Dynamic model of a non-equilibrium chemical composition formation in the shell
of single neutron stars

A.Yu.Ignatovskiy${}^{\text{{1,2,3,4,}}\asterisk}$, G.S.Bisnovatyi-
Kogan${}^{\text{{1,2,5}}}$

${}^{\text{{1}}}$Space research institute RAS
${}^{\text{{2}}}$Moscow institute of physics and technology
${}^{\text{{3}}}$Institute for theoretical and experimental physics, NRC
"Kurchatov Institute"
${}^{\text{{4}}}$National research center "Kurchatov Institute"
${}^{\text{{5}}}$Moscow engineering physics institute
${}^{\text{{*}}}$E-mail<EMAIL_ADDRESS>
Abstract–The process of a non-equilibrium chemical composition formation in
the crust of a new born neutron star is considered, during cooling due to
neutrino energy losses. A model is constructed for explaining accumulation of
a large quantity of nuclear energy, which can maintain the X-ray luminosity of
such compact objects for a long period of time. We studied numerically the
dependence of the final chemical composition on various parameters of the
model.

1\. INTRODUCTION
Neutron stars are the result of a massive star gravitational collapse at the
end of its evolution. Densities of neutron stars are of the order of the
atomic nucleus density $\sim 2\cdot 10^{14}$ g $\cdot$ $\mathrm{cm}^{-3}$ in
the inner parts and are $\sim 10^{9}$ g $\cdot$ $\mathrm{cm}^{-3}$ on the
outer shell boundary. Newly formed neutron stars with inner temperatures
$\mathrm{T_{9}\gg 10}$ (here and below $\mathrm{T_{9}=T/10^{9}\,K}$) cool
rapidly, in first seconds of life, due to neutrino energy losses.

The results of the chemical composition evolution in various systems are of
interest to all astrophysics. The answer is well known for the slow stellar
evolution, when, as a result of various nuclear reactions, elements of the
iron group are formed, which have the highest binding energy per nucleon
$\mathrm{\Delta E_{56Fe}\approx 8.55}$ MeV, and up to heavier elements such as
platinum and plumbum. The processes of nucleosynthesis were classified in the
work [1], where authors figured out that elements heavier than iron are formed
in the so-called s- and r-processes, the processes of slow and rapid capture
of neutrons by nuclei respectively, followed by beta decays [2, 3]. For the
s-process, the relation between the probabilities of a neutron capture and an
electron decay satisfies the inequality
$\mathrm{\lambda_{n\gamma}\ll\lambda_{\beta}}$, and its path runs along the
stable isotopes. For the r-process, the reverse inequality holds
$\mathrm{\lambda_{n\gamma}\gg\lambda_{\beta}}$ [4, 5, 6] and its path is far
from the stability valley of the chemical elements. No experimental nuclear
data on strongly neutron-rich nuclei are available. For the nuclei production
of the heaviest transuranic elements in the r-process, up to 150 free neutrons
are required per seed nucleus with a mass number of $\mathrm{A=50-100}$.
Astrophysical objects associated with the possible nucleosynthesis of
extremely heavy elements are neutron stars. The formation of heavy elements in
a neutron bunch was first considered in the work [7]. The conditions for the
r-process to occur are realized during the evolution of a close binary system
of neutron stars of different masses [8], during the merging of neutron stars
[9], and in a hot wind from the surface of neutron stars [10].

The aim of the current work is to carry out numerical calculations of the
chemical composition evolution in the shell of the hot, newly formed neutron
star that rapidly cools due to neutrino energy losses. This problem was first
formulated and discussed in the work [11]. The calculation for the relaxation
of the nuclear composition at some fixed temperature and density values had
been made earlier in the work [12].

2\. SCHEME OF A NON-EQUILIBRIUM LAYER FORMATION
A key role in the formation of the non-equilibrium layer belongs to a gradual
and non-simultaneous closure of various nuclear reaction channels during
cooling [13]. First, a fusion of heavy charged nuclei and protons capture by
nuclei are ceased because the probability of Coulomb barrier tunneling
decreases with a temperature drop [14]), and photonuclear reactions stop
later.

Initially, when a matter is at high temperature and density, nuclei, protons
and neutrons are non-degenerate $\mathrm{kT>\varepsilon_{fe}}$, non-
relativistic $\mathrm{p_{fe}<m_{p}c}$ and are described by Boltzmann gas.
Under these conditions, the matter is in equilibrium relative to strong
interaction reactions [15, 16]:

$\mathrm{\mu(A,Z)+\mu_{i}=\mu(A^{\prime},Z^{\prime})}.$ (1)

Here we denote $\mathrm{i=n,p}$. Using the expression for the chemical
potential $\mu$ of the Boltzmann gas, one can obtain the Saha equation [17]
that determines the concentration of each nucleus in the system as a function
of free neutrons and protons concentrations in the following form:

$\displaystyle\mathrm{n(A,Z)=g(A,Z)\frac{n_{p}^{Z}n_{n}^{A-Z}}{2^{A}}A^{\frac{3}{2}}\times}$
(2)
$\displaystyle\mathrm{\times\bigg{(}\frac{2\pi\hbar^{2}}{m_{\mu}kT}\bigg{)}^{\frac{3}{2}(A-1)}\exp\bigg{\\{}\frac{Q(A,Z)}{kT}\bigg{\\}}.}$

Here $\mathrm{Q(A,Z)=c^{2}[Zm_{p}+(A-Z)m_{n}-m(A,Z)]}$ is the nuclear binding
energy. Electrons are relativistic $\mathrm{p_{fe}\gg m_{e}c}$ [18] and
degenerate $\mathrm{kT<\varepsilon_{fe}}$, the degeneracy degree increases
with cooling $\mathrm{kT\ll\varepsilon_{fe}}$. Cooling occurs mainly due to
neutrino energy losses during the following reactions:

$\begin{cases}\mathrm{\;\;(A,Z)+e^{\pm}\rightarrow(A,Z\pm 1)+\nu^{\pm}},\\\
\mathrm{\;\;(A,Z\pm 1)\rightarrow(A,Z)+e^{\pm}+\nu^{\mp}},\end{cases}$ (3)

denoting $\mathrm{\nu^{\pm}}$ as antineutrino and neutrino, respectively. It
is necessary to take into account positrons in equilibrium at
$\mathrm{kT\gtrsim m_{e}c^{2}}$, what corresponds to temperatures
$\mathrm{T_{9}\gtrsim 6}$. The condition for the annihilation reaction
equilibrium of the pair $\mathrm{e^{-}+e^{+}\leftrightarrows 2\gamma}$ for
$\mu_{\gamma}=0$ leads to the relation $\mathrm{\mu_{e^{-}}+\mu_{e^{+}}=0}$.

As a result of a temperature drop, the reactions with protons freeze out and
positrons disappear. Only beta reactions with electrons as well as capture of
neutrons and photonuclear reactions remain at relatively low temperatures.
Stable nuclei rapidly capture free neutrons
($\mathrm{\tau_{n\gamma}\ll\tau_{\gamma n}}$ and
$\mathrm{\tau_{n\gamma}\ll\tau_{\beta}},\,\,\,\tau$ – characteristic reaction
time), moving away from the stability valley to the region of low binding
energy values of neutrons in the nuclei, until direct reactions of neutron
capture are balanced by photonuclear reactions
($\mathrm{\tau_{n\gamma}\sim\tau_{\gamma n}}$). As a result, formation of
heavy nuclei occurs, with large atomic numbers $\mathrm{A_{j}}$ for each
isotopic chain $\mathrm{Z_{i}}$.

Electrons degenerate and occupy the low energy levels under the Fermi surface.
The Pauli exclusion principle restrict beta reactions: the nucleus can either
decay at $\mathrm{Q_{\beta}=c^{2}[m(A,Z-1)-m(A,Z)-m_{e}]>\varepsilon_{fe}}$,
or capture electrons at $\mathrm{\varepsilon_{fe}>Q_{\beta}=\varepsilon_{\beta
i}}$, which is a threshold Fermi energy for electron capture. As a result of
these processes, nuclei accumulate in a non-equilibrium layer, limited in
$\mathrm{Z}$ and $\mathrm{A}$ [11].

Figure 1: Neutron drip line based on the finite range droplet model [21] for
$\mathrm{Z\in[0;\,83]}$.

In addition to the formation of a nuclear composition far from equilibrium
[19, 20], disequilibrium also consists in a large excess of free neutrons.
Free neutrons can neither decay ($\mathrm{\varepsilon_{fe}>Q_{\beta n}=0.78}$
MeV), nor join to neutron-rich nuclei with low neutron binding energies. As a
result, the matter does not reach the minimum energy state. The number of
allowed nuclei decreases with a temperature drop [11].

