Unimodular Gravity and General Relativity UV divergent contributions to the scattering of massive scalar particles

TL;DR

This paper compares UV divergent contributions from Unimodular Gravity and General Relativity to scalar particle scattering, revealing that physical scattering amplitudes are unaffected by differences in their beta functions.

Contribution

It demonstrates that UV divergences in scattering amplitudes are identical for both theories, challenging the physical significance of beta function differences.

Findings

UV divergent contributions are the same in both theories.

Differences in beta functions do not affect physical scattering amplitudes.

Asymptotic freedom claims from beta functions are not physically meaningful.

Abstract

We work out the one-loop and order $\kappa^2 m_\phi^2$ UV divergent contributions, coming from Unimodular Gravity and General Relativity, to the S matrix element of the scattering process $\phi + \phi\rightarrow \phi + \phi$ in a $\lambda \phi^4$ theory with mass $m_\phi$. We show that both Unimodular Gravity and General Relativity give rise to the same UV divergent contributions in Dimension Regularization. This seems to be at odds with the known result that in a multiplicative MS dimensional regularization scheme the General Relativity corrections, in the de Donder gauge, to the beta function $\beta_{\lambda}$ of the $\lambda$ coupling do not vanish, whereas the Unimodular Gravity corrections, in a certain gauge, do vanish. Actually, we show that the UV divergent contributions to the 1PI Feynman diagrams which give rise to those non-vanishing corrections to $\beta_{\lambda}$ do not contribute to the UV divergent behaviour of the S matrix element of $\phi + \phi\rightarrow \phi + \phi$ and this shows that any physical consequence --such existence of asymptotic freedom due to gravitational interactions-- drawn from the value of $\beta_{\lambda}$ is not physically meaningful.