Till now most of the results on interaction vertices for massless higher spin fields were obtained in a metric-like formalism using completely symmetric (spin-)tensors. In this, the Lagrangians turn out to be very complicated and the main reason is that the higher the spin one want to consider the more derivatives one has to introduce. In this paper we show that such investigations can be greatly simplified if one works in a frame-like formalism. As an illustration we consider massless spin 3 particle and reconstruct a number of vertices describing its interactions with lower spin 2, 1 and 0 ones. In all cases considered we give explicit expressions for the Lagrangians and gauge transformations and check that the algebra of gauge transformations is indeed closed.
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 …
Light-cone gauge formulation of relativistic dynamics of a continuous-spin field propagating in the flat space is developed. Cubic interaction vertices of continuous-spin massless fields and totally symmetric arbitrary spin massive fields are studied. We consider parity invariant cubic vertices that involve one continuous-spin massless field and two arbitrary spin massive fields and parity invariant cubic vertices that involve two continuous-spin massless fields and one arbitrary spin massive field. We construct the complete list of such vertices explicitly. Also we demonstrate that there are …
We derive a class of cubic interaction vertices for three higher spin fields, with integer spins $λ_1$, $λ_2$, $λ_3$, by closing commutators of the Poincaré algebra in four-dimensional flat spacetime. We find that these vertices exhibit an interesting factorization property which allows us to identify off-shell perturbative relations between them.
We construct a range of supersymmetric cubic vertices for three massless higher spin supermultiplets in the four-dimensional space. We use frame-like multispinor formalism, which allows to avoid most of the technical difficulties and provides a uniform description for bosons and fermions. Our work is based on the so-called Fradkin-Vasiliev formalism for construction of the cubic vertices, which requires the non-zero cosmological constant. Thus we first construct the vertices in AdS space and then consider the flat limit. We show that the AdS supersymmetric vertex is a sum of four elementary ve…
New method (arXiv:2104.11930) is applied to construct cubic interactions of massless spin 4 and real scalar fields. In contrast with the case of massless spin 3 fields (arXiv:2208.05700, 2209.03678, 2210.02842) the procedure requires to use an unconstrained formulation (arXiv:0702.161) for the Fronsdal theory. It is shown that in the unconstrained formulation there exists a four-parameter family of cubic interactions between massless spin 4 and real scalar fields, which contains four derivatives in vertices and is invariant with respect to the original gauge transformation. These vertices cont…
We present results of investigation into a problem of cubic interaction vertices of massless higher spin fields which transform in arbitrary irreps of the Poincare algebra
Using BRST-BV formulation of relativistic dynamics, we study arbitrary spin massive and massless fields propagating in flat space. Generating functions of gauge invariant off-shell cubic interaction vertices for mixed-symmetry and totally symmetric fields are obtained. For the case of totally symmetric fields, we derive restrictions on the allowed values of spins and the number of derivatives, which provide a classification of cubic interaction vertices for such fields. As by product, we present simple expressions for the Yang-Mills and gravitational interactions of massive totally symmetric a…
In the framework of light-cone formulation of relativistic dynamics, arbitrary spin massless fields propagating in the four-dimensional AdS space are studied. For such fields, the complete list of light-cone gauge cubic interaction vertices is obtained. Realization of relativistic symmetries on space of light-cone gauge massless AdS fields is also obtained. The light-cone gauge vertices for massless AdS fields take simple form similar to the one for massless fields in the flat space.
We study the interaction of massive spin-3/2 field with electromagnetic and gravitational fields in the four dimensional AdS space and construct the corresponding cubic vertices. Construction is based on generalization of Fradkin-Vasiliev formalism, developed for massless higher spin fields, to massive fermionic higher spin fields. The main ingredients of this formalism are the gauge invariant curvatures. We build such curvatures for the massive theory under consideration and show how the cubic vertices are written in their terms.
