# Pinning down the leptophobic $Z^{\prime}$ in leptonic final states with Deep
Learning

Tanumoy Mandal<EMAIL_ADDRESS>Indian Institute of Science Education
and Research Thiruvananthapuram, Vithura, Kerala, 695 551, India  Aniket
Masaye  Indian Institute of Science Education and Research Thiruvananthapuram,
Vithura, Kerala, 695 551, India  Subhadip Mitra<EMAIL_ADDRESS>Center for Computational Natural Sciences and Bioinformatics, International
Institute of Information Technology, Hyderabad 500 032, India  Cyrin Neeraj
<EMAIL_ADDRESS>Center for Computational Natural Sciences and
Bioinformatics, International Institute of Information Technology, Hyderabad
500 032, India  Naveen Reule<EMAIL_ADDRESS>Indian Institute of
Science Education and Research Thiruvananthapuram, Vithura, Kerala, 695 551,
India  Kalp Shah<EMAIL_ADDRESS>Center for Computational
Natural Sciences and Bioinformatics, International Institute of Information
Technology, Hyderabad 500 032, India

###### Abstract

A leptophobic $Z^{\prime}$ that does not couple with the Standard Model
leptons can evade the stringent bounds from the dilepton-resonance searches.
In our earlier paper [T. Arun _et al._ , Search for the $Z^{\prime}$ boson
decaying to a right-handed neutrino pair in leptophobic $U(1)$ models, Phys.
Rev. D, 106 (2022) 095035; arXiv:2204.02949], we presented two gauge anomaly-
free $U(1)$ models—one based on the Green-Schwarz (GS) anomaly cancellation
mechanism, and the other on a grand unified theory (GUT) framework with gauge
kinetic mixing—where a heavy leptophobic $Z^{\prime}$ is present along with
right-handed neutrinos ($N_{R}$). We pointed out the interesting possibility
of a correlated search for $Z^{\prime}$ and $N_{R}$ at the LHC through the
$pp\to Z^{\prime}\to N_{R}N_{R}$ channel. This channel can probe a part of the
$Z^{\prime}$ parameter space beyond the reach of the standard dijet resonance
searches. In this follow-up paper, we analyse the challenging monolepton final
state arising from the decays of the $N_{R}$ pair with Deep Learning. We
present the high-luminosity LHC discovery reaches for six different GUT
embeddings and a benchmark point in the GS setup. We also update our previous
estimates in the dilepton channel with Deep Learning. We identify parameter
regions that can be probed with the proposed channel but will remain
inaccessible to dijet searches at the HL-LHC.

††preprint: XXXXXX

##  I Introduction

Many extensions of the Standard Model (SM), like the grand unified theories
(GUTs) Langacker:1980js ; London:1986dk ; Hewett:1988xc , contain an
electrically neutral colour-singlet heavy gauge boson, $Z^{\prime}$. To get
such a particle in the spectrum in a bottom-up manner, one can simply add an
extra local $U(1)$ symmetry spontaneously broken by a complex scalar field
with a TeV-scale vacuum expectation value (vev). The phenomenology of a TeV-
scale $Z^{\prime}$ is well-explored in the literature (see, e.g., Leike:1998wr
; Langacker:2008yv ). One of the current programs of the LHC is to look for
the signature of $Z^{\prime}$. However, so far, the focus has been only on the
cases where it decays exclusively to the SM particles ATLAS:2019fgd ;
CMS:2019gwf ; ATLAS:2019erb ; CMS:2021ctt ; ATLAS:2016gzy ; CMS:2016kgr ;
ATLAS:2020fry ; CMS:2021klu ; CMS:2015fhb . The failure to find the
$Z^{\prime}$ so far motivates us to look for other possibilities with
_nonstandard_ decay modes of $Z^{\prime}$, i.e., where it can decay to beyond-
the-SM (BSM) particles as well.

The strongest exclusion limits on the $Z^{\prime}$ parameter space usually
come from the dilepton resonance searches ATLAS:2019erb ; CMS:2021ctt . For
instance, the current mass exclusion limit on a sequential $Z^{\prime}$ is
about $5$ TeV ATLAS:2019erb . The stringent dilepton exclusion limits can be
evaded if the $Z^{\prime}$ is leptophobic, i.e., it does not couple (or
couples very feebly) to the SM leptons. In the leptophobic case, the
nonstandard decay modes of the $Z^{\prime}$ can become important as the
branching ratios (BRs) to the new modes can be high. One interesting
possibility is when the $Z^{\prime}$ decays to a pair of right-handed
neutrinos (RHNs, $N_{R}$) Ferrari:2002ac ; Das:2017flq ; Das:2017deo ;
Cox:2017eme ; Chauhan:2021xus ; Das:2022rbl . While appending an additional
$U(1)$ gauge symmetry to the SM group, one must ensure the cancellation of all
gauge anomalies to maintain gauge invariance and renormalisability of the
theory. To cancel the gauge anomalies, one can include RHNs in the particle
spectrum Ekstedt:2016wyi  or rely on some special mechanism like the Green-
Schwarz (GS) mechanism Leontaris:1999wf ; Ekstedt:2017tbo . The minimal GS
mechanism does not require RHNs. However, the cancellation of pure gravity
anomaly motivates the existence of RHNs. Therefore, many anomaly-free $U(1)$
extensions of the SM contain RHNs since they can also help generate light
neutrino masses through various seesaw mechanisms.

In our previous paper Arun:2022ecj , we presented two theoretically well-
motivated leptophobic $Z^{\prime}$ scenarios where the $Z^{\prime}$ dominantly
decays to a pair of RHNs. Being leptophobic, these scenarios can easily bypass
the dilepton exclusion limits. At the same time, a large BR of the
$Z^{\prime}\to N_{R}N_{R}$ decay and the subsequent decays of $N_{R}$ to
$W\ell$, $Z\nu$, and $H\nu$ final states open up many interesting leptonic
final states (see, e.g., Fig. 1). The process $pp\to Z^{\prime}\to N_{R}N_{R}$
is important not only from the $Z^{\prime}$-search point of view but also for
the RHN searches. Since RHNs are SM singlets, producing them at the LHC is
difficult as their production cross sections are small, suppressed by the
light-heavy neutrino mixing angles. However, they can be copiously produced if
they come from the decays of another BSM particle Choudhury:2020cpm ;
Deka:2021koh ; ThomasArun:2021rwf ; Bhaskar:2023xkm .

We considered the $pp\to Z^{\prime}\to N_{R}N_{R}$ channel in a leptophobic
setup in Ref. Arun:2022ecj . We studied the dilepton channel (which is easy to
analyse with a cut-based analysis and shows a good sensitivity) to estimate
the HL-LHC reach. Interestingly, we found that a large chunk of the parameter
space beyond the reach of $pp\to Z^{\prime}\to jj$ can be probed through this
channel at the HL-LHC. The decay of the RHN pair can lead to multilepton final
states. The monolepton mode is complimentary but more challenging because of
the difficulty in the background reduction. In this follow-up paper, we
estimate the reach in the monolepton final state with a deep neural network
(DNN) model.

The RHNs we consider are around the TeV scale, much lighter than the standard
type-I seesaw scale $\sim 10^{14}$ GeV. To naturally realise the TeV-scale
RHNs, we consider inverse-seesaw mechanism (ISM) for neutrino mass generation
Mohapatra:1986aw ; Mohapatra:1986bd . As a result, unlike the Majorana-type
RHNs in the type-I seesaw, the RHNs here are pseudo-Dirac in nature. If
neutrino mass is generated using a type-I seesaw mechanism, because of the
Majorana nature of the RHNs, the collider consequences will be different from
the ISM. For example, from the $pp\to Z^{\prime}\to N_{R}N_{R}$ process, we
get a same-sign dilepton signature, which is not present in the ISM framework.
Earlier, in Refs. Das:2017kkm ; Jana:2019mez ; Das:2019pua ;
Bandyopadhyay:2020djh ; Das:2021nqj , different $U(1)$ extensions using the
ISM were considered in various contexts. Also, the future lepton colliders
could be the best testing ground for the heavy neutrinos Banerjee:2015gca ;
Chakraborty:2018khw ; Barducci:2022hll .

The rest of the paper is organised as follows. In Sec. II, we review the
leptophobic models; in Sec. III, we present our analysis of the monolepton
channel; in Sec. IV, we estimate the HL-LHC prospects of the monolepton
channel. We also update our estimate of the dilepton prospects with the DNN in
Sec. V. Finally, we conclude in Sec. VI.

Figure 1: Representative Feynman diagram for a leptophobic $Z^{\prime}$
decaying to two right-handed neutrinos.

##  II Leptophobic $Z^{\prime}$ models

We look at the two examples of anomaly-free gauge extensions of the SM from
Ref. Arun:2022ecj , where a leptophobic $Z^{\prime}$ with a substantial
$Z^{\prime}\to N_{R}N_{R}$ branching is present. In one example, the RHNs
cancel the mixed gauge-gravity anomaly while the GS mechanism cancels the
rest. In the other example, the leptophobia of $Z^{\prime}$ arises in a GUT
framework Babu:1996vt ; Lopez:1996ta ; Rizzo:1998ut ; Leroux:2001fx . Below,
we briefly review the essential details of these two constructions and
elaborate on some aspects for completeness.

