Theory of massive and massless Yang-Mills fields

Abstract

Introducing the Lagrangian multiplier field $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\chi}}(x)$, a canonical formalism for the Yang-Mills fields ${\stackrel{\ensuremath{\rightarrow}}{\mathrm{f}}}_{\ensuremath{\mu}}(x)$ with mass $M\ensuremath{\ge}0$ is proposed within the framework of an indefinite-metric quantum field theory. The formalism for the massive ${\stackrel{\ensuremath{\rightarrow}}{\mathrm{f}}}_{\ensuremath{\mu}}$ has a well-defined zero-mass limit, and the reduction of the physical components of ${\stackrel{\ensuremath{\rightarrow}}{\mathrm{f}}}_{\ensuremath{\mu}}$ as $M\ensuremath{\rightarrow}0$ is embodied in an elegant way. Using the field equation for $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\chi}}(x)$ and the path integral, we find that the "extra" factor in the amplitude due to the interaction of $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\chi}}(x)$ in the intermediate states is ${[\mathrm{det}(1+{(\ensuremath{\square}+{M}^{2})}^{\ensuremath{-}1}g{\stackrel{\ensuremath{\rightarrow}}{\mathrm{f}}}_{\ensuremath{\mu}}\ifmmode\times\else\texttimes\fi{}{\ensuremath{\partial}}^{\ensuremath{\mu}})]}^{\frac{\ensuremath{-}1}{2}}\ensuremath{\equiv}{{D}_{M}}^{\frac{\ensuremath{-}1}{2}}$ for the massive ${\stackrel{\ensuremath{\rightarrow}}{\mathrm{f}}}_{\ensuremath{\mu}}$, and that the extra factor is ${{D}_{M=0}}^{\ensuremath{-}1}$ for the massless ${\stackrel{\ensuremath{\rightarrow}}{\mathrm{f}}}_{\ensuremath{\mu}}$ because of their different degrees of observable freedom. Thus, the resultant rules for the Feynman diagrams for $M>0$ and $M=0$ are not smoothly connected. The theory is covariant, renormalizable, and unitary after the extra parts are removed from the amplitudes. The problems of unitarization and renormalizability are discussed.