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 diff --git a/doc/source/theory/pQCD.rst b/doc/source/theory/pQCD.rst index 2611c9876..93623d1cb 100644 --- a/doc/source/theory/pQCD.rst +++ b/doc/source/theory/pQCD.rst @@ -76,12 +76,58 @@ 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` +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_{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. +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: + +.. 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 ---------------- diff --git a/extras/uni-dglap/Makefile b/extras/uni-dglap/Makefile new file mode 100644 index 000000000..42eacb87e --- /dev/null +++ b/extras/uni-dglap/Makefile @@ -0,0 +1,9 @@ +all: uni-dglap.pdf + +%.pdf: %.tex + pdflatex $< + +clean: + rm -f *.aux + rm -f *.log + rm -f *.pdf diff --git a/extras/uni-dglap/uni-ad.nb b/extras/uni-dglap/uni-ad.nb new file mode 100644 index 000000000..486848a9a --- /dev/null +++ b/extras/uni-dglap/uni-ad.nb @@ -0,0 +1,5159 @@ +(* Content-type: 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"code", + "execution_count": 2, + "id": "c093db3d", + "metadata": {}, + "outputs": [], + "source": [ + "import sympy" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "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, 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" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "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*Pxps\n", + " res[2*i, 3] = ei2[i-1]*ed2*Pxps\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": "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": 6, + "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": 7, + "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": 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} + 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]$" + ], + "text/plain": [ + "Matrix([\n", + "[ P^x_gg*(3*e_d^2 + 3*e_u^2) + P_gg, P^x_g\\gamma*(3*e_d^2 + 3*e_u^2), 0.5*P^x_gq*e_d^2 + 0.5*P^x_gq*e_u^2 + 1.0*P_gq, -0.5*P^x_gq*e_d^2 + 0.5*P^x_gq*e_u^2],\n", + "[ P^x_\\gamma g*(3*e_d^2 + 3*e_u^2), P^x_\\gamma\\gamma, 0.5*P^x_\\gamma d*e_d^2 + 0.5*P^x_\\gamma u*e_u^2, -0.5*P^x_\\gamma d*e_d^2 + 0.5*P^x_\\gamma u*e_u^2],\n", + "[6*P^x_qg*e_d^2 + 6*P^x_qg*e_u^2 + 12*P_qg, 6*P^x_d\\gamma*e_d^2 + 6*P^x_u\\gamma*e_u^2, 0.5*P^x_+d*e_d^2 + 0.5*P^x_+u*e_u^2 + 0.25*P^x_ps*e_d^2**2 + 0.5*P^x_ps*e_d^2*e_u^2 + 0.25*P^x_ps*e_u^2**2 - 1.11022302462516e-16*P_+ + 1.0*P_qq, -0.5*P^x_+d*e_d^2 + 0.5*P^x_+u*e_u^2 - 0.25*P^x_ps*e_d^2**2 + 0.25*P^x_ps*e_u^2**2],\n", + "[ -6*P^x_qg*e_d^2 + 6.0*P^x_qg*e_u^2, -6*P^x_d\\gamma*e_d^2 + 6.0*P^x_u\\gamma*e_u^2, -0.5*P^x_+d*e_d^2 + 0.5*P^x_+u*e_u^2 - 0.25*P^x_ps*e_d^2**2 + 0.25*P^x_ps*e_u^2**2, 0.5*P^x_+d*e_d^2 + 0.5*P^x_+u*e_u^2 + 0.25*P^x_ps*e_d^2**2 - 0.5*P^x_ps*e_d^2*e_u^2 + 0.25*P^x_ps*e_u^2**2 + 1.0*P_+]])" + ] + }, + "execution_count": 8, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "P_ev_sing(6)" + ] + }, + { + "cell_type": "code", + "execution_count": 9, + "id": "5999a68e", + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "$\\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]$" + ], + "text/plain": [ + "Matrix([\n", + "[0.5*P^x_-d*e_d^2 + 0.5*P^x_-u*e_u^2 - 1.11022302462516e-16*P_- + 1.0*P_V, -0.5*P^x_-d*e_d^2 + 0.5*P^x_-u*e_u^2],\n", + "[ -0.5*P^x_-d*e_d^2 + 0.5*P^x_-u*e_u^2, 0.5*P^x_-d*e_d^2 + 0.5*P^x_-u*e_u^2 + 1.0*P_-]])" + ] + }, + "execution_count": 9, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "P_ev_val(6)" + ] + }, + { + "cell_type": "code", + "execution_count": 10, + "id": "525487e1", + "metadata": {}, + "outputs": [], + "source": [ + "def eD2_(nf):\n", + " nu = int(nf/2)\n", + " nd = nf - nu\n", + " return nd*eu2 + nu*ed2\n", + "def etam_(nf):\n", + " nu = int(nf/2)\n", + " nd = nf - nu\n", + " return 0.5*(eu2 - ed2)\n", + "def P_ev_sing2(nf):\n", + " es2=es2_(nf)\n", + " eD2=eD2_(nf)\n", + " etam=etam_(nf)\n", + " nu = int(nf/2)\n", + " nd = nf - nu\n", + " res = sympy.Matrix([\n", + " [Pgg + es2 * Pxgg, es2 * Pxgy, Pgq + es2/nf*Pxgq, 2*nu/nf*etam*Pxgq],\n", + " [es2 * Pxyg, Pxyy, nu/nf*eu2*Pxyu + nd/nf*ed2*Pxyd, 2*nu/nf*0.5*(eu2*Pxyu-ed2*Pxyd)],\n", + " [2*nf*Pqg + 2*es2*Pxqg, 2*(nu*eu2*Pxuy+nd*ed2*Pxdy), Pqq + (nu*eu2*Pxpu+nd*ed2*Pxpd)/nf +(es2/nf)**2*Pxps, 2*nu/nf*0.5*(eu2*Pxpu - ed2*Pxpd) +2*nu*etam*es2/nf**2*Pxps],\n", + " [4*nd*etam*Pxqg, 4*nd*0.5*(eu2*Pxuy - ed2*Pxdy), 2*nd/nf*0.5*(eu2*Pxpu - ed2*Pxpd) +2*nd*etam*es2/nf**2*Pxps, Pp + (nd*eu2*Pxpu + nu*ed2*Pxpd)/nf + 4*nu*nd/nf**2*etam**2*Pxps]\n", + " ])\n", + " return res\n", + "\n", + "def P_ev_val2(nf):\n", + " nu = int(nf/2)\n", + " nd = nf - nu\n", + " res = sympy.Matrix([\n", + " [Pv+(nu*eu2*Pxmu+nd*ed2*Pxmd)/nf, 2*nu/nf*0.5*(eu2*Pxmu - ed2*Pxmd)],\n", + " [2*nd/nf*0.5*(eu2*Pxmu - ed2*Pxmd), Pm + (nd*eu2*Pxmu + nu*ed2*Pxmd)/nf]\n", + " ])\n", + " return res" + ] + }, + { + "cell_type": "code", + "execution_count": 11, + "id": "cac66ba9", + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "$\\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 & 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]$" + ], + "text/plain": [ + "Matrix([\n", + "[0, 0, 8.88178419700125e-17*P^x_gq*e_d^2, 2.77555756156289e-17*P_gq],\n", + "[0, 0, 1.11022302462516e-16*P^x_\\gamma d*e_d^2, 0],\n", + "[0, 0, 1.11022302462516e-16*P^x_+d*e_d^2 + 3.5527136788005e-17*P^x_ps*e_u^2**2, -3.5527136788005e-17*P^x_ps*e_d^2*e_u^2 + 3.5527136788005e-17*P^x_ps*e_u^2**2 + 2.77555756156289e-17*P_+],\n", + "[0, 0, -1.11022302462516e-16*P^x_+d*e_d^2 + 1.11022302462516e-16*P^x_+u*e_u^2 + 5.32907051820075e-17*P^x_ps*e_d^2*e_u^2 + 3.5527136788005e-17*P^x_ps*e_u^2**2 + 5.55111512312578e-17*P_qq, e_u^2*(1.11022302462516e-16*P^x_+u - 5.55111512312578e-17*P^x_ps*e_d^2 + 5.55111512312578e-17*P^x_ps*e_u^2)]])" + ] + }, + "execution_count": 11, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "sympy.simplify(P_ev_sing(5)-P_ev_sing2(5))" + ] + }, + { + "cell_type": "code", + "execution_count": 12, + "id": "42a4712c", + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "$\\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]$" + ], + "text/plain": [ + "Matrix([\n", + "[0, 0, -1.11022302462516e-16*P_gq, 0],\n", + "[0, 0, 0, 0],\n", + "[0, 0, -1.11022302462516e-16*P_+, 0],\n", + "[0, 0, 0, -1.11022302462516e-16*P_+]])" + ] + }, + "execution_count": 12, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "sympy.simplify(P_ev_sing(6)-P_ev_sing2(6))" + ] + }, + { + "cell_type": "code", + "execution_count": 13, + "id": "ac0e82fc", + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "$\\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]$" + ], + "text/plain": [ + "Matrix([\n", + "[ 1.11022302462516e-16*P^x_-d*e_d^2, 2.77555756156289e-17*P_-],\n", + "[-1.11022302462516e-16*P^x_-d*e_d^2 + 1.11022302462516e-16*P^x_-u*e_u^2 + 5.55111512312578e-17*P_V, 1.11022302462516e-16*P^x_-u*e_u^2]])" + ] + }, + "execution_count": 13, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "sympy.simplify(P_ev_val(5)-P_ev_val2(5))" + ] + }, + { + "cell_type": "code", + "execution_count": 14, + "id": "3e3c29ce", + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "$\\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": 14, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "sympy.simplify(P_ev_val(6)-P_ev_val2(6))" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "121e1f75", + "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 +} diff --git a/extras/uni-dglap/uni-dglap.ipynb b/extras/uni-dglap/uni-dglap.ipynb new file mode 100644 index 000000000..e0d31c949 --- /dev/null +++ b/extras/uni-dglap/uni-dglap.ipynb @@ -0,0 +1,769 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "id": "4790b144-ceb5-4cba-87a5-ca2fd37e5912", + "metadata": {}, + "source": [ + "# Unified DGLAP" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "id": "1d7616d9-f4a3-447a-9ebf-e99db8126ffb", + "metadata": {}, + "outputs": [], + "source": [ + "import