5 this regularization procedure leads to the same result as Eq. (30). The details of calculations are given in Ap- pendix A. The final expression for the current becomes greatly exceed the scales relevant to the superconducting state, the corresponding contributions to the current co- incide with those in the normal state. One can show that j = ev 2 $_{F}$k ∫ ∞ 0 p dp 2 π    − 1 4 T cosh − 2 E$_{↑}$ 2 T − 1 4 T cosh − 2 E$_{↓}$ 2 T + ∣ u ∗ $_{↑}$u$_{↓}$ + v ∗ $_{↑}$v$_{↓}$ ∣ ∣$^{2}$ E$_{↑}$ − E$_{↓}$ ( tanh E$_{↑}$ 2 T − tanh E$_{↓}$ 2 T ) + 1 v$_{F}$ p − ∣ v ∗ $_{↑}$u$_{↓}$ − u ∗ $_{↑}$v$_{↓}$ ∣ ∣$^{2}$ E$_{↑}$ + E$_{↓}$ ( tanh E$_{↑}$ 2 T + tanh E$_{↓}$ 2 T )    . (31) The true wave functions satisfy It is worthwhile to note that the problem of spurious divergent terms in the expression for the supercurrent is rather general. In particular, it was discussed (and resolved similarly) for the superfluid excitonic current in graphene bilayers $^{18}$. We evaluate Eq. (31) for low temperatures, T ≪ | Δ | . Since E$_{↑}$, E$_{↓}$ > | Δ | , the first line in Eq. (31) vanishes at T = 0. The supercurrent becomes We assume v$_{F}$ k, µ ≪ E$_{0}$ . Let us expand the true wave function into the zero order orthonormal wave functions ˇ ψ (0) α , ˇ ψ$_{α}$ = ∑ β C$_{αβ}$ ˇ ψ (0) β , (36) ˇ H ˇ ψ$_{α}$ = E$_{α}$ ˇ ψ$_{α}$ where ˇ H = ˇ H (0) + ˇ H (1) and ˇ H (0) = ( v$_{F}$ ˆ σ · p | Δ | | Δ | − v$_{F}$ ˆ σ · p ) , ˇ H (1) = 1 $_{2}$v$_{F}$ ˆ σ · k µ 0 0 1 $_{2}$v$_{F}$ ˆ σ · k + µ ) . j = e k 2 π [ √ µ 2 + | Δ | 2 + | Δ | 2 | µ ln ( | µ | + √ µ 2 + | Δ | 2 | Δ | )] . (32) j = e | µ | k / 2 π . (33) j = e | Δ | k /π . (34) For µ ≫ | Δ | we have For µ ≪ | Δ | we find This result formally holds within the linear approxima- tion which assumes v$_{F}$ k ≪ µ . Therefore, in Eq. (34) one would have to put k → 0 first and then assume µ ≪ | Δ | . In the next section we demonstrate that Eq. (34) is in fact always valid provided | µ | ≪ | Δ | and v$_{F}$ k ≪ | Δ | irrespectively of the relation between v$_{F}$ k and µ . Consider now the limit of small µ when the zero order state is degenerate because E (0) 1 = E (0) 3 = E$_{0}$ and E (0) 2 = E (0) 4 = − E$_{0}$ where B. Low doping limit In what follows we demonstrate that despite the absence of the Doppler term in its usual form, there still is a finite linear in k supercurrent down to zero temperature (contrary to the result of Ref. 7). E$_{0}$ = √ ( v$_{F}$ p ) 2 + | Δ | 2 . (35) ˜ E$_{1}$ = √ ( µv$_{F}$ p E$_{0}$ − E$_{D}$ ) 2 + v 2 $_{F}$[ p × k ] 2 4 p 2 | Δ | 2 E 2 0 (39) u = 1 √$_{2}$ [ 1 + v$_{F}$ p E$_{0}$ ] 1 / 2 , v = 1 √$_{2}$ [ 1 − v$_{F}$ p E$_{0}$ ] 1 / 2 . (37) The zero-order wave functions have now the form of Eqs. (13), (15) with u$_{↑}$ = v$_{↓}$ ≡ u and u$_{↓}$ = v$_{↑}$ ≡ v where satisfying the zero-order equation ˇ H (0) ˇ ψ (0) α = E (0) α ˇ ψ (0) α . C$_{αγ}$ [ E$_{α}$ − E (0) γ ] = ∑ β C$_{αβ}$H$_{γβ}$ (38) The expansion coefficients satisfy where H$_{γβ}$ ≡ 〈 ˇ ψ (0)+ γ ˇ H (1) ˇ ψ (0) β 〉 . Consider the state α = 1. Since the difference E$_{1}$ − E (0) 1 = E$_{1}$ − E (0) 3 ≡ δE$_{1}$ in Eq. (38) is small, the coeffi- cients C$_{12}$ an C$_{14}$ are proportional to the perturbation, while C$_{11}$ and C$_{13}$ are of the order unity. The coefficients C (0) 11 , C (0) 13 in the leading approximation satisfy the sec- ular equations (B5), (B6) (see Appendix B) which yield δE$_{1}$$_{,}$$_{3}$ = ∓ ˜ E$_{1}$ ,