Search for an invisibly decaying Z ′ boson at Belle II in e + e − → μ + μ − ( e ± μ ∓ ) plus missing energy final states

Theories beyond the standard model often predict the existence of an additional neutral boson, the Z^{'}. Using data collected by the Belle II experiment during 2018 at the SuperKEKB collider, we perform the first searches for the invisible decay of a Z^{'} in the process e^{+}e^{-}→μ^{+}μ^{-}Z^{'} and of a lepton-flavor-violating Z^{'} in e^{+}e^{-}→e^{±}μ^{∓}Z^{'}. We do not find any excess of events and set 90% credibility level upper limits on the cross sections of these processes. We translate the former, in the framework of an L_{μ}-L_{τ} theory, into upper limits on the Z^{'} coupling constant at the level of 5×10^{-2}-1 for M_{Z^{'}}≤6  GeV/c^{2}.

Y. Yusa, 63 L. Zani, 88, 36 Z. Zhang, 106 V. Zhilich, 3, 64 Q. D. Zhou, 21 X. Y. Zhou, 1 and V. I. Zhukova 50 (Belle II Collaboration) Theories beyond the standard model often predict the existence of an additional neutral boson, the Z . Using data collected by the Belle II experiment during 2018 at the SuperKEKB collider, we perform the first searches for the invisible decay of a Z in the process e + e − → µ + µ − Z and of a lepton-flavor-violating Z in e + e − → e ± µ ∓ Z . We do not find any excess of events and set 90% credibility level upper limits on the cross sections of these processes. We translate the former, in the framework of an Lµ − Lτ theory, into upper limits on the Z coupling constant at the level of 5 × 10 −2 ÷ 1 for M Z ≤ 6 GeV/c 2 . The standard model (SM) is a successful and highly predictive theory of fundamental particles and interactions. However, it cannot be considered a complete description of nature, as it does not account for many phenomena, including dark matter.
The L µ − L τ extension of the SM [1,2] gauges the difference of the leptonic muon and tau number, giving rise to a new vector boson, the Z . The Z couples to the SM only through the µ, τ , ν µ and ν τ , with coupling constant g . The L µ − L τ model is potentially able to address important open issues in particle physics, including the anomalies in the b → sµ + µ − decays reported by the LHCb experiment [3], the anomaly in the muon anomalous magnetic moment (g −2) µ [4], and dark matter phenomenology, if extra matter is charged under L µ − L τ [1,5]. We investigate here, for the first time, the specific invisible decay topology e + e − → µ + µ − Z , Z → invisible, where the Z production occurs via radiation off a final state muon. The decay branching fractions (BF) to neutrinos are predicted to vary between 33% and 100% depending on the Z mass [5]. This model ("standard Z " in the following) is poorly constrained at low masses. Related searches have been performed by the BABAR and CMS experiments for a Z decaying to muons [6,7]. Our search is, therefore, the first to have sensitivity to Z masses m Z < 2m µ . If the Z is able to decay directly into a pair of dark matter particles χχ, one expects BF(Z → χχ) ≈ 1. We provide separate results for this scenario, which is not constrained by existing measurements.
The second scenario we consider postulates the existence of a lepton-flavor-violating (LFV) boson, either a scalar or a vector ("LFV Z " in the following), which couples to leptons [8,9]. We focus on the LFV e − µ coupling. While the presence of LFV mediators can be constrained by measurements of the forward-backward asymmetry in e + e − → µ + µ − [9,10], we present here a direct, model-independent search of e + e − → e ± µ ∓ Z , Z → invisible. The presence of missing energy decays make these searches especially suitable for an e + e − collider.
The Belle II detector [11] operates at the SuperKEKB electron-positron collider [12], located at the KEK laboratory in Tsukuba, Japan. Data were collected at the center-of-mass (CM) energy of the Υ (4S) resonance from April to July 2018. The energies of the electron and positron beams are 7 GeV and 4 GeV, respectively, resulting in a boost of βγ = 0.28 of the CM frame relative to the lab frame. The integrated luminosity used in this analysis amounts to 276 pb −1 [13].
