Light-Front Higher-Spin Theories in Flat Space
Dmitry Ponomarev, Evgeny Skvortsov,
Theoretical physics group, Blackett Laboratory,
Imperial College London, SW7 2AZ, U.K.
Arnold Sommerfeld Center for Theoretical Physics
Ludwig-Maximilians University Munich
Theresienstr. 37, D-80333 Munich, Germany
Lebedev Institute of Physics,
Leninsky ave. 53, 119991 Moscow, Russia
We revisit the problem of interactions of higher-spin fields in flat space. We argue that all no-go theorems can be avoided by the light-cone approach, which results in more interaction vertices as compared to the usual covariant approaches. It is stressed that there exist two-derivative gravitational couplings of higher-spin fields. We show that some reincarnation of the equivalence principle still holds for higher-spin fields — the strength of gravitational interaction does not depend on spin. Moreover, it follows from the results by Metsaev that there exists a complete chiral higher-spin theory in four dimensions. We give a simple derivation of this theory and show that the four-point scattering amplitude vanishes. Also, we reconstruct the quartic vertex of the scalar field in the unitary higher-spin theory, which turns out to be perturbatively local.
- 1 Introduction
- 2 Avoiding No-Go Theorems
- 3 Living on Light-Front
- 4 Complete Chiral Higher-Spin Theory
- 5 Quartic Hamiltonian of Unitary Higher-spin Theory
- 6 Higher-Spin Equivalence Principle
- 7 Conclusions and Discussion
- A Most General Form of Cubic Vertices
- B Triviality of
- C Metsaev Solution
Since the early days of quantum field theory there have been many no-go results that prevent non-trivial interacting theories with massless higher-spin fields to exist. Notable examples are the Weinberg low energy theorem  and the Coleman-Mandula theorem . One possible way out is to switch on the cosmological constant [3, 4, 5], which simultaneously avoids the no-go theorems that are formulated for QFT in flat space. Higher-spin theories in anti-de Sitter space later received a solid ground on the base of AdS/CFT correspondence [6, 7, 8] where higher-spin theories are supposed to be generic duals of free CFT’s [9, 10, 11, 12] with certain interacting ones accessible via an alternate choice  of boundary conditions [9, 11, 14, 12, 15].
The fate of higher-spin theories in flat space is still unclear and is a source of controversy. The no-go theorems are still true. Also, within the local field theory approach one immediately faces certain obstructions: Aragone-Deser argument forbids minimal gravitational interactions of massless higher-spin fields [16, 17] and, even if relaxing this assumption, it is still impossible to deform the gauge algebra [18, 19]. These results are based on the gauge invariant and manifestly Lorentz covariant field description in terms of Fronsdal fields , which suggests another possible way out.
Indeed, gauge symmetry can be thought of as just a redundancy of description, though it turns out to be exceptionally useful in many cases. Therefore, in order to look for higher-spin theories in flat space it can be useful to turn to methods that deal with physical degrees of freedom only and thereby avoid any problems that originate from specific field descriptions. One such method is the light-cone approach, which still allows one to have a local field theory.
It is in the light-cone approach that the first examples of non-trivial cubic interactions between higher-spin fields were found in [21, 22, 23]. The covariant results followed soon after [24, 25]. A detailed classification of cubic vertices within the light cone approach is now available in all dimensions for massive and massless fields of arbitrary spin and symmetry type [26, 27, 28, 29].
In this paper we revisit the problem of constructing higher-spin theories in flat space, specifically in four-dimensions. First of all, we argue that at least formally the most powerful no-go theorems are avoided by the light-cone approach. Also, we recall that there is a mismatch between the covariant cubic vertices and those found in [21, 22, 23] by the light-cone methods: there exist exceptional vertices not seen by some of the covariant methods. In particular, there does exist a two-derivative gravitational vertex for a field of any spin [30, 31, 32], which is also evident in the language of amplitudes [33, 34].
