# Novel imprint of a dark photon from the 3-3-1-1 model

Doan Minh Luong<EMAIL_ADDRESS>Phenikaa Institute for
Advanced Study, Phenikaa University, Yen Nghia, Ha Dong, Hanoi 100000, Vietnam
Phung Van Dong<EMAIL_ADDRESS>(corresponding author)
Phenikaa Institute for Advanced Study, Phenikaa University, Yen Nghia, Ha
Dong, Hanoi 100000, Vietnam  Nguyen Huy Thao<EMAIL_ADDRESS>Department of Physics, Hanoi Pedagogical University 2, Phuc Yen, Vinh Phuc,
Vietnam

###### Abstract

We investigate a dark photon that arises from the UV model based upon
$SU(3)_{C}\otimes SU(3)_{L}\otimes U(1)_{X}\otimes U(1)_{G}$ (3-3-1-1) gauge
symmetry, where the last three factors enlarge the electroweak symmetry
encompassing electric charge $Q=T_{3}-1/\sqrt{3}T_{8}+X$ and dark charge
$D=-2/\sqrt{3}T_{8}+G$. It is well-established that this model addresses the
questions of family number, neutrino mass, and dark matter. It is shown in
this work that if the 3-3-1-1 breaking scale is much bigger than the dark
charge breaking scale, the relevant dark gauge boson $Z^{\prime}$ is uniquely
imprinted at TeV, avoiding dangerous FCNC processes, obeying precision
electroweak measurements, as well as contributing to collider phenomena, even
if no kinetic mixing is presented. The dark matter observables are perhaps
governed by the dark charge breaking Higgs field instead of the dark photon.

##  I Introduction

The model based upon $SU(3)_{C}\otimes SU(3)_{L}\otimes U(1)_{X}$ (3-3-1)
gauge symmetry is well-motivated as predicting the family number to match that
of colors by anomaly cancellation [1, 2, 3, 4, 5]. The relevant 3-3-1 symmetry
has a special property as combining dark matter and normal matter in gauge
multiplets as well as coupling dark matter in pairs in minimal interactions
including the gauge interaction. In this model, the $B-L$ charge neither
commutes nor closes algebraically with gauge symmetry, which enlarges the
current gauge group to a complete symmetry $SU(3)_{C}\otimes SU(3)_{L}\otimes
U(1)_{X}\otimes U(1)_{N}$ (3-3-1-1), where $N$-charge is related to $B-L$ via
a $SU(3)_{L}$ charge [6, 7]. The setup of 3-3-1-1 symmetry provides a matter
parity as residual gauge symmetry stabilizing a dark matter candidate, since
dark matter carries a wrong $B-L$ number and is odd under the matter parity,
opposite to normal matter. This new model supplies neutrino mass via seesaw
mechanism [8, 9, 10, 11] or scotogenic mechanism [12, 13, 14, 15, 16].
Additionally, the cosmic inflation, baryon asymmetry, and kinetic mixing
effect are extensively investigated in [18, 21, 17, 19, 20, 22]. Since both
dark matter and normal matter carry a $B-L$ charge significantly governed by
known weak coupling, the dark matter observables are strictly constrained by
experimental detection [8, 9, 10, 11]. Another issue is that this kind of
model supplies dangerous FCNCs induced by both new neutral gauge bosons
$Z^{\prime},Z^{\prime\prime}$ [7, 8] confronting the current collider searches
(see, for instance, [23, 24]).

Alternative to $B-L$, the recent approach [25] proposes a dark charge $D$ for
dark matter, whereas normal matter carries no dark charge. In this case, the
dark charge neither commutes nor closes algebraically with gauge symmetry
enlarging it to $SU(3)_{C}\otimes SU(3)_{L}\otimes U(1)_{X}\otimes U(1)_{G}$
(called 3-3-1-1 too) for which $D=-2/\sqrt{3}T_{8}+G$. This interpretation of
dark charge yields naturally a scotogenic scheme for neutrino mass and dark
matter, instead of the canonical seesaw in $B-L$ setup. It is noteworthy that
this theory unifies a dark photon coupled to $D$ [26] with electroweak gauge
bosons in the same manner the electroweak theory [27, 28, 29] set for photon
and weak gauge bosons, called scotoelectroweak unification. Interestingly, if
the scale of scotoelectroweak unification is high, much bigger than the dark
charge scale at TeV, the 3-3-1-1 symmetry is reduced to $SU(3)_{C}\otimes
SU(2)_{L}\otimes U(1)_{Y}\otimes U(1)_{D}$, where $D$ as given is family
universal, similarly to $Y=-1/\sqrt{3}T_{8}+X$. This suppresses the large
FCNCs exchanged by new neutral gauge bosons, as well-established in 3-3-1 and
3-3-1-1 models, even if the dark photon associated with $D$ is at TeV.
Additionally, this unification of dark charge predicts precisely dark photon
couplings with normal fermions appropriate to the precision data, thus
predictive signals of dark photon at current colliders.

The rest of this work is organized as follows. In Sec. II, we reconsider the
scotoelectroweak model. In Sec. III, we obtain novel imprint of the dark
photon at TeV. In Sec. IV, we investigate the precision electroweak
measurements affected by dark photon. In Sec. V, we estimate dark photon
signals at LEPII and LHC. In Sec. VI, we revisit the issues of neutrino mass
and dark matter. Finally, we conclude this work in Section VII.

##  II  The scotoelectroweak unification

The standard model arranges left-handed fermions in doublets
$(\nu_{L},e_{L})\sim 2$ and $(u_{L},d_{L})\sim 2$ while it puts right-handed
fermions in singlets $e_{R}\sim 1$, $u_{R}\sim 1$, and $d_{R}\sim 1$.
Extending $SU(2)_{L}$ to $SU(3)_{L}$ correspondingly enlarges fermion doublets
to a fundamental triplet (3) or antitriplet ($3^{*}$), whereas fermion
singlets are retained. The $[SU(3)_{L}]^{3}$ anomaly cancellation requires the
number of triplets to equal that of antitriplets. This implies the family
number to be a multiple of color number, 3. Hence, the 3 families uniquely
exist ensuring QCD asymptotic freedom that limits the family number to be less
than or equal to 5.