Table 1: Particle representations and quantum numbers in the GS anomaly-cancellation setup. | $SU(3)_{c}$ | $SU(2)_{L}$ | $U(1)_{Y}$ | $U(1)_{z}$
---|---|---|---|---
$Q_{L}$ | $\mathbf{3}$ | $\mathbf{2}$ | $1/3$ | $z_{q}$
$u_{R}$ | $\overline{\mathbf{3}}$ | $\mathbf{1}$ | $4/3$ | $\alpha z_{q}$
$d_{R}$ | $\overline{\mathbf{3}}$ | $\mathbf{1}$ | $-2/3$ | $\beta z_{q}$
$\ell_{L}$ | $\mathbf{1}$ | $\mathbf{2}$ | $-1$ | $0$
$e_{R}$ | $\mathbf{1}$ | $\mathbf{1}$ | $-2$ | $0$
$N_{R}$ | $\mathbf{1}$ | $\mathbf{1}$ | $0$ | $z_{N}$
$H$ | $\mathbf{1}$ | $\mathbf{2}$ | $1$ | 0
$\phi$ | $\mathbf{1}$ | $\mathbf{1}$ | $0$ | $1$
$S_{L}$ | $\mathbf{1}$ | $\mathbf{1}$ | $0$ | $0$

###  II.1 Leptophobia with GS mechanism

In this model, the particle spectrum includes three RHNs (one for every
generation) equally charged under the extra $U(1)_{z}$. Even though the RHNs
are not essential for gauge anomaly cancellation with the GS mechanism
Ekstedt:2017tbo , the cancellation of the pure gravity anomaly motivates us to
introduce them. The light-neutrino masses are generated through the ISM, which
requires an extra chiral sterile fermion $S_{L}$ per generation.

We show the $U(1)_{z}$ charges of various particles in Table 1. The SM leptons
are uncharged under $U(1)_{z}$ to make the $Z^{\prime}$ leptophobic. The
nonzero $U(1)_{z}$ charges of the quarks ensure that the $Z^{\prime}$ can be
produced at the LHC through quark-quark fusion. The $U(1)_{z}$ charge
assignment is generation-independent, and we introduce two free parameters
$\alpha$ and $\beta$ to parametrise the right-handed up and down-type quark
charges, respectively ($\alpha=\beta=-1$ in Ref. Arun:2022ecj ). The SM Higgs
doublet is chargeless under $U(1)_{z}$ to minimise the mixing between $Z$ and
$Z^{\prime}$. The scalar $\phi$ with unit $U(1)_{z}$ charge is the flavon
field needed to generate the quark Yukawa interactions as shown in Eq. (5).
The fermion masses arise after $\phi$ and $H$ get vacuum expectation values.

The charge assignment in Table 1 leads to six anomalous triangle diagrams
proportional to the traces of the product of the generators:

  1. 1.

$[SU(3)_{C}]^{2}[U(1)_{z}]:\hskip 8.5359pt{\rm
tr}[\\{\mathcal{T}^{a},\mathcal{T}^{b}\\}z]=3z_{q}(2-\alpha-\beta)$,

  2. 2.

$[SU(2)_{L}]^{2}[U(1)_{z}]:\hskip 9.95863pt{\rm
tr}[\\{T^{a},T^{b}\\}z]=6z_{q}$,

  3. 3.

$[U(1)_{z}]^{3}:\hskip 45.52458pt{\rm
tr}[z^{3}]=3z_{q}^{3}(2-\alpha^{3}-\beta^{3})-z_{N}^{3}$,

  4. 4.

$[U(1)_{Y}]^{2}[U(1)_{z}]:\hskip 14.22636pt{\rm
tr}[Y^{2}z]=\dfrac{2z_{q}}{3}(1-8\alpha-2\beta)$,

  5. 5.

$[U(1)_{Y}][U(1)_{z}]^{2}:\hskip 14.22636pt{\rm
tr}[Yz^{2}]=2z_{q}^{2}(1-2\alpha^{2}+\beta^{2})$,

  6. 6.

$[R]^{2}[U(1)_{z}]:\hskip 34.14322pt{\rm tr}[z]=3z_{q}(2-\alpha-\beta)-z_{N}$.

Since gravity becomes prominent at high scales, it is natural to assume that
the mixed gauge-gravity anomaly vanishes at low energy without the GS
mechanism. Therefore, we set

$\displaystyle 3z_{q}(2-\alpha-\beta)-z_{N}=0\ $ (1)

by hand. This gives us a relation between the $U(1)_{z}$ charges of the quarks
and RHNs.

To cancel the other anomalies with the GS mechanism Anastasopoulos:2006cz ;
Anastasopoulos:2008jt , we add new gauge-dependent terms in the Lagrangian
with carefully chosen coefficients. The Peccei-Quinn (PQ) terms for the mixed
gauge anomalies and the $[U(1)_{z}]^{3}$ anomaly can be written as

$\displaystyle\mathcal{L}_{\text{PQ}}=$
$\displaystyle\dfrac{1}{96\pi^{2}}\left(\dfrac{\Theta}{M}\right)\varepsilon_{\mu\nu\rho\sigma}\Big{[}g_{z}^{2}\mathcal{C}_{zzz}F_{z}^{\mu\nu}F_{z}^{\rho\sigma}+g_{z}g^{\prime}\mathcal{C}_{zzy}F_{z}^{\mu\nu}F_{Y}^{\rho\sigma}$
$\displaystyle+g^{\prime
2}\mathcal{C}_{zyy}F_{Y}^{\mu\nu}F_{Y}^{\rho\sigma}+g^{2}\mathcal{D}_{2}\mathrm{tr}\left(F_{W}^{\mu\nu}F_{W}^{\rho\sigma}\right)$
$\displaystyle+g_{S}^{2}\mathcal{D}_{3}\mathrm{tr}\left(F_{S}^{\mu\nu}F_{S}^{\rho\sigma}\right)\Big{]}$
(2)

Under the new $U(1)_{z}$, the pseudoscalar (Goldstone) axion $\Theta$
transforms as $\Theta\rightarrow\Theta+Mg_{z}\theta_{z}$ and the new gauge
field transforms as $B^{\mu}_{z}\rightarrow
B^{\mu}_{z}-\partial^{\mu}\theta_{z}$. Here, $M$ is the scale at which
$U(1)_{z}$ breaks through the Stückelberg mechanism, ${F_{z},\ F_{Y},\ F_{W},\
F_{S}}$ are the field strengths and ${g_{z},\ g^{\prime},\ g,\ g_{S}}$ are the
coupling constants associated with the ${U(1)_{z},\ U(1)_{Y},\ SU(2)_{L},\
SU(3)_{c}}$ gauge groups, respectively. The generalised Chern-Simons (GCS)
terms for other mixed anomalies are given as

$\displaystyle\mathcal{L}_{\text{GCS}}=$
$\displaystyle\frac{1}{48\pi^{2}}\varepsilon_{\mu\nu\rho\sigma}\Big{[}\vphantom{\Omega_{S}^{\nu\rho\sigma}}g^{\prime
2}g_{z}\mathcal{E}_{zyy}B_{Y}^{\mu}B_{z}^{\nu}F_{Y}^{\rho\sigma}+g^{\prime}g_{z}^{2}\mathcal{E}_{zzy}B_{Y}^{\mu}B_{z}^{\nu}F_{z}^{\rho\sigma}$
$\displaystyle+g^{2}g_{z}\mathcal{K}_{2}B_{z}^{\mu}\Omega_{W}^{\nu\rho\sigma}+g_{S}^{2}g_{z}\mathcal{K}_{3}B_{z}^{\mu}\Omega_{S}^{\nu\rho\sigma}\Big{]}\
$ (3)

with

$\displaystyle\Omega_{S,W}^{\nu\rho\sigma}=$
$\displaystyle\frac{1}{3}\mathrm{tr}\Big{[}A_{S,W}^{\nu}\left(F_{S,W}^{\rho\sigma}-[A_{S,W}^{\rho},A_{S,W}^{\sigma}]\right)$
$\displaystyle+\left(\mathrm{cyclic~{}permutations~{}of~{}}\rho,\sigma,v\right)\Big{]}$

and $A_{X}=\left[G,W\right]$. The coefficients $\mathcal{C}$, $\mathcal{D}$,
$\mathcal{E}$, and $\mathcal{K}$ can be solved in terms of the $U(1)_{z}$
charges of the fermions as

