Search for $B^0$ Decays to Invisible Final States ($+\gamma$) at Belle

We report searches for $B^0\to\rm{invisible}$ and $B^0\to\rm{invisible}+\gamma$ decays, where the energy of the photon is required to be larger than 0.5 GeV. These results are obtained from a $711\,{\rm fb}^{-1}$ data sample that contains $772 \times 10^6 B\bar{B}$ pairs and was collected near the $\Upsilon\,(4S)$ resonance with the Belle detector at the KEKB $e^+ e^-$ collider. We observe no significant signal for either decay and set upper limits on their branching fractions at $90\%$ confidence level of $\mathcal{B}\,(B^0\to\rm{invisible})<7.8\times10^{-5}$ and $\mathcal{B}\,(B^0\to\rm{invisible}+\gamma)<1.6\times10^{-5}$.

The decays B 0 → invisible and B 0 → invisible + γ, with "invisible" defined as particles that leave no sig-nal in the Belle detector, are sensitive to new physics beyond the Standard Model (SM).For instance, mod-els with R-parity violation [1] or dark matter contributions [2] predict that the branching fraction of B 0 decays to an invisible final state could be as high as 10 −6 to 10 −7 .In the SM, such a decay is B 0 → (γ)ν ν, which proceeds through the Feynman diagrams in Fig. 1.The B 0 → ν ν decay is strongly helicity suppressed by a factor of (m ν /m B 0 ) 2 [3], and the estimated branching fraction is at the 10 −25 level.A recent calculation [4] predicts that a B 0 → ν νν ν decay, which has the same signature as B 0 → ν ν in the detector, also contributes to the invisible final state, and the estimated branching fraction is at the 10 −16 level.For the B 0 → γν ν decay, despite the removal of helicity suppression, the branching fraction predicted from the SM is of order 10 −9 [5], which is still too small to be observed by current experiments.A very low background from the SM indicates that a signal of B 0 → invisible (+γ) in the current B-factory data would indicate new physics.
Several experimental searches for B 0 → invisible (+γ) have been performed and no signal has been observed.The most stringent branching-fraction upper limits [6], B (B 0 → invisible) < 2.4 × 10 −5 and B (B 0 → invisible + γ) < 1.7 × 10 −5 , were provided by the BaBar Collaboration using the semileptonic tagging method and with 424 fb −1 of data.A previous search [7] from Belle with 606 fb −1 of data adopted a hadronic tagging method and reported the upper limit, B (B 0 → ν ν) < 1.2 × 10 −4 , a factor of five higher than the BaBar results.Here we report the updated results with the full Belle data set and improved hadronic tagging.These searches are based on a data sample that was collected with the Belle detector at the KEKB asymmetricenergy e + e − (3.5 on 8 GeV) collider [8].The sample contains 772 × 10 6 BB pairs accumulated at the Υ (4S) resonance, corresponding to an integrated luminosity of 711 fb −1 , and an additional 90 fb −1 of off-resonance data recorded at the center-of-mass (CM) energy 60 MeV below the Υ(4S) resonance.
The Belle detector is a large-solid-angle magnetic spec-trometer that consists of a silicon vertex detector (SVD), a 50-layer central drift chamber (CDC), an array of aerogel threshold Cherenkov counters, a barrel-like arrangement of time-of-flight scintillation counters, and an electromagnetic calorimeter (ECL) comprised of CsI (Tl) crystals located inside a superconducting solenoid coil that provides a 1.5 T magnetic field.Outside the coil, the K 0 L and muon detector (KLM), composed of alternating layers of charged particle detectors and iron plates, is instrumented to detect K 0 L mesons and to identify muons.The detector is described in detail elsewhere [9].Two inner detector configurations were used.A 2.0 cm radius beampipe and a 3-layer SVD were used for the first 140 fb −1 data sample, while a 1.5 cm radius beampipe, a 4-layer SVD and a small-cell inner CDC were used to record the remaining 571 fb −1 data sample [10].