Having the gravitational higher-spin vertex at our disposal we prove that fields of any spin couple to gravity universally, i.e. some form of the equivalence principle is still true for higher-spin fields. In fact, the strength of the gravitational coupling does not depend on spin at all.
A remarkable result obtained by Metsaev in [35, 36] is that one can fix the cubic vertex without having to perform the full quartic analysis. We present a simple derivation of this result, which clarifies the assumptions. Based on this solution, we note that there exists a consistent non-trivial higher-spin theory in flat space. This theory contains graviton, massless higher-spin fields, the two-derivative gravitational vertices as well as other vertices. The action terminates at cubic vertices. Like in the self-dual Yang-Mills theory the four-point scattering amplitude vanishes. The only feature is that it breaks parity and is non-unitary. Nevertheless, it provides a counterexample to a widespread belief that higher-spin theories in flat space do not exist at all.
Aiming at the unitary and parity preserving higher-spin theory in flat space we reconstruct the part of the quartic Hamiltonian that contains self-interactions of the scalar field, which can be regarded as the flat space counterpart of the result [37, 38].
The outline is as follows. In Section 2 we discuss how to avoid the famous no-go results. In Section 3 we review the basics of the light-cone approach with the main result being the classification of all possible couplings that was obtained in [21, 22, 23, 35, 36]. The relation to the Lorentz covariant classification is spelled out in Section 3.2. In Section 4 we present a complete chiral higher-spin theory with the details of the derivation of the Metsaev solution [35, 36] devoted to Appendix C. The scalar part of the quartic Hamiltonian of the unitary higher-spin theory is reconstructed in Section 5. The higher-spin equivalence principle is derived in Section 6. We conclude with some discussion of possible extensions of these results in Section 7.
2 Avoiding No-Go Theorems
In the distant past it was a common belief that higher-spin theories, i.e. the theories with massless fields with spin greater than two, are not consistent. The most notable examples of such no-go theorems are Weinberg low energy theorem , Coleman-Mandula theorem  and the Aragone-Deser argument . We briefly discuss them below, see also a very nice review , as to point out how all of them can be avoided.
Our conclusion is that there are still good chances to have nontrivial higher-spin theories in flat space. Moreover, we will present an example of consistent chiral theory in Section 4. However, it should be stressed that while higher-spin theories may avoid the assumptions of the no-go theorems they may not defy the spirit of these theorems: there are strong indications that -matrix should be trivial in some sense. For example, for the case of conformal higher-spin theories the -matrix is a combination of [40, 41, 32] and the AdS/CFT duals of unbroken higher-spin theories must be free CFT’s [42, 43, 44, 45, 46], which should be thought of as examples of trivial holographic -matrices.
Weinberg low energy theorem.
A serious restriction comes from the Weinberg low energy theorem  that eventually leads to too many conservation laws, when massless higher-spin fields are present. As a result of checking linearized gauge invariance or Lorentz invariance of the -particle amplitude with one soft spin- particle attached one finds
where is the coupling constant of the -th species to a spin- field. For one discovers that the total (electric) charge is conserved. For one finds a linear combination of momenta weighted by whose clash with the momentum conservation law can only be resolved by the equivalence principle, i.e. all fields must couple to gravity universally, .
For the higher-spin case one finds too many conservations laws, which is a rank tensorial expression, with the only solution given by permutations of momenta at the condition that all coupling constants are the same.
In the course of the proof of the theorem one makes an explicit use of Lorentz covariant vertices. In particular, the expressions are manifestly Lorentz covariant. This is not the case in the light-cone approach where the vertices do not have a manifestly Lorentz covariant form. It would be interesting to reconsider the Weinberg theorem as to see whether these assumptions can be weakened.111We are grateful to Sasha Zhiboedov for the useful discussion of this problem.
