Five-loop additive renormalization in the φ4 theory and amplitude functions of the minimally renormalized specific heat in three dimensions

We present an analytic five-loop calculation for the additive renormalization constant A(u,epsilon) and the associated renormalization-group function B(u) of the specific heat of the O(n) symmetric phi(4) theory within the minimal subtraction scheme. We show that this calculation does not require new five-loop integrations but can be performed on the basis of the previous five-loop calculation of the four-point vertex function combined with an appropriate identification of symmetry factors of vacuum diagrams. We also determine the amplitude function F+(u) of the specific heat in three dimensions for n = 1,2,3 above T-c and F-(u) for n = 1 below T-c up to five-loop order, without using the epsilon=4-d expansion. Accurate results are obtained from Borel resummations of B(tl) for n=1,2,3 and of the amplitude functions for n=1. Previous conjectures regarding the smallness of the resummed higher-order contributions are confirmed. Combining our results for B(u) and F+(U) for N = 1,2,3 with those of a recent three-loop calculation of F-(u) for general n in d= 3 dimensions we calculate Borel resummed universal amplitude ratios A(+)/A(-) for n = 1,2,3. Our result for A(+)/A(-) = 1.056 +/- 0.004 for n = 2 is significantly more accurate than the previous result obtained from the epsilon expansion up to O(epsilon(2)) and agrees well with the high-precision experimental result A(+)/A(-) = 1.054 +/- 0.001 for He-4 near the superfluid transition obtained from a recent experiment in space.