Probing FCNC couplings in single top quark production associated with a
neutral gauge boson at future lepton colliders

Sara<EMAIL_ADDRESS>Mehrnoosh
<EMAIL_ADDRESS>
†Department of Physics, University of Tehran, North Karegar Ave., Tehran
14395-547, Iran
‡Department of Physics, Isfahan University of Technology, Isfahan 84156-83111,
Iran

###### Abstract

In this paper, we consider a single top quark production in association with a
neutral gauge boson (Z boson or Photon) at the future electron-positron
colliders. We utilize these channels to probe the top quark Flavour Changing
Neutral Currents (FCNC) interactions in $tqZ$ and $tq\gamma$ vertices. We
perform two separate analyses for top-Z-jet and top-$\gamma$-jet channels. The
Standard Model Effective Field Theory (SMEFT) approach is employed to search
for these anomalous couplings. We consider parton showering, hadronization and
fast detector simulation in our study and use a cut-and-count technique to
separate the signal from the Standard Model (SM) background processes. The
upper limits on the FCNC couplings at $95\%$ confidence level for the
different integrated luminosities are obtained. It is shown that the future
lepton collider would be able to probe the FCNC branching fractions to
$\rm{Br}(t\rightarrow q\gamma)<10^{-4}$ and $\rm{Br}(t\rightarrow qZ)<10^{-4}$
with $3$ $\rm{ab}^{-1}$of integrated luminosity of data at a center of mass
energy of $350$ GeV.

##  1 Introduction

The Standard Model (SM) of particle physics has been found to be a successful
theory in describing particles and their interactions at $\,{\mathrm{TeV}}$
scale. So far, all experimental results are in a good agreement with the SM
predictions. However, the SM fails to answer some major experimental
observations like the need for the existence of Dark Matter (DM) in the
cosmos, the massive neutrinos, baryon asymmetry in the universe, etc. Hence,
particle physicists have looked for new physics for several decades.

Despite all the plentiful efforts at the Large Hadron Collider (LHC) at run I
and run II, no signal of new physics has been found yet. Therefore, it seems
that the new physics scale should be well separated from the electroweak scale
and the new degrees of freedom cannot be produced directly at the colliders.
So, in order to study the new physics, we should look for indirect signs such
as new interactions between the SM particles. Flavour changing neutral current
(FCNC) interactions are a good place to seek for these new effects since they
are forbidden at the tree level in the SM framework and they are also strongly
suppressed at the loop level by the Glashow-Iliopoulos-Maiani (GIM) mechanism
[1].

Particularly, the search for FCNC interactions in the top sector is well-
motivated, because of the especial properties of top quark. The SM predicts
very small branching ratios ($\mathcal{O}(10^{-14})$) for different top quark
FCNC decay modes [2], hence the current and future experiments would not be
able to probe such tiny values. Nevertheless, there are many Beyond Standard
Models (BSM) such as Technicolor, Minimal Supersymmetric Standard Model
(MSSM), two Higgs Doublet Models (2HDM) that anticipate for large branching
ratios for the top FCNC decay modes [3, 4, 5, 6, 7, 8, 9]. Therefore, any
footprint of the top FCNC interactions can prove the existence of the new
physics.

There are several phenomenological papers that studied the top FCNC couplings
at different colliders in the literature [10, 11, 12, 13, 14, 15, 16, 17, 18,
19, 20]. Furthermore, there are many experimental analyses that look for the
top FCNC interactions using different channels [21, 22, 23, 24, 25, 26, 27,
28, 29, 30, 31, 32]. For instance, the CMS collaboration has studied the top
quark FCNC interactions in the anomalous single top quark production in
association with a photon in proton-proton collision at 8 TeV [24]. Following
upper limits at the $95\%$ CL on the anomalous top branching ratios have been
found; $\rm{Br}(t\rightarrow u\gamma)<1.3\times 10^{-4}$ and
$\rm{Br}(t\rightarrow c\gamma)<1.7\times 10^{-3}$. Recently, more stringent
bounds on these branching ratios have been found by ATLAS detector at a center
of mass energy of 13 TeV; $\rm{Br}(t\rightarrow u\gamma)<6.1\times 10^{-5}$
and $\rm{Br}(t\rightarrow c\gamma)<18\times 10^{-5}$ [25]. In addition, the
ATLAS experiment has searched for the FCNC couplings in proton-proton
collision at 13 TeV by using top-quark pair events which one of them decaying
through $t\rightarrow qZ$ and another one decays through the standard mode
[26]. The collaboration has found the upper limits at the $95\%$ CL on the top
branching ratios as $\rm{Br}(t\rightarrow uZ)<1.7\times 10^{-4}$ and
$\rm{Br}(t\rightarrow cZ)<2.4\times 10^{-4}$. Some papers studied the FCNC
couplings in the HL-LHC, the reported values for branching ratios are
$\rm{Br}(t\rightarrow u\gamma)<4.6\times 10^{-5}$, $\rm{Br}(t\rightarrow
c\gamma)<3.4\times 10^{-4}$, $\rm{Br}(t\rightarrow
uZ)(\sigma_{\mu\nu})<8.1\times 10^{-4}$ and $\rm{Br}(t\rightarrow
uZ)(\gamma_{\mu})<1.7\times 10^{-3}$ [33, 34, 35].

There are several proposed future electron-positron colliders for very precise
measurements such as the future circular electron–positron collider (FCC-ee)
[36, 37], the Circular electron-positron Collider (CEPC) [38], the
International Linear Collider (ILC) [39, 40, 41], and the Compact Linear
Collider (CLIC) [42]. Since large amounts of production of top quarks would
happen at these colliders, the top quark couplings can be probed at high
precision. Our main purpose here is to study the top FCNC couplings in the
single top quark production in association with a neutral gauge boson (Z or
$\gamma$) at a future electron-positron collider. This analysis is a
complementary channel in addition to other channels to probe the anomalous
FCNC couplings, since the top-Z/$\gamma$-jet has a different signature at the
lepton colliders.

In this paper, these signals are studied separately. In the top-Z-jet
analysis, we consider both top quark and Z boson decay leptonically,
therefore, we expect to have three charged leptons, a b-jet, a light-jet, and
the missing energy in the final state. However, in the top-$\gamma$-jet
analysis, leptonic decay for the top quark is assumed, so the final state
comprises a charged lepton, a hard photon, a b-jet, a light-jet, and the
missing energy due to the neutrino. Furthermore, parton showering and
hadronization as well as the detector simulations are taken into account in
these studies. Then, we determine the upper limits on the top FCNC couplings
and the corresponding branching ratios in the both analyses.

