# Investigating the Relationship Between User Specialization and Toxicity on
Reddit: A Sentiment Analysis Approach

Abi Oppenheim1 Federico Albanese2,3 Esteban Feuerstein1,3
{aoppenheim, falbanese<EMAIL_ADDRESS>

1Departamento de Computación, Facultad de Ciencias Exactas y Naturales,
Universidad de Buenos Aires, Buenos Aires, Argentina. 2Instituto de Cálculo,
CONICET Universidad de Buenos Aires, Buenos Aires, Argentina. 3Instituto de
Ciencias de la Computación, CONICET Universidad de Buenos Aires, Buenos Aires,
Argentina.

###### Abstract

Online platforms host users with diverse interests, ranging from those with
focused interests to those engaging in a wide range of topics. This study
investigates the behavioral differences between these user types on Reddit,
focusing on the level of toxicity in their posts and associated sentiment
scores across nine emotional categories. By employing community embeddings to
represent users in a high-dimensional space, we measure activity diversity
using the GS score. The analysis utilizes a dataset of 16,291,992 posts from
4,926,237 users spanning 2019 to 2021, assessing toxicity and sentiment scores
for each post. Results indicate that subreddits characterized by users with
specialized interests exhibit heightened toxic behavior compared to those with
diverse interest users. Additionally, subreddits populated by users with
focused interests display elevated sentiments of sadness, annoyance, and
disappointment, while those inhabited by diverse interest users demonstrate
increased expressions of curiosity, admiration, and love.
These insights contribute to understanding user behavior on online platforms
and inform strategies for fostering healthier online communities.

Investigating the Relationship Between User Specialization and Toxicity on
Reddit: A Sentiment Analysis Approach

Abi Oppenheim1 and Federico Albanese2,3 and Esteban Feuerstein1,3 {aoppenheim,
falbanese<EMAIL_ADDRESS>1Departamento de Computación, Facultad de
Ciencias Exactas y Naturales, Universidad de Buenos Aires, Buenos Aires,
Argentina. 2Instituto de Cálculo, CONICET Universidad de Buenos Aires, Buenos
Aires, Argentina. 3Instituto de Ciencias de la Computación, CONICET
Universidad de Buenos Aires, Buenos Aires, Argentina.

##  1 Introduction

Social media platforms have become breeding grounds for toxic behaviors, hate
speech, and online harassment, posing significant challenges to digital
communities. A recent report Thomas et al. (2021) underscores the widespread
prevalence of online threats, with nearly half (48%) of individuals worldwide
reporting encounters with such menacing behavior. Marginalized communities, in
particular, bear a disproportionate burden of online harassment Golbeck
(2018); Massanari (2017), leading to detrimental effects on mental health and
civil discourse. Various studies have explored methods for detecting and
analyzing toxic texts and emotions, including sentiment analysis on COVID-19
vaccine-related discussions Melton et al. (2021), examining toxicity in
political contexts Albanese et al. (2023), analyzing toxic language usage on
Facebook in relation to political interest Kim et al. (2021), and studying
toxicity in conversations prompted by tweets from news outlets and political
candidatesSaveski et al. (2021). Building upon this research, our work
integrates these strands with the methodology proposed by Waller and Anderson
Waller and Anderson (2019). Their study investigates user behavior by
distinguishing between generalist and specialist tendencies and utilizes
community embeddings to represent online communities. They employ the
community2vec algorithm Martin (2017) to assign vectors to communities,
facilitating the quantification of similarity between communities based on
cosine similarity in the vector space. Additionally, they introduce the
Generalist-Specialist score (GS-score) to measure a user’s activity diversity,
where specialists have a low GS-score, contributing to a tight cluster of
communities, while generalists have a high GS-score, contributing to diverse,
distant communities.

However, up to our knowledge, the relationship between user activity diversity
and the level of toxicity in their online interactions remains unexplored. In
this study, we hypothesize that communities comprising users with a broader
array of interests, are less likely to have harmful behavior compared to
communities dominated by those who concentrate on a limited range of topics.
To test our hypothesis, we analyze a dataset of 16,291,992 posts from
4,926,237 users on Reddit, spanning the period from 2019 to 2021. By employing
community embeddings to represent users in a high-dimensional space, we
measure activity diversity using the GS score. We assess the degree of
toxicity and sentiment scores across 9 emotional categories for each post. Our
findings reveal two key observations.

  * •

Diversity of Interest and Toxic Behavior. Our investigation shows that
subreddits cultivated by individuals with specialized interests experience
more toxic behavior than those curated by individuals with diverse interests.

  * •

Emotions Across Communities Types. Subreddits shaped by specific interest
users show more sadness, annoyance, and disappointment, while those by diverse
interest users display higher levels of admiration, amusement, and affection.

These findings underscore a compelling link between the diversity of user
interests and a notable decrease in toxic behavior observed on social media
platforms.

##  2 Methodology

###  2.1 Dataset

We utilized the Pushshift API, a platform for collecting, analyzing, and
archiving social media data Baumgartner et al. (2020). The Pushshift dataset
encompasses submissions and comments posted on Reddit since June 2005,
offering extensive query limits and user-friendly access. To ensure
scalability and representativeness, we randomly selected 15,000 submissions
from each of the top 5,000 most popular subreddits, evenly distributed across
2019 to 2021. Our dataset comprises 16,291,992 posts from 4,926,237 users.

###  2.2 Generalist-Specialist Score (GS-score)

The GS-score quantifies user activity diversity, distinguishing between
specialized and generalized engagement patterns Waller and Anderson (2019).
Specialized users concentrate activity within specific areas of the community
space, while generalist users show broader dispersion.

To analyze user interactions across diverse communities, we first generate
community embeddings using the community2vec approach Waller and Anderson
(2021); Martin (2017). By treating communities as “words” and users submitting
in them as “contexts”, communities were embedded into a high-dimensional
vector space. The proximity of two communities in this space is determined by
the frequency of users posting in both communities. We trained the skip-gram
model with negative sampling using all pairs $(c_{i},u_{j})$ of users $u_{j}$
submitting in community $c_{i}$, to generate an initial community embedding.

Let user $u_{i}$ make $w_{j}$ contributions to community $c_{j}$, and let
$\vec{c_{j}}$ denote the normalized vector representation of $c_{j}$ in our
community embedding. The center of mass of $u_{i}$ is defined as
$\vec{\mu_{i}}=\sum(w_{j}*\vec{c_{j}})$. The diversity of $u_{i}$’s activity,
or GS-score, is the weighted average cosine similarity between $u_{i}$’s
communities and its center of mass:

$GS(u_{i})=\frac{1}{\sum
w_{j}}\sum_{j}w_{j}\frac{\vec{c_{j}}\cdot\vec{\mu_{i}}}{\|\vec{\mu_{i}}\|}$

The GS-score ranges from $-1$ to $1$, where $-1$ indicates extreme
generalization and $1$ indicates extreme specialization. For detailed
insights, refer to Waller and Anderson Waller and Anderson (2019). Once user
activity diversity is computed, we extend this analysis to understand
community activity diversity. The GS-score $GS(c_{i})$ of community $c_{i}$ is
the weighted average across its users:
$GS(c_{i})=\frac{1}{N}\sum_{j}w_{j}\cdot GS(u_{j})$, where $u_{j}$ contributes
$w_{j}$ times to $c_{i}$ and $N=\sum_{j}w_{j}$ total contributions.
Prioritizing community metrics enhances robustness and comprehensive
evaluation of submission sets within communities.

###  2.3 Emotion and Toxicity Analysis

For emotion classification, we utilized the pre-trained Language Model (LLM)
"SamLowe/roberta-base-go_emotions" Demszky et al. (2020), fine-tuned on the
GoEmotions dataset Liu et al. (2019), which consists of 58,000 Reddit comments
annotated across 27 emotional categories, including Neutral. This model’s
training on Reddit-specific data makes it well-suited for our analysis and has
been utilized in various scientific studies Kocoń et al. (2023); Davani et al.
(2022); Vu et al. (2021). Considering the vast range of emotions, we focused
on specific subsets relevant to our hypothesis, including admiration,
amusement, anger, annoyance, disappointment, gratitude, joy, love, and
sadness. This model was applied to over 200 pertinent subreddits within our
corpus.

To evaluate toxicity, we employed the compact Detoxify Model developed by
Unitary AI Hanu (2020), trained on a substantial dataset of Wikipedia comments
annotated for various forms of toxic behavior Lan et al. (2019). This model
outputs probabilities indicating the likelihood of input text containing
language associated with different toxic categories. Over 1100 relevant
subreddits were analyzed using this approach.

##  3 Results

###  3.1 Subreddit Distribution Analysis

To examine user behavior across subreddit types, we categorized the subreddits
into two groups: the bottom 10th percentile of GS-scores, representing
specialized-interest user communities, and the upper 10th percentile,
representing diverse-interest user communities, each comprising 20 subreddits.
To gain insights into the dataset’s distribution, Figure 1 presents a
histogram illustrating the distribution of GS-scores among the subreddits.

Figure 1: Distribution of GS-scores among subreddits

The blue bars represent subreddits within the lower 10th percentile of GS-
scores, while the orange bars represent those within the upper 10th
percentile. Notably, the GS-scores predominantly cluster within the range of
0.6 to 1, aligning with prior findings Waller and Anderson (2019).

###  3.2 Emotion and Toxicity Analysis

For each group, we calculated average emotion and toxicity scores by
determining the proportion of users expressing specific emotions relative to
the total number of users. To explore variations in emotional expression, we
employed the Mann-Whitney U test, suitable for comparing two independent
groups when normality assumptions are not met. Our findings indicate
significant differences between subreddits frequented by users with diverse
and specialized interests, as evidenced by p-values smaller than 0.05.
Subreddits with the highest GS-scores exhibited higher toxicity scores
compared to those with the lowest GS-scores ($\boldsymbol{r=.44}$).
Conversely, subreddits with the lowest GS-scores demonstrated higher scores
for emotions such as joy ($\boldsymbol{r=.48}$), anger ($\boldsymbol{r=.56}$),
gratitude ($\boldsymbol{r=.44}$), love ($\boldsymbol{r=.53}$), amusement
($\boldsymbol{r=.58}$), and admiration ($\boldsymbol{r=.4}$).Notably, all
statistically significant results align with our hypothesis. However,
disappointment ($p=.394,r=0.16$) and sadness ($p=.968,r=.01$) did not show
significant differences between the two types of subreddits. Figure 2 compares
emotion and toxicity scores between subreddit types.

Figure 2: Emotion and toxic scores comparison between subreddits with highest
and lowest GS-score

While the lowest GS-score subreddits demonstrated a higher score for anger
compared to the other subreddit type, it is essential to differentiate between
“angry” and “toxic” content in text analysis. The distinction between anger
and toxicity lies in their respective expressions: while anger typically
manifests as frustration towards situations, objects, or groups, toxicity
often involves directing abusive language or behavior towards individuals.

##  4 Conclusions

Our study delved into the connection between emotion, toxicity, and activity
diversity on Reddit. We found that subreddits characterized by users with
diverse interests tend to have lower toxicity and higher positive emotion
scores, including joy, gratitude, love, and admiration. Conversely, subreddits
shaped by users with more specific interests exhibit higher toxicity and
increased negative emotions like sadness, annoyance, and disappointment. These
findings underscore the significance of activity diversity in understanding
user behavior and its impact on online community dynamics. Our research
enhances comprehension of the factors influencing user conduct and offers
insights for fostering healthier online interactions. Future studies could
investigate causal links between activity diversity, emotion, and toxicity, as
well as explore interventions to encourage positive engagement and mitigate
toxicity on online platforms.

