Search for new light vector boson using $J/\Psi$ at BESIII and Belle II

We investigate various search strategies for light vector boson $X$ in $\mathcal{O}(10)~{\rm MeV}$ mass range using $J/\Psi$ associated channels at BESIII and Belle II: (i) $J/\Psi \to \eta_c X$ with $10^{10} J/\Psi$s at BESIII, (ii) $J/\Psi (\eta_c +X) +\ell \bar{\ell}$ production at Belle~II, and (iii) $J/\Psi +X$ with the displaced vertex in $X\to e^+e^-$ decay are analyzed and the future sensitivities at Belle II with 50 ${\rm ab}^{-1}$ luminosity are comprehensively studied. By requiring the displaced vertex to be within the beam pipe, the third method results in nearly background-free analysis, and the vector boson-electron coupling and the vector boson mass can be probed in the unprecedented range, $10^{-4}\leq |\varepsilon_e| \leq 10^{-3}$ and $9~{\rm MeV}\leq m_X\leq 100 {\rm MeV}$ with 50 ${\rm ab}^{-1}$ at Belle II. This covers the favored signal region of $^8{\rm Be}^*$ anomaly recently reported by Atomki experiment with $m_X \simeq 17~{\rm MeV}$.


I. INTRODUCTION
The Standard Model (SM) is a successful theory describing physics at least up to the electroweak scale, having survived more than 40 years of various experimental tests. However, there still remain a handful number of experimental and observational claims that indicate discrepancies from the SM predictions and consequently request extension of the SM: nonzero mass of neutrinos [1], anomalous magnetic moment of muon, (g − 2) µ [2,3], existence of dark matter (DM) [4], 1 and baryon vs. antibaryon asymmetry of the universe [6,7]. There have been discussions of extending the SM by gauging the lepton number, e.g. L µ − L τ or L e − L τ [8], intending to explain DM [9,10], the muon anomalous magnetic moment (g − 2) µ [11][12][13][14], and more recently EDGES 21cm anomaly [15]. The extension gives rise to a leptophilic light vector boson, dubbed as X in this paper. We note that the X boson may couple to the quarks via interactions with unknown heavy fermions that mix with SM quarks [16]. It may then be responsible for the recent anomaly from the KOTO experiment in K L → π 0 νν [17][18][19][20] and also the anomaly from the Atomki experiment in 8 Be * . The preferred mass of X for these cases is in sub-GeV range; in particular m X 17 MeV for Atomki [21,22].
High luminosity lepton colliders provide ideal environments to test for such light X boson. Thanks to less severe QCD backgrounds, the lepton colliders have definite advantages over hadron colliders even when the X boson has feeble couplings with the SM particles. In this paper, we take the lepton colliders, BESIII and Belle II, and study the search strategies of X. In particular, we focus on the channels in association with a J/Ψ meson, which will be enormously produced at BESIII and also at Belle II, thereby leaving the signals of X in various channels: • At BESIII, up to now, 10 10 J/Ψ events are collected, thus provide an excellent probe to study the J/Ψ rare decays to the X boson.
• At Belle II, even though less number of J/Ψ are expected, we use the process e + e − → + − J/Ψ → + − η c X → + − η c e + e − ( = e or µ), in which J/Ψ and η c are inferred by the recoil masses of + − and + − e + e − , respectively.
• At Belle II, we also use the channel e + e − → X + J/Ψ where the X bosons will leave signals with displaced vertices. Due to higher center-of-mass (CM) energy at Belle II, the X boson will be boosted and travel several millimeters before it decays into e + e − .
The rest of this paper is dedicated to studying the sensitivity reach of finding X boson taking realistic experimental situations into account under the effective field theory framework. This paper is organized as follows: we first set up our theoretical framework and introduce the effective interactions in Section II. The analysis for BESIII is carried out in Section III.
In Section IV and V, e + e − → + − J/Ψ and e + e − → X + J/Ψ with displaced vertex signal are analyzed, respectively for Belle II. Finally, our results are summarized in Section VI.

II. EFFECTIVE LAGRANGIAN
The vectorlike interactions of the X boson with the SM fermions, f , are introduced by the effective Lagrangian: where we regard the couplings ε f as free parameters without knowing the origin. In particular, we will assume four universal couplings, ε u , ε d , ε e , and ε ν , for up-type quarks, down-type quarks, charged leptons, and neutrinos, respectively, in our analysis below. We also note that the new interactions do not induce any axial anomaly by construction.
If the new boson X is responsible for the recent Atomki anomaly [22] via the process 8 Be + X → 8 Be + e + e − , its mass should be m X 17 MeV and the couplings with the first generation quarks should be in a particular window [23][24][25]: From the NA48/2 experiment for π 0 → Xγ, we require a protophobic condition [26]: Taking both relations in Eq. 2 and Eq. 3 into account, we finally get the preferred value for up-type and down-type quark couplings: which we will rely on below.
