Cubic interaction vertices for massive/massless continuous-spin fields and arbitrary spin fields

We use light-cone gauge formalism to study interacting massive and massless continuous-spin fields and finite component arbitrary spin fields propagating in the flat space. Cubic interaction vertices for such fields are considered. We obtain parity invariant cubic vertices for coupling of one continuous-spin field to two arbitrary spin fields and cubic vertices for coupling of two continuous-spin fields to one arbitrary spin field. Parity invariant cubic vertices for self-interacting massive/massless continuous-spin fields are also obtained. We find the complete list of parity invariant cubic vertices for continuous-spin fields and arbitrary spin fields.


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4 Parity invariant cubic vertices for one continuous-spin massive field and two massless fields 11 4.1 One continuous-spin massive field and two massless fields 11 4.2 One continuous-spin massive field, one massless field, and one massive field 14 4.3 One continuous-spin massive field and two massive fields 16 5 Parity invariant cubic vertices for two continuous-spin massive/massless fields and one arbitrary spin massive/massless field 18 5.1 One continuous-spin massive field, one continuous-spin massless field and one arbitrary spin massless field 19 5.2 Two continuous-spin massive fields with the same mass values and one massless field 21 5.3 Two continuous-spin massive field with different masses and one massless field 23 5.4 One continuous-spin massless field, one continuous-spin massive field, and one arbitrary spin massive field 25 5.5 Two continuous-spin massive fields and one massive field 27 6 Parity invariant cubic vertices for three continuous-spin fields 29 6.1 Two continuous-spin massless fields and one continuous-spin massive field 30 6.2 One continuous-spin massless field and two continuous-spin massive fields with equal masses 32 6.3 One continuous-spin massless field and two continuous-spin massive fields with nonequal masses 34 6.4 Three continuous-spin massive fields 36

Conclusions 38
A Notation and conventions 39 B Derivation of vertices p − [3]  Continuous-spin field propagating in flat space is associated with continuous-spin representation of the Poincaré algebra (for review, see refs. [1]- [3]). From the point of view of field theory, the continuous-spin field provides interesting example of relativistic dynamical system which involves infinite number of coupled finite component fields. In this respect the continuous-spin field theory has some features in common with string theory and higherspin theory. For example, we note that the continuous-spin field is decomposed into an infinite chain of coupled scalar, vector, and totally symmetric tensor fields which consists of every spin just once. We recall then a similar infinite chain of scalar, vector and totally symmetric fields appears in the theory of higher-spin gauge field in AdS space [4,5]. We note also the intriguing discussions about possible interrelations between continuous-spin field theory and the string theory in refs. [6,7]. 1 In view of just mentioned and other interesting features, the continuous-spin field theory has attracted some interest recently (see, e.g., refs. [9]- [17]). In ref. [18], we developed the light-cone gauge formulation of massless and massive continuous-spin fields propagating in the flat space R d−1,1 with arbitrary d ≥ 4. 2 Also, in ref. [18], we applied our formulation to study parity invariant cubic vertices for coupling of massless continuous-spin fields to massive arbitrary spin fields and obtained complete list of such cubic vertices. 3 Cubic vertices involving continuous-spin fields have one interesting and intriguing feature in common with interaction vertices of string theory and higher-spin theory. It turns out that the interaction vertices for coupling of continuous-spin fields to arbitrary spin fields involve infinite number of derivatives. It seems therefore highly likely that the continuous-spin field theory is the interesting and promising direction to go. This paper is a continuation of the investigation of cubic interaction vertices for continuous-spin fields and arbitrary spin field begun in ref. [18]. In ref. [18], we studied cubic vertices which involve massless continuous-spin fields and massive/massless arbitrary spin fields, while, in this paper, we study cubic vertices which involve massive/massless continuous-spin fields and massive/massless arbitrary spin fields. We recall that, in general, a continuous-spin field is labelled by mass parameter, which we denote by m, m 2 ≤ 0, and continuous-spin parameter, which we denote by κ, κ > 0. To indicate such continuousspin field we use the shortcut (m, κ) CSF . A finite-component arbitrary spin field is labeled by mass parameter, denoted by m, m 2 ≥ 0, and spin value denoted by integer s, s ≥ 0. Such finite component field will be denoted as (m, s). We now note that cubic vertices 1 Note that, in ref. [8], it was observed that the continuous-spin massless fields do not appear in perturbative string theory spectrum. We believe that continuous-spin fields could be appeared in the full-fledged string theory. 2 Light-cone gauge free continuous-spin massless fields in R 3,1 and R 4,1 were discussed in ref. [3]. For d ≥ 4, the discussion of UIR of the Poincaré algebra iso(d − 1, 1) may be found in ref. [1]. We think that our continuous-spin massive field is associated with the tachyonic UIR of the Poincaré algebra which is discussed in section 3 in ref. [1]. 3 We note that continuous-spin massless/massive fields are infinite component fields. In this paper, finite component massless/massive fields with arbitrary but fixed values of spin will be referred to as arbitrary spin fields.

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involving massless continuous-spin fields and massive/massless arbitrary spin fields can be classified as Cubic vertices with one massless continuous-spin field: Cubic vertices with two massless continuous-spin field:

6)
Cubic vertices with three massless continuous-spin field: Cubic vertices for fields in (1.1)-(1.6) were studied in ref. [18]. Namely, in ref. [18], we built the complete list of parity invariant cubic vertices for fields in (1.3), (1.4), (1.6). 4 Also, in ref. [18], we demonstrated that, there are no parity invariant vertices for fields in (1.1), (1.5). This is to say that, in the framework of so(d − 2) covariant light-cone gauge formalism, there are no parity invariant cubic vertices for coupling of continuous-spin massless fields to finite component arbitrary spin massless fields. It remains to investigate whether cubic vertices describing self interacting continuous-spin massless field (1.7) do exist. 5 In this paper, we study cubic vertices which involve, among other fields, at least one massive continuous-spin field. Such cubic vertices we separate in the following three groups.

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Cubic vertices with three continuous-spin fields: (0, κ 1 ) CSF -(0, κ 2 ) CSF -(m 3  In this paper, we build the complete list of parity invariant cubic vertices for fields in (1.8)- (1.19). 6 With exception of vertices (1.7), results in this paper together with the ones in ref. [18] provide the exhaustive solution to the problem of description of all parity invariant cubic vertices for massive/massless continuous spin fields and massive/massless arbitrary spin fields in (1.1)- (1.19). We now note the general features of our vertices. i) As in string field theory, our vertices involve infinite number of derivatives. Also, as in light-cone gauge string field theory, our vertices are nonlocal w.r.t. the spatial light-cone coordinate x − and expandable in derivatives w.r.t. the transverse spatial transverse coordinates x i . ii) Our vertices for massive fields are singular when the masses tend to zero. iii) In contrast to the massless continuous-spin fields, the massive continuous-spin fields admit coupling to arbitrary spin massless fields.
This paper is organized as follows. In section 2, we briefly review the manifestly so(d − 2) covariant light-cone gauge formulation of free continuous-spin massless and massive fields in R d−1,1 developed in ref. [18]. For the reader convenience, we also recall the textbook light-cone gauge formulation of finite-component arbitrary spin massless and massive fields. In section 3, we review the setup for studying cubic vertices which was developed in ref. [18]. We present the complete system of equations required to determine cubic vertices uniquely. For the case of parity invariant cubic vertices, we present the particular form of the complete system of equations which is convenient for our study.
In section 4, we present our result for parity invariant cubic vertices describing coupling of one continuous-spin massive field to two massive/massless arbitrary spin fields, while section 5 is devoted to study of cubic vertices for coupling of two massive/massless continuous-spin field to one massive/massless arbitrary spin field. Section 6 is devoted to cubic vertices for self-interacting massive/massless continuous-spin fields. Namely, we consider vertices for coupling of one massless continuous-spin field to two massive continuousspin fields and vertices for coupling of one massive continuous-spin field to two massless continuous-spin fields. Cubic vertices for self-interacting massive continuous-spin field are also studied.
In section 7, we summarize our conclusions and suggest directions for future research.
In appendix A, we describe notation we use in this paper. In appendices B, C, D we outline some details of the derivation of cubic vertices. Namely, in appendix B, we discuss 6 Lorentz covariant cubic vertex for coupling of massless continuous-spin field to two massive scalar fields (case (1.4) with s1 = s2 = 0) was studied in ref. [19], while Lorentz covariant cubic vertex for coupling of massive continuous-spin field to two massive scalar fields (case (1.10) with s1 = s2 = 0) was studied in ref. [20].

