QED - flavor basis

doc/source/development/Benchmarks.rst

@@ -58,7 +58,7 @@ APFEL
 (and FTDY as well).
 It has been used by the NNPDF collaboration up to NNPDF4.0
 
-|APFEL| solves |DGLAP| numerically in x-space up to |NNLO|. QED evolution is also available.
+|APFEL| solves |DGLAP| numerically in x-space up to |NNLO|. |QED| evolution is also available.
 The programs provides 3 different strategies, and in various theory setups (|FNS|, SV, IC ) as shown in the table.
 As |Eko|, |APFEL| can be interfaced with |lhapdf|.
 

doc/source/refs.bib

@@ -447,6 +447,34 @@ @article{Liu:2015fxa
     year = "2015"
 }
 
+@article{deFlorian:2015ujt,
+    author = "de Florian, Daniel and Sborlini, Germ\'an F. R. and Rodrigo, Germ\'an",
+    title = "{QED corrections to the Altarelli\textendash{}Parisi splitting functions}",
+    eprint = "1512.00612",
+    archivePrefix = "arXiv",
+    primaryClass = "hep-ph",
+    reportNumber = "ICAS-03-15, IFIC-15-81",
+    doi = "10.1140/epjc/s10052-016-4131-8",
+    journal = "Eur. Phys. J. C",
+    volume = "76",
+    number = "5",
+    pages = "282",
+    year = "2016"
+}
+
+@article{deFlorian:2016gvk,
+    author = "de Florian, Daniel and Sborlini, Germ\'an F. R. and Rodrigo, Germ\'an",
+    title = "{Two-loop QED corrections to the Altarelli-Parisi splitting functions}",
+    eprint = "1606.02887",
+    archivePrefix = "arXiv",
+    primaryClass = "hep-ph",
+    reportNumber = "ICAS-09-16, IFIC-15-88",
+    doi = "10.1007/JHEP10(2016)056",
+    journal = "JHEP",
+    volume = "10",
+    pages = "056",
+    year = "2016"
+}
 @article{Chetyrkin:2017bjc,
     author = "Chetyrkin, K. G. and Falcioni, G. and Herzog, F. and Vermaseren, J. A. M.",
     title = "{Five-loop renormalisation of QCD in covariant gauges}",

doc/source/shared/abbreviations.rst

@@ -49,6 +49,12 @@
 .. |RGE| replace::
    :abbr:`RGE (renormalization group equation)`
 
+.. |QCD| replace::
+   :abbr:`QCD (Quantum Chromodynamics)`
+
+.. |QED| replace::
+   :abbr:`QED (Quantum Electrodynamics)`
+
 .. external
 
 .. |yadism| replace::

doc/source/theory/FlavorSpace.rst

@@ -3,19 +3,19 @@ Flavor Space
 
 An |EKO| is a rank-4 operator acting both in Flavor Space :math:`\mathcal F`
 and momentum fraction space :math:`\mathcal X`.
-By Flavor Space :math:`\mathcal F` we mean the 13-dimensional function space that contains
+By Flavor Space :math:`\mathcal F` we mean the 14-dimensional function space that contains
 the different |PDF| flavor. Note, that there is an ambiguity concerning the
 word "Flavor Basis" which is sometimes referred to as an *abstract* basis
 in the Flavor Space, but often the specific basis described here below is meant.
 
 Flavor Basis
 ------------
 
-Here we use the raw quark flavors along with the gluon as they correspond to the
+Here we use the raw quark flavors along with the gluon and the photon, as they correspond to the
 operator in the Lagrange density:
 
 .. math ::
-    \mathcal F = \mathcal F_{fl} = \span(g, u, \bar u, d, \bar d, s, \bar s, c, \bar c, b, \bar b, t, \bar t)
+    \mathcal F = \mathcal F_{fl} = \span(\gamma, g, u, \bar u, d, \bar d, s, \bar s, c, \bar c, b, \bar b, t, \bar t)
 
 - we deliver the :class:`~eko.output.Output` in this basis, although the flavors are
   slightly differently arranged (Implementation: :data:`here <eko.basis_rotation.flavor_basis_pids>`).
@@ -35,17 +35,18 @@ that e.g. in the proton will carry most of the momentum at large x and :math:`q^
 sea quark distribution:
 
 .. math ::
-    \mathcal F \sim \mathcal F_{\pm} = \span(g, u^+, u^-, d^+, d^-, s^+, s^-, c^+, c^-, b^+, b^-, t^+, t^-)
+    \mathcal F \sim \mathcal F_{\pm} = \span(\gamma, g, u^+, u^-, d^+, d^-, s^+, s^-, c^+, c^-, b^+, b^-, t^+, t^-)
 