The Belle II detector consists of several subdetectors arranged around the beam pipe in a cylindrical structure. A superconducting solenoid, situated outside of the calorimeter, provides a 1.5 T magnetic field. Subdetectors relevant for this analysis are briefly described here; a description of the full detector is given in [11,14]. The innermost subdetector is the vertex detector (VXD), which includes two layers of silicon pixels and four outer layers of silicon strips. Only a single octant of the VXD was installed during the 2018 operations [15]. The main tracking device (CDC) is a large helium-based small-cell drift chamber. The electromagnetic calorimeter (ECL) consists of a barrel and two endcaps made of CsI(Tl) crystals. The z axis of the laboratory frame is along the detector solenoidal axis in the direction of the electron beam. "Longitudinal" and "transverse" are with respect to this direction, unless otherwise specified. The invisible Z signature is a peak in the distribution of the invariant mass of the system recoiling against a lepton pair. "Recoil" quantities such as mass and momentum refer to this system. They coincide with Z properties in the case of signal events and typically correspond to undetected SM particles in the case of background events. The analysis uses events with exactly two tracks, identified as µµ or eµ, and minimal other activity in the ECL. The standard Z selection is optimized using simulated events prior to examining data; the same criteria, aside from an electron in the final state, are used for the LFV Z search. The dominant backgrounds are SM final states with missing energy and two tracks identified as leptons. These are radiative muon pairs (e + e − → µ + µ − γ(γ)) with one or more photons which are not detected due to inefficiency or acceptance, e + e − → τ + τ − (γ), and e + e − → e + e − µ + µ − with electrons outside the acceptance. Control samples are used to check background rates predicted by simulation and to infer correction factors and related uncertainties. Upper limits on the standard Z cross section are computed with a counting technique in windows of the recoil mass distribution. For the LFV Z model-independent search, upper limits are interpreted in terms of signal efficiency times cross section. Details of each of these steps are described below.
Signal events are generated with MadGraph 5 [16] for standard Z masses ranging from 0.5 to 8 GeV/c 2 in steps of 0.5 GeV/c 2 .
The standard Z search uses the CDC two-track trigger, which selects events with at least two tracks with an azimuthal opening angle larger than 90 • . The LFV Z search uses the ECL trigger, which selects events with total energy in the barrel and part of the endcap above 1 GeV. Both triggers reject events that are consistent with being Bhabha scatterings.
To reject spurious tracks and beam induced background, "good" tracks are required to have transverse and longitudinal projections of the distance of closest approach with respect to the interaction point smaller than 0.5 cm and 2.0 cm, respectively. Photons are classified as ECL clusters with energy greater than 100 MeV, which are not associated with tracks. Quantities are defined in the laboratory frame unless specified otherwise. Events are required to pass the following selection criteria.
1. Exactly two oppositely charged good tracks, with polar angles in a restricted barrel ECL acceptance θ ∈ [37, 120] • and with azimuthal opening angle > 90 • , to match the CDC trigger requirement.
2. Recoil momentum pointing into the ECL barrel acceptance θ ∈ [32, 125] • , to exclude inefficient regions where photons from radiative backgrounds can pass undetected. This selection is applied only for recoil masses below 3 GeV/c 2 ; missed radiative photons are unlikely to produce higher masses.
3. An ECL-based particle identification (PID) selection: 0.15 < E < 0.4 GeV and E/pc < 0.4 for muons; 0.8 < E/pc < 1.2 and E > 1.5 GeV for electrons, where E is the energy of the ECL cluster associated to a track of momentum p.

No photons within a 15
• cone around the recoil momentum direction in the CM frame, to suppress radiative lepton pair backgrounds.
5. Total photon energy less than 0.4 GeV and no π 0 candidates (pairs of photons with invariant masses within 10 MeV/c 2 of the nominal π 0 value) After this selection, the background for recoil masses below 7 GeV/c 2 is dominated by e + e − → τ + τ − (γ) events with τ → µ, or τ → π where the pion is misidentified as a muon.
