a]INFN Sezione di Padova, Padova, Italy b]INFN Sezione di Torino, Torino,
Italy c]Università di Padova, Padova, Italy d]European Organization for
Nuclear Research, Geneva, Switzerland e]INFN Sezione di Trieste, Trieste,
Italy f]Università di Trieste, Trieste, Italy g]INFN Sezione di Roma, Roma,
Italy h]Università La Sapienza, Roma, Italy i]INFN Sezione di Pavia, Pavia,
Italy j]INFN Sezione di Bari, Bari, Italy k]Università di Bari, Bari, Italy
l]Laboratório de Instrumentação e Física Experimental de Partículas, Lisboa,
Portugal m]Stony Brook University, Stony Brook, United States n]University of
Birmingham, Birmingham, United Kingdom o]Fermi National Accelerator
Laboratory, Batavia, United States p]Université Catholique de Louvain,
Louvain-la-Neuve, Belgium q]Stockholms Universitet, Stockholm, Sweden
r]Lawrence Berkeley National Laboratory, Berkeley, United States ]

Contacts: Massimo Casarsa<EMAIL_ADDRESS>and Lorenzo Sestini
<EMAIL_ADDRESS>
# Higgs Physics at a $\sqrt{s}=3$ TeV Muon Collider with detailed detector
simulation

Paolo Andreetto  Nazar Bartosik  Laura Buonincontri  Daniele Calzolari  Vieri
Candelise  Massimo Casarsa  Luca Castelli  Mauro Chiesa  Anna Colaleo  Giacomo
Da Molin  Matthew Forslund  Luca Giambastiani  Alessio Gianelle  Karol Krizka
Sergo Jindariani  Anton Lechner  Donatella Lucchesi  Leo Mareso  Paola
Mastrapasqua  Patrick Meade  Alessandro Montella  Simone Pagan Griso  Nadia
Pastrone  Lorenzo Sestini  Rosamaria Venditti  Angela Zaza  Davide Zuliani  [
[  [  [  [  [  [  [  [  [  [  [  [  [  [  [  [  [  [

###### Abstract

The Muon Collider is one of the most promising future collider facilities with
the potential to reach multi-TeV center-of-mass energy and high luminosity.
Due to the significant Higgs boson production cross section in muon collisions
at these high energies, the collider can be considered a Higgs factory. It
holds the capability to significantly advance our understanding of the Higgs
sector to an unprecedented level of precision. However, the presence of beam-
induced background resulting from the decay of the beam muons poses unique
challenges for detector development and event reconstruction. In this paper,
the prospects for various measurements of the Higgs boson production cross
sections at a $\sqrt{s}=3$ TeV collider are presented using a detailed
detector simulation in a realistic environment. The study demonstrates the
feasibility of achieving high precision measurements of the Higgs boson
production cross sections with the current state-of-the-art detector design.
In addition, the paper discusses the detector requirements necessary for
obtaining such resolutions and for measuring the Higgs trilinear self-
coupling.

##  1 Introduction

The Higgs boson ($H$) is considered a portal to new physics, because it is
connected to some of the fundamental questions about the Universe [1],
including the mechanism of Electroweak Symmetry Breaking (EWSB), the origin of
the masses, the matter-antimatter asymmetry, and the nature of dark matter.
The EWSB [2, 3, 4, 5] is formulated via the scalar potential, which is written
below in a form that includes possible deviations from the Standard Model
(SM):

$V(h)=\frac{1}{2}m_{H}^{2}h^{2}+\lambda_{3}vh^{3}+\frac{1}{4}\lambda_{4}h^{4}+\mathcal{O}(h^{4}),$
(1)

where $h$ represents the field of the physical Higgs boson, which is what
remains after the EWSB. The Higgs boson mass is $m_{H}=\sqrt{2}\lambda v\simeq
125$ GeV111In this paper natural units where $\hbar=c=1$ are used.,
$v=\frac{1}{\sqrt{\sqrt{2}G_{F}}}\simeq 246$ GeV is the vacuum expectation
value, and $G_{F}$ is the Fermi constant. In the SM,
$\lambda_{3}=\lambda_{4}=\lambda=m_{H}^{2}/(2\,v^{2})$ represents the strength
of the coupling of the Higgs boson to itself. The couplings of the Higgs boson
to the elementary fermions and bosons generate the particle masses. The most
recent review of the expected experimental precision on such couplings,
including projections for the High-Luminosity LHC (HL-LHC), is presented in
the Snowmass Energy Frontier report [6]. Figure 1 summarises the expected
precision on the Higgs production cross sections and coupling modifiers
$\kappa_{i}$ [7] with 3000 fb-1 of data that will be collected by ATLAS and
CMS at the end of HL-LHC.

In the case of the Higgs self-interaction, the 95% C.L. limits on the coupling
modifier $\kappa_{\lambda_{3}}=\frac{\lambda_{3}}{\lambda_{3}^{\mathrm{SM}}}$
(with $\lambda_{3}^{\mathrm{SM}}$ being the expected SM value) are
$-0.6<\kappa_{\lambda_{3}}<6.6$ for ATLAS [8] and
$-1.24<\kappa_{\lambda_{3}}<6.49$ for CMS [9] with the current LHC data. With
HL-LHC data, a 50% precision on the SM value of the self-coupling is expected,
which is not sufficient to determine the potential shape.

Figure 1: Projections of the expected precision of the ATLAS and CMS
combination on the Higgs production cross sections (left) and $\kappa_{i}$
coupling modifiers (right) including HL-LHC [1] data.

In order to reveal new physics in the Higgs sector, it is necessary to
determine the Higgs couplings to elementary fermions and bosons with a
percent-level precision and achieve a few percent precision in the measurement
of the Higgs self-coupling. This underscores the need for a dedicated Higgs
factory.

Linear and circular $e^{+}e^{-}$ machines have been the subject of debate for
a long time as well as $\mu^{+}\mu^{-}$ colliders. Recently $\mu^{+}\mu^{-}$
collisions are regaining attention on the international scene, thanks to the
impressive physics potential and to new studies on technical feasibility [10].

A Muon Collider (MuC) is a proposed future accelerator in which $\mu^{+}$ and
$\mu^{-}$ beams collide at very high center-of-mass energies in a circular
machine, thanks to the negligible energy loss of muons due to synchrotron
radiation. The current proposal of the International Muon Collider
Collaboration (IMCC) [11] aims at 10 TeV center-of-mass energy collisions,
with a possible initial phase at 3 TeV [10]. A comprehensive review of the
physics potential of MuC experiments at both $\sqrt{s}=3$ and $\sqrt{s}=10$
TeV can be found in Refs. [12, 10, 13].

The determination of the Higgs couplings and self-coupling with an
unprecedented accuracy will be feasible with 10 TeV center-of-mass muon
collisions. However, the possibility of a $\sqrt{s}=3$ TeV run presents a
valuable opportunity for Higgs boson research. Indeed, a MuC can already be
viewed as a Higgs factory at $\sqrt{s}=3$ TeV, thanks to the high Higgs boson
production cross section and machine luminosity (Sec. 2) and to the detector
performance (Sec. 3).

This paper presents the results on the Higgs boson studies obtained at a
$\sqrt{s}=3$ TeV MuC with a detailed detector simulation and including the
beam-induced background (BIB). The BIB presents a challenge to extracting the
physics in a MuC experiment. However, as will be shown in the next sections,
the precision achieved is comparable to that claimed by MuC parametric
simulation studies that do not take the BIB into account. In these parametric
simulations, the detector response is described by functions defined with some
assumptions, discussed later in the paper, and applied directly to generator-
level quantities.

The experimental facility is introduced in Sec. 2, focusing on the interaction
region and the detector shielding. The paper, then, describes the generation
of physics and background samples, the detector model and the detector
simulation, and the event reconstruction in the presence of the BIB (Sec. 3).
The results obtained for the Higgs boson production cross sections times the
branching ratio for several decay channels are discussed in detail in Secs.
4.1-4.5, and are compared to other parametric simulation studies in Sec. 4.6.
The double Higgs production cross section and the self-coupling precision
determination are presented in Secs. 6.1 and 6.2, respectively. The lessons
learned on the detector requirements that are necessary for Higgs physics are
reported in Sec. 7. Finally, Sec. 8 summarizes the prospects for Higgs physics
with a MuC.

##  2 Experimental environment

A MuC facility consists of several sub-systems, which are described in detail
in Refs. [10] and [14]. Here, the main components of a $\sqrt{s}=3$ TeV MuC
complex, relevant for evaluating the background effects on the detector, are
briefly outlined. The structure includes the elements discussed in Ref. [10]
in the exact same sequence: a muon injector, an acceleration ring, and a
collision ring. Muons are produced in the decay of pions generated by a high-
intensity proton beam impinging on a target with high power (in the order of
megawatts). The original muon beam, which has large longitudinal and
transverse emittances, is organized in bunches and is directed into a cooling
channel. The choice of the technique for cooling the muon beams is dictated by
the short lifetime of the muon. The proposed technique is the ionization
cooling, currently under investigation [15]. The next step is the beam
acceleration. A ring will be dedicated to that operation with a size that can
vary depending on the final center-of-mass energy, on the magnet technology
available at the time of construction, and on the availability of existing
tunnels. Finally, the two muon beams are injected into the collider ring.

###  2.1 Interaction region and beam-induced background

Figure 2: Left: Arrival times of the main BIB components at the detector
relative to the bunch crossing. The light gray band between -1 and 15 ns
indicates the assumed readout window for the detector. Right: Energy spectra
of the BIB particles that reach the detector within the readout window. The
solid and dashed lines refer to the center-of-mass energy of 3 TeV and 1.5
TeV, respectively.

Interaction regions (IR) for a muon collider have been designed by the Muon
Accelerator Program (MAP) [16] in the US for different center-of-mass
energies: 125 GeV, 1.5 TeV, 3 TeV, and 6 TeV [17]. The studies presented in
this paper are based on MAP’s configuration at $\sqrt{s}=1.5$ TeV, since a BIB
sample for $\sqrt{s}=3$ TeV was not available at the start of the Higgs
analyses. Later in this section, a cross-check for validating this
approximation is presented.

This IR was optimized by MAP [18] to maximize the achievable luminosity while
minimizing the fluxes of background particles that are produced by the decay
of muons in the beams and reach the detector region. In fact, at the center-
of-mass energy of 1.5 TeV, the dominant source of background is constituted by
electromagnetic showers generated by the interactions of high-energy electrons
and positrons with the machine elements. The solution devised by MAP to shield
the detector from this irreducible background involves the use of two cone-
shaped tungsten absorbers (referred to as “nozzles” throughout the paper),
positioned inside the detector along the beam pipe on both sides of the
interaction point (IP). The dimensions, shape, and materials of the nozzles
have been carefully chosen to maximize the absorption of these particle fluxes
while maintaining a high detector acceptance.

Despite the nozzles, large fluxes of secondary and tertiary particles still
reach the detector, but their kinematical properties are such that their
effects on the detector can be managed with proper sensor design and advanced
event reconstruction methods. Figure 2 shows the main features of particles
produced by the in-flight decays of muons and subsequent interactions with the
machine, as they enter the detector outer surface. The detector surface is
defined as a box containing the detector, which also includes the nozzles and
beam pipe boundaries. The distributions are presented for the BIB generated by
MAP at $\sqrt{s}=1.5$ TeV with the MARS15 software [19]. In order to cross-
check the differences between the $\sqrt{s}=1.5$ TeV and $\sqrt{s}=3$ TeV
cases, another BIB sample has been generated at $\sqrt{s}=3$ TeV using the
same IR configuration optimized for $\sqrt{s}=1.5$ TeV. This second sample has
been generated with FLUKA [20], as described in Ref. [21], and is presented in
the same figure. Figure 2 (left) illustrates the arrival time of the
background particles at the detector for center-of-mass energies of 1.5 and 3
TeV: a significant fraction of them are outside the assumed detector readout
window of [-1, 15] ns (with $t=0$ as bunch-crossing time) and will therefore
be rejected. The energy spectra of the particles entering the detector within
the readout window are shown in the right panel of Fig. 2. These particles
predominantly have momenta below 1 GeV and are expected to primarily affect
the detector elements closer to the background entry points.

The differences in the background between the two center-of-mass energies are
minimal. The studies described in the remainder of the paper are conducted
using signal and physics background samples generated at $\sqrt{s}=3$ TeV.
However, as stated before, they utilize the nozzle configuration designed for
$\sqrt{s}=1.5$ TeV and the BIB samples generated at the same center-of-mass
energy. This is considered a conservative approximation, as preliminary
studies indicate the potential to reduce the volume of the nozzles while
maintaining the same level of beam-induced background in the detector.

The BIB sample was generated by MAP using a detector solenoidal magnetic field
of 3.57 T, which is hence employed also in the current detector model. The BIB
particles, given at the detector entry point with their initial kinematical
properties and spatial distributions, are propagated into the detector and
made interact with each sub-detector by GEANT4 [22]. The detector hits
resulting from the BIB particle interactions are stored on disk and then
overlaid onto the hits produced by the signals and physics backgrounds to
create realistic events. More details about the BIB generation and simulation
are reported in Ref. [10].

