# Explaining the $b\to s\ell^{+}\ell^{-}$ anomalies
in ${Z^{\prime}}$ scenarios with top-FCNC couplings

Xin-Qiang<EMAIL_ADDRESS>Meng<EMAIL_ADDRESS>Dong-Yang<EMAIL_ADDRESS>
Ya-Dong<EMAIL_ADDRESS>and Xing-Bo<EMAIL_ADDRESS>
1 Institute of Particle Physics and Key Laboratory of Quark and Lepton Physics
(MOE),
Central China Normal University, Wuhan, Hubei 430079, China
2 Department of Physics, Jiujiang University, Jiujiang 332005, China
3 School of Physics and Microelectronics, Zhengzhou University, Zhengzhou,
Henan 450001, China

###### Abstract

Motivated by the recent anomalies in $b\to s\ell^{+}\ell^{-}$ transitions, we
explore a minimal ${Z^{\prime}}$ scenario, in which the ${Z^{\prime}}$ boson
has a flavour-changing coupling to charm and top quarks and a flavour-
conserving coupling to muons. It is found that such a ${Z^{\prime}}$ boson can
explain the current $b\to s\ell^{+}\ell^{-}$ anomalies, while satisfying other
flavour and collider constraints simultaneously. The ${Z^{\prime}}$ boson can
be as light as few hundreds GeV. In this case, the $t\to c\mu^{+}\mu^{-}$
decay and the $t{Z^{\prime}}$ associated production at the LHC could provide
sensitive probes of such a ${Z^{\prime}}$ boson. As a special feature, the
${Z^{\prime}}$ contributions to all rare $B$\- and $K$-meson processes are
controlled by one parameter. This results in interesting correlations among
these processes, which could provide further insights into this scenario. In
addition, an extended scenario, in which the ${Z^{\prime}}$ boson interacts
with the $SU(2)_{L}$ fermion doublets with analogous couplings as in the
minimal scenario, is also investigated.

##  1 Introduction

The flavour-changing neutral current (FCNC) processes are sensitive to
possible contributions from heavy mediators and provide a complementary way to
direct searches for new physics (NP) at high-energy frontiers. While there is
so far no direct evidence for NP at the LHC, recent measurements of the rare
$b\to s\ell^{+}\ell^{-}$ decays exhibit several interesting discrepancies from
the Standard Model (SM) predictions of the branching ratios, the angular
distributions, and the lepton flavour universality (LFU) ratios [1, 2, 3, 4].

In this respect, the ratios defined as $R_{K^{(*)}}\equiv{\mathcal{B}}(B\to
K^{(*)}\mu^{+}\mu^{-})/{\mathcal{B}}(B\to K^{(*)}e^{+}e^{-})$ are of
particular interest, because the hadronic uncertainties largely cancel out
here [5]. Therefore, they provide a sensitive test of the LFU. In the SM,
these ratios are predicted to be close to unity up to tiny electromagnetic
corrections [5, 6, 7, 8]. Recently, the LHCb collaboration presented an
updated measurement of $R_{K}$ using the full data set of $9{\,\rm fb}^{-1}$
[9]:

$\displaystyle
R_{K}=0.846_{-0.039}^{+0.042}(\text{stat})_{-0.012}^{+0.013}(\text{syst}),\qquad\text{for
}1.1<q^{2}<6.0{\,\rm GeV}^{2},$ (1)

where $q^{2}$ denotes the dilepton invariant mass squared. This new result
confirms the previous LHCb measurement using a data set of $5{\,\rm fb}^{-1}$
[10]. However, the tension with the SM prediction has increased from
previously $2.5\sigma$ to now $3.1\sigma$, due to reduced experimental
uncertainties. LHCb has also reported a measurement of $R_{K^{*}}$ using the
full Run-I data set of $3{\,\rm fb}^{-1}$ [11]:

$\displaystyle R_{K^{*}}=\begin{cases}0.66_{-0.07}^{+0.11}(\text{stat})\pm
0.03(\text{syst}),&0.045<q^{2}<1.1{\,\rm GeV}^{2},\\\
0.69_{-0.07}^{+0.11}(\text{stat})\pm
0.05(\text{syst}),&1.1\hphantom{00}<q^{2}<6.0{\,\rm GeV}^{2},\end{cases}$ (2)

which are found to be about $2.1\sigma$ and $2.4\sigma$ lower than the SM
predictions, respectively. Very recently, the LHCb measurements of $R_{K_{S}}$
and $R_{K^{*+}}$ have also been reported [12]. Previous measurements from the
BaBar [13] and Belle [14, 15] experiments are also consistent with the LHCb
results, although with relatively large uncertainties. All these measurements
of $R_{K}$ and $R_{K^{*}}$ together provide intriguing hints of the LFU
violation.

Recently, the LHCb collaboration reported the most precise measurement of the
branching ratio of the $B_{s}\to\phi\mu^{+}\mu^{-}$ decay using the full Run-1
and Run-2 data sets [16],

$\displaystyle{\mathcal{B}}(B_{s}\to\phi\mu^{+}\mu^{-})=(2.88\pm 0.15\pm
0.05\pm 0.14)\times 10^{-8},\quad\text{for }1.1<q^{2}<6.0{\,\rm GeV}^{2},$ (3)

where the uncertainties are, in order, statistical, systematic, and from the
branching fraction of the normalization mode. This measurement is found to lie
$3.6\sigma$ below the SM prediction [17, 18, 19, 20]. Using the same full data
set, LHCb later presented an improved measurement of the branching ratio of
the rare $B_{s}\to\mu^{+}\mu^{-}$ decay [21],

$\displaystyle{\mathcal{B}}(B_{s}\to\mu^{+}\mu^{-})=\big{[}3.09_{-0.43}^{+0.46}(\text{stat})_{-0.11}^{+0.15}(\text{syst})\big{]}\times
10^{-9},$ (4)

which is statistically consistent with the previous world average [22]. This
new result is compatible with the SM expectation [23, 24] within $1\sigma$.
However, after combining all the measurements of the
$B_{s,d}\to\mu^{+}\mu^{-}$ decays from the ATLAS [25], CMS [26], and LHCb [21]
experiments, the total discrepancy with the SM is found to be at the level of
$2\sigma$ [27, 28]. In addition, the latest measurements of the $B^{0}\to
K^{*0}\mu^{+}\mu^{-}$ [29] and $B^{+}\to K^{*+}\mu^{+}\mu^{-}$ [30] decays
show tensions in the angular distributions with respect to the SM predictions.
The local discrepancies in the angular observables $P_{2}$ and
$P_{5}^{\prime}$ are observed to be at $2.5$-$3.0\sigma$ in two $q^{2}$ bins
[29, 30].

Although none of the individual deviations is statistically significant, and
further refinement of the hadronic uncertainties in some observables is still
an ongoing theoretical issue [31, 32, 33], the global tension in the $b\to
s\ell^{+}\ell^{-}$ decays has motivated a lot of NP interpretations [1, 2, 3,
4]. In this respect, one of the most popular NP explanations are models with
an extra heavy vector ${Z^{\prime}}$ boson. In these models, the
${Z^{\prime}}$ boson has couplings to quarks, as well as to either electrons
or muons. Depending on the quark couplings involved, these ${Z^{\prime}}$
models can be classified into two categories: (i) The ${Z^{\prime}}$ boson has
flavour-violating couplings to $b$ and $s$ quarks and the $b\to
s\ell^{+}\ell^{-}$ transitions receive contributions from tree-level
${Z^{\prime}}$ exchange [34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46,
47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58]. (ii) The ${Z^{\prime}}$ boson
has flavour-conserving couplings to top quark and affects the $b\to
s\ell^{+}\ell^{-}$ transitions via one-loop penguin diagrams [59, 60]. In this
paper, based on our previous works [61, 62], we will consider another
possibility, where the ${Z^{\prime}}$ boson has flavour-violating couplings to
top and charm quarks. This scenario does not suffer from the constraints from
$B_{s}-\bar{B}_{s}$ mixing, and the ${Z^{\prime}}$ boson contributes to the
$b\to s\ell^{+}\ell^{-}$ decays at the one-loop level. We will derive
constraints on the ${Z^{\prime}}$ mass and couplings from various flavour and
collider processes, and study the possibility of explaining the $b\to
s\ell^{+}\ell^{-}$ anomalies in such a scenario. Future prospects of searching
for such a ${Z^{\prime}}$ boson at the LHC will also be discussed.

This paper is organized as follows. In section 2, we introduce the
phenomenological ${Z^{\prime}}$ scenarios, which have the desired flavour-
changing couplings to explain the current $b\to s\ell^{+}\ell^{-}$ anomalies.
In section 3, we recapitulate the theoretical frameworks for various flavour
processes and discuss the ${Z^{\prime}}$ effects. In section 4, we give our
detailed numerical results and discussions. Our conclusions are finally made
in section 5.

##  2 ${Z^{\prime}}$ scenarios with top-quark FCNC couplings

We consider two phenomenological scenarios involving a ${Z^{\prime}}$ boson.
In the first scenario (denoted as scenario I), the ${Z^{\prime}}$ boson has
flavour-changing couplings to $c$ and $t$ quarks and a flavour-conserving
coupling to the $\mu$ lepton. Their interactions are described by the
effective Lagrangian

$\displaystyle\mathcal{L}_{Z^{\prime}}^{\rm
I}=\big{(}{X_{ct}^{L}}\,\bar{c}\gamma^{\mu}P_{L}t\,Z^{\prime}_{\mu}+{\rm
h.c.}\big{)}+{\lambda_{\mu\mu}^{L}}\,\bar{\mu}\gamma^{\mu}P_{L}\mu\,Z^{\prime}_{\mu},$
(5)

where $P_{L}=(1-\gamma_{5})/2$, and the fermion fields $c$, $t$ and $\mu$
refer to the mass eigenstates. Generally, the couplings ${X_{ct}^{L}}$ and
${\lambda_{\mu\mu}^{L}}$ are complex and real, respectively. In this scenario,
the ${Z^{\prime}}$ boson affects the $b\to s\mu^{+}\mu^{-}$ transitions via
one-loop penguin diagram and could explain the anomalies observed in $B\to
K^{(*)}\ell^{+}\ell^{-}$ decays. Furthermore, sizable contributions to the
$s\to d\mu^{+}\mu^{-}$ transitions could arise from the ${Z^{\prime}}$ penguin
diagram.

In the second scenario (denoted as scenario II), the ${Z^{\prime}}$ boson is
assumed to interact with the $SU(2)_{L}$ fermion doublets with similar
couplings as in scenario I. In the mass eigenstate basis, the effective
Lagrangian takes the form666In ref. [63], a similar ${Z^{\prime}}$ scenario
has been discussed, where the ${Z^{\prime}}$ boson has left-handed couplings
to the lepton doublets but only right-handed couplings to the up-type quarks.

$\displaystyle\mathcal{L}_{Z^{\prime}}^{\rm II}=$
$\displaystyle\big{(}X_{23}^{L}\,\bar{Q}_{L,2}\gamma^{\mu}Q_{L,3}\,Z_{\mu}^{\prime}+\text{h.c.}\big{)}+\lambda^{L}_{22}\,\bar{L}_{L,2}\gamma^{\mu}L_{L,2}\,Z_{\mu}^{\prime}$
(6) $\displaystyle=$
$\displaystyle\big{[}X_{23}^{L}\big{(}\bar{c}\gamma^{\mu}P_{L}t+\bar{s}\gamma^{\mu}P_{L}b\big{)}Z_{\mu}^{\prime}+\text{h.c.}\big{]}+\lambda^{L}_{22}\big{(}\bar{\mu}\gamma^{\mu}P_{L}\mu+\bar{\nu}_{\mu}\gamma^{\mu}P_{L}\nu_{\mu}\big{)}Z_{\mu}^{\prime},$

where $Q_{L,i}$ and $L_{L,i}$ denote the left-handed $SU(2)_{L}$ quark and
lepton doublets of the $i$-th generation, respectively. As in scenario I, the
couplings $X_{23}^{L}$ and $\lambda_{22}^{L}$ are generally complex and real,
respectively. In this scenario, the $b\bar{s}{Z^{\prime}}$ and the
$t\bar{c}{Z^{\prime}}$ interaction have the same coupling strength due to
$SU(2)_{L}$ invariance. As a consequence, the ${Z^{\prime}}$ contribution to
$b\to s$ processes is mainly induced by the $b\bar{s}{Z^{\prime}}$ interaction
at the tree level, and the phenomenology of these processes is similar to that
in the so-called minimal ${Z^{\prime}}$ scenario discussed in refs. [34, 45].
As in scenario I but contrary to the minimal ${Z^{\prime}}$ scenario, the
$s\to d$ transitions can receive the ${Z^{\prime}}$ contribution at the one-
loop level. Furthermore, the ${Z^{\prime}}$ boson also couples to the muon
neutrino due to $SU(2)_{L}$ invariance, which may affect the $b\to
s\nu\bar{\nu}$ and $s\to d\nu\bar{\nu}$ processes. The ${Z^{\prime}}$ effects
in various flavour processes will be discussed in detail in the next section.

