††institutetext: aDepartment of Physics, University of Toronto, Toronto,
Ontario, Canada M5S 1A7††institutetext: bInstitut für Experimentalphysik,
Universität Hamburg, Germany††institutetext: cInstitut für Theoretische
Physik, Universität Heidelberg, Germany

#  Unsupervised Hadronic SUEP at the LHC

Jared Barron,a<EMAIL_ADDRESS>David Curtin,a
<EMAIL_ADDRESS>Gregor Kasieczka,b<EMAIL_ADDRESS>Tilman Plehn,c<EMAIL_ADDRESS>and Aris Spourdalakisa
<EMAIL_ADDRESS>
###### Abstract

Confining dark sectors with pseudo-conformal dynamics produce SUEPs, or Soft
Unclustered Energy Patterns, at colliders: isotropic dark hadrons with soft
and democratic energies. We target the experimental nightmare scenario, SUEPs
in exotic Higgs decays, where all dark hadrons decay promptly to SM hadrons.
First, we identify three promising observables: the charged particle
multiplicity, the event ring isotropy, and the matrix of geometric distances
between charged tracks. Their patterns can be exploited through a cut-and-
count search, supervised machine learning, or an unsupervised autoencoder. We
find that the HL-LHC will probe exotic Higgs branching ratios at the per-cent
level, even without a detailed knowledge of the signal features. Our
techniques can be applied to other SUEP searches, especially the unsupervised
strategy, which is independent of overly specific model assumptions and the
corresponding precision simulations.

##  1 Introduction

Hidden sectors are one of the most interesting and generic paradigms for
physics beyond the Standard Model (BSM). These kinds of new particles and
forces are not only plausible from a bottom-up point of view, but also arise
in many top-down BSM theories Strassler:2006im ; Strassler:2008fv ;
Chacko:2005pe ; Schabinger:2005ei ; Patt:2006fw ; Espinosa:2007qk ; March-
Russell:2008lng ; Alimena:2019zri ; Curtin:2021alk ; Holdom:1985ag ;
Abel:2008ai ; Batell:2009yf ; Jaeckel:2010ni ; Foot:2014mia ; Feldman:2007wj ;
Pospelov:2007mp ; Dudas:2013sia ; An:2012va ; Kribs:2018ilo ; Knapen:2021eip ;
Cvetic:2002qa ; Hur:2007uz ; Bai:2013xga ; Grossman:2010iq . In most scenarios
of interest, hidden sector particles couple to SM fields via feeble
interactions or heavy messengers, and the nature of these portal couplings
determines their collider phenomenology. A case of special interest are hidden
valleys Strassler:2006im , referring to hidden sectors with a confining gauge
group which gives rise to rich infrared (IR) dynamics from very simple
ultraviolet (UV) theory structures. The production of dark quarks at the LHC
leads to a dark shower and high-multiplicity production of dark hadrons, in
analogy to QCD jets. Depending on the portal by which the dark hadrons are
produced and decay, these dark showers produce a wide variety of LHC
signatures, which have been the subject of intense theoretical and
experimental study in the last decade Han:2007ae ; Strassler:2008fv ;
Buschmann:2015awa ; Arkani-Hamed:2008kxc ; Ellis:2012zp ; Toro:2012sv ;
ATLAS:2018dfo ; Alimena:2019zri ; Cohen:2015toa ; Burdman:2018ehe ;
Schwaller:2015gea ; Cohen:2017pzm ; Cohen:2020afv .

We consider one of the most challenging varieties of dark showers, Soft
Unclustered Energy Patterns (SUEPs). If a hidden valley possesses a large
gauge coupling that is pseudo-conformal above its confinement scale, then
large-angle emission is unsuppressed for most of the parton shower evolution.
This means the dark hadrons are not arranged in narrow QCD-like jets, but
emitted approximately isotropically in the shower centre-of-mass frame
Strassler:2008bv ; Knapen:2016hky . This defines the SUEP final state as a
high-multiplicity spherically-symmetric shower of hidden sector states. While
existing searches can be sensitive to SUEPs produced at high energy scales, or
with dark hadrons that decay to sufficiently conspicuous final states like
leptons or Long-Lived Particles (LLPs) Knapen:2016hky ; Alimena:2019zri , no
dedicated SUEP searches exist to date. Furthermore, SUEPs _without_
conspicuous final states are not captured by any existing search and represent
an unusually cruel signal, since their soft, isotropic distributions can mimic
the ubiquitous pile-up produced by simultaneous LHC collisions.

To ensure that all types of SUEP signals can be discovered at the LHC, we
focus on a well-motivated SUEP nightmare scenario, where SUEP is produced in
exotic Higgs decays and the dark hadrons decay promptly and exclusively to SM
hadrons. The modest energy scale of exotic Higgs decays and the lack of
conspicuous final states forces us to rely on the kinematics of the resulting
SM hadrons to extract the signal from the overwhelming QCD background. This
production mode also allows us to side-step the problem of how to trigger on
SUEPs Knapen:2016hky  by using leptons in associated $Vh$ production. The
analysis techniques we develop will not only allow for the detection of this
SUEP nightmare scenario, but should also increase the LHC experiments’
sensitivity to all other SUEP possibilities.

An acute obstacle for SUEP searches is the lack of rigorous predictions and
simulations for the strongly coupled pseudo-conformal dark sectors. Rather
than hoping for conspicuous final states, we utilize the kinematics of the SM
hadrons without relying on fine details of the signal beyond the robust SUEP
characteristics of isotropic, soft, and democratic dark hadron energies. We
therefore simulate SUEP production using a simple QCD fireball model of
thermal dark hadron emission Knapen:2021eip , and find robust observables and
event representations. An important observable for SUEP searches is the inter-
particle distance matrix $\Delta R_{ij}$ between charged hadrons. This matrix
encodes the essential geometric differences between QCD-like and SUEP-like
hadron production. It forms the backbone of all our analysis strategies,
together with known variables like event isotropy Cesarotti:2020hwb  and the
charged particle multiplicity.

To demonstrate the distinguishing power of our observables, as well as the
drastic improvements from more sophisticated techniques, we examine three
strategies for systematic SUEP searches at the HL-LHC. First, we simulate a
simple cut-and-count analysis, which will turn out to allow for impressive
sensitivities to $\mathrm{Br}(h\to\mathrm{SUEP})\sim 1\%$ at the HL-LHC. We
anticipate that a realistic analysis with data-driven background estimation
will perform even better, since our study is limited by background simulation
statistics. To improve on this, we utilize supervised as well as unsupervised
machine learning (ML). Unsupervised analysis concepts along the lines of
autoencoder neural networks Rumelhart1986  have the potential to transform LHC
analyses Asadi:2017qon ; Metodiev:2017vrx ; Andreassen:2018apy ;
Collins:2018epr ; DeSimone:2018efk ; Heimel:2018mkt ; Farina:2018fyg ;
Roy:2019jae ; Cheng:2020dal ; Nachman:2020lpy ; MdAli:2020yzb ;
Bortolato:2021zic ; Dillon:2021nxw ; Finke:2021sdf ; Kasieczka:2021xcg ;
Aarrestad:2021oeb ; Cerri:2018anq , including searches for dark showers
Heimel:2018mkt . We point out how unsupervised methods are especially
appealing for difficult-to-simulate signals like SUEP since they only rely on
the known QCD background for training, while yielding at least several times
greater SUEP sensitivity than the cut-and-count analysis.

