Fast Kernel (FK) tables
The direct calculation of observables during a PDF fit is not very practical since it requires first solving the DGLAP evolution equation for each new boundary condition at the initial scale \(Q_0\) and then convoluting with the coefficient functions of the partonic cross-sections.
For this reason, we adopt the FK tables method [BDF+10] which is presented in this section.
In the framework of collinear QCD factorization cross section, such as the \(F_2\) DIS structure function, can be decomposed in terms of hard-scattering coefficient functions and PDFs as,
where \(\mathbf{C}(Q^2)\) are the process-dependent coefficient functions which can be computed perturbatively as an expansion in the QCD and QED couplings, \(\mathbf{E}(Q^2 \leftarrow Q_0^2)\) is an evolution operator, determined by the solutions of the DGLAP equations, which evolves the PDF from the initial parameterization scale \(Q_0^2\) up to the hard-scattering scale \(Q^2\), \(\mathbf{f}(Q^2_0)\) are the PDF at the parameterization scale, and \(\otimes\) denotes the Mellin convolution.
In the above equation (and in all equations from now on), the sum over flavors (running over the contributing quarks and anti-quarks, as well as over the gluon) is indicated by the bold font.
To evaluate the observable in a computationally more efficient way, it is better to precompute all the perturbative information: for the partonic coefficient functions \(\mathbf{C}\) we use pineappl grids and for the evolution operators \(\mathbf{E}\) we use eko.
Finally, we can arrive at
where all of the information about the partonic cross-sections and the DGLAP evolution operators is now encoded into the so-called FK table \(\mathbf{FK}\).