Weyl and transverse diffeomorphism invariant spin-2 models in D=2+1

There are two covariant descriptions of massless spin-2 particles in $D=3+1$ via a symmetric rank-2 tensor: the linearized Einstein-Hilbert (LEH) theory and the Weyl plus transverse diffeomorphism (WTDIFF) invariant model. From the LEH theory one can obtain the linearized New Massive Gravity (NMG) in $D=2+1$ via Kaluza-Klein dimensional reduction followed by a dual master action. Here we show that a similar route takes us from the WTDIFF model to a linearized scalar tensor NMG which belongs to a larger class of consistent spin-0 modifications of NMG. We also show that a traceless master action applied to a parity singlet furnishes two new spin-2 selfdual models. Moreover, we examine the singular replacement $h_{\mu\nu} \to h_{\mu\nu} - \eta_{\mu\nu}h/D$ and prove that it leads to consistent massive spin-2 models in $D=2+1$. They include linearized versions of unimodular topologically massive gravity (TMG) and unimodular NMG. Although the free part of those unimodular theories are Weyl invariant, we do not expect any improvement in the renormalizability. Both the linearized K-term (in NMG) and the linearized gravitational Chern-Simons term (in TMG) are invariant under longitudinal reparametrizations $\delta h_{\mu\nu} = \p_{\mu}\p_{\nu}\zeta$ which is not a symmetry of the WTDIFF Einstein-Hilbert term. Therefore, we still have one degree of freedom whose propagator behaves like $1/p^2$ for large momentum.

The WTDIFF model can be obtained from the usual LEH theory by the singular replacement h µν → h µν − η µν h/D. The reason why this replacement is successful is not obvious. An argument is given in [10]. Namely, we first introduce a harmless Stueckelberg scalar field altogether with a trivial Weyl symmetry in the LEH model via h µν → h µν +η µν φ, thus defining a conformal model. The Weyl symmetry can be broken by fixing the unitary gauge φ = 0 which leads us back to LEH. If we alternatively choose the gauge φ = −h/D, we keep the Weyl symmetry unbroken but the DIFF symmetry is reduced to TDIFF. Therefore, the LEH and the WTDIFF models correspond to two different gauges of the same conformal theory. This is not a rigorous proof of equivalence since the gauge fixing is implemented at action level. According to [11], the equivalence between a general gauge theory and its gauge fixed version (at action level) requires that the gauge condition be complete, which is not the case here. The key point is that the gauge fixed action leads to less equations of motion. This is not equivalent in general to first derive the full set of equations of motion and fix the gauge afterwards.
Mainly motivated by the accelerated expansion of the universe, but also as a matter of principle, we are interested here in massive gravitational theories. They have been a subject of intense work in the last decade, (see [12,13] and the review works [14,15]). The modern massive gravities are built up on the top of the massive FP model, so we might wonder whether massive WTDIFF models do exist or even before that, we must search for WTDIFF massive spin-2 theories. Naive addition of mass terms to the massless WTDIFF model breaks unitarity [2]. In [10] the reader can find a recent discussion in that direction via dimensional reduction.
Notice that the argument of [10] can not be used in order to derive a WTDIFF version of the massive FP model. The second gauge φ = −h/D is not allowed, since there is no vector symmetry to be fixed or partially fixed in the massive case. The best we can do is to replace h µν → h µν + η µν φ and carry out a field redefinition φ → φ − h/D. We end up with the Weyl symmetry δh µν = η µν Λ but φ is not pure gauge anymore. It remains in the theory as an extra degree of freedom [10]. This is the typical situation for massive WTDIFF models, extra fields are required in general.
In D = 2 + 1 the situation is different. We may have massive spin-2 models still invariant under vector symmetries. This is the case of the second, third and fourth order selfdual models of helicity +2 or -2 (parity singlets) and the linearized version of the new massive gravity (NMG) [16] with both helicities ±2 (parity doublet). This raises the question of defining WTDIFF versions of those models according to the argument of [10] and eventually building up unimodular versions of the corresponding massive gravitational theories. This issue is specially interesting from the point of view of renormalizability because the highest derivative term of topologically massive gravity (TMG) and of NMG is Weyl invariant at linearized level, contrary to the lower derivative term (Einstein-Hilbert). It would be interesting to have both terms Weyl invariant in order to make sure that all degrees of freedom have their large momentum behavior ruled by the highest derivative term. We examine that question here.
