# Neutrino Energy Loss Rates in 3-3-1 Models

D. T. Binh<EMAIL_ADDRESS>Institute for Theoretical and
Applied Research
Faculty of Natural Science, Duy Tan University
L. T. Hue<EMAIL_ADDRESS>Institute of Physics, Vietnam Academy of
Science and Technology, 10 Dao Tan, Ba Dinh, Hanoi, Vietnam  V. H. Binh
<EMAIL_ADDRESS>Institute of Physics, Vietnam Academy of Science and
Technology, 10 Dao Tan, Ba Dinh, Hanoi, Vietnam  Graduate University of
Science and Technology, Vietnam Academy of Science and Technology, 18 Hoang
Quoc Viet, Cau Giay, Hanoi, Vietnam  H. N. Long<EMAIL_ADDRESS>Institute of Physics, Vietnam Academy of Science and Technology, 10 Dao Tan,
Ba Dinh, Hanoi, Vietnam

###### Abstract

The stellar energy-loss rates $\mathcal{Q}$ due to the production of neutrino
pair in the framework of 3-3-1 models are presented. The energy loss rate
$\mathcal{Q}$ is evaluated for different values of
$\beta=\pm\frac{1}{\sqrt{3}},\pm\frac{2}{\sqrt{3}},\pm\sqrt{3}$ in which
$\beta$ is a parameter used to define the charge operator in the 3-3-1 models.
The correction to the rate which is compared with that of the Standard Model
($\delta\mathcal{Q}$) is also evaluated. We show that the correction does not
exceed 14% and gets the highest with $\beta=-\sqrt{3}$. The contribution of
dipole moment to the energy loss rate is small compared to the contribution of
new natural gauge boson $Z^{\prime}$ and this sets constraints for the mass of
Z’ $m_{Z^{\prime}}\leq 4000$ GeV. This mass range is within the searching
range for $Z^{\prime}$ boson at LHC.

###### pacs:

14.60.St, 13.40.Em, 12.15.Mm
Keywords: Non-standard-model neutrinos, Electric and Magnetic Moments, Neutral
Currents, models beyond the standard model.

##  I Introduction

Cooling rate is an important factor that affects the stellar evolution. In
their life time, stars can emit energy in the form of radiation, flux of
neutrinos Particle-Astro ; Axion-Eloss-Raffelt ; neutrino-Eloss-Beaudet ;
neutrino-Eloss-esposito2002 ; neutrino-Eloss-esposito2003 ; Neutrino-Gamow-1 ;
Neutrino-Gamow-2 ; Neutrino-Pontecorvo , gravitational wave Gwave-emission-1 ;
Gwave-emission-2  and axion Axion-Flux-1 ; Axion-Flux-2 ; Axion-Flux-3 . It is
well known that cooling by neutrino emission plays an important role in a
variety of stellar systems, from neutron stars and core-collapse supernovae
(CCSNe), from low-mass red giants (RG) and horizontal branch (HB) stars to
white dwarfs (WDs). Despite of the fact that neutrinos interact extremely
weakly, once produced they easily escape from stellar interiors carrying away
energy. The neutrino production is main process (99%) while photoproduction is
tiny one (1-2 %) in a massive star collapse.

There are mainly four interaction mechanisms for the energy loss due to
neutrino emissions: (i) pair annihilation
$e^{+}+e^{-}\rightarrow\nu+\bar{\nu}$; (ii) $\nu$ photoproduction
$\gamma+e^{\pm}\rightarrow e^{\pm}+\nu+\bar{\nu}$ ; (iii) plasmon decay
$\gamma^{*}\rightarrow\nu+\bar{\nu}$; and (iv) Bremsstrahlung on nuclei
$e^{\pm}+Z\rightarrow e^{\pm}+Z+\nu+\bar{\nu}$. Each of these processes will
give a dominant contribution to the energy loss rate ($\mathcal{Q}$) in a
particular region of temperature and density ( corresponding to a certain
evolution period of the star. The pair annihilation process dominates in a
high temperature ($T\geq 10^{9}$ K) and not too high density ($\rho$). The
$\nu$ photoproduction on the other hand give leading contribution in regions
where $10^{8}\textrm{K}\leq T\leq 10^{10}$ K and low densities ($\rho\leq
10^{5}\;\mathrm{gcm^{-3}}$). Finally, plasmon decay and bremsstrahlung on
nuclei are dominants process for large ($\rho\geq 10^{6}\;\mathrm{gcm^{-3}}$)
and very large ($\rho\geq$$10^{9}gcm^{-3}$) core densities respectively with
temperature in range $10^{8}\textrm{K}\leq T\leq 10^{10}$K.

In the Standard Model (SM) frame work SM-Glashow ; SM-Weinberg ; SM-Salam ,
the energy loss rate (ELR) due to neutrino emission ($\mathcal{Q}^{\nu}$)
receives contributions from both weak nuclear reactions and purely leptonic
processes. However, in many models beyond the SM (BSM) new interactions or new
contributions from new particles can change the rate at which neutrinos are
produced therefore the evolution of star may be modified. The stellar energy
loss rate was calculated in the frame works of the SM QSM-Dicus1972  and the
$U(1)_{B-L}$ extension model of SM QBSM-Hernandez .

In the SM, neutrinos are massless therefore neutrinos photon interaction at
tree level do not exits. However, neutrinos oscillation experiment SNO ; ATM
imply that neutrinos do have mass. In some beyond SM models, neutrino can be
massive. Consequently, there exist dipole moments. The interaction of neutrino
with photon via dipole moment can affect the ELR Dipole-Eloss-Heger2009 ;
Dipole-Eloss-Kerimov1992 ; QBSM-Hernandez .

New natural gauge boson $Z^{\prime}$ appears naturally in some extension
models of the SM such as the Left-Right symmetric model G.Senjanovic ;
G.Senjanovic1 , the model of composite boson Baur . One of the simplest and
attractive extension of the SM is the $SU(3)_{L}$ extension of the SM ppf-1 ;
ppf-2 ; ppf-3 ; r331-1 ; r331-2 ; r331-3 ; r331-4 ; r331-5 ; 331beta1 ;
331beta2 , where the SM fermion doublets are assigned as $SU(3)_{L}$ triplets
or antitriplets including new exotic fermions or positrons in the third
components of the $SU(3)_{L}$ (anti) triplets. In this work we pay attention
to the 3-3-1 model with an arbitrary parameter $\beta$ (3-3-1$\beta$)
containing exotic fermions with electric charges defined by the charge
operator characterized by $\beta$. In general the class of 331 models have the
same characteristics as follows: 1) The anomaly in 3-3-1 model is canceled
when all fermion generations are considered, 2) Peccei-Quinn (PQ) symmetry
Peccei-Quin  is a result of gauge invariance in the model 3) As the extension
of the gauge group there appears new natural gauge boson $Z^{\prime}$, 4) One
generation of quark is different from the other two ones, leading to the
appearance of the tree level Flavor Changing Neutral Current (FCNC) through
the mixing $Z-Z^{\prime}$. Also, the interactions of the $Z^{\prime}$ and
neutrinos will affect the production rates of neutrinos and modify the energy
loss rate predicted by the SM.

There are may works on the stellar energy loss in the frame work of the SM
QSM-Dicus1972  and extension models of the SM QBSM-Hernandez ; QBSM-Bugarin .
In this work we will investigate the effect of magnetic dipole moment and new
$Z^{\prime}$ boson on the ELR of a stellar. In our work, we will investigate
the ELR through the process
$e^{+}e^{-}\rightarrow(\gamma,W,Z,Z^{\prime})\rightarrow\nu\bar{\nu}$. We
investigate the energy loss rate of the $331\beta$ model and its relative
correction compared with the SM.

Our work is organized as follows: in section II we will briefly review the
$331\beta$. In section III we will calculate the
$e^{+}e^{-}\rightarrow\nu\overline{\nu}$ amplitude, derive its analytical
approximation in different limits. Lastly, section IV and V are the numerical
discussion and conclusion.