Alternatively, one can consider a ${Z^{\prime}}$ boson with flavour-changing
couplings to $u$ and $t$ instead of $c$ and $t$ quarks in the above scenarios.
Then, the ${Z^{\prime}}$ contributions to the top-quark production and FCNC
decays can be very different from that in scenarios I and II. For the
${Z^{\prime}}$ couplings to leptons, a right-handed
$\mu^{+}\mu^{-}{Z^{\prime}}$ interaction can also be added to simultaneously
accommodate the $b\to s\mu^{+}\mu^{-}$ and $(g-2)_{\mu}$ anomalies [64, 65].
Similarly, scenarios with an $e^{+}e^{-}{Z^{\prime}}$ instead of a
$\mu^{+}\mu^{-}{Z^{\prime}}$ coupling in the above scenarios can also be
considered. Such a ${Z^{\prime}}$ boson could be directly produced in
$e^{+}e^{-}$ colliders, but loses the possibility of explaining the
$(g-2)_{\mu}$ anomaly. We leave all these possibilities for future studies,
and consider in this paper only the above two scenarios that have the
couplings required to explain the $b\to s\ell^{+}\ell^{-}$ anomalies.

##  3 ${Z^{\prime}}$ effects in various flavour processes

In this section, we recapitulate the theoretical frameworks for various low-
energy flavour processes and discuss the ${Z^{\prime}}$ contributions to them
as well as to the top-quark physics.

###  3.1 $b\to s\mu^{+}\mu^{-}$ transitions

The rare decays $B_{s}\to\mu^{+}\mu^{-}$, $B\to K^{(*)}\mu^{+}\mu^{-}$, and
$B_{s}\to\phi\mu^{+}\mu^{-}$ are induced by the $b\to s\mu^{+}\mu^{-}$
transitions, and provide promising probes of NP effects. With the
${Z^{\prime}}$ contributions taken into account, the effective Hamiltonian for
the $b\to s\mu^{+}\mu^{-}$ transitions can be written as [66]

$\displaystyle\mathcal{H}_{\rm eff}^{b\to
s\mu^{+}\mu^{-}}=-\frac{4G_{F}}{\sqrt{2}}V_{tb}V_{ts}^{*}\sum_{i=1}^{10}{\mathcal{C}}_{i}{\mathcal{O}}_{i},$
(7)

where explicit definitions of the effective operators ${\mathcal{O}}_{1-8}$
can be found in ref. [66]. The operators most relevant to our study are,
however, the two semi-leptonic operators ${\mathcal{O}}_{9\ell}$ and
${\mathcal{O}}_{10\ell}$ defined by

$\displaystyle{\mathcal{O}}_{9\ell}=\frac{e^{2}}{16\pi^{2}}(\bar{s}\gamma_{\mu}P_{L}b)(\bar{\ell}\gamma^{\mu}\ell),\qquad\text{and}\qquad{\mathcal{O}}_{10\ell}=\frac{e^{2}}{16\pi^{2}}(\bar{s}\gamma_{\mu}P_{L}b)(\bar{\ell}\gamma^{\mu}\gamma_{5}\ell),$
(8)

respectively. In the SM, their Wilson coefficients have been calculated
including next-to-next-to-leading-order (NNLO) QCD [67, 68, 69] and next-to-
leading-order (NLO) electroweak corrections [70]. In the ${Z^{\prime}}$
scenario I, the $t\bar{c}{Z^{\prime}}$ vertex can affect the $b\to
s\mu^{+}\mu^{-}$ transitions through the ${Z^{\prime}}$-penguin diagram shown
in figure 1 (b), with the resulting NP Wilson coefficients given by [61, 62]

$\displaystyle{\mathcal{C}}_{9\mu}^{\rm NP,\,I}=-{\mathcal{C}}_{10\mu}^{\rm
NP,\,I}=\frac{1}{8\sqrt{2}G_{F}s_{W}^{2}}\frac{V_{cs}^{*}}{V_{ts}^{*}}\frac{{X_{ct}^{L}}{\lambda_{\mu\mu}^{L}}}{m_{Z^{\prime}}^{2}}f(x_{t}),$
(9)

with $s_{W}=\sin\theta_{W}$, $x_{t}=\bar{m}_{t}(\bar{m}_{t})^{2}/m_{W}^{2}$,
and $V_{ij}$ the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements [71, 72]. The
loop function $f(x)$ is defined as

$\displaystyle f(x)=\frac{3}{2}+x\log\frac{\mu^{2}}{m_{W}^{2}}+x-x\log x,$
(10)

which is obtained from the calculation of similar diagrams with anomalous
$t\bar{c}Z$ vertex in refs. [61, 62]. It is noted that the ${Z^{\prime}}$
contributions are enhanced by the CKM factor $V_{cs}/V_{ts}$. In the
${Z^{\prime}}$ scenario II, the ${Z^{\prime}}$ boson can also affect the $b\to
s\mu^{+}\mu^{-}$ transitions at tree level through the diagram in figure 1
(c). The total contributions to the NP Wilson coefficients are given by

$\displaystyle{\mathcal{C}}_{9\mu}^{\rm NP,\,II}=-{\mathcal{C}}_{10\mu}^{\rm
NP,\,II}=\frac{1}{\sqrt{2}G_{F}}\bigg{(}\frac{1}{8s_{W}^{2}}\frac{V_{cs}^{*}}{V_{ts}^{*}}f(x_{t})-\frac{\pi}{\alpha_{\rm
e}V_{tb}V_{ts}^{*}}\bigg{)}\frac{X_{23}^{L}\lambda_{22}^{L}}{m_{{Z^{\prime}}}^{2}}.$
(11)

Numerically, it is found that the tree-level contribution is dominant and the
loop-level contribution can be safely neglected. For $m_{Z^{\prime}}=1\,{\,\rm
TeV}$, ${\mathcal{C}}_{9\mu}^{\rm
NP,\,I}=(-4.4-0.1i){X_{ct}^{L}}{\lambda_{\mu\mu}^{L}}$ in scenario I, while
${\mathcal{C}}_{9\mu}^{\rm NP,\,II}=(585+11i)X_{23}^{L}\lambda_{22}^{L}$ in
scenario II.

Figure 1: Feynman diagrams for the $b\to s\mu^{+}\mu^{-}$ transitions,
including the selected SM penguin contributions (a), as well as the
${Z^{\prime}}$ effects from the scenarios I (b) and II (b, c).

A lot of effort has been put into the theoretical treatment of $B\to
K\mu^{+}\mu^{-}$, $B\to K^{*}\mu^{+}\mu^{-}$, and $B_{s}\to\phi\mu^{+}\mu^{-}$
decays [73, 74, 75, 76, 77, 19, 33]. Besides the LFU ratios $R_{K^{(*)}}$
introduced in section 1, the angular observables in these decays are also
known to provide valuable information of potential NP contributions, and hence
have been analyzed in detail in refs. [78, 79]. For these observables, the
main theoretical uncertainties come from the heavy-to-light transition form
factors. During recent years, significant progress has been made in
calculating these form factors from lattice QCD [18, 80] and light-cone sum
rules (LCSR) [81, 19, 82, 83, 84, 85]. For a recent review of the $b\to
s\mu^{+}\mu^{-}$ decays, we refer to refs. [86, 87, 88], where the SM
calculations, input parameters, form factors, and theoretical uncertainties
are discussed in detail.

###  3.2 $B_{s}-\bar{B}_{s}$ mixing

Figure 2: Feynman diagrams for the $B_{s}-\bar{B}_{s}$ mixing both within the
SM (a) and in the ${Z^{\prime}}$ scenario II (b, c). For each diagram, there
is also a second one obtained by a $90^{\circ}$ rotation.

The $B_{s}-\bar{B}_{s}$ mixing is induced by the $W$-box diagram in the SM,
and could receive contributions from the tree and penguin diagrams in the
${Z^{\prime}}$ scenario II, as shown in figure 2. The effective Hamiltonian
for $B_{s}-\bar{B}_{s}$ mixing can be written as [89]

$\displaystyle\mathcal{H}_{\rm eff}^{\Delta
B=2}=\frac{G_{F}^{2}}{16\pi^{2}}\,m_{W}^{2}(V_{tb}V_{ts}^{*})^{2}\,{\mathcal{C}}^{\rm
VLL}\,{\mathcal{O}}^{\rm VLL}+{\rm h.c.},$ (12)

with the effective operator ${\mathcal{O}}^{\rm
VLL}=(\bar{s}\gamma_{\mu}P_{L}b)(\bar{s}\gamma^{\mu}P_{L}b)$. Analytical
expression of the SM Wilson coefficient ${\mathcal{C}}_{\rm SM}^{\rm VLL}$ and
the QCD renormalization group evolution (RGE) can be found in refs. [66, 90]
and [89, 91], respectively. In the ${Z^{\prime}}$ scenario II, the
${Z^{\prime}}$-penguin diagram shown in figure 2 (c) is found to be
negligible, and the tree-level ${Z^{\prime}}$-exchange diagram shown in figure
2 (b) results in the NP contribution

$\displaystyle{\mathcal{C}}_{\rm NP,\,II}^{\rm
VLL}=\frac{8\pi^{2}}{G_{F}^{2}m_{W}^{2}}\frac{1}{(V_{tb}V_{ts}^{*})^{2}}\left(\frac{X_{23}^{L}}{m_{Z^{\prime}}}\right)^{2}.$
(13)

In the ${Z^{\prime}}$ scenario I, the ${Z^{\prime}}$ boson contributes to the
$B_{s}-\bar{B}_{s}$ mixing starting at the two-loop level, and its effects are
therefore expected to be small. Here we will take ${\mathcal{C}}_{\rm
NP,\,I}^{\rm VLL}=0$.

With the effective Hamiltonian in eq. (12), the off-diagonal mass matrix
element of $B_{s}-\bar{B}_{s}$ mixing is given by [89]

$\displaystyle M_{12}^{s}=\langle B_{s}|\mathcal{H}_{\rm eff}^{\Delta
B=2}|\bar{B}_{s}\rangle=\frac{G_{F}^{2}}{16\pi^{2}}\,m_{W}^{2}\big{(}V_{tb}V_{ts}^{*}\big{)}^{2}\,{\mathcal{C}}^{\rm
VLL}\,\langle B_{s}\left\lvert\mathcal{{\mathcal{O}}}^{\rm
VLL}\right\rvert\bar{B}_{s}\rangle\,,$ (14)

where ${\mathcal{C}}^{\rm VLL}={\mathcal{C}}_{\rm SM}^{\rm
VLL}+{\mathcal{C}}_{\rm NP}^{\rm VLL}$, and the most recent lattice
calculations of the hadronic matrix element $\langle
B_{s}\left\lvert\mathcal{{\mathcal{O}}}^{\rm
VLL}\right\rvert\bar{B}_{s}\rangle$ can be found in ref. [80]. Then, the mass
difference between the two mass eigenstates $B_{s}^{H}$ and $B_{s}^{L}$ and
the CP violation phase read [92]

$\displaystyle\Delta m_{s}=2|M_{12}^{s}|,\qquad\text{and}\qquad\phi_{s}=\arg
M_{12}^{s},$ (15)

respectively. In the case with a complex ${Z^{\prime}}$ coupling
${X_{23}^{L}}$, the phase $\phi_{s}$ can deviate from the SM prediction and
hence affects the CP violation $S_{\psi\phi}$ measured in the decay $B_{s}\to
J/\psi\phi$ [92].