##  3 Simulation and reconstruction for signal and background samples

The analysis strategy for measuring the single and double Higgs boson
production cross sections is driven by their respective production mechanisms.
These mechanisms determine the kinematic properties of the events as well the
contributing physics background processes.

In Sec. 3.1, the generation of physics signal and background processes is
described. The detector used in the detailed simulation is presented in Sec.
3.2, along with the simulation framework. Section 3.3 explains the
digitisation procedure, while Sec. 3.4 discusses the reconstruction algorithms
employed.

In this paper, a right-handed reference system is assumed with the origin at
the center of the detector, the nominal collision point: the $z$ axis is
parallel to the beam direction, the $y$ axis is vertical, and the $x$ axis
points to the center of the collider ring.

The MuC software framework [23], based on the ILCSoft [24] framework, has been
employed for the detector simulation studies in this paper.

###  3.1 Signal and background physics processes

####  3.1.1 Higgs boson production and decay

In muon collisions at a center-of-mass energy of 3 TeV with unpolarized beams,
the Higgs boson production is dominated by the vector boson fusion (VBF)
process, as discussed in Ref. [25]. As shown in Fig. 3, the $\mu^{+}\mu^{-}\to
W^{+}W^{-}\nu_{\mu}\bar{\nu}_{\mu}\rightarrow H\nu_{\mu}\bar{\nu}_{\mu}$
process provides the highest contribution, followed by the
$\mu^{+}\mu^{-}\rightarrow ZZ\mu^{+}\mu^{-}\rightarrow H\mu^{+}\mu^{-}$
process.

Figure 3: Cross sections for the most important single-Higgs production
processes as a function of the center-of-mass energy [25]. The $ZH$ and
$t\bar{t}H$ production proceeds via the $s$-channel $\mu^{+}\mu^{-}$
annihilation, while all the other production channels are vector-bosons-fusion
processes.

The Higgs decay modes considered in this paper are:

  * •

$H\rightarrow b\bar{b}$,

  * •

$H\rightarrow WW^{*}$,

  * •

$H\rightarrow ZZ^{*}$,

  * •

$H\rightarrow\mu^{+}\mu^{-}$,

  * •

$H\rightarrow\gamma\gamma$.

The sum of their branching ratios is approximately 82%. The
$H\rightarrow\tau^{+}\tau^{-}$, $H\rightarrow c\bar{c}$, $H\rightarrow
Z\gamma$ modes and the $t\bar{t}H$ production will be the subject of future
studies. Each decay mode is affected by different physics backgrounds, which
are discussed in the sections dedicated to the corresponding analyses.

The double-Higgs production at $\sqrt{s}=3$ TeV is illustrated by the Feynman
diagrams in Fig. 4, where the dominant tree-level processes are depicted. The
diagram on the left is directly related to the $\lambda_{3}$ parameter (see
Eq. 1). The two Higgs bosons in the final state are reconstructed in the decay
mode $H\rightarrow b\bar{b}$, therefore the major physics background
contribution comes from the inclusive process $\mu^{+}\mu^{-}\rightarrow
b\bar{b}b\bar{b}$.

Figure 4: Feynman tree-level diagrams representing the major contributions to
the double-Higgs production at $\sqrt{s}=3$ TeV. The first diagram on the left
represents the process directly related to the Higgs self-coupling.

####  3.1.2 Event generation

Process | Generator | Kinematical requirements
---|---|---
$\mu^{+}\mu^{-}\to H\nu_{\mu}\bar{\nu}_{\mu};\>H\to b\bar{b}$ | WHIZARD | -
$\mu^{+}\mu^{-}\to H\nu_{\mu}\bar{\nu}_{\mu};\>H\to WW^{*}\to q\bar{q}\mu\nu_{\mu}$ | WHIZARD |  $M(q,\bar{q})>10$ GeV
$\mu^{+}\mu^{-}\to H\nu_{\mu}\bar{\nu}_{\mu};\>H\to\tau^{+}\tau^{-}$ | WHIZARD | -
$\mu^{+}\mu^{-}\to H\nu_{\mu}\bar{\nu}_{\mu};\>H\to\mu^{+}\mu^{-}$ | MadGraph | -
$\mu^{+}\mu^{-}\to HH\nu_{\mu}\bar{\nu}_{\mu};\>H\to b\bar{b},H\to b\bar{b}$ | WHIZARD | -
$\mu^{+}\mu^{-}\to H\mu^{+}\mu^{-};\>H\to WW^{*}\to q\bar{q}\mu\nu_{\mu}$ | WHIZARD |  $M(q,\bar{q})>10$ GeV
$\mu^{+}\mu^{-}\to H\mu^{+}\mu^{-};\>H\to\mu^{+}\mu^{-}$ | MadGraph |  $M(\mu,\mu)>100$ GeV
$\mu^{+}\mu^{-}\to H\nu_{\mu}\bar{\nu}_{\mu};\>H\to\gamma\gamma$ | MadGraph |  $P_{\mathrm{T}}^{\gamma}>10$ GeV, $|\eta^{\gamma}|<2.436$
$\mu^{+}\mu^{-}\to ZZ\nu_{\mu}\bar{\nu}_{\mu};\>Z\to\mu^{+}\mu^{-},Z\to q\bar{q}$ | MadGraph |  $P_{\mathrm{T}}^{\mu}$ > 1 GeV, $|\eta^{\mu}|$ < 3
|  |  $P_{\mathrm{T}}^{q}$ > 5 GeV, $|\eta^{q}|$ < 3
|  |  $\Delta R(q,\bar{q})$ > 0.2, $\Delta R(q,\mu)>0.05$
$\mu^{+}\mu^{-}\to ZH\nu_{\mu}\bar{\nu}_{\mu};\>Z\to\mu^{+}\mu^{-},H\to b\bar{b}$ | MadGraph |  $P_{\mathrm{T}}^{\mu}$ > 1 GeV, $|\eta^{\mu}|$ < 3
|  |  $P_{\mathrm{T}}^{q}$ > 5 GeV, $|\eta^{q}|$ < 3
|  |  $\Delta R(q,\bar{q})$ > 0.2, $\Delta R(q,\mu)>0.05$
$\mu^{+}\mu^{-}\to W^{\pm}Z\mu\nu_{\mu};\>W^{\pm}\to q\bar{q},Z\to\mu^{+}\mu^{-}$ | MadGraph |  $P_{\mathrm{T}}^{\mu}$ > 10 GeV
|  |  $P_{\mathrm{T}}^{q}$ > 5 GeV, $|\eta^{q}|$ < 3
|  |  $\Delta R(q,\bar{q})$ > 0.2, $\Delta R(q,\mu)>0.05$
$\mu^{+}\mu^{-}\to q\bar{q}\nu_{\mu}\bar{\nu}_{\mu}$ | WHIZARD |  $M(q,\bar{q})>10$ GeV
$\mu^{+}\mu^{-}\to q\bar{q}l\nu$ | WHIZARD |  $M(q,\bar{q})>10$ GeV
$\mu^{+}\mu^{-}\to Hq\bar{q}\nu\bar{\nu};\>H\to b\bar{b}$ | WHIZARD |  $M(q,\bar{q})>10$ GeV
$\mu^{+}\mu^{-}\to q\bar{q}q\bar{q}\nu\bar{\nu}$ | WHIZARD |  $M(q,\bar{q})>10$ GeV
$\mu^{+}\mu^{-}\to HZ;\>H\to WW^{*}\to q\bar{q}\mu\nu_{\mu},Z\to\mu^{+}\mu^{-}$ | WHIZARD |  $M(q,\bar{q})>10$ GeV
$\mu^{+}\mu^{-}\to\nu_{\mu}\bar{\nu}_{\mu}\mu^{+}\mu^{-}q\bar{q}$ | WHIZARD | 10 GeV < $M(q,\bar{q})$ < 150 GeV
|  |  $|\eta^{\mu}|$ < 2.5, $|\eta^{q}|$ < 2.5
|  |  $P_{\mathrm{T}}^{\mu}$ > 5 GeV, $P_{\mathrm{T}}^{q}$ > 5 GeV
|  |  $\Delta R(q,\bar{q})$ > 0.3
$\mu^{+}\mu^{-}\to\mu^{+}\mu^{-}\nu_{\mu}\bar{\nu}_{\mu}$ | MadGraph |  $100<M(\mu,\mu)<150$ GeV,
|  | $|\eta^{\mu}|<2.66$
$\mu^{+}\mu^{-}\to\mu^{+}\mu^{-}\mu^{+}\mu^{-}$ | MadGraph |  $100<M(\mu,\mu)<150$ GeV,
|  | $|\eta^{\mu}|<2.66$
$\mu^{+}\mu^{-}\to t\bar{t};\>t\to Wb,W\to\mu\bar{\nu}_{\mu}$ | MadGraph |  $M(\mu,\mu)>50$ GeV
$\mu^{+}\mu^{-}\to\gamma\gamma\nu_{\mu}\bar{\nu}_{\mu}$ | Madgraph |  $P_{\mathrm{T}}^{\gamma}>10$ GeV, $|\eta^{\gamma}|<2.436$
$\mu^{+}\mu^{-}\to ll\gamma$ | Madgraph |  $P_{\mathrm{T}}^{\gamma}>10$ GeV, $|\eta^{\gamma}|<2.436$
$\mu^{+}\mu^{-}\to ll\gamma\gamma$ | Madgraph |  $P_{\mathrm{T}}^{\gamma}>10$ GeV, $|\eta^{\gamma}|<2.436$
$\mu^{+}\mu^{-}\to\gamma\gamma$ | Madgraph |  $P_{\mathrm{T}}^{\gamma}>10$ GeV, $|\eta^{\gamma}|<2.436$
Table 1: Summary of the generated samples indicating the Monte Carlo generator
used and the kinematical requirements applied to the final-state particles.
$M$ represents the invariant mass of the objects indicated in parentheses. The
symbol $q$ stands for $u$, $d$, $s$, $c$, or $b$ quarks, while $l$ stands for
electron or muon. If $q\bar{q}$ is present, all possible combination of quark
flavours, even different flavours, are considered by WHIZARD/Madgraph. The
line in the middle of the table separates the Higgs signal from background
processes. If the Higgs boson is not indicated in the process, Yukawa
couplings are switched off.

Signal and background samples were produced at leading order with the Monte
Carlo event generators MadGraph5_aMC@NLO [26], hereinafter referred to as
Madgraph, and WHIZARD222The program version 2.8.2 was used for WHIZARD, while
versions 3.X.X were used for Madgraph (differences between the versions are
not relevant for this paper). [27]. The particle hadronization and showering
are performed with PYTHIA version 8.2.0.0 [28].

Recently, WHIZARD authors made available a version with corrections at the
next-to-leading order [29] for multi-boson processes, but the VBF processes
are not yet ready to be used. An evaluation of the impact of such corrections
on the signal to noise ratio for the processes considered in this paper will
be the subject of future work.

In this paper, the $Z$-boson and $W$-boson fusion processes are not
distinguished experimentally, since such a separation would require the
capability to detect the muons of Z-boson fusion that are scattered in the
very forward regions of the detector. Moreover, with the requirements of Higgs
analyses, the event kinematics is compatible between $Z$-boson and $W$-boson
fusion productions [30]. For this reason, in most of the cases, events were
generated in the $W$-boson fusion channel, which dominates the production
cross sections at this center-of-mass energy. When relevant, the contribution
of the $Z$-boson fusion process is evaluated by scaling the process cross
section, as explained in Sec. 3.5.

Minimal requirements were applied to the sample generation, in order to speed-
up the process without biasing the kinematic properties of the events, whose
characteristics at this energy are not yet tested experimentally. Table 1
summarises the requirements applied at generator level for each final state
produced.

The generation requirements are applied to $P_{\mathrm{T}}^{i}$, the
transverse momentum with respect to the $z$-axis (where $i$ is the particle
type), to the invariant mass of two particles $M(i,j)$, to the
pseudorapidity333The pseudorapidity is defined as
$\eta=-\mathrm{log}\left(\mathrm{tan}\frac{\theta}{2}\right)$, where $\theta$
is the angle of the particle momentum with respect to the beam axis.
$\eta^{i}$ or to the distance between two particles in the $\eta$-$\phi$
space, $\Delta R(i,j)=\sqrt{(\Delta\eta)^{2}+(\Delta\phi)^{2}}$, where $\phi$
is the angle with respect to the $x$-axis in the $x$-$y$ plane.

###  3.2 Detector simulation

The detector model implemented in the detailed simulation is described in Ref.
[10], and here, the main characteristics relevant for the analysis are
summarised. The MuC detector is derived from the CLIC Collaboration’s detector
concept [31], optimized for $\sqrt{s}=3$ TeV $e^{+}e^{-}$ collisions, and has
been modified to incorporate the nozzles.

It consists of a silicon-based tracking system surrounded by electromagnetic
and hadronic calorimeters, all immersed in a 3.57 T magnetic field provided by
a superconducting solenoid. The magnet iron yoke, designed to contain the
return flux of the magnetic field, is instrumented with muon chambers.

Figure 5: The detector model used in the detailed simulation, view in the
$y$-$z$ plane. From the innermost to the outermost regions, it includes a
tracking system (green), an electromagnetic calorimeter (yellow), a hadronic
calorimeter (magenta), a superconducting solenoid (light blue), barrel (light
green) and endcap (blue) muon detectors. The nozzles are shown in cyan.

