# On Higgs-exchange DIS, physical evolution kernels and fourth-order splitting functions at large x

## TL;DR

This paper calculates third-order coefficient functions for Higgs-exchange deep-inelastic scattering and uses them to predict fourth-order splitting functions, revealing single-logarithmic behavior at large x and challenges at small x.

## Contribution

It provides the first third-order coefficient functions for scalar Higgs-exchange DIS and predicts fourth-order splitting functions using these results, highlighting their large-x behavior.

## Key findings

- Two-loop results enable construction of NNLO physical evolution kernels.
- Kernels show single-logarithmic enhancement at large x.
- Predictions for fourth-order splitting functions based on third-order calculations.

## Abstract

We present the coefficient functions for deep-inelastic scattering (DIS) via the exchange of a scalar phi directly coupling only to gluons, such as the Higgs boson in the limit of a very heavy top quark and n_f effectively massless light flavours, to the third order in perturbative QCD. The two-loop results are employed to construct the next-to-next-to-leading order physical evolution kernels for the system (F_2 F_phi) of flavour-singlet structure functions. The practical relevance of these kernels as an alternative to MSbar factorization is bedevilled by artificial double logarithms at small values of the scaling variable x, where the large top-mass limit ceases to be appropriate. However, they show an only single-logarithmic enhancement at large x. Conjecturing that this feature persists to the next order also in the present singlet case, the three-loop coefficient functions facilitate exact predictions (backed up by their particular colour structure) of the double-logarithmic contributions to the fourth-order singlet splitting functions, i.e., of the terms (1-x)^a ln^k(1-x) with k = 4, 5, 6 and k = 3, 4, 5, respectively, for the off-diagonal and diagonal quantities to all powers a in 1-x.