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Search for lepton flavour violating muon decay mediated by a new light particle in the MEG experiment

Search for lepton flavour violating muon decay mediated by a new light particle in the MEG... Eur. Phys. J. C manuscript No. (will be inserted by the editor) Search for lepton flavour violating muon decay mediated by a new light particle in the MEG experiment The MEG collaboration 1 a 2,3 4 ab 5 ab 5 a 6 ab A. M. Baldini , F. Berg , M. Biasotti , G. Boca , P. W. Cattaneo , G. Cavoto , 1 ab 1 ab 6 ab 7 ab 7 ab 5 ab F. Cei , M. Chiappini , G. Chiarello , C. Chiri , A. Corvaglia , A. de Bari , 4 a 1 a 1 a 4 ab 7 a M. De Gerone , M. Francesconi , L. Galli , F. Gatti , F. Grancagnolo , 1 a 8,9,10 2 2,3 11 M. Grassi , D. N. Grigoriev , M. Hildebrandt , Z. Hodge , K. Ieki , 8,10 11 11 11 2 12 F. Ignatov , R. Iwai , T. Iwamoto , S. Kobayashi , P.-R. Kettle , W. Kyle , 13 13 13 13 12 N. Khomutov , A. Kolesnikov , N. Kravchuk , N. Kuchinskiy , T. Libeiro , 12 13 11 6 ab 14 G. M. A. Lim , V. Malyshev , N. Matsuzawa , M. Meucci , S. Mihara , 12 11 2 *,11 11 W. Molzon , Toshinori Mori , A. Mtchedilishvili , M. Nakao , H. Natori , 1 ab 14 11 11 11 11 D. Nicolo ` , H. Nishiguchi , M. Nishimura , S. Ogawa , R. Onda , W. Ootani , 11 12 7 ab 2,1 ab 6 a 4 ab A. Oya , D. Palo , M. Panareo , A. Papa , V. Pettinacci , G. Pizzigoni , 8,10 6 a 2 13 5 a 11 A. Popov , F. Renga , S. Ritt , A. Rozhdestvensky , M. Rossella , R. Sawada , 2 1 a 2 7 a 11 P. Schwendimann , G. Signorelli , A. Stoykov , G. F. Tassielli , K. Toyoda , 11 11 6 a 11 8,10 Y. Uchiyama , M. Usami , C. Voena , K. Yanai , Yu.V. Yudin 1 a b INFN Sezione di Pisa ; Dipartimento di Fisica dell’Universita, ` Largo B. Pontecorvo 3, 56127 Pisa, Italy Paul Scherrer Institut PSI, 5232 Villigen, Switzerland Swiss Federal Institute of Technology ETH, 8093 Zurich, ¨ Switzerland 4 a b INFN Sezione di Genova ; Dipartimento di Fisica dell’Universita, ` Via Dodecaneso 33, 16146 Genova, Italy 5 a b INFN Sezione di Pavia ; Dipartimento di Fisica dell’Universita, ` Via Bassi 6, 27100 Pavia, Italy 6 a b INFN Sezione di Roma ; Dipartimento di Fisica dell’Universita ` “Sapienza”, Piazzale A. Moro, 00185 Roma, Italy 7 a b INFN Sezione di Lecce ; Dipartimento di Matematica e Fisica dell’Universita ` del Salento, Via per Arnesano, 73100 Lecce, Italy Budker Institute of Nuclear Physics of Siberian Branch of Russian Academy of Sciences, 630090 Novosibirsk, Russia Novosibirsk State Technical University, 630092 Novosibirsk, Russia Novosibirsk State University, 630090 Novosibirsk, Russia ICEPP, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan University of California, Irvine, CA 92697, USA Joint Institute for Nuclear Research, 141980 Dubna, Russia KEK, High Energy Accelerator Research Organization, 1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan Received: date / Accepted: date Abstract We present the first direct search for lepton fla- Contents vour violating muon decay mediated by a new light particle + + 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 1 X,  ! e X; X ! . This search uses a dataset result- 2 Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 ing from 7:5  10 stopped muons collected by the MEG 3 Search strategy . . . . . . . . . . . . . . . . . . . . . . . . 5 experiment at the Paul Scherrer Institut in the period 2009– 4 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 5 Event reconstruction . . . . . . . . . . . . . . . . . . . . . 6 2013. No significant excess is found in the mass region 20– 2 6 Dataset and event selection . . . . . . . . . . . . . . . . . . 11 45 MeV/c for lifetimes below 40 ps, and we set the most 7 Single event sensitivity . . . . . . . . . . . . . . . . . . . . 12 stringent branching ratio upper limits in the mass region of 8 Statistical treatment of background and signal . . . . . . . . 13 2 11 20–40 MeV/c , down to O(10 ) at 90% confidence level. 9 Results and discussion . . . . . . . . . . . . . . . . . . . . 15 10 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Appendix A Detailed results for di erent lifetimes . . . . . . . 16 Keywords Decay of muon, lepton flavour violation, flavour symmetry, long-lived particle, displaced vertex 1 Introduction The search for charged lepton flavour violating (CLFV) pro- Corresponding author: [email protected] cesses is one of the key tools to probe for physics beyond the arXiv:2005.00339v2 [hep-ex] 8 Nov 2020 2 τ = 10 ps Standard Model (SM) of elementary particles and interac- Crystal Box exclusion plot τ = 100 ps tions. The observation of neutrino oscillations [1–3] showed τ = 1 ns that lepton flavour is not conserved in nature. As a con- - 7 10 τ = 10 ns sequence, charged lepton flavour is violated, even though the rate is unobservably small (< 10 ) in an extension of the SM accounting for measured neutrino mass di erences - 8 and mixing angles [4, 5]. In the context of new physics, in 10 the framework of grand unified theories for example, CLFV processes can occur at an experimentally observable rate [6]. Therefore, such processes are free from SM physics back- - 9 grounds and a positive signal would constitute unambigu- ous evidence for physics beyond the SM. This motivates the e ort to search for evidence of new physics through CLFV processes [7, 8]. - 10 The MEG experiment at the Paul Scherrer Institut (PSI) 10 15 20 25 30 35 40 45 m (MeV/c ) in Switzerland searched for one such CLFV process,  ! e decay, with the highest sensitivity in the world. No evid- Figure 1 Upper limits on MEx2G decay estimated by converting the + + ence of the decay was found yet, leading to an upper limit on upper limits on  ! e from the Crystal Box experiment as a func- + + 13 tion of m . Lines with di erent markers and colours correspond to the branching ratio B( ! e ) < 4:2 10 at 90% con- di erent  . + + X fidence level (C.L.) [9]. Models that allow  ! e decay at an observable rate usually assume that CLFV couplings are introduced through an exchange of new particles much MEx2G decay if we assume that X is more likely to decay heavier than the muon. Negative results by CLFV searches into an e e pair. However, there is a possibility for X to leave open another possibility: new physics exists at a lighter be electrophobic, as pointed out in [29, 30], and searches for scale but with very weak coupling to SM particles. both decay modes can hint at the model behind these decay If a new particle X (with mass m and lifetime  ) X X modes. + + lighter than the muon exists, the CLFV two-body decay The current upper limit on the decay  ! e , + + 11 ! eX may be a good probe for such new physics. The B( ! e ) < 7:2  10 (90% C.L.) from the Crys- experimental signature depends on how the new particle X tal Box experiment [31] can be converted into an equivalent + + decays. In this paper, we report a search for  ! e X; X ! MEx2G upper limit by taking into account the di erence in (MEx2G) decay using the full dataset collected in the detector eciencies [32]; the converted limits are shown in MEG experiment. Here, we assume that X is an on-shell Fig. 1. scalar or pseudo-scalar particle. Axion-like particles [10– Axion-like particle searches from collider and beam 13], Majoron [14, 15], familon [16–19], flavon [20, 21], dump experiments and from supernova observations also flaxion [22, 23], hierarchion [24], and strongly interacting constrain the branching ratio X ! if the axion-like massive particles [25, 26] are candidates for X. particles are generated from coupling to photons [33]. Fig- A dedicated search for the MEx2G decay has never been ure 2 summarises the parameter regions excluded by these done, although some constraints on the X particle parameter experiments. A region with decay length c < 1 cm and space can be deduced by experimental results from both re- 2 m > 20 MeV/c still has room for the MEx2G decay. lated muon decay modes and non-muon experiments; these Based on limits discussed above, we define the target are discussed below. parameter space of this search in the  –m plane as shown X X Current upper limits on the inclusive decay  ! in Fig. 3. + 5 e X are given at O(10 ) for m in the range 13– 2 1 80 MeV/c [27]. However, the current limits do not impose any constraints on the MEx2G decay in the target region 2 Detector of this search. They are complementary, relevant for cases where X is either stable or decays invisibly. For X resulting The MEG detector is briefly presented in the following, em- from muon decays, the only kinematically allowed visible phasising aspects relevant to this search; a detailed descrip- decay channels are X ! e e and X ! . The former can tion is available in [37]. occur at tree level while the latter can occur via a fermion In this paper we adopt a Cartesian coordinate system + + + loop. The current upper limit on  ! e X; X ! e e (x; y; z) shown in Fig. 4 with the origin at the centre of the at a level of O(10 ) [28] give stringent constraints on the magnet. When necessary, we also refer to the cylindrical co- 1 + In these searches, only e is looked at. ordinate system (r; ; z) as well as the polar angle . + + BR(μ → e X,X→ γ γ ) 3 Positrons from the muon decays are detected with a magnetic spectrometer, called the COBRA (standing for COnstant Bending RAdius) spectrometer, consisting of a thin-walled superconducting magnet, a drift chamber array (DCH), and two scintillating timing counter (TC) arrays. The magnet [39] is made of a superconducting coil with three di erent radii. It generates a gradient magnetic field of 1.27 T at the centre and 0.49 T at each end. The diameter of an emitted e trajectory depends on the absolute momentum, independent of the polar angle due to the gradient field. This allows us to select e s within a specific momentum range by placing the TC detectors in a specific radial range; e s whose momenta are larger than 45 MeV/c fall into the ac- Figure 2 Excluded parameter regions for a scalar X with mass m and ceptance of the TC. Furthermore, the gradient field prevents coupling g to 2 s from collider, beam dumps, and supernova [34–36] + + e s emitted nearly perpendicular to the  beam direction (from [33]). In black we show contours of the boosted decay length c of X ! , assuming X to be produced from an at-rest muon from looping many times in the spectrometer. This results + + decay  ! e X. The solid black line corresponds to c = 0:01 cm, in a suppression of hit rates in the DCH. The thickness of the dotted one to 0.1 cm, the dashed one to 1 cm and the dot-dashed the central part of the magnet is 0.2 radiation length to max- line to 10 cm. imise transparency to ; 85% of the signal s penetrate the magnet without interaction and reach the photon detector. Positrons are tracked in the DCH [40]. It is composed of 16 independent modules. Each module has a trapezoidal shape with base lengths of 104 cm (at smaller radius, close Beamdumps to the stopping target) and 40 cm (at larger radius). These modules are installed in the bottom hemisphere in the mag- net at 10.5 intervals. The DCH covers the azimuthal region between 191.25 and 348.75 and the radial region between 19.3 cm and 27.9 cm. It is composed of low mass materials and helium-based gas (He : C H = 1 : 1) to suppress Cou- 2 6 MEG Acceptance lomb multiple scattering; 2:0  10 radiation length path Beamdumps + + + is achieved for the e from  ! e decay at energy of E + = 52:83 MeV (= m c =2, where m is the mass of 0 10 20 30 40 50 60 muon). m (MeV/c ) The TC [41, 42] is designed to measure precisely the Figure 3 Allowed X particle parameter space (white). The blue region e hit time. Fifteen scintillator bars are placed at each end has already been excluded [35] and the red shaded region on the right of the COBRA. They are made of 4  4  80 cm plastic (m & 45 MeV/c ) is inaccessible to MEG. scintillators with fine-mesh PMTs attached to both ends of the bars. The eciency of the spectrometer significantly depends Multiple intense  beams are available at the E5 chan- on E + as shown in Fig. 5. The e energy from the MEx2G nel in the 2.2-mA PSI proton accelerator complex. We use e + + decay is lower than that from  ! e depending on m , a beam of surface muons, produced by  decaying near X and the eciency is correspondingly lower. The large m the surface of a production target. The beam intensity is X + 7 search range is limited by this e ect as shown in Fig. 3. tuned to a  stopping rate of 3  10 , limited by the rate The photon detector is a homogeneous liquid-xenon capabilities of the tracking system and the rate of acci- + + (LXe) detector relying on scintillation light for energy, po- dental backgrounds in the  ! e search. The muons sition, and timing measurement [43, 44]. As shown in Fig. 4, at the production target are fully polarised (P = 1), it has a C-shaped structure fitting the outer radius of the and they reach a stopping target with a residual polarisation +0:05 magnet. The fiducial volume is 800 `, covering 11% of the P = 0:86 0:02 (stat) (syst) [38]. 0:06 solid angle viewed from the centre of the stopping target in The positive muons are stopped and decay in a thin target the radial range of 67:85 < r < 105:9 cm, corresponding to placed at the centre of the spectrometer at a slant angle of 14 radiation length. It is able to detect a 52.83-MeV with 20 from the  beam direction. The target is composed of a 205 m thick layer of polyethylene and polyester (density In the high rate MEG environment, only scintillation light with its fast 0:895 g/cm ). signal, is detected. cβ τ = 1 cm τ (ps) X 4 𝑥 𝑦 COBRA Magnet Drift Chamber e Timing Counter Drift Chamber Liquid Xenon Scintillation Detector Figure 4 The figure shows a schematic view of the MEG detector with a simulated MEx2G event emitted from the target. The top view is shown on the left, the view from downstream on the right. 1.2 100 0.8 0.6 0.4 0.2 40 42 44 46 48 50 52 54 56 E (MeV) 20 25 30 35 40 45 Figure 5 COBRA spectrometer relative eciency as a function of E + 2 m (MeV/c ) normalised to  + (52:83 MeV) = 1. Figure 6 Trigger direction match eciency for the MEx2G decay con- ditional to e and 2 detection as a function of m evaluated with a Monte Carlo simulation (Sect. 4). high eciency and to contain the electromagnetic shower induced by it. The scintillation light is detected by 846 2- inch PMTs submerged directly in the liquid xenon. They are placed on all six faces of the detector, with di erent PMT MEx2G events was neither foreseen nor implemented. Thus, coverage on di erent faces. On the inner face, which is the + + we rely on the  ! e triggered data in this search. densest part, the PMTs align at intervals of 6.2 cm. + + One of the distinctive features of the MEG experiment is The main  ! e trigger, with a prescaling of 1, used that it digitises and records all waveforms from the detectors the following observables: energy, time di erence between + + using the Domino Ring Sampler v4 (DRS4) chip [45]. The e and , and relative direction of e and . The DC was sampling speeds are set to 1.6 GSPS for TC and LXe photon not used in the trigger due to the slow drift velocity. The detector and 0.8 GSPS for DCH. This lower value for DCH condition on the relative direction is designed to select back- is selected to match the drift velocity and the required preci- to-back events. To calculate the relative direction, the PMT sion. that detects the largest amount of scintillation light is used The DAQ event rate was kept below 10 Hz in order to for the , while the hit position at the TC is used for the acquire the full waveform data ( 1 MB/event). It was ac- e . This direction match requirement results in inecient complished using a highly ecient online trigger system selection of the MEx2G signal because, unlike the  ! [46, 47]. e decay, the MEx2G decay has 2 s with a finite opening Several types of trigger logic were implemented and ac- angle, resulting in events often failing to satisfy the direction tivated during the physics data-taking each with its own trigger. The selection ineciency for MEx2G events is 10– prescaling factor. However, a dedicated trigger for the 50% depending on m as shown in Fig. 6. Efficiency (a.u.) Conditional efficiency (%) 5 Finally, the detector has been calibrated and monitored X → gg over all data-taking period with various methods [48,49], en- + + 𝒙 m → e X 1 suring that the detector performances have been under con- trol over the duration of the experiment. 