2018, January 20

S. Gonzalez-Martin and C. P. Martin

JCAP01(2018)028

## We work out the one-loop and order κ^{2} *m*_{}^{2} UV divergent contributions, coming from Unimodular Gravity and General Relativity, to the S matrix element of the scattering process + → + in a λ ^{4} theory with mass *m*_{}. We show that both Unimodular Gravity and General Relativity give rise to the same UV divergent contributions in Dimensional 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, β_{λ}, of the λ coupling do not vanish, whereas the Unimodular Gravity corrections, in a certain gauge, do vanish. Actually, by comparing the UV divergent contributions calculated in this paper with those which give rise to the non-vanishing gravitational corrections to β_{λ}, one readily concludes that the UV divergent contributions that yield the just mentioned non-vanishing gravitational corrections to β_{λ} do not contribute to the UV divergent behaviour of the S matrix element of + → + . This shows that any physical consequence—such as the existence of asymptotic freedom due to gravitational interactions—drawn from the value of β_{λ} is not physically meaningful.l