Towards frame-like gauge invariant formulation for massive mixed symmetry bosonic fields

Transcription

1 Nuclear Physics B ) Towards frame-like gauge invariant formulation for massive mixed symmetry bosonic fields Yu.M. Zinoviev Institute for High Energy Physics, Protvino, Moscow Region , Russia Received 29 September 2008; received in revised form 28 November 2008; accepted 1 December 2008 Available online 6 December 2008 Abstract In this paper, as a first step towards frame-like gauge invariant formulation for massive mixed symmetry bosonic fields, we consider mixed tensors, corresponding to Young tableau with two rows with k 2 boxes in the first row and only one box in the second row. We construct complete Lagrangian and gauge transformations describing massive particles in anti) de Sitter space time with arbitrary dimension d 4and investigate all possible massless and partially massless limits Elsevier B.V. All rights reserved. 0. Introduction As is well known, in d = 4 dimensions for the description of arbitrary spin particles it is enough to consider completely symmetric spin-)tensor fields only. At the same time, in dimensions greater than four in many cases like supergravity theories, superstrings and higher spin theories, one has to deal with mixed symmetry spin-)tensor fields [1 4]. There are different approaches to investigation of such fields both light-cone [5,6], as well as explicitly Lorentz covariant ones e.g. [7 13]). For the investigation of possible interacting theories for higher spin particles as well as of gauge symmetry algebras behind them it is very convenient to use socalled frame-like formulation [14 16] see also [17 19]) which is a natural generalization of well-known frame formulation of gravity in terms of veilbein e μ a and Lorentz connection ω μ ab. Till now, most of the papers on frame-like formulation for mixed symmetry fields deal with massless case [20 25] see however [26]). The aim of this work is to start an extension of frame-like formulation to the case of massive mixed symmetry fields. Namely, we will begin address: yurii.zinoviev@ihep.ru /$ see front matter 2008 Elsevier B.V. All rights reserved. doi: /j.nuclphysb

2 Yu.M. Zinoviev / Nuclear Physics B ) construction of gauge invariant formulation for such mixed symmetry massive fields in general A)dS d space times with non-zero cosmological constant and arbitrary space time dimension d 4. There are two general approaches to gauge invariant description of massive fields. One of them uses powerful BRST approach [11,12,27 30]. Another one, which we will follow in this work, [19,31 34] see also [9,26,35 37]) is a generalization to higher spin fields of well-known mechanism of spontaneous gauge symmetry breaking. In this, one starts with appropriate set of massless fields with all their gauge symmetries and obtain gauge invariant description of massive field as a smooth deformation. One of the nice feature of gauge invariant formulation for massive fields is that it allows us effectively use all known properties of massless fields serving as building blocks. There are two different frame-like formulations for massless mixed symmetry bosonic fields. For simplicity, let us restrict ourselves with mixed symmetry tensors corresponding to Young tableau with two rows. In what follows we will denote Yk,l) a tensor Φ a 1...a k,b 1...b l which is symmetric both on first k as well as last l indices, completely traceless on all indices and satisfies a constraint Φ a 1...a k,b 1 )b 2...b l = 0, where round brackets mean symmetrization. In the first approach [21 24] for the description of Yk,l) tensor k l) one use a one-form e Yk 1,l) μ as a main physical field. In this, only one of two gauge symmetries is realized explicitly and such approach is very well adapted for the A)dS spaces. Another formulation [25] uses two-form e Yk 1,l 1) μν as a main physical field in this, both gauge symmetries are realized explicitly. Such formalism works in flat Minkowski space while deformations into A)dS space requires introduction of additional fields [38]. In this paper we will use the second formalism. As we have already seen in all cases considered previously and we will see again in this paper, gauge invariant description of massive fields always allows smooth deformation into A)dS space without introduction of any additional fields besides those that are necessary in flat Minkowski space so that restriction mentioned above will not be essential for us. Mixed symmetry tensor fields have more gauge symmetries compared with well-known case of completely symmetric tensors and, as a result, gauge invariant formulation for them requires more additional fields making construction much more involved. In this paper, as a first step towards gauge invariant frame-like formulation of mixed symmetry bosonic fields, we consider Yk,1) tensors for arbitrary k 2. This case turns out to be special and anyway requires separate consideration. Indeed, in general case Yk,l), l>1, auxiliary field analogous to Lorentz connection has to be a two-form ω Yk 1,l 1,1) μν, while for the Yk,1) case one has to introduce one-form ω Yk 1,1,1) μ instead. Thus this case turns out to be a natural generalization of the simplest model for Y2, 1) tensor constructed by us before [26]. The structure of the paper is simple. In Section 1 we reproduce our results for simplest Y2, 1) tensor. Then, in Section 2 we consider more complicated case Y3, 1) which shows practically all general features. At last, in Section 3 we construct massive theory for general Yk,1) tensor field. In all cases we construct complete Lagrangian and gauge transformations describing massive particles in A)dS d spaces with arbitrary cosmological constant and arbitrary space time dimension d 4 and investigate all possible massless and partially massless limits [17,33,39 41]. 1. Tensor Y2, 1) In this case frame-like formulation requires two tensors [20]: two-form Φ μν a as a main physical field and one-form Ω μ abc, antisymmetric on abc, as analogue of Lorentz connection. To describe correct number of physical degrees of freedom, massless Lagrangian has to be invariant

