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# Experimentation at a muon collider

Massimo Casarsa  1 Donatella Lucchesi2 and Lorenzo Sestini3 1INFN Sezione di
Trieste, via Valerio 2, I-34127 Trieste, Italy; email:
<EMAIL_ADDRESS>2Department of Physics and Astronomy ”G. Galilei”
and INFN, via Marzolo 8, I-35131 Padova, Italy; email:
<EMAIL_ADDRESS>3INFN Sezione di Padova, via Marzolo 8, I-35131
Padova, Italy; email<EMAIL_ADDRESS>
###### Abstract

Experimental activities involving multi-TeV muon collisions are a relatively
recent endeavor. The community has limited experience in designing detectors
for lepton interactions at center-of-mass energies of 10 TeV and beyond. This
review provides a short overview of the machine characteristics and outlines
potential sources of beam-induced background that could impact the detector
performance.

The strategy for mitigating the effects of beam-induced background on the
detector at $\sqrt{s}=3$ TeV is discussed, focusing on the machine-detector
interface, detector design, and the implementation of reconstruction
algorithms.

The physics potential at this center-of-mass energy is evaluated using a
detailed detector simulation that incorporates the effects of beam-induced
background. This evaluation concerns the Higgs boson couplings and the Higgs
field potential sensitivity, that then are used to get confidence on the
expectations at 10 TeV.

The physics and detector requirements for an experiment at $\sqrt{s}=10$ TeV,
outlined here, form the foundation for the initial detector concept at that
center-of-mass energy .

###### doi:

10.1146/annurev-nucl-102622-011319

###### keywords:

muon collider, detector, multi-TeV muon collisions, Higgs boson, new physics

††journal: Xxxx. Xxx. Xxx. Xxx.

###### Contents

  1. 1 INTRODUCTION: YET ANOTHER COLLIDER?
    1. 1.1 Muon collider concept
    2. 1.2 The beauty of colliding muons
  2. 2 THE MUON COLLIDER SEEN FROM THE EXPERIMENTAL POINT OF VIEW
    1. 2.1 Background contributions in the detector region
    2. 2.2 Beam-induced background characteristics
  3. 3 DETECTOR PERFORMANCE IN THE PRESENCE OF THE BEAM-INDUCED BACKGROUND
    1. 3.1 Detector requirements and comparison with LHC experiments
    2. 3.2 Muon collider detector concept at $\sqrt{s}=3$ TeV
    3. 3.3 Beam-induced background mitigation in the tracking system
    4. 3.4 Beam-induced background mitigation in the calorimeter system
    5. 3.5 Physics objects reconstruction with particle flow
    6. 3.6 Luminosity measurement
  4. 4 THE GUARANTEED PHYSICS DISCOVERY: FULL DETERMINATION OF THE HIGGS BOSON PROPERTIES
    1. 4.1 The Higgs boson couplings determination
    2. 4.2 The Higgs boson self-couplings and the Higgs potential
  5. 5 THE EXPERIMENT AT THE 10 TEV FRONTIER
    1. 5.1 Quest for new physics
    2. 5.2 Physics and detector requirements for an experiment at $\sqrt{s}=10$ TeV
      1. 5.2.1 Tracking system and magnetic field at $\sqrt{s}=10$ TeV
      2. 5.2.2 Calorimeter systems at $\sqrt{s}=10$ TeV
      3. 5.2.3 Muon detection at $\sqrt{s}=10$ TeV
  6. 6 OUTLOOK

##  1 INTRODUCTION: YET ANOTHER COLLIDER?

A systematic study of multi-TeV muon collider started as a project in US in
2011 with the Muon Accelerator Program (MAP) [1, 2] to investigate the
feasibility of a facility where both neutrino physics and experiments based on
muon collisions could be possible. MAP designed colliders with center-of-mass
energies of 1.5, 3, and 6 TeV111In this paper natural units where $\hbar=c=1$
are used. and optimized the interaction region (IR) configuration for a
collider operating at $\sqrt{s}=1.5$ TeV. After the recommendations of The
2020 Update of the European Strategy for Particle Physics [3], an
International Muon Collider Collaboration (IMCC) [5] was formed currently
hosted at CERN with the aim of establishing a feasible design and defining the
necessary R&D for a multi-TeV muon collider. The Snowmass2021 process [4], a
community planning exercise in particle physics, has strengthened the
collaboration between US and Europe allowing to revise the studies performed
by MAP and to progress in the design of the facility. The current focus is on
the definition of the parameters of a muon collider at $\sqrt{s}=3$ and
$\sqrt{s}=10$ TeV, but the project still foresees a facility where also
neutrino experiments can be located.

###  1.1 Muon collider concept

The design of a muon collider facility is mainly driven by the short lifetime
of muons, $2.2~{}\mu$ at rest. A conceptual layout of such a facility is
presented in Figure 1. It foresees a muon injector and two rings, a large one
for muon acceleration and a small one for collisions. The reduced dimensions
of the collider are required to allow muons to pass through the interaction
point many times before decaying. The current rings dimensions [6] do not
consider any specific site, and by adjusting the machine configuration, they
can potentially exploit existing and planned infrastructure at CERN and
Fermilab. Preliminary parameters of the muon collider complex [6], relevant
for what will follow, are summarized in Table 1. It must be noticed that the
instantaneous luminosity increases significantly from $\sqrt{s}=3$ TeV to
$\sqrt{s}=10$ TeV. As demonstrated in [6], it grows approximately with the
square of the muon beam energy at fixed average bending magnetic field. This
is a feature of the muon collider due to the fact that the beam can be
recirculated many times through the interaction point (IP) thanks to the
negligible effect of the beamstrahlung on the focusing achievable at the
interaction point. The luminosity benefits also from having only one bunch of
$\mu^{+}$ and one of $\mu^{-}$ at the time in the collider ring.

Table 1: Preliminary muon collider parameters based on a revised MAP design.  Parameter | Unit of measure | Target value
---|---|---
Center-of-mass energy ($\sqrt{s}$) | TeV | 3 | 10
Luminosity |  $10^{34}$ cm-2 s-1 | 2 | 20
Collider circumference | km | 4.5 | 10
Muons/bunch | $10^{12}$ | 2.2 | 1.8
Beta function at IP | mm | 5 | 1.5
Figure 1: Conceptual layout of a multi-TeV muon collider.

Muon decays along the collider ring produce intense high energy neutrino
fluxes that can generate hadronic and electromagnetic showers when interacting
with the environment underground. In particular, in the straight sections of
the machine, intense and very collimated fluxes of high energy neutrinos are
produced. The effects at the Earth surface where they emerge, even hundreds of
kilometers away from the collider complex, are under study to ensure that the
impact on the environment is significantly below the legal limits and the ones
of LHC. The complete determination of such effects is a multi-parameter
problem: they depend on the machine center-of-mass energy, the collider
configuration and the geological characteristics of the soil.

The heart of the facility is the muon injector. Intense protons beams,
produced by a proton source, hit a multi-Watt class target to generate pions
that eventually decay to muons. The target is immersed in a high solenoidal
magnetic field to capture and guide pions into the decay channel, where muons
are captured in a bunch train, with a time dependent acceleration due to their
different energies.

Muon cooling process is critical to achieve the necessary beta function at IP
(see Table 1) for the high luminosity at any center-of-mass energy. The
challenge of the cooling process is the muon short lifetime, the cooling must
take place more quickly than any of the cooling methods presently in use. A
new technology, called “ionization cooling”, has been tested by the Muon
Ionization Cooling Experiment (MICE) [7]. The particle momentum is reduced
through ionization energy loss in absorbers and replenished only in the beam
direction through radio-frequency cavities. MICE experimentally demonstrated
the first part, it measured an increase in the phase-space density of the
beam, and the results agree with the simulations. The cooling in the direction
of the beam and the re-acceleration of the particles have to be proven even
though well predicted by the simulation tuned on MICE experimental results.

The first step toward a muon collider facility is the demonstration of the 6D
cooling at low emittance and the re-acceleration of the muon beam through
several cooling cells [8]. This requires, as highlighted in [6]:

  * -

design, construction and integration of the cooling cell;

  * -

a sufficiently intense proton beam that impinges on a target to produce pions,
typically with a momentum range of 100–300 MeV, that in turn decay into muons
having large transverse emittance;

  * -

an upstream instrumentation system to measure the muon beam properties and
then guide it to a certain number of cooling cells;

  * -

a downstream instrumentation apparatus that measures the beam characteristics.