$\displaystyle\mathcal{C}_{zzz}$
$\displaystyle=-3z_{q}^{3}\left(2-\alpha^{3}-\beta^{3}\right)+z_{N}^{3},$
$\displaystyle\mathcal{C}_{zyy}$
$\displaystyle=2z_{q}\left(2\beta-8\alpha-1\right)=\frac{3}{2}\mathcal{E}_{zyy},$
$\displaystyle\mathcal{C}_{zzy}$
$\displaystyle=6z_{q}\left(2\alpha^{2}-\beta^{2}-1\right)=3E_{zzy},$ (4)
$\displaystyle\mathcal{D}_{2}$
$\displaystyle=-18z_{q}=-\frac{3}{2}\mathcal{K}_{2},$
$\displaystyle\mathcal{D}_{3}$
$\displaystyle=9z_{q}\left(\alpha+\beta-2\right)=-\frac{3}{2}\mathcal{K}_{3},$

to cancel the anomalies. The $U(1)_{z}$ invariant Yukawa interactions are
written as

$\mathcal{L}_{Y}\supset-\lambda_{u}\left(\frac{\phi}{\Lambda}\right)^{2z_{q}}\overline{Q}_{L}\widetilde{H}u_{R}-\lambda_{d}\left(\frac{\phi}{\Lambda}\right)^{2z_{q}}\overline{Q}_{L}Hd_{R}+h.c.,$
(5)

where $\widetilde{H}=i\sigma_{2}H^{*}$, and $\lambda_{u}$ and $\lambda_{d}$
are the Yukawa couplings. After the spontaneous breaking of the $U(1)_{z}$
symmetry, $\phi$ acquires a vev, $v_{\phi}=\langle\phi\rangle$, leading to the
SM Yukawa terms. We set $z_{q}=1/2$ for our analysis. This fixes $z_{N}=6$ for
$\alpha=\beta=-1$.

####  II.1.1 Neutrino mass generation

The neutrino masses are generated by the ISM through the following higher-
dimensional operators,

$\displaystyle\mathcal{L}_{Y}\supset$ $\displaystyle\
-\lambda^{\nu}_{ij}\left(\frac{\phi^{\dagger}}{\Lambda}\right)^{z_{N}}\overline{L}^{i}_{L}\widetilde{H}N^{j}_{R}-{M_{R}}_{ij}\left(\frac{\phi}{\Lambda}\right)^{z_{N}}\overline{N}_{R}^{i}S_{L}^{j}$
$\displaystyle\ -\frac{\mu}{2}\overline{S}^{c}_{L}S_{L}+h.c.$ (6)

We get the mass matrix for the neutrinos after the sequential breaking of the
$U(1)_{z}$ and electroweak symmetries in the $(\nu_{L}^{c}\ N_{R}\
S_{L}^{c})^{T}$ basis as,

$\mathbf{M_{\nu}}=\begin{pmatrix}0&\mathbf{m_{D}}&0\\\
\mathbf{m_{D}^{T}}&0&\mathbf{M_{R}}\\\ 0&\mathbf{M_{R}^{T}}&\mathbf{\mu}\\\
\end{pmatrix},$ (7)

where $\mathbf{m_{D}}=\mathbf{\lambda^{\nu}}v_{h}$, $\mathbf{\mu}$, and
$\mathbf{M_{R}}$ are $3\times 3$ mass matrices in the generation space in
general. The parameter $\mathbf{\mu}$ is typically associated with lepton
number violation and restores the lepton number symmetry in the
$\mathbf{\mu}\rightarrow 0$ limit. After diagonalising the matrix in Eq. (7),
we get the active (light) neutrino mass matrix as Hettmansperger:2011bt ,

$\mathbf{m_{\nu}}=\mathbf{m_{D}}\mathbf{M^{-1}_{R}}\mathbf{\mu}(\mathbf{M_{R}^{T}})^{-1}\mathbf{m_{D}}+\mathcal{O}\left(\mu\frac{\mathbf{m_{d}^{4}}}{\mathbf{M^{4}_{R}}}\right).$
(8)

There is a double suppression of the light neutrino masses from the large
$\mathbf{M_{R}}$ (taken around the TeV scale) and the small $\mathbf{\mu}$.

Table 2:  Quantum numbers ($\overline{Q}_{\psi}\equiv 2\sqrt{6}Q_{\psi}$, $\overline{Q}_{\chi}\equiv 2\sqrt{10}Q_{\chi}$) of the particle contained in the $\mathbf{27}$ representation of $E_{6}$ for the six possible embeddings—the first one is called the standard embedding. Here, $\mathcal{G}_{\text{SM}}\equiv SU(3)_{c}\times SU(2)_{L}\times U(1)_{Y}$. The $\tan\theta$ and $\delta$ values needed to get the _leptophobic_ $Z^{\prime}$ boson in each embedding are shown in the bottom rows.  Particle | $\mathcal{G}_{\rm SM}$ | I | II | III | IV | V | VI
---|---|---|---|---|---|---|---
$\overline{Q}_{\psi}$ | $\overline{Q}_{\chi}$ | $\overline{Q}_{\psi}$ | $\overline{Q}_{\chi}$ | $\overline{Q}_{\psi}$ | $\overline{Q}_{\chi}$ | $\overline{Q}_{\psi}$ | $\overline{Q}_{\chi}$ | $\overline{Q}_{\psi}$ | $\overline{Q}_{\chi}$ | $\overline{Q}_{\psi}$ | $\overline{Q}_{\chi}$
$Q=(u,d)^{T}$ | $(\textbf{3},\textbf{2})_{1/3}$ | $1$ | $-1$ | $1$ | $-1$ | $1$ | $-1$ | $1$ | $-1$ | $1$ | $-1$ | $1$ | $-1$
$L=(\nu,e)^{T}$ | $(\textbf{1},\textbf{2})_{-1}$ | $1$ | $3$ | $-2$ | $-2$ | $1$ | $3$ | $-2$ | $-2$ | $1$ | $3$ | $-2$ | $-2$
$u^{c}$ | $(\overline{\textbf{3}},\textbf{1})_{-4/3}$ | $1$ | $-1$ | $1$ | $-1$ | $1$ | $3$ | $1$ | $3$ | $-2$ | $-2$ | $-2$ | $-2$
$d^{c}$ | $(\overline{\textbf{3}},\textbf{1})_{2/3}$ | $1$ | $3$ | $1$ | $3$ | $1$ | $-1$ | $1$ | $-1$ | $1$ | $-1$ | $1$ | $-1$
$e^{c}$ | $(\textbf{1},\textbf{1})_{2}$ | $1$ | $-1$ | $1$ | $-1$ | $1$ | $-5$ | $1$ | $-5$ | $4$ | $0$ | $4$ | $0$
$\nu^{c}$ | $(\textbf{1},\textbf{1})_{0}$ | $1$ | $-5$ | $1$ | $-5$ | $1$ | $-1$ | $1$ | $-1$ | $1$ | $-1$ | $1$ | $-1$
$\mathfrak{L}=(N,E)^{T}$ | $(\textbf{1},\textbf{2})_{-1}$ | $-2$ | $-2$ | $1$ | $3$ | $-2$ | $-2$ | $1$ | $3$ | $-2$ | $-2$ | $1$ | $3$
$\mathfrak{L}^{c}=(N^{c},E^{c})^{T}$ | $(\textbf{1},\textbf{2})_{1}$ | $-2$ | $2$ | $-2$ | $2$ | $-2$ | $2$ | $-2$ | $2$ | $-2$ | $2$ | $-2$ | $2$
$\mathfrak{D}$ | $(\textbf{3},\textbf{1})_{-2/3}$ | $-2$ | $2$ | $-2$ | $2$ | $-2$ | $2$ | $-2$ | $2$ | $-2$ | $2$ | $-2$ | $2$
$\mathfrak{D}^{c}$ | $(\overline{\textbf{3}},\textbf{1})_{2/3}$ | $-2$ | $-2$ | $-2$ | $-2$ | $-2$ | $-2$ | $-2$ | $-2$ | $1$ | $3$ | $1$ | $3$
$\widetilde{\nu}^{c}$ | $(\textbf{1},\textbf{1})_{0}$ | $4$ | $0$ | $4$ | $0$ | $4$ | $0$ | $4$ | $0$ | $1$ | $-5$ | $1$ | $-5$
$\tan\theta$ | $\sqrt{3/5}$ | $\sqrt{3/5}$ | $\sqrt{15}$ | $\sqrt{5/27}$ | $\sqrt{5/3}$ | $0$
$\delta$ | $-1/3$ | $-1/3$ | $-\sqrt{10}/3$ | $-\sqrt{5/12}$ | $-\sqrt{5/12}$ | $-\sqrt{10}/3$

###  II.2 Leptophobia in $E_{6}$ GUT models

A leptophobic $Z^{\prime}$ can be realised in some $E_{6}$ GUT models through
gauge-boson kinetic mixing Babu:1996vt  even though the fermion couplings to
the gauge bosons are not arbitrary free parameters. The $U(1)_{z}$ group in
such models can come from a symmetry-breaking chain, like,

$\begin{split}E_{6}&\rightarrow SO(10)\times U(1)_{\chi}\\\ &\rightarrow
SU(5)\times U(1)_{\chi}\times U(1)_{\psi}\\\ &\rightarrow SU(2)_{L}\times
U(1)_{Y}\times U(1)_{z}\\\ &\rightarrow SU(2)_{L}\times U(1)_{Y}.\end{split}$
(9)

The fundamental representation of $E_{6}$ is 27 dimensional, containing all
the SM fermions along with two colour-neutral isosinglets $\nu^{c}$ and
$\widetilde{\nu}^{c}$, a colour-singlet isodoublet $\mathfrak{L}=(N\ E)^{T}$
and its conjugate $\mathfrak{L}^{c}$, and a colour-triplet isosinglet
$\mathfrak{D}$ and its antiparticle $\mathfrak{D}^{c}$. The RHN can be
identified with either $\nu^{c}$ or $\widetilde{\nu}^{c}$. Under $SO(10)$, the
27 dimensional representation breaks into a 16, a 10 and an 1. These further
break into representations of $SU(5)$. In the _standard_ embedding (I in Table
2), we put all the SM fermions and the RHN into the 16, and the new fermions
in the 10 and the 1. There are five other ways one can embed these fermions,
leading to a total of six different embeddings Leroux:2001fx . In Table 2, we
show the new charges of all the fermions for the six different embeddings.