3 48 Yu.M. Zinoviev / Nuclear Physics B ) under the following gauge transformations: δφ μν a = [μ ξ ν] a + η μν a, δω μ abc = μ η abc, where η abc is completely antisymmetric. Note that ξ-transformations are reducible, i.e. 1) ξ μ a = μ χ a δφ μν a = 0. One of the advantages of frame-like formulation is the possibility to construct an object torsion ) out of first derivatives of main physical field Φ μν a which is invariant under ξ- transformations T μνα a = μ Φ να a + α Φ μν a + ν Φ αμ a = [μ Φ να]. To find a correct form of massless Lagrangian one can use the following simple trick. Let us consider an expression { μναβ abcd }Ω μ abc T ναβ d, { μναβ abcd }=δ[μ a δμ b δα c δβ] d and make a substitution T μνα a Ω [μ,να] a. We obtain { μναβ abcd }Ω μ abc T ναβ d { μναβ abcd }Ω μ abc Ω ν,αβ d Thus we will look for massless Lagrangian in the form { μν ab }Ω μ acd Ω ν bcd. L 0 = a 1 { μν ab }Ω μ acd Ω ν bcd + a 2 { μναβ abcd }Ω μ abc T ναβ d. It is by construction) invariant under the ξ-transformations, while invariance under η-transformations requires a 1 = 9a 2. We choose a 1 = 3, a 2 = 1 3 and obtain finally L 0 = 3{ μν ab }Ω μ acd Ω ν bcd +{ μναβ abcd }Ω μ abc ν Φ αβ d. All things are very simple in a flat Minkowski space, but if one tries to consider a deformation of this theory into A)dS space then it turns out to be impossible [38]. Thus we turn to the massive particle and consider the most general case massive particle in A)dS space with arbitrary cosmological constant. First of all, we have to determine which additional fields we need to construct gauge invariant formulation of such massive particle. In general, for each gauge transformation of main physical field we need appropriate Goldstone field but in most cases this Goldstone field turns out to be gauge field by itself so we need Goldstone fields of second order and so on. But for the mixed symmetry bosonic fields one has to take into account reducibility of gauge transformations. Let us illustrate this procedure on our present simplest) case [9]. Our main physical field Y2, 1) has two gauge transformations with parameters which are symmetric Y2, 0) and antisymmetric Y1, 1) tensors respectively. Thus we need two primary Goldstone fields corresponding to Y2, 0) and Y1, 1). Both have their own gauge transformations with vector parameter Y1, 0), but due to reducibility of gauge transformations of the main field, we have to introduce one secondary Goldstone field Y1, 0) only. This field has its own gauge transformation with parameter Y0, 0), but due to reducibility of gauge transformations of antisymmetric second rank tensor Y1, 1), the procedure stops here. It is natural to use frame-like formulation for all fields, so we introduce four pairs of tensors: Ω μ abc, Φ μν a ), ω μ ab, h μ a ), Ω abc, Φ μν ) and ω ab, h μ ). 2)

4 Yu.M. Zinoviev / Nuclear Physics B ) We start with the sum of kinetic terms for all fields L 0 = 3{ μν ab }Ω μ acd Ω ν bcd +{ μναβ abcd }Ω μ abc D ν Φ αβ d +{ μν ab }ω μ ac ω ν bc { μνα abc }ω μ ab D ν h α c Ω abc 2 +{ μνα abc }Ωabc D μ Φ να + ω ab 2 2{ μν ab }ωab D μ h ν 3) as well as appropriate set of initial gauge transformations δ 0 Φ a μν = D [μ ξ a ν] + η a μν, δ 0 Ω abc μ = D μ η abc, δ 0 h a μ = D μ ζ a + χ a μ, δ 0 ω ab μ = D μ χ ab, δ 0 Φ μν = D [μ ξ ν], δ 0 h μ = D μ ζ, 4) where all partial derivatives are replaced by A)dS covariant ones. Here and in what follows, we will use the following convention on covariant derivatives: [D μ,d ν ]ξ a = κ e a μ ξ ν e a ) 2Λ ν ξ μ, κ = 5) d 1)d 2). Note, that due to non-commutativity of covariant derivatives such Lagrangian is not invariant under the initial gauge transformations δ 0 L 0 = κ{ μν ab }[ 3d 3) 2Ω abc μ ξ c ν + η abc c Φ ) μν d 2) ω ab μ ζ ν χ ab )] h μν, so we have to take this non-invariance into account later on. Now to proceed with the construction of gauge invariant formulation for massive particle, we have to add to the Lagrangian all possible cross terms of order m i.e. with the coefficients having dimension of mass). Moreover, as our previous experience shows, we need to introduce cross terms for the nearest neighbours only, i.e. main field with primary Goldstone fields, primary fields with secondary ones and so on. For the case at hands all possible such terms could be written as follows: L 1 ={ μνα abc }[ a 1 ω ab μ Φ c να + a 2 Ω abc ] μ Φ να +{ μν ab }[ a 3 Ω abc μ h c ν + a 4 Ω abc c Φ ] μν +{ μν ab }[ a 5 ω μ ab h ν + a 6 ω ab Φ μν ] + a7 { μ a }ωab h ν b. 6) Non-invariance of these terms under the initial gauge transformations could be compensated by the following corrections to gauge transformations: δ 1 Φ a β 1 μν = 12d 3) e [μ a ζ ν] 3α 1 d 3) e [μ a ξ ν], δ 1 Ω abc β 1 μ = 6d 3) e μ [a χ bc], δ 1 h a μ = β 1 ξ a μ + 4ρ 0 d 2 e μ a ζ, δ 1 ω ab μ = β 1 2 η μ ab, δ 1 Φ μν = α 1 ξ [μ,ν], δ 1 Ω abc = 3α 1 η abc, δ 1 h μ = ρ 0 ζ μ + β 0 ξ μ, δ 1 ω ab = 2ρ 0 χ ab, 7) provided a 1 = a 3 = β 1 2, a 2 = a 4 = 3α 1, a 5 = a 7 = 4ρ 0, a 6 = β 0.