The paper is structured as follows. In section 2, we introduce the effective
Lagrangian which describes the top quark FCNC interactions with the neutral
gauge bosons. In section 3, we explain our analysis strategy like event
simulations and event selections for top-Z-jet signal and then present our
obtained limits on the new couplings and the corresponding branching ratios.
Analysis strategy for top-$\gamma$-jet signal and the related upper limits on
the anomalous couplings are described in section 4. Finally, we present our
summary in section 5.

##  2 Theoretical formalism

In this section, we explain the theoretical framework which we use to search
for FCNC couplings. There are two different approaches to study any new
physics at the colliders. The first approach is the direct searches for the
well-motivated BSM models. In another word, we look for the new degrees of
freedom which anticipated by these new models at the colliders. The second
approach is the indirect searches which mean we seek for the new interactions
between SM particles.

Since all the experimental measurements are in good agreement with the SM
predictions, the new physics scales seem to be well separated from the SM
scale. On the other hand, because of the presence of several new physics
scenarios and sometimes with the same signatures at the colliders, an
economical way to search for the new physics would be using a model
independent way. The Standard Model Effective Field Theory (SMEFT) approach
provides this model independent way to look for any sign of new physics [43].

In the SMEFT, all the new degrees of freedom are integrated out and the new
physics effects show up in the higher dimensional effective operators which
contain the new interactions between the SM field content. These higher
dimensional operators are suppressed by the inverse powers of the scale of new
physics and respect the Lorentz and SM gauge symmetries. The operators with
odd dimensions are not considered by assuming baryon and lepton number
conservation. Furthermore, the EFT approach is useful when the scale of new
physics is much larger than the energy scale of the process. Since the
operators are suppressed by the powers of $\Lambda$, one expects that the
smallest dimension provides the leading corrections to the SM amplitudes. The
effects of dimension eight and higher-order operators are suppressed by at
least $1/\Lambda^{4}$. So, the dimension six operators have dominant
contributions to the observables at the colliders. Worth mentioning that, if
the coefficients of dimension six operators, which can contribute to the
special process, are strongly constrained by current experiments, the
dimension eight operators have a main contribution to the process. The SMEFT
Lagrangian up to dimension six operators could be written as following [44]:

$\mathcal{L}_{\rm{SMEFT}}=\mathcal{L}_{\rm{SM}}+\sum_{i}\frac{C_{i}O_{i}}{\Lambda^{2}},$
(1)

here the first term is the SM Lagrangian with four dimensional operators and
$O_{i}$ demonstrates operators with dimension six. The new physics scale is
denoted by $\Lambda$ and $C_{i}$ are the Wilson couplings. The following
dimension six operators can contribute in the FCNC interactions of a top quark
and a Z boson/photon with an up-type quark (u or c quark) [45]:

$\displaystyle O_{uW}^{ij}$ $\displaystyle=$
$\displaystyle\left(\bar{q}_{Li}\sigma^{\mu\nu}\tau^{I}u_{Rj}\right)\tilde{\phi}W_{\mu\nu}^{I},$
$\displaystyle O_{uB\phi}^{ij}$ $\displaystyle=$
$\displaystyle\left(\bar{q}_{Li}\sigma^{\mu\nu}u_{Rj}\right)\tilde{\phi}B_{\mu\nu},$
$\displaystyle O_{\phi q}^{(3,ij)}$ $\displaystyle=$ $\displaystyle
i\left(\phi^{\dagger}\tau^{I}D_{\mu}\phi\right)\left(\bar{q}_{Li}\gamma^{\mu}\tau^{I}q_{Lj}\right),$
$\displaystyle O_{\phi q}^{(1,ij)}$ $\displaystyle=$ $\displaystyle
i\left(\phi^{\dagger}D_{\mu}\phi\right)\left(\bar{q}_{Li}\gamma^{\mu}q_{Lj}\right),$
$\displaystyle O_{\phi u}^{ij}$ $\displaystyle=$ $\displaystyle
i\left(\phi^{\dagger}D_{\mu}\phi\right)\left(\bar{u}_{Ri}\gamma^{\mu}u_{Rj}\right),$
(2)

where $q_{Li}$ and $u_{Rj}$ are left-handed quark doublet and right-handed
quark singlet, respectively, and $i,j$ are flavour indices. The Pauli matrices
are presented with $\tau^{I}$ and $D_{\mu}$ is covariant derivative. Moreover,
$\tilde{\phi}=i\tau^{2}\phi^{*}$ that $\phi$ is the SM Higgs doublet. Their
contribution to the FCNC interactions can be parametrized as follows,

$\displaystyle\mathcal{L}_{\rm{FCNC}}=\sum_{q=u,c}\\{$
$\displaystyle-\frac{g_{W}}{2c_{W}}\bar{q}\gamma^{\mu}\left(X_{tq}^{L}P_{L}+X_{tq}^{R}P_{R}\right)tZ_{\mu}+\frac{g_{W}}{4c_{W}m_{Z}}\bar{q}\sigma^{\mu\nu}\left(\kappa_{tq}^{L}P_{L}+\kappa_{tq}^{R}P_{R}\right)tZ_{\mu\nu}$
$\displaystyle+\frac{e}{2m_{t}}\bar{q}\sigma^{\mu\nu}\left(\lambda_{tq}^{L}P_{L}+\lambda_{tq}^{R}P_{R}\right)tA_{\mu\nu}\\}+h.c.,$
(3)

where the field strengths of the Z boson and photon are shown by $Z_{\mu\nu}$
and $A_{\mu\nu}$, respectively. Also, $P_{R}$ and $P_{L}$ demonstrate the
right and left-handed projection operators, respectively. The dimensionless
anomalous couplings between a top quark, a light up-type quark and a Z boson
are presented by $X_{tq}^{L}$, $X_{tq}^{R}$, $\kappa_{tq}^{L}$ and
$\kappa_{tq}^{R}$. Similarly, $\lambda_{tq}^{L}$ and $\lambda_{tq}^{R}$ are
dimensionless anomalous couplings between a top quark, a light up-type quark
and a photon. These dimensionless anomalous couplings have the following
relation with the Wilson coefficients of effective operators [45],

