∎

e1e-mail<EMAIL_ADDRESS>e2e-mail<EMAIL_ADDRESS>e3e-mail:
<EMAIL_ADDRESS>e4e-mail<EMAIL_ADDRESS>
11institutetext: Department of Physics, University of Delhi, Delhi, India.
22institutetext: School of Physics and Institute for Collider Particle
Physics, University of the Witwatersrand, Johannesburg, Wits 2050, South
Africa. 33institutetext: School of Basic Sciences, Indian Institute of
Information Technology Una, Una, Himachal Pradesh 177 209, India.
44institutetext: Regional Centre for Accelerator-based Particle Physics,
Harish-Chandra Research Institute, A CI of Homi Bhabha National Institute,
Chhatnag Road, Jhunsi, Prayagraj 211 019, India.

# Exploring dark $Z_{d}$-boson in future Large Hadron-electron collider

Ashok Goyale1,addr1  Mukesh Kumare2,addr2  Satendra Kumare3,addr3  Rafiqul
Rahamane4,addr4

(Received: date / Accepted: date)

###### Abstract

The interaction between the dark $U(1)_{d}$ sector with the visible Standard
Model (SM) sector takes place through the kinetic mixing between the dark
photon $U(1)_{d}$ field $Z_{d}^{\mu}$ and the SM $U(1)_{Y}$ gauge field
$B_{\mu}$. After the electroweak and $U(1)_{d}$ symmetry breaking, the dark
photon $Z_{d}^{\mu}$ acquires a mass and mixes with the SM neutral vector
boson $Z_{\mu}$. This mixing leads to parity-violating coupling between the
$Z_{d}^{\mu}$ and SM. The coupling between the dark photon and SM can be
explored in low energy phenomenology as well as in collider experiments. The
Lorentz structure of dark photon interaction with SM fermions is explored in
the proposed high energy future Large Hadron-electron collider, which would
provide efficient energy and a clean environment using cross-section and
asymmetries associated with polarisation observable of the dark photon in
leptons decay. A $\chi^{2}$-analysis is performed to compare the strength of
various variables for both the charge- and neutral-current processes. Based on
this analysis, $90\%$ confidence level (C.L.) contours in the
$\epsilon$-$m_{Z_{d}}$ and $\epsilon$-$g_{V}$ plane are obtained to put limits
on the $Z_{d}^{\mu}$ mass up to $100$ GeV, coupling strength $\epsilon$ and on
the Lorentz structure of dark photon coupling with the SM fermions ($g_{V}$)
at $\sqrt{s}\approx 1.3$ TeV.

##  1 Introduction

The existence of dark matter (DM) in the universe has been established by
several cosmological and astrophysical observations. Dark matter contributes
roughly $75\%$ of the entire matter existing in the universe. The nature of
the DM has, however, remained undetermined so far. The negative results of the
direct searches Arcadi:2017kky  have reached a sensitivity level where they
are in tension with the assumption that the DM has at least some common charge
under the SM gauge interactions. This has led to consider a scenario in which
there is a separate “dark sector” consisting of one or more particles that are
not charged under SM gauge group. The dark and visible SM sectors interact
through a portal - as it is commonly called. The portal consists of operators
of different forms depending upon the spin of the mediator. One can have a
scalar portal with Higgs boson, a pseudo scalar axion-like particle mediator,
a fermionic heavy (sterile) neutrino mediator, or a vector mediator. These
have been extensively studied, see for example Dark Sector 2016 Workshop
Community Report Alexander:2016aln  and Refs. Batell:2017cmf ; Blennow:2019fhy
; Fabbrichesi:2020wbt . In the case when the only interaction available to the
dark-sector particles with the visible sector is gravitational, the mediator
is a spin-2 graviton as in Kaluza-Klien (KK) warped extra-dimension models
Randall:1999ee ; Kraml:2017atm ; Lee:2013bua ; Lee:2014caa ; Rueter:2017nbk ;
Goyal:2019vsw .