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vertices for coupling of one continuous-spin field to two arbitrary spin fields, while, in appendix C, we discuss vertices for coupling of two continuous-spin fields to one arbitrary spin fields. In appendix D, we outline details of the derivation of vertices for self-interacting massive/massless continuous-spin fields. In appendix E, we study cubic vertex for one massless continuous-spin field, one arbitrary spin massless field, and one arbitrary spin massive field.
2 Light-cone gauge formulation of free continuous-spin fields and arbitrary spin massive and massless fields Detailed discussion of light-cone formulation to free continuous-spin fields and arbitrary spin massive/massless fields may be found in section 2 in ref. [18]. Here, in order to make our presentation self-contained, we briefly review the most important ingredients of the light-cone formulation.
Poincaré algebra in light-cone frame. We use the method in ref. [21] which reduces the problem of finding a new dynamical system to a problem of finding a new solution for commutators of a basic symmetry algebra. For the case under consideration, basic symmetries are associated with the Poincaré algebra. 7 Therefore we start with the presentation of a realization of the Poincaré algebra symmetries on a space of light-cone gauge fields. The commutation relations of the Poincaré algebra iso(d − 1, 1) are given by 8 where P µ are the translation generators, while J µν are generators of the so(d−1, 1) Lorentz algebra. The P µ taken to be hermitian, while the J µν are considered to be anti-hermitian. Flat metric η µν is taken to be mostly positive.
For the discussion of light-cone formulation, we usee, in place of the Lorentz basis coordinates x µ , the light-cone basis coordinates x ± , x i , where the so(d − 2) algebra vector indices i, j take values i, j = 1, . . . , d − 2, while the coordinates x ± are defined as The coordinate x + is considered as an evolution parameter. We note then that the so(d − 1, 1) Lorentz algebra vector X µ is decomposed as X + , X − , X i , while non vanishing elements of the flat metric are given by η +− = η −+ = 1, η ij = δ ij . We note also that, in light-cone approach, generators of the Poincaré algebra are separated into kinematical and dynamical generators defined as

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discussion of the field theoretical realization of the kinematical and dynamical generators we discuss light-cone gauge description of the continuous-spin field, and arbitrary spin massive and massless fields.
Field-theoretical realization of Poincaré algebra. To get a field theoretical realization of the Poincaré algebra we need the realization of generators (2.2), (2.3) in terms of differential operators acting on ket-vector |φ (2.7)-(2.9). The realization in term of differential operators is given by, 10 Kinematical generators: Dynamical generators: where, in (2.17), we explain our notation. Operator M ij (2.15) stands for a spin operator of the so(d − 2) algebra. Commutation relations for the operators M ij , M i take the form Realization of the operator M i on space of various ket-vectors is given by Continuous-spin field (massless, m 2 = 0, and massive, m 2 < 0): Arbitrary spin massive field, m 2 > 0: Arbitrary spin massless field, m = 0:

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For massive/massless continuous-spin fields, eigenvalues of the 2nd-and 4th-order Casimir operators of the Poincaré algebra are given by C 2 = m 2 , C 4 = κ 2 , while for arbitrary spin-s massive/massless fields one has the relations C 2 = m 2 , C 4 = m 2 s(s+d−3). Note that, in terms of the operators M i , M ij (2.18), (2.19), the 4th-order Casimir operator of the Poincaré algebra takes the form . Now we are ready to present a field theoretical realization of the Poincaré algebra generators. This is to say that, at quadratic order in fields, a field theoretical realization of the generators (2.2), (2.3) takes the form where G stands for the differential operators presented in (2.14)-(2.16), while G [2] stands for the field theoretical generators. Equal time commutator of G [2] (2.25) with |φ takes the form [|φ , G [2] ] = G|φ . Also we recall that the light-cone gauge action takes the form

Cubic interaction vertices and light-cone gauge dynamical principle
Detailed discussion of the setup we use for studying the light-cone gauge cubic vertices may be found in section 3 in ref. [18]. Here, to make our presentation self-contained, we just briefly review our result for complete system of equations required to determine the cubic vertices uniquely. In theories of interacting fields, the dynamical generators G dyn = P − , J −i (2.3) can be expanded as where G dyn [n] stands for a functional that has n powers of ket-vector |φ . At quadratic order in fields, contribution to G dyn is governed by G [2] (2.25), while, at cubic order in fields, contributions to G dyn are described by G dyn [3] . Our aim in this section is to describe the complete system of equations which allows us to determine G dyn [3] = P − [3] , J −i [3] uniquely. We start with the presentation of the expressions for dynamical generators P − [3] , J −i [3] given by 3)

