aainstitutetext: Institute for Particle Physics Phenomenology Durham
University, South Road, Durham, DH1 3LE, UKbbinstitutetext: Department of
Physics, Concordia University, 7141 Sherbrooke St. West, Montreal, Quebec H4B
1R6, Canadaccinstitutetext: Laboratoire de Physique Théorique et Hautes
Energies (LPTHE), UMR 7589, Sorbonne Université et CNRS, 4 place Jussieu,
75252 Paris Cedex 05, Franceddinstitutetext: School of Physics $\&$ Astronomy,
University of Southampton, Highfield, Southampton SO17 1BJ, UK

# Cross-fertilising extra gauge boson searches at the LHC

Jack Y. Araz b , Mariana Frank c , Benjamin Fuks d , Stefano Moretti b,d
 and Özer Özdal<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

For the purpose of cross-fertilising currently separate experimental
approaches, we connect results of LHC analyses attempting to access the
properties of additional $W^{\prime}$ and $Z^{\prime}$ bosons from Drell-Yan
processes. Under theoretical assumptions linking the two new gauge bosons, we
take into account that such possible states of nature are wide enough (in
relation to the leptonic mass resolution) for the corresponding signals be
significantly affected by interference effects with the background from the
Standard Model. The shape of the differential cross section may then no longer
be a standard Breit-Wigner distribution, and asymmetry observables would
become useful for characterisation (and, possibly, discovery) purposes. Under
such conditions we concentrate our analysis on specific widely-studied models:
the Sequential Standard Model, a model with an additional $SU(2)_{L}$ gauge
symmetry, as well as standard and alternative realisations of the Left-Right
Symmetric Model. We show how information gathered in $Z^{\prime}$ boson
searches in terms of cross section and/or asymmetry distributions can be used
to improve $W^{\prime}$ boson searches in terms of the LHC sensitivity, and
vice versa.

IPPP/21/04

##  1 Introduction

The Standard Model (SM) is very successful in explaining most of the available
experimental data, and it has survived all tests at colliders. Yet it appears
to be theoretically inconsistent or incomplete, and phenomena such as dark
matter, neutrino oscillations and matter-antimatter asymmetry remain
unanswered. A more complete description of nature, such as via Grand
Unification Theories (GUTs), supersymmetry, extra-dimensional models or
scenarios featuring dynamical symmetry breaking at a higher scale, is thus in
order and should introduce additional symmetries and particles. In particular,
extra gauge symmetries predict the presence of extra gauge bosons, with
properties similar to those of the usual $Z$ and $W$ bosons of the SM.
Specifically, $Z^{\prime}$ bosons point to the presence of at least an extra
$U(1)^{\prime}$ symmetry Hewett:1988xc ; Langacker:2008yv ; Accomando:2016eom
, while $W^{\prime}$ bosons are indicative of at least an extra
$SU(2)^{\prime}$ gauge group Mohapatra:1974gc ; Senjanovic:1975rk ; Beg:1977ti
; Ma:1986we .

Searches for these particles, thus, continue to be an important part of the
LHC physics programme. Consequently, an ability to improve the experimental
sensitivity in testing theoretical hypotheses is still desirable. Such heavy
resonances with significant couplings to quarks and leptons are most easily
seen at hadron colliders in the Drell-Yan production mode, wherein quark-
antiquark annihilations into a possibly off-shell $Z^{\prime}$ or $W^{\prime}$
boson is followed by the extra boson decay into either a dilepton or a lepton
plus missing transverse energy final state respectively ATLAS:2019erb ;
ATLAS:2019lsy ; Sirunyan:2021khd . The signals searched for in these Drell-Yan
channels are among the cleanest ones at high-energy hadron colliders.
Historically, both the charged current and neutral current channels have been
pursued separately without exploiting at the experimental level the
possibility that $Z^{\prime}$ and $W^{\prime}$ states could originate from a
single theoretical framework. This would however offer the possibility of
optimising searches for one state based on results concerning the other. While
such correlations could already be used at the level of total rates to
identify the underlying theoretical assumptions (that include at least
additional $SU(2)^{\prime}$ symmetries) Jezo:2012rm ; deBlas:2012qp , we push
forward this idea and assess the improvements that could arise from potential
measurements at the differential level.

The LHC collaborations have already looked for hints of additional gauge
bosons, albeit by separately considering neutral and charged states, and
imposed stringent constraints on their masses. These states are now required
to be heavier than 4–5 TeV, the exact limits depending on the specific new
gauge boson properties and those of the model in general (mass $M$, width
$\Gamma$, and couplings to the SM states). At such high masses, these
resonances have in particular a width that is comparable to the experimental
resolution in the invariant and the transverse masses of the final state which
are used to extract the new gauge boson signals. This is not only the case
when the extra bosons are broad, but also when they are narrow (i.e. for
$\Gamma/M\sim\alpha$, with $\alpha$ being the electromagnetic coupling
constant) or equivalently when the narrow-width approximation (NWA) holds.
Replacing the Breit-Wigner (BW) distribution by a $\delta$ function is
therefore an inaccurate way to characterise the production and decay of the
new states Accomando:2011eu ; Accomando:2013sfa ; Accomando:2016mvz ;
Accomando:2019ahs . Furthermore, in these circumstances, interference effects
with the SM background become visible in the aforementioned mass spectra
Accomando:2013sfa ; Kahlhoefer:2019vhz . A combined correlation of
$W^{\prime}$ and $Z^{\prime}$ analyses, including finite width effects, is so
far lacking. This is addressed in this work.

One way to take into account all correlations between $W^{\prime}$ and
$Z^{\prime}$ processes in full generality would be to consider a general
framework in which the interactions of new heavy resonances are parameterised
in a model-independent manner Fuks:2017vtl . In such a scenario, the
Lagrangian describing the interactions however includes many free parameters,
rendering this task unpractical. Alternatively, it may be desirable to
restrict this freedom and choose several simpler and well-motivated specific
models. For example, in the Sequential Standard Model (SSM)
Gopalakrishna:2010xm ; Han:2012vk , the $Z^{\prime}$ and $W^{\prime}$
couplings are assumed to be SM-like and the only free parameter is their mass,
possibly together with a global normalisation factor for the coupling
strengths. In models with additional $SU(2)^{\prime}$ gauge groups
Hsieh:2010zr , that include Left-Right Symmetric Models (LRSMs) Pati:1974yy ;
Mohapatra:1974gc ; Senjanovic:1975rk ; Mohapatra:1977mj ; Senjanovic:1978ev ;
Ma:2010us , the $W^{\prime}$ and $Z^{\prime}$ bosons are described in terms of
a minimal set of parameters. In this work, we follow this second approach and
we investigate the LHC phenomenology of a class of realistic theories Beyond
the SM (BSM), with the electroweak symmetry being extended to include an
additional $SU(2)^{\prime}$ gauge group. This predicts the existence of both a
$W^{\prime}$ and a $Z^{\prime}$ boson, and we will consider the best way to
discover those neutral and charged resonances through their Drell-Yan
production and decay and by using the invariant mass and transverse mass
distributions of the final-state products. We will consider not only total
rates, as done in previous works, but also differential distributions. The
signature of the production of a new vector boson and its subsequent decay
will indeed lead to a resonant peak in the invariant mass distribution of the
dilepton final state (for the $Z^{\prime}$ boson case) and to a Jacobian
shoulder in the transverse mass spectrum of the lepton-neutrino one (for the
$W^{\prime}$ boson case). We will also be looking at the asymmetries of the
final states, as the latter observables have already been proven to aid a
potential discovery of wide $W^{\prime}$ and $Z^{\prime}$ states
Accomando:2015cfa ; Accomando:2019ahs ; Fiaschi:2021okg .

In the next section we describe the Lagrangian terms relevant to $W^{\prime}$
and $Z^{\prime}$ production and decay at the LHC via Drell-Yan processes, and
detail specific BSM realisations featuring these extra gauge bosons
simultaneously (namely the SSM, a model with an additional $SU(2)_{L}$, and
other models with an additional $SU(2)_{R}$ symmetry). We also include, for
comparison, models featuring extra $U(1)$ gauge groups that are motivated by
an $E_{6}$ GUT symmetry or by gauging $B-L$ number conservation. We then
discuss experimental limits on $W^{\prime}$ and $Z^{\prime}$ bosons and next
present our numerical results in Section 3. We summarise our work and conclude
in Section 4.

##  2 Theoretical frameworks

In this section, we briefly introduce the various theoretical frameworks
relevant to our study. We begin with a general Lagrangian description, before
moving on with details on the various well-motivated models considered. In the
last part of this section, we introduce $U(1)^{\prime}$ extensions of the SM
which we will use as reference standards for the comparison of various LHC
signals.

