# Neutrino masses and self-interacting dark matter with mass mixing
$Z-Z^{\prime}$ gauge portal

Leon M.G. de la Vega<EMAIL_ADDRESS>Instituto de Física,
Universidad Nacional Autónoma de México, A.P. 20-364, Ciudad de México 01000,
México.  Eduardo Peinado<EMAIL_ADDRESS>Instituto de Física,
Universidad Nacional Autónoma de México, A.P. 20-364, Ciudad de México 01000,
México.  José Wudka<EMAIL_ADDRESS>Department of Physics and Astronomy,
UC Riverside, Riverside, California 92521-0413, USA.

###### Abstract

New light gauge bosons can affect several low-energy experiments, such as
atomic parity violation or colliders. Here, we explore the possibility that a
dark sector is charged under a new $U(1)$ gauge symmetry, and the portal to
the Standard Model is through a $Z-Z^{\prime}$ mass mixing. In our approach,
breaking the new gauge symmetry is crucial to generate neutrino masses. We
investigate the parameter space to reproduce neutrino masses, the correct dark
matter relic abundance, and to produce the observed core-like DM distribution
in galactic centers.

††preprint: APS/123-QED

##  I Introduction

The existence of dark matter and the non-zero neutrino masses are not
explained within the Standard Model (SM) of particle physics. Dark matter (DM)
is a component of the universe that accounts for $\sim 27\%$ of its total
matter-energy density [1]. No particle, fundamental or composite, in the SM
can account for it. A possible way to incorporate dark matter into the SM
framework is to extend the gauge group $G_{SM}=SU(3)_{C}\times SU(2)_{L}\times
U(1)_{Y}$ to include a dark gauge sector, under which the DM candidate is
charged. One of the simplest ways to extend the SM is with an extra abelian
gauge symmetry $U(1)_{D}$, which enlarges the gauge boson content of the SM.
The dark sector of such a theory may communicate with the SM particles via the
kinetic mixing term of the abelian subgroups $U(1)_{Y}$ and $U(1)_{D}$, the
mass mixing among the neutral gauge bosons, or the scalar sector. The
connection through the kinetic mixing is a popular and widely explored
paradigm known as the dark photon [2, 3, 4, 5, 6, 7], where the dark gauge
boson acquires vector couplings to the SM fermions. In the mass mixing case
[8, 9], the dark gauge boson acquires both vector and axial vector couplings
to the SM fermions, leading to possible signatures in parity-violation
experiments. This new gauge boson can provide a viable communication channel
between the DM and the SM, leading to the correct dark matter relic density
through the freeze-out while avoiding direct-detection constraints [10].
On the other hand, many neutrino mass generation mechanisms explain the
lightness of Majorana neutrinos, compared to the rest of SM fermions. Among
them, a popular class of models is the so-called seesaw mechanism, where a
large mass scale suppresses the electroweak scale in the neutrino masses,
giving rise to small neutrino masses. A popular type of these models requires
the introduction of right handed (RH) neutrinos, singlets of the SM. Different
mass models may be obtained, depending on the mass terms present in the
lagrangian after the breaking of $G_{SM}$ and any other additional symmetries
in the model, namely, the type-I seesaw [11, 12, 13, 14, 15], the linear
seesaw [16, 17, 18] or the inverse seesaw[19, 20]. Each of these models
results in different possible values for the heavy neutrino masses and active-
sterile neutrino mixings. In this work, we study the SM extended with an
anomaly-free $U(1)_{D}$ gauge symmetry. The fermions, charged under the new
gauge symmetry, will be identified with the right-handed neutrinos and the
dark matter candidate. An extra Higgs doublet, charged under the dark gauge
symmetry, generates a mass mixing among the dark gauge boson and the
electroweak neutral gauge boson. The right-handed neutrinos’ and Higgs fields’
charges shape the neutrino seesaw matrix [21]. Dark matter is connected to the
SM matter fields through the neutral gauge boson mass mixing, opening up
viable thermal freeze-out channels and signatures in direct detection
experiments.

##  II The model

$\begin{array}[]{||l|l|l|l|l|l|l|l|l|l||}\hline\cr&L&N&N^{\prime}&F&H_{1}&H_{2}&\phi&\chi_{L}&\chi_{R}^{c}\\\
\hline\cr SU(2)_{L}&2&1&1&1&2&2&1&1&1\\\ \hline\cr
U(1)_{Y}&-1/2&0&0&0&1/2&1/2&0&0&0\\\ \hline\cr U(1)_{D}&0&1&-1&0&0&1&-1&Q_{\tt
D}&-Q_{\tt D}\\\ \hline\cr\end{array}$

Table 1: Matter content of the dark matter with mass mixed $U(1)_{D}$ gauge
boson. The model contains two dark charges, we have chosen to absorb one of
them into the definition of the dark gauge coupling, leaving the dark matter
charge $Q_{D}$ free.

