a = cos φ − sin φ √$_{cos 2}$$_{θ}$$_{W}$ and b = cos φ + √ cos 2 θ$_{W}$ sin φ, (3) 1 2 × 1 2 |M| 2 = G 2 F 8 N $^{µν}$E$_{µν}$ , (4) N µν = 1 4 T r [( k /$_{2}$ + m$_{ν}$ )(1 + γ $^{5}$s /$_{2}$) γ $^{µ}$( a − bγ$_{5}$ )( k /$_{1}$ − m$_{ν}$ )(1 + γ $^{5}$s /$_{1}$) γ $^{ν}$( a − bγ$_{5}$ )] , (5) E$_{µν}$ = 1 4 T r [( p /$_{2}$ + m$_{e}$ ) γ$_{µ}$ ( ag e V − bg e $_{A}$γ$_{5}$ )( p /$_{1}$ − m$_{e}$ ) γ$_{ν}$ ( ag e V − bg e $_{A}$γ$_{5}$ )] . (6) s$_{i}$ · s$_{i}$ = − 1; s$_{i}$ · k$_{i}$ = 0; (7) s$_{i}$ ‖ λ$_{i}$ k$_{i}$ for i = 1 , 2 (8) k µ = ( E$_{ν}$, k ˆ k ) , (9) K µ 1 = k µ 1 + m$_{ν}$s µ $_{1}$; K µ 2 = k µ 2 − m$_{ν}$s µ $_{2}$. (10) 5 In conjunction with the properties given in Eqs. (7) and (8), these will allow us to write the amplitude squared for the process under consideration in a compact and physically holds, where and for a relativistic neutrino, the additional constraint Here s$_{1}$ and s$_{2}$ are the spin four-vectors associated with the antineutrino and neutrino respectively, while m$_{ν}$ and m$_{e}$ are the neutrino and electron mass. These spin vectors satisfy the Lorentz invariant conditions where where φ is the mixing parameter of the LRSM [26–28], u and v are the usual Dirac spinors, and the electron and positron helicity indexes have been suppressed since they will be aver- aged over. We then write with ˆ k being a unit vector along the three-momentum of the neutrino. We now introduce two four-vectors associated with the neutrino pair