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where we use the notation The ket-vectors |p − [3] and |j −i [3] appearing in (3.2), (3.3) can be presented as |p − [3] = p − [3] (P, β a , α a )|0 , |j −i [3] = j −i [3] (P, β a , α a )|0 , (3.6) where index a = 1, 2, 3 labels three fields entering dynamical generators (3.2), (3.3). Quantities p − [3] and j −i [3] in (3.6) are referred to as densities. Often the density p − [3] is referred to as cubic interaction vertex. The quantities β a (3.6) are three light-cone momenta (2.17), while the quantity α a (3.6) is a shortcut for the oscillators entering ket-vectors (2.7)-(2.9): for continuous-spin field, the α a stands for oscillators α i a , υ a , while, for massive and massless fields, the α a stands for oscillators α i a , ζ a and α i a respectively. A quantity P in (3.6) stands for a momentum P i defined by the relations Complete system of equations for cubic vertex. The complete system of equations for the cubic vertex p − [3] and the density j −i [3] discussed in ref. [18] takes the form Light-cone gauge dynamical principle: |p − [3] and |j −i [3] are expandable in P i ; (3.11) |p − [3] = P − |V , |V is expandable in P i ; (3.12) |p − [3] , |j −i [3] , |V are finite for P − = 0 , (3.13) where operators J +− , J ij , P − , J −i † appearing in (3.8)-(3.13) are given 14) Note also that, if cubic Hamiltonian P − [3] (3.2) involves continuous-spin field φ(p a , α a )|, then cubic vertex, besides variables in (3.19), depends on the oscillators υ a (3.2), while if cubic Hamiltonian P − [3] (3.2) involves massive field φ(p a , α a )|, then cubic vertex, besides variables in (3.19), depends on the oscillator ζ a . Restrictions imposed by J ij -symmetries (3.9) imply that the vertex p − [3] depends on invariants of the so(d − 2) algebra. The oscillators υ a , ζ a , and momenta β a are the invariants of the so(d − 2) algebra. Besides these invariants, in the problem under consideration, the remaining invariants of the so(d − 2) algebra can be constructed by using the P i , α i a , the delta-Kroneker δ ij , and the Levi-Civita symbol i 1 ...i d−2 . We ignore invariants that involve one Levi-Civita symbol. 11 This is to say that, in this paper, vertices that depend on the invariants given by 20) are referred to as parity invariant vertices. If P − [3] (3.2) involves continuous-spin field φ(p a , α a )|, or massless arbitrary spin field φ(p a , α a )|, then by virtue of the second constraints in (2.10), (2.12) the invariant α i a α i a does not contribute to the P − [3] , while, if P − [3] (3.2) involves massive field φ(p a , α a )|, then by virtue of the second constraint in (2.10), the contribution of α i a α i a , can be replaced by the contribution of (−ζ 2 a ). Also note that, by using field redefinitions, one can remove all terms in the vertex p − [3] that are proportional to P i P i (see appendix B in ref. [27]). This implies that, in the vertex p − [3] , we can drop down a dependence on the invariant P i P i . Taking the above-said into account, we note that cubic vertices in the list (1.8)-(1.19) take the following respective forms: Cubic vertices with one continuous-spin field: Cubic vertices with two continuous-spin fields: For finite-component fields, recent interesting discussion of parity-odd vertices may be found in ref. [34].
Generalization of our discussion to the case of light-cone gauge parity-odd vertices is straightforward. We do not discuss parity-odd vertices because their inclusion would make our discussion unwieldy. Note that full actions in light-cone gauge bosonic string field theories are formulated entirely in terms of the parity invariant vertices. We believe therefore that full actions in light-cone gauge bosonic continuous-spin field theories can also be formulated entirely in terms of the parity invariant vertices. For massless fields in 5d flat space, the pattern of the derivation of the light-cone gauge parity-odd cubic vertices may be found in section 8.1 in ref. [27].

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Cubic vertices with three continuous-spin fields: where we introduce the notation In other words, in place of the invariant α i a P i (3.20), we prefer to use the invariant B a (3.33). Using representations for cubic vertices in (3.21)-(3.32), we now present more convenient form of equations for the cubic vertices. Namely, using J −i (3.17), we find the following relation where explicit form of operators G a,P 2 , G β may be found in appendices B, C, D. Using explicit form of operators G a,P 2 , G β , we then conclude that equations (3.10), (3.13), (3.34) lead to the equations In turn, equations (3.34)-(3.36) and the ones in (3.10) imply the following expressions for density |j −i [3] corresponding to the respective vertices with one continuous-spin field (3.21)-(3.23), two continuous-spin field (3.24), (3.28) and three continuous-spin field (3.30) 39) where operator N a is defined in (A.7). We note also that, in terms of vertices presented in (3.21)-(3.32), equations given in (3.8) takes the form density |j −i [3] corresponding to three groups of cubic vertices in (3.21)-(3.32) are given by the respective three relations in (3.37)-(3.39). Equations (3.35), (3.36), (3.40) constitute the complete system of equations which to the best of our present understanding determine parity invariant vertices uniquely. Note that, if we consider the light-cone gauge formulation of the Einstein and Yang-Mills theories, then we can verify that eqs. (3.35), (3.36), (3.40) admit to fix the cubic interaction vertices unambiguously (see ref. [27]). Therefore it seems reasonable to use eqs.

Parity invariant cubic vertices for one continuous-spin massive field and two massless fields
In this section, we discuss parity invariant cubic vertices which involve one massive continuous-spin field and two finite component massless/massive fields. According to our classification, such vertices can be separated into three particular cases given in (1.8)-(1.10). Let us discuss these particular cases in turn.

One continuous-spin massive field and two massless fields
We now discuss parity invariant cubic vertices for one continuous-spin massive field and two arbitrary spin massless fields. This is to say that, using the shortcut (m, κ) CSF for a continuous-spin massive field and the shortcut (0, s) for a spin-s massless field, we study cubic vertices for the following three fields: two massless fields and one continuous-spin massive field. (4.1) Relation (4.1) tells us that the spin-s 1 and spin-s 2 massless fields carry the respective external line indices a = 1 and a = 2, while the continuous-spin massive field corresponds to a = 3. For fields (4.1), we find the following general solution to cubic vertex p − [3] (see appendix B) A , A (B 3 , α aa+1 ) . In (4.2), we introduce two vertices V (7) A labelled by the superscript A = +−. In (4.3) and (4.4), the arguments of the generic vertex p − [3] and the vertices V A are shown explicitly. The definition of the arguments B a and α ab may be found in (A.3). Various quantities U appearing in (4.2) are differential operators w.r.t. the B a and α aa+1 . These quantities will JHEP12(2018)055 be presented below. For two vertices V (7) A (4.4), we find the following solution: 1±z 3 2 V ± , V ± = V ± (α 12 , α 23 , α 31 ) , (4.5) where, in (4.5) and below, the F (α, β, γ; x) stands for the hypergeometric function. For the hypergeometric function, we use notation and convention in chapter 15 in ref. [35]. In (4.5), in place of the variable B 3 , we use new variable z 3 (4.7). Operator ν 3 is defined below. We see that the generic vertex p − [3] (4.3) depends on the ten variables, while, the vertices V A (4.5) depend only on the three variables. By definition, the vertices V A (4.5) are expandable in the three variables α 12 , α 23 , α 31 . The general solution (4.2) for the vertex p − [3] is expressed in terms of the operators U , ν 3 acting on the vertices V ± (4.5).To complete the description of the vertex p − [3] we now provide expressions for the operators U , ν 3 . These operators are given by where Γ (4.9), (4.13) stands for the Gamma-function. Quantitiesβ a , N Ba , N α ab , N a appearing in (4.8)-(4.14) are defined in (A.3)-(A.7). Expressions above-presented in (4.2)-(4.14) provide the complete generating form description of cubic interaction vertices for two chains of totally symmetric massless fields (2.13) with one continuous-spin massive field. Now our aim is to describe cubic vertices for one continuous-spin massive field and two totally symmetric massless fields with arbitrary but fixed spin-s 1 and spin-s 2 values. Using the first algebraic constraint for massless spin-s field in (2.12) it is easy to see that vertices we are interested in must satisfy the algebraic constraints (N αa − s a )|p − [3] = 0 , a = 1, 2.

