On higher spin interactions with matter

TL;DR

This work investigates cubic couplings between a complex scalar and an infinite tower of symmetric tensor gauge fields using generating functions and Weyl calculus. The key result is that the Noether interaction can be written succinctly as $S_1[\phi,h] = -\ll J \| h \gg = -\langle \phi| \hat{H} | \phi \rangle$, with $\hat{H} = \mathcal{W}[h]$, revealing a non-Abelian gauge structure at lowest order and enabling explicit computation of tree-level scalar amplitudes from the exchange of all higher-spin fields. The paper derives Feynman rules, obtains the closed-form sum over spins for the four-scalar scattering amplitude, and demonstrates that with suitable coupling generating functions $a(z)$ the high-energy behavior can be exponentially softened, while the long-range potential remains dominated by low-spin exchanges at low energies. The analysis connects Weyl quantization to higher-spin gauge symmetries and suggests that an infinite tower of higher-spin interactions could underpin ultraviolet softness in a manner reminiscent of string theory, though it also discusses fundamental caveats from S-matrix no-go theorems and higher-order consistency checks. Overall, the work provides a concrete framework for probing high-spin interactions with matter, their non-Abelian structure, and their potential implications for ultraviolet behavior and effective macroscopic forces.

Abstract

Cubic couplings between a complex scalar field and a tower of symmetric tensor gauge fields of all ranks are investigated. A symmetric conserved current, bilinear in the scalar field and containing r derivatives, is provided for any rank r>0 and is related to the corresponding rigid symmetry of Klein-Gordon's Lagrangian. Following Noether's method, the tensor gauge fields interact with the scalar field via minimal coupling to the conserved currents. The corresponding cubic vertex is written in a compact form by making use of Weyl's symbols. This enables the explicit computation of the non-Abelian gauge symmetry group, the current-current interaction between scalar particles mediated by any gauge field and the corresponding four-scalar elastic scattering tree amplitude. The exact summation of these amplitudes for an infinite tower of gauge fields is possible and several examples for a definite choice of the coupling constants are provided where the total amplitude exhibits fast (e.g. exponential) fall-off in the high-energy limit. Nevertheless, the long range interaction potential is dominated by the exchange of low-spin particles in the low-energy limit.