Measurement of $\mathcal{B}(J/\psi \to \eta' e^+ e^- $) and search for a dark photon

Using a data sample of $(1310.6\pm7.0)\times10^{6}$ $J/\psi$ decay events collected with the BESIII detector at BEPCII, we study the electromagnetic Dalitz decay $J/\psi \to \eta' e^+e^-$ with two dominant $\eta'$ decay modes, $\eta' \to \gamma \pi^+ \pi^-$ and $\eta' \to \pi^+\pi^-\eta$. The branching fraction is determined to be $\mathcal{B}(J/\psi \to \eta' e^+e^-) = (6.59\pm0.07\pm0.17) \times 10^{-5}$, which improves in precision by a factor of 2 over the previous BESIII measurement. A search for the dark photon ($\gamma '$) is performed via $J/\psi \to\eta' \gamma ', \gamma' \to e^{+}e^{-}$. Excluding the $\omega$ and $\phi$ mass regions, no significant signal is observed in the mass range from 0.1 to 2.1 GeV/$c^{2}$. We set upper limits at the 90\% confidence level on $\mathcal{B}(J/\psi \to \eta' \gamma ')\times\mathcal{B}(\gamma ' \to e^+e^-)$, $\mathcal{B}(J/\psi \to\eta' \gamma'$) and the mixing strength as a function of dark photon mass. This is among the first searches for dark photons in charmonium decays.

Except for gravitational effects, we still know very little about the constituents and interactions of dark matter. Many models beyond the Standard Model (SM) of particle physics have proposed the existence of a dark sector, which is being searched for by global efforts with different types of experiments [10]. The simple realizations of these models usually consist of an extra U(1) gauge group, with a corresponding massive vector boson force carrier, called a dark photon (γ ′ ), which is neutral under the SM gauge symmetries, but couples to the SM photon via kinetic mixing [11] and decays into SM particles. Such models provide a natural scenario for dark matter interactions. A dark photon with a mass in the MeV/c 2 to GeV/c 2 range can also be accommodated by observational astroparticle anomalies [12]. Low-energy electron-positron colliders offer an ideal environment to probe these low-mass dark sector models [13,14], and meson decays provide an important constraint on the mixing strength ε between the dark photon and SM photon [15,16]. The authors of Ref. [9] have estimated the achievable limits on the mixing strength in the processes J/ψ → P γ ′ (γ ′ → l + l − ) using the huge BESIII J/ψ data sample.
In this paper, we report on the updated branching fraction measurement of J/ψ → η ′ e + e − and a search for a dark photon through J/ψ → η ′ γ ′ with a sample of (1310.6±7.0)× 10 6 J/ψ events [17] collected by BESIII. It is the first time that the dark photon is searched for through the charmonium decay J/ψ → η ′ γ ′ , γ ′ → e + e − .

II. THE BESIII EXPERIMENT AND MONTE CARLO SIMULATION
BEPCII is a double ring e + e − collider running at the center-of-mass (c.m.) energy √ s from 2.0 to 4.6 GeV with a luminosity of 1×10 33 cm −2 s −1 at the ψ(3770) resonance. The BESIII detector [18], with a geometrical acceptance of 93% of the 4π stereo angle, operates in a magnetic field of 1.0 T (0.9 T in 2012) provided by a superconducting solenoid. It is composed of a helium-based main drift chamber (MDC) to measure the momentum and ionization energy loss (dE/dx) of charged particles, a plastic scintillator time-offlight (TOF) system for particle identification (PID) information, a CsI(Tl) electromagnetic calorimeter (EMC) to measure photon and electron energies and a multilayer resistive plate chamber muon detection system to identify muons.
Monte Carlo (MC) simulations are used to optimize the event selection, investigate background and determine the detection efficiency. The GEANT4-based [19] simulation includes the description of the geometry and material of the BESIII detector, the detector response, and digitization models and also tracks the detector running conditions and performance. An inclusive MC sample containing 1.225 × 10 9 J/ψ events is used to study potential backgrounds. The production of the J/ψ resonance is simulated by the MC event generator KKMC [20]. The known decay modes of the J/ψ are generated by EVTGEN [21] with branching fractions set at the world average values from the particle data group (PDG) [22], while the remaining unknown decays are generated by LUNDCHARM [23]. The analysis is performed in the framework of the BESIII offline software system which takes care of the detector calibration and event reconstruction.
