Fifth forces and broken scale symmetries in the Jordan frame

We study the origin of fifth forces in scalar-tensor theories of gravity in the so-called Jordan frame, where the modifications to the gravitational sector are manifest. We focus on theories of Brans-Dicke type in which an additional scalar field is coupled directly to the Ricci scalar of General Relativity. We describe how the necessary diffeomorphism invariance of the modified gravitational sector leads to a modification of the usual gauge fixing term (for the harmonic gauge), as compared to Einstein gravity. This allows us to perform a consistent linearization of the gravitational sector in the weak-field limit, which gives rise to a kinetic mixing between the non-minimally coupled scalar field and the graviton. It is through this mixing that a fifth force can arise between matter fields. We are then able to compute the matrix elements for fifth-force exchanges directly in the Jordan frame, without the need to perform a conformal transformation to the so-called Einstein frame, where the gravitational sector is of Einstein-Hilbert form. We obtain results that are in agreement with the equivalent Einstein-frame calculations and illustrate, still in the Jordan frame, the pivotal role that sources of explicit scale symmetry breaking in the matter sector play in admitting fifth-force couplings.