We consider the model for the mass mixing of a new gauge boson with the $Z$
boson, described by Table 1 111Note that the charge assignment under the new
$U(1)_{D}$ for $N$ and $N^{\prime}$ could take values different value as far
as they remain vector-like, but it is fixed by convenience. To have this, we
add to the Standard Model(SM) a new ”dark” gauge symmetry $U(1)_{D}$, a scalar
$SU(2)_{L}$ doublet $H_{2}$ charged under the new symmetry, and a scalar
singlet $\phi$ to trigger the symmetry breaking. The right handed neutrinos
are charged under $U(1)_{D}$. Therefore, at least two different sets of
vector-like fermions $N$ and $N^{\prime}$ are needed to cancel the anomalies.
We also need the inclusion of an extra fermion singlet under the new symmetry,
$F$, to break the lepton number.. The dark sector (stable after the SSB)
consists of a vector-like pair of fermions $\chi_{L}$ and $\chi_{R}$. Fermions
charged under $U(1)_{D}$ transform trivially under the SM gauge symmetry,
guaranteeing the cancellation of all mixed SM-dark anomalies. For each charged
fermion under $U(1)_{D}$, there is a fermion with an opposite charge, such
that the pure $U(1)_{D}$ and the $U(1)_{D}$-gravity anomalies vanish. The
right-handed neutrinos $N,~{}N^{\prime},~{}F$, participate in the seesaw
mechanism, with their $U(1)_{D}$ charges shaping the seesaw mass matrix. The
scalar sector will induce mass mixing among the electroweak and dark neutral
bosons, linking the SM fermions with the dark sector. The $\chi$ fields can
act as a dark matter candidate, interacting with SM fields through the mass-
mixed dark gauge boson. In this way, we show that the $U(1)_{D}$ can drive the
phenomenology of neutrino and dark matter. The neutral gauge boson mixing will
impact the quark and lepton physics, such as parity violation in polarized
electron-nucleon and electron-electron scattering.

###  II.1 Neutrino sector

The RH neutrinos are charged under the $U(1)_{D}$. To generate the Yukawa
Lagrangian, their charges must match that of the $H_{2}$ Higgs doublet. To
avoid extra Goldstone bosons, in the scalar sector we must have a term such as
$H_{2}H_{1}^{\dagger}\phi$ or $H_{2}H_{1}^{\dagger}\phi^{2}$. From these two
conditions, we conclude that the charge of $\phi$ equals one of the RH
neutrinos charges. The two RH neutrinos N and N’ have a Dirac mass term. In
contrast, there is no way to generate a Majorana mass for any of those fields
through the $\phi$ field. The only way to do so is to include an extra fermion
with no $U(1)_{D}$ charge. In this way, the Lagrangian density of the neutrino
sector is

$\mathcal{L}_{\nu}=Y^{\nu}_{1}\overline{L}\tilde{H}_{1}F+Y^{\nu}_{2}\overline{L}\tilde{H}_{2}N+M_{1}\overline{N^{c}}N^{\prime}+Y^{N}\overline{N^{c}}F\phi+Y^{N^{\prime}}\overline{N^{\prime
C}}F\phi^{*}+M_{F}\overline{F^{C}}F+h.c.$ (1)

After spontaneous symmetry breaking (SSB) the resulting neutrino mass matrix
in the $(\nu_{L},N,N^{\prime},F)$ basis is

$M=\begin{pmatrix}0&m^{D}_{2}&0&m^{D}_{1}\\\
(m^{D}_{2})^{T}&0&M_{1}&Y^{N}v_{\phi}\\\
0&(M_{1})^{T}&0&Y^{N^{\prime}}v_{\phi}\\\
(m^{D}_{1})^{T}&(Y^{N})^{T}v_{\phi}&(Y^{N^{\prime}})^{T}v_{\phi}&M_{F}\end{pmatrix},$
(2)

where $m^{D}_{i}=Y^{\nu}_{i}v_{i}$ are the Dirac mass matrices. The light
neutrino mass matrix is given by

$m_{light}=(m^{D}_{1})^{T}\alpha m^{D}_{1}+(m^{D}_{2})^{T}\beta
m^{D}_{1}+(m^{D}_{1})^{T}\delta m^{D}_{2}+(m^{D}_{2})^{T}\epsilon m^{D}_{2},$
(3)

where the $\alpha,\beta,\delta,\epsilon$ matrices are defined as

$\begin{split}\alpha=&(M_{F}+Y^{N}(M_{1}^{T})^{-1}(Y^{N^{\prime}})^{T}v_{\phi}^{2}+Y^{N^{\prime}}(M_{1})^{-1}(Y^{N})^{T}v_{\phi}^{2})^{-1},\\\
\\\
\beta=&((Y^{N})^{T}v_{\phi})^{-1}+((Y^{N})^{T}v_{\phi})^{-1}M_{1}(M_{1}^{T})^{-1}[Id+M_{1}^{T}(Y^{N}v_{\phi})^{-1}(Y^{N\prime}v_{\phi}M_{1}^{-1}(Y^{N}v_{\phi})^{-1}-M_{F})((Y^{N\prime}v_{\phi})^{T})^{-1}]^{-1},\\\
\\\
\delta=&-(M_{1}^{T})^{-1}(Y^{N\prime}v_{\phi})^{T}[M_{F}+Y^{N}(M_{1}^{T})^{-1}(Y^{N^{\prime}})^{T}v_{\phi}^{2}+Y^{N^{\prime}}(M_{1})^{-1}(Y^{N})^{T}v_{\phi}^{2}]^{-1},\\\
\\\
\epsilon=&-(Y^{N}v_{\phi})^{-1}[Y^{N\prime}v_{\phi}(M_{1}^{T})^{-1}[Id+M_{1}^{T}(Y^{N}v_{\phi})^{-1}(Y^{N\prime}v_{\phi}M_{1}^{-1}(Y^{N}v_{\phi})^{-1}-M_{F})((Y^{N\prime}v_{\phi})^{T})^{-1}]^{-1}+M_{F}\beta],\end{split}$
(4)

where $Id$ is the Identity Matrix. The minimal field content leading to two
massive light neutrinos is $(Nr(N)=2,Nr(N^{\prime})=2,Nr(F)=2)$. There are
several familiar limits to this framework:

  1. 1.

