Covariant Cubic Interacting Vertices for Massless and Massive Integer Higher Spin Fields

Abstract

We develop the BRST approach to construct the general off-shell local Lorentz covariant cubic interaction vertices for irreducible massless and massive higher spin fields on d-dimensional Minkowski space. We consider two different cases for interacting higher spin fields: with one massive and two massless; two massive, both with coinciding and with different masses and one massless field of spins $s_1,s_2,s_3$. Unlike the previous results on cubic vertices we extend our earlier result in (Buchbinder, I.L.; et al. Phys. Lett. B 2021, 820, 136470) for massless fields and employ the complete BRST operator, including the trace constraints, which is used to formulate an irreducible representation with definite integer spin. We generalize the cubic vertices proposed for reducible higher spin fields in (Metsaev, R.R. Phys. Lett. B 2013, 720, 237) in the form of multiplicative and non-multiplicative BRST-closed constituents and calculate the new contributions to the vertex, which contains the additional terms with a smaller number of space-time derivatives. We prove that without traceless conditions for the cubic vertices in (Metsaev, R.R. Phys. Lett. B 2013, 720, 237) it is impossible to provide the noncontradictory Lagrangian dynamics and find explicit traceless solution for these vertices. As the examples, we explicitly construct the interacting Lagrangians for the massive spin of the s field and the massless scalars, both with and without auxiliary fields. The interacting models with different combinations of triples higher spin fields: massive spin s with massless scalar and vector fields and with two vector fields; massless helicity $\lambda$ with massless scalar and massive vector fields; two massive fields of spins $s, 0$ and massless scalar is also considered.

1. Introduction

The construction of interacting higher spin field theory attracts significant attention both from a general theoretical point of view and in connection with the possibilities of discovering new approaches to describe gravity at the quantum level (see for a review, e.g., [1,2,3,4,5,6,7] and the references therein). The extension of General Relativity on a base of local supersymmetry principle up to the supergravity models [8] with improved quantum properties and a connection with (Super)string Field Theory permits one to include massless fields of spins $s>2$ in Higher Spin Gravity (see [9] and references therein) with respecting the string field theory properties, asymptotic safety and some others. The AdS/CFT correspondence gives strong indications that higher spin excitations can be significant to elaborate the quantum gravity challenges [10]. Interacting massive and massless higher spin fields in constant-curvature spaces provide another possible insight into the origin of Dark Matter and Dark Energy [11,12] beyond the models with vector massive fields [13] and sterile neutrinos [14] to be by reasonable candidates for Dark Matter, see for reviews [15,16,17].

The simplest of higher spin interactions, the cubic vertex for various fields with higher spins, has been studied by many authors with the use of different approaches (see, e.g., the recent papers [18,19,20,21,22,23,24,25,26,27,28,29,30,31] and the references therein) (A complete list of papers on a cubic vertex on constant curvature spaces contains dozens of papers. Here, we cite only the recent papers containing a full list of references). Note, the results on the structure of cubic vertices obtained in terms of physical degrees of freedom in a concise form in the light-cone approach in [31,32]. In the covariant metric-like form, the list of cubic vertices for reducible representations of the Poincare group with discrete spins (being consistent with [32]) are contained in [18], where the cubic vertices were derived using the constrained BRST approach, but without imposing on the vertex the algebraic constraints. The latter peculiarity leads to the violation of the irreducibility of the representation for interacting higher spin fields and, hence, to a possible change of the number of physical degrees of freedom (Without finding the solution for the vertex respecting the algebraic constraints, the number of physical degrees of freedom, which is determined by one of the independent initial data for the equations of motion for the interacting model is different (less) than as one from that for the undeformed model with vanishing algebraic constraints evaluated on respecting equations of motion, but with the deformed gauge symmetry not respecting these constraints). Also, we point out the constructions of cubic vertices within the BRST approach without the use of constraints responsible for trace conditions in the BRST charge (see e.g., [30] and the references therein). It means, in fact, that the vertex is obtained in terms of reducible higher spin fields (To avoid various misunderstandings, we emphasize that we use the term “unconstrained formulation” in the sense that all possible constraints are consequences of the Lagrangian equations of motion. No additional restrictions, separate from the equations of motion, are imposed).

In this paper, we derive the cubic vertices for irreducible massless and massive higher spin fields focusing on the manifest Lorentz covariance. The analysis is carried out within the BRST approach with complete BRST operator that extends our earlier approaches [33,34,35] and involves a converted set of operator constraints forming a first-class gauge algebra. The set of constraints includes on equal-footing the on-shell condition $l_0$ and constraints $l_1,l_{11}$, responsible for divergences and traces. Unlike our consideration, in the constrained BRST approach, the operator $l_{11}$ is imposed as a constraint on the set of fields and gauge parameters outside of the Lagrangian formulation for simplicity of calculations. Such an approach inherits the way of obtaining the Lagrangian formulation for higher spin fields from the tensionless limit [36] for (super)string theory with resulting in a BRST charge without the presence of the algebraic (e.g., trace) constraints. We have already noted [33] that this way of consideration is correct but the actual Lagrangian description of irreducible fields is achieved only after additional imposing the subsidiary conditions which are not derived from the Lagrangian. Of course, the Lagrangian formulations for the same free irreducible higher spin field in Minkowski space obtained in constrained and unconstrained BRST approaches are equivalent [37] (For irreducible massless and massive field representations with half-integer spin the Lagrangians with reducible gauge symmetries and compatible holonomic constraints, were firstly obtained therein). However, the corresponding equivalence has not yet been proved for interacting irreducible higher-spin fields as it was recently demonstrated for massless case [33,34] for cubic vertices. Aspects of the BRST approach with complete BRST operator for a Lagrangian description of various free and interacting massive higher spin field models in Minkowski and AdS spaces were developed in many works (e.g., see the papers [38,39,40,41,42,43,44,45], and the review [3]).

As a result, we face the problem when constructing the cubic vertex for irreducible massless and massive higher integer spin fields on d-dimensional flat space-time within metric-like formalism on the base of the complete BRST operator. It is exactly the problem that we intend to consider in the paper. We expect that the final cubic vertices will contain new terms (as compared with [18]) with the traces of the fields. Such new terms may evidently have significance when gauging away auxiliary gauge symmetry and fields to obtain a component Lagrangian formulation.

The aim of the paper is to present a complete solution of the above problem for the cubic vertices for unconstrained irreducible massless and massive higher spin fields within BRST approach and to obtain from general oscillator-like vertices explicit tensor representations for Lagrangian formulations with reducible gauge symmetry for some triples of interacting higher spin fields.

The paper has the following organization. Section 2 presents the basics of a BRST Lagrangian construction for a free massive higher spin field, with all constraints $l_0,l_1,l_{11}$ taken into account. In Section 3, we deduce a system of equations for a cubic (linear) deformation in fields of the free action (free gauge transformations). A solution for the deformed cubic vertices and gauge transformation is given in a Section 4 for one massive and two massless fields; for two massive with different and coinciding masses and one massless field. The number of examples for the fields with a special set of spins are presented in the Section 5. The main result of the work is that the cubic vertices and deformed reducible gauge transformations include both types of constraints: with derivative $l_1$ and one with trace $l_{11}$. In conclusion, a final summary with comments is given. A derivation of Singh–Hagen Lagrangian from free BRST Lagrangian formulation for the massive field of spin s presented in Appendix A. Appendix B and Appendix C contain results of calculations for component interacting Lagrangian and gauge transformations for massive field of spin s with massless scalars and with massless vector and scalar. In Appendix D we formulate conditions for the incomplete BRST operator, traceless constraints and cubic vertices to obtain non-contradictory Lagrangian dynamics for a model with interacting fields with given spins. We find the form of projectors ${\prod }_{j=1}^3{P}_{0|11}^{\left(j\right)}$ for .respective cubic vertices $|V_c^{\left(3\right)}{\rangle }_{{\left(s\right)}_3}^{{\left(m\right)}_3}$ from [18] to have the cubic vertices $|\overline{V}{}_c{}^{\left(3\right)}\rangle _{{\left(s\right)}_3}^{{\left(m\right)}_3}$, firstly determined by (A77) and (A79) for irreducible interacting fields. We use the usual definitions and notations from the work [33] for a metric tensor ${\eta }_{\mu \nu }=diag(+,-,\dots ,-)$ with Lorentz indices $\mu ,\nu =0,1,\dots ,d-1$ and the respective notation $ϵ\left(F\right)$, $gh\left(F\right)$, $[F,G\}$, $\left[x\right]$, ${\left(s\right)}_3$ for the values of Grassmann parity and ghost number of a homogeneous quantity F, as well as the supercommutator, ${\theta }_{k,l}$$=1\left(0\right)$, when $k>l(k\le l)$, ${\left(s\right)}_3$ is the Heaviside $\theta$-symbol.

2. Lagrangian Formulation for Free Massive Higher Spin Fields

Here, we present the basics of the BRST approach to free massive higher integer spin field theory for its following use to construct a general cubic interacting vertex.

The unitary massive irreducible representations of Poincare $ISO(1,d-1)$ group with integer spins s can be realized using the real-valued totally symmetric tensor fields ${\varphi }_{{\mu }_1\dots {\mu }_s}\left(x\right)\equiv {\varphi }_{\mu \left(s\right)}$ subject to the conditions

$$\begin{array}{ccc}& & \left({\partial }^{\nu }{\partial }_{\nu }+m^2,{\partial }^{{\mu }_1},{\eta }^{{\mu }_1{\mu }_2}\right){\varphi }_{\mu \left(s\right)}=(0,0,0)⟺\hfill \\ & & \left(l_0,l_1,l_{11},g_0-d/2\right)|\varphi \rangle =(0,0,0,s)|\varphi \rangle .\hfill \end{array}$$

(1)

The basic vectors $|\varphi \rangle$ and the operators $l_0,l_1,l_{11},g_0-d/2$ above are defined in the Fock space $\mathcal{H}$ with the Grassmann-even oscillators $a_{\mu },a_{\nu }^{+}$, ($[a_{\mu },a_{\nu }^{+}]=-{\eta }_{\mu \nu }$) as follows

$$\begin{array}{ccc}& & |\varphi \rangle =\sum _{s\ge 0}\frac{{ı}^s}{s!}{\varphi }^{\mu \left(s\right)}\prod _{i=1}^s{a}_{{\mu }_i}^{+}|0\rangle ,\hfill \\ & & \left(l_0,l_1,l_{11},g_0\right)=({\partial }^{\nu }{\partial }_{\nu }+m^2,-ıa^{\nu }{\partial }_{\nu },\frac{1}{2}{a}^{\mu }{a}_{\mu },-\frac{1}{2}\left\{a_{\mu }^{+},a^{\mu }\right\}).\hfill \end{array}$$

(2)

The free dynamics of the field with definite spin s in the framework of the BRST approach is described by the first-stage reducible gauge theory with the gauge invariant action given on the configuration space $M_{cl}^{\left(s\right)}$ whose dimension grows with the growth of “s”, thus, including the basic field ${\varphi }_{\mu \left(s\right)}$ with many auxiliary fields ${\varphi }_{1\mu (s-1)},\dots$ of lesser than s ranks. All these fields are incorporated into the vector ${|\chi \rangle }_s$ and the dynamics is encoded by the action

$$\begin{array}{c}\hfill {\mathcal{S}}_{0|s}^m[\varphi ,{\varphi }_1,\dots ]={\mathcal{S}}_{0|s}^m{\left[\right|\chi \rangle }_s]=\int d{\eta }_0{}_s\langle \chi \left|KQ\right|\chi \rangle _s,\end{array}$$

(3)

where ${\eta }_0$ and K are, respectively, a zero-mode ghost field and an operator defining the inner product. The action (3) is invariant under the reducible gauge transformations

$$\begin{array}{c}\hfill {\delta |\chi \rangle }_s={Q|\mathrm{\Lambda }\rangle }_s{,\delta |\mathrm{\Lambda }\rangle }_s=Q|{\mathrm{\Lambda }}^1{\rangle }_s,\delta {|{\mathrm{\Lambda }}^1\rangle }_s=0,\end{array}$$

(4)

with ${|\mathrm{\Lambda }\rangle }_s$, $|{\mathrm{\Lambda }}^1{\rangle }_s$ to be the vectors of zero-level and first-level gauge parameters of the abelian gauge transformations (4). The quantity Q in (3) is the BRST operator having the same structure as one for massless case [33] constructed on the base of the constraints $l_0,l_1,l{}_1{}^{+},l{}_{11},l{}_{11}{}^{+}=\frac{1}{2}{a}^{+\nu }{a}_{\nu }^{+}$ with the Grassmann-odd ghost operators ${\eta }_0,{\eta }_1^{+},{\eta }_1,{\eta }_{11}^{+},{\eta }_{11}$, $\mathcal{P}$, ${\mathcal{P}}_1$, ${\mathcal{P}}_1^{+},$${\mathcal{P}}_{11},{\mathcal{P}}_{11}^{+},$

$$\begin{array}{ccc}& & Q={\eta }_0{l}_0+{\eta }_1^{+}{\check{l}}_1+{\check{l}}_1^{+}{\eta }_1+{\eta }_{11}^{+}{\widehat{L}}_{11}+{\widehat{L}}_{11}^{+}{\eta }_{11}+ı{\eta }_1^{+}{\eta }_1{}_0{}^{\mathcal{P}},\hfill \end{array}$$

(5)

where

$$\begin{array}{ccc}\hfill & & \left({\check{l}}_1,{\check{l}}_1^{+}\right)=\left(l_1+md,l_1^{+}+m{d}^{+}\right),\left({\widehat{L}}_{11},{\widehat{L}}_{11}^{+}\right)=\left({\check{L}}_{11}+{\eta }_1{\mathcal{P}}_1,\check{L}{}_{11}{}^{+}+{\mathcal{P}}_1^{+}{\eta }_1^{+}\right).\hfill \end{array}$$

(6)

Here

$$\begin{array}{c}\hfill {\check{L}}_{11}=l_{11}-(1/2){\left(d\right)}^2+(b^{+}b+h)b,\check{L}{}_{11}{}^{+}=l_{11}^{+}-(1/2){\left(d^{+}\right)}^2+b^{+}\end{array}$$

(7)

and $(ϵ,gh)Q=(1,1).$ The algebra of the operators $l_0$, $l_1$, $l_1^{+},{\check{L}}_{11},{\check{L}}_{11}^{+},G_0$ looks like

$$[l_0,l_1^{(+)}]=0,[l_1,l_1^{+}]=l_0-m^2\mathrm{and}[{\check{L}}_{11},{\check{L}}_{11}^{+}]=G_0,[G_0,{\check{L}}_{11}^{+}]=2{\check{L}}_{11}^{+}$$

(8)

and their independent non-vanishing cross-commutators are $[l_1,{\check{L}}_{11}^{+}]=-l_1^{+}$, $[l_1,G_0]=l_1$.

