Phenomenology of muonium conversion

To determine the experimental observables related to oscillations, we recall that the treatment of the two-level system that represents muonium and antimuonium is similar to that of meson-antimeson oscillations [44-51]. However, there are several important differences. First, both the ortho- and para-muonium oscillate. Second, the SM oscillation probability is small, as it is related to the function of the neutrino masses. Any experimental indication of oscillation represents a sign of new physics.

In the presence of the interactions coupling M and , the time development of a muonium and anti-muonium states would be coupled, so it would be appropriate to consider their combined evolution,(2)The time evolution of evolution is governed by a Schrödinger-like equation,(3)where is a 2×2 Hamiltonian (mass matrix) with nonzero off-diagonal terms originating from the interactions. CPT invariance dictates that the masses and widths of the muonium and anti-muonium are the same, and thus, , . In what follows, we assume CP invariance of the interaction. A more general formalism without this assumption follows the same steps as those for or mixing [49, 50]. Then,(4)The off-diagonal matrix elements in Sect. 2.1 are related to the matrix elements of the effective operators, as discussed in [49, 50]. The related Lagrangian expressed in the effective operators can be found in Sect. 2.2.(5)where is the muonium mass and mi are the masses of the physical mass eigenstates as discussed below.

To determine the propagating states, the mass matrix must be diagonalized. The basis in which the mass matrix is diagonal is represented by the mass eigenstates , which are related to the flavor eigenstates M and as(6)where we employed a convention where . The mass and the width differences of the mass eigenstates are(7)where, mi (Γ_i) is the mass (width) of the physical mass eigenstates . It is interesting to see how Eq. (5) defines mass and lifetime differences. Because the first term in Eq. (5) is defined by a local operator, its matrix element does not develop an absorptive part, thus contributing to M₁₂, that is, the mass difference. The second term contains the bi-local contributions connected by physical intermediate states. This term has both real and imaginary parts and thus contributes to both M₁₂ and Γ₁₂.

It is often convenient to introduce dimensionless quantities,(8)where the average width . Note that Γ is defined by the standard model decay rate of the muon, and x and y are driven by the lepton-flavor violating interactions; we should expect that both .

The time evolution of flavor eigenstates follows from Eq. (3) [10, 48, 49],(9)where the coefficients are defined as(10)As we can expand Eq. (10) in power series in x and y to obtain(11)The most natural way to detect oscillations experimentally is to produce M state and look for the decay products of the CP-conjugated state . Denoting an amplitude for the M decay into a final state ‘f’ as and an amplitude for its decay into a CP-conjugated final state as , we can write the time-dependent decay rate of M into the ,(12)where N_f is a phase-space factor and we defined the oscillation rate as(13)Integrating over time and normalizing to we get the probability of M decaying as at some time t > 0,(14)Equation(14) generalizes the oscillation probability found in previous studies [10, 38, 40] by allowing for a nonzero lifetime difference in oscillations.

Muonium conversion and new physics beyond the Standard Model

The Standard Model Effective Field Theory (SMEFT) serves as a powerful framework for providing a model-independent characterization of new physics beyond the Standard Model [9, 12]. If we believe that the Standard Model acts as a low-energy approximation of a complete theory, the Standard Model Lagrangian can be interpreted as the leading term in a series expansion of the complete theory. Higher-order terms in this series expansion are effective forms of new physics, beyond the Standard Model. The Standard Model operators have a mass dimension of four, whereas higher-dimensional operators, which are scaled by inverse powers of the unknown new physics scale (NP scale), are incorporated into the SMEFT Lagrangian. Thus, the SMEFT Lagrangian can be expressed as a series expansion in terms of the inverse power of NP scale,(15)The leading term in the series is the Standard Model Lagrangian denoted as . In higher-order terms, each dimension-n effective operator is associated with a Wilson coefficient and scaled by the new physics scale Λ. Qualitatively, these higher-order terms encapsulate the potential new physics models through effective operators, whereas the Wilson coefficients represent the coupling strengths. The new physics scale Λ can be interpreted as a characteristic energy scale in the new physics, such as a combination of quantities with dimensions of mass involved in interactions beyond the Standard Model. Both the NP scale and Wilson coefficients will ultimately manifest in observables, allowing experiments to constrain the new physics scale or coupling strengths in a model-independent manner.

Dimension-6 operators are the lowest-order operators, where ΔL_μ=2 muonium-to-antimuonium conversion and other cLFV processes can be introduced at the tree level using a single SMEFT vertex [10, 52-54]. There are five independent dimension-6 four-fermion operators that leads to muonium-to-antimuonium conversion [10, 11, 25, 34],(16)These five operators contribute to the mass difference, as defined in Sect. 2.1. The width difference can arise from the following four-fermion operators,(17)However, the contribution of the width difference to the conversion probability is suppressed by a factor of [10, 34], which is orders of magnitude smaller than the contributions from the mass difference as long as and are not too large. Therefore, we only considered the mass difference contributions in the following calculations.

The effective Lagrangian for muonium-to-antimuonium conversion can be written as(18)The corresponding effective Hamiltonian in Eq. (5) is , which generates a muonium-to-antimuonium conversion, as shown in Fig. 1 [55].

Fig. 1Full-screen View

The SMEFT tree-level diagram for muonium-to-antimuonium conversion with one ΔL_μ=2 four-fermion effective vertex. The conversion probability is proportional to

According to the SMEFT Lagrangian, we calculated the amplitudes for muonium-to-antimuonium conversion. In the experiments, muonium atoms were produced in two spin states: spin-zero singlet states (para-muonium, denoted as M_P) and spin-one triplet states (ortho-muonium, denoted as , where ). The conversion probability is expressed in terms of dimensionless mass differences, that is, x in Sect. 2.1, under the assumption that the width differences are negligible. Each spin state has its own corresponding value of x. They are given by [25, 26](19)where(20)Here, is the reduced mass of muonium, α is the fine-structure constant, and Γ is the muonium width.

It is important to note that a non-zero external magnetic field will suppress the conversion probability [25, 56-58]. Since the decay kinematics of different spin states are indistinguishable in experiments like MACE, the total conversion probability is expressed as a weighted sum over all spin states, according to their populations,(21)where fP and denote the populations of para- and ortho-muonium states, respectively. PB denotes the conversion probability in an external magnetic field of magnitude B.

The conversion probabilities for different spin states exhibit distinct magnetic-field dependencies. For the m = ± 1 ortho-muonium states, the external magnetic field induces an energy splitting ΔEB, which suppresses their conversion probability(22)where [25]. The term suppresses the conversion probability by in a 0.1 T magnetic field. Consequently, contributions from the m = ± 1 states are quenched in the MACE.

In contrast, m=0 states are free from energy splitting, so the conversion probabilities are still sizeable in a practical external magnetic field. Instead, the m=0 para- and ortho-muonium states are mixed due to the magnetic field, and the dimensionless mass differences are written as(23)where [25]. Here, and represent the dimensionless mass differences in an external magnetic field of magnitude B.

The total magnetic-field-dependent conversion probability reads(24)If there exists some sizable magnetic field (typically larger than the geomagnetic field) and contributions from states are negligible as in MACE’s case, the total conversion probability can be simplified as(25)We can now estimate the new physics scale that MACE can probe in a B=0.1 T magnetic field by assuming f_P = 0.32, , and taking all Wilson coefficients to be 1. This leads to the result(26)If the experimental upper limit is set to , we conclude that MACE can probe new physics at a scale of 10–100 TeV.

Since processes are distinct from physics and they are not directly comparable to muonium-to-antimuonium conversion, it is essential to figure out which processes are sensitive to physics. They should involve fermions from two different generations and violate the lepton flavor in two units. An incomplete list of candidate processes for the experimental searches is provided in Table 1 processes. The current most stringent limit is set by the MACS experiment [8], whereas most of the other processes remain unconstrained. These unconstrained processes will likely be targeted for experimental searches or measurements in the coming decades.

The muonium-to-antimuonium conversion generating four-fermion operators can also lead to cLFV scattering processes and , which can be investigated in future lepton colliders. Recently, a muon collider called μTRISTAN was proposed [52]. The conceptual design utilizes an intensive muonium source produced from a multi-layer silica aerogel target to provide an ultra-cold muon source by ionizing muonium atoms. A collider-quality muon beam was produced by reaccelerating ultra-cold muons. The center-of-mass energies for and collisions are 346 GeV and 2 TeV, respectively, with estimated instantaneous luminosities of 5.7 × 10³² cm^-2 s^-1 and 4.6 × 10³³ cm^-2 s^-1. This high luminosity gives μTRISTAN the potential to provide stringent bounds for these two cLFV scattering processes. Based on previous phenomenological calculations on similar four-fermion operators, μTRISTAN can potentially push the new physics scale limit on four-fermion operators up to 50 TeV [59]. Substantial efforts are still required before the and colliders can be operational and accessible to physicists.

The MUonE experiment is a fixed-target experiment aimed at measuring the scattering differential cross section with unprecedented precision of [60]. MUonE is designed to utilize the intense 150 GeV muon beam available at CERN, incident on beryllium targets, to measure the scattering of muons on atomic electrons. The design aims to achieve the target statistical sensitivity within three years of running time with an integrated luminosity of 15 fb^-1. The limit on the cross section can be set by measuring anomalies in the scattering cross section. Unfortunately, this approach may not be sufficiently sensitive to heavy new physics to supersede the existing MACS limit owing to the low center-of-mass energy [61, 62].

The process (Fig. 2) is generated from pure four-fermion effective couplings, and its Lagrangian is the same as that of muonium-to-antimuonium conversion. Neglecting lepton masses, the differential cross section for the process is given by(27)where is the center-of-mass energy and θ is the angle between the incoming electron and the outgoing muon. The coefficients , , and are defined as follows,(28)The form of the differential cross section is similar to that for , but without contributions from Z boson or photon couplings [63]. For the heavy new physics mediators considered here, a higher center-of-mass energy leads to a larger cross section and increased sensitivity to new physics. The new physics scale that can be probed by GeV collisions with 16°<θ_lab<164° in the lab frame is estimated as(29)Assuming an integrated luminosity of 1 ab^-1 [59], the collider could probe new physics on the scale of 10–100 TeV. Operator-dependent new physics scales for the experiments are shown in Fig. 3, where the μTRISTAN result is projected from the background-free upper limit with a 90% confidence level (C.L.). We observed from this result that MACE, as a low-energy experiment, could probe a new physics scale comparable to or even higher than that of a future muon collider. This finding highlights the importance of low-energy experiments in the search for new TeV-scale physics.

Fig. 2Full-screen View

Charged lepton flavor violating scattering (a) and (b)

Fig. 3Full-screen View

(Color online) The new physics scale accessible to experiments searching for processes

Signal event signature

MACE aims to search for the muonium-to-antimuonium conversion process by detecting antimuonium decay events in vacuum. The detection scheme is identified the characteristic decay products of antimuonium. Upon the decay of antimuonium, a fast electron, slow positron, and two invisible neutrinos are produced. The fast electron originates from the μ^- decay with a kinetic energy of a few tens of MeV, while the slow positron comes from the atomic shell with a mean kinetic energy of 13.5 eV (considering 1s antimuonium, as shown in Fig. 4). By detecting these two opposite decay products, a few antimuonium decay events can be identified from the vast number of muons and muonium decays.

