Two different possibilities of constructing superfields of arbitrary spin content are considered. In one the superfield itself transforms according to some representation of the homogeneous Lorentz group (H.L.G.) while in the other it is a scalar but is a function of several four component Majorana spinors

Nilsson, J. S.

Tchrakian, D. H.

English

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7)4—1414November 26, 19714On the Higher Spin Content ofSuperfieldsbyJS, Nilsson and D.H Tchrakian*Partially supported y the Swedish Atomic ResearchCouncil contract 0310—019On leave from St Patrick’s College, Maynooth, IrelandInstitute of Theoretical PhysicsFackS-1402 20 GöTEBORG 5SwedenAbstractTwo different possibilities of constructing superfields of arbitraryspin content are considered. In one the superfield itself trans±ormsaccording torn some representation of the homogeneous Lorentz group(H.L.G.) while in the other it is a scalar but is a function of severalfour component Majorana spinors.1. Introduction(1,2,3). *An important aspect of a superfield is that it is a supermulti—plet field, describing particles of several different spins. Our aim in thispaper is to devise means of constructing superfields that contain particlesof arbitrarily high spans(1) (2The superfields defined by Wess and Zutnino and by Salam and Strathdee ‘are themselves Lorentz scalar fields defined over spacetame and the Majoranaspinors o which belong to the [(1,0) (0,1)] representation of the homogeneousLoretitz group (H.LG.). Denoting these by; ; Ci)the Majorana restriction is expressed by(2)where E is the two—component antisyinmetric symbol, or the metric spinor, andr simply denotes the complex conjugate of r, In addition these Majorana spinorsare taken to be anticommuting(3)so that no greater number than four factors of 0 can be non-vanishing (or twofactors of either or r)The scaler superfield thus defined can be expanded in powers of B(x,9). A(x) 944x) 1--k-9 F(x) + (go) G Xt) + D(ic).‘/ t (4-) A(x) R)2..in the notation of where A, F and D are scalar fields, G is a pseudoscalar field, A an axial vector field and and X are spin—- Dirac fields.Throughout the following, we shall enforce conditions (2) and (3) so thatthe spacetime translation induced by the supersyinmetry (2,3) transformation(should be realBelow, we introduce two types of superfields which can contain fields ofarbitrarily high spins.The first of these is the non—sealar, covariant superfield()transforming according to the (In general reducible) representation(A1,B)® (A2,B)® of the H L G This type of superfield was suggested b)rSalam and StrathdeeThe second type of superfield is a Lorentz scalar, defined over spacetimeand several anticouting four—component Majorana spinors (1), (2) •,o(n)Ii) f)(x)9OJ ) (6)As seen from the single S scalar superfield in (k), the spin content of thecomponent fields arises from the magnitude of the spin” representations of theHL’G carried byt the Majorana bases, Thus in U4) the highest spin field is theaxial vector A corresponding to the basis 8yy5 It follows then that fora superfield of the type (5), transforming according to an irreducible represen—tation (IR’), (A,B), of the H’L’G , the highest spin content will be that of(A,B) itself plus one, i.e. (A+B+1), while the lowest spin will be spin-Oor spin—( IA—BI—1), whichever is the greater.This means that with a judicious choice of (A,B), the superfield of spinsup to (A+B+1) will be a tower of particles starting from spin—O. On the other3hand, if A—B is not sufficiently small, for example for a Joos—Weinberg5,6)(J-w) representation where Aj, B0 with j > 1, the peculiar situation willoccur where the lowest spin will not be zero and the supermultiplet will occupy a spin band. This situation however need not arise, and by simply constructing superfields of reducible representations (5) the spin towers can be madeto start from spin-O,Another way of ensuring that the tower of spins starts from zero is toconstruct superfields of the type indicated by (5). This last type has the advantage that the superfield itself is a Lorentz scalar, but the disadvantagethat the Majorana bases are much more copious owing to the loss of the restrictions arising from the use of identical Majorana spinors.In the two following sections both approaches are presented respectively.II..2Covari&it Superfields2 1 The CovarianceAccording to the form (5), a covariant superfield labelled by the I.E.