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The Heavy Quark Form Factors at Two Loops

J. Ablinger, A. Behring, J. Blümlein, G. Falcioni, A. De Freitas, P. Marquard, N. Rana, C. Schneider

TL;DR

This work delivers a comprehensive two-loop QCD calculation of heavy-quark form factors for vector, axial-vector, scalar, and pseudoscalar currents, including terms up to $O(\varepsilon^2)$ to support higher-loop renormalization. The authors deploy two complementary computational strategies—conventional differential equations for master integrals and a novel difference-equation method—yielding analytic results expressed in harmonic polylogarithms and valid across key kinematic regimes: low-energy ($q^2\ll m^2$), high-energy ($|q^2|\gg m^2$), and threshold ($q^2\sim 4m^2$). They carefully treat renormalization in a mixed OS/$\overline{\rm MS}$ scheme and address the infrared structure via the massive cusp anomalous dimension, with non-singlet and singlet contributions handled according to the ABJ anomaly and Ward identities. The results, which agree with known limits and provide crucial inputs for three- and four-loop computations, have direct implications for precision heavy-quark phenomenology and top-quark production/decay predictions at current and future colliders.

Abstract

We compute the two-loop QCD corrections to the heavy quark form factors in case of the vector, axial-vector, scalar and pseudo-scalar currents up to second order in the dimensional parameter $ε= (4-D)/2$. These terms are required in the renormalization of the higher order corrections to these form factors.

The Heavy Quark Form Factors at Two Loops

TL;DR

This work delivers a comprehensive two-loop QCD calculation of heavy-quark form factors for vector, axial-vector, scalar, and pseudoscalar currents, including terms up to to support higher-loop renormalization. The authors deploy two complementary computational strategies—conventional differential equations for master integrals and a novel difference-equation method—yielding analytic results expressed in harmonic polylogarithms and valid across key kinematic regimes: low-energy (), high-energy (), and threshold (). They carefully treat renormalization in a mixed OS/ scheme and address the infrared structure via the massive cusp anomalous dimension, with non-singlet and singlet contributions handled according to the ABJ anomaly and Ward identities. The results, which agree with known limits and provide crucial inputs for three- and four-loop computations, have direct implications for precision heavy-quark phenomenology and top-quark production/decay predictions at current and future colliders.

Abstract

We compute the two-loop QCD corrections to the heavy quark form factors in case of the vector, axial-vector, scalar and pseudo-scalar currents up to second order in the dimensional parameter . These terms are required in the renormalization of the higher order corrections to these form factors.

Paper Structure

This paper contains 32 sections, 150 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The heavy quark form factors
  3. The vector and axial-vector current
  4. The scalar and pseudo-scalar current
  5. Anomaly and Ward identities
  6. Renormalization
  7. The infrared structure
  8. Details of the calculation
  9. The conventional differential equations method
  10. Calculation of the master integrals using difference equations
  11. Results
  12. Low energy region $0< q^2 \ll m^2$
  13. Vector form factor
  14. Axial-vector form factor
  15. Scalar form factor
  16. ...and 17 more sections

Figures (3)

  • Figure 1: The non-singlet topologies required for the calculation of two-loop form factors. Solid lines represent massive particles in external or internal lines, while dashed lines correspond to massless propagators. The external vector current can also be replaced by an axial-vector, scalar or pseudo-scalar. The master integrals associated to topologies (c) and (d) are a subset of the master integrals required for topologies (a) and (b).
  • Figure 2: The singlet topology.
  • Figure 3: The master integral $K_{14}$. Massive (massless) propagators are represented by solid (dashed) lines. The presence of a massless cut determines the asymptotic behaviour $K_{14}\sim \log(1-x)$.