The ghost operators satisfy the non-zero anticommuting relations

$$\{{\eta }_0,{\mathcal{P}}_0\}=ı,\{{\eta }_1,{\mathcal{P}}_1^{+}\}=\{{\eta }_1^{+},{\mathcal{P}}_1\}=\{{\eta }_{11},{\mathcal{P}}_{11}^{+}\}=\{{\eta }_{11}^{+},{\mathcal{P}}_{11}\}=1.$$

(9)

The theory is characterized by the spin operator $\sigma$, which is defined according to

$$\begin{array}{ccc}\hfill & & \sigma =G_0+{\eta }_1^{+}{\mathcal{P}}_1-{\eta }_1{\mathcal{P}}_1^{+}+2({\eta }_{11}^{+}{\mathcal{P}}_{11}-{\eta }_{11}{\mathcal{P}}_{11}^{+})\hfill \\ \hfill & & G_0=g_0+d^{+}d+2b^{+}b+\frac{1}{2}+h.\hfill \end{array}$$

(10)

Here, $d,d^{+}$, $b,b^{+}$ ($[d,d^{+}]=1$, $[b,b^{+}]=1$) are two pairs of auxiliary Grassmann-even oscillators. The operator $\sigma$ selects the vectors with definite spin value s

$$\begin{array}{ccc}\hfill & & \sigma {\left(\right|\chi \rangle }_s{,|\mathrm{\Lambda }\rangle }_s,|{\mathrm{\Lambda }}^1{\rangle }_s)=(0,0,0),\hfill \end{array}$$

(11)

where the standard distribution for Grassmann parities and the ghost numbers of these vectors are $(0,0)$, $(1,-1),$$(0,-2)$, respectively.

All the operators above act in a total Hilbert space with the scalar product of the vectors depending on all oscillators $(A;A^{+})$ = $(a^{\mu },b,d;a^{\mu +},b^{+},d^{+})$ and ghosts

$$\begin{array}{ccc}& & \langle \chi |\psi \rangle =\int d^{d}x\langle 0|{\chi }^{*}\left(A;{\eta }_0,{\eta }_1,{\mathcal{P}}_1,{\eta }_{11},{\mathcal{P}}_{11}\right)\psi \left(A^{+};{\eta }_0,{\eta }_1^{+},{\mathcal{P}}_1^{+},{\eta }_{11}^{+},{\mathcal{P}}_{11}^{+}\right)|0\rangle .\hfill \end{array}$$

(12)

The operators $Q, \sigma$ are supercommuting and Hermitian with respect to the scalar product (12) including the operator K (see e.g., [37,39,45]) being equal to 1 on Hilbert subspace not depending on auxiliary $b,b^{+}$ operators

$$\begin{array}{cc}{Q}^2={\eta }_{11}^{+}{\eta }_{11}\sigma ,\hfill & [Q,\sigma \}=0;\hfill \end{array}$$

(13)

$$\begin{array}{cc}{Q}^{+}K=KQ,\hfill & {\sigma }^{+}K=K\sigma ,\hfill \end{array}$$

(14)

$$\begin{array}{cc}K=1\otimes \sum _{n=0}^{\infty }\frac{1}{n!}\left(b^{+}\right)n|0\rangle \langle 0|b^{n}C(n,h\left(s\right)),\hfill & C(n,h\left(s\right))\equiv \prod _{i=0}^{n-1}(i+h\left(s\right))\hfill \end{array}$$

(15)

The BRST operator Q is nilpotent on the subspace with zero eigenvectors for the spin operator $\sigma$ (11).

The field ${|\chi \rangle }_s$, the zero ${|\mathrm{\Lambda }\rangle }_s$ and the first $|{\mathrm{\Lambda }}^1{\rangle }_s$ level gauge parameters labeled by the symbol $“s”$ as eigenvectors of the spin condition in (11) has the same decomposition as ones in [33] but with ghost-independent vectors $|{\mathrm{\Phi }}_{\dots }{\rangle }_{s-\dots }$, $|{\mathrm{\Xi }}_{\dots }{\rangle }_{s-\dots }$ instead of $|{\varphi }_{\dots }{\rangle }_{s-\dots }$, $|{\mathrm{\Xi }}_{\dots }{\rangle }_{s-\dots }$

$$\begin{array}{ccc}\hfill & & {|\chi \rangle }_s={|\mathrm{\Phi }\rangle }_s+{\eta }_1^{+}\left({\mathcal{P}}_1^{+}|{\mathrm{\Phi }}_2{\rangle }_{s-2}+{\mathcal{P}}_{11}^{+}|{\mathrm{\Phi }}_{21}{\rangle }_{s-3}+{\eta }_{11}^{+}{\mathcal{P}}_1^{+}{\mathcal{P}}_{11}^{+}{|{\mathrm{\Phi }}_{22}\rangle }_{s-6}\right)\hfill \\ \hfill & & \phantom{|{\chi }_c^0{\rangle }_s}+{\eta }_{11}^{+}\left({\mathcal{P}}_1^{+}|{\mathrm{\Phi }}_{31}{\rangle }_{s-3}+{\mathcal{P}}_{11}^{+}{|{\mathrm{\Phi }}_{32}\rangle }_{s-4}\right)+{\eta }_0({\mathcal{P}}_1^{+}|{\mathrm{\Phi }}_1{\rangle }_{s-1}+{\mathcal{P}}_{11}^{+}{|{\mathrm{\Phi }}_{11}\rangle }_{s-2}\hfill \\ \hfill & & \phantom{|{\chi }_c^0{\rangle }_s}+{\mathcal{P}}_1^{+}{\mathcal{P}}_{11}^{+}\left[{\eta }_1^{+}|{\mathrm{\Phi }}_{12}{\rangle }_{s-4}+{\eta }_{11}^{+}{|{\mathrm{\Phi }}_{13}\rangle }_{s-5}\right]),\hfill \end{array}$$

(16)

$$\begin{array}{ccc}\hfill & & {|\mathrm{\Lambda }\rangle }_s={\mathcal{P}}_1^{+}{|\mathrm{\Xi }\rangle }_{s-1}+{\mathcal{P}}_{11}^{+}|{\mathrm{\Xi }}_1{\rangle }_{s-2}+{\mathcal{P}}_1^{+}{\mathcal{P}}_{11}^{+}({\eta }_1^{+}{|{\mathrm{\Xi }}_{11}\rangle }_{s-4}\hfill \\ \hfill & & \phantom{|{\chi }^1{\rangle }_s}+{\eta }_{11}^{+}|{\mathrm{\Xi }}_{12}{\rangle }_{s-5})+{\eta }_0{\mathcal{P}}_1^{+}{\mathcal{P}}_{11}^{+}{|{\mathrm{\Xi }}_{01}\rangle }_{s-3},\hfill \end{array}$$

(17)

$$\begin{array}{ccc}\hfill & & |{\mathrm{\Lambda }}^1{\rangle }_s={\mathcal{P}}_1^{+}{\mathcal{P}}_{11}^{+}{|{\mathrm{\Xi }}^1\rangle }_{s-3}.\hfill \end{array}$$

(18)

Here,

$$\begin{array}{ccc}\hfill |{\mathrm{\Phi }}_{\dots }{\rangle }_{s-\dots }& =& \sum _{l=0}^{[s-../2]-l}\frac{{\left(b^{+}\right)}^l}{l!}\sum _{k=0}^{s-2l\dots }\frac{{\left(d^{+}\right)}^k}{k!}|{\varphi }_{\dots |l,k}\left(a^{+}\right){\rangle }_{s-k-2l\dots },\mathrm{for}|{\varphi }_{|0,0}\left(a^{+}\right){\rangle }_s\equiv {|\varphi \rangle }_s,\hfill \end{array}$$

(19)

$$\begin{array}{ccc}\hfill |{\mathrm{\Xi }}_{\dots }{\rangle }_{s-\dots }& =& \sum _{l=0}^{[s-\dots /2]-l}\frac{{\left(b^{+}\right)}^l}{l!}\sum _{k=0}^{s-2l\dots }\frac{{\left(d^{+}\right)}^k}{k!}{|{\mathrm{\Xi }}_{\dots |l,k}\left(a^{+}\right)\rangle }_{s-k-2l\dots }.\hfill \end{array}$$

(20)

We prove in Appendix A that after imposing the appropriate gauge conditions and eliminating the auxiliary fields with the help of the equations of motion, the theory under consideration is reduced to an ungauged form equivalent to the Singh–Hagen action [46] in terms of a totally symmetric traceful tensor field ${\varphi }^{\mu \left(s\right)}$ and auxiliary traceful ${\varphi }_1^{\mu (s-3)}$.

Now we turn to the interacting theory.

3. System of Equations for Cubic Vertex

Here, we follow the general scheme developed for massless case in [33] to find the cubic interaction vertices for the models with one massive and two massless higher spin fields, two massive and one massless higher spin field with different mass value distributions and derive the equations for these vertices.

To include the cubic interaction we introduce three vectors $|{\chi }^{\left(i\right)}{\rangle }_{s_i}$, gauge parameters $|{\mathrm{\Lambda }}^{\left(i\right)}{\rangle }_{s_i}$, $|{\mathrm{\Lambda }}^{\left(i\right)1}{\rangle }_{s_i}$ with corresponding vacuum vectors ${|0\rangle }^i$ and oscillators, where $i=1,2,3$. It permits to define the deformed action and the deformed gauge transformations as follows

$$\begin{array}{ccc}& & S_{{\left[1\right]|\left(s\right)}_3}^{{\left(m\right)}_3}[{\chi }^{\left(1\right)},{\chi }^{\left(2\right)},{\chi }^{\left(3\right)}]=\sum _{i=1}^3{\mathcal{S}}_{0|s_i}^{m_i}+g\int \prod _{e=1}^{3}d{\eta }_0^{\left(e\right)}\left({}_{s_e}\langle {\chi }^{\left(e\right)}{K}^{\left(e\right)}|V^{\left(3\right)}\rangle _{{\left(s\right)}_3}^{{\left(m\right)}_3}+h.c.\right),\hfill \end{array}$$

(21)

$$\begin{array}{ccc}& & {\delta }_{\left[1\right]}|{\chi }^{\left(i\right)}{\rangle }_{s_i}=Q^{\left(i\right)}|{\mathrm{\Lambda }}^{\left(i\right)}{\rangle }_{s_i}-g\int \prod _{e=1}^{2}d{\eta }_0^{(i+e)}({}_{s_{i+1}}\langle {\mathrm{\Lambda }}^{\left(i+1\right)}{K}^{(i+1)}|{}_{s_{i+2}}\langle {\chi }^{\left(i+2\right)}{K}^{(i+1)}|\hfill \\ & & \phantom{{\delta }_{\left[1\right]}|{\chi }^{\left(i\right)}{\rangle }_{s_i}}+(i+1\leftrightarrow i+2)\left)\right|\tilde{V}{}^{\left(3\right)}\rangle _{{\left(s\right)}_3}^{{\left(m\right)}_3},\hfill \end{array}$$

(22)

$$\begin{array}{ccc}& & {\delta }_{\left[1\right]}|{\mathrm{\Lambda }}^{\left(i\right)}{\rangle }_{s_i}=Q^{\left(i\right)}|{\mathrm{\Lambda }}^{\left(i\right)1}{\rangle }_{s_i}-g\int \prod _{e=1}^{2}d{\eta }_0^{(i+e)}({}_{s_{i+1}}\langle {\mathrm{\Lambda }}^{(i+1)1}{K}^{(i+1)}|{}_{s_{i+2}}\langle {\chi }^{\left(i+2\right)}{K}^{(i+1)}|\hfill \\ & & \phantom{{\delta }_{\left[1\right]}|{\chi }^{\left(i\right)}{\rangle }_{s_i}}+(i+1\leftrightarrow i+2))|\widehat{V}{}^{\left(3\right)}\rangle _{{\left(s\right)}_3}^{{\left(m\right)}_3}\hfill \end{array}$$

(23)

with some unknown three-vectors $|V^{\left(3\right)}{\rangle }_{{\left(s\right)}_3}^{{\left(m\right)}_3},|\tilde{V}{}^{\left(3\right)}{\rangle }_{{\left(s\right)}_3}^{{\left(m\right)}_3},|\widehat{V}{}^{\left(3\right)}\rangle _{{\left(s\right)}_3}^{{\left(m\right)}_3}.$ Here, ${\mathcal{S}}_{0|s_i}^{m_i}$ is the free action (3) for the field $|{\chi }^{\left(i\right)}{\rangle }_{s_i}$, $Q^{\left(i\right)}$ is the BRST charge corresponding to spin $s_i,i=1,2,3$, $K^{\left(i\right)}$ is the operator K (15) corresponding to spin $s_i,i=1,2,3$ for massive and with change $h\left(s\right)\to h\left(s\right)+1/2$ for massless field and g is a deformation parameter (called usually as a coupling constant). Also, we use the notation ${\left(m\right)}_3\equiv (m_1,m_2,m_3)$ and convention $[i+3\simeq i]$.