The type-I seesaw limit can be obtained when
${Y^{N},Y^{N^{\prime}}}\rightarrow 0$ or ${Y^{N},m^{D}_{2}}\rightarrow 0$ or
${Y^{N^{\prime}},m^{D}_{2}}\rightarrow 0$. The magnitude of light neutrino
masses is

$m_{\nu}\sim\frac{v_{1}^{2}}{M_{F}}.$ (5)

  2. 2.

When ${M_{F},m^{D}_{1}}\rightarrow 0$, the light neutrino masses take the form
of the inverse seesaw

$m_{\nu}\sim\frac{(m_{2}^{D})^{2}Y_{N^{\prime}}}{M_{1}Y_{N}}.$ (6)

  3. 3.

When ${M_{F},m^{D}_{2}}\rightarrow 0$, the light neutrino masses take a
inverse seesaw form

$m_{\nu}\sim\frac{(m_{D}^{1})^{2}M_{1}}{v_{\phi}^{2}Y_{N}Y_{N^{\prime}}}.$ (7)

.

  4. 4.

When ${Y_{N},m^{D}_{1}}\rightarrow 0$, the light neutrino masses take the form

$m_{\nu}\sim\frac{(m_{2}^{D})^{2}(Y_{N^{\prime}})^{2}v_{\phi}^{2}}{M_{1}^{2}M_{F}}.$
(8)

In this case, there is an extra suppression compared with the inverse seesaw
from the light (heavy) scale $v_{\phi}$ ($M_{F}$).

We will examine the viability of each limit, depending on the scale of
$v_{\phi}$ indicated by DM phenomenology in section (III.5).

###  II.2 Dark Sector

We choose as dark matter candidate a Dirac fermion $\chi=\chi_{L}+\chi_{R}$
with mass term

${\cal L}_{\chi}^{mass}=\frac{1}{2}M_{\chi}\overline{\chi}\chi.$ (9)

We choose $Q_{\tt D}$ so that Majorana mass terms are forbidden at any order
in perturbation theory, with the scalar content of Table 1. The condition to
keep the Dirac character of $\chi$ is

$Q_{\tt D}\not=\frac{m}{2},\quad m\in{\mathbbm{Z}}.$ (10)

Since $Q_{\tt D}\not=0$, $\chi$ couples to the dark gauge boson $X$; once the
gauge symmetry is broken this induces a coupling to both the physical $Z$
boson and the dark photon $Z^{\prime}$, see Eq. (18). For definiteness we will
choose $Q_{\tt D}=1/3$ that satisfies Eq. (10). A similar dark matter model is
described in [10].

###  II.3 Gauge sector

The $U(1)_{D}$ charges of the new fields, except for $\chi$, are equal in
magnitude. Therefore, we may redefine the gauge coupling and field charges
such that $Q=\pm 1$ for these fields. With this in mind, the kinetic terms of
the scalar fields in the model defined in Table 1 are

$\mathcal{L}_{SK}=\sum_{i=1}^{2}\left[(D_{\mu}H_{i})^{\dagger}(D^{\mu}H_{i})\right]+\left[(D_{\mu}\phi)^{\dagger}(D^{\mu}\phi)\right],$
(11)

where the covariant derivatives for the $SU(2)_{L}$ Higgs doublets are

$D_{\mu}H_{i}=(\partial_{\mu}+\frac{ig}{2}\vec{\tau}\cdot\vec{W}_{\mu}+\frac{ig^{\prime}}{2}B_{\mu}+ig_{\tt
D}Q_{i}X_{\mu})H_{i}\quad,$ (12)

with $Q_{1}=0$ and $Q_{2}=1$. The corresponding covariant derivative for the
$SU(2)_{L}$ scalar singlet is

$D_{\mu}\phi=(\partial_{\mu}-ig_{\tt D}X_{\mu})\phi\quad.$ (13)

After electroweak and dark symmetry breaking, $H_{1,2}$ and $\phi$ acquire
vacuum expectation values and we write

$H_{1}=\begin{pmatrix}H_{1}^{+}\cr(v_{1}+h_{1}+ia_{1})/\sqrt{2}\end{pmatrix}\,;\qquad
H_{2}=\begin{pmatrix}H_{2}^{+}\cr(v_{2}+h_{2}+ia_{2})/\sqrt{2}\end{pmatrix}\,;\qquad\phi=v_{\phi}+h_{\phi}+ia_{\phi}\,.$
(14)

The SM vacuum expectation value is $v_{\tt SM}^{2}=v_{1}^{2}+v_{2}^{2}$, and
we define $\tan\beta=v_{2}/v_{1}$. There are five vector bosons,
$W^{\pm}=(W_{1}\mp iW_{2})/\sqrt{2}$ correspond to the usual charged pair with
mass $gv_{\tt SM}/2$, one neutral gauge boson, $A=s_{\tt w}W_{3}+c_{\tt w}B$
(where $s_{\tt w}=\sin(\theta_{\tt w})$ and $\tan(\theta_{\tt
w})=g^{\prime}/g$) remains massless. For the remaining fields, let
$\tilde{Z}=c_{\tt w}W_{3}-s_{\tt w}B$, then the mass matrix for
$\\{\tilde{Z},\,X\\}$ becomes

$m_{\tilde{Z}\,X}^{2}=\frac{1}{4}\begin{pmatrix}g_{\tt z}^{2}v_{\tt
SM}^{2}&-2v_{2}^{2}\,g_{\tt D}g_{\tt z}\cr-2v_{2}^{2}\,g_{\tt D}g_{\tt
z}&4g_{\tt D}^{2}(v_{2}^{2}+v_{\phi}^{2}))\end{pmatrix},$ (15)

where $g_{\tt z}=\sqrt{g^{2}+g^{\prime 2}}$. The $\tilde{Z}$-$X$ mixing angle,
$\theta_{\tt X}$, is given by