In section II we show the consistency of the WTDIFF version of the linearized NMG model and of the second, third and fourth order spin-2 selfdual models SD n as well and comment on possible unimodular massive gravities and the issue of renormalizability. In section III, by means of a traceless master action approach we derive a new scalar-tensor selfdual model of second order NSD 2 and also a new fourth order model NSD 4 from NSD 2 . In section IV a traceless master action gives rise to a new scalar-tensor NMG model which is shown to be a specific subcase of a more general class of consistent spin-0 (scalar tensor) deformations of NMG. In section V we present our final comments.

WTDIF invariant models in
In order to point out the subtleties of gauge fixing procedure at action level, it is instructive to first look at the massless case. It is known that the Einstein-Hilbert theory has no propagating degrees of freedom in D = 2 + 1. At linearized level we have: where the transverse operators are such that The equations of motion E ρδ E γσ h δσ = 0 are equivalent (multiply by ǫ ρµν ǫ γαβ ) to a vanishing linearized Rieman curvature R L µναβ (h) = 0 (flat space). The general solution is pure gauge On the other hand, if we make the Stueckelberg replacement h µν → h µν + η µν φ in (1) followed by the gauge fixing φ = −h/3 at the action level, we have a WTDIFF invariant model S LEH [h µν ] whose equations of motion are traceless: Applying ∂ ρ we show that the linearized scalar curvature is an arbitrary constant, not necessarily vanishing anymore, i.e., R L = ∂ µ ∂ νh µν = c. The integration constant c can not be fixed by the equations of motion. Contracting (4) with ǫ ρµν ǫ γαβ we have a maximally symmetric space in general, not necessarily flat: The solution to (5) is given bȳ Except for the c-dependent term, the solution is pure gauge. So the number of propagating degrees of freedom still vanishes. We have only one (not one infinity) extra degree of freedom represented by c but the geometry has been changed as if we had a cosmological constant 2 . On the other hand, if after the Stueckelberg substitution in (1) we first derive the field equations with respect to h µν and φ and only afterwards we fix the gauge φ = −h/3 we would have obtained R L µναβ (h) = 0 and consequently R L = ∂ µ ∂ νh µν = 0 which corresponds to c = 0. We learn that the gauge fixing at action level is nontrivial, specially regarding gravitational theories. Thus, whenever we fix a gauge at action level, as in the next section, we must explicitly check the consistency of the resulting model. We can not take physical equivalence for granted. In the following subsections we turn to massive models which are still gauge invariant in D = 2 + 1 dimensions.

WTDIFF linearized New Massive Gravity (parity doublet)
The linearized version of the New Massive Gravity [16] can be written in the compact Fierz-Pauli form where the dual field is given by [18] and identically satisfies If we replace h * µν by h µν in (8) From (10) we see that θ ρσ h * ρσ = 0. Applying ∂ µ on (11) we have ∂ µh * µν [h] = 0. Using the identities (3) and (10) we see that (11) is equivalent to the Klein-Gordon equations is transverse, traceless and satisfies the Klein-Gordon equations. Moreover it is invariant under the WTDIFF gauge symmetry of S W LN M G , i.e., δh µν = ∂ µ ξ T ν + ∂ ν ξ T µ + η µν φ. So S W LN M G correctly describes massive spin-2 particles. From (10) and ∂ µh * µν = 0 we have ∂ µ h * = 0, so h * = η αβ h * αβ becomes an integration constant which plays no role from the point view of the particle content of S W LN M G but from the point of view of a linearized gravitational theory works like a cosmological constant.