###  2.1 Models with additional $W^{\prime}$ and $Z^{\prime}$ bosons

####  2.1.1 General Lagrangian description

In this study, we examine the LHC phenomenology of SM extensions featuring an
enlarged gauge group. We assume that the SM gauge symmetry is supplemented by
an additional generic $SU(2)^{\prime}$ group, which predicts the existence of
both $W^{\prime}$ and $Z^{\prime}$ bosons. By considering the couplings to the
fermions to be arbitrary, their form is only dictated by Lorentz and gauge
invariance. The most general charged current (CC) and neutral current (NC)
Lagrangians ${\cal L}_{\rm CC}^{W^{\prime}}$ and ${\cal L}_{\rm
NC}^{Z^{\prime}}$ respectively describing the interactions of a charged and a
neutral spin-one gauge boson with the SM fermions $f$ then read, after
including right-handed neutrino fields,

$\begin{split}{\cal L}_{\rm CC}^{W^{\prime}}=&\
\frac{g_{2}^{\prime}}{\sqrt{2}}\left[{\bar{u}}_{i}\gamma^{\mu}\left([\kappa_{q,L}^{W^{\prime}}]_{ij}P_{L}+[\kappa_{q,R}^{W^{\prime}}]_{ij}P_{R}\right)d_{j}+{\bar{\nu}}_{i}\gamma^{\mu}\left([\kappa_{\ell,L}^{W^{\prime}}]_{ij}P_{L}+[\kappa_{\ell,R}^{W^{\prime}}]_{ij}P_{R}\right)\ell_{j}\right]W^{\prime}_{\mu}+{\rm
H.c.}\,,\\\ {\cal L}_{\rm NC}^{Z^{\prime}}=&\
\frac{g_{2}^{\prime}}{\cos\theta_{W}}\left[{\bar{q}}_{i}\gamma^{\mu}\left([\kappa_{q,L}^{Z^{\prime}}]_{ij}P_{L}+[\kappa_{q,R}^{Z^{\prime}}]_{ij}P_{R}\right)q_{j}+{\bar{\ell}}_{i}\gamma^{\mu}\left([\kappa_{\ell,L}^{Z^{\prime}}]_{ij}P_{L}+[\kappa_{\ell,R}^{Z^{\prime}}]_{ij}P_{R}\right)\ell_{j}\right]Z^{\prime}_{\mu}+{\rm
H.c.}\,,\end{split}$ (1)

where a summation over quark and lepton flavours is understood for both
Lagrangians, and where we denote by $g_{2}^{\prime}$ the $SU(2)^{\prime}$
coupling constant. Here, the arbitrary parameters $\kappa$ are complex-valued
elements of $3\times 3$ matrices in the flavour space (with $q=u,d$ and $i,j$
being generation indices). These expressions for the CC and NC Lagrangians are
inspired by the SM, that we recover by setting

$\begin{split}[\kappa_{q,L}^{W^{\prime}}]_{ij}=V_{ij}^{\rm
CKM}\,,\qquad[\kappa_{\ell,L}^{W^{\prime}}]_{ij}=V_{ij}^{\rm
PMNS}\,,\qquad\kappa_{q(\ell),R}^{W^{\prime}}=0\,,\\\
[\kappa_{f,L}^{Z^{\prime}}]_{ij}=\left(T_{L}^{3,f}-Q_{f}\sin^{2}\theta_{W}\right)\delta_{ij}\,,\qquad[\kappa_{f,R}^{Z^{\prime}}]_{ij}=\left(-Q_{f}\sin^{2}\theta_{W}\right)\delta_{ij}\,,\end{split}$
(2)

after assuming that the $g_{2}^{\prime}$ coupling is the same as the weak
coupling constant $g_{W}$. In this notation, $V^{\rm CKM}$ and $V^{\rm PMNS}$
are the CKM and PMNS matrices, $T^{3}_{L}$ and $Q$ are the weak isospin and
electric charge operators and $\theta_{W}$ is the electroweak mixing angle.
Whereas in principle, gauge boson mixing can induce $W^{\prime}$ and
$Z^{\prime}$ couplings to a pair of SM weak bosons, such mixings are either
suppressed or forbidden in the new physics frameworks considered. They are
therefore ignored in the following.

These couplings encapsulate a large number of free parameters, and this still
holds even if we rely on sum rules, unitarity constraints and anomaly
cancellations to reduce the number of degrees of freedoms. As intimated, a
completely model-independent study of the associated collider phenomenology is
thus unfeasible practically, although attempts in this direction have been
made deBlas:2012qp . Furthermore, a detailed analysis of a particular chosen
model configuration, where we fix some of the parameters arbitrarily, may not
correspond to a theoretically-motivated framework and thus not be very
informative. For this reason, we rely in the following on specific classes of
models predicting $W^{\prime}$ and $Z^{\prime}$ bosons couplings parameterised
by a limited set of common free parameters. Our findings and our method, that
we will describe below, can however easily be generalised to other BSM
theories.

####  2.1.2 The Sequential Standard Model

The simplest approach to model the dynamics associated with additional gauge
bosons would be to extend the SM field content by two massive $W^{\prime}$ and
$Z^{\prime}$ vector fields that are charged and neutral respectively, as in
Section 2.1.1. The form of the $W^{\prime}$ and $Z^{\prime}$ chiral couplings
to all SM fermions is then imposed to be the same as in the SM, and their
possible interactions with other gauge and scalar bosons are omitted. This
approach, which is traditionally coined the SSM Gopalakrishna:2010xm ;
Han:2012vk , is described by the model Lagrangian (1) with the couplings (2).
Moreover, as no right-handed neutrinos are present in the SM, the
corresponding right-handed leptonic couplings are all zero. The $W^{\prime}$
and $Z^{\prime}$ bosons being introduced in an ad-hoc manner, their masses
(and sometimes their widths even if they are in principle calculable) are thus
the only new physics free parameters. In this work, we additionally take the
new physics interactions to be flavour-diagonal for simplicity.

####  2.1.3 Models with an additional $SU(2)_{L}$

A more consistent way to add simultaneously extra gauge bosons to the SM field
content is to enlarge the electroweak symmetry group. For this purpose, we
consider the group structure $SU(2)_{1}\times SU(2)_{2}\times U(1)_{X}$
related to the commonly called $G(221)$ models Hsieh:2010zr . Models of this
class all feature additional $W^{\prime}$ and $Z^{\prime}$ bosons, the various
possibilities differing through the quantum number assignments of the
different fields. We examine a configuration in which the $SU(2)_{L}$ symmetry
of the SM is doubled, so that $SU(2)_{1}\equiv SU(2)_{L}$ and $SU(2)_{2}\equiv
SU(2)_{L}^{\prime}$. The representation of the weak doublets of SM fermions
then read

$L_{L}=\begin{pmatrix}\nu_{L}\\\
\ell_{L}\end{pmatrix}\,\sim(\mathbf{2},\mathbf{2},X_{Q})\,,\qquad
Q_{L}=\begin{pmatrix}u_{L}\\\
d_{L}\end{pmatrix}\,\sim(\mathbf{2},\mathbf{2},X_{L})\,,$ (3)

where the $U(1)_{X}$ quantum numbers $X_{Q}$ and $X_{L}$ are fixed to recover
the correct hypercharge quantum numbers once the extended gauge symmetry is
broken down to the electroweak one, $SU(2)_{L}\times SU(2)_{L}^{\prime}\times
U(1)_{X}\to SU(2)_{L}\times U(1)_{Y}$. Here, the right-handed fermions are all
singlets under the two $SU(2)$. In a second step, the electroweak symmetry is
further broken to the electromagnetic gauge symmetry $U(1)_{\rm EM}$. Such a
two-stage breaking is achieved through a Higgs sector containing a scalar
bidoublet $\Phi$, that implies $SU(2)_{L}^{\prime}\times U(1)_{X}\to
U(1)_{Y}$, and the usual SM weak doublet $H$, that leads to $SU(2)_{L}\times
U(1)_{Y}\to U(1)_{\rm EM}$. Those scalar fields are given, in terms of their
component fields, by

$\Phi\equiv\frac{1}{\sqrt{2}}\begin{pmatrix}\phi^{0}&\sqrt{2}\phi^{+}\\\
\sqrt{2}\phi^{-}&\phi^{0}\end{pmatrix}\sim(\mathbf{2},\mathbf{2^{*}},\mathbf{1/2})\,,\qquad
H\equiv\begin{pmatrix}h^{+}\\\
h^{0}\end{pmatrix}\sim(\mathbf{1},\mathbf{2},\mathbf{1/2})\,,$ (4)

and the symmetry is broken once the neutral components of these fields get
non-vanishing vacuum expectation values (vevs), $\langle\phi^{0}\rangle=u$ and
$\langle h^{0}\rangle=v/\sqrt{2}$ with $u\gg v$ for consistency with
experimental data.