𝑙 𝒙 2 3 Search strategy vtx 𝑷 + 𝒙 + e 𝒙 The MEx2G signal results from the sequential decays of e 𝑡 + + 2 ! e X followed by X ! . The first part is a two- body decay of a muon at rest, signalled by a mono-energetic Figure 7 Decay kinematics and kinematic variables. e . The energy E + is determined by m : E + (m = 0) = e X e X 52:83 MeV and is a decreasing function of m . The sum of energies of the two s is also mono-energetic and an increas- Given the muon decay vertex, the two s’ energies and ing function of m . The momenta of the two s are Lorentz- positions, and m , the X decay vertex x can be computed. X vtx boosted along the direction of X, which increases the accept- Therefore, we reconstruct x by scanning the assumed vtx ance in the LXe photon detector compared to the three-body value of m (Sect. 5.3.1). If the final-state three particles do + + decay  ! e . The final-state three particles is expected not originate at a single muon decay vertex, these variables to have an invariant mass of 105.7 MeV=c (= m ) and the will be inconsistent with originating from a single point. total momentum vector equal to 0. After reconstructing x , the relative time and angles (mo- vtx A physics background that generates time-coincident menta) between X and e are tested for consistency with a + + + e in the final state is  ! e  ¯ . This mode has not muon decay (Sect. 5.3.2 and 5.3.3). yet been measured but exists in the SM. The branching ratio The MEx2G decay search analysis is performed within 2 2 2 is calculated to be  O(10 ) for the MEG detector con- the mass range 20 MeV/c < m < 45 MeV/c at 1 MeV/c figuration without any cut on E [50, 51]. Therefore, its step. This step is chosen small enough not to miss signals contribution is certainly negligible in this search where we in the gaps. Therefore, adjacent mass bins are not statist- apply cuts on E . e ically independent. The analysis was performed assuming The dominant background is the accidental pileup of lifetimes  = 5; 20, and 40 ps; the value a ects only the multiple  s decays. There are three types of accidental signal eciency. background events: We estimate the accidental background by using the data in which the particles are not time coincident. To reduce the + + Type 1: The e and one of the s originate from one  , and possibility of experimental bias, a blind analysis is adopted; the other from a di erent one. the blind region is defined in the plane of the relative times Type 2: The two s share the same origin, and the e is ac- of the three particles (Sect. 6). cidental. The signal eciency is evaluated on the basis of a Monte Type 3: All the particles are accidental. Carlo simulation (Sect. 4). Its tuning and validation are per- formed using pseudo-2 data as described in Sect. 4.1. The main source of a time-coincident e pair in type 1 is + + the radiative muon decay  ! e  ¯ [52]. The sources + + of time-coincident pairs in type 2 are e e ! (e + + 4 Simulation from  decay and e from material along the e trajectory), + + ! e  ¯ with an additional , e.g. by a bremsstrahlung + 3 The technical details of the program of Monte Carlo (MC) from the e , or a cosmic-ray induced shower. simulation are presented in [53] and an overview of the Figure 7 shows the decay kinematics and the kinematic physics and detector simulation is available in [37]. In the variables. The muon decay vertex and the momentum of the + + following we report a brief summary. e are obtained by reconstructing the e trajectory using the The first step of the simulation is the generation of the hits in DCH and TC and the intersection of the trajectory physics events. That is realised with custom written code for with the plane of the muon beam stopping target (Sect. 5.1). a large number of relevant physics channels. The MEx2G The interaction positions and times of the two s within the decay is simulated starting from a muon at rest in the tar- LXe photon detector and their energies are individually re- get; the decay products are generated in accordance with the constructed using the PMT charge and time information of decay kinematics for the given m and  . X X the LXe photon detector (Sect. 5.2). The muon beam transport, interaction in the target, and 3 + In the case of type 2, the e can have low energy and be undetected. propagation of the decay products in the detector are sim- 6 ulated with a MC program based on GEANT3.21 [54] that 0.55 describes the detector response. Between the detector simu- 0.5 lation and the reconstruction program, an intermediate pro- gram processes the MC information, adding readout simu- 0.45 lation and allowing event mixing to study the detector per- formance under combinatorial background events. Particu- 0.4 larly, the  beam, randomly distributed in time at a decay 7 + 1 rate of 3  10  s , is mixed with the MEx2G decay to 0.35 study the e spectrometer performance. The detectors’ op- erating condition, such as the active layers of DCH and the 0.3 applied high-voltages, are implemented with the known time 0.25 40 42 44 46 48 50 52 dependence. E (MeV) e+ In order to simulate the accidental activity in the LXe photon detector, data collected with a random-time trigger Figure 8 E + resolution as a function of E + . e e are used. A MC event and a random-trigger event are over- laid by summing the numbers of photo-electrons detected by account. After the first track fitting in DCH, the track is each PMT. propagated to the TC region to test matching with TC hits. The matched TC hits are connected to the track and then the track is refined using the TC hit time. Finally, the fit- 4.1 Pseudo two data ted track is propagated back to the stopping target, and the point of intersection with the target defines the muon decay To study the performance of the 2 reconstruction, we built vertex position (x ) and momentum vector that defines the pseudo 2 events using calibration data. The following -ray e + + + + + e emission angles ( ;  ). The e emission time (t ) is lines are obtained in calibration runs: e e e reconstructed from the TC hit time minus the e flight time. – 54.9 MeV and 82.9 MeV from  p !  n ! n reac- Positron tracks satisfying the following criteria are se- tion, lected: the number of hits in DCH is more than six, the 7 8 – 17.6 MeV and 14.6 MeV from Li(p; ) Be reaction, reduced chi-square of the track fitting is less than 12, the 11 12 – 11.7 MeV from B(p; 2 ) C reaction. track is matched with a TC hit, and the track is successfully The selection criteria for those calibration events are de- propagated back to the fiducial volume of the target. If mul- tiple tracks in an event pass the criteria, only one track is tailed in [48] and [55]. We take two events from the above calibration data and overlay them, summing the number of selected and passed to the following analysis, based on the covariance matrix of the track fitting as well as the number photo-electrons PMT by PMT. These pseudo 2 events are generated using both data and MC events. of hits and the reduced chi-square. The resolutions are evaluated based on the MC, tuned to data using double-turn events; tracks traversing DCH 5 Event reconstruction twice (two turns) are selected and reconstructed independ- ently by using hits belonging to each turn. The di erence We describe here the reconstruction methods and their per- in the reconstruction results by the two turns indicates the formance, focusing on high-level objects; descriptions of the resolution. The MC results are smeared so that the double- manipulation of low-level objects, including waveform ana- turn results become the same as those with the data. Fig- lysis and calibration procedures, are available in [9, 37]. The ure 8 shows the E + resolution as a function of E + . The e e e reconstruction (Sect. 5.1) is identical to that used in the angular resolutions also show a similar E dependence. + + ! e decay analysis in [9]. The 2 reconstruction was 2 The  + - and  + -resolutions for m = 20 (45) MeV/c are e e X developed originally for this analysis (Sect. 5.2). After re- 12 (15) mrad and   10 (11) mrad, respectively. +  + e e constructing the e and two s, the reconstructed variables The time resolution is   100 (130) ps. t + are combined to reconstruct the X decay vertex (Sect. 5.3). 5.2 Photon reconstruction 5.1 Positron reconstruction Positron trajectories in the DCH are reconstructed using the Coordinates (u; v; w) are used in the LXe photon detector Kalman filter technique [56, 57] based on the GEANE soft- local coordinate system rather than the global coordinates ware [58]. This technique takes the e ect of materials into (x; y; z): u coincides with z, v = r ( ) where r = in in E resolution (MeV) e+ 7 67:85 cm is the radius of the inner face, and w = r r where  (N ) = N = with  being the in pho;i pho;i PMT;i PMT;i pho;i is the depth measured from the inner face. product of quantum and collection eciencies of the PMT. This fitting is performed separately for each : first, the light distribution is fitted with fx ; M g as free paramet- pho;1 5.2.1 Multiple photon search ers, while the other parameters are fixed; next, the light dis- tribution is fitted withfx ; M g as free parameters, while pho;2 A peak search is performed based on the light distributions 2 the other parameters are fixed. on the LXe photon detector inner and outer faces by using TSpectrum2 [59, 60]. The threshold of the peak light yield is set to 200 photons. Events that have more than one peak Energy pre-fitting To improve the energy estimation, M are fitted while the other parameters are fixed. The are identified as multiple- events. pho;1(2) same  (Eq. (3)) is used but only with PMTs that detect more than 200 photo-electrons. 5.2.2 Position and energy The with the larger M is defined as and the second pho 1 largest one is defined as in the later analysis. Hereafter, only the multiple- events are analysed. When more than two s are found, we select the two with the Position and energy fitting At the final step, all the para- largest energy by performing the position-energy fitting de- meters are fitted simultaneously to eliminate the dependence scribed in this subsection on di erent combinations of two of the fitted positions on the value of M initially as- s. pho;1(2) sumed. The best-fit value of M is used to update R Figure 9 shows a typical event display of a 2 event. pho;1(2) 1 and calculations (1) and (2) are repeated again to obtain the Each PMT detects photons from the two s. The key point of final value of M . Finally, it is converted into E : the 2 reconstruction is how to divide the number of photons pho;1(2) 1(2) detected in each PMT into a contribution from each . E = U (x ) H(T ) S  M ; (4) pho;1(2) 1(2) 1(2) where U (x ) is a uniformity correction factor, H(T ) is a 1(2) Calculation of initial values First, the positions of the de- time variation correction factor with T being the calendar tected peaks in (u; v) are used as the initial estimate with time when the event was collected, and S is a factor to con- w = 1:5 cm. Given the interaction point of each within the vert the number of photons to energy. The functions U (x ) LXe photon detector, the contribution from each to each 1(2) and H(T ) are mainly derived from the 17.6-MeV line from PMT can be calculated as follows. Assuming the ratio of the 7 8 Li(p; ) Be reaction, which was measured twice per week. energy of to that of to be E : E = R : (1 R ) 1 2 1 1 1 2 The factor S is calibrated using the 54.9-MeV line from (0 < R < 1, at first R is set to 0.5), the fractions of the 1 1 decay, taken once per year. number of photons from is calculated as 1 1;i Energy-ratio correction Both the MC data and the pseudo- R = ; (1) 1;i + (1 R ) 1 1;i 1 2;i 2 data show an anti-correlation between the errors in E and E as shown in Fig. 10a, while their sum is not biased. where is the solid angle subtended by the i-th PMT from 1;i true Defining R as the R for true energies for MC data and the interaction point. The total number of photons gener- 1 that for energies reconstructed without the overlay for real ated by , M , is calculated from the ratio R and 1(2) pho;1(2) 1;i data, the reconstruction bias in both the MC data and the the number of photons at each PMT N as pho;i pseudo-2 data is apparent by the linear dependence of true all n R =R on R as shown in Fig. 10b. This bias is removed by 1 1 PMT X 1 applying a correction to the reconstructed energies; the cor- M = R  N : (2) pho;1(2) 1;i pho;i i rection coecients are evaluated from the pseudo-2 data with di erent combinations of calibration data. Then, R is updated to R = M =(M + M ) and 1 1 pho;1 pho;1 pho;2 calculations (1) and (2) are repeated with the updated R . Position correction Oblique incidence of s to the inner This procedure is iterated four times. face results in a bias of the fitted positions. This bias was checked and corrected for using the MC simulation. No bias Position pre-fitting Inner PMTs that detect more than 10 photons are selected to perform a position pre-fitting. The This fitting is performed by a grid search in x = (u; v; w) 1(2) 1(2) space for good stability, while subsequent fittings are performed with following quantity is minimised during the fitting: MINUIT [61] for better precision. selected 5 Error is defined as the di erence between the reconstructed energy PMT N M (x ) M (x ) pho;i pho;1 i pho;2 i 1 2 and the true energy deposit for MC data and between the reconstructed = ; (3) (N ) one with 2 and that with single for pseudo-2 data. pho;i i pho;i 8 1,𝑖 𝑖 −1 −thPMT 𝑖 −thPMT Figure 9 Event display of the LXe photon detector for a 2 event (in a development view). The red points show the interaction positions of the two s projected to each face. Each circular marker denotes a PMT. The colour indicates the measured light yield, which is the sum of photons from the two showers induced by the two s as depicted in the right figure. true is observed in the v direction while a significant bias is ob- E is a reconstructed energy, E is the true value, f is served in the u direction. This is because the s from the the fraction of the narrow component, A is a normalisation MEx2G decay enter the LXe photon detector almost per- parameter, E is the transition parameter between the Gaus- pendicularly in the x-y view but enter with angles in the z-r sian and exponential components, and  is the standard view. Since the u bias arises from the direction and the size deviation of the Gaussian component describing the width of the shower, it depends on the u coordinate and the energy. on the high-energy side. The parameters E and  are cor- t E Therefore, the correction function is prepared as a function related with each other, di erent for the narrow and wide true of u and E . components, and are dependent on E . Figure 11 shows 1(2) 1(2) true an example of the PDFs for 2 events with E = 55 MeV true Selection criteria To guarantee the quality of the reconstruc- and E = 12 MeV. tion, the following criteria are imposed on the reconstruc- tion results: the fits for both s converge; the two posi- Probability density functions for position The PDFs of tions are both within the detector fiducial volume defined as position are almost independent of E and hence (m ;  ). X X 1(2) juj < 25 cm ^ jvj < 71 cm; the distance between the two They are represented by double Gaussians with fractions of s on the inner face is d > 20 cm; E > 10 MeV; and uv tail components of  20%. The standard deviations of the 1(2) core E + E > 40 MeV. 1 2 core components are  = (5:4; 4:7; 6:5) mm in (u; v; w) 1(2) tail coordinates, those of the tail components are  = Probability density function for E The probability density 1(2) (29; 19; 45) mm. function (PDF) for E is evaluated by means of the MC 1(2) simulation. To tune the MC, the pseudo-2 data of MC and 5.2.3 Time data are used. It is asymmetric with a lower tail and mod- elled as follows: The interaction time of ( ) can be reconstructed using 1 2 true true narrow narrow P(E j E ) = f  F(E ; E ; E ;  ) the pulse time measured by each PMT (t ) by correcting t E PMT;i true wide wide for a delay time (t ) including the propagation time delay; ;i 1(2) + (1 f ) F(E ; E ; E ;  ); (5) of the light between the interaction point and the PMT and where the time-walk e ect, and a time o set due to the readout electronics (t ): o set;i true F(E ; E ; E ;  ) t E 8 ! t = t t t : (7) true ;i PMT;i delay; ;i o set;i > 1(2) 1(2) (E E ) true A exp E > E E > 2 t > 2 < E The single PMT time resolution  is approximately pro- ! t;i = ; (6) > p E E > t t true true portional to 1= N with  (N = 500)  500 ps, > pe; ;i t;i pe; ;i 1(2) 1(2) >A exp + (E E ) E  E E 2 t : 2 where N is the number of photo-electrons from ( ). pe; ;i 1 2 1(2) 9 5 800 0.59 ± 0.12 (a) narrow (a) E 1.29 ± 0.11 35 narrow 3 E 0.60 ± 0.07 wide E 2.34 ± 0.46 30 wide E 0.57 ± 0.25 −1 −2 40 45 50 55 60 65 70 −3 E (MeV) −4 −5 0 − 4 − 2 0 2 4 2000 f 0.43 ± 0.03 (b) narrow true-deposit E 0.49 ± 0.01 E − E (MeV) � � narrow 1 1 E 0.68 ± 0.10 wide 1500 σ 1.34 ± 0.05 1.1 14 wide E 0.61 ± 0.12 1.08 (b) 1.06 1.04 1.02 6 8 10 12 14 16 18 20 E (MeV) 6 γ 0.98 true 0.96 Figure 11 Energy response to MC 2 events with E = 55 MeV and true E = 12 MeV. The blue curves are the PDFs fit to the distributions. 