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Massive three-loop form factor in the planar limit

Johannes Henn, Alexander V. Smirnov, Vladimir A. Smirnov, Matthias Steinhauser

TL;DR

This paper delivers the first computation of massive three-loop QCD corrections to the quark-anti-quark-photon form factors $F_1$ and $F_2$ in the planar limit, with exact mass dependence expressed through Goncharov polylogarithms. The authors derive and analyze the infrared structure via the cusp anomalous dimension, provide detailed analytic and numerical results including low-energy, high-energy, and threshold expansions, and validate the results against known benchmarks and through multiple cross-checks. The work furnishes both exact expressions and practical expansions, enabling precise predictions for heavy-quark processes and contributing to the understanding of infrared behavior in QCD amplitudes. The findings have implications for matching, threshold resummation, and the reliability of higher-loop predictions in heavy-quark production and decay processes.

Abstract

We compute the three-loop QCD corrections to the massive quark-anti-quark-photon form factors $F_1$ and $F_2$ in the large-$N_c$ limit. The analytic results are expressed in terms of Goncharov polylogarithms. This allows for a straightforward numerical evaluation. We also derive series expansions, including power suppressed terms, for three kinematic regions corresponding to small and large invariant masses of the photon momentum, and small velocities of the heavy quarks.

Massive three-loop form factor in the planar limit

TL;DR

This paper delivers the first computation of massive three-loop QCD corrections to the quark-anti-quark-photon form factors and in the planar limit, with exact mass dependence expressed through Goncharov polylogarithms. The authors derive and analyze the infrared structure via the cusp anomalous dimension, provide detailed analytic and numerical results including low-energy, high-energy, and threshold expansions, and validate the results against known benchmarks and through multiple cross-checks. The work furnishes both exact expressions and practical expansions, enabling precise predictions for heavy-quark processes and contributing to the understanding of infrared behavior in QCD amplitudes. The findings have implications for matching, threshold resummation, and the reliability of higher-loop predictions in heavy-quark production and decay processes.

Abstract

We compute the three-loop QCD corrections to the massive quark-anti-quark-photon form factors and in the large- limit. The analytic results are expressed in terms of Goncharov polylogarithms. This allows for a straightforward numerical evaluation. We also derive series expansions, including power suppressed terms, for three kinematic regions corresponding to small and large invariant masses of the photon momentum, and small velocities of the heavy quarks.

Paper Structure

This paper contains 13 sections, 22 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Setup and calculation
  3. Infrared divergences of massive form factors
  4. Results
  5. Structure of results for form factors
  6. Structure of results for form factors
  7. Analytical results
  8. Low-energy: $s\ll m^2$ or $x\to1$
  9. High-energy: $s\ll m^2$ or $x\to0$
  10. Threshold: $s\to4m^2$ or $x\to-1$
  11. Numerical results
  12. Checks
  13. Conclusions and outlook

Figures (4)

  • Figure 1: Sample diagrams contributing to $F_1$ and $F_2$ at one-, two- and three-loop order. Solid, curly and wavy lines represent quarks, gluons and photons, respectively. In our calculation the closed fermion loops only involve massless quarks.
  • Figure 2: Real and imaginary parts of $\epsilon^0$ one-, two- and three-loop contribution of $F_1$ as a function of $x$. The leading high-energy term (i.e. $f_{1,\rm lar}^{(n,0)}$ from Eq. (\ref{['eq::F_i_lar']})) is subtracted so that $F_1$ is zero for $x=0$. The solid (black) lines show the exact result and the short-dashed (blue) lines represent the high-energy approximations including terms up to order $x^4$. The long-dashed (red) curves contain low-energy expansion terms up to order $(1-x)^4$. The number of light fermions is set to zero ($n_l=0$).
  • Figure 3: Real and imaginary parts of $\epsilon^0$ one-, two- and three-loop contribution of $F_2$ as a function of $x$. The same notation as in Figure \ref{['fig::F1_x']} has been used.
  • Figure 4: One-, two- and three-loop contribution of $F_1$ ($\epsilon^0$ terms) as a function of $\phi$ (with $x=e^{i\phi}$). The solid (black) lines shows the exact result and the dashed (blue) lines represent approximations including terms up to order $(1-x)^4$. The number of light fermions is set to zero ($n_l=0$). Note that in this region $F_1$ and $F_2$ have no imaginary parts.