Figure 5 shows the $y$-$z$ view of the detector including the nozzles. From
the innermost to the outermost regions with respect to the IP, it features the
following components:

  * •

a vertex detector made of double sensor layers of $25\times
25\leavevmode\nobreak\ \mu\mathrm{m}^{2}$ silicon pixels, composed of four
central barrel cylinders and four endcap disks on both sides of the barrel;

  * •

an inner tracker with three barrel layers and seven endcap disks on each side
made of silicon macropixels with a size of $50\leavevmode\nobreak\
\mu\mathrm{m}\times 1\leavevmode\nobreak\ \mathrm{mm}$;

  * •

an outer tracker composed of three barrel layers and four endcap disks per
side of silicon microstrips with a size of $50\leavevmode\nobreak\
\mu\mathrm{m}\times 10\leavevmode\nobreak\ \mathrm{mm}$;

  * •

an electromagnetic calorimeter (ECAL), formed by 40 alternating layers of
tungsten absorber and silicon sensors as active material, for a total of 22
radiation lengths, the cell granularity is $5\times 5$ mm2;

  * •

a hadronic calorimeter (HCAL) with 60 alternating layers of steel absorber and
scintillating-pad active material, for a total of 7.5 interactions lengths,
the cell size is $30\times 30$ mm2;

  * •

a superconductive solenoid generating a magnetic field of $B=3.57$ T;

  * •

an iron return yoke equipped with resistive-plate chambers for muon detection,
with seven layers in the barrel and six layers in the endcap, each cell having
an area of $30\times 30$ mm2.

The simulation of the detector response to particle passage is performed using
GEANT4, which is accessed through the MuC software framework [23].

###  3.3 Digitisation

A MuC event is defined as a generated signal or physics background process
with the BIB particles produced in a bunch crossing overlaid. The merging is
performed at the level of detector hits, ensuring that both contributions, the
signal or the physics background and the BIB, undergo hit digitization in each
sensor of the sub-detectors simultaneously. This simulates the process as it
would occur during data collection. Marlin [32], one of the packages of the
MuC software framework, performs these actions.

A digitisation algorithm is used for the tracker: hit positions and times are
smeared with Gaussian distributions representing the spatial and timing
resolutions. For ECAL, HCAL, and the Muon System, a digitization is applied
where the energies of hits in the same cell are summed up, and the time of the
earliest hit is assigned as the time of the digitized hit.

Only hits with arrival times within a specified range relative to the bunch-
crossing time ($t=0$) are retained for each sensor to reduce the impact of the
BIB component. These time windows are applied on the normalized hit time,
defined as $t_{N}=t-D/c$, where $t$ is the absolute time of the hit, $D$ is
the distance of the sensor from the interaction point, and $c$ is the speed of
light. The time windows are: $[-90,150]$ ps for the vertex detector and
$[-180,300]$ ps for the other tracking systems, while for ECAL, HCAL and Muon
System hits the range is $\pm 250$ ps. To meet these time requirements, an
excellent time resolution is essential. A resolution of 30 (60) ps for the
vertex (tracking) system and 100 ps for the calorimeters is assumed in the
detailed detector simulation. These time resolutions are expected to be
achievable with the technologies developed for HL-LHC, as described in Ref.
[10].

###  3.4 Event reconstruction

The reconstruction algorithms of the physics objects inherited from the
ILCSoft package [24] have been adapted for the MuC background conditions,
although a complete optimization is still in progress. The Marlin software is
employed for the reconstruction of the physics objects. Marlin also makes use
of other packages for specific reconstruction tasks, such as the PandoraPFA
package [33], which is used for particle flow techniques. A brief description
of the utilized algorithms is provided below:

  * •

Tracks: The trajectories of charged particles are reconstructed from the
positions of the hits in the silicon tracking stations. Hits are clustered
into tracks using two different algorithms. The first one is the Conformal
Tracking (CT) [34], the second one makes use of the Combinatorial Kalman
Filter, referred to here as the CKF algorithm [35]. The CT algorithm was
originally designed for the clean environment of electron-positron collisions
and has long running times (order of hours per event with the available
computational resources) in the MuC environment due to the presence of the BIB
hits. For this reason, it is usually applied to pre-selected hits in regions
of interests, _e.g._ inside calorimetric jet cones or around reconstructed
segments of the muon detectors. The CKF algorithm was developed for the
proton-proton collision environment and can be executed in the MuC environment
in a shorter time (order of minutes per event). The uncertainty on the
transverse momentum, $\Delta p_{\mathrm{T}}$, obtained with the CT and CKF
algorithms goes from $\Delta p_{\mathrm{T}}/p^{2}_{\mathrm{T}}\approx 1\cdot
10^{-1}$ $(5\cdot 10^{-3})$ GeV-1 for muons with momentum $p=1$ (100) GeV and
polar angle $\theta=13^{\circ}$, to $\Delta
p_{\mathrm{T}}/p^{2}_{\mathrm{T}}\approx 5\cdot 10^{-2}$ $(4\cdot 10^{-5})$
GeV-1 for muons with $p=1$ (100) GeV and $\theta=89^{\circ}$. The track
reconstruction efficiency with the CT algorithm is approximately 75% at
$\theta=13^{\circ}$ and close to 100% at $\theta=89^{\circ}$. In this paper,
the CT algorithm is used for the $H\to WW^{*}$ and $H\to\mu^{+}\mu^{-}$
analyses, whereas the CKF algorithm is employed for the $H\to b\bar{b}$, $H\to
ZZ^{*}$, $HH\to b\bar{b}b\bar{b}$, and Higgs width analyses.

  * •

Photons: Photons are defined as clusters of ECAL hits that are not matched to
tracks. The PandoraPFA algorithm, which is used for photon reconstruction,
takes tracks and calorimeter hits as inputs. Tracks are reconstructed with the
CKF algorithm and a track quality filter is applied [10]. An energy threshold
of 2 MeV is applied to the calorimeter hits to reduce the BIB contamination.
This threshold has a significant impact on the reconstruction of the jets, as
will be discussed later. The photon reconstruction efficiency is about 78% for
energies below 50 GeV and about 98% for energies above 300 GeV. The photon
energy resolution is approximately 13% for energies below 50 GeV and 1.5% for
energies above 400 GeV.

  * •

Muons: The PandoraPFA algorithm is also used for muon reconstruction and
identification: the tracks reconstructed with either the CT or CKF algorithm
are matched to stubs in the muon system. The reconstruction requirements
suppress the BIB contamination, at the cost of a low reconstruction and
identification efficiency in the forward and backward regions. The efficiency,
evaluated in this paper for the CKF tracking algorithm, is about 20$\%$ in the
angular regions $10^{\circ}<\theta<20^{\circ}$ and
$160^{\circ}<\theta<170^{\circ}$, while it is close to 100$\%$ for
$40^{\circ}<\theta<140^{\circ}$. The uncertainty on the muon transverse
momentum goes from $\Delta p_{\mathrm{T}}/p^{2}_{\mathrm{T}}\approx 2.1\cdot
10^{-3}$ GeV-1 for muons with $p_{\mathrm{T}}=2.5$ GeV to $\Delta
p_{\mathrm{T}}/p^{2}_{\mathrm{T}}\approx 1\cdot 10^{-4}$ GeV-1 for muons with
$p_{\mathrm{T}}>25$ GeV. The choice of the tracking algorithm employed in a
specific Higgs analysis, CT or CKF, depends mainly on the available computing
time for each study.

  * •

Jets: Hadronic jets are reconstructed with the following procedure:

    1. 1.

tracks are reconstructed as described above and filtered to remove
combinatorial background by requiring a minimum number of hits on different
tracking layers [10];

    2. 2.

calorimeter hits are selected requiring an energy threshold of 2 MeV to remove
spurious BIB hits, as in photon reconstruction;

    3. 3.

tracks and calorimeter hits are used as inputs to the PandoraPFA algorithm to
obtain the reconstructed physics objects;

    4. 4.

the $k_{t}$ jet algorithm [36] with a radius parameter of 0.5 is used to
cluster the reconstructed physics objects into jets.

This reconstruction procedure has a jet finding efficiency of $85\%$ for a jet
$p_{\mathrm{T}}$ around 20 GeV, and $90\%$ for a jet $p_{\mathrm{T}}$ around
200 GeV. On average, with a $p_{\mathrm{T}}$ threshold of 10 GeV, 13 fake jets
are reconstructed per event due to the BIB contribution. Each Higgs analysis
involving jets in the final state applies cleaning requirements to minimize
the fake rate. More information about fake jets and their mitigation can be
found in Ref. [10].

The jet 4-momentum is calculated by summing the 4-momenta of the clustered
physics objects, and the jet direction is defined by the jet momentum
direction. A correction is applied to the reconstructed jet energy to take
into account detector effects and inefficiencies. This correction is
calculated as a function of jet $p_{\mathrm{T}}$ and $\theta$ by comparing
reconstructed jets with corresponding truth-level jets (_i.e._ jets clustered
starting from generator-level stable particles). The jet $p_{\mathrm{T}}$
resolution is $35\%$ for a jet $p_{\mathrm{T}}$ around 20 GeV, and above
$20\%$ for a jet $p_{\mathrm{T}}$ around 200 GeV.

  * •

$\bm{b}$-tagging: The identification of jets originating from $b$ quarks and
their separation from $c$\- and light-quark jets is performed by searching for
displaced secondary vertices (SV) within the jets [37]. The CT algorithm is
used to reconstruct tracks inside the jet cone. A jet is identified as a $b$
jet if a secondary vertex is found with a proper lifetime $\tau=L\,m/P$
greater than 0.2 ps, where $L$ is the distance between the primary and the
secondary vertices, $m$ is the mass of the SV, and $P$ is the momentum
associated to the SV, calculated as the sum of the momenta of the tracks
associated to it. The $b$-jet identification efficiency and misidentification
rate have been determined as a function of the jet $p_{\mathrm{T}}$ and
$\theta$ with independent samples of $b\bar{b}$, $c\bar{c}$ and light-quark
jets [10]. The tagging algorithm has an efficiency of about $45\%$ for a
$b$-jet $p_{\mathrm{T}}$ around 20 GeV, and of about $70\%$ for a $b$-jet
$p_{\mathrm{T}}$ around 140 GeV with a misidentification rate of about $20\%$
for $c$-jets and from $1\%$ to $5\%$ for light-quark jets. It should be noted
that the fake rate in the case of tagged jets is negligible. The algorithm
used in the analyses presented in this paper is very simple, a much more
advanced algorithm based on artificial-intelligence methods is planned for
future updates.

A detailed discussion of the reconstruction performance of these objects can
be found in Ref. [10]. Other physics objects, like electrons or $\tau$
leptons, are not treated in this paper due to the lack of personnel. Studies
are planned to assess the statistical sensitivity of production cross-section
measurements for $H\to e^{+}e^{-}$ and $H\to\tau^{+}\tau^{-}$.

###  3.5 Analysis path

In multi-TeV muon collisions, Higgs bosons are mainly produced via $W$\- or
$Z$-boson fusion, _i.e._ $\mu^{+}\mu^{-}\rightarrow H\nu_{\mu}\bar{\nu}_{\mu}$
or $\mu^{+}\mu^{-}\rightarrow H\mu^{+}\mu^{-}$, respectively. A method to
experimentally distinguish the two production modes, which relies on the
identification of the forward-scattered muons of the ZZ fusion, is an item for
future studies and is not used in this paper. For this reason, in the
following analyses, the physics signals are referred to using the compact
notation $\mu^{+}\mu^{-}\rightarrow HX$, where $X=\nu_{\mu}\bar{\nu}_{\mu}$ or
$\mu^{+}\mu^{-}$.

The kinematic properties of $\mu^{+}\mu^{-}\rightarrow
H\nu_{\mu}\bar{\nu}_{\mu}$ and $\mu^{+}\mu^{-}\rightarrow H\mu^{+}\mu^{-}$ are
found to be compatible at $\sqrt{s}=3$ TeV. Hence, only the
$\mu^{+}\mu^{-}\rightarrow H\nu_{\mu}\bar{\nu}_{\mu}$ process has been
generated for the signal samples, the kinematic distributions of which are
considered representative of the entire signal. The total signal cross section
is then obtained by scaling the $W$-fusion cross section by the
$W$-fusion/$Z$-fusion cross-section ratio ($f$):

$\sigma_{H}=\sigma(\mu^{+}\mu^{-}\rightarrow HX)=\\\
\sigma(\mu^{+}\mu^{-}\rightarrow
H\nu_{\mu}\bar{\nu}_{\mu})\cdot\left(1+\frac{1}{f}\right).$ (2)

The factor $\left(1+\frac{1}{f}\right)$ is found to be 1.09 using both WHIZARD
and Madgraph. Exceptions to this procedure will be explained in the text.

The integrated luminosity is assumed to be $\mathcal{L}=1$ ab-1, which is
expected to be collected in 5 years of data-taking by a single experiment
[10]. It should also be noted that a longer run of 10 years is under
discussion, with the goal of collecting $\mathcal{L}=2$ ab-1. However, the
extrapolation of the results to this scenario is beyond the scope of this
paper.

