Note on recursion relations for the $\mathcal{Q}$-cut representation

TL;DR

This paper develops a recursion relation for the one-loop integrand in the $ ext{Q}$-cut representation by integrating BCFW deformation, demonstrating equivalence with traditional methods through explicit examples.

Contribution

It introduces a novel recursion relation for the one-loop integrand in the $ ext{Q}$-cut framework, connecting it with BCFW deformation and existing constructions.

Findings

Recursion relation expresses one-loop integrand using tree-level and lower-point integrands.

Explicit examples confirm equivalence with Feynman diagram calculations.

The method simplifies the computation of one-loop integrands in gauge theories.

Abstract

In this note, we study the $\mathcal{Q}$-cut representation by combining it with BCFW deformation. As a consequence, the one-loop integrand is expressed in terms of a recursion relation, i.e., $n$-point one-loop integrand is constructed using tree-level amplitudes and $m$-point one-loop integrands with $m\leq n-1$. By giving explicit examples, we show that the integrand from the recursion relation is equivalent to that from Feynman diagrams or the original $\mathcal{Q}$-cut construction, up to scale free terms.