Dimensional reduction of the massless limit of the linearized"New Massive Gravity"

The so called"New Massive Gravity"in $D=2+1$ consists of the Einstein-Hilbert action (with minus sign) plus a quadratic term in curvatures ($K$-term). Here we perform the Kaluza-Klein dimensional reduction of the linearized $K$-term to $D=1+1$. We end up with a fourth-order massive electrodynamics in $D=1+1$ described by a rank-2 tensor. Remarkably, there appears a local symmetry in $D=1+1$ which persists even after gauging away the Stueckelberg fields of the dimensional reduction. It plays the role of a $U(1)$ gauge symmetry. Although of higher-order in derivatives, the new $2D$ massive electrodynamics is ghost free, as we show here. It is shown, via master action, to be dual to the Maxwell-Proca theory with a scalar Stueckelberg field.


Introduction
The authors of [1] have suggested an invariant theory under general coordinate transformations which describes a massive spin-2 particle (graviton) in D = 2 + 1. The model contains the Einstein-Hilbert theory and an extra term of fourth-order in derivatives, quadratic in curvatures, the so called K-term which has been analyzed in [2], see also [3]. Since massless particles in D dimensions have the same number of degrees of freedom of massive particles in D − 1 dimensions, one might wonder whether the "New Massive Gravity" (NMG) theory might be regarded as a dimensional reduction of some fourth-order (in derivatives) massless spin-2 model in D = 3 + 1 which would be certainly interesting from the point of view of a renormalizable quantum gravity in D = 3 + 1. As far as we know there is no positive answer to that question so far. 1 As an attempt to gain more insight on that question we investigate here the dimensional reduction of the massless part of the linearized NMG theory, the linearized K-term. We show here that the linearized K-term is reduced to a kind of higher-derivative massive 2D electrodynamics, which is in agreement with the fact that the linearized K-term is dual to the Maxwell theory in 3D as shown in [5], see also [2] and [6]. However, it is remarkable that a new local symmetry shows up after dimensional reduction and plays the role of a U(1) symmetry not broken by the mass term. We also derive in section 4 a master action interpolating between the new (higher-order) massive 2D electrodynamics and the usual Maxwell-Proca theory with a Stueckelberg field. We emphasize that throughout this work we only deal with quadratic (linearized) free theories.
2 From 2 + 1 to 1 + 1 Here we take capital indices in three dimensions (M, N = 0, 1, 2) and greek small indices in two dimensions (µ, ν = 0, 1), except in the appendix. Expanding about a flat background, g M N = η M N + h M N , where η M N = (−, +, +), the K-term [1,2] becomes, in the quadratic approximation where we have the spin-1 projection operator while P T T is the spin-2 projection operator acting on symmetric rank-2 tensors in D = 3. It is given in formula (62) of the appendix for arbitrary dimensions.
Since projection operators of different spins are orthogonal, it is clear from (3) that there is a general spin-1 plus a general spin-0 local symmetry in the quadratic approximation for the K-term, i.e., The vector symmetry corresponds to linearized reparametrizations as expected from the general covariant form of the nonlinear theory (1). The Weyl symmetry is surprising from the point of view of the nonlinear version of (1) since it does not hold beyond the quadratic approximation. It leads to an awkward situation for perturbation theory, where a scalar degree of freedom is present in interacting vertices but it does not propagate, see comments in [2,7]. Indirectly the Weyl symmetry leads to an unexpected symmetry in the reduced theory as we show here.
In order to proceed with the dimensional reduction we consider the second space dimension (x 2 ≡ y) constrained to a circle of radius R = 1/m. Explicitly, the action S K becomes As usual for Kaluza-Klein reductions, we expand tensor fields with even (odd) number of indices in the y-dimension in terms of periodic even (odd) functions, see e.g. [8]. Using back in the action (6) we obtain the complicated action in D = 1 + 1: Following the same rationale already mentioned, we expand the parameters of the 3D symmetry (5) as follows Λ(x, y) = m π Λ(x) cos(m y) Back in (5) we deduce the 2D symmetry transformations which leave the reduced action (9) invariant, as we have explicitly checked, Before we go on, it is certainly welcome to simplify the long expression for S 2D in order to figure out its particle content. We use as a guide the known dimensional reductions of Mawell to Maxwell-Proca (spin-1) and from the massless to the massive Fierz-Pauli [9] theory (spin-2). Recall that in the first case we have whereÃ µ = A µ + ∂ µ φ/m is the only local combination involving the vector field A µ and the Stueckelberg scalar φ (stems from A D−1 ) which is invariant under the reduced gauge symmetry: δA µ = ∂ µ ǫ ; δφ = −m ǫ. Analogously, in the spin-2 case we have the massless and massive Fierz-Pauli theories respectively, Comparing (16) and (15) for spin-1 and (18) with (17) for spin-2, one infers the rather simple rule, see [10] which includes spin-3, for the Kaluza-Klein dimensional reduction: This suggests that the long expression (9) might be related with (2) via Whereh µν is some Stueckelberg combination invariant under (12)- (14). It turns out that there is no linear combination of the fields h µν , (12)- (14) if Λ = 0. The best we can do is to stick toh µν as given in (20). Under the 2D symmetries (12)- (14) we have δh µν = η µν − ∂µ∂ν m 2 Λ. Confirming our expectations and following the rules (21), the 2D action (9), including the Stueckelberg fields, can be rewritten in a rather simple way, compare with (2), It is now easy to check that (22) is invariant under δh µν = K −1 µν Λ which becomes exactly δh µν = η µν − ∂µ∂ν m 2 Λ after we take Λ → ( − m 2 )Λ. Note that the 2D symmetries (12)-(14) allow us to gauge away the Stueckelberg fields: Λ. This is rather surprising since local symmetries in dimensionally reduced massive theories usually disappear altogether with the Stueckelberg fields. In other words, the action S 2D [h µν ] is invariant under The above local symmetry seems to be technically related with the absence of a linear combination of the tensors h µν , (12), (13) and (14) with Λ = 0.

