# Updated constraints on $Z^{\prime}$ and $W^{\prime}$ bosons decaying into
bosonic and leptonic
final states using the Run 2 ATLAS data

P. Osland<EMAIL_ADDRESS>Department of Physics and Technology, University
of Bergen, Postboks 7803, N-5020 Bergen, Norway  A. A. Pankov<EMAIL_ADDRESS>The Abdus Salam ICTP Affiliated Centre, Technical University of Gomel, 246746
Gomel, Belarus  Institute for Nuclear Problems, Belarusian State University,
220030 Minsk, Belarus  Joint Institute for Nuclear Research, Dubna 141980
Russia  I. A. Serenkova<EMAIL_ADDRESS>The Abdus Salam ICTP
Affiliated Centre, Technical University of Gomel, 246746 Gomel, Belarus

###### Abstract

The full ATLAS Run 2 data set with time-integrated luminosity of 139 fb-1 in
the diboson and dilepton channels is used to probe benchmark models with
extended gauge sectors: the $E_{6}$-motivated Grand Unification models, the
left-right symmetric $LR$ and the sequential standard model (EGM). These all
predict neutral $Z^{\prime}$ vector bosons, decaying into lepton pairs,
$\ell\ell$, or into electroweak gauge boson pairs $WW$, where one $W$ in turn
decays semileptonically. 95% C.L. exclusion limits on the $Z^{\prime}$
resonance production cross section times branching ratio to electroweak gauge
boson pairs and to lepton pairs in the mass range of $\sim$ 1 – 6 TeV are
converted to constraints on the $Z$-$Z^{\prime}$ mixing parameter and the
heavy resonance mass. We present exclusion regions on the parameter space of
the $Z^{\prime}$ which are significantly extended compared to those obtained
from the previous analyses performed with LHC data collected at 7 and 8 TeV in
Run 1 as well as at 13 TeV in Run 2 at time-integrated luminosity of 36.1 fb-1
and are the most stringent bounds to date. Also presented, from a similar
analysis of electrically charged $W^{\prime}$ bosons arising in the EGM, which
can decay through $W^{\prime}\to WZ$ and $W^{\prime}\to\ell\nu$, are limits on
the $W$-$W^{\prime}$ mixing parameter and the charged $W^{\prime}$ vector
boson mass.

##  I Introduction

One of the main goals of the physics programme at the Large Hadron Collider
(LHC) is to search for new resonant or non-resonant phenomena that become
visible in high-energy proton-proton collisions. A prominent possible
signature of such phenomena would be the production of a heavy resonance with
its subsequent decay into a pair of leptons or into electroweak vector bosons.
Many scenarios beyond the Standard Model (SM) predict such signals. Possible
candidates are neutral and charged heavy gauge bosons which are commonly
referred to as $Z^{\prime}$ and $W^{\prime}$ bosons, respectively Zyla  _et
al._ (2020). Strong constraints have already been set on the production of
such new heavy particles.

At the LHC, heavy $Z^{\prime}$ and $W^{\prime}$ bosons could be observed
through their production as $s$-channel resonances with subsequent leptonic
decays

$pp\to Z^{\prime}X\to\ell^{+}\ell^{-}X,$ (1)

and

$pp\to W^{\prime}X\to\ell\nu X,$ (2)

respectively, where in what follows, $\ell=e,\mu$ unless otherwise stated. The
production of $Z^{\prime}$ and $W^{\prime}$ bosons at hadron colliders is
expected to be dominated by the Drell-Yan (DY) mechanism,
$q\bar{q}/q\bar{q}^{\prime}\to Z^{\prime}/W^{\prime}$. The Feynman diagrams
for the $Z^{\prime}$ ($W^{\prime}$) boson production at the parton level and
their dilepton and diboson decays are illustrated in Fig. 1.

Figure 1: Parton-level Feynman diagrams for $Z^{\prime}$ ($W^{\prime}$)
production with dilepton and diboson decays.

Leptonic final states provide a low-background and efficient experimental
signature that results in excellent sensitivity to new phenomena at the LHC.
Specifically, these processes (1) and (2) offer the simplest event topology
for the discovery of $Z^{\prime}$ and $W^{\prime}$ with a large production
rate and a clean experimental signature. These channels offer the most
promising discoveries at the LHC Aad  _et al._ (2019a); repository (2020a);
collaboration (2019); Zucchetta (2019); collaboration (2018a); Aad  _et al._
(2019b). There have also been many theoretical studies of $Z^{\prime}$ and
$W^{\prime}$ searches at the high-energy hadron colliders (see, e.g. Zyla  _et
al._ (2020); Altarelli  _et al._ (1989); Schmaltz and Spethmann (2011);
Grojean  _et al._ (2011); Jezo  _et al._ (2012); Cao  _et al._ (2012);
Dobrescu and Liu (2015); Langacker (2009); Erler  _et al._ (2009); Hewett and
Rizzo (1989); Leike (1999); Dittmar  _et al._ (2004); Osland  _et al._ (2009);
Godfrey and Martin (2013); Andreev  _et al._ (2014); Gulov  _et al._ (2018);
Bandyopadhyay  _et al._ (2018, 2019)).

In the simplest models such as the Sequential Standard Model (SSM) Altarelli
_et al._ (1989) new neutral $Z^{\prime}_{\rm SSM}$ and charged
$W^{\prime}_{\rm SSM}$ bosons have couplings to fermions that are identical to
those of the SM $Z$ and $W$ bosons, but for which the trilinear couplings
$Z^{\prime}WW$ and $W^{\prime}WZ$ are absent. The SSM has been used as a
reference for experimental $Z^{\prime}$ and $W^{\prime}$ searches for decades,
the results can be re-interpreted in the context of other models, it is
therefore useful for comparing the sensitivity of different experiments.
Another class of models considered here are those inspired by Grand Unified
Theories (GUT), which are motivated by gauge unification or a restoration of
the left–right symmetry violated by the weak interaction. Examples considered
in this paper include the $Z^{\prime}$ bosons of the $E_{6}$-motivated
Langacker (2009) theories containing $Z^{\prime}_{\psi}$, $Z^{\prime}_{\eta}$,
$Z^{\prime}_{\chi}$; and high-mass neutral bosons of the left-right (LR)
symmetric extensions of the SM, based on the $SU(2)_{L}\bigotimes
SU(2)_{R}\bigotimes U(1)_{B-L}$ gauge group, where $B-L$ refers to the
difference between baryon and lepton numbers.

The data we consider were collected with the ATLAS and CMS detectors during
the 2015–2018 running period of the LHC, referred to as Run 2 and
corresponding to a time-integrated luminosity of 139–140 fb-1. The ATLAS
experiment has presented the first search for dilepton resonances based on the
full Run 2 data set Aad  _et al._ (2019a, b) and set limits on the
$Z^{\prime}$ and $W^{\prime}$ production cross sections times branching
fraction in the processes (1) and (2), $\sigma(pp\to
Z^{\prime}X)\times\text{BR}(Z^{\prime}\to\ell^{+}\ell^{-})$ and $\sigma(pp\to
W^{\prime}X)\times\text{BR}(W^{\prime}\to\ell\nu)$, respectively, for
$M_{Z^{\prime}}$ and $M_{W^{\prime}}$ in the 0.25 TeV – 6 TeV and 0.15 TeV – 7
TeV ranges, correspondingly. Recently, similar searches have also been
presented by the CMS Collaboration using 140 fb-1 of data recorded at
$\sqrt{s}=13$ TeV collaboration (2019). The ATLAS and CMS collaborations set a
95% confidence level (CL) lower limit on the $Z^{\prime}$ mass of $\sim$ 4.6
TeV–5.2 TeV depending on the model Aad  _et al._ (2019a); collaboration
(2019), and 6.0 TeV for the $W^{\prime}_{\rm SSM}$ Aad  _et al._ (2019b).

Alternative $Z^{\prime}$ and $W^{\prime}$ search channels are the diboson
reactions

${pp\to Z^{\prime}X\to WWX,}$ (3)

and

${pp\to W^{\prime}X\to WZX.}$ (4)

The study of gauge boson pair production offers a powerful test of the
spontaneously broken gauge symmetry of the SM and can be used as a probe for
new phenomena beyond the SM. Specifically, in contrast to the DY processes (1)
and (2) diboson reactions are not the primary discovery channels, but can help
to understand the origin of new gauge bosons.

As mentioned above, heavy resonances that can decay to gauge boson pairs are
predicted in many scenarios of new physics, including extended gauge models
(EGM) Altarelli  _et al._ (1989); Eichten  _et al._ (1984), models of warped
extra dimensions Randall and Sundrum (1999); Davoudiasl  _et al._ (2001),
technicolour models Lane and Mrenna (2003); Eichten and Lane (2008) associated
with technirho and other technimesons, composite Higgs models Agashe  _et al._
(2005); Giudice  _et al._ (2007), and the heavy vector-triplet (HVT) model
Pappadopulo  _et al._ (2014), which generalises a large number of models that
predict spin-1 neutral ($Z^{\prime}$) and charged ($W^{\prime}$) resonances.
In the SSM, the coupling constants of the $Z^{\prime}(W^{\prime})$ boson with
SM fermions are the direct transcription of the corresponding SM couplings,
while the $Z^{\prime}(W^{\prime})$ coupling to $WW(WZ)$ is strongly
suppressed, $g_{Z^{\prime}WW}=0$ and $g_{W^{\prime}WZ}=0$. This suppression
may arise naturally in an EGM: if the new gauge bosons and the SM ones belong
to different gauge groups, a vertex such as $Z^{\prime}WW$ $(W^{\prime}WZ)$ is
forbidden. They can only be induced after symmetry breaking due to mixing of
the gauge eigenstates. Searches for exotic heavy particles that decay into
$WW$ or $WZ$ pairs are complementary to searches in the leptonic channels
$\ell^{+}\ell^{-}$ and $\ell\nu$ of the processes (1) and (2). Moreover, there
are models in which new gauge boson couplings to SM fermions are suppressed,
giving rise to a fermiophobic $Z^{\prime}$ and $W^{\prime}$ with an enhanced
coupling to electroweak gauge bosons Zyla  _et al._ (2020); He  _et al._
(2008). It is therefore important to search for $Z^{\prime}$ and $W^{\prime}$
bosons also in the $WW$ and $WZ$ final states.

