Analytical approximation schemes for solving exact renormalization group equations. II Conformal mappings

TL;DR

This paper introduces a new analytical approximation method using conformal mappings to efficiently solve boundary value problems in the exact renormalization group equations, providing accurate and fast results for scalar field models.

Contribution

The paper develops a novel conformal mapping-based approximation scheme for solving ODEs in the context of the exact renormalization group, improving accuracy and computational speed.

Findings

Accurate estimation of fixed points and eigenvalues for scalar field models.

The new method outperforms traditional shooting and analytical methods in speed and precision.

Applicable to complex systems of equations in renormalization group studies.

Abstract

We present a new efficient analytical approximation scheme to two-point boundary value problems of ordinary differential equations (ODEs) adapted to the study of the derivative expansion of the exact renormalization group equations. It is based on a compactification of the complex plane of the independent variable using a mapping of an angular sector onto a unit disc. We explicitly treat, for the scalar field, the local potential approximations of the Wegner-Houghton equation in the dimension $d=3$ and of the Wilson-Polchinski equation for some values of $d\in ] 2,3] $. We then consider, for $d=3$, the coupled ODEs obtained by Morris at the second order of the derivative expansion. In both cases the fixed points and the eigenvalues attached to them are estimated. Comparisons of the results obtained are made with the shooting method and with the other analytical methods available. The best accuracy is reached with our new method which presents also the advantage of being very fast. Thus, it is well adapted to the study of more complicated systems of equations.