# Multi-$Z^{\prime}$ signatures of spontaneously broken local $U(1)^{\prime}$
symmetry

Takaaki Nomura<EMAIL_ADDRESS>College of Physics, Sichuan University,
Chengdu 610065, China  Kei Yagyu<EMAIL_ADDRESS>Department
of Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan

###### Abstract

We discuss multi-$Z^{\prime}$ signatures coming from decays of Higgs bosons in
models with a spontaneously broken $U(1)^{\prime}$ symmetry, which can be
observed as “lepton jets” or multi-lepton final states depending on the mass
range of new bosons. We consider anomaly-free $U(1)^{\prime}$ models without
introducing new fermions except for right-handed neutrinos, in which the Higgs
sector is composed of an isospin doublet and a singlet fields with zero and
non-zero $U(1)^{\prime}$ charges, respectively. The multi-$Z^{\prime}$
signatures can then be obtained via the decays of the discovered (extra) Higgs
boson $h$ ($\phi$), i.e., $h\to Z^{\prime}Z^{\prime}$, $\phi\to
Z^{\prime}Z^{\prime}$ and/or $h\to\phi\phi\to 4Z^{\prime}$ as far as
kinematically allowed. We give the upper limit on the branching ratios of $h$
into $Z^{\prime}Z^{\prime}$ and $4Z^{\prime}$ from the current experimental
data in each model. We also show the deviation in the $hhh$ coupling from the
standard model prediction at one-loop level, and find that its amount is
typically smaller than 1%.

††preprint: OU-HET-1235

##  I Introduction

The electroweak (EW) gauge symmetry and its spontaneous breakdown have been
confirmed by various experiments to date. In particular, the discovery of the
Higgs boson at LHC has revealed the realization of the spontaneous symmetry
breaking via the Higgs mechanism. Although there is no contradiction between
the structure of $SU(2)_{L}\times U(1)_{Y}\to U(1)_{\rm em}$ implemented in
the Standard Model (SM) and current experimental data, one can regard such a
pattern of the symmetry breaking as a part of larger structures.

Models with a new $U(1)$ gauge symmetry, denoting $U(1)^{\prime}$, have often
been discussed as one of the simplest and well-motivated examples of such an
extended EW symmetry. In fact, grand unified theories such as $SO(10)$ and
$E_{6}$ models predict extra $U(1)$ gauge symmetries at the EW scale, see
e.g., Ref. [1] and references therein. In addition, gauging the $U(1)_{B-L}$
symmetry [2, 3, 4] can be regarded as a natural extension of the EW symmetry
because the global $U(1)_{B-L}$ accidentally appears in the SM. In models with
the gauged $U(1)_{B-L}$ symmetry, three right-handed neutrinos are inevitably
introduced to maintain the gauge anomaly cancellation, and they make left-
handed neutrinos massive via the seesaw mechanism [5, 6, 7, 8]. More
generally, one can consider the $U(1)_{X}$ gauge symmetry [9, 10] whose
charges of SM fields are expressed by two free parameters, and the well-known
examples, e.g., $U(1)_{B-L}$, $U(1)_{D}$ (dark photon) [11] and $U(1)_{R}$
(right-handed) [12, 13] correspond to its special cases. Furthermore, if we
allow a flavor dependence of the charge assignments of $U(1)^{\prime}$, three
possibilities, i.e., $U(1)_{L_{i}-L_{j}}$ ($L_{i}$ and $L_{j}$ being the
lepton number of $i$-th and $j$-th flavor, respectively) [14, 15], appear
under the anomaly free condition without introducing any exotic fermions111One
can also consider $U(1)_{Q_{i}-Q_{j}}$ symmetries as the anomaly free choice,
where $Q_{i}$ and $Q_{j}$ are the quark number of $i$-th and $j$-th flavor,
respectively. However, these models predict a diagonal form of the quark
Yukawa matrices for the case with one Higgs doublet without charged under
$U(1)_{Q_{i}-Q_{j}}$, and they cannot explain the CKM matrix. . In particular,
models with $U(1)_{L_{\mu}-L_{\tau}}$ have often been discussed as a solution
of the muon $g-2$ anomaly [16, 17, 18]. As the other possibilities, one can
also consider linear combinations of $U(1)_{B-L}$ and $U(1)_{L_{i}-L_{j}}$
symmetries, e.g., $U(1)_{B-3L_{i}}$ symmetries. Therefore, it is important to
comprehensively investigate scenarios with $U(1)^{\prime}$ to determine new
physics beyond the SM.

A common feature of models with $U(1)^{\prime}$ is the appearance of an extra
neutral gauge boson $Z^{\prime}$. We discuss a massive $Z^{\prime}$ case
because a massless $Z^{\prime}$ has highly been constrained by tests of
violation of the weak equivalence theorem [19], which give the bound on the
new gauge coupling constant, e.g., in the $U(1)_{B-L}$ case, to be smaller
than $10^{-24}$ [20]. Since we have already confirmed the spontaneous breaking
of the EW symmetry via the Higgs mechanism, it is quite natural to consider
that the $U(1)^{\prime}$ symmetry is also spontaneously broken by the Higgs
mechanism. 222Alternatively, one can introduce the $Z^{\prime}$ mass via the
Stueckelberg mechanism [21, 22] without introducing Higgs fields while keeping
the gauge invariance. However, this is nothing but a special case of the Higgs
mechanism where a new physical Higgs boson is decoupled from the theory and
the so-called Stueckelberg scalar field can be regarded as a Nambu-Goldstone
boson associated with the spontaneous $U(1)^{\prime}$ breaking.  The simplest
way to do that is to introduce a complex scalar field which is singlet under
the SM gauge symmetry. In this class of models, an additional physical neutral
Higgs boson $\phi$ also appears in addition to $Z^{\prime}$, and their masses
are naturally expected to be given in a similar order because they come from
the Vacuum Expectation Value (VEV) of the singlet scalar. Therefore, the
phenomenological impacts of two new bosons should be taken into account at the
same time. In particular, if the masses of $Z^{\prime}$ and $\phi$ are smaller
than the discovered Higgs boson ($h$), the latter can decay into a
$Z^{\prime}$ pair and/or $\phi$ pair. The $h\to Z^{\prime}Z^{\prime}$ mode has
been discussed in the dark photon model, i.e., the $U(1)_{D}$ model, in Refs.
[23, 24, 25, 26, 27, 28]. In addition, the decay channel of $h\to\phi\phi\to
4Z^{\prime}\to 8$ fermions has been discussed in the $U(1)_{D}$ model [25,
29]. Charged leptons produced via the decay of $Z^{\prime}$ can be observed as
“lepton jets” [30, 31] if there is a large mass difference between the
discovered Higgs boson and the masses of $Z^{\prime}/\phi$, especially around
GeV or less due to highly boosted and collimated charged leptons. As for the
other $U(1)^{\prime}$ models, Ref. [32] and Ref. [33] have discussed the
$Z^{\prime}$ production via the decay of $\phi$ in the
$U(1)_{L_{\mu}-L_{\tau}}$ model and that of $h$ in the $U(1)_{X}$ model,
respectively.

In this paper, we discuss the phenomenology of models with various
$U(1)^{\prime}$ extensions of the SM with a complex scalar field. We focus on
the impact of the collider phenomenology in the presence of $Z^{\prime}$ and
$\phi$ having a similar size of the masses, in which we classify three regions
of the masses, i.e., (i) small mass ${\cal O}(10)$ MeV, (ii) middle mass
${\cal O}(10)$ GeV and (iii) large mass ${\cal O}(100)$ GeV. We show the
branching ratios of the new decay channels of $h$ for the cases (i) and (ii)
as a function of the new gauge coupling ($g_{X}$) and the mixing angle of $h$
and $\phi$ ($\sin\alpha$) in each $U(1)^{\prime}$ model under the constraints
from flavor and collider experiments. We then clarify the upper limit on these
branching ratios in each model. For the case (iii), we show the branching
ratio of the new Higgs boson $\phi$ into a $Z^{\prime}$ pair as a function of
$g_{X}$ and $\sin\alpha$. In addition to these branching ratios, we also
analyze the deviation of the Higgs trilinear coupling $hhh$ from the SM
prediction at one-loop level, and find that the magnitude of the deviation is
typically less than $1\%$ level.

This paper is organized as follows. In Sec. II, we define our model with the
$U(1)^{\prime}$ gauge symmetry, and give the Lagrangian and relevant
interaction terms. In Sec. III, we discuss the decays of the $Z^{\prime}$
boson and the Higgs bosons. Sec. IV is devoted to calculating the one-loop
corrections to the $hhh$ coupling. In Sec. V, we discuss the collider
phenomenology of our scenario focusing on the multi-$Z^{\prime}$ signature
coming from the decay of the Higgs bosons. Summary and discussions are given
in Sec. VI. In Appendix A, we present details of the renormalization which is
necessary to calculate the one-loop correction to the $hhh$ coupling.

