1 Introduction

Colliders are microscopes that explore the structure and the interactions of particles at the shortest possible length scale. Their goal is not to chase discoveries that are inevitable or perceived as such based on current knowledge. On the contrary, their mission is to explore the unknown in order to acquire radically novel knowledge.

The current experimental and theoretical situation of particle physics is particularly favourable to collider exploration. No inevitable discovery diverts our attention from pure exploration, and we can focus on the basic questions that best illustrate our ignorance. Why is electroweak symmetry broken and what sets the scale? Is it really broken by the Standard Model Higgs or by a more rich Higgs sector? Is the Higgs an elementary or a composite particle? Is the top quark, in light of its large Yukawa coupling, a portal towards the explanation of the observed pattern of flavor? Is the Higgs or the electroweak sector connected with dark matter? Is it connected with the origin of the asymmetry between baryons and anti-baryons in the Universe?

The next collider should deepen our understanding of the questions above, and offer broad and varied opportunities for exploration to enable radically unexpected discoveries. A comprehensive exploration must exploit the complementarity between energy and precision. Precise measurements allow us to study the dynamics of the particles we already know, looking for the indirect manifestation of yet unknown new physics. With a very high energy collider we can access the new physics particles directly. These two exploration strategies are normally associated with two distinct machines, either colliding electrons/positrons (ee) or protons (pp).

With muons instead, both strategies can be effectively pursued at a single collider that combines the advantages of ee and of pp machines. Moreover, the simultaneous availability of energy and precision offers unique perspectives of indirect sensitivity to very heavy new physics, as well as unique perspectives for the characterisation of new heavy particles discovered at the muon collider itself.

This is the picture that emerges from the investigations of the muon colliders physics potential performed so far, to be reviewed in this document in Sects. 2 and 5. These studies identify a Muon Collider (MuC), with 10 TeV energy or more in the centre of mass and sufficient luminosity, as an ideal tool for a substantial ambitious jump ahead in the exploration of fundamental particles and interactions. Assessing its technological feasibility is thus a priority for the future of particle physics.

Muon collider concept

Initial ideas for muon colliders were proposed long ago [1,2,3,4,5,6]. Subsequent studies culminated in the Muon Accelerator Program (MAP) in the US (see [7,8,9,10] and [11, 12] for an overview). The MAP concept for the muon collider facility is displayed in Fig. 1. The proton complex produces a short, high-intensity proton pulse that hits the target and produces pions. The decay channel guides the pions and collects the muons from their decay into a bunching and phase rotator system to form a muon beam. Several cooling stages then reduce the longitudinal and transverse emittance of the beam using a sequence of absorbers and radiofrequency (RF) cavities in a high magnetic field. A system of a linear accelerators (linac) and two recirculating linacs accelerate the beams to 60 GeV. They are followed by one or more rings to accelerate them to higher energy, for instance one to 300 GeV and one to 1.5 TeV, in the case of a 3 TeV centre of mass energy MuC. In the 10 TeV collider an additional ring from 1.5 to 5 TeV follows. These rings can be either fast-pulsed synchrotrons or Fixed-Field Alternating gradients (FFA) accelerators. Finally, the beams are injected at full energy into the collider ring. Here, they will circulate to produce luminosity until they are decayed. Alternatively they can be extracted once the muon beam current is strongly reduced by decay. There are wide margins for the optimisation of the exact energy stages of the acceleration system, taking also into account the possible exploitation of the intermediate-energy muon beams for muon colliders of lower centre of mass energy.

Fig. 1
figure 1

A conceptual scheme of the muon collider

The concept developed by MAP provides the baseline for present and planned work on muon colliders, reviewed in Sect. 3. Three main reasons sparked this renewed interest in muon colliders. First, the focus on high collision energy and luminosity where the muon collider is particularly promising and offers the perspective of revolutionising particle physics. Second, the advances in technology and muon colliders design. Third, the difficulty of envisaging a radical jump ahead in the high-energy exploration with ee or pp colliders.

In fact, the required increase of energy and luminosity in future high-energy frontier colliders poses severe challenges [13, 14]. Without breakthroughs in concept and in technologies, the cost and use of land as well as the power consumption are prone to increase to unsustainable levels.

The muon collider promises to overcome these limitations and allow to push the energy frontier strongly. Circular electron-positron colliders are limited in energy by the emission of synchrotron radiation that increases strongly with energy. Linear colliders overcome this limitation but require the beam to be accelerated to full energy in a single pass through the main linac and allow to use the beams only in a single collision [15]. The high mass of the muon mitigates synchrotron radiation emission, allowing them to be accelerated in many passes through a ring and to collide repeatedly in another ring. This results in cost effectiveness and compactness combined with a luminosity per beam power that roughly increases linearly with energy. Protons can be also accelerated in rings and made to collide with very high energy. However, protons are composite particles and therefore only a small fraction of their collision energy is available to probe short-distance physics through the collisions of their fundamental constituents. The effective energy reach of a muon collider thus corresponds to the one of a proton collider of much higher centre of mass energy. This concept is illustrated more quantitatively in Sect. 2.1.

Currently, the limit of the energy reach for muon colliders has not been identified. Ongoing studies focus on a 10 TeV design with an integrated luminosity goal of \(10 ~ \mathrm ab^{-1}\). This goal is expected to provide a good balance between an excellent physics case and affordable cost, power consumption and risk. Once a robust design has been established at 10 TeV – including an estimate of the cost, power consumption and technical risk – other, higher energies will be explored.

The 2020 Update of the European Strategy for Particle Physics (ESPPU) recommended, for the first time in Europe, an R &D programme on muon colliders design and technology. This led to the formation of the International Muon Collider Collaboration (IMCC) [16] with the goal of initiating this programme and informing the next ESPPU process on the muon collider feasibility perspectives. This will enable the next ESPPU and other strategy processes to judge the scientific justification of a full Conceptual Design Report (CDR) and demonstration programme.

The European Roadmap for Accelerator R &D [17], published in 2021, includes the muon collider. The report is based on consultations of the community at large, combined with the expertise of a dedicated Muon Beams Panel. It also benefited from significant input from the MAP design studies and US experts. The report assessed the challenges of the muon collider and did not identify any insurmountable obstacle. However, the muon collider technologies and concepts are less mature than those of electron-positron colliders. Circular and linear electron-positron colliders already have been constructed and operated but the muon collider would be the first of its kind. The limited muon lifetime gives rise to several specific challenges including the need of rapid production and acceleration of the beam. These challenges and the solutions under investigation are detailed in Sect. 3.

The Roadmap describes the R &D programme required to develop the maturity of the key technologies and the concepts in the coming few years. This will allow the assessment of realistic luminosity targets, detector backgrounds, power consumption and cost scale, as well as whether one can consider implementing a MuC at CERN or elsewhere. Mitigation strategies for the key technical risks and a demonstration programme for the CDR phase will also be addressed. The use of existing infrastructure, such as existing proton facilities and the LHC tunnel, will also be considered. This will allow the next strategy process to make an informed choice on the future strategy. Based on the conclusions of the next strategy processes in the different regions, a CDR phase could then develop the technologies and the baseline design to demonstrate that the community can execute a successful MuC project.

Important progress in the past gives confidence that this goal can be achieved and that the programme will be successful. In particular, the developments of superconducting magnet technology has progressed enormously and high-temperature superconductors have become a practical technology used in industry. Similarly, RF technology has progressed in general and experiments demonstrated the solution of the specific muon collider challenge – operating RF cavities in very high magnetic fields – that previously has been considered a showstopper. Component designs have been developed that can cool the initially diffuse beam and accelerate it to multi-TeV energy on a time scale compatible with the muon lifetime. However, a fully integrated design has yet to be developed and further development and demonstration of technology are required.

The technological feasibility of the facility is one vital component of the muon collider programme, but the planning of the facility exploitation is equally important. This includes the assessment of the muon collider potential to address physics questions, as well as the design of novel detectors and reconstruction techniques to perform experiments with colliding muons.

The path to a new generation of experiments

The main challenge to operating a detector at a muon collider is the fact that muons are unstable particles. As such, it is impossible to study the muon interactions without being exposed to decays of the muons forming the colliding beams. From the moment the collider is turned on and the muon bunches start to circulate in the accelerator complex, the products of the in-flight decays of the muon beams and the results of their interactions with beam line material, or the detectors themselves, will reach the experiments contributing to polluting the otherwise clean collision environment. The ensemble of all these particles is usually known as “Beam Induced Backgrounds”, or BIB. The composition, flux, and energy spectra of the BIB entering a detector is closely intertwined with the design of the experimental apparatus, such as the beam optics that integrate the detectors in the accelerator complex or the presence of shielding elements, and the collision energy. However, two general features broadly characterise the BIB: it is composed of low-energy particles with a broad arrival time in the detector.

The design of an optimised muon collider detector is still in its infancy, but the work has initiated and it is reviewed in Sect. 4. It is already clear that the physics goals will require a fully hermetic detector able to resolve the trajectories of the outgoing particles and their energies. While the final design might look similar to those experiments taking data at the LHC, the technologies at the heart of the detector will have to be new. The large flux of BIB particles sets requirements on the need to withstand radiation over long periods of time, and the need to disentangle the products of the beam collisions from the particles entering the sensitive regions from uncommon directions calls for high-granularity measurements in space, time and energy. The development of these new detectors will profit from the consolidation of the successful solutions that were pioneered for example in the High Luminosity LHC upgrades, as well as brand new ideas. New solutions are being developed for use in the muon collider environment spanning from tracking detectors, calorimeters systems to dedicated muon systems. The whole effort is part of the push for the next generation of high-energy physics detectors, and new concepts targeted to the muon collider environment might end up revolutionising other future proposed collider facilities as well.

Together with a vibrant detector development program, new techniques and ideas needs to be developed in the interpretation of the energy depositions recorded by the instrumentation. The contributions from the BIB add an incoherent source of backgrounds that affect different detector systems in different ways and that are unprecedented at other collider facilities. The extreme multiplicity of energy depositions in the tracking detectors create a complex combinatorial problem that challenges the traditional algorithms for reconstructing the trajectories of the charged particles, as these were designed for collisions where sprays of particles propagate outwards from the centre of the detector. At the same time, the potentially groundbreaking reach into the high-energy frontier will lead to strongly collimated jets of particles that need to be resolved by the calorimeter systems, while being able to subtract with precision the background contributions. The challenging environment of the muon collider offers fertile ground for the development of new techniques, from traditional algorithms to applications of artificial intelligence and machine learning, to brand new computing technologies such as quantum computers.

Muon collider plans

The ongoing reassessment of the muon collider design and the plans for R &D allow us to envisage a possible path towards the realisation of the muon collider and a tentative technically-limited timeline, displayed in Fig. 12.

The goal [11, 12] is a muon collider with a centre of mass energy of 10 TeV or more (a \(10^+\) TeV MuC). Passing this energy threshold enables, among other things, a vast jump ahead in the search for new heavy particles relative to the LHC. The target integrated luminosity is obtained by considering the cross-section of a typical \(2\rightarrow 2\) scattering processes mediated by the electroweak interactions, \(\sigma \sim 1~{\textrm{fb}}\cdot (10~{\textrm{TeV}})^2/E_{{\textrm{cm}}}^2\). In order to measure such cross-sections with good (percent-level) precision and to exploit them as powerful probes of short distance physics, around ten thousand events are needed. The corresponding integrated luminosity is

$$\begin{aligned} \displaystyle {\mathfrak {L}}_{{\textrm{int}}}=10\,{\textrm{ab}}^{-1}\left( \frac{E_{{\textrm{cm}}}}{10\,{\textrm{TeV}}}\right) ^2. \end{aligned}$$
(1)

The luminosity requirement grows quadratically with the energy in order to compensate for the cross-section decrease. We will see in Sect. 3 that achieving this scaling is indeed possible at muon colliders.

Assuming a muon collider operation time of \(10^7\) seconds per year, and one interaction point, Eq. (1) corresponds to an instantaneous luminosity

$$\begin{aligned} \displaystyle {\mathfrak {L}}=\frac{5~{\textrm{years}}}{\textrm{time}} \left( \frac{E_{{\textrm{cm}}}}{10\,{\textrm{TeV}}}\right) ^22\cdot 10^{35}\,{\textrm{cm}}^{-2}{\textrm{s}}^{-1}. \end{aligned}$$
(2)

The current design target parameters (see Table 1) enable to collect the required integrated luminosity in a 5-year run, ensuring an appealingly compact temporal extension to the muon collider project even in its data taking phase. Furthermore this ambitious target leaves space to increase the integrated luminosity by running longer or by foreseeing a second interaction point. One could similarly compensate for a possible instantaneous luminosity reduction in the final design.

Muon collider stages

In order to design the path towards a \(10^+\) TeV MuC, one could exploit the possibility of building it in stages. In fact, the design of many elements of the facility is simply independent of the collider energy. Once built and exploited for a lower \(E_{{\textrm{cm}}}\) MuC, they can thus be reused for a higher energy collider. This applies to the muon source and cooling, and to the accelerator complex as well because energy staging is anyway required for fast acceleration. Only the final collision ring of the lower \(E_{{\textrm{cm}}}\) collider could not be reused. However because of its limited size it is a minor addition to the total cost.

A staged approach has several advantages. First, it spreads the total cost over a longer time period and reduces the initial investment. This could enable a faster financing for the first energy stage and accelerate the whole project. Furthermore the reduced energy of the first stage allows, if needed, to make compromises on technologies that might not yet be fully developed, avoiding potential delays. In particular completing construction in 2045 as foreseen in Fig. 12 could be optimistic for a \(10^+\) TeV MuC, but realistic for a first lower-energy collider at few TeV. A centre of mass energy \(E_{{\textrm{cm}}}=3\) TeV is being tentatively considered for the first stage. This matches, with a much more compact design, the maximal \(e^+e^-\) energy that could be achieved by the last stage of the CLIC linear collider [15].

The 3 TeV stage of the muon collider offers amazing opportunities for progress with respect to the LHC and its high-luminosity successor (HL-LHC) [18]. These opportunities include a determination of the Higgs trilinear coupling, extended sensitivity to Higgs and top quark compositeness and to extended Higgs sectors. A selected summary of available studies is reported in Sect. 5. On the other hand, the physics potential of the \(10^+\) TeV collider is much superior to the one of the 3 TeV collider. The higher energy stage will radically advance the knowledge acquired with the first stage operation. Additionally, the energy upgrade would enable to investigate new physics discoveries or tensions with the SM that might emerge at the first stage.

The 3 TeV stage, following Eq. (2), would collect \(0.9\simeq 1\) ab\(^{-1}\) luminosity (with one detector) in five years of full luminosity, after an initial ramp-up phase of two to three years. In the most optimistic scenario the construction of the first stage will proceed rapidly. The first stage will terminate after seven years to leave space to the second stage with radically improved physics performances. If the second stage is instead delayed, the one at 3 TeV could run longer. The optimistic and pessimistic scenarios thus foresee 1 and 2 \(\hbox {ab}^{-1}\) at 3 TeV, respectively.

Other muon colliders

The tentative staging scenario detailed above should serve as the baseline for future investigations of alternative plans. In particular, one could consider the possibility of a first stage of much lower energy than 3 TeV, to be possibly built on an even shorter time scale. However, it is worth remarking in this context that the quadratic luminosity scaling with energy in Eqs. (1) and (2) is not only the aspirational target, but it is also the natural scaling of the luminosity at muon colliders. By following the scaling for low \(E_{{\textrm{cm}}}\) one immediately realises that the luminosity of a muon collider at order 100 GeV energy can not be competitive with the one of an \(e^+e^-\) circular or linear collider. For instance this implies that while there is evidently a compelling physics case for a leptonic “Higgs factory” at around 250 GeV energy, a muon collider would be probably unable to collect the high luminosity needed for a successful program of Higgs coupling measurements, while this is possible for \(e^+e^-\) colliders. In general, the luminosity scaling suggests that a physics-motivated first stage of the muon collider should either exploit some peculiarity of the muons that make \(\mu ^+\mu ^-\) collisions more useful than \(e^+e^-\) collisions, or target the TeV energy that is hard to reach with \(e^+e^-\).

The possibility of operating a first muon collider at the Higgs pole \(E_{{\textrm{cm}}}=m_H=125\) GeV has been discussed extensively in the literature. The idea here is to exploit the large Yukawa coupling of the muon, much larger than the one of the electron, in order to produce the Higgs boson in the s-channel and study its lineshape. The physics potential of such Higgs-pole muon collider will be described in Sect. 5. The major results would be a rather precise and direct determination of the Higgs width and an astonishingly accurate measurement of the Higgs mass. However, the Higgs is a rather narrow particle, with a width over mass ratio \(\varGamma _H/m_H\) as small as \(3\cdot 10^{-5}\). The muon beams would thus need a comparably small energy spread \(\varDelta E/E\hspace{-2pt}=\hspace{-2pt}3\cdot 10^{-5}\) for the programme to succeed. Engineering such tiny energy spread might perhaps be possible. However it poses a challenge for the facility design that is peculiar to the Higgs-pole collider and of no relevance for higher energies, where a much higher spread is perfectly adequate for physics. For this reason, the Higgs-pole muon collider is not currently considered in the staging plan and further study is needed.