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Two constraints given in (4.15) tell us simply that the cubic vertex p − [3] should be degree-s 1 and degree-s 2 homogeneous polynomial in the oscillators α i 1 and α i 2 respectively. It is easy to check, that, in terms of the vertices V ± (4.5), algebraic constraints (4.15) take the form Vertices V A satisfy one and the same equations (4.16). Therefore to simplify our presentation we drop the superscript A and use a vertex V in place of the vertices V A , i.e., we use V = V A . Doing so, we note that the general solution to constraints (4.16) can be presented as (4.17) The integer n appearing in (4.17) is the freedom of our solution for the vertex V . In other words the integer n labels all possible cubic vertices that can be constructed for three fields in (4.1). In order for vertices (4.17) to be sensible, the integer n should satisfy the restrictions  Expressions for cubic interaction vertices given in (4.2)-(4.14), (4.17) and allowed values for n presented in (4.19) provide the complete description and classification of cubic interaction vertices that can be constructed for two spin-s 1 and spin-s 2 massless fields and one continuous-spin massive field. From (4.19), we find that, given spin values s 1 and s 2 , a number of cubic vertices V + (or V − ) which can be built for the fields in (4.1) is given by (4.20) Cubic vertex for continuous-spin massive and massless scalar fields. For illustration purposes we consider cubic vertex for one continuous-spin massive field and two massless scalar fields. For two scalar fields, we have spin values s 1 = 0, s 2 = 0, i.e., s min = 0. Plugging s min = 0 in (4.19), we get n = 0. This implies that there is only one vertex V + and one vertex V − . Plugging n = 0 in (4.17), we get V ± = 1. In turn, plugging V ± = 1 in (4.5), we get the vertices V ± :
4.2 One continuous-spin massive field, one massless field, and one massive field We discuss parity invariant cubic vertices for one continuous-spin massive field, one arbitrary spin massless field, and one arbitrary spin massive field. This is to say that, using the shortcut (m, κ) CSF for a continuous-spin massive field and the shortcut (m, s) for a mass-m and spin-s massive field, we study cubic vertices for the following three fields: one massless field, one massive field, and one continuous-spin massive field. (4.25) Relation (4.25) tells us that spin-s 1 massless and spin-s 2 massive fields carry the respective external line indices a = 1 and a = 2, while the continuous-spin massive field corresponds to a = 3. For fields (4.25), we find the following general solution to cubic vertex p − [3] (see appendix B): In (4.26), we introduce two vertices V (9) A labelled by the superscripts A = +, −. In (4.27) and (4.28), the arguments of the generic vertex p − [3] and the vertices V A are shown explicitly. The definition of the arguments B a and α ab may be found in (A.3). Various quantities U appearing in (4.26) are differential operators w.r.t. the B a and α aa+1 . These quantities will be presented below. For two vertices V (9) ± (4.28), we find the following solution: , (4.31)

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where the F (α, β, γ; x) stands for the hypergeometric function. In (4.29), in place of the variable B 3 , we use new variable z 3 (4.31). A quantity ν 3 is defined below. From (4.27), we see that the generic vertex p − [3] depends on the eleven variables, while, from (4.29), we learn that the vertices V A depend only on the four variables. Note also that, by definition, the vertices V A (4.29) are expandable in the variables B 2 , α 12 , α 23 , α 31 . From (4.26), we see that the general solution for the vertex p − [3] is expressed in terms of the operators U , ν 3 and the vertices V A (4.29). In order to complete the description of the vertex p − [3] we should provide expressions for the operators U , ν 3 . These operators are given by where quantitiesβ a , N Ba , N α ab , N a appearing in (4.32)-(4.40) are defined in (A.3)-(A.7). Expressions above-presented in (4.26)-(4.40) provide the complete generating form description of cubic vertices for coupling of one continuous-spin massive field to arbitrary spin massless and massive fields. Namely, these vertices describe an coupling of one continuousspin massive field to two chains of massless and massive fields (2.13). Now our aim is to describe cubic vertices for coupling of one continuous-spin massive field to massless and massive fields having the respective arbitrary but fixed spin-s 1 and spin-s 2 values. Using the first algebraic constraints in (2.11), (2.12) it is easy to see that vertices we are interested in must satisfy the algebraic constraints First constraint in (4.41) tells us that the p − [3] should be degree-s 1 homogeneous polynomial in the α i 1 , while, from the second constraint in (4.41), we learn that the p − [3] should be degree-

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s 2 homogeneous polynomial in the α i 2 , ζ 2 . In terms of the V ± (4.29), constraints (4.41) take the form Vertices V ± satisfy one and the same equations (4.42). Therefore to simplify our presentation we drop the superscripts ± and use a vertex V in place of the vertices V ± , i.e., we use V = V ± . Doing so, we note that the general solution to constraints (4.42) can be presented The integers n, k appearing in (4.43) are the freedom of our solution for the vertex V . In other words, the integers n, k label all possible cubic vertices that can be constructed for three fields in (4.25). In order for vertices (4.43) to be sensible, the integers n, k should satisfy the restrictions which amount to the requirement that the powers of all variables B 2 , α 12 , α 23 , α 31 in (4.43) be non-negative. Expressions for cubic interaction vertices given in (4.26)-(4.40), (4.43) and values of n, k which satisfy restrictions (4.44) provide the complete description and classification of cubic interaction vertices that can be constructed for one spin-s 1 massless field, one spin-s 2 massive field and one continuous-spin massive field.

One continuous-spin massive field and two massive fields
In this section, we discuss parity invariant cubic vertices for one continuous-spin massive field and two arbitrary spin massive fields. This is to say that, using the shortcut (m, κ) CSF for a continuous-spin massive field and the shortcut (m, s) for a spin-s massive field, we study cubic vertices for the following three fields: two massive fields and one continuous-spin massive field Relation (4.45) tells us that the spin-s 1 and spin-s 2 massive fields carry the respective external line indices a = 1, 2, while the continuous-spin massive field corresponds to a = 3. For fields (4.45), we find the following general solution to cubic vertex p − [3] (see appendix B) In (4.46), we introduce two vertices V (7) A labelled by the superscripts A = ±. In (4.47) and (4.48), the arguments of the generic vertex p − [3] and the vertices V A are shown explicitly. The definition of the arguments B a and α ab may be found in (A.3). Various quantities JHEP12(2018)055 U appearing in (4.46) are differential operators w.r.t. the B a and α aa+1 . These operators are given below. For two vertices V (7) A (4.48), we find the following solution: where the F (α, β, γ; x) stands for the hypergeometric function. In (4.49), in place of the variable B 3 , we use new variable z 3 (4.51). Quantities D, ν 3 are defined below. From (4.47), we see that the generic vertex p − [3] depends on the twelve variables, while, from (4.49), we learn that the vertices V ± depend only on the five variables. Note also that, by definition, the vertices V ± (4.49) are expandable in the five variables B 1 , B 2 , α 12 , α 23 , α 31 . From (4.46), we see that the general solution for the vertex p − [3] is expressed in terms of the operators U , ν 3 and the vertices V ± (4.49). In order to complete the description of the vertex p − [3] we should provide expressions for the operators U , ν 3 . These operators are given by

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Expressions above-presented in (4.46)-(4.62) provide the complete generating form description of cubic interaction vertices for coupling of one continuous-spin massive field to two chains of totally symmetric massive fields (2.13). Now our aim is to describe cubic vertices for coupling of one continuous-spin massive field to two totally symmetric massive fields having arbitrary but fixed spin-s 1 and spin-s 2 values. Using the first algebraic constraint in (2.11) it is easy to see that vertices we are interested in must satisfy the algebraic constraints Two constraints given in (4.63) tell us simply that the cubic vertex p − [3] should be degree-s 1 and degree-s 2 homogeneous polynomial in the oscillators α i 1 , ζ 1 and α i 2 , ζ 2 respectively. It is easy to check, that, in terms of the vertices V ± (4.49), algebraic constraints (4.63) take the form Vertices V A satisfy one and the same equations (4.64). Therefore to simplify our presentation we drop the superscript A and use a vertex V in place of the vertices V A , i.e., we use V = V A . Doing so, we note that the general solution to constraints (4.64) can be presented as The integers n 1 , n 2 , n 3 appearing in (4.65) are the freedom of our solution for the vertex V . In other words, the three integers n a label all possible cubic vertices that can be constructed for three fields in (4.45). In order for vertices (4.65) to be sensible, the integers n a should satisfy the restrictions which amount to the requirement that the powers of B 1 , B 2 , α aa+1 in (4.65) be nonnegative. Expressions for cubic vertices given in (4.46)-(4.60), (4.65) and restrictions on values of n a presented in (4.66) provide the complete description and classification of cubic interaction vertices that can be constructed for two spin-s 1 and spin-s 2 massive fields and one continuous-spin massive field.

