Bounds on Amplitudes in Effective Theories with Massive Spinning Particles

TL;DR

The paper develops a bootstrap-style framework to bound the high-energy growth of tree-level 2→2 amplitudes for a single massive particle with integer spin in an EFT with a finite derivative expansion. By constructing all on-shell cubic and quartic vertices and combining exchange and contact contributions into full amplitudes, the authors derive model-independent lower bounds on $A(E)$ scaling, enforcing locality, unitarity, and crossing. They find that for a single massive spin-2 particle the minimal growth is $E^6$, saturated only by the ghost-free theories dRGT massive gravity and the pseudolinear theory, with parity-odd amplitudes constrained as well; for spins 0 and 1 explicit bounds and structures are also analyzed, and a conjecture for higher spins is proposed. The work suggests a deep link between ghost freedom and slow high-energy growth, providing a holographic-type diagnostic for viable UV completions of massive-spin EFTs and offering a framework that could extend to more complex spectra and beyond four dimensions.

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.