JHEP05(2018)204 2) ≥ S Springer May 17, 2018 May 31, 2018 April 30, 2018 : : Ess Chiral, the El Formulated : E = 0

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JHEP05(2018)204 2) ≥ s Springer May 17, 2018 May 31, 2018 April 30, 2018 : : ess chiral, the el formulated : e = 0. Accepted Received Published W , 2912, U.S.A. the third one we provide the eractions of this theory with with linear superpotential. For st beyond supergravity ( urrents. As expected, a coupling 2) supermultiplet via higher spin de Fora, / ¯ l University, ΦΦ) and ( yk University, K + 1 . On the other hand, coupling to the Published for SISSA by https://doi.org/10.1007/JHEP05(2018)204 = , s W ¯ K Φ) and an arbitrary chiral superpotential + 1 , s and [email protected] (Φ and Konstantinos Koutrolikos K K d = 1) can always be found and we give the generating sylvester s , = 0) is possible only if s . 3 S. James Gates Jr. 1804.08539 a,b,c . However, we ﬁnd three exceptions, the theory of a free massl The Authors. W c Superspaces, Supersymmetric Eﬀective Theories

We consider a four dimensional generalized Wess-Zumino mod , [email protected] and K (Φ). A general analysis is given to describe the possible int [email protected] National Research Tomsk State University, Lenina avenue, Tomsk 634050, Russia Departamento de ICE, F´ısica, Universidade FederalCampus de de Universitário-Juiz Juiz Fora, 36036-900, MG,Department Brazil of Physics, BrownBox University, 1843, 182 Hope Street, Barus & Holley 545, Providence, RI 0 Department of Theoretical Physics,Tomsk StateKievskaya, Pedagogica Tomsk 634041, Russia Institute for Theoretical Physics andKotlarska, Astrophysics, 611 Masar 37 Brno, CzechE-mail: Republic b c e d a supercurrents. It is shown that such interactions do not exi vector supermultiplet ( theory of a free massive chiralthe and first the two, theory the of higher aexplicit spin free expressions. chiral supercurrents are We also known discuss andto the for (non-minimal) lower spin supergravity superc ( for any supercurrent and supertrace for arbitrary Keywords: ArXiv ePrint: W external higher spin gauge superfields of the ( in terms of an arbitrary potential Kähler Open Access Article funded by SCOAP Abstract: I.L. Buchbinder, Interaction of supersymmetric nonlinearwith sigma external models higher spinsupercurrents superfields via higher spin JHEP05(2018)204 8 8 9 ], 1 3 6 8 18 18 19 11 12 13 15 17 21 10 28 – 19 h [ d higher llows us to better ical phenomena. Neverthe- markable features, e.g. their extensive study. Despite the r studying and understanding heory. In many cases, interac- larizing the ultraviolet with an or flat spacetime at first order in – 1 – and great efforts it is not clear yet whether higher W 1 ] 18 – 7 , ]. 5 6 – , 3 2 , 1 by using a variety of techniques, such as light-cone approac g correspondence. Furthermore, studying higher spin fields a AdS/CFT Pedagogical review in refs. [ 6.2 For supergravity 6.3 For higher spin supermultiplet 5.2 Additional freedom 5.3 Conservation equation 5.4 Reality condition 6.1 For vector supermultiplet 4.2 Conservation equation 4.3 Reality condition 5.1 Improvement terms 4.1 Improvement terms spins 1 the spin fields play aless, role higher in spin the fieldscontribution attract description in much of attention the fundamental due softness phys infinite to of tower many of string re massive interactions states and by providing regu a framework fo understand the structure oftion interactions terms in for general higher gauge spinscoupling were t constant successfully constructed f various no-go theorems [ 1 Introduction Higher spin fields and their interactions are the subjects of 7 Summary and discussion 6 Turn on chiral superpotential 5 Coupling to higher spin supermultiplets 3 Coupling to vector supermultiplet 4 Coupling to (non-minimal) supergravity Contents 1 Introduction 2 Noether’s method for supersymmetric nonlinear sigma model an JHEP05(2018)204 ] ] ]. d ]) K ) a 52 55 53 s of 36 – ii , – 48 results 54 33 ¯ Φ) with , , However, 43 2 (Φ ]). However, , K 62 31 d supertrace for higher spin gauge theorem. ]). Such a model is a e search for the higher sure that the Noether ]. 62 our discussion and sim- or fields [ 61 ces have been constructed the cubic coupling of higher ned by analyzing tree level 2). We find that interactions consequence of the fact that ahler potential s have been obtained [ / ossibility of such interactions. to external higher spin gauge 2) are not possible for any use we do not know the fully y procedure. urrent multiplet of the theory. we have is to follow Noether’s l consider a theory of the chiral o take the functional derivative otion for higher spin fields [ nd a good candidate for explor- namics we will assume a nonlin- ting a higher spin supercurrent sed in [ d. For that case the calculation ns. It is natural to take the next ≥ perfield (see e.g. [ +1 oupling of the interacting matter s , s (Φ) (see e.g. [ +1 s ) a free massless chiral superfield, ( i W – 2 – ) a free chiral superfield with a linear superpotential. ], thus enhancing the connection between string theory iii 47 – ] (some of these results were later generalized in [ 45 32 – = 1 supersymmetric matter theory can be consistently couple 29 , where the conserved current is quadratic in the derivative N , 10 ]. In an intriguing manner, most of the previously mentioned 44 – ]. We add to this list the expressions for the supercurrent an 37 ] for supersymmetric generalizations) which are of the type 60 60 , – conserved current and thus extending the results of the no-go Coleman-Mandula 58 56 – × W In order to include both scalar and spinorial matter fields in It is known that any In this paper, we are following a Noether-type approach and w Among these interactions, the simplest class is provided by Examples of bypassing the Coleman-Mandula theorem are discus 56 2 and the addition of an arbitrary chiral superpotential For the first two,in the [ higher spin supercurrents and supertra free massive chiral superfield and ( with higher spin supermultiplets beyond supergravity ( ear supersymmetric sigma model described by an arbitrary K¨ step and consider thefield. coupling of In this interacting paper low we spin ask fields this question and investigateplify the the p technical details imposed bysupermultiplet supersymmetry described we by wil a chiral superfield Φ. For its dy the matter fields. Thewe cubic couple nature non-interacting (free) of matter these fields interactions to is higher a spi spin supercurrent multiplet that generates thetheory first with the