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We suggest a generalization of nonlinear $\sigma$-model for diffusivesuperconducting systems to account for magnetoelectric effects due tospin-orbit scattering. In the leading orders of spin-orbit strength andgradient expansion it includes two additional terms responsible for thespin-Hall effect and the spin-current swapping. First, assuming adelta-correlated disorder we derive the new terms from the Keldysh pathintegral representation of the generating functional. Then we arguephenomenologically that they exhaust all invariants allowed in the effectiveaction to the leading order in the spin-orbit coupling (SOC). Finally, theresults are confirmed by a direct derivation of the saddle-point (Usadel)equation from the quantum kinetic equations in the presence of randomlydistributed impurities with SOC. At this point we correct a recent derivationof the Usadel equation that includes magneto-electric effects and does notresort to the Born approximation.

Magnetoelectric effects in superconductors due to spin-orbit scattering: a non-linear $\sigma$-model description