Local Gauge Symmetry Restoration in Non-Linear -Models

Abstract

As is well known the massless photon in QED with covariant gauge fixing can be interpreted as a N ambu-Goldstone boson corresponding to the spontaneous breakdown of a local gauge symmetry.!) This local gauge symmetry is an invariance under the gauge transformation with gauge function A(x) linear in x", A(x )=E"X", E,,=const, which effects on the photon field A" a c-number shift oA"=a"A=E,,. In Refs. 2) and 3), motivated by this observation, we investigated a similar local gauge symmetry in nonAbelian gauge theories. Such a symmetry is spontaneously broken in perturbation theory but we showed that if it is restored dynamically, then the total confinement of color is realized. The purpose of this letter is to point out that an argument almost parallel to the one presented in Ref. 2) applies to non-linear 15models (NLSM) in 2 dimensions. These are renormalizable and asymptotically free quantum field theories and are known to have many dynamical aspects in common with gauge theories in 4 dimensions. 4 ) In Refs. 2) and 3) we emphasized on analogy between the restoration of the non-linear global symmetry in NLSM and that of the local gauge symmetry in gauge theories. However, the symmetry restoration in NLSM in 2 dimensions is a phenomenon of rather kinematical nature assured by Coleman's theorem) and occurs even in the Abelian model, while the local gauge symmetry in QED never restores. The analogy between the two theories presented below is more direct and we hope it will give some insight into the complicated dynamics of them.