(A,B) of HLG transforms, under weak transformation A and supersymmetry?trotationt? 5, according toU() ‘Le U1t\) : D° 1) (ø#I)(j b9) (T)U(a) c,e)U’) ‘x+ .;e, e÷) (8)where we have used the notation of ref0, ,(9)0 (A) _LO 0It is a straightforward matter to decompose the superfield specified by(T) and (8), in the manner of 4) by using all the available independentMe.jorana bases, whose coefficients then will be the fields of different spins.These bases are presented in the next subsectionS2 2 The Majorana BasesHere we construct the definite parity 2,3) bases from the spinor 0,subject to the conditions (2) and (3). The following basic spinor identitiesire used in the reduction of these bases.--(10)c’ + cv (11)ftI,P Dt11 4..-r$R’4..4’.bP?.•.ft 4.. p CDf:rj0 CD gq ro ‘aU9iCl) CD‘Iacba4’--çn.cfP.’1-.’Uf-JCl)F-’.Cl)HoF-’•Cl)IU](3c1o ø0CDçtCDCD-0.’CDU]3 U)I—’CD::5U)01-.’U)Cl)0Z3.’CDfry0)Cl))l-P.’P.’C)CDc-f--(13 P.’(-30HiiC11p.-<0 III0 oro Qo Ii c-F111H.11 0 Cl)l’.U I..,-I:-P-IC)‘a44’ (1‘1rP.’.-‘--q11U + $ q 4.‘J1bCD 1-I. P.’Cl) ci ci-— 4-CD U)0-‘Hrta)a)4)‘H‘H004)a)a)a)‘HL)a3_:‘r-—‘HçqH011Iq0pa)0C-,.‘cn0Hrci):1pCaaciHpcca-I-)0a)‘f-I4)4)C)-ds-f‘ri4)CardGia)s-I4)-HHp0O‘—U‘riçq(‘34a.,1-b64—a)a)ci)‘—‘‘H4)(1)C)4)ci)4)cf_Ia)0*-iP1ci)ci)CaU)::0-H-HCa4)‘rI‘)C)H‘H4)oHa)P-i05-4l-i‘HC4.4ci)U)0)a)a)ps-ai-H4)oa)C!)C/)rja)HLjH4)4)‘H-rIcf-i-l-0s:1Caa)ci)a)rcjCa-H(tI0a)s-I*C!)()bci)Q4-)I:iO(cf—I0‘rIrdCa1-Ha)44)ICa05--i4-)i:10Cas-I0)(‘35—iU)ci)(134)H0;‘--pa)Ca4)-Hs-I-pC,1I—!-H00I)4lb4-‘IIqa)$-Cd(‘3,-Cfl’d/1)U)*%p1_iCl)H-p0•4)04:10a)a)5-fC!)ci)5-4(1)0-H13:cf44-)—‘0c‘:-‘I*%I>qi1l:--‘I.0ifH1’1ci)7,2 3 Two ExiplesTo illustrate the construction and decomposition of the covariant super-fields givet. in 2 1 by means of the bases in 2 2, we consider two siirrpleexamples, one a Lorentz—vector superfield with AB-, and the other a Dirac—spinor superfield that transforms according to the (-,o) + (o,-) representation of the HL’G.Vector Superfield: Using the bases (iii.) we decompose (x,O ) as follows:\‘ (x, ) ‘4? + q ÷(iB) + (8) + (o)2’v.+ 7 + (ê’r 9) A, + (iO) ()d 4:(16)Here P is a pseudoscalar field, A an axial vector field containing a (spin)1 and a 0 particle, (i1,2,3) are vector fields each containing a1 and a 0+ particle, is a tensor field whose symmetric part includes a——+ (2)2 and a 0 partivle and its antisynmetric part a 1 particle. t. and )are spin- Dirac fields while and are unrestricted Rarita-Schwinger(R-S) fields, each containing a spin-3/2 and a spin— particle.As explained n Section 1, the maximum spin here is spin-(A+B+1)2, whilethe minimum is spin-C.In carrying out the decomposition (16) we have used the fact that the invariant tensors and the —inatrices in the Majorana bases serve to projectout the higher spin content of certain fields, for example the following termscontaining the B—S field and a third-rank tensor field T’,öi .4effectively contain the Dirac 4 and vector V fields respectIvely, and, are thensimply absorbeci into the corresponding field with the same Majorana basis.()which plar the role of Clebsch-Gordan coefficients here.8.Just for ordinary local fields, if the vector superfield V (x,) isjlsubjected to the Lorentz condition(17)its spin content diminishes appreciatly In fact, imposing (17) on (16) givesrise to the restrictions() (0+ ‘4’ 0ci.)+ 2 0to:0A a÷(18)It follows from the restrictions (18) that the number of spin— particlesis reduced by two, the number of scalars by three, the pseudoscalars by one,and the number of vectors by oneDirac Superfield: Using the basis (iii) we decompose the [(i-,o) (0,j)]superfield(x1 e) [ c, e) ., ; ., *in the following way:+ (9) 4 + (J+ e)p +(e) +fY9)+(iIr8)V ÷a) êe)(19,V÷ce)t4e) i?