In the vector-portal, the interaction between the dark and visible sectors
takes place through the kinetic mixing between the dark $U(1)_{d}$ and SM
$U(1)_{Y}$ gauge fields $Z_{d}^{\mu}$ which is called “dark photon" and
$B^{\mu}$, respectively. The kinetic mixing is given by Davoudiasl:2012ag

${\mathcal{L}}_{\rm
guage}=-\frac{1}{4}B_{\mu\nu}B^{\mu\nu}-\frac{1}{4}{Z_{d}}_{\mu\nu}Z^{\mu\nu}_{d}+\frac{1}{2}\frac{\epsilon}{\cos\theta_{W}}B_{\mu\nu}Z^{\mu\nu}_{d}$
(1)

with
$V_{\mu\nu}=\partial_{\mu}V_{\nu}-\partial_{\nu}V_{\mu},\leavevmode\nobreak\
V=Z_{d}/B$. A redefinition of fields leads to an effective parity conserving
induced coupling between the dark photon $Z_{d}$ and the electromagnetic
current as

${\mathcal{L}}_{J_{\mu}}=-\epsilon\,e\,J_{\mu}^{em}Z_{d}^{\mu}.$ (2)

The parameter $\epsilon$ is assumed to be small $\epsilon<{\cal
O}(10^{-3})-{\cal O}(10^{-1})$ but theoretically can be of order 1 and is not
required to be small. It can be calculated in some beyond the SM (BSM) models
to lie between ${\cal O}(10^{-12})<\epsilon<{\cal O}(10^{-3})$, see Refs.
Abel:2006qt ; Dienes:1996zr . After the electroweak symmetry breaking,
$Z_{\mu}$ obtains a nonzero mass $m_{Z}$ and depending on how $U(1)_{d}$
symmetry is broken, $Z_{d}^{\mu}$ will receive a nonzero mass $m_{Z_{d}}$ and
will cause mixing with the SM neutral vector boson $Z_{\mu}$. The $U(1)_{d}$
symmetry can be broken from a spontaneous symmetry breaking term through an SM
singlet scalar with a non-trivial $U(1)_{d}$ charge. Alternatively, $U(1)_{d}$
gauge boson $Z_{d}^{\mu}$ may acquire mass by means of the Stuckelberg
mechanism Battaglia:2003gb ; Gopalakrishna:2008dv ; Hewett:1988xc . The dark
$Z_{d}$ and SM $Z$ boson can now mix to give mass eigenstates leading to
parity-violating $Z_{d}$ coupling given by

${\mathcal{L}}_{\rm
int}=-\left(\epsilon\,e\,J_{\mu}^{em}+\epsilon_{Z}\frac{g}{\cos\theta_{W}}J_{\mu}^{\rm
NC}\right)Z_{d}^{\mu},$ (3)

where $\epsilon_{Z}$ depends on the $Z-Z_{d}$ mass mixing term
Davoudiasl:2012ag ; Davoudiasl:2013aya . The dark photon ($Z_{d}$)
interactions with the SM particles can be explored in low energy phenomenology
as well as in collider experiments. Dark photons of mass up to several MeV are
primarily constrained by the existing bounds from cosmology, astrophysics, and
accelerator (Beam-dump) experiments (see Dark sectors 2016 Workshops Report
Alexander:2016aln ). Depending on the mass of the dark photon, several
channels are available for the decay. For masses up to 100 GeV, the dark
photon can decay into a pair of SM fermions. Dark photon of mass of $1$ GeV
has been explored in Large Hadron-electron collider (LHeC) and Future Circular
Hadron electron collider (FCC-he) colliders where the signal is given by the
displaced decay of long-lived photon into the charged fermions. The non-
observation of the signal can exclude dark photons of mass $<1$ GeV
DOnofrio:2019dcp . There are also few studies on dark photon of mass $\sim 10$
GeV where LHCb provides stringent constraints on the kinetic mixing parameters
$\epsilon$. For dark photon mass $20\leq m_{Z_{d}}\leq 330$ GeV, the 2
$\sigma$ exclusion limits on the mixing parameter
$\epsilon\leq[10^{-3}-10^{-2}]$ can be explored at future $e^{+}e^{-}$
colliders He:2017zzr . The Lorentz structure of $Z_{d}^{\mu}\ell^{+}\ell^{-}$
interaction has been probed recently Lee:2020tpn  by studying the correlation
between $Z_{d}^{\mu}$ angle relative to $e^{+}e^{-}$ beam direction in the
$e^{+}e^{-}$ rest frame and the $\mu^{-}$ angle relative to the boost
direction of $Z_{d}^{\mu}$ in the $Z_{d}^{\mu}$ rest-frame based in Belle II
detector. It was shown that the detection of dark $Z_{d}^{\mu}$ decay into
muon-pairs in $e^{+}e^{-}$ colliders can probe the parity-violating couplings.

In the next sect. 2, a formalism for the analysis is discussed with
definitions and observable. Simulation and observable for the study are
presented in sect. 3. Sect. 4 includes the analysis details with the results
of this study. Summary and discussions are followed in sect. 5.

