Pauli-Fierz mass term in modified Plebanski gravity

David Beke 2 Giovanni Palmisano 0 Simone Speziale 1 0 Dipartimento di Fisica, Universita` La Sapienza , INFN Sez. Roma1, I-00185 Roma, Italy 1 Centre de Physique Theorique, Aix-Marseille University , CNRS-UMR 7332, 13288 Marseille, France 2 Department of Mathematical Analysis EA16, Ghent University , Galglaan 2, 9000 Ghent, Belgium We study SO(4) BF theory plus a general quadratic potential, which describes a bi-metric theory of gravity. We identify the profile of the potential leading to a Pauli-Fierz mass term for the massive graviton, thereby avoiding the linearized ghost. We include the Immirzi parameter in our analysis, and find that the mass of the second graviton depends on it. At the non-perturbative level, we find a situation similar to genuine bi-gravities: even choosing the Pauli-Fierz mass term, the ghost mode propagates through the interactions. We present some simple potentials leading to two and three degrees of freedom, and discuss the difficulties of finding a ghost-free bi-gravity with seven degrees of freedom. Finally, we discuss alternative reality conditions for the case of SO(3,1) BF theory, relevant for Lorentzian signature, and give a new solution to the compatibility equation. 1 Introduction Plebanski formalism and modified theories of gravity 2.1 Bi-metric interpretation Bi-flat background and linearized BF theory 3.1 Linearized on-shell connection 3.2 Linearized BF action 3.3 Bi-metric parametrization Pauli-Fierz mass term 5 Some simple singular potentials 5.1 Self-dual theory 5.2 Scalar-tensor theory 5.3 Scalar constraint and unimodular massive gravity 5.4 Genuine bi-gravity Lorentzian theory and reality conditions 6.1 Solving the compatibility condition Conclusions A SO(4) conventions and invariants A.1 Conventions A.2 Connection components A.3 Curvature components Introduction Topological theories of the BF type bear interesting relations with general relativity, see [1] for a recent review. A mechanism to relate the two theories is the use of the Plebanski action, which comes in either the self-dual [2, 3] or the non-chiral [36] formalism. In the latter, which is the subject of this paper, one starts with BF theory with the Lorentz group (or SO(4) in Euclidean signature) as local gauge group, and adds suitable simplicity constraints to recover the dynamics of general relativity. The resulting action is polynomial, and it plays an important role in the spin foam approach to quantum gravity [711]. Recently [12], building on previous work in the self-dual formalism [1316], it was shown that SO(4) BF theory alone, without constraints can be parametrized in terms of two metrics plus scalars fields. The metric interpretation is however completely artificial, and carries no physics, because of the large shift symmetry that makes the theory topological: any pair of metric is connected to the bi-flat one (e.g. for trivial topology) by a shift transformation, and no local degrees of freedom propagate. The situation changes when the simplicity constraints are added, and general relativity is recovered. The advantage of the bi-metric parametrization is to make the role of the constraints completely transparent: they freeze the scalars fields, and identify the two metrics with one another. An interesting set of theories can be constructed by adding a potential term to the BF action, instead of the simplicity constraints. Such modified non-chiral Plebanski actions were proposed initially in [17, 18] as models for grand unification, and in [13, 14, 19] for the self-dual case.1 The addition of the potential term breaks the shift symmetry, and propagating local degrees of freedom (DOFs) appear. A canonical analysis performed in [27] showed that for a generic potential, there are eight DOFs. Their interpretation was clarified in [12], where it was shown that the action describes a bi-metric theory of gravity plus scalars, with the two metrics interacting through the scalars and the potential. The eight degrees of freedom turned out to be, at the linearized level, the same of standard bi-metric theories [28, 29]. Namely, a massless and a massive graviton, plus a massive scalar. The scalar mode found in [12] is a ghost, and the theory is unstable around the bi-flat solution. This is a situation familiar from massive gravity, where a ghost is present unless the particular Pauli-Fierz form [30] for the mass term is chosen. The main result of the present paper is to complete the analysis of [12]: we study the most general quadratic potential of the modified Plebanski action, and identify the values of the coefficients leading to the Pauli-Fierz mass term. In doing so, we also extend the previous analysis to include the Immirzi parameter, and find that the mass of the second graviton depends on it. This might be surprising at first sight, but it is a natural consequence of giving a canonical form to the kinetic terms of the action, and it suggests an intriguing new role for the Immirzi parameter in such modified theories of gravity. We also give a more complete description of the linearized theory, studying the decomposition of the various fields in their irreducible representations, and unravel a simple relation between SO(4) BF theory and general relativity at this level: loosely speaking, second order SO(4) BF theory can be written as Einsteins action but with the connection as a functional of two metrics and scalar fields. This material appears in sections 2 to 4. The ghost absence at the linearized level is however not enough to guarantee the stability of the theory, as the unhealthy degree of freedom can reappear through interactions, as it is well-known from a classic result by Boulware and Deser [31]. While it has been argued that the theory is stable on cosmological or Lorentz-violating backgrounds [29, 3238], it is interesting to understand whether flat background stability is necessarily compromised by a graviton mass. In particular, recent literature [3942] claims that a specific, nonpolynomial potential is non-perturbatively ghost-free, although a debate on the proof still continues [4345]. One might hope that the situations is different in the context of the 1In the self-dual case, the modification turns out to be related to a class of generally covariant actions previously considered in [2022]. The modified self-dual theory has been studied quite extensively in the literature, e.g. [13, 14, 16, 2326]. modified Plebanski theory, due to the presence of additional scalar fields. However, this is not the case. In section 5, we show that the Pauli-Fierz mass term is again not a sufficient condition for the non-perturbative absence of the ghost. Although we are not able to provide a systematic study of all possible potentials, we discuss a few natural choises. While classes with two or three degrees of freedom can be easily identified, classes with seven degrees of freedom, corresponding to ghost-free bigravities, appear elu (...truncated)