Fig. 4Full-screen View

(Color online) Energy spectrum and the leading-order diagram of antimuonium decay . The energy spectrum of the fast decay e^- is accurate to next-to-leading-order, atomic shell e⁺ spectrum assumes 1s muonium [62]. a e^- energy spectrum; b Leading-order decay; c e⁺ energy spectrum

The signal signature in detection is clear: a signal event involves an energetic electron and slow positron. The energy spectra of the signals are shown in Fig. 4. The MACE design employed a silica aerogel target that produced muonium by injecting a high-intensity surface muon beam. The target was designed to be porous and perforated to maximize the diffusion of muonium into vacuum. Following the spontaneous conversion of muonium to antimuonium in vacuum, antimuonium decays into an energetic electron and a slow positron. The energetic electron may traverse the charged particle tracker with a sufficient energy threshold, whereas the positron is guided by the positron transport system towards a position-sensitive detector. Subsequently, the positron may annihilate the detector, producing a pair of 511 keV gamma rays.

Backgrounds

As a signal event involves an energetic electron and a low-energy positron, any event with such a signature might be misidentified as a signal, making them potential backgrounds. First, the possible sources of the background must be identified. Backgrounds can be grouped into two categories: physical and accidental backgrounds. Physical backgrounds may arise from the decay of muons or muonium atoms, whereas accidental backgrounds could result from two tracks coincident from distinct sources. The following discussion focuses on these two types of backgrounds.

The primary focus of MACE is muonium, raising the important question of whether muonium itself can introduce a physical backgrounds. In a minimal extension of the Standard Model that simply extends right-handed neutrinos, the M-to- conversion probability will experience GIM-like suppression because of the tiny neutrino masses, similar to cLFV decays Fig. 5, such as [64, 65] or [66, 67]. Consequently, the conversion probability is expected to be considerably low. Consequently, there is no irreducible physical backgrounds, and any background contribution is expected to originate from muon or muonium decay, as shown in Fig. 6.

Fig. 5Full-screen View

Muonium conversion to antimuonium by the Standard Model weak interaction. This process is strongly suppressed by the tiny neutrino masses

Fig. 6Full-screen View

Event topologies for signals and backgrounds. a M-to- conversion signal; b Background: SM allow rare decay ; c Background: muonium decay with Bhabha scattering; d Accidental backgrounds

MACE operates at an intensive muon beam; therefore, any muon-induced background should not be overlooked. The major decay mode or the radiative decay mode does not introduce extra electrons, and thus, they are generally safe. However, SM-allowed internal conversion (IC) decay introduces an electron in its final state with a branching ratio of 3.4×10^-5 (with a transverse momentum cut ) [68]. When the positron has a fairly low momentum and an electron is detected while another is not, the internal conversion decay can fake signals. The background event topology is shown in Fig. 6b and the energy spectra of the decay products are shown in Fig. 7. In fact, the internal conversion decay of muon was identified as one of the major background sources in the MACS experiment in PSI [8, 69]. To reduce this type of background, we can utilize the difference in kinematics between the antimuonium decay and muon IC decay.

Fig. 7Full-screen View

(Color online) Spectra of muon internal conversion (IC) decay. a e^- energy spectrum of from μ⁺ IC decay; b e⁺ energy spectrum from μ⁺ IC decay; c Joint energy spectrum of e⁺/e^- from μ⁺ IC decay; d Energy spectrum of e⁺ with the lowest energy from μ⁺ IC decay

As shown in the energy spectrum of the lowest positron energy (Fig. 7d), one can observe that the branching fraction decreased significantly at low energies. The signal positron typically has a kinetic energy of only a few tens of electronvolts, providing a strong discriminator between the signal and muon internal conversion decay background, as illustrated in Fig. 8. Selecting low-energy positrons and constraining their kinetic energy to be extremely low in the MACE detector system can effectively suppress the internal conversion decay background and improve the signal-to-background ratio. This approach requires the design of a transverse-momentum-selective positron transport solenoid system, as detailed in Sect. 8.1. However, the longitudinal momentum can be selected by the time-of-flight of the positron. Furthermore, the transverse momentum of the energetic electron from the same decay event is concentrated in the low momentum region, as shown in Fig. 9, in contrast to the signal electron spectrum shown in Fig. 4a. Therefore, selecting the transverse momentum of the energetic electron to be sufficient also aids in background suppression. With these signal purification strategies, the phase space of backgrounds can be strongly constrained while maintaining the majority of signals, eventually improving the signal-to-background ratio.

Fig. 8Full-screen View

(Color online) M-to- signal e⁺ kinetic energy spectrum of different conversion probability (blue) and background e⁺ kinetic energy spectrum from decay process (red)

Fig. 9Full-screen View

(Color online) Spectra of muon internal conversion (IC) decay with MACE Michel electron magnetic spectrometer (MMS) geometric acceptance. a Transverse momentum spectrum of e^- from μ⁺ IC decay (with detector geometric acceptance); b e⁺ energy spectrum from μ⁺ IC decay (with detector geometric acceptance); c Joint spectrum of e⁺ energy and e^- transverse momentum from μ⁺ IC decay; d Joint spectrum of e⁺ energy and e^- energy from μ⁺ IC decay

There is also a possibility that muonium decay could contribute to the physical background of MACE. Muonium shares most of its decay properties with muons, except for the presence of an atomic electron. When a Michel positron from muon decay interacts with an atomic electron and exchanges significant momentum, the muonium decay products behave as low-energy positrons and high-energy electrons, potentially leading to misidentification as antimuonium decay. The event topology is shown in Fig. 6c. This type of background was discussed in a previous theoretical study [38], and the branching ratio with MeV was estimated to be approximately 10^-10. In this scenario, a higher electron energy corresponds to a lower positron energy, indicating a higher momentum exchange and, consequently, a significantly reduced cross section when the final positron energy is sufficiently low. The background level is expected to be suppressed through an optimized detector design and a similar event selection scheme, as discussed above.

Accidental coincidences involving an energetic electron, potentially produced by Bhabha scattering, and a positron from muon or muonium decay represent another potential background source. If a Bhabha scattering event involving a Michel positron and an electron in the porous target material occurs near the material boundary, and the scattered positron is emitted from the material surface with low energy, it could fake a signal event. This process is similar to the Bhabha scattering of muonium decay final states, and similar strategies can be employed to suppress this type of background.

Furthermore, selection in an event-time window can improve the signal-to-background ratio [70]. All three types of background sources from muon or muonium decay are characterized by an exponentially decreasing time distribution proportional to , where τ_μ is the muon lifetime. Notably, the muonium-to-antimuonium conversion probability varies over time and is proportional to [25, 56]. This time-dependent conversion increases the ratio of antimuonium decay to muonium or muon decay by t², thereby contributing to the signal-to-background ratio with accumulated time. Background suppression can be achieved by selecting a late event time window between the arrival of two beam pulses in a high-repetition-rate muon beam if necessary. A 40–50-kHz muon beam would potentially be available at the China initiative Accelerator Driven System (CiADS) in Huizhou (see Sect. 4) or Shanghai HIgh repetitioN rate XFEL and Extreme light facility (SHINE) [71].

In a concise overview, the following potential background contributions should be considered in MACE.

The detector design should be optimized to enhance its ability to identify signals and background events. This includes improving the vertex resolution of the magnetic spectrometer, enhancing the resolution of low-energy positron time-of-flight measurements, and improving the energy resolution of the electromagnetic calorimeter to reduce accidental and physical backgrounds. Utilizing a state-of-the-art positron transport system to select the momentum of the transported particles can yield an optimal signal-to-background ratio. In addition, connecting to a high-repetition-rate muon source can further enhance the signal-to-background ratio. In subsequent sections, we discuss the experimental design and necessary specifications.

Accelerator and proton beam

The development of a superconducting linac for ADS in China began in 2011. The prototype front-end linac (CAFe), as shown in Fig. 10, was developed in stages and included an Electron Cyclotron Resonance (ECR) ion source, a Radio Frequency Quadrupole (RFQ), a superconducting acceleration section, and a 200-kW beam dump. The commissioning of the hundred kW beam started in 2018 and reached a milestone in early 2021 by producing a 20-MeV proton beam with an average current of 10 mA, demonstrating the feasibility of a superconducting linac in Continuous-Wave (CW) mode.

Fig. 10Full-screen View

(Color online) Schematic view of the development and commissioning history of CAFe [72]

The schematic diagram of the CiADS linac, illustrated in Fig. 11, consists of a normal conducting front-end, a superconducting acceleration section, and several High Energy Beam Transport (HEBT) lines. The civil construction of the linac is complete, and work on the experimental terminals is ongoing. The front-end, integrated in December 2022, includes an ECR ion source, a Low Energy Beam Transport (LEBT) with a fast chopper for beam pulse structuring and machine protection, an RFQ, and a Medium Energy Beam Transport (MEBT). The proton beam from the front-end was a CW beam with a current of 5.2 mA and an energy of 2.18 MeV. The superconducting section accelerated the beam from 2.1 MeV to 500 MeV. With a beam current of 50 μA on the target by 2025, the beam power will be 25 kW, and power ramping to 250 kW and 2.5 MW is expected by 2027 and 2029, respectively. The superconducting section houses three types of half-wave superconducting resonators (HWR010, HWR019, and HWR040) and two types of elliptical cavities (Ellip062 and Ellip082) in the 32 cryomodules. With the muon source proposal, the superconducting linac could potentially be upgraded to 600 MeV.

Fig. 11Full-screen View

(Color online) Schematic diagram of the CiADS linac [72]

Muon production and transport

4.2.1

Muon production target

The production target for a high-intensity muon source is extremely challenging because it requires addressing high-heat densities and a harsh irradiation environment. Arising from the two-body decay of the positive pion stopped close to the surface of the production target, the surface muon escaped from the target with a momentum ranging from 0 to 29.8 MeV/c. A new design based on a free-surface and sheet-shaped liquid lithium target is shown in Fig. 12 [72]. Liquid lithium passes through the lithium circuit and forms a sheet jet from a narrow nozzle. The proton beam is collimated to hit the lithium jet at a small angle, and the surface muons produced inside the lithium that escape from both sides of the jet are captured by the solenoids.

Fig. 12Full-screen View

(Color online) Schematic diagram of the free-surface liquid lithium target [72]

Liquid lithium has been used as a neutron production target, radionuclide production target, and ion beam charge stripper because of its low melting point, extremely low saturated vapor pressure, high heat capacity, and good compatibility with structural materials. Maintaining the stability of free-surface liquid lithium flakes in a jet is a challenging design aspect. Extensive research and development studies have been conducted to investigate the feasibility of producing free-surface liquid lithium films or sheets.

A liquid lithium target is particularly well suited for high-intensity muon sources driven by the CiADS linac. First, lithium has a low atomic number Z, and research performed at PSI has shown that the surface muon production efficiency is roughly proportional to . Second, the lithium target produces more low-energyπ⁺’s, resulting in more surface muons than the graphite target, with a significantly lower rate of positrons. Figure 13 shows the momentum spectra of the side-leaking μ⁺,π⁺ and e⁺ from lithium and graphite targets recorded by a detector next to the target.