One can estimate the range of densities that makes a non-equilibrium layer
formation possible. To evaluate the upper boundary, consider the evolution of
matter with increasing density and at low temperature. For the threshold
energy of electron capture, we approximately have $\mathrm{\varepsilon_{\beta
i}\approx Q_{p}(A,Z)-Q_{n}(A,Z-1)}$, where $\mathrm{Q_{i}}$ is the binding
energy of the particle i in the nucleus. As the density increases, the Fermi
energy of electrons $\mathrm{\varepsilon_{fe}\approx\hbar c(3\pi
n_{e})^{1/3}}$ increases also. Upon reaching
$\mathrm{\varepsilon_{fe}=\varepsilon_{\beta i}}$ nuclei capture electrons,
decreasing their atomic charge numbers $\mathrm{Z}$, until they reach the
neutron drip line $\mathrm{Q_{n}=0}$, at which $\mathrm{\varepsilon_{\beta
i}=Q_{p}-Q_{n}=Q_{p0}}$. The neutron drip line is illustrated in Fig.1, based
on the finite range droplet model [21] (see Section 3). At the moment the
nucleus reaches $\mathrm{Q_{n}=0}$, subsequent electron captures will lead to
the neutrons releasing from the nucleus [11]:

$\mathrm{(A,Z)+e^{-}\rightarrow(A-1,Z-1)+n+\nu^{-}}.$ (4)

The described process repeats until the nucleus reaches the maximum
$\mathrm{\varepsilon_{\beta max}=Q_{p0}^{(max)}\approx 32}$ MeV, which
corresponds to carbon $\mathrm{{}^{22}C}$. Fig.2 illustrates the proton
binding energies in nuclei on the neutron drip line $\mathrm{Q_{n}=0}$. Proton
binding energies in the used mass model are calculated as the following:

$\begin{split}\mathrm{Q_{p}(A,Z)}&=\mathrm{Q(A,Z)-Q(A-1,Z-1)}\\\
&\mathrm{\;-\mathinner{\\!\left\lvert
Q_{n}(A-1,Z-1)\right\rvert}}.\end{split}$ (5)

The correction for the negative neutron binding energies
$\mathrm{Q_{n}(A-1,Z-1)<0\,}$ of the nuclei is taken into account. With
further compression of the matter, beta reactions become non-equilibrium
$\mathrm{\varepsilon_{fe}>Q_{p0}^{(max)}}$ that can lead to reheating up to a
temperature of $\mathrm{T_{9}\sim 5}$, at which nuclear statistical
equilibrium is established. The maximum density at which matter consists of
nuclei located on the neutron drip line $\mathrm{Q_{n}=0}$ is:

$\displaystyle\rho_{\max}=\mu_{\mathrm{z}}\cdot
10^{6}\left(\frac{\varepsilon_{\mathrm{fe}}}{\mathrm{m}_{\mathrm{e}}\mathrm{c}^{2}}\right)^{3}=$
(6) $\displaystyle=\mu_{\mathrm{z}}\cdot
10^{6}\left(\frac{\mathrm{Q}_{\mathrm{p}0}^{(\max)}}{\mathrm{m}_{\mathrm{e}}\mathrm{c}^{2}}\right)^{3}\sim
10^{12}\text{ g $\cdot$ cm${}^{-3}$},$

where $\mathrm{\mu_{z}}\geq 4$ is the number of baryons per electron. The
paper [22] demonstrates that when one take into account pycnonuclear fusion
reactions with neutrons evaporation, the upper density limit of the non-
equilibrium layer can reach $\rho_{\max,p}\sim 5\cdot 10^{13}{\text{
g}}\cdot{\text{cm}}^{-3}$.

Figure 2: Proton binding energies of nuclei located on the neutron drip line
$\mathrm{Q_{n}=0}$ for $\mathrm{Z\in[0;\,83]}$. Numbers mark the atomic charge
$\mathrm{Z}$ at the top and the atomic mass $\mathrm{A}$ at the bottom.

To evaluate the minimal density, one can proceed as the following [11]. For
$\mathrm{Z>10}$, the proton binding energies $\mathrm{Q_{p0}}$ at the neutron
drip line $\mathrm{Q_{n}=0}$ can be approximated by the linear function [23]:

$\mathrm{\mathrm{Q_{p0}=29-\frac{Z}{8}}\text{ MeV}},$ (7)

where the relation $\mathrm{A\approx 3.4Z}$ is satisfied. We find an
approximate dependence of the chemical composition on density in the following
form:

$\mathrm{\mathrm{Z=8\bigg{[}29-0.511\bigg{(}10^{-6}\rho/\mu_{z}\bigg{)}^{1/3}\bigg{]}}}.$
(8)

According to the equation (8) it follows that both values $\mathrm{(A,Z)}$
grow with decreasing density. At high values of $\mathrm{(A,Z)}$ nuclei become
unstable with respect to fission and alpha decay reactions. As a result of
spontaneous fission, the total number of nuclei increases continuously until
all free neutrons are absorbed by them. According to the paper [23] for nuclei
at the neutron drip line $\mathrm{Q_{n}=0}$, the evaluated half-life is:

$\mathrm{\log T_{1/2}=157-0.93Z}.$ (9)

We find out that for $\mathrm{Z>153}$ the decay half-life is less than
$\mathrm{10^{7}}$ years. Using the expression (7) for $\mathrm{Z=153}$, we
obtain that $\mathrm{Q_{p0}^{(min)}\approx 8}$ MeV. Similarly to the
expression (LABEL:rhomax), we have the following estimation for the minimum
density:

$\mathrm{\rho_{min}=\mu_{z}\cdot
10^{6}\bigg{(}\frac{Q_{p0}}{m_{e}c^{2}}\bigg{)}^{3}\sim 10^{10}\text{ g
$\cdot$ cm${}^{-3}$}}.$ (10)

Here we consider formation of the non-equilibrium composition during cooling
for the densities $\rho=10^{11}$ and $\rho=10^{13}$ g $\cdot$ cm -3.

3\. NUCLEI PROPERTIES
Using the finite range droplet model [21] as a mass model, we obtain the
localization of the neutron drip line (Fig. 1) and the proton binding energies
at this boundary (Fig. 2). Within the framework of this model, the nucleus
binding energy is described as:

$\begin{split}&\mathrm{\Delta
E(A,Z)=Z(m_{p}-m_{\mu})c^{2}+a_{3}A^{1/3}B_{\mathrm{k}}}\\\
&\mathrm{+(A-Z)(m_{n}-m_{\mu})c^{2}+a_{0}A^{0}+c_{1}\frac{Z^{2}}{A^{1/3}}B_{3}}\\\
&\mathrm{+\left(-a_{1}+J\bar{\delta}^{2}-\frac{1}{2}K\bar{\varepsilon}^{2}\right)A-c_{2}Z^{2}A^{1/3}B_{\mathrm{r}}}\\\
&\mathrm{+\left(a_{2}B_{1}+\frac{9}{4}\frac{J^{2}}{Q}\bar{\delta}^{2}\frac{B_{\mathrm{s}}^{2}}{B_{1}}\right)A^{2/3}-c_{4}\frac{Z^{4/3}}{A^{1/3}}}\end{split}$
$\begin{split}&\mathrm{+W\left(|I|+\left\\{\begin{array}[]{ll}1/\mathrm{A},&\mathrm{Z}=\mathrm{N}\text{
and both odd}\\\ 0,&\text{even }\end{array}\right)\right.}\\\
&\mathrm{-c_{5}Z^{2}\frac{B_{\mathrm{W}}B_{\mathrm{s}}}{B_{1}}-c_{\mathrm{a}}(N-Z)+f_{0}\frac{Z^{2}}{A}-a_{\mathrm{el}}Z^{2.39}}\\\
&\mathrm{+\left\\{\begin{array}[]{ll}+\bar{\Delta}_{\mathrm{p}}+\bar{\Delta}_{\mathrm{n}}-\delta_{\mathrm{np}},&\mathrm{Z}\text{
and }\mathrm{N}\text{ odd}\\\ +\bar{\Delta}_{\mathrm{p}},&\mathrm{Z}\text{ odd
and }\mathrm{N}\text{ even}\\\ +\bar{\Delta}_{\mathrm{n}},&\mathrm{Z}\text{
even and }\mathrm{N}\text{ odd}\\\ +0,&\mathrm{Z}\text{ and }\mathrm{N}\text{
even},\end{array}\right.}\end{split}$ (11)

where $\mathrm{m_{\mu}=1.660539\cdot 10^{-24}}$ g is the atomic mass unit.
$\mathrm{B_{i}}$ – functions that take into account the finite range of
nuclear forces, the shape of the nucleus, relative Coulomb energy, curvature,
surface and volume energies; $\mathrm{I=(A-2Z)/A}$ – relative neutron excess;
$\mathrm{a_{i},\;c_{i},\;J,\;Q,\;L}$, C, $\gamma$, W, h, $\mathrm{r_{max}}$,
$\mathrm{f_{0}}$ – constants; parameters
$\mathrm{\bar{\Delta}_{n},\;\bar{\Delta}_{p},\;\bar{\varepsilon},\;\bar{\delta}}$
and $\mathrm{\delta_{np}}$ – functions that take into account the even-odd
nuclei properties. A detailed description of each function can be found in the
[21] paper.

4\. NUMERICAL SIMULATION
A Fortran program was developed to simulate the cooling of a hot matter. The
explicit Runge-Kutta’s fourth order method with automatic step selection was
used. For numerical calculation, it is convenient to use dimensionless
expressions of the involved values. Introducing concentrations, the change in
each isotope due to nuclear and beta reactions is written as follows:

$\begin{split}&\mathrm{\frac{dY(A,Z)}{dt}}=-\lambda_{\beta-}(\mathrm{A},\mathrm{Z})\mathrm{Y}(\mathrm{A},\mathrm{Z})\\\
&+\lambda_{\beta-}(\mathrm{A},\mathrm{Z}-1)\mathrm{Y}(\mathrm{A},\mathrm{Z}-1)\\\
&-\lambda_{\beta+}(\mathrm{A},\mathrm{Z})\mathrm{Y}(\mathrm{A},\mathrm{Z})\\\
&+\lambda_{\beta+}(\mathrm{A},\mathrm{Z}+1)\mathrm{Y}(\mathrm{A},\mathrm{Z}+1)\\\
&-\lambda_{\text{cap
}\beta-}(\mathrm{A},\mathrm{Z})\mathrm{Y}(\mathrm{A},\mathrm{Z})\\\
&+\lambda_{\text{cap
}\beta-}(\mathrm{A},\mathrm{Z}+1)\mathrm{Y}(\mathrm{A},\mathrm{Z}+1)\\\
&-\lambda_{\text{cap
}\beta+}(\mathrm{A},\mathrm{Z})\mathrm{Y}(\mathrm{A},\mathrm{Z})\\\
&+\lambda_{\mathrm{cap}\beta+}(\mathrm{A},\mathrm{Z}-1)\mathrm{Y}(\mathrm{A},\mathrm{Z}-1)\\\
&-\lambda_{\mathrm{n\gamma}}(\mathrm{A},\mathrm{Z})\mathrm{Y}(\mathrm{A},\mathrm{Z})\\\
&+\lambda_{\mathrm{n}\gamma}(\mathrm{A}-1,\mathrm{Z})\mathrm{Y}(\mathrm{A}-1,\mathrm{Z})\end{split}$
$\begin{split}&-\lambda_{\gamma\mathrm{n}}(\mathrm{A},\mathrm{Z})\mathrm{Y}(\mathrm{A},\mathrm{Z})\\\
&+\lambda_{\gamma\mathrm{n}}(\mathrm{A}+1,\mathrm{Z})\mathrm{Y}(\mathrm{A}+1,\mathrm{Z})\\\
&-\lambda_{\mathrm{p}\gamma}(\mathrm{A},\mathrm{Z})\mathrm{Y}(\mathrm{A},\mathrm{Z})\\\
&+\lambda_{\mathrm{p}\gamma}(\mathrm{A}-1,\mathrm{Z}-1)\mathrm{Y}(\mathrm{A}-1,\mathrm{Z}-1)\\\
&-\lambda_{\gamma\mathrm{p}}(\mathrm{A},\mathrm{Z})\mathrm{Y}(\mathrm{A},\mathrm{Z})\\\
&+\lambda_{\gamma\mathrm{p}}(\mathrm{A}+1,\mathrm{Z}+1)\mathrm{Y}(\mathrm{A}+1,\mathrm{Z}+1).\end{split}$
(12)