$\displaystyle X_{tq}^{L}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[C_{\phi q}^{(3,j3)}+C_{\phi
q}^{(3,3j)*}-C_{\phi q}^{(1,j3)}-C_{\phi
q}^{(1,3j)*}\right]\frac{v^{2}}{\Lambda^{2}},$ $\displaystyle X_{tq}^{R}$
$\displaystyle=$ $\displaystyle-\frac{1}{2}\left[C_{\phi u}^{j3}+C_{\phi
u}^{3j*}\right]\frac{v^{2}}{\Lambda^{2}},$ $\displaystyle\kappa_{tq}^{L}$
$\displaystyle=$
$\displaystyle\sqrt{2}\left[c_{W}C_{uW}^{3j*}-s_{W}C_{uB\phi}^{3j*}\right]\frac{v^{2}}{\Lambda^{2}},$
$\displaystyle\kappa_{tq}^{R}$ $\displaystyle=$
$\displaystyle\sqrt{2}\left[c_{W}C_{uW}^{j3}-s_{W}C_{uB\phi}^{j3}\right]\frac{v^{2}}{\Lambda^{2}},$
$\displaystyle\lambda_{tq}^{L}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{2}}{e}\left[s_{W}C_{uW}^{3j*}+c_{W}C_{uB\phi}^{3j*}\right]\frac{vm_{t}}{\Lambda^{2}},$
$\displaystyle\lambda_{tq}^{R}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{2}}{e}\left[s_{W}C_{uW}^{j3}+c_{W}C_{uB\phi}^{j3}\right]\frac{vm_{t}}{\Lambda^{2}},$
(4)

where $j$ can be $1$ and $2$ corresponding to u-quark and c-quark couplings,
respectively.

As mentioned before, we are interested to look for the single top production
in association with a Z boson or a photon and a light jet at a future
electron-positron collider. This process cannot be produced in the SM
framework, however, above FCNC interaction terms allow to generate such
processes. Some Feynman diagrams of these processes in presence of the FCNC
couplings are illustrated in Fig. 1. Diagrams $a$, $b$, and $c$ arise from the
anomalous couplings $tqZ$ ($X_{tq}$ or $\kappa_{tq}$) and diagrams $d$, $e$,
and $f$ occur in presence of the anomalous couplings $tq\gamma$
($\lambda_{tq}$). We should mention that the $e^{+}e^{-}\rightarrow
t\bar{t}\rightarrow tVj(V=Z~{}\text{or}~{}\gamma)$ process have contribution
in our final state and we considered it in our analysis as well (diagrams $c$
and $f$).

In the two next sections, we use the effective Lagrangian (Eq. 3) to generate
and analyse events for the top quark production with a neutral gauge boson
signal to study FCNC interactions. We explain extensively analysis for the
top-Z-jet signal in section 3 and analysis for the top-$\gamma$-jet signal in
4.

Figure 1: Representative Feynman diagrams for the single top quark production
in association with a neutral gauge boson in the presence of the FCNC
couplings are shown. Diagrams $a$, $b$, and $c$ arise from the anomalous
couplings $tZq$ ($X_{tq}$ or $\kappa_{tq}$) and diagrams $d$, $e$, and $f$
occur in presence of the anomalous couplings $t\gamma q$ ($\lambda_{tq}$).

##  3 Analysis Strategy for top-Z-jet signal

In this section, we study the single top quark production in association with
a Z boson to look for the $tqZ$ and $tq\gamma$ anomalous couplings at the
future electron-positron collider. In the first subsection, event generation
and simulation for both the signal and backgrounds are explained in detail.
Then, we describe our reconstruction method of the final state particles and
introduce some strategies to reduce the number of background events with
respect to the number of signal events. Eventually, the upper limit which can
be extracted from the top-Z-jet signal is presented.

###  3.1 Event Generation and Simulation

Now we are ready to define the top-Z-jet signal and related background
processes and describe the event generation and simulation method. Since the
mass of final state particles, the top quark and Z boson, are heavy to be
produced in the collider with $\sqrt{s}=240~{}\rm{GeV}$, we just focus on the
center of mass energy of $\sqrt{s}=350~{}\rm{GeV}$ in this analysis. The
signal process is single top production in association with a Z boson and a
light jet ($e^{-}~{}e^{+}\rightarrow t/\bar{t}~{}Z~{}j$), which both top quark
and Z boson decay leptonically. As a result, the final state consists of three
charged leptons (electron or muon), a b-jet, a light jet, and the missing
energy. The representative Feynman diagrams of the signal production are
presented in Fig. 1 ($a$, $b$ and $c$). Regarding generate of the signal
events, we insert the UFO module [46] in the MadGraph 5 package [47]. This UFO
module has been obtained from implementing the effective Lagrangian of Eq. 3
[48] in the FeynRules package [49].

Since all the $tqZ$ and $tq\gamma$ FCNC interactions produce our favorable
signal (top-Z-jet), the signal events are categorized into three scenarios. In
the first scenario, we choose a non zero value for
$\kappa_{tq}^{R}=\kappa_{tq}^{L}=\kappa_{tq}$ and put zero for other anomalous
couplings. In the second one, we put zero value for all FCNC couplings except
$X_{tq}^{R}=X_{tq}^{L}=X_{tq}$. And in the final case, just
$\lambda_{tq}^{R}=\lambda_{tq}^{L}=\lambda_{tq}$ coupling has a non zero
value. Also, it should be mentioned that we set the same values for the
u-quark and c-quark anomalous couplings.

According to the final state of the signal, the main background processes are
$W^{+}W^{-}jj$, $W^{\pm}Zjj$, $t\bar{t}$, and $ZZjj$. In this processes all
the W-boson, Z-boson, and top quark decay leptonically. Since tau lepton can
have a hadronic decay, it indicates a different signature in the detector, so
decay to electron and muon are just taken into account. The MadGraph 5 package
is used to generate all the background processes. At generator level,
preselection cuts like $p_{T}\geq 10$ GeV and $|\eta|<3$ are applied on all
objects in the final state.333 For applying cuts we use $p_{T}$ and $\eta$,
however, we should mention that at sub-TeV lepton colliders, energy and angle
are much more standard variables.  The angular distance between all the
charged leptons and jets has to be greater than $0.4$ ($\Delta
R=\sqrt{(\Delta\eta)^{2}+(\Delta\phi)^{2}}$ ). For both signal and
backgrounds, the Pythia package [50] is employed for parton showering and
hadronization. Then, the Delphes package [51] is utilized to model the
detector performance with using the ILD card [40]. For numerical calculation,
we use following the SM inputs: $m_{\text{top}}=173.3$ GeV, $m_{Z}=91.187$
GeV, $m_{W}=80.419$ GeV, $\alpha=1/127.90$, and $s^{2}_{W}=0.234$ [52].