The concrete construction of the cubic interaction means finding the concrete vectors $|V^{\left(3\right)}{\rangle }_{{\left(s\right)}_3}^{{\left(m\right)}_3}$, $|\tilde{V}{}^{\left(3\right)}\rangle _{{\left(s\right)}_3}^{{\left(m\right)}_3}$, $|\widehat{V}{}^{\left(3\right)}\rangle _{{\left(s\right)}_3}^{{\left(m\right)}_3}$. For this purpose, we can involve the set of fields, the constraints, and ghost operators related with spins $s_1,s_2,s_3$ and the respective conditions of gauge invariance of the deformed action under the deformed gauge transformations as well as the conservation of the form of the gauge transformations for the fields $|{\chi }^{\left(i\right)}{\rangle }_{s_i}$ under the gauge transformations ${\delta }_{\left[1\right]}|{\mathrm{\Lambda }}^{\left(i\right)}{\rangle }_{s_i}$ at the first power in g (In this connection, note also the results of recent works [47,48,49,50] obtained on the base of the deformation of general gauge theory [51,52,53,54]).

$$\begin{array}{ccc}& & g\int \prod _{e=1}^{3}d{\eta }_0^{\left(e\right)}{}_{s_j}\langle {\mathrm{\Lambda }}^{\left(j\right)}{K}^{\left(j\right)}|{}_{s_{j+1}}\langle {\chi }^{(j+1)}{K}^{(j+1)}|{}_{s_{j+2}}\langle {\chi }^{(j+2)}{K}^{(j+2)}|\mathcal{Q}(V^3,{\tilde{V}}^3)=0,\hfill \end{array}$$

(24)

$$\begin{array}{ccc}& & g\int \prod _{e=1}^{2}d{\eta }_0^{\left(e\right)}{}_{s_{j+1}}\langle {\mathrm{\Lambda }}^{(j+1)1}{K}^{(j+1)}|{}_{s_{j+2}}\langle {\chi }^{(j+2)}{K}^{(j+2)}|\left(\mathcal{Q}({\tilde{V}}^3,{\widehat{V}}^3)-Q^{(j+2)}|\widehat{V}{}^{\left(3\right)}\rangle \right)=0,\hfill \end{array}$$

(25)

where

$$\begin{array}{ccc}\hfill \phantom{g^1:}& & \mathcal{Q}(V^3,{\tilde{V}}^3)=\sum _{k=1}^3{Q}^{\left(k\right)}|\tilde{V}{}^{\left(3\right)}\rangle _{{\left(s\right)}_3}^{{\left(m\right)}_3}+Q^{\left(j\right)}\left(|V{}^{\left(3\right)}{\rangle }_{{\left(s\right)}_3}^{{\left(m\right)}_3}-|\tilde{V}{}^{\left(3\right)}\rangle _{{\left(s\right)}_3}^{{\left(m\right)}_3}\right),j=1,2,3.\hfill \end{array}$$

(26)

Following our results, [33,34] we choose coincidence for the vertices: $|\tilde{V}{}^{\left(3\right)}\rangle$ = $|V{}^{\left(3\right)}\rangle$ = $|\widehat{V}{}^{\left(3\right)}\rangle$, which provides the validity of the operator equations at the first order in g (the highest orders are necessary for finding the quartic and higher vertices)

$$Q^{tot}|V{}^{\left(3\right)}{\rangle }_{{\left(s\right)}_3}^{{\left(m\right)}_3}=0,{\sigma }^{\left(i\right)}|V{}^{\left(3\right)}\rangle _{{\left(s\right)}_3}^{{\left(m\right)}_3}=0,$$

(27)

jointly with the spin conditions as the consequence of the spin Equation (11) for each sample (with $|{\chi }^{\left(i\right)}{\rangle }_{s_i},|{\mathrm{\Lambda }}^{\left(i\right)}{\rangle }_{s_i},{|{\mathrm{\Lambda }}^{\left(i\right)1}\rangle }_{s_i}$) providing the nilpotency of total BRST operator $Q^{tot}\equiv {\sum }_i{Q}^{\left(i\right)}$ when evaluated on the vertex due to the Equations (13) and $\{Q^{\left(i\right)},Q^{\left(j\right)}\}=0$ for $i\ne j$.

A local dependence on space-time coordinates in the vertices $|V^{\left(3\right)}\rangle$, $|\tilde{V}{}^{\left(3\right)}\rangle$, $|\widehat{V}{}^{\left(3\right)}\rangle$ means

$$|V^{\left(3\right)}{\rangle }_{{\left(s\right)}_3}^{{\left(m\right)}_3}=\prod _{i=2}^3{\delta }^{\left(d\right)}\left(x_1-x_i\right)V^{\left(3\right)|}{}_{{\left(s\right)}_3}{}^{{\left(m\right)}_3}\prod _{j=1}^3{\eta }_0^{\left(j\right)}|0\rangle ,|0\rangle \equiv {\otimes }_{e=1}^3{|0\rangle }^e$$

(28)

(for $(ϵ,gh)V^{\left(3\right)|}{}_{{\left(s\right)}_3}{}^{{\left(m\right)}_3}=(0,0)$). We have the conservation law: ${\sum }_{i=1}^3{p}_{\mu }^{\left(i\right)}=0$, for the momenta associated with all vertices. Again as for the massless case [33], the deformed gauge transformations still form the closed algebra, that means after the simple calculations

$$\begin{array}{ccc}& & \left[{\delta }_{\left[1\right]}^{{\mathrm{\Lambda }}_1},{\delta }_{\left[1\right]}^{{\mathrm{\Lambda }}_2}\right]|{\chi }^{\left(i\right)}\rangle =-g{\delta }_{\left[1\right]}^{{\mathrm{\Lambda }}_3}|{\chi }^{\left(i\right)}\rangle ,\hfill \\ & & |{\mathrm{\Lambda }}_3\rangle \sim \int \prod _{e=1}^{2}d{\eta }_0^{(i+e)}\left(\langle {\mathrm{\Lambda }}_2^{(i+1)}K|\langle {\mathrm{\Lambda }}_1^{(i+2)}|K+\left(i+1\leftrightarrow i+2\right)\right)-\left({\mathrm{\Lambda }}_1\leftrightarrow {\mathrm{\Lambda }}_2\right)\}|V{}^{\left(3\right)}\rangle \hfill \end{array}$$

(29)

with the Grassmann-odd gauge parameter ${\mathrm{\Lambda }}_3$ being a function of the parameters ${\mathrm{\Lambda }}_1,{\mathrm{\Lambda }}_2,$${\mathrm{\Lambda }}_3={\mathrm{\Lambda }}_3({\mathrm{\Lambda }}_1,{\mathrm{\Lambda }}_2)$.

The Equations (27) (for coinciding vertices $V{}^{\left(3\right)}=\tilde{V}{}^{\left(3\right)}={\widehat{V}}^{\left(3\right)}$) together with the form of the commutator of the gauge transformations (29) determine the cubic interacting vertices for irreducible massive and massless totally symmetric higher spin fields.

4. Solution for Cubic Vertices

In this section, we will construct the general solution for the cubic vertices in the following cases for interacting higher spin fields: with one massive and two massless; with two massive and one massless of spins $s_1,s_2,s_3$ according to [18,31]: we introduce fourth order polynomial $D\equiv D(m_1,m_2,m_3)$ and quantity ${}_{ϵm}{}^P\equiv P(m_1,m_2,m_3)$:

$$\begin{array}{ccc}& & D=(m_1+m_2+m_3)(m_1-m_2+m_3)(m_1+m_2-m_3)(m_1-m_2-m_3),\hfill \end{array}$$

(30)

$$\begin{array}{ccc}& & {}_{ϵm}{}^P={ϵ}_1{m}_1+{ϵ}_2{m}_2+{ϵ}_3{m}_3,{ϵ}_i^2=1,i=1,2,3.\hfill \end{array}$$

(31)

With the help of these quantities, we have the classification

$$\begin{array}{cccccc}\hfill & m_1=m_2=0,m_3\ne 0\hfill & \hfill & \Rightarrow \hfill & \hfill & D(0,0,m_3)>0,\hfill \end{array}$$

(32)

$$\begin{array}{cccccc}\hfill & m_1=0,m_2=m_3=m\ne 0\hfill & \hfill & \Rightarrow \hfill & \hfill & D(0,m,m)=0,{}_{ϵm}{}^P=0\hfill \end{array}$$

(33)

$$\begin{array}{cccccc}\hfill & m_1=0,m_2\ne 0,m_3\ne 0,m_3\ne m_2\hfill & \hfill & \Rightarrow \hfill & \hfill & D(0,m_2,m_3)>0,\hfill \end{array}$$

(34)

$$\begin{array}{cccccc}\hfill & m_1\ne 0,m_2\ne 0,m_3\ne 0\hfill & \hfill & \Rightarrow \hfill & \hfill & D(m_1,m_2,m_3)>0,\hfill \end{array}$$

(35)

$$\begin{array}{cccccc}\hfill & m_1\ne 0,m_2\ne 0,m_3\ne 0,\hfill & \hfill & \Rightarrow \hfill & \hfill & D(m_1,m_2,m_3)<0,\hfill \end{array}$$

(36)

$$\begin{array}{cccccc}\hfill & m_1\ne 0,m_2\ne 0,m_3\ne 0,\hfill & \hfill & \Rightarrow \hfill & \hfill & D(m_1,m_2,m_3)=0,{}_{ϵm}{}^P=0.\hfill \end{array}$$

(37)

Cases (33)–(37) correspond to critical masses described in [31,55], respectively, for $d=4$ and $d=3$, on a base of use the conservation law for the momenta associated with vertices

$$\sum _{i=1}^3{p}_{\mu }^{\left(i\right)}=0.$$

(38)

and the process of decay of the massive particle ($i=1$) into the two massive particles ($i=2,3$) in the rest frame of the first particle

$$p_{\mu }^{\left(1\right)}=(m_1,\mathbf{0}).p_{\mu }^{\left(i\right)}=(E_i,{(-1)}^i\mathbf{p})\mathrm{with}{E}_i=\sqrt{m_i^2+{\mathbf{p}}^2},$$

(39)

from which it follows the well-known restrictions on masses and $(d-1)$-space momentum $\mathbf{p}$

$$\begin{array}{ccc}\hfill m_1>(=)m_2+m_3,\mathrm{for}\mathbf{p}\ne (=)0,& \hfill & \hfill {\mathbf{p}}^2=D/\left(4m_1^2\right).\end{array}$$

(40)

Note, the case of equal masses $m_1=m_2=m_3$ corresponds to (36) with $D(m,m,m)=-3m_1^4$, whereas the case of $m_i=m_{i+1}\ne m_3$ may satisfy to any from the relations (35)–(37). The latter cases related to real ($D>0$), virtual ($D<0$) processes, and real process ($D=0$) with vanishing transfer of momentum (Note, for the case (37) when ${ϵ}_i=1$: ${}_{ϵm}{}^P=m_1+m_2+m_3=0$, a consistent Lagrangian theory with reducible massive higher spin fields in $d=3$ flat space-time was derived in [56] in the light-cone approach).

4.1. Cubic Vertices for Two Massless Fields and One Massive Field

For the case (32) with $D>0$ we look for a general solution of the Equations (27) in the form of products of specific operators, homogenous in oscillators. As suggested in [18] two ways of vertex derivation known from the light-cone approach [32] as Minimal derivative scheme and Massive field strength scheme (however, due to the uniqueness of the interaction vertex with given order k of derivatives, we expect the vertex obtained by one scheme should differ from the vertex obtained by another scheme on BRST-exact terms) we will consider the first one.