Fig. 13Full-screen View

(Color online) Momentum spectra of μ⁺, π⁺ and e⁺ recorded by the virtual detector beside the target [72]

4.2.2

Muon beamline conceptual design

Through the interaction of the high-intensity proton beam with the muon target, the generated muons have a large transverse emittance and a significant longitudinal energy spread. A solenoid-based beamline concept was proposed to collect the large-emittance muon beam, as depicted in Fig. 14.

Fig. 14Full-screen View

Schematic diagram of the beamline, featuring dipoles, wien filters and solenoids, denoted with the labels

The surface muon rate from the target is expected to reach up to 5 × 10¹¹ μ⁺/s with a 5 mA proton beam. The closer the capture solenoid is to the muon target, the higher the transmission efficiency. However, owing to spatial constraints, the capture solenoid was positioned 200 mm away from the muon target. Drawing on design experience from PSI, we opted for a shorter solenoid with a larger aperture to facilitate particle transport and enhance the transmission efficiency. The beamline employed a solenoid with a length of 400 mm and diameter of 500 mm. Considering space utilization and shielding effectiveness, we set three dipoles to rotate the beam clockwise, clockwise, and counterclockwise.

During muon collection in the target area and the subsequent beamline transport, a significant number of positrons are generated, which can affect the accuracy of the subsequent experimental results. The primary mechanisms for positron production include the decay of π⁰ mesons into two photons, leading to the positron-electron pair production and the muon decay. Given that the decay length of muons is 170 m, the number of positrons produced during transport is relatively low; thus, the main source is π⁰ mesons in the target. Figure 15 shows the schematic diagram of Wien filter modeling with G4beamline. A long Wien filter is typically employed to remove positrons from the low-momentum muon beams. Figure 16 shows the comparisons of positron removal effects using a long Wien filter or two short Wien filters. Simulations using the G4beamline indicate that placing two short Wien filters can effectively eliminate positrons and other charged particles while maintaining the transmission efficiency μ⁺.

Fig. 15Full-screen View

(Color online) Schematic diagram of a Wien filter

Fig. 16Full-screen View

(Color online) Comparisons of the positron removal effects using one long Wien filter and two short Wien filters. a Particle distribution at the entrance; b Particle distribution at the exit of a single long Wien filter; c Particle distribution at the exit of the second of two short Wien filter

By externally invoking the TraceWin program and utilizing PSO (Particle Swarm Optimization) algorithm, we optimized each component of the beamline (hard edge model) to maximize overall transmission efficiency while minimizing beam spot size at the end. The optimized beam envelope plot is illustrated in Fig. 17. By externally invoking the G4beamline package and utilizing the Genetic Algorithm (GA) algorithm, we attempted to optimize each component of the beamline (real magnetic field). The transmission efficiency of the beamline was 3% with a transverse RMS size of approximately 20 mm. The surface muon beam rate will be up to 1.5×10¹⁰ μ⁺/s with a 5 mA proton beam. Through further optimization, higher transmission efficiency and surface muon beam rate can be achieved.

Fig. 17Full-screen View

(Color online) Beam envelope plot from the TraceWin software in the multi-particle mode with hard edge model

Introduction

The goal of MACE is to identify spontaneous muonium-to-antimuonium conversion events in vacuum. MACE cannot detect antimuonium decay events in materials for two reasons. First, atomic positrons are trapped in the material and are undetectable. Second, muonium-to-antimuonium conversion is a coherent process, and it is suppressed by decoherence owing to interactions between muonium and the material [73]. Therefore, enhancing the muonium yield is crucial for improving the signal-to-noise ratio. Consequently, increasing the sensitivity primarily involves maximizing the muonium yield in vacuum.

In the production of muonium in vacuum, a common approach involves directing a surface muon beam into a specific target material. A muon can spontaneously capture an electron to form a muonium atom. Subsequently, these muonium atoms diffuse within the material, with some escaping into vacuum. The ideal target materials for this purpose are porous and inert materials, such as silica powder or silica aerogel. A silica powder target can reach an efficiency of up to 8% for muonium emission into vacuum [74]. It was used in the MACS experiment, the most recent study on muonium-antimuonium conversion, where 5 × 10^-3 muonium atoms were produced in a vacuum per incident muon [8]. In a more contemporary approach, silica aerogel is employed as the target for muonium production. The self-supporting and porous properties of silica aerogel, together with its relatively high muonium production efficiency, make it suitable for MACE.

However, research has indicated that a significant portion of muonium atoms remain trapped within the target [75, 76], which presents challenges in enhancing muonium emission efficiency. To enhance the diffusion of muonium atoms from the aerogel into the vacuum, prior studies have demonstrated that a perforated surface downstream, created using a pulsed laser, effectively enhances the diffusion of muonium atoms within the target and their emission into the vacuum [75]. The configuration of these perforations is shown in Fig. 18.

Fig. 18Full-screen View

(Color online) The single-layer perforated silica aerogel target concept [79]

Laser ablation has also been demonstrated to be effective. Beare et al. conducted experiments at TRIUMF, investigating various parameters of the ablation structure, and achieved an emission efficiency (the ratio of muonium decay in a downstream vacuum to the total muonium formed) of up to 2% [77], representing an order of magnitude enhancement compared to the silica powder target. Additionally, Antognini et al. measured of muonium yields in a vacuum at PSI by employing a slow muon beam with momentum ranging from 11 to 13 MeV/c, achieving a yield (the ratio of muonium decay in a vacuum to the number of incident muons) of 6%. This yield was enhanced by utilizing a low beam momentum. Furthermore, the J-PARC muon g-2 experiment has designed to integrated a perforated silica aerogel target as a crucial component in muon cooling [78].

Design and optimization of single-layer target

In this study, we developed a Monte Carlo simulation method for muonium formation and diffusion in perforated silica aerogels. The independent simulation results were validated using experimental data. In this section, we summarize the geometrical design and simulation-guided optimization of the design.

For the benchmark design, a single-layer target geometry in the shape of a cuboid was selected, with its short edge aligned parallel to the beam direction (z-axis), and the remaining two longer edges of equal length. The downstream face of the target was perforated with blind-ended cylindrical holes perpendicular to the surface arranged in an equilateral lattice pattern. A schematic is shown in Fig. 18 The geometry was parameterized as follows:

The width and thickness of the target were determined using beam parameters. Specifically, the width is determined by the beam spot size, ensuring that it is sufficiently large to cover the entire beam spot for the maximum utilization of the beam flux. The target thickness was related to the beam momentum and momentum spread. It should be adjusted to maximize muonium yield, taking into account any additional materials, such as beam momentum degraders or beam monitors located upstream. In the benchmark design, the target dimensions were 60 mm × 60 mm (width) × 10 mm (thickness), with an aluminum beam degrader positioned upstream, whose thickness can be tuned.

Perforation geometry requires further consideration. Increasing the spacing between the holes results in more material, thereby leading to higher rates of muon stopping and increased muonium production. However, these muonium atoms may encounter difficulties in vacuum emission owing to obstruction from the material. Conversely, enlarging the hole diameter can enhance the emission efficiency but may decrease the overall muonium yield. Therefore, it is likely that a specific combination of geometric parameters yields the highest vacuum yield when the material and beam conditions are held constant.

It is essential to conduct a combined optimization of the geometric parameters. To perform this process, the first step involves defining the physical quantities relevant to muonium production in vacuum. Five quantities, , f_M, Y_tot, R_vac, and Y_vac, were used to characterize the formation and emission of muonium. These quantities represent the muonium formation fraction, muon-stopping fraction, total muonium yield, muonium emission efficiency, and muonium yield in vacuum, respectively. They are expressed as(30)In these equations, denotes the number of muons stopped at the target, denotes the total number of muons on the target, denotes the total number of muonium produced, and denotes the number of muonium in vacuum. Noteworthy relationships exist among these quantities, including(31)The value of f_M is predominantly influenced by the material properties and is associated with both material properties and beam conditions; both R_vac and are affected by the aerogel target geometry. A more detailed discussion on the relationships between these quantities can be found in Ref. [79].

Simulation-guided optimization was conducted as follows. The perforation structure is consistent with that shown in Fig. 18, featuring a target size of and ablation holes with varying geometric parameters within the 40 mm × 40 mm region at the center of the downstream surface. The temperature was fixed at 322 K, the aerogel density was set to 27 mg/cm³, and the muonium mean free path was 250 nm, all of which served as inputs for the Monte Carlo model parameters.

An aluminum degrader was positioned 5 mm in front of the target, where the muon beam traverses and loses energy. A fraction of the muons stops in the target and forms muonium atoms. The degrader, with varying thicknesses, produces different distributions of muon stopping positions, thereby influencing the emission of muonium into vacuum and resulting in varying vacuum yields. Therefore, optimizing the degrader thickness is crucial for maximizing the yield, as illustrated in the simulation results shown in Fig. 19. Based on these results, we found that the optimal degrader thicknesses for this single-layer target with momentum spreads of 2.5%, 5%, and 10% are 430 μm, 410 μm, and 370 μm, respectively.

Fig. 19Full-screen View

(Color online) Muonium production and emission from a flat silica aerogel target with different aluminium degraders (, and T=322 K) [79]. a Total yield Y_tot; b Emission efficiency R_vac; c Vacuum yield Y_vac

Using these optimal degrader configurations, the perforation geometry can be optimized, involving hole spacing ranging from 0 to 100 μm, diameters ranging from 40 to 360 μm, and depths of 1 mm, 2 mm, and 5 mm. The simulation results are presented in Fig. 20, and the optimal values are shown in Tab. 2. As illustrated in Fig. 20, maxima in the muonium emission efficiency and vacuum yield are observed. When the spacing and diameter deviate from their optimal values, the yield in vacuum gradually decreases, with significant suppression occurring when the spacing or diameter becomes excessively small.

Fig. 20Full-screen View

(Color online) Projection of total muonium yield, muonium emission efficiency, and vacuum muonium yield under different beam conditions and target geometries [79]. a Total muonium yield Y_tot (σ_p/p=2.5%, d=1 mm); b Emission efficiency R_vac (σ_p/p=2.5%, d=1 mm); c Muonium yield in a vacuum Y_vac (σ_p/p=2.5%, d=1 mm); d Total muonium yield Y_tot (σ_p/p=10%, d=5 mm); e Emission efficiency R_vac (σ_p/p=10%, d=5 mm); f Muonium yield in a vacuum Y_vac (σ_p/p=10%, d=5 mm)

In summary, for the design of a single-layer target, we recommend an appropriate configuration of perforation parameters that balances the total muonium yield (Y_tot) and muonium emission efficiency (R_vac) to achieve an optimal muonium yield in vacuum (Y_vac). This optimization can be realized through a combination of the simulation-guided optimization procedure proposed in Ref. [79] and data-driven experimental optimizations. These approaches are shared between single-layer and multi-layer target designs and will be employed in future technical designs to refine and determine optimal configurations.