Although LNMG can be obtained from the usual massive FP model via master action, see for instance [18], we have not been able to derive the WLNMG model from any second order theory via master action. The WLNMG model contains both helicities +2 and −2, in the next subsection we look at parity singlets of helicity +2 or −2 described in terms of a symmetric traceless tensor.

WTDIFF self-dual models (parity singlets)
Free helicity +2 or −2 states can be described by the so called spin-2 self-dual models (SDn), of n-th order in derivatives with n = 1, 2, 3, 4. The SDn model can be obtained from the SD(n-1) via a consecutive Noether gauge embedding procedure as shown in [18]. The equivalence among all those models can be proved by means of a master action approach [19], see [20,18], which also furnishes a dual map e µν → e * µν responsible for the equivalence of correlation functions of e µν in the SD1 model with correlation functions of e * µν in the higher order SDn models. All SDn models can be written in a compact way 3 similar to the first-order model of Aragone and Khoudeir [23], which was the first one to be suggested, namely where 3 A similar formula holds in the spin-1 case where n = 1, 2. The SD2 model is the Maxwell-Chern-Simons theory of [21] and SD1 was suggested in [22]. Namely, The equations of motion from (13) are then given by: Notice that (16) holds identically for the higher order cases n = 2, 3, 4 as a consequence of a local vector symmetry in those cases. Next, by acting with ǫ µνλ on (15) we conclude that e * [µν] = 0. If we take the trace of (15) we obtain e = 0. Therefore ∂ µ e * µν = 0. Then Now if we apply E µσ in (17) we obtain the Klein-Gordon equation for e * (µν) : SDn represent a massive particle of helicity +2 for all cases n = 1, 2, 3 and 4. The SDn models, with n = 2, 3, 4, are invariant under the following respective gauge transformations: where Λ [µν] = −Λ [νµ] stand for arbitrary antisymmetric shifts. If we replace e µν by its traceless partē µν = e µν −η µν e/3 in L SDn , the models will be invariant under transverse diffeomorphisms and Weyl transformations i.e.: with ∂ µ V T µ = 0. So we can define the models: which lead us to the following traceless equations of motion: Due to (16) which holds identically for n = 2, 3, 4, after applying ǫ µνβ on (22) it follows that e * αβ (h) = e * βα (h) which implies E αβ e * αβ (h) = 0. By applying ∂ µ on (22) we have ∂ µ e * = 0, so e * must be constant. Thus, the trace of the original equations of motion of the usual models L SDn , i.e. e * = 0 is recovered up to an integration constant. This is typical for WTDiff modifications of diffeomorphisms invariant theories. Notice however, that (22) is equivalent to (15) when e * µν is replaced byē * µν = e * µν − η µν e * /3. Consequently, we deduce the Klein-Gordon equations, the helicity equation (17) and the Fierz-Pauli conditions which assures that L W SDn have the same particle content of the L SDn models.

A note on renormalizability
The models S LN M G , SD3 and SD4 have gravitational nonlinear completions, they correspond respectively to NMG, topologically massive gravity (TMG) and higher derivative topologically massive gravity (HDTMG). In the cases of S LN M G and SD3 the highest derivative term, of fourth and third order respectively, is invariant under WDIFF δh µν = ∂ µ ξ ν + ∂ ν ξ µ + δ µν Λ while the lowest derivative term (linearized Einstein-Hilbert) is only invariant under DIFF. As argued in [24] this is an obstruction to the renormalizability of their nonlinear completions, since there will always be one metric degree o freedom (absent in the highest derivative term due to the Weyl symmetry) whose propagator is governed by the Einstein-Hilbert term and behaves unfortunately like 1/p 2 for large momentum.