In such a realisation, the physical gauge bosons are admixtures of the
$SU(2)_{L}^{\prime}$, $SU(2)_{L}$ and $U(1)_{X}$ bosons, and their masses are
not independent parameters. These can be derived from the vevs of the Higgs
fields and the gauge coupling constants $g_{1}\equiv g_{W}$, $g_{2}\equiv
g_{2}^{\prime}$ and $g_{X}$,

$\begin{split}M^{2}_{W^{\prime}}=&\
\frac{1}{4}(g_{1}^{2}+g_{2}^{2})u^{2}+\frac{\sin^{2}\phi}{4}g_{1}^{2}v^{2}+\sin^{4}\phi\frac{g^{2}_{1}g^{2}_{2}}{4(g_{1}^{2}+g_{2}^{2})}\frac{v^{4}}{u^{2}}\\\
M^{2}_{Z^{\prime}}=&\
\frac{1}{4}(g_{1}^{2}+g_{2}^{2})u^{2}+\frac{\sin^{2}\phi}{4}g_{1}^{2}v^{2}-\sin^{4}\phi\frac{g^{2}_{X}}{4e^{2}}\frac{g^{2}_{1}g^{2}_{2}}{(g_{1}^{2}+g_{2}^{2})}\frac{v^{4}}{u^{2}}\end{split}\qquad\text{with}\qquad\tan\phi=\frac{g_{X}}{g_{2}}\,.$
(5)

The extra gauge bosons are thus practically degenerate. The mixing relations
among them also fix their couplings with the SM fermions. They are written, in
the notation of Eq. (1), as

$\begin{split}\kappa_{q,L}^{W^{\prime}}=\frac{1}{\tan\theta_{W}}-\cos\theta_{W}\
\zeta\Big{(}\frac{\tan\theta_{W}}{M^{2}}\Big{)},\qquad&\kappa_{q,R}^{W^{\prime}}=0,\\\
\kappa_{\ell,L}^{W^{\prime}}=-\tan\theta_{W}-\cos\theta_{W}\
\zeta\Big{(}\frac{\tan\theta_{W}}{M^{2}}\Big{)},\qquad&\kappa_{\ell,R}^{W^{\prime}}=0,\\\
\kappa_{q,L}^{Z^{\prime}}=T_{3f}\frac{\cos\theta_{W}}{\tan\theta_{W}}+\left(T_{3f}+Q_{f}\sin^{2}\theta_{W}\right)\
\zeta\Big{(}\frac{\tan\theta_{W}}{M^{2}}\Big{)},\qquad&\kappa_{q,R}^{Z^{\prime}}=Q_{f}\sin^{2}\theta_{W}\
\zeta\Big{(}\frac{\tan\theta_{W}}{M^{2}}\Big{)},\\\
\kappa_{\ell,L}^{Z^{\prime}}=-T_{3f}\sin\theta_{W}+\left(T_{3f}+Q_{f}\sin^{2}\theta_{W}\right)\
\zeta\Big{(}\frac{\tan\theta_{W}}{M^{2}}\Big{)},\qquad&\kappa_{\ell,R}^{Z^{\prime}}=Q_{f}\sin^{2}\theta_{W}\
\zeta\Big{(}\frac{\tan\theta_{W}}{M^{2}}\Big{)}.\end{split}$ (6)

These couplings depend on the sum of the two weak isospin quantum numbers,
$T_{3f}=T_{L}^{3,f}+T_{L}^{\prime\,3,f}$, as well as on the electroweak mixing
angle defined by

$\tan\theta_{W}=\frac{g_{Y}}{g_{1}}\qquad\text{with}\qquad\frac{1}{g_{Y}^{2}}=\frac{1}{g_{2}^{2}}+\frac{1}{g_{X}^{2}}.$
(7)

Moreover, $M$ stands for the approximately degenerate mass of the $W^{\prime}$
and $Z^{\prime}$ bosons and the function $\zeta(\tan\theta_{W}/M^{2})$
involves subleading corrections of ${\cal O}(\tan\theta_{W}/M^{2})$. In this
class of models, the $SU(2)$ bosons only couple to the left-handed fermions.
The right-handed couplings are thus either zero (for the $W^{\prime}$ boson)
or strongly suppressed by mixing (for the $Z^{\prime}$ boson).

Electroweak precision tests constrain the masses of these bosons to be greater
than 2.5 TeV Hsieh:2010zr , and we will address limits originating from direct
searches at colliders below.

####  2.1.4 Left-Right Symmetric Models

In this section we still consider the $G(221)$ framework, but this time we
enforce the $SU(2)_{2}$ group to be an $SU(2)_{R}$ gauge symmetry, and the
$U(1)_{X}$ symmetry to be associated with $B-L$ conservation (with $B$ being
the baryon and $L$ the lepton number). The enlarged gauge group is thus
$SU(2)_{L}\times SU(2)_{R}\times U(1)_{B-L}$. This setup gives rise to the
commonly called LRSM Pati:1974yy ; Mohapatra:1974gc ; Senjanovic:1975rk ;
Mohapatra:1977mj ; Senjanovic:1978ev  in which the left-handed fermions of the
SM are organised into doublets of $SU(2)_{L}$ and are singlet under
$SU(2)_{R}$. In contrast, the right-handed fermions (including right-handed
neutrinos) are organised into doublets of $SU(2)_{R}$ and are singlet under
$SU(2)_{L}$. The $B-L$ quantum numbers are then the usual ones, namely 1/6 for
quark and $-1/2$ for lepton fields.

Similarly to the previously-discussed $SU(2)_{L}\times
SU(2)_{L}^{\prime}\times U(1)_{X}$ models, the extended gauge symmetry is
broken into two stages. The Higgs sector includes first two Higgs triplets
$\Delta_{L}$ and $\Delta_{R}$ respectively lying in the adjoint representation
of $SU(2)_{L}$ and $SU(2)_{R}$ and allowing for the preservation of the left-
right symmetry. In addition, it contains a Higgs bidoublet $\Phi$ state. The
former Higgs fields induce $SU(2)_{R}\times U(1)_{B-L}\to U(1)_{Y}$ breaking,
whereas the latter yields electroweak symmetry breaking. The Higgs field
representation under the extended gauge group and the electric charge
assignments of the various component fields are given by

$\begin{split}\Delta_{L}\equiv\begin{pmatrix}\delta_{L}^{+}/\sqrt{2}&\delta_{L}^{++}\\\
\delta_{L}^{0}&-\delta_{L}^{+}/\sqrt{2}\end{pmatrix}\sim(\mathbf{3},\mathbf{1},2)\,,\qquad\Delta_{R}\equiv\begin{pmatrix}\delta_{R}^{+}/\sqrt{2}&\delta_{R}^{++}\\\
\delta_{R}^{0}&-\delta_{R}^{+}/\sqrt{2}\end{pmatrix}\sim(\mathbf{1},\mathbf{3},2)\,,\\\
\Phi\equiv\begin{pmatrix}\phi_{1}^{0}&\phi_{2}^{+}\\\
\phi_{1}^{-}&\phi_{2}^{0}\end{pmatrix}\sim(\mathbf{2},\mathbf{2},0)\,.\hskip
113.81102pt\end{split}$ (8)

As in Section 2.1.3, the masses of the extra gauge bosons and their couplings
to the fermions are dictated by the gauge couplings and the vevs of the
neutral Higgs fields, $\langle\delta_{L}^{0}\rangle=v_{L}/\sqrt{2}$,
$\langle\delta_{R}^{0}\rangle=v_{R}/\sqrt{2}$,
$\langle\phi_{1}^{0}\rangle=v_{1}/\sqrt{2}$ and
$\langle\phi_{2}^{0}\rangle=v_{2}/\sqrt{2}$ (when ignoring potentially non-
zero phases that cannot be re-absorbed into field redefinitions). Experimental
data from the kaon and the $B$-meson sector enforce that $v_{R}\gg
v_{1},v_{2}\gg v_{L}$ and $\sqrt{v_{1}^{2}+v_{2}^{2}}=v=246$ GeV, and strongly
limit the mixing possibilities between the SM and the extra gauge bosons.
Under those assumptions, we obtain for the squared $W^{\prime}$ and
$Z^{\prime}$ boson masses

$\begin{split}M_{W^{\prime}}^{2}=&\
\frac{1}{4}\bigg{[}2g_{2}^{2}v_{R}^{2}+g_{2}^{2}v^{2}+2g_{2}g_{1}\frac{v^{2}_{1}v^{2}_{2}}{v_{R}^{2}}\bigg{]}\,,\\\
M^{2}_{Z^{\prime}}=&\
\frac{1}{4}\bigg{[}g_{1}^{2}v^{2}+2v_{R}^{2}(g_{2}^{2}+g_{X}^{2})+\sqrt{\Big{[}g_{1}^{2}v^{2}+2v_{R}^{2}(g_{2}^{2}+g_{X}^{2})\Big{]}^{2}-4g_{1}^{2}(g_{2}^{2}+2g_{X}^{2})v^{2}v_{R}^{2}}\bigg{]}\,,\end{split}$
(9)

with $g_{1}\equiv g_{W}$, $g_{2}\equiv g_{R}$, and $g_{X}\equiv g_{B-L}$.
Expressions for the couplings of the extra gauge bosons to the SM fermions
(including right-handed neutrinos) feature contributions from the three gauge
groups that are modulated by the various mixing angles related to the gauge
sector. They can be written in the form of Eq. (1), provided that all gauge
coupling constants are absorbed in the $\kappa$ parameters. We refer to the
works Roitgrund:2014zka ; Mattelaer:2016ynf  for their explicit derivation.