5 50 Yu.M. Zinoviev / Nuclear Physics B ) Fig. 1. General massive theory for Y2, 1) tensor. Thus we have δ 0 L 1 +δ 1 L 0 = 0 and this leaves us with variations of order m 2 taking into account non-invariance of kinetic terms due to non-commutativity of covariant derivatives) δ 0 L 0 + δ 1 L 1. In general, to compensate this non-invariance one has to introduce mass-like terms into the Lagrangian as well as appropriate corrections for gauge transformations. But in this case there are no possible mass-like terms the only possible term { μν ab }h μ a h b ν is forbidden by ζ -invariance). Note that if we go from frame-like to metric-like formulation by solving algebraic equations for auxiliary fields Ω abc μ, ω ab μ and Ω abc then we obtain mass-like terms exactly as in [9]. Nevertheless, it turns out to be possible to achieve complete invariance without any explicit mass-like terms just by adjusting the values for our four main parameters α 1, β 1, β 0 and ρ 0. We obtain 3d 2) ρ 0 = 4d 3) α 3d 2) 1, β 0 = 4d 3) β 1, β1 2 36α2 1 = 12κd 3). Now we are ready to analyze results obtained. First of all, recall that there is no strict definition of what is mass in A)dS space see e.g. discussion in [42]). Working with gauge invariant description of massive particles it is natural to define massless limit as a limit where all Goldstone fields decouple from the main gauge field. Such a limit, if it exists at all, leads to the particle having exactly the same number of physical degrees of freedom as massless particle in flat Minkowski space. To make analyze more transparent, let us give here a Fig. 1 showing the roles played by our four parameters. One can easily see that massless limit is a limit where both α 1 0 and β 1 0 simultaneously. But from the last relation above it is immediately follows that such a limit is possible in flat Minkowski space κ = 0) only. For non-zero values of cosmological constant one can obtain partially massless limits instead. 1 Indeed, in AdS space κ <0) one can put α 1 = 0 and this gives ρ 0 = 0). Then our system decomposes into two disconnected subsystems. One of them with the fields Φ μν a and h μ a describe partially massless theory [38] with the Lagrangian L = L 0 Φμν a ) + L 0 hμ a ) + β 1 2 {μνα abc }ω μ ab Φ να c + β 1 2 {μν ab }Ω μ abc h ν c, which is invariant under the following gauge transformations: 8) δφ a μν = D [μ ξ a ν] + η a β 1 μν + 12d 3) e [μ a ζ ν], δω abc μ = D μ η abc β 1 + 6d 3) e μ [a χ bc], 1 Here and in what follows we will call particle to be partially massless if its number of physical degrees of freedom lies between that of massless and massive one. Such particles correspond to irreducible representations of anti) de Sitter group that have no analogue among irreducible representations of Poincaré group.

6 Yu.M. Zinoviev / Nuclear Physics B ) δh μ a = D μ ζ a + χ μ a + β 1 ξ μ a, δω μ ab = D μ χ ab β 1 2 η μ ab, where β 1 2 = 12κd 3). In this, two other fields Φ μν and h μ provide gauge invariant description of massive antisymmetric second rank tensor field. In turn, in ds space κ >0) one can put β 1 = 0 and this gives β 0 = 0). In this case our system also decompose into two disconnected subsystems. One of them with the fields Φ μν a and Φ μν gives another example of partially massless theory with the Lagrangian 9) L = L 0 Φμν a ) + L 0 Φ μν ) 3α 1 { μνα abc }Ω μ abc Φ να 3α 1 { μν ab }Ωabc Φ μν c, 10) which is invariant under the following gauge transformations: δφ a μν = D [μ ξ a ν] + η a μν 3α 1 d 3) e [μ a ξ ν], δω abc μ = D μ η abc, δφ μν = D [μ ξ ν] + α 1 ξ [μ,ν], δω abc = 3α 1 η abc, where 3α 1 2 = κd 3). In this, two other fields h μ a and h μ provide gauge invariant description of partially massless spin 2 particle [17,19,26]. 2. Tensor Y3, 1) As we have already mentioned in the Introduction, frame-like formulation for massless Yk,1) tensors turns out to be special because it requires that auxiliary field be one form and not two form field. The Lagrangian for such massless Yk,1) tensors could be, in principle, extracted from the general formula 3.21) of [25], though the only explicit example given there deals with Y2, 1) case. Anyway, it can be easily constructed as a natural generalization of the simplest example given above. Namely, we introduce two-form Φ μν ab which is symmetric and traceless on ab as a main physical field as well as auxiliary one-form Ω μ abc,d which is completely antisymmetric on abc, traceless and satisfies a constraint Ω μ [abc,d] = 0. To provide correct number of physical degrees of freedom massless Lagrangian has to be invariant under the following gauge transformations: δφ μν ab = [μ ξ ν] ab + η μν a,b), δω μ abc,d = μ η abc,d. Here ξ μ ab is symmetric and traceless on ab, while η abc,d has the same properties on local indices as Ω μ abc,d. Note, that these gauge transformations are also reducible ξ μ ab = μ χ ab δφ μν ab = 0. To construct appropriate massless Lagrangian we will use the same trick as before. We introduce a torsion tensor T ab μνα = [μ Φ ab να] which is invariant under ξ-transformations, consider an expression { μναβ abcd }Ω μ abc,e T de ναβ and make a substitution T ab μνα Ω a,b) [μ,να]. We obtain { μναβ abcd }Ω μ abc,e T ναβ de { μναβ abcd }Ω μ abc,e Ω ν,αβ d,e + Ω ν,αβ e,d ) { μν ab }[ 3Ω μ acd,e Ω ν bcd,e + Ω μ cde,a Ω ν cde,b ]. Thus we will look for the massless Lagrangian in the form L 0 = a 1 { μν ab }[ 3Ω μ acd,e Ω ν bcd,e + Ω μ cde,a Ω ν cde,b ] + a 2 { μναβ abcd }Ω μ abc,e T ναβ de. 11) 12)

