Abstract
We consider a procedure for directly constructing general tree-level four-particle scattering amplitudes of massive spinning particles that are consistent with the usual requirements of Lorentz invariance, unitarity, crossing symmetry, and locality. There are infinitely many such amplitudes, but we can isolate interesting theories by bounding the high-energy growth of the tree amplitudes within the effective field theory. This allows us to set model-independent lower bounds on the growth of tree-level amplitudes in any effective field theory with a given particle content and any interaction terms with an arbitrary but finite number of derivatives. In certain common cases this corresponds to finding the highest possible strong coupling scale. When applied to spin 2, we show that the only amplitudes that saturate this bound are generated by the known ghost-free theories of a massive spin-2 particle, namely dRGT massive gravity and the pseudolinear theory. We also make a conjecture for the allowed growth of tree amplitudes in a theory with a single massive particle of any integer spin.
Highlights
Interacting massive higher-spin particles exist—both theoretically and in nature—as ingredients in consistent fundamental theories
This is the scale that sets the mass of the hadrons, so there is no regime in which they are well described by a point-particle effective field theory (EFT)
It was shown in Ref. [9] that if a theory is constructed by adding zeroderivative terms to the Einstein-Hilbert term, with no additional derivative interactions, the E6 behavior is the best possible for the amplitude; i.e., the dRGT choice produces amplitudes with the slowest growth among theories whose derivative interations are fixed to the Einstein-Hilbert form. This E6 behavior has not yet been shown to be a truly model-independent lower bound, because there is still the possibility that other derivative interactions, beyond those of the Einstein-Hilbert form, could be used to further slow the growth of the amplitude.1. We rule out this possibility and show that E6 is a true lower bound on the growth of amplitudes for EFTs of a single interacting massive spin-2 field with a finite number of interactions
Summary
Interacting massive higher-spin particles exist—both theoretically and in nature—as ingredients in consistent fundamental theories. In the case of the higher spin hadrons they become strongly coupled at the QCD scale, the intrinsic size of the hadrons, where they fail to be pointlike and require strong dynamics to complete the description This is the scale that sets the mass of the hadrons, so there is no regime in which they are well described by a point-particle effective field theory (EFT). In the Abelian case, the tree-level amplitude for four-point scattering of longitudinal modes grows with energy like E4 This growth comes with the scale Λ ∼ m=g, where g is a dimensionless coupling constant and m the mass of the spin-1 particle. This method of constructing amplitudes directly, without recourse to a Lagrangian, can be thought of as a kind of bootstrap procedure, and it may prove useful for other problems beyond the specific application of finding a lower bound on the growth that we focus on here
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