Cross-sections (in fb) | Signal | Background
---|---|---
Cuts | $tqZ(\sigma_{\mu\nu})$ | $tqZ(\gamma_{\mu})$ | $tq\gamma$ | $WWjj$ | $WZjj$ | $t\bar{t}$ | $ZZjj$
Preselection cuts | $13.5(\kappa_{tq})^{2}$ | $9.28(X_{tq})^{2}$ | $0.14(\lambda_{tq})^{2}$ | $10.12$ | $0.26$ | $9.25$ | $0.007$
$1~{}\rm{light~{}jet}+1\rm{b~{}jet}$ | $6.92(\kappa_{tq})^{2}$ | $4.76(X_{tq})^{2}$ | $0.066(\lambda_{tq})^{2}$ | $4.43$ | $0.012$ | $4.2$ | $0.001$
$3~{}\rm{leptons}$ | $3.55(\kappa_{tq})^{2}$ | $2.5(X_{tq})^{2}$ | $0.03(\lambda_{tq})^{2}$ | $0.00014$ | $0.006$ | $0.00024$ | $0.0007$
$\not{p}_{T}>30~{}\rm{GeV}$ | $2.6(\kappa_{tq})^{2}$ | $1.8(X_{tq})^{2}$ | $0.02(\lambda_{tq})^{2}$ | $0.00012$ | $0.004$ | $0.00022$ | $0.00001$
Table 1: Cross sections for the top-Z-jet signal and the SM background
processes before and after passing sequential preliminary cuts at the center
of mass energy of $\sqrt{s}$ = 350 GeV.

In order to select the signal events, we desire to have at least two jets with
$p_{T}\geq 20$ GeV and $|\eta|<2.4$, which at least one of them is required to
be tagged as a b-jet. We require the events with at least three charged
leptons with $p_{T}\geq 10$ GeV and $|\eta|<2.4$. Furthermore, we choose
events with the missing transverse momentum greater than $30$ GeV. The cross
sections including the branching ratios for three signal scenarios and the
corresponding SM backgrounds after passing sequential preliminary cuts at the
center of mass energy of 350 GeV are presented in Table 1. It is worth
mentioning that the reported cross sections for signal and background
processes are not included the QED ISR. Moreover, it’s noteworthy that the
gamma-induced processes or e-gamma processes can be the new sources of
background, however, we expect that by applying our preliminary and secondary
cuts their contribution will be strongly suppressed. So, we don’t consider
them in our analysis.

As it is clear, the cross section of the signal for $tq\gamma$ scenario is
significantly less than two others, so we expect that the extracting upper
limit for this coupling should be looser than two others. Also, the cross
section of the signal for $tqZ(\sigma_{\mu\nu})$ scenario is greater than
$tqZ(\gamma_{\mu})$, because of the presence of the gauge boson’s momentum in
the $tqZ(\sigma_{\mu\nu})$ interaction term. The preliminary cuts efficiencies
for $tqZ(\sigma_{\mu\nu})$, $tqZ(\gamma_{\mu})$, and $tq\gamma$ signal
scenarios are $0.19$, $0.19$, and $0.14$, respectively.

It is notable that, although the production cross sections for some background
processes like $WWjj$ and $t\bar{t}$ are large at parton level, they are
remarkably reduced after applying preliminary cuts, particularly, when we
require exactly three charged leptons at the final state. The preliminary cuts
efficiencies for $WWjj$, $WZjj$, $t\bar{t}$, and $ZZjj$ backgrounds are at the
order of $10^{-5}$, $10^{-2}$, $10^{-4}$, and $10^{-3}$, respectively.
Therefore, our main backgrounds are $WZjj$ and $ZZjj$ channels in this
analysis.

###  3.2 Separation of Signal from Background

For suppressing the contributions of background events, we look at different
kinematic distributions to find suitable secondary cuts. Therefore, we need to
reconstruct the top quark and Z boson of the final state. In order to
reconstruct the Z boson, we require two opposite charged leptons with an
invariant mass between 60 to 120 GeV. Then the top quark will be reconstructed
with the remaining charged lepton, the b-jet and the neutrino. As it is clear
from Table 1 that the signal cross sections for $\kappa_{tq}$ and $X_{tq}$
couplings are larger than the signal cross section for $\lambda_{tq}$, so we
first describe our secondary cuts for $tqZ$ interactions ($\sigma_{\mu\nu}$
and $\gamma_{\mu}$).

Figure 2: The kinematic distributions of the top-Z-jet signal (for
$tqZ(\sigma_{\mu\nu})$ and $tqZ(\gamma_{\mu})$ scenarios) and the main
backgrounds, $WZjj$ and $ZZjj$, at the center of mass energy of $\sqrt{s}=350$
GeV.

Fig. 2 illustrates some kinematic distributions of the top-Z-jet signal and
the main background processes, $WZjj$ and $ZZjj$, after preliminary cuts and
top quark and Z boson reconstruction. Figures upper right and lower right show
the mass and transverse momentum of the reconstructed top quark, respectively.
Figures upper left and lower left illustrate the transverse momentum of the
b-jet and the reconstructed Z boson, respectively.

To separate more background from signal events, we apply further cuts
according to the distributions in Fig. 2. From the top quark’s mass
distribution (upper right), it is clear that the signal has a prominent peak
around the nominal top mass. A window cut on the reconstructed top quark mass
($140<\rm{M}^{\rm{top}}<180$ GeV) causes to suppress backgrounds without a
real top quark in the final state. According to the top quark’s transverse
momentum distribution (lower right), the signal’s peak is around $20$ GeV,
however, the main backgrounds have more spread distributions which extend up
to around $120$ GeV. Therefore, the transverse momentum of the top quark is
required not to be greater than 30 GeV. Moreover, according to the b-jet’s
transverse momentum distribution (upper left), we accept events with
$p^{\rm{b-jet}}_{T}>30$ GeV. Also, we apply a cut on the Z boson’s transverse
momentum distribution (lower left) to alleviate the number of background
events which is required not to be greater than 80 GeV.

