Effective scalar field theory and reduction of couplings

Synthese, 1995

Much apprehension has been expressed by philosophers about the method of renormalisation in quantum field theory, as it apparently requires illegitimate procedure of infinite cancellation. This has lead to various speculations, in particular in . We examine Teller's discussion of perturbative renormalisation of quantum fields, and show why it is inadequate. To really approach the matter one needs to understand the ideas and results of the 'renormalisation group', so we give a simple but comprehensive account of this topic. With this in hand, we explain how renormalisation can and should be understood. One thing that is revealed is that apparently very successful theories such as quantum electro-dynamics cannot be universally true; resolving the tension between success and falsity leads to a picture in which any theory may be viewed as irreducibly phenomenological. We explain how, and argue that the support for this view is tenuous at best.

Physical Review D, 1996

The structure of the renormalization group equations for the low energy effective theory of gravity coupled to a scalar field is presented. An approximate solution to these equations with a finite number of independent renormalized parameters can be found when the mass scale characteristic of the fluctuations in the geometry is much smaller than the Planck mass. The cosmological constant problem is reformulated in this context and some conditions on the matter field content and interactions required in order to have a sufficiently small cosmological constant are identified. *

Physics Letters B, 1993

Massless φ 4-theory is investigated in zero and four space-time dimensions. Path-integral linearisation of the φ 4-interaction defines an effective theory, which is investigated in a loop-expansion around the mean field. In zero dimensions this expansion converges rapidly to the exact potential obtained numerically. In four dimensions its lowest order (mean-field approximation) produces a real and convex effective potential. Two phases are found. In one the renormalisation group improved one-loop effective potential is recovered as the leading contribution near the classical minimum. This phase, however, is unstable. The second (precarious) phase is found to have lower vacuum energy density. In this phase a dynamical mass is generated. The results are renormalisation group invariant.

2020

As applied to quantum theories, the program of renormalization is successful for ‘renormalizable models’ but fails for ‘nonrenormalizable models’. After some conceptual discussion and analysis, an enhanced program of renormalization is proposed that is designed to bring the ‘nonrenormalizable models’ under control as well. The new principles are developed by studying several, carefully chosen, soluble examples, and include a recognition of a ‘hard-core’ behavior of the interaction and, in special cases, an extremely elementary procedure to remove the source of all divergences. Our discussion provides the background for a recent proposal for a nontrivial quantization of nonrenormalizable scalar quantum field models, which is briefly summarized as well. Dedication: It is a pleasure to dedicate this article to the memory of Prof. Alladi Ramakrishnan who, besides his own important contributions to science, played a crucial role in the development of modern scientific research and educat...

Physics Letters B, 1996

The general structure of the renormalization group equations for the low energy effective field theory formulation of pure gravity is presented. The solution of these equations takes a particular simple form if the mass scale of the effective theory is much smaller than the Planck mass (a possibility compatible with the renormalization of the effective theory). A theory with just one free renormalized parameter is obtained when contributions suppressed by inverse powers of the Planck mass are neglected.

Studies in History and Philosophy of Science

The effective field theory (EFT) perspective on particle physics has yielded insight into the Standard Model. This paper investigates the epistemic consequences of the use of different variants of renormalization group (RG) methods as part of the EFT perspective on particle physics. RG methods are a family of formal techniques. While the semi-group variant of the RG has played a prominent role in condensed matter physics, the full-group variant has become the most widely applicable formalism in particle physics. We survey different construction techniques for EFTs in particle physics and analyze the role that semi-group and full-group variants of the RG play in each. We argue that the full-group variant is best suited to answering structural questions about relationships among EFTs at different scales, as well as explanatory questions, such as why the Standard Model has been empirically successful at low energy scales and why renormalizability was a successful criterion for constructing the Standard Model. We also present an account of EFTs in particle physics that is based on the full-RG. Our conclusion about the advantages of the full-RG is restricted to the particle physics case. We argue that a domain-specific approach to interpreting EFTs and RG methods is needed. Formal variations and flexibility in physical interpretation enable RG methods to support different explanatory strategies in condensed matter and particle physics. In particular, it is consistent to maintain that coarse-graining is an essential component of explanations in condensed matter physics, but not in particle physics.

Physics Letters B, 1995

Direct verification of the existence of an infinite set of multicritical non-perturbative FPs (Fixed Points) for a single scalar field in two dimensions, is in practice well outside the capabilities of the present standard approximate non-perturbative methods. We apply a derivative expansion of the exact RG (Renormalization Group) equations in a form which allows the corresponding FP equations to appear as non-linear eigenvalue equations for the anomalous scaling dimension η. At zeroth order, only continuum limits based on critical sine-Gordon models, are accessible. At second order in derivatives, we perform a general search over all η ≥ .02, finding the expected first ten FPs, and only these. For each of these we verify the correct relevant qualitative behaviour, and compute critical exponents, and the dimensions of up to the first ten lowest dimension operators. Depending on the quantity, our lowest order approximate description agrees with CFT (Conformal Field Theory) with an accuracy between 0.2% and 33%; this requires however that certain irrelevant operators that are total derivatives in the CFT are associated with ones that are not total derivatives in the scalar field theory. CERN-TH.7403/94 SHEP 94/95-04 hep-th/9410141 October, 1994. * On Leave from Southampton University, U.K. (Address after 1/10/94). Circumstantial evidence strongly suggests that there exists an infinite set of multicritical non-perturbative FPs for a single scalar field in two dimensions, corresponding to the universality classes of multicritical Ising models, equivalently to the diagonal invariants of the unitary minimal (p, p + 1) conformal models with p = 3, 4, · · · [1] [2], however direct verification of these facts is in practice well outside the capabilities of the standard approximate non-perturbative methods: lattice Monte Carlo, resummations of weak or strong coupling perturbation theory and the epsilon expansion. (The impracticableness of the epsilon expansion for higher p is covered in ref.[3], implying similar difficulties in weak coupling perturbation theory, while lattice methods suffer from difficulties of locating and accurately computing the multicritical points in the at least p − 2 dimensional bare coupling constant space)

Zeitschrift für Physik C Particles and Fields, 1994

Physical Review D, 1996

We show that general cutoff scalar field theories in four dimensions are perturbatively renormalizable through the use of diagrammatic techniques and an adapted BPH renormalization method. Weinberg's convergence theorem is used to show that operators in the Lagrangian with dimension greater than four, which are divided by powers of the cutoff, produce perturbatively only local divergences in the two-, three-, and four-point correlation functions. We also show that the renormalized Green's functions are the same as in ordinary Φ 4 theory up to corrections suppressed by inverse powers of the cutoff. These conclusions are consistent with those of existing proofs based on the renormalization group.

Physical Review D, 1995

We study constraint effective potentials for various strongly interacting φ4 theories. Renormalization-group (RG) equations for these quantities are discussed and a heuristic development of a commonly used RG approximation is presented which stresses the relationships among the loop expansion, the Schwinger-Dyson method, and the renormalization-group approach. We extend the standard RG treatment to account explicitly for finite lattice effects. Constraint effective potentials are then evaluated using Monte Carlo (MC) techniques and careful comparisons are made with RG calculations. An explicit treatment of finite lattice effects is found to be essential in achieving quantitive agreement with the MC effective potentials. Excellent agreement is demonstrated for d=3 and d=4, O(1) and O(2) cases in both symmetric and broken phases.