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Two-loop QCD Corrections to the Heavy Quark Form Factors: the Vector Contributions

W. Bernreuther, R. Bonciani, T. Gehrmann, R. Heinesch, T. Leineweber, P. Mastrolia, E. Remiddi

TL;DR

This work delivers closed analytic expressions for the two-loop QCD corrections to the heavy-quark vector vertex, keeping full mass and momentum dependence and expressing results in terms of 1D harmonic polylogarithms. The calculation combines a Laporta reduction to 17 master integrals, differential equations for these integrals, and a hybrid renormalization scheme (OS for heavy quark parameters and MSbar for αs). The authors provide unsubtracted and UV-renormalized form factors in the space-like region, along with their analytic continuations above threshold, threshold expansions, and high-energy asymptotics, and they verify consistency with existing threshold results and NNLO cross-section analyses. The results are relevant for NNLO corrections to forward-backward asymmetries and differential heavy-quark observables, and establish a framework for including axial-vector components in future work.

Abstract

We present closed analytic expressions of the electromagnetic vertex form factors for heavy quarks at the two-loop level in QCD for arbitrary momentum transfer. The calculation is carried out in dimensional regularization. The electric and magnetic form factors are expressed in terms of 1-dimensional harmonic polylogarithms of maximum weight 4.

Two-loop QCD Corrections to the Heavy Quark Form Factors: the Vector Contributions

TL;DR

This work delivers closed analytic expressions for the two-loop QCD corrections to the heavy-quark vector vertex, keeping full mass and momentum dependence and expressing results in terms of 1D harmonic polylogarithms. The calculation combines a Laporta reduction to 17 master integrals, differential equations for these integrals, and a hybrid renormalization scheme (OS for heavy quark parameters and MSbar for αs). The authors provide unsubtracted and UV-renormalized form factors in the space-like region, along with their analytic continuations above threshold, threshold expansions, and high-energy asymptotics, and they verify consistency with existing threshold results and NNLO cross-section analyses. The results are relevant for NNLO corrections to forward-backward asymmetries and differential heavy-quark observables, and establish a framework for including axial-vector components in future work.

Abstract

We present closed analytic expressions of the electromagnetic vertex form factors for heavy quarks at the two-loop level in QCD for arbitrary momentum transfer. The calculation is carried out in dimensional regularization. The electric and magnetic form factors are expressed in terms of 1-dimensional harmonic polylogarithms of maximum weight 4.

Paper Structure

This paper contains 18 sections, 65 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The QCD Form Factors
  3. Unsubtracted Contributions
  4. One-Loop Unsubtracted Form Factors
  5. Two-Loop Unsubtracted Form Factors
  6. Renormalization
  7. One-Loop Counterterms
  8. Two-Loop Counterterm
  9. Renormalized Form Factors
  10. One-Loop UV-Renormalized Form Factors
  11. Two-Loop UV-Renormalized Form Factors
  12. Form Factors for $\mu \neq m$
  13. Analytical Continuation above Threshold
  14. One-Loop Form Factors above Threshold
  15. Two-Loop Form Factors above Threshold
  16. ...and 3 more sections

Figures (3)

  • Figure 1: Tree-level and one-loop diagrams, involved in the calculation of the heavy-quark vertex form factors. The curly line represents a gluon; the double straight lines, quarks of mass $m$. The external fermion lines are on the mass-shell: $p_1^2 = p_2^2 = m^{2}$. The wavy line on the l.h.s. carries momentum $Q=p_{1}+p_{2}$, with the metrical convention $Q^{2}<0$ when $Q$ is space-like.
  • Figure 2: The two-loop vertex diagrams involved in the calculation of the form factors at order ${\mathcal{O}}(\alpha_{S}^{2})$. The curly lines are gluons; the double straight lines, quarks of mass $m$; the single straight lines, massless quarks and the dashed lines ghosts. The external fermion lines are on the mass-shell: $p_1^2 = p_2^2 = m^{2}$. The wavy line on the l.h.s. carries momentum $Q=p_{1}+p_{2}$, with the metrical convention $Q^{2}<0$ when $Q$ is space-like.
  • Figure 3: Counterterm diagrams. For the renormalization of the two-loop form factors we use diagrams (a)--(i). Diagram (j) is employed in the renormalization of the one-loop form factors.