FlavorSpace.rst

Flavor Space

An is a rank-4 operator acting both in Flavor Space ℱ and momentum fraction space 𝒳. By Flavor Space ℱ we mean the 14-dimensional function space that contains the different 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:

• we deliver the ~eko.output.Output in this basis, although the flavors are slightly differently arranged (Implementation: here <eko.basis_rotation.flavor_basis_pids>).
• most cross section programs as well as LHAPDF 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

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:

• this basis is not normalized with respect to the canonical Flavor Basis
• the basis transformation to the Flavor Basis is implemented in ~eko.evolution_operator.flavors.rotate_pm_to_flavor

QCD Evolution Basis

As the gluon is flavor-blind it is handy to solve not in the flavor basis, but in the Evolution Basis where instead we need to solve a minimal coupled system. This is the basis in which equations are solved when only corrections are taken into account. The new basis elements can be separated into two major classes: the singlet sector, consisting of the singlet distribution Σ 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 V₃…V₃₅, and third the singlet like distributions T₃…T₃₅. The mapping between the Evolution Basis and the +/- Basis is given by

• the associated numbers to the valence-like and singlet-like non-singlet distributions k follow the common group-theoretical notation k = n_f² − 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 ~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 Evolution Basis is not yet the most decoupled basis if we consider intrinsic evolution. The intrinsic distributions do not participate in the equation but instead evolve with a unity operator: this makes, e.g. T₁₅ a composite object in a evolution range below the charm mass. Instead, we will keep the non participating distributions here in their q^± representation. The Intrinsic 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):

where V₍₃₎ is not to be confused with the usual ( like) V₃.

• for n_f = 6 the Intrinsic Evolution Basis coincides with the Evolution Basis: ℱ_iev, 6 = ℱ_ev
• this basis is not normalized with respect to the canonical Flavor Basis
• the basis transformation from the Flavor Basis is implemented in ~eko.evolution_operator.flavors.pids_from_intrinsic_evol
• note that for the case of non-intrinsic component the higher elements in ℱ_ev do become linear dependent to other basis vectors (e.g. T₁₅|_c⁺ = 0 = Σ) but are non zero - instead in ℱ_iev, 3 this direction vanishes
• the photon is just a spectator and does not couple to anyone

Unified Evolution Basis

In presence of corrections to evolution equations, the Evolution basis does not decouple the distributions as it was for the pure evolution.

Defining the following combinations

we have that in this case the ⊗ evolution basis that performs the maximal decoupling is given by:

• this basis is not normalized with respect to the canonical Flavor Basis
• The singlet Σ is just the singlet
• The valence V is just the valence

Intrinsic Unified Evolution Basis

Again, we need the generalization to the presence of intrinsic (static) distributions. As can distinguish between up-like and down-like flavors the situation is again slightly more involved.

For n_f = 3 light flavors we find:

For n_f = 4 light flavors we find:

For n_f = 5 light flavors we find:

For n_f = 6 light flavors the Intrinsic Unified Evolution Basis coincides with the 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 ~eko.evolution_operator.flavors.pids_from_intrinsic_unified_evol
• the factors 3/2 in the definition of V_Δ, (5) and Σ_Δ, (5) are needed in order to have an orthogonal basis for n_f = 5

Other Bases

In an fitting environment sometimes yet different bases are used to enforce or improve positivity of the Candido:2020yat. E.g. Giele:2002hx uses

Operator Bases

An E is an operator in the Flavor Space ℱ mapping one vector onto an other:

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 theory/FlavorSpace:Flavor Basis both in the input and the output space
• the ~eko.output.Output is delivered in this basis
• this basis has (2n_f + 1)² = 13² = 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. E_S, E_ns, v, E_ns, +, E_ns, −
• this basis has 4 elements in , 6 elements in and its maximum 7 elements after
• 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 theory/FlavorSpace:Intrinsic QCD Evolution Bases both in the input and the output space
• this basis does not coincide with the theory/FlavorSpace:Operator Anomalous Dimension Basis as the decision on which operator of that basis is used is a non-trivial decision - see Matching
• this basis has 2n_f + 3 = 15 elements
• this basis can span arbitrary matching scales