Here we denote the reduced concentrations as
$\mathrm{Y}_{\mathrm{i}}=\mathrm{n}_{\mathrm{i}}/\left(\rho\cdot\mathrm{N}_{\mathrm{A}}\right)$,
$\mathrm{N_{A}}$ is Avogadro’s number. The two-particle reactions
$\mathrm{i}^{\eta}(\mathrm{j},\mathrm{\gamma})\mathrm{m}^{\nu}$ rates
$\lambda_{\mathrm{j\gamma}}$, where $\mathrm{j=n,p}$, and $\eta$ and $\nu$
take into account the states of the initial and final nuclei i and m,
respectively, are written in the form of
$\lambda_{\mathrm{j\gamma}}(\mathrm{A},\mathrm{Z})=\mathrm{Y}_{\mathrm{j}}\cdot\rho\cdot\mathrm{N}_{\mathrm{A}}\langle\sigma_{\mathrm{i}}\mathrm{v}\rangle$,
where $\mathrm{\sigma}$ is the reaction cross section and $\mathrm{v}$ is the
relative velocity of the interacting particles.

All isotopes in the range $\mathrm{Z\in[0;83]}$ with nuclear data available
are initially included in the system and distributed in accordance with (2)
for the given initial $\mathrm{\rho}$, $\mathrm{T_{9}}$ and the ratio of the
total number of protons to neutrons:

$\mathrm{R=\frac{n_{p}+\sum Z\cdot n(A,Z)}{n_{n}+\sum(A-Z)\cdot n(A,Z)}}.$
(13)

At each time step after a temperature change $\mathrm{T_{9}(t)}$ and constant
density $\mathrm{\rho}$, a new chemical potential of electrons
$\mathrm{\mu_{e^{-}}}$, which is equal in modulus and opposite in sign to
chemical potential of positrons, is calculated. New concentrations of
electrons and positrons are calculated after. In order to find them it is
necessary to solve numerically the system of the equations, using Newton’s
method:

$\begin{cases}\mathrm{n_{e^{\pm}}=\frac{1}{\pi^{2}}\left(\frac{\mathrm{kT}}{\mathrm{c}\hbar}\right)^{3}\int\limits_{0}^{\infty}\frac{\mathrm{x}^{2}\mathrm{dx}}{1+\exp\left(\sqrt{\mathrm{x}^{2}+\alpha^{2}}-\beta_{\pm}\right)}},\\\
\mathrm{n_{e^{-}}=n_{e^{+}}+\sum\limits_{A,Z}n(A,Z)\cdot Z},\end{cases}$ (14)

with respect to implicitly defined variables
$\mathrm{\beta_{\pm}=\mu_{e^{\pm}}/kT}$, $n_{\pm}$, and given parameters
$\mathrm{\alpha=m_{e}c^{2}/kT}$ and $\mathrm{x=cp/kT}$. At low temperatures,
with strongly degenerate electron gas and the absence of positrons, the
following relations [14] have been used:

$\begin{cases}\mathrm{n_{e^{-}}=\sum\limits_{A,Z}n(A,Z)\cdot Z},\\\
\mathrm{\mu_{\mathrm{e}^{-}}\approx\varepsilon_{\mathrm{fe}}\left[1-\frac{1}{3}\left(\frac{\pi\mathrm{kT}}{\varepsilon_{\mathrm{fe}}}\right)^{2}\right]}.\end{cases}$
(15)

If the chemical potentials of electrons and positrons are known, one can find
the rates of beta reactions $\mathrm{W}$ and neutrino energy loss
$\mathrm{\Theta}$ per nucleus [24] in reactions (3):

$\begin{split}&\mathrm{W}^{(\mathrm{a})}=\frac{\mathrm{g}_{\mathrm{z^{\prime}}}}{\mathrm{g_{z}}}\frac{\ln
2}{\mathrm{Ft}_{1/2}}\left(\frac{\mathrm{kT}}{\mathrm{m}_{\mathrm{e}}\mathrm{c}^{2}}\right)^{5}\mathrm{I}_{2},\\\
&\mathrm{W}^{(\mathrm{b})}=\frac{\ln
2}{\mathrm{Ft}_{1/2}}\left(\frac{\mathrm{kT}}{\mathrm{m}_{\mathrm{e}}\mathrm{c}^{2}}\right)^{5}\mathrm{I}_{2}^{\prime},\\\
&\mathrm{\mathrm{\Theta}^{(\mathrm{a})}=\frac{\mathrm{g}_{\mathrm{z^{\prime}}}}{\mathrm{g}_{\mathrm{z}}}\frac{\ln
2}{\mathrm{Ft}_{1/2}}\left(\frac{\mathrm{kT}}{\mathrm{m}_{\mathrm{e}}\mathrm{c}^{2}}\right)^{6}\mathrm{m}_{\mathrm{e}}\mathrm{c}^{2}\mathrm{I}_{3}},\\\
&\mathrm{\Theta}^{(\mathrm{b})}=\frac{\ln
2}{\mathrm{Ft}_{1/2}}\left(\frac{\mathrm{kT}}{\mathrm{m}_{\mathrm{e}}\mathrm{c}^{2}}\right)^{6}\mathrm{m}_{\mathrm{e}}\mathrm{c}^{2}\mathrm{I}_{3}^{\prime}.\end{split}$
(16)

The integrals $\mathrm{I_{k}}$ and $\mathrm{I^{\prime}_{k}}$ are defined as:

$\begin{split}&\mathrm{I_{k}=\int\limits_{0}^{\infty}\frac{x^{k}\left(x+x_{0}\right)\sqrt{\left(x+x_{0}\right)^{2}-\alpha^{2}}}{1+\exp\left(x+x_{0}-\beta_{\pm}\right)}dx},\\\
&\mathrm{I_{k}^{\prime}=\int\limits_{0}^{x_{0}-\alpha}\frac{\left(x_{0}-\alpha-x\right)^{k}(x+\alpha)\sqrt{x^{2}+2\alpha
x}}{1+\exp\left(\beta_{\pm}-x-\alpha\right)}dx},\end{split}$ (17)

where we denote $\mathrm{x_{0}=c^{2}[m(A,Z)-m(A,Z^{\prime})]/kT}$;
$\mathrm{g_{i}}$ are statistical weights. The constants $\mathrm{Ft_{1/2}}$
are determined by the experimentally measured half-lives. The values of
$\mathrm{Ft_{1/2}}$ differ considerably for various transitions to excited
states of the same nuclei. All integral expressions in this work are
calculated using the Gauss method and the similar ones, utilizing from 5 up to
11 points with the corresponding weights [26, 27]. In the case of strong
degeneracy at low temperatures, the integral expressions (16) turn to the
following ones [24]:

$\begin{split}&\mathrm{W}^{(\mathrm{a})}=\frac{\mathrm{g}_{\mathrm{z^{\prime}}}}{\mathrm{g_{z}}}\frac{\ln
2}{\mathrm{Ft}_{1/2}}\mathrm{\int\limits_{\delta}^{u_{fe}}u\sqrt{u^{2}-1}(u-\delta)^{2}du},\\\
&\mathrm{W}^{(\mathrm{b})}=\frac{\ln
2}{\mathrm{Ft}_{1/2}}\mathrm{\int\limits_{u_{fe}}^{\delta}u\sqrt{u^{2}-1}(\delta-u)^{2}du},\\\
&\mathrm{\mathrm{\Theta}^{(\mathrm{a})}=\frac{\mathrm{g}_{\mathrm{z^{\prime}}}}{\mathrm{g}_{\mathrm{z}}}\frac{\ln
2}{\mathrm{Ft}_{1/2}}\mathrm{m}_{\mathrm{e}}\mathrm{c}^{2}\mathrm{\int\limits_{\delta}^{u_{fe}}u\sqrt{u^{2}-1}(u-\delta)^{3}du}},\\\
&\mathrm{\mathrm{\Theta}^{(\mathrm{b})}=\frac{\ln
2}{\mathrm{Ft}_{1/2}}\mathrm{m}_{\mathrm{e}}\mathrm{c}^{2}\mathrm{\int\limits_{u_{fe}}^{\delta}u\sqrt{u^{2}-1}(\delta-u)^{3}du}},\end{split}$
(18)

which are calculated analytically [24, 25].

An approximate equation for the non-equilibrium layer cooling is used in the
following form:

$\mathrm{\frac{3}{2}k\frac{dT}{dt}\cdot\sum\limits_{(A,Z)}n(A,Z)=\sum\limits_{(A,Z)}\Theta(A,Z)\cdot
n(A,Z)}.$ (19)

Only the volume energy losses of the non-equilibrium layer due to neutrino
radiation are taken into account. Considering the heating of matter due to
non-equilibrium beta processes, one can obtain a self-consistent case in which
the rates of neutrino energy losses are compensated by the non-equilibrium
heating. Accurate accounting for non-equilibrium heating of matter is not
considered in this work. It is assumed that non-equilibrium heating
compensates for neutrino energy losses at a temperature of
$\mathrm{T_{9}=0.1}$ and the matter is not cooled rapidly further.

