The scaling functions of the free energy density and its derivatives for the 3d O(4) model

TL;DR

This paper derives and numerically determines the universal scaling functions of the 3d O(4) model's free energy density and its derivatives, providing precise amplitude ratios and insights relevant for comparisons with QCD.

Contribution

The paper introduces direct representations of the 3d O(4) model's scaling functions using expansions in the scaling variable, enabling high-precision analysis without interpolation.

Findings

Universal amplitude ratio A^+/A^- = 1.84(4) for the specific heat.

High-precision determination of the non-singular energy density epsilon_{ns}(T).

Validation of scaling properties with extensive lattice data.

Abstract

We derive direct representations of the scaling functions of the 3d O(4) model which are relevant for comparisons to other models, in particular QCD. This is done in terms of expansions in the scaling variable z= t/h^{1/Delta}. The expansions around z=0 and the corresponding asymptotic ones for z --> +- infinity overlap such that no interpolation is needed. The expansion coefficients are determined numerically from the data of a previous high statistics simulation of the O(4) model on a three-dimensional lattice of linear extension L=120. From the scaling function of the magnetization we calculate the leading asymptotic coefficients of the scaling function of the free energy density. As a result we obtain the universal amplitude ratio A^+/A^-=1.84(4) for the specific heat. Comparing the scaling function of the energy density to the data we find the non-singular part of the energy density epsilon_{ns}(T) with high precision and at the same time excellent scaling properties.