Multi-layer target design

The perforated single-layer target has the potential to achieve a high muonium yield in vacuum; however, there is still room for improvement. In the single-layer target design, the perforated surface faces downstream relative to the beam, and it is expected that nearly half of the muons will stop within the target, to maximize the number density of muons stopped near the perforated surface. This configuration implies that nearly half of the muon beam penetrates the target without being used. Additionally, the single-layer target favors a more concentrated momentum spread to minimize the spread of muon stopping positions; therefore, a high muonium yield in vacuum requires a low beam momentum spread. As observed in Sect. 5.2, an increased beam momentum spread significantly suppresses the muonium yield in vacuum, which is not expected because it amplifies the correlation between muonium yield and beam momentum spread. This reduces the design margins and may increase the systematic errors associated with muonium yield. Therefore, we expect a target design that can enhance the utilization of muon beam flux while tolerating a wider beam momentum spread. The design of a multilayer silica aerogel target has the potential to achieve this goal.

The original concept of a multi-layer silica aerogel target was proposed by Zhang et al. [80], with the aim of increasing the total muonium yield in vacuum to enhance the efficiency of converting a surface muon beam into a thermal muon beam by ionizing the muonium atoms produced in vacuum. This technology, known as muon cooling, shares the same objective of increasing the muonium yield in a vacuum as MACE. In the simulation work by Zhang et al., the multilayer target design was shown to increase the muonium yield in vacuum by a factor of 3.45.

The design horizontally aligns multiple silica aerogel targets in parallel, with the muon beam directed parallel to the layers. Each silica aerogel layer was perforated, allowing muonium atoms produced within the layer to diffuse and escape into the interlayer vacuum. If these muonium atoms are converted into antimuonium and decayed, low-energy positrons will be accelerated parallel to the target layer, exiting the target region, and guided to the positron detection system. This target concept is illustrated in Fig. 21 and Fig. 22. In scenarios where the beam may be well collimated, it can pass through the target via the spaces between the silica aerogel layers, potentially leading to reduced utilization of the muon beam flux. To address this, a beam degrader may be introduced in front of the multilayer target to scatter the beam if necessary. Using this design, the utilization of muon beams can be effectively enhanced. Furthermore, unlike the single-layer target design, the longitudinal length of the multilayer target can be extended without significantly compromising the muonium emission efficiency because muonium atoms diffuse and escape primarily in the transverse direction. Consequently, this design achieves the goal of increasing the muon beam utilization while tolerating a wider beam momentum spread.

Fig. 21Full-screen View

(Color online) The multi-layer silica aerogel target concept

Fig. 22Full-screen View

Simulated distributions of muonium formation (a) and decay vertices (b). A few percents of muonium atoms will diffuse out of the target before their decay

We conducted simulations of various multilayer target designs. We utilized a beam momentum of 24 MeV/c and an RMS spread of 1.35 MeV/c. The beam is assumed to be collimated, resulting in a beam spot of 40 mm×40 mm square, where the muons are uniformly distributed. Each layer had a longitudinal length of 60 mm, a height of 50 mm, and varying thicknesses of 2 mm, 3 mm, and 4 mm. The layers were perforated in a 50 mm×40 mm region at the center. The perforation parameters for the aerogel layers were inherited from the optimal parameters of the single-layer target design, that is, a hole spacing of 55 μm and a hole diameter of 184 μm. The spacing between adjacent layers was kept constant, varying from 2 mm, 3 mm, to 4 mm. The number of silica aerogel layers was selected to ensure that the overall height and width of the target shape matched.

In the simulation, 10⁶ μ⁺ were generated for each target design, and the results are summarized in Table 3. The definitions of the result parameters are consistent with those described in Sect. 5.2, with the exception that “in-vacuum muonium” is counted only in the vacuum region of interest, where atomic positrons from antimuonium decay can be accelerated and transported to the positron detection system. From the simulation results, we observed that the muonium yields in vacuum (Y_vac) are enhanced by factors of 2 to 4 compared to the single-layer target. This enhancement is primarily attributed to an increase in muonium emission efficiency (R_vac).

The optimal result is achieved with the smallest single-layer thickness and layer spacing, both at 2 mm, resulting in a muonium yield in vacuum of 4.94(2)%. Although the yield is high, the multilayer target has intrinsic limitations. The major concern raised by excessive muonium scattering between the silica aerogel layers would quench the muonium conversion probability if the layer spacing is too narrow. Therefore, a larger layer spacing is preferred, and we selected a muonium yield in a vacuum of 3.8% as the benchmark parameter. In conclusion, the results demonstrate the possibility of increasing the muonium yield in a vacuum to a few percent if the target geometry is properly designed.

Magnetic field and magnet

A solenoid magnet surrounds CDC and TTC to provide an axial magnetic field. Increasing the magnetic flux density can enhance the momentum resolution and reduce the charge misidentification rate of the MMS by increasing track curvature. However, the muonium conversion process is influenced by the magnetic field; because the target is positioned at the center of the MMS, this field affects the conversion process. A strong magnetic field suppresses the contributions of certain operators in the conversion process, creating a trade-off between the MMS performance and physical sensitivity. As discussed in Ref. [56, 57], muonium conversion induced by (V ± A) × (V ± A) and (S ± P) × (S ± P) effective couplings is significantly reduced when the magnetic field reaches 1 T or higher. To maintain sensitivity of the MACE to all effective operators and constrain the parameter space, the MMS operates at B=0.1 T. This magnetic field achieves balance, ensuring MMS performance while preserving reasonable conversion probabilities associated with (V ± A) × (V ± A) and (S ± P) × (S ± P) couplings. This configuration is consistent with the choices made in the MACS experiment [8]. Further details of the magnet design are provided in Sect. 8.1.

Cylindrical drift chamber

7.2.2

Wire configurations

Following these objectives, we considered CDC geometry consisting of near-squared cells. One cell had a 20 μm gold-plated tungsten sense wire at the center, surrounded by eight 80 μm aluminium field wires, as shown in Fig. 24. Cells are azimuthally arranged in sense layers, in which they share the same width, length, and stereo angle. Three adjacent sense layers are grouped into a super layer, in which the sense layers share the same stereo orientation, number of cells, and cell azimuth angular width. In a super layer, two adjacent layers are staggered by a half-cell to resolve the left-right ambiguity.

Fig. 24Full-screen View

(Color online) A 1/4 section view of CDC wires at z=0, where the red and blue circles are the field and sense wires, respectively. The wire radii were scaled 20 times for clarity. The CDC consists of concentric layers of wires with alternating directions, the odd-numbered super layers being stereo and the even-numbered super layers being axial

The current design contained seven super layers, with three sense layers in each super layer, resulting in a total of 21 layers. Each sense layer contains 124–212 cells according to the superlayer where they are located. This resulted in a total of 3540 cells, comprising 12980 field wires and 3540 sense wires. Among the seven super layers, three are axial layers whose cells orient axially, and the remaining four super layers are stereo layers. The stereo orientations are alternately arranged, with the first super layer being positively twisted, the third super layer being negatively twisted, etc., forming a UAVAUAV-twisted pattern. Detailed configurations are listed in Tables 4 and 5. Stereo layers have a hyperbolic profile as shown in Fig. 25. This introduces radial shrinkage toward the center, leaving a clear space between the next axial super layer in the region close to the central xy plane, as well as a clear space between the last axial layer in the region close to the endplate. Therefore, the field-wire layer is not shared between two adjacent super layers, resulting in cell geometries that are nearly identical.

Table 4Full-screen View

Geometric parameters of the cylindrical drift chamber

Parameter	Value
Inner radius (cm)	150
Outer radius (mm)	415
Inner length (mm)	1200
Outer length (mm)	1600
Number of layers	21
Number of cells	3536
Geometric acceptance	88.8%

Show more

Table 5Full-screen View

Wire configurations of MACE cylindrical drift chamber

Super layer ID	Sense layer ID	Cell ID	Radius at layer center (mm)	ϕstereo (mrad)	ϕ1st cell (mrad)	Cell width (mm)	Cell length along z (mm)
0	0	0–123	155.992	799.799	0	7.904	1222.530
	1	124–247	164.102	807.591	25.3354	8.315	1235.834
	2	248–371	172.633	815.790	0	8.747	1249.881
1	3	372–527	197.896	0	20.1384	7.971	1265.806
	4	528–683	206.030	0	0	8.298	1277.762
	5	684–839	214.499	0	20.1384	8.639	1290.209
2	6	840–1003	224.193	-611.622	0	8.589	1320.833
	7	1004–1167	232.950	-617.639	19.1561	8.925	1334.688
	8	1168–1331	242.049	-623.892	0	9.273	1349.115
3	9	1332–1495	265.361	0	19.1561	10.167	1365.356
	10	1496–1659	275.726	0	0	10.564	1380.606
	11	1660–1823	286.496	0	19.1561	10.976	1396.451
4	12	1824–1991	298.649	501.677	0	11.169	1428.736
	13	1992–2159	310.031	507.616	18.7000	11.595	1446.392
	14	2160–2327	321.847	513.782	0	12.037	1464.754
5	15	2328–2519	345.728	0	16.3625	11.314	1485.046
	16	2520–2711	357.230	0	0	11.690	1502.017
	17	2712–2903	369.115	0	16.3625	12.079	1519.553
6	18	2904–3115	381.894	-427.111	0	11.318	1552.408
	19	3116–3327	393.382	-431.826	14.8188	11.659	1570.086
	20	3328–3539	405.217	-436.683	0	12.010	1588.319

Show more

Fig. 25Full-screen View

(Color online) Perspective view of CDC wires, where golden wires denote sense wires and translucent-gray wires denote field wires. The wire radii were scaled 20 times to be clearly visible. A hyperbolic profile due to stereo wires in super layer 0, 2, 4, and 6 can be seen in the 1/4 clip-away view. a 1/4 clip-away perspective view; b The ID=2 super layer; c The ID=3 super layer

The stereo angle (ϕ_stereo) is defined as the angle at which the projection of the line on the xy plane spans the origin, and ranges from 430 mrad to 815 mrad. A large stereo angle provides excellent resolution along the axial direction. However, additional considerations are needed in future technical design. The cell widths range from 8 mm to 12 mm, offering good event rate tolerance under intensive muon decay event rates. The inner radius and length of the CDC are 150 mm and 1200 mm, respectively, while the outer radius and length are 415 mm and 1600 mm, respectively. Consequently, the geometric acceptance of the CDC was 88.8%. This also specifies the geometric acceptance of the tiled timing counter (TTC) to be discussed below. The overall acceptance of the MMS system is a combination of CDC and TTC.

Tiled timing counter

The goal of MACE is to identify spontaneous muonium-to-antimuonium conversion events in a vacuum. The tiled timing counter (TTC) is one of the core components of the MACE detector system, with its primary objective being to provide timing for particle tracks, particularly for the signal electron tracks. TTC triggers a data acquisition system by detecting positrons or electrons resulting from muon decay or signal events. It is essential for a tiled timing counter to have a high time resolution to accurately measure the arrival times of these particles. This capability enhances the performance of the MACE detector system in two key ways: (1) it improves the time-of-flight resolution for signal positrons by providing an accurate decay vertex time and (2) it increases the resolution of the CDC through enhanced drift time resolution, which assists in precise track reconstruction and minimizes the number of fake tracks.