##  II Models

| $~{}~{}q_{L}^{i}~{}~{}$ | $~{}~{}u_{R}^{i}~{}~{}$ | $~{}~{}d_{R}^{i}~{}~{}$ | $~{}~{}l_{L}^{e}~{}~{}$ | $~{}~{}l_{L}^{\mu}~{}~{}$ | $~{}~{}l_{L}^{\tau}~{}~{}$ | $~{}~{}e_{R}~{}~{}$ | $~{}~{}\mu_{R}~{}~{}$ | $~{}~{}\tau_{R}~{}~{}$ | $~{}~{}\nu_{R}^{e}~{}~{}$ | $~{}~{}\nu_{R}^{\mu}~{}~{}$ | $~{}~{}\nu_{R}^{\tau}~{}~{}$
---|---|---|---|---|---|---|---|---|---|---|---|---
$~{}~{}SU(2)_{L}~{}~{}$ | $\bf{2}$ | $\bf{1}$ | $\bf{1}$ | $\bf{2}$ | $\bf{2}$ | $\bf{2}$ | $\bf{1}$ | $\bf{1}$ | $\bf{1}$ | $\bf{1}$ | $\bf{1}$ | $\bf{1}$
$~{}~{}U(1)_{Y}~{}~{}$ | $\frac{1}{6}$ | $\frac{2}{3}$ | $-\frac{1}{3}$ | $-\frac{1}{2}$ | $-\frac{1}{2}$ | $-\frac{1}{2}$ | $-1$ | $-1$ | $-1$ | $0$ | $0$ | $0$
$~{}~{}U(1)^{\prime}~{}~{}$ | $X_{q_{L}}$ | $X_{u_{R}}$ | $X_{d_{R}}$ | $X_{l_{L}^{e}}$ | $X_{l_{L}^{\mu}}$ | $X_{l_{L}^{\tau}}$ | $X_{e_{R}}$ | $X_{\mu_{R}}$ | $X_{\tau_{R}}$ | $X_{\nu_{R}^{e}}$ | $X_{\nu_{R}^{\mu}}$ | $X_{\nu_{R}^{\tau}}$

Table 1: Charge assignments for fermions in a gauged $U(1)^{\prime}$ model,
where charges for quark fields ($q_{L}^{i}$, $u_{R}^{i}$ and $d_{R}^{i}$) are
taken to be flavor universal.

| $X_{q_{L}}$ | $X_{u_{R}}$ | $X_{d_{R}}$ | $X_{l_{L}^{e}}$ | $X_{l_{L}^{\mu}}$ | $X_{l_{L}^{\tau}}$ | $X_{e_{R}}$ | $X_{\mu_{R}}$ | $X_{\tau_{R}}$ | $X_{\nu_{R}^{e}}$ | $X_{\nu_{R}^{\mu}}$ | $X_{\nu_{R}^{\tau}}$
---|---|---|---|---|---|---|---|---|---|---|---|---
$~{}~{}U(1)_{B-L}~{}~{}$ | $\frac{1}{3}$ | $\frac{1}{3}$ | $\frac{1}{3}$ | $-1$ | $-1$ | $-1$ | $-1$ | $-1$ | $-1$ | $-1$ | $-1$ | $-1$
$~{}~{}U(1)_{L_{e}-L_{\mu}}~{}~{}$ | $0$ | $0$ | $0$ | $1$ | $-1$ | $0$ | $1$ | $-1$ | $0$ | $1$ | $-1$ | $0$
$~{}~{}U(1)_{L_{e}-L_{\tau}}~{}~{}$ | $0$ | $0$ | $0$ | $1$ | $0$ | $-1$ | $1$ | $0$ | $-1$ | $1$ | $0$ | $-1$
$~{}~{}U(1)_{L_{\mu}-L_{\tau}}~{}~{}$ | $0$ | $0$ | $0$ | $0$ | $1$ | $-1$ | $0$ | $1$ | $-1$ | $0$ | $1$ | $-1$
$~{}~{}U(1)_{D}~{}~{}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$

Table 2: Charges $X_{f}$ for a fermion $f$ in different $U(1)^{\prime}$
models.

We consider models with an extra $U(1)^{\prime}$ gauge symmetry in the minimal
extension of field contents where we introduce three right-handed neutrinos
$\nu_{R}^{i}(i=1-3)$ and a complex singlet scalar field $\Phi$. For
simplicity, we take the SM Higgs doublet $H$ to be neutral under
$U(1)^{\prime}$. Typical candidates of such $U(1)^{\prime}$ are then
$U(1)_{B-L}$, $U(1)_{L_{i}-L_{j}}$ and the dark $U(1)_{D}$ symmetry under the
anomaly cancellation condition. We summarize the $U(1)^{\prime}$ charges for
the SM fermions and right-handed neutrinos in Table 2 and Table 2.

The Lagrangian is generally written by

$\displaystyle\mathcal{L}=$
$\displaystyle\mathcal{L}_{\mathrm{SM}}-\frac{1}{4}X_{\mu\nu}X^{\mu\nu}-\frac{\epsilon}{2}B_{\mu\nu}X^{\mu\nu}+g_{X}X_{\mu}J^{\mu}_{X}+|D_{\mu}\Phi|^{2}-V(H,\Phi)+\mathcal{L}_{\nu_{R}},$
(1)

where $\mathcal{L}_{\mathrm{SM}}$ denotes the SM Lagrangian without the Higgs
potential, $X_{\mu\nu}$ ($B_{\mu\nu}$) is the field strength tensor for the
$U(1)^{\prime}$ (hypercharge) gauge field $X_{\mu}$ ($B_{\mu}$), $J_{\mu}^{X}$
is the current associated with $X^{\mu}$,
$D_{\mu}\Phi\equiv(\partial_{\mu}+ig_{X}X_{\Phi}X_{\mu})\Phi$ is the covariant
derivative for $\Phi$, $V$ is the Higgs potential, and $\mathcal{L}_{\nu_{R}}$
represents terms with right-handed neutrinos. In Eq. (1), we introduced
$\epsilon$ and $g_{X}$ denoting the kinetic mixing parameter and the gauge
coupling of $U(1)^{\prime}$, respectively.

Let us comment on the last term $\mathcal{L}_{\nu_{R}}$ which includes the
kinetic term for $\nu_{R}^{i}$ and the Yukawa interaction among $H$,
$l_{L}^{i}$ and $\nu_{R}^{i}$. Depending on the choice of the $U(1)^{\prime}$
symmetry and its charge assignments, $\mathcal{L}_{\nu_{R}}$ can also contain
Majorana masses of $\nu_{R}^{i}$ and/or the Yukawa interaction with $\Phi$
which generates the Majorana mass after the spontaneous breaking of
$U(1)^{\prime}$. Such a Majorana mass turns out to be masses of active
neutrinos via the type-I seesaw mechanism. In the following discussion, we
mainly focus on the phenomenology of the Higgs and $Z^{\prime}$ sector, and
details of the neutrino sector do not affect on the analyses given below. For
a concrete analysis, we fix $X_{\Phi}=1$ for all the $U(1)^{\prime}$ models
considered in this paper.

###  II.1 Higgs potential and Higgs boson masses

Here, we formulate eigenvalues for scalar bosons and corresponding eigenstates
after the spontaneous gauge symmetry breaking.

The Higgs potential $V$ is generally written by

$\displaystyle
V(H,\Phi)=-\mu_{H}^{2}H^{\dagger}H-\mu_{\Phi}^{2}\Phi^{*}\Phi+\frac{\lambda_{H}}{2}(H^{\dagger}H)^{2}+\frac{\lambda_{\Phi}}{2}(\Phi^{*}\Phi)^{2}+\lambda_{H\Phi}(H^{\dagger}H)(\Phi^{*}\Phi),$
(2)

where $\mu_{H}$ and $\mu_{\Phi}$ are the mass parameters, and
$\lambda_{H},~{}\lambda_{\Phi}$ and $\lambda_{H\Phi}$ are the quartic
couplings, respectively. We can parameterize the $H$ and $\Phi$ fields as

$\displaystyle H=\begin{pmatrix}G^{+}\\\
\frac{1}{\sqrt{2}}(v+\tilde{h}+iG)\end{pmatrix},\quad\Phi$
$\displaystyle=\frac{1}{\sqrt{2}}(v_{\Phi}+\tilde{\phi}+iG_{\Phi}),$ (3)

where $\tilde{h}$ and $\tilde{\phi}$ denote physical CP-even scalar components
while $G^{\pm}$, $G$ and $G_{\Phi}$ are the Nambu-Goldstone bosons that are
absorbed by the weak gauge bosons $Z,~{}W$ and extra $Z^{\prime}$ boson,
respectively. The mass squared matrix of the CP-even scalar bosons is given by

$\displaystyle
M^{2}_{\mathrm{even}}=\begin{pmatrix}\lambda_{H}v^{2}&\lambda_{H\Phi}vv_{\Phi}\\\
\lambda_{H\Phi}vv_{\Phi}&\lambda_{\Phi}v_{\Phi}^{2}\end{pmatrix},$ (4)

where we adopted the stationary conditions,

$\displaystyle\frac{\partial V}{\partial\tilde{h}}\Bigg{|}_{0}=\frac{\partial
V}{\partial\tilde{\phi}}\Bigg{|}_{0}=0,$ (5)

with $|_{0}$ denoting all the component fields to be zero. Then, we can
diagonalize the mass matrix by introducing an orthogonal matrix
$O_{\mathrm{even}}$,

$\displaystyle
O^{T}_{\mathrm{even}}M^{2}_{\mathrm{even}}O_{\mathrm{even}}=\mathrm{diag}(m_{h}^{2},m_{\phi}^{2}),$
(6)

where

$\displaystyle O_{\mathrm{even}}=\begin{pmatrix}\cos\alpha&-\sin\alpha\\\
\sin\alpha&\cos\alpha\end{pmatrix}.$ (7)

The mass eigenvalues are also given by

$\displaystyle m_{h}^{2}$
$\displaystyle=\lambda_{H}v^{2}\cos^{2}\alpha+\lambda_{\Phi}v_{\Phi}^{2}\sin^{2}\alpha+2\lambda_{H\Phi}vv_{\Phi}\sin\alpha\cos\alpha,$
(8a) $\displaystyle m_{\phi}^{2}$
$\displaystyle=\lambda_{\Phi}v_{\Phi}^{2}\cos^{2}\alpha+\lambda_{H}v^{2}\sin^{2}\alpha-2\lambda_{H\Phi}vv_{\Phi}\sin\alpha\cos\alpha.$
(8b)