The design of this experimental demonstration with the relative
instrumentation is part of the current activities of the IMCC. This muon beam
facility could be hosted in any place where an adequate proton source is
available. Such a facility could also host physics experiments as the muon
beam dump [9] project. The proposed study of new gauge forces would require a
beam with energy of the order of 100 GeV, in addition to the dedicated
detector. The feasibility of this experiment will be assessed during the
design phase of the demonstration facility.

###  1.2 The beauty of colliding muons

Muon collisions at high center-of-mass energies open a completely new physics
scenario. A muon collider offers the possibility to reach high center-of-mass
energy with high luminosity and also to perform high precision measurements
thanks to the fact that muons are elementary particles.

The direct particle production can proceed via $s-$channel $\mu^{+}\mu^{-}$
annihilation, in which heavy particles of mass $M$ can be produced with a
kinematical threshold at $M=\sqrt{s}/2$. As pointed out in [10], states such
as $Z^{\prime}$, $W^{\prime}$, heavy Higgses or other heavy particles could be
produced singly in association with soft/collinear vector boson like $\gamma$,
$Z^{0}$ and $W^{\pm}$ overcoming the kinematical limit. Comparisons between
proton-proton and muon colliders production cross sections for $2\rightarrow
2$ and $2\rightarrow 1$ processes are studied [11, 12] by taking into account
the differences of electroweak states and colored states to have insight of
the most appropriate collider to explore the very high energy regime. Such a
comparison is not trivial, because it can not represent the peculiarities of
the two machines and the complexity of the data produced in the two very
different environments.

Multi-TeV muon beams colliding at an energy which is much higher than the
electroweak scale, $\sim$100 GeV, have high probability to emit electroweak
bosons. A summary of the theoretical investigations performed so far on the
characteristics of such an emission is provided in [6], where it is also
mentioned that the experimental observation of these phenomena and their
comparison with the theoretical predictions is, by itself, a very interesting
measurement. At the center-of-mass energies considered, the vector boson
fusion (VBF) is the dominant production process and this is why the muon
collider is often referred to as a “vector boson collider”. The production
cross section of standard model (SM) processes via VBF are very high, see for
example the Higgs cross sections reported in Figure 7a, that, combined with
the high luminosity of the collider and the very low physics background
contribution, allow precise SM tests in a completely unexplored energy regime.
The precise knowledge of the SM processes at a multi-TeV energy scale gives
the possibility to probe new physics indirectly extending the reaches given by
the direct interactions.

##  2 THE MUON COLLIDER SEEN FROM THE EXPERIMENTAL POINT OF VIEW

The design of a detector able to operate and take data at a multi-TeV muon
collider has to take into account not only the physics requirements, but also
the IR conditions.

The machine-detector-interface (MDI) studies performed by MAP and recently by
the IMCC indicate that the final focusing magnets cannot be located more than
$\pm$6 meters from the IP not to degrade the instantaneous luminosity. Figure
2a shows the configuration of a possible IR, where the available space for the
detector along the beam axis is a total 12 meters. The vertical dimension can
be bigger, but it would not enhance the physics acceptance, particularly at
very high center-of-mass energy where several relevant physics process occur
forward respect to the beam line Section 5.

###  2.1 Background contributions in the detector region

Elementary particle collisions usually do not have secondary interactions
beyond the primary one as seen in hadron-hadron collisions. Nevertheless, muon
collider suffers from background coming from the machine, the main sources
expected to contribute in the detector region at $\sqrt{s}=3$ and 10 TeV are:

a) Muon decay: intense muon beams of energies of 1.5 and 5 TeV decay producing
positrons and electrons of several TeV and high energy photons from
synchrotron radiation of energetic electrons and positrons in the collider
magnetic fields. The majority of these particles directly reach the detector
and may compromise any physics measurements. The MAP project proposed to
insert two conical-shaped absorbers, referred to as nozzles [13], made of
tungsten and covered with borated polyethylene, inside the detector along the
beamline (Figure 2). This combination is effective in absorbing high-energy
particles and reducing their impact on the detector. This shielding structure
modifies the characteristics of the background arriving on the detector as
discussed in Section 2.2.

b) Incoherent $\bm{e^{+}e^{-}}$ production: the production cross section of
$e^{+}e^{-}$ pairs in high energy muon beams has been calculated in [14]
starting from the original formulation of Landau and Lifshitz [15]. Such a
background is synchronous with the beam collisions and is mainly produced
along the beam direction. The MAP project evaluated the contribution of this
source of background for beams of 750 GeV [16] and found it to be almost
negligible. This is primarily because the production cross-section is
suppressed at low muon energy, and the detector’s magnetic field effectively
traps $e^{+}e^{-}$ pairs near the beam line. As the beam energy increases, the
probability of incoherent positrons and electrons production rises.
Consequently, a stronger magnetic field is required to confine these particles
near the beam-line. This contribution to the background has not been studied
yet, a preliminary evaluation suggests that at $\sqrt{s}=10$ TeV, a solenoidal
magnetic field of about 5 T in the detector can significantly reduce the flux
of incoherent $e^{+}e^{-}$ pairs reaching the detector. However, given the
potential impact of these particles on the detector internal layers, it is
essential to conduct a detailed simulation study.

c) Beam halo: beam losses are unavoidable, but at the moment they are not
considered among the background sources in the detector region. It is believed
that an appropriately designed collimation system positioned upstream of IP
can effectively reduce these losses to a manageable level.

###  2.2 Beam-induced background characteristics

The nozzles inserted around the IP absorb the high energy electrons and
positrons generated by the muon decays, but are responsible for fluxes of
particles that eventually arrive to the detector. Electrons, positrons and
photons interacting with the tungsten material of the nozzles and the IR
machine elements generate electromagnetic showers. In addition, hadronic
particles, mainly neutrons with a minor contribution from charged hadrons, are
generated through photonuclear interactions with the IR material, including
the nozzles. The borated polyethylene cladding of the nozzles is designed to
absorb neutrons, but it also produces high fluxes of photons, which contribute
to background signals in the calorimeter system (Section 3.4). Muons instead
are produced in the Bethe-Heitler process [17] by energetic photons from
electromagnetic showers.

An assessment of the effects of the beam-induced background (BIB) on the
detector response involves a laborious procedure. It must be noticed that BIB
sample used in the detector studies until now was produced by MAP using the
MARS15 software [18, 19] at a center-of-mass energy of 1.5 TeV. The IR and the
nozzle dimensions and composition were optimized to minimize the occupancy on
the first layers of the tracking system.

The IMCC has setup a software procedure to simulate the beam induced
background [20]:

  * 1)

IR accelerator lattice and optics are modeled in the FLUKA code [21] via the
LineBuilder [22] package as well as the nozzles.

  * 2)

The detector solenoidal magnetic field is included in the IR description.

  * 3)

A primary beam is simulated starting from a given distance, $L_{BIB}$ from the
IP; the value of $L_{BIB}$ depends on the center-of-mass energy and it is
given by the point along the beam-line where the cumulative distribution of
particles exiting the machine start saturating to 100%. Given the symmetric
nature of $\mu^{+}$/$\mu^{-}$ only one primary beam is considered.

  * 4)

Particles exiting the machine, either from the nozzles or from any of the IR
components, are saved and then provided as an input to Geant4 [23] simulation
software to be tracked inside the detector.

The optimization path requires to execute all the actions from 1) to 4) with a
given IR and nozzle configuration, evaluate the occupancy in the detector
until an acceptable/optimal value is reached. The procedure has been executed
by using $\sqrt{s}=1.5$ TeV IR and the results compared with those obtained
with MARS15 simulation code showing a very good agreement [20], giving
robustness to the evaluation of the beam-induced background in the detector.

The determination of the characteristics of such a background at $\sqrt{s}=3$,
10 TeV is in its early stage. The IR at a center-of mass energy of 3 TeV [24]
comes from MAP, the preliminary design at $\sqrt{s}=10$ TeV has been done by
the IMCC [25]. By using the nozzles proposed by MAP [13], the beam-induced
background has been generated for both machine configurations. The solenoidal
magnetic field is 3.57 T. Figure 2b shows the energy distributions of
particles at $\sqrt{s}=3$ TeV and $\sqrt{s}=10$ TeV for each particle type
arriving on the detector in a time window of [-1, 15] ns with respect to the
collision time [26]. The two center-of-mass energies show a similar behavior,
which is also similar to the one at $\sqrt{s}=1.5$ TeV [6]. Given the wide
range of energies considered, this demonstrates that the energy distribution
of the beam-induced background is determined by the nozzles. Figure 2c
illustrates the particles arrival time on the detector still for the same
center-of-mass energies. The distributions, similar in the two energies cases,
show a sizeable contribution out of the time window considered as adequate for
data taking. The similar behavior is expected due to the similar energy
spectrum. It can be noticed that the absolute value at $\sqrt{s}=10$ TeV is
slightly higher than that at $\sqrt{s}=3$ TeV, mainly for two reasons: the
different center-of-mass energy and IR configuration. An important effort is
ongoing to optimize the IR and the nozzles shape and materials for the center-
of-mass energy of 3 and 10 TeV.

a

b

c

Figure 2: (a) Top view of the IR, the $x$ axis represents the beam direction.
The outer shape of the detector is a cylinder with a radius of 6 m. The space
between the outer shape and the nozzles is considered as a perfect particle
absorber. (b) Energy distribution of particles arriving at the detector for
$\sqrt{s}=3$ TeV (dashed line) and $\sqrt{s}=10$ TeV (solid line). Different
colors represent different types of particles. (c) Distribution of the arrival
time of the variuos particle types for the two center-of-mass energies and
with the same legend as used in figure (b).