At each time step with a constant density $\mathrm{\rho}$, beta reactions
change the temperature $\mathrm{T_{9}(t)}$ and the ratio of the total number
of protons to neutrons $\mathrm{R(t)}$. With a new $\mathrm{T}$ and
$\mathrm{R}$, considering nuclear statistical equilibrium conditions (2), the
implicit equations (2) and (14) are solved simultaneously, using Newton’s
method. New nuclear composition, values $\mathrm{\beta_{\pm}=\mu_{e^{-}}/kT}$
and $n_{\pm}$ are found. When the temperature drops to $\mathrm{T_{9}\sim
3-5}$ and below, the equilibrium (2) is violated due to the Coulomb barrier.
Equilibrium for protons and neutrons (2) is replaced with equilibrium for
neutrons only. Similarly to the Saha expression (2), one derive equation for
adjacent atomic mass numbers $\mathrm{A}$ in isotopic chain with fixed atomic
charge numbers $\mathrm{Z}$ as function of free neutrons concentration:

$\begin{split}&\mathrm{\frac{n(A,Z)}{n(A+k,Z)}=\bigg{(}\frac{A}{A+k}\bigg{)}^{\frac{3}{2}}\bigg{(}\frac{2}{n_{n}}\bigg{)}^{k}}\\\
&\mathrm{\times\bigg{(}\frac{m_{n}kT}{2\pi\hbar^{2}}\bigg{)}^{\frac{3}{2}k}\frac{g(A,Z)}{g(A+k,Z)}}\\\
&\mathrm{\times\exp\bigg{(}\frac{Q(A,Z)-Q(A+k,Z)}{kT}\bigg{)}}.\end{split}$
(20)

The criterion for terminating the calculation is that the concentrations of
elements and most parameters reach constant values. It occurs in
$\mathrm{t\sim 2\cdot 10^{2}}$ s for the problem considered.

Degenerate electrons make the key contribution to the pressure for the chosen
conditions. We evaluate the pressure according to [28] in the following form:

$\mathrm{P\approx 1.2\cdot
10^{15}\bigg{(}\frac{\rho}{\mu_{z}}\bigg{)}^{4/3}}.$ (21)

A pressure difference at the boundaries of the non-equilibrium layer between
the densities $\mathrm{\rho_{max}}$ and $\mathrm{\rho_{min}}$ is:

$\mathrm{\Delta P=P_{max}-P_{min}\sim 10^{31}\text{ Ba}}.$ (22)

A mass evaluation of the non-equilibrium layer in the crust of the neutron
star envelope $\mathrm{M_{shell}}$ is found using the forces balance equation
in the envelope of a neutron star with mass $\mathrm{M_{NS}}$. That gives in a
flat layer approximation the following estimation: [11]

$\mathrm{M_{shell}=\frac{4\pi R^{4}}{GM_{NS}}\Delta P\approx 0.1\Delta P\sim
10^{30}\text{ g}}.$ (23)

To evaluate the nuclear energy stored in the non-equilibrium layer relative to
the equilibrium nuclear composition let us assume that $\mathrm{\Delta E\sim
7\cdot 10^{-3}\,m_{n}c^{2}}$ energy is released per nucleon [11] during the
transition to equilibrium. For the total energy stored we get:

$\mathrm{E\sim\frac{M_{shell}}{m_{\mu}}\cdot\Delta E\sim 10^{48}\text{ erg}}.$
(24)

The presence of such large nuclear energy reserves in the non-equilibrium
layer is sufficient to maintain the X-ray luminosity at the level of
$\mathrm{L=10^{36-37}}$ erg/s for tens of thousands of years, and can also
lead to powerful nuclear explosions accompanying starquakes of neutron stars
[29, 30, 31]. Taking into account pycnonuclear reactions that increase the
upper density boundary of the non-equilibrium layer by about an order of
magnitude [22], as well as lower mass neutron stars with a thicker non-
equilibrium layer [31], the nuclear energy reserve in the non-equilibrium
layer increases by more than an order of magnitude.

Figure 3: Time evolution of different parameter $\mathrm{R}$ values at density
of $\mathrm{log\rho=11}$ during cooling.

As a result of various neutron star activity processes, such as starquakes,
the matter of the non-equilibrium layer can be moved from inner layers into
the outer ones. Under conditions of low density, the degree of electron
degeneracy decreases, and the Pauli principle does not restrict beta reactions
anymore, during the time scale of which the stored nuclear energy can be
released in an explosive manner. The trigger of such an explosion can play
nuclear reactions, considered in [32], in the ejected part of the superheavy
nuclei matter that lead to chain nuclear fission, followed by the explosion.

5\. RESULTS AND DISCUSSION
Let us consider the results of a numerical calculation of the chemical
composition evolution. The initial temperature for all calculating tracks is
$\mathrm{T_{9}=10}$. For each density value $\mathrm{log\rho=11,\,13}$,
several values of the $\mathrm{R}$ parameter are selected. Only the most
common elements with mass fractions $\mathrm{x_{j}>10^{-4}}$ are presented on
the graphs that display the initial and final isotope distributions. Figures 3
and 11 correspond to different trajectories of the parameter $\mathrm{R}$ at
densities $\mathrm{log\rho=11}$ and $\mathrm{log\rho=13}$ respectively.

Figures 4 and 12 correspond to the Fermi energy trajectories
$\mathrm{\varepsilon_{fe}}$ for different parameter $\mathrm{R}$ and density
$\mathrm{\rho}$ values. Figures 5 \- 10 and 13 \- 18 represents the initial
and final chemical compositions of the shell layer for different parameter
$\mathrm{R_{0}}$ initial values at the densities $\mathrm{log\rho=11}$ and
$\mathrm{log\rho=13}$ respectively. All possible isotopes
$\mathrm{Z\in[0;\;83]}$ that are in the equilibrium described by the Saha
equation (2) were chosen as seed nuclei. The density does not change during
the calculations. The captions next to the labels in the figures mark the
atomic mass numbers $\mathrm{A}$ of the isotopes.

Figure 4: Time evolution of the Fermi energies $\mathrm{\varepsilon_{fe}}$
corresponding to different parameter $\mathrm{R}$ initial values at density of
$\mathrm{log\rho=11}$ during cooling. Figure 5: Initial chemical composition
at $\mathrm{log\rho=11}$, $\mathrm{T_{9}=10}$ and $\mathrm{R_{0}=0.05}$. Each
isotope is marked by the atomic mass number $\mathrm{A}$. Figure 6: Chemical
composition at the last calculated point $\mathrm{t=200}$ s for
$\mathrm{log\rho=11}$, $\mathrm{T_{9}=10}$ and $\mathrm{R_{0}=0.05}$. Each
isotope is marked by the atomic mass number $\mathrm{A}$. Figure 7: Initial
chemical composition at $\mathrm{log\rho=11}$, $\mathrm{T_{9}=10}$ and
$\mathrm{R_{0}=0.20}$. Each isotope is marked by the atomic mass number
$\mathrm{A}$. Figure 8: Chemical composition at the last calculated point
$\mathrm{t=200}$ s for $\mathrm{log\rho=11}$, $\mathrm{T_{9}=10}$ and
$\mathrm{R_{0}=0.20}$. Each isotope is marked by the atomic mass number
$\mathrm{A}$. Figure 9: Initial chemical composition at $\mathrm{log\rho=11}$,
$\mathrm{T_{9}=10}$ and $\mathrm{R_{0}=0.33}$. Each isotope is marked by the
atomic mass number $\mathrm{A}$. Figure 10: Chemical composition at the last
calculated point $\mathrm{t=200}$ s for $\mathrm{log\rho=11}$,
$\mathrm{T_{9}=10}$ and $\mathrm{R_{0}=0.33}$. Each isotope is marked by the
atomic mass number $\mathrm{A}$. Figure 11: Time evolution of different
parameter $\mathrm{R}$ values at density of $\mathrm{log\rho=13}$ during
cooling. Figure 12: Time evolution of the Fermi energies
$\mathrm{\varepsilon_{fe}}$ corresponding to different parameter $\mathrm{R}$
initial values at density of $\mathrm{log\rho=13}$ during cooling. Figure 13:
Initial chemical composition at $\mathrm{log\rho=13}$, $\mathrm{T_{9}=10}$ and
$\mathrm{R_{0}=0.20}$. Each isotope is marked by the atomic mass number
$\mathrm{A}$. Figure 14: Chemical composition at the last calculated point
$\mathrm{t=200}$ s for $\mathrm{log\rho=13}$, $\mathrm{T_{9}=10}$ and
$\mathrm{R_{0}=0.20}$. Each isotope is marked by the atomic mass number
$\mathrm{A}$. Figure 15: Initial chemical composition at
$\mathrm{log\rho=13}$, $\mathrm{T_{9}=10}$ and $\mathrm{R_{0}=0.33}$. Each
isotope is marked by the atomic mass number $\mathrm{A}$. Figure 16: Chemical
composition at the last calculated point $\mathrm{t=200}$ s for
$\mathrm{log\rho=13}$, $\mathrm{T_{9}=10}$ and $\mathrm{R_{0}=0.33}$. Each
isotope is marked by the atomic mass number $\mathrm{A}$. Figure 17: Initial
chemical composition at $\mathrm{log\rho=13}$, $\mathrm{T_{9}=10}$ and
$\mathrm{R_{0}=0.40}$. Each isotope is marked by the atomic mass number
$\mathrm{A}$. Figure 18: Chemical composition at the last calculated point
$\mathrm{t=200}$ s for $\mathrm{log\rho=13}$, $\mathrm{T_{9}=10}$ and
$\mathrm{R_{0}=0.40}$. Each isotope is marked by the atomic mass number
$\mathrm{A}$.

As a result of calculations, a non-equilibrium composition of matter at a
given density was obtained that is formed during cooling. The most prevalent
nuclei have $\mathrm{A\geq 3Z}$ and are located next to the neutron drip line
with low neutron binding energies $\mathrm{Q_{n}\approx 0}$, and
$\mathrm{Q_{\beta}\approx\varepsilon_{fe}}$. The variety of resulting nuclei
decreases with a temperature drop. The final composition of matter depends on
the relative neutrons number of the initial composition.

