Flavor Space

An EKO is a rank-4 operator acting both in Flavor Space \(\mathcal F\) and momentum fraction space \(\mathcal X\). By Flavor Space \(\mathcal F\) 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:

\[\mathcal F = \mathcal F_{fl} = \span(\gamma, g, u, \bar u, d, \bar d, s, \bar s, c, \bar c, b, \bar b, t, \bar t)\]

  • we deliver the Output in this basis, although the flavors are slightly differently arranged (Implementation: here).

  • most cross section programs as well as LHAPDF [BFL+15] 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

\[q^\pm = q \pm \bar q\]

as this is closer to the actual physics: \(q^-\) corresponds to the valence quark distribution that e.g. in the proton will carry most of the momentum at large x and \(q^+\) effectively is the sea quark distribution:

\[\mathcal F \sim \mathcal F_{\pm} = \span(\gamma, g, u^+, u^-, d^+, d^-, s^+, s^-, c^+, c^-, b^+, b^-, t^+, t^-)\]

  • this basis is not normalized with respect to the canonical Flavor Basis

  • the basis transformation to the Flavor Basis is implemented in 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 \(\Sigma\) and the gluon distribution \(g\), and the non-singlet sector. The non-singlet sector can be again subdivided into three groups: first the full valence distribution \(V\), second the valence-like distributions \(V_3 \ldots V_{35}\), and third the singlet like distributions \(T_3 \ldots T_{35}\). The mapping between the Evolution Basis and the +/- Basis is given by

\[\begin{split}\Sigma &= \sum\limits_{j}^6 q_j^+\\ V &= \sum\limits_{j}^6 q_j^-\\ V_3 &= u^- - d^-\\ V_8 &= u^- + d^- - 2 s^-\\ V_{15} &= u^- + d^- + s^- - 3 c^-\\ V_{24} &= u^- + d^- + s^- + c^- - 4 b^-\\ V_{35} &= u^- + d^- + s^- + c^- + b^- - 5 t^-\\ T_3 &= u^+ - d^+\\ T_8 &= u^+ + d^+ - 2 s^+\\ T_{15} &= u^+ + d^+ + s^+ - 3 c^+\\ T_{24} &= u^+ + d^+ + s^+ + c^+ - 4 b^+\\ T_{35} &= u^+ + d^+ + s^+ + c^+ + b^+ - 5 t^+\\ \mathcal F \sim \mathcal F_{ev} &= \span(\gamma, g, \Sigma, V, V_{3}, V_{8}, V_{15}, V_{24}, V_{35}, T_{3}, T_{8}, T_{15}, T_{24}, T_{35})\end{split}\]

  • the associated numbers to the valence-like and singlet-like non-singlet distributions \(k\) follow the common group-theoretical notation \(k = n_f^2 - 1\) where \(n_f\) 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 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. \(T_{15}\) a composite object in a evolution range below the charm mass. Instead, we will keep the non participating distributions here in their \(q^\pm\) representation. The Intrinsic QCD Evolution Bases will explicitly depend on the number of light flavors \(n_f\). For \(n_f=3\) we define (the other cases are defined analogously):

\[\begin{split}\Sigma_{(3)} &= u^+ + d^+ +s^+\\ V_{(3)} &= u^- + d^- + s^-\\ \mathcal F \sim \mathcal F_{iev,3} &= \span(\gamma, g, \Sigma_{(3)}, V_{(3)}, V_3, V_8, T_3, T_8, c^+, c^-, b^+, b^-, t^+, t^-)\end{split}\]

where \(V_{(3)}\) is not to be confused with the usual (QCD like) \(V_3\).

  • for \(n_f=6\) the Intrinsic QCD Evolution Basis coincides with the QCD Evolution Basis: \(\mathcal F_{iev,6} = \mathcal F_{ev}\)

  • this basis is not normalized with respect to the canonical Flavor Basis

  • the basis transformation from the Flavor Basis is implemented in pids_from_intrinsic_evol()

  • note that for the case of non-intrinsic component the higher elements in \(\mathcal F_{ev}\) do become linear dependent to other basis vectors (e.g. \(\left. T_{15}\right|_{c^+ = 0} = \Sigma\)) but are non zero - instead in \(\mathcal F_{iev,3}\) 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

\[\begin{split}\Sigma_u & = u^+ + c^+ + t^+ \\ \Sigma_d & = d^+ + s^+ + b^+ \\ V_u & = u^- + c^- + t^- \\ V_d & = d^- + s^- + b^- \\\end{split}\]

we have that in this case the QED \(\otimes\) QCD evolution basis that performs the maximal decoupling is given by:

\[\begin{split}\Sigma &= \Sigma_u + \Sigma_d \\ \Sigma_{\Delta} &= \Sigma_u - \Sigma_d \\ V &= V_u + V_d \\ V_{\Delta} &= V_u - V_d \\ T_3^u &=u^+ - c^+ \\ T_8^u &=u^+ + c^+ - 2t^+ \\ T_3^d &=d^+ - s^+ \\ T_8^d &=d^+ + s^+ - 2b^+ \\ V_3^u &=u^- - c^- \\ V_8^u &=u^- + c^- - 2t^- \\ V_3^d &=d^- - s^- \\ V_8^d &=d^- + s^- - 2b^- \\ \mathcal F \sim \mathcal F_{uni,ev} &= \span(\gamma, g, \Sigma, \Sigma_{\Delta}, V, V_{\Delta}, T_3^u, T_8^u, T_3^d, T_8^d, V_3^u, V_8^u, V_3^d, V_8^d)\end{split}\]