Further work is also needed to assess the possible relevance of a muon collider at the \(t{{\overline{t}}}\) production threshold \(E_{{\textrm{cm}}}\simeq 343\) GeV, aimed at measuring the top quark mass with precision. The top threshold could be reached also with \(e^+e^-\) colliders. However the \(e^+e^-\) Higgs factory at 250 GeV, to be possibly built before the muon collider, might not be easily and quickly upgradable to 343 GeV. Moreover, the naturally small (permille-level) beam energy spread and the reduction of initial state radiation effects give an advantage to muons over electrons for the threshold scan. Such “first muon collider” was proposed long ago [19, 20]. Its modern relevance stems from the need of an improved top mass determination for establishing the possible instability of the SM Higgs potential [21]. We will not discuss this option further and we refer the reader to the literature.

This review

A muon collider could be a sustainable innovative device for a big ambitious jump ahead in fundamental physics exploration. It is a long-term project, but with a tight schedule and a narrow temporal window of opportunity. The initiated work must continue in the next decade, fostered by a positive recommendation of the 2023 US Particle Physics Prioritization Panel (P5) and the next Update of the European Strategy for Particle Physics foreseen in 2026/2027. Progress must be made by then on the perspectives for a muon collider to be built and operated, for the outcome of its collisions to be recorded, interpreted and exploited to advance physics knowledge. This offers stimulating challenges for accelerator, experimental and theoretical physics.

Muon colliders require innovative research in each of these three directions. The novelty of the theme and the lack of established solutions enable a high rate of progress, but it also requires that the three directions advance simultaneously because progress in one motivates and supports work in the others. Furthermore, exploiting synergies between accelerator, experimental and theoretical physics is of utmost importance at this initial stage of the muon collider project design.

Fig. 2
figure 2

Equivalent proton collider energy. The left plot [11], assumes that qq and gg partonic initial states both contribute to the production. In the right panel [23], production from qq and from gg are considered separately

In this spirit, the present Review summarises the state of the art and the recent progress in all these three areas, and outlines directions for future work. The aim is to provide a global perspective on muon colliders.

This Review is organised as follows. Section 2 summarises the key exploration opportunities offered by very high energy muon colliders and illustrates the potential for progress on selected physics questions. We also outline the challenges for the theoretical predictions needed to exploit this potential. These challenges constitute in fact a tremendous opportunity to advance knowledge of SM physics in a regime where the electroweak bosons are relatively light particles, entailing the emergence of the novel phenomenon of electroweak radiation. Section 3 describes the challenges and the opportunities of muon colliders for accelerator physics. It reviews the basic principles for the design of the muon production, cooling and fast acceleration systems. The required R &D, and a tentative staging plan and timeline, are also outlined. Section 4 describes the experimental conditions that are expected at the muon collider and the ongoing work on the design of the detector and of the event reconstruction software. We devote Sect. 5 to selected muon collider sensitivity projection studies. The \(10^+\) TeV MuC is the main focus, but some opportunities of the 3 TeV stage are also described, as well as those of the Higgs-pole collider. A summary of the perspectives and opportunities for future work on muon colliders is reported in Sect. 6.

2 Physics opportunities

2.1 Why muons?

Muons, like protons, can be made to collide with a centre of mass energy of 10 TeV or more in a relatively compact ring, without fundamental limitations from synchrotron radiation. However, being point-like particles, unlike protons, their nominal centre of mass collision energy \(E_{{\textrm{cm}}}\) is entirely available to produce high-energy reactions that probe length scales as short as \(1/E_{{\textrm{cm}}}\). The relevant energy for proton colliders is instead the centre of mass energy of the collisions between the partons that constitute the protons. The partonic collision energy is distributed statistically, and approaches a significant fraction of the proton collider nominal energy with very low probability. A muon collider with a given nominal energy and luminosity is thus evidently way more effective than a proton collider with comparable energy and luminosity.

This concept is made quantitative in Fig. 2. The figure displays the center of mass energy \({\sqrt{s\,}}_{\hspace{-2pt}p}\) that a proton collider must possess to be “equivalent” to a muon collider of a given energy \(E_{{\textrm{cm}}}=\sqrt{s\,}_{\hspace{-2pt}\mu }\). Equivalence is defined [11, 22, 23] in terms of the pair production cross-section for heavy particles, with mass close to the muon collider kinematical threshold of \(\sqrt{s\,}_{\hspace{-2pt}\mu }/2\). The equivalent \({\sqrt{s\,}}_{\hspace{-2pt}p}\) is the proton collider centre of mass energy for which the cross-sections at the two colliders are equal.

The estimate of the equivalent \({\sqrt{s\,}}_{\hspace{-2pt}p}\) depends on the relative strength \(\beta \) of the heavy particle interaction with the partons and with the muons. If the heavy particle only possesses electroweak quantum numbers, \(\beta =1\) is a reasonable estimate because the particles are produced by the same interaction at the two colliders. If instead it also carries QCD colour, the proton collider can exploit the QCD interaction to produce the particle, and a ratio of \(\beta =10\) should be considered owing to the large QCD coupling and colour factors. The orange line on the left panel of Fig. 2, obtained with \(\beta =1\), is thus representative of purely electroweak particles. The blue line, with \(\beta =10\), is instead a valid estimate for particles that also possess QCD interactions, as it can be verified in concrete examples.

The general lesson we learn from the left panel of Fig. 2 (orange line) is that at a proton collider with around 100 TeV energy the cross-section for processes with an energy threshold of around 10 TeV is quite smaller than the one of a muon collider (MuC) operating at \(E_{{\textrm{cm}}}={\sqrt{s\,}}_{\hspace{-2pt}\mu }\) \(\sim 10\) TeV. The gap can be compensated only if the process dynamics is different and more favourable at the proton collider, like in the case of QCD production. The general lesson has been illustrated for new heavy particles production, where the threshold is provided by the particle mass. But it also holds for the production of light SM particles with energies as high as \(E_{{\textrm{cm}}}\), which are very sensitive indirect probes of new physics. This makes exploration by high energy measurements more effective at muon than at proton colliders, as we will see in Sect. 2.4. Moreover the large luminosity for high energy muon collisions produces the copious emission of effective vector bosons. In turn, they are responsible at once for the tremendous direct sensitivity of muon colliders to “Higgs portal” type new physics and for their excellent perspectives to measure single and double Higgs couplings precisely as we will see in Sects. 2.2 and 2.3, respectively.

On the other hand, no quantitative conclusion can be drawn from Fig. 2 on the comparison between the muon and proton colliders discovery reach for the heavy particles. That assessment will be performed in the following section based on available proton colliders projections.

2.2 Direct reach

The left panel of Fig. 3 displays the number of expected events, at a 10 TeV MuC with 10 \(\hbox {ab}^{-1}\) integrated luminosity, for the pair production due to electroweak interactions of Beyond the Standard Model (BSM) particles with variable mass M. The particles are named with a standard BSM terminology, however the results do not depend on the detailed BSM model (such as Supersymmetry or Composite Higgs) in which these particles emerge, but only on their Lorentz and gauge quantum numbers. The dominant production mechanism at high mass is the direct \(\mu ^+\mu ^-\) annihilation, whose cross-section flattens out below the kinematical threshold at \({\textrm{M}}=5\) TeV. The cross-section increase at low mass is due to the production from effective vector boson annihilation.

Fig. 3
figure 3

Left panel: the number of expected events (from Ref. [24]) at a 10 TeV MuC, with 10 ab\(^{-1}\) luminosity, for several BSM particles. Right panel: \(95\%\) CL mass reach, from Ref. [25], at the HL-LHC (solid bars) and at the FCC-hh (shaded bars). The tentative discovery reach of a 10, 14 and 30 TeV MuC are reported as horizontal lines

Fig. 4
figure 4

Left panel: exclusion and discovery mass reach on Higgsino and Wino dark matter candidates at muon colliders from disappearing tracks, and at other facilities. The plot is adapted from Ref. [47]. Right: exclusion contour [23] for a scalar singlet of mass \(m_\phi \) mixed with the Higgs boson with strength \(\sin \gamma \). More details in Sect. 5.1

The figure shows that with the target luminosity of 10 \(\hbox {ab}^{-1}\) a 10 TeV MuC can produce the BSM particles abundantly. If they decay to energetic and detectable SM final states, the new particles can be definitely discovered up to the kinematical threshold. Taking into account that the entire target integrated luminosity will be collected in 5 years, a few month run could be sufficient for a discovery. Afterwards, the large production rate will allow us to observe the new particles decaying in multiple final states and to measure kinematical distributions. We will thus be in the position of characterising the properties of the newly discovered states precisely. Similar considerations hold for muon colliders with higher \(E_{{\textrm{cm}}}\), up to the fact that the kinematical mass threshold obviously grows to \(E_{{\textrm{cm}}}/2\). Notice however that the production cross-section decreases as \(1/E_{{\textrm{cm}}}^2\).Footnote 1 Therefore, we obtain as many events as in the left panel of Fig. 3 only if the integrated luminosity grows quadratically with the energy as in Eq. (1). A luminosity that is lower than this by a factor of around 10 would not affect the discovery reach, but it might reduce the potential for characterising the discoveries.

The direct reach of muon colliders vastly and generically exceeds the sensitivity of the High-Luminosity LHC (HL-LHC). This is illustrated by the solid bars on the right panel of Fig. 3, where we report the projected HL-LHC mass reach [25] on several BSM states. The \(95\%\) CL exclusion is reported, instead of the discovery, as a quantification of the physics reach. Specifically, we consider Composite Higgs fermionic top-partners T (e.g., the \(X_{5/3}\) and the \(T_{2/3}\)) and supersymmetric particles such as stops \({{\widetilde{t}}}\), charginos \({{\widetilde{\chi }}}_1^\pm \), stau leptons \({{\widetilde{\tau }}}\) and squarks \({{\widetilde{q}}}\). For each particle we report the highest possible mass reach, as obtained in the configuration for the BSM particle couplings and decay chains that maximises the hadron colliders sensitivity. The reach of a 100 TeV proton-proton collider (FCC-hh) is shown as shaded bars on the same plot. The muon collider reach, displayed as horizontal lines for \(E_{{\textrm{cm}}}=10\), 14 and 30 TeV, exceeds the one of the FCC-hh for several BSM candidates and in particular, as expected, for purely electroweak charged states. It should be noted that detailed muon collider sensitivity projections for the BSM candidates in Fig. 3 have not been performed yet. In general, a relatively limited literature exists on direct new physics searches at the MuC [26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44]. More studies would be desirable also to offer targets to the design of the detector.

Several interesting BSM particles do not decay to easily detectable final states, and an assessment of their observability requires dedicated studies. A clear case is the one of minimal WIMP Dark Matter (DM) candidates. The charged state in the DM electroweak multiplet is copiously produced, but it decays to the invisible DM plus a soft undetectable pion, owing to the small mass-splitting. WIMP DM can be studied at muon colliders in several channels (such as mono-photon) without directly observing the charged state [45, 46]. Alternatively, one can instead exploit the disappearing tracks produced by the charged particle [47]. The result is displayed on the left panel of Fig. 4 for the simplest candidates, known as Higgsino and Wino. A 10 TeV muon collider reaches the “thermal” mass, marked with a dashed line, for which the observed relic abundance is obtained by thermal freeze out. Other minimal WIMP candidates become kinematically accessible at higher muon collider energies [45, 46]. Muon colliders could actually even probe some of these candidates when they are above the kinematical threshold, by studying their indirect effects on high-energy SM processes [48, 49]. A more extensive overview of the muon collider potential to probe WIMP DM is provided in Sect. 5.2.

Fig. 5
figure 5

Left panel: schematic representation of vector boson fusion or scattering processes. The collinear V bosons emitted from the muons participate to a process with hardness \(\sqrt{{\hat{s}}}\ll E_{{\textrm{cm}}}\). Right panel: number of expected events for selected SM processes at a muon collider with variable \(E_{{\textrm{cm}}}\) and luminosity scaling as in Eq. (1)

New physics particles are not necessarily coupled to the SM by gauge interaction. One setup that is relevant in several BSM scenarios (including models of baryogenesis, dark matter, and neutral naturalness; see Sect. 5.1) is the “Higgs portal” one, where the BSM particles interact most strongly with the Higgs field. By the Goldstone Boson Equivalence Theorem, Higgs field couplings are interactions with the longitudinal polarisations of the SM massive vector bosons W and Z, which enable Vector Boson Fusion (VBF) production of the new particles. A muon collider is extraordinarily sensitive to VBF production, owing to the large luminosity for effective vector bosons. This is illustrated on the right panel of Fig. 4, in the context of a benchmark model [23, 26] (see also [27, 28]) where the only new particle is a real scalar singlet with Higgs portal coupling. The coupling strength is traded for the strength of the mixing with the Higgs particle, \(\sin \gamma \), that the interaction induces. The scalar singlet is the simplest extension of the Higgs sector. Extensions with richer structure, such as involving a second Higgs doublet, are a priori easier to detect as one can exploit the electroweak production of the new charged Higgs bosons, as well as their VBF production. See Refs. [50,51,52,53,54] for dedicated studies, and Sect. 5.1 for a review.

In several cases the muon collider direct reach compares favourably to the one of the most ambitious future proton collider project. This is not a universal statement, in particular at a muon collider it is obviously difficult to access heavy particles that carry only QCD interactions. One might also expect a muon collider of 10 TeV to be generically less effective than a 100 TeV proton collider for the detection of particles that can be produced singly. For instance, for additional \(Z'\) massive vector bosons, that can be probed at the FCC-hh well above the 10 TeV mass scale. We will see in Sect. 2.4 that the situation is slightly more complex and that, in the case of \(Z'\)s, a 10 TeV MuC sensitivity actually exceeds the one of the FCC-hh in most of the parameter space (see the right panel of Fig. 7).

2.3 A vector bosons collider

When two electroweak charged particles like muons collide at an energy much above the electroweak scale \(m_{{{\textsc {w}}}}\sim 100~\)GeV, they have a high probability to emit electroWeak (EW) radiation. There are multiple types of EW radiation effects that can be observed at a muon collider and play a major role in muon collider physics. Actually we will argue in Sect. 2.5 that the experimental observation and the theoretical description of these phenomena emerges as a self-standing reason of interest in muon colliders.

Here we focus on the practical implications [11, 22,23,24, 55,56,57] of the collinear emission of nearly on-shell massive vector bosons, which is the analog in the EW context of the Weizsäcker–Williams emission of photons. The vector bosons V participate, as depicted in Fig. 5, to a scattering process with a hard scale \(\sqrt{{\hat{s}}}\) that is much lower than the muon collision energy \(E_{{\textrm{cm}}}\). The typical cross-section for VV annihilation processes is thus enhanced by \(E_{{\textrm{cm}}}^2/{{\hat{s}}}\), relative to the typical cross-section for \(\mu ^+\mu ^-\) annihilation, whose hard scale is instead \(E_{{\textrm{cm}}}\). The emission of the V bosons from the muons is suppressed by the EW coupling, but the suppression is mitigated or compensated by logarithms of the separation between the EW scale and \(E_{{\textrm{cm}}}\) (see [22, 23, 55] for a pedagogical overview). The net result is a very large cross-section for VBF processes that occur at \(\sqrt{{\hat{s}}}\sim m_{{{\textsc {w}}}}\), with a tail in \(\sqrt{{\hat{s}}}\) up to the TeV scale.

Fig. 6
figure 6

Left panel: \(1\sigma \) sensitivities (in %) from a 10-parameter fit in the \(\kappa \)-framework at a 10 TeV MuC with 10 ab\(^{-1}\), compared with HL-LHC. The effect of measurements from a 250 GeV \(e^+e^-\) Higgs factory is also reported. Right panel: sensitivity to \(\delta \kappa _\lambda \) for different \(E_{{\textrm{cm}}}\). The luminosity is as in Eq. (1) for all energies, apart from \(E_{{\textrm{cm}}}\hspace{-2pt}=\hspace{-2pt}3\) TeV, where doubled luminosity (of 2 ab\(^{-1}\)) is assumed. More details in Sect. 5.1

We already emphasised (see Fig. 3) the importance of VBF for the direct production of new physics particles. The relevance of VBF for probing new physics indirectly simply stems for the huge rate of VBF SM processes, summarised on the right panel of Fig. 5. In particular we see that a 10 TeV muon collider produces ten million Higgs bosons, which is around 10 times more than future \(e^+e^-\) Higgs factories. Since the Higgs bosons are produced in a relatively clean environment, without large physics backgrounds from QCD, a 10 TeV muon collider (over-)qualifies as a Higgs factory [23, 56,57,58,59]. Unlike \(e^+e^-\) Higgs factories, a muon collider also produces Higgs pairs copiously, enabling accurate and direct measurements of the Higgs trilinear coupling [22, 24, 56] and possibly also of the quadrilinear coupling [60].

The opportunities for Higgs physics at a muon collider are summarised extensively in Sect. 5.1. In Fig. 6 we report for illustration the results of a 10-parameter fit to the Higgs couplings in the \(\kappa \)-framework at a 10 TeV MuC, and the sensitivity projections on the anomalous Higgs trilinear coupling \(\delta \kappa _\lambda \). The table shows that a 10 TeV MuC will improve significantly and broadly our knowledge of the properties of the Higgs boson. The combination with the measurements performed at an \(e^+e^-\) Higgs factory, reported on the third column, does not affect the sensitivity to several couplings appreciably, showing the good precision that a muon collider alone can attain. However, it also shows complementarity with an \(e^+e^-\) Higgs factory program.

