# On-Shell Gauge Invariant Three-Point Amplitudes

## TL;DR

This paper derives gauge-invariant three-point amplitudes for massless bosons, revealing constraints, formalism mismatches, and duality limitations in interactions, advancing understanding of gauge theories in various dimensions.

## Contribution

It provides a comprehensive derivation of all possible 3-point interactions for tensor fields with various symmetries, highlighting gauge invariance constraints and formalism discrepancies.

## Key findings

- Derived all gauge-invariant 3-point amplitudes for tensor fields.
- Identified mismatches between covariant and light-cone formalisms in 4D.
- Showed duality between scalar and antisymmetric tensor fields does not extend to interactions.

## Abstract

Assuming locality, Lorentz invariance and parity conservation we obtain a set of differential equations governing the 3-point interactions of massless bosons, which in turn determines the polynomial ring of these amplitudes. We derive all possible 3-point interactions for tensor fields with polarisations that have total symmetry and mixed symmetry under permutations of Lorentz indices. Constraints on the existence of gauge-invariant cubic vertices for totally symmetric fields are obtained in general spacetime dimensions and are compared with existing results obtained in the covariant and light-cone approaches. Expressing our results in spinor helicity formalism we reproduce the perhaps mysterious mismatch between the covariant approach and the light cone approach in 4 dimensions. Our analysis also shows that there exists a mismatch, in the 3-point gauge invariant amplitudes corresponding to cubic self-interactions, between a scalar field $\phi$ and an antisymmetric rank-2 tensor field $A_{\mu\nu}$. Despite the well-known fact that in 4 dimensions rank-2 anti-symmetric fields are dual to scalar fields in free theories, such duality does not extend to interacting theories.