$\tan 2\theta_{\tt X}=\frac{g_{\tt z}g_{\tt D}v_{2}^{2}}{\frac{1}{4}g_{\tt
z}^{2}v_{\tt SM}^{2}-g_{\tt D}^{2}(v_{2}^{2}+v_{\phi}^{2})}.$ (16)

Denoting the mass eigenstates by $Z$ and $Z^{\prime}$, the corresponding
masses are given by

$M^{2}_{Z/Z^{\prime}}=\frac{1}{8}g_{\tt z}^{2}v_{\tt SM}^{2}+\frac{1}{2}g_{\tt
D}^{2}(v_{2}^{2}+v_{\phi}^{2})\pm\\\ \frac{1}{8}\left\\{\left[g_{\tt
z}^{2}v_{\tt SM}^{2}-4g_{\tt
D}^{2}(v_{2}^{2}+v_{\phi}^{2})\right]^{2}+\left(4g_{\tt z}g_{\tt
D}v_{2}^{2}\right)^{2}\right\\}^{1/2}.$ (17)

The $\tilde{Z}$-$X$ mixing induces a coupling of the $Z^{\prime}$ to the
electroweak neutral current $J_{\tt NC}$, and a coupling of the $Z$ to the
dark current $J_{\tt DC}$ proportional to $\sin\theta_{\tt X}$; explicitly,

$\mathcal{L}_{\tt NC}=-eJ^{\mu}_{\tt EM}A^{\mu}-Z_{\mu}(\cos\theta_{\tt
X}\frac{g_{\tt z}}{2}J^{\mu}_{\tt NC}+\sin\theta_{\tt X}g_{\tt D}J^{\mu}_{\tt
DC})-Z^{\prime}_{\mu}(-\sin\theta_{\tt X}\frac{g_{\tt z}}{2}J^{\mu}_{\tt
NC}+\cos\theta_{\tt X}g_{\tt D}J^{\mu}_{\tt DC}),$ (18)

where $e=gs_{\tt w}$. The currents in Eq. ( 18) are

$J_{\tt EM}^{\mu}=\sum_{r}Q^{\tt
EM}_{r}\overline{f_{r}}\gamma^{\mu}f_{r}\quad,\quad J_{\tt
NC}^{\mu}=\sum_{r}t^{3}_{L}(r)\overline{f}_{r}\gamma^{\mu}(1-\gamma^{5})f_{r}-2s_{W}^{2}J^{\mu}_{\tt
EM}\quad,\quad J^{\mu}_{\tt DC}=\frac{g_{\tt
D}}{3}\overline{\chi}\gamma^{\mu}\chi+g_{\tt
D}(\overline{N}\gamma^{\mu}N-\overline{N^{\prime}}\gamma^{\mu}N^{\prime}),$
(19)

where $Q^{\tt EM}_{r}$ is the EM charge of the $f_{r}$ fermion and
$t_{L}^{3}(r)$ its weak isospin.

###  II.4 Scalar Sector.

The scalar potential for the model in Table 1 is given by

$\begin{split}V_{i}=&\mu_{1}^{2}H_{1}^{\dagger}H_{1}+\mu_{2}^{2}H_{2}^{\dagger}H_{2}+\mu_{\phi}^{2}\phi^{*}\phi+\kappa\phi^{*}H_{1}^{\dagger}H_{2}+\lambda_{1}(H_{1}^{\dagger}H_{1})^{2}+\lambda_{2}(H_{2}^{\dagger}H_{2})^{2}+\lambda_{3}(\phi^{*}\phi)^{2}\\\
&+\lambda_{4}(H_{1}^{\dagger}H_{1})(H_{2}^{\dagger}H_{2})+\lambda_{5}(H_{2}^{\dagger}H_{2})(\phi^{*}\phi)+\lambda_{6}(H_{1}^{\dagger}H_{1})(\phi^{*}\phi)+\lambda_{7}(H_{1}^{\dagger}H_{2})(H_{2}^{\dagger}H_{1}),\end{split}$
(20)

where the only complex coupling is $\kappa$, however by a field redefinition
it can be made real. From the 10 real scalar degrees of freedom, four
goldstone bosons are absorbed in the vector boson masses; the remaining six
correspond to a charged pair $H^{\pm}$, a pseudoscalar $A$, and three neutral
scalars. Using the notation of Eq. (14) the first three and their masses are
given by

$\displaystyle H^{+}\sim\sin\beta\,H_{1}^{+}-\cos\beta\,H_{2}^{+}\,,\hskip
91.04872ptM_{H^{+}}^{2}=\frac{1}{2}sv_{\phi}^{2}-\frac{1}{2}\lambda_{7}v_{\tt
SM}^{2},$ (21) $\displaystyle A\sim
v_{\phi}\sin\beta\,a_{1}-v_{\phi}\cos\beta\,a_{2}+\frac{1}{2}v_{\tt
SM}\sin(2\beta)\,a_{\phi}\,,\quad
M_{A}^{2}=\frac{1}{2}sv_{\phi}^{2}+\frac{\sin^{2}(2\beta)}{8}sv_{\tt SM}^{2},$
(22)

while in the $\\{h_{1},\,h_{2},\,h_{\phi}\\}$ basis the CP-even mass matrix is
given by