## References

  * Albanese et al. (2023) Federico Albanese, Esteban Feuerstein, Gabriel Kessler, and Juan Manuel Ortiz de Zárate. 2023.  Aprendizaje automático para el análisis crossplataforma de la comunicación política: Gobierno y oposición argentinos en facebook, instagram y twitter.  _Cuadernos. info_ , (55):256–280.
  * Baumgartner et al. (2020) Jason Baumgartner, Savvas Zannettou, Brian Keegan, Megan Squire, and Jeremy Blackburn. 2020.  The pushshift reddit dataset.  In _Proceedings of the international AAAI conference on web and social media_ , volume 14, pages 830–839.
  * Davani et al. (2022) Aida Mostafazadeh Davani, Mark Díaz, and Vinodkumar Prabhakaran. 2022.  Dealing with disagreements: Looking beyond the majority vote in subjective annotations.  _Transactions of the Association for Computational Linguistics_ , 10:92–110.
  * Demszky et al. (2020) Dorottya Demszky, Dana Movshovitz-Attias, Jeongwoo Ko, Alan Cowen, Gaurav Nemade, and Sujith Ravi. 2020.  Goemotions: A dataset of fine-grained emotions.  _arXiv preprint arXiv:2005.00547_.
  * Golbeck (2018) Jennifer Golbeck. 2018.  _Online harassment: A research challenge for HCI_.  Springer.
  * Hanu (2020) Laura; Unitary team Hanu. 2020.  Detoxify.  _https://github.com/unitaryai/detoxify_.
  * Kim et al. (2021) Jin Woo Kim, Andrew Guess, Brendan Nyhan, and Jason Reifler. 2021.  The distorting prism of social media: How self-selection and exposure to incivility fuel online comment toxicity.  _Journal of Communication_ , 71(6):922–946.
  * Kocoń et al. (2023) Jan Kocoń, Igor Cichecki, Oliwier Kaszyca, Mateusz Kochanek, Dominika Szydło, Joanna Baran, Julita Bielaniewicz, Marcin Gruza, Arkadiusz Janz, Kamil Kanclerz, et al. 2023.  Chatgpt: Jack of all trades, master of none.  _Information Fusion_ , 99:101861.
  * Lan et al. (2019) Zhenzhong Lan, Mingda Chen, Sebastian Goodman, Kevin Gimpel, Piyush Sharma, and Radu Soricut. 2019.  Albert: A lite bert for self-supervised learning of language representations.  _arXiv preprint arXiv:1909.11942_.
  * Liu et al. (2019) Yinhan Liu, Myle Ott, Naman Goyal, Jingfei Du, Mandar Joshi, Danqi Chen, Omer Levy, Mike Lewis, Luke Zettlemoyer, and Veselin Stoyanov. 2019.  Roberta: A robustly optimized bert pretraining approach.  _arXiv preprint arXiv:1907.11692_.
  * Martin (2017) Trevor Martin. 2017.  community2vec: Vector representations of online communities encode semantic relationships.  In _Proceedings of the Second Workshop on NLP and Computational Social Science_ , pages 27–31.
  * Massanari (2017) Adrienne Massanari. 2017.  # gamergate and the fappening: How reddit’s algorithm, governance, and culture support toxic technocultures.  _New media & society_, 19(3):329–346.
  * Melton et al. (2021) Chad A Melton, Olufunto A Olusanya, Nariman Ammar, and Arash Shaban-Nejad. 2021.  Public sentiment analysis and topic modeling regarding covid-19 vaccines on the reddit social media platform: A call to action for strengthening vaccine confidence.  _Journal of Infection and Public Health_ , 14(10):1505–1512.
  * Saveski et al. (2021) Martin Saveski, Brandon Roy, and Deb Roy. 2021.  The structure of toxic conversations on twitter.  In _Proceedings of the Web Conference 2021_ , pages 1086–1097.
  * Thomas et al. (2021) Kurt Thomas, Devdatta Akhawe, Michael Bailey, Dan Boneh, Elie Bursztein, Sunny Consolvo, Nicola Dell, Zakir Durumeric, Patrick Gage Kelley, Deepak Kumar, et al. 2021.  Sok: Hate, harassment, and the changing landscape of online abuse.  In _2021 IEEE Symposium on Security and Privacy (SP)_ , pages 247–267. IEEE.
  * Vu et al. (2021) Tu Vu, Brian Lester, Noah Constant, Rami Al-Rfou, and Daniel Cer. 2021.  Spot: Better frozen model adaptation through soft prompt transfer.  _arXiv preprint arXiv:2110.07904_.
  * Waller and Anderson (2019) Isaac Waller and Ashton Anderson. 2019.  Generalists and specialists: Using community embeddings to quantify activity diversity in online platforms.  In _The World Wide Web Conference_ , pages 1954–1964.
  * Waller and Anderson (2021) Isaac Waller and Ashton Anderson. 2021.  Quantifying social organization and political polarization in online platforms.  _Nature_ , 600(7888):264–268.


††thanks<EMAIL_ADDRESS> 0009-0005-7748-7085††thanks<EMAIL_ADDRESS> 0000-0002-3143-1976††thanks<EMAIL_ADDRESS> 0000-0002-3291-1701

# Higgs Width and Couplings at High Energy Muon Colliders with Forward Muon
Detection

Peiran Li  Zhen Liu Kun-Feng Lyu School of Physics and Astronomy, University
of Minnesota, Minneapolis, MN 55455, USA

###### Abstract

We propose a novel method using the $ZZ$-fusion channel and forward muon
detection at high-energy muon colliders to address the challenge of the Higgs
coupling-width degeneracy. Our approach enables inclusive Higgs rate
measurement to 0.75% at 10 TeV muon collider, breaking the coupling-width
degeneracy. Results indicate the potential to refine Higgs coupling to sub-
percent levels and estimate its total width within (-0.41%, +2.1%). Key
insights include the effectiveness of forward muon tagging in signal-
background separation despite broad recoil mass distribution due to muon
energy reconstruction and beam energy spread. The study emphasizes the
significance of muon rapidity coverage up to $|\eta(\mu)|<6$, enhancing
measurement precision. Our findings highlight the unique capabilities of high-
energy lepton colliders for model-independent Higgs coupling determination and
lay the groundwork for future advancements in muon collider technology and
Higgs physics research.

††preprint: UMN-TH-4309/24

###### Contents

  1. I Introduction
  2. II $ZZ$-Fusion for the Inclusive Rate
  3. III Higgs Precision
  4. IV Conclusion
  5. A $ZZ$-fusion for the Inclusive Rate at 3 TeV Muon Collider

##  I Introduction

The core quest of the next collider is to study the Higgs boson properties and
reveal the many profound fundamental puzzles around this new boson
deBlas:2019rxi ; EuropeanStrategyforParticlePhysicsPreparatoryGroup:2019qin ;
Dawson:2022zbb ; Narain:2022qud . Many future colliders are proposed, and
their physics potentials are thoroughly understood, such as CEPC An:2018dwb ;
CEPCStudyGroup:2018ghi ; CEPCPhysicsStudyGroup:2022uwl , ILC Asner:2013psa ;
ILCInternationalDevelopmentTeam:2022izu , FCC-ee Benedikt:2651299 ;
Bernardi:2022hny , CLIC CLICdp:2018cto and C3 Vernieri:2022fae . One key
question is to pin down the Higgs overall coupling scale, which coincides with
the Higgs width measurement. Besides a high signal background ratio that helps
measure Higgs property, these future $e^{+}e^{-}$ colliders measure the
overall coupling strength through an inclusive Higgs measurement in the
$e^{+}e^{-}\rightarrow ZH$ process, without dependence on the Higgs exclusive
decay modes, reconstructed using the recoil mass derived from the associated
$Z$ boson four-momentum.

Future high-energy muon collider generates vigorous interest and promises due
to their direct access to ten-TeV scale physics. The high-energy muon collider
can also copiously produce the Higgs boson mainly through the vector boson
fusion (VBF) process, dominated by $WW$-fusion. We generally understand its
Higgs performance at various stages Buttazzo:2020uzc ; Han:2020pif ;
Han:2021lnp ; Forslund:2022xjq ; Black:2022cth ; Chen:2022yiu ;
Ruhdorfer:2023uea ; Liu:2023yrb ; Forslund:2023reu ; Han:2023njx ;
Celada:2023oji . However, high energy muon collider alone cannot pin down the
Higgs coupling scale as no inclusive Higgs measurement anticipated. In this
work, we point out that studying the $ZZ$-fusion channel with forward muon
tagging would enable this hallmark of Higgs property measurement, namely, the
absolute scale of its coupling and its width.

We quantify many essential aspects to establish this channel and show its
critical impact on the Higgs width, coupling determination, and global fit.
The paper is organized as follows. In section II, we first analyze the signal
and background and then make a simulation so that the precision on $hZZ$
coupling measurement can be extracted. Then, in section III, we develop global
fits combined with other channels and show the sensitivity on various Higgs
couplings. We summarize the key of this study in section IV.

##  II $ZZ$-Fusion for the Inclusive Rate

For high-energy lepton colliders, the dominant single Higgs production channel
is from the VBF process, especially the $WW$ fusion channel. At the high
energy, the parton distribution function of gauge boson Han:2020uid ;
Han:2021kes ; Kane:1984bb ; Dawson:1984gx ; Fornal:2018znf ; Chen:2016wkt ;
Ruiz:2021tdt split from the incoming muons can be significantly enhanced. For
the $WW$ fusion, the associated particles to the Higgs boson are the invisible
neutrinos emitted in the forward direction. In the $ZZ$ fusion, the particles
associated with the Higgs boson are the forward muons. We can separate the two
primary channels if one can tag the forward muons. To quantify the projected
sensitivity of this channel, we use the 10 TeV muon collider benchmark with an
integrated luminosity of $10\leavevmode\nobreak\ \text{ab}^{-1}$. We also show
the distribution and analysis results for a 3 TeV muon collider, which can be
an essential stage. The $WW$-fusion,
$\mu^{+}\mu^{-}\rightarrow\bar{\nu}_{\mu}\nu_{\mu}h$, has signal rate close to
1 pb. The $ZZ$-fusion of $\mu^{+}\mu^{-}\rightarrow\mu^{+}\mu^{-}h$ follows,
with around one order smaller cross-section. The $WW$ fusion leads to the
invisible neutrinos, while the $ZZ$ fusion is accompanied by the potentially
detectable muon in the forward direction. If the forward muons can be tagged
by placing the detector in the high $\eta$ region, one can exploit the
information of such muons.

In this study, we quantify the behavior of forward muons and the physics gain
in Higgs precision from such detection. The recoil mass is defined by

$\text{M}^{2}_{\rm
recoil}=\left[p_{1}(\mu^{+})+p_{2}(\mu^{-})-k_{1}(\mu^{+})-k_{2}(\mu^{-})\right]^{2}$
(1)

where $p_{1}$ and $p_{2}$ are the momenta of the incoming muons, $k_{1}$ and
$k_{2}$ refer to the momenta of outgoing forward muons. Such dimuon quantity
could help us extract the inclusive scattering information of neutral gauge
boson scattering, regardless of the final states from the scattering.

Figure 1: The recoil mass distributions of parton level (dashed) and
reconstruction level (solid) at a 10 TeV muon collider. Two solid lines
represent 10% and 20% momentum resolution at the reconstruction level. The
negative value of recoil mass $\text{M}_{\text{recoil}}$ is defined when
$\text{M}_{\text{recoil}}^{2}<0$. At the parton level, the signal should be a
Breit-Wigner peak center at the Higgs mass of 0.125 TeV.