Using the light-cone gauge approach to relativistic field dynamics, we study arbitrary spin fermionic and bosonic fields propagating in flat space of dimension greater than or equal to four. Generating functions of parity invariant cubic interaction vertices for totally symmetric and mixed-symmetry massive and massless fields are obtained. For the case of totally symmetric fields, we derive restrictions on the allowed values of spins and the number of derivatives. These restrictions provide a complete classification of parity invariant cubic interaction vertices for totally symmetric fermionic…
In the framework of light-cone gauge formulation, massless arbitrary spin N=1 supermultiplets in four-dimensional flat space are considered. We study both the integer (super)spin and half-integer (super)spin supermultiplets. For such supermultiplets, formulation in terms of unconstrained light-cone gauge superfields defined on chiral momentum superspace is used. Superfield representation for all cubic interaction vertices of the supermultiplets is obtained. Representation of the cubic vertices in terms of component fields is derived. Realization of relativistic symmetries of N=1 Poincare super…
Based purely on symmetry considerations, we derive the following result: in momentum space, the coefficient of the cubic interaction vertex for a spin $λ$ field is equal to the corresponding Yang-Mills (spin 1) coefficient, raised to the power $λ$. This result is valid for all $λ$ for Lagrangians that contain a cubic interaction vertex of the form $λ$-$λ$-$λ$, in four-dimensional flat spacetime. For $λ=3$, we present an additional derivation of this result.
In this paper we provide a complete list of spin-2 cubic interaction vertices with two derivatives. We work in (anti) de Sitter space with dimension d >= 4 and arbitrary value of cosmological constant and use simple metric formalism without any auxiliary or Stueckelberg fields. We separately consider cases with one, two and three different spin-2 fields entering the vertex where each field may be massive, massless or partially massless one. The connection of our results with massive (bi)gravity theories is also briefly discussed.
In this paper we construct a number of cubic interaction vertices for massless bosonic and fermionic higher spin fields in flat four dimensional space. First of all, we construct these cubic vertices in AdS_4 space using a so-called Fradkin-Vasiliev approach, which works only for the non-zero cosmological constant. Then we consider a flat limit taking care on all the higher derivative terms which FV-approach generates. We restrict ourselves with the four dimensions because this allows us to use the frame-like multispinor formalism which greatly simplifies all calculations and provides a descri…
Using the light-cone formulation of relativistic dynamics, we develop various methods for constructing cubic interaction vertices and apply these methods to the study of higher spin fields propagating in flat space of dimension greater than or equal to four. Generating functions of parity invariant cubic interaction vertices for massive and massless higher spin fields of arbitrary symmetry are obtained. We derive restrictions on the allowed values of spins and the number of derivatives, which provide a classification of cubic interaction vertices for totally symmetric fields. As an example of …
Using the BRST approach to higher spin field theories we develop a generic technique for constructing the cubic interaction vertices for N=1 supersymmetric massless higher spin fields on four, six and ten dimensional flat backgrounds. Such an approach allows formulation of the equations for cubic vertices including bosonic and fermionic higher spin fields, and the problem of finding the vertices is reduced to finding the consistent solutions to these equations. As a realization of this procedure, we present the particular solutions for the vertices where the fields obey some off-shell constrai…
Metric-like Cubic Vertices for Massless Bosonic Higher-Spin Fields in AdS$_3$
University of Sussex（萨塞克斯大学）；Ludwig-Maximilians-Universität München（慕尼黑大学）；Max Planck Institute for Astrophysics（马克斯·普朗克天体物理研究所）；Abdus Salam Centre for Theoretical Physics, Imperial College London（帝国理工学院阿卜杜斯·萨拉姆理论物理中心）
Cubic vertices for symmetric higher-spin gauge fields of integer spins in $(A)dS_d$ are analyzed. $(A)dS_d$ generalization of the previously known action in $AdS_4$, that describes cubic interactions of symmetric massless fields of all integer spins $s\geq 2$, is found. A new cohomological formalism for the analysis of vertices of higher-spin fields of any symmetry and/or order of nonlinearity is proposed within the frame-like approach. Using examples of spins two and three it is demonstrated how nontrivial vertices in $(A)dS_d$, including Einstein cubic vertex, can result from the $AdS$ defor…
This work completes the classification of the cubic vertices for arbitrary spin massless bosons in three dimensions started in a previous companion paper by constructing parity-odd vertices. Similarly to the parity-even case, there is a unique parity-odd vertex for any given triple $s_1\geq s_2\geq s_3\geq 2$ of massless bosons if the triangle inequalities are satisfied ($s_1<s_2+s_3$) and none otherwise. These vertices involve two (three) derivatives for odd (even) values of the sum $s_1+s_2+s_3$. A non-trivial relation between parity-even and parity-odd vertices is found. Similarly to the pa…
We study the exchange diagrams in the computation of four-point functions of all chiral primary operators in D=4, $\mathcal{N}=4$ Super-Yang-Mills using AdS/CFT correspondence. We identify all supergravity fields that can be exchanged and compute the cubic couplings. As a byproduct, we also rederive the normalization of the quadratic action of the exchanged fields. The cubic couplings computed in this paper and the propagators studied extensively in the literature can be used to compute almost all the exchange diagrams explicitly. Some issues in computing the complete four-point function in th…
Using light-cone gauge formulation, massive arbitrary spin irreducible fields and massless (scalar and one-half spin) fields in three-dimensional flat space are considered. Both the integer spin and half-integer spin fields are studied. For such fields, we provide classification for cubic interactions and obtain explicit expressions for all cubic interaction vertices. We study two forms of the cubic interaction vertices which we refer to as first-derivative form and higher-derivative form. All cubic interaction vertices are built by using the first-derivative form.