$M_{E}^{2}=\begin{pmatrix}2\lambda_{1}v_{1}^{2}-\frac{\kappa
v_{\phi}}{\sqrt{2}}\tan\beta&v_{1}v_{2}(\lambda_{4}+\lambda_{7})+\frac{\kappa
v_{\phi}}{\sqrt{2}}&v_{1}v_{\phi}\lambda_{6}+\frac{\kappa v_{2}}{\sqrt{2}}\\\
v_{1}v_{2}(\lambda_{4}+\lambda_{7})+\frac{\kappa
v_{\phi}}{\sqrt{2}}&2\lambda_{2}v_{2}^{2}-\frac{\kappa
v_{\phi}}{\sqrt{2}}\cot\beta&v_{2}v_{\phi}\lambda_{5}+\frac{\kappa
v_{1}}{\sqrt{2}}\\\ v_{1}v_{\phi}\lambda_{6}+\frac{\kappa
v_{2}}{\sqrt{2}}&v_{2}v_{\phi}\lambda_{5}+\frac{\kappa
v_{1}}{\sqrt{2}}&2\lambda_{3}v_{\phi}^{2}-\frac{\kappa
v_{1}v_{2}}{\sqrt{2}v_{\phi}}\end{pmatrix}.$ (23)

To simplify the expressions we defined

$s=-\frac{\sqrt{8}\,\kappa}{v_{\phi}\sin(2\beta)},$ (24)

which is positive since in our conventions $\kappa<0$. The scalar potential
must be bounded from below, leading to restrictions on the scalar couplings.
We have collected these restrictions in Appendix A. We note the existence of a
decoupling limit, where the scalar masses become much heavier than the
electroweak scale, save for the Higgs seen at LHC. This limit is achieved when
$v_{2}\rightarrow 0$ and $\kappa\rightarrow 0$, with $\mu_{2}^{2}$ setting the
heavy scalar scale. The decoupling limit drives the gauge boson mixing to
zero, making the $Z^{\prime}$ and dark matter invisible.

##  III Phenomenology

###  III.1 General constraints on light dark $Z^{\prime}$ bosons

The induced couplings of the dark $Z^{\prime}$ boson to the SM fermions can be
probed by a variety of experiments [8, 9, 23, 24, 25, 26]. The observables
measured by those experiments, constrain the parameter space in the
$\theta_{\tt X}-M_{Z^{\prime}}$ plane. The most relevant experimental
constraints are the following:

  * •

Atomic Parity Violation. As noted above, the mass mixing among the SM and $X$
bosons induces a $Z^{\prime}$ couplings to SM fermions (cf. Eq. (19)), which
inherit the parity violating nature of the SM $Z$ couplings. The $Z^{\prime}$
parity violating couplings may induce observable effects on low energy
experiments, when the mass of the $Z^{\prime}$ is comparable to the energy
scale of the experiment [27]. This parity-violating interaction of quarks and
leptons mediated by the $Z^{\prime}$ has been probed in atomic transitions of
Yb, Cs, Tl, Pb, and Bi. The measurement of the nuclear weak charge in these
experiments can be used to constrain the $Z^{\prime}$ couplings to the SM
fermions as a function of its mass [8]. The resulting constraint is
approximately [25, 26, 28]

$\sin\theta_{\tt X}\lesssim 5\times
10^{-5},~{}\text{for}~{}M_{Z^{\prime}}<40\text{
MeV}\,;\quad\text{and}\quad\frac{\sin\theta_{\tt
X}}{M_{Z^{\prime}}}\lesssim\frac{10^{-6}}{\text{
MeV}},~{}\text{for}~{}40\text{ MeV}<M_{Z^{\prime}}<100\text{ GeV}.$ (25)

  * •

Collider searches. As the $Z^{\prime}$ couples to the SM fermions, it can be
produced in a multitude of collider experiments. The $Z^{\prime}$-mediated
Drell-Yan production of muons in hadron colliders yield some of the strongest
constraints on the $Z^{\prime}$ couplings for masses below $M_{Z}$. Neutral
gauge boson production and decay to leptons, in association with photon
production has been searched for in $e^{+}e^{-}$ collisions, at BaBaR [29],
LHCb [30], among others. The constraint from colliders in the mass region,
$100\text{ MeV}<M_{Z^{\prime}}<80\text{GeV}$, is roughly [25, 26]

$\sin\theta_{\tt X}\lesssim 5\times 10^{-3}.$ (26)

  * •

Beam dump experiments. The production of neutral bosons in electron
bremsstrahlung processes in beam dumps has been probed, for example at the
NA64 [31], E141 [32], E137 [33] and E774 [34] experiments. The beam dump
limits on $\sin\theta_{\tt X}$ in the $Z^{\prime}$ gauge boson mass region,
$1\text{ MeV}<M_{Z^{\prime}}<500\text{ MeV}$ are roughly [25, 26]

$\sin\theta_{\tt X}\lesssim 3\times 10^{-8}\text{ or }\frac{\sin\theta_{\tt
X}}{(M_{Z^{\prime}}/\text{GeV})^{-1.2}}\gtrsim 3.3\times 10^{-7}.$ (27)

###  III.2 Dark matter relic density

We consider the thermal freeze-out to determine the DM relic density,
$\Omega_{\chi}$. We identify three scenarios which can result in a relic
density of dark matter in accordance with cosmological measurements,
$\Omega_{\chi}h^{2}\leq 0.1198$:

  1. 1.