Our investigation establishes that SUEP searches need not rely on conspicuous
SM final states for excellent sensitivity at the HL-LHC. The unique dark
hadron kinematics, which robustly follows from their origin in pseudo-
conformal strongly coupled dynamics, allows for the SUEP final state to be
distinguished from its overwhelming QCD background. Our techniques can be
applied to all SUEP searches to dramatically enhance their sensitivity,
regardless of energy scale or SM final state.

This paper is structured as follows. In Section 2, we briefly review the SUEP
theory and define the benchmark scenario for our study. Signal and background
simulation is discussed in in Section 3. In Section 4 we define the relevant
observables to distinguish SUEP from QCD and discuss supervised and
unsupervised machine learning techniques. In Section 5 we present our results,
including projections for the $\mathrm{Br}(h\to\mathrm{SUEP})$ sensitivity at
the HL-LHC, and we conclude in Section 6.

##  2 Theory of SUEPs

Hidden valley models are a large class of BSM theories in which the SM is
extended by additional gauge groups under which SM particles are neutral. New
particles that are charged under non-Abelian extensions can give rise to a
wide range of interesting hidden sector dynamics Strassler:2006im ;
Strassler:2008fv ; Han:2007ae  and various challenging SM signatures. These
models appear in the context of many top-down constructions Morrissey:2009tf ,
including string theory, of course Cvetic:2002qa , and are compatible with
various potential resolutions to the hierachy problem such as supersymmetry
Arkani-Hamed:2005zuc , little Higgs models, TeV extra dimensions and Randall-
Sundrum scenarios Strassler:2006im ; Arkani-Hamed:2001nha ; Randall:1999ee ;
Randall:1999vf . In Neutral Naturalness scenarios, hidden valleys actually
solve the little hierarchy problem directly Chacko:2005pe . Models containing
a hidden valley have also been studied in the context of dark matter
Hur:2007uz , matter-antimatter asymmetry Bai:2013xga  and the origin of
neutrino masses Grossman:2010iq .

In most hidden sector scenarios with a confining gauge force, the dark parton
shower is qualitatively similar to QCD: the asymptotic freedom of the running
gauge coupling enhances soft and collinear emission, resulting in the
production of hidden sector states in collimated jets. If the hidden ’t Hooft
coupling is large ($\lambda\equiv g^{2}N_{c}\gg 1$) and approximately constant
over a significant energy range, the distribution of the produced dark hadron
momenta will be much more democratic than the hierarchical jet-like behaviour
we see in QCD. This is the SUEP class of signals, characterized by relatively
soft, isotropic emission of dark hadrons Strassler:2008bv ; Knapen:2016hky .
It is worth keeping in mind that strongly coupled hidden sector dynamics is
not the only scenario that can lead to SUEPs. Similar final states can be
produced in many-step cascade decays in the hidden sector Elor:2015tva , or in
theories with extra spatial dimensions (see e.g. Cesarotti:2020uod ;
Costantino:2020msc ; Park:2012fe ). We now describe the general features of
production, evolution and decay that constitute the SUEP signal, and define a
benchmark scenario for our study.

###  2.1 Dark Hadron Production in Exotic Higgs Decays

Production of SUEP can occur through various portals coupling SM particles to
SM-singlet states charged under a dark gauge group. Higgs and vector boson
portals are commonly studied examples. Alternatively, a new particle charged
under both SM and the dark gauge group could be produced Knapen:2021eip . We
assume that hidden sector states $\psi_{D}$ can be produced via the Higgs
portal:

$\mathcal{O}_{\text{production}}\sim|H|^{2}\psi_{D}\bar{\psi}_{D}\;.$ (1)

$\psi_{D}$ could represent a fermion or scalar dark quark charged under the
hidden gauge group. In the fermion case, the above operator is actually
dimension 5 with a coupling ${\sim}1/Lambda$ for some UV scale $\Lambda$. This
operator will give rise to the production of hidden states in exotic Higgs
decays, as long as the dark hadrons are kinematically accessible. The dark
quarks hadronize into a large multiplicity of dark hadrons.

For a confining gauge theory, the evolution of the system from the hard scale
$Q$ at which the first partons charged under the gauge group are generated to
the IR confinement scale $\Lambda$ governs the hadron multiplicity generated
during showering. $Q$ is of the order $m_{h}$ in our case. For QCD jets, this
evolution can be reliably simulated as a parton shower, but for SUEPs
different strategies are required.

The average hadron multiplicity is given by

$\langle n(Q)\rangle=\int_{0}^{1}F(x,Q)dx$ (2)

where $F$ is the fragmentation function describing the final state
distribution of momenta Ellis:1996mzs , and $x$ is the momentum fraction of a
given splitting. Calculating its evolution from the scale $Q$ down to an IR
scale $\Lambda$ involves resumming the divergent contributions from the
anomalous dimensions, with the leading contribution obtained from just the
first ($j=1$) Mellin transform of the anomalous dimension. If the theory is
conformal between $Q$ and $\Lambda$, the running of the coupling can (by
definition) be neglected. In the limit where the ’t Hooft coupling is large
$\lambda\gg(1-x)$ it was first shown in Ref. Hatta:2008tn  that

$\langle n\rangle\sim\left(\frac{Q}{\Lambda}\right)^{2\gamma_{T}(1)}\;,$ (3)

where $\gamma_{T}(1)$ is the first Mellin-Moment of the time-like anomalous
dimension of the fragmentation function. Further expanding to zeroth order in
the coupling yields

$\langle
n\rangle\sim\left(\frac{Q}{\Lambda}\right)^{1+\mathcal{O}(1/\sqrt{\lambda})}$
(4)

Note that the small momentum fraction $x$ carried by each individual splitting
follows from $\lambda\gg(1-x)$. In this strong regime branching is expected to
yield emissions with $x\sim\Lambda/Q$ that are relatively isotropic in
direction and democratic in momentum Hatta:2008tn ; Knapen:2016hky . Thus,
with a large enough scale separation $Q\gg\Lambda$, low-$x$, high-multiplicity
final states are generated. Branching terminates after
$N_{\text{final}}\sim\mathrm{log}{\langle n\rangle}$ splittings at
$Q_{N_{\text{final}}}\sim Q/2^{N_{\text{final}}}\sim\Lambda$, at which point
hadronization takes over.

