Search for the dark photon in B0 → A′A′, A′ → e+e−, μ+μ−, and π+π− decays at Belle

We present a search for the dark photon A′ in the B0 → A′A′ decays, where A′ subsequently decays to e+e−, μ+μ−, and π+π−. The search is performed by analyzing 772 × 106BB¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ B\overline{B} $$\end{document} events collected by the Belle detector at the KEKB e+e− energy-asymmetric collider at the ϒ(4S) resonance. No signal is found in the dark photon mass range 0.01 GeV/c2 ≤ mA′ ≤ 2.62 GeV/c2, and we set upper limits of the branching fraction of B0 → A′A′ at the 90% confidence level. The products of branching fractions, ℬB0→A′A′×ℬA′→e+e−2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathrm{\mathcal{B}}\left({B}^0\to A^{\prime }A^{\prime}\right)\times \mathrm{\mathcal{B}}{\left(A\prime \to {e}^{+}{e}^{-}\right)}^2 $$\end{document} and ℬB0→A′A′×ℬA′→μ+μ−2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathrm{\mathcal{B}}\left({B}^0\to A^{\prime }A^{\prime}\right)\times \mathrm{\mathcal{B}}{\left(A\prime \to {\mu}^{+}{\mu}^{-}\right)}^2 $$\end{document}, have limits of the order of 10−8 depending on the A′ mass. Furthermore, considering A′ decay rate to each pair of charged particles, the upper limits of ℬB0→A′A′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathrm{\mathcal{B}}\left({B}^0\to A^{\prime }A^{\prime}\right) $$\end{document} are of the order of 10−8–10−5. From the upper limits of ℬB0→A′A′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathrm{\mathcal{B}}\left({B}^0\to A^{\prime }A^{\prime}\right) $$\end{document}, we obtain the Higgs portal coupling for each assumed dark photon and dark Higgs mass. The Higgs portal couplings are of the order of 10−2–10−1 at mh′≃mB0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {m}_{h\prime}\simeq {m}_{B^0} $$\end{document} ± 40 MeV/c2 and 10−1–1 at mh′≃mB0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {m}_{h\prime}\simeq {m}_{B^0} $$\end{document} ± 3 GeV/c2.

E-mail: yjkwon63@yonsei.ac.kr Abstract: We present a search for the dark photon A in the B 0 → A A decays, where A subsequently decays to e + e − , µ + µ − , and π + π − . The search is performed by analyzing 772 × 10 6 BB events collected by the Belle detector at the KEKB e + e − energyasymmetric collider at the Υ(4S) resonance. No signal is found in the dark photon mass range 0.01 GeV/c 2 ≤ m A ≤ 2.62 GeV/c 2 , and we set upper limits of the branching fraction of B 0 → A A at the 90% confidence level. The products of branching fractions,

Introduction
The validity of the Standard Model (SM) has been confirmed by various experimental measurements [1], but it is also known that the SM is incomplete and cannot explain several phenomena occurring in nature, e.g. neutrino oscillations [2,3] and the baryon asymmetry [4]. A possible way to explain the above problems while keeping the internal structure of the SM unaffected is to introduce a dark sector [5] that interacts with the SM particles only very weakly. For example, a vector mediator of hypothetical U (1) gauge interaction in the dark sector, the so-called dark photon, may interact with matter through various portals with a small coupling strength [6][7][8]. Such a model of the dark sector with portal interaction to the SM could explain the muon g−2 anomaly [9][10][11][12], baryogenesis [13], and high energy positron fraction anomaly in cosmic rays [14][15][16][17][18].
In this paper, we report a search for the dark photon A , in the decays of B 0 mesons by analyzing the e + e − collision data from the Belle experiment. In particular, we study B 0 decays into a pair of dark photons, B 0 → A A , which are mediated by an off-shell dark Higgs h [5] (figure 1), wherein we scan the A mass range between 0.01 GeV/c 2 and 2.62 GeV/c 2 in 10 MeV/c 2 (m A < 1.1 GeV/c 2 ) and 20 MeV/c 2 (m A > 1.1 GeV/c 2 ) intervals. Throughout the paper, the charge-conjugate modes are always implied. In this paper, we restrict ourselves to the hypothesis that all dark-sector particles coupling to A are heavier than A , therefore the latter can only decay to SM particles. Moreover, we assume that the A decays promptly. In the kinematic range of this analysis, the allowed A decay are to e + e − , µ + µ − , or hadronic final states. Lepton-flavor-violating decays [19,20] A → e ± µ ∓ are not considered in this analysis.

