Minimal renormalization without \epsilon-expansion: Three-loop amplitude functions of the O(n) symmetric \phi^4 model in three dimensions below T_c

TL;DR

This paper provides a three-loop analytic calculation of thermodynamic amplitude functions in the O(n) symmetric  theory in three dimensions below T_c, avoiding -expansion and improving accuracy for critical universality predictions.

Contribution

It introduces a minimal renormalization approach at fixed dimension d=3 for three-loop amplitude functions, bypassing -expansion, and achieves more precise universal amplitude ratios.

Findings

Resummed amplitude functions slightly larger than one-loop results.

Excellent agreement with experimental data for He(4) specific heat ratio.

More accurate universal amplitude ratios than previous -expansion results.

Abstract

We present an analytic three-loop calculation for thermodynamic quantities of the O(n) symmetric \phi^4 theory below T_c within the minimal subtraction scheme at fixed dimension d=3. Goldstone singularities arising at an intermediate stage in the calculation of O(n) symmetric quantities cancel among themselves leaving a finite result in the limit of zero external field. From the free energy we calculate the three-loop terms of the amplitude functions f_phi, F+ and F- of the order parameter and the specific heat above and below T_c, respectively, without using the \epsilon=4-d expansion. A Borel resummation for the case n=2 yields resummed amplitude functions f_phi and F- that are slightly larger than the one-loop results. Accurate knowledge of these functions is needed for testing the renormalization-group prediction of critical-point universality along the \lambda-line of superfluid He(4). Combining the three-loop result for F- with a recent five-loop calculation of the additive renormalization constant of the specific heat yields excellent agreement between the calculated and measured universal amplitude ratio A+/A- of the specific heat of He(4). In addition we use our result for f_phi to calculate the universal combination R_C of the amplitudes of the order parameter, the susceptibility and the specific heat for n=2 and n=3. Our Borel-resummed three-loop result for R_C is significantly more accurate than the previous result obtained from the \epsilon-expansion up to O(\epsilon^2).