Abstract
For a general renormalizable $\mathcal{N}=1$ supersymmetric gauge theory with a simple gauge group we verify the ultraviolet (UV) finiteness of the two-loop matter contribution to the triple gauge-ghost vertices. These vertices have one leg of the quantum gauge superfield and two legs corresponding to the Faddeev--Popov ghost and antighost. By an explicit calculation made with the help of the higher covariant derivative regularization we demonstrate that the sum of the corresponding two-loop supergraphs containing a matter loop is not UV divergent in the case of using a general $\ensuremath{\xi}$-gauge. In the considered approximation this result confirms the recently proved theorem that the triple gauge-ghost vertices are UV finite in all orders, which is an important ingredient of the all-loop perturbative derivation of the Novikov-Shifman-Vainshtein-Zakharov relation.
Highlights
Possible ultraviolet divergences in supersymmetric theories are restricted by some nonrenormalization theorems
It is reasonable to consider the exact Novikov-Shifman-Vainshtein-Zakharov (NSVZ) β-function [2,3,4,5] as a nonrenormalization theorem, because it relates the renormalization of the gauge coupling constant to the renormalization of chiral matter superfields
The NSVZ scheme is obtained if a theory is regularized by the higher covariant derivative method [30,31] in the superfield version [33,34] and the renormalization is made by minimal subtractions of logarithms [27]
Summary
Possible ultraviolet divergences in supersymmetric theories are restricted by some nonrenormalization theorems. The NSVZ scheme is obtained if a theory is regularized by the higher covariant derivative method [30,31] (which includes introducing the Pauli–Villars determinants for removing one-loop divergences [32]) in the superfield version [33,34] and the renormalization is made by minimal subtractions of logarithms [27] This renormalization prescription is usually called HD þ MSL [35,36].2. Villars determinant removes one-loop divergences produced by a matter loop It is given by the functional integral over the (commuting) chiral superfields Φi in a representation RPV which admits a gauge invariant mass term such that MikMÃkj 1⁄4 M2δij, DetðPV; MÞ−1 1⁄4 DΦ expðiSΦÞ; ð18Þ. This implies that the considered version of the regularization can be constructed only if the gauge anomalies are absent, so that no contradiction appears
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