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Two-loop soft anomalous dimensions with massive and massless quarks

Nikolaos Kidonakis

TL;DR

The work computes two-loop soft anomalous dimensions for processes with massive and massless quarks within the eikonal, dimensionally regularized framework to enable NNLL soft-gluon resummation in heavy-quark production. By evaluating a comprehensive set of heavy-quark eikonal diagrams (both vertex and self-energy) and isolating UV poles, the authors obtain explicit expressions for $\Gamma_S^{(1)}$ and $\Gamma_S^{(2)}$ in terms of the heavy-quark velocity parameter $\beta$ (or cusp angle $\gamma$) and special functions like polylogarithms. Key findings include the mass dependence of $\Gamma_S^{(2)}$, the small- and large-$\beta$ behaviors, and the fact that the simple massless relation $\Gamma_S^{(2)} = (K/2)\Gamma_S^{(1)}$ does not hold in general; in the massless limit, this relation is recovered. The results underpin NNLL resummation for $t\bar t$ production and related processes, with cross-checks against the heavy-quark form factor and cusp anomalous dimension literature. The paper also discusses mixed massive-massless cases and provides approximate, accurate representations for $\Gamma_S^{(2)}$ across the full kinematic range.

Abstract

I present results for two-loop soft anomalous dimensions, which are derived from dimensionally regularized diagrams with eikonal quark lines and control soft-gluon emission in hard-scattering processes. Detailed results for the UV poles of the eikonal integrals are shown for massive quarks, and the massless limit is also taken. The construction of soft anomalous dimensions at two-loops allows soft-gluon resummations at NNLL accuracy.

Two-loop soft anomalous dimensions with massive and massless quarks

TL;DR

The work computes two-loop soft anomalous dimensions for processes with massive and massless quarks within the eikonal, dimensionally regularized framework to enable NNLL soft-gluon resummation in heavy-quark production. By evaluating a comprehensive set of heavy-quark eikonal diagrams (both vertex and self-energy) and isolating UV poles, the authors obtain explicit expressions for and in terms of the heavy-quark velocity parameter (or cusp angle ) and special functions like polylogarithms. Key findings include the mass dependence of , the small- and large- behaviors, and the fact that the simple massless relation does not hold in general; in the massless limit, this relation is recovered. The results underpin NNLL resummation for production and related processes, with cross-checks against the heavy-quark form factor and cusp anomalous dimension literature. The paper also discusses mixed massive-massless cases and provides approximate, accurate representations for across the full kinematic range.

Abstract

I present results for two-loop soft anomalous dimensions, which are derived from dimensionally regularized diagrams with eikonal quark lines and control soft-gluon emission in hard-scattering processes. Detailed results for the UV poles of the eikonal integrals are shown for massive quarks, and the massless limit is also taken. The construction of soft anomalous dimensions at two-loops allows soft-gluon resummations at NNLL accuracy.

Paper Structure

This paper contains 12 sections, 77 equations, 6 figures.

Table of Contents

  1. Introduction
  2. One-loop diagrams
  3. Two-loop vertex diagrams
  4. $I_{2a}$ and $I_{2b}$
  5. $I_{2c}$
  6. $I_{2d}$ and $I_{2e}$
  7. $I_{2f}$
  8. Two-loop heavy-quark self-energy diagrams
  9. $I_{3a}$
  10. $I_{3b}$
  11. $I_{3c}$
  12. Two-loop soft anomalous dimension

Figures (6)

  • Figure 1: One-loop diagrams with heavy-quark eikonal lines.
  • Figure 2: Two-loop vertex diagrams with heavy-quark eikonal lines.
  • Figure 3: Two-loop heavy-quark self-energy diagrams with eikonal lines.
  • Figure 4: Detail of the diagrams of Fig. 3(a).
  • Figure 5: The two-loop soft anomalous dimension $\Gamma_S^{(2)}$.
  • ...and 1 more figures