##  II  The model 3-3-1$\beta$

The model 3-3-1$\beta$ is constructed based on the gauge group
$SU(3)_{c}\times SU(3)_{L}\times U(1)_{X}$. One common feature of the class of
$SU(3)_{L}$ model is that the extension of the gauge group from
$SU(2)_{L}\rightarrow SU(3)_{L}$ requires new fermions. Normaly, the left-
handed fermions are arranged into the third components of the triplets, while
the right ones are in the $SU(3)_{L}$ singlets. The anomaly cancellation
requires that the number of fermion triplets equals the number of fermion
antitriplets, leading to the consequence that one quark family must have the
same $SU(3)_{L}$ representation as the three lepton families and different
from the remaining quark families. The electric charges of all particles in
the 3-3-1$\beta$ are determined by the following charge operator

$Q=I_{3}+\beta\,I_{8}+X\,,$ (1)

where $I_{3}$, $I_{8}$ are the $SU(3)$ generators. The models are
characterized by the parameter $\beta$ in the charge operator $Q$. The lepton
representation can be represented as follows 331beta1 ; 331beta2 :

$\displaystyle L^{\prime}_{aL}=\left(\begin{array}[]{c}e^{\prime}_{a}\\\
-\nu^{\prime}_{a}\\\ E^{\prime}_{a}\\\
\end{array}\right)_{L}\sim\left(1,3^{*}~{},-\frac{1}{2}+\frac{\beta}{2\sqrt{3}}\right),\hskip
14.22636pta=1,2,3,$ (5) $\displaystyle
e^{\prime}_{aR}\sim\left(1,1~{},-1\right),\hskip
14.22636pt\nu^{\prime}_{aR}\sim\left(1,~{}1~{},0\right),\hskip
14.22636ptE^{\prime}_{aR}\sim\left(1,~{}1~{},-\frac{1}{2}+\frac{\sqrt{3}\beta}{2}\right).$
(6)

In particular, the left-handed leptons are assigned to anti-triplets while the
right-handed leptons to singlets. The model predicts three exotic leptons
$E^{\prime a}_{L,R}$ which are much heavier than the ordinary leptons. The
right-handed neutrinos $\nu^{\prime}_{aR}$ is needed to generate Dirac mass
for active neutrinos The prime denotes flavor states to be distinguished with
mass eigenstates will be introduced later. The numbers in the parentheses are
to label the representation of $SU(3)_{C}\otimes SU(3)_{L}\otimes U(1)_{X}$
group.

For our purpose of this work, the quark sector is irrelevant therefore we do
not present it here. It has been discussed in details in many previous works
331beta1 ; 331beta2 ; 331beta3 ; Buras:2012jb .

The detail calculation of gauge and Higgs interactions has been shown in
331beta1 ; 331beta2 ; 331beta3 ; Buras:2012jb . Within nine EW gauge bosons,
the covariant derivative is defined as follows

$D_{\mu}\equiv\partial_{\mu}-igI^{a}W^{a}_{\mu}-ig_{X}XI^{9}X_{\mu},$ (7)

where $I^{9}=\frac{\mathbb{1}}{\sqrt{6}}$, $g$ and $g_{X}$ are coupling
constants corresponding to the two groups $SU(3)_{L}$ and $U(1)_{X}$,
respectively. The matrix $W^{a}I^{a}$ for a triplet can be written as

$\displaystyle
W^{a}_{\mu}I^{a}=\frac{1}{2}\left(\begin{array}[]{ccc}W^{3}_{\mu}+\frac{1}{\sqrt{3}}W^{8}_{\mu}&\sqrt{2}W^{+}_{\mu}&\sqrt{2}Y^{+A}_{\mu}\\\
\sqrt{2}W^{-}_{\mu}&-W^{3}_{\mu}+\frac{1}{\sqrt{3}}W^{8}_{\mu}&\sqrt{2}V^{+B}_{\mu}\\\
\sqrt{2}Y^{-A}_{\mu}&\sqrt{2}V^{-B}_{\mu}&-\frac{2}{\sqrt{3}}W^{8}_{\mu}\\\
\end{array}\right),$ (11)

where we have denoted the charged gauge bosons as

$\displaystyle W^{\pm}_{\mu}=\frac{1}{\sqrt{2}}\left(W^{1}_{\mu}\mp
iW^{2}_{\mu}\right),$ $\displaystyle Y^{\pm
A}_{\mu}=\frac{1}{\sqrt{2}}\left(W^{4}_{\mu}\mp iW^{5}_{\mu}\right),$
$\displaystyle V^{\pm B}_{\mu}=\frac{1}{\sqrt{2}}\left(W^{6}_{\mu}\mp
iW^{7}_{\mu}\right).$ (12)

From (1), the electric charges of the gauge bosons are given by

$\displaystyle A=\frac{1}{2}+\beta\frac{\sqrt{3}}{2},\hskip
14.22636ptB=-\frac{1}{2}+\beta\frac{\sqrt{3}}{2}.$ (13)

The scalar sector contains three scalar triplets as follows

$\displaystyle\chi=\left(\begin{array}[]{c}\chi^{+A}\\\ \chi^{+B}\\\
\chi^{0}\\\
\end{array}\right)\sim\left(1,3~{},\frac{\beta}{\sqrt{3}}\right),\hskip
14.22636pt\eta=\left(\begin{array}[]{c}\eta^{0}\\\ \eta^{-}\\\ \eta^{-A}\\\
\end{array}\right)\sim\left(1,3~{},-\frac{1}{2}-\frac{\beta}{2\sqrt{3}}\right)$
(20) $\displaystyle\rho=\left(\begin{array}[]{c}\rho^{+}\\\ \rho^{0}\\\
\rho^{-B}\\\
\end{array}\right)\sim\left(1,3~{},\frac{1}{2}-\frac{\beta}{2\sqrt{3}}\right),$
(24)

where $A,B$ denote electric charges as determined in (13). Only the vacuum
expectation values (VEV) of the neutral Higgs components are non zero and
defined as follows: $\langle\chi^{0}\rangle=\omega/\sqrt{2}$,
$\langle\rho^{0}\rangle=v/\sqrt{2}$, and $\langle\eta^{0}\rangle=u/\sqrt{2}$.
They are enoungh to generate masses for all particles in the model.

As usual, the symmetry breaking happens in two steps: $SU(3)_{L}\otimes
U(1)_{X}\xrightarrow{\omega}SU(2)_{L}\otimes
U(1)_{Y}\xrightarrow{v,u}U(1)_{Q}$. Therefore, it is reasonable to assume that
$\omega\gg v,u$. There are well-known relations between the gauge couplings of
the 3-3-1$\beta$ model and the SM, namely

$g_{2}=g,\hskip
14.22636pt\frac{g_{X}^{2}}{g^{2}}=\frac{6s_{W}^{2}}{1-(1+\beta^{2})s_{W}^{2}},,$
(25)

where $g_{2}$ and $g_{1}$ are the couplings corresponding to $SU(2)_{L}$ and
$U(1)_{Y}$ subgroups, respectively. The weak mixing angle is defined as
$\sin\theta_{W}\equiv s_{W}$, $\tan\theta_{W}\equiv
t_{W}=\frac{g_{1}}{g_{2}}$, and so forth.

The equation in (25) leads to an interesting constraint of the parameter
$\beta$:

$\displaystyle|\beta|\leq\sqrt{3}\,.$ (26)

With the above VEVs, the charged gauge boson masses are

$\displaystyle m^{2}_{Y^{\pm A}}=\frac{g^{2}}{4}(\omega^{2}+u^{2}),\hskip
14.22636ptm^{2}_{V^{\pm B}}=\frac{g^{2}}{4}(\omega^{2}+v^{2}),\hskip
14.22636ptm^{2}_{W}=\frac{g^{2}}{4}(v^{2}+u^{2})\,.$ (27)

Let us now discuss the mixings of leptons. In the 3-3-1$\beta$ with heavy
exotic leptons, we will ignore the mixing betweent the SM leptons and these
new leptons in the case that they have the same electric charges. The Yukawa
interations related to the above mentioned mixings given by

$\displaystyle-\mathcal{L}^{\text{Yuk}}_{\mathrm{lepton}}=Y^{e}_{ab}\overline{L^{\prime}}_{aL}\eta^{*}e^{\prime}_{bR}+Y^{\nu}_{ab}\overline{L^{\prime}}_{aL}\rho^{*}\nu^{\prime}_{bR}+Y^{E}_{ab}\overline{L^{\prime}}_{aL}\chi^{*}E^{\prime}_{bR}+\mathrm{H.c.},$
(28)

where $a,b=e,\mu,\tau$ are family indices. This Largangian generates
consistent masses for leptons. Hereafter, without loss of generality, ones
will work in the basis where the SM charged leptons are in their mass
eigenstates. Ones can therefore set $Y^{e}_{ab}$ to be diagonal and
$e^{\prime}=e$ in Eqs. (28). The SM lepton masses are
$m_{e_{a}}=Y^{e}_{aa}u/\sqrt{2}$. The mass term and mixing of the Dirac
neutrinos are derived as follows:

$\displaystyle-\mathcal{L}^{\text{mass}}_{\nu}=\frac{Y^{\nu}_{ab}v}{\sqrt{2}}\overline{\nu^{\prime}}_{aL}\nu^{\prime}_{bR}+\mathrm{H.c.},\hskip
14.22636pt\nu^{\prime}_{aL}=U^{L}_{ab}\nu_{bL},\hskip
14.22636pt\nu^{\prime}_{aR}=U^{R}_{ab}\nu_{bR},$ (29)

where $U^{L,R}$ are $3\times 3$ unitary matrices the neutrinos, respectively.
It can be identified that $U^{L}=U_{\text{PMNS}}$ are the well-known lepton
mixing matrix.

