Off-shell cubic hypermultiplet couplings to $\mathcal{N}=2$ higher spin gauge superfields

TL;DR

This paper constructs off-shell cubic couplings between hypermultiplets and higher spin gauge superfields in 4D, N=2 supersymmetry, revealing how these couplings depend on the spin's parity and the hypermultiplet's mass and internal symmetries.

Contribution

It introduces a manifestly supersymmetric and gauge-invariant framework for cubic couplings of hypermultiplets to higher spin gauge superfields, including massive and massless cases, with detailed symmetry considerations.

Findings

Cubic couplings are constructed for all integer spins.

Odd spins require a U(1) internal symmetry for couplings.

Massless limits of odd spins vanish unless hypermultiplet is massless.

Abstract

We construct manifestly $4D, \mathcal{N}=2$ supersymmetric and gauge invariant off-shell cubic couplings of matter hypermultiplets to the higher integer spin gauge $\mathcal{N}=2$ multiplets introduced in arXiv:2109.07639 [hep-th]. The hypermultiplet is described by an analytic harmonic $4D, \mathcal{N}=2$ superfield $q^{+}$ with the physical component spins ${\bf s} = (\frac{1}{2}\,, \;0)$ and an infinite number of auxiliary fields. The cubic coupling constructed has the schematic structure $q^+ \hat{{\cal H}}^{++}_{(s)} q^+$, where $\hat{{\cal H}}^{++}_{(s)}$ is a differential analytic operator of the highest degree $({\bf s} - 1)$ accommodating the massless gauge $\mathcal{N}=2$ multiplet with the highest spin ${\bf s}$. For odd ${\bf s}$ the gauge group generators and couplings are proportional to ${\rm U}(1)_{PG}$ generator of the internal ${\rm SU}(2)_{PG}$ symmetry of the hypermultiplet and so do not exist if ${\rm SU}(2)_{PG}$ is unbroken. If this ${\rm U}(1)_{PG}$ is identified with the central charge of $ 4D, \mathcal{N}=2$ supersymmetry, a mass for the hypermultiplet is generated and the odd ${\bf s}$ couplings vanish in the proper massless limit. For even ${\bf s}$ the higher-spin gauge transformations and cubic superfield couplings can be defined for both massive and massless (central-charge neutral) hypermultiplets without including ${\rm U}(1)_{PG}$ generator. All these features directly extend to the case of $n$ hypermultiplets with the maximal internal symmetry ${\rm USp}(2n) \times {\rm SU}(2)$.