Cubic interaction vertices for N=1 arbitrary spin massless supermultiplets in flat space

In the framework of light-cone gauge formulation, massless arbitrary spin N=1 supermultiplets in four-dimensional flat space are considered. We study both the integer (super)spin and half-integer (super)spin supermultiplets. For such supermultiplets, formulation in terms of unconstrained light-cone gauge superfields defined on chiral momentum superspace is used. Superfield representation for all cubic interaction vertices of the supermultiplets is obtained. Representation of the cubic vertices in terms of component fields is derived. Realization of relativistic symmetries of N=1 Poincare superalgebra on space of interacting superfields is also found.


Introduction
The light-cone gauge approach [1] offers considerable simplifications of approaches to many problems of quantum field theory and superstring. This approach hides Lorentz symmetries but eventually turns out to be effective. Exploring this approach for analysis of ultraviolet finiteness of N = 4 Yang-Mills theory may be found in Refs. [2,3]. Light-cone gauge superstring field theories are studied in Refs. [4], while string bit models for superstring and super p-branes in the framework of light-cone gauge formulation are considered in Ref. [5] and [6] respectively. Application of light-cone gauge formalism for studying the interacting continuous-spin fields in flat space may be found in Refs. [7,8]. Various applications of light-cone gauge approach to field theory like QCD are discussed in [9]. Methods for building Lorentz covariant formulation by using light-cone gauge formulation are discussed in Ref. [10]. In the framework of light-cone gauge approach, study of free continuous-spin field in AdS space may be found in Refs. [11,12].
One interesting application of light-cone gauge approach is a higher-spin massless field theory. In Refs. [13,14], a wide class of cubic interaction vertices for higher-spin massless fields in 4d flat space was constructed, while, in Ref. [15], the full list of cubic interaction vertices for arbitrary spin massless fields in 4d flat space was obtained. Extension of results in Ref. [15] to the case of massless and massive arbitrary spin bosonic and fermionic fields in R d−1,1 , d-arbitrary, may be found in Ref. [16,17]. Our aim in this paper is to provide the full list of cubic vertices for N = 1 integer and half-integer spin supermultiplets in the flat 4d space. Doing so, we provide, among other things, the supersymmetric extension for all cubic interaction vertices for massless fields in the 4d flat space presented in Ref. [15]. 1 To this end we use superfields defined on a light-cone momentum superspace. The light-cone momentum superspace has successfully been used in many interesting studies of superstring and supergravity theories. For example, we mention the use of the momentum superspace in superstring field theories in Refs. [4] and 10d extended supergravity in Ref. [18]. 2 The momentum superspace was also adapted for the studying light-cone gauge 11d supergravity in Ref. [20]. Using the momentum superspace, we collect the N = 1 integer and half-integer spin massless supermultiplets into a suitable unconstrained superfields and use such superfields for building cubic interaction vertices. It is the use of the light-cone gauge unconstrained superfields that allows us to build a simple representation for all cubic interaction vertices of the N = 1 integer and half-integer spin massless supermultiplets and provide the complete classification for such vertices.
This paper is organized as follows.
In Sec.2, we review the well known description of N = 1 integer spin and half-integer spin supermultiplets in terms of light-cone gauge components fields. We introduce the field content and describe a realization of the Poincaré superalgebra on space of the component fields.
In Sec.3, we introduce a chiral momentum superspace and describe light-cone gauge unconstrained superfields defined on such superspace. Also we describe a realization of the Poincaré superalgebra on space of our light-cone gauge unconstrained superfields.
In Sec.4, we describe a general structure of n-point interaction vertices. Namely, we present restrictions imposed by kinematical symmetries of the Poincaré superalgebra on n-point interaction vertices.
In Sec.5, we study cubic vertices. First, we present restrictions imposed by kinematical and 1 In the framework of light-cone superspace formalism, a scalar superfield that describes arbitrary N -extended supermultiplets and involves fields with helicities − 1 4 N ≤ λ ≤ 1 4 N ( 1 4 N -integer) was studied in Ref. [14]. For such scalar superfield, a cubic vertex that involves 1 4 N derivatives was obtained in Ref. [14]. 2 Recent interesting discussion of 10d Yang-Mills theory in light-cone superspace may be found in Ref. [19]. dynamical symmetries of the Poincaré superalgebra on cubic vertices. Second, we formulate lightcone gauge dynamical principle and present complete system of equations required to determine the cubic vertices uniquely.
