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Effects of impurity scatterings on the strong coupling (SC) contribution,stabilizing the ABM (axial) pairing state, to the quartic term of theGinzburg-Landau (GL) free energy of superfluid $^3$He are theoretically studiedto examine recent observations suggestive of an anomalously small SC effect insuperfluid $^3$He in aerogels. To study the SC corrections, two approaches areused. One is based on a perturbation in the short-range repulsive interaction,and the other is a phenomenological approach used previously for the bulkliquid by Sauls and Serene [Phys.Rev.B 24, 183 (1981)]. It is found that theimpurity scattering favors the BW pairing state and shrinks the region of theABM pairing state in the T-P phase diagram. In the phenomenological approach,the resulting shrinkage of the ABM region is especially substantial and, ifassuming an anisotropy over a large scale in aerogel, leads to justifying thephase diagrams determined experimentally.

Strong Coupling Correction in Superfluid $^3$He in Aerogel