sympy" + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "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": 6, + "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": 7, + "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": 8, + "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": 9, + "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": 10, + "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": 10, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "P_ev(2)" + ] + }, + { + "cell_type": "code", + "execution_count": 11, + "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": 11, + "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": 1, + "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=es2_(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": 2, + "id": "a47dc3b2", + "metadata": {}, + "outputs": [ + { + "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": [ + "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]$" + ], + "text/plain": [ + "Matrix([\n", + "[0, 0, 0, 0],\n", + "[0, 0, 0, 0],\n", + "[0, 0, 0, 0],\n", + "[0, 0, 0, 0]])" + ] + }, + "execution_count": 328, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "sympy.simplify(P_ev_sing(4)-P_ev_sing2(4))" + ] + }, + { + "cell_type": "code", + "execution_count": 329, + "id": "e49bef49", + "metadata": {}, + "outputs": 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0\\\\0 & - 1.11022302462516 \\cdot 10^{-16} P_{-}\\end{matrix}\\right]$" + ], + "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": "26d06926", + "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 +} diff --git a/extras/uni-dglap/uni-dglap.tex b/extras/uni-dglap/uni-dglap.tex new file mode 100644 index 000000000..a5c82d077 --- /dev/null +++ b/extras/uni-dglap/uni-dglap.tex @@ -0,0 +1,174 @@ +\documentclass[a4paper,oneside]{article} +\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=1.5cm,bottom=1.5cm,left=1.5cm,right=1.5cm]{geometry} + + +\title{} +\author{} + +\date{} + +\begin{document} + +%\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*} +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*} +\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}_{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 \\ +\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*} +\item Decoupled sector: +\begin{align*} +\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} diff --git a/src/eko/anomalous_dimensions/__init__.py b/src/eko/anomalous_dimensions/__init__.py index bdf7d67ef..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 as1, 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/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 new file mode 100644 index 000000000..20e2b8dad --- /dev/null +++ b/src/eko/anomalous_dimensions/aem2.py @@ -0,0 +1,327 @@ +# -*- coding: utf-8 -*- +""" +This file contains the O(aem2) Altarelli-Parisi splitting kernels. +""" + +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(aem2) photon-photon singlet anomalous dimension. + + Implements Eq. (68) of :cite:`deFlorian:2016gvk`. + + Parameters + ---------- + + Returns + ------- + gamma_gg : complex + O(aem2) photon-photon singlet anomalous dimension + :math:`\\gamma_{\\gamma \\gamma}^{(0,2)}(N)` + + """ + + nu = constants.uplike_flavors(nf) + nd = nf - nu + return ( + constants.NC + * (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(aem2) quark-photon anomalous dimension for up quarks. + + Implements Eq. (55) 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_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. + + Implements Eq. (55) 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_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. + + Implements Eq. (56) 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_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] + 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,u1,c16[:])", cache=True) +def gamma_phd(N, nf, sx): + """Computes the O(aem2) photon-quark anomalous dimension