The scalar mixing angle $\alpha$ is obtained as

$\displaystyle\tan
2\alpha=\frac{2\lambda_{H\Phi}vv_{\Phi}}{\lambda_{H}v^{2}-\lambda_{\Phi}v_{\Phi}^{2}}.$
(9)

The corresponding mass eigenstates are written by

$\displaystyle\begin{pmatrix}h\\\
\phi\end{pmatrix}=O^{T}_{\mathrm{even}}\begin{pmatrix}\tilde{h}\\\
\tilde{\phi}\end{pmatrix}.$ (10)

where we identify $h$ with the discovered Higgs boson with a mass of 125 GeV.
From Eqs. (8) and (9), the quartic couplings can be expressed in terms of the
masses and the mixing as

$\displaystyle\lambda_{H}$
$\displaystyle=\frac{1}{2v^{2}}\big{[}m_{h}^{2}+m_{\phi}^{2}+(m_{h}^{2}-m_{\phi}^{2})\cos
2\alpha\big{]},$ (11a) $\displaystyle\lambda_{\Phi}$
$\displaystyle=\frac{1}{2v_{\Phi}^{2}}\big{[}m_{h}^{2}+m_{\phi}^{2}-(m_{h}^{2}-m_{\phi}^{2})\cos
2\alpha\big{]},$ (11b) $\displaystyle\lambda_{H\Phi}$
$\displaystyle=\frac{1}{2vv_{\Phi}}(m_{h}^{2}-m_{\phi}^{2})\sin 2\alpha.$
(11c)

Notice that $\lambda_{H}$ and $\lambda_{\Phi}$ are always positive due to
positive mass squared of $h$ and $\phi$ while $\lambda_{H\Phi}$ can be
negative.

The trilinear couplings for the scalar bosons are then expressed in their mass
bases as follows

$\displaystyle\mathcal{L}\supset$
$\displaystyle\left(\frac{m_{Z}^{2}}{v}Z_{\mu}Z^{\mu}+\frac{2m_{W}^{2}}{v}W^{+}_{\mu}W^{-\mu}-\sum_{f_{\rm
SM}}\frac{m_{f_{\rm SM}}}{v}\overline{f_{\rm SM}}f_{\rm
SM}\right)(h\cos\alpha-\phi\sin\alpha)$
$\displaystyle+\frac{m_{Z^{\prime}}^{2}}{v_{\Phi}}Z^{\prime}_{\mu}Z^{\prime\mu}(\phi\cos\alpha+h\sin\alpha)+\sum_{\rm
scalars}\lambda_{\varphi_{i}\varphi_{j}\varphi_{k}}\varphi_{i}\varphi_{j}\varphi_{k},$
(12)

where $m_{f_{\rm SM}}$ represents the mass of the SM fermions $f_{\rm SM}$
($f_{\rm SM}\neq\nu_{R}$). The last term shows the scalar trilinear terms, and
the relevant couplings are given as follows:

$\displaystyle\lambda_{hhh}$
$\displaystyle=-\frac{m_{h}^{2}(v_{\Phi}\cos^{3}\alpha+v\sin^{3}\alpha)}{2vv_{\Phi}},$
(13) $\displaystyle\lambda_{\phi hh}$ $\displaystyle=\frac{\sin
2\alpha(2m_{h}^{2}+m_{\phi}^{2})(v_{\Phi}\cos\alpha-v\sin\alpha)}{4vv_{\Phi}},$
(14) $\displaystyle\lambda_{\phi\phi h}$ $\displaystyle=-\frac{\sin
2\alpha(m_{h}^{2}+2m_{\phi}^{2})(v\cos\alpha+v_{\Phi}\sin\alpha)}{4vv_{\Phi}},$
(15) $\displaystyle\lambda_{G_{\Phi}G_{\Phi}h}$
$\displaystyle=-\frac{m_{h}^{2}\sin\alpha}{2v_{\Phi}}.$ (16)

###  II.2 $Z^{\prime}$ mass and interactions

Because of the existence of the kinetic mixing term in Eq. (1), we first need
to diagonalize the kinetic terms of $X_{\mu}$ and $B_{\mu}$ by the
transformation

$\left(\begin{array}[]{c}X^{3}_{\mu}\\\ B_{\mu}\\\
\end{array}\right)=\left(\begin{array}[]{cc}r&0\\\ -\epsilon r&1\\\
\end{array}\right)\left(\begin{array}[]{c}\tilde{Z}^{\prime}_{\mu}\\\
\tilde{B}_{\mu}\\\ \end{array}\right),$ (17)

where $r=1/\sqrt{1-\epsilon^{2}}$. After the spontaneous breaking of the EW
and the $U(1)^{\prime}$ gauge symmetries, the mass terms of electrically
neutral gauge fields are induced as follows

$\displaystyle\mathcal{L}_{M}=\frac{1}{2}(\tilde{Z}_{\mu},\tilde{Z}_{\mu}^{\prime})\begin{pmatrix}m^{2}_{Z_{\rm
SM}}&\epsilon r\sin\theta_{W}m^{2}_{Z_{\rm SM}}\\\ \epsilon
r\sin\theta_{W}m^{2}_{Z_{\rm
SM}}&r^{2}\left(g_{X}^{2}v_{\Phi}^{2}+\epsilon^{2}m^{2}_{Z_{\rm
SM}}\sin^{2}\theta_{W}\right)\end{pmatrix}\begin{pmatrix}\tilde{Z}^{\mu}\\\
\tilde{Z}^{\prime\mu}\end{pmatrix},$ (18)

where
$\tilde{Z}_{\mu}=\cos\theta_{W}W^{3}_{\mu}-\sin\theta_{W}\tilde{B}_{\mu}$ and
$m^{2}_{Z_{\rm SM}}=v^{2}g^{2}/(4\cos^{2}\theta_{W})$ with $\theta_{W}$ and
$g$ being the Weinberg angle and the $SU(2)_{L}$ gauge coupling, respectively.
The above mass matrix can be diagonalized by introducing the following
orthogonal transformation:

$\displaystyle\begin{pmatrix}\tilde{Z}_{\mu}\\\
\tilde{Z}_{\mu}^{\prime}\end{pmatrix}=\begin{pmatrix}\cos\chi&\sin\chi\\\
-\sin\chi&\cos\chi\end{pmatrix}\begin{pmatrix}Z_{\mu}\\\
Z_{\mu}^{\prime}\end{pmatrix},$ (19)

where $Z_{\mu}$ can be identified with the observed $Z$ boson. The mixing
angle $\chi$ is written by

$\displaystyle\sin 2\chi=-\frac{2\epsilon r\sin\theta_{W}m_{Z_{{\rm
SM}}}^{2}}{m_{Z}^{2}-m_{Z^{\prime}}^{2}}.$ (20)

Taking a tiny kinetic mixing parameter $\epsilon\ll 1$, deviation of the $Z$
boson mass from the SM value is negligible and the $Z^{\prime}$ boson mass is
written by

$\displaystyle m_{Z^{\prime}}$
$\displaystyle=g_{X}v_{\Phi}[1+\mathcal{O}(\epsilon^{2})].$ (21)

The gauge current $J_{X}^{\mu}$ is given by

$\displaystyle J^{\mu}_{X}=\sum_{f}X_{f}\bar{f}\gamma^{\mu}f,$ (22)

where $f$ indicates the fermions in the models. The relevant interaction
Lagrangian for $Z^{\prime}$ and fermions $f$ is given by

$\displaystyle\mathcal{L}_{Z^{\prime}f\bar{f}}=g_{X}\bar{f}\gamma^{\mu}(v^{\prime}_{f}-a^{\prime}_{f}\gamma_{5})fZ^{\prime}_{\mu},$
(23)

where

$\displaystyle v^{\prime}_{f}$
$\displaystyle=X_{f}r\cos\chi+\frac{e}{g_{X}}Q_{f}\epsilon
r\cos\theta_{W}\cos\chi$
$\displaystyle-\frac{g}{g_{X}\cos\theta_{W}}\left(\frac{T_{3f}}{2}-Q_{f}\sin^{2}\theta_{W}\right)(\sin\chi+\epsilon
r\sin\theta_{W}\cos\chi),$ (24a) $\displaystyle a^{\prime}_{f}$
$\displaystyle=-\frac{g}{g_{X}\cos\theta_{W}}\frac{T_{3f}}{2}(\sin\chi+\epsilon
r\sin\theta_{W}\cos\chi),$ (24b)

where $T_{3f}$ is the diagonal $SU(2)_{L}$ generator, $Q_{f}$ is the electric
charge, and $X_{f}$ is the $U(1)^{\prime}$ charge given in Table 2,
respectively. We note that the dark photon ($X_{f}=0$) interaction is simply
proportional to the charge $Q_{f}$, i.e., $\mathcal{L}_{Z^{\prime}ff}\simeq
eQ_{f}\epsilon r\cos\theta_{W}\bar{f}\gamma^{\mu}fZ^{\prime}_{\mu}$ for
$m_{Z^{\prime}}\ll m_{Z}$.

##  III Decays of the Higgs bosons and $Z^{\prime}$

We discuss the decays of the Higgs bosons $\phi$ and $h$ as well as those of
$Z^{\prime}$.