From the chain in Eq. (9), the $U(1)_{z}$ charge $Q_{z}$ can be expressed as
$Q_{z}=Q_{\psi}\cos\theta-Q_{\chi}\sin\theta$ where $\theta$ is the $E_{6}$
mixing angle. However, there is no solution for $\theta$ where the
lepton-$Z^{\prime}$ couplings vanish. Therefore, we look at the
$U(1)_{Y}\leftrightarrow U(1)_{z}$ gauge-kinetic mixing terms allowed by the
gauge invariance,

$\mathcal{L}_{kin}\supset-\frac{1}{4}\widetilde{F}_{Y}^{\mu\nu}\widetilde{F}_{Y\,\mu\nu}-\frac{1}{4}\widetilde{F}_{z}^{\mu\nu}\widetilde{F}_{z\,\mu\nu}-\frac{\sin\chi}{2}\widetilde{F}_{Y\,\mu\nu}\widetilde{F}^{\mu\nu}_{z}.$
(10)

The kinetic mixing, parametrised by $\sin\chi$, can be removed by the
following transformation Rizzo:1998ut :

$\tilde{B}^{\mu}_{Y}=B_{Y}^{\mu}-\tan\chi\
B_{z}^{\mu},\quad\tilde{B}_{z}^{\mu}=\frac{B_{z}^{\mu}}{\cos\chi}.$ (11)

However, this cancellation is valid only at a particular scale, implying that
the mixing term can regenerate at higher orders. This makes the couplings
energy-dependent. Hence, they must be evaluated at the TeV scale before
relating to the experiments. After the above rotation, the couplings of the
abelian gauge bosons with the fermions are given as Rizzo:1998ut ,

$\displaystyle\mathcal{L}_{int}\supset-\overline{\psi}\gamma_{\mu}\left[g^{\prime}\dfrac{Y}{2}B^{\mu}_{Y}+g_{z}\left(Q_{z}+\sqrt{\frac{3}{5}}\delta\dfrac{Y}{2}\right)B_{z}^{\mu}\right]\psi,$
(12)

where $\delta=-\tilde{g}_{Y}\sin\chi/\tilde{g}_{z}$. Now, the fermion
couplings are functions of two free parameters $\theta$ and $\delta$; we can,
in general, make the couplings of any two fields vanish. For example, for the
_standard_ embedding (Table 2), leptophobia corresponds to
$\theta=\tan^{-1}\sqrt{3/5}$ and $\delta=-1/3$ Leroux:2001fx . Different
combinations of $\theta$ and $\delta$ work in different embeddings, as shown
in Table 2.

((a)) (a)

((b)) (b)

((c)) (c)

((d)) (d)

((e)) (e)

((f)) (f)

((g)) (g)

((h)) (h)

((i)) (i)

Figure 2: Normalised density plots of the key features after passing through the cuts $\mathfrak{C}_{1}$ to $\mathfrak{C}_{5}$ for four benchmarks points: (a) BP1 $\equiv(M_{Z^{\prime}}=5.0,~{}M_{N_{R}}=2.0)$ TeV, (b) BP2 $\equiv(M_{Z^{\prime}}=3.0,~{}M_{N_{R}}=0.5)$ TeV, (c) BP3 $\equiv(M_{Z^{\prime}}=4.0,~{}M_{N_{R}}=1.0)$ TeV, and (d) BP4 $\equiv(M_{Z^{\prime}}=4.0,~{}M_{N_{R}}\approx 2.0)$ TeV. BP1 distributions are highlighted in green as we use this benchmark to tune the DNN model. Here, the fatjets are mass-ordered and the jets are $p_{T}$-ordered.  Table 3: Effective $U(1)_{z}$ charges for various $E_{6}$ embeddings and the GS model. For the six $E_{6}$ embeddings, we compute the charges using the leptophobic $\tan\theta$ and $\delta$ combinations from Table 2. For the GS model, we use the benchmark point mentioned in Sec. II.1. Model | $z_{u}$ | $z_{d}$ | $z_{N}$
---|---|---|---
$E_{6}$ embedding |  |  |
I | $1.242$ | $0.786$ | $-1.667$
II | $1.242$ | $0.786$ | $-1.667$
III | $0.393$ | $0.248$ | $-0.527$
IV | $0.962$ | $0.481$ | $-1.936$
V | $0.481$ | $0.481$ | $-1.936$
VI | $0.393$ | $0.248$ | $-0.527$
GS | $0.707$ | $0.707$ | $6$
Table 4: Projected number of events at the $14$ TeV HL-LHC at different stages of analysis in the monolepton channel for the two benchmark points, BP1 $\equiv(M_{Z^{\prime}}=5.0,~{}M_{N_{R}}=2.0)$ TeV and BP2 $\equiv(M_{Z^{\prime}}=3.0,~{}M_{N_{R}}=0.5)$ TeV (see Fig. 2) with $g_{z}=0.3$. Here, $S_{1}$ and $S_{2}$ denote the number of signal events for BP1 and BP2, respectively. We obtain DNN(BP1) and DNN(BP2) after training the network on BP1 and BP2 training data, respectively. For the signal, we assume a $K$ factor of $1.3$ Eichten:1984eu  and BR$(Z^{\prime}\to N_{R}N_{R})=50\%$. | Signal | Background |  |
---|---|---|---|---
| $S_{1}$ | $S_{2}$ | $t_{h}t_{\ell}$ | $t_{\ell}t_{\ell}$ | $W_{\ell}+j$ | $W_{\ell}W_{h}+j$ | $Z_{h}W_{\ell}+j$ | $S_{1}/B$ | $S_{2}/B$
Initial | $235$ | $12409$ | $276832$ | $4382$ | $392206$ | $291684$ | $22729$ | $2.37\times 10^{-4}$ | $1.26\times 10^{-2}$
$\mathfrak{C}_{1}$ | $234$ | $11764$ | $262336$ | $4035$ | $379874$ | $284631$ | $22168$ | $2.46\times 10^{-4}$ | $1.23\times 10^{-2}$
$\mathfrak{C}_{2}$ | $186$ | $9431$ | $165438$ | $1006$ | $271623$ | $209486$ | $16168$ | $2.80\times 10^{-4}$ | $1.42\times 10^{-2}$
$\mathfrak{C}_{3}$ | $184$ | $9158$ | $164646$ | $964$ | $257573$ | $205999$ | $15973$ | $2.85\times 10^{-4}$ | $1.41\times 10^{-2}$
$\mathfrak{C}_{4}$ | $152$ | $5224$ | $101562$ | $711$ | $105221$ | $99886$ | $7559$ | $4.83\times 10^{-4}$ | $1.65\times 10^{-2}$
$\mathfrak{C}_{5}$ | $151$ | $5212$ | $96071$ | $683$ | $98984$ | $94865$ | $7171$ | $5.07\times 10^{-3}$ | $1.66\times 10^{-2}$
DNN(BP1) | $147$ | — | $\sim 11$ | $\sim 1$ | $\sim 136$ | $\sim 213$ | $\sim 15$ | $0.39$ | —
DNN(BP2) | — | $4534$ | $\sim 8$ | $\sim 1$ | $\sim 82$ | $\sim 160$ | $\sim 10$ | — | $17.37$
Table 5: Kinematic variables chosen to analyse the monolepton signature; $\tau$’s are the n-subjettiness ratios. Variable | Description
---|---
$H_{T}$ | Sum of transverse momenta of hadronic objects.
$p_{T_{i}}$ | Transverse Momenta of all reconstructed objects.
$\not{E}_{T}$ | Missing transverse energy.
$m_{J_{i}}$ | Invariant Masses of the reconstructed fatjets.
$\Delta R_{ik}$ | Distances between any two reconstructed objects.
$m_{J_{i}\mu}$ | Invariant masses of the fatjet-lepton pairs.
$m_{j_{i}j_{k}}$ | Invariant masses of any two reconstructed AK-4 jets.
$\tau_{21}^{\beta=1}$ |  $\tau_{2}/\tau_{1}$, $\beta=1.0$ for each fatjet.
$\tau_{21}^{\beta=2}$ |  $\tau_{2}/\tau_{1}$, $\beta=2.0$ for each fatjet.
$\tau_{32}^{\beta=1}$ |  $\tau_{3}/\tau_{2}$, $\beta=1.0$ for each fatjet.
$\tau_{32}^{\beta=2}$ |  $\tau_{3}/\tau_{2}$, $\beta=2.0$ for each fatjet.

