Massive Color-Kinematics Duality and Double-Copy for Kaluza-Klein Scattering Amplitudes

TL;DR

This paper develops a framework for deriving and understanding massive Kaluza-Klein scattering amplitudes using double-copy relations, extending massless amplitude techniques to include massive states from compactified extra dimensions.

Contribution

It introduces a shifting method to derive massive KK amplitudes from massless ones and extends double-copy relations to massive KK states, including new group theory insights.

Findings

Explicit four- and five-point KK scattering amplitudes derived.

Established connections between KK amplitudes and double-copy relations.

Proposed a group theory approach to validate KK theories under toroidal compactification.

Abstract

We study the scattering amplitudes of massive Kaluza-Klein (KK) states under toroidal compactification. We present a shifting method to quantitatively derive the scattering amplitudes of massive KK gauge bosons and KK gravitons from the corresponding massless amplitudes in the noncompactified higher dimensional theories. With these we construct the massive KK scattering amplitudes by extending the double-copy relations of massless scattering amplitudes within the field theory framework, including both the BCJ and CHY methods, and build up their connections to the massive KK KLT relations. We present the massive BCJ-type double-copy construction of the $N$-point KK gauge boson/graviton scattering amplitudes, and as the applications we derive explicitly the four-point KK scattering amplitudes as well as the five-point KK scattering amplitudes. We further study the nonrelativistic limit of these massive scattering amplitudes with the heavy external KK states and discuss the impact of the compactified extra dimensions on the low energy gravitational potential. Finally, we analyze the four-point and $N$-point mass spectral conditions and newly propose a novel group theory approach to prove that only the KK theories under toroidal compactification can satisfy these conditions for directly realizing massive double-copy in the field theory framework.