In Figure 1, we show the recoil mass distribution at the parton level (dashed
line) and after reconstruction (solid line) under two different momenta
resolutions; the details are provided in later sections. Ideally, parton-level
events have a sharp recoil mass peak at the Higgs mass pole. However, energy
resolution for the high energy and forward muons erodes this information to be
much less powerful. One also omits the information that is further
reconstructable from the scattering events, which requires careful
determination of the observational possibility. We first classify the signal
and background. The parton level events are generated using MadGraph5_aMC
Alwall:2014hca . We implement a parton-level pre-selection111to avoid IR
singularities, and as detector reconstruction-level will be more complicated
than those when generating both signal and background samples:

$p_{T}(\ell,j)>5\leavevmode\nobreak\ \text{GeV},\leavevmode\nobreak\
p_{T}(\gamma)>1\leavevmode\nobreak\ \text{GeV},\leavevmode\nobreak\
0<\eta(\ell)<10,\leavevmode\nobreak\ \Delta R(jj,j\ell,\ell\ell)>0.2.$

Then the parton level events are passed to the interfaced Pythia
Bierlich:2022pfr and Delphes deFavereau:2013fsa ; Mertens:2015kba to simulate
the reconstructed events we measure from the (forward) detectors.

###  II.1 Signal and Background Considerations

Figure 2: The parton level distributions of Higgs kinematics after parton-
level pre-selection.

The signal is the $ZZ$-fusion Higgs production,
$\mu^{+}\mu^{-}\rightarrow\mu^{+}\mu^{-}h$. Under the forward muons
requirement (included in the pre-selection at the parton level) the primary
contribution is $ZZ$-fusion process with the cross-section around 86.7 fb as
shown in Table 1. The parton-level kinematics of the Higgs boson are shown in
Figure 2. Due to the unpolarized initial muon beams, the produced Higgs boson
is symmetric in $\eta$ and $\phi$. We can also see that the Higgs boson
$p_{T}$ is dominant at around 100 GeV, the typical energy scale of the
production, and spreads up to TeV.

The kinematics of muon is the main object in this study, and its truth level
(including QED shower) information is shown as the red histogram in Figure 3.
We display the differential distribution over the minimum energy of the muon
$E_{\text{min}}(\mu)$, the dimuon transverse momentum sum $p_{T}(\mu\mu)$, the
pseudorapidity of each muon $\eta(\mu)$ and the recoil mass
$\text{M}_{\text{recoil}}$. The primary change is that the parton shower
spreads the recoil mass distribution with a long tail instead of the delta-
like peak at $m_{h}$.

Type | Scattering process |  cross-section $\sigma$ (pb)
---|---|---
VBF | $\mu^{+}\mu^{-}\to\mu^{+}\mu^{-}h$ | $8.67\times 10^{-2}$
$t$-channel | $\mu^{+}\mu^{-}\to\mu^{+}\mu^{-}$ | $1.12\times 10^{4}$
$t$-channel | $\mu^{+}\mu^{-}\to\mu^{+}\mu^{-}\gamma$ | $7.55\times 10^{2}$
VBS | $\mu^{+}\mu^{-}\to\mu^{+}\mu^{-}\ell^{+}\ell^{-}$ | 3.96
VBS | $\mu^{+}\mu^{-}\to\mu^{+}\mu^{-}jj$ | 2.06
VBS | $\mu^{+}\mu^{-}\to\mu^{+}\mu^{-}\nu_{\ell}\bar{\nu}_{\ell}$ | 1.68
VBS | $\mu^{+}\mu^{-}\to\mu^{+}\mu^{-}W^{+}W^{-}$ | $0.939$
Table 1: Cross-section for both signal and background after parton-level pre-
selection.

Figure 3: Truth level (including parton shower) muon kinematics distributions
with cuts of $2.5<|\eta(\mu)|<8$, $\eta(\mu^{-})\cdot\eta(\mu^{+})<0$, and
$p_{T}(\mu)>20\leavevmode\nobreak\ $GeV.

The background events include $\mu^{+}\mu^{-}\rightarrow\mu^{+}\mu^{-}+X$ with
$X$ to be any particles except the Higgs boson. The major background channels
are listed in the following:

  1. 1.

$\mu^{+}\mu^{-}\rightarrow\mu^{+}\mu^{-}$ and
$\mu^{+}\mu^{-}\rightarrow\mu^{+}\mu^{-}\gamma$

  2. 2.

$\mu^{+}\mu^{-}\rightarrow\mu^{+}\mu^{-}f\bar{f}$ ($f=l,\nu,j$)

  3. 3.

$\mu^{+}\mu^{-}\rightarrow\mu^{+}\mu^{-}W^{+}W^{-}$.

Their cross-section and truth level (including parton shower) distributions
for both the signal and these backgrounds are shown in Table 1 and Figure 3.

The first class is the elastic Bhabha scattering and those with one (visible)
photon emission process. At the parton level, this is a 2-to-2 scattering
process whose peak is at the ultra-forward region due to the $t$-channel
enhancement. However, this is only a tree-level description. Furthermore, one
must consider the real photon emission in the forward region plus the higher-
order QED correction to cancel the infrared (IR) and collinear divergence. One
feasible way to handle this is to merge the Parton shower sample events of the
Bhabha scattering with the process with one additional photon
$\mu^{+}\mu^{-}\rightarrow\mu^{+}\mu^{-}\gamma$, which will be elaborated
afterward. We ignore more photon events due to the QED suppression.

The second class of background is the forward muon pairs plus SM fermion
pairs. The Feynman diagrams can be further divided into two categories. The
forward muon scattering could have initial- and final-state radiations of (on-
or off-shell) SM gauge bosons that further split into SM fermions. The other
category is the neutral VBS process that produces fermion pairs through
$s$-channel or $t$-channel process.

The final states for the third class are
$\mu^{+}\mu^{-}\to\mu^{+}\mu^{-}+ff\bar{f}\bar{f}$ where the four fermions do
not need to be the same flavor and are mainly from the on-shell $W$ bosons
decay. The contribution from on-shell $Z$ boson decay is much smaller. The
lower rate is caused by the suppressed “hard” scattering cross-section for
$ZZ\rightarrow ZZ$ in which only intermediate Higgs boson can be exchanged.
While for $W^{+}W^{-}$ boson final states, both gamma and $Z$ boson can serve
as initial states, leading to a much higher cross-section. It turns out that
the cross-section of $\mu^{+}\mu^{-}\to\mu^{+}\mu^{-}ZZ$ is much smaller than
$\mu^{+}\mu^{-}\to\mu^{+}\mu^{-}W^{+}W^{-}$ by a factor of $10^{4}$
approximately hence we ignore it.

Figure 4: The truth level $p_{T}(\mu\mu)$ distribution after QED parton
shower, under the cuts of $2.5<|\eta(\mu)|<8$, $\eta(\mu^{+})\eta(\mu^{-})<0$
and $p_{T}(\mu)>20\leavevmode\nobreak\ $GeV. The right panel zooms in on the
left plot under $10\leavevmode\nobreak\ $GeV. The sharp cliff around 1 GeV is
caused by a $p_{T}(\gamma)>1\leavevmode\nobreak\ $GeV cut.

All backgrounds are generated via MadGraph and Pythia. The primary background
$\mu^{+}\mu^{-}\rightarrow\mu^{+}\mu^{-}(\gamma)$, whose event number is
around 4 or 5 orders larger, needs to be carefully analyzed. The matrix
element is reliable for the larger transverse momentum $p_{T}$ of
$\mu^{+}\mu^{-}$ pair or the photon. For a lower $p_{T}$ value, one has to
consider the Infrared (IR) effects in which resummation is needed. We turn on
the QED parton shower for both Bhabha scattering and
$\mu^{+}\mu^{-}\to\mu^{+}\mu^{-}\gamma$, and include Initial State Radiation
(ISR) and Final State Radiation (FSR). Furthermore, double counting must be
avoided due to an overlap of phase space (final states) through matching and
event merging. When the photon of a $\mu^{+}\mu^{-}\to\mu^{+}\mu^{-}\gamma$
process is soft/collinear, it is already counted inside the final states of
Bhabha scattering after the QED shower. Similarly, a Bhabha scattering
process, whose final state has a hard photon coming out after QED shower, is
included in the phase space of $\mu^{+}\mu^{-}\to\mu^{+}\mu^{-}\gamma$.
Therefore, when plotting the distribution of $p_{T}(\mu\mu)$, we expect there
is a matching scale where both their cross-section and trend are the same.
This feature is checked in truth level as shown in Figure 4. By observing a
matching scale around $0.8$ GeV, the lower $p_{T}$ region is dominated by
Bhabha scattering, and the higher $p_{T}$ region should be contributed by
$\mu^{+}\mu^{-}\to\mu^{+}\mu^{-}\gamma$. The $\mu^{+}\mu^{-}\gamma$ process
has a cliff at $1$ GeV due to the parton level cut $p_{T}(\gamma)>1$ GeV. We
will apply a cut of $p_{T}(\mu\mu)>50$ GeV at the detector level so that only
$\mu^{+}\mu^{-}\to\mu^{+}\mu^{-}\gamma$ process survive. Given the large
separation of scales we make, we can safely ignore the theoretical uncertainty
on the pure Bhabha backgrounds.

###  II.2 Forward Muon Tagging

As an unstable particle, muon colliders have a unique source of background
which is from the decay products and interactions of them, called Beam Induced
Background (BIB) Ally:2022rgk ; MuonCollider:2022glg ; MuonCollider:2022ded .
To ensure the delivery of high-precision central physics, the muon collider
detector designs have two tungsten cone-shaped shields (nozzles) around the
beampipe. This shield limits the available angular coverage of the detected
interaction region to $\theta>10^{\circ}$, corresponding to $\eta<2.4$.

The tungsten nozzles are initially designed for
$\sqrt{s}=1.5\leavevmode\nobreak\ $TeV to shield beam-induced background. For
a 10 TeV muon collider, sharper-cone nozzles can be further optimized. The
potential of forward muon detectors has already been highlighted in several
muon collider reviews AlAli:2021let ; Accettura:2023ked . TeV muons can
penetrate the shielding nozzle. Then, tagging and measuring forward muon with
momentum can be exploited by adding a tracking station in the forward region,
such as the design of FCC-hh FCC:2018vvp . We note here that the forward muon
tagging does not require high precision in energy reconstruction. Hence, the
energy loss for muons transpassing the tungsten nozzles does not affect the
physics performance for the Higgs physics discussed here.

###  II.3 Reconstruction Level Analysis

Both the detector effect and beam resolution are considered at the
reconstruction level. We produce the truth-level events from Pythia and let
them pass through Delphes for a fast detector simulation. The detector
configuration is set by the muon collider card in Delphes’s package
Mertens:2015kba ; deFavereau:2013fsa ; Delphes:muon_card . The forward muon
reconstruction efficiency is approximately 95%, and the momentum smearing is
10%. For a 10 TeV muon collider, each beam energy resolution is expected to be
$\delta=0.1\%\times 5\leavevmode\nobreak\ $TeV. In order to simulate beam
resolution, we generate three different sets of samples with total energy
$\\{10\leavevmode\nobreak\ \text{TeV}-\sqrt{2}\delta,\leavevmode\nobreak\
10\leavevmode\nobreak\ \text{TeV},\leavevmode\nobreak\ 10\leavevmode\nobreak\
\text{TeV}+\sqrt{2}\delta\\}$, and combine them with
$\\{0.25,\leavevmode\nobreak\ 0.5,\leavevmode\nobreak\ 0.25\\}$ weights. Our
full reconstruction level analysis is under the following pre-selection cuts
before applying our analysis for different detector coverage considerations:

  * •

Only two visible muons in the forward region ($2.5<|\eta(\mu)|<8.0$).