In Sec.6, we present our general solution for all cubic vertices which describe interactions of arbitrary spin massless supermultiplets. First, we present superfield form of the cubic vertices. After that, we discuss the cubic vertices in terms of the component fields. Sec.7 is devoted to our conclusions. In Appendix A, we describe our basic notation and conventions for Grassmann algebra we use in this paper. In Appendix B, we discuss properties of our unconstrained superfields. In appendix C, we present details of derivation of the cubic vertices.
2 Light-cone gauge formulation of free massless N = 1 supermultiplets Poincaré superalgebra in light-cone frame. A method suggested in Ref. [1] tells us that the problem of finding a light-cone gauge dynamical system amounts to a problem of finding a light cone gauge solution for commutation relations of a symmetry algebra. For supersymmetric theories in the flat space R 3,1 , basic symmetries are associated with the Poincaré superalgebra. Therefore, in this section, we review a realization of the Poincaré superalgebra on a space of massless supermultiplets and present well known formulation of free N = 1 supersymmetric multiplets in terms of the light-cone gauge component fields.
For the flat space R 3,1 , the Poincaré superalgebra is spanned by the four translation generators P µ , the six generators of the so(3, 1) Lorentz algebra J µν , and four Majorana supercharges Q α . We assume the following (anti)commutators: where η µν stands for the mostly positive flat metric tensor. In place of the Lorentz basis coordinates x µ , µ = 0, 1, 2, 3 we introduce the light-cone basis coordinates x ± , x R , x L defined as where the coordinate x + is considered as an evolution parameter. In the light-cone basis (2.3), the so(3, 1) Lorentz algebra vector X µ is decomposed as X + , X − , X R , X L , while a scalar product of the so(3, 1) Lorentz algebra vectors X µ and Y µ is decomposed as From (2.4), we learn that, in the light-cone basis, non-vanishing elements of the flat metric are given by η +− = η −+ = 1, η RL = η LR = 1. This implies that the covariant and contravariant components of vector X µ are related as In light-cone basis (2.3), generators of the Poincaré superalgebra are separated into two groups: P + , P R , P L , J +R , J +L , J +− , J RL , Q +R , Q +L , kinematical generators; (2.5) We recall that, for x + = 0, in a field realization, generators (2.5) are quadratic in fields 3 , while, generators (2.6) involve quadratic and higher order terms in fields.
In the light-cone basis, commutators of the Poincaré algebra are obtained from (2.1) simply by using the flat metric η µν which has non-vanishing elements η +− = η −+ = 1, η RL = η LR = 1. We now present the light-cone form of the (anti)commutators given in (2.2), Hermitian conjugation rules for the generators are assumed to be as follows In order to provide a field theoretical realization of generators of the Poincaré superalgebra on massless fields, we use a light-cone gauge formulation. To this end we start with a description of field content we use in this paper and review the well known light-cone gauge formulation of arbitrary spin massless fields. Field content. To discuss supersymmetric field theories we use light-cone gauge massless fields considered in helicity basis. Using a label λ to denote a helicity of a massless field, we introduce the following set of complex-valued massless fields: where fields (2.14),(2.15) depend on space time-coordinates x ± , x R,L (2.3). Fields (2.14),(2.15) satisfy the following hermitian conjugation rules We now collect fields (2.14),(2.15) into integer and half-integer spin supermultiplets given by spin-s supermultiplets, s = 1, 2, . . . , ∞; (2.17) Fields in (2.17)- (2.19) constitute the field content in our approach. In our field content, each helicity occurs twice. Our motivation for the use of such field content is discussed in Sec. 6.
In what follows, we prefer to use fields which obtained from the ones in (2.14)-(2.19) by using the Fourier transform with respect to the coordinates x − , x R , and x L , where the argument p of fields φ λ (x + , p), ψ λ (x + , p) stands for the momenta β, p R , p L . In terms of the fields φ λ (x + , p), ψ λ (x + , p) , the hermicity conditions (2.16),(2.20) take the form where in (2.22) and below dependence of the fields on the light-cone time x + is implicit. Field-theoretical realization of the Poincaré superalgebra. We now review a field theoretical realization of the Poincaré superalgebra on the space of massless supermultiplets. First, we consider even elements of the Poincaré superalgebra (2.1). Realizations of the Poincaré algebra (2.1) in terms of differential operators acting on the fields φ λ (p) and ψ λ (p) is given by the well known expressions.
Realizations on space of φ λ (p) and ψ λ (p) : where the notation for partial derivatives and the definition of symbol e λ are given by In (2.30), we present relations which follow from the definition of e λ given in (2.29).