Hadron production in QCD at high temperatures is a close analogue to our
situation, and statistical models have consistently shown that hadron
multiplicities follow a thermal distribution Fermi:1950jd ; Hagedorn:1965st ;
PhysRevD.1.1416 ; Blanchard:2004du ; Hatta:2008tn ; Becattini:2001fg ;
Cleymans:2012wm ; Becattini:2008tx ; Becattini:2010sk ; Becattini:2009ee ;
Ferroni:2011fh . We use this picture as a toy model of dark hadron production
in SUEP, modelling the distribution of dark meson momenta as a relativistic
Boltzmann distribution

$\frac{dN}{d^{3}\bf{p}}\sim\exp\left(-\frac{\sqrt{{\bf{p}}^{2}+m_{D}^{2}}}{T_{D}}\right)\
,$ (5)

where $m_{D}$ is the mass of the final dark states and $T_{D}$ acts as the
Hagedorn temperature of the hidden confining gauge force, with
$T_{D}\sim\Lambda$ Knapen:2016hky ; Blanchard:2004du . This temperature
controls the kinetic energy of the dark hadron distribution.

###  2.2 Dark Hadron Decay

Generically the decay of the dark hadrons $\phi_{D}$ into the SM will occur
through some effective coupling of the form

$\mathcal{O}_{\text{decay}}=\phi_{D}\mathcal{O}_{\text{SM}}$ (6)

where $\mathcal{O}_{\text{SM}}$ contains fields charged under the SM gauge
group. The phenomenology of the SUEP signal is determined to a large extent by
the dominant decay portal. For example, if the dark hadrons decay to massive
dark photons that mix with the SM photon, then the SM final state contains
both hadrons and leptons in roughly gauge-ordered proportions, though for
various dark photon masses in the GeV-regime, hadrons can dominate
Knapen:2016hky . Alternatively, if the dark hadrons decay through the Higgs
portal, the final state will contain hadrons and leptons in roughly Yukawa-
ordered proportions. For sufficiently small portal coupling, the decay length
of the dark hadrons may also be macroscopic, resulting in LLP signatures
Alimena:2019zri .

It is also possible for the hidden sector states to decay purely hadronically,
which is the most experimentally challenging case at the LHC. A very simple
example is the gluon portal as described in Knapen:2021eip :

$\mathcal{L}\supset-\frac{1}{2}m_{a}^{2}a^{2}-\frac{\alpha_{2}}{8\pi}\frac{1}{f_{a}}aG_{\mu\nu}\tilde{G}^{\mu\nu}-iy_{\psi_{D}}a\psi_{D}\psi_{D}^{*}\
.$ (7)

Here, $a$ is a heavy elementary pseudo-scalar in the dark sector and
$\psi_{D}$ is the dark quark. Dark hadrons, which are bound states of
$\psi_{D}$, could then decay to SM hadrons via an effective operator
$\phi_{D}G\tilde{G}$. Another example is the hadrophilic (or leptophobic)
$Z^{\prime}$ portal Bernreuther:2019pfb , where a new heavy gauge boson
couples to SM quarks but not leptons, allowing dark hadrons to decay via an
effective operator like $\phi_{D}q\bar{q}$.

###  2.3 Prompt Hadronic Benchmark Scenario

The production of hidden sector states via the Higgs portal generally and
exotic Higgs decays in particular is one of the most motivated and plausible
discovery scenarios for new physics Curtin:2013fra . It is therefore vital
that our experimental search strategies cover all possibilities for a signal
at the LHC. This is especially urgent since for final states that are not
covered by existing searches, branching fractions of ${\sim}10\%$ are easily
allowed by current measurements of Higgs couplings and invisible decays
ATLAS:2018bnv ; ATLAS:2019cid ; ATLAS:2019nkf ; CMS:2018yfx ;
Biekoetter:2018ypq .

We therefore focus on the experimental worst-case scenario for SUEP produced
in exotic Higgs decays: purely hadronic and prompt decays, with a particular
interest in low dark hadron masses that make resonance searches Pierce:2017taw
; ATLAS:2020ahi  or applications of jet substructure techniques Park:2017rfb
challenging. While the simplest gluon portal scenarios suggest that dark
hadrons lighter than ${\sim}10\;\mathrm{GeV}$ have macroscopic decay lengths
Knapen:2021eip  (which could allow for the use of long-lived particle search
techniques Schwaller:2015gea ; Alimena:2019zri ), other possibilities can
easily realize prompt, purely hadronic decays over a much wider range of dark
hadron masses. For example, the hadrophilic (leptophobic) vector portal
Bernreuther:2019pfb  with a hypothetical confining sector where the lightest
dark hadron is a dark-rho-like vector $\rho_{D}$ would allow dark hadrons
lighter than a GeV to decay promptly into SM hadrons. In focusing on the
prompt case we can develop techniques that allow SUEP production to be
identified using only the geometrical and momentum distribution of its SM
final states. These techniques will enhance our sensitivity for any kind of
general SUEP, in addition to whatever other features of the final state, like
displaced decays, leptons, or photons, can also be exploited.

Figure 1: Cartoon of the our benchmark scenario for SUEP produced in Higgs
decays with prompt decay of dark hadrons into SM hadrons. The SUEP description
applies in the purple area, for $T$ within a factor of a few of the dark
hadron mass. In the green region, dark hadron production is not thermal, but
described by processes more analogous to chiral QCD. In the orange and green
regions, searches for Higgs decays to a few resonances would be sensitive to
this dark sector. The parameter space region indicated with the darker purple
rectangle is the focus of our analysis. Our cuts are optimized for the most
archetypically SUEP-like final states, schematically indicated by the lower-
left corner of this rectangle, demarcated with the diagonal line.

The parameter space of our benchmark scenario is shown schematically in Figure
1. The Higgs production portal sets the high scale for the event to
$Q=125\;\mathrm{GeV}$. With this scale fixed, the distribution of final states
is in principle determined by the dark hadron mass $m_{D}$ and the dark
Hagedorn temperature $T_{D}$ of Eq. (5), shown on the vertical and horizontal
axes of Figure 1. In reality, there may be different dark hadrons with
different spins, decays, and distributions, but this simplified description is
sufficient for our purposes. On the left and right, the relevant parameter
space is kinematically bounded by the red-hatched areas. Dark hadron
production in Higgs decays requires $m_{D}<m_{h}/2$. As we explain below, we
focus on dark hadrons that decay to SM hadrons, which requires
$m_{D}>(2-3)m_{\pi}$, depending on the exact decay portal.

Even within that mass range, hidden sectors with pseudo-conformal dynamics do
not always manifest as SUEP signatures in exotic Higgs decays. If
$m_{D}>{m_{h}}/{3}$ (blue area in Figure 1), then dark hadrons are only pair-
produced in Higgs decays, making this scenario equivalent to standard exotic
Higgs decays to pairs of various new particles (see e.g. Curtin:2013fra ). If
the dark hadron mass is fairly large, $m_{D}\sim m_{h}/(\mathrm{few})$ (yellow
area), or the dark Hagedorn temperature is comparable to or above the Higgs
mass $T\gtrsim m_{h}$ (green area), then exotic Higgs decay would produce only
a small multiplicity of dark hadrons that are either fairly hard or fairly
heavy. In both cases, dark hadron production is not thermal but is described
by processes more akin to chiral QCD. These regions are likely accessible
through modified searches for resonances in exotic Higgs decays Pierce:2017taw
; ATLAS:2020ahi , and we do not focus on them here. The gray area where
$T/m\ll 1$ is not expected to be realized by any pseudo-conformal hidden
sector, since the dark hadron mass and temperature are both related to the
strong coupling scale $\Lambda$. On the other hand, $T/m\gg 1$ is possible in
what we call the “dark pion regime”, where the dark hadrons are pseudo-
Goldstone bosons of an approximate symmetry, meaning their mass can be much
smaller than $\Lambda$. We do not focus on this region, but it would be an
interesting target for future investigations.