The $SU(3)_{L}$ multiplets decompose as $3=2\oplus 1$ and $3^{*}=2^{*}\oplus
1$ under $SU(2)_{L}$. Enlarging usual fermion doublets ($2/2^{*}$) implies new
fermion singlets (1’s) being at bottom of $3/3^{*}$, such as

$\displaystyle\psi_{aL}=\begin{pmatrix}\nu_{aL}\\\ e_{aL}\\\
N_{aL}\end{pmatrix}\sim 3,\hskip 14.22636ptQ_{\alpha
L}=\begin{pmatrix}d_{\alpha L}\\\ -u_{\alpha L}\\\ D_{\alpha
L}\end{pmatrix}\sim 3^{*},\hskip 14.22636ptQ_{3L}=\begin{pmatrix}u_{3L}\\\
d_{3L}\\\ U_{3L}\end{pmatrix}\sim 3,$ (1)

where $a=1,2,3$ and $\alpha=1,2$ are family indices. Right-handed fermions are
$SU(3)_{L}$ singlets,

$e_{aR}\sim 1,\hskip 14.22636ptN_{aR}\sim 1,\hskip 14.22636ptu_{aR}\sim
1,\hskip 14.22636ptd_{aR}\sim 1,\hskip 14.22636ptD_{\alpha R}\sim 1,\hskip
14.22636ptU_{3R}\sim 1.$ (2)

We assume that $N_{a}$, $D_{\alpha}$, and $U_{3}$ possess a dark charge $D=1$,
$-1$, and 1, respectively, whereas all the normal fields have no dark charge,
i.e. $D=0$, as all collected in Tab. 1 [25]. Additionally, we assume that
$N_{a}$, $D_{\alpha}$, and $U_{3}$ have an electric charge $Q=0$, $-1/3$, and
$2/3$, respectively, like those of the 3-3-1 model with right-handed
neutrinos.

Particle | $\nu_{a}$ | $e_{a}$ | $N_{a}$ | $u_{a}$ | $d_{a}$ | $D_{\alpha}$ | $U_{3}$ | $\chi$ | $\eta_{1,2}$ | $\rho_{1,2}$ | $\varphi_{3}$ | $\eta_{3}$ | $\rho_{3}$ | $\varphi_{1,2}$ | $\phi$ | gluon | $\gamma$ | $Z$ | $Z^{\prime}$ | $Z^{\prime\prime}$ | $W$ | $X^{0}$ | $Y^{-}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$D$ | 0 | 0 | 1 | 0 | 0 | $-1$ | $1$ | 1 | 0 | 0 | 0 | $1$ | $1$ | $-1$ | $-2$ | 0 | 0 | 0 | 0 | 0 | 0 | $-1$ | $-1$
$P_{D}$ | $+$ | $+$ | $-$ | $+$ | $+$ | $-$ | $-$ | $-$ | $+$ | $+$ | $+$ | $-$ | $-$ | $-$ | $+$ | $+$ | $+$ | $+$ | $+$ | $+$ | $+$ | $-$ | $-$
Table 1:  Dark charge ($D$) and dark parity ($P_{D}$).

We derive $Q=\mathrm{diag}(0,-1,0)$ and $D=\mathrm{diag}(0,0,1)$, for a lepton
triplet $\psi_{L}$, which neither commute nor close algebraically with
$SU(3)_{L}$. New abelian charges $X$ and $G$ arise by symmetry principle,
yielding a 3-3-1-1 gauge symmetry,

$SU(3)_{C}\otimes SU(3)_{L}\otimes U(1)_{X}\otimes U(1)_{G},$ (3)

where the color group is also included, and $X,G$ determine $Q,D$, such as

$Q=T_{3}-\frac{1}{\sqrt{3}}T_{8}+X,\hskip
14.22636ptD=-\frac{2}{\sqrt{3}}T_{8}+G,$ (4)

where $T_{n}$ ($n=1,2,3,\cdots,8$) denotes $SU(3)_{L}$ charges.

The fermion representations under the 3-3-1-1 symmetry are given by

$\displaystyle\psi_{aL}\sim(1,3,-1/3,1/3),\hskip 14.22636ptQ_{\alpha
L}\sim(3,3^{*},0,-1/3),\hskip 14.22636ptQ_{3L}\sim(3,3,1/3,1/3),$ (5)
$\displaystyle e_{aR}\sim(1,1,-1,0),\hskip
14.22636ptN_{aR}\sim(1,1,0,1),\hskip 14.22636ptu_{aR}\sim(3,1,2/3,0),$ (6)
$\displaystyle d_{aR}\sim(3,1,-1/3,0),\hskip 14.22636ptD_{\alpha
R}\sim(3,1,-1/3,-1),\hskip 14.22636ptU_{3R}\sim(3,1,2/3,1).$ (7)

It is shown that all other anomalies vanish, as desirable [25].

The gauge symmetry breaking is derived by scalar fields,

$\displaystyle\eta$ $\displaystyle=$
$\displaystyle\begin{pmatrix}\eta^{0}_{1}\\\ \eta^{-}_{2}\\\
\eta^{0}_{3}\end{pmatrix}\sim(1,3,-1/3,1/3),\hskip
14.22636pt\rho=\begin{pmatrix}\rho^{+}_{1}\\\ \rho^{0}_{2}\\\
\rho^{+}_{3}\end{pmatrix}\sim(1,3,2/3,1/3),$ (8) $\displaystyle\varphi$
$\displaystyle=$ $\displaystyle\begin{pmatrix}\varphi^{0}_{1}\\\
\varphi^{-}_{2}\\\ \varphi^{0}_{3}\end{pmatrix}\sim(1,3,-1/3,-2/3),\hskip
14.22636pt\phi\sim(1,1,0,-2),$ (9)

which possess a VEV,

$\displaystyle\langle\eta\rangle$ $\displaystyle=$
$\displaystyle\begin{pmatrix}\frac{u}{\sqrt{2}}\\\ 0\\\ 0\end{pmatrix},\hskip
14.22636pt\langle\rho\rangle=\begin{pmatrix}0\\\ \frac{v}{\sqrt{2}}\\\
0\end{pmatrix},\hskip 14.22636pt\langle\varphi\rangle=\begin{pmatrix}0\\\ 0\\\
\frac{w}{\sqrt{2}}\end{pmatrix},\hskip
14.22636pt\langle\phi\rangle=\frac{\Lambda}{\sqrt{2}}.$ (10)

The dark charge values of scalar components are shown in Tab. 1.