In the case of a low density $\mathrm{log\rho=11}$, heavy nuclei are formed.
The initial Fermi energies $\mathrm{\varepsilon_{fe}}$ (Fig. 4) are less than
the beta decays energies of nuclei at the neutron drip line. As the
temperature drops, the nuclei capture neutrons and, passing to the
$\mathrm{Q_{n}\approx 0}$ boundary, undergo beta decay at
$\mathrm{Q_{\beta}(A,Z)>\varepsilon_{fe}}$, increasing their charge atomic
numbers $\mathrm{Z}$. The resulting nuclei also absorb neutrons, passing to
the neutron drip line and decaying. The parameter $\mathrm{R}$ value
increases. The Fermi energy increases (Fig. 3, 4) until it equals
$\mathrm{Q_{\beta}(A,Z)}$. Beta decays stop at this point. The final
composition strongly depends on the parameter $\mathrm{R_{0}}$ initial value,
which determines the ratio of the nuclei number to free neutrons number at the
transition from Saha equilibrium described by equation (2) to quasi-
equilibrium described by expression (20), when the total number of nuclei does
not change anymore. The less parameter $\mathrm{R_{0}}$ value
($\mathrm{R_{0}=0.05}$ case, Fig. 5 and 6), the heavier nuclei are formed
$\mathrm{A>230}$. The final chemical compositions for
$\mathrm{R_{0}=0.20,\,0.33}$ (Fig. 8 and 10) do not reach neutron drip line
and match $\mathrm{A\approx 3Z}$, what refers to $\mathrm{R\approx 0.5}$ (Fig.
3). The reason is that there is insufficient amount of free neutrons remaining
after the nuclear statistical equilibrium stage described by (2). The lower
free neutrons concentration $\mathrm{n_{n}}$, the fewer nuclei close to the
neutron drip line are formed (20).

In contrast, at high density $\mathrm{log\rho=13}$ light elements are formed.
The initial Fermi energy is $\mathrm{\varepsilon_{fe}>Q_{\beta}^{(max)}}$ and
only electron captures are available to nuclei in degenerate matter. As a
result, the nuclei decrease their charge atomic number $\mathrm{Z}$, passing
to the neutron drip line $\mathrm{Q_{n}\approx 0}$ and emitting neutrons. The
parameter $\mathrm{R}$ value decreases (Fig. 11) as well as $\mathrm{(A,Z)}$
until the reducing Fermi energy of electrons equals the threshold electron
capture energies of light nuclei
$\mathrm{\varepsilon_{fe}\approx\varepsilon_{\beta i}}$. As a result, the
final nuclei distribution consists of free neutrons, helium
$\mathrm{{}^{8}He}$, beryllium $\mathrm{{}^{14}Be}$, boron $\mathrm{{}^{19}B}$
and carbon $\mathrm{{}^{22}C}$ nuclei in different proportions that depend on
the initial $\mathrm{R_{0}}$ (Fig. 14, 16 and 18). In our calculations, the
neutrons have been considered in equilibrium with nuclei according to (20). So
the final composition consists of several elements near the neutron drip line.
A temperature drop below our limit of $\mathrm{T_{9}=0.1}$ leads to the
situation when neutrons photodetachment can not occur. In this case, the final
composition consists of only one element on the neutron drip line with
$\mathrm{\varepsilon_{fe}\approx Q_{p0}}$ [11]. Maximum value of
$\mathrm{Q_{p0}}$ in the used nucleus model corresponds to carbon
$\mathrm{{}^{22}C}$ with $\mathrm{Q_{p}(22,6)\approx 32}$ MeV.

6\. CONCLUSION
The analysis of the calculation results reveals that the problem of the
chemical composition evolution of a hot neutron star crust at subnuclear
densities, cooling due to neutrino energy loss, should be considered taking
into account the previous history of the system. To obtain real initial
conditions, it is necessary to have information on the dynamics of a
collapsing supernova core and the parameters of a hot neutron star being born.

The properties of the non-equilibrium layer obtained within the framework of
this model indicate the presence of a large reserve of neutron stars nuclear
energy, capable of maintaining the X-ray luminosity for a long period of time.

As a result of various stellar activity processes, such as starquakes,
elements from the shell can be carried outward, leading to the appearance of
explosive transient processes, as well as forming heavy elements, such as
transuranium elements, that are part of the next generation objects, including
the Earth. ACKNOWLEDGEMENTS
The authors are grateful to I. V. Panov for providing the nuclear data used in
this article. The work of G. S. Bisnovatyi-Kogan was carried out with the
partial support of the RFBR grant 20-02-00455, the work of A. Yu. Ignatovskiy
was supported by the RSF grant 21-12-00061.

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aainstitutetext: Institute for Nuclear Research of the Russian Academy of
Sciences,
117312 Moscow, Russiabbinstitutetext: Department of Physics, TU Dortmund,
D-44221 Dortmund, Germany

# A Light Mediator Relating Neutrino Reactions

I. Alikhanov b and E. A. Paschos<EMAIL_ADDRESS><EMAIL_ADDRESS>dortmund.de

###### Abstract

The extension of the standard model with a multiplicative $U(1)_{R}$ factor is
consistent with a light vector boson. In its simplest realization, only right-
handed particles carry charges of the new group. In this model, there is a
residual $\tau_{3R}$ symmetry and one new coupling constant which correlates
neutrino interactions. We compute new contributions to antineutrino–electron
scattering and coherent scattering on nuclei, and compare them with the
XENON1T result.

###### Keywords:

Neutrino, Beyond Standard Model, Spontaneous Symmetry Breaking

##  1 Introduction

In the standard model, right-handed neutrinos are very inert. They are
introduced in the couplings of Higgs particles and are observable in
interactions where scalar particles are present. There is also the possibility
that they modify relations of the standard model. In fact, this happens in
models where right-handed neutrinos are embedded in a larger group. A typical
and popular extension is based on the group $SU(2)_{L}\times U(1)_{Y}\times
U(1)_{R}$ with fermions carrying charges of the three groups. There are many
such extensions Appelquist:2002mw ; Fayet:1989mq ; Dutta:2019fxn ;
Dutta:2020enk ; Lindner:2018kjo ; DelleRose:2017xil ; Lindner:2020kko ;
Dev:2021otb ; our_e . They differ in the number of Higgs particles and many of
them include new fermions Dutta:2019fxn ; Dutta:2020enk  identified with dark
matter.

The simplest extension, considered here, includes one new gauge boson
$X_{\mu}$ and a singlet scalar particle $\sigma(x)$. It is shown that the
model is self-consistent and contains a residual $\tau_{3R}$ symmetry. Related
are models based on the same group and an enlarged Higgs sector Dutta:2019fxn
; Dutta:2020enk ; Lindner:2020kko . These models have additional interactions
and fermions identified with dark matter.

Masses for the particles are generated by the standard model Higgs $\phi(x)$
and the new scalar $\sigma(x)$ being charged in $U(1)_{R}$. Both scalars
contribute to the masses of the gauge bosons and mix the neutral bosons. A
consequence is the presence of neutral current interactions, which now also
include couplings with right-handed neutrinos, proportional to their
$U(1)_{R}$ charges. In addition, the new scalar $\sigma(x)$ generates the mass
for the $X_{\mu}$ boson.

Dirac masses for fermions of the standard model are generated by the vacuum
expectation value (VEV) of $\phi(x)$. For fermions with the same charge, only
the $SU(2)_{L}$ contributes to the Dirac mass term producing mass matrices
proportional to the couplings of the neutral Higgs mesons. Thus, Dirac mass
matrices and neutral Higgs couplings can be diagonalized at the same time,
eliminating flavor changing neutral currents. Majorana masses are introduced
for right-handed neutrinos by the VEV of $\sigma(x)$. Finally, all fermions
appear in the interactions with their specific helicities producing chiral
interactions. In explicit calculations, it is helpful to use Weyl spinors for
them.

An additional requirement for an ultraviolet complete theory is the
cancellation of triangle anomalies. The relations summarized in Table 1 are
necessary, but not sufficient, conditions for the theory to be ultraviolet
finite. In our case, the left-handed neutrinos do not couple to the $U(1)_{R}$
term and we select their charges to be zero: $Y_{e}=Y_{e_{L}}=0$ and
$Y_{u}=Y_{u_{L}}=0$. Applying these conditions, a $\tau_{3R}$ symmetry
survives for quarks and leptons, as shown in Section 3. An explicit
classification of the triangle diagrams demonstrates that the theory is
ultraviolet finite.

The presence of one coupling constant in the $U(1)_{R}$ group relates several
reactions. The elastic antineutrino–electron scattering still allows a small
new contribution. The TEXONO experiment gives data integrated over the
electron recoiling energy, but still as a function of the incident energy.
Integrating the cross section with the propagator of the mediator brings a
logarithmic dependence on the mass of the mediator. The bound we obtain for
the coupling constant at very low energies is consistent with the value
expected when the XENON1T events are produced by neutrinos originating in the
Sun. The third process is the coherent scattering of neutrinos on atomic
nuclei.

The content of the article is the following. Sections 2 and 3 describe the
model and the cancellation of triangle anomalies. The description concentrates
to one generation of fermions and must be repeated for three. Section 4 shows
how right-handed neutrinos couple to the light boson mediator through their
$U(1)_{R}$ charge. Finally, we discuss three neutral current reactions, where
low energies are preferable because propagator effects are more important.