The coupling to the leptons, especially to electron and electron-neutrino, are stringently constrained by the beam dump experiment SLAC E141 [27], the anomalous magnetic moment of the electron g − 2 [28], and neutrino-electron scattering experiment [29]: When a small coupling for neutrino ε ν 10 −6 is assumed, we do not worry about constraints from neutrinos.
The normalized differential widths for partial decay widths of J/Ψ → η c γ * → η c e + e − and J/Ψ → η c X * → η c e + e − , respectively for off-shell photon and X boson are obtained using a common factor F V P (q 2 ) [30]: where the kinematic window is given as (2m e ) 2 ≤ q 2 = m 2 e + e − ≤ (m J/Ψ − m ηc ) 2 . The precise expression for the factor F QED is found in Ref. [30] where the factor is found to include the amplitude square and phase space factor for off-shell photon [30]. Analogous expression for F X is obtained: by replacing the couplings and propagator from F QED . Assuming that ε ν 10 −6 and ε e 10 −3 , and the quark channels are kinematically forbidden with m X ≤ 2m π , the X boson dominantly decay to electrons with the width which is narrow Γ X m X . After performing the integration of q 2 , we can obtain the partial decay width Γ ηcX * . By inserting favoured coupling values ε c = ε u = 3.7 × 10 −3 , ε e = 10 −3 and fixing m X = 17 MeV for 8 Be * anomaly, it gives Γ ηcX * = 9.59 × 10 −3 keV and implies Br(J/Ψ → η c X * → η c e + e − ) = 1.64 × 10 −6 ε c 10 −2
The e + e − invariant-mass-squared distributions for signal (η c X * ) and background (η c γ * ) are compared in Fig. 1. The different features are clearly displayed: the signal has a peak at q 2 = m 2 X and the background is broadly distributed. Therefore our task now is to efficiently extract the signal near the peak and suppress the background.
We first impose a kinematic condition for signal: where σ m is the e + e − mass resolution which is roughly of the same order of magnitude as the energy resolution σ E . The signal yield S and background yield B are now obtained as where N J/Ψ is the total number of J/Ψ produced in the collision, and Γ J/Ψ = 92.2 keV is the total decay width of J/Ψ. The BESIII experiment, which has collected 10 10 J/Ψ events in the resonance process e + e − → J/Ψ [30], plans to increase the size of J/Ψ sample to 10 11 events in the near Br(η c → 2(π + π − π 0 )) = (15.2 ± 1.8)% 3. 07%  TABLE II. For N J/Ψ = 10 11 , and chose favoured parameters ε c = ε u = 3.7 × 10 −3 , ε e = 10 −3 , m X = 17 MeV for 8 Be * anomaly, the significances of signal to background from J/Ψ → η c e + e − with various energy resolutions of detector and 1.23% η c reconstruction efficiency. future. The energy resolution of BESIII is σ E /E 0.005 for the final-state electron, which smears the invariant mass distribution of e + e − by σ m 1 MeV. In order to exclusively reconstruct the J/Ψ → η c e + e − decays, we have to consider η c decay modes that can be fully reconstructed with reasonable background contamination. Table I below lists the branching fractions of a few such η c decay modes along with the corresponding efficiencies [35].  The overall efficiency of the above three η c modes is obtained by adding the individual efficiencies weighted by their corresponding branching fractions: = 1.23%. Given these, and taking 17 MeV X boson for 8 Be * anomaly to be real, we list, in Table II, the expected significances of J/Ψ → η c X → η c e + e − with N J/Ψ = 10 11 at BESIII under the assumption of ε c = ε u , for various σ m values.
For general light vector boson searches through J/Ψ → η c e + e − , the variation of the expected significance over (m X , ε c , ε e ) are shown in Fig. 2. With the present value of N J/Ψ = 10 10 at BESIII, the region of sensitivity is |ε c | 5 × 10 −3 at m X 17 MeV as shown in the upper left panel of Fig. 2 and left panel of Fig. 3. The sensitivity slightly improves as m X increases, because of the reduction of background (see Fig. 1), and reaches the maximal sensitivity |ε c | 3 × 10 −3 at m X 60 MeV. But as m X approaches m J/Ψ − m ηc , the sensitivity becomes weaker due to the phase space suppression. The two right panels of Fig. 2 show that the significance is independent of the ε e as we expect from the narrow width approximation. For N J/Ψ = 10 11 which is expected in the near future, the projected sensitivity becomes |ε c | 3 × 10 −3 at m X 17 MeV as shown in the bottom-left panel of Fig. 2 and the right panel of Fig. 3, whereby the entire favored region of 8 Be * anomaly can be probed.
