, 0, 0, 0], #c+\n", + " [0, 0, theta(nf-6)*Pp, 0, 0, 0, theta(nf-6)*2*Pqg, 0, 0, 0, 0, 0, 0, 0], #t+\n", + " [0, 0, 0, Pp, 0, 0, 2 * Pqg, 0, 0, 0, 0, 0, 0, 0], #d+\n", + " [0, 0, 0, 0, Pp, 0, 2 * Pqg, 0, 0, 0, 0, 0, 0, 0], #s+\n", + " [0, 0, 0, 0, 0, theta(nf-5)*Pp, theta(nf-5)*2*Pqg, 0, 0, 0, 0, 0, 0, 0], #b+\n", + " [Pgq, theta(nf-4)*Pgq, theta(nf-6)*Pgq, Pgq, Pgq, theta(nf-5)*Pgq, Pgg, 0, 0, 0, 0, 0, 0, 0], #g\n", + " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], #\\gamma\n", + " [0, 0, 0, 0, 0, 0, 0, 0, theta(nf-5)*Pm, 0, 0, 0, 0, 0], #b-\n", + " [0, 0, 0, 0, 0, 0, 0, 0, 0, Pm, 0, 0, 0, 0], #s-\n", + " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, Pm, 0, 0, 0], #d-\n", + " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, theta(nf-6)*Pm, 0, 0], #t-\n", + " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, theta(nf-4)*Pm, 0], #c-\n", + " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, Pm]]) #u-\n", + "\n", + "def Ps_qcd(nf): #\\Sigma_u,\\Sigma_d,g,\\gamma,V_d,V_u,T_3^d,V_3^d,T_3^u,V_3^u,T_8^d,V_8^d,T_8^u,V_8^u\n", + " return sympy.Matrix([[Pqq - Pp, Pqq - Pp, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],#u+\n", + " [theta(nf-4)*(Pqq - Pp),theta(nf-4)*(Pqq - Pp), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],#c+\n", + " [theta(nf-6)*(Pqq - Pp),theta(nf-6)*(Pqq - Pp), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], #t+\n", + " [Pqq - Pp, Pqq - Pp, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], #d+\n", + " [Pqq - Pp, Pqq - Pp, 0, 0, 0,0, 0, 0, 0, 0, 0, 0, 0, 0], #s+\n", + " [theta(nf-5)*(Pqq - Pp),theta(nf-5)*(Pqq - Pp), 0, 0, 0,0, 0, 0, 0, 0, 0, 0, 0, 0], #b+\n", + " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], #g\n", + " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], #\\gamma\n", + " [0, 0, 0, 0, theta(nf-5)*(Pv - Pm), theta(nf-5)*(Pv - Pm), 0, 0, 0, 0, 0, 0, 0, 0], #b-\n", + " [0, 0, 0, 0, Pv - Pm, Pv - Pm, 0, 0, 0, 0, 0, 0, 0, 0], #s-\n", + " [0, 0, 0, 0, Pv - Pm, Pv - Pm, 0, 0, 0, 0, 0, 0, 0, 0], #d-\n", + " [0, 0, 0, 0, theta(nf-6)*(Pv - Pm), theta(nf-6)*(Pv - Pm), 0, 0, 0, 0, 0, 0, 0, 0], #t-\n", + " [0, 0, 0, 0, theta(nf-4)*(Pv - Pm), theta(nf-4)*(Pv - Pm), 0, 0, 0, 0, 0, 0, 0, 0], #c-\n", + " [0, 0, 0, 0, Pv - Pm, Pv - Pm, 0, 0, 0, 0, 0, 0, 0, 0]]) / nf #u-" + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "id": "287b8cd2", + "metadata": {}, + "outputs": [], + "source": [ + "#QED\n", + "\n", + "def P_qed(nf):\n", + " return sympy.Matrix([[eu2 * Pxp, 0, 0, 0, 0, 0, 2 * eu2 * Pxqg, 2 * eu2 * Pxqy, 0, 0, 0, 0, 0, 0], #u+\n", + " [0, theta(nf-4)*eu2 * Pxp, 0, 0, 0, 0, theta(nf-4)*2 * eu2 * Pxqg, theta(nf-4)*2 * eu2 * Pxqy, 0, 0, 0, 0, 0, 0], #c+\n", + " [0, 0, theta(nf-6)*eu2 * Pxp, 0, 0, 0, theta(nf-6)*2 * eu2 * Pxqg, theta(nf-6)*2 * eu2 * Pxqy, 0, 0, 0, 0, 0, 0], #t+\n", + " [0, 0, 0, ed2 * Pxp, 0, 0, 2 * ed2 * Pxqg, 2 * ed2 * Pxqy, 0, 0, 0, 0, 0, 0], #d+\n", + " [0, 0, 0, 0, ed2 * Pxp, 0, 2 * ed2 * Pxqg, 2 * ed2 * Pxqy, 0, 0, 0, 0, 0, 0], #s+\n", + " [0, 0, 0, 0, 0, theta(nf-5)*ed2 * Pxp, theta(nf-5)*2 * ed2 * Pxqg, theta(nf-5)*2 * ed2 * Pxqy, 0, 0, 0, 0, 0, 0], #b+\n", + " [eu2 * Pxgq, theta(nf-4)*eu2 * Pxgq, theta(nf-6)*eu2 * Pxgq, ed2 * Pxgq, ed2 * Pxgq, theta(nf-5)*ed2 * Pxgq, es2 * Pxgg, es2 * Pxgy, 0, 0, 0, 0, 0, 0], #g\n", + " [eu2 * Pxyq, theta(nf-4)*eu2 * Pxyq, theta(nf-6)*eu2 * Pxyq, ed2 * Pxyq, ed2 * Pxyq, theta(nf-5)*ed2 * Pxyq, es2 * Pxyg, es2 * Pxyy, 0, 0, 0, 0, 0, 0], #\\gamma\n", + " [0, 0, 0, 0, 0, 0, 0, 0, theta(nf-5)*ed2 * Pxm, 0, 0, 0, 0, 0], #b-\n", + " [0, 0, 0, 0, 0, 0, 0, 0, 0, ed2 * Pxm, 0, 0, 0, 0],#s-\n", + " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ed2 * Pxm, 0, 0, 0],#d-\n", + " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, theta(nf-6)*eu2 * Pxm, 0, 0], #t-\n", + " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, theta(nf-4)*eu2 * Pxm, 0], #c-\n", + " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, eu2 * Pxm]]) #u-\n", + "def Ps_qed(nf): #\\Sigma_u,\\Sigma_d,g,\\gamma,V_d,V_u,T_3^d,V_3^d,T_3^u,V_3^u,T_8^d,V_8^d,T_8^u,V_8^u\n", + " return sympy.Matrix([[(Pxqq - Pxp)*eu2**2, eu2 * ed2 * (Pxqq - Pxp), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],#u+\n", + " [theta(nf-4)*(Pxqq - Pxp)*eu2**2, theta(nf-4)*eu2 * ed2 * (Pxqq - Pxp), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],#c+\n", + " [theta(nf-6)*(Pxqq - Pxp)*eu2**2, theta(nf-6)*eu2 * ed2 * (Pxqq - Pxp), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], #t+\n", + " [eu2 * ed2 * (Pxqq - Pxp), ed2**2 * (Pxqq - Pxp), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], #d+\n", + " [eu2 * ed2 * (Pxqq - Pxp), ed2**2 * (Pxqq - Pxp), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], #s+\n", + " [theta(nf-5)*eu2 * ed2 * (Pxqq - Pxp), theta(nf-5)*ed2**2 * (Pxqq - Pxp), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], #b+\n", + " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], #g\n", + " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], #\\gamma\n", + " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], #b-\n", + " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], #s-\n", + " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], #d-\n", + " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], #t-\n", + " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], #c-\n", + " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]]) / nf #u-" + ] + }, + { + "cell_type": "code", + "execution_count": 8, + "id": "57bb7421", + "metadata": {}, + "outputs": [], + "source": [ + "def P_uni(nf):\n", + " return P_qcd(nf)+P_qed(nf)\n", + "\n", + "def Ps_uni(nf):\n", + " return Ps_qcd(nf)+Ps_qed(nf)" + ] + }, + { + "cell_type": "code", + "execution_count": 9, + "id": "cd1200b7", + "metadata": {}, + "outputs": [], + "source": [ + "def rot_fl_to_ev(nf):\n", + " if nf==3 :\n", + " nu=1\n", + " nd=2\n", + " if nf==4 :\n", + " nu=2\n", + " nd=2\n", + " if nf==5 :\n", + " nu=2\n", + " nd=3\n", + " if nf==6 :\n", + " nu=3\n", + " nd=3\n", + " return sympy.Matrix([[0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0], #g\n", + " [0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0], #\\gamma\n", + " [1, theta(nf-4), theta(nf-6), 1, 1, theta(nf-5), 0, 0, 0, 0, 0, 0, 0, 0], #\\Sigma\n", + " [nd/nu, theta(nf-4)*nd/nu, theta(nf-6)*nd/nu, -1, -1, -theta(nf-5), 0, 0, 0, 0, 0, 0, 0, 0], #\\Delta_\\Sigma\n", + " [0, 0, 0, 0, 0, 0, 0, 0, theta(nf-5), 1, 1, theta(nf-6), theta(nf-4), 1], #V\n", + " [0, 0, 0, 0, 0, 0, 0, 0, -theta(nf-5), -1, -1, nd/nu*theta(nf-6), nd/nu*theta(nf-4), nd/nu], #\\Delta_V\n", + " [0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0], #T_3^d\n", + " [0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0], #V_3^d\n", + " [theta(nf-4), theta(3-nf)-theta(nf-4), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], #T_3^u / c+\n", + " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, theta(3-nf)-theta(nf-4), theta(nf-4)], #V_3^u / c-\n", + " [0, 0, 0, theta(nf-5), theta(nf-5), theta(4-nf)-2*theta(nf-5), 0, 0, 0, 0, 0, 0, 0, 0], #T_8^d / b+\n", + " [0, 0, 0, 0, 0, 0, 0, 0, theta(4-nf)-2*theta(nf-5), theta(nf-5), theta(nf-5), 0, 0, 0], #V_8^d / b-\n", + " [theta(nf-6), theta(nf-6), theta(5-nf)-2*theta(nf-6), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], #T_8^u / t+\n", + " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, theta(5-nf)-2*theta(nf-6), theta(nf-6), theta(nf-6)]]) #V_8^u / t-" + ] + }, + { + "cell_type": "code", + "execution_count": 10, + "id": "77982ede", + "metadata": {}, + "outputs": [], + "source": [ + "def rot_ev_to_fl(nf):\n", + " return rot_fl_to_ev(nf).inv()" + ] + }, + { + "cell_type": "code", + "execution_count": 11, + "id": "5ab19866", + "metadata": {}, + "outputs": [], + "source": [ + "def rot_sin_to_ev(nf):\n", + " if nf==3 :\n", + " nu=1\n", + " nd=2\n", + " if nf==4 :\n", + " nu=2\n", + " nd=2\n", + " if nf==5 :\n", + " nu=2\n", + " nd=3\n", + " if nf==6 :\n", + " nu=3\n", + " nd=3\n", + " return sympy.Matrix([[0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], #g\n", + " [0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], #\\gamma\n", + " [1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], #Sigma\n", + " [nd/nu, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], #\\Delta_\\Sigma\n", + " [0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0], #V\n", + " [0, 0, 0, 0, -1, nd/nu, 0, 0, 0, 0, 0, 0, 0, 0], #\\Delta_V\n", + " [0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0], #T_3^d\n", + " [0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0], #V_3^d\n", + " [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0], #T_3^u / c+\n", + " [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0], #V_3^u / c-\n", + " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0], #T_8^d / b+\n", + " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0], #V_8^d / b-\n", + " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0], #T_8^u / t+\n", + " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]]) #V_8^u / t-" + ] + }, + { + "cell_type": "code", + "execution_count": 12, + "id": "601fe92c", + "metadata": {}, + "outputs": [], + "source": [ + "def rot_ev_to_sin(nf):\n", + " return rot_sin_to_ev(nf).inv()" + ] + }, + { + "cell_type": "code", + "execution_count": 13, + "id": "ebfaa9ad", + "metadata": {}, + "outputs": [], + "source": [ + "def P_ev(nf):\n", + " res = rot_fl_to_ev(nf) * P_uni(nf) * rot_ev_to_fl(nf) + rot_fl_to_ev(nf) * Ps_uni(nf) * rot_ev_to_sin(nf)\n", + " return res" + ] + }, + { + "cell_type": "code", + "execution_count": 20, + "id": "6f5d3f7c", + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\left[\\begin{array}{cccccccccccccc}P^{x}_{gg} e^{2}_{\\Sigma} + P_{gg} & P^{x}_{g\\gamma} e^{2}_{\\Sigma} & 0.666666666666667 P^{x}_{gq} e^{2}_{d} + 0.333333333333333 P^{x}_{gq} e^{2}_{u} + 1.0 P_{gq} & - 0.333333333333333 P^{x}_{gq} e^{2}_{d} + 0.333333333333333 P^{x}_{gq} e^{2}_{u} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\P^{x}_{\\gamma g} e^{2}_{\\Sigma} & P^{x}_{\\gamma\\gamma} e^{2}_{\\Sigma} & 0.666666666666667 P^{x}_{\\gamma q} e^{2}_{d} + 0.333333333333333 P^{x}_{\\gamma q} e^{2}_{u} & - 0.333333333333333 P^{x}_{\\gamma q} e^{2}_{d} + 0.333333333333333 P^{x}_{\\gamma q} e^{2}_{u} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\4 P^{x}_{qg} e^{2}_{d} + 2 P^{x}_{qg} e^{2}_{u} + 6 P_{qg} & 4 P^{x}_{q\\gamma} e^{2}_{d} + 2 P^{x}_{q\\gamma} e^{2}_{u} & 0.666666666666667 P^{x}_{+} e^{2}_{d} + 0.333333333333333 P^{x}_{+} e^{2}_{u} + 1.0 P_{qq} + 0.444444444444444 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.444444444444444 e^{2}_{d} e^{2}_{u} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.111111111111111 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & - 0.333333333333333 P^{x}_{+} e^{2}_{d} + 0.333333333333333 P^{x}_{+} e^{2}_{u} - 0.222222222222222 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.111111111111111 e^{2}_{d} e^{2}_{u} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.111111111111111 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\- 4 P^{x}_{qg} e^{2}_{d} + 4.0 P^{x}_{qg} e^{2}_{u} & - 4 P^{x}_{q\\gamma} e^{2}_{d} + 4.0 P^{x}_{q\\gamma} e^{2}_{u} & - 0.666666666666667 P^{x}_{+} e^{2}_{d} + 0.666666666666667 P^{x}_{+} e^{2}_{u} - 0.444444444444444 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.222222222222222 e^{2}_{d} e^{2}_{u} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.222222222222222 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & 0.333333333333333 P^{x}_{+} e^{2}_{d} + 0.666666666666667 P^{x}_{+} e^{2}_{u} + 1.0 P_{+} + 0.222222222222222 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) - 0.444444444444444 e^{2}_{d} e^{2}_{u} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.222222222222222 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0.666666666666667 P^{x}_{-} e^{2}_{d} + 0.333333333333333 P^{x}_{-} e^{2}_{u} + 1.0 P_{V} & - 0.333333333333333 P^{x}_{-} e^{2}_{d} + 0.333333333333333 P^{x}_{-} e^{2}_{u} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & - 0.666666666666667 P^{x}_{-} e^{2}_{d} + 0.666666666666667 P^{x}_{-} e^{2}_{u} & 0.333333333333333 P^{x}_{-} e^{2}_{d} + 0.666666666666667 P^{x}_{-} e^{2}_{u} + 1.0 P_{-} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{+} e^{2}_{d} + 1.0 P_{+} & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{-} e^{2}_{d} + 1.0 P_{-} & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\end{array}\\right]$" + ], + "text/plain": [ + "Matrix([\n", + "[ P^x_gg*e_\\Sigma^2 + P_gg, P^x_g\\gamma*e_\\Sigma^2, 0.666666666666667*P^x_gq*e_d^2 + 0.333333333333333*P^x_gq*e_u^2 + 1.0*P_gq, -0.333333333333333*P^x_gq*e_d^2 + 0.333333333333333*P^x_gq*e_u^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ P^x_\\gamma g*e_\\Sigma^2, P^x_\\gamma\\gamma*e_\\Sigma^2, 0.666666666666667*P^x_\\gamma q*e_d^2 + 0.333333333333333*P^x_\\gamma q*e_u^2, -0.333333333333333*P^x_\\gamma q*e_d^2 + 0.333333333333333*P^x_\\gamma q*e_u^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[4*P^x_qg*e_d^2 + 2*P^x_qg*e_u^2 + 6*P_qg, 4*P^x_q\\gamma*e_d^2 + 2*P^x_q\\gamma*e_u^2, 0.666666666666667*P^x_+*e_d^2 + 0.333333333333333*P^x_+*e_u^2 + 1.0*P_qq + 0.444444444444444*e_d^2**2*(-P^x_+ + P^x_qq) + 0.444444444444444*e_d^2*e_u^2*(-P^x_+ + P^x_qq) + 0.111111111111111*e_u^2**2*(-P^x_+ + P^x_qq), -0.333333333333333*P^x_+*e_d^2 + 0.333333333333333*P^x_+*e_u^2 - 0.222222222222222*e_d^2**2*(-P^x_+ + P^x_qq) + 0.111111111111111*e_d^2*e_u^2*(-P^x_+ + P^x_qq) + 0.111111111111111*e_u^2**2*(-P^x_+ + P^x_qq), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ -4*P^x_qg*e_d^2 + 4.0*P^x_qg*e_u^2, -4*P^x_q\\gamma*e_d^2 + 4.0*P^x_q\\gamma*e_u^2, -0.666666666666667*P^x_+*e_d^2 + 0.666666666666667*P^x_+*e_u^2 - 0.444444444444444*e_d^2**2*(-P^x_+ + P^x_qq) + 0.222222222222222*e_d^2*e_u^2*(-P^x_+ + P^x_qq) + 0.222222222222222*e_u^2**2*(-P^x_+ + P^x_qq), 0.333333333333333*P^x_+*e_d^2 + 0.666666666666667*P^x_+*e_u^2 + 1.0*P_+ + 0.222222222222222*e_d^2**2*(-P^x_+ + P^x_qq) - 0.444444444444444*e_d^2*e_u^2*(-P^x_+ + P^x_qq) + 0.222222222222222*e_u^2**2*(-P^x_+ + P^x_qq), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0.666666666666667*P^x_-*e_d^2 + 0.333333333333333*P^x_-*e_u^2 + 1.0*P_V, -0.333333333333333*P^x_-*e_d^2 + 0.333333333333333*P^x_-*e_u^2, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, -0.666666666666667*P^x_-*e_d^2 + 0.666666666666667*P^x_-*e_u^2, 0.333333333333333*P^x_-*e_d^2 + 0.666666666666667*P^x_-*e_u^2 + 1.0*P_-, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 1.0*P^x_+*e_d^2 + 1.0*P_+, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 1.0*P^x_-*e_d^2 + 1.0*P_-, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]])" + ] + }, + "execution_count": 20, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "P_ev(3)" + ] + }, + { + "cell_type": "code", + "execution_count": 17, + "id": "28ecee8c", + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\left[\\begin{array}{cccccccccccccc}P^{x}_{gg} e^{2}_{\\Sigma} + P_{gg} & P^{x}_{g\\gamma} e^{2}_{\\Sigma} & 0.5 P^{x}_{gq} e^{2}_{d} + 0.5 P^{x}_{gq} e^{2}_{u} + 1.0 P_{gq} & - 0.5 P^{x}_{gq} e^{2}_{d} + 0.5 P^{x}_{gq} e^{2}_{u} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\P^{x}_{\\gamma g} e^{2}_{\\Sigma} & P^{x}_{\\gamma\\gamma} e^{2}_{\\Sigma} & 0.5 P^{x}_{\\gamma q} e^{2}_{d} + 0.5 P^{x}_{\\gamma q} e^{2}_{u} & - 0.5 P^{x}_{\\gamma q} e^{2}_{d} + 0.5 P^{x}_{\\gamma q} e^{2}_{u} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\4 P^{x}_{qg} e^{2}_{d} + 4 P^{x}_{qg} e^{2}_{u} + 8 P_{qg} & 4 P^{x}_{q\\gamma} e^{2}_{d} + 4 P^{x}_{q\\gamma} e^{2}_{u} & 0.5 P^{x}_{+} e^{2}_{d} + 0.5 P^{x}_{+} e^{2}_{u} + 1.0 P_{qq} + 0.25 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.5 e^{2}_{d} e^{2}_{u} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.25 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & - 0.5 P^{x}_{+} e^{2}_{d} + 0.5 P^{x}_{+} e^{2}_{u} - 0.25 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.25 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\- 4 P^{x}_{qg} e^{2}_{d} + 4.0 P^{x}_{qg} e^{2}_{u} & - 4 P^{x}_{q\\gamma} e^{2}_{d} + 4.0 P^{x}_{q\\gamma} e^{2}_{u} & - 0.5 P^{x}_{+} e^{2}_{d} + 0.5 P^{x}_{+} e^{2}_{u} - 0.25 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.25 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & 0.5 P^{x}_{+} e^{2}_{d} + 0.5 P^{x}_{+} e^{2}_{u} + 1.0 P_{+} + 0.25 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) - 0.5 e^{2}_{d} e^{2}_{u} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.25 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0.5 P^{x}_{-} e^{2}_{d} + 0.5 P^{x}_{-} e^{2}_{u} + 1.0 P_{V} & - 0.5 P^{x}_{-} e^{2}_{d} + 0.5 P^{x}_{-} e^{2}_{u} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & - 0.5 P^{x}_{-} e^{2}_{d} + 0.5 P^{x}_{-} e^{2}_{u} & 0.5 P^{x}_{-} e^{2}_{d} + 0.5 P^{x}_{-} e^{2}_{u} + 1.0 P_{-} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{+} e^{2}_{d} + 1.0 P_{+} & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{-} e^{2}_{d} + 1.0 P_{-} & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{+} e^{2}_{u} + 1.0 P_{+} & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{-} e^{2}_{u} + 1.0 P_{-} & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\end{array}\\right]$" + ], + "text/plain": [ + "Matrix([\n", + "[ P^x_gg*e_\\Sigma^2 + P_gg, P^x_g\\gamma*e_\\Sigma^2, 0.5*P^x_gq*e_d^2 + 0.5*P^x_gq*e_u^2 + 1.0*P_gq, -0.5*P^x_gq*e_d^2 + 0.5*P^x_gq*e_u^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ P^x_\\gamma g*e_\\Sigma^2, P^x_\\gamma\\gamma*e_\\Sigma^2, 0.5*P^x_\\gamma q*e_d^2 + 0.5*P^x_\\gamma q*e_u^2, -0.5*P^x_\\gamma q*e_d^2 + 0.5*P^x_\\gamma q*e_u^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[4*P^x_qg*e_d^2 + 4*P^x_qg*e_u^2 + 8*P_qg, 4*P^x_q\\gamma*e_d^2 + 4*P^x_q\\gamma*e_u^2, 0.5*P^x_+*e_d^2 + 0.5*P^x_+*e_u^2 + 1.0*P_qq + 0.25*e_d^2**2*(-P^x_+ + P^x_qq) + 0.5*e_d^2*e_u^2*(-P^x_+ + P^x_qq) + 0.25*e_u^2**2*(-P^x_+ + P^x_qq), -0.5*P^x_+*e_d^2 + 0.5*P^x_+*e_u^2 - 0.25*e_d^2**2*(-P^x_+ + P^x_qq) + 0.25*e_u^2**2*(-P^x_+ + P^x_qq), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ -4*P^x_qg*e_d^2 + 4.0*P^x_qg*e_u^2, -4*P^x_q\\gamma*e_d^2 + 4.0*P^x_q\\gamma*e_u^2, -0.5*P^x_+*e_d^2 + 0.5*P^x_+*e_u^2 - 0.25*e_d^2**2*(-P^x_+ + P^x_qq) + 0.25*e_u^2**2*(-P^x_+ + P^x_qq), 0.5*P^x_+*e_d^2 + 0.5*P^x_+*e_u^2 + 1.0*P_+ + 0.25*e_d^2**2*(-P^x_+ + P^x_qq) - 0.5*e_d^2*e_u^2*(-P^x_+ + P^x_qq) + 0.25*e_u^2**2*(-P^x_+ + P^x_qq), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0.5*P^x_-*e_d^2 + 0.5*P^x_-*e_u^2 + 1.0*P_V, -0.5*P^x_-*e_d^2 + 0.5*P^x_-*e_u^2, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, -0.5*P^x_-*e_d^2 + 0.5*P^x_-*e_u^2, 0.5*P^x_-*e_d^2 + 0.5*P^x_-*e_u^2 + 1.0*P_-, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 1.0*P^x_+*e_d^2 + 1.0*P_+, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 1.0*P^x_-*e_d^2 + 1.0*P_-, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 1.0*P^x_+*e_u^2 + 1.0*P_+, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1.0*P^x_-*e_u^2 + 1.0*P_-, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]])" + ] + }, + "execution_count": 17, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "P_ev(4)" + ] + }, + { + "cell_type": "code", + "execution_count": 18, + "id": "0d6f55c2", + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\left[\\begin{array}{cccccccccccccc}P^{x}_{gg} e^{2}_{\\Sigma} + P_{gg} & P^{x}_{g\\gamma} e^{2}_{\\Sigma} & 0.6 P^{x}_{gq} e^{2}_{d} + 0.4 P^{x}_{gq} e^{2}_{u} + 1.0 P_{gq} & - 0.4 P^{x}_{gq} e^{2}_{d} + 0.4 P^{x}_{gq} e^{2}_{u} + 5.55111512312578 \\cdot 10^{-17} P_{gq} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\P^{x}_{\\gamma g} e^{2}_{\\Sigma} & P^{x}_{\\gamma\\gamma} e^{2}_{\\Sigma} & 0.6 P^{x}_{\\gamma q} e^{2}_{d} + 0.4 P^{x}_{\\gamma q} e^{2}_{u} & - 0.4 P^{x}_{\\gamma q} e^{2}_{d} + 0.4 P^{x}_{\\gamma q} e^{2}_{u} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\6 P^{x}_{qg} e^{2}_{d} + 4 P^{x}_{qg} e^{2}_{u} + 10 P_{qg} & 6 P^{x}_{q\\gamma} e^{2}_{d} + 4 P^{x}_{q\\gamma} e^{2}_{u} & 0.6 P^{x}_{+} e^{2}_{d} + 0.4 P^{x}_{+} e^{2}_{u} + 1.0 P_{qq} + 0.36 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.48 e^{2}_{d} e^{2}_{u} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.16 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & - 0.4 P^{x}_{+} e^{2}_{d} + 0.4 P^{x}_{+} e^{2}_{u} + 5.55111512312578 \\cdot 10^{-17} P_{+} - 0.24 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.08 e^{2}_{d} e^{2}_{u} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.16 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\- 6 P^{x}_{qg} e^{2}_{d} + 6.0 P^{x}_{qg} e^{2}_{u} & - 6 P^{x}_{q\\gamma} e^{2}_{d} + 6.0 P^{x}_{q\\gamma} e^{2}_{u} & - 0.6 P^{x}_{+} e^{2}_{d} + 0.6 P^{x}_{+} e^{2}_{u} + 5.55111512312578 \\cdot 10^{-17} P_{qq} - 0.36 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.12 e^{2}_{d} e^{2}_{u} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.24 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & 0.4 P^{x}_{+} e^{2}_{d} + 0.6 P^{x}_{+} e^{2}_{u} + 1.0 P_{+} + 0.24 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) - 0.48 e^{2}_{d} e^{2}_{u} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.24 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0.6 P^{x}_{-} e^{2}_{d} + 0.4 P^{x}_{-} e^{2}_{u} + 1.0 P_{V} & - 0.4 P^{x}_{-} e^{2}_{d} + 0.4 P^{x}_{-} e^{2}_{u} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & - 0.6 P^{x}_{-} e^{2}_{d} + 0.6 P^{x}_{-} e^{2}_{u} - 1.11022302462516 \\cdot 10^{-16} P_{-} + 1.11022302462516 \\cdot 10^{-16} P_{V} & 0.4 P^{x}_{-} e^{2}_{d} + 0.6 P^{x}_{-} e^{2}_{u} + 1.0 P_{-} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{+} e^{2}_{d} + 1.0 P_{+} & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{-} e^{2}_{d} + 1.0 P_{-} & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{+} e^{2}_{u} + 1.0 P_{+} & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{-} e^{2}_{u} + 1.0 P_{-} & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{+} e^{2}_{d} + 1.0 P_{+} & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{-} e^{2}_{d} + 1.0 P_{-} & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\end{array}\\right]$" + ], + "text/plain": [ + "Matrix([\n", + "[ P^x_gg*e_\\Sigma^2 + P_gg, P^x_g\\gamma*e_\\Sigma^2, 0.6*P^x_gq*e_d^2 + 0.4*P^x_gq*e_u^2 + 1.0*P_gq, -0.4*P^x_gq*e_d^2 + 0.4*P^x_gq*e_u^2 + 5.55111512312578e-17*P_gq, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ P^x_\\gamma g*e_\\Sigma^2, P^x_\\gamma\\gamma*e_\\Sigma^2, 0.6*P^x_\\gamma q*e_d^2 + 0.4*P^x_\\gamma q*e_u^2, -0.4*P^x_\\gamma q*e_d^2 + 0.4*P^x_\\gamma q*e_u^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[6*P^x_qg*e_d^2 + 4*P^x_qg*e_u^2 + 10*P_qg, 6*P^x_q\\gamma*e_d^2 + 4*P^x_q\\gamma*e_u^2, 0.6*P^x_+*e_d^2 + 0.4*P^x_+*e_u^2 + 1.0*P_qq + 0.36*e_d^2**2*(-P^x_+ + P^x_qq) + 0.48*e_d^2*e_u^2*(-P^x_+ + P^x_qq) + 0.16*e_u^2**2*(-P^x_+ + P^x_qq), -0.4*P^x_+*e_d^2 + 0.4*P^x_+*e_u^2 + 5.55111512312578e-17*P_+ - 0.24*e_d^2**2*(-P^x_+ + P^x_qq) + 0.08*e_d^2*e_u^2*(-P^x_+ + P^x_qq) + 0.16*e_u^2**2*(-P^x_+ + P^x_qq), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ -6*P^x_qg*e_d^2 + 6.0*P^x_qg*e_u^2, -6*P^x_q\\gamma*e_d^2 + 6.0*P^x_q\\gamma*e_u^2, -0.6*P^x_+*e_d^2 + 0.6*P^x_+*e_u^2 + 5.55111512312578e-17*P_qq - 0.36*e_d^2**2*(-P^x_+ + P^x_qq) + 0.12*e_d^2*e_u^2*(-P^x_+ + P^x_qq) + 0.24*e_u^2**2*(-P^x_+ + P^x_qq), 0.4*P^x_+*e_d^2 + 0.6*P^x_+*e_u^2 + 1.0*P_+ + 0.24*e_d^2**2*(-P^x_+ + P^x_qq) - 0.48*e_d^2*e_u^2*(-P^x_+ + P^x_qq) + 0.24*e_u^2**2*(-P^x_+ + P^x_qq), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0.6*P^x_-*e_d^2 + 0.4*P^x_-*e_u^2 + 1.0*P_V, -0.4*P^x_-*e_d^2 + 0.4*P^x_-*e_u^2, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, -0.6*P^x_-*e_d^2 + 0.6*P^x_-*e_u^2 - 1.11022302462516e-16*P_- + 1.11022302462516e-16*P_V, 0.4*P^x_-*e_d^2 + 0.6*P^x_-*e_u^2 + 1.0*P_-, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 1.0*P^x_+*e_d^2 + 1.0*P_+, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 1.0*P^x_-*e_d^2 + 1.0*P_-, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 1.0*P^x_+*e_u^2 + 1.0*P_+, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1.0*P^x_-*e_u^2 + 1.0*P_-, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1.0*P^x_+*e_d^2 + 1.0*P_+, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1.0*P^x_-*e_d^2 + 1.0*P_-, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]])" + ] + }, + "execution_count": 18, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "P_ev(5)" + ] + }, + { + "cell_type": "code", + "execution_count": 19, + "id": "c11fad98", + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\left[\\begin{array}{cccccccccccccc}P^{x}_{gg} e^{2}_{\\Sigma} + P_{gg} & P^{x}_{g\\gamma} e^{2}_{\\Sigma} & 0.5 P^{x}_{gq} e^{2}_{d} + 0.5 P^{x}_{gq} e^{2}_{u} + 1.0 P_{gq} & - 0.5 P^{x}_{gq} e^{2}_{d} + 0.5 P^{x}_{gq} e^{2}_{u} + 5.55111512312578 \\cdot 10^{-17} P_{gq} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\P^{x}_{\\gamma g} e^{2}_{\\Sigma} & P^{x}_{\\gamma\\gamma} e^{2}_{\\Sigma} & 0.5 P^{x}_{\\gamma q} e^{2}_{d} + 0.5 P^{x}_{\\gamma q} e^{2}_{u} & - 0.5 P^{x}_{\\gamma q} e^{2}_{d} + 0.5 P^{x}_{\\gamma q} e^{2}_{u} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\6 P^{x}_{qg} e^{2}_{d} + 6 P^{x}_{qg} e^{2}_{u} + 12 P_{qg} & 6 P^{x}_{q\\gamma} e^{2}_{d} + 6 P^{x}_{q\\gamma} e^{2}_{u} & 0.5 P^{x}_{+} e^{2}_{d} + 0.5 P^{x}_{+} e^{2}_{u} - 1.11022302462516 \\cdot 10^{-16} P_{+} + 1.0 P_{qq} + 0.25 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.5 e^{2}_{d} e^{2}_{u} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.25 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & - 0.5 P^{x}_{+} e^{2}_{d} + 0.5 P^{x}_{+} e^{2}_{u} + 5.55111512312578 \\cdot 10^{-17} P_{+} - 0.25 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.25 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\- 6 P^{x}_{qg} e^{2}_{d} + 6.0 P^{x}_{qg} e^{2}_{u} & - 6 P^{x}_{q\\gamma} e^{2}_{d} + 6.0 P^{x}_{q\\gamma} e^{2}_{u} & - 0.5 P^{x}_{+} e^{2}_{d} + 0.5 P^{x}_{+} e^{2}_{u} + 5.55111512312578 \\cdot 10^{-17} P_{qq} - 0.25 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.25 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & 0.5 P^{x}_{+} e^{2}_{d} + 0.5 P^{x}_{+} e^{2}_{u} + 1.0 P_{+} + 0.25 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) - 0.5 e^{2}_{d} e^{2}_{u} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.25 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0.5 P^{x}_{-} e^{2}_{d} + 0.5 P^{x}_{-} e^{2}_{u} - 1.11022302462516 \\cdot 10^{-16} P_{-} + 1.0 P_{V} & - 0.5 P^{x}_{-} e^{2}_{d} + 0.5 P^{x}_{-} e^{2}_{u} - 5.55111512312578 \\cdot 10^{-17} P_{-} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & - 0.5 P^{x}_{-} e^{2}_{d} + 0.5 P^{x}_{-} e^{2}_{u} - 5.55111512312578 \\cdot 10^{-17} P_{V} & 0.5 P^{x}_{-} e^{2}_{d} + 0.5 P^{x}_{-} e^{2}_{u} + 1.0 P_{-} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{+} e^{2}_{d} + 1.0 P_{+} & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{-} e^{2}_{d} + 1.0 P_{-} & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{+} e^{2}_{u} + 1.0 P_{+} & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{-} e^{2}_{u} + 1.0 P_{-} & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{+} e^{2}_{d} + 1.0 P_{+} & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{-} e^{2}_{d} + 1.0 P_{-} & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{+} e^{2}_{u} + 1.0 P_{+} & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{-} e^{2}_{u} + 1.0 P_{-}\\end{array}\\right]$" + ], + "text/plain": [ + "Matrix([\n", + "[ P^x_gg*e_\\Sigma^2 + P_gg, P^x_g\\gamma*e_\\Sigma^2, 0.5*P^x_gq*e_d^2 + 0.5*P^x_gq*e_u^2 + 1.0*P_gq, -0.5*P^x_gq*e_d^2 + 0.5*P^x_gq*e_u^2 + 5.55111512312578e-17*P_gq, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ P^x_\\gamma g*e_\\Sigma^2, P^x_\\gamma\\gamma*e_\\Sigma^2, 0.5*P^x_\\gamma q*e_d^2 + 0.5*P^x_\\gamma q*e_u^2, -0.5*P^x_\\gamma q*e_d^2 + 0.5*P^x_\\gamma q*e_u^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[6*P^x_qg*e_d^2 + 6*P^x_qg*e_u^2 + 12*P_qg, 6*P^x_q\\gamma*e_d^2 + 6*P^x_q\\gamma*e_u^2, 0.5*P^x_+*e_d^2 + 0.5*P^x_+*e_u^2 - 1.11022302462516e-16*P_+ + 1.0*P_qq + 0.25*e_d^2**2*(-P^x_+ + P^x_qq) + 0.5*e_d^2*e_u^2*(-P^x_+ + P^x_qq) + 0.25*e_u^2**2*(-P^x_+ + P^x_qq), -0.5*P^x_+*e_d^2 + 0.5*P^x_+*e_u^2 + 5.55111512312578e-17*P_+ - 0.25*e_d^2**2*(-P^x_+ + P^x_qq) + 0.25*e_u^2**2*(-P^x_+ + P^x_qq), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ -6*P^x_qg*e_d^2 + 6.0*P^x_qg*e_u^2, -6*P^x_q\\gamma*e_d^2 + 6.0*P^x_q\\gamma*e_u^2, -0.5*P^x_+*e_d^2 + 0.5*P^x_+*e_u^2 + 5.55111512312578e-17*P_qq - 0.25*e_d^2**2*(-P^x_+ + P^x_qq) + 0.25*e_u^2**2*(-P^x_+ + P^x_qq), 0.5*P^x_+*e_d^2 + 0.5*P^x_+*e_u^2 + 1.0*P_+ + 0.25*e_d^2**2*(-P^x_+ + P^x_qq) - 0.5*e_d^2*e_u^2*(-P^x_+ + P^x_qq) + 0.25*e_u^2**2*(-P^x_+ + P^x_qq), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0.5*P^x_-*e_d^2 + 0.5*P^x_-*e_u^2 - 1.11022302462516e-16*P_- + 1.0*P_V, -0.5*P^x_-*e_d^2 + 0.5*P^x_-*e_u^2 - 5.55111512312578e-17*P_-, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, -0.5*P^x_-*e_d^2 + 0.5*P^x_-*e_u^2 - 5.55111512312578e-17*P_V, 0.5*P^x_-*e_d^2 + 0.5*P^x_-*e_u^2 + 1.0*P_-, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 1.0*P^x_+*e_d^2 + 1.0*P_+, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 1.0*P^x_-*e_d^2 + 1.0*P_-, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 1.0*P^x_+*e_u^2 + 1.0*P_+, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1.0*P^x_-*e_u^2 + 1.0*P_-, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1.0*P^x_+*e_d^2 + 1.0*P_+, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1.0*P^x_-*e_d^2 + 1.0*P_-, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1.0*P^x_+*e_u^2 + 1.0*P_+, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1.0*P^x_-*e_u^2 + 1.0*P_-]])" + ] + }, + "execution_count": 19, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "P_ev(6)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "21fe7814", + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3 (ipykernel)", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": 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+\usepackage[utf8]{inputenc} +\usepackage{xcolor} +\usepackage{amsmath} +\usepackage{amssymb} +\usepackage{amsfonts} + +%\usepackage[a4paper,top=3cm,bottom=3cm,left=3cm,right=3cm]{geometry} +\usepackage[a4paper,top=1cm,bottom=1cm,left=0.0cm,right=1cm]{geometry} + + +\title{} +\author{} + +\date{} + +\begin{document} + +\maketitle +\begin{itemize} +\item Singlet sector: + +\begin{equation*} +\mu^2\frac{d}{d\mu^2} +\begin{pmatrix} +g \\ +\gamma \\ +\Sigma \\ +\Delta_\Sigma +\end{pmatrix} += +\begin{pmatrix} + P_{gg}+e_\Sigma^2 \tilde{P}_{gg} & e_\Sigma^2 \tilde{P}_{g\gamma} & P_{gq} + \frac{e_\Sigma^2}{n_f}\tilde{P}_{gq} & 2\frac{n_u}{n_f}\eta^- \tilde{P}_{gq} \\ + e_\Sigma^2 \tilde{P}_{\gamma g} & e_\Sigma^2 \tilde{P}_{\gamma \gamma} & \frac{e_\Sigma^2}{n_f}\tilde{P}_{\gamma q} & 2\frac{n_u}{n_f}\eta^- \tilde{P}_{\gamma q} \\ + 2n_f P_{qg} +2e_\Sigma^2 \tilde{P}_{qg} & 2 e_\Sigma^2 \tilde{P}_{q\gamma} & P_{qq}+ \frac{e_\Sigma^2}{n_f}\tilde{P}_{+} \Bigl(\frac{e_\Sigma^2}{n_f}\Bigr)^2(\tilde{P}_{qq}-\tilde{P}_{+}) & 2\frac{n_u}{n_f}\eta^-\tilde{P}_{+}+2\frac{\eta^-e_\Sigma^2}{n_f^2}(\tilde{P}_{qq}-\tilde{P}_{+}) \\ + 4\frac{n_d}{n_f}\eta^-\tilde{P}_{qg} & 4\frac{n_d}{n_f}\eta^-\tilde{P}_{q\gamma} & \eta^- \tilde{P}_{+} +2\frac{\eta^-e_\Sigma^2}{n_f^2}(\tilde{P}_{qq}-\tilde{P}_{+}) & P_+ +\frac{e_\Delta^2}{n_f}\tilde{P}_{+} +4\frac{n_u n_d}{n_f^2}(\eta^-)^2(\tilde{P}_{qq}-\tilde{P}_{+}) +\end{pmatrix} +\begin{pmatrix} +g \\ +\gamma \\ +\Sigma \\ +\Delta_\Sigma +\end{pmatrix} +\end{equation*} +with $e_\Delta^2=n_u e_d^2+n_d e_u^2$ + +\item Valence sector: +\begin{equation*} +\mu^2\frac{d}{d\mu^2} +\begin{pmatrix} +V \\ +\Delta_V +\end{pmatrix} += +\begin{pmatrix} +P_V+\frac{e_\Sigma^2}{n_f} \tilde{P}_{-} & 2\frac{n_u}{n_f}\eta^- \tilde{P}_{-}\\ + 2\frac{n_d}{n_f}\eta^- \tilde{P}_{-}& P_-+\frac{e_\Delta^2}{n_f} \tilde{P}_{-} +\end{pmatrix} +\begin{pmatrix} +V \\ +\Delta_V +\end{pmatrix} +\end{equation*} + + +\end{itemize} +\end{document} diff --git a/notebook/uni-dglap.ipynb b/notebook/uni-dglap.ipynb new file mode 100644 index 000000000..a29b8dc6f --- /dev/null +++ b/notebook/uni-dglap.ipynb @@ -0,0 +1,259 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "id": "4790b144-ceb5-4cba-87a5-ca2fd37e5912", + "metadata": {}, + "source": [ + "# Unified DGLAP" + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "id": "1d7616d9-f4a3-447a-9ebf-e99db8126ffb", + "metadata": {}, + "outputs": [], + "source": [ + "import sympy" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "id": "cb36381d-57b5-4972-b028-cf4b6300938f", + "metadata": {}, + "outputs": [], + "source": [ + "# QCD\n", + "Pv, Pp, Pm, Pqq, Pqg, Pgq, Pgg = sympy.symbols(\"P_V P_+ P_- P_qq P_qg P_gq P_gg\")\n", + "nf = sympy.symbols(\"n_f\")\n", + "P_qcd = sympy.Array([[Pp, 0, 2 * Pqg, 0, 0, 0],\n", + " [0, Pp, 2 * Pqg, 0, 0, 0],\n", + " [Pgq, Pgq, Pgg, 0, 0, 0],\n", + " [0, 0, 0, 0, 0, 0],\n", + " [0, 0, 0, 0, Pm, 0],\n", + " [0, 0, 0, 0, 0, Pm]])\n", + "Ps_qcd = sympy.Array([[Pqq - Pp, Pqq - Pp, 0, 0, 0, 0],\n", + " [Pqq - Pp, Pqq - Pp, 0, 0, 0, 0],\n", + " [0, 0, 0, 0, 0, 0],\n", + " [0, 0, 0, 0, 0, 0],\n", + " [0, 0, 0, 0, Pv - Pm, Pv - Pm],\n", + " [0, 0, 0, 0, Pv - Pm, Pv - Pm]]) / nf\n", + "# QED\n", + "Pxv, Pxp, Pxm, Pxqq, Pxqg, Pxgq, Pxgg = sympy.symbols(\"P^x_V P^x_+ P^x_- P^x_qq P^x_qg P^x_gq P^x_gg\")\n", + "Pxqy, Pxyq, Pxyg, Pxgy, Pxyy = sympy.symbols(\"P^x_q\\gamma P^x_\\gamma\\q P^x_\\gamma\\g P^x_g\\gamma P^x_\\gamma\\gamma\")\n", + "eu2, ed2, es2 = sympy.symbols(\"e_u^2 e_d^2 e_\\Sigma^2\") # charges\n", + "P_qed = sympy.Array([[eu2 * Pxp, 0, 2 * eu2 * Pxqg, 2 * eu2 * Pxqy, 0, 0],\n", + " [0, ed2 * Pxp, 2 * ed2 * Pxqg, 2 * ed2 * Pxqy, 0, 0],\n", + " [eu2 * Pxgq, ed2 * Pxgq, es2 * Pxgg, es2 * Pxgy, 0, 0],\n", + " [eu2 * Pxyq, ed2 * Pxyq, es2 * Pxyg, es2 * Pxyy, 0, 0],\n", + " [0, 0, 0, 0, ed2 * Pxm, 0],\n", + " [0, 0, 0, 0, 0, eu2 * Pxm]])\n", + "Ps_qed = sympy.Array([[eu2**2 * (Pxqq - Pxp), eu2 * ed2 * (Pxqq - Pxp), 0, 0, 0, 0],\n", + " [eu2 * ed2 * (Pxqq - Pxp), ed2**2 * (Pxqq - Pxp), 0, 0, 0, 0],\n", + " [0, 0, 0, 0, 0, 0],\n", + " [0, 0, 0, 0, 0, 0],\n", + " [0, 0, 0, 0, 0, 0],\n", + " [0, 0, 0, 0, 0, 0]]) / nf" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "id": "1be60cdd-e84d-4230-ba24-cbe7c59b86b4", + "metadata": {}, + "outputs": [], + "source": [ + "rot3 = sympy.Array([[1, 1, 1, 0, 0, 0, 0, 0],\n", + " [2, -1, -1, 0, 0, 0, 0, 0],\n", + " [0, 0, 0, 1, 1, 1, 0, 0],\n", + " [0, 0, 0, 0, 1, -1, 0, 0],\n", + " [0, 1, -1, 0, 0, 0, 0, 0],\n", + " [0, 0, 0, 2, -1, -1, 0, 0],\n", + " [0, 0, 0, 0, 0, 0, 1, 0],\n", + " [0, 0, 0, 0, 0, 0, 0, 1]])" + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "id": "bec96505-4cdb-4c5e-b107-6d3abdaaf2e8", + "metadata": {}, + "outputs": [], + "source": [ + "P3 = {}\n", + "ns, s, qed, qcd = \"ns\", \"s\", \"qed\", \"qcd\"\n", + "P3[ns, qcd] = sympy.Array.zeros(8,8).as_mutable()\n", + "P3[ns, qed] = sympy.Array.zeros(8,8)\n", + "P3[s, qcd] = sympy.Array.zeros(8,8)\n", + "P3[s, qed] = sympy.Array.zeros(8,8)" + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "id": "82dc84f7-fa49-443e-a36b-7e241038b7a7", + "metadata": {}, + "outputs": [], + "source": [ + "P3[ns, qcd][0, 0] = Pxyy" + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "id": "ea5e0bfb-d3ea-4ec5-aeb3-ca11a1f5ddad", + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\left[\\begin{matrix}P^{x}_{\\gamma\\gamma} & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\end{matrix}\\right]$" + ], + "text/plain": [ + "[[P^x_\\gamma\\gamma, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0]]" + ] + }, + "execution_count": 7, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "P3[ns, qcd]" + ] + }, + { + "cell_type": "code", + "execution_count": 69, + "id": "eea413d6-90b9-4f5d-bf96-c8dbc0d22256", + "metadata": {}, + "outputs": [], + "source": [ + "def pqcd(n):\n", + " res = sympy.Array.zeros(8,8).as_mutable()\n", + " \n", + " res[-2, -2] = Pgg\n", + " res[-1, -1] = 0\n", + " return res" + ] + }, + { + "cell_type": "code", + "execution_count": 70, + "id": "35df2f71-58b0-4e28-bfff-4efd2149abbe", + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\left[\\begin{matrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & P_{gg} & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\end{matrix}\\right]$" + ], + "text/plain": [ + "[[0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, P_gg, 0], [0, 0, 0, 0, 0, 0, 0, 0]]" + ] + }, + "execution_count": 70, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "pqcd(3)" + ] + }, + { + "cell_type": "code", + "execution_count": 71, + "id": "4d2a139a-4c02-47c8-a79d-8b74e29998f0", + "metadata": {}, + "outputs": [], + "source": [ + "def pqed(n):\n", + " res = sympy.Array.zeros(8,8).as_mutable()\n", + " \n", + " res[-2, -2] = Pxgg\n", + " res[-1, -1] = Pxyy\n", + " return res" + ] + }, + { + "cell_type": "code", + "execution_count": 72, + "id": "2daa47a5-e000-4d40-90f2-1cb11863e871", + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\left[\\begin{matrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & P^{x}_{gg} & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & P^{x}_{\\gamma\\gamma}\\end{matrix}\\right]$" + ], + "text/plain": [ + "[[0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, P^x_gg, 0], [0, 0, 0, 0, 0, 0, 0, P^x_\\gamma\\gamma]]" + ] + }, + "execution_count": 72, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "pqed(3)" + ] + }, + { + "cell_type": "code", + "execution_count": 74, + "id": "dd594f07-85d0-48c7-940b-a47a4e0ed75d", + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\left[\\begin{matrix}1 & 1 & 1 & 0 & 0 & 0 & 0 & 0\\\\2 & -1 & -1 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 1 & 1 & 1 & 0 & 0\\\\0 & 0 & 0 & 0 & 1 & -1 & 0 & 0\\\\0 & 1 & -1 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 2 & -1 & -1 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\end{matrix}\\right]$" + ], + "text/plain": [ + "[[1, 1, 1, 0, 0, 0, 0, 0], [2, -1, -1, 0, 0, 0, 0, 0], [0, 0, 0, 1, 1, 1, 0, 0], [0, 0, 0, 0, 1, -1, 0, 0], [0, 1, -1, 0, 0, 0, 0, 0], [0, 0, 0, 2, -1, -1, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 0, 0, 1]]" + ] + }, + "execution_count": 74, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "rot3" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "9764233f-9c53-4860-b7c3-2be12fbec857", + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3 (ipykernel)", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + 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zuWv6GSF_tGM9r?RYoty3YJ85j>cZe+8#;tPd~LW(gV5jpWLfj);;l>KhNkGtJAaNo znB6ZMI;v3ctPhjY@o<6v%&wYmOZt}osM>!dyMM3h$<|x4O-HtWqAe@oZn*H?ji)Eu zZpkL+=m_s1m&1kkE+{?W{wKgYvh{G;y{kh{cHENP6|H?M+7B1KS&KI%Tlpi&u48O9 zT<~VO>NtK?iaJ3=q)$vI$0}O#o>~BWv3_DKT&WPF1S}AdV=E<1*hSHw{!WX zY`5)Urkl;Ux3YCGIp-?f(tm|p2crM-mh@-d1N|2_s@~78kc!!zdRh481jS%69px>Z z|Ly(<1WY%Im#A$4Y*H~rF+(&)H83+WI5b2>GeJZ)H8(IfGB7tpL^U%xHAFrjJTXNv zLo`M;Ff%eZG(<%+K}0q+H!wFcFgHX*H8VLiL_S>#FHB`_XLM*FF*P?Jli*7zeLZTt(YQFrMdvXEb>wtg) z6sn2ni;jpzTlB<0bj4T<#faQK?RF}hNQtz_h^)v7PxvA)3Zf;7gtxGj5-~qpm5Fs| z6%f0XRfRY@t159lSk;L8|AUbDL|m-uWc0F%$mPkZK_)k=rkIhN2gv;!KhLTZxC&)% oWOH Date: Wed, 16 Mar 2022 16:20:53 +0100 Subject: [PATCH 03/71] Fix doc of Singlet basis in QCDxQED_DGLAP.ipynb --- notebook/QCDxQED_DGLAP.ipynb | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/notebook/QCDxQED_DGLAP.ipynb b/notebook/QCDxQED_DGLAP.ipynb index 9a945cf3b..75095534f 100644 --- a/notebook/QCDxQED_DGLAP.ipynb +++ b/notebook/QCDxQED_DGLAP.ipynb @@ -18,7 +18,7 @@ "## Flavor basis :\n", "### [u+, c+, t+, d+, s+, b+, g, \\gamma, b-, s-, d-, t-, c-, u-]\n", "## Singlet basis :\n", - "### [\\Sigma_u, \\Sigma_d, g, \\gamma, V_u, V_d, T_3^d, V_3^d, T_3^u / c+, V_3^u / c-, T_8^d / b+, V_8^d / b-, T_8^u / t+, V_8^u / t-]\n", + "### [\\Sigma_u, \\Sigma_d, g, \\gamma, V_d, V_u, T_3^d, V_3^d, T_3^u / c+, V_3^u / c-, T_8^d / b+, V_8^d / b-, T_8^u / t+, V_8^u / t-]\n", "## Intrinsic evolution basis :\n", "### [g, \\gamma, \\Sigma, \\Delta_\\Sigma, V, \\Delta_V, T_3^d, V_3^d, T_3^u / c+, V_3^u / c-, T_8^d / b+, V_8^d / b-, T_8^u / t+, V_8^u / t-]" ] From 907f983abd02b933121b36296b9327d8214bf048 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Niccol=C3=B2=20Laurenti?