The famous Coleman-Mandula theorem  prevents -matrix from having symmetry generators, beyond those of the Poincare group, that transform under the Lorentz group. Under assumptions of non-triviality of the symmetry action, discrete mass spectrum and the analyticity of the -matrix in Mandelstam invariants, it can be shown that the symmetry algebra can only be a product of the Poincare group and a group of internal symmetries whose generators are Lorentz scalars. It does not apply to the case of QFT, where only forward/backward scattering is possible, so -matrix must have scattering angles and thereby it is not analytic. The essence of the proof is that the scattering process is a map from one set of momenta to another one and the momenta are restricted by energy-momentum conservation, which is a Lorentz vector equation. Existence of some other charges that transform non-trivially under the spacetime symmetry would impose tensorial equations on momenta, e.g. like in Weinberg theorem, which would restrict possible processes to exchanges of momenta like in or trivialize the scattering completely. One way the original Coleman-Mandula theorem can be avoided is by assuming that symmetry generators transform as spinors, which leads to supersymmetry.
One of the assumptions of the theorem is to have a finite number of particles below any mass-shell. This is certainly not true in higher-spin theories where the spectrum should contain infinitely many massless particles [47, 25, 48, 44]. It would be interesting to weaken the assumptions of the theorem .
Aragone-Deser argument/No canonical gravity coupling.
Contrary to the Weinberg and Coleman-Mandula theorems, this argument is local and is attached to specific field variables [16, 17]. It says that the canonical way of putting fields on a curved background by replacing partial derivatives with covariant ones does not work for massless higher-spin fields. Indeed, in checking the gauge invariance of the action we have to commute derivatives, which brings the Riemann tensor:
Unlike low-spin examples, we find the full four-index Riemann tensor — the structure that cannot be compensated by any modifications of the action/gauge transformations. For the action is manifestly gauge invariant, while for we find not the full Riemann tensor but its trace, the Ricci tensor, which allows to overcome the problem by going to supergravities.
The argument above makes use of the specific field variables and of the manifestly Lorentz covariant methods. Obviously, this is avoided by the light-cone approach. We will emphasize in Section 3.2 that there exists in fact a two-derivative gravitational coupling of massless higher-spin fields to gravity [21, 22, 23], which is not captured by covariant studies [50, 51, 52].
A relatively new no-go type result came from the BCFW approach [53, 34, 33, 54, 55, 56]. However, higher-spin theories are clearly different from Yang-Mills theory and even gravity and are not expected to have an -matrix that is analytic. Moreover, BCFW approach is essentially based on the assumption of certain behavior of amplitudes for infinite BCFW shifts. It is not a priori clear whether these assumptions can be justified in the higher-spin case. Some works towards weakening these assumptions include [53, 34, 33, 54, 55].
3 Living on Light-Front
In this Section we review the light-cone approach to relativistic dynamics. Next, we discuss the classification of cubic vertices that results from the light-cone dynamics and confront it with the covariant methods. The main lesson is that there are more vertices in the light-cone approach. In particular there are two-derivative interaction vertices of a spin- field and a graviton, which can be called gravitational. The reader not interested in the somewhat boring details222Nice, pedagogical exposition of the light-cone approach can also be found in [28, 63]. can jump directly to Section 3.2. It is worth stressing that the Yang-Mills theory, when rewritten in the light-cone approach, is a theory of scalar fields in the adjoint of the global symmetry group. Similarly, gravity is a theory of two scalar fields with no symmetries like diffeomorphisms whatsoever.
Quantum field theory in flat space in its most rigorous definition requires a Hilbert space endowed with the unitary action of the Poincare algebra, i.e. the generators of Lorentz transformations and translations should be realized as to obey:333It is convenient to choose the mostly plus convention for and .
In free theory the generators are quadratic in the quantum fields and have to receive certain corrections when interactions are switched on.
Canonical quantization begins with postulating the canonical commutation relations of fields and momenta at some fixed time, which encodes the choice of the Cauchy surface for evolution. As was pointed out by Dirac  there are different quantization schemes depending on the choice of the quantization surface.444Let us note that there are several different things that bear almost the same name: light-cone gauge, light-front (or light-cone) quantization and one can also combine the two by quantizing a theory on a light-front with the light-cone gauge imposed. The difference is in the stability group that preserves the surface. The generators associated with the stability group, called kinematical, do not receive any quantum corrections and stay quadratic in the fields on the Cauchy surface. The left-over generators, called dynamical, do deform.
