Amplitudes, Form Factors and the Dilatation Operator in $\mathcal{N}=4$ SYM Theory

TL;DR

This paper explores the connection between form factors, integrability, and the dilatation operator in $ ext{N}=4$ SYM theory, deriving the one-loop dilatation operator from field theory and linking it to scattering amplitudes.

Contribution

It provides a field-theoretic derivation of the one-loop dilatation operator from form factors and relates it to scattering amplitudes, highlighting integrability aspects.

Findings

Form factor at tree level matches spin-chain integrability.

Computed one-loop correction using unitarity cuts.

Derived the complete one-loop dilatation operator from UV divergences.

Abstract

We study the form factor of a generic gauge-invariant local composite operator in $\mathcal{N}=4$ SYM theory. At tree level and for a minimal number of external on-shell super fields, we find that the form factor precisely yields the spin-chain picture of integrability in the language of scattering amplitudes. Moreover, we compute the cut-constructible part of the one-loop correction to this minimal form factor via generalised unitarity. From its UV divergence, we obtain the complete one-loop dilatation operator of $\mathcal{N}=4$ SYM theory. Thus, we provide a field-theoretic derivation of a relation between the one-loop dilatation operator and the four-point tree-level amplitude which was observed earlier. We also comment on the implications of our findings in the context of integrability.