New channel to search for dark matter at Belle II

We propose a new ``disappearing positron track'' channel at Belle II to search for dark matter, in which a positron that is produced at the primary interaction vertex scatters with the electromagnetic calorimeter to produce dark matter particles. Such scatterings can occur via either annihilation with atomic electrons, or the bremsstrahlung process with target nuclei. The main backgrounds are due to photons and neutrons that are produced in the same scatterings and then escape detection. We require a large missing energy and further veto certain activities in the KLM detector to suppress such backgrounds. To illustrate the sensitivity of the new channel, we consider a new physics model where dark matter interacts with the standard model via a dark photon, which decays predominantly to dark matter; we find that our proposed channel can probe some currently unexplored parameter space, surpassing both the mono-photon channel at Belle II and the NA64 constraints.

In this paper, we propose a new channel to search for DM at colliders where DM are produced in collisions between SM particles and the detector, instead of at the primary collision vertex of the collider.For concreteness, in this analysis we take as an example the Belle II experiment, the electron-positron collider operated at SuperKEKB.Belle II is expected to accumulate at least 50 ab −1 data and has a hermetic electromagnetic calorimeter (ECL) [32], making it an ideal machine to search for light DM as well as other new light particles [33][34][35][36][37][38][39][40][41][42][43][44][45][46][47].
Electrons and positrons are copiously produced at Belle II via the Bhabha scattering process, leading to O(10 12 ) positrons expected with 50 ab −1 .These final state positrons can then interact with the ECL detector to produce DM, as shown in Fig. (1).The interactions between the positron and the ECL can occur either via annihilations with atomic electrons in the ECL (the annihilation process) or via scatterings with target nuclei in the ECL (the bremsstrahlung process) where χ is the DM particle, e − A is the atomic electron in the ECL, and N is the target nucleus, which can be either Cs or I in the ECL.The Feynman diagrams of these two processes are shown in Fig. (2).The DM particles then escape the Belle II detectors, resulting in a missing energy signature.We note that this channel is analogous to that in electron fixed-target experiments (e.g., NA64 [48]), with the ECL detector as the target.Unlike DM produced at the primary collision vertex, the missing energy in this new channel is preceded by a charged track in the central drift chamber (CDC) and a small amount of energy deposited in the ECL.We thus refer to this new channel as the "disappearing positron track" channel. 1 Moreover, the disappearing positron is accompanied by an electron that has an opposite momentum to the positron track in the center of mass A → χχ (left), and the bremsstrahlung process e + N → e + N χχ (right).The process with χχ radiated from the initial state e + is included in the analysis, but not shown here.
(c.m.) frame, a clear CDC track, and an energy deposition in the ECL that is consistent with the Bhabha scattering.
The collisions between positrons and the ECL can also produce photons and neutrons, which have a non-zero probability to penetrate all the sub-detectors of Belle II without leaving a trace, mimicking the signal events of DM.We find that a large missing energy in the ECL and a veto on a cluster or a track in the KLM (K L -muon detector) [32,46] are instrumental in controlling these backgrounds.
To illustrate the capability of this new channel in probing DM, we consider the dark photon (DP) model in which the DP predominantly decays into DM.The main results are summarized in Fig. (3).For the DP mass in the vicinity of 66 MeV, the new channel can probe unexplored parameter space, surpassing both the mono-photon channel at Belle II and the NA64 experiment.We note that, despite our efforts to reproduce the Belle II conditions to the best of our ability, the limits presented in this analysis should be revised by Belle II physicists using a full simulation of the detectors and inputs from data control samples.In particular, the various efficiency factors associated with the ECL and KLM detectors, which play a pivotal role in estimating the background, will be updated with a better understanding as the experiment progresses.