With the use of the notations

$$\widehat{p}{}_{\mu }{}^{\left(i\right)}=p_{\mu }^{(i+1)}-p_{\mu }^{(i+2)},\widehat{\mathcal{P}}{}_0{}^{\left(i\right)}={\mathcal{P}}_0^{(i+1)}-{\mathcal{P}}_0^{(i+2)},\mathrm{a}{}_{\mu }{}^{\left(3\right)+}=a_{\mu }^{\left(3\right)+}-\frac{p_{\mu }^{\left(3\right)}}{m_3}{d}^{\left(3\right)+}$$

(41)

the massive field strength scheme corresponds to the set of monomials given on the constrained Fock space ${\mathcal{H}}_{tot|c}$

$$\begin{array}{ccc}& & L^{\left(i\right)}=\widehat{p}{}_{\mu }{}^{\left(i\right)}a^{\left(i\right)\mu +}-ı\widehat{\mathcal{P}}{}_0{}^{\left(i\right)}\eta _1^{\left(i\right)+},i=1,2;L^{\left(3\right)}=\widehat{p}{}_{\mu }{}^{\left(3\right)}\mathrm{a}{}^{\left(3\right)\mu +},\hfill \end{array}$$

(42)

$$\begin{array}{ccc}& & L_{11}^{\left(12\right)+}=a^{\left(1\right)\mu +}{a}_{\mu }^{\left(2\right)+}+\frac{1}{2m_3^2}{L}^{\left(1\right)}{L}^{\left(2\right)}-\frac{1}{2}{\mathcal{P}}_1^{\left(1\right)+}{\eta }_1^{\left(2\right)+}-\frac{1}{2}{\mathcal{P}}_1^{\left(2\right)+}{\eta }_1^{\left(1\right)+}\hfill \end{array}$$

(43)

$$\begin{array}{ccc}& & L_{11}^{\left(i3\right)+}=a^{\left(i\right)\mu +}{\mathrm{a}}_{\mu }^{\left(3\right)+}+{(-1)}^i\frac{1}{m_3^2}{L}^{\left(i\right)}p{}_{\mu }{}^{\left(i\right)}\mathrm{a}{}^{\left(3\right)\mu +},i=1,2.\hfill \end{array}$$

(44)

In turn, the minimal derivative scheme contains the monomials

$$\begin{array}{ccc}& & L^{\left(i\right)}=\widehat{p}{}_{\mu }{}^{\left(i\right)}a^{\left(i\right)\mu +}-ı\widehat{\mathcal{P}}{}_0{}^{\left(i\right)}\eta _1^{\left(i\right)+},i=1,2,3;\hfill \end{array}$$

(45)

$$\begin{array}{ccc}& & L_{11}^{\left(12\right)+}=a^{\left(1\right)\mu +}{a}_{\mu }^{\left(2\right)+}+\frac{1}{2m_3^2}{L}^{\left(1\right)}{L}^{\left(2\right)}-\frac{1}{2}{\mathcal{P}}_1^{\left(1\right)+}{\eta }_1^{\left(2\right)+}-\frac{1}{2}{\mathcal{P}}_1^{\left(2\right)+}{\eta }_1^{\left(1\right)+};\hfill \end{array}$$

(46)

$$\begin{array}{ccc}& & L_{11}^{\left(23\right)+}=a^{\left(2\right)\mu +}{a}_{\mu }^{\left(3\right)+}-\frac{1}{2m_3^2}{L}^{\left(2\right)}{L}^{\left(3\right)}+\frac{1}{2m_3}{L}^{\left(2\right)}{d}^{\left(3\right)+}-\frac{1}{2}{\mathcal{P}}_1^{\left(2\right)+}{\eta }_1^{\left(3\right)+}-\frac{1}{2}{\mathcal{P}}_1^{\left(3\right)+}{\eta }_1^{\left(2\right)+};\hfill \end{array}$$

(47)

$$\begin{array}{ccc}& & L_{11}^{\left(31\right)+}=a^{\left(3\right)\mu +}{a}_{\mu }^{\left(1\right)+}-\frac{1}{2m_3^2}{L}^{\left(3\right)}{L}^{\left(1\right)}-\frac{1}{2m_3}{d}^{\left(3\right)+}{L}^{\left(1\right)}-\frac{1}{2}{\mathcal{P}}_1^{\left(1\right)+}{\eta }_1^{\left(3\right)+}-\frac{1}{2}{\mathcal{P}}_1^{\left(3\right)+}{\eta }_1^{\left(1\right)+};\hfill \end{array}$$

(48)

$$\begin{array}{ccc}& & Z=L_{11}^{\left(12\right)+}{L}^{\left(3\right)}+L_{11}^{\left(23\right)+}{L}^{\left(1\right)}+L_{11}^{\left(31\right)+}{L}^{\left(2\right)}.\hfill \end{array}$$

(49)

The operators above do not introduce the divergences into the vertices, are Grassmann-even with vanishing ghost number and have the distributions in powers of creation oscillators $A^{\left(i\right)+}$ and momenta

	$L^{\left(i\right)}$	$L_{11}^{\left(12\right)+}$	$L_{11}^{\left(23\right)+}$	$L_{11}^{\left(31\right)+}$	${\widehat{L}}_{11}^{\left(i\right)+}$	$Z$
${\mathrm{deg}}_{A^{\left(j\right)+}}$	${\delta }_{ij}$	$\left(1,1,0\right)$	$\left(0,1,1\right)$	$\left(1,0,1\right)$	$2{\delta }_{ij}$	$(1,1,1)$
${\mathrm{deg}}_p$	1	$\le 2$	$\le 2$	$\le 2$	0	$\le 3$

Note, first, that for massless case the latter row (${\mathrm{deg}}_p$) for massless analogs of operators is filled as: $\left(1,0,0,0,0,1\right)$. Second, the operators (42) $L^{\left(i\right)}$ for $i=1,2$ are not BRST-closed with respect to the constrained BRST operator $Q_c^{tot}\equiv Q^{tot}{|}_{{\eta }_{11}^{(+)i}=0}$ as compared to $L^{\left(3\right)}$, $Q_c^{tot}{L}^{\left(3\right)}|0\rangle =0$. Namely, we have:

$$\begin{array}{ccc}& & Q_c^{tot}{L}^{\left(i\right)}|0\rangle ={(-1)}^i{m}_3^2{\eta }_1^{\left(i\right)+}|0\rangle \ne 0,i=1,2.\hfill \end{array}$$

(50)

and, therefore, the operator Z (49) is not $Q_c^{tot}$- BRST-closed

$$\begin{array}{ccc}& & Q_c^{tot}Z|0\rangle \ne 0\Rightarrow Q^{tot}Z|0\rangle \ne 0\hfill \end{array}$$

(51)

In turn, the operators $L_{11}^{(ii+1)+}$, ${\widehat{L}}_{11}^{\left(i\right)+}$ are $Q_c^{tot}$- BRST-closed, but not $Q^{tot}$- BRST-closed due to

$$\begin{array}{ccc}& & \widehat{L}{}_{11}{}^{\left(i\right)}L_{11}^{\left(ij\right)+}|0\rangle =0,i,j=1,2,3\hfill \end{array}$$

(52)

$$\begin{array}{ccc}& & \widehat{L}{}_{11}{}^{\left(i\right)}\left(L_{11}^{\left(ij\right)+}\right)^2|0\rangle \ne 0\hfill \end{array}$$

(53)

Then, following [33] we have the respective trace operators (massless for $i=1,2$ and massive $i=3$)

$$U_{j_i}^{\left(s_i\right)}\left({\eta }_{11}^{\left(i\right)+},{\mathcal{P}}_{11}^{\left(i\right)+}\right):={(\widehat{L}{}_{11}{}^{\left(i\right)+})}^{(j_i-2)}\left\{{(\widehat{L}{}_{11}{}^{+\left(i\right)})}^2-j_i(j_i-1){\eta }_{11}^{\left(i\right)+}{\mathcal{P}}_{11}^{\left(i\right)+}\right\},i=1,2,3.$$

(54)

Indeed, the $Q^{tot}$- BRST closeness for the operator $L^{\left(3\right)}$ is reduced to the fulfillment of the equations at the terms linear in ${\eta }_{11}^{\left(3\right)+}$

$$\begin{array}{ccc}& & \widehat{L}{}_{11}{}^{\left(i\right)}\left(L^{(i+j)}\right)^k|0\rangle \equiv 0,j=1,2,\forall k\in \mathbb{N},\hfill \end{array}$$

(55)

$$\begin{array}{ccc}& & \widehat{L}{}_{11}{}^{\left(i\right)}L^{\left(i\right)}|0\rangle =\left(-\widehat{p}{}_{\mu }{}^{\left(i\right)}a^{\left(i\right)\mu }+ı{\widehat{\mathcal{P}}}_0^{\left(i\right)}{\eta }_1^{\left(i\right)}+L^{\left(i\right)}{\widehat{L}}_{11}^{\left(i\right)}\right)|0\rangle =0,\hfill \end{array}$$

(56)

$$\begin{array}{ccc}& & \widehat{L}{}_{11}{}^{\left(i\right)}\left(L^{\left(i\right)}\right)^2|0\rangle ={\eta }^{\mu \nu }\widehat{p}{}_{\mu }{}^{\left(i\right)}\widehat{p}_{\nu }^{\left(i\right)}|0\rangle ={(\widehat{p}{}^{\left(i\right)})}^2|0\rangle \ne 0,i=1,2,3.\hfill \end{array}$$

(57)

The last relations and ones for ${\left(L^{\left(3\right)}\right)}^k$ do not vanish under the sign of inner products and justify the introduction of the BRST-closed forms, first for $k\le 5$

$$\begin{array}{ccc}& & {\mathcal{L}}_1^{\left(3\right)}=L^{\left(3\right)}-\left[\widehat{L}{}_{11}{}^{\left(3\right)},L^{\left(3\right)}\right\}\frac{b^{\left(3\right)+}}{h^{\left(3\right)}},{\mathcal{L}}_2^{\left(3\right)}={\left(L^{\left(3\right)}\right)}^2-{\left({\widehat{p}}^{\left(3\right)}\right)}^2\frac{b^{\left(3\right)+}}{h^{\left(3\right)}},\hfill \end{array}$$

(58)

$$\begin{array}{ccc}& & {\mathcal{L}}_3^{\left(3\right)}=L^{\left(i\right)}\left({\left(L^{\left(3\right)}\right)}^2-3{\left({\widehat{p}}^{\left(3\right)}\right)}^2\frac{b^{\left(3\right)+}}{h^{\left(3\right)}}\right),\hfill \end{array}$$

(59)

$$\begin{array}{ccc}& & {\mathcal{L}}_4^{3)}={\left(L^{\left(3\right)}\right)}^2\left({\left(L^{\left(3\right)}\right)}^2-6{\left({\widehat{p}}^{\left(3\right)}\right)}^2\frac{b^{\left(3\right)+}}{h^{\left(3\right)}}\right)+3{\left({\widehat{p}}^{\left(3\right)}\right)}^4\frac{{\left(b^{\left(3\right)+}\right)}^2}{h^{\left(3\right)}(h^{\left(3\right)}+1)},\hfill \end{array}$$

(60)

$$\begin{array}{ccc}& & {\mathcal{L}}_5^{\left(3\right)}={\left(L^{\left(3\right)}\right)}^3\left({\left(L^{\left(3\right)}\right)}^2-10{\left({\widehat{p}}^{\left(3\right)}\right)}^2\frac{b^{\left(3\right)+}}{h^{\left(3\right)}}\right)+\frac{3·10}{2}{L}^{\left(3\right)}{\left({\widehat{p}}^{\left(3\right)}\right)}^4\frac{{\left(b^{\left(3\right)+}\right)}^2}{h^{\left(3\right)}(h^{\left(3\right)}+1)},\hfill \end{array}$$

(61)

then, by induction for arbitrary $k\in \mathbb{N}$

$$\begin{array}{ccc}& & {\mathcal{L}}_k^{\left(3\right)}=\sum _{j=0}^{[k/2]}{(-1)}^j{\left(L^{\left(3\right)}\right)}^{k-2j}{\left({\widehat{p}}^{\left(3\right)}\right)}^{2j}\frac{k!}{j!2^j(k-2j))!}\frac{{\left(b^{\left(3\right)+}\right)}^j}{C(j,h^{\left(3\right)}\left(s\right))}.\hfill \end{array}$$

(62)

(the equivalent polynomial representation for BRST-closed operator ${\mathcal{L}}_k^{\left(3\right)}$ is also found, see Section 4.1). For the same reason, any power of the forms $L_{11}^{(ii+1)+}$, (hence ${\left(L_{11}^{(ii+1)+}\right)}^k$, $k>1$) are not BRST-closed as well due to Equation (65),

$$\begin{array}{ccc}& & {\eta }_{11}^{\left(i\right)+}\widehat{L}{}_{11}{}^{\left(i\right)}\left(L_{11}^{(i+1i+2)+}\right)^k|0\rangle \equiv 0,\forall k\in \mathbb{N},\hfill \end{array}$$

