Minimal renormalization without ϵ-expansion: Three-loop amplitude functions of the 0(n) symmetric Φ4 theory in three dimensions below Tc Original Research Article

We present an analytic three-loop calculation for thermodynamic quantities of the O(n) symmetric Φ4 theory below Tc 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 ƒΦ, F+ and F− of the order parameter and the specific heat above and below Tc, respectively, without using the e = 4-d expansion. A Borel resummation for the case n = 2 yields resummed amplitude functions fΦ 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 4He. 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 4He. In addition we use our result for fΦ to calculate the universal combination Rc 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 Rc is significantly more accurate than the previous result obtained from the ϵ-expansion up to O(ϵ2.