The decay J/ψ → η ′ e + e − is simulated according to the Lorentz-invariant amplitude, taking into account of the J/ψ polarization state in the e + e − annihilation system, where a monopole form factor with pole mass Λ =3.686 GeV/c 2 is assumed. Then the subsequent decay mode of η ′ → γπ + π − is simulated with the ρ-ω interference and box anomaly effects [24,25]. The decays of η ′ → π + π − η and η → γγ are generated with a phase space model. The decays of J/ψ → η ′ V (V represents ρ, ω, φ) and J/ψ → η ′ γ ′ are generated by a helicity amplitude model, and the decay of γ ′ → e + e − is modeled with a vector meson decaying to a lepton pair [21].
Charged tracks in the BESIII detector are reconstructed from ionization deposits in the MDC. We select good charged tracks passing within ±10 cm from the interaction point (IP) in the beam direction and within 1 cm in the plane perpendicular to the beam. The polar angle of the track is required to satisfy | cos θ| < 0.93. Four candidate charged tracks are required, and their net charge must be equal to zero. The combined information of the energy loss dE/dx from the MDC and the time of flight from the TOF is used to calculate the PID confidence levels (C.L.) for the e, π and K hypotheses. Both the electron and positron require the electron hypothesis to have the highest PID C.L., and the other two charged tracks are treated as π + π − candidates without any PID requirement. The four charged tracks π + π − e + e − must satisfy a common vertex constrained fit to ensure they originate from the interaction point.
Electromagnetic showers are reconstructed from clusters of energy deposits in the EMC. The shower energy of photon candidates in the EMC must be greater than 25 MeV in the barrel region (| cos θ| < 0.80) or 50 MeV in the end-cap region (0.86 < | cos θ| < 0.92). Showers located between the barrel and end-cap regions are excluded due to worse reconstruction. Showers are required to be separated from the extrapolated positions of any charged track by more than 10 • . To suppress electronic noise and energy deposits unrelated to the event, the time in the EMC of showers must be less than 700 ns after the event start time. We require at least one (two) candidate photon(s) for mode-I (II).
A kinematic fit with an energy-momentum constraint (4C) is performed to improve the resolution and suppress background. For candidate events with extra photon candidates, the combination with the smallest χ 2 4C is selected, and the χ 2 4C is required to be less than 100. The χ 2 4C requirement removes more than 12% and 30% backgrounds for mode-I and II, respectively, and results in a signal efficiency loss around 7% for both modes.
In the branching fraction measurement, the distance from the reconstructed vertex of the e + e − pair to the IP in the xy projection, δ xy , is used to distinguish γ conversion events from J/ψ → γη ′ , where γ converts into an e + e − pair when it interacts with material in the detector. Here δ xy = R 2 x + R 2 y , and R x and R y are the coordinates of the reconstructed vertex in the x and y directions [26]. The scatter plot of R y versus R x of the J/ψ → γη ′ (η ′ → γπ + π − ) MC sample, and the δ xy distributions of data and various MC samples are shown in Fig. 1. The peaks around δ xy = 3 and 6.5 cm in the δ xy distribution match the positions of the beam pipe and inner wall of the MDC. The γ conversion background is suppressed with the requirement δ xy < 2 cm, and the number of remaining events, estimated from MC simulation according to the corresponding branching fractions from the PDG [22], is 202.2 ± 7.3 (70.6 ± 2.5) in mode-I (II).
The black crosses are data, the blue solid line is the total contribution from the MC and η ′ sideband, the green dot-dashed line is signal MC, the dashed line is J/ψ → γη ′ (η ′ → γπ + π − ) MC and the shaded area is η ′ sideband.
Other common peaking background originates from J/ψ → V η ′ decays, where V is a vector meson ρ, ω, or φ decaying to e + e − or π + π − . These backgrounds are studied with high statistics MC samples, and the numbers of background events (N bkg ), as summarized in Table I, are normalized according to the branching fractions from the PDG [22].  [22].