+ +()( 9) F(i9)9.As expected the highest spin occuring in this superniultiplet is thespin—3/2 content of the fl—S field p’’, while the lowest is spin—C. Thenomenclature used in (19) is the same as that used in (18). Thus the bosoncontent is, two scalars, two pseudos calars, two vectors, two axial vectors,and four pure-spin—i fields (two of either parity signature). The fermion content is more subtle and needs comment. Apart from the four spin—1- Dirac fieldsexhibited, the unrestricted fl-S field contains a spin— 1- Dirac field in addition.Now it might appear that in the exapansion (19), the terms(‘p)e . (20)are omitted - It turns out that these terms are not independent of each other,and from (ii) it follows that(‘) Ice • U -(c) e (ëe) (21)which would simply be obsorbed into the appropriate terms in (19), whileV5—G zc9iY&) q) (22)where 4(1-) is the spin— 1- part of the unrestricted fl-S field given by(23)which is already account. for by 4? in (19).Other terms like (oe)(oipe = j(io)2p are already accounted for./ Using the wellknown suinor identities = 2 ZL10The Dirac superfield (19) has the seine H L G transformation propertiesas the ordinary Dirac field, as well as the property that under parity itstwo—component spinor parts will undergo the transformationsa(ijG) XV(x,9)Just as in the case of the free Dirac field, and X of the free super-field are related through the Dirac equation(25)Finally therefore, imposition of the conditions (25) gives rise to thefollowing restrictions on the fields contained in(x,8 ),‘a 4S’÷4F, 4G’. (v’ \2) y)(26)both for 1=1 and 1=2 (cf labels in (19)). This means that out of the three setsof fields, one scalar (pseudoscalar), one vector (axial vector) and one pure—spin—one, only one, say the vector (axial vector) can be chosen as independent,thus diminishing the content of the supermultiplet [‘The above decomposition (19) of the Dirac superfield is the simplest caseof a J—W 2(2jt4)—component superfield 8) decomposition, to which there willcorespond the more general parity constraints/) _btb)(0dJ)’)( ) 0 (27)where the 2(2,5+1) x 2(2,5+1) spin—matrices “1’ ‘2J as well as their relevantproperties are given in detail in ref (6)11,2.)4 Choice of RepresentationThe question of the choice of the HLG representation according to whicha superfield transforms is an open one just like in the case of space-time—local fields.As remarked in Section 1, for a covariant—superfield to contain particlesof spins starting from spin—zero, it must transform according to an IR ofHLG satisfying }A-Bj 1.Another aspect we comment on is that due to the HL’G transformation properties of the Majoraria bases (1)4), it is impossible to avoid component—fieldsof the tensor and (generalised) R-S type, even if the superfield itself mightbe a J—W type, for example the Dirac superfield considered above. Converselya tensor orR—S type superfield will involve component fields of the J-W types,for example in the vector superfield above the field has an antisymmetricpart which is essentially a [(i,o) (0,1)) J—W representation, namely‘iO) O1O’) t %(o,1)‘ (4)’ (28)where the symbol 1 stands for a Clebsch—G-ordan coefficients12.3. Scalar Several—p Superfields3.1 The FieldsThe following construction yields Lorentz scalar superfields that are super—multiplets of fields of particles with spins ranging from zero to arbitrarilyhigh values.We extend the definition of a superfield, by considering fields that arefunctions of space—time and. in general of n four—conrponent Majorana spinors(i) o(2),o, ,, which anticoinniutett) Cj)(29)for any i and j, from 1,.,n, We actually require that the corresponding genera(1) (2) (i) (n)tors S , S ,..., S ,..., S of the supersymmetry transformations sa.tis—fy the following anticomniutation relations‘3,1S ,S 3 a ,(30)We further require that each satisfy the same commutation relations withthe generators P, and of the Poincar group as given in Refs.