On the Dimensional Regularization of QCD Helicity Amplitudes With Quarks

TL;DR

<p>The paper tackles the problem of computing QCD helicity amplitudes with external quarks in dimensional regularization, focusing on HV and FDH schemes. It introduces a systematic embedding of four-dimensional external fermions into integer-dimensional spaces via Clifford-algebra decomposition and a normalized partial-trace operation, enabling analytic control of the $D_s$ dependence. The authors derive explicit, scheme-differentiated decompositions of one- and two-loop amplitudes by particle content, showing how to obtain HV/FDH results from a single $D_s$-dependent calculation. They also discuss connections to the FDF formulation, propose improvements for multi-quark processes, and report an implementation in BlackHat validating the approach against known NLO QCD results. The work provides a practical, numerically friendly framework to extend dimensionally regulated helicity-amplitude computations to multi-quark final states.

Abstract

We study QCD helicity amplitudes with an arbitrary number of (massive) quarks, keeping unobserved (loop) particles in fixed integer $D_s$ dimensions. We find a suitable embedding of external four-dimensional fermion states into higher dimensional spaces. This allows to identify the $D_s$ dependence of amplitudes with external quarks at one and two loops, permitting an analytic continuation in $D_s$. Explicitly we focus on 't Hooft-Veltman and four-dimensional helicity schemes for which we provide a compact prescription for the computation of one- and two-loop amplitudes amenable for numerical implementation.

Figures (7)

• Figure 1: A schematic subdiagram for a contribution from a generic spinor line entering the loop. Bold lines and vertices represent objects in $D_s$ dimensions. Blobs represent parts of the diagram which explicit form is not important here.
• Figure 2: Examples of diagrams with quarks in the loop. Bold lines and vertices represent objects in $D_s$ dimensions.
• Figure 3: Diagrams contributing to the $(D_s-6)^2$ coefficient. Quark-scalar vertices originating from equal indices are represented by the same shape.
• Figure 4: Diagrams contributing to the $(D_s-6)$ coefficient.
• Figure 5: Diagrams contributing to both $(D_s-6)$ and $(D_s-6)^2$ coefficients. The diagram \ref{['subfloat:sk']} represents a skeleton diagram, where any number of 6d gluons can be inserted on fermion or scalar lines. The diagram \ref{['subfloat:db']} is an example of such an insertion. Quark-scalar vertices originating from equal indices are represented by the same shape.