##  2 The Model

###  2.1 The Gauge Sector

The new field $X_{\mu}$ couples to the standard Higgs doublet $\varphi(x)$
through the covariant derivative

$D_{\mu}\varphi(x)=\left(\partial_{\mu}+ig\frac{\tau_{3}}{2}\hat{W}^{3}_{\mu}+ig^{\prime}\frac{Y}{2}\hat{B}_{\mu}+ig_{X}\frac{Y_{\varphi}}{2}\hat{X}_{\mu}\right)\varphi(x).$
(1)

Mass terms for the three bosons are generated when $\varphi(x)$ acquires a
vacuum expectation value. An additional scalar Higgs $\sigma(x)$ is also
introduced to provide another mass term $(1/4)g_{X}^{2}Y^{\prime 2}v_{0}^{2}$
for the $X_{\mu}$ field with $Y^{\prime}$ being the charge of $\sigma(x)$ in
the $U(1)_{R}$ group. This brings a non-zero mass for the new gauge boson.
After symmetry breaking the square of

the mass matrix for the fields $\hat{W}^{3}_{\mu}$, $\hat{B}_{\mu}$ and
$\hat{X}_{\mu}$ has the form Fayet:1989mq ; Appelquist:2002mw ;
DelleRose:2017xil ; Dev:2021otb

$M^{2}=\frac{1}{4}\left(\begin{array}[]{ccc}g^{2}&-gg^{\prime}&-gg_{X}Y_{\varphi}\\\
-gg^{\prime}&g^{\prime 2}&g^{\prime}g_{X}Y_{\varphi}\\\
-gg_{X}Y_{\varphi}&g^{\prime}g_{X}Y_{\varphi}&g_{X}^{2}{\tilde{Q}_{X}}^{2}\\\
\end{array}\right)\frac{v^{2}}{2},$ (2)

where $Y_{X}=Y_{\varphi}$ is the hypercharge of the Higgs doublet in
$U(1)_{R}$ and

$M^{2}_{33}=\frac{1}{4}g_{X}^{2}{\tilde{Q}_{X}}^{2}\frac{v^{2}}{2}=\frac{1}{4}\left(g_{X}^{2}Y_{\varphi}^{2}+g_{X}^{2}Y^{\prime
2}\frac{v_{0}^{2}}{v^{2}}\right)\frac{v^{2}}{2}.$ (3)

In this specific extension, the photon remains massless and electromagnetism
is not affected.

The diagonalization of the mass matrix is well known, with the electromagnetic
field retaining its form in the standard model,

$A_{\mu}(x)=s_{W}\hat{W}^{3}_{\mu}(x)+c_{W}\hat{B}_{\mu}(x)$ (4)

and the charge operator being

$Q=\frac{\tau_{3}}{2}+\frac{Y}{2}.$ (5)

For the computation of the neutral current, it is useful to express the
initial gauge fields in terms of the physical fields $A_{\mu}$, $Z_{\mu}$ and
$X_{\mu}$:

$\hat{W}^{3}_{\mu}=s_{W}A_{\mu}+c_{\alpha}c_{W}Z_{\mu}+s_{\alpha}c_{W}X_{\mu},$
(6)

$\hat{B}_{\mu}=c_{W}A_{\mu}-c_{\alpha}s_{W}Z_{\mu}-s_{\alpha}s_{W}X_{\mu},$
(7)

$\hat{X}_{\mu}=-s_{\alpha}Z_{\mu}+c_{\alpha}X_{\mu},$ (8)

where $s_{W}=\sin\theta_{W}$, $c_{W}=\cos\theta_{W}$ with $\theta_{W}$ being
the Weinberg angle and $s_{\alpha}=\sin\alpha$, $c_{\alpha}=\cos\alpha$ define
a new mixing angle $\alpha$ given by Fayet:1989mq

$c_{\alpha}^{2}=\frac{g^{2}+g^{\prime 2}}{g^{2}+g^{\prime
2}+g_{X}^{2}Q_{X}^{2}}\approx\frac{g^{2}+g^{\prime 2}}{g^{2}+g^{\prime
2}+g_{X}^{2}Y_{\varphi}^{2}}.$ (9)

The diagonalization of the mass matrix (2)—in the limit $(v_{0}^{2}/v^{2})\ll
1$—provides the relation

$g_{X}c_{\alpha}Y_{\varphi}=\frac{gs_{\alpha}}{c_{W}}$ (10)

which will be useful later on.

| $Y_{SM}$ | $U(1)_{R}$
---|---|---
$\nu_{L}$, $e_{L}$ | -1 | $Y_{e}$
$u_{L}$, $d_{L}$ | 1/3 | $Y_{u}$
$N_{M}$ | 0 | $Y_{N_{R}}=Y_{e}+Y_{\varphi}$
$e_{R}$ | -2 | $Y_{e_{R}}=Y_{e}-Y_{\varphi}$
$u_{R}$ | 4/3 | $Y_{u_{R}}=Y_{u}+Y_{\varphi}$
$d_{R}$ | -2/3 | $Y_{d_{R}}=Y_{u}-Y_{\varphi}$
$\varphi$ | 1 | $Y_{\varphi}$
Table 1: $U(1)_{R}$ charges for leptons and quarks.

###  2.2 Fermion Sector

Fermion masses are generated by Yukawa couplings of the Higgs doublet $H(x)$
and the singlet $\sigma(x)$

$\mathcal{L}_{Y}=Y^{ij}_{e}\bar{l}_{Li}\varphi
e_{Rj}+Y^{ij}_{\nu}\bar{l}_{Li}\tilde{\varphi}\nu^{\prime}_{Rj}+Y^{ij}_{\sigma}\overline{{\nu^{\prime}}_{Ri}^{C}}\nu^{\prime}_{Rj}\sigma+h.c.$
(11)

with $Y_{e}$, $Y_{\nu}$ and $Y_{\sigma}$ being $3\times 3$ matrices. When the
neutral Higgs doublet acquires vacuum expectation value, it produces Dirac
masses. Since it is the only scalar contributing to Dirac masses, there are no
flavor changing interactions (NFCI). The two contributions produce a mass
matrix of the form

$\left(\bar{\nu}_{L}\,\,\,\overline{{\nu^{\prime}}_{R}^{C}}\right)\left(\begin{array}[]{cc}0&\tilde{m}_{D}\\\
\tilde{m}_{D}^{T}&M_{R}\\\
\end{array}\right)\left(\begin{array}[]{c}\nu^{C}_{L}\\\ {\nu^{\prime}}_{R}\\\
\end{array}\right)$ (12)

It has the structure of the seesaw matrix of type-I. The primed neutrino,
$\nu^{\prime}_{R}$, will be soon replaced by another state. The Majorana mass
matrix is symmetric and is diagonalized by a unitary matrix Doi:1980yb ;
Joshipura:2001ya .

The invariance of the Dirac mass terms under $U(1)_{R}$ transformations supply
relations among the charges of left- and right-handed leptons and quarks. The
sum of charges in each trilinear must be zero, thus giving the relations
Fayet:1989mq

$Y_{u_{R}}=Y_{u_{L}}+Y_{\varphi},$ (13)

$Y_{d_{R}}=Y_{u_{L}}-Y_{\varphi},$ (14)

$Y_{e_{R}}=Y_{e_{L}}-Y_{\varphi}.$ (15)

In our case, a Dirac mass for the neutrinos is generated from the Yukawa
coupling to the right-handed neutrino, giving

$Y_{N_{R}}=Y_{e_{L}}+Y_{\varphi}.$ (16)

In this manner, there are three independent $U(1)_{R}$ fermionic charges for
each generation summarized in Table 1.

In the table, we also included the standard model hypercharges denoted by
$Y_{SM}=2(Q-\tau_{3}/2).$ We mention that whenever all charges are active, the
charge assignments in Table 1, together with the relation $Y_{e}+3Y_{u}=0$,
are sufficient for the cancellation of anomalies.

The symmetric matrix $M_{R}$ is diagonalized by a unitary matrix $U_{R}$
Joshipura:2001ya

$U^{*}_{R}M_{R}U^{\dagger}_{R}=D=\mathrm{diag}(M_{1},M_{2},M_{3}).$ (17)

Now, redefining the right-handed neutrinos $\nu_{R}=U_{R}\nu^{\prime}_{R}$ and
the Dirac matrices $m_{D}=\tilde{m}_{D}U_{R}$, we obtain

$\left(\bar{\nu}_{L}\,\,\,\overline{{\nu}_{R}^{C}}\right)\left(\begin{array}[]{cc}0&m_{D}\\\
m_{D}^{T}&D\\\ \end{array}\right)\left(\begin{array}[]{c}\nu^{C}_{L}\\\
{\nu}_{R}\\\ \end{array}\right).$ (18)

Equation (12) may now be used to find the eigenvalues (masses) and
eigenfunctions of the mass matrix, because the $D$ matrix is diagonal. We
computed the secular equation for the two-family case. In this case, the
entire matrix is a $4\times 4$ matrix. A straightforward computation produces
a $4^{\text{th}}$ degree polynomial in $\lambda$. Its form has the structure

$\lambda f\left(\lambda,m_{Dij},D_{i}\right)+\mathrm{det}(m_{D}^{T}m_{D})=0$
(19)

with $f\left(\lambda,m_{Dij},D_{i}\right)$ being a cubic polynomial in
$\lambda$ with the coefficients being functions of elements of the Dirac
matrix $m_{Dij}$ and of $D_{i}$. Thus, whenever $\mathrm{det}(m_{D})=0$, then
one neutrino mass is zero, independent of values for the elements of the
Majorana matrix.

The eigenvalue equation (19) has the same structure also in models with more
than two families. This follows from the structure of the secular equation

$\mathrm{det}\left|\begin{array}[]{cc}-\lambda&m_{D}\\\ m_{D}^{T}&D-\lambda\\\
\end{array}\right|=0$ (20)

and the definition of a determinant.

In order to account for the very small masses of neutrinos, it is usually
assumed that all elements of the matrix $D$ are very large. This is not always
the case, because one eigenvalue of $m_{D}$ may be very small. For instance,
when one eigenvalue of $m_{D}$ is zero, the determinant of $m_{D}^{T}m_{D}$ is
also zero for any values of the elements in $D$. In fact, it has been shown
that in seesaw models of type-I one sterile neutrino can have a mass in the eV
range Branco:2019avf . Thus, light sterile neutrinos are possible.

##  3 Cancellation of Anomalies

In addition to the conditions required for the generation of fermion masses,
there are restrictions on the hypercharges from the cancellation of triangle
anomalies. In the $U(1)_{R}$ extension of the standard model there are general
conditions Appelquist:2002mw , which are consistent with the conditions in
Table 1. In our case, the left-handed states of quarks and leptons do not
couple to the $U(1)_{R}$ bosons and we must set $Y_{e}=Y_{u}=0$. This reduces
the three independent parameters of the table to one and introduces the
$\tau_{3R}$ symmetry

$Y_{e_{R}}=Y_{d_{R}}=-Y_{\varphi}\,\,\,\text{and}\,\,\,Y_{N_{R}}=Y_{u_{R}}=Y_{\varphi}.$
(21)

In addition, the Majorana couplings are "axial-vector" and in the loops when
the momenta of integrations are much larger than the masses, their
contributions are equivalent to those for Dirac fermions with a
$\gamma^{5}\gamma^{\mu}$ vertex.

