Search for Lepton Flavor Violating τ Decays to a Lepton and an Invisible Boson at Belle II

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We search for lepton-flavor-violating τ − → e − α and τ − → µ − α decays, where α is an invisible spin-0 boson. The search uses electron-positron collisions at 10.58 GeV center-of-mass energy with an integrated luminosity of 62.8 fb −1 , produced by the SuperKEKB collider and collected with the Belle II detector. We search for an excess in the lepton-energy spectrum of the known τ − → e −ν eντ and τ − → µ −ν µντ decays. We report 95% confidence-level upper limits on the branching-fraction ratio B(τ − → e − α)/B(τ − → e −ν eντ ) in the range (1.1 − 9.7) × 10 −3 and on B(τ − → µ − α)/B(τ − → µ −ν µντ ) in the range (0.7 − 12.2) × 10 −3 for α masses between 0 and 1.6 GeV/c 2 . These results provide the most stringent bounds on invisible boson production from τ decays. The standard model of particle physics successfully describes the electromagnetic, weak, and strong interactions and classifies all known elementary particles. However, phenomena such as neutrino oscillations, longstanding discrepancies between expectations and observations such as the muon magnetic-moment anomaly, and indirect evidence of dark matter clearly indicate that the standard model is incomplete. Many extensions of the standard model that attempt to incorporate these phenomena require new bosons that are candidates for dark matter or that explain the muon's anomalous magnetic moment [1].
Decays of τ leptons into final states involving light, beyond-the-standard-model bosons that are not directly detectable (invisible) are predicted in models with, e.g., axionlike particles [2][3][4][5]. These bosons are collectively referred to as α in this work. A direct search in τ − → − α, where − indicates e − or µ − , can probe theories beyond the standard model with high sensitivity (chargeconjugated decays are implied throughout). This process was previously searched for by the MARK III [6] and ARGUS [7] collaborations. The current best upper limits on the τ − → − α branching fractions, relative to the corresponding standard-model leptonic decays, are at the 95% confidence level (C.L.) where the range indicates their dependence on the α mass in the (0-1.6) GeV/c 2 [7] range.
We report a search for the lepton-flavor-violating τ − → − α decay involving a spin-0 α boson, where both the α and its decay products are invisible. We use data from e + e − collisions produced in 2019 and 2020 by the Su-perKEKB collider at KEK [8]. The data, corresponding to an integrated luminosity of 62.8 fb −1 [9], were recorded by the Belle II detector [10] at a center-of-mass energy of √ s = 10.58 GeV and contain 57.7 million e + e − → τ + τ − events.
The Belle II detector consists of several subdetectors arranged in a cylindrical structure around the e + e − interaction point. The z axis of the laboratory frame is defined as the detector solenoid axis, with the positive direction in the direction of the electron beam. The polar angle θ and the transverse plane are defined according to this axis. Charged-particle trajectories are reconstructed by a tracking system consisting of a two-layer silicon-pixel detector, surrounded by a four-layer double-sided siliconstrip detector and then a central drift chamber (CDC). Only 15% of the second pixel layer was installed when the data used in this work were collected. Outside the CDC, time-of-propagation and aerogel ring-imaging Cherenkov detectors cover the 31 • < θ < 128 • and 14 • < θ < 30 • polar ranges, respectively. The electromagnetic calorimeter (ECL) that serves to reconstruct photons and identify electrons fills the remaining volume inside a superconducting solenoid that generates a 1.5 T uniform, axial magnetic field. A dedicated subdetector to identify K 0 L mesons and muons is located at the outermost radius of the detector.
In the center-of-mass frame, τ pairs are produced back to back; thus, decay products of each τ are isolated from the others and contained in opposite hemispheres. The boundary between those hemispheres is the plane perpendicular to the τ direction, which can be experimentally approximated by the thrust axist, i.e., the vector that maximizes the thrust value |t · p c.m.
i is the momentum of each final-state particle in the center-of-mass frame [11,12]. We define the tag hemisphere as that containing three charged particles, which originate from τ − → h − h + h − ν τ (h = π, K), and require that the other hemisphere, the signal hemisphere, contain only one charged particle.
