# Toward Designing Social Human-Robot Interactions for Deep Space Exploration

Huili Chen  MIT Media LabMIT Media Lab<EMAIL_ADDRESS>and  Cynthia
Breazeal  MIT Media Lab<EMAIL_ADDRESS>
(2021)

###### Abstract.

In planning for future human space exploration, it is important to consider
how to design for uplifting interpersonal communications and social dynamics
among crew members. What if embodied social robots could help to improve the
overall team interaction experience in space? On Earth, social robots have
been shown effective in providing companionship, relieving stress and anxiety,
fostering connection among people, enhancing team performance, and mediating
conflicts in human groups. In this paper, we introduce a set of novel research
questions exploring social human-robot interactions in long-duration space
exploration missions.

Human-robot Interaction, Long-duration Human Space Mission, Space Robot
Companion, Social Connection

††journalyear: 2021††copyright: acmlicensed††conference: The 2021 ACM CHI
Virtual Conference on Human Factors in Computing Systems; May 8-13, 2021;
Yokohama, Japan (Online Virtual Conference)††booktitle: CHI ’21 workshop
SpaceCHI: Human-Computer Interaction for Space Exploration

##  1\. Introduction

###  1.1. Human Factors in LDSE Missions

NASA is actively planning for long-duration space exploration (LDSE) missions
such as an anticipated manned mission to Mars in the 2030s (NASA, [n.d.]). In
planning for future LDSE missions, one major risk area pertaining to human
factors concerns the ability of astronauts to adapt to the isolated, confined,
and extreme (ICE) conditions (NASA, 2020). Living in ICE conditions for a long
duration is highly stressful. The psycho-environmental factors of living and
working in such ICE environments, as documented by studies of these ICE
analogues and evidence from past spaceflights, include crowding, lack of
privacy, social isolation, and sensory restriction (Palinkas and Browner,
1995; Raybeck, 1991). The sense of isolation will become further heightened
with longer distances from Earth and communication lag with Earth (NASA,
2020).

To ensure the success of a LDSE mission, adaptation is needed for both
individual astronauts and the crew team as a whole. On the individual level,
astronauts will need to cope with a variety of observed behavioral,
physiological and psychological problems including anger, anxiety,
interpersonal conflict, social withdrawal, sleep deprivation, decrease in
group cohesion, and decrease in motivation (Flynn, 2005). On the team level,
factors related to interpersonal communications and group dynamics among
astronauts also decisively impact mission success. For example, astronauts
will have to live in an ICE condition for the entirety of the mission,
necessitating group living skills to combat potential interpersonal problems
(Galarza and Holland, 1999). Their diverse cultural backgrounds may
additionally impact coping in an ICE environment and interplanetary crew’s
behavior (Tafforin and Giner Abati, 2017). Unlike shorter-duration space
missions, LDSE missions require astronauts to have an unprecedented level of
autonomy, leading to greater importance of interpersonal communication among
crew members for mission success. The real-time support and interventions from
human specialists on Earth (e.g., psychologist, doctor, conflict mediator) are
reduced to minimal due to costly communication and natural delays in
communication between space and Earth. All these human factor risks could lead
to serious detrimental impacts on a LDSE mission if left unmitigated. To
address them, socially interactive technologies such as social robots offer
great opportunities of delivering effective interventions.

###  1.2. Human-robot Interaction (HRI) in Space Exploration

In the past decades, most research on HRI in space exploration focused on the
engineering and cognitive aspects of space robotics. Space robots were often
envisioned with advanced cognitive systems (e.g.,CARACaS) capable of model
building, continuous planning/re-planning, self-diagnosis, and novel situation
understanding (Huntsberger and Stoica, 2010). These space robots could assist
crew members with a variety of physical and cognitive tasks, but they did not
necessarily support social and affective communications with crew members. In
recent years, sociable space robots started to emerge, able to display and
recognize social cues. An in-cabin flying robot named Astronaut Assistant
Robot (AAR) (Liu et al., 2016) allows astronauts to use hand gestures to
communicate with it face-to-face (Gao et al., 2017). As the first free-
floating, sphere-shaped interactive companion robot in space, Crew Interactive
Mobile Companion (CIMON) designed by NASA and IBM (Murphy, 2018) can display
human-like facial expressions on its screen and respond to voice questions or
directions without the need for a tablet or computer when assisting astronauts
in their daily work.

The potential benefits of social robots in space are also suggested by prior
HRI work conducted on Earth, which found social robots effective in improving
people’s mental health and well-being (Scoglio et al., 2019) as well as human
group dynamics and team performance (Sebo et al., 2020c). In LDSE missions,
the needs of astronauts for social robots will very likely become heightened
given the LDSE-related human factor risks listed in Section 1.1, thereby
providing a greater motivation for more research on social HRI in space. In
this paper, we focus on the opportunity of using social robots as a multimodal
medium for ameliorating interpersonal communications and group dynamics of
crew members in LDSE missions.

##  2\. Social HRI on Earth

###  2.1. Robot Social Roles in Human Groups

Social robots can play a variety of roles ranging from highly knowledgeable
agents to social-emotional companions across human group interaction contexts.
When engaging with a human group as a highly knowledgeable agent and expert in
a task area, a social robot, for example, can provide counseling to couples
(Utami and Bickmore, 2019), offer direction guide in public space (Fraune et
al., 2019), host group game activities (Żarkowski, 2019), and allocate turns
to players (Claure et al., 2020). In contrast to an expert role, a robot can
take a peer role that strategically displays its vulnerability or emotions
with the goal of ameliorating human group dynamics. For example, a robot Kip1
was designed as a peripheral conversation companion that promotes non-
aggressive conversation between people by expressing either curious interest
or fear via its gesture cues (Hoffman et al., 2015). Similarly, vulnerable
statements made by a robot in a human group can positively impact the dynamics
and collaboration among the group members (e.g., more equal conversation
distribution and more positive group perception (Traeger et al., 2020)).

In addition to the expert and social-emotional companion roles, a robot can
take a mediator role to directly resolve human team conflicts. For example, a
social robot can improve people’s interpersonal conflict resolution skills by
flagging a conflict onset and offering prompts for conflict resolution (Shen
et al., 2018). Mediation via the Telenoid robot was also found to produce more
agreements and more integrative agreements among human teammates in comparison
with both a screen mediator and a human mediator (Druckman et al., 2020). The
last social role category that a robot often takes in a human group is a
moderator/supporter role. When taking this role, a robot could improve a human
group’s social dynamics (Short et al., 2016) and a dyadic human team’s
performance on a collaborative task by posing questions (Strohkorb et al.,
2016).

Overall, a robot’s social role in a human group interaction distinctly shapes
how humans interact with each other and how they perceive the robot.

###  2.2. Robot Social Behaviors in Human Groups

Both of a robot’s verbal and nonverbal social behaviors can positively
influence human groups when displayed appropriately, though in their unique
ways. Specifically, a robot’s verbal communication can support its expressions
of emotion (Leite et al., 2013; Correia et al., 2018) and share informational
content (Sabelli and Kanda, 2016) with humans. It can also improve a variety
of affective phenomena in human-human interactions such as trust building
(Correia et al., 2018), group engagement (Matsuyama et al., 2015),
psychological safety of outgroup team members (Sebo et al., 2020b), equality
and positivity within conversational groups (Traeger et al., 2020), and the
inclusion of human members within a team (Sebo et al., 2020a).

In addition to its verbal utterances, a robot’s nonverbal behaviors such as
gestures (Liu et al., 2013), gaze (Mutlu et al., 2009; Skantze, 2017)
navigation (Kidokoro et al., 2013; Mavrogiannis et al., 2019), and physical
orientation (Vázquez et al., 2017; Shiomi et al., 2010) can influence people’s
responses in a group interaction and their perception of the group (Sebo et
al., 2020c). For example, robots were designed, in some interactions (Tennent
et al., 2019; Fink et al., 2014), as peripheral or passive entities that
shaped human group dynamics via implicit nonverbal behaviors without eliciting
users’ awareness. Often inspired by Ju’s theory of implicit interactions (Ju
and Leifer, 2008), this alternative design situates a robot as a passive
entity in a group. For example, a robotic microphone that exhibits implicit
engagement behaviors (e.g., nonverbal backchanneling) in a group problem
solving context encouraged more active participation from passive human
members and promoted more effective group collaboration performance (Tennent
et al., 2019). As shown above, a robot’s social influence on human groups can
be achieved through both its use of speech and nonverbal social cues.

Nevertheless, an excessive or inappropriate display of social cues may
distract or interrupt humans in some circumstances (Kennedy et al., 2015;
Yadollahi et al., 2018). For example, the frequency and length of a robot’s
speech affected humans’ interaction outcomes more significantly than its
content due to the disturbing effects of the inappropriate speech display
(Jung et al., 2015; Short and Mataric, 2017). Penalizing high frequency of
speech as the “communication cost” can yield more effective human-robot
communication (Unhelkar et al., 2020). For nonverbal behaviors, a robot’s
congruent display of social cues, e.g., contingent gaze and pointing gestures,
can elicit greater human participation in an interaction (Lohan et al., 2012),
while the asynchrony of social cues, e.g. robot head-gazes and pointing
gestures made in different directions, may lead to interference effects and
slow down human processing time (Langton, 2000; Langton and Bruce, 2000).