Having presented realization of the Poincaré algebra in terms of differential operators in (2.23)-(2.27) we are able to provide a field representation for generators in (2.1). To quadratic order in fields, a field representation of the Poincaré algebra generators (2.1) is given by , where G diff denotes the differential operators presented in (2.23)-(2.27), while G [2] denotes the field representation for the generators (2.1). For the odd elements of the Poincaré superalgebra (supercharges Q ±R,L ), a field representation G [2] takes the form as in (2.31), where Q +R (s) The fields φ λ , ψ λ satisfy the Poisson-Dirac equal-time commutation relations Using relations given in (2.31)-(2.33), we verify the standard equal-time commutation relations between the fields and the even generators while using expressions in (2.34)-(2.37), we find the following equal-time (anti)commutation relations between the supercharges and the fields

Superfield formulation
In order to discuss a light-cone gauge superfield formulation we introduce a Grassmann-odd momentum denoted by p θ . Now we introduce a superspace parametrized by the light-cone time x + , the momenta p R , p L , β and the Grassmann momentum p θ , x + , p R , p L , β , p θ . , Ψ −s defined on the superspace (3.1) in the following way: where, in (3.2),(3.3), s = 1, . . . , ∞.
Our basic observation which considerably simplifies our analysis of theory of interacting superfields is that the unconstrained superfields (3.2)-(3.4) can be collected into unconstrained superfields denoted as Θ λ , where, depending on λ, the superfield Θ λ is identified with the ones in (3.2)-(3.4) as follows: We note the following property of the superfield Θ λ . Using the notation GP(Θ λ ) for the Grassmann parity of the superfield Θ λ and taking into account definition of e λ (2.29), we note the relation, We see that, for integer λ, the superfield Θ λ is Grassmann even, while, for half-integer λ, the superfield Θ λ is Grassmann odd.
Realizations on superfield Θ λ . Realization of the Poincaré superalgebra in terms of differential operators acting on the superfield Θ λ (p, p θ ) is given by 10) where the symbol e λ is defined in (2.29), while a quantity ∂ p θ stands for left derivative w.r.t the Grassmann momenta p θ (see Appendix A).
In addition to the superfields Θ λ , we find it convenient to use other superfields denoted by Θ * λ . The superfields Θ * λ are constructed out of the hermitian conjugated fields φ † λ , ψ † λ and defined as follows. First, we define superfields Φ * λ , Ψ * λ by the relations for spin-1 2 supermultiplet; (3.19) where, in (3.17),(3.18), s = 1, . . . ∞. Second, we introduce superfields Θ * λ defined for all λ, where, depending on λ, the superfield Θ * λ is identified with the ones in (3.17)-(3.19) as Obviously the new superfields Θ * λ are not independent of the superfields Θ λ . Namely, in view of hermitian conjugation rules given in (2.22), we find the relation where e λ is defined in (2.29) and we show explicitly momentum arguments p, p θ entering the superfields. For integer λ, the superfield Θ * λ is Grassmann even while, for half-integer λ, the superfield Θ * λ is Grassmann odd. In other words, for the Grassmann parity of the superfield Θ * λ , one has the relation GP(Θ * λ ) = e λ . (3.23) Using the realization of the Poincaré superalgebra in terms of differential operators in (3.9)-(3.16), we can present a superfield representation for generators in (2.5),(2.6). To quadratic order in the superfields Θ λ , a superfield representation of Poincaré superalgebra generators (2.5), (2.6) is given by where realization of G diff, λ on space of Θ λ is given in (3.9)-(3.16). A realization of G diff, λ on space of the superfield Θ * λ may be found in Appendix B. The superfields Θ λ , Θ * λ satisfy the Poisson-Dirac equal-time commutation relations Using relations given in (3.24),(3.25), we verify the standard equal-time (anti)commutation relation between the superfields and the generators In light-cone gauge Lagrangian approach, the light-cone gauge action takes the form where ∂ − ≡ ∂/∂x + and P − int is a light-cone gauge Hamiltonian describing interactions. Internal symmetry can be incorporated by analogy with the Chan-Paton method used in string theory (see Sec.6).