##  III Collider Analysis

For collider analysis, we use the following simplified Lagrangian,

$\displaystyle\mathcal{L}\supset\dfrac{g_{z}}{2}\left(z_{u}\overline{u}_{i}\gamma^{\mu}u_{i}+z_{d}\overline{d}_{i}\gamma^{\mu}d_{i}+z_{N}\overline{N}_{R}\gamma^{\mu}N_{R}\right)Z^{\prime}_{\mu}\
,$ (13)

where $z_{u,d}$ are the effective $U(1)_{z}$ charges of up and down type
quarks (for any quark $q$, $z_{q}^{2}=z_{q_{L}}^{2}+z_{q_{R}}^{2}$). We show
$z_{u}$ and $z_{d}$ for the six different $E_{6}$ embeddings and the GS model
in Table 3. Between the two colour-singlet isosinglets in Table 2, we identify
the one with higher absolute $U(1)_{z}$ charge with the RHN, i.e.,
$z_{N}=~{}\mbox{Max}(z_{\nu},z_{\widetilde{\nu}})$.

We use FeynRules Alloul:2013bka  to build the above model and obtain the
Universal FeynRules Output (UFO). We simulate the hard scattering in MadGraph5
Alwall:2014hca , showering and hadronisation in Pythia Sjostrand:2014zea  and
the LHC detector environment in Delphes3 deFavereau:2013fsa . The events are
generated at $\sqrt{s}=14$ TeV. Jets are clustered using the anti-$k_{T}$
algorithm Cacciari:2008gp  with $R=0.4$ (we call them AK-4 jet). For the
electron, the DeltaRMax parameter in the Delphes card (distance between the
other isolated objects and the identified electron) for isolation has been
modified to $0.2$ from $0.5$.

###  III.1 Event selection criteria: Monolepton channel

We show the signal topology in Fig. 1. In Ref. Arun:2022ecj , we performed a
cut-based analysis of the dilepton signature. Here, in this paper, we mainly
focus on the monolepton signature, which arises when one of the RHNs decays to
a hadronically decaying $W$ boson and a lepton (muon) while the other decays
to a $Z/H$ boson and a neutrino:

$pp\to Z^{\prime}\to N_{R}N_{R}\to\mu+\not{E}+\text{jets}.$

As explained earlier, probing the monolepton channel is more challenging than
the dilepton case as the backgrounds (see Table 4) are harder to tame with
simple kinematic cuts. Hence, we employ a DNN model to enhance the signal
sensitivity. We look at $Z^{\prime}$ masses between $3-7$ TeV and RHNs ranging
from $0.5$ to $3.5$ TeV. From the topology and the distributions of the final
state, we design the following basic selection criteria for the signal events:

  * $\mathfrak{C}_{1}$:

$H_{T}>750$ GeV, where $H_{T}$ is the scalar sum of the transverse momenta of
all hadronic objects in an event.

  * $\mathfrak{C}_{2}$:

Exactly one muon, with $p_{T}>120$ GeV with $|\eta|<2.5$. (We consider the
muon as it has the best detector sensitivity among the leptons.)

  * $\mathfrak{C}_{3}$:

At least two AK-4 jets. We also demand that the leading jet has $p_{T}>120$
GeV and the subleading one has $p_{T}>40$ GeV.

  * $\mathfrak{C}_{4}$:

Exactly two fatjets, clustered using the anti-$k_{T}$ algorithm with $R=0.7$
and $p_{T}>200$ GeV with mass $M_{J}\in[40,165]$.

  * $\mathfrak{C}_{5}$:

The invariant mass between the reconstructed muon and missing transverse
energy ($\not{E}_{T}$) must be greater than $140$ GeV, i.e.,
$M_{\mu\not{E}_{T}}>140$ GeV.

We pick the fatjet radius $R=0.7$ after optimising to tag a boosted two-
pronged jet. The constraints on fatjet mass ($\mathfrak{C}_{4}$) are chosen to
include reconstructed $W,\ Z$ and $H$ fatjets. Since the missing energy and
muon come from the decay of $W$-boson in all the backgrounds, we put a high
cut on the invariant mass of reconstructed lepton and missing transverse
energy pair in $\mathfrak{C}_{5}$.

((a)) (a) ((b)) (b)

Figure 3: The (a) discovery ($5\sigma$) and (b) exclusion ($\sim 2\sigma$)
contours for the monolepton (1-muon) channel of $pp\to Z^{\prime}\to
N_{R}N_{R}$ at HL-LHC.

((a))

((b))

((c))

((d))

((e))

Figure 4: HL-LHC reaches for the GS model and the $E_{6}$ embeddings in the
monolepton channel for $g_{z}=0.72$. The green regions (dotted contours) are
discoverable ($5\sigma$), and the red regions are excludable ($2\sigma$). The
$U(1)_{z}$ charges are found in Table 3 .

((a))

((b))

((c))

((d))

Figure 5: Regions where the monolepton signal achieves at least $2\sigma$
significance at the HL-LHC but are beyond the projected reach in the dijet
channel (see text).

((a)) (a) ((b)) (b)

Figure 6: The (a) discovery ($5\sigma$) and (b) exclusion ($\sim 2\sigma$)
contours for the dilepton (2-muon) channel of $pp\to Z^{\prime}\to N_{R}N_{R}$
at HL-LHC.

((a))

((b))

((c))

((d))

((e))

Figure 7: HL-LHC reaches for the GS model and the $E_{6}$ embeddings in the
dilepton channel for $g_{z}=0.72$. The green regions (dotted contours) are
discoverable ($5\sigma$), and the red regions are excludable ($2\sigma$). The
$U(1)_{z}$ charges are found in Table 3 .

###  III.2 Background processes

The major backgrounds considered for the analysis are listed in Table 4, along
with the cut efficiencies and the total number of events at the HL-LHC
luminosity, $3$ ab-1. We generate the background events with some generation-
level cuts to reduce the computation time. The _$W^{\pm}$ \+ jets_ process is
the leading background before the cuts; the demand of a high-$p_{T}$ jet
($\mathfrak{C}_{3}$) and two fatjets ($\mathfrak{C}_{4}$) reduces this
background significantly. The contributions from the _$W^{+}W^{-}$ \+ jets_
process drop drastically with the demand of two fatjets ($\mathfrak{C}_{4}$).
The fatjet criterion ($\mathfrak{C}_{4}$) also cuts the _$t\bar{t}$ \+ jets_
background as most of these events have just one fatjet. The hadronically-
decaying top is sometimes reconstructed as the fatjet in a fraction of
_$t\bar{t}$ \+ jets_ events; the demand on the fatjet mass cuts down these
events. After the event selection criteria are enforced, _$W^{+}W^{-}$ \+
jets_ and _$t\bar{t}$ \+ jets_ become the leading backgrounds.

###  III.3 Kinematic picture and the Deep Learning model

We use the kinematic information of the identified objects, i.e., one muon,
three AK-4 jets, and two fatjets, along with some derived quantities to create
the input set of variables for the multivariate analysis—the complete list of
variables is given in Table 5 where $J$ is used to denote a fatjet whereas $j$
is used to identify an AK-4 jet.

We show the distributions of the key features of the signal and the background
for three benchmark parameters in Fig. 2. The $H_{T}$ (scalar sum of all
hadronic objects) distributions show a clear separation between the signals
and background due to the boosted nature of the final state; see Fig. 2(a).
Moreover, for the signals, the transverse momentum distributions of the tagged
muon, AK-4 jets, and the fatjet and the $\not{E}_{T}$ distribution peak at
considerably higher values than those of the background, see Figs. 2(b), 2(c),
2(d), 2(e). We also consider the $n$-subjettiness variables Thaler:2010tr
which check for two-pronged ($\tau_{2}/\tau_{1}$) and three-pronged
($\tau_{3}/\tau_{2}$) nature of a fatjet for various values of $\beta$
($\beta\in\left\\{1,2,3\right\\}$). Since our fatjet parameters are tuned to
identify a $W$-like (two-pronged) jet, the $\tau_{2}/\tau_{1}$ ratios are the
key variables. Of these, the $\tau_{2}/\tau_{1}$ for $\beta=2$ gives a clear
separation between the signals and the background [Fig. 2(f)]. The
$\tau_{3}/\tau_{2}$ ratios do not offer clear separations, but, nevertheless,
we include them due to the inclusive nature of the analysis cuts on the number
of fatjets. The variable $m_{J_{i}\ell}$ (invariant mass of a fatjet and the
tagged muon) reconstructs the mass of the RHN very well—the trend can be
clearly seen in Fig. 2(g). The $\Delta R_{J_{i}\not{E}_{T}}$ distributions in
Fig. 2(h) can also distinguish the signals and the background well, as the
final state objects come from the decay of an on-shell $Z^{\prime}$.

