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We work out the one-loop and order $\kappa^2 m_\phi^2$ UV divergentcontributions, coming from Unimodular Gravity and General Relativity, to the Smatrix element of the scattering process $\phi + \phi\rightarrow \phi + \phi$in a $\lambda \phi^4$ theory with mass $m_\phi$. We show that both UnimodularGravity and General Relativity give rise to the same UV divergent contributionsin Dimension Regularization. This seems to be at odds with the known resultthat in a multiplicative MS dimensional regularization scheme the GeneralRelativity corrections, in the de Donder gauge, to the beta function$\beta_{\lambda}$ of the $\lambda$ coupling do not vanish, whereas theUnimodular Gravity corrections, in a certain gauge, do vanish. Actually, weshow that the UV divergent contributions to the 1PI Feynman diagrams which giverise to those non-vanishing corrections to $\beta_{\lambda}$ do not contributeto 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.

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