4 General structure of n-point dynamical generators of the Poincaré superalgebra We now describe a general structure of the dynamical generators of the Poincaré superalgebra. For theories of interacting fields, the Poincaré superalgebra dynamical generators receive corrections having higher powers of fields. In general, one has the following expansion for the dynamical generators where G dyn [n] in (4.1) stands for a functional that has n powers of superfields Θ * . In this Section, for arbitrary n ≥ 3, we describe restrictions imposed on the dynamical generators G dyn [n] by the kinematical symmetries of the Poincaré superalgebra. We discuss the restrictions in turn. Kinematical P R,L , P + , Q +L symmetries.. Using (anti)commutation relations between the dynamical generators given in (2.6) and the kinematical generators P R , P L , P + , Q +L , we verify that the dynamical generators G dyn [n] with n ≥ 3 take the following form: , , (4.5) , (4.6) where we use the notation , and Θ [n] |j −R,L [n] stand for shortcuts defined as To simplify our presentation, the quantities p − λ 1 ...λn , q −R,L λ 1 ...λn , and j −R,L λ 1 ...λn appearing in (4.12)-(4.14), will simply be denoted as g λ 1 ...λn , We refer to quantities g λ 1 ...λn (4.16) as n-point densities. We note that n-point densities g λ 1 ...λn (4.16) depend on the momenta p R a , p L a , β a , Grassmann momenta p θa , and helicities λ a , a = 1, 2 . . . , n, g λ 1 ...λn = g λ 1 ...λn (p a , p θa ) . Note that we use the indices a, b = 1, . . . , n to label superfields entering n-point interaction vertex. Also note that, in (4.2)-(4.6), the differential operators X R,L [n] , X [n] θ are acting only on the arguments of the superfields. For example, the expression X R Θ * [n] |g [n] should read as Note that the argument p a in (4.8) stands for the momenta p R a , p L a , and β a . In what follows, the density p − [n] will often be referred to as an n-point interaction vertex, while, for n = 3, we refer to density p − [3] as cubic interaction vertex. J +− -symmetry equations. Commutation relations between the dynamical generators P − , Q −R,L , J −R,L and the kinematical generator J +− amount to equations for the densities given by: J RL -symmetry equations. Commutation relations between the dynamical generators P − , Q −R,L , J −R,L and the kinematical generator J RL amount to equations for the densities given by J +R , J +L , Q +R -symmetry equations. Using (anti)commutation relations between the dynamical generators P − , Q −R,L , J −R,L and the kinematical generators J +R , J +L , and Q +R , we find that the densities g λ 1 ...λn (4.16) depend on the momenta p R,L a and the Grassmann momenta p θa through new momentum variables P R,L ab and P θ ab defined by the relations This is to say that our densities g λ 1 ...λn (4.16) turn out to be functions of P R,L ab and P θ ab in place of p R,L a , p θa , We now summarize our study of restrictions imposed on n-point densities by kinematical symmetries of the Poincaré superalgebra as follows. i) (Anti)commutation relations between the dynamical generators P − , Q −R,L , J −R,L and the kinematical generators P R,L , P + , Q +L lead to delta-functions in (4.8),(4.9) and hence imply the conservation laws for the momenta p R,L a , β a and the Grassmann momenta p θa . ii) (Anti)commutation relations between the dynamical generators P − , Q −R,L , J −R,L and the kinematical generators J +− , J RL lead to the differential equations given in (4.19)-(4.25). iii) (Anti)commutation relations between the dynamical generators P − , Q −R,L , J −R,L and the kinematical generators J +R,L , Q +R tell us that the n-point densities turn out to be dependent of the momenta P R,L ab , P θab (4.26) in place of the respective momenta p R,L a , p θa . iv) Using the conservation laws for the momenta p R a , β a it is easy to check that there are only n − 2 independent momenta P R ab (4.26). For example, for n = 3, there is only one independent P R (see relations below). The same holds true for the momenta P L ab and P ab θ .

