Kernels

The coefficient functions consist of the partonic cross-sections, that have to be convolved with the PDFs in order to obtain the hadronic structure functions, according to the factorization theorem:

\[F(x) = \sum_i f_i \otimes c_i (x) = \int_x^1 \frac{dz}{z} f_i\left(\frac{x}{z}\right) c_i(z)\]

So actually there is a coefficient function for each PDF (i.e. for each flavor).

The actual physical combination are not as many as flavors, indeed QCD is flavor blind (in the massless case, and the only difference in the massive case are the value of the masses themselves).

In particular there are 3 relevant combinations, for polarized and unpolarized DIS as described in Coefficient Functions.

Todo

• describe kernel structure

• stress that there are four kinds:

  • in non-singlet channel allows to reuse a single integration for multiple channels, that there the line directly connects to the EW boson

  • in gluon and singlet the incoming line is always decoupled by a gluon from the EW and thus naturally appears the charge average (that depend on the number of flavors running in the loops)

  • that intrinsic is treated separately (heavy-initiated)

• stress that Charged Current are slightly different from Neutral ones, and point to the corresponding theory section

• describe that kernels are defined as (partons, coefficient) and turned into (partons, operator, operator-error) by convolutions

Singlet & Non-singlet

For the distinction about singlet and non-singlet coefficient function, see Non-singlet definition.