P Renormalizability of Effective Scalar Field Theory

We present a comprehensive discussion of the consistency of the effective quantum field theory of a single Z2 symmetric scalar field. The theory is constructed from a bare Euclidean action which at a scale much greater than the particle’s mass is constrained only by the most basic requirements; stability, finiteness, analyticity, naturalness, and global symmetry. We prove to all orders in perturbation theory the boundedness, convergence, and universality of the theory at low energy scales, and thus that the theory is perturbatively renormalizable in the sense that to a certain precision over a range of such scales it depends only on a finite number of parameters. We then demonstrate that the effective theory has a well defined unitary and causal analytic S–matrix at all energy scales. We also show that redundant terms in the Lagrangian may be systematically eliminated by field redefinitions without changing the S–matrix, and discuss the extent to which effective field theory and analytic S–matrix theory are actually equivalent. All this is achieved by a systematic exploitation of Wilson’s exact renormalization group flow equation, as used by Polchinski in his original proof of the renormalizability of conventional φ-theory. CERN-TH.7067/93 OUTP-93-23P November 1993 An effective quantum field theory is any quantum field theory designed to give a description of the physics of a particular set of particles over a limited range of scales, without requiring detailed knowledge of any further physics outside of this range. Consider the following familiar examples: very soft photons scattering electromagnetically at energies below the electron mass; soft pions scattering strongly at energies below the scale of chiral symmetry breaking; electroweak processes involving leptons, pions and kaons; deep inelastic scattering at energies below the mass of a heavy quark; weak processes involving heavy quarks with masses below that of the intermediate vector bosons; electroweak processes involving intermediate vector bosons at energies below the mass of the Higgs boson (or the supersymmetry scale, or the technicolour scale, or whatever); the supersymmetric standard model below the scale of supersymmetry breaking; the evolution of standard model couplings below the scale of grand unification; quantum gravity below the Planck scale. Most of the physics we know, or even hope to know, seems to be described by effective quantum field theories, which are useful only until the scale of some new physical process is reached. Despite the wide variety of increasingly useful applications, from a formal point of view effective quantum field theories are still relatively poorly understood. Most of the original work on the consistency of quantum field theory dealt only with idealized theories, supposedly fundamental in the sense that they attempted to describe physics at arbitrarily high energies. In particular the S–matrix of the theory had to exist, be manifestly finite and well defined, and then satisfy the usual physical requirements of Lorentz invariance, unitarity, cluster decomposition and causality; in short, the theory had to be ‘renormalizable’. Only a rather small class of such theories was discovered; this was regarded as a virtue, since it helped guide the physicist to the ‘correct’ theory — the standard model. Now however we have become more discerning; from the examples given above it is clear that in many situations an effective theory can in practice be more useful than the underlying more fundamental theory which gives rise to it, and conversely that any theory we at present like to think of as fundamental might eventually turn out to be itself an effective theory. It thus becomes important to find out just how consistent an effective quantum field theory can be made to be; does it have a finite, well defined S–matrix, to what extent does this S–matrix depend on the unknown (and thus in principle arbitrary) physics at higher scales, and what about unitarity and causality? In other words, to what extent is effective field theory ‘renormalizable’? The purpose of this article, and a number of succeeding articles, is to address this question, using the exact renormalization group pioneered by Wilson[1]. The techniques necessary to prove conventional perturbative renormalizability of conventional scalar field theories using renormalization group flow were developed by Polchinski[2]; we will show how to extend his ideas to effective field theories, in such a way as to permit a thorough investigation of their consistency. In this article we treat only the special case of a Z2 symmetric scalar field propagating in 3 + 1 dimensions, using this as a theoretical model with which to develop the necessary techniques. In a number of subsequent articles we hope to consider more interesting (and indeed physically relevant) theories.