This rest of the paper is organized as follows.In section 2, we discuss the "disappearing positron track" signature at Belle II.In section 3, we discuss various SM backgrounds for the new signature.In section 4, we compute the signal events in the new channel for the invisible dark photon model, and further compare its sensitivity to other experimental searches, including the mono-photon channel at Belle II and the missing energy channel at NA64.In section 5, we summarize our findings.In the appendix, we provide the positron track length distribution for completeness, explore the dependence of the signal on the coupling constant, present our calculations on the NA64 constraints, discuss the backgrounds due to charged particles, compute the sensitivity on the scalar mediator model, and explore the sensitivity of the radiative Bhabha process.

Disappearing positron track
The energy of the final state positron is measured by the ECL detector, which has a barrel region and two endcap regions.In our analysis, we only consider the positrons in the barrel region (with a polar angle between 32.2 • and 128.7 • in the lab frame), due to the following three reasons.First, there are less non-instrumented setups, such as magnetic wires and pole tips, between ECL and KLM in the barrel region as compared to the endcap regions [51].This leads to a better KLM veto efficiency in the barrel region, which is essential in controlling the SM background.Second, the barrel region has a better hermiticity: Gaps between ECL crystals in the barrel region are non-projective to the collision point; however, some gaps in the endcap regions are projective so that particles can escape the ECL detector without being noticed when traversing them [52,53].Third, the endcap regions have more beam backgrounds [54].
Although in the signal process the positron cannot deposit all its energy in the ECL due to the production of DM, its transverse momentum can be measured in the CDC with a good resolution, e.g., δp T /p T ≃ 0.4% for p T ≃ 3 GeV [55].Using the CDC measurements (both the transverse momentum and the angular information), we can compute the positron energy, which is then required to be equal to the electron energy (measured both in the ECL and CDC) in the c.m. frame.To suppress the backgrounds (especially those from neutrons), we further require a large missing energy such that the final state positron only deposits at most 5% of its energy in the ECL, and veto KLM activities including multi-hits or a cluster.
We next compute the event number of positrons.Positrons at Belle II are mainly produced at the primary interaction point, via the Bhabha scattering process with the cross section [56] dσ where θ * is the polar angle of the final state positron in the c.m. frame, s is the square of the center of mass energy, and α ≃ 1/137 is the fine structure constant.Because SuperKEKB is an asymmetric collider, which collides 7 GeV electrons with 4 GeV positrons [32], the differential cross section of the Bhabha scattering in the lab frame is given by where E is the energy of the final state positron in the lab frame, β = 3/11, E * = √ s/2, and cos θ * = ( 1 − β 2 E/E * − 1)/β.In the lab frame, the energy of the final state positron E is related to its polar angle θ via E = E * 1 − β 2 /(1 − β cos θ).Thus, the minimum and maximum energy of the positron at the barrel region in the lab frame are E min ≃ 4.35 GeV (for θ = 128.7 • ) and E max ≃ 6.62 GeV (for θ = 32.2• ) respectively.The total number of positrons in the barrel region is about 6 × 10 11 with the total luminosity of 50 ab −1 .

Standard Model Backgrounds
SM backgrounds arise when the SM particles that are produced in the collision between the final state positron and the ECL detector escape detection.Charged particles (such as electron and muon) are likely to be detected by the ECL and KLM detectors: The probability for positrons to penetrate the ECL is very small; the KLM detector, which consists of an alternating sandwich of 4.7 cm thick iron plates and active detector elements [32], is very sensitive to the muon tracks, leading to negligible muon backgrounds via the KLM veto. 2 On the other hand, neutral particles (such as photon, neutron, and neutrino) have a significant probability to traverse the ECL and KLM detectors without being detected.Backgrounds due to neutrinos are found to be negligible, due to the large W/Z masses.Thus, the main backgrounds are due to photons and neutrons, which we discuss below.

