Background_field_abbott - Vol 813(1982 ACTA PHYSICA...

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Vol. 813 (1982) ACTA PHYSICA POLONICA No 1-2 INTRODUCTION TO THE BACKGROUND FIELD METHOD* By L. F. ABBoTT** CERN, Geneva (Received July 20, 1981) The background field approach to calculations in gauge field theories is presented. Conventional functional techniques are reviewed and the background field method is intro- duced. Feynman rules and renormalization are discussed and, as an example, the Yang- -Mills f3 function is computed. PACS numbers: 11.10.Np, 11.10.Gh 1. Introduction The background field method is a technique for quantizing gauge field theories without losing explicit gauge invariar~ce. It makes gauge theories easier to understand and greatly simplifies computations. In this review I will present the formalism of this method and show how it is applied to gauge theory calculations. The background field method was introduced by DeWitt [I, 2] in a formalism which was applicable to one-loop processes. The extension to multi-loop calculatior.s, which involved a reformulation of the method, was first made by 't Hooft [3] and then discussed in more detail by DeWitt [4], Boulware [5] and by myself [6]. It is this extension, valid to all orders ofperturbation theory, which I will present here. The background field method is used extensively in gravity [7] and supergravity [8] theories. In addition, it has been used to derive light-particle effective field theories from grand unified models [9], to com- pute the Yang-Mills fJ fULction up to two loops [6] and to perform calculatioLs in lattice gauge theories [10]. In all of these applications, the great simplifications introduced by the method playa key role. Any formulation of a gauge field theory begins with a gauge invariant Lagrangian. However, in order to quantize the theory a gauge must be chosen. In the conventional formulation, this means that the Lagrangian you actually use to derive Feynman rules and perform calculations, consisting of the classical Lagrangian plus gauge-fixing and ghost terms, is not gauge invariant. Of course, any physical quantity calculated will be * Presented at the XXI Cracow School ofTheoretical Physics, Paszk6wka, May 29 June 9, 1981. ** Address: Physics Department, Brandeis University, Waltham, MA 02254, USA. (33)
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34 gauge invariant and independent of the particular gauge chosen, but quantities with no direct physical interpretation like off-shell Green's functions or divergent counterterms will not be gauge invariaTtt. Green's functions in the conventional formulation do not directly reflect the underlying gauge invariance of the theory but rather obey complicated Slavnov-Taylor identities (I I] resultir:g from BRS invariar:ce (12]. In the background field approach, one arranges things so that explicit gauge invariar:ce, present in the original Lagrangian, is still present OLce gauge-fixing and ghost terms have been added. As a result, in this formalism, Green's functions obey the naive Ward identities of gauge invariance and even unphysical quantities like divergerit counterterms take a gauge invariant form.
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Background_field_abbott - Vol 813(1982 ACTA PHYSICA...

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