On Perturbation theory improved by Strong coupling expansion

TL;DR

This paper introduces a new class of smooth interpolating functions combining two perturbative expansions in physics, providing a criterion to identify the best approximation and demonstrating its effectiveness across various models.

Contribution

It proposes a novel interpolating function framework that unifies existing methods and offers a criterion to select the optimal approximation without knowing the exact solution.

Findings

The criterion successfully identifies the best interpolating function in multiple models.

The method generalizes and includes Padé approximant and Sen's fractional power polynomial as special cases.

Applications to systems with phase transitions are discussed.

Abstract

In theoretical physics, we sometimes have two perturbative expansions of physical quantity around different two points in parameter space. In terms of the two perturbative expansions, we introduce a new type of smooth interpolating function consistent with the both expansions, which includes the standard Pad\'e approximant and fractional power of polynomial method constructed by Sen as special cases. We point out that we can construct enormous number of such interpolating functions in principle while the "best" approximation for the exact answer of the physical quantity should be unique among the interpolating functions. We propose a criterion to determine the "best" interpolating function, which is applicable except some situations even if we do not know the exact answer. It turns out that our criterion works for various examples including specific heat in two-dimensional Ising model, average plaquette in four-dimensional SU(3) pure Yang-Mills theory on lattice and free energy in c=1 string theory at self-dual radius. We also mention possible applications of the interpolating functions to system with phase transition.