, there are mainly two kinds of non-peaking backgrounds. One is J/ψ → π + π − η/π 0 with η/π 0 → γe + e − , which has the same final state γπ + π − e + e − as the signal process. The contribution from the η case is negligible from a MC study. To reject the background with a π 0 intermediate state, candidates with the γe + e − invariant mass M (γe + e − ) in the π 0 mass window [0.10, 0.16] GeV/c 2 are removed. The other backgrounds are from J/ψ decays with multiple pions in the final state, where a pion pair is misidentified as an electron-positron pair. Such backgrounds produce a smooth shape on the γπ + π − invariant mass M (γπ + π − ) distribution around the η ′ mass. For mode-II J/ψ → η ′ e + e − (η ′ → ηπ + π − , η → γγ), the background is low. To reconstruct η mesons, the γγ invariant mass M (γγ) of candidates is required to be within [0.48, 0.60] GeV/c 2 . The potential peaking background from the process J/ψ → η ′ π + π − is estimated with a MC sample generated according to the amplitude as reported in Ref. [27]. Another potential peaking background from the two-photon process [28] e + e − → e + e − η ′ is studied with an independent data sample taken at c.m. energy of 3.773 GeV and corresponding to an integrated luminosity of 2.93 fb −1 [29,30]. The contributions are found to be negligible for both processes.
Unbinned maximum likelihood (ML) fits are performed on the M (γπ + π − ) and M (γγπ + π − ) distributions to determine the signal yields. In the fits, the signal probability density function (PDF) is described by an signal MC simulated shape convolved with a Gaussian function, which takes into account the resolution difference between data and MC simulation. The major peaking backgrounds from γ conversion J/ψ → γη ′ and J/ψ → φη ′ (φ → e + e − ) are described with MC shapes, and the magnitudes are fixed to the expected values. The other three minor peaking background sources, listed in Table I, are directly subtracted from the fitted η ′ events. The non-peaking backgrounds in mode-I and II are described with second and first order Chebychev polynomial functions, respectively. The fit results are shown in Fig. 2, which give the goodness of fit χ 2 /ndf = 74.8/35 and 34.3/17 for Fig. 2(a) and (b), respectively. The branching fraction of J/ψ → η ′ e + e − is determined by where N sig is the number of signal events, which are 6436.9± 87.1 and 2494.4 ± 51.3 events for mode-I and II, respectively, N J/ψ is the number of J/ψ events, B η ′ →F is the intermediate branching fraction of the η ′ decay, (28.9 ± 0.5)% for B(η ′ → γπ + π − ) in mode-I and (16.9 ± 0.3)% for B(η ′ → ηπ + π − ) × B(η → γγ)) in mode-II [22], and E is the detection efficiency (25.01 ± 0.06)% for mode-I and (16.82 ± 0.06)% for mode-II. Using Eq. (1) and taking into account the systematic uncertainties, which are discussed in Sec. IV, B(J/ψ → η ′ e + e − ) for mode-I and mode-II are calculated to be (6.80 ± 0.09 ± 0.35) × 10 −5 and (6.74 ± 0.14 ± 0.42)× 10 −5 , respectively, where the first uncertainties are statistical and the second are systematic. The results from the two η ′ decay modes are consistent with each other within the statistical and uncorrelated systematic uncertainties. With the weighted least squares method taking into account the correlated and uncorrelated uncertainties [31], the weighted average branching fraction from the two decay modes is B(J/ψ → η ′ e + e − ) = (6.79 ± 0.08 ± 0.35) × 10 −5 . The combined result is consistent with the previous BESIII measurement [8], and the precision is improved.  2. (color online). Spectra of (a) M (γπ + π − ) and (b) M (γγπ + π − ). Black crosses are data, the blue solid line is the total fitting projection, the red dashed line is signal, the green dotdashed line is non-peaking background, the pink cross-hatched area is γ conversion J/ψ → γη ′ and the yellow solid area is J/ψ → η ′ φ, φ → e + e − .

