Higher-Spin Fermionic Gauge Fields and Their Electromagnetic Coupling

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.


Outline
• EM coupling of massless spin 3/2: simple but nontrivial. We start with free theory and perform cohomological reformulation of the gauge system. We employ the BRST deformation scheme to construct consistent parity-preserving off-shell cubic vertices.
• We generalize to arbitrary spin, s = n+1/2, coupled to EM. There appear restrictions on the gauge parameter and the field. These actually make easy the search for consistent interactions! • Comparative study of the vertices with known results.

Massless Rarita-Schwinger Field
Coupled to Electromagnetism Step 0: Free Gauge Theory • The free theory contains a photon A µ and a massless spin-3/2 Rarita-Schwinger field ψ µ , described by the action: • It enjoys two abelian the gauge invariances: • Bosonic gauge parameter: λ, fermionic gauge parameter: ε • Curvature for the fermionic field: Step 1: Introduce Ghosts • For each gauge parameter, we introduce a ghost field, with the same algebraic symmetries but opposite Grassmann parity: • Grassmann-odd bosonic ghost: C • Grassmann-even fermionic ghost: ξ • The original fields and ghosts are collectively called fields: • Introduce the grading: pure ghost number, pgh, which is • 1 for the ghost fields • 0 for the original fields Step 2: Introduce Antifields • One introduces, for each field and ghost, an antifield Φ A * , with the same algebraic symmetries but opposite Grassmann parity. Each antifield has 0 pure ghost number: pgh(Φ A * )=0.
• Introduce the grading: antighost number, agh, which is 0 for the fields and non-zero for the antifields: Step 3: Define Antibracket • On the space of fields and antifields, one defines an odd symplectic structure, called the antibracket: • Here R and L respectively mean right and left derivatives.
Step 4: Construct Master Action • The master action S 0 is an extension of the original action; it includes terms involving ghosts and antifields.
• Because of Noether identities, it solves the master equation: • The antifields appear as sources for the "gauge" variations, with gauge parameters replaced by corresponding ghosts.
Step 5: BRST Differential • S 0 is the generator of the BRST differential s of the free theory • Then the free master equation means: S 0 is BRST-closed.
• Graded Jacobi identity of the antibracket gives: • The free master action S 0 is in the cohomology of s, in the local functionals of the fields, antifields and their derivatives. Locality calls for a finite number of derivatives.
• The BRST differential decomposes into two differentials:

s = Γ + Δ
• Δ is the Koszul-Tate differential. It implements the equations of motion by acting only on the antifields. It decreases the agh by one unit while keeping unchanged the pgh.
• Γ is the longitudinal derivative along the gauge orbits. It acts only on the original fields to produce the gauge transformations. It increases the pgh by one unit without modifying the agh.
• All Γ, Δ, s increase the ghost number, gh, by one unit, where

gh = pgh -agh
Step 6 • Any consistent deformation of the theory corresponds to:
• Coupling constant expansion gives, up to O(g 2 ): • The first equation is fulfilled by assumption.
• The second equation says S 1 is BRST-closed: • First order non-trivial consistent local deformations: S 1 = ∫ a are in one-to-one correspondence with elements of H 0 ( s|d )the cohomology of the free BRST differential s, modulo total derivative d, at ghost number 0. One has the cocycle condition: • A cubic deformation with 0 ghost number cannot have agh>2.
Thus one can expand a in antighost number:

a = a 0 + a 1 + a 2 , agh( a i ) = i
• a 0 is the deformation of the Lagrangian. a 1 and a 2 encode information about the deformations of the gauge transformations and the gauge algebra respectively.
• Then the cocycle condition reduces, by s = Γ + Δ, to a cascade • The cubic vertex will deform the gauge algebra if and only if a 2 is in the cohomology of Γ.
In this case, if a 1 is in the cohomology of Γ, the vertex deforms the gauge transformations.
• If this is also not the case, we can take a 1 = 0, so that the vertex is abelian, i.e. a 0 is in the cohomology of Γ modulo d.
• The cohomology of Δ is also relevant in that the Lagrangian deformation a 0 is Δ-closed, whereas trivial interactions are given by Δ-exact terms.
Step 7: Cohomology of Γ Cohomology of Γ isomorphic to the space of functions of: • These are nothing but "gauge-invariant" objects, that themselves are not "gauge variation" of something else.
• Note: Fronsdal tensor is already included in this list.
Step 8: Non-Abelian Vertices • Recall that a 2 must be Grassmann even, satisfying: • The most general parity-even Lorentz scalar solution is: • It is a linear combination of two independent terms: one that contains C, another that contains C * . The former one potentially gives rise to minimal coupling, while the latter could produce dipole interactions (look at the cascade and count derivatives).
• Each of the terms can be lifted to an a 1 : • The ambiguity is in the cohomology of Γ : • The Δ variation of none of the unambiguous pieces is Γ-exact modulo d. The Δ variation of the ambiguity must kill, modulo d, the non-trivial part, so that Δa 1 could be Γ-exact modulo d: • Any element of the cohomology of Γ at antighost number 1 contains at least 1 derivative, so that such a cancellation is not possible for the would-be minimal coupling, simply because the ambiguity contains too many derivatives.
• Thus minimal coupling is ruled out, and we must set g 0 = 0.