Gravitational Contact Interactions and the Physical Equivalence of Weyl Transformations in Effective Field Theory

TL;DR

This paper demonstrates that contact interactions in scalar-tensor gravity theories are equivalent to operators generated by Weyl transformations, confirming the physical equivalence of different action representations both classically and quantum mechanically.

Contribution

It explicitly shows the equivalence of non-minimal and minimal Einstein-Hilbert actions via contact terms and Weyl transformations in effective field theories of gravity.

Findings

Contact terms arise from graviton exchange in non-minimal scalar-gravity theories.

Weyl transformations relate non-minimal and minimal gravitational actions.

Choosing the minimal Einstein-Hilbert form avoids hidden contact terms.

Abstract

Theories of scalars and gravity, with non-minimal interactions, $\sim (M_P^2 +F(\phi) )R +L(\phi)$, have graviton exchange induced contact terms. These terms arise in single particle reducible diagrams with vertices $\propto q^2$ that cancel the Feynman propagator denominator $1/q^2$ and are familiar in various other physical contexts. In gravity these lead to additional terms in the action such as $\sim F(\phi) T_\mu^\mu(\phi)/M_P^2$ and $F(\phi)\partial^2 F(\phi)/M_P^2$. The contact terms are equivalent to induced operators obtained by a Weyl transformation that removes the non-minimal interactions, leaving a minimal Einstein-Hilbert gravitational action. This demonstrates explicitly the equivalence of different representations of the action under Weyl transformations, both classically and quantum mechanically. To avoid such "hidden contact terms" one is compelled to go to the minimal Einstein-Hilbert representation.