Scattering equations and BCJ relations for gauge and gravitational amplitudes with massive scalar particles

TL;DR

This work extends color-kinematic duality and CHY representations to tree-level amplitudes involving a pair of massive scalars and massless gauge or gravitational states. By generalizing the scattering equations with symmetric mass terms $\Delta_{ab}$ and constructing CHY-type formulas for a double-color theory, gauge theory, and gravity, the authors derive BCJ relations for amplitudes with two massive fundamentals and provide explicit expressions in terms of $\det'\Phi(\sigma)$ and related Pfaffians. The results unify massive and massless cases within a single CHY framework and illustrate the double-copy structure in gravity via $E(\sigma)$ and $\tilde{E}(\sigma)$ factors, with concrete checks at $n=4$–$6$. These developments offer a path toward loop-level generalizations and potential connections to soft-graviton theorems, while expanding applicability to non-adjoint matter in gauge theories. Key formulas include the massive scattering equations $f_a=\sum_{b\neq a} \frac{k_a\cdot k_b + \Delta_{ab}}{\sigma_a-\sigma_b}=0$, the CHY amplitudes ${\cal A}_{\rm scalar}$, ${\cal A}_{\rm gauge}$, ${\cal A}_{\rm grav}$, and the fundamental massive BCJ relation $0=\sum_{a=3}^n(- m_\psi^2+\sum_{b=a}^n s_{2b}) A(1_\psi,3,\dots, a-1,2,a,\dots,n_{\bar{\psi}})$.

Abstract

We generalize the scattering equations to include both massless and massive particles. We construct an expression for the tree-level n-point amplitude with n-2 gluons or gravitons and a pair of massive scalars in arbitrary spacetime dimension as a sum over the (n-3)! solutions of the scattering equations, a la Cachazo, He, and Yuan. We derive the BCJ relations obeyed by these massive amplitudes.