The fields $\eta$, $\rho$, and $\varphi$ break $SU(3)_{L}\otimes
U(1)_{X}\otimes U(1)_{G}$ down to $U(1)_{Q}\otimes U(1)_{D}$, while the field
$\phi$ breaks $U(1)_{D}$ down to a dark parity $P_{D}$ [25],

$P_{D}=(-1)^{D}=(-1)^{-\frac{2}{\sqrt{3}}T_{8}+G}.$ (11)

Notice that $\eta^{0}_{3}$ and $\varphi^{0}_{1}$ do not develop a VEV as
prevented by dark parity conservation. A scalar field $\xi\sim(1,1,0,1)$ that
is odd under $P_{D}$ was included in [25] for scotogenic neutrino mass
generation, but it does not affect the current consideration and result and is
thus skipped. Given that $w\gg\Lambda\gg u,v$, the scheme of symmetry breaking
is

$SU(3)_{C}\otimes SU(3)_{L}\otimes U(1)_{X}\otimes U(1)_{G}$
---
$\downarrow w$
$SU(3)_{C}\otimes SU(2)_{L}\otimes U(1)_{Y}\otimes U(1)_{D}$
$\downarrow\Lambda$
$SU(3)_{C}\otimes SU(2)_{L}\otimes U(1)_{Y}\otimes P_{D}$
$\downarrow u,v$
$SU(3)_{C}\otimes U(1)_{Q}\otimes P_{D}$

Here, when $w\gg\Lambda$, the 3-3-1-1 model presents a UV-completion of the
often-studied dark charge, while $\Lambda\gg u,v$ ensures a consistency with
the standard model.

The total Lagrangian has the form,
$\mathcal{L}=\mathcal{L}_{\mathrm{kin}}+\mathcal{L}_{\mathrm{Yuk}}-V$. The
kinetic term is

$\displaystyle\mathcal{L}_{\mathrm{kin}}$ $\displaystyle=$
$\displaystyle\sum_{F}\bar{F}i\gamma^{\mu}D_{\mu}F+\sum_{S}(D^{\mu}S)^{\dagger}(D_{\mu}S)-\frac{1}{4}\sum_{A}A_{\mu\nu}A^{\mu\nu},$
(12)

where $F,S,A$ sum over fermion, scalar, gauge multiplets, respectively, for
which the covariant derivative $D_{\mu}$ is defined by

$D_{\mu}=\partial_{\mu}+ig_{s}t_{n}G_{n\mu}+igT_{n}A_{n\mu}+ig_{X}XB_{\mu}+ig_{G}GC_{\mu},$
(13)

where ($g_{s},\ g,\ g_{X},\ g_{G}$), ($G_{n\mu},A_{n\mu},B_{\mu},C_{\mu}$),
and ($t_{n},\ T_{n},\ X,\ G$) are couplings, gauge bosons, and charges
according to 3-3-1-1 groups, respectively.

The Yukawa Lagrangian is

$\displaystyle\mathcal{L}_{\mathrm{Yuk}}$ $\displaystyle=$ $\displaystyle
h^{e}_{ab}\bar{\psi}_{aL}\rho e_{bR}+h^{N}_{ab}\bar{\psi}_{aL}\varphi
N_{bR}+\frac{1}{2}h^{\prime N}_{ab}\bar{N}^{c}_{aR}N_{bR}\phi$ (14)
$\displaystyle+h^{d}_{\alpha a}\bar{Q}_{\alpha L}\eta^{*}d_{aR}+h^{u}_{\alpha
a}\bar{Q}_{\alpha L}\rho^{*}u_{aR}+h^{D}_{\alpha\beta}\bar{Q}_{\alpha
L}\varphi^{*}D_{\beta R}$ $\displaystyle+h^{u}_{3a}\bar{Q}_{3L}\eta
u_{aR}+h^{d}_{3a}\bar{Q}_{3L}\rho d_{aR}+h^{U}_{33}\bar{Q}_{3L}\varphi
U_{3R}+H.c.,$

while the scalar potential is

$\displaystyle V$ $\displaystyle=$
$\displaystyle\mu^{2}_{1}\rho^{\dagger}\rho+\mu^{2}_{2}\varphi^{\dagger}\varphi+\mu^{2}_{3}\eta^{\dagger}\eta+\lambda_{1}(\rho^{\dagger}\rho)^{2}+\lambda_{2}(\varphi^{\dagger}\varphi)^{2}+\lambda_{3}(\eta^{\dagger}\eta)^{2}$
(15)
$\displaystyle+\lambda_{4}(\rho^{\dagger}\rho)(\varphi^{\dagger}\varphi)+\lambda_{5}(\rho^{\dagger}\rho)(\eta^{\dagger}\eta)+\lambda_{6}(\varphi^{\dagger}\varphi)(\eta^{\dagger}\eta)$
$\displaystyle+\lambda_{7}(\rho^{\dagger}\varphi)(\varphi^{\dagger}\rho)+\lambda_{8}(\rho^{\dagger}\eta)(\eta^{\dagger}\rho)+\lambda_{9}(\varphi^{\dagger}\eta)(\eta^{\dagger}\varphi)+(f\epsilon^{ijk}\eta_{i}\rho_{j}\varphi_{k}+H.c.)$
$\displaystyle+\mu^{2}\phi^{\dagger}\phi+\lambda(\phi^{\dagger}\phi)^{2}+\lambda_{10}(\phi^{\dagger}\phi)(\rho^{\dagger}\rho)+\lambda_{11}(\phi^{\dagger}\phi)(\varphi^{\dagger}\varphi)+\lambda_{12}(\phi^{\dagger}\phi)(\eta^{\dagger}\eta).$

Above, $h$’s and $\lambda$’s are dimensionless couplings, whereas $\mu$’s and
$f$ are mass parameters.

##  III  Novel imprint of the dark photon

The new leptons $N_{1,2,3}$ gain a pseudo-Dirac mass at $w$ scale due to
$w\gg\Lambda$, while the new quarks $D_{1,2}$ and $U_{3}$ get a Dirac mass
also at $w$ scale. They are all integrated out. Concerning the scalar sector,
given that $f\sim w\gg\Lambda\gg u,v$, all new Higgs fields receive a large
mass at $w$ scale being thus integrated out, except a new neutral Higgs boson
$H^{\prime}\sim\Re(\phi)$ associated with dark charge breaking has a mass at
$\Lambda$ scale, the standard model Higgs boson $H\sim
u\Re(\eta^{0}_{1})+v\Re(\rho^{0}_{2})$ with a mass at weak scale, and massless
Goldstone bosons $G_{Z^{\prime}}\sim\Im(\phi)$, $G_{Z}\sim
u\Im(\eta^{0}_{1})-v\Im(\rho^{0}_{2})$, and $G^{\pm}_{W}\sim
u\eta^{\pm}_{2}-v\rho^{\pm}_{1}$ associated with $Z^{\prime}$, $Z$, and
$W^{\pm}$, respectively (cf. [8]).

