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Therefore, the up-spin wave functions ˆ u (1 , 2) contain only corrections with the down-spin components ˆ u (3 , $^{4)}$, and vice versa. Expansion Eq. (23) yields also the corrections to the eigenenergies which coincide with Eq. (22). 

Using Eq. (23) one can show that the quasiparticle current j$_{p}$ is zero. The supercurrent Eq. (4) takes the form of Eq. (5), j = j$_{e}$ , which can be written as 

where 

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The term j$_{K}$ ( p ) is the contribution from the valley K in the Brillouin zone while j$_{−}$$_{K}$ ( p ) is the contribution from valley − K . Therefore, Eq. (26) in fact collects contribu- tions from the vicinity of the Dirac points at the opposite corners of the entire Brillouin zone [shaded sectors in Fig. 1(b) or (c)]. 

Using Eq. (23) we obtain from Eq. (5) in the linear approximation 

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Equation (29) contains the terms which diverge for large v$_{F}$ p ≫ | Δ | , T because of the contributions from the Fermi sea of the states with negative energies, which extend over the entire Brillouin zone including regions far from the Dirac point. This divergence is spurious and can be eliminated using two equivalent methods. 

First, we note that the divergence of this kind is caused simply by the fact that the overall shift of the particle momentum in the Brillouin zone creates corrections to the wave functions which do not decay as functions of the momentum far from the Dirac points. Let us consider a change in the particle momentum p → p + δ p everywhere in the Brillouin zone. It will lead to the shift p → p + δ p in the functions u and, at the same time, to the shift p → p − δ p in v , because the functions v are associated with the complex conjugated wave functions u taken at the point − K . In this way, the wave functions used in Eq. (7) contain corrections associated with the overall shift p → p + k / 2 in the Brillouin zone. It is thus legitimate to simultaneously change the momentum under the integral in Eq. (26) or Eq. (29) back to its original value p , i.e., to change the blind integration variable p → p − k / 2 in the first and p → p + k / 2 in second term. Excluding this momentum shift we thus remove the diverging part, which is not relevant to the supercurrent. 

Within the linear approximation, it is sufficient to shift the momenta in the zero-order term which comes from 

the first line of Eq. (29). The zero-order term yields 

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Here j (0) K ( p ) and j (0) − $_{K}$( p ) are the currents Eqs. (27) and (28) within the zero-order approximation in k , i.e., with the functions ˆ u (0) α and ˆ v (0) α and the energies E (0) α of states Eqs. (12)–(17) without a current. For these states j (0) K ( p ) + j (0) − $_{K}$( p ) = 0. As a result 

Here we transformed into the surface integral over a re- mote sphere in the momentum space. At these momenta and energies, the current in Eq. (30) does not contain any information on the superconducting properties of the material. This term compensates the divergence of the corrections in the second line of that equation. 

Another way to remove the divergence in the second line of Eq. (29) would be to directly subtract from it the normal-state current which is identically zero for the wave functions specified by Eq. (7). Indeed, as we already mentioned, the diverging contributions to the current come from the regions far from the Dirac point. Since, at these quasiparticle momenta the energies 

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