On the right panel of the figure we see that the performances of muon colliders in the measurement of \(\delta \kappa _\lambda \) are similar or much superior to the one of the other future colliders where this measurement could be performed. In particular, CLIC measures \(\delta \kappa _\lambda \) at the \(10\%\) level [61], and the FCC-hh sensitivity ranges from 3.5 to \(8\%\) depending on detector assumptions [62]. A determination of \(\delta \kappa _\lambda \) that is way more accurate than the HL-LHC projections is possible already at a low energy stage of a muon collider with \(E_{{\textrm{cm}}}=3\) TeV as discussed in Sect. 5.1.

The potential of a muon collider as a vector boson collider has not been explored fully. In particular a systematic investigation of vector boson scattering processes, such as \(WW\hspace{-3pt}\rightarrow \hspace{-3pt} WW\), has not been performed. The key role played by the Higgs boson to eliminate the energy growth of the corresponding Feynman amplitudes could be directly verified at a muon collider by means of differential measurements that extend well above 1 TeV for the invariant mass of the scattered vector bosons. Along similar lines, differential measurements of the \(WW\hspace{-3pt}\rightarrow \hspace{-3pt} HH\) process has been studied in [24, 56] (see also [22]) as an effective probe of the composite nature of the Higgs boson, with a reach that is comparable or superior to the one of Higgs coupling measurements. A similar investigation was performed in [22, 23] (see also [22]) for \(WW\hspace{-3pt}\rightarrow \hspace{-3pt} t{{\overline{t}}}\), aimed at probing Higgs-top interactions.

2.4 High-energy measurements

Direct \(\mu ^+\mu ^-\) annihilation, such as HZ and \(t{{\overline{t}}}\) production, displays a number of expected events of the order of several thousands, reported in Fig. 5. These are much less than the events where a Higgs or a \(t{{\overline{t}}}\) pair are produced from VBF, but they are sharply different and easily distinguishable. The invariant mass of the particles produced by direct annihilation is indeed sharply peaked at the collider energy \(E_{{\textrm{cm}}}\), while the invariant mass rarely exceeds one tenth of \(E_{{\textrm{cm}}}\) in the VBF production mode.

Fig. 7
figure 7

Left panel: \(95\%\) reach on the Composite Higgs scenario from high-energy measurements in di-boson and di-fermion final states [63]. The green contour display the sensitivity from “Universal” effects related with the composite nature of the Higgs boson and not of the top quark. The red contour includes the effects of top compositeness. Right panel: sensitivity to a minimal \(Z'\) [63]. Discovery contours at \(5\sigma \) are also reported in both panels

The good statistics and the limited or absent background thus enables few-percent level measurements of SM cross sections for hard scattering processes of energy \(E_{{\textrm{cm}}}=10\) TeV at the 10 TeV MuC. An incomplete list of the many possible measurements is provided in Ref. [63], including the resummed effects of EW radiation on the cross section predictions. It is worth emphasising that also charged final states such as WH or \(\ell \nu \) are copiously produced at a muon collider. The electric charge mismatch with the neutral \(\mu ^+\mu ^-\) initial state is compensated by the emission of soft and collinear W bosons, which occurs with high probability because of the large energy.

High energy scattering processes are as unique theoretically as they are experimentally [11, 24, 63]. They give direct access to the interactions among SM particles with 10 TeV energy, which in turn provide indirect sensitivity to new particles at the 100 TeV scale of mass. In fact, the effects on high-energy cross sections of new physics at energy \(\varLambda \gg E_{{\textrm{cm}}}\) generically scale as \((E_{{\textrm{cm}}}/\varLambda )^2\) relative to the SM. Percent-level measurements thus give access to \(\varLambda \sim 100\) TeV. This is an unprecedented reach for new physics theories endowed with a reasonable flavor structure. Notice in passing that high-energy measurements are also useful to investigate flavor non-universal phenomena, as we will see in Sect. 5.3.

This mechanism is not novel. Major progress in particle physics always came from raising the available collision energy, producing either direct or indirect discoveries. Among the most relevant discoveries that did not proceed through the resonant production of new particles, there is the one of the inner structure of nucleons. This discovery could be achieved [64] only when the transferred energy in electron scattering could reach a significant fraction of the proton compositeness scale \(\varLambda _{{{\textsc {qcd}}}}=1/r_{p}=300\) MeV. Proton-compositeness effects became sizeable enough to be detected at that energy, precisely because of the quadratic enhancement mechanism we described above.

Figure 7 illustrates the tremendous reach on new physics of a 10 TeV MuC with 10 ab\(^{-1}\) integrated luminosity. The left panel (green contour) is the sensitivity to a scenario that explains the microscopic origin of the Higgs particle and of the scale of EW symmetry breaking by the fact that the Higgs is a composite particle. In the same scenario the top quark is likely to be composite as well, which in turn explains its large mass and suggest a “partial compositeness” origin of the SM flavour structure. Top quark compositeness produces additional signatures that extend the muon collider sensitivity up to the red contour. The sensitivity is reported in the plane formed by the typical coupling \(g_*\) and of the typical mass \(m_*\) of the composite sector that delivers the Higgs. The scale \(m_*\) physically corresponds to the inverse of the geometric size of the Higgs particle. The coupling \(g_*\) is limited from around 1 to \(4\pi \), as in the figure. In the worst case scenario of intermediate \(g_*\), a 10 TeV MuC can thus probe the Higgs radius up to the inverse of 50 TeV, or discover that the Higgs is as tiny as (35 TeV\()^{-1}\). The sensitivity improves in proportion to the centre of mass energy of the muon collider.

The figure also reports, as blue dash-dotted lines denoted as “Others”, the envelop of the \(95\%\) CL sensitivity projections of all the future collider projects that have been considered for the 2020 update of the European Strategy for Particle Physics, summarised in Ref. [25]. These lines include in particular the sensitivity of very accurate measurements at the EW scale performed at possible future \(e^+e^-\) Higgs, electroweak and Top factories. These measurements are not competitive because new physics at \(\varLambda \sim 100\) TeV produces unobservable one part per million effects on 100 GeV energy processes. High-energy measurements at a 100 TeV proton collider are also included in the dash-dotted lines. They are not competitive either, because the effective parton luminosity at high energy is much lower than the one of a 10 TeV MuC, as explained in Sect. 2.1. For example the cross-section for the production of an \(e^+e^-\) pair with more than 9 TeV invariant mass at the FCC-hh is only 40 ab, while it is 900 ab at a 10 TeV muon collider. Even with a somewhat higher integrated luminosity, the FCC-hh just does not have enough statistics to compete with a 10 TeV MuC.

The right panel of Fig. 7 considers a simpler new physics scenario, where the only BSM state is a heavy \(Z'\) spin-one particle. The “Others” line also includes the sensitivity of the FCC-hh from direct \(Z'\) production. The line exceeds the 10 TeV MuC sensitivity contour (in green) only in a tiny region with \(M_{Z'}\) around 20 TeV and small \(Z'\) coupling. This result substantiates our claim in Sect. 2.2 that a reach comparison based on the \(2\rightarrow 1\) single production of the new states is simplistic. Single \(2\rightarrow 1\) production couplings can produce indirect effect in \(2\rightarrow 2\) scattering by the virtual exchange of the new particle, and the muon collider is extraordinarily sensitive to these effects. Which collider wins is model-dependent. In the simple benchmark \(Z'\) scenario, and in the motivated framework of Higgs compositeness that future colliders are urged to explore, the muon collider is just a superior device.

We have seen that high energy measurements at a muon collider enable the indirect discovery of new physics at a scale in the ballpark of 100 TeV. However the muon collider also offers amazing opportunities for direct discoveries at a mass of several TeV, and unique opportunities to characterise the properties of the discovered particles, as emphasised in Sect. 2.2. High energy measurements will enable us take one step further in the discovery characterisation, by probing the interactions of the new particles well above their mass. For instance in the Composite Higgs scenario one could first discover Top Partner particles of few TeV mass, and next study their dynamics and their indirect effects on SM processes. This might be sufficient to pin down the detailed theoretical description of the newly discovered sector, which would thus be both discovered and theoretically characterised at the same collider. Higgs coupling determinations and other precise measurements that exploit the enormous luminosity for vector boson collisions, described in Sect. 2.3, will also play a major role in this endeavour.

We can dream of such glorious outcome of the project, where an entire new sector is discovered and characterised in details at the same machine, only because energy and precision are simultaneously available at a muon collider.

2.5 Electroweak radiation

The novel experimental setup offered by lepton collisions at 10 TeV energy or more outlines possibilities for theoretical exploration that are at once novel and speculative, yet robustly anchored to reality and to phenomenological applications.

The muon collider will probe for the first time a new regime of EW interactions, where the scale \(m_{{{\textsc {w}}}}\hspace{-2pt}\sim \hspace{-2pt}100~\)GeV of EW symmetry breaking plays the role of a small IR scale, relative to the much larger collision energy. This large scale separation triggers a number of novel phenomena that we collectively denote as “EW radiation” effects. Since they are prominent at muon collider energies, the comprehension of these phenomena is of utmost importance not only for developing a correct physical picture but also to achieve the needed accuracy of the theoretical predictions.

The EW radiation effects that the muon collider will observe, which will play a crucial role in the assessment of its sensitivity to new physics, can be broadly divided in two classes.

The first class includes the emission of low-virtuality vector bosons from the initial muons. It effectively makes the muon collider a high-luminosity vector boson collider, on top of a very high-energy lepton-lepton machine. The compelling associated physics studies described in Sect. 2.3 pose challenges for fixed-order theoretical predictions and Monte Carlo event generation even at tree-level, owing to the sharp features of the Monte Carlo integrand induced by the large scale separation and the need to correctly handle QED and weak radiation at the same time, respecting EW gauge invariance. Strategies to address these challenges are available in WHIZARD [65], they have been recently implemented in MadGraph5_aMC@NLO [22, 66] and applied to several phenomenological studies in the muon collider context. Dominance of such initial-state collinear radiation will eventually require a systematic theoretical modelling in terms of EW Parton Distribution Function where multiple collinear radiation effects are resummed. First studies show that EW resummation effects can be significant at a 10 TeV MuC [55].

The second class of effects are the virtual and real emissions of soft and soft-collinear EW radiation. They affect most strongly the measurements performed at the highest energy, described in Sect. 2.4, and impact the corresponding cross-section predictions at order one [63]. They also give rise to novel processes such as the copious production of charged hard final states out of the neutral \(\mu ^+\mu ^-\) initial state, and to new opportunities to detect new short distance physics by studying, for one given hard final state, different patterns of radiation emission [63]. The deep connection with the sensitivity to new physics makes the study of EW radiation an inherently multidisciplinary enterprise that overcomes the traditional separation between “SM background” and “BSM signal” studies.

At very high energies EW radiation displays similarities with QCD and QED radiation, but also remarkable differences that pose profound theoretical challenges.

First, being EW symmetry broken at low energy, different particles in the same EW multiplet – i.e., with different “EW color” like the W and the Z – are distinguishable. In particular the beam particles (e.g., charged left-handed leptons) carry definite colour thus violating the KLN theorem assumptions. Therefore, no cancellation takes place between virtual and real radiation contributions, regardless of the final state observable inclusiveness [67, 68]. Furthermore, the EW colour of the final state particles can be measured, and it must be measured for a sufficiently accurate exploration of the SM and BSM dynamics. For instance, distinguishing the top from the bottom quark or the W from the Z boson (or photon) is necessary to probe accurately and comprehensively new short-distance physical laws that can affect the dynamics of the different particles differently. When dealing with QCD and QED radiation only, it is sufficient instead to consider “safe” observables where QCD/QED radiation effects can be systematically accounted for and organised in well-behaved (small) corrections. The relevant observables for EW physics at high energy are on the contrary dramatically affected by EW radiation effects.

Second, in analogy with QCD and unlike QED, for EW radiation the IR scale is physical. However, at variance with QCD, the theory is weakly-coupled at the IR scale, and the EW “partons” do not “hadronise”. EW showering therefore always ends at virtualities of order 100 GeV, where heavy EW states (tWZH) coexist with light SM ones, and then decay. Having a complete and consistent description of the evolution from high virtualities where EW symmetry is restored, to the weak scale where it is broken, to GeV scales, including also leading QED/QCD effects in all regimes is a new challenge [69].

Such a strong phenomenological motivation, and the peculiarities of the problem, stimulate work and offer a new perspective on resummation and showering techniques, or more in general trigger theoretical progress on IR physics. Fixed-order calculations will also play an important role. Indeed while the resummation of the leading logarithmic effects of radiation is mandatory at muon collider energies [63, 70], subleading logarithms could perhaps be included at fixed order. Furthermore one should eventually develop a description where resummation is merged with fixed-order calculations in an exclusive way, providing the most accurate predictions in the corresponding regions of the phase space, as currently done for QCD computations.

A significant literature on EW radiation exists, starting from the earliest works on double-logarithm resummations based on Asymptotic Dynamics [67, 68] or on the IR evolution equation [71, 72]. The factorisation of virtual massive vector boson emissions, leading to the notion of effective vector boson is also known since long [73,74,75,76]. More recent progress includes resummation at the next to leading log in the Soft-Collinear Effective Theory framework [77,78,79,80,81], the operatorial definition of the distribution functions for EW partons [70, 82, 83] and the calculation of the corresponding evolution, as well as the calculation of the EW splitting functions [84] for EW showering and the proof of collinear EW emission factorisation [85,86,87]. Additionally, fixed-order virtual EW logarithms are known for generic process at the 1-loop order [88, 89] and are implemented in Sherpa [90] and MadGraph5_aMC@NLO [91]. Exact EW corrections at NLO are available in an automatic form for arbitrary processes in the SM, for example in the MadGraph5_aMC@NLO [92] and in the Sherpa+Recola [93] packages or using WHIZARD+Recola [94]. Implementations of EW showering are also available through a limited set of splittings in Pythia 8 [95, 96] and in a complete way in Vincia [97].

While we are still far from an accurate systematic understanding of EW radiation, the present-day knowledge is sufficient to enable rapid progress in the next few years. The outcome will be an indispensable toolkit for muon collider predictions. Moreover, while we do expect that EW radiation phenomena can in principle be described by the Standard Model, they still qualify as “new phenomena” until when we will be able to control the accuracy of the predictions and verify them experimentally. Such investigation is a self-standing reason of scientific interest in the muon collider project.

2.6 Muon-specific opportunities

In the quest for generic exploration, engineering collisions between muons and anti-muons is in itself a unique opportunity. The concept can be made concrete by considering scenarios where the sensitivity to new physics stems from colliding muons, rather than electrons or other particles. An overview of such “muon-specific” opportunities is provided in Sect. 5.3 based on the available literature [23, 29,30,31,32,33,34,35,36,37,38, 51, 98,99,100,101,102,103,104,105,106,107,108,109,110,111,112]. A brief discussion is reported below.

It is worth emphasising in this context that lepton flavour universality is not a fundamental property of Nature. Therefore new physics could exist, coupled to muons, that we could not yet discover using electrons. In fact, it is not only conceivable, but even expected that new physics could couple more strongly to muons than to electrons. Even in the SM lepton flavour universality is violated maximally by the Yukawa interaction with the Higgs field, which is larger for muons than for electrons. New physics associated to the Higgs or to flavour could follow the same pattern, offering a competitive advantage to muon over electron collisions at similar energies. The comparison with proton colliders is less straightforward. By the same type of considerations one expects larger couplings with quarks, especially with the ones of the second and third generation. This expectation should be folded in with the much lower luminosity for heavier quarks at proton colliders than for muons at a muon collider. The perspectives of muon versus proton colliders are model-dependent and of course strongly dependent on the energy of the muon and of the proton collider.

Recently experimental anomalies in g-2 and in B-meson physics measurements triggered numerous studies of muon-philic new physics. These results provide interesting quantitative illustrations of the generic added value for exploration of a collider that employs second-generation particles. They show the muon collider potential to probe new physics that is presently untested because it couples mostly to muons. These models, and others with the same property, will still exist – though in a slightly different region of their parameter space – even if the anomalies will be explained by SM physics as the most recent LHCb results suggest for the B-meson anomalies [113, 114].

Illustrative results are reported in Fig. 8, displaying the minimal muon collider energy that is needed to probe different types of new physics potentially responsible for the g-2 anomaly. The solid lines correspond to limits on contact interaction operators due to unspecified new physics, that contribute at the same time to the muon g-2 and to high-energy scattering processes. Semi-leptonic muon-charm (muon-top) interactions that can account for the g-2 discrepancy can be probed by di-jets at a 3 TeV (10 TeV) MuC, whereas a 30 TeV collider could even probe a tree-level contribution to the muon electromagnetic dipole operator directly through \(\mu \mu \rightarrow h\gamma \). These sensitivity estimates are agnostic on the specific new physics model responsible for the anomaly. Explicit models typically predict light particles that can be directly discovered at the muon collider, and correlated deviations in additional observables. We will see in Sect. 5.3 that a complete coverage of several models that accommodate the current discrepancy is possible already at a 3 TeV MuC, and a collider of tens of TeV could provide a full-fledged no-lose theorem.