Similarly, gauge bosons obtain a mass from
$\mathcal{L}\supset\sum_{S}(D^{\mu}\langle S\rangle)^{\dagger}(D_{\mu}\langle
S\rangle)$ for which substituting the VEVs, we get physical non-Hermitian
fields

$W^{\pm}=\frac{A_{1}\mp iA_{2}}{\sqrt{2}},\hskip
14.22636ptX^{0,0*}=\frac{A_{4}\mp iA_{5}}{\sqrt{2}},\hskip
14.22636ptY^{\mp}=\frac{A_{6}\mp iA_{7}}{\sqrt{2}},$ (16)

with respective masses,

$m^{2}_{W}=\frac{g^{2}}{4}(u^{2}+v^{2}),\hskip
14.22636ptm^{2}_{X}=\frac{g^{2}}{4}(u^{2}+w^{2}),\hskip
14.22636ptm^{2}_{Y}=\frac{g^{2}}{4}(v^{2}+w^{2}).$ (17)

Here, $W$ is identical to that of the standard model and $u^{2}+v^{2}=(246\
\mathrm{GeV})^{2}$, whereas $X,Y$ are heavy at $w$ scale and are thus
integrated out.

Concerning neutral gauge bosons, we first identify the photon field that is
coupled to the electric charge $Q=T_{3}-1/\sqrt{3}T_{8}+X$ to be

$A=s_{W}A_{3}+c_{W}\left(-\frac{t_{W}}{\sqrt{3}}A_{8}+\sqrt{1-\frac{t^{2}_{W}}{3}}B\right),$
(18)

where the sine of the Weinberg angle is matched by
$s_{W}=e/g=\sqrt{3}t_{X}/\sqrt{3+4t^{2}_{X}}$ with $t_{X}=g_{X}/g$, and notice
that the field in parentheses is coupled to the hypercharge
$Y=-1/\sqrt{3}T_{8}+X$ [30]. We next identify the standard model $Z$ boson,

$Z=c_{W}A_{3}-s_{W}\left(-\frac{t_{W}}{\sqrt{3}}A_{8}+\sqrt{1-\frac{t^{2}_{W}}{3}}B\right),$
(19)

which is orthogonal to $A$, as usual, while the 3-3-1 model neutral gauge
boson,

$\mathcal{Z}^{\prime}=\sqrt{1-\frac{t^{2}_{W}}{3}}A_{8}+\frac{t_{W}}{\sqrt{3}}B,$
(20)

is given/orthogonal to the hypercharge field.

The photon $A$ is massless and decoupled as a physical field. By contrast, $Z$
and $\mathcal{Z}^{\prime}$ mix with $C$-boson of $U(1)_{G}$ via a mass matrix,

$\displaystyle
M^{2}=\left(\begin{array}[]{ccc}m^{2}_{Z}&m^{2}_{Z\mathcal{Z}^{\prime}}&m^{2}_{ZC}\\\
m^{2}_{Z\mathcal{Z}^{\prime}}&m^{2}_{\mathcal{Z}^{\prime}}&m^{2}_{\mathcal{Z}^{\prime}C}\\\
m^{2}_{ZC}&m^{2}_{\mathcal{Z}^{\prime}C}&m^{2}_{C}\end{array}\right)=$ (24)
$\displaystyle\frac{g^{2}}{2}\left(\begin{array}[]{ccc}\frac{(3+4t_{X}^{2})(u^{2}+v^{2})}{2(3+t_{X}^{2})}&\frac{\sqrt{3+4t_{X}^{2}}((3-2t_{X}^{2})u^{2}-(3+4t_{X}^{2})v^{2})}{6(3+t_{X}^{2})}&\frac{\sqrt{3+4t_{X}^{2}}t_{G}(u^{2}-v^{2})}{3\sqrt{3+t_{X}^{2}}}\\\
\frac{\sqrt{3+4t_{X}^{2}}((3-2t_{X}^{2})u^{2}-(3+4t_{X}^{2})v^{2})}{6(3+t_{X}^{2})}&\frac{(3-2t_{X}^{2})^{2}u^{2}+(3+4t_{X}^{2})^{2}v^{2}+4(3+t_{X}^{2})^{2}w^{2}}{18(3+t_{X}^{2})}&\frac{t_{G}((3-2t_{X}^{2})u^{2}+(3+4t_{X}^{2})v^{2}+4(3+t_{X}^{2})w^{2})}{9\sqrt{3+t_{X}^{2}}}\\\
\frac{\sqrt{3+4t_{X}^{2}}t_{G}(u^{2}-v^{2})}{3\sqrt{3+t_{X}^{2}}}&\frac{t_{G}((3-2t_{X}^{2})u^{2}+(3+4t_{X}^{2})v^{2}+4(3+t_{X}^{2})w^{2})}{9\sqrt{3+t_{X}^{2}}}&\frac{2}{9}t_{G}^{2}(u^{2}+v^{2}+4(w^{2}+9\Lambda^{2}))\\\
\end{array}\right),$ (28)

given in the $(Z,\mathcal{Z}^{\prime},C)$ basis, where $t_{G}=g_{G}/g$.
Because of $u^{2},v^{2}\ll\Lambda^{2}\ll w^{2}$, the field $Z$ is light and
decoupled from $\mathcal{Z}^{\prime},C$ to be a physical field with mass,

$m^{2}_{Z}\simeq\frac{g^{2}}{4c^{2}_{W}}(u^{2}+v^{2}),$ (29)

whereas $\mathcal{Z}^{\prime},C$ significantly mix, yielding physical fields,

$Z^{\prime}=c_{\theta}\mathcal{Z}^{\prime}-s_{\theta}C,\hskip
14.22636ptZ^{\prime\prime}=s_{\theta}\mathcal{Z}^{\prime}+c_{\theta}C,$ (30)

with mixing angle and respective masses,

$t_{\theta}\simeq\frac{\sqrt{3+t^{2}_{X}}}{2t_{G}},\hskip
14.22636ptm^{2}_{Z^{\prime}}\simeq\frac{4g^{2}_{G}(3+t_{X}^{2})}{4t^{2}_{G}+3+t^{2}_{X}}\Lambda^{2},\hskip
14.22636ptm^{2}_{Z^{\prime\prime}}\simeq\frac{g^{2}}{9}(4t^{2}_{G}+3+t^{2}_{X})w^{2}.$
(31)

The $Z^{\prime\prime}$ boson is associated with 3-3-1-1 symmetry, heavy at $w$
scale and integrated out. The $Z^{\prime}$ boson is coupled to the dark charge
$D=-2/\sqrt{3}T_{8}+G$, having a mass at $\Lambda\sim$ TeV scale.

In summary, the new physics at TeV scale is only governed by $U(1)_{D}$ with
its gauge field $Z^{\prime}$ and Higgs field $H^{\prime}$, while all other new
particles ($N_{a}$, $U_{3}$, $D_{\alpha}$, $X$, $Y$, $Z^{\prime\prime}$, and
the rest of new Higgs fields) are superheavy as manifestly integrated out. The
relevant interactions of $Z^{\prime}$ with normal fermions are
straightforwardly computed as

$\mathcal{L}\supset-\frac{2}{3}g_{G}s_{\theta}t^{2}_{W}\sum_{f}\bar{f}\gamma^{\mu}YfZ^{\prime}_{\mu},$
(32)

where $f$ runs over usual fermion doublets and singlets. These couplings are
proportional to the hypercharge $Y$, resulting from a mixing between
$\mathcal{Z}^{\prime}$ and $C$ due to symmetry breaking, not a kinetic mixing
as in the normal sense. For convenience, the couplings are collected in Tab. 2
too.

