Higher-spin fermionic gauge fields and their electromagnetic coupling
Marc Henneaux
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Gustavo Lucena Gomez
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Rakibur Rahman
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Physique Theorique et Mathematique & International Solvay Institutes, Universite Libre de Bruxelles
, Campus Plaine C.P. 231, B-1050 Bruxelles,
Belgium
We study the electromagnetic coupling of massless higher-spin fermions in flat space. Under the assumptions of locality and Poincare invariance, we employ the BRST-BV cohomological methods to construct consistent parity-preserving off-shell cubic 1 s s vertices. Consistency and non-triviality of the deformations not only rule out minimal coupling, but also restrict the possible number of derivatives. Our findings are in complete agreement with, but derived in a manner independent from, the light-cone-formulation results of Metsaev and the string-theory-inspired results of Sagnotti-Taronna. We prove that any gauge-algebra-preserving vertex cannot deform the gauge transformations. We also show that in a local theory, without additional dynamical higher-spin gauge fields, the non-abelian vertices are eliminated by the lack of consistent second-order deformations.
1 Introduction
1.1 Conventions & notations
1.2 Results
2 The BRST deformation scheme
2.1 The cohomology of
3 EM coupling of massless spin 3/2
3.1 Gauge-algebra deformation
3.2 Deformation of gauge transformations
3.3 Lagrangian deformation
3.4 Abelian vertices
4.1 Non-Abelian vertices
4.2 Abelian vertices
5 Arbitrary spin: s = n + 12
6 Abelian vertices preserve gauge symmetries
7 Comparative study of vertices 7.1 1-3/2-3/2 vertices
7.2 1-s-s vertices: s 5/2
8 Second-order consistency
9 Remarks & future perspectives A Curvatures & equations of motion A.1 The photon A.2 Spin 3/2
B The cohomology of
B.1 The curvatures
B.2 The antifields
B.3 The ghosts & curls thereof
B.4 The Fronsdal tensor
Consistent interacting theories of massless higher-spin fields in flat space are difficult to
construct. Severe restrictions arise from powerful no-go theorems [15], which prohibit,
in Minkowski space, minimal coupling to gravity, when the particles spin s 25 , as well
as to electromagnetism (EM), when s 32 . However, these particles may still interact
through gravitational and EM multipoles. Indeed, N = 2 SUGRA [6, 7] allows massless
gravitini to have dipole and higher-multipole couplings, but forbids a non-zero U(1) charge
in flat space. Gravitational and EM multipole interactions also show up, for example, as
the 2 s s and 1 s s trilinear vertices constructed in [8, 9] for bosonic fields.1
These cubic vertices are but special cases of the general form s s s, that involves
massless fields of arbitrary spins. Metsaevs light-cone formulation [10, 11] puts restrictions
on the number of derivatives in these vertices, and thereby provides a way of classifying
them. For bosonic fields, while the complete list of such vertices was given in [12], Noether
procedure has been employed in [1315] to explicitly construct off-shell vertices, which
do obey the number-of-derivative restrictions. Also, the tensionless limit of string theory
gives rise to a set of cubic vertices, which are in one-to-one correspondence with the ones of
Metsaev, as has been noticed by Sagnotti-Taronna in [16, 17], where generating functions
for off-shell trilinear vertices for both bosonic and fermionic fields were presented.
In this paper, we consider the coupling of a massless fermion of arbitrary spin to a
U(1) gauge field, in flat spacetime of dimension D 4. Such a study is important in that
fermionic fields are required by supersymmetry, which plays a crucial role in string theory,
which in turn involves higher-spin fields. This fills a gap in the higher-spin literature, most
of which is about bosons only (with [11, 1619] among the exceptions). We do not
consider mixed-symmetry fields, and restrict our attention to totally symmetric Dirac fermions
1...n , of spin s = n + 21 . For these fields, we employ the powerful machinery of BRST-BV
cohomological methods [20, 21] to construct systematically consistent interaction vertices,2
with the underlying assumptions of locality, Poincare invariance and conservation of parity,
and without relying on other methods. The would-be off-shell 1 s s cubic vertices will
complement their bosonic counterparts constructed in [9].
The organization of the paper is as follows. We clarify our conventions and notations,
and present our main results in the next two subsections. In section 2, we briefly recall the
BRST deformation scheme [20, 21] for irreducible gauge theories. With this knowledge, we
then move on to constructing consistent off-shell 1 s s vertices in the following three
sections. In particular, section 3 considers the massless Rarita-Schwinger field, while section 4
pertains to s = 25 , and section 5 generalizes the results, rather straightforwardly, to
arbitrary spin, s = n + 21 . In section 6, we prove an interesting property of the vertices under
study: an abelian 1 s s vertex, i.e., a 1 s s vertex tha (...truncated)