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CONVERGENT PERTURBATION THEORY FOR STUDYING PHASE TRANSITIONS M. Yu. Nalimov∗ and A. V. Ovsyannikov∗

We propose a method for constructing a perturbation theory with a finite radius of convergence for a rather wide class of quantum field models traditionally used to describe critical and near-critical behavior in problems in statistical physics. For the proposed convergent series, we use an instanton analysis to find the radius of convergence and also indicate a strategy for calculating their coefficients based on the diagrams in the standard (divergent) perturbation theory. We test the approach in the example of the standard stochastic dynamics A-model and a matrix model of the phase transition in a system of nonrelativistic fermions, where its application allows explaining the previously observed quasiuniversal behavior of the trajectories of a first-order phase transition.

Keywords: renormalization group, instanton analysis, convergent perturbation theory, superconductivity, critical behavior DOI: 10.1134/S004057792008005X

1. Introduction In quantum field theory (QFT) models, concrete expressions for physical quantities are most often sought in the framework of a perturbation theory in the form of a power series in some coupling constant. As a rule, such expansions do not have radii of convergence and are asymptotic series in the Poincar´e sense. In elementary particle physics, we always encounter two characteristic situations. Either the nonperturbative contributions are negligibly small as a result of an extremely small expansion parameter (the fine structure constant α  1/137 in quantum electrodynamics), in which case segments of the divergent power series obtained from lower orders of the perturbation theory are a reliable basis for numerical approximations, or the perturbative approach is inapplicable (in passing to the low-energy region in quantum chromodynamics, the coupling constant α grows without bound). The perturbation theory expansions work excellently in the first case (and will continue to work for a long time) and are unsuitable for a wide energy spectrum in the second case. The problem of studying a quantum field perturbation theory in the region of large coupling constant values presents a separate interest. For example, it arises in studying the behavior of an effective charge in the neighborhood of an ultraviolet-stable (infrared-stable) fixed point [1]. In applying quantum field ∗

St. Petersburg State University, St. Petersburg, Russia, e-mail: [email protected] (corresponding author), [email protected], [email protected]. This research was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics “BASIS” (Grant No. 19-1-1-35-1). Prepared from an English manuscript submitted by the authors; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 204, No. 2, pp. 226–241, August, 2020. Received January 15, 2020. Revised March 9, 2020. Accepted March 30, 2020. c 2020 Pleiades Publishing, Ltd. 0040-5779/20/2042-1033 

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