Photon-induced backgrounds
We first discuss the photon-induced backgrounds.Photon energy can be measured in the ECL detector, which is made up of CsI crystals with the length of 16 X 0 [57], where X 0 = 1.86 cm is the radiation length of CsI [58].The energy distribution of photons that are produced in the collision between a positron with energy E and the ECL can be well approximated by [59] where x γ = E γ /E with E γ being the energy of the photon, and tX 0 is the position of the photon in the ECL detector.Therefore, the probability of a photon carrying more than 95% of the positron energy to escape the ECL detector is given by3 This leads to ∼2.8×104 potential background events after the ECL detector, for the 6×10 11 positrons.Although the probability for GeV-scale photons to penetrate the whole KLM (consisting of at least 60 cm iron plates in total [32]) without producing KLM clusters is negligibly small, photons can also be absorbed by some non-instrumented setups (for example the magnet coil) between the ECL and the KLM [51].For that reason the veto power of the KLM on photons is limited.To take into account such effects, we adopt the IFR veto efficiency at BABAR, which is about 4.5 × 10 −4 in the barrel region [61], as the conservative estimate of the KLM veto efficiency, since the KLM veto efficiency is expected to be better than the IFR [62]. 4 This then leads to 13 background events due to photons, for the 6 × 10 11 positrons.We note that a small fraction of secondary photons can leak into the endcap region, where the veto efficiency is somewhat reduced; in our analysis, we have neglected this small effect.We also note that if the KLM veto efficiency can be improved by one order of magnitude as compared to the IFR, the photon backgrounds will decrease to be about a single event.On the other hand, if the KLM veto efficiency turns out to be inferior to that of the IFR, an increase in photon-induced background events is expected.To account for this possibility, in section 4, we compute the Belle II sensitivity both with the background events analyzed in our current analysis and with a more conservative background level.The sensitivities under these different background assumptions are given in Fig. (3).

Neutron-induced backgrounds
We next discuss the neutron-induced backgrounds.Neutrons with energy of several GeV are mainly produced by photo-nuclear reactions between the positron and the ECL detector [58].To estimate such backgrounds, we simulate collisions of 10 9 positrons with 4. 35 GeV energy onto a CsI target with one X 0 , by using GEANT4 (version 11.0) [63] with the FTFP_BERT physics list.Our choice of the positron energy in the simulation is motivated by the fact that ∼ 50% of the positrons are in the first tenth of the entire energy range in the barrel region, according to Eq. (2.2).We only simulate a fraction of positrons with a thin CsI target (with one X 0 ) because the full simulation with a 16-X 0 CsI target is extremely time-consuming.The simulation results with the thin CsI target are acceptable for our purpose, because neutrons with significant energy are mainly produced within the first radiation length, which are confirmed in our simulations with a 2-X 0 CsI target.
To ensure that the missing energy is mainly caused by neutrons, we only select the GEANT4 simulated events that contain at least one neutron with energy exceeding 3 GeV.There are 4950 events in the 10 9 simulations that satisfy this selection cut.We then compute the total energy deposited in the ECL, by taking into account both the deposited energy in the first X 0 calculated by GEANT4, and the kinetic energy of e ± and γ.We further include the kinetic energy of protons with momentum less than 0.6 GeV, because such protons have a gyroradius radius ≲ 1.3 m in the ECL where B = 1.5 T [32], and thus can deposit the kinetic energy when orbiting around.We do not add the kinetic energy of π ± to the deposited energy, because π ± decays primarily into a neutrino and a muon which deposit negligible energy to the ECL.We then require the deposited energy in the ECL to be less than 5% of the energy of the positron; there are 100 events after this detector cut.We further veto events that contain protons or π ± with momentum exceeding 0.6 GeV, because these charged hadrons can either deposit significant energy in the ECL and/or produce tracks in the KLM.There are 64 events after this veto.