The Higgs sector does not involve our work. The mass and eigenstates of the
all Higgs in the 3-3-1$\beta$ model were given detailedly previously, for
example ref. 331beta2 . Hence, we will not repeat here. We stress that the
scalar sector contains six charged Higgs bosons, one neutral pseudoscalar
Higgs and three neutral scalar Higgs bosons, which one of them can be
identified with the standard model-like Higgs found by experiments at LHC.

The neutral currents mediated by $Z$ and $Z^{\prime}$ bosons relating with the
lepton sector used in our calculation are given by:

$i\,L_{int}^{Z}=\frac{ig}{2c_{W}}Z^{\mu}\sum_{\ell=e,\mu,\tau}\left[{\bar{\nu}}_{\ell\,L}\gamma_{\mu}\nu_{\ell
L}-(1-2s_{W}^{2}){\bar{\ell}}_{L}\gamma_{\mu}\ell_{L}+2s_{W}^{2}{\bar{\ell}}_{R}\gamma_{\mu}\ell_{R}\right],$
(30)

and

$\displaystyle i\,L_{int}^{Z^{\prime}}=$ $\displaystyle
i\frac{gZ^{\prime\mu}}{2\sqrt{3}c_{W}\sqrt{1-(1+\beta^{2})s_{W}^{2}}}$
$\displaystyle\times\sum_{\ell=e,\mu,\tau}\left\\{\left[1-(1+\sqrt{3}\beta)s_{W}^{2}\right]\left({\bar{\nu}}_{\ell\,L}\gamma_{\mu}\nu_{\ell\,L}+{\bar{\ell}}_{L}\gamma_{\mu}\ell_{L}\right)-2\sqrt{3}\beta
s_{W}^{2}{\bar{\ell}}_{R}\gamma_{\mu}\ell_{R}\right\\}.$ (31)

The common $V-A$ form of the interaction of the Z bosons with $e,\nu_{e}$ are
331beta3

$\mathcal{L}_{Z^{i}ff}=\frac{g}{2c_{W}}\bar{f}\gamma_{\mu}[g^{Z^{i}}_{V}(f)-g^{Z^{i}}_{A}(f)\gamma_{5}]fZ_{i}^{\mu}$
(32)

where $Z_{i}=Z,Z^{\prime}$ and the $g^{Z^{i}}_{V}(f)$, $g^{Z^{i}}_{A}(f)$ are
given in Table 1.

The common $V-A$ form of the Lagrangian of the interactions of the neutral
gauge bosons with $e,\nu_{e}$ are

$\mathcal{L}_{Z^{i}ff}=\frac{g}{2c_{W}}\bar{f}\gamma^{\mu}[g^{Z^{i}}_{V}(f)-g^{Z^{i}}_{A}(f)\gamma_{5}]fZ^{i}_{\mu}\,,$
(33)

where $Z^{i}=Z,Z^{\prime}$. The $g^{Z^{i}}_{V}(f)$, $g^{Z^{i}}_{A}(f)$ are
given in Table 1.

$f|Z^{i}$ | $g^{Z}_{V}(f)$ | $g^{Z}_{A}(f)$ | $g^{Z^{\prime}}_{V}(f)$ | $g^{Z^{\prime}}_{A}(f)$
---|---|---|---|---
$\nu$ | $\frac{1}{2}$ | $\frac{1}{2}$ | $\frac{1}{2\sqrt{3}}\frac{[1-(1+\sqrt{3}\beta)s_{w}^{2}]}{\sqrt{1-(1+\beta^{2})s_{w}^{2}}}$ | $\frac{1}{2\sqrt{3}}\frac{[1-(1+\sqrt{3}\beta)s_{w}^{2}]}{\sqrt{1-(1+\beta^{2})s_{w}^{2}}}$
$e$ | $-\frac{1}{2}+2s^{2}_{W}$ | $-\frac{1}{2}$ | $\frac{1}{2\sqrt{3}}\frac{[1-(1+3\sqrt{3}\beta)s_{w}^{2}]}{\sqrt{1-(1+\beta^{2})s_{w}^{2}}}$ | $\frac{1}{2\sqrt{3}}\frac{[1-(1-\sqrt{3}\beta)s_{w}^{2}]}{\sqrt{1-(1+\beta^{2})s_{w}^{2}}}$

Table 1: The couplings of $Z$ and $Z^{\prime}$ with $e,\nu_{e}$.

##  III Stellar ELRs through annihilation of electron-positron pair into
electron neutrino and antineutrino

Let us consider the process inside star, namely the annihilation of electron-
positron pair into electron neutrino and antineutrino:

$e^{+}\left(p_{1}\right)e^{-}\left(p_{2}\right)\rightarrow(\gamma,W,Z_{a})\to\nu_{i}\left(k_{1},\lambda_{3}\right)\bar{\nu}_{i}\left(k_{2},\lambda_{4}\right).$
(34)

The Feynman diagrams contributing to the process given by Eq. (34) are shown
in Fig.1 where $Z_{a}=Z,Z^{\prime}$ and
$\nu_{i}=\nu_{e},\nu_{\mu},\nu_{\tau}$, respectively. Here, $p_{1},p_{2}$ are
the momentum of the incoming electron, positron and $k_{1},k_{2}$ are the
momentum of the outgoing $\nu\bar{\nu}$ pair while $\lambda_{3,4}$ is the
neutrino helicity.

Figure 1: Feynman diagrams contributing to $e^{+}e^{-}\rightarrow\nu\bar{\nu}$

In Fig.1, the coupling between neutrinos and photon arises at the quantum
level from loop diagrams. The general expression for the electromagnetic form
factors of a massive neutrino is as follows Neutrino-FFactor-Kayser ;
Neutrino-FFactor-Nieves ; Neutrino-FFactor-Broggini ; EMproperties-Neutrino

$\Gamma^{\mu}(q)=eF_{1}(q^{2})\gamma^{\mu}+\frac{ie}{2m_{\nu}}F_{2}(q^{2})\sigma^{\mu\nu}q_{\nu}+\frac{e}{2m_{\nu}}F_{3}(q^{2})\gamma_{5}\sigma^{\mu\nu}q_{\nu}+eF_{4}(q^{2})\gamma_{5}\left(\gamma^{\mu}-\frac{q\hbox
to0.0pt{\hss/}q^{\mu}}{q^{2}}\right),$ (35)

where $q^{\mu}$ is the photon momentum, and $F_{i}(q^{2}),\,,\,i=1,2,3,4$ are
the electromagnetic form factors of the neutrino. In this analysis, we are
interested in the anomalous magnetic moment (AMM) and the electric dipole
moment (EDM) of the neutrino, which are defined in terms of the $F_{2}$ and
$F_{3}$ form factors at $q^{2}=0$ as follows:

$\mu_{\nu}=\frac{m_{e}}{m_{\nu}}F_{2}|_{q^{2}=0},\hskip
14.22636ptd_{\nu}=\frac{e}{2m_{\nu}}F_{3}|_{q^{2}=0}\,.$ (36)

###  III.1 Low energy amplitudes

For our purpose, the low energy limit where the propagator of gauge bosons
takes a form $\frac{-ig_{\mu\nu}}{M^{2}}$ is applicable. In the mentioned
limit, the amplitudes are thus given by:

$\displaystyle i\mathcal{M}^{\gamma}_{fi}$ $\displaystyle=$
$\displaystyle\bar{u}\left(k_{2},\lambda_{4}\right)\Gamma^{\mu}v\left(k_{1},\lambda_{3}\right)\frac{g_{\mu\nu}}{\left(p_{1}+p_{2}\right)^{2}}\bar{v}\left(p_{1}\right)e\gamma^{\nu}u\left(p_{2}\right),$
(37) $\displaystyle i\mathcal{M}^{Z}_{fi}$ $\displaystyle=$
$\displaystyle-\left(\frac{g^{2}}{8\cos^{2}{\theta_{W}}M_{Z}^{2}}\right)\bar{u}\left(k_{2},\lambda_{4}\right)\gamma^{\mu}\left[1-\gamma_{5}\right]v\left(k_{1},\lambda_{3}\right)$
(38) $\displaystyle\times$
$\displaystyle\bar{v}\left(p_{1}\right)\gamma_{\mu}\left[g_{V}^{Z}(e)-{g_{A}^{Z}(e)}\gamma_{5}\right]u\left(p_{2}\right)$
$\displaystyle i\mathcal{M}^{W}_{fi}$ $\displaystyle=$
$\displaystyle-\left(\frac{g^{2}}{8M_{W}^{2}}\right)\bar{u}\left(k_{2},\lambda_{4}\right)\gamma^{\mu}\left[1-\gamma_{5}\right]u\left(k_{1},\lambda_{3}\right)$
(39) $\displaystyle\times$
$\displaystyle\bar{v}\left(p_{1}\right)\gamma_{\mu}\left[1-\gamma_{5}\right]v\left(p_{2}\right)$
$\displaystyle i\mathcal{M}^{Z^{\prime}}_{fi}$ $\displaystyle=$
$\displaystyle-\left(\frac{g^{2}}{4\cos^{2}{\theta_{W}}M_{Z^{\prime}}^{2}}\right)\bar{u}\left(k_{2},\lambda_{4}\right)\gamma^{\mu}\left[g_{V}^{Z^{\prime}}(\nu)-{g_{A}^{Z^{\prime}}(\nu)}\gamma_{5}\right]v\left(k_{1},\lambda_{3}\right)$
(40) $\displaystyle\times$
$\displaystyle\bar{v}\left(p_{1}\right)\gamma_{\mu}\left[g_{V}^{Z^{\prime}}(e)-{g_{A}^{Z^{\prime}}(e)}\gamma_{5}\right]u\left(p_{2}\right)\,.$