Hence, understanding the relations between the display of robot social
behaviors and interaction contexts is crucial for designing successful multi-
party HRI.

###  2.3. Human Trust in Robots

Trust is a result of dynamic interactions (Mayer et al., 1995). Successful
interactions will lead to feelings of security, trust and optimism, while
failed interactions may result in human’s unsecured feelings or mistrust. Much
empirical evidence shows trust essential for successful human-human
interactions (McAllister, 1995). In human-robot interactions, humans’ trust in
robots also plays a critical role such as influencing their willingness to
accept information from robots (Hancock et al., 2011) and to cooperate with
robots (Freedy et al., 2007). A person’s trust in a robot is often intertwined
with the robot’s task performance, display of nonverbal behaviors, and
interaction personalization. A single error of the robot can impact humans’
trust of it, especially in critical situations (Robinette et al., 2017). If a
robot displays nonverbal signals that humans often exhibit to indicate
distrust, the robot would also be perceived as less trustworthy (DeSteno et
al., 2012). In the context of workplace-based long-term HRI, a personalized
human-robot discussion was found to increase the person’s rapport and
cooperation with the robot as compared with a social but not personalized
discussion (Lee et al., 2012). Humans’ trust in robots, given its importance
in HRIs, is often used as an evaluation measure on robot decision making (Chen
et al., 2018) and human-robot team effectiveness (Freedy et al., 2007) across
contexts and tasks.

##  3\. Future Social HRI in Space Exploration

Despite extensive HRI research conducted on Earth, we barely know to what
extent these prior findings can be readily applied to the deep space
exploration context. Contextualizing the HRI research in LDSE missions would
unlock the full potential of using social robots to foster positive
interpersonal communications and social dynamics among crew members in future
space exploration. Hence, we propose a set of key design questions (DQ)
pertaining to social HRI in space.

###  3.1. Robot Social Role Design

When interacting with a human group, social robots have been taking a variety
of roles from resource providers to listeners, each of which offers unique
benefits contingent on the interaction context. However, barely any prior work
focused on the design and impact of robot roles on the group dynamics and
processes of space crew teams, posing a new urgency to investigate the nuanced
challenges for social HRI in space. For example, a robot mediator has been
empirically shown to resolve people’s interpersonal conflicts more effectively
than a screen mediator or a human mediator (Druckman et al., 2020), but the
design considerations for a robot mediation in astronaut-astronaut
interactions remain unexplored. Understanding astronauts’ perception,
acceptance and needs of social robots in different LDSE-related interaction
contexts would help design for more astronaut-centered HRI. Therefore, we
propose the first two research questions as follows:

  * •

DQ1: What robot interactions would astronauts perceive to be socially,
cognitively and affectively beneficial to the crew team in a LDSE mission?

  * •

DQ2: How would individual characteristics and cultural backgrounds of
astronauts affect their perception, acceptance and preference of social HRI in
space?

In LDSE missions, crew members engage in a variety of group interactions
ranging from highly-critical team cooperation and urgent problem-solving
meetings to leisure and recreational activities. In these activities, the crew
team’s social dynamics may vary as the social roles of the members are
adaptive to the context, e.g., superior-subordinate communication and peer
interaction. The way a social robot engages with the team should thus be
contingent on the team’s social dynamics. Designing a diverse set of robot
social roles customized to different group interaction contexts could
potentially promote the overall positive astronaut-robot interaction
experience in space. We summarize this research topic as follows:

  * •

DQ3: What robot role(s) could be designed for each crew team interaction
context in a LDSE mission?

Since all group interactions take place in an ICE environment with limited
physical space and resources for a long duration, different robot roles
designed for specific group contexts, e.g., coach, peer and mediator, may
likely have to share a robot hardware embodiment rather than each owning a
different physical embodiment. This constraint on robot hardware resources
further poses additional design challenges pertaining to robot role and
identity switching, as summarized below.

  * •

DQ4: Should a robot’s social identity (e.g., name, memory and personality)
remain consistent while its social role switches across interaction contexts,
as illustrated in Figure 1(a)? Should its social identity switch across
contexts so that each robot role is uniquely associated with a distinct robot
identity, as illustrated in Figure 1(b)?

  * •

DQ5: How should a physical robot embodiment identify group contexts and
dynamics, and proactively switch its social role or identity to effectively
promote positive crew team experience?

  * •

DQ6: How would the aforementioned role-switching and identity-switching
approaches influence astronauts’ perception and acceptance of robot
interactions such as their trust in robots?

(a) Robot Role Switching

(b) Robot Identity Switching

Figure 1. Two design approaches to address the constraint on robot hardware
resources in LDSE missions.

###  3.2. Robot Social Behavior Design

Robots can positively influence a human group via diverse combinations of
social cues, from actively utilizing animated speech to solely using nonverbal
cues without eliciting human awareness. In order to be interpreted correctly
and efficiently, robot social cues need be contingent on specific interactions
as well as congruent with other social cues being utilized. Unlike the Earth
environments where the most prior HRI studies were conducted, outer space
environment imposes much more stressful challenges on human bodies and minds,
resulting in a variety of expected physiological, psychological, cognitive and
physical changes of astronauts. Investigating how astronauts encode and decode
multimodal social cues when interacting with a robot in various physical and
mental conditions would help design for space robots that can adapt their
social-affective expression styles based on both physical environment and in-
the-moment states of astronauts to improve human-robot and human-human
communications. Thus, we summarize this design question on robot social cues
as follows:

  * •

DQ7: How would different gravity conditions, e.g., microgravity, lunar and
Martian gravity, impact astronauts’ social, cognitive and affective reactions
to a variety of robot social cues, e.g., gaze sharing, body motion and speech,
in both group and single-person HRI contexts? What social cues would
astronauts prefer to use for human-robot communications in different gravity
conditions?

Living in ICE conditions for a long duration exposes astronauts to long-term
sensory restriction. Unaffected by ICE conditions, social robots hence have
great potential of delivering high-quality consistent multi-sensory
intervention to astronauts. This design question is summarized as follows:

  * •

DQ8: How could different robot social cues and behaviors be designed to
provide positive multi-sensory stimulation to the crew team as well as enhance
multimodal social-affective communications among crew members, e.g., rapport,
social touch, and social reciprocity?

###  3.3. Robot Social Adaptation Design

Human adaptability for deep space environment is crucial for mission success.
In addition to traditional coping and resource strategies for astronauts’
adaptation (Bartone et al., 2019), a space robot’s social adaptability to
astronauts could also facilitate their adaptation to deep space with the long-
term goal of maintaining their health and productivity. Multiple social-
affective signal modalities of astronauts could be leveraged to develop a
robot social adaptation system, including their physiological signals (e.g.,
blood pressure, sleep patterns), physical conditions (e.g., malnutrition), as
well as their psychological and cognitive states (e.g., depressive mood,
anxiety, loneliness), alongside the robot’s audiovisual observation of the
naturalistic crew interactions. For example, if a robot can monitor each
astronaut’s sleep quality, it can adapt its social role and social cues to
minimize the negative effect of astronauts’ sleep deprivation on team decision
making and other cognitive processes. The access to a more diverse set of
astronauts’ social-affective signal modalities would enable a robot’s more
strategic and timely adaptation. However, a robot’s overaggressive data-driven
adaptation may increase the risk of exacerbating astronauts’ perceived lack of
privacy in an ICE environment and weakening their interpersonal trust in the
robot and the team, potentially resulting in lower overall crew team dynamics
and productivity. Investigating the optimal robot adaptation in LDSE missions
requires an astronaut-centered design approach. Therefore, we propose two
astronaut-centered design questions as follows:

  * •

DQ9: What social-affective signal modalities of astronauts could a robot have
access to for its long-term social adaptation?

  * •

DQ10: How to design a robot’s social adaptation system that can maintain and
even foster crew members’ perceived privacy and interpersonal trust in the
robot and other members in LDSE missions?

##  4\. Conclusion

This paper introduces novel opportunities for social human-robot interactions
in deep space exploration. We believe that exploring the design questions
listed in the paper would help shed light on how social robotics could be
potentially used to promote interpersonal communications and facilitate group
interactions among crew members in long-duration space exploration missions.

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aainstitutetext: Theory Center, IPNS, KEK,
1-1 Oho, Tsukuba, Ibaraki 305-0801, Japanbbinstitutetext: The Graduate
University of Advanced Studies (Sokendai),
1-1 Oho, Tsukuba, Ibaraki 305-0801, Japanccinstitutetext: Kavli IPMU (WPI),
University of Tokyo,
5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8583, Japan

# Spectral Analysis of Jet Substructure with Neural Networks: Boosted Higgs
Case

Sung Hak Lim a,b,c and Mihoko M. Nojiri<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Jets from boosted heavy particles have a typical angular scale which can be
used to distinguish them from QCD jets. We introduce a machine learning
strategy for jet substructure analysis using a spectral function on the
angular scale. The angular spectrum allows us to scan energy deposits over the
angle between a pair of particles in a highly visual way. We set up an
artificial neural network (ANN) to find out characteristic shapes of the
spectra of the jets from heavy particle decays. By taking the Higgs jets and
QCD jets as examples, we show that the ANN of the angular spectrum input has
similar performance to existing taggers. In addition, some improvement is seen
when additional extra radiations occur. Notably, the new algorithm
automatically combines the information of the multi-point correlations in the
jet.