0.94 2 See text for the formula of the PDFs. 0.92 0.9 0 0.5 0.6 0.7 0.8 0.9 vals; each assumed mass results in a di erent reconstructed X ! vertex position. Figure 10 (a) Scatter plot of the energy reconstruction errors (MC). truedeposit E is the MC true value of the energy deposited in the LXe. (b) 1(2) 5.3.1 X decay vertex Dependence of the reconstructed energy ratio bias as a function of the reconstructed energy ratio (MC). A maximum likelihood fit is used in the reconstruction, with the following observables: These individual PMT measurements are combined to + + + X = (E ; E ; x ; x ; x ;  ;  ): (10) e e e 1 2 1 2 obtain the best estimate of the interaction time of ( ) 1 2 (t ). The following  is minimised: The fit parameters are the following: 1(2) selected 2 = (cos  ;  ; x ); (11) rest rest vtx PMT t t ;i 1(2) 1(2) = : (8) time 2 where  is the emission angle in the X rest frame,  is rest rest (N ) pe; ;i 1(2) i t;i the angle of the photons in the X rest frame with respect to We use PMTs whose light yield from ( ) is 5 times higher 1 2 the X momentum direction in the MEG coordinate system, than that from ( ) excluding PMTs whose light yield is 2 1 and x is the X decay vertex position. The function L() is vtx less than 100 photons or which give large  contribution in defined as follows: the fitting. L() = P(E j cos  ; m ) rest X The E -dependent time resolution for single event is P(E j cos  ; m ) evaluated with the calibration runs and corrected for 2 rest X events using the MC: P(x j cos  ;  ; x ; x +; m ) rest rest vtx e X P(x j cos  ;  ; x ; x ; m ) 2 2 rest rest vtx e X = 338 =E (MeV) + 45 (ps): (9) 1(2) 1(2) P( + j x ; x + ) e vtx e + + P( j x ; x ) e vtx e 5.3 Combined reconstruction P(l j x ; x ;  ; m ); (12) X vtx e X X true In this section, we present the reconstruction method for where l is the X decay length. The term P(x + j x ) is X e + true the X ! vertex assuming a value for m in the recon- omitted by approximating x by x to reduce the fitting X + e 2 2 struction. We scan m in 20–45 MeV/c at 1 MeV/c inter- parameters. true true-deposit R / R 1 1 E − E (MeV) � � 2 2 Entries Entries 10 The energy dependence of the E PDF Eq. (5) is 1(2) 1500 modelled with a morphing technique [62] using two quasi- monoenergetic calibration lines: the 11.7 MeV line from 11 12 the nuclear reaction of B(p; 2 ) C and the 54.9 MeV line from  decay. The PDFs of the position are approximated as double Gaussians to fit better tails in the PDF. The positron angles are compared with those of the flipped direction of the X momentum ((x x )) with vtx e PDFs approximated as single Gaussians. 6 0 0 The decay length is defined as l = jx x j. Under X vtx e 10 20 30 40 Assumed m (MeV/c ) the approximation  ! 0, the PDF is x + 1 l X Figure 12  distribution for the MC signal events at (m ;  ) = X X vtx P(l j x ; x ;  ; m ) =  exp ; (13) X e vtx X X c c X X (30 MeV=c ; 20 ps) as a function of m assumed in the reconstruction. which is defined and normalised for l  0. The approxima- tion is justified because the transverse component of  is x + 2 7 Figure 12 shows the dependence of  on the assumed vtx 1–2 mm [55] while the longitudinal component is largely value of m for the MC signal events, providing another ra- driven by the target thickness ( 0.2 mm), which is to be tionale for Eq. (14). When the assumed value is the same compared with c ranging between  6–30 mm. as the true value (m = 30 MeV/c in this case), the res- We fix  = 20 ps since the vertex reconstruction per- ultant  becomes minimum on average. The e ective m vtx formance is almost independent of  in the assumed range. resolution is  2:5 MeV/c . This likelihood term e ectively penalizes non-zero decay lengths using a scale that is fixed to the average expected 5.3.2 Momentum decay length of 20 ps. The x resolution of the maximum-likelihood fit is vtx Given the vertex position, the momentum of each can be evaluated via the MC to be  = (8; 12) mm in the trans- vtx calculated. The sum of the final-state three particles mo- verse and longitudinal directions. menta, We define an expression to quantify the goodness of the P  P + + P + P ; (15) vertex fit as sum e 1 2 0 1 0 1 2 2 best best X X B E E C B x x C B C B C should be 0 for the MEx2G events. 2 B C B C B C B C = B C + B C vtx @ A @ A E x = ; = ; 1 2 1 2 5.3.3 Relative time 0 1 0 1 0 1 2 2 2 best best best B  C B  C B l C X X B C B C B C X X X B C B C B C B C B C B C + + + : (14) @ A @ A @ A X The time di erence between the 2 s at the X vertex is cal- X X culated as The variables with the superscript “best” indicate the best- ! ! fitted parameters in the maximum likelihood fit and the vari- l l 1 2 t = t t ; (16) 1 2 ables with no superscript indicate the measured ones. Here, c c ( ;  ) = (  +;  +  + ) is the direction opposite to X X e e where l is the distance between the interaction point 1(2) 1(2) + + ( ;  ). e e in the LXe photon detector and the X vertex position, l = 1(2) The  of each variable is the corresponding resolution jx x j. The relative position of the vertices is such that vtx 1(2) when the distribution is approximated as a single Gaussian. this definition is identical to the signed distance defined ac- This expression is not expected to follow a  distribution cording to the X direction and therefore the distribution is because the PDFs of the variables are not in general Gaus- centred at 0 for MEx2G events. sian. The last term is quadratic by analogy with the other The time di erence between and e at the muon ver- terms and its expression has been found to be e ective in tex is calculated as separating signal from background. The rationale for using 1 X Eq. (14) is to provide a powerful discriminator between sig- t + = t t +: (17) e e 1 1 c c nal and background as shown later in Fig. 14f. With the unsigned definition of l the distribution is slightly From the fifth and sixth terms in Eq. (12), the reason for using the o set with respect to 0 for MEx2G events as visible in absolute value of l rather than the signed value with the sign of (x x + )  P + becomes apparent. If the signed value of the de- Fig. 14d. vtx e e cay length were negative, the X angle would be flipped by  and the 7 2 penalty would be huge preventing the fit to succeed. For better display, we take the square root of  here. vtx vtx 11 + + ! e decays, the time of the LXe photon detector is re- Signal (MC) constructed with PMTs around the largest peak found in the peak search (Sect. 5.2.1). This retained16% of the dataset, + + on which the full event reconstruction for the  ! e de- cay analysis was performed. Before processing the MEx2G dedicated reconstruction, we applied an additional event se- C A C + + 100 lection using the  ! e reconstruction results. It was based on the existence of multiple ( 2) s and the total en- 8 + y ergy of the s and e (E ) being jE m j < 0:2m . total total B 0 This selection reduces the dataset by an additional factor −1 of  300. We applied the MEx2G dedicated reconstruction (Sect. 5.2.2, 5.2.3, and 5.3) to this selected dataset. −2 y A blind region was defined containing the events satis- fying the cuts jt +j < 1 ns ^ jt j < 1 ns. This blind region 20 e −3 C A C is large enough to hide the signal. Those events were sent in −4 0 a separated data-stream and were not used in the definition −4 −2 0 2 4 of the analysis strategy including cuts; background events t (ns)  in the signal region were estimated without using events in x x x C A C the blind region. After the analysis strategy was defined, the BG (data) blind region was opened and events in this region were ad- 4 120 ded to perform the last step of analysis. The accidental background can be estimated from the 100 o -time sideband regions defined in Fig. 13. There are three such regions: A, B, and C; each containing a di erent com- bination of the types of background as defined in Sect. 3. The outer boundary of the time sidebands, jt j < 3:5 ns ^ jt j < 3:5 ns, are determined so that the background distri- 0 60 bution is not deformed by the time-coincidence trigger con- dition. The widths x and y in Fig. 13 are the same as the A B −1 outer boundary of the signal region defined depending on −2 m by the signal selection criteria described below. 20 The following seven variables are used for the signal se- −3 lection: −4 0 1. E : the e energy. −4 −2 0 2 4 2. E : the total energy of the three particles. Blind region t (ns) sum  Signal region 3. jP j: the magnitude of the sum of the three particles’ sum momenta. Figure 13 Top: Signal (MC) and bottom: background (sideband data) event distributions in the t + –t plane before the signal selection cri- 4. d : the distance between the 2 positions on the LXe e uv teria are applied. (20 MeV/c , 20 ps) case is shown as an example. The photon detector inner face. time sideband regions (A, B, C) and the signal region (the red box) are 5. t : the time di erence between and e calculated in e 1 also shown. Eq. (17). 6. t : the time di erence between 2 s calculated in 6 Dataset and event selection Eq. (16). 7.  : the goodness of vertex fitting calculated in Eq. (14). vtx We use the full MEG dataset, collected in 2009–2013, as First, we fix the E + selection to require jE + E j < + + e e + was used in the  ! e search reported in [9]. As de- X + 1 MeV, where E is the e energy for the MEx2G decay + + + scribed in Sect. 2, the  ! e trigger data are used in this with m . This selection is also used in the Michel normal- 14 + analysis. In total, 7:5 10  s were stopped on the target. isation described in Sect. 7. A pre-selection was applied at the first stage of the Next, we optimise the cut thresholds for the other vari- + + ! e decay analysis, requiring that at least one posi- ables to maximise the experimental outcomes. Distributions tron track is reconstructed and the time di erence between signals in the LXe photon detector and TC is in the range At this stage, the sum of the multiple s’ energy is reconstructed 6:9 < t < 4:4 ns. At this stage, aiming to select the without being separated into each . LXeTC t (ns) t (ns) + +  e  e 1 1 12 0.1 0.1 0.2 (a) (b) (c) 0.08 0.08 0.15 0.06 0.06 0.1 0.04 0.04 0.05 0.02 0.02 0 0 0 90 95 100 105 110 115 0 5 10 15 20 25 30 20 30 40 50 60 70 80 90 d (cm) E (MeV) |P | (MeV/c) sum sum uv 0.04 0.35 (d) (e) (f) 0.04 0.3 0.03 0.25 0.03 Signal 5042 events 0.2 0.02 Background 99741 events 0.02 0.15 0.1 0.01 0.01 0.05 0 0 0 - 1 - 0.5 0 0.5 1 - 1 - 0.5 0 0.5 1 0 10 20 30 40 50 60 70 80 t + (ns) t (ns) γ e χ γ γ 1 vtx Figure 14 Distributions of variables used in the event selection for (m ;  ) = (20 MeV/c , 20 ps) case. The hatched histograms show the distri- X X bution of MC signal events while the blank histograms that of background events; each histogram is normalised to 1. The vertical lines show the optimised thresholds. (a) The peak value of the signal distribution is at m with FWHM = 2.7 MeV. (c) Cut-o at 20 cm in the background Esum distribution comes from one of the 2 reconstruction conditions. (e) The threshold lines are not visible because they are set to1 ns. For a detailed definition of the variables see Sect. 6. of these variables for the signal and background at a para- cess, we approximate N to  , a selection eciency for BG BG meter set (20 MeV/c , 20 ps) are shown in Fig. 14. All other the background events calculated using the time sideband selection criteria, such as trigger and reconstruction condi- samples selected up to this point. Because of this approxim- tions as well as the E selection, are applied. The time side- ation, the first step leads to suboptimal criteria. band events are used for the background distribution, while In the second step, after all other selection criteria are MC samples are used for the signal distribution. applied, the threshold for  is optimised to give the highest vtx Punzi’s expression [63] is used as a figure of merit F . In this step, to estimate N from the low statistics Punzi BG in the sideband regions, we use a kernel-density-estimation selection F = ; (18) Punzi p p method [64] to model the continuous event distribution. 2 2 b + 2a N + b b + 4a N + 4N BG BG BG 2 The cut thresholds are optimised at 5 MeV=c intervals where a and b are the significance and the power of a test, in m , while the same thresholds are used for di erent X X for each m . The optimised thresholds for m = 20 MeV=c respectively,  is the selection eciency for the sig- selection X X nal, and N is the expected number of background events. are shown as black lines in Fig. 14. These cuts result in BG = 67% (m = 45 MeV/c ) – 51% (m = 20 The values of a and b should be defined before the analysis, selection X X and we set a = 3; b = 1:28 (= 90%), where b is set to the MeV/c ). value appropriate to the confidence level being used to set the upper limit when a non-significant result is obtained. 7 Single event sensitivity The optimisation process is divided into two steps. In the first step, we optimise the cut thresholds of variables 2–6, in- The single event sensitivity of the MEx2G decay s is defined dependently for each variable in order to maintain high stat- as follows: istics in the sidebands. Because the absolute value of N BG does not make sense in this independent optimisation pro- B = s N ; (19) MEx2G MEx2G Probability / (0.02 ns) Probability / (0.36 MeV) Probability / (0.02 ns) Probability / (0.30 MeV/c) Probability / (2.00 ) Probability / (1.60 cm) 13 3.5 where N is the expected number of signal events in the MEx2G + 3 signal region. We calculate it using Michel decay ( ! + + + 2.5 e  ¯ ) events taken at the same time with the  ! e trigger. This Michel normalisation is beneficial for the fol- 1.5 lowing reasons. First, systematic uncertainties coming from the muon beam are cancelled because beam instability is in- 0.5 + + cluded in both Michel triggered and the  ! e triggered + 20 25 30 35 40 45 events. Moreover, we do not need to know the  stopping m (MeV/c ) rate nor the live DAQ time. Second, most of the systematic uncertainties coming from e detection are also cancelled. Figure 15  (see the text for the definition) versus m for  = 20 ps. 2 X X The absolute value of e eciency is not needed. The number of Michel events is given by 97% (m = 20 MeV/c ) increasing monotonically with m . X X B  f Michel Michel The estimate of  is shown in Fig. 15;  = 0:6% (m = 45 N = N +   A   ; (20) 2 2 X Michel  Michel Michel p  p 2 2 Michel correction MeV/c ) – 2.9% (m = 20 MeV/c ), decreasing monoton- where ically with m . This dependence comes mainly from the 2 acceptance: for increasing m , the opening angle between + X N + : the number of stopped  s; the 2 s becomes larger, resulting in a decreasing eciency. B : branching ratio of the Michel decay ( 1); Michel The systematic uncertainties are summarised in Table 1. f : branching fraction of the selected energy region Michel The uncertainty in the 2 detection eciency and that in the (7%–10% depending on m ); 7 MC smearing parameters are the dominant components. p : prescaling factor of the Michel trigger (= 10 ); Michel The estimated value of SES is s = (2:9  0:3)  10 p : correction factor of p depending on the correction Michel 2 10 2 (20 MeV/c ) – (6:31:1)10 (45 MeV/c ) for  = 20 ps muon beam intensity; increasing monotonically with m . The e eciency is  = X e A : geometrical acceptance of the spectrometer for Michel 2 2 + 1% (45 MeV/c ) – 36% (20 MeV/c ) decreasing mono- Michel e s; + tonically with m , estimated with the MC, although