##  2 Framework

In this study, we explore the Lorentz structure and the magnitude of parity
violation of dark $Z_{d}$ coupling to the SM fermions. We consider a generic
structure:

${\mathcal{L}}_{Z_{d}f\bar{f}}=-\epsilon\,e\,\bar{f}\left(g_{V}\gamma_{\mu}+g_{A}\gamma_{\mu}\gamma_{5}\right)fZ_{d}^{\mu},$
(4)

for the dark $Z_{d}$ interaction with the SM fermions, where
$g_{V}^{2}+g_{A}^{2}=1$. We base our study on LHeC environment, which employs
the $E_{p}=7$ TeV proton beam of the Large Hadron Collider (LHC) and electrons
from an Energy Recovery Linac (ERL) being developed for the LHeC. The choice
of an ERL energy of $E_{e}=60$ (120) GeV with an available proton energy
$E_{p}=7$ TeV would provide a centre of mass energy of $\sqrt{s}\approx
1.3\,(1.8)$ TeV at the LHeC using the LHC protons AbelleiraFernandez:2012cc ;
Bruening:2013bga ; AbelleiraFernandez:2012ty . Since $Z_{d}$ couples to SM
particles via the interactions in Lagrangian (4), the production of $Z_{d}$ in
$e^{-}p$ collider follows through charged (neutral (NC)) currents (CC):
$e^{-}p\to\nu_{e}(e^{-})Z_{d}\,j$. To probe the mass range of $Z_{d}$ as a
function of $\epsilon$ in this setup, we use cross-section and asymmetries
associated with polarisation observables Boudjema:2009fz ; Rahaman:2016pqj  of
$Z_{d}$ in the decay $Z_{d}\to\ell^{+}\ell^{-}$.

For a spin-$s$ particle one can construct a total of $(2s+1)^{2}-1=4s(s+1)$
polarisation observables. In our case, it should be eight such combinations
since $s=1$, which are related to the angular distributions of its daughter
($\ell^{-}$) in its rest frame. The normalised decay angular distribution of
the lepton $\ell^{-}$ from the $Z_{d}$ in the rest frame of $Z_{d}$ is given
by Boudjema:2009fz ,

$\displaystyle\frac{1}{\sigma}\frac{d\sigma}{d\Omega_{\ell^{-}}^{\star}}=\frac{3}{8\pi}\left[\left(\frac{2}{3}-(1-3\delta)\
\frac{T_{zz}}{\sqrt{6}}\right)+\alpha\
p_{z}\cos\theta_{\ell^{-}}^{\star}\right.$
$\displaystyle+\sqrt{\frac{3}{2}}(1-3\delta)\
T_{zz}\cos^{2}\theta_{\ell^{-}}^{\star}$ $\displaystyle+\left(\alpha\
p_{x}+2\sqrt{\frac{2}{3}}(1-3\delta)\
T_{xz}\cos\theta_{\ell^{-}}^{\star}\right)\sin\theta_{l^{-}}^{\star}\
\cos\phi_{\ell^{-}}^{\star}$ $\displaystyle+\left(\alpha\
p_{y}+2\sqrt{\frac{2}{3}}(1-3\delta)\
T_{yz}\cos\theta_{\ell^{-}}^{\star}\right)\sin\theta_{\ell^{-}}^{\star}\
\sin\phi_{l^{-}}^{\star}$
$\displaystyle+(1-3\delta)\left(\frac{T_{xx}-T_{yy}}{\sqrt{6}}\right)\sin^{2}\theta_{\ell^{-}}^{\star}\cos
2\phi_{\ell^{-}}^{\star}$ $\displaystyle+\left.\sqrt{\frac{2}{3}}(1-3\delta)\
T_{xy}\ \sin^{2}\theta_{\ell^{-}}^{\star}\ \sin
2\phi_{\ell^{-}}^{\star}\right].$ (5)

Here $\theta_{\ell^{-}}^{\star}$, $\phi_{\ell^{-}}^{\star}$ are the polar and
the azimuthal orientation of $\ell^{-}$ from the spin-$1$ particle in the rest
frame of the particle ($Z_{d}$) with its momentum along the $z$-direction. The
quantity $\alpha$ is called analysing power, and it is related to the decay
vertex structure of the particle. The quantity $\delta=0$ for massless
leptonic decay. The vector polarisations $\vec{p}$ and independent tensor
polarisations $T_{ij}$ are calculable from the asymmetries constructed from
the decay angular distributions above.