This leaves us with the actual SUEP regime for dark hadron production in
exotic Higgs decays, indicated by the light purple area. In this
investigation, we will focus on dark hadron masses below 8 GeV and $T/m$ in
the reasonable range of $\sim$ 0.25 to 4. This target SUEP parameter space is
marked out as the darker purple rectangle. Our cuts will be particularly
optimized for the lower-left region of the rectangle, demarcated with the
thick diagonal line. This is the region of low dark hadron mass and/or
temperature, corresponding to the softest SM final states that are most
difficult to search for using existing techniques.

##  3 Simulation

We briefly outline how we simulate our SUEP signal and the most important QCD
backgrounds, where the latter is necessary to develop our analysis techniques,
even though a realistic experimental analysis would rely on data-driven
background estimation.

###  3.1 Signal

We generate event samples for exotic Higgs decay into SUEP using the
SUEP_Generator plugin Knapen:2021eip  for Pythia 8.243 Sjostrand:2014zea ,
which models the dark shower as a spherical distribution of dark pseudo-scalar
mesons with momenta drawn from the relativistic Boltzmann distribution Eq.
(5). As in Ref. Knapen:2016hky , we make the simplifying assumption that there
is only one flavor of dark meson produced in the exotic Higgs decay. The free
parameters of the SUEP shower are the dark hadron mass $m_{D}$ and the
effective temperature $T_{D}$, setting the energy scale at which dark hadrons
are produced dominantly.

We simulate associated Higgs production at the 14 TeV HL-LHC, $pp\to
Vh,V\to\ell\ell/\ell\nu$, in Pythia 8. The Higgs is decayed to the SUEP final
state of dark mesons, which then decay directly to a $u\bar{u}$ quark pair
that in turn undergoes SM hadronization. The exact choice of hadronic decay
mode does not significantly affect our analysis, so we use this single channel
as a stand-in for other purely hadronic portals. The events are then passed
through the simplified detector simulation code Delphes 3 deFavereau:2013fsa
with CMS detector settings. The simulated detector-level objects output by
Delphes are used for our analysis. Our SUEP search will only use charged-track
information. Since charged tracks can be traced back to the primary vertex,
they are very robust with respect to pile-up contamination. We can therefore
neglect the effects of pile-up in the remainder of our study.

In our event samples we cover $m_{D}$ from 400 MeV to 8 GeV, and $T_{D}/m_{D}$
from 0.25 to 4. The lower bound on the dark hadron mass ensures that the dark
mesons are kinematically allowed to decay to two pions. The upper bound is
chosen to show where our search loses sensitivity. The range of temperatures
is chosen to satisfy $T_{D}\sim m_{D}$, which is the regime where the thermal
picture of SUEP production is valid. The signal cross section is
Cepeda:2019klc

$\displaystyle\sigma(pp\to e\nu/\mu\nu+\ \mathrm{SUEP})$ $\displaystyle=$
$\displaystyle(0.34{\ \rm pb})\cdot\mathrm{Br}(h\to\ \mathrm{SUEP})$ (8)
$\displaystyle\sigma(pp\to ee/\mu\mu+\ \mathrm{SUEP})$ $\displaystyle=$
$\displaystyle(0.066{\ \rm pb})\cdot\mathrm{Br}(h\to\ \mathrm{SUEP})$

for SUEP production in exotic Higgs decays in association with leptonic
$W/Z$-bosons that decay into electrons or muons. In total, we generate
$2.0\times 10^{5}$ $Zh$ and $Wh$ signal events, proportional to their
respective cross section, for each set of signal parameters
$(m_{D},T_{D}/m_{D})$.

###  3.2 Background

The dominant background to our signal is production of one or two leptons in
association with any number of QCD jets. It is highly challenging to model
reliably, and in a realistic study, data-driven background estimation would be
employed, see Section 4.4. However, for the purpose of developing our analysis
techniques, we simulate QCD+leptons background samples using MadGraph5_aMC@NLO
2.6.6 and Pythia 8.243.

Ideally, one should simulate fully matched multi-jet $+$ $\ell\ell/\ell\nu$
samples to capture the background distribution as closely as possible.
However, due to the large statistics needed for our analysis, and the fact
that such a simulation is anyway unlikely to be a perfect representation of
the detailed hadronic distributions at the relevant high multiplicities and
relatively low energy scale of the Higgs mass, this approach is not practical.
Instead, we simulate $nj+\ell\ell/\ell\nu$, where $n=2,3,4$ without jet
matching and $p_{T}>15$ GeV at generator level, to determine the effect of jet
multiplicity at the hard event level on our analysis. We find that $n>2$ leads
to lower cross section while being _more distinguishable_ from the SUEP signal
using the analysis techniques we develop here. Therefore, to be conservative,
we simulate $2j+\ell\ell/\ell\nu$ as our background samples for $Zh$ and $Wh$
production and decay into SUEP, respectively. In total, we use $10^{8}$
background events to represent the

$\sigma(\mathrm{QCD}\ +\ \ell\ell/\ell\nu)\approx 3.7\times 10^{3}{\ \rm pb}$
(9)

lowest-order MadGraph5 cross section for this background. While this is
sufficient to develop our analysis techniques, the Monte Carlo background
sample has $\sim 1/100$ the statistics of the full HL-LHC dataset. This is
important for the interpretation of our results in Section 5.

##  4 Analysis

The goal of this paper is to devise strategies for extracting SUEP signals
from a large background events without relying on the details of the simulated
signal. We first describe our trigger assumptions and baseline cuts, and
define a cut-based classifier to establish how sensitive such a simple
approach can be. Section 4.2 introduces a supervised neural network
classifier, to demonstrate both the advantages and limitations of the
supervised approach for our physics problem. We then introduce our primary
tool in Section 4.3 — an unsupervised neural network that we employ as an
anomaly detector for SUEP.

###  4.1 SUEP Observables

We define our trigger pre-selection by requiring that all events have at least
one charged electron or muon with $p_{T}\geq 40$ GeV, or two opposite sign
charged leptons with $p_{T}\geq 30(20)$ GeV. We also require that the scalar
$p_{T}$-sum of hadronic charged tracks from Delphes is above $30$ GeV. Both
signal and background have a trigger efficiency of ${\approx}40\%$, relative
to the cross sections in Eq. (8) and Eq. (9).

We focus on the most challenging region of SUEP parameter space, with either
low dark hadron masses or low dark shower temperatures. This gives rise to the
most archetypically SUEP-like final states with a high multiplicity of
isotropically distributed, relatively soft SM hadrons. The three observables
that best capture the characteristics of this signal are the charged particle
multiplicity $N_{\mathrm{charged}}$, the event isotropy $\mathcal{I}$
Cesarotti:2020hwb , and the interparticle distance. In all steps of the
analysis that follow, we only use charged particle tracks with $p_{T}\geq 300$
MeV from the primary vertex, excluding the one or two hard leptons associated
with the decaying gauge boson.