= Date: Wed, 16 Mar 2022 16:21:53 +0100 Subject: [PATCH 04/71] Reimplement basis rotation in uni-dglap.ipynb --- notebook/uni-dglap.ipynb | 374 ++++++++++++++++++++++++++++----------- 1 file changed, 275 insertions(+), 99 deletions(-) diff --git a/notebook/uni-dglap.ipynb b/notebook/uni-dglap.ipynb index a29b8dc6f..6af20fe1e 100644 --- a/notebook/uni-dglap.ipynb +++ b/notebook/uni-dglap.ipynb @@ -10,7 +10,7 @@ }, { "cell_type": "code", - "execution_count": 1, + "execution_count": 144, "id": "1d7616d9-f4a3-447a-9ebf-e99db8126ffb", "metadata": {}, "outputs": [], @@ -20,216 +20,392 @@ }, { "cell_type": "code", - "execution_count": 3, + "execution_count": 145, "id": "cb36381d-57b5-4972-b028-cf4b6300938f", "metadata": {}, "outputs": [], "source": [ "# QCD\n", "Pv, Pp, Pm, Pqq, Pqg, Pgq, Pgg = sympy.symbols(\"P_V P_+ P_- P_qq P_qg P_gq P_gg\")\n", - "nf = sympy.symbols(\"n_f\")\n", - "P_qcd = sympy.Array([[Pp, 0, 2 * Pqg, 0, 0, 0],\n", - " [0, Pp, 2 * Pqg, 0, 0, 0],\n", - " [Pgq, Pgq, Pgg, 0, 0, 0],\n", - " [0, 0, 0, 0, 0, 0],\n", - " [0, 0, 0, 0, Pm, 0],\n", - " [0, 0, 0, 0, 0, Pm]])\n", - "Ps_qcd = sympy.Array([[Pqq - Pp, Pqq - Pp, 0, 0, 0, 0],\n", - " [Pqq - Pp, Pqq - Pp, 0, 0, 0, 0],\n", - " [0, 0, 0, 0, 0, 0],\n", - " [0, 0, 0, 0, 0, 0],\n", - " [0, 0, 0, 0, Pv - Pm, Pv - Pm],\n", - " [0, 0, 0, 0, Pv - Pm, Pv - Pm]]) / nf\n", "# QED\n", "Pxv, Pxp, Pxm, Pxqq, Pxqg, Pxgq, Pxgg = sympy.symbols(\"P^x_V P^x_+ P^x_- P^x_qq P^x_qg P^x_gq P^x_gg\")\n", - "Pxqy, Pxyq, Pxyg, Pxgy, Pxyy = sympy.symbols(\"P^x_q\\gamma P^x_\\gamma\\q P^x_\\gamma\\g P^x_g\\gamma P^x_\\gamma\\gamma\")\n", - "eu2, ed2, es2 = sympy.symbols(\"e_u^2 e_d^2 e_\\Sigma^2\") # charges\n", - "P_qed = sympy.Array([[eu2 * Pxp, 0, 2 * eu2 * Pxqg, 2 * eu2 * Pxqy, 0, 0],\n", - " [0, ed2 * Pxp, 2 * ed2 * Pxqg, 2 * ed2 * Pxqy, 0, 0],\n", - " [eu2 * Pxgq, ed2 * Pxgq, es2 * Pxgg, es2 * Pxgy, 0, 0],\n", - " [eu2 * Pxyq, ed2 * Pxyq, es2 * Pxyg, es2 * Pxyy, 0, 0],\n", - " [0, 0, 0, 0, ed2 * Pxm, 0],\n", - " [0, 0, 0, 0, 0, eu2 * Pxm]])\n", - "Ps_qed = sympy.Array([[eu2**2 * (Pxqq - Pxp), eu2 * ed2 * (Pxqq - Pxp), 0, 0, 0, 0],\n", - " [eu2 * ed2 * (Pxqq - Pxp), ed2**2 * (Pxqq - Pxp), 0, 0, 0, 0],\n", - " [0, 0, 0, 0, 0, 0],\n", - " [0, 0, 0, 0, 0, 0],\n", - " [0, 0, 0, 0, 0, 0],\n", - " [0, 0, 0, 0, 0, 0]]) / nf" + "Pxqy, Pxyq, Pxyg, Pxgy, Pxyy = sympy.symbols(\"P^x_q\\gamma P^x_\\gamma\\ q P^x_\\gamma\\ g P^x_g\\gamma P^x_\\gamma\\gamma\")\n", + "eu2, ed2, es2 = sympy.symbols(\"e_u^2 e_d^2 e_\\Sigma^2\") # charges" + ] + }, + { + "cell_type": "markdown", + "id": "bdf7ad8c", + "metadata": {}, + "source": [ + "## Flavor basis :\n", + "### [g, \\gamma, u+, u-, d+, d-, s+, s-, c+, c-, b+, b-, t+, t-]\n", + "## Singlet basis :\n", + "### [g, \\gamma, \\Sigma_u, \\Sigma_d, V_u, V_d, T_3^d, V_3^d, T_3^u / c+, V_3^u / c-, T_8^d / b+, V_8^d / b-, T_8^u / t+, V_8^u / t-]\n", + "## Intrinsic evolution basis :\n", + "### [g, \\gamma, \\Sigma, \\Delta_\\Sigma, V, \\Delta_V, T_3^d, V_3^d, T_3^u / c+, V_3^u / c-, T_8^d / b+, V_8^d / b-, T_8^u / t+, V_8^u / t-]\n", + "### [T_3^u, V_3^u, T_8^d, V_8^d, T_8^u, V_8^u] is:\n", + "### [c+, c-, b+, b-, t+, t-] in nf=3,\n", + "### [T_3^u, V_3^u, b+, b-, t+, t-] in nf=4,\n", + "### [T_3^u, V_3^u, T_8^d, V_8^d, t+, t-] in nf=5,\n", + "### [T_3^u, V_3^u, T_8^d, V_8^d, T_8^u, V_8^u] in nf=6" ] }, { "cell_type": "code", - "execution_count": 4, - "id": "1be60cdd-e84d-4230-ba24-cbe7c59b86b4", + "execution_count": 146, + "id": "9764233f-9c53-4860-b7c3-2be12fbec857", "metadata": {}, "outputs": [], "source": [ - "rot3 = sympy.Array([[1, 1, 1, 0, 0, 0, 0, 0],\n", - " [2, -1, -1, 0, 0, 0, 0, 0],\n", - " [0, 0, 0, 1, 1, 1, 0, 0],\n", - " [0, 0, 0, 0, 1, -1, 0, 0],\n", - " [0, 1, -1, 0, 0, 0, 0, 0],\n", - " [0, 0, 0, 2, -1, -1, 0, 0],\n", - " [0, 0, 0, 0, 0, 0, 1, 0],\n", - " [0, 0, 0, 0, 0, 0, 0, 1]])" + "P = {}\n", + "ns, s, qed, qcd = \"ns\", \"s\", \"qed\", \"qcd\"\n", + "P[ns, qcd] = sympy.Array.zeros(14,14).as_mutable()\n", + "P[ns, qed] = sympy.Array.zeros(14,14).as_mutable()\n", + "P[s, qcd] = sympy.Array.zeros(14,14).as_mutable()\n", + "P[s, qed] = sympy.Array.zeros(14,14).as_mutable()\n", + "\n", + "ei2=[eu2, ed2, ed2, eu2, ed2, eu2]\n", + "\n", + "def P_qcd(nf):\n", + " res = sympy.Matrix.zeros(14,14).as_mutable()\n", + " res[0, 0]=Pgg\n", + " for i in range(1, nf+1):\n", + " res[0, 2*i] = Pgq #g q+\n", + " res[2*i, 0] = 2 * Pqg #q+ g\n", + " res[2*i,2*i] = Pp #q+ q+\n", + " res[1 + 2*i,1 + 2*i] = Pm #q- q-\n", + " return res\n", + "\n", + "def P_qed(nf):\n", + " res = sympy.Matrix.zeros(14,14).as_mutable()\n", + " res[0, 0]=es2*Pxgg\n", + " res[1, 1]=es2*Pxyy\n", + " res[0, 1]=es2*Pxgy\n", + " res[1, 0]=es2*Pxyg\n", + " for i in range(1, nf+1):\n", + " res[0, 2*i] = ei2[i-1]*Pxgq\n", + " res[2*i, 0] = 2*ei2[i-1]*Pxqg\n", + " res[1, 2*i] = ei2[i-1]*Pxyq\n", + " res[2*i, 1] = 2*ei2[i-1]*Pxqy\n", + " res[2*i,2*i] = ei2[i-1]*Pxp \n", + " res[1 + 2*i,1 + 2*i] = ei2[i-1]*Pxm\n", + " return res\n", + "\n", + "def Ps_qcd(nf):\n", + " res = sympy.Matrix.zeros(14,14).as_mutable()\n", + " for i in range(1, nf+1):\n", + " res[2*i, 2] = Pqq - Pp\n", + " res[2*i, 3] = Pqq - Pp\n", + " res[1 + 2*i, 4] = Pv - Pm\n", + " res[1 + 2*i, 5] = Pv - Pm\n", + " return res/nf\n", + "\n", + "def Ps_qed(nf):\n", + " res = sympy.Matrix.zeros(14,14).as_mutable()\n", + " for i in range(1, nf+1):\n", + " res[2*i, 2] = ei2[i-1]*eu2*(Pxqq - Pxp)\n", + " res[2*i, 3] = ei2[i-1]*ed2*(Pxqq - Pxp)\n", + " return res/nf\n", + "\n", + "def P_uni(nf):\n", + " return P_qcd(nf)+P_qed(nf)\n", + "\n", + "def Ps_uni(nf):\n", + " return Ps_qcd(nf)+Ps_qed(nf)" ] }, { "cell_type": "code", - "execution_count": 5, - "id": "bec96505-4cdb-4c5e-b107-6d3abdaaf2e8", + "execution_count": 147, + "id": "3a376dcf", "metadata": {}, "outputs": [], "source": [ - "P3 = {}\n", - "ns, s, qed, qcd = \"ns\", \"s\", \"qed\", \"qcd\"\n", - "P3[ns, qcd] = sympy.Array.zeros(8,8).as_mutable()\n", - "P3[ns, qed] = sympy.Array.zeros(8,8)\n", - "P3[s, qcd] = sympy.Array.zeros(8,8)\n", - "P3[s, qed] = sympy.Array.zeros(8,8)" + "def rot_fl_to_ev(nf):\n", + " nu = int(nf/2)\n", + " nd = nf - nu\n", + " res = sympy.Matrix.zeros(14,14).as_mutable()\n", + " res[0, 0]=1\n", + " res[1, 1]=1\n", + " for i in range(2 + 2*nf, 14): \n", + " res[i,i] = 1\n", + " for i in range(1,nf+1): #Sigma and V\n", + " res[2, 2*i] = 1\n", + " res[4, 1 + 2*i] = 1\n", + " for i in [1, 4, 6]:#loop on up quarks\n", + " if i <= nf:\n", + " res[3, 2*i] = nd/nu\n", + " res[5,1 + 2*i] = nd/nu\n", + " for i in [2, 3, 5]:#loop on down quarks\n", + " if i <= nf:\n", + " res[3, 2*i] = -1\n", + " res[5, 1 + 2*i] = -1\n", + " if nf >= 3 :\n", + " res[6, 4] = 1\n", + " res[6, 6] = -1\n", + " res[7, 5] = 1\n", + " res[7, 7] = -1\n", + " if nf >= 4 :\n", + " res[8, 2] = 1\n", + " res[8, 8] = -1\n", + " res[9, 3] = 1\n", + " res[9, 9] = -1\n", + " if nf >= 5 :\n", + " res[10, 4] = 1\n", + " res[10, 6] = 1\n", + " res[10, 10] = -2\n", + " res[11, 5] = 1\n", + " res[11, 7] = 1\n", + " res[11, 11] = -2\n", + " if nf == 6 :\n", + " res[12, 2] = 1\n", + " res[12, 8] = 1\n", + " res[12, 12] = -2\n", + " res[13, 3] = 1\n", + " res[13, 9] = 1\n", + " res[13, 13] = -2\n", + " return res\n", + "\n", + "def rot_ev_to_fl(nf):\n", + " return rot_fl_to_ev(nf).inv()" + ] + }, + { + "cell_type": "code", + "execution_count": 148, + "id": "b5747379", + "metadata": {}, + "outputs": [], + "source": [ + "def rot_sin_to_ev(nf):\n", + " nu = int(nf/2)\n", + " nd = nf - nu\n", + " res = sympy.Matrix.zeros(14,14).as_mutable()\n", + " res[0, 0]=1\n", + " res[1, 1]=1\n", + " res[2,2]=1\n", + " res[2,3]=1\n", + " res[3,2]=nd/nu\n", + " res[3,3]=-1\n", + " res[4,4]=1\n", + " res[4,5]=1\n", + " res[5,4]=nd/nu\n", + " res[5,5]=-1\n", + " for i in range(6,14):\n", + " res[i,i]=1\n", + " return res\n", + "\n", + "def rot_ev_to_sin(nf):\n", + " return rot_sin_to_ev(nf).inv()" ] }, { "cell_type": "code", - "execution_count": 6, - "id": "82dc84f7-fa49-443e-a36b-7e241038b7a7", + "execution_count": 149, + "id": "45fb3a16", "metadata": {}, "outputs": [], "source": [ - "P3[ns, qcd][0, 0] = Pxyy" + "def P_ev(nf):\n", + " res = rot_fl_to_ev(nf) * P_uni(nf) * rot_ev_to_fl(nf) + rot_fl_to_ev(nf) * Ps_uni(nf) * rot_ev_to_sin(nf)\n", + " return res" ] }, { "cell_type": "code", - "execution_count": 7, - "id": "ea5e0bfb-d3ea-4ec5-aeb3-ca11a1f5ddad", + "execution_count": 150, + "id": "9b5050aa", "metadata": {}, "outputs": [ { "data": { "text/latex": [ - "$\\displaystyle \\left[\\begin{matrix}P^{x}_{\\gamma\\gamma} & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\end{matrix}\\right]$" + "$\\displaystyle \\left[\\begin{array}{cccccccccccccc}P^{x}_{gg} e^{2}_{\\Sigma} + P_{gg} & P^{x}_{g\\gamma} e^{2}_{\\Sigma} & 0.5 P^{x}_{gq} e^{2}_{d} + 0.5 P^{x}_{gq} e^{2}_{u} + 1.0 P_{gq} & - 0.5 P^{x}_{gq} e^{2}_{d} + 0.5 P^{x}_{gq} e^{2}_{u} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\P^{x}_{\\gamma g} e^{2}_{\\Sigma} & P^{x}_{\\gamma\\gamma} e^{2}_{\\Sigma} & 0.5 P^{x}_{\\gamma q} e^{2}_{d} + 0.5 P^{x}_{\\gamma q} e^{2}_{u} & - 0.5 P^{x}_{\\gamma q} e^{2}_{d} + 0.5 P^{x}_{\\gamma q} e^{2}_{u} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\2 P^{x}_{qg} e^{2}_{d} + 2 P^{x}_{qg} e^{2}_{u} + 4 P_{qg} & 2 P^{x}_{q\\gamma} e^{2}_{d} + 2 P^{x}_{q\\gamma} e^{2}_{u} & 0.5 P^{x}_{+} e^{2}_{d} + 0.5 P^{x}_{+} e^{2}_{u} + 1.0 P_{qq} + 0.25 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.5 e^{2}_{d} e^{2}_{u} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.25 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & - 0.5 P^{x}_{+} e^{2}_{d} + 0.5 P^{x}_{+} e^{2}_{u} - 0.25 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.25 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\- 2 P^{x}_{qg} e^{2}_{d} + 2.0 P^{x}_{qg} e^{2}_{u} & - 2 P^{x}_{q\\gamma} e^{2}_{d} + 2.0 P^{x}_{q\\gamma} e^{2}_{u} & - 0.5 P^{x}_{+} e^{2}_{d} + 0.5 P^{x}_{+} e^{2}_{u} - 0.25 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.25 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & 0.5 P^{x}_{+} e^{2}_{d} + 0.5 P^{x}_{+} e^{2}_{u} + 1.0 P_{+} + 0.25 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) - 0.5 e^{2}_{d} e^{2}_{u} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.25 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0.5 P^{x}_{-} e^{2}_{d} + 0.5 P^{x}_{-} e^{2}_{u} + 1.0 P_{V} & - 0.5 P^{x}_{-} e^{2}_{d} + 0.5 P^{x}_{-} e^{2}_{u} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & - 0.5 P^{x}_{-} e^{2}_{d} + 0.5 P^{x}_{-} e^{2}_{u} & 0.5 P^{x}_{-} e^{2}_{d} + 0.5 P^{x}_{-} e^{2}_{u} + 1.0 P_{-} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\end{array}\\right]$" ], "text/plain": [ - "[[P^x_\\gamma\\gamma, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0]]" + "Matrix([\n", + "[ P^x_gg*e_\\Sigma^2 + P_gg, P^x_g\\gamma*e_\\Sigma^2, 0.5*P^x_gq*e_d^2 + 0.5*P^x_gq*e_u^2 + 1.0*P_gq, -0.5*P^x_gq*e_d^2 + 0.5*P^x_gq*e_u^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ P^x_\\gamma g*e_\\Sigma^2, P^x_\\gamma\\gamma*e_\\Sigma^2, 0.5*P^x_\\gamma q*e_d^2 + 0.5*P^x_\\gamma q*e_u^2, -0.5*P^x_\\gamma q*e_d^2 + 0.5*P^x_\\gamma q*e_u^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[2*P^x_qg*e_d^2 + 2*P^x_qg*e_u^2 + 4*P_qg, 2*P^x_q\\gamma*e_d^2 + 2*P^x_q\\gamma*e_u^2, 0.5*P^x_+*e_d^2 + 0.5*P^x_+*e_u^2 + 1.0*P_qq + 0.25*e_d^2**2*(-P^x_+ + P^x_qq) + 0.5*e_d^2*e_u^2*(-P^x_+ + P^x_qq) + 0.25*e_u^2**2*(-P^x_+ + P^x_qq), -0.5*P^x_+*e_d^2 + 0.5*P^x_+*e_u^2 - 0.25*e_d^2**2*(-P^x_+ + P^x_qq) + 0.25*e_u^2**2*(-P^x_+ + P^x_qq), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ -2*P^x_qg*e_d^2 + 2.0*P^x_qg*e_u^2, -2*P^x_q\\gamma*e_d^2 + 2.0*P^x_q\\gamma*e_u^2, -0.5*P^x_+*e_d^2 + 0.5*P^x_+*e_u^2 - 0.25*e_d^2**2*(-P^x_+ + P^x_qq) + 0.25*e_u^2**2*(-P^x_+ + P^x_qq), 0.5*P^x_+*e_d^2 + 0.5*P^x_+*e_u^2 + 1.0*P_+ + 0.25*e_d^2**2*(-P^x_+ + P^x_qq) - 0.5*e_d^2*e_u^2*(-P^x_+ + P^x_qq) + 0.25*e_u^2**2*(-P^x_+ + P^x_qq), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0.5*P^x_-*e_d^2 + 0.5*P^x_-*e_u^2 + 1.0*P_V, -0.5*P^x_-*e_d^2 + 0.5*P^x_-*e_u^2, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, -0.5*P^x_-*e_d^2 + 0.5*P^x_-*e_u^2, 0.5*P^x_-*e_d^2 + 0.5*P^x_-*e_u^2 + 1.0*P_-, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]])" ] }, - "execution_count": 7, + "execution_count": 150, "metadata": {}, "output_type": "execute_result" } ], "source": [ - "P3[ns, qcd]" - ] - }, - { - "cell_type": "code", - "execution_count": 69, - "id": "eea413d6-90b9-4f5d-bf96-c8dbc0d22256", - "metadata": {}, - "outputs": [], - "source": [ - "def pqcd(n):\n", - " res = sympy.Array.zeros(8,8).as_mutable()\n", - " \n", - " res[-2, -2] = Pgg\n", - " res[-1, -1] = 0\n", - " return res" + "P_ev(2)" ] }, { "cell_type": "code", - "execution_count": 70, - "id": "35df2f71-58b0-4e28-bfff-4efd2149abbe", + "execution_count": 151, + "id": "07b1874a", "metadata": {}, "outputs": [ { "data": { "text/latex": [ - "$\\displaystyle \\left[\\begin{matrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & P_{gg} & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\end{matrix}\\right]$" + "$\\displaystyle \\left[\\begin{array}{cccccccccccccc}P^{x}_{gg} e^{2}_{\\Sigma} + P_{gg} & P^{x}_{g\\gamma} e^{2}_{\\Sigma} & 0.666666666666667 P^{x}_{gq} e^{2}_{d} + 0.333333333333333 P^{x}_{gq} e^{2}_{u} + 1.0 P_{gq} & - 0.333333333333333 P^{x}_{gq} e^{2}_{d} + 0.333333333333333 P^{x}_{gq} e^{2}_{u} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\P^{x}_{\\gamma g} e^{2}_{\\Sigma} & P^{x}_{\\gamma\\gamma} e^{2}_{\\Sigma} & 0.666666666666667 P^{x}_{\\gamma q} e^{2}_{d} + 0.333333333333333 P^{x}_{\\gamma q} e^{2}_{u} & - 0.333333333333333 P^{x}_{\\gamma q} e^{2}_{d} + 0.333333333333333 P^{x}_{\\gamma q} e^{2}_{u} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\4 P^{x}_{qg} e^{2}_{d} + 2 P^{x}_{qg} e^{2}_{u} + 6 P_{qg} & 4 P^{x}_{q\\gamma} e^{2}_{d} + 2 P^{x}_{q\\gamma} e^{2}_{u} & 0.666666666666667 P^{x}_{+} e^{2}_{d} + 0.333333333333333 P^{x}_{+} e^{2}_{u} + 1.0 P_{qq} + 0.444444444444444 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.444444444444444 e^{2}_{d} e^{2}_{u} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.111111111111111 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & - 0.333333333333333 P^{x}_{+} e^{2}_{d} + 0.333333333333333 P^{x}_{+} e^{2}_{u} - 0.222222222222222 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.111111111111111 e^{2}_{d} e^{2}_{u} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.111111111111111 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\- 4 P^{x}_{qg} e^{2}_{d} + 4.0 P^{x}_{qg} e^{2}_{u} & - 4 P^{x}_{q\\gamma} e^{2}_{d} + 4.0 P^{x}_{q\\gamma} e^{2}_{u} & - 0.666666666666667 P^{x}_{+} e^{2}_{d} + 0.666666666666667 P^{x}_{+} e^{2}_{u} - 0.444444444444444 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.222222222222222 e^{2}_{d} e^{2}_{u} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.222222222222222 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & 0.333333333333333 P^{x}_{+} e^{2}_{d} + 0.666666666666667 P^{x}_{+} e^{2}_{u} + 1.0 P_{+} + 0.222222222222222 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) - 0.444444444444444 e^{2}_{d} e^{2}_{u} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.222222222222222 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0.666666666666667 P^{x}_{-} e^{2}_{d} + 0.333333333333333 P^{x}_{-} e^{2}_{u} + 1.0 P_{V} & - 0.333333333333333 P^{x}_{-} e^{2}_{d} + 0.333333333333333 P^{x}_{-} e^{2}_{u} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & - 0.666666666666667 P^{x}_{-} e^{2}_{d} + 0.666666666666667 P^{x}_{-} e^{2}_{u} & 0.333333333333333 P^{x}_{-} e^{2}_{d} + 0.666666666666667 P^{x}_{-} e^{2}_{u} + 1.0 P_{-} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{+} e^{2}_{d} + 1.0 P_{+} & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{-} e^{2}_{d} + 1.0 P_{-} & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\end{array}\\right]$" ], "text/plain": [ - "[[0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, P_gg, 0], [0, 0, 0, 0, 0, 0, 0, 0]]" + "Matrix([\n", + "[ P^x_gg*e_\\Sigma^2 + P_gg, P^x_g\\gamma*e_\\Sigma^2, 0.666666666666667*P^x_gq*e_d^2 + 0.333333333333333*P^x_gq*e_u^2 + 1.0*P_gq, -0.333333333333333*P^x_gq*e_d^2 + 0.333333333333333*P^x_gq*e_u^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ P^x_\\gamma g*e_\\Sigma^2, P^x_\\gamma\\gamma*e_\\Sigma^2, 0.666666666666667*P^x_\\gamma q*e_d^2 + 0.333333333333333*P^x_\\gamma q*e_u^2, -0.333333333333333*P^x_\\gamma q*e_d^2 + 0.333333333333333*P^x_\\gamma q*e_u^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[4*P^x_qg*e_d^2 + 2*P^x_qg*e_u^2 + 6*P_qg, 4*P^x_q\\gamma*e_d^2 + 2*P^x_q\\gamma*e_u^2, 0.666666666666667*P^x_+*e_d^2 + 0.333333333333333*P^x_+*e_u^2 + 1.0*P_qq + 0.444444444444444*e_d^2**2*(-P^x_+ + P^x_qq) + 0.444444444444444*e_d^2*e_u^2*(-P^x_+ + P^x_qq) + 0.111111111111111*e_u^2**2*(-P^x_+ + P^x_qq), -0.333333333333333*P^x_+*e_d^2 + 0.333333333333333*P^x_+*e_u^2 - 0.222222222222222*e_d^2**2*(-P^x_+ + P^x_qq) + 0.111111111111111*e_d^2*e_u^2*(-P^x_+ + P^x_qq) + 0.111111111111111*e_u^2**2*(-P^x_+ + P^x_qq), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ -4*P^x_qg*e_d^2 + 4.0*P^x_qg*e_u^2, -4*P^x_q\\gamma*e_d^2 + 4.0*P^x_q\\gamma*e_u^2, -0.666666666666667*P^x_+*e_d^2 + 0.666666666666667*P^x_+*e_u^2 - 0.444444444444444*e_d^2**2*(-P^x_+ + P^x_qq) + 0.222222222222222*e_d^2*e_u^2*(-P^x_+ + P^x_qq) + 0.222222222222222*e_u^2**2*(-P^x_+ + P^x_qq), 0.333333333333333*P^x_+*e_d^2 + 0.666666666666667*P^x_+*e_u^2 + 1.0*P_+ + 0.222222222222222*e_d^2**2*(-P^x_+ + P^x_qq) - 0.444444444444444*e_d^2*e_u^2*(-P^x_+ + P^x_qq) + 0.222222222222222*e_u^2**2*(-P^x_+ + P^x_qq), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0.666666666666667*P^x_-*e_d^2 + 0.333333333333333*P^x_-*e_u^2 + 1.0*P_V, -0.333333333333333*P^x_-*e_d^2 + 0.333333333333333*P^x_-*e_u^2, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, -0.666666666666667*P^x_-*e_d^2 + 0.666666666666667*P^x_-*e_u^2, 0.333333333333333*P^x_-*e_d^2 + 0.666666666666667*P^x_-*e_u^2 + 1.0*P_-, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 1.0*P^x_+*e_d^2 + 1.0*P_+, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 1.0*P^x_-*e_d^2 + 1.0*P_-, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]])" ] }, - "execution_count": 70, + "execution_count": 151, "metadata": {}, "output_type": "execute_result" } ], "source": [ - "pqcd(3)" + "P_ev(3)" ] }, { "cell_type": "code", - "execution_count": 71, - "id": "4d2a139a-4c02-47c8-a79d-8b74e29998f0", + "execution_count": 152, + "id": "c9f75de1", "metadata": {}, - "outputs": [], + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\left[\\begin{array}{cccccccccccccc}P^{x}_{gg} e^{2}_{\\Sigma} + P_{gg} & P^{x}_{g\\gamma} e^{2}_{\\Sigma} & 0.5 P^{x}_{gq} e^{2}_{d} + 0.5 P^{x}_{gq} e^{2}_{u} + 1.0 P_{gq} & - 0.5 P^{x}_{gq} e^{2}_{d} + 0.5 P^{x}_{gq} e^{2}_{u} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\P^{x}_{\\gamma g} e^{2}_{\\Sigma} & P^{x}_{\\gamma\\gamma} e^{2}_{\\Sigma} & 0.5 P^{x}_{\\gamma q} e^{2}_{d} + 0.5 P^{x}_{\\gamma q} e^{2}_{u} & - 0.5 P^{x}_{\\gamma q} e^{2}_{d} + 0.5 P^{x}_{\\gamma q} e^{2}_{u} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\4 P^{x}_{qg} e^{2}_{d} + 4 P^{x}_{qg} e^{2}_{u} + 8 P_{qg} & 4 P^{x}_{q\\gamma} e^{2}_{d} + 4 P^{x}_{q\\gamma} e^{2}_{u} & 0.5 P^{x}_{+} e^{2}_{d} + 0.5 P^{x}_{+} e^{2}_{u} + 1.0 P_{qq} + 0.25 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.5 e^{2}_{d} e^{2}_{u} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.25 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & - 0.5 P^{x}_{+} e^{2}_{d} + 0.5 P^{x}_{+} e^{2}_{u} - 0.25 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.25 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\- 4 P^{x}_{qg} e^{2}_{d} + 4.0 P^{x}_{qg} e^{2}_{u} & - 4 P^{x}_{q\\gamma} e^{2}_{d} + 4.0 P^{x}_{q\\gamma} e^{2}_{u} & - 0.5 P^{x}_{+} e^{2}_{d} + 0.5 P^{x}_{+} e^{2}_{u} - 0.25 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.25 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & 0.5 P^{x}_{+} e^{2}_{d} + 0.5 P^{x}_{+} e^{2}_{u} + 1.0 P_{+} + 0.25 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) - 0.5 e^{2}_{d} e^{2}_{u} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.25 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0.5 P^{x}_{-} e^{2}_{d} + 0.5 P^{x}_{-} e^{2}_{u} + 1.0 P_{V} & - 0.5 P^{x}_{-} e^{2}_{d} + 0.5 P^{x}_{-} e^{2}_{u} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & - 0.5 P^{x}_{-} e^{2}_{d} + 0.5 P^{x}_{-} e^{2}_{u} & 0.5 P^{x}_{-} e^{2}_{d} + 0.5 P^{x}_{-} e^{2}_{u} + 1.0 P_{-} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{+} e^{2}_{d} + 1.0 P_{+} & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{-} e^{2}_{d} + 1.0 P_{-} & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{+} e^{2}_{u} + 1.0 P_{+} & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{-} e^{2}_{u} + 1.0 P_{-} & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\end{array}\\right]$" + ], + "text/plain": [ + "Matrix([\n", + "[ P^x_gg*e_\\Sigma^2 + P_gg, P^x_g\\gamma*e_\\Sigma^2, 0.5*P^x_gq*e_d^2 + 0.5*P^x_gq*e_u^2 + 1.0*P_gq, -0.5*P^x_gq*e_d^2 + 0.5*P^x_gq*e_u^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ P^x_\\gamma g*e_\\Sigma^2, P^x_\\gamma\\gamma*e_\\Sigma^2, 0.5*P^x_\\gamma q*e_d^2 + 0.5*P^x_\\gamma q*e_u^2, -0.5*P^x_\\gamma q*e_d^2 + 0.5*P^x_\\gamma q*e_u^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[4*P^x_qg*e_d^2 + 4*P^x_qg*e_u^2 + 8*P_qg, 4*P^x_q\\gamma*e_d^2 + 4*P^x_q\\gamma*e_u^2, 0.5*P^x_+*e_d^2 + 0.5*P^x_+*e_u^2 + 1.0*P_qq + 0.25*e_d^2**2*(-P^x_+ + P^x_qq) + 0.5*e_d^2*e_u^2*(-P^x_+ + P^x_qq) + 0.25*e_u^2**2*(-P^x_+ + P^x_qq), -0.5*P^x_+*e_d^2 + 0.5*P^x_+*e_u^2 - 0.25*e_d^2**2*(-P^x_+ + P^x_qq) + 0.25*e_u^2**2*(-P^x_+ + P^x_qq), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ -4*P^x_qg*e_d^2 + 4.0*P^x_qg*e_u^2, -4*P^x_q\\gamma*e_d^2 + 4.0*P^x_q\\gamma*e_u^2, -0.5*P^x_+*e_d^2 + 0.5*P^x_+*e_u^2 - 0.25*e_d^2**2*(-P^x_+ + P^x_qq) + 0.25*e_u^2**2*(-P^x_+ + P^x_qq), 0.5*P^x_+*e_d^2 + 0.5*P^x_+*e_u^2 + 1.0*P_+ + 0.25*e_d^2**2*(-P^x_+ + P^x_qq) - 0.5*e_d^2*e_u^2*(-P^x_+ + P^x_qq) + 0.25*e_u^2**2*(-P^x_+ + P^x_qq), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0.5*P^x_-*e_d^2 + 0.5*P^x_-*e_u^2 + 1.0*P_V, -0.5*P^x_-*e_d^2 + 0.5*P^x_-*e_u^2, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, -0.5*P^x_-*e_d^2 + 0.5*P^x_-*e_u^2, 0.5*P^x_-*e_d^2 + 0.5*P^x_-*e_u^2 + 1.0*P_-, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 1.0*P^x_+*e_d^2 + 1.0*P_+, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 1.0*P^x_-*e_d^2 + 1.0*P_-, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 1.0*P^x_+*e_u^2 + 1.0*P_+, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1.0*P^x_-*e_u^2 + 1.0*P_-, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]])" + ] + }, + "execution_count": 152, + "metadata": {}, + "output_type": "execute_result" + } + ], "source": [ - "def pqed(n):\n", - " res = sympy.Array.zeros(8,8).as_mutable()\n", - " \n", - " res[-2, -2] = Pxgg\n", - " res[-1, -1] = Pxyy\n", - " return res" + "P_ev(4)" ] }, { "cell_type": "code", - "execution_count": 72, - "id": "2daa47a5-e000-4d40-90f2-1cb11863e871", + "execution_count": 153, + "id": "ce40d361", "metadata": {}, "outputs": [ { "data": { "text/latex": [ - "$\\displaystyle \\left[\\begin{matrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & P^{x}_{gg} & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & P^{x}_{\\gamma\\gamma}\\end{matrix}\\right]$" + "$\\displaystyle \\left[\\begin{array}{cccccccccccccc}P^{x}_{gg} e^{2}_{\\Sigma} + P_{gg} & P^{x}_{g\\gamma} e^{2}_{\\Sigma} & 0.6 P^{x}_{gq} e^{2}_{d} + 0.4 P^{x}_{gq} e^{2}_{u} + 1.0 P_{gq} & - 0.4 P^{x}_{gq} e^{2}_{d} + 0.4 P^{x}_{gq} e^{2}_{u} + 2.77555756156289 \\cdot 10^{-17} P_{gq} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\P^{x}_{\\gamma g} e^{2}_{\\Sigma} & P^{x}_{\\gamma\\gamma} e^{2}_{\\Sigma} & 0.6 P^{x}_{\\gamma q} e^{2}_{d} + 0.4 P^{x}_{\\gamma q} e^{2}_{u} & - 0.4 P^{x}_{\\gamma q} e^{2}_{d} + 0.4 P^{x}_{\\gamma q} e^{2}_{u} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\6 P^{x}_{qg} e^{2}_{d} + 4 P^{x}_{qg} e^{2}_{u} + 10 P_{qg} & 6 P^{x}_{q\\gamma} e^{2}_{d} + 4 P^{x}_{q\\gamma} e^{2}_{u} & 0.6 P^{x}_{+} e^{2}_{d} + 0.4 P^{x}_{+} e^{2}_{u} + 1.0 P_{qq} + 0.36 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.48 e^{2}_{d} e^{2}_{u} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.16 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & - 0.4 P^{x}_{+} e^{2}_{d} + 0.4 P^{x}_{+} e^{2}_{u} + 2.77555756156289 \\cdot 10^{-17} P_{+} - 0.24 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.08 e^{2}_{d} e^{2}_{u} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.16 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\- 6 P^{x}_{qg} e^{2}_{d} + 6.0 P^{x}_{qg} e^{2}_{u} & - 6 P^{x}_{q\\gamma} e^{2}_{d} + 6.0 P^{x}_{q\\gamma} e^{2}_{u} & - 0.6 P^{x}_{+} e^{2}_{d} + 0.6 P^{x}_{+} e^{2}_{u} + 5.55111512312578 \\cdot 10^{-17} P_{qq} - 0.36 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.12 e^{2}_{d} e^{2}_{u} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.24 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & 0.4 P^{x}_{+} e^{2}_{d} + 0.6 P^{x}_{+} e^{2}_{u} + 1.0 P_{+} + 0.24 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) - 0.48 e^{2}_{d} e^{2}_{u} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.24 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0.6 P^{x}_{-} e^{2}_{d} + 0.4 P^{x}_{-} e^{2}_{u} + 1.0 P_{V} & - 0.4 P^{x}_{-} e^{2}_{d} + 0.4 P^{x}_{-} e^{2}_{u} + 2.77555756156289 \\cdot 10^{-17} P_{-} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & - 0.6 P^{x}_{-} e^{2}_{d} + 0.6 P^{x}_{-} e^{2}_{u} + 5.55111512312578 \\cdot 10^{-17} P_{V} & 0.4 P^{x}_{-} e^{2}_{d} + 0.6 P^{x}_{-} e^{2}_{u} + 1.0 P_{-} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{+} e^{2}_{d} + 1.0 P_{+} & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{-} e^{2}_{d} + 1.0 P_{-} & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{+} e^{2}_{u} + 1.0 P_{+} & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{-} e^{2}_{u} + 1.0 P_{-} & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{+} e^{2}_{d} + 1.0 P_{+} & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{-} e^{2}_{d} + 1.0 P_{-} & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\end{array}\\right]$" ], "text/plain": [ - "[[0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, P^x_gg, 0], [0, 0, 0, 0, 0, 0, 0, P^x_\\gamma\\gamma]]" + "Matrix([\n", + "[ P^x_gg*e_\\Sigma^2 + P_gg, P^x_g\\gamma*e_\\Sigma^2, 0.6*P^x_gq*e_d^2 + 0.4*P^x_gq*e_u^2 + 1.0*P_gq, -0.4*P^x_gq*e_d^2 + 0.4*P^x_gq*e_u^2 + 2.77555756156289e-17*P_gq, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ P^x_\\gamma g*e_\\Sigma^2, P^x_\\gamma\\gamma*e_\\Sigma^2, 0.6*P^x_\\gamma q*e_d^2 + 0.4*P^x_\\gamma q*e_u^2, -0.4*P^x_\\gamma q*e_d^2 + 0.4*P^x_\\gamma q*e_u^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[6*P^x_qg*e_d^2 + 4*P^x_qg*e_u^2 + 10*P_qg, 6*P^x_q\\gamma*e_d^2 + 4*P^x_q\\gamma*e_u^2, 0.6*P^x_+*e_d^2 + 0.4*P^x_+*e_u^2 + 1.0*P_qq + 0.36*e_d^2**2*(-P^x_+ + P^x_qq) + 0.48*e_d^2*e_u^2*(-P^x_+ + P^x_qq) + 0.16*e_u^2**2*(-P^x_+ + P^x_qq), -0.4*P^x_+*e_d^2 + 0.4*P^x_+*e_u^2 + 2.77555756156289e-17*P_+ - 0.24*e_d^2**2*(-P^x_+ + P^x_qq) + 0.08*e_d^2*e_u^2*(-P^x_+ + P^x_qq) + 0.16*e_u^2**2*(-P^x_+ + P^x_qq), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ -6*P^x_qg*e_d^2 + 6.0*P^x_qg*e_u^2, -6*P^x_q\\gamma*e_d^2 + 6.0*P^x_q\\gamma*e_u^2, -0.6*P^x_+*e_d^2 + 0.6*P^x_+*e_u^2 + 5.55111512312578e-17*P_qq - 0.36*e_d^2**2*(-P^x_+ + P^x_qq) + 0.12*e_d^2*e_u^2*(-P^x_+ + P^x_qq) + 0.24*e_u^2**2*(-P^x_+ + P^x_qq), 0.4*P^x_+*e_d^2 + 0.6*P^x_+*e_u^2 + 1.0*P_+ + 0.24*e_d^2**2*(-P^x_+ + P^x_qq) - 0.48*e_d^2*e_u^2*(-P^x_+ + P^x_qq) + 0.24*e_u^2**2*(-P^x_+ + P^x_qq), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0.6*P^x_-*e_d^2 + 0.4*P^x_-*e_u^2 + 1.0*P_V, -0.4*P^x_-*e_d^2 + 0.4*P^x_-*e_u^2 + 2.77555756156289e-17*P_-, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, -0.6*P^x_-*e_d^2 + 0.6*P^x_-*e_u^2 + 5.55111512312578e-17*P_V, 0.4*P^x_-*e_d^2 + 0.6*P^x_-*e_u^2 + 1.0*P_-, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 1.0*P^x_+*e_d^2 + 1.0*P_+, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 1.0*P^x_-*e_d^2 + 1.0*P_-, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 1.0*P^x_+*e_u^2 + 1.0*P_+, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1.0*P^x_-*e_u^2 + 1.0*P_-, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1.0*P^x_+*e_d^2 + 1.0*P_+, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1.0*P^x_-*e_d^2 + 1.0*P_-, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]])" ] }, - "execution_count": 72, + "execution_count": 153, "metadata": {}, "output_type": "execute_result" } ], "source": [ - "pqed(3)" + "P_ev(5)" ] }, { "cell_type": "code", - "execution_count": 74, - "id": "dd594f07-85d0-48c7-940b-a47a4e0ed75d", + "execution_count": 154, + "id": "3f014fa2", "metadata": {}, "outputs": [ { "data": { "text/latex": [ - "$\\displaystyle \\left[\\begin{matrix}1 & 1 & 1 & 0 & 0 & 0 & 0 & 0\\\\2 & -1 & -1 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 1 & 1 & 1 & 0 & 0\\\\0 & 0 & 0 & 0 & 1 & -1 & 0 & 0\\\\0 & 1 & -1 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 2 & -1 & -1 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\end{matrix}\\right]$" + "$\\displaystyle \\left[\\begin{array}{cccccccccccccc}P^{x}_{gg} e^{2}_{\\Sigma} + P_{gg} & P^{x}_{g\\gamma} e^{2}_{\\Sigma} & 0.5 P^{x}_{gq} e^{2}_{d} + 0.5 P^{x}_{gq} e^{2}_{u} + 1.0 P_{gq} & - 0.5 P^{x}_{gq} e^{2}_{d} + 0.5 P^{x}_{gq} e^{2}_{u} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\P^{x}_{\\gamma g} e^{2}_{\\Sigma} & P^{x}_{\\gamma\\gamma} e^{2}_{\\Sigma} & 0.5 P^{x}_{\\gamma q} e^{2}_{d} + 0.5 P^{x}_{\\gamma q} e^{2}_{u} & - 0.5 P^{x}_{\\gamma q} e^{2}_{d} + 0.5 P^{x}_{\\gamma q} e^{2}_{u} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\6 P^{x}_{qg} e^{2}_{d} + 6 P^{x}_{qg} e^{2}_{u} + 12 P_{qg} & 6 P^{x}_{q\\gamma} e^{2}_{d} + 6 P^{x}_{q\\gamma} e^{2}_{u} & 0.5 P^{x}_{+} e^{2}_{d} + 0.5 P^{x}_{+} e^{2}_{u} - 1.11022302462516 \\cdot 10^{-16} P_{+} + 1.0 P_{qq} + 0.25 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.5 e^{2}_{d} e^{2}_{u} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.25 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & - 0.5 P^{x}_{+} e^{2}_{d} + 0.5 P^{x}_{+} e^{2}_{u} - 0.25 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.25 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\- 6 P^{x}_{qg} e^{2}_{d} + 6.0 P^{x}_{qg} e^{2}_{u} & - 6 P^{x}_{q\\gamma} e^{2}_{d} + 6.0 P^{x}_{q\\gamma} e^{2}_{u} & - 0.5 P^{x}_{+} e^{2}_{d} + 0.5 P^{x}_{+} e^{2}_{u} - 0.25 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.25 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & 0.5 P^{x}_{+} e^{2}_{d} + 0.5 P^{x}_{+} e^{2}_{u} + 1.0 P_{+} + 0.25 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) - 0.5 e^{2}_{d} e^{2}_{u} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.25 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0.5 P^{x}_{-} e^{2}_{d} + 0.5 P^{x}_{-} e^{2}_{u} - 1.11022302462516 \\cdot 10^{-16} P_{-} + 1.0 P_{V} & - 0.5 P^{x}_{-} e^{2}_{d} + 0.5 P^{x}_{-} e^{2}_{u} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & - 0.5 P^{x}_{-} e^{2}_{d} + 0.5 P^{x}_{-} e^{2}_{u} & 0.5 P^{x}_{-} e^{2}_{d} + 0.5 P^{x}_{-} e^{2}_{u} + 1.0 P_{-} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{+} e^{2}_{d} + 1.0 P_{+} & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{-} e^{2}_{d} + 1.0 P_{-} & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{+} e^{2}_{u} + 1.0 P_{+} & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{-} e^{2}_{u} + 1.0 P_{-} & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{+} e^{2}_{d} + 1.0 P_{+} & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{-} e^{2}_{d} + 1.0 P_{-} & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{+} e^{2}_{u} + 1.0 P_{+} & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{-} e^{2}_{u} + 1.0 P_{-}\\end{array}\\right]$" ], "text/plain": [ - "[[1, 1, 1, 0, 0, 0, 0, 0], [2, -1, -1, 0, 0, 0, 0, 0], [0, 0, 0, 1, 1, 1, 0, 0], [0, 0, 0, 0, 1, -1, 0, 0], [0, 1, -1, 0, 0, 0, 0, 0], [0, 0, 0, 2, -1, -1, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 0, 0, 1]]" + "Matrix([\n", + "[ P^x_gg*e_\\Sigma^2 + P_gg, P^x_g\\gamma*e_\\Sigma^2, 0.5*P^x_gq*e_d^2 + 0.5*P^x_gq*e_u^2 + 1.0*P_gq, -0.5*P^x_gq*e_d^2 + 0.5*P^x_gq*e_u^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ P^x_\\gamma g*e_\\Sigma^2, P^x_\\gamma\\gamma*e_\\Sigma^2, 0.5*P^x_\\gamma q*e_d^2 + 0.5*P^x_\\gamma q*e_u^2, -0.5*P^x_\\gamma q*e_d^2 + 0.5*P^x_\\gamma q*e_u^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[6*P^x_qg*e_d^2 + 6*P^x_qg*e_u^2 + 12*P_qg, 6*P^x_q\\gamma*e_d^2 + 6*P^x_q\\gamma*e_u^2, 0.5*P^x_+*e_d^2 + 0.5*P^x_+*e_u^2 - 1.11022302462516e-16*P_+ + 1.0*P_qq + 0.25*e_d^2**2*(-P^x_+ + P^x_qq) + 0.5*e_d^2*e_u^2*(-P^x_+ + P^x_qq) + 0.25*e_u^2**2*(-P^x_+ + P^x_qq), -0.5*P^x_+*e_d^2 + 0.5*P^x_+*e_u^2 - 0.25*e_d^2**2*(-P^x_+ + P^x_qq) + 0.25*e_u^2**2*(-P^x_+ + P^x_qq), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ -6*P^x_qg*e_d^2 + 6.0*P^x_qg*e_u^2, -6*P^x_q\\gamma*e_d^2 + 6.0*P^x_q\\gamma*e_u^2, -0.5*P^x_+*e_d^2 + 0.5*P^x_+*e_u^2 - 0.25*e_d^2**2*(-P^x_+ + P^x_qq) + 0.25*e_u^2**2*(-P^x_+ + P^x_qq), 0.5*P^x_+*e_d^2 + 0.5*P^x_+*e_u^2 + 1.0*P_+ + 0.25*e_d^2**2*(-P^x_+ + P^x_qq) - 0.5*e_d^2*e_u^2*(-P^x_+ + P^x_qq) + 0.25*e_u^2**2*(-P^x_+ + P^x_qq), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0.5*P^x_-*e_d^2 + 0.5*P^x_-*e_u^2 - 1.11022302462516e-16*P_- + 1.0*P_V, -0.5*P^x_-*e_d^2 + 0.5*P^x_-*e_u^2, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, -0.5*P^x_-*e_d^2 + 0.5*P^x_-*e_u^2, 0.5*P^x_-*e_d^2 + 0.5*P^x_-*e_u^2 + 1.0*P_-, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 1.0*P^x_+*e_d^2 + 1.0*P_+, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 1.0*P^x_-*e_d^2 + 1.0*P_-, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 1.0*P^x_+*e_u^2 + 1.0*P_+, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1.0*P^x_-*e_u^2 + 1.0*P_-, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1.0*P^x_+*e_d^2 + 1.0*P_+, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1.0*P^x_-*e_d^2 + 1.0*P_-, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1.0*P^x_+*e_u^2 + 1.0*P_+, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1.0*P^x_-*e_u^2 + 1.0*P_-]])" ] }, - "execution_count": 74, + "execution_count": 154, "metadata": {}, "output_type": "execute_result" } ], "source": [ - "rot3" + "P_ev(6)" ] }, { "cell_type": "code", "execution_count": null, - "id": "9764233f-9c53-4860-b7c3-2be12fbec857", + "id": "53626511", "metadata": {}, "outputs": [], "source": [] From c3f5d07c818a46a55b4a6c1c1b59aa4248abe59b Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Niccol=C3=B2=20Laurenti?= Date: Wed, 16 Mar 2022 16:38:02 +0100 Subject: [PATCH 05/71] Fix singlet matrix in pdf --- notebook/QCDxQED_DGLAP.pdf | Bin 94027 -> 94029 bytes notebook/QCDxQED_DGLAP.tex | 2 +- 2 files changed, 1 insertion(+), 1 deletion(-) diff --git a/notebook/QCDxQED_DGLAP.pdf b/notebook/QCDxQED_DGLAP.pdf index 44017e6ccd0ef150cf688f0a14a8f340d9633df7..46e12d0d350d283613ee4492474bb0818bc61c88 100644 GIT binary patch delta 2526 zcmai$Rag^@0*2`rk{clD=+R>v1I7eLcMR!J5Qa4WM%SdIafF0`0#d?{C@4rM(jX;W z5)w*H8bJ}x`TyHKk059bTd(iBch;Su2JRBH3>E(G-%ceeX_&NLBK| 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z8C@=pAaXo5GuqWbdL@NaG$ntcUh8>+$@L%;OA^DN{>n5v&D+Vx&gkbJm2m8K-ioB+ zpSxYX-d+@X6FU-jJgzr-k;26nshI+gBf`1jI8b>`Z|@_tW}*!t&(v6d`4KvZQs8%3 z|2{uLZ*ynsSq)45Wb(?}iHcn++SyuCw(A3-<1K05{bLe>wVs##! zbYSwzleB<*2G$V?i`uzgsq-X&_%Yt%Aw zl1VS*l$x`SQqDB)j=HES>f(#kWscakpirpLAth-!@UnVm_zlO;{}VPGg$iYEWOHLo-1#I7LB0FhM~^HZ(>6lMng73H9lPmFHB`_XLM*FF*PzElhNKNe;v#@4nY9`gy9+Mvp$}E-wUlC6KydNEzuQy(Gg=Y6eDtb^>!*^ zA}$glDN-UW0+A6}krPdkCxW|Q3dDTrDiW)kD Date: Wed, 16 Mar 2022 19:30:22 +0100 Subject: [PATCH 07/71] Test valence sector --- notebook/uni-dglap.ipynb | 326 ++++++++++++++++++++++++++++++++++----- 1 file changed, 290 insertions(+), 36 deletions(-) diff --git a/notebook/uni-dglap.ipynb b/notebook/uni-dglap.ipynb index 0201f91c1..d4cdcb709 100644 --- a/notebook/uni-dglap.ipynb +++ b/notebook/uni-dglap.ipynb @@ -10,7 +10,7 @@ }, { "cell_type": "code", - "execution_count": 155, + "execution_count": 312, "id": "1d7616d9-f4a3-447a-9ebf-e99db8126ffb", "metadata": {}, "outputs": [], @@ -20,7 +20,7 @@ }, { "cell_type": "code", - "execution_count": 156, + "execution_count": 313, "id": "cb36381d-57b5-4972-b028-cf4b6300938f", "metadata": {}, "outputs": [], @@ -53,7 +53,7 @@ }, { "cell_type": "code", - "execution_count": 157, + "execution_count": 314, "id": "9764233f-9c53-4860-b7c3-2be12fbec857", "metadata": {}, "outputs": [], @@ -66,7 +66,7 @@ "P[s, qed] = sympy.Array.zeros(14,14).as_mutable()\n", "\n", "ei2=[eu2, ed2, ed2, eu2, ed2, eu2]\n", - "def e2s(nf):\n", + "def es2_(nf):\n", " nu = int(nf/2)\n", " nd = nf - nu\n", " return nu*eu2 + nd*ed2\n", @@ -82,7 +82,7 @@ " return res\n", "\n", "def P_qed(nf):\n", - " es2=e2s(nf)\n", + " es2=es2_(nf)\n", " res = sympy.Matrix.zeros(14,14).as_mutable()\n", " res[0, 0]=es2*Pxgg\n", " res[1, 1]=es2*Pxyy\n", @@ -122,7 +122,7 @@ }, { "cell_type": "code", - "execution_count": 158, + "execution_count": 315, "id": "3a376dcf", "metadata": {}, "outputs": [], @@ -178,7 +178,7 @@ }, { "cell_type": "code", - "execution_count": 159, + "execution_count": 316, "id": "b5747379", "metadata": {}, "outputs": [], @@ -207,7 +207,7 @@ }, { "cell_type": "code", - "execution_count": 160, + "execution_count": 317, "id": "45fb3a16", "metadata": {}, "outputs": [], @@ -219,7 +219,7 @@ }, { "cell_type": "code", - "execution_count": 161, + "execution_count": 318, "id": "9b5050aa", "metadata": {}, "outputs": [ @@ -246,7 +246,7 @@ "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]])" ] }, - "execution_count": 161, + "execution_count": 318, "metadata": {}, "output_type": "execute_result" } @@ -257,7 +257,7 @@ }, { "cell_type": "code", - "execution_count": 162, + "execution_count": 319, "id": "07b1874a", "metadata": {}, "outputs": [ @@ -284,7 +284,7 @@ "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]])" ] }, - "execution_count": 162, + "execution_count": 319, "metadata": {}, "output_type": "execute_result" } @@ -295,7 +295,7 @@ }, { "cell_type": "code", - "execution_count": 163, + "execution_count": 320, "id": "c9f75de1", "metadata": {}, "outputs": [ @@ -322,7 +322,7 @@ "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]])" ] }, - "execution_count": 163, + "execution_count": 320, "metadata": {}, "output_type": "execute_result" } @@ -333,7 +333,7 @@ }, { "cell_type": "code", - "execution_count": 164, + "execution_count": 321, "id": "ce40d361", "metadata": {}, "outputs": [ @@ -360,7 +360,7 @@ "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]])" ] }, - "execution_count": 164, + "execution_count": 321, "metadata": {}, "output_type": "execute_result" } @@ -371,7 +371,7 @@ }, { "cell_type": "code", - "execution_count": 165, + "execution_count": 322, "id": "3f014fa2", "metadata": {}, "outputs": [ @@ -398,7 +398,7 @@ "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1.0*P^x_-*e_u^2 + 1.0*P_-]])" ] }, - "execution_count": 165, + "execution_count": 322, "metadata": {}, "output_type": "execute_result" } @@ -409,45 +409,45 @@ }, { "cell_type": "code", - "execution_count": 204, + "execution_count": 323, "id": "53626511", "metadata": {}, "outputs": [], "source": [ - "def e2D(nf):\n", + "def eD2_(nf):\n", " nu = int(nf/2)\n", " nd = nf - nu\n", " return nd*eu2 + nu*ed2\n", "def etam_(nf):\n", " nu = int(nf/2)\n", " nd = nf - nu\n", - " return 0.5*(eu2 - ed2)\n", - "\n", - "def P02(nf):\n", - " return Pgq + e2s(nf)/nf*Pxgq" + " return 0.5*(eu2 - ed2)" ] }, { "cell_type": "code", - "execution_count": 205, - "id": "0534aba3", + "execution_count": 324, + "id": "9c9ce615", "metadata": {}, "outputs": [], "source": [ "def P_ev_sing(nf):\n", - " return P_ev(nf)[:4,:4]" + " return P_ev(nf)[:4,:4]\n", + "\n", + "def P_ev_val(nf):\n", + " return P_ev(nf)[4:6,4:6]" ] }, { "cell_type": "code", - "execution_count": 227, - "id": "c4afbebe", + "execution_count": 325, + "id": "5267ab16", "metadata": {}, "outputs": [], "source": [ "def P_ev_sing2(nf):\n", - " es2=e2s(nf)\n", - " e2d=e2D(nf)\n", + " es2=es2_(nf)\n", + " eD2=eD2_(nf)\n", " etam=etam_(nf)\n", " nu = int(nf/2)\n", " nd = nf - nu\n", @@ -455,15 +455,139 @@ " [Pgg + es2 * Pxgg, es2 * Pxgy, Pgq + es2/nf*Pxgq, 2*nu/nf*etam*Pxgq],\n", " [es2 * Pxyg, es2 * Pxyy, es2/nf*Pxyq, 2*nu/nf*etam*Pxyq],\n", " [2*nf*Pqg + 2*es2*Pxqg, 2*es2 * Pxqy, Pqq + es2/nf*Pxp +(es2/nf)**2*(Pxqq - Pxp), 2*nu/nf*etam*Pxp +2*nu*etam*es2/nf**2*(Pxqq - Pxp)],\n", - " [4*nd*etam*Pxqg, 4*nd*etam*Pxqy, 2*nd/nf*etam*Pxp +2*nd*etam*es2/nf**2*(Pxqq - Pxp), Pp + e2d/nf*Pxp + 4*nu*nd/nf**2*etam**2*(Pxqq - Pxp)]\n", + " [4*nd*etam*Pxqg, 4*nd*etam*Pxqy, 2*nd/nf*etam*Pxp +2*nd*etam*es2/nf**2*(Pxqq - Pxp), Pp + eD2/nf*Pxp + 4*nu*nd/nf**2*etam**2*(Pxqq - Pxp)]\n", + " ])\n", + " return res\n", + "\n", + "def P_ev_val2(nf):\n", + " es2=e2s(nf)\n", + " eD2=eD2_(nf)\n", + " etam=etam_(nf)\n", + " nu = int(nf/2)\n", + " nd = nf - nu\n", + " res = sympy.Matrix([\n", + " [Pv+es2/nf*Pxm, 2*nu/nf*etam*Pxm],\n", + " [2*nd/nf*etam*Pxm, Pm + eD2/nf*Pxm]\n", " ])\n", " return res" ] }, { "cell_type": "code", - "execution_count": 232, - "id": "4b6c1c1a", + "execution_count": 326, + "id": "a47dc3b2", + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\left[\\begin{matrix}0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\end{matrix}\\right]$" + ], + "text/plain": [ + "Matrix([\n", + "[0, 0, 0, 0],\n", + "[0, 0, 0, 0],\n", + "[0, 0, 0, 0],\n", + "[0, 0, 0, 0]])" + ] + }, + "execution_count": 326, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "sympy.simplify(P_ev_sing(2)-P_ev_sing2(2))" + ] + }, + { + "cell_type": "code", + "execution_count": 327, + "id": "6fddcd7c", + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\left[\\begin{matrix}0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\\\0 & 0 & 0 & - 1.11022302462516 \\cdot 10^{-16} P_{+}\\end{matrix}\\right]$" + ], + "text/plain": [ + "Matrix([\n", + "[0, 0, 0, 0],\n", + "[0, 0, 0, 0],\n", + "[0, 0, 0, 0],\n", + "[0, 0, 0, -1.11022302462516e-16*P_+]])" + ] + }, + "execution_count": 327, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "sympy.simplify(P_ev_sing(3)-P_ev_sing2(3))" + ] + }, + { + "cell_type": "code", + "execution_count": 328, + "id": "f151ecdd", + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\left[\\begin{matrix}0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\end{matrix}\\right]$" + ], + "text/plain": [ + "Matrix([\n", + "[0, 0, 0, 0],\n", + "[0, 0, 0, 0],\n", + "[0, 0, 0, 0],\n", + "[0, 0, 0, 0]])" + ] + }, + "execution_count": 328, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "sympy.simplify(P_ev_sing(4)-P_ev_sing2(4))" + ] + }, + { + "cell_type": "code", + "execution_count": 329, + "id": "e49bef49", + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\left[\\begin{matrix}0 & 0 & 8.88178419700125 \\cdot 10^{-17} P^{x}_{gq} e^{2}_{d} & 2.77555756156289 \\cdot 10^{-17} P_{gq}\\\\0 & 0 & 8.88178419700125 \\cdot 10^{-17} P^{x}_{\\gamma q} e^{2}_{d} & 0\\\\0 & 0 & 7.105427357601 \\cdot 10^{-17} P^{x}_{+} e^{2}_{d} - 3.5527136788005 \\cdot 10^{-17} P^{x}_{+} \\left(e^{2}_{u}\\right)^{2} + 3.5527136788005 \\cdot 10^{-17} P^{x}_{qq} \\left(e^{2}_{u}\\right)^{2} & 4.16333634234434 \\cdot 10^{-17} P^{x}_{+} e^{2}_{d} e^{2}_{u} - 2.77555756156289 \\cdot 10^{-17} P^{x}_{+} \\left(e^{2}_{u}\\right)^{2} - 4.16333634234434 \\cdot 10^{-17} P^{x}_{qq} e^{2}_{d} e^{2}_{u} + 2.77555756156289 \\cdot 10^{-17} P^{x}_{qq} \\left(e^{2}_{u}\\right)^{2} + 2.77555756156289 \\cdot 10^{-17} P_{+}\\\\0 & 0 & - 5.55111512312578 \\cdot 10^{-17} P^{x}_{+} e^{2}_{d} e^{2}_{u} - 1.11022302462516 \\cdot 10^{-16} P^{x}_{+} e^{2}_{d} - 5.55111512312578 \\cdot 10^{-17} P^{x}_{+} \\left(e^{2}_{u}\\right)^{2} + 1.11022302462516 \\cdot 10^{-16} P^{x}_{+} e^{2}_{u} + 5.55111512312578 \\cdot 10^{-17} P^{x}_{qq} e^{2}_{d} e^{2}_{u} + 5.55111512312578 \\cdot 10^{-17} P^{x}_{qq} \\left(e^{2}_{u}\\right)^{2} + 5.55111512312578 \\cdot 10^{-17} P_{qq} & e^{2}_{u} \\left(8.88178419700125 \\cdot 10^{-17} P^{x}_{+} e^{2}_{d} - 4.44089209850063 \\cdot 10^{-17} P^{x}_{+} e^{2}_{u} + 8.88178419700125 \\cdot 10^{-17} P^{x}_{+} - 8.88178419700125 \\cdot 10^{-17} P^{x}_{qq} e^{2}_{d} + 4.44089209850063 \\cdot 10^{-17} P^{x}_{qq} e^{2}_{u}\\right)\\end{matrix}\\right]$" + ], + "text/plain": [ + "Matrix([\n", + "[0, 0, 8.88178419700125e-17*P^x_gq*e_d^2, 2.77555756156289e-17*P_gq],\n", + "[0, 0, 8.88178419700125e-17*P^x_\\gamma q*e_d^2, 0],\n", + "[0, 0, 7.105427357601e-17*P^x_+*e_d^2 - 3.5527136788005e-17*P^x_+*e_u^2**2 + 3.5527136788005e-17*P^x_qq*e_u^2**2, 4.16333634234434e-17*P^x_+*e_d^2*e_u^2 - 2.77555756156289e-17*P^x_+*e_u^2**2 - 4.16333634234434e-17*P^x_qq*e_d^2*e_u^2 + 2.77555756156289e-17*P^x_qq*e_u^2**2 + 2.77555756156289e-17*P_+],\n", + "[0, 0, -5.55111512312578e-17*P^x_+*e_d^2*e_u^2 - 1.11022302462516e-16*P^x_+*e_d^2 - 5.55111512312578e-17*P^x_+*e_u^2**2 + 1.11022302462516e-16*P^x_+*e_u^2 + 5.55111512312578e-17*P^x_qq*e_d^2*e_u^2 + 5.55111512312578e-17*P^x_qq*e_u^2**2 + 5.55111512312578e-17*P_qq, e_u^2*(8.88178419700125e-17*P^x_+*e_d^2 - 4.44089209850063e-17*P^x_+*e_u^2 + 8.88178419700125e-17*P^x_+ - 8.88178419700125e-17*P^x_qq*e_d^2 + 4.44089209850063e-17*P^x_qq*e_u^2)]])" + ] + }, + "execution_count": 329, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "sympy.simplify(P_ev_sing(5)-P_ev_sing2(5))" + ] + }, + { + "cell_type": "code", + "execution_count": 330, + "id": "82b3e343", "metadata": {}, "outputs": [ { @@ -479,7 +603,7 @@ "[0, 0, 0, -1.11022302462516e-16*P_+]])" ] }, - "execution_count": 232, + "execution_count": 330, "metadata": {}, "output_type": "execute_result" } @@ -488,10 +612,140 @@ "sympy.simplify(P_ev_sing(6)-P_ev_sing2(6))" ] }, + { + "cell_type": "code", + "execution_count": 331, + "id": "a9f61d31", + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\left[\\begin{matrix}0 & 0\\\\0 & 0\\end{matrix}\\right]$" + ], + "text/plain": [ + "Matrix([\n", + "[0, 0],\n", + "[0, 0]])" + ] + }, + "execution_count": 331, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "sympy.simplify(P_ev_val(2)-P_ev_val2(2))" + ] + }, + { + "cell_type": "code", + "execution_count": 332, + "id": "3e91130a", + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\left[\\begin{matrix}0 & 0\\\\0 & - 1.11022302462516 \\cdot 10^{-16} P_{-}\\end{matrix}\\right]$" + ], + "text/plain": [ + "Matrix([\n", + "[0, 0],\n", + "[0, -1.11022302462516e-16*P_-]])" + ] + }, + "execution_count": 332, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "sympy.simplify(P_ev_val(3)-P_ev_val2(3))" + ] + }, + { + "cell_type": "code", + "execution_count": 333, + "id": "6f5e4417", + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\left[\\begin{matrix}0 & 0\\\\0 & 0\\end{matrix}\\right]$" + ], + "text/plain": [ + "Matrix([\n", + "[0, 0],\n", + "[0, 0]])" + ] + }, + "execution_count": 333, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "sympy.simplify(P_ev_val(4)-P_ev_val2(4))" + ] + }, + { + "cell_type": "code", + "execution_count": 334, + "id": "768a92b2", + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\left[\\begin{matrix}8.88178419700125 \\cdot 10^{-17} P^{x}_{-} e^{2}_{d} & 2.77555756156289 \\cdot 10^{-17} P_{-}\\\\- 1.11022302462516 \\cdot 10^{-16} P^{x}_{-} e^{2}_{d} + 1.11022302462516 \\cdot 10^{-16} P^{x}_{-} e^{2}_{u} + 5.55111512312578 \\cdot 10^{-17} P_{V} & 8.88178419700125 \\cdot 10^{-17} P^{x}_{-} e^{2}_{u}\\end{matrix}\\right]$" + ], + "text/plain": [ + "Matrix([\n", + "[ 8.88178419700125e-17*P^x_-*e_d^2, 2.77555756156289e-17*P_-],\n", + "[-1.11022302462516e-16*P^x_-*e_d^2 + 1.11022302462516e-16*P^x_-*e_u^2 + 5.55111512312578e-17*P_V, 8.88178419700125e-17*P^x_-*e_u^2]])" + ] + }, + "execution_count": 334, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "sympy.simplify(P_ev_val(5)-P_ev_val2(5))" + ] + }, + { + "cell_type": "code", + "execution_count": 335, + "id": "82565536", + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\left[\\begin{matrix}- 1.11022302462516 \\cdot 10^{-16} P_{-} & 0\\\\0 & - 1.11022302462516 \\cdot 10^{-16} P_{-}\\end{matrix}\\right]$" + ], + "text/plain": [ + "Matrix([\n", + "[-1.11022302462516e-16*P_-, 0],\n", + "[ 0, -1.11022302462516e-16*P_-]])" + ] + }, + "execution_count": 335, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "sympy.simplify(P_ev_val(6)-P_ev_val2(6))" + ] + }, { "cell_type": "code", "execution_count": null, - "id": "7f9154a3", + "id": "26d06926", "metadata": {}, "outputs": [], "source": [] From f615f62e2765e346320665841d85869f75b8edad Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Niccol=C3=B2=20Laurenti?= Date: Fri, 18 Mar 2022 11:20:20 +0100 Subject: [PATCH 08/71] Change eSigma2 -> nf --- notebook/QCDxQED_DGLAP.pdf | Bin 94061 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a/notebook/QCDxQED_DGLAP.tex +++ b/notebook/QCDxQED_DGLAP.tex @@ -21,7 +21,7 @@ \item Singlet sector: \begin{equation*} -\mu^2\frac{d}{d\mu^2} +\hspace*{-0.4cm}\mu^2\frac{d}{d\mu^2} \begin{pmatrix} g \\ \gamma \\ @@ -30,10 +30,10 @@ \end{pmatrix} = \begin{pmatrix} - P_{gg}+e_\Sigma^2 \tilde{P}_{gg} & e_\Sigma^2 \tilde{P}_{g\gamma} & P_{gq} + \frac{e_\Sigma^2}{n_f}\tilde{P}_{gq} & 2\frac{n_u}{n_f}\eta^- \tilde{P}_{gq} \\ - e_\Sigma^2 \tilde{P}_{\gamma g} & e_\Sigma^2 \tilde{P}_{\gamma \gamma} & \frac{e_\Sigma^2}{n_f}\tilde{P}_{\gamma q} & 2\frac{n_u}{n_f}\eta^- \tilde{P}_{\gamma q} \\ - 2n_f P_{qg} +2e_\Sigma^2 \tilde{P}_{qg} & 2 e_\Sigma^2 \tilde{P}_{q\gamma} & P_{qq}+ \frac{e_\Sigma^2}{n_f}\tilde{P}_{+} + \Bigl(\frac{e_\Sigma^2}{n_f}\Bigr)^2(\tilde{P}_{qq}-\tilde{P}_{+}) & 2\frac{n_u}{n_f}\eta^-\tilde{P}_{+}+2n_u \frac{\eta^-e_\Sigma^2}{n_f^2}(\tilde{P}_{qq}-\tilde{P}_{+}) \\ - 4n_d\eta^-\tilde{P}_{qg} & 4n_d\eta^-\tilde{P}_{q\gamma} & 2\frac{n_d}{n_f}\eta^- \tilde{P}_{+} +2n_d \frac{\eta^-e_\Sigma^2}{n_f^2}(\tilde{P}_{qq}-\tilde{P}_{+}) & P_+ +\frac{e_\Delta^2}{n_f}\tilde{P}_{+} +4\frac{n_u n_d}{n_f^2}(\eta^-)^2(\tilde{P}_{qq}-\tilde{P}_{+}) + P_{gg}+n_f \langle e^2\rangle \tilde{P}_{gg} & n_f \langle e^2\rangle \tilde{P}_{g\gamma} & P_{gq} + \langle e^2\rangle \tilde{P}_{gq} & \frac{n_u}{n_f}e^- \tilde{P}_{gq} \\ + n_f \langle e^2\rangle \tilde{P}_{\gamma g} & n_f \langle e^2\rangle \tilde{P}_{\gamma \gamma} & \langle e^2\rangle \tilde{P}_{\gamma q} & \frac{n_u}{n_f}e^- \tilde{P}_{\gamma q} \\ + 2n_f (P_{qg} +\langle e^2\rangle \tilde{P}_{qg} )& 2 n_f \langle e^2\rangle \tilde{P}_{q\gamma} & P_{qq}+ \langle e^2\rangle \tilde{P}_{+} + \langle e^2\rangle^2(\tilde{P}_{qq}-\tilde{P}_{+}) & \frac{n_u}{n_f}\Bigl(e^-\tilde{P}_{+}+ e^- \langle e^2\rangle (\tilde{P}_{qq}-\tilde{P}_{+}) \Bigr)\\ + 2n_de^-\tilde{P}_{qg} & 2n_de^-\tilde{P}_{q\gamma} & \frac{n_d}{n_f}\Bigl(e^- \tilde{P}_{+} + e^- \langle e^2\rangle (\tilde{P}_{qq}-\tilde{P}_{+}) \Bigr)& P_+ +\frac{e_\Delta^2}{n_f}\tilde{P}_{+} +\frac{n_u n_d}{n_f^2}(e^-)^2(\tilde{P}_{qq}-\tilde{P}_{+}) \end{pmatrix} \begin{pmatrix} g \\ @@ -44,9 +44,9 @@ \end{equation*} with \begin{align*} -e_\Sigma^2&=n_u e_u^2+n_d e_d^2 \\ + \langle e^2\rangle&=\frac{n_u e_u^2+n_d e_d^2}{n_f} \\ e_\Delta^2&=n_u e_d^2+n_d e_u^2 \\ -\eta^\pm &= \frac{1}{2}(e_u^2 \pm e_d^2) +e^- &= e_u^2 -e_d^2 \end{align*} \item Valence sector: @@ -58,8 +58,8 @@ \end{pmatrix} = \begin{pmatrix} -P_V+\frac{e_\Sigma^2}{n_f} \tilde{P}_{-} & 2\frac{n_u}{n_f}\eta^- \tilde{P}_{-}\\ - 2\frac{n_d}{n_f}\eta^- \tilde{P}_{-}& P_-+\frac{e_\Delta^2}{n_f} \tilde{P}_{-} +P_V+\langle e^2\rangle \tilde{P}_{-} & \frac{n_u}{n_f}e^- \tilde{P}_{-}\\ + \frac{n_d}{n_f}e^- \tilde{P}_{-}& P_-+\frac{e_\Delta^2}{n_f} \tilde{P}_{-} \end{pmatrix} \begin{pmatrix} V \\ From 778806cc3a51b4a589673be9b4e41da3d01307c8 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Niccol=C3=B2=20Laurenti?= Date: Fri, 18 Mar 2022 11:32:28 +0100 Subject: [PATCH 09/71] Collect e- in 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z31*B&o>^#0e@#a5nT#pZe(+Ga%Ev{3T19&Z(?c+b97;Hba--QW(qhq NF)%U;B_%~qMhZRnQWO9H diff --git a/notebook/QCDxQED_DGLAP.tex b/notebook/QCDxQED_DGLAP.tex index 63334ff13..d6866481a 100644 --- a/notebook/QCDxQED_DGLAP.tex +++ b/notebook/QCDxQED_DGLAP.tex @@ -32,8 +32,8 @@ \begin{pmatrix} P_{gg}+n_f \langle e^2\rangle \tilde{P}_{gg} & n_f \langle e^2\rangle \tilde{P}_{g\gamma} & P_{gq} + \langle e^2\rangle \tilde{P}_{gq} & \frac{n_u}{n_f}e^- \tilde{P}_{gq} \\ n_f \langle e^2\rangle \tilde{P}_{\gamma g} & n_f \langle e^2\rangle \tilde{P}_{\gamma \gamma} & \langle e^2\rangle \tilde{P}_{\gamma q} & \frac{n_u}{n_f}e^- \tilde{P}_{\gamma q} \\ - 2n_f (P_{qg} +\langle e^2\rangle \tilde{P}_{qg} )& 2 n_f \langle e^2\rangle \tilde{P}_{q\gamma} & P_{qq}+ \langle e^2\rangle \tilde{P}_{+} + \langle e^2\rangle^2(\tilde{P}_{qq}-\tilde{P}_{+}) & \frac{n_u}{n_f}\Bigl(e^-\tilde{P}_{+}+ e^- \langle e^2\rangle (\tilde{P}_{qq}-\tilde{P}_{+}) \Bigr)\\ - 2n_de^-\tilde{P}_{qg} & 2n_de^-\tilde{P}_{q\gamma} & \frac{n_d}{n_f}\Bigl(e^- \tilde{P}_{+} + e^- \langle e^2\rangle (\tilde{P}_{qq}-\tilde{P}_{+}) \Bigr)& P_+ +\frac{e_\Delta^2}{n_f}\tilde{P}_{+} +\frac{n_u n_d}{n_f^2}(e^-)^2(\tilde{P}_{qq}-\tilde{P}_{+}) + 2n_f (P_{qg} +\langle e^2\rangle \tilde{P}_{qg} )& 2 n_f \langle e^2\rangle \tilde{P}_{q\gamma} & P_{qq}+ \langle e^2\rangle \Bigl(\tilde{P}_{+} + \langle e^2\rangle(\tilde{P}_{qq}-\tilde{P}_{+})\Bigr) & \frac{n_u}{n_f}e^-\Bigl(\tilde{P}_{+}+ \langle e^2\rangle (\tilde{P}_{qq}-\tilde{P}_{+}) \Bigr)\\ + 2n_de^-\tilde{P}_{qg} & 2n_de^-\tilde{P}_{q\gamma} & \frac{n_d}{n_f}e^-\Bigl( \tilde{P}_{+} + \langle e^2\rangle (\tilde{P}_{qq}-\tilde{P}_{+}) \Bigr)& P_+ +\frac{e_\Delta^2}{n_f}\tilde{P}_{+} +\frac{n_u n_d}{n_f^2}(e^-)^2(\tilde{P}_{qq}-\tilde{P}_{+}) \end{pmatrix} \begin{pmatrix} g \\ From 002007136b4f2facb9f5615539f9c726a75471a1 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Niccol=C3=B2=20Laurenti?