  * •

Back-to-back muons: $\eta(\mu^{-})\cdot\eta(\mu^{+})<0$.

  * •

Minimal muon momentum: $p_{T}(\mu)>20\leavevmode\nobreak\ $GeV.

Note that depending on the forward coverage of the detector, the minimal
momentum $p_{T}(\mu)$ can be redundant.

Figure 5: Reconstruction level $p_{T}(\mu\mu)$ distribution under pre-
selection cuts. The left panel compares Bhabha scattering and
$\mu^{+}\mu^{-}\to\mu^{+}\mu^{-}\gamma$ process. The right panel shows the
$p_{T}(\mu\mu)$ distribution of all processes.

In the previous section, we argue that a cut of $p_{T}(\mu\mu)>50$ GeV would
remove all the pure Bhabha scattering process. Here, we further check the
merging pattern between Bhabha scattering and
$\mu^{+}\mu^{-}\to\mu^{+}\mu^{-}\gamma$ under reconstruction level on the
right panel of Figure 5, and observe these two processes crossover at around
$35\leavevmode\nobreak\ $GeV, which is still below our choice the transverse
momentum cut. Hence, we are further convinced of the safety of ignoring the
pure Bhabha scattering after a cut of $p_{T}(\mu\mu)>50$ GeV. The right plot
in Figure 5 shows the $p_{T}(\mu\mu)$ distribution of all the processes
considered.

Process | Pre-selection |  $p_{T}(\mu\mu)>50\leavevmode\nobreak\ $GeV |  $E(\mu)>3000\leavevmode\nobreak\ $GeV & $p_{T,\text{min}}(\mu)<300\leavevmode\nobreak\ $GeV
---|---|---|---
$\mu^{+}\mu^{-}\to\mu^{+}\mu^{-}h$ | 73.3% | 65.7% | 56.4% (0.0489 pb)
$\mu^{+}\mu^{-}\to\mu^{+}\mu^{-}\gamma$ | 13.1% | 0.38% | 0.12% (0.906 pb)
$\mu^{+}\mu^{-}\to\mu^{+}\mu^{-}f\bar{f}$ | 8.13% | 4.69% | 2.58% (0.199 pb)
$\mu^{+}\mu^{-}\to\mu^{+}\mu^{-}W^{+}W^{-}$ | 40.0% | 34.9% | 22.0% (0.207 pb)

Table 2: Cutflow table for both signal and background events for 10 TeV muon
collider. All processes before the pre-selection cuts are set to 100%. In the
last column, $p_{T,\text{min}}(\mu)$ stands for the transverse momentum of the
less energetic forward muon.

After the $p_{T}(\mu\mu)$ cut, we show several other kinematics distributions
in Figure 6, and add a cut of $E(\mu)>3000\leavevmode\nobreak\ $GeV and
$p_{T,\text{min}}(\mu)<300\leavevmode\nobreak\ $GeV. A cut-flow table is shown
in Table 2. Our cut-based analysis keeps around half of the signal, equal to
around 0.05 pb in the signal cross-section. While for the three categories of
background, only per mil level of $\mu^{+}\mu^{-}\gamma$ is left with a size
of around 0.9 pb. For the other two types of background, the remaining cross-
sections are each around 0.2 pb. Note that we performed a cut-and-count
analysis here for simplicity and robustness of the result, which also leaves
room for improvements with multi-variable analysis that leads to better
results. For instance, although the recoil mass distribution is wide due to
the beam energy spread and reconstruction smearing, we can improve our result
by around 5% by imposing a recoil mass cut.

Figure 6: Muon kinematic distributions under reconstruction level. Except for the recoil mass distribution at the right bottom, all other three distributions are after pre-selection and $p_{T}(\mu\mu)>50\leavevmode\nobreak\ $GeV. The recoil mass distribution is after all the cuts in Table 2. The $p_{T,\text{min}}(\mu)$ stands for the transverse momentum of the forward muon, which has lower energy.  Benchmark | $|\eta(\mu)|<4$ | $|\eta(\mu)|<6$ | $|\eta(\mu)|<8$
---|---|---|---
$\Delta\sigma/\sigma$ | 15% | 0.75% | 0.74%
Table 3: The 68% projected sensitivity on the Higgs inclusive rate from $ZZ$
fusion at 10 TeV muon collider with different forward detector scenarios.

We carry out the same study for the other two possible forward detector
coverage: $|\eta(\mu)|<6$ and $|\eta(\mu)|<4$. We perform the study for three
possible forward muon detector configurations, which cover $|\eta(\mu)|<8$
(ultra-forward), $|\eta(\mu)|<6$ (forward), and $|\eta(\mu)|<4$ (moderately
forward), respectively. The precision results are shown in Table 3. For the
first narrow range with $|\eta(\mu)|<4$, the $1\sigma$ precision on the $ZZ$
fusion signal rate is just around $15\%$. While for the other two coverage
$|\eta(\mu)|<6$ and $|\eta(\mu)|<8$, the corresponding precision are $0.75\%$
and $0.74\%$. The result of $|\eta(\mu)|<6$ is much better than
$|\eta(\mu)|<4$. The reason can be seen clearly from the $\eta(\mu)$
distribution in Figure 6. For a 10 TeV muon collider, most of the signals are
accumulated within $|\eta(\mu)|<6$, as one can estimate the signal typical
rapidity to be approximately $\log(m_{h}/4/E_{\rm beam})\simeq 5.1$ or
$\log(m_{W}/2/E_{\rm beam})\simeq 4.8$. This highlights a target forward
coverage for a 10 TeV muon collider of $|\eta(\mu)|<6$.

Interestingly, a 3 TeV muon collider would demand less forward coverage. One
can perform a similar study and analysis and see that the muons for the signal
process are slightly more central. Hence, moderate forward coverage
$|\eta(\mu)|<4$ does not significantly differ from the forward muon coverage
case of $|\eta(\mu)|<6$. However, the precision would be at around $5\%$ level
due to the lower signal rate and luminosity. We show the corresponding results
and distributions in Appendix Appendix A.

##  III Higgs Precision

Our study’s primary physics motivation is to highlight that this new,
inclusive Higgs channel will enable the high-energy muon collider alone
222Without relying on any other future Higgs factories, whose timelines are
uncertain at this point. 333Another muon collider possibility is to have a
resonant Higgs factory as a first stage, which can also uniquely determine
Higgs width and muon Yukawas to a percent-level precision Barger:1996jm ;
Han:2012rb ; deBlas:2022aow . As shown in Ref. Black:2022cth , before knowing
the possibility of the proposed inclusive channel in this paper, the muon
collider Higgs fit does not close in a general setup. to probe Higgs precisely
and (by and large) model independently. 444We also note here that it is
possible to use off-shell Higgs rate to constraint the width Kauer:2012hd ;
Forslund:2023reu ; Caola:2013yja ; Campbell:2015vwa , which is complementary
to our work, but it does depend on the assumption of Higgs coupling scaling at
different energy scales Logan:2014ppa .

In this section, we perform the global fit to get the projected sensitivity on
the Higgs couplings and discuss many new features that emerged from such a new
input from this new inclusive channel.

One general and intuitive way to parametrize the precision of the Higgs
coupling is the so-called $\kappa$ framework. Compared with the SMEFT
framework, this parameterization characterizes the most relevant Higgs
coupling deviations while keeping simplicity for describing on-shell Higgs
properties. Of course, using the SMEFT would allow one to incorporate the more
general comparisons using non-Higgs inputs, such as electroweak precision, and
compare Higgs properties across scales Buchmuller:1985jz ; Grzadkowski:2010es
; DiVita:2017vrr ; Barklow:2017suo ; deBlas:2019rxi ; Dawson:2020oco ;
Ethier:2021bye . The main point of this study, on the importance of this
inclusive channel with forward muon coverage, holds in all Higgs precision
frameworks where the Higgs total width is effectively a free parameter
Dixon:2003yb ; Peskin:2012we ; Dobrescu:2012td ; Kauer:2012hd ; Dixon:2013haa
; Caola:2013yja ; Han:2015ofa ; Coradeschi:2015tna ; Campbell:2017rke . Hence,
for simplicity, we show our results in the $\kappa$ framework and leave the
general SMEFT global fit for future studies. The potential deviation of the SM
coupling is expressed as

$\kappa_{f}=\dfrac{y(h\bar{f}f)}{y_{\rm
SM}(h\bar{f}f)}\,,\quad\quad\kappa_{V}=\dfrac{g(hVV)}{g_{\rm SM}(hVV)}$ (2)

with $\kappa=1$ corresponding to the SM prediction. For instance, the major
Higgs production channels at high-energy muon colliders are $WW$-fusion and
$ZZ$-fusion. One can parameterize the total signal rate under the Narrow Width
Approximation (NWA) as

$\sigma(W^{*}W^{*}\rightarrow h)\times\text{Br}(h\rightarrow
f\bar{f})=\dfrac{\kappa_{W}^{2}\kappa_{f}^{2}}{\kappa_{\Gamma}}\sigma_{\rm
SM}(W^{*}W^{*}\rightarrow h)\times\text{Br}_{\rm SM}(h\rightarrow f\bar{f})$
(3)

where $\kappa_{\Gamma}=\Gamma_{h}/\Gamma_{h,\rm SM}$ being the Higgs total
width as a free parameter. For the loop-induced process like
$h\rightarrow\gamma\gamma$ that dependents on $\kappa_{t}$ and $\kappa_{W}$
from the SM loop computation, we instead choose to parameterize in terms of
the effective coupling $\kappa_{\gamma}$ and $\kappa_{Z\gamma}$, which
corresponds to a linear combination of the $H^{\dagger}HB^{\mu\nu}B_{\mu\nu}$
and $H^{\dagger}HW^{\mu\nu}W_{\mu\nu}$ operators. In general, new physics
considerations and the Higgs portal to the dark sector would induce Higgs
invisible decays. Hence, as a typical choice of Higgs global fit, we also
include $\Gamma_{\rm inv}$ as a free parameter, which can be parameterized as
branching ratio $\text{BR}_{\rm inv}^{\rm BSM}$. Note that there is “scaling
degeneracy” of the NWA cross-section, which is the origin of why many studies
focused on determining the Higgs total width Dixon:2003yb ; Peskin:2012we ;
Dobrescu:2012td ; Kauer:2012hd ; Dixon:2013haa ; Caola:2013yja ; Han:2015ofa ;
Coradeschi:2015tna ; Campbell:2017rke . The exclusive Higgs cross-sections
Equation 3 are invariant under the scaling of the related coupling by a factor
of $t$ and the total decay width scaled by $t^{4}$. This coupling-width
scaling degeneracy for exclusive Higgs observables represents an interesting
flat-direction in Higgs global fit, which hinders our extraction of Higgs
couplings when projected into space of individual coupling strengths. We
emphasize that the overall Higgs property is accessible piece-by-piece in the
precision differential cross-section measurements. However, it would appear to
be “lost-in-translation"555Full information is still kept with the additional
correlation matrix. when expressed in the kappa or SMEFT framework when
allowing for general BSM decays that make Higgs width a free parameter. It is
unnecessary to overly emphasize this flat direction, as different BSM models
would provide additional constraints to enable stronger ones once the
underlying model is specified. Our study on the inclusive $ZZ$ fusion channel
can provide isolated constraints on $\kappa_{Z}$, thus breaking this flat
direction.

To see the achievable Higgs precision, we take by now a well-established
choice in the following $\kappa$ parameters 666This is close to the kappa-2
and kappa-3 scenarios defined in Ref. deBlas:2019rxi .