The event isotropy observable $\mathcal{I}\in(0,1)$ Cesarotti:2020hwb
quantifies the energy mover’s distance between a collider event and an
idealized isotropic event with uniform energy distribution, so $\mathcal{I}=0$
indicates a fully isotropic event. Originally, three different definitions of
the event isotropy are laid out, utilizing different geometries — spherical,
cylindrical, or in a two-dimensional ring. We compute the _ring isotropy_ of
the set of charged hadronic tracks of each event, since at a $pp$ collider we
have no way to know the longitudinal boost of the Higgs that decays to SUEP.
Since the SUEP is isotropic in the Higgs rest frame, we boost the hadronic
charged track system of each event into its transverse rest-frame before
computing the ring isotropy. To do this we assume that all hadronic charged
tracks belong to particles with the pion mass, but this is sufficient to
significantly separate signal and background events.

The variable that we introduce for the specific purpose of studying SUEP
events is the interparticle distance matrix $\Delta R_{ij}$ for charged hadron
tracks in the lab frame. It captures the unique topology of SUEP events while
being very suitable for machine-learning applications, since it is invariant
under re-definitions of the azimuthal angle around the beam axis. It is also
useful to define the mean $\overline{\Delta R}$ of all matrix entries.

Figure 2 shows distributions of $N_{\text{charged}}$, $\mathcal{I}$, and
$\overline{\Delta R}$ for the QCD background and a variety of SUEP benchmark
points after trigger selection. The separation between signal and background
is clear, with SUEP having higher multiplicity, more isotropic distribution of
tracks, and a significantly wider spread of inter-particle distances. To
understand how these observables change across SUEP parameter space, we show
their average values (across the whole sample after trigger selection) as a
function of $m_{D}$ and $T_{D}$ in Figure 3. The pairwise correlations between
each of these observables are included in the Appendix in Figure 8,
demonstrating that each of these three variables encode distinct information
about each event.

Figure 2: Comparison of QCD and SUEP distributions of selected observables.

In addition to the trigger requirements, we therefore impose the following
pre-selection cuts on all events:

$N_{\text{charged}}\geq 70\ \ \ ,\ \ \ \mathcal{I}<0.07\ \ \ ,\ \ \
\overline{\Delta R}<3\ .$ (10)

This cut targets the most SUEP-like parts of signal parameter space with low
dark Hagedorn temperature and/or low dark hadron mass (see Fig. 1). All but
2.2% of the post-trigger background is eliminated, while leaving $31.8\%$ of
signal for $m_{D}=0.4$ GeV and $T_{D}=0.4$ GeV. These requirements are less
optimal for larger dark hadron masses or temperatures — for example, the
signal efficiency is only $1.1\%$ for $m_{D}=5$ GeV, $T_{D}=20$ GeV. However,
larger temperatures and masses generally lead to higher-energy final states or
separable resonances, and are therefore not the focus of our present analysis.

Figure 3: Average values of charged particle multiplicity, event isotropy and
mean-interparticle-distance as a function of $m_{D}$ and $T_{D}$ for SUEP. QCD
average values: $\langle{N}_{\text{charged}}\rangle=51$,
$\langle{\mathcal{I}}\rangle=0.25$, $\langle\overline{\Delta R}\rangle=2.7$.

We now consider three options for SUEP searches at the HL-LHC, using events
which pass the baseline pre-selection as a starting point:

  1. 1.

The simplest strategy is a cut-and-count analysis using high-level
observables. It will serve as a baseline for more sophisticated machine-
learning techniques, and can be implemented very easily and effectively with a
stricter cut on $\overline{\Delta R}$ compared to Eq. (10). Varying the cut
threshold yields a significance improvement curve (SIC) of the signal
efficiency vs the background efficiency for each point in signal parameter
space, which we will then be able to compare to results using machine learning
methods. As we will see, this already yields very promising SUEP
sensitivities.

  2. 2.

A supervised ML-classifier requires detailed knowledge of the signal, since it
is trained on signal and background. This makes supervised techniques unlikely
to be a realistic analysis candidate for broad SUEP searches. However, we
perform a simple supervised study in Section 4.2 to demonstrate the best-case
scenario for SUEP sensitivity if the signal was very well understood.

  3. 3.

In Section 4.3, we use an unsupervised autoencoder trained only on the
background as an anomaly detector to improve on the sensitivity of the cut-
and-count analysis. This is likely to be a realistic analysis candidate since
it can be performed using data-driven background estimation techniques without
precise knowledge of the signal.

###  4.2 Supervised ML-Classifier

While a supervised classifier is explicitly dependent on the characteristics
of the signal (and background) simulation used in its training, which we
cannot trust in detail, it can still provide a useful comparison to an
unsupervised network, and give an indication of how sensitive a search based
on similar methods could be with improved modelling of the SUEP signal. An
additional limitation is that even with reliable signal simulation, the
parameters of the real SUEP are unknown, and a supervised network trained to
recognize SUEP with one set of parameters may fail if the true parameters
change. As we will see, this is indeed the case.

The supervised network architecture we choose is a dynamical graph
convolutional neural network wang2019dynamic ; Bernreuther:2021gds . We
implement the network in PyTorch. The input feature representation for both
the supervised and unsupervised networks is the interparticle distance matrix,
with the redundant left-lower half set to zero and track $p_{T}$ information
added to the diagonal:

$\Delta\tilde{R}_{ij}\equiv\left\\{\begin{array}[]{lll}\Delta R_{ij}&&i>j\\\
p_{T,i}/\mathrm{GeV}&\mathrm{for}&i=j\\\ 0&&i<j\end{array}\right.$ (11)

All events are required to have at least $N_{\text{charged}}\geq 70$ tracks,
and all events are truncated to keep only the $70$ highest-$p_{T}$ tracks to
ensure each event has the same dimensionality in the analysis.

The input matrix $\Delta R_{ij}$ is used to generate graph edges between each
particle (node) and its $k=7$ nearest neighbours in $\Delta R$ space. The node
features for each particle are the 70-dimensional vector of $\Delta R$
distances to all other particles in the event. The graph network has two
EdgeConv blocks, each comprising a three-layer perceptron with leaky ReLU
activation Qu:2019gqs . The EdgeConv operation updates the node features
$x_{i}$ of each particle as

$\vec{x}_{i}^{{}^{\prime}}=\frac{1}{k}\sum_{j=1}^{k}\vec{h}_{\theta}(\vec{x}_{i},\vec{x}_{i}-\vec{x}_{j})\;,$
(12)

where $k$ is the number of neighbours assigned to each node, and $h_{\theta}$
is a non-linear function of learned parameters $\theta$, implemented as a
three-layer perceptron. The graph edges are re-computed between the first and
second blocks using the Euclidean distance between the feature vectors of each
node to determine its $14$ nearest neighbours. The first block has feature
dimension 64, while the second block has feature dimension 32. Batch
normalization follows each layer. The output of the graph layers is averaged,
then passed through two fully connected layers, first expanding to dimension
128, then down to output dimension 2. The loss function is the cross-entropy
loss between the output and the true class label of each event.