###### Keywords:

Jets, QCD phenomenology

††preprint: KEK-TH-2060

##  1 Introduction

At multi TeV $pp$ colliders such as the LHC, boosted heavy particles can be
produced and form a single collimated cluster of particles. Such a localized
cluster is distinguished from a QCD jet from a hard quark or gluon by the
substructures of the cluster Butterworth:2008iy . For this purpose, consistent
definitions of substructures of jets have been studied extensively. There are
various methods for identifying the jet substructures, such as strategies
based on cluster decomposition Butterworth:2008iy ; Thaler:2008ju ;
Kaplan:2008ie ; Plehn:2009rk ; Plehn:2010st ; Soper:2011cr ; Soper:2012pb ;
Dasgupta:2013ihk ; Soper:2014rya ; Larkoski:2014wba  and shape variables
Gallicchio:2010sw ; Thaler:2010tr ; Gallicchio:2011xq ; Chien:2013kca ;
Larkoski:2013eya ; Larkoski:2014gra ; Moult:2016cvt . These methods focus on
different features of jet substructures to maximize the discrimination power.
For the case of Higgs, W, and Z boson decaying hadronically into two quarks, a
critical feature is a two-prong substructure inside. Because the key features
depend on the nature of the parent particle of a jet, there are several
frameworks that can be applied to jets Jankowiak:2011qa ; Datta:2017rhs ;
Chakraborty:2017mbz ; Aguilar-Saavedra:2017rzt ; Komiske:2017aww ;
Chien:2018dfn .

We propose a spectral analysis in order to identify originating partons of a
jet using a spectral function inspired by the proton nuclear magnetic
(${}^{1}\mathrm{H}$-NMR) spectroscopy of organic molecules. The organic
molecules consist mostly of a carbon skeleton and hydrogen atoms. These
substructures can be identified by the ${}^{1}\mathrm{H}$-NMR spectrum, which
records the resonant frequency of the hydrogen nuclei under an external
magnetic field. The resonant frequency depends on an induced magnetic field
generated by the rest of the molecular substructure; hence, the molecular
structures can be determined by investigating the spectrum, as shown in Figure
1. In particular, we use shift and splitting of the resonance frequency, where
the big shift comes from the electron cloud, whose density is determined by
the skeletal structure, and small splittings are from the electromagnetic
interaction between the hydrogen nuclei. Similarly, we develop a spectral
function for jet substructure study, where the big structure of the spectrum
is made from initiating hard partons while small structures come from QCD
interaction from the hard partons. The spectral function that we propose is
similar to the angular structure function Jankowiak:2011qa ; Jankowiak:2012na
; Larkoski:2012eh . The resulting spectrum contains useful information for
identifying the nature of a given jet.

C (-[:210]H) (¡[:70]H) (¡:[:110]H) (-[:330]C( (¡[:290]H) (¡:[:250]H) )-[:30]OH) |  | 0$\mathrm{O{\color[rgb]{0,0.6,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.6,0}\pgfsys@color@cmyk@stroke{0.64}{0}{0.95}{0.40}\pgfsys@color@cmyk@fill{0.64}{0}{0.95}{0.40}{H}}}$0$\mathrm{C{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{H_{2}}}}$00$\mathrm{C{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{H_{3}}}}$01234chemical shift
---|---|---
Figure 1:  Molecular structure of ethanol and its ${}^{1}\mathrm{H}$-NMR
spectrum. The intensity, location and splitting of peaks allow us to identify
the original molecular structure. The chemical shift is the resonant frequency
of a hydrogen nucleus relative to a reference frequency.

An artificial neural network (ANN) is a useful model for analyzing the
spectrum. Jet substructure analyses based on ANN are gaining attention
recently and have been studied in various contexts. The analyses are
categorized mainly into two groups with different inputs. One group utilizes
special-purpose observables and uses ANN to identify a correlation between
substructures Datta:2017rhs ; Aguilar-Saavedra:2017rzt ; Datta:2017lxt
similar to analyses with a boosted decision tree Bhattacherjee:2016bpy ;
Chien:2017xrb . The other group uses the jet constituents directly and uses
ANN to find out particular substructures in a jet. This group is again
categorized into two subgroups depending on how you interpret jet
constituents. One can interpret jet constituents as an image Cogan:2014oua
and use image recognition techniques Almeida:2015jua ; deOliveira:2015xxd ;
Baldi:2016fql ; Komiske:2016rsd ; Kasieczka:2017nvn ; Choi:2018dag . The other
strategies are to interpret a jet as a sequence of data, such as clustering
sequence of the jet algorithms Bartel:1986ua ; Catani:1993hr ; Ellis:1993tq ;
Dokshitzer:1997in ; Wobisch:1998wt ; Cacciari:2008gp , and utilize ANN for
sequential data anlysis Louppe:2017ipp ; Cheng:2017rdo ; Egan:2017ojy  Our
approach is different from these approaches, namely our network is requested
to analyze an event-by-event spectrum of a jet. This approach reduces the
inputs of the ANN significantly but it can still learn characteristic non-
local correlations in a jet from a heavy particle like the non-local neural
networks for video classification NonLocal2018 . We will show that our
approach improves the separation between Higgs jets and QCD jets in natural
manner.

This paper is organized as follows. In Sec. 2, we define a spectral function
$S_{2}(R)$ and describe its nature. We also explain the setup of our Monte-
Carlo simulations. In Sec. 3, we show a cut-based classification of the Higgs
jet vs QCD jet using the spectrum $S_{2}(R)$ and compare the performance with
the ratio of the enegy correlation function, $D_{2}$ Larkoski:2014gra , for
two-prong substructures. In Sec. 4, we introduce the spectral analysis of jet
substructure with ANN. Sec. 5 is devoted for summary and discussions.

##  2 A Spectral Function for Jet Substructure

The jet substructure analysis has similarities with the ${}^{1}\mathrm{H}$-NMR
analysis of organic molecular structures. An organic molecule has a carbon
skeleton surrounded by hydrogens, which ${}^{1}\mathrm{H}$-NMR analyzes to
find out the topological skeleton. On the other hand, a jet of our interest
arises from hard partons, which can be regarded as a topological skeleton of
the momentum distribution of jet constituents. The jet constituents radiated
from hard partons are something analogous to the hydrogens in a organic
molecule.

In the ${}^{1}\mathrm{H}$-NMR analysis, we record interactions between
hydrogen nuclei and the rest of the molecular structure by their resonant
frequencies to measure the molecular structure. Likewise, we focus on
correlations between pairs of the constituents of the jet based on distance to
identify the originating partons. A popular choice of the distance is angular
distance $R_{ij}=\sqrt{\eta^{2}_{ij}+\phi^{2}_{ij}}$, where $\eta_{ij}$ is
pseudorapidity and $\phi_{ij}$ is azimuthal angle between the jet constituents
$i$ and $j$. Hence, we define a binned spectral function of angular distance
$R$ as follows,

$\displaystyle S_{2}(R;\Delta R)$ $\displaystyle=$
$\displaystyle\frac{1}{\Delta R}\sum_{\begin{subarray}{c}i,j\in\mathrm{jet}\\\
R_{ij}\in[R,R+\Delta R)\end{subarray}}p_{T,i}p_{T,j},$ (1)

where $\Delta R$ is a bin width, and $p_{T,i}$ and $p_{T,j}$ are transverse
momenta of jet constituents $i$ and $j$. In a continuum limit $\Delta
R\rightarrow 0$,111For more formal description, see Tkachov:1995kk . this
binned spectral function turns into,

$\displaystyle S_{2}(R)$ $\displaystyle=$ $\displaystyle\int
d\vec{R}_{1}d\vec{R}_{2}\;P_{T}(\vec{R}_{1})P_{T}(\vec{R}_{2})\cdot\delta(R-R_{12}),$
(2) $\displaystyle P_{T}(\vec{R})$ $\displaystyle=$
$\displaystyle\sum_{\begin{subarray}{c}i\in\mathrm{jet}\end{subarray}}p_{T,i}\delta(\vec{R}-\vec{R}_{i})$
(3)

where $d\vec{R}\,P_{T}(\vec{R})$ is a $p_{T}$ sum of constituents in a
neighborhood $d\vec{R}$ of $\vec{R}$, and $\delta(x)$ is the Dirac $\delta$
function. Integrating $S_{2}(R)$ over the bin range $[R,R+\Delta R)$ returns
the binned spectral function,

$S_{2}(R;\Delta R)=\frac{1}{\Delta R}\int_{R}^{R+\Delta R}dR\,S_{2}(R).$ (4)

The two-point correlation spectral function may be easily generalized to
three-point or multi-point correlation spectral function, analogous to the
energy correlation functions Larkoski:2013eya  and the energy flow polynomials
Komiske:2017aww . However, those generalizations are out of the scope of this
paper.

