ELSEVIER 12 January 1995 PhysicsLetters B 342 (1995) 277-283 PHYSICS LETTERS B On the relation between the generalized Pauli-Villars and the covariant regularization E. Elizalde Department of Physics, Faculty of Science, Hiroshima University, Higashi-Hiroshima 724, Japan 1 Received 17 August 1994 Editor: M. Dine Abstract Some aspects of the comparison that has been established recently by Fujikawa between the generalized Pauli-Villars regularization and the covariant regularization of composite current operators are investigated, in particular, the fundamental question of the choice of regulator, satisfying appropriate conditions. The notion of zeta function of the operators is basic in the discussion. While developing the method, some new formulas that are very useful in physical applications of the theory are given. The aim of this note is to show how deeply the zeta function regularization method pervades all the different, alternative regularization procedures that are being used in gauge theory. We see this in the example provided by the very interesting scheme recently proposed by Frolov and Slavnov of a generalized Pauli-Villars (P-V) regularization of chiral gauge theory [1] (see also [2,3] ). As clearly pointed out by Fujikawa, a formal introduction of an infinite number of regulator fields in the Lagrangian does not specify the method completely and a most fundamental issue in this new regularization is to define how to sum over the contributions coming from the infinite number of regulator fields [4]. By reformulating the generalized P-V regularization as a regularization of composite current operators, Fujikawa has proven that an explicit choice for the sum of contributions of the infinite fields results, essentially, in a corresponding specific selection of a regulator, in the language of covariant regularization [5 ]. (Let us recall that the calculational scheme of covariant anomalies was introduced as an original, conveniently simple method in the path integral formulation of anomalous identities but it is not implemented at the Lagrangian level.) The chiral gauge theory to regularize has for Lagrangian i_ /2 = ~¢ D( 1 + ys)~h, (1) l Address June-September1994,e-mail: eli@aso.sci.hiroshima-u.ac.jp. On leave of absence from: Centerfor AdvancedStudy CEAB, CSIC, and Department ECM and 1FAE,Universityof Barcelona,Diagonal 647, 08028 Barcelona,Spain. 0370-2693/95/$09.50 (~) 1995 ElsevierScienceB.V. All rights reserved SSD10370-2693(94)01418-3