The signal and background selection efficiencies ($\epsilon$) are defined as
the the ratio between the number of events passing the selection requirements
and the corresponding number of generated events. The number
$N_{\mathrm{exp}}$ of expected signal and background events is determined with
the formula:

$N_{\mathrm{exp}}=\epsilon\cdot\sigma\cdot\cal{L}\ ,$ (3)

where $\sigma$ is the production cross section given by the event generator.

The following studies on the Higgs boson cross sections estimate the
statistical sensitivity on the production cross section times the branching
fraction $\sigma_{H}\times BF(H\to xx)$, where $xx$ denotes any considered
final state. The quantity $\sigma_{H}\times BF(H\to xx)$ is indicated with
$\sigma(H\to xx)$, and the corresponding uncertainty with $\Delta\sigma(H\to
xx)$. It is important to note that in the studies the systematic uncertainties
on the luminosity and efficiency are considered negligible: only the
statistical sensitivity, resulting from the number of events collected in the
data taking, is evaluated.

In several Higgs analyses, machine-learning methods such as Boosted Decision
Trees (BDTs) [38] and Multi Layer Perceptrons (MLPs) are employed to enhance
the separation between the signal and the physics backgrounds. The BDTs and
MLPs have been trained and applied with the TMVA package [39] from the ROOT
software suite [40]. Their configuration parameters are set to the default
values as defined by TMVA.

##  4 Determination of Higgs boson cross sections times branching ratios

###  4.1 $H\to b\bar{b}$ cross section

This section is dedicated to the determination of the statistical sensitivity
on the measurement of the $H\rightarrow b\bar{b}$ cross section. Samples of
$\mu^{+}\mu^{-}\rightarrow H(\rightarrow b\bar{b})X$ signal events and
$\mu^{+}\mu^{-}\to q_{h}\bar{q_{h}}X$, $q_{h}=b,c$ background events have been
generated following the prescription of Tab. 1.

Jets are reconstructed as explained in Sec. 3.4 using the CKF algorithm for
the input tracks. Jets with $p_{\mathrm{T}}>40$ GeV and $|\eta|<2.5$ are
selected to maximize the reconstruction performance and minimize the BIB
effects. Both jets from the Higgs decay are required to be identified as
$b$-jets, _i.e._ they have to be associated to a displaced SV. However,
applying this requirement directly to the signal and background samples would
significantly affect their statistics. Instead, the identification
efficiencies and misidentification rates have been determined as a function of
jet $p_{\mathrm{T}}$ and $\theta$ (Sec. 3.4) and applied to the signal and
background samples as event weights. The light-quark jet misidentification
rate has been found negligible, and for this reason light-quark jets have not
been included in the background processes. Also the fake jet rate is
negligible after the SV requirement.

The expected number of signal and background events is determined with Eq. 3:
59.5k $H\rightarrow b\bar{b}$ events and 65.4k background events are estimated
to be collected in 1 ab-1, as reported in Tab. 2. The di-jet invariant-mass
distributions for signal and background are fitted with a double-Gaussian
function to obtain models for signal and background.

Process | $\epsilon[\%]$ | $\sigma\ [\mathrm{fb}]$ | $N_{exp}$
---|---|---|---
$\mu^{+}\mu^{-}\to H(\to b\bar{b})X$ | $19.3$ | $308$ | $59500$
$\mu^{+}\mu^{-}\to q_{h}\bar{q}_{h}X,\>q_{h}=b,c$ | $11.2$ | $584$ | $65400$
Table 2: Selection efficiency, theoretical cross section, and expected events
for the signal and background processes in the $H\to b\bar{b}$ analysis for
$\mu^{+}\mu^{-}$ collisions at 3 TeV and $\mathcal{L}=1$ ab-1.

The signal and background models are used to build a likelihood function and
generate samples of pseudo-data according to the expected number of events. An
unbinned maximum-likelihood fit is then performed on the invariant-mass
distributions of the pseudo-experiments, allowing the signal and background
yields to float. In this way, the average $H\rightarrow b\bar{b}$ yield and
its uncertainty are extracted. An example of one of the fits is shown in Fig.
6. Due to the limited jet energy resolution, the di-jet invariant mass
distribution shows a significant overlap between the Higgs signal and the
background, consisting mainly of $Z$ boson decays to $b\bar{b}$ and $c\bar{c}$
pairs. In any case, the number of events is sufficiently high to disentangle
the two components on a statistical basis. The $H/Z$ separation will be
improved by utilizing a jet reconstruction algorithm that is specifically
tuned and optimized for MuC.

The cross section is estimated as follows:

$\sigma(H\to b\bar{b})=\frac{N_{S}}{\epsilon\cdot\cal{L}}\ ,$ (4)

where $N_{S}$ is the expected number of signal events and $\epsilon$ is the
total signal selection efficiency. Assuming the uncertainties on the
efficiency and the luminosity negligible, the statistical uncertainty on the
cross section is dominated by the uncertainty on the signal yield. By
averaging the results of several pseudo-experiments, the estimated uncertainty
on the production cross section is:

$\frac{\Delta\sigma(H\to b\bar{b})}{\sigma(H\to b\bar{b})}=0.75\%\ .$ (5)

Figure 6: Example of the di-jet invariant mass fit for one of the pseudo-
experiments used to extract the $H\rightarrow b\bar{b}$ yield and uncertainty.
Pseudo-data are obtained assuming $\mu^{+}\mu^{-}$ collisions at 3 TeV and
$\mathcal{L}=1$ ab-1 [41].

###  4.2 $H\rightarrow WW^{*}$ cross section

Figure 7: Distributions of the BDT outputs to separate the $H\to WW^{*}$
signal from the background processes without a Higgs bosons (left) and with a
Higgs boson (right). Signal and backgrounds are presented with stacked
histograms, normalized to the expected number of events. The black vertical
lines indicate the threshold for achieving the optimal signal significance.

The $H\to WW^{*}$ decay is reconstructed in the semileptonic final state,
where one $W$ boson decays into a muon and a neutrino and the other one decays
hadronically into two jets ($H\to WW^{*}\to qq^{\prime}\mu\nu$) [30]. This
final-state has been chosen for its good signal-over-background ratio. In
fact, the selection of a high-momentum muon significantly reduces the
background from full hadronic final states. On the other hand, the $W$ boson
decay to jets has a high branching ratio, hence the semileptonic final state
has a higher statistics than the full leptonic final state.

Signal and background samples have been generated as summarized in Tab. 1.
Among the physics background processes that could mimic the signal signature,
only those with significant cross sections have been considered. They have
been grouped into two sets, depending on the presence of the Higgs boson along
the process:

  1. 1.

background processes without a Higgs boson: $\mu^{+}\mu^{-}\to\mu\nu
qq^{\prime}$, $\mu^{+}\mu^{-}\to qq^{\prime}l^{+}l^{-}$ and $\mu^{+}\mu^{-}\to
qq^{\prime}\nu\bar{\nu}$, with $q=u,d,c,s$ and $l=e,\mu,\tau$;

  2. 2.

background processes with a Higgs boson: $\mu^{+}\mu^{-}\to H(\to WW^{*}\to
qq^{\prime}qq^{\prime})X$, $\mu^{+}\mu^{-}\to H(\to b\bar{b})X$ and
$\mu^{+}\mu^{-}\to H(\to\tau^{+}\tau^{-})X$.

In this analysis, jets are first clustered using calorimeter hits, then tracks
are reconstructed with the CT algorithm in regions of interest defined by the
calorimetric jet cones. Tracks obtained in this way, together with calorimeter
clusters, are given as input to the final jet reconstruction. Muons are
reconstructed using CT tracks as input. In order to speed-up the muon
reconstruction, BIB hits are neglected for this specific task: this
approximation does not bias the result since the muon reconstruction
requirements suppress the BIB contribution, at the cost of a low efficiency in
the forward and backward regions (Sec. 3.4). Signal candidates are formed by
combining two reconstructed jets and one muon. The muon is required to have
$p_{\mathrm{T}}>10$ GeV and $|\eta|<2.44$ (corresponding to
$10^{\circ}<\theta<170^{\circ}$). The two jets must be central with
$|\eta|<2.5$ and $p_{\mathrm{T}}>20$ GeV. Requirements are applied to remove
fake jets from the analysis: the number of the jet constituents must be
greater than two, and the maximum fraction of the jet momentum carried by a
single constituent is required to be greater than 0.8. Further details on this
prescription can be found in Ref. [30].

In order to separate signal from backgrounds, two BDTs have been trained: the
first to separate the $H\to WW^{*}$ signal from the background processes
without a Higgs bosons (listed above in bullet 1), the second to separate the
signal from the processes with a Higgs boson (bullet 2). The BDTs have the
following observables as inputs:

  * •

the $p_{T}$ and $\theta$ of the selected muon;

  * •

the invariant mass, $p_{T}$, and $\theta$ of the $W\to qq^{\prime}$ candidate
formed by two jets;

  * •

the invariant mass, $p_{T}$, and $\theta$ of the Higgs boson candidate;

  * •

the missing transverse momentum, defined as
$P^{\mathrm{miss}}_{T}=-\sqrt{P_{x}^{2}+P_{y}^{2}}$ with
$P_{x,y}=\sum_{i}p_{x,y}^{\mu_{i}}+\sum_{i}p_{x,y}^{\mathrm{jet}_{i}}$, where
the momentum components $p_{x}$ and $p_{y}$ are summed over all reconstructed
muons and jets in the event;

  * •

the number of isolated muons in the event, _i.e._ the muons with $\Delta
R=\sqrt{\Delta\phi^{2}+\Delta\eta^{2}}>0.5$ with respect to any jet in the
event;

  * •

three acollinearity angles, defined between the momenta $\vec{p}_{i}$ and
$\vec{p}_{j}$ of two particles as:

$\theta_{\mathrm{acoll}}(i,j)=\pi-\mathrm{acos}\left(\frac{\vec{p}_{i}\cdot\vec{p}_{j}}{p_{i}p_{j}}\right)\
,$ (6)

where $p_{i,j}=|\vec{p}_{i,j}|$ and $i$ and $j$ are: the Higgs boson
candidate, the $W\to qq^{\prime}$ candidate, and the selected muon. Hence, the
three acollinearity angles are $\theta_{\mathrm{acoll}}(H,\mu)$,
$\theta_{\mathrm{acoll}}(H,W\to qq^{\prime})$, and
$\theta_{\mathrm{acoll}}(W\to qq^{\prime},\mu)$.

More details on the BDT input variables can be found in Ref. [30]. The output
of the BDTs for the signal and backgrounds is shown in Fig. 7. The outputs of
the two BDTs are required to be above thresholds determined by maximizing the
signal significance

$\mathcal{S}=\frac{S}{\sqrt{S+B}}\ ,$ (7)

where $S$ and $B$ are the expected number of signal and background events,
respectively, calculated using Eq. 3. Table 3 summarises the efficiencies,
theoretical cross sections, and expected number of events passing the
selection described. After the event selection, the dominant physics
background is the process $\mu^{+}\mu^{-}\to qq^{\prime}\mu\nu$.

Process | $\epsilon[\%]$ | $\sigma\ [\mathrm{fb}]$ | $N_{exp}$
---|---|---|---
$\mu^{+}\mu^{-}\to H(\to WW^{*}\to qq^{\prime}\mu\nu)X$ | $14.1$ | $17.3$ | $2430$
$\mu^{+}\mu^{-}\to qq^{\prime}\mu\nu$ | $0.05$ | 5020 | 2600
$\mu^{+}\mu^{-}\to qq^{\prime}l^{+}l^{-}$ | $<0.01$ | 1040 | $<100$
$\mu^{+}\mu^{-}\to qq^{\prime}\nu\bar{\nu}$ | $<0.01$ | 1560 | $<100$
$\mu^{+}\mu^{-}\to HX\to WW^{*}X\to qq^{\prime}qq^{\prime}X$ | $<0.01$ | $108$ | $<10$
$\mu^{+}\mu^{-}\to HX\to b\bar{b}X$ | $<0.05$ | $313$ | $<150$
$\mu^{+}\mu^{-}\to HX\to\tau^{+}\tau^{-}X$ | $<0.01$ | $34.3$ | $<4$
Table 3: Selection efficiencies, theoretical cross sections, and expected
number of events for signal and background processes in the $H\to WW^{*}$
analysis, with $\mu^{+}\mu^{-}$ collisions at 3 TeV and $\mathcal{L}=1$ ab-1.
The upper limits are determined at $68\%$ Confidence Level.