7.3.2

Conceptual design

Figure 26 illustrates the geometric design of TTC. Each unit in this design measured 50 mm × 30 mm × 5 mm. A total of 9,780 units were organized in an overlapping arrangement across 60 layers (along the z-axis) and 163 columns (along the azimuthal direction) at a tilt of 30°. This overlapping configuration facilitates two-to-three-tile coincidence detection, ensuring that most high-momentum charged tracks traverse at least two scintillators while minimizing the occurrence of accidental coincidences from the background particles. Compared to a design with two complete layers of scintillators, this approach reduces the number of channels and cabling complexity. The TTC configuration has a radius of 480 mm and total length along the z-axis of 60 × 30 mm = 1.8 m. Consequently, the geometric acceptance of the TTC system was calculated as 88.2%, which is consistent with that of the CDC. The parameters can be summarized by Table 7.

Fig. 26Full-screen View

(Color online) Perspective and zoom-in view of the tiled timing counter (green)

Table 7Full-screen View

Geometric parameters of the tiled timing counter system

Parameter	Value
Layers along the z-axis	60
Tiles per layer	163
Total number of tiles	9780
Scintillator size (mm3)	50 × 30 × 5
Tilt angle (°)	30
Total length (mm)	1800
Radius (mm)	480
Geometric acceptance	88.2%

Show more

The tilt of each scintillator tile was intentionally oriented in the direction where the largest surface was more perpendicular to the electron track. Because the TTC system is placed in an axial magnetic field, electrons and positrons are bent in different directions. This tilt design naturally favors tracks that are more perpendicular to the large surfaces of the tiles, resulting in the positively tilted layers being more sensitive to electrons, which are the decay products of anti-muonium and of particular interest to MACE. Differences in detection efficiency can be observed in the efficiency curves shown in Fig. 27, and the results are discussed in Sect. 7.4. The 30-degree tilt angle was calculated to balance the spacing between adjacent tiles within a single layer and the overlap of their projected areas.

Fig. 27Full-screen View

(Color online) MMS tracking efficiencies for electrons and positrons at two different TTC coincidence thresholds. Reconstruction efficiencies are not included

Silicon photomultipliers (SiPMs) are known for their excellent time resolution and photon detection efficiency (PDE), making them well suited for MACE timing counter. Two SiPMs are attached to each scintillator tile at its two smallest faces (measuring 10 mm × 100 mm). This double-ended readout reduces the timing bias owing to the different hit positions and improves the single-tile time resolution. The Hamamatsu S13360 series MPPC was considered because of its ~100 ps time resolution and 40–50% PDE [87]. The packaging scheme and reflector type of TTC scintillators are still under investigation. In conclusion, the TTC geometry and detector design are expected to satisfy the requirements of MACE.

Performance

In this section, we present the simulation results of the detection efficiency of the MMS system. As discussed earlier, the MMS consists of CDC and TTC, and their geometric acceptance is consistent with the design. A noteworthy characteristic of the TTC design is the overlapping arrangement of scintillator tiles, which results in differing expected coincidence probabilities for electrons and positrons owing to the influence of a magnetic field. In the fast simulation, we selected tracks that had a number of hit cells equal to or greater than the number of CDC layers and that hit at least two or three TTC tiles. The full track reconstruction algorithm was not applied; therefore, the results should be interpreted in terms of the geometric effects on the tracking efficiency. Complete tracking efficiency can be considered as the geometric efficiency multiplied by the reconstruction efficiency. Electrons and positrons with isotropic momentum ranging from 1 keV/c to 52.8 MeV/c were generated from the center of the MMS, and the efficiencies for events with 2-tile or 3-tile TTC coincidences are shown in Fig. 27, respectively. The direction-dependent tracking efficiencies for electrons and positrons are shown in Fig. 28.

Fig. 28Full-screen View

MMS tracking efficiencies for electrons (a) and positrons (b) with 2-tile TTC coincidence. Reconstruction efficiencies are not included

As expected, at the 2-tile coincidence threshold, the efficiencies for both electrons and positrons increase with momentum, although the increase in positron efficiency is slower and consistently lower than that of electrons. In the high-momentum region, the electron and positron efficiencies converge at maximum values of 88% and 86%, respectively, and are limited mainly by geometric acceptance.

For the 3-tile coincidence threshold, while the positron efficiency remains lower than that of the electron, the efficiencies are no longer monotonic with respect to momentum. For electrons, the 3-tile efficiency initially increases, reaching a maximum around 20 MeV/c, before gradually decreasing as the momentum increases further. This initial increase can be attributed to the increasing gyration radius, whereas the subsequent decrease is due to the greater track curvature. If the track curvature becomes too large, an electron track is more likely to traverse two TTC tiles instead of three because of the overlapping design. In the case of positrons, the trend is similar during the initial increase, but the situation reverses at high momentum. The decrease at a high momentum can be attributed to a reason similar to that of electron. The decrease in efficiency observed between 10 and 20 MeV/c is likely because the track gyration radii match well with the TTC radius and tilt angle, leading to an optimal momentum at which tracks align with the TTC tile orientations, resulting in a lower coincidence probability.

MMS is responsible for tracking electrons and positrons from the decays of muon, muonium, or antimuonium. The decay momenta ranged from 0 to 53 MeV/c, with the majority falling within the high-momentum region. The momentum spectra of electrons from the antimuonium decay and positrons from the muon decay in the MMS are shown in Fig. 29. The momentum spectra for electrons and positrons are slightly different owing to differences in the tracking efficiency, as discussed earlier. According to the simulation results, the tracking efficiency (excluding the reconstruction efficiency) for Michel electrons and positrons was 84.6% and 81.8%, respectively. These efficiencies are close to the full geometric acceptance because efficiency losses are mostly contributed by the low-momentum region, where the Michel spectrum is low. Assuming a reconstruction efficiency of approximately 80%, the effective tracking efficiency for Michel electrons and positrons is approximately 68% and 65%, respectively, as listed in Table 8.

Fig. 29Full-screen View

(Color online) Michel positron momentum spectrum in MMS without momentum resolution. a Momentum spectrum; b Transverse momentum spectrum

Generally, the 2-tile TTC coincidence is suitable when the accidental coincidence rate is not excessively high, allowing for an optimal detection efficiency. Meanwhile, the 3-tile coincidence can be employed if accidental coincidences need to be suppressed, although it results in reduced positron efficiency, while the electron efficiency remains reasonable. In conclusion, the MMS design was optimized for resolutions and tracking efficiencies, and the system allowed for adjustable coincidence configurations for different background rates. The conceptual design of the MMS system can meet the requirements for the physical goals of MACE.

Magnet and transport solenoid

As shown in Fig. 30, the solenoid system in the PTS has three components: the Michel electron magnetic spectrometer (MMS) solenoid, transport solenoid, and electromagnetic calorimeter (ECAL) solenoid. The MMS solenoid is cylindrical in shape, 2400 mm long, and 700 mm in radius, designed to bend charged tracks and confine low-energy positrons, surrounded by a 50 mm thick iron yoke. The S-shaped solenoid transports slow positrons to the MCP in the middle of the ECAL. The transport solenoid comprises five sections: three straight solenoids, T1, T2, and T3, and two toroid solenoids, B1 and B2, make up the system. Each solenoid had identical coils with a length of 30 mm, inner diameter of 120 mm, and outer diameter of 180 mm. T1 and T3 are 150 mm long and T2 is 1314.5 mm long. The arcs of B1 and B2 are each 90°, with a radius of 250 mm and opposite rotation directions, forming an S-shape. The transport solenoid is covered by a 30 mm thick iron yoke and connects the centers of the MMS solenoid and ECAL solenoid. The cylindrical ECAL solenoid is 1200 mm in length and 650 mm in radius and is surrounded by a 50 mm thick iron yoke, which can constrain the positrons after transportation.

Fig. 30Full-screen View

(Color online) Concept of the MACE positron transport system (PTS). The outer gray shell represents the iron yoke, while the interior copper-colored component denotes the solenoid. Key elements and overall dimensions are indicated in the picture

The S-shaped transport solenoid was designed to reduce the background signals. Low-energy positrons with small magnetic stiffness can be considered to follow the magnetic field lines during transport. In contrast, high-energy particles, which have greater magnetic stiffness, tend to collide with the solenoid at S-shaped turns, effectively acting as a momentum selector. In addition, as positrons are transported to the MCP, their transverse spatial coordinates and impact times can be measured. A position-mapping algorithm can then be employed to reconstruct the decay positions of particles in the target region. In this way, the PTS acts as a link between the spatial and timing information of events in the PDS and MMS, demonstrating how the MMS, PTS, and PDS collaborate to coincide with antimuonium events.

To transport positrons with a high spatial resolution, it is crucial to maintain a consistent magnetic field in the transportation region of the PTS. This requires that the magnetic fields produced by the MMS solenoid and ECAL solenoid have a magnitude similar to that of the transport solenoid, as this affects the magnetic field within the transport solenoid. In our experiment, we used a yoke to shield against the magnetic field interference between different regions. We then controlled the magnetic field and its distribution by adjusting the current through various components.

The magnetic field distribution is shown in Fig. 31. The arrangement of the magnetic flux lines confirms that the magnetic field is uniform. By adjusting the current in the different solenoids, we could control the magnetic field to approximately 0.1 T. The decrease in the magnetic field at the end of the MMS was attributed to the fringe field of the solenoid, whereas the increase was due to the connection between the MMS yoke and T1. The interaction between the MMS and T1, along with the enhancing effect of the yoke, contributed to the magnetic field reaching 0.114 T in this region. The same applies to T3; however, because the ECAL solenoid is smaller than the MMS solenoid, the edge effect is less pronounced. Because the maximum magnetic field intensity caused by the edge effect does not exceed the predetermined 12%, and the affected region is very short, this edge effect has little impact on the overall transmission performance according to the simulation results.

Fig. 31Full-screen View

(Color online) The magnetic field distribution. a The distribution of the overall magnetic field across the cross-section of the PTS simulation. Magnetic field lines are also shown. The corresponding regions in (b) have been marked in the figure; b Distribution of the magnetic field along the central axis of the PTS solenoid. The reference line of 0.1T represents the ideal case

Electrostatic accelerator

Signals in MACE are low-energy positrons produced by antimuonium decays. The kinetic energy spectrum is shown in Fig. 4c. The momenta of the positrons are isotropic, which means that they cannot be directed solely by a magnetic field toward the MCP. Additionally, their kinetic energy, typically a few tens of eV, is insufficient to activate MCP. However, accelerating the positrons along the axis by a few hundred volts can address both these challenges.

In this design, an electrostatic accelerator is used for acceleration. This device generates a uniform electric field in the acceleration section, enhancing the precision of the position mapping. Unlike typical designs that place the accelerator outside the beam pipe, the MACE electrostatic accelerator is positioned inside the beam pipe, which will be grounded to prevent the accumulation of charge on it owing to this high-intensity but low accelerator energy situation.