The spectral function is infrared and collinear (IRC) safe, namely invariant
under soft and collinear radiations. If the IRC safety is not satisfied,
Kinoshita-Lee-Nauenberg theorem Kinoshita:1962ur ; Lee:1964is  is not
applicable, and the resulting spectrum is hard to be estimated from
perturbative QCD calculations. Soft radiation does not change $S_{2}(R)$
because the soft radiation has a zero transverse momentum, which has no impact
on $P_{T}(\vec{R})$ as well as $S_{2}(R)$. Collinear radiation does not change
$S_{2}(R)$ because the products stay at the original $\vec{R}$ coordinate. The
momenta of the products are added together at $\vec{R}$; therefore,
$P_{T}(\vec{R})$ and $S_{2}(R)$ are invariant.

The IRC safety of the spectral function can be understood easily by explicit
examples such as a jet with a single constituent. Suppose that $a$ is the only
jet constituent. This jet has the only angular scale $R_{aa}=0$ and its binned
spectrum is given as follows,

$\displaystyle S_{2}(R;\Delta R)$ $\displaystyle=$
$\displaystyle\frac{1}{\Delta R}\begin{cases}p_{T,a}^{2}&\mathrm{if}\;R=0,\\\
0&\mathrm{if}\;R\neq 0.\end{cases}$ (5)

Invariance of this $S_{2}(R;\Delta R)$ under soft radiation is trivial, and
hence, we consider collinear splitting of $a$. Suppose that $b$ and $c$ are
the jet constituents from the collinear splitting of $a$. Then all the pairs
$bb$, $bc$, $cb$, and $cc$ have the same angular scale $R_{ij}=0$. If $b$ and
$c$ carry $z$ and $1-z$ fraction of the transverse momenta $p_{T,a}$,
$S_{2}(R;\Delta R)$ turns into

$\displaystyle S_{2}(R;\Delta R)$ $\displaystyle=$
$\displaystyle\frac{1}{\Delta
R}\begin{cases}z^{2}p_{T,a}^{2}+z(1-z)p_{T,a}^{2}+(1-z)zp_{T,a}^{2}+(1-z)^{2}p_{T,a}^{2}&\mathrm{if}\;R=0,\\\
0&\mathrm{if}\;R\neq 0,\end{cases}$ (6) $\displaystyle=$
$\displaystyle\frac{1}{\Delta R}\begin{cases}p_{T,a}^{2}&\mathrm{if}\;R=0,\\\
0&\mathrm{if}\;R\neq 0.\end{cases}$ (7)

Therefore, the binned spectrum is IRC safe. Note that summing the
autocorrelation term $p_{T,i}^{2}$ in Eq. 1 is necessary to achieve IRC safety
at $R=0$, because the crossing term $z(1-z)p_{T,a}^{2}$ after the splitting is
originated from the autocorrelation term $p_{T,a}^{2}$.

We show another example of a jet with two constituents $a$ and $b$, so that it
has a non-zero spectrum at $R=R_{ab}>0$. Now the binned spectrum has cross-
correlation terms at non-zero angular scale,

$\displaystyle S_{2}(R;\Delta R)$ $\displaystyle=$
$\displaystyle\frac{1}{\Delta
R}\begin{cases}p_{T,a}^{2}+p_{T,b}^{2}&\mathrm{if}\;R=0,\\\
0&\mathrm{otherwise},\end{cases}+\frac{1}{\Delta
R}\begin{cases}2p_{T,a}p_{T,b}&\mathrm{if}\;R_{ab}\in[R,R+\Delta R),\\\
0&\mathrm{otherwise}.\end{cases}$ (8)

Suppose that a collinear splitting of $b$ produces two partons with transverse
momenta $zp_{T,b}$ and $(1-z)p_{T,b}$ respectively. Then the binned spectrum
turns into

$\displaystyle S_{2}(R;\Delta R)$ $\displaystyle=$
$\displaystyle\frac{1}{\Delta
R}\begin{cases}p_{T,a}^{2}+\left[z^{2}p_{T,b}^{2}+z(1-z)p_{T,b}^{2}+(1-z)zp_{T,b}^{2}+(1-z)^{2}p_{T,b}^{2}\right]&\mathrm{if}\;R=0,\\\
0&\mathrm{otherwise,}\end{cases}$ (9) $\displaystyle+\frac{1}{\Delta
R}\begin{cases}2zp_{T,a}p_{T,b}+2(1-z)p_{T,a}p_{T,b}&\mathrm{if}\;R_{ab}\in[R,R+\Delta
R),\\\ 0&\mathrm{otherwise},\end{cases}$ $\displaystyle=$
$\displaystyle\frac{1}{\Delta
R}\begin{cases}p_{T,a}^{2}+p_{T,b}^{2}&\mathrm{if}\;R=0,\\\
0&\mathrm{otherwise},\end{cases}+\frac{1}{\Delta
R}\begin{cases}2p_{T,a}p_{T,b}&\mathrm{if}\;R_{ab}\in[R,R+\Delta R),\\\
0&\mathrm{otherwise}.\end{cases}$ (10)

Therefore, the binned spectrum is IRC safe. In general, this IRC safety is
achieved by the bilinear term of $p_{T,i}$ and $p_{T,j}$ in Eq. 1 like the
other jet substructure variables directly built from jet constituents
Gallicchio:2010sw ; Thaler:2010tr ; Gallicchio:2011xq ; Jankowiak:2011qa ;
Larkoski:2013eya ; Larkoski:2014gra ; Moult:2016cvt ; Komiske:2017aww .

The IRC safety of the spectral function $S_{2}(R)$ is also understood in the
context of $C$-correlators Tkachov:1995kk ; Komiske:2017aww . The $S_{2}(R)$
is a special case of $C$-correlators with an unbounded non-smooth angular
weighting function $f_{2}(\hat{p}_{1},\hat{p}_{2})=\delta(R-R_{12})$. If we
replace the Dirac $\delta$ function to a bounded smoooth function, for
example, $\delta(x)\rightarrow(|a|\sqrt{\pi})^{-1}e^{-(x/a)^{2}}$, the Taylor
expansion of $\delta(x)$ transforms $S_{2}(R)$ into a series of IRC safe
energy flow polynomials Komiske:2017aww  with two vertices. The series
converges to $S_{2}(R)$ in the limit $a\rightarrow 0$, and the IRC safety of
the spectral function is understood asymptotically.

Note that the spectral function $S_{2}(R)$ is a basis of bilinear
$C$-correlators $F_{2}$ Tkachov:1995kk  with an angular weighting function
$f_{2}(R_{ij})$ of the angular distance $R_{ij}$,

$F_{2}=\int
d\vec{R}_{1}d\vec{R}_{2}\;P_{T}(\vec{R}_{1})P_{T}(\vec{R}_{2})\cdot
f_{2}(R_{12})=\int_{0}^{\infty}dR\;S_{2}(R)\,f_{2}(R).$ (11)

For example, the zeroth and the second moment of $S_{2}(R)$ are the one-point
and the two-point energy correlation functions Larkoski:2013eya , which are
approximately the transverse momentum $p_{T,\mathrm{jet}}$ and the mass
$m_{\mathrm{jet}}$ of the jet respectively,

$\displaystyle\int_{0}^{\infty}dR\,S_{2}(R)\,\hphantom{R^{2}}$
$\displaystyle=$
$\displaystyle\makebox[0.0pt][l]{$\displaystyle\left(\sum_{i\in\mathrm{jet}}p_{T,i}\right)^{2}$}\phantom{\sum_{i,j\in\mathrm{jet}}p_{T,i}p_{T,j}R_{ij}^{2}}\approx
p_{T,\mathrm{jet}}^{2},$ (12)
$\displaystyle\int_{0}^{\infty}dR\,S_{2}(R)\,R^{2}$ $\displaystyle=$
$\displaystyle\sum_{i,j\in\mathrm{jet}}p_{T,i}p_{T,j}R_{ij}^{2}\approx
2m_{\mathrm{jet}}^{2}.$ (13)

These integrals help to interpret $S_{2}(R)$. The spectral densities
$S_{2}(R)$ and $S_{2}(R)\,R^{2}$ measure contribution to
$p_{T,\mathrm{jet}}^{2}$ and $m_{\mathrm{jet}}^{2}$ from the pairs of jet
constituents at the angular scale $R$, respectively.

We perform a Monte Carlo study of Higgs jets vs QCD jets classification using
the spectrum $S_{2}(R)$. We generate $pp\rightarrow Zj$ events and
$pp\rightarrow Zh$ events followed by $h\rightarrow b\bar{b}$, and use the
leading jet of the events as training sample of one prong and two prong jets
respectively. Each sample is generated at the leading order in QCD using
MadGraph5_aMC@NLO 2.6.1 Alwall:2014hca  with parton distribution function
(PDF) set NNPDF 2.3 LO at $\alpha_{S}(m_{Z})=0.130$ Ball:2012cx . $Z$ bosons
are forced to decay into neutrinos so that they are not detected. The events
are showered and hadronized by Pythia 8.226 Sjostrand:2014zea  with Monash
tune Skands:2014pea . We include effects of underlying events such as multi-
parton interaction and beam remnant treatment but we do not take pile-ups into
account.

