D=4 superspace formulation for higher spin fields

Volume 193, number 1

PHYSICSLETTERSB

9 July 1987

D--4 SUPERSPACE FORMULATION FOR HIGHER SPIN FIELDS M. BELLON a and S. OUVRY b,a a LPTHE ~, Universit~Paris-VI, Tour 16, 4, placeJussieu, F-75230Paris Cedex 05, France b IPN, Division de Physique Th~orique ~, B.P. I, F-91406 Orsay Cedex, France

Received 26 January 1987

A D = 4 superspaceformulation for covariant gauge fields of any spin and any permutation symmetryof their Lorentz indices is given. The superspacelagrangian contains a kinetic D-term plus an F-term needed to compensatefor the gauge variation of the former. As an example,a detailed account of the "16 + 16" off-shellsupermultiplet describingthe case (1, ½) is given.

In a previous publication [1], we addressed the problem of the generalization of the Wess-Zumino supermatter multiplet to the case of a supermultiplet containing massless gauge fields of any spin and any permutation symmetry of their Lorentz indices. The basic ingredient of the construction was to consider these fields as elements of the Fock space associated to annihilation and creation classical operators carrying Lorentz indices (a~,, a~t). In this way, one can define bosonic as well as fermionic classical free fields of any spin and any "Young tableau" symmetry. In each sector, a covariant formulation, valid in any spacetime dimension, was derived by introducing supplementary fields [ 1,2 ]. The latter fields can be conveniently described by means of ghost and antighost creation and annihilation operators (b*~, ~ , bn, ~ ) with {bn, Et~}=t~n,,. These ghosts operators define an SU(1,1 ) algebra with generators ( T+, T_,/'3), which allows for a convenient description of the classical and supplementary fields in terms of their isospin [3]: 10, 0 ) for the former and 1½, - ½) for the latter. It was then quite natural [ 1 ] to derive in the D = 4 case a supersymmetry which connects both sectors and mimics the usual supermatter case. Some extra auxiliary fields were necessary for the off-shell closure of the supersymmetry. In this short note, we derive the superspace formulation of this theory. New fields as well as new gauge parameters have to be introduced, which corn-

plete the supermultiplets of fields and gauge parameters already present in ref. [ 1 ]. We will study as an example the case of the "16 + 16" supermultiplet which describes fields carrying at most one Lorentz index and analyse its reducibility. In ref. [ 1 ], the closure of the supersymmetry algebra is realized only modulo a gauge transformation: this indicates that a special, non-supersymmetric gauge has been chosen. As a first step towards a superspace formulation one has to relax this implicit gauge fixing condition: the situation is here quite analogous to the formulation of the vector supermultiplet in the Wess-Zumino gauge [4], in which a particular gauge fixing eliminates the fields which are pure gauge. Let us denote as in ref. [ 1 ] the classical 10, 0 ) real bosonic fields by IA) and IB), the classical 10, 0) Majorana fermionic field by I ~ ) . The supplementary 1½,- ½) fields necessary for a covariant formulation in each sector are respectively I n ) , IN) and I ~ ) . Finally, IF) and IH> denote the auxiliary 10, 0) fields needed for off-shell supersymmetry. IF) and IH) are gauge inert, whereas the gauge transformations on the other fields are given by:

t Laboratoiresassoci~sau CNRS.

where G+ = Y~(p.a,,)b+~ + h.c. is an operator defined

81A>=G+Ia>,

81n)=p21a),

81B> = G + Ib>,

8IN> = p 2 l b > ,

81~>=G+18>,

81@>=:le>,

0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

(1)

67

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in the Fock space of classical and ghost oscillators generalizing the usual differentiation (~A u =pua) and grading the SU(1,1) algebra. Here la) and Ib) are both purely imaginary whereas I* ) is anti-Majorana. Let us first recall the supersymmetry transformations with an anti-Majorana parameter r/acting on these fields: 8[A>=#I~>,

81B> = f f 2 : l ~ > ,

~1~> =P(IA) + 7 5 l B > ) q + (1~> + ~ 5 l H > ) q ,

9 July 1987

the notations and conventions of Wess and Bagger [ 5 ]. In order to describe these supermultiplets, we introduce a chiral 10, 0 ) superfield I ~ ¢ ) = ( 1 ~ ) , I ~ ) , I~:)) and an antichiral 1½,-½) superfield I~ > = ( [~ >, cba >, IJ¢> ), as well as their hermitian conjugates <~¢1 and <~l: <~¢l=(<~l, < ~al, <~=]), ( : l = ( ( :1, ( ~b~l, (Jgl). The gauge parameters are described by the chiral superfield laO = (laO, l e , ) , / ) ) of isospin 1½, - ½). It is obvious that the gauge transformation on [~¢) should read

8IF) =ff(/~l~ ) - G + I ~ ) ) ,

~1~¢) =G+ la~) ,

81H) = f f T s ( # l ~ ) - G + 1 ~ ) ) ,

in order to reproduce the gauge transformations on the classical fields which are all of the form (field) = G+ (parameter). As far as the superfield /N ) is concerned, it is anti-chiral and should trans-

8 1 ~ ) = ( [ M ) + ~: I N ) )q , 81M=gDI~),

8IN) = r 7 7 s P l ~ ) .