####  2.1.5 Alternative Left-Right Model

The Alternative Left-Right Symmetric Model (ALRSM) relies on the same gauge
group as in Section 2.1.4, but the fields are organised differently into
multiplets of the gauge group. Such an option emerges from the breaking of an
$E_{6}$ GUT group into its $SU(3)\times SU(3)\times SU(3)$ maximal subgroup.
One of these $SU(3)$’s remains unbroken and is associated with the SM strong
interaction group, while the two others $SU(3)$ groups further break into
$SU(2)_{1}\times SU(2)_{2}\times U(1)_{X}$. There are several options for this
breaking to occur, and they respectively lead to the Inert Doublet Model
Deshpande:1977rw ; Ma:2006km ; Barbieri:2006dq , the usual LRSM of the
previous section, or the ALRSM Ma:1986we  considered in this section.

In the ALRSM, the $SU(2)_{2}\equiv SU(2)^{\prime}_{R}$ partner of the right-
handed up-quark is an exotic down-type quark and the $SU(2)^{\prime}_{R}$
partner of the right-handed charged lepton is a new neutral lepton dubbed a
scotino. The more common right-handed neutrino and down-type quark thus remain
singlets under both the $SU(2)_{1}\equiv SU(2)_{L}$ and $SU(2)^{\prime}_{R}$
groups. Furthermore, the model includes $SU(2)_{L}$ singlet counterparts to
the new states, i.e. $SU(2)_{L}$ scotinos and exotic down-type quarks. Whereas
the field including the SM states have standard $B-L$ quantum numbers, the
exotic singlets are assigned $B-L$ quantum numbers of 0 and $-2/3$ for the
scotino and the down-type quarks respectively.

The breaking of the extended gauge symmetry is again performed in two steps.
It relies, this time, on a pair of Higgs doublets $\chi_{L}$ and $\chi_{R}$
that break $SU(2)^{\prime}_{R}\times U(1)_{B-L}$ into the hypercharge group,
and on a Higgs bidoublet $\Phi$ that ensures electroweak symmetry breaking.
The representation and charge assignments of the various component fields are
given by

$\begin{split}\chi_{L}\equiv\begin{pmatrix}\chi_{L}^{+}\\\
\chi_{L}^{0}\end{pmatrix}\sim(\mathbf{2},\mathbf{1},1)\,,\qquad\chi_{R}\equiv\begin{pmatrix}\chi_{R}^{+}\\\
\chi_{R}^{0}\end{pmatrix}\sim(\mathbf{1},\mathbf{2},1)\,,\qquad\Phi\equiv\begin{pmatrix}\phi_{1}^{0}&\phi_{2}^{+}\\\
\phi_{1}^{-}&\phi_{2}^{0}\end{pmatrix}\sim(\mathbf{2},\mathbf{2},0)\,.\end{split}$
(10)

Once again, the masses and mixing of the extra gauge bosons are not free
parameters and depend on the vevs of the neutral components of the Higgs
states, $\langle\chi_{L}^{0}\rangle=v_{L}/\sqrt{2}$,
$\langle\chi_{R}^{0}\rangle=v_{R}/\sqrt{2}$ and
$\langle\phi^{0}_{2}\rangle=k/\sqrt{2}$. In the ALRSM, the conservation of a
generalised lepton number protects $\phi^{0}_{1}$ from acquiring any vev, and
forbids the mixing of the SM quarks with the exotic quarks, as well as the one
that could occur between the charged gauge bosons. The extra gauge boson
masses read

$\begin{split}M_{W^{\prime}}=&\ \frac{1}{2}g_{2}\sqrt{k^{2}+v_{R}^{2}}\ ,\\\
M_{Z^{\prime}}=&\
\frac{1}{2}\sqrt{g_{X}^{2}v_{L}^{2}{\sin^{2}\phi_{W}}+\frac{g_{2}^{2}k^{2}(\cos^{4}{\phi_{W}}+v_{R}^{2})}{\cos^{2}{\phi_{W}}}}\
,\end{split}$ (11)

where as in Section 2.1.4, $g_{X}\equiv g_{B-L}$ and $g_{2}\equiv g_{R}$. The
mixing angle $\phi_{W}$ that appear in the expression for the $Z^{\prime}$
boson mass describes the mixing of the $U(1)_{B-L}$ gauge boson with the
neutral $SU(2)_{R}^{\prime}$ gauge boson. Such a mixing gives rise to the
hypercharge gauge boson and is defined by

$\sin{\phi_{W}}=\frac{g_{X}}{\sqrt{g_{X}^{2}+g_{2}^{2}}}\equiv\frac{g_{Y}}{g_{2}}\,.$
(12)

It is hence connected to the hypercharge coupling constant $g_{Y}$.

As for the LRSM, the coupling of the new gauge bosons to fermions can be
written in the form of Eq. (1) once we absorb the full gauge coupling
dependence into the $\kappa$ parameters. We refer to the work of Refs.
Ashry:2013loa ; Frank:2019nid  for details.

###  2.2 Models with an additional $Z^{\prime}$ boson

In our study we compare predictions for models containing both a $W^{\prime}$
and a $Z^{\prime}$ boson also with those associated with models solely
featuring an extra gauge boson $Z^{\prime}$. In this case, the absence of any
anomaly, in either cross section or charge asymmetry in the $\ell\nu$ final
state which would correlate to signal evidences in the $\ell^{+}\ell^{-}$ one,
could be indicative of the existence of such a $U(1)^{\prime}$ extra
$Z\textquoteright$ boson. In this section, we briefly review two of the most
popular classes of models that feature an extra Abelian gauge symmetry (and
that are thus based on the extended $SU(2)_{L}\times U(1)_{Y}\times
U(1)^{\prime}$ gauge symmetry), namely models that emerge from the breaking of
a GUT group (such as $E_{6}$) and $U(1)_{B-L}$ models.

####  2.2.1 $E_{6}$ motivated $U(1)^{\prime}$ models

| $2\sqrt{10}Q^{{}^{\prime}}_{\chi}$ | $2\sqrt{6}Q^{{}^{\prime}}_{\psi}$ | $2\sqrt{15}Q^{{}^{\prime}}_{\eta}$ | $2\sqrt{15}Q^{{}^{\prime}}_{S}$ | $2Q^{{}^{\prime}}_{I}$ | $2\sqrt{10}Q^{{}^{\prime}}_{N}$
---|---|---|---|---|---|---
$\theta_{E_{6}}$ | $-\pi/2$ | $0$ | $\arccos\sqrt{\frac{5}{8}}-\pi$ | $\arctan\sqrt{\frac{15}{9}}-\frac{\pi}{2}$ | $\arccos\sqrt{\frac{5}{8}}+\frac{\pi}{2}$ | $\arctan\sqrt{15}-\frac{\pi}{2}$
$Q_{Q_{L},u_{R},e_{R}}$ | $-1$ | 1 | $-2$ | $-1/2$ | 0 | 1
$Q_{d_{R},L_{L}}$ | 3 | 1 | 1 | 4 | $-1$ | 2
$Q_{N}$ | $-5$ | 1 | $-5$ | $-5$ | 1 | 0
$Q_{H_{u}}$ | 2 | $-2$ | 4 | 1 | 0 | 2
$Q_{H_{d}}$ | $-2$ | $-2$ | 1 | $-7/2$ | 1 | $-3$
$Q_{S}$ | 0 | 4 | $-5$ | 5/2 | $-1$ | 5
Table 1: $U(1)^{\prime}$ charges of the quarks, leptons, and Higgs states in
six commonly-studied $U(1)^{\prime}$ extensions of the SM that could arise
from the breaking of a $E_{6}$ GUT symmetry. We additionally present the value
of the mixing angle $\theta_{E_{6}}\in[-\pi,\pi]$.

GUT models based on the breaking of the exceptional group $E_{6}$
Gursey:1975ki ; Achiman:1978vg  have been well motivated from developments in
string theories Hewett:1988xc  and as generators of models featuring
additional $U(1)$ symmetries. The so-called $U(1)^{\prime}$ models can be seen
as arising from the breaking of $E_{6}$ into $SO(10)\times U(1)$, rather than
directly into the $SU(3)\times SU(3)\times SU(3)$ subgroup as in Sections
2.1.4 and 2.1.5. The most commonly-used parameterisation to define the extra
$U(1)^{\prime}$ symmetry emerges from a linear combination of the
$U(1)^{\prime}_{\psi}$ and $U(1)^{\prime}_{\chi}$ groups resulting from the
breaking $E_{6}\to SO(10)\times U(1)^{\prime}_{\psi}\to(SU(5)\times
U(1)^{\prime}_{\chi})\times U(1)^{\prime}_{\psi}$. In this context, the
corresponding charge operator $Q^{\prime}$ is written as an admixture of the
initial $U(1)^{\prime}_{\psi}$ and $U(1)^{\prime}_{\chi}$ charge operators
$Q^{\prime}_{\psi}$ and $Q^{\prime}_{\chi}$,

$Q^{\prime}=Q^{\prime}_{\psi}\cos\theta_{E_{6}}-Q^{\prime}_{\chi}\sin\theta_{E_{6}}\,,$
(13)

in which we introduce the mixing angle $\theta_{E_{6}}$. Several benchmark
models with various mixing angle and quantum numbers for the SM fields have
been considered, and are presented in Table 1. Variants exist and generally
differ by their (possibly secluded and extended) Higgs sector.