##  3 DETECTOR PERFORMANCE IN THE PRESENCE OF THE BEAM-INDUCED BACKGROUND

The design of an experiment at a multi-TeV muon collider has many features in
common with the experiments at other proposed multi-TeV machines [27, 28, 29,
30], but also presents several novel challenges due to the unique background
conditions related to the unstable nature of muons. The first constraints come
from the machine layout and the detector shielding requirements. As discussed
in Section 2, the available space around the interaction region is limited by
the collider final focusing magnets and the MDI, in particular the nozzles
reduce the angular acceptance in the forward and backward regions of the
detector. In addition, each sub-detector must deal with the beam-induced
background to ensure optimal physics performance.

In the following sections only one detector concept is discussed. Given that
the muon collider can have two interaction points, different decisions can be
made during the actual design phase.

###  3.1 Detector requirements and comparison with LHC experiments

The presence of the BIB makes the detector requirements more similar to those
for proton-proton than for positron-electron collisions. In addition, detector
technology R&D is currently focused on the High Luminosity LHC (HL-LHC) [32],
making it interesting to have LHC [31], with its experiments, and HL-LHC as
references.

The starting point is the comparison of the beam dimensions at the muon
collider with the nominal LHC configuration [33]: the parameters at the
interaction point are reported in Table 2.

Table 2: Comparison of the beam size at IP for LHC and muon collider. | LHC | muon collider
---|---|---
$\sqrt{s}$ | 14 TeV | 3 TeV | 10 TeV
bunch length | 7.7 cm | 5 mm | 1.5 mm
transversal bunch size | 16.7 $\mu$m | 3 $\mu$m | 0.9 $\mu$m

At the IP, the bunch of the muon collider is shorter and narrower compared to
that of the LHC. Consequently, the first detector layer can be positioned
closer to the beam line with respect to the ATLAS [34] and CMS [35] detectors,
to enhance the resolution on the impact parameter and the displaced vertices
reconstruction. On the other hand, this is the region where the BIB flux is
more intense, and large detector occupancies are expected. Another significant
difference with respect to LHC is that the muon collider is expected to
operate in single bunch mode and thus the bunch crossing rate coincides with
the beam revolution frequency. This results in an expected bunch crossing rate
below 100 kHz (with respect to 40 MHz at LHC) and implies a longer time
between two collisions for processing and reading out the detector signals.

Similar to the upgraded ATLAS and CMS experiments for the HL-LHC, where it
will be crucial to precisely measure time to disentangle the pile-up
interactions, at the muon collider time measurements are necessary to mitigate
the effects of the out-of-time BIB particles. For this purpose, sensors with
time resolutions in the 30-100 ps range should be developed. It is clear that
the R&D path towards such precise devices can benefit from synergies with the
development of HL-LHC detectors.

Studies on the radiation environment due the BIB at $\sqrt{s}=1.5$ TeV are
documented in [11]. Table 3 presents the approximate expected radiation levels
per year of data taking and compares them to HL-LHC estimates. Notably, the
maximum total integrated dose is of about 10 Mrad in the region close to the
interaction point (distance $R=2.2$ cm), while at HL-LHC it is higher, around
100 Mrad. Conversely, the maximum fluence in the calorimeter region ($R=150$
cm) is $\sim$$10^{14}$ 1 MeV neutron equivalent per cm2 at the muon collider,
while it is $\sim$$10^{13}$ 1 MeV neutron equivalent per cm2 at the HL-LHC.
Despite some differences in the radiation environment, similar radiation
hardness requirements could be expected for the muon collider and the HL-LHC
detectors.

Table 3: Radiation levels per year at a $\sqrt{s}=1.5$ TeV muon collider and HL-LHC. | muon collider | HL-LHC
---|---|---
maximum dose at $R=2.2$ cm | 10 Mrad | 100 Mrad
maximum dose at $R=150$ cm | 0.1 Mrad | 0.1 Mrad
maximum fluence at $R=2.2$ cm |  $10^{15}$ 1 MeV-neq/cm2 |  $10^{15}$ 1 MeV-neq/cm2
maximum fluence at $R=150$ cm |  $10^{14}$ 1 MeV-neq/cm2 |  $10^{13}$ 1 MeV-neq/cm2

###  3.2 Muon collider detector concept at $\sqrt{s}=3$ TeV

The design of the first muon collider detector concept at $\sqrt{s}=3$ TeV was
based on CLIC detector [36], which had been optimized for $e^{+}e^{-}$
collisions up to 3 TeV. However, modifications were necessary to adapt it to
the muon collider MDI, described in Section 2, and to mitigate the effects of
the BIB.

The detector, illustrated in Figure 3, consists of a tracking system and a
calorimeter system, both immersed in a uniform solenoidal magnetic field, with
an external muon detector. From the innermost to the outermost regions with
respect to the IP, it features the following components:

  * -

Vertex detector made of double layers of $25\times 25~{}\mu\mathrm{m}^{2}$
silicon pixels. It features four central barrel cylinders and four endcap
disks on both sides of the barrel.

  * -

Inner tracker with three barrel layers and seven endcap disks on each side
made of silicon macropixels with a size of $50~{}\mu\mathrm{m}\times
1~{}\mathrm{mm}$.

  * -

Outer tracker composed of three barrel layers and four endcap disks per side
of silicon microstrips with a size of $50~{}\mu\mathrm{m}\times
10~{}\mathrm{mm}$.

  * -

Electromagnetic calorimeter (ECAL), composed of 40 alternating layers of
tungsten absorber and silicon sensors as active material, for a total of 22
radiation lenghts ($X_{0}$). The cell granularity is $5\times 5$ mm2.

  * -

Hadronic calorimeter (HCAL), that has 60 alternating layers of steel absorber
and scintillating-pads active material, for a total of 7.5 interactions
lengths ($\lambda_{I}$) and with cells of size $30\times 30$ mm2.

  * -

Superconductive solenoid generates a magnetic field of $B=3.57$ T; the
magnetic field value has been chosen to be consistent with that used in the
generation of the BIB sample.

  * -

Return yoke equipped with resistive plate chambers (RPCs) for muon detection,
with seven layers in the barrel and six layers in the endcap. Each cell has an
area of $30\times 30$ mm2.

The radius of the tracker barrel is 1.5 m, while the calorimeters barrel have
outer radii of 1.7 m and 3.3 m, for ECAL and HCAL respectively. The total
radial dimension of the detector, including the return yoke, is 6.5 m, and the
detector lenght along the beam direction is $\pm 5.6$ m with respect to the
IP.

Figure 3: Muon collider detector concept. From the innermost to the outermost
regions, it includes the tracking system (green), the electromagnetic
calorimeter (yellow), the hadronic calorimeter (magenta), the superconducting
solenoid (light blue), the barrel (light green) and endcap (blue) muon
detectors. The nozzles are shown in cyan.

In order to study the impact of BIB on the physics performance, the
interactions of these particles with the detector are simulated using the muon
collider software framework [37]. This software is a branched version of
iLCSoft [38], a framework developed for electron-positron colliders. The
detector description is implemented with DD4hep [39] and the interaction of
the BIB with the detector is simulated with Geant4 [23]. The BIB sample used
to study the detector response was generated with MARS15 at $\sqrt{s}=1.5$
TeV. As discussed in Section 2.2, beam-induced background samples generated
with FLUKA at $\sqrt{s}=1.5$ and 3 TeV have become available only recently and
show good compatibility. The particles produced by the primary muon-muon
collisions (i.e. the physics signal) are generated with different programs
like MadGraph5_aMC@NLO [41] or WHIZARD [40]. The interactions of final state
particles with the detector are then simulated with Geant4. The digitization
of the detector hits and the reconstruction of the physics object are
performed by the Marlin software [42]. Marlin also overlays the hits produced
by the BIB particles onto the physics signal hits on an event-by-event basis.