Next we classify the remaining events according to the number of neutrons that have kinetic energy exceeding 280 MeV, the energy threshold for hadronic showers [64].There are 13 events with a single neutron and 51 events with more than 2 neutrons.We compute the probability for a neutron to penetrate a target with length L via [65,66] where λ 0 is the hadronic interaction length.The KLM has ∼ 3.9λ 0 , and the ECL has ∼ 0.8λ 0 [32].We note that Eq. (3.3) can yield consistent results on neutron-induced backgrounds as compared with GEANT4 simulations in the context of the NA64 experiment [66].Thus, the probability for a neutron to penetrate the remaining 15 X 0 's of the ECL and the KLM is P ≃ 0.01. 5 Rescaling this to the 6 × 10 11 positrons, one expects ∼81 background events due to neutrons in total.We note that there is another source of neutrons from the beam backgrounds (dominated by 10-100 keV neutrons), which can also produce KLM hits [69] and thus complicates the situation.Because of the beam backgrounds, one cannot veto events with any hits in the KLM [51].Fortunately, unlike neutrons with kinetic energy above 280 MeV which are expected to produce multi-hits or a cluster in the KLM, a single beam background neutron is usually absorbed in one scintillator strip [68].For that reason, in our analysis, we only select neutrons above 280 MeV, which can be well controlled by the veto on multi-hits or a cluster in the KLM.However, since there is already a neutron with energy above 3 GeV in our selected events, including another neutron below 280 MeV would further suppress the background, leading to an even smaller neutron background.Thus, our analysis serves as a conservative estimate of the neutron backgrounds.
Taking into account backgrounds from both neutrons and photons, one expects at most ∼94 background events for the 6 × 10 11 positrons.

Sensitivity on dark matter in dark photon models
To show the sensitivity of the "disappearing positron track" channel on DM, we consider a new physics model in which DM interacts with the SM through a DP [70,71].In new physics scenarios where the SM gauge group is extended by an additional U (1), DP can naturally arise, either via the kinetic mixing portal [72,73], or via the Stueckelberg mass mixing portal [74][75][76][77][78][79][80].For both portals, field redefinitions are necessary to recast the kinetic terms into canonical forms and to diagonalize the mass matrix for the neutral gauge bosons.These processes lead to DP interactions with matter fields both in the SM sector and in the dark sector.In the case of a small mixing parameter, the interaction Lagrangian can be parameterized as follows [71,78] where A ′ µ is the DP with mass m A ′ , χ is the Dirac DM with mass m χ , f denotes the SM fermion with electric charge Q f , ϵ is the small mixing parameter, e is the QED coupling constant, and g χ is the hidden coupling constant.In our analysis we fix m χ = m A ′ /3 such that A ′ decays dominantly into DM in the parameter space of interest where g χ ≫ eϵ.
We compute the signal events from both diagrams shown in Fig. (2).We first compute the annihilation process e + e − A → A ′ → χ χ; the cross section is given by where α D = g 2 χ /4π, Γ A ′ is the decay width of the DP, and s = 2m e E ′ + 2m 2 e = 2m e E A ′ with E ′ being the energy of the positron at the collision point and E A ′ = E ′ + m e being the energy of A ′ .Note that we have E ′ ≤ E where E is the positron energy before entering ECL.The partial decay width of the DP into DM is Because the invisible decay width is much larger than the visible ones in the parameter space of interest, we use Γ A ′ ≃ Γ(A ′ → χχ) in our analysis.The signal events in the annihilation process can be computed by [48,81,82] where L = 50 ab −1 is the integrated luminosity, n e is the number density of the electron in CsI, and dσ B /dE is given in Eq. (2.2).Here T e (E ′ , E, L T ) is the positron differential tracklength distribution [59,[82][83][84] where L T = 16X 0 is the thickness of the ECL target.The expression of T e is given in appendix A. The integration of E A ′ is performed for E A ′ > 0.95E so that the positron deposits less than 5% of its original energy in the ECL.