We adopt a configuration in which the Higgs sector contains two Higgs doublets
$H_{u}$ and $H_{d}$, as well as a Higgs singlet $S$ (that is charged under the
extra $U(1)^{\prime}$ symmetry). This choice is a minimal option that is
consistent with the supersymmetric version of the model. The representation of
those fields are given by

$H_{u}=\left(\begin{array}[]{c}H_{u}^{+}\\\
H_{u}^{0}\end{array}\right)\sim(\mathbf{2},1/2,Q_{H_{u}})\,,\qquad
H_{d}=\left(\begin{array}[]{c}H_{d}^{0}\\\
H_{d}^{-}\end{array}\right)\sim(\mathbf{2},-1/2,Q_{H_{d}})\,,\qquad
S\sim(\mathbf{1},0,Q_{S})\ ,\ $ (14)

their $U(1)^{\prime}$ quantum numbers being provided in Table 1 for the
specific $U(1)^{\prime}$ models considered. The model additionally includes
right-handed neutrinos.

After the spontaneous breaking of the symmetry group down to electromagnetism,
the $W$, $Z$ and $Z^{\prime}$ bosons get massive while the photon stays
massless. In general, the SM $Z$ and the extra $Z^{\prime}$ eigenstates mix,
although such a mixing is strongly constrained by electroweak precision data.
In our study, we consider an important limit in which the extra gauge boson is
much heavier than the SM weak bosons and does not mix. Its mass can then be
simply related to the weak coupling constant $g_{W}$, the quantum numbers of
the Higgs states, and their vevs $\langle H_{u}^{0}\rangle=v_{u}/\sqrt{2}$,
$\langle H_{d}^{0}\rangle=v_{d}/\sqrt{2}$ and $\langle
S\rangle=v_{s}/\sqrt{2}$. It is written as

$M_{Z^{\prime}}^{2}=g_{W}^{2}\Big{(}Q_{H_{u}}v_{u}^{2}+Q_{H_{d}}v_{d}^{2}+Q_{S}^{2}v_{s}^{2}\Big{)}\,,$
(15)

and generalisations from this limiting case are outlined in Ref.
Langacker:1984dp .

####  2.2.2 Models with an extra $U(1)_{B-L}$

As a second class of models featuring an additional neutral gauge boson
relative to the SM, we consider frameworks in which the SM gauge symmetry is
extended by gauging the $B-L$ number. The gauge symmetry of the model is thus
based on the $SU(2)_{L}\times U(1)_{Y}\times U(1)_{B-L}$ group Basso:2008iv ;
Basso:2009hf , and its dynamics is similar to the other models treated in
Section 2.2.1. Differences lie at the level of the $U(1)$ charges of the
various fields and concern the details of the Higgs sector.

In terms of its field content, the model includes additional right-handed
neutrinos relatively to the SM, so that gauge and gravity anomalies cancel.
Moreover, it includes an extra Higgs singlet $S$ that allows for the
spontaneous breaking of the $B-L$ symmetry. The quantum numbers of all scalar
fields are given by

$H=\left(\begin{array}[]{c}h^{+}\\\
h^{0}\end{array}\right)\sim(\mathbf{2},1/2,0)\,,\qquad S\sim(\mathbf{1},0,2)\
.\ $ (16)

In addition to the SM fermion masses, symmetry breaking also yields the
generation of neutrino masses through the implementation of a seesaw
mechanism. As usual, gauge boson masses can be written in terms of the gauge
couplings and the vevs of the two Higgs bosons, $\langle
h^{0}\rangle=v/\sqrt{2}$ and $\langle S\rangle=v_{s}/\sqrt{2}$. We obtain, for
the heaviest $Z^{\prime}$ state,

$M_{Z^{\prime}}=2g_{B-L}v_{s}\ ,$ (17)

where we ignored $Z-Z^{\prime}$ mixing for simplicity.

###  2.3 Constraints from direct searches for additional gauge bosons

All extra gauge bosons studied in this work couple to a pair of SM quarks,
with the exception of the $W^{\prime}$ boson in ALRSM scenarios that features
mixed couplings to one SM quark and one exotic quark. Limits can therefore be
extracted from ATLAS and CMS dijet searches for new resonances in most cases
considered. The analysis of the entire ATLAS ATLAS:2019fgd  and CMS
CMS:2019gwf  Run 2 dataset yields respective bounds of 3.6–4 TeV on SSM
$W^{\prime}$ bosons, and of 2.7–2.9 TeV on the corresponding $Z^{\prime}$
states. These $W^{\prime}$ limits can readily be extended to models featuring
either an extra $SU(2)_{L}^{\prime}$ symmetry or an extra $SU(2)_{R}$
symmetry, at least in situations where all $SU(2)$ gauge couplings are equal
($g_{1}=g_{2}\equiv g_{W}$).

As dijet bounds on extra $Z^{\prime}$ bosons are quite weak, more stringent
bounds could originate from other channels, like the cleaner dilepton probes.
The shape analysis of the high-mass dilepton spectrum indeed allows, by making
use of the entire LHC Run 2 dataset, for the exclusion of SSM $Z^{\prime}$
masses ranging up to 5.1 TeV Sirunyan:2021khd ; ATLAS:2019erb . Those limits
are only slightly reduced in the framework of $E_{6}$-inspired $U(1)^{\prime}$
models, with the $Z^{\prime}$ mass being enforced to be larger than about 4.56
TeV in $U(1)^{\prime}_{\psi}$ models for instance. Leptophobic $Z^{\prime}$
bosons, that we do not consider in this study, are therefore much less
constrained Araz:2017wbp . Here, limits originating from resonance searches
through top-antitop pair production become more relevant. The analysis of 36.1
fb-1 of LHC data constraints (narrow) $Z^{\prime}$ boson masses to be greater
than 3.1–3.8 TeV, the exact bound depending on the details of the model
Aaboud:2019roo ; CMS:2018rkg .

Bounds have also been extracted from searches for $Z^{\prime}$ bosons in the
ditau channel. They are, however, weaker. The analysis of a luminosity of
$36~{}{\rm fb}^{-1}$ of LHC Run 2 data hence leads to a mass limit of about
2.4 TeV Aaboud:2017sjh .

Finally, several studies are dedicated to LRSM scenarios. They consider two
specific signatures of a resonantly-produced $W_{R}$ boson, a first one when
it decays into a $tb$ system Sirunyan:2017vkm ; ATLAS:2018wmg , and a second
one when it decays into a lepton and a right-handed neutrino ATLAS:2018dcj ;
ATLAS:2019isd ; CMS:2018agk , the latter new physics state further decaying
into a lepton plus dijet system. In the former channel, limits on the mass of
the $W_{R}$ boson have been found to be of 3.2–3.6 TeV after relying on 36
fb-1 of LHC data. In the latter channel, the analysis of about 36 fb-1 of Run
2 LHC data (and 80 fb-1 in the boosted regime) reveals constraints on $W_{R}$
boson masses lying in the 3.5–4.4 TeV range for right-handed neutrino masses
below about 3 TeV.

##  3 Phenomenological investigations

In this section, we investigate various signals of additional (heavy) gauge
bosons at the LHC, as well as the correlations that could emerge from the
simultaneous presence of additional $W^{\prime}$ and $Z^{\prime}$ states in
the model. Our aim is to get indications on the potential non-existence of one
of these bosons from the non-observation of the other, or conversely, if given
the observation of one of them we could get guidance in the search for the
other one. In other words, we aim to unravel the phenomenological connection
between the two extra bosons. We provide information on our computational
setup in Section 3.1, and study correlations among cross sections and
asymmetries in Sections 3.2 and 3.3 respectively.

###  3.1 Computational setup

To perform our numerical analysis, we have implemented the models presented in
Section 2 into FeynRules (version 2.3.35) Alloul:2013bka  and converted them
into leading-order UFO libraries Christensen:2009jx ; Degrande:2011ua . Our
new implementations are based on already publicly available ones
Mattelaer:2016ynf ; Fuks:2017vtl ; Borah:2018yxd ; Frank:2018ifw ;
Frank:2019nid , that we have harmonised with each other in the context of
simplified versions of the various models (that are sufficient for our
purpose). Hard-scattering parton-level events have then been generated with
MG5aMC (version 3.0.3) Alwall:2014hca , and have been analysed with
MadAnalysis 5 Conte:2012fm ; Conte:2014zja ; Conte:2018vmg  (version 1.8.58).
In our simulations, we convoluted the relevant hard-scattering matrix elements
with the leading-order set of NNPDF 3.1 Ball:2017nwa  parton distribution
functions (PDF), that has been accessed through LHAPDF Buckley:2014ana .

###  3.2 Production cross sections

Figure 1: $Z^{\prime}$ production cross sections, as a function of the
$Z^{\prime}$ boson mass and at the $\sqrt{s}=13$ TeV LHC. We consider a
$Z^{\prime}$ decay into a dilepton final state, and we show predictions for
various scenarios and models. These predictions are compared to current CMS
and ATLAS limits. In this figure, we ignore the uncertainties on the theory
results for clarity, as those are discussed in Figure 3.