Fig. 8
figure 8

Summary, from Sect. 5.3 of the muon collider perspectives to probe the muon g-2 anomaly

3 Facility

The Muon Accelerator Programme (MAP) has developed a concept for the muon collider, shown in Fig. 9. This concept serves as the starting point for the baseline concept and a seed for the tentative parameters in the design studies initiated by the International Muon Collider Collaboration (IMCC) [16]. The tentative target parameters are reported in Table 1.

This section describes the physical principles that motivate the present baseline design and outlines promising avenues that may yield improved performances or efficiency. Technical issues are also highlighted. Section 3.1 provides an overview of the overall design concept. An approximate expression for luminosity is derived in some detail, as this motivates many of the design choices. Consideration of attainable energy, facility scale and power requirements are described. Possible upgrade schemes and timescales are outlined. The detailed description of design concepts and technical issues surrounding each subsystem are reported in Sects. 3.23.7, describing the proton source, target, front end, muon cooling system, acceleration and collider in turn. The concepts and technologies developed for the muon collider will require technical demonstration, to be achieved by a number of demonstrator facilities described in Sect. 3.8. Section 3.10 is devoted to the synergies between the R &D programme for the muon collider and the one of other muon-beam facilities. We summarise our conclusions in Sect. 3.11.

Fig. 9
figure 9

A conceptual scheme for a muon collider

Table 1 Tentative target parameters for MuCs of different energies based on the MAP design with modifications

3.1 Design overview

Most muon collider designs foresee that muons are created as a product of a high power proton beam incident on a target. Most muons are produced by decay of pions created in the target. In order to capture a large number of negatively and positively charged secondary particles over a broad range of momenta, a solenoid focusing system is used rather than the more conventional horn-type focusing. Following the target the beam is cleaned and most pions decay leaving a beam composed mostly of muons. The muons are captured longitudinally in a sequence of RF cavities arranged to manipulate the single short bunch with large energy spread into a series of bunches each with a much smaller energy spread.

Following creation of a bunch train, the beam is split into the different charge species and each charge species is cooled separately by a 6D ionisation cooling system. Ionisation cooling increases the beam brightness and hence luminosity. An initial cooling line reduces the muon phase space volume sufficiently that each bunch train can be remerged into a single bunch. Further cooling then reduces the longitudinal and transverse beam size. Final cooling systems for each charge species results in a beam suitable for acceleration and collision.

Ionisation cooling is chosen as it operates on a time scale that is competitive with the muon lifetime. Despite the short time scale, a significant number of muons are lost due to muon decay as well as transmission losses. Nonetheless the increased beam brightness provided by the cooling system yields a significant increase in luminosity.

The beam is then accelerated. Rapid acceleration is required in order to maintain an acceleration time that is much shorter than the muon lifetime in the laboratory frame. Satisfactory yields may be achieved by leveraging the muon lifetime increase during acceleration due to Lorentz time dilation to maximise the acceleration efficiency \(\eta _\tau \). The number of muons N changes with time t according to

$$\begin{aligned} \frac{dN}{dt} = -\frac{m_\mu N c^2}{E \tau _\mu }, \end{aligned}$$
(3)

where \(m_\mu \), E and \(\tau _\mu \) are the muon mass, energy and lifetime respectively. Assuming the muons are travelling, near to the speed of light c, through a mean field gradient \({\bar{V}}\), their energy changes as \(dE/dt=e{\bar{V}}c\), where e is the muon charge. We thus find

$$\begin{aligned} \frac{dN}{dE} = - \frac{N}{\delta _\tau E} , \end{aligned}$$
(4)

where \(\delta _\tau = e{\bar{V}} \tau _\mu /m_\mu c\) is the mean change in energy in one muon lifetime, normalised to the muon rest energy. Integrating yields the acceleration efficiency

$$\begin{aligned} \eta _\tau = \frac{N_\pm }{N_{0\pm }} = \prod _i \left( \frac{E_{i+1}}{E_{i}}\right) ^{-1/\delta _{\tau ,i}}, \end{aligned}$$
(5)

where the product is taken over all accelerator subsystems. Mean gradients \({\bar{V}} \sim O(1{-}10)\) MV/m are possible, with higher gradients available in the early parts of the accelerator chain, yielding \(\delta _\tau \sim O(10)\gg 1\). Muons can thus be accelerated faster than they decay, entailing a limited loss of muons during the acceleration process.

At low energy rapid acceleration is achieved using a linear accelerator, in order to maximise the average accelerating gradient in this relatively short section. At higher energies recirculation may be used to improve the system efficiency, for example in a dogbone recirculator. Finally, a sequence of pulsed synchrotrons bring the beam up to final energy. Synchrotrons that employ a combination of fixed high-field, superconducting dipoles and lower field pulsed dipoles are under study. The muons are eventually transferred into a low circumference collider ring where collisions occur.

Luminosity

The muon collider benefits from significant luminosity even at high energies. Many of the design parameters for a muon collider are driven by the need to achieve a good luminosity. An approximate expression for luminosity may be derived to inform design choices and highlight the critical parameters for optimisation. In particular proton sources are a relatively well-known technology, with examples such as SNS and JPARC in a similar class to the proton driver required for the muon collider. Muon beam facilities comparable to the muon collider have instead never been constructed. In order to quantify the required performance for a muon collider facility, it is convenient to express the luminosity in terms of the proton source parameters and muon facility performance indicators, for example the final muon energy, muon collection efficiency and muon beam quality.

In each beam crossing in a collider the integrated luminosity increases by [115]

$$\begin{aligned} \varDelta {\mathfrak {L}} = \frac{N_{+,j} N_{-,j}}{4\pi \sigma ^2_\perp }, \end{aligned}$$
(6)

where \(N_{\pm ,j}\) are the number of muons in each positively and negatively charged bunch on the \(j^{th}\) crossing and \(\sigma _\perp \) is the geometric mean of the horizontal (x) and vertical (y) RMS beam sizes, assumed to be the same for both charge species.

The number of particles in each beam on the \(j^{th}\) crossing decreases due to muon decay as

$$\begin{aligned} N_{\pm ,j} = N_{\pm } \exp (-2 \pi R j/(c\gamma \tau _\mu )), \end{aligned}$$
(7)

where R is the collider radius and \(\gamma \) the Lorentz factor of the muons. If the facility has a repetition rate of \(f_r\) acceleration cycles per second and \(n_b\) bunches circulate in the collider, the luminosity will be

$$\begin{aligned} {\mathfrak {L}} = f_r n_b \frac{N_{+} N_{-}}{4\pi \sigma ^2_\perp } \sum ^{j_\text {max}}_{j=0} \exp \left( -\frac{4\pi R}{\gamma c \tau _\mu } j\right) . \end{aligned}$$
(8)

For the designs discussed here the muon passes around the collider ring many times (\(j_\text {max}\rightarrow \infty \)) so we can sum the geometric series. Furthermore, \(2 \pi R/(c\gamma \tau _\mu )\ll 1\), therefore to a good approximation

$$\begin{aligned} {\mathfrak {L}} \approx f_r n_b \frac{N_{+} N_{-}}{(4\pi )^2\sigma ^2_\perp } \frac{\gamma c \tau _\mu }{R}. \end{aligned}$$
(9)

The average collider radius R, in terms of the average bending field \({\bar{B}}\), is \(R = p/(e{\bar{B}}) \approx \gamma m_\mu c / (e{\bar{B}})\) and

$$\begin{aligned} {\mathfrak {L}} \approx f_r n_b \frac{N_{+} N_{-}}{(4\pi )^2\sigma ^2_\perp } \frac{\tau _\mu e {\bar{B}}}{m_\mu }. \end{aligned}$$
(10)

The transverse beam size \(\sigma _\perp \) may be expressed in terms of the beam quality (emittance) and the focusing provided by the magnets. \(\varepsilon _\perp \) and \(\varepsilon _l\) are the normalised emittances in transverse and longitudinal coordinates; a small \(\varepsilon \) indicates a beam occupying a small region in position and momentum phase space. To a good approximation \(\varepsilon \) is conserved during acceleration. The degree to which the beam is focused is denoted by the lattice Twiss parameter \(\beta ^*_\perp \). For a short bunch

$$\begin{aligned} \sigma _\perp = \sqrt{\frac{m_\mu c \beta ^*_\perp \varepsilon _\perp }{p}}. \end{aligned}$$
(11)

Stronger lenses create a tighter focus and make the beam size smaller at the interaction point, reducing \(\beta ^*_\perp \). The minimum beam size is practically limited by the “hourglass effect”; when the focal length of the lensing system is much shorter than the length of the beam itself, the average beam size at the crossing is dominated by particles that are not at the focus [116]. For example, when the RMS bunch length is not zero, but \(\sigma _z = \beta ^*_\perp \), Eq. (11) is replaced by

$$\begin{aligned} \sigma _\perp = \sqrt{\frac{m_\mu c \sigma _z \varepsilon _\perp }{p f_{hg}}}, \end{aligned}$$
(12)

with a hourglass factor \(f_{hg} \approx 0.76\). The RMS longitudinal emittance is \(\varepsilon _l = \gamma m_\mu c^2 \sigma _\delta \sigma _z\) where \(\sigma _\delta \) is the fractional RMS energy spread, so the luminosity may be expressed as

$$\begin{aligned} {\mathfrak {L}} \approx \frac{e \tau _\mu }{(4 \pi m_\mu c)^2} \frac{f_{hg} \sigma _\delta {\bar{B}}}{\varepsilon _\perp \varepsilon _L} {E_\mu }^2 N_+ N_- n_b f_r , \end{aligned}$$
(13)

where \(E_\mu =\gamma m_\mu c^2\) is the energy of the colliding muons.

Naively, the number of muons reaching the accelerator may be obtained from the number and energy of protons, i.e. from the proton beam power. This assumes proton energy is fully converted to pions and the capture and beam cooling systems have no losses. In reality pion production is more complicated; practical constraints such as pion reabsorption, other particle production processes and geometrical constraints in the target have a significant effect. Decay and transmission losses occur in the ionisation cooling system that significantly degrades the efficiency.

The final number of muons per bunch in the collider, \(N_\pm \), can be related to the proton beam power on target \(P_p\) and the conversion efficiency per proton per unit energy \(\eta _\pm \) by

$$\begin{aligned} N_{\pm } =\frac{\eta _\tau \eta _\pm P_p}{n_b f_r}. \end{aligned}$$
(14)

Overall the luminosity may be expressed as

$$\begin{aligned} {\mathfrak {L}} \approx \underbrace{ \frac{e \tau _\mu }{(4 \pi m_\mu c)^2} }_{K_L} \frac{f_{hg} \sigma _\delta {\bar{B}}}{\varepsilon _\perp \varepsilon _L n_b f_r} \underbrace{ \eta _+ \eta _- (\eta _\tau P_p E_\mu )^2}_{P_+ P_-}, \end{aligned}$$
(15)

where \(K_L = 4.38 \times 10^{36} \mathrm { \, MeV \, MW^{-2} \, T^{-1} \, s^{-2}} \) and \(P_\pm \) is the muon beam power per species.

This luminosity dependence yields a number of consequences. The luminosity improves approximately with the square of energy at fixed average bending field. We thus find the desired scaling in Eq. (1) that entails, as discussed in the previous section, a constant rate for very massive particles pair-production, as well as a growing VBF rate for precision measurements. The quadratic scaling of the luminosity with energy is peculiar of muon colliders and it is not present, for example, in a linear collider. This is because the beam can be recirculated many times through the interaction point and beamstrahlung has a negligible affect on the focusing that may be achieved at the interaction point of the muon collider. This yields an improvement in power efficiency with energy.

The luminosity is highest for collider rings having strong dipole fields (large \({\bar{B}}\)), so that the circumference is smaller and muons can pass through the interaction region many times before decaying. For this reason a separate collider ring with the highest available dipole fields is proposed after the final acceleration stage, as in Fig. 9.

The luminosity is highest for a small number of very high intensity bunches. The MAP design demanded a single muon bunch of each charge, which yields the highest luminosity per detector. Such a design would enable detectors to be installed at two interaction points.

The luminosity decreases linearly with the facility repetition rate, assuming a fixed proton beam power. For the baseline design, a low repetition rate has been chosen relative to equivalent pulsed proton sources.

The luminosity decreases with the product of the transverse and longitudinal emittance. It is important to achieve a low beam emittance in order to deliver satisfactory luminosity, while maintaining the highest possible efficiency \(\eta _\pm \) of converting protons to muons.

Based on these considerations, an approximate guide to the luminosity normalised to beam power is shown in Fig. 10 and compared with the one of CLIC.

Facility size

The geometric dimensions of the MuC depend on future technology and design choices. Some indication of the dimensions can be estimated. The facility scale is expected to be driven by the pulsed synchrotrons in the acceleration system.

Fig. 10
figure 10

MuC luminosity normalised to the muon beam power and compared to CLIC, for different beam energies

The rapid pulsing required in the synchrotrons precludes the use of high-field ramped superconducting magnets such as those used in the LHC. Static high-field superconducting dipoles are proposed combined with rapidly pulsed low-field dipoles. As the beam accelerates the pulsed dipoles are ramped, enabling variation of the mean dipole field. The static dipoles provide a relatively compact and efficient bend.

Preliminary estimates indicate that acceleration up to 3 TeV centre-of-mass energy, assuming 10 T static dipoles and pulsed dipoles with a field swing ±1.8 T, would require a ring of circumference around 10 km. Around 60% of the ring is estimated to be required for pulsed and static bending dipoles. Acceleration up to 10 TeV centre-of-mass energy would require 16 T static dipoles, which are only expected to become available later in the century, and an approximately 70% dipole packing fraction. A 10 TeV facility could be implemented as an upgrade to the 3 TeV facility, as discussed below. Estimates indicate that a ring circumference of up to 35 km may be required. Tuning to accommodate the beam into existing infrastructure such as the 26.7 km circumference LHC tunnel is possible, for example by changing the energy swing in each ring so that more or less space is required for pulsed dipoles. Options that have a fixed dipole field that varies radially in the same superconducting magnet are under study, which may enable an increased average dipole field to be considered.

Wall-plug power requirements

The power usage of future accelerator facilities, often referred to as the ‘wall-plug power’, is of great concern and in future may be a stronger practical constraint than the financial cost. The goal is to remain at a wall-plug power consumption for the 10 TeV MuC well below the level estimated for CLIC at 3 TeV (550 MW) or FCC-hh (560 MW). This seems readily achievable; the facility length is considerably shorter than other proposed colliders. Reuse of magnets and RF should make the facility more efficient than linear colliders while the lower energy requirement should result in lower power requirements than hh colliders. The design has to advance more to assess the power consumption scale in a robust fashion.

Muon beam production imposes a fixed power consumption requirement. In particular, the muon cooling system requires several GeV of acceleration in normal conducting cavities at high gradient. Additional power consumption arises from the proton source and cooling plant for the target and other cryogenic systems.

A number of key components drive the power consumption as one extends to high energy:

  • The power loss in the fast-ramping magnets of the pulsed synchrotron and their power converter.

  • The cryogenics system that cools the superconducting magnets in the collider ring. This depends on the efficiency of shielding the magnets from the muon decay-induced heating.

  • The cryogenics power to cool the superconducting magnets and RF cavities in the pulsed synchrotrons.

  • The power to provide the RF for accelerating cavities in the pulsed synchrotrons.

The first contribution requires particular study as it depends on unprecedented large-scale fast ramping systems. The second and third contributions require optimisation of the volume reserved for shielding as compared to magnetised volume. The contributions can be estimated reliably following a suitable design and optimisation of the relevant equipment.

Fig. 11
figure 11

A schematic of a possible staged approach to a muon collider. The first stage, shown on the left, would produce collisions at 3 TeV center-of-mass energy while the second would produce collisions at 10 TeV centre-of-mass energy. Sections of the facility that are not required are shown in grey

In summary power consumption follows a relation

$$\begin{aligned} P = P_{src} + P_{linac} + P_{rf} + P_{rcs}+P_{coll}, \end{aligned}$$
(16)

where \(P_{src}\) is the power needed for the muon source, which is constant with energy, \(P_{linac}\) is the power requirement for the linac, which is fixed by the transition energy between the linacs and RCS (Rapid Cycling Synchrotron). \(P_{rf}\) is the power requirement for RCS and collider RF cavities, which is approximately proportional to the beam energy, \(P_{rcs}\) is the power requirement for the RCS magnets which is likely to rise slowly with energy due to the lower losses associated with slower ramping at high energy and \(P_{coll}\) is the power requirement for the collider ring, which is approximately proportional to the collider length i.e. proportional to the beam energy. Overall, the power requirement is expected to rise slightly less than linearly with energy and hence the wall-plug power is expected to be approximately proportional to the beam power.

Upgrade scheme

The muon collider can be implemented as a staged concept providing a road toward higher energies. One such staging scenario is shown in Fig. 11. The accelerator chain can be expanded by an additional accelerator ring for each energy stage. A new collider ring is required for each energy stage. It may be possible to reuse the magnets and other equipment of the previous collider ring, for example in the new accelerator ring.

