Unitarity and Different Phases of the Extended New Massive Gravity

We study the New Massive Gravity extended by the curvature cubed invariant and the Cosmological Constant using the tree-level scattering amplitudes on a maximally symmetric space-time. We identify the parameter space, consisting of the ratio between C.C. and graviton mass squared and the relative coupling strength between curvature squared and cubed invariants, for which the massive spin-2 is a ghost. Different phases of this model are also discussed.


Introduction and Summary
Higher curvature modifications of General Relativity (GR) are important in order to understand the quantum nature of gravity. It has been long known that GR in 3 + 1 dimensions is perturbatively non-renormalizable [1]. Adding terms that are quadratic in curvature solves this problem at the expense of introducing a massive scalar and a massive spin-2 ghost [2].
In 2 + 1 dimensional space-time GR does not suffer the non-renormalizability problem, simply because it does not propagate any degrees of freedom (d.o.f.) [3]. The massive spin-2 generated through higher curvature modifications can be healthy if one chooses the massless mode to be ghost. The wrong sign kinetic term of the massless spin-2 does not lead to any problems (not considering non-positive central charges for the dual CFTs [4]) since in 3D this mode is a pure gauge, but in this case the unitarity is violated through the massive scalar. The conformal mode in GR is a ghost, but closer look reveals that it's not propagating. Addition of higher curvature terms to the action modifies the dynamics of the conformal mode as well. In general, its equation of motion becomes of quartic order and a massive scalar gets generated. The kinetic terms of massive scalar and the massless spin-2 have the same sign, so if one chooses massless spin-2 to be a ghost, so has to be the massive scalar. In general it seems that one can not build a unitary theory in 2 + 1 dimensions by introducing higher curvature invariants.
Ten years ago, Bergshoeff, Hohm & Townsend found a curvature squared invariant that, linearly, is equivalent to Fierz-Pauli and does not violate unitarity [5]. Sinha found its curvature cubed extension [6] (see also [7]). Within this model, now referred as "Extended New Massive Gravity (ENMG)", the conformal mode of the metric has the structure of a galileon [8], so its equation of motion is of second order. As a consequence the massive scalar is absent and one is free to choose the massless spin-2 to have the wrong sign kinetic term. The authors of [9] addressed the non-linear unitarity for the curvature squared term and proved that the model was ghost free to all orders. In [10] it is argued that full ENMG is free of scalar ghosts. There are other works addressing the causality problem for this model [11,12]. The author of this paper is not familiar with any previous works done that addresses the complete unitarity of ENMG.
While writing this paper, I found out that some of my results coincide with [10], so I feel it's necessary to point out the differences. First of all, our common results agree with each other. The disjoint part is that they are including the square of Cotton tensor, which makes the spectrum richer, but necessarily leading to ghosts and tachions. My analysis is restricted to (2.1), but it's more detailed. Their analysis relies on the Lagrangian formalism, while I'll be using the equations of motion and classical scattering amplitudes. I'll also identify additional phases missed in [10].
This paper is structured as follows: In Section 2 I will argue that a model containing any power of Ricci tensor can propagate a maximum of three degrees of freedom in 3D. Then I will argue that the "Extended New Massive Gravity" defined by (2.1) propagates only two degrees of freedom. In Section 3 I'll do linear analysis of (2.1) on a maximally symmetric background and show that the model does indeed propagate only one massive spin-2, i.e. 2 d.o.f. In Section 4 different phases of (2.1) are discussed. I will argue that there exists a choice of parameters, within the limits of EFT, for which the massive spin-2 is a ghost.