$\mathcal{L}(x,\text{class})=-\log\frac{x[\text{class}]}{\sum_{j}x[j]}$ (13)

To test how the performance of the model depends on the choice of training
parameters, we train twelve neural networks on twelve different choices of
$(m_{D},T_{D}/m_{D})$, with $m_{D}=0.5,1,2\;\mathrm{GeV}$ and
$T_{D}/m_{D}=0.5,1,2,4$, and evaluate their effectiveness over the whole
signal parameter space. We also evaluate the efficacy of a ‘cocktail approach’
Aguilar-Saavedra:2017rzt ; Knapp:2020dde ; Bernreuther:2021gds ; Baldi:2016fzo
by training on a mixed sample including signal events from each of the twelve
parameter choices. A conditional training Louppe:2016ylz ; CMS:2019dqq  on the
signal parameters would be a similar approach. Each network is trained for 10
epochs with a decaying learning rate. Longer training periods were tested and
found to be unnecessary. Loss values for each test sample event are obtained
for the model realizations of the last 5 training epochs before being
averaged.

###  4.3 Unsupervised Autoencoder

Supervised ML-classification is an extremely powerful analysis tool, but it
only works for signals with well-defined and universal features and
corresponding precision simulations. At the LHC, this is not always the case,
and SUEP with its toy shower is a perfect example for a more broadly defined
signal. Here we prefer not to train a classifier on signal simulations.
Instead, we employ anomaly detection methods, where we train a network only on
the well-understood background dataset, so that it can flag events that are
anomalous in comparison. We use an autoencoder Rumelhart1986 ; Heimel:2018mkt
; Farina:2018fyg  and train it on data without class labels, with a loss
function that incentivizes its output to be as close as possible to the input.
The intermediate network layers have a restricted number of nodes compared to
the dimension of the input and output, forcing the network to compress the
information in the input, and then decompress it to recover the output. The
principle of the autoencoder’s use as an anomaly detector is that it should
fail to accurately reconstruct events that are anomalous compared to the
dataset it was trained on. A high reconstruction loss flags an event as being
potentially anomalous, or in collider physics parlance, a signal (SUEP)
candidate event.

The autoencoder uses the same modified $\Delta\tilde{R}_{ij}$ matrix input as
used by the supervised network, see Eq. (11). Other representations were
tested, including the matrices of both $k_{T}$ and anti-$k_{T}$ distances
$d_{ij}=\min(k_{ti}^{\pm 2},k_{tj}^{\pm 2})(\Delta\tilde{R}_{ij})^{2}/R$
between particles Cacciari:2008gp , the high-level observables used in the
pre-selection cuts, as well as the raw $p_{T},\phi,\eta$ values for each
particle; all yielded results that were much less useful than the analysis we
present here. This emphasizes the importance of choosing the correct input
representation over sophisticated network architecture for SUEP searches.

The matrix $\Delta\tilde{R}_{ij}$ is flattened into a vector of length
$N_{\text{charged}}^{2}$ and fed into the autoencoder. The neural network
comprises five fully connected layers, with the number of nodes decreasing to
the bottleneck size in the third layer, then increasing back to
$N_{\text{charged}}^{2}=4900$. An alternative 3-layer network performs only
slightly worse. For the bottleneck size we find that larger bottlenecks
consistently lead to better performance than smaller ones, so we use
$N_{\text{bottleneck}}=1000$. Each layer of the network has a leaky ReLU
activation with slope -0.2 for negative values of x, except the final layer
which has a ReLU activation. The loss function of the network measures the
difference between the network’s input and output as

$\mathcal{L}(y^{\text{in}},y^{\text{out}})=\frac{1}{N_{\text{charged}}}\sum_{i}\frac{|y^{\text{out}}_{i}-\sigma(y^{\text{in}}_{i})|^{m}}{|\sigma(y^{\text{in}}_{i})|^{n}}$
(14)

where $\sigma(x)=1/(1+e^{-x})$. Among different values of $m$ and $n$,
starting with the usual mean squared error $m=2,n=0$, the best-performing is
$m=3,n=0$. The sigmoid normalization of the input is essential to the
network’s success. Without it, the autoencoder encodes SUEP events with
slightly _lower_ loss than the QCD background on which it was trained, and
completely fails to identify anomalous events. We hypothesize that this is
because, unlike many experimental signatures, SUEP is less complex than its
QCD background, and it has smaller values of $\Delta R$ than QCD. This
complexity bias has been noted before Heimel:2018mkt ; Dillon:2021nxw ;
Finke:2021sdf . The sigmoid function reduces sensitivity to large values of
$\Delta R$ and $p_{T}$ by mapping them to values very close to 1, while
remaining approximately linear for small input values, but offset to a minimum
value of 0.5. These effects make it easier to accurately reconstruct QCD
events while enhancing the network’s sensitivity to deviations from the input
on the SUEP events with characteristically smaller absolute values of the
input features.

Out of $8.8\times 10^{5}$ background Monte Carlo events that pass the pre-
selection cuts, $2.4\times 10^{5}$ are used for training, $5\times 10^{4}$ for
validation when tuning network hyperparameters, and $5.9\times 10^{5}$ for
testing. The number of signal events that pass the cuts varies with $m_{D}$
and $T_{D}$, but generally a few $\times 10^{4}$ events remain at each
parameter point to be used for testing. SUEP events with $m_{D}=1$ GeV,
$T_{D}=0.5$ GeV are used for validation purposes.

The network is trained for 15 epochs with a decaying learning rate. Longer
training periods were tested and found to be unnecessary. Loss values for each
test sample event are obtained for the model realizations of the last 5
training epochs before being averaged.

Other architectures were investigated, including variational autoencoders
kingma2014autoencoding  and a graph convolutional autoencoder utilizing the
same EdgeConv operations as the supervised network, which itself led to the
use of the $\Delta\tilde{R}_{ij}$ event representation. Interestingly, the
simpler, fully connected architecture consistently delivered much better
background rejection than any of the more sophisticated graph networks in the
unsupervised approach.

###  4.4 Data-driven Background Estimation

Following the logic of this section further, we briefly describe how a data-
driven background sample could be derived for use in a realistic experimental
analysis based on our study. The total background cross-section can be
measured directly (and compared against simulation), since the leptons+QCD
signal region is completely background dominated for SUEP production in exotic
Higgs decays. The background efficiency of the classifier, whether it is based
on cuts, an unsupervised neural network, or a supervised one, can be estimated
by evaluating it on a control region.

A variety of control regions are possible, but perhaps the most promising is
defined by replacing the lepton criterion by a mono-photon criterion. Such a
sample should be free of signal contamination, and its hadronic content is
extremely similar to the hadronic background to our SUEP search, since it is
unconstrained in its detailed production channel except that it recoils off a
hard electroweak boson. The QCD distributions in this mono-photon-plus-jets
sample should therefore closely match those in the $Z$+jets signal region,
especially if the control region sample is reweighted to match the shape of
the photon $p_{T}$ spectrum to the dilepton $p_{T}$ spectrum in the $Z$+jets
signal sample. A variant of this approach can likely be adapted to estimate
the background in the $W$+jets channel as well, by using simulation to compute
a transfer function from $p_{T,W}$ to $p_{T,\ell}$ and applying it to
$p_{T,\gamma}$ in the control region before applying the $p_{T,\ell}$
reweighing.