  • this basis is not normalized with respect to the canonical Flavor Basis

  • The singlet \(\Sigma\) is just the QCD singlet

  • The valence \(V\) 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 \(n_f=3\) light flavors we find:

\[\begin{split}\Sigma_{(3)} &= u^+ + d^+ + s^+\\ \Sigma_{\Delta,(3)} &= 2u^+ - d^+ - s^+ \\ V_{(3)} &= u^- + d^- + s^-\\ V_{\Delta,(3)} &= 2u^- - d^- - s^-\\ T_3^d &=d^+ - s^+ \\ V_3^d &=d^- - s^- \\ \mathcal F \sim \mathcal F_{uni,iev,3} &= \span(\gamma, g, \Sigma_{(3)}, \Sigma_{\Delta,(3)}, V_{(3)}, V_{\Delta,(3)}, T_3^d, V_3^d, c^+, c^-, b^+, b^-, t^+, t^-)\end{split}\]

For \(n_f=4\) light flavors we find:

\[\begin{split}\Sigma_{(4)} &= u^+ + d^+ + s^+ + c^+\\ \Sigma_{\Delta,(4)} &= u^+ + c^+ - d^+ - s^+\\ V_{(4)} &= u^- + d^- + s^- + c^-\\ V_{\Delta,(4)} &= u^- + c^- - d^- - s^-\\ T_3^u &=u^+ - c^+ \\ T_3^d &=d^+ - s^+ \\ V_3^u &=u^- - c^- \\ V_3^d &=d^- - s^- \\ \mathcal F \sim \mathcal F_{uni,iev,4} &= \span(\gamma, g, \Sigma_{(4)}, \Sigma_{\Delta,(4)}, V_{(4)}, V_{\Delta,(4)}, V_3^d, T_3^d, V_3^u, T_3^u, b^+, b^-, t^+, t^-)\end{split}\]

For \(n_f=5\) light flavors we find:

\[\begin{split}\Sigma_{(5)} &= u^+ + d^+ + s^+ + c^+ + b^+\\ \Sigma_{\Delta,(5)} &= \frac{3}{2}u^+ + \frac{3}{2}c^+ - d^+ -s^+ - b^+\\ V_{(5)} &= u^- + d^- + s^- + c^- + b^-\\ V_{\Delta,(5)} &= \frac{3}{2}u^- + \frac{3}{2}c^- - d^- -s^- - b^-\\ T_3^u &=u^+ - c^+ \\ T_3^d &=d^+ - s^+ \\ V_3^u &=u^- - c^- \\ V_3^d &=d^- - s^- \\ T_8^d &=d^+ + s^+ - 2b^+ \\ V_8^d &=d^- + s^- - 2b^- \\ \mathcal F \sim \mathcal F_{uni,iev,5} &= \span(\gamma, g, \Sigma_{(5)}, \Sigma_{\Delta,(5)}, V_{(5)}, V_{\Delta,(5)}, V_3^d, T_3^d, V_3^u, T_3^u, T_8^d, V_8^d, t^+, t^-)\end{split}\]

For \(n_f=6\) light flavors the Intrinsic Unified Evolution Basis coincides with the 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 pids_from_intrinsic_unified_evol()

  • the factors 3/2 in the definition of \(V_{\Delta,(5)}\) and \(\Sigma_{\Delta,(5)}\) are needed in order to have an orthogonal basis for \(n_f=5\)

Other Bases

In an PDF fitting environment sometimes yet different bases are used to enforce or improve positivity of the PDF [CFH20]. E.g. [G+02] uses

\[u_v = u^-, d_v = d^-, L_+ = 2(\bar u + \bar d), L_- = \bar d - \bar u, s^+, c^+, b^+, g\]

Operator Bases

An EKO \(\mathbf E\) is an operator in the Flavor Space \(\mathcal F\) mapping one vector onto an other:

\[\mathbf E \in \mathcal F \otimes \mathcal F\]

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 Flavor Basis both in the input and the output space

  • the Output is delivered in this basis

  • this basis has \((2n_f+ 1)^2 = 13^2 = 169\) 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. \(\mathbf{E}_S, E_{ns,v}, E_{ns,+}, E_{ns,-}\)

  • 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