 - this basis is *not* normalized with respect to the canonical Flavor Basis
 - the basis transformation to the Flavor Basis is implemented in
   :meth:`~eko.evolution_operator.flavors.rotate_pm_to_flavor`
 
-Evolution Basis
----------------
+QCD Evolution Basis
+-------------------
 
 As the gluon is flavor-blind it is handy to solve |DGLAP| not in the flavor basis,
-but in the Evolution Basis where instead we need to solve a minimal coupled system.
+but in the |QCD| Evolution Basis where instead we need to solve a minimal coupled system.
+This is the basis in which |DGLAP| equations are solved when only |QCD| corrections are taken into account.
 The new basis elements can be separated into two major classes: the singlet sector, consisting of the
 singlet distribution :math:`\Sigma` and the gluon distribution :math:`g`, and the non-singlet
 sector. The non-singlet sector can be again subdivided into three groups: first the full
@@ -66,7 +67,7 @@ The mapping between the Evolution Basis and the +/- Basis is given by
     T_{15} &= u^+ + d^+ + s^+ - 3 c^+\\
     T_{24} &= u^+ + d^+ + s^+ + c^+ - 4 b^+\\
     T_{35} &= u^+ + d^+ + s^+ + c^+ + b^+ - 5 t^+\\
-    \mathcal F \sim \mathcal F_{ev} &= \span(g, \Sigma, V, V_{3}, V_{8}, V_{15}, V_{24}, V_{35}, T_{3}, T_{8}, T_{15}, T_{24}, T_{35})
+    \mathcal F \sim \mathcal F_{ev} &= \span(\gamma, g, \Sigma, V, V_{3}, V_{8}, V_{15}, V_{24}, V_{35}, T_{3}, T_{8}, T_{15}, T_{24}, T_{35})
 
 
 - the associated numbers to the valence-like and singlet-like non-singlet distributions
@@ -75,30 +76,125 @@ The mapping between the Evolution Basis and the +/- Basis is given by
 - this basis is *not* normalized with respect to the canonical Flavor Basis
 - the basis transformation from the Flavor Basis is implemented in
   :data:`~eko.basis_rotation.rotate_flavor_to_evolution`
+- the photon is just a spectator and does not couple to anyone
 
-Intrinsic Evolution Bases
--------------------------
+Intrinsic QCD Evolution Bases
+-----------------------------
 
-However, the Evolution Basis is not yet the most decoupled basis if we consider intrinsic evolution.
+However, the |QCD| Evolution Basis is not yet the most decoupled basis if we consider intrinsic evolution.
 The intrinsic distributions do *not* participate in the |DGLAP| equation but instead evolve with a unity operator:
 this makes, e.g. :math:`T_{15}` a composite object in a evolution range below the charm mass.
 Instead, we will keep the non participating distributions here in their :math:`q^\pm` representation.
-The Intrinsic Evolution Bases will explicitly depend on the number of light flavors :math:`n_f`.
+The Intrinsic |QCD| Evolution Bases will explicitly depend on the number of light flavors :math:`n_f`.
 For :math:`n_f=3` we define (the other cases are defined analogously):
 
 .. math ::
-    \mathcal F \sim  \mathcal F_{iev,3} = \span(g, \Sigma_{(3)}, V_{(3)}, V_3, V_8, T_3, T_8, c^+, c^-, b^+, b^-, t^+, t^-)
+  \Sigma_{(3)} &= u^+ + d^+ +s^+\\
+  V_{(3)} &= u^- + d^- + s^-\\
+  \mathcal F \sim  \mathcal F_{iev,3} &= \span(\gamma, g, \Sigma_{(3)}, V_{(3)}, V_3, V_8, T_3, T_8, c^+, c^-, b^+, b^-, t^+, t^-)
 
-where we defined :math:`\Sigma_{(3)} = \sum\limits_{j=1}^3 q_j^+` and :math:`V_{(3)} = \sum\limits_{j=1}^3 q_j^-`
-(not to be confused with the usual :math:`V_3`).
+where :math:`V_{(3)}` is not to be confused with the usual (|QCD| like) :math:`V_3`.
 