Equations of motion
After gauging away the Stueckelberg fields, the fourth-order equations of motion of S 2D [h µν ] are given by where we have defined the vector field From the trace of (25) we have It is convenient to fix the gauge of the new scalar symmetry (24) in a such way that (27) is reduced to second-order. Namely, we choose the scalar gauge condition From (27) and (28) we have The gauge condition (28) and all equations so far are invariant under residual symmetry transformations (24) with the restriction ( − m 2 )Λ = 0. In particular, we have Under such residual transformations we have δ h = (2 m 2 − )Λ = m 2 Λ. Therefore we can use the residual symmetry to get rid of the trace From ∂ µ on (25) and (25) itself we deduce At this point it is convenient to recall that the particle content of the massive Fierz-Pauli theory in D = 1 + 1 is zero. It is the same content of the massless FP theory in D = 2 + 1. This amounts to say that we have the trivial identity in D = 1 + 1: The reader can check that (35) is not a dynamic equation and vanishes identically for each of its components 2 . From (29),(32),(33) and (35) the equation (34) becomes Therefore, we can consider ( − m 2 )V µ = 0 and ∂ · V = 0 as our primary dynamic equations and V µ as our fundamental vector field. Thus, we have the same particle content of the Maxwell-Proca theory as expected from the equivalence of the linearized K-term and the Maxwell theory in D = 2 + 1, see [5], see also [2] and [6]. In the next subsection we confirm the particle content of the 2D theory via the analytic structure of the propagator.