$\kappa_{b},\kappa_{t},\kappa_{c},\kappa_{g},\kappa_{W},\kappa_{\tau},\kappa_{Z},\kappa_{\gamma},\kappa_{\mu},\text{Br}_{\text{inv}}^{\text{BSM}},\kappa_{\Gamma}$
(4)

with the only requirement $\text{Br}_{\text{inv}}^{\text{BSM}}\geq 0$ and all
other $\kappa$ parameters can fluctuate around unity. $\kappa_{\Gamma}$ is a
free parameter because one can generally anticipate Higgs has decay channels
that are not actively searched for or buried under the background. Generally,
any Higgs exotic decays that are not well-constrained or probed can contribute
Curtin:2013fra ; Liu:2016zki ; Jung:2021tym . An equivalent choice is to trade
the total width free parameter $\kappa_{\Gamma}$ by these “untagged” branching
fractions $\text{Br}_{\text{unt}}^{\text{BSM}}$ via

$\kappa_{\Gamma}=\frac{\sum\kappa_{i}^{2}\text{Br}_{i}^{\text{SM}}}{1-\text{Br}_{\text{inv}}^{\text{BSM}}-\text{Br}_{\text{unt}}^{\text{BSM}}}\
.$ (5)

Given that this allows for general, unspecified, new physics contributions,
sometimes people call such fitting schemes “model-independent” fit. There are
no global fits that are not dependent on any assumption, but this scheme does
relax the assumptions on Higgs properties in a significant way and covers BSM
models broadly.

There have been many exclusive Higgs sensitivity studies at high-energy muon
colliders. We refer to the simulated precision results on various channels
adapted in our study as shown in Table 4, where the effective constrained
parameters $\mu$ are defined based on the on-shell Higgs boson production and
decay

$\mu_{aa}^{bb}=\frac{\kappa_{a}^{2}\kappa_{b}^{2}}{\kappa_{\Gamma}}.$ (6)

In this table, the first channel is from Ref. Liu:2023yrb for the indirect
probe of the Top Yukawa coupling through the $VV\rightarrow t\bar{t}$ process.
One can also find a compatible, but without all channels and parameterization,
study in Ref. Chen:2022yiu . The last channel shown in red is the inclusive
channel in this study. The second to last input is from the Higgs invisible
study at muon collider in Ref. Ruhdorfer:2023uea . The rest of the input
precisions on Higgs exclusive decay channels are collected from Ref
Forslund:2022xjq . We show the constraint results for the 11 free parameters
under both $\kappa$ frameworks at 10 TeV muon collider in Table 5. We verified
that the results are the same for all the fitting parameters at two
significant digit levels for these two fitting choices. Hence, we incorporate
all the results into one column. The two branching ratio values are presented
to be the exclusion rate at 95% C.L. We mainly focus on the forward muon
result at $\eta(\mu)<6$ as this provides almost optimal performance, and in
the following discussions about the results, we focus on this scenario. We
also provide the global fit result with $\eta(\mu)<4$ for comparison, and as
we can see, the result is generally about one order of magnitude worse. Hence,
we should design a forward muon detection with coverage up to $\eta(\mu)$ of 6
for Higgs physics purposes.

$\mu_{\text{production}}^{\text{decay}}$ | $\mu_{VV}^{tt}$ | $\mu_{WW}^{bb}$ | $\mu_{WW}^{cc}$ | $\mu_{WW}^{gg}$ | $\mu_{WW}^{\tau\tau}$ | $\mu_{WW}^{WW}$ | $\mu_{WW}^{ZZ}$ | $\mu_{WW}^{\gamma\gamma}$ | $\mu_{WW}^{\mu\mu}$
---|---|---|---|---|---|---|---|---|---
$\Delta\sigma/\sigma(\%)$ | $2.8$ | $\leavevmode\nobreak\ 0.22\leavevmode\nobreak\ $ | $\leavevmode\nobreak\ 3.6\leavevmode\nobreak\ $ | $\leavevmode\nobreak\ 0.79\leavevmode\nobreak\ $ | $\leavevmode\nobreak\ 1.1\leavevmode\nobreak\ $ | $\leavevmode\nobreak\ 0.40\leavevmode\nobreak\ $ | $\leavevmode\nobreak\ 3.2\leavevmode\nobreak\ $ | $\leavevmode\nobreak\ 1.7\leavevmode\nobreak\ $ | $\leavevmode\nobreak\ 5.7\leavevmode\nobreak\ $
$\mu_{\text{production}}^{\text{decay}}$ | $\mu_{ZZ}^{bb}$ | $\mu_{ZZ}^{cc}$ | $\mu_{ZZ}^{gg}$ | $\mu_{ZZ}^{\tau\tau}$ | $\mu_{ZZ}^{WW}$ | $\mu_{ZZ}^{ZZ}$ | $\mu_{ZZ}^{\gamma\gamma}$ | $\mu_{ZZ}^{\text{inv}}$ | $\mu_{ZZ}^{H}$
$\Delta\sigma/\sigma(\%)$ | $\leavevmode\nobreak\ 0.77\leavevmode\nobreak\ $ | $\leavevmode\nobreak\ 17\leavevmode\nobreak\ $ | $\leavevmode\nobreak\ 3.3\leavevmode\nobreak\ $ | $\leavevmode\nobreak\ 4.8\leavevmode\nobreak\ $ | $\leavevmode\nobreak\ 1.8\leavevmode\nobreak\ $ | $\leavevmode\nobreak\ 11\leavevmode\nobreak\ $ | $\leavevmode\nobreak\ 4.8\leavevmode\nobreak\ $ | $\leavevmode\nobreak\ 0.05\leavevmode\nobreak\ $ |  0.75
Table 4: The input precision used in this study for the Higgs precision fit for a 10 TeV High Energy Muon Collider at 10 ab-1, and the $ZZ$-fusion channel assumes forward coverage. The inclusive Higgs channel studied here is shown in red. The cross-section precision including Top Yukawa coupling determination Liu:2023yrb , various Higgs decay channel Forslund:2022xjq , Higgs invisible decay study Ruhdorfer:2023uea and our study of inclusive Higgs rate at 10 TeV Muon Collider.  | $|\eta(\mu)|<4$ | $|\eta(\mu)|<6$
---|---|---
| MuC@10TeV | +HL-LHC |  +$e^{+}e^{-}$ | MuC@10TeV | +HL-LHC |  +$e^{+}e^{-}$
$\kappa_{b}(\%)$ | $+7.5$ | $+1.7$ | $+0.25$ | $+0.56$ | $+0.53$ | $+0.24$
$-0.25$ | $-0.24$ | $-0.18$ | $-0.23$ | $-0.23$ | $-0.17$
$\kappa_{t}(\%)$ | $+1.4$ | $+1.3$ | $+1.3$ | $+1.4$ | $+1.3$ | $+1.3$
$-7.1$ | $-1.6$ | $-1.2$ | $-1.4$ | $-1.2$ | $-1.2$
$\kappa_{c}(\%)$ | $+7.8$ | $+2.6$ | $+0.91$ | $+1.8$ | $+1.8$ | $+0.89$
$-2.1$ | $-2.1$ | $-0.91$ | $-1.8$ | $-1.8$ | $-0.89$
$\kappa_{g}(\%)$ | $+7.5$ | $+1.7$ | $+0.38$ | $+0.67$ | $+0.63$ | $+0.35$
$-0.52$ | $-0.50$ | $-0.35$ | $-0.45$ | $-0.44$ | $-0.32$
$\kappa_{W}(\%)$ | $+7.5$ | $+1.7$ | $+0.17$ | $+0.51$ | $+0.48$ | $+0.16$
$-0.15$ | $-0.13$ | $-0.099$ | $-0.10$ | $-0.10$ | $-0.090$
$\kappa_{\tau}(\%)$ | $+7.5$ | $+1.8$ | $+0.33$ | $+0.76$ | $+0.71$ | $+0.32$
$-0.62$ | $-0.57$ | $-0.27$ | $-0.56$ | $-0.55$ | $-0.27$
$\kappa_{Z}(\%)$ | $+7.3$ | $+1.9$ | $+0.13$ | $+0.37$ | $+0.37$ | $+0.12$
$-1.4$ | $-0.93$ | $-0.058$ | $-0.25$ | $-0.25$ | $-0.056$
$\kappa_{\gamma}(\%)$ | $+7.6$ | $+1.8$ | $+0.66$ | $+0.97$ | $+0.86$ | $+0.65$
$-0.83$ | $-0.71$ | $-0.64$ | $-0.82$ | $-0.71$ | $-0.64$
$\kappa_{\mu}(\%)$ | $+9.1$ | $+3.8$ | $+2.3$ | $+2.9$ | $+2.5$ | $+1.9$
$-5.0$ | $-3.6$ | $-2.4$ | $-2.9$ | $-2.5$ | $-2.0$
$\text{Br}_{\text{inv}}^{95\%}(\%)$ | $+0.64$ | $+0.63$ | $+0.13$ | $+0.10$ | $+0.10$ | $+0.080$
0 | 0 | 0 | 0 | 0 | 0
$\text{Br}_{\text{unt}}^{95\%}(\%)$ | $+27$ | $+6.6$ | $+0.57$ | $+2.0$ | $+1.9$ | $+0.54$
0 | 0 | 0 | 0 | 0 | 0
$\kappa_{\Gamma}(\%)$ | $+34$ | $+6.9$ | $+0.69$ | $+2.1$ | $+1.9$ | $+0.65$
$-0.45$ | $-0.43$ | $-0.31$ | $-0.41$ | $-0.40$ | $-0.29$
Table 5: The $1\sigma$ upper and lower bound of the 11 parameters in the Higgs
global fit under two different forward coverage at 10 TeV MuC. The constraint
values for $\text{Br}_{\text{inv}}$ and $\text{Br}_{\text{unt}}$ are the
exclusion rate at 95% C.L.

Except for the two branching ratios, which can be only larger or equal to
zero, all the other $\kappa$ parameters can be on either side of unity. In
other words, $\Delta\kappa$ can take either positive or negative values.
However, the constraint results in Table 5 for these $\Delta\kappa$ are not
symmetric around zero due to the asymmetric boundary conditions. The negative
branch can achieve better precision due to scaling degeneracy and physical
boundary conditions. As we stated before, the observable $\mu$ is invariant
under the scaling $\kappa_{f}\rightarrow t\kappa_{f},\kappa_{V}\rightarrow
t\kappa_{V}$ as well as $\kappa_{\Gamma}\rightarrow t^{4}\kappa_{\Gamma}$. As
shown in Equation 5, $\kappa_{\Gamma}$ scales like $\kappa_{f/V}^{2}$. If
$\kappa_{f/V}>1$, the degeneracy condition requires
$\kappa_{\Gamma}\sim\kappa_{f/V}^{4}$, which can be compensated by positive
$\text{Br}_{\text{unt}}$ value. On the contrary, if $\kappa_{f/V}<1$,
$\kappa_{\Gamma}$ is smaller than the quadratic scaling, which requires
negative $\text{Br}_{\text{unt}}$ value. However, physically, the branching
ratio can only be equal to or larger than zero. Hence, the degeneracy
condition cannot be satisfied for the $\kappa$ parameters shown in the highly
precise observables. As a result, on the negative branch, the 68% allowed
$\kappa$ parameters must be determined by the relative error of the $\mu$
observables. This generally enhances the corresponding projected sensitivity
for the $\kappa_{f/V}$ compared to the positive branch, except the
$\kappa_{t}$ under $|\eta(\mu)|<4$ scenario. From the Top Yukawa coupling
determination Liu:2023yrb , the $\Delta\chi^{2}$ only depends on

$O(10^{3})\times(\kappa_{V}\kappa_{t}-1)^{n}\approx
O(10^{3})\times(\Delta\kappa_{V}+\Delta\kappa_{t})^{n}.$ (7)

The sum of $\Delta\kappa_{V}$ and $\Delta\kappa_{t}$ need to be small, which
implies that once the deviation of $\Delta\kappa_{V}$ is large, it will
dominantly determine the opposite deviation of $\Delta\kappa_{t}$. It is the
case that, under $\eta<4$ scenario, both $\Delta\kappa_{Z}$ and
$\Delta\kappa_{W}$ can positively deviate to $\sim 7\%$ due to the poor
precision shown in Table 3.