The properties of possible $Z^{\prime}$ and $W^{\prime}$ bosons are also
constrained by measurements of electroweak (EW) processes at low energies,
i.e., at energies much below their masses. Such bounds on the $Z$-$Z^{\prime}$
($W$-$W^{\prime}$) mixing are mostly due to the constraints on deviation in
$Z$ ($W$) properties from the SM predictions. In particular, limits from
direct hadron production with subsequent diboson decay at the Tevatron
Aaltonen  _et al._ (2010) and from virtual effects at LEP, through
interference or mixing with the $Z$ boson, imply that any new $Z^{\prime}$
boson is rather heavy and mixes very little with the $Z$ boson. At LEP and the
SLC, the mixing angle is strongly constrained by very high-precision $Z$ pole
experiments Schael  _et al._ (2006). These include measurements of the $Z$
line shape, the leptonic branching ratios as well as leptonic forward-backward
asymmetries. The measurements show that the mixing angles, referred to as
$\xi_{Z\text{-}Z^{\prime}}$ and $\xi_{W\text{-}W^{\prime}}$, between the gauge
eigenstates must be smaller than about $10^{-3}$ and $10^{-2}$, respectively
Zyla  _et al._ (2020); Erler  _et al._ (2009).

Previous analyses of the $Z$-$Z^{\prime}$ and $W$-$W^{\prime}$ mixing Osland
_et al._ (2017); Bobovnikov  _et al._ (2018); Serenkova  _et al._ (2019) were
carried out using the diboson and dilepton production data sets corresponding
to the time-integrated luminosity of $\sim$ 36 fb-1 collected in 2015 and 2016
with the ATLAS and CMS collaborations at $\sqrt{s}=$ 13 TeV where, in the
former case, electroweak $Z$ and $W$ gauge bosons decay into the semileptonic
channel Aaboud  _et al._ (2018) or into the dijet final state Sirunyan  _et
al._ (2018a). The results of the present analysis benefit from the increased
size of the data sample, now amounting to an integrated luminosity of 139 fb-1
recorded by the ATLAS detector in Run 2 Aad  _et al._ (2019c); repository
(2020b); Aad  _et al._ (2020); repository (2020c), almost four times larger
than what was available for the previous study.111In the current analysis, we
utilize the full Run 2 ATLAS data set on diboson resonance production Aad  _et
al._ (2019c, 2020); repository (2020c), rather than that of CMS, as the latter
one is unavailable so far. In addition, further improvement in placing limits
on the $Z^{\prime}$ and $W^{\prime}$ mass and $Z$-$Z^{\prime}$ and
$W$-$W^{\prime}$ mixing parameters can be achieved in semileptonic $WW/WZ$
final states in which one vector boson decays leptonically
($Z\to\ell\ell,\nu\nu$, $W\to\ell\nu$) while the other decays hadronically
($Z/W\to qq$).222To simplify notation, antiparticles are denoted by the same
symbol as the corresponding particles. Also, here we extend our analysis
presented in Pankov  _et al._ (2020) where we utilized the full Run 2 ATLAS
data set for EGM (SSM) to various $Z^{\prime}$ models, including $E_{6}$-based
$Z_{\chi}$, $Z_{\psi}$, $Z_{\eta}$, and also $Z_{\rm LR}$ boson appearing in
models with left-right symmetry. Thus, our present analysis is complementary
to the previous studies Pankov  _et al._ (2020).

We present results as constraints on the relevant $Z$-$Z^{\prime}$
($W$-$W^{\prime}$) mixing angle, $\xi_{Z\text{-}Z^{\prime}}$
($\xi_{W\text{-}W^{\prime}}$), and on the mass $M_{Z^{\prime}}$
$(M_{W^{\prime}}$) and display the combined allowed parameter space for the
benchmark $Z^{\prime}$ ($W^{\prime}$) models, showing also indirect
constraints from electroweak precision data. Previous direct search
constraints from the Tevatron and from the LHC with 7 and 8 TeV in Run 1
(where available) are compared to those obtained from the LHC at 13 TeV with
the full ATLAS Run 2 data set of time-integrated luminosity of 139 fb-1 in the
semileptonic Aad  _et al._ (2020); repository (2020c) and fully hadronic
($qqqq$) Aad  _et al._ (2019c) final states.

The paper is structured as follows. In Sect. II we present the theoretical
framework, then, in Sect. III we summarize the relevant cross sections for the
diboson and dilepton production processes (3) and (1) in the narrow-width
approximation (NWA). Next, we discuss the relevant $Z^{\prime}$ widths and
branching ratios within the considered benchmark models. Further, we present
an analysis of bounds on $Z$-$Z^{\prime}$ mixing from constraints on diboson
and dilepton production in the context of the benchmark models with extended
gauge sector, employing the most recent searches recorded by the ATLAS (139
fb-1) detector in the semileptonic Aad  _et al._ (2020); repository (2020c)
and fully hadronic (referred to as $qqqq$) Aad  _et al._ (2019c); repository
(2020b) final states at the LHC. Then, we show the resulting constraints on
the $M_{Z^{\prime}}-\xi_{Z\text{-}Z^{\prime}}$ parameter space obtained from
these processes. Further, we collect and compare the indirect constraints
obtained from electroweak precision data, direct search constraints derived
from the LHC in Run 1 and early Run 2 data. In Sect. IV we present the
corresponding analysis of bounds on $W$-$W^{\prime}$ mixing, performed in a
similar fashion as for the $Z^{\prime}$, from constraints on diboson and
dilepton production processes (4) and (2) in the context of the EGM. Sect. V
presents some concluding remarks.

##  II Mixing and parameters

We consider $Z$–$Z^{\prime}$ mixing within the framework of models with
extended gauge sector such as the $E_{6}$ models, the LR model and the EGM
(see, e.g. Altarelli  _et al._ (1989); Langacker (2009); Erler  _et al._
(2009); Hewett and Rizzo (1989); Leike (1999)). The mass eigenstates $Z$ and
$Z^{\prime}$ are admixtures of the weak eigenstates $Z^{0}$ of $SU(2)\times
U(1)$ and $Z^{0\prime}$ of the extra $U(1)^{\prime}$, respectively:

$\displaystyle Z$ $\displaystyle=Z^{0}\cos\phi+Z^{0\prime}\sin\phi\;,$ (5a)
$\displaystyle Z^{\prime}$
$\displaystyle=-Z^{0}\sin\phi+Z^{0\prime}\cos\phi\;.$ (5b)

For each type of $Z^{\prime}$ boson, defined by its gauge couplings, there are
three classes of models, which differ in the assumptions concerning the
quantum numbers of the Higgs fields which generate the $Z$-boson mass matrix
Langacker (2009); Erler  _et al._ (2009). In each case there is a relation
between the $Z^{0}$-$Z^{0\prime}$ mixing angle $\phi$ and the masses $M_{Z}$
and $M_{Z^{\prime}}$ Langacker (2009):

$\tan^{2}\phi={\frac{M_{Z^{0}}^{2}-M_{Z}^{2}}{M_{Z^{\prime}}^{2}-M_{Z^{0}}^{2}}}\simeq\frac{2M_{Z^{0}}\Delta
M}{M_{Z^{\prime}}^{2}}\;,$ (6)

where the downward shift $\Delta M=M_{Z^{0}}-M_{Z}>0$, and ${M_{Z^{0}}}$ is
the mass of the $Z$ boson in the absence of mixing, i.e., for $\phi=0$, given
by

${M_{Z^{0}}}=\frac{M_{W}}{\sqrt{\rho_{0}}\cos\theta_{W}}.$ (7)

The mixing angle $\phi$ will play an important role in our analysis. Such
mixing effects reflect the underlying gauge symmetry and/or the Higgs sector
of the model as the $\rho_{0}$ parameter depends on the ratios of Higgs vacuum
expectation values and on the total and third components of weak isospin of
the Higgs fields. We set $\rho_{0}=1$ here, this corresponds to a Higgs sector
with only $SU(2)$ doublets and singlets Langacker (2009). Once we assume the
mass $M_{Z}$ to be determined experimentally, the mixing depends on two free
parameters, which we identify as $\phi$ and $M_{Z^{\prime}}$, a
parametrization that we will adopt throughout the paper.

