Structure constants of planar N =4 Yang Mills at one loop

TL;DR

This paper derives a simple formula for one-loop corrections to structure constants in planar N=4 Yang-Mills theory, revealing their relation to anomalous dimensions and conformal invariance restoration.

Contribution

It introduces a renormalization group invariant formula for structure constant corrections and connects them to the anomalous dimension Hamiltonian in the scalar sector.

Findings

One-loop corrections are determined by the anomalous dimension Hamiltonian.

Corrections can be expressed as derivatives on a four-point scalar function.

Conformal invariance is restored by combining diagrams using a differential equation.

Abstract

We study structure constants of gauge invariant operators in planar N=4 Yang-Mills at one loop with the motivation of determining features of the string dual of weak coupling Yang-Mills. We derive a simple renormalization group invariant formula characterizing the corrections to structure constants of any primary operator in the planar limit. Applying this to the scalar SO(6) sector we find that the one loop corrections to structure constants of gauge invariant operators is determined by the one loop anomalous dimension Hamiltonian in this sector. We then evaluate the one loop corrections to structure constants for scalars with arbitrary number of derivatives in a given holomorphic direction. We find that the corrections can be characterized by suitable derivatives on the four point tree function of a massless scalar with quartic coupling. We show that individual diagrams violating conformal invariance can be combined together to restore it using a linear inhomogeneous partial differential equation satisfied by this function.