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 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 and the photon, as they correspond to the operator in the Lagrange density: .. math :: \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 `). - most cross section programs as well as `LHAPDF `_ :cite:`Buckley:2014ana` use this basis - we will consider this basis as the canonical basis +/- Basis --------- Instead of using the raw flavors, we recombine the quark flavors into .. math :: q^\pm = q \pm \bar q as this is closer to the actual physics: :math:`q^-` corresponds to the valence quark distribution that e.g. in the proton will carry most of the momentum at large x and :math:`q^+` effectively is the sea quark distribution: .. math :: \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` QCD Evolution Basis ------------------- As the gluon is flavor-blind it is handy to solve |DGLAP| not in the flavor basis, 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 valence distribution :math:`V`, second the valence-like distributions :math:`V_3 \ldots V_{35}`, and third the singlet like distributions :math:`T_3 \ldots T_{35}`. The mapping between the Evolution Basis and the +/- Basis is given by .. math :: \Sigma &= \sum\limits_{j}^6 q_j^+\\ V &= \sum\limits_{j}^6 q_j^-\\ V_3 &= u^- - d^-\\ V_8 &= u^- + d^- - 2 s^-\\ V_{15} &= u^- + d^- + s^- - 3 c^-\\ V_{24} &= u^- + d^- + s^- + c^- - 4 b^-\\ V_{35} &= u^- + d^- + s^- + c^- + b^- - 5 t^-\\ T_3 &= u^+ - d^+\\ T_8 &= u^+ + d^+ - 2 s^+\\ 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(\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 :math:`k` follow the common group-theoretical notation :math:`k = n_f^2 - 1` where :math:`n_f` denotes the incorporated number of quark flavors - 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 QCD Evolution Bases ----------------------------- 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 |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 :: \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 :math:`V_{(3)}` is not to be confused with the usual (|QCD| like) :math:`V_3`. - 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_{(5)}, \Sigma_{\Delta,(5)}, V_{(5)}, V_{\Delta,(5)}, 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_unified_evol` - the factors 3/2 in the definition of :math:`V_{\Delta,(5)}` and :math:`\Sigma_{\Delta,(5)}` are needed in order to have an orthogonal basis for :math:`n_f=5` Other Bases ----------- In an |PDF| fitting environment sometimes yet different bases are used to enforce or improve positivity of the |PDF| :cite:`Candido:2020yat`. E.g. :cite:`Giele:2002hx` uses .. math :: u_v = u^-, d_v = d^-, L_+ = 2(\bar u + \bar d), L_- = \bar d - \bar u, s^+, c^+, b^+, g Operator Bases -------------- An |EKO| :math:`\mathbf E` is an operator in the Flavor Space :math:`\mathcal F` mapping one vector onto an other: .. math :: \mathbf E \in \mathcal F \otimes \mathcal F since evolution can (and will) mix flavors. To specify the basis for these operators we need to specify the basis for both the input and output space. Operator Flavor Basis ^^^^^^^^^^^^^^^^^^^^^ - here we mean :ref:`theory/FlavorSpace:Flavor Basis` both in the input and the output space - the :class:`~eko.output.Output` is delivered in this basis - this basis has :math:`(2n_f+ 1)^2 = 13^2 = 169` elements - this basis can span arbitrary matching scales Operator Anomalous Dimension Basis ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ - here we mean the true underlying physical basis where elements correspond to the different splitting functions, i.e. :math:`\mathbf{E}_S, E_{ns,v}, E_{ns,+}, E_{ns,-}` - this basis has 4 elements in |LO|, 6 elements in |NLO| and its maximum 7 elements after |NNLO| - 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 QCD Evolution Basis ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ - 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 - this basis can span arbitrary matching scales