Loop-Level Double-Copy for Massive Quantum Particles

TL;DR

This work tackles loop-level scattering in massive scalar QCD by developing a constructive bootstrap based on factorization and color-kinematics duality. By carefully organizing amplitudes into color-dual cubic graphs and exploiting unitarity cuts, the authors build one-loop four-point and five-point integrands from tree-level building blocks, even with massive external states. The results yield explicit loop integrands for two distinct massive scalars and for configurations with an emitted gluon, and the same color-dual structures naturally double-copy to the corresponding ${ m N}=0$ supergravity predictions. This framework demonstrates that loop-level color-kinematics duality extends to massive theories and offers a streamlined path to gravity via double-copy, with clear implications for classical/gravitational observable calculations and potential extensions to massive higher-spin amplitudes.

Abstract

We find that scattering amplitudes in massive scalar QCD can manifest the duality between color and kinematics at loop-level. Specifically we construct the one-loop integrands for four-point scattering between two distinct massive scalars, and the five-point process encoding the first correction to massive scalar scattering with gluonic radiation. We find that factorization and the color-kinematics duality are sufficient principles to entirely bootstrap these calculations, allowing us to construct all contributions ultimately from the three-point tree-level amplitudes which are themselves entirely constrained by symmetry. Double-copy construction immediately provides the associated predictions for massive scalars scattering in the so called N=0 supergravity theory.

Figures (13)

• Figure 1: Color-weights of cubic-vertices. The solid (red) lines represent massive scalars charged potentially under the fundamental representation of the group.
• Figure 2: Pictorial representation of the relations between color weights and potentially kinematic weights of graphs. The solid (red) lines represent same-mass scalar particles.
• Figure 3: Graph manipulation operators that rearrange one internal edge's connectivity. This is useful for specifying Jacobi-like relations, e.g. $c(g) = c(\hat{t}g) + c(\hat{u}g)$.
• Figure 4: The four-point tree amplitudes in scalar QCD.
• Figure 5: The five-point tree amplitudes in scalar QCD.