(63)

$$\begin{array}{ccc}& & \sum _i{\eta }_{11}^{\left(i\right)+}\widehat{L}{}_{11}{}^{\left(i\right)}L_{11}^{\left(12\right)+}|0\rangle =\sum _i{\eta }_{11}^{\left(i\right)+}(-a^{\left(2\right)\mu +}{a}_{\mu }^{\left(i\right)}+\frac{1}{2m_3^2}\left(-{\widehat{p}}^{\left(i\right)}{a}_{\mu }^{\left(i\right)}-ı\widehat{\mathcal{P}}{}_0{}^{\left(i\right)}\eta _1^{\left(i\right)}\right)L^{\left(2\right)}\hfill \\ & & -\frac{1}{2}{\mathcal{P}}_1^{\left(2\right)+}{\eta }_1^{\left(i\right)}-\frac{1}{2}{\eta }_1^{\left(2\right)+}{\mathcal{P}}_1^{\left(i\right)})|0\rangle =0,\hfill \end{array}$$

(64)

$$\begin{array}{ccc}& & \sum _i{\eta }_{11}^{\left(i\right)+}\widehat{L}{}_{11}{}^{\left(i\right)}\left(L_{11}^{\left(12\right)+}\right)^2|0\rangle =\sum _i{\eta }_{11}^{\left(i\right)+}[\left[\widehat{L}{}_{11}{}^{\left(i\right)},L_{11}^{\left(12\right)+}\right\},L_{11}^{\left(12\right)+}\}|0\rangle \ne 0,\hfill \\ & & \sum _{i=1}^2{\eta }_{11}^{\left(i\right)+}[\left[\widehat{L}{}_{11}{}^{\left(i\right)},L_{11}^{\left(12\right)+}\right\},L_{11}^{\left(12\right)+}\}=\left[{\eta }_{11}^{\left(1\right)+}\right(a^{\left(2\right)\mu +}{a}_{\mu }^{\left(2\right)+}+\frac{1}{2m_3^2}\left({\widehat{p}}^{\left(1\right)}{a}_{\mu }^{\left(2\right)+}+\frac{ı}{2}\widehat{\mathcal{P}}{}_0{}^{\left(1\right)}\eta _1^{\left(2\right)+}\right)L^{\left(2\right)}\hfill \\ & & +\frac{1}{2}{\mathcal{P}}_1^{\left(2\right)+}{\eta }_1^{\left(2\right)+})+{\eta }_{11}^{\left(2\right)+}\left(a^{\left(1\right)\mu +}{a}_{\mu }^{\left(1\right)+}+\frac{1}{2m_3^2}\left({\widehat{p}}^{\left(2\right)}{a}_{\mu }^{\left(1\right)+}+\frac{ı}{2}\widehat{\mathcal{P}}{}_0{}^{\left(2\right)}\eta _1^{\left(1\right)+}\right)L^{\left(1\right)}+\frac{1}{2}{\mathcal{P}}_1^{\left(1\right)+}{\eta }_1^{\left(1\right)+}\right)],\hfill \end{array}$$

(65)

To compensate for this term in $L_{11}^{\left(12\right)+}$ we add the modified summand for it and find BRST-closed completion in the form

$$\begin{array}{ccc}& & {\mathcal{L}}_{11|1}^{\left(12\right)+}=L_{11}^{\left(12\right)+}-\sum _{i_0}{W}_{\left(12\right)|0}^{\left(i_0\right)}\frac{b^{\left(i_0\right)+}}{h^{\left(i_0\right)}}+\frac{1}{2}(\sum _{i_0\ne j_0}\left[\widehat{L}{}_{11}{}^{\left(j_0\right)},W_{\left(12\right)|0}^{\left(i_0\right)}\right\}\frac{b^{\left(i_0\right)+}}{h^{\left(i_0\right)}}\frac{b^{\left(j_0\right)+}}{h^{\left(j_0\right)}}\hfill \\ & & +\sum _{i_0}\left[\widehat{L}{}_{11}{}^{\left(i_0\right)},W_{\left(12\right)|0}^{\left(i_0\right)}\right\}\frac{{\left(b^{\left(i_0\right)+}\right)}^2}{h^{\left(i_0\right)}(h^{\left(i_0\right)}+1)}),\hfill \end{array}$$

(66)

for ${\left(L_{11}^{\left(12\right)+}\right)}^2$

$$\begin{array}{ccc}& & {\mathcal{L}}_{11|2}^{\left(12\right)+}=L_{11}^{\left(12\right)+}{\mathcal{L}}_{11|1}^{\left(12\right)+}-\sum _{i_1}{W}_{\left(12\right)|1}^{\left(i_1\right)}\frac{b^{\left(i_1\right)+}}{h^{\left(i_1\right)}}+\frac{1}{2}(\sum _{i_1\ne j_1}\left[\widehat{L}{}_{11}{}^{\left(j_1\right)},W_{\left(12\right)|1}^{\left(i_1\right)}\right\}\frac{b^{\left(i_1\right)+}}{h^{\left(i_1\right)}}\frac{b^{\left(j_1\right)+}}{h^{\left(j_1\right)}}\hfill \\ & & +\sum _{i_1}\left[\widehat{L}{}_{11}{}^{\left(i_1\right)},W_{\left(12\right)|1}^{\left(i_1\right)}\right\}\frac{{\left(b^{\left(i_1\right)+}\right)}^2}{h^{\left(i_1\right)}(h^{\left(i_1\right)}+1)})-\frac{1}{6}(3\sum _{i_1\ne j_0}\left[\widehat{L}{}_{11}{}^{\left(j_1\right)},\left[\widehat{L}{}_{11}{}^{\left(i_1\right)},W_{\left(12\right)|1}^{\left(i_1\right)}\right\}\right\}\times \hfill \\ & & \times \frac{b^{\left(j_1\right)+}}{h^{\left(j_1\right)}}\frac{(b^{\left(i_1\right){+)}^2}}{h^{\left(i_1\right)}(h^{\left(i_1\right)}+1)}+\sum _{i_1}\left[\widehat{L}{}_{11}{}^{\left(i_1\right)},\left[\widehat{L}{}_{11}{}^{\left(i_1\right)},W_{\left(12\right)|2}^{\left(i_1\right)}\right\}\right\}\frac{{\left(b^{\left(i_1\right)+}\right)}^3}{C(3,h^{\left(i_1\right)}\left(s\right))}),\hfill \end{array}$$

(67)

then for ${\left(L_{11}^{\left(12\right)+}\right)}^3$

$$\begin{array}{ccc}& & {\mathcal{L}}_{11|3}^{\left(12\right)+}=L_{11}^{\left(12\right)+}{\mathcal{L}}_{11|2}^{\left(12\right)+}-\sum _{i_2}{W}_{\left(12\right)|2}^{\left(i_2\right)}\frac{b^{\left(i_2\right)+}}{h^{\left(i_2\right)}}+\frac{1}{2}(\sum _{i_2\ne j_1}\left[\widehat{L}{}_{11}{}^{\left(j_2\right)},W_{\left(12\right)|2}^{\left(i_2\right)}\right\}\frac{b^{\left(i_2\right)+}}{h^{\left(i_2\right)}}\frac{b^{\left(j_2\right)+}}{h^{\left(j_2\right)}}\hfill \\ & & +\sum _{i_2}\left[\widehat{L}{}_{11}{}^{\left(i_2\right)},W_{\left(12\right)|2}^{\left(i_2\right)}\right\}\frac{{\left(b^{\left(i_2\right)+}\right)}^2}{h^{\left(i_2\right)}(h^{\left(i_2\right)}+1)})-\frac{1}{6}(3\sum _{i_2\ne j_2}\left[\widehat{L}{}_{11}{}^{\left(j_2\right)},\left[\widehat{L}{}_{11}{}^{\left(i_2\right)},W_{\left(12\right)|2}^{\left(i_2\right)}\right\}\right\}\hfill \\ & & \times \frac{b^{\left(j_2\right)+}}{h^{\left(j_2\right)}}\frac{(b^{\left(i_2\right){+)}^2}}{h^{\left(i_2\right)}(h^{\left(i_2\right)}+1)}+\sum _{i_2}\left[\widehat{L}{}_{11}{}^{\left(i_2\right)},\left[\widehat{L}{}_{11}{}^{\left(i_2\right)},W_{\left(12\right)|2}^{\left(i_2\right)}\right\}\right\}\frac{{\left(b^{\left(i_2\right)+}\right)}^3}{{\prod }_{p=0}^2(h^{\left(i_2\right)}+p)})\hfill \\ & & +\frac{1}{4!}(4\sum _{i_2\ne j_2}{\mathrm{ad}}_{{\widehat{L}}_{11}^{\left(j_2\right)}}{\mathrm{ad}}_{{\widehat{L}}_{11}^{\left(i_2\right)}}^2{W}_{\left(12\right)|2}^{\left(i_2\right)}\frac{b^{\left(j_2\right)+}}{h^{\left(j_2\right)}}\frac{{\left(b^{\left(i_2\right)+}\right)}^3}{C(3,h^{\left(i_2\right)})}+3\sum _{i_2\ne j_2}{\mathrm{ad}}_{{\widehat{L}}_{11}^{\left(j_2\right)}}^2{\mathrm{ad}}_{{\widehat{L}}_{11}^{\left(i_2\right)}}{W}_{\left(12\right)|2}^{\left(i_2\right)}\hfill \\ & & \times \frac{{\left(b^{\left(j_2\right)+}\right)}^2}{C(2,h^{j_2)})}\frac{{\left(b^{\left(i_2\right)+}\right)}^2}{C(2,h^{\left(i_2\right)})}+\sum _{i_2}{\mathrm{ad}}_{{\widehat{L}}_{11}^{\left(i_2\right)}}^3{W}_{\left(12\right)|2}^{\left(i_2\right)}\frac{{\left(b^{\left(i_2\right)+}\right)}^4}{C(4,h^{\left(i_2\right)})}),\hfill \end{array}$$

(68)

and by induction for ${\left(L_{11}^{\left(12\right)+}\right)}^{k+1}$ at $k=1,\dots ,min(s_1,s_2)-1$:

$$\begin{array}{ccc}& & {\mathcal{L}}_{11|k+1}^{\left(12\right)+}=L_{11}^{\left(12\right)+}{\mathcal{L}}_{11|k}^{\left(12\right)+}-\sum _{i_k}{W}_{\left(12\right)|k}^{\left(i_k\right)}\frac{b^{\left(i_k\right)+}}{h^{\left(i_k\right)}}+\frac{1}{2}(\sum _{i_k\ne j_k}\left[\widehat{L}{}_{11}{}^{\left(j_k\right)},W_{\left(12\right)|k}^{\left(i_k\right)}\right\}\frac{b^{\left(i_k\right)+}}{h^{\left(j_k\right)}}\frac{b^{\left(j_k\right)+}}{h^{\left(j_k\right)}}\hfill \\ & & +\sum _{i_k}\left[\widehat{L}{}_{11}{}^{\left(i_k\right)},W_{\left(12\right)|k}^{\left(i_k\right)}\right\}\frac{{\left(b^{\left(i_k\right)+}\right)}^2}{{\prod }_{p=0}^1(h^{\left(i_k\right)}+p)})+\sum _{e=3}^{k+2}\frac{{(-1)}^e}{e!}\sum _{l=0,j\ne i}^{[e/2]}\frac{e!}{[e/(e-l\left)\right](e-l)!l!}\times \hfill \\ & & \times {\mathrm{ad}}_{{\widehat{L}}_{11}^{\left(j_k\right)}}^l{\mathrm{ad}}_{{\widehat{L}}_{11}^{\left(i_k\right)}}^{e-l-1}{W}_{\left(12\right)|k}^{\left(i_k\right)}\frac{{\left(b^{\left(j_k\right)+}\right)}^l}{C(l,h^{\left(j_k\right)})}\frac{{\left(b^{\left(i_k\right)+}\right)}^{e-l}}{C(e-l,h^{\left(i_k\right)})}.\hfill \end{array}$$

(69)

In (66)–(69) the indices $i_0,\dots ,i_k$ are running two values: $1,2$; and we have used the notations

$$\begin{array}{ccc}& & \left[\widehat{L}{}_{11}{}^{\left(i_0\right)},L_{11}^{\left(12\right)+}\right\}\equiv W_{\left(12\right)|0}^{\left(i_0\right)},\left[\widehat{L}{}_{11}{}^{\left(i_k\right)},L_{11}^{\left(12\right)+}{\mathcal{L}}_{11|k}^{\left(12\right)+}\right\}\equiv W_{\left(12\right)|k}^{\left(i_k\right)},k=1,2,\dots ,\hfill \\ & & {\mathrm{ad}}_{{\widehat{L}}_{11}^{\left(j_k\right)}}^{l+1}B\equiv [\widehat{L}{}_{11}{}^{\left(j_k\right)},\stackrel{l\mathrm{times}}{\overbrace{\left[\widehat{L}{}_{11}{}^{\left(j_k\right)},\dots ,\left[\widehat{L}{}_{11}{}^{\left(j_k\right)},B\right\}\right\}}}\}.\hfill \end{array}$$

(70)

By construction the calligraphic operators are traceless

$$\begin{array}{ccc}& & \left[\widehat{L}{}_{11}{}^{\left(i\right)},{\mathcal{L}}_{11|k+1}^{\left(12\right)+}\right\}=0,k=0,1,\dots ,min(s_1,s_2)-1,\hfill \end{array}$$

(71)

because of the last terms in (66)–(69) in front of the maximal power in $b^{\left(i\right)+}$, i.e., $[\widehat{L}{}_{11}{}^{\left(j_0\right)},W_{\left(12\right)|0}^{\left(i_0\right)}\}$, …, ${\mathrm{ad}}_{{\widehat{L}}_{11}^{\left(j_k\right)}}^l{\mathrm{ad}}_{{\widehat{L}}_{11}^{\left(i_k\right)}}^{k-l+1}{W}_{\left(12\right)|k}^{\left(i_k\right)}$ depend on only $a_{\mu }^{\left(i\right)},d^{\left(i\right)},\eta {/}_1^{\left(i\right)},{\mathcal{P}}_1^{\left(i\right)}$ (annihilation) oscillators, and therefore, the compensation procedure is finalized.