The statistical sensitivity on the cross section is determined considering
this case as a counting experiment. Thus, assuming that the uncertainties on
the efficiency and the integrated luminosity are negligible, the error on the
$H\to WW^{*}$ cross section is given by:

$\frac{\Delta\sigma(H\to WW^{*})}{\sigma(H\to WW^{*})}=\frac{\sqrt{S+B}}{S}\
,$ (8)

where the formula is derived considering the properties of the Poisson
statistics. The statistical sensitivity resulting from the numbers in Tab. 3
is:

$\frac{\Delta\sigma(H\to WW^{*})}{\sigma(H\to WW^{*})}=2.9\%\ .$ (9)

###  4.3 $H\rightarrow ZZ^{*}$ cross section

As with the $H\to WW^{*}$ channel, the $H\to ZZ^{*}$ production has been
studied in the semileptonic final state, where one $Z$ boson decays into two
muons and the other into two jets, _i.e._ $\mu^{+}\mu^{-}\to H(\to ZZ^{*})X\to
q\bar{q}\mu^{+}\mu^{-}X$. Differently from the $H\to WW^{*}$ analysis, the
$Z\to\mu\mu$ decay is fully reconstructed, therefore the background processes
in Tab. 3 are considered negligible after applying the selection requirements
described below. On the other hand, the $H\to ZZ^{*}$ branching ratio is 10
times lower than $H\to WW^{*}$, and in this measurement the irreducible
background $\mu^{+}\mu^{-}\to qq\mu^{+}\mu^{-}X$ becomes the dominant one.

The input tracks of jets and muons have been reconstructed by employing the
CKF tracking algorithm. To minimize the number of fake jets, at least one
track is required in the jets. Jets must have transverse momentum
$p_{\mathrm{T}}>$15 GeV and $|\eta|<1.32$ (corresponding to
$30^{\circ}<\theta<150^{\circ}$). Muons are selected with $p_{\mathrm{T}}>10$
GeV. A signal candidate is formed by the combination of two jets and two
opposite-charge muons. In order to remove background processes where muons are
produced in the jet fragmentation, the distance $\Delta R$ between muons and
jets for each combination, which is defined as the distance in the
$\eta$-$\phi$ space, must be greater than 0.5. If more than one combination
per event is found, the candidate with the invariant mass closer to the Higgs
boson mass is selected.

A BDT is used to remove the irreducible background $\mu^{+}\mu^{-}\to
qq\mu^{+}\mu^{-}X$. This BDT uses the following inputs: the invariant mass of
the Higgs and $Z$ bosons candidates; the angle between the two $Z$ bosons in
the laboratory frame; the angles between the muons boosted in the $Z$ boson
frame and the flight direction of the $Z$ boson in the laboratory frame; the
angles between the $Z$ bosons boosted in the Higgs boson frame and the flight
direction of the Higgs boson in the laboratory frame. The simulated signal and
background samples are used in the training.

A threshold on the BDT output is determined in order to maximize the signal
significance of Eq. 7. The number of expected events after the preselection
and the BDT threshold requirement is reported in Tab. 4.

Process | $\epsilon[\%]$ | $\sigma\ [\mathrm{fb}]$ | $N_{exp}$
---|---|---|---
$\mu^{+}\mu^{-}\to H(\to ZZ^{*})X\to q\bar{q}\mu^{+}\mu^{-}X$ | $15.9$ | $0.376$ | $60$
$\mu^{+}\mu^{-}\to qq\mu^{+}\mu^{-}X$ | $0.69$ | $6.18$ | $42$
Table 4: Selection efficiency, theoretical cross section, and expected events
for signal and background processes in the $H\to ZZ^{*}$ analysis, with
$\mu^{+}\mu^{-}$ collisions at 3 TeV and $\mathcal{L}=1$ ab-1.

As with the $H\to WW^{*}$ analysis, the counting experiment procedure is
followed, assuming that the uncertainties on the efficiency and the integrated
luminosity are negligible. This yields:

$\frac{\Delta\sigma(H\to ZZ^{*})}{\sigma(H\to ZZ^{*})}=\frac{\sqrt{S+B}}{S}\
.$ (10)

Finally, the estimated statistical sensitivity is:

$\frac{\Delta\sigma(H\to ZZ^{*})}{\sigma(H\to ZZ^{*})}=17\%\ .$ (11)

###  4.4 $H\rightarrow\mu^{+}\mu^{-}$ cross section

This section presents the study of the statistical sensitivity on the
$H\to\mu^{+}\mu^{-}$ production cross section [42]. The Monte Carlo samples
used in the analysis were produced with the MadGraph generator. Two production
processes of the Higgs boson are generated: $\mu^{+}\mu^{-}\rightarrow
H(\rightarrow\mu^{+}\mu^{-})\nu_{\mu}\bar{\nu}_{\mu}$ and
$\mu^{+}\mu^{-}\rightarrow H(\rightarrow\mu^{+}\mu^{-})\mu^{+}\mu^{-}$
featuring two muons and two muonic neutrinos and four muons in the final
state, respectively. The main background processes with the same final states
as the sought signals are the inclusive channels
$\mu^{+}\mu^{-}\rightarrow\mu^{+}\mu^{-}\nu_{\mu}\bar{\nu}_{\mu}$ and
$\mu^{+}\mu^{-}\rightarrow\mu^{+}\mu^{-}\mu^{+}\mu^{-}$, from which the
contributions from the signal are removed, and $\mu^{+}\mu^{-}\rightarrow
t\bar{t}\rightarrow W^{+}W^{-}b\bar{b}$
($W^{\pm}\rightarrow\mu^{\pm}\nu_{\mu}[\bar{\nu}_{\mu}]$). In order to reduce
the computational time of the event simulation and reconstruction, the BIB was
not overlaid to the physical processes, since it was shown to have a minimal
impact on the muon reconstruction [10]. A cross-check was performed to verify
this assumption, as explained later in this section.

The $H\to\mu^{+}\mu^{-}$ analysis proceeds in two steps. First, a loose
preselection is applied to identify events with a potential Higgs boson
candidate and to suppress part of the low-energy background, then a final
event selection is performed using two BDTs. In order to remove a fraction of
the physics backgrounds, two opposite-charge muons are required with
$p_{\mathrm{T}}>5$ GeV in the angular region $|\eta|<2.44$. Further
requirements are applied to the dimuon invariant mass and momenta:
$105<m_{\mu\mu}<145$ GeV, $p_{\mathrm{T}}(\mu^{+}\mu^{-})>30$ GeV, and
$p_{\mathrm{T}}(\mu^{+})+p_{\mathrm{T}}(\mu^{-})>50$ GeV.

Figure 8: Stack of the $m_{\mu\mu}$ distributions for the selected signal and
background events [41]. The distributions have been normalized to the expected
number of events with $\mu^{+}\mu^{-}$ collisions at 3 TeV and $\mathcal{L}=1$
ab-1, reported in Tab.5.

After this preselection, a BDT classifier is trained to discriminate the total
signal from the background
$\mu^{+}\mu^{-}\rightarrow\mu^{+}\mu^{-}\nu_{\mu}\bar{\nu}_{\mu}$. A second
BDT classifier is trained independently for separating the signal from the
background $\mu^{+}\mu^{-}\rightarrow\mu^{+}\mu^{-}\mu^{+}\mu^{-}$. The BTDs
exploit observables of the two final-state muons (the $p_{\mathrm{T}}$ scalar
sum, the angular separation $\Delta R=\sqrt{\Delta\phi^{2}+\Delta\theta^{2}}$,
the cosine of the angle between the three-momenta, the cosine of the angle of
$\mu^{-}$ momentum in the dimuon rest frame with respect to the dimuon-system
direction in the laboratory frame), of the dimuon system (the boost
$\beta_{\mu\mu}$, the transverse momentum, the polar angle and the modulus of
the three-momentum, the recoil mass), and event observables (the total visible
energy, the total visible energy minus the dimuon-system energy, the missing
transverse momentum). A cut on the BDT scores is used to select the final
samples. The cut values are chosen to maximize the signal significance
$\mathcal{S}$, as defined in Eq. 7, and events are required to satisfy at
least one of the two BDT requirements.

Process | $\epsilon[\%]$ | $\sigma\ [\mathrm{fb}]$ | $N_{exp}$
---|---|---|---
$\mu^{+}\mu^{-}\rightarrow H(\rightarrow\mu^{+}\mu^{-})\nu_{\mu}\bar{\nu}_{\mu}$ | $22.12$ | 0.109 | 24.2
$\mu^{+}\mu^{-}\rightarrow H(\rightarrow\mu^{+}\mu^{-})\mu^{+}\mu^{-}$ | $16.31$ | 0.010 | 1.63
$\mu^{+}\mu^{-}\rightarrow\mu^{+}\mu^{-}\nu\bar{\nu}_{\mu}$ | $5.74$ | 11.09 | 637
$\mu^{+}\mu^{-}\rightarrow\mu^{+}\mu^{-}\mu^{+}\mu^{-}$ | $0.160$ | 297.40 | 476
$\mu^{+}\mu^{-}\rightarrow t\bar{t}\rightarrow W^{+}W^{-}b\bar{b},$ $W^{\pm}\rightarrow\mu^{\pm}\nu_{\mu}(\bar{\nu}_{\mu})$ | $0.34$ | 0.32 | 1.1
Table 5: Selection efficiency, theoretical cross section, and expected events
for signal and background processes in the $H\to\mu^{+}\mu^{-}$ analysis, with
$\mu^{+}\mu^{-}$ collisions at 3 TeV and $\mathcal{L}=1$ ab-1.

In Fig. 8 is shown the distribution of the dimuon mass $m_{\mu\mu}$ for the
expected signal and background events in 1 ab-1 of data. The corresponding
yields are listed in Tab. 5.

The number of signal events $N_{S}$, which is used to calculate the production
cross section in Eq .4, can be determined with a fit to the $m_{\mu\mu}$
distribution. The expected uncertainty on $N_{S}$ is estimated with a toy
Monte Carlo study. The signal and background models are build from a
parameterization of the distributions of Fig. 8. $N_{S}$ is then determined
with an unbinned extended maximum-likelihood fit in 10000 pseudo experiments:
the $N_{S}$ distribution is centered around 25.8 with an RMS of 9.9. Assuming
negligible uncertainties on the selection efficiency and on the integrated
luminosity, the relative uncertainty on $N_{S}$ can be considered as the
statistical uncertainty of the cross-section measurement:

$\frac{\Delta\sigma(H\rightarrow\mu^{+}\mu^{-})}{\sigma(H\rightarrow\mu^{+}\mu^{-})}=38\%\
.$ (12)

It is worth noting that the possibility of identifying the $ZZ$-fusion
processes, as outlined in Sec. 3.5, would significantly reduce the
$\mu^{+}\mu^{-}\rightarrow\mu^{+}\mu^{-}\mu^{+}\mu^{-}$ background and improve
the cross-section sensitivity. To get a rough assessment of the BIB impact to
this study, the signal sample $\mu^{+}\mu^{-}\rightarrow
H\nu_{\mu}\bar{\nu}_{\mu}$ was reconstructed with the BIB superimposed and
compared to the case without BIB at the preselection level of the analysis.
The $H\rightarrow\mu^{+}\mu^{-}$ selection efficiency results reduced by about
2%, while the width of the dimuon mass peak degrades only by a few percent
which is within the reached precision.

###  4.5 $H\to\gamma\gamma$ cross section

The experimental signature for the process $\mu\mu\to H(\to\gamma\gamma)X$
consists of two high-$p_{\mathrm{T}}$ photons with an invariant mass
compatible with the Higgs boson mass. The signal and physics background
processes are generated with MadGraph. The list of the considered backgrounds
and their production cross sections is reported in Tab. 6, where only
processes with photons in the final state and the highest cross sections are
taken into account. The processes with leptons in the final state can pass the
event selection because of photon misidentification.

Events are selected with at least two reconstructed photons having $E>15$ GeV,
$p_{\mathrm{T}}>10$ GeV, and $|\eta|<2.436$. The most energetic photon must
have a $p_{\mathrm{T}}>40$ GeV. If more than two photons in the event fulfill
these requirements, the two with the highest energy are selected. The
invariant mass of the two photons must satisfy the condition
$m_{\gamma\gamma}>40$ GeV.

After the preselection, signal and background events are classified using a
BDT. The following variables are used for the classification: the invariant
mass $m_{\gamma\gamma}$ and the transverse momentum
$p_{\mathrm{T}}(\gamma\gamma)$ of the di-photon system; $p_{\mathrm{T}}$ of
the two photons; $\Delta\theta$ and $\Delta\phi$ between the two photons; the
acollinearity between the two photons (Eq. 6); the acollinearity between each
photon and the Higgs candidate. The BDT is trained by using the signal sample
and a mixture of the background samples, merged together with equal weights.
The distributions of the BDT output are shown in Fig. 9.

Figure 9: Distribution of the BDT output of the signal and major physics
background process. The distributions are stack and normalized to the expected
number of events with $\mu^{+}\mu^{-}$ collisions at 3 TeV and $\mathcal{L}=1$
ab-1 after the preselection cuts. The black vertical line indicates the value
of the BDT cut.

A cut on the BDT output is applied to maximize the significance of Eq.7. The
expected number of signal and background events after the BDT requirement are
reported in Tab. 6. Fig. 10 shows the invariant mass of the Higgs candidates,
with no physics background, after the preselection and the BDT cut. A Gaussian
fit of the invariant mass distribution gives a mass resolution of 3.2 GeV for
the signal sample.