An alternative way to explicitly reconstructing η c in J/Ψ → η c e + e − at BESIII is to use the recoil of e + e − . As the e ± carries low energy around 50 MeV, it gets difficult to distinguish e ± tracks from π ± background. With an improvement of low-energy electron identification in the future, the BESIII with N J/Ψ = 10 11 can reach the sensitivity of |ε c | 10 −3 .
For vector meson J/Ψ, the partial width to e + e − is given by where g J/Ψee = 8.2048 × 10 −3 [7] is the coupling strength in the effective interaction g J/Ψee [ēγ µ e](J/Ψ) µ that matches the measured value Γ J/Ψ→e + e − = 5.53 keV [7]. Then the cross sections to + − J/Ψ where = e, or µ at Belle II are obtained via e + e − → γ * J/Ψ: σ(e + e − → γ * + J/Ψ → e + e − J/Ψ) = 286 fb, With the design integrated luminosity L = 50 ab −1 , we estimate N J/Ψ = 1.75 × 10 7 events for e + e − → γ * + J/Ψ → + − J/Ψ at Belle II. This N J/Ψ is applied to Eq. 11, along with Eq. 12, to give estimates of S and B: Therefore, the estimated S/ √ B is too small at this level so that we will improve the analysis by a more realistic MC study below.
For event generation, we use MG5 aMC@NLO [37] for both signal and background with FeynRules v2.0 [38] model for J/Ψ, η c mesons and X boson couplings and X couples to the leptons. We generate with the E beam1,2 = 5.2941 GeV in the CM frame, which is boosted by β = 0.2732 with respect to the lab frame. The amplitude of the electromagnetic Dalitz decay, V → P e + e − can be written in a Lorentz-invariant form [30], MeV. Here, we normalized to 10 5 events for each channel.
and we can obtain the interaction Lagrangian as, where g Xc , g Xe , and g eV are the effective coupling constants, whose numerical values are to be obtained by experiments. In the left panel of Fig. 5, the invariant mass distributions of e + e − are plotted, where samples of an equal number of events (= 10 5 ) for each of the signal and background at the parton level have been used before any selection cuts. We give the Gaussian smearing effect with the momentum resolution σ p ± /p ± = 0.005 on the parton level data for our event analysis.
To simulate the effects of the Belle II detector, we apply the following baseline cuts: |η * ± | ≤ 1.60 in the CM frame [14,39], |E µ ± | ≥ 0.6 GeV, and |E e ± | ≥ 0.06 GeV 2 in the lab frame [40]. In addition, we apply kinematic requirements: |M ee − m X | ≤ 2 MeV, whereby the dominant background from the process shown in the right panel of Fig. 4 are suppressed. We also require |M rec ee − m ηc | ≤ 200 MeV to eliminate the sub-dominant background + − e + e − (γ). In Table III, we summarize the cumulative effects of the baseline cuts and the kinematic requirements. For the energy-momentum 4-vector of η c , we use the energy and momentum recoiling against e + e − + − . The signal and background distributions of the recoil mass M rec ee , smeared by the charged-track momentum resolution, are displayed in the right panel of Fig. 5. The signal events clearly show a peak at M rec ee m ηc , while the background is mostly populated at M rec ee 0. The sensitivities of + − e + e − search at Belle II are derived from the requirement of S/ √ B = 2. Combining Eq. (14) and Table III, we obtain the corresponding values of ε c with respect to luminosities of 50, 100, and 200 ab −1 in Table IV. They are about 5 times larger than the estimates from current BESIII sensitivity |ε c | 5 × 10 −3 in Section III.  The X boson produced in the process e + e − → X + J/Ψ travels several millimeters before decaying into e + e − in the Belle II detector and leaves displaced vertex. In particular, when the distance of the flight is between 2 mm and 8 mm, which is inside the beam pipe, and outside the interaction region, Belle II has excellent power to reconstruct the displaced vertex and makes the signal almost free from the SM background [41]. Therefore, we propose to use the clean displaced e + e − vertex from the X boson decay (with prompt + − from J/Ψ).
The leading-order Feynman diagram of the relevant process is shown in the left panel of Fig. 6. A typical event with displaced vertex at Belle II detector with the + − from J/Ψ decay is schematically shown in the right panel of Fig. 6.
We note that compared with other lighter vector mesons, the heavier mass of J/Ψ helps to induce larger scattering angle in such a way that more events from X → e + e − will satisfy the cut |η * ± | ≤ 1.60. Furthermore, the electron and positron from X → e + e − carry energy above GeV, which make them easier to be distinguished from charged-pion backgrounds.