Counting the Degrees of Freedom
In what follows, I will call the "Extended New Massive Gravity" to the model defined by the following action: In the limit when Λ → 0, Minkowski space-time is a solution to the equations of motion. When linearized on this background the spectrum consists of one massless and one massive (with mass = m) spin-2 modes. In this section I will argue that this is the case for an arbitrary background. A general diffeomorphism invariant action involving any degree of curvatures 1 can propagate a maximum of three degrees of freedom in 2 + 1 dimensions. In order to see this let's decompose the metric in the following way: Hereḡ µν is arbitrary solution to (2.1) and∇ is covariant derivative with respect to this background. h µν , A µ and ϕ are tensor, vector and scalar perturbations on this metric (linear response to matter). The last piece, exp (2π), is the conformal mode. Because of the diffeomorphism invariance of the action, neither A µ or ϕ can be physical. The equations of motion, in general, involve quartic time derivatives, therefore there are two spin-2 modes and one massive scalar. The massless spin-2 in 2 + 1 dimensions is a pure gauge, so we are left with two degrees of freedom associated with massive spin-2 and one d.o.f. coming from the massive conformal scalar. The combination (2.1) is special in a sense that the conformal mode turns out to be a galileon [8]. The equations of motion corresponding to the variation of π are of second order (w.r.t π) and the massive scalar disappears from the spectrum, so the model defined by (2.1) propagates only two degrees of freedom. In three dimensions this is the unique combination with non-trivial conformal mode having this property, any higher order (> 3) curvature invariant should either have a trivial conformal mode or will propagate three degrees of freedom and will not be unitary. 2 1 Terms involving the derivatives of curvatures are not considered. 2 In 2+1 dimensions the square of Cotton tensor √ −gC µνρ C µνρ has a trivial conformal mode. This term is sometimes considered together with (2.1) [10], I will ignore it since it contains the derivatives of curvature.
The overall sign of the action in (2.1) guarantees that, when linearized around flat background (Λ = 0), the kinetic term of the massive spin-2 has the right sign. In the following sections I will positively answer the following question: Is there a choice of parameters (Λ/m 2 ; κ), within the limits of EFT, on which the kinetic terms of spin-2 modes flip the sign?
3 Analysis of the Linear Perturbations

A Toy Model for Linear Perturbations
Working with tensorial structures often shadows the physical picture. The analysis gets even more complicated when calculations are done on a curved background. In the next section, while calculating scattering amplitudes, I will rely on the equations of motion and will avoid dealing with Lagrangians. In order to gain some intuition for the conclusions I'm going to make, it is useful to first consider the following toy model of a scalar field: Here a, b, c are dimensionfull constants. An arbitrary φ = φ 0 = const is a solution of (3.1). Let's introduce some source ρ and study the linear response on this background (φ = φ 0 + χ): Solution to this Lagrangian and the scattering amplitude between two sources are respectively given by: Mediators are massless and massive scalars (see Figure 1), one of them is necessarily a ghost. When a = 0 the amplitude has a peculiar overall factor due to the renormalization of the coupling strength on a non-trivial background. The matter of which mode is ghost depends on the sign of this factor. To see this more clearly let's introduce a Lagrange multiplier and rewrite (3.2) in an equivalent form: Making a field redefinition, χ = ω + ϕ, unmixes the two modes from each other: From this expression we see that the factor (1 + aφ 0 ) does indeed decide which field should be a ghost. When (1 + aφ 0 ) = 0, the model is infinitely strongly coupled. Figure 1. Diagrammatic representation of the scattering amplitude (3.4) Disclaimer: The model defined by (3.1) is an EFT with some cutoff. Whether the values of φ 0 for which (1 + aφ 0 ) flips sign are within the range of EFT is another story. I will ignore this issue for the present case and will come back to it for ENMG in section 4.

Linearized GR on a Maximally Symmetric Space-Time
It is instructive to consider the linearized General Relativity on a maximally symmetric background [13] before linearizing (2.1). The Einstein-Hilbert action and vacuum equations of motion are: Riemann and Ricci curvatures of a three dimensional maximally symmetric space-time satisfy:R µρνσ =R 6 ḡ µνḡρσ −ḡ µσḡρν ,R µν =R 3ḡ µν . (3.8) From (3.7) we also haveR = Λ. Hereḡ µν is the background metric and the expressions with a bar are defined with respect to it. On the next step let's define the operator that will simplify expressions:¯ ∆ L stands for the Lichnerowicz Laplacian [14] (up to a factor of −1) acting on scalar, vector and tensor as: The covariant derivative∇ µ and the background metricḡ µν commute with¯ . The linear response of spin-2 to matter on this background: Fixing the gauge as: gives the following solution: A covariantly conserved energy momentum tensor is consistent with linear perturbations around a metric that is solution to the Einstein equation in vacuum, meaning is a gauge invariant amplitude. Even though the pole is shifted, from the tensorial structure it is easy to see that we are dealing with one massless spin-2 particle. Interested reader is invited to check [13] to see that¯ does indeed have the right properties when acted on transverse and traceless tensors.