First, the decay rates of $\phi\to Z^{\prime}Z^{\prime}$ and $\phi\to hh$ are
given by

$\displaystyle\Gamma(\phi\to Z^{\prime}Z^{\prime})$
$\displaystyle=\frac{m_{Z^{\prime}}^{4}\cos^{2}\alpha}{8\pi
v_{\Phi}^{2}m_{\phi}}\beta(x_{Z^{\prime}})\left[2+\frac{1}{4x_{Z^{\prime}}^{2}}\left(1-2x_{Z^{\prime}}\right)^{2}\right],$
(25a) $\displaystyle\Gamma(\phi\to hh)$ $\displaystyle=\frac{\lambda_{\phi
hh}^{2}}{8\pi m_{\phi}}\beta(x_{h}),$ (25b)

where $x_{i}=m_{i}^{2}/m_{\phi}^{2}$ and $\beta(x)=\sqrt{1-4x}$. The scalar
trilinear coupling $\lambda_{\phi hh}$ is given in Eq. (14). The decay rates
of $\phi$ into a pair of SM particles (except for $h$) are obtained from the
corresponding expression of the SM Higgs boson by multiplying the factor of
$\sin^{2}\alpha$ and replacing $m_{h}\to m_{\phi}$.

Second, the decay rates for the $h\to Z^{\prime}Z^{\prime}$ and $h\to\phi\phi$
modes are given by

$\displaystyle\Gamma(h\to Z^{\prime}Z^{\prime})$
$\displaystyle=\frac{m_{Z^{\prime}}^{4}\sin^{2}\alpha}{8\pi
v_{\Phi}^{2}m_{h}}\beta(z_{Z^{\prime}})\left[2+\frac{1}{4z_{Z^{\prime}}^{2}}\left(1-2z_{Z^{\prime}}\right)^{2}\right],$
(26a) $\displaystyle\Gamma(h\to\phi\phi)$
$\displaystyle=\frac{\lambda_{\phi\phi h}^{2}}{8\pi m_{h}}\beta(z_{\phi}),$
(26b)

where $z_{i}=m_{i}^{2}/m_{h}^{2}$ and $\lambda_{\phi\phi h}$ is given in Eq.
(15). The other decay modes of $h$ into SM particles are the same as the SM
ones except for the overall extra factor of $\cos^{2}\alpha$. Thus, the
branching ratios of $h$ into a pair of SM particles do not change so much from
the SM predictions as long as the new decay channels given in Eq. (26) do not
become large.

The decay rates of $Z^{\prime}$ into a fermion pair are calculated as

$\displaystyle\Gamma(Z^{\prime}\to
f\bar{f})=g_{X}^{2}\frac{m_{Z^{\prime}}}{12\pi}\beta\left(\frac{m_{f}^{2}}{m_{Z^{\prime}}^{2}}\right)\left[{v^{\prime}_{f}}^{2}+{a^{\prime}_{f}}^{2}+2({v^{\prime}_{f}}^{2}-3{a^{\prime}_{f}}^{2})\frac{m_{f}^{2}}{m_{Z^{\prime}}^{2}}\right],$
(27)

where $v^{\prime}_{f}$ and $a^{\prime}_{f}$ are given by Eq. (24).

Figure 1: Branching ratios for the Higgs boson decay $h\to
Z^{\prime}Z^{\prime}$ and $h\to\phi\phi$ as a function of $\sin\alpha$ where
the masses of $Z^{\prime}$ and $\phi$ are fixed to be 20 (0.02) GeV and 50
(0.05) GeV in the left (right) plot. We also change $g_{X}$ as 0.1, 0.01 and
0.001 for each plot as indicated by curves with black, red and blue colors.

In Fig. 1, we show the branching ratios of the Higgs boson $h$ into the new
modes $h\to\phi\phi$ and $h\to Z^{\prime}Z^{\prime}$ as a function of
$\sin\alpha$ by the solid and dashed curves respectively, where we fix the
$Z^{\prime}$ and $\phi$ masses to be 20 (0.02) GeV and 50 (0.05) GeV for the
left (right) plot. The value of $g_{X}$ is taken to be 0.1, 0.01 and 0.001 for
each plot as indicated by the black, red and blue curves. We see that the
branching ratios of $h\to\phi\phi$ and $h\to Z^{\prime}Z^{\prime}$ are going
to be similar values for smaller $\sin\alpha$ and/or smaller masses of $\phi$
and $Z^{\prime}$. This behavior can be understood by the following approximate
formulae:

$\displaystyle\begin{split}&\Gamma(h\to
Z^{\prime}Z^{\prime})=\frac{m_{h}^{3}\alpha^{2}}{32\pi
v_{\Phi}^{2}}+\mathcal{O}(\alpha^{4}),\\\
&\Gamma(h\to\phi\phi)=\frac{m_{h}^{3}\alpha^{2}}{32\pi
v_{\Phi}^{2}}\left(1+\frac{2m_{\phi}^{2}}{m_{h}^{2}}\right)^{2}\left(1+2\alpha\frac{v_{\Phi}}{v}\right)+\mathcal{O}(\alpha^{4}).\end{split}$
(28)

Thus, the above decay rates become the same if we neglect the
$m_{\phi}^{2}/m_{h}^{2}$ term and the ${\cal O}(\alpha^{3})$ contribution.
This can also be understood by noticing that the $h\to Z^{\prime}Z^{\prime}$
mode becomes one with the corresponding Goldstone boson in the final state,
i.e., $h\to G_{\Phi}G_{\Phi}$ for $m_{Z^{\prime}}/m_{h}\ll 1$ due to the
equivalence theorem. In the limit of $\alpha\ll 1$ and $m_{\phi}/m_{h}\ll 1$,
we obtain the relation among the couplings $\lambda_{\phi\phi
h}=\lambda_{G_{\Phi}G_{\Phi}h}=-v\lambda_{H\Phi}/2(=-\alpha
m_{h}^{2}/2v_{\Phi})$, so that these two decay rates coincide with each other.
It is also seen in Fig. 1 that smaller masses of $Z^{\prime}$ and $\phi$ tend
to give larger branching ratios of $h\to Z^{\prime}Z^{\prime}$ and
$h\to\phi\phi$ for fixed values of $\sin\alpha$ and $g_{X}$. This is because
the smaller masses correspond to the smaller VEV $v_{\Phi}$ from the relation
$m_{Z^{\prime}}\simeq g_{X}v_{\Phi}$, and the decay rates of these modes are
proportional to $1/v_{\Phi}^{2}$ as seen in Eq. (28).

##  IV One-loop corrections to the $hhh$ vertex

We discuss the one-loop corrections to the Higgs trilinear vertex $hhh$. In
order to obtain the finite results, we perform the renormalization of
parameters, particularly those of the Higgs potential.

We first shift the relevant bare parameters into the renormalized ones and the
counterterms as follows:

$\displaystyle T_{\varphi}\to\delta T_{\varphi},~{}~{}m_{\varphi}^{2}\to
m_{\varphi}^{2}+\delta m_{\varphi}^{2},~{}~{}v\to v+\delta v,~{}~{}v_{\Phi}\to
v_{\Phi}+\delta v_{\Phi},~{}~{}\alpha\to\alpha+\delta\alpha+\delta\alpha_{\rm
PT},$ (29)

where $\varphi=h$ or $\phi$. For the shift of the mixing angle $\alpha$, we
introduce not only the ordinal counterterm $\delta\alpha$ but also the one
coming from the pinch-term [34, 35] $\delta\alpha_{\rm PT}$ which is required
to maintain the gauge independent result [36, 37]333It has been known that the
on-shell renormalization scheme, that is applied in this paper, generally
gives rise to a gauge dependence in the renormalization of mixing parameter
[38]. . In Eq. (29), $\delta T_{\varphi}$ represent the counterterms for the
tadpole of $\varphi$. There are two ways for the treatment of $\delta
T_{\varphi}$, i.e., the so-called standard tadpole scheme [39] and the
alternative tadpole scheme [40]. In the former scheme, tadpole counterterms
are determined by requiring that renormalized scalar one-point functions
vanish. On the other hand, in the latter, tadpole counterterms are not
introduced, but all contributions from one-particle irreducible (1PI) diagrams
are modified to include those with tadpole-inserted diagrams. We analytically
confirm that there is no difference between the renormalized $hhh$ vertex
obtained by using the standard tadpole scheme and that by using the
alternative tadpole scheme. In the following, the alternative tadpole scheme
is applied to clarify the discussion. In addition to the parameter shift given
in Eq. (29), the bare fields are shifted as

$\displaystyle\begin{pmatrix}h\\\
\phi\end{pmatrix}\to\begin{pmatrix}1+\frac{1}{2}\delta Z_{h}&\delta
Z_{h\phi}\\\ \delta Z_{\phi h}&1+\frac{1}{2}\delta
Z_{\phi}\end{pmatrix}\begin{pmatrix}h\\\ \phi\end{pmatrix},$ (30)

where $\delta Z_{\phi h}$ and $\delta Z_{h\phi}$ can be decomposed as $\delta
Z_{h\phi}=\delta\alpha+\delta C_{h}$ and $\delta Z_{\phi
h}=-\delta\alpha+\delta C_{h}$ with $\delta C_{h}$ being the wavefunction
renormalization factor [41].