Figure 10: Distribution of reconstructed $m_{\gamma\gamma}$ for the selected $H\to\gamma\gamma$ events, with no physics background, overlaid with a Gaussian fit. The distribution is normalized to the expected number of events with $\mu^{+}\mu^{-}$ collisions at 3 TeV and $\mathcal{L}=1$ ab-1. Process | $\epsilon[\%]$ | $\sigma\ [\mathrm{fb}]$ | $N_{exp}$
---|---|---|---
$\mu^{+}\mu^{-}\to H(\to\gamma\gamma)X$ | $43.9$ | 0.91 | $396$
$\mu^{+}\mu^{-}\to\nu_{\mu}\bar{\nu}_{\mu}\gamma\gamma$ | $1.1$ | 81.98 | $442$
$\mu^{+}\mu^{-}\to l^{+}l^{-}\gamma\gamma$ | $0.3$ | 4.41 | $11$
$\mu^{+}\mu^{-}\to l^{+}l^{-}\gamma$ | $0.06$ | 159.01 | $31$
$\mu^{+}\mu^{-}\to\gamma\gamma$ | $<0.02$ | 60.15 | $<12$
Table 6: Selection efficiencies, theoretical cross sections, and expected
events for signal and physics background processes that have a final state
with two high energy photons in $\mu^{+}\mu^{-}$ collisions at 3 TeV and
$\mathcal{L}=1$ ab-1.

Assuming that the signal yield is obtained by subtracting the expected number
of background events from the observed number of events, after the BDT cut,
and assuming a negligible uncertainty on the luminosity and on the signal
selection efficiency, the statistical sensitivity on the measurement of the
$H\to\gamma\gamma$ cross section can be evaluated as:

$\frac{\Delta\sigma(H\to\gamma\gamma)}{\sigma(H\to\gamma\gamma)}=\frac{\sqrt{S+B}}{S}\
.$ (13)

With the expected number of events reported in Tab. 6, the estimated
statistical sensitivity is:

$\frac{\Delta\sigma(H\to\gamma\gamma)}{\sigma(H\to\gamma\gamma)}=7.6\%\ .$
(14)

###  4.6 Comparison of results obtained with detailed and parametric
simulations

The results presented in the preceding sections are compared with those
derived from a parametric description of the detector response and the effects
of the BIB, whenever feasible. Given that the BIB represents the primary
constraint for physics measurements, it is crucial to demonstrate that
suitable detector design and reconstruction algorithms can overcome this
limitation.

Furthermore, performing a detailed detector simulation for each physics
channel of interest may prove challenging due to the extensive computational
times and the consequently significant resources required. This validation
provides confidence that the outcomes obtained from parametric simulations
remain valid for other physics channels exhibiting similar final states.

To date, the most comprehensive study of Higgs physics at MuC utilizing
parametric simulations can be found in Ref. [25]. The detector response has
been reproduced by means of the DELPHES [43] program by using parametric
functions in the “DELPHES card”. The MuC detector card in DELPHES is a hybrid
between CLIC and FCC-hh cards, it is not derived from MuC detector studies.
The parametric study considers both the signal and the primary physics
background channels, with fiducial requirements adjusted to the detector
acceptance to account for the presence of the nozzles.

For the Higgs cross sections, the expected statistical uncertainties have been
obtained for $\sqrt{s}=3$ TeV and $\sqrt{s}=10$ TeV collisions, assuming a
dataset of 1 and 10 ab-1, respectively. The former are compared with the
results obtained with the detailed simulation and described in this section.
The comparison is presented in Tab. 7.

Process | Detailed sim. | Parametric sim.
---|---|---
| $\frac{\Delta\sigma}{\sigma}[\%]$ | $\frac{\Delta\sigma}{\sigma}[\%]$
$H\to b\bar{b}$ | 0.75 | 0.76
$H\to WW^{*}$ | 2.9 | 1.7
$H\to ZZ^{*}$ | 17 | 11
$H\to\mu^{+}\mu^{-}$ | 38 | 40
$H\to\gamma\gamma$ | 7.6 | 6.1
Table 7: Comparison of statistical sensitivities for Higgs boson production
cross sections at a $\sqrt{s}=3$ TeV Muon Collider with $\mathcal{L}=1$ ab-1,
obtained with a detailed detector simulation in this paper and a parametric
simulation in Ref. [25].

It should be noted that the $H\to b\bar{b}$, $H\to\mu^{+}\mu^{-}$ and
$H\to\gamma\gamma$ analyses in Ref. [25] have similar requirements with
respect to the detailed simulation studies, and the cross section
uncertainties are very close. The $H\to WW^{*}$ and $H\to ZZ^{*}$ analyses in
Ref. [25] include additional final states of the considered processes, while
this paper focuses only on the final state that contributes most to the
sensitivity. Even with these differences, the $H\to WW^{*}$ and $H\to ZZ^{*}$
results are comparable in both frameworks. These comparisons indicate that the
proposed detector under realistic conditions, even before full optimization,
is already capable of achieving results that are consistent with those of the
parametric simulations.

##  5 Higgs boson width

The Higgs width ($\Gamma_{H}$) is a fundamental parameter of Higgs boson
physics, since it is closely connected to the broader picture of the Higgs
couplings. The expected value in the SM is $\Gamma_{H}=4.07$ MeV for a 125 GeV
Higgs boson [44], a deviation from this value would be a clear indication of
new physics in the Higgs sector. Moreover, a model-independent measurement of
$\Gamma_{H}$ is an essential input to the Higgs couplings fit, which considers
the probability of Higgs boson decays to invisible final states as a free
parameter. The proposed method to measure the Higgs width at the HL-LHC is to
extract $\Gamma_{H}$ from the interference between the off-shell Higgs boson
production and the full amplitude in the $ZZ\to 4l$ channel [1]. The
projection for the combined result of the ATLAS and CMS experiments is
$\Gamma_{H}=4.1^{+0.7}_{-0.8}$ MeV with about a 17% accuracy [45]. At
$e^{+}e^{-}$ colliders, the $e^{+}e^{-}\to HZ$ process with full reconstructed
$Z$ bosons allows for the determination of the Higgs boson kinematics
independently of the Higgs decay channel [1]. This process provides the
primary contribution to the sensitivity on the $\Gamma_{H}$ parameter in
global Higgs fits at $e^{+}e^{-}$ colliders. The FCC-ee program, combined with
the HL-LHC, expects a $1\%$ precision on the $\Gamma_{H}$ measurement [46],
while the expected precision for ILC is at the 2$\%$ level with only the
$\sqrt{s}=250$ GeV stage [47]. The projection for the full CLIC program is a
sensitivity of 3.5% [48].

At a 3 TeV MuC, the $HZ$ production cross section is relatively small (see
Fig.3), therefore the proposed method for the $e^{+}e^{-}$ colliders is not a
viable option. Instead, the approach considered for HL-LHC, where $\Gamma_{H}$
is obtained by studying the kinematics of the full $ZZ$ process, is used in
this paper [49]. Hence, the pair production of on-shell $Z$ bosons via the VBF
process $\mu^{+}\mu^{-}\rightarrow\nu_{\mu}\bar{\nu}_{\mu}ZZ$ is simulated in
the detector and reconstructed for the determination of $\Gamma_{H}$. This
final state has a contribution coming from the decay of an off-shell Higgs
boson, as shown in Fig. 11, therefore the process is sensitive to the
couplings of the Higgs boson to $Z$ bosons.

Figure 11: Feynman diagram for on-shell $ZZ$ production via VBF, occurring
with an off-shell Higgs propagator.

Thanks to the contribution of this process, the production of a pair of $Z$
bosons can be used directly to extract the modifier $\kappa$ of the couplings
between Higgs boson and vector bosons, assuming it to be the same for both $Z$
and $W^{\pm}$:

$\kappa=\frac{g_{HZZ}}{g_{HZZ}^{SM}}=\frac{g_{HWW}}{g_{HWW}^{SM}}\ .$ (15)

To obtain $\Gamma_{H}$, the measurement of the $H\rightarrow ZZ^{*}$ cross
section is also needed (Sec. 4.3). Indeed, the following equation holds [50]:

$\frac{\sigma(H\rightarrow ZZ^{*})}{\sigma^{SM}(H\rightarrow
ZZ^{*})}\propto\frac{\kappa^{4}}{\Gamma_{H}/\Gamma_{H}^{SM}}\ ,$ (16)

where $\sigma^{SM}(H\rightarrow ZZ^{*})$ and $\Gamma_{H}^{SM}$ are the
expected SM values for the $W^{+}W^{-}\rightarrow H\rightarrow ZZ^{*}$ cross
section and Higgs boson width, respectively. With $\kappa$ obtained from the
$ZZ$ production and the measurement of the $H\rightarrow ZZ^{*}$ cross section
(Sec. 4.3), the determination of $\Gamma_{H}$ is finally possible.

The approach considered for measuring $\kappa$ is to compare the data with
different simulation samples generated by varying $\kappa$, and subsequently
find the best $\kappa$ value with a likelihood scan. The
$\mu^{+}\mu^{-}q\bar{q}$ final state, where $q$ is any quark, has been chosen
as a compromise between event counts and precision in the reconstruction of
the final state. The relevant processes come from VBF di-boson production with
the $\mu^{+}\mu^{-}qq^{\prime}$ final state, namely:
$\mu^{+}\mu^{-}\rightarrow\nu_{\mu}\bar{\nu}_{\mu}Z(\rightarrow\mu^{+}\mu^{-})Z(\rightarrow
q\bar{q})$, $\mu^{+}\mu^{-}\rightarrow\nu_{\mu}\bar{\nu}_{\mu}H(\rightarrow
b\bar{b})Z(\rightarrow\mu^{+}\mu^{-})$, and
$\mu^{+}\mu^{-}\rightarrow\nu_{\mu}\mu^{\pm}W^{\pm}(\rightarrow
qq^{\prime})Z(\rightarrow\mu^{+}\mu^{-})$. The first two are $WW$-fusion
processes and are the ones with the largest dependence on $\kappa$. The latter
is a $WZ$-boson fusion process, where the first muon is usually undetected,
since it is emitted at very small polar angles. In Ref. [51], it has been
verified that the $ZZ$-fusion contribution to the $\Gamma_{H}$ sensitivity is
negligible, therefore it is not considered in this analysis.

Several high-statistics simulation samples corresponding to different $\kappa$
values should be produced for determining the uncertainty on $\kappa$. The
task of performing the detailed simulation for all these samples is
excessively computing intensive and time demanding with the resources
available for this study. For this reason a hybrid fast-simulation/full-
simulation approach has been used:

  * •

A sample of $\mu^{+}\mu^{-}qq^{\prime}$ events has been generated with
Madgraph with $k=1.0$ and processed with the detailed detector simulation.
This sample has been used to determine the reconstruction efficiencies and
momentum resolutions for jets and muons, as well as their dependence on the
kinematics.

  * •

$\mu^{+}\mu^{-}qq^{\prime}$ samples have been generated with Madgraph,
corresponding to 17 values of $\kappa$ ranging form $-0.2$ to $2.0$, to which
the resolutions and efficiencies obtained with the detailed simulation have
been applied. Resolutions are applied as smearing factors and efficiencies as
event weights. The dependence on the kinematics is taken into account.

  * •

These fast-simulation samples are then used to perform the likelihood scan for
the determination of the uncertainty on $\kappa$.

The cross sections, calculated with Madgraph, of the main processes that
contribute to $\mu^{+}\mu^{-}qq^{\prime}$ are shown in Fig. 13 as a function
of $\kappa$. It is evident that
$\mu^{+}\mu^{-}\rightarrow\nu_{\mu}\bar{\nu}_{\mu}Z(\rightarrow\mu^{+}\mu^{-})Z(\rightarrow
q\bar{q})$ is the most sensitive process to $\kappa$.

Figure 12: Cross sections obtained with Madgraph for the processes considered
in the determination of the Higgs coupling modifier $\kappa$.

In this analysis, jets and muons are reconstructed as in Sec. 4.3. Events are
selected requiring at least two opposite-charge muons with $p_{\mathrm{T}}>15$
GeV and at least two jets with $p_{\mathrm{T}}>20$ GeV. The positive and
negative muons with the highest $p_{\mathrm{T}}$ are selected and required to
have an invariant mass between 85 and 97 GeV. Jets with a $\Delta R<0.5$ with
respect to any of the two selected muons are discarded. To remove fake jets
from the BIB, jets are required to contain at least one track. After this
requirement fake jets are assumed to be negligible and they are not considered
in the analysis. Finally, the two jets with the highest $p_{\mathrm{T}}$ are
selected and required to have an invariant mass greater than 50 GeV. The
invariant mass $M_{\mu\mu qq}$ of all selected final state objects ($\mu^{+}$,
$\mu^{-}$ and the two jets) is required to be between 180 and 1800 GeV.

Likelihood values are calculated comparing 2-dimensional distributions of
$M_{\mu\mu qq}$ and $\theta_{ZZ}$ (_i.e._ the angle between the reconstructed
$Z$ bosons in the laboratory frame) between data and simulation. Pseudo-
datasets are generated from the distribution with $\kappa=1$. For each pseudo-
dataset, the likelihood difference $-2\Delta\log(L)$ as a function of $\kappa$
is obtained. These values are fitted to a 6th degree polynomial, and the
$1\,\sigma$ interval is estimated as the interval around $\kappa=1$ where the
fitted polynomial has a value below 1. For each pseudo-dataset, upper and
lower bounds are found, and the final result is given by the average of 50
pseudo-datasets. The obtained likelihood scans are shown in Fig. 13. The final
result on the $\kappa$ uncertainty interval is $\kappa=1^{+17\%}_{-12\%}$.