###  3.3 $b\to s\nu\bar{\nu}$ decays

Figure 3: Feynman diagrams for the $b\to s\nu\bar{\nu}$ transition, including
the selected SM diagram (a), and the ${Z^{\prime}}$ contributions in the
scenario II (b, c).

The rare decays $B\to X_{s}\nu\bar{\nu}$ and $B\to K^{(*)}\nu\bar{\nu}$ are
all induced by the quark-level $b\to s\nu\bar{\nu}$ transition. With the
${Z^{\prime}}$ effects taken into account, the effective Hamiltonian governing
the $b\to s\nu\bar{\nu}$ decays can be written as [93, 94]

$\displaystyle\mathcal{H}_{\rm eff}^{b\to
s\nu\bar{\nu}}=\frac{4G_{F}}{\sqrt{2}}\,\frac{\alpha_{\rm e}}{2\pi
s_{W}^{2}}\,V_{tb}V_{ts}^{*}\sum_{\ell=e,\mu,\tau}{\mathcal{C}}_{\ell}\big{(}\bar{s}\gamma_{\mu}P_{L}b\big{)}\big{(}\bar{\nu}_{\ell}\gamma^{\mu}P_{L}\nu_{\ell}\big{)}+{\rm
h.c.},$ (16)

where $\ell$ denotes the neutrino flavour. In the SM, the Wilson coefficient
${\mathcal{C}}_{\ell}^{\rm SM}(\mu_{W})=X(x_{t})$ are incuded by the
$Z$-penguin and $W$-box diagrams, and are lepton flavour universal. Analytical
expression of the Inami-Lim function $X(x_{t})$ can be found in refs. [95,
96]. Numerically, we obtain ${\mathcal{C}}_{\ell}^{\rm SM}(\mu_{W})=1.481\pm
0.009$ [97] after including the two-loop electroweak corrections [98]. In the
${Z^{\prime}}$ scenario II, the tree-level and the one-loop penguin diagram
shown in figure 3 contribute to the $b\to s\nu_{\mu}\bar{\nu}_{\mu}$
transition, resulting in

$\displaystyle{\mathcal{C}}_{\mu}^{\rm
NP,\,II}(\mu_{W})=-\frac{1}{\sqrt{2}G_{F}}\bigg{(}\frac{1}{8}\frac{V_{cs}^{*}}{V_{ts}^{*}}f(x_{t})-\frac{\pi
s_{W}^{2}}{\alpha_{\rm
e}V_{tb}V_{ts}^{*}}\bigg{)}\frac{X_{23}^{L}\lambda_{22}^{L}}{m_{{Z^{\prime}}}^{2}},$
(17)

with the loop function $f(x_{t})$ defined already by eq. (10). Numerically,
the tree-level ${Z^{\prime}}$-exchange contribution dominates over that from
the one-loop ${Z^{\prime}}$-penguin diagram, as in $b\to s\mu^{+}\mu^{-}$
decays. As there are no couplings of the ${Z^{\prime}}$ boson to $\nu_{e}$ and
$\nu_{\tau}$ neutrinos, ${\mathcal{C}}_{e}^{{\rm NP},\,\rm
II}={\mathcal{C}}_{\tau}^{{\rm NP},\,\rm II}=0$. In the ${Z^{\prime}}$
scenario I, there is no direct coupling of the ${Z^{\prime}}$ boson to
neutrinos. The ${Z^{\prime}}$ boson contributes to the $b\to s\nu\bar{\nu}$
transition at least at the two-loop level and hence can be safely neglected,
i.e. ${\mathcal{C}}_{\ell}^{\rm NP,\,I}=0$.

The inclusive decay $B\to X_{s}\nu\bar{\nu}$ is the theoretically cleanest
rare $B$-meson decay [99]. With the effective Hamiltonian in eq. (16), its
differential decay width can be written as [93]

$\displaystyle\frac{d\Gamma(B\to X_{s}\nu\bar{\nu})}{ds_{b}}=$ $\displaystyle
N(m_{b},m_{s})\,\kappa(0)\big{[}3s_{b}\left(1+\hat{m}_{s}^{2}-s_{b}\right)+\lambda\left(1,\hat{m}_{s}^{2},s_{b}\right)\big{]}\sum_{\ell=e,\mu,\tau}\big{\lvert}{\mathcal{C}}_{\ell}(\mu_{W})\big{\rvert}^{2},$
(18)

with the overall factor given by

$\displaystyle
N(m_{1},m_{2})=\frac{G_{F}^{2}\alpha_{e}^{2}m_{1}^{5}}{384\pi^{5}s_{W}^{4}}\big{\lvert}V_{tb}V_{ts}^{*}\big{\rvert}^{2}\lambda^{1/2}\bigg{(}1,\frac{m_{2}^{2}}{m_{1}^{2}},\frac{q^{2}}{m_{1}^{2}}\bigg{)},$
(19)

where $\lambda(x,y,z)=x^{2}+y^{2}+z^{2}-2(xy+yz+zx)$,
$\hat{m}_{i}=m_{i}/m_{b}$, and $s_{b}=q^{2}/m_{b}^{2}$ with $q^{2}$ the
invariant mass squared of the neutrino pair. The factor $\kappa(0)=0.83$
contains the virtual and bremsstrahlung QCD corrections to the $b\to
s\nu\bar{\nu}$ matrix element [100, 101].

For the exclusive decay $B\to K\nu\bar{\nu}$, the dineutrino invariant mass
distribution reads [102, 93]

$\displaystyle\frac{d\Gamma(B\to
K\nu\bar{\nu})}{ds_{B}}=\frac{1}{2}N(m_{B},m_{K})\,\lambda\left(1,\tilde{m}_{K}^{2},s_{B}\right)\left[f_{+}^{B\to
K}\left(s_{B}\right)\right]^{2}\sum_{\ell=e,\mu,\tau}\big{\lvert}{\mathcal{C}}_{\ell}(\mu_{W})\big{\rvert}^{2},$
(20)

where $\tilde{m}_{K}=m_{K}/m_{B}$, $s_{B}=q^{2}/m_{B}^{2}$, and $q^{2}$
denotes the dineutrino invariant mass squared. The $B\to K$ form factor
$f_{+}^{B\to K}(s_{B})$ is the main source of theoretical uncertainties.
Recent lattice QCD and LCSR results for $f_{+}^{B\to K}(s_{B})$ can be found
in refs. [103, 104] and [105, 82], respectively.

For the $B\to K^{*}\nu\bar{\nu}$ decay, the dineutrino invariant mass spectrum
can be written as [93]

$\displaystyle\frac{d\Gamma(B\to
K^{*}\nu\bar{\nu})}{ds_{B}}=\frac{1}{8}N(m_{B},m_{K^{*}})\,s_{B}\sum_{\ell=e,\mu,\tau}\Big{(}\left|A^{\ell}_{\perp}(s_{B})\right|^{2}+\left|A^{\ell}_{\|}(s_{B})\right|^{2}+\left|A^{\ell}_{0}(s_{B})\right|^{2}\Big{)},$
(21)

with the three transversity amplitudes given, respectively, as

$\displaystyle A_{\perp}(s_{B})$
$\displaystyle=-2\sqrt{2}\,\lambda^{1/2}(1,\tilde{m}_{K^{*}}^{2},s_{B})\,{\mathcal{C}}_{\ell}(\mu_{W})\frac{V(s_{B})}{1+\tilde{m}_{K^{*}}},$
(22) $\displaystyle A_{\|}(s_{B})$
$\displaystyle=2\sqrt{2}\,(1+\tilde{m}_{K^{*}})\,{\mathcal{C}}_{\ell}(\mu_{W})\,A_{1}(s_{B}),$
$\displaystyle A_{0}(s_{B})$
$\displaystyle=\frac{{\mathcal{C}}_{\ell}(\mu_{W})}{\widetilde{m}_{K^{*}}\sqrt{s_{B}}}\left[(1-\tilde{m}_{K^{*}}^{2}-s_{B})(1+\tilde{m}_{K^{*}})A_{1}(s_{B})-\lambda(1,\tilde{m}_{K^{*}}^{2},s_{B})\frac{A_{2}(s_{B})}{1+\tilde{m}_{K^{*}}}\right],$

where $\tilde{m}_{K^{*}}=m_{K^{*}}/m_{B}$. Here, the main theoretical
uncertainties arise from the three $B\to K^{*}$ transition form factors
$V(s_{B}),A_{1}(s_{B})$ and $A_{2}(s_{B})$. Recent lattice QCD and LCSR
calculations of these form factors can be found in refs. [106, 80] and refs.
[19, 82, 84], respectively.

###  3.4 $s\to d\mu^{+}\mu^{-}$ transition

Figure 4: Feynman diagrams for the $s\to d\mu^{+}\mu^{-}$ transition,
including the selected SM diagrams (a, b), and the ${Z^{\prime}}$
contributions (c, d).

For the $s\to d\mu^{+}\mu^{-}$ transition, the most relevant decay modes are
the $K_{L,S}\to\mu^{+}\mu^{-}$ decays. After taking into account the
${Z^{\prime}}$ effects, the effective Hamiltonian inducing the short-distance
(SD) contribution to $K_{L,S}\to\mu^{+}\mu^{-}$ decays can be written as [107]

$\displaystyle\mathcal{H}_{\rm eff}^{s\to
d\mu^{+}\mu^{-}}=\frac{4G_{F}}{\sqrt{2}}\frac{\alpha_{\rm e}}{4\pi
s_{W}^{2}}\,\big{(}V_{ts}^{*}V_{td}\,Y+V_{cs}^{*}V_{cd}\,Y_{\rm
NL}\big{)}\big{(}\bar{s}\gamma_{\mu}P_{L}d\big{)}\big{(}\bar{\mu}\gamma^{\mu}\gamma_{5}\mu\big{)}+{\rm
h.c.}.$ (23)

In the SM, the function $Y=Y(x_{t})$ describes contributions from the penguin
diagrams with internal top quark [96, 95, 69, 70], while $Y_{\rm NL}$ involves
the charm-quark contributions [108]. In both the ${Z^{\prime}}$ scenarios I
and II, the $s\to d\mu^{+}\mu^{-}$ transition is induced by the penguin
diagrams shown in figure 4, which result in the NP contributions

$\displaystyle Y_{\rm NP,I}$
$\displaystyle=\frac{1}{8\sqrt{2}G_{F}}\bigg{(}\frac{V_{cd}}{V_{td}}\frac{{X_{ct}^{L}}^{*}{\lambda_{\mu\mu}^{L}}}{m_{Z^{\prime}}^{2}}+\frac{V_{cs}^{*}}{V_{ts}^{*}}\frac{{X_{ct}^{L}}{\lambda_{\mu\mu}^{L}}}{m_{{Z^{\prime}}}^{2}}\bigg{)}f(x_{t}),$
(24) $\displaystyle Y_{\rm NP,II}$
$\displaystyle=\frac{1}{8\sqrt{2}G_{F}}\bigg{(}\frac{V_{cd}}{V_{td}}\frac{{X_{23}^{L}}^{*}{\lambda_{22}^{L}}}{m_{{Z^{\prime}}}^{2}}+\frac{V_{cs}^{*}}{V_{ts}^{*}}\frac{{X_{23}^{L}}{\lambda_{22}^{L}}}{m_{{Z^{\prime}}}^{2}}\bigg{)}f(x_{t}),$
(25)

with the loop function $f(x_{t})$ defined already in eq. (10). Here, the NP
contributions in both of the two ${Z^{\prime}}$ scenarios are of the same
magnitude, which is different from what is observed in the $b\to
s\mu^{+}\mu^{-}$ transitions. Numerically,
$Y=0.94-(1.03+0.02i){X_{ct}^{L}}{\lambda_{\mu\mu}^{L}}-(0.99+0.45i){X_{ct}^{L}}^{*}{\lambda_{\mu\mu}^{L}}$
for $m_{Z^{\prime}}=1{\,\rm TeV}$ in scenario I.

