I nm 1 = m n + m +8 e 6(2 π ) 5 [ 3 G − n $_{2}$G + m 2 + 2 G − n +1 2 G + m +1 2 + G − n$_{−}$ 1 2 G + m$_{−}$ 1 2 + 4 9 ( G − n +1 2 − G − n$_{−}$ 1 2 ) ( G + m +1 2 − G + m$_{−}$ 1 2 )] , (31) I nm 3 = m n + m +6 e (2 π ) 5 [ G − n$_{−}$ 1 2 G + m$_{−}$ 1 2 + G − n $_{2}$G + m 2 ] . (32) Q [1] ν ¯ ν = G 2 $_{F}$( a 2 + b $^{2}$) [ a $^{2}$( g e $_{V}$) 2 + b $^{2}$( g e $_{A}$) 2 + 4 a $^{2}$b2 a 2 + b $_{2}$g e $_{V}$g e A ] [ I 10 1 + I 01 1 ] , (33) Q [2] ν ¯ ν = G 2 $_{F}$( a 2 + b $^{2}$) [ a $^{2}$( g e $_{V}$) 2 + b $^{2}$( g e $_{A}$) 2 − 4 a $^{2}$b2 a 2 + b $_{2}$g e $_{V}$g e A ] [ I 10 2 + I 01 2 ] , (34) Q [3] ν ¯ ν = G 2 $_{F}$( a 2 + b $^{2}$) [ a $^{2}$( g e $_{V}$) 2 − b $^{2}$( g e $_{A}$) 2 ] m 2 e [ I 10 3 + I 01 3 ] . (35) Q LRSM ν ¯ ν ( φ, β ) = Q [1] ν ¯ ν ( φ, β ) + Q [2] ν ¯ ν ( φ, β ) + Q [3] ν ¯ ν ( φ, β ) , (36) − 1 . 6 × 10 − 3 ≤ φ ≤ 1 . 1 × 10 − $^{3}$, (37) 9 For the mixing angle φ of the left-right symmetric model, we use the reported data of A. Guti´ errez-Rodr´ıguez, et al. [37]: The numerical result on the emissivity of neutrino pair production via the process e $^{+}$e − → ν ¯ ν as a function of the mixing angle φ and the generation parameter β are present in this section. For our analysis we consider the following data: the Fermi constant G$_{F}$ = 1 . 166 × 10 − 5 GeV − $^{2}$, angle of Weinberg sin$^{2}$θ$_{W}$ = 0 . 223 and the electron mass m$_{e}$ = 0 . 51 MeV , thereby obtaining the emissivity of the neutrinos Q LRSM ν ¯ ν = Q LRSM ν ¯ ν ( φ, β ). where the dependence of the φ mixing parameter of the LRSM is contained in the constants a and b , while the dependence of the β degeneration parameter is contained in the Fermi integrals G ± s . Finally, the expression for the emissivity of neutrino pair production via the process e $^{+}$e − → ν ¯ ν in the context of a left-right symmetric model is given by IV. RESULTS AND CONCLUSIONS Therefore, Eqs. (18), (22) and (23) are explicitly