# Form factor recursion relations at loop level

## TL;DR

This paper develops a new recursive method for calculating loop-level form factor integrands in planar $ ext{N}=4$ SYM theory using a periodic kinematic setup, extending known amplitude techniques.

## Contribution

It introduces a novel prescription for defining form factor integrands at loop level and formulates recursion relations specific to form factors, highlighting differences from amplitude recursion.

## Key findings

- Form factor integrands at loop level can be constructed using BCFW and all-line shifts.
- The method is validated through explicit one-loop examples.
- The approach connects form factors at strong and weak coupling regimes.

## Abstract

We introduce a prescription to define form factor integrands at loop level in planar $\mathcal{N}=4$ supersymmetric Yang-Mills theory. This relies on a periodic kinematic configuration that has been instrumental to describe form factors at strong coupling in terms of periodic Wilson loops. With this prescription, we are able to formulate loop-level recursion relations for planar form factor integrands, using a two-line (BCFW) and an all-line shift. We also point out important differences with the known recursion relations of integrands of planar loop amplitudes. We present a number of concrete one-loop examples to illustrate and validate our prescription for form factor integrands.