In deriving (66)–(69) we have used the permutation properties

$$\begin{array}{ccc}& & \left[\widehat{L}{}_{11}{}^{\left(i_3\right)},W_{\left(12\right)|k}^{\left(j_3\right)}\right\}=\left[\widehat{L}{}_{11}{}^{\left(j_3\right)},W_{\left(12\right)|k}^{\left(i_3\right)}\right\},\hfill \end{array}$$

(72)

$$\begin{array}{ccc}& & \left[\widehat{L}{}_{11}{}^{\left(i_3\right)},\left[\widehat{L}{}_{11}{}^{\left(j_3\right)},W_{\left(12\right)|k}^{\left(i_3\right)}\right\}\right\}=\left[\widehat{L}{}_{11}{}^{\left(j_3\right)},\left[\widehat{L}{}_{11}{}^{\left(i_3\right)},W_{\left(12\right)|k}^{\left(i_3\right)}\right\}\right\}=\left[\widehat{L}{}_{11}{}^{\left(i_3\right)},\left[\widehat{L}{}_{11}{}^{\left(i_3\right)},W_{\left(12\right)|k}^{\left(j_3\right)}\right\}\right\},\hfill \end{array}$$

(73)

$$\begin{array}{ccc}& & {\mathrm{ad}}_{{\widehat{L}}_{11}^{\left(i_{k-1}\right)}}^p{\mathrm{ad}}_{{\widehat{L}}_{11}^{\left(j_{k-1}\right)}}^l{\mathrm{ad}}_{{\widehat{L}}_{11}^{\left(i_{k-1}\right)}}^e{W}_{\left(12\right)|k}^{\left(i_{k-1}\right)}={\mathrm{ad}}_{{\widehat{L}}_{11}^{\left(j_{k-1}\right)}}^l{\mathrm{ad}}_{{\widehat{L}}_{11}^{\left(i_{k-1}\right)}}^{e+p}{W}_{\left(12\right)|k}^{\left(i_{k-1}\right)},\hfill \end{array}$$

(74)

which follow from the Jacobi identity, first, for triple ${\widehat{L}}_{11}^{\left(i_3\right)}$, ${\widehat{L}}_{11}^{\left(j_3\right)}$, $W_{\left(12\right)|k}^{\left(i_3\right)}$, second, of its repeated application for ${\widehat{L}}_{11}^{\left(i_3\right)}$, ${\widehat{L}}_{11}^{\left(j_3\right)}$, ${\mathrm{ad}}_{{\widehat{L}}_{11}^{\left(i_{k-1}\right)}}^p{W}_{\left(12\right)|k}^{\left(i_{k-1}\right)}$ with account for commuting of two first trace operators, e.g.,

$${\mathrm{ad}}_{{\widehat{L}}_{11}^{\left(i_{k-1}\right)}}{\mathrm{ad}}_{{\widehat{L}}_{11}^{\left(j_{k-1}\right)}}{\mathrm{ad}}_{{\widehat{L}}_{11}^{\left(i_{k-1}\right)}}^e{W}_{\left(12\right)|k}^{\left(i_{k-1}\right)}={\mathrm{ad}}_{{\widehat{L}}_{11}^{\left(j_{k-1}\right)}}{\mathrm{ad}}_{{\widehat{L}}_{11}^{\left(i_{k-1}\right)}}^{e+1}{W}_{\left(12\right)|k}^{\left(i_{k-1}\right)}.$$

(75)

Thus, all calligraphic operators ${\mathcal{L}}_{11|k+1}^{\left(12\right)+}$ are BRST-closed.

Analogously, we have the same BRST-closed completions for ${\mathcal{L}}_{11|1}^{\left(23\right)+}$ and ${\mathcal{L}}_{11|1}^{\left(31\right)+}$, and respective BRST-closed forms ${\mathcal{L}}_{11|k+1}^{\left(23\right)+}$, ${\mathcal{L}}_{11|k+1}^{\left(31\right)+}$ to be uniquely written as follows, for $i=2,3$:

$$\begin{array}{ccc}& & {\mathcal{L}}_{11|1}^{(ii+1)+}=L_{11}^{(ii+1)+}-\sum _{i_0=2}^3{W}_{(ii+1)|0}^{\left(i_0\right)}\frac{b^{\left(i_0\right)+}}{h^{\left(i_0\right)}}+\frac{1}{2}(\sum _{i_0\ne j_0}\left[\widehat{L}{}_{11}{}^{\left(j_0\right)},W_{(ii+1)|0}^{\left(i_0\right)}\right\}\frac{b^{\left(i_0\right)+}}{h^{\left(i_0\right)}}\frac{b^{\left(j_0\right)+}}{h^{\left(j_0\right)}}\hfill \\ & & +\sum _{i_0}\left[\widehat{L}{}_{11}{}^{\left(i_0\right)},W_{(ii+1)|0}^{\left(i_0\right)}\right\}\frac{{\left(b^{\left(i_0\right)+}\right)}^2}{h^{\left(i_0\right)}(h^{\left(i_0\right)}+1)}),\hfill \end{array}$$

(76)

$$\begin{array}{ccc}& & {\mathcal{L}}_{11|k+1}^{(ii+1)+}=L_{11}^{(ii+1)+}{\mathcal{L}}_{11|k}^{(ii+1)+}-\sum _{i_k}{W}_{(ii+1)|k}^{\left(i_k\right)}\frac{b^{\left(i_k\right)+}}{h^{\left(i_k\right)}}+\frac{1}{2}(\sum _{i_k\ne j_k}\left[\widehat{L}{}_{11}{}^{\left(j_k\right)},W_{(ii+1)|k}^{\left(i_k\right)}\right\}\hfill \\ & & \times \frac{b^{\left(i_k\right)+}}{h^{\left(j_k\right)}}\frac{b^{\left(j_k\right)+}}{h^{\left(j_k\right)}}+\sum _{i_k}\left[\widehat{L}{}_{11}{}^{\left(i_k\right)},W_{(ii+1)|k}^{\left(i_k\right)}\right\}\frac{{\left(b^{\left(i_k\right)+}\right)}^2}{{\prod }_{p=0}^1(h^{\left(i_k\right)}+p)})+\sum _{e=3}^{k+2}\frac{{(-1)}^e}{e!}\times \hfill \\ & & \times \sum _{l=0,j\ne i}^{[e/2]}\frac{e!}{[e/(e-l\left)\right](e-l)!l!}{\mathrm{ad}}_{{\widehat{L}}_{11}^{\left(j_k\right)}}^l{\mathrm{ad}}_{{\widehat{L}}_{11}^{\left(i_k\right)}}^{e-l-1}{W}_{(ii+1)|k}^{\left(i_k\right)}\frac{{\left(b^{\left(j_k\right)+}\right)}^l}{C(l,h^{\left(j_k\right)})}\frac{{\left(b^{\left(i_k\right)+}\right)}^{e-l}}{C(e-l,h^{\left(i_k\right)})}\hfill \end{array}$$

(77)

(for $k=1,\dots ,min(s_i,s_{i+1})-1$). Note, in the expressions for BRST-closed forms ${\mathcal{L}}_{11|k}^{\left(23\right)+}$, ${\mathcal{L}}_{11|k}^{\left(31\right)+}$ indices $i_0,j_0,\dots ,i_{k-1},j_{k-1}$ are ranging, respectively, from $\{2,3\}$ and $\{3,1\}$.

As a result, the solution for the parity invariant vertex (given by Figure 1), has the form

$$\begin{array}{ccc}\hfill |V{}^{\left(3\right)}\rangle _{{\left(s\right)}_3}^m& =& |V{}^{M\left(3\right)}\rangle _{{\left(s\right)}_3}^m+\sum _{(j_1,j_2,j_3)>0}^{([s_1/2],[s_2/2],[s_3/2])}{U}_{j_3}^{\left(s_1\right)}{U}_{j_3}^{\left(s_2\right)}{U}_{j_3}^{\left(s_3\right)}{|V{}^{M\left(3\right)}\rangle }_{{\left(s\right)}_3-2{\left(j\right)}_3}^m,\hfill \end{array}$$

(78)

where the vertex $|V{}^{M\left(3\right)}\rangle _{{\left(s\right)}_3-2{\left(j\right)}_3}^m$ was defined in Metsaev’s paper [18] with account for (28) but with modified forms ${\mathcal{L}}_k^{\left(3\right)}$, (62) and ${\mathcal{L}}_{11|{\sigma }^{(i+2)}}^{(ii+1)+}$ instead of ${\left(L_{11}^{(ii+1)+}\right)}^{{\sigma }^{(i+2)}}$ (46)–(48)

$$\begin{array}{ccc}\hfill V_{{\left(s\right)}_3-2{\left(j\right)}_3}^{M\left(3\right)|m}& =& \sum _k{\mathcal{L}}_k^{\left(3\right)}\prod _{i=1}^3{\mathcal{L}}_{11|{\sigma }_{i+2}}^{(ii+1)+},(s,J)=\sum _i\left(s_i,j_i\right),\hfill \end{array}$$

(79)

and is $(3+1)$-parameters family to be enumerated by the natural parameters ${\left(j\right)}_3$, and k subject to the relations

$$\begin{array}{ccc}& & {\sigma }_i=\frac{1}{2}(s-2J-k)-s_i,i=1,2;{\sigma }_3=\frac{1}{2}(s+k)-s_3,\hfill \end{array}$$

(80)

$$\begin{array}{ccc}& & max(0,(s_3-2j_3)-\sum _{i=1}^2(s_i-2j_i))\le k\le s_3-2j_3-|s_1-2j_1-(s_2-2j_2)|,\hfill \end{array}$$

(81)

$$\begin{array}{ccc}& & 0\le 2j_i\le 2[s_i/2],s-2J-k=2p,p\in {\mathbb{N}}_0.\hfill \end{array}$$

(82)

Note, that the vertex in the unconstrained formulation depends on 3 additional parameters ${\left(j\right)}_3$ enumerated the number of traces in the respective set of fields and usual one k respecting the minimal order of derivatives in $V_{{\left(s\right)}_3-2{\left(j\right)}_3}^{M\left(3\right)|m}$. For vanishing ${\left(j\right)}_3$, the remaining parameters correspond to one in the constrained BRST formulation [18].

Trace-Deformed Vertex Generalization

The standard trace restriction $\widehat{L}{}_{11}{}^{\left(i\right)}|{\chi }^{\left(i\right)}\rangle =\widehat{L}{}_{11}{}^{\left(i\right)}|{\mathrm{\Lambda }}^{\left(i\right)}\rangle =0$ imposed off-shell in constrained BRST approach may be deformed (in the scheme with complete BRST operator) on the interacting level when deriving from resolution of deformed equations of motion and gauge transformations by following $Q^{tot}$-closed modification of the ${\mathcal{L}}_k^{\left(3\right)}$ as compared to (58)–(62)

$$\begin{array}{ccc}& & {\mathcal{L}}_1^{\left(3\right)}=L^{\left(3\right)}-\left[\widehat{L}{}_{11}{}^{\left(3\right)},L^{\left(3\right)}\right\}\frac{b^{\left(3\right)+}}{h^{\left(3\right)}},\hfill \end{array}$$

(83)

$$\begin{array}{ccc}& & \tilde{\mathcal{L}}{}_2{}^{\left(3\right)}={\left({\mathcal{L}}^{\left(3\right)}\right)}^2-i\widehat{\mathcal{P}}{}_0{}^{\left(3\right)}\eta _{11}^{\left(3\right)+}-\widehat{l}{}_0{}^{\left(3\right)}\frac{b^{\left(3\right)+}}{h^{\left(3\right)}},\hfill \end{array}$$

(84)

$$\begin{array}{ccc}& & \tilde{\mathcal{L}}{}_{2k}{}^{\left(3\right)}={\left({\tilde{\mathcal{L}}}_2^{\left(3\right)}\right)}^k,\tilde{\mathcal{L}}{}_{2k-1}{}^{\left(3\right)}={\left({\tilde{\mathcal{L}}}_2^{\left(3\right)}\right)}^{k-1}{\mathcal{L}}_1^{\left(3\right)},\hfill \end{array}$$

(85)

Note, the representation (83)–(85) contains the term linear in ${\eta }_{11}^{\left(3\right)+}$ without ${\mathcal{P}}_{11}^{\left(3\right)+}$. Thus, the other (more general) solution for the vertex $|V{}^{\left(3\right)}\rangle _{{\left(s\right)}_3}^m$ (78) is obtained after the substitution of new ${\tilde{\mathcal{L}}}_k^{\left(3\right)}$ instead of old ones ${\mathcal{L}}_k^{\left(3\right)}$. It is used in Section 5.1.2 for the example with the vertex $|V{}^{\left(3\right)}\rangle _{(1,0,s)}^m$ of interacting massive field of spin s with massless scalar and vector fields.