order higher c spin supermultiplets of type ( we find three exceptions to this rule and these are ( interacting theory at present.method The in only order to alternative constructHowever, option directly in the higher the spinprocedure case superc leads of to the coupling same to supercurrent as supergravity the we supergravit should be this procedure is not applicable for higher spin theory beca good parametrization of many interactinging matter theories the a possible interactionsmultiplet. with higher spins by construc spin fields with(and low [ spin matterfield fields, such as scalar and spin and BRST [ together with some newamplitudes of interaction (super)strings vertices [ haveand been higher obtai spin fields.which For eventually (A)dS led to backgrounds the similar fully result interacting equations of m Noether’s procedure [ to supergravity with the helpof of the the conserved supercurrent gravitational is superfiel of straightforward. the One interacting has action t with respect the gravitational su the third theory. JHEP05(2018)204 ] : is g φ, h (2.4) (2.1) (2.3) (2.2) [ S l and ontribution or the presence such that the chiral oether’s method and ϕ ld, most of the structure viously, unlike the higher ..., → a conserved current out of e vector supermultiplet. gauge invariance. Noether’s quirement of preserving the .... ary and sufficient conditions ..., equirement order by order in on-minimal supergravity can or supergravity and similarly ] + that correspond to a different sections we had the chiral su- tions are still not understood. e focus on the vector supermul- ] + ] + 1 e review and discuss our results. S φ, h [ . h, ζ φ, ξ 2 [ [ interaction vertices. In addition, we S 2 and the requirement of invariance for 2 2 δ δ g = 0 1 2 g 2 . In this approach the full action S g 0 g h 0 S Φ = 0) and we explored the consequences δ ] + 1 ] + 1 ] + δ ]. On the other hand, interactions with the δh ˙ δS φ, h α [ 63 h, ζ φ, ξ ¯ – 3 – [ 1 [ D + 1 1 gS φ gδ gδ 1 δ = 1 higher spin supermultiplets and the conservation ] + 0 . Indeed, this follows from our analysis and we get ] + ] + φ ξ ζ δφ [ N [ [ δS are expanded in a power series of a coupling constant W 0 0 0 S δ δ D, = = and φ, h corresponding to the free theory of a chiral superfield. In th = 0) and the potential Kähler depends only in the product of ] = 0 K δh δφ S W φ, h [ S interaction terms are given by . g 0 turned of. In section six, we turn it back on and consider its c S )). These conditions can be understood as the requirements f W ϕϕ ( ¯ K = ) for the choice of ] we demonstrated that for the case of a single chiral superfie Φ is fixed by the chiral requirement ( K We also consider lower spin supercurrents. As mentioned pre The paper is organized as follows. In section two, we review N higher spins 2.4 1 ( 57 δ ¯ of ( starting action paper we want to explore if there are interaction terms ϕϕ superpotential vanishes ( of spin case, coupling ofalways the be theory found under consideration forexpression with compatible any n with thevector results supermultiplet of do not [ alwaysare exist. the We find existence the of necess a redefinition of the chiral superfield Φ In [ of a global U(1) symmetry which is usually associated with th review the description of free 4 its application for the construction of first order in equation of the supercurrent multiplet.tiplet In and section go three, through w the the chiral requirements theory. in Inin order section section to four, five we construct for repeatperpotential higher the spin procedure supermultiplets. f In previous to the supercurrent multiplets. Finally, in section seven w 2 Noether’s method for supersymmetric nonlinear sigma mode The fundamental principles thatHence, govern higher the spin only interac guidingpropagating principle degrees one of freedom. has,method is is This the the is framework physical manifested wherea re one through perturbative organizes expansion the around invariance a r starting point this order gives: The first order in and transformation of fields JHEP05(2018)204 . J (2.8) (2.6) (2.9) (2.7) ¯ Φ) and (2.10a) (2.11a) (2.10b) (2.11b) . , s ± (Φ , K 2)) , 2) have two de- − 2) and s / / ( 1)) are described by a α ) ˙ . − . The same notation is + 1 s + 1 4 s k ( ( ) . s 1) α ( α tarting point a super- ¯ Φ) (2.5) , s − ) ˙ , s ± Λ ( s s . 1) and it uses the pair of ( ( 2 . . . α 1 ) ¯ α α / 2 − s W + 1 s ( ≥ α L Φ = 0 α α s 1 ¯ he super-Poincarégroup in s z alrpotential ähler ]. A quick synopsis of the , ( ˙ s 1 +1) ˙ an be written as higher spin + 1 6 α α s 1) ˙ δ ¯ ( ˙ ( D d 73 s − ¯ rved supercurrent multiplet α Φ) ¯ s – D Φ re defined modulo the relations ( ! Λ Z 1)! 1 α ¯ s Λ( 69 W g: . ¯ 1 with helicity +1 L − s with the following zero order gauge z s − corresponds to the set of superfields ˙ α 6 s α ) ( d 1) s ¯ h D ( (Φ) + − α s + 3 Φ Z + D ( + = 1 free, massless, higher spin supermul- W α 1)) ˙ 1) W + 1) 1) z 1) ˙ − − N − 6 s − s − = s ( s ( d ( s ( α h ¯ α ( 2 supermultiplets ( Φ) + Λ(Φ) + α Φ α ) ˙ 0 ¯ ) ˙ α D, / , – 4 – L ) ˙ s Z δ s ]. Later, a superfield formulation was introduced s ( s K V ( ( ]. (Φ) + constant 2 α α α (Φ 1) supermultiplets ( J α ( 65 + 1 ¯ L 64 D L with the following zero order gauge transformations [ L D 2 s ≥ 2 s ! ¯ and Φ) + 1 D α ¯ s 1) s ) becomes: D , ∼ K ∼ W Φ + = 5 ( − 1 − s 1) δ s (Φ = = ( 2.4 Y ¯ Φ) − α = D = ) K (Φ) Φ s , ) ˙ = s ( 1) s ( ( K α z 1) 1) − W α ) ˙ (Φ α 8 Superspace s Y ) ˙ s − − ( s d ( χ s s K ( α ( ( α ) ˙ (Φ) , z α α α s ) Z ) ˙ 8 ( s s 1) ˙ H ( W d α ( 0 α − = α χ ) ˙ δ s 0 s ( Z 0 Ψ ( δ α S 0 α ) is a shorthand for k undotted symmetric indices V δ k 0 H ( δ α ] and further developments can be found in [ 68 – pair of superfields Ψ transformations scriptions. The first is called the transverse formulation ( superfields In order to be as general as possible, we will consider as our s The massless, higher spin irreducible representations of t 66 The notation We follow the conventions of On-shell they describe the propagation of degrees of freedom 1. The integer superspin 2. The half-integer superspin 5 3 4 used for the dotted indices. a chiral superpotential and the invariance requirement ( in [ description of higher spin supermultiplets is the followin symmetric nonlinear sigma model described by an arbitrary K where the potential Kähler and the chiral superpotential a that participate in the description of the 4 The above expression is symbolic in the sense that The on-shell equation of motion for this system is four dimensions were first described in [ tiplets and also we havegauge assumed superfields that times the corresponding interaction elements of terms the c conse JHEP05(2018)204 . ¯ J L 2 , (2.16) (2.17) (2.15) (2.13) (2.14) ercur- (2.12a) iplet (2.12b) + D 2)) − . L s ( 1) 2 ˙ 1) α − ¯ ˙ D β s − ) ( . s 1) s ( α ( ) = 1)) − ) ˙ α s α s s ( − ( ( 2) which is being 1) ˙ s 1) ˙ V α 2) α ( α ) ˙ / 0 − , − / s α s 1 δ 1) ˙ L s 2) 2) ( ( ( T , 1 / / α − ˙ α ) makes obvious that 1)) α β ¯ s − ¯ ¯ T J ( + 1 (1 s − ¯ J ˙ D s α α ) 2.4 ( s + 1 + 1 s T = = , s ¯ α ( D α s ) ˙ ( ) s α s α , s , s s 1) α ( ( D . 