The asymmetries for the polarisations are constructed as follows
Rahaman:2016pqj :

$\displaystyle A_{i}=\frac{\sigma\left({\cal N}_{i}>0\right)-\sigma\left({\cal
N}_{i}<0\right)}{\sigma\left({\cal N}_{i}>0\right)+\sigma\left({\cal
N}_{i}<0\right)},$ (6)

where $A_{i}$ and the corresponding observable ${\cal N}_{i}$ are shown in
Table 1.

|  Asymmetry ($A_{i})$ |  Observable (${\cal N}_{i}$)
---|---|---
1 | $A_{x}=\frac{3}{4}\alpha p_{x}$ | ${\cal N}_{x}=\cos\phi_{\ell^{-}}^{\star}$
2 | $A_{y}=\frac{3}{4}\alpha p_{y}$ | ${\cal N}_{y}=\sin\phi_{\ell^{-}}^{\star}$
3 | $A_{z}=\frac{3}{4}\alpha p_{z}$ | ${\cal N}_{z}=\cos\theta_{\ell^{-}}^{\star}$
4 | $A_{x^{2}-y^{2}}=\frac{1}{\pi}\sqrt{\frac{2}{3}}(1-3\delta)(T_{xx}-T_{yy})$ | ${\cal N}_{x^{2}-y^{2}}=\cos 2\phi_{\ell^{-}}^{\star}$
5 | $A_{xy}=\frac{2}{\pi}\sqrt{\frac{2}{3}}(1-3\delta)T_{xy}$ | ${\cal N}_{xy}=\sin 2\phi_{\ell^{-}}^{\star}$
6 | $A_{xz}=\frac{2}{\pi}\sqrt{\frac{2}{3}}(1-3\delta)T_{xz}$ | ${\cal N}_{xz}=\cos\theta\cos\phi_{\ell^{-}}^{\star}$
7 | $A_{yz}=\frac{2}{\pi}\sqrt{\frac{2}{3}}(1-3\delta)T_{yz}$ | ${\cal N}_{yz}=\cos\theta\sin\phi_{\ell^{-}}^{\star}$
8 | $A_{zz}=\frac{3}{8}\sqrt{\frac{3}{2}}(1-3\delta)T_{zz}$ | ${\cal N}_{zz}=\sin 3\theta_{\ell^{-}}^{\star}$
Table 1: The expression for asymmetry and corresponding observable defined in
eq. (6).

In addition to these polarization asymmetries, $A_{pol}$, other asymmetries
can be defined in lab-frame, $A_{Lab}$, with all possible observable. This
study considered $\cos\theta_{j}$, $\Delta\Phi\left(\nu_{e},j\right)$,
$\Delta\Phi\left(\nu_{e},Z_{d}\right)$, and $\Delta\Phi\left(j,Z_{d}\right)$
for CC process. Similarly for NC process observable with $e^{-}$ such as
$\cos\theta_{e^{-}}$, $\Delta\Phi\left({e^{-}},j\right)$, and
$\Delta\Phi\left({e^{-}},Z_{d}\right)$ in addition with $\cos\theta_{j}$,
$\Delta\Phi\left(j,Z_{d}\right)$ are considered. Here $j$ is the scattered
jet, and $\Delta\Phi(m,n)$ is the azimuthal-angle difference between ($m,n$).
Further, a robust $\chi^{2}$ analysis is performed to find the limits on
$\epsilon(m_{Z_{d}})$ using these observable and discussed in the next
section.

Figure 1:  Leading order representative Feynman diagrams at matrix-element
level for single dark $Z_{d}$ production in CC (through $W^{\pm}$-mediator)
and NC (through $Z$ and $\gamma$-mediators) processes through deep inelastic
electron-proton collisions. The dark-$Z_{d}$ can radiate from either initial
beams (vertex 1 or 2) or final scattered fermions (vertex 3 or 4). Here,
$q,q^{\prime}$ $\equiv$ $u,\bar{u}$, $d,\bar{d}$, $c,\bar{c}$, $s,\bar{s}$, or
$b,\bar{b}$.