Finally, we simulate detector response using Delphes 3.3.3 deFavereau:2013fsa
with their default ATLAS configuration. Jets are reconstucted from calorimeter
towers using anti-$k_{T}$ algorithm Cacciari:2008gp  with a jet radius
parameter $R_{\mathrm{jet}}=1.0$ implemented in fastjet 3.3.0 Cacciari:2011ma
; Cacciari:2005hq . We study substructures of the leading jets with
$p_{T,\mathrm{jet}}\in[300,400]$ GeV and $m_{\mathrm{jet}}\in[100,150]$. The
characteristic angle between two $b$ quarks from the boosted Higgs boson is
then $R_{b\bar{b}}\gtrsim 2m_{h}/p_{T,\mathrm{jet}}\approx 0.83$. Hence, the
choice of the jet radius is enough to catch parton showers from those two $b$
quarks efficiently. For $Zh$ events, we additionally require that at least
there is one $b$ parton whose momentum in matrix element level is located
within $R\leq 1$ from the jet center to remove events with hard initial state
radiations. After these preselections, we have 256691 $Zh$ and $Zj$ events for
training ANN. When we test ANN, we use a testing sample generated
independently to the training sample.

Figure 2:  The pixelated jet image (left), two-point spectrum $S_{2}(R;0.1)$
(center), and the trimmed spectrum $S_{2}(R;0.1)$ (right) of a typical Higgs
jet (top) and a typical quark jet (bottom). The sum of the bins of each
spectrum is normalized to 1. The red reversed triangles, blue triangles, and
green plus symbols in the pixelated jet images indicate direction of light
quarks, $b$ quarks and gluons obtained from the matrix element including order
$\alpha_{S}$ radiations. We show the Higgs-like probabilities of the jets,
$p_{h}(Y_{X})$ of an ANN model with various input $X$, defined in Sec. 4, in
the central plots.

We show typical pixelated jet images and spectra $S_{2}(R;0.1)$ of a Higgs
boson jet and a quark jet in Figure 2. For the boosted Higgs boson, the
$S_{2}(R;0.1)$ distribution has two prominent peaks at $R=0$ and
$R_{b\bar{b}}$ which correspond to autocorrelation and cross-correlation of
two $b$ partons respectively,

$\displaystyle p_{T}(\vec{R})$ $\displaystyle=$ $\displaystyle
p_{T,b}\cdot\delta(\vec{R}-\vec{R}_{b})+p_{T,\bar{b}}\cdot\delta(\vec{R}-\vec{R}_{\bar{b}}),$
(14) $\displaystyle S_{2}(R)$ $\displaystyle=$
$\displaystyle\left(p_{T,b}^{2}+p_{T,\bar{b}}^{2}\right)\cdot\delta(R)+2p_{T,b}p_{T,\bar{b}}\cdot\delta(R-R_{b\bar{b}}).$

Because the Higgs boson decays spherically, the two peaks tend to have
comparable intensities, $p_{T,b}^{2}+p_{T,\bar{b}}^{2}\simeq
2p_{T,b}p_{T,\bar{b}}$, as shown in Figure 2. Parton shower develops along the
partons. Each splitting of a parton is characterized by the angle betwen
daughter partons and their momenta. The spectral density $S_{2}(R)$ sums up
those individual splittings. The peaks in $R$ is smeared by the parton shower
and hadronization, but it does not change the initial radiation pattern. In
Figure 2, we also show the quark jet spectrum. The spectrum does not have
distinctive peaks significantly. Instead, it is gradually decreasing as $R$
increases.

Note that the spectral function $S_{2}(R)$ is not completely independent to
the angular structure function $\Delta\mathcal{G}(R)$ in Jankowiak:2011qa . We
have a relation between $S_{2}(R)$ and $\Delta\mathcal{G}(R)$ as follows,

$\displaystyle\mathcal{G}(R)$ $\displaystyle=$
$\displaystyle\frac{\int_{0}^{R}dR^{\prime}\,S_{2}(R^{\prime})\,R^{\prime
2}}{\int_{0}^{\infty}dR^{\prime}\,S_{2}(R^{\prime})\,R^{\prime 2}},$ (15)
$\displaystyle\Delta\mathcal{G}(R)$ $\displaystyle=$
$\displaystyle\frac{d\log\mathcal{G}(R)}{d\log R}=\frac{R\cdot
S_{2}(R)\,R^{2}}{\int_{0}^{R}dR^{\prime}\,S_{2}(R^{\prime})\,R^{\prime 2}}.$
(16)

This $\Delta\mathcal{G}(R)$ is a Higuchi’s fractal dimension HIGUCHI1988277
of $\mathcal{G}(R)$ which measures irregularity of $\mathcal{G}(R)$ over $R$.
QCD jets have a uniform $\Delta\mathcal{G}(R)$ distribution on average because
of approximate scale invariance of QCD Jankowiak:2011qa ; Jankowiak:2012na ;
Larkoski:2012eh . On the other hand, $\mathcal{G}(R)$ of the multi-prong jet
shows sharp peaks at some angular scales Jankowiak:2011qa . Hence, a number of
peaks and peak heights in $\Delta\mathcal{G}(R)$ can be used as a classifier
of the jets.

In the following sections, we will show two Higgs jet classifiers using the
spectrum $S_{2}(R)$. In Sec. 3, we show a cut-based analysis using $S_{2}(R)$.
In Sec. 4, we introduce neural network classifiers using $S_{2}(R)$.

##  3 Cut-Based Spectral Analysis

Before presenting our neural network classifier for the Higgs jets, we show a
cut-based analysis using $S_{2}(R)$ to get a quick insight on the spectrum.
Namely, we introduce a ratio of the activity on the characteristic angular
scale $\hat{R}_{b\bar{b}}$ of a Higgs jet and that of the surrounding angular
scales,

$R_{S_{2}}=\frac{\int_{a\hat{R}_{b\bar{b}}}^{\min[a^{\prime}\hat{R}_{b\bar{b}},R_{\mathrm{jet}}]}dR\,S_{2}(R)}{\int_{0}^{a\hat{R}_{b\bar{b}}}dR\,S_{2}(R)+\int_{\min[a^{\prime}\hat{R}_{b\bar{b}},R_{\mathrm{jet}}]}^{\infty}dR\,S_{2}(R)},$
(17)

where $\hat{R}_{b\bar{b}}=2\cdot 125\,\mathrm{GeV}/{p_{T,\mathrm{jet}}}$,
$a=0.75$, and $a^{\prime}=1.25$. For the upper boundary, we take the minimum
between the boundary $a^{\prime}\hat{R}_{b\bar{b}}$ and the jet radius
$R_{\mathrm{jet}}$. The angular scale beyond $R_{\mathrm{jet}}$ is mainly
covered by large angle radiations rather than soft and collinear radiations
from the $b$ partons. The $b$ partons from the Higgs boson do not emit parton
shower in large angle because the whole system is color neutral. Therefore, we
restrict the upper bound of the integral in the numerator up to
$R_{\mathrm{jet}}$ while include the integral beyond the upper bound to the
denominator.

The probability that a QCD jet emits another hard parton at
$\hat{R}_{b\bar{b}}$ is small; hence, this ratio works as a classifier. Even
if a QCD jet accidentally has substructure from parton splittings, the ratio
of the momenta of two partons are different from that of Higgs jets. It is
typically small,

$R_{S_{2}}=\frac{2p_{T,i}p_{T,j}}{p_{T,i}^{2}+p_{T,j}^{2}}=\frac{2z(1-z)}{1-2z(1-z)},$
(18)

where $z$ is a fraction of momemtum of the one of the partons, i.e.,
$p_{T,i}=zp_{T,q/g}$ and $p_{T,j}=(1-z)p_{T,q/g}$. The fraction $z$ tends to
be much smaller than 1 and suppresses the numerator. In contrast, the phase-
space of the Higgs boson decay is symmetric; hence, the two terms in Eq. 14
are in a similar order and $R_{S_{2}}$ is approximately 1. Therefore, the
typical Higgs jets and QCD jets have different $R_{S_{2}}$ values as in Figure
3 and the ratio works as a classifier.

Figure 3:  Histogram of $R_{S_{2}}$ of the Higgs jets and the QCD jets in the
training sample. The bin height is the probability distribution function (PDF)
of the given bin.