On the other hand, in the last subsections we have shown that WLNMG and WSD3 correctly describe free massive spin-2 particles. They are obtained from LNMG and SD3 by the replacement h µν →h µν ≡ h µν − η µν h/3 which assures that the Weyl symmetry is present in all sectors of the Lagrangian. In fact, they are invariant under WTDIFF transformations. The nonlinear version of such replacement, i.e., g µν →ĝ µν ≡ g µν /(−g) 1/3 leads to unimodular theoriesĝ = −1 which are invariant under Weyl transformations and volume preserving diffeomorphisms (∇ µ ξ µ =0). Now we can be sure that both highest and lowest derivative terms in the quadratic part of the action are invariant under WTDIFF by construction. So we may hope that all degrees of freedom behave like 1/P 4 in the case of unimodular NMG or 1/P 3 for unimodular topologically massive gravity respectively. However, there is a subtlety. Due to their Weyl symmetry, the highest derivative terms are unchanged by the replacement h µν →h µν . So, they remain invariant under full WDIFF while the Einstein-Hilbert term is only invariant under WTDIFF. Consequently, the linearized K-term (NMG case) and the linearized gravitational Chern-Simons term (TMG case) still have one more local symmetry than the EH term, namely, they are invariant under longitudinal diffeomorphisms: δ h µν = ∂ µ ∂ ν ζ. Indeed, such symmetry can be used in order to obtain the WSD4 model, which is equivalent to SD4, from the WSD3 model via Noether gauge embedding just like the Weyl symmetry is used to get from SD3 to SD4 as shown in [18]. Therefore, the pure longitudinal sector of the metric will behave like 1/P 2 . So there is no improvement in the renormalizability as we go to the unimodular theories.
The case of HDTMG [18,25], i.e., the nonlinear completion of SD4, is even worse from the point of view of perturbative quantum field theory. Both terms of the quadratic (free) piece of HDTMG, i.e., the linearized K-term and the linearized gravitational Chern-Simons term are invariant under linearized WDIFF while the cubic and higher vertices are only invariant under DIFF. Thus, there is one metric degree of freedom which only appears in the vertices without any free propagator. At quantum level it gives rise to a nonlinear constraint whose role is unclear. A similar problem also appears in the massless limit of NMG as discussed in [26]. The replacement g µν →ĝ µν , which amounts to h µν →h µν in the quadratic (O(h 2 )) piece of the theory, turns the DIFF symmetry into Weyl plus volume preserving diffeomorphisms or WTDIFF at linearized level. However, the quadratic theory is invariant under the larger WDIFF transformations, so the pure longitudinal degree of freedom (∂ µ ∂ ν ζ) of the metric only appears in the vertices leading us to an awkward constraint again. The only hope is to start with the SD4 model and examine the addition of cubic and higher vertices invariant under the full set of WDIFF.
3 New massive spin-2 models via a traceless master action 3.1 Selfdual models Let us consider the first order self-dual model originally proposed by [23]: We can split the non-symmetrical field f µν into its traceless and trace full parts by making f µν =f µν + η µν φ where φ is a fundamental scalar field. After that we have: The traceless Chern-Simons like term is invariant under δf µν = ∂ µ ξ T ν with ∂ ν ξ T ν = 0. Moreover, it is possible to show that it is trivial, it has no particle content by itself. This fact tells us that it might be used as a mixing term in order to construct a master action: Then it might be possible to interpolate between the first-order self-dual model and alternative traceless descriptions. In order to implement it we define a generating functional by adding a source term to the field f µν , where one can easily see that after the shiftē µν →ē µν +f µν we have basically the first order self-dual model, since we end up with a completely decoupled Chern-Simons trivial term. On the other hand without any shifts, we would have: After functionally integrating overf µν and shifting: we can obtain the alternative second order self-dual model given by: where we have neglected quadratic contributions in the source term, which lead us to contact terms when we are comparing correlation functions between SD1 and ASD2. As a byproduct we have obtained the following dual maps: the model we have found in (29) is invariant under the gauge transformations δē µν = ∂ µ ξ T ν and δφ = 0. Surprisingly one can also note that the set of second order terms in (29) are (all together) invariant under the gauge transformations: besides the same second order sector is altogether free of particle content, which can be seen by means of a hamiltonian analysis and also by studying its correspondent propagator. Then we can now use this set of terms as mixing terms in order to construct another master action with the following structure: which after the shiftsf µν →f µν −ē µν and χ → χ − φ take us back to the ASD2 model thanks to the triviality of the second order sector. On the other hand we have: After functionally integrating overē µν and the scalar φ and then defining f µν =f µν + η µν χ we arrive at an alternative, and unusual, new self-dual model which contains second, third and fourth order terms in derivatives: where we have defined the dual field: One can check that correlation functions of e µν in the first order self-dual model of [23] coincide with correlation functions of the dual field f * µν in the model S ASD4 up to contact terms. The model (34) is invariant under the gauge transformation δf µν = ∂ µ A ν + ∂ ν B µ which leaves f * µν also invariant.