We next compute the bremsstrahlung process.In the parameter space of interest, the signal is dominated by the on-shell produced A ′ .Thus, the signal events are given by where n N is the number density of I (or Cs).Here dσ bre /dE A ′ are the differential cross section of the on-shell produced A ′ [85][86][87][88], where x ≡ E A ′ /E ′ , and ϕ N denotes the effective flux of photons from nucleus N [85]: with , and d = 0.164A −2/3 GeV 2 .We use Z = 53 (55) and A = 127 (133) for I (Cs).Here we only consider the dominant elastic form factor.
Fig. (3) shows the expected 90% CL limits on the dark photon coupling parameter ϵ from the "disappearing positron track" channel, as a function of the DP mass, where we take m A ′ = 3m χ , L = 50 ab −1 , and α D = 0.001.We compute the 90% CL limits by using N s / √ N b = √ 2.71 [45] where N s = N ann + N bre and N b = 94.We find that in the narrow mass window, 66 MeV ≲ m A ′ ≲ 82 MeV, the annihilation process with the atomic electrons dominates; outside this region, the bremsstrahlung process dominates.The narrow mass window can be explained by the Breit-Wigner resonance in Eq. is in the range of 4.35-6.62GeV, which leads to 66 MeV ≲ m A ′ ≲ 82 MeV if the < 5% energy difference between secondary positrons and the incident position is neglected.We thus refer to the narrow mass window where the annihilation process can proceed via the Breit-Wigner resonance of the mediator as the resonance region.The Belle II sensitivity in the resonance is significantly enhanced, as shown in Fig. (3). 6he DP models can also be searched for in the mono-photon channel e + e − → γA ′ at Belle II [32] and by the missing energy signature at NA64 [48,60].The Belle II sensitivity from the mono-photon channel with 50 ab −1 , shown in Fig. (3), is rescaled from the result with 20 fb −1 in Ref. [32], assuming that the limit on ϵ is proportional to L −1/4 .7 Fig. (3) shows the NA64 constraints with 2.84×10 11 electrons on target in the α D = 0.001 case.The NA64 constraints away from the resonance region, which is ∼ (200 − 300) MeV, are from Ref. [60].We compute the NA64 constraints in the resonance region by taking into account both the annihilation process and the bremsstrahlung process; the detailed calculations are given in appendix C, where we also carry out a comparison between our analytic method and the results given in Ref. [48], to demonstrate the accuracy of our calculation.For the α D = 0.001 case, if we take N b = 94, we find that the best limit on ϵ from the new "disappearing positron track" channel is ϵ ≲ 1.7 × 10 −5 , which occurs in the vicinity of m A ′ ≃ 66 MeV, surpassing both the mono-photon channel at Belle II and the NA64 constraints.As α D increases, the limits on ϵ are slightly weakened; for example, the best limit is ϵ ≲ 2.6 × 10 −5 for the case of α D = 0.1.See appendix B for the detailed dependence of the limits on α D .
In our benchmark model point, we have used the mass relation m A ′ = 3m χ .We note that the limits have a weak dependence on m χ , if it is decreased to even smaller values.For example, in the m A ′ = 66 MeV case, the number of events is only decreased by ∼ 0.2%, if we change m A ′ = 3m χ to m A ′ = 5m χ .This is because for the invisible dark photon model the signal process can be well approximated by the production of an on-shell dark photon.
We note that, despite our efforts to reproduce the Belle II conditions to the best of our ability, the photon-and neutron-induced backgrounds estimated in our analysis could still be subject to large uncertainties.To illustrate the impacts of these potential uncertainties on the sensitivity, we also compute the sensitivity with a background that is about one order of magnitude larger than our initial analysis.This is shown as the blue dashed curve with N b = 1000 in Fig. (3).We find that the sensitivity on ϵ is weakened by a factor of ≃ 1.8, if the background is increased from N b = 94 to N b = 1000.This is because the new physics signal is proportional to ϵ 2 , while the expected limit on the new physics signal is proportional to √ N b .Even with N b = 1000, we find that the sensitivity from the "disappearing positron track" in the resonance region, 66 MeV ≲ m A ′ ≲ 82 MeV, remains stronger than the NA64 constraint and can still be comparable to the mono-photon sensitivity from Belle II.
