ρ < ∼ 10 5 g cm − $^{3}$. These are the typical ranges for very massive stars in their late evolution. Our main objetive in this paper is to provide suitable expressions for the emissivity of pair production of neutrinos via the process e $^{+}$e − → ν ¯ ν in the context of a Left-Right Symmetric Model (LRSM) [21–28] and in a form which can be easily incorporated into realistic supernova models to evaluate the energy lost in the form of neutrinos. The amplitude of transition M in the context of the LRSM can be written as a function of the mixing angle φ between W 3 $_{L}$, W 3 R and B bosons of the model to give the physical Z$_{1}$ and Z$_{2}$ and the photon, being φ the only extra parameter besides the standard model parameters. For which in this paper we choice the Left-Right symmetric model [26–28] to calculate the emissivity of neutrinos in supernova. This paper is organized as follows: In Sect. II we present the calculation of the transition amplitude of the process e $^{+}$e − → ν ¯ ν in the context of a left-right symmetric model. In Sect. III we calculate the emissivity and, finally, we give our results and conclusions in Sec. IV. In this section we obtain the cross section for the Z exchange process i.e., in the limit of a four-fermion electroweak interaction no electromagnetic radiative cor- rections. Here the k$_{i}$ and p$_{i}$ are the particle momenta and λ is the helicity of the neutrino. We recall that within the context of the standard theory, a neutrino interaction eigenstate ( ν$_{L}$ or ν$_{R}$ ) is a superposition of helicity ( λ ) eigenstates ν ± , where λ = σ · p = ± 1. For a relativistic particle, this translates into the statement that a ν$_{L}$ is predominantly in the λ = − 1 state and a ν$_{R}$ is predominantly in the λ = +1 state, with small admixtures of the opposite helicity of the order m/E$_{ν}$ . The amplitude of transition for the process (1) is given by where the constant a and b depend only on the parameters of the LRSM model [26–28] 4 M = g 2 Z 2 M 2 Z [ ¯ u ( k$_{2}$, λ$_{2}$ ) γ $_{µ}$1 2 ( ag ν V − bg ν $_{A}$γ$_{5}$ ) v ( k$_{1}$, λ$_{1}$ ) ] [ ¯ v ( p$_{1}$ ) γ$_{µ}$ 1 2 ( ag e V − bg e $_{A}$γ$_{5}$ ) u ( p$_{2}$ ) ] , (2) e $^{+}$( p$_{1}$ ) + e $^{−}$( p$_{2}$ ) → ¯ ν ( k$_{1}$, λ$_{1}$ ) + ν ( k$_{2}$, λ$_{2}$ ) , (1) II. CROSS SECTION OF THE PROCESS e + + e − → ν + ¯ ν