B. Search for the dark photon through
In the search for the dark photon, candidate events with the η ′ e + e − final state are selected with the same selection criteria as described in Sec. III A but without the γ conversion veto criteria. Since γ conversion events distribute mainly in the low M (e + e − ) region below 70 MeV/c 2 , the candidate events with M (e + e − ) > 70 MeV/c 2 are retained. The mass window [0.93, 0.98] GeV/c 2 is used for M (γπ + π − ) and M (γγπ + π − ) to select η ′ mesons. The remaining events are mainly from the EM Dalitz decay J/ψ → η ′ e + e − and a few from J/ψ → V η ′ with V → e + e − . Electron-positron pairs from ω and φ meson decays will contaminate the possible dark photon signal, and a dedicated phenomenological study shows that the dark photon width grows significantly with mass around these vector mesons, which results in a significant drop of the sensitivity [13]. Therefore, the mass ranges [0.74, 0.84] GeV/c 2 and [1.00, 1.04] GeV/c 2 , corresponding to the regions of the ω and φ mesons, respectively, are excluded in the dark photon search.
A series of signal MC samples, which are used to determine the signal PDF and detection efficiency, are generated according to the decay chain J/ψ → η ′ γ ′ , γ ′ → e + e − , with different m γ ′ values, ranging from 0.1 to 2.0 GeV/c 2 with a step of 0.1 GeV/c 2 . The dark photon width, suppressed by a factor of ε 2 and expected to be far below the experimental resolution, is set to be zero in the MC generation. The dark photon signal PDF is parameterized by the sum of two Crystal Ball (CB) functions with a common mean value, where the parameters are determined by fitting the signal MC samples. The resolution, which is evaluated by weighting the width of two CBs according to their ratio, grows as a function of m γ ′ from 2 MeV/c 2 to 8 MeV/c 2 as m γ ′ increases. The detection efficiency shown in Fig. 3 ranges from 35% to 41% and from 22% to 27% depending on m γ ′ in mode-I and II, respectively.
The continuum shape of the M (e + e − ) distribution in the range from 70 MeV/c 2 to 2.13 GeV/c 2 is described by a PDF of the sum of a second-order polynomial function and an The corresponding global significance is less than 1σ evaluated by using a large number of pseudoexperiments [32]. Thus, in conclusion, no significant dark photon signal is observed within the searched range.

IV. SYSTEMATIC UNCERTAINTIES
Most of the systematic uncertainties from the event selection are same for the branching fraction measurement and the dark photon search. The sources of uncertainties for the branching fraction measurement are divided into correlated and uncorrelated terms between the two different η ′ reconstruction modes. The uncertainties for MDC tracking, photon detection, PID, number of J/ψ events, and the γ conversion veto are considered as correlated sources. Those for an additional photon in η ′ → π + π − η(γγ), the 4C kinematic fit, η reconstruction, the form factor, signal shape, fit range, background shape and magnitude and η ′ branching fractions are considered as the uncorrelated sources. The systematic uncertainties are discussed below and summarized in Table II. Some of the uncertainties are estimated in a similar way as described in Ref. [8]. The uncertainties are 1.2% associated with the MDC tracking and 0.6% associated with PID per electron by comparing the efficiency difference between data and MC simulation for a control sample of radiative Bhabha events e + e − → γe + e − . The uncertainty associated with the tracking efficiency of the charged pion is 1.0% obtained from a study of the control samples of ψ(3686) → π + π − J/ψ, J/ψ → l + l − and J/ψ → π + π − π 0 [33].
The systematic uncertainty due to the photon detection efficiency is studied with a control sample of J/ψ → π + π − π 0 , π 0 → γγ. The uncertainty is determined to be 0.5% or 1.5% for a photon in the EMC barrel or end-cap region, respectively. The average uncertainty, 0.6% per photon, is calculated according to the ratio of the number of photons in the two parts of the EMC, obtained from signal MC simulations.
The uncertainty associated with the 4C kinematic fit is estimated with a high purity control sample of J/ψ → π + π − π 0 , π 0 → γe + e − . The efficiency difference between data and MC simulation is 0.5%, which is taken as the systematic uncertainty. This control sample is also used to estimate the uncertainty due to the γ conversion veto criterion δ xy < 2 cm. The difference in efficiency between data and MC simulation is 1% and is taken as the uncertainty.
For the η reconstruction via its γγ decay, the systematic uncertainty is determined to be 1.0% from a study of the control sample J/ψ → ppη [34].