Parity invariant cubic vertices for two continuous-spin massive/massless fields and one arbitrary spin massive/massless field
In this section, we discuss parity invariant cubic vertices which involve two continuous-spin massive/massless fields and one finite component massive/massless field. According to our classification, such vertices can be separated into five particular cases given in (1.11)-(1.15). Let us discuss these particular cases in turn.

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5.1 One continuous-spin massive field, one continuous-spin massless field and one arbitrary spin massless field We start with discussion of vertices involving one continuous-spin massive field, one continuous-spin massless spin and one arbitrary spin massless field. This is to say that, using the shortcuts (m, κ) CSF and (0, κ) CSF for the respective continuous-spin massive and massless fields and the shortcut (0, s) for a spin-s massless field, we study cubic vertices for the following three fields: one continuous-spin massive field, one continuous-spin massless field, and one massless field Relation (5.1) tells us that massive and massless continuous-spin fields carry the respective external line indices a = 1 and a = 2, while the massless spin-s 3 field corresponds to a = 3.
For fields (5.1), we find the following general solution to cubic vertex p − [3] (see appendix C) AB labelled by the superscripts A, B. In (5.3) and (5.4), the arguments of the generic vertex p − [3] and the vertices V where F (α, β, γ; x) stands for the hypergeometric function, while I ν and K ν are the modified Bessel functions. In (5.6), in place of the variables B 1 and B 2 we use the respective variables z 1 and z 2 (5.8). Operators ν 1 , ν 2 are defined below.
From (5.3), we see that the generic vertex p − [3] depends on the eleven variables, while, from (5.5), we learn that the vertices V AB depend only on the three variables. Note also that, by definition, the vertices V AB (5.5) are expandable in the three variables α 12 , α 23 , α 31 . We now present the explicit expressions for the operators U , ν 1 , ν 2 . These operators JHEP12(2018)055 are given by where quantitiesβ a , N Ba , N α ab , N a appearing in (5.9)-(5.18) are defined in (A.3)-(A.7). Expressions (5.2)-(5.18) provide the complete generating form description of cubic vertices for coupling of continuous-spin massless and massive fields to arbitrary spin massless field. Namely, these vertices describe coupling of two continuous-spin fields to chain of massless fields (2.13). Now our aim is to describe cubic vertices for coupling of continuousspin massless and massive fields to massless field with arbitrary but fixed spin-s 3 value. Using the first algebraic constraint in (2.12), it is easy to see that vertices we are interested in must satisfy the algebraic constraint which implies that the cubic vertex p − [3] should be degree-s 3 homogeneous polynomial in the α i 3 . It is easy to check, that, in terms of the vertices V AB (5.5), algebraic constraint (5.19) takes the form 20) where to simplify the notation we drop the superscripts AB in V AB . Obviously, general solution to constraints (5.20) can be presented as

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The integers n, k appearing in (5.21) are the freedom of our solution for the vertex V , i.e., the integers n, k label all possible cubic vertices that can be constructed for three fields in (5.1). In order for vertices (5.21) to be sensible, the integers should satisfy the restrictions in (5.22), which amount to the requirement that the powers of all variables α 12 , α 23 , α 31 in (5.21) be non-negative. From (5.22), we see that allowed values of n, k are given by Expressions for cubic vertices given in (5.2)-(5.18), (5.21) and allowed values for n and k presented in (5.23) provide the complete description and classification of cubic interaction vertices that can be constructed for one continuous-spin massive field, one continuous-spin massless field, and one spin-s 3 massless field.

Two continuous-spin massive fields with the same mass values and one massless field
In this section, we discuss parity invariant cubic vertices for two continuous-spin massive fields having the same masses and one arbitrary spin massless field. This is to say that, using the shortcut (m, κ) CSF for a continuous-spin massive field and the shortcut (0, s) for a spin-s massless field, we study cubic vertices for the following three fields: two continuous-spin massive fields with the same masses and one massless field.
(5.24) Relation (5.24) tells us that the massive continuous-spin fields carry the respective external line indices a = 1, 2, while the massless spin-s 3 field corresponds to a = 3. For fields (5.24), we find the following general solution to cubic vertex p − [3] (see appendix C) AB , A, B = 1, 2, . . . , 6 , In (5.25), we introduce vertices V AB labelled by the superscripts A, B. In (5.26) and (5.27), the arguments of the generic vertex p − [3] and the vertices V AB are shown explicitly. The definition of the arguments B a and α ab may be found in (A.3). Operators U appearing in (5.25) are defined below. For the vertices V

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where a = 1, 2, N Z = Z∂ Z . The generic vertex p − [3] (5.26) depends on eleven variables, while, the vertices V AB (5.29) depend only on three variables. By definition, the vertices V AB (5.29) are expandable in the three variables B 3 , Z, α 12 . In order to complete the description of the vertex p − [3] we now present the explicit form of the operators U . These operators are given by Expressions above-presented in (5.25)-(5.43) provide the complete generating form description of cubic vertices for coupling of two continuous-spin massive fields to chain of massless fields (2.13). Now our aim is to describe cubic vertices for coupling of two continuous-spin massive fields to massless field having arbitrary but fixed spin-s 3 value. Using the first algebraic constraint in (2.12), it is easy to see that vertices we are interested in must satisfy the algebraic constraint which implies that the vertex p − [3] should be degree-s 3 homogeneous polynomial in the oscillators α i 3 . In terms of the vertices V AB (5.29), algebraic constraint (5.44) takes the form

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where to simplify the notation we drop the superscripts AB in V AB . General solution to constraint (5.45) can be presented as The integers n and k appearing in (5.46) are the freedom of our solution for the vertex V . In other words, these integers label all possible cubic vertices that can be constructed for three fields in (5.24). In order for vertices (5.46) to be sensible, the integers n, k should satisfy the restrictions in (5.47) which amount to the requirement that the powers of all variables B 3 , Z and α 12 in (5.46) be non-negative. From (5.47), we see that allowed values of n, k are given by Expressions for cubic interaction vertices given in (5.25)-(5.43), (5.46) and values for n, k presented in (5.48) provide the complete description and classification of cubic vertices that can be constructed for two continuous-spin massive fields and one spin-s 3 massless field (5.24).

