  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.