We classify the diagrams according to the number of external $X_{\mu}$ fields.

(1) Diagrams with three external fields are shown in Figure 1. They cancel by
virtue of the condition

Figure 1: Triangle diagrams with three $X_{\mu}$ external fields.

$Y_{N_{R}}^{3}+Y_{e_{R}}^{3}=(-Y_{\varphi})^{3}+Y_{\varphi}^{3}=0\,\,\,\,\text{and}\,\,\,\,Y_{u_{R}}^{3}+Y_{d_{R}}^{3}=0.$
(22)

(2) For the case $Y_{SM}X^{2}$, the cancellation of anomalies requires the
condition

$V_{e_{R}}Y_{e_{R}}^{2}+3(V_{u_{R}}Y_{u_{R}}^{2}+V_{d_{R}}Y_{d_{R}}^{2})=-2Y_{e_{R}}^{2}+3\left(\frac{4}{3}Y_{u_{R}}^{2}-\frac{2}{3}Y_{d_{R}}^{2}\right)=0$
(23)

as it follows from Figure 2.

Figure 2: Triangle diagrams for the case $Y_{SM}X^{2}$.

(3) For $Y_{SM}^{2}X$, the conditions required for the cancellation are

$V_{e_{R}}^{2}Y_{e_{R}}+3(V_{u_{R}}^{2}Y_{u_{R}}+V_{d_{R}}^{2}Y_{d_{R}})=4Y_{e_{R}}+\left(\frac{16}{3}Y_{u_{R}}+\frac{4}{3}Y_{d_{R}}\right)=0.$
(24)

The corresponding diagrams are shown in Figure 3.

Figure 3: Triangle diagrams for the case $Y_{SM}^{2}X$.

(4) For $(QCD)^{2}X$, the cancellations require

$Y_{u_{R}}+Y_{d_{R}}=0.$ (25)

(5) For $(Gravity)^{2}X$, the diagrams in Figure 4 lead to the condition

$Y_{N_{R}}+Y_{e_{R}}+3\left(Y_{u_{R}}+Y_{d_{R}}\right)=0.$ (26)

Figure 4: Triangle diagrams for the case $(Gravity)^{2}X$.

As we mentioned earlier, the charge assignments in Table 1 with the condition
$Y_{e}+3Y_{u}=0$ are sufficient for the cancellation of anomalies. Our
approach satisfies this requirement with additional condition $Y_{e}=Y_{u}=0$.

##  4 Structure of the Neutral Currents

Neutral current interactions for left-handed and right-handed quarks and
leptons are prescribed by the Lagrangian

$\mathcal{L}_{NC}=\bar{\Psi}_{L}i\gamma^{\mu}\left[ig\frac{\tau_{3}}{2}\hat{W}^{3}_{\mu}+ig^{\prime}\frac{Y}{2}\hat{B}_{\mu}\right]\Psi_{L}+\bar{\Psi}_{R}i\gamma^{\mu}\left[ig^{\prime}\frac{Y}{2}\hat{B}_{\mu}+ig_{X}\frac{Y_{\Psi}}{2}\hat{X}_{\mu}\right]\Psi_{R}.$
(27)

The hats on the fields denote flavor fields that are replaced, with the help
of (6)–(8), by physical gauge fields. The complete neutral current interaction
is now written in matrix form:

$\left(\begin{array}[]{c}\mathcal{L}_{NC}(Z)\\\ \mathcal{L}_{NC}(X)\\\
\end{array}\right)=-\left(\begin{array}[]{cc}Z_{\mu}\frac{gc_{\alpha}}{c_{W}}&\,\,-Z_{\mu}g_{X}s_{\alpha}\\\
X_{\mu}\frac{gs_{\alpha}}{c_{W}}&\,\,X_{\mu}g_{X}c_{\alpha}\\\
\end{array}\right)\left(\begin{array}[]{c}\bar{\Psi}_{L}\gamma^{\mu}\frac{\tau_{3}}{2}\Psi_{L}-s_{W}^{2}\sum\limits_{\Psi}\bar{\Psi}\gamma^{\mu}Q\Psi\\\
\sum\limits_{\Psi_{R}}\bar{\Psi}_{R}\gamma^{\mu}\frac{Y_{\Psi}}{2}\Psi_{R}\\\
\end{array}\right).$ (28)

The upper component in the column matrix on the right-hand side has the
structure of the standard model and the lower component is the new
contribution from the initial $\hat{X}_{\mu}$ field DelleRose:2017xil . The
$2\times 2$ matrix emphasizes the mixing between the gauge bosons. We notice
that the two contributions for the $X_{\mu}$ couplings are related through
(10) and are of the same magnitude. For the first generation of quarks, the
left-handed state is $\Psi_{L}^{T}=(u_{L}\,\,d_{L})$ and the right-handed
states are $u_{R}$ and $d_{R}$. The summation $\sum\limits_{\Psi}$ is over the
charged states and $\sum\limits_{\Psi_{R}}$ runs over $u_{R}$ and $d_{R}$
carrying the $U(1)_{R}$ charges $Y_{\varphi}$ and $-Y_{\varphi}$,
respectively. Introducing them in (28), we obtain the $\mathcal{L}_{NC}(X)$
interaction for quarks

$\displaystyle-\mathcal{L}_{NC}(X)=X_{\mu}\frac{g_{X}c_{\alpha}}{4}Y_{\varphi}\left[\bar{u}\gamma^{\mu}(1-\gamma^{5})u-\bar{d}\gamma^{\mu}(1-\gamma^{5})d-s_{W}^{2}\left(\frac{8}{3}\bar{u}\gamma^{\mu}u-\frac{4}{3}\bar{d}\gamma^{\mu}d\right)\right]$
$\displaystyle+X_{\mu}\frac{g_{X}c_{\alpha}}{4}Y_{\varphi}\left[\bar{u}\gamma^{\mu}(1+\gamma^{5})u-\bar{d}\gamma^{\mu}(1+\gamma^{5})d\right]$
$\displaystyle=X_{\mu}\frac{g_{X}c_{\alpha}}{4}Y_{\varphi}\left[2(\bar{u}\gamma^{\mu}u-\bar{d}\gamma^{\mu}d)-s_{W}^{2}\left(\frac{8}{3}\bar{u}\gamma^{\mu}u-\frac{4}{3}\bar{d}\gamma^{\mu}d\right)\right].$
(29)

The final couplings are vector in ordinary space and their isospin content is
mostly isovector.

For the leptons, we use left-handed states, and for the right-handed states,
we introduce the right-handed electron and the Majorana neutrino, obtaining
the couplings to $X_{\mu}$:

$\displaystyle-\mathcal{L}^{\text{leptons}}_{NC}(X)=X_{\mu}\frac{g_{X}c_{\alpha}}{4}Y_{\varphi}\left[\bar{u}_{\nu}\gamma^{\mu}(1-\gamma^{5})u_{\nu}-\bar{u}_{e}\gamma^{\mu}(1-\gamma^{5})u_{e}+4s_{W}^{2}\bar{u}_{e}\gamma^{\mu}u_{e}\right.$
$\displaystyle\left.+(\bar{u}_{N}\gamma^{\mu}\gamma^{5}u_{N}-\bar{u}_{e}\gamma^{\mu}(1+\gamma^{5})u_{e})\right]$
$\displaystyle=X_{\mu}\frac{g_{X}c_{\alpha}}{4}Y_{\varphi}\left[\bar{u}_{N}\gamma^{\mu}\gamma^{5}u_{N}+\bar{u}_{\nu}\gamma^{\mu}(1-\gamma^{5})u_{\nu}-2(1-2s_{W}^{2})\bar{u}_{e}\gamma^{\mu}u_{e}\right].$
(30)

There are two "axial-vector" terms, one for the Majorana particle and the
other for the normal neutrino. When both of them are present in a beam, they
contribute incoherently. The term
$\bar{u}_{\nu}\gamma^{\mu}(1-\gamma^{5})u_{\nu}$ is for the traditional
neutrino and is generated from the mixing of $X_{\mu}$ with the $Z_{\mu}$
boson.

##  5 A Bound for the Mixing Angle

The model is subject to constrains. The eigenvalues of the mass matrix (2) are
complicated algebraic functions, but the leading terms are simple and
sufficient for the analysis. The $Z$ boson mass has a new contribution

$m_{Z}=\frac{1}{2}v\sqrt{g^{2}+g^{\prime 2}+g_{X}^{2}Y_{\varphi}^{2}}.$ (31)

A limit on the mixing angle is obtained from the $\rho$ parameter, which is
modified

$\frac{m_{W}^{2}}{m_{Z}^{2}}=\frac{g^{2}}{g^{2}+g^{\prime
2}+g_{X}^{2}Y_{\varphi}^{2}}=c_{W}^{2}c_{\alpha}^{2}.$ (32)

The experimental value for the ratio

$\rho=\frac{m_{W}^{2}}{m_{Z}^{2}c_{W}^{2}}=c_{\alpha}^{2}=1.00037\pm 0.00023$
(33)

restricts the mixing angle to within two standard deviations.

$0.99991\leq c_{\alpha}^{2}\leq 1$ (34)

restricting $s_{\alpha}^{2}<8\times 10^{-5}$ or $s_{\alpha}\leq 9\times
10^{-3}\approx 0.01$. The value of $c_{\alpha}^{2}$ is very close to unity so
that the strengths of the neutral relative to the charged current coupling are
for practical purposes the same.

Combining (34) with the relation (10) leads to the upper bound

$g_{X}c_{\alpha}Y_{\varphi}<2.3\times 10^{-3}.$ (35)

We show later on that the upper bound produces substantial effects on
antineutrino–electron scattering and must be restricted even more.

##  6 Experimental Bounds

The fact that the model depends on a single new parameter
$(g_{X}c_{\alpha}/4)Y_{\varphi}$ relates several processes. We discuss three
reactions that are accessible to running experiments.