Currently, the focus of studies is on 10 TeV. This energy is well beyond the 3 TeV of the third and highest energy stage of CLIC, the highest energy \(\mathrm {e^+e^-}\) collider proposal to reach a mature design. A potential intermediate energy of 3 TeV is envisaged at this moment. Its physics case is similar to the final stage of CLIC and it is expected that this stage would roughly cost half as much as the 10 TeV stage [14].

A 3 TeV stage is less demanding in several technological areas. It may not be necessary to implement a mechanical neutrino flux mitigation system in the collider ring arcs; moving the beam inside of the magnet apertures may be sufficient. The final focus magnets require an aperture and a gradient comparable to the values for HL-LHC. In general, it will be possible to implement larger margins in the design at 3 TeV. The operational experience will then allow to accept smaller margins at 10 TeV.

The strength of the collider ring dipoles is crucial for determining the ring size and luminosity. The cost optimum is given by the magnet and tunnel cost; it is possible that cheaper, well established magnet technology – such as NbTi at this moment – would result in a lower cost even if the tunnel has to be longer. For fixed beam current, the luminosity is proportional to the magnet field and is one aspect of the optimisation.

The Snowmass 2021 international Collider Implementation Task Force has undertaken direct comparison of all future collider proposals and concepts in terms of luminosity, environmental impact, power consumption, R &D duration, time to construct, and cost, and its report [14] indicates that a multi-TeV MuC is potentially the most promising facility for a 10+ TeV energy range, with, among other factors, the lowest cost range. Once the cost scale of the muon collider concept is more precisely known and once the mitigation of the technical challenges are well defined, the energy staging may be reviewed taking into account the physics case and additional considerations from the site and reuse of existing equipment and infrastructure, such as the LHC tunnel, making possible a high-impact and sustainable physics programme with each upgrade manageable in terms of cost and technical feasibility.

Fig. 12
figure 12

A technically limited timeline for the muon collider R &D programme that would see a 3 TeV muon collider constructed in the 2040s

Timeline

A muon collider with a centre-of-mass energy around 3 TeV could be delivered on a time scale compatible with the end of operation of the HL-LHC. A technically limited timeline is shown in Fig. 12 and discussed in greater detail in [117]. The muon collider R &D programme will consist of the initial phase followed by the conceptual and the technical design phases. The initial phase will establish the potential of the muon collider and the required R &D programme for the subsequent phases.

The performance and cost of the facility would be established in detail. A programme of test stands and prototyping of equipment would be performed over a five-year period, including a cooling cell prototype and the possibility of beam tests in a cooling demonstrator. This programme is expected to be consistent with the development of high field solenoid and dipole magnets that could be exploited for both the final stages of cooling and the collider ring development. A technical design phase would follow in the early 2030s with a continuing programme focusing on prototyping and pre-series development before production for construction begins in the mid-2030s, to enable delivery of a 3 TeV MuC by 2045. The programme is flexible, in order to match the prioritisation and timescales defined by the next ESPPU, the Particle Physics Projects Prioritization Panel (P5) in the US and equivalent processes.

Principal technical challenges

The timeline described above is technically limited by the time required to address a number of key technical challenges.

  • The collider can potentially produce a high neutrino flux that might lead to neutron showering far from the collider. A scheme is under study to ensure that the effect is negligible.

  • Beam impurities such as products of muon decay may strike the detector causing beam-induced background. The detector and machine need to be simultaneously optimised in order to ensure that the physics reach is not limited by this effect.

  • The collider ring and the acceleration system that follows the muon cooling can limit the energy reach. These systems have not been studied for 10 TeV or higher energy. The collider ring design impacts the neutrino flux and the design of the machine-detector interface.

  • The production of a high-quality muon beam is required to achieve the desired luminosity. Optimisation and improved integration are required to achieve the performance goal, while maintaining low power consumption and cost. The source performance also impacts the high-energy design.

The technology options and mitigation measures that can address these challenges are described in some detail below. Further dedicated studies of the key technologies would provide a robust quantitative assessment of their maturity and technical risk, for example based on Technical Readiness Levels (TRL) as described in [14].

3.2 Proton driver

The MuC proton driver has similarities with existing and proposed high intensity proton facilities such as those used for neutron and neutrino production. The main parameters of the MuC proton source are listed in Table 2. The technology choices for the MuC, in particular for acceleration and bunch compression, have not yet been determined.

The main part of the proton source follows existing pulsed proton driver designs, for example similar to JPARC, Fermilab or ISIS. H\(^-\) ions are created in an ion source, accelerated through a radiofrequency quadrupole followed by a series of drift-tube linacs. Acceleration proceeds through a linear accelerator using conventional RF cavities before the ions are injected into a ring using charge-exchange injection and phase space painting. In some designs, the protons are further accelerated using a Rapid Cycling Synchrotron or Fixed Field Alternating Gradient accelerator while in others protons are accelerated to the top energy in the linac and the final ring is used only for accumulation of the protons. Uniquely for the muon collider, it is desirable to compress the protons into a very short bunch with RMS length 1–3 ns, which may require an additional ring. However, bunches with lengths of the order of 30 ns would reduce the produced muon yield only by up to 50% [118, 119]. The bunch is then extracted and transferred onto the pion production target where the short proton bunch in turn creates a short pion bunch, which is important for capture of the resultant beam.

Table 2 Typical proton source parameters. The parameters are indicative

Technical issues and required R &D

The technology choice for the MuC proton driver relies on successfully managing the heat deposition in the injection foil in the accumulator ring, limits on beam intensity due to space charge and the successful compression of the proton bunch. The MuC uses a beam that has a higher intensity at lower repetition rate than comparable machines, resulting in a space-charge dominated facility. At higher energy space charge limitations may be relaxed, but care must be taken to avoid uncontrolled stripping of the H\(^-\) beam and excessive heat deposition that can damage the foil.

In order to achieve the high current and short bunches in the presence of space charge, the MAP scheme used a high harmonic lattice with a series of extraction lines to bring multiple proton bunches onto the target, each extraction line having a different time-of-flight, enabling the required proton beam current to be brought onto the target within a short time.

The possibility to stack bunches in longitudinal space, for example using an FFA ring that has a naturally large momentum acceptance [120], may enable repetition rates lower than the baseline 5 Hz. Time-compression of a beam having larger momentum spread would be more challenging.

Bunch compression has been performed, for example at ISIS and the SPS, yielding short bunches, but not as short as those required for the muon collider. Simulations indicate that such a compression is possible [121,122,123].

In order to deliver a self-consistent design, simulations must be performed for each parameter set. Many potential host sites have existing facilities which could be reused, given appropriate consideration of the muon collider requirements.

3.3 Pion production target and active handling region

The MuC design calls for a target immersed in a magnetic field of 15–20 T where muons, pions, kaons and other secondary particles are produced [119]. Unlike more conventional horn focusing arrangements the solenoid field captures secondary particles of both signs over a wide range of momenta. The pions and kaons decay to produce muons. In order to create a beam having the smallest emittance, a very short proton bunch is required having a small spot size. Typical proton bunch sizes considered are a few mm across with lengths 1–3 ns RMS. The resultant secondary particles have a large transverse and longitudinal momentum spread, but initially the position spread is the same as the proton beam. The time spread of the beam is approximately given by the proton bunch length. Some non-relativistic secondary particles are produced which will pass through the target region more slowly than the protons, resulting in a slight bunch lengthening.

Following the target the magnetic field is tapered to values sustainable over long distances. If the field strength is tapered adiabatically along the beam line, the relativistic magnetic moment

$$\begin{aligned} I_0 = \frac{p_\perp ^2}{m_\mu qB}, \end{aligned}$$
(17)

is an invariant. As the field B is reduced, the transverse momentum also reduces so that \(I_0\) does not change. Magnetic fields conserve total momentum, so reduction in transverse momentum \(p_\perp \) must lead to an appropriate increase in longitudinal momentum. For particles that initially have small transverse momentum, the longitudinal momentum growth will be small, while for those with large transverse momentum, the growth will be large. Overall, the longitudinal momentum spread increases while the transverse momentum spread decreases. This is equivalent to an exchange from transverse emittance to longitudinal emittance. The longitudinal emittance is already large, so the relative increase is small, while the decrease in transverse emittance is beneficial and results in a large flux of muons compared to horn geometries.

The beam leaving the target area has a large longitudinal and transverse emittance. A number of undesirable particles are captured in the beam that could be lost in an uncontrolled manner further downstream. In order to handle these unwanted particles, a beam cleaning system is foreseen. A solenoid-style double chicane is used to remove particles having too high momentum. The chicane has a toroidal solenoid field which induces a vertical dispersion in the beam. The vertical dispersion is exactly cancelled by a reverse bend yielding, for particles lower than the maximum momentum, a virtually unperturbed beam. High momentum particles strike the walls of the chicane where they are collected on a dedicated collimator system.

A number of low momentum beam impurities remain in the beamline. Protons arising from spallation of the target would create an unacceptable radiation hazard if they were lost in subsequent systems. A Beryllium plug is placed immediately after the chicane to remove these low momentum protons. The mean energy loss is given by [124]

$$\begin{aligned} \frac{dE}{dx} = \frac{Z}{A}\frac{K q^2}{\beta ^2 \rho } \left[ 0.5\,\ln \left( 2 m_e \beta ^2 \gamma ^2 \frac{T_{max}}{I^2}\right) - \beta ^2\right] . \end{aligned}$$
(18)

Due to their larger mass, protons have much lower \(\beta \) and less kinetic energy compared to muons following the chicane and so stop in a shorter distance. By choosing an appropriate thickness, protons are stopped while most muons are able to pass through. A low Z material such as beryllium is considered as it causes less scattering in the muon beam for a given proton stopping power.

Alternative schemes

In addition to a long graphite target of the type used in pion production targets for neutrino beams, moving targets have been considered. The required dimensions of the target make a rotating wheel such as the one in use at PSI challenging to implement. Liquid eutectic targets or fluidised powder jets have been considered, travelling coaxially with the proton beam.

Focusing using a horn may also be considered, as used in other pion production targets. A horn target is a well-known technology but it is only capable of capturing a single sign of pion. Two targets would likely be required, leading to an increased demand on the proton driver.

Technical issues and required R &D

The target region is technically challenging. The high beam power and short proton pulse length will create a significant instantaneous shock in the target, which may lead to mechanical damage. The target will become heated by the proton beam and active cooling is expected to be required. Long term radiation damage will degrade the mechanical qualities of the target necessitating regular replacement. Calculations indicate that a graphite target, similar to the targets used at existing neutrino sources, should survive for an adequate period in the adverse conditions.

The high field in the target region, in the presence of a radiation source, is challenging to achieve. Thick shielding will be required to prevent damage to the insulation and superconductor in the coil and excessive heating of the cryogenic materials. A large aperture is required to accommodate the shielding, and this in turn poses challenges for the magnet. A magnet bore diameter in the range of 2 m would be required in case of coils built with low-temperature superconductor, and operated with liquid helium. Alternative conductor configurations, based on high-temperature superconductor with large operating margin may be operated at higher temperature with improved efficiency. This may provide means to significantly decrease the size and cost of the system. The target magnet would be in a similar class to those proposed for fusion facilities.

Handling of the spent proton beam requires significant care. The remnant beam power is expected to be beyond the capabilities of a collimation scheme to manage. A system for removing these primary protons to a beam dump must be considered. Such a system has to extract the protons without adversely affecting the pion and muon transport in the line.

Even following the removal of primary protons, significant radiation will impinge on the chicane aperture. In this region only modest fields are required but nonetheless an appropriate shielding and collimation scheme must be designed.

The Low EMittance Muon Accelerator (LEMMA) is an alternative scheme to produce a muon beam with a very small emittance [125,126,127]. An injector complex produces a high-current positron beam [128]. The positrons impact a target with an energy of 45 GeV, sufficient to produce muon pairs by annihilating with the electrons of the target. This scheme can produce small emittance muon beams. However, it is difficult to achieve a high muon beam current and hence competitive luminosity. Novel ideas are required to overcome this limitation.

3.4 Muon front end

Following the target, the muon front end system captures the beam into a train of bunches suitable for ionisation cooling. This is achieved in three stages. First the beam is allowed to drift longitudinally. Faster particles reach the subsequent RF system first. Once a suitable time-energy correlation has developed, RF cavities are placed sequentially with adiabatically increasing voltage as the beam passes along the line. Because the beam is not captured longitudinally, the RF frequency of subsequent cavities is lower so that the evolving buckets are correctly phased. This adiabatic capture results in microbunches forming in the bunch train. Once suitable microbunches have been formed, the cavities are dephased so that higher energy, early bunches, are at a decelerating phase. Lower energy, later bunches are at an accelerating phase. At the end of the capture system the resulting microbunches have the same energy, with spacings appropriate for 325 MHz RF cavities.

Following the capture system a charge selection system is required to split the beam into a positive and negative muon line. A second solenoid chicane system is envisaged. Unlike in the beam cleaning system, the bunch structure in the beam would be maintained by this system, requiring appropriate RF cavity placement.

Alternative schemes

Direct capture into 44 MHz RF buckets has been considered. These lower frequency RF systems may have a larger muon energy and time acceptance than those at few 100 MHz, despite the lower voltages which may be achieved before breakdown occurs. Use of a single RF frequency would be simpler to construct and operate than the system described above, requiring only a single RF frequency. However, such a system would yield bunches having a larger longitudinal emittance, which would need to be cooled. The efficacy of the cooling system is directly related to the voltage available, which would be lower for 44 MHz than 200–300 MHz proposed in the design described above.

Technical issues and required R &D

The front end system design itself is mature, although re-optimisation is likely to be required as the other facility parameters are developed. The charge selection system has received only preliminary conceptual work and a full design is required. Use of many RF frequencies would require a dedicated power source for each frequency, which may be costly to implement.

3.5 Muon cooling

The beam arising from the muon front end occupies a large volume in position-momentum phase space. According to Liouville’s theorem, in a non-dissipative system, phase space volume is conserved. In order to achieve a satisfactory luminosity it is necessary to reduce the phase space volume of the muon beam, a process known as beam cooling. Typical cooling techniques require time scales that are not competitive with the muon life time. Ionisation cooling, a relatively novel technique, is proposed to reduce the phase space volume of the muon beam.

Transverse cooling

In muon ionisation cooling, muons are passed through an energy-absorbing material. The transverse and longitudinal momentum of the muon beam is reduced, reducing the phase space volume occupied by the beam in the direction transverse to the beam motion. The muon beam is subsequently re-accelerated in RF cavities. The cooling effect is reduced by multiple Coulomb scattering of the muons off nuclei in the absorber, which tends to increase the transverse momentum of the beam. By using a material having low atomic number and focusing the beam tightly, the effect of scattering can be minimised.

The Muon Ionisation Cooling Experiment (MICE) demonstrated the principle of transverse ionisation cooling [129]. In MICE, the particles were passed through a solenoid focusing system. The particles were incident on various configurations of absorbers and focusing arrangements. Configurations with a lithium hydride cylinder and a liquid hydrogen-filled thin-walled vessel were compared with configurations having an empty vessel and no absorber at all. The particle position and momenta were measured upstream and downstream of the focus and the distance from the beam cores were calculated in normalised phase space coordinates. By examining the behaviour of the ensemble of particles, an increase in phase space density in the beam core could be identified when an absorber was installed, indicating ionisation cooling had taken place. No such increase was measured when no absorber was installed, as expected. The results were consistent with simulation.

Phase space volume is conveniently quantified by the beam RMS emittance, given by

$$\begin{aligned} \varepsilon = \root n \of {|{\textbf{V}}|}/m_\mu , \end{aligned}$$
(19)

where \({\textbf{V}}\) is the matrix of covariances of the phase space vector, having elements \(v_{ij} = \textrm{Cov}(u_i, u_j)\). \(\vec {u}\) is the n-dimensional phase space vector under consideration. Typically either the four-dimensional transverse vector, \((x, p_x, \)\( y, p_y)\) or the two-dimensional longitudinal vector, \((c(t-t_0), \delta )\), is considered, where x and y are the horizontal and vertical positions relative to the beam axis, \(p_x\) and \(p_y\) are the corresponding momenta, \(t-t_0\) is the time relative to some reference trajectory’s time \(t_0\) and \(\delta = E-E_0\) is the energy relative to the reference trajectory’s energy \(E_0\). This quantity is proportional to the content of an elliptical contour in phase space that is one standard deviation from the reference trajectory. In the limit that the beam is paraxial, it is a conserved quantity in the absence of dissipative forces.

