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

R R Metsaev

doi:10.1007/jhep11(2017)197

Abstract

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 no cubic vertices describing consistent interaction of continuous-spin massless fields with arbitrary spin massless fields.

Highlights

In this paper, we study interacting continuous-spin fields.1 Namely, our major aim in this paper is to study interaction of continuous-spin fields with arbitrary spin fields which propagate in flat space
First, we develop the light-cone gauge so(d − 2) covariant formulation of free continuous-spin fields propagating in the flat space Rd−1,1 with arbitrary d ≥ 4.2 Second, using such formulation and adopting the method for constructing cubic interaction vertices developed in refs. [30,31,32] for arbitrary spin massive and massless fields, we find cubic vertices describing interaction of continuous-spin fields with arbitrary spin massive fields propagating in Rd−1,1, d ≥ 4
From the commutators of the dynamical generators (2.5) with the kinematical generators P i and P +, we find that the dynamical generators Gd[ny]n with n ≥ 3 can be cast into the following form: P[−n] = dΓn Φ[n]||p−[n], (3.2)

Summary

Equations for parity invariant cubic interaction vertices

We recall that we study parity invariant cubic vertices for one continuous-spin massless field and two arbitrary spin massive fields and parity invariant cubic vertices for two continuous-spin massless fields and one arbitrary spin massive field. To consider vertices for one continuous-spin massless field and arbitrary but fixed spin-s1 and spin-s2 massive fields (5.1) we note that the ket-vectors for massive spin-sa fields |φsa are the respective degree-sa homogeneous polynomials in the oscillators αai , ζa, a = 1, 2, (2.16) This implies that the vertices we are interested in must satisfy the equations (5.22). Expressions given in (6.2)–(6.27) provide the complete description of the cubic vertex describing interaction of two continuous-spin massless fields with one infinite chain of massive fields. To consider vertices for two continuousspin massless fields and one arbitrary but fixed spin-s3 massive field (6.1), we note that the ket-vector for massive field |φs is a degree-s3 homogeneous polynomial in the oscillators α3i , ζ3 (2.16) This implies that the vertices we are interested in must satisfy the equation (Nα3 + Nζ3 − s3)|p−[3] = 0 ,. In appendix F, by analysing equations for cubic vertices, we demonstrate explicitly that cubic vertices describing interaction of two continuous-spin massless fields with one arbitrary spin massless field are not consistent

Conclusions

A Notation and conventions

B Continuous-spin field in helicity basis

M p RL R β