Muonium to Antimuonium Conversion Search Using an LGAD-Based Detector System: A Simulation Study

Yu-ting Wang¹²³, Chen Wu¹², Yu Bao¹², Ji-feng Hu³, Yu-hang Guo¹², You Lv¹², Cong Chen¹²

¹Spallation Neutron Source Science Center, Dongguan 523803, China
²Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China
³Institute of Quantum Matter, South China Normal University, Guangzhou 510006, China

Yu Bao
yubao@ihep.ac.cn

Chen Wu
wuchen@ihep.ac.cn

Abstract

Muonium to antimuonium conversion research would greatly benefit from an intense pulsed muon source. However, high-intensity muon beams induce elevated pile-up rates in detector systems, thereby reducing overall efficiency. Here we propose a Low Gain Avalanche Detector (LGAD)-based system to search for the muonium to antimuonium conversion process. We design and simulate the detector system, accounting for LGAD dead time to evaluate pulse pile-up-induced efficiency loss for signal events under various beam intensities. The upper limit on the conversion probability as a function of beam intensity has been assessed. Our results indicate that at a muon beam intensity of 10⁶ μ⁺/pulse, the upper limit on the conversion probability can reach 4.67×10⁻¹² (at 90% C.L.) with approximately 32.97% total efficiency. Although detection efficiency decreases due to pile-up, the experimental sensitivity increases with beam intensity. When the beam intensity reaches 10⁸ μ⁺/pulse, the sensitivity can achieve 5.93×10⁻¹⁴ (at 90% C.L.), while the detector efficiency begins to decline rapidly.

Keywords: muonium to antimuonium conversion; LGAD; muon physics; cLFV

1 Introduction

The phenomenon of neutrino oscillation provides compelling evidence for neutrino mass and lepton flavor violation in the neutral lepton sector of the Standard Model (SM). This discovery suggests that similar lepton flavor violations (LFV) may also occur in the charged lepton sector. Within the SM framework, charged lepton flavor violation (cLFV) processes are heavily suppressed by the GIM mechanism and the extremely small neutrino masses, yielding branching ratios as tiny as 10⁻⁵⁴ that render their detection practically impossible. However, theories beyond the SM naturally accommodate cLFV processes at rates potentially accessible to experiments. Discovering any cLFV process would constitute strong evidence for new physics beyond the SM, driving extensive research into various cLFV channels. Searches have been conducted in high-energy collisions, including Z boson rare decays at LEP \cite{1}, τ rare decays at LHCb and BELLE II \cite{2-4}, J/ψ rare decays at BESIII \cite{5-6}, and prospective future lepton collider studies of e⁺e⁻ → ℓ⁺ℓ⁻ \cite{7}. Nevertheless, many of the most stringent limits on cLFV have emerged from low-energy muon experiments. For instance, the MEG II experiment aims for sensitivity of O(10⁻¹⁴) for the μ → eγ process \cite{8}, the Mu3e collaboration at PSI targets O(10⁻¹⁶) sensitivity for μ → eee \cite{9}, and the COMET \cite{10} and Mu2e \cite{11} experiments probe μ⁺ + Z → e⁺ + Z conversion with sensitivities of O(10⁻¹⁷).

Muonium to antimuonium conversion represents another cLFV process that violates lepton flavor number by two units. Muonium is a hydrogen-like atom composed of an electron and a positively charged muon. Due to its simple structure governed solely by electroweak interactions, the muonium system serves as an excellent platform for testing quantum electrodynamics (QED) and exploring new physics beyond the SM, including Lorentz and CPT violations in the Standard-Model Extension (SME) \cite{12}, antimatter gravity research \cite{13}, dark sector searches \cite{14}, and neutrino-mediated forces \cite{15}. The most recent muonium to antimuonium conversion experiment was conducted at PSI in 1999, establishing an upper limit on the conversion probability of P < 8.3×10⁻¹¹ at 90% confidence level \cite{16}. The Muonium-to-Antimuonium Conversion Experiment (MACE) \cite{17} aims to improve this limit by two orders of magnitude, and another experiment has been proposed at the Japan Proton Accelerator Research Complex (J-PARC) \cite{18}.