Two continuous-spin massive field with different masses and one massless field
In this section, we discuss parity invariant cubic vertices for two continuous-spin massive fields having different masses and one arbitrary spin massless field. This is to say that, using the shortcut (m, κ) CSF for a continuous-spin massive field and the shortcut (0, s) for a spin-s massless field, we study cubic vertices for the following three fields: two continuous-spin massive fields with different masses and one massless field.
(5.49) Relation (5.49) tells us that two continuous-spin massive fields carry the external line indices a = 1, 2, while the spin-s 3 massless field corresponds to a = 3. For fields (5.49), we find the following general solution to cubic vertex p − [3] (see appendix C) In (5.50), we introduce four vertices V AB labelled by the superscripts A, B = ±. In (5.51) and (5.52), the arguments of the generic vertex p − [3] and the vertices V will be presented below. For four vertices V (8) AB (5.52), we find the following solution: where the F (α, β, γ; x) stands for the hypergeometric function. In (5.53), in place of the variables B 1 , B 2 , we use new variables z 1 , z 2 defined in (5.55). Operators ν a are given below.
The generic vertex p − [3] (5.51) depends on the eleven variables, while, the vertices V AB (5.53) depend only on the three variables. Note also that, by definition, the vertices V AB (5.53) are expandable in the three variables α 12 , α 23 , α 31 . In order to complete the description of the vertex p − [3] we now provide expressions for the operators U , ν 1 , ν 2 . These operators are given by but fixed spin-s 3 value. Using the first algebraic constraint in (2.12), it is easy to see that vertices we are interested in must satisfy the algebraic constraint 65) which implies that the cubic vertex p − [3] should be degree-s 3 homogeneous polynomial in the oscillators α i 3 . In terms of the vertices V AB (5.53), algebraic constraint (5.65) takes the form General solution to constraint (5.66) can be presented as

5.4
One continuous-spin massless field, one continuous-spin massive field, and one arbitrary spin massive field In this section, we discuss parity invariant cubic vertices for one continuous-spin massless field, one continuous-spin massive field, and one arbitrary spin massive field. This is to say that, using the shortcut (m, κ) CSF for a continuous-spin mass-m field and the shortcut (m, s) for the mass-m and spin-s field, we study cubic vertices for the following three fields: one continuous-spin massless field, one continuous-spin massive field, and one massive field.
Relation (5.70) tells us that two continuous-spin massless and massive fields carry the external line indices a = 1, 2, while the spin-s 3 massive field corresponds to a = 3. For fields (5.70), we find the following general solution to cubic vertex p − [3] (see appendix C) AB (B a , α aa+1 ) .

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In (5.71), we introduce four vertices V (8) AB labelled by the superscripts AB = ±. In (5.72) and (5.73), the arguments of the generic vertex p − [3] and the vertices V (8) AB are shown explicitly. The definition of the arguments B a and α ab may be found in (A.3). Various quantities U appearing in (5.71) are differential operators w.r.t. the B a and α aa+1 . These quantities will be presented below. For four vertices V (8) AB (5.73), we find the following solution: where F (α, β, γ; x) is the hypergeometric function, while I ν , K ν are the modified Bessel functions. In (5.74), in place of B 1 , B 2 , we use z 1 (5.75) and z 2 (5.77). Operators ν a are given below.
The generic vertex p − [3] (5.72) depends on the twelve variables, while, the vertices V AB (5.74) depend only on the four variables. Note also that, by definition, the vertices V AB (5.74) are expandable in the four variables B 3 , α 12 , α 23 , α 31 . In order to complete the description of the vertex p − [3] we now provide expressions for the operators U , ν 1 , ν 2 . These operators are given by  Expressions above-presented in (5.71)-(5.88) provide the complete generating form description of cubic vertices for coupling of two continuous-spin fields to chain of massive fields (2.13). Now our aim is to describe cubic vertices for coupling of two continuous-spin fields to massive field having arbitrary but fixed spin-s 3 value. Using the first algebraic constraint (2.11), it is easy to see that vertices we are interested in must satisfy the algebraic constraint which implies that the vertex p − [3] should be degree-s 3 homogeneous polynomial in the oscillators α i 3 , ζ 3 . In terms of the vertices V AB (5.74), algebraic constraint (5.89) takes the form

Two continuous-spin massive fields and one massive field
In this section, we discuss parity invariant cubic vertices for two continuous-spin massive fields and one arbitrary spin massive field. This is to say that, using the shortcut (m, κ) CSF for a continuous-spin massive field and the shortcut (m, s) for mass-m and spin-s massive field, we study cubic vertices for the following three fields:

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For fields (5.93), we find the following general solution to cubic vertex p − [3] (see appendix C) In (5.94), we introduce four vertices V AB labelled by the superscripts A, B. In (5.95) and (5.96), the arguments of the generic vertex p − [3] and the vertices V AB are shown explicitly. The definition of the arguments B a and α ab may be found in (A.3). Various quantities U appearing in (5.94) are differential operators w.r.t. the B a and α aa+1 . These quantities will be presented below. For four vertices V where the F (α, β, γ; x) stands for the hypergeometric function. In (5.97), in place of the variables B a , we use new variables z a defined in (5.99). A quantity D and operators ν a are defined below. The generic vertex p − [3] (5.95) depends on the twelve variables, while, the vertices V AB (5.97) depend only on the four variables. Note also that, by definition, the vertices V AB (5.97) are expandable in the four variables B 3 , α 12 , α 23 , α 31 . To complete the description of the vertex p − [3] we provide expressions for the operators U , ν a . These operators are given by Γ(ν +n) 4 n n!Γ(ν +2n) W n , (5.107) Expressions above-presented in (5.94)-(5.111) provide the complete generating form description of cubic vertices for coupling of two continuous-spin massive fields to chain of massive fields (2.13). Now our aim is to describe cubic vertices for coupling of two continuous-spin massive fields to one massive field having arbitrary but fixed spin-s 3 value. Using the first algebraic constraint (2.11), is easy to see that vertices we are interested in must satisfy the algebraic constraint which implies that the cubic vertex p − [3] should be degree-s 3 homogeneous polynomial in the oscillators α i 3 , ζ 3 . In terms of the vertices V AB (5.97), algebraic constraint (5.112) takes the form

Parity invariant cubic vertices for three continuous-spin fields
In this section, we discuss parity invariant cubic vertices which involve three continuous-spin fields. According to our classification, such vertices can be separated into four particular cases given in (1.16)-(1.19). Let us discuss these particular cases in turn.

Two continuous-spin massless fields and one continuous-spin massive field
We start with discussion of parity invariant cubic vertices for two continuous-spin massless fields and one continuous-spin massive field. This is to say that, using the shortcut (m, κ) CSF for a mass-m continuous-spin field, we study cubic vertices for the following three fields: two continuous-spin massless fields and one continuous-spin massive field.
(6.1) Relation (6.1) tells us that the massless continuous-spin fields carry the external line indices a = 1, 2, while the massive continuous-spin field corresponds to a = 3.
For fields (6.1), we find the following general solution to cubic vertex p − [3] (see appendix D) In (6.2), we introduce eight vertices V ABC labelled by the superscripts A, B, C. In (6.3) and (6.4), the arguments of the generic vertex p − [3] and the vertices V ABC are shown explicitly. The definition of the arguments B a and α ab may be found in (A.3). Various quantities U appearing in (6.2) are differential operators w.r.t. the B a and α aa+1 . These quantities will be presented below. For eight vertices V ABC (6.4), we find the following solution: , a = 1, 2 , (6.6) where, I ν (x) and K ν (x) (6.6) are the modified Bessel functions, while the F (α, β, γ; x) (6.7) is the hypergeometric function. In (6.5), in place of the variables B 1 , B 2 , B 3 , we use new variables z 1 , z 2 , z 3 defined in (6.6) and (6.8). Operators ν a are defined below.
The generic vertex p − [3] (6.3) depends on the twelve variables, while, the vertices V ABC (6.5) depend only on the three variables. Note also that, by definition, the vertices V ABC (6.5) are expandable in the three variables α 12 , α 23 , α 31 . To complete the description of the vertex p − [3] we provide expressions for the operators U , ν a . These operators are
As we have already said, the vertices V ABC (6.5) should be expandable in the three variables α 12 , α 23 , α 31 . Using the simplified notation V = V ABC , we then note that a general representative of the vertex V can be chosen to be V = V (n 1 , n 2 , n 3 ) , V (n 1 , n 2 , n 3 ) = α n 3 12 α n 1 23 α n 2 31 , (6.19) n a ≥ 0, a = 1, 2, 3 . (6.20) The three integers n a in (6.19) are the freedom of our solution for the vertex V . In other words, the integers n a label all possible cubic vertices that can be constructed for three fields in (6.1). In order for vertices (6.19) to be sensible, the integers n a should satisfy the restrictions (6.20) which amount to the requirement that the powers of all variables α 12 , α 23 , α 31 in (6.19) be non-negative. Relations given in (6.2)-(6.18), (6.19), and (6.20) provide the complete description and classification of cubic interaction vertices that can be constructed for fields in (6.1).