As shown in Fig. 32, the electrostatic accelerator of a PTS consists of multiple circular thin sheets arranged at equal distances. It was situated within the beryllium beam tube along the axis of the MMS solenoid. The electrostatic accelerator has a length of 380 mm, an inner radius of 50 mm, and an outer radius of 60 mm, enveloping the 60 mm × 50 mm × 50 mm target region. The upper end of the accelerator was positioned 110 mm front from the center of the MMS solenoid and the target region. The highest potential is 780 V, the distance between adjacent accelerator sheets is 20 mm, and the potential difference is 41.6 V. The resulted electric field distribution is ploted by Fig. 33.

Fig. 32Full-screen View

(Color online) Concept of the electrostatic accelerator

Fig. 33Full-screen View

(Color online) Accelerator electric field distribution. a Simulated electric field distribution along the axis of the electrostatic accelerator where the blank area represents the region with the reversed electric field; b The electric potential distribution in the axial beam direction. The orange-yellow shaded area is the location of the target area, the black line is the electric field in the beam direction, and the red line is the electric potential

Performance

After the positron was transported to the ECAL solenoid, its transverse position was measured by the MCP. The transverse projection of the positron can be reconstructed using the solenoid position mapping algorithm. The distributions of the difference between the coordinates of the positron transported to the MCP and the coordinates of the antimuonium decay position in the Geant4 simulation results are shown in Fig. 34. In the plot, x₁ and y₁ represent the coordinates of the particle hitting the virtual detector; x₀ and y₀ represent the initial coordinates of the particle; and the horizontal and vertical coordinates x and y in the picture are the differences between the two, indicating the offset of the particle during transportation, not the beam spot. Virtual detectors 0 to 3 represent the positions of the virtual detectors in the beam direction: after acceleration, after B1, before B2, and in the middle of the MCP. The average values of the horizontal and vertical coordinates represent the change in relative position during particle transportation. The root mean square represents the spatial resolution of particle transportation, measuring 0.088(1) mm × 0.102(1) mm, and the offset of the transport was 0.315(1) mm in the x direction and 0.003(1) mm in the y direction. Our simulation included 10⁷ initial incident particles. The signal transmission efficiency was 65.8%, as calculated from the ratio of events detected by each detector to the number of initial incident particles.

Fig. 34Full-screen View

(Color online) The relative displacement of the positron

Beam vertical plane position mapping was performed on the 2D position detected by Virtual Detector 3. The position mapping of the positrons before and after transportation is shown in Fig. 35. In the analysis, the horizontal and vertical coordinates of the particles were considered separately. The coordinates of the transported particles were highly correlated with the corresponding initial coordinates, whereas the calculated coordinates of the transported particles had a low correlation with other initial coordinates, such as , with a correlation coefficient of less than 0.01. Therefore, it can be assumed that the position mapping function before and after the transportation of the positron has a linear relationship, and the slope is approximately 1, with almost no stretching, and only a partial offset in the vertical beam direction. This offset is consistent with the offset shown in Fig. 34; therefore, in the current simulation stage, it can be considered that the coordinate transformation before and after the positron transportation is only a position offset, with an offset of 0.315 mm in the x direction and 0.003 mm in the y direction.

Fig. 35Full-screen View

(Color online) PTS position mapping analysis, the coordinates before transportation are marked as , and after transportation are marked as (x, y). The red line is the fitting line, and the parameters are given in the legend

Because low-energy positrons are produced over the entire target region, the acceleration potential and flight distance vary. Therefore, the overall flight time, that is, the time it takes for the positrons to be transported from the target region to the MCP, is approximately 322.4 ns, and the flight time width is 6.9 ns.

As shown in Fig. 36, the PTS system simulates the presence or absence of a collimator and differently charged particles e⁺, e^-, μ^-, μ⁺, π⁺, and π^-. Because π⁺ and π^- promptly decay and cannot be transported to the MCP, they are not shown in the figure, and the number of μ^- and μ⁺ events is too small, so only μ⁺ is given as a reference. It can be seen that the transmission efficiency is very low for charged particles other than electrons or positrons, and the PTS system is highly energy-selective. As the high-energy charged particles in MACE are mainly positrons from muon decay and their secondaries, the momentum direction can be regarded as isotropic. The following figure can be used to refer to the PTS system’s ability to select the momentum.

Fig. 36Full-screen View

(Color online) Simulation results of signal transmission efficiency for different charged particles at different energies with and without a collimator. a The simulated particle has an isotropic initial momentum and the upper limit of the simulated energy is 1 keV; b The simulated particle has an isotropic initial momentum and the upper limit of the simulated energy is 1 MeV; c The simulated particle has an initial momentum in the axial direction and the upper limit of the simulated energy is 1 keV; d The simulated particle has an initial momentum in the axial direction and the upper limit of the simulated energy is 1 MeV

It can be seen that the transmission efficiency of the particles decreases significantly when the collimator is present, especially for high-momentum particles, and the filtering capability of the collimator is outstanding. For positrons with small initial momenta (initial momentum ≤100 eV), the transmission efficiency was ~75%. At this kinetic energy, the transmission efficiency of the electrons is extremely low because the momentum cannot overcome the reverse electric field. Low-energy μ⁺ cannot be transported because of its limited lifetime, so the transmission efficiency is also extremely low. When the initial kinetic energy is greater than 500 eV, the kinetic energy of electrons can overcome the electric field of the electrostatic accelerator, and the transmission efficiency curves of the positrons and electrons gradually approach each other. The initial momentum of μ is sufficient to allow some of them to reach the MCP before decay; however, the maximum transmission efficiency does not exceed 1%. When the initial kinetic energy continued to increase, the transmission efficiency of μ dropped to 10^-6 at 20 keV, and the transmission efficiency of electrons and positrons also decreased to 1%.

In summary, the physical design of the PTS effectively satisfied the requirements of the MACE system. The PTS achieved a high spatial resolution of 0.088(1) mm × 0.102(1) mm, showing excellent transmission capability. In terms of background suppression, the primary sources of the background are Michel positrons from muonium decay and muons in the beam. When the kinetic energy exceeds 20 keV, the transmission efficiency for Michel positrons decreases to 10^-2, whereas that for muons is 10^-6. Consequently, the current design demonstrated robust momentum selection capabilities, maintaining a signal transmission efficiency of 65.8%. The next step involves exploring the engineering implementation of this design.

Microchannel plate

Nowadays, MCP, as a type of electron multipliers based on the secondary electron emission, has been widely used in high energy physics [82, 88] and other relevant fields [89-96] due to its excellent spatial and time resolution. MACE, as an experiment searching for a rare process beyond the standard model, also uses MCP as a part of the positron detector, expecting to better suppress background events using precise position and time information. After the decay of antimuonium, positrons are transported along the solenoid to the PDS and annihilate within the material once it hits the MCP, producing a pair of 511 keV gamma rays, which will soon be detected by the electromagnetic calorimeter. Before annihilation, some initial free electrons are produced by the ionization of the positron, amplified inside the channels of the MCP, and finally detected. The energy of the positrons to be detected is several hundred electron volts and the required time resolution is approximately . The spatial resolution of the MCP depends mainly on the distance between channels, which is typically <10 μm. The strict requirement of time resolution excluded most of the mass-produced MCPs, so further R&D is needed. To make the design and optimization more convenient, a simulation program for MCP has been developed, which is still independent and can be combined into the MACE software system in the future. However, we still face challenges before the combination, such as the heavy burden of computing the electron avalanche.

The simulation of electron multipliers, such as the MCP, has always been difficult. On the one hand, the millions of electrons produced in the electron avalanche will definitely be a challenge for computing if they are simulated just track-by-track. However, the precision of the simulation cannot be guaranteed if phenomenological models are used because of the necessary approximation and simplification. Furthermore, the mechanism of secondary electron emission cannot be described accurately because of the large uncertainty in the dynamics of particles and the fluctuation of statistics. Thus far, much effort has been put into the simulation of electron multiplier devices worldwide based on multiple models [97-100]. Several commercial software programs have been developed for the general simulation of electron multipliers, including CST Studio Suite^TM [101] and SIMION^TM [102]. However, they are not so easy to be combined into the software system of MACE and difficult to be modified according the real case of the experiment, so that we developed an open-source software based on Geant4 [103] using Furman model [104].

The implementation of the Furman model into Geant4 can be found in detail in Ref. [105, 106]. It is difficult to obtain the real secondary electron yield (SEY) and the energy spectrum. In the simulation, the SEY distribution with respect to the energy of the incident electrons for lead glass was obtained by fitting the measured data, as shown in Fig. 37.

Fig. 37Full-screen View

The measured SEY of lead glass overlaid with the best fit

The MCP gains at various applied voltages were measured using the device shown in Fig. 38. As illustrated in Fig. 38, a tantalum filament, the MCP to be measured, and an anode were placed sequentially inside a vacuum device. A voltage of U₀ is applied to the MCP. When the system is working, a current I_s flows through the tantalum filament, and some of the electrons emitted from the filament enter the input pore of the MCP, of which the input current I_in is measured by the galvanometer GI. After the avalanche amplification in the MCP channel, the output current I_out is measured by the anode and the MCP gain at the applied voltage is then calculated by(34)

Fig. 38Full-screen View

The schematic diagram of the device used to measure the MCP gain

The energy spectrum of the secondary electrons for lead glass is adjusted from the measured spectrum of other materials and iterated in the simulation to approximate reality. Figure 39 shows the angular distribution and kinetic energy distribution of the secondary electrons when the energy of incident electron is 100 eV.

Fig. 39Full-screen View

The angular (a) and kinetic energy distribution (b) of the secondary electrons produced by an incident electron with 100 eV energy. The red solid line is the theoretical prediction and the black points with error bars are the simulation results

After the implementation of the secondary electron emission process, it will be brilliant to study the amplitude and shape of the signal when a certain electron from the initial ionization enters a channel because the response of each channel will be the same or at least similar to each other. Once the signal has been known in detail, the response of a single electron can be accessed by simple sampling so that the response of the entire MCP can be simulated quickly and precisely. We built a simplified geometry of MCP with only seven channels for the simulation of single-electron response, considering the complexity of constructing an MCP with millions of channels, as shown in Fig. 40. Limited by the rendering power of the computer, only 5000 tracks were drawn in this figure. However, amplification of the secondary electrons can be clearly observed.

Fig. 40Full-screen View

(Color online) Simulation events using the simple MCP model

To further validate our simulation, we simulated the gain of the two MCPs versus the applied voltages and compared them with the measured results. The geometries of #1 and #2 MCP were established in Geant4 based on actual measurements and are provided in Table 9. In the simulation, the kinematic information of all the particles that are able to reach the exit of the channel is recorded without considering the detection efficiency, fluctuation, or other relevant issues of the readout system; thus, the MCP gain is simply calculated by dividing the number of electrons at the exit by the number of electrons shot towards the MCP. The two pieces of MCPs are simulated at different applied voltages from 700 V to 1200 V with a step of 100 V using the same SEY and energy spectra of the secondary electrons. For each voltage, the gain was simulated 1000 times with random primary incident electrons, and the average was determined to be the gain at this voltage. The gains with respect to the voltages are shown in Fig. 41a and b.