For the $K_{L,S}\to\mu^{+}\mu^{-}$ decays, only the SD part of a dispersive
contribution can be reliably calculated. The branching ratio of
$K_{L}\to\mu^{+}\mu^{-}$ can be written as [107]

$\displaystyle{\mathcal{B}}(K_{L}\to\mu^{+}\mu^{-})_{\rm
SD}=\kappa_{\mu}\left[\frac{{\rm Re}(\lambda_{t}Y)}{\lambda^{5}}+\frac{{\rm
Re}\lambda_{c}}{\lambda}P_{c}(Y)\right]^{2},$ (26)

with $\lambda_{t}=V_{ts}^{*}V_{td}$, $\lambda_{c}=V_{cs}^{*}V_{cd}$, and
$\lambda\approx V_{us}$ denoting the Wolfenstein parameter. The factor
$\kappa_{\mu}$ contains the relevant hadronic matrix element that can be
extracted from the $K^{+}\to\mu^{+}\nu_{\mu}$ decay, and numerically we have
$\kappa_{\mu}=(2.009\pm 0.017)\times 10^{-9}\,(\lambda/0.225)^{8}$ [107, 108].
The charm contribution $P_{c}(Y)$ is found to be $P_{c}(Y)=0.115\pm 0.017$ at
the NNLO in QCD [108]. For the $K_{S}\to\mu^{+}\mu^{-}$ decay, the SD and
long-distance contributions add incoherently in the total rate [109, 110, 111,
112]. The SD part of the branching ratio is given by [109, 110, 111, 112]

$\displaystyle{\mathcal{B}}(K_{S}\to\mu^{+}\mu^{-})_{\rm
SD}=\tau_{K_{S}}\frac{G_{F}^{2}\alpha_{\rm
e}^{2}}{8\pi^{3}s_{W}^{4}}m_{K}f_{K}^{2}\bigg{(}1-\frac{m_{\mu}^{2}}{m_{K}^{2}}\bigg{)}^{1/2}m_{\mu}^{2}\,{\rm
Im}^{2}\big{(}\lambda_{t}Y\big{)},$ (27)

with $\tau_{K_{S}}$ the lifetime of $K_{S}$ and $f_{K}$ the decay constant.

###  3.5 $s\to d\nu\bar{\nu}$ decays

The $K^{+}\to\pi^{+}\nu\bar{\nu}$ and $K_{L}\to\pi^{0}\nu\bar{\nu}$ decays are
induced by the $s\to d\nu\bar{\nu}$ transition. They are both characterized by
the theoretically clean virtue, since the relevant hadronic matrix elements
can be extracted with the help of isospin symmetry from the leading semi-
leptonic $K_{\ell 3}$ decays [113]. With the ${Z^{\prime}}$ effects taken into
account, the $s\to d\nu\bar{\nu}$ decays are governed by the effective
Hamiltonian [95, 96]

$\displaystyle\mathcal{H}_{\rm eff}^{s\to
d\nu\bar{\nu}}=\frac{4G_{F}}{\sqrt{2}}\frac{\alpha_{\rm e}}{2\pi
s_{W}^{2}}\sum_{\ell=e,\mu,\tau}\big{(}V_{ts}^{*}V_{td}X^{\ell}+V_{cs}^{*}V_{cd}X_{\rm
NL}^{\ell}\big{)}\big{(}\bar{s}\gamma_{\mu}P_{L}d\big{)}\big{(}\bar{\nu}_{\ell}\gamma^{\mu}P_{L}\nu_{\ell}\big{)}+{\rm
h.c.}.$ (28)

In the SM, similar to the $b\to s\nu\bar{\nu}$ transition, the $Z$-penguin and
$W$-box diagrams with internal top quark result in a flavour-universal Wilson
coefficient $X_{\rm SM}^{\ell}=X(x_{t})$, with the Inami-Lim function
$X(x_{t})$ introduced already in eq. (16). The contribution from internal
charm quark is represented by the function $X_{\rm NL}^{\ell}$. In the
${Z^{\prime}}$ scenario II, the ${Z^{\prime}}$ boson can only interact with
the $\nu_{\mu}$ neutrino. Its contributions to the operators with $\nu_{e}$
and $\nu_{\tau}$ neutrinos arise from two-loop level and can be therefore
neglected, i.e., $X_{{\rm NP},\rm II}^{e}=X_{{\rm NP},\rm II}^{\tau}=0$.
However, the ${Z^{\prime}}$-penguin diagrams similar to the one shown in
figure 4 contribute to the Wilson coefficient $X^{\mu}$, and we find $X_{\rm
NP,II}^{\mu}=-Y_{\rm NP}^{\rm II}$. In the ${Z^{\prime}}$ scenario I, as in
the case of the $b\to s\nu\bar{\nu}$ transition, the ${Z^{\prime}}$ effects
arise firstly at two-loop level and are neglected, i.e., $X_{\rm
NP,I}^{\ell}=0$.

In the SM, with the help of isospin symmetry, the branching ratio of
$K^{+}\to\pi^{+}\nu\bar{\nu}$ decay after summing over the three neutrino
flavours is given by [113, 114, 115]

${\mathcal{B}}(K^{+}\to\pi^{+}\nu\bar{\nu})=\frac{1}{3}\kappa_{+}(1+\Delta_{\rm
EM})\sum_{\ell=e,\mu,\tau}\bigg{[}\bigg{(}\frac{{\rm
Im}[\lambda_{t}X^{\ell}]}{\lambda^{5}}\bigg{)}^{2}+\bigg{(}\frac{{\rm
Re}\lambda_{c}}{\lambda}P_{c}(X)+\frac{{\rm
Re}[\lambda_{t}X^{\ell}]}{\lambda^{5}}\bigg{)}^{2}\,\bigg{]}.$ (29)

The parameter $\kappa_{+}$ contains the relevant hadronic matrix elements
extracted from the $K_{\ell 3}$ decays, and numerically we have
$\kappa_{+}=(5.173\pm 0.025)\times 10^{-11}\,(\lambda/0.225)^{8}$ [97]. The
parameter $\Delta_{\rm EM}=-0.003$ describes the isospin breaking correction
[114, 115]. The parameter $P_{c}(X)$ summarizes the SD charm contribution [95,
107, 116, 117] and the long-distance contributions [118]. Numerically, it is
found that $P_{c}(X)=0.404\pm 0.024$ [97].

The rare decay $K_{L}\to\pi^{0}\nu\bar{\nu}$ proceeds in the SM almost
entirely through direct CP violation [119, 120]. It is completely dominated by
the SD loop diagrams with top-quark exchanges, and the charm contribution can
be fully neglected [113]. With the help of isospin symmetry, the branching
ratio of $K_{L}\to\pi^{0}\nu\bar{\nu}$ decay after summing over three neutrino
flavours reads [113, 121]

$\displaystyle{\mathcal{B}}(K_{L}\to\pi^{0}\nu\bar{\nu})=\frac{1}{3}\kappa_{L}(1-\delta_{\epsilon})\sum_{\ell=e,\mu,\tau}\left(\frac{1}{\lambda^{5}}\text{Im}\big{[}\lambda_{t}X^{\ell}\big{]}\right)^{2},$
(30)

where $\kappa_{L}$ encodes the hadronic matrix element extracted from the
$K_{\ell 3}$ data [114, 122], and numerically $\kappa_{L}=(2.231\pm
0.013)\times 10^{-10}\,(\lambda/0.225)^{8}$. The parameter $\delta_{\epsilon}$
denotes the indirectly CP-violating contribution, and is highly suppressed by
the $K^{0}-\bar{K}^{0}$ mixing parameter $|\epsilon|$ [120].

###  3.6 $t\to c\mu^{+}\mu^{-}$ decay

In the SM, the rare FCNC decay $t\to c\mu^{+}\mu^{-}$ is highly suppressed by
the Glashow-Iliopoulos-Maiani mechanism [123], with a branching ratio of
$\mathcal{O}(10^{-10})$ [124, 125]. However, this process could be
significantly enhanced by the ${Z^{\prime}}$ boson through the tree-level
diagrams shown in figure 5. Let us discuss the ${Z^{\prime}}$ effects in the
cases $m_{Z^{\prime}}>m_{t}$ and $m_{Z^{\prime}}<m_{t}$, respectively.

Figure 5: Feynman diagrams for the $t\to c\mu^{+}\mu^{-}$ (left) and
${Z^{\prime}}\to\bar{b}cW^{-}$ (right) decays in the ${Z^{\prime}}$ scenarios.

####  3.6.1 $m_{Z^{\prime}}>m_{t}$

The branching ratio of the ${Z^{\prime}}$-mediated $t\to c\mu^{+}\mu^{-}$
decay can be written as

$\displaystyle{\mathcal{B}}(t\to c\mu^{+}\mu^{-})=\frac{\Gamma(t\to
c\mu^{+}\mu^{-})}{\Gamma_{t}},$ (31)

where $\Gamma_{t}$ is the total width of the top quark. The differential decay
width can be calculated from the left tree-level diagram shown in figure 5,
with the result in the ${Z^{\prime}}$ scenario I given by

$\displaystyle\frac{d\Gamma_{0}^{\rm I}(t\to c\mu^{+}\mu^{-})}{dq^{2}}$
$\displaystyle=\frac{\left|{X_{ct}^{L}}\lambda^{L}_{\mu\mu}\right|^{2}}{768\pi^{3}m_{t}^{3}}\frac{(m_{t}^{2}-q^{2})(m_{t}^{4}+q^{2}m_{t}^{2}-2q^{4})}{(q^{2}-m_{{Z^{\prime}}}^{2})^{2}+m_{Z^{\prime}}^{2}\Gamma_{Z^{\prime}}^{2}},$
(32)

where $q^{2}$ denotes the dilepton invariant mass squared, and
$\Gamma_{{Z^{\prime}}}$ the finite decay width of the ${Z^{\prime}}$ boson.
Since the ${Z^{\prime}}$ boson cannot be on-shell in the case of
$m_{Z^{\prime}}>m_{t}$, we can safely neglect the finite-width effect of the
${Z^{\prime}}$ boson here. After including the NLO QCD correction, the
differential decay width of $t\to c\mu^{+}\mu^{-}$ can be rewritten as

$\displaystyle\frac{d\Gamma_{\rm NLO}^{\rm I}(t\to
c\mu^{+}\mu^{-})}{dq^{2}}=\frac{d\Gamma_{0}^{\rm I}(t\to
c\mu^{+}\mu^{-})}{dq^{2}}\big{[}1+f_{\rm NLO}(q^{2})\big{]},$ (33)

where the factor $f_{\rm NLO}(q^{2})$ can be obtained from the NLO QCD
calculation of the $t\to cZ$ decay [126, 127, 128], and is given explicitly by

$\displaystyle f_{\rm NLO}(q^{2})=\frac{4\alpha_{s}}{3\pi}\biggl{[}$
$\displaystyle-\frac{1}{2}\pi^{2}-3-\frac{1}{2}\log\frac{\mu^{2}}{m_{t}^{2}}+\frac{12+135\beta^{2}-43\beta^{4}}{12\beta^{2}(3-\beta^{2})}+\left(2\log\beta-\frac{9-\beta^{2}}{3-\beta^{2}}\right)\log\beta$
$\displaystyle+\frac{(1-\beta^{2})(1-4\beta^{2}+\beta^{4})}{\beta^{4}\left(3-\beta^{2}\right)}\log(1-\beta^{2})+\mathrm{Li}_{2}(\beta^{2})+\mathrm{Li}_{2}(1-\beta^{-2})\biggr{]},$
(34)

with $\beta=\sqrt{1-q^{2}/m_{t}^{2}}$. Numerically, the NLO QCD corrections
decrease the LO width by $6.4\%\sim 9.1\%$ for $m_{t}<m_{Z^{\prime}}<1{\,\rm
TeV}$. Expressions in scenario II can be obtained from the above formulas with
the replacement
$({X_{ct}^{L}},{\lambda_{\mu\mu}^{L}})\to(X_{23}^{L},\lambda_{22}^{L})$. As
will be shown in subsection 4.2.2, the branching ratio ${\mathcal{B}}(t\to
c\mu^{+}\mu^{-})$ is predicted to be below ${\mathcal{O}}(10^{-5})$ in the two
${Z^{\prime}}$ scenarios. Therefore, their contributions to the top-quark
total width $\Gamma_{t}$ can be safely neglected.

