Hypergeometric Gevrey-0 approximation for the Gevrey-k divergent series with application to eight-loop renormalization group functions of the O(N)-symmetric field model

Abouzeid M. Shalaby; Hamdi M. Abdelhamid; I. S. Elkamash

doi:10.1140/epjp/s13360-024-05373-y

2023 Impact factor 2.8

EPJ PLUS - Condensed Matter and Complex Systems

Open Access

Eur. Phys. J. Plus (2024) 139: 592
https://doi.org/10.1140/epjp/s13360-024-05373-y

Regular Article

Hypergeometric Gevrey-0 approximation for the Gevrey-k divergent series with application to eight-loop renormalization group functions of the O(N)-symmetric field model

Abouzeid M. Shalaby¹^a, Hamdi M. Abdelhamid²^,3 and I. S. Elkamash²

¹ Physics and Materials Sciences Department, College of Arts and Sciences, Qatar University, P.O. Box 2713, Doha, Qatar
² Physics Department, Faculty of Science, Mansoura University, 35516, Mansoura, Egypt
³ Physics Department, Faculty of Science, New Mansoura University, New Mansoura City, Egypt

^a amshalab@qu.edu.qa

Received: 14 December 2023
Accepted: 14 June 2024
Published online: 7 July 2024

Abstract

Mera et al. (Phys Rev Lett 115:143001, 2015) discovered that the hypergeometric function $_{2} F_{1} (a_{1}, a_{2} ; b_{1} ; ω g)$ ${}_2F_1(a_1,a_2;b_1; \omega g)$ can serve as an accurate approximant for a divergent Gevrey-1 type of series with an asymptotic large-order behavior of the form $n! n^{b} σ^{n}$ $n! n^b \sigma ^n$ . What is strange about this approximant is that it has a series expansion with the wrong large-order behavior (Gevrey-0 type). In this work, we extend this discovery to Gevrey-k series where we show that the hypergeometric approximants and its extension to the generalized hypergeometric approximants are not only able to approximate divergent (Gevrey-1) series but also able to approximate strongly-divergent series of Gevrey-k type with $k = 2, 3, \dots$ $k=2,3,\ldots$ . Moreover, we show that these hypergeometric approximants are able to predict accurate results for the non-perturbative strong-coupling and large-order parameters from weak-coupling data as input. Examples studied here are the ground-state energy for the $x^{n}$ $$x^n$$ anharmonic oscillators. The hypergeometric approximants are also used to approximate the recent eight-loop series ( g-expansion) of the renormalization group functions for the O(N)-symmetric $ϕ^{4}$ $\phi ^4$ scalar field model. Form these functions for $N = 0, 1, 2$ $$N=0, 1, 2$$ , and 3, critical exponents are extracted which are very competitive to results from more sophisticated approximation techniques.

© The Author(s) 2024

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