When $M_{\chi}>M_{Z^{\prime}}$, the annihilation t-channel,
$\bar{\chi}\chi\rightarrow Z^{\prime}Z^{\prime}$, is kinematically allowed.
This leads to a relic density which only depends on the $Z^{\prime}$ boson
mass $M_{Z^{\prime}}$, the dark gauge coupling $g_{\tt D}$, and the dark
matter mass $M_{\chi}$. Numerically we find that for each value of
$M_{Z^{\prime}}$ there is a minimum value of $g_{\tt D}$ for which there is no
dark matter overabundance. We show this behavior in Figure 2. This scenario is
well-studied and is known in the literature as Secluded WIMP Dark Matter [35,
36, 37].

  2. 2.

When $M_{\chi}>M_{f}$, $f$ being a SM fermion, the Z-Z’ mixing allows the
s-channel annihilation of dark matter into $f$,
$\bar{\chi}\chi\rightarrow\bar{f}f$. In the resonant regions $M_{\chi}\sim
M_{Z/Z^{\prime}}/2$ the annihilation cross section can be enhanced enough to
reach the required value to result in an allowed relic density, while keeping
the value of $\theta_{\tt X}$ in the allowed region discussed in section III.1
[38]. In this channel in addition to $M_{Z/Z^{\prime}}$, $M_{\chi}$ and
$g_{\tt D}$, the Z-Z’ mixing angle $\theta_{\tt X}$ is a crucial parameter.

  3. 3.

When $2M_{\chi}>M_{Z(Z^{\prime})}+M_{S}$, where $S$ is one of the four neutral
scalars in the model, the s-channel Higgsstrahlung channel
($\bar{\chi}\chi\rightarrow SZ(Z^{\prime})$) is kinematically allowed. In this
channel, the scalar masses and mixing angles become relevant to the relic
density calculation. For dark matter masses above the $W$ boson mass, the
channel $\bar{\chi}\chi\rightarrow WW$ is open and can contribute
significantly to the dark matter relic density.

We illustrate the tree-level Feynman diagrams of these processes in Figure 1.

(a) t-channel annihilation into $Z^{\prime}Z^{\prime}$

(b) Z/Z’ mediated resonant annihilation into $\bar{f}f$

(c) Higgstrahlung annihilation

Figure 1: Dark matter annihilation channels for the determination of the
freeze-out relic density.

To calculate the relic density for this model, we have implemented the model
in SARAH [39] and micrOmegas [40, 41], scanning over a range of the free
parameters. We study the first two cases, as they lead to an interesting
interplay between the dark sector, the light gauge boson parameters and the
neutrino sector.

Figure 2: Case 1: t-channel annihilation into a $Z^{\prime}$ pair. Parameter
space in the $M_{\chi}-g_{\chi}$ plane (where $g_{\chi}=g_{\tt
D}\,\cos\theta_{\tt X}\approx g_{\tt D}$ ) excluded by relic density
overabundance. Area shaded in green results in relic density overabundance,
while the green line corresponds to $\Omega_{DM}=0.1195$. The area indicated
in Magenta with the label ”SIDM” corresponds to the region where dark matter
self-interactions are consistent with astrophysical observations. Within this
area, the blue region corresponds to a mediator mass $M_{Z^{\prime}}$ of 10
MeV, considering the uncertainty as estimated in the text.

###  III.3 Dark matter direct detection

Figure 3: Spin Independent direct detection cross section ($\sigma_{SI}$) as a
function of dark matter mass, for the $Z^{\prime}$ resonant case (s-channel).
In the green lines the product $sin\theta_{\tt X}g_{\tt D}$ is fixed to values
between $10^{-8}$ and $10^{-3}$ as indicated in the caption. The purple lines
show the experimental limit on $\sigma_{SI}$ set by the LZ [42] and CRESST-III
[43] experiments. The dashed magenta line shows the neutrino floor on Argon
[44]. Note that the LZ experiment target nucleus is Xenon, and it has not yet
reached the Xenon neutrino floor.

After excluding the parameter space where the relic density of $\chi$ does not
correspond to that of dark matter, we look at the Spin-Independent cross
section of dark matter with nucleons. The $Z-Z^{\prime}$ mass mixing leads to
tree-level dark matter-nucleon elastic scattering, a process searched for in
direct detection experiments. The scattering is mediated by $Z$ and
$Z^{\prime}$ exchange.
In the case where relic density is determined by t-channel
$\bar{\chi}\chi\rightarrow Z^{\prime}Z^{\prime}$ annihilations, there is no
correlation between dark matter relic density and the direct detection cross
section, as the relic annihilation cross section is independent of the
$Z-Z^{\prime}$ mixing angle at leading order. However, a bound on the Z-Z’
mixing angle may be derived from the limits on this cross section. For the
$M_{Z^{\prime}}^{2}<<M_{Z}^{2}$ limit, the spin independent $\chi$-nucleus
elastic scattering cross section is approximately [45]

$\sigma_{SI}^{Z,A}=\frac{\mu^{2}_{\chi N}\sin^{2}2\theta_{\tt
X}g_{X}^{2}Q_{\chi}^{2}}{4\pi
M_{Z^{\prime}}^{4}}\left[Z(2g_{SM}^{Zu}+g_{SM}^{Zd})+(A-Z)(g_{SM}^{Zu}+2g_{SM}^{Zd})\right],$
(28)

where $\mu^{2}_{\chi N}$ is the reduced $\chi$-nucleus mass, $g_{SM}^{Zq}$ is
the vector coupling of $q=u,d$ to the Z boson in the SM, and $Z$ and $A$ are
the electric charge and atomic mass number of the nucleon respectively. Using
the most stringent limits on dark matter direct detection we can list the
following constraints

  * •

$\frac{|g_{X}\sin 2\theta_{\tt X}Q_{\chi}|}{2}\leq\sim 10^{-10}$ at
$M_{Z^{\prime}}=10$ GeV and $M_{\chi}=30$ GeV from LZ [42].

