Supersymmetric Regularization, Two-Loop QCD Amplitudes and Coupling Shifts

TL;DR

The paper introduces and formalizes the four-dimensional helicity (FDH) regularization for two-loop QCD amplitudes, designed to preserve the boson–fermion state balance and hence supersymmetry. It leverages a unitarity-cut framework and a color-primitive decomposition to construct and test identical-helicity two-loop amplitudes, verifying supersymmetry Ward identities in SUSY QCD and providing cross-checks with non-supersymmetric QCD results. A detailed comparison between FDH/DR and MS-bar schemes yields a two-loop coupling shift and a corresponding three-loop beta function in the FDH/DR schemes. The work also clarifies the scheme dependence via a continuous δ_R parameter and discusses exactness properties for N=2, with implications for higher-loop consistency and cross-scheme translations of amplitudes and couplings, offering a robust toolkit for precision multi-loop QCD/IPM computations in supersymmetric contexts.

Abstract

We present a definition of the four-dimensional helicity (FDH) regularization scheme valid for two or more loops. This scheme was previously defined and utilized at one loop. It amounts to a variation on the standard 't Hooft-Veltman scheme and is designed to be compatible with the use of helicity states for "observed" particles. It is similar to dimensional reduction in that it maintains an equal number of bosonic and fermionic states, as required for preserving supersymmetry. Supersymmetry Ward identities relate different helicity amplitudes in supersymmetric theories. As a check that the FDH scheme preserves supersymmetry, at least through two loops, we explicitly verify a number of these identities for gluon-gluon scattering (gg to gg) in supersymmetric QCD. These results also cross-check recent non-trivial two-loop calculations in ordinary QCD. Finally, we compute the two-loop shift between the FDH coupling and the standard MS-bar coupling, alpha_s. The FDH shift is identical to the one for dimensional reduction. The two-loop coupling shifts are then used to obtain the three-loop QCD beta function in the FDH and dimensional reduction schemes.

Figures (11)

• Figure 1: Sample diagrams contributing to: (a) a gluon circulating in the loop, (b) a fermion circulating in the loop, and (c) a scalar circulating in the loop.
• Figure 2: Representative diagrams contributing to the supersymmetric amplitudes (a) ${\cal A}_4^{\rm vector}$, (b) ${\cal A}_4^{\rm matter (1)}$, (c) ${\cal A}_4^{\rm matter (2)}$, and (d) ${\cal A}_4^{\rm Yukawa}$ in eqs. (\ref{['N=1fundSchematicAmplitude']}) and (\ref{['N=1adjSchematicAmplitude']}). The curly, solid without arrows, solid with arrows, and dotted lines represent gluons, gluinos, quarks, and scalars, respectively.
• Figure 3: Two examples of $s$-channel double two-particle cuts of a two-loop amplitude, which separate it into a product of three tree amplitudes. The dashed lines represent the generalized cuts.
• Figure 4: The standard three-particle cut of a two-loop amplitude.
• Figure 5: Parent graphs for the pure gluon $G$ contributions.