On tree amplitudes with gluons coupled to gravitons

TL;DR

This work constructs and interrelates tree-level amplitudes where gluons couple to gravitons, showing that COR, $U(1)$-decoupling, and KK relations extend from pure-gluon to mixed amplitudes via BCFW recursion, and developing a disk relation that expresses $(n,1)$ mixed amplitudes in terms of $(n+2)$-point pure-gluon amplitudes. The disk relation is proven in two forms and then used to re-derive the standard amplitude relations for mixed amplitudes, highlighting how gravity does not alter color structure but introduces nontrivial kinematic factors, notably in BCJ-like relations. The results bridge gauge and gravity sectors at tree level, offering a systematic way to obtain mixed amplitudes from well-studied pure-gluon amplitudes and clarifying the role of helicity and momentum configurations in these mappings. The findings pave the way for further exploration of multi-graviton disks and refined BCJ-like relations in gauge–gravity interactions, with potential applications in perturbative gravity and string-inspired field theories.

Abstract

In this paper, we study the tree amplitudes with gluons coupled to gravitons. We first study the relations among the mixed amplitudes. With BCFW on-shell recursion relation, we will show the color-order reversed relation, $U(1)$-decoupling relation and KK relation hold for tree amplitudes with gluons coupled to gravitons. We then study the disk relation which expresses mixed amplitudes by pure gluon amplitudes. More specifically we will prove the disk relation for mixed amplitudes with gluons coupled to one graviton. Using the disk relation and the properties of pure gluon amplitudes, the color-order reversed relation, $U(1)$-decoupling relation and KK relation for mixed amplitudes can also be proved. Finally, we give some brief discussions on BCJ-like relation for mixed amplitudes.