Enhancing the sensitivity of muonium to antimuonium conversion searches requires increasing muon beam intensity. Additionally, a pulsed muon source can significantly suppress background due to the time-dependent nature of the conversion process \cite{19}. However, higher beam intensities challenge detection systems, as pile-up in trackers complicates particle detection and reconstruction, particularly for pulsed beams. This study proposes adopting Low Gain Avalanche Detectors (LGADs) to address the pile-up problem. LGADs, with their superior time resolution and short dead time, can substantially improve the tracker’s ability to mitigate pile-up \cite{20}.

The objective of this paper is to evaluate the potential of an LGAD-based detection system for searching for muonium to antimuonium conversion using a pulsed muon beam. The muon beam parameters, including momentum, time, and spatial distributions, are based on the surface muon beam of the MELODY project \cite{21}.

2 Experimental Setup

Muonium atoms are produced by directing a positive muon beam onto a target. These surface muons exhibit a low-energy distribution with narrow spread, making them favorable for muonium production. Upon collision, many stopped muons capture electrons from the material to form muonium, but only a small fraction diffuses into vacuum before decaying. Several target materials, including hot tungsten, silica powders, and silica aerogels, have proven effective for producing thermal muonium in vacuum. Among these, laser-ablated aerogel has demonstrated an enhanced vacuum yield of 3% per stopped muon \cite{22}, which this study adopts for muonium generation.

Muonium to antimuonium conversion yields a distinctive decay signature: an atomic shell positron, an energetic electron from the Michel decay of the μ⁻, and two neutrinos. Detection of the Michel electron with energy up to 53 MeV and an atomic shell positron with average kinetic energy of 13.5 eV, both originating from the same vertex, defines the signal for muonium to antimuonium conversion.

[FIGURE:1] (left) depicts the experimental setup for the muonium to antimuonium conversion experiment modeled in Geant4 \cite{23}. The apparatus comprises three principal components: the Michel Electron Tracker (MET), a transport region, and the Track Electron Detector (TED), all situated within a 0.1 T axial magnetic field.

The MET, located on the left side of the detection system, consists of five layers of barrel-shaped LGAD detectors, with each layer composed of multiple LGAD pixels as shown in [FIGURE:1] (right). The radii of the tracker layers are approximately 64 mm, 128 mm, 192 mm, 256 mm, and 320 mm, with lengths of 360 mm, 470 mm, 580 mm, 690 mm, and 800 mm, respectively. The LGAD pixel size is 1.3 × 1.3 mm². The muonium production target is a silica aerogel piece at the MET center that stops muons and generates muonium in vacuum. An axial electric field accelerates atomic shell positrons to 8 keV. Adjacent to the MET lies a transport region consisting of two straight tubes and one 90-degree bent tube. Particle momentum is selected by a bending magnetic field, while a collimator selects the positron's transverse momentum. A Track Electron Detector (TED) with an LGAD-based detector surrounded by CsI scintillators for photon detection is positioned at the transport region's endpoint for coincidence measurement.

Two primary background sources exist. The first involves the rare decay μ⁺ → e⁺νₑν̄_μe⁺e⁻ with branching ratio (3.4 ± 0.4) × 10⁻⁵, producing a high-energy electron, a low-energy positron, and an undetected positron. The second arises from accidental coincidences where Michel positrons undergo Bhabha scattering near the target region, transferring momentum to become low-energy positrons and creating a false signature when paired with the scattered electron.

In both processes, positrons typically have higher energy than orbital positrons, resulting in different flight times compared to signal events. Additionally, particles with different energies experience varying degrees of trajectory deflection in the transport region, further distinguishing them from signal particles in vertex reconstruction. Effective background suppression under high-intensity conditions requires excellent timing and spatial resolution.

3 Simulation

Increasing beam intensity is crucial for enhancing search sensitivity, as the number of muonium atoms emitted into vacuum scales with muon beam intensity. This relationship is expressed as:

N_MU = N_in · P_stop · P_vac, [Eq.1]

where N_in is the number of incident muons per pulse, P_stop is the probability that incident muons stop in the silica aerogel target, and P_vac is the probability that stopped muons form vacuum muonium atoms. Based on literature \cite{22}, this simulation adopts P_stop = 50% and P_vac = 3%. However, high beam intensity causes pile-up in the MET, as multiple beam noise events overlap with genuine signals and affect their detection. These noise events originate from decays of stopped muons and muonium atoms in the target material and surrounding vacuum, with total number approximately half the incident positive muon count. Most muons in these events, including those bound in muonium atoms, decay into two neutrinos and a high-energy positron with branching ratio approaching 100%.