####  3.6.2 $m_{Z^{\prime}}<m_{t}$

In the case of $m_{Z^{\prime}}<m_{t}$, the branching ratio of the
${Z^{\prime}}$-mediated $t\to c\mu^{+}\mu^{-}$ decay can also be written as
eqs. (32) and (33). However, since the intermediate ${Z^{\prime}}$ boson can
be on-shell in this case, it is necessary to consider the ${Z^{\prime}}$
finite-width effect in eq. (32). When $m_{W}<m_{Z^{\prime}}<m_{t}$, the main
${Z^{\prime}}$ decay modes in scenario I are ${Z^{\prime}}\to\mu^{+}\mu^{-}$
and ${Z^{\prime}}\to b\bar{c}W^{+}$ decays. Therefore, the total width of the
${Z^{\prime}}$ boson can be written as

$\displaystyle\Gamma_{Z^{\prime}}^{\rm I}=\Gamma^{\rm
I}({Z^{\prime}}\to\mu^{+}\mu^{-})+2\Gamma^{\rm I}({Z^{\prime}}\to
b\bar{c}W^{+}),$ (35)

where the decay width of ${Z^{\prime}}\to\mu^{+}\mu^{-}$ is given by

$\displaystyle\Gamma^{\rm I}({Z^{\prime}}\to\mu^{+}\mu^{-})$
$\displaystyle=\frac{1}{24\pi}\left|\lambda^{L}_{\mu\mu}\right|^{2}m_{Z^{\prime}},$
(36)

and the width of the top-quark mediated ${Z^{\prime}}\to b\bar{c}W^{+}$ decay
reads

$\displaystyle\Gamma^{\rm I}({Z^{\prime}}\to b\bar{c}W^{+})=$
$\displaystyle\frac{G_{F}|V_{tb}|^{2}|X_{ct}|^{2}}{128\sqrt{2}\,\pi^{3}}\frac{m_{t}^{8}}{m_{Z^{\prime}}^{5}}\biggl{\\{}4y_{W}^{3}y_{Z^{\prime}}^{3}\log\frac{y_{Z^{\prime}}}{y_{W}}+\frac{1}{12}(y_{Z^{\prime}}-y_{W})\Big{[}60+48y_{W}^{2}y_{Z^{\prime}}^{2}$
$\displaystyle+20y_{W}y_{Z^{\prime}}+(30-39y_{W}y_{Z^{\prime}})(y_{W}+y_{Z^{\prime}})-88(y_{W}^{2}+y_{Z^{\prime}}^{2})+9(y_{W}^{3}+y_{Z^{\prime}}^{3})\Big{]}$
$\displaystyle+\Big{[}5+9y_{W}^{2}y_{Z^{\prime}}^{2}-9\big{(}y_{W}^{2}+y_{Z^{\prime}}^{2}\big{)}-4y_{W}^{3}y_{Z^{\prime}}^{3}+4\big{(}y_{W}^{3}+y_{Z^{\prime}}^{3}\big{)}\Big{]}\log\frac{\beta_{Z^{\prime}}^{2}}{\beta_{W}^{2}}\biggr{\\}},$
(37)

with $\beta_{W}=\sqrt{1-m_{W}^{2}/m_{t}^{2}}$,
$\beta_{Z^{\prime}}=\sqrt{1-m_{Z^{\prime}}^{2}/m_{t}^{2}}$,
$y_{W}=m_{W}^{2}/m_{t}^{2}$, and
$y_{Z^{\prime}}=m_{Z^{\prime}}^{2}/m_{t}^{2}$.

In scenario II, the ${Z^{\prime}}$ boson can also decay into the additional
channels ${Z^{\prime}}\to\nu_{\mu}\bar{\nu}_{\mu}$ and ${Z^{\prime}}\to
b\bar{s}$. The total width can then be written as

$\displaystyle\Gamma_{Z^{\prime}}^{\rm II}=\Gamma^{\rm
II}({Z^{\prime}}\to\mu^{+}\mu^{-})+\Gamma^{\rm
II}({Z^{\prime}}\to\nu_{\mu}\bar{\nu}_{\mu})+2\Gamma^{\rm
II}({Z^{\prime}}\to\bar{c}bW^{+})+2\Gamma^{\rm II}({Z^{\prime}}\to b\bar{s}),$
(38)

with

$\displaystyle\Gamma^{\rm
II}({Z^{\prime}}\to\nu_{\mu}\bar{\nu}_{\mu})=\frac{1}{24\pi}\left|\lambda^{L}_{22}\right|^{2}m_{Z^{\prime}}\qquad\text{and},\qquad\Gamma^{\rm
II}({Z^{\prime}}\to
b\bar{s})=\frac{1}{8\pi}\left|X_{23}^{L}\right|^{2}m_{Z^{\prime}}.$ (39)

The decay widths $\Gamma^{\rm II}({Z^{\prime}}\to\mu^{+}\mu^{-})$ and
$\Gamma^{\rm II}({Z^{\prime}}\to\bar{c}bW^{+})$ can be obtained from eqs. (36)
and (3.6.2) with the replacement
$({\lambda_{\mu\mu}^{L}},{X_{ct}^{L}})\to({\lambda_{22}^{L}},{X_{23}^{L}})$.

In the SM, $t\to bW$ is the main decay channel of the top quark. When
$m_{Z^{\prime}}<m_{t}$, the top quark can also decay into an on-shell
${Z^{\prime}}$, which contributes to the top-quark total width as

$\displaystyle\Gamma_{t}\simeq\Gamma(t\to bW)+\Gamma(t\to c{Z^{\prime}}).$
(40)

After including the NLO QCD correction, we can write the decay width of $t\to
bW$ as [129, 130, 131]

$\displaystyle\Gamma(t\to bW)=$ $\displaystyle\Gamma_{0}(t\to
bW)\Bigg{\\{}1+\dfrac{2\alpha_{s}}{3\pi}\left[2\left(\dfrac{(1-\beta_{W}^{2})(2\beta_{W}^{2}-1)(\beta_{W}^{2}-2)}{\beta_{W}^{4}(3-2\beta_{W}^{2})}\right)\log(1-\beta_{W}^{2})\right.$
$\displaystyle\left.-\dfrac{9-4\beta_{W}^{2}}{3-2\beta_{W}^{2}}\log\beta_{W}^{2}+2\text{Li}_{2}\beta_{W}^{2}-2\text{Li}_{2}(1-\beta_{W}^{2})-\dfrac{6\beta_{W}^{4}-3\beta_{W}^{2}-8}{2\beta_{W}^{2}(3-2\beta_{W}^{2})}-\pi^{2}\right]\Bigg{\\}},$
(41)

with the LO result given by

$\displaystyle\Gamma_{0}(t\to
bW)=\frac{G_{F}m^{3}_{t}}{8\sqrt{2}\,\pi}|V_{tb}|^{2}\beta^{4}_{W}(3-2\beta^{2}_{W}).$
(42)

For the $t\to c{Z^{\prime}}$ decay in scenario I, the decay width including
the NLO QCD correction can be written as

$\displaystyle\Gamma_{\rm NLO}^{\rm I}(t\to c{Z^{\prime}})=\Gamma_{0}^{\rm
I}(t\to c{Z^{\prime}})\big{[}1+f_{\rm NLO}(m_{Z^{\prime}}^{2})\big{]},$ (43)

with the tree-level result given by

$\displaystyle\Gamma_{0}^{\rm I}(t\to c{Z^{\prime}})=\frac{m_{t}^{3}}{32\pi
m_{Z^{\prime}}^{2}}|X_{ct}|^{2}\beta_{Z^{\prime}}^{4}\big{(}3-2\beta_{Z^{\prime}}^{2}\big{)}.$
(44)

The factor $f_{\rm NLO}(q^{2})$ is obtained from the calculation of NLO QCD
correction to the $t\to cZ$ decay [126, 127, 128] and its explicit expression
has been given in eq. (3.6.1). Numerically, we find
$f(m_{Z^{\prime}}^{2})\approx-10.7\%\sim+5.3\%$ when $m_{Z^{\prime}}$ varies
in the range $105{\,\rm GeV}<m_{Z^{\prime}}<m_{t}$. Expressions in the
${Z^{\prime}}$ scenario II can be obtained from the above formulas with the
replacement
$({X_{ct}^{L}},{\lambda_{\mu\mu}^{L}})\to({X_{23}^{L}},{\lambda_{22}^{L}})$.
Since the decay rate of $t\to c{Z^{\prime}}$ is proportional to
$|X_{ct}^{L}|^{2}$ in scenario I, the top-quark width can provide a unique
constraint on the $t\bar{c}{Z^{\prime}}$ coupling in the case of
$m_{Z^{\prime}}<m_{t}$.

##  4 Numerical results and discussions

Input | Value | Unit | Ref.
---|---|---|---
$m_{t}^{\text{pole}}$ | $173.76\pm 0.30$ | GeV | [132]
$m_{b}(m_{b})$ | $4.195\pm 0.014$ | GeV | [133]
$m_{c}(m_{c})$ | $1.273\pm 0.01$ | GeV | [133]
$|V_{cb}|$(semi-leptonic) | $(41.1\pm 0.3\pm 0.5)\times 10^{-3}$ |  | [134]
$|V_{ub}|$(semi-leptonic) | $(3.91\pm 0.08\pm 0.21)\times 10^{-3}$ |  | [134]
$|V_{us}|f^{K\to\pi}_{+}$(0) | $0.2165\pm 0.0004$ |  | [134]
$\gamma$ | $72.1^{+5.4}_{-5.7}$ | ∘ | [134]
$f^{K\to\pi}_{+}(0)$ | $0.9681\pm 0.0014\pm 0.0022$ |  | [134]
Table 1: Main input parameters used in our numerical analysis.

In this section, we proceed to present our numerical analysis of the
${Z^{\prime}}$ scenarios presented in section 2. In table 1, we list the main
input parameters used throughout this work. Table 2 summaries the SM
predictions and the current experimental data for the various processes
discussed in the previous section.

Observable | SM | Exp. | Ref.
---|---|---|---
$\mathcal{B}(B_{s}\to\mu^{+}\mu^{-})$ | $(3.46\pm 0.11)\times 10^{-9}$ | $(3.09^{+0.46+0.15}_{-0.43-0.11})\times 10^{-9}$ | [21]
$\Delta m_{s}$ [GeV] | $(1.18\pm 0.07)\times 10^{-11}$ | $(1.1688\pm 0.0014)\times 10^{-11}$ | [135]
$S_{\psi\phi}$ | $0.041\pm 0.003$ | $0.033\pm 0.033$ | [135]
$\mathcal{B}(B\to X_{s}\nu\bar{\nu})$ | $(3.03\pm 0.08)\times 10^{-5\hphantom{1}}$ | $<6.4\times 10^{-4}$ | [136]
$\mathcal{B}(B^{+}\to K^{+}\nu\bar{\nu})$ | $(4.15\pm 0.56)\times 10^{-6\hphantom{1}}$ | $<1.9\times 10^{-5}$ | [137]
$\mathcal{B}(B^{0}\to K^{0}\nu\bar{\nu})$ | $(3.85\pm 0.52)\times 10^{-6\hphantom{1}}$ | $<2.6\times 10^{-5}$ | [137]
$\mathcal{B}(B^{+}\to K^{*+}\nu\bar{\nu})$ | $(9.71\pm 0.93)\times 10^{-6\hphantom{1}}$ | $<6.1\times 10^{-5}$ | [137]
$\mathcal{B}(B^{0}\to K^{*0}\nu\bar{\nu})$ | $(9.01\pm 0.87)\times 10^{-6\hphantom{1}}$ | $<1.8\times 10^{-5}$ | [137]
$\mathcal{B}(K_{L}\to\mu^{+}\mu^{-})$ | $(7.34\pm 1.21)\times 10^{-9\hphantom{1}}$ | $(6.84\pm 0.11)\times 10^{-9}$ | [132]
$\mathcal{B}(K_{S}\to\mu^{+}\mu^{-})$ | $(5.18\pm 1.57)\times 10^{-12}$ | $(0.94^{+0.72}_{-0.64})\times 10^{-10}$ | [138]
$\mathcal{B}(K^{+}\to\pi^{+}\nu\bar{\nu})$ | $(8.38\pm 0.62)\times 10^{-11}$ | $(10.6^{+4.0}_{-3.4}\pm 0.9)\times 10^{-11}$ | [139]
$\mathcal{B}(K_{L}\to\pi^{0}\nu\bar{\nu})$ | $(3.46\pm 0.46)\times 10^{-11}$ | $<3.0\times 10^{-9}$ | [140]
Table 2: Our SM predictions and the experimental measurements of the relevant
observables used in our global fit. Upper limits are all at $95\%$ confidence
level (CL) .