By exploiting the detailed detector simulation, several strategies have been
developed to mitigate the effects of BIB. Dedicated reconstruction algorithms
have been employed to achieve high performance physics measurements. In the
following sections, these techniques are described for the tracking and
calorimeter systems (Sections 3.3 and 3.4, respectively). The discussion also
covers the physics objects reconstruction performance achieved by combining
information from the different sub-detectors (Section 3.5).

###  3.3 Beam-induced background mitigation in the tracking system

The BIB is responsible for a high hit occupancy in the tracking system,
especially in the first layers of the vertex detector. Hit densities of 3.68
mm-2 and 0.51 mm-2 are expected in the first and second pixel layers,
respectively [6]. These values are almost one order of magnitude higher than
those predicted for the HL-LHC experiments. Nevertheless, it has to be
remarked that the bunch crossing rate at muon collider is expected much lower
than LHC allowing more time for data processing.

The primary strategies for mitigating the BIB in the tracking system include:

  * -

Exploiting the hit times in the silicon sensors.

  * -

Utilizing directional information of the incoming particles.

  * -

Employing pulse shape analysis of the signal.

  * -

Analyzing the shape of hit clusters.

As explained in Section 2, a significant fraction of the BIB is out-of-time
with respect to the bunch crossing. Therefore, this component can be
significantly reduced by requiring the hit times to be compatible with the
timing of a particle coming from the IP. A time resolution of 30 ps is assumed
for the vertex detector sensors, while a time resolution of 60 ps is
considered for the inner and outer trackers.

Most of the BIB particles arriving at the detector originate far from the IP,
so their incoming direction does not point directly to the interaction region.
The double-layer configuration of the vertex detector can be exploited to
estimate the direction of the particle by correlating the hit positions on
adjacent layers. The effective hit occupancy reduction in the vertex detector,
achieved by applying requirements on time and direction, is presented in
Figure 4a. Notably, the hit density is reduced by more than an order of
magnitude when using the double layer requirement. A further reduction can be
achieved by considering the pulse shape at the sensor level and the shape of
the hit clusters, but these studies are ongoing.

The hits that survive the BIB suppression are then utilized as input for the
track reconstruction procedure. The primary algorithm employed for this
purpose makes use of the Combinatorial Kalman Filter, often referred to as the
CKF algorithm [43]. This algorithm reconstructs the trajectory of charged
particles in the magnetic field (tracks), allowing the measurement of their
transverse momentum ($p_{\mathrm{T}}$). The tracking algorithm was tested on
samples of muons generated at the primary vertex. Figure 4b shows the track
reconstruction efficiency as a function of muon $p_{\mathrm{T}}$. To
understand the impact of the BIB, these efficiencies are compared with those
obtained without the BIB overlay. It is evident that the BIB leads to a drop
in efficiency at low $p_{\mathrm{T}}$, below 4 GeV, while the tracking
efficiency remains close to 1 for muons with higher $p_{\mathrm{T}}$. The
estimated uncertainty on the transverse momentum is $\Delta
p_{\mathrm{T}}/p^{2}_{T}\approx 1\cdot 10^{-1}\ (5\cdot 10^{-3})$ GeV-1 for
muons with momentum $p=1\ (100)$ GeV and $\theta=13^{\circ}$, and $\Delta
p_{\mathrm{T}}/p^{2}_{T}\approx 5\cdot 10^{-2}\ (4\cdot 10^{-5})$ GeV-1 for
muons with $p=1\ (100)$ GeV and $\theta=89^{\circ}$ [6].

a

b

Figure 4: (a) Hit density as a function of the vertex detector layer, with the
reduction obtained with time requirements and double layer (DL) filters. (b)
Muon track reconstruction efficiencies as a function of the $p_{\mathrm{T}}$,
with and without the BIB overlaid [6].

###  3.4 Beam-induced background mitigation in the calorimeter system

At the muon collider, the internal surface of the electromagnetic calorimeter
(ECAL) is exposed to a flux of about 300 BIB particles per cm2 at each bunch
crossing. The majority of these particles are photons ($96\%$) with an average
energy of 1.7 MeV and the remaining $4\%$ are neutrons. Neutrons that don’t
interact with the ECAL reach the HCAL where on the first layer, the occupancy
is approximately one order of magnitude lower than that of the first ECAL
layer [6]. The following features can be exploited to suppress the BIB in the
calorimeter:

  * -

Arrival time of the particles in the calorimeter cells.

  * -

Calorimeter granularity.

  * -

Energy thresholds to reject the soft component.

  * -

Calorimeter segmentation for the measurement of the longitudinal energy
distribution.

Similarly to the tracking system, time information plays an important role
also in the calorimeters. As shown in Figure 5a, the times of a consistent
fraction of the BIB hits of the ECAL is out of the time window of the physics
signal (e.g. hadronic jets). Assuming a calorimeter hit time resolution of
approximately 100 ps, it is possible to apply a time window requirement of
$\pm$250 ps, which efficiently suppresses a significant portion of BIB hits.

A high granularity and a fine longitudinal segmentation of the calorimeter
could help in separating the showers initiated by signal particles from those
due to the BIB. This is the reason why cells of $5\times 5$ mm2 are considered
for ECAL.

To suppress the low energy deposits associated with the BIB, a threshold of 2
MeV is set on the energy of the calorimeter cells. This threshold, while
relatively high, represents a compromise between performance, BIB suppression
and computation time. Nonetheless, new handles can be explored in the future
to lower this threshold and enhance performance. For instance, the analysis of
the longitudinal distribution of the deposited energy in the calorimeter can
be leveraged to differentiate signal from BIB showers, as distinct profiles
are expected.

The photon energy resolution achieved with the current calorimeter
configuration and the aforementioned requirements is presented in Figure 5b.
The resolution is approximately $13\%$ for low energy photons below 50 GeV,
and around $1.5\%$ for photons with energy above 400 GeV. By comparing the
resolution with that obtained without the BIB overlaid, it becomes evident
that the BIB leads to a degradation in performance.

a

b

Figure 5: (a) Distribution of the ECAL hit times with respect to the bunch
crossing for signal (hadronic jets) and BIB particles. (b) Photon energy
resolution for low- and high-energy photons, obtained with and without the BIB
overlaid [6].

###  3.5 Physics objects reconstruction with particle flow

Several physics objects are crucial for the success of the muon collider
physics program. The reconstruction and identification of muons, photons,
electrons and jets have been studied with the detailed detector simulation. As
explained in the following sections, these particles are present in the final
states of the most important Higgs decay channels and may be part of the
signatures for new physics searches.

The reconstruction of these physical objects involves combining information
from all the detector sub-systems. Tracks, calorimeter clusters, and muon
detector hits are combined to achieve optimal performance in terms of
identification efficiency, background rejection, and momentum resolution. For
this purpose, a particle flow algorithm, PandoraPFA [44], is employed.
PandoraPFA is an algorithm based on modern particle flow techniques and has
been used for object reconstruction tasks. Initial studies have demonstrated
its effectiveness in the muon collider environment, although its initial
configuration was optimized for electron-positron colliders. It is evident
that there is room for further improvements and, in the future, a dedicated
optimization of the PandoraPFA settings for the muon collider can be pursued.

The reconstruction of the following physics objects have been studied, along
with the resulting performance:

  * -

Muons are identified by matching tracks with hits in the muon detector system.
This sub-detector is not significantly affected by the BIB, except for the
forward region with respect to the beam direction. In Figure 6a the muon
reconstruction and identification efficiency is presented as a function of the
muon production angle with respect to the beamline. It is evident that the low
angle region suffers from the BIB presence while the central region exhibits
high efficiency.

  * -

Electrons and photons are reconstructed by matching tracks with ECAL clusters.
Isolated clusters are classified as photons, while clusters matched with a
track are classified as electrons. The energy of electrons and photons is
corrected to take into account inefficiencies and radiation losses. The
electron reconstruction efficiency as a function of the energy is presented in
Figure 6b: it reaches around 95$\%$ for high energy, but the typical
efficiency drop at low energy caused by the BIB is evident.

  * -

Hadronic jets: particles reconstructed by PandoraPFA are given in input to the
jet clustering algorithm, that groups together particles belonging to the same
fragmentation process, by exploiting their correlations. The jet algorithm
used is the $k_{t}$ algorithm [45, 46] with a radius parameter of 0.5. The
resulting jet energy is corrected to recover the losses due to particles
escaping the detector, detector inefficiencies, and also for the BIB
contamination. The jet reconstruction algorithm has been tested with simulated
samples of different jet flavours: gluons and $b$, $c$, $u$, $d$, and $s$
quarks. The reconstruction efficiencies are illustrated in Figure 6c. Despite
the presence of BIB throughout the detector, the efficiency ranges between
80$\%$ and 95$\%$, with a negligible fake jet probability.