Using Fierz transformation generalized-Fierz  and summing over (38) and (39)
we have

$\displaystyle i\mathcal{M}^{SM}_{fi}$ $\displaystyle=$
$\displaystyle-\left(\frac{g^{2}}{8\cos^{2}{\theta_{W}}M_{Z}^{2}}\right)\bar{u}\left(k_{2},\lambda_{4}\right)\gamma^{\mu}\left[1-\gamma_{5}\right]v\left(k_{1},\lambda_{3}\right)$
(41) $\displaystyle\times$
$\displaystyle\bar{v}\left(p_{1}\right)\gamma_{\mu}\left[C_{V}(e)-C_{A}(e)\gamma_{5}\right]u\left(p_{2}\right)$

where $C_{V}(e)=1+g_{V}^{Z}(e)=\frac{1}{2}+2\sin^{2}{\theta_{W}}$ and
$C_{A}(e)=1+g_{A}^{Z}(e)=\frac{1}{2}$.

The final amplitude is

$\mathcal{M}_{fi}^{331\beta}=\mathcal{M}_{fi}^{SM}+\mathcal{M}_{fi}^{\gamma}+\mathcal{M}_{fi}^{Z^{\prime}}\,.$
(42)

Squaring (42), summing and and taking average over spin of particles in the
final and initial states, respectively, we get:

$\displaystyle\frac{1}{4}\sum_{s}\left|\mathcal{M}_{fi}\right|^{2}=\frac{\left(4\pi\alpha\right)^{2}}{\sin^{4}{2\theta_{W}}}\Bigg{\\{}\frac{\sin^{4}{2\theta_{W}}}{\left(4\pi\alpha\right)}(\mu_{\nu}^{2}+d_{\nu}^{2})\Big{[}\frac{(p_{1}\cdot
p_{2}+m_{e}^{2})(p_{1}\cdot k_{2}+m_{e}^{2})}{s}\Big{]}$
$\displaystyle+\Bigg{[}\bigg{(}\frac{1}{M_{Z}^{2}}\big{[}C_{V}(e)-C_{A}(e)\big{]}+\frac{1}{M_{Z^{\prime}}^{2}}\big{[}g_{V}^{Z^{\prime}}(e)-g_{A}^{Z^{\prime}}(e)\big{]}\big{[}g_{V}^{Z^{\prime}}(\nu)+g_{A}^{Z^{\prime}}(\nu)\big{]}\bigg{)}^{2}$
$\displaystyle+\bigg{(}\frac{1}{M_{Z^{\prime}}^{2}}\big{[}g_{V}^{Z^{\prime}}(e)+g_{A}^{Z^{\prime}}(e)\big{]}\big{[}g_{V}^{Z^{\prime}}(\nu)-g_{A}^{Z^{\prime}}(\nu)\big{]}\bigg{)}^{2}\Bigg{]}\times\left(p_{1}\cdot
k_{1}\right)\left(p_{2}\cdot k_{2}\right)$
$\displaystyle+\Bigg{[}\bigg{(}\frac{1}{M_{Z}^{2}}\big{[}C_{V}(e)+C_{A}(e)\big{]}+\frac{1}{M_{Z^{\prime}}^{2}}\big{[}g_{V}^{Z^{\prime}}(e)+g_{A}^{Z^{\prime}}(e)\big{]}\big{[}g_{V}^{Z^{\prime}}(\nu)+g_{A}^{Z^{\prime}}(\nu)\big{]}\bigg{)}^{2}$
$\displaystyle+\bigg{(}\frac{1}{M_{Z^{\prime}}^{2}}\big{[}g_{V}^{Z^{\prime}}(e)-g_{A}^{Z^{\prime}}(e)\big{]}\big{[}g_{V}^{Z^{\prime}}(\nu)-g_{A}^{Z^{\prime}}(\nu)\big{]}\bigg{)}^{2}\Bigg{]}\times\left(p_{1}\cdot
k_{2}\right)\left(p_{2}\cdot k_{1}\right)$
$\displaystyle+2\Bigg{[}\frac{1}{2M_{Z}^{4}}\Big{[}\left(C_{V}(e)\right)^{2}-\left(C_{A}(e)\right)^{2}\Big{]}+\frac{1}{M_{Z^{\prime}}^{4}}\Big{[}\left(g_{V}^{Z^{\prime}}(e)\right)^{2}-\left(g_{A}^{Z^{\prime}}(e)\right)^{2}\Big{]}$
$\displaystyle\times\Big{[}\left(g_{V}^{Z^{\prime}}(\nu)\right)^{2}+\left(g_{A}^{Z^{\prime}}(\nu)\right)^{2}\Big{]}$
$\displaystyle+\frac{2}{M_{Z}^{2}M_{Z^{\prime}}^{2}}\left[C_{V}(e)g_{V}^{Z^{\prime}}(e)-C_{A}(e)g_{A}^{Z^{\prime}}(e)\right]\left[\frac{1}{2}g_{V}^{Z^{\prime}}(\nu)+\frac{1}{2}(\nu)g_{A}^{Z^{\prime}}(\nu)\right]\Bigg{]}$
$\displaystyle\times\left(m_{e}^{2}\right)\left(k_{1}\cdot
k_{2}\right)\Bigg{\\}}$ (43)

Denoting new coefficients defined as follows:

$\displaystyle\mathcal{C}_{1}$ $\displaystyle=$
$\displaystyle\Bigg{[}\bigg{(}\frac{1}{M_{Z}^{2}}\big{[}C_{V}(e)-C_{A}(e)\big{]}+\frac{1}{M_{Z^{\prime}}^{2}}\big{[}g_{V}^{Z^{\prime}}(e)-g_{A}^{Z^{\prime}}(e)\big{]}\big{[}g_{V}^{Z^{\prime}}(\nu)+g_{A}^{Z^{\prime}}(\nu)\big{]}\bigg{)}^{2}$
(44) $\displaystyle+$
$\displaystyle\bigg{(}\frac{1}{M_{Z^{\prime}}^{2}}\big{[}g_{V}^{Z^{\prime}}(e)+g_{A}^{Z^{\prime}}(e)\big{]}\big{[}g_{V}^{Z^{\prime}}(\nu)-g_{A}^{Z^{\prime}}(\nu)\big{]}\bigg{)}^{2}\Bigg{]}$
$\displaystyle\mathcal{C}_{2}$ $\displaystyle=$
$\displaystyle\Bigg{[}\bigg{(}\frac{1}{M_{Z}^{2}}\big{[}C_{V}(e)+C_{A}(e)\big{]}+\frac{1}{M_{Z^{\prime}}^{2}}\big{[}g_{V}^{Z^{\prime}}(e)+g_{A}^{Z^{\prime}}(e)\big{]}\big{[}g_{V}^{Z^{\prime}}(\nu)+g_{A}^{Z^{\prime}}(\nu)\big{]}\bigg{)}^{2}$
(45) $\displaystyle+$
$\displaystyle\bigg{(}\frac{1}{M_{Z^{\prime}}^{2}}\big{[}g_{V}^{Z^{\prime}}(e)-g_{A}^{Z^{\prime}}(e)\big{]}\big{[}g_{V}^{Z^{\prime}}(\nu)-g_{A}^{Z^{\prime}}(\nu)\big{]}\bigg{)}^{2}\Bigg{]}$
$\displaystyle\mathcal{C}_{3}$ $\displaystyle=$ $\displaystyle
2\Bigg{[}\frac{1}{2M_{Z}^{4}}\Big{[}\left(C_{V}(e)\right)^{2}-\left(C_{A}(e)\right)^{2}\Big{]}$
(46) $\displaystyle+$
$\displaystyle\frac{1}{M_{Z^{\prime}}^{4}}\Big{[}\left(g_{V}^{Z^{\prime}}(e)\right)^{2}-\left(g_{A}^{Z^{\prime}}(e)\right)^{2}\Big{]}\Big{[}\left(g_{V}^{Z^{\prime}}(\nu)\right)^{2}+\left(g_{A}^{Z^{\prime}}(\nu)\right)^{2}\Big{]}$
$\displaystyle+$
$\displaystyle\frac{2}{M_{Z}^{2}M_{Z^{\prime}}^{2}}\big{[}C_{V}(e)g_{V}^{Z^{\prime}}(e)-C_{A}(e)g_{A}^{Z^{\prime}}(e)\big{]}\big{[}\frac{1}{2}g_{V}^{Z^{\prime}}(\nu)+\frac{1}{2}g_{A}^{Z^{\prime}}(\nu)\big{]}\Bigg{]}\,,$

