No conformal anomaly in unimodular gravity Enrique A ´ lvarez 1,2, * and Mario Herrero-Valea 2,† 1 Physics Department, Theory Unit, CERN 1211 Gene `ve 23, Switzerland 2 Instituto de Fı ´sica Teo ´rica UAM/CSIC and Departamento de Fı ´sica Teo ´rica, Universidad Auto ´noma de Madrid, E-28049 Madrid, Spain (Received 23 January 2013; published 23 April 2013) The conformal invariance of unimodular gravity survives quantum corrections, even in the presence of conformal matter. Unimodular gravity can actually be understood as a certain truncation of the full Einstein-Hilbert theory, where in the Einstein frame the metric tensor has unit determinant. Our result is compatible with the idea that the corresponding restriction in the functional integral is consistent as well. DOI: 10.1103/PhysRevD.87.084054 PACS numbers: 04.50.Kd I. INTRODUCTION A radical approach towards explaining why (the zero mode of) the vacuum energy seems to violate the equiva- lence principle (the active cosmological constant problem) is just to eliminate the direct coupling in the action between the potential energy and the gravitational field [1]. This leads us to consider unimodular theories, where the metric tensor is constrained to be unimodular g E j det g E j¼ 1 (1.1) in the Einstein frame. This equality only stands in those reference frames obtained from the Einstein one by an area preserving diffeomorphism. Those are by definition the ones that enjoy unit Jacobian, and g is a singlet under them. We shall represent the absolute value of the determinant of the metric tensor in an arbitrary frame as g instead of jgj in order to simplify the corresponding formulas. We work in arbitrary dimension n in order to be able to employ dimensional regularization as needed. The simplest non- trivial such unimodular action [1] reads S U M n2 Z d n xR E þ S matt ¼M n2 Z d n xg 1 n R þ ðn 1Þðn 2Þ 4n 2 g r gr g g 2 þ S matt ; (1.2) where the n-dimensional Planck mass is related to the n-dimensional Newton constant through M n2 1 16G ; (1.3) and S matt is the matter contribution to the action. This theory is conformally (Weyl) invariant under ~ g ¼ 2 ðxÞg ðxÞ (1.4) (the Einstein metric is inert under those) as well as under area preserving (transverse) diffeomorphisms, id est, those that enjoy unit Jacobian, thereby preserving the Lebesgue measure. We shall speak always of conformal invariance in the above sense. The aim of this paper is to explore whether this gauge symmetry is anomalous or survives when one-loop quan- tum corrections are taken into account. The result we have found is that, given the fact that this theory can be thought of as a partial gauge fixed sector of a conformal upgrading of general relativity, there is no conformal anomaly for unimodular gravity, even when conformal matter is included. Other interesting viewpoints on the cosmological constant from the point of view of unimodular gravity are presented in Refs. [2,3]. In this last reference Smolin suggested the absence of conformal anomaly for related theories. We will proceed as follows. First, we will define a more general scalar-tensor theory by introducing a spurion field . This theory is diffeomorphism as well as conformal invariant and unimodular gravity is no more than a partial gauge fixed sector of it. This happens to be, also, the same theory that ’t Hooft proposed [4] in order to solve some special issues of black hole complementarity. Consequently, in Sec. III we will explore ’t Hooft’s approach in order to obtain the divergent part of the one- loop effective action of such theory. In Sec. IV , however, we will show how the scalar-tensor action can be written in a more useful and manifestly conformal invariant form and we will use it to easily compute the one-loop gravitational counterterm in Sec. V . Our result is not only consistent with ’t Hooft’s computations but also shows in a very clear way how the anomaly vanishes. II. A MORE GENERAL SCALAR-TENSOR THEORY It is technically quite complicated to gauge fix a theory invariant under area preserving (transverse) diffeomor- phisms only, because the theory is reducible, id est, the corresponding gauge parameters are not independent. * enrique.alvarez@uam.es † mario.herrero@estudiante.uam.es PHYSICAL REVIEW D 87, 084054 (2013) 1550-7998= 2013=87(8)=084054(9) 084054-1 Ó 2013 American Physical Society