$f$ | $g^{Z^{\prime}}_{V}(f)$ | $g^{Z^{\prime}}_{A}(f)$
---|---|---
$\nu_{a}$ | $\frac{-c_{\theta}s^{2}_{W}}{2\sqrt{3-4s^{2}_{W}}}$ | $\frac{-c_{\theta}s^{2}_{W}}{2\sqrt{3-4s^{2}_{W}}}$
$e_{a}$ | $\frac{-3c_{\theta}s^{2}_{W}}{2\sqrt{3-4s^{2}_{W}}}$ | $\frac{c_{\theta}s^{2}_{W}}{2\sqrt{3-4s^{2}_{W}}}$
$u_{a}$ | $\frac{5c_{\theta}s^{2}_{W}}{6\sqrt{3-4s^{2}_{W}}}$ | $\frac{-c_{\theta}s^{2}_{W}}{2\sqrt{3-4s^{2}_{W}}}$
$d_{a}$ | $\frac{-c_{\theta}s^{2}_{W}}{6\sqrt{3-4s^{2}_{W}}}$ | $\frac{c_{\theta}s^{2}_{W}}{2\sqrt{3-4s^{2}_{W}}}$
Table 2:  Couplings of $Z^{\prime}$ dark photon with fermions.

A final remark is that the dark photon does not flavor change, although it
originates from a family-dependent gauge charge $T_{8}$. This avoids the
neutral meson mixing constraint, opposite to those in 3-3-1 and 3-3-1-1 models
[7, 8, 23, 24].

##  IV  EWPT affected by dark photon

The precision data consist of low-energy weak neutral current, $Z$-pole
physics, and others. The weak neutral current is affected by $Z^{\prime}$
exchange, mainly sensitive to $Z^{\prime}$ mass. The $Z$-pole physics measured
at LEP and SLC is mainly sensitive to $Z$-$(\mathcal{Z}^{\prime},C)$ mixing,
modifying $Z$ mass and $Zf\bar{f}$ couplings, by contrast. The precision data
strongly limit $Z$-$(\mathcal{Z}^{\prime},C)$ mixing angles, say
$\mathcal{E}\sim 10^{-3}$, extracted by comparing coupling measurements with
standard model prediction (cf. [31]). This also limits new neutral gauge boson
mass but generally weaker than high energy collider, e.g. LEPII and LHC, which
is above $Z$ pole and insensitive to mixing angles $\mathcal{E}$.
Alternatively, global fit to full electroweak data set yields a comparable
bound as $\mathcal{E}\sim 10^{-3}$ [32].

###  IV.1 Atomic parity violation

It is clear from (32) that the dark photon couplings to usual fermions violate
parity at a considerable level because the left-handed and right-handed
fermions including neutrinos have different hypercharges. In the conventional
dark photon model, the couplings of dark photon with fermions are vectorlike,
conserving parity, by contrast. This is a new observation of the work,
allowing discriminating the current model in experiment.

Unlike the conventional dark photon, the current field $Z^{\prime}$
contributes to atomic parity violation via an effective interaction,

$\mathcal{L}_{\mathrm{eff}}\supset\frac{G_{F}}{\sqrt{2}}(\bar{e}\gamma_{\mu}\gamma_{5}e)(C^{\prime}_{1u}\bar{u}\gamma^{\mu}u+C^{\prime}_{1d}\bar{d}\gamma^{\mu}d),$
(33)

where

$\frac{G_{F}}{\sqrt{2}}=\frac{1}{2(u^{2}+v^{2})},\hskip
14.22636ptC^{\prime}_{1u}=-\frac{5c^{2}_{\theta}s^{4}_{W}}{6(3-4s^{2}_{W})}\frac{m^{2}_{Z}}{m^{2}_{Z^{\prime}}},\hskip
14.22636ptC^{\prime}_{1d}=\frac{c^{2}_{\theta}s^{4}_{W}}{6(3-4s^{2}_{W})}\frac{m^{2}_{Z}}{m^{2}_{Z^{\prime}}}.$
(34)

A parity violation that couples electrons as vector and quarks as axial-vector
due to $Z^{\prime}$ exists too, but it is prevented for heavy atoms due to its
dependence on spins other than charges, analogous to the case of $Z$, which
would be neglected.

The weak charge deviation from the standard model computed for an atom that
composes $Z$ protons and $N$ neutrons is

$\Delta
Q_{W}(Z,N)=-2\left[Z\left(2C^{\prime}_{1u}+C^{\prime}_{1d}\right)+N\left(C^{\prime}_{1u}+2C^{\prime}_{1d}\right)\right]=\frac{(3Z+N)c^{2}_{\theta}s^{4}_{W}}{3-4s^{2}_{W}}\frac{m^{2}_{Z}}{m^{2}_{Z^{\prime}}}.$
(35)

Taking Cesium with $Z=55$ and $N=78$, the current measurement and the standard
model value for Cs weak charge make a bound $\Delta Q_{W}(\mathrm{Cs})<0.61$
[33], implying $m_{Z^{\prime}}>291c_{\theta}$ GeV, for $s^{2}_{W}=0.231$ and
$m_{Z}=91.1876$ GeV. Since $c_{\theta}\leq 1$, this bound is easily evaded by
the following constraint, i.e. $Z^{\prime}$ negligibly contributes to the
parity violation in Cs.

###  IV.2 $\rho$-parameter

The mass matrix of $(Z,\mathcal{Z}^{\prime},C)$ given above has a seesaw form
due to $u^{2},v^{2}\ll\Lambda^{2}\ll w^{2}$ for which $Z$ is light, while
$\mathcal{Z}^{\prime},C$ are heavy. With the aid of the seesaw
diagonalization, we obtain the mass of the physical light state (called
$Z_{1}$), such as

$m^{2}_{Z_{1}}\simeq
m^{2}_{Z}-\mathcal{E}\begin{pmatrix}m^{2}_{Z\mathcal{Z}^{\prime}}\\\
m^{2}_{ZC}\end{pmatrix},\hskip
14.22636pt\mathcal{E}\equiv\begin{pmatrix}m^{2}_{Z\mathcal{Z}^{\prime}}&m^{2}_{ZC}\end{pmatrix}\begin{pmatrix}m^{2}_{\mathcal{Z}^{\prime}}&m^{2}_{\mathcal{Z}^{\prime}C}\\\
m^{2}_{\mathcal{Z}^{\prime}C}&m^{2}_{C}\end{pmatrix}.$ (36)

Then, this mass contributes to the $\rho$-parameter at the tree level, such as

$\Delta\rho=\rho-1=\frac{m^{2}_{W}}{c^{2}_{W}m^{2}_{Z_{1}}}-1\simeq\frac{t^{4}_{W}(u^{2}+v^{2})}{36\Lambda^{2}}.$
(37)

It is noted that the tree-level terms of $(u^{2},v^{2})/w^{2}$ order are
negligible and omitted.

