content_image/347007.jpg content_image/347008.jpg content_image/347009.jpg content_image/347010.jpg content_image/347011.jpg
<img src='content_image/347007.jpg'>

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 

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] 

<img src='content_image/347008.jpg'>

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). 

From the transition amplitude Eq. (16) and the formula of the emissivity Eq. (17) we obtain 

<img src='content_image/347009.jpg'>

where I$_{1}$ is explicitly given by 

<img src='content_image/347010.jpg'>

The intregration can be performed by using the Lenard formula, namely [36] 

<img src='content_image/347011.jpg'>

thus Eq. (19) takes the form 

