Effective scalar field theory and reduction of couplings

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, 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.

The Legacy of Alladi Ramakrishnan in the Mathematical Sciences, 2010

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 education in his native India. Besides a number of recent informative discussions during his yearly visits to the University of Florida, the present author had the pleasure much earlier of hosting Prof. Alladi during his visit and lecture at Bell Telephone Laboratories.

arXiv: Mathematical Physics, 2016

The problem of renormalization in perturbative quantum field theory (pQFT) can be described in a rigorous way through the theory of extension of distributions. In the framework of pQFT a certain type of distribution appears, given by products of Green functions which act by integration with a test function. They present ultraviolet divergences, whenever any pair of arguments coincide on one point of spacetime, and therefore, they are not defined everywhere. In this work we have studied the necessary and sufficient conditions for the extension (or regularization) of this type of distribution. Moreover, we have constructed such extensions explicitly, satisfying a series of physically relevant axioms, such as the axiom of causality.

Studies in History and Philosophy of Modern Physics, 2019

In this paper, I propose a general framework for understanding renormalization by drawing on the distinction between effective and continuum Quantum Field Theories (QFTs), and offer a comprehensive account of perturbative renormalization on this basis. My central claim is that the effective approach to renormalization provides a more physically perspicuous, conceptually coherent and widely applicable framework to construct perturbative QFTs than the continuum approach. I also show how a careful comparison between the two approaches: (i) helps to dispel the mystery surrounding the success of the renormalization procedure; (ii) clarifies the various notions of renormalizability; and (iii) gives reasons to temper Butterfield and Bouatta's claim that some continuum QFTs are ripe for metaphysical inquiry (Butterfield and Bouatta, 2014).

SciPost Physics, 2021

We compute the one-loop renormalisation group running of the bosonic Standard Model effective operators to order v^4/\Lambda^4v4/Λ4, with v\sim 246v∼246 GeV being the electroweak scale and \LambdaΛ the unknown new physics threshold. We concentrate on the effects triggered by pairs of the leading dimension-six interactions, namely those that can arise at tree level in weakly-coupled ultraviolet completions of the Standard Model. We highlight some interesting consequences, including the interplay between positivity bounds and the form of the anomalous dimensions; the non-renormalisation of the SS and UU parameters; or the importance of radiative corrections to the Higgs potential for the electroweak phase transition. As a byproduct of this work, we provide a complete Green basis of operators involving only the Higgs and derivatives at dimension-eight, comprising 13 redundant interactions.

Physical Review D, 2012

The purpose of this work is to rewrite the generating functional of φ 4 theory for the n = 0 and n = 4 correlation functions as the inner product of a state with an observable, as we did in a previous work, for the twopoints correlation function. The observables are defined through the external sources and the states are defined through the correlation function itself. In this sense, the divergences of Quantum Field Theory (QFT) appear in the reduced state by taking the partial trace of the state with respect to the internal vertices that appear in the perturbation expansion. From this viewpoint, the renormalization can be substituted by applying a projector on the internal quantum state. The advantage of this new insight is that we can obtain finite contributions to the correlation functions without introducing counterterms in the Lagrangian or by manipulating complex divergent quantities.