According to Monte Carlo simulations presented below, 94.5% of these positrons enter the MET, further hindering signal event detection. In contrast, the number of particles reaching the TED is quite small due to transport region limitations such as magnetic fields, curved beam pipes, and collimators. Consequently, pile-up effects in the TED are considered negligible in this study.

To facilitate detection efficiency calculations in subsequent sections, this section provides a detailed description of beam noise and signal event generation, along with their respective hit distributions within the MET.

3.1 Beam Noise Simulation

Since vacuum-emitted muonium atoms account for only 3% of beam noise events, we focus on simulating the decay of stopped muons within the target. The simulation involves a pulsed muon beam impinging on a 7 mm thick silica aerogel target with density 29 mg/cm³, with parameters derived from literature \cite{22}. Beam parameters follow the MELODY design \cite{21}.

Stopped muons subsequently decay via μ⁺ → e⁺νₑν̄_μ, and the resulting energetic positrons are detected by the MET while the two neutrinos remain undetected. [FIGURE:2] schematically illustrates beam noise event simulation in Geant4, where the yellow line represents the incident positive muon, green lines denote the two neutrinos from decay, and the blue line indicates the energetic Michel positron.

3.2 Signal Simulation

The signal process comprises muonium to antimuonium conversion in vacuum near the target, followed by antimuonium decay (e⁻e⁺ → e⁻νₑν̄_μe⁺). This simulation begins with the final-state electron and positron, whose spatial distributions and kinematic properties are determined as described below.

Neglecting muonium diffusion within the target, the initial muonium position can be approximated as a Gaussian-distributed beam spot (5 mm standard deviation) localized on the silica aerogel target surface. Within the target, muonium atoms undergo thermal motion, and their velocities upon vacuum emission follow the Maxwell-Boltzmann distribution at 300 K. For antimuonium, the decay time distribution follows \cite{19}:

exp(-t/τ_M̄M), where δ = G_MM / (G_F · a_0³) [Eq.2]

where a_0 is the Bohr radius, n is the principal quantum number of the atomic state, G_MM is the effective four-fermion coupling constant, and G_F is the Fermi coupling constant. The initial time in Equation 2 corresponds to muonium formation time. Stochastic sampling of these combined distributions yields the spatial decay distributions of antimuonium atoms ([FIGURE:3]), where the x-direction is horizontal, y-direction is vertical, and z-direction is the muon beam direction, with the target surface center at x = 0 mm, y = 0 mm, and z = 0 mm. Subfigures (a) and (b) display the x- and y-directional position distributions. Given the vacuum beam tube between target and MET, antimuonium atoms reaching the tube wall are assumed to stop and decay there, concentrating decay events near the wall at z = 60 mm. Subfigure (c) shows the decay position distribution along the beam direction, revealing that antimuonium atoms can propagate up to 10 cm from the target surface.

For final-state particles produced by antimuonium decay, the electron energy follows the Michel distribution, while the orbital positron kinetic energy is 13.5 eV. Both particles' momentum directions are simulated as isotropic.

3.3 Hit Distributions in MET

Based on the simulation procedures described above, we obtained the hit distributions for beam noise and signal events in the MET. [FIGURE:4] shows the hit position distributions for beam noise positrons (left) and signal electrons (right), with both simulations using 10 million events. The horizontal axis represents ring ID (as defined in the right panel of [FIGURE:1]), with ring numbers increasing sequentially from left to right within each detector layer. Color-coded curves indicate hit position distributions across different layers. Observed hits predominantly cluster near each layer's central region, consistent with the target's geometric alignment at the detector center. Geometric divergence effects produce progressively more uniform hit distributions at larger radial distances from the target. Due to detector size constraints, particle information beyond the MET's outermost edge is not recorded, causing abrupt drops at both ends of the position hit curves.