Branching fraction of dark photon decay
In order to obtain B(B 0 → A A ) from the analysis of the decays into the final states considered, we need to know the branching fractions of A to a particular final state. Below the τ + τ − threshold, the branching fraction of the dark photon that is consistent with our hypothesis is obtained as where = e or µ. Following ref. [21], we write down the partial widths to + − and hadrons as with the square of the total center-of-mass (CM) frame energy s, the kinetic mixing parameter ε mix , and R(s) = e + e − →hadrons / e + e − →µ + µ − which is determined by various experiments [1]. The branching fraction of A → π + π − is then obtained as [22]:

JHEP04(2021)191 2 The Belle detector
Our analysis is based on the full 711 fb −1 integrated luminosity of the Υ(4S) data set from the Belle detector [26,27] at KEKB e + e − energy-asymmetric collider [28,29]. The Belle detector consists of seven subdetectors with 1.5 T magnetic field along the beam axis. Inside the coil, there are the silicon vertex detector, the central drift chamber (CDC), the aerogel threshold Cherenkov counters (ACC), the time-of-flight scintillation counters (TOF), and the electromagnetic calorimeter (ECL). In the return yoke outside the coil, a K 0 L meson and muon detector (KLM) is instrumented. We perform a blind search in this analysis, for which we generate Monte Carlo (MC) simulation samples using EvtGen [30] for event generation and GEANT3 [31] for detector simulation. Signal efficiencies are determined from the signal MC set, where one million events are generated for each signal mode and dark photon mass. The event shape and amount of the background events are studied by using generic MC samples simulating e + e − → Υ(4S) → BB and e + e − → qq (q = u, d, s, c) ('continuum') processes. The size of MC samples for Υ(4S) and continuum simulation corresponds to 10 and 6 times that of real data, respectively.

Signal event selection
To select signal events, we retain tracks satisfying the following track reconstruction quality requirements. Because we assume prompt dark photon decays, all tracks are required to originate from near the interaction point (IP). In particular, each track should satisfy the following conditions on the impact parameters in the transverse and longitudinal directions, dr < 0.2 cm and |dz| < 3.0 cm, respectively. The impact parameters are calculated using the beam IP and track helix, and the z-axis is aligned opposite the direction of positron beam. We also require a good track fit based upon χ 2 per degree of freedom (N d.o.f. ) by accepting only the tracks with The species of the charged particles are identified by considering the likelihood ratios. Muons are identified by requiring L µ /(L µ + L K + L π ) > 0.9, where the likelihood L j (j = µ, K, π) [32] is constructed by the hit position and penetration in the KLM. Electrons are required to meet L e /(L e + L not-e ) > 0.9 where the likelihood L j (j = e, not-e) [33] is determined by dE/dx from the CDC, ratio of the ECL cluster energy to the matched track momentum, shower shape of the ECL cluster, and the ACC photoelectron response. Charged pions and kaons are identified by the likelihood [34] using the dE/dx from the CDC, the ACC photoelectron response, and the time-of-flight information from the TOF. The tracks with L π /(L K + L π ) > 0.4 are identified as pions.
To recover energy losses by e ± candidates due to bremsstrahlung, radiative photons are added to the electron momentum if they fall within a 0.05 radian cone around the e ± direction. We require these photons to exceed an energy threshold that depends on the ECL region: E γ > 50 (barrel), 100 (forward endcap), and 150 (backward endcap) MeV.
The dark photon candidate is reconstructed in the following modes: A → e + e − , µ + µ − , and π + π − . For B 0 → e + e − e + e − and µ + µ − µ + µ − modes, we have an ambiguity between JHEP04(2021)191 , where the lepton pair from a single A decay is indicated in parentheses. To find a single dark photon combination per event, we choose that corresponding to the smallest invariant mass difference of dark photon candidates, ∆M A .
Finally, B 0 candidates are reconstructed from two dark photon candidates. To extract signal events from data, we use the following five variables, defined in the CM frame: M bc , is the difference between the B 0 -candidate energy and the beam energy (= √ s/2), and E miss is the missing energy, E miss ≡ √ s − j E j where the index j is for all charged and neutral particles in the event. The missing energy is useful to reduce combinatorial background due to multiple semileptonic decays from b → c −ν and c → (s, d) + ν for both B and B. For the two dark photon candidates in an event, we calculate