Further, at the one-loop level, the $\rho$-parameter receives a contribution
from $(X,Y)$ gauge vector doublet [34, 35, 36]. However, this radiative
correction is proportional to the doublet mass splitting
$(m^{2}_{X}-m^{2}_{Y})/m^{2}_{X,Y}\sim(u^{2}-v^{2})/w^{2}$, which is
infinitesimal and suppressed too.

From the global fit, the $\rho$-parameter is given by $\rho=1.00038$, which is
2 $\sigma$ above the standard model prediction $\rho=1$ [33]. Hence, we
require $\Delta\rho<0.00038$, leading to $\Lambda>631$ GeV. This translates to
$m_{Z^{\prime}}=(3gc_{W}/\sqrt{3-4s^{2}_{W}})c_{\theta}\Lambda>749c_{\theta}$
GeV for $g=0.65$, which is stronger than the parity violation bound.

###  IV.3 $Zf\bar{f}$ couplings

The above seesaw diagonalization determines a mixing between $Z$ and
$\mathcal{Z}^{\prime},C$, such as

$\begin{pmatrix}Z\\\ \mathcal{Z}^{\prime}\\\
C\end{pmatrix}=\begin{pmatrix}1&\mathcal{E}_{1}&\mathcal{E}_{2}\\\
-\mathcal{E}_{1}&1&0\\\
-\mathcal{E}_{2}&0&1\end{pmatrix}\begin{pmatrix}Z_{1}\\\ Z_{2}\\\
Z_{3}\end{pmatrix},$ (38)

where $\mathcal{E}\equiv(\mathcal{E}_{1},\mathcal{E}_{2})$ is defined in (36)
and evaluated as

$\mathcal{E}_{1}=-\frac{t^{2}_{W}\sqrt{3-4s^{2}_{W}}(u^{2}+v^{2})}{36c^{2}_{W}\Lambda^{2}},\hskip
14.22636pt\mathcal{E}_{2}=\frac{t^{2}_{W}(u^{2}+v^{2})}{24c_{W}t_{G}\Lambda^{2}},$
(39)

where notice that $|\mathcal{E}_{1,2}|\ll 1$ due to
$u^{2},v^{2}\ll\Lambda^{2}$. It is clear that
$Z_{1}=Z-\mathcal{E}_{1}\mathcal{Z}^{\prime}-\mathcal{E}_{2}C\simeq Z$,
$Z_{2}=\mathcal{E}_{1}Z+\mathcal{Z}^{\prime}\simeq\mathcal{Z}^{\prime}$, and
$Z_{3}=\mathcal{E}_{2}Z+C\simeq C$ approximate the original states due to
$|\mathcal{E}|\ll 1$.

$\mathcal{Z}^{\prime}$ and $C$ modify the well-measured couplings of $Z$ with
fermions by amounts $\mathcal{E}_{1,2}$, respectively. Demanding
$|\mathcal{E}_{1,2}|<10^{-3}$ gives $\Lambda>972$ and $929/\sqrt{t_{G}}$ GeV,
respectively. Taking the former bound translates to
$m_{Z^{\prime}}>1.15c_{\theta}$ TeV, which is stronger than the
$\rho$-parameter bound.

##  V  Dark photon signal at colliders

The precision data hint to a dark photon mass beyond the electroweak scale.
Hence, it appropriately looks for a dark photon signal at high energy
colliders. Since $Z^{\prime}$ couples to leptons and quarks, it may be
produced at both LEPII and LHC.

###  V.1 LEPII

The LEPII experiment studied a new neutral gauge boson $Z^{\prime}$ that
mediates the process $e^{+}e^{-}\to f\bar{f}$ for $f=\mu,\tau$. The new
physics contribution to the process can be determined by integrating
$Z^{\prime}$ out, yielding an effective interaction

$\displaystyle\mathcal{L}_{\mathrm{eff}}$ $\displaystyle=$
$\displaystyle\frac{g^{2}}{c^{2}_{W}m^{2}_{Z^{\prime}}}[\bar{e}\gamma^{\mu}(a^{Z^{\prime}}_{L}(e)P_{L}+a^{Z^{\prime}}_{R}(e)P_{R})e][\bar{f}\gamma_{\mu}(a^{Z^{\prime}}_{L}(f)P_{L}+a^{Z^{\prime}}_{R}(f)P_{R})f]$
(40) $\displaystyle=$
$\displaystyle\frac{g^{2}[a^{Z^{\prime}}_{L}(e)]^{2}}{c^{2}_{W}m^{2}_{Z^{\prime}}}(\bar{e}\gamma^{\mu}P_{L}e)(\bar{f}\gamma_{\mu}P_{L}f)+(LR)+(RL)+(RR),$

since the charged lepton couplings for $f=\mu,\tau$ equal those of electron,
and these chiral couplings $a^{Z^{\prime}}_{L,R}=(g^{Z^{\prime}}_{V}\pm
g^{Z^{\prime}}_{A})/2$ can directly be extracted from Tab. 2.

The LEPII searched for such chiral interactions as in (40), providing
respective bounds on the couplings, typically [37]

$\frac{g^{2}[a^{Z^{\prime}}_{L}(e)]^{2}}{c^{2}_{W}m^{2}_{Z^{\prime}}}<\frac{1}{(6\
\mathrm{TeV})^{2}}.$ (41)

This implies $m_{Z^{\prime}}>(3gs^{2}_{W}/c_{W}\sqrt{3-4s^{2}_{W}})c_{\theta}\
\mathrm{TeV}=356c_{\theta}$ GeV, which is more smaller than the
$\rho$-parameter and $f$-fermion coupling bounds. In other words, the dark
photon with mass at TeV easily escapes the LEPII detection.