Kinematical and dynamical restrictions on cubic vertices and light-cone gauge dynamical principle
We now restrict our attention to cubic vertices. First, we represent kinematical J +− , J RL symmetry equations (4.19)-(4.25) in terms of the momenta P R,L ab and P ab θ . Second, we find restrictions imposed by dynamical symmetries. Finally, we formulate light-cone gauge dynamical principle and present the complete system equations required to determine the cubic vertices uniquely. Kinematical symmetries of the cubic densities. Taking into account the momentum conservation laws we verify that P R,L 12 , P R,L 23 , P R,L 31 and Grassmann momenta P θ 12 , P θ 23 , P θ 31 are expressed in terms of new momenta P R,L , P θ , P R,L 12 = P R,L 23 = P R,L 31 = P R,L , P θ 12 = P θ 23 = P θ 31 = P θ , where the new momenta P R,L and P θ are defined as We find it convenient to use the momenta (5.3) because these momenta are manifestly invariant under cyclic permutations of the external line indices 1, 2, 3. Therefore, using the simplified notation for the densities, we note then the our cubic densities p − [3] , q −R,L [3] , and j −R,L [3] are functions of the momenta β a , P R,L , the Grassmann momentum P θ and the helicities λ 1 , λ 2 , λ 3 , p − [3] = p − λ 1 λ 2 λ 3 (P R , P L , P θ , β a ) , q −R,L [3] = q −R,L λ 1 λ 2 λ 3 (P R , P L , P θ , β a ) , j −R,L [3] = j −R,L λ 1 λ 2 λ 3 (P R , P L , P θ , β a ) . (5.5) Thus we see that the momenta p R,L a and p θa enter cubic densities (5.5) through the respective momenta P R,L and P θ . This feature of the cubic densities simplifies considerably the study of kinematical symmetry equations presented in (4.19)-(4.25). We now represent equations (4.19)-(4.25) in terms of cubic densities (5.5). J +− -symmetry equations: Using representation for cubic densities in (5.5), we find that, for n = 3, equations (4.19),(4.20) take the form where J +− stands for an operator defined as J RL -symmetry equations: Using the representation for the cubic densities in (5.5), we find that, for n = 3, equations (4.21)-(4.25) take the form where J RL stands for an operator defined as and we use the notation in (5.8).
We now proceed with studying the restrictions imposed by dynamical symmetries. Dynamical symmetries of the cubic densities. In this paper, restrictions on the interaction vertices imposed by (anti)commutation relations between the dynamical generators will be referred to as dynamical symmetry restrictions. We now discuss restrictions imposed on cubic interaction vertices by the dynamical symmetries of the Poincaré superalgebra. In other words, we consider the (anti)commutators Let us first consider the commutation relations given in (5.11). In the cubic approximation, commutation relations given in (5.11) can be represented as [P − [2] , J −R [3] ] + [P − [3] , J −R [2] ] = 0 , [P − [2] , Q −R,L [3] ] + [P − [3] , Q −R,L [2] ] = 0 . (5.14) Using equations (5.14), we find the following representation for the densities q −R,L [3] and j −R,L [3] in terms of the cubic vertex p − [3] , where J −Rt , J −Lt stand for operators defined by the relations Using expressions for q −R,L [3] and j −R,L [3] given in (5.15),(5.16), we verify that all the kinematical symmetry equations for q −R,L [3] and j −R,L [3] given in (5.6),(5.9) are satisfied automatically provided the vertex p − [3] satisfies the kinematical symmetry equations for p − [3] in (5.6),(5.9). Using expressions for q −R,L [3] and j −R,L [3] given in (5.15),(5.16), we also verify that, in the cubic approximation, all (anti)commutation relations given in (5.12),(5.13) are satisfied automatically. This is to say that, in the cubic approximation, we checked that the kinematical symmetry equations for p − [3] in (5.6),(5.9) and the equations (5.15),(5.16), provide the complete list of equations which are obtainable from all the (anti)commutation relations of the Poincaré superalgebra. Light-cone gauge dynamical principle. The kinematical symmetry equations for p − [3] in (5.6),(5.9) and the equations (5.15),(5.16) do not admit to determine the cubic vertex p − [3] uniquely. In order to determine the cubic vertex p − [3] uniquely we should impose some additional restrictions on the cubic vertex p − [3] . We will refer to these additional restrictions as light-cone gauge dynamical principle. The light-cone gauge dynamical principle is formulated as follows: i) The densities p − [3] , q −R,L [3] , j −R,L [3] are required to be polynomial in the momenta P R , P L ; ii) The density p − [3] is required to satisfy the restriction p − [3] = P R P L W , W is polynomial in P R , P L .

