Massive , Topologically Massive , Models

In three dimensions, there are two distinct mass-generating mechanisms for gauge vector and tensor fields: adding the usual Proca/Pauli-Fierz, or the more esoteric Chern-Simons (CS), terms. Here, we analyze the three-term models where both types are present, and their various limits. Surprisingly, in the tensor case, these seemingly innocuous systems are physically unacceptable. If the sign of the Einstein term is “wrong” as required in the CS case, then the excitation masses are always complex; with the usual sign, there is a (known) region of the two mass parameters where reality is restored, but a ghost problem arises. For the two-term system without Einstein term, complex masses are unavoidable. This contrasts with the smooth behavior of the corresponding vector models. Separately, we show that “partial masslessness” exhibited by (plain) massive spin-2 models in de Sitter backgrounds is formally shared by the three-term system: it enjoys a reduced local gauge invariance when this mass parameter is tuned to the cosmological constant. e-mail: deser, [email protected] Topologically massive vector (TME) and tensor (TMG) gauge theories in D = 3 are well-understood models, whose linearized versions describe single massive but gaugeinvariant excitations [1]. We analyze here the augmented, 3-term, systems that break the invariance through explicit mass terms. We will concentrate on the tensor case, massive topologically massive gravity (MTMG), the vectors are briefly reviewed for contrast. We were motivated by two quite separate developments: In the first, it was shown in terms of the propagator structure that the mass eigenvalues of a particular MTMG form are solutions of a cubic equation, two of whose roots become complex in a range of the underlying two “mass” parameter space [2]. In contrast, the vector system’s masses solve a quadratic equation with everywhere real, positive roots [3]. We will analyze the excitations and mass counts for generic signs and values of the parameters as well as both sign choices for the Einstein term, as physically required here. We will uncover not only the objectionable complex masses, but also exhibit the unavoidable presence of ghosts and tachyon excitations in limiting to the two-term models. Our second topic is the study of MTMG in a constant curvature, rather than flat, background. Here we follow recent results which discuss the “partial masslessness” of ordinary massive gravity at a value of the mass tuned to the cosmological constant, where a residual gauge invariance eliminates the helicity zero mode but also leads to non-unitary regions in (m,Λ) plane [5]. It is natural to ask whether this phenomenon persists for MTMG ( it has no vector analog), given the common gauge covariance of the two systems’ kinetic terms, and we will see that it does. The action we consider here is the sum of Einstein, (third derivative order) Chern– Simons, and standard Pauli-Fierz mass term,