Remarkably, the model S ASD4 can be written in the form (13) with help of (35). Although the fourth order term of (34) has no particle content, we have not been able to produce any higher (than four) selfdual model out of S ASD4 . It seems that the highest number of derivatives in spin-2 models in D = 2 + 1 is indeed four.

Scalar-tensor New Massive Gravities
One way of obtaining the New Massive Gravity of [16] is to start with the massless linearized Einstein-Hilbert (LEH) theory in D = 3 + 1 and perform its Kaluza-Klein dimensional reduction leading to the massive Fierz-Pauli theory in D = 2 + 1 from which we obtain NMG as a dual model via a master action technique [19] where the mixing term between old and new (dual) fields is the full Einstein-Hilbert theory, see [18]. If we replace the LEH by the WTDIFF model in D = 3 + 1, the KK dimensional reduction leads to a massive model where one of the Stueckelberg fields can not be gauged away, see [10]. We may choose to end up with a lower dimensional theory which corresponds to the FP model after the replacement h µν →h µν + η µν φ. This is physically equivalent to the usual FP model since it could have been obtained by first introducing a scalar Stueckelberg field h µν → h µν + η µν φ, altogether with a Weyl symmetry, followed by the harmless shift φ → φ − h/3. This new form of the FP theory inspires us to define a new master action with a traceless mixing term: Since the traceless LEH theory has no propagating degree of freedom, after the shiftf µν → f µν +h µν the fields decouple and it is clear that the particle content of (36) is the same one of the massive FP model, i.e., one helicity doublet +2 and −2. On the other hand, after integrating overh µν in (36) we have a quadratic scalar tensor model depending upon (φ,f µν ). If we suppose that such theory comes from the singular replacement (gauge fixing at action level) f µν →f µν of a full reparametrization invariant model, its simplest nonlinear completion would be where g µν = η µν + f µν and The model (37) is a scalar modification of NMG. This becomes clearer after introducing an auxiliary scalar field which allows us, using (38), to rewrite (37) in the local form: (39) Following [16] we can eventually introduce an auxiliary symmetric field and bring (39) to a fully second order form.
The NMG itself corresponds to (37) with (r, s) = (1, 3 m 2 ). In what follows we perform an analysis of the analytic structure of the linearized version of (37) in search for other viable (unitary and non tachyonic) scalar deformations of NMG. The linearized version of (37), using the more common notation g µν = η µν + h µν , can be conveniently written as where s( ) and r( ) are now arbitrary functions of the d'Alembertian while A is an arbitrary constant. In the case of (38) we had A = 1/12m 2 . We have used with the spin-2 and spin-0 (for later use) projection operators given by After Gaussian integrating the scalar field, we rewrite the lagrangian as follows where The Lagrangian (43) can be further written in terms of a four indices differential operator L SN M G ≡ h µν G µναβ h αβ . The inverse G −1 does not exist due to DIFF symmetry. After adding the de Donder gauge fixing term L GF = λ (∂ µ h µν − ∂ ν h/2) 2 and suppressing the indices we have where dots stand for terms which vanish when we saturate G −1 with conserved sources and build up a gauge invariant amplitude. The NMG case is recovered at F ( ) = 0. The two point amplitude in the momentum space is given by Where G −1 (k) = G −1 (∂ µ → i k µ ) and k µ T µν = 0. More explicitly we have With k 2 = k µ k µ and where T = η µν T µν = −T 00 + T ii is the trace of the source in momentum space. The analytic structure of A(k) determines the particle content of the theory. Physical particles correspond to simple poles with residues with positive imaginary part. First we look at the massless pole k 2 = 0. Since both S (2) and