For the canonical equal time choice the stability subgroup of the Poincare group ( in dimensions) consists of spacial rotations and translations, while boosts and time translations do not preserve the surface. Therefore, there are four generators ( in the case of dimensions) that receive corrections due to interactions.
The light-front is the light-like quantization surface. The canonical choice is , so that is treated as the time direction and is the Hamiltonian.555In the light-front coordinates , etc., and . Also, in one can replace with two complex conjugate variables , , so that the metric is . As a result only generators need to be deformed, which is the least number possible. A somewhat unfortunate feature of any non-covariant quantization, including the light-cone one, is that due to the manifest Lorentz symmetry breaking we have to deal with many more generators whose total number is the same. The ten generators of can be split into kinematical (K) and dynamical (D) as follows:
The time evolution of any operator is determined by the Hamiltonian . Therefore, if the Poincare algebra relations are satisfied at the initial light-cone time , then they will be satisfied at all times. This has a useful consequence that some of the generators having explicit dependence
should be declared to be kinematical, as we did above, since the dynamical part vanishes at . The -dependence can then be reconstructed by virtue of the equations of motion.
Let us now list the commutation relations and the consequences thereof. A generator can be split into its free part and an interacting part , the latter being absent for the kinematical generators. The kinematical generators are fixed once and for all times. As for dynamical generators the procedure is that there are commutators that simply constrain the dynamical generators to have certain dependence on the kinematical variables. Also, there are few other relations that represent nontrivial equations to be solved for the dynamical generators.
The kinematical generators do not receive any corrections, so this part stays unchanged and is of no use, which is why we list them here-below for completeness:
This set of relations splits into two parts. First one is -type relations that immediately restrict the dynamical generators. The second one are -type commutators, which imply that the interacting part of commutes to the given , i.e. , which is due to and the right-hand side being taken into account by free fields, .
These relations are similar to the previous ones and constrain the dynamical generators to behave nicely under the light-front symmetries:
This class consists of the actual equations to be solved and constitutes the main problem of the light-cone approach:
There are three dynamical generators: two boosts and the Hamiltonian :
where we split them into the free and interacting parts and moreover symbolically extract the dependence of on . The commutation relations imply that is a centralizer of several kinematical generators:
Initially, commutes only to , . The shift by cancels in , which becomes the -type relation for . Therefore, we find
The -type relations, when written for the deformations, give
All the constraints above can be explicitly solved and one is left with (3.10), of which only needs to be solved, as we explain below.
3.1.1 Free Field Realization
We have just discussed which commutation relations need to be solved. Further progress can only be made for specific theories. The general comment is that the quantization on the light-front leads to second-class constraints.666See very nice books [65, 66] for quantization of field theories with constraints. Indeed, the kinetic term , when written in the light-cone coordinates, , is linear in the velocity and hence the momenta, i.e. the primary constraints, cannot be solved for . Therefore, the bracket is the Dirac bracket.
From now on we confine ourselves to live in the four-dimensional world. The nice feature of the world is that all massless spinning particles have two degrees of freedom, i.e. made of two scalar fields except for the spin-zero particle, which equals one scalar field. A spin- particle has two states with helicities and can be described as two fields that are complex conjugate. It is convenient to work with the fields that are Fourier transformed with respect to and transverse coordinates :
In the world the equal time commutation relations that follow from the Dirac bracket are:
From now on we set and will omit the arguments in most of the cases. It is very easy to find the kinematical generators of the Poincare algebra in the Fourier space:777Following the light-cone commandments we rename otherwise the paper will not be understandable at all.
where is the Euler operator, idem. for , and we sometimes use , etc. The generators are supposed to act on . The dynamical generators at the free level are:
The Poincare charges can be built in a standard way:
where is the generator of the Poincare algebra associated with a Killing vector . We draw reader’s attention to the fact that the integration measure is . The Poincare algebra is then realized via commutators
Due to the nontrivial integration measure the conjugate operators are defined as
where the transposed operator is defined via integration by parts as usual, e.g. , . The generators of the Poincare algebra given above are Hermitian, . In particular, we find . With the help of (3.18) and
one can verify all the commutation relations:
(3.24) follows from a more general formula for the action of on an arbitrary functional :
which we will immediately apply to read off the constraints imposed by kinematical generators on the dynamical ones.