(a)

(b)

Figure 2:  The production cross-section of $Z_{d}$ in CC and NC channels including leptonic decay of $Z_{d}$ as a function of (a) $m_{Z_{d}}$ for three choices of coupling $g_{V}=0.0$, $0.4$ and $0.7$, (b) $g_{V}$ for $m_{Z_{d}}=10$, $50$ and $100$ GeV at $e^{-}p$ collider with $E_{e}=60$ GeV and $E_{p}=7$ TeV, respectively. $m_{Z_{d}}$ (GeV) |  $m_{\ell\ell}\in[m_{\ell\ell}^{min},m_{\ell\ell}^{max}]$ (GeV)
---|---
1 | [0, 5]
5 | [0, 10]
10 | [0, 20]
20 | [5, 35]
30 | [10, 50]
40 | [15, 65]
50 | [20, 80]
60 | [25, 85]
70 | [35, 85]
80 | [40, 88]
90 | [50, 130]
100 | [95, 140]
Table 2: The invariant-dilepton mass cuts as discussed in the text to optimize
the signal vs backgrounds for a particular mass of dark-$Z_{d}$ production.

##  3 Simulation and observable

In order to explore the limits of dark-$Z_{d}$ we first build a model file for
the Lagrangian given in eq. (4) using the package FeynRules Alloul:2013bka
and then simulate the $Z_{d}$ production in CC: $pe^{-}\to\nu_{e}Z_{d}j$ and
NC: $pe^{-}\to e^{-}Z_{d}j$, with $Z_{d}\to\ell^{+}\ell^{-}$ pairs, in the
LHeC set up as stated previously. Here we consider $\ell^{\pm}=e^{\pm}$ and
$\mu^{\pm}$ only. The representative Feynman diagram for the production of
$Z_{d}$ is shown in Fig. 1. For the generation of events, we use the Monte
Carlo event generator package MadGraph5_aMC@NLO Alwall:2011uj . The
factorization and normalization scales are set to be dynamic scales for both
signal and potential backgrounds. For this study, $e^{-}$ polarization is
assumed to be $-80$%. The initial requirements on cuts are $p_{T}^{\ell,j}>10$
GeV, $|\eta_{\ell,j}|<5$ and no cuts on missing energy.

We initiate our analysis by estimating the cross-section ($\sigma$) of $Z_{d}$
production as a function of its mass $m_{Z_{d}}$, shown in Fig. 2a for three
choices of coupling, $g_{V}=0.0$, $0.4$, and $0.7$. The production cross-
section falls as the mass of $Z_{d}$ increases for both CC and NC cases due to
less available phase space energy. Variation of the cross-section due to the
coupling $g_{V}$ is imperceptible in NC case, while it is much prominent in
the case of CC production. The cross-section decreases from $g_{V}=0$ and
becomes minimum around $g_{V}=0.7$ and increases till $g_{V}=1.0$ because of
the constraint $g_{V}^{2}+g_{A}^{2}=1$. Further, the cross-section is smaller
in CC case than in NC case, see Fig. 2b.

Next, in order to optimize the significance of the signal over all the
backgrounds we chose the nominal selection cuts of $p_{T}^{j}>20$ GeV,
$p_{T}^{\ell}>10$ GeV, $-1<\eta_{j}<5$ and $-1<\eta_{\ell}<3$. The mass
dependent cut on invariant-mass distribution of di-leptonic final state (which
originate from $Z_{d}\to\ell^{+}\ell^{-}$) $m_{\ell\ell}(m_{Z_{d}})$ shall be
an appropriate observable to distinguish the signal from backgrounds. Note
that for our signal, the background from $Z$-boson production is the dominant
one, and $m_{\ell\ell}(m_{Z_{d}})$ cut reduces it significantly. We show the
applied cuts for a range of $Z_{d}$ mass in Table 2. In Fig. 3, the normalized
distribution of $m_{\ell\ell}$ is shown for masses 10 to 100 GeV (in the
interval of 10 GeV) with the dominant background. The peaks at $m_{Z_{d}}$ for
signals and $m_{Z}$ for the background clearly show the efficacy of choosing
the $m_{\ell\ell}$ cuts.

Angular observables and hence the asymmetries defined in eq. (6) are also
significant in this study. The two angular variables
($\cos\phi_{\ell^{-}}^{\star}$, $\cos\theta_{\ell^{-}}^{\star}$) for the
polarization of $Z_{d}$ (Fig. 4) along with two other angular variables
independent of the polarization of $Z_{d}$ are depicted in Fig. 5 for few
benchmark of $m_{Z_{d}}$ and $g_{V}=0.4$ in the CC process as representative
distribution. The asymmetry for the $\cos\phi_{\ell^{-}}^{\star}$ becomes more
positive as $m_{Z_{d}}$ increases, although not much change is seen for
$\cos\theta_{\ell^{-}}^{\star}$. On the other hand, asymmetries for
$\cos\theta_{j}$ and $\cos\Delta\phi(j,Z_{d})$ become more negative as
$m_{Z_{d}}$ increases. Asymmetries for these angular variables, thus, possess
the potential ability to constrain on $m_{Z_{d}}$.

