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

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 ƒ Φ, 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 f Φ 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 f Φ 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 R c is significantly more accurate than the previous result obtained from the ϵ-expansion up to O( ϵ 2.