3.1.2 Kinematical Constraints
An appropriate ansatz for the Hamiltonian and dynamical boosts reads:888The derivatives can also act both on and wave functions, which is equivalent to redefining .
where the delta function imposes the conservation of the total and transverse momenta , which is a consequence of the translation invariance imposed by and , (3.12), (3.13). The rest of the kinematical generators imposes the following constraints:
where means an equality up to an overall delta-function .
In practice it is tedious to keep all delta-functions unresolved and it is more convenient to choose some independent momenta as basic variables. Moreover, (3.28a)-(3.28b) imply that everything depends on specific combinations of momenta :
There are such independent variables for -point function. In the case we have
Therefore, we assume that some variables out of all ’s have been chosen and
The rest of the system of kinematical constraints can be rewritten as
The above conditions are very simple homogeneity constraints and need no further comments.
3.1.3 Cubic Vertices
The first nontrivial dynamical constraints arise at the cubic order. First of all, the kinematics of three -dimensional momenta restricted by the conservation delta-function is very simple. There is one independent variable since . Therefore, in we have just and . It is advantageous to represent it in a manifestly cyclic-invariant way:
Therefore, belongs to the totally anti-symmetric representation of . There is an identity that is of utter importance for the cubic approximation:
Also, at the three-point level we find
Now we proceed to the dynamical constraints. The first one is restricted to the cubic order in fields :
which, after using the magic identity (3.36), can be shown to lead to
where the transposed generators are
Now one can make an appropriate ansatz for that solves the kinematical constraints (3.33), act with and read off and up to possible redefinitions. The most general case is studied in Appendix A, while below we simply quote the representation given by Metsaev in [35, 36]. The first results on cubic interactions of HS fields were obtained in [21, 22, 23] in a slightly different base.
At the interaction level there is always a problem of fixing the field redefinitions. The light-cone approach is not free of this ambiguity too. At the cubic order redefinitions allow one to eliminate powers of , but not each of the two separately. Therefore, the most natural choice of the redefinition frame is to have purely holomorphic vertices. It is worth stressing that this is not the most natural choice in the covariant approaches. The vertices are [35, 36]:
Here and are two sets of coupling constants which are a priori independent. For dimensional reasons we have to introduce a parameter with the dimension of length as to compensate for the higher powers of momenta, as was noted as early as [21, 22, 23]:
In higher-spin theories the parameter will be naturally associated with the Planck length as the Einstein-Hilbert vertex is a part of the set above and corresponds to .
The light-cone locality implies that the powers of , must be non-negative or whenever we should have . The latter is due to the fact that has one power of or less. The exception is when all , which is the scalar self-interaction vertex, since it leads to , which is implied in (3.41).
Let us stress that the light-cone approach deals only with physical degrees of freedom, so the light-cone gauge is a unitary gauge, but it is not an on-shell method. Nevertheless, there is a striking relation between the on-shell amplitude methods and the light-cone approach [67, 68, 69]. One can introduce
so that the basic building blocks of cubic vertices can be found in
and analogously one can define . As a result, the cubic vertices, i.e. Hamiltonian density , can be rewritten in a more suggestive form:
3.2 Light-Cone vs. Covariant Vertices
On one hand, the general formula for cubic vertices (3.41) is given above. On another hand, the classification of cubic vertices in covariant approaches is also available.999There is a vast literature on cubic vertices in covariant approaches. We give a minimalistic list of references [3, 70, 51, 52, 71, 18, 72, 53, 73, 19] that allows one to trace all the initial results and further developments by following references therein/thereon, the accent being put on the diversity of approaches. For our purposes it is sufficient to confront the classification of the light-cone vertices [27, 28, 29] with some of the covariant results [52, 18]. Remarkably, by confronting the light-cone vertices and covariant ones we observe a mismatch in the number of local interactions, see also [33, 34, 31, 32], which is due to the difference between locality in light-cone and covariant approaches.