1) ˙ . + 1 − D α 1 α D s − ) ˙ . − s ( L s n conditions: 1) + 1 + 1 s 1 3)) s ( ( s α n of the higher spin superfields. 1) − α s s − α α α hey define the supercurrent mul- s s ( ˙ 2) and it includes the superfields 1) ˙ s the higher spin gauge superfields: − ( ˙ 1) − ( ( ¯ J s Ψ ¯ − D α 2 ( − ¯ s ( α D s ) ˙ ≥ s / ! ( α 1 s ( ! + 2 ( 2 ( 1) ˙ 1 ( s s α + α s 1) ˙ / / − α ! − T 1) s 1) ˙ − s + 1 χ ( 1) − , s s − − ( α s ) − s as expressed in ( + 1 + 1 s ( α s s + 1) ( J ( ( − s s α g 2 T α ) − = ) ˙ α α ¯ s , s − T ) ˙ s ) ) we must have 1) ( 1) ( s ( s = = s α Y ( 1)) ˙ α − ˙ α 1) − α α ) ˙ , s , s ( ˙ s s α − s − ( ( Y Y J 2 2 ¯ ( s D ¯ 1)) ˙ – 5 – s D L α ( α / / ) ( α 1) s − s α α 1) ˙ s ( α J − 1)) ˙ ( ¯ with the gauge transformation 2)! L ) α s − α s ( + 1 + 1 D s 1)) ˙ − s ( 1 ( 1) ˙ s 1 α T α − − s s V ( α ) ˙ ( α s − − s ( ( ) ˙ s α s s ( s α s ( D ˙ ( V ( ( T ( α α α ! . α 2 α 1 T ¯ + s D s s D + ) Ψ s χ ¯ 2 D s H ( α s ¯ +¯ ( = = D = = α α with ) ˙ ! ( z D 1 s ) z s Y Y ! ( 8 s 2) 8 D 2) 1 ( s α d ! d − ¯ α − 1 (with appropriate index structures) are the higher spin sup = s s ) ˙ J s ( s ( ) = Z ( Z α s α T ) ( = = α 1) ˙ s 1) ˙ α = 0 ) we get a conservation condition on the supercurrent mult ∼ ( ∼ ) ) ) ˙ − H − s s α s s 0 0 1 ( ( ) ˙ s ( 1 ( Φ and ( δ s α α α S δ α ( S δS ) ˙ ) ˙ α s s α J χ J χ ( ( s 0 J ˙ α α , δ α s ) ˙ J J ¯ s α D s ( ˙ ¯ D α α ) ˙ 2 ¯ s D ( D α = 0 case corresponds to the well known vector supermultiplet The second one is the longitudinal formulation ( H s 1. For integer superspin 2. For transverse half-integer superspin 2. For transverse half-integer superspin 1. For integer superspin 3. For longitudinal half-integer superspin The invariance of the action up to first order in The superfields if we go on-shell ( The which is controlled by theUsing zeroth order the gauge expressions transformatio above, we find the precise conservatio rent and higher spin supertrace respectivelytiplet which and generate together the t first order interaction terms with described by a real scalar superfield JHEP05(2018)204 that then (2.18) (2.19) 1 1) (2.20c) (2.20a) (2.20b) − 2)) s ( − α s 1) ( , ˙ 1) ˙ α − ˙ s − β 1) ( s ( α − 1)) α s 1) ˙ ( − 2) s X α − / ( s k 1) ˙ ( α 2). Furthermore, in α − T ˙ + 1 s T β ( ≥ 1 α ¯ D − , s ¯ s s s X ˙ α α 1)) ( . = 0] and another one to go − ¯ + 1 + D s D d all the contributing factors ( s 1 2) 1) 1) α 1) permultiplets are dual to each ting interactions with half inte- − = 0 (vector supermultiplet) to for the next three sections and − s a choice of − s − − s s s ( 1)) ˙ s α s , ( ( satisfy the following conservation ( ( ˙ , α 2 ( − W α ) kind via higher spin supercurrent α ire more than one chiral supefields. α will be examined in section six. ) are not independent. They are s ¯ / D 1) ˙ ( = 0]. This is a manifestation of the 1)) 1) 1) ˙ 1) 1) ˙ 1) ˙ 1 α − ! Φ is linear in derivatives of Φ − − − − − − s W s 1) s s s 1 s s ( s 2.17 + 1 X ! ( ( ( ( ( 2.15 − ( δ ⊥ − α s s α α α α α ). This will be done in the following sec- s s α s α ˆ ( χ ¯ T . It is straight forward to show that if ( 1) ˙ T 1) ˙ X α 1)) ˙ 1 = − − 1) − D + − )[ 1) ˙ − s s s 2.14 s ( s − ( ) = 0 (3.1) ( − k Y s s α 1) s α α + 1 α ) gets simplified to s ( ( ˙ ( α ( ) and ( − ¯ T 1) α k s α D ¯ X ¯ s – 6 – J 2.15 α D ) ˙ X ( 2) of the ( − ˆ 2 ! 1) ˙ s T 1) s s α / ( s + + 2.14 ( ¯ − α 1 D ! − α 2.14 ( ˙ s α s )[ and s ( 1)) ˙ ( 1) 1) J ¯ +1 D α ) − α − − 1)) ˙ s s s 1) + s s ( ( 2) ˙ X α − ( ( , s k ) − 2.14 ( α s α α α − s s ) ˙ ( = 0] to ( ( s ( ⊥ T s D α ( 1) ˙ 1) ˙ ( 2 α α ! +1 ] we know that if ) ˙ α T α − − s 1) s ¯ s 1) ˙ s 1 D ! s s χ ( 57 ( ( − s α s . The contributions of − ⊥ k H α s ( α α α s +¯ ( ( ( K − J T T ⊥ D α α z = 0] to ( D 2 1) ˙ T 8 ! = = = ¯ 1 D d s − , 1) ! s ) ) 1 ( − s s s 1) 1) + k ( ( s Z α ( − − α α T s s ) ˙ ) ˙ α = ( ( s s ∼ ( ( α α 1) ˙ ) )[ α α 1 s − 1) ˙ 1) ˙ ˆ ( s S J J ( − − α ⊥ ) ˙ s s α ( ( 2.14 s ⊥ k ( T α α α ˆ ˆ T T )[ J s ˙ α ¯ 2.15 D Based on the results of [ 3. For longitudinal half-integer superspin = 1 (supergravity) and then to higher spin supermultiplets ( Furthermore, conservation equations ( 3 Coupling to vectorIn supermultiplet this case, the conservation equation ( order to avoid unnecessary complexity we will turn off interactions with integer superspin supermultipletsTherefore, requ in this paper weger will superspin focus supermultiplets our ( efforts in construc other. fact that the two formulations of half-integer superspin su then the hatted superfields equation consider only the effects of tions. However, in order towe get will some not intuition start and with understan the arbitrary spin case but from multiplets that satisfy conservation equation ( related via an improvement term from ( s the superfields satisfy exactly the same conservation equation. So, there i will convert ( JHEP05(2018)204 ¯ φ ¯ Φ = , + ϕ φ (3.8) (3.9) (3.7) (3.6) (3.4) (3.2) (3.3) (3.5) K . Due (3.10) 2 = J ¯ D G . A detailed φ ave a global ). This con- K = ϕϕ an define a new as a power series ( ¯ J (Φ) under which the K Φ J 1 − K = must depend on Φ . F iλ K J (Φ) Φ ) gives: p + d 6 f , φ R . 3.1 e vector supermultiplet. = → ϕϕ same form as before ( l, scalar supercurrent must be of the form φ ) = = 0 Φ le to express p,q ϕ J ( K A # . q ln ¯ Φ . p,q (Φ) = p,q ¯ A Φ q F ) and the supercurrent is = 0), we must have q (Φ) A φ ¯ X ¯ φ K 1 . q ¯ Φ = Φ 7 − + ¯ ¯ ϕ Φ ¯ Φ) ϕ ϕ K Φ ], the supercurrent p ϕ φ q F ( ⇒ K ( 2 K X K Φ ¯ iλ ϕ F 57 ¯ K D Φ e ϕ d " q = = ¯ = 2 – 7 – X = = 0 → (Φ) ¯ K ϕ D Z Φ p by definition has to be real therefore we must have p ϕ f J , K p K =0 p ¯ X φ ϕ Φ Φ J Φ + K exp = (Φ) 2 p φ p ¯ D F X = J