Cross-sections (in fb) | Signal | Background
---|---|---
Cuts |  $tqZ$ $(\sigma_{\mu\nu})$ |  $tqZ$ $(\gamma_{\mu})$ | $WZjj$ | $ZZjj$
$p^{\rm{top}}_{T}<30~{}\rm{GeV}$ | $2.17(\kappa_{tq})^{2}$ | $1.54(X_{tq})^{2}$ | $0.001$ | $4.2\times 10^{-6}$
$p^{Z}_{T}<80~{}\rm{GeV}$ | $2.14(\kappa_{tq})^{2}$ | $1.52X_{tq})^{2}$ | $0.0009$ | $4\times 10^{-6}$
$p^{\rm{b-jet}}_{T}>30~{}\rm{GeV}$ | $1.9(\kappa_{tq})^{2}$ | $1.37(X_{tq})^{2}$ | $0.0007$ | $3.1\times 10^{-6}$
$140<\rm{M}^{\rm{top}}<180~{}\rm{GeV}$ | $1.7(\kappa_{tq})^{2}$ | $1.14(X_{tq})^{2}$ | $0.0003$ | $1.7\times 10^{-6}$
Table 2: Cross sections for the top-Z-jet signal (for $tqZ(\sigma_{\mu\nu})$
and $tqZ(\gamma_{\mu})$ scenarios) and the SM main background processes
($WZjj$ and $ZZjj$) after passing sequential secondary cuts at the center of
mass energy of $\sqrt{s}=350$ GeV.

The cross sections for the signal scenarios and the corresponding SM
backgrounds after passing sequential secondary cuts at $\sqrt{s}=350$ GeV are
shown in Table 2. The secondary cuts efficiencies for $tqZ$
$(\sigma_{\mu\nu})$ and $tqZ$ $(\gamma_{\mu})$ signal scenarios are $0.13$ and
$0.12$ while those for $WZjj$ and $ZZjj$ backgrounds are $0.001$ and $0.0002$,
respectively.

The same method is used to find the secondary cuts and final efficiencies for
$tq\gamma$ signal scenario. Form the same kinematic distribution as before, we
extract following cuts: $p^{\rm{top}}_{T}<80~{}\rm{GeV}$,
$p^{Z}_{T}<80~{}\rm{GeV}$, $p^{\rm{b-jet}}_{T}>30~{}\rm{GeV}$, and
$140<\rm{M}^{\rm{top}}<180~{}\rm{GeV}$.Then, we find the corresponding
efficiencies as; $\epsilon_{tq\gamma}=0.084$, $\epsilon_{WZjj}=0.002$, and
$\epsilon_{ZZjj}=0.0004$.

###  3.3 Sensitivity Estimation

In this subsection, we present the upper limits on the FCNC couplings ($tqZ$
and $tq\gamma$) at a confidence level (CL) of $95\%$ by analyzing the single
top quark production in association with a Z boson. To estimate the
sensitivity of the future electron-positron collider to these anomalous
couplings, we employ a Bayesian approach using a flat prior distribution [52,
53]. A Poisson distribution is assumed for the probability of observing
$n_{\mathrm{obs}}$ events,

$L\left(n_{\mathrm{obs}},n_{\mathrm{S}},n_{\mathrm{B}}\right)=\frac{\left(n_{\mathrm{S}}+n_{\mathrm{B}}\right)^{n_{\mathrm{obs}}}}{n_{\mathrm{obs}}!}e^{-\left(n_{\mathrm{S}}+n_{\mathrm{B}}\right)}.$
(5)

Notice that in the above equation, $n_{\mathrm{S}}$ is defined by
$n_{\mathrm{S}}=\epsilon_{\mathrm{S}}\times\mathcal{L}\times\sigma_{\mathrm{S}}$,
that $\epsilon_{\mathrm{S}}$ is the efficiency of the signal after all cuts
and $\mathcal{L}$ is the integrated luminosity. For a given integrated
luminosity and backgrounds cross section (Table 1, first row), the number of
expected background events are given by
$n_{\mathrm{B}}=\epsilon_{\mathrm{B}}\times\mathcal{L}\times\sigma_{\mathrm{B}}$,
which $\epsilon_{\mathrm{B}}$ is the efficiency of the background processes
after all the preliminary and secondary cuts. The upper limit on the number of
signal process of $e^{-}e^{+}\rightarrow tZj$ can be obtained from the
following equation at $95\%$ CL:

$\frac{95}{100}=\frac{\int_{0}^{n_{\text{limit
}}}L\left(n_{\mathrm{obs}},n_{\mathrm{S}},n_{\mathrm{B}}\right)\mathrm{d}n_{\mathrm{S}}}{\int_{0}^{\infty}L\left(n_{\mathrm{obs}},n_{\mathrm{S}},n_{\mathrm{B}}\right)\mathrm{d}n_{\mathrm{S}}}.$
(6)

We assume that the observed number of events at the detectors are consistent
with the expected number of background events and solve above equation to find
the upper limit on the number of signal events. This limit can be translated
to the upper limit on the signal cross section and then to the upper limits on
the FCNC couplings.

Integrated luminosity | Coupling | Branching ratio
---|---|---
| $\kappa_{tq}<0.07$ | $\rm{Br}(t\rightarrow qZ)\left(\sigma_{\mu\nu}\right)<1.8\times 10^{-3}$
$300~{}\mathrm{fb}^{-1}$ | $X_{tq}<0.09$ | $\rm{Br}(t\rightarrow qZ)\left(\gamma_{\mu}\right)<3.8\times 10^{-3}$
| $\lambda_{tq}<0.91$ | $\rm{Br}(t\rightarrow q\gamma)<3.5\times 10^{-1}$
| $\kappa_{tq}<0.04$ | $\rm{Br}(t\rightarrow qZ)\left(\sigma_{\mu\nu}\right)<5.8\times 10^{-4}$
$1~{}\mathrm{ab}^{-1}$ | $X_{tq}<0.05$ | $\rm{Br}(t\rightarrow qZ)\left(\gamma_{\mu}\right)<1.2\times 10^{-3}$
| $\lambda_{tq}<0.50$ | $\rm{Br}(t\rightarrow q\gamma)<1.1\times 10^{-1}$
| $\kappa_{tq}<0.028$ | $\rm{Br}(t\rightarrow qZ)\left(\sigma_{\mu\nu}\right)<2.8\times 10^{-4}$
$3~{}\mathrm{ab}^{-1}$ | $X_{tq}<0.03$ | $\rm{Br}(t\rightarrow qZ)\left(\gamma_{\mu}\right)<4.2\times 10^{-4}$
| $\lambda_{tq}<0.33$ | $\rm{Br}(t\rightarrow q\gamma)<4.6\times 10^{-2}$
Table 3: The upper limits and the corresponding branching ratios at 95$\%$ CL
on the $tqZ$ ($\sigma_{\mu\nu}$ and $\gamma_{\mu}$) and $tq\gamma$ at the
center of mass energies $\sqrt{s}=350$ GeV for different integrated luminosity
of $300~{}\rm{fb}^{-1}$, $1~{}\rm{ab}^{-1}$ and $3~{}\rm{ab}^{-1}$ by studying
the top-Z-jet signal.