As discussed in section 2, the relevant model parameters in ${Z^{\prime}}$
scenario I (II) contain the complex couplings ${X_{ct}^{L}}$ and
${\lambda_{\mu\mu}^{L}}$ ($X_{23}^{L}$ and $\lambda_{22}^{L}$), and the
${Z^{\prime}}$ mass $m_{Z^{\prime}}$. When performing a global fit on these
parameters and studying their effects, we consider the following parameter
space:

S.I:
$\displaystyle\lvert{X_{ct}^{L}}\rvert<2.0,\qquad\lvert{\lambda_{\mu\mu}^{L}}\rvert<2.0,\qquad
105{\,\rm GeV}<m_{Z^{\prime}}<1000{\,\rm GeV},$ (45) S.II:
$\displaystyle\lvert{X_{23}^{L}}\rvert<2.0,\qquad\lvert{\lambda_{22}^{L}}\rvert<2.0,\qquad
105{\,\rm GeV}<m_{Z^{\prime}}<1000{\,\rm GeV},$

where the upper limits on the magnitudes of the couplings correspond to about
half of the perturbativity bound $\sqrt{4\pi}$. Here the lower bound of
$m_{Z^{\prime}}$ is chosen to avoid the potentially strong bound from the
branching ratio ${\mathcal{B}}(t\to cZ)$ [141] (see subsection 4.2.2 for
further details).

In our numerical analysis, we follow the approach used in ref. [86] and
construct a likelihood function that only depends on the Wilson coefficients
as

$\displaystyle-2\log
L(\bm{{\mathcal{C}}})=\sum_{i}\bm{x_{i}}^{T}(\bm{{\mathcal{C}}})\big{[}V_{\rm
exp}+V_{\rm th}(\bm{\theta})]^{-1}\bm{x_{i}}(\bm{{\mathcal{C}}}),$ (46)

with the definition

$\displaystyle\bm{x}_{i}(\bm{{\mathcal{C}}})=\bm{{\mathcal{O}}}_{i}^{\rm
exp}-\bm{{\mathcal{O}}}_{i}^{\rm th}(\bm{{\mathcal{C}}},\bm{\theta}).$

Here, $\bm{{\mathcal{O}}}_{i}^{\rm exp}$ are the central values of the
experimental measurements, while $\bm{{\mathcal{O}}}_{i}^{\rm th}$ denote the
central values of the theoretical predictions, which depend on the Wilson
coefficients $\bm{{\mathcal{C}}}$ and the input parameters $\bm{\theta}$.
$V_{\rm exp}$ is the covariance matrix of the experimental measurements, while
$V_{\rm th}$ denotes the covariance matrix of the theoretical predictions,
which contains all the theoretical uncertainties and their correlations.
Generally, the covariance matrix $V_{\rm th}$ depends on both the Wilson
coefficients $\bm{{\mathcal{C}}}$ and the input parameters $\bm{\theta}$.
However, as all the experimental measurements do not show large deviations
from the SM predictions, we have neglected in our numerical analysis the
dependence of $V_{\rm th}$ on the ${Z^{\prime}}$ contributions to the Wilson
coefficients $\bm{{\mathcal{C}}}$, i.e., $V_{\rm th}$ is evaluated with the
Wilson coefficients $\bm{{\mathcal{C}}}$ fixed to their SM values.
Furthermore, the theoretical uncertainties are approximated as Gaussian and
obtained by randomly sampling the observables with the input parameters
$\bm{\theta}$ distributed according to their probability density functions.
With eq. (46), the $\Delta\chi^{2}$ function, which depends on the Wilson
coefficients $\bm{{\mathcal{C}}}$, can be written as
$\Delta\chi^{2}(\bm{{\mathcal{C}}})=-2\log L(\bm{{\mathcal{C}}})/L_{\rm max}$,
where $L_{\rm max}$ denotes the maximum of $L$ at different values of the
Wilson coefficients. For more details on the fit methodology, we refer to
refs. [86, 142]. Our global fit is performed using an extended version of the
package flavio [142].

###  4.1 Flavour constraints

As discussed in section 3, the ${Z^{\prime}}$ contributions to the $b\to
s\mu^{+}\mu^{-}$, $s\to d\mu^{+}\mu^{-}$, $b\to s\nu\bar{\nu}$, and $s\to
d\nu\bar{\nu}$ transitions are all controlled by the products
${X_{ct}^{L}}{\lambda_{\mu\mu}^{L}}/m_{Z^{\prime}}^{2}$ in scenario I or
${X_{23}^{L}}{\lambda_{22}^{L}}/m_{Z^{\prime}}^{2}$ in scenario II. These two
products are complex in general and constraints on them will be derived in
terms of their real and imaginary parts in the following analysis.

For the $b\to s\mu^{+}\mu^{-}$ transitions, as in refs. [143, 144, 145, 27],
we consider the following experimental data in the numerical analysis: 1) the
branching ratios of the decays $B\to K\mu^{+}\mu^{-}$ [146], $B\to
K^{*}\mu^{+}\mu^{-}$ [147, 146, 148, 149], $B_{s}\to\phi\mu^{+}\mu^{-}$ [16],
$\Lambda_{b}\to\Lambda\mu^{+}\mu^{-}$ [150], $B\to X_{s}\mu^{+}\mu^{-}$ [151],
and $B_{s}\to\mu^{+}\mu^{-}$ [21]. 2) the angular distributions in $B\to
K\mu^{+}\mu^{-}$ [147, 152], $B\to K^{*}\mu^{+}\mu^{-}$ [148, 153, 154, 155,
29, 30], $B_{s}\to\phi\mu^{+}\mu^{-}$ [156], and
$\Lambda_{b}\to\Lambda\mu^{+}\mu^{-}$ [157] decays. 3) the LFU ratios
$R_{K^{(*)}}$ [13, 11, 14, 15, 10, 9]. In scenarios I and II, the best fit
regions of the ${Z^{\prime}}$ parameters in terms of their real and imaginary
parts are shown in figure 6. The best fit values in scenario I (II) read ${\rm
Re}{X_{ct}^{L}}{\lambda_{\mu\mu}^{L}}/m_{Z^{\prime}}^{2}=0.0991$ and ${\rm
Im}{X_{ct}^{L}}{\lambda_{\mu\mu}^{L}}/m_{Z^{\prime}}^{2}=-0.184$ (${\rm
Re}{X_{23}^{L}}{\lambda_{22}^{L}}/m_{Z^{\prime}}^{2}=-0.0007$ and ${\rm
Im}{X_{23}^{L}}{\lambda_{22}^{L}}/m_{Z^{\prime}}^{2}=0.0018$) in unit of
${\,\rm TeV}^{-2}$ with $\text{Pull}_{\rm SM}=4.91\sigma\,(4.96\sigma)$, which
is defined as $\sqrt{\Delta\chi^{2}}$ between the best fit point and the SM
value [145]. We observe that both the ${Z^{\prime}}$ scenarios I and II allow
a good description of the current $b\to s\mu^{+}\mu^{-}$ measurements. This is
not surprising, because the ${Z^{\prime}}$ scenarios considered induce NP
effects along the direction ${\mathcal{C}}_{9}^{\rm
NP}=-{\mathcal{C}}_{10}^{\rm NP}$, which is strongly preferred in the model-
independent global fit [143, 144, 158, 159, 145, 31, 160, 27, 28, 161]. In the
survived parameter space of the scenarios I and II, the magnitude of
${X_{ct}^{L}}{\lambda_{\mu\mu}^{L}}/m_{Z^{\prime}}^{2}$ and
${X_{23}^{L}}{\lambda_{22}^{L}}/m_{Z^{\prime}}^{2}$ are of
${\mathcal{O}}(10^{-1}){\,\rm TeV}^{-2}$ and ${\mathcal{O}}(10^{-3}){\,\rm
TeV}^{-2}$, respectively. In both of these two scenarios, the real parts of
${X_{ct}^{L}}{\lambda_{\mu\mu}^{L}}/m_{Z^{\prime}}^{2}$ and
${X_{23}^{L}}{\lambda_{22}^{L}}/m_{Z^{\prime}}^{2}$ are bounded from below,
while the imaginary parts are loosely constrained. It is noted that the $b\to
s\mu^{+}\mu^{-}$ decays provide the most stringent constraints on the
${Z^{\prime}}$ parameters among all the flavour processes, as will be
discussed in the following.

Figure 6: Allowed regions of the real and imaginary parts of
${X_{ct}^{L}}{\lambda_{\mu\mu}^{L}}/m_{Z^{\prime}}^{2}$ and
$X_{23}^{L}\lambda^{L}_{22}/m_{Z^{\prime}}^{2}$ by the $b\to s\mu^{+}\mu^{-}$
processes in the ${Z^{\prime}}$ scenarios I (left) and II (right). The dark
and the light regions correspond to the $68\%$ and the $95\%$ CL,
respectively.

Figure 7: Same as in figure 6 but now under the constraints from the $s\to
d\mu^{+}\mu^{-}$ processes.

The rare decays $K_{L}\to\mu^{+}\mu^{-}$ and $K_{S}\to\mu^{+}\mu^{-}$ are
induced by the $s\to d\mu^{+}\mu^{-}$ transition. In the ${Z^{\prime}}$
scenarios, the constraints from the branching ratios
${\mathcal{B}}(K_{L,S}\to\mu^{+}\mu^{-})_{\rm SD}$ are shown in figure 7. As
discussed in subsection 3.3, the ${Z^{\prime}}$ boson in scenarios I and II
affects the $s\to d\mu^{+}\mu^{-}$ transition via the same penguin diagram.
Therefore, the constraints on the ${Z^{\prime}}$ parameters are identical in
the two scenarios, as shown in figure 7. It can also be seen that the bounds
on the imaginary parts of
${X_{ct}^{L}}{\lambda_{\mu\mu}^{L}}/m_{Z^{\prime}}^{2}$ and
$X_{23}^{L}\lambda_{22}^{L}/m_{Z^{\prime}}^{2}$ are quite weak. In addition,
the constraints from the $s\to d\mu^{+}\mu^{-}$ decays, while being relatively
weaker compared to that from the $b\to s\mu^{+}\mu^{-}$ decays, are still
compatible with the parameter regions required to explain the $b\to
s\mu^{+}\mu^{-}$ anomalies shown in figure 6.

Figure 8: Allowed regions of the real and imaginary parts of
$X_{23}^{L}\lambda^{L}_{22}/m_{Z^{\prime}}^{2}$ by the $b\to s\nu\bar{\nu}$
(left) and $s\to d\nu\bar{\nu}$ processes (right) in scenario II. The colour
captions are the same as in figure 6.

Since the ${Z^{\prime}}$ boson does not couple directly to the neutrinos in
the ${Z^{\prime}}$ scenario I, the $b\to s\nu\bar{\nu}$ and $s\to
d\nu\bar{\nu}$ decays are not relevant for this scenario. In the
${Z^{\prime}}$ scenario II, on the other hand, the ${Z^{\prime}}$ boson
affects the $b\to s\nu\bar{\nu}$ and $s\to d\nu\bar{\nu}$ transitions at the
tree and the one-loop level, respectively. In order to constrain the
${Z^{\prime}}$ parameters, we consider five $b\to s\nu\bar{\nu}$ processes
(i.e., the decays $B\to X_{s}\nu\bar{\nu}$, $B^{0}\to K^{(*)0}\nu\bar{\nu}$,
and $B^{+}\to K^{(*)+}\nu\bar{\nu}$), and two $s\to d\nu\bar{\nu}$ processes
(i.e., the decays $K^{+}\to\pi^{+}\nu\bar{\nu}$ and
$K_{L}\to\pi^{0}\nu\bar{\nu}$). We show in figure 8 the allowed regions of the
real and imaginary parts of $X_{23}^{L}\lambda_{22}^{L}/m_{Z^{\prime}}^{2}$ by
the branching ratios of these decays. It can be seen that the bound from the
$s\to d\nu\bar{\nu}$ decays is much weaker than that from the $b\to
s\nu\bar{\nu}$ transitions. This can be understood by the fact that the
${Z^{\prime}}$ contributions to the former is suppressed by
${\mathcal{O}}(\lambda^{3})$ while its contributions to the latter do not
suffer any CKM suppression. In addition, the constraints from the $b\to
s\nu\bar{\nu}$ and $s\to d\nu\bar{\nu}$ decays are consistent with the best
fit region from the $b\to s\mu^{+}\mu^{-}$ processes.