To perform the classification task, we use a fully connected DNN model, the
details of which are described in Appendix A. We pick the benchmark point
$(M_{Z^{\prime}},M_{N_{R}})=(5.0,2.0)$ TeV to optimise the hyperparameters
using a grid search. The performance metric is the signal significance
($\mathcal{Z}$ score),

$\mathcal{Z}=\sqrt{2\left(N_{S}+N_{B}\right)\ln\left(\frac{N_{S}+N_{B}}{N_{B}}\right)-2N_{S}},$
(14)

where $N_{S}$ and $N_{B}$ are the numbers of signal and background events the
network allows at $3$ ab-1 luminosity.

##  IV Monolepton Channel Prospects

Fig. 3 shows the HL-LHC discovery ($5\sigma$) and exclusion ($\sim 2\sigma$)
reaches through the monolepton channel in a model-independent manner. Assuming
a fixed BR($Z^{\prime}\to N_{R}N_{R}$) $=50\%$, we mark the contours of the
$U(1)_{z}$ gauge coupling, $g_{z}$ [Eq. (II.1)], needed to achieve $5\sigma$
and $2\sigma$ significances on the $M_{Z^{\prime}}-M_{N_{R}}$ plane. To
calculate the number of events, $N_{S}$ and $N_{B}$, we have used the final
efficiencies of the DNN model at the working point with DNN response $=0.95$.
The $g_{z}$ contours are broadly insensitive to the mass of the RHN. However,
because of the high boosts, the event selection efficiency around
$M_{N_{R}}=0.5$ TeV (or less) is lower than those at larger values. Hence, one
needs a larger $g_{z}$ to achieve a high signal significance. This can be seen
from the slightly deformed contours between $M_{N_{R}}=1.0$ TeV and
$M_{N_{R}}=0.5$ TeV.

In particular models, however, the $Z^{\prime}$ BR will not be fixed. For
example, in Fig. 5, we show the $5\sigma$ and $2\sigma$ contours for the GS
model and the six $E_{6}$ embeddings for $g_{z}=0.72$ ($\approx g_{ew}$); the
corresponding $U(1)_{z}$ charges are found in Table 3. As the
$M_{N_{R}}/M_{Z^{\prime}}$ ratio approaches the threshold value, $1/2$,
BR($Z^{\prime}\to N_{R}N_{R}$) drops. This is why the contours in Fig. 5 turn
inwards with increasing $M_{N_{R}}$ near the kinematic threshold (top-left
edge).

In Fig. 5, we show the regions that will be beyond the reach of the dijet
channel at the HL-LHC (approximately) but can be reached ($\gtrsim 2\sigma$)
by the monolepton signal. To estimate the projected dijet reach, we have
simply scaled down the current ATLAS dijet exclusion limits by the square root
of the luminosity ratio ($\sqrt{3000/140}\approx 4.65$), ignoring the
difference between the centre-of-mass energies.

##  V Update: Dilepton Channel Prospects

We update our earlier cut-based estimates of the dilepton prospects
Arun:2022ecj  with the current DNN model. The dileptonic signal comes from the
following process,

$pp\to Z^{\prime}\to
N_{R}N_{R}\to\left(W^{+}\mu^{-}\right)\>\left(W^{-}\mu^{-}\right),$ (15)

where the $W$ bosons decay hadronically. See Ref. Arun:2022ecj  for a
discussion on the important background processes. The event-selection criteria
remain similar to those in Sec. III.1 with some minor changes: in
$\mathfrak{C}_{2}$, we now demand two muons with $p_{T}>120$ GeV and the new
fatjet window in $\mathfrak{C}_{4}$ is now $\left[40,120\right]$ to only tag
$W$-like fatjets. To suppress the huge Drell-Yan dilepton background, a Z-veto
cut is necessary; we also change $\mathfrak{C}_{5}$ to demand that the
invariant mass of the muon pair $(m_{\ell\ell})$ be greater than 140 GeV.

We show the model-independent HL-LHC discovery ($5\sigma$) and exclusion
($\sim 2\sigma$) reaches through the dilepton channel in Fig. 7 and the
$5\sigma$ and $2\sigma$ contours for the GS model and the six $E_{6}$
embeddings with $g_{z}=0.72$ in Fig. 7. Comparing Fig. 7 with Fig. 4 of Ref.
Arun:2022ecj , we see the DNN has enhanced the reaches (discovery and
exclusion) significantly.

##  VI Conclusions and Discussions

In this paper, we studied the prospects of a correlated search of a heavy
neutral gauge boson $Z^{\prime}$ and the right-handed neutrino $N_{R}$ in
leptophobic $U(1)$ extensions. In particular, we analysed an experimentally
unexplored channel $pp\to Z^{\prime}\to N_{R}N_{R}$ where the $N_{R}$ pair
decays to a monomuon final state. While in Ref. Arun:2022ecj  we only
considered the GS model and the standard $E_{6}$ embedding, here, we study all
possible embeddings of $E_{6}$ GUT models where a leptophobic $Z^{\prime}$ can
dominantly decay to TeV-scale RHNs in large parts of the parameter space.
Moreover, unlike the dilepton signature considered in our previous paper
Arun:2022ecj , probing the monolepton signature is highly challenging due to
the huge background from the SM processes, which are hard to reduce with a
cut-based analysis. Hence, in this paper, we used a DNN model to isolate the
signal from the SM background, providing orders of magnitude gains in
significance scores. With this, we found both mono and dilepton channels to be
comparable and highly promising; for an order-electroweak coupling $g_{z}$,
the leptophobic $Z^{\prime}$ with mass up to $6-7$ TeV can be discovered at
the $14$ TeV HL-LHC. Therefore, this follow-up paper improves upon our
previous study on several counts: it covers more model possibilities with a
more sophisticated analysis. It also covers the complimentary signatures—mono
and dilepton channels—to present a comprehensive view of the prospects of this
exciting but experimentally unexplored process.

Our findings point to a few directions for follow-up investigations. Studying
the phenomenology of the Yukawa, scalar, and neutrino sectors of the GS and
GUT models we consider will be interesting. It will also be interesting to
explore the parameter space where the RHNs can produce displaced vertices (as
we pointed out in Ref. Arun:2022ecj ) or form $N_{R}$-jets Mitra:2016kov . On
the other hand, our analysis is generic as it mainly relies on the kinematic
features of the mono and dilepton final states produced from RHN pairs. Hence,
the results—presented in Figs. 3 and 7 in a mostly model-independent
manner—can be mapped to other BSM scenarios not necessarily related to the
leptophobic $Z^{\prime}$. For example, we could consider
$(\overline{q}\,\Gamma\,q)(\overline{N}_{R}\,\Gamma\,N_{R})$-type operators
(where $\overline{\psi}\,\Gamma\,\psi$ is a Dirac bilinear) in the
$N_{R}$-effective field theory framework delAguila:2008ir ;
Bhattacharya:2015vja ; Liao:2016qyd ; Alcaide:2019pnf ; Mitra:2022nri  to
produce RHN pairs at the LHC. Similarly, a pair of RHNs can also come from a
leptoquark pair Bhaskar:2023xkm :
$pp\rightarrow\ell_{q}\ell_{q}\rightarrow(N_{R}j)(N_{R}j)$. In such cases,
where RHNs are resonantly produced in pairs, we can directly recast our
results for comparable kinematics. For example, we can consider the simplest
case of a charge-$1/3$ singlet scalar leptoquark. For $M_{\ell_{q}}=1.5$ TeV,
$M_{N_{R}}=0.5$ TeV (the kinematics is roughly comparable to the benchmark
point BP2, defined in Fig. 2) and BR($\ell_{q}\to N_{R}j$) $=100\%$, both the
monolepton and the dilepton channels give $\mathcal{Z}>5$ (i.e., they are
discoverable). The prospects are better than those reported in Ref.
Bhaskar:2023xkm , which used no machine learning techniques.

Finally, some studies on the leptophobic $Z^{\prime}$ in supersymmetric models
are available in the literature Langacker:1998tc ; Araz:2017wbp ;
Frank:2020kvp . Generally, in such models, the $Z^{\prime}$ can decay to a
pair of charginos, producing a dilepton plus missing energy signal at the LHC
Frank:2020kvp :

$p~{}p\to
Z^{\prime}\to\widetilde{\chi}^{\pm}\widetilde{\chi}^{\mp}\to\ell^{+}\ell^{-}+\not{E}_{T}.$

Since there is no fatjet in the final states, our analysis is not directly
applicable to this signal, but one can suitably modify the cuts to investigate
it further. However, if a leptophobic $Z^{\prime}$ decays to a pair of
neutralinos, those can decay to $W$ bosons and charginos, producing the
lepton(s) plus fatjets signature. In that case, the signal will resemble ours,
allowing recast, even though no RHN is involved in the process.

##  VII Acknowledgements

We thank T. Rizzo for insightful discussions on the $E_{6}$ embeddings. C. N.
acknowledges the Department of Science and Technology (DST)-Inspire for his
fellowship. We also thank Jai Bardhan for helping us speed up the input-
feature generation pipeline and for helpful discussions.