Fig. 41Full-screen View

The gain of two MCPs versus applied voltage. The blue “×” markers denote the experimental measurements and the green points denote the simulated results. a #1 MCP b #2 MCP

According to the simulation results shown in Fig. 41a and b, the gain will increase exponentially with increasing applied voltage, which is consistent with the experimental measurements and other studies [107], whereas the measured gain is obviously lower than the simulated result and tends to stabilize to a constant value when the applied voltages are sufficiently high because the multiplication is suppressed by the saturation effect [108, 109] which is not considered in the simulation at present. The distribution of the time when the secondary electrons arrive at the electronics for #2 MCP is shown in Fig. 42.

Fig. 42Full-screen View

Time distribution of secondary electrons received by electronics. The horizontal axis denotes the time after the initial electron enters the channel

The simulation program for electron multipliers such as MCP has been summarized into an independent software named “MCPSim” [105]. In the future, a new MCP will be developed to satisfy the requirements of MACE. MCPSim will be used in development and optimization. Meanwhile, the simulation of MCP will be implemented in the software system of MACE with more proper consideration of the saturation and space charge effects.

Electromagnetic calorimeter

9.2.2

Conceptual design

To meet the aforementioned requirements (e.g., excellent energy resolution, high detection efficiency, and good geometrical acceptance), we proposed a Goldberg polyhedron geometry design of the ECAL (see Fig. 43). The Goldberg polyhedron consists of pentagons and hexagons, with every three faces meeting at the same vertex [110]. It is similar in appearance to a sphere as an inpolyhedron and possesses icosahedral rotational symmetry. A Class I GP(8, 0) Goldberg polyhedron geometric layout was introduced. It consists of 642 faces, including 12 regular pentagons and 630 irregular hexagons, which can be categorized into nine types. The modules created from the different types of faces were identified as Type-PEN and Type-HEX (01–09). There were 19 modules at the beam entrance and one module at the exit removed, resulting in a total of 622 modules and achieving 97% coverage of the 4π solid angle [111].

Fig. 43Full-screen View

(Color online) 3D rendered image of the ECAL geometry [111]

The ECAL should consider the particle energy in sub-MeV and several tens of MeV simultaneously, owing to the different physics goals in MACE Phase-I and MACE. In the 0.511 MeV energy regime of interest, a total absorption homogeneous calorimeter with inorganic scintillators generally provides excellent energy resolution, which is expected for MACE. Three types of crystals were considered for the MACE calorimeter: bismuth germanium oxide (BGO), thallium-doped cesium iodide (CsI:Tl), and cerium-doped lutetium-yttrium oxyorthosilicate (LYSO:Ce). The CsI(Tl) crystal was widely employed by the electromagnetic calorimeter of various experiments due to its high light yield [81, 82, 112]. However, it is characterized by a slow emission time, which may cause a loss of efficiency. The short radiation length also increases the cost of materials outside the ECAL. BGO combines an acceptable light output and a fast decay time, and is widely used in positron emission tomography (PET), which also investigates the signal of annihilation γ-rays. In addition, LYSO could be an appealing option because it is brighter and faster than BGO. The properties of these crystals are summarized in Table 10 [113].

Table 10Full-screen View

Properties of BGO, CsI(Tl), and LYSO(Ce) scintillation crystal [37, 113]

Parameter	ρ (gcm3)	X0 cm	RM (cm)	dE/dx (MeV/cm)	τdecay (ns)	λmax (nm)	n	Relative output	Hygro- scopicity
BGO	7.13	1.12	2.23	9.0	300	480	2.15	21	no
CsI(Tl)	4.51	1.86	3.57	5.6	1220	550	1.79	165	slight
LYSO(Ce)	7.40	1.14	2.07	9.6	40	420	1.82	85	no

Show more

The technical specifications of the photonsensor (e.g., the active area and photon detection efficiency) determine the efficiency of light collection. Photomultiplier Tubes (PMTs) and Silicon Photomultipliers (SiPMs) are commonly used photosensors in current particle and nuclear physics experiments. The active area of the PMTs was relatively larger than that of the SiPMs. However, they are characterized by a limited quantum efficiency (~20%), high operating voltage, and shielding requirements in magnetic fields. On the other hand, SiPMs offer excellent photon detection efficiency (~50%) and tolerance to magnetic fields. However, a single SiPM has small areas (usually 6 mm at maximum). Thus, SiPM arrays with large active areas are considered to enhance light collection efficiency.

9.2.3

Simulation

A beam of 10⁸ μ⁺ events with a momentum of 26.3 MeV/c entering the MACE detector has been simulated to evaluate the signal efficiency of ECAL. Following coincidence with MCP, a clustering algorithm was applied to search the seed modules and their adjacent modules. The energy of each triggered module was smeared, assuming an energy resolution of 10% at 0.511 MeV. We then estimated the energy sum and triggered time of the cluster, and applied an energy difference cut of ΔE_recon<0.1 MeV to the reconstructed annihilation events. The resulting ECAL response spectra are shown in Fig. 44. Events that fell within the interval of the full-energy peak were identified as signal events.

Fig. 44Full-screen View

(Color online) Spectra of a single Type-PEN module (blue line), a single Type-HEX module (green line) and reconstructed annihilation signals (red line) of M-to- conversion by simulation

Performance

The overall signal efficiency of the PDS can be derived from the MCP and ECAL results. First, the detection efficiency of MCP depends on the kinetic energy of the incident positron. According to the simulation results, the MCP efficiency for the positrons accelerated by the PTS was 32.6%. Second, we found that the primary components of the MCP, lead glass, and aluminum may induce multiple Compton scattering of 0.511 MeV photons. This effect resulted in a 36.6% efficiency loss, leading to a 63.4% incident efficiency of annihilation events for the ECAL. Additionally, the geometric and reconstruction efficiencies of ECAL were considered to be 95.3% and 94%, respectively. The total signal efficiency can be estimated as(35)The values of these efficiencies are summarized in Table 11. Consequently, this analysis yielded a signal efficiency of 18.5% for the PDS. A higher efficiency can be achieved using MCP with a low-material-budget substrate.

Introduction

The significance of offline software has increased in high-precision frontier experiments. The MACE offline software is responsible for the physical event generation, detector simulation, event reconstruction, offline event display, and analysis. In the conceptual design stage, offline software supports Monte Carlo simulations of physical and background events to guide the detector system design and optimization. During the engineering stage, offline software plays a crucial role in detector alignment, and can enhance the understanding of real circumstances through detailed simulations. Once the physical run begins, the offline software serves as a bridge that connects the experimental data to the physical results. A high-quality offline software system can simplify the workflow of researchers and potentially reduce systematic errors by accommodating more advanced technologies and algorithms, leading to accelerated research progress and increased physical sensitivity.

There are some objectives to ensure the expected functionality and performance of offline software.

Framework

The Mustard framework is the most fundamental library of the MACE offline software. Mustard is a generic offline software framework for particle physics experiments [114], which provides a modern, high-performance architecture for distributed computing capabilities, unified geometric interfaces, and a high-level abstraction layer of data models out of the box, accelerating the development and performance of applications and libraries. The Mustard framework was formerly developed within the MACE offline software and is now extracted as a general proposed framework. It is designed as a set of commonly used interfaces and functionalities in fixed-target experimental simulation, reconstruction, and analysis, as shown in Fig. 45.

Fig. 45Full-screen View

The architecture of MACE offline software

The framework was mainly developed with a concept-based object-oriented paradigm and template metaprogramming paradigm, utilizing the language constructs and standard library features of C++20. This approach enables the development of performance-critical utilities and interfaces without introducing extra runtime overheads. Following the technology roadmap, a low-level component inspired by the principles of the C++ standard library involves a set of useful concepts, type traits, functions, and an optimized numeric algorithm. The low-level components are predominantly found in muc [115] and in the Concept and Utility components within Mustard.

Built on top of low-level infrastructure, Mustard provides a set of interfaces and utilities for distributed computing, simulation, customization of data models and distributed data processing, parameterized geometry construction and serialization, command argument parsing, logging, etc. An environmental component manages the resources and variables required for these functionalities. Mustard directly depends on MPI [116], ROOT [117], Geant4 [103, 118, 119], Eigen [120], and other libraries to form a high-level abstraction of these functionalities while maintaining compatibility with the common workflow of these underlying frameworks. A detailed description of these functionalities is provided in subsequent sections.

Parallel computing

MACE will be operated with a muon beam with a flux of 10⁸/s and will collect data from 3×10¹⁵ muon or muonium decay events. Consequently, distributed parallel computing is essential for handling data processing requirements and carrying out simulations during the design stage. Two parallel computing models are investigated:

Both models demonstrated excellent scalability and performance, although they differed in practice. The hybrid model is found to be more memory-efficient and provides greater flexibility owing to its capability to finely control intra-node parallelism. However, it shows a slightly lower speedup owing to increased thread synchronization and potentially poorer CPU cache locality, as well as increased complexity in coding. On the other hand, the pure message passing model is slightly more memory consuming because of data (e.g., Geant4 data) duplication for each process within a node, but it offers superior speedup and reduced technical complexity. These are the results of potentially better CPU cache locality and absence of thread competition. Consequently, the pure message passing model was selected as the parallel computing model for the MACE offline software.

The message passing model is implemented based on the Message Passing Interface (MPI) [116]. MPI offers a scalable interface for distributed computing on high-performance computing (HPC) clusters and high-throughput computing (HTC) clusters. Within the Mustard framework, a generic executor based on MPI is provided for dynamically scheduling computing tasks among all the executing processes. The executor handles scheduling and execution while users customize the actual tasks. Building upon this extensible functionality is a Geant4 simulation run manager and an event-by-event data processor as solutions to distributed Geant4 simulation and distributed event reconstruction or analysis. In the executor, a master/slave pattern communication algorithm is employed for task division and assignment, and the overlap between computation and communication is maximized to minimize latency and enhance performance. A speedup test was performed and the results are shown in Fig. 46. The results demonstrate a linear speedup on 1920 cores (80 computing nodes) in the event loop of a benchmark simulation example, and no significant performance degradation was observed. Thus, it is concluded that the distributed computing infrastructure can meet the requirements for the future development of MACE offline software.

Fig. 46Full-screen View

Speedup test at Tianhe-2 supercomputer. Each node has 24 cores. Results shows a linear speedup, and no significant performance degradation is observed

Event data model

The event data model is in the central position of the offline software system. It conceptualizes how data are produced and stored, data streamlines, and how data should be processed. Therefore, it conceptualizes the large functional parts of offline software. The Mustard framework provides an interface for building a static event data model, encoding field information in the C++ type system with the help of a template metaprogramming infrastructure. This enables the checking of data model consistency and usage in user code at compile time. With the names and types of all fields known at the compile time, Mustard also provides a zero-overhead interface that maps the data model to a tuple-like type. This interface serves as a bridge between the data model and the actual data object, which ensures consistency and is practical in any operation, such as reading/writing data from/to a disk or applying an algorithm to process the data. Based on the data model interface, the MACE offline software predefined some commonly used data models for simulation, reconstruction, and analysis applications. It is also simple to extend data models to developers and users. However, owing to the static nature of the event data model interface and architecture, it remains a challenge to dynamically define the data model via configuration files, such as JSON or YAML. This issue will be addressed in future studies.