7 52 Yu.M. Zinoviev / Nuclear Physics B ) This Lagrangian is by construction) invariant under ξ-transformations, while invariance under η-transformations requires a 1 = 3a 2. We choose a 1 = 1, a 2 = 1 3 and finally obtain L 0 Φμν ab ) ={ μν ab }[ 3Ω μ acd,e Ω ν bcd,e + Ω μ cde,a Ω ν cde,b ] { μναβ abcd }Ω μ abc,e ν Φ αβ de. 13) As in the previous case, it is not possible to deform this massless Lagrangian into A)dS space without introduction of additional fields. Thus we turn to the general case massive particle in A)dS space with arbitrary cosmological constant. First of all, we have to determine the set of additional fields which is necessary for gauge invariant description of such massive particle. Our main gauge field Y3, 1) has two gauge transformations combined into one ξ μ ab transformation in frame-like formalism) with parameters corresponding to Y2, 1) and Y3, 0). Recall that these transformations are reducible with the reducibility parameter Y2, 0). Thus we have to introduce two primary Goldstone fields Y2, 1) and Y3, 0). The first one also has two gauge transformations with parameters Y1, 1) and Y2, 0) with the reducibility Y1, 0), while the second field has one gauge transformation Y2, 0) only. Taking into account reducibility of main field gauge transformations it is enough to introduce two secondary fields Y1, 1) and Y2, 0) only. Both have gauge transformations with parameters Y1, 0), but due to reducibility of gauge transformations for Y2, 1) field it is enough to introduce one additional field Y1, 0). It has its own gauge transformation Y0, 0), but due to reducibility of gauge transformations for Y1, 1) field, the procedure stops here. Thus we need six fields Yl,1), Yl,0), 1 l 3. Again we will use frame-like formalism for the description of all fields and introduce six pairs: Ω μ abc,d, Φ μν ab ), ω μ a,bc, h μ ab ), Ω μ abc, Φ μν a ), ω μ ab, h μ a ), Ω abc, Φ μν ) and ω ab, h μ ). Note, that here and in what follows we use the same conventions for the frame-like formulation of Yk,0) fields as in [19]. We start with the sum of kinetic terms for all six fields L 0 ={ μν ab }[ 3Ω acd,e μ Ω bcd,e ν + Ω cde,a cde,b μ Ω ] ν { μναβ abcd }Ω μ abc,e de D ν Φ αβ { μν ab } [ 1 2 ω μ a,cd ω ν b,cd + ω μ c,ad ω ν c,bd 3{ μν ab }Ω μ acd Ω ν bcd +{ μναβ abcd }Ω μ abc D ν Φ αβ d +{ μν ab }ω μ ac ω ν bc { μνα abc }ω μ ab D ν h α c ] + 2{ μνα abc }ω μ a,bd D ν h α cd Ω abc 2 +{ μνα abc }Ωabc μ Φ να ω ab 2 { μν ab }ωab μ h ν 14) as well as with appropriate set of initial gauge transformations δ 0 Φ μν ab = D [μ ξ ν] ab + η μν a,b), δ 0 Ω μ abc,d = D μ η abc,d, δ 0 Φ μν a = D [μ ξ ν] a + η μν a, δ 0 Ω μ abc = D μ η abc, δ 0 h μ ab = D μ ζ ab + χ μ ab, δ 0 ω μ a,bc = D μ χ a,bc, δ 0 h μ a = D μ ζ a + χ μ a, δ 0 ω μ ab = D μ χ ab, δ 0 Φ μν = D [μ ξ ν], δ 0 h μ = D μ ζ, where all derivatives are now A)dS covariant ones. As usual, due to non-commutativity of covariant derivatives this Lagrangian is not invariant under the initial gauge transformations 15)