  * -

Heavy flavour jets: in order to identify the jets originating from heavy
quarks, secondary vertices compatible with the decays of $b$\- and $c$-hadrons
are reconstructed by combining tracks. A jet is identified as a heavy-flavour
jet if one of these secondary vertices is reconstructed within the jet cone.
The $b$-jet identification probability, called $b$-tagging efficiency, is
shown in Figure 6d. It is of approximately 45$\%$ at low $p_{\mathrm{T}}$ (20
GeV) and increases to 70$\%$ at 120 GeV. This level of performance is
comparable to what is currently achieved at hadron colliders. At lepton
colliders, a higher efficiency is expected. The current algorithm is mainly
limited by the presence of the BIB. In the future, artificial-intelligence-
based methods will be implemented to minimize the effects of BIB and further
enhance efficiency.

a

b

c

d

Figure 6: (a) Muon reconstruction efficiency as a function muon
$p_{\mathrm{T}}$. (b) Electron reconstruction efficiency as a function of the
electron energy, with and without the BIB overlay. (c) Jet reconstruction
efficiency as a function of jet $p_{\mathrm{T}}$, for different jet flavours.
(d) $b$-tagging probability as a function of the jet $p_{\mathrm{T}}$. The
above figures have been reproduced from [6].

###  3.6 Luminosity measurement

Accurate knowledge of the absolute luminosity is crucial in collider
experiments, as it is essential for determining the production cross sections.
Uncertainty below $1\%$ has been achieved by the ATLAS [48] and BelleII [49]
experiments. In general, the luminosity measurement methods rely on the direct
proportionality between process rates observed in the detector and luminosity.
Two different approaches are typically used for hadron and lepton colliders:

  * -

At hadron colliders, the rate $R$ of a given process is measured and related
to the integrated luminosity by $\mathcal{L}_{int}=R/\sigma_{vis}$. The
visible cross section $\sigma_{vis}$ is defined as
$\sigma_{vis}=2\pi\,A_{eff}/(N_{1}N_{2})$, where $N_{1,2}$ is the average
number of particles per bunch provided by the machine and $A_{eff}$ is the
effective transverse area of the luminous region. In this method, a precise
calibration of the visible cross section is crucial. At the LHC, it is
achieved with dedicated van der Meer scans [50], in which the beams are
displaced against each other.

  * -

Experiments at lepton colliders count the number of events $N_{ev}$ of a
reference process, whose cross section $\sigma$ is sufficiently high and
theoretically know with very good precision, and determine the luminosity by
$\mathcal{L}_{int}=N_{ev}/(\epsilon\,\sigma)$, where $\epsilon$ is the total
selection efficiency for the considered events. Electron-positron experiments
exploit the abundant $e^{+}e^{-}\to e^{+}e^{-}$ elastic scattering (Bhabha
scattering), for which the cross section is known with a precision below a
percent at $\sqrt{s}=1$-10 GeV [51].

In both cases, the rate of the reference processes is measured by dedicated
detectors, called luminometers, that are installed at very small polar angles
on both sides of the interaction point, where the observed rates are higher.
In a muon collider experiment, the presence of the nozzles hinders the
installation of the luminometers and an alternative technique must be utilized
to determine the luminosity. A possible approach could be to count muons
scattered elastically at large angles, similar to what was done in [54].

A preliminary study verified the feasibility of such a method [52]. The Pythia
Monte Carlo generator [53] was used to produce a sample of
$\mu^{+}\mu^{-}\to\mu^{+}\mu^{-}$ Bhabha events with an emission angle of
$30^{\circ}<\theta_{\mu}<150^{\circ}$ with respect to the beam direction at a
center-of-mass energy of 1.5 TeV. The sample was processes with the detector
simulation and the muon collider reconstruction software. The event
requirements are quite simple: two opposite-charge muons with a transverse
momentum $p_{T}>130$ GeV and an invariant mass $1440<m_{\mu\mu}<1560$ GeV.
Background contributions from other physics processes are found to be
negligible as well as possible effects from the beam-induced background. The
statistical uncertainty on the integrated luminosity estimated at
$\sqrt{s}=1.5$ TeV is:

$\frac{\Delta N_{Bhabha}}{N_{Bhabha}}=\frac{1}{\sqrt{N_{Bhabha}}}=0.002\ .$
(1)

This approach requires the knowledge of the cross section of the $\mu$-Bhabha
scattering at large angles and a center-of-mass energy of several TeV, which
is currently not available. Assuming this, the method seems promising.

##  4 THE GUARANTEED PHYSICS DISCOVERY: FULL DETERMINATION OF THE HIGGS BOSON
PROPERTIES

The Higgs boson plays a fundamental role in the standard model and could hold
the key to a deeper understanding of the universe. Its observation has been
the key step in establishing the mechanism of electroweak symmetry breaking
and mass generation [55]. Precisely determining the Higgs boson properties and
understanding its nature are therefore of the utmost importance in the
particle physics field and are central to the physics programs of all the
proposed next-generation colliders.

A high energy muon collider represents an ideal machine to study in detail the
Higgs boson properties, since it is expected to provide large samples of
single Higgs bosons as well as unprecedented samples of multi-Higgs bosons in
a relatively clean experimental environment. The cross sections of the most
important production channels are shown in Figure 7a as a function of the muon
collider center-of-mass energy $\sqrt{s}$. At multi-TeV lepton collisions,
Higgs bosons are produced via two fundamental processes: the vector boson
fusion and the muon-antimuon annihilation. The cross sections of the latter
decrease as $1/s$ with increasing collision energies, whereas the former
displays cross sections that increase logarithmically with $s$ and eventually
become dominant at high energies.

A 3 TeV muon collider with a dataset of 1 ab-1 is expected to produce a sample
of approximately $5.5\times 10^{5}$ single Higgs bosons and a sample of about
1000 double Higgs events. A muon collider operating at 3 TeV would undoubtedly
advance our understanding of the Higgs boson’s properties significantly, but
it is at higher energies that the full potential of a muon collider emerges.
At a 10 TeV collider with a dataset of 10 ab-1, approximately 9.3 million of
single Higgs bosons and a sample of about 38000 double-Higgs events are
expected. The 1 ab-1 and 10 ab-1 datasets correspond to the integrated
luminosity targets set in the IMCC’s studies [6].

a

b

c

d

Figure 7: (a) Cross sections of the main Higgs boson production channels as a
function of the muon collider center-of-mass energy $\sqrt{s}$, from Ref.
[56]. (b) Invariant mass of the two jets reconstructed in the $H\to b\bar{b}$
channel: the reconstructed mass peak of the Higgs boson is indicated in red,
while the background is in green. (c) Invariant mass of the two muons in the
$H\to\mu^{+}\mu^{-}$ analysis: the backgrounds (in yellow and brown) are
stacked on top of the signal (dark blue). (d) Precision on the Higgs boson
couplings at a 3 TeV and 10 TeV muon collider, with a dataset of 1 ab-1 and 10
ab-1, respectively. The light-colored histograms indicate the expected
precision of muon collider alone, while the dark-colored ones show the
precision when combining with the HL-LHC. The plotted values are taken from
[64].

###  4.1 The Higgs boson couplings determination

A first campaign of studies to estimate the precision on the Higgs cross
sections in different final states was carried out at $\sqrt{s}=3$ TeV with a
detailed detector simulation that includes the beam-induced background. Given
the unique nature of such a background, it was not possible to model or
parameterize its impact on the detector performance, and hence the physics
reach, relying on prior experience with machine backgrounds at current or past
colliders.

The physics objects described in Section 3.5 were used to reconstruct the
Higgs boson decays into the final states $f=b\bar{b}$, $WW^{\ast}$,
$ZZ^{\ast}$, $\gamma\gamma$, and $\mu^{+}\mu^{-}$ to estimate the statistical
sensitivity on the production cross sections multiplied by the decay branching
ratios: $\sigma_{H}\times BR(H\to f)$.

The Higgs and physics background samples were generated at leading order with
WHIZARD or MadGraph5_aMC@NLO and Pythia8 was used for the final state
hadronization. The samples were then processed with the detailed detector
simulation and reconstructed with the muon collider software. The results are
outlined below. A dataset of 1 ab-1 and a single interaction point are
assumed. A natural reference to compare to are the corresponding results by
CLIC at $\sqrt{s}=3$ TeV [57] with the caveat that CLIC is assuming a dataset
of 2 ab-1 and the number of reconstructed final states for each Higgs boson
decay mode are in some cases different.