= Date: Fri, 18 Mar 2022 19:16:11 +0100 Subject: [PATCH 10/71] Moved uni-dglap.ipynb and pdf into extras --- extras/uni-dglap.ipynb | 775 +++++++++++++++++++++++++++++++++++++++++ extras/uni-dglap.pdf | Bin 0 -> 70508 bytes extras/uni-dglap.tex | 74 ++++ 3 files changed, 849 insertions(+) create mode 100644 extras/uni-dglap.ipynb create mode 100644 extras/uni-dglap.pdf create mode 100644 extras/uni-dglap.tex diff --git a/extras/uni-dglap.ipynb b/extras/uni-dglap.ipynb new file mode 100644 index 000000000..d4cdcb709 --- /dev/null +++ b/extras/uni-dglap.ipynb @@ -0,0 +1,775 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "id": "4790b144-ceb5-4cba-87a5-ca2fd37e5912", + "metadata": {}, + "source": [ + "# Unified DGLAP" + ] + }, + { + "cell_type": "code", + "execution_count": 312, + "id": "1d7616d9-f4a3-447a-9ebf-e99db8126ffb", + "metadata": {}, + "outputs": [], + "source": [ + "import sympy" + ] + }, + { + "cell_type": "code", + "execution_count": 313, + "id": "cb36381d-57b5-4972-b028-cf4b6300938f", + "metadata": {}, + "outputs": [], + "source": [ + "# QCD\n", + "Pv, Pp, Pm, Pqq, Pqg, Pgq, Pgg = sympy.symbols(\"P_V P_+ P_- P_qq P_qg P_gq P_gg\")\n", + "# QED\n", + "Pxv, Pxp, Pxm, Pxqq, Pxqg, Pxgq, Pxgg = sympy.symbols(\"P^x_V P^x_+ P^x_- P^x_qq P^x_qg P^x_gq P^x_gg\")\n", + "Pxqy, Pxyq, Pxyg, Pxgy, Pxyy = sympy.symbols(\"P^x_q\\gamma P^x_\\gamma\\ q P^x_\\gamma\\ g P^x_g\\gamma P^x_\\gamma\\gamma\")\n", + "eu2, ed2, es2 = sympy.symbols(\"e_u^2 e_d^2 e_\\Sigma^2\") # charges" + ] + }, + { + "cell_type": "markdown", + "id": "bdf7ad8c", + "metadata": {}, + "source": [ + "## Flavor basis :\n", + "### [g, \\gamma, u+, u-, d+, d-, s+, s-, c+, c-, b+, b-, t+, t-]\n", + "## Singlet basis :\n", + "### [g, \\gamma, \\Sigma_u, \\Sigma_d, V_u, V_d, T_3^d, V_3^d, T_3^u / c+, V_3^u / c-, T_8^d / b+, V_8^d / b-, T_8^u / t+, V_8^u / t-]\n", + "## Intrinsic evolution basis :\n", + "### [g, \\gamma, \\Sigma, \\Delta_\\Sigma, V, \\Delta_V, T_3^d, V_3^d, T_3^u / c+, V_3^u / c-, T_8^d / b+, V_8^d / b-, T_8^u / t+, V_8^u / t-]\n", + "### [T_3^u, V_3^u, T_8^d, V_8^d, T_8^u, V_8^u] is:\n", + "### [c+, c-, b+, b-, t+, t-] in nf=3,\n", + "### [T_3^u, V_3^u, b+, b-, t+, t-] in nf=4,\n", + "### [T_3^u, V_3^u, T_8^d, V_8^d, t+, t-] in nf=5,\n", + "### [T_3^u, V_3^u, T_8^d, V_8^d, T_8^u, V_8^u] in nf=6" + ] + }, + { + "cell_type": "code", + "execution_count": 314, + "id": "9764233f-9c53-4860-b7c3-2be12fbec857", + "metadata": {}, + "outputs": [], + "source": [ + "P = {}\n", + "ns, s, qed, qcd = \"ns\", \"s\", \"qed\", \"qcd\"\n", + "P[ns, qcd] = sympy.Array.zeros(14,14).as_mutable()\n", + "P[ns, qed] = sympy.Array.zeros(14,14).as_mutable()\n", + "P[s, qcd] = sympy.Array.zeros(14,14).as_mutable()\n", + "P[s, qed] = sympy.Array.zeros(14,14).as_mutable()\n", + "\n", + "ei2=[eu2, ed2, ed2, eu2, ed2, eu2]\n", + "def es2_(nf):\n", + " nu = int(nf/2)\n", + " nd = nf - nu\n", + " return nu*eu2 + nd*ed2\n", + "\n", + "def P_qcd(nf):\n", + " res = sympy.Matrix.zeros(14,14).as_mutable()\n", + " res[0, 0]=Pgg\n", + " for i in range(1, nf+1):\n", + " res[0, 2*i] = Pgq #g q+\n", + " res[2*i, 0] = 2 * Pqg #q+ g\n", + " res[2*i,2*i] = Pp #q+ q+\n", + " res[1 + 2*i,1 + 2*i] = Pm #q- q-\n", + " return res\n", + "\n", + "def P_qed(nf):\n", + " es2=es2_(nf)\n", + " res = sympy.Matrix.zeros(14,14).as_mutable()\n", + " res[0, 0]=es2*Pxgg\n", + " res[1, 1]=es2*Pxyy\n", + " res[0, 1]=es2*Pxgy\n", + " res[1, 0]=es2*Pxyg\n", + " for i in range(1, nf+1):\n", + " res[0, 2*i] = ei2[i-1]*Pxgq\n", + " res[2*i, 0] = 2*ei2[i-1]*Pxqg\n", + " res[1, 2*i] = ei2[i-1]*Pxyq\n", + " res[2*i, 1] = 2*ei2[i-1]*Pxqy\n", + " res[2*i,2*i] = ei2[i-1]*Pxp \n", + " res[1 + 2*i,1 + 2*i] = ei2[i-1]*Pxm\n", + " return res\n", + "\n", + "def Ps_qcd(nf):\n", + " res = sympy.Matrix.zeros(14,14).as_mutable()\n", + " for i in range(1, nf+1):\n", + " res[2*i, 2] = Pqq - Pp\n", + " res[2*i, 3] = Pqq - Pp\n", + " res[1 + 2*i, 4] = Pv - Pm\n", + " res[1 + 2*i, 5] = Pv - Pm\n", + " return res/nf\n", + "\n", + "def Ps_qed(nf):\n", + " res = sympy.Matrix.zeros(14,14).as_mutable()\n", + " for i in range(1, nf+1):\n", + " res[2*i, 2] = ei2[i-1]*eu2*(Pxqq - Pxp)\n", + " res[2*i, 3] = ei2[i-1]*ed2*(Pxqq - Pxp)\n", + " return res/nf\n", + "\n", + "def P_uni(nf):\n", + " return P_qcd(nf)+P_qed(nf)\n", + "\n", + "def Ps_uni(nf):\n", + " return Ps_qcd(nf)+Ps_qed(nf)" + ] + }, + { + "cell_type": "code", + "execution_count": 315, + "id": "3a376dcf", + "metadata": {}, + "outputs": [], + "source": [ + "def rot_fl_to_ev(nf):\n", + " nu = int(nf/2)\n", + " nd = nf - nu\n", + " res = sympy.Matrix.zeros(14,14).as_mutable()\n", + " res[0, 0]=1\n", + " res[1, 1]=1\n", + " for i in range(2 + 2*nf, 14): \n", + " res[i,i] = 1\n", + " for i in range(1,nf+1): #Sigma and V\n", + " res[2, 2*i] = 1\n", + " res[4, 1 + 2*i] = 1\n", + " for i in [1, 4, 6]:#loop on up quarks\n", + " if i <= nf:\n", + " res[3, 2*i] = nd/nu\n", + " res[5,1 + 2*i] = nd/nu\n", + " for i in [2, 3, 5]:#loop on down quarks\n", + " if i <= nf:\n", + " res[3, 2*i] = -1\n", + " res[5, 1 + 2*i] = -1\n", + " if nf >= 3 :\n", + " res[6, 4] = 1\n", + " res[6, 6] = -1\n", + " res[7, 5] = 1\n", + " res[7, 7] = -1\n", + " if nf >= 4 :\n", + " res[8, 2] = 1\n", + " res[8, 8] = -1\n", + " res[9, 3] = 1\n", + " res[9, 9] = -1\n", + " if nf >= 5 :\n", + " res[10, 4] = 1\n", + " res[10, 6] = 1\n", + " res[10, 10] = -2\n", + " res[11, 5] = 1\n", + " res[11, 7] = 1\n", + " res[11, 11] = -2\n", + " if nf == 6 :\n", + " res[12, 2] = 1\n", + " res[12, 8] = 1\n", + " res[12, 12] = -2\n", + " res[13, 3] = 1\n", + " res[13, 9] = 1\n", + " res[13, 13] = -2\n", + " return res\n", + "\n", + "def rot_ev_to_fl(nf):\n", + " return rot_fl_to_ev(nf).inv()" + ] + }, + { + "cell_type": "code", + "execution_count": 316, + "id": "b5747379", + "metadata": {}, + "outputs": [], + "source": [ + "def rot_sin_to_ev(nf):\n", + " nu = int(nf/2)\n", + " nd = nf - nu\n", + " res = sympy.Matrix.zeros(14,14).as_mutable()\n", + " res[0, 0]=1\n", + " res[1, 1]=1\n", + " res[2,2]=1\n", + " res[2,3]=1\n", + " res[3,2]=nd/nu\n", + " res[3,3]=-1\n", + " res[4,4]=1\n", + " res[4,5]=1\n", + " res[5,4]=nd/nu\n", + " res[5,5]=-1\n", + " for i in range(6,14):\n", + " res[i,i]=1\n", + " return res\n", + "\n", + "def rot_ev_to_sin(nf):\n", + " return rot_sin_to_ev(nf).inv()" + ] + }, + { + "cell_type": "code", + "execution_count": 317, + "id": "45fb3a16", + "metadata": {}, + "outputs": [], + "source": [ + "def P_ev(nf):\n", + " res = rot_fl_to_ev(nf) * P_uni(nf) * rot_ev_to_fl(nf) + rot_fl_to_ev(nf) * Ps_uni(nf) * rot_ev_to_sin(nf)\n", + " return res" + ] + }, + { + "cell_type": "code", + "execution_count": 318, + "id": "9b5050aa", + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\left[\\begin{array}{cccccccccccccc}P^{x}_{gg} \\left(e^{2}_{d} + e^{2}_{u}\\right) + P_{gg} & P^{x}_{g\\gamma} \\left(e^{2}_{d} + e^{2}_{u}\\right) & 0.5 P^{x}_{gq} e^{2}_{d} + 0.5 P^{x}_{gq} e^{2}_{u} + 1.0 P_{gq} & - 0.5 P^{x}_{gq} e^{2}_{d} + 0.5 P^{x}_{gq} e^{2}_{u} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\P^{x}_{\\gamma g} \\left(e^{2}_{d} + e^{2}_{u}\\right) & P^{x}_{\\gamma\\gamma} \\left(e^{2}_{d} + e^{2}_{u}\\right) & 0.5 P^{x}_{\\gamma q} e^{2}_{d} + 0.5 P^{x}_{\\gamma q} e^{2}_{u} & - 0.5 P^{x}_{\\gamma q} e^{2}_{d} + 0.5 P^{x}_{\\gamma q} e^{2}_{u} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\2 P^{x}_{qg} e^{2}_{d} + 2 P^{x}_{qg} e^{2}_{u} + 4 P_{qg} & 2 P^{x}_{q\\gamma} e^{2}_{d} + 2 P^{x}_{q\\gamma} e^{2}_{u} & 0.5 P^{x}_{+} e^{2}_{d} + 0.5 P^{x}_{+} e^{2}_{u} + 1.0 P_{qq} + 0.25 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.5 e^{2}_{d} e^{2}_{u} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.25 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & - 0.5 P^{x}_{+} e^{2}_{d} + 0.5 P^{x}_{+} e^{2}_{u} - 0.25 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.25 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\- 2 P^{x}_{qg} e^{2}_{d} + 2.0 P^{x}_{qg} e^{2}_{u} & - 2 P^{x}_{q\\gamma} e^{2}_{d} + 2.0 P^{x}_{q\\gamma} e^{2}_{u} & - 0.5 P^{x}_{+} e^{2}_{d} + 0.5 P^{x}_{+} e^{2}_{u} - 0.25 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.25 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & 0.5 P^{x}_{+} e^{2}_{d} + 0.5 P^{x}_{+} e^{2}_{u} + 1.0 P_{+} + 0.25 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) - 0.5 e^{2}_{d} e^{2}_{u} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.25 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0.5 P^{x}_{-} e^{2}_{d} + 0.5 P^{x}_{-} e^{2}_{u} + 1.0 P_{V} & - 0.5 P^{x}_{-} e^{2}_{d} + 0.5 P^{x}_{-} e^{2}_{u} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & - 0.5 P^{x}_{-} e^{2}_{d} + 0.5 P^{x}_{-} e^{2}_{u} & 0.5 P^{x}_{-} e^{2}_{d} + 0.5 P^{x}_{-} e^{2}_{u} + 1.0 P_{-} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\end{array}\\right]$" + ], + "text/plain": [ + "Matrix([\n", + "[ P^x_gg*(e_d^2 + e_u^2) + P_gg, P^x_g\\gamma*(e_d^2 + e_u^2), 0.5*P^x_gq*e_d^2 + 0.5*P^x_gq*e_u^2 + 1.0*P_gq, -0.5*P^x_gq*e_d^2 + 0.5*P^x_gq*e_u^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ P^x_\\gamma g*(e_d^2 + e_u^2), P^x_\\gamma\\gamma*(e_d^2 + e_u^2), 0.5*P^x_\\gamma q*e_d^2 + 0.5*P^x_\\gamma q*e_u^2, -0.5*P^x_\\gamma q*e_d^2 + 0.5*P^x_\\gamma q*e_u^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[2*P^x_qg*e_d^2 + 2*P^x_qg*e_u^2 + 4*P_qg, 2*P^x_q\\gamma*e_d^2 + 2*P^x_q\\gamma*e_u^2, 0.5*P^x_+*e_d^2 + 0.5*P^x_+*e_u^2 + 1.0*P_qq + 0.25*e_d^2**2*(-P^x_+ + P^x_qq) + 0.5*e_d^2*e_u^2*(-P^x_+ + P^x_qq) + 0.25*e_u^2**2*(-P^x_+ + P^x_qq), -0.5*P^x_+*e_d^2 + 0.5*P^x_+*e_u^2 - 0.25*e_d^2**2*(-P^x_+ + P^x_qq) + 0.25*e_u^2**2*(-P^x_+ + P^x_qq), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ -2*P^x_qg*e_d^2 + 2.0*P^x_qg*e_u^2, -2*P^x_q\\gamma*e_d^2 + 2.0*P^x_q\\gamma*e_u^2, -0.5*P^x_+*e_d^2 + 0.5*P^x_+*e_u^2 - 0.25*e_d^2**2*(-P^x_+ + P^x_qq) + 0.25*e_u^2**2*(-P^x_+ + P^x_qq), 0.5*P^x_+*e_d^2 + 0.5*P^x_+*e_u^2 + 1.0*P_+ + 0.25*e_d^2**2*(-P^x_+ + P^x_qq) - 0.5*e_d^2*e_u^2*(-P^x_+ + P^x_qq) + 0.25*e_u^2**2*(-P^x_+ + P^x_qq), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0.5*P^x_-*e_d^2 + 0.5*P^x_-*e_u^2 + 1.0*P_V, -0.5*P^x_-*e_d^2 + 0.5*P^x_-*e_u^2, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, -0.5*P^x_-*e_d^2 + 0.5*P^x_-*e_u^2, 0.5*P^x_-*e_d^2 + 0.5*P^x_-*e_u^2 + 1.0*P_-, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]])" + ] + }, + "execution_count": 318, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "P_ev(2)" + ] + }, + { + "cell_type": "code", + "execution_count": 319, + "id": "07b1874a", + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\left[\\begin{array}{cccccccccccccc}P^{x}_{gg} \\left(2 e^{2}_{d} + e^{2}_{u}\\right) + P_{gg} & P^{x}_{g\\gamma} \\left(2 e^{2}_{d} + e^{2}_{u}\\right) & 0.666666666666667 P^{x}_{gq} e^{2}_{d} + 0.333333333333333 P^{x}_{gq} e^{2}_{u} + 1.0 P_{gq} & - 0.333333333333333 P^{x}_{gq} e^{2}_{d} + 0.333333333333333 P^{x}_{gq} e^{2}_{u} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\P^{x}_{\\gamma g} \\left(2 e^{2}_{d} + e^{2}_{u}\\right) & P^{x}_{\\gamma\\gamma} \\left(2 e^{2}_{d} + e^{2}_{u}\\right) & 0.666666666666667 P^{x}_{\\gamma q} e^{2}_{d} + 0.333333333333333 P^{x}_{\\gamma q} e^{2}_{u} & - 0.333333333333333 P^{x}_{\\gamma q} e^{2}_{d} + 0.333333333333333 P^{x}_{\\gamma q} e^{2}_{u} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\4 P^{x}_{qg} e^{2}_{d} + 2 P^{x}_{qg} e^{2}_{u} + 6 P_{qg} & 4 P^{x}_{q\\gamma} e^{2}_{d} + 2 P^{x}_{q\\gamma} e^{2}_{u} & 0.666666666666667 P^{x}_{+} e^{2}_{d} + 0.333333333333333 P^{x}_{+} e^{2}_{u} + 1.0 P_{qq} + 0.444444444444444 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.444444444444444 e^{2}_{d} e^{2}_{u} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.111111111111111 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & - 0.333333333333333 P^{x}_{+} e^{2}_{d} + 0.333333333333333 P^{x}_{+} e^{2}_{u} - 0.222222222222222 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.111111111111111 e^{2}_{d} e^{2}_{u} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.111111111111111 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\- 4 P^{x}_{qg} e^{2}_{d} + 4.0 P^{x}_{qg} e^{2}_{u} & - 4 P^{x}_{q\\gamma} e^{2}_{d} + 4.0 P^{x}_{q\\gamma} e^{2}_{u} & - 0.666666666666667 P^{x}_{+} e^{2}_{d} + 0.666666666666667 P^{x}_{+} e^{2}_{u} - 0.444444444444444 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.222222222222222 e^{2}_{d} e^{2}_{u} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.222222222222222 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & 0.333333333333333 P^{x}_{+} e^{2}_{d} + 0.666666666666667 P^{x}_{+} e^{2}_{u} + 1.0 P_{+} + 0.222222222222222 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) - 0.444444444444444 e^{2}_{d} e^{2}_{u} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.222222222222222 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0.666666666666667 P^{x}_{-} e^{2}_{d} + 0.333333333333333 P^{x}_{-} e^{2}_{u} + 1.0 P_{V} & - 0.333333333333333 P^{x}_{-} e^{2}_{d} + 0.333333333333333 P^{x}_{-} e^{2}_{u} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & - 0.666666666666667 P^{x}_{-} e^{2}_{d} + 0.666666666666667 P^{x}_{-} e^{2}_{u} & 0.333333333333333 P^{x}_{-} e^{2}_{d} + 0.666666666666667 P^{x}_{-} e^{2}_{u} + 1.0 P_{-} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{+} e^{2}_{d} + 1.0 P_{+} & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{-} e^{2}_{d} + 1.0 P_{-} & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\end{array}\\right]$" + ], + "text/plain": [ + "Matrix([\n", + "[ P^x_gg*(2*e_d^2 + e_u^2) + P_gg, P^x_g\\gamma*(2*e_d^2 + e_u^2), 0.666666666666667*P^x_gq*e_d^2 + 0.333333333333333*P^x_gq*e_u^2 + 1.0*P_gq, -0.333333333333333*P^x_gq*e_d^2 + 0.333333333333333*P^x_gq*e_u^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ P^x_\\gamma g*(2*e_d^2 + e_u^2), P^x_\\gamma\\gamma*(2*e_d^2 + e_u^2), 0.666666666666667*P^x_\\gamma q*e_d^2 + 0.333333333333333*P^x_\\gamma q*e_u^2, -0.333333333333333*P^x_\\gamma q*e_d^2 + 0.333333333333333*P^x_\\gamma q*e_u^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[4*P^x_qg*e_d^2 + 2*P^x_qg*e_u^2 + 6*P_qg, 4*P^x_q\\gamma*e_d^2 + 2*P^x_q\\gamma*e_u^2, 0.666666666666667*P^x_+*e_d^2 + 0.333333333333333*P^x_+*e_u^2 + 1.0*P_qq + 0.444444444444444*e_d^2**2*(-P^x_+ + P^x_qq) + 0.444444444444444*e_d^2*e_u^2*(-P^x_+ + P^x_qq) + 0.111111111111111*e_u^2**2*(-P^x_+ + P^x_qq), -0.333333333333333*P^x_+*e_d^2 + 0.333333333333333*P^x_+*e_u^2 - 0.222222222222222*e_d^2**2*(-P^x_+ + P^x_qq) + 0.111111111111111*e_d^2*e_u^2*(-P^x_+ + P^x_qq) + 0.111111111111111*e_u^2**2*(-P^x_+ + P^x_qq), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ -4*P^x_qg*e_d^2 + 4.0*P^x_qg*e_u^2, -4*P^x_q\\gamma*e_d^2 + 4.0*P^x_q\\gamma*e_u^2, -0.666666666666667*P^x_+*e_d^2 + 0.666666666666667*P^x_+*e_u^2 - 0.444444444444444*e_d^2**2*(-P^x_+ + P^x_qq) + 0.222222222222222*e_d^2*e_u^2*(-P^x_+ + P^x_qq) + 0.222222222222222*e_u^2**2*(-P^x_+ + P^x_qq), 0.333333333333333*P^x_+*e_d^2 + 0.666666666666667*P^x_+*e_u^2 + 1.0*P_+ + 0.222222222222222*e_d^2**2*(-P^x_+ + P^x_qq) - 0.444444444444444*e_d^2*e_u^2*(-P^x_+ + P^x_qq) + 0.222222222222222*e_u^2**2*(-P^x_+ + P^x_qq), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0.666666666666667*P^x_-*e_d^2 + 0.333333333333333*P^x_-*e_u^2 + 1.0*P_V, -0.333333333333333*P^x_-*e_d^2 + 0.333333333333333*P^x_-*e_u^2, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, -0.666666666666667*P^x_-*e_d^2 + 0.666666666666667*P^x_-*e_u^2, 0.333333333333333*P^x_-*e_d^2 + 0.666666666666667*P^x_-*e_u^2 + 1.0*P_-, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 1.0*P^x_+*e_d^2 + 1.0*P_+, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 1.0*P^x_-*e_d^2 + 1.0*P_-, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]])" + ] + }, + "execution_count": 319, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "P_ev(3)" + ] + }, + { + "cell_type": "code", + "execution_count": 320, + "id": "c9f75de1", + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\left[\\begin{array}{cccccccccccccc}P^{x}_{gg} \\left(2 e^{2}_{d} + 2 e^{2}_{u}\\right) + P_{gg} & P^{x}_{g\\gamma} \\left(2 e^{2}_{d} + 2 e^{2}_{u}\\right) & 0.5 P^{x}_{gq} e^{2}_{d} + 0.5 P^{x}_{gq} e^{2}_{u} + 1.0 P_{gq} & - 0.5 P^{x}_{gq} e^{2}_{d} + 0.5 P^{x}_{gq} e^{2}_{u} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\P^{x}_{\\gamma g} \\left(2 e^{2}_{d} + 2 e^{2}_{u}\\right) & P^{x}_{\\gamma\\gamma} \\left(2 e^{2}_{d} + 2 e^{2}_{u}\\right) & 0.5 P^{x}_{\\gamma q} e^{2}_{d} + 0.5 P^{x}_{\\gamma q} e^{2}_{u} & - 0.5 P^{x}_{\\gamma q} e^{2}_{d} + 0.5 P^{x}_{\\gamma q} e^{2}_{u} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\4 P^{x}_{qg} e^{2}_{d} + 4 P^{x}_{qg} e^{2}_{u} + 8 P_{qg} & 4 P^{x}_{q\\gamma} e^{2}_{d} + 4 P^{x}_{q\\gamma} e^{2}_{u} & 0.5 P^{x}_{+} e^{2}_{d} + 0.5 P^{x}_{+} e^{2}_{u} + 1.0 P_{qq} + 0.25 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.5 e^{2}_{d} e^{2}_{u} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.25 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & - 0.5 P^{x}_{+} e^{2}_{d} + 0.5 P^{x}_{+} e^{2}_{u} - 0.25 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.25 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\- 4 P^{x}_{qg} e^{2}_{d} + 4.0 P^{x}_{qg} e^{2}_{u} & - 4 P^{x}_{q\\gamma} e^{2}_{d} + 4.0 P^{x}_{q\\gamma} e^{2}_{u} & - 0.5 P^{x}_{+} e^{2}_{d} + 0.5 P^{x}_{+} e^{2}_{u} - 0.25 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.25 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & 0.5 P^{x}_{+} e^{2}_{d} + 0.5 P^{x}_{+} e^{2}_{u} + 1.0 P_{+} + 0.25 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) - 0.5 e^{2}_{d} e^{2}_{u} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.25 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0.5 P^{x}_{-} e^{2}_{d} + 0.5 P^{x}_{-} e^{2}_{u} + 1.0 P_{V} & - 0.5 P^{x}_{-} e^{2}_{d} + 0.5 P^{x}_{-} e^{2}_{u} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & - 0.5 P^{x}_{-} e^{2}_{d} + 0.5 P^{x}_{-} e^{2}_{u} & 0.5 P^{x}_{-} e^{2}_{d} + 0.5 P^{x}_{-} e^{2}_{u} + 1.0 P_{-} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{+} e^{2}_{d} + 1.0 P_{+} & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{-} e^{2}_{d} + 1.0 P_{-} & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{+} e^{2}_{u} + 1.0 P_{+} & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{-} e^{2}_{u} + 1.0 P_{-} & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\end{array}\\right]$" + ], + "text/plain": [ + "Matrix([\n", + "[ P^x_gg*(2*e_d^2 + 2*e_u^2) + P_gg, P^x_g\\gamma*(2*e_d^2 + 2*e_u^2), 0.5*P^x_gq*e_d^2 + 0.5*P^x_gq*e_u^2 + 1.0*P_gq, -0.5*P^x_gq*e_d^2 + 0.5*P^x_gq*e_u^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ P^x_\\gamma g*(2*e_d^2 + 2*e_u^2), P^x_\\gamma\\gamma*(2*e_d^2 + 2*e_u^2), 0.5*P^x_\\gamma q*e_d^2 + 0.5*P^x_\\gamma q*e_u^2, -0.5*P^x_\\gamma q*e_d^2 + 0.5*P^x_\\gamma q*e_u^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[4*P^x_qg*e_d^2 + 4*P^x_qg*e_u^2 + 8*P_qg, 4*P^x_q\\gamma*e_d^2 + 4*P^x_q\\gamma*e_u^2, 0.5*P^x_+*e_d^2 + 0.5*P^x_+*e_u^2 + 1.0*P_qq + 0.25*e_d^2**2*(-P^x_+ + P^x_qq) + 0.5*e_d^2*e_u^2*(-P^x_+ + P^x_qq) + 0.25*e_u^2**2*(-P^x_+ + P^x_qq), -0.5*P^x_+*e_d^2 + 0.5*P^x_+*e_u^2 - 0.25*e_d^2**2*(-P^x_+ + P^x_qq) + 0.25*e_u^2**2*(-P^x_+ + P^x_qq), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ -4*P^x_qg*e_d^2 + 4.0*P^x_qg*e_u^2, -4*P^x_q\\gamma*e_d^2 + 4.0*P^x_q\\gamma*e_u^2, -0.5*P^x_+*e_d^2 + 0.5*P^x_+*e_u^2 - 0.25*e_d^2**2*(-P^x_+ + P^x_qq) + 0.25*e_u^2**2*(-P^x_+ + P^x_qq), 0.5*P^x_+*e_d^2 + 0.5*P^x_+*e_u^2 + 1.0*P_+ + 0.25*e_d^2**2*(-P^x_+ + P^x_qq) - 0.5*e_d^2*e_u^2*(-P^x_+ + P^x_qq) + 0.25*e_u^2**2*(-P^x_+ + P^x_qq), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0.5*P^x_-*e_d^2 + 0.5*P^x_-*e_u^2 + 1.0*P_V, -0.5*P^x_-*e_d^2 + 0.5*P^x_-*e_u^2, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, -0.5*P^x_-*e_d^2 + 0.5*P^x_-*e_u^2, 0.5*P^x_-*e_d^2 + 0.5*P^x_-*e_u^2 + 1.0*P_-, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 1.0*P^x_+*e_d^2 + 1.0*P_+, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 1.0*P^x_-*e_d^2 + 1.0*P_-, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 1.0*P^x_+*e_u^2 + 1.0*P_+, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1.0*P^x_-*e_u^2 + 1.0*P_-, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]])" + ] + }, + "execution_count": 320, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "P_ev(4)" + ] + }, + { + "cell_type": "code", + "execution_count": 321, + "id": "ce40d361", + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\left[\\begin{array}{cccccccccccccc}P^{x}_{gg} \\left(3 e^{2}_{d} + 2 e^{2}_{u}\\right) + P_{gg} & P^{x}_{g\\gamma} \\left(3 e^{2}_{d} + 2 e^{2}_{u}\\right) & 0.6 P^{x}_{gq} e^{2}_{d} + 0.4 P^{x}_{gq} e^{2}_{u} + 1.0 P_{gq} & - 0.4 P^{x}_{gq} e^{2}_{d} + 0.4 P^{x}_{gq} e^{2}_{u} + 2.77555756156289 \\cdot 10^{-17} P_{gq} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\P^{x}_{\\gamma g} \\left(3 e^{2}_{d} + 2 e^{2}_{u}\\right) & P^{x}_{\\gamma\\gamma} \\left(3 e^{2}_{d} + 2 e^{2}_{u}\\right) & 0.6 P^{x}_{\\gamma q} e^{2}_{d} + 0.4 P^{x}_{\\gamma q} e^{2}_{u} & - 0.4 P^{x}_{\\gamma q} e^{2}_{d} + 0.4 P^{x}_{\\gamma q} e^{2}_{u} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\6 P^{x}_{qg} e^{2}_{d} + 4 P^{x}_{qg} e^{2}_{u} + 10 P_{qg} & 6 P^{x}_{q\\gamma} e^{2}_{d} + 4 P^{x}_{q\\gamma} e^{2}_{u} & 0.6 P^{x}_{+} e^{2}_{d} + 0.4 P^{x}_{+} e^{2}_{u} + 1.0 P_{qq} + 0.36 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.48 e^{2}_{d} e^{2}_{u} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.16 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & - 0.4 P^{x}_{+} e^{2}_{d} + 0.4 P^{x}_{+} e^{2}_{u} + 2.77555756156289 \\cdot 10^{-17} P_{+} - 0.24 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.08 e^{2}_{d} e^{2}_{u} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.16 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\- 6 P^{x}_{qg} e^{2}_{d} + 6.0 P^{x}_{qg} e^{2}_{u} & - 6 P^{x}_{q\\gamma} e^{2}_{d} + 6.0 P^{x}_{q\\gamma} e^{2}_{u} & - 0.6 P^{x}_{+} e^{2}_{d} + 0.6 P^{x}_{+} e^{2}_{u} + 5.55111512312578 \\cdot 10^{-17} P_{qq} - 0.36 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.12 e^{2}_{d} e^{2}_{u} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.24 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & 0.4 P^{x}_{+} e^{2}_{d} + 0.6 P^{x}_{+} e^{2}_{u} + 1.0 P_{+} + 0.24 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) - 0.48 e^{2}_{d} e^{2}_{u} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.24 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0.6 P^{x}_{-} e^{2}_{d} + 0.4 P^{x}_{-} e^{2}_{u} + 1.0 P_{V} & - 0.4 P^{x}_{-} e^{2}_{d} + 0.4 P^{x}_{-} e^{2}_{u} + 2.77555756156289 \\cdot 10^{-17} P_{-} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & - 0.6 P^{x}_{-} e^{2}_{d} + 0.6 P^{x}_{-} e^{2}_{u} + 5.55111512312578 \\cdot 10^{-17} P_{V} & 0.4 P^{x}_{-} e^{2}_{d} + 0.6 P^{x}_{-} e^{2}_{u} + 1.0 P_{-} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{+} e^{2}_{d} + 1.0 P_{+} & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{-} e^{2}_{d} + 1.0 P_{-} & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{+} e^{2}_{u} + 1.0 P_{+} & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{-} e^{2}_{u} + 1.0 P_{-} & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{+} e^{2}_{d} + 1.0 P_{+} & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{-} e^{2}_{d} + 1.0 P_{-} & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\end{array}\\right]$" + ], + "text/plain": [ + "Matrix([\n", + "[ P^x_gg*(3*e_d^2 + 2*e_u^2) + P_gg, P^x_g\\gamma*(3*e_d^2 + 2*e_u^2), 0.6*P^x_gq*e_d^2 + 0.4*P^x_gq*e_u^2 + 1.0*P_gq, -0.4*P^x_gq*e_d^2 + 0.4*P^x_gq*e_u^2 + 2.77555756156289e-17*P_gq, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ P^x_\\gamma g*(3*e_d^2 + 2*e_u^2), P^x_\\gamma\\gamma*(3*e_d^2 + 2*e_u^2), 0.6*P^x_\\gamma q*e_d^2 + 0.4*P^x_\\gamma q*e_u^2, -0.4*P^x_\\gamma q*e_d^2 + 0.4*P^x_\\gamma q*e_u^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[6*P^x_qg*e_d^2 + 4*P^x_qg*e_u^2 + 10*P_qg, 6*P^x_q\\gamma*e_d^2 + 4*P^x_q\\gamma*e_u^2, 0.6*P^x_+*e_d^2 + 0.4*P^x_+*e_u^2 + 1.0*P_qq + 0.36*e_d^2**2*(-P^x_+ + P^x_qq) + 0.48*e_d^2*e_u^2*(-P^x_+ + P^x_qq) + 0.16*e_u^2**2*(-P^x_+ + P^x_qq), -0.4*P^x_+*e_d^2 + 0.4*P^x_+*e_u^2 + 2.77555756156289e-17*P_+ - 0.24*e_d^2**2*(-P^x_+ + P^x_qq) + 0.08*e_d^2*e_u^2*(-P^x_+ + P^x_qq) + 0.16*e_u^2**2*(-P^x_+ + P^x_qq), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ -6*P^x_qg*e_d^2 + 6.0*P^x_qg*e_u^2, -6*P^x_q\\gamma*e_d^2 + 6.0*P^x_q\\gamma*e_u^2, -0.6*P^x_+*e_d^2 + 0.6*P^x_+*e_u^2 + 5.55111512312578e-17*P_qq - 0.36*e_d^2**2*(-P^x_+ + P^x_qq) + 0.12*e_d^2*e_u^2*(-P^x_+ + P^x_qq) + 0.24*e_u^2**2*(-P^x_+ + P^x_qq), 0.4*P^x_+*e_d^2 + 0.6*P^x_+*e_u^2 + 1.0*P_+ + 0.24*e_d^2**2*(-P^x_+ + P^x_qq) - 0.48*e_d^2*e_u^2*(-P^x_+ + P^x_qq) + 0.24*e_u^2**2*(-P^x_+ + P^x_qq), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0.6*P^x_-*e_d^2 + 0.4*P^x_-*e_u^2 + 1.0*P_V, -0.4*P^x_-*e_d^2 + 0.4*P^x_-*e_u^2 + 2.77555756156289e-17*P_-, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, -0.6*P^x_-*e_d^2 + 0.6*P^x_-*e_u^2 + 5.55111512312578e-17*P_V, 0.4*P^x_-*e_d^2 + 0.6*P^x_-*e_u^2 + 1.0*P_-, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 1.0*P^x_+*e_d^2 + 1.0*P_+, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 1.0*P^x_-*e_d^2 + 1.0*P_-, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 1.0*P^x_+*e_u^2 + 1.0*P_+, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1.0*P^x_-*e_u^2 + 1.0*P_-, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1.0*P^x_+*e_d^2 + 1.0*P_+, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1.0*P^x_-*e_d^2 + 1.0*P_-, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]])" + ] + }, + "execution_count": 321, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "P_ev(5)" + ] + }, + { + "cell_type": "code", + "execution_count": 322, + "id": "3f014fa2", + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\left[\\begin{array}{cccccccccccccc}P^{x}_{gg} \\left(3 e^{2}_{d} + 3 e^{2}_{u}\\right) + P_{gg} & P^{x}_{g\\gamma} \\left(3 e^{2}_{d} + 3 e^{2}_{u}\\right) & 0.5 P^{x}_{gq} e^{2}_{d} + 0.5 P^{x}_{gq} e^{2}_{u} + 1.0 P_{gq} & - 0.5 P^{x}_{gq} e^{2}_{d} + 0.5 P^{x}_{gq} e^{2}_{u} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\P^{x}_{\\gamma g} \\left(3 e^{2}_{d} + 3 e^{2}_{u}\\right) & P^{x}_{\\gamma\\gamma} \\left(3 e^{2}_{d} + 3 e^{2}_{u}\\right) & 0.5 P^{x}_{\\gamma q} e^{2}_{d} + 0.5 P^{x}_{\\gamma q} e^{2}_{u} & - 0.5 P^{x}_{\\gamma q} e^{2}_{d} + 0.5 P^{x}_{\\gamma q} e^{2}_{u} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\6 P^{x}_{qg} e^{2}_{d} + 6 P^{x}_{qg} e^{2}_{u} + 12 P_{qg} & 6 P^{x}_{q\\gamma} e^{2}_{d} + 6 P^{x}_{q\\gamma} e^{2}_{u} & 0.5 P^{x}_{+} e^{2}_{d} + 0.5 P^{x}_{+} e^{2}_{u} - 1.11022302462516 \\cdot 10^{-16} P_{+} + 1.0 P_{qq} + 0.25 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.5 e^{2}_{d} e^{2}_{u} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.25 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & - 0.5 P^{x}_{+} e^{2}_{d} + 0.5 P^{x}_{+} e^{2}_{u} - 0.25 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.25 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\- 6 P^{x}_{qg} e^{2}_{d} + 6.0 P^{x}_{qg} e^{2}_{u} & - 6 P^{x}_{q\\gamma} e^{2}_{d} + 6.0 P^{x}_{q\\gamma} e^{2}_{u} & - 0.5 P^{x}_{+} e^{2}_{d} + 0.5 P^{x}_{+} e^{2}_{u} - 0.25 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.25 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & 0.5 P^{x}_{+} e^{2}_{d} + 0.5 P^{x}_{+} e^{2}_{u} + 1.0 P_{+} + 0.25 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) - 0.5 e^{2}_{d} e^{2}_{u} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.25 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0.5 P^{x}_{-} e^{2}_{d} + 0.5 P^{x}_{-} e^{2}_{u} - 1.11022302462516 \\cdot 10^{-16} P_{-} + 1.0 P_{V} & - 0.5 P^{x}_{-} e^{2}_{d} + 0.5 P^{x}_{-} e^{2}_{u} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & - 0.5 P^{x}_{-} e^{2}_{d} + 0.5 P^{x}_{-} e^{2}_{u} & 0.5 P^{x}_{-} e^{2}_{d} + 0.5 P^{x}_{-} e^{2}_{u} + 1.0 P_{-} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{+} e^{2}_{d} + 1.0 P_{+} & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{-} e^{2}_{d} + 1.0 P_{-} & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{+} e^{2}_{u} + 1.0 P_{+} & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{-} e^{2}_{u} + 1.0 P_{-} & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{+} e^{2}_{d} + 1.0 P_{+} & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{-} e^{2}_{d} + 1.0 P_{-} & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{+} e^{2}_{u} + 1.0 P_{+} & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{-} e^{2}_{u} + 1.0 P_{-}\\end{array}\\right]$" + ], + "text/plain": [ + "Matrix([\n", + "[ P^x_gg*(3*e_d^2 + 3*e_u^2) + P_gg, P^x_g\\gamma*(3*e_d^2 + 3*e_u^2), 0.5*P^x_gq*e_d^2 + 0.5*P^x_gq*e_u^2 + 1.0*P_gq, -0.5*P^x_gq*e_d^2 + 0.5*P^x_gq*e_u^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ P^x_\\gamma g*(3*e_d^2 + 3*e_u^2), P^x_\\gamma\\gamma*(3*e_d^2 + 3*e_u^2), 0.5*P^x_\\gamma q*e_d^2 + 0.5*P^x_\\gamma q*e_u^2, -0.5*P^x_\\gamma q*e_d^2 + 0.5*P^x_\\gamma q*e_u^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[6*P^x_qg*e_d^2 + 6*P^x_qg*e_u^2 + 12*P_qg, 6*P^x_q\\gamma*e_d^2 + 6*P^x_q\\gamma*e_u^2, 0.5*P^x_+*e_d^2 + 0.5*P^x_+*e_u^2 - 1.11022302462516e-16*P_+ + 1.0*P_qq + 0.25*e_d^2**2*(-P^x_+ + P^x_qq) + 0.5*e_d^2*e_u^2*(-P^x_+ + P^x_qq) + 0.25*e_u^2**2*(-P^x_+ + P^x_qq), -0.5*P^x_+*e_d^2 + 0.5*P^x_+*e_u^2 - 0.25*e_d^2**2*(-P^x_+ + P^x_qq) + 0.25*e_u^2**2*(-P^x_+ + P^x_qq), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ -6*P^x_qg*e_d^2 + 6.0*P^x_qg*e_u^2, -6*P^x_q\\gamma*e_d^2 + 6.0*P^x_q\\gamma*e_u^2, -0.5*P^x_+*e_d^2 + 0.5*P^x_+*e_u^2 - 0.25*e_d^2**2*(-P^x_+ + P^x_qq) + 0.25*e_u^2**2*(-P^x_+ + P^x_qq), 0.5*P^x_+*e_d^2 + 0.5*P^x_+*e_u^2 + 1.0*P_+ + 0.25*e_d^2**2*(-P^x_+ + P^x_qq) - 0.5*e_d^2*e_u^2*(-P^x_+ + P^x_qq) + 0.25*e_u^2**2*(-P^x_+ + P^x_qq), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0.5*P^x_-*e_d^2 + 0.5*P^x_-*e_u^2 - 1.11022302462516e-16*P_- + 1.0*P_V, -0.5*P^x_-*e_d^2 + 0.5*P^x_-*e_u^2, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, -0.5*P^x_-*e_d^2 + 0.5*P^x_-*e_u^2, 0.5*P^x_-*e_d^2 + 0.5*P^x_-*e_u^2 + 1.0*P_-, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 1.0*P^x_+*e_d^2 + 1.0*P_+, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 1.0*P^x_-*e_d^2 + 1.0*P_-, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 1.0*P^x_+*e_u^2 + 1.0*P_+, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1.0*P^x_-*e_u^2 + 1.0*P_-, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1.0*P^x_+*e_d^2 + 1.0*P_+, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1.0*P^x_-*e_d^2 + 1.0*P_-, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1.0*P^x_+*e_u^2 + 1.0*P_+, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1.0*P^x_-*e_u^2 + 1.0*P_-]])" + ] + }, + "execution_count": 322, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "P_ev(6)" + ] + }, + { + "cell_type": "code", + "execution_count": 323, + "id": "53626511", + "metadata": {}, + "outputs": [], + "source": [ + "def eD2_(nf):\n", + " nu = int(nf/2)\n", + " nd = nf - nu\n", + " return nd*eu2 + nu*ed2\n", + "def etam_(nf):\n", + " nu = int(nf/2)\n", + " nd = nf - nu\n", + " return 0.5*(eu2 - ed2)" + ] + }, + { + "cell_type": "code", + "execution_count": 324, + "id": "9c9ce615", + "metadata": {}, + "outputs": [], + "source": [ + "def P_ev_sing(nf):\n", + " return P_ev(nf)[:4,:4]\n", + "\n", + "def P_ev_val(nf):\n", + " return P_ev(nf)[4:6,4:6]" + ] + }, + { + "cell_type": "code", + "execution_count": 325, + "id": "5267ab16", + "metadata": {}, + "outputs": [], + "source": [ + "def P_ev_sing2(nf):\n", + " es2=es2_(nf)\n", + " eD2=eD2_(nf)\n", + " etam=etam_(nf)\n", + " nu = int(nf/2)\n", + " nd = nf - nu\n", + " res = sympy.Matrix([\n", + " [Pgg + es2 * Pxgg, es2 * Pxgy, Pgq + es2/nf*Pxgq, 2*nu/nf*etam*Pxgq],\n", + " [es2 * Pxyg, es2 * Pxyy, es2/nf*Pxyq, 2*nu/nf*etam*Pxyq],\n", + " [2*nf*Pqg + 2*es2*Pxqg, 2*es2 * Pxqy, Pqq + es2/nf*Pxp +(es2/nf)**2*(Pxqq - Pxp), 2*nu/nf*etam*Pxp +2*nu*etam*es2/nf**2*(Pxqq - Pxp)],\n", + " [4*nd*etam*Pxqg, 4*nd*etam*Pxqy, 2*nd/nf*etam*Pxp +2*nd*etam*es2/nf**2*(Pxqq - Pxp), Pp + eD2/nf*Pxp + 4*nu*nd/nf**2*etam**2*(Pxqq - Pxp)]\n", + " ])\n", + " return res\n", + "\n", + "def P_ev_val2(nf):\n", + " es2=e2s(nf)\n", + " eD2=eD2_(nf)\n", + " etam=etam_(nf)\n", + " nu = int(nf/2)\n", + " nd = nf - nu\n", + " res = sympy.Matrix([\n", + " [Pv+es2/nf*Pxm, 2*nu/nf*etam*Pxm],\n", + " [2*nd/nf*etam*Pxm, Pm + eD2/nf*Pxm]\n", + " ])\n", + " return res" + ] + }, + { + "cell_type": "code", + "execution_count": 326, + "id": "a47dc3b2", + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\left[\\begin{matrix}0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\end{matrix}\\right]$" + ], + "text/plain": [ + "Matrix([\n", + "[0, 0, 0, 0],\n", + "[0, 0, 0, 0],\n", + "[0, 0, 0, 0],\n", + "[0, 0, 0, 0]])" + ] + }, + "execution_count": 326, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "sympy.simplify(P_ev_sing(2)-P_ev_sing2(2))" + ] + }, + { + "cell_type": "code", + "execution_count": 327, + "id": "6fddcd7c", + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\left[\\begin{matrix}0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\\\0 & 0 & 0 & - 1.11022302462516 \\cdot 10^{-16} P_{+}\\end{matrix}\\right]$" + ], + "text/plain": [ + "Matrix([\n", + "[0, 0, 0, 0],\n", + "[0, 0, 0, 0],\n", + "[0, 0, 0, 0],\n", + "[0, 0, 0, -1.11022302462516e-16*P_+]])" + ] + }, + "execution_count": 327, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "sympy.simplify(P_ev_sing(3)-P_ev_sing2(3))" + ] + }, + { + "cell_type": "code", + "execution_count": 328, + "id": "f151ecdd", + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\left[\\begin{matrix}0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\end{matrix}\\right]$" + ], + "text/plain": [ + "Matrix([\n", + "[0, 0, 0, 0],\n", + "[0, 0, 0, 0],\n", + "[0, 0, 0, 0],\n", + "[0, 0, 0, 0]])" + ] + }, + "execution_count": 328, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "sympy.simplify(P_ev_sing(4)-P_ev_sing2(4))" + ] + }, + { + "cell_type": "code", + "execution_count": 329, + "id": "e49bef49", + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\left[\\begin{matrix}0 & 0 & 8.88178419700125 \\cdot 10^{-17} P^{x}_{gq} e^{2}_{d} & 2.77555756156289 \\cdot 10^{-17} P_{gq}\\\\0 & 0 & 8.88178419700125 \\cdot 10^{-17} P^{x}_{\\gamma q} e^{2}_{d} & 0\\\\0 & 0 & 7.105427357601 \\cdot 10^{-17} P^{x}_{+} e^{2}_{d} - 3.5527136788005 \\cdot 10^{-17} P^{x}_{+} \\left(e^{2}_{u}\\right)^{2} + 3.5527136788005 \\cdot 10^{-17} P^{x}_{qq} \\left(e^{2}_{u}\\right)^{2} & 4.16333634234434 \\cdot 10^{-17} P^{x}_{+} e^{2}_{d} e^{2}_{u} - 2.77555756156289 \\cdot 10^{-17} P^{x}_{+} \\left(e^{2}_{u}\\right)^{2} - 4.16333634234434 \\cdot 10^{-17} P^{x}_{qq} e^{2}_{d} e^{2}_{u} + 2.77555756156289 \\cdot 10^{-17} P^{x}_{qq} \\left(e^{2}_{u}\\right)^{2} + 2.77555756156289 \\cdot 10^{-17} P_{+}\\\\0 & 0 & - 5.55111512312578 \\cdot 10^{-17} P^{x}_{+} e^{2}_{d} e^{2}_{u} - 1.11022302462516 \\cdot 10^{-16} P^{x}_{+} e^{2}_{d} - 5.55111512312578 \\cdot 10^{-17} P^{x}_{+} \\left(e^{2}_{u}\\right)^{2} + 1.11022302462516 \\cdot 10^{-16} P^{x}_{+} e^{2}_{u} + 5.55111512312578 \\cdot 10^{-17} P^{x}_{qq} e^{2}_{d} e^{2}_{u} + 5.55111512312578 \\cdot 10^{-17} P^{x}_{qq} \\left(e^{2}_{u}\\right)^{2} + 5.55111512312578 \\cdot 10^{-17} P_{qq} & e^{2}_{u} \\left(8.88178419700125 \\cdot 10^{-17} P^{x}_{+} e^{2}_{d} - 4.44089209850063 \\cdot 10^{-17} P^{x}_{+} e^{2}_{u} + 8.88178419700125 \\cdot 10^{-17} P^{x}_{+} - 8.88178419700125 \\cdot 10^{-17} P^{x}_{qq} e^{2}_{d} + 4.44089209850063 \\cdot 10^{-17} P^{x}_{qq} e^{2}_{u}\\right)\\end{matrix}\\right]$" + ], + "text/plain": [ + "Matrix([\n", + "[0, 0, 8.88178419700125e-17*P^x_gq*e_d^2, 2.77555756156289e-17*P_gq],\n", + "[0, 0, 8.88178419700125e-17*P^x_\\gamma q*e_d^2, 0],\n", + "[0, 0, 7.105427357601e-17*P^x_+*e_d^2 - 3.5527136788005e-17*P^x_+*e_u^2**2 + 3.5527136788005e-17*P^x_qq*e_u^2**2, 4.16333634234434e-17*P^x_+*e_d^2*e_u^2 - 2.77555756156289e-17*P^x_+*e_u^2**2 - 4.16333634234434e-17*P^x_qq*e_d^2*e_u^2 + 2.77555756156289e-17*P^x_qq*e_u^2**2 + 2.77555756156289e-17*P_+],\n", + "[0, 0, -5.55111512312578e-17*P^x_+*e_d^2*e_u^2 - 1.11022302462516e-16*P^x_+*e_d^2 - 5.55111512312578e-17*P^x_+*e_u^2**2 + 1.11022302462516e-16*P^x_+*e_u^2 + 5.55111512312578e-17*P^x_qq*e_d^2*e_u^2 + 5.55111512312578e-17*P^x_qq*e_u^2**2 + 5.55111512312578e-17*P_qq, e_u^2*(8.88178419700125e-17*P^x_+*e_d^2 - 4.44089209850063e-17*P^x_+*e_u^2 + 8.88178419700125e-17*P^x_+ - 8.88178419700125e-17*P^x_qq*e_d^2 + 4.44089209850063e-17*P^x_qq*e_u^2)]])" + ] + }, + "execution_count": 329, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "sympy.simplify(P_ev_sing(5)-P_ev_sing2(5))" + ] + }, + { + "cell_type": "code", + "execution_count": 330, + "id": "82b3e343", + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\left[\\begin{matrix}0 & 0 & - 1.11022302462516 \\cdot 10^{-16} P_{gq} & 0\\\\0 & 0 & 0 & 0\\\\0 & 0 & - 1.11022302462516 \\cdot 10^{-16} P_{+} & 0\\\\0 & 0 & 0 & - 1.11022302462516 \\cdot 10^{-16} P_{+}\\end{matrix}\\right]$" + ], + "text/plain": [ + "Matrix([\n", + "[0, 0, -1.11022302462516e-16*P_gq, 0],\n", + "[0, 0, 0, 0],\n", + "[0, 0, -1.11022302462516e-16*P_+, 0],\n", + "[0, 0, 0, -1.11022302462516e-16*P_+]])" + ] + }, + "execution_count": 330, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "sympy.simplify(P_ev_sing(6)-P_ev_sing2(6))" + ] + }, + { + "cell_type": "code", + "execution_count": 331, + "id": "a9f61d31", + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\left[\\begin{matrix}0 & 0\\\\0 & 0\\end{matrix}\\right]$" + ], + "text/plain": [ + "Matrix([\n", + "[0, 0],\n", + "[0, 0]])" + ] + }, + "execution_count": 331, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "sympy.simplify(P_ev_val(2)-P_ev_val2(2))" + ] + }, + { + "cell_type": "code", + "execution_count": 332, + "id": "3e91130a", + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\left[\\begin{matrix}0 & 0\\\\0 & - 1.11022302462516 \\cdot 10^{-16} P_{-}\\end{matrix}\\right]$" + ], + "text/plain": [ + "Matrix([\n", + "[0, 0],\n", + "[0, -1.11022302462516e-16*P_-]])" + ] + }, + "execution_count": 332, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "sympy.simplify(P_ev_val(3)-P_ev_val2(3))" + ] + }, + { + "cell_type": "code", + "execution_count": 333, + "id": "6f5e4417", + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\left[\\begin{matrix}0 & 0\\\\0 & 0\\end{matrix}\\right]$" + ], + "text/plain": [ + "Matrix([\n", + "[0, 0],\n", + "[0, 0]])" + ] + }, + "execution_count": 333, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "sympy.simplify(P_ev_val(4)-P_ev_val2(4))" + ] + }, + { + "cell_type": "code", + "execution_count": 334, + "id": "768a92b2", + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\left[\\begin{matrix}8.88178419700125 \\cdot 10^{-17} P^{x}_{-} e^{2}_{d} & 2.77555756156289 \\cdot 10^{-17} P_{-}\\\\- 1.11022302462516 \\cdot 10^{-16} P^{x}_{-} e^{2}_{d} + 1.11022302462516 \\cdot 10^{-16} P^{x}_{-} e^{2}_{u} + 5.55111512312578 \\cdot 10^{-17} P_{V} & 8.88178419700125 \\cdot 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+\begin{document} + +\maketitle +\begin{itemize} +\item Singlet sector: + +\begin{equation*} +\hspace*{-0.4cm}\mu^2\frac{d}{d\mu^2} +\begin{pmatrix} +g \\ +\gamma \\ +\Sigma \\ +\Delta_\Sigma +\end{pmatrix} += +\begin{pmatrix} + P_{gg}+n_f \langle e^2\rangle \tilde{P}_{gg} & n_f \langle e^2\rangle \tilde{P}_{g\gamma} & P_{gq} + \langle e^2\rangle \tilde{P}_{gq} & \nu_ue^2_-\tilde{P}_{gq} \\ + n_f \langle e^2\rangle \tilde{P}_{\gamma g} & n_f \langle e^2\rangle \tilde{P}_{\gamma \gamma} & \langle e^2\rangle \tilde{P}_{\gamma q} & \nu_ue^2_-\tilde{P}_{\gamma q} \\ + 2n_f (P_{qg} +\langle e^2\rangle \tilde{P}_{qg} )& 2 n_f \langle e^2\rangle \tilde{P}_{q\gamma} & P_{qq}+ \langle e^2\rangle \Bigl(\tilde{P}_{+} + \langle e^2\rangle(\tilde{P}_{qq}-\tilde{P}_{+})\Bigr) & \nu_ue^-\Bigl(\tilde{P}_{+}+ \langle e^2\rangle (\tilde{P}_{qq}-\tilde{P}_{+}) \Bigr)\\ + 2n_f \nu_d e^-\tilde{P}_{qg} & 2n_f \nu_d e^-\tilde{P}_{q\gamma} & \nu_de^-\Bigl( \tilde{P}_{+} + \langle e^2\rangle (\tilde{P}_{qq}-\tilde{P}_{+}) \Bigr)& P_+ +e_\Delta^2 \tilde{P}_{+} +\nu_u \nu_d (e^-)^2(\tilde{P}_{qq}-\tilde{P}_{+}) +\end{pmatrix} +\begin{pmatrix} +g \\ +\gamma \\ +\Sigma \\ +\Delta_\Sigma +\end{pmatrix} +\end{equation*} +with +\begin{align*} + \langle e^2\rangle&=\frac{n_u e_u^2+n_d e_d^2}{n_f} \\ +e_\Delta^2&=\frac{n_u e_d^2+n_d e_u^2}{n_f} \\ +e^2_-&= e_u^2 -e_d^2 \\ +\nu_u &= \frac{n_u}{n_f}\\ +\nu_d &= \frac{n_d}{n_f} +\end{align*} + +\item Valence sector: +\begin{equation*} +\mu^2\frac{d}{d\mu^2} +\begin{pmatrix} +V \\ +\Delta_V +\end{pmatrix} += +\begin{pmatrix} +P_V+\langle e^2\rangle \tilde{P}_{-} & \nu_ue^2_-\tilde{P}_{-}\\ + \nu_de^2_-\tilde{P}_{-}& P_-+e_\Delta^2 \tilde{P}_{-} +\end{pmatrix} +\begin{pmatrix} +V \\ +\Delta_V +\end{pmatrix} +\end{equation*} + + +\end{itemize} +\end{document} From bdb48da6076d6b7cdc6c12e5941609bee9c8daa7 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Niccol=C3=B2=20Laurenti?