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,ε) and the associated renormalization-group function B(u) of the specific heat of the O(n) symmetric φ⁴ 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 ε=4-d expansion. Accurate results are obtained from Borel resummations of B(u) 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 ε expansion up to O(ε²) and agrees well with the high-precision experimental result A⁺/A^-=1.054±0.001 for ⁴He near the superfluid transition obtained from a recent experiment in space.