Skrinsky et al. [5] first introduced the concept of ionisation cooling. Neuffer [6] derived equations that characterise the emittance change on passing through an absorber in terms of the beam energy, the normalised beam size and the properties of the energy absorbing material. The change in transverse emittance on passing through an axially symmetric absorber with radiation length \(L_R\) is

$$\begin{aligned} \frac{d\varepsilon _n}{dz} \approx \frac{1}{\beta ^2 E} \left<\frac{dE}{dz}\right>(1-\frac{g_L}{2})\varepsilon _n + \frac{(13.6 \,{\textrm{MeV}})^2}{2\,m_\mu L_R} \frac{\beta _\perp }{\beta ^3 E}. \nonumber \\ \end{aligned}$$
(20)

Longitudinal cooling, discussed below, may be achieved by arranging for a correlation between energy and energy loss and is parameterised by \(g_L\), the longitudinal partition function. In the absence of longitudinal cooling \(g_L\) may be assumed to be 0. Assuming that beam energy is continuously replaced by RF cavities the emittance change is 0 at the equilibrium emittance

$$\begin{aligned} \varepsilon _{n,eqm} \approx \frac{1}{2m_\mu } \frac{(13.6 \,{\textrm{MeV}})^2}{L_R} \frac{\beta _\perp }{\beta \left<\frac{dE}{dz}\right>(1-\frac{g_L}{2})}, \end{aligned}$$
(21)

where emittance growth due to scattering and emittance reduction due to ionisation are equal.

Two principal cooling stages are proposed for the MuC. In the first stage, known as rectilinear cooling, muons are cooled both transversely and longitudinally. In the second stage, known as final cooling, muons are cooled transversely and heated longitudinally.

Both cooling systems use solenoids with approximately cylindrically symmetric beams in order to provide the tight focus required for satisfactory cooling performance. The focusing is characterised by the beam transverse \(\beta _\perp \) function. This is the variance of the beam size, normalised to the beam emittance

$$\begin{aligned} \beta _\perp = \frac{p_z \textrm{Var}(x)}{m_\mu c\, \varepsilon _n} = \frac{p_z \textrm{Var}(y)}{m_\mu c \,\varepsilon _n}. \end{aligned}$$
(22)

In a solenoid, \(\beta _\perp \) evolves according to

$$\begin{aligned} 2 \beta _\perp \beta _\perp '' - \beta _\perp '^2 + 4 \beta _\perp ^2 \kappa ^2 - 4(1+{\mathcal {L}}^2) = 0. \end{aligned}$$
(23)

Here \({\mathcal {L}}\) is the normalised canonical angular momentum. \(\kappa ^2\) is the solenoid focusing strength, where

$$\begin{aligned} \kappa = \frac{qc B_z(r=0, z)}{2p_z}. \end{aligned}$$
(24)

Particles in the solenoid may be considered as oscillators with angular phase advance of the oscillator over a distance \(z_0\)

$$\begin{aligned} \phi = \int _0^{z_0} \frac{1}{\beta _\perp }dz. \end{aligned}$$
(25)

Oscillations are known as betatron oscillations.

Wang and Kim [130] showed that for periodic systems, solutions to Eq. (23) can be related to the Fourier coefficients \(\vartheta _n\) of \(\kappa \)

$$\begin{aligned} \beta _\perp (z=0) \approx \frac{z_0}{\pi } \frac{\sin (\sqrt{\vartheta _0}\pi )}{\sqrt{\vartheta _0}\sin \mu } \left[ 1+\sum ^{\infty }_{n=1}\frac{Re[\vartheta _n]}{n^2-\vartheta _0}\right] . \end{aligned}$$
(26)

Explicitly \(\vartheta _n\) are defined by

$$\begin{aligned} \left( \frac{L}{\pi }\right) ^2 \kappa ^2(z) = \sum ^{\infty }_{-\infty } \vartheta _n e^{i 2n\pi s/L}. \end{aligned}$$
(27)

There are values of \(\kappa \) where the solenoid focusing is unstable known as stop-bands. In these regions, particle motion follows hyperbolae in phase space and the phase advance is complex. These regions can be expressed in terms of the Fourier coefficients

$$\begin{aligned} \left| \sqrt{\vartheta _0}-n+\frac{5}{16}\left| \frac{\vartheta _n}{\vartheta _0}\right| ^2\right| <\frac{1}{2}\left| \frac{\vartheta _n}{\vartheta _0}\right| . \end{aligned}$$
(28)

Ionisation cooling systems are often characterised in terms of the phase advance and the nearby stop bands. The pass band in which the cooling cell operates can be selected for properties of acceptance and achievable \(\beta _\perp \). The pass band is determined in practice by scaling the average \(B^2_z\) and cell length in relation to the beam central \(p_z\). The properties can be tuned by adjusting the harmonic content of \(B^2_z(z)\). The rectilinear cooling system has been developed with particular attention to the resonance structure in order to minimise \(\beta _\perp \).

Longitudinal cooling

The process described above only reduces transverse emittance. In order to reach suitable luminosity, both transverse and longitudinal beam emittance must be reduced. This may be achieved by exchanging emittance from longitudinal to transverse phase space.

Emittance exchange is achieved in two steps. First a position-energy correlation is introduced into the muon beam using a dipole. Higher energy particles have a larger bending radius, so that a correlation is introduced between transverse position and energy. An appropriately arranged wedge-shaped absorber is used so that the higher energy particles pass through a thicker part of the wedge, losing more energy. In this way the energy spread is reduced and the position spread is increased; the emittance is moved from longitudinal space to transverse space.

The horizontal dispersion can be related to the beam by defining

$$\begin{aligned} D_x = \frac{1}{p}\textrm{Var}(x, E), \end{aligned}$$
(29)

with a similar relationship for vertical dispersion. The change in longitudinal emittance is

$$\begin{aligned} \frac{d\varepsilon _L}{dz} = -\frac{g_L}{\beta ^2 E} \frac{dE}{dz} \varepsilon _L + \frac{\beta _\phi }{2} \frac{d Var(E)}{dz}, \end{aligned}$$
(30)

with \(\beta _\phi \) the longitudinal Twiss parameter,

$$\begin{aligned} g_L=\frac{D_x}{\rho (x=0)} \frac{d\rho }{dx}, \end{aligned}$$
(31)

and \(\rho \) the effective line density of the absorber at different transverse positions. \(\rho \) may be adjusted by using variable density materials but more commonly wedge-shaped absorbers are assumed.

Rectilinear cooling

The rectilinear cooling channel employs solenoids with weak dipoles that yield the tight focusing and dispersion required to provide a satisfactory cooling performance. In this lattice the focusing effect of the solenoids is dominant compared to weak dipole focusing and the transverse optics approach discussed above is appropriate.

In order to create a cooling channel having a suitable performance, it is necessary to maintain a sufficient dynamic aperture (DA), so that the beam is not lost, while decreasing the emittance as quickly as possible using low \(\beta _\perp \) in order to prevent significant losses through muon decay. Typically, the requirement for large DA and small \(\beta _\perp \) are in tension. The process of tapering the cooling channel is employed where the DA and absorber \(\beta _\perp \) is successively reduced as the emittance decreases in order to provide a satisfactory cooling performance.

Two working points have been chosen, known as A-type and B-type lattices. A-type lattices operate in the first stability region, with phase advance below the \(\pi \) stop-band. These regions typically are characterised by larger dynamic aperture and larger \(\beta _\perp \) at the absorber, which is suitable for the initial part of the cooling channel.

B-type lattices operate in the second stability region. In order to increase the phase advance, stronger solenoids are required. The harmonic content of the lattice is chosen to optimise the positioning of \(\pi \) and \(2\pi \) stop bands; initially the stop bands are placed far apart, in order to maximise the momentum acceptance and transverse DA. As the transverse and longitudinal beam emittance is reduced the harmonic content of the lattice is adjusted to move the stop bands closer together in order to optimise the focusing strength. It will be further noted that \(\beta _\perp \) scales with BL. By reducing the cell length and employing stronger magnets a lower equilibrium emittance may be achieved.

In [131] a dipole field is introduced by small tilts to the focusing solenoids. [132] proposed using solenoids with additional dipole coils to achieve the same effect. The dipole field introduces a weak dispersion which, when combined with wedge-shaped absorbers, leads to longitudinal emittance reduction.

Bunch merge

Between the A-type and B-type lattices, a bunch merge system is employed. The beam emittance is sufficiently small that the train of 21 bunches created in the front end can be compressed into one single bunch which can then be further cooled. This improves the luminosity, which is inversely proportional to the number of bunches as previously shown.

Bunch merge is achieved by first rotating the bunch in time-energy space using a sequence of RF cavities having different voltage and frequency. As in the phase rotation scheme described above, the early bunch is slowed while the late bunch is accelerated. The bunches then naturally merge, yielding seven bunches.

Each of the bunches is kicked into one of seven separate beamlines using a kicker that has a rotating transverse field. Each beamline has a different length so that the bunches, when they reach the end, are coincident. The beamlines are terminated by a funnel so that each individual bunch is arranged adjacent to the other six bunches in phase space, yielding a single combined bunch having large longitudinal and transverse emittance.

Final cooling

Final cooling is achieved by means of a series of high field solenoids, in which very tight focusing may be achieved. Liquid hydrogen absorbers in the solenoids provide cooling. Particles experience many betatron oscillations in each solenoid. Deviations in momentum and transverse parameters can change the number of betatron oscillations, which causes emittance growth. To minimise this emittance growth the beam within the uniform solenoid field is matched so that \(\beta _\perp '\), and hence \(\beta _\perp ''\), is close to 0. By reference to Eq. (23)

$$\begin{aligned} \beta _\perp = \frac{\sqrt{1+{\mathcal {L}}^2}}{\kappa }, \end{aligned}$$
(32)

so the equilibrium transverse emittance that may be reached is inversely proportional to \(B_z\) and proportional to \(p_z\).

The transverse emittance that is reached in the final cooling system determines the overall luminosity of the complex. The smallest \(\beta _\perp \), and hence highest fields, must be used. In order to provide the tightest focus possible the beam momentum is decreased to approximately 70 MeV/c. As the beam momentum decreases, longitudinal emittance growth due to the natural curvature of the Bethe-Bloch relationship becomes stronger. This effect would be further enhanced by a large energy spread in the incident beam. Maintaining a large momentum acceptance is very challenging in lattices with such a large phase advance per cell. For these reasons the energy spread must be minimised.

In order to minimise the longitudinal emittance growth and maintain satisfactory transmission, the beam is lengthened and energy spread is reduced using a phase rotation system between each solenoid. A relatively weaker, higher aperture solenoid field maintains transverse containment. The beam is allowed to drift longitudinally so that faster particles move ahead of the beam and slower particles lag behind. The beam passes through accelerating electric fields where slower, later particles undergo an accelerating field and faster, earlier particles undergo a decelerating field. The phase rotation enables the energy spread to stay roughly constant and a time spread increase.

Some longitudinal emittance growth inevitably occurs in the final cooling system. Later cooling cells require low frequency RF or induction-based acceleration in order to contain the full beam.

Alternative concepts

Several alternatives to the scheme outlined above have been proposed. The rectilinear cooling scheme is itself an enhanced version of earlier ring cooler concepts. Ring coolers were rejected owing to issues surrounding injection and extraction of large emittance beams. Rings cannot take advantage of the \(\beta _\perp \) tapering. Small circumference rings require tight bending fields that significantly perturb the transverse optics and are challenging to extract and inject from. Higher circumferences are possible, but only a small number of turns may be achieved before the beam approaches equilibrium emittance, making the additional challenge of a ring geometry less appealing. Higher bending radius ‘Guggenheim’ [133] and ‘helical’ cooling schemes [134] have been considered, with concomitantly higher dispersion. The scheme presented here achieved the best performance.

A dual-sign precooler, known as an ‘HFoFo’ lattice, has also been considered. In the HFoFo lattice, positive and negative muons have dispersion that is partially aligned even in the same magnetic lattice. A wedge shaped absorber may be placed in such a lattice that cools both positively- and negatively- charged muons at the same time. Such a scheme is attractive to consider as it can cool the beam before charge separation. Charge separation is likely to be easier for lower beam emittances. Collective effects such as beam loading will be more severe in such a lattice due to the higher number of particles passing through the system and must be carefully considered.

In Parametric Resonance Ionisation Cooling (PIC), very low \(\beta _\perp \) is achieved by driving the beam near to resonances rather than using high-field solenoids. The DA would normally be poor in such a lattice, but higher order corrections may enable an enhanced DA.

Frictional cooling is of interest both for a muon collider and also for existing muon facilities. At low energies, below \(\beta \gamma \approx 0.1\), the energy loss becomes greater for higher energy particles. This leads directly to longitudinal cooling in addition to transverse cooling. Additional consideration must be made to account for losses that may occur due to \(\mu ^+\) and \(\mu ^-\) capture on electrons and nuclei respectively. A muon cooling scheme operating at these energies has been demonstrated [135].

Technical issues and R &D

Rectilinear cooling In order to maintain the tight focusing required to yield good cooling performance in the rectilinear cooling system, a compact lattice is required with large real-estate RF gradient. This results in a short lattice with large forces between adjacent solenoids and RF operating near to the break down limit while immersed in a strong solenoid field. Integration of the various components of a cooling cell, considering the necessary mechanical support and the many vacuum and cryogenic interfaces, will pose significant engineering challenges.

The operation of RF cavities in a solenoid field poses specific challenges. The solenoid field guides electrons that are emitted at one location of the cavity surface to another location on the opposing wall and leads to localised heating that can result in breakdown and cavity damage. Operation of copper cavities in 3 T field showed a maximum useable gradient of only 10 MV/m.

Three approaches to overcome this obstacle are known:

  • Use of lower-Z materials such as beryllium to limit the energy loss density.

  • The use of high-pressure hydrogen gas inside the cavity. In this case the mean free path of the electrons is limited and does not allow them to gain enough energy to ionise the gas or to produce a breakdown.

  • The use of very short RF pulses to limit the duration of the heat load in the cavity.

The first two techniques have been experimentally verified in MUCOOL with a field of about 3 T (limited by the solenoid). They yielded a gradient of 50 MV/m in a beryllium cavity under vacuum and 65 MV/m in a molybdenum cavity with hydrogen [136, 137], demonstrating no degradation in achievable field in the presence of an applied magnetic field.

Systematic studies in a new test stand are required to further develop the technologies. It will also be important to test the cavities in the actual field configuration of the cooling cell, which differs from a homogeneous longitudinal field. Finally, given the number of components required, the system optimisation will require designs that require minimal amount of material and suitable for medium scale production. Compact solenoid coil windings are hence of specific interest for this part of the complex.

Final cooling In the final muon cooling system, solenoids with the highest practical field are needed. A design based on 30 T solenoids – a value that has already been exceeded in a user facility using high-temperature superconductor – demonstrated that an emittance about a factor two above the target can be achieved [138]; it should be noted that the study aimed at this larger target. Several options to improve the emittance will be studied. Solenoids with field of 30 T are close to become commercially available, and fields above 40T are planned at several high magnetic field user facilities. This development would benefit the muon collider, and activities are directed towards conceptual design of cooling solenoids in this range of field and higher, exploring the performance limits in the operating conditions of an accelerator. Operating the cooling at lower beam energy may also make cooling to lower emittance possible; preliminary studies indicate that 30 T may be sufficient to reach the emittance target.

The use of liquid hydrogen absorbers is technically challenging in the presence of high beam currents. The instantaneous beam power is large enough to cause significant heating. The associated pressure increase may damage the absorber windows. The use of solid absorbers or hybrid absorbers may mitigate this issue.

3.6 Acceleration

The baseline design for acceleration is to use a series of linacs and recirculating linacs at sub-100 GeV energies followed by a series of pulsed synchrotrons.

Linacs and recirculating linacs

Initial acceleration is performed using a sequence of linear accelerators. The beam at the end of the final cooling system has low energy and large time spread owing to the phase rotation system discussed previously. This makes rapid acceleration challenging. A design for this acceleration region does not exist and is being pursued.

As the beam becomes relativistic a conventional linac may be used. A linac can give very high real-estate gradients enabling rapid acceleration, which is crucial in the initial stages where the muon beam is not sufficiently time-dilated.

Above a few GeV, such a linac would become very costly. Efficient use of the RF system may be made by recirculating the beam through the linac using recirculating arcs, yielding a “dogbone” shaped accelerator [139] (Fig. 9). The arcs have fixed magnetic fields. A conventional focusing solution requires a different arc for each beam momentum. FFA-type focusing may be employed so that beams having different focusing strengths are directed along the same arc. Muons of different momenta pass through quadrupole magnets, offset from the quadrupole centre so that the magnets deliver a bending field that is stronger for higher momenta particles.

Care must be taken to ensure that the muon beam remains synchronised with RF cavities, which places constraints on the length of the arcs. Negatively and positively charged beams must be injected with a half-wave phase delay relative to each other so that they are correctly phased for acceleration, and the arcs must return the bunch with a half-wave phase delay so that, when travelling in the opposite direction the beam is still accelerated. The transverse focusing must also have appropriate symmetries so that both positively and negatively charge muons are contained.

A similar system has been demonstrated in practice at CBETA [140] using an electron beam. CBETA used a racetrack-style layout, and only electrons were accelerated. In CBETA the beam was brought into a linear accelerator. A spreader magnet diverted beams of different momenta into short delay lines on each turn. The beams were subsequently recombined and recirculated through an FFA magnet system. A further spreader, delay line and recombiner was placed prior to entry back into the RF linac. The beam recirculated in this way for four turns during which the beam was accelerated and four further turns during which the beam was decelerated.