To determine our signal efficiency and optimize event selection criteria, we use Monte Carlo (MC) simulated events.All MC samples in the analysis are generated by the EvtGen package [11], with the detector response simulated by the Geant3 package [12].Ten million B 0 → ν ν and B 0 → γν ν signal events are generated with a phase-space decay model.However, for the B 0 → γν ν search, a phase-space decay model is not appropriate to describe the process.Thus, the signal efficiency is reweighted according to theoretical calculations [5], in which the "quark constituent model" is assumed and differential branching fraction as a function of squared missing mass (M 2 miss ) is given.M 2 miss is defined as: where P beam , P Btag and P γ are the four-momenta of the e + e − system, the other B meson and the photon for a B 0 → γν ν signal event, respectively.In addition, a second model-independent binned analysis is performed in five different M 2 miss regions using the signal MC sample generated with the phase-space decay model: Since the signal-side particles, except for the photon, cannot be detected, a technique that fully reconstructs the other B meson (tag-side B tag meson) is used.The signature of B 0 → invisible or a photon for B 0 → invisible + γ is then identified in the remaining part of the event.
The hadronic full reconstruction is a hierarchical process for reconstructing the B tag meson [13].The B 0 candidates are reconstructed from 489 decay channels in which B 0 mesons decay to hadrons.The process consists of four stages, starting from an initial selection of charged tracks, photons, K 0 S , and π 0 , followed by two stages of forming intermediate particles, (D ± (s) , D 0 , J/ψ) and (D * ± (s) , D * 0 ), and ending at the stage of reconstructing the B 0 meson from its daughter products.The neural network (NN) package, NeuroBayes [14], is used to assign a signal probability (P FR ) to the reconstructed particle at each individual stage.The NN at each stage is trained with the P FR of the daughter particles and properties of the candidate, such as invariant mass and the opening angle between daughters.If there are multiple B 0 meson candidates in an event, the candidate with the highest P FR is selected as the B tag .
From the previous study [13], the number of correctly reconstructed B tag in the full data set is 1.4 × 10 6 .In the case of B 0 → ν ν and B 0 → γν ν signal MC simulation, the reconstruction efficiencies of the B tag are 0.41% and 0.47%, respectively.In this analysis, a loose preselection on the beam-energy-constrained B tag mass, M bc,tag > 5.26 GeV/c 2 , is applied.This mass is cal- where the E beam is the beam energy in the e + e − CM frame, and the P Btag is also defined in this frame.
For B 0 → invisible+ γ, at least one photon is required.The signal photon is detected by the ECL and an energy threshold of 0.5 GeV in the e + e − CM frame is applied in order to eliminate the huge number of photons from the beam background.Furthermore, we require that the corresponding ECL cluster does not match with a track in the CDC, and that the fraction of energy detected in the inner 3 × 3 array of crystals relative to the 5 × 5 array of crystals centered on the crystal with the maximum energy exceeds 0.9.In the case that more than one photon satisfies the selection criteria, the one with the highest energy is selected as the signal photon.
After the reconstruction of B tag , and selecting the photon for B 0 → invisible + γ, events are rejected that have extra tracks, π 0 , or K 0 L because no extra detectable particles except photons are expected in the event.Extra tracks are defined as those passing the loose impact parameter selections dr < 4 cm and |dz| < 35 cm, where dr and dz are the shortest distance from the track to the interaction point (IP) on the transverse plane and along the beam axis, respectively.The loose requirement aims to include low-momentum tracks that are illreconstructed and tracks not produced around the IP.Extra π 0 candidates are reconstructed from photon pairs passing the following requirements: each photon has energy larger than 40 MeV; the absolute cosine value of the angle between a photon direction and the boost direction of the lab system in the π 0 rest frame smaller than 0.9; 120 MeV/c 2 < M π 0 < 145 MeV/c 2 , which corresponds to a window within 1.5 standard deviations (σ) of the nominal mass [15].Extra K 0 L candidates are detected in the KLM detector, where a minimum of two hit layers is required.
A powerful variable to identify B 0 → invisible and B 0 → invisible + γ signal is E ECL , which is defined as the sum of all the remaining energies of ECL clusters that are not associated with tag-side B daughter particles.For B 0 → invisible + γ, the signal photon is also excluded.In the E ECL calculation, in order to reduce a contribution from beam background, only the ECL clusters that satisfy the following energy thresholds are included: E cluster > 0.05, 0.10 and 0.15 GeV for the barrel region (32.2 • < θ < 128.7 • ), forward endcap (θ < 32.2 • ) and backward endcap (θ > 128.7 • ), respectively, where θ is the polar angle in the lab frame.Since the distribution for signal events peaks at zero, the E ECL signal box is defined as E ECL < 0.3 GeV, and the E ECL sideband is defined as 0.3 GeV < E ECL < 1.2 GeV.
