Minimal renormalization without ϵ-expansion: Amplitude functions for O(n) symmetric systems in three dimensions below Tc Original Research Article

Massive field theory at fixed dimension d < 4 is combined with the minimal subtraction scheme to calculate the amplitude functions of thermodynamic quantities for the O(n) symmetric Φ4 model below Tc in two-loop order. Goldstone singularities arising at an intermediate stage in the calculation of O(n) symmetric quantities are shown to cancel among themselves leaving a finite result in the limit of zero external field. From the free energy we calculate the amplitude functions in zero field for the order parameter, specific heat and helicity modulus (superfluid density) in three dimensions. We also calculate the q2 part of the inverse of the wavenumber-dependent transverse susceptibility χT(q) which provides an independent check of our result for the helicity modulus. The two-loop contributions to the superfluid density and specific heat below Tc turn out to be comparable in magnitude to the one-loop contributions, indicating the necessity of higher-order calculations and Padé-Borel type resummations.