this : e eciency for Michel events within the geomet- Michel quantity is not necessary for the normalisation. The over- rical acceptance of the spectrometer. all eciency for the MEx2G events conditional to the e in The number of MEx2G events is given by the geometrical acceptance of the spectrometer is therefore 5 2 3 2 B  = 2:0 10 (45 MeV/c ) – 4:7 10 (20 MeV/c ) MEx2G MEx2G + + + N = N   A         ; (21) MEx2G  e e 2 DM selection decreasing monotonically with m . p X MEG where + + p : prescaling factor of the  ! e trigger (=1); MEG 8 Statistical treatment of background and signal A + : geometrical acceptance of the spectrometer for MEx2G e s; In the following, we describe how we estimate the expec- + + + : e eciency for MEx2G events conditional to e s in ted number of background events in the signal region (N ) BG the geometrical acceptance of the spectrometer; from the numbers of events observed in sidebands A, B, and : the product of 2 geometrical acceptance and 2 trig- obs obs obs C (N , N , and N ). A B C ger, detection, and reconstruction eciency, conditional There are three types of accidental background events to the e detection; defined in Sect. 3. The expected number of background : the trigger direction match eciency conditional to DM events in the signal region is given by the e and 2 detection (Fig. 6); : the signal selection eciency. selection N = N + N + N ; (23) BG 1 2 3 Using Eqs. (19)–(21), an estimate of the SES (s ) is where N ; N ; N are the expected numbers of background 1 2 3 given by events in the signal region from the types 1, 2, and 3, re- 1 p  p Michel correction spectively. Sideband A has the contributions from types 2 s = N Michel B  f p Michel Michel MEG and 3, B has the contributions from types 1 and 3, and C has A +  + e e the contribution from type 3. : (22) 2 DM selection Michel Michel Figure 16 shows the time distributions in the sideband The geometrical acceptance of the spectrometer is com- regions. A peak of type 2 on a flat component of type 3 is mon, hence A =A = 1; the estimate of the relative observed in the t distribution, while a peak of type 1 is e Michel + 2 e eciency is  += = 89% (m = 45 MeV/c ) not clearly visible in the t + distribution. The uniformity of e Michel X e ∈ (%) 2γ 14 Table 1 Systematic uncertainties in the single event sensitivity ( = 20 ps). m (MeV/c ) 20 25 30 35 40 45 Michel e counting 0.99% 1.1% 1.1% 0.93% 1.6% 2.8% Relative e eciency 1.4% 0.18% 0.31% 0.55% 0.89% 1.3% 2 acceptance 1.3% 2.0% 3.4% 2.9% 5.4% 1.9% trigger eciency 0.98% 0.32% 0.26% 0.52% 1.4% 3.2% 2 detection eciency 7.9% 7.9% 7.9% 7.9% 7.9% 7.9% MC statistics 1.8% 1.9% 2.2% 3.1% 1.9% 4.7% MC smearing 4.8% 3.4% 3.8% 5.3% 3.3% 14% Total 9.7% 9.1% 9.7% 11% 10% 17% tion are thus negligibly small compared with the statistical (a) obs obs obs uncertainties in N ; N ; N . A B C Using N ; N ; N , the expected numbers of events in 1 2 3 sidebands A, B, C can be calculated as follows: |t | < 1 ns γ γ 2y 2y exp C C |t | > 1 ns N = N  + N  ; (24) γ γ 1000 2 3 y y B B 2x 2x exp C C N = N  + N  + N  f ; (25) 1 3 2 escape x x A A 2y 2x 2y exp C C C N = N   + N  f  ; (26) 3 2 escape y x y B A B - 3 - 2 - 1 0 1 2 3 t (ns) γ e where x and y are the sizes of the signal regions A(C) B(C) (sideband regions) in t and t + , respectively, as defined in (b) Fig. 13, and f = 0:171 0:003 is the fraction of type 2 escape events in jt j > 1 ns. 2500 The likelihood function for N is given from the Pois- BG |t | > 1 ns γ e son statistics as, |t | < 1 ns γ e (scaled) obs obs obs L (N j N ; N ; N ) BG A B C exp exp exp obs obs obs = P (N j N )P (N j N )P (N j N ): (27) Poi Poi Poi A B C A B C The best estimate of N can be obtained by maxim- BG ising Eq. (27) (listed in Table 2). However, we do not use - 3 - 2 - 1 0 1 2 3 t (ns) γ γ this estimated N in the inference of the signal but use BG obs obs obs 2 (N ; N ; N ) as discussed in the following. Figure 16 Time distributions in the sideband regions for (20 MeV/c , A B C 20 ps). (a) t distributions for jt j < 1 ns (red open circles) and Our goal is to estimate the branching ratio of the MEx2G for 1 < jt j < 3:5 ns (black closed circles). (b) t distributions for decay (B ). The likelihood function Eq. (27) is exten- MEx2G + + jt j < 1 ns (red open circles) and for 1 < jt j < 3:5 ns scaled by the e e 1 1 ded to include B as a parameter and the number of MEx2G ratio of the time ranges (black closed circles). A loose cut is applied: obs mX events in the signal region (N ) as an observable. In ad- jE + E j < 1 MeV^ E < 115 MeV^jP j < 30 MeV=c^ d < e + sum sum uv S 90 cm^  < 80. dition, to incorporate the uncertainty in the SES into the vtx B estimation, the estimated SES (s ) and the true value MEx2G 0 (s) are included into the likelihood function: the accidental backgrounds is examined using these distri- obs obs obs obs L(B ; N ; s j N ; N ; N ; N ; s ): (28) butions; the number of events in (jt +j < 1 ns ^ 1 < jt j < MEx2G BG 0 S A B C 3:5 ns) is compared to the the number of events interpol- Using N ; N ; N and a Gaussian PDF for the inverse of SES, 1 2 3 ated from the region (1 < jt +j < 3:5 ns ^ 1 < jt j < it can be written as, 3:5 ns) scaled by the ratio of the widths of the time ranges obs obs obs obs (2 ns/5 ns). They agree within 1.7% (the central part, includ- L (B ; N ; N ; N ; s j N ; N ; N ; N ; s ) MEx2G 1 2 3 0 S A B C exp exp exp obs obs obs ing type 1, is 1.7% larger than the interpolation). In Fig. 16b, = P (N j N )P (N j N )P (N j N ) Poi Poi Poi S S A A B B t the distribution for 1 < jt j < 3:5 ns is superimposed exp obs 1 1 P (N j N )P (s j s ); (29) Poi Gaus C 0 on that forjt +j < 1 ns after scaling by the time range ratio. exp The tail component of type 2 is consistent in these regions. where N = N + N + N + B =s is the expected 1 2 3 MEx2G The errors on the background estimations by the interpola- number of events in the signal region. Number of events Number of events 15 The best estimated values of the parameter set fB , Table 2 The number of observed events in the sideband regions and MEx2G the signal region and the expected number of background events in the N , N , N , sg are obtained by maximising Eq. (29). 1 2 3 signal region. Among them, only B is the interesting parameter, MEx2G 2 obs obs obs obs while the others are regarded as nuisance parameters  = m (MeV/c ) N N N N N X BG A B C S +0:202 (N ; N ; N ; s). 1 2 3 20 0 0 1 0:048 1 0:046 A frequentist test of the null (background-only) hypo- +0:198 21 0 0 3 0:146 0 0:084 thesis is performed with the following profile likelihood ra- +0:211 22 1 0 5 0:292 0 0:140 tio  as the test statistic [3]: +0:425 23 3 0 3 0:622 0 0:330 ˆ +0:346 L(B ;  ˆ ) MEx2G 24 2 0 1 0:414 1 0:260 (B ) = ; (30) p MEx2G +0:346 25 2 0 3 0:414 0 L(B ;  ˆ ) MEx2G 0:261 ˆ ˆ +0:189 where B and  ˆ are the best-estimated values, and  ˆ 26 0 0 3 0:150 0 MEx2G 0:091 +0:200 is the value of  that maximises the likelihood at the fixed 27 0 0 1 0:050 0 0:049 +0:202 B . The systematic uncertainties of the background es- MEx2G 28 0 0 1 0:048 0 0:046 +0:202 timation and the SES are incorporated into the test by profil- 29 0 0 1 0:048 1 0:046 ing the likelihood about . The local significance is quan- +0:170 30 0 0 0 0:000 0 0:000 tified by the p-value p , defined as the probability to find local +0:202 31 0 0 1 0:048 0 0:046 that is equally or less compatible with the null hypothesis +0:170 32 0 0 0 0:000 0 0:000 than that observed with the data when the signal does not ex- +0:210 33 0 0 0 0:000 0 0:000 ist. +0:210 34 0 0 0 0:000 1 0:000 Since m is unknown, we need to take the look- +0:210 35 0 0 0 0:000 2 0:000 elsewhere e ect [3] into account to calculate the global sig- +0:210 nificance. We estimate this e ect following the approaches 36 0 0 0 0:000 2 0:000 +0:517 in [65, 66], in which the trial factor of the search is estimated 37 1 0 0 0:400 1 0:301 +0:183 using an asymptotic property of  , obeying the chi-square 38 0 0 2 0:168 0 0:105 +0:201 distribution. The smallest p in the m scan is converted 39 0 0 1 0:084 0 local X 0:084 +0:210 into the global p-value p assuming that the signal can 40 0 0 0 0:000 0 global 0:000 appear only at one m . +0:210 41 0 0 0 0:000 0 0:000 The range of B at 90% C.L. is constructed based MEx2G +0:210 42 0 0 0 0:000 0 0:000 on the Feldman–Cousins unified approach [67] extended to +0:201 43 0 0 1 0:084 0 0:084 use the profile-likelihood ratio as the ordering statistic in or- +0:210 44 0 0 0 0:000 0 0:000 der to incorporate the systematic uncertainties [68]. +0:210 45 0 0 0 0:000 0 0:000 9 Results and discussion adopt the Feldman–Cousins unified approach, a one-sided or Table 2 summarises the numbers of events in the signal re- two-sided interval is automatically determined according to gion and the sidebands as well as the expected number of the data. Therefore, lower limits can be set in m regions background events in the signal region. We observe non-zero where non-zero events are observed with small N . BG events in the signal region for some masses. Note that the ad- The statistical significance of the excesses is tested jacent m bins are not statistically independent. Summing against the null hypothesis. Figure 18 shows p versus local up the observed events gives nine events but five of them are m . We observe the lowest p = 0:012 at m = X local X unique events. One event appears in four bins (m = 34, 35, 35 MeV/c , which corresponds to 2.2 significance. The 36, 37 MeV/c ) and another event appears in two bins (m = global p-value is calculated to be p  0:10 by taking global 35, 36 MeV/c ). the look-elsewhere e ect into account. This corresponds to We discuss the results for  = 20 ps below. The results 1.3, that is not statistically significant. for other  are similar, with small changes in the eciency. Owing to the large statistics of the MEG dataset, the The results are presented in detail in Appendix A. branching ratio upper limits have been reduced to the level Figure 17 shows 90% confidence intervals on B MEx2G of O(10 ). Our results improves the upper limits from the obtained from this analysis together with the sensitivities Crystal Box experiment for m < 40 MeV=c , by a factor of and the previous upper limits due to Crystal Box. The sensit- 60 at most. ivities are evaluated by the mean of the branching ratio lim- This publication reports results from the full MEG data- its at 90% C.L. under the null hypothesis. Note that since we set. Hence, new experiments will be needed for further ex- Assuming that the signal is at the assumed m . ploration of this decay, e.g. to test whether the small excess X 16 - 7 Table 3 Results for  = 5 ps. LL and UL denote the lower limit and 10 X upper limit of the 90% confidence interval. - 8 m (MeV/c ) s LL UL X 0 12 13 11 Crystal Box 20 (3:29 0:31) 10 4:60 10 1:28 10 - 9 12 12 21 (3:56 0:44) 10 – 7:78 10 Sensitivity 12 12 22 (3:73 0:44) 10 – 8:30 10 - 10 12 12 10 23 (3:96 0:45) 10 – 7:72 10 This search 12 11 24 (4:26 0:49) 10 – 1:43 10 12 11 25 (5:41 0:55) 10 – 1:08 10 - 11 12 11 26 (5:16 0:61) 10 – 1:13 10 12 11 27 (5:81 0:70) 10 – 1:39 10 - 12 12 11 28 (6:62 0:81) 10 – 1:58 10 12 13 11 29 (7:67 0:95) 10 8:53 10 2:93 10 - 13 12 11 30 (8:62 0:85) 10 – 2:26 10 20 25 30 35 40 45 m (MeV/c ) 11 11 X 31 (1:07 0:14) 10 – 2:54 10 11 11 32 (1:29 0:17) 10 – 3:20 10 Figure 17 Confidence intervals (90% C.L.) onB (blue band) for MEx2G 11 11 33 (1:59 0:21) 10 – 4:00 10 = 20 ps. The red broken line shows the expected upper limits under 11 12 11 34 (1:97 0:26) 10 1:97 10 7:66 10 the null hypothesis and the yellow line shows the limits extracted by 11 11 10 35 (2:60 0:31) 10 1:65 10 1:40 10 Crystal Box analysis. 11 11 10 36 (3:18 0:42) 10 1:98 10 1:72 10 11 10 37 (4:12 0:53) 10 – 1:43 10 1 11 10 38 (5:43 0:70) 10 – 1:27 10 11 10 39 (7:25 0:93) 10 – 1:79 10 11 10 40 (9:01 0:93) 10 – 2:20 10 10 10 41 (1:35 0:19) 10 – 3:37 10 1σ 10 10 42 (1:88 0:27) 10 – 4:72 10 - 1 10 10 10 43 (2:66 0:39) 10 – 6:47 10 10 9 44 (3:81 0:57) 10 – 1:01 10 10 9 45 (6:25 0:88) 10 – 1:59 10 2σ - 2 10 Conclusions 20 25 30 35 40 45 We have searched for a lepton-flavour-violating muon decay m (MeV/c ) + + mediated by a new light particle,  ! e X; X ! decay, Figure 18 Local p-value under null hypothesis as a function of as- for the first time using the full dataset (2009–2013) of the sumed m . MEG experiment. No significant excess was found in the mass range m = 20–45 MeV/c and  < 40 ps, and we X X observed in this search grows. An upgraded experiment, set new branching ratio upper limits in the mass range m = MEG II, is currently being prepared [69]. A brief prospect 20–40 MeV/c . In particular, the upper limits are lowered to 11 2 for improved sensitivity to MEx2G in MEG II is discussed the level of O(10 ) for m = 20–30 MeV/c . The result is below. In this analysis the sensitivity worsens with increas- up to 60 times more stringent than the bound converted from ing m , mainly due to the 2 acceptance and direction match the previous experiment, Crystal Box. eciencies. The acceptance is determined by the geometry of the LXe photon detector and is not changed by the up- grade. The direction match eciency can even worsen if we Appendix A Detailed results for di erent lifetimes + + only consider the  ! e search; the position resolution is expected to improve by a factor two, which enables tight- The detailed results for di erent lifetimes  are summar- ening the direction match trigger condition. However, the ised in Tables 3–5. MEG II trigger development is underway and the trigger ef- ficiency for high mass can be improved up to a factor 2 if a dedicated trigger is prepared. Basically, MEG II will collect Acknowledgments ten times more  decays and the resolutions of each kin- ematic variable will improve by roughly a factor two, lead- We are grateful for the support and co-operation provided ing to higher eciency while maintaining low background. by PSI as the host laboratory and to the technical It is therefore possible to improve the sensitivity by one or- and engineering sta of our institutes. This work is der of magnitude. supported by DOE DEFG02-91ER40679 (USA); INFN BR(μ→ eX, X→γ γ ) Local p-value 17 Table 4 Results for  = 20 ps. LL and UL denote the lower limit and Table 5 Results for  = 40 ps. LL and UL denote the lower limit and X X upper limit of the 90% confidence interval. upper limit of the 90% confidence interval. 2 2 m (MeV/c ) s LL UL m (MeV/c ) s LL UL X 0 X 0 12 13 11 12 13 11 20 (2:92 0:28) 10 3:94 10 1:10 10 20 (3:02 0:29) 10 3:78 10 1:19 10 12 12 12 12 21 (3:18 0:39) 10 – 7:54 10 21 (3:28 0:39) 10 – 7:44 10 12 12 12 12 22 (3:35 0:39) 10 – 7:36 10 22 (3:44 0:39) 10 – 7:80 10 12 12 12 12 23 (3:57 0:40) 10 – 7:14 10 23 (3:66 0:40) 10 – 7:41 10 12 11 12 11 24 (3:86 0:43) 10 – 1:33 10 24 (3:94 0:43) 10 – 1:43 10 12 11 12 11 25 (4:74 0:43) 10 – 1:06 10 25 (4:83 0:43) 10 – 1:04 10 12 11 12 11 26 (4:71 0:53) 10 – 1:15 10 26 (4:76 0:54) 10 – 1:09 10 12 11 12 11 27 (5:31 0:62) 10 – 1:33 10 27 (5:35 0:61) 10 – 1:33 10 12 11 12 11 28 (6:07 0:72) 10 – 1:58 10 28 (6:09 0:71) 10 – 1:51 10 12 12 11 12 13 11 29 (7:04 0:85) 10 1:04 10 2:70 10 29 (7:03 0:83) 10 9:22 10 2:71 10 12 11 12 11 30 (7:94 0:78) 10 – 2:07 10 30 (7:87 0:78) 10 – 1:88 10 12 11 12 11 31 (9:86 1:26) 10 – 2:33 10 31 (9:78 1:21) 10 – 2:33 10 11 11 11 11 32 (1:19 0:15) 10 – 3:14 10 32 (1:18 0:15) 10 – 2:92 10 11 11 11 11 33 (1:46 0:19) 10 – 3:92 10 33 (1:44 0:18) 10 – 3:57 10 11 12 11 11 12 11 34 (1:82 0:23) 10 2:25 10 7:10 10 34 (1:78 0:22) 10 1:97 10 6:81 10 11 11 10 11 11 10 35 (2:38 0:25) 10 1:53 10 1:31 10 35 (2:30 0:23) 10 1:47 10 1:26 10 11 11 10 11 11 10 36 (2:93 0:37) 10 1:85 10 1:56 10 36 (2:84 0:34) 10 1:78 10 1:54 10 11 10 11 10 37 (3:79 0:47) 10 – 1:29 10 37 (3:67 0:43) 10 – 1:28 10 11 10 11 10 38 (4:99 0:62) 10 – 1:16 10 38 (4:80 0:56) 10 – 1:12 10 11 10 11 10 39 (6:65 0:83) 10 – 1:66 10 39 (6:37 0:75) 10 – 1:51 10 11 10 11 10 40 (8:20 0:87) 10 – 2:04 10 40 (7:94 0:78) 10 – 1:90 10 10 10 10 10 41 (1:23 0:20) 10 – 3:15 10 41 (1:17 0:18) 10 – 2:84 10 10 10 10 10 42 (1:71 0:28) 10 – 4:44 10 42 (1:62 0:25) 10 – 4:28 10 10 10 10 10 43 (2:41 0:41) 10 – 5:98 10 43 (2:27 0:37) 10 – 5:94 10 10 10 10 10 44 (3:44 0:61) 10 – 8:25 10 44 (3:23 0:54) 10 – 8:31 10 10 9 10 9 45 (6:29 1:12) 10 – 1:63 10 45 (5:65 0:97) 10 – 1:53 10 (Italy); MIUR Montalcini D.M. 2014 n. 975 (Italy); 6. 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Search for lepton flavour violating muon decay mediated by a new light particle in the MEG experiment

