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On the Casimir scaling violation in the cusp anomalous dimension at small angle

Andrey Grozin, Johannes Henn, Maximilian Stahlhofen

TL;DR

The paper computes the four-loop nf C_{F,4} contribution to the QCD cusp anomalous dimension at small cusp angle φ, showing a genuine Casimir-scaling violation. Using a HQET-based approach, it evaluates non-planar four-loop Feynman integrals, reduces them to master integrals, and extracts the φ^2 and φ^4 coefficients, revealing an analytic form that diverges from a prior all-order conjecture but is numerically close. As a by-product, it determines the corresponding nf C_{F,4} term in the four-loop HQET heavy-quark field anomalous dimension, providing a cross-check for future calculations. The results have implications for the all-order structure of Γ_cusp and for matching to the static potential and light-like limits.

Abstract

We compute the four-loop $n_f$ contribution proportional to the quartic Casimir of the QCD cusp anomalous dimension as an expansion for small cusp angle $φ$. This piece is gauge invariant, violates Casimir scaling, and first appears at four loops. It requires the evaluation of genuine non-planar four-loop Feynman integrals. We present results up to ${\mathcal O}(φ^4)$. One motivation for our calculation is to probe a recent conjecture on the all-order structure of the cusp anomalous dimension. As a byproduct we obtain the four-loop HQET wave function anomalous dimension for this color structure.

On the Casimir scaling violation in the cusp anomalous dimension at small angle

TL;DR

The paper computes the four-loop nf C_{F,4} contribution to the QCD cusp anomalous dimension at small cusp angle φ, showing a genuine Casimir-scaling violation. Using a HQET-based approach, it evaluates non-planar four-loop Feynman integrals, reduces them to master integrals, and extracts the φ^2 and φ^4 coefficients, revealing an analytic form that diverges from a prior all-order conjecture but is numerically close. As a by-product, it determines the corresponding nf C_{F,4} term in the four-loop HQET heavy-quark field anomalous dimension, providing a cross-check for future calculations. The results have implications for the all-order structure of Γ_cusp and for matching to the static potential and light-like limits.

Abstract

We compute the four-loop contribution proportional to the quartic Casimir of the QCD cusp anomalous dimension as an expansion for small cusp angle . This piece is gauge invariant, violates Casimir scaling, and first appears at four loops. It requires the evaluation of genuine non-planar four-loop Feynman integrals. We present results up to . One motivation for our calculation is to probe a recent conjecture on the all-order structure of the cusp anomalous dimension. As a byproduct we obtain the four-loop HQET wave function anomalous dimension for this color structure.

Paper Structure

This paper contains 4 sections, 26 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Calculation
  3. The HQET heavy-quark field anomalous dimension
  4. Result for the cusp anomalous dimension

Figures (2)

  • Figure 1: Diagrams contributing to the $n_f \mathcal{C}_{F,4}$ term in the vertex function $V(\phi)$. Double lines represent Wilson lines, wavy lines gluons and single lines the $n_f$ light quarks. Left-right mirror graphs and the diagrams with reversed light fermion flow are not displayed. The $n_f \mathcal{C}_{F,4}$ contributions of the latter equal the ones of their relatives shown here.
  • Figure 2: Independent integral topologies (1,2,3) for the small angle expansion of $V(\phi)$. The number $i$ for each (double) line refers to the corresponding propagator with power $a_i$ in $G_{a_1,\ldots,a_{14}}$ according to eq. \ref{['eq:Gdef']}. Propagators 1,2,3 are (linear) Wilson line (heavy quark) propagators.