Propagator and absence of ghosts
After gauging away (h µν → h µν ) the Stueckelberg fields in (22) we can define the differential operator G µναβ via S 2D [h µν ] = d 2 xh µν G µναβ h αβ . Due to the symmetry (24) we need a gauge fixing term in order to obtain G −1 . We can use the same gauge condition (28) and add a gauge fixing term Suppressing the four indices, the operator G −1 , in momentum space, can be written in terms of the operators defined in the appendix as follows Notice that although there is no transverse traceless symmetric tensor in D = 1 + 1, see (35), we have included the operator P T T in order to bookkeep the massless poles since P T T is singular at k 2 = 0, we come back to that point later.
After adding a source term S source = d 2 x h µν T µν and integrating over the fields h µν in the path integral we obtain the two point function saturated with external sources: The particle content of the theory is obtained from the poles of A 2 (k). Because of the symmetry (24) we need δS source = 0 which imposes a constraint on the sources, in momentum space we have where T = η µν T µν = −T 00 + T 11 . Using the constraint (40), suppressing some indices on the left handed side, we obtain Back in (39) and (38) we end up with The dependence on the arbitrary gauge parameter λ has canceled out, as expected, as well as the massless pole k 2 = 0 present in the operators P (s) IJ . We are left apparently with a dangerous double pole at k 2 = −m 2 . Double poles indicate ghosts, see comment in [11]. In order to compute the imaginary part of the residue (I −m 2 ) at k 2 = −m 2 we introduce the 2D vector k µ = (m, ǫ) which implies k 2 + m 2 = ǫ 2 and take the limit So we only need to compute the numerator of A 2 (k) up to the order ǫ 2 . Back in the constraint (40) we can eliminate T 11 = −2 ǫ T 01 /m + O(ǫ 3 ). Consequently, Back in (45) altogether we have Therefore the particle content of the higher order massive electrodynamics S 2D consists solely of a massive physical particle, confirming the classical analysis of the previous subsection. The apparent double pole was in fact a simple pole. Precisely the same result for I −m 2 could have been obtained by dropping the contribution of the P T T term in (38). This is in agreement with the fact, already used in the last subsection, that P (2) T T is identically zero if k 2 = 0. Moreover, if we look at (43) and (44) back in (38) we conclude that the spin-0 operators have furnished no contribution to the massive pole, which is not obvious if we only look at (38). So we conclude that the only relevant contribution stems from the spin-1 operator P (1) SS . The potentially dangerous double pole in the denominator of P (2) T T in (38) has vanishing residue due to the absence of massive symmetric, transverse and traceless rank-2 tensors in 2D, see (35). This is a relic of the remarkable analytic structure of the propagator of the K-term in D = 2 + 1, see [3]. Although dangerous poles show up in the K-term and its 2D descendant here, their residues are harmless basically due to the low dimensionality of the space-time.

