Amplitudes for massive vector and scalar bosons in spontaneously-broken gauge theory from the CHY representation

TL;DR

This paper derives formulas for calculating tree-level amplitudes involving massive vector and scalar bosons in spontaneously-broken gauge theories using the CHY representation, by dimensional reduction from massless amplitudes.

Contribution

It introduces a method to obtain massive particle amplitudes from massless CHY amplitudes via dimensional reduction, linking them to spontaneously-broken gauge theories.

Findings

Massive amplitudes are obtained by replacing dot products with a shifted version involving internal momentum.

The approach applies to gauge theories with spontaneous symmetry breaking and Higgs mechanism.

Explicit formulas for tree-level amplitudes with massive bosons are provided.

Abstract

In the formulation of Cachazo, He, and Yuan, tree-level amplitudes for massless particles in gauge theory and gravity can be expressed as rational functions of the Lorentz invariants $k_a \cdot k_b$, $\epsilon_a \cdot k_b$, and $\epsilon_a \cdot \epsilon_b$, valid in any number of spacetime dimensions. We use dimensional reduction of higher-dimensional amplitudes of particles with internal momentum $\kappa$ to obtain amplitudes for massive particles in lower dimensions. In the case of gauge theory, we argue that these massive amplitudes belong to a theory in which the gauge symmetry is spontaneously broken by an adjoint Higgs field. Consequently, we show that tree-level $n$-point amplitudes containing massive vector and scalar bosons in this theory can be obtained by simply replacing $k_a \cdot k_b$ with $k_a \cdot k_b - \kappa_a \kappa_b $ in the corresponding massless amplitudes, where the masses of the particles are given by $|\kappa_a|$.