  * •

$\frac{|g_{X}\sin 2\theta_{\tt X}Q_{\chi}|}{2}\leq\sim 10^{-8}$ at
$M_{Z^{\prime}}=100$ MeV and $M_{\chi}=1$ GeV from CRESST-III [43].

For the resonant Z’ channel case we observe a clear correlation between the
parameter product $g_{\tt D}sin\theta_{\tt X}$ and the SI cross section in Fig
3.

###  III.4 Dark Matter Self-Interactions

The dark gauge interaction provides a possible avenue for dark matter self-
interactions, which can account for several contradictions between
astronomical observations and the collisionless cold dark matter paradigm
[46]. Self-Interacting Dark Matter (SIDM) can explain the too big to fail,
core-cusp, diversity and missing satellites problems. Dark matter self-
interactions in this model consist of $\chi\chi\rightarrow\chi\chi$
scatterings mediated by the $Z^{\prime}$ boson. Defining

$\beta_{\chi}=\sqrt{1-\frac{4M_{\chi}^{2}}{s}}\,,\quad g_{\chi}=g_{\tt
D}\cos\theta_{\tt X},$ (29)

we obtain the self-interaction cross section $\sigma_{\tt SIDM}$ [47]

$\displaystyle\frac{\sigma_{\tt SIDM}}{M_{\chi}}$
$\displaystyle=\frac{g_{\chi}^{4}}{4\pi
sM_{\chi}}\Biggl{\\{}\frac{(2s+3M_{Z^{\prime}}^{2})s\beta_{\chi}^{2}+2(M_{Z^{\prime}}^{2}+2M_{\chi}^{2})^{2}}{2M_{Z^{\prime}}^{2}(M_{Z^{\prime}}^{2}+s\beta_{\chi}^{2})}$
(30) $\displaystyle\hskip
72.26999pt-\frac{(s\beta_{\chi}^{2}+2M_{Z^{\prime}}^{2})(3M_{Z^{\prime}}^{2}+4M_{\chi}^{2})+2(M_{Z^{\prime}}^{2}+2M_{\chi}^{2})^{2}-4M_{\chi}^{4}}{s\beta_{\chi}^{2}\left(2M_{Z^{\prime}}^{2}+s\beta_{\chi}^{2}\right)}\ln\left(1+\frac{s\beta_{\chi}^{2}}{M_{Z^{\prime}}^{2}}\right)\Biggr{\\}}\,;$
(31)

The requirements for SIDM are met in this model when this cross section is
enhanced by the kinematic condition $M_{\chi}\gg M_{Z^{\prime}}$, in the small
relative velocity $\beta_{\chi}$ regime.The SIDM cross section magnitude is
determined by astrophysical data, namely dwarf and low surface brightness
galaxies. A velocity-dependence of the cross section is obtained by the
different typical velocities of dark matter in each environment. The central
values of the cross sections and velocities are [48]

$\left.\frac{\sigma_{\tt SIDM}}{M_{\chi}}\right|_{\rm
galaxy}=1.9\frac{\text{cm}^{2}}{\text{gr}}\,,~{}\left.\frac{\sigma_{\tt
SIDM}}{M_{\chi}}\right|_{\rm
cluster}=0.1\frac{\text{cm}^{2}}{\text{gr}}\,;\quad\left.\beta_{\chi}\right|_{\rm
galaxy}=3.3\times 10^{-4}\,,~{}\left.\beta_{\chi}\right|_{\rm
cluster}=5.4\times 10^{-3}\,.$ (33)

Fitting to these values we find

$M_{Z^{\prime}}=\frac{M_{\chi}}{566}\,,\qquad
g_{\chi}=\left(\frac{M_{\chi}}{75\,\text{GeV}}\right)^{3/4}\,.$ (34)

We estimate the errors on the numerical coefficients of these expresions to be
a factor of 2.5 (e.g., that the first coefficient ranges from $566/2.5$ to
$2.5*566$). We find that these conditions can be met when dark matter relic
density is obtained through annihilations to $Z^{\prime}$ pairs. In Figure 2
we show the band where SIDM is viable. We note that there is an overlap
between the SIDM band and the line where $\chi$ accounts for dark matter
completely. In this scenario, the mass of the $Z^{\prime}$ is of order $\sim
10$ MeV, which is where low energy experiments are most sensitive.

###  III.5 Neutrino masses and $U(1)_{D}$ breaking scale

From the constraints on the dark sector parameters from the dark matter
phenomenology, we can infer the following possibilities for the neutrino mass
mechanism.
For all dark matter relic density channels, the inverse seesaw-like scenario

$m_{\nu}\sim\frac{(m_{D}^{2})^{2}Y_{N^{\prime}}}{M_{1}Y_{N}},$ (35)

and the type-I seesaw scenario

$m_{l}\sim\frac{v_{1}^{2}}{M_{F}},$ (36)

are viable with heavy neutrinos of canonical seesaw scale.
Of special interest are the cases where the neutrino masses are linked to the
$U(1)_{D}$ breaking scale, namely the limits considered in eqs. (7) and (8).
In the limit considered in eq. (8)

$m_{\nu}\sim\frac{(m_{D}^{2})^{2}Y_{N^{\prime}}v_{\phi}^{2}}{M_{1}^{2}M_{F}},$
(37)

a light neutrino masses of the order $\mathcal{O}(\text{eV})$ can be achieved
with $\mathcal{O}(\text{TeV})$ heavy neutrinos, as in the canonical inverse
seesaw. In this way the smallness of neutrino masses is linked to the
smallness of the $Z^{\prime}$ boson mass.