4.2. Cubic Vertices for One Massless Field and Two Massive Fields

In this section, we consider the cases of coinciding and different masses for massive fields

4.2.1. One Massless and Two Massive Fields with Coinciding Masses

In the case (33) with $D=0$, $P_{ϵm}=0$ of massive fields of the same masses $m_2=m_3=m\ne 0$ (critical case) there are the following $Q_c^{tot}$ BRST-closed operators (in minimal derivative scheme)

$$\begin{array}{ccc}& & \check{L}{}^{\left(i\right)}=\widehat{p}{}_{\mu }{}^{\left(i\right)}a^{\left(i\right)\mu +}+{(-1)}^{i}m{d}^{\left(i\right)+}{\theta }_{i,1}-ı{\widehat{\mathcal{P}}}_0^{\left(i\right)}{\eta }_1^{\left(i\right)+},i=1,2,3\hfill \end{array}$$

(86)

$$\begin{array}{ccc}& & L_{11}^{\left(23\right)+}=a^{\left(2\right)\mu +}{a}_{\mu }^{\left(3\right)+}+\frac{1}{2m}(d^{\left(2\right)+}{L}^{\left(3\right)}-d^{\left(3\right)+}{L}^{\left(2\right)})+d^{\left(2\right)+}{d}^{\left(3\right)+}\hfill \\ & & \phantom{L_{11}^{\left(23\right)+}=}-\frac{1}{2}{\mathcal{P}}_1^{\left(2\right)+}{\eta }_1^{\left(3\right)+}-\frac{1}{2}{\mathcal{P}}_1^{\left(3\right)+}{\eta }_1^{\left(2\right)+},\hfill \end{array}$$

(87)

$$\begin{array}{ccc}& & Z={\tilde{L}}_{11}^{\left(12\right)+}{\check{L}}^{\left(3\right)}+cycl.perm.(1,2,3).\hfill \end{array}$$

(88)

$$\begin{array}{ccc}& & {\tilde{L}}_{11}^{\left(23\right)+}=L_{11}^{\left(23\right)+}-\frac{1}{2m}\left(d^{2)+}{\check{L}}^{\left(3\right)}\right)-d^{3)+}{\check{L}}^{\left(2\right)})\hfill \end{array}$$

(89)

$$\begin{array}{ccc}& & {\tilde{L}}_{11}^{(ii+1)+}=a^{\left(i\right)\mu +}{a}_{\mu }^{(i+1)+}-\frac{1}{2}{\mathcal{P}}_1^{\left(i\right)+}{\eta }_1^{(i+1)+}-\frac{1}{2}{\mathcal{P}}_1^{(i+1)+}{\eta }_1^{\left(i\right)+},i=1,3.\hfill \end{array}$$

(90)

The trace operators ${\widehat{L}}_{11}^{\left(i\right)+}$, ${\widehat{L}}_{11}^{\left(i\right)}$ look for massless field

$$\begin{array}{ccc}& & \left(\widehat{L}{}_{11}{}^{\left(1\right)+},{\widehat{L}}_{11}^{\left(1\right)}\right)=\left(l_{11}^{\left(1\right)+}+b^{\left(1\right)+}+{\mathcal{P}}_1^{\left(1\right)+}{\eta }_1^{\left(1\right)+},l_{11}^{\left(1\right)}+(b^{\left(1\right)+}{b}^{\left(1\right)}+h^{\left(1\right)})b^{\left(1\right)}+{\eta }_1^{\left(1\right)}{\mathcal{P}}_1^{\left(1\right)}\right)\hfill \end{array}$$

(91)

and according to (6) and (7) for massive case when $i=2,3$

$$\begin{array}{ccc}& & \widehat{L}{}_{11}{}^{\left(i\right)+}=l_{11}^{\left(i\right)+}-(1/2){\left(d^{\left(i\right)+}\right)}^2+b^{\left(i\right)+}+{\mathcal{P}}_1^{\left(i\right)+}{\eta }_1^{\left(i\right)+},\hfill \end{array}$$

(92)

$$\begin{array}{ccc}& & \widehat{L}{}_{11}{}^{\left(i\right)}=l_{11}^{\left(i\right)}-(1/2){\left(d^{\left(i\right)}\right)}^2+(b^{\left(i\right)+}{b}^{\left(i\right)}+h^{\left(i\right)})b^{\left(i\right)}+{\eta }_1^{\left(i\right)}{\mathcal{P}}_1^{\left(i\right)}.\hfill \end{array}$$

(93)

The general solution for the parity invariant cubic vertex describing interaction for irreducible massless field with helicity $s_1$ and two massive with spins $s_2$, $s_3$ with the same masses ${\left(\overline{m}\right)}_2=(0,m,m)$ is shown in Figure 2, and has the form

$$\begin{array}{ccc}\hfill |V{}^{\left(3\right)}\rangle _{{\left(s\right)}_3}^{{\left(\overline{m}\right)}_2}& =& |V{}^{M\left(3\right)}\rangle _{{\left(s\right)}_3}^{{\left(\overline{m}\right)}_2}+\sum _{(j_3,j_3,j_3)>0}^{([s_1/2],[s_2/2],[s_3/2])}{U}_{j_3}^{\left(s_1\right)}{U}_{j_3}^{\left(s_2\right)}{U}_{j_3}^{\left(s_3\right)}{|V{}^{M\left(3\right)}\rangle }_{{\left(s\right)}_3-2{\left(j\right)}_3}^{{\left(\overline{m}\right)}_2},\hfill \end{array}$$

(94)

where the vertex $|V{}^{M\left(3\right)}\rangle _{{\left(s\right)}_3-2{\left(j\right)}_3}^{{\left(\overline{m}\right)}_2}$ was defined in Metsaev’s paper [18] for ${\left(j\right)}_3=0$ with account for (86)–(90) but with modified forms ${\check{\mathcal{L}}}_{k_i}^{\left(i\right)}$,

$$\begin{array}{ccc}& & {\check{\mathcal{L}}}_{k_i}^{\left(i\right)}=\sum _{j=0}^{[k_i/2]}{(-1)}^j{\left({\check{L}}^{\left(i\right)}\right)}^{k_i-2j}{\left({\widehat{p}}^{\left(i\right)}\right)}^{2j}\frac{k_i!}{j!2^j(k_i-2j))!}\frac{{\left(b^{\left(i\right)+}\right)}^j}{C(j,h^{\left(i\right)})},\hfill \\ & & \mathrm{for}{\left({\widehat{p}}^{\left(i\right)}\right)}^2={\eta }^{\mu \nu }(p_{\mu }^{(i+1)}-p_{\mu }^{(i+2)})(p_{\nu }^{(i+1)}-p_{\nu }^{(i+2)})-{(-1)}^i{m}^2{\theta }_{i,1},i=1,2,3.\hfill \end{array}$$

(95)

and forms ${\mathcal{L}}_{11|{\sigma }_{(i+2)}}^{(ii+1)+}$ instead of polynomials ${\left(L_{11}^{(ii+1)+}\right)}^{{\sigma }_{(i+2)}}$ given in (89), (90) for $i=1,2,3$:

$$\begin{array}{ccc}& & {\mathcal{L}}_{11|1}^{(ii+1)+}=L_{11}^{(ii+1)+}-\sum _{i_0=i}^{i+1}{W}_{(ii+1)|0}^{\left(i_0\right)}\frac{b^{\left(i_0\right)+}}{h^{\left(i_0\right)}}+\frac{1}{2}(\sum _{i_0\ne j_0}\left[\widehat{L}{}_{11}{}^{\left(j_0\right)},W_{(ii+1)|0}^{\left(i_0\right)}\right\}\frac{b^{\left(i_0\right)+}}{h^{\left(i_0\right)}}\frac{b^{\left(j_0\right)+}}{h^{\left(j_0\right)}}\hfill \\ & & +\sum _{i_0}\left[\widehat{L}{}_{11}{}^{\left(i_0\right)},W_{(ii+1)|0}^{\left(i_0\right)}\right\}\frac{{\left(b^{\left(i_0\right)+}\right)}^2}{h^{\left(i_0\right)}(h^{\left(i_0\right)}+1)}),\hfill \end{array}$$

(96)

$$\begin{array}{ccc}& & {\mathcal{L}}_{11|k+1}^{(ii+1)+}=L_{11}^{(ii+1)+}{\mathcal{L}}_{11|k}^{(ii+1)+}-\sum _{i_k=i}^{i+1}{W}_{(ii+1)|k}^{\left(i_k\right)}\frac{b^{\left(i_k\right)+}}{h^{\left(i_k\right)}}+\frac{1}{2}(\sum _{i_k\ne j_k}\left[\widehat{L}{}_{11}{}^{\left(j_k\right)},W_{(ii+1)|k}^{\left(i_k\right)}\right\}\hfill \\ & & \times \frac{b^{\left(i_k\right)+}}{h^{\left(j_k\right)}}\frac{b^{\left(j_k\right)+}}{h^{\left(j_k\right)}}+\sum _{i_k}\left[\widehat{L}{}_{11}{}^{\left(i_k\right)},W_{(ii+1)|k}^{\left(i_k\right)}\right\}\frac{{\left(b^{\left(i_k\right)+}\right)}^2}{{\prod }_{p=0}^1(h^{\left(i_k\right)}+p)})+\sum _{e=3}^{k+2}\frac{{(-1)}^e}{e!}\times \hfill \\ & & \times \sum _{l=0,j\ne i}^{[e/2]}\frac{e!}{[e/(e-l\left)\right](e-l)!l!}{\mathrm{ad}}_{{\widehat{L}}_{11}^{\left(j_k\right)}}^l{\mathrm{ad}}_{{\widehat{L}}_{11}^{\left(i_k\right)}}^{e-l-1}{W}_{(ii+1)|k}^{\left(i_k\right)}\frac{{\left(b^{\left(j_k\right)+}\right)}^l}{C(l,h^{\left(j_k\right)})}\frac{{\left(b^{\left(i_k\right)+}\right)}^{e-l}}{C(e-l,h^{\left(i_k\right)})}.\hfill \end{array}$$

(97)

(for $k=1,\dots ,min(s_i,s_{i+1})-1$) and with $Q^{tot}$- BRST-closed polynomials ${\mathcal{Z}}^{\lambda }$, for $\lambda =1,2,\dots ,min(s_1,s_2,s_3)$ constructed from BRST-closed forms ${\tilde{\mathcal{L}}}_{11|1}^{(ii+1)+}$ according to (96), subject to change of ${\mathcal{L}}_{11|1}^{(ii+1)+}$ on one with tilde and ${\tilde{\mathcal{L}}}_1^{\left(i\right)}$:

$$\begin{array}{ccc}& & \mathcal{Z}={\tilde{\mathcal{L}}}_{11|1}^{\left(12\right)+}{\check{\mathcal{L}}}_1^{\left(3\right)}+cycl.perm.(1,2,3).\mathrm{with}{\tilde{\mathcal{L}}}_1^{\left(i\right)}=\check{L}{}^{\left(i\right)}-\left[\widehat{L}{}_{11}{}^{\left(i\right)},{\check{L}}^{\left(i\right)}\right\}\frac{b^{\left(3\right)+}}{h^{\left(3\right)}}.\hfill \end{array}$$

(98)

Explicitly, the vertex $V_{{\left(s\right)}_3-2{\left(j\right)}_3}^{M\left(3\right)|{\left(\overline{m}\right)}_2}$ is determined by

$$\begin{array}{ccc}\hfill V_{{\left(s\right)}_3-2{\left(j\right)}_3}^{M\left(3\right)|{\left(\overline{m}\right)}_2}& =& \sum _{\sigma ,\lambda }{\mathcal{L}}_{11|\sigma }^{\left(23\right)+}{\mathcal{Z}}^{\lambda }\prod _{i=1}^3{\check{\mathcal{L}}}_{k_i}^{\left(i\right)},\mathrm{for}0\le k_1\le s_1-2j_1\hfill \end{array}$$

(99)

and is $(3+2)$-parameter family to be enumerated by the natural parameters (corresponding for traces) ${\left(j\right)}_3$, and $k_{min}$, $k_{max}$ (corresponding for order of derivatives) subject to the equations

$$\begin{array}{ccc}& & k_1=k_{min},k_2=k_{max}-k_{min}-(s_3-2j_3),k_3=k_{max}-k_{min}-(s_3-2j_3),\hfill \end{array}$$

(100)

$$\begin{array}{ccc}& & \sigma =s-2(s_1-2j_1)-k_{max}+2k_{min},\lambda =s_1-2j_1-k_{min},\hfill \\ & & \left[max(s_2-2j_2,s_3-2j_3)\right]\le k_{max}-k_{min}\le \left[s-2(s_1-2j_1)+k_{min}\right].\hfill \end{array}$$

(101)

The representation for the vertex (94) and (99) for irreducible massless and massive fields with the same masses presents the basic results of this subsection. For vanishing ${\left(j\right)}_3$ remaining parameters correspond to ones in constrained BRST formulation [18].