$\bm{H\to b\bar{b}:}$

The $b\bar{b}$ channel [58] is reconstructed selecting two central
high-$p_{T}$ jets satisfying: $p_{T}^{\textsuperscript{jet}}>40$ GeV and
$|\eta_{\textsuperscript{jet}}|<2.5$. Once the $b$-flavor identification is
applied to both jets the only significant background that survives is form
$Z^{0}$ boson decays into $b\bar{b}$ and $c\bar{c}$ pairs. A total number of
59500 signal events and 65400 background events are estimated. Figure 7b shows
the reconstructed dijet invariant mass. The dijet mass resolution is critical
to separate the $H$ and $Z^{0}$ peaks. The current value of the Higgs boson
mass resolution is of about $18\%$, being dominated by BIB effects, i.e. the
energy thresholds set on the calorimeter hits (see Section 3.4). The estimated
sensitivity on $\sigma_{H}\times BR(H\to b\bar{b})$ is 0.75%, to be compared
with 0.3% of CLIC.

$\bm{H\to WW^{\ast}:}$

The $WW$ channel [59] is reconstructed in the semileptonic final state
$qq^{\prime}\mu\nu_{\mu}$, which provides a good signal-to-background ratio.
Events are selected with at least two reconstructed jets, having
$p_{T}^{\textsuperscript{jet}}>20$ GeV and
$|\eta_{\textsuperscript{jet}}|<2.5$, and one muon with $p_{T}^{\mu}>10$ GeV
and $10^{\circ}<\theta_{\mu}<170^{\circ}$. The estimated number of Higgs
candidates is 2430 over a background of 2600 events. The resulting statistical
uncertainty on the production cross section is 2.9%. Reconstructing the full
hadronic channel and the semileptonic channels with both muons and electrons,
CLIC achieves a sensitivity of 0.7%.

$\bm{H\to ZZ^{\ast}:}$

The $ZZ^{\ast}$ channel [60] is reconstructed in the semileptonic final state
$q\bar{q}\mu^{+}\mu^{-}$. Events are selected with at least two reconstructed
jets and two opposite-charge muons, having $p_{T}^{\textsuperscript{jet}}>15$
GeV and $30^{\circ}<\theta_{\textsuperscript{jet}}<150^{\circ}$,
$p_{T}^{\mu}>10$ GeV and $10^{\circ}<\theta_{\mu}<170^{\circ}$. The
sensitivity estimated on the cross section with 55 signal events and 39
background events is 17%. CLIC analyses also the semileptonic channel with
electrons and gets 3.9%.

$\bm{H\to\gamma\gamma:}$

The $\gamma\gamma$ analysis [61] searches for events with at least two
reconstructed photons, featuring $E_{\gamma}>15$ GeV, $p_{T}^{\gamma}>10$ GeV
and $10^{\circ}<\theta_{\gamma}<170^{\circ}$. The reconstruction of high
energy photons is not significantly affected by the BIB. In fact, the diphoton
invariant mass exhibits a very good resolution of 3.2 GeV. The estimated
numbers of signal and background events are 396 and 484, respectively, which
result in a relative uncertainty on the production cross section of 7.6%, to
be compared with CLIC’s 10%.

$\bm{H\to\mu^{+}\mu^{-}:}$

The $\mu\mu$ channel [62] is reconstructed selecting events with two opposite-
charge muons in the kinematical region $p_{T}^{\mu}>5$ GeV and
$10^{\circ}<\theta_{\mu}<170^{\circ}$. The BIB impact on the reconstruction of
muons is found to be negligible. The reconstructed dimuon mass is shown in
Figure 7c, where the signal peak has a width of 0.4 GeV. After the final
selection, 26 signal events and 1114 background events are left, which result
in a sensitivity on the production cross section of 38%. For this specific
channel, CLIC’s ability to veto events with electrons scattered at very low
angles significantly reduces background contamination, resulting in a
sensitivity of 25%.

A precise measurement of the the observables $\sigma_{H}\times BR(H\to f)$
reflects into a precise determination of the Higgs couplings to $f$, where $f$
represents each standard model boson and fermion. The precision achievable on
the couplings is usually estimated with a fit to the cross sections within the
so-called $\kappa$-framework222The parameters $\kappa_{i}$ are coupling
modifiers, defined as the ratio between the measured and the standard model
values. [63]. The studies carried out so far at the muon collider do not cover
all the most relevant decay modes of the Higgs boson, mainly due to a lack of
personnel; therefore, the coupling fit is not yet meaningful. Nonetheless,
this exercise was extremely useful as it allowed to benchmark parametric
studies with which it is possible to estimate the sensitivity on the Higgs
couplings of an experiment at the muon collider at $\sqrt{s}=3$ TeV. In fact,
the parametric study in [64] presents results for the Higgs boson production
cross sections consistent with those for the available channels with the
detailed simulation. In the same study, it is shown how the identification of
the Higgs production mechanism, $WW$ or $Z^{0}Z^{0}$ fusion, could help in
improving the coupling sensitivity. However, this could be done mainly through
the identification of the forward muons emitted outside the detector
acceptance. Prospects on their detection are discussed in Section 5.2.3. The
output of the fit of the couplings is reported in Figure 7d in the
$\kappa$-framework for a muon collider of 3 TeV (1 ab-1) and 10 TeV (10 ab-1)
independently and in combination with the HL-LHC estimates, in the assumption
that there are no BSM decays of the Higgs boson. While the HL-LHC results are
complementary to those of a 3 TeV muon collider, in the 10 TeV scenario the
combination is definitely dominated by the muon collider precision, with the
only exception of the top quark coupling.

###  4.2 The Higgs boson self-couplings and the Higgs potential

One of the strengths of the muon collider physics program is the possibility
to determine the Higgs field potential. In the standard model [55], its
configuration is parameterized in terms of the vacuum expectation value of the
Higgs field $v=1/\sqrt{\sqrt{2}\,G_{F}}=246$ GeV, with $G_{F}$ the Fermi
constant, the strength of the Higgs self-coupling $\lambda$, and the Higgs
boson mass $m_{H}=\sqrt{2\lambda}\,v$. Processes in which a virtual Higgs
boson produces two or three Higgs bosons, governed by the so-called trilinear
and quartic Higgs self-coupling, are of the utmost important to test whether
the Higgs potential is consistent with that predicted by the standard model.
To be as general as possible and to account for deviations from the standard
model, the Higgs potential is formulated with two different values for
trilinear ($\lambda_{3}$) and quartic ($\lambda_{4}$) self-couplings:

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

where $\lambda_{3}=\lambda_{4}=\lambda=m_{h}^{2}/2v^{2}$ in the standard model
and the scalar field, $h$, is an expansion around $v/\sqrt{2}$, describes a
physical Higgs boson.

The gateway to the trilinear self-coupling is the double Higgs production
cross section. Its measurement was studied with the detailed detector
simulation in the hadronic final state $HH\to b\bar{b}b\bar{b}$ for a 3 TeV
collider with an integrated luminosity of 1 ab-1 [58]. Events are selected
with at least four reconstructed jets having
$p_{T}^{\textsuperscript{jet}}>20$ GeV. The requirement of $b$-tagging
suppresses most of the backgrounds from light quarks. A final number of 77
signal events and 1422 background events is expected, which leads to an
estimated sensitivity on the production cross section of 33%. Including
additional final states and assuming an 80% polarization of the electron beam,
CLIC quotes a precision of 22% with 2 ab-1 of data. From this study, a
preliminary estimate in the order of 20-30% is derived on the sensitivity on
the trilinear self-coupling [60]. It will be possible to determine the final
sensitivity with the detailed detector simulation once all the relevant final
states are included and the reconstruction algorithms, particularly the
$b-$jet tagging, are fully optimized

Since the design of the detector at $\sqrt{s}=10$ TeV is not ready yet, a
study by using a parametric description of the physics objects is performed.
In [65] the sensitivity at 3 TeV center-of-mass energy has been evaluated with
selections that mimic those applied in the analysis with the detailed
simulation. The uncertainty on the trilinear self-coupling is around $25\%$.
The exercise is repeated at 10 TeV center-of-mass energy finding an
uncertainty of $5.6\%$ assuming 10 ab-1 of data.

The muon collider offers a unique opportunity to measure the quartic self-
coupling. A parametric study indicate that the quartic self-coupling can be
probed to an accuracy of tens of percent [66] with 20 ab-1 of data. Here no
background has been considered. The reconstruction of events with 6 jets, some
of them in the forward region will require dedicated algorithms. In addition,
at the moment the generation of the background with 6 $b-$jets final state is
not viable due to high computational time required.