Here, we provide some approximation to help the reader understand the result,
comment on special features unique to muon colliders, and avoid viewing the
Higgs global fit result as a black box. Let us first focus on the positive
branch of the global fit. The 68% precision of $\Delta\kappa_{Z}$ equals half
of $\mu_{ZZ}^{H}$, as this is the key and leading constraints to confine the
overall scale of the Higgs couplings, exactly as what we anticipated and
motivated this study. The $\kappa_{\Gamma}$ can be constrained to be around
2%. We can get a reasonable estimation on $\kappa_{\Gamma}$ via picking up
four observables dependent on $\kappa_{b},\kappa_{W}$,$\kappa_{Z}$ and
$\kappa_{\Gamma}$. Their correlation gives

$\kappa_{\Gamma}=\dfrac{\left(\mu_{ZZ}^{H}\right)^{2}}{\mu_{WW}^{WW}}\left(\dfrac{\mu_{WW}^{bb}}{\mu^{bb}_{ZZ}}\right)^{2}$
(8)

which gives

$\Delta\kappa_{\Gamma}=\left[4\left(\Delta\mu_{ZZ}^{H}\right)^{2}+\left(\Delta\mu_{WW}^{WW}\right)^{2}+4(\Delta\mu_{WW}^{bb})^{2}+4(\Delta\mu_{ZZ}^{bb})\right]^{1/2}=2.2\%$
(9)

Incorporating other observables and parameters would further improve the
sensitivity of $\kappa_{\Gamma}$. On the other hand, the size of
$\Delta\kappa_{W}$ and $\Delta\kappa_{b}$ are to be one-quarter of
$\Delta\kappa_{\Gamma}$. For instance, the leading constraints on $\kappa_{W}$
can be derived from

$\kappa_{W}^{4}=(\mu_{WW}^{WW})\kappa_{\Gamma}={\left(\mu_{ZZ}^{H}\right)^{2}}\left(\dfrac{\mu_{WW}^{bb}}{\mu^{bb}_{ZZ}}\right)^{2},$
(10)

implying

$\Delta\kappa_{W}=\frac{1}{4}\left[4\left(\Delta\mu_{ZZ}^{H}\right)^{2}+4(\Delta\mu_{WW}^{bb})^{2}+4(\Delta\mu_{ZZ}^{bb})^{2}\right]^{1/2}=0.55\%.$
(11)

Similarly, the leading constraints on $\kappa_{b}$ can be derived from

$\kappa_{b}^{2}=\frac{\mu_{WW}^{bb}\kappa_{W}^{2}}{\mu_{WW}^{WW}}=\frac{\mu_{ZZ}^{H}(\mu_{WW}^{bb})^{2}}{\mu^{bb}_{ZZ}\mu_{WW}^{WW}},$
(12)

implying

$\Delta\kappa_{b}=\frac{1}{2}\left[\left(\Delta\mu_{ZZ}^{H}\right)^{2}+4(\Delta\mu_{WW}^{bb})^{2}+(\Delta\mu_{ZZ}^{bb})^{2}+(\Delta\mu_{WW}^{WW})\right]^{1/2}=0.61\%.$
(13)

This is due to the two most precise observables $\mu_{WW}^{bb}$ and
$\mu_{WW}^{WW}$. The constraint for $\kappa_{\mu},\kappa_{c},\kappa_{\gamma}$
is roughly to half of the precision of $\mu_{WW}^{\mu\mu},\mu_{WW}^{cc}$ and
$\mu_{WW}^{\gamma\gamma}$ as the corresponding exclusive cross-section
dominant the uncertainty. In other channels, such as $\kappa_{\tau}$ and
$\kappa_{g}$, one can use a similar derivation as $\kappa_{b}$ above to see
the leading contribution, where uncertainties of $\kappa_{W}$ and
$\mu_{WW}^{WW}$ contributes. The $\kappa_{t}$ is from an interference
determination, where $\kappa_{W}$ and $\kappa_{Z}$ also enter.

The results from the above reveal an interesting feature: the extracted
coupling precision for Higgs is generally better than half of the total width
precision. This sharply contrasts the results from many future Lepton collider
Higgs factories, such as CEPC, FCC-ee, ILC, and C3. For these colliders, the
main channel to break the scaling degeneracy is the inclusive $Z$ of the
Higgs-associated production, which is also the major production channel of the
Higgs, but the process of $\mu_{ZH}^{ZZ}$ cannot be determined to be much
better than $\kappa_{\Gamma}$. The resulting coupling precision is generally
larger than half of $\kappa_{\Gamma}$, through
$\kappa_{x}=\kappa_{\Gamma}\mu_{ZH}^{xx}/\mu_{ZH}$. For high-energy muon
colliders, the major production coupling $\kappa_{W}$ can be determined via
the chain in Equation 10. One can then use cross-section ratio with
$\mu_{WW}^{WW}$ and $\kappa_{W}^{2}$ to get rid of direct dependence on
$\kappa_{\Gamma}$, which yields a better precision than one would naively
anticipate. One can summarize the situation: the high precision in different
production modes at high-energy muon colliders enables a better Higgs
precision through diversification. One also observes similar effects and
realizes the importance of the $WW$-fusion run at the 360 GeV electron-
positron colliders.

The constraint results differ when switching to the negative branch of the
Higgs precision. The scaling degeneracy is broken due to the non-negative
value of BSM decay partial widths. The boundary of the Higgs precision is
mostly set by minimal total width case, where all BSM decay particle widths
are turned off. Similar to the positive branch discussion, while the global
fit is performed in the fully general setup, we can have some analytical
insights to understand the results. The first set of constraints is mainly
from $\mu_{WW}^{bb}$ and $\mu_{WW}^{WW}$. The lower boundary on $\kappa_{W}$
is set to be within uncertainty of $\mu_{WW}^{WW}$, which would simultaneously
require the minimization of $\kappa_{b}$, as the Higgs branching to $b\bar{b}$
in SM is 57.8%, and to $WW$ is 21.6%, dominating the Higgs width. The lower
boundary on $\kappa_{W}$ also needs to be within the uncertainty of
$\mu_{WW}^{bb}$, which would require maximizing $\kappa_{b}$. Noticing that,

$\operatorname{Br}_{b}^{\rm SM}\mu_{WW}^{bb}+\operatorname{Br}_{W}^{\rm
SM}\mu_{WW}^{WW}=\kappa_{W}^{2}\frac{\kappa_{b}^{2}\operatorname{Br}_{b}^{\rm
SM}+\kappa_{W}^{2}\operatorname{Br}_{W}^{\rm
SM}}{\kappa_{b}^{2}\operatorname{Br}_{b}^{\rm
SM}+\kappa_{W}^{2}\operatorname{Br}_{W}^{\rm SM}+...}\simeq\kappa_{W}^{2},$
(14)

if one makes the approximation to ignore (and the variations) all other SM
partial widths, one can find, $\Delta\kappa_{W}\simeq-0.11\%$. Note that the
sum of the rest of SM branching would not vary much as each coupling would
vary less than the % level (as one can derive from the general equations when
discussing the results in the positive branch), so this is a somewhat
reasonable approximation. We note here that the discussion on the negative
branch of results is less rigorous as its boundary condition is more complex
and involves all Higgs couplings. On the other hand, such an approximation is
no longer valid when discussing the positive side of Higgs precision.
$\kappa_{b}$ can be approximately understood by using $\mu_{WW}^{WW}$ and the
lower bound on $\kappa_{W}$ derived earlier. One can also obtain the results
by performing a two-parameter, $\kappa_{W}$ and $\kappa_{b}$ with input
precision of $\mu_{WW}^{WW}$ and $\mu_{WW}^{bb}$, assuming all other width
variations can be ignored to obtain similar results above.

Figure 7:  11-parameter Higgs global fit where the constraint on total width
$\kappa_{\Gamma}$ is determined from the previous 11 parameters. The muon
collider results are under the forward detection $|\eta(\mu)|<6$. The light-
colored bars stand for the results without combining with HL-LHC. The global
fit at CEPC without HL-LHC is under a 10-parameter fit (excluding
$\kappa_{t}$) to avoid a flat direction.

In Figure 7, we show the results for different high energy muon collider
scenarios, $e^{+}e^{-}$ Higgs factories, and combined with HL-LHC777The HL-LHC
precision results (Scenario 2) are taken from
https://twiki.cern.ch/twiki/bin/view/LHCPhysics/GuidelinesCouplingProjections2018.
and $e^{+}e^{-}$ Higgs factories888Here we use the CEPC input precision
CEPCStudyGroup:2018ghi ; An:2018dwb as a representative precision for
$e^{+}e^{-}$ Higgs factories, where in details ILC, C3, FCC-ee and CEPC differ
slightly.. To avoid the complexities of asymmetric uncertainties and as many
inputs for other colliders are missing, we show the symmetrized uncertainty
here, averaging the absolute precision value in the positive and negative
directions. The last bar displays the constraint for the $\kappa_{\Gamma}$,
highlighting its signature and importance. One can extract many interesting
messages from this plot. With our proposed channel and forward coverage, one
can obtain a closed fit at a 3 TeV muon collider alone. The general achievable
precision is at % level, a factor of a few worse than a typical $e^{+}e^{-}$
Higgs factory. On the other hand, a 10 TeV muon collider alone would achieve
sub-percent level precision for many couplings, comparable or, in some cases,
better than $e^{+}e^{-}$ Higgs factories. We found that combining with the HL-
LHC would not improve the Higgs precision much for a 10 TeV muon collider or
$e^{+}e^{-}$ Higgs factories, which is different for the 3 TeV muon collider.
The feature of $e^{+}e^{-}$ Higgs factories lies in the measurement of the
Higgsstrahlung process, which can limit $\kappa_{Z}$ as low as around 0.1%.
Due to the center-of-mass energy limit, $e^{+}e^{-}$ Higgs factories (except
for CLIC or C3) cannot constrain $\kappa_{t}$, manifested by the light yellow
bar. After combining with HL-LHC simulated observable, $\kappa_{t}$ can be
measured to be around 3%, similar to 3 TeV muon collider. While the 10 TeV
muon collider can probe $\Delta\kappa_{t}$ to around 1.3%, benefiting from the
indirect channels and higher luminosity. Notably, although the Higgs total
width precision at a 10 TeV muon collider, enabled by our proposed channel
with forward muon coverage, is worse than that of $e^{+}e^{-}$ Higgs
factories, the resulting coupling precision is in many cases better, in
particular $\kappa_{b}$ and $\kappa_{W}$. Lastly, as shown in the cyan bars,
combining high energy muon collider and $e^{+}e^{-}$ machine would improve
many precisions significantly, more than naive statistical additions. This
demonstrates the complementarity between different colliders and shows us the
importance of producing and detecting Higgs in many channels. Our proposed
inclusive measurement relies on a non-trivial development of a forward muon
detector but delivers a robust Higgs width and coupling determination on-
shell. Note that one can also determine the Higgs overall coupling through the
on-shell, off-shell Higgs measurements Kauer:2012hd ; Forslund:2023reu ;
Caola:2013yja ; Campbell:2015vwa at muon colliders Forslund:2023reu . Such a
determination of Higgs width and coupling strength benefits from not relying
on forward muon tagging. However, it depends on the assumption of Higgs’ boson
coupling modification being invariant across the scales of the measurements
Logan:2014ppa . In the same fitting framework, our determination of Higgs
couplings in many cases is a factor of few better than this complementary
measurement (see Fig.7 of Ref. Forslund:2023reu ).