Detector geometry and field

Geometric system are extensively used in both simulation and reconstruction to describe and define the geometry, material, and field of the detector system. The task of geometric framework is to describe the information required for both humans and machines. This process involves two aspects. A human-readable geometry description is generally a characteristic geometric parameter such as the inner radius of the CDC and the number of collimator foils. The machine needs a full description of the entire geometry, such as the position and transformation of every CDC wire. Therefore, a gap exists between the real human-readable parameters and exact geometry definition.

The geometric interface of the Mustard framework aims to address this discrepancy. It separates the human-readable detector description from the detailed definition of detector geometry. In this architecture, the detector definitions are responsible for the exact construction of the detector geometry and material, depending on the parameters specified in the corresponding detector descriptions. The detector description does nothing but provides an interface for defining and fetching these parameters. This architecture separates the technical parts from the physical parts during detector construction. The flow chart of detector construction is shown in Fig. 47.

Fig. 47Full-screen View

Flow chart of geometry management

The electromagnetic field is another essential subject in simulations and reconstruction. The Mustard framework provides an interface for users to define a set of reusable fields. The defined field can be used in both the simulation and reconstruction through the corresponding wrappers. The Mustard framework also supports the definition of interpolated electromagnetic fields from discrete field data to fulfill the requirement of importing the MACE field map.

Continuous integration

Continuous integration (CI) is the practice of frequently integrating software code, building, deploying and testing its functionality. CI can keep the software in an available state and efficiently incorporate newly developed functionalities into practice or production. CI also serves as a practice for observing and fixing errors at an early stage. The development branch of the MACE offline software has been continuously building and deploying on the Tianhe-2 supercomputer utilizing the container technique via an Apptainer (formerly known as Singularity) [121]. An autonomous workflow was developed to build fully optimized Apptainer images for different MPI applications. Currently, the image size ranges from approximately 4 GB for images with Geant4 data and approximately 2 GB for images without Geant4 data. They were uploaded to the Tianhe-2 supercomputer and updated on demand. The CI workflow is still under investigation, and it is planned to deploy a CI workflow on local or cloud servers to automatically perform the CI of the MACE offline software in the future.

Event display

Event display software has become increasingly important in experimental particle and nuclear physics research. This software was used to visualize the detector response information and the reconstructed information for the particles that passed through the detectors. The importance of this software is threefold: First, it is essential for outreach and educational purposes. Students and new researchers can use event displays to gain a more intuitive understanding of how particles interact with detectors and how data are collected and analyzed. This visual approach helps to bridge the gap between theoretical knowledge and practical applications, making it easier to grasp complex concepts. Second, by providing a clear and detailed visualization of the event data, researchers may identify inconsistencies or errors in the data processing pipeline that might not be obvious through numerical data alone, allowing them to identify potential issues more easily in data analysis. Third, it is crucial for the online monitoring of data acquisition to ensure data quality. Real-time event displays allow researchers to monitor the data as they are being collected, quickly identifying any issues with the detectors or data acquisition system. This immediate feedback is vital for maintaining high data quality and ensuring that any problems are addressed promptly, thereby minimizing the risk of data loss or corruption.

For MACE, we need to develop event display software for similar purposes. We envision that the software should have the following contents and functionalities:

Game engines are suitable candidates for meeting these requirements, as they are designed to handle complex graphics, real-time rendering, and interactive environments, making them ideal for developing sophisticated event display software. Among the available game engines, Unreal Engine 5 [122] stands out as an optimal choice for several reasons.

Currently, the event display software development is in its early stages. We successfully imported the 3D models for the essential from Geant4 into Unreal Engine 5 (Fig. 48a). For simulated events, the software was already capable of displaying several key elements. With simple keyboard inputs, it can visualize the energy deposition on the ECAL modules for different events, showing how and where particles interact with the calorimeter (Fig. 48b). Additionally, the software can display electron tracks within the spectrometer, allowing researchers to observe the electron trajectories as they move through the magnetic field and eventually produce signals on the TTCs. Furthermore, the software can identify and display the drift cells activated by these particles, providing a detailed view of the particle interactions within the detector (Fig. 48c).

Fig. 48Full-screen View

(Color online) a 3D modules imported from Geant4 into the Unreal Engine, including ECAL, MCP, PTS, TTC and CDC. Sensitive detectors will be made semi-transparent during event displays for better signal visualization. b An event display for the ECAL. Energy deposition in the ECAL modules was displayed using a rainbow color scheme, with purple indicating low energy and red indicating high energy. c A event display for the CDC. The highlighted curved tracks depict the electron trajectory, whereas the semi-transparent colored cylinders indicate the drift cells that were fired. The radii of the cylinders represent drift distances. The electron eventually hits the TTC, leaving detectable signals on several adjacent modules. d An overview of the entire event

Physical backgrounds

a. Muon internal conversion decay . As stated in Sect. 3.2, the SM-allowed muon rare decay is a major source of physical background. We generated 10⁹ events using the Metropolis-Hasting algorithm based on the leading-order squared amplitude of [123]. Kinematic cuts corresponding the MMS geometry were applied to reduce the variance in the simulation results. Assuming a 10⁸ μOT/s muon-on-target (μOT) flux, the generated internal conversion decay events correspond to 114.12 years of data, predicted by leading-order QED calculations based on McMule [124]. After event selection, the background rate was estimated to be 0.29±0.02/(10⁸ μOT/s·1 yr).

Accidental backgrounds

b. Beam-related Backgrounds. The muon beam produced by stopping π⁺ on the surface of a target typically contains positrons or other particles of the same momentum [125]. These backgrounds are considered accidental coincidences. As shown in Sect. 4.2.2, the ratio of positrons to muons will reach the level of 1%. Assuming a 10⁸ μOT/s muon-on-target flux, the upper limit of the background rate is estimated to be 0.07/(10⁸ μOT/s·1 yr) at 90% C.L.

c. Cosmic-Ray Muon Backgrounds. The EcoMug generator [126] was employed to generate cosmic-ray muon events for the simulation. The ECAL was placed at the center of a hypothetical tunnel surrounded by a copper solenoid and iron shield. Cosmic-ray muons are set to be emitted from a plane source at sea level. No event passed the event selection, and the 90% C.L. upper limit of the background rates was estimated to be 2/(10⁸ μOT/s·1 yr). Generally, a veto system can be installed outside the ECAL to further exclude the background contribution of cosmic-ray muons [2, 127].

Introduction

Taking advantage of the optimized MACE detector system and the construction of high-intensity muon beams in China, the search for other cLFV processes involving muons could be forthcoming. Therefore, we propose the MACE Phase-I experiment aimed at further physical goals. This will help evaluate the performance of particular sub-detectors before the full construction of MACE. Additionally, a measurement of muon polarization, similar to that performed in the MEG experiment [128], can be conducted to characterize the energy and angular distributions of the muon decay products.

Figure 50 presents the layout of Phase-I detector system. The detector system is based on the ECAL and an extra inner tracker system, which consists of a scintillating fiber (SciFi) tracker and a multigap resistive plate chamber (MRPC) hodoscope. The tracker system is designed to provide the hit position and hit time of e⁺ which could be a signal or background, depending on specific processes. The photons barely trigger the SciFi and MRPC, such that they can only be reconstructed by ECAL. The cLFV processes in the detector’s capacity may include μ⁺e^- scattering and the rare decay of muons and muonium. To better emphasize the necessity of the Phase-I experimental stage, efforts to search for such processes in history are reviewed as follows.

Fig. 50Full-screen View

(Color online) The MACE Phase-I detector concept (an event is displayed for example)

Muon beam

An estimation of the requirements for a muon beam is presented in Tab. 14. The beam intensity is primarily limited by the sensitivity and rate of accidental coincidences. An excessively low intensity is insufficient to meet the sensitivity goals of Phase-I, whereas an excessively high intensity would lead to an increased rate of accidental coincidences, which would not contribute further to enhancing the sensitivity. As a conservative estimation, the optimal beam intensity ranged from 10⁶ to 10⁷ μ⁺/s. In the next few years, the surface muon beam provided by CiADS is expected to achieve the required beam parameters.

Design of the detector system

13.3.2

Tracker system

The timing and spatial resolution of ECAL are limited. More importantly, ECAL can hardly discriminate between e⁺ and γ by itself. An inner tracker system was designed to identify e⁺s and measure their arrival times and tracks. First, three layers of scintillating fibers were arranged along the surface of the cylinder outside the beam pipe. It reconstructs the track and hit times of incoming e⁺. Second, the outer layer is an array of multigap resistive plate chamber (MRPC), which offers extremely high timing precision. This combination ensures a great enhancement of sensitivity by reducing any accidental and physical background as a result of increased tracking and timing precision.

a. Scintillating fiber tracker A scintillating fiber (SciFi) tracker for high-precision timing and tracking was designed. It is expected to have a time resolution of ns and a high detection efficiency for distinguishing the signal from the background. In the current design, the SciFi tracker comprises three layers: two helical layers and a transverse layer. The helical fibers were arranged at a tilt angle to ensure that they rotated 360° in the azimuthal angle ϕ from one side of the veto to the other side, whereas the transverse fibers extended parallel to the axial direction. Each layer contains 200 fibers and extends 325.4 mm along the axial direction with an average thickness of 1.3 mm, as shown in Fig. 51. The radii of each layer were 45 mm, 50 mm and 55 mm, respectively. This design resulted in a geometric acceptance of 95.6%. The 1 mm width single cladding square scintillating fibers produced by Kuraray were considered in the benchmark design [132]. Each fiber was mechanically held in place independently. The scintillation light emitted in the fibers is transported by light guide fibers and read out by SiPMs, where the signals of each fiber are output individually. The total number of channels was 600 in the current design.

Fig. 51Full-screen View

(Color online) a Overall view of the SciFi Tracker. The radius of the detector is approximately 5.5 cm, the length is approximately 32.5 cm. b Details of the SciFi tracker. In this figure, One of the three layers is made up of two layers of fibers. Each Scintillating fiber (green) is connected with an optical fiber (blue) and a SiPM (orange) is tied at the end of the optical fiber

b. Multigap resistive plate chamber hodoscope Although the SciFi tracker can provide good spatial resolution, better timing capability is required to further suppress potential physical and accidental backgrounds. MRPC is typically used in high energy physics experiments as a time-of-flight detector owing to its high performance, simplicity, and low cost. The detector is mainly made of glass and plastic, and is finally housed in an aluminum alloy gas chamber filled with working gas. MRPC can achieve a detection efficiency exceeding 95% for charged particles, along with an excellent timing resolution of approximately 50 ps and mm spatial resolution [133]. The specification perfectly matches the requirements of the MACE Phase-I. Under the current design, eight MRPC modules were situated between the SciFi tracker and the ECAL, fully covering the radial direction. The MRPC tags the charged particle along with the SciFi and provides high-precision time information for event reconstruction.