8 Yu.M. Zinoviev / Nuclear Physics B ) δ 0 L 0 = 3κd 2){ μν ab } 2Ω abc,d μ ξ cd ν η abc,d cd Φ ) μν + 3κd 1) ω μ,ab μ ζ ab χ μ,ab ab h ) μ + κ{ μν ab }[ 3d 3) 2Ω abc μ ξ c ν + η abc c Φ ) μν d 2) ω ab μ ζ ν χ ab )] h μν, but we will take this non-invariance into account later on. To construct gauge invariant description of massive particles we proceed by adding cross terms of order m i.e. terms with the coefficients with dimension of mass) to the Lagrangian. As we have already noted, one has to introduce such cross terms for the nearest neighbours only, i.e. main gauge field with primary ones, primary with secondary and so on. To simplify the presentation we consider these terms step by step. Φ ab μν Φ a μν,h ab μ. In this case additional terms to the Lagrangian could be written in the following form: L 1 ={ μνα abc }[ a 1 Ω abc,d μ Φ d να + a 2 Φ ad μν Ω bcd α + a 3 Φ ad b,cd μν ω ] α +{ μν 16) ab }a 4Ω abc,d μ h cd ν. As usual, their non-invariance under the initial gauge transformations could be compensated by appropriate corrections to gauge transformations δ 1 Φ ab μν = 4α 2 d 2 δ 1 Ω μ abc,d = α 2 d + [e [μ a ξ ν] b) + 2d gab ξ [μ,ν] ] + [ 3e μ d η abc + e μ [a η bc]d 4 β 2 3d 3) β 2 6d 2) e [μ a ζ b) ν], ] d 2) gd[a η bc] μ ], [ e μ [a χ b,c]d 1 d 2 gd[a χ b,c] μ δ 1 Φ μν a = α 2 ξ [μ,ν] a, δ 1 Ω μ abc = 4α 2 η abc μ, δh ab μ = β 2 ξ ab μ, δ 1 ω a,bc μ = β 2 2 η μ ab,c), provided a 1 = 4α 2, a 2 = 3α 2, a 3 = a 4 = β 2. Φ a μν,h ab μ Φ μν,h a μ. Now additional terms to the Lagrangian have the form L 1 ={ μνα abc }[ a 5 ω ab μ Φ c να + a 6 Ω abc ] μ Φ να +{ μν ab }[ a 7 Ω abc μ h c ν + a 8 Ω abc c Φ ] μν +{ μν 18) ab }[ a 9 ω a,bc μ h c ν + a 10 h ac bc μ ω ] ν. To compensate their non-invariance under the initial gauge transformations we introduce the following corrections to gauge transformations: δ 1 Φ a β 1 μν = 12d 3) e [μ a ζ ν] 3α 1 d 3 e [μ a ξ ν], δ 1 Ω abc β 3 μ = 6d 3) e μ [a χ bc], δ 1 h ab μ = ρ [ 1 e μ a d 1 ζ b) 2 d gab ζ μ ], δ 1 ω a,bc μ = ρ [ 1 χ ab e c) μ + 1 2g bc χ a μ g ab c) χ )] μ, d d 1 17)

9 54 Yu.M. Zinoviev / Nuclear Physics B ) δ 1 h a μ = β 1 ξ a μ + ρ 1 ζ a μ, δ 1 ω ab μ = β 1 2 η μ ab + ρ 1 χ [a,b] μ, δ 1 Φ μν = α 1 ξ [μ,ν], δ 1 Ω abc = 3α 1 η abc, where a 5 = a 7 = β 1 /2, a 6 = a 8 = 3α 1, a 9 = a 10 = 2ρ 1. Φ μν,h μ a Φ μν,h μ. Finally, we add to the Lagrangian terms we already familiar with) 19) L 1 ={ μν ab }[ a 11 ω μ ab h ν + a 12 ω ab Φ μν ] + a13 ω ab h ab 20) and corresponding corrections to gauge transformations δh a μ = 2ρ 0 d 2 e μ a ζ, δh μ = ρ 0 ζ μ + β 0 ξ μ, δω ab = 2ρ 0 χ ab, where a 11 = a 13 = 2ρ 0, a 12 = β 0 /2. Collecting all pieces together, we obtain complete set of cross terms 21) L 1 ={ μνα abc }[ 4α 2 Ω μ abc,d Φ να d + 3α 2 Φ μν ad Ω α bcd β 2 Φ μν ad ω α b,cd ] β 2 { μν ab }Ω μ abc,d h ν cd +{ μνα abc } [ β1 +{ μν ab } [ β1 2 Ω μ abc h ν c 3α 1 Ω abc Φ μν c + 2ρ 0 ω μ ab h ν + β 0 2 ωab Φ μν ] + 2ρ 0 ω ab h ab. 2 ω μ ab Φ c να 3α 1 Ω abc μ Φ να ] [ +{ μν ab } 2ρ 1 ω a,bc μ h c ν 2ρ 1 h ac bc μ ω ν Now, as we have achieved cancellation of all variations of order m, i.e.δ 0 L 1 + δ 1 L 0 = 0, we have to take care on variations of order m 2 including contribution from kinetic terms due to non-commutativity of covariant derivatives) δ 0 L 0 + δ 1 L 1. As in the previous case, there are no any explicit mass-like terms allowed here, but complete invariance of the Lagrangian could be achieved just by adjusting the values of remaining free parameters α 1,2, β 0,1,2 and ρ 0,1 d 1 6d 1) β 1 = 2 d 2 β 2, β 0 = d 3 β 2, d 1 ρ 1 = 2 d 2 α 3d 2) 2, ρ 0 = 2d 3) α 1, 24α2 2 β2 2 = 6d 2)κ, 12d + 1)α2 2 3dα2 1 = dd + 1)κ. The role that each of the parameters plays could be easily seen from Fig. 2. Now we are ready to analyze the results obtained. First of all, note that the massless limit i.e. decoupling of main gauge fields from all others) requires α 2 = β 2 = 0. As the first of last two relations clearly shows this is possible in flat Minkowski space κ = 0) only. In this, for non-zero values of cosmological constant there exists a number of partially massless limits. In ds space κ >0) one can put β 2 = 0 and this simultaneously gives β 1 = β 0 = 0). In this complete system decompose into two disconnected subsystems, as shown on Fig. 3. One of them, with the fields Φ ab μν, Φ a μν and Φ μν gives new example of partially massless theory with the Lagrangian ] 22)