This $Z^{0}$-$Z^{0\prime}$ mixing induces a change in the couplings of the two
bosons to fermions. From Eq. (5), one obtains the vector and axial-vector
couplings of the $Z$ and $Z^{\prime}$ bosons to fermions:

$\displaystyle v_{f}$
$\displaystyle=v^{0}_{f}\cos\phi+v^{0\prime}_{f}\sin\phi\;,$
$\displaystyle\quad a_{f}$
$\displaystyle=a^{0}_{f}\cos\phi+a^{0\prime}_{f}\sin\phi\;,$ (8a)
$\displaystyle v^{\prime}_{f}$
$\displaystyle=v_{f}^{0\prime}\cos\phi-v^{0}_{f}\sin\phi\;,$
$\displaystyle\quad a^{\prime}_{f}$
$\displaystyle=a_{f}^{0\prime}\cos\phi-a^{0}_{f}\sin\phi\;,$ (8b)

with unprimed and primed couplings referring to $Z^{0}$ and $Z^{0\prime}$,
respectively, and found, e.g. in Leike (1999).

An important property of the models under consideration is that the gauge
eigenstate $Z^{0\prime}$ does not couple to the $W^{+}W^{-}$ pair since it is
neutral under $SU(2)$. Therefore the $W$-pair production is sensitive to a
$Z^{\prime}$ only in the case of a non-zero $Z^{0}$–$Z^{0\prime}$ mixing. From
Eq. (5), one obtains:

$\displaystyle g_{WWZ}=\cos\phi\;g_{WWZ^{0}}\;,$ (9a) $\displaystyle
g_{WWZ^{\prime}}=-\sin\phi\;g_{WWZ^{0}}\;,$ (9b)

where $g_{WWZ^{0}}=e\cot\theta_{W}$. Also, $g_{WW\gamma}=e$.

In many extended models, while the couplings to fermions do not differ much
from those of the SM, the $Z^{\prime}WW$ coupling is substantially suppressed
with respect to that of the SM. In fact, in the extended gauge models the SM
trilinear gauge boson coupling strength, $g_{WWZ^{0}}$, is replaced by
$g_{WWZ^{0}}\rightarrow\xi_{Z\text{-}Z^{\prime}}\cdot g_{WWZ^{0}}$, where
$\xi_{Z\text{-}Z^{\prime}}\equiv|\sin\phi|$ (see Eq. (9b)) is the mixing
factor333For weak mixing, $\xi_{Z\text{-}Z^{\prime}}\simeq|\phi|$, and is
therefore often referred to as a mixing “angle”.. We will set cross section
limits on such $Z^{\prime}$ as functions of the mass $M_{Z^{\prime}}$ and
$\xi$.

In addition, we study $W$–$W^{\prime}$ mixing in the process (4) within the
framework of the EGM model Altarelli  _et al._ (1989); Eichten  _et al._
(1984). At the tree-level mass mixing may be induced between the electrically
charged gauge bosons. The physical (mass) eigenstates of $W$ and $W^{\prime}$
are admixtures of the weak eigenstates denoted as $\hat{W}$ and
$\hat{W^{\prime}}$, respectively, and obtained by a rotation of those fields
Zyla  _et al._ (2020); Dobrescu and Liu (2015):

$\displaystyle W^{\pm}$ $\displaystyle=$
$\displaystyle\hat{W}^{\pm}\cos\theta+\hat{W^{\prime}}^{\pm}\sin\theta,$ (10a)
$\displaystyle W^{\prime\pm}$ $\displaystyle=$
$\displaystyle-\hat{W}^{\pm}\sin\theta+\hat{W^{\prime}}^{\pm}\cos\theta,$
(10b)

in analogy with Eq. (5). Upon diagonalization of their mass matrix, the
couplings of the observed $W$ boson are shifted from the SM values.

The properties of possible $Z^{\prime}$ and $W^{\prime}$ bosons, apart from
collider experiments, are also constrained by measurements of electroweak (EW)
processes at low energies, i.e., at energies much below their masses. Such
bounds on the $Z$-$Z^{\prime}$ ($W$-$W^{\prime}$) mixing are mostly due to the
constraints on the deviation in $Z$ ($W$) properties compared to the SM
predictions. These measurements show that the mixing angles
$\xi_{Z\text{-}Z^{\prime}}$ and
$\xi_{W\text{-}W^{\prime}}\,(\equiv|\sin\theta|)$ between the gauge
eigenstates must be smaller than about $\sim 10^{-3}$ and $10^{-2}$ Zyla  _et
al._ (2020), respectively.

##  III $Z^{\prime}$ production and decay in $pp$ collision

We shall first consider $Z^{\prime}$ production in some detail, and
subsequently turn to the $W^{\prime}$ case. In some sense the $Z^{\prime}$
sector is richer than the $W^{\prime}$ sector, different models predict
different ratios of the vector and axial-vector couplings. The $W^{\prime}$
models on the other hand, will all be restricted in the choice of pure left-
handed couplings to fermions. Among the $Z^{\prime}$ models, we start out with
a detailed discussion of the $\psi$ model.

###  III.1 $Z^{\prime}$ resonant production cross section

The $Z^{\prime}$ production and subsequent decay into $WW$ in proton-proton
collisions occurs via quark-antiquark annihilation in the $s$-channel. The
cross section of the process (3) can at the LHC be observed through resonant
pair production of gauge bosons $WW$. Using the narrow width approximation
(NWA), one can factorize the process (3) into the $Z^{\prime}$ production and
its subsequent decay,

$\sigma(pp\to Z^{\prime}X\to WWX)=\sigma(pp\to
Z^{\prime}X)\times\text{BR}(Z^{\prime}\to WW)\;.$ (11)

Here, $\sigma(pp\to Z^{\prime}X)$ is the total (theoretical) $Z^{\prime}$
production cross section and $\text{BR}(Z^{\prime}\to
WW)=\Gamma_{Z^{\prime}}^{WW}/\Gamma_{Z^{\prime}}$ with $\Gamma_{Z^{\prime}}$
the total width of the $Z^{\prime}$. “Narrow” refers to the assumption that
the natural width of the resonance is smaller than the typical experimental
resolution of 5% of its mass Aaboud  _et al._ (2016); Sirunyan  _et al._
(2017). This is valid for a large fraction of the parameter space of the
considered models.

###  III.2 The $Z^{\prime}$ width

In the calculation of the total width $\Gamma_{Z^{\prime}}$ we consider the
following channels: $Z^{\prime}\to f\bar{f}$, $W^{+}W^{-}$, and $ZH$ Salvioni
_et al._ (2009); Osland  _et al._ (2017); Bobovnikov  _et al._ (2018); Pankov
_et al._ (2020), where $H$ is the SM Higgs boson and $f$ refers to the SM
fermions ($f=l,\nu,q$). Throughout the paper we shall ignore the couplings of
the $Z^{\prime}$ to any beyond-SM particles such as right-handed neutrinos,
which we take to be heavier than $M{{}_{Z}^{\prime}}/2$, as well as to SUSY
partners and any other exotic fermions. Such additional states may all
together increase the width of the $Z^{\prime}$ by up to about a factor of
five Kang and Langacker (2005) and hence lower the branching ratio into a
$W^{+}W^{-}$ pair by the same factor.

The total width $\Gamma_{Z^{\prime}}$ of the $Z^{\prime}$ boson can then be
written as follows:

$\Gamma_{Z^{\prime}}=\sum_{f}\Gamma_{Z^{\prime}}^{ff}+\Gamma_{Z^{\prime}}^{WW}+\Gamma_{Z^{\prime}}^{ZH}.$
(12)

The two last terms, which are often neglected in studies at low and moderate
values of $M_{Z^{\prime}}$, are due to $Z$-$Z^{\prime}$ mixing. For the range
of $M_{Z^{\prime}}$ values below $\sim 3-4$ TeV, the dependence of
$\Gamma_{Z^{\prime}}$ on the values of $\xi_{Z\text{-}Z^{\prime}}$ (within its
allowed range) is unimportant. Therefore, in this mass range, one can
approximate the total width as
$\Gamma_{Z^{\prime}}\approx\sum_{f}\Gamma_{Z^{\prime}}^{ff}$, where the sum
runs over SM fermions only. The ratios of
$\Gamma_{Z^{\prime}}^{ff}/M_{Z^{\prime}}$ for the benchmark models are
summarized in Table 1. One can appreciate the narrowness of the
$Z_{Z^{\prime}}$ pole from this Table 1.

Table 1: Ratio $\Gamma_{Z^{\prime}}^{ff}/M_{Z^{\prime}}$ for the $\chi,\psi,\eta$, ${\rm LR}$ and EGM models. $Z^{\prime}$ |  $\Gamma_{Z^{\prime}}^{ff}/M_{Z^{\prime}}$ [%]
---|---
$\chi$ | 1.2
$\psi$ | 0.5
$\eta$ | 0.6
${\rm LR}$ | 2.0
EGM | 3.0

However, for larger $Z^{\prime}$ masses, $M_{Z^{\prime}}>4$ TeV, there is an
enhancement in the coupling that cancels the suppression due to the tiny
$Z$-$Z^{\prime}$ mixing parameter $\xi_{Z\text{-}Z^{\prime}}$ Salvioni  _et
al._ (2009). We note that the “Equivalence theorem” Chanowitz and Gaillard
(1985) suggests a value for $\text{BR}(Z^{\prime}\to ZH)$ comparable to
$\text{BR}(Z^{\prime}\to W^{+}W^{-})$, up to electroweak symmetry breaking
effects and phase-space factors. Throughout this paper, for definiteness, we
adopt a scenario where both partial widths are comparable,
$\Gamma_{Z^{\prime}}^{ZH}\simeq\Gamma_{Z^{\prime}}^{WW}$ for heavy
$M_{Z^{\prime}}$ Barger and Whisnant (1987); Barger  _et al._ (2009); Dib and
Gilman (1987).

