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

Abstract

We present an analytic three-loop calculation for thermodynamic quantities of the O(n) symmetric ϕ^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 ε=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 λ-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 ε-expansion up to O(ε^2).