Relation between leading divergences in nonrenormalizable $4D$ supersymmetric theories
Ali Lakhal, Konstantin Stepanyantz
TL;DR
The work addresses leading power divergences in a nonrenormalizable N=1 SUSY gauge theory with a quartic chiral superpotential and demonstrates that, using Slavnov’s higher covariant derivative regularization, these divergences appear as integrals of double total derivatives in loop momenta. The authors show that the leading quadratically divergent corrections to the gauge coupling and to the chiral matter kinetic term are linked by an NSVZ-like relation, with explicit expressions for the relevant integrals and a vacuum-graph method that reproduces the same structure. Specifically, they establish $\Delta d^{-1}|_{p\to 0} = - \frac{1}{3\pi r} \lambda^*_{0ijkn} \lambda_0^{ijkm} C(R)_m{}^n \int d^4K d^4L \frac{1}{K^2 F_K L^2 F_L (K+L)^2 F_{K+L}}$ and $\Delta G_i{}^j|_{p\to 0} = \frac{2}{3} \lambda^*_{0imnp} \lambda_0^{jmnp} \int d^4K d^4L \frac{1}{K^2 F_K L^2 F_L (K+L)^2 F_{K+L}}$, which satisfy $\Delta d^{-1}|_{p\to 0} = - \frac{1}{2\pi r} C(R)_i{}^j \Delta G_j{}^i|_{p\to 0}$. The vacuum-graph analysis confirms the double-total-derivative mechanism in this nonrenormalizable setting and suggests that an NSVZ-like relation governs the leading divergences in such theories, with implications for calculational schemes and the understanding of gauge–matter coupling interrelations beyond renormalizable models.
Abstract
We consider an ${\cal N}=1$ nonrenormalizable supersymmetric gauge theory with the superpotential quartic in the chiral matter superfields. With the help of the Slavnov's higher covariant derivative regularization it is demonstrated that (in the lowest nontrivial order) the leading power divergent quantum correction to the gauge coupling constant is given by an integral of double total derivatives with respect to the loop momenta. The result obtained after calculating this integral turned out to be proportional to the corresponding quantum correction to the kinetic term of the matter superfields. More exactly, in the considered approximation the quadratically divergent contributions to the gauge coupling and to the kinetic term of the chiral matter superfields are related by an equation analogous to the exact NSVZ $β$-function for the renormalizable case.