10 Yu.M. Zinoviev / Nuclear Physics B ) Fig. 2. General massive theory for Y3, 1) tensor. Fig. 3. Partially massless limit in ds space. L = L 0 Φμν ab ) + L 0 Φμν a ) + L 0 Φ μν ) Fig. 4. Non-unitary partially massless theory. + α 2 { μνα abc }[ 4Ω abc,d μ Φ d να + 3Φ ad bcd μν Ω ] α [ 3α 1 { μνα 23) abc }Ω μ abc Φ να +{ μν ab }Ωabc c Φ ] μν, where 4α2 2 = d 2)κ, 3dα2 1 = 2d + 1)d 3)κ, which is invariant under the following gauge transformations for simplicity we reproduce here gauge transformations for physical fields only): δφ ab μν = D [μ ξ ab ν] + η a,b) μν 4α 2 d 2 e [μ a ξ b) ν], δφ a μν = D [μ ξ a ν] + η a μν + α 2 ξ a [μ,ν] 3α 1 d 3 e [μ a ξ ν], δφ μν = D [μ ξ ν] + α 1 ξ [μ,ν]. At the same time, three other fields h ab μ, h a μ and h μ give gauge invariant description of partially massless spin 3 particle [17,19]. One more example of partially massless theory appears if one put α 1 = 0 and hence ρ 0 = 0). In this, complete system also decompose into two disconnected subsystems as shown on Fig. 4. Note, however that in this case we obtain 12α2 2 = dκ, β2 2 = 4d 3)κ, so that such theory as is often to be the case) lives inside unitary forbidden region. From the other hand, in AdS space κ <0) one can put α 2 = 0 and hence ρ 1 = 0). In this, decomposition into two subsystems looks as shown on Fig. 5. Thus we obtain one more example 24)

11 56 Yu.M. Zinoviev / Nuclear Physics B ) Fig. 5. Partially massless limit in AdS space. of partially massless theory with two fields Φ μν ab and h μ ab. The Lagrangian L = L 0 Φμν ab ) + L 0 hμ ab ) β 2 { μνα abc }Φ μν ad ω α b,cd β 2 { μν ab }Ω μ abc,d h ν cd, 25) where β2 2 = 6d 2)κ, is invariant under the following gauge transformations: δφ ab μν = D [μ ξ ab ν] + η a,b) β 2 μν + 6d 2) e [μ a ζ b) ν], δh ab μ = D μ ζ ab + χ ab μ + β 2 ξ ab μ. In this, four remaining fields Φ μν a, h μ a, Φ μν and h μ just gives the same gauge invariant massive theory as in the previous section. 3. Tensor Yk,1) For the description of massless particles we introduce main physical field two-form Φ a 1...a k 1 μν = Φ k 1) μν here and in what follows we will use the same condensed notations for tensor objects as in [19]) which is completely symmetric and traceless on local indices and auxiliary one-form Ω abc,k 2) μ which is completely antisymmetric on abc, traceless on all local indices and satisfies a constraint Ω [abc,d]k 3) μ = 0. To have correct number of physical degrees of freedom massless Lagrangian has to be invariant under the following gauge transformations: δφ k 1) μν = [μ ξ k 1) ν] + η 1,k 2) μν, δω abc,k 2) μ = μ η abc,k 2), 27) where properties of parameters ξ and η correspond to that of Φ μν and Ω μ. To find appropriate Lagrangian, we introduce a tensor T k 1) μνα = [μ Φ k 1) να], which is invariant under ξ-transformations, consider an expression { μναβ abcd }Ω μ abc,k 2) T dk 2) ναβ and make a substitution T k 1) μνα Ω 1,k 2) [μ,να]. We obtain { μναβ abcd }Ω μ abc,k 2) T ναβ dk 2) 26) { μναβ abcd }Ω μ abc,k 2) Ω ν,αβ d,k 2) { μν ab }[ 3Ω μ acdk 2) Ω ν bcd,k 2) + k 2)Ω μ cde,ak 3) Ω ν cde,bk 3) ]. Thus we will look for massless Lagrangian in the form L 0 = a 1 { μν ab }[ 3Ω μ acd,k 2) Ω ν bcd,k 2) + k 2)Ω μ cde,ak 3) Ω ν cde,bk 3) ] + a 2 { μναβ abcd }Ω μ abc,k 2) T ναβ dk 2).

12 Yu.M. Zinoviev / Nuclear Physics B ) It is by construction invariant under the ξ-transformations, while invariance under the η- transformations requires a 1 = 3a 2. We choose a 1 = 1) k 1, a 2 = 1) k 1 /3 and obtain finally 1) k 1 L 0 ={ μν ab }[ 3Ω acd,k 2) μ Ω bcd,k 2) ν + k 2)Ω cde,ak 3) cde,bk 3) μ Ω ] ν { μναβ 28) abcd }Ω μ abc,k 2) ν Φ dk 2) αβ. As in the previous cases, it is not possible to deform this massless theory into A)dS space without introduction of additional fields, so we will turn to the general case massive particle in A)dS space with arbitrary cosmological constant. Our first task to determine the set of additional fields which are necessary for gauge invariant description of such massive particle. Our main gauge field Yk,1) has two gauge transformations with parameters Yk 1, 1) and Yk,0) with the reducibility Yk 1, 0), thus we need two primary Goldstone fields Yk 1, 1) and Yk,0). The first one has two own gauge transformations with parameters Yk 2, 1) and Yk 1, 0) with reducibility Yk 2, 0), while the second one has one gauge transformation with parameter Yk 1, 0) only. So we need two secondary fields Yk 2, 1) and Yk 1, 0) and so on. As in the previous cases, this procedure stops at vector field Y1, 0), thus we totally have to introduce fields Yl,1) and Yl,0) with 1 l k. Let us start with the sum of kinetic terms for all these fields L 0 = k l=2 + L 0 Φμν l 1) ) Ω abc 2 +{ μνα abc }Ωabc D μ Φ να k l=2 L 0 hμ l 1) ) ω ab 2 { μν ab }ωab D μ h ν, 1) l L 0 Φμν l) ) ={ μν ab }[ 3Ω μ acd,l 1) Ω ν bcd,l 1) + l 1)Ω μ cde,al 2) Ω ν cde,bl 2) ] { μναβ abcd }Ω μ abc,l 1) D ν Φ dl 1) αβ, 1) l l) L ) [ 0 hμ = { μν ab } ω c,al 1) μ ω c,bl 1) ν + 1 ] l ω μ a,l) b,l) ω ν + 2{ μνα abc }ω μ a,bl 1) D ν h cl 1) α, as well as appropriate set of initial gauge transformations δφ l) μν = D [μ ξ l) ν] + η 1,l 1) μν, δω abc,l 1) μ = D μ η abc,l 1), δφ μν = D [μ ξ ν], δh l) μ = D μ ζ l) + χ l) μ, δω a,l) μ = D μ χ a,l), δh μ = D μ ζ, 30) where all derivatives are now A)dS covariant ones. Due to non-commutativity of covariant derivatives this Lagrangian is not invariant under the initial gauge transformations k [ δ 0 L 0 = 1) l κ{ μν ab } 3d + l 4) 2Ω abc,l 1) μ ξ cl 1) ν + η abc,l 1) cl 1) Φ ) μν l=2 + 2d + l 3) ω a,bl 1) μ ζ l 1) ν l + 1 χ μ,l) l) h μ l but we will take this non-invariance into account later on. )], 29)