Table 3 presents the upper limits at 95$\%$ CL on the $tqZ$ ($\sigma_{\mu\nu}$
and $\gamma_{\mu}$) and $tq\gamma$ couplings at the center of mass energies
$350$ GeV for different integrated luminosity of $300~{}\rm{fb}^{-1}$,
$1~{}\rm{ab}^{-1}$ and $3~{}\rm{ab}^{-1}$ by studying the top-Z-jet channel.
As we expected before, the upper limit on the $\lambda_{tq}$ is looser than
other anomalous couplings. Then, these limits are translated to the
corresponding branching which are shown in the table 3 for different
integrated luminosities of data. It is worth mentioning that these upper
limits on the anomalous coupling can be translated to the bounds on
$\Lambda/\sqrt{C_{i}}$ by using Eq. 4. According to our upper limits on the
anomalous coupling, we can roughly say $\Lambda/\sqrt{C_{i}}$ should be
greater than $\sim 1.5$ TeV.

##  4 Analysis Strategy for top-$\gamma$-jet signal

In this section, we analyze the single top quark production in association
with a photon process to search for the $tq\gamma$ and $tqZ$ FCNC couplings at
the future lepton collider. First of all, we describe event generation and
simulation for the signal and background processes. Then, we look at some
kinematic distribution to separate the background events from the signal
events. Finally, the upper limits which can be obtained from the
top-$\gamma$-jet signal are given.

###  4.1 Event Generation and Simulation

Here, we define the top-$\gamma$-jet signal and related SM background
processes and describe the event generation and simulation method. In this
analysis, we focus on the electron-positron collisions at the center of mass
energy of $\sqrt{s}=350$ GeV. The signal is defined as a single top production
in association with a photon and a light jet ($e^{-}~{}e^{+}\rightarrow
t/\bar{t}~{}\gamma~{}j$), that top quark decays leptonically. Consequently,
the final state includes a charged leptons (electron or muon), a hard photon,
a b-jet, a light jet, and the missing energy due to the neutrino. Since the
photon in the final state is a hard photon, we can handle it to suppress
background contributions. Some related Feynman diagrams for top-$\gamma$-jet
signal are demonstrated in Fig. 1 ($d$, $e$ and $f$). Since our desirable
signal (top-$\gamma$-jet) is raised up by all the $tqZ$ and $tq\gamma$ FCNC
interactions, again we employ three different scenarios to generate signal
events; 1) $\lambda_{tq}^{R}=\lambda_{tq}^{L}=\lambda_{tq}\neq 0$, 2)
$\kappa_{tq}^{R}=\kappa_{tq}^{L}=\kappa_{tq}\neq 0$, and 3)
$X_{tq}^{R}=X_{tq}^{L}=X_{tq}\neq 0$.

Cross-sections (in fb) | Signal | Background
---|---|---
Cuts | $tq\gamma$ | $tqZ(\sigma_{\mu\nu})$ | $tqZ(\gamma_{\mu})$ | $Wjj\gamma$ | $Zjj\gamma$ | $t\bar{t}\gamma$
Preselection cuts | $182(\lambda_{tq})^{2}$ | $230.6(\kappa_{tq})^{2}$ | $29.3(X_{tq})^{2}$ | $84$ | $2.9$ | $0.0057$
$1~{}\rm{light~{}jet}+1\rm{b~{}jet}$ | $98.16(\lambda_{tq})^{2}$ | $125.9(\kappa_{tq})^{2}$ | $15.57(X_{tq})^{2}$ | $4.9$ | $0.46$ | $0.0045$
$1~{}\rm{Photon}$ | $47.83(\lambda_{tq})^{2}$ | $33.3(\kappa_{tq})^{2}$ | $4.5(X_{tq})^{2}$ | $1.54$ | $0.15$ | $3\times 10^{-5}$
$1~{}\rm{lepton}$ | $38.26(\lambda_{tq})^{2}$ | $26.34(\kappa_{tq})^{2}$ | $3.6(X_{tq})^{2}$ | $1.2$ | $0.06$ | $3\times 10^{-6}$
$\not{p}_{T}>30~{}\rm{GeV}$ | $28.4(\lambda_{tq})^{2}$ | $19.5(\kappa_{tq})^{2}$ | $2.6X_{tq})^{2}$ | $0.8$ | $0.002$ | $2\times 10^{-6}$
Table 4: Cross sections for the top-$\gamma$-jet signal and the SM background
processes before and after passing sequential preliminary cuts at center of
mass energy of $\sqrt{s}$ = 350 GeV.

According to the final state, we consider the following background processes;
$W^{\pm}jj\gamma$, $Zjj\gamma$, and $t\bar{t}\gamma$. The W-boson and Z-boson
decay leptonically in all the above backgrounds, however, for $t\bar{t}\gamma$
semi-leptonic decay is assumed. In all cases, just decay to electron and muon
are taken into account. To generate and simulate the signal and background
events, we follow the procedure which explained in subsection 3.1.

In order to select the events, at least two jets with $p_{T}\geq 20$ GeV and
$|\eta|<2.4$ are required, which at least one of them should be tagged as a
b-jet. Events including at least one isolated photon with $p_{T}\geq 30$ GeV
and $|\eta|<2.4$ are selected. Furthermore, we require the events with exactly
one charged leptons with $p_{T}\geq 10$ GeV and $|\eta|<2.4$. Also, we choose
events with missing transverse momentum greater than $30$ GeV.