= Date: Fri, 18 Mar 2022 19:20:24 +0100 Subject: [PATCH 11/71] Remove folder notebook --- notebook/QCDxQED_DGLAP.ipynb | 470 --------------------- notebook/QCDxQED_DGLAP.pdf | Bin 86474 -> 0 bytes notebook/QCDxQED_DGLAP.tex | 72 ---- notebook/uni-dglap.ipynb | 775 ----------------------------------- 4 files changed, 1317 deletions(-) delete mode 100644 notebook/QCDxQED_DGLAP.ipynb delete mode 100644 notebook/QCDxQED_DGLAP.pdf delete mode 100644 notebook/QCDxQED_DGLAP.tex delete mode 100644 notebook/uni-dglap.ipynb diff --git a/notebook/QCDxQED_DGLAP.ipynb b/notebook/QCDxQED_DGLAP.ipynb deleted file mode 100644 index 75095534f..000000000 --- a/notebook/QCDxQED_DGLAP.ipynb +++ /dev/null @@ -1,470 +0,0 @@ -{ - "cells": [ - { - "cell_type": "code", - "execution_count": 2, - "id": "e9589ad4", - "metadata": {}, - "outputs": [], - "source": [ - "import sympy" - ] - }, - { - "cell_type": "markdown", - "id": "83c8a5aa", - "metadata": {}, - "source": [ - "## Flavor basis :\n", - "### [u+, c+, t+, d+, s+, b+, g, \\gamma, b-, s-, d-, t-, c-, u-]\n", - "## Singlet basis :\n", - "### [\\Sigma_u, \\Sigma_d, g, \\gamma, V_d, V_u, T_3^d, V_3^d, T_3^u / c+, V_3^u / c-, T_8^d / b+, V_8^d / b-, T_8^u / t+, V_8^u / t-]\n", - "## Intrinsic evolution basis :\n", - "### [g, \\gamma, \\Sigma, \\Delta_\\Sigma, V, \\Delta_V, T_3^d, V_3^d, T_3^u / c+, V_3^u / c-, T_8^d / b+, V_8^d / b-, T_8^u / t+, V_8^u / t-]" - ] - }, - { - "cell_type": "markdown", - "id": "34f7316a", - "metadata": {}, - "source": [ - "### [T_3^u, V_3^u, T_8^d, V_8^d, T_8^u, V_8^u] is: \n", - "### [c+, c-, b+, b-, t+, t-] in nf=3, \n", - "### [T_3^u, V_3^u, b+, b-, t+, t-] in nf=4, \n", - "### [T_3^u, V_3^u, T_8^d, V_8^d, t+, t-] in nf=5, \n", - "### [T_3^u, V_3^u, T_8^d, V_8^d, T_8^u, V_8^u] in nf=6" - ] - }, - { - "cell_type": "code", - "execution_count": 3, - "id": "d17e49bd", - "metadata": {}, - "outputs": [], - "source": [ - "#QCD\n", - "Pv, Pp, Pm, Pqq, Pqg, Pgq, Pgg = sympy.symbols(\"P_V P_+ P_- P_qq P_qg P_gq P_gg\")\n", - "nf = sympy.symbols(\"n_f\")" - ] - }, - { - "cell_type": "code", - "execution_count": 4, - "id": "48aa792c", - "metadata": {}, - "outputs": [], - "source": [ - "# QED\n", - "Pxv, Pxp, Pxm, Pxqq, Pxqg, Pxgq, Pxgg = sympy.symbols(\"P^x_V P^x_+ P^x_- P^x_qq P^x_qg P^x_gq P^x_gg\")\n", - "Pxqy, Pxyq, Pxyg, Pxgy, Pxyy = sympy.symbols(\"P^x_q\\gamma P^x_\\gamma\\ q P^x_\\gamma\\ g P^x_g\\gamma P^x_\\gamma\\gamma\")\n", - "eu2, ed2, es2 = sympy.symbols(\"e_u^2 e_d^2 e_\\Sigma^2\") # charges" - ] - }, - { - "cell_type": "markdown", - "id": "ea0536a9", - "metadata": {}, - "source": [ - "# Unified Evolution for generic n_f" - ] - }, - { - "cell_type": "code", - "execution_count": 5, - "id": "b48ccd4b", - "metadata": {}, - "outputs": [], - "source": [ - "def theta(x):\n", - " if x>=0 :\n", - " return 1\n", - " else:\n", - " return 0" - ] - }, - { - "cell_type": "code", - "execution_count": 6, - "id": "6c5d10c1", - "metadata": {}, - "outputs": [], - "source": [ - "#QCD\n", - "\n", - "def P_qcd(nf): \n", - " return sympy.Matrix([[Pp , 0, 0, 0, 0, 0, 2 * Pqg, 0, 0, 0, 0, 0, 0, 0], #u+\n", - " [0, theta(nf-4)*Pp, 0, 0, 0, 0, theta(nf-4)*2*Pqg, 0, 0, 0, 0, 0, 0, 0], #c+\n", - " [0, 0, theta(nf-6)*Pp, 0, 0, 0, theta(nf-6)*2*Pqg, 0, 0, 0, 0, 0, 0, 0], #t+\n", - " [0, 0, 0, Pp, 0, 0, 2 * Pqg, 0, 0, 0, 0, 0, 0, 0], #d+\n", - " [0, 0, 0, 0, Pp, 0, 2 * Pqg, 0, 0, 0, 0, 0, 0, 0], #s+\n", - " [0, 0, 0, 0, 0, theta(nf-5)*Pp, theta(nf-5)*2*Pqg, 0, 0, 0, 0, 0, 0, 0], #b+\n", - " [Pgq, theta(nf-4)*Pgq, theta(nf-6)*Pgq, Pgq, Pgq, theta(nf-5)*Pgq, Pgg, 0, 0, 0, 0, 0, 0, 0], #g\n", - " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], #\\gamma\n", - " [0, 0, 0, 0, 0, 0, 0, 0, theta(nf-5)*Pm, 0, 0, 0, 0, 0], #b-\n", - " [0, 0, 0, 0, 0, 0, 0, 0, 0, Pm, 0, 0, 0, 0], #s-\n", - " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, Pm, 0, 0, 0], #d-\n", - " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, theta(nf-6)*Pm, 0, 0], #t-\n", - " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, theta(nf-4)*Pm, 0], #c-\n", - " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, Pm]]) #u-\n", - "\n", - "def Ps_qcd(nf): #\\Sigma_u,\\Sigma_d,g,\\gamma,V_d,V_u,T_3^d,V_3^d,T_3^u,V_3^u,T_8^d,V_8^d,T_8^u,V_8^u\n", - " return sympy.Matrix([[Pqq - Pp, Pqq - Pp, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],#u+\n", - " [theta(nf-4)*(Pqq - Pp),theta(nf-4)*(Pqq - Pp), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],#c+\n", - " [theta(nf-6)*(Pqq - Pp),theta(nf-6)*(Pqq - Pp), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], #t+\n", - " [Pqq - Pp, Pqq - Pp, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], #d+\n", - " [Pqq - Pp, Pqq - Pp, 0, 0, 0,0, 0, 0, 0, 0, 0, 0, 0, 0], #s+\n", - " [theta(nf-5)*(Pqq - Pp),theta(nf-5)*(Pqq - Pp), 0, 0, 0,0, 0, 0, 0, 0, 0, 0, 0, 0], #b+\n", - " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], #g\n", - " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], #\\gamma\n", - " [0, 0, 0, 0, theta(nf-5)*(Pv - Pm), theta(nf-5)*(Pv - Pm), 0, 0, 0, 0, 0, 0, 0, 0], #b-\n", - " [0, 0, 0, 0, Pv - Pm, Pv - Pm, 0, 0, 0, 0, 0, 0, 0, 0], #s-\n", - " [0, 0, 0, 0, Pv - Pm, Pv - Pm, 0, 0, 0, 0, 0, 0, 0, 0], #d-\n", - " [0, 0, 0, 0, theta(nf-6)*(Pv - Pm), theta(nf-6)*(Pv - Pm), 0, 0, 0, 0, 0, 0, 0, 0], #t-\n", - " [0, 0, 0, 0, theta(nf-4)*(Pv - Pm), theta(nf-4)*(Pv - Pm), 0, 0, 0, 0, 0, 0, 0, 0], #c-\n", - " [0, 0, 0, 0, Pv - Pm, Pv - Pm, 0, 0, 0, 0, 0, 0, 0, 0]]) / nf #u-" - ] - }, - { - "cell_type": "code", - "execution_count": 7, - "id": "287b8cd2", - "metadata": {}, - "outputs": [], - "source": [ - "#QED\n", - "\n", - "def P_qed(nf):\n", - " return sympy.Matrix([[eu2 * Pxp, 0, 0, 0, 0, 0, 2 * eu2 * Pxqg, 2 * eu2 * Pxqy, 0, 0, 0, 0, 0, 0], #u+\n", - " [0, theta(nf-4)*eu2 * Pxp, 0, 0, 0, 0, theta(nf-4)*2 * eu2 * Pxqg, theta(nf-4)*2 * eu2 * Pxqy, 0, 0, 0, 0, 0, 0], #c+\n", - " [0, 0, theta(nf-6)*eu2 * Pxp, 0, 0, 0, theta(nf-6)*2 * eu2 * Pxqg, theta(nf-6)*2 * eu2 * Pxqy, 0, 0, 0, 0, 0, 0], #t+\n", - " [0, 0, 0, ed2 * Pxp, 0, 0, 2 * ed2 * Pxqg, 2 * ed2 * Pxqy, 0, 0, 0, 0, 0, 0], #d+\n", - " [0, 0, 0, 0, ed2 * Pxp, 0, 2 * ed2 * Pxqg, 2 * ed2 * Pxqy, 0, 0, 0, 0, 0, 0], #s+\n", - " [0, 0, 0, 0, 0, theta(nf-5)*ed2 * Pxp, theta(nf-5)*2 * ed2 * Pxqg, theta(nf-5)*2 * ed2 * Pxqy, 0, 0, 0, 0, 0, 0], #b+\n", - " [eu2 * Pxgq, theta(nf-4)*eu2 * Pxgq, theta(nf-6)*eu2 * Pxgq, ed2 * Pxgq, ed2 * Pxgq, theta(nf-5)*ed2 * Pxgq, es2 * Pxgg, es2 * Pxgy, 0, 0, 0, 0, 0, 0], #g\n", - " [eu2 * Pxyq, theta(nf-4)*eu2 * Pxyq, theta(nf-6)*eu2 * Pxyq, ed2 * Pxyq, ed2 * Pxyq, theta(nf-5)*ed2 * Pxyq, es2 * Pxyg, es2 * Pxyy, 0, 0, 0, 0, 0, 0], #\\gamma\n", - " [0, 0, 0, 0, 0, 0, 0, 0, theta(nf-5)*ed2 * Pxm, 0, 0, 0, 0, 0], #b-\n", - " [0, 0, 0, 0, 0, 0, 0, 0, 0, ed2 * Pxm, 0, 0, 0, 0],#s-\n", - " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ed2 * Pxm, 0, 0, 0],#d-\n", - " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, theta(nf-6)*eu2 * Pxm, 0, 0], #t-\n", - " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, theta(nf-4)*eu2 * Pxm, 0], #c-\n", - " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, eu2 * Pxm]]) #u-\n", - "def Ps_qed(nf): #\\Sigma_u,\\Sigma_d,g,\\gamma,V_d,V_u,T_3^d,V_3^d,T_3^u,V_3^u,T_8^d,V_8^d,T_8^u,V_8^u\n", - " return sympy.Matrix([[(Pxqq - Pxp)*eu2**2, eu2 * ed2 * (Pxqq - Pxp), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],#u+\n", - " [theta(nf-4)*(Pxqq - Pxp)*eu2**2, theta(nf-4)*eu2 * ed2 * (Pxqq - Pxp), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],#c+\n", - " [theta(nf-6)*(Pxqq - Pxp)*eu2**2, theta(nf-6)*eu2 * ed2 * (Pxqq - Pxp), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], #t+\n", - " [eu2 * ed2 * (Pxqq - Pxp), ed2**2 * (Pxqq - Pxp), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], #d+\n", - " [eu2 * ed2 * (Pxqq - Pxp), ed2**2 * (Pxqq - Pxp), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], #s+\n", - " [theta(nf-5)*eu2 * ed2 * (Pxqq - Pxp), theta(nf-5)*ed2**2 * (Pxqq - Pxp), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], #b+\n", - " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], #g\n", - " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], #\\gamma\n", - " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], #b-\n", - " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], #s-\n", - " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], #d-\n", - " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], #t-\n", - " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], #c-\n", - " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]]) / nf #u-" - ] - }, - { - "cell_type": "code", - "execution_count": 8, - "id": "57bb7421", - "metadata": {}, - "outputs": [], - "source": [ - "def P_uni(nf):\n", - " return P_qcd(nf)+P_qed(nf)\n", - "\n", - "def Ps_uni(nf):\n", - " return Ps_qcd(nf)+Ps_qed(nf)" - ] - }, - { - "cell_type": "code", - "execution_count": 9, - "id": "cd1200b7", - "metadata": {}, - "outputs": [], - "source": [ - "def rot_fl_to_ev(nf):\n", - " if nf==3 :\n", - " nu=1\n", - " nd=2\n", - " if nf==4 :\n", - " nu=2\n", - " nd=2\n", - " if nf==5 :\n", - " nu=2\n", - " nd=3\n", - " if nf==6 :\n", - " nu=3\n", - " nd=3\n", - " return sympy.Matrix([[0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0], #g\n", - " [0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0], #\\gamma\n", - " [1, theta(nf-4), theta(nf-6), 1, 1, theta(nf-5), 0, 0, 0, 0, 0, 0, 0, 0], #\\Sigma\n", - " [nd/nu, theta(nf-4)*nd/nu, theta(nf-6)*nd/nu, -1, -1, -theta(nf-5), 0, 0, 0, 0, 0, 0, 0, 0], #\\Delta_\\Sigma\n", - " [0, 0, 0, 0, 0, 0, 0, 0, theta(nf-5), 1, 1, theta(nf-6), theta(nf-4), 1], #V\n", - " [0, 0, 0, 0, 0, 0, 0, 0, -theta(nf-5), -1, -1, nd/nu*theta(nf-6), nd/nu*theta(nf-4), nd/nu], #\\Delta_V\n", - " [0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0], #T_3^d\n", - " [0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0], #V_3^d\n", - " [theta(nf-4), theta(3-nf)-theta(nf-4), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], #T_3^u / c+\n", - " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, theta(3-nf)-theta(nf-4), theta(nf-4)], #V_3^u / c-\n", - " [0, 0, 0, theta(nf-5), theta(nf-5), theta(4-nf)-2*theta(nf-5), 0, 0, 0, 0, 0, 0, 0, 0], #T_8^d / b+\n", - " [0, 0, 0, 0, 0, 0, 0, 0, theta(4-nf)-2*theta(nf-5), theta(nf-5), theta(nf-5), 0, 0, 0], #V_8^d / b-\n", - " [theta(nf-6), theta(nf-6), theta(5-nf)-2*theta(nf-6), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], #T_8^u / t+\n", - " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, theta(5-nf)-2*theta(nf-6), theta(nf-6), theta(nf-6)]]) #V_8^u / t-" - ] - }, - { - "cell_type": "code", - "execution_count": 10, - "id": "77982ede", - "metadata": {}, - "outputs": [], - "source": [ - "def rot_ev_to_fl(nf):\n", - " return rot_fl_to_ev(nf).inv()" - ] - }, - { - "cell_type": "code", - "execution_count": 11, - "id": "5ab19866", - "metadata": {}, - "outputs": [], - "source": [ - "def rot_sin_to_ev(nf):\n", - " if nf==3 :\n", - " nu=1\n", - " nd=2\n", - " if nf==4 :\n", - " nu=2\n", - " nd=2\n", - " if nf==5 :\n", - " nu=2\n", - " nd=3\n", - " if nf==6 :\n", - " nu=3\n", - " nd=3\n", - " return sympy.Matrix([[0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], #g\n", - " [0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], #\\gamma\n", - " [1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], #Sigma\n", - " [nd/nu, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], #\\Delta_\\Sigma\n", - " [0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0], #V\n", - " [0, 0, 0, 0, -1, nd/nu, 0, 0, 0, 0, 0, 0, 0, 0], #\\Delta_V\n", - " [0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0], #T_3^d\n", - " [0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0], #V_3^d\n", - " [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0], #T_3^u / c+\n", - " [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0], #V_3^u / c-\n", - " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0], #T_8^d / b+\n", - " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0], #V_8^d / b-\n", - " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0], #T_8^u / t+\n", - " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]]) #V_8^u / t-" - ] - }, - { - "cell_type": "code", - "execution_count": 12, - "id": "601fe92c", - "metadata": {}, - "outputs": [], - "source": [ - "def rot_ev_to_sin(nf):\n", - " return rot_sin_to_ev(nf).inv()" - ] - }, - { - "cell_type": "code", - "execution_count": 13, - "id": "ebfaa9ad", - "metadata": {}, - "outputs": [], - "source": [ - "def P_ev(nf):\n", - " res = rot_fl_to_ev(nf) * P_uni(nf) * rot_ev_to_fl(nf) + rot_fl_to_ev(nf) * Ps_uni(nf) * rot_ev_to_sin(nf)\n", - " return res" - ] - }, - { - "cell_type": "code", - "execution_count": 20, - "id": "6f5d3f7c", - "metadata": {}, - "outputs": [ - { - "data": { - "text/latex": [ - "$\\displaystyle \\left[\\begin{array}{cccccccccccccc}P^{x}_{gg} e^{2}_{\\Sigma} + P_{gg} & P^{x}_{g\\gamma} e^{2}_{\\Sigma} & 0.666666666666667 P^{x}_{gq} e^{2}_{d} + 0.333333333333333 P^{x}_{gq} e^{2}_{u} + 1.0 P_{gq} & - 0.333333333333333 P^{x}_{gq} e^{2}_{d} + 0.333333333333333 P^{x}_{gq} e^{2}_{u} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\P^{x}_{\\gamma g} e^{2}_{\\Sigma} & P^{x}_{\\gamma\\gamma} e^{2}_{\\Sigma} & 0.666666666666667 P^{x}_{\\gamma q} e^{2}_{d} + 0.333333333333333 P^{x}_{\\gamma q} e^{2}_{u} & - 0.333333333333333 P^{x}_{\\gamma q} e^{2}_{d} + 0.333333333333333 P^{x}_{\\gamma q} e^{2}_{u} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\4 P^{x}_{qg} e^{2}_{d} + 2 P^{x}_{qg} e^{2}_{u} + 6 P_{qg} & 4 P^{x}_{q\\gamma} e^{2}_{d} + 2 P^{x}_{q\\gamma} e^{2}_{u} & 0.666666666666667 P^{x}_{+} e^{2}_{d} + 0.333333333333333 P^{x}_{+} e^{2}_{u} + 1.0 P_{qq} + 0.444444444444444 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.444444444444444 e^{2}_{d} e^{2}_{u} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.111111111111111 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & - 0.333333333333333 P^{x}_{+} e^{2}_{d} + 0.333333333333333 P^{x}_{+} e^{2}_{u} - 0.222222222222222 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.111111111111111 e^{2}_{d} e^{2}_{u} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.111111111111111 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\- 4 P^{x}_{qg} e^{2}_{d} + 4.0 P^{x}_{qg} e^{2}_{u} & - 4 P^{x}_{q\\gamma} e^{2}_{d} + 4.0 P^{x}_{q\\gamma} e^{2}_{u} & - 0.666666666666667 P^{x}_{+} e^{2}_{d} + 0.666666666666667 P^{x}_{+} e^{2}_{u} - 0.444444444444444 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.222222222222222 e^{2}_{d} e^{2}_{u} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.222222222222222 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & 0.333333333333333 P^{x}_{+} e^{2}_{d} + 0.666666666666667 P^{x}_{+} e^{2}_{u} + 1.0 P_{+} + 0.222222222222222 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) - 0.444444444444444 e^{2}_{d} e^{2}_{u} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.222222222222222 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0.666666666666667 P^{x}_{-} e^{2}_{d} + 0.333333333333333 P^{x}_{-} e^{2}_{u} + 1.0 P_{V} & - 0.333333333333333 P^{x}_{-} e^{2}_{d} + 0.333333333333333 P^{x}_{-} e^{2}_{u} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & - 0.666666666666667 P^{x}_{-} e^{2}_{d} + 0.666666666666667 P^{x}_{-} e^{2}_{u} & 0.333333333333333 P^{x}_{-} e^{2}_{d} + 0.666666666666667 P^{x}_{-} e^{2}_{u} + 1.0 P_{-} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{+} e^{2}_{d} + 1.0 P_{+} & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{-} e^{2}_{d} + 1.0 P_{-} & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\end{array}\\right]$" - ], - "text/plain": [ - "Matrix([\n", - "[ P^x_gg*e_\\Sigma^2 + P_gg, P^x_g\\gamma*e_\\Sigma^2, 0.666666666666667*P^x_gq*e_d^2 + 0.333333333333333*P^x_gq*e_u^2 + 1.0*P_gq, -0.333333333333333*P^x_gq*e_d^2 + 0.333333333333333*P^x_gq*e_u^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[ P^x_\\gamma g*e_\\Sigma^2, P^x_\\gamma\\gamma*e_\\Sigma^2, 0.666666666666667*P^x_\\gamma q*e_d^2 + 0.333333333333333*P^x_\\gamma q*e_u^2, -0.333333333333333*P^x_\\gamma q*e_d^2 + 0.333333333333333*P^x_\\gamma q*e_u^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[4*P^x_qg*e_d^2 + 2*P^x_qg*e_u^2 + 6*P_qg, 4*P^x_q\\gamma*e_d^2 + 2*P^x_q\\gamma*e_u^2, 0.666666666666667*P^x_+*e_d^2 + 0.333333333333333*P^x_+*e_u^2 + 1.0*P_qq + 0.444444444444444*e_d^2**2*(-P^x_+ + P^x_qq) + 0.444444444444444*e_d^2*e_u^2*(-P^x_+ + P^x_qq) + 0.111111111111111*e_u^2**2*(-P^x_+ + P^x_qq), -0.333333333333333*P^x_+*e_d^2 + 0.333333333333333*P^x_+*e_u^2 - 0.222222222222222*e_d^2**2*(-P^x_+ + P^x_qq) + 0.111111111111111*e_d^2*e_u^2*(-P^x_+ + P^x_qq) + 0.111111111111111*e_u^2**2*(-P^x_+ + P^x_qq), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[ -4*P^x_qg*e_d^2 + 4.0*P^x_qg*e_u^2, -4*P^x_q\\gamma*e_d^2 + 4.0*P^x_q\\gamma*e_u^2, -0.666666666666667*P^x_+*e_d^2 + 0.666666666666667*P^x_+*e_u^2 - 0.444444444444444*e_d^2**2*(-P^x_+ + P^x_qq) + 0.222222222222222*e_d^2*e_u^2*(-P^x_+ + P^x_qq) + 0.222222222222222*e_u^2**2*(-P^x_+ + P^x_qq), 0.333333333333333*P^x_+*e_d^2 + 0.666666666666667*P^x_+*e_u^2 + 1.0*P_+ + 0.222222222222222*e_d^2**2*(-P^x_+ + P^x_qq) - 0.444444444444444*e_d^2*e_u^2*(-P^x_+ + P^x_qq) + 0.222222222222222*e_u^2**2*(-P^x_+ + P^x_qq), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, 0.666666666666667*P^x_-*e_d^2 + 0.333333333333333*P^x_-*e_u^2 + 1.0*P_V, -0.333333333333333*P^x_-*e_d^2 + 0.333333333333333*P^x_-*e_u^2, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, -0.666666666666667*P^x_-*e_d^2 + 0.666666666666667*P^x_-*e_u^2, 0.333333333333333*P^x_-*e_d^2 + 0.666666666666667*P^x_-*e_u^2 + 1.0*P_-, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, 0, 0, 1.0*P^x_+*e_d^2 + 1.0*P_+, 0, 0, 0, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, 0, 0, 0, 1.0*P^x_-*e_d^2 + 1.0*P_-, 0, 0, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]])" - ] - }, - "execution_count": 20, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "P_ev(3)" - ] - }, - { - "cell_type": "code", - "execution_count": 17, - "id": "28ecee8c", - "metadata": {}, - "outputs": [ - { - "data": { - "text/latex": [ - "$\\displaystyle \\left[\\begin{array}{cccccccccccccc}P^{x}_{gg} e^{2}_{\\Sigma} + P_{gg} & P^{x}_{g\\gamma} e^{2}_{\\Sigma} & 0.5 P^{x}_{gq} e^{2}_{d} + 0.5 P^{x}_{gq} e^{2}_{u} + 1.0 P_{gq} & - 0.5 P^{x}_{gq} e^{2}_{d} + 0.5 P^{x}_{gq} e^{2}_{u} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\P^{x}_{\\gamma g} e^{2}_{\\Sigma} & P^{x}_{\\gamma\\gamma} e^{2}_{\\Sigma} & 0.5 P^{x}_{\\gamma q} e^{2}_{d} + 0.5 P^{x}_{\\gamma q} e^{2}_{u} & - 0.5 P^{x}_{\\gamma q} e^{2}_{d} + 0.5 P^{x}_{\\gamma q} e^{2}_{u} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\4 P^{x}_{qg} e^{2}_{d} + 4 P^{x}_{qg} e^{2}_{u} + 8 P_{qg} & 4 P^{x}_{q\\gamma} e^{2}_{d} + 4 P^{x}_{q\\gamma} e^{2}_{u} & 0.5 P^{x}_{+} e^{2}_{d} + 0.5 P^{x}_{+} e^{2}_{u} + 1.0 P_{qq} + 0.25 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.5 e^{2}_{d} e^{2}_{u} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.25 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & - 0.5 P^{x}_{+} e^{2}_{d} + 0.5 P^{x}_{+} e^{2}_{u} - 0.25 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.25 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\- 4 P^{x}_{qg} e^{2}_{d} + 4.0 P^{x}_{qg} e^{2}_{u} & - 4 P^{x}_{q\\gamma} e^{2}_{d} + 4.0 P^{x}_{q\\gamma} e^{2}_{u} & - 0.5 P^{x}_{+} e^{2}_{d} + 0.5 P^{x}_{+} e^{2}_{u} - 0.25 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.25 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & 0.5 P^{x}_{+} e^{2}_{d} + 0.5 P^{x}_{+} e^{2}_{u} + 1.0 P_{+} + 0.25 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) - 0.5 e^{2}_{d} e^{2}_{u} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.25 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0.5 P^{x}_{-} e^{2}_{d} + 0.5 P^{x}_{-} e^{2}_{u} + 1.0 P_{V} & - 0.5 P^{x}_{-} e^{2}_{d} + 0.5 P^{x}_{-} e^{2}_{u} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & - 0.5 P^{x}_{-} e^{2}_{d} + 0.5 P^{x}_{-} e^{2}_{u} & 0.5 P^{x}_{-} e^{2}_{d} + 0.5 P^{x}_{-} e^{2}_{u} + 1.0 P_{-} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{+} e^{2}_{d} + 1.0 P_{+} & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{-} e^{2}_{d} + 1.0 P_{-} & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{+} e^{2}_{u} + 1.0 P_{+} & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{-} e^{2}_{u} + 1.0 P_{-} & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\end{array}\\right]$" - ], - "text/plain": [ - "Matrix([\n", - "[ P^x_gg*e_\\Sigma^2 + P_gg, P^x_g\\gamma*e_\\Sigma^2, 0.5*P^x_gq*e_d^2 + 0.5*P^x_gq*e_u^2 + 1.0*P_gq, -0.5*P^x_gq*e_d^2 + 0.5*P^x_gq*e_u^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[ P^x_\\gamma g*e_\\Sigma^2, P^x_\\gamma\\gamma*e_\\Sigma^2, 0.5*P^x_\\gamma q*e_d^2 + 0.5*P^x_\\gamma q*e_u^2, -0.5*P^x_\\gamma q*e_d^2 + 0.5*P^x_\\gamma q*e_u^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[4*P^x_qg*e_d^2 + 4*P^x_qg*e_u^2 + 8*P_qg, 4*P^x_q\\gamma*e_d^2 + 4*P^x_q\\gamma*e_u^2, 0.5*P^x_+*e_d^2 + 0.5*P^x_+*e_u^2 + 1.0*P_qq + 0.25*e_d^2**2*(-P^x_+ + P^x_qq) + 0.5*e_d^2*e_u^2*(-P^x_+ + P^x_qq) + 0.25*e_u^2**2*(-P^x_+ + P^x_qq), -0.5*P^x_+*e_d^2 + 0.5*P^x_+*e_u^2 - 0.25*e_d^2**2*(-P^x_+ + P^x_qq) + 0.25*e_u^2**2*(-P^x_+ + P^x_qq), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[ -4*P^x_qg*e_d^2 + 4.0*P^x_qg*e_u^2, -4*P^x_q\\gamma*e_d^2 + 4.0*P^x_q\\gamma*e_u^2, -0.5*P^x_+*e_d^2 + 0.5*P^x_+*e_u^2 - 0.25*e_d^2**2*(-P^x_+ + P^x_qq) + 0.25*e_u^2**2*(-P^x_+ + P^x_qq), 0.5*P^x_+*e_d^2 + 0.5*P^x_+*e_u^2 + 1.0*P_+ + 0.25*e_d^2**2*(-P^x_+ + P^x_qq) - 0.5*e_d^2*e_u^2*(-P^x_+ + P^x_qq) + 0.25*e_u^2**2*(-P^x_+ + P^x_qq), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, 0.5*P^x_-*e_d^2 + 0.5*P^x_-*e_u^2 + 1.0*P_V, -0.5*P^x_-*e_d^2 + 0.5*P^x_-*e_u^2, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, -0.5*P^x_-*e_d^2 + 0.5*P^x_-*e_u^2, 0.5*P^x_-*e_d^2 + 0.5*P^x_-*e_u^2 + 1.0*P_-, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, 0, 0, 1.0*P^x_+*e_d^2 + 1.0*P_+, 0, 0, 0, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, 0, 0, 0, 1.0*P^x_-*e_d^2 + 1.0*P_-, 0, 0, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, 0, 0, 0, 0, 1.0*P^x_+*e_u^2 + 1.0*P_+, 0, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1.0*P^x_-*e_u^2 + 1.0*P_-, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]])" - ] - }, - "execution_count": 17, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "P_ev(4)" - ] - }, - { - "cell_type": "code", - "execution_count": 18, - "id": "0d6f55c2", - "metadata": {}, - "outputs": [ - { - "data": { - "text/latex": [ - "$\\displaystyle \\left[\\begin{array}{cccccccccccccc}P^{x}_{gg} e^{2}_{\\Sigma} + P_{gg} & P^{x}_{g\\gamma} e^{2}_{\\Sigma} & 0.6 P^{x}_{gq} e^{2}_{d} + 0.4 P^{x}_{gq} e^{2}_{u} + 1.0 P_{gq} & - 0.4 P^{x}_{gq} e^{2}_{d} + 0.4 P^{x}_{gq} e^{2}_{u} + 5.55111512312578 \\cdot 10^{-17} P_{gq} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\P^{x}_{\\gamma g} e^{2}_{\\Sigma} & P^{x}_{\\gamma\\gamma} e^{2}_{\\Sigma} & 0.6 P^{x}_{\\gamma q} e^{2}_{d} + 0.4 P^{x}_{\\gamma q} e^{2}_{u} & - 0.4 P^{x}_{\\gamma q} e^{2}_{d} + 0.4 P^{x}_{\\gamma q} e^{2}_{u} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\6 P^{x}_{qg} e^{2}_{d} + 4 P^{x}_{qg} e^{2}_{u} + 10 P_{qg} & 6 P^{x}_{q\\gamma} e^{2}_{d} + 4 P^{x}_{q\\gamma} e^{2}_{u} & 0.6 P^{x}_{+} e^{2}_{d} + 0.4 P^{x}_{+} e^{2}_{u} + 1.0 P_{qq} + 0.36 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.48 e^{2}_{d} e^{2}_{u} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.16 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & - 0.4 P^{x}_{+} e^{2}_{d} + 0.4 P^{x}_{+} e^{2}_{u} + 5.55111512312578 \\cdot 10^{-17} P_{+} - 0.24 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.08 e^{2}_{d} e^{2}_{u} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.16 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\- 6 P^{x}_{qg} e^{2}_{d} + 6.0 P^{x}_{qg} e^{2}_{u} & - 6 P^{x}_{q\\gamma} e^{2}_{d} + 6.0 P^{x}_{q\\gamma} e^{2}_{u} & - 0.6 P^{x}_{+} e^{2}_{d} + 0.6 P^{x}_{+} e^{2}_{u} + 5.55111512312578 \\cdot 10^{-17} P_{qq} - 0.36 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.12 e^{2}_{d} e^{2}_{u} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.24 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & 0.4 P^{x}_{+} e^{2}_{d} + 0.6 P^{x}_{+} e^{2}_{u} + 1.0 P_{+} + 0.24 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) - 0.48 e^{2}_{d} e^{2}_{u} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.24 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0.6 P^{x}_{-} e^{2}_{d} + 0.4 P^{x}_{-} e^{2}_{u} + 1.0 P_{V} & - 0.4 P^{x}_{-} e^{2}_{d} + 0.4 P^{x}_{-} e^{2}_{u} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & - 0.6 P^{x}_{-} e^{2}_{d} + 0.6 P^{x}_{-} e^{2}_{u} - 1.11022302462516 \\cdot 10^{-16} P_{-} + 1.11022302462516 \\cdot 10^{-16} P_{V} & 0.4 P^{x}_{-} e^{2}_{d} + 0.6 P^{x}_{-} e^{2}_{u} + 1.0 P_{-} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{+} e^{2}_{d} + 1.0 P_{+} & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{-} e^{2}_{d} + 1.0 P_{-} & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{+} e^{2}_{u} + 1.0 P_{+} & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{-} e^{2}_{u} + 1.0 P_{-} & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{+} e^{2}_{d} + 1.0 P_{+} & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{-} e^{2}_{d} + 1.0 P_{-} & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\end{array}\\right]$" - ], - "text/plain": [ - "Matrix([\n", - "[ P^x_gg*e_\\Sigma^2 + P_gg, P^x_g\\gamma*e_\\Sigma^2, 0.6*P^x_gq*e_d^2 + 0.4*P^x_gq*e_u^2 + 1.0*P_gq, -0.4*P^x_gq*e_d^2 + 0.4*P^x_gq*e_u^2 + 5.55111512312578e-17*P_gq, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[ P^x_\\gamma g*e_\\Sigma^2, P^x_\\gamma\\gamma*e_\\Sigma^2, 0.6*P^x_\\gamma q*e_d^2 + 0.4*P^x_\\gamma q*e_u^2, -0.4*P^x_\\gamma q*e_d^2 + 0.4*P^x_\\gamma q*e_u^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[6*P^x_qg*e_d^2 + 4*P^x_qg*e_u^2 + 10*P_qg, 6*P^x_q\\gamma*e_d^2 + 4*P^x_q\\gamma*e_u^2, 0.6*P^x_+*e_d^2 + 0.4*P^x_+*e_u^2 + 1.0*P_qq + 0.36*e_d^2**2*(-P^x_+ + P^x_qq) + 0.48*e_d^2*e_u^2*(-P^x_+ + P^x_qq) + 0.16*e_u^2**2*(-P^x_+ + P^x_qq), -0.4*P^x_+*e_d^2 + 0.4*P^x_+*e_u^2 + 5.55111512312578e-17*P_+ - 0.24*e_d^2**2*(-P^x_+ + P^x_qq) + 0.08*e_d^2*e_u^2*(-P^x_+ + P^x_qq) + 0.16*e_u^2**2*(-P^x_+ + P^x_qq), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[ -6*P^x_qg*e_d^2 + 6.0*P^x_qg*e_u^2, -6*P^x_q\\gamma*e_d^2 + 6.0*P^x_q\\gamma*e_u^2, -0.6*P^x_+*e_d^2 + 0.6*P^x_+*e_u^2 + 5.55111512312578e-17*P_qq - 0.36*e_d^2**2*(-P^x_+ + P^x_qq) + 0.12*e_d^2*e_u^2*(-P^x_+ + P^x_qq) + 0.24*e_u^2**2*(-P^x_+ + P^x_qq), 0.4*P^x_+*e_d^2 + 0.6*P^x_+*e_u^2 + 1.0*P_+ + 0.24*e_d^2**2*(-P^x_+ + P^x_qq) - 0.48*e_d^2*e_u^2*(-P^x_+ + P^x_qq) + 0.24*e_u^2**2*(-P^x_+ + P^x_qq), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, 0.6*P^x_-*e_d^2 + 0.4*P^x_-*e_u^2 + 1.0*P_V, -0.4*P^x_-*e_d^2 + 0.4*P^x_-*e_u^2, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, -0.6*P^x_-*e_d^2 + 0.6*P^x_-*e_u^2 - 1.11022302462516e-16*P_- + 1.11022302462516e-16*P_V, 0.4*P^x_-*e_d^2 + 0.6*P^x_-*e_u^2 + 1.0*P_-, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, 0, 0, 1.0*P^x_+*e_d^2 + 1.0*P_+, 0, 0, 0, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, 0, 0, 0, 1.0*P^x_-*e_d^2 + 1.0*P_-, 0, 0, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, 0, 0, 0, 0, 1.0*P^x_+*e_u^2 + 1.0*P_+, 0, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1.0*P^x_-*e_u^2 + 1.0*P_-, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1.0*P^x_+*e_d^2 + 1.0*P_+, 0, 0, 0],\n", - "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1.0*P^x_-*e_d^2 + 1.0*P_-, 0, 0],\n", - "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]])" - ] - }, - "execution_count": 18, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "P_ev(5)" - ] - }, - { - "cell_type": "code", - "execution_count": 19, - "id": "c11fad98", - "metadata": {}, - "outputs": [ - { - "data": { - "text/latex": [ - "$\\displaystyle \\left[\\begin{array}{cccccccccccccc}P^{x}_{gg} e^{2}_{\\Sigma} + P_{gg} & P^{x}_{g\\gamma} e^{2}_{\\Sigma} & 0.5 P^{x}_{gq} e^{2}_{d} + 0.5 P^{x}_{gq} e^{2}_{u} + 1.0 P_{gq} & - 0.5 P^{x}_{gq} e^{2}_{d} + 0.5 P^{x}_{gq} e^{2}_{u} + 5.55111512312578 \\cdot 10^{-17} P_{gq} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\P^{x}_{\\gamma g} e^{2}_{\\Sigma} & P^{x}_{\\gamma\\gamma} e^{2}_{\\Sigma} & 0.5 P^{x}_{\\gamma q} e^{2}_{d} + 0.5 P^{x}_{\\gamma q} e^{2}_{u} & - 0.5 P^{x}_{\\gamma q} e^{2}_{d} + 0.5 P^{x}_{\\gamma q} e^{2}_{u} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\6 P^{x}_{qg} e^{2}_{d} + 6 P^{x}_{qg} e^{2}_{u} + 12 P_{qg} & 6 P^{x}_{q\\gamma} e^{2}_{d} + 6 P^{x}_{q\\gamma} e^{2}_{u} & 0.5 P^{x}_{+} e^{2}_{d} + 0.5 P^{x}_{+} e^{2}_{u} - 1.11022302462516 \\cdot 10^{-16} P_{+} + 1.0 P_{qq} + 0.25 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.5 e^{2}_{d} e^{2}_{u} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.25 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & - 0.5 P^{x}_{+} e^{2}_{d} + 0.5 P^{x}_{+} e^{2}_{u} + 5.55111512312578 \\cdot 10^{-17} P_{+} - 0.25 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.25 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\- 6 P^{x}_{qg} e^{2}_{d} + 6.0 P^{x}_{qg} e^{2}_{u} & - 6 P^{x}_{q\\gamma} e^{2}_{d} + 6.0 P^{x}_{q\\gamma} e^{2}_{u} & - 0.5 P^{x}_{+} e^{2}_{d} + 0.5 P^{x}_{+} e^{2}_{u} + 5.55111512312578 \\cdot 10^{-17} P_{qq} - 0.25 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.25 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & 0.5 P^{x}_{+} e^{2}_{d} + 0.5 P^{x}_{+} e^{2}_{u} + 1.0 P_{+} + 0.25 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) - 0.5 e^{2}_{d} e^{2}_{u} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.25 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0.5 P^{x}_{-} e^{2}_{d} + 0.5 P^{x}_{-} e^{2}_{u} - 1.11022302462516 \\cdot 10^{-16} P_{-} + 1.0 P_{V} & - 0.5 P^{x}_{-} e^{2}_{d} + 0.5 P^{x}_{-} e^{2}_{u} - 5.55111512312578 \\cdot 10^{-17} P_{-} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & - 0.5 P^{x}_{-} e^{2}_{d} + 0.5 P^{x}_{-} e^{2}_{u} - 5.55111512312578 \\cdot 10^{-17} P_{V} & 0.5 P^{x}_{-} e^{2}_{d} + 0.5 P^{x}_{-} e^{2}_{u} + 1.0 P_{-} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{+} e^{2}_{d} + 1.0 P_{+} & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{-} e^{2}_{d} + 1.0 P_{-} & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{+} e^{2}_{u} + 1.0 P_{+} & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{-} e^{2}_{u} + 1.0 P_{-} & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{+} e^{2}_{d} + 1.0 P_{+} & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{-} e^{2}_{d} + 1.0 P_{-} & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{+} e^{2}_{u} + 1.0 P_{+} & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{-} e^{2}_{u} + 1.0 P_{-}\\end{array}\\right]$" - ], - "text/plain": [ - "Matrix([\n", - "[ P^x_gg*e_\\Sigma^2 + P_gg, P^x_g\\gamma*e_\\Sigma^2, 0.5*P^x_gq*e_d^2 + 0.5*P^x_gq*e_u^2 + 1.0*P_gq, -0.5*P^x_gq*e_d^2 + 0.5*P^x_gq*e_u^2 + 5.55111512312578e-17*P_gq, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[ P^x_\\gamma g*e_\\Sigma^2, P^x_\\gamma\\gamma*e_\\Sigma^2, 0.5*P^x_\\gamma q*e_d^2 + 0.5*P^x_\\gamma q*e_u^2, -0.5*P^x_\\gamma q*e_d^2 + 0.5*P^x_\\gamma q*e_u^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[6*P^x_qg*e_d^2 + 6*P^x_qg*e_u^2 + 12*P_qg, 6*P^x_q\\gamma*e_d^2 + 6*P^x_q\\gamma*e_u^2, 0.5*P^x_+*e_d^2 + 0.5*P^x_+*e_u^2 - 1.11022302462516e-16*P_+ + 1.0*P_qq + 0.25*e_d^2**2*(-P^x_+ + P^x_qq) + 0.5*e_d^2*e_u^2*(-P^x_+ + P^x_qq) + 0.25*e_u^2**2*(-P^x_+ + P^x_qq), -0.5*P^x_+*e_d^2 + 0.5*P^x_+*e_u^2 + 5.55111512312578e-17*P_+ - 0.25*e_d^2**2*(-P^x_+ + P^x_qq) + 0.25*e_u^2**2*(-P^x_+ + P^x_qq), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[ -6*P^x_qg*e_d^2 + 6.0*P^x_qg*e_u^2, -6*P^x_q\\gamma*e_d^2 + 6.0*P^x_q\\gamma*e_u^2, -0.5*P^x_+*e_d^2 + 0.5*P^x_+*e_u^2 + 5.55111512312578e-17*P_qq - 0.25*e_d^2**2*(-P^x_+ + P^x_qq) + 0.25*e_u^2**2*(-P^x_+ + P^x_qq), 0.5*P^x_+*e_d^2 + 0.5*P^x_+*e_u^2 + 1.0*P_+ + 0.25*e_d^2**2*(-P^x_+ + P^x_qq) - 0.5*e_d^2*e_u^2*(-P^x_+ + P^x_qq) + 0.25*e_u^2**2*(-P^x_+ + P^x_qq), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, 0.5*P^x_-*e_d^2 + 0.5*P^x_-*e_u^2 - 1.11022302462516e-16*P_- + 1.0*P_V, -0.5*P^x_-*e_d^2 + 0.5*P^x_-*e_u^2 - 5.55111512312578e-17*P_-, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, -0.5*P^x_-*e_d^2 + 0.5*P^x_-*e_u^2 - 5.55111512312578e-17*P_V, 0.5*P^x_-*e_d^2 + 0.5*P^x_-*e_u^2 + 1.0*P_-, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, 0, 0, 1.0*P^x_+*e_d^2 + 1.0*P_+, 0, 0, 0, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, 0, 0, 0, 1.0*P^x_-*e_d^2 + 1.0*P_-, 0, 0, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, 0, 0, 0, 0, 1.0*P^x_+*e_u^2 + 1.0*P_+, 0, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1.0*P^x_-*e_u^2 + 1.0*P_-, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1.0*P^x_+*e_d^2 + 1.0*P_+, 0, 0, 0],\n", - "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1.0*P^x_-*e_d^2 + 1.0*P_-, 0, 0],\n", - "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1.0*P^x_+*e_u^2 + 1.0*P_+, 0],\n", - "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1.0*P^x_-*e_u^2 + 1.0*P_-]])" - ] - }, - "execution_count": 19, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "P_ev(6)" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "21fe7814", - "metadata": {}, - "outputs": [], - "source": [] - } - ], - "metadata": { - "kernelspec": { - "display_name": "Python 3 (ipykernel)", - "language": "python", - "name": "python3" - }, - "language_info": { - "codemirror_mode": { - "name": "ipython", - "version": 3 - }, - "file_extension": ".py", - "mimetype": "text/x-python", - "name": "python", - "nbconvert_exporter": "python", - "pygments_lexer": "ipython3", - "version": "3.8.12" - } - }, - "nbformat": 4, - 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z{ANvSq%qvM?+opfztNw#NNLoO43HkPZk|u--Ct8+$jeAg!U(tshn{Dq6+eHtF%`aE z&~x`~{$-T&tQMkVZMQ)>)W>L`nvQI4$zzdLtos+gQg=t!oCSk1Wb(<@3 z#)?znM6z~1XWD_LlCx)A_HVzB$?Y4wGFvbm-2nR9c0js|$fWAuHeDcIffBcOKO%N& zM6CwP5C1SE=`?2rQyq)G7ly#pL=KC8^&}5o&Uw|FrZ@6Kesl@jU+E{0@XE1lEH)GC z8d&XO(PLnv^wV&pnPQt+^$f>g`8}Hydiay&5;=JEA`i`~aq=cM;dO_&f3SGF>+9DF zy)qU-?RDvs*iXK{pA+?azmMl)d+XH3hgv0i_kdF_y5$6Q47=%C=_cZTMWJOV=neK0 z-X5FUzUWnEt*(zzY0t&74R+`)mt=RIAj@CmK3GpDzii7r4=CR!vxcTS`ObyFPq57@ zwX%2j*K*lN`~!9VeHiE;z~y3(!}Dud1bE?S00soWKv;gQvp#ejXUDJQ3}C>3HUQ?> z?1@Aw07Yu2%Yd5UT-cZyIBNnYQf7ozOR4WqsWjlOUw= 3 :\n", - " res[6, 4] = 1\n", - " res[6, 6] = -1\n", - " res[7, 5] = 1\n", - " res[7, 7] = -1\n", - " if nf >= 4 :\n", - " res[8, 2] = 1\n", - " res[8, 8] = -1\n", - " res[9, 3] = 1\n", - " res[9, 9] = -1\n", - " if nf >= 5 :\n", - " res[10, 4] = 1\n", - " res[10, 6] = 1\n", - " res[10, 10] = -2\n", - " res[11, 5] = 1\n", - " res[11, 7] = 1\n", - " res[11, 11] = -2\n", - " if nf == 6 :\n", - " res[12, 2] = 1\n", - " res[12, 8] = 1\n", - " res[12, 12] = -2\n", - " res[13, 3] = 1\n", - " res[13, 9] = 1\n", - " res[13, 13] = -2\n", - " return res\n", - "\n", - "def rot_ev_to_fl(nf):\n", - " return rot_fl_to_ev(nf).inv()" - ] - }, - { - "cell_type": "code", - "execution_count": 316, - "id": "b5747379", - "metadata": {}, - "outputs": [], - "source": [ - "def rot_sin_to_ev(nf):\n", - " nu = int(nf/2)\n", - " nd = nf - nu\n", - " res = sympy.Matrix.zeros(14,14).as_mutable()\n", - " res[0, 0]=1\n", - " res[1, 1]=1\n", - " res[2,2]=1\n", - " res[2,3]=1\n", - " res[3,2]=nd/nu\n", - " res[3,3]=-1\n", - " res[4,4]=1\n", - " res[4,5]=1\n", - " res[5,4]=nd/nu\n", - " res[5,5]=-1\n", - " for i in range(6,14):\n", - " res[i,i]=1\n", - " return res\n", - "\n", - "def rot_ev_to_sin(nf):\n", - " return rot_sin_to_ev(nf).inv()" - ] - }, - { - "cell_type": "code", - "execution_count": 317, - "id": "45fb3a16", - "metadata": {}, - "outputs": [], - "source": [ - "def P_ev(nf):\n", - " res = rot_fl_to_ev(nf) * P_uni(nf) * rot_ev_to_fl(nf) + rot_fl_to_ev(nf) * Ps_uni(nf) * rot_ev_to_sin(nf)\n", - " return res" - ] - }, - { - "cell_type": "code", - "execution_count": 318, - "id": "9b5050aa", - "metadata": {}, - "outputs": [ - { - "data": { - "text/latex": [ - "$\\displaystyle \\left[\\begin{array}{cccccccccccccc}P^{x}_{gg} \\left(e^{2}_{d} + e^{2}_{u}\\right) + P_{gg} & P^{x}_{g\\gamma} \\left(e^{2}_{d} + e^{2}_{u}\\right) & 0.5 P^{x}_{gq} e^{2}_{d} + 0.5 P^{x}_{gq} e^{2}_{u} + 1.0 P_{gq} & - 0.5 P^{x}_{gq} e^{2}_{d} + 0.5 P^{x}_{gq} e^{2}_{u} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\P^{x}_{\\gamma g} \\left(e^{2}_{d} + e^{2}_{u}\\right) & P^{x}_{\\gamma\\gamma} \\left(e^{2}_{d} + e^{2}_{u}\\right) & 0.5 P^{x}_{\\gamma q} e^{2}_{d} + 0.5 P^{x}_{\\gamma q} e^{2}_{u} & - 0.5 P^{x}_{\\gamma q} e^{2}_{d} + 0.5 P^{x}_{\\gamma q} e^{2}_{u} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\2 P^{x}_{qg} e^{2}_{d} + 2 P^{x}_{qg} e^{2}_{u} + 4 P_{qg} & 2 P^{x}_{q\\gamma} e^{2}_{d} + 2 P^{x}_{q\\gamma} e^{2}_{u} & 0.5 P^{x}_{+} e^{2}_{d} + 0.5 P^{x}_{+} e^{2}_{u} + 1.0 P_{qq} + 0.25 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.5 e^{2}_{d} e^{2}_{u} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.25 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & - 0.5 P^{x}_{+} e^{2}_{d} + 0.5 P^{x}_{+} e^{2}_{u} - 0.25 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.25 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\- 2 P^{x}_{qg} e^{2}_{d} + 2.0 P^{x}_{qg} e^{2}_{u} & - 2 P^{x}_{q\\gamma} e^{2}_{d} + 2.0 P^{x}_{q\\gamma} e^{2}_{u} & - 0.5 P^{x}_{+} e^{2}_{d} + 0.5 P^{x}_{+} e^{2}_{u} - 0.25 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.25 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & 0.5 P^{x}_{+} e^{2}_{d} + 0.5 P^{x}_{+} e^{2}_{u} + 1.0 P_{+} + 0.25 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) - 0.5 e^{2}_{d} e^{2}_{u} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.25 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0.5 P^{x}_{-} e^{2}_{d} + 0.5 P^{x}_{-} e^{2}_{u} + 1.0 P_{V} & - 0.5 P^{x}_{-} e^{2}_{d} + 0.5 P^{x}_{-} e^{2}_{u} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & - 0.5 P^{x}_{-} e^{2}_{d} + 0.5 P^{x}_{-} e^{2}_{u} & 0.5 P^{x}_{-} e^{2}_{d} + 0.5 P^{x}_{-} e^{2}_{u} + 1.0 P_{-} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\end{array}\\right]$" - ], - "text/plain": [ - "Matrix([\n", - "[ P^x_gg*(e_d^2 + e_u^2) + P_gg, P^x_g\\gamma*(e_d^2 + e_u^2), 0.5*P^x_gq*e_d^2 + 0.5*P^x_gq*e_u^2 + 1.0*P_gq, -0.5*P^x_gq*e_d^2 + 0.5*P^x_gq*e_u^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[ P^x_\\gamma g*(e_d^2 + e_u^2), P^x_\\gamma\\gamma*(e_d^2 + e_u^2), 0.5*P^x_\\gamma q*e_d^2 + 0.5*P^x_\\gamma q*e_u^2, -0.5*P^x_\\gamma q*e_d^2 + 0.5*P^x_\\gamma q*e_u^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[2*P^x_qg*e_d^2 + 2*P^x_qg*e_u^2 + 4*P_qg, 2*P^x_q\\gamma*e_d^2 + 2*P^x_q\\gamma*e_u^2, 0.5*P^x_+*e_d^2 + 0.5*P^x_+*e_u^2 + 1.0*P_qq + 0.25*e_d^2**2*(-P^x_+ + P^x_qq) + 0.5*e_d^2*e_u^2*(-P^x_+ + P^x_qq) + 0.25*e_u^2**2*(-P^x_+ + P^x_qq), -0.5*P^x_+*e_d^2 + 0.5*P^x_+*e_u^2 - 0.25*e_d^2**2*(-P^x_+ + P^x_qq) + 0.25*e_u^2**2*(-P^x_+ + P^x_qq), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[ -2*P^x_qg*e_d^2 + 2.0*P^x_qg*e_u^2, -2*P^x_q\\gamma*e_d^2 + 2.0*P^x_q\\gamma*e_u^2, -0.5*P^x_+*e_d^2 + 0.5*P^x_+*e_u^2 - 0.25*e_d^2**2*(-P^x_+ + P^x_qq) + 0.25*e_u^2**2*(-P^x_+ + P^x_qq), 0.5*P^x_+*e_d^2 + 0.5*P^x_+*e_u^2 + 1.0*P_+ + 0.25*e_d^2**2*(-P^x_+ + P^x_qq) - 0.5*e_d^2*e_u^2*(-P^x_+ + P^x_qq) + 0.25*e_u^2**2*(-P^x_+ + P^x_qq), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, 0.5*P^x_-*e_d^2 + 0.5*P^x_-*e_u^2 + 1.0*P_V, -0.5*P^x_-*e_d^2 + 0.5*P^x_-*e_u^2, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, -0.5*P^x_-*e_d^2 + 0.5*P^x_-*e_u^2, 0.5*P^x_-*e_d^2 + 0.5*P^x_-*e_u^2 + 1.0*P_-, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]])" - ] - }, - "execution_count": 318, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "P_ev(2)" - ] - }, - { - "cell_type": "code", - "execution_count": 319, - "id": "07b1874a", - "metadata": {}, - "outputs": [ - { - "data": { - "text/latex": [ - "$\\displaystyle \\left[\\begin{array}{cccccccccccccc}P^{x}_{gg} \\left(2 e^{2}_{d} + e^{2}_{u}\\right) + P_{gg} & P^{x}_{g\\gamma} \\left(2 e^{2}_{d} + e^{2}_{u}\\right) & 0.666666666666667 P^{x}_{gq} e^{2}_{d} + 0.333333333333333 P^{x}_{gq} e^{2}_{u} + 1.0 P_{gq} & - 0.333333333333333 P^{x}_{gq} e^{2}_{d} + 0.333333333333333 P^{x}_{gq} e^{2}_{u} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\P^{x}_{\\gamma g} \\left(2 e^{2}_{d} + e^{2}_{u}\\right) & P^{x}_{\\gamma\\gamma} \\left(2 e^{2}_{d} + e^{2}_{u}\\right) & 0.666666666666667 P^{x}_{\\gamma q} e^{2}_{d} + 0.333333333333333 P^{x}_{\\gamma q} e^{2}_{u} & - 0.333333333333333 P^{x}_{\\gamma q} e^{2}_{d} + 0.333333333333333 P^{x}_{\\gamma q} e^{2}_{u} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\4 P^{x}_{qg} e^{2}_{d} + 2 P^{x}_{qg} e^{2}_{u} + 6 P_{qg} & 4 P^{x}_{q\\gamma} e^{2}_{d} + 2 P^{x}_{q\\gamma} e^{2}_{u} & 0.666666666666667 P^{x}_{+} e^{2}_{d} + 0.333333333333333 P^{x}_{+} e^{2}_{u} + 1.0 P_{qq} + 0.444444444444444 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.444444444444444 e^{2}_{d} e^{2}_{u} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.111111111111111 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & - 0.333333333333333 P^{x}_{+} e^{2}_{d} + 0.333333333333333 P^{x}_{+} e^{2}_{u} - 0.222222222222222 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.111111111111111 e^{2}_{d} e^{2}_{u} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.111111111111111 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\- 4 P^{x}_{qg} e^{2}_{d} + 4.0 P^{x}_{qg} e^{2}_{u} & - 4 P^{x}_{q\\gamma} e^{2}_{d} + 4.0 P^{x}_{q\\gamma} e^{2}_{u} & - 0.666666666666667 P^{x}_{+} e^{2}_{d} + 0.666666666666667 P^{x}_{+} e^{2}_{u} - 0.444444444444444 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.222222222222222 e^{2}_{d} e^{2}_{u} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.222222222222222 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & 0.333333333333333 P^{x}_{+} e^{2}_{d} + 0.666666666666667 P^{x}_{+} e^{2}_{u} + 1.0 P_{+} + 0.222222222222222 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) - 0.444444444444444 e^{2}_{d} e^{2}_{u} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.222222222222222 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0.666666666666667 P^{x}_{-} e^{2}_{d} + 0.333333333333333 P^{x}_{-} e^{2}_{u} + 1.0 P_{V} & - 0.333333333333333 P^{x}_{-} e^{2}_{d} + 0.333333333333333 P^{x}_{-} e^{2}_{u} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & - 0.666666666666667 P^{x}_{-} e^{2}_{d} + 0.666666666666667 P^{x}_{-} e^{2}_{u} & 0.333333333333333 P^{x}_{-} e^{2}_{d} + 0.666666666666667 P^{x}_{-} e^{2}_{u} + 1.0 P_{-} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{+} e^{2}_{d} + 1.0 P_{+} & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{-} e^{2}_{d} + 1.0 P_{-} & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\end{array}\\right]$" - ], - "text/plain": [ - "Matrix([\n", - "[ P^x_gg*(2*e_d^2 + e_u^2) + P_gg, P^x_g\\gamma*(2*e_d^2 + e_u^2), 0.666666666666667*P^x_gq*e_d^2 + 0.333333333333333*P^x_gq*e_u^2 + 1.0*P_gq, -0.333333333333333*P^x_gq*e_d^2 + 0.333333333333333*P^x_gq*e_u^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[ P^x_\\gamma g*(2*e_d^2 + e_u^2), P^x_\\gamma\\gamma*(2*e_d^2 + e_u^2), 0.666666666666667*P^x_\\gamma q*e_d^2 + 0.333333333333333*P^x_\\gamma q*e_u^2, -0.333333333333333*P^x_\\gamma q*e_d^2 + 0.333333333333333*P^x_\\gamma q*e_u^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[4*P^x_qg*e_d^2 + 2*P^x_qg*e_u^2 + 6*P_qg, 4*P^x_q\\gamma*e_d^2 + 2*P^x_q\\gamma*e_u^2, 0.666666666666667*P^x_+*e_d^2 + 0.333333333333333*P^x_+*e_u^2 + 1.0*P_qq + 0.444444444444444*e_d^2**2*(-P^x_+ + P^x_qq) + 0.444444444444444*e_d^2*e_u^2*(-P^x_+ + P^x_qq) + 0.111111111111111*e_u^2**2*(-P^x_+ + P^x_qq), -0.333333333333333*P^x_+*e_d^2 + 0.333333333333333*P^x_+*e_u^2 - 0.222222222222222*e_d^2**2*(-P^x_+ + P^x_qq) + 0.111111111111111*e_d^2*e_u^2*(-P^x_+ + P^x_qq) + 0.111111111111111*e_u^2**2*(-P^x_+ + P^x_qq), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[ -4*P^x_qg*e_d^2 + 4.0*P^x_qg*e_u^2, -4*P^x_q\\gamma*e_d^2 + 4.0*P^x_q\\gamma*e_u^2, -0.666666666666667*P^x_+*e_d^2 + 0.666666666666667*P^x_+*e_u^2 - 0.444444444444444*e_d^2**2*(-P^x_+ + P^x_qq) + 0.222222222222222*e_d^2*e_u^2*(-P^x_+ + P^x_qq) + 0.222222222222222*e_u^2**2*(-P^x_+ + P^x_qq), 0.333333333333333*P^x_+*e_d^2 + 0.666666666666667*P^x_+*e_u^2 + 1.0*P_+ + 0.222222222222222*e_d^2**2*(-P^x_+ + P^x_qq) - 0.444444444444444*e_d^2*e_u^2*(-P^x_+ + P^x_qq) + 0.222222222222222*e_u^2**2*(-P^x_+ + P^x_qq), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, 0.666666666666667*P^x_-*e_d^2 + 0.333333333333333*P^x_-*e_u^2 + 1.0*P_V, -0.333333333333333*P^x_-*e_d^2 + 0.333333333333333*P^x_-*e_u^2, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, -0.666666666666667*P^x_-*e_d^2 + 0.666666666666667*P^x_-*e_u^2, 0.333333333333333*P^x_-*e_d^2 + 0.666666666666667*P^x_-*e_u^2 + 1.0*P_-, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, 0, 0, 1.0*P^x_+*e_d^2 + 1.0*P_+, 0, 0, 0, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, 0, 0, 0, 1.0*P^x_-*e_d^2 + 1.0*P_-, 0, 0, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]])" - ] - }, - "execution_count": 319, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "P_ev(3)" - ] - }, - { - "cell_type": "code", - "execution_count": 320, - "id": "c9f75de1", - "metadata": {}, - "outputs": [ - { - "data": { - "text/latex": [ - "$\\displaystyle \\left[\\begin{array}{cccccccccccccc}P^{x}_{gg} \\left(2 e^{2}_{d} + 2 e^{2}_{u}\\right) + P_{gg} & P^{x}_{g\\gamma} \\left(2 e^{2}_{d} + 2 e^{2}_{u}\\right) & 0.5 P^{x}_{gq} e^{2}_{d} + 0.5 P^{x}_{gq} e^{2}_{u} + 1.0 P_{gq} & - 0.5 P^{x}_{gq} e^{2}_{d} + 0.5 P^{x}_{gq} e^{2}_{u} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\P^{x}_{\\gamma g} \\left(2 e^{2}_{d} + 2 e^{2}_{u}\\right) & P^{x}_{\\gamma\\gamma} \\left(2 e^{2}_{d} + 2 e^{2}_{u}\\right) & 0.5 P^{x}_{\\gamma q} e^{2}_{d} + 0.5 P^{x}_{\\gamma q} e^{2}_{u} & - 0.5 P^{x}_{\\gamma q} e^{2}_{d} + 0.5 P^{x}_{\\gamma q} e^{2}_{u} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\4 P^{x}_{qg} e^{2}_{d} + 4 P^{x}_{qg} e^{2}_{u} + 8 P_{qg} & 4 P^{x}_{q\\gamma} e^{2}_{d} + 4 P^{x}_{q\\gamma} e^{2}_{u} & 0.5 P^{x}_{+} e^{2}_{d} + 0.5 P^{x}_{+} e^{2}_{u} + 1.0 P_{qq} + 0.25 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.5 e^{2}_{d} e^{2}_{u} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.25 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & - 0.5 P^{x}_{+} e^{2}_{d} + 0.5 P^{x}_{+} e^{2}_{u} - 0.25 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.25 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\- 4 P^{x}_{qg} e^{2}_{d} + 4.0 P^{x}_{qg} e^{2}_{u} & - 4 P^{x}_{q\\gamma} e^{2}_{d} + 4.0 P^{x}_{q\\gamma} e^{2}_{u} & - 0.5 P^{x}_{+} e^{2}_{d} + 0.5 P^{x}_{+} e^{2}_{u} - 0.25 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.25 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & 0.5 P^{x}_{+} e^{2}_{d} + 0.5 P^{x}_{+} e^{2}_{u} + 1.0 P_{+} + 0.25 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) - 0.5 e^{2}_{d} e^{2}_{u} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.25 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0.5 P^{x}_{-} e^{2}_{d} + 0.5 P^{x}_{-} e^{2}_{u} + 1.0 P_{V} & - 0.5 P^{x}_{-} e^{2}_{d} + 0.5 P^{x}_{-} e^{2}_{u} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & - 0.5 P^{x}_{-} e^{2}_{d} + 0.5 P^{x}_{-} e^{2}_{u} & 0.5 P^{x}_{-} e^{2}_{d} + 0.5 P^{x}_{-} e^{2}_{u} + 1.0 P_{-} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{+} e^{2}_{d} + 1.0 P_{+} & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{-} e^{2}_{d} + 1.0 P_{-} & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{+} e^{2}_{u} + 1.0 P_{+} & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{-} e^{2}_{u} + 1.0 P_{-} & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\end{array}\\right]$" - ], - "text/plain": [ - "Matrix([\n", - "[ P^x_gg*(2*e_d^2 + 2*e_u^2) + P_gg, P^x_g\\gamma*(2*e_d^2 + 2*e_u^2), 0.5*P^x_gq*e_d^2 + 0.5*P^x_gq*e_u^2 + 1.0*P_gq, -0.5*P^x_gq*e_d^2 + 0.5*P^x_gq*e_u^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[ P^x_\\gamma g*(2*e_d^2 + 2*e_u^2), P^x_\\gamma\\gamma*(2*e_d^2 + 2*e_u^2), 0.5*P^x_\\gamma q*e_d^2 + 0.5*P^x_\\gamma q*e_u^2, -0.5*P^x_\\gamma q*e_d^2 + 0.5*P^x_\\gamma q*e_u^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[4*P^x_qg*e_d^2 + 4*P^x_qg*e_u^2 + 8*P_qg, 4*P^x_q\\gamma*e_d^2 + 4*P^x_q\\gamma*e_u^2, 0.5*P^x_+*e_d^2 + 0.5*P^x_+*e_u^2 + 1.0*P_qq + 0.25*e_d^2**2*(-P^x_+ + P^x_qq) + 0.5*e_d^2*e_u^2*(-P^x_+ + P^x_qq) + 0.25*e_u^2**2*(-P^x_+ + P^x_qq), -0.5*P^x_+*e_d^2 + 0.5*P^x_+*e_u^2 - 0.25*e_d^2**2*(-P^x_+ + P^x_qq) + 0.25*e_u^2**2*(-P^x_+ + P^x_qq), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[ -4*P^x_qg*e_d^2 + 4.0*P^x_qg*e_u^2, -4*P^x_q\\gamma*e_d^2 + 4.0*P^x_q\\gamma*e_u^2, -0.5*P^x_+*e_d^2 + 0.5*P^x_+*e_u^2 - 0.25*e_d^2**2*(-P^x_+ + P^x_qq) + 0.25*e_u^2**2*(-P^x_+ + P^x_qq), 0.5*P^x_+*e_d^2 + 0.5*P^x_+*e_u^2 + 1.0*P_+ + 0.25*e_d^2**2*(-P^x_+ + P^x_qq) - 0.5*e_d^2*e_u^2*(-P^x_+ + P^x_qq) + 0.25*e_u^2**2*(-P^x_+ + P^x_qq), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, 0.5*P^x_-*e_d^2 + 0.5*P^x_-*e_u^2 + 1.0*P_V, -0.5*P^x_-*e_d^2 + 0.5*P^x_-*e_u^2, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, -0.5*P^x_-*e_d^2 + 0.5*P^x_-*e_u^2, 0.5*P^x_-*e_d^2 + 0.5*P^x_-*e_u^2 + 1.0*P_-, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, 0, 0, 1.0*P^x_+*e_d^2 + 1.0*P_+, 0, 0, 0, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, 0, 0, 0, 1.0*P^x_-*e_d^2 + 1.0*P_-, 0, 0, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, 0, 0, 0, 0, 1.0*P^x_+*e_u^2 + 1.0*P_+, 0, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1.0*P^x_-*e_u^2 + 1.0*P_-, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]])" - ] - }, - "execution_count": 320, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "P_ev(4)" - ] - }, - { - "cell_type": "code", - "execution_count": 321, - "id": "ce40d361", - "metadata": {}, - "outputs": [ - { - "data": { - "text/latex": [ - "$\\displaystyle \\left[\\begin{array}{cccccccccccccc}P^{x}_{gg} \\left(3 e^{2}_{d} + 2 e^{2}_{u}\\right) + P_{gg} & P^{x}_{g\\gamma} \\left(3 e^{2}_{d} + 2 e^{2}_{u}\\right) & 0.6 P^{x}_{gq} e^{2}_{d} + 0.4 P^{x}_{gq} e^{2}_{u} + 1.0 P_{gq} & - 0.4 P^{x}_{gq} e^{2}_{d} + 0.4 P^{x}_{gq} e^{2}_{u} + 2.77555756156289 \\cdot 10^{-17} P_{gq} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\P^{x}_{\\gamma g} \\left(3 e^{2}_{d} + 2 e^{2}_{u}\\right) & P^{x}_{\\gamma\\gamma} \\left(3 e^{2}_{d} + 2 e^{2}_{u}\\right) & 0.6 P^{x}_{\\gamma q} e^{2}_{d} + 0.4 P^{x}_{\\gamma q} e^{2}_{u} & - 0.4 P^{x}_{\\gamma q} e^{2}_{d} + 0.4 P^{x}_{\\gamma q} e^{2}_{u} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\6 P^{x}_{qg} e^{2}_{d} + 4 P^{x}_{qg} e^{2}_{u} + 10 P_{qg} & 6 P^{x}_{q\\gamma} e^{2}_{d} + 4 P^{x}_{q\\gamma} e^{2}_{u} & 0.6 P^{x}_{+} e^{2}_{d} + 0.4 P^{x}_{+} e^{2}_{u} + 1.0 P_{qq} + 0.36 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.48 e^{2}_{d} e^{2}_{u} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.16 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & - 0.4 P^{x}_{+} e^{2}_{d} + 0.4 P^{x}_{+} e^{2}_{u} + 2.77555756156289 \\cdot 10^{-17} P_{+} - 0.24 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.08 e^{2}_{d} e^{2}_{u} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.16 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\- 6 P^{x}_{qg} e^{2}_{d} + 6.0 P^{x}_{qg} e^{2}_{u} & - 6 P^{x}_{q\\gamma} e^{2}_{d} + 6.0 P^{x}_{q\\gamma} e^{2}_{u} & - 0.6 P^{x}_{+} e^{2}_{d} + 0.6 P^{x}_{+} e^{2}_{u} + 5.55111512312578 \\cdot 10^{-17} P_{qq} - 0.36 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.12 e^{2}_{d} e^{2}_{u} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.24 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & 0.4 P^{x}_{+} e^{2}_{d} + 0.6 P^{x}_{+} e^{2}_{u} + 1.0 P_{+} + 0.24 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) - 0.48 