Based on the assumption that such a strategy can be implemented, we therefore
will use benchmark estimates of $1\%$ and $10\%$ for the systematic background
uncertainty in our analysis to estimate the final physics reach.

##  5 Results

|
---|---
|
Figure 4: Some examples of Significance Improvement Curves (SIC), relative to trigger selection, of the autoencoder (black solid), cut on $\overline{\Delta R}$ (black dashed), and supervised graph networks (black dash-dotted and colored) for SUEP with test parameters of $(m_{D},T_{D}/m_{D})=$ (1.04, 0.435), (1.04, 1), (0.56, 0.435), (0.56, 1). For the autoencoder, the peak sensitivity improvement we are able to probe reliably with the statistics of our Monte Carlo samples (see text) corresponds to signal and background efficiencies of $(1.6\times 10^{-2},1\times 10^{-7})$, $(3.7\times 10^{-4},1.5\times 10^{-7})$, $(6.1\times 10^{-2},1\times 10^{-7})$ and $(2.2\times 10^{-2},1\times 10^{-7})$ relative to trigger selection, respectively.  |
---|---
|
Figure 5: Same SIC curves as Fig. 5, but with background efficiency on the
horizontal axis. This demonstrates that with the full HL-LHC dataset,
significantly harder cuts could increase sensitivity beyond the limitations of
what we can demonstrate with our simulated background dataset.

Figure 4 shows significance improvement curves for the cut on
$\overline{\Delta R}$, the autoencoder, and a selection of supervised networks
trained on signal samples with a variety of different dark hadron masses and
temperatures $m_{\text{train}}$ and $T_{\text{train}}$. The horizontal axis
shows either signal efficiency (after triggering) or number of remaining
signal events at the HL-LHC as the threshold is varied. The vertical axis
shows signal significance improvement relative to the trigger sample. From
these specific examples, a few general features are apparent:

  * •

The autoencoder consistently outperforms the simple $\overline{\Delta R}$ cut
significantly;

  * •

The supervised networks outperform the autoencoder for signal parameters close
to their training values, but can perform much worse for different parameters;

  * •

Our analysis is not optimized for larger dark hadron masses and temperatures;

  * •

For lower dark hadron masses or temperatures, both the cut-and-count and
autoencoder analysis strategies are very powerful, yielding orders of
magnitude improvement in signal significance compared to the baseline
preselection cuts of Eq. (10).

In fact, the statistics of our QCD background sample is insufficient to
capture the true power of our analysis techniques. We would expect the
significance improvement to increase as the signal efficiency decreases, but
only until the SI curve turns around when the cut becomes too harsh. This is
clearly the case in the top-right panel of Fig. 4, for signal parameters to
which our analysis is not optimized. However, for the three other examples, we
never reach this turn-over point before running out of simulated background
events. It is not even clear if this turn-over is reached with the full
statistics of the HL-LHC (100 times greater than our simulated background
sample).

To understand how much harder we might be able to cut on the background, Fig.
5 shows the same SIC curves but with background efficiency on the vertical
axis. Naively extrapolating, we can anticipate that a realistic autoencoder
search with a fully data-driven background sample at the HL-LHC might be able
to reach the very-low background regime while retaining enough signal to probe
$\mathrm{Br}(h\to\ \mathrm{SUEP})$ as much as an order of magnitude smaller
than the sensitivity estimates we can rigorously derive here.

As a result, the reach projections we present in this paper will be very
conservative. Furthermore, the limited statistics of our background sample
means that reach projections will be very similar for the three analysis
methods despite their obvious differences in performance in Figs. 4 and 5. It
is therefore important to additionally evaluate our classifiers using a
somewhat orthogonal metric.

The area under the curve (AUC) of the signal efficiency as a function of
background rejection can be computed as a function of the number of events
kept after the baseline preselection cuts, see Eq. (10). This metric is
standard to measure the performance classifiers independently of the choice of
threshold. Fig. 6 shows the AUC achieved by the cut-based classifier, the
fully connected autoencoder, and the supervised graph network trained on a
cocktail of signal events, where for the latter the signal parameters are
indicated with red dots in the $(m_{D},T_{D}/m_{D})$-plane). Unsurprisingly,
the highest AUC values are achieved at low $m_{D}$ or $T_{D}/m_{D}$, with the
supervised graph network modestly outperforming the autoencoder, which
outperforms the simple cut, across the SUEP parameter space. The performance
of supervised networks trained on single signal parameter points are presented
in the Appendix.

(a)blablablablablablabla(b)blablablablablablabla(c)bla

Figure 6:  AUC for (a) cut on $\overline{\Delta R}$, (b) fully connected
autoencoder, (c) supervised graph network trained using cocktail of signal
parameter choices (training parameters indicated with red dots). These plots
illustrate the significant performance improvements of the autoencoder
relative to the simple $\overline{\Delta R}$ cut, and of the supervised
cocktail approach relative to the autoencoder.

(a)blablablablablablabla(b)blablablablablablabla(c)bla

Figure 7:  Minimum excludable $\mathrm{Br}(h\to\mathrm{SUEP})$ at the HL-LHC,
assuming $1\%$ systematic uncertainty on QCD background for (a) cut on
$\overline{\Delta R}$, (b) fully connected autoencoder, (c) supervised graph
network trained using cocktail of signal parameter choices (indicated with red
dots). Note that the limited statistics of our QCD background sample leads
these projections to be very conservative while also de-emphasizing the
performance differences between the three methods.

A key physics result is the actual sensitivity of HL-LHC searches to the SUEP
final state. We therefore extract the smallest $\mathrm{Br}(h\to\
\mathrm{SUEP})$ branching ratio for which $S/\sqrt{B+u_{sys}B^{2}}>2$, where
$u_{sys}$ gives the systematic uncertainty on the background. Because of the
high degree of background rejection required to be sensitive to relevant Higgs
branching ratios, and the limited size of our background dataset, very few or
no simulated background events remain when the classifier’s threshold is set
to maximize sensitivity to low branching ratios. This introduces a significant
statistical uncertainty to the estimated LHC reach. We can decouple our
estimate somewhat from these limitations by demanding the statistical
uncertainty from the limited size of the background dataset to remain below
$50\%$. This is conservative, since a realistic analysis will employ even
harsher cuts with better significance improvement. For the same reason, the
difference in reach between our three methods is likely underestimated. The
performance gaps between the $\overline{\Delta R}$ cut, autoencoder, and
supervised network are more clearly shown by the individual significance
improvement curves and the AUC differences.

Finally, Fig. 7 shows the SUEP sensitivity achievable by the $\overline{\Delta
R}$ cut, the autoencoder, and the cocktail-trained supervised network, all
assuming 1% systematic uncertainty on the background. Both the cut and the
autoencoder can probe %-level branching ratios. In the Appendix we show
sensitivity projections assuming a much larger 10% systematic background
uncertainty, with only very modest degradation in reach. This shows that the
overwhelming QCD background has been reduced to low enough levels to make the
search statistics limited, and speaks to the robustness of our results.