##  IV Conclusion

We aim to solve the shortcoming of high energy muon collider, which is
anticipated not to be able to break the Higgs coupling-width degeneracy or
often simplified as incapable of determining the Higgs boson width. We show
that through the $ZZ$-fusion channel with a forward muon detection, one can
measure the Higgs rate inclusively. Such a measurement breaks the coupling-
width degeneracy and enables the determination of the Higgs coupling to sub-
percent level and the total width to $(-0.41\%,+2.1\%)$ at 10 TeV muon
collider alone. The results are summarized in Table 5 and Figure 7. Here, we
highlight a few key findings. Due to the multiple Higgs production modes and
low background, a high-energy muon collider is also highly complementary to
other Higgs factories, combining them to improve the Higgs precision
significantly. Note that we performed a cut-and-count analysis here, which can
be improved with multi-variable analysis that leads to better results.

Forward muon tagging is critical for such inclusive Higgs measurements. While
one naively considers the recoil mass to provide a sharp peak to separate
signal and background, the muon energy reconstruction and beam energy spread
render such distribution rather broad, invalidating the potential strong usage
of such a variable. However, being a lepton collider, the muon collider is
still clean and enjoys a low background. We found that a signal-background
ratio of around 1/30 is achievable. The critical implication is that one
should be able to tag the high-energy muons without high demand on the
accuracy of energy reconstruction and the energy loss in the shielding region.

For a 10 TeV muon collider, most forward muons reside with $|\eta(\mu)|<6$.
Covering this region improves the precision by a factor of 20, as shown in
Table 3. For a 3 TeV muon collider, the muons are slightly less forward but
still benefit from having a forward muon detector, as shown in Table 6.

The inclusive Higgs measurement is unique for high-energy lepton colliders; it
enables often-dubbed “model-independent” Higgs coupling determination where
the total width is a free parameter. New studies, including forward muon
detector design and performance estimation, precision timing, beam-induced
backgrounds, and precision calculation of the signal and background rates, are
all highly motivated future steps to fully realize the high energy muon
collider’s Higgs physics potential.

###### Acknowledgements.

We thank Matthew Forslund, Simone Pagan Griso, Tao Han, Donatella Lucchesi,
Patrick Meade, and Andrea Wulzer for helpful discussions. This study was
supported in part by the DOE Grant DE-SC0022345 and DE-SC0011842. Z.L. and
K.F.L acknowledge KITP for hosting the muon collider workshop, where this
study was inspired, which was supported in part by the National Science
Foundation under Grant No. NSF PHY-1748958. Z.L. and K.F.L. acknowledge the
support of the Aspen Center for Physics, supported by National Science
Foundation grant PHY-2210452, where part of this work was completed. The data
associated with the figures in this paper can be accessed via Github .

##  Appendix A $ZZ$-fusion for the Inclusive Rate at 3 TeV Muon Collider

Figure 8: The reconstruction level $\eta(\mu)$ distribution are after pre-
selection and $p_{T}(\mu\mu)>50\leavevmode\nobreak\ $GeV at 3 TeV muon
collider.

Benchmark | $|\eta(\mu)|<4$ | $|\eta(\mu)|<6$ | $|\eta(\mu)|<8$
---|---|---|---
$\Delta\sigma/\sigma$ | 6.2% | 3.9% | 3.9%

Table 6: The 68% projected sensitivity on the Higgs inclusive rate from $ZZ$ fusion at 3 TeV Muon Collider with different forward detector scenarios. | $|\eta(\mu)|<4$ | $|\eta(\mu)|<6$
---|---|---
| MuC@3TeV | +HL-LHC |  +$e^{+}e^{-}$ | MuC@3TeV | +HL-LHC |  +$e^{+}e^{-}$
$\kappa_{b}(\%)$ | $+6.5$ | $+2.7$ | $+0.37$ | $+2.5$ | $+2.0$ | $+0.36$
$-0.90$ | $-0.81$ | $-0.30$ | $-0.86$ | $-0.79$ | $-0.29$
$\kappa_{t}(\%)$ | $+9.1$ | $+3.5$ | $+2.8$ | $+9.2$ | $+3.2$ | $+2.8$
$-8.6$ | $-2.8$ | $-2.8$ | $-6.4$ | $-2.8$ | $-2.8$
$\kappa_{c}(\%)$ | $+9.2$ | $+6.9$ | $+1.0$ | $+6.3$ | $+6.1$ | $+1.0$
$-6.9$ | $-6.9$ | $-1.0$ | $-6.3$ | $-6.3$ | $-1.0$
$\kappa_{g}(\%)$ | $+6.7$ | $+3.0$ | $+0.52$ | $+2.8$ | $+2.3$ | $+0.51$
$-1.9$ | $-1.4$ | $-0.49$ | $-1.6$ | $-1.3$ | $-0.48$
$\kappa_{W}(\%)$ | $+6.6$ | $+2.7$ | $+0.30$ | $+2.3$ | $+1.9$ | $+0.29$
$-0.46$ | $-0.39$ | $-0.26$ | $-0.37$ | $-0.35$ | $-0.25$
$\kappa_{\tau}(\%)$ | $+6.8$ | $+2.9$ | $+0.44$ | $+3.1$ | $+2.2$ | $+0.44$
$-2.3$ | $-1.3$ | $-0.38$ | $-2.1$ | $-1.3$ | $-0.38$
$\kappa_{Z}(\%)$ | $+3.1$ | $+2.5$ | $+0.13$ | $+2.0$ | $+1.8$ | $+0.13$
$-2.7$ | $-1.2$ | $-0.063$ | $-1.0$ | $-0.82$ | $-0.063$
$\kappa_{\gamma}(\%)$ | $+7.2$ | $+2.9$ | $+0.99$ | $+3.9$ | $+2.3$ | $+0.99$
$-3.2$ | $-1.4$ | $-0.98$ | $-3.2$ | $-1.4$ | $-0.98$
$\kappa_{\mu}(\%)$ | $+20$ | $+5.4$ | $+2.6$ | $+13$ | $+5.0$ | $+2.6$
$-23$ | $-5.0$ | $-2.7$ | $-15$ | $-4.8$ | $-2.6$
$\text{Br}_{\text{inv}}^{95\%}(\%)$ | $+0.57$ | $+0.56$ | $+0.13$ | $+0.56$ | $+0.56$ | $+0.13$
0 | 0 | 0 | 0 | 0 | 0
$\text{Br}_{\text{unt}}^{95\%}(\%)$ | $+23$ | $+10$ | $+0.58$ | $+9.0$ | $+7.5$ | $+0.59$
0 | 0 | 0 | 0 | 0 | 0
$\kappa_{\Gamma}(\%)$ | $+28$ | $+11$ | $+0.88$ | $+9.6$ | $+7.9$ | $+0.87$
$-1.6$ | $-1.4$ | $-0.56$ | $-1.5$ | $-1.3$ | $-0.54$
Table 7: The $1\sigma$ upper and lower bound of the 11 parameters in the Higgs
global fit under two different, forward coverage at 3 TeV MuC. The constraint
values for $\text{Br}_{\text{inv}}$ and $\text{Br}_{\text{unt}}$ are the
exclusion rate at 95% C.L.

In this appendix, Table 6 shows the result of a similar analysis at 3 TeV Muon
Collider with an integrated luminosity of $1\leavevmode\nobreak\
\text{ab}^{-1}$. Note that 3 TeV Muon Collider performs better than 10 TeV
under the scenario of $|\eta(\mu)|<4$. The reason comes from the different
behavior of the signal process. The rapidity of forward muon at $\sqrt{s}=3$
TeV (shown in Figure 8) is more central. Nevertheless, forward muon coverage
can improve precision by more than 30%, from 6.2% to 3.9%. We show the Higgs
precision global fit results with 3 TeV muon collider in Table 7 for two
forward muon detector coverages.