13 58 Yu.M. Zinoviev / Nuclear Physics B ) To proceed with the construction of gauge invariant description of massive particle, we have to add to the Lagrangian cross terms of order m i.e. with the coefficients having dimension of mass). As we have already noted above, one has to introduce such cross terms for the nearest neighbours only i.e. main field with primary ones, primary with secondary and so on). For the case at hands, this means introduction cross terms between pairs Yl + 1, 1), Yl + 2, 0) and Yl,1), Yl + 1, 0). Moreover, due to symmetry and tracelessness properties of the fields, there exists such terms for three possible cases Yl+ 1, 1) Yl,1), Yl+ 1, 1) Yl+ 1, 0), Yl+ 2, 0) Yl+ 1, 0) only. We consider these three possibilities in turn. Ω μ abc,l 1),Φ μν l) Ω μ abc,l 2),Φ μν l 1). Here additional terms to the Lagrangian could be written as follows: L 1 = 1) l { μνα abc }[ a 1l Ω μ abc,l 1) Φ να l 1) + a 2l Ω μ abd,l 2) Φ να cdl 2) ]. Their non-invariance under the initial gauge transformations could be compensated by the following corrections to gauge transformations: [ e 1 [μ ξ l 1) ν] + provided δ 1 Φ l) l + 2)α l μν = l 1)d + l 4) α l δ 1 Ω abc,l 1) μ = l 1)d δ 1 Φ μν l 1) = α l ξ [μ,ν] l 1), a 1l = l + 2) l 1) α l, a 2l = 3α l. ] 2 d + 2l 4 ξ [μ,ν] l 2 g 12), [ 3η abc,l 2 e 1) μ + e μ [a η bc]1,l 2) Tr ], δ 1 Ω μ abc,l 2) = l + 2)α l l 1 ηabc,l 2) μ, Ω abc,l 1) μ,φ l) μν ω a,l) μ,h l) μ. This time additional terms to the Lagrangian have the form L 1 = 1) l[ a 3l { μν 33) ab }Ω μ abc,l 1) h cl 1) ν + a 4l { μνα abc }ω μ a,bl 1) cl 1) Φ ] να and their non-invariance under the initial gauge transformations could be compensated by δ 1 Φ l) β l μν = 6d + l 4) e [μ 1 ζ l 1) ν], δ 1 Ω abc,l 1) β l [ μ = [a eμ χ b,c]l 1) Tr ], 3d 3) δ 1 h l) μ = β l ξ l) μ, δ 1 ω a,l) μ = β l 2 η μ a1,l 1), 34) provided a 3l = a 4l = β l. ω a,l+1) μ,h l+1) μ ω a,l) μ,h l) μ. This case that has already been considered in [19]) requires additional terms to the Lagrangian in the form L 1 = 1) l { μν ab }[ a 5l ω a,bl) μ h l) ν + a 6l ω a,l) bl) μ h ] ν 35) as well as the following corrections to gauge transformations δ 1 h l+1) μ = l + 1)ρ l [ 1 eμ ξ l) Tr ], δ 1 ω a,l+1) μ = l + 1)ρ l [ η a,l e 1) μ Tr ], ld + l 2) ld + l 1) 31) 32)