S (0) are Lorentz invariant we can choose the convenient frame k µ = (k, ǫ, k), at the end we take ǫ → 0. In [27] we have shown that in such frame, up to terms of order ǫ and higher, we may write Therefore, requiring the imaginary part of the residue at k 2 = 0 vanishes and we get rid of the massless pole, The same mechanism works in the NMG case, see [28]. Now we look at possible massive poles k 2 = −m 2 . We choose the rest frame k µ = (m, 0, 0). From k µ T µν = 0 one can show [27] that, up to terms of order ǫ and higher, We see that S (2) > 0 while S (0) has no definite sign. Thus, if we have any massive pole coming from [1 − 4 k 2 F (−k 2 )] = 0, withm = m, its residue will be proportional to S (0) and we are doomed to have a ghost. It is impossible to have a physical massive scalar particle with m = m. The case k 2 =m 2 = m 2 is subtler since the residue acquires contribution from both spin-2 and spin-0 sectors. Let us suppose that with some continuous function G(k 2 ). Consequently, we have the imaginary part of the residue: If we take an arbitrary real constant a, we see from (53) and (54) that S (2) + a S (0) > 0 whenever 0 ≤ a ≤ 1, consequently we must have G(−m 2 ) ≤ −1/m 2 . On the other hand, from (51) and (55) we get G(0) = 1/m 2 . From those two points and the continuity of G(k 2 ) it is clear that G(−bm 2 ) = 0 with some 0 < b < 1. However, as we have argued before, we are not allowed to have a massive scalar particle withm = m. So (55) can not be true and there can not be any contribution to the residue at k 2 = −m 2 coming from the denominator of S (0) in A(k). Thus, we are left with I m = S (2) > 0 and we are left with only one massive spin-2 particle in the spectrum just like the NMG case. The previous arguments amount to require that the numerator of the function be independent of . Thus, the polynomials r( ) and s( ) must be such that Where A is an arbitrary constant and s 0 = s( = 0) After integration over the scalar field in (37) using (58), we have the following class of spin-0 nonlocal deformations of NMG: The case s( ) = s 0 corresponds to the NMG [16]. The reader can check that r( ), s( ) and A given in (38) and in the text after (40) respectively, satisfy (58). Another special case is s 0 = 0 where the function H( ) vanishes. Such momentum independent singularity in G −1 indicates the presence of a spin-0 local symmetry, in fact we have a Weyl symmetry. The corresponding model has been found before in our previous work [27]. It corresponds to make the Stueckelberg substitution h µν → h µν + η µν φ in the LNMG and then build up its simplest nonlinear completion. Since this is not equivalent to first take the nonlinear NMG and then make g µν → e φ g µν , we expect that the linearized unitarity of the s 0 = 0 case breaks down at nonlinear level, since φ stops being a pure gauge degree of freedom at nonlinear level, so the Weyl symmetry only exists in the linear theory.
Regarding the other solutions to (58), since they are not associated with any local symmetry it is not clear whether the unitarity of the linearized model is broken in the nonlinear theory (37).

Conclusion
Here we have examined different issues regarding the Weyl and transverse diffeomorphism (WTDIFF) symmetry in D = 2+1 massive spin-2 theories as well as their nonlinear analogues (unimodular theories).
Although WTDIFF theories correspond to gauge fixed versions of DIFF theories, the issue of gauge fixing at action level is nontrivial, see [11]. In particular, the triviality of Einstein-Hilbert gravity in D = 2 + 1 is lost in its unimodular (g = −1) version as we have briefly commented in the beginning of section II using the linearized theory. Instead of flat space we have now a maximally symmetric space in general which may include BTZ black holes [29] in the nonlinear case, depending on the sign of an integration constant which plays the role of a cosmological constant.