The renormalized scalar two-point functions are then expressed as

$\displaystyle\hat{\Gamma}_{\varphi\varphi}(p^{2})$
$\displaystyle=(p^{2}-m_{\varphi}^{2})(1+\delta Z_{\varphi})-\delta
m_{\varphi}^{2}+\Gamma_{\varphi\varphi}^{\rm
1PI}(p^{2}),~{}~{}(\varphi=h,~{}\phi)$ (31) $\displaystyle\hat{\Gamma}_{\phi
h}(p^{2})$ $\displaystyle=(p^{2}-m_{h}^{2})\delta
Z_{h\phi}+(p^{2}-m_{\phi}^{2})\delta Z_{\phi h}+\Gamma_{\phi h}^{\rm
1PI}(p^{2}),$ (32)

where $\Gamma_{XY}^{\rm 1PI}$ denotes the contribution from 1PI diagrams to
the two-point function for $X$ and $Y$ particles. We impose the on-shell
conditions as

$\displaystyle\Re[\hat{\Gamma}_{\varphi\varphi}(m_{\varphi}^{2})]=0,\quad\Re\left[\frac{d}{dp^{2}}\hat{\Gamma}_{\varphi\varphi}(p^{2})\right]_{p^{2}=m_{\varphi}^{2}}=1,\quad\Re[\hat{\Gamma}_{\phi
h}(m_{h}^{2})]=\Re[\hat{\Gamma}_{\phi h}(m_{\phi}^{2})]=0.$ (33)

These lead to

$\displaystyle\delta m_{\varphi}^{2}$
$\displaystyle=\Re[\Gamma_{\varphi\varphi}^{\rm
1PI}(m_{\varphi}^{2})],\quad\delta
Z_{\varphi}=-\Re\left[\frac{d}{dp^{2}}\Gamma_{\varphi\varphi}^{\rm
1PI}(p^{2})\right]_{p^{2}=m_{\varphi}^{2}},$ (34) $\displaystyle\delta Z_{\phi
h}$ $\displaystyle=-\frac{1}{m_{h}^{2}-m_{\phi}^{2}}\Re[\Gamma_{\phi h}^{\rm
1PI}(m_{h}^{2})],\quad\delta
Z_{h\phi}=\frac{1}{m_{h}^{2}-m_{\phi}^{2}}\Re[\Gamma_{\phi h}^{\rm
1PI}(m_{\phi}^{2})].$ (35)

The unpinched counterterm for the mixing angle $\delta\alpha$ is then given by

$\displaystyle\delta\alpha$ $\displaystyle=\frac{\delta Z_{h\phi}-\delta
Z_{\phi h}}{2}=\frac{1}{m_{\phi}^{2}-m_{h}^{2}}\Re\left[\Gamma_{\phi h}^{\rm
1PI}(m_{h}^{2})+\Gamma_{\phi h}^{\rm 1PI}(m_{\phi}^{2})\right].$ (36)

The pinch-term part is given by [37]

$\displaystyle\delta\alpha_{\rm
PT}=\frac{1}{m_{\phi}^{2}-m_{h}^{2}}\Re\left[\Gamma_{\phi h}^{\rm
PT}(m_{h}^{2})+\Gamma_{\phi h}^{\rm PT}(m_{\phi}^{2})\right],$ (37)

where

$\displaystyle\Gamma_{\phi h}^{\rm PT}(p^{2})=\frac{g^{2}}{64\pi^{2}}\sin
2\alpha(2p^{2}-m_{h}^{2}-m_{\phi}^{2})\left[B_{0}(p^{2};m_{W},m_{W})+\frac{1}{2c_{W}^{2}}B_{0}(p^{2};m_{Z},m_{Z})\right],$
(38)

with $B_{0}$ being the Passarino-Veltman’s scalar $B$ function [42].

The counterterms for the VEVs $\delta v$ and $\delta v_{\Phi}$ can be
determined from the renormalization of the EW parameters [39] which can
significantly be different from those in the SM if we introduce the non-zero
$Z$-$Z^{\prime}$ mixing. Although the $Z$-$Z^{\prime}$ mixing can play an
important role in the decay of $Z^{\prime}$ even for a small mixing angle, its
impact on the radiative correction to the $hhh$ vertex is negligibly small. We
thus neglect the effect of the $Z$-$Z^{\prime}$ mixing, i.e., $\chi\to 0$ or
equivalently $\epsilon\to 0$, on the renormalization of the $hhh$ vertex. In
this setup, the renormalization of the EW parameters can be performed in a
similar way as in the SM. Since the $Z$-$Z^{\prime}$ mixing is neglected, the
transverse part of the renormalized two-point function for $Z^{\prime}$ is
simply expressed as

$\displaystyle\hat{\Gamma}_{Z^{\prime}Z^{\prime}}(p^{2})=\Gamma_{Z^{\prime}Z^{\prime}}(p^{2})+(p^{2}-m_{Z^{\prime}}^{2})(1+\delta
Z_{Z^{\prime}})+\delta m_{Z^{\prime}}^{2},$ (39)

where $\delta m_{Z^{\prime}}^{2}$ and $\delta Z_{Z^{\prime}}$ are respectively
the counterterms for the mass and wavefunction of $Z^{\prime}$. Imposing the
on-shell conditions, i.e.,

$\displaystyle\Re[\hat{\Gamma}_{Z^{\prime}Z^{\prime}}(m_{Z^{\prime}}^{2})]=0,\quad\Re\left[\frac{d}{dp^{2}}\hat{\Gamma}_{Z^{\prime}Z^{\prime}}(p^{2})\right]_{p^{2}=m_{Z^{\prime}}^{2}}=1,$
(40)

we obtain

$\displaystyle\delta m_{Z^{\prime}}^{2}=\Re[\Gamma^{\rm
1PI}_{Z^{\prime}Z^{\prime}}(m_{Z^{\prime}}^{2})],\quad\delta
Z_{Z^{\prime}}=-\Re\left[\frac{d}{dp^{2}}\Gamma^{\rm
1PI}_{Z^{\prime}Z^{\prime}}(p^{2})\right]_{p^{2}=m_{Z^{\prime}}^{2}}.$ (41)

The counterterm of the singlet VEV $\delta v_{\Phi}$ is then determined via
the tree level relation, $v_{\Phi}=m_{Z^{\prime}}/g_{X}$:

$\displaystyle\frac{\delta v_{\Phi}}{v_{\Phi}}$ $\displaystyle=\frac{\delta
m_{Z^{\prime}}^{2}}{2m_{Z^{\prime}}^{2}}-\frac{\delta
g_{X}}{g_{X}}=\frac{\Gamma^{\rm
1PI}_{Z^{\prime}Z^{\prime}}(m_{Z^{\prime}}^{2})}{2m_{Z^{\prime}}^{2}}-\frac{\delta
g_{X}}{g_{X}}.$ (42)

The counterterm for the $U(1)^{\prime}$ gauge coupling $\delta g_{X}$ is
determined via the renormalization of the $\bar{f}fZ_{\mu}^{\prime}$ vertex as
follows

$\displaystyle\frac{\delta
g_{X}}{g_{X}}=\frac{1}{2}\frac{d}{dp^{2}}\Gamma_{Z^{\prime}Z^{\prime}}(p^{2})\Big{|}_{p^{2}=0}.$
(43)

See Appendix A for details of the renormalization of the $\bar{f}fZ^{\prime}$
vertex.

Now, we determine all the necessary counterterms for the computation of the
renormalized $hhh$ vertex which is expressed as

$\displaystyle\hat{\Gamma}_{hhh}$ $\displaystyle=\Gamma_{hhh}^{\rm
tree}+\Gamma_{hhh}^{\rm 1PI}+\delta\Gamma_{hhh},$ (44)

where the first, second and third terms respectively denote the contributions
from the tree level diagram, 1PI diagrams and the counterterms. The tree-level
contribution is given by

$\displaystyle\Gamma_{hhh}^{\rm tree}=3!\lambda_{hhh},$ (45)

where $\lambda_{hhh}$ is given in Eq. (13). The counterterm contribution is
given by

$\displaystyle\delta\Gamma_{hhh}$
$\displaystyle=3!\left(\delta\lambda_{hhh}+\frac{3}{2}\lambda_{hhh}\delta
Z_{h}+\lambda_{\phi hh}\delta Z_{\phi h}\right),$ (46)

where

$\displaystyle\delta\lambda_{hhh}$
$\displaystyle=\frac{m_{h}^{2}\cos^{3}\alpha}{2v}\frac{\delta
v}{v}+\frac{m_{h}^{2}\sin^{3}\alpha}{2v_{\Phi}}\frac{\delta
v_{\Phi}}{v_{\Phi}}-\frac{1}{2vv_{\Phi}}(v\sin^{3}\alpha+v_{\Phi}\cos^{3}\alpha)\delta
m_{h}^{2}$ $\displaystyle+\frac{3m_{h}^{2}}{4vv_{\Phi}}\sin
2\alpha(v_{\Phi}\cos\alpha-v\sin\alpha)(\delta\alpha+\delta\alpha_{\rm PT}).$
(47)

We will see in the next section that the deviation of the $hhh$ coupling from
the SM prediction is very small such as $\sim 1\%$ or smaller. This is because
the contribution to the $hhh$ vertex from the $Z^{\prime}$ loop is suppressed
by $(g_{X}^{2}\sin\alpha)^{3}$ which is negligibly small for $g_{X}\ll 1$. In
addition, the contribution from the $\phi$ loop is proportional to
$\lambda_{\phi\phi h}^{3}$ which is suppressed by $\sin^{3}\alpha$ for
$|\sin\alpha|\ll 1$. Thus, the situation is completely different from models
with singlet-scalar extensions without a $U(1)^{\prime}$ symmetry, where the
latter can predict ${\cal O}(100)$ % deviation from the SM, see e.g., [43].
This can be realized because more parameters appear in the potential and the
$\lambda_{\phi\phi h}$ coupling can independently be taken of the mixing
angle.

##  V Phenomenology

In this section, we discuss multi-$Z^{\prime}$ signatures via the decays of
$h$ and $\phi$ in our models with the spontaneously broken $U(1)^{\prime}$
symmetry at the LHC. There are five new independent parameters in our
scenario, which are chosen to be
$\\{m_{\phi},m_{Z^{\prime}},\sin\alpha,g_{X},\epsilon\\}$. For simplicity, we
take the kinetic mixing parameter $\epsilon$ to be zero except for the
$U(1)_{D}$ scenario. In the latter, the phenomenology does not depend on the
value of $\epsilon$ as far as $\epsilon\ll 1$444For instance, we can consider
the case with $e\epsilon\sim 10^{-4}$ which is allowed by current experimental
data [44].  because we consider the $Z^{\prime}$ production only via the Higgs
decay and the branching ratio of $Z^{\prime}$ is simply determined by the
electric charge of SM fermions, see Sec. II.2. For the masses of $\phi$ and
$Z^{\prime}$, we consider three reference sets that are (i) small mass case
$\\{m_{Z^{\prime}},m_{\phi}\\}=\\{0.02,0.05\\}$ GeV, (ii) middle mass case
$\\{m_{Z^{\prime}},m_{\phi}\\}=\\{20,50\\}$ GeV, and (iii) large mass case
$\\{m_{Z^{\prime}},m_{\phi}\\}=\\{200,800\\}$ GeV, all of which the $\phi\to
Z^{\prime}Z^{\prime}$ mode is kinematically allowed. We then discuss the
phenomenology on the parameter space of $\\{g_{X},\sin\alpha\\}$.