14 Yu.M. Zinoviev / Nuclear Physics B ) δ 1 h l) μ = ρ l ξ l) μ, δ 1 ω a,l) μ = ρ l [ al) ημ + l + 1)η a,l) μ Tr ], l where a 5l = a 6l = 2l+1) l ρ l. Collecting all pieces together, we obtain finally k 1 [ L 1 = 1) l l=2 { μνα abc }α l [ ] l + 2 l 1 Ω μ abc,l 1) Φ l 1) να + 3Ω abd,l 2) cdl 2) μ Φ να β l [ { μν ab }Ω μ abc,l 1) h ν cl 1) +{ μνα abc }ω μ a,bl 1) Φ να cl 1) ] 2l + 1) + ρ l { μν l ab }[ ω a,bl) μ h l) ν + ω a,l) bl) μ h ]] ν +{ μν ab }[ 3α 1 Ω abc Φ c μν + 2ρ 0 ω ab μ h ν + β 0 ω ab ] Φ μν 36) 3α 1 { μνα 37) abc }Ω μ abc Φ να + 2ρ 0 ω ab h ab. As for the corrections to gauge transformations, we once again restrict ourselves with the transformations for physical fields only δ 1 Φ l) μν = α l+1 ξ l) l + 2)α l [ [μ,ν] 1 e[μ ξ l 1) ν] Tr ] l 1)d + l 4) β l + 6d + l 4) e [μ 1 ζ l 1) ν], δ 1 Φ a μν = α 2 ξ a [μ,ν] 3α 1 d 3 e [μ a β 1 ξ ν] + 6d 3) e [μ a ζ ν], δ 1 Φ μν = α 1 ξ [μ,ν], δ 1 h l) μ = β l ξ l) μ + ρ l ζ l) lρ l 1 [ μ + 1 eμ ζ l 1) Tr ], l 1)d + l 3) δ 1 h a μ = β 1 ξ a μ + ρ 1 ζ a μ + ρ 0 38) d 2 e μ a ζ, δ 1 h μ = β 0 ξ μ + ρ 0 ζ μ. Now, having achieved cancellation of all variations of order mδ 0 L 1 + δ 1 L 0 = 0, we have to take care on variations of order m 2 including contribution of kinetic terms due to noncommutativity of covariant derivatives) δ 0 L 0 + δ 1 L 1. As in the previous cases, complete invariance of the Lagrangian could be achieved without introduction of any explicit mass-like terms into the Lagrangian and appropriate corrections to gauge transformations). Indeed, rather long calculations give four relations α l β l 1 = β l ρ l 1, l + 1)d + l 3)β l ρ l = l + 3)d + l 2)α l+1 β l+1, 6l + 3)d + l 4)d + 2l) 6l + 2) ld + l 3)d + 2l 2) α2 l+1 + l 1 α2 l βl 2 = 6κd + l 4), d + l 3 2l + 1)d + l 3)d + 2l) 3d + l 4) β2 l + ld + l 2)d + 2l 2) ρ2 l 2l l 1 ρ2 l 1 = 2κd + l 3). To solve these relations we proceed as follows. From the first one we get ρ l = β l β l+1 α l+1.

15 60 Yu.M. Zinoviev / Nuclear Physics B ) Fig. 6. General massive Yk,1) theory. Putting this relation into the second one, we obtain recurrent relation on parameters β β 2 l = l + 3)d + l 2) l + 1)d + l 3) β2 l+1. This allows us to express all parameters β l in terms of β k 1 β 2 l = kk + 1)d + k 4) l + 1)l + 2)d + l 3) β2 k 1. In this, one can show that fourth equation is equivalent to third one. When all parameters β are known, the third equation becomes recurrent relation on parameters α and this allows us taking into account that α k = 0) to express all α l in terms of α k 1. Let us introduce a notation M 2 = kk+1) k 2 α2 k 1, then the expression for α l could be written as follows: α 2 l = l 1)d + k + l 3) [ M 2 k l 1)d + k + l 4)κ ]. l + 1)l + 2)d + 2l 2) Thus we are managed to express all parameters in terms of two main ones β k 1 and M or α k 1 ), in this the following relation must hold: 6M 2 kβk 1 2 = 6kd + k 5)κ. Now we are ready to analyze the results obtained. In complete theory we have three sets of parameters α, β and ρ and the roles they play could be easily seen from Fig. 6. First of all note, that massless limit that requires M 0 and β k 1 0 simultaneously) is indeed possible in flat Minkowski space only, while for non-zero values of cosmological constant we can obtain a number of partially massless theories. In AdS space κ <0) one can put α k 1 = 0 and this gives ρ k 2 = 0), in this two fields Φ μν k 1) and h μ k 1) decouple and describe partially massless theory with the Lagrangian Fig. 7) L = L 0 Φμν k 1) ) + L 0 hμ k 1) ) + 1) k β k 1 [ { μν ab }Ω μ abc,k 2) h ν ck 2) +{ μνα abc }ω μ a,bk 2) Φ να ck 2) ], which is invariant under the following gauge transformations: δφ k 1) μν = D [μ ξ k 1) ν] + η 1,k 2) β k 1 μν + 6d + k 5) e [μ 1 ζ k 2) ν], δh k 1) μ = D μ ζ k 1) + χ k 1) μ + β k 1 ξ k 1) μ. Such particle corresponds to irreducible representation of anti-de Sitter group in complete agreement with general discussion in [38]. Indeed, a pair of representations Yk,1) and Yk,0) of the 39) 40)

16 Yu.M. Zinoviev / Nuclear Physics B ) Fig. 7. Partially massless limit in AdS space. Fig. 8. Partially massless limit in ds space. Fig. 9. Example of non-unitary partially massless theory. Lorentz group perfectly combine into one Yk,1) representation of anti-de Sitter group compare with Eq. 43) of [38]). As we have already mentioned, massless Yk,1) particle in flat Minkowski space has two gauge transformations with parameters corresponding to Yk 1, 1) and Yk,0). If one goes to metric-like formulation by solving algebraic equations for auxiliary fields, then one can see that in the absence of h field only gauge symmetry with the parameter Yk 1, 1) survives. At the same time, all other fields besides Φ μν k 1) and h μ k 1) just give gauge invariant description of massive Φ μν k 2) tensor. On the other hand, in ds space κ >0) one can put β k 1 = 0 and this results in β l = 0for all l). In this case complete system decompose into two disconnected subsystems Fig. 8). One subsystem with the fields Φ μν l),0 l k 1 gives new example of partially massless theory. Again if one goes to metric-like formulation one can see that in this case only gauge symmetry with the parameter Yk,0) survives. At the same time partially massless theory described by the second subsystem h μ k),0 l k 1 is already known [17,19]. Besides, a number of nonunitary) partially massless theories appears then one put one of the α l = 0 and hence ρ l 1 = 0). In this, complete system also decompose into two disconnected subsystems. One of them gives partially massless theory with the fields Φ μν n), h μ n), l n k 1Fig. 9), while the rest of fields just give massive theory for the Φ μν l 1) tensor.

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