then Eq. (43) can be rewritten as

$\displaystyle\frac{1}{4}\sum_{s}\left|\mathcal{M}_{fi}\right|^{2}=\frac{\left(4\pi\alpha\right)^{2}}{\sin^{4}{2\theta_{W}}}\Bigg{\\{}\frac{\sin^{4}{2\theta_{W}}}{s\left(4\pi\alpha\right)}\left(\mu_{\nu}^{2}+d_{\nu}^{2}\right)\left[\left(p_{1}\cdot
p_{2}+m_{e}^{2}\right)\left(p_{1}\cdot k_{2}+m_{e}^{2}\right)\right]$
$\displaystyle+\mathcal{C}_{1}\times\left(p_{1}\cdot
k_{1}\right)\left(p_{2}\cdot
k_{2}\right)+\mathcal{C}_{2}\times\left(p_{1}\cdot
k_{2}\right)\left(p_{2}\cdot
k_{1}\right)+\mathcal{C}_{3}\left(m_{e}^{2}\right)\left(k_{1}\cdot
k_{2}\right)\Bigg{\\}}\,.$ (47)

where $s=\left(p_{1}+p_{2}\right)^{2}=\left(k_{1}+2_{2}\right)^{2}$. For
future presentation, let us concrete the notations: four momentum is
$p_{i}=(E_{i},\vec{p}_{i}),i=1,2$ and $k_{i}=(E_{i},\vec{k}_{i}),i=1,2$.

###  III.2 Stellar energy loss rates

The ionized gas in thermal equilibrium with a temperature $T$ and density
$\rho$ is suggested to be exist in the star content.

Due to Fermi-Dirac distributions, the number densities of the electrons and
positrons are given by

$n_{\pm}=\int
dn_{\pm}=\frac{2}{(2\pi)^{3}}\int\frac{d^{3}p}{\left(e^{\frac{E\mp\mu}{kT}}+1\right)}\,,$
(48)

where $\mu$ is the electron chemical potential.Then, the plasma mass density
is given as:

$n_{0}=n_{-}-n_{+}=N\frac{\rho}{\mu_{e}}$ (49)

where $N$ is Avogadro’s number.

The expression for the stellar ELRs for pair-annihilation process in (34) is
determined by Dipole-Eloss-Kerimov1992 ; neutrino-Eloss-esposito2002 ; Dipole-
Eloss-Heger2009 ; QSM-Dicus1972 :

$\displaystyle\mathcal{Q}_{\nu\bar{\nu}}^{331\beta}$
$\displaystyle=\frac{4}{\left(2\pi\right)^{6}}\int_{m_{e}}^{\infty}{\frac{d^{3}\mathbf{p}_{1}}{\left[e^{\left(E_{1}-\mu\right)/T}+1\right]}\frac{d^{3}\mathbf{p}_{2}}{\left[e^{\left(E_{2}+\mu\right)/T}+1\right]}\left(E_{1}+E_{2}\right)}v\sigma_{tot}^{331\beta},$
(50)
$\displaystyle=\frac{16}{(2\pi)^{4}}\int_{m_{e}}^{\infty}{\frac{|\mathbf{p}_{1}|E_{1}dE_{1}}{\left[e^{\left(E_{1}-\mu\right)/T}+1\right]}\frac{|\mathbf{p}_{2}|E_{2}dE_{2}}{\left[e^{\left(E_{2}+\mu\right)/T}+1\right]}\left(E_{1}+E_{2}\right)}v\sigma_{tot}^{331\beta},$

where $\sigma_{tot}$ is the process cross-section,
$\left[\exp\left({\left(E_{1,2}\pm\mu\right)/T}\right)+1\right]^{-1}$ is the
Fermi-Dirac distribution functions for electron/positron, $\mu$ is the
electron chemical potential, $T$ is the stellar temperature and $v$ is the
electron-positron relative velocity $\frac{1}{2}[s(s-4m_{e}^{2})]^{1/2}$ .

The quantity $E_{1}E_{2}v\sigma_{tot}$ is given by Dipole-Eloss-Heger2009

$\displaystyle E_{1}E_{2}v\sigma_{tot}^{331\beta}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\int{\frac{d^{3}k_{1}d^{3}k_{2}}{\left(2\pi\right)^{3}2E_{3}\left(2\pi\right)^{3}2E_{4}}}\left|\mathcal{M}_{fi}\right|^{2}\left(2\pi\right)^{4}\delta^{4}\left(p_{1}+p_{2}-k_{1}-k_{2}\right)$
$\displaystyle=$
$\displaystyle\frac{\pi\alpha^{2}}{3\sin^{4}{2\theta_{W}}}\Bigg{\\{}\frac{\sin^{4}\theta_{W}}{2\pi\alpha}\left(\mu_{\nu}^{2}+d_{\nu}^{2}\right)\left(2m_{e}^{2}+p_{1}\cdot
p_{2}\right)+\Big{(}\mathcal{C}_{1}^{331\beta}\Big{)}\Big{[}m_{e}^{4}$
$\displaystyle+$ $\displaystyle 3m_{e}^{2}\left(p_{1}\cdot
p_{2}\right)+2\left(p_{1}\cdot
p_{2}\right)^{2}\Big{]}+12\Big{(}\mathcal{C}_{2}^{331\beta}\Big{)}\Big{[}m_{e}^{4}+m_{e}^{2}\left(p_{1}\cdot
p_{2}\right)\Big{]}\Bigg{\\}}\,,$

where the coefficients $\mathcal{C}_{1,2}^{331\beta}$ are defined as:

$\displaystyle\mathcal{C}_{1}^{331\beta}=\dfrac{1}{2M_{Z}^{4}}\Big{[}C^{2}_{V}(e)+C^{2}_{A}(e)\Big{]}+\dfrac{1}{M_{Z^{\prime}}^{4}}\Big{[}\left(g_{V}^{Z^{\prime}}(e)\right)^{2}+\left(g_{A}^{Z^{\prime}}(e)\right)^{2}\Big{]}$
$\displaystyle\times\Big{[}\left(g_{V}^{Z^{\prime}}(\nu)\right)^{2}+\left(g_{A}^{Z^{\prime}}(\nu)\right)^{2}\Big{]}+\dfrac{1}{M_{Z}^{2}M_{Z^{\prime}}^{2}}\big{[}C_{V}(e)g_{V}^{Z^{\prime}}(e)+C_{A}(e)g_{A}^{Z^{\prime}}(e)\big{]}$
$\displaystyle\times\big{[}g_{V}^{Z^{\prime}}(\nu)+g_{A}^{Z^{\prime}}(\nu)\big{]}\,,$
$\displaystyle\mathcal{C}_{2}^{331\beta}=\dfrac{1}{2M_{Z}^{4}}\Big{[}C^{2}_{V}(e)-C^{2}_{A}(e)\Big{]}+\dfrac{1}{M_{Z^{\prime}}^{4}}\Big{[}\left(g_{V}^{Z^{\prime}}(e)\right)^{2}-\left(g_{A}^{Z^{\prime}}(e)\right)^{2}\Big{]}$
$\displaystyle\times\Big{[}\left(g_{V}^{Z^{\prime}}(\nu)\right)^{2}+\left(g_{A}^{Z^{\prime}}(\nu)\right)^{2}\Big{]}+\dfrac{1}{M_{Z}^{2}M_{Z^{\prime}}^{2}}\big{[}C_{V}(e)g_{V}^{Z^{\prime}}(e)-C_{A}(e)g_{A}^{Z^{\prime}}(e)\big{]}$
$\displaystyle\times\big{[}g_{V}^{Z^{\prime}}(\nu)+g_{A}^{Z^{\prime}}(\nu)\big{]}\,.$
(52)

The calculation of the stellar ELRs given by (50) can be fulfill by expressing
in terms of the Fermi integral defined as

$G_{n}^{\pm}(\lambda,\eta,x)=\lambda^{{3+2n}}\int_{\lambda^{-1}}^{\infty}{x^{2n+1}\frac{\sqrt{x^{2}-\lambda^{-2}}}{\left(e^{\left(x\pm\eta\right)}+1\right)}}dx,$
(53)

where dimensionless variables $\lambda$ and $\eta$ are defined as:

$\lambda=\frac{k_{B}T}{m_{e}},\hskip 14.22636pt\eta=\frac{\mu_{e}}{k_{B}T}.$
(54)

Assuming $k_{B}=1$ for the Boltzmann constant and from (48) and (49) ones
have:

$G_{s}^{\pm}(\lambda,\eta,E)=\frac{1}{m_{e}^{3+2s}}\int_{m_{e}/T}^{\infty}{E^{2s+1}\frac{\sqrt{E^{2}-m_{e}^{2}}}{\left(e^{\left(E\pm\mu_{e}\right)/T}+1\right)}}dE\,,$
(55)

and

$N\frac{\rho}{\mu_{e}}=\frac{m_{e}^{3}}{\pi^{2}}[G_{0}^{-}-G_{0}^{+}]\,.$ (56)