for down quarks. + + Implements Eq. (56) 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_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] + 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(aem2) singlet-like non-singlet anomalous dimension for up quarks. + + Implements sum of 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_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) + nd = nf - nu + eSigma2 = constants.NC * (nu * constants.eu2 + nd * constants.ed2) + tmp = ( + 2 + * (-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 + ) * 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(aem2) singlet-like non-singlet anomalous dimension for down quarks. + + Implements sum of 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_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) + nd = nf - nu + eSigma2 = constants.NC * (nu * constants.eu2 + nd * constants.ed2) + tmp = ( + 2 + * (-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 + ) * eSigma2 + 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) valence-like non-singlet anomalous dimension for up quarks. + + 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(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) + nd = nf - nu + eSigma2 = constants.NC * (nu * constants.eu2 + nd * constants.ed2) + tmp = ( + 2 + * (-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 + ) * 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) valence-like non-singlet anomalous dimension for down quarks. + + 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(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) + nd = nf - nu + eSigma2 = constants.NC * (nu * constants.eu2 + nd * constants.ed2) + tmp = ( + 2 + * (-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 + ) * 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 O(aem2) pure-singlet quark-quark anomalous dimension. + + Implements Eq. (59) of :cite:`deFlorian:2016gvk`. + + Parameters + ---------- + N : complex + Mellin moment + nf : int + Number of active flavors + + Returns + ------- + gamma_ps : complex + O(aem2) pure-singlet quark-quark anomalous dimension + :math:`\\gamma_{ps}^{(0,2)}(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 * constants.CA * result diff --git a/src/eko/anomalous_dimensions/as1aem1.py b/src/eko/anomalous_dimensions/as1aem1.py new file mode 100644 index 000000000..d9dc92709 --- /dev/null +++ b/src/eko/anomalous_dimensions/as1aem1.py @@ -0,0 +1,359 @@ +# -*- coding: utf-8 -*- +""" +This file contains the O(as1aem1) Altarelli-Parisi splitting kernels. +""" + +import numba as nb +import numpy as np + +from .. import constants +from . import harmonics + + +@nb.njit("c16(c16,c16[:])", cache=True) +def gamma_phq(N, sx): + """Computes the O(as1aem1) photon-quark anomalous dimension + + Implements Eq. (36) of :cite:`deFlorian:2015ujt`. + + Parameters + ---------- + N : complex + Mellin moment + 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. (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)` + + """ + 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 ( + 2 + * nf + * constants.CA + * 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. (27) of :cite:`deFlorian:2015ujt`. + + 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 + * 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)) + ) + + +@nb.njit("c16(c16)", cache=True) +def gamma_phg(N): + """Computes the O(as1aem1) photon-gluon anomalous dimension + + Implements Eq. (30) of :cite:`deFlorian:2015ujt`. + + 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 / 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. + + Implements Eq. (29) of :cite:`deFlorian:2015ujt`. + + 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 / 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. + + Implements Eq. (35) of :cite:`deFlorian:2015ujt`. + + Parameters + ---------- + N : complex + Mellin moment + 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, sx) + + +@nb.njit("c16(u1)", cache=True) +def