Self-similar sequence transformation for critical exponents

TL;DR

The paper introduces a self-similar sequence transformation method for calculating critical exponents in quantum field theory, demonstrating its simplicity and accuracy compared to existing techniques.

Contribution

It presents a novel self-similar factor transformation approach for summing asymptotic series, simplifying the calculation of critical exponents in the $O(N)$-symmetric $^4$ theory.

Findings

Method is regular and effective for asymptotic series

Achieves comparable accuracy with simpler calculations

Successfully applied to critical exponents in three dimensions

Abstract

Self-similar sequence transformation is an original type of nonlinear sequence transformations allowing for defining effective limits of asymptotic sequences. The method of self-similar factor transformations is shown to be regular. This method is applied for calculating the critical exponents of the $O(N)$-symmetric $\varphi^4$ theory in three dimensions by summing asymptotic $\varepsilon$ expansions. It is shown that this method is straightforward and essentially simpler than other summation techniques involving complicated numerical calculations, while enjoying comparable accuracy.