###  V.2 LHC

At the LHC, the dark photon $Z^{\prime}$ may be produced and then decayed to a
dilepton signal, $\sigma(pp\to Z^{\prime}\to l\bar{l})=\sigma(pp\to
Z^{\prime})\mathrm{Br}(Z^{\prime}\to l\bar{l})$. This process was particularly
studied in [25] for a case with $t_{G}=1$ or a corresponding $c_{\theta}$
value following (31) since $t_{X}=\sqrt{3}s_{W}/\sqrt{3-4s^{2}_{W}}$ is
already fixed by the Weinberg angle. We now generalize the result for a
variation of $c_{\theta}$ and obtain the corresponding bound as plotted in
Fig. 1 (red line), which implies $m_{Z^{\prime}}>2.18$ TeV for $c_{\theta}=1$,
for a reference. Notice, hereafter, that the allowed regime is given above the
relevant curve.

Further, we also include the atomic parity violation, $\rho$-parameter,
$\mathcal{E}_{1,2}$-mixings, and LEPII bounds to Fig. 1, for comparison. For
each value of $c_{\theta}$, the LHC makes the strongest bound on the
$Z^{\prime}$ mass as compared with the other constraints.

Figure 1:  The $Z^{\prime}$ dark photon mass for each value of $c_{\theta}$
constrained by the LHC dilepton search [38] (top curve), while the other
curves counted from the bottom up present the bounds by atomic parity
violation, LEPII, $\rho$-parameter, and $\mathcal{E}_{1,2}$ mixing,
respectively.

##  VI  Neutrino mass and dark matter revisited

Would neutrino mass generation and dark matter stability be determined by the
$U(1)_{D}$ dark physics at TeV regime? Although the issues are still addressed
by the model in itself in the case $w\gg\Lambda$ like the current setup as
shown in [25], this work gives an alternative answer.

The question may be simply answered by introducing three right-handed
neutrinos

$\nu_{aR}\sim(1,1,0,0),$ (42)

which both couple to $\psi_{bL}\eta$ and possess a Majorana mass
$(m_{R})_{ab}\nu_{aR}\nu_{bR}$, inducing the observed neutrino masses via a
canonical seesaw, as well as a vectorlike dark fermion

$\chi\sim(1,1,0,1),$ (43)

which provides the lightest component of dark fields ($P_{D}=-1$) responsible
for dark matter. The canonical seesaw works the same normal case [39], while
the fermion dark matter observables are set by $H^{\prime}$ portal rather than
$Z^{\prime}$ portal. Let us study the latter, governed by interactions

$\mathcal{L}\supset\frac{3gc_{\theta}c_{W}}{2\sqrt{3-4s^{2}_{W}}}\bar{\chi}\gamma^{\mu}\chi
Z^{\prime}_{\mu}+\left(\mu_{\chi}\bar{\chi}_{L}\chi_{R}+\frac{1}{2}h_{L}\chi_{L}\chi_{L}\phi+\frac{1}{2}h_{R}\chi_{R}\chi_{R}\phi+H.c.\right),$
(44)

where $\phi=(\Lambda+H^{\prime}+iG_{Z^{\prime}})/\sqrt{2}$, as mentioned,
while $\mu_{\chi}$ and $h_{L,R}$ are a mass parameter and dimensionless
couplings, respectively. Notice that $\chi$ may insignificantly couple to a
superheavy lepton $N$, which is suppressed by a $Z_{2}$ as $\chi\to-\chi$,
without loss of generality.

Because of $h_{L,R}$ contributions, $\chi$ is generically separated into two
physical Majorana fields $\chi_{1,2}$ by diagonalizing, such as

$\displaystyle\mathcal{L}$ $\displaystyle\supset$
$\displaystyle-\frac{1}{2}\begin{pmatrix}\bar{\chi}_{L}&\bar{\chi}^{c}_{R}\end{pmatrix}\begin{pmatrix}-h_{L}\frac{\Lambda}{\sqrt{2}}&-\mu_{\chi}\\\
-\mu_{\chi}&-h_{R}\frac{\Lambda}{\sqrt{2}}\end{pmatrix}\begin{pmatrix}\chi^{c}_{L}\\\
\chi_{R}\end{pmatrix}+H.c.$ (45) $\displaystyle=$
$\displaystyle-\frac{1}{2}m_{\chi_{1}}\chi^{2}_{1}-\frac{1}{2}m_{\chi_{2}}\chi^{2}_{2}+H.c.,$

where we define $t_{2\zeta}=\frac{2\sqrt{2}\mu_{\chi}}{(h_{R}-h_{L})\Lambda}$,
and

$\displaystyle\chi_{1}=c_{\zeta}\chi_{L}-s_{\zeta}\chi^{c}_{R},\hskip
14.22636pt\chi_{2}=s_{\zeta}\chi_{L}+c_{\zeta}\chi^{c}_{R},$ (46)
$\displaystyle
m_{\chi_{1,2}}=\frac{1}{2\sqrt{2}}\left[-(h_{L}+h_{R})\Lambda\mp\sqrt{(h_{L}-h_{R})^{2}\Lambda^{2}+8\mu^{2}_{\chi}}\right].$
(47)

We suppose $\chi_{1}$ to be lightest, i.e. $|m_{\chi_{1}}|<|m_{\chi_{2}}|$.
Thus, $\chi_{1}$ is stabilized by $P_{D}$ to be a dark matter candidate. It
possesses the following couplings to $Z^{\prime},H^{\prime}$,

$\mathcal{L}\supset-\frac{3gc_{\theta}c_{W}}{4\sqrt{3-4s^{2}_{W}}}\bar{\chi}_{1}\gamma^{\mu}\gamma_{5}\chi_{1}Z^{\prime}_{\mu}+\left(\frac{1}{2}h_{1}\chi_{1}\chi_{1}H^{\prime}+H.c.\right),$
(48)

where $h_{1}=(c^{2}_{\zeta}h_{L}+s^{2}_{\zeta}h_{R})/\sqrt{2}$, and
$Z^{\prime}$ has only axial-vector current to Majorana field $\chi_{1}$. Due
to the Majorana nature, the annihilation of $\chi_{1}$ via $Z^{\prime}$ portal
is helicity suppressed, which starts only from $p$-wave contributions.
Further, $\chi_{1}$ has only spin-dependent scattering cross-section with
nuclei by $Z^{\prime}$ portal, which is suppressed too (cf. [25]). Thai said,
the Higgs portal $H^{\prime}$ would dominantly contribute to the dark matter
observables, as shown below.

Figure 2:  Annihilation of Majorana fermion dark matter to usual Higgs fields.