Figure 13: Likelihood for th determination of $\kappa$, fitted with a 6th
degree polynomial. Likelihood scans for 50 pseudo-experiments are overlaid
(red lines), the points represent the average and standard deviations of
$-2\Delta\log(L)$ for the considered $\kappa$ values.

Starting from Eq. 16, it can be written as:

$\Delta\Gamma_{H}=\Delta\sigma(H\rightarrow ZZ^{*})\oplus(4\,\Delta\kappa)\ ,$
(17)

where the $\oplus$ symbol means squared sum. Taking $\Delta\sigma(H\rightarrow
ZZ^{*})$ from Sec. 4.3, the final result is found to be

$\frac{\Gamma_{H}}{\Gamma_{H}^{SM}}=1^{+71\%}_{-88\%}\ .$ (18)

As was done for the Higgs boson cross sections in Sec. 4.6, the result on the
Higgs width is compared with that obtained with a parametric simulation. In
particular, it has been compared with the study presented in Ref. [51], where
the detector effects are taken into account with a fast simulation based on
DELPHES, which uses the MuC detector card discussed in Sec. 4.6. In the
article cited above, the precision on the Higgs width, obtained with a fully
general $\kappa$-fit, is 24$\%$. However, the contribution obtained by
exploiting only the $\mu^{+}\mu^{-}\rightarrow\nu_{\mu}\bar{\nu}_{\mu}ZZ$ and
$\mu^{+}\mu^{-}\rightarrow\nu_{\mu}\bar{\nu}_{\mu}W^{+}W^{-}$ channels is
58$\%$ [52]. The latter result includes different final states ($4q$, $2q2l$,
and $l\nu qq$) and is of the same order as the one presented in this section
($\Delta\Gamma_{H}\sim 80\%$), which uses only $2q2\mu$ events and is based on
the detailed detector simulation. Furthermore, the aforementioned paper [51]
obtained an uncertainty of $\Delta\Gamma_{H}=3.4\%$ for the case of
$\sqrt{s}=10$ TeV muon collisions: future simulation studies with the
$\sqrt{s}=10$ TeV detector will be necessary to confirm these statements. We
notice also that the study presented in Ref. [50] shows how the introduction
of a system for tagging forward muons up to $|\eta|<6$ could enable a Higgs
width precision up to $1^{+2.1\%}_{-0.4\%}$ at a $\sqrt{s}=10$ TeV muon
collider, which is competitive with the expectation from FCC-ee combined with
HL-LHC [46]. Again, these results should be validated with the detector
simulation that includes a forward muon tagging system.

##  6 Trilinear coupling

In this section, the measurement of the trilinear coupling at a $\sqrt{s}=3$
TeV MuC is studied with the detailed detector simulation. The state-of-the-art
projections on the $\kappa_{\lambda_{3}}$ sensitivity obtained with MuC
parametric simulations are reported in Refs. [13, 10]. In those studies, the
68$\%$ confidence-level bound, obtained with $\mathcal{L}=1$ ab-1 of data
collected at $\sqrt{s}=3$ TeV, is composed of two intervals, corresponding to
two different likelihood minima: $0.73<\kappa_{\lambda_{3}}<1.35$ and
$1.85<\kappa_{\lambda_{3}}<1.94$. It is also stated that the second minimum
can be removed by collecting $\mathcal{L}=2$ ab-1, leading to a 68$\%$
confidence level of $0.85<\kappa_{\lambda_{3}}<1.16$. In the following, the
case with $\mathcal{L}=1$ ab-1 is considered for the detailed simulation, but
just the first minimum, closer to the SM value, is studied and compared with
the parametric simulation.

Sec. 6.1 describes the measurement of the double Higgs production ($HH$),
which is the first step towards a measurement of the trilinear coupling. Then,
the trilinear coupling extraction is presented in Sec. 6.2.

###  6.1 $HH\rightarrow b\bar{b}b\bar{b}$ cross section determination

The $HH$ production is studied using the $H\to b\bar{b}$ decay for both Higgs
bosons, since this final state exhibits the highest branching ratio, and the
channel with four $b$ quarks provides the highest statistics [53, 54]. The
contribution of other final states, like $HH\to b\bar{b}\gamma\gamma$, will be
investigated in future studies.

In this analysis, at least four reconstructed jets are required in the event
with a minimum $p_{\mathrm{T}}$ of 20 GeV. Input tracks for jets are obtained
with the CKF algorithm. The $b$-jet identification efficiencies and the
misidentification rates presented in Sec. 3.4, have been applied to the
reconstructed jets as a function of the jet $p_{\mathrm{T}}$ and taking into
account the flavour composition of the final state, as with the $H\to
b\bar{b}$ analysis in Sec. 4.1. This procedure replaces the direct application
of the tagging algorithms to the simulated samples with the BIB overlaid,
which requires computing resources not currently available.

To reconstruct the $HH$ events, all possible two-jet combinations are formed,
in which at least one jet is requested to be identified as a $b$-jet. Two
Higgs boson candidates are then built from the two jet pairs whose invariant
masses $m_{12}$ and $m_{34}$ minimize the figure of merit:

$F=\sqrt{(m_{12}-m_{H})^{2}+(m_{34}-m_{H})^{2}}\ ,$ (19)

where $m_{H}$ is the nominal Higgs boson mass.

The main physics background contribution comes from processes with four heavy-
quark jets in the final state, $\mu^{+}\mu^{-}\to
q_{h}\bar{q}_{h}q_{h}\bar{q}_{h}X$ ($q_{h}=b$ or $c$), which comprise multiple
intermediate electroweak gauge bosons. The other important source of
background is the process $\mu^{+}\mu^{-}\to Hq_{h}\bar{q}_{h}X\to
b\bar{b}q_{h}\bar{q}_{h}X$ where the $HHH$ vertex is not present. Physics
backgrounds resulting from light quarks and fake jets are assumed negligible.
The rationale for this assumption is that advanced tagging methods, based on
machine learning techniques, should be capable of rejecting these events while
keeping the signal, as discussed later in Sec. 6.2. A summary table presenting
the signal and background processes, along with their cross sections and
expected event yields, is provided in Tab. 8.

Process | $\epsilon[\%]$ | $\sigma\ [\mathrm{fb}]$ | $N_{exp}$
---|---|---|---
$\mu^{+}\mu^{-}\to HHX\to b\bar{b}b\bar{b}X$ | $27.5$ | 0.31 | $84$
$\mu^{+}\mu^{-}\to H(\to b\bar{b})q_{h}\bar{q}_{h}X$ | $24.72$ | 3.1 | $761$
$\mu^{+}\mu^{-}\to q_{h}\bar{q}_{h}q_{h}\bar{q}_{h}X$ | $18.1$ | 5.9 | $1066$
Table 8: Selection efficiencies, theoretical cross sections, and expected
events for signal and background processes in the $HH\rightarrow
b\bar{b}b\bar{b}$ analysis, with $\mu^{+}\mu^{-}$ collisions at 3 TeV and
$\mathcal{L}=1$ ab-1.

A Multilayer Perceptron (MLP) [39] has been trained on twelve observables to
separate the signal events from the physics background $\mu^{+}\mu^{-}\to
q_{h}\bar{q}_{h}q_{h}\bar{q}_{h}X$:

  * •

the invariant masses of the jet pairs with the highest (leading candidate) and
the lowest (sub-leading candidate) transverse momentum;

  * •

the module of the vectorial sum of the four jet momenta;

  * •

the sum of the four jet energies;

  * •

the angle between the two jets relative to the leading candidate;

  * •

the maximum separation angle between the jets in the event;

  * •

the angle between the highest-$p_{\mathrm{T}}$ jet in the pair relative to the
leading candidate, and the angle between the highest-$p_{\mathrm{T}}$ jet in
the pair relative to the sub-leading candidate, with respect to the $z$-axis;

  * •

the four jet transverse momenta.

The MLP output distributions for the $HH$ signal and background samples are
shown in Fig. 14. It can be observed that a good separation between the $HH$
and $q_{h}\bar{q}_{h}q_{h}\bar{q}_{h}X$ processes is achieved. It has been
verified that an additional MLP to separate the signal from the
$\mu^{+}\mu^{-}\to H(\to b\bar{b})q_{h}\bar{q}_{h}X$ background does not
improve significally the measurement, since the kinematic properties of this
process are very similar to those of the signal.

Using the MLP output distributions, pseudo-datasets have been generated
according to the expected number of events. The yield of the $HH$ signal is
extracted by fitting the MLP distribution: an average statistical uncertainty
of approximately 33% is obtained on the signal yield. The $HH$ cross section
is calculated with Eq. 4. Therefore, assuming negligible uncertainties on the
selection efficiency and integrated luminosity, the statistical uncertainty on
the double-Higgs production cross section is:

$\frac{\Delta\sigma(HH\rightarrow b\bar{b}b\bar{b})}{\sigma(HH\rightarrow
b\bar{b}b\bar{b})}=33\%\ .$ (20)

Figure 14: Distributions of the MLP output for the $HH$ signal and the main
background contributions. The distributions are normalized to unit area.

At this time, there is no estimate of the statistical sensitivity on the $HH$
cross section based on a parametric simulation of the detector. Reference [55]
reports only the expected uncertainty on the trilinear coupling without
considering detector effects, as explained in the next section.

###  6.2 Trilinear Higgs coupling precision determination

The evaluation of the uncertainty on the trilinear Higgs boson self-coupling
has been conducted using the simulated sample $\mu^{+}\mu^{-}\to HHX\to
b\bar{b}b\bar{b}X$, described in Sec. 6.1. The total double-Higgs production
cross section is sensitive to the value of the trilinear self-coupling: Fig. 4
shows the three Feynman diagrams that primarily contribute to the double-Higgs
production. The one on the left, where two Higgs bosons are produced via an
off-shell Higgs ($H^{*}$), is the only process directly influenced by the
value of the trilinear Higgs self-coupling. The different kinematics of the
$HH$ events coming from $H^{*}$ is exploited to extract the value of
$\lambda_{3}$. The strategy employed to estimate the uncertainty on the
trilinear Higgs self-coupling is as follows:

  * •

Samples of double-Higgs events are generated with WHIZARD for different values
of $\kappa_{\lambda_{3}}=\lambda_{3}/\lambda_{3}^{SM}$: 0.2, 0.4, 0.6, 0.8,
0.9, 1.0, 1.1, 1.2, 1.4, 1.6, 1.8. Subsequently, the samples are simulated and
reconstructed with the procedure described in Sec. 6.1. The
$\kappa_{\lambda_{3}}=1$ case represents the SM signal. Tab. 9 shows the
expected number of $HH$ events for different $\kappa_{\lambda_{3}}$
hypotheses;

$\kappa_{\lambda_{3}}$ |  $\sigma$ [fb] |  $\epsilon$ [%] | Nevents
---|---|---|---
0.2 | 0.52 | 30.1 | 169
0.4 | 0.48 | 29.3 | 141
0.6 | 0.41 | 28.1 | 116
0.8 | 0.35 | 27.5 | 97
0.9 | 0.33 | 27.6 | 92
1.0 | 0.31 | 27.2 | 84
1.1 | 0.29 | 25.7 | 74
1.2 | 0.27 | 25.7 | 71
1.4 | 0.25 | 26.5 | 68
1.6 | 0.24 | 25.1 | 61
1.8 | 0.25 | 24.8 | 61
Table 9: Theoretical cross section values, selection efficiencies, and
expected number of $HH\to b\bar{b}b\bar{b}$ events for all
$\kappa_{\lambda_{3}}$ hypotheses, assuming 1 ab-1 of integrated luminosity at
$\sqrt{s}=3$ TeV.

  * •

Two MLPs are trained independently:

    1. 1.

The first MLP is the same as the one used for the $HH$ cross section
determination in Sec. 6.1 to separate the SM signal ($\kappa_{\lambda_{3}}=1$)
from the physics background.

    2. 2.

The other MLP is trained on four variables to differentiate between the
$\mu^{+}\mu^{-}\to HH\nu_{\mu}\bar{\nu}_{\mu}\to
b\bar{b}b\bar{b}\nu_{\mu}\bar{\nu}_{\mu}$ process, where the Higgs pair is
produced exclusively via an off-shell $H^{*}$ (left diagram in Fig. 4), and
the other two contributions. The rationale behind this approach is to
construct a MLP that leverages the kinematic properties specific to the
production of Higgs pairs via an off-shell Higgs boson, which is indicative of
the trilinear Higgs self-coupling. This allows for the separation of this
process from other double-Higgs production processes. The variables that allow
the best separation are: the angle between the two Higgs boson momenta in the
laboratory frame, the angle between the highest-$p_{T}$ jet momenta of each
pair with respect to the $z$ axis, and the helicity angle of the two Higgs
boson candidates.

  * •

The scores of the two MLPs for the considered samples have been arranged in
2-dimensional histograms. To obtain the expected data distribution,
2-dimensional templates of the signal and background components are built for
each $\kappa_{\lambda_{3}}$ hypothesis. Signal and background yields
calculated in Tabs. 8 and 9 are used to weight the templates.