For all $M_{Z^{\prime}}$ values of interest for the LHC the width of the
$Z^{\prime}$ boson is considerably smaller than the experimental mass
resolution $\Delta M$ for which we adopt the parametrization in reconstructing
the diboson invariant mass of the $W^{+}W^{-}$ system, $\Delta M/M\approx
5\%$, as proposed, e.g., in Aaboud  _et al._ (2016); Sirunyan  _et al._
(2017).444This $\Delta M$ should not be confused with that of Eq. (6).

The partial width of the $Z^{\prime}\to W^{+}W^{-}$ decay channel can be
written as Altarelli  _et al._ (1989):

$\Gamma_{Z^{\prime}}^{WW}=\frac{\alpha_{\rm
em}}{48}\cot^{2}\theta_{W}\,M_{Z^{\prime}}\left(\frac{M_{Z^{\prime}}}{M_{W}}\right)^{4}\left(1-4\,\frac{M_{W}^{2}}{M_{Z^{\prime}}^{2}}\right)^{3/2}\left[1+20\left(\frac{M_{W}}{M_{Z^{\prime}}}\right)^{2}+12\left(\frac{M_{W}}{M_{Z^{\prime}}}\right)^{4}\right]\cdot\xi_{Z\text{-}Z^{\prime}}^{2}.$
(13)

For a fixed mixing factor $\xi_{Z\text{-}Z^{\prime}}$ and at large
$M_{Z^{\prime}}$ where $\Gamma_{Z^{\prime}}^{WW}$ dominates over
$\sum_{f}\Gamma_{Z^{\prime}}^{ff}$ the total width increases rapidly with the
mass $M_{Z^{\prime}}$ because of the quintic dependence of the $W^{+}W^{-}$
mode on the $Z^{\prime}$ mass as shown in Eq. (13). In this case, the
$W^{+}W^{-}$ mode (together with $Z^{\prime}\to ZH$) becomes dominant and
$\text{BR}(Z^{\prime}\to W^{+}W^{-})\to 0.5$ (this value arises from the
assumption $\Gamma_{Z^{\prime}}^{ZH}=\Gamma_{Z^{\prime}}^{WW}$), while the
fermionic decay channels ($\Gamma_{Z^{\prime}}^{ff}\propto M_{Z^{\prime}}$)
are increasingly suppressed. These features are illustrated in Fig. 2, where
we plot $\text{BR}(Z^{\prime}\to W^{+}W^{-})$ and $\text{BR}(Z^{\prime}\to
e^{+}e^{-})$ vs $M_{Z^{\prime}}$ for the $Z^{\prime}_{\psi}$ model.

Figure 2:  Branching ratio $\text{BR}(Z^{\prime}\to W^{+}W^{-})$ (solid) and
$\text{BR}(Z^{\prime}\to e^{+}e^{-})$ (dashed) vs. $M_{Z^{\prime}}$ in the
$Z^{\prime}_{\psi}$ model for $Z$-$Z^{\prime}$ mixing factor
$\xi_{Z\text{-}Z^{\prime}}=0$ and $\xi_{Z\text{-}Z^{\prime}}=3\cdot 10^{-3}$.
It is assumed that $\Gamma_{Z^{\prime}}^{ZH}=\Gamma_{Z^{\prime}}^{WW}$. Figure
3:  Observed $95\%$ C.L. upper limits on the production cross section times
the branching fraction, $\sigma_{95\%}\times\text{BR}(Z^{\prime}\to
W^{+}W^{-})$, as a function of the $Z^{\prime}$ mass, $M_{Z^{\prime}}$,
showing ATLAS data for the semileptonic (thin solid) Aad  _et al._ (2020);
repository (2020c) and fully hadronic (thick solid) Aad  _et al._ (2019c)
final states for $139~{}\text{fb}^{-1}$. Theoretical production cross sections
$\sigma(pp\to Z^{\prime}_{\psi}+X)\times\text{BR}(Z^{\prime}_{\psi}\to
W^{+}W^{-})$ are shown for mixing factors $\xi_{Z\text{-}Z^{\prime}}$ ranging
from $3\cdot 10^{-3}$ down to $3\cdot 10^{-4}$. Also, the cross section solid
line labeled $\xi^{\rm EW}_{\psi}$ corresponds to the mixing parameter
$\xi^{\rm EW}_{Z^{\prime}}$ indicated in Table 2 for the $Z^{\prime}_{\psi}$
model. The area lying below the long-dashed curve labelled NWA corresponds to
the region where the $Z^{\prime}$ resonance width is predicted to be less than
5% of the resonance mass, in which the narrow-width assumption is satisfied.
The lower boundary of the region excluded by the unitarity constraints is also
indicated.

###  III.3 Hadron production and diboson decay of $Z^{\prime}$

In Fig. 3, we consider the full ATLAS Run 2 data set of time integrated
luminosity of 139 fb-1 and show the observed $95\%$ C.L. upper limits on the
production cross section times the branching fraction,
$\sigma_{95\%}\times\text{BR}(Z^{\prime}\to W^{+}W^{-})$, as a function of the
$Z^{\prime}$ mass, obtained from the semileptonic Aad  _et al._ (2020);
repository (2020c) and fully hadronic ($qqqq$) Aad  _et al._ (2019c);
repository (2020b) final states. This allows for a comparison of the
sensitivities of the data to mixing parameters and new gauge boson mass. This
comparison demonstrates the dominating sensitivity to $Z^{\prime}$ of the
semileptonic channel with respect to the fully hadronic one, over almost the
whole $Z^{\prime}$ mass range.

Then, for $Z^{\prime}_{\psi}$ we compute the LHC production cross section
multiplied by the branching ratio into two $W$ bosons, $\sigma(pp\to
Z^{\prime}_{\psi}X)\times{\rm BR}(Z^{\prime}_{\psi}\to W^{+}W^{-})$, as a
function of the two parameters ($M_{Z^{\prime}}$,
$\xi_{Z\text{-}Z^{\prime}}$), and compare it with the limits established by
the ATLAS experiment, $\sigma_{95\%}\times{\rm BR}(Z^{\prime}\to W^{+}W^{-})$.
The SM backgrounds have been carefully evaluated by the experimental
collaborations and accounted for in $\sigma_{95\%}\times{\rm BR}(Z^{\prime}\to
W^{+}W^{-})$. Therefore, in our analysis we simulate only the
$Z^{\prime}_{\psi}$ signal.

In Fig. 3, the theoretical production cross section $\sigma(pp\to
Z^{\prime}_{\psi})\times\text{BR}(Z^{\prime}_{\psi}\to W^{+}W^{-})$ for
$Z^{\prime}_{\psi}$ boson is calculated from a dedicated modification of
PYHTHIA 8.2 Sjöstrand  _et al._ (2015). As mentioned above, higher-order QCD
corrections to the signal were estimated using a $K$-factor, for which we
adopt a mass-independent value of 1.9 Frixione (1993); Agarwal  _et al._
(2010); Gehrmann  _et al._ (2014). These theoretical curves for the cross
sections, in descending order, correspond to values of the $Z$-$Z^{\prime}$
mixing factor $\xi_{Z\text{-}Z^{\prime}}$ ranging from $3\cdot 10^{-3}$ and
down to $3\cdot 10^{-4}$. The intersection points of the measured upper limits
on the production cross section with this theoretical cross section for
various values of $\xi_{Z\text{-}Z^{\prime}}$ give the corresponding lower
bounds on ($M_{Z^{\prime}}$, $\xi_{Z\text{-}Z^{\prime}}$), presented in Fig.
4.

Figure 4: The $Z^{\prime}_{\psi}$ model: 95%C.L. exclusion regions in the two-
dimensional ($M_{Z^{\prime}}$, $\xi_{Z\text{-}Z^{\prime}}$) plane obtained
after incorporating indirect constraints from electroweak precision data
(horizontal dashed straight line labeled “EW”), and direct search constraints
from the LHC search for $pp\to Z^{\prime}\to WW$ in semileptonic final states
using the full Run 2 ATLAS data set. Limits obtained from the hadronic channel
$qqqq$ are overlaid for comparison. The region above each curve for the $WW$
channel is excluded. The steep curve labelled “excluded by
$Z^{\prime}_{\psi}\to\ell\ell$” shows the exclusion based on the dilepton
channel $pp\to Z^{\prime}_{\psi}\to\ell\ell+X$. The unitarity limit is shown
as the dot-dashed curve. The overall allowed region is shown as a yellow area.

Different bounds on the $Z^{\prime}$ parameter space are collected in Fig. 4
for the $Z^{\prime}_{\psi}$ model, showing that at high masses, the limits on
$\xi_{Z\text{-}Z^{\prime}}$ obtained from the full Run 2 data set collected at
$\sqrt{s}=13$ TeV and recorded by the ATLAS detector are substantially
stronger than that derived from the global analysis of the precision
electroweak data Erler  _et al._ (2009), which is also displayed. Limits
obtained separately from the individual semileptonic channel $\ell\nu qq$ and
the fully hadronic cannel $qqqq$ are shown for comparison. It turns out that
the semileptonic channel dominates the sensitivity over almost the whole
resonance mass range 0.5 TeV $\leq M_{Z^{\prime}}\leq$ 5 TeV, while in the
rather narrow mass range 2.2 TeV $\leq M_{Z^{\prime}}\leq$ 2.5 TeV the all-
hadronic channel is most sensitive.