Figure 8: Schematic for the DNN model with four hidden layers. The number of
nodes in a layer is shown on the top.

##  Appendix A Neural-network model

A neural network is a series of perceptron blocks with a non-linear activation
function. A perceptron is a linear transformation of the input vector of the
form:

$f(\textbf{x})=\sigma(\textbf{W}\cdot\textbf{x}+\textbf{b}),$

where $\sigma$ is an activation function. A DNN can effectively approximate
any real continuous function provided the dimensions of hidden layers are
sufficiently large. Here, we use a feed-forward DNN of four hidden layers with
$50$, $35$, $20$, and $10$ nodes respectively (see Fig. 8), with a ReLU
activation function and batch normalisation 2015arXiv150203167I . We implement
it using the TensorFlow module tensorflow2015-whitepaper  with the AdamW
optimiser 2017arXiv171105101L  with a learning rate of $0.001$. The layers use
The final layer contains a single node with a sigmoid activation function,
which outputs a value between $0$ and $1$. We set a cut on this output value
while performing the classification task and estimate the signal and
background efficiencies from the fraction of the events passing the threshold
value. We incorporate dropouts JMLR:v15:srivastava14a  with a probability of
$0.1$ for training regularisation. For each mass point, we train the network
for $200$ epochs with a batch size of $4096$. We use a basic binary cross-
entropy loss while training. For tuning the network, a grid scan is performed
for the benchmark point $(M_{Z^{\prime}},M_{N_{R}})$ = $(5.0,2.0)$ TeV on the
number of hidden layers, the number of nodes in each hidden layer and the
learning rate.

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11institutetext: CERN

# Key4hep, a framework for future HEP experiments and its use in FCC

Gerardo Ganis 11 Clément Helsens 11 Valentin Völkl 11

((Submitted to EPJ+ special issue: A future Higgs and Electroweak factory
(FCC): Challenges towards discovery, Focus on FCC-ee))

###### Abstract

The road map to the FCC Feasibility Study Report, for submission to the next
Update of the European Strategy for Particle Physics, will require detailed
simulation and advanced reconstruction algorithms to explore and maximise the
physics reach of proposed detector solutions. The optimisation process will
require maximal flexibility in changing detector geometries, materials and
sensitive areas, and efficient tools to quantify the overall performance. To
synergise such developments the CEPC, CLIC, FCC, ILC and SCT communities have
engaged in the commissioning of a ‘Turnkey Software Stack’ (Key4hep), which
would provide all the necessary ingredients, from simulation to analysis, for
future experiments. This approach is based on the positive experience of the
linear collider projects ILC and CLIC, which have developed and used a common
software stack (iLCSoft) over the last decade. Key4hep aims to cover most, if
not all, future linear and circular machines colliding leptons (electrons,
muons), and hadrons. The common software ecosystem will facilitate writing
specific components for experiments ensuring coherency and maximising the re-
use of established solutions. Project-specific software frameworks will
require adaptation to fully profit from the common software base. In this
essay we present the status and plans for re-framing the FCC software
framework, FCCSW, around Key4hep and discuss the challenges associated with
the transition.

††offprints:

##  1 Introduction

The road map to deepen the understanding of the physics potential of the
Future Circular Collider (FCC), due for the next Update of the European
Strategy for Particle Physics (UESPP), will require detailed simulation and
advanced reconstruction algorithms to explore and maximise the physics reach
of proposed detector solutions. The results submitted to the 2019 UESPP were
obtained using FCCSW, an experiment software based on the LHC experience
integrated with the results of developing common software projects, such as
DD4hep, which served the initial needs of the FCC community fccswchep  well.
The next phase of the studies will require further developments of the
software, in particular concerning the flexibility in sub-detector choices,
including the simulation and reconstruction of their response. The interplay
between reconstruction algorithms and detector geometry, for example in
particle flow clustering, means that the detector hardware cannot be developed
and designed independently from the software. At the same time, developing and
validating sophisticated algorithms, including accounting for a large number
of edge cases, requires a significant amount of resources. Thus the
communities for future experiments – CEPC cepc , CLIC Charles:2018vfv , FCC
Abada:2019lih , ILC Bambade:2019fyw , SCT sct_summary , STC Luo:2019xqt  –
came to the agreement that the development of a common software solution would
benefit everyone kickBO ; kickHK . Software development is a collaborative
effort, and one way Key4hep facilitates writing specific components for
experiments is by maximising the re-use of established solutions and packages
to benefit from existing community developments, for example, ROOT ROOT ,
Geant4 Allison:2006ve , DD4hep Gaede:2020tui , Gaudi Clemencic:2017qgi  and
PODIO Gaede:2020frr . By using and contributing to other experiment-
independent software packages, an ‘ecosystem’ of HEP software is fostered.

In this essay we describe the proposal of the turnkey software stack, we
outline the requirements and planned ingredients, and showcase the evolution
of FCCSW towards this common software solution. We focus on the functionality
provided by the ecosystem because we believe that matching the needs of the
FCC project is the real challenge ahead. We expect other aspects, such as the
one of computing performance, which will certainly become very important for
the FCC operations, to be addressed, when relevant, at the level of the
components that Key4hep brings together.

##  2 Key4hep: the Turnkey Software Stack for Future Colliders

The turnkey software stack, nicknamed Key4hep, aims to encompass all the
libraries needed for generation, simulation, reconstruction, and analysis of
events at the participating future colliders. The project effectively started
in February 2020 after two kick-off events which took place in Bologna kickBO
and Hong-Kong kickHK . A more detailed description of the vision and goals of
the project can be found in Ref. keyhepchep . In this section we will recall
the main ingredients.

###  2.1 Ingredients

A software ecosystem for HEP data processing comprises several components
which need to be chosen carefully. The most important are the following:

  1. 1.

A data processing framework, which provides the structure for all dependent
components. Despite the fact that the linear collider community has used
Marlin Gaede:2006pj  for many years very successfully, the choice has fallen
on Gaudi because it is adopted by LHC experiments and it has a large user and
developer community, offering - or planning to offer - support for access to
heterogeneous resources, different architectures, and task-oriented
concurrency.111A selected number of features provided by Marlin and currently
not available in Gaudi, e.g. capability to automatically generate steering
file templates, will be contributed back to Gaudi as part of the Key4hep
project.

  2. 2.

A geometry description tool; for this purpose all of the communities concerned
already use DD4hep, which offers a complete detector description for
simulation, reconstruction and analysis.222DD4hep is not only of interest for
future experiments community but also for LHC’s: indeed it will be used in
production, as of run 3, by CMS and LHCb. This will entail a consolidation and
development process highly beneficial for the future experiments, in
particular with respect to aspects more relevant for running experiments, e.g.
alignment and calibration. In addition, support for common digitisation
modules is planned as part of the research and development program AIDAInnova
AIDAInnova  with the aim of facilitating the translation of the energy
deposits produced by the full simulation into something which can be used to
reconstruct physics objects.

  3. 3.

A common event data model (see Section 2.2).

  4. 4.

A common build infrastructure. For the ease of use by librarians and
developers, one needs to be able to build any and all pieces of the stack with
minimum effort. Most packages have a similar build system based on CMake and
follow common good practices which already reduces the maintenance burden, but
the large number of packages requires additional tooling to automate the
installation and ensure a consistently linked set of packages with correct
dependency resolution. The investigations of the HEP Software Foundation
packaging working group have identified the scientific package manager Spack
Spack  as a suitable solution, and it has since been successfully used for
prototype builds and established as the main build tool for Key4hep software
spackchep . In addition, not only does Spack allow sharing of the build
results but also the build recipes with a wider community.

###  2.2 EDM4hep: the Event Data Model

The event data model defines the structures needed to describe the required
event information in the persistent store. While, in principle, this can be
different from what is used in the transient store seen by the data processing
algorithms, and also from algorithm to algorithm, adopting the same event
model everywhere reduces the need for conversions, enhances generality and
improves inter-operability.

Since the beginning of the FCC physics potential studies, the choice has been
for an event data model managed by PODIO Gaede:2020frr , a toolkit that
generates the data model implementation from templates. This allows for a
separation of the high level description in YAML files using plain-old-data
simple types, and the low level persistency layer that can be optimised
according to the back-end. Once the required classes are defined in the YAML
file, the PODIO tool creates the source code automatically. For Key4hep the
same technology has been chosen, with an event data model, referred to as
EDM4hep, initially based on the LCIO and FCC-edm classes, and improved from
there as needed.

An underlying assumption or challenge is that the same event data model can be
used for all types of HEP experiments, in particular for hadron and lepton
colliders, a case which is particularly applicable to the integrated FCC
programme, which has a lepton collision phase followed by a hadron collision
phase, similar to what happened with LEP and LHC; this seems to be the case
with EDM4hep and it will be further discussed in Section 3.

###  2.3 Development strategy

During the kick-off meetings kickBO ; kickHK  it became clear that to maximise
the chances of an early adoption a deliver-early-deliver-often approach needed
to be adopted. Furthermore, early identification of key components, to be made
quickly available in usable form, was required in order to plan and make
progress with the evaluation process.