###  6.1 Elastic Neutrino–Electron Scattering

A new contribution arises in elastic scattering of neutrinos or antineutrinos
on electrons mediated by the exchange of the $X_{\mu}$ boson. The amplitude
for the process includes the $X_{\mu}$ propagator with $T_{e}$ the kinetic
energy of the recoiling electron

$\displaystyle\mathcal{M}_{e}=i\left(\frac{g_{X}c_{\alpha}}{4}\right)^{2}Y_{\varphi}^{2}\frac{2(1-2s_{W}^{2})}{m_{X}^{2}+2T_{e}m_{e}}\bar{u}_{\nu}\gamma^{\mu}(1-\gamma^{5})u_{\nu}\bar{u}_{e}\gamma_{\mu}u_{e}.$
(36)

The amplitude modifies the vector coupling to electrons. Several experiments
measured the process Vilain:1993kd ; Deniz:2009mu ; Park:2015eqa  including
the energy spectrum of the recoiling electrons. The TEXONO collaboration
investigated the region of lower energies $3<E_{\nu}<8$ MeV and reported the
results integrated over the recoiling energy of the electrons Deniz:2009mu .
The error bars for the integrated cross section are $25\%$. We demand that the
interference of the new amplitude with the amplitude from the standard model
is less than 20 %.

This gives the condition

$\frac{\dfrac{\sqrt{2}}{G}\lambda\int_{0}^{1}[(g_{V}-g_{A})+(g_{V}+g_{A})(1-x)^{2}]\dfrac{1}{m_{X}^{2}+sx}dx}{(g_{V}-g_{A})^{2}+\frac{1}{3}(g_{V}+g_{A})^{2}}\leq
0.1.$ (37)

In the relation we kept only the interference term and integrated over $x$.
For $m_{X}^{2}\leq s$, we use the exact integral, and for $m_{X}^{2}\gg s$, we
approximate the propagator by $1/m_{X}^{2}$. A consequence of the condition is
the upper bound for the coupling constant as a function of $m_{X}$. The bound
for the coupling constant

$\sqrt{\lambda}=\frac{g_{X}c_{\alpha}}{4}Y_{\varphi}$ (38)

as a function of $m_{X}$ and for $E_{\nu}=8$ MeV is shown in Figure 5, denoted
by elastic $\bar{\nu}e$ scattering. It is important to keep the mass $m_{X}$
because it defines the contribution for the lower limit of integration. We
note that the dependence of the coupling constant on the mass of the mediator
for $m_{X}^{2}\ll 2m_{e}T_{e}$ is smoother without a sharp break. For the
TEXONO experiment, we take $2m_{e}T_{e}=7$ MeV2, thus for $m_{X}\ll 1$ MeV the
bound for the coupling constant is almost independent of $m_{X}$.

The recoiling spectrum of the electrons will remain an important channel for
investigations beyond the standard model Ge:2017mcq ; for this reason there
are updates for the standard model parameters Canas:2016vxp  and also
extensive studies of radiative corrections Tomalak:2019ibg .

The XENON1T experiment Aprile:2020tmw  reported an excess of events at
recoiling energies $2.5~{}\text{keV}<T_{e}<30~{}\text{keV}$. The origin of the
excess has been interpreted in various ways. One of them attributes the excess
to the scattering of solar neutrinos on atomic electrons Lindner:2020kko ;
Boehm:2020ltd ; AristizabalSierra:2020edu ; Khan:2020vaf . Since the events
are at very low energies they can be produced by the light mediator discussed
in this article. Thus we extended our calculation our_e  of the
neutrino–electron scattering in Figure 5 down to 10 keV with an upper limit
for the coupling constant $\sqrt{\lambda}\leq 8\times 10^{-7}$. This is in
agreement with the range $8\times 10^{-7}<\sqrt{g_{e}g_{V}}<10^{-6}$ given in
the articles Lindner:2020kko ; Boehm:2020ltd ; AristizabalSierra:2020edu ;
Khan:2020vaf  for $10~{}\text{keV}\leq m_{X}\leq 100~{}\text{keV}$.

Figure 5: Upper limits for the coupling constant as functions of $m_{X}$
obtained from various experiments. The upper curve is from the
$\rho$-parameter; middle curve from COHERENT Akimov:2017ade  and lower curve
from elastic antineutrino–electron scattering Deniz:2009mu .

###  6.2 Coherent Elastic Neutrino Scattering on Nuclei

Many years after the original suggestion Freedman:1973yd , coherent elastic
scattering of neutrinos (CE$\nu$NS) on atomic nucleus was observed
Akimov:2017ade . The process is sensitive to new interactions and to the value
of the Weinberg angle Scholberg:2015 . A new measurement for the so-called
quenching factor of CsI(Na) Collar:2019ihs  reduced the experimental error and
motivated theoretical studies Papoulias:2019txv ; Khan:2019cvi ;
Giunti:2019xpr  that reanalyzed the data and obtained the value
$\sin^{2}{\theta_{W}}=0.238\pm 0.045$ for the Weinberg angle. In the previous
section, we used this value for determining the vector coupling $g_{V}$.

The dominant term for coherent scattering of neutrinos on atomic nuclei is
proportional to the baryonic current, which contributes coherently over the
entire nucleus. The new amplitude is

$\mathcal{M}_{C}=i\frac{2}{3}\frac{\lambda
s_{W}^{2}}{m_{X}^{2}+2MT}\bar{u}_{\nu}\gamma_{\mu}(1-\gamma^{5})u_{\nu}\left[\bar{u}\gamma^{\mu}u+\bar{d}\gamma^{\mu}d\right].$
(39)

This amplitude contributes coherently to the cross section, producing

$\frac{d\sigma}{dT}=\left[1+\frac{4\sqrt{2}}{3G}\frac{A}{Q_{W}}\frac{\lambda
s_{W}^{2}}{m_{X}^{2}+2MT}\right]^{2}\left(\frac{d\sigma}{dT}\right)_{\text{SM}}.$
(40)

Here, $(d\sigma/dT)_{\text{SM}}$ is the standard model cross section with the
recoiling kinetic energy of the nucleus $T$, $Q_{W}=[N-(1-4s_{W}^{2})Z]$ and
$A$ is the atomic number of the nucleus. The coupling constant $\lambda$ is
the same as in (38). The neutrinos in the experiment Akimov:2017ade ;
Scholberg:2015  are in the energy range
$16~{}\text{MeV}<E_{\nu}<53~{}\text{MeV}$ and we will use the median value
$E_{\nu}=30$ MeV, which gives

$2MT=4E_{\nu}^{2}=3.6\times 10^{3}~{}\text{MeV}^{2}.$ (41)

Allowing $1\sigma$ uncertainty in the Weinberg angle brings a change on the
value of the cross section larger than $20\%$. Thus, assuming the contribution
from the new amplitude to be smaller than $10\%$ of the amplitude from the
standard model restricts the coupling constant to be below the curve denoted
as COHERENT in Figure 5. Non-standard neutrino couplings to a light vector
mediator have been studied by other groups Kosmas:2017tsq ; Billard:2018jnl ;
Lindner:2016wff ; Abdullah:2018ykz  and limits were obtained for
phenomenological couplings. Their results are comparable to our bounds in
Figure 5. The limits from CE$\nu$NS are less stringent than those coming from
$\nu-e$ scattering; however, they are new and there is a lot of room for
significant improvements Kosmas:2017tsq ; Billard:2018jnl .

##  7 Summary

Neutrinos produced in laboratories or coming from astrophysical sources induce
reactions compatible with a light mediator. There are also many theoretical
models where a mediator is present. A popular family of models is based on the
group $SU(2)_{L}\times U(1)_{Y}\times U(1)_{R}$. These minimal extensions have
an additional gauge boson as the mediator and one or more scalar particles.

One such model was described in Sections 2 and 3 of the article borrowing
results from earlier publications Appelquist:2002mw ; Dutta:2019fxn ;
Lindner:2018kjo ; Lindner:2020kko ; Dev:2021otb , especially our_e . The model
has a new gauge boson, $X_{\mu}$, which mixes with the standard $Z_{\mu}$. We
selected a special assignment of quantum numbers: all left-handed fermions are
singlet in $U(1)_{R}$ and only right-handed fermions carry $U(1)_{R}$ charges.
As a result, there is only one new coupling constant (Equation (38)). Thus,
several reactions are related to each other, producing a fertile ground for
investigations. It is also natural to generate a mass matrix for neutrinos
with a see-saw mechanism of type-I. Furthermore, the mixing of the gauge
bosons produces a neutral current whose left-handed components have the
structure of the standard model, plus a new component from right-handed
states, including right-handed neutrinos (Equation (28)). When the two
components (left-handed and right-handed) refer to the same fermion, they add
up and produce a vector interaction.

Several neutral current reactions are modified due to the exchange of the
light mediator. Elastic neutrino–electron or antineutrino–electron scattering
experiments have an accuracy of 25 %, which provides an upper bound for the
coupling constant as a function of $m_{X}$. The new interaction contributes to
the vector coupling of the electron and modifies the recoiling spectrum of
electrons our_e . It has been proposed that the XENON1T signal may be
attributed to solar neutrinos from the $p-p$ process Lindner:2020kko ;
Boehm:2020ltd ; AristizabalSierra:2020edu ; Khan:2020vaf . In this case, there
is a value for the coupling from the $10~{}\text{keV}<m_{X}<100~{}\text{keV}$
region. Adopting this value for the coupling constant makes the cross sections
at recoiling energies of MeV very small.

A third process is coherent scattering of neutrinos on atomic nuclei
Akimov:2017ade ; Scholberg:2015 . The corresponding amplitude has a baryonic
current coupled to quarks, which enhances the cross section. The dependence of
the cross section on the small recoiling energy is the same as in the standard
model, and the signature will be a change in the magnitude of the cross
Section Kosmas:2017tsq ; Billard:2018jnl ; Lindner:2016wff ; Abdullah:2018ykz
.

This work is an improvement of the reference our_e  [arXiv:1902.09950]
indicating that

(i) chiral gauge invariance is maintained whenever the initial fermions are
chiral, and

(ii) the relevance of the XENON1T result.

###### Acknowledgements.

One of us (I.A.) was partly supported by the Program of fundamental scientific
research of the Presidium of the Russian Academy of Sciences "Physics of
fundamental interactions and nuclear technologies".

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