###  III.4 $Z\text{-}Z^{\prime}$ mixing effects in dilepton decay of
$Z^{\prime}\to\ell\ell$

The above analysis was for the diboson process (3), employing one of the most
recent ATLAS searches for semileptonic Aad  _et al._ (2020); repository
(2020c) and fully hadronic Aad  _et al._ (2019c) final states. Next, we turn
to the dilepton production process (1), this process gives valuable
complementary information.

We compute the $Z^{\prime}$ theoretical production cross section at the LHC,
$\sigma(pp\to Z^{\prime}X)$, multiplied by the branching ratio into two
leptons, $\ell\ell$ ($\ell=e,\mu$), i.e., $\sigma(pp\to
Z^{\prime}X)\times\text{BR}(Z^{\prime}\to\ell\ell)$, as a function of
$M_{Z^{\prime}}$, and compare it with the upper limits established by the
experiment Aad  _et al._ (2019a) for $139~{}\text{fb}^{-1}$. We make use of
the relevant set of tables and figures (including additional results for
dielectron and dimuon channels) available at the Durham HepData repository
repository (2020a).

Results for $\sigma_{95\%}\times{\rm BR}(Z^{\prime}\to\ell\ell)$ are shown in
Fig. 5. To account for next-to-next-to-leading order (NNLO) effects in the QCD
strong coupling constant, the leading order (LO) cross sections calculated
with PYHTHIA 8.2 Sjöstrand  _et al._ (2015) are multiplied by a mass-
independent $K$-factor. The value of the $K$-factor is estimated at a dilepton
invariant mass of $\sim 3.0-4.5$ TeV and found to be consistent with unity
Aaboud  _et al._ (2017); Sirunyan  _et al._ (2018b).

Figure 5:  Solid: observed $95\%$ C.L. upper bound on the $Z^{\prime}$
production cross section times branching ratio to two leptons,
$\sigma_{95\%}\times\text{BR}(Z^{\prime}\to\ell\ell)$, where $\ell=e,\mu$,
obtained at the LHC with integrated luminosity ${\cal L}_{\rm int}$=139 fb-1
by the ATLAS collaboration Aad  _et al._ (2019a); repository (2020a). Results
are shown for the combined dilepton channel. Dashed lines: theoretical
production cross section $\sigma(pp\to Z^{\prime})\times{\rm
BR}(Z^{\prime}\to\ell\ell)$ for the $Z^{\prime}$ boson in the $\psi$ model,
calculated from PYHTHIA 8.2 with an NNLO $K$ factor. These curves in
descending order correspond to values of the $Z$-$Z^{\prime}$ mixing factor
$\xi_{Z\text{-}Z^{\prime}}$ from 0 to 0.005.

For illustrative purposes we show theoretical production cross sections
$\sigma(pp\to Z^{\prime}X)\times\text{BR}(Z^{\prime}\to\ell\ell)$ for the
$\psi$ model $Z^{\prime}$, given by the dashed curves in Fig. 5. These curves,
in descending order, correspond to values of the mixing factor
$\xi_{Z\text{-}Z^{\prime}}$ from 0 to $5\cdot 10^{-3}$. Qualitatively, the
decrease of the theoretical cross section with increasing values of
$\xi_{Z\text{-}Z^{\prime}}$ can be understood as follows: For increasing
$\xi_{Z\text{-}Z^{\prime}}$, the $Z^{\prime}\to W^{+}W^{-}$ mode will at high
mass $M_{Z^{\prime}}$ become more dominant (as illustrated in Fig. 2), and
$\text{BR}(Z^{\prime}\to\ell\ell)$ will decrease correspondingly. Notice also,
that applying a mass-dependent $K$-factor (which for this process is less than
1.04), the $\psi$ model mass limit of the $Z^{\prime}$ changes by only
$\sim{\cal O}$(50 GeV), justifying the use of the simpler mass-independent
$K$-factor Aaboud  _et al._ (2017); Sirunyan  _et al._ (2018b).

Table 2: Observed 95% C.L. lower mass limits on $M_{Z^{\prime}}$ for different $Z^{\prime}$ gauge models from $pp\to Z^{\prime}\to\ell\ell X$ taking into account the effect of potential $Z\text{-}Z^{\prime}$ mixing. Model | mixing parameter |  $M_{Z^{\prime}}$ (TeV) lower limits
---|---|---
$Z_{\psi}^{\prime}$ | no mixing | 4.5
| $\xi_{Z\text{-}Z^{\prime}}^{\rm EW}=1.8\cdot 10^{-3}$ | 3.8
$Z_{\eta}^{\prime}$ | no mixing | 4.6
| $\xi_{Z\text{-}Z^{\prime}}^{\rm EW}=4.7\cdot 10^{-3}$ | 3.3
$Z_{\chi}^{\prime}$ | no mixing | 4.8
| $\xi_{Z\text{-}Z^{\prime}}^{\rm EW}=1.6\cdot 10^{-3}$ | 4.2
$Z_{\rm LR}^{\prime}$ | no mixing | 4.9
| $\xi_{Z\text{-}Z^{\prime}}^{\rm EW}=1.3\cdot 10^{-3}$ | 4.5
$Z_{\rm EGM}^{\prime}$ | no mixing | 5.1
| $\xi_{Z\text{-}Z^{\prime}}^{\rm EW}=2.6\cdot 10^{-3}$ | 4.4

Figure 6: The $Z^{\prime}_{\eta}$ model: top-left, top-right, bottom-left,
bottom-right panels: analogous to Figs. 2, 3, 4, 5, respectively.

Figure 7: The $Z^{\prime}_{\chi}$ model: top-left, top-right, bottom-left,
bottom-right panels: analogous to Figs. 2, 3, 4, 5, respectively.

Figure 8: The $Z^{\prime}_{\rm LR}$ model: top-left, top-right, bottom-left,
bottom-right panels: analogous to Figs. 2, 3, 4, 5, respectively.

Figure 9: The $Z^{\prime}_{\rm EGM}$ model: top-left, top-right, bottom-left,
bottom-right panels: analogous to Figs. 2, 3, 4, 5, respectively. Bottom-left
panel: Also shown are exclusion regions obtained after incorporating direct
search constraints from the CDF and D0 collaborations which are referred to as
Tevatron (the dark shaded area) in $p\bar{p}\to W^{+}W^{-}X$ as well as those
derived from the LHC measurement of $p{p}\to WWX$ in Run 1 (the gray area)
Pankov  _et al._ (2020). For comparison, we also show the expected exclusion
from Run 3 (300 $\text{fb}^{-1}$) and the HL-LHC option (3000
$\text{fb}^{-1}$), see text.

Comparison of $\sigma(pp\to
Z^{\prime}X)\times\text{BR}(Z^{\prime}\to\ell\ell)$ vs
$\sigma_{95\%}\times{\rm BR}(Z^{\prime}\to\ell\ell)$ displayed in Fig. 5
permits us to read off an allowed mixing for a given mass value, higher masses
are allowed for smaller mixing, for the reason stated above. This analysis of
$Z$-$Z^{\prime}$ mixing, illustrated here for the $\psi$ model, can also be
performed for the other benchmark models. The results of the numerical
analysis for these tested models are presented in Figs. 6–9. Mass limits are
calculated as the intersection between the observed limits with the model
prediction. Table 2 lists the mass limits for two representative cases, namely
for vanishing mixing ($\xi_{Z\text{-}Z^{\prime}}=0$) and for the mixing
$\xi_{Z\text{-}Z^{\prime}}^{\rm EW}$ derived from the electroweak precision
data Erler  _et al._ (2009). The former are consistent with those derived in
Refs. Aad  _et al._ (2019a); collaboration (2019) whereas the mass limits at
$\xi_{Z\text{-}Z^{\prime}}^{\rm EW}$ are weaker by $\sim 10-30\%$.