Figure 3: The normalised distribution for the invariant mass of
$\ell^{+}\ell^{-}$ pair ($m_{\ell\ell}$) are shown for the CC process for
background as well as for the signal with a representative value of
$g_{V}=0.4$.

(a)

(b)

Figure 4:  The normalized distribution of $\cos\phi^{\star}_{\ell^{-}}$ and
$\cos\theta^{\star}_{\ell^{-}}$ for the charged-current process with
$m_{Z_{d}}=10$, $30$, $50$, $70$, and $90$ GeV and $g_{V}=0.4$.

(a)

(b)

Figure 5:  The normalized distribution of $\cos\theta_{j}$ and
$\cos\Delta\phi(j,Z_{d})$ are shown for the charged-current process with
$m_{Z_{d}}=10$, $30$, $50$, $70$, and $90$ GeV and $g_{V}=0.4$.

##  4 Analysis and results

After having a preliminary understanding of cross-sections of signal and
backgrounds, its optimization through $m_{\ell\ell}$, and with different
potential angular observables, we follow analysis using an efficient
$\chi^{2}$-formula and describe the limits on $\epsilon$ in this section. We
use observable comprising the cross-section, eight polarization asymmetries
and the lab-frame asymmetries as defined in eq. (6) to form a total $\chi^{2}$
and obtain limits on the $\epsilon(m_{Z_{d}})$ (see eq. (4)). The total
$\chi^{2}$ is defined as:

$\displaystyle\chi^{2}=\frac{\left(\sigma(g_{f})-\sigma\right)^{2}}{\delta\sigma^{2}}+\sum_{j}\frac{\left(A_{j}(g_{f})-A_{j}\right)^{2}}{\delta
A_{j}^{2}}.$ (7)

The errors $\delta\sigma$ and $\delta A_{j}$ in eq. (7) are obtained from the
SM backgrounds and given by

$\displaystyle\delta\sigma=\sqrt{\frac{\sigma}{{\cal
L}}+\left(\epsilon_{\sigma}\sigma\right)^{2}},\quad\delta
A_{j}=\sqrt{\frac{1-A_{j}^{2}}{{\sigma\cal
L}}+\left(\epsilon_{A_{j}}\right)^{2}},$ (8)

which comprised of the integrated luminosity ${\cal L}$ dependent statistical
errors in the first term and systematic errors
$\epsilon_{\sigma}(\epsilon_{A_{j}})$ for the cross-section (asymmetries) in
the second term. We have chosen $\epsilon_{\sigma}=0.02$ and
$\epsilon_{A}=1\leavevmode\nobreak\ \%$ as benchmark for our analysis.

(a)

(b)

Figure 6:  The $\chi^{2}$ of cross-section ($\sigma$) and all asymmetries
($A_{pol}$ , $A_{Lab}$) are shown as a function of $m_{Z_{d}}$ for $g_{V}=0.4$
with (a) $\epsilon=5\times 10^{-2}$ for the CC process and (b)
$\epsilon=10^{-2}$ for the NC process with integrated luminosity of ${\cal
L}=1$ ab-1.

###  4.1 Comparing the variables based on $\chi^{2}$

In order to compare the strength of various variables, we estimate the
$\chi^{2}$ (eq. 7) for the cross-section and all the angular variables
separately as well as in various combinations in both CC and NC processes. The
$\chi^{2}$ values for different variables are shown in Fig. 6 as a function of
$m_{Z_{d}}$ with $g_{V}=0.4$ for an integrated luminosity of ${\cal L}=1$
ab-1. The polarization asymmetries ($A_{pol}$) performs better than cross-
section in CC process, while it performs poorer in NC process after
$m_{Z_{d}}=30$ GeV. The lab frame asymmetries ($A_{Lab}$), however, perform
better than both polarization asymmetries and cross-section in both processes.
The $\chi^{2}$ values decreases as $m_{Z_{d}}$ increases (since the production
cross-section decreases with increasing values of $m_{Z_{d}}$ in both
processes) and it shows a dip around $m_{Z_{d}}\approx 90$ GeV because of
$m_{\ell\ell}\approx m_{Z}$ invariant mass cut from the background.