##  5 THE EXPERIMENT AT THE 10 TEV FRONTIER

The discovery potential of the multi-TeV muon collider can be exploited by
looking for a large variety of signatures [6]. The possibility of having
collisions at $\sqrt{s}=10$ TeV and beyond enables the search for new
particles at an energy scale not accessible by other proposed $e^{+}e^{-}$
colliders.

Although the importance of exploring high-mass particles is evident, the
capability to measure low-momentum signatures should not be overlooked. As an
example, scenarios with Higgs composite models or where dark matter is a
particle charged under the electroweak interaction could be probed at the muon
collider in a wide range of final state momenta. As will be discussed in this
Section, the standard model Higgs physics can be considered low momentum
physics at $\sqrt{s}=10$ TeV, as well as various other standard model
measurements. The measurements done at $\sqrt{s}=3$ TeV could be used to
improve the LHC exclusion bounds for several models [67] and to perform
precision standard model physics tests. Nevertheless, the muon collider
potential will be fully unleashed with very high energy collisions.

###  5.1 Quest for new physics

As previously noted, muon collisions at $\sqrt{s}=10$ TeV or higher offer an
opportunity to investigate new physics scenarios at the energy frontier and to
test several beyond standard model hypotheses. An experimental access to the
signatures of many of these scenarios is closely tied to specific capabilities
of the experimental apparatus. Below, such a connection is illustrated for
some representative physics cases.

An overview of dark matter candidates that can be searched for at a
$\sqrt{s}=10$ muon collider can be found in [75]. Within the framework of
weakly interacting massive particles, this study demonstrates that an
experiment at a muon collider has the potential to explore models where new
electroweak particles are created in the early universe. This analysis is
based on inclusive experimental signatures such as mono-photon, mono-muon, and
dimuon events. A study performed with a detailed detector simulation of mono-
photon signatures for searches related to axion-like particles [76]
demonstrates the critical importance of detecting objects such as photons with
energies in the TeV range. This is crucial for advancing the understanding of
new physics, in particular the dark matter candidates.

New physics scenarios could manifest themselves with unconventional
experimental signatures including, but are not limited to, long-lived
particles, disappearing tracks, and emerging jets.

Disappearing tracks refer to charged long-lived particles that travel a few
centimeters in the detector before decaying into undetectable particles,
resulting in track stubs that do not traverse the entire tracker. The spurious
hits produced by the beam-induced background in the tracker can potentially
compromise the identification of this type of signatures.

The study presented in [73] demonstrates that the BIB can be effectively
mitigated with an appropriate detector configuration and analysis strategy.

This study has been performed using the detector concept presented in Section
3. Further optimization aimed specifically at the 10 TeV center-of-mass energy
configuration could improve the results.

The physics possibilities at a $\sqrt{s}\geq 10$ TeV muon collider extend
beyond Higgs presented in Section 4 and exotic signatures. The fact that
vector bosons collisions dominate the inclusive $\mu^{+}\mu^{-}$ interactions
at these energies results in unprecedented precise measurements of electroweak
processes thanks to their high production cross section and the relatively
clean environment. This offers a mean for indirectly probing new physics by
identifying any deviations from the SM predictions. Measurements of vector
boson scattering, the anomalous muon magnetic moment, $b$-hadron decays, and
other possibilities can be explored in this context [6]. Further studies based
on detailed detector simulations will be necessary in the future to determine
the optimal detector configuration and performance for this types of
measurements.

###  5.2 Physics and detector requirements for an experiment at $\sqrt{s}=10$
TeV

a

b

c

d

Figure 8: (a) $b$-quark $p_{\mathrm{T}}$ distributions from the $H\rightarrow
b\bar{b}$ decays at $\sqrt{s}=3$ and 10 TeV, as obtained at generator level.
(b) Distribution of leptons and jets $p_{\mathrm{T}}$ from the decay of a
$Z^{\prime}$ produced at $\sqrt{s}=10$ TeV with a mass of 9.5 TeV, as obtained
at generator level. (c) $p_{\mathrm{T}}$ resolution for a muon track with
$p_{\mathrm{T}}=5$ TeV as a function of the tracker radius ($R_{max}$) for
different values of the detector magnetic field. (d) Minimum $p_{\mathrm{T}}$
required for a charged particle to reach the outermost tracker layer as a
function of the magnetic field $B$ for different tracker radius values
($R_{\mathrm{max}}$). Figures taken from [68].

The detector design for an experiment at a muon collider at $\sqrt{s}=10$ TeV
is currently under study. In this section, requirements are discussed and
guidelines are drawn, but further studies with the detailed simulation are
necessary to validate the detector concept at $\sqrt{s}=10$ TeV. This includes
assessing the BIB at detector level, utilizing the $\sqrt{s}=10$ TeV
generation described in Section 2.2.

The detector design must aim to provide as much as possible good efficiencies
and precision for new physics signatures from low to high momentum,
conventional and unconventional. It is also important to ensure capability to
study Higgs physics, as demonstrated with the $\sqrt{s}=$ 3 TeV detector and
predicted for $\sqrt{s}=$ 10 TeV (Section 4). In the following, as for the
case of 3 TeV center-of-mass energy, only one concept detector is discussed.
The possibility to have two interaction points may lead to change the strategy
in the future.

Figure 8a presents the $b$-quark $p_{\mathrm{T}}$ distributions from the
$H\rightarrow b\bar{b}$ decays at $\sqrt{s}=3$ and 10 TeV, as obtained at
generator level with MadGraph5_aMC@NLO. Despite the higher center-of-mass
energy, it is evident that the distributions are quite similar. Therefore the
considerations made for Higgs physics at 3 TeV stand also at 10 TeV center-of-
mass energy. It can be stated that Higgs physics lives in the low
$p_{\mathrm{T}}$ regime even for the $\sqrt{s}=10$ TeV muon collider. Another
important feature is that Higgs bosons at $\sqrt{s}=10$ TeV are emitted at
smaller polar angles compared to 3 TeV, therefore the forward region of the
detector should be designed carefully.

Going from the low energy regime to the high energy scale, Figure 8b shows the
distributions of leptons and jets $p_{\mathrm{T}}$ from the decays of a
$Z^{\prime}$ produced at $\sqrt{s}=10$ TeV with a mass of 9.5 TeV, as obtained
at truth-level with MadGraph5_aMC@NLO: this distributions exhibit a peak at
$p_{\mathrm{T}}\sim 5$ TeV, and a long tail at lower $p_{\mathrm{T}}$. A
general consideration is that combinatorial and standard model backgrounds are
expected to have a minor impact at high invariant mass compared to low
invariant mass. It is crucial to maintain a good momentum resolution at low
$p_{\mathrm{T}}$, while at high $p_{\mathrm{T}}$ a moderate resolution should
be sufficient.

####  5.2.1 Tracking system and magnetic field at $\sqrt{s}=10$ TeV

The handles for improving the track momentum resolution are:

  * -

Tracker transverse dimension.

  * -

Number of tracking detector layers.

  * -

Position resolution of the sensor.

  * -

Magnetic field intensity.

To understand the possible tracker configurations, Figure 8c shows the
$p_{\mathrm{T}}$ resolution as a function of the tracker radius
($R_{\mathrm{max}}$) and magnetic field strength ($B$), as obtained by scaling
analytically the detector’s performance presented in Section 3. The curves are
obtained with the same position resolution assumed for the $\sqrt{s}=3$ TeV
detector, as well as the number of tracking layers. The resolution on the hit
position depends on the technology of the sensor, and various options have
been explored for the muon collider experiment in [69].

The choice of the magnetic field is crucial: its magnitude determines the
size, construction cost, operational cost, and the technological feasibility
of the superconductive solenoid. As a rule of thumb, the energy density of the
magnetic field is proportional to $B^{2}$, and the solenoid cost has
approximately the same scaling. Very high magnetic fields require the
development of challenging technologies to maintain the mechanical and
electrical stability of the superconducting coil, and obtain the uniformity of
the field. For these reasons, it is important to aim for the lowest possible
magnetic field strength, as long as it allows the desired physics performance
to be achieved.

Increasing the magnetic field intensity enhances the tracking momentum
resolution, but there is also a trade-off at low $p_{\mathrm{T}}$: the minimum
$p_{\mathrm{T}}$ required for a charged particle to reach the outermost layer
of the tracker is proportional to $B$. In Figure 8d the minimum
$p_{\mathrm{T}}$ as a function of $B$ for different tracker radius hypotheses
is shown. As an example, with $B=5$ T and $R_{\mathrm{max}}=1.6$ m, the
minimum $p_{\mathrm{T}}$ is 1.2 GeV. Below this threshold a deterioration of
efficiency and/or resolution is expected. Preliminary results [68] indicate
that going from $B=3.57$ T to $B=5$ T, assuming the detector configuration of
Section 3, results in an efficiency loss of about 10$\%$ for reconstructing
tracks originating from the $H\rightarrow b\bar{b}$ decay.