Lets consider now the limit in eq. (7). For the t-channel annihilation case,
the correct relic density is determined for values of $g_{\tt D}$ larger than
$3\times 10^{-4}$ for a dark matter mass of $\sim 1$ MeV, or $g_{\tt D}$
larger than $2\times 10^{-1}$ for a dark matter mass of $\sim 100$ GeV (see
Figure 2). In the light $Z^{\prime}$ paradigm, with small $Z-Z^{\prime}$
mixing we have

$M_{Z^{\prime}}>g_{\tt D}v_{\phi},$ (38)

and the t-channel dominated annihilation has the kinematic condition

$M_{\chi}>M_{Z^{\prime}}.$ (39)

These two equations rule out the inverse seesaw-like limit of neutrino masses

$m_{\nu}\sim\frac{(m_{D}^{1})^{2}M_{1}}{v_{\phi}^{2}Y_{N}Y_{N^{\prime}}},$
(40)

as the low scale of $v_{\phi}$ needed to obtain a light $M_{Z^{\prime}}$ would
result in either $\sim 1\text{GeV}$ scale sterile neutrinos with large mixings
with the active neutrinos, or neutrino Yukawa couplings much smaller than the
electron Yukawa coupling of the SM, calling into question the necessity of the
seesaw scheme.

##  IV Conclusions

Abelian gauge extensions of the SM with light gauge mediators have become
popular recently because they lead to interesting implications in low energies
experiments and observables. In this work we have studied a scenario where the
coupling of such a light gauge boson with the SM is generated through a mass
mixing with the $Z$ boson. We extended the model by including a DM fermion
charged under the new $U(1)_{D}$ symmetry which is automatically stable
without the inclusion of extra symmetries. In this way the DM phenomenology
further constrains the new gauge sector. The DM self-interactions are also
discussed and we present the parameter space where the cusp-core problem is
resolved. Focusing on the mass region for the $Z^{\prime}$ gauge boson where
low energy parity violation experiments are more sensitive, $M_{Z}\sim
10~{}\text{MeV}$ we have found the parameter space for the dark gauge
coupling, $g_{D}$ and mass mixing parameter to reproduce the DM relic
abundance and search for the region where the DM self-interaction explain the
approximately constant dark matter density in the inner parts of galaxies. We
have found an overlap to explain correctly the DM relic abundance and the DM
self-interaction in the DM mass region $\sim 2-7~{}\text{GeV}$ with $g_{D}$
above $\sim 0.01$. We also identify a scenario where there is no significant
DM self-interactions, but DM can be seen in direct detection experiments. The
neutrino masses are generated through the seesaw mechanism. In some cases, the
$U(1)_{D}$ breaking scale plays a crucial role in the neutrino mass
generation. In these scenarios it is possible to generate neutrino masses
through a low-energy seesaw.

##  Appendix A Tree-level stability of the potential

Let

$x_{1}=\sqrt{2}\,|H_{1}|^{2}\,,\quad x_{2}=\sqrt{2}\,|H_{2}|^{2}\,,\quad
x_{3}=\sqrt{2}\,|\phi|^{2},$ (41)

then the quartic part of the potential takes the form

$V_{4}=\frac{1}{2}\sum_{i=1}^{3}\lambda_{i}x_{i}+\eta_{3}x_{1}x_{2}+\eta_{1}x_{2}x_{3}+\eta_{2}x_{3}x_{1},$
(42)

where
$\eta_{1}=\lambda_{5}/2,\,\eta_{2}=\lambda_{6}/2,\,\eta_{3}=(\lambda_{4}+\zeta^{2}\lambda_{7})/2$
with $\zeta=|H^{\dagger}_{1}H_{2}|/(|H_{1}|\,|H_{2}|)$ (note that
$0\leq\zeta\leq 1$). The tree level stability conditions are then:
$\lambda_{i}>0$ and $\eta_{i}>-\sqrt{\lambda_{j}\lambda_{k}}$ with
$\\{i,\,j,\,k\\}$ a cyclic permutations of $\\{1,\,2,\,3\\}$; in addition,

  * •

If $\eta_{i}>0\,,~{}\,\eta_{j},\,\eta_{k}<0$, then
$\lambda_{i}\eta_{i}>\eta_{j}\eta_{k}-\sqrt{(\lambda_{i}\lambda_{j}-\eta_{k}^{2})(\lambda_{i}\lambda_{k}-\eta_{j}^{2})}$.

  * •

If $\eta_{i,j,k}<0$ then
$\lambda_{1}\lambda_{2}\lambda_{3}+2\eta_{1}\eta_{2}\eta_{3}>\lambda_{1}\eta_{1}^{2}+\lambda_{2}\eta_{2}^{2}+\lambda_{3}\eta_{3}^{2}$.

## Acknowledgements

The authors thank Luis J. Flores for kindly providing the neutrino floor data
and R. Ferro-Hernandez for useful discussions. LMGDLV thanks CONACYT for the
funding of his PhD studies. This work has been supported by the University of
California Institute for Mexico and the United States (UC MEXUS) (CN 18-128)
and the Consejo Nacional de Ciencia y Tecnología (CONACYT) (CN 18-128), by the
German-Mexican research collaboration grant SP 778/4-1 (DFG) and 278017
(CONACYT), the CONACYT CB-2017-2018/A1-S-13051, and DGAPA-PAPIIT IN107621.

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