Table 3: Upper limits on mixing parameters $\xi_{Z\text{-}Z^{\prime}}$ and $\xi_{W\text{-}W^{\prime}}$ at 95% C.L. in different models, processes and experiments (past - Tevatron, present - EW and LHC, future - ILC). We also compare with the expected ILC reach.  collider, process | $\xi_{Z\text{-}Z^{\prime}}^{\psi}$ | $\xi_{Z\text{-}Z^{\prime}}^{\eta}$ | $\xi_{Z\text{-}Z^{\prime}}^{\chi}$ | $\xi_{Z\text{-}Z^{\prime}}^{\rm LR}$ | $\xi_{Z\text{-}Z^{\prime}}^{\rm EGM}$ | $\xi_{W\text{-}W^{\prime}}^{\rm EGM}$ | @$M_{V}^{\prime}$ (TeV)
---|---|---|---|---|---|---|---
Tevatron, $p\bar{p}\to Z^{\prime}/W^{\prime}\to WW/WZ\,(\to\ell\nu\,qq)$ | … | … | … | … | $2\cdot 10^{-2}$ | $2\cdot 10^{-2}$ | 0.4–0.9
Aaltonen  _et al._ (2010) |  |  |  |  |  |  |
electroweak (EW) data Erler  _et al._ (2009); Zyla  _et al._ (2020) | $1.8\cdot 10^{-3}$ | $4.7\cdot 10^{-3}$ | $1.6\cdot 10^{-3}$ | $1.3\cdot 10^{-3}$ | $2.6\cdot 10^{-3}$ | $\sim 10^{-2}$ | …
LHC@13 TeV, 139 fb-1: Run 2 (this work) |  |  |  |  |  |  |
$p{p}\to Z^{\prime}/W^{\prime}\to WW/WZ\,(\to qqqq)$ | $2{.}9\cdot 10^{-4}$ | $2{.}7\cdot 10^{-4}$ | $3{.}8\cdot 10^{-4}$ | $3{.}0\cdot 10^{-4}$ | $3{.}1\cdot 10^{-4}$ | $4.3\cdot 10^{-4}$ | 1.3–5.0
$p{p}\to Z^{\prime}\to WW\,(\to\ell\nu\,qq)$ | $2{.}5\cdot 10^{-4}$ | $2{.}4\cdot 10^{-4}$ | $3{.}3\cdot 10^{-4}$ | $2{.}7\cdot 10^{-4}$ | $2{.}8\cdot 10^{-4}$ | … | 0.5–5.0
$p{p}\to W^{\prime}\to WZ\,(\to\ell\nu\,/\ell\ell/\nu\nu\,\,qq)$ | … | … | … | … | … | $2.9\cdot 10^{-4}$ | 0.5–5.0
<EMAIL_ADDRESS>TeV, 0.5 ab-1, $e^{+}e^{-}\to W^{+}W^{-}$ Andreev  _et al._ (2012) | $2.3\cdot 10^{-3}$ | $1.6\cdot 10^{-3}$ | $1.5\cdot 10^{-3}$ | $1.4\cdot 10^{-3}$ | $1.2\cdot 10^{-3}$ | … |  $\geq$ 3
<EMAIL_ADDRESS>TeV, 1.0 ab-1, $e^{+}e^{-}\to W^{+}W^{-}$ Andreev  _et al._ (2012) | $0.6\cdot 10^{-3}$ | $0.5\cdot 10^{-3}$ | $0.4\cdot 10^{-3}$ | $0.4\cdot 10^{-3}$ | $0.3\cdot 10^{-3}$ | … |  $\geq$ 3

As described above, both the diboson mode and the dilepton process yield
limits on the ($M_{Z^{\prime}}$, $\xi_{Z\text{-}Z^{\prime}}$) parameter space.
These are rather complementary, as shown in Fig. 4, where we collect these
limits for the $\psi$ model. The limits arising from the diboson channel are
basically excluding large values of $\xi_{Z\text{-}Z^{\prime}}$, strongest at
intermediate masses $M_{Z^{\prime}}\sim 2-4~{}\text{TeV}$. The limits arising
from the dilepton channel, on the other hand, basically exclude masses
$M_{Z^{\prime}}\mathrel{\hbox to0.0pt{\raise 2.5pt\hbox{$<$}\hss}\lower
2.5pt\hbox{$\sim$}}4.5~{}\text{TeV}$, with only a weak dependence on
$\xi_{Z\text{-}Z^{\prime}}$. For reference, we plot also a curve labelled
“Unitarity limit” that corresponds to the unitarity bound Alves  _et al._
(2009); Bobovnikov  _et al._ (2018). In Alves  _et al._ (2009), it was shown
that the saturation of unitarity in the elastic scattering $W^{+}W^{-}\to
W^{+}W^{-}$ leads to the constraint
$(g_{Z^{\prime}WW})_{\text{max}}=g_{ZWW}\cdot(M_{Z}/\sqrt{3}M_{Z^{\prime}})$
that was adopted here.

For comparison, we show in Fig. 9 (for the EGM model) also the exclusion reach
expected at the end of Run 3 (300 $\text{fb}^{-1}$) at the LHC, as well as at
the HL-LHC (3000 $\text{fb}^{-1}$) Cid Vidal  _et al._ (2019) which
illustrates the corresponding extension of the excluded limits on
$\xi_{Z-Z^{\prime}}$ down to $2.0\cdot 10^{-4}$ and $1.1\cdot 10^{-4}$,
respectively, whithin the $Z^{\prime}$ mass range under study. Furthermore,
the expected lower limit on the $Z^{\prime}$ mass can be set from the dilepton
production at higher luminosity. The current $Z_{\rm EGM}^{\prime}$ mass limit
of 5.1 TeV at $\xi_{Z-Z^{\prime}}=0$ obtained using 139 fb-1 of data will
extend to 6.7 TeV collaboration (2018b).

In Table 3, we collect our limits on the $Z^{\prime}$ parameters for the
benchmark models. Also shown in Table 3 are the limits on the $Z-Z^{\prime}$
($W-W^{\prime}$) mixing parameter $\xi_{Z-Z^{\prime}}$ ($\xi_{W-W^{\prime}}$)
from studies of diboson $WW$ ($WZ$) pair production at the Tevatron. Table 3
shows that the limits on $\xi_{Z-Z^{\prime}}$ from the EW precision data are
generally competitive with the future collider<EMAIL_ADDRESS>TeV, but in many
cases, they are stronger than those from the Tevatron.

The diboson production at the LHC@13 TeV allows to place stringent constraints
on the $Z$-$Z^{\prime}$ mixing angle and $Z^{\prime}$ mass, $M_{Z^{\prime}}$.
We imposed limits on the mass and the $Z$-$Z^{\prime}$ mixing angle of the
$Z^{\prime}$ bosons by using data comprised of $pp$ collisions at
$\sqrt{s}=13$ TeV and recorded by the ATLAS detectors at the CERN LHC, with
integrated luminosities of $\sim$ 139 fb-1 from Run 2 data taking.

Also, we show that the derived constraints on the $Z$-$Z^{\prime}$ mixing
angle for the benchmark models are of the order of a few$\,\times 10^{-4}$ and
they are greatly improved with respect to those derived from the global
analysis of electroweak data, EW. In addition, we demonstrated in Fig. 9 that
further improvement on the constraining of this mixing can be achieved from
the analysis of data to be collected at higher luminosity expected in the Run
3 and HL-LHC options. We also show that only the future $e^{+}e^{-}$ linear
collider ILC with polarized beams and with very high energy and luminosity,
$\sqrt{s}=1$ TeV and ${\cal L}_{\rm int}=1\,{\rm ab}^{-1}$, may have a chance
to compete with the current LHC sensitivity to the mixing angle in Run 2 but
will not reach the levels of the Run 3 and HL-LHC options.

##  IV $W^{\prime}$ production and decay in $pp$ collision

In contrast to the rich spectrum of $Z^{\prime}$ models considered above, with
different vector and axial-vector couplings, for $W^{\prime}$ we consider only
$V-A$ couplings to fermions.

###  IV.1 $W^{\prime}$ resonant production cross section

We consider the simplest EGM model which predicts charged heavy gauge bosons.
The analysis of $W\text{-}W^{\prime}$ mixing in diboson and dilepton pair
production which will be performed below is quite analogous to that carried
out in previous sections for $Z\text{-}Z^{\prime}$ mixing. At lowest order in
the EGM, $W^{\prime}$ production and decay into $WZ$ in proton-proton
collisions occurs through quark-antiquark annihilation in the $s$-channel.
Using the NWA, one can factorize the process (4) into the $W^{\prime}$
production and the $W^{\prime}$ decay,

$\sigma(pp\to W^{\prime}X\to WZX)=\sigma(pp\to
W^{\prime}X)\times\text{BR}(W^{\prime}\to WZ)\;.$ (14)

Here, $\sigma(pp\to W^{\prime}X)$ is the total (theoretical) $W^{\prime}$
production cross section and $\text{BR}(W^{\prime}\to
WZ)=\Gamma_{W^{\prime}}^{WZ}/\Gamma_{W^{\prime}}$ with $\Gamma_{W^{\prime}}$
the total width of the $W^{\prime}$.

###  IV.2 The $W^{\prime}$ width

In the EGM the $W^{\prime}$ bosons can decay into SM fermions, gauge bosons
($WZ$), or $WH$. In the calculation of the total width $\Gamma_{W^{\prime}}$
we consider the following channels: $W^{\prime}\to f{\bar{f}}^{\prime}$, $WZ$,
and $WH$, where $f$ is a SM fermion ($f=\ell,\nu,q$) 555Here, in contrast to
the $Z^{\prime}$ case, the $\ell$ includes $\tau$ leptons.. Only left-handed
neutrinos are considered, possible right-handed neutrinos are assumed to be
kinematically unavailable as final states. Also, like for the $Z^{\prime}$
case, we shall ignore the couplings to other beyond-SM particles such as SUSY
partners and exotic fermions. As a result, the total decay width of the
$W^{\prime}$ boson is taken to be

$\Gamma_{W^{\prime}}=\sum_{f}\Gamma_{W^{\prime}}^{f\bar{f}^{\prime}}+\Gamma_{W^{\prime}}^{WZ}+\Gamma_{W^{\prime}}^{WH}.$
(15)

Like for the $Z^{\prime}$ case, the presence of the last two decay channels,
which are often neglected at low and moderate values of $M_{W^{\prime}}$, is
due to $W$-$W^{\prime}$ mixing which is constrained to be tiny. In particular,
for the range of $M_{W^{\prime}}$ values below $\sim 1.0-1.5$ TeV, the
dependence of $\Gamma_{W^{\prime}}$ on the values of
$\xi_{W\text{-}W^{\prime}}$ (within its allowed range) induced by
$\Gamma_{W^{\prime}}^{WZ}$ and $\Gamma_{W^{\prime}}^{WH}$ is unimportant
because $\sum_{f}\Gamma_{W^{\prime}}^{f\bar{f^{\prime}}}$ dominates over the
diboson partial widths. Therefore, in this mass range, one can approximate the
total width as
$\Gamma_{W^{\prime}}\approx\sum_{f}\Gamma_{W^{\prime}}^{f\bar{f^{\prime}}}=3.5\%\times
M_{W^{\prime}}$ Serenkova  _et al._ (2019), where the sum runs over SM
fermions only.