In subsequent steps of the analysis, events are grouped into windows of recoil mass. The width of these windows is ±2σ, where σ is the recoil mass resolution. It is obtained by fitting each Z recoil mass distribution with a sum of a Crystal Ball (CB) [24][25][26] and a Gaussian function with coincident peaks. The resolution is computed as the sum in quadrature of the CB and Gaussian widths weighted according to their contributions. The choice of ±2σ maximizes a figure of merit (FOM) [27] over the full spectrum. Mass window widths vary from 1150 MeV/c 2 at M Z = 0.5 GeV/c 2 to a minimum of 51 MeV/c 2 at M Z = 6.9 GeV/c 2 . There are in total 69 mass windows below 8 GeV/c 2 .
A final selection, denoted as "τ suppression", exploits the kinematics of the Z production, which occurs radiatively from a final state muon, to further suppress τ + τ − events in which the missing momentum arises from neutrinos from both τ decays. The variables, defined in the CM frame, are: the transverse recoil momentum with respect to the lepton with the higher momentum p T,lmax rec ; with respect to the lower momentum p T,lmin rec ; the transverse momentum of the dilepton pair (p T µµ or p T eµ ).  ) is the transverse recoil momentum with respect to the direction of the muon with maxiumum (minimum) momentum in the CM frame. The optimal separation line is superimposed.
For the standard Z search, a linear cut is imposed in the p T,lmax rec -p T,lmin rec plane and a selection p T µµ > p T cut where the cut values are determined using an optimization procedure that numerically maximizes the FOM in each recoil mass window. p T cut is typically 1.5÷2.0 GeV/c and is effective in suppressing the remaining µ + µ − (γ) and e + e − µ + µ − backgrounds. For masses higher than 7 GeV/c 2 , signal and background overlap in the p T,lmax rec p T,lmin rec plane and effective separation lines are not found. The same values are used for the LFV Z search. Trigger, tracking and particle identification efficiencies are studied on control samples. The performance of the CDC two-track trigger is studied on data samples, mostly radiative Bhabha scattering events, selected by means of the ECL trigger. The efficiency is (79 ± 5)% when both tracks are within the acceptance of selection 1; the un-certainty is systematic and is due to kinematic dependencies. The performance of the ECL trigger is studied using e + e − → µ + µ − γ events with E γ > 1 GeV that are selected with the CDC two-track trigger. The efficiency is found to be uniformly (96 ± 1)% in the ECL barrel region.
The tracking efficiency for data is compared to simulation using radiative Bhabha and e + e − → τ + τ − events. Differences are found to be 10% for two-track final states, with a 4% systematic uncertainty due to kinematic dependencies.
The PID efficiency for data is compared to simulation using samples of four-lepton events from two-photon mediated processes. Discrepancies at the level of 2% per track are found, resulting in a systematic uncertainty of 4%.
The dimuon recoil mass resolution of data is compared to simulation using e + e − → µ + µ − γ events that are consistent with the full event energy, and which satisfy selections 1-5 except selection 4, which they are required to fail (µµγ sample). The two-dimensional muon momentum distributions are reweighted to produce analogous distributions for e + e − → µ + µ − Z events with Z masses up to 3 GeV/c 2 . The recoil mass widths for data and simulation are consistent, and no systematic uncertainty is assigned.
