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The singular behavior of massive QCD amplitudes

A. Mitov, S. Moch

TL;DR

Mitov and Moch develop a universal, small-mass factorization framework for on-shell QCD amplitudes with massive partons, yielding an all-order exponentiation of soft poles and large mass logarithms and a process-independent relation between massive and massless amplitudes via $M^{(m)} = \prod_i (Z^{(m|0)}_{[i]})^{1/2} M^{(m=0)}$. They derive an all-order Sudakov exponentiation for the heavy-quark form factor $\mathcal{F}$ in terms of cusp-related functions $A$, $G$, and $K$, and provide fixed-order expansions up to three loops, including decoupling and scheme matching. The formalism extends to general $2\to n$ processes, enabling NNLO predictions of both singular and finite (mass-independent) terms and clarifying connections to perturbative fragmentation functions. Overall, the work offers a practical, universal toolkit to relate and predict massive amplitudes from known massless results and to organize higher-order QCD corrections involving massive partons.

Abstract

We discuss the structure of infrared singularities in on-shell QCD amplitudes with massive partons and present a general factorization formula in the limit of small parton masses. The factorization formula gives rise to an all-order exponentiation of both, the soft poles in dimensional regularization and the large collinear logarithms of the parton masses. Moreover, it provides a universal relation between any on-shell amplitude with massive external partons and its corresponding massless amplitude. For the form factor of a heavy quark we present explicit results including the fixed-order expansion up to three loops in the small mass limit. For general scattering processes we show how our constructive method applies to the computation of all singularities as well as the constant (mass-independent) terms of a generic massive n-parton QCD amplitude up to the next-to-next-to-leading order corrections.

The singular behavior of massive QCD amplitudes

TL;DR

Mitov and Moch develop a universal, small-mass factorization framework for on-shell QCD amplitudes with massive partons, yielding an all-order exponentiation of soft poles and large mass logarithms and a process-independent relation between massive and massless amplitudes via . They derive an all-order Sudakov exponentiation for the heavy-quark form factor in terms of cusp-related functions , , and , and provide fixed-order expansions up to three loops, including decoupling and scheme matching. The formalism extends to general processes, enabling NNLO predictions of both singular and finite (mass-independent) terms and clarifying connections to perturbative fragmentation functions. Overall, the work offers a practical, universal toolkit to relate and predict massive amplitudes from known massless results and to organize higher-order QCD corrections involving massive partons.

Abstract

We discuss the structure of infrared singularities in on-shell QCD amplitudes with massive partons and present a general factorization formula in the limit of small parton masses. The factorization formula gives rise to an all-order exponentiation of both, the soft poles in dimensional regularization and the large collinear logarithms of the parton masses. Moreover, it provides a universal relation between any on-shell amplitude with massive external partons and its corresponding massless amplitude. For the form factor of a heavy quark we present explicit results including the fixed-order expansion up to three loops in the small mass limit. For general scattering processes we show how our constructive method applies to the computation of all singularities as well as the constant (mass-independent) terms of a generic massive n-parton QCD amplitude up to the next-to-next-to-leading order corrections.

Paper Structure

This paper contains 9 sections, 67 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Factorization of QCD amplitudes
  3. The Sudakov form factor in QCD
  4. Fixed-order expansions and resummation coefficients
  5. The factor $Z^{(m\vert0)}$
  6. The form factor ${\cal F}$
  7. Applications
  8. Summary
  9. Appendix

Figures (1)

  • Figure 1: Feynman diagrams contributing to the vertex $\gamma^{\,\ast}\!qq$ as examples for the flavor classes $ll$, $hl$, $lh$ and $hh$ discussed in the text. Curly lines denote gluons, double straight lines quarks of mass $m$ and single straight lines massless quarks.