To compare the performance of $R_{S_{2}}$ with other observables, we consider
an observable $D_{2}$ Larkoski:2014gra  for the two-prong substurcture
identification. The variable $D_{2}$ is defined by a ratio of two-point and
three-point energy correlation functions $e_{2}^{\beta}$ and $e_{3}^{\beta}$
as follows,

$\displaystyle e_{2}^{\beta}$ $\displaystyle=$
$\displaystyle\frac{1}{p_{T,\mathrm{jet}}^{2}}\sum_{\begin{subarray}{c}i,j\in\mathrm{jet}\\\
i<j\end{subarray}}p_{T,i}p_{T,j}R_{ij}^{\beta},$ (19) $\displaystyle
e_{3}^{\beta}$ $\displaystyle=$
$\displaystyle\frac{1}{p_{T,\mathrm{jet}}^{3}}\sum_{\begin{subarray}{c}i,j,k\in\mathrm{jet}\\\
i<j<k\end{subarray}}p_{T,i}p_{T,j}p_{T,k}R_{ij}^{\beta}R_{jk}^{\beta}R_{ki}^{\beta},$
(20) $\displaystyle D_{2}^{\beta}$ $\displaystyle=$
$\displaystyle\frac{e_{3}^{\beta}}{(e_{2}^{\beta})^{3}},$ (21)

where the summations of $e_{2}^{\beta}$ and $e_{3}^{\beta}$ run over all jet
constituents. We consider angular exponents $\beta=0,5$, $1$, $2$ and $4$ for
further discussion, but we focus on $\beta=2$ when we discuss analyses with a
single $D_{2}$. The Higgs jets have a small $D_{2}$ value because
$e_{3}^{\beta}$ is suppressed by collinear and soft radiations while
$e_{2}^{\beta}$ is large because the pairs of jet constituents with
$R_{ij}\sim R_{b\bar{b}}$ dominate. The QCD jets do not have such suppression,
and hence, the Higgs jets and the QCD jets cover different regions of $D_{2}$.

Since $R_{S_{2}}$ and $D_{2}^{\beta=2}$ are sensitive to the two-prong
substructure, moderate correlation between them is expected. We show
histograms of $(D_{2}^{\beta=2},R_{S_{2}})$ of the training sample, in Figure
4. The Higgs jets prefer small $D_{2}$ and large $R_{S_{2}}$, while the QCD
jets prefer large $D_{2}$ and small $R_{S_{2}}$. Hence, there is correlation.

Figure 4:  Histograms of $(D_{2}^{\beta=2},R_{S_{2}})$ of the Higgs jets and
the QCD jets in the training sample.  Figure 5:  ROC curves of $D_{2}$ and
$R_{S_{2}}$ for classifying Higgs jets and QCD jets.

We show the receiver operating characteristic (ROC) curves of $R_{S_{2}}$ and
$D_{2}^{\beta=2}$ in Figure 5 to compare the classification performance. Since
$R_{S_{2}}$ and $D_{2}^{\beta=2}$ are correlated, they show similar
performance. At the Higgs tagging efficiency 0.2 (0.4), the QCD jet mistag
rate is 0.0506 (0.135) for $D_{2}^{\beta=2}$ and 0.0525 (0.145) for
$R_{S_{2}}$.

Note that $R_{S_{2}}$ is not a unique classifier. However, building the
sophisticated variable from $S_{2}(R)$ is not the scope of this paper.
Instead, we will let the neural network build an optimized variable from
$S_{2}(R)$ for the Higgs jet classification.

##  4 Spectral Analysis with Artificial Neural Networks

We now feed the event-by-event binned spectra $S_{2}(R;\Delta R)$ to our ANN
and build a neural network classifier between the Higgs jets and QCD jets.
First, we prepare an equal number of Higgs jets and QCD jets to avoid
overfitting from unbalanced data. We use TFLearn tflearn2016  with backend
TensorFlow tensorflow2015-whitepaper  for the ANN analysis. An input set we
consider includes $S_{2}(R;0.1)$ up to angular scale $R<2$,

$\\{x_{i}\\}_{S_{2}}=\\{p_{T,\mathrm{jet}},m_{\mathrm{jet}},S_{2}(0;0.1),\cdots,S_{2}(1.9;0.1)\\}.$
(22)

Note that $R=2$ is the diameter of our jet definition. All the input data
$\\{x_{i}\\}$ are standardized, i.e.,
$x_{i}\rightarrow(x_{i}-\bar{x}_{i})/\sigma(x_{i})$, where $\bar{x}_{i}$ and
$\sigma(x_{i})$ are the mean and the standard deviation of $x_{i}$ of the
whole training sample including both Higgs jets and QCD jets. The network is
configured with four hidden layers having $(400,300,200,100)$ nodes with the
ReLU activation functions, $f(x)=\max(0,x)$, and an output layer with two
nodes having the softmax activations which map inputs to a Higgs-like score
$y_{S_{2}}$. To avoid overtraining, we insert dropout layers
JMLR:v15:srivastava14a  with rate 20% between each hidden layer. The network
is trained by Adam optimizer ADAM  with learning rate 0.001, $\beta_{1}=0.99$
and $\beta_{2}=0.999$ minimizing a categorical cross-entropy as a loss
function,

$L=\frac{1}{N_{\mathrm{events}}}\sum_{\mathrm{events}}\begin{cases}-\log[y_{S_{2}}(\\{x_{i}\\})]&\mathrm{Higgs\,jets},\\\
-\log[1-y_{S_{2}}(\\{x_{i}\\})]&\mathrm{QCD\,jets},\end{cases}$ (23)

where $N_{\mathrm{events}}$ is the number of events in the training event set.
We call this network as $\mathcal{N}_{S_{2}}$. In the trained network, Higgs
jets have scores near 1, while QCD jets have scores near 0. We validate
$\mathcal{N}_{S_{2}}$ using the testing samples.

We compare the performance of $\mathcal{N}_{S_{2}}$ with that of a network
trained with $D_{2}$. We prepare another neural network which maps following
inputs to the Higgs-like score $y_{D_{2}}$,

$\\{x_{i}\\}_{D_{2}}=\\{p_{T,\mathrm{jet}},m_{\mathrm{jet}},D_{2}^{\beta=2}\\}.$
(24)

Again, the input data is normalized to $[0,1]$, i.e.,
$x_{i}\rightarrow(x_{i}-\min x_{i})/(\max x_{i}-\min x_{i})$, where $\max
x_{i}$ and $\min x_{i}$ are the maximum and the minimum of $x_{i}$ in the
training sample respectively. We use smaller hidden layers $(100,100)$ ReLU
nodes because the number of inputs is smaller. The other ANN setups are
identical to the $S_{2}(R)$ analysis. We call this network as
$\mathcal{N}_{D_{2}}$.

In our approach, we do not use individual pixels as inputs; therefore, the
binned spectrum is affected by both soft and hard calorimeter activities. To
make ANN learn a hierarchy between soft and hard radiations, variables after
jet trimming Krohn:2009th  are useful. To obtain trimmed quantities, we first
reconstruct $k_{T}$ subjets Catani:1993hr ; Ellis:1993tq  with
$R_{\mathrm{sub}}=0.2$ from constituents of the jet and remove subjets having
transverse momentum $p_{T,\mathrm{subjet}}<f_{\mathrm{cut}}\cdot
p_{T,\mathrm{jet}}$, where $f_{\mathrm{cut}}=0.05$. In the right panel of
Figure 2, we show typical two-point correlation spectra of the trimmed jet
constituents, $S_{2,tr}(R)$, of a Higgs jet and a QCD jet. The spectra
$S_{2}(R)$ before trimming are shown in the central panel. The two-prong
substructure of a Higgs jet is consisted by hard activities, and the double
peak structure appears both in $S_{2}(R)$ and $S_{2,tr}(R)$. On the other
hand, the spectrum of a QCD jet is significantly changed after trimming, which
means that soft activities dominate the spectrum $S_{2}(R)$. This shows a
difference between $S_{2}(R)$ and $S_{2,tr}(R)$ contains useful information
for the classification.

We then prepare three networks $\mathcal{N}_{D_{2}+tr}$,
$\mathcal{N}_{D_{2}^{\beta}+tr}$ and $\mathcal{N}_{S_{2}+tr}$ taking inputs
$\\{x_{i}\\}_{D_{2}+tr}$, $\\{x_{i}\\}_{D_{2}^{\beta}+tr}$ and
$\\{x_{i}\\}_{S_{2}+tr}$ respectively,

$\displaystyle\\{x_{i}\\}_{D_{2}+tr}$ $\displaystyle=$
$\displaystyle\\{x_{i}\\}_{D_{2}}\cup\\{p_{T,\mathrm{jet},tr},m_{\mathrm{jet},tr},D_{2,tr}^{\beta=2}\\},$
(25) $\displaystyle\\{x_{i}\\}_{D_{2}^{\beta}+tr}$ $\displaystyle=$
$\displaystyle\\{x_{i}\\}_{D_{2}+tr}\cup\\{D_{2}^{\beta=0.5},D_{2}^{\beta=1},D_{2}^{\beta=2},D_{2}^{\beta=4},D_{2,tr}^{\beta=0.5},D_{2,tr}^{\beta=1},D_{2,tr}^{\beta=2},D_{2,tr}^{\beta=4}\\},$
(26) $\displaystyle\\{x_{i}\\}_{S_{2}+tr}$ $\displaystyle=$
$\displaystyle\\{x_{i}\\}_{S_{2}}\cup\\{p_{T,\mathrm{jet},tr},m_{\mathrm{jet},tr},S_{2,tr}(0.0;0.1),\cdots,S_{2,tr}(1.9;0.1)\\},$
(27)

where the variables calculated after trimming have a subscript $tr$. The other
ANN setups are the same as that of $\mathcal{N}_{D_{2}}$ and
$\mathcal{N}_{S_{2}}$. The networks $\mathcal{N}_{D_{2}+tr}$,
$\mathcal{N}_{D_{2}^{\beta}+tr}$ and $\mathcal{N}_{S_{2}+tr}$ give us Higgs-
like scores $y_{D_{2}+tr}$, $y_{D_{2}^{\beta}+tr}$ and $y_{S_{2}+tr}$. Note
that $\mathcal{N}_{D_{2}^{\beta}+tr}$ takes the $D_{2}^{\beta}$ with various
angular exponents $\beta=0.5$, $1$, $2$, and $4$ as inputs.