Hence, the stellar ELRs can be expressed as:

$\displaystyle\mathcal{Q}^{331\beta}=\frac{\alpha^{2}m_{e}^{9}}{9\pi^{3}\sin^{4}{2\theta_{W}}}\Bigg{\\{}\frac{3\sin^{4}{2\theta_{W}}}{2\pi\alpha
m_{e}^{2}}\left(\mu_{\nu}^{2}+d_{\nu}^{2}\right)$
$\displaystyle\times\bigg{[}2\Big{(}G_{-1/2}^{-}G_{0}^{+}+G_{0}^{-}G_{-1/2}^{+}\Big{)}+G_{0}^{-}G_{1/2}^{+}+G_{1/2}^{-}G_{0}^{+}\bigg{]}$
$\displaystyle+4\Big{(}\mathcal{C}_{1}^{331\beta}\Big{)}\bigg{[}5\Big{(}G_{-1/2}^{-}G_{0}^{+}+G_{0}^{-}G_{-1/2}^{+}\Big{)}+7\Big{(}G_{0}^{-}G_{1/2}^{+}+G_{1/2}^{-}G_{0}^{+}\Big{)}$
$\displaystyle-2\Big{(}G_{1}^{-}G_{-1/2}^{+}+G_{-1/2}^{-}G_{1}^{+}\Big{)}+8\Big{(}G_{1}^{-}G_{1/2}^{+}+G_{1/2}^{-}G_{1}^{+}\Big{)}\bigg{]}$
$\displaystyle+36\Big{(}\mathcal{C}_{2}^{331\beta}\Big{)}\bigg{[}G_{-1/2}^{-}G_{0}^{+}+G_{0}^{-}G_{-1/2}^{+}+G_{0}^{-}G_{1/2}^{+}+G_{1/2}^{-}G_{0}^{+}\bigg{]}\Bigg{\\}}.$
(57)

To investigate the effects of dipole moments and new contribution of
$Z^{\prime}$ boson on the ELR we have to evaluate the relative correction for
the star ELR between 3-3-1$\beta$ model and that of the SM
($\mathcal{Q}^{SM}$) QSM-Dicus1972  given as:

$\displaystyle\mathcal{Q}^{SM}=\frac{G_{F}^{2}m_{e}^{9}}{18\pi}\Biggl{\\{}(7C_{V}(e)^{2}-2C_{A}(e)^{2})[G_{0}^{-}G_{1/2}^{+}+G_{-1/2}^{-}G_{0}^{+}]$
$\displaystyle+9C_{V}(e)^{2}[G_{1/2}^{-}G_{0}^{+}+G_{0}^{-}G_{1/2}^{+}]+(C_{V}(e)^{2}+C_{A}(e)^{2})[4G_{1}^{-}G_{1/2}^{+}+G_{1/2}^{-}G_{1}^{+}$
$\displaystyle-
G_{1}^{-}G_{-1/2}^{+}-G_{1/2}^{-}G_{0}^{+}-G_{0}^{-}G_{1/2}^{+}-G_{-1/2}^{-}G_{1}^{+}]\Biggr{\\}}\,,$
(58)

Hence

$\frac{\delta\mathcal{Q}^{331\beta}}{\mathcal{Q}^{SM}}=\frac{\mathcal{Q}^{331\beta}\left(\mu_{\nu},d_{\nu},\beta,M_{Z^{\prime}},\eta\right)-\mathcal{Q}^{SM}\left(\eta\right)}{\mathcal{Q}^{SM}\left(\eta\right)}.$
(59)

Since the functions $G^{\pm}(\lambda,\eta)$ can only be defined analytically
in some limiting regions of parameters $\lambda$ and $\eta$ therefore we will
investigate the correction (59) in five regions

####  III.2.1 Region I: $\lambda\ll 1$ and $\eta\ll 1/\lambda$

In this region, temperature and densities vary between $3\times
10^{8}\,\textrm{K}\leq\textrm{T}\leq 3\times 10^{9}\,\textrm{K}$ and $\rho\leq
10^{5}gr/cm^{3}$, respectively.

The Fermi integral is

$G_{n}^{\pm}(\lambda,\eta,x)=\lambda^{{3+2n}}\int_{\lambda^{-1}}^{\infty}{x^{2n+1}\frac{\sqrt{x^{2}-\lambda^{-2}}}{\left(e^{\left(x\pm\eta\right)}+1\right)}}dx\,.$
(60)

Changing variable $x=z+\lambda^{-1}$ yields

$G_{n}^{\pm}=\sqrt{2}\lambda^{3/2}e^{-\lambda^{-1}}e^{\mp\eta}\int_{0}^{\infty}dz(\lambda
z+1)^{2n+1}z^{1/2}\left(1+\frac{\lambda z}{2}\right)^{1/2}e^{-z}$ (61)

For every $\lambda$ satisfying $\lambda z\approx\epsilon$ we have

$(\lambda z+1)^{2n+1}\hskip 14.22636pt[1+(2n+1)\lambda z],\hskip
14.22636pt\left(1+\frac{\lambda z}{2}\right)^{1/2}\approx 1+\frac{\lambda
z}{4}$

Neglecting the second order in $\lambda z$ ones get

$\displaystyle G_{n}^{\pm}$
$\displaystyle=\sqrt{2}\lambda^{3/2}e^{-\lambda^{-1}}e^{\mp\eta}\left[\int_{0}^{\infty}dzz^{1/2}e^{-z}+\left(2n+\frac{5}{4}\right)\lambda\int_{0}^{\infty}dzz^{3/2}e^{-z}\right]$
(62)
$\displaystyle=\sqrt{2}\lambda^{3/2}e^{-\lambda^{-1}}e^{\mp\eta}\left[\Gamma(3/2)+\left(2n+\frac{5}{4}\lambda\right)\Gamma(5/2)\right]$
$\displaystyle=\sqrt{\frac{\pi}{2}}\lambda^{3/2}e^{-\lambda^{-1}}e^{\mp\eta}\left[1+\frac{3}{2}\left(2n+\frac{5}{4}\right)\lambda\right]\,,$

where $\Gamma(n)=\int_{0}^{\infty}z^{n-1}e^{-z}dz$.

In the case $1\ll\lambda^{-1}$ ones have

$G_{0}^{\pm}\approx\sqrt{\frac{\pi}{2}}\lambda^{3/2}e^{-\lambda^{-1}}e^{\mp\eta}$
(63)

and

$G_{n}^{-}\approx
G_{0}^{-}=\left(\frac{\pi}{2}\right)^{\frac{1}{2}}\lambda^{\frac{3}{2}}e^{-1/\lambda}e^{\eta}.$
(64)

Hence, we get

$\displaystyle\mathcal{Q}_{I}^{331\beta}$
$\displaystyle=\frac{\alpha^{2}m_{e}^{9}}{\pi^{3}\sin^{4}{2\theta_{W}}}\Bigg{[}\frac{\sin^{4}{2\theta_{W}}}{\pi\alpha
m_{e}^{2}}\left(\mu_{\nu}^{2}+d_{\nu}^{2}\right)+16(\mathcal{C}_{1}^{331\beta}+\mathcal{C}_{2}^{331\beta})\Bigg{]}G_{0}^{-}G_{0}^{+}$
(65)
$\displaystyle=\frac{\alpha^{2}m_{e}^{9}}{\pi^{3}\sin^{4}{2\theta_{W}}}\Bigg{[}\frac{\sin^{4}{2\theta_{W}}}{\pi\alpha
m_{e}^{2}}\left(\mu_{\nu}^{2}+d_{\nu}^{2}\right)+16(\mathcal{C}_{1}^{331\beta}+\mathcal{C}_{2}^{331\beta})\Bigg{]}$
$\displaystyle\times\left(\frac{\pi}{2}\right)^{3/2}\left(\frac{T}{m_{e}}\right)^{3}e^{-2m_{e}/T}$

and

$\displaystyle\mathcal{Q}_{I}^{SM}$
$\displaystyle=\frac{G_{F}^{2}m_{e}^{9}}{18\pi^{5}}\times
36C^{2}_{V}(e)G_{0}^{-}G_{0}^{+}$ (66)
$\displaystyle=\frac{\alpha^{2}m_{e}^{9}}{\pi^{3}\sin^{4}{2\theta_{W}}}\frac{16}{m_{Z}^{4}}C_{V}^{2}(e)G_{0}^{-}G_{0}^{+}$

Then, the correction is given by:

$\frac{\delta\mathcal{Q}_{I}^{331\beta}}{\mathcal{Q}_{I}^{SM}}=\frac{\Bigg{[}\frac{\sin^{4}{2\theta_{W}}}{\pi\alpha
m_{e}^{2}}\left(\mu_{\nu}^{2}+d_{\nu}^{2}\right)+16(\mathcal{C}_{1}^{331\beta}+\mathcal{C}_{2}^{331\beta})\Bigg{]}-\frac{16}{m_{Z}^{4}}C_{V}^{2}(e)}{\frac{16}{m_{Z}^{4}}C_{V}^{2}(e)}$
(67)

####  III.2.2 Region II: $\lambda\ll 1$ and
$\frac{1}{\lambda}\ll\eta\ll\frac{2}{\lambda}$

This region is nonrelativistic and mildly degenerate, with temperatures $T\leq
10^{8}$ K and densities between
$10^{4}gr/cm^{3}\,\leq\rho\leq\,10^{6}gr/cm^{3}$.