We begin by investigating, in Figure 1, the existing and relevant $Z^{\prime}$
limits originating from LHC searches at a centre-of-mass energy of
$\sqrt{s}=13$ TeV. We compute cross section predictions associated with the
production of a dileptonically-decaying $Z^{\prime}$ boson in each model
considered. Our predictions are then compared with the most recent ATLAS
ATLAS:2019erb  and CMS Sirunyan:2021khd  bounds, that we display for various
assumptions on the width-over-mass ratio of the $Z^{\prime}$ boson,
$\Gamma_{Z^{\prime}}/M_{Z^{\prime}}$. All adopted values of this ratio are
broadly compatible with those emerging from the BSM scenarios under
consideration. In the figure, we abusively make use of the $\sigma\times{\rm
BR}$ notation for the title of the $y$-axis for simplicity, although our
calculations are related to the full process
$pp\to\gamma,Z,Z^{\prime}\to\ell^{+}\ell^{-}$. The cross section therefore
includes both the SM contribution (squared), the new physics contribution
(squared), and their interference. The pure SM component of the cross section
(representing the SM background) is generally small, especially for the
dilepton invariant mass ($M_{\ell\ell}$) intervals typically considered in the
corresponding experimental analyses. Moreover, the new physics states are
generically quite narrow (see Table 2 for examples), so that a factorised
approach in the NWA would give production rates times branching ratio values
compatible with those arising from the complete calculation. Whilst this is
true for integrated cross sections, this may not hold at the differential
cross section level, as distributions remain subject to non-negligible
interference effects enabled by a finite value of the new gauge boson width.

The lowest $Z^{\prime}$ mass limits are found for the $E_{6}$-inspired $\psi$
and $\eta$ models (defined in Table 1), in which we get $M_{Z^{\prime}}\gtrsim
5$ TeV. In a left-right-symmetric setup (i.e. for the ALRSM and LRSM models),
we have found slightly stronger bounds with $M_{Z^{\prime}}\gtrsim 5.6$ TeV.
Finally, models including an additional $SU(2)_{L}$ symmetry or a $B-L$ extra
gauge symmetry feature larger cross sections, which yield even stronger limits
with $M_{Z^{\prime}}\gtrsim 6$ TeV and 6.7 TeV respectively. The hierarchy
between those different bounds show that there is no generic impact of the
presence of an additional charged gauge boson in the model. The bounds are
indeed model-dependent and cannot be generically extracted. This justifies our
model-by-model approach.

Figure 2: $W^{\prime}$ production cross section as a function of the
$W^{\prime}$ boson mass and at the $\sqrt{s}=13$ TeV LHC. We consider a
$W^{\prime}$ boson decaying into a charged lepton plus neutrino final state,
and we show predictions for various scenarios in the SSM (left) and for models
with an additional $SU(2)_{L}$ symmetry (right). The shaded areas include a
combination in quadrature of the PDF and scale uncertainties, and for models
with an additional $SU(2)_{L}$ symmetry, the colour code indicates, when it is
used, the mass of the accompanying $Z^{\prime}$ state for each considered
$M_{W^{\prime}}$ value. In both figures, the predictions are compared to the
current experimental bounds. The two pairs of curves compare the bounds
obtained when a full calculation of the matrix element related to the $pp\to
W^{\prime}\to\ell\nu$ is used, and when the NWA is used (so that the
production cross section factorises from the decay matrix element).

We repeat the exercise in Figure 2 for $W^{\prime}$ boson production at the
LHC, when the extra gauge boson decays into an associated charged-
lepton/neutrino pair. We consider SSM predictions (left panel), as well as
predictions in models featuring an additional $SU(2)_{L}$ symmetry (right
panel), and the predictions are compared in both cases to current ATLAS bounds
ATLAS:2019lsy . We do not show any result for the LRSM and ALRSM scenarios, as
we cannot extract limits on the $W^{\prime}$ properties from the CC Drell-Yan
channel in those models. The $W^{\prime}$ boson indeed always decays into a
final state different from the $\ell\nu$ one.

In the setup featuring an extra $SU(2)_{L}$ symmetry, the $W^{\prime}$ boson
is always accompanied by a $Z^{\prime}$ boson whose mass, that is not an
independent parameter (see Eq. (5)), is represented through a colour code for
each $M_{W^{\prime}}$ value. This is one of the relevant points emphasised in
this study: we advocate that bounds on one gauge boson can imply potentially
stronger bounds (like here) on the other one (for models featuring both new
gauge bosons). Here, direct searches for an $SU(2)_{L}$ $Z^{\prime}$ gauge
boson set limits of 6 TeV on its mass, while the associated $W^{\prime}$
searches imply more stringent bounds of $M_{Z^{\prime}}\gtrsim 6.4$ TeV.
Likewise, if with an increased luminosity an excess were to be found in either
of the two search channels, the search in the other channel would be in the
ideal position to optimise the mass selection to evidence the presence of the
companion new gauge state. Clearly, the same approach cannot be exploited in
the case of the SSM in which the two masses are independent parameters by
construction.

Figure 2 also compares the bounds that could be derived when relying on a
computation of the full matrix element associated with the $pp\to
W^{\prime}\to\ell\nu$ process with those obtained when this matrix element is
approximated by its NWA form. In this last case, off-shell effects are lost
and the cross section gets reduced by a factor of a few in the large-mass
regime. Consequently, the bounds on the extra gauge boson are weaker.
Moreover, allowing the $W^{\prime}$ boson to be offshell implies that a phase-
space region associated with a smaller Bjorken-$x$ could contribute to the
cross section, even dominantly like here due to larger PDF values. This
additionally yields, in this case, smaller PDF uncertainties. Handling a full
calculation offers hence a possibility to extract stronger and more robust
bounds through a greater cross section and a better control of the theory
errors. For that reason, all the results below are extracted from predictions
for the full process.

Figure 3: Same as in Figure 1 but for individual models considered separately.
We display results for BSM setups with an additional $SU(2)_{L}$ symmetry (top
row), and for the LRSM (bottom left) and ALRSM (bottom right). Our predictions
are provided together with the associated PDF and scale uncertainties given as
shaded areas, and the colour code associated with each $Z^{\prime}$ mass
choice refers to the mass of the accompanying $W^{\prime}$ state.

In Figure 3, we revisit the $Z^{\prime}$ bounds investigated in Figure 1, this
time including theory errors on the predictions and considering all models
that feature both a $W^{\prime}$ and a $Z^{\prime}$ boson. We display
information about the experimental limits on the $Z^{\prime}$ boson, and about
the mass of the associated $W^{\prime}$ boson that we provide through a colour
code. As before, the obtained $Z^{\prime}$ bounds allow for the extraction of
a limit on the $W^{\prime}$ mass. For example, $Z^{\prime}$ mass limits in the
ALRSM and the LRSM scenarios imply that the corresponding $W^{\prime}$ boson
mass has to be of at least 4.8 TeV and 3.5 TeV respectively. This therefore
provides strong $W^{\prime}$ bounds despite the absence of any direct search
limits in the ALRSM and LRSM.

The constraining power in the high mass regime is found to be poorer than
expected because of the theoretical errors, and in particular because of the
PDF errors that grow quite significantly with $M_{Z^{\prime}}$. Such an issue
was also present in the CC case (as illustrated in Figure 2) as well as for
$U(1)^{\prime}$ models for which predictions for $Z^{\prime}$ production are
similar to those shown in Figure 3. The impact of this growth of the errors
may however be tamed down by the different masses of the two extra bosons. An
assessment of the likelihood for the heavier boson to exist in LHC data can
indeed accurately be ascertain through a search for the lighter boson, as in
this case the parton density errors inherent to the calculation are smaller.
These larger PDF errors in the large Bjorken-$x$ regime are connected to the
way the NNPDF sets are extracted, and the usage of CT Hou_2021  or MSHT
Bailey_2021  densities would lead to a more reasonable high-mass behaviour.
This however comes at the cost of a theoretical bias on the functional form of
the individual PDF, as already pointed out in the context of the Minimal
Supersymmetric Standard Model and for leptoquark scenarios Frixione:2019fxg ;
Borschensky:2020hot ; Borschensky:2021hbo . As it is conceivable that the
modelling of the structure functions will see improvements in the years to
come, it is quite likely that both channels will benefit from them in a
similar way. The advocated cross-fertilising approach will thus be even more
in order and valid.