X is a function of the product ¯ # ] where the connection between the action of a real linear ϕ = Φ as found in [ = 64 1 K p,q δ J J A 2 q ¯ D ¯ Φ in presented. takes the form (Φ). However, ϕ q p J X f the global U(1) is realized as a shift symmetry p as follows: " Φ φ 2 ϕ ¯ p D P can be understood as the demand for the potential Kähler to h are a set of constants. The conservation equation ( 6= 0 which is the non-trivial case we are being interested, we c K (Φ) is a function of Φ. Hence, we conclude that p,q (Φ) = p f A F ) and the structure of (Φ) F 2.9 For the variable Equivalently, one can define another chiral superfield 7 6 discussion of this can be found in [ and the reality condition takes the simpler expression 0), the supercurrent This can be satisfied only if ¨he potentialKähler is a function of the sum For this new variable the on-shell equation of motion has the which can be gauged in order to generate interactions with th chiral superfield For to ( and the supercurrent multiplet has only one element, the rea where but crucially not in their derivatives. Hence we should be ab and the action of a chiral where where Furthermore, because it must hold on-shell ( straint in U(1) symmetry expressed by the phase shift of JHEP05(2018)204 ), ). 4.1 (4.7) (4.6) (4.1) (4.2) (4.5) (4.3) 2.19 (4.4a) (4.4b) ) is not Φ 2)] which K 4.1 / α 3 D , 2 ¯ D ¯ Φ y in the transverse K )Φ α κ ) defined by: D parameter can be used ˆ − ˙ T α , r β ¯ ¯ Φ ˙ D α . α ¯ r. In this case, there is an K Φ − ¯ ¯ ¯ Φ Φ Φ ¯ ˆ ¯ Φ Φ α J , γ ¯ ˙ K Φ K K α K K D α ˙ α α α ˙ α ng ansatz: ¯ supermultiplet (2 K α D 2 plet that generates interactions ˙ U ¯ α D D ¯ ¯ D . +(2 D 2 ) ˙ α ¯ α D α ¯ ¯ Φ ¯ h ΦD ¯ ¯ Φ Φ ΦD D ¯ ¯ D ˙ Φ ˙ ˙ α K α α ¯ Φ K Φ + K ˙ Λ, then the α ¯ ˙ ) D ¯ ¯ ¯ 2 α ¯ . Φ D D ΦD K a ¯ ¯ ¯ ΦD c Φ ¯ U ¯ ¯ Φ ˙ ( Φ Φ ) via conservation equation ( ¯ ¯ α ˙ D D ˙ a b c ˙ α α a c T α − − ¯ α K T α − D becomes h α ¯ α ˙ D − , − α − h − − D ˙ ˙ ¯ ( Λ D α α ¯ + Φ U Φ Φ Φ + ¯ 2 Φ d ∂ Φ Φ α Φ α α d ∂ 2 α ). This can also be extracted from ( − id Φ K ¯ K K α J J K D K K ¯ K D D ˙ + D ˙ ˙ + α ˙ α α K α α α α + U ˙ α α 4.1 r ¯ Φ – 8 – ¯ = ¯ Φ α D ¯ D ¯ Φ D D D D ¯ D Φ ˙ ˙ . α ˙ K α K α α ¯ α = K Φ ˙ Φ ¯ α ¯ Φ α ΦD ) ¯ Φ κ + D D ¯ D K α α 2 e Φ α ¯ Φ J D ˙ K α ˙ ˙ + 2D ˙ α ¯ U + D ΦD α D α Φ α 2 D ΦD ¯ Φ + α Φ α α ¯ 2 ¯ D D ) κ J T K α β γ α β γ ¯ α h D + δ ∂ + + e + ¯ δ ∂ + + + ΦD α = = ˙ + α α D ˙ = = α a = ˆ ¯ ¯ T D ΦD + α ˙ ˙ ˆ α (2 α id c c T J α iδ α ) to determine the consequences of conservation equation ( + − − + J J such that the supercurrent multiplet ( 4.3 = ( α ). Hence the ansatz for T U α 4.2 D 2 ) and ( ¯ D in ( 4.6 − b ˙ α α J ˙ α ) and the demand that the hat supercurrent and supertrace sta ¯ D 0 = 2.20a The result is: ( unique. There are improvementarbitrary terms superfield that one should conside satisfies the conservation equation 4.2 Conservation equation Now we use ( Next, we attempt the construction ofwith the supercurrent non-minimal multi supergravity [transverse formulation of 4 Coupling to (non-minimal) supergravity For the supercurrent and supertrace we consider the followi satisfies the same conservation equation ( where Λ is the prepotential of chiral superfield Φ = formulation. If we select to eliminate 4.1 Improvement terms The definition of the supercurrent multiplet ( JHEP05(2018)204 (4.9) (4.8) (4.11) (4.10) (4.13) (4.14) (4.16) (4.17) (4.15) , Φ , . K α D = 0 = 0 ˙ α ¯ ¯ Φ Φ ¯ D ¯ Φ Φ . Φ ] K Φ , ¯ Φ K ¯ ¯ ¯ γ Φ they are independent, we Φ Φ there exist a supercurrent ¯ ΦΦ Φ ∗ + ΦΦ ΦΦ K K ∗ iβ Φ ) K K γ ial K + ¯ ¯ an be satisfied only if ˙ Φ Φ )Φ α . ˙ ˙ − α α γ . For this it will be useful to take ¯ D . Φ ˙ ΦΦ ¯ ¯ α β D D α , β , − K α K ] (4.12) Φ Φ Φ J β = 0 + + ( α α Φ K ΦD are: δ ΦΦ ¯ γ )Φ ˙ Φ α D + ( 2 β K iγ β ¯ D = T ¯ ΦΦ − + D Φ . ] − − + + Φ Φ γ , ¯ K ¯ Φ Φ Φ α γ Φ α Φ ¯ Φ K ΦΦ Φ Φ β , = and iγ – 9 – Φ K − K K K − K K α ˙ ˙ α α − )Φ K + δ + (2 ¯ Φ α iδ Φ Φ γ = D = , , , ¯ , β D γ is arbitrary, meaning it does not have special prop- ˙ ˙ Φ ¯ ¯ α α Φ J Φ[ ∗ − ˙ K α ˙ ˙ − α α α α K T iβ Φ α K α ) ¯ ¯ α β = 2 = 0 = 0 = 0 = 0 = D i∂ i∂ D δ J β ( ) in the following manner = D ¯ c e Φ[ d a κ h Φ = = = ˙ + − α α Φ[ α α 4.9 α ˙ ¯ γ Φ Φ Φ α Φ ∂ 2 α + Φ K K K + +D ∂ ˙ ˙ α α α Φ K − ¯ ¯ D D D K = ) α ˙ α α α ˙ α Φ ¯ − D D ˙ α α − α J iδ ∗ as expressed above demands: − α δ ∂ ˙ α α = = ( J ˙ α T α J ) )( . . 1 2 which can be used to re-write ( and the expressions for the conserved into account the following expressions The last condition we must impose is the reality of 4.3 Reality condition erties that relate someconclude of that the the terms coefficient above of with each each term other must vanish: and Assuming that the potential Kähler The reality of For an arbitrary potential, Kähler the above constraints c multiplet Therefore, we conclude that for an arbitrary potent Kähler JHEP05(2018)204 ) (5.1) (5.2) 4.4a (4.18) ( , 2) with . ˆ T ¯ / Φ ¯ ¯ ¯ Φ ¯ ¯ Φ Φ Φ Φ ]. K K K K K K + 1 D ¯ 63 ¯ D D D ]. However, it 1) ¯ ¯ DD D 2) DD D , s − − 1) 63 1) p 2) 2) p , − − − − − − + 1 p s is not zero and for p p p ) and we write the s ( ( 57 − − − − s [ ∂ s s s s ) to the longitudinal T ∂ ( ( ( ( ∂ ¯ ∂ ∂ ∂ Φ 2.14 ¯ Φ ( and non-minimal super- ¯ D ¯ ¯ ¯ ¯ ΦΦ), which includes the ¯ ¯ Φ Φ Φ D Φ ¯ ) Φ ) ( ) ) ) ) 2.20a p K nce on the workings of our p p p p p ( tiplet ( K ( ( ( ( ( K that any theory can be cou- ∂ ∂ ∂ ∂ ∂ ∂ one. This is possible only if ¯ Φ p = p canonical p p p p a he property 1) b ntial c g s f over the results of [ ) via ( 1 − is 1 1 K 8 2 2 2 s =0 − =0 ( − =0 − =0 =0 − − =0 − s p X s p X s p nd their symmetrization when necessary. X s s p p s p X X X the supertrace T} 4.17 − − , − h ∂ + + + ˙ ) give the supercurrent multiplet that K ) between a supersymmetric nonlinear α minimal ¯ ΦΦ, Φ Φ Φ Φ Φ Φ g α + K K K K K K Φ spacetime derivatives. ¯ {J ¯ 4.17 Φ D ¯ D ¯ D D is a K p ) and ( ¯ Φ ) such that the improved supertrace K 1) ¯ ¯ D DD 2) DD D } = 0 − K Φ Φ − 1) 1) 2) 2) 0 p p 4.5 , − 1) − ˆ 4.16 − − − ¯ T – 10 – Φ ˙ − p p s p p α − ) s ( ) and ( s − s α ( − − − ( ( s ˆ K ∼ s ∂ s s ∂ ( ( J ( ( { ∂ ∂ ∂ ∂ 4.16 d ∂ κ ∂ DΦ Φ Φ Φ Φ DΦ ) ) ) + ) ) ) p p p + p p p ( ( ( ( ( ( Φ ∂ ∂ ∂ ∂ ∂ ∂ K K p p p p p p 1) α γ β ℓ ζ ξ Φ ) − 1 1 1 2 2 2 s s ( ( =0 =0 =0 =0 =0 =0 − − − − − − s s s s s s p p p p p p X X X X X X + + + + δ ∂ + + e ∂ = = is an abbreviation for a string of ) s 1) ) ( p − ( α s ) ˙ ∂ ( s ( α α 1) ˙ J − s ( α T 2. For this case the conservation equation we must satisfy is For simplicity, we omit to write explicitly the free indices a 8 ≥ then there is an improvement term of type ( the potential Kähler is a function of the product free theory. Furthermore, by converting ( model of a single chiral superfield described by pote Kähler 5 Coupling to higherBased spin on the supermultiplets two previousmethod lower spin and examples it we is build time confide to generalize itfollowing to ansatz for higher the spin supercurrent supermul and the supertrace: formulation of supergravity (minimal supergravity) we rec is straight forward to see that if the potential Kähler has t will be zero generates the linearized interaction (first order in Also the symbol gravity. Additionally, we observe thatthat for case arbitrary the supercurrent multiplet defined by pled to supergravity. Expressions, ( Conceptually, this result was to be expected because we know and the new supercurrent multiplet s JHEP05(2018)204 (5.5) (5.3) (5.4) (5.6a) (5.6b) ¯ Φ , 2 as well. K ≥ 1)) . ) − s ¯ DD s s ( ( 2) α α this equivalence terms. Also, the ) ˙ − 1) s 1) ˙ ws: p ( p er spin supertrace . 