The Fermi integrals satisfy the following conditions QBSM-Hernandez ; QBSM-
Hernandez ; QSM-Dicus1972 : $G_{0}^{-}\gg G_{0}^{+}$ and $G_{n}^{-}\approx
G_{0}^{-}$ and

$G_{n}^{+}\approx
G_{0}^{+}=\left(\frac{\pi}{2}\right)^{\frac{1}{2}}\lambda^{\frac{3}{2}}e^{-1/\lambda}e^{-\eta},\qquad
G_{n}^{-}\approx
G_{0}^{-}=\left(\frac{\rho}{\mu_{e}}\right)\frac{\pi^{2}}{m_{e}^{3}}N_{A}.$
(68)

Then ones have:

$\displaystyle\mathcal{Q}_{II}^{331\beta}$
$\displaystyle=\frac{\alpha^{2}m_{e}^{9}}{\pi^{3}\sin^{4}{2\theta_{W}}}\Bigg{[}\frac{\sin^{4}{2\theta_{W}}}{\pi\alpha
m_{e}^{2}}\left(\mu_{\nu}^{2}+d_{\nu}^{2}\right)+16(\mathcal{C}_{1}^{331\beta}+\mathcal{C}_{2}^{331\beta})\Bigg{]}G_{0}^{-}G_{0}^{+}$
(69)
$\displaystyle=\frac{\alpha^{2}m_{e}^{6}}{\pi\sin^{4}{2\theta_{W}}}\left(\frac{\rho}{\mu_{e}}N_{A}\right)\left(\frac{\pi}{2}\right)^{1/2}\left(\frac{T}{m_{e}}\right)^{3/2}e^{-(m_{e}+\mu_{e})/T}$
$\displaystyle\times\Bigg{[}\frac{\sin^{4}{2\theta_{W}}}{\pi\alpha
m_{e}^{2}}\left(\mu_{\nu}^{2}+d_{\nu}^{2}\right)+16(\mathcal{C}_{1}^{331\beta}+\mathcal{C}_{2}^{331\beta})\Bigg{]}$

and

$\displaystyle\mathcal{Q}_{II}^{SM}$
$\displaystyle=\frac{G_{F}^{2}m_{e}^{9}}{18\pi^{5}}\times
36C^{2}_{V}(e)G_{0}^{-}G_{0}^{+}$ (70)
$\displaystyle=\frac{\alpha^{2}m_{e}^{9}}{\pi^{3}\sin^{4}{2\theta_{W}}}\frac{16}{m_{Z}^{4}}C_{V}^{2}(e)G_{0}^{-}G_{0}^{+}$
$\displaystyle=\frac{\alpha^{2}m_{e}^{9}}{\pi^{3}\sin^{4}{2\theta_{W}}}\frac{16}{m_{Z}^{4}}C_{V}^{2}(e)\left(\frac{\rho}{\mu_{e}}N_{A}\frac{\pi^{2}}{m_{e}^{3}}\right)\left(\frac{\pi}{2}\right)^{1/2}\left(\frac{T}{m_{e}}\right)^{3/2}e^{-(m_{e}+\mu_{e})/T}\,.$

Therefore the correction is given as exactly as in (67) which is equal as in
region I.

####  III.2.3 Region III: $\lambda\ll 1$ and $1\ll\lambda\eta$

The considered region represents the relativistic and degenerate case and is
valid for temperatures $T>6\times 10^{7}$ K and densities
$\rho>10^{7}gr/cm^{3}$. The Fermi integrals result:

$G_{n}^{+}\approx
G_{0}^{+}=\left(\frac{\pi}{2}\right)^{\frac{1}{2}}\lambda^{\frac{3}{2}}e^{-1/\lambda}e^{-\eta},\hskip
14.22636ptG_{n}^{-}=\left(\frac{3}{2n+3}\right)\left(\lambda\eta\right)^{2n}G_{0}^{-}.$
(71)

Then to highest power in $\lambda\eta$ ones have

$\displaystyle\mathcal{Q}_{III}^{331\beta}$
$\displaystyle=\frac{8\alpha^{2}m_{e}^{9}}{3\pi^{3}\sin^{4}{2\theta_{W}}}\mathcal{C}_{1}^{331\beta}G_{1}^{-}G_{0}^{+}$
(72)
$\displaystyle=\frac{8\alpha^{2}m_{e}^{9}}{3\pi^{3}\sin^{4}{2\theta_{W}}}\frac{3}{5}\left(\frac{1}{m_{e}}\frac{\mu_{e}}{T}\right)^{2}\left(\frac{\rho}{\mu_{e}}\right)\frac{\pi^{2}}{\mu^{3}_{e}}N_{A}\left(\frac{\pi}{2}\right)^{1/2}\left(\frac{T}{m_{e}}\right)^{3/2}$
$\displaystyle\times e^{-({m_{e}+\mu_{e}})/{T}}\mathcal{C}_{1}^{331\beta},$

and

$\displaystyle\mathcal{Q}^{SM}=\frac{4}{3}\frac{\alpha^{2}m_{e}^{9}}{\pi^{3}\sin^{4}{2\theta_{W}}m^{4}_{Z}}[C_{V}^{2}(e)+C^{2}_{A}(e)]G_{1}^{-}G_{0}^{+}\,.$
(73)

The relativistic correction is given by:

$\displaystyle\frac{\delta\mathcal{Q}^{331\beta}_{III}}{\mathcal{Q}^{SM}_{III}}$
$\displaystyle=$
$\displaystyle\frac{2m^{4}_{Z}\mathcal{C}_{1}^{331\beta}}{[C_{V}^{2}(e)+C^{2}_{A}(e)]}-1\,.$
(74)

Since the approximation for this region only considers the terms of dominant
powers, so there is no dependence on the AMM and/or EDM of the neutrino.

####  III.2.4 Region IV: $\lambda\gg 1$ and $\eta\ll 1$

The relativistic and nondegenerate case holds for densities
$\rho>10^{7}gr/cm^{3}$. In this region we may ignore the chemical potential.
Considering the dominance of the highest orders in $\lambda$, we get

$G_{n}^{\pm}\approx\lambda^{2n+3}\Gamma\left(2n+3\right)\sum_{S=1}^{\infty}\frac{\left(-1\right)^{S+1}}{S^{2n+3}}.$
(75)

Then, the stellar ELRs for this region is given by

$\displaystyle\mathcal{Q}_{IV}^{331\beta}$
$\displaystyle=\frac{64\alpha^{2}m_{e}^{9}}{9\pi^{3}\sin^{4}{2\theta_{W}}}\mathcal{C}_{1}^{331\beta}G_{1}^{-}G_{1/2}^{+}$
(76)
$\displaystyle=\frac{64\alpha^{2}m_{e}^{9}}{9\pi^{3}\sin^{4}{2\theta_{W}}}\mathcal{C}_{1}^{331\beta}\left(\frac{T}{m_{e}}\right)^{9}\Gamma(5)\xi(5)\Gamma(4)\xi(4)$

where $\Gamma$ and $\xi$ are the Gamma and Riemann zeta functions,
respectively.

