On-shell Diagrams, Gra{\ss}mannians and Integrability for Form Factors

TL;DR

This paper extends on-shell and integrability techniques from scattering amplitudes to tree-level form factors in N=4 SYM, providing new constructions, representations, and insights into their integrable structure and symmetries.

Contribution

It systematically constructs on-shell diagrams and Gra{ ext}mannian integral representations for form factors, revealing their integrable properties and eigenstate structure under the spin-chain transfer matrix.

Findings

Form factors can be represented by Gra{ ext}mannian integrals in various variables.

Form factors are eigenstates of the integrable spin-chain transfer matrix.

Integrable properties extend to loop-level form factors and certain singularities.

Abstract

We apply on-shell and integrability methods that have been developed in the context of scattering amplitudes in N=4 SYM theory to tree-level form factors of this theory. Focussing on the colour-ordered super form factors of the chiral part of the stress-tensor multiplet as an example, we show how to systematically construct on-shell diagrams for these form factors with the minimal form factor as further building block in addition to the three-point amplitudes. Moreover, we obtain analytic representations in terms of Gra{\ss}mannian integrals in spinor helicity, twistor and momentum twistor variables. While Yangian invariance is broken by the operator insertion, we find that the form factors are eigenstates of the integrable spin-chain transfer matrix built from the monodromy matrix that yields the Yangian generators. Constructing them via the method of R operators allows to introduce deformations that preserve the integrable structure. We finally show that the integrable properties extend to minimal tree-level form factors of generic composite operators as well as certain leading singularities of their n-point loop-level form factors.