Given the current constraints on the $W^{\prime}$ and $Z^{\prime}$ masses, it
is rather unlikely that the upcoming Run 3 of the LHC can yield significant
advances relative to where we stand today. We therefore decide to make use of
the expected increased luminosity of the future high-luminosity phase of the
LHC (i.e. the HL-LHC Gianotti:2002xx ; CidVidal:2018eel ) to establish whether
cross-correlations between the $W^{\prime}$ and $Z^{\prime}$ bounds may exist
in the case of various asymmetries, whose extraction requires more substantial
event rates. We benchmark our forthcoming results within the scope of the LHC
operating at 14 TeV and with a total luminosity of 3000 fb-1, and we rely on
this tenfold increase in event rates with respect to those achievable during
the LHC Run 3 to study asymmetries in the considered Drell-Yan channels. We
examine those asymmetries in the next subsection, and study them as a function
of the invariant mass $M_{\ell\ell}$ of the lepton pair in the NC case, and as
a function of the transverse mass $M_{T}$ of the $\ell\nu$ system in the CC
case.

###  3.3 Asymmetries

Models | SSM | LRSM | ALRSM | $SU(2)_{L}^{\prime}$ | $U(1)_{\psi}$ | $U(1)_{\eta}$
---|---|---|---|---|---|---
$M_{Z^{\prime}}$ | 6 TeV | 6 TeV | 6 TeV | 6 TeV | 5.5 TeV | 5.5 TeV
$\Gamma_{Z^{\prime}}/M_{Z^{\prime}}$ | 2.95% | 2.05% | 1.34% | 0.75% | 0.59% | 1.02%
$M_{W^{\prime}}$ | 6.5 TeV | 6 TeV | 5 TeV | 6.5 TeV | - | -
$\Gamma_{W^{\prime}}/M_{W^{\prime}}$ | 0.30% | 0.39% | 0.34% | 0.30% | - | -
Table 2: Extra gauge boson masses and widths adopted in our study.

In this section, we study various asymmetries arising from extra gauge boson
production through Drell-Yan processes at the LHC. We fix the new gauge boson
masses in the different scenarios as shown in Table 2, these masses being
chosen as the lowest allowed ones in a given model according to the
constraints depicted in Sections 2.3 and 3.2. The widths $\Gamma_{Z^{\prime}}$
and $\Gamma_{W^{\prime}}$ are then evaluated analytically.

We begin by focusing on the NC Drell-Yan channel for which we consider the
forward-backward asymmetry $A_{\rm FB}^{\rm True}$ that we define by

$A^{\rm True}_{\rm FB}=\frac{\frac{{\rm d}\sigma}{{\rm
d}M_{\ell\ell}}(\cos\theta_{\ell^{-}}>0)-\frac{{\rm d}\sigma}{{\rm
d}M_{\ell\ell}}(\cos\theta_{\ell^{-}}<0)}{\frac{{\rm d}\sigma}{{\rm
d}M_{\ell\ell}}(\cos\theta_{\ell^{-}}>0)+\frac{{\rm d}\sigma}{{\rm
d}M_{\ell\ell}}(\cos\theta_{\ell^{-}}<0)}\,,$ (18)

where $\theta_{\ell^{-}}$ is the negatively-charged lepton polar angle
relative to the initial-quark direction axis. While such an angle allows for
the identification of the forward and backward hemispheres, the quark
direction cannot be obtained without relying on Monte Carlo information. For
this reason, we additionally consider the reconstructed forward-backward
asymmetry $A_{\rm FB}^{\rm Rec}$ (sometimes noted as $A{{}^{*}_{\rm FB}}$)
that is defined from the rapidity of the dilepton system $y_{\ell\ell}$,

$y_{\ell{\ell}}=\frac{1}{2}\ln\left[\frac{E_{\ell\ell}+p^{z}_{\ell\ell}}{E_{\ell\ell}-p^{z}_{\ell\ell}}\right].$
(19)

In this expression, $E_{\ell\ell}$ and $p^{z}_{\ell\ell}$ stand for the energy
and the momentum in the $z$-direction of the lepton pair respectively. The
$y_{\ell\ell}$ variable provides a measure of the boost induced by the initial
state partonic configuration, and its sign allows for a definition of the
direction of the initial quark. The polar angle is then extracted in the
dilepton restframe. In the following, we examine the validity of the
approximation $A_{\rm FB}^{\rm Rec}\approx A^{\rm True}_{\rm FB}$, that gets
better once an additional cut on $|y_{\ell\ell}|$ is implemented. We consider
several options for the corresponding lower thresholds, that are varied from 0
(no cut) to 0.6.

For every bin in $M_{\ell\ell}$ we compute the associated statistical
uncertainty $\delta A_{\rm FB}$ on the asymmetry,

$\delta A_{\rm
FB}=\sqrt{\frac{4}{\mathcal{L}}\frac{\sigma_{F}\sigma_{B}}{(\sigma_{F}+\sigma_{B})^{3}}}=\sqrt{\frac{1-A_{\rm
FB}^{2}}{\sigma\mathcal{L}}}=\sqrt{\frac{1-A_{\rm FB}^{2}}{N}},$ (20)

where $\mathcal{L}$ is the integrated luminosity, $N$ is the total number of
events populating the considered $M_{\ell\ell}$ bin (associated with the cross
section $\sigma$), and $A_{\rm FB}$ equivalently stands for the true or
reconstructed forward-backward asymmetry. Moreover, we denote by $\sigma_{F}$
and $\sigma_{B}$ the forward and backward cross sections in the bin. Our
results are given in Figure 4 for the SSM, and in Figure 5 for all models
featuring both a $W^{\prime}$ and a $Z^{\prime}$ boson (left panel), and for
$U(1)^{\prime}$ models (right panel). Figure 4 additionally includes the
corresponding SM expectation as a reference.

Figure 4: Distribution in the invariant mass $M_{\ell\ell}$ of the lepton pair
produced in SM and SSM NC Drell-Yan processes at the HL-LHC (left panel). The
dashed lines represent events with $\cos\theta_{\ell^{-}}>0$ and
$\cos\theta_{\ell^{-}}<0$, and the solid areas are associated with the spectra
obtained after the implementation of various $|y_{\ell\ell}|$ cuts.
Corresponding true and reconstructed forward-backward asymmetries are shown in
the right panel of the figure. We refer to the text for more details.

We provide, in the left panel of Figure 4, predictions for the $M_{\ell\ell}$
distributions in the SM and the SSM after several cuts on the dilepton
rapidity $|y_{\ell\ell}|$ (as filled histograms). Our results show that a too
strict cut on the $|y_{\ell\ell}|$ variable is statistically not an option.
Too few events would indeed survive the cut, implying that there is not enough
statistics to meaningfully compute $A_{\rm FB}$ asymmetries, and therefore
verify that $A_{\rm FB}^{\rm True}\approx A_{\rm FB}^{\rm Rec}$ after the cut.
Such a cut, although potentially useful, is therefore ignored in the
following. We additionally present, for the SSM, the resulting invariant mass
distributions when we solely consider events populating the forward
($\cos\theta_{\ell^{-}}>0$; orange dashes) and backward
($\cos\theta_{\ell^{-}}<0$; brown dashes) hemispheres. This clearly
demonstrates the existence of an asymmetry, that is studied in the right panel
of the figure.

Figure 5: Dependence of the forward-backward asymmetry on the invariant mass
$M_{\ell\ell}$ of the lepton pair produced in NC Drell-Yan processes at the
HL-LHC, in models featuring both a $W^{\prime}$ and a $Z^{\prime}$ boson (left
panel), and in $U(1)^{\prime}$ models (right panel). Predictions are shown for
the true (top row) and reconstructed (bottom row) asymmetries. In the lower
panels of each subfigure, we show the number of events populating a given bin,
which is related to the associated uncertainty according to Eq. (20).

The asymmetry results are generally plagued with very large uncertainties,
that we only show for the SSM case without any $|y_{\ell\ell}|$ cut for
clarity. This consists in the most optimistic situation from the point of view
of the errors. The SM background is indeed by definition extremely low, as one
lies in a very large $M_{\ell\ell}$ regime, and the SSM signal is quite
reduced once cuts are implemented. By taking errors into account, we can
observe that the $A_{\rm FB}^{\rm True}$ and $A_{\rm FB}^{\rm Rec}$
distributions are almost compatible with each other, at least when no
$|y_{\ell\ell}|$ cut is included. The issue with the size of the errors can be
addressed by focusing only on the $Z^{\prime}$ peak where statistics is more
significant. This allows for a precise (enough) determination of the shape of
the asymmetry distribution, so that it could be used for signal
characterisation and to distinguish various $Z^{\prime}$ signals at a given
mass (see below). While we could in principle also compare our new physics
signal to the SM predictions, we refrain from doing so. The SM asymmetry is
indeed plagued with enormous statistical errors (not shown on the figure),
preventing us from making any meaningful conclusive statement. This is
neverthless not crucial as the invariant mass spectrum (left panel of Figure
4) is sufficient to observe a sign of an extra gauge boson relative to the SM.