e^{2}_{d} e^{2}_{u} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.24 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0.6 P^{x}_{-} e^{2}_{d} + 0.4 P^{x}_{-} e^{2}_{u} + 1.0 P_{V} & - 0.4 P^{x}_{-} e^{2}_{d} + 0.4 P^{x}_{-} e^{2}_{u} + 2.77555756156289 \\cdot 10^{-17} P_{-} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & - 0.6 P^{x}_{-} e^{2}_{d} + 0.6 P^{x}_{-} e^{2}_{u} + 5.55111512312578 \\cdot 10^{-17} P_{V} & 0.4 P^{x}_{-} e^{2}_{d} + 0.6 P^{x}_{-} e^{2}_{u} + 1.0 P_{-} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{+} e^{2}_{d} + 1.0 P_{+} & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{-} e^{2}_{d} + 1.0 P_{-} & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{+} e^{2}_{u} + 1.0 P_{+} & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{-} e^{2}_{u} + 1.0 P_{-} & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{+} e^{2}_{d} + 1.0 P_{+} & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{-} e^{2}_{d} + 1.0 P_{-} & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\end{array}\\right]$" - ], - "text/plain": [ - "Matrix([\n", - "[ P^x_gg*(3*e_d^2 + 2*e_u^2) + P_gg, P^x_g\\gamma*(3*e_d^2 + 2*e_u^2), 0.6*P^x_gq*e_d^2 + 0.4*P^x_gq*e_u^2 + 1.0*P_gq, -0.4*P^x_gq*e_d^2 + 0.4*P^x_gq*e_u^2 + 2.77555756156289e-17*P_gq, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[ P^x_\\gamma g*(3*e_d^2 + 2*e_u^2), P^x_\\gamma\\gamma*(3*e_d^2 + 2*e_u^2), 0.6*P^x_\\gamma q*e_d^2 + 0.4*P^x_\\gamma q*e_u^2, -0.4*P^x_\\gamma q*e_d^2 + 0.4*P^x_\\gamma q*e_u^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[6*P^x_qg*e_d^2 + 4*P^x_qg*e_u^2 + 10*P_qg, 6*P^x_q\\gamma*e_d^2 + 4*P^x_q\\gamma*e_u^2, 0.6*P^x_+*e_d^2 + 0.4*P^x_+*e_u^2 + 1.0*P_qq + 0.36*e_d^2**2*(-P^x_+ + P^x_qq) + 0.48*e_d^2*e_u^2*(-P^x_+ + P^x_qq) + 0.16*e_u^2**2*(-P^x_+ + P^x_qq), -0.4*P^x_+*e_d^2 + 0.4*P^x_+*e_u^2 + 2.77555756156289e-17*P_+ - 0.24*e_d^2**2*(-P^x_+ + P^x_qq) + 0.08*e_d^2*e_u^2*(-P^x_+ + P^x_qq) + 0.16*e_u^2**2*(-P^x_+ + P^x_qq), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[ -6*P^x_qg*e_d^2 + 6.0*P^x_qg*e_u^2, -6*P^x_q\\gamma*e_d^2 + 6.0*P^x_q\\gamma*e_u^2, -0.6*P^x_+*e_d^2 + 0.6*P^x_+*e_u^2 + 5.55111512312578e-17*P_qq - 0.36*e_d^2**2*(-P^x_+ + P^x_qq) + 0.12*e_d^2*e_u^2*(-P^x_+ + P^x_qq) + 0.24*e_u^2**2*(-P^x_+ + P^x_qq), 0.4*P^x_+*e_d^2 + 0.6*P^x_+*e_u^2 + 1.0*P_+ + 0.24*e_d^2**2*(-P^x_+ + P^x_qq) - 0.48*e_d^2*e_u^2*(-P^x_+ + P^x_qq) + 0.24*e_u^2**2*(-P^x_+ + P^x_qq), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, 0.6*P^x_-*e_d^2 + 0.4*P^x_-*e_u^2 + 1.0*P_V, -0.4*P^x_-*e_d^2 + 0.4*P^x_-*e_u^2 + 2.77555756156289e-17*P_-, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, -0.6*P^x_-*e_d^2 + 0.6*P^x_-*e_u^2 + 5.55111512312578e-17*P_V, 0.4*P^x_-*e_d^2 + 0.6*P^x_-*e_u^2 + 1.0*P_-, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, 0, 0, 1.0*P^x_+*e_d^2 + 1.0*P_+, 0, 0, 0, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, 0, 0, 0, 1.0*P^x_-*e_d^2 + 1.0*P_-, 0, 0, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, 0, 0, 0, 0, 1.0*P^x_+*e_u^2 + 1.0*P_+, 0, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1.0*P^x_-*e_u^2 + 1.0*P_-, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1.0*P^x_+*e_d^2 + 1.0*P_+, 0, 0, 0],\n", - "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1.0*P^x_-*e_d^2 + 1.0*P_-, 0, 0],\n", - "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]])" - ] - }, - "execution_count": 321, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "P_ev(5)" - ] - }, - { - "cell_type": "code", - "execution_count": 322, - "id": "3f014fa2", - "metadata": {}, - "outputs": [ - { - "data": { - "text/latex": [ - "$\\displaystyle \\left[\\begin{array}{cccccccccccccc}P^{x}_{gg} \\left(3 e^{2}_{d} + 3 e^{2}_{u}\\right) + P_{gg} & P^{x}_{g\\gamma} \\left(3 e^{2}_{d} + 3 e^{2}_{u}\\right) & 0.5 P^{x}_{gq} e^{2}_{d} + 0.5 P^{x}_{gq} e^{2}_{u} + 1.0 P_{gq} & - 0.5 P^{x}_{gq} e^{2}_{d} + 0.5 P^{x}_{gq} e^{2}_{u} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\P^{x}_{\\gamma g} \\left(3 e^{2}_{d} + 3 e^{2}_{u}\\right) & P^{x}_{\\gamma\\gamma} \\left(3 e^{2}_{d} + 3 e^{2}_{u}\\right) & 0.5 P^{x}_{\\gamma q} e^{2}_{d} + 0.5 P^{x}_{\\gamma q} e^{2}_{u} & - 0.5 P^{x}_{\\gamma q} e^{2}_{d} + 0.5 P^{x}_{\\gamma q} e^{2}_{u} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\6 P^{x}_{qg} e^{2}_{d} + 6 P^{x}_{qg} e^{2}_{u} + 12 P_{qg} & 6 P^{x}_{q\\gamma} e^{2}_{d} + 6 P^{x}_{q\\gamma} e^{2}_{u} & 0.5 P^{x}_{+} e^{2}_{d} + 0.5 P^{x}_{+} e^{2}_{u} - 1.11022302462516 \\cdot 10^{-16} P_{+} + 1.0 P_{qq} + 0.25 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.5 e^{2}_{d} e^{2}_{u} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.25 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & - 0.5 P^{x}_{+} e^{2}_{d} + 0.5 P^{x}_{+} e^{2}_{u} - 0.25 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.25 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\- 6 P^{x}_{qg} e^{2}_{d} + 6.0 P^{x}_{qg} e^{2}_{u} & - 6 P^{x}_{q\\gamma} e^{2}_{d} + 6.0 P^{x}_{q\\gamma} e^{2}_{u} & - 0.5 P^{x}_{+} e^{2}_{d} + 0.5 P^{x}_{+} e^{2}_{u} - 0.25 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.25 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & 0.5 P^{x}_{+} e^{2}_{d} + 0.5 P^{x}_{+} e^{2}_{u} + 1.0 P_{+} + 0.25 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) - 0.5 e^{2}_{d} e^{2}_{u} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.25 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0.5 P^{x}_{-} e^{2}_{d} + 0.5 P^{x}_{-} e^{2}_{u} - 1.11022302462516 \\cdot 10^{-16} P_{-} + 1.0 P_{V} & - 0.5 P^{x}_{-} e^{2}_{d} + 0.5 P^{x}_{-} e^{2}_{u} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & - 0.5 P^{x}_{-} e^{2}_{d} + 0.5 P^{x}_{-} e^{2}_{u} & 0.5 P^{x}_{-} e^{2}_{d} + 0.5 P^{x}_{-} e^{2}_{u} + 1.0 P_{-} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{+} e^{2}_{d} + 1.0 P_{+} & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{-} e^{2}_{d} + 1.0 P_{-} & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{+} e^{2}_{u} + 1.0 P_{+} & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{-} e^{2}_{u} + 1.0 P_{-} & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{+} e^{2}_{d} + 1.0 P_{+} & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{-} e^{2}_{d} + 1.0 P_{-} & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{+} e^{2}_{u} + 1.0 P_{+} & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 P^{x}_{-} e^{2}_{u} + 1.0 P_{-}\\end{array}\\right]$" - ], - "text/plain": [ - "Matrix([\n", - "[ P^x_gg*(3*e_d^2 + 3*e_u^2) + P_gg, P^x_g\\gamma*(3*e_d^2 + 3*e_u^2), 0.5*P^x_gq*e_d^2 + 0.5*P^x_gq*e_u^2 + 1.0*P_gq, -0.5*P^x_gq*e_d^2 + 0.5*P^x_gq*e_u^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[ P^x_\\gamma g*(3*e_d^2 + 3*e_u^2), P^x_\\gamma\\gamma*(3*e_d^2 + 3*e_u^2), 0.5*P^x_\\gamma q*e_d^2 + 0.5*P^x_\\gamma q*e_u^2, -0.5*P^x_\\gamma q*e_d^2 + 0.5*P^x_\\gamma q*e_u^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[6*P^x_qg*e_d^2 + 6*P^x_qg*e_u^2 + 12*P_qg, 6*P^x_q\\gamma*e_d^2 + 6*P^x_q\\gamma*e_u^2, 0.5*P^x_+*e_d^2 + 0.5*P^x_+*e_u^2 - 1.11022302462516e-16*P_+ + 1.0*P_qq + 0.25*e_d^2**2*(-P^x_+ + P^x_qq) + 0.5*e_d^2*e_u^2*(-P^x_+ + P^x_qq) + 0.25*e_u^2**2*(-P^x_+ + P^x_qq), -0.5*P^x_+*e_d^2 + 0.5*P^x_+*e_u^2 - 0.25*e_d^2**2*(-P^x_+ + P^x_qq) + 0.25*e_u^2**2*(-P^x_+ + P^x_qq), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[ -6*P^x_qg*e_d^2 + 6.0*P^x_qg*e_u^2, -6*P^x_q\\gamma*e_d^2 + 6.0*P^x_q\\gamma*e_u^2, -0.5*P^x_+*e_d^2 + 0.5*P^x_+*e_u^2 - 0.25*e_d^2**2*(-P^x_+ + P^x_qq) + 0.25*e_u^2**2*(-P^x_+ + P^x_qq), 0.5*P^x_+*e_d^2 + 0.5*P^x_+*e_u^2 + 1.0*P_+ + 0.25*e_d^2**2*(-P^x_+ + P^x_qq) - 0.5*e_d^2*e_u^2*(-P^x_+ + P^x_qq) + 0.25*e_u^2**2*(-P^x_+ + P^x_qq), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, 0.5*P^x_-*e_d^2 + 0.5*P^x_-*e_u^2 - 1.11022302462516e-16*P_- + 1.0*P_V, -0.5*P^x_-*e_d^2 + 0.5*P^x_-*e_u^2, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, -0.5*P^x_-*e_d^2 + 0.5*P^x_-*e_u^2, 0.5*P^x_-*e_d^2 + 0.5*P^x_-*e_u^2 + 1.0*P_-, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, 0, 0, 1.0*P^x_+*e_d^2 + 1.0*P_+, 0, 0, 0, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, 0, 0, 0, 1.0*P^x_-*e_d^2 + 1.0*P_-, 0, 0, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, 0, 0, 0, 0, 1.0*P^x_+*e_u^2 + 1.0*P_+, 0, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1.0*P^x_-*e_u^2 + 1.0*P_-, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1.0*P^x_+*e_d^2 + 1.0*P_+, 0, 0, 0],\n", - "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1.0*P^x_-*e_d^2 + 1.0*P_-, 0, 0],\n", - "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1.0*P^x_+*e_u^2 + 1.0*P_+, 0],\n", - "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1.0*P^x_-*e_u^2 + 1.0*P_-]])" - ] - }, - "execution_count": 322, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "P_ev(6)" - ] - }, - { - "cell_type": "code", - "execution_count": 323, - "id": "53626511", - "metadata": {}, - "outputs": [], - "source": [ - "def eD2_(nf):\n", - " nu = int(nf/2)\n", - " nd = nf - nu\n", - " return nd*eu2 + nu*ed2\n", - "def etam_(nf):\n", - " nu = int(nf/2)\n", - " nd = nf - nu\n", - " return 0.5*(eu2 - ed2)" - ] - }, - { - "cell_type": "code", - "execution_count": 324, - "id": "9c9ce615", - "metadata": {}, - "outputs": [], - "source": [ - "def P_ev_sing(nf):\n", - " return P_ev(nf)[:4,:4]\n", - "\n", - "def P_ev_val(nf):\n", - " return P_ev(nf)[4:6,4:6]" - ] - }, - { - "cell_type": "code", - "execution_count": 325, - "id": "5267ab16", - "metadata": {}, - "outputs": [], - "source": [ - "def P_ev_sing2(nf):\n", - " es2=es2_(nf)\n", - " eD2=eD2_(nf)\n", - " etam=etam_(nf)\n", - " nu = int(nf/2)\n", - " nd = nf - nu\n", - " res = sympy.Matrix([\n", - " [Pgg + es2 * Pxgg, es2 * Pxgy, Pgq + es2/nf*Pxgq, 2*nu/nf*etam*Pxgq],\n", - " [es2 * Pxyg, es2 * Pxyy, es2/nf*Pxyq, 2*nu/nf*etam*Pxyq],\n", - " [2*nf*Pqg + 2*es2*Pxqg, 2*es2 * Pxqy, Pqq + es2/nf*Pxp +(es2/nf)**2*(Pxqq - Pxp), 2*nu/nf*etam*Pxp +2*nu*etam*es2/nf**2*(Pxqq - Pxp)],\n", - " [4*nd*etam*Pxqg, 4*nd*etam*Pxqy, 2*nd/nf*etam*Pxp +2*nd*etam*es2/nf**2*(Pxqq - Pxp), Pp + eD2/nf*Pxp + 4*nu*nd/nf**2*etam**2*(Pxqq - Pxp)]\n", - " ])\n", - " return res\n", - "\n", - "def P_ev_val2(nf):\n", - " es2=e2s(nf)\n", - " eD2=eD2_(nf)\n", - " etam=etam_(nf)\n", - " nu = int(nf/2)\n", - " nd = nf - nu\n", - " res = sympy.Matrix([\n", - " [Pv+es2/nf*Pxm, 2*nu/nf*etam*Pxm],\n", - " [2*nd/nf*etam*Pxm, Pm + eD2/nf*Pxm]\n", - " ])\n", - " return res" - ] - }, - { - "cell_type": "code", - "execution_count": 326, - "id": "a47dc3b2", - "metadata": {}, - "outputs": [ - { - "data": { - "text/latex": [ - "$\\displaystyle \\left[\\begin{matrix}0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\end{matrix}\\right]$" - ], - "text/plain": [ - "Matrix([\n", - "[0, 0, 0, 0],\n", - "[0, 0, 0, 0],\n", - "[0, 0, 0, 0],\n", - "[0, 0, 0, 0]])" - ] - }, - "execution_count": 326, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "sympy.simplify(P_ev_sing(2)-P_ev_sing2(2))" - ] - }, - { - "cell_type": "code", - "execution_count": 327, - "id": "6fddcd7c", - "metadata": {}, - "outputs": [ - { - "data": { - "text/latex": [ - "$\\displaystyle \\left[\\begin{matrix}0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\\\0 & 0 & 0 & - 1.11022302462516 \\cdot 10^{-16} P_{+}\\end{matrix}\\right]$" - ], - "text/plain": [ - "Matrix([\n", - "[0, 0, 0, 0],\n", - "[0, 0, 0, 0],\n", - "[0, 0, 0, 0],\n", - "[0, 0, 0, -1.11022302462516e-16*P_+]])" - ] - }, - "execution_count": 327, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "sympy.simplify(P_ev_sing(3)-P_ev_sing2(3))" - ] - }, - { - "cell_type": "code", - "execution_count": 328, - "id": "f151ecdd", - "metadata": {}, - "outputs": [ - { - "data": { - "text/latex": [ - "$\\displaystyle \\left[\\begin{matrix}0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\end{matrix}\\right]$" - ], - "text/plain": [ - "Matrix([\n", - "[0, 0, 0, 0],\n", - "[0, 0, 0, 0],\n", - "[0, 0, 0, 0],\n", - "[0, 0, 0, 0]])" - ] - }, - "execution_count": 328, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "sympy.simplify(P_ev_sing(4)-P_ev_sing2(4))" - ] - }, - { - "cell_type": "code", - "execution_count": 329, - "id": "e49bef49", - "metadata": {}, - "outputs": [ - { - "data": { - "text/latex": [ - "$\\displaystyle \\left[\\begin{matrix}0 & 0 & 8.88178419700125 \\cdot 10^{-17} P^{x}_{gq} e^{2}_{d} & 2.77555756156289 \\cdot 10^{-17} P_{gq}\\\\0 & 0 & 8.88178419700125 \\cdot 10^{-17} P^{x}_{\\gamma q} e^{2}_{d} & 0\\\\0 & 0 & 7.105427357601 \\cdot 10^{-17} P^{x}_{+} e^{2}_{d} - 3.5527136788005 \\cdot 10^{-17} P^{x}_{+} \\left(e^{2}_{u}\\right)^{2} + 3.5527136788005 \\cdot 10^{-17} P^{x}_{qq} \\left(e^{2}_{u}\\right)^{2} & 4.16333634234434 \\cdot 10^{-17} P^{x}_{+} e^{2}_{d} e^{2}_{u} - 2.77555756156289 \\cdot 10^{-17} P^{x}_{+} \\left(e^{2}_{u}\\right)^{2} - 4.16333634234434 \\cdot 10^{-17} P^{x}_{qq} e^{2}_{d} e^{2}_{u} + 2.77555756156289 \\cdot 10^{-17} P^{x}_{qq} \\left(e^{2}_{u}\\right)^{2} + 2.77555756156289 \\cdot 10^{-17} 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--git a/extras/uni-dglap.tex b/extras/uni-dglap.tex index 9ca892030..b0e9dd514 100644 --- a/extras/uni-dglap.tex +++ b/extras/uni-dglap.tex @@ -16,7 +16,30 @@ \begin{document} -\maketitle +%\maketitle +Defining +\begin{align*} +\Sigma_u&=\sum_{k=1}^{n_u}u_k^+, \quad \Sigma_d=\sum_{k=1}^{n_d}d_k^+ \\ +V_u&=\sum_{k=1}^{n_u}u_k^-, \quad V_d=\sum_{k=1}^{n_d}d_k^- +\end{align*} + +our basis is +\begin{align*} +g & \\ +\gamma & \\ +\Sigma &= \Sigma_u + \Sigma_d \\ +\Delta_\Sigma & = \frac{n_d}{n_u}\Sigma_u - \Sigma_d \\ +V & = V_u + V_d \\ +\Delta_V & = \frac{n_d}{n_u}V_u - V_d \\ +T_3^d &=d^+ - s^+ \\ +V_3^d &=d^- - s^- \\ +T_3^u &=u^+ - c^+ \\ +V_3^u &=u^- - c^- \\ +T_8^d &=d^+ + s^+ - 2b^+ \\ +V_8^d &=d^- + s^- - 2b^- \\ +T_8^u &=u^+ + c^+ - 2t^+ \\ +V_8^u &=u^- + c^- - 2t^- +\end{align*} \begin{itemize} \item Singlet sector: @@ -68,7 +91,11 @@ \Delta_V \end{pmatrix} \end{equation*} - +\item Decoupled sector: +\begin{align*} +\mu^2\frac{d}{d\mu^2}T^{u/d}_{3/8} & = (P_{+} + e_i^2 \tilde{P}_{+}) T^{u/d}_{3/8} \\ +\mu^2\frac{d}{d\mu^2}V^{u/d}_{3/8} & = (P_{-} + e_i^2 \tilde{P}_{-} )V^{u/d}_{3/8} +\end{align*} \end{itemize} \end{document} From 99cd0415b896004a87f5597f6e6022066643bd4f Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Niccol=C3=B2=20Laurenti?= Date: Mon, 28 Mar 2022 10:25:39 +0200 Subject: [PATCH 17/71] Simplify again AD for unified evolution --- extras/uni-ad.nb | 3917 ++++++++++++++++++---------------------------- 1 file changed, 1543 insertions(+), 2374 deletions(-) diff --git a/extras/uni-ad.nb b/extras/uni-ad.nb index 522940e7a..139accdcf 100644 --- a/extras/uni-ad.nb +++ b/extras/uni-ad.nb @@ -10,10 +10,10 @@ NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] -NotebookDataLength[ 247518, 6556] -NotebookOptionsPosition[ 237760, 6391] -NotebookOutlinePosition[ 238158, 6407] -CellTagsIndexPosition[ 238115, 6404] +NotebookDataLength[ 215011, 5725] +NotebookOptionsPosition[ 205026, 5558] +NotebookOutlinePosition[ 205466, 5575] 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file changed, 407 insertions(+) create mode 100644 src/eko/anomalous_dimensions/as1aem1.py diff --git a/src/eko/anomalous_dimensions/as1aem1.py b/src/eko/anomalous_dimensions/as1aem1.py new file mode 100644 index 000000000..d08819242 --- /dev/null +++ b/src/eko/anomalous_dimensions/as1aem1.py @@ -0,0 +1,407 @@ +# -*- coding: utf-8 -*- +""" +This file contains the O(as1aem1) Altarelli-Parisi splitting kernels. + +These expression have been obtained using the procedure described in the +`wiki `_ +involving ``FormGet`` :cite:`Hahn:2016ebn`. +""" + +import numba as nb +import numpy as np + +from .. import constants +from . import harmonics + + +@nb.njit("c16(c16,u1,c16[:])", cache=True) +def gamma_phq(N, nf, sx): + """ + Computes the O(as1aem1) photon-quark anomalous dimension + + Implements Eq. (2.5) of :cite:`Carrazza:2015dea`. + + Parameters + ---------- + N : complex + Mellin moment + nf : int + Number of active flavors + sx : np array + List of harmonic sums + + Returns + ------- + gamma_phq : complex + O(as1aem1) photon-quark anomalous dimension :math:`\\gamma_{\\gamma q}^{(1,1)}(N)` + """ + + S1 = sx[0] + S2 = sx[1] + tmp_const = ( + 2 + * (-4 - 12 * N - N**2 + 28 * N**3 + 43 * N**4 + 30 * N**5 + 12 * N**6) + / ((-1 + N) * N**3 * (1 + N) ** 3) + ) + tmp_S1 = ( + -4 + * (10 + 27 * N + 25 * N**2 + 13 * N**3 + 5 * N**4) + / ((-1 + N) * N * (1 + N) ** 3) + ) + tmp_S12 = 4 * (2 + N + N**2) / ((-1 + N) * N * (1 + N)) + tmp_S2 = 4 * (2 + N + N**2) / ((-1 + N) * N * (1 + N)) + + return constants.CF * (tmp_const + tmp_S1 * S1 + tmp_S12 * S1**2 + tmp_S2 * S2) + + +@nb.njit("c16(c16,u1,c16[:])", cache=True) +def gamma_qph(N, nf, sx): + """ + Computes the O(as1aem1) quark-photon anomalous dimension + + Implements Eq. (2.5) of :cite:`Carrazza:2015dea`. + But adding the :math:`N_C` and the :math:`2n_f` factors from :math:`\\theta` inside the + definition of :math:`\\gamma_{q \\gamma}^{(0)}(N)`. + + Parameters + ---------- + N : complex + Mellin moment + nf : int + Number of active flavors + sx : np array + List of harmonic sums + + Returns + ------- + gamma_qph : complex + O(as1aem1) quark-photon anomalous dimension :math:`\\gamma_{q \\gamma}^{(1,1)}(N)` + """ + S1 = sx[0] + S2 = sx[1] + tmp_const = ( + -2 + * ( + 4 + + 8 * N + + 25 * N**2 + + 51 * N**3 + + 36 * N**4 + + 15 * N**5 + + 5 * N**6 + ) + / (N**3 * (1 + N) ** 3 * (2 + N)) + ) + tmp_S1 = 8 / N**2 + tmp_S12 = -4 * (2 + N + N**2) / (N * (1 + N) * (2 + N)) + tmp_S2 = 4 * (2 + N + N**2) / (N * (1 + N) * (2 + N)) + return constants.CF * (tmp_const + tmp_S1 * S1 + tmp_S12 * S1**2 + tmp_S2 * S2) + + +@nb.njit("c16(c16)", cache=True) +def gamma_gph(N): + """ + Computes the O(as1aem1) gluon-photon anomalous dimension + + Implements Eq. (2.5) of :cite:`Carrazza:2015dea`. + But adding the :math:`N_C` and the :math:`2n_f` factors from :math:`\\theta` inside the + definition of :math:`\\gamma_{q \\gamma}^{(0)}(N)`. + + Parameters + ---------- + N : complex + Mellin moment + + Returns + ------- + gamma_qph : complex + O(as1aem1) gluon-photon anomalous dimension :math:`\\gamma_{g \\gamma}^{(1,1)}(N)` + """ + + return ( + constants.CF + * (8 * (-4 + N * (-4 + N * (-5 + N * (-10 + N + 2 * N**2 * (2 + N)))))) + / (N**3 * (1 + N) ** 3 * (-2 + N + N**2)) + ) + + +@nb.njit("c16(c16)", cache=True) +def gamma_phg(N): + """ + Computes the O(as1aem1) photon-gluon anomalous dimension + + Implements Eq. (2.5) of :cite:`Carrazza:2015dea`. + But adding the :math:`N_C` and the :math:`2n_f` factors from :math:`\\theta` inside the + definition of :math:`\\gamma_{q \\gamma}^{(0)}(N)`. + + Parameters + ---------- + N : complex + Mellin moment + + Returns + ------- + gamma_qph : complex + O(as1aem1) photon-gluon anomalous dimension :math:`\\gamma_{\\gamma g}^{(1,1)}(N)` + """ + + return constants.TR / constants.CF * gamma_gph(N) + + +@nb.njit("c16(c16,u1,c16[:])", cache=True) +def gamma_qg(N, nf, sx): + """ + Computes the O(as1aem1) quark-gluon singlet anomalous dimension. + + Implements Eq. (3.7) of :cite:`Vogt:2004mw`. + + Parameters + ---------- + N : complex + Mellin moment + nf : int + Number of active flavors + sx : np array + List of harmonic sums + + Returns + ------- + gamma_qg : complex + O(as1aem1) quark-gluon singlet anomalous dimension + :math:`\\gamma_{qg}^{(1,1)}(N)` + """ + + return constants.TR / constants.CF * gamma_qph(N, nf, sx) + + +@nb.njit("c16(c16,u1,c16[:])", cache=True) +def gamma_gq(n, nf, sx): + """ + Computes the O(as1aem1) gluon-quark singlet anomalous dimension. + + Implements Eq. (3.8) of :cite:`Vogt:2004mw`. + + Parameters + ---------- + N : complex + Mellin moment + nf : int + Number of active flavors + sx : np array + List of harmonic sums + + Returns + ------- + gamma_gq : complex + O(as1aem1) gluon-quark singlet anomalous dimension + :math:`\\gamma_{gq}^{(1,1)}(N)` + """ + + return gamma_phq(n, nf, sx) + + +@nb.njit("c16(c16,u1)", cache=True) +def gamma_phph(): + """ + Computes the O(as1aem1) photon-photon singlet anomalous dimension. + + Implements Eq. (3.9) of :cite:`Vogt:2004mw`. + + Parameters + ---------- + + Returns + ------- + gamma_gg : complex + O(as1aem1) photon-photon singlet anomalous dimension + :math:`\\gamma_{\\gamma \\gamma}^{(1,1)}(N)` + """ + + return 4 * constants.CF + + +@nb.njit("c16(c16,u1)", cache=True) +def gamma_gg(): + """ + Computes the O(as1aem1) gluon-gluon singlet anomalous dimension. + + Implements Eq. (3.9) of :cite:`Vogt:2004mw`. + + Parameters + ---------- + + Returns + ------- + gamma_gg : complex + O(as1aem1) gluon-gluon singlet anomalous dimension + :math:`\\gamma_{gg}^{(1,1)}(N)` + """ + + return 4 * constants.TR + + +@nb.njit("c16(c16,u1,c16[:])", cache=True) +def gamma_nsp(N, nf, sx): + """ + Computes the O(as1aem1) singlet-like non-singlet anomalous dimension. + + Implements Eq. (3.5) of :cite:`Moch:2004pa`. + + Parameters + ---------- + N : complex + Mellin moment + nf : int + Number of active flavors + sx : np array + List of harmonic sums + + Returns + ------- + gamma_nsp : complex + O(as1aem1) singlet-like non-singlet anomalous dimension + :math:`\\gamma_{ns,+}^{(1)}(N)` + """ + S1 = sx[0] + S2 = sx[1] + S3 = sx[2] + S1h = harmonics.harmonic_S1(N / 2) + S2h = harmonics.harmonic_S2(N / 2) + S3h = harmonics.harmonic_S3(N / 2) + S1p1h = harmonics.harmonic_S1((N + 1.0) / 2) + S2p1h = harmonics.harmonic_S2((N + 1) / 2) + S3p1h = harmonics.harmonic_S3((N + 1) / 2) + g3n = harmonics.mellin_g3(N) + zeta2 = harmonics.zeta2 + zeta3 = harmonics.zeta3 + result = ( + +32.0 * zeta2 * S1h + + 8.0 / (N + N**2) * S2h + + 24 + + 16 / (N + N**2) * S2 + - 8.0 / (N + N**2) * S2p1h + + S1 + * ( + 16 * (3 / N**2 - 3 / (1 + N) ** 2 + 2 * zeta2) + - 16 * S2h + - 32 * S2 + + 16 * S2p1h + ) + + ( + -8 + + N + * ( + -32 + + N + * ( + -8 * (1 + np.pi**2) + + N + * ( + 64 * g3n * (1 + N) ** 3 + - 3 * (3 + N) * (3 + N**2) + - 8 * (2 + N) * np.pi**2 + ) + ) + ) + ) + / (N**3 * (1 + N) ** 3) + + 16.0 / 3 * np.pi**2 * S1p1h + - 4 * S3h + - 32 * S3 + + 4 * S3p1h + - 16 * zeta3 + ) + return constants.CF * result + + +@nb.njit("c16(c16,u1,c16[:])", cache=True) +def gamma_nsm(N, nf, sx): + """ + Computes the O(as1aem1) singlet-like non-singlet anomalous dimension. + + Implements Eq. (3.5) of :cite:`Moch:2004pa`. + + Parameters + ---------- + N : complex + Mellin moment + nf : int + Number of active flavors + sx : np array + List of harmonic sums + + Returns + ------- + gamma_nsm : complex + O(as1aem1) singlet-like non-singlet anomalous dimension + :math:`\\gamma_{ns,-}^{(1,1)}(N)` + """ + S1 = sx[0] + S2 = sx[1] + S3 = sx[2] + S1h = harmonics.harmonic_S1(N / 2) + S2h = harmonics.harmonic_S2(N / 2) + S3h = harmonics.harmonic_S3(N / 2) + S1p1h = harmonics.harmonic_S1((N + 1.0) / 2) + S2p1h = harmonics.harmonic_S2((N + 1) / 2) + S3p1h = harmonics.harmonic_S3((N + 1) / 2) + g3n = harmonics.mellin_g3(N) + zeta2 = harmonics.zeta2 + zeta3 = harmonics.zeta3 + result = ( + -32.0 * zeta2 * S1h + - 8.0 / (N + N**2) * S2h + + (24 + 16 / (N + N**2)) * S2 + + 8.0 / (N + N**2) * S2p1h + + S1 + * ( + -16 * (3 / N**2 - 3 / (1 + N) ** 2 - 2 * zeta2) + + 16 * S2h + - 32 * S2 + - 16 * S2p1h + ) + + ( + 72 + + N + * ( + 96 + - 3 + * N + * (8 + 64 * g3n * N * (1 + N) ** 3 + 3 * N * (3 + N) * (3 + N**2)) + + 8 * N * (1 + N) ** 2 * np.pi**2 + ) + ) + / (3.0 * N**3 * (1 + N) ** 3) + + 16.0 / 3 * np.pi**2 * S1p1h + + 4 * S3h + - 32 * S3 + - 4 * S3p1h + - 16 * zeta3 + ) + return constants.CF * result + + +@nb.njit("c16(c16,u1,c16[:])", cache=True) +def gamma_nsV(N, nf, sx): + """ + Computes the O(as1aem1) valence non-singlet anomalous dimension. + + Implements Eq. (3.5) of :cite:`Moch:2004pa`. + + Parameters + ---------- + N : complex + Mellin moment + nf : int + Number of active flavors + sx : np array + List of harmonic sums + + Returns + ------- + gamma_nsV : complex + O(as1aem1) singlet-like non-singlet anomalous dimension + :math:`\\gamma_{ns,V}^{(1,1)}(N)` + """ + + return gamma_nsm(N, nf, sx) From bf775e32cf56c4abdeacbf2d30378284c2a11f31 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Niccol=C3=B2=20Laurenti?= Date: Wed, 30 Mar 2022 17:40:26 +0200 Subject: [PATCH 19/71] Fixed color factors in O(as1aem1) AD and tested gluon mom conserv --- src/eko/anomalous_dimensions/as1aem1.py | 13 ++++++++---- tests/eko/test_ad_as1aem1.py | 28 +++++++++++++++++++++++++ 2 files changed, 37 insertions(+), 4 deletions(-) create mode 100644 tests/eko/test_ad_as1aem1.py diff --git a/src/eko/anomalous_dimensions/as1aem1.py b/src/eko/anomalous_dimensions/as1aem1.py index d08819242..bd82e851b 100644 --- a/src/eko/anomalous_dimensions/as1aem1.py +++ b/src/eko/anomalous_dimensions/as1aem1.py @@ -95,7 +95,11 @@ def gamma_qph(N, nf, sx): tmp_S1 = 8 / N**2 tmp_S12 = -4 * (2 + N + N**2) / (N * (1 + N) * (2 + N)) tmp_S2 = 4 * (2 + N + N**2) / (N * (1 + N) * (2 + N)) - return constants.CF * (tmp_const + tmp_S1 * S1 + tmp_S12 * S1**2 + tmp_S2 * S2) + return ( + constants.CA + * constants.CF + * (tmp_const + tmp_S1 * S1 + tmp_S12 * S1**2 + tmp_S2 * S2) + ) @nb.njit("c16(c16)", cache=True) @@ -120,6 +124,7 @@ def gamma_gph(N): return ( constants.CF + * constants.CA * (8 * (-4 + N * (-4 + N * (-5 + N * (-10 + N + 2 * N**2 * (2 + N)))))) / (N**3 * (1 + N) ** 3 * (-2 + N + N**2)) ) @@ -145,7 +150,7 @@ def gamma_phg(N): O(as1aem1) photon-gluon anomalous dimension :math:`\\gamma_{\\gamma g}^{(1,1)}(N)` """ - return constants.TR / constants.CF * gamma_gph(N) + return constants.TR / constants.CF / constants.CA * gamma_gph(N) @nb.njit("c16(c16,u1,c16[:])", cache=True) @@ -171,7 +176,7 @@ def gamma_qg(N, nf, sx): :math:`\\gamma_{qg}^{(1,1)}(N)` """ - return constants.TR / constants.CF * gamma_qph(N, nf, sx) + return constants.TR / constants.CF / constants.CA * gamma_qph(N, nf, sx) @nb.njit("c16(c16,u1,c16[:])", cache=True) @@ -217,7 +222,7 @@ def gamma_phph(): :math:`\\gamma_{\\gamma \\gamma}^{(1,1)}(N)` """ - return 4 * constants.CF + return 4 * constants.CF * constants.CA @nb.njit("c16(c16,u1)", cache=True) diff --git a/tests/eko/test_ad_as1aem1.py b/tests/eko/test_ad_as1aem1.py new file mode 100644 index 000000000..811879659 --- /dev/null +++ b/tests/eko/test_ad_as1aem1.py @@ -0,0 +1,28 @@ +# -*- coding: utf-8 -*- +# Test O(as1aem1) splitting functions +import numpy as np +from test_ad_nnlo import get_sx + +import eko.anomalous_dimensions.aem1 as aem1 +import eko.anomalous_dimensions.as1 as as1 +import eko.anomalous_dimensions.as1aem1 as as1aem1 +from eko import constants +from eko.anomalous_dimensions import harmonics + +NF = 5 +ND = 3 +NU = 2 + + +def test_gluon_momentum_conservation(): + # gluon momentum + N = complex(2.0, 0.0) + sx = get_sx(N) + # import pdb; pdb.set_trace() + np.testing.assert_almost_equal( + +2 * NU * constants.eu2 * as1aem1.gamma_qg(N, NF, sx) + + 2 * ND * constants.ed2 * as1aem1.gamma_qg(N, NF, sx) + + (NU * constants.eu2 + ND * constants.ed2) * as1aem1.gamma_phg(N) + + (NU * constants.eu2 + ND * constants.ed2) * as1aem1.gamma_gg(), + 0, + ) From 0583db095bf2c61ee804c20d13fb8a194bf6e8e0 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Niccol=C3=B2=20Laurenti?= Date: Wed, 30 Mar 2022 17:44:22 +0200 Subject: [PATCH 20/71] Test photon momentum conservation --- tests/eko/test_ad_as1aem1.py | 14 ++++++++++++++ 1 file changed, 14 insertions(+) diff --git a/tests/eko/test_ad_as1aem1.py b/tests/eko/test_ad_as1aem1.py index 811879659..9b7b87a02 100644 --- a/tests/eko/test_ad_as1aem1.py +++ b/tests/eko/test_ad_as1aem1.py @@ -26,3 +26,17 @@ def test_gluon_momentum_conservation(): + (NU * constants.eu2 + ND * constants.ed2) * as1aem1.gamma_gg(), 0, ) + + +def test_photon_momentum_conservation(): + # photon momentum + N = complex(2.0, 0.0) + sx = get_sx(N) + # import pdb; pdb.set_trace() + np.testing.assert_almost_equal( + +2 * NU * constants.eu2 * as1aem1.gamma_qph(N, NF, sx) + + 2 * ND * constants.ed2 * as1aem1.gamma_qph(N, NF, sx) + + (NU * constants.eu2 + ND * constants.ed2) * as1aem1.gamma_phph() + + (NU * constants.eu2 + ND * constants.ed2) * as1aem1.gamma_gph(N), + 0, + ) From 7d1ee487ce35ad11cb0f36ebde1f4351cd998602 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Niccol=C3=B2=20Laurenti?= Date: Fri, 1 Apr 2022 10:29:11 +0200 Subject: [PATCH 21/71] Test number conservation --- src/eko/anomalous_dimensions/as1aem1.py | 22 +++++++++--------- tests/eko/test_ad_as1aem1.py | 30 ++++++++++++++++++++++++- 2 files changed, 40 insertions(+), 12 deletions(-) diff --git a/src/eko/anomalous_dimensions/as1aem1.py b/src/eko/anomalous_dimensions/as1aem1.py index bd82e851b..26a49e96d 100644 --- a/src/eko/anomalous_dimensions/as1aem1.py +++ b/src/eko/anomalous_dimensions/as1aem1.py @@ -180,7 +180,7 @@ def gamma_qg(N, nf, sx): @nb.njit("c16(c16,u1,c16[:])", cache=True) -def gamma_gq(n, nf, sx): +def gamma_gq(N, nf, sx): """ Computes the O(as1aem1) gluon-quark singlet anomalous dimension. @@ -202,7 +202,7 @@ def gamma_gq(n, nf, sx): :math:`\\gamma_{gq}^{(1,1)}(N)` """ - return gamma_phq(n, nf, sx) + return gamma_phq(N, nf, sx) @nb.njit("c16(c16,u1)", cache=True) @@ -276,14 +276,14 @@ def gamma_nsp(N, nf, sx): S1p1h = harmonics.harmonic_S1((N + 1.0) / 2) S2p1h = harmonics.harmonic_S2((N + 1) / 2) S3p1h = harmonics.harmonic_S3((N + 1) / 2) - g3n = harmonics.mellin_g3(N) + g3N = harmonics.mellin_g3(N) + g3Np2 = harmonics.mellin_g3(N + 2) zeta2 = harmonics.zeta2 zeta3 = harmonics.zeta3 result = ( +32.0 * zeta2 * S1h + 8.0 / (N + N**2) * S2h - + 24 - + 16 / (N + N**2) * S2 + + (24 + 16 / (N + N**2)) * S2 - 8.0 / (N + N**2) * S2p1h + S1 * ( @@ -350,17 +350,18 @@ def gamma_nsm(N, nf, sx): S1p1h = harmonics.harmonic_S1((N + 1.0) / 2) S2p1h = harmonics.harmonic_S2((N + 1) / 2) S3p1h = harmonics.harmonic_S3((N + 1) / 2) - g3n = harmonics.mellin_g3(N) + g3N = harmonics.mellin_g3(N) + g3Np2 = harmonics.mellin_g3(N + 2) zeta2 = harmonics.zeta2 zeta3 = harmonics.zeta3 result = ( - -32.0 * zeta2 * S1h + -16.0 / 3 * np.pi**2 * S1h - 8.0 / (N + N**2) * S2h + (24 + 16 / (N + N**2)) * S2 + 8.0 / (N + N**2) * S2p1h + S1 * ( - -16 * (3 / N**2 - 3 / (1 + N) ** 2 - 2 * zeta2) + 16 * (-3 / N**2 + 3 / (1 + N) ** 2 + np.pi**2) / 3 + 16 * S2h - 32 * S2 - 16 * S2p1h @@ -370,11 +371,10 @@ def gamma_nsm(N, nf, sx): + N * ( 96 - - 3 - * N - * (8 + 64 * g3n * N * (1 + N) ** 3 + 3 * N * (3 + N) * (3 + N**2)) + - 3 * N * (8 + 3 * N * (3 + N) * (3 + N**2)) + 8 * N * (1 + N) ** 2 * np.pi**2 ) + - 96 * N**3 * (1 + N) ** 3 * (g3N + g3Np2) ) / (3.0 * N**3 * (1 + N) ** 3) + 16.0 / 3 * np.pi**2 * S1p1h diff --git a/tests/eko/test_ad_as1aem1.py b/tests/eko/test_ad_as1aem1.py index 9b7b87a02..44a85acbc 100644 --- a/tests/eko/test_ad_as1aem1.py +++ b/tests/eko/test_ad_as1aem1.py @@ -14,11 +14,17 @@ NU = 2 +def test_number_conservation(): + # number + N = complex(1.0, 0.0) + sx = get_sx(N) + np.testing.assert_almost_equal(+as1aem1.gamma_nsm(N, NF, sx), 0, decimal=4) + + def test_gluon_momentum_conservation(): # gluon momentum N = complex(2.0, 0.0) sx = get_sx(N) - # import pdb; pdb.set_trace() np.testing.assert_almost_equal( +2 * NU * constants.eu2 * as1aem1.gamma_qg(N, NF, sx) + 2 * ND * constants.ed2 * as1aem1.gamma_qg(N, NF, sx) @@ -40,3 +46,25 @@ def test_photon_momentum_conservation(): + (NU * constants.eu2 + ND * constants.ed2) * as1aem1.gamma_gph(N), 0, ) + + +# def test_gammansp(): +# N = complex(2.0, 0.0) +# sx = get_sx(N) +# # import pdb; pdb.set_trace() +# np.testing.assert_almost_equal( +# as1aem1.gamma_nsp(N,NF,sx) , +# -6.67306, +# ) + + +# def test_quark_momentum_conservation(): +# # quark momentum +# N = complex(2.0, 0.0) +# sx = get_sx(N) +# np.testing.assert_almost_equal( +# - 6.67306 +# + as1aem1.gamma_gq(N, NF, sx) +# + as1aem1.gamma_phq(N, NF, sx), +# 0, +# ) From 52868541c2ce4abc5febc1a1418c0231fb40c9e0 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Niccol=C3=B2=20Laurenti?= Date: Fri, 1 Apr 2022 11:02:36 +0200 Subject: [PATCH 22/71] Test quark momentum conservation --- src/eko/anomalous_dimensions/as1aem1.py | 21 ++++++----------- tests/eko/test_ad_as1aem1.py | 31 +++++++++---------------- 2 files changed, 18 insertions(+), 34 deletions(-) diff --git a/src/eko/anomalous_dimensions/as1aem1.py b/src/eko/anomalous_dimensions/as1aem1.py index 26a49e96d..652924813 100644 --- a/src/eko/anomalous_dimensions/as1aem1.py +++ b/src/eko/anomalous_dimensions/as1aem1.py @@ -281,13 +281,16 @@ def gamma_nsp(N, nf, sx): zeta2 = harmonics.zeta2 zeta3 = harmonics.zeta3 result = ( - +32.0 * zeta2 * S1h + +16 / 3 * np.pi**2 * S1h + - 16.0 / 3 * np.pi**2 * S1p1h + 8.0 / (N + N**2) * S2h + - 4 * S3h + (24 + 16 / (N + N**2)) * S2 + - 32 * S3 - 8.0 / (N + N**2) * S2p1h + S1 * ( - 16 * (3 / N**2 - 3 / (1 + N) ** 2 + 2 * zeta2) + +16 * (9 / N**2 - 9 / (1 + N) ** 2 + np.pi**2) / 3 - 16 * S2h - 32 * S2 + 16 * S2p1h @@ -298,21 +301,11 @@ def gamma_nsp(N, nf, sx): * ( -32 + N - * ( - -8 * (1 + np.pi**2) - + N - * ( - 64 * g3n * (1 + N) ** 3 - - 3 * (3 + N) * (3 + N**2) - - 8 * (2 + N) * np.pi**2 - ) - ) + * (-8 - 3 * N * (3 + N) * (3 + N**2) - 8 * (1 + N) ** 2 * np.pi**2) ) + + 32 * N**3 * (1 + N) ** 3 * (g3N + g3Np2) ) / (N**3 * (1 + N) ** 3) - + 16.0 / 3 * np.pi**2 * S1p1h - - 4 * S3h - - 32 * S3 + 4 * S3p1h - 16 * zeta3 ) diff --git a/tests/eko/test_ad_as1aem1.py b/tests/eko/test_ad_as1aem1.py index 44a85acbc..c066e1587 100644 --- a/tests/eko/test_ad_as1aem1.py +++ b/tests/eko/test_ad_as1aem1.py @@ -48,23 +48,14 @@ def test_photon_momentum_conservation(): ) -# def test_gammansp(): -# N = complex(2.0, 0.0) -# sx = get_sx(N) -# # import pdb; pdb.set_trace() -# np.testing.assert_almost_equal( -# as1aem1.gamma_nsp(N,NF,sx) , -# -6.67306, -# ) - - -# def test_quark_momentum_conservation(): -# # quark momentum -# N = complex(2.0, 0.0) -# sx = get_sx(N) -# np.testing.assert_almost_equal( -# - 6.67306 -# + as1aem1.gamma_gq(N, NF, sx) -# + as1aem1.gamma_phq(N, NF, sx), -# 0, -# ) +def test_quark_momentum_conservation(): + # quark momentum + N = complex(2.0, 0.0) + sx = get_sx(N) + np.testing.assert_almost_equal( + +as1aem1.gamma_nsp(N, NF, sx) + + as1aem1.gamma_gq(N, NF, sx) + + as1aem1.gamma_phq(N, NF, sx), + 0, + decimal=4, + ) From 66fa17ff560855805e6f7c39639249b9c3d5bb4f Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Niccol=C3=B2=20Laurenti?= Date: Fri, 1 Apr 2022 11:41:49 +0200 Subject: [PATCH 23/71] Test gamma_nsV --- tests/eko/test_ad_as1aem1.py | 1 + 1 file changed, 1 insertion(+) diff --git a/tests/eko/test_ad_as1aem1.py b/tests/eko/test_ad_as1aem1.py index c066e1587..25fbc2c0b 100644 --- a/tests/eko/test_ad_as1aem1.py +++ b/tests/eko/test_ad_as1aem1.py @@ -19,6 +19,7 @@ def test_number_conservation(): N = complex(1.0, 0.0) sx = get_sx(N) np.testing.assert_almost_equal(+as1aem1.gamma_nsm(N, NF, sx), 0, decimal=4) + np.testing.assert_almost_equal(+as1aem1.gamma_nsV(N, NF, sx), 0, decimal=4) def test_gluon_momentum_conservation(): From 681cdc8015fcf94bdfe5e56acfe9fcece4b9d5f3 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Niccol=C3=B2=20Laurenti?= Date: Fri, 1 Apr 2022 12:39:29 +0200 Subject: [PATCH 24/71] Remove gamma_nsV --- src/eko/anomalous_dimensions/as1aem1.py | 56 ++++++------------------- tests/eko/test_ad_as1aem1.py | 8 +--- 2 files changed, 14 insertions(+), 50 deletions(-) diff --git a/src/eko/anomalous_dimensions/as1aem1.py b/src/eko/anomalous_dimensions/as1aem1.py index 652924813..39d6c113c 100644 --- a/src/eko/anomalous_dimensions/as1aem1.py +++ b/src/eko/anomalous_dimensions/as1aem1.py @@ -19,7 +19,7 @@ def gamma_phq(N, nf, sx): """ Computes the O(as1aem1) photon-quark anomalous dimension - Implements Eq. (2.5) of :cite:`Carrazza:2015dea`. + Implements Eq. (36) of :cite:`deFlorian:2015ujt`. Parameters ---------- @@ -59,9 +59,7 @@ def gamma_qph(N, nf, sx): """ Computes the O(as1aem1) quark-photon anomalous dimension - Implements Eq. (2.5) of :cite:`Carrazza:2015dea`. - But adding the :math:`N_C` and the :math:`2n_f` factors from :math:`\\theta` inside the - definition of :math:`\\gamma_{q \\gamma}^{(0)}(N)`. + Implements Eq. (26) of :cite:`deFlorian:2015ujt`. Parameters ---------- @@ -107,9 +105,7 @@ def gamma_gph(N): """ Computes the O(as1aem1) gluon-photon anomalous dimension - Implements Eq. (2.5) of :cite:`Carrazza:2015dea`. - But adding the :math:`N_C` and the :math:`2n_f` factors from :math:`\\theta` inside the - definition of :math:`\\gamma_{q \\gamma}^{(0)}(N)`. + Implements Eq. (27) of :cite:`deFlorian:2015ujt`. Parameters ---------- @@ -135,9 +131,7 @@ def gamma_phg(N): """ Computes the O(as1aem1) photon-gluon anomalous dimension - Implements Eq. (2.5) of :cite:`Carrazza:2015dea`. - But adding the :math:`N_C` and the :math:`2n_f` factors from :math:`\\theta` inside the - definition of :math:`\\gamma_{q \\gamma}^{(0)}(N)`. + Implements Eq. (30) of :cite:`deFlorian:2015ujt`. Parameters ---------- @@ -158,7 +152,7 @@ def gamma_qg(N, nf, sx): """ Computes the O(as1aem1) quark-gluon singlet anomalous dimension. - Implements Eq. (3.7) of :cite:`Vogt:2004mw`. + Implements Eq. (29) of :cite:`deFlorian:2015ujt`. Parameters ---------- @@ -184,7 +178,7 @@ def gamma_gq(N, nf, sx): """ Computes the O(as1aem1) gluon-quark singlet anomalous dimension. - Implements Eq. (3.8) of :cite:`Vogt:2004mw`. + Implements Eq. (35) of :cite:`deFlorian:2015ujt`. Parameters ---------- @@ -205,12 +199,12 @@ def gamma_gq(N, nf, sx): return gamma_phq(N, nf, sx) -@nb.njit("c16(c16,u1)", cache=True) +@nb.njit("c16()", cache=True) def gamma_phph(): """ Computes the O(as1aem1) photon-photon singlet anomalous dimension. - Implements Eq. (3.9) of :cite:`Vogt:2004mw`. + Implements Eq. (28) of :cite:`deFlorian:2015ujt`. Parameters ---------- @@ -225,12 +219,12 @@ def gamma_phph(): return 4 * constants.CF * constants.CA -@nb.njit("c16(c16,u1)", cache=True) +@nb.njit("c16()", cache=True) def gamma_gg(): """ Computes the O(as1aem1) gluon-gluon singlet anomalous dimension. - Implements Eq. (3.9) of :cite:`Vogt:2004mw`. + Implements Eq. (31) of :cite:`deFlorian:2015ujt`. Parameters ---------- @@ -250,7 +244,7 @@ def gamma_nsp(N, nf, sx): """ Computes the O(as1aem1) singlet-like non-singlet anomalous dimension. - Implements Eq. (3.5) of :cite:`Moch:2004pa`. + Implements sum of Eqs. (33-34) of :cite:`deFlorian:2015ujt`. Parameters ---------- @@ -317,7 +311,7 @@ def gamma_nsm(N, nf, sx): """ Computes the O(as1aem1) singlet-like non-singlet anomalous dimension. - Implements Eq. (3.5) of :cite:`Moch:2004pa`. + Implements difference between Eqs. (33-34) of :cite:`deFlorian:2015ujt`. Parameters ---------- @@ -377,29 +371,3 @@ def gamma_nsm(N, nf, sx): - 16 * zeta3 ) return constants.CF * result - - -@nb.njit("c16(c16,u1,c16[:])", cache=True) -def gamma_nsV(N, nf, sx): - """ - Computes the O(as1aem1) valence non-singlet anomalous dimension. - - Implements Eq. (3.5) of :cite:`Moch:2004pa`. - - Parameters - ---------- - N : complex - Mellin moment - nf : int - Number of active flavors - sx : np array - List of harmonic sums - - Returns - ------- - gamma_nsV : complex - O(as1aem1) singlet-like non-singlet anomalous dimension - :math:`\\gamma_{ns,V}^{(1,1)}(N)` - """ - - return gamma_nsm(N, nf, sx) diff --git a/tests/eko/test_ad_as1aem1.py b/tests/eko/test_ad_as1aem1.py index 25fbc2c0b..6a985c539 100644 --- a/tests/eko/test_ad_as1aem1.py +++ b/tests/eko/test_ad_as1aem1.py @@ -3,11 +3,8 @@ import numpy as np from test_ad_nnlo import get_sx -import eko.anomalous_dimensions.aem1 as aem1 -import eko.anomalous_dimensions.as1 as as1 -import eko.anomalous_dimensions.as1aem1 as as1aem1 from eko import constants -from eko.anomalous_dimensions import harmonics +from eko.anomalous_dimensions import as1aem1, harmonics NF = 5 ND = 3 @@ -18,8 +15,7 @@ def test_number_conservation(): # number N = complex(1.0, 0.0) sx = get_sx(N) - np.testing.assert_almost_equal(+as1aem1.gamma_nsm(N, NF, sx), 0, decimal=4) - np.testing.assert_almost_equal(+as1aem1.gamma_nsV(N, NF, sx), 0, decimal=4) + np.testing.assert_almost_equal(as1aem1.gamma_nsm(N, NF, sx), 0, decimal=4) def test_gluon_momentum_conservation(): From c24bd710ff34f7de1de44934c1b2581588383d17 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Niccol=C3=B2=20Laurenti?= Date: Fri, 1 Apr 2022 14:23:02 +0200 Subject: [PATCH 25/71] Changed import in test_ad_as1aem1.py --- src/eko/anomalous_dimensions/__init__.py | 2 +- tests/eko/test_ad_as1aem1.py | 26 ++++++++++++------------ 2 files changed, 14 insertions(+), 14 deletions(-) diff --git a/src/eko/anomalous_dimensions/__init__.py b/src/eko/anomalous_dimensions/__init__.py index bdf7d67ef..e9c8fbd06 100644 --- a/src/eko/anomalous_dimensions/__init__.py +++ b/src/eko/anomalous_dimensions/__init__.py @@ -21,7 +21,7 @@ import numpy as np from .. import basis_rotation as br -from . import as1, as2, as3, harmonics +from . import aem1, as1, as1aem1, as2, as3, harmonics @nb.njit("Tuple((c16[:,:],c16,c16,c16[:,:],c16[:,:]))(c16[:,:])", cache=True) diff --git a/tests/eko/test_ad_as1aem1.py b/tests/eko/test_ad_as1aem1.py index 6a985c539..dd9d27ea5 100644 --- a/tests/eko/test_ad_as1aem1.py +++ b/tests/eko/test_ad_as1aem1.py @@ -3,8 +3,8 @@ import numpy as np from test_ad_nnlo import get_sx +from eko import anomalous_dimensions as ad from eko import constants -from eko.anomalous_dimensions import as1aem1, harmonics NF = 5 ND = 3 @@ -15,7 +15,7 @@ def test_number_conservation(): # number N = complex(1.0, 0.0) sx = get_sx(N) - np.testing.assert_almost_equal(as1aem1.gamma_nsm(N, NF, sx), 0, decimal=4) + np.testing.assert_almost_equal(ad.as1aem1.gamma_nsm(N, NF, sx), 0, decimal=4) def test_gluon_momentum_conservation(): @@ -23,10 +23,10 @@ def test_gluon_momentum_conservation(): N = complex(2.0, 0.0) sx = get_sx(N) np.testing.assert_almost_equal( - +2 * NU * constants.eu2 * as1aem1.gamma_qg(N, NF, sx) - + 2 * ND * constants.ed2 * as1aem1.gamma_qg(N, NF, sx) - + (NU * constants.eu2 + ND * constants.ed2) * as1aem1.gamma_phg(N) - + (NU * constants.eu2 + ND * constants.ed2) * as1aem1.gamma_gg(), + 2 * NU * constants.eu2 * ad.as1aem1.gamma_qg(N, NF, sx) + + 2 * ND * constants.ed2 * ad.as1aem1.gamma_qg(N, NF, sx) + + (NU * constants.eu2 + ND * constants.ed2) * ad.as1aem1.gamma_phg(N) + + (NU * constants.eu2 + ND * constants.ed2) * ad.as1aem1.gamma_gg(), 0, ) @@ -37,10 +37,10 @@ def test_photon_momentum_conservation(): sx = get_sx(N) # import pdb; pdb.set_trace() np.testing.assert_almost_equal( - +2 * NU * constants.eu2 * as1aem1.gamma_qph(N, NF, sx) - + 2 * ND * constants.ed2 * as1aem1.gamma_qph(N, NF, sx) - + (NU * constants.eu2 + ND * constants.ed2) * as1aem1.gamma_phph() - + (NU * constants.eu2 + ND * constants.ed2) * as1aem1.gamma_gph(N), + 2 * NU * constants.eu2 * ad.as1aem1.gamma_qph(N, NF, sx) + + 2 * ND * constants.ed2 * ad.as1aem1.gamma_qph(N, NF, sx) + + (NU * constants.eu2 + ND * constants.ed2) * ad.as1aem1.gamma_phph() + + (NU * constants.eu2 + ND * constants.ed2) * ad.as1aem1.gamma_gph(N), 0, ) @@ -50,9 +50,9 @@ def test_quark_momentum_conservation(): N = complex(2.0, 0.0) sx = get_sx(N) np.testing.assert_almost_equal( - +as1aem1.gamma_nsp(N, NF, sx) - + as1aem1.gamma_gq(N, NF, sx) - + as1aem1.gamma_phq(N, NF, sx), + ad.as1aem1.gamma_nsp(N, NF, sx) + + ad.as1aem1.gamma_gq(N, NF, sx) + + ad.as1aem1.gamma_phq(N, NF, sx), 0, decimal=4, ) From 845251bf9946f62b83382c244f49e7bbc006c875 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Niccol=C3=B2=20Laurenti?= Date: Fri, 1 Apr 2022 15:24:16 +0200 Subject: [PATCH 26/71] Clean notebook --- extras/uni-ad.nb | 5207 +++++++++++++++++++++++----------------------- 1 file changed, 2558 insertions(+), 2649 deletions(-) diff --git a/extras/uni-ad.nb b/extras/uni-ad.nb index 139accdcf..c0dd98555 100644 --- a/extras/uni-ad.nb +++ b/extras/uni-ad.nb @@ -10,10 +10,10 @@ NotebookFileLineBreakTest NotebookFileLineBreakTest 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"Output",ExpressionUUID->"f9c613c3-7d00-454b-b525-aafaf0c61187"] }, Open ]], -Cell[204870, 5553, 152, 3, 30, "Input",ExpressionUUID->"64bc49a7-9093-4878-9054-ca87783816aa"] +Cell[205588, 5478, 152, 3, 30, "Input",ExpressionUUID->"64bc49a7-9093-4878-9054-ca87783816aa"] } ] *) - -(* End of internal cache information *) From 8bcf8e23f70db986391fefc5cb7f6e3c8c40c236 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Niccol=C3=B2=20Laurenti?= Date: Mon, 4 Apr 2022 10:03:51 +0200 Subject: [PATCH 27/71] Change np.pi**2 -> 6*zeta2 --- src/eko/anomalous_dimensions/as1aem1.py | 37 +++++++++++++------------ 1 file changed, 19 insertions(+), 18 deletions(-) diff --git a/src/eko/anomalous_dimensions/as1aem1.py b/src/eko/anomalous_dimensions/as1aem1.py index 39d6c113c..348991cbc 100644 --- a/src/eko/anomalous_dimensions/as1aem1.py +++ b/src/eko/anomalous_dimensions/as1aem1.py @@ -14,8 +14,8 @@ from . import harmonics -@nb.njit("c16(c16,u1,c16[:])", cache=True) -def gamma_phq(N, nf, sx): +@nb.njit("c16(c16,c16[:])", cache=True) +def gamma_phq(N, sx): """ Computes the O(as1aem1) photon-quark anomalous dimension @@ -94,7 +94,9 @@ def gamma_qph(N, nf, sx): tmp_S12 = -4 * (2 + N + N**2) / (N * (1 + N) * (2 + N)) tmp_S2 = 4 * (2 + N + N**2) / (N * (1 + N) * (2 + N)) return ( - constants.CA + 2 + * nf + * constants.CA * constants.CF * (tmp_const + tmp_S1 * S1 + tmp_S12 * S1**2 + tmp_S2 * S2) ) @@ -174,7 +176,7 @@ def gamma_qg(N, nf, sx): @nb.njit("c16(c16,u1,c16[:])", cache=True) -def gamma_gq(N, nf, sx): +def gamma_gq(N, sx): """ Computes the O(as1aem1) gluon-quark singlet anomalous dimension. @@ -196,7 +198,7 @@ def gamma_gq(N, nf, sx): :math:`\\gamma_{gq}^{(1,1)}(N)` """ - return gamma_phq(N, nf, sx) + return gamma_phq(N, sx) @nb.njit("c16()", cache=True) @@ -239,8 +241,8 @@ def gamma_gg(): return 4 * constants.TR -@nb.njit("c16(c16,u1,c16[:])", cache=True) -def gamma_nsp(N, nf, sx): +@nb.njit("c16(c16,c16[:])", cache=True) +def gamma_nsp(N, sx): """ Computes the O(as1aem1) singlet-like non-singlet anomalous dimension. @@ -275,8 +277,8 @@ def gamma_nsp(N, nf, sx): zeta2 = harmonics.zeta2 zeta3 = harmonics.zeta3 result = ( - +16 / 3 * np.pi**2 * S1h - - 16.0 / 3 * np.pi**2 * S1p1h + +32 * zeta2 * S1h + - 32 * zeta2 * S1p1h + 8.0 / (N + N**2) * S2h - 4 * S3h + (24 + 16 / (N + N**2)) * S2 @@ -284,7 +286,7 @@ def gamma_nsp(N, nf, sx): - 8.0 / (N + N**2) * S2p1h + S1 * ( - +16 * (9 / N**2 - 9 / (1 + N) ** 2 + np.pi**2) / 3 + +16 * (3 / N**2 - 3 / (1 + N) ** 2 + 2 * zeta2) - 16 * S2h - 32 * S2 + 16 * S2p1h @@ -294,8 +296,7 @@ def gamma_nsp(N, nf, sx): + N * ( -32 - + N - * (-8 - 3 * N * (3 + N) * (3 + N**2) - 8 * (1 + N) ** 2 * np.pi**2) + + N * (-8 - 3 * N * (3 + N) * (3 + N**2) - 48 * (1 + N) ** 2 * zeta2) ) + 32 * N**3 * (1 + N) ** 3 * (g3N + g3Np2) ) @@ -306,8 +307,8 @@ def gamma_nsp(N, nf, sx): return constants.CF * result -@nb.njit("c16(c16,u1,c16[:])", cache=True) -def gamma_nsm(N, nf, sx): +@nb.njit("c16(c16,c16[:])", cache=True) +def gamma_nsm(N, sx): """ Computes the O(as1aem1) singlet-like non-singlet anomalous dimension. @@ -342,13 +343,13 @@ def gamma_nsm(N, nf, sx): zeta2 = harmonics.zeta2 zeta3 = harmonics.zeta3 result = ( - -16.0 / 3 * np.pi**2 * S1h + -32.0 * zeta2 * S1h - 8.0 / (N + N**2) * S2h + (24 + 16 / (N + N**2)) * S2 + 8.0 / (N + N**2) * S2p1h + S1 * ( - 16 * (-3 / N**2 + 3 / (1 + N) ** 2 + np.pi**2) / 3 + 16 * (-1 / N**2 + 1 / (1 + N) ** 2 + 2 * zeta2) + 16 * S2h - 32 * S2 - 16 * S2p1h @@ -359,12 +360,12 @@ def gamma_nsm(N, nf, sx): * ( 96 - 3 * N * (8 + 3 * N * (3 + N) * (3 + N**2)) - + 8 * N * (1 + N) ** 2 * np.pi**2 + + 48 * N * (1 + N) ** 2 * zeta2 ) - 96 * N**3 * (1 + N) ** 3 * (g3N + g3Np2) ) / (3.0 * N**3 * (1 + N) ** 3) - + 16.0 / 3 * np.pi**2 * S1p1h + + 32.0 * zeta2 * S1p1h + 4 * S3h - 32 * S3 - 4 * S3p1h From 60dc87be25617ad09ddce5a13b3df147b874397e Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Niccol=C3=B2=20Laurenti?