Linearized ENMG
In this section we discuss the linearized version of the Extended Massive Gravity (2.1) on a maximally symmetric space-time. We can write the equation of motion as: where the K µν and Q µν are given in the Appendix A. To the lowest order we assume the matter to be absent. Since we are dealing with maximally symmetric space-time, the background satisfies a scalar equation: After fixing the gauge (3.12), the linear response to matter becomes: with: Note that the trace of (3.17) is of second order as a consequence of the conformal mode being a galileon: Since the full theory is gauge invariant, LHS of the equation (3.17) is consistent with covariantly conserved energy-momentum tensor. Solution to (3.17) and gauge invariant amplitude are respectively: From the last expression we see that the spectrum consists of massless and massive spin-2 modes. As we saw in the toy model, the matter of which one is ghost and which one is particle depends on the sign of β.

Different Phases of ENMG
Let's discuss how ENMG behaves for the different choice of parameters m 2 , Λ and κ. Since the massless mode does not propagate we can ignore it and concentrate on the massive spin-2. Causality and absence of dynamical ghosts require β γ m 2 > 0 and β > 0 respectively. First of all let's focus on the case when κ → ∞, i.e. New Massive Gravity amended by the C.C. For this case the above conditions translate to (see Figure 2a): Lower bound comes by demanding (3.16) to have real roots. Λ = −6m 2 is special, at this point α = 0 and from (3.19) we see that so does T . Linear model aquires scale invariance and diffeomorphism transformations get enhanced by a Weyl piece [15]: Since in the NMG limit γ = 1, causality condition is simply m 2 > 0. When the C.C. is absent, β = 1 or β = 3 and around a maximally symmetric background model is free of dynamical ghosts and is causal (m 2 > 0). As we can see from (4.1) this is not true for general Λ. Let's see what changes if we regard (2.1) as an EFT. The lowest cutoff of this EFT is (after recovering the Planck mass) Λ 5/2 = ( M p m 2 ) 2/5 [9] 3 . Recovering the Planck mass does not affect (4.1). The values of Λ for which β < 0 are within the EFT regime, Λ ≪ Λ 2 5/2 , as long as 10 4 m ≪ M p . The latter condition seems reasonable and is expected to hold, therefore (2.1) with κ → ∞ can host a massive spin-2 ghost.
The phase diagram for finite κ is shown on Figure 2b. Once again, there exist parameter choices within the EFT for which the massive spin-2 is a ghost.
For Λ = −8κm 2 there exists a solution of (3.16) for which γ = 0 and we get an exotic phase. The equation of motion (3.17) becomes second order and massive spin-2 disappears from the spectrum. From the identity: To summarize: Red and Orange regions are healthy (kinetic term has the right sign, but it can be tachyonic), for Blue and Purple regions we get ghosts. On the boundary between red and purple or orange and blue the model is infinitely strongly coupled.

A Appendix: Equations of Motion
The variation of the curvature squared and curvature cubed parts of (3.17) are respectively: where: Linearized versions of the different parts of (3.15): Here we used the background equation (3.16). After fixing the gauge (3.12), we recover (3.17). Setting γ = 0, (A.9) reduces to linearized Einstein equation (3.11) with rescaled Planck mass.
Since we are only focusing on the ∂ 2 h∂ 4 h interactions, we can set C.C. and the background curvature to zero Λ =R = 0 and linearize this expression around the Minkowski space-time. I will work in the transverse and traceless gauge ∂ ν h ν µ = 0 = h. Relevant part of Q µν : Equation for the next order corrections contains quartic derivatives. Even though when γ = 0 the linear equations are of second order and as a consequence the massive mode disappears, it will reappear in the next order perturbations. This shows that γ = 0 phase is infinitely strongly coupled.