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

A bstract We provide a new construction of superfield collinear twist-2 operators as infinite-dimensional, irreducible representations of the collinear superconformal algebra in $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 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 $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 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 $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 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 $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 1 SYM theory that may be a pivotal guide for the search of such a solution.