Supersymmetric regularization, two-loop QCD amplitudes, and coupling shifts

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 $\left(g\overrightarrow{g}gg\right)$ in supersymmetric QCD. These results also cross-check recent nontrivial two-loop calculations in ordinary QCD. Finally, we compute the two-loop shift between the FDH coupling and the standard modified minimal subtraction $\left(\overline{\mathrm{MS}}\right)$ scheme 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.