Quantum equivalence principle violations in scalar-tensor theories

We study the equivalence principle and its violations by quantum effects in scalar-tensor theories that admit a conformal frame in which matter only couples to the spacetime metric. These theories possess Ward identities that guarantee the validity of the weak equivalence principle to all orders in the matter coupling constants. These Ward identities originate from a broken Weyl symmetry under which the scalar field transforms by a shift, and from the symmetry required to couple a massless spin two particle to matter (diffeomorphism invariance). But the same identities also predict violations of the weak equivalence principle relatively suppressed by at least two powers of the gravitational couplings, and imply that quantum corrections do not preserve the structure of the action of these theories. We illustrate our analysis with a set of specific examples for spin zero and spin half matter fields that show why matter couplings do respect the equivalence principle, and how the couplings to the gravitational scalar lead to the weak equivalence principle violations predicted by the Ward identities.

Figure 1

Scalar and graviton exchange between two different matter species. Continuous lines denote matter fields (bosonic or fermionic), dashed lines label the scalar $\varphi$, and wiggly lines label the graviton.

Figure 2

A light scalar (dashed) and a massless graviton mediate long-ranged interactions through the interchange of a single quantum. Each blob in a vertex represents the sum of all 1PI diagrams with the corresponding number of external lines, and external propagators stripped off. Each blob with two external lines represents the full propagator, the sum of all (connected) diagrams with the corresponding type of external lines.

Figure 3

Diagrammatic expression of Eq. (48) to leading order in the gravitational couplings $M_P^{-1}$ and $M^{-1}$. The irreducible vertex for scalar emission equals the trace of that for graviton emission plus or minus corrections terms. In each correction term, the blob represents the sum of all 1PI diagrams with the corresponding number of external lines and the vertex insertion marked by a dot.

Figure 4

One-loop corrections to the vertex for scalar emission by matter.

Figure 5

One-loop correction to the self-energy of matter.

Figure 6

One-loop corrections to the vertex for scalar emission at order $1/M^3$. Continuous lines denote matter fields, while dashed lines label the scalar $\varphi$.

Figure 7

Self-energy of matter to order $1/M^2$.

Figure 8

Diagrams with two external lines and the insertion of the two vertices in Eq. (89).

Figure 9

One-loop corrections to the vertex for graviton emission to order $1/M^2$.