Massive Gauge Theories from Consistency Conditions of Amplitudes

TL;DR

This work develops an amplitude-centric framework to derive consistent massive gauge theories by imposing two principles—on-shell gauge symmetry and strong massive-massless continuation—grounded in Lorentz invariance. By formulating a 5-component vector-boson formalism and building 3-point amplitudes from massless limits, then enforcing the two conditions and consistent factorization to assemble 4-point amplitudes, it shows that almost all couplings are fixed by particle masses and Lie-group structure, with scalar self-couplings remaining free. The analysis demonstrates that for $n_V\geq 3$ the only elementary-particle theory compatible with these conditions is Yang–Mills theory with spontaneous symmetry breaking, while for smaller $n_V$ alternative realizations such as massive QED with S.S.B or Stueckelberg mechanisms can arise under specific mass relations. Scalar self-interactions remain the primary ambiguity, reflecting the undetermined Higgs potential shape; the results therefore provide amplitude-level justification for mass generation and gauge structure without relying on a particular Lagrangian.

Abstract

Based on the general principles of Lorentz symmetry and unitary, we introduce two consistency conditions -- on-shell gauge symmetry and strong massive-massless continuation -- in constructing amplitudes of massive gauge theory with elementary particles. In particular we argue on-shell gauge symmetry can be understood as a consequence of Lorentz symmetry, through mixture of a vector boson and a scalar with degenerate mass spectrum. Based on the two conditions, combined with the little group transformation and consistent factorization, we construct 3-point and 4-point vector boson/scalar amplitudes, then analyze the underlying physical models. Given the particle masses, almost all possible vertices, including those involving Goldstone modes, are uniquely fixed. The only exceptions are triple and quartic scalar self-couplings. In addition, all particle masses must have the same physical origin. If the number of vector bosons is smaller than 3, the underlying theories for the amplitudes are either massive gauge theories with spontaneous symmetry breaking (S.S.B) or Stueckelberg theory. The necessary condition for the latter is that the scalars have equal masses. We also discuss different models depending on the number of scalars involved. If the number of vector bosons is larger than 3, the underlying theory must be Yang-Mills theory with S.S.B. In both abelian and non-abelian cases, the specific shape of Higgs potential cannot be determined, which explains the fact that scalar self-couplings are undetermined, and the relations between the masses are generally not linear.