Figure 6:  ROC curves of ANN classifiers for Higgs jets and QCD jets with
inputs Eq. 22, Eq. 24, Eq. 25, Eq. 26, and Eq. 27. We overlay the cut-based
ROC curves of $R_{S_{2}}$ shown in Figure 5 as a gray dashed line for a
reference.

To compare the information contained in $D_{2}$ and $S_{2}(R)$, we show the
ROC curves of our ANN analyses with $\mathcal{N}_{D_{2}+tr}$,
$\mathcal{N}_{D_{2}+tr}$, $\mathcal{N}_{D_{2}^{\beta}+tr}$,
$\mathcal{N}_{S_{2}}$, and $\mathcal{N}_{S_{2}+tr}$ in Figure 6. The ROC
curves show that the ANN with $S_{2}(R;\Delta R)$ rejects more QCD jets for a
fixed Higgs jet efficiency. At the Higgs tagging efficiency 0.4 (0.2), the QCD
jet mistag rate of $\mathcal{N}_{S_{2}+tr}$, which shows the best performance
among the classifiers, is 0.0246 (0.0807). The mistag rate is reduced by 51.4%
(40.2%) compared to that of the cut-based analysis of $R_{S_{2}}$ while the
mistag rate is still reduced by 17.1% (10.5%) compared to that of
$\mathcal{N}_{D_{2}^{\beta}+tr}$. This is expected because
$\mathcal{N}_{S_{2}+tr}$ uses two-point energy correlation from
$S_{2}(R;\Delta R)$ and infers three-point energy correlation from
correlations between different angular scales. For example, $S_{2}(R;\Delta
R)$ of a three-prong jet having angular scales $R_{1}$, $R_{2}$, and $R_{3}$
has three peaks away from $R=0$. The intensity of each peak gives a three-
point energy correlation function, $e_{3}^{\beta}\approx
p_{T,\mathrm{jet}}^{-1}\cdot\sqrt{(\Delta R)^{3}S_{2}(R_{1};\Delta
R)S_{2}(R_{2};\Delta R)S_{2}(R_{3};\Delta
R)}R_{1}^{\beta}R_{2}^{\beta}R_{3}^{\beta}$. Hence, $\mathcal{N}_{S_{2}}$ and
$\mathcal{N}_{S_{2}+tr}$ have better discrimation power than
$\mathcal{N}_{D_{2}+tr}$, $\mathcal{N}_{D_{2}+tr}$ and
$\mathcal{N}_{D_{2}^{\beta}+tr}$. Also, adding trimmed observables allows ANN
to learn hard and soft substructures separately. Hence, the ANN solve
degeneracy in the variables before trimming and reject QCD jets better. One
interesting feature is that $\mathcal{N}_{D_{2}^{\beta}+tr}$ suppress QCD jets
about as equal as $\mathcal{N}_{S_{2}}$ in the region of high tagging
efficiency, but the difference between $\mathcal{N}_{D_{2}^{\beta}+tr}$ and
$\mathcal{N}_{S_{2}}$ is large in the region of low tagging efficiency. This
gap in the ROC curves implies that $S_{2}(R)$ has additional information,
which we will be discussed in the later part.

Note that the ANN inputs include $p_{T,\mathrm{jet}}$; as a result, the ANN
partially use the jet $p_{T}$ distribution of Higgs jets and QCD jets for the
classification. To remove the $p_{T}$ dependence, we may resample the training
set whose signal and background $p_{T}$ distribution is same, or use a
weighted loss function for an imbalanced training set, or train adversarial
neural networks Louppe:2016ylz ; Shimmin:2017mfk . For the comparison of the
ANN’s with different inputs, we do not need to use these techniques because
$p_{T,\mathrm{jet}}$ is a common input for all the ANN analyses. We have to
pay attention to the bias from $p_{T,\mathrm{jet}}$ distribution of signal and
background when this analysis is applied to the experimental data.

|
---|---
|
|
Figure 7:  The distributions of Higgs jets and QCD jets in the Higgs-like
probability $p_{h}(Y_{D_{2}+tr})$ and inputs in $\\{x_{i}\\}_{D_{2}+tr}$
planes. For each pair of figures, a left panel is the distribution of Higgs
jets and a right panel is the distribution of QCD jets.

The network $\mathcal{N}_{D_{2}+tr}$ uses a small number of inputs; therefore,
it is easy to check the ANN outputs of the input parameters
$m_{\mathrm{jet}}$, $p_{T,\mathrm{jet}}$, and $D_{2}$. We show distributions
of the events in the Higgs-like probability $p_{h}(Y_{D_{2}+tr})$ and one of
the inputs $\\{x_{i}\\}_{D_{2}+tr}$ plane in Figure 7. The Higgs-like
probability of $\mathcal{N}_{X}$, $p_{h}(Y_{X})$, is defined by the
probability of getting a score $y_{X}$ less than $Y_{X}$ in the Higgs jet
samples,

$p_{h}(Y_{X})=\mathrm{Pr}(y_{X}<Y_{X}\,|\,\mathrm{Higgs}\;\mathrm{jets}),$
(28)

where $\mathrm{Pr}(C|H)$ represents the conditional probability of $C$ given
$H$. A large $p_{h}(Y_{X})$ means that the given jet is more Higgs-like in
$\mathcal{N}_{X}$. Figure 7 shows that $\mathcal{N}_{D_{2}+tr}$ tries to
select jets having $m_{\mathrm{jet}}$ around 125 GeV up to energy loss, and
small $D_{2}$ for capturing two-prong jets. Note that the
$\mathcal{N}_{D_{2}+tr}$ is trained by the QCD jets and Higgs jets with
$m_{h}=125\;\mathrm{GeV}$. The $m_{\mathrm{jet}}$ and trimmed
$m_{\mathrm{jet}}$ of Higgs jet is tend to be higher (lower) than the mass of
Higgs boson and a jet with $m_{\mathrm{jet}}\sim 120\,\mathrm{GeV}$ is
regarded most likely as a Higgs jet. These shifts of the mass from the input
Higgs mass are due to contamination of other activities, or large angle
radiations from $b$ jets. The mistagged QCD jets with high
$p_{h}(Y_{D_{2}+tr})$ cover similar phase-space, where $D_{2,tr}$ is small,
$m_{\mathrm{jet},tr}\sim 115\;\mathrm{GeV}$, and $m_{\mathrm{jet}}\sim
125\;\mathrm{GeV}$. Meanwhile, the $m_{\mathrm{jet}}$ and
$m_{\mathrm{jet},tr}$ distributions of QCD jets in $p_{h}(Y_{D_{2}+tr})<0.2$
and $p_{h}(Y_{D_{2}+tr})<0.3$ in Figure 7 show relatively low probability for
$m_{\mathrm{jet}}\sim 130\;\mathrm{GeV}$ and
$m_{\mathrm{jet},tr}>100\;\mathrm{GeV}$, repsectively. This means,
$\mathcal{N}_{D_{2}+tr}$ labels events in the particular mass window as Higgs
jet; hence, $\mathcal{N}_{D_{2}+tr}$ rejects QCD jets in a similar fashion
that a cut-based approach rejects QCD events.

Figure 8:  The distributions of Higgs-like probability in
$p_{h}(Y_{S_{2}+tr})$ and $p_{h}(Y_{D_{2}+tr})$ planes. The numbers in upper
left and lower right area represent the fraction of events in the area.

We compare the Higgs-like probabilities in $\mathcal{N}_{D_{2}+tr}$ and
$\mathcal{N}_{S_{2}+tr}$ in Figure 8 to find out the origin of improvement.
The events are widely spreading around the line
$p_{h}(Y_{D_{2}+tr})=p_{h}(Y_{S_{2}+tr})$, which means that those two analyses
have different selection criteria. To quantify the residual of the
anticorrelation of $\mathcal{N}_{D_{2}+tr}$ and $\mathcal{N}_{S_{2}+tr}$, we
show the fractions of the events in the upper triangular region
$p_{h}(Y_{D_{2}+tr})>p_{h}(Y_{S_{2}+tr})$ and the lower triangular region
$p_{h}(Y_{D_{2}+tr})<p_{h}(Y_{S_{2}+tr})$. For Higgs jets, the lower
triangular region contains more events compared with the upper triangular
region, 51.3% of the total events. For QCD jets, the lower triangular region
contains less events, 43.1%. Hence, $\mathcal{N}_{S_{2}+tr}$ improves signal
and background ratio $S/B$ from $\mathcal{N}_{D_{2}+tr}$.