However, observing a $Z^{\prime}$ state at a given mass (of 6 TeV here) does
not guarantee that we can get handles on the underlying theory. In the
following, we try to examine how using asymmetries in the $M_{\ell\ell}$
region close to the peak (where we have in principle enough statistics) could
help. In Figure 5, we show the forward-backward asymmetries resulting from
Drell-Yan dilepton production in models featuring additional $W^{\prime}$ and
$Z^{\prime}$ bosons (left panel), and in $U(1)^{\prime}$ models (right panel).
We provide our BSM predictions together with the associated statistical
uncertainties. The overarching message emerging from the results is that
alongside the total rate, the $A_{\rm FB}$ observable is useful for the
separation of one model from another. While this is clear for most pairs of
models when relying on the true $A_{\rm FB}^{\rm True}$ spectra, the
discriminating power remains intact for several pairs of models when using
reconstructed $A_{\rm FB}^{\rm Rec}$ spectra. However, due to the limited
number of events, some care should be given to optimising the binning.
Statistical uncertainties are indeed rather large away from the $Z^{\prime}$
mass position, while they can well be controllable near it. In the light of
this, it will be important to also combine ATLAS and CMS data samples to
increase the statistics and reduce the uncertainties.

Figure 6: Distribution in the transverse mass $M_{T}$ of the $\ell\nu$ system
produced in CC Drell-Yan processes at the HL-LHC (left panel). We show
predictions for the SM (olive), as well as for the SSM (red) and for models
with an additional $SU(2)_{L}$ symmetry (blue). In the case of the SM, we
distinguish the production of a positively-charged (purple) and negatively-
charged (yellow) $\ell\nu$ system. The corresponding charge asymmetries are
shown in the right panel of the figure, together with the associated
statistical uncertainties. In the lower panel of the right subfigure, we show
the number of events populating a given bin, which is related to the
associated uncertainty according to Eq. (20).

To conclude, we have shown that in the case of an excess in the NC Drell-Yan
channel, its nature could be diagnosed in terms of the chiral couplings of the
underlying $Z^{\prime}$ state through the usage of asymmetries. This could
then in turn be used (through the existence of $W^{\prime}$ and $Z^{\prime}$
correlations in models containing both bosons) to predict the shape of the
asymmetry that could emerge in the CC Drell-Yan mode $pp\to\ell\nu$, to which
we turn next. We consider in this case the lepton charge asymmetry $A_{\rm
C}$,

$A_{\rm C}=\frac{\frac{{\rm d}\sigma}{{\rm
d}\eta_{\ell}}(pp\to\ell^{+}\nu)-\frac{{\rm d}\sigma}{{\rm
d}\eta_{\ell}}(pp\to\ell^{-}\bar{\nu})}{\frac{{\rm d}\sigma}{{\rm
d}\eta_{\ell}}(pp\to\ell^{+}\nu)+\frac{{\rm d}\sigma}{{\rm
d}\eta_{\ell}}(pp\to\ell^{-}\bar{\nu})}\,,$ (21)

in which the variable $\eta_{\ell}$ refers to the final-state charged lepton
pseudorapidity. We additionally include statistical uncertainties in the
calculation of the charge asymmetry, that we derive similarly to Eq. (20).

The results are presented for the relevant models (i.e. for the SSM and
$SU(2)_{L}^{\prime}$ models) in Figure 6, in which we focus on the
distribution in the transverse mass $M_{T}$ of the $\ell\nu$ system (left
panel) and on the corresponding charge asymmetries (right panel). In the left
panel, we present $M_{T}$ spectra for the SM (olive histogram), as well as for
the SSM (red) and the $SU(2)_{L}^{\prime}$ case (blue). As already found in
the context of the $M_{\ell\ell}$ distribution for neutral currents, the
$M_{T}$ distribution arising in charged currents is sufficient to observe
deviations from the SM through the presence of a significant peak around
$M_{W}^{\prime}$, such a peak being absent in the SM. By virtue of the parton
densities, a charge asymmetry exist, as shown in the SM through the purple
(the $pp\to\ell^{-}\bar{\nu}$ process) and yellow (the $pp\to\ell^{+}\nu$
process) histograms. The difference between these two distributions, that is
similar in the BSM configurations considered and therefore not shown for the
SSM and $SU(2)_{L}^{\prime}$ predictions, results from the different parton
densities relevant for the two hard-scattering processes. Consequently, we
expect the lepton charge asymmetry to be quite similar in all models, as
testified by the right panel of the figure. While the event rate is typically
larger than in the NC case (so that the resulting statistical uncertainties
are smaller), the discriminative power is typically (much) lower. Therefore,
separating the SSM hypothesis from the one assuming a model with an additional
$SU(2)_{L}$ would require to rely essentially on NC events.

##  4 Summary and conclusions

In this work, we have tried to establish a connection between analyses
searching for $W^{\prime}$ and $Z^{\prime}$ boson production and decay at the
LHC. While only neutral gauge bosons are required by models with additional
$U(1)$ groups, both charged and neutral vector states are present in theories
with additional $SU(2)$ factors. Correlations may thus exist. Of course,
searches for such new gauge bosons have been undertaken already at the LHC,
and stringent constraints have been imposed on their properties. The results
of these searches can be translated into mass bounds on the extra bosons
currently falling in the 5–6 TeV region for typically weak gauge couplings.
What is currently missing, though, are correlated searches for charged and
neutral gauge bosons. Within a specific model, relevant information about one
type of boson can indeed yield information about the other, possibly avoiding
the necessity for separate searches or opening the door to more sensitive
combined analyses.

Unlike previous literature, we have chosen specific (rather than general or
simplified) model implementations to illustrate this point. The advantage is
that we can keep the number of parameters to a minimum while taking advantage
of the correlations between masses and/or couplings of the two new gauge
bosons. Among models with extra charged and neutral gauge bosons, we have
selected those with an additional $SU(2)_{L}$ group, as well as those with an
additional $SU(2)_{R}$ group (i.e. the LRSM and ALRSM). We have moreover
compared the associated predictions with those obtained in the SM and the SSM.
In our calculations, we have included interference effects with the SM
background for all new resonances, taking into account that such high
$W^{\prime}$ and $Z^{\prime}$ masses can yield a width comparable to the
experimental resolution in the invariant and transverse masses of the NC and
CC Drell-Yan final states respectively. Furthermore, we have compared our
findings with models that include only a single $Z^{\prime}$ gauge boson,
concentrating on those generated by the breaking of the $E_{6}$ group, as well
as on $U(1)_{B-L}$.

After briefly describing the models considered, we have studied current LHC
constraints on the related new charged and neutral gauge bosons. We have then
proceeded to study the correlations that could arise from present $W^{\prime}$
and $Z^{\prime}$ analyses. In this connection, we have shown how limits on
$W^{\prime}$ signals can be interpreted as constraints on the existence of a
$Z^{\prime}$ boson which are more stringent than those obtained from direct
searches. We have further moved on to analyse their Drell-Yan production and
decay at the 14 TeV LHC, focusing on a large integrated luminosity of 3000
fb-1 like the one attainable at the HL-LHC. Whereas the distributions in both
the dilepton invariant mass (for NCs) and the $\ell\nu$ transverse mass (for
CCs) differ amongst the considered BSM scenarios, they present themselves in
the form of a rather universal Breit-Wigner shape. Getting information on the
underlying theory therefore requires a deeper study.

We have proposed to combine the analysis of the differential rates of the
cross section with that of the corresponding forward-backward (in the NC case)
and lepton charge (in the CC case) asymmetries. We have shown that the
asymmetry lineshapes emerging in the different BSM scenarios considered are
notably different, in particular for the $Z^{\prime}$ analysis. We have
therefore concluded that, for both discovery and characterisation purposes,
information gathered from $A_{\rm FB}$ studies could be beneficial, providing
handles on the underlying model. In our work, we have employed a variable bin
size and advocated the use of combined future ATLAS and CMS data samples at
the HL-LHC in order to minimise the statistical uncertainties. The limited
event rates stemming from the fact that accessible $W^{\prime}$ and
$Z^{\prime}$ masses are rather heavy indeed hinders somewhat the scope of our
procedure at present.

Cross-correlating cross sections and asymmetries appears to be a promising way
to distinguish among the possible BSM models embedding both a $W^{\prime}$ and
a $Z^{\prime}$ boson, and thus mirror the underlying BSM gauge structure. We
regard our results as a useful proof-of-concept of a synergetic approach,
which validation will require dedicated experimental analyses across different
ATLAS and CMS working groups currently engaged in parallel, but not connected,
searches for neutral and charged exotic particles. In this respect, while our
choice of new gauge boson masses near their experimentally allowed minimum
values at present was done to maximise the sensitivity at the HL-LHC, we are
confident that, even after the LHC Run 3, the scope of the advocated analyses
will not be significantly reduced. We then consider these as an additional
motivation for the deployment of the HL-LHC, though the limited event rate
remains a concern which will need to be compensated by ever improving the
phenomenological analyses, combining the advances coming from more precise
theoretical predictions with those stemming from enhanced experimental
efficiencies.

## Code availability

The code in this work can be found in:
https://github.com/oozdal/Asymmetries_MA5

###### Acknowledgements.

M.F. and Ö.Ö. thank NSERC for partial financial support under grant number
SAP105354. The work of SM is supported in part through the NExT Institute and
the STFC Consolidated Grant No. ST/L000296/1. Parts of the numerical
calculations reported in this paper were performed using High Performance
Computing (HPC) managed by Calcul Quebec and Compute Canada, and the IRIDIS
High Performance Computing Facility and associated support services at the
University of Southampton.

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