Minimal renormalization without epsilon-expansion: Three-loop amplitude functions of the O(n) symmetric phi(4) theory in three dimensions below T-c

We present an analytic three-loop calculation for thermodynamic quantities of the O(n) symmetric phi(4) theory below T-c within the minimal subtractio n scheme at fixed dimension d = 3. Goldstone singularities arising at an in termediate stage in the calculation of O(n) symmetric quantities cancel amo ng 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 fun ctions f(phi), F+. and F- of the order parameter and the specific heat abov e 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 k nowledge of these functions is needed for testing the renormalization-group prediction of critical-point universality along the Aline of superfluid He -4. Combining the three-loop result for F- with a recent five-loop calculat ion of the additive renormalization constant of the specific heat yields ex cellent agreement between the calculated and measured universal amplitude r atio 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)). (C) 1999 Elsevier Science B.V.