# 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:

## +/- 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 Basisthe 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

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 Basisthe 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):

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 Basisthe 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

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

this basis is

*not*normalized with respect to the canonical Flavor BasisThe 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:

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 Unified Evolution Basis.

this basis is

*not*normalized with respect to the canonical Flavor Basisthe 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

## Operator Bases

An EKO \(\mathbf E\) is an operator in the Flavor Space \(\mathcal F\) 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 Flavor Basis both in the input and the output space

the

`Output`

is delivered in this basisthis 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 flavorsall actual computations are done in this basis

### Operator Intrinsic QCD Evolution Basis

here we mean Intrinsic QCD Evolution Bases both in the input and the output space

this basis does

**not**coincide with the Operator Anomalous Dimension Basis as the decision on which operator of that basis is used is a non-trivial decision - see Matching Conditions on Crossing Thresholdsthis basis has \(2n_f+ 3 = 15\) elements

this basis can span arbitrary matching scales