###  V.1 Small mass case: $\\{m_{Z^{\prime}},m_{\phi}\\}=\\{0.02,0.05\\}$ GeV

Figure 2: Contour plots of the branching ratios BR$(h\to 4Z^{\prime})$ and
BR$(h\to 2Z^{\prime})$ in the $U(1)^{\prime}$ models where DP stands for dark
photon, i.e., the $U(1)_{D}$ model. The gray region labeled by $h\to$invisible
is excluded by the search for $h\to$invisible signal at the LHC. The orange
region is excluded by $K^{+}\to\pi^{+}\phi$ search at NA62.

In this case, we focus on the $h$ decay into $Z^{\prime}Z^{\prime}$ and
$4Z^{\prime}$ final states and estimate the branching fraction of these modes.
In principle, the singlet-like scalar $\phi$ can be produced via the gluon
fusion process, but its decay products have too small energies to be detected
at LHC. We thus do not consider the gluon fusion production of $\phi$. On the
other hand, we need to take into account the production of $\phi$ via meson
decays through the scalar mixing effect. In particular, we impose the
strongest bound on the scalar mixing angle coming from the
$K^{+}\to\pi^{+}\phi$ search at NA62 [45]. Constraints from the discovered
Higgs boson decay should also be considered. Here, we note that 2 or 4 SM
fermion pairs produced from the decay chains of $h\to
Z^{\prime}Z^{\prime}\to(f\bar{f})(f\bar{f})$ and
$h\to\phi\phi\to(Z^{\prime}Z^{\prime})(Z^{\prime}Z^{\prime})\to(f\bar{f}f\bar{f})(f\bar{f}f\bar{f})$
are highly collimated due to the lightness of $\phi$ and $Z^{\prime}$. Thus,
typical constraints from multi-lepton final states cannot be applied directly.
Instead, we consider the bounds from “$h\to{\rm invisible}$” [46, 47] and
“$h\to{\rm BSM}$” [48] where “BSM” is any BSM final states. Among them the
strongest bound is BR$(h\to{\rm invisible})<0.107$ [47], so that we impose the
following constraint as a conservative limit

${\rm BR}(h\to Z^{\prime}Z^{\prime})+{\rm BR}(h\to\phi\phi)<0.107.$ (48)

In addition, we impose constraints on the new gauge coupling $g_{X}$ from
various experiments that can test $Z^{\prime}$ interaction with SM fermions.
For the $Z^{\prime}$ coupling, the strongest constraints are obtained by the
Texono experiment [49] in the $U(1)_{B-L}$, $U(1)_{L_{e}-L_{\mu}}$ and
$U(1)_{L_{e}-L_{\tau}}$ models and by the Borexino experiment [50] in the
$U(1)_{L_{\mu}-L_{\tau}}$ model, which test interactions between $Z^{\prime}$
and neutrinos. Bounds on the $Z^{\prime}$ coupling and mass are
comprehensively summarized in Refs. [51, 44, 52, 53, 54].

In Fig. 2, we show the contour plots for BR$(h\to Z^{\prime}Z^{\prime})$ and
BR$(h\to 4Z^{\prime})$ on the $\\{g_{X},\sin\alpha\\}$ plane in each model.
The gray region is excluded by the constraint from the branching ratio of
$h\to``{\rm invisible}"$. The light cyan regions for the $U(1)_{B-L}$,
$U(1)_{L_{e}-L_{\mu}}$ and $U(1)_{L_{e}-L_{\tau}}$ cases are excluded by the
Texono experiment. The light yellow region for the $U(1)_{L_{\mu}-L_{\tau}}$
case is excluded by the Borexino experiment. In addition, the light orange
region is excluded by the $K^{+}\to\pi^{+}\phi$ search at NA62. We see that
$\sin\alpha$ is highly restricted to be smaller than $\sim 10^{-4}$ due to the
constraints from $K^{+}\to\pi^{+}\phi$ and BR$(h\to
Z^{\prime}Z^{\prime})\simeq\ $BR$(h\to\phi\phi\to 4Z^{\prime})$ as we
discussed below Eq. (28). It is thus found that sizable branching ratios
cannot be obtained when SM fermions are charged under $U(1)^{\prime}$. On the
other hand, the branching ratio of BR$(h\to
Z^{\prime}Z^{\prime}/4Z^{\prime})\lesssim 0.05$ can be realized for the
$U(1)_{D}$ case since the new gauge coupling is not directly constrained from
the experiments which have a sensitivity to the $Z^{\prime}f\bar{f}$
interactions. Therefore, it is possible to have a large number of events at
the LHC for the $U(1)_{D}$ case, where $Z^{\prime}$ decays into $e^{+}e^{-}$
pair via the kinetic mixing with almost $100\%$. Thus, the signal processes
are

$h\to Z^{\prime}Z^{\prime}\to(e^{+}e^{-})(e^{+}e^{-}),\quad
h\to\phi\phi\to(Z^{\prime}Z^{\prime})(Z^{\prime}Z^{\prime})\to(e^{+}e^{-}e^{+}e^{-})(e^{+}e^{-}e^{+}e^{-}),$
(49)

where the particles inside the parentheses are highly collimated. Thus, these
signals can be seen as electron-jets where we need dedicated simulation
studies to clarify their detectability at collider experiments. We can also
have $\mathcal{O}(10^{-4})$ for BR$(h\to Z^{\prime}Z^{\prime}/4Z^{\prime})$ in
the $U(1)_{L_{\mu}-L_{\tau}}$ case but $Z^{\prime}$ only decays into neutrinos
as long as we take the kinetic mixing to be zero while $Z^{\prime}\to
e^{+}e^{-}$ is possible if a non-zero kinetic mixing is taken as in the
$U(1)_{D}$ case. For the $U(1)_{L_{e}-L_{\mu}}$ and $U(1)_{L_{e}-L_{\tau}}$
cases, we can have BR$(h\to Z^{\prime}Z^{\prime}/4Z^{\prime})\lesssim 10^{-6}$
while BR$(Z^{\prime}\to e^{+}e^{-})$ is given to be $\sim 50\%$. Therefore,
the $U(1)_{D}$ case is the most promising in this mass scale to search for the
signal at the LHC.

###  V.2 Middle mass case: $\\{m_{Z^{\prime}},m_{\phi}\\}=\\{20,50\\}$ GeV

| $~{}e^{+}e^{-}~{}$ | $~{}\mu^{+}\mu^{-}~{}$ | $~{}\tau^{+}\tau^{-}~{}$ | $~{}\nu\bar{\nu}~{}$ | $~{}jj~{}$ | $~{}b\bar{b}~{}$
---|---|---|---|---|---|---
$U(1)_{D}$ | 0.15 | 0.15 | 0.15 | 0 | 0.50 | 0.049
$U(1)_{B-L}$ | 0.16 | 0.16 | 0.16 | 0.24 | 0.22 | 0.054
$U(1)_{L_{e}-L_{\mu}}$ | 0.33 | 0.33 | 0 | 0.33 | 0 | 0
$U(1)_{L_{e}-L_{\tau}}$ | 0.33 | 0 | 0.33 | 0.33 | 0 | 0
$U(1)_{L_{\mu}-L_{\tau}}$ | 0 | 0.33 | 0.33 | 0.33 | 0 | 0
Table 3: The BRs of $Z^{\prime}\to f\bar{f}$ decays for $m_{Z^{\prime}}=20$
GeV where kinetic mixing is ignored except for the $U(1)_{D}$ case.

Figure 3: Contour plots of the branching ratios BR$(h\to 4Z^{\prime})$ and
BR$(h\to 2Z^{\prime})$ in the $U(1)^{\prime}$ models. The gray region is
excluded by the search for $h\to 4\ell$ signals at the LHC. The light green
region is excluded by the multilepton search at the LHC where leptons come
from $\phi\to Z^{\prime}Z^{\prime}\to 4\ell$ decay after $\phi$ production via
gluon fusion. The light magenta region is excluded by LHCb measurement for the
$U(1)_{B-L}$ case. The light blue regions are excluded by LHC
$\mu^{+}\mu^{-}Z^{\prime}(\to\mu^{+}\mu^{-})$ searches for
$U(1)_{L_{e(\mu)-L_{\mu(\tau)}}}$ case. The light cyan region is excluded by
Texono experiment for $U(1)_{L_{e}-L_{\tau}}$. The dashed contours indicate
values of deviation of the $hhh$ coupling from the SM at one-loop level.