In the light-cone approach, the vertices can be arranged by the number of derivatives for a given triplet of spins . The power counting is easy in the light-cone approach too: one counts the total power of the transverse momenta, or of , which is the same. Therefore, vertices (3.41) have derivatives, where we note that the helicities can be negative. In covariant approaches one can distinguish between the following classes of vertices, though this classification is incomplete:
- Current Interaction.
The vertex has derivatives and corresponds to the usual current interaction where a spin- current built of two scalar fields is contracted with the Fronsdal field . This is the simplest vertex that involves one higher-spin field and for corresponds to the current interaction while for to the coupling of the stress-tensor to gravity.
- Non-abelian Vertices.
For every spin the vertex has derivatives and drives the non-abelian deformation of the gauge algebra in the covariant approach . In there can be more than one non-abelian self-interaction, but in this seems to be the only one. In particular, is the Yang-Mills vertex and is the Einstein-Hilbert vertex. Having such vertices activated is important for non-triviality of the theory. There is a covariant vertex with derivatives that can be called gravitational, but it certainly cannot result from replacement in the action due to its higher derivative nature for .
- Abelian Vertices.
It is also possible to construct the -derivative vertex . It does not induce any deformation of the gauge algebra and therefore cannot be used as a seed of any interesting theory, while such vertices can be required for consistency at the quartic order. This indeed happens for higher-spin theory, but does not happen for Yang-Mills and Einstein theory, where and vertices can be dropped (or have an independent coupling constant in front of them).
As a result, there is a mismatch between the covariant and the light-cone dictionaries. Indeed, vertex in can have or derivatives , i.e. one can have two vertices at most. On contrary, the light-cone vertices exist for any triplet of helicities, i.e. there can be up to four independent complex vertices (3.41). When the reality condition is imposed, , this number still reduces to three vertices at most. For example, there exists an exceptional series of vertices , , that have less derivatives (transverse momenta) and is absent in covariant approaches. In particular, this exceptional series contains a two-derivative gravitational vertex!101010The existence of such an vertex was stressed in , though it is certainly present in [23, 35, 36, 27]. The case corresponds to the usual Einstein-Hilbert vertex and does not look strange anymore.
The existence of such vertex seems paradoxical in view of the simplest no-go result, known as the Aragone-Deser argument . As we discussed in Section 2 the argument is explicitly Lorentz covariant and is formulated in terms of specific field content, Fronsdal fields, rather than in terms of physical degrees of freedom and therefore is avoided by the light-cone approach.
More generally, within the covariant approaches the statement that some interaction does not exist depends heavily on the field content. Few examples include: local electromagnetic interactions mediated by will look non-locally in terms of ; the formulation of self-dual fields may require an infinite number of auxiliary fields ; there may be the need for some compensator or other auxiliary fields, see e.g. ; a seeming breaking of Lorentz symmetries might be needed, e.g. .
Another result that seems to be in tension with the existence of the two-derivative vertex is the Weinberg low energy theorem. As we discussed in Section 2, the light-cone approach seems to avoid the assumptions of the theorem.
3.3 Quartic Analysis and Beyond
The main result of the cubic approximation is the list of all possible cubic vertices that can be used for constructing any theory. As is usual for the cubic approximation, the coefficients and in front of the cubic vertices are completely free and will be fixed by the quartic analysis, which we will now proceed to.
After an appropriate ansatz for and that solves the kinematical constraints (3.33) is chosen, one has to solve