gamma_phph(nf): + """Computes the O(as1aem1) photon-photon singlet anomalous dimension. + + Implements Eq. (28) of :cite:`deFlorian:2015ujt`. + + Parameters + ---------- + nf : int + Number of active flavors + + Returns + ------- + 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) + + +@nb.njit("c16()", cache=True) +def gamma_gg(): + """Computes the O(as1aem1) gluon-gluon singlet anomalous dimension. + + Implements Eq. (31) of :cite:`deFlorian:2015ujt`. + + 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,c16[:])", cache=True) +def gamma_nsp(N, 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 + 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) + g3Np2 = harmonics.mellin_g3(N + 2) + zeta2 = harmonics.zeta2 + zeta3 = harmonics.zeta3 + result = ( + +32 * zeta2 * S1h + - 32 * zeta2 * 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 * S2h + - 32 * S2 + + 16 * S2p1h + ) + + ( + -8 + + N + * ( + -32 + + N * (-8 - 3 * N * (3 + N) * (3 + N**2) - 48 * (1 + N) ** 2 * zeta2) + ) + ) + / (N**3 * (1 + N) ** 3) + + 32 * (g3N + g3Np2) + + 4 * S3p1h + - 16 * zeta3 + ) + return constants.CF * result + + +@nb.njit("c16(c16,c16[:])", cache=True) +def gamma_nsm(N, sx): + """Computes the O(as1aem1) valence-like non-singlet anomalous dimension. + + Implements difference between Eqs. (33-34) of :cite:`deFlorian:2015ujt`. + + Parameters + ---------- + N : complex + Mellin moment + 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) + 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 + + 8.0 / (N + N**2) * S2p1h + + S1 + * ( + 16 * (-1 / N**2 + 1 / (1 + N) ** 2 + 2 * zeta2) + + 16 * S2h + - 32 * S2 + - 16 * S2p1h + ) + + ( + 72 + + N + * ( + 96 + - 3 * N * (8 + 3 * N * (3 + N) * (3 + N**2)) + + 48 * N * (1 + N) ** 2 * zeta2 + ) + ) + / (3.0 * N**3 * (1 + N) ** 3) + - 32 * (g3N + g3Np2) + + 32.0 * zeta2 * S1p1h + + 4 * S3h + - 32 * S3 + - 4 * S3p1h + - 16 * zeta3 + ) + return constants.CF * result diff --git a/src/eko/constants.py b/src/eko/constants.py index a9bb89027..d45b47f00 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 +39,23 @@ 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 not in range(2, 6 + 1): + raise NotImplementedError("Selected nf is not implemented") + nu = nf // 2 + return nu 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 + ) diff --git a/tests/eko/test_ad_aem2.py b/tests/eko/test_ad_aem2.py new file mode 100644 index 000000000..f975578b2 --- /dev/null +++ b/tests/eko/test_ad_aem2.py @@ -0,0 +1,56 @@ +# -*- coding: utf-8 -*- +# Test O(as1aem1) splitting functions +import numpy as np +from test_ad_as3 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 in range(2, 6 + 1): + 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 = constants.uplike_flavors(NF) + ND = NF - NU + 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 = constants.uplike_flavors(NF) + ND = NF - NU + 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, + ) 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_as1aem1.py b/tests/eko/test_ad_as1aem1.py new file mode 100644 index 000000000..ddd486a89 --- /dev/null +++ b/tests/eko/test_ad_as1aem1.py @@ -0,0 +1,62 @@ +# -*- coding: utf-8 -*- +# Test O(as1aem1) splitting functions +import numpy as np +import pytest +from test_ad_as3 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) + 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) + for NF in range(2, 6 + 1): + NU = constants.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) + + (NU * constants.eu2 + ND * constants.ed2) * ad.as1aem1.gamma_phg(N) + + (NU * constants.eu2 + ND * constants.ed2) * ad.as1aem1.gamma_gg(), + 0, + ) + with pytest.raises(NotImplementedError): + constants.uplike_flavors(7) + + +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 = constants.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) + + 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, + ) 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