1) − − α ξ − s s − ¯ s ( Φ Φ ( s U ( α ( α 2 K K ∂ constrained superfield α ¯ terms and additionally . This means that we U ravity case, and all it D 2)) ˙ s 1) ˙ ¯ s p 1) Φ ˙ 1) − α ) − s α s − p ( ˙ s − ( ¯ ¯ ¯ s D DD DD ( ( s (1) ( α t identically vanish in both ¯ ) also hold for ( D α reduce the unknown param- ∂ α α P T ! + and p 1 1) ˙ 1 1) ˙ s ¯ ¯ g − D + D + − p 4.4a − s 1) s 2 g s ( α D D − − ( ( (3) =0 − can be expanded in the following α s (3) α ) s p X ( s D P α ( are not uniquely defined but up to 2) 2) 1) ) ˙ 2 ¯ + α . Φ s ,P − − − ¯ ( Φ ¯ ¯ 1)! p p s Φ D 1) K Φ Φ 1) ( α 1)) ˙ − − K 1 − K α − K K s s − − U s + ¯ coefficients. Therefore, we can ignore it Φ ( ( s ( s s 1) ˙ ( ¯ ¯ s ( Φ D ¯ ∂ ∂ α Φ α 1) p 1) ( α − α ∂ ¯ s D DD s 1) ˙ − 1) − ¯ D 1) ˙ ¯ ( U Φ Φ s s − + − 2) 2) ( ) ) + 2 α ( − s s p p s α ( − − ∂ T ( ¯ ( ( ( D Φ 1) p p – 11 – α (2) s s 1) ˙ and α ∂ ∂ + 1 − − − + T K α s − s s ( h ∂ s p ( s ( ( Φ 1 1 2 2 ( α Φ ,P D ∂ ∂ (2) α − − − − + K + ∂ 1) ˙ , ξ s s p p ! 1) and P 1 p Φ Φ − Φ s 2 − 1) ) ) Φ s ) s 2 2 ( p p K s ( − ( ( 2) 1) ) due to the D algebra. For + ( α =1 =1 − − s , g α ( s s p p ∂ ∂ X X − − α p +D ) Φ ) ˙ s s terms because they can be converted to α 2) ˙ s i i ( ( s p p ( 1) ( − 1) ˙ p ξ ζ 2.14 ∼ T α − ∂ − ∂ s α − ) ˙ ℓ ( − 2 2 s s (1) s J α ( ( 1) =0 =0 ( − − = = s s α p p X X P α − s J T κ, h, ζ K ( 2, the first term of + κ ∂ + α 1) = = 1) ˙ ≥ = − s ) − ( s 1) s s ( 1) ( ∂ − α α − s ) ˙ ( s s T ( ( α α α 1) ˙ ˆ 1) ˙ J − and define the improved supercurrent and supertrace as follo − s terms can be disregarded. Therefore, the ansatz for the high ( s ( α 1) p α ˆ T − s T s ( α ) ˙ s = 0). Moreover, both terms can be ignored because they can be converted to ( α p e 5.1 Improvement terms Now we consider various improvementeters terms in that the will above further expressions. The arguments that led to ( takes the form: all the an equivalence class. Thisleft is and due right hand to sides the of presence ( of terms tha ( f manner: U Thus, the improvement terms we have are parametrized by an un can immediately ignore the relation is the following: does is to redefine the Hence, this term is not independent any more, as in the superg for arbitrary superfields However, because JHEP05(2018)204 ). . Φ Φ 5.1 (5.7) (5.8) (5.9) K K (5.10) (5.11) ¯ ¯ D D 1) D . ¯ − Φ 1) ¯ Φ p − K − K p s D ( ¯ − terms in ( D s ∂ 1) ( p D − b ∂ supercurrent and . 1) p escribe the freedom in − − DΦ Φ s p ) ) ( = 0 p p − , . ( ( ∂ s ( ∂ ∂ ¯ ¯ Φ Φ ¯ ∂ Φ ∗ p ∗ p K K ¯ Φ ¯ D Φ ¯ Φ ¯ iρ iρ ) D ) Φ p K ΦΦ p ( ¯ D DD 2 2 ( K Φ ¯ D K ∂ =0 =0 ∂ 2) − − 2) ¯ D K s s p p X X p − − ¯ p Φ ¯ p p a D c ΦD + + ¯ D − − 1 1 i s s DΦ ¯ ¯ Φ Φ ( =0 − ( =0 − s p X s p X ∂ 1) ∂ ΦD + K K − 1) ¯ − Φ D ¯ − s ¯ Λ Λ ( − ¯ ¯ 1) Φ K DD Φ Φ s Φ D D ∂ ( ) ) − ¯ DΦ DΦ in order to eliminate the Φ 1) K K K p K p ∗ ∂ p ( ( ∂ − ¯ r ¯ p D − D ¯ p ∗ ∂ ∂ Λ s ρ ¯ Φ r ( 1) ¯ − D DD − p p D 2) s in the following way ∂ − K ( σ ρ 1) – 12 – 1) ¯ − p Φ 2) − 1) ∂ s − − ¯ 1) − − Φ ¯ − ( and ¯ Φ 2 2 K Φ p p s s s ) − ( ¯ ¯ ( ( =0 =0 − − D Φ − s − s r K i∂ s s ) ) p p ( X X s ( s ∂ ∂ ( p p ( α ( ( ¯ ΦD ) ˙ = ∂ + + r ∂ ∂ = ¯ s ∂ ∂ DD 9 d ∂ ¯ ( D DΦ 1) Φ α Φ p p = 1) ) ¯ ) 1) ) Φ + − p U p p − iρ iρ s − ( ( ( 1) s ( 1) Φ s ( ∂ ( ∂ ∂ α − − 2 2 , then α K s s p p p ( =0 =0 p − − ( 1) ˙ 1)) ˙ s s p p X X α β γ α r ∂ Φ − ) ˙ − , σ ) 1 1 1 s s s + + + r ∂ s p ( ( ( ( =0 =0 =0 − − − . α α α s s s p p p X X X 1) = Z U Z r, ρ − + + δ ∂ + s ) s terms do not participate in the result. That is because they d ( s α ( ( α p ) ˙ = α s σ ) ˙ D ( ) s α s ( ( U α α ) ˙ s ( − J α ) 2 there is some additional freedom in defining the higher spin J s ( α ≥ ) ˙ s s ( Notice that the α 9 ˆ J for some parameters As a result, we can select parameters the definition of Thus we get One can show that if we select supertrace. Let us consider the following quantity: For 5.2 Additional freedom It is straight forward to prove that JHEP05(2018)204 , 1 c (5.13) (5.14) (5.12) (5.15) , . , ¯ . Hence, Φ ¯ ¯ Φ 1) Φ 2 K c = 0 − K K s , ( p D ¯ D α , whereas the 2 1) iζ ) ¯ D DD s − 1)) ˙ ( − 1) p 2) − α − , ) ˙ − − − s 1 s s p ( p ( ( − term by α − − p α ∂ s ¯ s = 0 1 Z ( ( J iζ s − 2 ¯ ∂ ∂ Φ α s , . . . , s ce, we can enhance the − ( ˙ − ¯ by adding the following 1 a D s ¯ ¯ Φ Φ ) ¯ , 1 D ) can be set to zero by ) ) p iζ s p p ( − 1) l parts for the higher spin ( quation. If we expand these ( α p ∂ − ( 5.5 ∂ ¨he potentialKähler is arbitrary, − th the conservation equation. ∂ = 0 s iξ ( p D 2 spin supercurrent multiplet: p p α a c − g 3 + s 1) ˙ 2 1 c 2 =0 − − =0 iξ − =0 − s s p X s p + X s p X ( , p + α s − + 1 1) − term in ( + . T κ − s Φ Φ Φ Φ Φ h s ( = 0 − , K K K K K α p and similarly the ¯ ¯ p and D ¯ D 1) D 1)) ˙ γ iξ ¯ 3 Φ − 1) ¯ ) ¯ D DD DD D c − s s − s ( ( K s + + + 1 1) 1) 2) 2) p ( α α − s ) ˙ − α − − − ¯ – 13 – Φ p 1) ˙ s p s p p ) ( Z ( − − s − − − s α ( s s s ∂ s s s s 1 ( α ( + 1 + 1 ( ( ( J ( α − ∂ ∂ ∂ ∂ p p , s D d ∂ Z γ s 2 DΦ Φ Φ Φ Φ Φ ) 1 α ) ) + ) ) ( ˙ p p p c + p ) we get that the p p − ( K ( ( ( ( β ¯ +1 Φ D 1 ∂ ∂ p ∂ + ∂ ∂ Φ − K 5.3 2 β − p p s p p p c 1) 1) α β γ ξ ζ β Φ − − − ) 2 1 1 2 2 + p s s s , − ( ( ( =0 =0 =0 =0 =0 , . . . , s − − − − − ) s α s s s s s s p p p p p X X X X X p − ( 1) ˙ s s 1 α terms are identically zero they do not change p + + δ ∂ + + + κ ∂ − = 0 ) ˙ = 1 − ) can be set to zero by s s 3 s ( ( κ p c = = α 1 2 α 5.9 ) + − − s 1) s p p ( − α α α iδ α and ∼ T ∼ J s ) ˙ ( s 2 ) ( α s 1) c ( α 1) ˙ − α J s ) ˙ − ( term in ( s s ) ) ) ) ( α ( . . . . 