We have explicitly checked that WTDIFF versions of massive spin-2 theories (one and two helicities) are fully consistent. In the special cases of the third and fourth order (in derivatives) selfdual (one helicity) theories, they correspond to linearized versions of unimodular topologically massive gravity and unimodular higher derivative topologically massive gravity. Likewise, in the case of a parity doublet we have a linearized version of a unimodular New Massive Gravity.
At the end of section II we have examined the issue of renormalizability and Weyl symmetry. We argue that although both highest and lowest derivative terms in the free (quadratic) sector of unimodular TMG and unimodular NMG are Weyl invariant, we still have a mismatch of local symmetries which is dangerous for renormalizability as pointed out in [24]. The Einstein-Hilbert theory is only invariant under WTDIFF (linearized theory) while the highest derivative term (gravitational Chern-Simons term or the the K-term) is invariant under full WDIFF. Thus, the pure longitudinal degree of freedom h µν ∼ ∂ µ ∂ ν φ only appears in the Einstein-Hilbert term. Consecutively, it propagates like 1/p 2 at large momentum and no improvement is achieved for renormalizability in unimodular theories. The mismatch between the symmetries of the highest derivative term and the lower one seems to be unavoidable. In [30] we have pointed out that there exists a massive spin-2 model in D = 2 + 1 described by a nonsymmetric tensor e µν , see [31], where both the second and fourth order terms are Weyl invariant, however only the fourth order one is invariant under antisymmetric shifts. The mismatch also occurs in the higher dimensional analogue of the linearized NMG, see [32] and [33]. This is the higher derivative analogue of the usual breakdown of gauge symmetries by mass terms as in the Proca (s=1) and massive Fierz-Pauli (s=2) theories. The only hope is to find a theory where the lowest derivative term has already more than two derivatives.
It is known that massive theories in D dimensions can be obtained from D + 1 massless theories via Kaluza-Klein dimensional reduction. From the massless spin-2 linearized Einstein-Hilbert theory in D = 3 + 1 one can obtain the massive spin-2 Fierz-Pauli theory in D = 2 + 1. From the later theory one can derive, via the master action approach of [19], the fourth order linearized New Massive Gravity theory [16]. A key point in this approach is the absence of propagating degrees of freedom of the Einstein-Hilbert theory in D = 2 + 1 which works like a mixing term between old and new (dual) fields in the master action approach. If however, we replace the linearized EH theory in D = 3 + 1 as starting point by the the WTDIFF massless spin-2 theory, its dimensional reduction 4 [10] leads to the massive FP theory with the replacement h µν →h µν + η µν φ. In section III, starting from the latter theory we have defined a noncanonical (traceless) master action where the mixing term is the EH action for the traceless fieldh µν . This leads us to a scalar tensor modification of the NMG theory. We have shown it belongs to a more general class of consistent (unitary at quadratic level) scalar tensor modifications of NMG. The consistency of their nonlinear completion (37) demands further investigations.
From the point of view of dimensional reduction the appearance of an extra scalar field in the massive model is a consequence of the fact that the gauge parameter of the massless higher dimensional theory satisfies the scalar constrain ∂ M ξ T M = 0 with M = 0, · · · , D. This makes the lower dimensional gauge parameters not independent, consequently we are not able to eliminate all the Stueckelberg fields and we may choose to remain with one scalar Stueckelberg field, see [10]. This is similar to the massive spin-3 theory which requires an extra scalar field besides a totally symmetric rank-3 tensor φ αβγ due to the constrained symmetry of the higher dimensional massless theory δφ αβγ = ∂ (αξβγ) where η µν ξ µν = 0.
Still in section IV we have also applied the noncanonical master action approach on the first order selfdual model of [23] and derived a new second order model (NSD2) which, on its turn, has given rise to a new fourth order model (NSD4). The unusual NSD4 model contains second, third and fourth order terms. Remarkably, the SDn models and also NSD4 can all be written in the compact form (13) which also works in the spin-1 case (see footnote (3)). We believe that such compact formulas may exist also for higher spins which might help us in filling some gaps in the chain of spin-3 and of even higher spin selfdual models.