As in the small mass case, we also focus on the new decay modes of the Higgs
boson $h$. In this case, we consider the constraints from the
$h\to(\ell^{+}\ell^{-})(\ell^{+}\ell^{-})$ searches [55, 56] and the non-
resonant multi-lepton searches [57] at the LHC where the latter can be induced
from the gluon fusion production of $\phi$ followed by the decay chain of
$\phi\to Z^{\prime}Z^{\prime}\to(\ell^{+}\ell^{-})(\ell^{+}\ell^{-})$. The
cross section of $gg\to\phi$ is estimated by CalcHEP_3.8 [58] implementing the
effective interaction for $\phi$-gluon-gluon, and we get
$\sigma(gg\to\phi)\simeq\sin^{2}\alpha\times 250$ pb applying K-factor 1.6
[59] for $m_{\phi}=$ 50 GeV and $\sqrt{s}=13$ TeV case. Here, we use the
branching ratios of $Z^{\prime}$ given in Table 3 for the estimation of the
bounds. In addition, we apply to the constraints on the gauge coupling $g_{X}$
from $Z^{\prime}$ searches where the strongest constraint comes from LHCb [60]
for $U(1)_{B-L}$, $pp\to\mu^{+}\mu^{-}Z^{\prime}(\to\mu^{+}\mu^{-})$ searches
at the LHC [61, 62] for $U(1)_{L_{e(\mu)}-L_{\mu(\tau)}}$ and neutrino
scattering measurement at Texono [49] for $U(1)_{L_{e}-L_{\tau}}$.

In Fig. 3, we show the contour plots for BR$(h\to Z^{\prime}Z^{\prime})$ and
BR$(h\to 4Z^{\prime})$ on the $\\{g_{X},\sin\alpha\\}$ plane in each model.
The gray and green regions are excluded by the
$h\to(\ell^{+}\ell^{-})(\ell^{+}\ell^{-})$ searches and the non-resonant
multi-lepton searches, respectively. The light magenta region is excluded by
the LHCb measurement for the $U(1)_{B-L}$ case. Also, the light blue regions
are excluded by the $\mu^{+}\mu^{-}Z^{\prime}(\to\mu^{+}\mu^{-})$ searches at
LHC for the $U(1)_{L_{e(\mu)}-L_{\mu(\tau)}}$ cases. In addition, the light
cyan region is excluded by the Texono experiment for the
$U(1)_{L_{e}-L_{\tau}}$ case. We find that BR$(h\to
Z^{\prime}Z^{\prime})\simeq\text{BR}(h\to 4Z^{\prime})\lesssim 3\times
10^{-5}$ is allowed for the $U(1)_{D}$ case and the value of $\lesssim
10^{-5}$ is allowed for the $L_{e}-L_{\tau}$ case. On the other hand, the
allowed value of the branching ratio is much smaller for the $U(1)_{B-L}$ and
$U(1)_{L_{e(\mu)}-L_{\mu(\tau)}}$ cases than that of the $U(1)_{D}$ or
$U(1)_{L_{e}-L_{\tau}}$ case due to the stronger constraint from LHCb and
$\mu^{+}\mu^{-}Z^{\prime}$ searches. From the maximal branching ratio of
$\mathcal{O}(10^{-5})$, we expect the cross section of the $gg\to h\to
Z^{\prime}Z^{\prime}/4Z^{\prime}$ processes to be ${\cal O}(1)$ fb which is
obtained by $\sigma_{gg\to h}\sim 50$ pb for the collision energy of 13 and 14
TeV. The signals from the multi-$Z^{\prime}$ production depend on
$U(1)^{\prime}$ via branching ratios of $Z^{\prime}$ as shown in Table 3. In
addition, we show the deviation of the $hhh$ coupling from the SM at one-loop
level, denoting $\Delta\lambda_{hhh}$, by partially modifying the H-COUP
package [63, 64, 65]. We find that the deviation is almost independent of
$g_{X}$ except for the region with $g_{X}=\mathcal{O}(1)$, and the magnitude
of the deviation can be $\mathcal{O}(1)\%$ for $\sin\alpha=\mathcal{O}(0.1)$.

###  V.3 Large mass case: $\\{m_{Z^{\prime}},m_{\phi}\\}=\\{200,800\\}$ GeV

Figure 4: Contours of the branching ratios BR$(\phi\to 2Z^{\prime})$ are shown
by the red dashed curves. The purple dash-dotted contours represent $\phi$
production cross section via gluon fusion with $\sqrt{s}=13$ TeV. The light
orange region is excluded by the LEP experiment. The dashed contours indicate
values of deviation of the $hhh$ coupling from the SM at one-loop level.

In this case, the Higgs boson $h$ cannot decay into $\phi\phi$ or
$Z^{\prime}Z^{\prime}$. We thus consider the production and decay of $\phi$.
By using CalcHEP_3.8, we obtain $\sigma(gg\to\phi)\simeq\sin^{2}\alpha\times
350$ fb for $m_{\phi}=$ 800 GeV and $\sqrt{s}=13$ TeV.

In Fig. 4, BR$(\phi\to Z^{\prime}Z^{\prime})$ is shown as the red dashed lines
on the $\\{g_{X},\sin\alpha\\}$ plane. The lower limit on
$m_{Z^{\prime}}/g_{X}$ is given by the LEP experiment as
$m_{Z^{\prime}}/g_{X}>6.9\,(5.8)$ TeV for the $U(1)_{B-L}$
($U(1)_{L_{e}-L_{\mu}}$ and $U(1)_{L_{e}-L_{\tau}}$) case [66], which is
translated into the upper limit on $g_{X}$ shown by the light orange color in
the figure. We also show the values of the gluon fusion cross section by the
purple dot-dashed horizontal lines. It is found that the cross section for the
$gg\to\phi\to Z^{\prime}Z^{\prime}$ process can maximally be ${\cal O}(1)$ fb
and ${\cal O}(0.1)$ fb for the $U(1)_{D}$ and $U(1)_{L_{\mu}-L_{\tau}}$ cases
and the $U(1)_{B-L}$, $U(1)_{L_{e}-L_{\mu}}$ and $U(1)_{L_{e}-L_{\tau}}$
cases, respectively. We note that the decay branching ratios of $Z^{\prime}$
into the SM fermions are almost the same as those shown in Table 3. In
addition, we show the deviation in the $hhh$ coupling from the SM which has
almost the same behavior as the middle mass case in Fig. 3.

In Table 4, we summarize the approximate upper bounds on BR$(h\to
Z^{\prime}Z^{\prime}/\phi\phi)$ and the signatures expected from the
multi-$Z^{\prime}$ production in each model. For each mass case, the
$U(1)_{D}$ model can provide the largest number of events due to the weaker
constraint on the gauge coupling $g_{X}$, in which $Z^{\prime}$ does not decay
into neutrinos so that the final states of the signature do not include
sizable missing transverse energy unlike all the other $U(1)^{\prime}$ models.
In each model, we expect the final state with multi-jets and/or multi-leptons
for the middle and large mass cases, where the fractions of jets and leptons
depend on the model via the branching ratio of $Z^{\prime}$ (see Table. 3 for
example). Thus, detailed analyses of the multi-$Z^{\prime}$ signature can
determine the $U(1)^{\prime}$ model. Further simulation studies are beyond the
scope of this work, and we leave them as future works.

(i) Small mass case
---
|  $U(1)_{D}$ |  $U(1)_{B-L}$ |  $U(1)_{L_{e}-L_{\mu}}$ |  $U(1)_{L_{e}-L_{\tau}}$ |  $U(1)_{L_{\mu}-L_{\tau}}$
BR$(h\to Z^{\prime}Z^{\prime}/\phi\phi)$ | $\lesssim 0.05$ | $\lesssim 10^{-6}$ | $\lesssim 10^{-6}$ | $\lesssim 10^{-6}$ | $\lesssim 10^{-4}$
Signal of Multi-$Z^{\prime}$ |  e-jets |  e-jets (+$\not{E}_{T}$) |  e-jets ($+\not{E}_{T}$) |  e-jets ($+\not{E}_{T}$) |  $\not{E}_{T}$ [+e-jets]
(ii) Middle mass case
|  $U(1)_{D}$ |  $U(1)_{B-L}$ |  $U(1)_{L_{e}-L_{\mu}}$ |  $U(1)_{L_{e}-L_{\tau}}$ |  $U(1)_{L_{\mu}-L_{\tau}}$
BR$(h\to Z^{\prime}Z^{\prime}/\phi\phi)$ | $\lesssim 10^{-5}$ | $\lesssim 10^{-7}$ | $\lesssim 10^{-5}$ | $\lesssim 10^{-7}$ | $\lesssim 10^{-7}$
Signal of Multi-$Z^{\prime}$ |  multi-$j/\ell$ |  multi-$j/\ell$ ($+\not{E}_{T}$) |  multi-$\ell$ ($+\not{E}_{T}$) |  multi-$\ell$ ($+\not{E}_{T}$) |  multi-$\ell$ ($+\not{E}_{T}$)
(iii) Large mass case
Signal of Multi-$Z^{\prime}$ |  multi-$j/\ell$ |  multi-$j/\ell$ ($+\not{E}_{T}$) |  multi-$\ell$ ($+\not{E}_{T}$) |  multi-$\ell$ ($+\not{E}_{T}$) |  multi-$\ell$ ($+\not{E}_{T}$)
Table 4: Summary of upper bounds of BR$(h\to Z^{\prime}Z^{\prime}/\phi\phi)$
[No upper bound of BR$(\phi\to Z^{\prime}Z^{\prime})$ for case (iii)] and
signatures in each model where e-jets stand for “electron-jet”, $\not{E}_{T}$
is missing transverse energy, and multi-$\ell/j$ denotes multi-leptons/jets
signal. The e-jets for $U(1)_{L_{\mu}-L_{\tau}}$ in case (i) can be obtained
via kinetic mixing effect.

##  VI Summary and discussion

In this paper, we have explored models with a spontaneously broken
$U(1)^{\prime}$ gauge symmetry, and have shown possibilities for obtaining
multi-$Z^{\prime}$ signatures from decays of the Higgs bosons. In particular,
we have focused on anomaly-free $U(1)^{\prime}$ models without introducing new
fermions (except for right-handed neutrinos), in which the SM Higgs field is
not charged under the symmetry. More specifically, we have considered the
models with the $U(1)_{B-L}$, $U(1)_{L_{i}-L_{j}}$ and hidden $U(1)_{D}$
symmetries where the last one is also known as the dark photon model. The
$U(1)^{\prime}$ symmetry is spontaneously broken by the VEV of a new scalar
field $\Phi$ which is singlet under the SM gauge symmetry.