The cross sections including the branching ratios for three signal scenarios
and the corresponding SM backgrounds after passing sequential preliminary cuts
at $\sqrt{s}$ = 350 GeV are presented in Table 4. The preliminary cuts
efficiencies for $tq\gamma$, $tqZ(\sigma_{\mu\nu})$, and $tqZ(\gamma_{\mu})$
signal scenarios are $0.15$, $0.08$, and $0.09$, respectively. The primary
cuts efficiencies for $Wjj\gamma$, $Zjj\gamma$, and $t\bar{t}\gamma$
backgrounds are at the order of $10^{-2}$, $10^{-4}$, and $10^{-4}$,
respectively. It is notable that these cuts can reduce significantly the
number of background events.

It is worth mentioning that there is another source for background for
top-$\gamma$-jet channel. Processes like $W^{\pm}jj$, $Zjj$, and $t\bar{t}$
which don’t have any real photon can mimic the signal since their jets would
be misidentified as photons in the detector. The production cross sections for
these background processes are large with respect to the other considered
backgrounds, however, after above requirements, their efficiencies remarkably
reduce. As a result, here we neglect this source of background nevertheless in
a more realistic detector simulation their contribution should be taken into
account.

###  4.2 Separation of Signal over Background

In order to alleviate the number of the main SM background events, we follow
the same way as described in subsection 3.2. Therefore, we focus on some
kinematic distributions to find the suitable secondary cuts. Again at the
first step, we need to reconstruct the top quark of the final state. Since we
have a clean final state, it is straightforward to reconstruct the top quark.
The top quark will be reconstructed by combining the charged lepton, the high
energetic b-jet and the neutrino. We show the kinematic distributions and
describe our obtained secondary cuts and limits for $\lambda_{tq}$ anomalous
coupling. Then, we will follow the same procedure to find the upper limits on
the $\kappa_{tq}$ and $X_{tq}$ FCNC couplings.

Figure 3: The kinematic distributions of the top-$\gamma$-jet signal (for
$tq\gamma$ scenario) and the main backgrounds; $Wjj\gamma$ and $Zjj\gamma$ at
the center of mass energy of $\sqrt{s}=350$ GeV.

Fig. 3 demonstrates different kinematic distributions of the top-$\gamma$-jet
signal and some SM background processes ($Wjj\gamma$ and $Zjj\gamma$) after
preselection cuts and top quark reconstruction. Figures upper right and lower
right demonstrate mass and the transverse momentum of the reconstructed top
quark, respectively. Figures upper left and lower left illustrate the
transverse momentum of the light-jet and the photon, respectively.

For suppressing more background events, we apply further cuts according to
distributions in Fig. 3. The top quark mass distribution (upper right) for the
signal shows a clear peak around $170$ GeV. Therefore, we applied a window cut
on the reconstructed top quark mass ($140<\rm{M}^{\rm{top}}<180$ GeV). From
the top quark transverse momentum distribution (lower right), the signal’s
peak is around $20$ GeV, however, the main backgrounds have more spread
distributions which extended up to around $160$ GeV. So, the transverse
momentum of the top quark is required not to be greater than 40 GeV.

Cross-sections (in fb) | Signal | Background
---|---|---
Cuts | $tq\gamma$ | $Wjj\gamma$ | $Zjj\gamma$ | $t\bar{t}\gamma$
$p^{\rm{top}}_{T}<40~{}\rm{GeV}$ | $18.4(\lambda_{tq})^{2}$ | $0.2$ | $5.7\times 10^{-4}$ | $1\times 10^{-6}$
$p^{\gamma}_{T}>40~{}\rm{GeV}$ | $17(\lambda_{tq})^{2}$ | $0.13$ | $3.3\times 10^{-4}$ | $4.6\times 10^{-7}$
$p^{\rm{light-jet}}_{T}>40~{}\rm{GeV}$ | $15(\lambda_{tq})^{2}$ | $0.07$ | $1.8\times 10^{-4}$ | $2.6\times 10^{-7}$
$140<\rm{M}^{\rm{top}}<180~{}\rm{GeV}$ | $12.6(\lambda_{tq})^{2}$ | $0.02$ | $2.3\times 10^{-5}$ | $2\times 10^{-7}$
Table 5: Cross sections for the top-$\gamma$-jet signal (for $tq\gamma$
scenario) and the SM background processes after passing sequential secondary
cuts at the center of mass energy of $\sqrt{s}=350$ GeV.

Furthermore, according to the light-jet’s transverse momentum distribution
(upper left), we desire events with $p^{\rm{light-jet}}_{T}>40$ GeV. Also, we
apply a cut on the hard photon transverse momentum distribution (lower left)
to reduce more number of background events which is required to be greater
than 40 GeV. In addition, according to the jet and b-jet multiplicities, we
choose events with number of jets and b-jet less than 3 and 2, respectively.
This requirement can suppress a large number of background events with a high
number of jets.

The cross sections for the $tq\gamma$ signal scenarios and the corresponding
SM backgrounds after passing sequential secondary cuts are shown in Table 5.
The final efficiencies for the $tq\gamma$ signal scenario is $0.07$ while
those for $Wjj\gamma$, $Zjj\gamma$, and $t\bar{t}\gamma$ backgrounds are at
the order of $10^{-4}$, $10^{-6}$, and $10^{-5}$, respectively. The same
method is used to find the secondary cuts and final efficiencies for
$tqZ(\sigma_{\mu\nu})$ and $tqZ(\gamma_{\mu})$ signal scenarios. In the
following, these final efficiencies are used to estimate the upper limits on
the anomalous couplings.

###  4.3 Sensitivity Estimation

In this part, the obtained upper limits on the FCNC couplings ($tqZ$ and
$tq\gamma$) at $95\%$ CL by analyzing the single top quark production in
association with photon are presented. Again, we follow the same procedure
which explained in subsection 3.3.

