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MT: a Mathematica package to compute convolutions

Maik Hoeschele, Jens Hoff, Alexey Pak, Matthias Steinhauser, Takahiro Ueda

TL;DR

Problem addressed: collinear divergences in initial-state radiation require convolutions of partonic cross sections with splitting functions. Approach: MT implements a Mellin-space algorithm using regularized derivatives to handle harmonic polylogarithms (HPLs), plus distributions, and polynomials, enabling analytic and numeric convolutions up to weight eight. Key contributions: a comprehensive Mathematica package MT with Mellin-transform pipelines and utilities for HPLs and harmonic sums, plus explicit demonstrations for Higgs production and Drell–Yan up to N^3LO, including new Drell–Yan results and ε-expansion terms. Significance: provides a practical tool for NNLO/N^3LO phenomenology and downloadable data/resources for collaborations.

Abstract

We introduce the \prog{Mathematica} package \prog{MT} which can be used to compute, both analytically and numerically, convolutions involving harmonic polylogarithms, polynomials or generalized functions. As applications contributions to next-to-next-to-next-to leading order Higgs boson production and the Drell-Yan process are discussed.

MT: a Mathematica package to compute convolutions

TL;DR

Problem addressed: collinear divergences in initial-state radiation require convolutions of partonic cross sections with splitting functions. Approach: MT implements a Mellin-space algorithm using regularized derivatives to handle harmonic polylogarithms (HPLs), plus distributions, and polynomials, enabling analytic and numeric convolutions up to weight eight. Key contributions: a comprehensive Mathematica package MT with Mellin-transform pipelines and utilities for HPLs and harmonic sums, plus explicit demonstrations for Higgs production and Drell–Yan up to N^3LO, including new Drell–Yan results and ε-expansion terms. Significance: provides a practical tool for NNLO/N^3LO phenomenology and downloadable data/resources for collaborations.

Abstract

We introduce the \prog{Mathematica} package \prog{MT} which can be used to compute, both analytically and numerically, convolutions involving harmonic polylogarithms, polynomials or generalized functions. As applications contributions to next-to-next-to-next-to leading order Higgs boson production and the Drell-Yan process are discussed.

Paper Structure

This paper contains 6 sections, 23 equations, 1 figure, 4 tables.

Table of Contents

  1. Introduction
  2. Systematic approach to convolution integrals
  3. Description of MT
  4. Example 1: collinear singularities for Higgs boson production at the LHC
  5. Example 2: collinear singularities for Drell-Yan production
  6. Summary

Figures (1)

  • Figure 1: Sample Feynman diagrams up to N$^3$LO contributing to the Drell-Yan process. The wiggly lines denote a generic vector boson, i.e., $\gamma$, $W$, or $Z$. The dashed lines attached to crosses mark the considered cut(s) through a massive vector boson and additional massless particles. The captions state the perturbative order, the channel and the type of contributions ("r" for real, "v" for virtual, or their interference).