Abstract

Self-similar approximation theory is shown to be a powerful tool for describing phase transitions in quantum field theory. Self-similar approximants present the extrapolation of asymptotic series in powers of small variables to the arbitrary values of the latter, including the variables tending to infinity. The approach is illustrated by considering three problems: (i) The influence of the coupling parameter strength on the critical temperature of the O(N)-symmetric multicomponent field theory. (ii) The calculation of critical exponents for the phase transition in the O(N)-symmetric field theory. (iii) The evaluation of deconfinement temperature in quantum chromodynamics. The results are in good agreement with the available numerical calculations, such as Monte Carlo simulations, Padé-Borel summation, and lattice data.

Highlights

  • Phase transitions in field theory are known to be a very interesting physical problem, whose description usually confronts complicated calculational challenges

  • Suppose we study the region of asymptotically weak coupling in QCD corresponding to high temperature. Is it possible to extract the information on the existence of the confinement-deconfinement phase transition from these asymptotic expansions? Below we show that extrapolating asymptotic series by self-similar approximants allows us to predict the deconfinement phase transition and correctly estimate the transition temperature

  • We have shown that self-similar approximation theory is a powerful tool for extrapolating asymptotic series in small variables to the whole region of the latter from zero to infinity

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Summary

Introduction

Phase transitions in field theory are known to be a very interesting physical problem, whose description usually confronts complicated calculational challenges (see, e.g., [1,2,3,4,5]). Our aim in this report is to show that the description of phase transitions can be efficiently done by an original technique called self-similar approximation theory This theory allows us to find analytical expressions for the sought solutions, it is rather simple, involving only low-cost calculations, and at the same time, it is very accurate, being comparable in accuracy with heavy numerical calculations. We show that self-similar approximation theory overcomes the problems of asymptotic perturbation theory, making it possible to find analytic approximate solutions that are valid for the whole range of variables between zero to infinity. This theory combines simplicity with good accuracy.

Main ideas of self-similar approximation theory
Influence of coupling-parameter strength on critical temperature
Calculation of critical exponents in φ4 field theory
Estimation of deconfinement temperature in quantum chromodynamics
Conclusion
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