After the signal event selections, e + e − → q q (q = u, d, s, c) continuum events are the dominant background, followed by B B decay with a b → c transition (generic B background).Two separate NN implemented using the NeuroBayes package are used in order to reduce the former.The first NN focuses on rejecting fake B tag , the input variables are those related to the B tag reconstruction qualities: P FR of the B tag ; M bc,tag ; ∆E tag , which is defined as the energy difference between the reconstructed B tag meson and the beam energy at the e + e − CM frame.The second NN focuses on the jet-like topology of continuum events.The input variables are the sum of the transverse momentum, M 2 miss , which is defined in Eq. 1 without the P γ term, and sixteen modified Fox-Wolfram moments [16].For B 0 → invisible + γ, the signal photon is excluded in all the momentum-related calculations in order to reduce model dependence.Outputs of the two NN (O tag and O shape respectively) are continuous variables within the range (−1, 1), and larger (smaller) values correspond to events more (less) likely to be signal.We find that O tag and O shape are also effective at distinguishing the generic B background from the signal.The O tag and O shape distributions for signal and both kinds of the background are shown in Fig. 2.
Thresholds for O tag and O shape are determined by maximizing a figure of merit (FOM) separately for the modes, B 0 → invisible and B 0 → invisible+γ, and the five M 2 miss bins.The optimization is done in the E ECL signal box and the FOM has the form [17]: where ǫ sig is the signal efficiency in MC simulation and N bkg is the number of background events reconstructed as signal in MC.Here n σ is the number of σ in a onetailed Gaussian test, where n σ = 1.28 corresponds to the choice of a 90% confidence level.The optimized NN output thresholds, O tag > 0.7 and of signal events are kept.
The signal yield for B 0 → invisible is extracted from data through fitting variables E ECL and cos θ T , where cos θ T is the cosine of the angle between the two thrust axes in the e + e − CM frame.The thrust axis is defined as the direction that maximizes the sum of the longitudinal momenta of particles, and here one of the axes is constructed using B tag final-state particles, while the other is from the remaining part of the event.The latter is composed of photons and charged tracks that survive the extra-track rejection.In case there is no particle in the remaining part, the beam axis replaces the second thrust axis.In data and the signal MC simulation, this occurs in less than 1% of events.
Beside generic B and continuum backgrounds, background from rare B B decays (i.e., with a b → u, b → d, or b → s transition) and from e + e − → τ + τ − are also considered.From MC simulation, it is found that the rare B B decay background has cos θ T and E ECL distributions similar to those of generic B background, and thus those two background sources are combined.In addition, the continuum and e + e − → τ + τ − background also have similar cos θ T distributions, and their E ECL combined distribution can be described by the off-resonance data.As a result, continuum and e + e − → τ + τ − backgrounds are combined and referred to as non-B background.
An extended unbinned maximum likelihood fit is ap-plied with the form: where i is the events identifier, n j is the number of event belonging to the jth category.P j (E ECL , cos θ T ) is a direct product of the probability density functions (PDFs) P j (E ECL ) and P j (cos θ T ).With the exception that the E ECL distribution for the non-B component is obtained from the off-resonance data, all the other PDFs are obtained from MC simulation.In order to enhance the statistics when constructing PDFs, the O tag threshold is removed after verifying that no correlation exists between O tag and the fitting variables.From the MC simulation, the proportions of the continuum background among the non-B background are (83 ± 5)% and (75 ± 1)% before and after removing the O tag threshold, respectively, consistent within 1.6σ uncertainty.Second-order Legendre polynomials are used to describe cos θ T , while histogram PDFs are used for the E ECL distributions.No correlation is found between the fitting variables in background components.However, a small but nonnegligible correlation between variables exists for signal events.The direct product between PDFs is used nonetheless, and the corresponding systematic uncertainty is determined by generating an ensemble according to two-dimensional histogram PDFs, and then fitting with the product of one-dimensional PDFs.The validity of the E ECL PDFs for background is checked using the sideband samples excluded by the O tag threshold.Comparison between sideband data and the combined distribution of non-B and generic B background according to the MC ratio shows consistency, as shown in Fig. 3.In the comparison, the correctness of the MC ratio between background components is further verified by fitting cos θ T in the sideband sample, which is shown in Fig. 4. In this fit, there are (23 ± 8)% of generic B events among the combined background, which is consistent with the proportion of (25 ± 1)% from MC simulation.