We then have discussed decays of scalar bosons $h$ (SM-like) and $\phi$
(singlet-like) and a new gauge boson $Z^{\prime}$. We have seen that the new
decay channels $h\to Z^{\prime}Z^{\prime}$ and $h\to\phi\phi$ can be
significant for a larger mixing angle $\sin\alpha$ and/or a larger new gauge
coupling $g_{X}$. Furthermore, we have evaluated one-loop corrections to the
Higgs trilinear vertex by applying the on-shell renormalization scheme without
gauge dependence.

Finally, multi-$Z^{\prime}$ signatures from the scalar boson decays have been
discussed, where we have chosen three reference points for the masses of
$\phi$ and $Z^{\prime}$, i.e., $\\{m_{Z^{\prime}},m_{\phi}\\}$ to be (i)
$\\{0.02,0.05\\}$ GeV, (ii) $\\{20,50\\}$ GeV, and (iii) $\\{200,800\\}$ GeV.
For case (i), we have shown the branching ratios of $h\to
Z^{\prime}Z^{\prime}$ and $h\to\phi\phi\,(\phi\to Z^{\prime}Z^{\prime})$ modes
taking into account constraints on the new gauge coupling and the scalar
mixing from $h\to\text{invisible}$ decay, $K^{+}\to\pi^{+}\phi$ process and
searches for $Z^{\prime}$ interactions with SM fermions by various
experiments. For the dark photon case, it has been found that we can have the
sizable branching ratios for $h\to
Z^{\prime}Z^{\prime}\to(e^{+}e^{-})(e^{+}e^{-})$ and
$h\to\phi\phi\to(Z^{\prime}Z^{\prime})(Z^{\prime}Z^{\prime})\to(e^{+}e^{-}e^{+}e^{-})(e^{+}e^{-}e^{+}e^{-})$
where particles inside bracket will be highly collimated, while those of
$h\to\phi\phi/Z^{\prime}Z^{\prime}$ are much smaller in the other
$U(1)^{\prime}$ models than the dark photon model. Thus, we can expect
“electron-jets” signals from the processes at collider experiments. For case
(ii), we have also shown the Higgs decay branching ratios. In this case, we
have discussed constraints from multi-lepton signals from the decays of $h$
and $\phi$ at LHC. The new gauge coupling is also constrained from searches
for $Z^{\prime}$ by flavor experiments. It has been found that BR$(h\to
2Z^{\prime})\simeq$BR$(h\to 2\phi\to 4Z^{\prime})\sim\mathcal{O}(10^{-5})$ is
allowed in the dark photon and $U(1)_{L_{e}-L_{\tau}}$ cases while the
branching ratios have to be smaller by orders in the other cases. In these
cases, we obtain the cross section of the $gg\to h\to
Z^{\prime}Z^{\prime}/4Z^{\prime}$ process to be ${\cal O}$(1-0.1) fb which can
be detected by future LHC experiments. In addition, we have shown the
deviation of $hhh$ coupling from the SM, and found that its magnitude is less
than 1%. For case (iii), the SM-like Higgs boson does not decay into $\phi$ or
$Z^{\prime}$, and we have discussed production and decays of $\phi$. In this
mass range, $g_{X}$ and $\sin\alpha$ are not strongly constrained by
experiments compared with the other mass ranges. We have then shown the
production cross section of $\phi$, BR$(\phi\to Z^{\prime}Z^{\prime})$ and the
deviation of the $hhh$ coupling. In this case, $\phi$ production induces
multi-fermion signals via $Z^{\prime}$ where the fraction of signals depends
on the $U(1)^{\prime}$ type.

In summary, we have found possibilities for obtaining a detectable number of
multi-$Z^{\prime}$ events from the decay of $h$ and the production/decay of
$\phi$, where a type of signals depends on the mass scale of new bosons as
well as that of the $U(1)^{\prime}$ symmetry. Exploration of such signals is
important to test spontaneously broken $U(1)^{\prime}$ models. Detailed
analysis of signals is required for further testing models and it will be done
in upcoming works.

## Acknowledgments

The work was supported by the Fundamental Research Funds for the Central
Universities (T. N.).

##  Appendix A Renormalization of the $\bar{f}fZ_{\mu}^{\prime}$ vertex

The renormalized $\bar{f}fZ^{\prime}_{\mu}$ vertex can be decomposed into the
following form factors:

$\displaystyle\hat{\Gamma}_{Z^{\prime}ff}^{\mu}=g_{X}\gamma^{\mu}(\hat{\Gamma}_{Z^{\prime}ff}^{V}-\gamma_{5}\hat{\Gamma}_{Z^{\prime}ff}^{A})+i\sigma^{\mu\nu}q_{\nu}\hat{\Gamma}_{Z^{\prime}ff}^{M},$
(50)

where $q^{\mu}$ is the outgoing four-momentum for $Z^{\prime}$, and

$\displaystyle\sigma^{\mu\nu}=\frac{i}{2}[\gamma^{\mu},\gamma^{\nu}].$ (51)

Each form factor is further decomposed into the tree, 1PI and counterterm
contributions as

$\displaystyle\Gamma_{Z^{\prime}ff}^{i}=\Gamma_{Z^{\prime}ff}^{i,{\rm
tree}}+\Gamma_{Z^{\prime}ff}^{i,{\rm
1PI}}+\delta\Gamma_{Z^{\prime}ff}^{i},~{}~{}(i=V,~{}A,~{}M),$ (52)

where the tree-contribution is given by $\Gamma_{Z^{\prime}ff}^{V,{\rm
tree}}=v_{f}^{\prime}$, $\Gamma_{Z^{\prime}ff}^{A,{\rm tree}}=a_{f}^{\prime}$
and $\Gamma_{Z^{\prime}ff}^{M,{\rm tree}}=0$ with $v_{f}^{\prime}$ and
$a_{f}^{\prime}$ given in Eq. (24). The counterterm is given by

$\displaystyle\delta\Gamma_{Z^{\prime}ff}^{V}$
$\displaystyle=v_{f}^{\prime}\left(\frac{\delta g_{X}}{g_{X}}+\frac{\delta
Z_{Z^{\prime}}}{2}+\delta Z_{V}^{f}\right)+a_{f}^{\prime}\delta Z_{A}^{f},$
(53) $\displaystyle\delta\Gamma_{Z^{\prime}ff}^{A}$
$\displaystyle=a_{f}^{\prime}\left(\frac{\delta g_{X}}{g_{X}}+\frac{\delta
Z_{Z^{\prime}}}{2}+\delta Z_{V}^{f}\right)+v_{f}^{\prime}\delta Z_{A}^{f},$
(54)

where $\delta Z_{V}^{f}$ and $\delta Z_{A}^{f}$ are the wavefunction
renormalization factors for a fermion $f$. These are expressed as

$\displaystyle\delta Z_{V}^{f}=\frac{\delta Z_{L}^{f}+\delta
Z_{R}^{f}}{2},\quad\delta Z_{A}^{f}=\frac{\delta Z_{L}^{f}-\delta
Z_{R}^{f}}{2},$ (55)

with $\delta Z_{L}^{f}$ ($\delta Z_{R}^{f}$) being the wavefunction
renormalization factors for a left-handed (right-handed) fermion. Due of the
Ward-Takahashi identity, the 1PI contributions to the vector and axial-vector
parts take the following form in the limit of $q^{2}\to 0$:

$\displaystyle\lim_{q^{2}\to 0}\Gamma_{Z^{\prime}ff}^{V,\text{1PI}}$
$\displaystyle=-v_{f}^{\prime}\delta Z_{V}^{f}-a_{f}^{\prime}\delta
Z_{A}^{f},$ (56) $\displaystyle\lim_{q^{2}\to
0}\Gamma_{Z^{\prime}ff}^{A,\text{1PI}}$ $\displaystyle=-a_{f}^{\prime}\delta
Z_{V}^{f}-v_{f}^{\prime}\delta Z_{A}^{f}.$ (57)

Therefore, the renormalized form factors can be rewritten as

$\displaystyle\hat{\Gamma}_{Z^{\prime}ff}^{V}=v_{f}^{\prime}+\Delta\Gamma_{Z^{\prime}ff}^{V}(q^{2})+v_{f}^{\prime}\left(\frac{\delta
g_{X}}{g_{X}}+\frac{\delta Z_{Z^{\prime}}}{2}\right),$ (58)
$\displaystyle\hat{\Gamma}_{Z^{\prime}ff}^{A}=a_{f}^{\prime}+\Delta\Gamma_{Z^{\prime}ff}^{A}(q^{2})+a_{f}^{\prime}\left(\frac{\delta
g_{X}}{g_{X}}+\frac{\delta Z_{Z^{\prime}}}{2}\right),$ (59)

where
$\Delta\Gamma_{Z^{\prime}ff}^{X}(q^{2})=\Gamma_{Z^{\prime}ff}^{X,\text{1PI}}(q^{2})-\Gamma_{Z^{\prime}ff}^{X,\text{1PI}}(0)$
with $X=V,A$.

Now, we impose the renormalization condition as

$\displaystyle\hat{\Gamma}_{Z^{\prime}ff}^{V}(q^{2})\Big{|}_{q^{2}=0}=v_{f}^{\prime},$
(60)

by which we obtain

$\displaystyle\frac{\delta g_{X}}{g_{X}}=-\frac{\delta Z_{Z^{\prime}}}{2}.$
(61)

Combing Eqs. (42) and (61), we can determine the counterterm of the singlet
VEV $\delta v_{\Phi}/v_{\Phi}$.

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