-- for :math:`n_f=6` the Intrinsic Evolution Basis coincides with the Evolution Basis: :math:`\mathcal F_{iev,6} = \mathcal F_{ev}`
+- for :math:`n_f=6` the Intrinsic |QCD| Evolution Basis coincides with the |QCD| Evolution Basis: :math:`\mathcal F_{iev,6} = \mathcal F_{ev}`
 - this basis is *not* normalized with respect to the canonical Flavor Basis
 - the basis transformation from the Flavor Basis is implemented in
   :meth:`~eko.evolution_operator.flavors.pids_from_intrinsic_evol`
 - note that for the case of non-intrinsic component the higher elements in :math:`\mathcal F_{ev}` do become linear dependent
   to other basis vectors (e.g. :math:`\left. T_{15}\right|_{c^+ = 0} = \Sigma`) but are non zero - instead in :math:`\mathcal F_{iev,3}`
   this direction vanishes
+- the photon is just a spectator and does not couple to anyone
+
+
+Unified Evolution Basis
+-----------------------
+
+In presence of |QED| corrections to |DGLAP| evolution equations,
+the |QCD| Evolution basis does not decouple the distributions
+as it was for the pure |QCD| evolution.
+
+Defining the following combinations
+
+.. math ::
+  \Sigma_u & = u^+ + c^+ + t^+ \\
+  \Sigma_d & = d^+ + s^+ + b^+ \\
+  V_u & = u^- + c^- + t^- \\
+  V_d & = d^- + s^- + b^- \\
+
+we have that in this case the |QED| :math:`\otimes` |QCD| evolution basis that performs the maximal decoupling is given by:
+
+.. math ::
+  \Sigma &= \Sigma_u + \Sigma_d \\
+  \Sigma_{\Delta} &= \Sigma_u - \Sigma_d \\
+  V &= V_u + V_d \\
+  V_{\Delta} &= V_u - V_d \\
+  T_3^u &=u^+ - c^+ \\
+  T_8^u &=u^+ + c^+ - 2t^+ \\
+  T_3^d &=d^+ - s^+ \\
+  T_8^d &=d^+ + s^+ - 2b^+ \\
+  V_3^u &=u^- - c^- \\
+  V_8^u &=u^- + c^- - 2t^- \\
+  V_3^d &=d^- - s^- \\
+  V_8^d &=d^- + s^- - 2b^- \\
+  \mathcal F \sim  \mathcal F_{uni,ev} &= \span(\gamma, g, \Sigma, \Sigma_{\Delta}, V, V_{\Delta}, T_3^u, T_8^u, T_3^d, T_8^d, V_3^u, V_8^u, V_3^d, V_8^d)
+
+
+- this basis is *not* normalized with respect to the canonical Flavor Basis
+- The singlet :math:`\Sigma` is just the |QCD| singlet
+- The valence :math:`V` is just the |QCD| valence
+
+
+Intrinsic Unified Evolution Basis
+---------------------------------
+
+Again, we need the generalization to the presence of intrinsic (static) distributions.
+As |QED| can distinguish between up-like and down-like flavors the situation is again slightly
+more involved.
+
+For :math:`n_f=3` light flavors we find:
+
+.. math ::
+  \Sigma_{(3)} &= u^+ + d^+ + s^+\\
+  \Sigma_{\Delta,(3)} &= 2u^+ - d^+ - s^+ \\
+  V_{(3)} &= u^- + d^- + s^-\\
+  V_{\Delta,(3)} &= 2u^- - d^- - s^-\\
+  T_3^d &=d^+ - s^+ \\
+  V_3^d &=d^- - s^- \\
+  \mathcal F \sim  \mathcal F_{uni,iev,3} &= \span(\gamma, g, \Sigma_{(3)}, \Sigma_{\Delta,(3)}, V_{(3)}, V_{\Delta,(3)}, T_3^d, V_3^d, c^+, c^-, b^+, b^-, t^+, t^-)
+
+For :math:`n_f=4` light flavors we find:
+
+.. math ::
+  \Sigma_{(4)} &= u^+ + d^+ + s^+ + c^+\\
+  \Sigma_{\Delta,(4)} &= u^+ + c^+ - d^+ - s^+\\
+  V_{(4)} &= u^- + d^- + s^- + c^-\\
+  V_{\Delta,(4)} &= u^- + c^- - d^- - s^-\\
+  T_3^u &=u^+ - c^+ \\
+  T_3^d &=d^+ - s^+ \\
+  V_3^u &=u^- - c^- \\
+  V_3^d &=d^- - s^- \\
+  \mathcal F \sim  \mathcal F_{uni,iev,4} &= \span(\gamma, g, \Sigma_{(4)}, \Sigma_{\Delta,(4)}, V_{(4)}, V_{\Delta,(4)}, V_3^d, T_3^d, V_3^u, T_3^u, b^+, b^-, t^+, t^-)
+
+For :math:`n_f=5` light flavors we find:
+
+.. math ::
+  \Sigma_{(5)} &= u^+ + d^+ + s^+ + c^+ + b^+\\
+  \Sigma_{\Delta,(5)} &= \frac{3}{2}u^+ + \frac{3}{2}c^+ - d^+ -s^+ - b^+\\
+  V_{(5)} &= u^- + d^- + s^- + c^- + b^-\\
+  V_{\Delta,(5)} &= \frac{3}{2}u^- + \frac{3}{2}c^- - d^- -s^- - b^-\\
+  T_3^u &=u^+ - c^+ \\
+  T_3^d &=d^+ - s^+ \\
+  V_3^u &=u^- - c^- \\
+  V_3^d &=d^- - s^- \\
+  T_8^d &=d^+ + s^+ - 2b^+ \\
+  V_8^d &=d^- + s^- - 2b^- \\
+  \mathcal F \sim  \mathcal F_{uni,iev,5} &= \span(\gamma, g, \Sigma_{(4)}, \Sigma_{\Delta,(4)}, V_{(4)}, V_{\Delta,(4)}, V_3^d, T_3^d, V_3^u, T_3^u, T_8^d, V_8^d, t^+, t^-)
+
+For :math:`n_f=6` light flavors the Intrinsic Unified Evolution Basis coincides with the :ref:`theory/FlavorSpace:Unified Evolution Basis`.
+
+- this basis is *not* normalized with respect to the canonical Flavor Basis
+- the basis transformation from the Flavor Basis is implemented in
+  :meth:`~eko.evolution_operator.flavors.pids_from_intrinsic_evol`
+- the factors 3/2 in the definition of :math:`V_{0,(5)}` and :math:`T_{0,(5)}` are needed in order to have an orthogonal basis for :math:`n_f=5`
 