For heavier $W^{\prime}$ bosons, the diboson decay channels, $WZ$ and $WH$,
start to play an important role, and we are no longer able to ignore them
Serenkova  _et al._ (2019); Pankov  _et al._ (2020). To be specific, in
analogy with the $Z^{\prime}$ case, we assume that both partial widths are
comparable, $\Gamma_{W^{\prime}}^{WH}\simeq\Gamma_{W^{\prime}}^{WZ}$ for heavy
$M_{W^{\prime}}$, as required by the Equivalence theorem Chanowitz and
Gaillard (1985).

Figure 10:  Left panel: Branching ratio $\text{BR}(W^{\prime}\to WZ)$ (solid)
vs $M_{W^{\prime}}$ in the EGM for $W$-$W^{\prime}$ mixing factor
$\xi_{W\text{-}W^{\prime}}=10^{-2}$. Dashed: $\text{BR}(W^{\prime}\to e\nu)$
for $\xi_{W\text{-}W^{\prime}}=0$ ($W^{\prime}_{\rm SSM}$) and
$\xi_{W\text{-}W^{\prime}}=0.01$. Right panel: $95\%$ C.L. upper limits on
$\sigma_{95\%}\times\text{BR}(W^{\prime}\to WZ)$, showing ATLAS data on the
fully hadronic and semileptonic final states for $139~{}\text{fb}^{-1}$ Aad
_et al._ (2019c, 2020). The theoretical production cross sections
$\sigma(pp\to W^{\prime}X)\times\text{BR}(W^{\prime}\to WZ)$ for the ${\rm
EGM}$ are calculated from PYTHIA with a $W^{\prime}$ mass-dependent
$K$-factor, given by solid curves, for mixing factor
$\xi_{W\text{-}W^{\prime}}$ ranging from $10^{-2}$ and down to $3\cdot
10^{-4}$. The NWA and unitarity constraints are also shown Alves  _et al._
(2009); Serenkova  _et al._ (2019).

The expression for the partial width of the $W^{\prime}\to WZ$ decay channel
in the EGM can be written as Altarelli  _et al._ (1989); Serenkova  _et al._
(2019):

$\displaystyle\Gamma_{W^{\prime}}^{WZ}$ $\displaystyle=$
$\displaystyle\frac{\alpha_{\rm
em}}{48}\cot^{2}\theta_{W}\,M_{W^{\prime}}\frac{M_{W^{\prime}}^{4}}{M_{W}^{2}M_{Z}^{2}}\left[\left(1-\frac{M_{Z}^{2}-M_{W}^{2}}{M_{W^{\prime}}^{2}}\right)^{2}-4\,\frac{M_{W}^{2}}{M_{W^{\prime}}^{2}}\right]^{3/2}$
$\displaystyle\times\left[1+10\left(\frac{M_{W}^{2}+M_{Z}^{2}}{M_{W^{\prime}}^{2}}\right)+\frac{M_{W}^{4}+M_{Z}^{4}+10M_{W}^{2}M_{Z}^{2}}{M_{W^{\prime}}^{4}}\right]\cdot\xi_{W\text{-}W^{\prime}}^{2}.$

For a fixed mixing factor $\xi_{W\text{-}W^{\prime}}$ and at large
$M_{W^{\prime}}$, the total width increases rapidly with the $W^{\prime}$ mass
because of the quintic dependence of the $WZ$ mode on the $W^{\prime}$ mass
$\Gamma_{W^{\prime}}^{WZ}\propto
M_{W^{\prime}}\left[{M_{W^{\prime}}^{4}}/({M_{W}^{2}M_{Z}^{2}})\right]$,
corresponding to the production of longitudinally polarized $W$ and $Z$ in the
channel $W^{\prime}\to W_{L}Z_{L}$ Altarelli  _et al._ (1989); Serenkova  _et
al._ (2019). In this case, the $WZ$ mode (as well as $WH$) becomes dominant
and $\text{BR}(W^{\prime}\to WZ)\to 0.5$, while the fermionic decay channels,
$\sum_{f}\Gamma_{W^{\prime}}^{f\bar{f^{\prime}}}\propto M_{W^{\prime}}$, are
increasingly suppressed, as illustrated in Fig. 10 (left panel).

###  IV.3 Hadron production and diboson decay of $W^{\prime}$

Our analysis employs the recent searches for diboson processes in semileptonic
final states provided by ATLAS Aad  _et al._ (2020) with the full Run 2 data
set with time-integrated luminosity of 139 fb-1 as well as, for the sake of
comparison, in the fully hadronic ($qqqq$) final states Aad  _et al._ (2019c).

In Fig. 10 (right panel), we show the observed $95\%$ C.L. upper limits on the
production cross section times the branching fraction,
$\sigma_{95\%}\times\text{BR}(W^{\prime}\to WZ)$, as a function of the
$W^{\prime}$ mass.

Then, for $W^{\prime}$ we compute the LHC theoretical production cross section
multiplied by the branching ratio into $WZ$ bosons, $\sigma(pp\to
W^{\prime}X)\times{\rm BR}(W^{\prime}\to WZ)$, as a function of the two
parameters ($M_{W^{\prime}}$, $\xi_{W\text{-}W^{\prime}}$) Serenkova  _et al._
(2019), and compare it with the limits established by the ATLAS experiment,
$\sigma_{95\%}\times{\rm BR}(W^{\prime}\to WZ)$. The simulation of signals for
the EGM $W^{\prime}$ is based on a suitably adapted version of the leading
order PYHTHIA 8.2 event generator Sjöstrand  _et al._ (2015). A mass-dependent
$K$ factor is adopted to rescale the LO PYTHIA prediction to the the NNLO one,
using the ZWPROD Hamberg  _et al._ (1991) software. The result is presented as
solid curves in the right panel for a mixing factor
$\xi_{W\text{-}W^{\prime}}$ ranging from $10^{-2}$ and down to $3\cdot
10^{-4}$. The factorization and renormalization scales are both set to the
$W^{\prime}$ mass.

The area below the long-dashed curve labelled “NWA” corresponds to the region
where the $W^{\prime}$ resonance width is predicted to be less than 5% of its
mass, corresponding to the best detector resolution of the searches, where the
narrow-width assumption is satisfied. We also show a curve labelled “Unitarity
limit” that corresponds to the unitarity bound (see, e.g. Alves  _et al._
(2009) and references therein). It was shown that the saturation of unitarity
in the elastic scattering $W^{\pm}Z\to W^{\pm}Z$ leads to the constraint
$(g_{W^{\prime}WZ})_{\text{max}}=g_{WWZ}\cdot
M_{Z}^{2}/(\sqrt{3}\,M_{W^{\prime}}\,M_{W})$ that was adopted in plotting this
bound. The constraint was obtained under the assumption that the couplings of
the $W^{\prime}$ to quarks and to gauge bosons have the same Lorentz structure
as those of the SM but with rescaled strength.

The theoretical curves for the cross sections $\sigma(pp\to
W^{\prime}X)\times{\rm BR}(W^{\prime}\to WZ)$, in descending order, correspond
to values of the $W$-$W^{\prime}$ mixing factor $\xi_{W\text{-}W^{\prime}}$
from 0.01 to 0.0003. The intersection points of the measured upper limits on
the production cross section with these theoretical cross sections for various
values of $\xi_{W\text{-}W^{\prime}}$ give the corresponding lower bounds on
($M_{W^{\prime}}$, $\xi_{W\text{-}W^{\prime}}$), displayed in Fig. 11, left
panel.

Figure 11:  Left panel: 95% C.L. exclusion regions in the two-dimensional
($M_{W^{\prime}}$, $\xi_{W\text{-}W^{\prime}}$) plane obtained from the
precision electroweak data (horizontal dashed straight line labeled “EW”),
direct search constraints from the Tevatron in $p\bar{p}\to WZX$ (dark shaded
area) as well as from the LHC searches for $p{p}\to WZX$ at 7 TeV and 8 TeV
(Run 1) (gray area) and at 13 TeV from diboson $W^{\prime}\to WZ$ production
in hadronic and semileptonic final states using the full Run 2 ATLAS data set.
The region above each curve for the $WZ$ channel is excluded. The steep curve
labelled “excluded by $W^{\prime}\to\ell\nu$” shows the exclusion based on the
dilepton channel $pp\to\ell\nu X$. The overall allowed region for the EGM
$W^{\prime}$ boson is shown as the yellow area. Right panel: Solid: observed
$95\%$ C.L. upper bound on the $W^{\prime}$ production cross section times
branching ratio to two leptons,
$\sigma_{95\%}\times\text{BR}(W^{\prime}\to\ell\nu)$, obtained in the combined
channels (electron and muon) at the LHC with integrated luminosity ${\cal
L}_{\rm int}$=139 fb-1 by the ATLAS collaboration Aad  _et al._ (2019b).
Dashed lines: theoretical production cross section $\sigma(pp\to
W^{\prime})\times{\rm BR}(W^{\prime}\to\ell\nu)$ for the EGM $W^{\prime}$
boson, calculated from PYHTHIA 8.2 with an NNLO $K$ factor. These curves, in
descending order, correspond to values of the $W$-$W^{\prime}$ mixing factor
$\xi$ from 0 to 0.01.