= Date: Mon, 4 Apr 2022 10:04:43 +0200 Subject: [PATCH 28/71] Test nf=6 --- tests/eko/test_ad_as1aem1.py | 52 ++++++++++++++++++++++++++++++++++++ 1 file changed, 52 insertions(+) diff --git a/tests/eko/test_ad_as1aem1.py b/tests/eko/test_ad_as1aem1.py index dd9d27ea5..52d9a0647 100644 --- a/tests/eko/test_ad_as1aem1.py +++ b/tests/eko/test_ad_as1aem1.py @@ -11,6 +11,58 @@ NU = 2 +def test_number_conservation(): + # number + N = complex(1.0, 0.0) + sx = get_sx(N) + np.testing.assert_almost_equal(ad.as1aem1.gamma_nsm(N, NF, sx), 0, decimal=4) + + +def test_gluon_momentum_conservation(): + # gluon momentum + N = complex(2.0, 0.0) + sx = get_sx(N) + np.testing.assert_almost_equal( + 2 * NU * constants.eu2 * ad.as1aem1.gamma_qg(N, NF, sx) + + 2 * ND * constants.ed2 * ad.as1aem1.gamma_qg(N, NF, sx) + + (NU * constants.eu2 + ND * constants.ed2) * ad.as1aem1.gamma_phg(N) + + (NU * constants.eu2 + ND * constants.ed2) * ad.as1aem1.gamma_gg(), + 0, + ) + + +def test_photon_momentum_conservation(): + # photon momentum + N = complex(2.0, 0.0) + sx = get_sx(N) + # import pdb; pdb.set_trace() + np.testing.assert_almost_equal( + 2 * NU * constants.eu2 * ad.as1aem1.gamma_qph(N, NF, sx) + + 2 * ND * constants.ed2 * ad.as1aem1.gamma_qph(N, NF, sx) + + (NU * constants.eu2 + ND * constants.ed2) * ad.as1aem1.gamma_phph() + + (NU * constants.eu2 + ND * constants.ed2) * ad.as1aem1.gamma_gph(N), + 0, + ) + + +def test_quark_momentum_conservation(): + # quark momentum + N = complex(2.0, 0.0) + sx = get_sx(N) + np.testing.assert_almost_equal( + ad.as1aem1.gamma_nsp(N, NF, sx) + + ad.as1aem1.gamma_gq(N, NF, sx) + + ad.as1aem1.gamma_phq(N, NF, sx), + 0, + decimal=4, + ) + + +NF = 6 +ND = 3 +NU = 3 + + def test_number_conservation(): # number N = complex(1.0, 0.0) From 79b415a23a513708d93b01c0d10b042b853cbb2b Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Niccol=C3=B2=20Laurenti?= Date: Mon, 4 Apr 2022 10:13:50 +0200 Subject: [PATCH 29/71] Fix test_ad_as1aem1.py --- tests/eko/test_ad_as1aem1.py | 32 ++++++++++++++++---------------- 1 file changed, 16 insertions(+), 16 deletions(-) diff --git a/tests/eko/test_ad_as1aem1.py b/tests/eko/test_ad_as1aem1.py index 52d9a0647..6d87bf23f 100644 --- a/tests/eko/test_ad_as1aem1.py +++ b/tests/eko/test_ad_as1aem1.py @@ -15,7 +15,7 @@ def test_number_conservation(): # number N = complex(1.0, 0.0) sx = get_sx(N) - np.testing.assert_almost_equal(ad.as1aem1.gamma_nsm(N, NF, sx), 0, decimal=4) + np.testing.assert_almost_equal(ad.as1aem1.gamma_nsm(N, sx), 0, decimal=4) def test_gluon_momentum_conservation(): @@ -23,8 +23,8 @@ def test_gluon_momentum_conservation(): N = complex(2.0, 0.0) sx = get_sx(N) np.testing.assert_almost_equal( - 2 * NU * constants.eu2 * ad.as1aem1.gamma_qg(N, NF, sx) - + 2 * ND * constants.ed2 * ad.as1aem1.gamma_qg(N, NF, sx) + constants.eu2 * ad.as1aem1.gamma_qg(N, NU, sx) + + constants.ed2 * ad.as1aem1.gamma_qg(N, ND, sx) + (NU * constants.eu2 + ND * constants.ed2) * ad.as1aem1.gamma_phg(N) + (NU * constants.eu2 + ND * constants.ed2) * ad.as1aem1.gamma_gg(), 0, @@ -37,8 +37,8 @@ def test_photon_momentum_conservation(): sx = get_sx(N) # import pdb; pdb.set_trace() np.testing.assert_almost_equal( - 2 * NU * constants.eu2 * ad.as1aem1.gamma_qph(N, NF, sx) - + 2 * ND * constants.ed2 * ad.as1aem1.gamma_qph(N, NF, sx) + constants.eu2 * ad.as1aem1.gamma_qph(N, NU, sx) + + constants.ed2 * ad.as1aem1.gamma_qph(N, ND, sx) + (NU * constants.eu2 + ND * constants.ed2) * ad.as1aem1.gamma_phph() + (NU * constants.eu2 + ND * constants.ed2) * ad.as1aem1.gamma_gph(N), 0, @@ -50,9 +50,9 @@ def test_quark_momentum_conservation(): N = complex(2.0, 0.0) sx = get_sx(N) np.testing.assert_almost_equal( - ad.as1aem1.gamma_nsp(N, NF, sx) - + ad.as1aem1.gamma_gq(N, NF, sx) - + ad.as1aem1.gamma_phq(N, NF, sx), + ad.as1aem1.gamma_nsp(N, sx) + + ad.as1aem1.gamma_gq(N, sx) + + ad.as1aem1.gamma_phq(N, sx), 0, decimal=4, ) @@ -67,7 +67,7 @@ def test_number_conservation(): # number N = complex(1.0, 0.0) sx = get_sx(N) - np.testing.assert_almost_equal(ad.as1aem1.gamma_nsm(N, NF, sx), 0, decimal=4) + np.testing.assert_almost_equal(ad.as1aem1.gamma_nsm(N, sx), 0, decimal=4) def test_gluon_momentum_conservation(): @@ -75,8 +75,8 @@ def test_gluon_momentum_conservation(): N = complex(2.0, 0.0) sx = get_sx(N) np.testing.assert_almost_equal( - 2 * NU * constants.eu2 * ad.as1aem1.gamma_qg(N, NF, sx) - + 2 * ND * constants.ed2 * ad.as1aem1.gamma_qg(N, NF, sx) + constants.eu2 * ad.as1aem1.gamma_qg(N, NU, sx) + + constants.ed2 * ad.as1aem1.gamma_qg(N, ND, sx) + (NU * constants.eu2 + ND * constants.ed2) * ad.as1aem1.gamma_phg(N) + (NU * constants.eu2 + ND * constants.ed2) * ad.as1aem1.gamma_gg(), 0, @@ -89,8 +89,8 @@ def test_photon_momentum_conservation(): sx = get_sx(N) # import pdb; pdb.set_trace() np.testing.assert_almost_equal( - 2 * NU * constants.eu2 * ad.as1aem1.gamma_qph(N, NF, sx) - + 2 * ND * constants.ed2 * ad.as1aem1.gamma_qph(N, NF, sx) + constants.eu2 * ad.as1aem1.gamma_qph(N, NU, sx) + + constants.ed2 * ad.as1aem1.gamma_qph(N, ND, sx) + (NU * constants.eu2 + ND * constants.ed2) * ad.as1aem1.gamma_phph() + (NU * constants.eu2 + ND * constants.ed2) * ad.as1aem1.gamma_gph(N), 0, @@ -102,9 +102,9 @@ def test_quark_momentum_conservation(): N = complex(2.0, 0.0) sx = get_sx(N) np.testing.assert_almost_equal( - ad.as1aem1.gamma_nsp(N, NF, sx) - + ad.as1aem1.gamma_gq(N, NF, sx) - + ad.as1aem1.gamma_phq(N, NF, sx), + ad.as1aem1.gamma_nsp(N, sx) + + ad.as1aem1.gamma_gq(N, sx) + + ad.as1aem1.gamma_phq(N, sx), 0, decimal=4, ) From f5ce97e4a57c208bb1d12b34ce1b7acfe2f7889b Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Niccol=C3=B2=20Laurenti?= Date: Mon, 4 Apr 2022 10:20:44 +0200 Subject: [PATCH 30/71] Fix numba for gamma_gq(N, sx) --- src/eko/anomalous_dimensions/as1aem1.py | 2 +- tests/eko/test_ad_as1aem1.py | 2 -- 2 files changed, 1 insertion(+), 3 deletions(-) diff --git a/src/eko/anomalous_dimensions/as1aem1.py b/src/eko/anomalous_dimensions/as1aem1.py index 348991cbc..fad29cdd6 100644 --- a/src/eko/anomalous_dimensions/as1aem1.py +++ b/src/eko/anomalous_dimensions/as1aem1.py @@ -175,7 +175,7 @@ def gamma_qg(N, nf, sx): return constants.TR / constants.CF / constants.CA * gamma_qph(N, nf, sx) -@nb.njit("c16(c16,u1,c16[:])", cache=True) +@nb.njit("c16(c16,c16[:])", cache=True) def gamma_gq(N, sx): """ Computes the O(as1aem1) gluon-quark singlet anomalous dimension. diff --git a/tests/eko/test_ad_as1aem1.py b/tests/eko/test_ad_as1aem1.py index 6d87bf23f..c9018f8ea 100644 --- a/tests/eko/test_ad_as1aem1.py +++ b/tests/eko/test_ad_as1aem1.py @@ -35,7 +35,6 @@ def test_photon_momentum_conservation(): # photon momentum N = complex(2.0, 0.0) sx = get_sx(N) - # import pdb; pdb.set_trace() np.testing.assert_almost_equal( constants.eu2 * ad.as1aem1.gamma_qph(N, NU, sx) + constants.ed2 * ad.as1aem1.gamma_qph(N, ND, sx) @@ -87,7 +86,6 @@ def test_photon_momentum_conservation(): # photon momentum N = complex(2.0, 0.0) sx = get_sx(N) - # import pdb; pdb.set_trace() np.testing.assert_almost_equal( constants.eu2 * ad.as1aem1.gamma_qph(N, NU, sx) + constants.ed2 * ad.as1aem1.gamma_qph(N, ND, sx) From b6a0a271abea39876c233ba21eccc0584a443998 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Niccol=C3=B2=20Laurenti?= Date: Tue, 5 Apr 2022 09:30:59 +0200 Subject: [PATCH 31/71] Implement function number_of_flavors(nf) --- src/eko/constants.py | 34 ++++++++++++++++++++++++++++++++++ 1 file changed, 34 insertions(+) diff --git a/src/eko/constants.py b/src/eko/constants.py index a9bb89027..9055aed45 100644 --- a/src/eko/constants.py +++ b/src/eko/constants.py @@ -37,3 +37,37 @@ def update_colors(nc): NC = int(nc) CA = float(NC) CF = float((NC * NC - 1.0) / (2.0 * NC)) + + +def number_of_flavors(nf): + """ + Computes the number of up (nu) and down (nd) flavors + + Parameters + ---------- + nf : int + Number of active flavors + + Returns + ------- + nu : int + nd : int + """ + if nf == 2: + nu = 1 + nd = 1 + elif nf == 3: + nu = 1 + nd = 2 + elif nf == 4: + nu = 2 + nd = 2 + elif nf == 5: + nu = 2 + nd = 3 + elif nf == 6: + nu = 3 + nd = 3 + else: + raise NotImplementedError("Selected nf is not implemented") + return nu, nd From 8438b8197f66b2321a58beac23f2fee0423cc694 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Niccol=C3=B2=20Laurenti?= Date: Tue, 5 Apr 2022 09:32:22 +0200 Subject: [PATCH 32/71] Move (nu*eu2+nd*ed2) inside def of gamma_phph --- src/eko/anomalous_dimensions/as1aem1.py | 9 ++++++--- tests/eko/test_ad_as1aem1.py | 4 ++-- 2 files changed, 8 insertions(+), 5 deletions(-) diff --git a/src/eko/anomalous_dimensions/as1aem1.py b/src/eko/anomalous_dimensions/as1aem1.py index fad29cdd6..03fc69126 100644 --- a/src/eko/anomalous_dimensions/as1aem1.py +++ b/src/eko/anomalous_dimensions/as1aem1.py @@ -201,8 +201,8 @@ def gamma_gq(N, sx): return gamma_phq(N, sx) -@nb.njit("c16()", cache=True) -def gamma_phph(): +@nb.njit("c16(u1)", cache=True) +def gamma_phph(nf): """ Computes the O(as1aem1) photon-photon singlet anomalous dimension. @@ -210,6 +210,8 @@ def gamma_phph(): Parameters ---------- + nf : int + Number of active flavors Returns ------- @@ -218,7 +220,8 @@ def gamma_phph(): :math:`\\gamma_{\\gamma \\gamma}^{(1,1)}(N)` """ - return 4 * constants.CF * constants.CA + nu, nd = constants.number_of_flavors(nf) + return 4 * constants.CF * constants.CA * (nu * constants.eu2 + nd * constants.ed2) @nb.njit("c16()", cache=True) diff --git a/tests/eko/test_ad_as1aem1.py b/tests/eko/test_ad_as1aem1.py index c9018f8ea..4af5ed1a4 100644 --- a/tests/eko/test_ad_as1aem1.py +++ b/tests/eko/test_ad_as1aem1.py @@ -38,7 +38,7 @@ def test_photon_momentum_conservation(): np.testing.assert_almost_equal( constants.eu2 * ad.as1aem1.gamma_qph(N, NU, sx) + constants.ed2 * ad.as1aem1.gamma_qph(N, ND, sx) - + (NU * constants.eu2 + ND * constants.ed2) * ad.as1aem1.gamma_phph() + + ad.as1aem1.gamma_phph(NF) + (NU * constants.eu2 + ND * constants.ed2) * ad.as1aem1.gamma_gph(N), 0, ) @@ -89,7 +89,7 @@ def test_photon_momentum_conservation(): np.testing.assert_almost_equal( constants.eu2 * ad.as1aem1.gamma_qph(N, NU, sx) + constants.ed2 * ad.as1aem1.gamma_qph(N, ND, sx) - + (NU * constants.eu2 + ND * constants.ed2) * ad.as1aem1.gamma_phph() + + ad.as1aem1.gamma_phph(NF) + (NU * constants.eu2 + ND * constants.ed2) * ad.as1aem1.gamma_gph(N), 0, ) From 06d67669498562c568098abee13bae22e1995c42 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Niccol=C3=B2=20Laurenti?= Date: Tue, 5 Apr 2022 11:21:47 +0200 Subject: [PATCH 33/71] Move function number_of_flavors from constants.py to as1aem1.py and rename number_of_up_flavors --- src/eko/anomalous_dimensions/as1aem1.py | 32 ++++++++++++++++++++++- src/eko/constants.py | 34 ------------------------- 2 files changed, 31 insertions(+), 35 deletions(-) diff --git a/src/eko/anomalous_dimensions/as1aem1.py b/src/eko/anomalous_dimensions/as1aem1.py index 03fc69126..7524ee065 100644 --- a/src/eko/anomalous_dimensions/as1aem1.py +++ b/src/eko/anomalous_dimensions/as1aem1.py @@ -14,6 +14,35 @@ from . import harmonics +@nb.njit("u1(u1)", cache=True) +def number_of_up_flavors(nf): + """ + Computes the number of up flavors + + Parameters + ---------- + nf : int + Number of active flavors + + Returns + ------- + nu : int + """ + if nf == 2: + nu = 1 + elif nf == 3: + nu = 1 + elif nf == 4: + nu = 2 + elif nf == 5: + nu = 2 + elif nf == 6: + nu = 3 + else: + raise NotImplementedError("Selected nf is not implemented") + return nu + + @nb.njit("c16(c16,c16[:])", cache=True) def gamma_phq(N, sx): """ @@ -220,7 +249,8 @@ def gamma_phph(nf): :math:`\\gamma_{\\gamma \\gamma}^{(1,1)}(N)` """ - nu, nd = constants.number_of_flavors(nf) + nu = number_of_up_flavors(nf) + nd = nf - nu return 4 * constants.CF * constants.CA * (nu * constants.eu2 + nd * constants.ed2) diff --git a/src/eko/constants.py b/src/eko/constants.py index 9055aed45..a9bb89027 100644 --- a/src/eko/constants.py +++ b/src/eko/constants.py @@ -37,37 +37,3 @@ def update_colors(nc): NC = int(nc) CA = float(NC) CF = float((NC * NC - 1.0) / (2.0 * NC)) - - -def number_of_flavors(nf): - """ - Computes the number of up (nu) and down (nd) flavors - - Parameters - ---------- - nf : int - Number of active flavors - - Returns - ------- - nu : int - nd : int - """ - if nf == 2: - nu = 1 - nd = 1 - elif nf == 3: - nu = 1 - nd = 2 - elif nf == 4: - nu = 2 - nd = 2 - elif nf == 5: - nu = 2 - nd = 3 - elif nf == 6: - nu = 3 - nd = 3 - else: - raise NotImplementedError("Selected nf is not implemented") - return nu, nd From dfc9bc2beae81e11ff90f1d5cad38280f5eae0c9 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Niccol=C3=B2=20Laurenti?= Date: Tue, 5 Apr 2022 11:23:10 +0200 Subject: [PATCH 34/71] Test correctly both NF=5 and NF=6 --- tests/eko/test_ad_as1aem1.py | 83 ++++++++---------------------------- 1 file changed, 18 insertions(+), 65 deletions(-) diff --git a/tests/eko/test_ad_as1aem1.py b/tests/eko/test_ad_as1aem1.py index 4af5ed1a4..1bf786fca 100644 --- a/tests/eko/test_ad_as1aem1.py +++ b/tests/eko/test_ad_as1aem1.py @@ -22,77 +22,30 @@ def test_gluon_momentum_conservation(): # gluon momentum N = complex(2.0, 0.0) sx = get_sx(N) - np.testing.assert_almost_equal( - constants.eu2 * ad.as1aem1.gamma_qg(N, NU, sx) - + constants.ed2 * ad.as1aem1.gamma_qg(N, ND, sx) - + (NU * constants.eu2 + ND * constants.ed2) * ad.as1aem1.gamma_phg(N) - + (NU * constants.eu2 + ND * constants.ed2) * ad.as1aem1.gamma_gg(), - 0, - ) + for NF, ND in ((5, 3), (6, 3)): + NU = NF - ND + np.testing.assert_almost_equal( + constants.eu2 * ad.as1aem1.gamma_qg(N, NU, sx) + + constants.ed2 * ad.as1aem1.gamma_qg(N, ND, sx) + + (NU * constants.eu2 + ND * constants.ed2) * ad.as1aem1.gamma_phg(N) + + (NU * constants.eu2 + ND * constants.ed2) * ad.as1aem1.gamma_gg(), + 0, + ) def test_photon_momentum_conservation(): # photon momentum N = complex(2.0, 0.0) sx = get_sx(N) - np.testing.assert_almost_equal( - constants.eu2 * ad.as1aem1.gamma_qph(N, NU, sx) - + constants.ed2 * ad.as1aem1.gamma_qph(N, ND, sx) - + ad.as1aem1.gamma_phph(NF) - + (NU * constants.eu2 + ND * constants.ed2) * ad.as1aem1.gamma_gph(N), - 0, - ) - - -def test_quark_momentum_conservation(): - # quark momentum - N = complex(2.0, 0.0) - sx = get_sx(N) - np.testing.assert_almost_equal( - ad.as1aem1.gamma_nsp(N, sx) - + ad.as1aem1.gamma_gq(N, sx) - + ad.as1aem1.gamma_phq(N, sx), - 0, - decimal=4, - ) - - -NF = 6 -ND = 3 -NU = 3 - - -def test_number_conservation(): - # number - N = complex(1.0, 0.0) - sx = get_sx(N) - np.testing.assert_almost_equal(ad.as1aem1.gamma_nsm(N, sx), 0, decimal=4) - - -def test_gluon_momentum_conservation(): - # gluon momentum - N = complex(2.0, 0.0) - sx = get_sx(N) - np.testing.assert_almost_equal( - constants.eu2 * ad.as1aem1.gamma_qg(N, NU, sx) - + constants.ed2 * ad.as1aem1.gamma_qg(N, ND, sx) - + (NU * constants.eu2 + ND * constants.ed2) * ad.as1aem1.gamma_phg(N) - + (NU * constants.eu2 + ND * constants.ed2) * ad.as1aem1.gamma_gg(), - 0, - ) - - -def test_photon_momentum_conservation(): - # photon momentum - N = complex(2.0, 0.0) - sx = get_sx(N) - np.testing.assert_almost_equal( - constants.eu2 * ad.as1aem1.gamma_qph(N, NU, sx) - + constants.ed2 * ad.as1aem1.gamma_qph(N, ND, sx) - + ad.as1aem1.gamma_phph(NF) - + (NU * constants.eu2 + ND * constants.ed2) * ad.as1aem1.gamma_gph(N), - 0, - ) + for NF, ND in ((5, 3), (6, 3)): + NU = NF - ND + np.testing.assert_almost_equal( + constants.eu2 * ad.as1aem1.gamma_qph(N, NU, sx) + + constants.ed2 * ad.as1aem1.gamma_qph(N, ND, sx) + + ad.as1aem1.gamma_phph(NF) + + (NU * constants.eu2 + ND * constants.ed2) * ad.as1aem1.gamma_gph(N), + 0, + ) def test_quark_momentum_conservation(): From bb6110bcbf575b1deff76b2e46569f2c92b9b692 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Niccol=C3=B2=20Laurenti?= Date: Tue, 5 Apr 2022 12:15:40 +0200 Subject: [PATCH 35/71] Fix test_ad_as1aem1.py --- tests/eko/test_ad_as1aem1.py | 4 ---- 1 file changed, 4 deletions(-) diff --git a/tests/eko/test_ad_as1aem1.py b/tests/eko/test_ad_as1aem1.py index 1bf786fca..168e2421c 100644 --- a/tests/eko/test_ad_as1aem1.py +++ b/tests/eko/test_ad_as1aem1.py @@ -6,10 +6,6 @@ from eko import anomalous_dimensions as ad from eko import constants -NF = 5 -ND = 3 -NU = 2 - def test_number_conservation(): # number From 99d65656320fb5c715f18a2168f8dcc2e12a408b Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Niccol=C3=B2=20Laurenti?= Date: Tue, 5 Apr 2022 12:16:30 +0200 Subject: [PATCH 36/71] Implement O(aem2) AD --- src/eko/anomalous_dimensions/__init__.py | 2 +- src/eko/anomalous_dimensions/aem2.py | 262 +++++++++++++++++++++++ 2 files changed, 263 insertions(+), 1 deletion(-) create mode 100644 src/eko/anomalous_dimensions/aem2.py diff --git a/src/eko/anomalous_dimensions/__init__.py b/src/eko/anomalous_dimensions/__init__.py index e9c8fbd06..e764bbad2 100644 --- a/src/eko/anomalous_dimensions/__init__.py +++ b/src/eko/anomalous_dimensions/__init__.py @@ -21,7 +21,7 @@ import numpy as np from .. import basis_rotation as br -from . import aem1, as1, as1aem1, as2, as3, harmonics +from . import aem1, aem2, as1, as1aem1, as2, as3, harmonics @nb.njit("Tuple((c16[:,:],c16,c16,c16[:,:],c16[:,:]))(c16[:,:])", cache=True) diff --git a/src/eko/anomalous_dimensions/aem2.py b/src/eko/anomalous_dimensions/aem2.py new file mode 100644 index 000000000..7fa9f6e6e --- /dev/null +++ b/src/eko/anomalous_dimensions/aem2.py @@ -0,0 +1,262 @@ +# -*- coding: utf-8 -*- +""" +This file contains the O(aem2) Altarelli-Parisi splitting kernels. + +These expression have been obtained using the procedure described in the +`wiki `_ +involving ``FormGet`` :cite:`Hahn:2016ebn`. +""" + +import numba as nb +import numpy as np + +from .. import constants +from . import as1aem1, harmonics + + +@nb.njit("c16(c16,u1)", cache=True) +def gamma_phph(N, nf): + """ + Computes the O(as1aem1) photon-photon singlet anomalous dimension. + + Implements Eq. (28) of :cite:`deFlorian:2015ujt`. + + Parameters + ---------- + + Returns + ------- + gamma_gg : complex + O(as1aem1) photon-photon singlet anomalous dimension + :math:`\\gamma_{\\gamma \\gamma}^{(1,1)}(N)` + """ + + nu = as1aem1.number_of_up_flavors(nf) + nd = nf - nu + return (nu * constants.eu2**2 + nd * constants.ed2**2) * ( + as1aem1.gamma_gph(N) / constants.CF / constants.CA + 4 + ) + + +@nb.njit("c16(c16,u1,c16[:])", cache=True) +def gamma_uph(N, nf, sx): + """ + Computes the O(as1aem1) quark-photon anomalous dimension + + Implements Eq. (26) of :cite:`deFlorian:2015ujt`. + + Parameters + ---------- + N : complex + Mellin moment + nf : int + Number of active flavors + sx : np array + List of harmonic sums + + Returns + ------- + gamma_qph : complex + O(as1aem1) quark-photon anomalous dimension :math:`\\gamma_{q \\gamma}^{(1,1)}(N)` + """ + + return constants.eu2 * as1aem1.gamma_qph(N, nf, sx) / constants.CF + + +@nb.njit("c16(c16,u1,c16[:])", cache=True) +def gamma_dph(N, nf, sx): + """ + Computes the O(as1aem1) quark-photon anomalous dimension + + Implements Eq. (26) of :cite:`deFlorian:2015ujt`. + + Parameters + ---------- + N : complex + Mellin moment + nf : int + Number of active flavors + sx : np array + List of harmonic sums + + Returns + ------- + gamma_qph : complex + O(as1aem1) quark-photon anomalous dimension :math:`\\gamma_{q \\gamma}^{(1,1)}(N)` + """ + + return constants.ed2 * as1aem1.gamma_qph(N, nf, sx) / constants.CF + + +@nb.njit("c16(c16,c16[:])", cache=True) +def gamma_phu(N, nf, sx): + """ + Computes the O(as1aem1) photon-quark anomalous dimension + + Implements Eq. (36) of :cite:`deFlorian:2015ujt`. + + Parameters + ---------- + N : complex + Mellin moment + nf : int + Number of active flavors + sx : np array + List of harmonic sums + + Returns + ------- + gamma_phq : complex + O(as1aem1) photon-quark anomalous dimension :math:`\\gamma_{\\gamma q}^{(1,1)}(N)` + """ + + nu = as1aem1.number_of_up_flavors(nf) + nd = nf - nu + S1 = sx[0] + tmp = (-16 * (-16 - 27 * N - 13 * N**2 - 8 * N**3)) / ( + 9.0 * (-1 + N) * N * (1 + N) ** 2 + ) - 16 * (2 + 3 * N + 2 * N**2 + N**3) / ( + 3.0 * (-1 + N) * N * (1 + N) ** 2 + ) * S1 + eSigma2 = constants.NC * (nu * constants.eu2 + nd * constants.ed2) + return constants.eu2 * as1aem1.gamma_phq(N, sx) / constants.CF + eSigma2 * tmp + + +@nb.njit("c16(c16,c16[:])", cache=True) +def gamma_phd(N, nf, sx): + """ + Computes the O(as1aem1) photon-quark anomalous dimension + + Implements Eq. (36) of :cite:`deFlorian:2015ujt`. + + Parameters + ---------- + N : complex + Mellin moment + nf : int + Number of active flavors + sx : np array + List of harmonic sums + + Returns + ------- + gamma_phq : complex + O(as1aem1) photon-quark anomalous dimension :math:`\\gamma_{\\gamma q}^{(1,1)}(N)` + """ + + nu = as1aem1.number_of_up_flavors(nf) + nd = nf - nu + S1 = sx[0] + tmp = (-16 * (-16 - 27 * N - 13 * N**2 - 8 * N**3)) / ( + 9.0 * (-1 + N) * N * (1 + N) ** 2 + ) - 16 * (2 + 3 * N + 2 * N**2 + N**3) / ( + 3.0 * (-1 + N) * N * (1 + N) ** 2 + ) * S1 + eSigma2 = constants.NC * (nu * constants.eu2 + nd * constants.ed2) + return constants.ed2 * as1aem1.gamma_phq(N, sx) / constants.CF + eSigma2 * tmp + + +@nb.njit("c16(c16,u1,c16[:])", cache=True) +def gamma_nspu(N, nf, sx): + """ + Computes the O(as1aem1) singlet-like non-singlet anomalous dimension. + + Implements sum of Eqs. (33-34) of :cite:`deFlorian:2015ujt`. + + Parameters + ---------- + N : complex + Mellin moment + nf : int + Number of active flavors + sx : np array + List of harmonic sums + + Returns + ------- + gamma_nsp : complex + O(as1aem1) singlet-like non-singlet anomalous dimension + :math:`\\gamma_{ns,+}^{(1)}(N)` + """ + + S1 = sx[0] + S2 = sx[1] + nu = as1aem1.number_of_up_flavors(nf) + nd = nf - nu + eSigma2 = constants.NC * (nu * constants.eu2 + nd * constants.ed2) + tmp = ( + 2 + * (-12 + 20 * N + 107 * N**2 + 126 * N**3 + 63 * N**4) + / (9.0 * N**2 * (1 + N) ** 2) + - 80 / 9 * S1 + + 16 / 3 * S2 + ) * eSigma2 + return constants.eu2 * as1aem1.gamma_nsp(N, sx) / constants.CF / 2 + tmp + + +@nb.njit("c16(c16,u1,c16[:])", cache=True) +def gamma_nspd(N, nf, sx): + """ + Computes the O(as1aem1) singlet-like non-singlet anomalous dimension. + + Implements sum of Eqs. (33-34) of :cite:`deFlorian:2015ujt`. + + Parameters + ---------- + N : complex + Mellin moment + nf : int + Number of active flavors + sx : np array + List of harmonic sums + + Returns + ------- + gamma_nsp : complex + O(as1aem1) singlet-like non-singlet anomalous dimension + :math:`\\gamma_{ns,+}^{(1)}(N)` + """ + + S1 = sx[0] + S2 = sx[1] + nu = as1aem1.number_of_up_flavors(nf) + nd = nf - nu + eSigma2 = constants.NC * (nu * constants.eu2 + nd * constants.ed2) + tmp = ( + 2 + * (-12 + 20 * N + 107 * N**2 + 126 * N**3 + 63 * N**4) + / (9.0 * N**2 * (1 + N) ** 2) + - 80 / 9 * S1 + + 16 / 3 * S2 + ) * eSigma2 + return constants.ed2 * as1aem1.gamma_nsp(N, sx) / constants.CF / 2 + tmp + + +@nb.njit("c16(c16,u1)", cache=True) +def gamma_ps(N, nf): + """ + Computes the |NLO| pure-singlet quark-quark anomalous dimension. + + Implements Eq. (3.6) of :cite:`Vogt:2004mw`. + + Parameters + ---------- + n : complex + Mellin moment + nf : int + Number of active flavors + + Returns + ------- + gamma_ps : complex + |NLO| pure-singlet quark-quark anomalous dimension + :math:`\\gamma_{ps}^{(1)}(N)` + """ + + result = ( + -4 + * (2 + N * (5 + N)) + * (4 + N * (4 + N * (7 + 5 * N))) + / ((-1 + N) * N**3 * (1 + N) ** 3 * (2 + N) ** 2) + ) + return 2 * nf * result From 42ffe8a8063c52cc0f954a2fbc9a6b3a82c9b0da Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Niccol=C3=B2=20Laurenti?= Date: Tue, 5 Apr 2022 12:17:25 +0200 Subject: [PATCH 37/71] Generalize DGLAP eqs for O(aem2) --- extras/uni-dglap-aem2.ipynb | 432 ++++++++++++++++++++++++++++++++++++ 1 file changed, 432 insertions(+) create mode 100644 extras/uni-dglap-aem2.ipynb diff --git a/extras/uni-dglap-aem2.ipynb b/extras/uni-dglap-aem2.ipynb new file mode 100644 index 000000000..9ddcabb8d --- /dev/null +++ b/extras/uni-dglap-aem2.ipynb @@ -0,0 +1,432 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": 1, + "id": "c093db3d", + "metadata": {}, + "outputs": [], + "source": [ + "import sympy" + ] + }, + { + "cell_type": "code", + "execution_count": 14, + "id": "66408b17", + "metadata": {}, + "outputs": [], + "source": [ + "# QCD\n", + "Pv, Pp, Pm, Pqq, Pqg, Pgq, Pgg = sympy.symbols(\"P_V P_+ P_- P_qq P_qg P_gq P_gg\")\n", + "# QED\n", + "Pxv, Pxp, Pxpu, Pxpd, Pxm, Pxmu, Pxmd, Pxqq, Pxqg, Pxgq, Pxgg = sympy.symbols(\"P^x_V P^x_+ P^x_+u P^x_+d P^x_- P^x_-u P^x_-d P^x_qq P^x_qg P^x_gq P^x_gg\")\n", + "Pxqy, Pxuy, Pxdy, Pxyq, Pxyu, Pxyd, Pxyg, Pxgy, Pxyy = sympy.symbols(\"P^x_q\\gamma P^x_u\\gamma P^x_d\\gamma P^x_\\gamma\\ q P^x_\\gamma\\ u P^x_\\gamma\\ d P^x_\\gamma\\ g P^x_g\\gamma P^x_\\gamma\\gamma\")\n", + "eu2, ed2, es2 = sympy.symbols(\"e_u^2 e_d^2 e_\\Sigma^2\") # charges\n", + "eu4, ed4 = sympy.symbols(\"e_u^4 e_d^4\") # charges" + ] + }, + { + "cell_type": "code", + "execution_count": 16, + "id": "558d64a5", + "metadata": {}, + "outputs": [], + "source": [ + "P = {}\n", + "ns, s, qed, qcd = \"ns\", \"s\", \"qed\", \"qcd\"\n", + "P[ns, qcd] = sympy.Array.zeros(14,14).as_mutable()\n", + "P[ns, qed] = sympy.Array.zeros(14,14).as_mutable()\n", + "P[s, qcd] = sympy.Array.zeros(14,14).as_mutable()\n", + "P[s, qed] = sympy.Array.zeros(14,14).as_mutable()\n", + "\n", + "ei2=[eu2, ed2, ed2, eu2, ed2, eu2]\n", + "ei4=[eu4, ed4, ed4, eu4, ed4, eu4]\n", + "def es2_(nf):\n", + " nu = int(nf/2)\n", + " nd = nf - nu\n", + " return nu*eu2 + nd*ed2\n", + "\n", + "def es4_(nf):\n", + " nu = int(nf/2)\n", + " nd = nf - nu\n", + " return nu*eu4 + nd*ed4\n", + "\n", + "def P_qcd(nf):\n", + " res = sympy.Matrix.zeros(14,14).as_mutable()\n", + " res[0, 0]=Pgg\n", + " for i in range(1, nf+1):\n", + " res[0, 2*i] = Pgq #g q+\n", + " res[2*i, 0] = 2 * Pqg #q+ g\n", + " res[2*i,2*i] = Pp #q+ q+\n", + " res[1 + 2*i,1 + 2*i] = Pm #q- q-\n", + " return res\n", + "\n", + "def P_qed(nf):\n", + " es2=es2_(nf)\n", + " es4=es4_(nf)\n", + " res = sympy.Matrix.zeros(14,14).as_mutable()\n", + " res[0, 0]=es2*Pxgg\n", + " res[1, 1]=Pxyy #the factor es2 ar O(aem1) and O(as1aem1) and the factor es4 at O(aem2) are inside Pxyy\n", + " res[0, 1]=es2*Pxgy\n", + " res[1, 0]=es2*Pxyg\n", + " for i in [1, 4, 6]:\n", + " if i <= nf:\n", + " res[0, 2*i] = ei2[i-1]*Pxgq\n", + " res[2*i, 0] = 2*ei2[i-1]*Pxqg\n", + " res[1, 2*i] = ei2[i-1]*Pxyu #a factor of eu^2 at O(aem2) is inside Pxyu\n", + " res[2*i, 1] = 2*ei2[i-1]*Pxuy #a factor of eu^2 at O(aem2) is inside Pxuy\n", + " res[2*i,2*i] = ei2[i-1]*Pxpu #a factor of eu^2 at O(aem2) is inside Pxpu\n", + " res[1 + 2*i,1 + 2*i] = ei2[i-1]*Pxmu #a factor of eu^2 at O(aem2) is inside Pxmu\n", + " for i in [2, 3, 5]:\n", + " if i <= nf:\n", + " res[0, 2*i] = ei2[i-1]*Pxgq\n", + " res[2*i, 0] = 2*ei2[i-1]*Pxqg\n", + " res[1, 2*i] = ei2[i-1]*Pxyd #a factor of ed^2 at O(aem2) is inside Pxyd\n", + " res[2*i, 1] = 2*ei2[i-1]*Pxdy #a factor of ed^2 at O(aem2) is inside Pxdy\n", + " res[2*i,2*i] = ei2[i-1]*Pxpd #a factor of ed^2 at O(aem2) is inside Pxpd\n", + " res[1 + 2*i,1 + 2*i] = ei2[i-1]*Pxmd #a factor of ed^2 at O(aem2) is inside Pxmd\n", + " return res\n", + "\n", + "def Ps_qcd(nf):\n", + " res = sympy.Matrix.zeros(14,14).as_mutable()\n", + " for i in range(1, nf+1):\n", + " res[2*i, 2] = Pqq - Pp\n", + " res[2*i, 3] = Pqq - Pp\n", + " res[1 + 2*i, 4] = Pv - Pm\n", + " res[1 + 2*i, 5] = Pv - Pm\n", + " return res/nf\n", + "\n", + "def Ps_qed(nf):\n", + " res = sympy.Matrix.zeros(14,14).as_mutable()\n", + " for i in range(1, nf+1):\n", + " res[2*i, 2] = ei2[i-1]*eu2*(Pxqq - Pxp)\n", + " res[2*i, 3] = ei2[i-1]*ed2*(Pxqq - Pxp)\n", + " return res/nf\n", + "\n", + "def P_uni(nf):\n", + " return P_qcd(nf)+P_qed(nf)\n", + "\n", + "def Ps_uni(nf):\n", + " return Ps_qcd(nf)+Ps_qed(nf)" + ] + }, + { + "cell_type": "code", + "execution_count": 17, + "id": "9242fd63", + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\left[\\begin{array}{cccccccccccccc}P^{x}_{gg} \\left(3 e^{2}_{d} + 3 e^{2}_{u}\\right) & P^{x}_{g\\gamma} \\left(3 e^{2}_{d} + 3 e^{2}_{u}\\right) & P^{x}_{gq} e^{2}_{u} & 0 & P^{x}_{gq} e^{2}_{d} & 0 & P^{x}_{gq} e^{2}_{d} & 0 & P^{x}_{gq} e^{2}_{u} & 0 & P^{x}_{gq} e^{2}_{d} & 0 & P^{x}_{gq} e^{2}_{u} & 0\\\\P^{x}_{\\gamma g} \\left(3 e^{2}_{d} + 3 e^{2}_{u}\\right) & P^{x}_{\\gamma\\gamma} & P^{x}_{\\gamma u} e^{2}_{u} & 0 & P^{x}_{\\gamma d} e^{2}_{d} & 0 & P^{x}_{\\gamma d} e^{2}_{d} & 0 & P^{x}_{\\gamma u} e^{2}_{u} & 0 & P^{x}_{\\gamma d} e^{2}_{d} & 0 & P^{x}_{\\gamma u} e^{2}_{u} & 0\\\\2 P^{x}_{qg} e^{2}_{u} & 2 P^{x}_{u\\gamma} e^{2}_{u} & P^{x}_{+u} e^{2}_{u} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & P^{x}_{-u} e^{2}_{u} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\2 P^{x}_{qg} e^{2}_{d} & 2 P^{x}_{d\\gamma} e^{2}_{d} & 0 & 0 & P^{x}_{+d} e^{2}_{d} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & P^{x}_{-d} e^{2}_{d} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\2 P^{x}_{qg} e^{2}_{d} & 2 P^{x}_{d\\gamma} e^{2}_{d} & 0 & 0 & 0 & 0 & P^{x}_{+d} e^{2}_{d} & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & P^{x}_{-d} e^{2}_{d} & 0 & 0 & 0 & 0 & 0 & 0\\\\2 P^{x}_{qg} e^{2}_{u} & 2 P^{x}_{u\\gamma} e^{2}_{u} & 0 & 0 & 0 & 0 & 0 & 0 & P^{x}_{+u} e^{2}_{u} & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & P^{x}_{-u} e^{2}_{u} & 0 & 0 & 0 & 0\\\\2 P^{x}_{qg} e^{2}_{d} & 2 P^{x}_{d\\gamma} e^{2}_{d} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & P^{x}_{+d} e^{2}_{d} & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & P^{x}_{-d} e^{2}_{d} & 0 & 0\\\\2 P^{x}_{qg} e^{2}_{u} & 2 P^{x}_{u\\gamma} e^{2}_{u} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & P^{x}_{+u} e^{2}_{u} & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & P^{x}_{-u} e^{2}_{u}\\end{array}\\right]$" + ], + "text/plain": [ + "Matrix([\n", + "[ P^x_gg*(3*e_d^2 + 3*e_u^2), P^x_g\\gamma*(3*e_d^2 + 3*e_u^2), P^x_gq*e_u^2, 0, P^x_gq*e_d^2, 0, P^x_gq*e_d^2, 0, P^x_gq*e_u^2, 0, P^x_gq*e_d^2, 0, P^x_gq*e_u^2, 0],\n", + "[P^x_\\gamma g*(3*e_d^2 + 3*e_u^2), P^x_\\gamma\\gamma, P^x_\\gamma u*e_u^2, 0, P^x_\\gamma d*e_d^2, 0, P^x_\\gamma d*e_d^2, 0, P^x_\\gamma u*e_u^2, 0, P^x_\\gamma d*e_d^2, 0, P^x_\\gamma u*e_u^2, 0],\n", + "[ 2*P^x_qg*e_u^2, 2*P^x_u\\gamma*e_u^2, P^x_+u*e_u^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, P^x_-u*e_u^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 2*P^x_qg*e_d^2, 2*P^x_d\\gamma*e_d^2, 0, 0, P^x_+d*e_d^2, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, P^x_-d*e_d^2, 0, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 2*P^x_qg*e_d^2, 2*P^x_d\\gamma*e_d^2, 0, 0, 0, 0, P^x_+d*e_d^2, 0, 0, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, P^x_-d*e_d^2, 0, 0, 0, 0, 0, 0],\n", + "[ 2*P^x_qg*e_u^2, 2*P^x_u\\gamma*e_u^2, 0, 0, 0, 0, 0, 0, P^x_+u*e_u^2, 0, 0, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, P^x_-u*e_u^2, 0, 0, 0, 0],\n", + "[ 2*P^x_qg*e_d^2, 2*P^x_d\\gamma*e_d^2, 0, 0, 0, 0, 0, 0, 0, 0, P^x_+d*e_d^2, 0, 0, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, P^x_-d*e_d^2, 0, 0],\n", + "[ 2*P^x_qg*e_u^2, 2*P^x_u\\gamma*e_u^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, P^x_+u*e_u^2, 0],\n", + "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, P^x_-u*e_u^2]])" + ] + }, + "execution_count": 17, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "P_qed(6)" + ] + }, + { + "cell_type": "code", + "execution_count": 18, + "id": "58ce83af", + "metadata": {}, + "outputs": [], + "source": [ + "def rot_fl_to_ev(nf):\n", + " nu = int(nf/2)\n", + " nd = nf - nu\n", + " res = sympy.Matrix.zeros(14,14).as_mutable()\n", + " res[0, 0]=1\n", + " res[1, 1]=1\n", + " for i in range(2 + 2*nf, 14): \n", + " res[i,i] = 1\n", + " for i in range(1,nf+1): #Sigma and V\n", + " res[2, 2*i] = 1\n", + " res[4, 1 + 2*i] = 1\n", + " for i in [1, 4, 6]:#loop on up quarks\n", + " if i <= nf:\n", + " res[3, 2*i] = nd/nu\n", + " res[5,1 + 2*i] = nd/nu\n", + " for i in [2, 3, 5]:#loop on down quarks\n", + " if i <= nf:\n", + " res[3, 2*i] = -1\n", + " res[5, 1 + 2*i] = -1\n", + " if nf >= 3 :\n", + " res[6, 4] = 1\n", + " res[6, 6] = -1\n", + " res[7, 5] = 1\n", + " res[7, 7] = -1\n", + " if nf >= 4 :\n", + " res[8, 2] = 1\n", + " res[8, 8] = -1\n", + " res[9, 3] = 1\n", + " res[9, 9] = -1\n", + " if nf >= 5 :\n", + " res[10, 4] = 1\n", + " res[10, 6] = 1\n", + " res[10, 10] = -2\n", + " res[11, 5] = 1\n", + " res[11, 7] = 1\n", + " res[11, 11] = -2\n", + " if nf == 6 :\n", + " res[12, 2] = 1\n", + " res[12, 8] = 1\n", + " res[12, 12] = -2\n", + " res[13, 3] = 1\n", + " res[13, 9] = 1\n", + " res[13, 13] = -2\n", + " return res\n", + "\n", + "def rot_ev_to_fl(nf):\n", + " return rot_fl_to_ev(nf).inv()" + ] + }, + { + "cell_type": "code", + "execution_count": 19, + "id": "28585067", + "metadata": {}, + "outputs": [], + "source": [ + "def rot_sin_to_ev(nf):\n", + " nu = int(nf/2)\n", + " nd = nf - nu\n", + " res = sympy.Matrix.zeros(14,14).as_mutable()\n", + " res[0, 0]=1\n", + " res[1, 1]=1\n", + " res[2,2]=1\n", + " res[2,3]=1\n", + " res[3,2]=nd/nu\n", + " res[3,3]=-1\n", + " res[4,4]=1\n", + " res[4,5]=1\n", + " res[5,4]=nd/nu\n", + " res[5,5]=-1\n", + " for i in range(6,14):\n", + " res[i,i]=1\n", + " return res\n", + "\n", + "def rot_ev_to_sin(nf):\n", + " return rot_sin_to_ev(nf).inv()" + ] + }, + { + "cell_type": "code", + "execution_count": 22, + "id": "a0c9e31d", + "metadata": {}, + "outputs": [], + "source": [ + "def P_ev(nf):\n", + " res = rot_fl_to_ev(nf) * P_uni(nf) * rot_ev_to_fl(nf) + rot_fl_to_ev(nf) * Ps_uni(nf) * rot_ev_to_sin(nf)\n", + " return res\n", + "\n", + "def P_ev_sing(nf):\n", + " return P_ev(nf)[:4,:4]\n", + "\n", + "def P_ev_val(nf):\n", + " return P_ev(nf)[4:6,4:6]" + ] + }, + { + "cell_type": "code", + "execution_count": 23, + "id": "75bfa63a", + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\left[\\begin{matrix}P^{x}_{gg} \\left(3 e^{2}_{d} + 3 e^{2}_{u}\\right) + P_{gg} & P^{x}_{g\\gamma} \\left(3 e^{2}_{d} + 3 e^{2}_{u}\\right) & 0.5 P^{x}_{gq} e^{2}_{d} + 0.5 P^{x}_{gq} e^{2}_{u} + 1.0 P_{gq} & - 0.5 P^{x}_{gq} e^{2}_{d} + 0.5 P^{x}_{gq} e^{2}_{u}\\\\P^{x}_{\\gamma g} \\left(3 e^{2}_{d} + 3 e^{2}_{u}\\right) & P^{x}_{\\gamma\\gamma} & 0.5 P^{x}_{\\gamma d} e^{2}_{d} + 0.5 P^{x}_{\\gamma u} e^{2}_{u} & - 0.5 P^{x}_{\\gamma d} e^{2}_{d} + 0.5 P^{x}_{\\gamma u} e^{2}_{u}\\\\6 P^{x}_{qg} e^{2}_{d} + 6 P^{x}_{qg} e^{2}_{u} + 12 P_{qg} & 6 P^{x}_{d\\gamma} e^{2}_{d} + 6 P^{x}_{u\\gamma} e^{2}_{u} & 0.5 P^{x}_{+d} e^{2}_{d} + 0.5 P^{x}_{+u} e^{2}_{u} - 1.11022302462516 \\cdot 10^{-16} P_{+} + 1.0 P_{qq} + 0.25 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.5 e^{2}_{d} e^{2}_{u} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.25 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & - 0.5 P^{x}_{+d} e^{2}_{d} + 0.5 P^{x}_{+u} e^{2}_{u} - 0.25 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.25 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right)\\\\- 6 P^{x}_{qg} e^{2}_{d} + 6.0 P^{x}_{qg} e^{2}_{u} & - 6 P^{x}_{d\\gamma} e^{2}_{d} + 6.0 P^{x}_{u\\gamma} e^{2}_{u} & - 0.5 P^{x}_{+d} e^{2}_{d} + 0.5 P^{x}_{+u} e^{2}_{u} - 0.25 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.25 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & 0.5 P^{x}_{+d} e^{2}_{d} + 0.5 P^{x}_{+u} e^{2}_{u} + 1.0 P_{+} + 0.25 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) - 0.5 e^{2}_{d} e^{2}_{u} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.25 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right)\\end{matrix}\\right]$" + ], + "text/plain": [ + "Matrix([\n", + "[ P^x_gg*(3*e_d^2 + 3*e_u^2) + P_gg, P^x_g\\gamma*(3*e_d^2 + 3*e_u^2), 0.5*P^x_gq*e_d^2 + 0.5*P^x_gq*e_u^2 + 1.0*P_gq, -0.5*P^x_gq*e_d^2 + 0.5*P^x_gq*e_u^2],\n", + "[ P^x_\\gamma g*(3*e_d^2 + 3*e_u^2), P^x_\\gamma\\gamma, 0.5*P^x_\\gamma d*e_d^2 + 0.5*P^x_\\gamma u*e_u^2, -0.5*P^x_\\gamma d*e_d^2 + 0.5*P^x_\\gamma u*e_u^2],\n", + "[6*P^x_qg*e_d^2 + 6*P^x_qg*e_u^2 + 12*P_qg, 6*P^x_d\\gamma*e_d^2 + 6*P^x_u\\gamma*e_u^2, 0.5*P^x_+d*e_d^2 + 0.5*P^x_+u*e_u^2 - 1.11022302462516e-16*P_+ + 1.0*P_qq + 0.25*e_d^2**2*(-P^x_+ + P^x_qq) + 0.5*e_d^2*e_u^2*(-P^x_+ + P^x_qq) + 0.25*e_u^2**2*(-P^x_+ + P^x_qq), -0.5*P^x_+d*e_d^2 + 0.5*P^x_+u*e_u^2 - 0.25*e_d^2**2*(-P^x_+ + P^x_qq) + 0.25*e_u^2**2*(-P^x_+ + P^x_qq)],\n", + "[ -6*P^x_qg*e_d^2 + 6.0*P^x_qg*e_u^2, -6*P^x_d\\gamma*e_d^2 + 6.0*P^x_u\\gamma*e_u^2, -0.5*P^x_+d*e_d^2 + 0.5*P^x_+u*e_u^2 - 0.25*e_d^2**2*(-P^x_+ + P^x_qq) + 0.25*e_u^2**2*(-P^x_+ + P^x_qq), 0.5*P^x_+d*e_d^2 + 0.5*P^x_+u*e_u^2 + 1.0*P_+ + 0.25*e_d^2**2*(-P^x_+ + P^x_qq) - 0.5*e_d^2*e_u^2*(-P^x_+ + P^x_qq) + 0.25*e_u^2**2*(-P^x_+ + P^x_qq)]])" + ] + }, + "execution_count": 23, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "P_ev_sing(6)" + ] + }, + { + "cell_type": "code", + "execution_count": 24, + "id": "5999a68e", + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\left[\\begin{matrix}0.5 P^{x}_{-d} e^{2}_{d} + 0.5 P^{x}_{-u} e^{2}_{u} - 1.11022302462516 \\cdot 10^{-16} P_{-} + 1.0 P_{V} & - 0.5 P^{x}_{-d} e^{2}_{d} + 0.5 P^{x}_{-u} e^{2}_{u}\\\\- 0.5 P^{x}_{-d} e^{2}_{d} + 0.5 P^{x}_{-u} e^{2}_{u} & 0.5 P^{x}_{-d} e^{2}_{d} + 0.5 P^{x}_{-u} e^{2}_{u} + 1.0 P_{-}\\end{matrix}\\right]$" + ], + "text/plain": [ + "Matrix([\n", + "[0.5*P^x_-d*e_d^2 + 0.5*P^x_-u*e_u^2 - 1.11022302462516e-16*P_- + 1.0*P_V, -0.5*P^x_-d*e_d^2 + 0.5*P^x_-u*e_u^2],\n", + "[ -0.5*P^x_-d*e_d^2 + 0.5*P^x_-u*e_u^2, 0.5*P^x_-d*e_d^2 + 0.5*P^x_-u*e_u^2 + 1.0*P_-]])" + ] + }, + "execution_count": 24, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "P_ev_val(6)" + ] + }, + { + "cell_type": "code", + "execution_count": 42, + "id": "525487e1", + "metadata": {}, + "outputs": [], + "source": [ + "def eD2_(nf):\n", + " nu = int(nf/2)\n", + " nd = nf - nu\n", + " return nd*eu2 + nu*ed2\n", + "def etam_(nf):\n", + " nu = int(nf/2)\n", + " nd = nf - nu\n", + " return 0.5*(eu2 - ed2)\n", + "def P_ev_sing2(nf):\n", + " es2=es2_(nf)\n", + " eD2=eD2_(nf)\n", + " etam=etam_(nf)\n", + " nu = int(nf/2)\n", + " nd = nf - nu\n", + " res = sympy.Matrix([\n", + " [Pgg + es2 * Pxgg, es2 * Pxgy, Pgq + es2/nf*Pxgq, 2*nu/nf*etam*Pxgq],\n", + " [es2 * Pxyg, Pxyy, nu/nf*eu2*Pxyu + nd/nf*ed2*Pxyd, 2*nu/nf*0.5*(eu2*Pxyu-ed2*Pxyd)],\n", + " [2*nf*Pqg + 2*es2*Pxqg, 2*(nu*eu2*Pxuy+nd*ed2*Pxdy), Pqq + (nu*eu2*Pxpu+nd*ed2*Pxpd)/nf +(es2/nf)**2*(Pxqq - Pxp), 2*nu/nf*0.5*(eu2*Pxpu - ed2*Pxpd) +2*nu*etam*es2/nf**2*(Pxqq - Pxp)],\n", + " [4*nd*etam*Pxqg, 4*nd*0.5*(eu2*Pxuy - ed2*Pxdy), 2*nd/nf*0.5*(eu2*Pxpu - ed2*Pxpd) +2*nd*etam*es2/nf**2*(Pxqq - Pxp), Pp + (nd*eu2*Pxpu + nu*ed2*Pxpd)/nf + 4*nu*nd/nf**2*etam**2*(Pxqq - Pxp)]\n", + " ])\n", + " return res\n", + "\n", + "def P_ev_val2(nf):\n", + " nu = int(nf/2)\n", + " nd = nf - nu\n", + " res = sympy.Matrix([\n", + " [Pv+(nu*eu2*Pxmu+nd*ed2*Pxmd)/nf, 2*nu/nf*0.5*(eu2*Pxmu - ed2*Pxmd)],\n", + " [2*nd/nf*0.5*(eu2*Pxmu - ed2*Pxmd), Pm + (nd*eu2*Pxmu + nu*ed2*Pxmd)/nf]\n", + " ])\n", + " return res" + ] + }, + { + "cell_type": "code", + "execution_count": 43, + "id": "cac66ba9", + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\left[\\begin{matrix}0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\end{matrix}\\right]$" + ], + "text/plain": [ + "Matrix([\n", + "[0, 0, 0, 0],\n", + "[0, 0, 0, 0],\n", + "[0, 0, 0, 0],\n", + "[0, 0, 0, 0]])" + ] + }, + "execution_count": 43, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "sympy.simplify(P_ev_sing(2)-P_ev_sing2(2))" + ] + }, + { + "cell_type": "code", + "execution_count": 47, + "id": "ac0e82fc", + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\left[\\begin{matrix}- 1.11022302462516 \\cdot 10^{-16} P_{-} & 0\\\\0 & - 1.11022302462516 \\cdot 10^{-16} P_{-}\\end{matrix}\\right]$" + ], + "text/plain": [ + "Matrix([\n", + "[-1.11022302462516e-16*P_-, 0],\n", + "[ 0, -1.11022302462516e-16*P_-]])" + ] + }, + "execution_count": 47, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "sympy.simplify(P_ev_val(6)-P_ev_val2(6))" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "3e3c29ce", + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3 (ipykernel)", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.8.12" + } + }, + "nbformat": 4, + "nbformat_minor": 5 +} From 98104c26c9d367f027b4d79b17872b7667404c16 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Niccol=C3=B2=20Laurenti?= Date: Tue, 5 Apr 2022 12:21:17 +0200 Subject: [PATCH 38/71] Clean notebook --- extras/uni-ad.nb | 2283 ++++++++++++++++------------------------------ 1 file changed, 800 insertions(+), 1483 deletions(-) diff --git a/extras/uni-ad.nb b/extras/uni-ad.nb index c0dd98555..81a08ec09 100644 --- a/extras/uni-ad.nb +++ b/extras/uni-ad.nb @@ -10,10 +10,10 @@ NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] -NotebookDataLength[ 215025, 5636] -NotebookOptionsPosition[ 205744, 5483] -NotebookOutlinePosition[ 206184, 5500] -CellTagsIndexPosition[ 206141, 5497] +NotebookDataLength[ 193463, 4953] +NotebookOptionsPosition[ 183667, 4793] +NotebookOutlinePosition[ 184107, 4810] +CellTagsIndexPosition[ 184064, 4807] WindowFrame->Normal*) (* Beginning of Notebook Content *) @@ -75,68 +75,68 @@ University of Zurich\"\>"], "Print", CellChangeTimes->{3.857443465944676*^9, 3.857554067233485*^9, 3.8576223614575357`*^9, 3.857701955928193*^9, 3.8577135075316963`*^9, 3.857724749123518*^9, 3.857725602969599*^9, 3.857786853169505*^9, 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"Input",ExpressionUUID->"8015a7b0-6c5e-4ea4-9d35-0ea868417989"], +Cell[182252, 4752, 1399, 38, 64, "Output",ExpressionUUID->"ee35f642-60e4-4819-94d1-f7a6096458f0"] +}, Open ]] } ] *) From 0547e32aca02060a1dad86ff6d355e321e774042 Mon Sep 17 00:00:00 2001 From: Alessandro Candido Date: Tue, 5 Apr 2022 12:21:34 +0200 Subject: [PATCH 39/71] Ignore notebooks in language stats --- .gitattributes | 2 ++ 1 file changed, 2 insertions(+) create mode 100644 .gitattributes diff --git a/.gitattributes b/.gitattributes new file mode 100644 index 000000000..f63135eb4 --- /dev/null +++ b/.gitattributes @@ -0,0 +1,2 @@ +*.ipynb linguist-generated +*.nb linguist-generated From 86b8f4abf1fa0cb4177b859986819d0881aac2d4 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Niccol=C3=B2=20Laurenti?= Date: Tue, 5 Apr 2022 12:21:53 +0200 Subject: [PATCH 40/71] Change number_of_up_flavors -> uplike_flavors --- extras/uni-dglap.ipynb | 50 +++++++++++-------------- src/eko/anomalous_dimensions/aem2.py | 10 ++--- src/eko/anomalous_dimensions/as1aem1.py | 4 +- 3 files changed, 29 insertions(+), 35 deletions(-) diff --git a/extras/uni-dglap.ipynb b/extras/uni-dglap.ipynb index d4cdcb709..e0d31c949 100644 --- a/extras/uni-dglap.ipynb +++ b/extras/uni-dglap.ipynb @@ -10,7 +10,7 @@ }, { "cell_type": "code", - "execution_count": 312, + "execution_count": 3, "id": "1d7616d9-f4a3-447a-9ebf-e99db8126ffb", "metadata": {}, "outputs": [], @@ -20,7 +20,7 @@ }, { "cell_type": "code", - "execution_count": 313, + "execution_count": 5, "id": "cb36381d-57b5-4972-b028-cf4b6300938f", "metadata": {}, "outputs": [], @@ -53,7 +53,7 @@ }, { "cell_type": "code", - "execution_count": 314, + "execution_count": 6, "id": "9764233f-9c53-4860-b7c3-2be12fbec857", "metadata": {}, "outputs": [], @@ -122,7 +122,7 @@ }, { "cell_type": "code", - "execution_count": 315, + "execution_count": 7, "id": "3a376dcf", "metadata": {}, "outputs": [], @@ -178,7 +178,7 @@ }, { "cell_type": "code", - "execution_count": 316, + "execution_count": 8, "id": "b5747379", "metadata": {}, "outputs": [], @@ -207,7 +207,7 @@ }, { "cell_type": "code", - "execution_count": 317, + "execution_count": 9, "id": "45fb3a16", "metadata": {}, "outputs": [], @@ -219,7 +219,7 @@ }, { "cell_type": "code", - "execution_count": 318, + "execution_count": 10, "id": "9b5050aa", "metadata": {}, "outputs": [ @@ -246,7 +246,7 @@ "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]])" ] }, - "execution_count": 318, + "execution_count": 10, "metadata": {}, "output_type": "execute_result" } @@ -257,7 +257,7 @@ }, { "cell_type": "code", - "execution_count": 319, + "execution_count": 11, "id": "07b1874a", "metadata": {}, "outputs": [ @@ -284,7 +284,7 @@ "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]])" ] }, - "execution_count": 319, + "execution_count": 11, "metadata": {}, "output_type": "execute_result" } @@ -440,7 +440,7 @@ }, { "cell_type": "code", - "execution_count": 325, + "execution_count": 1, "id": "5267ab16", "metadata": {}, "outputs": [], @@ -460,7 +460,7 @@ " return res\n", "\n", "def P_ev_val2(nf):\n", - " es2=e2s(nf)\n", + " es2=es2_(nf)\n", " eD2=eD2_(nf)\n", " etam=etam_(nf)\n", " nu = int(nf/2)\n", @@ -474,26 +474,20 @@ }, { "cell_type": "code", - "execution_count": 326, + "execution_count": 2, "id": "a47dc3b2", "metadata": {}, "outputs": [ { - "data": { - "text/latex": [ - "$\\displaystyle \\left[\\begin{matrix}0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\end{matrix}\\right]$" - ], - "text/plain": [ - "Matrix([\n", - "[0, 0, 0, 0],\n", - "[0, 0, 0, 0],\n", - "[0, 0, 0, 0],\n", - "[0, 0, 0, 0]])" - ] - }, - "execution_count": 326, - "metadata": {}, - "output_type": "execute_result" + "ename": "NameError", + "evalue": "name 'sympy' is not defined", + "output_type": "error", + "traceback": [ + "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", + "\u001b[0;31mNameError\u001b[0m Traceback (most recent call last)", + "\u001b[0;32m/var/folders/9_/b6wkhsvj63jg9wr71d3_kgqw0000gn/T/ipykernel_2455/3287418665.py\u001b[0m in \u001b[0;36m\u001b[0;34m\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0msympy\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0msimplify\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mP_ev_sing\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m2\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m-\u001b[0m\u001b[0mP_ev_sing2\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m2\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m", + "\u001b[0;31mNameError\u001b[0m: name 'sympy' is not defined" + ] } ], "source": [ diff --git a/src/eko/anomalous_dimensions/aem2.py b/src/eko/anomalous_dimensions/aem2.py index 7fa9f6e6e..f169eacc1 100644 --- a/src/eko/anomalous_dimensions/aem2.py +++ b/src/eko/anomalous_dimensions/aem2.py @@ -31,7 +31,7 @@ def gamma_phph(N, nf): :math:`\\gamma_{\\gamma \\gamma}^{(1,1)}(N)` """ - nu = as1aem1.number_of_up_flavors(nf) + nu = as1aem1.uplike_flavors(nf) nd = nf - nu return (nu * constants.eu2**2 + nd * constants.ed2**2) * ( as1aem1.gamma_gph(N) / constants.CF / constants.CA + 4 @@ -110,7 +110,7 @@ def gamma_phu(N, nf, sx): O(as1aem1) photon-quark anomalous dimension :math:`\\gamma_{\\gamma q}^{(1,1)}(N)` """ - nu = as1aem1.number_of_up_flavors(nf) + nu = as1aem1.uplike_flavors(nf) nd = nf - nu S1 = sx[0] tmp = (-16 * (-16 - 27 * N - 13 * N**2 - 8 * N**3)) / ( @@ -144,7 +144,7 @@ def gamma_phd(N, nf, sx): O(as1aem1) photon-quark anomalous dimension :math:`\\gamma_{\\gamma q}^{(1,1)}(N)` """ - nu = as1aem1.number_of_up_flavors(nf) + nu = as1aem1.uplike_flavors(nf) nd = nf - nu S1 = sx[0] tmp = (-16 * (-16 - 27 * N - 13 * N**2 - 8 * N**3)) / ( @@ -181,7 +181,7 @@ def gamma_nspu(N, nf, sx): S1 = sx[0] S2 = sx[1] - nu = as1aem1.number_of_up_flavors(nf) + nu = as1aem1.uplike_flavors(nf) nd = nf - nu eSigma2 = constants.NC * (nu * constants.eu2 + nd * constants.ed2) tmp = ( @@ -219,7 +219,7 @@ def gamma_nspd(N, nf, sx): S1 = sx[0] S2 = sx[1] - nu = as1aem1.number_of_up_flavors(nf) + nu = as1aem1.uplike_flavors(nf) nd = nf - nu eSigma2 = constants.NC * (nu * constants.eu2 + nd * constants.ed2) tmp = ( diff --git a/src/eko/anomalous_dimensions/as1aem1.py b/src/eko/anomalous_dimensions/as1aem1.py index 7524ee065..ccd0c2624 100644 --- a/src/eko/anomalous_dimensions/as1aem1.py +++ b/src/eko/anomalous_dimensions/as1aem1.py @@ -15,7 +15,7 @@ @nb.njit("u1(u1)", cache=True) -def number_of_up_flavors(nf): +def uplike_flavors(nf): """ Computes the number of up flavors @@ -249,7 +249,7 @@ def gamma_phph(nf): :math:`\\gamma_{\\gamma \\gamma}^{(1,1)}(N)` """ - nu = number_of_up_flavors(nf) + nu = uplike_flavors(nf) nd = nf - nu return 4 * constants.CF * constants.CA * (nu * constants.eu2 + nd * constants.ed2) From 2d990865c922df7bd97e5a3783b43cfc86f1c8af Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Niccol=C3=B2=20Laurenti?