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Respecting the conservation laws of momentum and energy in a many body theoryis very important for understanding the transport phenomena. The previousconserving approximation requires that the self-energy of a single particle canbe written as a functional derivative of a full dressed Green's function. Thiscondition can not be satisfied in the $G_0G$ t-matrix or pair fluctuationtheory which emphasizes the fermion pairing with a stronger than theBardeen-Cooper-Schrieffer (BCS) attraction. In the previouswork\cite{stressWI}, we have shown that when the temperature is above thesuperfluid transition temperature $T_c$, the $G_0G$ t-matrix theory can be putinto a form that satisfies the stress tensor Ward identity (WI) or local formof conservation laws by introducing a new type of vertex correction. In thispaper, we will extend the above conservation approximation to the superfluidphase in the BCS mean field level. To establish the stress tensor WI, we haveto include the fluctuation of the order parameter or the contribution from theGoldstone mode. The result will be useful for understanding the transportproperties such as the behavior of the viscosity of Fermionic gases in thesuperfluid phases.

Conserving Approximation of Pairing theories in Fermionic superfluid phase