The activities started from the definition of the common event data model
EDM4hep, which defines a sort of common language underlying the inter-
operation of the various components. Once a usable version of
EDM4hep333Currently version 0.3.2, https://github.com/key4hep/EDM4hep. was
available, the core component of the framework, k4FWCore444Currently version
1.0pre06, https://github.com/key4hep/k4FWCore., which provides the interfaces
and the data service, needed to be made available; k4FWCore is largely based
on the corresponding components of FCCSW. The operative availability of
EDM4hep and k4FWCore enabled the development and integration of data
processing components, in the form of Gaudi algorithms and tasks.

It was soon realised that having a full workflow producing ‘usable’ events in
EDM4hep format would allow acceleration of the process. For this purpose the
choice has been to integrate Delphes delphes , a popular tool for super-fast
parameterised detector response. The component k4SimDelphes, derived from a
similar one existing in FCCSW, provides a Key4hep interface to Delphes both in
terms of standalone applications, running on top of any of the input formats
supported by Delphes and a Gaudi algorithm.

In parallel, to facilitate the transition and interplay with iLCSoft, a Gaudi-
based wrapper around Marlin processes, which are the equivalent of the Gaudi
algorithms, has been developed, enabling the complete iLCSoft reconstruction
via Gaudi555Currently version 0.3.2, https://github.com/key4hep/k4LCIOReader.
to be run. The picture here is completed by the provision of an on-the-fly
converter LCIO–EDM4hep–LCIO666Currently version 0.3.1,
https://github.com/key4hep/k4MarlinWrapper. enabling the possibility of
running an algorithm developed for Marlin and LCIO, for example the flavor
tagging code LCFIplus, inside a full Gaudi application. By design, EDM4hep
allows a one-to-one conversion to LCIO and back without loss of information.
In addition, the PODIO-generated interfaces allow simple ad-hoc user data
extensions of the data model should that be required.

The basic components described above enable the development of specific new
elements or the adoption of existing ones – a process which is now ongoing for
generation, full simulation and reconstruction capabilities.

##  3 FCCSW and Key4hep

The FCC software framework, FCCSW, has a lot of commonalities with Key4hep,
stemming from the fact that they are both based on Gaudi. The main differences
are the event data model and the structure governing the components, which in
Key4hep is flat while FCCSW features a hierarchical structure with sub-
components organised in categories.

For FCCSW moving to Key4hep conceptually means using common components
available from Key4hep as much as possible, but FCC-specific components will
be kept under FCC responsibility.

From the technical point of view, the following chronologically-ordered steps
were identified:

  1. 1.

Replacement of FCC-edm with EDM4hep;

  2. 2.

Identification and migration of non-FCC-specific FCCSW components to Key4hep;

  3. 3.

Completion of the ecosystem, either in Key4hep or in the components remaining
in FCCSW

    1. (a)

Addition of missing sub-components, sub-detectors; optimisation/replacement of
full/fast simulation techniques, reconstruction algorithms, etc.

###  3.1 Current status

####  3.1.1 Migration to EDM4hep

The replacement of FCC-edm with EDM4hep has been completed without any major
problem for all components, i.e. including those specific to FCC-hh. This
means that all the data structures needed to describe hadronic collisions
could be handled with the structures provided by EDM4hep.

####  3.1.2 General and FCC-specific FCCSW components

As mentioned above, Key4hep and FCCSW are conceptually very similar. The FCCSW
components are therefore natural candidates to be generalised and become parts
of upstream Key4hep. Table 1 summarises the current status of the main
components.

Table 1: Original FCCSW components and their fate in the new structures, as well as additional information on the current progress in transitioning them to use Key4hep interfaces (”status”) and whether they will be ultimately hosted by Key4hep (”migration”) . See text for more details about dual–readout. old FCCSW (version $\leq$ 0.16) | Key4hep | new FCCSW | status | migration
---|---|---|---|---
FWCore | k4FWCore |  | done | yes
Sim/SimDelphesInterface | k4SimDelphes |  | done | yes
Generation |  | k4Gen | done | under evaluation
Sim |  | k4SimGeant4 | done | under evaluation
Reconstruction/Rec[…]Calorimeter |  | k4RecCalorimeter | done | under evaluation
Reconstruction/RecDriftChamber |  | to be determined |  |
Detector |  | FCCDetectors | done | no, FCC specific
| to be determined | dual–readout |  | under evaluation

As can be seen from the table, in preparing the migration to Key4hep, a
general restructuring of the FCCSW repository has taken place, resulting in a
flat structure similar to that of Key4hep, with sub-repositories promoted to
separate repositories. Some of the components, FWCore and SimDelphesInterface,
have been polished, renamed and migrated, and are already extensively used.
For some others the preparation work has been completed and they are being
analysed for migration. The detector descriptions have moved to a separate
repository, FCCDetectors, which will not be migrated, as it is FCC-specific.
The dual–readout module includes Geant4 simulation and reconstruction of the
IDEA dual readout calorimeter; it has always been hosted in the FCC project as
a separate component, i.e. never part of FCCSW. Finally, the RecDriftChamber
module implemented an approximate reconstruction of the IDEA drift chamber and
will not be migrated, since a detailed simulation and reconstruction for such
a detector is being developed in Key4hep by the IDEA working groups ideatrack
.

####  3.1.3 Completion of the ecosystem

The completion of the ecosystem consists firstly of identifying the missing
pieces in the findings of the other FCC working groups, in particular the
Physics Performance and Detector Concepts groups; secondly in creating the
appropriate conditions for the provision of the required software module.

Although this step will only take place once, the existing code is properly
re-framed within Key4hep or FCCSW following the new paradigm. Some
developments have already been enabled and can serve as an example of these
activities. The first example is the ongoing process of integration of the
simulation and reconstruction code for the IDEA drift chamber and dual-readout
calorimeter. For both of these important and innovative detectors the IDEA
collaboration developed standalone programs which are now being reviewed for
inclusion in Key4hep. The process is well advanced and required some
modification of EDM4hep. The second example relates to track-related
reconstruction algorithms, such as vertex finders, jet clustering and alike.
The Delphes tool has recently been improved with a detailed simulation of the
reconstructed parameters of electrically charged particles, with an output
similar to that of full simulation and reconstruction. This allows algorithms
to be developed which can serve as an advanced starting point when the tool is
available.

The completion of the ecosystem also includes a set of interfaces with
external packages which are typically being developed as part of general
purpose research and development programs. Examples of these packages, for
which the investigation work has already started, are the advanced tracking
reconstruction system ACTS Gessinger:2020nne , the particle flow package
PandoraPFA Marshall:2015rfa  and the clustering algorithm CLUE cluepaper . One
of the main challenges affecting the integration process is ensuring that the
event data model contains all of the information required.

###  3.2 The plug-and-play or LEGO approach to detector composition and
simulation

The optimisation of detector design to support the full physics potential of
FCC-ee adds the challenge of finding the best combination of detection
technologies. The fact that at least two – but possibly four – interaction
regions are planned, opens the way to more solutions, potentially fine-tuned
on specific physics aspects.

An infrastructure supporting the possibility to easily interchange sub-
detector components is required for a broad investigation of all solutions.
The DD4hep and Gaudi frameworks chosen for detector description and workflow
execution support this through a plugin system which allows switching of both
subdetectors and workflow components at runtime. The process has already been
applied when adapting the liquid Argon calorimeter solution developed for FCC-
hh to the FCC-ee case. Modular simulation workflows that allow the use of
different fast and full simulation components in the same job, are being
implemented. The extension and subsequent validation of this approach, which
for obvious reasons has been nicknamed LEGO, is the goal and challenge of this
exercise.

##  4 Conclusions and Outlook

The turnkey software stack, Key4hep, aims to create a complete data processing
ecosystem for the benefit of HEP experiments, with immediate application to
future collider designs and physics potential studies. The ecosystem will be
built on established solutions such as ROOT, Geant4, DD4hep and Gaudi, and
features a new event data model EDM4hep based on PODIO. Key4hep already
provides enough components and modules to enable the migration of existing
frameworks based on Gaudi. In this essay we described the status of the
migration of FCCSW which, at the time of writing, is close to completion. The
Key4hep projects seem to be on the right path to find the equilibrium to serve
the needs of the several participating projects well, with the appropriate
balance between common infrastructure and project-specific developments, and a
clear path to promote the latter to the common repositories.

The challenge ahead is to maintain the current momentum and to create the
conditions to attract and support people who will develop and implement their
tools in the Key4hep context, thereby creating a critical mass of the best
tools to cover most of the needs of HEP experiments. This task includes the
provision of sufficiently extensive documentation, training and adequate
advertising. Ultimately this will depend on the capability to guarantee an
adequate workforce and funding.

## Acknowledgements

This work beneﬁted from support by the CERN Strategic R&D Programme on
Technologies for Future Experiments (https://cds.cern.ch/record/2649646/,
CERN-OPEN-2018-006).

## References

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