The uncertainty of the monopole form factor used in the MC generation is estimated with the alternative signal MC samples generated with the parameters Λ=3.0 or 4.0 GeV/c 2 , and the largest efficiency difference 0.4% (0.2%)with respective to the nominal one is taken as the uncertainty for mode-I (II).
The uncertainty of the signal shape is 0.4% (0.2%) for mode-I (II), evaluated by comparing the signal yields with and without the Gaussian function convolution in the fit. We select alternative fit ranges and a higher order Chebychev polynomial function for non-peaking background shapes to estimate the related uncertainty. The largest difference of the signal yield with respect to the nominal one, 0.6% (1.3%), is taken as the uncertainty for mode-I (II). The uncertainty of peaking background is 0.3% for both modes, evaluated by adjusting the number of peaking background events by one standard deviation.
The uncertainty of the number of J/ψ events is determined to be 0.5% [17] and those of the η ′ branching fractions are taken as 1.7% for both modes [22].
For the dark photon search, the systematic uncertainties are divided into additive and multiplicative terms. The additive systematic uncertainties arise from the fit bias, signal and background PDF, while the multiplicative uncertainties come from the number of J/ψ events, η ′ branching fractions and detection efficiencies, which have been discussed in the branching fraction measurement. To incorporate these uncertainties, we take the additive systematic uncertainty into consideration by performing the same fit procedure with the different combinations of the nominal and alternative fit ranges, signal shapes and background shapes. The maximum number of signal events between the different fit scenarios corresponding to 90% of the likelihood function is taken as the upper limit of the signal yield N sig at the 90% C.L. This procedure is performed for mode-I and mode-II separately. The multiplicative systematic uncertainties in the search of dark photon are listed in Table II. Most of them come from differences in the selection efficiency between data and MC simulation. When deriving the upper limit of B(J/ψ → η ′ γ ′ ), one of the dominant systematic uncertainties originates from the theoretical branching fraction of B(γ ′ → e + e − ), which is 0 ∼ 14% depending on m γ ′ according to the Ref. [13] and mainly comes from the R value measurement.

V. DARK PHOTON SEARCH RESULT
We compute the upper limit on the branching fractions B(J/ψ → η ′ γ ′ ) × B(γ ′ → e + e − ) and B(J/ψ → η ′ γ ′ ) at the 90% C.L. using a Bayesian method [22]. The expected number of signal events observed in the i th mode is calculated is the branching fraction of η ′ decay to final state F, B(J/ψ → γ ′ η ′ ) and B(γ ′ → e + e − ) are the branching fractions and E i is the detection efficiency determined from the signal MC simulation. The likelihood value (L), as a function of product branching fraction B(J/ψ → η ′ γ ′ ) × B(γ ′ → e + e − ), is calculated as a product of L from mode-I and mode-II with the method described in Ref. [35]. The systematic uncertainties, which have been discussed in Section IV, are separately incorporated into the likelihood distribution as correlated and uncorrelated terms. The upper limit B UP on the product branching fraction at the 90% C.L. is determined from the The values of B UP are plotted as a function of m γ ′ in Fig. 5(a). We also obtain the likelihood value as a function of B(J/ψ → η ′ γ ′ ) by taking into account the branching fraction B(γ ′ → e + e − ) and its corresponding uncertainty [13], and compute the upper limit on the B(J/ψ → η ′ γ ′ ) at the 90% C.L. with a similar method, which is shown in Fig. 5 (b). The upper limit at the 90% C.L. on the branching fraction B(J/ψ → η ′ γ ′ ) × B(γ ′ → e + e − ) ranges from 1.8 × 10 −8 to 2.0 × 10 −7 and that on B(J/ψ → η ′ γ ′ ) ranges from 6.0 × 10 −8 to 7.8 × 10 −7 .
The mixing strength ε coupling the dark photon γ ′ and SM photon is determined from the ratio of the branching fraction B(J/ψ → η ′ γ ′ ) and the radiative process B(J/ψ → η ′ γ) TABLE II. Sources of systematic uncertainties for the branching fraction measurement and multiplicative terms for the dark photon search (in %). The correlated sources between η ′ → γπ + π − and η ′ → π + π − η(γγ) modes are marked with an asterisk.
The result is compatible with the previous BESIII measurement [8] and the statistical uncertainty is reduced.