= Date: Tue, 5 Apr 2022 12:42:36 +0200 Subject: [PATCH 41/71] Fix numba in aem2.py --- src/eko/anomalous_dimensions/aem2.py | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/src/eko/anomalous_dimensions/aem2.py b/src/eko/anomalous_dimensions/aem2.py index f169eacc1..b40d3f729 100644 --- a/src/eko/anomalous_dimensions/aem2.py +++ b/src/eko/anomalous_dimensions/aem2.py @@ -88,7 +88,7 @@ def gamma_dph(N, nf, sx): return constants.ed2 * as1aem1.gamma_qph(N, nf, sx) / constants.CF -@nb.njit("c16(c16,c16[:])", cache=True) +@nb.njit("c16(c16,u1,c16[:])", cache=True) def gamma_phu(N, nf, sx): """ Computes the O(as1aem1) photon-quark anomalous dimension @@ -122,7 +122,7 @@ def gamma_phu(N, nf, sx): return constants.eu2 * as1aem1.gamma_phq(N, sx) / constants.CF + eSigma2 * tmp -@nb.njit("c16(c16,c16[:])", cache=True) +@nb.njit("c16(c16,u1,c16[:])", cache=True) def gamma_phd(N, nf, sx): """ Computes the O(as1aem1) photon-quark anomalous dimension From 42dbce2b3a8e7be4ec25ee124f8d0074488a3748 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Niccol=C3=B2=20Laurenti?= Date: Tue, 5 Apr 2022 12:45:21 +0200 Subject: [PATCH 42/71] Fix documentation in as1aem1.py --- src/eko/anomalous_dimensions/as1aem1.py | 8 -------- 1 file changed, 8 deletions(-) diff --git a/src/eko/anomalous_dimensions/as1aem1.py b/src/eko/anomalous_dimensions/as1aem1.py index ccd0c2624..4abf11d5a 100644 --- a/src/eko/anomalous_dimensions/as1aem1.py +++ b/src/eko/anomalous_dimensions/as1aem1.py @@ -54,8 +54,6 @@ def gamma_phq(N, sx): ---------- N : complex Mellin moment - nf : int - Number of active flavors sx : np array List of harmonic sums @@ -215,8 +213,6 @@ def gamma_gq(N, sx): ---------- N : complex Mellin moment - nf : int - Number of active flavors sx : np array List of harmonic sums @@ -285,8 +281,6 @@ def gamma_nsp(N, sx): ---------- N : complex Mellin moment - nf : int - Number of active flavors sx : np array List of harmonic sums @@ -351,8 +345,6 @@ def gamma_nsm(N, sx): ---------- N : complex Mellin moment - nf : int - Number of active flavors sx : np array List of harmonic sums From 29b1acf8d62b98efddd07af0ba0cccd374713f2a Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Niccol=C3=B2=20Laurenti?= Date: Tue, 5 Apr 2022 12:53:45 +0200 Subject: [PATCH 43/71] Fix documentation in aem2.py --- src/eko/anomalous_dimensions/aem2.py | 110 ++++++++++++++++++++++----- 1 file changed, 93 insertions(+), 17 deletions(-) diff --git a/src/eko/anomalous_dimensions/aem2.py b/src/eko/anomalous_dimensions/aem2.py index b40d3f729..bd21c7bc1 100644 --- a/src/eko/anomalous_dimensions/aem2.py +++ b/src/eko/anomalous_dimensions/aem2.py @@ -17,9 +17,9 @@ @nb.njit("c16(c16,u1)", cache=True) def gamma_phph(N, nf): """ - Computes the O(as1aem1) photon-photon singlet anomalous dimension. + Computes the O(aem2) photon-photon singlet anomalous dimension. - Implements Eq. (28) of :cite:`deFlorian:2015ujt`. + Implements Eq. (68) of :cite:`deFlorian:2016gvk`. Parameters ---------- @@ -41,9 +41,9 @@ def gamma_phph(N, nf): @nb.njit("c16(c16,u1,c16[:])", cache=True) def gamma_uph(N, nf, sx): """ - Computes the O(as1aem1) quark-photon anomalous dimension + Computes the O(aem2) quark-photon anomalous dimension - Implements Eq. (26) of :cite:`deFlorian:2015ujt`. + Implements Eq. (55) of :cite:`deFlorian:2016gvk` for q=u. Parameters ---------- @@ -66,9 +66,9 @@ def gamma_uph(N, nf, sx): @nb.njit("c16(c16,u1,c16[:])", cache=True) def gamma_dph(N, nf, sx): """ - Computes the O(as1aem1) quark-photon anomalous dimension + Computes the O(aem2) quark-photon anomalous dimension - Implements Eq. (26) of :cite:`deFlorian:2015ujt`. + Implements Eq. (55) of :cite:`deFlorian:2016gvk` for q=d. Parameters ---------- @@ -91,9 +91,9 @@ def gamma_dph(N, nf, sx): @nb.njit("c16(c16,u1,c16[:])", cache=True) def gamma_phu(N, nf, sx): """ - Computes the O(as1aem1) photon-quark anomalous dimension + Computes the O(aem2) photon-quark anomalous dimension - Implements Eq. (36) of :cite:`deFlorian:2015ujt`. + Implements Eq. (56) of :cite:`deFlorian:2016gvk` for q=u. Parameters ---------- @@ -125,9 +125,9 @@ def gamma_phu(N, nf, sx): @nb.njit("c16(c16,u1,c16[:])", cache=True) def gamma_phd(N, nf, sx): """ - Computes the O(as1aem1) photon-quark anomalous dimension + Computes theO(aem2) photon-quark anomalous dimension - Implements Eq. (36) of :cite:`deFlorian:2015ujt`. + Implements Eq. (56) of :cite:`deFlorian:2016gvk` for q=d. Parameters ---------- @@ -159,9 +159,9 @@ def gamma_phd(N, nf, sx): @nb.njit("c16(c16,u1,c16[:])", cache=True) def gamma_nspu(N, nf, sx): """ - Computes the O(as1aem1) singlet-like non-singlet anomalous dimension. + Computes the O(aem2) singlet-like non-singlet anomalous dimension. - Implements sum of Eqs. (33-34) of :cite:`deFlorian:2015ujt`. + Implements sum of Eqs. (57-58) of :cite:`deFlorian:2016gvk` for q=u. Parameters ---------- @@ -197,9 +197,9 @@ def gamma_nspu(N, nf, sx): @nb.njit("c16(c16,u1,c16[:])", cache=True) def gamma_nspd(N, nf, sx): """ - Computes the O(as1aem1) singlet-like non-singlet anomalous dimension. + Computes the O(aem2) singlet-like non-singlet anomalous dimension. - Implements sum of Eqs. (33-34) of :cite:`deFlorian:2015ujt`. + Implements sum of Eqs. (57-58) of :cite:`deFlorian:2016gvk` for q=d. Parameters ---------- @@ -232,16 +232,92 @@ def gamma_nspd(N, nf, sx): return constants.ed2 * as1aem1.gamma_nsp(N, sx) / constants.CF / 2 + tmp +@nb.njit("c16(c16,u1,c16[:])", cache=True) +def gamma_nsmu(N, nf, sx): + """ + Computes the O(aem2) singlet-like non-singlet anomalous dimension. + + Implements difference between Eqs. (57-58) of :cite:`deFlorian:2016gvk` for q=u. + + Parameters + ---------- + N : complex + Mellin moment + nf : int + Number of active flavors + sx : np array + List of harmonic sums + + Returns + ------- + gamma_nsp : complex + O(as1aem1) singlet-like non-singlet anomalous dimension + :math:`\\gamma_{ns,+}^{(1)}(N)` + """ + + S1 = sx[0] + S2 = sx[1] + nu = as1aem1.uplike_flavors(nf) + nd = nf - nu + eSigma2 = constants.NC * (nu * constants.eu2 + nd * constants.ed2) + tmp = ( + 2 + * (-12 + 20 * N + 107 * N**2 + 126 * N**3 + 63 * N**4) + / (9.0 * N**2 * (1 + N) ** 2) + - 80 / 9 * S1 + + 16 / 3 * S2 + ) * eSigma2 + return constants.eu2 * as1aem1.gamma_nsm(N, sx) / constants.CF / 2 + tmp + + +@nb.njit("c16(c16,u1,c16[:])", cache=True) +def gamma_nsmd(N, nf, sx): + """ + Computes the O(aem2) singlet-like non-singlet anomalous dimension. + + Implements difference between Eqs. (57-58) of :cite:`deFlorian:2016gvk` for q=d. + + Parameters + ---------- + N : complex + Mellin moment + nf : int + Number of active flavors + sx : np array + List of harmonic sums + + Returns + ------- + gamma_nsp : complex + O(as1aem1) singlet-like non-singlet anomalous dimension + :math:`\\gamma_{ns,+}^{(1)}(N)` + """ + + S1 = sx[0] + S2 = sx[1] + nu = as1aem1.uplike_flavors(nf) + nd = nf - nu + eSigma2 = constants.NC * (nu * constants.eu2 + nd * constants.ed2) + tmp = ( + 2 + * (-12 + 20 * N + 107 * N**2 + 126 * N**3 + 63 * N**4) + / (9.0 * N**2 * (1 + N) ** 2) + - 80 / 9 * S1 + + 16 / 3 * S2 + ) * eSigma2 + return constants.ed2 * as1aem1.gamma_nsm(N, sx) / constants.CF / 2 + tmp + + @nb.njit("c16(c16,u1)", cache=True) def gamma_ps(N, nf): """ - Computes the |NLO| pure-singlet quark-quark anomalous dimension. + Computes the O(aem2) pure-singlet quark-quark anomalous dimension. - Implements Eq. (3.6) of :cite:`Vogt:2004mw`. + Implements Eq. (59) of :cite:`deFlorian:2016gvk`. Parameters ---------- - n : complex + N : complex Mellin moment nf : int Number of active flavors From f7d8708af29731dc966205be722166b31ba28cc0 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Niccol=C3=B2=20Laurenti?= Date: Tue, 5 Apr 2022 14:39:34 +0200 Subject: [PATCH 44/71] Fix aem2.gamma_nsmd and test number conservation --- src/eko/anomalous_dimensions/aem2.py | 4 ++-- tests/eko/test_ad_aem2.py | 18 ++++++++++++++++++ 2 files changed, 20 insertions(+), 2 deletions(-) create mode 100644 tests/eko/test_ad_aem2.py diff --git a/src/eko/anomalous_dimensions/aem2.py b/src/eko/anomalous_dimensions/aem2.py index bd21c7bc1..120354ee9 100644 --- a/src/eko/anomalous_dimensions/aem2.py +++ b/src/eko/anomalous_dimensions/aem2.py @@ -262,7 +262,7 @@ def gamma_nsmu(N, nf, sx): eSigma2 = constants.NC * (nu * constants.eu2 + nd * constants.ed2) tmp = ( 2 - * (-12 + 20 * N + 107 * N**2 + 126 * N**3 + 63 * N**4) + * (-12 + 20 * N + 47 * N**2 + 6 * N**3 + 3 * N**4) / (9.0 * N**2 * (1 + N) ** 2) - 80 / 9 * S1 + 16 / 3 * S2 @@ -300,7 +300,7 @@ def gamma_nsmd(N, nf, sx): eSigma2 = constants.NC * (nu * constants.eu2 + nd * constants.ed2) tmp = ( 2 - * (-12 + 20 * N + 107 * N**2 + 126 * N**3 + 63 * N**4) + * (-12 + 20 * N + 47 * N**2 + 6 * N**3 + 3 * N**4) / (9.0 * N**2 * (1 + N) ** 2) - 80 / 9 * S1 + 16 / 3 * S2 diff --git a/tests/eko/test_ad_aem2.py b/tests/eko/test_ad_aem2.py new file mode 100644 index 000000000..f2d4aab6d --- /dev/null +++ b/tests/eko/test_ad_aem2.py @@ -0,0 +1,18 @@ +# -*- coding: utf-8 -*- +# Test O(as1aem1) splitting functions +import numpy as np +from test_ad_nnlo import get_sx + +from eko import anomalous_dimensions as ad +from eko import constants + + +def test_number_conservation(): + # number + N = complex(1.0, 0.0) + sx = get_sx(N) + for NF, ND in ((5, 3), (6, 3)): + # NU = NF - ND + # import pdb; pdb.set_trace() + np.testing.assert_almost_equal(ad.aem2.gamma_nsmu(N, NF, sx), 0, decimal=4) + np.testing.assert_almost_equal(ad.aem2.gamma_nsmd(N, NF, sx), 0, decimal=4) From 00ea2eb728cdc3958da9c494ed3b0ed81c842125 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Niccol=C3=B2=20Laurenti?= Date: Tue, 5 Apr 2022 16:06:32 +0200 Subject: [PATCH 45/71] Test photon mom conservation --- src/eko/anomalous_dimensions/aem2.py | 6 ++++-- tests/eko/test_ad_aem2.py | 19 +++++++++++++++++-- tests/eko/test_ad_as1aem1.py | 13 +++++++++---- 3 files changed, 30 insertions(+), 8 deletions(-) diff --git a/src/eko/anomalous_dimensions/aem2.py b/src/eko/anomalous_dimensions/aem2.py index 120354ee9..20521c2f5 100644 --- a/src/eko/anomalous_dimensions/aem2.py +++ b/src/eko/anomalous_dimensions/aem2.py @@ -33,8 +33,10 @@ def gamma_phph(N, nf): nu = as1aem1.uplike_flavors(nf) nd = nf - nu - return (nu * constants.eu2**2 + nd * constants.ed2**2) * ( - as1aem1.gamma_gph(N) / constants.CF / constants.CA + 4 + return ( + constants.NC + * (nu * constants.eu2**2 + nd * constants.ed2**2) + * (as1aem1.gamma_gph(N) / constants.CF / constants.CA + 4) ) diff --git a/tests/eko/test_ad_aem2.py b/tests/eko/test_ad_aem2.py index f2d4aab6d..3a5d27014 100644 --- a/tests/eko/test_ad_aem2.py +++ b/tests/eko/test_ad_aem2.py @@ -11,8 +11,23 @@ def test_number_conservation(): # number N = complex(1.0, 0.0) sx = get_sx(N) - for NF, ND in ((5, 3), (6, 3)): + for NF in range(2, 6 + 1): # NU = NF - ND - # import pdb; pdb.set_trace() np.testing.assert_almost_equal(ad.aem2.gamma_nsmu(N, NF, sx), 0, decimal=4) np.testing.assert_almost_equal(ad.aem2.gamma_nsmd(N, NF, sx), 0, decimal=4) + + +def test_photon_momentum_conservation(): + # photon momentum + N = complex(2.0, 0.0) + sx = get_sx(N) + for NF in range(2, 6 + 1): + NU = ad.as1aem1.uplike_flavors(NF) + ND = NF - NU + # import pdb; pdb.set_trace() + np.testing.assert_almost_equal( + constants.eu2 * ad.aem2.gamma_uph(N, NU, sx) + + constants.ed2 * ad.aem2.gamma_dph(N, ND, sx) + + ad.aem2.gamma_phph(N, NF), + 0, + ) diff --git a/tests/eko/test_ad_as1aem1.py b/tests/eko/test_ad_as1aem1.py index 168e2421c..a86cefb21 100644 --- a/tests/eko/test_ad_as1aem1.py +++ b/tests/eko/test_ad_as1aem1.py @@ -1,6 +1,7 @@ # -*- coding: utf-8 -*- # Test O(as1aem1) splitting functions import numpy as np +import pytest from test_ad_nnlo import get_sx from eko import anomalous_dimensions as ad @@ -18,8 +19,9 @@ def test_gluon_momentum_conservation(): # gluon momentum N = complex(2.0, 0.0) sx = get_sx(N) - for NF, ND in ((5, 3), (6, 3)): - NU = NF - ND + for NF in range(2, 6 + 1): + NU = ad.as1aem1.uplike_flavors(NF) + ND = NF - NU np.testing.assert_almost_equal( constants.eu2 * ad.as1aem1.gamma_qg(N, NU, sx) + constants.ed2 * ad.as1aem1.gamma_qg(N, ND, sx) @@ -27,14 +29,17 @@ def test_gluon_momentum_conservation(): + (NU * constants.eu2 + ND * constants.ed2) * ad.as1aem1.gamma_gg(), 0, ) + with pytest.raises(NotImplementedError): + ad.as1aem1.uplike_flavors(7) def test_photon_momentum_conservation(): # photon momentum N = complex(2.0, 0.0) sx = get_sx(N) - for NF, ND in ((5, 3), (6, 3)): - NU = NF - ND + for NF in range(2, 6 + 1): + NU = ad.as1aem1.uplike_flavors(NF) + ND = NF - NU np.testing.assert_almost_equal( constants.eu2 * ad.as1aem1.gamma_qph(N, NU, sx) + constants.ed2 * ad.as1aem1.gamma_qph(N, ND, sx) From 44a1ec556e659ccbdb54bf1242cb61c0703f866f Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Niccol=C3=B2=20Laurenti?= Date: Tue, 5 Apr 2022 17:32:31 +0200 Subject: [PATCH 46/71] Test quark mom conservation at aem2 --- src/eko/anomalous_dimensions/aem2.py | 6 +++--- tests/eko/test_ad_aem2.py | 27 ++++++++++++++++++++++++++- 2 files changed, 29 insertions(+), 4 deletions(-) diff --git a/src/eko/anomalous_dimensions/aem2.py b/src/eko/anomalous_dimensions/aem2.py index 20521c2f5..5c11032b5 100644 --- a/src/eko/anomalous_dimensions/aem2.py +++ b/src/eko/anomalous_dimensions/aem2.py @@ -188,7 +188,7 @@ def gamma_nspu(N, nf, sx): eSigma2 = constants.NC * (nu * constants.eu2 + nd * constants.ed2) tmp = ( 2 - * (-12 + 20 * N + 107 * N**2 + 126 * N**3 + 63 * N**4) + * (-12 + 20 * N + 47 * N**2 + 6 * N**3 + 3 * N**4) / (9.0 * N**2 * (1 + N) ** 2) - 80 / 9 * S1 + 16 / 3 * S2 @@ -226,7 +226,7 @@ def gamma_nspd(N, nf, sx): eSigma2 = constants.NC * (nu * constants.eu2 + nd * constants.ed2) tmp = ( 2 - * (-12 + 20 * N + 107 * N**2 + 126 * N**3 + 63 * N**4) + * (-12 + 20 * N + 47 * N**2 + 6 * N**3 + 3 * N**4) / (9.0 * N**2 * (1 + N) ** 2) - 80 / 9 * S1 + 16 / 3 * S2 @@ -337,4 +337,4 @@ def gamma_ps(N, nf): * (4 + N * (4 + N * (7 + 5 * N))) / ((-1 + N) * N**3 * (1 + N) ** 3 * (2 + N) ** 2) ) - return 2 * nf * result + return 2 * nf * constants.CA * result diff --git a/tests/eko/test_ad_aem2.py b/tests/eko/test_ad_aem2.py index 3a5d27014..ced16b5c0 100644 --- a/tests/eko/test_ad_aem2.py +++ b/tests/eko/test_ad_aem2.py @@ -24,10 +24,35 @@ def test_photon_momentum_conservation(): for NF in range(2, 6 + 1): NU = ad.as1aem1.uplike_flavors(NF) ND = NF - NU - # import pdb; pdb.set_trace() np.testing.assert_almost_equal( constants.eu2 * ad.aem2.gamma_uph(N, NU, sx) + constants.ed2 * ad.aem2.gamma_dph(N, ND, sx) + ad.aem2.gamma_phph(N, NF), 0, ) + + +def test_quark_momentum_conservation(): + # quark momentum + N = complex(2.0, 0.0) + sx = get_sx(N) + NF = 6 + NU = ad.as1aem1.uplike_flavors(NF) + ND = NF - NU + # import pdb; pdb.set_trace() + np.testing.assert_almost_equal( + ad.aem2.gamma_nspu(N, NF, sx) + + constants.eu2 * ad.aem2.gamma_ps(N, NU) + + constants.ed2 * ad.aem2.gamma_ps(N, ND) + + ad.aem2.gamma_phu(N, NF, sx), + 0, + decimal=4, + ) + np.testing.assert_almost_equal( + ad.aem2.gamma_nspd(N, NF, sx) + + constants.eu2 * ad.aem2.gamma_ps(N, NU) + + constants.ed2 * ad.aem2.gamma_ps(N, ND) + + ad.aem2.gamma_phd(N, NF, sx), + 0, + decimal=4, + ) From 50480209b2fe6a3928b8fe699409e4885dd30b93 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Niccol=C3=B2=20Laurenti?= Date: Tue, 5 Apr 2022 22:07:22 +0200 Subject: [PATCH 47/71] Move uplike_flavors to constants.py --- src/eko/anomalous_dimensions/as1aem1.py | 31 +------------------------ src/eko/constants.py | 31 +++++++++++++++++++++++++ 2 files changed, 32 insertions(+), 30 deletions(-) diff --git a/src/eko/anomalous_dimensions/as1aem1.py b/src/eko/anomalous_dimensions/as1aem1.py index 4abf11d5a..5f4b5469d 100644 --- a/src/eko/anomalous_dimensions/as1aem1.py +++ b/src/eko/anomalous_dimensions/as1aem1.py @@ -14,35 +14,6 @@ from . import harmonics -@nb.njit("u1(u1)", cache=True) -def uplike_flavors(nf): - """ - Computes the number of up flavors - - Parameters - ---------- - nf : int - Number of active flavors - - Returns - ------- - nu : int - """ - if nf == 2: - nu = 1 - elif nf == 3: - nu = 1 - elif nf == 4: - nu = 2 - elif nf == 5: - nu = 2 - elif nf == 6: - nu = 3 - else: - raise NotImplementedError("Selected nf is not implemented") - return nu - - @nb.njit("c16(c16,c16[:])", cache=True) def gamma_phq(N, sx): """ @@ -245,7 +216,7 @@ def gamma_phph(nf): :math:`\\gamma_{\\gamma \\gamma}^{(1,1)}(N)` """ - nu = uplike_flavors(nf) + nu = constants.uplike_flavors(nf) nd = nf - nu return 4 * constants.CF * constants.CA * (nu * constants.eu2 + nd * constants.ed2) diff --git a/src/eko/constants.py b/src/eko/constants.py index a9bb89027..723a2c585 100644 --- a/src/eko/constants.py +++ b/src/eko/constants.py @@ -4,6 +4,8 @@ """ +import numba as nb + NC = 3 """the number of colors""" @@ -37,3 +39,32 @@ def update_colors(nc): NC = int(nc) CA = float(NC) CF = float((NC * NC - 1.0) / (2.0 * NC)) + + +@nb.njit("u1(u1)", cache=True) +def uplike_flavors(nf): + """ + Computes the number of up flavors + + Parameters + ---------- + nf : int + Number of active flavors + + Returns + ------- + nu : int + """ + if nf == 2: + nu = 1 + elif nf == 3: + nu = 1 + elif nf == 4: + nu = 2 + elif nf == 5: + nu = 2 + elif nf == 6: + nu = 3 + else: + raise NotImplementedError("Selected nf is not implemented") + return nu From 0d00df6e41555dc44828ab4d73cd45cb9a1a9a5d Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Niccol=C3=B2=20Laurenti?= Date: Tue, 5 Apr 2022 22:11:13 +0200 Subject: [PATCH 48/71] Rename test_ad_lo.py -> test_ad_as1.py ecc --- tests/eko/{test_ad_lo.py => test_ad_as1.py} | 27 ------------------- tests/eko/{test_ad_nlo.py => test_ad_as2.py} | 0 tests/eko/{test_ad_nnlo.py => test_ad_as3.py} | 0 3 files changed, 27 deletions(-) rename tests/eko/{test_ad_lo.py => test_ad_as1.py} (64%) rename tests/eko/{test_ad_nlo.py => test_ad_as2.py} (100%) rename tests/eko/{test_ad_nnlo.py => test_ad_as3.py} (100%) diff --git a/tests/eko/test_ad_lo.py b/tests/eko/test_ad_as1.py similarity index 64% rename from tests/eko/test_ad_lo.py rename to tests/eko/test_ad_as1.py index d08e79e32..e93fa62dc 100644 --- a/tests/eko/test_ad_lo.py +++ b/tests/eko/test_ad_as1.py @@ -24,10 +24,6 @@ def test_quark_momentum_conservation(): ad_as1.gamma_ns(N, s1) + ad_as1.gamma_gq(N), 0, ) - np.testing.assert_almost_equal( - ad_aem1.gamma_ns(N, s1) + ad_aem1.gamma_phq(N), - 0, - ) def test_gluon_momentum_conservation(): @@ -39,12 +35,6 @@ def test_gluon_momentum_conservation(): ) -def test_photon_momentum_conservation(): - # gluon momentum - N = complex(2.0, 0.0) - np.testing.assert_almost_equal(ad_aem1.gamma_qph(N, NF) + ad_aem1.gamma_phph(NF), 0) - - def test_gamma_qg_0(): N = complex(1.0, 0.0) res = complex(-20.0 / 3.0, 0.0) @@ -62,20 +52,3 @@ def test_gamma_gg_0(): s1 = harmonics.harmonic_S1(N) res = complex(5.195725159621, 10.52008856962) np.testing.assert_almost_equal(ad_as1.gamma_gg(N, s1, NF), res) - - -def test_gamma_phq_0(): - N = complex(0.0, 1.0) - res = complex(4.0, -4.0) / 3.0 / 4 * 3 - np.testing.assert_almost_equal(ad_aem1.gamma_phq(N), res) - - -def test_gamma_qph_0(): - N = complex(1.0, 0.0) - res = complex(-20.0 / 3.0, 0.0) * 3 / 0.5 - np.testing.assert_almost_equal(ad_aem1.gamma_qph(N, NF), res) - - -def test_gamma_phph_0(): - res = complex(2.0 / 3 * 3 * 2 * NF, 0.0) - np.testing.assert_almost_equal(ad_aem1.gamma_phph(NF), res) diff --git a/tests/eko/test_ad_nlo.py b/tests/eko/test_ad_as2.py similarity index 100% rename from tests/eko/test_ad_nlo.py rename to tests/eko/test_ad_as2.py diff --git a/tests/eko/test_ad_nnlo.py b/tests/eko/test_ad_as3.py similarity index 100% rename from tests/eko/test_ad_nnlo.py rename to tests/eko/test_ad_as3.py From 0f34629a0d46867269e39a297fbe82b35a7c21d5 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Niccol=C3=B2=20Laurenti?= Date: Tue, 5 Apr 2022 22:12:37 +0200 Subject: [PATCH 49/71] Move aem1 test from test_ad_lo to test_ad_aem1 --- tests/eko/test_ad_aem1.py | 32 ++++++++++++++++++++++++++++++++ 1 file changed, 32 insertions(+) create mode 100644 tests/eko/test_ad_aem1.py diff --git a/tests/eko/test_ad_aem1.py b/tests/eko/test_ad_aem1.py new file mode 100644 index 000000000..8ec5f5f3d --- /dev/null +++ b/tests/eko/test_ad_aem1.py @@ -0,0 +1,32 @@ +# -*- coding: utf-8 -*- +# Test LO splitting functions +import numpy as np + +from eko import anomalous_dimensions as ad +from eko import constants + + +def test_number_conservation(): + # number + N = complex(1.0, 0.0) + s1 = ad.harmonics.harmonic_S1(N) + np.testing.assert_almost_equal(ad.aem1.gamma_ns(N, s1), 0) + + +def test_quark_momentum_conservation(): + # quark momentum + N = complex(2.0, 0.0) + s1 = ad.harmonics.harmonic_S1(N) + np.testing.assert_almost_equal( + ad.aem1.gamma_ns(N, s1) + ad.aem1.gamma_phq(N), + 0, + ) + + +def test_photon_momentum_conservation(): + # photon momentum + N = complex(2.0, 0.0) + for NF in range(2, 6 + 1): + np.testing.assert_almost_equal( + ad.aem1.gamma_qph(N, NF) + ad.aem1.gamma_phph(NF), 0 + ) From 600022afb99d84d6bf4617821fd3cdc2e8be667f Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Niccol=C3=B2=20Laurenti?= Date: Tue, 5 Apr 2022 22:14:44 +0200 Subject: [PATCH 50/71] Fix scope of uplike_flavors and test_ad_nnlo --- extras/uni-ad.nb | 497 +++++++++++++++++++++------ src/eko/anomalous_dimensions/aem2.py | 14 +- tests/eko/test_ad_aem2.py | 6 +- tests/eko/test_ad_as1aem1.py | 8 +- 4 files changed, 411 insertions(+), 114 deletions(-) diff --git a/extras/uni-ad.nb 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constants.uplike_flavors(nf) nd = nf - nu return ( constants.NC @@ -112,7 +112,7 @@ def gamma_phu(N, nf, sx): O(as1aem1) photon-quark anomalous dimension :math:`\\gamma_{\\gamma q}^{(1,1)}(N)` """ - nu = as1aem1.uplike_flavors(nf) + nu = constants.uplike_flavors(nf) nd = nf - nu S1 = sx[0] tmp = (-16 * (-16 - 27 * N - 13 * N**2 - 8 * N**3)) / ( @@ -146,7 +146,7 @@ def gamma_phd(N, nf, sx): O(as1aem1) photon-quark anomalous dimension :math:`\\gamma_{\\gamma q}^{(1,1)}(N)` """ - nu = as1aem1.uplike_flavors(nf) + nu = constants.uplike_flavors(nf) nd = nf - nu S1 = sx[0] tmp = (-16 * (-16 - 27 * N - 13 * N**2 - 8 * N**3)) / ( @@ -183,7 +183,7 @@ def gamma_nspu(N, nf, sx): S1 = sx[0] S2 = sx[1] - nu = as1aem1.uplike_flavors(nf) + nu = constants.uplike_flavors(nf) nd = nf - nu eSigma2 = constants.NC * (nu * constants.eu2 + nd * constants.ed2) tmp = ( @@ -221,7 +221,7 @@ def gamma_nspd(N, nf, sx): S1 = sx[0] S2 = sx[1] - nu = as1aem1.uplike_flavors(nf) + nu = constants.uplike_flavors(nf) nd = nf - nu eSigma2 = constants.NC * (nu * constants.eu2 + nd * constants.ed2) tmp = ( @@ -259,7 +259,7 @@ def gamma_nsmu(N, nf, sx): S1 = sx[0] S2 = sx[1] - nu = as1aem1.uplike_flavors(nf) + nu = constants.uplike_flavors(nf) nd = nf - nu eSigma2 = constants.NC * (nu * constants.eu2 + nd * constants.ed2) tmp = ( @@ -297,7 +297,7 @@ def gamma_nsmd(N, nf, sx): S1 = sx[0] S2 = sx[1] - nu = as1aem1.uplike_flavors(nf) + nu = constants.uplike_flavors(nf) nd = nf - nu eSigma2 = constants.NC * (nu * constants.eu2 + nd * constants.ed2) tmp = ( diff --git a/tests/eko/test_ad_aem2.py b/tests/eko/test_ad_aem2.py index ced16b5c0..71f858496 100644 --- a/tests/eko/test_ad_aem2.py +++ b/tests/eko/test_ad_aem2.py @@ -1,7 +1,7 @@ # -*- coding: utf-8 -*- # Test O(as1aem1) splitting functions import numpy as np -from test_ad_nnlo import get_sx +from test_ad_as3 import get_sx from eko import anomalous_dimensions as ad from eko import constants @@ -22,7 +22,7 @@ def test_photon_momentum_conservation(): N = complex(2.0, 0.0) sx = get_sx(N) for NF in range(2, 6 + 1): - NU = ad.as1aem1.uplike_flavors(NF) + NU = constants.uplike_flavors(NF) ND = NF - NU np.testing.assert_almost_equal( constants.eu2 * ad.aem2.gamma_uph(N, NU, sx) @@ -37,7 +37,7 @@ def test_quark_momentum_conservation(): N = complex(2.0, 0.0) sx = get_sx(N) NF = 6 - NU = ad.as1aem1.uplike_flavors(NF) + NU = constants.uplike_flavors(NF) ND = NF - NU # import pdb; pdb.set_trace() np.testing.assert_almost_equal( diff --git a/tests/eko/test_ad_as1aem1.py b/tests/eko/test_ad_as1aem1.py index a86cefb21..ddd486a89 100644 --- a/tests/eko/test_ad_as1aem1.py +++ b/tests/eko/test_ad_as1aem1.py @@ -2,7 +2,7 @@ # Test O(as1aem1) splitting functions import numpy as np import pytest -from test_ad_nnlo import get_sx +from test_ad_as3 import get_sx from eko import anomalous_dimensions as ad from eko import constants @@ -20,7 +20,7 @@ def test_gluon_momentum_conservation(): N = complex(2.0, 0.0) sx = get_sx(N) for NF in range(2, 6 + 1): - NU = ad.as1aem1.uplike_flavors(NF) + NU = constants.uplike_flavors(NF) ND = NF - NU np.testing.assert_almost_equal( constants.eu2 * ad.as1aem1.gamma_qg(N, NU, sx) @@ -30,7 +30,7 @@ def test_gluon_momentum_conservation(): 0, ) with pytest.raises(NotImplementedError): - ad.as1aem1.uplike_flavors(7) + constants.uplike_flavors(7) def test_photon_momentum_conservation(): @@ -38,7 +38,7 @@ def test_photon_momentum_conservation(): N = complex(2.0, 0.0) sx = get_sx(N) for NF in range(2, 6 + 1): - NU = ad.as1aem1.uplike_flavors(NF) + NU = constants.uplike_flavors(NF) ND = NF - NU np.testing.assert_almost_equal( constants.eu2 * ad.as1aem1.gamma_qph(N, NU, sx) From b07ec48b3366aa91cf3259ce1ce78214c8598e89 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Niccol=C3=B2=20Laurenti?= Date: Tue, 5 Apr 2022 22:36:53 +0200 Subject: [PATCH 51/71] Clean notebook --- extras/uni-ad.nb | 576 ++++++++++++++++++++--------------------------- 1 file changed, 239 insertions(+), 337 deletions(-) diff --git a/extras/uni-ad.nb 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| 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/src/eko/anomalous_dimensions/as1aem1.py b/src/eko/anomalous_dimensions/as1aem1.py index 5f4b5469d..e454fb50d 100644 --- a/src/eko/anomalous_dimensions/as1aem1.py +++ b/src/eko/anomalous_dimensions/as1aem1.py @@ -296,9 +296,9 @@ def gamma_nsp(N, sx): -32 + N * (-8 - 3 * N * (3 + N) * (3 + N**2) - 48 * (1 + N) ** 2 * zeta2) ) - + 32 * N**3 * (1 + N) ** 3 * (g3N + g3Np2) ) / (N**3 * (1 + N) ** 3) + + 32 * (g3N + g3Np2) + 4 * S3p1h - 16 * zeta3 ) @@ -358,9 +358,9 @@ def gamma_nsm(N, sx): - 3 * N * (8 + 3 * N * (3 + N) * (3 + N**2)) + 48 * N * (1 + N) ** 2 * zeta2 ) - - 96 * N**3 * (1 + N) ** 3 * (g3N + g3Np2) ) / (3.0 * N**3 * (1 + N) ** 3) + - 32 * (g3N + g3Np2) + 32.0 * zeta2 * S1p1h + 4 * S3h - 32 * S3 From b1ac4f720c89aa425e6740a380d7817c68ac4dca Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Niccol=C3=B2=20Laurenti?= Date: Wed, 6 Apr 2022 10:15:42 +0200 Subject: [PATCH 53/71] Change alpha -> alpha/4pi in AD documentation --- doc/source/theory/pQCD.rst | 5 +++-- 1 file changed, 3 insertions(+), 2 deletions(-) diff --git a/doc/source/theory/pQCD.rst b/doc/source/theory/pQCD.rst index 2611c9876..1751209bd 100644 --- a/doc/source/theory/pQCD.rst +++ b/doc/source/theory/pQCD.rst @@ -76,9 +76,10 @@ where :math:`\mathbf{\tilde{P}}` are the usual |QCD| splitting kernels defined i while :math:`\mathbf{\bar{P}}` are given by .. math :: - \mathbf{\bar{P}} = \alpha \mathbf{P}^{(0,1)} + \alpha_s \alpha \mathbf{P}^{(1,1)} + - \alpha^2 \mathbf{P}^{(0,2)} + \dots + \mathbf{\bar{P}} = a \mathbf{P}^{(0,1)} + a_s a \mathbf{P}^{(1,1)} + + a^2 \mathbf{P}^{(0,2)} + \dots +where :math:`a = \alpha/(4\pi)`. The expression of the pure |QED| and of the mixed |QED| :math:`\otimes` |QCD| splitting kernels are given in :cite:`deFlorian:2015ujt,deFlorian:2016gvk` From e0a553911a26814e55bb0f1a47d991b18e3a69cf Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Niccol=C3=B2=20Laurenti?= Date: Wed, 6 Apr 2022 11:45:48 +0200 Subject: [PATCH 54/71] Remove uni-dglap.pdf and move the rest in uni-dglap/ --- extras/uni-dglap.pdf | Bin 83536 -> 0 bytes extras/{ => uni-dglap}/uni-ad.nb | 0 extras/{ => uni-dglap}/uni-dglap-aem2.ipynb | 0 extras/{ => uni-dglap}/uni-dglap.ipynb | 0 extras/{ => uni-dglap}/uni-dglap.tex | 0 5 files changed, 0 insertions(+), 0 deletions(-) delete mode 100644 extras/uni-dglap.pdf rename extras/{ => uni-dglap}/uni-ad.nb (100%) rename extras/{ => uni-dglap}/uni-dglap-aem2.ipynb (100%) rename extras/{ => uni-dglap}/uni-dglap.ipynb (100%) rename extras/{ => uni-dglap}/uni-dglap.tex (100%) diff --git a/extras/uni-dglap.pdf b/extras/uni-dglap.pdf deleted file mode 100644 index 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to extras/uni-dglap/uni-ad.nb diff --git a/extras/uni-dglap-aem2.ipynb b/extras/uni-dglap/uni-dglap-aem2.ipynb similarity index 100% rename from extras/uni-dglap-aem2.ipynb rename to extras/uni-dglap/uni-dglap-aem2.ipynb diff --git a/extras/uni-dglap.ipynb b/extras/uni-dglap/uni-dglap.ipynb similarity index 100% rename from extras/uni-dglap.ipynb rename to extras/uni-dglap/uni-dglap.ipynb diff --git a/extras/uni-dglap.tex b/extras/uni-dglap/uni-dglap.tex similarity index 100% rename from extras/uni-dglap.tex rename to extras/uni-dglap/uni-dglap.tex From 3e7250a3cb9f0795dff66f0a4cff890da3134e4f Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Niccol=C3=B2=20Laurenti?= Date: Wed, 6 Apr 2022 11:50:04 +0200 Subject: [PATCH 55/71] Fix documentation of aem1.py as1aem1.py aem2.py --- src/eko/anomalous_dimensions/aem1.py | 4 ++++ src/eko/anomalous_dimensions/aem2.py | 4 ---- src/eko/anomalous_dimensions/as1aem1.py | 4 ---- 3 files changed, 4 insertions(+), 8 deletions(-) diff --git a/src/eko/anomalous_dimensions/aem1.py b/src/eko/anomalous_dimensions/aem1.py index de620655a..7ec1960fb 100644 --- a/src/eko/anomalous_dimensions/aem1.py +++ b/src/eko/anomalous_dimensions/aem1.py @@ -1,4 +1,8 @@ # -*- coding: utf-8 -*- +""" +This file contains the O(aem1) Altarelli-Parisi splitting kernels. +""" + import numba as nb from .. import constants diff --git a/src/eko/anomalous_dimensions/aem2.py b/src/eko/anomalous_dimensions/aem2.py index 04ae848ed..be53c6aef 100644 --- a/src/eko/anomalous_dimensions/aem2.py +++ b/src/eko/anomalous_dimensions/aem2.py @@ -1,10 +1,6 @@ # -*- coding: utf-8 -*- """ This file contains the O(aem2) Altarelli-Parisi splitting kernels. - -These expression have been obtained using the procedure described in the -`wiki `_ -involving ``FormGet`` :cite:`Hahn:2016ebn`. """ import numba as nb diff --git a/src/eko/anomalous_dimensions/as1aem1.py b/src/eko/anomalous_dimensions/as1aem1.py index e454fb50d..7b7e20661 100644 --- a/src/eko/anomalous_dimensions/as1aem1.py +++ b/src/eko/anomalous_dimensions/as1aem1.py @@ -1,10 +1,6 @@ # -*- coding: utf-8 -*- """ This file contains the O(as1aem1) Altarelli-Parisi splitting kernels. - -These expression have been obtained using the procedure described in the -`wiki `_ -involving ``FormGet`` :cite:`Hahn:2016ebn`. """ import numba as nb From 4d2bf8908d72396275525db81f2767d0ce7ee9e8 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Niccol=C3=B2=20Laurenti?= Date: Wed, 6 Apr 2022 12:01:00 +0200 Subject: [PATCH 56/71] Specify distinction between P and \tilde{P} --- extras/uni-dglap/uni-dglap.tex | 12 ++++++++++-- 1 file changed, 10 insertions(+), 2 deletions(-) diff --git a/extras/uni-dglap/uni-dglap.tex b/extras/uni-dglap/uni-dglap.tex index b0e9dd514..b782e89ac 100644 --- a/extras/uni-dglap/uni-dglap.tex +++ b/extras/uni-dglap/uni-dglap.tex @@ -6,7 +6,7 @@ \usepackage{amsfonts} %\usepackage[a4paper,top=3cm,bottom=3cm,left=3cm,right=3cm]{geometry} -\usepackage[a4paper,top=1cm,bottom=1cm,left=-0.2cm,right=1cm]{geometry} +\usepackage[a4paper,top=1cm,bottom=1cm,left=0.5cm,right=1cm]{geometry} \title{} @@ -40,11 +40,19 @@ T_8^u &=u^+ + c^+ - 2t^+ \\ V_8^u &=u^- + c^- - 2t^- \end{align*} +The unified splitting functions can be split as +\begin{equation*} +P^{uni}_{ij} = P_{ij}+\tilde{P}_{ij} +\end{equation*} +where $P_{ij}$ are the usual pure QCD splitting functions, while $\tilde{P}_{ij}$ contain the pure QED and the mixed QCD$\otimes$QED contributions, i.e.\ +\begin{equation*} +\tilde{P}_{ij} = \alpha P^{(0,1)}_{ij} +\alpha_s \alpha P^{(1,1)}_{ij} + \alpha^2 P^{(0,2)}_{ij} + \dots +\end{equation*} \begin{itemize} \item Singlet sector: \begin{equation*} -\hspace*{-0.4cm}\mu^2\frac{d}{d\mu^2} +\hspace*{-1.2cm}\mu^2\frac{d}{d\mu^2} \begin{pmatrix} g \\ \gamma \\ From 84dcf297139a282ca702fda457fd9a47a50a4851 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Niccol=C3=B2=20Laurenti?= Date: Wed, 6 Apr 2022 12:04:43 +0200 Subject: [PATCH 57/71] Fix documentation of gamma_nsmu and gamma_nsmd --- src/eko/anomalous_dimensions/aem2.py | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) diff --git a/src/eko/anomalous_dimensions/aem2.py b/src/eko/anomalous_dimensions/aem2.py index be53c6aef..075f94298 100644 --- a/src/eko/anomalous_dimensions/aem2.py +++ b/src/eko/anomalous_dimensions/aem2.py @@ -157,7 +157,7 @@ def gamma_phd(N, nf, sx): @nb.njit("c16(c16,u1,c16[:])", cache=True) def gamma_nspu(N, nf, sx): """ - Computes the O(aem2) singlet-like non-singlet anomalous dimension. + Computes the O(aem2) singlet-like non-singlet anomalous dimension for up quarks. Implements sum of Eqs. (57-58) of :cite:`deFlorian:2016gvk` for q=u. @@ -195,7 +195,7 @@ def gamma_nspu(N, nf, sx): @nb.njit("c16(c16,u1,c16[:])", cache=True) def gamma_nspd(N, nf, sx): """ - Computes the O(aem2) singlet-like non-singlet anomalous dimension. + Computes the O(aem2) singlet-like non-singlet anomalous dimension for down quarks. Implements sum of Eqs. (57-58) of :cite:`deFlorian:2016gvk` for q=d. @@ -233,7 +233,7 @@ def gamma_nspd(N, nf, sx): @nb.njit("c16(c16,u1,c16[:])", cache=True) def gamma_nsmu(N, nf, sx): """ - Computes the O(aem2) singlet-like non-singlet anomalous dimension. + Computes the O(aem2) valence-like non-singlet anomalous dimension for up quarks. Implements difference between Eqs. (57-58) of :cite:`deFlorian:2016gvk` for q=u. @@ -271,7 +271,7 @@ def gamma_nsmu(N, nf, sx): @nb.njit("c16(c16,u1,c16[:])", cache=True) def gamma_nsmd(N, nf, sx): """ - Computes the O(aem2) singlet-like non-singlet anomalous dimension. + Computes the O(aem2) valence-like non-singlet anomalous dimension for down quarks. Implements difference between Eqs. (57-58) of :cite:`deFlorian:2016gvk` for q=d. From bcdc7cb6067880fff785cbc83e376b8e27a82c66 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Niccol=C3=B2=20Laurenti?= Date: Wed, 6 Apr 2022 12:08:09 +0200 Subject: [PATCH 58/71] Add quark type in doc string for phu phd uph dph --- src/eko/anomalous_dimensions/aem2.py | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) diff --git a/src/eko/anomalous_dimensions/aem2.py b/src/eko/anomalous_dimensions/aem2.py index 075f94298..4b83eda64 100644 --- a/src/eko/anomalous_dimensions/aem2.py +++ b/src/eko/anomalous_dimensions/aem2.py @@ -39,7 +39,7 @@ def gamma_phph(N, nf): @nb.njit("c16(c16,u1,c16[:])", cache=True) def gamma_uph(N, nf, sx): """ - Computes the O(aem2) quark-photon anomalous dimension + Computes the O(aem2) quark-photon anomalous dimension for up quarks. Implements Eq. (55) of :cite:`deFlorian:2016gvk` for q=u. @@ -64,7 +64,7 @@ def gamma_uph(N, nf, sx): @nb.njit("c16(c16,u1,c16[:])", cache=True) def gamma_dph(N, nf, sx): """ - Computes the O(aem2) quark-photon anomalous dimension + Computes the O(aem2) quark-photon anomalous dimension for down quarks. Implements Eq. (55) of :cite:`deFlorian:2016gvk` for q=d. @@ -89,7 +89,7 @@ def gamma_dph(N, nf, sx): @nb.njit("c16(c16,u1,c16[:])", cache=True) def gamma_phu(N, nf, sx): """ - Computes the O(aem2) photon-quark anomalous dimension + Computes the O(aem2) photon-quark anomalous dimension for up quarks. Implements Eq. (56) of :cite:`deFlorian:2016gvk` for q=u. @@ -123,7 +123,7 @@ def gamma_phu(N, nf, sx): @nb.njit("c16(c16,u1,c16[:])", cache=True) def gamma_phd(N, nf, sx): """ - Computes theO(aem2) photon-quark anomalous dimension + Computes theO(aem2) photon-quark anomalous dimension for down quarks. Implements Eq. (56) of :cite:`deFlorian:2016gvk` for q=d. From bea06e44d8e4e832cddbd3bd5a2b2bb89d44dd41 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Niccol=C3=B2=20Laurenti?= Date: Wed, 6 Apr 2022 12:19:31 +0200 Subject: [PATCH 59/71] Fix return part in doc string --- src/eko/anomalous_dimensions/aem2.py | 44 ++++++++++++------------- src/eko/anomalous_dimensions/as1aem1.py | 2 +- 2 files changed, 23 insertions(+), 23 deletions(-) diff --git a/src/eko/anomalous_dimensions/aem2.py b/src/eko/anomalous_dimensions/aem2.py index 4b83eda64..9c35d947e 100644 --- a/src/eko/anomalous_dimensions/aem2.py +++ b/src/eko/anomalous_dimensions/aem2.py @@ -23,8 +23,8 @@ def gamma_phph(N, nf): Returns ------- gamma_gg : complex - O(as1aem1) photon-photon singlet anomalous dimension - :math:`\\gamma_{\\gamma \\gamma}^{(1,1)}(N)` + O(aem2) photon-photon singlet anomalous dimension + :math:`\\gamma_{\\gamma \\gamma}^{(0,2)}(N)` """ nu = constants.uplike_flavors(nf) @@ -54,8 +54,8 @@ def gamma_uph(N, nf, sx): Returns ------- - gamma_qph : complex - O(as1aem1) quark-photon anomalous dimension :math:`\\gamma_{q \\gamma}^{(1,1)}(N)` + gamma_uph : complex + O(aem2) quark-photon anomalous dimension :math:`\\gamma_{u \\gamma}^{(0,2)}(N)` """ return constants.eu2 * as1aem1.gamma_qph(N, nf, sx) / constants.CF @@ -79,8 +79,8 @@ def gamma_dph(N, nf, sx): Returns ------- - gamma_qph : complex - O(as1aem1) quark-photon anomalous dimension :math:`\\gamma_{q \\gamma}^{(1,1)}(N)` + gamma_dph : complex + O(aem2) quark-photon anomalous dimension :math:`\\gamma_{d \\gamma}^{(0,2)}(N)` """ return constants.ed2 * as1aem1.gamma_qph(N, nf, sx) / constants.CF @@ -104,8 +104,8 @@ def gamma_phu(N, nf, sx): Returns ------- - gamma_phq : complex - O(as1aem1) photon-quark anomalous dimension :math:`\\gamma_{\\gamma q}^{(1,1)}(N)` + gamma_phu : complex + O(aem2) photon-quark anomalous dimension :math:`\\gamma_{\\gamma u}^{(0,2)}(N)` """ nu = constants.uplike_flavors(nf) @@ -138,8 +138,8 @@ def gamma_phd(N, nf, sx): Returns ------- - gamma_phq : complex - O(as1aem1) photon-quark anomalous dimension :math:`\\gamma_{\\gamma q}^{(1,1)}(N)` + gamma_phd : complex + O(aem2) photon-quark anomalous dimension :math:`\\gamma_{\\gamma d}^{(0,2)}(N)` """ nu = constants.uplike_flavors(nf) @@ -172,9 +172,9 @@ def gamma_nspu(N, nf, sx): Returns ------- - gamma_nsp : complex - O(as1aem1) singlet-like non-singlet anomalous dimension - :math:`\\gamma_{ns,+}^{(1)}(N)` + gamma_nspu : complex + O(aem2) singlet-like non-singlet anomalous dimension + :math:`\\gamma_{ns,+,u}^{(0,2)}(N)` """ S1 = sx[0] @@ -210,9 +210,9 @@ def gamma_nspd(N, nf, sx): Returns ------- - gamma_nsp : complex - O(as1aem1) singlet-like non-singlet anomalous dimension - :math:`\\gamma_{ns,+}^{(1)}(N)` + gamma_nspd : complex + O(aem2) singlet-like non-singlet anomalous dimension + :math:`\\gamma_{ns,+,d}^{(0,2)}(N)` """ S1 = sx[0] @@ -249,8 +249,8 @@ def gamma_nsmu(N, nf, sx): Returns ------- gamma_nsp : complex - O(as1aem1) singlet-like non-singlet anomalous dimension - :math:`\\gamma_{ns,+}^{(1)}(N)` + O(aem2) valence-like non-singlet anomalous dimension + :math:`\\gamma_{ns,-,u}^{(0,2)}(N)` """ S1 = sx[0] @@ -287,8 +287,8 @@ def gamma_nsmd(N, nf, sx): Returns ------- gamma_nsp : complex - O(as1aem1) singlet-like non-singlet anomalous dimension - :math:`\\gamma_{ns,+}^{(1)}(N)` + O(aem2) valence-like non-singlet anomalous dimension + :math:`\\gamma_{ns,-,d}^{(0,2)}(N)` """ S1 = sx[0] @@ -323,8 +323,8 @@ def gamma_ps(N, nf): Returns ------- gamma_ps : complex - |NLO| pure-singlet quark-quark anomalous dimension - :math:`\\gamma_{ps}^{(1)}(N)` + O(aem2) pure-singlet quark-quark anomalous dimension + :math:`\\gamma_{ps}^{(0,2)}(N)` """ result = ( diff --git a/src/eko/anomalous_dimensions/as1aem1.py b/src/eko/anomalous_dimensions/as1aem1.py index 7b7e20661..5cc10fe57 100644 --- a/src/eko/anomalous_dimensions/as1aem1.py +++ b/src/eko/anomalous_dimensions/as1aem1.py @@ -304,7 +304,7 @@ def gamma_nsp(N, sx): @nb.njit("c16(c16,c16[:])", cache=True) def gamma_nsm(N, sx): """ - Computes the O(as1aem1) singlet-like non-singlet anomalous dimension. + Computes the O(as1aem1) valence-like non-singlet anomalous dimension. Implements difference between Eqs. (33-34) of :cite:`deFlorian:2015ujt`. From 89425e29382b49fd9710494c9859aebdc091045d Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Niccol=C3=B2=20Laurenti?