The other advantage of this channel is that the signal strength only depends on the ε e coupling since only X-e + -e − vertices are involved at tree level. For 0.3 × 10 −3 ≤ ε e ≤ 0.8 × 10 −3 , it yields a few mm transverse flight distance d xy which is defined as the distance between the beam axis and the X decay vertex. The left-panel of Fig. 7 shows the distribution of d xy corresponding to several values of ε e .    With the baseline cuts and 2 mm ≤ d xy ≤ 8 mm, we estimate the signal sensitivity by considering two cases: (i) explicitly reconstructing J/Ψ → + − ('e + e − + − channel'), and (ii) using the recoil mass of X → e + e − to infer J/Ψ ('e + e − channel'). Tables V and VI show, respectively for the e + e − + − and e + e − channels, the signal efficiencies and expected significances for various assumed values of ε e , according to the 50 ab −1 luminosity at Belle II and the e + e − → X + J/Ψ cross section The final result of expected sensitivity with Belle II at 50 ab −1 luminosity is shown in Fig. 8. Also displayed in Fig. 8 are the expected results with the η c -related studies at Belle II and BESIII as discussed in Sections III and IV. The displaced e + e − vertex searches can probe the 17 MeV X boson in the region 2.5 × 10 −4 ≤ ε e ≤ 8.0 × 10 −4 with significance larger than 2 by assuming near-zero background, and it covers the ε e region preferred by Atomki. While we expect less than one signal event with the currently available Belle data sample of 1 ab −1 , we can start exploring the Atomki preferred region within a few years once Belle II accumulates data sample of 10 ab −1 or more.
Our study with the displaced vertex is extended for wider mass range of X-like boson, whereby we determine the region of sensitivity with Belle II at 50 ab −1 , as displayed in Fig. 8. Two cases for the SM background are considered: B = 0.1 and B = 1, where B is the number of background events expected. We then use the Poisson probability of the expected background to fluctuate upward, to calculate the p-values and the corresponding significances. The 2σ significance region with B = 0.1 (B = 1) is shown by the green (darkgreen) contour in Fig. 8. The dark (light)-green-shaded area corresponds to the region with S B=1 (S B=0.1 ) > 2σ, where S B=1 (S B=0.1 ) is the expected significance with the background level of 1 (0.1) event. This study can probe the parameter region of 5 MeV ≤ m X ≤ 100 MeV and 1.0×10 −4 ≤ ε e ≤ 3×10 −3 , which have not been constrained by any existing experiments. The upper edge of this region is determined by the condition 2 mm ≤ d xy , while the lower edge is limited by the statistics. Therefore, with even higher luminosity of Belle II exceeding the target 50 ab −1 , the lower edge of the sensitivity region can be extended further.

VI. SUMMARY AND CONCLUSION
In summary, we propose several studies using J/Ψ at lepton colliders such as Belle II and BESIII, to search for light vector boson around the mass range suggested by the 8 Be * anomaly of the ATOMKI experiment. At BESIII, the J/Ψ → η c X → η c e + e − channel can be used to constrain the vector boson and charm quark coupling, ε c . With the currently available sample of N J/Ψ = 10 10 and effective η c reconstruction efficiency of 1.23%, we can exclude the region |ε c | 5 × 10 −3 for m X = 17 MeV. If N J/Ψ = 10 11 is produced at BESIII in the near future, exclusion of the region |ε c | 3 × 10 −3 might be achieved. If universal coupling to up-type quarks is assumed, we expect that the entire favored signal region from the 8 Be * anomaly could then be covered.
On the other hand at Belle II, with higher CM energy of 10.59 GeV, we propose to study the process e + e − → + − J/Ψ followed by J/Ψ → η c X → η c e + e − . Using the recoil mass against + − e + e − , we perform MC study and find that the expected production of J/Ψ events is about three orders of magnitude smaller than that of BESIII, thus yielding the sensitivity of |ε c | 1.8 × 10 −2 at m X = 17 MeV. Alternatively, we can study the process e + e − → X + J/Ψ → e + e − + − at Belle II and directly constrain the X boson-electron coupling. The X boson is boosted by the higher CM energy and heavy mass of J/Ψ, producing displaced vertex of X → e + e − which is longer than several millimeters. Particularly, it is almost background free when the transverse flight distance is 2 mm ≤ d xy ≤ 8 mm. Selecting this window and requiring > 2σ significance, it gives the sensitivity 2.0 × 10 −4 ≤ |ε e | ≤ 8.0 × 10 −4 at m X = 17 MeV for 50 ab −1 luminosity and covers most of the favored signal region from the claimed 8 Be * anomaly. Extending the range of the X boson mass, this method can cover the unprecedented parameter space of 9 MeV ≤ m X ≤ 100 MeV and 1.0 × 10 −4 ≤ |ε e | ≤ 10 −3 .