  * •

Pseudo-datasets are generated with the total 2D template for the
$\kappa_{\lambda_{3}}=1$ hypothesis. For each pseudo-experiment, the
likelihood difference $-\Delta\mathrm{log}(L)$ is calculated as a function of
$\kappa_{\lambda_{3}}$ by comparing the pseudo-data distribution to the
$\kappa_{\lambda_{3}}$ templates.

  * •

The log-likelihood profile has been fitted with a polynomial function of
fourth degree. The uncertainty on $\kappa_{\lambda_{3}}$ at 68% C.L. is
estimated as the interval around $\kappa_{\lambda_{3}}=1$ where the fitted
polynomial has a value below 0.5.

As shown in Tab. 9, the expected production cross sections for
$\kappa_{\lambda_{3}}>1$ are very similar. Distinguishing between them
requires a signal-to-noise separation better than what is currently
achievable. Therefore, the result achievable with the detailed detector
simulation, including the BIB overlay, and the current version of the
reconstruction algorithms, is significantly biased due to the non-optimized
algorithms. This result does not provide an adequate representation of the
true potential of MuC for this measurement. Although many improvements are
currently in progress, they are not yet ready to be applied to the analysis.
Nevertheless, it is possible to assess the impact of the new algorithms on the
measurement, as discussed in the following:

  * •

With a dedicated machine-learning algorithm that exploits the jet sub-
structure, it is possible to identify $b$-jets with a performance comparable
to that of hadron colliders. Machine-learning algorithms that exploit
information about the jet sub-structure are currently employed at LHC to
enhance the performance of jet flavor identification both in the central
region of $pp$ collisions, like the DeepJet algorithm [56] at CMS, and in the
forward region at LHCb [57]. At a MuC, $b$-jets from $HH$ are expected to be
produced predominantly in the forward region, and an excellent vertex detector
is considered for the MuC detector, as the one of LHCb that is optimized for
having the best vertex reconstruction performance. For these reasons, the LHCb
tagging capabilities are assumed for the MuC expectation: the $b$-tagging
efficiency and $c$-mistag are taken from Ref. [57], while light jet mistag is
assumed negligible, as expected from LHCb data. A working point for the $HH$
analysis has been determined by maximizing the significance ($S/\sqrt{S+B}$)
of $HH$ events across various configurations of LHCb $b$-tagging and
$c$-mistag performance. The optimal configuration, yielding the highest
significance, requires: 3 tagged jet, a total $b$-tagging efficiency of 76$\%$
and a $c$-mistag rate of 20$\%$. Simulated events for $HH$ signal and
background processes are re-weighted according to these efficiencies.

  * •

Several developments are currently underway to minimize the impact of the BIB
on both jet reconstruction efficiency and jet energy resolution, aiming to
reduce it to a negligible level. For example, the number of BIB hits in the
tracker system could be reduced by applying a cutting-edge clustering
algorithm capable of identifying and rejecting hits associated with low-
momentum BIB particles, which tend to deposit more ionization energy. This
strategy aims to decrease the occurrence of spurious tracks, thereby enhancing
the overall tracking performance. The impact of such an algorithm is evaluated
by studying the jet reconstruction performance without overlaying the BIB in
the analysis, and reducing the calorimeter energy thresholds from 2 MeV to 200
KeV. The last assumption leads to an improvement in the jet energy resolution,
as already demonstrated in Ref. [10]. The impact of the jet energy resolution
improvements on the $HH$ analysis can be understood by comparing the invariant
mass distribution of the leading Higgs candidate in $HH$ events to that of
$q_{h}\bar{q}_{h}q_{h}\bar{q}_{h}$ events, both without the BIB overlay, Fig.
15 (left). The improved separation between the Higgs and $Z$ boson resonances
with respect to what has been obtained in Fig. 6 is evident. The jet momentum
resolution in the case without the BIB overlay is in the order of 10%, while
the resolution with the BIB is of about 15%. This 10% resolution is used in
the analysis to evaluate the trilinear coupling precision.

Figure 15: Left: Invariant mass of the leading Higgs candidate compared
between $HH$ events and $q_{h}\bar{q}_{h}q_{h}\bar{q}_{h}$ events, with
samples reconstructed without the BIB. Right: $\Delta LL$ as a function of
$\kappa$ hypothesis for samples reconstructed without the BIB, selected by
assuming LHCb tagging performance ($b$-tagging efficiency of 76$\%$ and
$c$-mistag rate of 20$\%$), by requiring three tagged jets in the event.

The likelihood scan obtained with the previous assumptions on the tagging
efficiencies and jet momentum resolution is shown in Fig. 15 (right). The
uncertainty on the trilinear Higgs self-coupling is found to be
$0.81<\kappa_{\lambda_{3}}<1.44$ at 68 $\%$ confidence level. It could be
noticed that the upper bound is higher since the number of signal events as a
function of $\kappa_{\lambda_{3}}$ varies slowly for
$\kappa_{\lambda_{3}}>1.0$, as can be seen in Tab. 9.

The result obtained with the detailed detector simulation can be compared with
the first minimum determined with the parametric simulation at the same
luminosity of $\mathcal{L}=1$ ab-1 [13, 10], $0.73<\kappa_{\lambda_{3}}<1.35$.
The two results are of the same order.

##  7 Lesson learned: requirements for Higgs measurements

The Higgs studies presented in this paper allow to establish the detector
requirements needed to improve Higgs knowledge. Sec. 4.6 demonstrates that the
results obtained for a $\sqrt{s}=3$ TeV MuC with the detailed detector
simulation are compatible with the ones determined with the MuC DELPHES card
in Ref. [25]. In the same article, it is shown that including those results
from parametric simulations in the Higgs couplings fit, together with HL-LHC
measurements, improves the overall precision with respect to the HL-LHC-only
case.

In general, Sec. 4 has shown how the detector and reconstruction algorithms,
discussed in Sec. 3, are sufficient to achieve the precision on single Higgs
cross sections obtained with parametric simulations. This has been
demonstrated for final states with jets, photons, and muons.

On the other hand, Sec. 6.2 has shown how several improvements on jet
reconstruction and identification are needed to improve the precision on the
Higgs trilinear coupling with respect to HL-LHC. In fact, the projection for
HL-LHC is a precision of about 50% on $\kappa_{\lambda_{3}}$ [1], while
$0.81<\kappa_{\lambda_{3}}<1.44$ is expected with the $\sqrt{s}=3$ TeV MuC
detector and $\mathcal{L}=1$ ab-1.

With the previous considerations, the requirements, reported in Tab. 10, are
determined in the following way:

  * •

For muons, photons, jets, and $b$-jets the resolution requirements are taken
from the performance of the reconstruction described in Sec. 3, and evaluated
for objects with $p_{\mathrm{T}}=100$ GeV. The value of $p_{\mathrm{T}}=100$
GeV is chosen because it is the peak of Higgs daughters $p_{\mathrm{T}}$
distribution in 2-body Higgs decays (Fig. 16). For $b$-jets the identification
efficiencies and the $c$-mistag rate are also quoted. The motivation for
determining the requirements in this way is that these performance is
sufficient to achieve results on the cross-section statistical sensitivity
comparable with those obtained with the parametric simulations.

  * •

For the measurement of $\lambda_{3}$, the performance of Sec. 3 is not enough
to achieve a result that surpasses HL-LHC and of the same order of the
parametric simulation. Hence, just for this measurement, the requirements on
$b$-jets $p_{\mathrm{T}}$ resolution and identification are taken from the
performance described in Sec. 6.2, obtained with reasonable assumptions and
that allows a competitive precision.

In Tab, 10, the performance requirements are summarized.

Object | Requirements
---|---
muons | $\frac{\Delta p_{T}}{p_{T}}=0.4\%$
photons | $\frac{\Delta E}{E}=3\%$
jets | $\frac{\Delta p_{T}}{p_{T}}=15\%$
$b$-jets | $\frac{\Delta p_{T}}{p_{T}}=15\%$
|  $b$ efficiency = 60 %
|  $c$ mistag = 20 %
$b$-jets | $\frac{\Delta p_{T}}{p_{T}}=10\%$
(for $\lambda_{3}$) |  $b$ efficiency = 76 %
|  $c$ mistag = 20%
Table 10: Detector and reconstruction requirements for Higgs physics at the
muon collider, determined with the detailed detector simulation studies. The
values have been obtained for objects with $p_{T}$ of about 100 GeV, which is
the average transverse momentum of the particles from the 2-body decay of the
Higgs boson with $\sqrt{s}=3$ TeV. The first set of requirements is intended
for single Higgs boson measurements, while more stringent requirements are
presented for the trilinear coupling determination with the four-$b$-jet final
state.

The detector requirements determined for a $\sqrt{s}=3$ TeV MuC case are also
fundamental for designing the $\sqrt{s}=10$ TeV detector. Fig. 16 presents the
transverse momentum of $b$-quarks from $H\to b\bar{b}$ decay at $\sqrt{s}=3$
and $\sqrt{s}=10$ TeV: it is evident that the distributions are very similar,
with a peak around $p_{\mathrm{T}}\sim 100$ GeV. Therefore, for Higgs physics,
similar requirements are expected for $\sqrt{s}=3$ TeV and $\sqrt{s}=10$ TeV
cases.

Figure 16: Transverse momenta of $b$-quarks from the $H\rightarrow b\bar{b}$
decay, at $\sqrt{s}=3$ and $\sqrt{s}=10$ TeV muon-muon collisions, determined
with Madgraph [58].

##  8 Outlook

The studies presented in this paper demonstrate that at a $\sqrt{s}=3$ TeV
MuC, a state-of-the-art detector enables high-precision measurements of Higgs
cross sections, including that of double Higgs production. This allows for the
determination of the Higgs trilinear self-coupling with an accuracy that
surpasses that of the HL-LHC, with only one experiment and 1 ab-1 of data. The
estimates have been performed using a detailed detector simulation, which
includes the beam-induced background [21]. For each analysis, the events of
the Higgs boson signal and corresponding physics backgrounds have been fully
simulated and reconstructed (Sec. 3). The statistical sensitivities on the
Higgs cross sections measurements (Sec. 4) for the considered Higgs decay
channels ($H\rightarrow b\bar{b}$, $H\rightarrow WW^{*}$, $H\rightarrow
ZZ^{*}$, $H\rightarrow\mu^{+}\mu^{-}$, and $H\rightarrow\gamma\gamma$) are
consistent with those obtained using parametric detector simulations. This
consistency underscores the capability of the detector design and
reconstruction algorithms to effectively manage the effects of the beam-
induced background. The 3 TeV center-of-mass energy is not optimal for
measuring the Higgs width, as indicated by Refs. [51, 50] and the analysis
presented here in Sec. 5. The limitation is due to the production mechanism
and therefore cannot be overcome.

The estimated precision on the Higgs trilinear self-coupling (discussed in
Sec. 6) is currently affected by the performance of the simplified algorithms
employed in event reconstruction. The impact of planned improvements have been
assessed, bringing the precision to a level compatible with that obtained with
the parametric studies in Refs. [13, 10]. This also provides confidence in the
prospects obtained in Ref. [13] for a measurement of the trilinear self-
coupling at a center-of-mass energy of 10 TeV. Running a MuC at $\sqrt{s}=10$
TeV for 5 years is expected to achieve an uncertainty of approximately 4% with
10 ab-1, which represents the best precision attainable by any future
collider.

Another important outcome of this paper is the determination of detector
requirements for Higgs physics (Sec. 7), derived from detailed analyses that
utilize various physics objects. These requirements (Tab. 10), found to be
applicable also at higher center-of-mass energies, serve as the foundation for
detector design.

In fact, the average transverse momentum of particles from Higgs decays is of
the same order ($\sim$$100$ GeV) at both $\sqrt{s}=3$ TeV and $\sqrt{s}=10$
TeV [58]. Moreover, the impact of beam-induced background on the detector at
$\sqrt{s}=10$ TeV is expected to be similar to that at $\sqrt{s}=3$ TeV [21].
For these reasons, the detector is anticipated to provide the desired
performance at the higher center-of-mass energy, where a MuC can achieve a
precision on the Higgs sector unreachable by other proposed facilities.

Future work with the detailed detector simulation should consider: the decay
channels not yet studied (_e.g._ $H\to\tau^{+}\tau^{-}$), for the
determination of additional Higgs couplings with fermions and bosons; an
optimization of the algorithms for the measurement of the trilinear Higgs
self-coupling; a first study on the measurement of the quartic self-coupling
at $\sqrt{s}=10$ TeV.

## Acknowledgments

We are grateful to the International Muon Collider Collaboration and the US
Muon Accelerator Program for their support. In particular, we thank Anthony
Badea, Tova Holmes, Federico Meloni, and Andrea Wulzer for their valuable
comments that helped improve the paper. We acknowledge the financial support
of the Italian National Institute for Nuclear Physics (INFN), the University
of Padua, and the European Organization for Nuclear Research (CERN). This work
was supported by the European Union’s Horizon 2020 and Horizon Europe Research
and Innovation programs through the Marie Skłodowska-Curie RISE Grant
Agreement No. 101006726, the Research Infrastructures INFRADEV Grant Agreement
No. 101094300, and the EXCELLENT SCIENCE - Research Infrastructures Research
Innovation Grant Agreement No. 101004761.

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