2) and m bin
A is the nominal A mass for a particular bin of m A . For the signal event selection, we require M bc > 5.27 GeV/c 2 and E miss < 3.5 GeV for all modes. Considering the energy loss from e ± , ∆E requirements are chosen separately for different modes: We use ∆M A and δM A to set the search window for each m bin A and the final-state mode. The requirements on these variables depend on both m bin A and the number of electrons in the final state. For m bin The above conditions are determined so that if we consider the distribution of ∆M A the upper edge of the accepted region has a value of nearly 3-5% of the peak value. In addition, we make use of the approximately linear increase of the ∆M A width as a function of m bin A . We choose the same selection for δM A since the distribution is almost the same as ∆M A . For m bin A ≤ 0.1 GeV/c 2 , we apply slightly different selection conditions for ∆M A and δM A , while requirements on M bc and ∆E remain the same as for m bin A > 0.1 GeV/c 2 . We do not use E miss for m bin A ≤ 0.1 GeV/c 2 , because for such low-mass dark photons, little background is expected from generic B decays. For m bin A ≤ 0.1 GeV/c 2 , the resolutions of both ∆M A and δM A are nearly independent of m bin A . Therefore, we require ∆M A < 0.02 GeV/c 2 and δM A < 0.02 GeV/c 2 for all m A ≤ 0.1 GeV/c 2 . From the MC study, our ∆M A selections in A → µ + µ − and π + π − cover roughly ±2.5 times the mass resolution. In case of A → e + e − , the mass resolution is worse, and our selections correspond to ±(1.7 − 2.5) times the mass resolution, depending on m A . For instance, the M A resolution of the 1.5 GeV dark photon is about 5 MeV for A → µ + µ − or π + π − , while for A → e + e − it is about 20 MeV. The union of the search windows determined using ∆M A and δM A for all m bin A covers the entire dark photon mass range of our study without any gap. For the charged pion pairs, there is strong background from light mesons, such as K 0 S , ρ 0 , and f 0 (980). Because of possible K-π misidentification, K * 0 , φ and so on are also a source of possible background. Since production of such mesons is copious, especially that of ρ 0 mesons, we reject the 0.45 GeV/c 2 < M π + π − < 1.1 GeV/c 2 . Another source of pion pairs is D 0 meson. Two decay channels, D 0 → π + π − and D 0 → π + K − are considered. A direct D 0 veto is applied by removing π + π − combinations which satisfy 1.85 GeV/c 2 < M π + π − < 1.88 GeV/c 2 . The other decay channel, D 0 → π + K − , can mimic the signal via K-π misidentification. We reject these events by discarding the 1.85 GeV/c 2 < M π + K − < 1.88 GeV/c 2 mass range.
After signal selection, most of the combinatorial background is in the B 0 → + − π + π − mode, coming from the continuum processes e + e − → qq (q = u, d, s or c). In the four-lepton mode, on the other hand, there is almost no background left. The continuum background is suppressed via multivariate analysis (MVA) using the Fisher discriminant [35] method in the TMVA [36] package. We make use of 16 event shape variables: the cosine of angle between the beam axis and B 0 momentum (cos θ B ), the cosine of angle between the thrust axis of the B 0 daughters and that of the rest of the event (cos θ T ), and the Fisher discriminant components of modified Fox-Wolfram moments [37]. The MVA training is performed for the + − π + π − final state for each m bin A , using the signal and continuum MC. We apply MVA selection creteria to retain from 75% to 90% of signal and from 10% to 30% of continuum background, depending on m A and final state.