In τ decays from τ -pair production, the τ − → − α has an identical visible topology to that of τ − → −ν ν τ , the latter thus being an irreducible background. However, in τ − → − α the magnitude of the lepton momentum depends only on the α mass. The two-body decay provides a distinctive signature for our signal above the spectrum of τ − → −ν ν τ , where the lepton momentum has a broad distribution.
We cannot determine the τ momentum from the observed particles directly; we can, however, approximate its energy in the center-of-mass frame as √ s/2 (neglecting the initial-state radiation) and its direction as being opposite to the three hadrons on the tag side, p τ ≈ − p 3h /| p 3h |. From these we construct the τ pseudo rest frame [7].
We search for τ − → − α by looking for an excess of events above the spectrum of τ − → −ν ν τ normalized lepton energy [13] x where E * is the energy of the charged lepton in the τ pseudo rest frame. We then measure the branching- The search uses an online event selection (trigger) that requires events with at least three localized energy deposits in the ECL (clusters). One of these clusters must have an energy above 0.3 GeV and the rest above 0.1 GeV. In addition, we require a topology inconsistent with Bhabha scattering. The efficiency of this trigger is measured in experimental data with respect to independent triggers based on the number of particles reconstructed in the CDC; simulated distributions are then scaled by this efficiency. The trigger efficiencies are on average 97% and 86% for the electron and muon channels, respectively.
We select τ -pair candidates by requiring the event to contain exactly four charged particles with zero total charge, each displaced from the average interaction space point by less than 3 cm in the z-axis direction and less than 1 cm in the transverse plane. The particle in the signal hemisphere must be identified as either an electron or a muon by combining the information from all subdetectors into a global discriminator similar to a likelihood ratio. Each charged particle in the tag hemisphere must satisfy the condition E ECL /p ≤ 0.8 to reject electron contamination. Here, E ECL is the particle energy measured in the ECL and p the magnitude of its momentum measured in the tracker, both in the laboratory frame.
Backgrounds from e + e − → qq are suppressed by rejecting events containing neutral pions and photons. Photons used in π 0 reconstruction are ECL clusters with energy deposits of at least 0.1 GeV, which must be within the CDC acceptance to ensure they are not matched to any charged particle. Displaced clusters from secondary hadronic interactions and multiple clusters deposited by low-momentum charged particles are challenging to model correctly; photons that deposit less than 0.4 GeV in the ECL must be at least 40 cm from the nearest charged particle at the inner surface of the ECL to suppress these contributions. Neutral pions are identified as photon pairs with masses within [115, 152] MeV/c 2 . Events containing photons satisfying the above conditions, but not used in π 0 reconstruction and with energy greater than 0.2 GeV, are also rejected. We reject events from e + e − → + − γ, e + e − + − , and e + e − h + h − , characterized by low-momentum tag-side charged particles, by sorting the three charged particles in the tag hemisphere by increasing transverse momentum and requiring that they exceed, respectively, 0.08, 0.30, and 0.70 GeV/c for τ − → e − α candidates and 0.04, 0.17, and 0.47 GeV/c for τ − → µ − α candidates. We further reject events from e + e − → e + e − γ and qq by restricting the thrust value to ranges consistent with that of τ + τ − events and from e + e − → e + e − + − by requiring the event missing momentum, i.e., the negative vector sum of the momenta of all reconstructed particles in the event, to be within a polar angle (θ miss ) range where the process can be accurately simulated. In addition we suppress all types of backgrounds by requiring that charged particle trajectories in the tag hemisphere are consistent with a common origin, and that these particles have a mass M 3h and center-of-mass-frame energy E c.m. 3h consistent with τ decay kinematics. The ranges for the thrust, θ miss , E c.m. 3h , and M 3h selections are listed in Table I. The reconstruction efficiencies in the normalization channels are 12.7% for τ − → e −ν e ν τ and 16.2% for τ − → µ −ν µ ν τ decays, while the purities are 95.9% and 92.0% respectively. The reconstruction efficiency of τ − → e − α decays depends on the α mass and varies between 9.4% and 13.9%. Likewise the efficiency for τ − → µ − α decays varies between 9.1% and 17.4%. The parameter space defined by the selection criteria is referred to as the signal region. We perform the analysis in a closed-box approach; before examining the x distribution of experimental data in the signal region, we validate the simulation using variables that are insensitive to the presence of τ − → − α decay, and study control regions defined by accepting events containing neutral pions or photons, rather than rejecting them. The distributions of x for events belonging to the signal region are shown in Fig. 1.