We use the Fradkin-Vasiliev procedure to construct the full set of non-abelian cubic vertices for totally symmetric higher spin gauge fields in anti-de Sitter space. The number of such vertices is given by a certain tensor-product multiplicity. We discuss the one-to-one relation between our result and the list of non-abelian gauge deformations in flat space obtained elsewhere via the cohomological approach. We comment about the uniqueness of Vasiliev's simplest higher-spin algebra in relation with the (non)associativity properties of the gauge algebras that we classified. The gravitational int…
Totally symmetric arbitrary spin conformal fields propagating in the flat space of even dimension greater than or equal to four are studied. For such fields, we develop a general ordinary-derivative light-cone gauge formalism and obtain restrictions imposed by the conformal algebra symmetries on interaction vertices. We apply our formalism for the detailed study of conformal scalar and vector fields. For such fields, all parity-even cubic interaction vertices are obtained. The cubic vertices obtained are presented in terms of dressing operators and undressed vertices. We show that the undresse…
We derive light-cone cubic interaction vertices involving fermions and bosons of arbitrary spin by demanding closure of the Poincaré algebra. We derive the three-point scattering amplitude corresponding to these interaction vertices and find that they possess interesting factorization properties, identical to the case of three boson scattering.
We provide a formalism to calculate the cubic interaction vertices of the stable string bit model, in which string bits have $s$ spin degrees of freedom but no space to move. With the vertices, we obtain a formula for one-loop self-energy, i.e., the $\mathcal{O}\left(1/N^{2}\right)$ correction to the energy spectrum. A rough analysis shows that, when the bit number $M$ is large, the ground state one-loop self-energy $ΔE_{G}$ scale as $M^{5-s/4}$ for even $s$ and $M^{4-s/4}$ for odd $s$. Particularly, in $s=24$, we have $ΔE_{G}\sim 1/M$, which resembles the Poincaré invariant relation $P^{-}\si…
We continue the program of constructing cubic interactions between matter and higher spin supermultiplets. In this work we consider a complex linear superfield and we find that it can have cubic interactions only with supermultiplets with propagating spins $j=s+1$, $j=s+1/2$ for any non-negative integer $s$ (half-integer superspin supermultiplets). We construct the higher spin supercurrent and supertrace, these compose the canonical supercurrent multiplet which generates the cubic interactions. We also prove that for every $s$ there exist an alternative minimal supercurrent multiplet, with van…
The problem of determining all consistent non-Abelian local interactions is reviewed in flat space-time. The antifield-BRST formulation of the free theory is an efficient tool to address this problem. Firstly, it allows to compute all on-shell local Killing tensor fields, which are important because of their deep relationship with higher-spin algebras. Secondly, under the sole assumptions of locality and Poincare invariance, all non-trivial consistent deformations of a sum of spin-three quadratic actions deforming the Abelian gauge algebra were determined. They are compared with lower-spin cas…
N-extended massless arbitrary integer and half-integer spin supermultiplets in four dimensional flat space are studied in the framework of light-cone gauge formalism. For such multiplets, by using light-cone momentum superspace, we build unconstrained light-cone gauge superfield formulation. The superfield formulation is used to develop a superspace representation for all cubic interactions vertices of the N-extended massless supermultiplets. Our suitable treatment of the light-cone gauge superfields allows us to obtain attractively simple superspace representation for the cubic interaction ve…
Parity-even cubic vertices of massless bosons of arbitrary spins in three dimensional Minkowski space are classified in the metric-like formulation. As opposed to higher dimensions, there is at most one vertex for any given triple $s_1,s_2,s_3$ in three dimensions. All the vertices with more than three derivatives are of the type $(s,0,0)$, $(s,1,1)$ and $(s,1,0)$ involving scalar and/or Maxwell fields. All other vertices contain two (three) derivatives, when the sum of the spins is even (odd). Minimal coupling to gravity, $(s,s,2)$, has two derivatives and is universal for all spins (equivale…
We demonstrate how to reconstruct standard cubic vertices for N=1 supersymmetric Yang-Mills and Supergravities using various techniques adopted for the description of cubic interactions between higher spin fields.