Master action and duality
Since S 2D is physically equivalent to a gauge invariant massive electrodynamics, one might wonder whether there could exist a master action, see [12], interpolating between S 2D and a Maxwell-Proca theory with a Stueckelberg scalar field, leading eventually to a dual map between gauge invariants. Indeed, the key point is to consider a second-order version of the K-term [1], where f M N = f N M is an auxiliary tensor field and G M N (h) is the linearized Einstein tensor in 3D. Integrating over f M N leads to the K-term given in (1) while integrating over h M N leads to the pure gauge solution f M N = ∂ M A N + ∂ N A M whose substitution in (50) leads to the 3D Maxwell theory. Using (7) and (8) and the decomposition the dimensional reduction of (50) furnishes whereh µν is defined in (20) and The quantity G µν [h, m 2 ] is the linearized 2D Einstein tensor after the shift → − m 2 . If we perform the Gaussian integral over f µν , ψ µ and Φ we recover (22). Since the 2D Einstein tensor vanishes identically, see (35), it turns out that G µν [h, m 2 ] = m 2 (h µν − h η µν )/2. Back in (53) and Gaussian integrating over f µν we have the master action where we have gauged away the Stueckelberg fields (h µν → h µν ) and introduced a source term. The master action is invariant under the U(1) gauge symmetry On one hand, if we integrate over h µν in the path integral we derive the Maxwell-Proca theory with a Stueckelberg field On the other hand, if we first integrate over ψ µ and Φ in the master action we have the dual new massive electrodynamics The higher derivative massive electrodynamics in (58) is exactly the same one given in (9) as one can check by use of the identity (35). So we have demonstrated the duality between the Maxwell-Proca-Stueckelberg theory (57) and S 2D . Moreover, from derivatives of (57) and (58) with respect to the source J µ we show that correlation functions of B µ [h] in the dual massive electrodynamics agree, up to contact terms, with correlation functions of the vector field ψ µ /m − ∂ µ Φ/m 2 in the Maxwell-Proca-Stueckelberg theory. So we have the dual map In particular, B µ is invariant under the U(1) transformation (56) just like the right handed side of (60). From the point of view of U(1) transformations the map (60) is consistent with the identification of ψ µ with m (∂ α h αµ − ∂ µ h) and Φ with −( + m 2 )h + ∂ α ∂ β h αβ /2. Thus, the three degrees of freedom h µν are somehow mapped into the three variables (ψ µ , Φ). Moreover, we notice that the identification of the propagating massive vector field with ∂ α h αµ in subsection 3.1 is consistent with (59) and the gauge condition (28). Now two remarks are in order. First, if we take D−1 → D + m 2 in (16), (18) and (22) we derive the corresponding massless higher dimensional theories (15),(17) and (2) (for D = 3) respectively. However, if we try the same (non-rigorous) inverse dimensional reduction with the linearized "New massive gravity" of [1] it turns out that we do not get rid of the m 2 in the corresponding 4D theory. Moreover, we have a tachyon at k 2 = m 2 . Thus, there seems to be no simple Kaluza-Klein reduction of a fourth-order spin-2 massless model in 4D which might lead to the "new massive gravity" of [1], see however the footnote in the introduction. The results of [13] and [14] suggest us that one should try to obtain [1] from the dimensional reduction of an extra discrete dimension, see also [15,16], this is under investigation. Second, it is worth commenting that, although the K-term has a nonlinear gravitational completion in D = 2 + 1, see (1), there is no such completion for S 2D since there is no local vector symmetry whatsoever in S 2D .

Conclusion
By performing a Kaluza-Klein dimensional reduction of the massless limit of the linearized "New Massive Gravity" (linearized K-term) we have obtained a new 2D massive electrodynamics of fourth-order in derivatives. This is in agreement with the equivalence of the linearized K-term with the Maxwell-theory [5], see also [2] and [6]. However, it is remarkable that the reduced 2D theory, although massive, has local U(1) gauge symmetry even after gauging away the Stueckelberg fields of the dimensional reduction. The U(1) symmetry (24) seems to be a consequence of the lack of a Stueckelberg version of the fundamental field h µν invariant under both linearized reparametrizations and Weyl transformations, see comment after (24).
We have also noticed that the dimensional reduction of the K-term follows the same simple pattern of the usual spin-1 (Maxwell to Maxwell-Proca) and spin-2 (massless Fierz-Pauli to massive Fierz-Pauli) cases, namely, we have the practical rule D → D−1 − m 2 altogether with the replacement of the fundamental field by its Stueckelberg version h M N → h µν +(∂ µ φ ν + ∂ ν φ µ )/m − ∂ µ ∂ ν H/m 2 .
We have made a classical and quantum analysis of the particle content of the reduced theory, confirming that, although of 4th-order in derivatives, it is ghost free and contains only a massive vector field in the spectrum. In particular, we have found a master action interpolating between the new 2D massive electrodynamics and the Maxwell-Proca theory with a scalar Stueckelberg field and identified a dual map between gauge invariant vector fields in both theories, see (60). A possible non-Abelian extension of the new 2D electrodynamics and the issue of unitarity in the context of the Schwinger mass generation when coupled to fermions are under investigation.

Appendix A
In this appendix we use small Greek indices in D-dimensions for both D = 3 and D = 2. Using the spin-0 and spin-1 projection operators acting on vector fields, respectively, as building blocks, one can define the projection and transition operators in D dimensions acting on symmetric rank-2 tensors, They satisfy the symmetric closure relation