We model each x spectrum as a sum of contributions from the signal decay, the standard-model leptonic decay, and all other sources of background, where the probability density functions f α , f νν , and f b are binned distributions taken from simulations, N νν and N b are the observed yields, and α / νν is the effi-  Figure 1: Spectra of x for electrons (top) and muons (bottom) in simulation and experimental data. Simulated spectra for standard-model processes are shown stacked, with the gray band indicating the total uncertainty, which is dominated by the lepton-identification efficiency uncertainty. Remaining background processes other than τ − → −ν ντ contributing to the spectrum are combined together and collectively referred to as "other". The distributions for τ − → − α are shown for three α masses assuming branching-fraction ratios of 5%. ciency of observing τ − → − α decays relative to that for observing τ − → −ν ν τ .
We use RooStats [31] and HistFactory [32] to fit our model to binned data using extended maximum likelihoods that are functions of the branching-fraction ratio B α /B νν and of N νν and N b .
The leading systematic uncertainties originate from the corrections to the lepton-identification efficiency and particle misidentification rate, based on comparison of calibration samples in data and simulated events. These corrections depend on the momentum and polar angle; their typical ranges are summarized in Table II. The resulting uncertainties are asymmetric and strongly depend on x ; their ranges and averaged values over the standard-model yields are also reported in the same table. The contribution from lepton-identification efficiency partially cancels in the ratio between signal and normalization channels; while the contribution from particle misidentification rates does not, as it affects only other background sources. Uncertainties from the trigger and π 0 reconstruction efficiency corrections are also taken into account. Trigger uncertainties range in 0.1%-4% for the electron channel and in 0.2%-1.5% for the muon channel, depending on x . Neutral pion reconstruction efficiency is evaluated from studies on independent samples to be 0.914 ± 0.020. Each of these systematic uncertainties is included in the likelihood as an additional shape-correlated nuisance parameter that is assumed to follow a Gaussian distribution. Other sources of uncertainty from track reconstruction efficiency, beam-energy determination, relative reconstruction efficiency, and momentum-scale correction have negligible impact on the results.
Inspection of events in the signal region shows that asymmetrical uncertainties yield unreliable results. We, therefore, revise our definitions and symmetrize their distributions using their greater variation in each bin.
We observe no significant signal and determine upper limits using the CL S method [34], a modified frequentist approach based on a profile likelihood ratio [35]. Figure 2 shows the 95% C.L. upper limits as well as expectations calculated assuming the background-only hypotheses, ranging in (1.1 − 9.7) × 10 −3 for the electron channel and in (0.7 − 12.2) × 10 −3 for the muon channel. Systematic uncertainties degrade on average our upper limit sensitivity by approximately 35% in both channels.
The fit results and upper limits are summarized in Table III. The corresponding absolute upper limits for B(τ − → − α), computed using standard-model worldaverage branching fractions for the reference channel [33], are also provided for convenience. Our 95% C.L. limits are 2.2-14 times more stringent than the best previous bounds in [7], depending on the value of the α mass.
In conclusion, we search for the lepton-flavor-violating decay τ − → − α using data collected by the Belle II detector at an e + e − center-of-mass energy of 10.58 GeV, corresponding to an integrated luminosity of 62.8 fb −1 . We observe no statistically significant signal and set 90% and 95% confidence-level upper limits on the branchingfraction ratios B(τ − → − α)/B(τ − → −ν ν τ ). These constitute the most stringent limits on invisible spin-0 boson production from τ lepton decays, allowing one to directly constrain standard model extensions (see, e.g., Ref. [3]) in ways not otherwise possible outside of collider experiments.
This work, based on data collected using the Belle II detector, which was built and commissioned prior to