Spin-fluctuation theory of strong coupling corrections to the free energy in superfluid 3He

Previous spin-fluctuation theories yielding the strong coupling corrections -_i to the five coefficients of the fourth-order invariants in the free energy functional are extended. First, the superfluid part of the susceptibility is calculated up to order ⁴ for all momenta and frequencies and the contribution arising from p-wave fluctuations of the order parameter is included. Then the frequency sums yielding the -_i are calculated by taking into account the full momentum and frequency dependence of the superfluid susceptibility and the spin fluctuation propagator. The results for the -_i are plotted vs. a cutoff q_c on the momentum integration for spin-fluctuation parameters N(0)I = 0.75 and = 0.95. The cutoff takes into account in a rough way the effect of additional terms in the free energy functional which were neglected in previous theories. These additional terms are due to the implicit dependence of the superfluid susceptibility on the spin-fluctuation parameter I via the gap parameter . The gap equation providing the relation between and I is derived in the weak coupling approximation. The cutoffs obtained by fitting the experimental values of the three combinations of -_i (arising from the measured specific heat discontinuities on the melting curve) are comparable to the cutoff obtained from the spin-fluctuation contribution to the weak coupling freeenergy [q_c = 0.3(2k_F) for = 0.75]. The corrections due to the momentum and frequency dependence of the superfluid susceptibility and the spinfluctuation propagator are large and point in the direction of better agreement with experiment: The ratio % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiOuamaaBa% aaleaacaaIXaaabeaakiabg2da9iabfs5aejqbek7aIzaaraWaaSba% aSqaaiaaiwdaaeqaaOGaai4laiaacIcacqqHuoarcuaHYoGygaqeam% aaBaaaleaacaaIYaaabeaakiabgUcaRiabfs5aejqbek7aIzaaraWa% aSbaaSqaaiaaisdaaeqaaOGaaiykaaaa!47E0![R_1 = Delta bar beta _5 /(Delta bar beta _2 + Delta bar beta _4 )]decreases from 2 to about 1.2. However, in view of the large discrepancy with experiment (experimentally, R₁= 0.14), we conclude that spin-fluctuation theory in its present form (without taking into account renormalization effects) cannot account quantitatively for the measured specific heat discontinuities on the melting curve.