Superfield twist-$2$ operators in $\mathcal{N} = 1$ SCFTs and their renormalization-group improved generating functional in $\mathcal{N} = 1$ SYM theory

Abstract

We provide a new construction of superfield collinear twist-$2$ operators as infinite-dimensional, irreducible representations of the collinear superconformal algebra in $\mathcal{N}=1$ superconformal field theories. As an application, we realize the above representations in terms of free superfields, in a manifestly gauge-invariant and supersymmetric-covariant fashion, in the zero coupling limit of $\mathcal{N}=1$ supersymmetric Yang-Mills (SYM) theory. This realization makes manifest their mixing and renormalization properties at one loop. We also extend to the superfield formalism the perturbative and nonperturbative techniques in [1-7] to a large class of supersymmetric theories that are superconformal in the zero-coupling limit. Specifically, we compute the generating functional of superfield twist-$2$ operators in $\mathcal{N}=1$ SU($N$) SYM theory in the zero coupling limit. We also work out in a closed form the corresponding asymptotic renormalization-group improved generating functional in Euclidean superspace and its planar and leading nonplanar large-$N$ expansion. We verify -- as originally predicted in [5] and verified in the component formalism [3, 4, 6, 7] -- that the leading nonplanar asymptotic RG-improved generating functional matches the structure of logarithm of a functional superdeterminant of the corresponding nonperturbative object, which it should be asymptotic to at short distances because of the asymptotic freedom. Hence, our large-$N$ computation sets strong ultraviolet asymptotic constraints on the nonperturbative solution of large-$N$ $\mathcal{N} = 1$ SYM theory that may be a pivotal guide for the search of such a solution.