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Abstract

Eur. Phys. J. C manuscript No. (will be inserted by the editor) Search for lepton flavour violating muon decay mediated by a new light particle in the MEG experiment The MEG collaboration 1 a 2,3 4 ab 5 ab 5 a 6 ab A. M. Baldini , F. Berg , M. Biasotti , G. Boca , P. W. Cattaneo , G. Cavoto , 1 ab 1 ab 6 ab 7 ab 7 ab 5 ab F. Cei , M. Chiappini , G. Chiarello , C. Chiri , A. Corvaglia , A. de Bari , 4 a 1 a 1 a 4 ab 7 a M. De Gerone , M. Francesconi , L. Galli , F. Gatti , F. Grancagnolo , 1 a 8,9,10 2 2,3 11 M. Grassi , D. N. Grigoriev , M. Hildebrandt , Z. Hodge , K. Ieki , 8,10 11 11 11 2 12 F. Ignatov , R. Iwai , T. Iwamoto , S. Kobayashi , P.-R. Kettle , W. Kyle , 13 13 13 13 12 N. Khomutov , A. Kolesnikov , N. Kravchuk , N. Kuchinskiy , T. Libeiro , 12 13 11 6 ab 14 G. M. A. Lim , V. Malyshev , N. Matsuzawa , M. Meucci , S. Mihara , 12 11 2 *,11 11 W. Molzon , Toshinori Mori , A. Mtchedilishvili , M. Nakao , H. Natori , 1 ab 14 11 11 11 11 D. Nicolo ` , H. Nishiguchi , M. Nishimura , S. Ogawa , R. Onda , W. Ootani , 11 12 7 ab 2,1 ab 6 a 4 ab A. Oya , D. Palo , M. Panareo , A. Papa , V. Pettinacci , G. Pizzigoni , 8,10 6 a 2 13 5 a 11 A. Popov , F. Renga , S. Ritt , A. Rozhdestvensky , M. Rossella , R. Sawada , 2 1 a 2 7 a 11 P. Schwendimann , G. Signorelli , A. Stoykov , G. F. Tassielli , K. Toyoda , 11 11 6 a 11 8,10 Y. Uchiyama , M. Usami , C. Voena , K. Yanai , Yu.V. Yudin 1 a b INFN Sezione di Pisa ; Dipartimento di Fisica dell’Universita, ` Largo B. Pontecorvo 3, 56127 Pisa, Italy Paul Scherrer Institut PSI, 5232 Villigen, Switzerland Swiss Federal Institute of Technology ETH, 8093 Zurich, ¨ Switzerland 4 a b INFN Sezione di Genova ; Dipartimento di Fisica dell’Universita, ` Via Dodecaneso 33, 16146 Genova, Italy 5 a b INFN Sezione di Pavia ; Dipartimento di Fisica dell’Universita, ` Via Bassi 6, 27100 Pavia, Italy 6 a b INFN Sezione di Roma ; Dipartimento di Fisica dell’Universita ` “Sapienza”, Piazzale A. Moro, 00185 Roma, Italy 7 a b INFN Sezione di Lecce ; Dipartimento di Matematica e Fisica dell’Universita ` del Salento, Via per Arnesano, 73100 Lecce, Italy Budker Institute of Nuclear Physics of Siberian Branch of Russian Academy of Sciences, 630090 Novosibirsk, Russia Novosibirsk State Technical University, 630092 Novosibirsk, Russia Novosibirsk State University, 630090 Novosibirsk, Russia ICEPP, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan University of California, Irvine, CA 92697, USA Joint Institute for Nuclear Research, 141980 Dubna, Russia KEK, High Energy Accelerator Research Organization, 1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan Received: date / Accepted: date Abstract We present the first direct search for lepton fla- Contents vour violating muon decay mediated by a new light particle + + 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 1 X,  ! e X; X ! . This search uses a dataset result- 2 Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 ing from 7:5  10 stopped muons collected by the MEG 3 Search strategy . . . . . . . . . . . . . . . . . . . . . . . . 5 experiment at the Paul Scherrer Institut in the period 2009– 4 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 5 Event reconstruction . . . . . . . . . . . . . . . . . . . . . 6 2013. No significant excess is found in the mass region 20– 2 6 Dataset and event selection . . . . . . . . . . . . . . . . . . 11 45 MeV/c for lifetimes below 40 ps, and we set the most 7 Single event sensitivity . . . . . . . . . . . . . . . . . . . . 12 stringent branching ratio upper limits in the mass region of 8 Statistical treatment of background and signal . . . . . . . . 13 2 11 20–40 MeV/c , down to O(10 ) at 90% confidence level. 9 Results and discussion . . . . . . . . . . . . . . . . . . . . 15 10 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Appendix A Detailed results for di erent lifetimes . . . . . . . 16 Keywords Decay of muon, lepton flavour violation, flavour symmetry, long-lived particle, displaced vertex 1 Introduction The search for charged lepton flavour violating (CLFV) pro- Corresponding author: [email protected] cesses is one of the key tools to probe for physics beyond the arXiv:2005.00339v2 [hep-ex] 8 Nov 2020 2 τ = 10 ps Standard Model (SM) of elementary particles and interac- Crystal Box exclusion plot τ = 100 ps tions. The observation of neutrino oscillations [1–3] showed τ = 1 ns that lepton flavour is not conserved in nature. As a con- - 7 10 τ = 10 ns sequence, charged lepton flavour is violated, even though the rate is unobservably small (< 10 ) in an extension of the SM accounting for measured neutrino mass di erences - 8 and mixing angles [4, 5]. In the context of new physics, in 10 the framework of grand unified theories for example, CLFV processes can occur at an experimentally observable rate [6]. Therefore, such processes are free from SM physics back- - 9 grounds and a positive signal would constitute unambigu- ous evidence for physics beyond the SM. This motivates the e ort to search for evidence of new physics through CLFV processes [7, 8]. - 10 The MEG experiment at the Paul Scherrer Institut (PSI) 10 15 20 25 30 35 40 45 m (MeV/c ) in Switzerland searched for one such CLFV process,  ! e decay, with the highest sensitivity in the world. No evid- Figure 1 Upper limits on MEx2G decay estimated by converting the + + ence of the decay was found yet, leading to an upper limit on upper limits on  ! e from the Crystal Box experiment as a func- + + 13 tion of m . Lines with di erent markers and colours correspond to the branching ratio B( ! e ) < 4:2 10 at 90% con- di erent  . + + X fidence level (C.L.) [9]. Models that allow  ! e decay at an observable rate usually assume that CLFV couplings are introduced through an exchange of new particles much MEx2G decay if we assume that X is more likely to decay heavier than the muon. Negative results by CLFV searches into an e e pair. However, there is a possibility for X to leave open another possibility: new physics exists at a lighter be electrophobic, as pointed out in [29, 30], and searches for scale but with very weak coupling to SM particles. both decay modes can hint at the model behind these decay If a new particle X (with mass m and lifetime  ) X X modes. + + lighter than the muon exists, the CLFV two-body decay The current upper limit on the decay  ! e , + + 11 ! eX may be a good probe for such new physics. The B( ! e ) < 7:2  10 (90% C.L.) from the Crys- experimental signature depends on how the new particle X tal Box experiment [31] can be converted into an equivalent + + decays. In this paper, we report a search for  ! e X; X ! MEx2G upper limit by taking into account the di erence in (MEx2G) decay using the full dataset collected in the detector eciencies [32]; the converted limits are shown in MEG experiment. Here, we assume that X is an on-shell Fig. 1. scalar or pseudo-scalar particle. Axion-like particles [10– Axion-like particle searches from collider and beam 13], Majoron [14, 15], familon [16–19], flavon [20, 21], dump experiments and from supernova observations also flaxion [22, 23], hierarchion [24], and strongly interacting constrain the branching ratio X ! if the axion-like massive particles [25, 26] are candidates for X. particles are generated from coupling to photons [33]. Fig- A dedicated search for the MEx2G decay has never been ure 2 summarises the parameter regions excluded by these done, although some constraints on the X particle parameter experiments. A region with decay length c < 1 cm and space can be deduced by experimental results from both re- 2 m > 20 MeV/c still has room for the MEx2G decay. lated muon decay modes and non-muon experiments; these Based on limits discussed above, we define the target are discussed below. parameter space of this search in the  –m plane as shown X X Current upper limits on the inclusive decay  ! in Fig. 3. + 5 e X are given at O(10 ) for m in the range 13– 2 1 80 MeV/c [27]. However, the current limits do not impose any constraints on the MEx2G decay in the target region 2 Detector of this search. They are complementary, relevant for cases where X is either stable or decays invisibly. For X resulting The MEG detector is briefly presented in the following, em- from muon decays, the only kinematically allowed visible phasising aspects relevant to this search; a detailed descrip- decay channels are X ! e e and X ! . The former can tion is available in [37]. occur at tree level while the latter can occur via a fermion In this paper we adopt a Cartesian coordinate system + + + loop. The current upper limit on  ! e X; X ! e e (x; y; z) shown in Fig. 4 with the origin at the centre of the at a level of O(10 ) [28] give stringent constraints on the magnet. When necessary, we also refer to the cylindrical co- 1 + In these searches, only e is looked at. ordinate system (r; ; z) as well as the polar angle . + + BR(μ → e X,X→ γ γ ) 3 Positrons from the muon decays are detected with a magnetic spectrometer, called the COBRA (standing for COnstant Bending RAdius) spectrometer, consisting of a thin-walled superconducting magnet, a drift chamber array (DCH), and two scintillating timing counter (TC) arrays. The magnet [39] is made of a superconducting coil with three di erent radii. It generates a gradient magnetic field of 1.27 T at the centre and 0.49 T at each end. The diameter of an emitted e trajectory depends on the absolute momentum, independent of the polar angle due to the gradient field. This allows us to select e s within a specific momentum range by placing the TC detectors in a specific radial range; e s whose momenta are larger than 45 MeV/c fall into the ac- Figure 2 Excluded parameter regions for a scalar X with mass m and ceptance of the TC. Furthermore, the gradient field prevents coupling g to 2 s from collider, beam dumps, and supernova [34–36] + + e s emitted nearly perpendicular to the  beam direction (from [33]). In black we show contours of the boosted decay length c of X ! , assuming X to be produced from an at-rest muon from looping many times in the spectrometer. This results + + decay  ! e X. The solid black line corresponds to c = 0:01 cm, in a suppression of hit rates in the DCH. The thickness of the dotted one to 0.1 cm, the dashed one to 1 cm and the dot-dashed the central part of the magnet is 0.2 radiation length to max- line to 10 cm. imise transparency to ; 85% of the signal s penetrate the magnet without interaction and reach the photon detector. Positrons are tracked in the DCH [40]. It is composed of 16 independent modules. Each module has a trapezoidal shape with base lengths of 104 cm (at smaller radius, close Beamdumps to the stopping target) and 40 cm (at larger radius). These modules are installed in the bottom hemisphere in the mag- net at 10.5 intervals. The DCH covers the azimuthal region between 191.25 and 348.75 and the radial region between 19.3 cm and 27.9 cm. It is composed of low mass materials and helium-based gas (He : C H = 1 : 1) to suppress Cou- 2 6 MEG Acceptance lomb multiple scattering; 2:0  10 radiation length path Beamdumps + + + is achieved for the e from  ! e decay at energy of E + = 52:83 MeV (= m c =2, where m is the mass of 0 10 20 30 40 50 60 muon). m (MeV/c ) The TC [41, 42] is designed to measure precisely the Figure 3 Allowed X particle parameter space (white). The blue region e hit time. Fifteen scintillator bars are placed at each end has already been excluded [35] and the red shaded region on the right of the COBRA. They are made of 4  4  80 cm plastic (m & 45 MeV/c ) is inaccessible to MEG. scintillators with fine-mesh PMTs attached to both ends of the bars. The eciency of the spectrometer significantly depends Multiple intense  beams are available at the E5 chan- on E + as shown in Fig. 5. The e energy from the MEx2G nel in the 2.2-mA PSI proton accelerator complex. We use e + + decay is lower than that from  ! e depending on m , a beam of surface muons, produced by  decaying near X and the eciency is correspondingly lower. The large m the surface of a production target. The beam intensity is X + 7 search range is limited by this e ect as shown in Fig. 3. tuned to a  stopping rate of 3  10 , limited by the rate The photon detector is a homogeneous liquid-xenon capabilities of the tracking system and the rate of acci- + + (LXe) detector relying on scintillation light for energy, po- dental backgrounds in the  ! e search. The muons sition, and timing measurement [43, 44]. As shown in Fig. 4, at the production target are fully polarised (P = 1), it has a C-shaped structure fitting the outer radius of the and they reach a stopping target with a residual polarisation +0:05 magnet. The fiducial volume is 800 `, covering 11% of the P = 0:86 0:02 (stat) (syst) [38]. 0:06 solid angle viewed from the centre of the stopping target in The positive muons are stopped and decay in a thin target the radial range of 67:85 < r < 105:9 cm, corresponding to placed at the centre of the spectrometer at a slant angle of 14 radiation length. It is able to detect a 52.83-MeV with 20 from the  beam direction. The target is composed of a 205 m thick layer of polyethylene and polyester (density In the high rate MEG environment, only scintillation light with its fast 0:895 g/cm ). signal, is detected. cβ τ = 1 cm τ (ps) X 4 𝑥 𝑦 COBRA Magnet Drift Chamber e Timing Counter Drift Chamber Liquid Xenon Scintillation Detector Figure 4 The figure shows a schematic view of the MEG detector with a simulated MEx2G event emitted from the target. The top view is shown on the left, the view from downstream on the right. 1.2 100 0.8 0.6 0.4 0.2 40 42 44 46 48 50 52 54 56 E (MeV) 20 25 30 35 40 45 Figure 5 COBRA spectrometer relative eciency as a function of E + 2 m (MeV/c ) normalised to  + (52:83 MeV) = 1. Figure 6 Trigger direction match eciency for the MEx2G decay con- ditional to e and 2 detection as a function of m evaluated with a Monte Carlo simulation (Sect. 4). high eciency and to contain the electromagnetic shower induced by it. The scintillation light is detected by 846 2- inch PMTs submerged directly in the liquid xenon. They are placed on all six faces of the detector, with di erent PMT MEx2G events was neither foreseen nor implemented. Thus, coverage on di erent faces. On the inner face, which is the + + we rely on the  ! e triggered data in this search. densest part, the PMTs align at intervals of 6.2 cm. + + One of the distinctive features of the MEG experiment is The main  ! e trigger, with a prescaling of 1, used that it digitises and records all waveforms from the detectors the following observables: energy, time di erence between + + using the Domino Ring Sampler v4 (DRS4) chip [45]. The e and , and relative direction of e and . The DC was sampling speeds are set to 1.6 GSPS for TC and LXe photon not used in the trigger due to the slow drift velocity. The detector and 0.8 GSPS for DCH. This lower value for DCH condition on the relative direction is designed to select back- is selected to match the drift velocity and the required preci- to-back events. To calculate the relative direction, the PMT sion. that detects the largest amount of scintillation light is used The DAQ event rate was kept below 10 Hz in order to for the , while the hit position at the TC is used for the acquire the full waveform data ( 1 MB/event). It was ac- e . This direction match requirement results in inecient complished using a highly ecient online trigger system selection of the MEx2G signal because, unlike the  ! [46, 47]. e decay, the MEx2G decay has 2 s with a finite opening Several types of trigger logic were implemented and ac- angle, resulting in events often failing to satisfy the direction tivated during the physics data-taking each with its own trigger. The selection ineciency for MEx2G events is 10– prescaling factor. However, a dedicated trigger for the 50% depending on m as shown in Fig. 6. Efficiency (a.u.) Conditional efficiency (%) 5 Finally, the detector has been calibrated and monitored X → gg over all data-taking period with various methods [48,49], en- + + 𝒙 m → e X 1 suring that the detector performances have been under con- trol over the duration of the experiment. 