We develop an approach to constructing the manifestly Lorentz covariant cubic interaction vertices for the four-dimensional massless higher spin bosonic fields with two-component dotted and undotted spinor indices. Such fields automatically satisfy the traceless conditions what simplify form of the equations determining the irreducible massless representation of the Poincar'e group with given helicity. The cubic vertex is formulated in the framework of the BRST approach to higher spin field theory. Use of the above spin-tensor fields allows to simplify a form of the BRST-charge and hence to fi…
In is shown explicitly that the Witten's interaction 3-vertex is a solution to the comma overlap equations; hence establishing the equivalence between the conventional and the "comma" formulation of interacting string theory at the level of vertices.
We study the cubic vertices for Maxwell-like higher-spins in flat and (A)dS background spaces of any dimension. Reducibility of their free spectra implies that a single cubic vertex involving any three fields subsumes a number of couplings among different particles of various spins. The resulting vertices do not involve traces of the fields and in this sense are simpler than their Fronsdal counterparts. We propose an extension of both the free theory and of its cubic deformation to a more general class of partially reducible systems, that one can obtain from the original theory upon imposing t…
Cubic interactions of higher-spin gauge fields in (A)dS are studied in the metric-like approach. Making use of the traceless and transverse constraints together with the ambient-space formalism, all consistent parity-invariant cubic vertices are obtained for d>3 in closed form pointing out the key role of their flat-space counterparts.
In this paper we elaborate on higher spin cubic interactions for massless, massive and partially massless fields. We work in the gauge invariant frame-like multispinor formalism, combining Lagrangian and unfolded formulations.
The problem of finding consistent cubic AdS interactions of massless mixed-symmetry higher-spin fields is recast into a system of partial differential equations that can be solved for given spins of the particles entering the cubic vertices. For simplicity, we consider fields with two families of indices for which some examples of interactions are explicitly discussed.
We consider scattering of massless higher-spin particles in the eikonal regime in four dimensions. By demanding the absence of asymptotic superluminality, corresponding to positivity of the eikonal phase, we place constraints on the possible cubic couplings which can appear in the theory. The cubic couplings come in two types: lower-derivative non-abelian vertices, and higher-derivative abelian vertices made out of gauge-invariant curvature tensors. We find that the abelian couplings between massless higher spins lead to an asymptotic time advance for certain choices of polarizations, indicati…
We make an explicit link between the cubic interactions of off-shell fields and the on-shell three-point amplitudes in four dimensions. Both the cubic interactions and the on-shell three-point amplitudes had been independently classified in the literature, but their relation has not been made explicit. The aim of this note is to provide such a relation and discuss similarities and differences of their constructions. For the completeness of our analysis, we also derive the covariant form of all parity-odd massless vertices.
The Bern-Kosower formalism, developed around 1990 as a novel way of obtaining QCD amplitudes as the limit of infinite string tension of the corresponding string amplitudes, was originally designed as an on-shell formalism. Building on early work by Strassler, the authors have recently shown that this `string-inspired formalism' is extremely efficient also as a tool for the study of off-shell amplitudes in QCD, and in particular for achieving compact form factor decompositions of the N-gluon vertices. Among other things, this formalism allows one to achieve a manifestly gauge invariant decompos…
We consider in detail the structure of quartic vertices for massless higher spin fields on Minkowski background, and study the consistency conditions imposed on cubic and quartic interactions by symmetries of the S--matrix. We discuss the possibility of generalizing the construction of quartic vertices to D-dimensional anti-de Sitter space.
New homotopy approach to the analysis of nonlinear higher-spin equations is developed. It is shown to directly reproduce the previously obtained local vertices. Simplest cubic (quartic in Lagrangian nomenclature) higher-spin interaction vertices in four dimensional theory are examined from locality perspective by the new approach and shown to be local. The results are obtained in a background independent fashion.
We investigate the cubic interactions of a massless higher-spin fermion with gravity in flat space and present covariant 2-s-s vertices, compatible with the gauge symmetries of the system, preserving parity. This explicit construction relies on the BRST deformation scheme that assumes locality and Poincare invariance. Consistent nontrivial cubic deformations exclude minimal gravitational coupling and may appear only with a number of derivatives constrained in a given range. Derived in an independent manner, our results do agree with those obtained from the light-cone formulation or inspired by…
The descent relations between string field theory (SFT) vertices are characteristic relations of the operator formulation of SFT and they provide self-consistency of this theory. The descent relations <V_2|V_1> and <V_3|V_1> in the NS fermionic string field theory in the kappa and discrete bases are established. Different regularizations and schemes of calculations are considered and relations between them are discussed.