(2)

It is clear from (1) and (2) that IF), IH), IM), and IN) are purely imaginary whereas I¢~ ) is a Majorana field. One notices that the supersymmetry transformation acting on IF> (or I H ) ) is not exactly in the usual form but contains a piece - G + ( g I ~ ) ) which can be considered as a gauge transformation. On the other hand the fields I n ) and IN) transform as the last components ofa chiral multiplet, of which the field I¢~ ) is the fermionic component, whereas the first components of the multiplet are missing. These two anomalies can be cured if one introduces two new 10, 0 ) real fields ]P) and IQ) that have to be pure gauge, e.g. 8 IP> = If), 8 IQ > = Ih >, where If) and [h > are two new real gauge parameters, with for supersymmetry transformations 8[P> = ~l ~ >, 81Q > = ~' 51~ >. The supersymmetry transformation on I~> becomes 8 1 ~ > = p ( [ P > + ~ 5 l Q > ) q + ( [ g > +yS IS> )r/, whereas IF> and IH> are now gauge transformed as 8 IF> = G+ ~r) and 8 l H > = G + l h > . One has thus two "matter supermultiplets", the first one ofisospin 10, 0> (IA >, [B>, I~U>,lF> IH>) containing the classical fields, the second one ofisospin 1½, - ½) (IP), IQ), 1¢~>, IM), IN)) containing the supplementary fields. Moreover, the gauge parameters can now be ranged in the gauge supermultiplet ([a), Ib), ]e ), LD, I h ) ) o f i s o spin 1½, - ½). What about an invariant lagrangian? Let us first reformulate these results in the superspace. We use 68

6(~¢1= (~IG+,

(3)

f o r n l as

81~)=-]D2[~)

,

(4) This last equation implies that 81,g)=-p2[~), 8 l ~ a > = ( a " p , , l e > ) ~ and 8 1 ~ ) = [ / ) . This is exactly the type of gauge transformations that are needed for these supplementary fields as can be seen from eq. (1) and the fact that IP) and IQ) should be pure gauge. The next step consists in building a gauge-invariant superspace lagrangian that reproduces the one derived in ref. [1]. It reads

H'= -- ~ [D2D2((~gld) -- (:IT+ ~ ) )

+4D2(dlG+~)+4D2(~dG+~¢)], where T+=7.n(b*~bn).This lagrangian is

(5)

indeed invariant under the gauge transformations (3), (4). In the component formulation it becomes Aa= ( ~ ¢ l p 2 d ) - ( ~ l T + p 2 ~g'? - ( .~: [.~ )

+ (.gl T+.g) + (~¢IG+.g) + (.giG+ ~ ) + (:[G÷ ~ ) + ( ~IG+ • > + ( ~/[~rnpm~[J)

-<~lT÷ampm~>-<~PlG+ ~> - (O[G+ ~u).

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Volume 193, number 1

PHYSICS LETTERS B

9 July 1987

Iz¢> = (1/~/2)( IA> +ilB> ),

in the case of interest D = 4, a single oscillator is sufficient to describe all possible on-shell multiplets. Let us work out in detail the case "16 + 16". From now on, we will use the notations of ref. [ 1 ]. Let us first express explicitly all Lorentz indices, in the lagrangian and the gauge and supersymmetry transformations:

I ~ > = - ( 1 / x / 2 ) ( I M > +ilN> ) , I O > = 7 5 1 0 > ,

~=A.p2A ~ +B.p2B. - Pp~t'- OreQ

This lagrangian is identical to the one derived in ref. [ 1 ] except for the new terms involving the field I~ > previously set equal to zero by a special gauge choice. This identity is transparent once the following field redefinitions are performed:

I~r> = (1/x/~)(IF> +ilH> ) , [~ > = (1/v/2)(IP> + i t Q > ) .

- F ~,P' - HuH' + M* M + N* N - AupUM - M*pUAu (7)

One should also notice that there is a factor i between the ?-matrices of Wess and Bagger and the ones of ref. [1]. The structure of eq. (5) is quite remarkable because of its simplicity considered the infinity of massless fields of any spin that it describes. It contains kinetic pieces for the classical and supplementary superfields which look like the usual one in the supermatter case. But unlike this simple ease and due to the gauge transformations acting on these superfields, an extra piece, which is of the F-type, is needed in order to compensate for the gauge variation of the kinetic terms. It would be interesting to proceed further and look for a superpotential which could give mass to certain of the fields: this problem is related to the definition of a proper interaction [ 6 ] which could lead to a generalized super-Higgs effect. A last comment has also to be made concerning eq. (5): it constitutes certainly a good starting point for the coupling of the theory to supergravity. One has to substitute the global chiral projector by a local one, as well as to define an appropriate chiral density and to covariantize the operator G+ which appears in the F-term. This project is presently under study. One can compute the total number of off-shell bosonic and fermionic degrees of freedom present at each eigenvalue N of the level operator N=Z(a*,a,+b*,g,+E*,b,). For N = 0 , one has the usual " 4 + 4 " Wess-Zumino supermultiplet. For N= 1 one describes the case (~¢u, ~u) which has "16 + 16" degrees of freedom. In the case of one single oscillator a general value of N corresponds to the completely symmetric fields (~lu,...uu, ~tu,...u~). The counting gives then a " Y + A / " structure with Y = ~ ( N + 1 ) (N 2 + 2N+ 3). One should stress that