Figure 9:  The pixelated jet image (left), and two-point spectrum
$S_{2}(R;0.1)$ (right) of a Higgs jet. The event is Higgs-like in the neural
networks with $S_{2}(R)$, $\mathcal{N}_{S_{2}}$ and $\mathcal{N}_{S_{2}+tr}$,
but it is not in those with $D_{2}$, $\mathcal{N}_{D_{2}}$ and
$\mathcal{N}_{D_{2}+tr}$. The sum of the bins of the spectrum is normalized to
1. The blue triangles and green plus symbols in the pixelated jet images
indicate direction of $b$ quarks and gluons obtained from the matrix element
including order $\alpha_{S}$ radiations. We show the Higgs-like probability
$p_{h}(Y_{X})$ in an ANN model $\mathcal{N}_{X}$ defined by Eq. 28.  Figure
10:  Distributions of Higgs jets (left) and QCD jets (right) in $D_{2}$ and
$D_{2,tr}$, classified as Higgs jets in $\mathcal{N}_{S_{2}+tr}$ but QCD jets
in $\mathcal{N}_{D_{2}+tr}$. We select events with Higgs-like probabilities
$p_{h}(Y_{D_{2,tr}})<30\%$ and $p_{h}(Y_{S_{2,tr}})>70\%$.

To figure out how $\mathcal{N}_{S_{2}+tr}$ accepts more Higgs jets while
rejecting more QCD jets compared to $\mathcal{N}_{D_{2}+tr}$, we will show
three examples of the events located on the off-diagonal regions in Figure 8.
We show a Higgs jet in Figure 9, which is Higgs-like in
$\mathcal{N}_{S_{2}+tr}$ but regarded as a QCD jet in
$\mathcal{N}_{D_{2}+tr}$, $p_{h}(Y_{D_{2}+tr})=19.6\%$ and
$p_{h}(Y_{S_{2}+tr})=80.7\%$. This jet has a moderate wide-angle radiation on
top of two-prong substructure which increases $D_{2}$ significantly. Remind
that a Higgs jet originates from a color singlet particle while a QCD jet
originates from a colored parton. Such wide-angle radiation is easily
generated from a colored parton compared to a color singlet particle.
$\mathcal{N}_{D_{2}+tr}$ is distracted by a large $D_{2}$ and assigns this jet
as a QCD jet even though the jet has small trimmed $D_{2}$.
$\mathcal{N}_{S_{2}+tr}$ must have determined the jet as Higgs-like from the
information of microscopic radiation patterns in $S_{2}(R)$ which shows a
clear double peak structure. Figure 10 shows $D_{2}$ and $D_{2,tr}$
distributions in events having $p_{h}(Y_{D_{2,tr}})<30\%$ and
$p_{h}(Y_{S_{2,tr}})>70\%$. We can see that some, but not all, events with
large $D_{2}$ but small $D_{2,tr}$ fall into this region.

Figure 11:  The Pixelated jet image (left), and two-point spectrum
$S_{2}(R;0.1)$ (right) of a Higgs jet. The event is Higgs-like in the neural
networks with $D_{2}$, $\mathcal{N}_{D_{2}}$ and $\mathcal{N}_{D_{2}+tr}$, but
it is not in those with $S_{2}(R)$, $\mathcal{N}_{S_{2}}$ and
$\mathcal{N}_{S_{2}+tr}$. The sum of the bins of the spectrum is normalized to
1. The red triangles and green plus symbols in the pixelated jet images
indicate direction of light quarks and gluons obtained from the matrix element
including order $\alpha_{S}$ radiations. We show the Higgs-like probability
$p_{h}(Y_{X})$ in an ANN model $\mathcal{N}_{X}$ defined by Eq. 28.

The second example in Figure 11 is a jet classified as a Higgs jet in
$\mathcal{N}_{D_{2}+tr}$ but categorized as a QCD jet in
$\mathcal{N}_{S_{2}+tr}$. This jet has evident two-prong substructure, and
hence, $\mathcal{N}_{D_{2}+tr}$ classifies this jet as a Higgs jet. However,
the two subjets are asymmetric in $p_{T}$. Such events appear frequently in
QCD jet samples. We did not give subjet momenta to $\mathcal{N}_{D_{2}+tr}$,
and the ANN classify the jet as a Higgs jet. In contrast, $S_{2}(R)$ knows the
$p_{T}$ asymmetry by comparing the peak intensity; see Eq. 14. Hence,
$\mathcal{N}_{S_{2}+tr}$ avoids these $p_{T}$ asymmetric events which appear
often among QCD jets while $\mathcal{N}_{S_{2}+tr}$ finds the cut on the
subjet $p_{T}$ from the training samples. In the mass-drop tagger
Butterworth:2008iy , the events with asymmetric $p_{T}$ subjets are removed by
a cut.

Figure 12:  The Pixelated jet image (left), two-point spectrum $S_{2}(R;0.1)$
(center), and trimmed spectrum $S_{2}(R;0.1)$ (right) of a Higgs jet which is
Higgs-like only in the $\mathcal{N}_{S_{2}+tr}$. The sum of the bins of each
spectrum is normalized to 1. The blue triangles and green plus symbols in the
pixelated jet images indicate direction of $b$ quarks and gluons obtained from
the matrix element including order $\alpha_{S}$ radiations. We show the Higgs-
like probability $p_{h}(Y_{X})$ in an ANN model $\mathcal{N}_{X}$ defined by
Eq. 28.

The third example in Figure 12 is the case where only
$\mathcal{N}_{S_{2}+tr}$, which takes into account trimmed $S_{2}(R)$,
classifies the jet as a Higgs jet. This jet has a two-prong substructure but
deeply buried in radiations compared to Figure 11. As a result, $D_{2}$ is
large, and $S_{2}(R)$ spectrum is falling toward high angular scale $R$ as in
Figure 2. Trimming helps $\mathcal{N}_{S_{2}+tr}$ this time because
$\mathcal{N}_{S_{2}+tr}$ recognizes hard and soft substructure separately by
comparing $S_{2}(R)$ and $S_{2,tr}(R)$. Trimming does not change the tail of
$S_{2}(R)$ distribution, which means the substructure at large $R$ is hard.

##  5 Discussion and Conclusion

In this paper, we have introduced a spectral analysis of jet substructure with
the artificial neural networks (ANN). Unlike the other ANN approach, our
algorithm use the spectral function $S_{2}(R)$ constructed from $p_{T}$ and
$R$ of the pair of particles in the jet. The spectrum $S_{2}(R)$ is useful in
describing substructures with large angular separation by relatively small
inputs. ANN can learn non-local correlations in jets from the spectrum. To
show this, we have constructed ANN from $S_{2}(R)$, $\mathcal{N}_{S_{2}}$, and
compare it with ANN from $D_{2}$, $\mathcal{N}_{D_{2}}$. We have shown that
$\mathcal{N}_{S_{2}}$ discriminates between boosted Higgs jets and QCD jets
with better performance compared to $\mathcal{N}_{D_{2}}$. Introducing
trimming to $S_{2}(R)$ further helps ANN separate hard and soft substructures,
and the ANN with trimmed observable outperforms the ANN without trimming. The
improvement comes from the better handling of the cases with radiation from
$b$ parton or with contamination of other hadronic activities.

The improvement we observe is not large, because $D_{2}$ catches the two-prong
substructure of the Higgs jet efficiently, but ANN analysis with $S_{2}(R)$
has much wider application. One of the merits of $\mathcal{N}_{S_{2}}$ and
$\mathcal{N}_{S_{2}+tr}$ is that the analyses automatically take care of
radiations from the $b$ jet. Note that the existence of radiation has to be
taken care of even in the original mass drop tagger by Butterworth:2008iy .
The $S_{2}(R)$ has information on three-point correlation and higher
simultaneously and additional selections are not required. Consequently,
$S_{2}(R)$ can be used in a cascade decay of a heavy particle, especially the
top quark. We also note that $S_{2}(R)$ is sensitive to the color of the
boosted heavy particle. We will show in a separate publication that a modified
$\mathcal{N}_{S_{2}}$ discriminate color octet resonance and color singlet
resonance efficiently octet .

###### Acknowledgements.

The authors would like to thank Amit Chakraborty, Patrick T. Komiske, Devdatta
Majumder, and Tilman Plehn for useful discussions. This work was supported by
the Grant-in-Aid for Scientific Research on Scientific Research B
(No.16H03991, 17H02878 [MMN]) and Innovative Areas (16H06492 [MMN]), and by
World Premier International Research Center Initiative (WPI Initiative), MEXT,
Japan. The work of SHL was supported in part by MEXT KAKENHI Grant Number
JP16K21730.

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