Spontaneously Broken Gauge Symmetries. I. Preliminaries

This is the first of a series of papers addressed to the renormalizability question of spontaneously broken gauge theories. We give a brief outline of the motivation for such an investigation and describe the manner in which the renormalizability of such theories will be proved in the sequel. Put briefly, we will show that in an appropriate gauge, ultraviolet divergences of a spontaneously broken gauge theory are removed completely by the gauge-invariant counterterms in the Lagrangian which would make the Greon's functions of the corresponding unbroken gauge theory finite, that the $S$ matrix computed in this gauge is unitary, and that the $S$ matrix is independent of the gauge chosen. In this paper, the renormalizability question of the unbroken gauge theory is considered. We derive the Ward-Takahashi identities of the theory. We discuss several ways of regulating divergent Feynman integrals of the theory without destroying gauge invariance. Infrared divergences are avoided by the device of intermediate renormalization, wherein we choose as subtraction points some points where external momenta are Euclidean. This suffices to establish that the Bogoliubov-Parasiuk-Hepp renormalization will give renormalized Green's functions which satisfy the Ward-Takahashi identities. The existence of finite, renormalized Green's functions satisfying the Ward-Takahashi identities provides us with the means of proving the renormalizability of the spontaneously broken symmetry case. The Ward-Takahashi identities were previously derived for the gauge bosons by Slavnov. We present here a new derivation. The discussions on regularization methods and intermediate renormalization procedure and the renormalization conditions for matter fields, we believe, are new contributions of the present paper.