content_image/348548.jpg content_image/348549.jpg content_image/348550.jpg content_image/348546.jpg content_image/348547.jpg content_image/348551.jpg content_image/348552.jpg
Therefore, Eqs. (18), (22) and (23) are explicitly 

<img src='content_image/348548.jpg'>

<img src='content_image/348549.jpg'>

<img src='content_image/348550.jpg'>

<img src='content_image/348546.jpg'>

<img src='content_image/348547.jpg'>

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 

<img src='content_image/348551.jpg'>

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 . 

## IV. RESULTS AND CONCLUSIONS 

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 ). 

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]: 

<img src='content_image/348552.jpg'>