Systematic uncertainties
We determine the branching fraction of B 0 → A A as where B 0 is the branching fraction of Υ(4S) → B 0 B 0 , of which the current world-average value is 0.486 ± 0.006 [1], N obs is the yield, N bkg is the number of expected background events determined from MC, is the signal reconstruction efficiency considering branching fraction of A subdecays, and N BB = (772 ± 11) × 10 6 is the number of BB pairs which are collected by the Belle detector. The most important source of systematic uncertainties is the signal reconstruction efficiency, which is obtained by MC. The sources of uncertainty include the statistical error in the signal MC, track reconstruction efficiency, particle identification (PID) efficiency, and uncertainties in the MVA method used to suppress continuum background for + − π + π − final states. The uncertainties for N BB and B 0 also contribute to systematics. Relative error  The uncertainties due to background estimation are very small compared to other systematic uncertainties.
Track reconstruction efficiency is studied using the decay chain D * + → D 0 π + , D 0 → K 0 S π + π − , and K 0 S → π + π − where we tag all the charged tracks in the chain but one from K 0 S decays ('test track') then try to find the test track. We compare the tracking efficiency difference of the test track for both data and MC. The error is 1.4%, independent of the dark photon mass and final state.
The PID correction is applied to each daughter electron, muon, and pion. The lepton (pion) identification correction is studied using the difference between MC and data for the
The MVA correction factor and uncertainty are studied using the control mode, B 0 → J/ψK * 0 → e(µ) + e(µ) − π − K + . We apply MVA training results for the continuum suppression of + − π + π − modes for each assumed value of m A to B 0 → J/ψK * 0 MC and data. We then calculate the double ratio (N data,A /N data,B )/(N MC,A /N MC,B ), where N data(MC),B and N data(MC),A are the number of signal candidates in data(MC) before and after MVA training application, respectively. The systematic uncertainty due to MVA training is taken from the uncertainties in the double ratio, and these uncertainties are approximately 2% at all values of m A .
After multiplying all correction factors, signal efficiencies are mostly 5 − 20%. The efficiencies increase as the A mass approaches 0 or m B 0 /2, in which case both e ± (µ ± ) from the A decays are more likely to exceed the energy threshold for ECL (KLM) detection. The summary of signal-efficiency-related systematic uncertainties is shown in figure 2, and the total systematic uncertainties are 7.5-10% for e + e − e + e − and µ + µ − µ + µ − final states and 5-7.5% for e + e − µ + µ − , e + e − π + π − , and µ + µ − π + π − final states. Figure 3 shows the number of B 0 → A A candidate events. There are no events observed in any bin in the e + e − µ + µ − and µ + µ − µ + µ − mode, while we find N obs ≤ 2 events for e + e − e + e − , e + e − π + π − , and µ + µ − π + π − modes. The yields are consistent with the expected number of background events and we set the upper limits at 90% C.L.

Results
For the limits of B(B 0 → A A ), we combine the number of expected background events, signal candidates in data, and signal reconstruction efficiencies of the five final states. The combined numbers of expected background events and signal candidates in data are calculated by simply adding the results for the individual final states. For the signal efficiencies, we first obtain the ratio where f is each final state, using eq. (1.1). In case of e + e − µ + µ − , for example, The graph of F f is presented in figure 4. With this ratio F f , the combined efficiency is determined as f f F f where f is the signal efficiency of the final state f . The upper limits are calculated using the POLE program [38], which is based on the Feldman-Cousins unified approach [39]. We report the limits on the products of branching fractions distributions for each final state and dark photon mass. e + e − π + π − and µ + µ − π + π − distributions are almost the same for the whole region. e + e − e + e − and µ + µ − µ + µ − distributions are the same and e + e − µ + µ − distribution is twice that of four-electron or four-muon final states in the region m A > 0.5 GeV/c 2 .  eq. (1.1), the upper limits near the masses of ρ 0 and φ mesons are less restrictive than others. Table 1 lists the signal efficiency, the expected number of backgrounds and number of observed events (N obs ) for some of m A .

Conclusions
In summary, we have searched for B 0 → A A decays for the first time using the full data set of 772 × 10 6 BB events of Belle. We restrict our study to the case where A decays promptly to e + e − , µ + µ − , or hadronic final states, and consider five final states of B 0 which are e + e − e + e − , e + e − µ + µ − , µ + µ − µ + µ − , e + e − π + π − , and µ + µ − π + π − . From the branching fraction of A , the five B 0 final states are merged to determine the branching fraction of B 0 → A A . We find no significant signal in any assumed A mass and decay mode, so With minor modifications our analysis can be used to set limits on the other new physics models which include prompt B 0 → XX and X → + − /π + π − decays. We expect to have much more stringent results from the Belle II experiment [40,41], with nearly two orders of magnitude increase in statistics, in the future.