To verify the E ECL PDF obtained from the signal MC simulation, B 0 → D * − l + ν (l = e, µ, D * − → D0 π − , D0 → K + π − ) is used as a control sample.In these events, B tag is fully reconstructed, and the other B meson is identified by decays to D ( * ) lν from the remaining part of the event (double tagging).To mimic the invisible final state, particles used in the signal-side reconstruction are excluded, such as in the E ECL and the shape variables calculations.Event selections are done in the same manner as in the B 0 → invisible study.The extra tracks, π 0 , and K 0 L vetoes are demanded after removing particles involved in the reconstruction of B tag and B sig .The O tag and O shape are also based on the algorithms established before.Additional selections include: 1.855 GeV/c 2 < M D 0 <1.885GeV/c 2 (1.8σ window); 0.143 GeV/c 2 < ∆M D <0.148 GeV/c 2 (2.2σ window), where ∆M D is the difference between the reconstructed D * − and D0 masses; −0.5 GeV 2 /c 4 < M 2 miss < 0.5 GeV 2 /c 4 (1.5σ window), where M 2 miss is defined in Eq. 1 with P γ replaced by P D * − l .After the double tagging, background for the B 0 → D * − l + ν becomes negligible.Comparison of the E ECL distribution between the doubly tagged data and the B 0 → ν ν MC simulation shows excellent agreement as seen in Fig. 3.  Non-B −3.9 +19.5 The systematic uncertainty due to the statistical error of the E ECL and cos θ T PDFs modelling is estimated by varying the content of each bin in the histogram PDFs and parameters of the Legendre polynomials by ±1σ respectively and repeating the fit on data.All of the systematic uncertainties of signal yields are listed in Table II, and the total systematic uncertainty is the sum in quadrature of all terms.The significance of the signal yield is defined as −2ln(L 0 /L s ), where L 0 and L s are the maximized likelihood values when the signal yield is constrained to zero and floated, respectively.The systematic uncertainty is taken into consideration by convolving the likelihood function with a Gaussian function whose width equals to the systematic uncertainty.The signal significance thus obtained for B 0 → invisible is 1.2σ.
Since few events are expected in data for B 0 → invisible + γ and in the binned analysis, an approach that counts events in the E ECL signal region and then subtracts the background is employed to measure any signal.The number of background events in the signal box (N data bkg, box ) is estimated from the E ECL sideband data (N data bkg, s.b. ) by multiplying by a parameter N MC bkg, box /N MC bkg, s.b. : where the N MC bkg, box and N MC bkg, s.b. denote the number of background events in the E ECL signal box and sidebands from MC simulation.
Uncertainties of N data bkg, box come from the statistical error of the first term and the systematic error of the second term in the right-hand side of Eq. 4. The latter is estimated by a control sample B 0 → D − l + ν (l = e, µ, D − → K + π − π − ).Similar to the case of B 0 → D * − l + ν , the double tagging, M D − requirements, extra particles vetoes, O tag and O shape thresholds are applied.In the control sample, background numbers in the E ECL signal box and sideband are obtained through fitting the M 2 miss distribution to data, which is shown in Fig. 6.The ratio of the background yields in the two regions is compared with the ratio in the control sample MC simulation.The difference and the statistical uncertainty of fitting, which is between 16-20%, are added in quadrature and taken as the systematic uncertainty.For B 0 → invisible + γ, the uncertainty is 33% and for the binned cases, the uncertainties are between 23-30%.The counting results in the E ECL signal box are shown for B 0 → invisible + γ and the binned analysis in Table III.Figure 7 shows the M 2 miss and E ECL distributions of data and the expected background for B 0 → invisible + γ.The observed numbers of events are all consistent within uncertainties with the expected backgrounds.Taking the data-MC difference in selection rates into account, the signal efficiencies are calibrated through the formula: where ǫ data sig and ǫ MC sig are the signal efficiencies from data and MC, respectively, C FR , C tr , C π 0 , C K 0 L and C NN are calibration factors due to the full reconstruction process, the extra tracks, π 0 , K 0 L vetoes and the NN output thresholds, respectively.The C FR factor has been studied [18] using charmed semileptonic signalside B decays, and its value depends on the P FR of the B tag and the