In the context of higher-spin holography, we compare the classification of cubic interaction vertices for higher-spin gravity theories in three dimensions to the possible three-point correlation functions of conserved higher-spin currents in two-dimensional conformal field theories. In both cases, the allowed structures are governed by triangle inequalities for the involved spins. It is established that higher-order correlators satisfy similar polygon inequalities and that the same inequalities are valid for higher-order continuations of cubic vertices in the three-dimensional higher-spin grav…
We determine the form of a significantly large class of gauge-invariant quartic vertices for symmetric higher-spin fields (in Vasiliev formalism) in AdS5 space in the rectangular limit, (defined in the paper) using the higher-spin vertex operators in superstring theory, which construction is based on the generators of the higher-spin symmetry algebra, enveloping the AdS isometries. In this limit, a particular simplification is that the quartic vertices do not receive corrections from worldsheet renormalization group flows of the cubic terms, so their structure is fixed directly by the 4-point …
We constrain theories of a massive spin-2 particle coupled to a massless spin-2 particle by demanding the absence of a time advance in eikonal scattering. This is an $S$-matrix consideration that leads to model-independent constraints on the cubic vertices present in the theory. Of the possible cubic vertices for the two spin-2 particles, the requirement of subluminality leaves a particular linear combination of cubic vertices of the Einstein--Hilbert type. Either the cubic vertices must appear in this combination or new physics must enter at a scale parametrically the same as the mass of the …
A special class of higher curvature theories of gravity, Ricci Cubic Gravity (RCG), in general d dimensional space-time has been investigated in this paper. We have used two different approaches, the linearized equations of motion and auxiliary field formalism to study the massive and massless graviton propagating modes of the AdS background. Using the auxiliary field formalism, we have found the renormalized boundary stress tensor to compute the mass of Schwarzschild AdS and Lifshitz black holes in RCG theory.
We investigate cubic interactions between a chiral superfield and higher spin superfield corresponding to irreducible representations of the $4D,\, \mathcal{N}=1$ super-Poincaré algebra. We do this by demanding an invariance under the most general transformation, linear in the chiral superfield. Following Noether's method we construct an infinite tower of higher spin supercurrent multiplets which are quadratic in the chiral superfield and include higher derivatives. The results are that a single, massless, chiral superfield can couple only to the half-integer spin supermultiplets $(s+1,s+1/2)$…
Utilizing the worldline formalism we study the effects of demanding local interactions on the corresponding vertex factor. We begin by reviewing the familiar case of a relativistic particle in Minkowksi space, showing that localization gives rise to the standard conservation of momentum at each vertex. A generalization to curved geometry is then studied and a notion of covariant Fourier transform is introduced to aid in the analysis. The vertex factor is found to coincide with the one derived for flat spacetime. Next, we apply this formalism to a loop immersed in a gravitational field, demonst…
We introduce a first order description of linearized non-minimal ($n=-1$) supergravity in superspace, using the unconstrained prepotential superfield instead of the conventionally constrained super one forms. In this description, after integrating out the connection-like auxiliary superfield of first-order formalism, the superspace action is expressed in terms of a single superfield which combines the prepotential and compensator superfields. We use this description to construct the supersymmetric cubic interaction vertex $3/2-3/2-1/2$ which describes the electromagnetic interaction between tw…
We rewrite the Martin-Siggia-Rose (MSR) formalism for the statistical dynamics of classical fields in a covariant second order form appropriate for the statistical dynamics of relativistic field theory. This second order formalism is related to a rotation of Schwinger's closed time path (CTP) formalism for quantum dynamics, with the main difference being that certain vertices are absent in the classical theory. These vertices are higher order in an $\hbar$ expansion. The structure of the second order formulation of the Schwinger Dyson (S-D) equations is identical to that of the rotated CTP for…
We consider a massless higher spin field theory within the BRST approach and construct a general off-shell cubic vertex corresponding to irreducible higher spin fields of helicities $s_1, s_2, s_3$. Unlike the previous works on cubic vertices, which do not take into account of the trace constraints, we use the complete BRST operator, including the trace constraints that describe an irreducible representation with definite integer helicity. As a result, we generalize the cubic vertex found in [arXiv:1205.3131 [hep-th]] and calculate the new contributions to the vertex, which contain additional …