One continuous-spin massless field and two continuous-spin massive fields with equal masses
In this section, we discuss parity invariant cubic vertices for two continuous-spin massive fields having equal masses and one continuous-spin massless field. This is to say that, using the shortcut (m, κ) CSF for a continuous-spin field, we study cubic vertices for the following three fields: two continuous-spin massive fields with equal masses and one continuous-spin massless field.
Relation (6.21) tells us that the massive continuous-spin fields having equal masses carry the external line indices a = 1, 2, while the continuous-spin massless field corresponds to a = 3. For fields (6.21), we find the following general solution to cubic vertex p − [3] (see appendix D) ABC , A, B = 1, . . . , 6 , C = +, − , In (6.22), we introduce vertices V ABC labelled by the superscripts A, B, C. In (6.23) and (6.24), the arguments of the generic vertex p − [3] and the vertices V ABC are shown explicitly. The definition of the arguments B a and α ab may be found in (A.3). Various quantities U appearing in (6.22) are differential operators w.r.t. the B a and α aa+1 . These quantities will be presented below. For the vertices V where a = 1, 2, while operators ν a and Ω 3 are defined below. Note that, for κ 2 a in (6.28), the σ a is purely imaginary.
The generic vertex p − [3] (6.23) depends on the twelve variables, while, the vertices V ABC (6.25) depend only on the three variables. Note also that, by definition, the vertices V ABC (6.25) are expandable in the three variables α 12 , α 23 , α 31 . To complete the description of the vertex p − [3] (6.22) we should provide expressions for the operators U , ν a , JHEP12(2018)055 Ω 3 . These operators are given by where quantitiesβ a , N Ba , N α ab , N a appearing in (6.31)-(6.43) are defined in (A.3)-(A.7). Using simplified notation for the vertices V = V ABC , we note that the general expression for the V can be presented as V = V (n 1 , n 2 , n 3 ) , V (n 1 , n 2 , n 3 ) = α n 3 12 α n 1 23 α n 2 31 , (6.44) n a ≥ 0, a = 1, 2, 3 . The three integers n a in (6.44) are the freedom of our solution for the vertex V . In other words these integers label all possible cubic vertices that can be constructed for three fields in (6.21). In order for vertices (6.44) to be sensible, the n a should satisfy the restrictions (6.45) which amount to the requirement that the powers of all variables α 12 , JHEP12(2018)055 α 23 , α 31 in (6.44) be non-negative. Relations given in (6.22)-(6.43), (6.44), and (6.45) provide the complete description and classification of cubic interaction vertices that can be constructed for fields in (6.21).

One continuous-spin massless field and two continuous-spin massive fields with nonequal masses
In this section, we discuss parity invariant cubic vertices for two continuous-spin massive fields having different masses and one continuous-spin massless field. This is to say that, using the shortcut (m, κ) CSF for a mass-m continuous-spin field, we study cubic vertices for the following three fields: two continuous-spin massive fields with nonequal masses and one continuous-spin massless field.
(6.46) Relation (6.46) tells us that the massive continuous-spin fields carry the external line indices a = 1, 2, while the continuous-spin massless field corresponds to a = 3. For fields (6.46), we find the following general solution to cubic vertex p − [3] (see appendix D) ABC (B a , α aa+1 ) . (6.49) In (6.47), we introduce eight vertices V ABC labelled by the superscripts A, B, C = ±. In (6.48) and (6.49), the arguments of the generic vertex p − [3] and the vertices V ABC are shown explicitly. The definition of the arguments B a and α ab may be found in (A.3). Various quantities U appearing in (6.47) are differential operators w.r.t. the B a and α aa+1 . These quantities will be presented below. For eight vertices V (7) ABC (6.49), we find the following solution: where F (α, β, γ; x) stands for the hypergeometric function, while I ν (x) and K ν (x) are the modified Bessel functions. In (6.50), in place of the variables B a , we use new variables z a defined in (6.52), (6.53). Operators ν a are defined below.

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From (6.48), we see that the generic vertex p − [3] depends on the twelve variables, while, from (6.50), we learn that the vertices V ABC depend only on the three variables. Note also that, by definition, the vertices V ABC (6.50) are expandable in the three variables α 12 , α 23 , α 31 . To complete our description of the vertex p − [3] (6.47) we should provide expressions for the operators U , ν a . These operators are given by Γ(ν +n) 4 n n!Γ(ν +2n) W n , (6.62) where quantitiesβ a , N Ba , N α ab , N a appearing in (6.54)-(6.64) are defined in (A.3)-(A.7). Using the notation for the vertices V = V ABC , we note that the general V can be presented as V (n 1 , n 2 , n 3 ) = α n 3 12 α n 1 23 α n 2 31 , (6.65) n a ≥ 0, a = 1, 2, 3 . The three integers n a in (6.65) are the freedom of our solution for the vertex V . These integers label all possible cubic vertices that can be constructed for three fields in (6.46). In order for vertices (6.65) to be sensible, the integers n a should satisfy the restrictions (6.66) which amount to the requirement that the powers of all variables α 12 , α 23 , α 31 in (6.65)

Three continuous-spin massive fields
Finally, we discuss parity invariant cubic vertices for three continuous-spin massive fields. Using shortcut (m, κ) CSF for a continuous-spin massive field, we study cubic vertices for the following three fields: For fields (6.67), we find the following general solution to cubic vertex p − [3] (see appendix D) In (6.68), we introduce eight vertices V ABC labelled by the superscripts A, B, C. In (6.69) and (6.70), the arguments of the generic vertex p − [3] and the vertices V ABC are shown explicitly. The definition of the arguments B a and α ab may be found in (A.3). Various quantities U appearing in (6.68) are differential operators w.r.t. the B a and α aa+1 . These operators are given below. For eight vertices V (7) ABC (6.70), we find the following solution: where the F (α, β, γ; x) is the hypergeometric function. In (6.71), in place of the variables B a , we use new variables z a defined in (6.73). Quantities ν a and D are defined below in (6.82), (6.84).
The generic vertex p − [3] (6.69) depends on the twelve variables, while, the vertices V ABC (6.71) depend only on the three variables. Note also that, by definition, the vertices V ABC (6.71) are expandable in the three variables α 12 , α 23 , α 31 . To complete our description of the vertex p − [3] (6.69) we should provide expressions for the operators U , ν a .

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These operators are given by Γ(ν +n) 4 n n!Γ(ν +2n) W n , (6.80) The three integers n a in (6.85) are the freedom of our solution for the vertex V . These integers label all possible cubic vertices that can be constructed for three fields in (6.67). In order for vertices (6.85) to be sensible, the integers n a should satisfy the restrictions (6.86) which amount to the requirement that the powers of all variables α 12 , α 23 , α 31 in (6.85) be non-negative. Relations given in (6.68)-(6.84), (6.85), and (6.86) provide the complete description and classification of cubic interaction vertices that can be constructed for fields in (6.67).