𝑙 𝒙 2 3 Search strategy vtx 𝑷 + 𝒙 + e 𝒙 The MEx2G signal results from the sequential decays of e 𝑡 + + 2 ! e X followed by X ! . The first part is a two- body decay of a muon at rest, signalled by a mono-energetic Figure 7 Decay kinematics and kinematic variables. e . The energy E + is determined by m : E + (m = 0) = e X e X 52:83 MeV and is a decreasing function of m . The sum of energies of the two s is also mono-energetic and an increas- Given the muon decay vertex, the two s’ energies and ing function of m . The momenta of the two s are Lorentz- positions, and m , the X decay vertex x can be computed. X vtx boosted along the direction of X, which increases the accept- Therefore, we reconstruct x by scanning the assumed vtx ance in the LXe photon detector compared to the three-body value of m (Sect. 5.3.1). If the final-state three particles do + + decay  ! e . The final-state three particles is expected not originate at a single muon decay vertex, these variables to have an invariant mass of 105.7 MeV=c (= m ) and the will be inconsistent with originating from a single point. total momentum vector equal to 0. After reconstructing x , the relative time and angles (mo- vtx A physics background that generates time-coincident menta) between X and e are tested for consistency with a + + + e in the final state is  ! e  ¯ . This mode has not muon decay (Sect. 5.3.2 and 5.3.3). yet been measured but exists in the SM. The branching ratio The MEx2G decay search analysis is performed within 2 2 2 is calculated to be  O(10 ) for the MEG detector con- the mass range 20 MeV/c < m < 45 MeV/c at 1 MeV/c figuration without any cut on E [50, 51]. Therefore, its step. This step is chosen small enough not to miss signals contribution is certainly negligible in this search where we in the gaps. Therefore, adjacent mass bins are not statist- apply cuts on E . e ically independent. The analysis was performed assuming The dominant background is the accidental pileup of lifetimes  = 5; 20, and 40 ps; the value a ects only the multiple  s decays. There are three types of accidental signal eciency. background events: We estimate the accidental background by using the data in which the particles are not time coincident. To reduce the + + Type 1: The e and one of the s originate from one  , and possibility of experimental bias, a blind analysis is adopted; the other from a di erent one. the blind region is defined in the plane of the relative times Type 2: The two s share the same origin, and the e is ac- of the three particles (Sect. 6). cidental. The signal eciency is evaluated on the basis of a Monte Type 3: All the particles are accidental. Carlo simulation (Sect. 4). Its tuning and validation are per- formed using pseudo-2 data as described in Sect. 4.1. The main source of a time-coincident e pair in type 1 is + + the radiative muon decay  ! e  ¯ [52]. The sources + + of time-coincident pairs in type 2 are e e ! (e + + 4 Simulation from  decay and e from material along the e trajectory), + + ! e  ¯ with an additional , e.g. by a bremsstrahlung + 3 The technical details of the program of Monte Carlo (MC) from the e , or a cosmic-ray induced shower. simulation are presented in [53] and an overview of the Figure 7 shows the decay kinematics and the kinematic physics and detector simulation is available in [37]. In the variables. The muon decay vertex and the momentum of the + + following we report a brief summary. e are obtained by reconstructing the e trajectory using the The first step of the simulation is the generation of the hits in DCH and TC and the intersection of the trajectory physics events. That is realised with custom written code for with the plane of the muon beam stopping target (Sect. 5.1). a large number of relevant physics channels. The MEx2G The interaction positions and times of the two s within the decay is simulated starting from a muon at rest in the tar- LXe photon detector and their energies are individually re- get; the decay products are generated in accordance with the constructed using the PMT charge and time information of decay kinematics for the given m and  . X X the LXe photon detector (Sect. 5.2). The muon beam transport, interaction in the target, and 3 + In the case of type 2, the e can have low energy and be undetected. propagation of the decay products in the detector are sim- 6 ulated with a MC program based on GEANT3.21 [54] that 0.55 describes the detector response. Between the detector simu- 0.5 lation and the reconstruction program, an intermediate pro- gram processes the MC information, adding readout simu- 0.45 lation and allowing event mixing to study the detector per- formance under combinatorial background events. Particu- 0.4 larly, the  beam, randomly distributed in time at a decay 7 + 1 rate of 3  10  s , is mixed with the MEx2G decay to 0.35 study the e spectrometer performance. The detectors’ op- erating condition, such as the active layers of DCH and the 0.3 applied high-voltages, are implemented with the known time 0.25 40 42 44 46 48 50 52 dependence. E (MeV) e+ In order to simulate the accidental activity in the LXe photon detector, data collected with a random-time trigger Figure 8 E + resolution as a function of E + . e e are used. A MC event and a random-trigger event are over- laid by summing the numbers of photo-electrons detected by account. After the first track fitting in DCH, the track is each PMT. propagated to the TC region to test matching with TC hits. The matched TC hits are connected to the track and then the track is refined using the TC hit time. Finally, the fit- 4.1 Pseudo two data ted track is propagated back to the stopping target, and the point of intersection with the target defines the muon decay To study the performance of the 2 reconstruction, we built vertex position (x ) and momentum vector that defines the pseudo 2 events using calibration data. The following -ray e + + + + + e emission angles ( ;  ). The e emission time (t ) is lines are obtained in calibration runs: e e e reconstructed from the TC hit time minus the e flight time. – 54.9 MeV and 82.9 MeV from  p !  n ! n reac- Positron tracks satisfying the following criteria are se- tion, lected: the number of hits in DCH is more than six, the 7 8 – 17.6 MeV and 14.6 MeV from Li(p; ) Be reaction, reduced chi-square of the track fitting is less than 12, the 11 12 – 11.7 MeV from B(p; 2 ) C reaction. track is matched with a TC hit, and the track is successfully The selection criteria for those calibration events are de- propagated back to the fiducial volume of the target. If mul- tiple tracks in an event pass the criteria, only one track is tailed in [48] and [55]. We take two events from the above calibration data and overlay them, summing the number of selected and passed to the following analysis, based on the covariance matrix of the track fitting as well as the number photo-electrons PMT by PMT. These pseudo 2 events are generated using both data and MC events. of hits and the reduced chi-square. The resolutions are evaluated based on the MC, tuned to data using double-turn events; tracks traversing DCH 5 Event reconstruction twice (two turns) are selected and reconstructed independ- ently by using hits belonging to each turn. The di erence We describe here the reconstruction methods and their per- in the reconstruction results by the two turns indicates the formance, focusing on high-level objects; descriptions of the resolution. The MC results are smeared so that the double- manipulation of low-level objects, including waveform ana- turn results become the same as those with the data. Fig- lysis and calibration procedures, are available in [9, 37]. The ure 8 shows the E + resolution as a function of E + . The e e e reconstruction (Sect. 5.1) is identical to that used in the angular resolutions also show a similar E dependence. + + ! e decay analysis in [9]. The 2 reconstruction was 2 The  + - and  + -resolutions for m = 20 (45) MeV/c are e e X developed originally for this analysis (Sect. 5.2). After re- 12 (15) mrad and   10 (11) mrad, respectively. +  + e e constructing the e and two s, the reconstructed variables The time resolution is   100 (130) ps. t + are combined to reconstruct the X decay vertex (Sect. 5.3). 5.2 Photon reconstruction 5.1 Positron reconstruction Positron trajectories in the DCH are reconstructed using the Coordinates (u; v; w) are used in the LXe photon detector Kalman filter technique [56, 57] based on the GEANE soft- local coordinate system rather than the global coordinates ware [58]. This technique takes the e ect of materials into (x; y; z): u coincides with z, v = r ( ) where r = in in E resolution (MeV) e+ 7 67:85 cm is the radius of the inner face, and w = r r where  (N ) = N = with  being the in pho;i pho;i PMT;i PMT;i pho;i is the depth measured from the inner face. product of quantum and collection eciencies of the PMT. This fitting is performed separately for each : first, the light distribution is fitted with fx ; M g as free paramet- pho;1 5.2.1 Multiple photon search ers, while the other parameters are fixed; next, the light dis- tribution is fitted withfx ; M g as free parameters, while pho;2 A peak search is performed based on the light distributions 2 the other parameters are fixed. on the LXe photon detector inner and outer faces by using TSpectrum2 [59, 60]. The threshold of the peak light yield is set to 200 photons. Events that have more than one peak Energy pre-fitting To improve the energy estimation, M are fitted while the other parameters are fixed. The are identified as multiple- events. pho;1(2) same  (Eq. (3)) is used but only with PMTs that detect more than 200 photo-electrons. 5.2.2 Position and energy The with the larger M is defined as and the second pho 1 largest one is defined as in the later analysis. Hereafter, only the multiple- events are analysed. When more than two s are found, we select the two with the Position and energy fitting At the final step, all the para- largest energy by performing the position-energy fitting de- meters are fitted simultaneously to eliminate the dependence scribed in this subsection on di erent combinations of two of the fitted positions on the value of M initially as- s. pho;1(2) sumed. The best-fit value of M is used to update R Figure 9 shows a typical event display of a 2 event. pho;1(2) 1 and calculations (1) and (2) are repeated again to obtain the Each PMT detects photons from the two s. The key point of final value of M . Finally, it is converted into E : the 2 reconstruction is how to divide the number of photons pho;1(2) 1(2) detected in each PMT into a contribution from each . E = U (x ) H(T ) S  M ; (4) pho;1(2) 1(2) 1(2) where U (x ) is a uniformity correction factor, H(T ) is a 1(2) Calculation of initial values First, the positions of the de- time variation correction factor with T being the calendar tected peaks in (u; v) are used as the initial estimate with time when the event was collected, and S is a factor to con- w = 1:5 cm. Given the interaction point of each within the vert the number of photons to energy. The functions U (x ) LXe photon detector, the contribution from each to each 1(2) and H(T ) are mainly derived from the 17.6-MeV line from PMT can be calculated as follows. Assuming the ratio of the 7 8 Li(p; ) Be reaction, which was measured twice per week. energy of to that of to be E : E = R : (1 R ) 1 2 1 1 1 2 The factor S is calibrated using the 54.9-MeV line from (0 < R < 1, at first R is set to 0.5), the fractions of the 1 1 decay, taken once per year. number of photons from is calculated as 1 1;i Energy-ratio correction Both the MC data and the pseudo- R = ; (1) 1;i + (1 R ) 1 1;i 1 2;i 2 data show an anti-correlation between the errors in E and E as shown in Fig. 10a, while their sum is not biased. where is the solid angle subtended by the i-th PMT from 1;i true Defining R as the R for true energies for MC data and the interaction point. The total number of photons gener- 1 that for energies reconstructed without the overlay for real ated by , M , is calculated from the ratio R and 1(2) pho;1(2) 1;i data, the reconstruction bias in both the MC data and the the number of photons at each PMT N as pho;i pseudo-2 data is apparent by the linear dependence of true all n R =R on R as shown in Fig. 10b. This bias is removed by 1 1 PMT X 1 applying a correction to the reconstructed energies; the cor- M = R  N : (2) pho;1(2) 1;i pho;i i rection coecients are evaluated from the pseudo-2 data with di erent combinations of calibration data. Then, R is updated to R = M =(M + M ) and 1 1 pho;1 pho;1 pho;2 calculations (1) and (2) are repeated with the updated R . Position correction Oblique incidence of s to the inner This procedure is iterated four times. face results in a bias of the fitted positions. This bias was checked and corrected for using the MC simulation. No bias Position pre-fitting Inner PMTs that detect more than 10 photons are selected to perform a position pre-fitting. The This fitting is performed by a grid search in x = (u; v; w) 1(2) 1(2) space for good stability, while subsequent fittings are performed with following quantity is minimised during the fitting: MINUIT [61] for better precision. selected 5 Error is defined as the di erence between the reconstructed energy PMT N M (x ) M (x ) pho;i pho;1 i pho;2 i 1 2 and the true energy deposit for MC data and between the reconstructed = ; (3) (N ) one with 2 and that with single for pseudo-2 data. pho;i i pho;i 8 1,𝑖 𝑖 −1 −thPMT 𝑖 −thPMT Figure 9 Event display of the LXe photon detector for a 2 event (in a development view). The red points show the interaction positions of the two s projected to each face. Each circular marker denotes a PMT. The colour indicates the measured light yield, which is the sum of photons from the two showers induced by the two s as depicted in the right figure. true is observed in the v direction while a significant bias is ob- E is a reconstructed energy, E is the true value, f is served in the u direction. This is because the s from the the fraction of the narrow component, A is a normalisation MEx2G decay enter the LXe photon detector almost per- parameter, E is the transition parameter between the Gaus- pendicularly in the x-y view but enter with angles in the z-r sian and exponential components, and  is the standard view. Since the u bias arises from the direction and the size deviation of the Gaussian component describing the width of the shower, it depends on the u coordinate and the energy. on the high-energy side. The parameters E and  are cor- t E Therefore, the correction function is prepared as a function related with each other, di erent for the narrow and wide true of u and E . components, and are dependent on E . Figure 11 shows 1(2) 1(2) true an example of the PDFs for 2 events with E = 55 MeV true Selection criteria To guarantee the quality of the reconstruc- and E = 12 MeV. tion, the following criteria are imposed on the reconstruc- tion results: the fits for both s converge; the two posi- Probability density functions for position The PDFs of tions are both within the detector fiducial volume defined as position are almost independent of E and hence (m ;  ). X X 1(2) juj < 25 cm ^ jvj < 71 cm; the distance between the two They are represented by double Gaussians with fractions of s on the inner face is d > 20 cm; E > 10 MeV; and uv tail components of  20%. The standard deviations of the 1(2) core E + E > 40 MeV. 1 2 core components are  = (5:4; 4:7; 6:5) mm in (u; v; w) 1(2) tail coordinates, those of the tail components are  = Probability density function for E The probability density 1(2) (29; 19; 45) mm. function (PDF) for E is evaluated by means of the MC 1(2) simulation. To tune the MC, the pseudo-2 data of MC and 5.2.3 Time data are used. It is asymmetric with a lower tail and mod- elled as follows: The interaction time of ( ) can be reconstructed using 1 2 true true narrow narrow P(E j E ) = f  F(E ; E ; E ;  ) the pulse time measured by each PMT (t ) by correcting t E PMT;i true wide wide for a delay time (t ) including the propagation time delay; ;i 1(2) + (1 f ) F(E ; E ; E ;  ); (5) of the light between the interaction point and the PMT and where the time-walk e ect, and a time o set due to the readout electronics (t ): o set;i true F(E ; E ; E ;  ) t E 8 ! t = t t t : (7) true ;i PMT;i delay; ;i o set;i > 1(2) 1(2) (E E ) true A exp E > E E > 2 t > 2 < E The single PMT time resolution  is approximately pro- ! t;i = ; (6) > p E E > t t true true portional to 1= N with  (N = 500)  500 ps, > pe; ;i t;i pe; ;i 1(2) 1(2) >A exp + (E E ) E  E E 2 t : 2 where N is the number of photo-electrons from ( ). pe; ;i 1 2 1(2) 9 5 800 0.59 ± 0.12 (a) narrow (a) E 1.29 ± 0.11 35 narrow 3 E 0.60 ± 0.07 wide E 2.34 ± 0.46 30 wide E 0.57 ± 0.25 −1 −2 40 45 50 55 60 65 70 −3 E (MeV) −4 −5 0 − 4 − 2 0 2 4 2000 f 0.43 ± 0.03 (b) narrow true-deposit E 0.49 ± 0.01 E − E (MeV) � � narrow 1 1 E 0.68 ± 0.10 wide 1500 σ 1.34 ± 0.05 1.1 14 wide E 0.61 ± 0.12 1.08 (b) 1.06 1.04 1.02 6 8 10 12 14 16 18 20 E (MeV) 6 γ 0.98 true 0.96 Figure 11 Energy response to MC 2 events with E = 55 MeV and true E = 12 MeV. The blue curves are the PDFs fit to the distributions. 