For signal events, a complete signal track requires hits in all five MET layers, yielding a signal event geometric acceptance efficiency in the MET of 80.3%. While hit position distributions for signal and beam noise events are similar ([FIGURE:4], left vs. right), causing significant pulse pile-up in the MET at high beam intensities, a clear distinction exists in their hit time distributions, as shown in [FIGURE:5]. The red curve represents signal event hit times, while the blue curve shows beam noise events, with time zero defined as the moment the muon pulse reaches the target. Neglecting the ~1.2 ns flight time of energetic particles in the MET, the beam noise hit time distribution can be treated as the convolution of the beam pulse shape with the muon lifetime. Similarly, the signal time distribution represents the convolution of the beam pulse shape with the antimuonium decay time. The difference in decay times between positive muon decay and antimuonium decay, described by Equation 2, creates a noticeable distinction in arrival times between beam noise and signal hits, reducing the likelihood of signal hits being overlapped by beam noise and thereby mitigating efficiency loss. Furthermore, selecting an appropriate time window can minimize both beam noise events and the dominant background from positive muon decay, highlighting the advantages of using a pulsed muon beam.

4 Pile-up and Total Sensitivity

As beam intensity increases, signal event detection efficiency is significantly reduced by pulse pile-up in the MET. This reduction primarily arises from spatial and temporal overlaps between signal hits and beam noise hits when both arrive at the same pixel during the LGAD's dead time. This section systematically investigates pulse pile-up-induced efficiency loss for signal events.

We define detection efficiency as ε_det, representing the probability that signal tracks in the MET are not piled up with other tracks. Calculating this efficiency requires evaluating how beam noise events interfere with signal events at specific beam intensities. The LGAD produces a pulse signal with full width of approximately 3 ns \cite{20}, necessitating consideration of a 3 ns dead time. We therefore define pile-up as the occurrence of signal hits and noise hits within 3 ns in a single LGAD unit. The detailed workflow for calculating ε_det comprises the following steps:

First, based on [FIGURE:4] (left), we determine the total number of beam noise hits for each ring in different layers. The timing of these hits is modeled as a convolution of the beam pulse width with the muon decay time. Using the total beam noise hit count and convolution function, we identify the number of beam noise hits within 3 ns time intervals at any given moment in ring i (where i is the ring number from [FIGURE:1]), denoted N_bn,i.

Next, to establish the relationship between beam noise hit count N_bn,i and efficiency ε_i (representing the probability that signal hits avoid pile-up in ring i), we conduct Monte Carlo simulations: (1) For a given N_bn,i, beam noise hits are uniformly distributed across all pixels in ring i, as particles are emitted isotropically; (2) Available pixels are identified by counting those with zero hits, representing candidate positions capable of detecting signal hits in this 3 ns interval; (3) Instantaneous detection efficiency for a signal hit in ring i is calculated by dividing available pixels by the total number of pixels in the ring; (4) Steps 1–3 are repeated to obtain the statistically averaged efficiency ε_i.

The final conversion relationship between N_bn,i and ε_i is illustrated in [FIGURE:6], where the five lines correspond to efficiencies of different layers from first to fifth. Using these curves, N_bn,i can be converted into the detection efficiency ε_i of ring i, and the single-event efficiency is derived from the product of individual layer efficiencies ε_i.

Finally, the overall efficiency ε_det at a given beam intensity is determined by statistically averaging over all simulated signal events. [FIGURE:7] (left) shows ε_det as a function of beam intensity N_in. Different colored points represent two simulation scenarios: red squares indicate antimuonium atoms decaying according to Equation 2, while blue triangles represent exponential decay, consistent with both beam noise and dominant background decay forms.

As beam intensity increases, ε_det initially remains stable but subsequently decreases. Pile-up occurs in the MET when beam intensity exceeds 10⁷ μ⁺/pulse, reducing efficiency. The decline rate is further influenced by the decay time structure of signal and beam noise events. After pulsed beam arrival, beam noise events decay exponentially, so pile-up subsides over time, favoring detection of later-decaying signal antimuonium atoms. For the red points, antimuonium decay probability is concentrated around two muon lifetimes, yielding higher detection likelihood than exponentially decaying blue points at the same beam intensity. Consequently, ε_det for the red points decreases more gradually.