In the early universe, $\chi_{1}$ annihilates to the standard model Higgs
boson $H$ via the $H^{\prime}$ portal which sets the relic density, as
despited in Fig. 2. It is noted that $H^{\prime}$ couples to $H$ such as
$\mathcal{L}\supset\frac{1}{2}\bar{\lambda}\Lambda H^{\prime}H^{2}$, where
$\bar{\lambda}=(\lambda_{10}v^{2}+\lambda_{12}u^{2})/(v^{2}+u^{2})$.
Therefore, the dark matter annihilation cross-section is straightforwardly
computed as

$\langle\sigma
v\rangle_{\chi_{1}}=\frac{1}{128\pi}\frac{h^{2}_{1}\bar{\lambda}^{2}\Lambda^{2}}{(4m^{2}_{\chi_{1}}-m^{2}_{H^{\prime}})^{2}}\sqrt{1-\frac{m^{2}_{H}}{m^{2}_{H^{\prime}}}}.$
(49)

Figure 3:  Dark matter relic density plotted as function of its mass, where
the measured density is also shown.

We take $m_{H}=125$ GeV, $h_{1}=0.5$, $\bar{\lambda}=0.5$, and $\Lambda=2$
TeV, where the last one is appropriate to the strongest bound coming from the
LHC. We plot the dark matter relic density $\Omega_{\chi_{1}}h^{2}\simeq 0.1\
\mathrm{pb}/\langle\sigma v\rangle_{\chi_{1}}$ as a function of the dark
matter mass for a fixed value of the $H^{\prime}$ mass, say $m_{H^{\prime}}=2$
TeV, as given in Fig. 3. We also include the experimental value
$\Omega_{\chi_{1}}h^{2}=0.12$ [33] to this figure. It is clear from the figure
that the $H^{\prime}$ resonance is crucial to set the relic density. A correct
abundance, i.e. $\Omega_{\chi_{1}}h^{2}\leq 0.12$, requires
$m_{\chi_{1}}=930$–1066 GeV.

Although $H^{\prime}$ does not couple to quark, as well as $H$ does not couple
to $\chi_{1}$, it is evident that $\chi_{1}$ can interact with quark through a
$H$-$H^{\prime}$ mixing given by the scalar potential, since $\Lambda$ is not
much bigger than $u,v$. Let us parametrize the mixing,
$H_{1}=c_{\delta}H-s_{\delta}H^{\prime}$ and
$H_{2}=s_{\delta}H+c_{\delta}H^{\prime}$, which $H_{1}$ now becomes the
standard model Higgs like boson, while $H_{2}$ acts as a new heavy Higgs
boson. It is easily derived that $\delta\sim(u,v)/\Lambda$ which arises from
the scalar potential. Hence, such mixing gives rise to an effective coupling
between $\chi_{1}$ and quark to be
$\mathcal{L}_{\mathrm{eff}}\supset\alpha^{S}_{q}(\bar{\chi}^{c}_{1}\chi_{1})(\bar{q}q)$,
where

$\alpha^{S}_{q}=-\frac{s_{\delta}c_{\delta}h_{1}m_{q}}{v_{\mathrm{w}}m^{2}_{H_{1}}},$
(50)

which is dominantly contributed by $H_{1}$, where
$v_{\mathrm{w}}=\sqrt{u^{2}+v^{2}}=246$ GeV is the weak scale.

Hence, the scattering cross-section of $\chi_{1}$ with nucleon $N=p,n$ is [40]

$\sigma^{\mathrm{SI}}_{\chi_{1}-N}=\frac{4m^{2}_{N}}{\pi}(f^{N})^{2},$ (51)

where the form factor is

$\frac{f^{N}}{m_{N}}=\sum_{q=u,d,s}\frac{\alpha^{S}_{q}}{m_{q}}f^{N}_{Tq}+\frac{2}{27}f^{N}_{TG}\sum_{q=c,b,t}\frac{\alpha^{S}_{q}}{m_{q}}\simeq-\frac{0.35s_{\delta}c_{\delta}h_{1}}{v_{\mathrm{w}}m^{2}_{H_{1}}},$
(52)

where $f^{N}_{TG}=1-\sum_{q=u,d,s}f^{N}_{Tq}$, $f^{p}_{Tu}=0.02$,
$f^{p}_{Td}=0.026$, $f^{p}_{Ts}=0.118$, $f^{n}_{Tu}=f^{p}_{Td}$,
$f^{n}_{Td}=f^{p}_{Tu}$, and $f^{n}_{Ts}=f^{p}_{Ts}$ [41]. With the aid of
$m_{N}=1$ GeV and $m_{H_{1}}=125$ GeV, the SI cross-section becomes

$\sigma^{\mathrm{SI}}_{\chi_{1}-N}\simeq\frac{0.49m^{4}_{N}s^{2}_{\delta}h^{2}_{1}}{\pi
v^{2}_{\mathrm{w}}m^{4}_{H_{1}}}\simeq\left(\frac{s_{\varphi}}{0.01}\right)^{2}\left(\frac{h_{1}}{0.5}\right)^{2}\times
10^{-46}\ \mathrm{cm}^{2}.$ (53)

This cross-section agrees with the latest data
$\sigma^{\mathrm{SI}}_{\chi_{1}-N}\sim 10^{-46}\ \mathrm{cm}^{2}$ [42] for the
dark matter mass $m_{\chi_{1}}\sim$ TeV, the mixing angle $s_{\varphi}\sim
0.01$, and the coupling constant $h_{1}\sim 0.5$.

##  VII Conclusion

The 3-3-1 and 3-3-1-1 models have been well established. As a result of
anomaly cancelation, one of quark families transforms differently from the
other two. This leads to dangerous FCNCs due to nonuniversal couplings of
$Z^{\prime},Z^{\prime\prime}$ with quarks, confronting the current collider
bounds. This work show that if the 3-3-1-1 unification is at a very high
energy, it imprints at TeV a new neutral gauge boson, $Z^{\prime}$, conserving
flavors, avoiding the above issue. The interest is associated with a dark
photon $Z^{\prime}$ obtained by assigning the dark charge
$D=-2/\sqrt{3}T_{8}+G$ to exotic quarks and leptons, whereas the usual
fermions possess $D=0$.

Different from the usual setup of dark physics, the current dark photon
$Z^{\prime}$ imprinted by 3-3-1-1 breaking has chiral couplings with normal
fermions proportional to hypercharge, i.e. $\sim
c_{\theta}\bar{f}\gamma^{\mu}YfZ^{\prime}_{\mu}$, without necessity of any
kinetic mixing. Studying various bounds yield that the dark photon should have
a mass in TeV regime.

Additionally, the dark photon $Z^{\prime}$ negligibly contributes to dark
matter observables. Instead, the Higgs field associated with it, i.e. the one
($H^{\prime}$) that breaks dark charge induces dark photon mass, manifestly
governs the dark matter relic density and direct detection. Last, but not
least, the $H^{\prime}$ resonance is necessarily to set the correct relic
density. Consequently, the dark matter mass is suitably given at TeV regime.

## Acknowledgements

This research is funded by Vietnam National Foundation for Science and
Technology Development (NAFOSTED) under grant number 103.01-2023.50.

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