1 α α 1) ˙ 3 4 2 1 T − J − s s ( α α term is not zero but does not contribute in the conservation e T 1 terms in a manner similar to ( c the Because the we should consider the following expressions for the higher equivalence class in the definition of It vanishes due to the symmetrization of the two DΦ terms. Hen terms we obtain the following system of conditions: 5.3 Conservation equation The above streamlined expressionssupercurrent and do supertrace and not are theAfter include ones a we any lengthy should calculation use trivia wi and assuming once again that the JHEP05(2018)204 , , , , 2 2 2 1 coef- − − − − (5.18) (5.17) (5.19) (5.16) δ , Φ K ) we realize , . . . , s , . . . , s , . . . , s , . . . , s ), the ¯ DD 4.9 1) 4.9 = 0 = 0 = 0 = 0 − p p p − s , p ( , , ∂ ) ) i = 1 case ( i ) Φ i Φ γ ) γ s i i p γ ( K i Φ 1) 1) ∂ urrents. In ( K 1) − − p ¯ ¯ DD , , , ( D − ( γ ( 2 2 2) 2 1) 1 1 − − 1 . Specifically we get the following p =0 − − =0 − p − − =0 p s p p i X X i − s =0 X − i s γ − X s s + ( ( . ∂ ∂ Φ Φ + 1) + 1) . and K + 1) Φ K s s , . . . , s , . . . , s ) , . . . , s 1 ¯ p s , ¯ D p D DΦ ( ) − D + ( D p ∂ + ( = 0 = 0 ( = 0 , , β + ( i 1) i 2) = 0 p p ∂ i 1 α β − ζ − – 14 – i α i 0 β p p p i 2 − − iζ − 1) α 1) =0 − s s 1) s p ( X 2 ( , . . . , s − − + , p ∂ , p − ( , p =0 ( − ∂ , p + ( s p X 1 Φ Φ = 0 Φ 1 ) p + ) − =0 =0 = 0 p = 0 , . . . , s = 0 i p i − s X K =0 X s ( + 1 ( i s Φ X p p p ∂ ∂ + 1) s − − Φ K ig ig − = 0 ig p i i p p 1) i ( β ξ Φ α α + + − − , p 0 i ) i α 1 2 s s i γ ( ( =0 =0 − − +1 1) 1) s s p p X X , p 1) − = 0 − − β s s − ( + δ ∂ κ ∂ + ( s , p + 1 + 1 + 1 p ( , − c s 1 = 0 s p 1 p 2 ) with the corresponding expression for the = = 2 ) − 1 − =0 =0 + = 0 i p ) i p − s X =0 X = 0 − i s 1) s X 0 p p p s p p ( ( ( 5.18 − − c c d β c a a a α ( s ) ˙ 1 1 ( s s 1 ( ( α − , − − p s α p 1) ˙ s s J s − 1) 1) α s s 1) = 0 ) ) ) ) ) ) ) ) ( − − ...... , , − α ( ( p 8 9 5 6 7 i s ( i i c T 10 11 12 = 0 ======0 p p p p δ d κ ξ ζ g a that there is a qualitative difference between the two superc and the conserved supercurrent multiplet we get is: If we compare ( These uniquely fix allsolutions: parameters except JHEP05(2018)204 ) ) ) s s ( ( α α ) ˙ ) ˙ s 5.17 s 1 ( ( (5.21) (5.22) (5.20) (D) α α p − J J s ), we find that make 0 . K 4.12 iβ ad the special ) with spinorial ¯ ¯ Φ Φ ¯ Φ tion equation that Φ ΦΦ ΦΦ 5.20 ¯ Φ K K K Φ ) p ¯ ¯ K ( Φ Φ = 1, whereas it could be ∂ ¯ ¯ D D ity, where all coefficients ¯ Φ s ¯ Φ ¯ thus we can examine them D ch for real supercurrents of ) rms in ( ¯ p ΦΦ does that. DΦ − s 1) = ) + DΦ ( − tinction is useful because terms ¯ Φ s ). The reason for that is exactly − ( ∂ ¯ Φ p Φ K α terms, as we did in ( ) ˙ Φ − ΦΦ 1 K s s ( ( =1 − K 5.16 K Φ (D) s p X α ∂ 0 ¯ Φ J Φ + iβ i∂ i∂ + . To simplify this step we split without spinorial derivatives and DΦ ) ) p ) s p ) by making the corresponding term vanish, s ( 1 ( ( ¯ 1) , γ 1) Φ α ∂ ) α p p ) ˙ − − − ) ˙ s s p p ( p s ( D) s 5.16 ✚ ( ( ¯ , β – 15 – Φ − α − ∂ α α p s s Φ ΦΦ ( ( J 2 0 α + J 2 the same coefficient is fixed and given by ( ∂ K ∂ K =0 − term was independent in iγ s p X = p p Φ ≥ Φ ) K ) ) s + iγ iβ e ΦΦ p ( p s ( ( α Φ K ) ˙ ∂ ∂ ΦΦ s K . It will be interesting to consider potentials Φ ( p p K ¯ Φ Φ α Φ ) ) ) 0 β γ Φ p p p J ) ( K 1 1 iγ s ). The − − ∂ ( =0 =0 − − s s D s s p p ( ( X X + 2 5.3 Φ ∂ ∂ ¯ ), we can write Φ ) D δ ∂ + + p K Φ Φ − ) ) Φ = s p p δ ( 4.11 ( ( 1) ) ∂ s ∂ ∂ − ( s α ( 1 1 1 ) ˙ Φ ∂ are the contributions to =1 s − ) 2. However, if the potential Kähler was not arbitrary, but h =1 =1 − − ( s p X s s s p p X X ) ( α s ≥ ( + + + ∂ J α s ) ˙ ). Using ( s = ( D) ✚ ( ) α ) are unconstrained. The corresponding term in the conserva s ( J 5.18 α ) ˙ s 5.18 ( D) ✚ ( α J type ( 5.4 Reality condition The last step in order to complete our construction is to sear into two pieces this term vanish identically. Notice that the free theory where derivatives we get: the conditions imposed on coefficients in ( controls this is which was the outcome ofwhat the was third mentioned (3) in condition ( in ( then by accident we will be in the same situation as supergrav ignored in property to remove the third condition in ( are the terms which includefrom spinorial one derivatives. piece This canseparately. dis not contribute Therefore, to if the we reality ignore of for the the other, moment all the te After distributing the derivatives and collecting similar ficient was not fixed, whereas for JHEP05(2018)204 (5.23) (5.24) , , 1 1 1 − − − q p 10 − , . s , . . . , s , . . . , s . 2 2 2 2 , , p − − β 1 = 1 = 1 1 ¯ Φ − − ivatives we get the constraints: can be written in the following q p q p ΦΦ , . . . , s , . . . , s ) s 1 − 2 2 − K ( to track the origin of these conditions. , , ) s s α p q − ¯ 1 ) ˙ Φ ( ¯ s Φ − s ( p ∂ p = 1 = 1 p (D) α ΦΦ ΦΦ ¯ iγ − Φ ¯ iβ Φ 0 J s K Φ . K ¯ α D α p K ¯ p Φ 1) β ΦΦ β 1 Φ p ¯ − Φ ¯ Φ q K α K Φ Φ − ) ) − q ¯ ¯ q p Φ Φ K ( q q K ( – 16 – ) ) − ¯ ¯ ∂ p D D ∂ p s ¯ , ( Φ gives the following conditions: + ( ( 1) ∂ ¯ ¯ Φ ¯ ∂ D ∂ p Φ Φ ) − K ) Φ q p 1) ¯ q DΦ Φ ¯ = 0 Φ q − − − − K ¯ 1) , p D s 0 p ¯ p p D − ( − 0 p p β − − − 1) = 0, the terms in p ∂ s s s α − − Φ DΦ ( − ( ( ∗ ∗ − s s p ) = p s ∂ ∂ ∂ DΦ p γ β β ( , p , , , p , p ( − 0 s ∂ 1) γ − − ∂ Φ ( ¯ Φ Φ = ) − ) ∂ p p ¯ p Φ DΦ Φ DΦ 2 0 D = = = 0 = ( = 0 = 0 = 0 = 0 ( ) ) ) − − β p p p ∂ 1) p p ∂ s δ 0 1 p p p ( ( ( =1 ( γ β − q β X − α α − 1 ∂ ∂ ∂ DΦ 1 1 ∂ s s s ( − − β 2 2 2 2 2 p − ∂ =1 p 2 =1 ) for an arbitrary q X =1 =1 =1 =1 =1 − − − − − q − X =1 − − q p s s s s s p p p p p s X X X X X s s p X 2 2 − + + + + + + 5.22 =1 − =1 − s s p X s p X = DΦ + + p ) s β ( α ) ˙ s ( (D) α J We have underlined the relevant terms in order for the reader 10 The reality of ( This is because by using manner: Doing the same for the reality of the terms with spinorial der JHEP05(2018)204 = s take K (5.25) (5.26) ¯ Φ 1 ]. ΦΦ accidents K − 57 ¯ ercurrents. After Φ 1 , q p r arbitrary Kähler 1 − ΦΦ 56 − ) q p K s − e is a special Kähler q ( 1 q p − then maybe the higher ∂ p can imagine that if the − is: ercurrent all coefficients s s β he free theory the term − ¯ (Φ) in order to study its Φ s β s ¯ p r supermultiplet, where the D K W ¯ Φ α 1) p en that is possible. We must ¯ Φ − β ΦΦ . r ¯ D ¯ Φ s and conservation equation. Ad- K − Φ ) q q ¯ − ( K Φ p p ) 1 ∂ ¯ q D β ) vanishes, hence there is no incom- − ( s Φ DΦ − ¯ 1) ( ∂ Φ 1) − q ∂ = 0) and the on-shell equation of motion ¯ ¯ Φ . D q ¯ 5.17 Φ − r s − − ¯ ( D ΦΦ W p W p ∂ ˙ DΦ − = 0 α 1) DΦ K in ( ) ) and examining the terms responsible for the s r ( ¯ − − D ¯ higher spin symmetry 1) + 1) Φ become relevant now. All these ( q δ ∂ ) becomes identically zero for the free theory. So – 17 – s − − ∂ − 1 q p s ΦΦ which holds true only for the free theory p 5.25 0 − − − Φ − s K p s α ) s DΦ 5.10 ( p ( α ) 1 − ( ( ∂ s q ∂ ( ( ∂ ¯ − Φ 1) ¯ ∂ ∂ 2 Φ Φ q − p − ¯ s D DΦ Φ Φ q K ( ) ) ) − constant α − =1 p p p 1) r p X ( ( ( s 1) ˙ − − ∂ ∂ ∂ = s − s ( s p . 