In the ${Z^{\prime}}$ scenario I, as explained in subsection 3.2, the
$B_{s}-\bar{B}_{s}$ mixing cannot provide any relevant constraint on the
${Z^{\prime}}$ parameters. In the ${Z^{\prime}}$ scenario II, the
${Z^{\prime}}$ contribution is proportional to the product
$({X_{23}^{L}})^{2}/m_{Z^{\prime}}^{2}$ due to the tree-level ${Z^{\prime}}$
exchange. After taking into account the constraints from the mass difference
$\Delta m_{s}$ and the CP-violating observable $S_{\psi\phi}$, we obtain the
allowed regions in the plane $\big{(}{\rm Re}{X_{23}^{L}}/m_{Z^{\prime}},{\rm
Im}{X_{23}^{L}}/m_{Z^{\prime}}\big{)}$, which are shown in figure 9. It can be
seen that the mass difference $\Delta m_{s}$ provides a strong bound on ${\rm
Re}{X_{ct}^{L}}/m_{Z^{\prime}}$. For ${\rm Im}{X_{23}^{L}}/m_{Z^{\prime}}$,
the bound arises mainly from the CP-violating observable $S_{\psi\phi}$ and is
weaker compared to the one on ${\rm Re}{X_{23}^{L}}/m_{Z^{\prime}}$, due to
the currently larger experimental uncertainty of $S_{\psi\phi}$. The phase of
the maximum value of $|{X_{23}^{L}}|/m_{Z^{\prime}}$ also matches the CKM
phase $\beta_{s}$ [134].

Figure 9: Allowed regions in the plane $\big{(}{\rm
Re}{X_{23}^{L}}/m_{Z^{\prime}},{\rm Im}{X_{23}^{L}}/m_{Z^{\prime}}\big{)}$ by
the $B_{s}-\bar{B}_{s}$ mixing in the ${Z^{\prime}}$ scenario II. The dark and
the light regions correspond to the $68\%$ and the $95\%$ CL, respectively.

Summarizing the numerical analysis made above, we can see that the strongest
constraints are provided by the $b\to s\mu^{+}\mu^{-}$ processes in the
${Z^{\prime}}$ scenario I. Therefore, after considering all the flavour
processes, we can obtain the combined allowed regions from figure 6, with the
numerical results given by

$\displaystyle 0.04<\frac{{\rm
Re}{X_{ct}^{L}}{\lambda_{\mu\mu}^{L}}}{m_{Z^{\prime}}^{2}}<0.21{\,\rm
TeV}^{-2},\quad\quad-0.44<\frac{{\rm
Im}{X_{ct}^{L}}{\lambda_{\mu\mu}^{L}}}{m_{Z^{\prime}}^{2}}<0.41{\,\rm
TeV}^{-2},$ (47)

at the $95\%$ CL. The above lower bound on ${\rm
Re}{X_{ct}^{L}}{\lambda_{\mu\mu}^{L}}/m_{Z^{\prime}}^{2}$, together with the
parameter space chosen in eq. (45), implies an upper bound on the
${Z^{\prime}}$ mass, $m_{Z^{\prime}}<9.0{\,\rm TeV}$.

Turning to the ${Z^{\prime}}$ scenario II, the $b\to s\mu^{+}\mu^{-}$
processes also provide the most stringent constraints on the ${Z^{\prime}}$
parameter, as in scenario I. Therefore, after considering all the $B$\- and
$K$-meson FCNC decays, the combined constraints can be obtained from figure 6,
and numerically

$\displaystyle-0.0015<\frac{{\rm
Re}{X_{23}^{L}}{\lambda_{22}^{L}}}{m_{Z^{\prime}}^{2}}<-0.0003{\,\rm
TeV}^{-2},\quad\quad-0.0029<\frac{{\rm
Im}{X_{23}^{L}}{\lambda_{22}^{L}}}{m_{Z^{\prime}}^{2}}<+0.0031{\,\rm
TeV}^{-2},$ (48)

at the $95\%$ CL. However, different from the scenario I, the
$B_{s}-\bar{B}_{s}$ mixing also provides an independent bound on the parameter
product ${X_{23}^{L}}/m_{Z^{\prime}}$. Numerically, the combined constraints
shown in figure 9 correspond to

$\displaystyle-0.004<\frac{{\rm Re}{X_{23}^{L}}}{m_{Z^{\prime}}}<+0.004{\,\rm
TeV}^{-1},\quad-0.005<\frac{{\rm Im}{X_{23}^{L}}}{m_{Z^{\prime}}}<+0.005{\,\rm
TeV}^{-1},$ (49)

at the $95\%$ CL. In the case of
${\lambda_{22}^{L}}/m_{Z^{\prime}}\sim{\mathcal{O}}(1){\,\rm TeV}^{-1}$,
constraints on ${X_{23}^{L}}/m_{Z^{\prime}}$ from the $b\to s\mu^{+}\mu^{-}$
processes and the $B_{s}-\bar{B}_{s}$ mixing are of the same order.
Furthermore, since ${\rm Re}{X_{23}^{L}}{\lambda_{22}^{L}}/m_{Z^{\prime}}^{2}$
is lower bounded by the $b\to s\mu^{+}\mu^{-}$ processes, a lower bound,
${\lambda_{22}^{L}}/m_{Z^{\prime}}>0.07{\,\rm TeV}^{-1}$, can be derived in
order to simultaneously satisfy the constraints from the $B_{s}-\bar{B}_{s}$
mixing. Therefore, the ${Z^{\prime}}$ couplings in scenario II exhibit the
hierarchy ${\lambda_{22}^{L}}\gg{X_{23}^{L}}$. Finally, our numerical analysis
has shown that the main constraints in scenario II are obtained from the
processes affected by the tree-level ${Z^{\prime}}$ contributions. Therefore,
the flavour phenomenology of the ${Z^{\prime}}$ scenario II is almost
identical to that of the minimal ${Z^{\prime}}$ scenario discussed in refs.
[34, 45].

###  4.2 Collider constraints

In this subsection, we discuss the collider constraints on the ${Z^{\prime}}$
scenarios in the case of both $m_{Z^{\prime}}>m_{t}$ and
$m_{Z^{\prime}}<m_{t}$. For recent collider studies of similar ${Z^{\prime}}$
scenarios with top-FCNC couplings, we refer to refs. [162, 163, 164].

####  4.2.1 $m_{Z^{\prime}}>m_{t}$

In the case of $m_{Z^{\prime}}>m_{t}$, constraints on the ${Z^{\prime}}$
parameters could be obtained from the decay $t\to c\mu^{+}\mu^{-}$. However,
the current LHC searches for the decay $t\to c\mu^{+}\mu^{-}$ have been only
performed at the $Z$ peak, with $|m_{\mu^{+}\mu^{-}}-m_{Z}|<15\,{\,\rm GeV}$,
and interpreted as a bound on ${\mathcal{B}}(t\to cZ)$ [165, 166, 167, 141].
No dedicated searches for non-resonant (outside the $Z$ peak) $t\to
c\mu^{+}\mu^{-}$ decays have been performed yet. In ref. [168], an upper bound
on ${\mathcal{B}}(t\to c\mu^{+}\mu^{-})$ was estimated by using the
experimental bounds on ${\mathcal{B}}(t\to cZ)$ and
${\mathcal{B}}(Z\to\mu^{+}\mu^{-})=3.37\%$ [132]. With such an approach, an
upper bound ${\mathcal{B}}(t\to c\mu^{+}\mu^{-})<4.4\times 10^{-6}$ can be
derived from the current bound ${\mathcal{B}}(t\to cZ)<1.3\times 10^{-4}$ set
by the ATLAS experiment with an integrated luminosity of $139{\,\rm fb}^{-1}$
at $13{\,\rm TeV}$ [141]. In this way, constraints on the ${Z^{\prime}}$
parameters can be derived from the bound ${\mathcal{B}}(t\to
c\mu^{+}\mu^{-})<4.4\times 10^{-6}$. However, we find that such a constraint
is at least one order of magnitude weaker than that obtained from the low-
energy flavour processes discussed in the last subsection.

Within the effective field theory (EFT) framework, by using different signal
regions of the LHC searches for the rare FCNC decay $t\to cZ$, an improved
approach has been developed in ref. [169]. However, in the case when
$m_{Z^{\prime}}$ is not far from $m_{t}$, the framework with four-fermion
effective operators is not appropriate to describe the ${Z^{\prime}}$
contributions to the decay $t\to c\mu^{+}\mu^{-}$.

####  4.2.2 $m_{Z^{\prime}}<m_{t}$

Figure 10: Allowed regions in the plane $(\lvert
X_{ct(23)}^{L}\rvert,m_{Z^{\prime}})$ from the top-quark width $\Gamma_{t}$ in
the ${Z^{\prime}}$ scenario I (II). The colour captions are the same as in
figure 6.

In the case of $m_{Z^{\prime}}<m_{t}$, the $t\to c\mu^{+}\mu^{-}$ decay
involves the resonant ${Z^{\prime}}$ contribution, and is expected to provide
strong constraints on the ${Z^{\prime}}$ parameters. However, similar to the
case of $m_{Z^{\prime}}>m_{t}$, no experimental searches for the $t\to
c\mu^{+}\mu^{-}$ decay mediated by the ${Z^{\prime}}$ resonance has been
performed yet. Detailed analysis of the signal shape could be used to derive
constraints from the current ${\mathcal{B}}(t\to cZ)$ bound, as performed in
ref. [169]. Especially, for $\lvert m_{Z^{\prime}}-m_{Z}\rvert<15\,{\,\rm
GeV}$, the signal regions of the $t\to cZ$ and $t\to c{Z^{\prime}}$ decays can
be largely overlapped, and stringent constraints on the ${Z^{\prime}}$
parameters could be therefore derived. We leave these possibilities for
further work, and concentrate on the mass region $105{\,\rm
GeV}<m_{Z^{\prime}}<m_{t}$ to avoid the potentially strong bound from
${\mathcal{B}}(t\to cZ)$ [141].

As discussed in subsection 3.6.2, in the case of $m_{Z^{\prime}}<m_{t}$, the
decay $t\to c{Z^{\prime}}$ can contribute to the top-quark width. Compared to
the SM prediction $\Gamma_{t}^{\rm SM}=1.3\,{\,\rm GeV}$ [170], the current
measurement $\Gamma_{t}=(1.42_{-0.15}^{+0.19})\,{\,\rm GeV}$ [132] leaves
${\mathcal{O}}(20\%)$ room for the ${Z^{\prime}}$ effects. We show in figure
10 the allowed regions in the plane $(\lvert
X_{ct(23)}^{L}\rvert,m_{Z^{\prime}})$ in the scenario I (II). For scenario I,
it is noted that the top-quark width provides a unique bound on the coupling
${X_{ct}^{L}}$. For scenario II, the bound on ${X_{23}^{L}}$ is much weaker
than that obtained from the $B_{s}-\bar{B}_{s}$ mixing.