4.2.2. One Massless Field and Two Massive Fields with Different Masses

For the case (34) with $D>0$ with different non-vanishing masses $m_3\ne m_2$ we start from $Q_c^{tot}$ BRST-closed operators, with except for $L^{\left(1\right)}$ (in minimal derivative scheme)

$$\begin{array}{ccc}& & L^{\left(i\right)}=\widehat{p}{}_{\mu }{}^{\left(i\right)}a^{\left(i\right)\mu +}+{(-1)}^i\frac{{\delta }_{i2}{m}_3^2+{\delta }_{i3}{m}_2^2}{m_i}{d}^{\left(i\right)+}-ı\widehat{\mathcal{P}}{}_0{}^{\left(i\right)}\eta _1^{\left(i\right)+},i=1,2,3;\hfill \end{array}$$

(102)

$$\begin{array}{ccc}& & L_{11}^{\left(23\right)+}=a^{\left(2\right)\mu +}{a}_{\mu }^{\left(3\right)+}+\frac{d^{\left(2\right)+}}{2m_2}{L}^{\left(3\right)}-\frac{d^{\left(3\right)+}}{2m_3}{L}^{\left(2\right)}+\frac{m_2^2+m_3^2}{2m_2{m}_3}{d}^{\left(2\right)+}{d}^{\left(3\right)+}\hfill \\ & & \phantom{L_{11}^{\left(23\right)+}}-\frac{1}{2}{\mathcal{P}}_1^{\left(2\right)+}{\eta }_1^{\left(3\right)+}-\frac{1}{2}{\mathcal{P}}_1^{\left(3\right)+}{\eta }_1^{\left(2\right)+},\hfill \end{array}$$

(103)

$$\begin{array}{ccc}& & L_{11}^{\left(12\right)+}=a^{\left(1\right)\mu +}{a}_{\mu }^{\left(2\right)+}-\left(\frac{d^{\left(2\right)+}}{2m_2}+\frac{L^{\left(2\right)}}{2(m_3^2-m_2^2)}\right)L^{\left(1\right)}-\frac{1}{2}{\mathcal{P}}_1^{\left(1\right)+}{\eta }_1^{\left(2\right)+}-\frac{1}{2}{\mathcal{P}}_1^{\left(2\right)+}{\eta }_1^{\left(1\right)+},\hfill \end{array}$$

(104)

$$\begin{array}{ccc}& & L_{11}^{\left(31\right)+}=a^{\left(3\right)\mu +}{a}_{\mu }^{\left(1\right)+}+\left(\frac{d^{\left(3\right)+}}{2m_3}+\frac{L^{\left(3\right)}}{2(m_3^2-m_2^2)}\right)L^{\left(1\right)}-\frac{1}{2}{\mathcal{P}}_1^{\left(3\right)+}{\eta }_1^{\left(1\right)+}-\frac{1}{2}{\mathcal{P}}_1^{\left(1\right)+}{\eta }_1^{\left(3\right)+},\hfill \end{array}$$

(105)

The general solution for the parity invariant cubic vertex describing interaction for irreducible massless field with helicity $s_1$ and two massives with spins $s_2$, $s_3$ with different masses is shown by Figure 3 and has the representation in terms of the product of BRST-closed (with respective complete BRST operator $Q^{tot}$ in question) forms

$$\begin{array}{ccc}\hfill |V{}^{\left(3\right)}\rangle _{{\left(s\right)}_3}^{{\left(m\right)}_2}& =& |V{}^{M\left(3\right)}\rangle _{{\left(s\right)}_3}^{{\left(m\right)}_2}+\sum _{(j_3,j_3,j_3)>0}^{([s_1/2],[s_2/2],[s_3/2])}{U}_{j_3}^{\left(s_1\right)}{U}_{j_3}^{\left(s_2\right)}{U}_{j_3}^{\left(s_3\right)}{|V{}^{M\left(3\right)}\rangle }_{{\left(s\right)}_3-2{\left(j\right)}_3}^{{\left(m\right)}_2},\hfill \end{array}$$

(106)

where the vertex $|V{}^{M\left(3\right)}\rangle _{{\left(s\right)}_3-2{\left(j\right)}_3}^{{\left(m\right)}_2}$ was given in [18] with account for (102)–(105) but with modified forms ${\mathcal{L}}_{k_i}^{\left(i\right)}$, (108) and ${\mathcal{L}}_{11|{\sigma }_{(j+2)}}^{(jj+1)+}$ (110), (77) instead of ${\left(L_{11}^{(jj+1)+}\right)}^{{\sigma }_{(j+2)}}$

$$\begin{array}{ccc}\hfill V_{{\left(s\right)}_3-2{\left(j\right)}_3}^{M\left(3\right)|{\left(m\right)}_2}& =& \sum _{{\tau }_2,{\tau }_3}\prod _{i=2}^3{\mathcal{L}}_{{\tau }_i}^{\left(i\right)}\prod _{r=1}^3{\mathcal{L}}_{11|{\sigma }_{r+2}}^{(rr+1)+},\hfill \end{array}$$

(107)

were the ranges for ${\tau }_2,{\tau }_3,{\sigma }_1,{\sigma }_2,{\sigma }_3$ are specified below. Here,

$$\begin{array}{ccc}& & {\mathcal{L}}_{k_i}^{\left(i\right)}=\sum _{j=0}^{[k_i/2]}{(-1)}^j{\left(L^{\left(i\right)}\right)}^{k_i-2j}{\left({\widehat{p}}^{\left(i\right)}\right)}^{2j}\frac{k_i!}{j!2^j(k_i-2j))!}\frac{{\left(b^{\left(i\right)+}\right)}^j}{C(j,h^{\left(i\right)})},\hfill \end{array}$$

(108)

$$\begin{array}{ccc}& & \mathrm{for}{\left({\widehat{p}}^{\left(i\right)}\right)}^2={\eta }^{\mu \nu }(p_{\mu }^{(i+1)}-p_{\mu }^{(i+2)})(p_{\nu }^{(i+1)}-p_{\nu }^{(i+2)})-\frac{{({\delta }_{i2}{m}_3^2+{\delta }_{i3}{m}_2^2)}^2}{m_i^2},i=2,3.\hfill \end{array}$$

(109)

and for mixed trace operators, when $i=1,2,3$

$$\begin{array}{ccc}& & {\mathcal{L}}_{11|1}^{(ii+1)+}=L_{11}^{(ii+1)+}-\sum _{i_0=i}^{i+1}{W}_{(ii+1)|0}^{\left(i_0\right)}\frac{b^{\left(i_0\right)+}}{h^{\left(i_0\right)}}+\frac{1}{2}(\sum _{i_0\ne j_0}\left[\widehat{L}{}_{11}{}^{\left(j_0\right)},W_{(ii+1)|0}^{\left(i_0\right)}\right\}\frac{b^{\left(i_0\right)+}}{h^{\left(i_0\right)}}\frac{b^{\left(j_0\right)+}}{h^{\left(j_0\right)}}\hfill \\ & & +\sum _{i_0}\left[\widehat{L}{}_{11}{}^{\left(i_0\right)},W_{(ii+1)|0}^{\left(i_0\right)}\right\}\frac{{\left(b^{\left(i_0\right)+}\right)}^2}{h^{\left(i_0\right)}(h^{\left(i_0\right)}+1)}),\hfill \end{array}$$

(110)

$$\begin{array}{ccc}& & {\mathcal{L}}_{11|k+1}^{(ii+1)+}=L_{11}^{(ii+1)+}{\mathcal{L}}_{11|k}^{(ii+1)+}-\sum _{i_k=i}^{i+1}{W}_{(ii+1)|k}^{\left(i_k\right)}\frac{b^{\left(i_k\right)+}}{h^{\left(i_k\right)}}+\frac{1}{2}(\sum _{i_k\ne j_k}\left[\widehat{L}{}_{11}{}^{\left(j_k\right)},W_{(ii+1)|k}^{\left(i_k\right)}\right\}\hfill \\ & & \times \frac{b^{\left(i_k\right)+}}{h^{\left(j_k\right)}}\frac{b^{\left(j_k\right)+}}{h^{\left(j_k\right)}}+\sum _{i_k}\left[\widehat{L}{}_{11}{}^{\left(i_k\right)},W_{(ii+1)|k}^{\left(i_k\right)}\right\}\frac{{\left(b^{\left(i_k\right)+}\right)}^2}{{\prod }_{p=0}^1(h^{\left(i_k\right)}+p)})+\sum _{e=3}^{k+2}\frac{{(-1)}^e}{e!}\times \hfill \\ & & \times \sum _{l=0,j\ne i}^{[e/2]}\frac{e!}{[e/(e-l\left)\right](e-l)!l!}{\mathrm{ad}}_{{\widehat{L}}_{11}^{\left(j_k\right)}}^l{\mathrm{ad}}_{{\widehat{L}}_{11}^{\left(i_k\right)}}^{e-l-1}{W}_{(ii+1)|k}^{\left(i_k\right)}\frac{{\left(b^{\left(j_k\right)+}\right)}^l}{C(l,h^{\left(j_k\right)})}\frac{{\left(b^{\left(i_k\right)+}\right)}^{e-l}}{C(e-l,h^{\left(i_k\right)})}\hfill \end{array}$$

(111)

(for $k=1,\dots ,min(s_i,s_{i+1})-1$). The vertex (107) represents the $(3+2)$-parameter family to be enumerated by the natural parameters ${\left(j\right)}_3$, and ${\tau }_2,{\tau }_3$ subject to the equations

$$\begin{array}{ccc}& & {\sigma }_i=\frac{1}{2}[s-2J-2(s_i-2j_i)-{(-1)}^{{\delta }_{i1}}{\tau }_2-{(-1)}^{{\delta }_{i2}}{\tau }_3],i=1,2,3;\hfill \end{array}$$

(112)

$$\begin{array}{ccc}& & |\sum _{i=2}^3{(-1)}^i(s_i-2j_i-{\tau }_i)|\le s_1-2j_1\le \sum _{i=2}^3(s_i-2j_i-{\tau }_i),\hfill \end{array}$$

(113)

$$\begin{array}{ccc}& & s-2J-{\tau }_2-{\tau }_3=2p,p\in {\mathbb{N}}_0.\hfill \end{array}$$

(114)

The relations for the vertex (106) and (107) for irreducible massless and massive fields with different masses present our basic results in the subsection. Again, for vanishing ${\left(j\right)}_3$ the remaining parameters correspond to the ones in the constrained BRST formulation [18].

5. Examples for HS Fields with Special Spin Values

Here, we consider ghost-independent and component (tensor) forms of the cubic vertices for special cases of interacting higher spin fields.

5.1. Vertices for Fields with $(m,s)$, $(0,{\lambda }_1)$, $(0,{\lambda }_2)$

In the subsection, we derive the cubic vertices $V_{{\left(s\right)}_3}^{M\left(3\right)|m}$ for one massive HS field with $(m,s)$ and two massless HS fields with $(0,{\lambda }_1)$, $(0,{\lambda }_2)$ with small values of two spin parameters.

5.1.1. Case $(m,s)$, $(0,{\lambda }_i)$ for ${\lambda }_i=0$

First, for the interaction of 2 massless scalars with a massive HS field we have, according to (78) and (79), the j-parameter family of vertices for $j=1,\dots ,[s/2]$ restoring the dimensional coupling constants $t_j$ ($dim{t}_j$=$s+d/2-3-2j$, in metric units providing a dimensionless of the action)

$$\begin{array}{ccc}\hfill V_{(0,0,s)}^{\left(3\right)|m}& =& \sum _{j\ge 0}^{[s/2]}{t}_j{U}_j^{\left(s\right)}{\mathcal{L}}_{s-2j}^{\left(3\right)}=\sum _{j\ge 0}^{[s/2]}{t}_j{U}_j^{\left(s\right)}\sum _{i=0}^{\left[\right(s-2j)/2]}{(-1)}^i{\left(L^{\left(3\right)}\right)}^{s-2j-2i}\times \hfill \\ & & \times {\left({\widehat{p}}^{\left(3\right)}\right)}^{2i}\frac{(s-2j)!}{i!2^i(s-2j-2i)!}\frac{{\left(b^{\left(3\right)+}\right)}^i}{C(i,h^{\left(3\right)})},\hfill \end{array}$$

(115)

with following decomposition in powers of ${\eta }_1^{\left(3\right)+}$ for the operators

$$\begin{array}{ccc}& & {\left(L^{\left(3\right)}\right)}^k={\left(L^{\left(3\right)}\right)}_0^k-ık{\widehat{\mathcal{P}}}_0^{\left(3\right)}{\eta }_1^{\left(3\right)+}{\left(L^{\left(3\right)}\right)}_0^{k-1}\equiv {\left(\widehat{p}{}_{\mu }{}^{\left(3\right)}a^{\left(i\right)\mu +}\right)}^{k-1}\left(\widehat{p}{}_{\mu }{}^{\left(3\right)}a^{\left(3\right)\mu +}-ık{\widehat{\mathcal{P}}}_0^{\left(3\right)}{\eta }_1^{\left(3\right)+}\right),\hfill \end{array}$$

(116)