(5.22)
We note that the requirement for p − [3] in (5.22) is related to the freedom of field redefinitions. We recall that upon field redefinitions the p − [3] is changed by terms proportional to P R P L . This implies that ignoring requirement (5.22) leads to cubic vertices which can be removed by field redefinitions. As we are interested in the cubic vertices p − [3] that cannot be removed by field redefinitions, we impose the requirement (5.22). Note also the assumption i) is the light-cone counterpart of locality condition commonly used in gauge invariant and Lorentz covariant formulations. Complete system of equations for cubic vertex. To summarize the discussion in this section, we note that, for the cubic vertex given by p − [3] = p − λ 1 λ 2 λ 3 (P R , P L , P θ , β a ) (5.23) the complete system of equations which remain to be analysed is given by

Light-cone gauge dynamical principle:
p − [3] , q −R,L [3] , j −R,L [3] are polynomial in P R , P L ; Equations given in (5.24)-(5.29) constitute our basic complete system of equations which allow us to determine the cubic vertex p − [3] and densities q −R,L [3] , j −R,L [3] uniquely. Differential operators J +− , J RL , J −R,Lt and quantity ǫ entering our basic equations are given in (5.7),(5.10), (5.18),(5.19) and (5.17) respectively. Considering the super Yang-Mills and supergravity theories, we can verify that our basic equations in (5.24)-(5.29) allow us to determine the cubic interaction vertices of those theories unambiguously (up to coupling constants). It seems then reasonable to use our equations for studying the cubic vertices of arbitrary spin supersymmetric theories.

Cubic interaction vertices
We now present the solution to our basic equations for densities given (5.24)-(5.29).
iii) Requiring the cubic Hamiltonian P − [3] to be hermitian, we get the restrictions for coupling constants given in (6.11).
To make our results more transparent and pragmatic we now discuss an explicit representation for the cubic Hamiltonian P − [3] in terms of component fields (2.17)- (2.19). Doing so, we demonstrate explicitly number of momenta appearing in our cubic vertices. Cubic vertices in terms of component fields. Our aim is to find a representation for the cubic vertex in terms of the component fields by using superfield representation for cubic vertex above obtained. To this end we restrict our attention to interaction of three superfields Θ λ 1 , Θ λ 2 , Θ λ 3 and represent the corresponding cubic Hamiltonian P − [3] (4.2) with p − λ 1 λ 2 λ 3 (6.1) as follows where the integration measures dΓ p [3] , dΓ p θ [3] are obtained by setting n = 3 in (4.8), (4.9). It is the vertex V Θ λ 1 Θ λ 2 Θ λ 3 (6.18) that we refer to as the vertex in terms of the component fields.
Three integer spin supermultiplets : ΦΦΦvertices in (2a) (6.31) Two integer and one half-integer spin supermultiplets : ΦΦΨ-vertices in (1a)(3a)(3b) (6.32) Two half integer and one integer spin supermultiplets : ΦΨΨ-vertices in (2b)(2c) (6.33) Three half-integer spin supermultiplets : ΨΨΨ-vertices in (1b)(3c) (6.34) To illustrate our result for vertices let us consider particular cases from the list in (6.23)-(6.30). i) For the particular case s 1 = 0, s 2 = 0, s 3 = 0, the vertex (6.24) takes the form (P L ) 2 , supergravity; (6.37) iii) On the one hand, considering the cubic vertices for the integer spin supermultiplets in (6.25) for the case s 1 = 1, s 2 = 1, s 3 = 1 and for the case s 1 = 2, s 2 = 2, s 3 = 2, we do not find vertices with three momenta for spin-1 field and vertices with six momenta for spin-2 fields. This reflects the well known fact that Poincaré supersymmetries forbid supersymmetric extension of F 3 terms for massless spin-1 theory and R 3 terms for massless spin-2 theory. The same happens if we consider cubic vertices for the half-integer spin supermultiplets (6.24),(6.30). On the other hand, if we consider cubic vertices that involves both the integer and half-integer spin supermultiplets then we can find supersymmetric extension of F 3 and R 3 terms. In other words, the F 3 and R 3 terms can be supersymmetrized in the respective theories involving two different massless spin-1 fields (super YM-like theory) and two different massless spin-2 fields (supergravity-like theory).
(P L ) 6 , R 3 supergravity-like theory; (6.39) 4 In the framework of Lorentz covariant approach, the recent extensive study of higher-spin interacting supermultiplets may be found in Refs. [21]- [25]. In Refs. [24,25], by using gauge invariant supercurrents, the cubic vertices of type s − Y − Y and (s + 1 2 ) − Y − Y have been constructed. To our knowledge, such type of cubic vertices are the most general cubic vertices that are available at the present time in the framework of the Lorentz covariant approach. Lorentz covariant superfield formulations of free N = 1 supermultiplets in 4d flat space were studied in Refs. [26].