###  4.3 Predictions

As mentioned in the last subsection, experimental searches for the $t\to
c\mu^{+}\mu^{-}$ decay off the $Z$ pole have not been performed yet. Using the
allowed parameter space derived in the global fit, we can make prediction on
the branching ratio ${\mathcal{B}}(t\to c\mu^{+}\mu^{-})$ in the two
${Z^{\prime}}$ scenarios as a function of the ${Z^{\prime}}$ mass, which is
shown in figure 11. Although there are several studies of the expected
sensitivities to the $t\to cZ$ decay at the LHC [171] and other future
colliders [172, 173, 174], to our knowledge, detailed analysis of the $t\to
c\mu^{+}\mu^{-}$ decay at the LHC has not been performed yet. In order to
estimate the current (future) experimental sensitivity to ${\mathcal{B}}(t\to
c\mu^{+}\mu^{-})$, we adopt as a benchmark the product of
${\mathcal{B}}(Z\to\mu^{+}\mu^{-})=3.37\%$ [132] and the current (future
projected) experimental limit on ${\mathcal{B}}(t\to cZ)$. With the current
ATLAS limit ${\mathcal{B}}(t\to cZ)<1.3\times 10^{-4}$ based on the full Run-2
data [141] and the future expected sensitivity $5\times 10^{-5}$ at the HL-LHC
with $3\,\mathrm{ab}^{-1}$ [171], the current and the future sensitivity to
${\mathcal{B}}(t\to c\mu^{+}\mu^{-})$ are estimated to be $4.4\times 10^{-6}$
and $1.7\times 10^{-6}$ respectively, as shown by the dashed and the dotted
line in figure 11. Here we make the following two observations:

  * •

In the case of $m_{Z^{\prime}}<m_{t}$, the branching ratio of $t\to
c\mu^{+}\mu^{-}$ decay in the two ${Z^{\prime}}$ scenarios is strongly
enhanced by the resonance effect, which is several orders of magnitude higher
than the one in the case of $m_{Z^{\prime}}>m_{t}$. In the mass region
$105{\,\rm GeV}<m_{Z^{\prime}}<m_{t}$, the predicted ${\mathcal{B}}(t\to
c\mu^{+}\mu^{-})$ in scenario I (II) is higher (lower) than the current and
future estimated bounds. Therefore, experimental searches for the $t\to
c\mu^{+}\mu^{-}$ decay are expected to probe a ${Z^{\prime}}$ mass window of
$[105{\,\rm GeV},m_{t}]$.

  * •

In the case of $m_{Z^{\prime}}>m_{t}$, the predicted ${\mathcal{B}}(t\to
c\mu^{+}\mu^{-})$ in scenario I is compatible with the estimated
sensitivities, while the one in scenario II is several orders of magnitude
lower the estimated future sensitivity. Therefore, direct searches for the
decay in scenario II with a heavy ${Z^{\prime}}$ boson could be very
challenging for the LHC and its high-luminosity upgrade.

Figure 11: The branching ratio of $t\to c\mu^{+}\mu^{-}$ decay as a function
of $m_{Z^{\prime}}$ in the ${Z^{\prime}}$ scenarios I (left) and II (right).
The dark regions are allowed at the $95\%$ CL.

Figure 12: Correlations between the observables of $B$\- and $K$-meson decays
in the two ${Z^{\prime}}$ scenarios. The $1\sigma$ range of the experimental
measurements and the SM predictions are shown, except the data on
${\mathcal{B}}(B^{0}\to K^{*0}\nu\nu)$, which corresponds to the $90\%$ CL
limit.

Besides the $t\to c\mu^{+}\mu^{-}$ decay, which should be searched in $pp\to
t\bar{t}$ production, the associated production $pp\to t{Z^{\prime}}$ could
also be considered and may provide more relevant constraints for a heavy
${Z^{\prime}}$ boson. Recently, the associated production of a single top with
dileptons has been investigated in the EFT framework [175], and a $95\%$ CL
bound on the scale of the effective operator
$(\bar{t}\gamma^{\mu}P_{L}c)(\bar{\mu}\gamma_{\mu}P_{L}\mu)$,
$\Lambda_{tc\mu\mu}=1.1\,(1.5)\,{\,\rm TeV}$, has been obtained at the LHC
with an integrated luminosity of $140\,(3000){\,\rm fb}^{-1}$. In the
${Z^{\prime}}$ scenario I (II) with $m_{Z^{\prime}}=1\,{\,\rm TeV}$, the
$95\%$ CL allowed parameter space in our global fit results in
$1.4\,(17.0)<\Lambda_{tc\mu\mu}<4.8\,(55.2){\,\rm TeV}$. This implies that the
${Z^{\prime}}$ scenario I reaches the sensitivity of the HL-LHC even for a
TeV-scale ${Z^{\prime}}$ boson.777In the ${Z^{\prime}}$ scenario II, the
$b\bar{s}{Z^{\prime}}$ coupling makes the $pp\to\mu^{+}\mu^{-}$ process
sensitive to the ${Z^{\prime}}$ boson. For more details, we refer to ref.
[52]. Detailed collider simulation is still necessary to obtain more reliable
estimation of the experimental sensitivity, which will be our future work. We
also encourage our experimental colleagues to carry out relevant searches for
such a ${Z^{\prime}}$ boson.

An important feature of our ${Z^{\prime}}$ scenarios is that the
${Z^{\prime}}$ contributions to all the FCNC decays are controlled by the same
product, ${X_{ct}^{L}}{\lambda_{\mu\mu}^{L}}/m_{Z^{\prime}}^{2}$ in scenario I
or ${X_{23}^{L}}{\lambda_{22}^{L}}/m_{Z^{\prime}}^{2}$ in scenario II.
Therefore, all the $B$\- and $K$-meson processes are strongly correlated. As
an illustration, we show in figure 12 the correlations among some observables.
From these plots, we make the following two observations:

  * •

The branching ratio ${\mathcal{B}}(K_{L}\to\mu^{+}\mu^{-})$ is suppressed by
the ${Z^{\prime}}$ effects in scenario I. However, its SM value remain almost
unchanged in scenario II, due to the strongly constrained ${Z^{\prime}}$
couplings. For the $B^{0}\to K^{*0}\mu^{+}\mu^{-}$ decay, the situation is
opposite: the branching ratio is enhanced in scenario II, while remaining
unchanged in scenario I. These correlations can be used to distinguish the two
${Z^{\prime}}$ scenarios. In addition, the ${Z^{\prime}}$ effects on the
branching ratios of $K_{S}\to\mu^{+}\mu^{-}$, $K_{L}\to\pi\nu\bar{\nu}$ and
$K^{+}\to\pi^{+}\nu\bar{\nu}$ decays are all found to be negligible in the two
scenarios and will be not shown here.

  * •

The correlations between $R_{K}^{[1.1,\,6.0]}$ and $R_{K^{*}}^{[1.1,\,6.0]}$,
which are the LFU ratios in the $q^{2}$ bin $[1.1,\,6.0]\,{\,\rm GeV}^{2}$, as
well as the correlations between $R_{K^{*}}^{[1.1,\,6.0]}$ and
$P_{5}^{\prime}$, in the two ${Z^{\prime}}$ scenarios are almost identical.
The reason is that the best fit regions for the Wilson coefficients
${\mathcal{C}}_{9}$ and ${\mathcal{C}}_{10}$ are numerically almost the same
in the two scenarios. As expected, the ${Z^{\prime}}$ effects make
$R_{K^{(*)}}^{[1.1,\,6.0]}$, $P_{5}^{\prime}$, and
${\mathcal{B}}(B_{s}\to\phi\mu^{+}\mu^{-})$ move closer to the experimental
measurements.

With high-precision measurements at the HL-LHC and Belle II [176], these
interesting correlation could provide further insights into our ${Z^{\prime}}$
scenarios.

In the case of $m_{Z^{\prime}}\gg m_{t}$, the ${Z^{\prime}}$ contributions to
both the $t\to c\mu^{+}\mu^{-}$ and $b\to s\mu^{+}\mu^{-}$ processes are
controlled by the same products,
${X_{ct}^{L}}{\lambda_{\mu\mu}^{L}}/m_{Z^{\prime}}^{2}$ in scenario I or
${X_{23}^{L}}{\lambda_{22}^{L}}/m_{Z^{\prime}}^{2}$ in scenario II, since they
are described by the effective $t\bar{c}\mu^{+}\mu^{-}$ operator. Thus, we
show in figure 13 the correlation between $R_{K^{*}}^{[1.1,\,6.0]}$ and
${\mathcal{B}}(t\to c\mu^{+}\mu^{-})$ for $m_{Z^{\prime}}=1\,{\,\rm TeV}$. It
can be seen that ${\mathcal{B}}(t\to c\mu^{+}\mu^{-})$ approaches to its
maximum value when $R_{K^{*}}^{[1.1,\,6.0]}\approx 0.85$. The predicted upper
limit is of the same order as the estimated future LHC sensitivity in scenario
I, while being quite below the estimated sensitivity in scenario II.

Figure 13: Correlation between $R_{K^{*}}$ and ${\mathcal{B}}(t\to
c\mu^{+}\mu^{-})$ in the $Z^{\prime}$ scenarios I (left) and II (right) for
$m_{Z^{\prime}}=1\,{\,\rm TeV}$ with real (gray) and complex (red) couplings.

##  5 Conclusions

Motivated by the recent anomalies in the $b\to s\ell^{+}\ell^{-}$ transitions,
we have considered a phenomenological ${Z^{\prime}}$ scenario (denoted as
scenario I), in which a heavy ${Z^{\prime}}$ boson couples only to $t\bar{c}$
and $\mu^{+}\mu^{-}$ with left-handed chirality. The ${Z^{\prime}}$ effects on
the $b\to s\mu^{+}\mu^{-}$ processes automatically induce opposite
contributions to the effective operators ${\mathcal{O}}_{9}$ and
${\mathcal{O}}_{10}$, which is favored by the model-independent analyses of
the $b\to s\ell^{+}\ell^{-}$ anomalies.

We have performed a global fit to all relevant experimental data. It is found
that such a minimal ${Z^{\prime}}$ scenario can address the current $b\to
s\ell^{+}\ell^{-}$ anomalies, while simultaneously satisfying other flavour
and collider constraints. The mass of the ${Z^{\prime}}$ boson can be less
than $1\,{\,\rm TeV}$. In the region $105{\,\rm GeV}<m_{Z^{\prime}}<m_{t}$,
the $t\to c\mu^{+}\mu^{-}$ decay is significantly enhanced by resonance
effects and can serve as a sensitive probe of such a ${Z^{\prime}}$ boson. As
an important feature of this scenario, all the low-energy flavour observables
are controlled by the product
${X_{ct}^{L}}{\lambda_{\mu\mu}^{L}}/m_{Z^{\prime}}^{2}$. We have found
interesting correlations among the various flavour observables, which could
provide further insights into this scenario.

We have also considered an extended scenario (denoted as scenario II), in
which the ${Z^{\prime}}$ boson interacts with the $SU_{L}(2)$ fermion doublets
with analogous couplings as in scenario I. Due to tree-level ${Z^{\prime}}$
contributions, the ${Z^{\prime}}$ couplings suffer from severe constraints
from the $b\to s\ell^{+}\ell^{-}$ processes and the $B_{s}-\bar{B}_{s}$
mixing, which makes the collider signals of $t\to c\mu^{+}\mu^{-}$ decay below
the estimated sensitivity at the HL-LHC. The correlations between flavour
observables in scenario II are different from that observed in scenario I, and
can be therefore used to distinguish between the two scenarios.

Our scenarios can be modified by replacing $t\bar{c}{Z^{\prime}}$ with
$t\bar{u}{Z^{\prime}}$ coupling. Then the $K$-meson rare decays could become
more relevant due to the large CKM factors involved, and the $t{Z^{\prime}}$
associated production at the LHC may play crucial role in searching for such a
${Z^{\prime}}$ boson. A right-handed $\mu^{+}\mu^{-}{Z^{\prime}}$ coupling can
also be taken into account to simultaneously accommodate the $(g-2)_{\mu}$
anomaly. In addition, if the $\mu^{+}\mu^{-}{Z^{\prime}}$ coupling is replaced
by the $e^{+}e^{-}{Z^{\prime}}$ coupling, the direct production of the
${Z^{\prime}}$ boson at $e^{+}e^{-}$ collider should then be taken into
account. It is noted that, although our ${Z^{\prime}}$ scenarios are not
realistic, our studies have shown that a vector boson with top-quark FCNC
interactions can explain the current $b\to s\ell^{+}\ell^{-}$ anomalies and
may provide new avenues for model buildings. We also encourage our
experimental colleagues to carry out relevant searches for such a
${Z^{\prime}}$ boson at the LHC and its high-luminosity upgrade.

## Acknowledgements

This work is supported by the National Natural Science Foundation of China
under Grant Nos. 12135006, 12075097, 11675061, 11775092, and 11805077, as well
as by the self-determined research funds of CCNU from the colleges’ basic
research and operation of MOE under Grant Nos. CCNU19TD012 and CCNU20TS007. XY
is also supported in part by the Startup Research Funding from CCNU.

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