Scattering of fermions in the Yukawa theory coupled to Unimodular Gravity

TL;DR

This paper calculates quantum corrections to fermion scattering in Yukawa theory coupled to Unimodular Gravity and General Relativity, finding identical UV divergences and unexpected finiteness at certain orders, suggesting similar quantum behavior.

Contribution

It demonstrates that Unimodular Gravity and General Relativity produce the same UV divergent contributions in fermion scattering, indicating similar quantum properties when coupled to matter.

Findings

Both theories yield identical UV divergences in dimensional regularization.

Unexpected UV divergence cancellations lead to finiteness at specific orders.

Results support the idea that Unimodular Gravity and General Relativity are equivalent at the quantum level under certain conditions.

Abstract

We compute the lowest order gravitational UV divergent radiative corrections to the S matrix element of the $fermion + fermion\rightarrow fermion + fermion$ scattering process in the massive Yukawa theory, coupled either to Unimodular Gravity or to General Relativity. We show that both Unimodular Gravity and General Relativity give rise to the same UV divergent contribution in Dimensional Regularization. This is a nontrivial result, since in the classical action of Unimodular Gravity coupled to the Yukawa theory, the graviton field does not couple neither to the mass operator nor to the Yukawa operator. This is unlike the General Relativity case. The agreement found points in the direction that Unimodular Gravity and General Relativity give rise to the same quantum theory when coupled to matter, as long as the Cosmological Constant vanishes. Along the way we have come across another unexpected cancellation of UV divergences for both Unimodular Gravity and General Relativity, resulting in the UV finiteness of the one-loop and $\kappa g^2$ order of the vertex involving two fermions and one graviton only.