Two-loop renormalization of the matter superfields and finiteness of mathcal{N} = 1 supersymmetric gauge theories regularized by higher derivatives

A E Kazantsev,K V Stepanyantz

doi:10.1007/jhep06(2020)108

Abstract

The two-loop anomalous dimension of the chiral matter superfields is calculated for a general mathcal{N} = 1 supersymmetric gauge theory regularized by higher covariant derivatives. We obtain both the anomalous dimension defined in terms of the bare couplings, and the one defined in terms of the renormalized couplings for an arbitrary renormalization prescription. For the one-loop finite theories we find a simple relation between the higher derivative regulators under which the anomalous dimension defined in terms of the bare couplings vanishes in the considered approximation. In this case the one-loop finite theory is also two-loop finite in the HD+MSL scheme. Using the assumption that with the higher covariant derivative regularization the NSVZ equation is satisfied for RGFs defined in terms of the bare couplings, we construct the expression for the three-loop β-function. Again, the result is written both for the β-function defined in terms of the bare couplings and for the one defined in terms of the renormalized couplings for an arbitrary renormalization prescription.

Highlights

In our notation T A are the generators of the representation to which the matter superfields belong
The two-loop anomalous dimension of the chiral matter superfields is calculated for a general N = 1 supersymmetric gauge theory regularized by higher covariant derivatives
We obtain both the anomalous dimension defined in terms of the bare couplings, and the one defined in terms of the renormalized couplings for an arbitrary renormalization prescription

Summary

The theory under consideration

In terms of the bare couplings, the definitions of the β-function and the anomalous dimension of the chiral matter superfields read as β(α0, λ0) = dα0 ; d ln Λ α,λ=const (γφ)ij(α0, λ0) = − d(ln Zφ)ij . For calculating the two-loop anomalous dimension of the matter superfields, one needs the one-loop relation between the bare and renormalized couplings. [7], the superpotential does not receive divergent radiative corrections This implies that the renormalization of the Yukawa couplings is related to the renormalization of the matter superfields by the equation λijk = ( Zφ)li( Zφ)mj ( Zφ)nkλl0mn. In the HD+MSL renormalization scheme all constants bi and gi are set to 0

Two-loop anomalous dimension

Two-loop anomalous dimension defined in terms of the renormalized couplings

The β-function defined in terms of the bare couplings

The β-function defined in terms of the renormalized couplings

Conclusion

A Calculation of the loop integrals giving the two-loop anomalous dimension