- B~p~N - N*pUBu + F~upUP+ PpUFu

_ ~upU ~ + ~pu ~tu. 5Au=pua,

8Bu=pub,

5Hu=Puh,

8P=f,

6M=p2a,

5N=p2b,

8~Fu=p,e,

~q~=lbe,

(8) 5Fu=puf,

8Q=h,

(9)

~. =.o(a,, + 758.),i+ (F. + ?~.),7, 5 • =/~(P+ 75 Q) tl+ (M+ 75N) tl,

8F.=#:~., 8P=ff~,

8H~ = # 7 5 ~ ,

8Q=ff?5~,

5M=FllbO, 5N=ff?5/~O.

(I0)

This supermultiplet contains the (1, ~) and (1, ½) supermultiplets. We want to address the problem of its complete reducibility. In order to get rid of the (1, ½) part of the supermultiplet, one can impose on • the gauge-invariant condition 0 =?~ W# .

(I I)

To see the full set of consequences of this constraint, we first work out its supersymmetric variation. It 69

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gives a condition of the form X~l=O, where X is a 4 × 4 matrix. When expanded on the Dirac basis, it leads to M=puAU, Fu=puP ,

N=puBU, Hu=puQ,

p(uA~) - ½e~papPB ° = 0 .

(12) (I 3)

Eqs. (12) are simply the Lagrange equations with respect to the fields M, N, Fu and Hu. Eq. (13) is a duality relation between A u and Bu: in fact it is the only possible parity preserving relation between these two fields. Using the Bianchi identity on the field strength of B u, we see that eq. (13) implies the Maxwell equation on Au. It follows then from (12) that A u and Bu are on-shell. Thus all bosonic fields are onshell, since the Lagrange equations of P and Q can also be deduced from the ones of Fu and H u. Besides, a further application of the supersymmetry transformation on the Lagrange equation with respect to Fu gives the Rarita-Schwinger equation of the gravitino field ~u. It follows that all the fields are on-shell. The field content is now the following: there is a gravitino field ~u and a vector field which can be described by either Au or Bu. Thus the off-shell supermultiplet has been reduced to a pure (1, ~) multiplet, but doing so we have been driven on-shell. To complete this analysis, we would like to obtain the (1, ½) supermultiplet. We may begin by eliminating one of the spin-1 fields, via a constraint orthogonal to (13): P(uA~) + ½euupapPB a = 0 .

(14)

Let us proceed in the same way as for the (1, ~) case. The supersymmetry variation of (13) leads to P(u ~ ) + ½8uz'p~)jSP p ~r./a----0.

70

(15)

9 July 1987

This equation does not imply the Rarita-Schwinger equation for the gravitino field. In fact, through contractions with Dirac matrices, one can only deduce the condition ap~pp~u~= 0. However, if we consider eq. (15) altogether with the Lagrange equation with respect to ~u, we obtain that its "field-strength" p(u~u ) can be expressed as a function of the derivative of the gauge-invariant spin-½ field ~ ' =@ - Y ~ ' u . One can indeed verify that p ( u ~ ) = ½Y(uP~)q~' is the solution for both equations. ~ ' then satisfies the massless Dirac equation. It follows that the fermionic sector of the theory describes only a massless Majorana field. Thus the '° 16 + 16" supermultiplet has been reduced on-shell to an (1, ½) supermultiplet. The "16 + 16" supermultiplet appears to be completely irreducible off-shell and to split into a (I, ~) and a (1, ½) supermultiplet on-shell via the imposition of gauge invariant constraints. In the general case, it would, however, be nice to describe a pure (N, N + ½) supermultiplet. It is quite possible that the coupling of the theory to supergravity could provide the necessary constraints as field equations.

References

[ 1] M. Bellon and S. Ouvry, Phys. Lett. B 187 (1987) 93. [2] S. Ouvry and J. Stern, Phys. Lett. B 177 (1986) 335. [3] W. Siegel and B. Zwiebach, Nucl. Phys. B 263 (1986) 105; A. Neveu, H. Nicolai and P.C. West,Phys. Lett. B 167 (1986) 307; Y. Kazama, A. Neveu, H. Nicolai and P.C. West, Nucl. Phys. B, to be published. G.D. Dat6 et al., Phys. Lett. B 171 (1986) 182. [4] J. Wcssand B. Zumino, Nucl. Phys. B 78 (1974) I. [5] J. Wess and J. Bagger, Supersymmetry and supergravity (Princeton U.P., Princeton, 1983). [6] I,G. Koh and S. Ouvry, Phys. Lett. B 179 (1986) 115.