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 Λ3 = (m2MPl)1/3 cutoff. We construct explicitly the Λ3 decoupling limit of this theory and show that it is equivalent to a bi-Galileon extension of the standard Λ3 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.
Highlights
The first and most important example is the relationship between Yang-Mills theory and gravity amplitudes [2]
In this paper we initiate the application of the double copy paradigm to the scattering amplitudes of massive Yang-Mills theory, i.e. the low energy effective field theory of YangMills coupled to a heavy Higgs field which spontaneously breaks the gauge symmetry in a way that all of the gauge bosons acquire the same mass
We find that the double copy paradigm automatically leads to a theory in which the interactions of the massive spin-2 field are described by the de RhamGabadadze-Tolley (dRGT) massive gravity [60], at least to quartic order
Summary
The action of massive Yang-Mills theory comes from the low energy effective action of Yang-Mills theory with a Higgs field in which the Higgs particles are integrated out. (∂μAaν − ∂ν Aaμ)2 − Λ2tr(∂μV ∂μV −1) This encodes straightforwardly the content of the ‘Goldstone equivalence theorem’ that the leading interactions for the helicity-0 modes of the massive spin-1 particle are determined by the effective theory for the Goldsones described by (2.4). These extensions just correspond to the addition of further irrelevant operators to the nonlinear sigma model Lagrangian, which have been considered in the double copy context for example in [20,21,22,23]. These tree amplitudes are most conveniently computed in unitary gauge (2.1). We will show that it continues to hold up to quartic order
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