Comparison of sensitivities of the process (4) to $W^{\prime}$ with different
decay channels, e.g., $VV\to\ell\nu qq$ and $qqqq$, can be performed by the
matching of $95\%$ C.L. upper limits on the production cross section times the
branching fraction, $\sigma_{95\%}\times\text{BR}(W^{\prime}\to WZ)$, which
includes the SM branching fractions of the electroweak bosons to the final
states in the analysis channel, effects from detector acceptance, as well as
reconstruction and selection efficiencies. ATLAS bounds were included
acccording to the Durham HEPdata repository repository (2020b, c).

From a comparison of the upper limits on the production cross section times
the branching fraction for semileptonic vs. fully hadronic decay channels, one
can conclude that the sensitivity of the semileptonic channel dominates over
the fully hadronic one within the whole range of the $W^{\prime}$ mass, from
0.5 TeV to 5 TeV. These features are illustrated in Fig. 10 (right panel) and
Fig. 11 (left panel).

For reference, we display limits on the $W^{\prime}$ parameters from the
Tevatron (CDF and D0) as well as from ATLAS and CMS obtained at 7 and 8 TeV of
LHC data taking in Run 1 denoted “LHC Run 1” Serenkova  _et al._ (2019). Fig.
11 (left panel) shows that the experiments CDF and D0 at the Tevatron exclude
EGM $W^{\prime}$ bosons with $\xi_{W\text{-}W^{\prime}}\mathrel{\raise
1.29167pt\hbox{$>$\kern-7.5pt\lower 4.30554pt\hbox{$\sim$}}}2\cdot 10^{-2}$ in
the resonance mass range 0.25 TeV $<M_{W^{\prime}}<$ 1 TeV at the $95\%$ C.L.,
whereas LHC in Run 1 improved those constraints, excluding $W^{\prime}$ boson
parameters at $\xi_{W\text{-}W^{\prime}}\mathrel{\raise
1.29167pt\hbox{$>$\kern-7.5pt\lower 4.30554pt\hbox{$\sim$}}}2\cdot 10^{-3}$ in
the mass range 0.2 TeV $<M_{W^{\prime}}<$ 2 TeV.

As expected, the increase of the time-integrated luminosity up to 139 fb-1
leads to dominant sensitivity of the semileptonic channel over the whole
resonance mass range of 0.5 TeV $<M_{W^{\prime}}<$ 5 TeV and it allows to set
stronger constraints on the mixing angle $\xi_{W\text{-}W^{\prime}}$,
excluding $\xi_{W\text{-}W^{\prime}}>2.3\cdot 10^{-4}$ as shown in Fig. 11.
Our results extend the sensitivity beyond the corresponding CDF Tevatron
results Aaltonen  _et al._ (2010) as well as the ATLAS and CMS sensitivity
attained at 7 and 8 TeV. Also, for the first time, we set $W^{\prime}$ limits
as functions of the mass $M_{W^{\prime}}$ and mixing factor
$\xi_{W\text{-}W^{\prime}}$ from the study of the diboson production and
subsequent decay into semileptonic final states at the LHC at 13 TeV with the
full ATLAS Run 2 data set. The exclusion region obtained in this way on the
parameter space of the $W^{\prime}$ naturally supersedes the corresponding
exclusion area obtained for time-integrated luminosity of 36.1 fb-1 in the
semileptonic channel as reported in Serenkova  _et al._ (2019). The limits on
the $W^{\prime}$ parameters presented in this section obtained from the
diboson $WZ$ production in semileptonic final states, corresponding to a time-
integrated luminosity of 139 fb-1, are the best to date.

###  IV.4 $W\text{-}W^{\prime}$ mixing effects in dilepton decay of
$W^{\prime}\to\ell\nu$

The above analysis was for the diboson process (4), employing one of the most
recent ATLAS searches Aad  _et al._ (2020, 2019c). Next, we turn to the
dilepton production process (2), this process gives valuable complementary
information. Unlike the SSM, where there is no $W$-$W^{\prime}$ mixing, in the
EGM we consider a non-zero mixing $\xi_{W\text{-}W^{\prime}}$ in the analysis
of the $W^{\prime}\to\ell\nu$ process. As described in Sec. IV.2, this results
in a modification of $\text{BR}(W^{\prime}\to e\nu)$.

We compute the $W^{\prime}$ production cross section at LO with PYTHIA 8.2
Sjöstrand  _et al._ (2015), $\sigma(pp\to W^{\prime})$, multiplied by the
branching ratio into two leptons, $\ell\nu$ (here $\ell=e$), i.e.,
$\sigma(pp\to W^{\prime})\times{\rm BR}(W^{\prime}\to e\nu)$, as a function of
$M_{W^{\prime}}$. A mass-dependent $K$ factor is applied, based on NNLO QCD
cross sections as calculated with FEWZ 3.1 Gavin  _et al._ (2011); Li and
Petriello (2012). The $K$ factor varies approximately from 1.3 to 1.1 for the
range of $W^{\prime}$ masses studied in this analysis, namely from 0.5 to 6.0
TeV. The NNLO corrections decrease with increasing $W^{\prime}$ mass up to
around 4.5 TeV Khachatryan  _et al._ (2017). For higher $W^{\prime}$ masses,
the $K$ factor increases again and becomes similar to the low-mass values.

The product of the NNLO $W^{\prime}$ theoretical production cross section and
branching fraction, $\sigma(pp\to W^{\prime})\times{\rm BR}(W^{\prime}\to
e\nu)$, for the $W^{\prime}$ boson for EGM strongly depends on the
$W^{\prime}$ mass, and is given by dashed curves, in descending order,
corresponding to values of the mixing factor $\xi$ from 0.0 to 0.01, as
displayed in Fig. 11 (right panel).

Comparison of $\sigma(pp\to W^{\prime})\times{\rm BR}(W^{\prime}\to e\nu)$ vs
$\sigma_{95\%}\times\text{BR}(W^{\prime}\to\ell\nu)$ displayed in Fig. 11
(right panel) allows us to read off an allowed mixing for a given mass value,
higher masses are allowed for smaller mixing, for the reason stated above.
That comparison can be translated into constraints on the two-dimensional
$M_{W^{\prime}}$-$\xi_{W\text{-}W^{\prime}}$ parameter plane, as shown in Fig.
11 (left panel).

The above results are based on data corresponding to an integrated luminosity
of 139 fb-1 taken by the ATLAS collaboration at $\sqrt{s}=13$ TeV in Run 2 Aad
_et al._ (2019b). The corresponding lower limits on the $W^{\prime}$ boson
mass of 6 TeV (at $\xi_{W\text{-}W^{\prime}}=0$) was set at 95% C.L. from
combination of the electron and muon channels. Notice that, similar to the
case of $Z^{\prime}$ bosons, at $\xi_{W\text{-}W^{\prime}}^{\rm EW}=10^{-2}$
these limits become weaker, reaching $\sim$ 4.4 TeV, as illustrated in Fig. 11
(left and right panels).

##  V Concluding remarks

Examination of the diboson, $WW$ and $WZ$, and dilepton, $\ell\ell$ and
$\ell\nu$, production at the LHC with the 13 TeV data set allows to place
stringent constraints on the $Z$-$Z^{\prime}$ and $W$-$W^{\prime}$ mixing
parameters as well as on the $Z^{\prime}$ and $W^{\prime}$ masses for
benchmark extended models, respectively. We derived such limits by using the
full ATLAS Run 2 data set recorded at the CERN LHC, with integrated luminosity
of 139 fb-1. The constraints are summarized in Table 3. We note that in a
situation when the limit is dominated by statistical errors, the $K$-factor
plays a role similar to integrated luminosity. Thus, if we had adopted the
same $K$-factor for the $WW$ ($WZ$) channel as for the dilepton channel, the
bounds on $\xi$ would have been slightly weaker, by a factor $K^{1/4}\simeq
1.17$ for the $Z^{\prime}$ case Bobovnikov  _et al._ (2018).

By comparing the experimental limits to the theoretical predictions for the
total cross section of the $Z^{\prime}$ and $W^{\prime}$ resonant production
and their subsequent decays into $WW$ or $WZ$ pairs, we show that the derived
constraints on the mixing parameters, $\xi_{Z\text{-}Z^{\prime}}$ and
$\xi_{W\text{-}W^{\prime}}$, are substantially improved with respect to those
obtained from the global analysis of low energy electroweak data, as well as
compared to the diboson production study performed at the Tevatron, and to
those published previously and based on the LHC Run 1 as well as at 13 TeV in
Run 2 at time-integrated luminosity of $\sim 36$ fb-1 and are the most
stringent bounds to date. Further constraining of this mixing can be achieved
from the analysis of data to be collected in Run 3 as well as at the next
options of hadron colliders such as HL-LHC and HE-LHC Cid Vidal  _et al._
(2019); Abada  _et al._ (2019).

## Acknowledgements

It is a pleasure to thank Dr. Anurag Tripathi for discussions on $K$-factors.
This research has been partially supported by the Abdus Salam ICTP (TRIL
Programme) and by the Belarusian Republican Foundation for Fundamental
Research, F20MC-004 and F20MC-005. The work of PO has been supported by the
Research Council of Norway.

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