Soldering spin-3 opposite helicities in D=2+1

Abstract

Here we present the "soldering" of opposite helicity states of a spin-3 particle, in D = 2 + 1, into one parity doublet. The starting points may be either the sixth-or the fifth-order (in derivatives) spin-3 self-dual models of opposite helicities. The high number of derivatives avoids the use of auxiliary fields which has been so far an obstacle for a successful soldering procedure. The resulting doublet model is a new Lagrangian with six orders in derivatives and no auxiliary field. It may be regarded as a spin-3 analogue of the linearized "New Massive Gravity". We check its particle content via a gauge invariant and Lorentz covariant analysis of the analytic structure of the twopoint amplitude with the help of spin-3 analogues of the Barnes and Rivers projection operators. The particle content is alternatively confirmed in a specific non-covariant gauge by a decomposition in helicity variables. The soldered model is ghost free and contains two physical states as expected for a parity doublet. *

Unitarity of higher derivative spin-3 models

“…Such result is quite similar to the one we have found in [11] for a sixth order doublet model which do not depend on auxiliary fields. In fact, we can observe that, the effect of the auxiliary fields on the present model is codified in the last term of (59), which do not affect the particle content of the model.…”

“…Then we can state that the transition amplitude ( 59) is also free of ghosts propagating a unique massive spin-3 particle. In addition notice that in [11] the set of symmetries which keeps the sixth order term invariant are larger than here, once in the present case the gauge parameters have restrictions such as the tracelessness and the traversity, while there they are full. In a certain way, we see that the role of the auxiliary fields has been played by the increase of symmetries in the sixth order term [11].…”

“…where λ 1 is a gauge fixing parameter and ∂ • h = ∂ µ h µ . This is exactly the same gauge fixing term we have used in [11] in D = 2 + 1. By its turn it can be rewritten in terms of the spin-projection operators as:…”

“…It is then natural to search for higher-spin extensions of general relativity and NMG, taking advantage of the geometrical formulation provided by de Wit and Freedman [5] and by Deser and Damour [6]. In this sense some progress has been reached in the spin-3 and spin-4 cases [7,8,9,10,11]. Specially in [9] we have obtained fourth and sixth order in derivatives parity preserving models describing massive spin ±3 particles, which we have argued in terms of symmetries and by counting degrees of freedom to be free of ghosts.…”

“…A consensus on what would be the spin-3 analog of NMG has been discussed in [11], there in the sixth order in derivatives action, the auxiliary fields are not necessary, and the sixth order term is invariant under reparametrization and Weyl symmetries much like the fourth order spin-2 counterpart, named the K term in [2]. However, differently of the spin-2 case, such model can not be obtained from the second order fundamental SH model as the linearized NMG can be obtained from the second order fundamental FP model, as we have demonstrated in [12].…”