0.94 2 See text for the formula of the PDFs. 0.92 0.9 0 0.5 0.6 0.7 0.8 0.9 vals; each assumed mass results in a di erent reconstructed X ! vertex position. Figure 10 (a) Scatter plot of the energy reconstruction errors (MC). truedeposit E is the MC true value of the energy deposited in the LXe. (b) 1(2) 5.3.1 X decay vertex Dependence of the reconstructed energy ratio bias as a function of the reconstructed energy ratio (MC). A maximum likelihood fit is used in the reconstruction, with the following observables: These individual PMT measurements are combined to + + + X = (E ; E ; x ; x ; x ;  ;  ): (10) e e e 1 2 1 2 obtain the best estimate of the interaction time of ( ) 1 2 (t ). The following  is minimised: The fit parameters are the following: 1(2) selected 2 = (cos  ;  ; x ); (11) rest rest vtx PMT t t ;i 1(2) 1(2) = : (8) time 2 where  is the emission angle in the X rest frame,  is rest rest (N ) pe; ;i 1(2) i t;i the angle of the photons in the X rest frame with respect to We use PMTs whose light yield from ( ) is 5 times higher 1 2 the X momentum direction in the MEG coordinate system, than that from ( ) excluding PMTs whose light yield is 2 1 and x is the X decay vertex position. The function L() is vtx less than 100 photons or which give large  contribution in defined as follows: the fitting. L() = P(E j cos  ; m ) rest X The E -dependent time resolution for single event is P(E j cos  ; m ) evaluated with the calibration runs and corrected for 2 rest X events using the MC: P(x j cos  ;  ; x ; x +; m ) rest rest vtx e X P(x j cos  ;  ; x ; x ; m ) 2 2 rest rest vtx e X = 338 =E (MeV) + 45 (ps): (9) 1(2) 1(2) P( + j x ; x + ) e vtx e + + P( j x ; x ) e vtx e 5.3 Combined reconstruction P(l j x ; x ;  ; m ); (12) X vtx e X X true In this section, we present the reconstruction method for where l is the X decay length. The term P(x + j x ) is X e + true the X ! vertex assuming a value for m in the recon- omitted by approximating x by x to reduce the fitting X + e 2 2 struction. We scan m in 20–45 MeV/c at 1 MeV/c inter- parameters. true true-deposit R / R 1 1 E − E (MeV) � � 2 2 Entries Entries 10 The energy dependence of the E PDF Eq. (5) is 1(2) 1500 modelled with a morphing technique [62] using two quasi- monoenergetic calibration lines: the 11.7 MeV line from 11 12 the nuclear reaction of B(p; 2 ) C and the 54.9 MeV line from  decay. The PDFs of the position are approximated as double Gaussians to fit better tails in the PDF. The positron angles are compared with those of the flipped direction of the X momentum ((x x )) with vtx e PDFs approximated as single Gaussians. 6 0 0 The decay length is defined as l = jx x j. Under X vtx e 10 20 30 40 Assumed m (MeV/c ) the approximation  ! 0, the PDF is x + 1 l X Figure 12  distribution for the MC signal events at (m ;  ) = X X vtx P(l j x ; x ;  ; m ) =  exp ; (13) X e vtx X X c c X X (30 MeV=c ; 20 ps) as a function of m assumed in the reconstruction. which is defined and normalised for l  0. The approxima- tion is justified because the transverse component of  is x + 2 7 Figure 12 shows the dependence of  on the assumed vtx 1–2 mm [55] while the longitudinal component is largely value of m for the MC signal events, providing another ra- driven by the target thickness ( 0.2 mm), which is to be tionale for Eq. (14). When the assumed value is the same compared with c ranging between  6–30 mm. as the true value (m = 30 MeV/c in this case), the res- We fix  = 20 ps since the vertex reconstruction per- ultant  becomes minimum on average. The e ective m vtx formance is almost independent of  in the assumed range. resolution is  2:5 MeV/c . This likelihood term e ectively penalizes non-zero decay lengths using a scale that is fixed to the average expected 5.3.2 Momentum decay length of 20 ps. The x resolution of the maximum-likelihood fit is vtx Given the vertex position, the momentum of each can be evaluated via the MC to be  = (8; 12) mm in the trans- vtx calculated. The sum of the final-state three particles mo- verse and longitudinal directions. menta, We define an expression to quantify the goodness of the P  P + + P + P ; (15) vertex fit as sum e 1 2 0 1 0 1 2 2 best best X X B E E C B x x C B C B C should be 0 for the MEx2G events. 2 B C B C B C B C = B C + B C vtx @ A @ A E x = ; = ; 1 2 1 2 5.3.3 Relative time 0 1 0 1 0 1 2 2 2 best best best B  C B  C B l C X X B C B C B C X X X B C B C B C B C B C B C + + + : (14) @ A @ A @ A X The time di erence between the 2 s at the X vertex is cal- X X culated as The variables with the superscript “best” indicate the best- ! ! fitted parameters in the maximum likelihood fit and the vari- l l 1 2 t = t t ; (16) 1 2 ables with no superscript indicate the measured ones. Here, c c ( ;  ) = (  +;  +  + ) is the direction opposite to X X e e where l is the distance between the interaction point 1(2) 1(2) + + ( ;  ). e e in the LXe photon detector and the X vertex position, l = 1(2) The  of each variable is the corresponding resolution jx x j. The relative position of the vertices is such that vtx 1(2) when the distribution is approximated as a single Gaussian. this definition is identical to the signed distance defined ac- This expression is not expected to follow a  distribution cording to the X direction and therefore the distribution is because the PDFs of the variables are not in general Gaus- centred at 0 for MEx2G events. sian. The last term is quadratic by analogy with the other The time di erence between and e at the muon ver- terms and its expression has been found to be e ective in tex is calculated as separating signal from background. The rationale for using 1 X Eq. (14) is to provide a powerful discriminator between sig- t + = t t +: (17) e e 1 1 c c nal and background as shown later in Fig. 14f. With the unsigned definition of l the distribution is slightly From the fifth and sixth terms in Eq. (12), the reason for using the o set with respect to 0 for MEx2G events as visible in absolute value of l rather than the signed value with the sign of (x x + )  P + becomes apparent. If the signed value of the de- Fig. 14d. vtx e e cay length were negative, the X angle would be flipped by  and the 7 2 penalty would be huge preventing the fit to succeed. For better display, we take the square root of  here. vtx vtx 11 + + ! e decays, the time of the LXe photon detector is re- Signal (MC) constructed with PMTs around the largest peak found in the peak search (Sect. 5.2.1). This retained16% of the dataset, + + on which the full event reconstruction for the  ! e de- cay analysis was performed. Before processing the MEx2G dedicated reconstruction, we applied an additional event se- C A C + + 100 lection using the  ! e reconstruction results. It was based on the existence of multiple ( 2) s and the total en- 8 + y ergy of the s and e (E ) being jE m j < 0:2m . total total B 0 This selection reduces the dataset by an additional factor −1 of  300. We applied the MEx2G dedicated reconstruction (Sect. 5.2.2, 5.2.3, and 5.3) to this selected dataset. −2 y A blind region was defined containing the events satis- fying the cuts jt +j < 1 ns ^ jt j < 1 ns. This blind region 20 e −3 C A C is large enough to hide the signal. Those events were sent in −4 0 a separated data-stream and were not used in the definition −4 −2 0 2 4 of the analysis strategy including cuts; background events t (ns)  in the signal region were estimated without using events in x x x C A C the blind region. After the analysis strategy was defined, the BG (data) blind region was opened and events in this region were ad- 4 120 ded to perform the last step of analysis. The accidental background can be estimated from the 100 o -time sideband regions defined in Fig. 13. There are three such regions: A, B, and C; each containing a di erent com- bination of the types of background as defined in Sect. 3. The outer boundary of the time sidebands, jt j < 3:5 ns ^ jt j < 3:5 ns, are determined so that the background distri- 0 60 bution is not deformed by the time-coincidence trigger con- dition. The widths x and y in Fig. 13 are the same as the A B −1 outer boundary of the signal region defined depending on −2 m by the signal selection criteria described below. 20 The following seven variables are used for the signal se- −3 lection: −4 0 1. E : the e energy. −4 −2 0 2 4 2. E : the total energy of the three particles. Blind region t (ns) sum  Signal region 3. jP j: the magnitude of the sum of the three particles’ sum momenta. Figure 13 Top: Signal (MC) and bottom: background (sideband data) event distributions in the t + –t plane before the signal selection cri- 4. d : the distance between the 2 positions on the LXe e uv teria are applied. (20 MeV/c , 20 ps) case is shown as an example. The photon detector inner face. time sideband regions (A, B, C) and the signal region (the red box) are 5. t : the time di erence between and e calculated in e 1 also shown. Eq. (17). 6. t : the time di erence between 2 s calculated in 6 Dataset and event selection Eq. (16). 7.  : the goodness of vertex fitting calculated in Eq. (14). vtx We use the full MEG dataset, collected in 2009–2013, as First, we fix the E + selection to require jE + E j < + + e e + was used in the  ! e search reported in [9]. As de- X + 1 MeV, where E is the e energy for the MEx2G decay + + + scribed in Sect. 2, the  ! e trigger data are used in this with m . This selection is also used in the Michel normal- 14 + analysis. In total, 7:5 10  s were stopped on the target. isation described in Sect. 7. A pre-selection was applied at the first stage of the Next, we optimise the cut thresholds for the other vari- + + ! e decay analysis, requiring that at least one posi- ables to maximise the experimental outcomes. Distributions tron track is reconstructed and the time di erence between signals in the LXe photon detector and TC is in the range At this stage, the sum of the multiple s’ energy is reconstructed 6:9 < t < 4:4 ns. At this stage, aiming to select the without being separated into each . LXeTC t (ns) t (ns) + +  e  e 1 1 12 0.1 0.1 0.2 (a) (b) (c) 0.08 0.08 0.15 0.06 0.06 0.1 0.04 0.04 0.05 0.02 0.02 0 0 0 90 95 100 105 110 115 0 5 10 15 20 25 30 20 30 40 50 60 70 80 90 d (cm) E (MeV) |P | (MeV/c) sum sum uv 0.04 0.35 (d) (e) (f) 0.04 0.3 0.03 0.25 0.03 Signal 5042 events 0.2 0.02 Background 99741 events 0.02 0.15 0.1 0.01 0.01 0.05 0 0 0 - 1 - 0.5 0 0.5 1 - 1 - 0.5 0 0.5 1 0 10 20 30 40 50 60 70 80 t + (ns) t (ns) γ e χ γ γ 1 vtx Figure 14 Distributions of variables used in the event selection for (m ;  ) = (20 MeV/c , 20 ps) case. The hatched histograms show the distri- X X bution of MC signal events while the blank histograms that of background events; each histogram is normalised to 1. The vertical lines show the optimised thresholds. (a) The peak value of the signal distribution is at m with FWHM = 2.7 MeV. (c) Cut-o at 20 cm in the background Esum distribution comes from one of the 2 reconstruction conditions. (e) The threshold lines are not visible because they are set to1 ns. For a detailed definition of the variables see Sect. 6. of these variables for the signal and background at a para- cess, we approximate N to  , a selection eciency for BG BG meter set (20 MeV/c , 20 ps) are shown in Fig. 14. All other the background events calculated using the time sideband selection criteria, such as trigger and reconstruction condi- samples selected up to this point. Because of this approxim- tions as well as the E selection, are applied. The time side- ation, the first step leads to suboptimal criteria. band events are used for the background distribution, while In the second step, after all other selection criteria are MC samples are used for the signal distribution. applied, the threshold for  is optimised to give the highest vtx Punzi’s expression [63] is used as a figure of merit F . In this step, to estimate N from the low statistics Punzi BG in the sideband regions, we use a kernel-density-estimation selection F = ; (18) Punzi p p method [64] to model the continuous event distribution. 2 2 b + 2a N + b b + 4a N + 4N BG BG BG 2 The cut thresholds are optimised at 5 MeV=c intervals where a and b are the significance and the power of a test, in m , while the same thresholds are used for di erent X X for each m . The optimised thresholds for m = 20 MeV=c respectively,  is the selection eciency for the sig- selection X X nal, and N is the expected number of background events. are shown as black lines in Fig. 14. These cuts result in BG = 67% (m = 45 MeV/c ) – 51% (m = 20 The values of a and b should be defined before the analysis, selection X X and we set a = 3; b = 1:28 (= 90%), where b is set to the MeV/c ). value appropriate to the confidence level being used to set the upper limit when a non-significant result is obtained. 7 Single event sensitivity The optimisation process is divided into two steps. In the first step, we optimise the cut thresholds of variables 2–6, in- The single event sensitivity of the MEx2G decay s is defined dependently for each variable in order to maintain high stat- as follows: istics in the sidebands. Because the absolute value of N BG does not make sense in this independent optimisation pro- B = s N ; (19) MEx2G MEx2G Probability / (0.02 ns) Probability / (0.36 MeV) Probability / (0.02 ns) Probability / (0.30 MeV/c) Probability / (2.00 ) Probability / (1.60 cm) 13 3.5 where N is the expected number of signal events in the MEx2G + 3 signal region. We calculate it using Michel decay ( ! + + + 2.5 e  ¯ ) events taken at the same time with the  ! e trigger. This Michel normalisation is beneficial for the fol- 1.5 lowing reasons. First, systematic uncertainties coming from the muon beam are cancelled because beam instability is in- 0.5 + + cluded in both Michel triggered and the  ! e triggered + 20 25 30 35 40 45 events. Moreover, we do not need to know the  stopping m (MeV/c ) rate nor the live DAQ time. Second, most of the systematic uncertainties coming from e detection are also cancelled. Figure 15  (see the text for the definition) versus m for  = 20 ps. 2 X X The absolute value of e eciency is not needed. The number of Michel events is given by 97% (m = 20 MeV/c ) increasing monotonically with m . X X B  f Michel Michel The estimate of  is shown in Fig. 15;  = 0:6% (m = 45 N = N +   A   ; (20) 2 2 X Michel  Michel Michel p  p 2 2 Michel correction MeV/c ) – 2.9% (m = 20 MeV/c ), decreasing monoton- where ically with m . This dependence comes mainly from the 2 acceptance: for increasing m , the opening angle between + X N + : the number of stopped  s; the 2 s becomes larger, resulting in a decreasing eciency. B : branching ratio of the Michel decay ( 1); Michel The systematic uncertainties are summarised in Table 1. f : branching fraction of the selected energy region Michel The uncertainty in the 2 detection eciency and that in the (7%–10% depending on m ); 7 MC smearing parameters are the dominant components. p : prescaling factor of the Michel trigger (= 10 ); Michel The estimated value of SES is s = (2:9  0:3)  10 p : correction factor of p depending on the correction Michel 2 10 2 (20 MeV/c ) – (6:31:1)10 (45 MeV/c ) for  = 20 ps muon beam intensity; increasing monotonically with m . The e eciency is  = X e A : geometrical acceptance of the spectrometer for Michel 2 2 + 1% (45 MeV/c ) – 36% (20 MeV/c ) decreasing mono- Michel e s; + tonically with m , estimated with the MC, although this : e eciency for Michel events within the geomet- Michel quantity is not necessary for the normalisation. The