+ [ a $^{2}$( g e $_{V}$) 2 + b $^{2}$( g e $_{A}$) − 4 a $^{2}$b2 ( a 2 + b $^{2}$) g e $_{V}$g e A ] ( p$_{1}$ · k$_{2}$ )( p$_{2}$ · k$_{1}$ ) + [ a $^{2}$( g e $_{V}$) 2 − b $^{2}$( g e $_{A}$) ] m 2 $_{e}$( k$_{1}$ · k$_{2}$ ) } , (16) Q$_{ν}$$_{¯}$ ν = 4 (2 π ) 8 ∫ d $^{3}$p$_{1}$ 2 E$_{1}$ d $^{3}$p$_{2}$ 2 E$_{2}$ d $^{3}$k$_{1}$ 2 ϵ$_{1}$ d $^{3}$k$_{2}$ 2 ϵ$_{2}$ ( E$_{1}$ + E$_{2}$ ) F$_{1}$F$_{2}$δ $^{(4)}$( p$_{1}$ + p$_{2}$ − k$_{1}$ − k$_{2}$ ) |M| $^{2}$, (17) Q [1] ν ¯ ν = G 2 $_{F}$( a 2 + b $^{2}$) [ a $^{2}$( g e $_{V}$) 2 + b $^{2}$( g e $_{A}$) 2 + 4 a $^{2}$b2 a 2 + b $_{2}$g e $_{V}$g e A ] I$_{1}$, (18) I$_{1}$ = 4 (2 π ) 8 ∫ d $^{3}$p$_{1}$ 2 E$_{1}$ d $^{3}$p$_{2}$ 2 E$_{2}$ d $^{3}$k$_{1}$ 2 ϵ$_{1}$ d $^{3}$k$_{2}$ 2 ϵ$_{2}$ ( E$_{1}$ + E$_{2}$ ) F$_{1}$F$_{2}$δ $^{(4)}$( p$_{1}$ + p$_{2}$ − k$_{1}$ − k$_{2}$ )( p$_{1}$ · k$_{1}$ )( p$_{2}$ · k$_{2}$ ) . (19) ∫ d $^{3}$k$_{1}$ 2 ϵ$_{1}$ d $^{3}$k$_{2}$ 2 ϵ$_{2}$ k α $_{1}$k β $_{2}$δ$^{(4)}$( p$_{1}$ + p$_{2}$ − k$_{1}$ − k$_{2}$ ) = π 24 [ g $^{αβ}$( p$_{1}$ + p$_{2}$ ) 2 + 2( p α 1 + p α $_{2}$)( p β 1 + p β $_{2}$) ] · Θ ( p$_{1}$ + p$_{2}$ ) 2 ] , (20) 7 thus Eq. (19) takes the form The intregration can be performed by using the Lenard formula, namely [36] where I$_{1}$ is explicitly given by From the transition amplitude Eq. (16) and the formula of the emissivity Eq. (17) we obtain where the quantities F$_{1}$$_{,}$$_{2}$ = [1 + exp( E$_{e}$ − ± µ$_{e}$ − ) /T ] − 1 are the Fermi-Dirac distribution functions for e $^{±}$, µ$_{e}$ is the chemical potential for the electrons and T is the temperature (we take K$_{B}$ = 1 for the Boltzmann constant). In this section, we calculate the emissivity associated with neutrino pair production by using Eq. (16). The formula of the emissivity is given by [29–31, 33, 36] where the contribution of the parameters of the LRSM is contained in the constants a and b . Upon evaluating the limit when the mixing angle φ = 0 and a = b = 1, Eq. (16) is thus reduced to the expression to the amplitude given in Refs. [29–35]. III. CALCULATION OF EMISSIVITY