We emphasize once more that vertex (6.38) describes interaction of two different massless spin-1 fields (and their superpartners), while vertex (6.39) describes interaction of two different massless spin-2 fields (and their superpartners). Namely vertex (6.38) describes interaction of two integer supermultiplets (1, 1 2 ) and one half integer spin supermultiplet ( 3 2 , 1), while vertex (6.39) describes interaction of two integer supermultiplets (2, 3 2 ) and one half integer spin supermultiplet ( 5 2 , 2). Interrelations between number of derivatives in light-cone gauge and covariant approaches. To make our results more transparent and helpful for those readers who prefer Lorentz covariant formulations we now discuss a correspondence between number of momenta (transverse derivatives) appearing in our light-cone gauge cubic vertices and number of momenta (derivatives) appearing in the corresponding Lorentz covariant theory. Using shortcuts B and F for the respective massless bosonic and massless fermionic fields, we can write symbolically a cubic Lagrangian of Lorentz covariant theory L cov and related light-cone gauge cubic Lagrangian L lc as follows In (6.40), P stands for momenta (derivatives), while K cov BBB and K cov F F B denote numbers of momenta P (derivatives) entering cubic vertices in metric-like Lorentz covariant formulation. Accordingly in (6.41), the P stands for the momenta P R , P L (transverse derivatives), while K lc BBB and K lc F F B denote numbers of momenta P (transverse derivatives) entering our light-cone gauge cubic vertices. We note the following relations for the numbers of the momenta (derivatives) 5 Now we use (6.42) to relate the classification for light-cone gauge vertices to the one for covariant vertices. We classify covariant vertices focusing on number of integer spin supermultiplets (s, s − 1 2 ) (superfield Φ) and half-integer spin supermultiplets (s + 1 2 , s) (superfield Ψ) appearing in the vertices. This is to say that we represent the classification of light-cone gauge vertices (6.23)- (6.30) in terms of the corresponding covariant cubic vertices as follows: Three integer spin supermultiplets (ΦΦΦ − vertex) : Two integer and one half-integer spin supermultiplets (ΦΦΨ − vertex) :

46)
Two half integer and one integer spin supermultiplets (ΦΨΨ − vertex) : Three half-integer spin supermultiplets (ΨΨΨ − vertex) : In the left column in (6.43)-(6.50), we use the labels to show explicitly the correspondence between the classification for covariant vertices in (6.43)-(6.50) and the one for the light-cone gauge vertices in (6.23)-(6.30).
Motivation for study of both the integer and half-integer spin supermultiplets in flat space.
We can try to restrict our attention to the study of supersymmetric higher-spin theory that involves only integer spin supermultiplets. It turns out that such theory does not exist. Our arguments are as follows. Consider vertex (6.25) with s 1 = 4, s 2 = 4, s 3 = 4. In Ref. [29], for the case of higher-spin bosonic theories, we demonstrated that in order to respect restrictions on the coupling constants which appear in the quartic approximation (Jacobi constraints) one needs to use vertices that involves (P L ) s 1 +s 2 +s 3 -terms and (P L ) s 1 −s 2 −s 3 -terms. From the expressions in (6.23), (6.28)-(6.30), we see however that, in our supersymmetric theory, such terms can be build only if we use both the integer and half-integer spin supermultiplets. Thus, the N = 1 supersymmetry in higherspin theory in flat space requires the use of both the integer and half-integer spin supermultiplets.
In other words, we should use the chain of fields that involves each helicity twice. Appearance of such chain of fields in higher-spin theory in AdS space is the well known fact [27] (see also [28]). Finally, we conjecture that the solution for coupling constants C λ 1 λ 2 λ 3 in Refs. [29,30] can be generalized to the case of N = 1 supersymmetric higher-spin theory considered in this paper as follows where e λ is defined in (2.29). In (6.51), the g is a coupling constant, while k is some dimensionfull complex-valued parameter. The g and k do not depend on the helicities. For the supersymmetric theory with hermitian Hamiltonian, the constantsC λ 1 ,λ 2 ,λ 3 are fixed by the relation in (6.11), while, for supersymmetric generalization of the chiral higher-spin theory in Ref. [31], we should setC λ 1 ,λ 2 ,λ 3 = 0. For three bosonic fields entering the cubic vertex, solution (6.51) amounts to the one in Refs. [29,30]. Solution (6.51) can be used for discussion at least the following two supersymmetric higherspin field models in the flat space.
i) Field content of the first model is given in (2.17)- (2.19). For this model the superfields Θ λ (3.5) are considered to be matrices of the internal symmetry o(N) algebra. The superfields Θ λ are subject to the algebraic constraint in (6.14). ii) In the second model, the superfields Θ λ are singlets of the o(N) algebra and we use the set of superfields given by where the summation is performed over those values of λ in (3.5) that satisfy restriction given in (6.52). In terms of the superfields Φ λ , Ψ λ , (3.6),(3.7), the set of superfields in (6.52) can be presented as In terms of the component fields, using notation (s, s − 1 2 ) and (s + 1 2 , s) for the respective supermultiplets in (2.17) and (2.18), (2.19), we represent the field content of the second model as Appearance of such two models in the N = 1 supersymmetric higher-spin theory in AdS space is the well known fact.