Integrated luminosity | Coupling | Branching ratio
---|---|---
| $\kappa_{tq}<0.065$ | $\rm{Br}(t\rightarrow qZ)\left(\sigma_{\mu\nu}\right)<1.5\times 10^{-3}$
$300~{}\mathrm{fb}^{-1}$ | $X_{tq}<0.15$ | $\rm{Br}(t\rightarrow qZ)\left(\gamma_{\mu}\right)<1.1\times 10^{-2}$
| $\lambda_{tq}<0.04$ | $\rm{Br}(t\rightarrow q\gamma)<6.8\times 10^{-4}$
| $\kappa_{tq}<0.047$ | $\rm{Br}(t\rightarrow qZ)\left(\sigma_{\mu\nu}\right)<8\times 10^{-4}$
$1~{}\mathrm{ab}^{-1}$ | $X_{tq}<0.11$ | $\rm{Br}(t\rightarrow qZ)\left(\gamma_{\mu}\right)<5.7\times 10^{-3}$
| $\lambda_{tq}<0.03$ | $\rm{Br}(t\rightarrow q\gamma)<3.87\times 10^{-4}$
| $\kappa_{tq}<0.035$ | $\rm{Br}(t\rightarrow qZ)\left(\sigma_{\mu\nu}\right)<4.5\times 10^{-4}$
$3~{}\mathrm{ab}^{-1}$ | $X_{tq}<0.085$ | $\rm{Br}(t\rightarrow qZ)\left(\gamma_{\mu}\right)<3.4\times 10^{-3}$
| $\lambda_{tq}<0.02$ | $\rm{Br}(t\rightarrow q\gamma)<1.7\times 10^{-4}$
Table 6: The upper limits and the corresponding branching ratios at 95$\%$ CL
on the $tqZ$ ($\sigma_{\mu\nu}$ and $\gamma_{\mu}$) and $tq\gamma$ at the
center of mass energies $\sqrt{s}=350$ GeV for different integrated luminosity
of $300~{}\rm{fb}^{-1}$, $1~{}\rm{ab}^{-1}$ and $3~{}\rm{ab}^{-1}$ by studying
the top-$\gamma$-jet signal.

Table 6 shows the upper limits and the corresponding branching ratios at
95$\%$ CL on the $tqZ$ ($\sigma_{\mu\nu}$ and $\gamma_{\mu}$) and $tq\gamma$
at the center of mass energy of 350 GeV for different integrated luminosities
of 300 $\rm{fb}^{-1}$, 1 $\rm{ab}^{-1}$ and 3 $\rm{ab}^{-1}$ using
top-$\gamma$-jet signal. Also, we do the same study on top-$\gamma$-jet
channel at the center of mass energy of 240 GeV for the integrated luminosity
of $10~{}\rm{ab}^{-1}$ and find upper limits at $95\%$ CL on the branching
ratio as follows; $\rm{Br}(t\rightarrow q\gamma)<2.2\times 10^{-2}$. It is
worth mentioning that the cross section of $t\bar{t}$ at the threshold is too
small by a factor of three. So we consider the threshold resummation effect on
the $t\bar{t}$ cross section, however, these effects change our result by less
than 5%.

##  5 Summary

Flavour Changing Neutral Current (FCNC) interactions in the top quark sector
are highly suppressed in the Standard Model (SM) framework. The predicted
branching ratios for different top FCNC decay modes are of
$\mathcal{O}(10^{-14})$ in the SM, while there are some Beyond SM models that
anticipate large values for these branching ratios. Therefore, any sign of top
quark FCNC interaction would be a promising indication of the existence of new
physics. On the other hand, future electron-positron colliders with high
luminosity are able to produce large numbers of top quark and give us this
opportunity to probe top quark properties with high precision.

In this paper, we study the single top quark production in association with a
neutral gauge boson (Z boson or photon) to search for $tqZ$ and $tq\gamma$
anomalous couplings at the future electron-positron collider. Since many BSMs
predict the FCNC interactions with the same signatures at particle colliders
in many cases, using the model independent approach is more useful. So, we
employ the Standard Model Effective Field Theory (SMEFT) approach in this
study. We perform two separate analyses for top-Z-jet and top-$\gamma$-jet
channels.

In the top-Z-jet study, we consider leptonic decay of top quark and Z boson.
As a result, we have three charged leptons (electron or muon), a light-jet, a
b-jet and missing energy in the final state. Since all the FCNC couplings
$tqZ(\sigma_{\mu\nu})$, $tqZ(\gamma_{\mu})$ and $tq\gamma$ can produce our
favourable signal, we categorize our signal in three scenarios. We consider
all the main SM background processes for this channel. Also, the parton
showering and hadronization are taken into account as well as fast detector
simulation. Then, we use a cut-and-count technique to separate the signal from
the background events. The upper limits at $95\%$ CL are obtained on the
anomalous couplings and the corresponding branching ratios for different the
integrated luminosity at the center of mass energy of $350$ GeV. For example,
it is found at the $95\%$ CL with $3~{}\rm{ab}^{-1}$ luminosity of data;
$\rm{Br}(t\rightarrow qZ)\left(\sigma_{\mu\nu}\right)<2.8\times 10^{-4}$,
$\rm{Br}(t\rightarrow qZ)\left(\gamma_{\mu}\right)<4.2\times 10^{-4}$, and
$\rm{Br}(t\rightarrow q\gamma)<4.6\times 10^{-2}$.

In the top-$\gamma$-jet study, leptonically decay of top quark is assumed. So,
the final state consists of one charged leptons (electron or muon), a hard
photon, a light-jet, a b-jet and the missing energy. The upper limits at
$95\%$ CL are obtained for $3~{}\rm{ab}^{-1}$ luminosity of data at the center
of mass energy $350$ GeV are as follows; $\rm{Br}(t\rightarrow
qZ)\left(\sigma_{\mu\nu}\right)<4.5\times 10^{-4}$, $\rm{Br}(t\rightarrow
qZ)\left(\gamma_{\mu}\right)<3.4\times 10^{-3}$, and $\rm{Br}(t\rightarrow
q\gamma)<1.7\times 10^{-4}$. Also, it is worth mentioning here that by using
polarized beams the obtained upper limits can be improved [54]. Our obtained
results would be improved significantly by using the statistical techniques
for separating more signal from background events such as multivariate
analysis. The combination of the obtained upper limits from the top quark
production associated with a neutral gauge boson with other channels also can
lead to stringent bounds on the FCNC couplings($tqZ$ ana $tq\gamma$),
particularly tighter limits would be expected from combination with limits
coming from the top+jet final state.

## Acknowledgments

Authors would like to especially thank Mojtaba Mohammadi Najafabadi and Hamzeh
Khanpour for many useful discussions and their invaluable comments on the
draft. Authors are also thankful School of Particles and Accelerators,
Institute for Research in Fundamental Sciences (IPM) for hospitality.

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