= Date: Wed, 6 Apr 2022 12:40:17 +0200 Subject: [PATCH 60/71] Fix docstring layout --- src/eko/anomalous_dimensions/aem2.py | 49 ++++++++++--------------- src/eko/anomalous_dimensions/as1aem1.py | 47 ++++++++++-------------- 2 files changed, 40 insertions(+), 56 deletions(-) diff --git a/src/eko/anomalous_dimensions/aem2.py b/src/eko/anomalous_dimensions/aem2.py index 9c35d947e..20e2b8dad 100644 --- a/src/eko/anomalous_dimensions/aem2.py +++ b/src/eko/anomalous_dimensions/aem2.py @@ -12,8 +12,7 @@ @nb.njit("c16(c16,u1)", cache=True) def gamma_phph(N, nf): - """ - Computes the O(aem2) photon-photon singlet anomalous dimension. + """Computes the O(aem2) photon-photon singlet anomalous dimension. Implements Eq. (68) of :cite:`deFlorian:2016gvk`. @@ -25,6 +24,7 @@ def gamma_phph(N, nf): gamma_gg : complex O(aem2) photon-photon singlet anomalous dimension :math:`\\gamma_{\\gamma \\gamma}^{(0,2)}(N)` + """ nu = constants.uplike_flavors(nf) @@ -38,8 +38,7 @@ def gamma_phph(N, nf): @nb.njit("c16(c16,u1,c16[:])", cache=True) def gamma_uph(N, nf, sx): - """ - Computes the O(aem2) quark-photon anomalous dimension for up quarks. + """Computes the O(aem2) quark-photon anomalous dimension for up quarks. Implements Eq. (55) of :cite:`deFlorian:2016gvk` for q=u. @@ -56,15 +55,14 @@ def gamma_uph(N, nf, sx): ------- gamma_uph : complex O(aem2) quark-photon anomalous dimension :math:`\\gamma_{u \\gamma}^{(0,2)}(N)` - """ + """ return constants.eu2 * as1aem1.gamma_qph(N, nf, sx) / constants.CF @nb.njit("c16(c16,u1,c16[:])", cache=True) def gamma_dph(N, nf, sx): - """ - Computes the O(aem2) quark-photon anomalous dimension for down quarks. + """Computes the O(aem2) quark-photon anomalous dimension for down quarks. Implements Eq. (55) of :cite:`deFlorian:2016gvk` for q=d. @@ -81,15 +79,14 @@ def gamma_dph(N, nf, sx): ------- gamma_dph : complex O(aem2) quark-photon anomalous dimension :math:`\\gamma_{d \\gamma}^{(0,2)}(N)` - """ + """ return constants.ed2 * as1aem1.gamma_qph(N, nf, sx) / constants.CF @nb.njit("c16(c16,u1,c16[:])", cache=True) def gamma_phu(N, nf, sx): - """ - Computes the O(aem2) photon-quark anomalous dimension for up quarks. + """Computes the O(aem2) photon-quark anomalous dimension for up quarks. Implements Eq. (56) of :cite:`deFlorian:2016gvk` for q=u. @@ -106,8 +103,8 @@ def gamma_phu(N, nf, sx): ------- gamma_phu : complex O(aem2) photon-quark anomalous dimension :math:`\\gamma_{\\gamma u}^{(0,2)}(N)` - """ + """ nu = constants.uplike_flavors(nf) nd = nf - nu S1 = sx[0] @@ -122,8 +119,7 @@ def gamma_phu(N, nf, sx): @nb.njit("c16(c16,u1,c16[:])", cache=True) def gamma_phd(N, nf, sx): - """ - Computes theO(aem2) photon-quark anomalous dimension for down quarks. + """Computes the O(aem2) photon-quark anomalous dimension for down quarks. Implements Eq. (56) of :cite:`deFlorian:2016gvk` for q=d. @@ -140,8 +136,8 @@ def gamma_phd(N, nf, sx): ------- gamma_phd : complex O(aem2) photon-quark anomalous dimension :math:`\\gamma_{\\gamma d}^{(0,2)}(N)` - """ + """ nu = constants.uplike_flavors(nf) nd = nf - nu S1 = sx[0] @@ -156,8 +152,7 @@ def gamma_phd(N, nf, sx): @nb.njit("c16(c16,u1,c16[:])", cache=True) def gamma_nspu(N, nf, sx): - """ - Computes the O(aem2) singlet-like non-singlet anomalous dimension for up quarks. + """Computes the O(aem2) singlet-like non-singlet anomalous dimension for up quarks. Implements sum of Eqs. (57-58) of :cite:`deFlorian:2016gvk` for q=u. @@ -175,8 +170,8 @@ def gamma_nspu(N, nf, sx): gamma_nspu : complex O(aem2) singlet-like non-singlet anomalous dimension :math:`\\gamma_{ns,+,u}^{(0,2)}(N)` - """ + """ S1 = sx[0] S2 = sx[1] nu = constants.uplike_flavors(nf) @@ -194,8 +189,7 @@ def gamma_nspu(N, nf, sx): @nb.njit("c16(c16,u1,c16[:])", cache=True) def gamma_nspd(N, nf, sx): - """ - Computes the O(aem2) singlet-like non-singlet anomalous dimension for down quarks. + """Computes the O(aem2) singlet-like non-singlet anomalous dimension for down quarks. Implements sum of Eqs. (57-58) of :cite:`deFlorian:2016gvk` for q=d. @@ -213,8 +207,8 @@ def gamma_nspd(N, nf, sx): gamma_nspd : complex O(aem2) singlet-like non-singlet anomalous dimension :math:`\\gamma_{ns,+,d}^{(0,2)}(N)` - """ + """ S1 = sx[0] S2 = sx[1] nu = constants.uplike_flavors(nf) @@ -232,8 +226,7 @@ def gamma_nspd(N, nf, sx): @nb.njit("c16(c16,u1,c16[:])", cache=True) def gamma_nsmu(N, nf, sx): - """ - Computes the O(aem2) valence-like non-singlet anomalous dimension for up quarks. + """Computes the O(aem2) valence-like non-singlet anomalous dimension for up quarks. Implements difference between Eqs. (57-58) of :cite:`deFlorian:2016gvk` for q=u. @@ -251,8 +244,8 @@ def gamma_nsmu(N, nf, sx): gamma_nsp : complex O(aem2) valence-like non-singlet anomalous dimension :math:`\\gamma_{ns,-,u}^{(0,2)}(N)` - """ + """ S1 = sx[0] S2 = sx[1] nu = constants.uplike_flavors(nf) @@ -270,8 +263,7 @@ def gamma_nsmu(N, nf, sx): @nb.njit("c16(c16,u1,c16[:])", cache=True) def gamma_nsmd(N, nf, sx): - """ - Computes the O(aem2) valence-like non-singlet anomalous dimension for down quarks. + """Computes the O(aem2) valence-like non-singlet anomalous dimension for down quarks. Implements difference between Eqs. (57-58) of :cite:`deFlorian:2016gvk` for q=d. @@ -289,8 +281,8 @@ def gamma_nsmd(N, nf, sx): gamma_nsp : complex O(aem2) valence-like non-singlet anomalous dimension :math:`\\gamma_{ns,-,d}^{(0,2)}(N)` - """ + """ S1 = sx[0] S2 = sx[1] nu = constants.uplike_flavors(nf) @@ -308,8 +300,7 @@ def gamma_nsmd(N, nf, sx): @nb.njit("c16(c16,u1)", cache=True) def gamma_ps(N, nf): - """ - Computes the O(aem2) pure-singlet quark-quark anomalous dimension. + """Computes the O(aem2) pure-singlet quark-quark anomalous dimension. Implements Eq. (59) of :cite:`deFlorian:2016gvk`. @@ -325,8 +316,8 @@ def gamma_ps(N, nf): gamma_ps : complex O(aem2) pure-singlet quark-quark anomalous dimension :math:`\\gamma_{ps}^{(0,2)}(N)` - """ + """ result = ( -4 * (2 + N * (5 + N)) diff --git a/src/eko/anomalous_dimensions/as1aem1.py b/src/eko/anomalous_dimensions/as1aem1.py index 5cc10fe57..d9dc92709 100644 --- a/src/eko/anomalous_dimensions/as1aem1.py +++ b/src/eko/anomalous_dimensions/as1aem1.py @@ -12,8 +12,7 @@ @nb.njit("c16(c16,c16[:])", cache=True) def gamma_phq(N, sx): - """ - Computes the O(as1aem1) photon-quark anomalous dimension + """Computes the O(as1aem1) photon-quark anomalous dimension Implements Eq. (36) of :cite:`deFlorian:2015ujt`. @@ -28,8 +27,8 @@ def gamma_phq(N, sx): ------- gamma_phq : complex O(as1aem1) photon-quark anomalous dimension :math:`\\gamma_{\\gamma q}^{(1,1)}(N)` - """ + """ S1 = sx[0] S2 = sx[1] tmp_const = ( @@ -50,8 +49,7 @@ def gamma_phq(N, sx): @nb.njit("c16(c16,u1,c16[:])", cache=True) def gamma_qph(N, nf, sx): - """ - Computes the O(as1aem1) quark-photon anomalous dimension + """Computes the O(as1aem1) quark-photon anomalous dimension Implements Eq. (26) of :cite:`deFlorian:2015ujt`. @@ -68,6 +66,7 @@ def gamma_qph(N, nf, sx): ------- gamma_qph : complex O(as1aem1) quark-photon anomalous dimension :math:`\\gamma_{q \\gamma}^{(1,1)}(N)` + """ S1 = sx[0] S2 = sx[1] @@ -98,8 +97,7 @@ def gamma_qph(N, nf, sx): @nb.njit("c16(c16)", cache=True) def gamma_gph(N): - """ - Computes the O(as1aem1) gluon-photon anomalous dimension + """Computes the O(as1aem1) gluon-photon anomalous dimension Implements Eq. (27) of :cite:`deFlorian:2015ujt`. @@ -112,8 +110,8 @@ def gamma_gph(N): ------- gamma_qph : complex O(as1aem1) gluon-photon anomalous dimension :math:`\\gamma_{g \\gamma}^{(1,1)}(N)` - """ + """ return ( constants.CF * constants.CA @@ -124,8 +122,7 @@ def gamma_gph(N): @nb.njit("c16(c16)", cache=True) def gamma_phg(N): - """ - Computes the O(as1aem1) photon-gluon anomalous dimension + """Computes the O(as1aem1) photon-gluon anomalous dimension Implements Eq. (30) of :cite:`deFlorian:2015ujt`. @@ -138,15 +135,14 @@ def gamma_phg(N): ------- gamma_qph : complex O(as1aem1) photon-gluon anomalous dimension :math:`\\gamma_{\\gamma g}^{(1,1)}(N)` - """ + """ return constants.TR / constants.CF / constants.CA * gamma_gph(N) @nb.njit("c16(c16,u1,c16[:])", cache=True) def gamma_qg(N, nf, sx): - """ - Computes the O(as1aem1) quark-gluon singlet anomalous dimension. + """Computes the O(as1aem1) quark-gluon singlet anomalous dimension. Implements Eq. (29) of :cite:`deFlorian:2015ujt`. @@ -164,15 +160,14 @@ def gamma_qg(N, nf, sx): gamma_qg : complex O(as1aem1) quark-gluon singlet anomalous dimension :math:`\\gamma_{qg}^{(1,1)}(N)` - """ + """ return constants.TR / constants.CF / constants.CA * gamma_qph(N, nf, sx) @nb.njit("c16(c16,c16[:])", cache=True) def gamma_gq(N, sx): - """ - Computes the O(as1aem1) gluon-quark singlet anomalous dimension. + """Computes the O(as1aem1) gluon-quark singlet anomalous dimension. Implements Eq. (35) of :cite:`deFlorian:2015ujt`. @@ -188,15 +183,14 @@ def gamma_gq(N, sx): gamma_gq : complex O(as1aem1) gluon-quark singlet anomalous dimension :math:`\\gamma_{gq}^{(1,1)}(N)` - """ + """ return gamma_phq(N, sx) @nb.njit("c16(u1)", cache=True) def gamma_phph(nf): - """ - Computes the O(as1aem1) photon-photon singlet anomalous dimension. + """Computes the O(as1aem1) photon-photon singlet anomalous dimension. Implements Eq. (28) of :cite:`deFlorian:2015ujt`. @@ -210,8 +204,8 @@ def gamma_phph(nf): gamma_gg : complex O(as1aem1) photon-photon singlet anomalous dimension :math:`\\gamma_{\\gamma \\gamma}^{(1,1)}(N)` - """ + """ nu = constants.uplike_flavors(nf) nd = nf - nu return 4 * constants.CF * constants.CA * (nu * constants.eu2 + nd * constants.ed2) @@ -219,8 +213,7 @@ def gamma_phph(nf): @nb.njit("c16()", cache=True) def gamma_gg(): - """ - Computes the O(as1aem1) gluon-gluon singlet anomalous dimension. + """Computes the O(as1aem1) gluon-gluon singlet anomalous dimension. Implements Eq. (31) of :cite:`deFlorian:2015ujt`. @@ -232,15 +225,14 @@ def gamma_gg(): gamma_gg : complex O(as1aem1) gluon-gluon singlet anomalous dimension :math:`\\gamma_{gg}^{(1,1)}(N)` - """ + """ return 4 * constants.TR @nb.njit("c16(c16,c16[:])", cache=True) def gamma_nsp(N, sx): - """ - Computes the O(as1aem1) singlet-like non-singlet anomalous dimension. + """Computes the O(as1aem1) singlet-like non-singlet anomalous dimension. Implements sum of Eqs. (33-34) of :cite:`deFlorian:2015ujt`. @@ -256,6 +248,7 @@ def gamma_nsp(N, sx): gamma_nsp : complex O(as1aem1) singlet-like non-singlet anomalous dimension :math:`\\gamma_{ns,+}^{(1)}(N)` + """ S1 = sx[0] S2 = sx[1] @@ -303,8 +296,7 @@ def gamma_nsp(N, sx): @nb.njit("c16(c16,c16[:])", cache=True) def gamma_nsm(N, sx): - """ - Computes the O(as1aem1) valence-like non-singlet anomalous dimension. + """Computes the O(as1aem1) valence-like non-singlet anomalous dimension. Implements difference between Eqs. (33-34) of :cite:`deFlorian:2015ujt`. @@ -320,6 +312,7 @@ def gamma_nsm(N, sx): gamma_nsm : complex O(as1aem1) singlet-like non-singlet anomalous dimension :math:`\\gamma_{ns,-}^{(1,1)}(N)` + """ S1 = sx[0] S2 = sx[1] From fcf7832683ab7abeab9e64e08488bd9502d0c32d Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Niccol=C3=B2=20Laurenti?= Date: Wed, 6 Apr 2022 12:42:01 +0200 Subject: [PATCH 61/71] Remove unecessary comments --- tests/eko/test_ad_aem2.py | 2 -- 1 file changed, 2 deletions(-) diff --git a/tests/eko/test_ad_aem2.py b/tests/eko/test_ad_aem2.py index 71f858496..f975578b2 100644 --- a/tests/eko/test_ad_aem2.py +++ b/tests/eko/test_ad_aem2.py @@ -12,7 +12,6 @@ def test_number_conservation(): N = complex(1.0, 0.0) sx = get_sx(N) for NF in range(2, 6 + 1): - # NU = NF - ND np.testing.assert_almost_equal(ad.aem2.gamma_nsmu(N, NF, sx), 0, decimal=4) np.testing.assert_almost_equal(ad.aem2.gamma_nsmd(N, NF, sx), 0, decimal=4) @@ -39,7 +38,6 @@ def test_quark_momentum_conservation(): NF = 6 NU = constants.uplike_flavors(NF) ND = NF - NU - # import pdb; pdb.set_trace() np.testing.assert_almost_equal( ad.aem2.gamma_nspu(N, NF, sx) + constants.eu2 * ad.aem2.gamma_ps(N, NU) From 8f3c04028948ad76fa3ca3df6b6679210f4fe25d Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Niccol=C3=B2=20Laurenti?= Date: Wed, 6 Apr 2022 15:18:02 +0200 Subject: [PATCH 62/71] Start section on sum rules --- doc/source/theory/pQCD.rst | 37 +++++++++++++++++++++++++++++++++++++ 1 file changed, 37 insertions(+) diff --git a/doc/source/theory/pQCD.rst b/doc/source/theory/pQCD.rst index 1751209bd..309e98676 100644 --- a/doc/source/theory/pQCD.rst +++ b/doc/source/theory/pQCD.rst @@ -83,6 +83,43 @@ where :math:`a = \alpha/(4\pi)`. The expression of the pure |QED| and of the mixed |QED| :math:`\otimes` |QCD| splitting kernels are given in :cite:`deFlorian:2015ujt,deFlorian:2016gvk` +Sum Rules +--------- + +The Altarelli-Parisi Splitting functions have to satisfy certain sum rules. In fact |QED|:math:`\otimes` |QCD| +interactions preserve fermion number, therefore + +.. math :: + \int_0^1dx P_{ns,q}^-(x)=0 + +Moreover, the conservation of the proton's momentum implies that + +.. math :: + \int_0^1dx x (2n_dP_{dg}(x)+2n_uP_{ug}(x)+P_{\gamma g}(x)+P_{gg}(x))=0 + +.. math :: + \int_0^1dx x (2n_dP_{d\gamma}(x)+2n_uP_{u\gamma}(x)+P_{\gamma \gamma}(x)+P_{g\gamma}(x))=0 + +.. math :: + \int_0^1dx x \Bigl(\sum_{i=1}^{n_f} P_{q_iq_j}(x)+P_{\gamma q_j}(x)+P_{gq_j}(x)\Bigr)=0 + +Using the definition of anomalous dimensions the sum rules are written as: + +.. math :: + \gamma_{ns}^-(N=1)=0 + +.. math :: + \bigl(2n_d\gamma_{dg}+2n_u\gamma_{ug}+\gamma_{\gamma g}+\gamma_{gg}\bigr)(N=2)=0 + +.. math :: + \bigl(2n_d \gamma_{d\gamma}+2n_u \gamma_{u\gamma}+ \gamma_{\gamma \gamma}+ \gamma_{g\gamma})(N=2)=0 + +.. math :: + \Bigl(\gamma_{ns,q}^+ +2n_u\gamma^S_{uq}+2n_d\gamma^S_{dq} + \gamma_{\gamma q}+\gamma_{gq}\Bigr)(N=2)=0 + +that must be satisfied order by order in perturbation theory. + + Scale Variations ---------------- From 7992ea799da8d0f7eea285ee35410ba5e6a7a602 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Niccol=C3=B2=20Laurenti?= Date: Wed, 6 Apr 2022 15:35:00 +0200 Subject: [PATCH 63/71] Fix quark mom conservation in documentation --- doc/source/theory/pQCD.rst | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/doc/source/theory/pQCD.rst b/doc/source/theory/pQCD.rst index 309e98676..02af16e87 100644 --- a/doc/source/theory/pQCD.rst +++ b/doc/source/theory/pQCD.rst @@ -86,7 +86,7 @@ The expression of the pure |QED| and of the mixed |QED| :math:`\otimes` |QCD| sp Sum Rules --------- -The Altarelli-Parisi Splitting functions have to satisfy certain sum rules. In fact |QED|:math:`\otimes` |QCD| +The Altarelli-Parisi Splitting functions have to satisfy certain sum rules. In fact |QED| :math:`\otimes` |QCD| interactions preserve fermion number, therefore .. math :: @@ -101,7 +101,7 @@ Moreover, the conservation of the proton's momentum implies that \int_0^1dx x (2n_dP_{d\gamma}(x)+2n_uP_{u\gamma}(x)+P_{\gamma \gamma}(x)+P_{g\gamma}(x))=0 .. math :: - \int_0^1dx x \Bigl(\sum_{i=1}^{n_f} P_{q_iq_j}(x)+P_{\gamma q_j}(x)+P_{gq_j}(x)\Bigr)=0 + \int_0^1dx x \Bigl(\sum_{q_i=q,\bar{q}} P_{q_iq_j}(x)+P_{\gamma q_j}(x)+P_{gq_j}(x)\Bigr)=0 Using the definition of anomalous dimensions the sum rules are written as: From f33e0201eb6fdf39958cbee6d8f7cf1acde6c86f Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Niccol=C3=B2=20Laurenti?= Date: Wed, 6 Apr 2022 15:44:38 +0200 Subject: [PATCH 64/71] Add short discussion about sum rules --- doc/source/theory/pQCD.rst | 4 ++++ 1 file changed, 4 insertions(+) diff --git a/doc/source/theory/pQCD.rst b/doc/source/theory/pQCD.rst index 02af16e87..115ac4abe 100644 --- a/doc/source/theory/pQCD.rst +++ b/doc/source/theory/pQCD.rst @@ -103,6 +103,10 @@ Moreover, the conservation of the proton's momentum implies that .. math :: \int_0^1dx x \Bigl(\sum_{q_i=q,\bar{q}} P_{q_iq_j}(x)+P_{\gamma q_j}(x)+P_{gq_j}(x)\Bigr)=0 +The reason why multiple conservation equations follow from a single conserved quantity (i.e. proton's momentum) +is that one is free to choose a border condition in which there is only one parton, e.g. the gluon, and the +momentum should be preserved. + Using the definition of anomalous dimensions the sum rules are written as: .. math :: From 070da1315d439b4d06c205fb20687470a8348625 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Niccol=C3=B2=20Laurenti?= Date: Wed, 6 Apr 2022 16:13:46 +0200 Subject: [PATCH 65/71] Add Makefile for compiling uni-dglap.tex --- extras/uni-dglap/Makefile | 10 ++++++++++ 1 file changed, 10 insertions(+) create mode 100644 extras/uni-dglap/Makefile diff --git a/extras/uni-dglap/Makefile b/extras/uni-dglap/Makefile new file mode 100644 index 000000000..08f4e8098 --- /dev/null +++ b/extras/uni-dglap/Makefile @@ -0,0 +1,10 @@ +FILE=uni-dglap.tex +SHELL=/bin/bash + +all: + pdflatex $(FILE) + +clean: + rm -f *.aux + rm -f *.log + rm -f *.pdf From aab9af44f0b1862241967576a710e8c246c69490 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Niccol=C3=B2=20Laurenti?= Date: Wed, 6 Apr 2022 16:24:21 +0200 Subject: [PATCH 66/71] Simplify function uplike_flavors --- src/eko/constants.py | 17 ++++------------- 1 file changed, 4 insertions(+), 13 deletions(-) diff --git a/src/eko/constants.py b/src/eko/constants.py index 723a2c585..d45b47f00 100644 --- a/src/eko/constants.py +++ b/src/eko/constants.py @@ -43,8 +43,7 @@ def update_colors(nc): @nb.njit("u1(u1)", cache=True) def uplike_flavors(nf): - """ - Computes the number of up flavors + """Computes the number of up flavors Parameters ---------- @@ -54,17 +53,9 @@ def uplike_flavors(nf): Returns ------- nu : int + """ - if nf == 2: - nu = 1 - elif nf == 3: - nu = 1 - elif nf == 4: - nu = 2 - elif nf == 5: - nu = 2 - elif nf == 6: - nu = 3 - else: + if nf not in range(2, 6 + 1): raise NotImplementedError("Selected nf is not implemented") + nu = nf // 2 return nu From 7e42c547e8739b483391e80a22bdff619b27eb4c Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Niccol=C3=B2=20Laurenti?= Date: Wed, 6 Apr 2022 16:41:00 +0200 Subject: [PATCH 67/71] Modify Makefile --- extras/uni-dglap/Makefile | 7 +++---- 1 file changed, 3 insertions(+), 4 deletions(-) diff --git a/extras/uni-dglap/Makefile b/extras/uni-dglap/Makefile index 08f4e8098..42eacb87e 100644 --- a/extras/uni-dglap/Makefile +++ b/extras/uni-dglap/Makefile @@ -1,8 +1,7 @@ -FILE=uni-dglap.tex -SHELL=/bin/bash +all: uni-dglap.pdf -all: - pdflatex $(FILE) +%.pdf: %.tex + pdflatex $< clean: rm -f *.aux From 2c01845c9b8e6904abebd2d8ec4815d0281f212a Mon Sep 17 00:00:00 2001 From: Alessandro Candido Date: Wed, 6 Apr 2022 18:17:55 +0200 Subject: [PATCH 68/71] Expand sum rules for operators description --- doc/source/theory/pQCD.rst | 10 +++++++--- 1 file changed, 7 insertions(+), 3 deletions(-) diff --git a/doc/source/theory/pQCD.rst b/doc/source/theory/pQCD.rst index 115ac4abe..93623d1cb 100644 --- a/doc/source/theory/pQCD.rst +++ b/doc/source/theory/pQCD.rst @@ -103,9 +103,13 @@ Moreover, the conservation of the proton's momentum implies that .. math :: \int_0^1dx x \Bigl(\sum_{q_i=q,\bar{q}} P_{q_iq_j}(x)+P_{\gamma q_j}(x)+P_{gq_j}(x)\Bigr)=0 -The reason why multiple conservation equations follow from a single conserved quantity (i.e. proton's momentum) -is that one is free to choose a border condition in which there is only one parton, e.g. the gluon, and the -momentum should be preserved. +The reason why multiple conservation equations follow from a single conserved +quantity (i.e. proton's momentum) is that one is free to choose a border +condition in which there is only one parton, e.g. the gluon, and the momentum +should be preserved. +This is just a simple way to consider that anomalous dimensions are actually +operators, and the conservation thus apply element by element in the first +dimension (summing over the second one only). Using the definition of anomalous dimensions the sum rules are written as: From 77037715ef74b6a9c7a191c2b0c434d843e2f9a4 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Niccol=C3=B2=20Laurenti?= Date: Thu, 7 Apr 2022 11:29:49 +0200 Subject: [PATCH 69/71] Modify .tex --- extras/uni-dglap/uni-dglap.tex | 79 +++++++++++++++++++++++++++++++--- 1 file changed, 72 insertions(+), 7 deletions(-) diff --git a/extras/uni-dglap/uni-dglap.tex b/extras/uni-dglap/uni-dglap.tex index b782e89ac..a5c82d077 100644 --- a/extras/uni-dglap/uni-dglap.tex +++ b/extras/uni-dglap/uni-dglap.tex @@ -1,4 +1,4 @@ -\documentclass[a4paper,twoside]{article} +\documentclass[a4paper,oneside]{article} \usepackage[utf8]{inputenc} \usepackage{xcolor} \usepackage{amsmath} @@ -6,7 +6,7 @@ \usepackage{amsfonts} %\usepackage[a4paper,top=3cm,bottom=3cm,left=3cm,right=3cm]{geometry} -\usepackage[a4paper,top=1cm,bottom=1cm,left=0.5cm,right=1cm]{geometry} +\usepackage[a4paper,top=1.5cm,bottom=1.5cm,left=1.5cm,right=1.5cm]{geometry} \title{} @@ -52,7 +52,7 @@ \item Singlet sector: \begin{equation*} -\hspace*{-1.2cm}\mu^2\frac{d}{d\mu^2} +\mu^2\frac{d}{d\mu^2} \begin{pmatrix} g \\ \gamma \\ @@ -63,8 +63,8 @@ \begin{pmatrix} P_{gg}+n_f \langle e^2\rangle \tilde{P}_{gg} & n_f \langle e^2\rangle \tilde{P}_{g\gamma} & P_{gq} + \langle e^2\rangle \tilde{P}_{gq} & \nu_ue^2_-\tilde{P}_{gq} \\ n_f \langle e^2\rangle \tilde{P}_{\gamma g} & n_f \langle e^2\rangle \tilde{P}_{\gamma \gamma} & \langle e^2\rangle \tilde{P}_{\gamma q} & \nu_ue^2_-\tilde{P}_{\gamma q} \\ - 2n_f (P_{qg} +\langle e^2\rangle \tilde{P}_{qg} )& 2 n_f \langle e^2\rangle \tilde{P}_{q\gamma} & P_{qq}+ \langle e^2\rangle \Bigl(\tilde{P}_{+} + \langle e^2\rangle(\tilde{P}_{qq}-\tilde{P}_{+})\Bigr) & \nu_ue^2_-\Bigl(\tilde{P}_{+}+ \langle e^2\rangle (\tilde{P}_{qq}-\tilde{P}_{+}) \Bigr)\\ - 2n_f \nu_d e^2_-\tilde{P}_{qg} & 2n_f \nu_d e^2_-\tilde{P}_{q\gamma} & \nu_de^2_-\Bigl( \tilde{P}_{+} + \langle e^2\rangle (\tilde{P}_{qq}-\tilde{P}_{+}) \Bigr)& P_+ +e_\Delta^2 \tilde{P}_{+} +\nu_u \nu_d (e^2_-)^2(\tilde{P}_{qq}-\tilde{P}_{+}) + 2n_f (P_{qg} +\langle e^2\rangle \tilde{P}_{qg} )& 2 n_f \langle e^2\rangle \tilde{P}_{q\gamma} & P_{qq}+ \langle e^2\rangle \Bigl(\tilde{P}_{+} + \langle e^2\rangle\tilde{P}_{ps}\Bigr)& \nu_ue^2_-\Bigl(\tilde{P}_{+}+ \langle e^2\rangle \tilde{P}_{ps} \Bigr)\\ + 2n_f \nu_d e^2_-\tilde{P}_{qg} & 2n_f \nu_d e^2_-\tilde{P}_{q\gamma} & \nu_de^2_-\Bigl( \tilde{P}_{+} + \langle e^2\rangle \tilde{P}_{ps} \Bigr)& P_+ +e_\Delta^2 \tilde{P}_{+} +\nu_u \nu_d (e^2_-)^2\tilde{P}_{ps} \end{pmatrix} \begin{pmatrix} g \\ @@ -101,9 +101,74 @@ \end{equation*} \item Decoupled sector: \begin{align*} -\mu^2\frac{d}{d\mu^2}T^{u/d}_{3/8} & = (P_{+} + e_i^2 \tilde{P}_{+}) T^{u/d}_{3/8} \\ -\mu^2\frac{d}{d\mu^2}V^{u/d}_{3/8} & = (P_{-} + e_i^2 \tilde{P}_{-} )V^{u/d}_{3/8} +\mu^2\frac{d}{d\mu^2}T^{u/d}_{3/8} & = (P_{+} + e_{u/d}^2 \tilde{P}_{+}) T^{u/d}_{3/8} \\ +\mu^2\frac{d}{d\mu^2}V^{u/d}_{3/8} & = (P_{-} + e_{u/d}^2 \tilde{P}_{-} )V^{u/d}_{3/8} +\end{align*} +\end{itemize} + + +Observe that starting from $\mathcal{O}(\alpha^2)$ it is no longer possible to write that $\tilde{P}_{u \gamma}=e_u^2 \tilde{P}_{q \gamma}$ where $\tilde{P}_{q \gamma}$ is independent of the flavor of the quark. For this reason even if we define the QED splitting function factorizing a factor of $e_q^2$, we are left with a dependence on the flavor of the quark, i.e.\ $\tilde{P}_{u \gamma} \rightarrow e_u^2 \tilde{P}_{u \gamma}$. Moreover, we can no longer factorize a term $n_f \langle e^2\rangle=n_u e_u^2+n_d e_d^2$ out of $\tilde{P}_{\gamma \gamma}$ since at $\mathcal{O}(\alpha^2)$ it is proportional to $n_f \langle e^4\rangle=n_u e_u^4+n_d e_d^4$. For this reason it is better to reinsert both $n_f \langle e^2\rangle$ and $n_f \langle e^4\rangle$ inside $\tilde{P}_{\gamma \gamma}^{(1,1)}$ and $\tilde{P}_{\gamma \gamma}^{(0,2)}$. +Therefore a more general expression of the DGLAP equations is: +\begin{itemize} +\item Singlet sector: +\begin{align*} +&\mu^2\frac{d}{d\mu^2} +\begin{pmatrix} +g \\ +\gamma \\ +\Sigma \\ +\Delta_\Sigma +\end{pmatrix} += \\ +&\begin{pmatrix} + P_{gg}+n_f \langle e^2\rangle \tilde{P}_{gg} & n_f \langle e^2\rangle \tilde{P}_{g\gamma} & P_{gq} + \langle e^2\rangle \tilde{P}_{gq} & \nu_ue^2_-\tilde{P}_{gq} \\ + n_f \langle e^2\rangle \tilde{P}_{\gamma g} & \tilde{P}_{\gamma \gamma} & \langle \tilde{P}_{\gamma q} \rangle& \nu_u \tilde{P}_{\gamma \Delta q} \\ + 2n_f (P_{qg} +\langle e^2\rangle \tilde{P}_{qg} )& 2 n_f \langle \tilde{P}_{q \gamma} \rangle& P_{qq}+ \langle \tilde{P}^+_{q} \rangle+ \langle e^2\rangle^2\tilde{P}_{ps}& \nu_u\tilde{P}^+_{ \Delta q}+ \nu_ue_-^2\langle e^2\rangle \tilde{P}_{ps}\\ + 2n_f \nu_d e^2_-\tilde{P}_{qg} & 2n_f \nu_d \tilde{P}_{\Delta q\gamma} & \nu_d\tilde{P}^+_{ \Delta q}+ \nu_d e^2_-\langle e^2\rangle \tilde{P}_{ps}& P_+ + \{ \tilde{P}^+_{q} \}+\nu_u \nu_d (e^2_-)^2\tilde{P}_{ps} +\end{pmatrix} +\begin{pmatrix} +g \\ +\gamma \\ +\Sigma \\ +\Delta_\Sigma +\end{pmatrix} \end{align*} +\item Valence sector: +\begin{equation*} +\mu^2\frac{d}{d\mu^2} +\begin{pmatrix} +V \\ +\Delta_V +\end{pmatrix} += +\begin{pmatrix} +P_V+\langle \tilde{P}^-_{q} \rangle & \nu_u\tilde{P}^-_{\Delta q}\\ + \nu_d\tilde{P}^-_{\Delta q}& P_-+\{ \tilde{P}^-_{q} \} +\end{pmatrix} +\begin{pmatrix} +V \\ +\Delta_V +\end{pmatrix} +\end{equation*} +\item Decoupled sector: +\begin{align*} +\mu^2\frac{d}{d\mu^2}T^{u/d}_{3/8} & = (P_{+} + e_{u/d}^2 \tilde{P}^{+}_{u/d}) T^{u/d}_{3/8} \\ +\mu^2\frac{d}{d\mu^2}V^{u/d}_{3/8} & = (P_{-} + e_{u/d}^2 \tilde{P}^{-}_{u/d} )V^{u/d}_{3/8} +\end{align*} \end{itemize} +with +\begin{align*} + \langle \tilde{P}_{\gamma q} \rangle &= \nu_u e_u^2 \tilde{P}_{\gamma u}+\nu_d e_d^2 \tilde{P}_{\gamma d} \\ + \langle \tilde{P}_{q \gamma} \rangle &= \nu_u e_u^2 \tilde{P}_{u \gamma}+\nu_d e_d^2 \tilde{P}_{d \gamma} \\ + \langle \tilde{P}^+_{q} \rangle &= \nu_u e_u^2 \tilde{P}^+_{u}+\nu_d e_d^2 \tilde{P}^+_{d} \\ + \langle \tilde{P}^-_{q} \rangle &= \nu_u e_u^2 \tilde{P}^-_{u}+\nu_d e_d^2 \tilde{P}^-_{d} \\ + \{ \tilde{P}^+_{q} \}&= \nu_d e_u^2 \tilde{P}^+_{u}+\nu_u e_d^2 \tilde{P}^+_{d} \\ %awful notation + \{ \tilde{P}^-_{q} \}&= \nu_d e_u^2 \tilde{P}^-_{u}+\nu_u e_d^2 \tilde{P}^-_{d} \\ %awful notation + \tilde{P}_{\gamma \Delta q}&=e_u^2 \tilde{P}_{\gamma u}-e_d^2 \tilde{P}_{\gamma d} \\ + \tilde{P}_{\Delta q\gamma}&=e_u^2 \tilde{P}_{u\gamma}-e_d^2 \tilde{P}_{d\gamma} \\ + \tilde{P}^+_{\Delta q}&=e_u^2 \tilde{P}^+_{ u}-e_d^2 \tilde{P}^+_{d} \\ + \tilde{P}^-_{\Delta q}&=e_u^2 \tilde{P}^-_{ u}-e_d^2 \tilde{P}^-_{d} +\end{align*} + \end{document} From 09ee5ec9485fe6a4fe561791c791bc72becb195b Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Niccol=C3=B2=20Laurenti?= Date: Thu, 7 Apr 2022 11:34:14 +0200 Subject: [PATCH 70/71] Clean uni-dglap-aem2.ipynb --- extras/uni-dglap/uni-dglap-aem2.ipynb | 132 +++++++++++++++----------- 1 file changed, 74 insertions(+), 58 deletions(-) diff --git a/extras/uni-dglap/uni-dglap-aem2.ipynb b/extras/uni-dglap/uni-dglap-aem2.ipynb index 9ddcabb8d..c6b2066e1 100644 --- a/extras/uni-dglap/uni-dglap-aem2.ipynb +++ b/extras/uni-dglap/uni-dglap-aem2.ipynb @@ -2,7 +2,7 @@ "cells": [ { "cell_type": "code", - "execution_count": 1, + "execution_count": 2, "id": "c093db3d", "metadata": {}, "outputs": [], @@ -12,7 +12,7 @@ }, { "cell_type": "code", - "execution_count": 14, + "execution_count": 3, "id": "66408b17", "metadata": {}, "outputs": [], @@ -28,7 +28,7 @@ }, { "cell_type": "code", - "execution_count": 16, + "execution_count": 4, "id": "558d64a5", "metadata": {}, "outputs": [], @@ -113,45 +113,7 @@ }, { "cell_type": "code", - "execution_count": 17, - "id": "9242fd63", - "metadata": {}, - "outputs": [ - { - "data": { - "text/latex": [ - "$\\displaystyle \\left[\\begin{array}{cccccccccccccc}P^{x}_{gg} \\left(3 e^{2}_{d} + 3 e^{2}_{u}\\right) & P^{x}_{g\\gamma} \\left(3 e^{2}_{d} + 3 e^{2}_{u}\\right) & P^{x}_{gq} e^{2}_{u} & 0 & P^{x}_{gq} e^{2}_{d} & 0 & P^{x}_{gq} e^{2}_{d} & 0 & P^{x}_{gq} e^{2}_{u} & 0 & P^{x}_{gq} e^{2}_{d} & 0 & P^{x}_{gq} e^{2}_{u} & 0\\\\P^{x}_{\\gamma g} \\left(3 e^{2}_{d} + 3 e^{2}_{u}\\right) & P^{x}_{\\gamma\\gamma} & P^{x}_{\\gamma u} e^{2}_{u} & 0 & P^{x}_{\\gamma d} e^{2}_{d} & 0 & P^{x}_{\\gamma d} e^{2}_{d} & 0 & P^{x}_{\\gamma u} e^{2}_{u} & 0 & P^{x}_{\\gamma d} e^{2}_{d} & 0 & P^{x}_{\\gamma u} e^{2}_{u} & 0\\\\2 P^{x}_{qg} e^{2}_{u} & 2 P^{x}_{u\\gamma} e^{2}_{u} & P^{x}_{+u} e^{2}_{u} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & P^{x}_{-u} e^{2}_{u} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\2 P^{x}_{qg} e^{2}_{d} & 2 P^{x}_{d\\gamma} e^{2}_{d} & 0 & 0 & P^{x}_{+d} e^{2}_{d} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & P^{x}_{-d} e^{2}_{d} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\2 P^{x}_{qg} e^{2}_{d} & 2 P^{x}_{d\\gamma} e^{2}_{d} & 0 & 0 & 0 & 0 & P^{x}_{+d} e^{2}_{d} & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & P^{x}_{-d} e^{2}_{d} & 0 & 0 & 0 & 0 & 0 & 0\\\\2 P^{x}_{qg} e^{2}_{u} & 2 P^{x}_{u\\gamma} e^{2}_{u} & 0 & 0 & 0 & 0 & 0 & 0 & P^{x}_{+u} e^{2}_{u} & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & P^{x}_{-u} e^{2}_{u} & 0 & 0 & 0 & 0\\\\2 P^{x}_{qg} e^{2}_{d} & 2 P^{x}_{d\\gamma} e^{2}_{d} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & P^{x}_{+d} e^{2}_{d} & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & P^{x}_{-d} e^{2}_{d} & 0 & 0\\\\2 P^{x}_{qg} e^{2}_{u} & 2 P^{x}_{u\\gamma} e^{2}_{u} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & P^{x}_{+u} e^{2}_{u} & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & P^{x}_{-u} e^{2}_{u}\\end{array}\\right]$" - ], - "text/plain": [ - "Matrix([\n", - "[ P^x_gg*(3*e_d^2 + 3*e_u^2), P^x_g\\gamma*(3*e_d^2 + 3*e_u^2), P^x_gq*e_u^2, 0, P^x_gq*e_d^2, 0, P^x_gq*e_d^2, 0, P^x_gq*e_u^2, 0, P^x_gq*e_d^2, 0, P^x_gq*e_u^2, 0],\n", - "[P^x_\\gamma g*(3*e_d^2 + 3*e_u^2), P^x_\\gamma\\gamma, P^x_\\gamma u*e_u^2, 0, P^x_\\gamma d*e_d^2, 0, P^x_\\gamma d*e_d^2, 0, P^x_\\gamma u*e_u^2, 0, P^x_\\gamma d*e_d^2, 0, P^x_\\gamma u*e_u^2, 0],\n", - "[ 2*P^x_qg*e_u^2, 2*P^x_u\\gamma*e_u^2, P^x_+u*e_u^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[ 0, 0, 0, P^x_-u*e_u^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[ 2*P^x_qg*e_d^2, 2*P^x_d\\gamma*e_d^2, 0, 0, P^x_+d*e_d^2, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, 0, P^x_-d*e_d^2, 0, 0, 0, 0, 0, 0, 0, 0],\n", - "[ 2*P^x_qg*e_d^2, 2*P^x_d\\gamma*e_d^2, 0, 0, 0, 0, P^x_+d*e_d^2, 0, 0, 0, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, 0, 0, 0, P^x_-d*e_d^2, 0, 0, 0, 0, 0, 0],\n", - "[ 2*P^x_qg*e_u^2, 2*P^x_u\\gamma*e_u^2, 0, 0, 0, 0, 0, 0, P^x_+u*e_u^2, 0, 0, 0, 0, 0],\n", - "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, P^x_-u*e_u^2, 0, 0, 0, 0],\n", - "[ 2*P^x_qg*e_d^2, 2*P^x_d\\gamma*e_d^2, 0, 0, 0, 0, 0, 0, 0, 0, P^x_+d*e_d^2, 0, 0, 0],\n", - "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, P^x_-d*e_d^2, 0, 0],\n", - "[ 2*P^x_qg*e_u^2, 2*P^x_u\\gamma*e_u^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, P^x_+u*e_u^2, 0],\n", - "[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, P^x_-u*e_u^2]])" - ] - }, - "execution_count": 17, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "P_qed(6)" - ] - }, - { - "cell_type": "code", - "execution_count": 18, + "execution_count": 5, "id": "58ce83af", "metadata": {}, "outputs": [], @@ -207,7 +169,7 @@ }, { "cell_type": "code", - "execution_count": 19, + "execution_count": 8, "id": "28585067", "metadata": {}, "outputs": [], @@ -236,7 +198,7 @@ }, { "cell_type": "code", - "execution_count": 22, + "execution_count": 9, "id": "a0c9e31d", "metadata": {}, "outputs": [], @@ -254,7 +216,7 @@ }, { "cell_type": "code", - "execution_count": 23, + "execution_count": 11, "id": "75bfa63a", "metadata": {}, "outputs": [ @@ -271,7 +233,7 @@ "[ -6*P^x_qg*e_d^2 + 6.0*P^x_qg*e_u^2, -6*P^x_d\\gamma*e_d^2 + 6.0*P^x_u\\gamma*e_u^2, -0.5*P^x_+d*e_d^2 + 0.5*P^x_+u*e_u^2 - 0.25*e_d^2**2*(-P^x_+ + P^x_qq) + 0.25*e_u^2**2*(-P^x_+ + P^x_qq), 0.5*P^x_+d*e_d^2 + 0.5*P^x_+u*e_u^2 + 1.0*P_+ + 0.25*e_d^2**2*(-P^x_+ + P^x_qq) - 0.5*e_d^2*e_u^2*(-P^x_+ + P^x_qq) + 0.25*e_u^2**2*(-P^x_+ + P^x_qq)]])" ] }, - "execution_count": 23, + "execution_count": 11, "metadata": {}, "output_type": "execute_result" } @@ -308,7 +270,7 @@ }, { "cell_type": "code", - "execution_count": 42, + "execution_count": 13, "id": "525487e1", "metadata": {}, "outputs": [], @@ -347,37 +309,91 @@ }, { "cell_type": "code", - "execution_count": 43, + "execution_count": 20, "id": "cac66ba9", "metadata": {}, "outputs": [ { "data": { "text/latex": [ - "$\\displaystyle \\left[\\begin{matrix}0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\end{matrix}\\right]$" + "$\\displaystyle \\left[\\begin{matrix}0 & 0 & 8.88178419700125 \\cdot 10^{-17} P^{x}_{gq} e^{2}_{d} & 2.77555756156289 \\cdot 10^{-17} P_{gq}\\\\0 & 0 & 1.11022302462516 \\cdot 10^{-16} P^{x}_{\\gamma d} e^{2}_{d} & 0\\\\0 & 0 & - 3.5527136788005 \\cdot 10^{-17} P^{x}_{+} \\left(e^{2}_{u}\\right)^{2} + 1.11022302462516 \\cdot 10^{-16} P^{x}_{+d} e^{2}_{d} + 3.5527136788005 \\cdot 10^{-17} P^{x}_{qq} \\left(e^{2}_{u}\\right)^{2} & 3.5527136788005 \\cdot 10^{-17} P^{x}_{+} e^{2}_{d} e^{2}_{u} - 3.5527136788005 \\cdot 10^{-17} P^{x}_{+} \\left(e^{2}_{u}\\right)^{2} - 3.5527136788005 \\cdot 10^{-17} P^{x}_{qq} e^{2}_{d} e^{2}_{u} + 3.5527136788005 \\cdot 10^{-17} P^{x}_{qq} \\left(e^{2}_{u}\\right)^{2} + 2.77555756156289 \\cdot 10^{-17} P_{+}\\\\0 & 0 & - 5.32907051820075 \\cdot 10^{-17} P^{x}_{+} e^{2}_{d} e^{2}_{u} - 3.5527136788005 \\cdot 10^{-17} P^{x}_{+} \\left(e^{2}_{u}\\right)^{2} - 1.11022302462516 \\cdot 10^{-16} P^{x}_{+d} e^{2}_{d} + 1.11022302462516 \\cdot 10^{-16} P^{x}_{+u} e^{2}_{u} + 5.32907051820075 \\cdot 10^{-17} P^{x}_{qq} e^{2}_{d} e^{2}_{u} + 3.5527136788005 \\cdot 10^{-17} P^{x}_{qq} \\left(e^{2}_{u}\\right)^{2} + 5.55111512312578 \\cdot 10^{-17} P_{qq} & e^{2}_{u} \\left(5.55111512312578 \\cdot 10^{-17} P^{x}_{+} e^{2}_{d} - 5.55111512312578 \\cdot 10^{-17} P^{x}_{+} e^{2}_{u} + 1.11022302462516 \\cdot 10^{-16} P^{x}_{+u} - 5.55111512312578 \\cdot 10^{-17} P^{x}_{qq} e^{2}_{d} + 5.55111512312578 \\cdot 10^{-17} P^{x}_{qq} e^{2}_{u}\\right)\\end{matrix}\\right]$" ], "text/plain": [ "Matrix([\n", - "[0, 0, 0, 0],\n", - "[0, 0, 0, 0],\n", - "[0, 0, 0, 0],\n", - "[0, 0, 0, 0]])" + "[0, 0, 8.88178419700125e-17*P^x_gq*e_d^2, 2.77555756156289e-17*P_gq],\n", + "[0, 0, 1.11022302462516e-16*P^x_\\gamma d*e_d^2, 0],\n", + "[0, 0, -3.5527136788005e-17*P^x_+*e_u^2**2 + 1.11022302462516e-16*P^x_+d*e_d^2 + 3.5527136788005e-17*P^x_qq*e_u^2**2, 3.5527136788005e-17*P^x_+*e_d^2*e_u^2 - 3.5527136788005e-17*P^x_+*e_u^2**2 - 3.5527136788005e-17*P^x_qq*e_d^2*e_u^2 + 3.5527136788005e-17*P^x_qq*e_u^2**2 + 2.77555756156289e-17*P_+],\n", + "[0, 0, -5.32907051820075e-17*P^x_+*e_d^2*e_u^2 - 3.5527136788005e-17*P^x_+*e_u^2**2 - 1.11022302462516e-16*P^x_+d*e_d^2 + 1.11022302462516e-16*P^x_+u*e_u^2 + 5.32907051820075e-17*P^x_qq*e_d^2*e_u^2 + 3.5527136788005e-17*P^x_qq*e_u^2**2 + 5.55111512312578e-17*P_qq, e_u^2*(5.55111512312578e-17*P^x_+*e_d^2 - 5.55111512312578e-17*P^x_+*e_u^2 + 1.11022302462516e-16*P^x_+u - 5.55111512312578e-17*P^x_qq*e_d^2 + 5.55111512312578e-17*P^x_qq*e_u^2)]])" ] }, - "execution_count": 43, + "execution_count": 20, "metadata": {}, "output_type": "execute_result" } ], "source": [ - "sympy.simplify(P_ev_sing(2)-P_ev_sing2(2))" + "sympy.simplify(P_ev_sing(5)-P_ev_sing2(5))" ] }, { "cell_type": "code", - "execution_count": 47, + "execution_count": 19, + "id": "42a4712c", + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\left[\\begin{matrix}0 & 0 & - 1.11022302462516 \\cdot 10^{-16} P_{gq} & 0\\\\0 & 0 & 0 & 0\\\\0 & 0 & - 1.11022302462516 \\cdot 10^{-16} P_{+} & 0\\\\0 & 0 & 0 & - 1.11022302462516 \\cdot 10^{-16} P_{+}\\end{matrix}\\right]$" + ], + "text/plain": [ + "Matrix([\n", + "[0, 0, -1.11022302462516e-16*P_gq, 0],\n", + "[0, 0, 0, 0],\n", + "[0, 0, -1.11022302462516e-16*P_+, 0],\n", + "[0, 0, 0, -1.11022302462516e-16*P_+]])" + ] + }, + "execution_count": 19, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "sympy.simplify(P_ev_sing(6)-P_ev_sing2(6))" + ] + }, + { + "cell_type": "code", + "execution_count": 15, "id": "ac0e82fc", "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\left[\\begin{matrix}1.11022302462516 \\cdot 10^{-16} P^{x}_{-d} e^{2}_{d} & 2.77555756156289 \\cdot 10^{-17} P_{-}\\\\- 1.11022302462516 \\cdot 10^{-16} P^{x}_{-d} e^{2}_{d} + 1.11022302462516 \\cdot 10^{-16} P^{x}_{-u} e^{2}_{u} + 5.55111512312578 \\cdot 10^{-17} P_{V} & 1.11022302462516 \\cdot 10^{-16} P^{x}_{-u} e^{2}_{u}\\end{matrix}\\right]$" + ], + "text/plain": [ + "Matrix([\n", + "[ 1.11022302462516e-16*P^x_-d*e_d^2, 2.77555756156289e-17*P_-],\n", + "[-1.11022302462516e-16*P^x_-d*e_d^2 + 1.11022302462516e-16*P^x_-u*e_u^2 + 5.55111512312578e-17*P_V, 1.11022302462516e-16*P^x_-u*e_u^2]])" + ] + }, + "execution_count": 15, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "sympy.simplify(P_ev_val(5)-P_ev_val2(5))" + ] + }, + { + "cell_type": "code", + "execution_count": 16, + "id": "3e3c29ce", + "metadata": {}, "outputs": [ { "data": { @@ -390,7 +406,7 @@ "[ 0, -1.11022302462516e-16*P_-]])" ] }, - "execution_count": 47, + "execution_count": 16, "metadata": {}, "output_type": "execute_result" } @@ -402,7 +418,7 @@ { "cell_type": "code", "execution_count": null, - "id": "3e3c29ce", + "id": "121e1f75", "metadata": {}, "outputs": [], "source": [] From 246d84ec4487576577e6ed0b0ede3c15e7c2337d Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Niccol=C3=B2=20Laurenti?= Date: Thu, 7 Apr 2022 11:44:38 +0200 Subject: [PATCH 71/71] Modify (Pxqq-Pxp)->Pxps --- extras/uni-dglap/uni-dglap-aem2.ipynb | 60 +++++++++++++-------------- 1 file changed, 30 insertions(+), 30 deletions(-) diff --git a/extras/uni-dglap/uni-dglap-aem2.ipynb b/extras/uni-dglap/uni-dglap-aem2.ipynb index c6b2066e1..4a40583bb 100644 --- a/extras/uni-dglap/uni-dglap-aem2.ipynb +++ b/extras/uni-dglap/uni-dglap-aem2.ipynb @@ -20,7 +20,7 @@ "# QCD\n", "Pv, Pp, Pm, Pqq, Pqg, Pgq, Pgg = sympy.symbols(\"P_V P_+ P_- P_qq P_qg P_gq P_gg\")\n", "# QED\n", - "Pxv, Pxp, Pxpu, Pxpd, Pxm, Pxmu, Pxmd, Pxqq, Pxqg, Pxgq, Pxgg = sympy.symbols(\"P^x_V P^x_+ P^x_+u P^x_+d P^x_- P^x_-u P^x_-d P^x_qq P^x_qg P^x_gq P^x_gg\")\n", + "Pxv, Pxp, Pxpu, Pxpd, Pxm, Pxmu, Pxmd, Pxqq, Pxqg, Pxgq, Pxgg, Pxps= sympy.symbols(\"P^x_V P^x_+ P^x_+u P^x_+d P^x_- P^x_-u P^x_-d P^x_qq P^x_qg P^x_gq P^x_gg P^x_ps\")\n", "Pxqy, Pxuy, Pxdy, Pxyq, Pxyu, Pxyd, Pxyg, Pxgy, Pxyy = sympy.symbols(\"P^x_q\\gamma P^x_u\\gamma P^x_d\\gamma P^x_\\gamma\\ q P^x_\\gamma\\ u P^x_\\gamma\\ d P^x_\\gamma\\ g P^x_g\\gamma P^x_\\gamma\\gamma\")\n", "eu2, ed2, es2 = sympy.symbols(\"e_u^2 e_d^2 e_\\Sigma^2\") # charges\n", "eu4, ed4 = sympy.symbols(\"e_u^4 e_d^4\") # charges" @@ -100,8 +100,8 @@ "def Ps_qed(nf):\n", " res = sympy.Matrix.zeros(14,14).as_mutable()\n", " for i in range(1, nf+1):\n", - " res[2*i, 2] = ei2[i-1]*eu2*(Pxqq - Pxp)\n", - " res[2*i, 3] = ei2[i-1]*ed2*(Pxqq - Pxp)\n", + " res[2*i, 2] = ei2[i-1]*eu2*Pxps\n", + " res[2*i, 3] = ei2[i-1]*ed2*Pxps\n", " return res/nf\n", "\n", "def P_uni(nf):\n", @@ -169,7 +169,7 @@ }, { "cell_type": "code", - "execution_count": 8, + "execution_count": 6, "id": "28585067", "metadata": {}, "outputs": [], @@ -198,7 +198,7 @@ }, { "cell_type": "code", - "execution_count": 9, + "execution_count": 7, "id": "a0c9e31d", "metadata": {}, "outputs": [], @@ -216,24 +216,24 @@ }, { "cell_type": "code", - "execution_count": 11, + "execution_count": 8, "id": "75bfa63a", "metadata": {}, "outputs": [ { "data": { "text/latex": [ - "$\\displaystyle \\left[\\begin{matrix}P^{x}_{gg} \\left(3 e^{2}_{d} + 3 e^{2}_{u}\\right) + P_{gg} & P^{x}_{g\\gamma} \\left(3 e^{2}_{d} + 3 e^{2}_{u}\\right) & 0.5 P^{x}_{gq} e^{2}_{d} + 0.5 P^{x}_{gq} e^{2}_{u} + 1.0 P_{gq} & - 0.5 P^{x}_{gq} e^{2}_{d} + 0.5 P^{x}_{gq} e^{2}_{u}\\\\P^{x}_{\\gamma g} \\left(3 e^{2}_{d} + 3 e^{2}_{u}\\right) & P^{x}_{\\gamma\\gamma} & 0.5 P^{x}_{\\gamma d} e^{2}_{d} + 0.5 P^{x}_{\\gamma u} e^{2}_{u} & - 0.5 P^{x}_{\\gamma d} e^{2}_{d} + 0.5 P^{x}_{\\gamma u} e^{2}_{u}\\\\6 P^{x}_{qg} e^{2}_{d} + 6 P^{x}_{qg} e^{2}_{u} + 12 P_{qg} & 6 P^{x}_{d\\gamma} e^{2}_{d} + 6 P^{x}_{u\\gamma} e^{2}_{u} & 0.5 P^{x}_{+d} e^{2}_{d} + 0.5 P^{x}_{+u} e^{2}_{u} - 1.11022302462516 \\cdot 10^{-16} P_{+} + 1.0 P_{qq} + 0.25 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.5 e^{2}_{d} e^{2}_{u} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.25 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & - 0.5 P^{x}_{+d} e^{2}_{d} + 0.5 P^{x}_{+u} e^{2}_{u} - 0.25 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.25 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right)\\\\- 6 P^{x}_{qg} e^{2}_{d} + 6.0 P^{x}_{qg} e^{2}_{u} & - 6 P^{x}_{d\\gamma} e^{2}_{d} + 6.0 P^{x}_{u\\gamma} e^{2}_{u} & - 0.5 P^{x}_{+d} e^{2}_{d} + 0.5 P^{x}_{+u} e^{2}_{u} - 0.25 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.25 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) & 0.5 P^{x}_{+d} e^{2}_{d} + 0.5 P^{x}_{+u} e^{2}_{u} + 1.0 P_{+} + 0.25 \\left(e^{2}_{d}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) - 0.5 e^{2}_{d} e^{2}_{u} \\left(- P^{x}_{+} + P^{x}_{qq}\\right) + 0.25 \\left(e^{2}_{u}\\right)^{2} \\left(- P^{x}_{+} + P^{x}_{qq}\\right)\\end{matrix}\\right]$" + "$\\displaystyle \\left[\\begin{matrix}P^{x}_{gg} \\left(3 e^{2}_{d} + 3 e^{2}_{u}\\right) + P_{gg} & P^{x}_{g\\gamma} \\left(3 e^{2}_{d} + 3 e^{2}_{u}\\right) & 0.5 P^{x}_{gq} e^{2}_{d} + 0.5 P^{x}_{gq} e^{2}_{u} + 1.0 P_{gq} & - 0.5 P^{x}_{gq} e^{2}_{d} + 0.5 P^{x}_{gq} e^{2}_{u}\\\\P^{x}_{\\gamma g} \\left(3 e^{2}_{d} + 3 e^{2}_{u}\\right) & P^{x}_{\\gamma\\gamma} & 0.5 P^{x}_{\\gamma d} e^{2}_{d} + 0.5 P^{x}_{\\gamma u} e^{2}_{u} & - 0.5 P^{x}_{\\gamma d} e^{2}_{d} + 0.5 P^{x}_{\\gamma u} e^{2}_{u}\\\\6 P^{x}_{qg} e^{2}_{d} + 6 P^{x}_{qg} e^{2}_{u} + 12 P_{qg} & 6 P^{x}_{d\\gamma} e^{2}_{d} + 6 P^{x}_{u\\gamma} e^{2}_{u} & 0.5 P^{x}_{+d} e^{2}_{d} + 0.5 P^{x}_{+u} e^{2}_{u} + 0.25 P^{x}_{ps} \\left(e^{2}_{d}\\right)^{2} + 0.5 P^{x}_{ps} e^{2}_{d} e^{2}_{u} + 0.25 P^{x}_{ps} \\left(e^{2}_{u}\\right)^{2} - 1.11022302462516 \\cdot 10^{-16} P_{+} + 1.0 P_{qq} & - 0.5 P^{x}_{+d} e^{2}_{d} + 0.5 P^{x}_{+u} e^{2}_{u} - 0.25 P^{x}_{ps} \\left(e^{2}_{d}\\right)^{2} + 0.25 P^{x}_{ps} \\left(e^{2}_{u}\\right)^{2}\\\\- 6 P^{x}_{qg} e^{2}_{d} + 6.0 P^{x}_{qg} e^{2}_{u} & - 6 P^{x}_{d\\gamma} e^{2}_{d} + 6.0 P^{x}_{u\\gamma} e^{2}_{u} & - 0.5 P^{x}_{+d} e^{2}_{d} + 0.5 P^{x}_{+u} e^{2}_{u} - 0.25 P^{x}_{ps} \\left(e^{2}_{d}\\right)^{2} + 0.25 P^{x}_{ps} \\left(e^{2}_{u}\\right)^{2} & 0.5 P^{x}_{+d} e^{2}_{d} + 0.5 P^{x}_{+u} e^{2}_{u} + 0.25 P^{x}_{ps} \\left(e^{2}_{d}\\right)^{2} - 0.5 P^{x}_{ps} e^{2}_{d} e^{2}_{u} + 0.25 P^{x}_{ps} \\left(e^{2}_{u}\\right)^{2} + 1.0 P_{+}\\end{matrix}\\right]$" ], "text/plain": [ "Matrix([\n", - "[ P^x_gg*(3*e_d^2 + 3*e_u^2) + P_gg, P^x_g\\gamma*(3*e_d^2 + 3*e_u^2), 0.5*P^x_gq*e_d^2 + 0.5*P^x_gq*e_u^2 + 1.0*P_gq, -0.5*P^x_gq*e_d^2 + 0.5*P^x_gq*e_u^2],\n", - "[ P^x_\\gamma g*(3*e_d^2 + 3*e_u^2), P^x_\\gamma\\gamma, 0.5*P^x_\\gamma d*e_d^2 + 0.5*P^x_\\gamma u*e_u^2, -0.5*P^x_\\gamma d*e_d^2 + 0.5*P^x_\\gamma u*e_u^2],\n", - "[6*P^x_qg*e_d^2 + 6*P^x_qg*e_u^2 + 12*P_qg, 6*P^x_d\\gamma*e_d^2 + 6*P^x_u\\gamma*e_u^2, 0.5*P^x_+d*e_d^2 + 0.5*P^x_+u*e_u^2 - 1.11022302462516e-16*P_+ + 1.0*P_qq + 0.25*e_d^2**2*(-P^x_+ + P^x_qq) + 0.5*e_d^2*e_u^2*(-P^x_+ + P^x_qq) + 0.25*e_u^2**2*(-P^x_+ + P^x_qq), -0.5*P^x_+d*e_d^2 + 0.5*P^x_+u*e_u^2 - 0.25*e_d^2**2*(-P^x_+ + P^x_qq) + 0.25*e_u^2**2*(-P^x_+ + P^x_qq)],\n", - "[ -6*P^x_qg*e_d^2 + 6.0*P^x_qg*e_u^2, -6*P^x_d\\gamma*e_d^2 + 6.0*P^x_u\\gamma*e_u^2, -0.5*P^x_+d*e_d^2 + 0.5*P^x_+u*e_u^2 - 0.25*e_d^2**2*(-P^x_+ + P^x_qq) + 0.25*e_u^2**2*(-P^x_+ + P^x_qq), 0.5*P^x_+d*e_d^2 + 0.5*P^x_+u*e_u^2 + 1.0*P_+ + 0.25*e_d^2**2*(-P^x_+ + P^x_qq) - 0.5*e_d^2*e_u^2*(-P^x_+ + P^x_qq) + 0.25*e_u^2**2*(-P^x_+ + P^x_qq)]])" + "[ P^x_gg*(3*e_d^2 + 3*e_u^2) + P_gg, P^x_g\\gamma*(3*e_d^2 + 3*e_u^2), 0.5*P^x_gq*e_d^2 + 0.5*P^x_gq*e_u^2 + 1.0*P_gq, -0.5*P^x_gq*e_d^2 + 0.5*P^x_gq*e_u^2],\n", + "[ P^x_\\gamma g*(3*e_d^2 + 3*e_u^2), P^x_\\gamma\\gamma, 0.5*P^x_\\gamma d*e_d^2 + 0.5*P^x_\\gamma u*e_u^2, -0.5*P^x_\\gamma d*e_d^2 + 0.5*P^x_\\gamma u*e_u^2],\n", + "[6*P^x_qg*e_d^2 + 6*P^x_qg*e_u^2 + 12*P_qg, 6*P^x_d\\gamma*e_d^2 + 6*P^x_u\\gamma*e_u^2, 0.5*P^x_+d*e_d^2 + 0.5*P^x_+u*e_u^2 + 0.25*P^x_ps*e_d^2**2 + 0.5*P^x_ps*e_d^2*e_u^2 + 0.25*P^x_ps*e_u^2**2 - 1.11022302462516e-16*P_+ + 1.0*P_qq, -0.5*P^x_+d*e_d^2 + 0.5*P^x_+u*e_u^2 - 0.25*P^x_ps*e_d^2**2 + 0.25*P^x_ps*e_u^2**2],\n", + "[ -6*P^x_qg*e_d^2 + 6.0*P^x_qg*e_u^2, -6*P^x_d\\gamma*e_d^2 + 6.0*P^x_u\\gamma*e_u^2, -0.5*P^x_+d*e_d^2 + 0.5*P^x_+u*e_u^2 - 0.25*P^x_ps*e_d^2**2 + 0.25*P^x_ps*e_u^2**2, 0.5*P^x_+d*e_d^2 + 0.5*P^x_+u*e_u^2 + 0.25*P^x_ps*e_d^2**2 - 0.5*P^x_ps*e_d^2*e_u^2 + 0.25*P^x_ps*e_u^2**2 + 1.0*P_+]])" ] }, - "execution_count": 11, + "execution_count": 8, "metadata": {}, "output_type": "execute_result" } @@ -244,7 +244,7 @@ }, { "cell_type": "code", - "execution_count": 24, + "execution_count": 9, "id": "5999a68e", "metadata": {}, "outputs": [ @@ -259,7 +259,7 @@ "[ -0.5*P^x_-d*e_d^2 + 0.5*P^x_-u*e_u^2, 0.5*P^x_-d*e_d^2 + 0.5*P^x_-u*e_u^2 + 1.0*P_-]])" ] }, - "execution_count": 24, + "execution_count": 9, "metadata": {}, "output_type": "execute_result" } @@ -270,7 +270,7 @@ }, { "cell_type": "code", - "execution_count": 13, + "execution_count": 10, "id": "525487e1", "metadata": {}, "outputs": [], @@ -292,8 +292,8 @@ " res = sympy.Matrix([\n", " [Pgg + es2 * Pxgg, es2 * Pxgy, Pgq + es2/nf*Pxgq, 2*nu/nf*etam*Pxgq],\n", " [es2 * Pxyg, Pxyy, nu/nf*eu2*Pxyu + nd/nf*ed2*Pxyd, 2*nu/nf*0.5*(eu2*Pxyu-ed2*Pxyd)],\n", - " [2*nf*Pqg + 2*es2*Pxqg, 2*(nu*eu2*Pxuy+nd*ed2*Pxdy), Pqq + (nu*eu2*Pxpu+nd*ed2*Pxpd)/nf +(es2/nf)**2*(Pxqq - Pxp), 2*nu/nf*0.5*(eu2*Pxpu - ed2*Pxpd) +2*nu*etam*es2/nf**2*(Pxqq - Pxp)],\n", - " [4*nd*etam*Pxqg, 4*nd*0.5*(eu2*Pxuy - ed2*Pxdy), 2*nd/nf*0.5*(eu2*Pxpu - ed2*Pxpd) +2*nd*etam*es2/nf**2*(Pxqq - Pxp), Pp + (nd*eu2*Pxpu + nu*ed2*Pxpd)/nf + 4*nu*nd/nf**2*etam**2*(Pxqq - Pxp)]\n", + " [2*nf*Pqg + 2*es2*Pxqg, 2*(nu*eu2*Pxuy+nd*ed2*Pxdy), Pqq + (nu*eu2*Pxpu+nd*ed2*Pxpd)/nf +(es2/nf)**2*Pxps, 2*nu/nf*0.5*(eu2*Pxpu - ed2*Pxpd) +2*nu*etam*es2/nf**2*Pxps],\n", + " [4*nd*etam*Pxqg, 4*nd*0.5*(eu2*Pxuy - ed2*Pxdy), 2*nd/nf*0.5*(eu2*Pxpu - ed2*Pxpd) +2*nd*etam*es2/nf**2*Pxps, Pp + (nd*eu2*Pxpu + nu*ed2*Pxpd)/nf + 4*nu*nd/nf**2*etam**2*Pxps]\n", " ])\n", " return res\n", "\n", @@ -309,24 +309,24 @@ }, { "cell_type": "code", - "execution_count": 20, + "execution_count": 11, "id": "cac66ba9", "metadata": {}, "outputs": [ { "data": { "text/latex": [ - "$\\displaystyle \\left[\\begin{matrix}0 & 0 & 8.88178419700125 \\cdot 10^{-17} P^{x}_{gq} e^{2}_{d} & 2.77555756156289 \\cdot 10^{-17} P_{gq}\\\\0 & 0 & 1.11022302462516 \\cdot 10^{-16} P^{x}_{\\gamma d} e^{2}_{d} & 0\\\\0 & 0 & - 3.5527136788005 \\cdot 10^{-17} P^{x}_{+} \\left(e^{2}_{u}\\right)^{2} + 1.11022302462516 \\cdot 10^{-16} P^{x}_{+d} e^{2}_{d} + 3.5527136788005 \\cdot 10^{-17} P^{x}_{qq} \\left(e^{2}_{u}\\right)^{2} & 3.5527136788005 \\cdot 10^{-17} P^{x}_{+} e^{2}_{d} e^{2}_{u} - 3.5527136788005 \\cdot 10^{-17} P^{x}_{+} \\left(e^{2}_{u}\\right)^{2} - 3.5527136788005 \\cdot 10^{-17} P^{x}_{qq} e^{2}_{d} e^{2}_{u} + 3.5527136788005 \\cdot 10^{-17} P^{x}_{qq} \\left(e^{2}_{u}\\right)^{2} + 2.77555756156289 \\cdot 10^{-17} P_{+}\\\\0 & 0 & - 5.32907051820075 \\cdot 10^{-17} P^{x}_{+} e^{2}_{d} e^{2}_{u} - 3.5527136788005 \\cdot 10^{-17} P^{x}_{+} \\left(e^{2}_{u}\\right)^{2} - 1.11022302462516 \\cdot 10^{-16} P^{x}_{+d} e^{2}_{d} + 1.11022302462516 \\cdot 10^{-16} P^{x}_{+u} e^{2}_{u} + 5.32907051820075 \\cdot 10^{-17} P^{x}_{qq} e^{2}_{d} e^{2}_{u} + 3.5527136788005 \\cdot 10^{-17} P^{x}_{qq} \\left(e^{2}_{u}\\right)^{2} + 5.55111512312578 \\cdot 10^{-17} P_{qq} & e^{2}_{u} \\left(5.55111512312578 \\cdot 10^{-17} P^{x}_{+} e^{2}_{d} - 5.55111512312578 \\cdot 10^{-17} P^{x}_{+} e^{2}_{u} + 1.11022302462516 \\cdot 10^{-16} P^{x}_{+u} - 5.55111512312578 \\cdot 10^{-17} P^{x}_{qq} e^{2}_{d} + 5.55111512312578 \\cdot 10^{-17} P^{x}_{qq} e^{2}_{u}\\right)\\end{matrix}\\right]$" + "$\\displaystyle \\left[\\begin{matrix}0 & 0 & 8.88178419700125 \\cdot 10^{-17} P^{x}_{gq} e^{2}_{d} & 2.77555756156289 \\cdot 10^{-17} P_{gq}\\\\0 & 0 & 1.11022302462516 \\cdot 10^{-16} P^{x}_{\\gamma d} e^{2}_{d} & 0\\\\0 & 0 & 1.11022302462516 \\cdot 10^{-16} P^{x}_{+d} e^{2}_{d} + 3.5527136788005 \\cdot 10^{-17} P^{x}_{ps} \\left(e^{2}_{u}\\right)^{2} & - 3.5527136788005 \\cdot 10^{-17} P^{x}_{ps} e^{2}_{d} e^{2}_{u} + 3.5527136788005 \\cdot 10^{-17} P^{x}_{ps} \\left(e^{2}_{u}\\right)^{2} + 2.77555756156289 \\cdot 10^{-17} P_{+}\\\\0 & 0 & - 1.11022302462516 \\cdot 10^{-16} P^{x}_{+d} e^{2}_{d} + 1.11022302462516 \\cdot 10^{-16} P^{x}_{+u} e^{2}_{u} + 5.32907051820075 \\cdot 10^{-17} P^{x}_{ps} e^{2}_{d} e^{2}_{u} + 3.5527136788005 \\cdot 10^{-17} P^{x}_{ps} \\left(e^{2}_{u}\\right)^{2} + 5.55111512312578 \\cdot 10^{-17} P_{qq} & e^{2}_{u} \\left(1.11022302462516 \\cdot 10^{-16} P^{x}_{+u} - 5.55111512312578 \\cdot 10^{-17} P^{x}_{ps} e^{2}_{d} + 5.55111512312578 \\cdot 10^{-17} P^{x}_{ps} e^{2}_{u}\\right)\\end{matrix}\\right]$" ], "text/plain": [ "Matrix([\n", - "[0, 0, 8.88178419700125e-17*P^x_gq*e_d^2, 2.77555756156289e-17*P_gq],\n", - "[0, 0, 1.11022302462516e-16*P^x_\\gamma d*e_d^2, 0],\n", - "[0, 0, -3.5527136788005e-17*P^x_+*e_u^2**2 + 1.11022302462516e-16*P^x_+d*e_d^2 + 3.5527136788005e-17*P^x_qq*e_u^2**2, 3.5527136788005e-17*P^x_+*e_d^2*e_u^2 - 3.5527136788005e-17*P^x_+*e_u^2**2 - 3.5527136788005e-17*P^x_qq*e_d^2*e_u^2 + 3.5527136788005e-17*P^x_qq*e_u^2**2 + 2.77555756156289e-17*P_+],\n", - "[0, 0, -5.32907051820075e-17*P^x_+*e_d^2*e_u^2 - 3.5527136788005e-17*P^x_+*e_u^2**2 - 1.11022302462516e-16*P^x_+d*e_d^2 + 1.11022302462516e-16*P^x_+u*e_u^2 + 5.32907051820075e-17*P^x_qq*e_d^2*e_u^2 + 3.5527136788005e-17*P^x_qq*e_u^2**2 + 5.55111512312578e-17*P_qq, e_u^2*(5.55111512312578e-17*P^x_+*e_d^2 - 5.55111512312578e-17*P^x_+*e_u^2 + 1.11022302462516e-16*P^x_+u - 5.55111512312578e-17*P^x_qq*e_d^2 + 5.55111512312578e-17*P^x_qq*e_u^2)]])" + "[0, 0, 8.88178419700125e-17*P^x_gq*e_d^2, 2.77555756156289e-17*P_gq],\n", + "[0, 0, 1.11022302462516e-16*P^x_\\gamma d*e_d^2, 0],\n", + "[0, 0, 1.11022302462516e-16*P^x_+d*e_d^2 + 3.5527136788005e-17*P^x_ps*e_u^2**2, -3.5527136788005e-17*P^x_ps*e_d^2*e_u^2 + 3.5527136788005e-17*P^x_ps*e_u^2**2 + 2.77555756156289e-17*P_+],\n", + "[0, 0, -1.11022302462516e-16*P^x_+d*e_d^2 + 1.11022302462516e-16*P^x_+u*e_u^2 + 5.32907051820075e-17*P^x_ps*e_d^2*e_u^2 + 3.5527136788005e-17*P^x_ps*e_u^2**2 + 5.55111512312578e-17*P_qq, e_u^2*(1.11022302462516e-16*P^x_+u - 5.55111512312578e-17*P^x_ps*e_d^2 + 5.55111512312578e-17*P^x_ps*e_u^2)]])" ] }, - "execution_count": 20, + "execution_count": 11, "metadata": {}, "output_type": "execute_result" } @@ -337,7 +337,7 @@ }, { "cell_type": "code", - "execution_count": 19, + "execution_count": 12, "id": "42a4712c", "metadata": {}, "outputs": [ @@ -354,7 +354,7 @@ "[0, 0, 0, -1.11022302462516e-16*P_+]])" ] }, - "execution_count": 19, + "execution_count": 12, "metadata": {}, "output_type": "execute_result" } @@ -365,7 +365,7 @@ }, { "cell_type": "code", - "execution_count": 15, + "execution_count": 13, "id": "ac0e82fc", "metadata": {}, "outputs": [ @@ -380,7 +380,7 @@ "[-1.11022302462516e-16*P^x_-d*e_d^2 + 1.11022302462516e-16*P^x_-u*e_u^2 + 5.55111512312578e-17*P_V, 1.11022302462516e-16*P^x_-u*e_u^2]])" ] }, - "execution_count": 15, + "execution_count": 13, "metadata": {}, "output_type": "execute_result" } @@ -391,7 +391,7 @@ }, { "cell_type": "code", - "execution_count": 16, + "execution_count": 14, "id": "3e3c29ce", "metadata": {}, "outputs": [ @@ -406,7 +406,7 @@ "[ 0, -1.11022302462516e-16*P_-]])" ] }, - "execution_count": 16, + "execution_count": 14, "metadata": {}, "output_type": "execute_result" }