 Other Bases
 -----------
@@ -137,10 +233,10 @@ Operator Anomalous Dimension Basis
 - this basis can *not* span any threshold but can only be used for a *fixed* number of flavors
 - all actual computations are done in this basis
 
-Operator Intrinsic Evolution Basis
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+Operator Intrinsic QCD Evolution Basis
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
 
-- here we mean :ref:`theory/FlavorSpace:Intrinsic Evolution Bases` both in the input and the output space
+- here we mean :ref:`theory/FlavorSpace:Intrinsic QCD Evolution Bases` both in the input and the output space
 - this basis does **not** coincide with the :ref:`theory/FlavorSpace:Operator Anomalous Dimension Basis` as the decision on which operator of that
   basis is used is a non-trivial decision - see :doc:`Matching`
 - this basis has :math:`2n_f+ 3 = 15` elements

doc/source/theory/Matching.rst

@@ -16,13 +16,13 @@ present :math:`\left(\mu_{h}^2 < Q_0^2 < Q_1^2 < \mu_{h+1}^2\right)`, in :doc:`M
     \tilde{\mathbf{f}}^{(n_f)}(Q^2_1)= \tilde{\mathbf{E}}^{(n_f)}(Q^2_1\leftarrow Q^2_0) \tilde{\mathbf{f}}^{(n_f)}(Q^2_0)
 
 The bold font indicates the vector space spanned by the :doc:`flavor space <FlavorSpace>` and the equations decouple mostly
-in the :ref:`Intrinsic Evolution Basis <theory/FlavorSpace:Intrinsic Evolution Bases>`.
+in the :ref:`Intrinsic Evolution Basis <theory/FlavorSpace:Intrinsic QCD Evolution Bases>`.
 
 If a single threshold :math:`\left(\mu_{h-1}^2 < Q_0^2 < \mu_{h}^2 < Q_1^2 < \mu_{h+1}^2\right)` is present
 we decompose the matching into two independent steps:
-first, the true QCD induced |OME| :math:`\mathbf{A}^{(n_f)}(\mu_{h}^2)` that are given by perturbative calculations and expressed in the flavor space,
+first, the true |QCD| induced |OME| :math:`\mathbf{A}^{(n_f)}(\mu_{h}^2)` that are given by perturbative calculations and expressed in the flavor space,
 and, second, the necessary :doc:`flavor space rotation <FlavorSpace>` :math:`\mathbf{R}^{(n_f)}` to fit the
-new :ref:`Intrinsic Evolution Basis <theory/FlavorSpace:Intrinsic Evolution Bases>`.
+new :ref:`Intrinsic Evolution Basis <theory/FlavorSpace:Intrinsic QCD Evolution Bases>`.
 We can then denote the solution as
 