Ultimately, based on the pulse pile-up-affected detection efficiency, MET geometric acceptance, and estimated efficiencies for muonium formation, track reconstruction, TED detection, shell-positron transport, DAQ, and signal event selection, we derive the optimistic upper limit of the detectable conversion probability P at a given beam intensity. Assuming no signal events are observed within the region of interest, the 90% confidence level (C.L.) upper limit for P is calculated via Poisson statistics as:

P(0.1 T) < 2.44 / (N_Mu · ε_det · ε_acc · ε_TED · ε_trans · ε_DAQ · ε_cut · ε_recon · T) [Eq.3]

where N_Mu represents the number of produced vacuum muonium atoms from Equation 1, ε_det is the efficiency for recording signal tracks without pile-up, ε_acc is the geometric acceptance efficiency (determined to be 80.3% from Section 3.2 signal simulations), ε_TED and ε_trans are the TED detection and shell-positron transport efficiencies (estimated as 80% and 79% based on the MACS experiment \cite{16}), ε_DAQ and ε_cut represent data acquisition and event selection efficiencies (approximated as 90% in this study), and ε_recon is the reconstruction efficiency (estimated at 80%). Implementing a time window starting 2 μs after pulsed beam arrival significantly suppresses beam noise events due to distinct time structures of muonium and antimuonium decay \cite{19}, alleviating track reconstruction demands. For instance, at 10⁸ μ⁺/pulse beam intensity, the MET receives approximately 275 tracks within the LGAD's 30 ps time resolution after 2 μs, under which conditions the reconstruction efficiency is expected to reach 80%. The beam repetition rate can reach 5 Hz according to MELODY design specifications. Total data acquisition time T is set to 5000 hours for calculations.

The upper limit of conversion probability P as a function of pulse intensity is shown in [FIGURE:7] (right). Red data points correspond to the signal simulation process detailed in Section 3.2. At the MELODY design specification of 10⁶ μ⁺/pulse, the upper limit on conversion probability becomes 4.67×10⁻¹², with 99.6% detection efficiency and 32.9% total efficiency. As beam intensity continues increasing, sensitivity to muonium to antimuonium conversion still improves, but an inflection point appears in the sensitivity curve at approximately 10⁸ μ⁺/pulse. Using a beam intensity of 10⁸ μ⁺/pulse, this LGAD-based detector system could achieve sensitivity of 5.93×10⁻¹⁴, representing a three-order-of-magnitude improvement over PSI's MACS experiment. This advancement primarily stems from the increased number of available muonium atoms: the simulated beam intensity of 5×10⁸ μ⁺/s exceeds MACS's average intensity of 1.8×10⁶ μ⁺/s by approximately 277 times; the simulated vacuum muonium formation efficiency is 0.015 per incident μ⁺, three times higher than PSI's reported efficiency of 5×10⁻³ per incident μ⁺; and the simulated acquisition time extends to 5000 hours, three times longer than MACS's effective runtime of 1730 hours. These cumulative improvements yield a three-order-of-magnitude increase in available muonium atoms for experimental observation.

Beyond 10⁸ μ⁺/pulse, the rate of sensitivity enhancement slows due to significant pulse pile-up-induced efficiency loss. However, blue data points representing signals following exponential decay laws exhibit distinct behavior: once beam intensity surpasses 10⁸ μ⁺/pulse, the upper limit on conversion probability P stabilizes. This discrepancy arises from differences in decay time structures between signal and beam noise processes.

5 Summary

Detection of muonium to antimuonium conversion would provide compelling evidence for new physics beyond the Standard Model. LGAD detectors, with their high time resolution and short dead time, offer significant advantages for searching for muonium to antimuonium conversion at high beam intensities. This study explored the potential of an LGAD-based detection system for such searches. Our simulations show that sensitivity increases continuously with beam intensity, though pile-up increases correspondingly. At the MELODY muon beam intensity of 10⁶ μ⁺/pulse, the upper limit on conversion probability can reach 4.67×10⁻¹² with approximately 32.97% total efficiency. Furthermore, when beam intensity reaches approximately 10⁸ μ⁺/pulse, an inflection point emerges in the sensitivity enhancement curve, corresponding to an upper limit of 5.93×10⁻¹⁴—three orders of magnitude better than MACS results. Beyond this intensity, the rate of sensitivity enhancement slows significantly due to pulse pile-up-induced efficiency loss. The improved sensitivity demonstrates that the LGAD-based detection system is particularly well-suited for exploring new physics even under high-intensity conditions.

Acknowledgements: This work was supported by the Guangdong Basic and Applied Basic Research Foundation (Grant No. 2023B1515120096).

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