tag-side reconstructed channel.For B 0 → invisible, B 0 → invisible + γ, and the binned analysis, the C FR factor lies between 0.64 to 0.70.On the other hand, C tr , C π 0 , C K 0 L , and C NN are estimated through control samples, in which the signal efficiencies before and after each selection on data and MC simulation are compared.The control samples include six modes, with the signal side decaying respectively through: B , where l = e or µ.The events are doubly tagged with the selections on M D 0 , M 2 miss , and ∆M D the same as mentioned before.In addition, we require −0.5 GeV 2 /c 4 < M 2 miss < 0 (0.4σ window) for B 0 → D − l + ν, E ECL < 0.4 GeV for all the control sample modes and the difference between the reconstructed D * 0 and D 0 masses to lie within 0.138 GeV/c 2 to 0.146 GeV/c 2 (2.4σ window) for B − → D * 0 l − ν.The averaged calibration factors obtained from the six modes are used to calibrate the B 0 → invisible (+γ) signal efficiencies.Results for C tr , C π 0 , and C K 0 L are 0.98, 0.96, and 1.06, respectively.For the C NN , values vary between 0.90 to 0.95 according to the different O tag and O shape thresholds for B 0 → invisible, B 0 → invisible + γ, and the binned analysis.
Since the signal yield is not significant for both B 0 → invisible and B 0 → invisible + γ (whole range or the five M 2 miss bins), upper limits at 90% confidence level on the branching fraction (B UL ) are calculated.For B 0 → invisible, the upper limit is obtained by solving the equation: where B is the assumed branching fraction, and L (B) is the corresponding maximized likelihood from the fit on data.The 1.4% uncertainty on the number of produced B-meson pairs, systematic uncertainties of signal yield and efficiency are taken into consideration by convolving the likelihood function with a Gaussian function whose width equals the total systematic uncertainty.The result is: B (B 0 → invisible) < 7.8 × 10 −5 at 90% C.L.
For B 0 → invisible + γ, a frequentist style limit evaluated in the TRolke package [20] is used to obtain upper limits on the branching fraction.The method is based on the profile likelihood with the uncertainties on background and signal efficiency taken into account.The upper limits of the branching fraction are shown in Table IV.In summary, we have searched for the decays B 0 → invisible and B 0 → invisible + γ and find no evidence for them.For the latter decay, the energy of the photon is required to be greater than 0.5 GeV.We set upper limits on the branching fractions B (B 0 → invisible) < 7.8 × 10 −5 and B (B 0 → invisible + γ) < 1.6 × 10 −5 at 90% confidence level.We improve upon the previous Belle limit [7] on B 0 → invisible, and the limit obtained for B 0 → invisible + γ is the most stringent.

6 FIG. 3 .FIG. 4 .
FIG. 3. Left: Comparison of EECL distributions between background MC simulation and data in the Otag sideband.The black points are data.The blue crosses with a shaded error band are the background MC simulation.Right: Comparison of EECL distributions between B 0 → ν ν signal MC simulation and B 0 → D * − l + ν data.The black points are data.The red and shaded distribution is signal MC simulation.Histograms are normalized such that the sum of all bins equals one.

FIG. 6 .
FIG. 6. Fitting result of the control sample B 0 → D − l + ν in the EECL signal box (left) and sideband (right).Selections are based on B 0 → invisible + γ.Dots with error bars are data, black-solid line is the combined fit result, red-dashed line is the signal component and blue dashed-dotted line is background.

10 FIG. 7
FIG. 7. M 2 miss (left) and EECL (right) distributions of data and the expected background for B 0 → invisible + γ.The M 2 miss distribution is plotted in the EECL signal box and the EECL distribution is plotted in the whole M 2 miss region.The black points with error are data.The gray crosses with a shaded error band are the expected background.

Table I .
No significant signal is observed.Projections of the fit result on cos θT (left) and EECL (right) for B 0 → invisible.Dots with error bars are data, black-solid line is the fit result, red-dotted line is the signal component, green short-dashed line is the generic B background component and blue dashed-dotted line is the non-B background component.

TABLE II .
Summary of systematic uncertainties on fitting yield.

TABLE III .
Estimated number of background events in the signal box and the number of events in the signal box (N data box ) for B 0 → invisible + γ and M 2 miss bins.