Rapid cycling synchrotrons

At higher energy a RCS becomes possible and, owing to the larger number of recirculations through each cavity, is cost effective compared to recirculating linacs. Four RCS are envisaged, accelerating to around 300, 750, 1500 and 5000 GeV respectively. Even at these energies, the muon lifetime constrains the ramp times to be a few ms or less.

In order to maintain a large average dipole field, the higher energy rings are designed with a combination of pulsed dipoles and superconducting fixed field dipoles. In this arrangement, care must be taken to ensure that the beam excursion in the fixed dipoles maintains the beam within the aperture and the path length deviation is not large enough to cause the beam to lose phase with the RF cavities.

Alternative concepts

Acceleration using Fixed Field Alternating Gradient accelerators (FFAs) is an interesting alternative to RCS. FFAs have a dipole field that increases with position, either vertically or radially, rather than with time and so do not require fast ramping, reducing the overall power consumption. As the field increases spatially, FFA magnets are naturally focusing (bending) or defocusing (reverse bending) in the horizontal plane with the opposite focusing properties in the vertical plane. In order to provide alternating gradient, a mix of bending and reverse bending magnets are required. FFAs can be designed that move the beam horizontally or vertically; a vertical orbit excursion FFA would have a path length that does not vary with energy and so is isochronous in the relativistic limit. In addition, non-linear FFA lattices that employ gradient, edge and weak focusing to achieve simultaneous control of betatron tunes and time of flight have also been studied recently. So far, however, these have not been applied to muon acceleration for a muon collider. Acceleration of electrons was demonstrated in the EMMA FFA as a scaled representation of muon acceleration [141].

FFAs often use reverse bending to enable vertical focusing, which can decrease the efficiency in their use of dipoles. Horizontal orbit excursion FFAs must have a limited orbit excursion so that the time-of-flight does not vary significantly during the acceleration cycle in order that the beam remains phased with the RF.

The beam moves across the RF cavity during acceleration, which limits the size, frequency and efficiency of the cavities. Fixed field or pulsed dispersion suppressors have been proposed to reduce the impact, but no design yet exists.

Technical issues and required R &D

The RCS must be able to ramp extremely quickly while maintaining synchronisation with the RF system. Acceleration cycles as short as a few 100 \(\upmu \)s and ramp rates O(kT/s) are envisaged, around 100 times faster than existing RCS. The design of the resistive pulsed magnets is optimized to obtain a minimum stored energy, thus reducing the power required for ramping. This still reaches values of the order of several tens of GW. Fast pulsed power supplies may be considered that can ramp on this time scale, but in order to maintain power and cost efficiency resonant circuits are required. The total large power of a single accelerator stage is expected to be divided into several independent sectors. Since resonant circuits discharges can have large uncertainties, additional active power converters may be required to guarantee the controllability of the field ramps among sectors and follow the approximately constant energy change produced by the acceleration system. Efficient energy recovery is a must, implying that magnet losses should be kept to small fraction of the stored energy. This can be achieved using thin laminations of low-hysteresis and high-resistivity magnetic alloys, whereby material characterization in the representative range of frequencies and field swings will still be needed.

Heating of the magnets arising from muon decay products must be successfully managed. This may require a modest amount of shielding in the bore of the superconducting dipoles, in turn increasing the magnet bore requirement.

3.7 Collider

The collider ring itself must be as low circumference as possible in order to maintain the highest possible luminosity. High field bending magnets are desirable. In order to deliver a collider in a timely manner, 10 T dipoles are assumed for a 3 TeV collider. 11 T dipoles are planned for the HL LHC upgrade. 16 T dipoles, under study for FCC-hh, are assumed for a 10 TeV collider. At present the strongest accelerator-style dipoles have a field of 14.5 T.

The collider ring requires a small beta-function at the collision point, resulting in significant chromaticity that needs to be compensated. A short bunch is required to reduce the “hour-glass-’ effect, previously discussed, whereby the \(\beta ^*\) of the bunch is degraded to be the average \(\beta ^*\) along the bunch at collision.

The bunch length may be exchanged with energy spread using RF cavities, respecting longitudinal emittance conservation. Additional energy spread makes focusing at the interaction point more challenging; in general higher energy particles have a longer focal length than lower energy particles in a quadrupole focusing system, a feature known as chromatic aberration. This limitation may be mitigated by using sextupole magnets that have stronger focusing on one side of the magnet than the other. By arranging dipoles near the sextupoles, such that higher energy particles are aligned with the stronger focusing region, correction of these chromatic aberrations is possible.

The minimum bunch length is limited by momentum compaction. Particles having a larger momentum may have a different path length l than particles having a lower momentum. Hence the higher energy particles have a longer time-of-flight around the ring, and this practically limits the minimum bunch length. In order to deliver sufficiently short bunches, the momentum compaction factor, dl/dp, must be almost 0. This can be achieved by careful consideration of dispersion and focusing around the collider.

Overall, a solution for 3 TeV has been developed and successfully addressed these challenges. A design of 10 TeV is one of the key ongoing efforts.

Alternative concepts

Use of combined function skew quadrupole magnets for bending has been proposed to reduce the momentum compaction factor in the ring. Such an arrangement would introduce dispersion in the vertical plane by appropriate matching of the dipole and skew-quadrupole fields, enabling significant reduction in the momentum compaction factor. Further reduction may be achieved by including higher order multipoles.

Technical issues and required R &D

Neutrino flux The decay of muons in the collider ring produces neutrinos that will exit from the ground hundreds of kilometers away from the collider. Since they are very energetic, these neutrinos have a non-negligible probability to interact in material near to the Earth’s surface producing secondary particle showers. A study is underway to ensure that this effect does not entail any noticeable addition to natural radioactivity and that the environmental impact of the muon collider is negligible, similar, for instance, to the impact from the LHC.

The flux density arising from the collider ring arcs will be reduced to a negligible level by deforming the muons trajectory, achieving a wide enough angular spread of the neutrinos. Wobbling of the muon beam within the beam pipe would be enough for 1.5 TeV muon beam energy. At 5 TeV muon beam energy, the beam line components in the arcs may have to be placed on movers to deform the ring periodically in small steps such that the muon beam direction would change over time. Studies must be performed to address the mechanical aspects of the solution and its impact on the beam operation. Similarly the flux densities arising from the straight sections at the interaction points are addressed in the study by optimising the location and orientation of the collider.

In order to predict the environmental impact and to design suitable methods for demonstrating compliance, detailed studies of the expected neutrino and secondary-particle fluxes are being performed with the FLUKA Monte Carlo particle transport code [142, 143]. The preliminary results confirm that the effective doses generated by the neutrinos are dominated by neutrons and electromagnetic showers produced in the material located close to the ground level. By contrast, the radiological impact due to activation of materials in the area of concern, such as soil, water, or air, has been found to be negligible.

Accidental beam loss will have a significant impact on the surrounding equipment but also may create a particle shower. The impact of accidental beam loss can be mitigated by placing the tunnel sufficiently deep. In this case a lost muon beam would not be able to penetrate the Earth sufficiently to escape from the surface.

Beam induced background Muon beam decay produces a significant flux of secondary and tertiary particles in the detector. The current solution to mitigate such a background flux, initially proposed by MAP, consists of two tungsten cone-shaped shields (nozzles) around the beampipe, with the origin in proximity of the interaction point. A study of the impact of the background on the detector performance has been performed and is discussed in Sect. 4.1.

Further optimisation is foreseen of the Interaction Region (IR) together with the shielding elements to reduce their dimensions, therefore increasing the detector acceptance in the forward region. A proper combined optimisation of the system of detector, shielding and IR will enable reduction of peak background increasing the detector performance in the forward region. The lessons learned in design of the 3 TeV system will serve as a starting point for the 10 TeV case.

Magnets and shielding The high-energy electrons and positrons arising from muon decay and striking the collider ring magnets can cause radiation damage and unwanted heat load. The decay electrons and positrons mostly strike the inner wall of the chamber due to their lower magnetic rigidity. Synchrotron radiation is emitted by the electrons and positrons and the resultant photons strike both sides of the aperture. The heating can be mitigated with sufficient tungsten shielding; a successful design has been developed at 3 TeV. First studies at 10 TeV indicate that the effect is comparable to 3 TeV, since the power per unit length of the particle loss remains similar.

The shielding requires a substantial aperture in the superconducting magnets. The limit for the dipole field is thus given by the maximum stress that the conductor can withstand (mechanics), and by the stored magnetic energy (quench protection), rather than by the maximum field that it can support. Novel concepts such as stress-managed coils will allow mechanical challenges to be addressed. Demonstration of such concepts is hence crucial to the feasibility of the collider magnets. Alternative schemes are considered, based on high-temperature superconductors, exploiting the same compact winding features planned to be developed for the high- and ultra-high-field cooling solenoids. Such development would benefit from similar activities in the field of superconducting motors and generators, whose pole windings have similar geometry, albeit at reduced dimension.

3.8 Technical demonstrators

Demonstrations are required both for the muon source and the high energy complex.

  • The compact nature of the muon cooling system, high gradients and solenoids of relatively high field poses some unique challenges that require demonstration.

  • The high-power target presents a number of challenges that should be evaluated using irradiation facilities or single impact beam tests.

  • The issues in the high energy complex arise from the muon lifetime. Fast acceleration systems and appropriate handling of decay products result in unique challenges for the equipment.

The following new facilities are foreseen.

Fig. 13
figure 13

Schematic of the muon cooling demonstrator

Ionisation cooling demonstrator MICE has delivered the seminal demonstration of transverse ionisation cooling. In order to prove the concept for a muon collider further tests must be performed to demonstrate the 6D cooling principle at low emittance and including re-acceleration through several cooling cells.

A schematic of a Muon Cooling Demonstrator is shown in Fig. 13. A proton beam strikes a target creating pions. Pions with momenta 100–300 MeV/c are brought into a beam preparation line where RF cavities are used to develop a pulsed beam and muons having large transverse emittance are scraped from the beam using collimators. The resultant high emittance beam is brought back onto the beamline and transported through a high-fidelity instrumentation system before passing through a number of cooling cells. Finally the beam is delivered onto a downstream instrumentation system.

Many of the challenges in delivering such a facility are associated with integration issues of the magnets, absorbers and RF cavities. For example, operation of normal conducting cavities near to superconducting magnets may compromise the cryogenic performance of the magnet. Installation of absorbers, particularly using liquid hydrogen, may be challenging in such compact assemblies. In order to understand and mitigate the associated risks, an offline prototype cooling system will be required. Such a system will require an assembly and testing area, with access to RF power and support services. This could be integrated with the demonstrator facility, which will need an area for staging and offline testing of equipment prior to installation on the beamline.

The possibility to perform intensity studies with a muon beam are limited. In the first instance such effects will be studied using simulation tools. If such studies reveal potential technical issues, beam studies in the presence of a high intensity source will be necessary, for example using a proton beam.

Ionisation cooling RF development The cooling systems require normal-conducting RF cavities that can operate with high gradient in strong magnetic fields without breakdown. No satisfactory theory exists to model the breakdown. Considerable effort was made by MAP to develop high-gradient RF cavities. Two test cavities have been developed. The first cavity was filled with gas at very high pressure. The second cavity used beryllium walls. Both tests presented promising results. Operation of normal-conducting RF cavities at liquid nitrogen temperature has been demonstrated to reduce multipacting. In order to test the concepts above and others further, a dedicated test facility is required. An RF source having high peak power at the appropriate frequency and a large aperture solenoid that can house the RF cavity will be needed. No such facility exists at present.

Cooling magnet tests In order to improve the rectilinear cooling channel performance, high field magnets are required with opposing-polarity coils very close together. The possibility to implement high-field magnets (including those based on HTS) needs to be investigated, with appropriate design studies leading to the construction of high-field solenoid magnets having fields in the range 20 T to 25 T. Very high field magnets are required for the final cooling system. In this system, the ultimate transverse emittance is reached using focusing in the highest-field magnets. As a first step, a 30 T magnet, corresponding to the MAP baseline, would be designed and constructed. Feasibility studies towards a 50 T magnet would also be desirable, which may include material electro-mechanical characterisation at very high field as well as technology demonstration at reduced scale. These very demanding magnets are envisaged to be developed separately to the cooling demonstrator. Eventually they could be tested in beam if it was felt to be a valuable addition to the programme. In order to support this magnet R &D, appropriate facilities will be required. Testing of conductors requires a suitable test installation, comprising high field magnets, variable temperature cryogenics and high-current power supplies. Magnet development and test will also require these facilities in addition to access to appropriate coil and magnet manufacturing capabilities.

Acceleration RCS magnets Acceleration within the short muon lifetime is rather demanding. The baseline calls for magnets that can cycle through several T on a time scale of a few ms. A resonant circuit is the best suited solution to power the magnets for good energy storage efficiency. The design of the magnet and powering system will be highly integrated, and work on scaled prototypes is anticipated. Superconducting RCS magnets may offer higher field reach than normal-conducting magnets, but are challenging to realise owing to heating arising from energy dissipation in the conductor during cycling (AC loss). This heating can lead to demands on the cryogenic systems that outweigh the benefits over normal-conducting magnets. Recent prototypes have been developed using HTS that can operate at higher temperatures, and in configurations leading to lower AC losses, yielding improved performance [144, 145]. In order to continue this research, magnet tests with rapid pulsed power supplies and cryogenic infrastructure will be required.

Effects of radiation in material The high beam power incident on the target and its surroundings is very demanding. Practical experience from existing facilities coupled with numerical studies indicate that there will be challenges in terms of target temperature and lifetime. Instantaneous shock load on the target will also be significant. Tests are foreseen to study behaviour of target material under beam in this instance. Tests are desirable both for instantaneous shock load and target lifetime studies. Additionally, the effect of radiation in the target region on the superconducting materials (LTS and HTS) and insulators is an important parameter. Additional studies may be required taking into account the magnet arrangement, conductor design and estimates of radiation levels. In order to realise such tests, facilities having both instantaneous power and integrated protons on target equivalent to the proton beam parameters assumed for this study are desirable. The database of radiation effects on superconductors (HTS) and insulators also requires an extension to cover the projected conditions in the target area.

Superconducting RF cavities Development of efficient superconducting RF with large accelerating gradient is essential for the high energy complex. Initially, the work will focus on cavity design; however eventually a high gradient prototype at an appropriate frequency will be required. In order to realise such a device, appropriate superconducting cavity production and test facilities will be required including surface preparation techniques and a capability for high power tests.

3.9 Start-to-end facility simulations

As part of the design process for the muon collider, detailed studies of the various subsystems would be followed by an assessment of the overall performance of the whole facility. Individual components will be modelled to understand the impact of misalignments and imperfections. Such simulations would help to build a comprehensive understanding of the effects induced by deviations from the ideal running conditions. These tolerance specifications would inform and guide the technical R &D programme.

The assessment may be refined by modelling the whole system in a start-to-end simulations, from the proton driver to the collider ring. This could be performed using a set of codes and a systematic procedure to transfer the results from one code to another including data format conversions and coordinate system transformations.

3.10 Synergies with other concepts or existing facilities

The ambitious programme of R &D necessary to deliver the muon collider has the potential to enhance the science that can be done at other muon-beam facilities. The progress in other accelerator facilities will also benefit the design and construction of the muon collider in the future.

nuSTORM [146] and ENUBET [147] offer world-leading precision in the measurement of neutrino cross sections and exquisite sensitivity to sterile neutrinos and physics beyond the Standard Model. nuSTORM in particular will require capture and storage of a high-power pion and muon beam and management of the resultant radiation near to superconducting magnets. In nuSTORM, multi-GeV pions are brought from a target and injected into a racetrack-shaped storage ring. The storage ring is tuned to capture muons arising from pions that decay in the first straight. Remnant pions are extracted to a beam dump, while the muons are circulated many times. The apparatus would deliver a ‘flash’ of neutrinos from the initial pion decay followed by a well-characterised neutrino beam arising from the decay of the circulating muon beam. The momentum of the circulating muon beam, and hence the resultant neutrinos, would be tunable, enabling characterisation of neutrino interactions with the detector over a broad range of momenta.

The muon rate and energy for nuSTORM as compared to the muon collider and other muon beamlines is shown in Fig. 14. The target and capture system for nuSTORM and ENUBET may also provide a testing ground for the technologies required at the muon collider and as a possible source of beams for the essential 6D cooling-demonstration experiment, for example as in the schematic shown in Fig. 15.

Fig. 14
figure 14

Muon energy and rate of different muon facilities

Fig. 15
figure 15

Schematic showing nuSTORM including the muon cooling demonstrator for the muon collider

The ongoing LBNF and T2HK projects and their future upgrades will develop graphite targets to sustain the bombardment of MW-level proton beams, which may also lead to a solution for the muon production target for the muon collider. The next generation searches for charged lepton flavour violation exploit high-power proton beams impinging on a solid target placed within a high-field solenoid, such as COMET at J-PARC and Mu2e at FNAL. The technological issues of target and muon capture for these experiments are similar to those present in the muon collider design.

The potential to deliver high quality muon beams could enhance the capabilities of muon sources such as those at PSI, J-PARC and ISIS. The use of frictional cooling to deliver ultra-cold positive and negative muon beams is under study at PSI and may be applicable to the muon collider.

