Where is the conformal coupling in Palatini gravity?

Abstract

We study the conformal coupling in the torsion-free Palatini formulation with an ordinary scalar kinetic term. Tuning the curvature coupling cannot restore Weyl invariance in a Palatini theory with an ordinary scalar kinetic term because, for an independent and inert connection, the scalar-curvature term is separately Weyl-invariant, and no finite curvature coupling cancels the Weyl variation of the kinetic term. This can be demonstrated by evaluating the energy-momentum trace on the scalar equation of motion and noting that it is explicitly independent of the Palatini coupling. Eliminating the connection, we construct a map between the curvature coupling and the effective coupling for the equivalent metric theory, and demonstrate that the scalar continues to propagate even if the reduced kinetic coefficient vanishes. We also show that it becomes a Weyl compensator with a scale-invariant potential only in the rescaled infinite-coupling limit, which in four dimensions is directly related to Palatini ${\cal R}^2$ gravity. We calculate the running of the couplings at one loop, and we find that the conformal shift in the metric-curvature coupling's beta function generates a metric-curvature coupling on a fixed metric-affine background, whereas the independent-curvature coupling runs homogeneously. Furthermore, we follow an alternate prescription in which the connection is allowed to respond to the scalar fluctuations, which results in a different scalar Hessian and therefore different beta functions. Nonetheless, we find that both prescriptions admit the formal infrared limit $(ξ_{eff},λ_{eff})\to(1/6,0)$ through different limits of the original couplings in the case of positive perturbative quartics. Finally, we note that trajectories starting on the positive gravitational branch encounter a vanishing gravitational coefficient at a finite scale before this endpoint.