2 2 2 2 ( ∂ Φ DΦ ¯ Φ β Φ α ) − − − − s Φ p is a realization of p p p p ¯ =1 =1 =1 =1 Φ we find that a necessary condition for Z ( q q q q W X X X X − − − − K ¯ ¯ ∂ s Φ D p s s s s is a chiral superfield ( = 2 2 2 2 K α 2 ΦΦ =1 =1 =1 =1 − − − − =1 − W Φ s s s s ) is true and for that case we recover the results of [ p p p p X X X X s p X leading to the fixing of K + + + + + ×K +DΦ and 2 Φ ¯ 5.26 p D such that we can construct non-trivial real, higher spin sup K β s . D 2 must vanish. Therefore, there is no non-trivial solution fo K K ¯ D p Φ , γ p However, an interesting question one can ask is whether ther 1) − , β s p ( ¯ spin supercurrent exist. Going back to ( all, we have seen potentialKähler this behavior must in have the aspecial coupling U(1) property with the symmetry. of vecto Therefore, one potential place only if ( potential 6 Turn on theIn chiral this superpotential section,contribution we to turn the back higherkeep on in spin mind the supercurrent that chiral multiplet, wh superpotential patibility between the non-trivial valuesditionally, of the the quantity parameter the parameters we removed, such as vanishing of The conclusion isα that, in order to get a real, higher spin sup now becomes This is equivalent to ΦΦ. Furthermore,∂ this condition is consistent because for t JHEP05(2018)204 ed (6.6) (6.1) (6.9) (6.8) (6.7) (6.3) (6.5) (6.4) F Λ α D ˙ α ¯ D = 0 (6.2) 4 , c ¯ Φ ¯ F ¯ + W 2 ¯ Λ which appears in the F ˙ , ¯ α D α ) Φ ¯ ¯ Λ D ¯ D F α , , . ˙ ∗ W α ¯ c Λ D ¯ ( D Λ. It is straight forward to ∗ 4 α + = 1 = 2 = 0 derstood as the fact that the 2 c . its Taylor expansion does not Λ D ¯ ¯ Φ 3 1 2 4 D ˙ ion terms α 3 ¯ Φ ¯ c W ¯ D ¯ ¯ ˙ β F − W + ˙ α ¯ ¯ Λ D ¯ ) + D ¯ F Λ ∗ . . ˙ α c α β F Φ ¯ D + F (Λ (Φ) ∗ ¯ ˙ ΛD α ΛD Φ c ˙ F ) can be generalized to include an arbitrary α ¯ . ∗ 3 D W + Φ c ¯ + ) that generates interactions with the super- α D ¯ F Λ ¯ Φ D 2 F ¯ 4.17 Λ – 18 – c c ¯ ¯ 4.17 W ) we get a non-trivial solution − J F − = + + ˙ ¯ . Hence, the conclusion is that in the presence of α Λ ), ( + 2 Φ (Φ) = Φ ). Additionally, it relates c Φ 2 Φ ¯ 4.1 D ), ( F K ¯ F D ˙ W K W α 2.7 ¯ 4.16 Λ Λ ∗ ¯ . D ˙ α c α 4.16 , d + Λ ¯ ) and we modify it in order to include the chiral superpo- ¯ D = Φ F D 1 Φ , d , c + α ¯ α ∗ 2 Λ − Φ 3.8 J c D , there can be no conserved supercurrent no matter what the −K 2 in the following manner: = = 1 = 0 d W 1 (Φ) is a chiral superfield and a holomorphic function of Φ defin W c = D = 1 2 3 F ¯ + c c c F F − ˙ )Φ α + T c α F ¯ = Λ Φ J Λ α K F 1 ˙ D α ˙ d α ¯ D ¯ D + = (1 + breaks the global U(1) symmetry of section Φ ∗ 1 c α J W 2 − −K ¯ D = D = ˙ α T α J Imposing the conservation equation ( where Λ is the prepotential of the chiral superfield, Φ = equation of motion with ¨he potentialKähler is, even for thepresence free of theory. This can be un prove that the conservation equation of can not be satisfiedany for chiral superpotential any value of chiral superpotential gravity supermultiplet we consider the following modificat 6.2 For supergravity For the supercurrent multiplet ( tential information while preserving its reality: We start with supercurrent ( 6.1 For vector supermultiplet This definition holds forinclude any the chiral constant superpotential term because due to ( as Thus, the supercurrent multiplet ( In the above terms, JHEP05(2018)204 (6.11) (6.12) (6.13) (6.14f) (6.14e) (6.14c) (6.14a) (6.14g) (6.14d) (6.14b) (6.10b) minimal , must take ¯ Φ , , . ¯ , 1 1 1 W W Φ , ). Hence, the ¯ ¯ Φ s D 1 − − − W ent multiplet is ¯ D 1) Φ . D − − 1) p W 1) e of − − − p 2) = 0 s generated terms are: p − ( , . . . , s , . . . , s , . . . , s − s − p 1 1 1 1) ∂ ( ], there is the s , , , , . . . , s ( − W − ∂ ¯ Λ dependent terms gives: s ¯ s Λ ∂ 57 ( ( α ∂ , D he = 0 = 0 = 0 = 1 ) ¯ Φ (6.10a) DΦ p ¯ 1)) ˙ p p ) 56 ) DΛ ( p ) p − al superpotential ∂ ( p s − ( ( s ∂ ¯ ∗ p W DDΛ ( ∂ α , p ) δ ∂ T p p 1 p 1 ( s δ − =0 − ∂ α Φ + 1 − p s ( p 1 X ) γ p s p p =0 − D ( s p p σ X − 2 2 ∂ 2 ¯ ¯ s Φ D − + 2 =0 − s p ! ¯ s p X 1 s , s W Φ s p ¯ p Φ 1) + − W − ¯ − 1) p Φ ) W p 1) p – 19 – s γ − ( − − 1) W ¯ 1) ( s p ΦΦ. For that case [ 1 α ( − ) ˙ 1 − 1) p − s ∂ s =0 − − ( ( = − ( − s W p X α p s p ¯ ∂ Λ ( s s s =0 − J 2 ) K p X ∂ s ), the cancellation of the Λ and s i ( ˙ s α − ¯ DD ¯ ∂ − ) ¯ Λ DΛ ) ¯ , p i ( ) D , s p p 6.13 1 D ( Λ 2 ( − ic , ) ) + − ( ∂ − − p ∂ p p + c ( ) s ( ∗ s p γ p γ ∂ ∂ ( +1 ξ γ ) in ( p = = 0 s 1 α p , p p ) ˙ 1 1 γ ) s ζ γ s 0 − =0 =0 ( − − s 1) − free γ 1 1 ( s s p p X X α s − 6.12 1 α =0 =0 − − 1 s s ) ˙ J + 1 s s p p ( X + X − − s s + s , s ( α ˙ − p ), ( α free γ α ∗ p p = = 1) ˙ s ¯ D 1 γ γ J = − ) s , s 1) s − 6.11 ( 1 ( − free 0 − , α we showed that the construction of the higher spin supercurr s α s 1 − + 1 + 1 s s γ ) ˙ p p T ( γ s 5 s s p 1 ( γ α s 1 W = iσ α 1) ˙ is a real proportionality constant (it may depend on the valu − − − − 2 J − = + c s − = = = = ( p s 0 W α p p p 0 δ iσ ξ iσ ζ iσ ζ T and the conservation equation they must satisfy is place in the same configuration. The most general ansatz for t possible only for the free theory, Substituting ( In section multiplet 6.3 For higher spin supermultiplet where consideration of contributions due to the presence of a chir JHEP05(2018)204 ) Φ + 1) W 6.17 l (6.19) (6.20) (6.17) (6.15) (6.18) 1) , (6.16c) (6.16a) (6.16b) − y chiral ¯ p Λ = 2 . − s s 2 ( ¯ ( DD Φ ∂ rrent and su- − ) 1) s W − s D p ( DΦ ) will lead to the γ ) 1) ∂ p ( − + s , . . . , s +1 ∂ ( 6.15 s 1 ) ∂ , will vanish and we get ) we reach to a differ- +1 i ) p Φ − Φ γ ent with each other. For = 0 p tion ( dd values of W γ 6.16c 0 ¯ DDΛ γ +1)( + parameters. Of course, ( s 2) + 1 s e constraints − ( − p p , p +1 s γ 1 ( p ¯ 2 fc γ ∂ − − s s ond exception exist and corresponds p − ) ) p i i − ( + 1) + 1 − − ¯ 1 s DΛ ( s p + 1 s p c ( . . D − s 2 p 2 2 2 Φ 1) s Φ ) reveals two exceptions exist for some special f − − s p + ]. The cancellation of the Φ dependent terms s p . s ( fc p ∼ Φ ¯ = Λ 57 ∂ – 20 – s 6.15 s + 1 1) − W s +1 + 1 − s Λ+ p W s ) are no longer independent and they can be com- W s + 1) ) s 2 ( 1) i s ∂ DΦ − ( p , s s p p 6.15 1) ( +1 ) with derivatives acting on 1) s − ∂ 1) s ) +1)( − ( i ( − + 1) s +1 , ∂ ( s s ( 6.15 . − + 1 s . 