Derivation of gauge invariance from high-energy unitarity bounds on theSmatrix

Abstract

A systematic search is made for all renormalizable theories of heavy vector bosons. It is argued that in any renormalizable Lagrangian theory high-energy unitarity bounds should not be violated in perturbation theory (apart from logarithmic factors in the energy). This leads to the specific requirement of "tree unitarity": the $N$-particle $S$-matrix elements in the tree approximation must grow no more rapidly than ${E}^{4\ensuremath{-}N}$ in the limit of high energy ($E$) at fixed, nonzero angles (i.e., at angles such that all invariants ${p}_{i}\ifmmode\cdot\else\textperiodcentered\fi{}{p}_{j}$, $i\ensuremath{\ne}j$, grow like ${E}^{2}$). We have imposed this tree-unitarity criterion on the most general scalar, spinor, and vector Lagrangian with terms of mass dimension less than or equal to four; a certain class of nonpolynomial Lagrangians is also considered. It is proved that any such theory is tree-unitary if and only if it is equivalent under a point transformation to a spontaneously broken gauge theory, possibly modified by the addition of mass terms for vectors associated with invariant Abelian subgroups. Our result suggests that gauge theories are the only renormalizable theories of massive vector particles and that the existence of Lie groups of internal symmetries in particle physics can be traced to the requirement of renormalizability.