Stellar energy loss rates beyond the standard model

Abstract

It is known that the dipole moments of the neutrino lead to important astrophysical and cosmological effects. In this regard, within the context of a $U(1)_\mathrm{B-L}$ model, we develop and present novel analytical formulas to assess the effects of the anomalous magnetic moment and electric dipole moment of the neutrino on the stellar energy loss rates through some common physical process of pair-annihilation $e^+e^-\rightarrow (\gamma , Z, Z^{\prime })\rightarrow \nu \bar{\nu }$. Our results show that the stellar energy loss rates strongly depend on the effective magnetic moment of the neutrino, but also on the parameters which characterize the adopted $U(1)_\mathrm{B-L}$ model.

Appendices

Appendix A: Lagrangian of the $U(1)_{B-L}$ model

For the Lagrangian of the $U(1)_\mathrm{B-L}$ model, the terms for the interactions between neutral gauge bosons $Z, Z^{\prime }$ and a pair of fermions of the SM can be written as [21, 39, 40]:

$$\begin{aligned} \mathcal{L}_{NC}=\frac{-ig}{\cos \theta _W}\sum _f{{\bar{f}}}\gamma ^\mu \frac{1}{2}\left( g^f_V- g^f_A\gamma ^5\right) f Z_\mu + \frac{-ig}{\cos \theta _W}\sum _f{{\bar{f}}}\gamma ^\mu \frac{1}{2}\left( g^{\prime f}_V- g^{\prime f}_A\gamma ^5\right) f Z^{\prime }_\mu .\nonumber \\ \end{aligned}$$

(A1)

Thus, the expressions for the new couplings between the $Z, Z^{\prime }$ bosons and the SM fermions are presented in Table 1. As usual, the SM couplings are recovered in the limit when $\theta _\mathrm{B-L}=0$ and $g^{\prime }_1=0$,

Table 1 New couplings of the $Z, Z^{\prime }$ bosons with the SM fermions

Appendix B: Couplings constants

In Eq. (18), we have redefined the coupling constants of the $U\left( 1\right) _\mathrm{B-L}$ model as:

$$\begin{aligned} g_{1}^\mathrm{[B-L]}&=\Bigg [\dfrac{1}{M_Z^4}\Big ( \left( g_V^{e}\right) ^2 +\left( g_A^{e}\right) ^2 \Big )\Big ( \left( g_V^{\nu }\right) ^2 +\left( g_A^{\nu }\right) ^2 \Big )+\dfrac{1}{M_{Z^\prime }^4}\Big ( \left( g_V^{\prime e}\right) ^2 +\left( g_A^{\prime e}\right) ^2 \Big )\nonumber \\&\quad \times \Big ( \left( g_V^{\prime \nu }\right) ^2 +\left( g_A^{\prime \nu }\right) ^2 \Big ) +\dfrac{2}{M_Z^2M_{Z^\prime }^2}\big (g_V^{e}g_V^{\prime e}+g_A^{e}g_A^{\prime e}\big )\big ( g_V^{\nu }g_V^{\prime \nu }+g_A^{\nu }g_A^{\prime \nu }\big )\Bigg ], \end{aligned}$$

(B1)

$$\begin{aligned} g_{2}^\mathrm{[B-L]}&=\Bigg [\dfrac{1}{M_Z^4}\Big ( \left( g_V^{e}\right) ^2 -\left( g_A^{e}\right) ^2 \Big )\Big ( \left( g_V^{\nu }\right) ^2 +\left( g_A^{\nu }\right) ^2 \Big )+\dfrac{1}{M_{Z^\prime }^4}\Big ( \left( g_V^{\prime e}\right) ^2 -\left( g_A^{\prime e}\right) ^2 \Big )\nonumber \\&\quad \times \Big ( \left( g_V^{\prime \nu }\right) ^2 +\left( g_A^{\prime \nu }\right) ^2 \Big ) +\dfrac{2}{M_Z^2M_{Z^\prime }^2}\big (g_V^{e}g_V^{\prime e}-g_A^{e}g_A^{\prime e}\big )\big ( g_V^{\nu }g_V^{\prime \nu }+g_A^{\nu }g_A^{\prime \nu }\big )\Bigg ]. \end{aligned}$$

(B2)