Massive Gravity from Double Copy

TL;DR

This work extends the BCJ double copy to massive Yang–Mills theory, deriving a double-copy effective theory that includes a massive spin-2, spin-1, and spin-0 sector. Up to quartic order, the spin-2 interactions reproduce ghost-free dRGT massive gravity, with a Λ3=(m^2 M_Pl)^{1/3} cutoff, and the Λ3 decoupling limit yields a bi-Galileon theory, revealing that decoupling and double copy do not commute due to how kinematic factors scale. The authors construct explicit amplitudes and the full Lagrangian, discuss the role of Stückelberg fields, and demonstrate how the decoupling limit structures emerge from the double-copy construction, while also outlining limitations at higher points and potential resolutions. The work clarifies how Galileon-like theories arise in decoupling limits of double-copy constructions and points to future directions for achieving a consistent massive-double-copy framework beyond quartic order and in loop settings.

Abstract

We consider the double copy of massive Yang-Mills theory in four dimensions, whose decoupling limit is a nonlinear sigma model. The latter may be regarded as the leading terms in the low energy effective theory of a heavy Higgs model, in which the Higgs has been integrated out. The obtained double copy effective field theory contains a massive spin-2, massive spin-1 and a massive spin-0 field, and we construct explicitly its interacting Lagrangian up to fourth order in fields. We find that up to this order, the spin-2 self interactions match those of the dRGT massive gravity theory, and that all the interactions are consistent with a $Λ_3= (m^2 M_{Pl})^{1/3}$ cutoff. We construct explicitly the $Λ_3$ decoupling limit of this theory and show that it is equivalent to a bi-Galileon extension of the standard $Λ_3$ massive gravity decoupling limit theory. Although it is known that the double copy of a nonlinear sigma model is a special Galileon, the decoupling limit of massive Yang-Mills theory is a more general Galileon theory. This demonstrates that the decoupling limit and double copy procedures do not commute and we clarify why this is the case in terms of the scaling of their kinematic factors.