Based on the above considerations, a magnetic field strength between 4 and 5 T
and a tracker radius larger than that of the $\sqrt{s}=3$ TeV detector could
be a good compromise at $\sqrt{s}=10$ TeV. For instance, a radius of
$R_{\mathrm{max}}=1.6$ m could be a suitable choice.

####  5.2.2 Calorimeter systems at $\sqrt{s}=10$ TeV

Showers produced by photons, electrons, and hadrons with energies of the order
of TeV need be contained in the calorimeters volume to achieve an adequate
resolution on the energy measurement. For example, a photon with an energy of
1 TeV deposits less than 95% of its energy in 22 $X_{0}$ of ECAL [74], the
depth of the electromagnetic calorimeter described in Section 3. The remaining
energy spills into the HCAL. It is evident that the detector at $\sqrt{s}=10$
TeV should have deeper calorimeters compared to the $\sqrt{s}=3$ TeV
configuration. Additionally, alternative materials with higher $X_{0}$ values
should be considered. In fact, it is important to remark that the dimensions
of the detector are constrained by the cavern dimension, accelerator
components, MDI and the magnet structure.

Another aspect to consider is the position of the solenoid with respect to the
calorimeters. Placing the solenoid in front of the ECAL could shield the
calorimeter from the low-energy BIB photons, but could also absorb the photons
from $H\rightarrow\gamma\gamma$ with lower energies or the electromagnetic
component of $b$-jets from $H\rightarrow b\bar{b}$, degrading the performance
in the energy range where a precise resolution is crucial. On the other hand,
positioning a high-field solenoid outside HCAL might be costly due to its
large dimensions. A compromise could be placing the solenoid between ECAL and
HCAL. However, defining the final configuration requires detailed simulation
studies and the assessment of BIB impact.

Several calorimeter technologies that meet the requirements of a multi-TeV
muon collider are currently under study. These technologies are not only being
explored through simulations, but are also characterized with experimental
tests. For examples, an ECAL made of $PbF_{2}$ crystals has been proposed
[70], and the evaluation of Micro Pattern Gas detectors for the active layers
of HCAL is ongoing [71].

####  5.2.3 Muon detection at $\sqrt{s}=10$ TeV

Identifying and measuring muons with momenta in the TeV range present
significant challenges. Promising technologies are discussed in [72], where
gas detectors with excellent time resolution and high rate capability are
presented. In the muon detector, the bending due to the residual magnetic
field alone is not sufficient for the precise determination of muon
$p_{\mathrm{T}}$. Nevertheless, the muon detector can still be used to tag
events with high momentum muons.

Additionally, high-granularity imaging calorimeters [77] could play an
important role in detecting high-momentum muons since significante radiative
losses are expected for these particles with $p_{\mathrm{T}}$ in the TeV
range. Therefore, integrating information from all subsystems, including
tracker, calorimeters and muon detectors, will be of paramount importance for
muon reconstruction. It is essential to develop algorithms optimized for the
TeV scale.

The detection of forward muons, i.e muons that are scattered at very low
angles with respect to the beam direction and pass through the nozzles, would
be an important feature of the muon collider detector. This capability would
make it possible to distinguish between the $WW$ and $Z^{0}Z^{0}$ fusion
processes. The former is accompanied by forward neutrinos, while the latter
has forward muons. At $\sqrt{s}=10$ TeV, the muons from $Z^{0}Z^{0}$-fusion
process are expected to be even more forward compared to $\sqrt{s}=3$ TeV.
Consequently, the detector acceptance may cover just a small fraction of the
angular distribution. The feasibility of instrumenting the region close to the
beam line to tag these muons is currently under investigation.

##  6 OUTLOOK

The muon collider facility, initially proposed by MAP, has been revised during
the Accelerator R&D Roadmap [78] and by the IMCC, revealing no fundamental
showstoppers. Using the same scheme as MAP for muon production, a preliminary
design of a machine at a center-of-mass energy of 10 TeV was possible, opening
up uncharted territory for lepton collisions.

For the first time, extensive studies have been conducted on detector
performance in the presence of beam-induced background. Currently, the BIB
simulation sample generated by MAP is still used, due to the complexity of the
generation process. Given the critical impact of BIB on the detector,
extensive validations were done by using a different simulation software
leading to consistent results. This validation gives confidence and robustness
in the predictions of the beam-induced background. The proposed detector
concept for $\sqrt{s}=3$ TeV and the associated physics object reconstruction
algorithms have demonstrated to be able to handle the effects of the beam-
induced background, although neither the detector nor the algorithms have been
optimized yet.

The physics performance assessed at $\sqrt{s}=3$ TeV with the detailed
detector simulation, including the effects of the beam-induced background,
pertains mainly the Higgs physics. It was imperative to demonstrate that such
a detector in such an environment has excellent capabilities to study an
existing particle. Other studies are currently in development but have not yet
been carried out due to lack of personnel.

Nonetheless, this has demonstrated that an initial stage of a muon collider
operating at a center-of-mass energy of 3 TeV can achieve a performance
comparable to that of the extensively studied CLIC detector at the same
collision energy. While this energy is seen as the ultimate limit for
$e^{+}e^{-}$ colliders due to beamstrhalung radiation, it is considered the
first stage for the muon collider.

The results obtained at $\sqrt{s}=3$ TeV were used to benchmark parametric
studies of the Higgs processes not included in the studies with the detailed
detector simulation. Such studies were carried out at the same center-of-mass
energy and at $\sqrt{s}=10$ TeV and beyond. The possibility to reach high
energy muon collisions allows to probe new physics by investigating various
models including the precise determination of the Higgs potential.

A detector concept for muon collisions at 10 TeV center-of-mass energy is
currently in preparation, the requirements were studied as discussed in this
review for the first time. At $\sqrt{s}=10$ TeV, the detector shielding
structure needs to be designed, currently the nozzles optimized by MAP for
$\sqrt{s}=1.5$ TeV are used across all energies. Designing a new nozzle
requires expertise that had been lost after MAP shutdown, only recently IMCC
recovered it. Preliminary studies indicate that the amount of BIB particles
arriving on the detector is not significantly larger than at $\sqrt{s}=3$ TeV,
the aim is to reduce the dimension of the tungsten material to enhance the
detector acceptance in the forward region. This region is one of the open
questions for a detector at muon collider. In fact, there are physics
processes, as for example the Higgs physics, that at high energy,
$\sqrt{s}\geq 10$ TeV, are produced with higher cross section in the forward
region. It would be important to have at least the possibility to tag forward
muons to distinguish between $WW$ from $Z^{0}Z^{0}$ boson fusion production.

The design of the detector at $\sqrt{s}\geq 10$ TeV has demonstrated that the
requirements for the technologies needed to achieve the desired performance
could be satisfied through the R&D processes already ongoing for HL-LHC and
future colliders. The muon collider community is actively engaged in the
Detector Research and Development collaborations - DRD [79] in Europe and in
the Coordinating Panel for Advanced Detectors - CPAD [80] in the United
States. Participation in the development of new detector designs and related
test beams for a novel detector concept is of crucial importance to engage and
sustain the young community.

On a shorter time scale, the muon collider community may have to design and
operate a facility to demonstrate the full muon ionization cooling chain.
Experts in accelerator and detector technologies will need to collaborate
together to address challenges previously not encountered. Moreover, both the
detector R&D and the demonstration facility will play a vital role in actively
involving the community in experimental activities, thereby preventing the
loss of valuable expertise and knowledge.

## DISCLOSURE STATEMENT

The authors are not aware of any affiliations, memberships, funding, or
financial holdings that might be perceived as affecting the objectivity of
this review.

## ACKNOWLEDGMENTS

We would like to thank the many people who contributed with their hard work,
useful discussions, and valuable advice: Chiara Aimé, Paolo Andreetto, Nazar
Bartosik, Laura Buonincontri, Daniele Calzolari, Luca Castelli, Giacomo Da
Molin, Luca Giambastiani, Alessio Gianelle, Sergo Jindariani, Karol Kritza,
Anton Lechner, Alessandro Montella, Mark Palmer, Simone Pagan Griso, Nadia
Pastrone, Cristina Riccardi, Ilaria Vai, Davide Zuliani. We are grateful to
the International Muon Collider Collaboration and the US Muon Accelerator
Program for their support. We acknowledge the financial support of the Italian
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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