over- rical acceptance of the spectrometer. all eciency for the MEx2G events conditional to the e in The number of MEx2G events is given by the geometrical acceptance of the spectrometer is therefore 5 2 3 2 B  = 2:0 10 (45 MeV/c ) – 4:7 10 (20 MeV/c ) MEx2G MEx2G + + + N = N   A         ; (21) MEx2G  e e 2 DM selection decreasing monotonically with m . p X MEG where + + p : prescaling factor of the  ! e trigger (=1); MEG 8 Statistical treatment of background and signal A + : geometrical acceptance of the spectrometer for MEx2G e s; In the following, we describe how we estimate the expec- + + + : e eciency for MEx2G events conditional to e s in ted number of background events in the signal region (N ) BG the geometrical acceptance of the spectrometer; from the numbers of events observed in sidebands A, B, and : the product of 2 geometrical acceptance and 2 trig- obs obs obs C (N , N , and N ). A B C ger, detection, and reconstruction eciency, conditional There are three types of accidental background events to the e detection; defined in Sect. 3. The expected number of background : the trigger direction match eciency conditional to DM events in the signal region is given by the e and 2 detection (Fig. 6); : the signal selection eciency. selection N = N + N + N ; (23) BG 1 2 3 Using Eqs. (19)–(21), an estimate of the SES (s ) is where N ; N ; N are the expected numbers of background 1 2 3 given by events in the signal region from the types 1, 2, and 3, re- 1 p  p Michel correction spectively. Sideband A has the contributions from types 2 s = N Michel B  f p Michel Michel MEG and 3, B has the contributions from types 1 and 3, and C has A +  + e e the contribution from type 3. : (22) 2 DM selection Michel Michel Figure 16 shows the time distributions in the sideband The geometrical acceptance of the spectrometer is com- regions. A peak of type 2 on a flat component of type 3 is mon, hence A =A = 1; the estimate of the relative observed in the t distribution, while a peak of type 1 is e Michel + 2 e eciency is  += = 89% (m = 45 MeV/c ) not clearly visible in the t + distribution. The uniformity of e Michel X e ∈ (%) 2γ 14 Table 1 Systematic uncertainties in the single event sensitivity ( = 20 ps). m (MeV/c ) 20 25 30 35 40 45 Michel e counting 0.99% 1.1% 1.1% 0.93% 1.6% 2.8% Relative e eciency 1.4% 0.18% 0.31% 0.55% 0.89% 1.3% 2 acceptance 1.3% 2.0% 3.4% 2.9% 5.4% 1.9% trigger eciency 0.98% 0.32% 0.26% 0.52% 1.4% 3.2% 2 detection eciency 7.9% 7.9% 7.9% 7.9% 7.9% 7.9% MC statistics 1.8% 1.9% 2.2% 3.1% 1.9% 4.7% MC smearing 4.8% 3.4% 3.8% 5.3% 3.3% 14% Total 9.7% 9.1% 9.7% 11% 10% 17% tion are thus negligibly small compared with the statistical (a) obs obs obs uncertainties in N ; N ; N . A B C Using N ; N ; N , the expected numbers of events in 1 2 3 sidebands A, B, C can be calculated as follows: |t | < 1 ns γ γ 2y 2y exp C C |t | > 1 ns N = N  + N  ; (24) γ γ 1000 2 3 y y B B 2x 2x exp C C N = N  + N  + N  f ; (25) 1 3 2 escape x x A A 2y 2x 2y exp C C C N = N   + N  f  ; (26) 3 2 escape y x y B A B - 3 - 2 - 1 0 1 2 3 t (ns) γ e where x and y are the sizes of the signal regions A(C) B(C) (sideband regions) in t and t + , respectively, as defined in (b) Fig. 13, and f = 0:171 0:003 is the fraction of type 2 escape events in jt j > 1 ns. 2500 The likelihood function for N is given from the Pois- BG |t | > 1 ns γ e son statistics as, |t | < 1 ns γ e (scaled) obs obs obs L (N j N ; N ; N ) BG A B C exp exp exp obs obs obs = P (N j N )P (N j N )P (N j N ): (27) Poi Poi Poi A B C A B C The best estimate of N can be obtained by maxim- BG ising Eq. (27) (listed in Table 2). However, we do not use - 3 - 2 - 1 0 1 2 3 t (ns) γ γ this estimated N in the inference of the signal but use BG obs obs obs 2 (N ; N ; N ) as discussed in the following. Figure 16 Time distributions in the sideband regions for (20 MeV/c , A B C 20 ps). (a) t distributions for jt j < 1 ns (red open circles) and Our goal is to estimate the branching ratio of the MEx2G for 1 < jt j < 3:5 ns (black closed circles). (b) t distributions for decay (B ). The likelihood function Eq. (27) is exten- MEx2G + + jt j < 1 ns (red open circles) and for 1 < jt j < 3:5 ns scaled by the e e 1 1 ded to include B as a parameter and the number of MEx2G ratio of the time ranges (black closed circles). A loose cut is applied: obs mX events in the signal region (N ) as an observable. In ad- jE + E j < 1 MeV^ E < 115 MeV^jP j < 30 MeV=c^ d < e + sum sum uv S 90 cm^  < 80. dition, to incorporate the uncertainty in the SES into the vtx B estimation, the estimated SES (s ) and the true value MEx2G 0 (s) are included into the likelihood function: the accidental backgrounds is examined using these distri- obs obs obs obs L(B ; N ; s j N ; N ; N ; N ; s ): (28) butions; the number of events in (jt +j < 1 ns ^ 1 < jt j < MEx2G BG 0 S A B C 3:5 ns) is compared to the the number of events interpol- Using N ; N ; N and a Gaussian PDF for the inverse of SES, 1 2 3 ated from the region (1 < jt +j < 3:5 ns ^ 1 < jt j < it can be written as, 3:5 ns) scaled by the ratio of the widths of the time ranges obs obs obs obs (2 ns/5 ns). They agree within 1.7% (the central part, includ- L (B ; N ; N ; N ; s j N ; N ; N ; N ; s ) MEx2G 1 2 3 0 S A B C exp exp exp obs obs obs ing type 1, is 1.7% larger than the interpolation). In Fig. 16b, = P (N j N )P (N j N )P (N j N ) Poi Poi Poi S S A A B B t the distribution for 1 < jt j < 3:5 ns is superimposed exp obs 1 1 P (N j N )P (s j s ); (29) Poi Gaus C 0 on that forjt +j < 1 ns after scaling by the time range ratio. exp The tail component of type 2 is consistent in these regions. where N = N + N + N + B =s is the expected 1 2 3 MEx2G The errors on the background estimations by the interpola- number of events in the signal region. Number of events Number of events 15 The best estimated values of the parameter set fB , Table 2 The number of observed events in the sideband regions and MEx2G the signal region and the expected number of background events in the N , N , N , sg are obtained by maximising Eq. (29). 1 2 3 signal region. Among them, only B is the interesting parameter, MEx2G 2 obs obs obs obs while the others are regarded as nuisance parameters  = m (MeV/c ) N N N N N X BG A B C S +0:202 (N ; N ; N ; s). 1 2 3 20 0 0 1 0:048 1 0:046 A frequentist test of the null (background-only) hypo- +0:198 21 0 0 3 0:146 0 0:084 thesis is performed with the following profile likelihood ra- +0:211 22 1 0 5 0:292 0 0:140 tio  as the test statistic [3]: +0:425 23 3 0 3 0:622 0 0:330 ˆ +0:346 L(B ;  ˆ ) MEx2G 24 2 0 1 0:414 1 0:260 (B ) = ; (30) p MEx2G +0:346 25 2 0 3 0:414 0 L(B ;  ˆ ) MEx2G 0:261 ˆ ˆ +0:189 where B and  ˆ are the best-estimated values, and  ˆ 26 0 0 3 0:150 0 MEx2G 0:091 +0:200 is the value of  that maximises the likelihood at the fixed 27 0 0 1 0:050 0 0:049 +0:202 B . The systematic uncertainties of the background es- MEx2G 28 0 0 1 0:048 0 0:046 +0:202 timation and the SES are incorporated into the test by profil- 29 0 0 1 0:048 1 0:046 ing the likelihood about . The local significance is quan- +0:170 30 0 0 0 0:000 0 0:000 tified by the p-value p , defined as the probability to find local +0:202 31 0 0 1 0:048 0 0:046 that is equally or less compatible with the null hypothesis +0:170 32 0 0 0 0:000 0 0:000 than that observed with the data when the signal does not ex- +0:210 33 0 0 0 0:000 0 0:000 ist. +0:210 34 0 0 0 0:000 1 0:000 Since m is unknown, we need to take the look- +0:210 35 0 0 0 0:000 2 0:000 elsewhere e ect [3] into account to calculate the global sig- +0:210 nificance. We estimate this e ect following the approaches 36 0 0 0 0:000 2 0:000 +0:517 in [65, 66], in which the trial factor of the search is estimated 37 1 0 0 0:400 1 0:301 +0:183 using an asymptotic property of  , obeying the chi-square 38 0 0 2 0:168 0 0:105 +0:201 distribution. The smallest p in the m scan is converted 39 0 0 1 0:084 0 local X 0:084 +0:210 into the global p-value p assuming that the signal can 40 0 0 0 0:000 0 global 0:000 appear only at one m . +0:210 41 0 0 0 0:000 0 0:000 The range of B at 90% C.L. is constructed based MEx2G +0:210 42 0 0 0 0:000 0 0:000 on the Feldman–Cousins unified approach [67] extended to +0:201 43 0 0 1 0:084 0 0:084 use the profile-likelihood ratio as the ordering statistic in or- +0:210 44 0 0 0 0:000 0 0:000 der to incorporate the systematic uncertainties [68]. +0:210 45 0 0 0 0:000 0 0:000 9 Results and discussion adopt the Feldman–Cousins unified approach, a one-sided or Table 2 summarises the numbers of events in the signal re- two-sided interval is automatically determined according to gion and the sidebands as well as the expected number of the data. Therefore, lower limits can be set in m regions background events in the signal region. We observe non-zero where non-zero events are observed with small N . BG events in the signal region for some masses. Note that the ad- The statistical significance of the excesses is tested jacent m bins are not statistically independent. Summing against the null hypothesis. Figure 18 shows p versus local up the observed events gives nine events but five of them are m . We observe the lowest p = 0:012 at m = X local X unique events. One event appears in four bins (m = 34, 35, 35 MeV/c , which corresponds to 2.2 significance. The 36, 37 MeV/c ) and another event appears in two bins (m = global p-value is calculated to be p  0:10 by taking global 35, 36 MeV/c ). the look-elsewhere e ect into account. This corresponds to We discuss the results for  = 20 ps below. The results 1.3, that is not statistically significant. for other  are similar, with small changes in the eciency. Owing to the large statistics of the MEG dataset, the The results are presented in detail in Appendix A. branching ratio upper limits have been reduced to the level Figure 17 shows 90% confidence intervals on B MEx2G of O(10 ). Our results improves the upper limits from the obtained from this analysis together with the sensitivities Crystal Box experiment for m < 40 MeV=c , by a factor of and the previous upper limits due to Crystal Box. The sensit- 60 at most. ivities are evaluated by the mean of the branching ratio lim- This publication reports results from the full MEG data- its at 90% C.L. under the null hypothesis. Note that since we set. Hence, new experiments will be needed for further ex- Assuming that the signal is at the assumed m . ploration of this decay, e.g. to test whether the small excess X 16 - 7 Table 3 Results for  = 5 ps. LL and UL denote the lower limit and 10 X upper limit of the 90% confidence interval. - 8 m (MeV/c ) s LL UL X 0 12 13 11 Crystal Box 20 (3:29 0:31) 10 4:60 10 1:28 10 - 9 12 12 21 (3:56 0:44) 10 – 7:78 10 Sensitivity 12 12 22 (3:73 0:44) 10 – 8:30 10 - 10 12 12 10 23 (3:96 0:45) 10 – 7:72 10 This search 12 11 24 (4:26 0:49) 10 – 1:43 10 12 11 25 (5:41 0:55) 10 – 1:08 10 - 11 12 11 26 (5:16 0:61) 10 – 1:13 10 12 11 27 (5:81 0:70) 10 – 1:39 10 - 12 12 11 28 (6:62 0:81) 10 – 1:58 10 12 13 11 29 (7:67 0:95) 10 8:53 10 2:93 10 - 13 12 11 30 (8:62 0:85) 10 – 2:26 10 20 25 30 35 40 45 m (MeV/c ) 11 11 X 31 (1:07 0:14) 10 – 2:54 10 11 11 32 (1:29 0:17) 10 – 3:20 10 Figure 17 Confidence intervals (90% C.L.) onB (blue band) for MEx2G 11 11 33 (1:59 0:21) 10 – 4:00 10 = 20 ps. The red broken line shows the expected upper limits under 11 12 11 34 (1:97 0:26) 10 1:97 10 7:66 10 the null hypothesis and the yellow line shows the limits extracted by 11 11 10 35 (2:60 0:31) 10 1:65 10 1:40 10 Crystal Box analysis. 11 11 10 36 (3:18 0:42) 10 1:98 10 1:72 10 11 10 37 (4:12 0:53) 10 – 1:43 10 1 11 10 38 (5:43 0:70) 10 – 1:27 10 11 10 39 (7:25 0:93) 10 – 1:79 10 11 10 40 (9:01 0:93) 10 – 2:20 10 10 10 41 (1:35 0:19) 10 – 3:37 10 1σ 10 10 42 (1:88 0:27) 10 – 4:72 10 - 1 10 10 10 43 (2:66 0:39) 10 – 6:47 10 10 9 44 (3:81 0:57) 10 – 1:01 10 10 9 45 (6:25 0:88) 10 – 1:59 10 2σ - 2 10 Conclusions 20 25 30 35 40 45 We have searched for a lepton-flavour-violating muon decay m (MeV/c ) + + mediated by a new light particle,  ! e X; X ! decay, Figure 18 Local p-value under null hypothesis as a function of as- for the first time using the full dataset (2009–2013) of the sumed m . MEG experiment. No significant excess was found in the mass range m = 20–45 MeV/c and  < 40 ps, and we X X observed in this search grows. An upgraded experiment, set new branching ratio upper limits in the mass range m = MEG II, is currently being prepared [69]. A brief prospect 20–40 MeV/c . In particular, the upper limits are lowered to 11 2 for improved sensitivity to MEx2G in MEG II is discussed the level of O(10 ) for m = 20–30 MeV/c . The result is below. In this analysis the sensitivity worsens with increas- up to 60 times more stringent than the bound converted from ing m , mainly due to the 2 acceptance and direction match the previous experiment, Crystal Box. eciencies. The acceptance is determined by the geometry of the LXe photon detector and is not changed by the up- grade. The direction match eciency can even worsen if we Appendix A Detailed results for di erent lifetimes + + only consider the  ! e search; the position resolution is expected to improve by a factor two, which enables tight- The detailed results for di erent lifetimes  are summar- ening the direction match trigger condition. However, the ised in Tables 3–5. MEG II trigger development is underway and the trigger ef- ficiency for high mass can be improved up to a factor 2 if a dedicated trigger is prepared. Basically, MEG II will collect Acknowledgments ten times more  decays and the resolutions of each kin- ematic variable will improve by roughly a factor two, lead- We are grateful for the support and co-operation provided ing to higher eciency while maintaining low background. by PSI as the host laboratory and to the technical It is therefore possible to improve the sensitivity by one or- and engineering sta of our institutes. This work is der of magnitude. supported by DOE DEFG02-91ER40679 (USA); INFN BR(μ→ eX, X→γ γ ) Local p-value 17 Table 4 Results for  = 20 ps. LL and UL denote the lower limit and Table 5 Results for  = 40 ps. LL and UL denote the lower limit and X X upper limit of the 90% confidence interval. upper limit of the 90% confidence interval. 2 2 m (MeV/c ) s LL UL m (MeV/c ) s LL UL X 0 X 0 12 13 11 12 13 11 20 (2:92 0:28) 10 3:94 10 1:10 10 20 (3:02 0:29) 10 3:78 10 1:19 10 12 12 12 12 21 (3:18 0:39) 10 – 7:54 10 21 (3:28 0:39) 10 – 7:44 10 12 12 12 12 22 (3:35 0:39) 10 – 7:36 10 22 (3:44 0:39) 10 – 7:80 10 12 12 12 12 23 (3:57 0:40) 10 – 7:14 10 23 (3:66 0:40) 10 – 7:41 10 12 11 12 11 24 (3:86 0:43) 10 – 1:33 10 24 (3:94 0:43) 10 – 1:43 10 12 11 12 11 25 (4:74 0:43) 10 – 1:06 10 25 (4:83 0:43) 10 – 1:04 10 12 11 12 11 26 (4:71 0:53) 10 – 1:15 10 26 (4:76 0:54) 10 – 1:09 10 12 11 12 11 27 (5:31 0:62) 10 – 1:33 10 27 (5:35 0:61) 10 – 1:33 10 12 11 12 11 28 (6:07 0:72) 10 – 1:58 10 28 (6:09 0:71) 10 – 1:51 10 12 12 11 12 13 11 29 (7:04 0:85) 10 1:04 10 2:70 10 29 (7:03 0:83) 10 9:22 10 2:71 10 12 11 12 11 30 (7:94 0:78) 10 – 2:07 10 30 (7:87 0:78) 10 – 1:88 10 12 11 12 11 31 (9:86 1:26) 10 – 2:33 10 31 (9:78 1:21) 10 – 2:33 10 11 11 11 11 32 (1:19 0:15) 10 – 3:14 10 32 (1:18 0:15) 10 – 2:92 10 11 11 11 11 33 (1:46 0:19) 10 – 3:92 10 33 (1:44 0:18) 10 – 3:57 10 11 12 11 11 12 11 34 (1:82 0:23) 10 2:25 10 7:10 10 34 (1:78 0:22) 10 1:97 10 6:81 10 11 11 10 11 11 10 35 (2:38 0:25) 10 1:53 10 1:31 10 35 (2:30 0:23) 10 1:47 10 1:26 10 11 11 10 11 11 10 36 (2:93 0:37) 10 1:85 10 1:56 10 36 (2:84 0:34) 10 1:78 10 1:54 10 11 10 11 10 37 (3:79 0:47) 10 – 1:29 10 37 (3:67 0:43) 10 – 1:28 10 11 10 11 10 38 (4:99 0:62) 10 – 1:16 10 38 (4:80 0:56) 10 – 1:12 10 11 10 11 10 39 (6:65 0:83) 10 – 1:66 10 39 (6:37 0:75) 10 – 1:51 10 11 10 11 10 40 (8:20 0:87) 10 – 2:04 10 40 (7:94 0:78) 10 – 1:90 10 10 10 10 10 41 (1:23 0:20) 10 – 3:15 10 41 (1:17 0:18) 10 – 2:84 10 10 10 10 10 42 (1:71 0:28) 10 – 4:44 10 42 (1:62 0:25) 10 – 4:28 10 10 10 10 10 43 (2:41 0:41) 10 – 5:98 10 43 (2:27 0:37) 10 – 5:94 10 10 10 10 10 44 (3:44 0:61) 10 – 8:25 10 44 (3:23 0:54) 10 – 8:31 10 10 9 10 9 45 (6:29 1:12) 10 – 1:63 10 45 (5:65 0:97) 10 – 1:53 10 (Italy); MIUR Montalcini D.M. 2014 n. 975 (Italy); 6. 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High Energy Physics - Experiment – arXiv (Cornell University)

Published: May 1, 2020

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