(a)

(b)

Figure 7:  The $90\%$ C.L. contours are shown in the $\epsilon$-$m_{Z_{d}}$
plane with the observable $\sigma$, $A$ and $\sigma+A$ for $g_{V}=0.4$ for (a)
CC and (b) NC process with integrated luminosity of ${\cal L}=1$ ab-1.

(a)

(b)

Figure 8: The $90\%$ C.L. contours are shown in the $\epsilon$-$g_{V}$ plane
with the observable $\sigma+A$ for $m_{Z_{d}}=10$, $30$, $50$, $70$, and $90$
GeV for (a) CC and (b) NC processes with integrated luminosity of ${\cal L}=1$
ab-1.

(a)

(b)

Figure 9:  The $90\%$ C.L. contours are shown in the $\epsilon$-$m_{Z_{d}}$
plane with the observable $\sigma$, $A$ and $\sigma+A$ for $g_{V}=0.4$ with
$Z_{d}\to b\bar{b}$ for both (a) charged current and (b) neutral current
process with integrated luminosity of ${\cal L}=1$ ab-1.  Figure 10: The
$90\%$ C.L. contours in the $\epsilon$-$m_{Z_{d}}$ plane are shown with the
observable $\sigma$, $A$ and $\sigma+A$ for $g_{V}=0.4$ for both charged
current process as well as neutral current process with an integrated
luminosity of ${\cal L}=1$ ab-1. Corresponding available limits from various
experiments such as LHCb, BaBar, and Bell-II are also shown for comparison.

###  4.2 Limits on $\epsilon$ as function of $m_{Z_{d}}$ and $g_{V}$

We obtain two parameter limits on the $\epsilon$-$m_{Z_{d}}$ and
$\epsilon$-$g_{V}$ plane by fixing the values of $g_{V}$ and $m_{Z_{d}}$
respectively. In case of $\epsilon$-$m_{Z_{d}}$ plane we consider the
observable as (a) cross-section, (b) all asymmetries together
($A\equiv\sum_{i}A_{i}+A_{pol}+A_{Lab}$), and (c) cross-section along with
asymmetries in both CC and NC processes. The $90\%$ confidence level (C.L.)
contours ($\chi^{2}=4.61$ Cowan:2010js ) for two parameter in the
$\epsilon$-$m_{Z_{d}}$ plane are shown in Fig. 7 with $g_{V}=0.4$ for both CC
(Fig. 7a) and NC (Fig. 7b) processes with an integrated luminosity of ${\cal
L}=1$ ab-1. We scanned the mass of $Z_{d}$ in the range of $0.1$ \- $100$ GeV
for cross-section. For the asymmetries, however, we scanned $m_{Z_{d}}$ in the
range of $10-100$ GeV (due to the limitation in event generation). The limits
on $\epsilon$ get weaker with increasing $m_{Z_{d}}$ up to $90$ GeV as
expected from the Fig. 6. The asymmetry observables provide better limits on
$\epsilon$ compared to the cross-section for the studied mass range. Also the
limits in NC process are better compared to CC process because of larger
cross-sections (see Fig. 2).

Similarly in Fig. 8, we show the limits on $\epsilon$ as a function of $g_{V}$
for fixed $m_{Z_{d}}=10$, $30$, $50$, $70$ and $90$ GeV by considering cross-
section along with all asymmetries in both CC and NC processes. Since the
cross-section for CC process is minimal around $g_{V}\approx 0.7$ (Fig. 2a),
limits on $\epsilon$ get weaker as it approaches to $g_{V}\approx 0.7$ and get
stronger afterwards. Due to the cross-section behaviour seen in Fig. 2b for NC
process there is almost no fluctuation on $\epsilon$. Since the value of total
cross-section decreases as a function of increasing $m_{Z_{d}}$, the limits on
$\epsilon$ also get weaker as $m_{Z_{d}}$ goes higher in both CC and NC
processes for full range of $g_{V}$.

###  4.3 Analysis in $Z_{d}\to b\bar{b}$ channel

At this point, it is prudent to ask if the limits obtained in
$Z_{d}\to\ell^{+}\ell^{-}$ decay modes are comparable with $Z_{d}\to jj$.
However, in this decay mode, the polarization observable can not be
constructed. So we only consider cross-section and the lab frame asymmetries
for the $Z_{d}\to b\bar{b}$ decay in both CC and NC processes. And we repeat
the analysis to extract limits on the new physics parameter ($m_{Z_{d}}$,
$\epsilon$, $g_{V}$) using the $\chi^{2}$-method (eq. 7) in this channel.