## References

  * (1) J. de Blas et al., “Higgs Boson Studies at Future Particle Colliders,” JHEP 01 (2020) 139, arXiv:1905.03764 [hep-ph].
  * (2) R. K. Ellis et al., “Physics Briefing Book: Input for the European Strategy for Particle Physics Update 2020,” arXiv:1910.11775 [hep-ex].
  * (3) S. Dawson et al., “Report of the Topical Group on Higgs Physics for Snowmass 2021: The Case for Precision Higgs Physics,” in Snowmass 2021.  9, 2022.  arXiv:2209.07510 [hep-ph].
  * (4) M. Narain et al., “The Future of US Particle Physics – The Snowmass 2021 Energy Frontier Report,” arXiv:2211.11084 [hep-ex].
  * (5) F. An et al., “Precision Higgs physics at the CEPC,” Chin. Phys. C 43 (2019) no. 4, 043002, arXiv:1810.09037 [hep-ex].
  * (6) CEPC Study Group Collaboration, M. Dong et al., “CEPC Conceptual Design Report: Volume 2 - Physics & Detector,” arXiv:1811.10545 [hep-ex].
  * (7) CEPC Physics Study Group Collaboration, H. Cheng et al., “The Physics potential of the CEPC. Prepared for the US Snowmass Community Planning Exercise (Snowmass 2021),” in Snowmass 2021.  5, 2022.  arXiv:2205.08553 [hep-ph].
  * (8) D. M. Asner et al., “ILC Higgs White Paper,” in Snowmass 2013: Snowmass on the Mississippi.  10, 2013.  arXiv:1310.0763 [hep-ph].
  * (9) ILC International Development Team Collaboration, A. Aryshev et al., “The International Linear Collider: Report to Snowmass 2021,” arXiv:2203.07622 [physics.acc-ph].
  * (10) M. Benedikt, A. Blondel, O. Brunner, M. Capeans Garrido, F. Cerutti, J. Gutleber, P. Janot, J. M. Jimenez, V. Mertens, A. Milanese, K. Oide, J. A. Osborne, T. Otto, Y. Papaphilippou, J. Poole, L. J. Tavian, and F. Zimmermann, “FCC-ee: The Lepton Collider: Future Circular Collider Conceptual Design Report Volume 2. Future Circular Collider,” Tech. Rep. 2, CERN, Geneva, 2019.  https://cds.cern.ch/record/2651299.
  * (11) G. Bernardi et al., “The Future Circular Collider: a Summary for the US 2021 Snowmass Process,” arXiv:2203.06520 [hep-ex].
  * (12) CLICdp, CLIC Collaboration, T. K. Charles et al., “The Compact Linear Collider (CLIC) - 2018 Summary Report,” arXiv:1812.06018 [physics.acc-ph].
  * (13) C. Vernieri et al., “Strategy for Understanding the Higgs Physics: The Cool Copper Collider,” JINST 18 (2023) no. 07, P07053, arXiv:2203.07646 [hep-ex].
  * (14) D. Buttazzo, R. Franceschini, and A. Wulzer, “Two Paths Towards Precision at a Very High Energy Lepton Collider,” JHEP 05 (2021) 219, arXiv:2012.11555 [hep-ph].
  * (15) T. Han, D. Liu, I. Low, and X. Wang, “Electroweak couplings of the Higgs boson at a multi-TeV muon collider,” Phys. Rev. D 103 (2021) no. 1, 013002, arXiv:2008.12204 [hep-ph].
  * (16) T. Han, W. Kilian, N. Kreher, Y. Ma, J. Reuter, T. Striegl, and K. Xie, “Precision test of the muon-Higgs coupling at a high-energy muon collider,” JHEP 12 (2021) 162, arXiv:2108.05362 [hep-ph].
  * (17) M. Forslund and P. Meade, “High precision higgs from high energy muon colliders,” JHEP 08 (2022) 185, arXiv:2203.09425 [hep-ph].
  * (18) K. M. Black et al., “Muon Collider Forum Report,” arXiv:2209.01318 [hep-ex].
  * (19) M. Chen and D. Liu, “Top Yukawa Coupling at the Muon Collider,” arXiv:2212.11067 [hep-ph].
  * (20) M. Ruhdorfer, E. Salvioni, and A. Wulzer, “Invisible Higgs boson decay from forward muons at a muon collider,” Phys. Rev. D 107 (2023) no. 9, 095038, arXiv:2303.14202 [hep-ph].
  * (21) Z. Liu, K.-F. Lyu, I. Mahbub, and L.-T. Wang, “Top Yukawa Coupling Determination at High Energy Muon Collider,” arXiv:2308.06323 [hep-ph].
  * (22) M. Forslund and P. Meade, “Precision Higgs Width and Couplings with a High Energy Muon Collider,” arXiv:2308.02633 [hep-ph].
  * (23) T. Han, D. Liu, I. Low, and X. Wang, “Electroweak Scattering at the Muon Shot,” arXiv:2312.07670 [hep-ph].
  * (24) E. Celada, T. Han, W. Kilian, N. Kreher, Y. Ma, F. Maltoni, D. Pagani, J. Reuter, T. Striegl, and K. Xie, “Probing Higgs-muon interactions at a multi-TeV muon collider,” arXiv:2312.13082 [hep-ph].
  * (25) T. Han, Y. Ma, and K. Xie, “High energy leptonic collisions and electroweak parton distribution functions,” Phys. Rev. D 103 (2021) no. 3, L031301, arXiv:2007.14300 [hep-ph].
  * (26) T. Han, Y. Ma, and K. Xie, “Quark and gluon contents of a lepton at high energies,” JHEP 02 (2022) 154, arXiv:2103.09844 [hep-ph].
  * (27) G. L. Kane, W. W. Repko, and W. B. Rolnick, “The Effective W+-, Z0 Approximation for High-Energy Collisions,” Phys. Lett. B 148 (1984) 367–372.
  * (28) S. Dawson, “The Effective W Approximation,” Nucl. Phys. B 249 (1985) 42–60.
  * (29) B. Fornal, A. V. Manohar, and W. J. Waalewijn, “Electroweak Gauge Boson Parton Distribution Functions,” JHEP 05 (2018) 106, arXiv:1803.06347 [hep-ph].
  * (30) J. Chen, T. Han, and B. Tweedie, “Electroweak Splitting Functions and High Energy Showering,” JHEP 11 (2017) 093, arXiv:1611.00788 [hep-ph].
  * (31) R. Ruiz, A. Costantini, F. Maltoni, and O. Mattelaer, “The Effective Vector Boson Approximation in high-energy muon collisions,” JHEP 06 (2022) 114, arXiv:2111.02442 [hep-ph].
  * (32) J. Alwall, R. Frederix, S. Frixione, V. Hirschi, F. Maltoni, O. Mattelaer, H. S. Shao, T. Stelzer, P. Torrielli, and M. Zaro, “The automated computation of tree-level and next-to-leading order differential cross sections, and their matching to parton shower simulations,” JHEP 07 (2014) 079, arXiv:1405.0301 [hep-ph].
  * (33) C. Bierlich et al., “A comprehensive guide to the physics and usage of PYTHIA 8.3,” arXiv:2203.11601 [hep-ph].
  * (34) DELPHES 3 Collaboration, J. de Favereau, C. Delaere, P. Demin, A. Giammanco, V. Lemaître, A. Mertens, and M. Selvaggi, “DELPHES 3, A modular framework for fast simulation of a generic collider experiment,” JHEP 02 (2014) 057, arXiv:1307.6346 [hep-ex].
  * (35) A. Mertens, “New features in Delphes 3,” J. Phys. Conf. Ser. 608 (2015) no. 1, 012045.
  * (36) D. Ally, L. Carpenter, T. Holmes, L. Lee, and P. Wagenknecht, “Strategies for Beam-Induced Background Reduction at Muon Colliders,” arXiv:2203.06773 [physics.ins-det].
  * (37) Muon Collider Collaboration, S. Jindariani et al., “Promising Technologies and R&D Directions for the Future Muon Collider Detectors,” arXiv:2203.07224 [physics.ins-det].
  * (38) Muon Collider Collaboration, N. Bartosik et al., “Simulated Detector Performance at the Muon Collider,” arXiv:2203.07964 [hep-ex].
  * (39) H. Al Ali et al., “The muon Smasher’s guide,” Rept. Prog. Phys. 85 (2022) no. 8, 084201, arXiv:2103.14043 [hep-ph].
  * (40) C. Accettura et al., “Towards a muon collider,” Eur. Phys. J. C 83 (2023) no. 9, 864, arXiv:2303.08533 [physics.acc-ph].
  * (41) FCC Collaboration, A. Abada et al., “FCC-hh: The Hadron Collider: Future Circular Collider Conceptual Design Report Volume 3,” Eur. Phys. J. ST 228 (2019) no. 4, 755–1107.
  * (42) M. Selvaggi, “Delphes card for muon collider.” https://indico.cern.ch/event/957299/contributions/4023467/attachments/2106044/3541874/delphes_card_mucol_mdi_.pdf, 2020\.
  * (43) V. D. Barger, M. S. Berger, J. F. Gunion, and T. Han, “Higgs Boson physics in the s channel at $\mu^{+}\mu^{-}$ colliders,” Phys. Rept. 286 (1997) 1–51, arXiv:hep-ph/9602415.
  * (44) T. Han and Z. Liu, “Potential precision of a direct measurement of the Higgs boson total width at a muon collider,” Phys. Rev. D 87 (2013) no. 3, 033007, arXiv:1210.7803 [hep-ph].
  * (45) J. de Blas, J. Gu, and Z. Liu, “Higgs boson precision measurements at a 125 GeV muon collider,” Phys. Rev. D 106 (2022) no. 7, 073007, arXiv:2203.04324 [hep-ph].
  * (46) N. Kauer and G. Passarino, “Inadequacy of zero-width approximation for a light Higgs boson signal,” JHEP 08 (2012) 116, arXiv:1206.4803 [hep-ph].
  * (47) F. Caola and K. Melnikov, “Constraining the Higgs boson width with ZZ production at the LHC,” Phys. Rev. D 88 (2013) 054024, arXiv:1307.4935 [hep-ph].
  * (48) J. M. Campbell and R. K. Ellis, “Higgs Constraints from Vector Boson Fusion and Scattering,” JHEP 04 (2015) 030, arXiv:1502.02990 [hep-ph].
  * (49) H. E. Logan, “Hiding a Higgs width enhancement from off-shell gg(→h*)→ZZ measurements,” Phys. Rev. D 92 (2015) no. 7, 075038, arXiv:1412.7577 [hep-ph].
  * (50) W. Buchmuller and D. Wyler, “Effective Lagrangian Analysis of New Interactions and Flavor Conservation,” Nucl. Phys. B 268 (1986) 621–653.
  * (51) B. Grzadkowski, M. Iskrzynski, M. Misiak, and J. Rosiek, “Dimension-Six Terms in the Standard Model Lagrangian,” JHEP 10 (2010) 085, arXiv:1008.4884 [hep-ph].
  * (52) S. Di Vita, G. Durieux, C. Grojean, J. Gu, Z. Liu, G. Panico, M. Riembau, and T. Vantalon, “A global view on the Higgs self-coupling at lepton colliders,” JHEP 02 (2018) 178, arXiv:1711.03978 [hep-ph].
  * (53) T. Barklow, K. Fujii, S. Jung, R. Karl, J. List, T. Ogawa, M. E. Peskin, and J. Tian, “Improved Formalism for Precision Higgs Coupling Fits,” Phys. Rev. D 97 (2018) no. 5, 053003, arXiv:1708.08912 [hep-ph].
  * (54) S. Dawson, S. Homiller, and S. D. Lane, “Putting standard model EFT fits to work,” Phys. Rev. D 102 (2020) no. 5, 055012, arXiv:2007.01296 [hep-ph].
  * (55) SMEFiT Collaboration, J. J. Ethier, G. Magni, F. Maltoni, L. Mantani, E. R. Nocera, J. Rojo, E. Slade, E. Vryonidou, and C. Zhang, “Combined SMEFT interpretation of Higgs, diboson, and top quark data from the LHC,” JHEP 11 (2021) 089, arXiv:2105.00006 [hep-ph].
  * (56) L. J. Dixon and M. S. Siu, “Resonance continuum interference in the diphoton Higgs signal at the LHC,” Phys. Rev. Lett. 90 (2003) 252001, arXiv:hep-ph/0302233.
  * (57) M. E. Peskin, “Comparison of LHC and ILC Capabilities for Higgs Boson Coupling Measurements,” arXiv:1207.2516 [hep-ph].
  * (58) B. A. Dobrescu and J. D. Lykken, “Coupling Spans of the Higgs-like Boson,” JHEP 02 (2013) 073, arXiv:1210.3342 [hep-ph].
  * (59) L. J. Dixon and Y. Li, “Bounding the Higgs Boson Width Through Interferometry,” Phys. Rev. Lett. 111 (2013) 111802, arXiv:1305.3854 [hep-ph].
  * (60) T. Han, Z. Liu, Z. Qian, and J. Sayre, “Improving Higgs Coupling Measurements Through ZZ Fusion at the ILC,” Phys. Rev. D 91 (2015) 113007, arXiv:1504.01399 [hep-ph].
  * (61) F. Coradeschi, D. de Florian, L. J. Dixon, N. Fidanza, S. Höche, H. Ita, Y. Li, and J. Mazzitelli, “Interference effects in the $H(\rightarrow\gamma\gamma)+2$ jets channel at the LHC,” Phys. Rev. D 92 (2015) no. 1, 013004, arXiv:1504.05215 [hep-ph].
  * (62) J. Campbell, M. Carena, R. Harnik, and Z. Liu, “Interference in the $gg\rightarrow h\rightarrow\gamma\gamma$ On-Shell Rate and the Higgs Boson Total Width,” Phys. Rev. Lett. 119 (2017) no. 18, 181801, arXiv:1704.08259 [hep-ph]. [Addendum: Phys.Rev.Lett. 119, 199901 (2017)].
  * (63) D. Curtin et al., “Exotic decays of the 125 GeV Higgs boson,” Phys. Rev. D 90 (2014) no. 7, 075004, arXiv:1312.4992 [hep-ph].
  * (64) Z. Liu, L.-T. Wang, and H. Zhang, “Exotic decays of the 125 GeV Higgs boson at future $e^{+}e^{-}$ lepton colliders,” Chin. Phys. C 41 (2017) no. 6, 063102, arXiv:1612.09284 [hep-ph].
  * (65) S. Jung, Z. Liu, L.-T. Wang, and K.-P. Xie, “Probing Higgs boson exotic decays at the LHC with machine learning,” Phys. Rev. D 105 (2022) no. 3, 035008, arXiv:2109.03294 [hep-ph].