For the SM the ELR is given by:

$\displaystyle\mathcal{Q}^{SM}=\frac{32\alpha^{2}m_{e}^{9}}{9\pi^{3}\sin^{4}{2\theta_{W}}m^{4}_{Z}}(C^{2}_{V}(e)+C^{2}_{A}(e))G_{1}^{-}G_{1/2}^{+}\,.$
(77)

Thus the relativistic correction is the same as in the previous region, namely

$\displaystyle\frac{\delta\mathcal{Q}^{331\beta}_{IV}}{\mathcal{Q}^{SM}_{IV}}$
$\displaystyle=$
$\displaystyle\frac{2m_{Z}^{4}\mathcal{C}_{1}^{331\beta}}{[C_{V}^{2}(e)+C^{2}_{A}(e)]}-1.$
(78)

####  III.2.5 Region V: $\lambda\gg 1$ and $\eta\gg 1$

This degenerate relativistic region holds for densities greater than
$\rho>10^{8}gr/cm^{3}$ with temperatures of $T\approx 10^{10}$ K at the lowest
density, extendable to a range between $10^{10}$ K and $10^{11}$ K at a
density of $\rho>10^{10}gr/cm^{3}$. Here $G_{n}^{-}\gg G_{n}^{+}$, then

$G_{n}^{+}\approx\lambda^{2n+3}\left(2n+2\right)!e^{-\eta}\hskip
14.22636ptG_{n}^{-}\approx\left(\frac{3}{2n+3}\right)\left(\lambda\eta\right)^{2n}\left(\frac{\rho}{\mu_{e}}\right)\frac{\pi^{2}}{m_{e}^{3}}N_{A}.$
(79)

Restricting the calculation to the higher powers in $\lambda\eta$, the stellar
ELRs result:

$\displaystyle\mathcal{Q}^{331\beta}_{V}$
$\displaystyle=\frac{32\alpha^{2}m_{e}^{9}}{9\pi^{3}\sin^{4}{2\theta_{W}}}\mathcal{C}_{1}^{331\beta}G_{1}^{-}G_{1/2}^{+}$
(80)
$\displaystyle=\frac{64\alpha^{2}m_{e}^{6}}{5\pi\sin^{4}{2\theta_{W}}}\mathcal{C}_{1}^{331\beta}\left(\frac{\rho}{\mu_{e}}N_{A}\right)\left(\frac{T}{m_{e}}\right)^{6}\left(\frac{\mu_{e}}{T}\right)^{2}e^{-\mu_{e}/{T}}$

and

$\displaystyle\mathcal{Q}^{SM}_{V}$
$\displaystyle=\frac{32\alpha^{2}m_{e}^{9}}{9\pi^{3}\sin^{4}{2\theta_{W}}m_{Z}^{4}}[C_{V}^{2}(e)+C_{A}^{2}(e)]G_{1}^{-}G_{1/2}^{+}$

and the relativistic correction is given by

$\displaystyle\frac{\delta\mathcal{Q}^{331\beta}_{V}}{\mathcal{Q}^{SM}_{V}}$
$\displaystyle=$
$\displaystyle\frac{2m_{Z}^{4}\mathcal{C}_{1}^{331\beta}}{[C_{V}^{2}(e)+C^{2}_{A}(e)]}-1\,.$
(82)

The above result is equal to that in Region III. Again, it becomes clear that
there is an indistinguishability of treating with nondegenerate or degenerate
electrons.

##  IV Numerical analysis

Let us investigate the pair annihilation neutrino ELR in the context of the
$SU(3)_{C}\times SU(3)_{L}\times U(1)_{X}$ models. The process in (34) is one
of the main mechanisms of neutrino pair production relevant for the neutrino
luminosity. We investigate for both degenerate and nondegenerate Fermi gas. In
the context of beyond Standard Model, at loop level the non-vanishing of AMM
and EDM of neutrinos and the appearance of new neural gauge bosons with V-A
interaction can contribute to the process of energy loss.

Before investigate the effect of temperature and densities on the ELR we will
investigate the rate of correction of 3-3-1 models compared with the SM
because with the same value of temperature and densities the parameters that
distinguish models is the value of parameter of the model (the parameter
$\beta$) and the mass of the new gauge bosons $m_{Z^{\prime}}$. We plot the
$\mathcal{Q}$ correction for regions I, II in Figs.2 and 3. and for regions
II, IV, V in Figs.4 and 5

Figure 2: Q correction regions I,II with $\beta>0$ Figure 3: Q correction
regions I,II with $\beta<0$ Figure 4: Q correction regions III,IV,V with
$\beta>0$ Figure 5: Q correction regions III,IV,V with $\beta<0$

The correction is plotted for all values of parameter $\beta$. For negative
value of $\beta$ the correction is up to $15\%$ while positive value of
$\beta$ give very small correction ($0.2\%$). For all negative $\beta$, the
correction decreases with the increase of the mass of Z’ boson. For
$\beta=-\frac{1}{\sqrt{3}},-\frac{2}{\sqrt{3}}$, the correction is
approximated to 0% when $m_{Z^{\prime}}\geq 2000$ GeV and only increase
significantly when $m_{Z^{\prime}}\approx 200$ GeV which is around the energy
region of the $Z$ boson. In the case $\beta=-\sqrt{3}$, the correction is
higher from 1% up to 15% in the mass range from 4000 GeV to 1000 GeV. This
case is more interesting when showing that the contribution of the
$Z^{\prime}$ boson in the 3-3-1 model is distinguishable with the SM.
Therefore put constraints on the mass range of the $Z^{\prime}$ boson which is
in agreement with searching mass range for the $Z^{\prime}$ boson at LHC
PDG2016 ; Z' resonance . In the followings detail numerical analysis, we will
work with the case where $\beta=-\sqrt{3}$.

In Fig.6, we plot the energy loss as a function of $M_{Z^{\prime}}$ and
compare it with those in the SM. The value of temperature is set $T=3*10^{9}$K
and $\mu_{\nu}=3*10^{-12}(\mu_{B})$. We plot the contribution of dipole moment
and gauge boson $Z^{\prime}$ separately. In Fig.7 we plot the energy loss rate
as a function of magnetic moment.

Figure 6: Energy loss rate for region I as a function of $m_{Z^{\prime}}$
Figure 7: Energy loss rate for region I as a function of magnetic dipole
moment Figure 8: Energy loss rate for region II as a function of
$m_{Z^{\prime}}$

To investigate the effect of only of the dipole moment we take the limit where
the value of $m_{Z^{\prime}}$ very large ($\infty$). In this limit the
contribution of the gauge boson $Z^{\prime}$ is small. The magnitude of
$\mathcal{Q}$ increase with the value of $\mu_{\nu}$, however with current
bound $\mu_{\nu}<3*10^{-12}\mu_{B}$ the value of $\mathcal{Q}<2.6675*10^{18}$
which is approximate the magnitude of those in the SM as in Fig.6 or the
$\mathcal{Q}$ correction for region I-II is less than 0.1%( Fig.9) comparing
with 14% of the combine contribution of both dipole moment and the
$Z^{\prime}$ boson. Hence the effect of dipole moment is not clear in this
case.

Figure 9: Energy loss rate for region I as a function of magnetic dipole
moment

The Fig.10 is the case for $\mu_{\nu}=0,d_{\nu}=0$ and the value of
temperature $T=3*10^{9}$ K. Plot $\mathcal{Q}$ as function of
$m_{Z^{\prime}}$, for mass range of $m_{Z^{\prime}}<2000$ GeV the magnitude of
energy loss $\mathcal{Q}\approx 2.8*10^{18}$ therefore the effect of gauge
boson $Z^{\prime}$ is much stronger than that of dipole moment.

Figure 10: Energy loss rate for region I as a function of $m_{Z^{\prime}}$,
for $\mu_{\nu}=0,d_{\nu}=0$, $T=3*10^{9}$ K Figure 11: Energy loss rate for
region I as a function of temperature

Next we plot the dependence of the energy loss on the temperature as in Fig.
11. We plot for three values of the mass of the gauge boson
$m_{Z^{\prime}}=500,100,2000$ GeV. The ELR increases with the increase of the
temperature. The dependence of $\mathcal{Q}$ on dimensionless parameter $\eta$
is depicted as in Fig. 12. As in the figure, $\mathcal{Q}$ decreases with the
increase of $\eta$. This is what one would expect since
$\eta=\frac{\mu_{e}}{k_{B}T}$ is inverse proportional to the temperature T.
The dependence of $\mathcal{Q}$ on the temperature and density are plotted in
Fig.13. $\mathcal{Q}$ increases in the region of high temperature and high
density.

Figure 12: Energy loss rate for region II as a function of $\eta$ Figure 13:
Energy loss rate for region II as a function of temperature and density

##  V Conclusions

Quantifying stellar loss energy is a priority in astrophysics and cosmology.
One of most interesting possibilities is to use stars and its physical process
to put set constraints on physics beyond Standard Model.

We have evaluated the stellar ELR in the frameworks of the 3-3-1$\beta$ model.
The energy loss rate is in the form of neutrino emission assessed in the pair
annihilation $e^{+}e^{-}\xrightarrow{\gamma,W,Z,Z^{\prime}}\nu\bar{\nu}$. We
obtained the approximated formula for energy loss ($\mathcal{Q}$) and the
correction $\mathcal{Q}$ in comparing with that of the SM.

We evaluated the $\mathcal{Q}$ correction for different value of
$\beta=\pm\frac{1}{\sqrt{3}},\pm\frac{2}{\sqrt{3}},\pm\sqrt{3}$. The negative
value of $\beta$ give higher value compared with positive value and up to 14%.
We have shown that the contribution of dipole moment is small compared with
that of the $Z^{\prime}$ boson. The $\mathcal{Q}$ gives the constraints on the
mass range of the $Z^{\prime}$ boson $m_{Z^{\prime}}\leq 4000$ GeV which is in
agreement with current searching the mass range of the $Z^{\prime}$ at LHC.

##  Acknowledgments

We acknowledge the financial support of the International Centre of Physics at
the Institute of Physics, VAST under grant No: ICP.2021.02

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