Conclusions
In this paper, we used light-cone gauge formalism for studying the N = 1 integer spin and halfinteger supermultiplets in the flat 4d space. For such supermultiplets, we developed the light-cone gauge formulation in terms of the unconstrained superfields. We used our superfield formulation to build the full list of the cubic vertices that describes all interactions of massless integer and half-integer spin supermultiplets. Taking into account orders of derivatives appearing in our cubic interaction vertices, we concluded that the integer spin supermultiplets alone are not enough for studying full theory of massless N = 1 interacting supermultiplets in the flat 4d space. For studying the full N = 1 supersymmetric theory of higher-spin massless fields in the flat 4d space one needs to use both integer and half-integer spin supermultiplets. In other words, as compared to bosonic theory of massless higher-spin fields, in supersymmetric theory of higher-spin fields, one needs to use the double set of fields (each helicity occurs twice). In this respect, the supersymmetric theory of higher-spin massless fields in the flat space and the one in the AdS space are similar. We believe that results in this paper might be helpful for the following generalizations and applications.
i) In this paper, we studied supersymmetric massless higher-spin theory in flat space. Generalization of our results to the case of supersymmetric massive fields in flat space could be of interest. We note that all parity invariant cubic vertices for massless and massive arbitrary spin fields in flat space time R d−1,1 , d-arbitrary, were built in Refs. [16,17,32]. Namely, in Ref. [16,17], we built all parity invariant cubic vertices for massless and massive bosonic and fermionic fields in the framework of light-cone gauge formalism, while, in Ref. [32], we built all parity invariant cubic vertices for massless and massive bosonic fields in the framework of BRST-BV approach. 6 We expect that light-cone gauge cubic vertices in Refs. [16,17] will be helpful for studying supersymmetric theories of massless and massive fields. Discussion of supermultiplets in various dimensions may be found, e.g., in Ref. [39]. Study of N = 1 higher-spin massless supermultiplets via BRST approach may be found in Ref. [40], while the N = 1 massive supermultiplets are investigated in Ref. [41]. ii) In this paper, we restricted our attention to supersymmetric massless higher-spin theory in the flat space. Gauge invariant formulation of higher-spin theory in AdS space is well known [42]. Various aspects of supersymmetric higher-spin gauge field theory in AdS space have extensively been studied in the past (see, e.g., Refs. [27,28,43]). Generalization of our results to the case of light-cone gauge supersymmetric massless higher-spin fields in AdS space could of great interest. Light-cone gauge cubic vertices of higher-spin massless fields in AdS 4 space have recently been obtained in Ref. [44]. We believe therefore that result in this paper and the one in Ref. [44] provide a good starting point for the studying light-cone gauge supersymmetric massless fields in AdS 4 space. 7 iii) In recent time there has been increasing interest in studying various higher-spin theories in three-dimensional flat and AdS spaces (see, e.g., Refs. [47]- [53] and references therein). We think that the light-cone gauge approach will provide considerable simplification in whole analysis of higher-spin theories in three dimensions. Light-cone gauge formulation of fields in flat space is well known (see, e.g., Ref. [10]), while the light-cone gauge formulation of fields in AdS 3 was developed in Refs. [54,55]. Ordinary derivative light-cone gauge formulation of free conformal fields was developed in Ref. [56]. We expect that use of the light-cone formulation in Refs. [55,56] might be helpful for better understanding of various theories in three dimensions. iv) As discussed in Ref. [57], the chiral higher-spin model [31] is free of one-loop divergencies. Also, the general arguments were given for cancellation of all loop divergencies. Loop diagrams in the chiral higher-spin theory are subset of the ones in the full (non-chiral) higher-spin theory. Therefore, the result in Ref. [57] is a good sign for the quantum finiteness of full (non-chiral) higher-spin theory. The study of quantum properties of full (non-chiral) higher-spin in flat space theory may be found in Ref. [58]. We believe that the superfield formulation of N = 1 higherspin theory suggested in this paper will brings new interesting novelty in the studying quantum properties of higher-spin theory.