The selection criteria before the τ suppression are studied using signal-free control samples in data and simulation. The µµγ sample is useful for the low recoil mass region. Similar eeγ and eµγ control samples are used for consistency checks. We also select µµ and eµ samples that satisfy requirements 1-5, but which fail the p T,lmax rec p T,lmin rec requirement. These studies indicate that the efficiency before the τ suppression is 35% lower for µ + µ − events in data than in simulation, and 10% lower for e ± µ ∓ events. The latter is explained by tracking inefficiency, leaving a −25% unexplained deficit in dimuon events. A variety of studies failed to uncover the source of this discrepancy, which is consistently found to be independent of all checked quantities, including the recoil mass. The background predictions from simulation and the signal efficiency are thus corrected with scaling factors of 0.65 for µ + µ − events and 0.9 for e ± µ ∓ events. The background level before the τ suppression selection is measured with a 2% statistical uncertainty in both samples [28], which is used as a systematic uncertainty. This is a strong constraint for the standard Z signal efficiency as well, as the topology of background and signal events (a pair of muons and missing energy) is identical for signal and background and the discrepancy in the measured yield is found not to depend on kinematic quantities (see above). We nevertheless conservatively assign a systematic uncertainty of 12.5% on the correction factor to the signal efficiency for the dimuon sample, half the size of the observed discrepancy.
To study the τ suppression, we use an e + e − sample selected using the same analysis criteria, but with both tracks satisfying the electron criteria in selection 3. The resulting sample includes e + e − γ, e + e − e + e − and τ + τ − events where both leptons decay to electrons. The latter has the same kinematic features of the most relevant background source to both searches. Agreement between data and simulation is found after the τ suppression, within a 22% statistical uncertainty. This is taken as a systematic uncertainty on the background; no systematic uncertainty due to this effect is considered for the signal, as the selection has a high efficiency (around 50%, slightly depending on the Z mass), and the distributions on which it is based are well reproduced in simulation. After the corrections for the two-track trigger efficiency and for the data/simulation discrepancy in µ + µ − events, signal efficiencies are found to range between 2.6% and 4.9% for Z masses below 7 GeV/c 2 . Signal efficiencies are interpolated from the generated Z masses to the center of each recoil mass window. An additional binning scheme is introduced with a shift of a half bin, to cover hypothetical signals located at the border of two contiguous bins, where the signal efficiency is reduced. Systematic uncertainties are summarized in Table I.   The final recoil mass spectrum of the µ + µ − sample is shown in Fig. 2, together with background simulations. We look for the presence of possible local excesses by calculating for each recoil mass window the probability to obtain a yield greater or equal to that obtained in data given the predicted background, including statistical and systematic uncertainties. No anomalies are observed, with all results below 3σ local significance in both the normal and shifted-binning options [28]. A Bayesian procedure [29] is used to compute 90% credibility level (CL) upper limits on the standard Z cross section. We assume flat priors for all positive values of the cross section, while Poissonian likelihoods are assumed for the number of observed and simulated events. Gaussian smearing is used to model the systematic uncertainties. Results are cross-checked with log-flat priors and with a frequentist procedure based on the Feldman-Cousins approach [30] and are found to be compatible in both cases [28]. Cross section results are translated into 90% CL upper limits on the coupling constant g . These are shown in Fig. 3, where only values g ≤ 1 are displayed.  The final recoil mass spectrum of the e ± µ ∓ sample is shown in Fig. 4, together with background simulations. Again, no anomalies are observed above 3σ local significance [28]. Model-independent 90% CL upper limits on the LFV Z efficiency times cross section are computed using the Bayesian procedure described above and crosschecked with a frequentist Feldman-Cousins procedure (Fig. 5). Additional plots and numerical results can be found in the supplemental material [28].
In summary, we have searched for an invisibly decaying Z boson in the process e + e − → µ + µ − Z and for a  LFV Z in the process e + e − → e ± µ ∓ Z , using 276 pb −1 of data collected by Belle II at SuperKEKB in 2018. We find no significant excess and set for the first time 90% CL upper limits on the coupling constant g in the range 5 × 10 −2 to 1 for the former case and to the efficiency times cross section around 10 fb for the latter. The full Belle II data set, with better muon identification, a deeper knowledge of the detector, and the use of multivariate analysis techniques should enable the full (g − 2) µ band to be probed in the future.
We thank the SuperKEKB group for the excellent operation of the accelerator; the KEK cryogenics group for the efficient operation of the solenoid; and the KEK computer group for on-site computing support. This work was supported by the following funding sources: