Scheme transformations as the gauge group of DGLAP: sum rules, classification and solution

Abstract

A parton density depends on the renormalization scheme, and any two schemes are connected by a transformation that preserves the DGLAP form of the evolution. In Mellin space such a transformation acts on the evolution kernels as a gauge transformation, and we take this point of view throughout the discussion. For the unpolarized densities we first ask which transformations also preserve the charge-conjugation and flavor symmetries of the evolution kernels and the momentum and valence number sum rules. The symmetries reduce the transformation to a few blocks, and the sum rules constrain only two Mellin moments, where they leave a single freedom, the split of momentum between the quark singlet and the gluon, and freeze the valence numbers. The resulting classification is complete, because with asymptotic freedom as a boundary condition a change of scheme is determined by its action on the kernels. Inside this group, the transformations that preserve positivity form a semigroup, and the same argument covers the polarized densities. A transformation that fails only the sum rules is not lost either, since a normalization that the sum rules themselves determine repairs it, and we use this to define collinear densities as integrals of transverse-momentum-dependent densities without giving up the sum rules. We then use the same gauge freedom to solve the evolution. Gauging the kernel away reduces DGLAP at any order to a single second-order linear equation plus a quadrature, and it organizes the estimate of missing higher orders in a way that respects the sum rules exactly.