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

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

TL;DR

The paper computes the axial-vector heavy-quark form factors $G_1$ and $G_2$ for the $ZQQ$ vertex at two-loop order in QCD, keeping full dependence on the heavy-quark mass and arbitrary momentum transfer, while excluding triangle-diagram contributions. It provides complete analytic results for unsubtracted and UV-renormalized form factors as Laurent expansions in $\epsilon$, with expressions in terms of 1-dimensional harmonic polylogarithms up to weight 4, and analyzes both spacelike and timelike kinematics, including analytical continuation, threshold expansions, and high-energy asymptotics. The renormalization uses an $\overline{\text{MS}}$ coupling with an on-shell mass and wave function for the heavy quark, and the work systematically accounts for counterterms and scale dependence when $\mu \neq m$. The results are directly applicable to NNLO QCD corrections to heavy-quark electroproduction and forward-backward asymmetries, and provide insight into the infrared/collinear structure of massive-quark amplitudes.

Abstract

We consider the Z Q Qbar vertex to second order in the QCD coupling for an on-shell massive quark-antiquark pair and for arbitrary momentum transfer of the Z boson. We present closed analytic expressions for the two parity-violating form factors of that vertex at the two-loop level in QCD, excluding the contributions from triangle diagrams. These 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: Axial Vector Contributions

TL;DR

The paper computes the axial-vector heavy-quark form factors and for the vertex at two-loop order in QCD, keeping full dependence on the heavy-quark mass and arbitrary momentum transfer, while excluding triangle-diagram contributions. It provides complete analytic results for unsubtracted and UV-renormalized form factors as Laurent expansions in , with expressions in terms of 1-dimensional harmonic polylogarithms up to weight 4, and analyzes both spacelike and timelike kinematics, including analytical continuation, threshold expansions, and high-energy asymptotics. The renormalization uses an coupling with an on-shell mass and wave function for the heavy quark, and the work systematically accounts for counterterms and scale dependence when . The results are directly applicable to NNLO QCD corrections to heavy-quark electroproduction and forward-backward asymmetries, and provide insight into the infrared/collinear structure of massive-quark amplitudes.

Abstract

We consider the Z Q Qbar vertex to second order in the QCD coupling for an on-shell massive quark-antiquark pair and for arbitrary momentum transfer of the Z boson. We present closed analytic expressions for the two parity-violating form factors of that vertex at the two-loop level in QCD, excluding the contributions from triangle diagrams. These form factors are expressed in terms of 1-dimensional harmonic polylogarithms of maximum weight 4.

Paper Structure

This paper contains 15 sections, 53 equations, 5 figures.

Table of Contents

  1. Introduction
  2. The Axial Vector Form Factors
  3. Renormalization Scheme
  4. One-loop Unsubtracted and UV-Renormalized Form Factors
  5. Two-Loop Unsubtracted and Renormalized Form Factors
  6. Two-Loop Unsubtracted Form Factors
  7. Contributions from the Counterterms
  8. UV-Renormalized Two-Loop Form Factors
  9. Form Factors for $\mu \not{\! \! =} \, m$
  10. Analytical Continuation above Threshold
  11. One-Loop Form Factors above Threshold
  12. Two-Loop Form Factors above Threshold
  13. Threshold Expansions
  14. Asymptotic Expansions
  15. Summary

Figures (5)

  • Figure 1: Triangle diagram contributions to $V^\mu$. Crossed diagrams are not drawn. The external dashed line refers to an incoming $Z$ boson, the curly lines to gluons, the double straight lines to the massive quark and the simple straight lines to massless quarks.
  • Figure 2: One-loop QCD contribution to $V^\mu_{c_1 c_2}(q)$.
  • Figure 3: Subtraction term for the one-loop renormalization.
  • Figure 4: Two-loop QCD contributions to $V^\mu_{c_1 c_2}(p_1,p_2)$. The inner dashed line refers to ghosts. Straight lines represent massless quarks and the double straight line the massive quark (c.f. Fig. \ref{['triangle-graph']}).
  • Figure 5: Subtraction terms for the two-loop renormalization. Note that the diagrams (a)-(b) and (c)-(d) yield the same contribution, respectively.