The corresponding $90\%$ C.L. contours in the $\epsilon$-$m_{Z_{d}}$ plane are
shown in Fig. 9 with $g_{V}=0.4$ for the same integrated luminosity of ${\cal
L}=1$ ab-1 for both CC and NC processes. The order of $\epsilon(m_{Z_{d}})$
remains roughly the same with a little variation of similar nature as in Fig.
7. But the limits are much weaker in this channel because of the large $Z\to
b\bar{b}$ background compared to the $Z_{d}\to\ell^{-}\ell^{+}$ channel.

##  5 Summary and discussions

In this article, we consider a vector-portal interaction in which the dark and
visible sectors interact through kinetic mixing, leading to a so-called heavy
dark photon, $Z_{d}$. To explore the Lorentz structure and strength of $Z_{d}$
coupling to fermions we assume a generic structure involving two set of
couplings $\epsilon,\,g_{V}$ (where $g_{V}^{2}+g_{A}^{2}=1$) and explore the
limits for $m_{Z_{d}}\in[\sim 1-100]$ GeV in LHeC environment. A particular
decay mode $Z_{d}\to\ell^{+}\ell^{-}$ has been taken for the study in which an
invariant mass $m_{\ell\ell}$ cut significantly able to reduce the dominant
background from $Z$-boson. In order to obtain limits on $\epsilon$-$m_{Z_{d}}$
and $\epsilon$-$g_{V}$ planes, a $\chi^{2}$-method is used in which different
observable $viz.$ cross-section, polarisation asymmetries, and Lab-frame
asymmetries are taken as input. Interestingly the charged-current process
shows large variations in cross-section with $g_{V}$ in comparison to neutral-
current process at the LHeC $\sqrt{s}\approx 1.3$ TeV considered in this
study. Though the higher cross-section of NC channel leads to stronger limits
on $\epsilon$ in comparison to CC channel.

A summary plot on limits of $\epsilon$-$m_{Z_{d}}$ at 90% C.L. is shown in
Fig. 10, where the limits are compared and depicted with existing experiments
(for available dark photon mass) from LHCb Aaij:2019bvg , Belle-II Kou:2018nap
and BaBar Lees:2014xha . The limits from CMS Sirunyan:2019wqq  are not shown
in this plot since it originates beyond $m_{Z_{d}}>100$ GeV. For a low mass of
order $m_{Z_{d}}<10$ GeV, the limits of $\epsilon$ are nearly within the
limits of BaBar and LHCb experiments. But for the higher masses $m_{Z_{d}}>10$
GeV the LHeC with $\sqrt{s}\approx 1.3$ TeV results weaker limits of
$\epsilon$-$m_{Z_{d}}$ in comparison to these experiments.

To summarise our study, we also compared the limits on $\epsilon$-$m_{Z_{d}}$
plane for $Z_{d}\to b\bar{b}$ which gives a little weaker limits in comparison
to $Z_{d}\to\ell^{+}\ell^{-}$ due to the large background. Interestingly it is
noted that in the $\epsilon$-$m_{Z_{d}}$ plane, the limits get weaker as
$m_{Z_{d}}$ approaches the $Z$-boson mass and get stronger after crossing this
mass in all cases. And thus, the SM background due to $Z$-boson plays a very
significant role within $m_{Z_{d}}\approx 100$ GeV.

To our knowledge, this is the first study we carried out in the LHeC
environment to obtain limits on the searches of high mass ($>1$ GeV) dark
photon with sensitive observable, where without evading the theoretical
constraint on the limit of $\epsilon\lesssim\mathcal{O}(1)$, $m_{Z_{d}}$ up to
$100$ GeV can be explored with $\sqrt{s}\approx 1.3$ TeV energy. By increasing
the energy of electron sufficiently in the LHeC, higher masses of
$m_{Z_{d}}>100$ GeV can be studied, but then the single Higgs-boson (of mass
$125$ GeV) production in both charged- and neutral-current will be the
dominant SM backgrounds in $Z_{d}\to b\bar{b}$ decay, and hence appropriate
$m_{\ell\ell}$ or $m_{jj}$ cut will help to reduce this background and limits
on $\epsilon$-$m_{Z_{d}}/g_{V}$ can be obtained accordingly.

## Acknowledgment

AG thanks SERB, G.O.I. under CRG/2018/004889.

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