JHEP12(2018)055 7 Conclusions
In this paper, we used the light-cone gauge approach to construct the parity invariant cubic vertices for continuous-spin massive/massless fields and arbitrary spin massive/massless fields. We investigated three types of the parity invariant cubic vertices: a) vertices describing coupling of one continuous-spin massive/massless field to two arbitrary spin massive/massless fields; b) vertices describing coupling of two continuous-spin massive/massless fields to one arbitrary spin massive/massless field; c) vertices for self-interacting massive/massless continuous-spin fields. We obtained the complete list of such cubic vertices. With exception of cubic vertices for massless self-interacting continuous-spin field, results in this paper together with the ones in ref. [18] provide the exhaustive solution to the problem of description of all parity invariant cubic vertices for coupling of massive/massless continuous-spin fields to arbitrary spin massive/massless fields as well as vertices for selfinteracting massive/massless continuous-spin fields. Results in this paper might have the following applications and generalizations.
i) Sometimes light-cone gauge formulation turns out to be good starting point for deriving Lorentz covariant formulations. It is the parity invariant light-cone gauge vertices that turn out to convenient for deriving of Lorentz covariant and BRST gauge invariant formulations [36]. For example, all light-cone gauge vertices for massive/massless fields in ref. [27] have straightforwardly been cast into BRST gauge invariant form in ref. [37]. 12 Various BRST formulations of free continuous-spin fields were discussed in refs. [43]- [45]. We expect therefore that our results in this paper may be good starting point for deriving Lorentz covariant and BRST gauge invariant formulations of vertices for interacting continuous-spin fields. Various applications of BRST approach for studying interacting finite-component fields may be found, e.g., in refs. [46][47][48][49].
ii) From the perspective of investigation of interrelations between theory of continuousspin fields and string theory it is important to extend our study to the case of mixed-symmetry fields. During last time, various interesting descriptions of mixedsymmetry fields were obtained in refs. [50]- [53]. For example, light-cone gauge description in ref. [50] and methods in this paper provide opportunity for investigation of interacting mixed-symmetry continuous-spin fields. Also, for finite-component mixed-symmetry fields, we mention interesting formulations developed in refs. [54][55][56][57][58][59][60][61][62][63][64]. Generalization of these formulations to the case continuous-spin fields could also be of some interest.
iii) In this paper, we investigated cubic vertices for light-cone gauge continuous-spin fields propagating in flat space. Extension of our investigation to the case of continuousspin fields in AdS space [14,15] could be of some interest. In this respect we note that light-cone gauge formulation of free continuous-spin AdS fields was recently obtained in ref. [50], while for finite component light-cone gauge AdS fields, the systematic method for building cubic vertices was developed in ref. [65]. We think therefore
iv) Generalization of our results to the case of interacting supersymmetric continuousspin field theories could be of great interest. As is known, for finite component fields, supersymmetry imposes additional constraints and leads to more simple interacting vertices. We expect therefore that, for continuous-spin field, supersymmetry might simplify a structure of interactions vertices. Supermultiplets for continuous-spin representations are considered in refs. [3,16]. For finite-component fields, recent study of higher-spin supersymmetric theories can be found in refs. [82][83][84][85]. Use of twistor-like variables for discussion of supersymmetric theories turns out to be helpful. Recent various interesting applications of twistor-like variables may be found in refs. [86][87][88][89][90].
Finally, we note that investigation of various algebraic aspects of continuous-spin field theory along the line in refs. [91][92][93] could also be very interesting.

Acknowledgments
This work was supported by Russian Science Foundation grant 14-42-00047.

A Notation and conventions
Unless otherwise specified, the vector indices of the so(d − 2) algebra i, j, k, l run over 1, . . . , d−2. We refer to creation operators α i , υ, ζ and the respective annihilation operators α i ,ῡ,ζ as oscillators. Our conventions for the commutation relations, the vacuum |0 , and hermitian conjugation rules are as follows Throughout this paper we use the following definitions for momentum P i and quantities B a , α ab

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where β a ≡ β a+3 . Our notation for the scalar product of the oscillators and various quantities constructed out of the B a , α ab and derivatives of the B a , α ab are as follows for massless continuous-spin field (A.11) , for massive continuous-spin field (A.12) Γ(ν +n) 4 n n!Γ(ν +2n) W n , (A.14) Our notation for the quantities constructed out of the masses m a and continuous-spin parameters κ a are as follows B Derivation of vertices p − [3] (4.2), (4.26), (4.46) We consider three vertices in (3.21)-(3.23). For these vertices, we outline the derivation of the respective solutions in (4.2), (4.26), (4.46). We split our derivation in several steps.
For ν a (A.8) and W -operators (C.28), we note the relations which admit us to get G 1 , G 2 in (C.30), Equations G 1 V (8) = 0 and G 2 V (8) = 0 with G 1 and G 2 as in (C.31) constitute a system of two decoupled second-order differential equations w.r.t. B 1 and B 2 . Four independent solutions of these equations are given in (5.5).
Step 3. We make the transformation

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Realization of G 1 , G 2 (C.39) on V (5) (C.41) takes the form Step 4. We make the transformation where N Z = Z∂ Z , while e a are given in (A.17). Realization of G 1 , G 2 (C.42) on V (6) (C.43) takes the form Step 5. We make the transformations where we use notation in (C.44), (C.46), (A.14). Realization of G a (C.45) on V (7) (C.47) takes the form For ω a (C.44) and W -operators (C.46), we note the relations which admit us to get G a in (C.48), Equations G 1 V (7) = 0 and G 2 V (7) = 0 with G 1 and G 2 as in (C.48) constitute a system of two decoupled second-order differential equations w.r.t. B 1 and B 2 . All independent solutions of these equations are given in (5.29).

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Dependence of V (3) on β a . We fix dependence of V (3) on β a . To this end we use eqs. (3.36), (3.40) and relation (D.7) to note the following equations for the vertex V (3) : We now use results above-obtained for the study of four vertices in (3.29) in turn.
Step 2. We make the transformation
E One continuous-spin massless field, one arbitrary spin massless field, and one arbitrary spin massive field We use the shortcut (0, κ) CSF for a continuous-spin massless field, while the shortcuts (0, s) and (m, s) are used for the respective spin-s massless and spin-s massive field. We now consider cubic vertices for the following three fields: (0, s 1 )-(m 2 , s 2 )-(0, κ 3 ) CSF m 2 2 > 0 , one massless field, one massive field, and one continuous-spin massless field. (E.1) Relation (E.1) tells us that spin-s 1 massless and spin-s 2 massive fields carry the respective external line indices a = 1 and a = 2, while the continuous-spin massless field corresponds to a = 3.

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Expressions (E.2)-(E.15) provide the complete generating form description of cubic vertices for coupling of one continuous-spin massless field to two chains of massless and massive fields (2.13). Now we describe cubic vertices for coupling of one continuous-spin massless field to massless and massive fields having the respective arbitrary but fixed spins 1 and spin-s 2 values. Using the first algebraic constraints in (2.11), (2.12), it is easy to see that vertices we are interested in must satisfy the algebraic constraints 16) In terms of the vertices V ± (E.5), constraints (E.16) take the form where to simplify our presentation we drop the superscripts ± and use a vertex V in place of the vertices V ± , i.e., we use V = V ± . Doing so, we note that the general solution to constraints (E.17) can be presented as Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.