 .. math ::
@@ -38,7 +38,7 @@ The matching matrices :math:`\mathbf{A}^{(n_f)}(\mu_{h+1}^2)` mediate between :m
 and :math:`\mathcal F_{iev,n_f}^{(n_f+1)}`, i.e. they transform the basis vectors of the :math:`n_f`-flavors space
 in a :math:`n_f`-flavor scheme to the :math:`(n_f+1)`-flavor scheme. Hence, the supscript refers to the flavor scheme
 with a smaller number of active flavors. To compute the matrices in a minimal coupled system we decompose the
-:ref:`Intrinsic Evolution Basis <theory/FlavorSpace:Intrinsic Evolution Bases>` :math:`\mathcal F_{iev,n_f}` into
+:ref:`Intrinsic Evolution Basis <theory/FlavorSpace:Intrinsic QCD Evolution Bases>` :math:`\mathcal F_{iev,n_f}` into
 several subspaces (of course irrespective of the |FNS|):
 
 .. math ::

doc/source/theory/pQCD.rst

@@ -6,7 +6,7 @@ Strong Coupling
 
 Implementation: :class:`~eko.strong_coupling.StrongCoupling`.
 
-We use perturbative QCD with the running coupling
+We use perturbative |QCD| with the running coupling
 :math:`a_s(\mu_R^2) = \alpha_s(\mu_R^2)/(4\pi)` given at 5-loop by
 :cite:`Herzog:2017ohr,Luthe:2016ima,Baikov:2016tgj,Chetyrkin:2017bjc,Luthe:2017ttg`
 
@@ -47,10 +47,10 @@ In particular, the matching involved in the change from :math:`n_f` to :math:`n_
 is presented in equation 3.1 of :cite:`Schroder:2005hy` for |MSbar| masses, while the
 same expression for POLE masses is reported in Appendix A.
 
-Splitting Functions
--------------------
+QCD Splitting Functions
+-----------------------
 
-The Altarelli-Parisi splitting kernels can be expanded in powers of the strong
+In the case in which only the |QCD| corrections are considered, the Altarelli-Parisi splitting kernels can be expanded in powers of the strong
 coupling :math:`a_s(\mu^2)` and are given by :cite:`Moch:2004pa,Vogt:2004mw`
 
 .. math ::
@@ -59,6 +59,24 @@ coupling :math:`a_s(\mu^2)` and are given by :cite:`Moch:2004pa,Vogt:2004mw`
 
 Note the additional minus in the definition of :math:`\gamma`.
 
+Unified Splitting Functions
+---------------------------
+
+When the |QED| corrections are taken into account, |DGLAP| equation take the form
+
+.. math ::
+    \mathbf{P}=\mathbf{\tilde{P}}+\mathbf{\bar{P}}
+
+where :math:`\mathbf{\tilde{P}}` are the usual |QCD| splitting kernels defined in the previous section,
+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
+
+The expression of the pure |QED| and of the mixed |QED| :math:`\otimes` |QCD| splitting kernels are given in
+:cite:`deFlorian:2015ujt,deFlorian:2016gvk`
+
 Scale Variations
 ----------------
 
@@ -83,7 +101,7 @@ corresponds to schemes A and B in :cite:`AbdulKhalek:2019ihb`.
 Heavy Quark Masses
 ------------------
 
-In QCD also the heavy quark masses (:math:`m_{c}, m_{b}, m_{t}`) follow a |RGE|
+In |QCD| also the heavy quark masses (:math:`m_{c}, m_{b}, m_{t}`) follow a |RGE|
 and their values depend on the energy scale at which the quark is probed.
 Masses do not play any role in a single flavour patch, but are important in
 |VFNS| when more flavour schemes need to be joined (see :doc:`matching
@@ -144,7 +162,7 @@ In doing so EKO takes advantage of the monotony of the |RGE| solution
 Now, being able to evaluate :math:`a_s(\mu_{h,0}^2)`, there are two ways of
 solving the previous integral and finally compute the evolved
 :math:`m_{\overline{MS},h}`. In fact, the function :math:`\gamma_m(a_s)` is the
-anomalous QCD mass dimension and, as the :math:`\beta` function, it can be evaluated
+anomalous |QCD| mass dimension and, as the :math:`\beta` function, it can be evaluated
 perturbatively in :math:`a_s` up to :math:`\mathcal{O}(a_s^4)`:
 
 .. math ::