##  6 Conclusion

SUEPs represent a highly plausible but extremely challenging experimental
signature of confining hidden sectors, which typically results in a high
multiplicity of soft SM final states. To date, there are no targeted LHC
searches for SUEP, and most existing searches have limited or no sensitivity.
Furthermore, since SUEP is produced by hidden sectors featuring fairly
strongly-coupled, approximately conformal dynamics with a wide variety of
possible dark hadronization scenarios, modelling the detailed production of
SUEP is fraught with uncertainties. Existing proposals for SUEP searches
Knapen:2016hky ; Alimena:2019zri  target conspicuous, qualitative features of
the final state, such as displaced vertices or leptons. Their existence is a
prediction for some decay portals of the dark hadrons, and targeting them with
searches is fairly robust with respect to modelling uncertainties.

An important but previously unaddressed question is how well we can look for
SUEP using only its basic kinematic features, without any conspicuous SM final
states. This is not only an experimental challenge, the design of such a
search also has to be very mindful of uncertainties in the signal modelling.
We investigated this worst-case scenario by studying SUEPs from dark hadrons
produced in exotic Higgs decays, which decay promptly and purely hadronically.
Our choice of model is motivated by the Higgs portal being one of the most
plausible production modes for new physics. The modest energy scale of the
Higgs decay also eliminates high-energy or high-mass observables as
discriminants. Finally, Higgs production in association with a $W/Z$
circumvents the problem of triggering on the SUEP final state directly
Knapen:2016hky .

Our first results are observables which capture the essential SUEP features
from the track momenta. We focused on the charged particle multiplicity
$N_{\text{charged}}$, the event ring-isotropy Cesarotti:2020hwb
$\mathcal{I}$, and the interparticle distance $\Delta R_{ij}$. All three are
robust with respect to modelling uncertainties of the SUEP final state, since
they capture the essential model features: high multiplicity of final states,
and dark hadron production that is more isotropic and democratic in momentum
than the QCD background. The charged track interparticle distance matrix
$\Delta R_{ij}$ is particularly suitable for ML applications.

Based on these observables, we devised three strategies for $h\to\
\mathrm{SUEP}$ searches. The first is based on a simple cut on the
interparticle distance matrix. The second assumes that our naive signal
simulation tools can be trusted, and uses supervised ML techniques. The third
is an unsupervised ML approach using a fully connected autoencoder trained
only on simulated QCD events as an anomaly detector. Both ML approaches used
the slightly modified interparticle distance matrix of Eq. (11) as the event
representation. The cut-and-count approach serves as a simple baseline, over
which the unsupervised ML approach represents a significant improvement, even
without detailed knowledge of the signal. This is to be compared to the higher
sensitivity of the supervised machine learning approach, which is unlikely to
be robust with respect to modeling uncertainties or choice of training
parameters.

All three approaches will probe exotic Higgs decays to prompt, hadronic SUEP
at the HL-LHC for branching fractions at the percent level, with both the
autoencoder and supervised approaches probing rates as low as 1%. We assumed a
systematic background uncertainty of one percent, but increasing this to ten
percent only modestly decreases sensitivity, signaling that our analyses have
sufficient differentiation power to reduce the enormous QCD background to a
statistics-dominated level. These estimates are conservative, since our
simulated background samples were still too small to fully cover the
exceedingly large background rejection. A realistic search combining the
autoencoder with data-driven background estimates will achieve significantly
higher sensitivities.

Our results show that even without a detailed theoretical description of the
SUEP showering process, an analysis using an unsupervised neural network can
be highly sensitive to exotic Higgs decays to SUEP. Our observables and
techniques can equally well be used in SUEP searches using leptons, photons or
displaced vertices, to significantly enhance their sensitivity based on the
inherently SUEP-y kinematics. Generalizations of our methods, for instance
including searches for explicit dark hadron resonances, should allow for
sensitivity to SUEPs with higher dark hadron masses or dark Hagedorn
temperatures than those targeted by our analysis. Our new, unsupervised search
strategy can be applied to a wide range of LHC scenarios to discover new
physics, even if the true BSM model should differ from our exact theory
expectations.

#### Acknowledgements

The authors would like to thank Anja Butter, Simon Knapen, Jessie Shelton, and
Jennifer Thompson for helpful conversations. The research of JB and DC is
supported in part by a Discovery Grant from the Natural Sciences and
Engineering Research Council of Canada, and by the Canada Research Chair
program. JB also acknowledges funding from a Postgraduate Doctoral Scholarship
(PGS D) provided by the Natural Sciences and Engineering Research Council of
Canada. GK acknowledges the support of the Deutsche Forschungsgemeinschaft
(DFG, German Research Foundation) under Germany’s Excellence Strategy – EXC
2121 “Quantum Universe” – 390833306. The research of TP is supported by the
Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under grant
396021762 – TRR 257 Particle Physics Phenomenology after the Higgs Discovery.

Computations were performed on the Niagara supercomputer at the SciNet HPC
Consortium Niagara ; SciNetLessons . SciNet is funded by: the Canada
Foundation for Innovation; the Government of Ontario; Ontario Research Fund -
Research Excellence; and the University of Toronto. This research was enabled
in part by support provided by Compute Canada (www.computecanada.ca).

##  Appendix A Additional Results

Figure 8: Two-dimensional unit-normalized histograms showing pairwise
correlations between $N_{charged}$, $\mathcal{I}$, and $\overline{\Delta R}$
for the background sample and the same four choices of signal parameters as in
Figure 4. Figure 9: AUC for supervised graph networks trained using different
signal parameter choices. Red dots indicate the training parameters in each
plot. Figure 10: Minimum branching ratio excludable by supervised graph
networks trained on different choices of ($m_{D}$,$T_{D}$). Red dots indicate
the training parameters in each plot.

Figure 8 shows the two-dimensional correlations between each pair of the
observables $N_{charged}$, $\mathcal{I}$, and $\overline{\Delta R}$. While the
average values of $N_{charged}$ and $\overline{\Delta R}$ vary in a similar
fashion with $m_{D}$ and $T_{D}$, their distributions for a fixed choice of
signal parameters are not highly correlated.

Figure 9 shows the AUC for each of the twelve supervised graph networks
trained on a single sample of signal events, with parameters indicated by the
red dot in each plot. This vividly demonstrates how the the parameter choice
of the signal dataset on which the supervised network is trained has a large
impact on the regime of parameter space where it can achieve high AUC,
providing a strong argument for using the cocktail approach.

The sensitivity projection for these supervised networks with one percent
background systematic is shown in Fig. 10. The peak sensitivity for each
network is also at the 1%-level, similar to the other approaches, but for
potentially a different range of SUEP parameters, depending greatly on the
training parameters of each network.

Figures 11 and 12 show sensitivity projections for 10% systematic background
uncertainty.

(a)blablablablablablabla(b)blablablablablablabla(c)bla

Figure 11:  Same as Fig. 7 but for 10% systematic background uncertainty.
Figure 12: Same as Fig. 10 but for 10% systematic background uncertainty.

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