High-power proton accelerators are in use throughout the world, accelerating protons using linacs and accumulation in fixed energy rings or prior to further acceleration using rapid cycling synchrotrons. Proton drivers having power ranging from hundreds of kW to multiple MW are used as spallation neutron sources at SNS, J-PARC, ESS, PSI, ISIS and CSNS and neutrino sources at FNAL and J-PARC proton accelerator complexes as well accelerator-driven systems such as CiADS and MYRRHA. Many of the accelerator technologies required for the muon collider proton beam and for rapid acceleration are in use or under development at these facilities. For example FFAs have been proposed as a route to attain high proton beam power for secondary particle sources such as neutron spallation sources, owing to the potential for high repetition rate and lower wall plug power compared to other accelerator schemes.

The underlying technologies required for the muon collider are also of interest in many scientific fields. The delivery of high field solenoid magnets is of great interest to fields as wide ranging as particle physics, accelerator science and imaging technology. Operation of RF cavities with high gradient is of interest to the accelerator community.

3.11 Outlook

The muon collider presents enormous potential for fundamental physics at the energy frontier. Previous studies have demonstrated feasibility of many critical components of the facility. Several proof-of-principle experiments and component tests like MICE, CBETA, EMMA and the MUCOOL programme, have been carried out to practically demonstrate the underlying technologies. Bright muon beams are also the basis of the nuSTORM facility. This experiment could share a large part of the complex with a cooling demonstrator.

The muon collider is a novel concept and is not as mature as the other high-energy lepton collider options. However, it promises a unique opportunity to deliver physics reach at the energy frontier on a cost, power consumption and time scale that might improve significantly on other energy-frontier colliders. At this stage, building upon significant prior work, no insurmountable technological issues were identified. Therefore a development path can address the major challenges and deliver a 3 TeV muon collider by 2045.

A global assessment has identified the R &D effort that is essential to address these challenges before the next regional strategy processes to a level that allows estimation of the performance, cost and power consumption with adequate certainty. Execution of this R &D is required in order to maintain the timescale described in this document. Ongoing developments in underlying technologies will be exploited as they arise in order to ensure the best possible performance. This R &D effort will allow future strategy processes to make fully informed recommendations. Based on the subsequent decisions, a significant ramp-up of resources could be made to accomplish construction by 2045 and exploit the enormous potential of the muon collider.

4 Particle detectors and event reconstruction

The unstable nature of muons makes the beam-induced background (BIB) a much more challenging issue at a muon collider than it is at facilities that use stable-particle beams. For instance, with \(2.2 \cdot 10^{12}\) muons per bunch a \({1.5}\,\textrm{TeV}\) muon beam leads to about \(2 \cdot 10^5\) muon decays per meter in a 3 TeV MuC with parameters as in Table 1. The interactions of the decay products with the accelerator lattice produce even larger amounts of particles that eventually reach the detector, making the reconstruction of clean \(\mu ^+ \mu ^-\) collision events nearly impossible without a dedicated BIB mitigation.

Muon collider detectors and event reconstruction techniques therefore need to be designed specifically to cope with the presence of the continuous flux of secondary and tertiary particles from the BIB. This section reviews the state-of-the-art design studies, and it is organised as follows. In Sect. 4.1 we describe the muon collision environment based on simulations of the BIB fluxes and composition reaching the detector. A tentative detector model is employed. The software setup used for the detector response simulation is described in Sect. 4.2. Section 4.3 presents promising technologies that could be employed in the tracking detector, the calorimeter systems, and dedicated muon spectrometers. General considerations regarding trigger systems and data acquisition are also discussed. Section 4.4 describes the status of development of the reconstruction algorithms and their expected performance for the basic objects needed to carry out a comprehensive physics programme. The reconstruction of other objects, as \(\tau \) leptons or missing momentum, is still in progress, but it is expected to pose challenges similar to the ones that have been already solved. A discussion of the special challenges and opportunities for progress that are posed by the forward region of the detector is reported in Sect. 4.5. The summary and conclusions are presented in Sect. 4.6.

4.1 Collision environment

The BIB creates a large particle flux that interacts with the detector elements. On top of a detector model, its detailed simulation would require the design of the machine interaction region and of the Machine-Detector Interface (MDI). In fact, the BIB emerges from a chain of interactions with the material that composes these elements, entailing a strong dependence on their configuration of the BIB composition, flux, and energy spectra [58, 148,149,150]. The interaction region and MDI design also offers opportunities for the mitigation of the level of BIB that reaches the detector.

Since no final design of these elements nor of the detector is available, and even the conceptual design of the collider facility is ongoing, current studies are based on tentative configurations, described below.

Detector model

The design of a dedicated experiment is still in its preliminary phase, but some general conclusions can be already drawn. Given the breadth of the expected physics programme, a hermetic detector with angular coverage as close as possible to \(4\pi \) is required. The detector will feature a cylindrical layout and will include: an inner tracking detector immersed in a magnetic field; a set of calorimeter systems designed to fully contain the products of the muon collisions; and an external muon spectrometer.

The tentative detector model we consider, referred to as Muon Collider Detector (MCD), is based on the CLICdet concept [151,152,153,154]. The innermost system consists of a full-silicon tracking detector divided in three sub-detectors: the Vertex Detector, the Inner and the Outer Tracker. The tracking detector is surrounded by a calorimeter system that consists of an electromagnetic calorimeter (ECAL) and a hadronic calorimeter (HCAL), and is immersed in a magnetic field of 3.57 T provided by a solenoid with an inner bore of 3.5 m. Finally, the outermost part of the detector features a magnet iron yoke designed to contain the return flux of the magnetic field and is instrumented with muon chambers. The full detector is shown in Fig. 16. The most relevant modifications to the CLICdp detector are in the tracker topology. They are introduced for the installation of two double-cone shielding absorbers made of tungsten with a borated polyethylene (BCH2) coating and having an opening angle of \(10^{\circ }\), referred to as “nozzles”. The nozzles are located inside the detector in the forward regionsFootnote 2 along the beam axis in the region between 6 and 600 cm away from the Interaction Point (IP), as displayed in Fig. 17.

Fig. 16
figure 16

Illustration of the full detector, from the Geant4 model. Different colours represent different sub-detector systems: the innermost region, highlighted in the yellow shade, represents the tracking detectors. The green and red elements represent the calorimeter system, while the blue outermost shell represents the magnet return yoke instrumented with muon chambers. The space between the calorimeters and the return yoke is occupied by a 3.57 T solenoid magnet

Fig. 17
figure 17

Cross-sectional view of the MDI as designed by the MAP collaboration for a \(\sqrt{s} = \)1.5 TeV MuC and visualised with FLUKA. Distinct colours represent different materials of the MDI: tungsten (green), borated polyethylene (dark magenta), iron (dark yellow), and concrete (gray). The black box in the center encloses the detector volume, which is excluded from the standalone BIB simulation process. Dimensions are reported in centimeters

The installation of the nozzles was proposed by the MAP collaboration [148] in order to mitigate the BIB effects. These nozzles, assisted by the magnetic field induced by a solenoidal magnet encasing the innermost detector region, could trap most of the electrons arising from muon decays close to the IP, as well as most of incoherent \(e^{+}e^{-}\) pairs generated at the IP. With this sophisticated shielding in the MDI region, a total BIB reduction of more than three orders of magnitude was obtained [148]. The exact shape and positioning of the nozzles, including the \({10}^\circ \) opening angle and \({12}\,\textrm{cm}\) distance between the tips, was optimised specifically for the MAP design of a MuC with \(\sqrt{s} =~{1.5}\,\textrm{TeV}\) energy in the centre of mass. Their re-optimisation will be an important component of future work on the design of the MDI for the 3 and 10 TeV colliders.

Table 3 Multiplicities of different types of particles after the shielding structure, therefore arriving on the detector surface. A single bunch crossing with \(2\cdot 10^{12}\) muons is considered. In all cases, the MAP 1.5 TeV collider design and optimised MDI is assumed

Characterisation of BIB

Detailed BIB simulations were first performed in the context of the MAP studies, employing the MARS15 [155] Monte Carlo software. These are based on the accelerator lattice and interaction regions designed by MAP for a \({1.5}\,\textrm{TeV}\) MuC. The previously-mentioned optimised MDI design was based on these simulations. A Higgs-pole muon collider with \({62.5}\,\textrm{GeV}\) energy beams was also considered by MAP.

The BIB simulation for the \({1.5}\,\textrm{TeV}\) collider have been repeated in [156], using the Monte Carlo multi-particle transport code FLUKA [142, 143]. The complex FLUKA geometry was assembled by means of the LineBuilder program [157] using the optics file provided by the MAP collaboration. The accelerator elements have been defined in the FLUKA Elements Database following the information contained in this file and in MAP publications [158, 159]. The particles induced by the muon decays are collected at the outer surface of the MDI and before entering the detector volume, which is represented by a black box on Fig. 17. This will allow to later simulate their interaction with the detector together with particles from the \(\mu ^+\mu ^-\) collision. The “BIB sample” that we describe here thus refers to the collection of particles originating from the muon decays before any interaction with the detector material.

The results obtained by FLUKA are in good agreement with the ones from MAP as shown in Table 3 and discussed in Ref. [156] in more detail. The FLUKA simulation is then repeated with the same setup for the higher energy beams of the 3 and 10 TeV MuC [160]. This corresponds to assuming that the interaction region and the MDI are the same at all energies, which is not fully realistic but sufficient for a first assessment of the BIB levels dependence on the collider energy. Furthermore, the findings of [156] confirm the major role that is played by the nozzles in determining the particles fluxes that arrive on the detector surface. Their optimisation for the 3 and 10 TeV MuC could thus reduce the estimated BIB levels strongly. Studies for the 10 TeV collider option showed that lattice design choices such as combined function magnets in the final focus region or a larger \(L^{*}\) provide instead only a limited potential for reducing the BIB [160].

Table 3 (see also [156]) displays a moderate dependence of the BIB multiplicities on the collider energy. In what follows we will thus employ \({1.5}\,\textrm{TeV}\) BIB simulation results, being confident that no dramatic changes are expected at higher energies. The results below are obtained for a single beam travelling counterclockwise starting \({200}\,\textrm{m}\) away from the IP. The other beam will have the mirrored effect owing to the symmetric nature of the BIB due to \(\mu ^+\) and \(\mu ^-\) decays.

The most important BIB property is that it is composed of a large number of particles with low energy, thanks to the MDI mitigation action, and it is characterised by a broad arrival time in the detector. More specifically, around \(4 \cdot 10^{8}\) low-momentum particles exit the MDI in a single bunch crossing depositing energy to the detector in a diffused manner. There is a substantial spread in the arrival time of the BIB particles with respect to the bunch crossing, ranging from a few nanoseconds for electrons and photons to microseconds for neutrons, due to their smaller velocity.

Each of these aspects has different implications for the BIB signatures in different parts of the detector, which depend on the position, spatial granularity and timing capabilities of the corresponding sensitive elements. Thus, a careful choice of detector technologies and reconstruction techniques allows to mitigate the negative BIB effects, as demonstrated in later sections.

The time at which the BIB particles exit the machine in the interaction region is spread over several tens of ns, but the major concentration is around the beam crossing time (\(t = 0\)), as shown by the left panel of Fig. 18. This distribution suggests that the use of time-sensitive detectors would allow to suppress a large fraction of the background. The right panel of Fig. 18 reports the longitudinal distribution of primary \(\mu ^-\) decays generating the most relevant BIB components: neutrons, photons and electrons/positrons. Simulations show that to correctly account for the secondary \(\mu ^\pm \), it is necessary to consider primary decays up \(\sim {100}\,\textrm{m}\) from the IP.

Fig. 18
figure 18

Time distribution of BIB particles exiting the machine (left) and longitudinal distribution of primary \(\mu ^-\) decay generating BIB particles exiting the machine (right). The results are based on the FLUKA simulation, considering the primary \(\mu ^-\) decays within \({100}\,\textrm{m}\) from the IP

Fig. 19
figure 19

Lethargy plot (left) and longitudinal exit coordinate distribution (right) of BIB particles, by particle type. No time cut is applied to distributions represented in dotted lines while in solid lines only particles exiting the machine between − 1 and \({15}\,\textrm{ns}\) are considered. The results are based on the FLUKA simulation, considering primary \(\mu ^-\) within \({100}\,\textrm{m}\) from the IP

The kinetic energy distribution of most relevant BIB particle types is reported in Fig. 19. Energy cutoffs have been applied in the simulation at 100 keV for \(\gamma \), \(e^\pm \), \(\mu ^\pm \), charged hadrons and at 10\(^{-14}\) GeV for neutrons. The shielding nozzles strongly suppress the high energy BIB component, making the fraction of particles entering the detector volume with kinetic energy above few GeVs negligible. Only charged hadrons and secondary muons can reach higher energies, but their rate is of the order of 10\(^{4}\) and 10\(^{3}\), with respect to 10\(^{7}\) photons, neutrons and 10\(^{5}\) electrons, positrons. The longitudinal exit coordinate distribution displayed in the figure shows that most BIB particles enter the detector with a large longitudinal displacement from the collision region. This suggest that detectors with excellent pointing capabilities would allow to strongly suppress these background contributions.

Fig. 20
figure 20

Map of the 1-MeV-neq fluence in the detector region, shown as a function of the position along the beam axis and the radius. The map is normalised to 1 year of operation (200 days/year) and a collision rate of 100 kHz

Fig. 21
figure 21

Map of the TID in the detector region, shown as a function of the position along the beam axis and the radius. The map is normalised to 1 year of operation (200 days/year) and a collision rate of 100 kHz

Radiation levels

The BIB distributions and rates are crucial to quantify the radiation levels and in turn the requirements on the detector components. The FLUKA BIB sample and the detector model previously described are employed. The simulation [156] used in fact a simplified detector geometry. The calorimeters, magnetic coils, and the return yoke were approximated with cylindrical elements with densities and material composition based on the averages from the full geometry. The magnetic field was assumed to be uniform. The silicon layers composing the inner tracker were instead included with exact dimensions.

Figures 20 and 21 display respectively the expected 1 MeV neutron equivalent fluence (1-MeV-neq) and the total ionising dose (TID) in the detector region, shown as a function of the beam axis z and the radial distance r from the beam axis. The normalisation for the dose maps is computed considering that the muon collisions are expected to happen at the maximum rate of 100 kHz, corresponding to the minimum time between crossings of 10 \(\upmu \)s. With a single bunch collider operation scheme, this in turn corresponds to a minimal collider ring length of 2.5 km. Assuming 200 days of operation during a year, the 1-MeV-neq fluence is expected to be \(\sim 10^{14{-}15}\) cm\(^{-2}\)y\(^{-1}\) in the region of the tracking detector and of \(\sim 10^{14}\) cm\(^{-2}\)y\(^{-1}\) in the electromagnetic calorimeter, with a steeply decreasing radial dependence beyond it. The total ionising dose is \(\sim 10^{-3}\) Grad/y on the tracking system and \(\sim 10^{-4}\) Grad/y on the electromagnetic calorimeter.

4.2 Detector simulation software

The full simulation of a \(\mu ^+ \mu ^-\) collision event involves several stages going from the generation of input particles, the simulation of their interactions with the detector material and of the detector response.

The first stage corresponds to the generation of all particles entering the detector. This stage is handled by standalone software, such as FLUKA or MARS15 for the BIB particles as previously described and Monte Carlo event generators for the \(\mu ^+ \mu ^-\) scattering process.

The input particles are then propagated through the detector material and their interactions with the passive and sensitive material of the detector are simulated with the Geant4 [161] software. The iLCSoft framework [162], previously used by CLIC [163] and now forked for developments of muon collider studies [164], is used for this and all further processing stages.

The detector response and event reconstruction are handled inside the modular Marlin framework [165]. The detector geometry is defined using the DD4hep detector description toolkit [166], which provides a consistent interface with both the Geant4 and Marlin environments. The response of each sensitive detector element to the corresponding energy deposits returned by Geant4 is simulated by dedicated digitisation modules implemented as individual Marlin processors.

The tracking detectors use Gaussian smearing functions to account for the spatial and time resolutions of the hits registered on the sensor surface. Acceptance time intervals, individually configured for each detector, are used for replicating the finite readout time windows in the electronics of a real detector and to reject hits from from out-of-time BIB particles.

The result of this simplified approach is a one-to-one correspondence between the Geant4 hits and digitised hits, which ignores the effect of charge distribution across larger area due to the Lorentz drift and shallow crossing angles with respect to the sensor surface. These effects are taken into account in the more realistic tracker digitisation software that is currently under development and will allow stronger BIB suppression based on cluster-shape analysis.

The ECAL and HCAL detectors are digitised using realistic segmentation of sensitive layers into cells by summing all energy deposits in a single cell over the configured integration time of \({\pm 250}\, \textrm{ps}\). The time of the earliest energy deposit is consequently assigned to the whole digitised hit. The same digitisation approach is used also for the Muon Detector.

More details about the software structure and computational optimisation methods used for simulating the very large number of BIB particles are given in Ref. [167]. For the interested user, pointers to the documentation of the software stack, tutorials and other tools are available at the MuC software project page [164].

4.3 Detector technologies

The simulation workflow described in the previous sections enables a first assessment of the challenges for the various detector systems and of the required technologies, which are described in the present section.

Tracking systems

The abi