1 ) ( s +1 ( i s s +1 − 2 fc Φ ) s s and using the recursive equation ( 1 i ) ) +1 p + i i s W − 0 ) − s s ) 1) s ( γ +1) i − − D +1)( γ ( ( ( ( c s − s α 2) c c c ( ( ) ˙ ( − ). The corresponding supercurrent multiplet is: ( − s ( p ¯ = = = 1 free +1 fc fc α − than the one required. The conclusion is that for an arbitrar s . The first one is for s ) 0 1 p s , there is no higher spin supercurrent multiplet. − J − − ( i 1 γ + 1) γ − 6.16a ∂ p s + 1 − W W − s = = s ] were the mass term contributions to the higher spin supercu γ ( + ( p Φ c − γ ) s 1) 57 ( s +1 ( +1 − s α p +1) s 2 2 ) ) ˙ ( p γ i s =0 =0 ( − − ( α ( s s p p X X ∂ c α 1) ˙ J + + × − s Nevertheless, careful examination of ( ( α T 0 = superpotential corresponds to the massanalysis term of of [ the chiralpertrace superfield were and calculated. equa The main result was that only the o superpotential bined, resulting to a different set of constraints for the ent value of gives: and it exists for all values of In this case, the various terms of ( Therefore, for an arbitrary chiral superpotential we get th These are exactly the conditions found in [ will lead to consistent, interactions.to Additionally, a a sec linear superpotential only condition ( In that case all the terms of ( One can easily checkexample, that starting the with above equations are not consist JHEP05(2018)204 K can 1) − s ( W α 1) ˙ (Φ). The states and − t multiplet and s W ( and a higher α K ¯ ) l there are no ing of both T out s tion is that we s ( ˙ α α ) ˙ ¯ D s , ( ) and s the poten- Kähler ¯ -matrix. We consid- ΦΦ)] and the chiral s ) e gauging procedure α ( ( S ¯ J α F in K y our approach because ¯ 1) ˙ Λ ultiplet. ( − = s α . rds, one can not construct ( em. We have also assumed D α 1) K gher spin supermultiplets of ˙ α χ − of free ¯ ly define s D rfield Φ with its dynamics been + ¯ ( er spin gauge fields with matter α 1) ¯ ΦΦ [ ) + − 1)) ˙ s − ( F s α ( α Φ (Λ 1) ˙ = 0. The constraints in ˙ T α − K s s ¯ , ( D J α ( 2 α α ¯ ¯ F Φ) and a chiral superpotential T ¯ D D = Φ D , s ¯ Λ 2 α ¯ − J D (Φ D ! – 21 – + 2 1 Φ s K 1) K − F ˙ s = α ( ) ¯ α D s ) ˙ . Furthermore, this result is consistent with the results ( + Λ s ( α Φ (Φ) = 0]. For these cases, the supercurrent is F ) ˙ α allows us to parametrize some very complicated, strong α s ( χ W −K α + = Φ W J ) = = D s s ˙ α ( ˙ 1) α W T α ¯ α − D ) ˙ and s s J ( ( we can construct the following supercurrent multiplet ) defined by a real higher spin supercurrent α α K 1) ˙ 1) J W ) − − Λ and s s s ( 2 2), if they exist, are generated by a higher spin supercurren ( ( α / ¯ α α D ) ˙ s and T 1) ˙ ( 2 (higher spin supermultiplets) there is a severe constrain α − + 1 K s . For almost any potential Kähler and chiral superpotentia ( = 1 (non-minimal supergravity supermultiplet) the expecta ≥ H α = 0 (vector supermultiplet) there are no interactions unles , s ]. W s s s ,T z 63 ) 8 + 1 s ( d s α be understood as thewhich global will U(1) generate symmetry the interactions requirement with for the th vector superm and satisfies the conservation equation superpotential vanishes [ tial can be written as a function of the product and interactions with higher spin supermultiplets. In other wo where Φ = for any should always be able to find interactions. This was verified b of [ ) ˙ Z s ) For ) For ) For ( ∼ i α ii ( 1 iii ( J ( S spin supertrace ( and on-shell satisfy the conservation equation thus evading the consequencesthat of the the first Coleman-Mandula theor ordertype interactions ( of the matter theory with hi ered an interacting matter theory describeddescribed by an a chiral arbitrary supe potential Kähler In this paper, we havetheory investigated fields the independently interactions from of the high ability to have a proper 7 Summary and discussion The results we find are: arbitrariness of both interactions which can disrupt the conventional definition JHEP05(2018)204 , ¯ Λ ¯ , DD ]. 1) − ommons s ( ]. For ex- . However ∂ 6 1) SPIRE , − +1 IN 5 s [ s ( ) on (2) α i ge conservation 1) ˙ − − ¯ DDΛ s eorems such as the ( , chapter 24, , 2) α − T stry of Education and ech Republic. K. K. is + 1)( s ( s , section 13.1, Cambridge let have been constructed ∂ ( ions. (1964) B1049 s ¯ ]. fc ty. The work of K. K. was ) R grant, project No. 18-02- ed by the endowment of the redited. i ( − 135 Supergravity and the S matrix 1 SPIRE ¯ DΛ − s IN 2 D s 1) and supertrace . ]. − ) s ¯ s fc Λ ( ( ]. ∂ Phys. Rev. 1) α ) ˙ , − s s SPIRE ( Λ + All possible generators of supersymmetries of the ( +1 α s IN ∂ 1) SPIRE ) J i − – 22 – IN s [ ( +1 s ∂ ]. ) ]. i + 1)( : free, massive, chiral superfield − +1 ] and its extensions to supersymmetry [ s s ( ( ) 4 2 i All possible symmetries of the S matrix 2 SPIRE SPIRE Φ fc Φ : free, chiral superfield with linear superpotential ), which permits any use, distribution and reproduction in IN IN [ f m [ (1975) 257 + s + 1) ) + 1)( = =0 : free, massless, chiral superfield = s s ( s ( ( α ) ˙ W W W B 88 s ( ¯ fc free fc α , , , J − − CC-BY 4.0 (1977) 996 (1967) 1251 = = ¯ ¯ ¯ ΦΦ ΦΦ ΦΦ This article is distributed under the terms of the Creative C ) Photons and gravitons in S matrix theory: derivation of char The quantum theory of fields. Volume III: supersymmetry The quantum theory of fields. Volume I: foundations s 1) ( = = = − D 15 159 α ]. To this list we add the supercurrent multiplet for excepti s ) ˙ Nucl. Phys. ( K K K s ( , α 58 α – 1) ˙ ) ) ) J − . . . 56 s 2 1 3 ( α T there are three exceptions: ceptions (1) and (3) thein corresponding [ supercurrent multip a non-trivial higher spin supercurrent These exceptions areColeman-Mandula consistent theorem with [ various known no-go th and equality of gravitational and inertial mass Cambridge University Press, Cambridge U.K., (2000) [ University Press, Cambridge U.K., (1995) [ Phys. Rev. S matrix Phys. Rev. [1] S. Weinberg, [5] R. Haag, J.T. Lopuszanski and M. Sohnius, [6] S. Weinberg, [2] S. Weinberg, [3] M.T. Grisaru, H.N. Pendleton and P. van Nieuwenhuizen, [4] S.R. Coleman and J. Mandula, supported by the grantthankful P201/12/G028 to of Rikard the von Unge Grant and agency Linus ofOpen Wulff Cz for Access. useful discuss Acknowledgments The research of I.Science, L. project B. No. was00153 3.1386.2017. supported for He partial in is support. parts alsoFord The by grateful Foundation research Professorship to Russian of RFB of Mini S. Physics J. G. at is Brown support Universi Attribution License ( any medium, provided the original author(s) and source areReferences c JHEP05(2018)204 , ]. n ons , , , in SPIRE gauge IN [ (1979) 161 2 , / , Yaroslavl 5 ) (1993) 39 massless fields (2008) 065016 3 ]. B 86 2012 in flat space ymmetry massless f the Poincaré , ]. flat space D 78 B 309 On spin (1995) 4660 SPIRE (2012) 987 fields and higher helicity IN and spin- 3 ][ SPIRE 84 2 Phys. Lett. ]. IN D 52 [ huizen, , ]. Phys. Rev. , Phys. Lett. , , ns SPIRE IN SPIRE [ ]. IN gauge field couplings Phys. Rev. ][ (1993) L39 ]. ]. 2 , ]. SPIRE Rev. Mod. 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