Continued functions and perturbation series: Simple tools for convergence of diverging series in $O(n)$-symmetric $\phi^4$ field theory at weak coupling limit

Abstract

We determine universal critical exponents that describe the continuous phase transitions in different dimensions of space. We use continued functions without any external unknown parameters to obtain analytic continuation for the recently derived 7-loop $\epsilon$ expansion from $O(n)$-symmetric $\phi^4$ field theory. Employing a new blended continued function, we obtain critical exponent $\alpha=-0.01211$ for the phase transition of superfluid helium which matches closely with the most accurate experimental value. This result addresses the long-standing discrepancy between the theoretical predictions and precise experimental result of $O(2)$ $\phi^4$ model known as "$\lambda$-point specific heat experimental anomaly". Further we have also examined the applicability of such continued functions in other examples of field theories.

Continued functions and Borel-Leroy transformation: Resummation of six-loop ε-expansions from different universality classes

“…To test the applicability of CE with BL transformation, we implement it to calculate the correction-to-scaling exponent ω from O(n)-symmetric theory. Especially in 3d system for the O(4) φ 4 model we previously used continued fraction to obtain the estimate ω = 0.7896(1) [8], which is only compatible with recent calculations from self-consistent resummation algorithm [16] where ω = 0.7863(9) was obtained. It is also comparable with estimates from Borel with conformal mapping (BCM) and conformal bootstrap calculations, ω = 0.794( 9) [1] and ω = 0.817(30) [17], respectively.…”

“…Coefficients {b i } and {c i } can be solved up to arbitary order i from the perturbative coefficients {q i } by Taylor expansion of CE and CEF as shown previously [8]. These affine transformations generally enlarge the region of convergence for Q( ) by a different mapping.…”

“…We have already checked the applicability of CE and CEF on perturbative expansions from O(n)symmetric field theory [8]. To test the applicability of CE with BL transformation, we implement it to calculate the correction-to-scaling exponent ω from O(n)-symmetric theory.…”

“…This was achieved using Hypergeometric-Meijer resummation to solve for critical exponents [3,4]. The issue commonly called as "λ-point specific heat experimental anomaly" was also addressed by using a new blended, continued exponential fraction [8]. We solved for the critical exponents of arbitrary n-component field in different spatial dimensions with the help of other continued functions such as continued exponential and continued fraction in the O(n)-vector model.…”

From Asymptotic Series to Self-Similar Approximants

“…In addition to the considered above combinations of different summation methods, one can use other combinations. For example, the combination of exponential approximants and continued fractions has been employed [95].…”