(2,3). Thesupersyminetry transformationto 8 4 (31a)then causes to be translated according to)+ 4. (3m)Following the procedure in Ref. (3), the variations due to infinitesi—13.ma]. versions o± the transformations (9), and hence also the form of the“covariant derivative&’, can be calculated. The “covariant derivative” ofeah is clearly then of the same form as that for the single 8 super—field.3.2 The Highest Spin ContentHere we do not give complete expansions in a Majorana basis as we didfor covariant—superfields, mainly because the procedure is similar., straightforward and cumbersome Instead we examine the maximum spin content of theseveral—6 superfields.The Majorana bases will consist of a set of the form (1)4) for each 0(1),in addition to the bases constructed from several different spinors‘i), (‘# j ) Compared to (1)4) these new sets of bases will havemore independent members as the restrictions ensuring from the use of identical spinors 0 will b relaxed. To start with, the vector bases willbe nonzero for i jThe maximum power of in the expansion of this superfield will be fourfor each type of 0, namely 1n.Just as in Section 2, each here is taken as being subject to the Majo—rana conditions (3), which results in halving the nuxiber of independent elementsin the (otherwise spin— j) 0’, thus the highest (irreducible) spin basis thatcan be constructed from n types of 0’ is n. In particular for a single 0scalar superfield123), the highest spin bases is spin—one. We give a demonstration of this below for a two—B scalar superfield For simplicity, we use the two—component formalism• Denoting the two spinors bye[ •-j .we consider the following bases constructed from both 0’s:B (3.a(32b)(32c)It follows from the properties (2), (3) and (29) of 0 and 81, and identities (10) and (11) that3 CS’7’)(.) (33a)(33b)c)C&’) +(J)(;u’) (330)Clearly (33a) carries the HRLG transformations of a particle of only spin—zero,(33b) being antisymmetric in ‘i and v carries only spin—one, while (33c) cancarry the transformation of a particle of spin—2Thus using a two—8 scalar superfield we can describe particles of spin-2while with a one—0 scalar superfield one had only up to spin—L That no higherspin basis can be constructed with only two 0e8 is easily checked For examplemultiplying the spin—2 basis B3 by (flaAfl) results (cf (33b)) in antisynmietrybetween all three indices i, , A, and such a tensor cannot carry a totally sym—metric third rank spin—3 tensor field Multiplring B3 by (an’) on the otherhand results (cf (33a)) in (ia,n) which could also not carry a spin—3 fiela.We make a final remarks Amongst the bases that can be constructed with n0’s are the J—Wbases transforming like the IR (j,o) of HeL,G, for all j up15.to For n2 these are, the (1,0) basesJand its parity con,5ugate, the (0,1) basisL’ 7owith only n tes of 0(i) it is impossible to construct in this manner a nonvanishing (1(n+1 ) ,o) basis This follows from (29), the symmetry of fi - i] ain the interchange of a and at, and the properties of the angular momentum re—coupling coefficients.We are very grateful. to Professor L.S. O’Raifeartaigh for helpful clarifying discussions. One of us (J.S.N) would like to thank Professors L. O’Raifeartaigh and J Spelman for the generous hospitality at the D I A S and. atSt Patrick’s College (Ma ynooth) respectively, and the other CD Tch ) wouldlike to thank the Chairman of the Institute of Theoretical Physics in Goteborgfor his hospitality where part of this work was done.iS.Pference s1. S. Ferrari, J. Wess and B. Zumino, CERN preprint TH-1863 (197i).2. A. Salam and J. Strathdee, I.C.T.P., Trieste, preprint IC/714/11.3. A. Salam and J. Strathdee, I.C.T.P., Trieste, preprint IC/T)4/112.l.. Weinberg, Phys. Rev. 181, 1893 (1969).5. H. Joos, Fortschr. der Phys. 10, 65 (1962).6, S. Weinberg, Phys. Rev, 133, B1318 (196)4).7, W. Rarita and J. Schvinger, Phys. Rev. So, Si (19)41).8. L.S. O’Raifeartaigh, Dublin preprint D.I.A.S.—TP-”Weight Diagrams forSuperfields”.

On the Higher Spin Content of Superfields

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On the Higher Spin Content of Superfields

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