Here we discuss the possibility of using the "disappearing electron track" as a control sample.In this control sample, the electron in the final state of the Bhabha scattering interacts with the ECL to produce DM, with the positron being fully reconstructed.The control sample should yield a similar signal in the bremsstrahlung process as the "disappearing positron track", but its signal in the annihilation process is very small.One might expect a null result in the annihilation process for the "disappearing electron track" because there are no atomic positrons in the ECL.However, secondary positrons can be generated when an electron traverses the ECL.By comparing the differential track lengths of positrons for incident electrons and positrons, which are given in Appendix A, we find that the secondary positrons that carry > 95% energy of the incident electron are smaller than those originating from an incident positron by a factor of ∼ 5 × 10 −4 .Thus, the annihilation process in the control sample is expected to be smaller than the "disappearing positron track" by the same reduction factor.Because backgrounds are expected to be the same for both the "disappearing electron track" and the "disappearing positron track" channels, one can use the control sample to cross check the background analysis.The most interesting parameter space in our analysis occurs at m A ′ ≃ 66 MeV and 2 × 10 −5 ≲ ϵ ≲ 10 −4 , where the sensitivity is dominated by the annihilation between positrons and atomic electrons in the ECL.If excess events beyond the SM backgrounds were to be observed in this region for the "disappearing positron track", one could cross check the signal with the control sample, which is expected to yield a signal that is a factor of ∼ 5 × 10 −4 smaller.

Conclusions
In this paper, we propose a new "disappearing positron track" channel at Belle II to search for DM, where DM are generated via collisions between positrons and the ECL.The major backgrounds are due to photons and neutrons produced in the same collisions.We design a set of detector cuts to reconstruct such a new signal from the Belle II data, as well as to suppress various SM backgrounds.We compute the sensitivity of the new channel on the invisible dark photon model.We find that the new channel at Belle II can probe the dark photon coupling parameter ϵ ≃ 1.7 × 10 −5 with 50 ab −1 data for dark photon mass at ∼66.66 MeV, surpassing both the mono-photon channel at Belle II and the missing energy channel at NA64.We note that the disappearing track signal on positrons can be further extended to other SM particles at different particle colliders, thus presenting an opportunity to probe various new physics models with diverse interactions.

A Track-length distribution of positrons and electrons
For an incident positron with an initial energy E to enter a target with thickness L T , the differential track-length distribution of positrons as a function of the positron energy E ′ can be computed by [82,84] where X 0 is the radiation length of the target.Here I e (E ′ , E, t) is the energy distribution of E ′ at the depth tX 0 , which can be computed iteratively such that I e = i I (i) e where I (i) e denotes the i-th generation positrons [59].We adopt the analytical model of Ref. [59] up to second-generation positrons, which are found to be in good agreement with simulations in Ref. [82].The contributions from the first two generations are [59] where b 1 = 4/3, b 2 = 7/9, v = E ′ /E.We note that the analytical expression for the positron track length (with the first two generations) can yield consistent results with GEANT4 simulations; see e.g., figure 4 of Ref. [82] for the comparison in the case of an aluminum target.
Note that Eqs.(A.1, A.2, A.3) can also be used to compute the track length distribution of electrons with an incident electron.We compute the track length distribution of positrons (electrons) with an incident electron (positron) via e (E ′ , E, t)dt, (A.4) where we have only used e .This is because e describes the effects of bremsstrahlung, and I (2) e subsequently describes the production of the electron and positron pair due to the emitted photon [59].

B Dependence of signal on α D
Here we investigate the dependence of the dark photon constraint from the "disappearing positron track" channel at Belle II on the gauge coupling constant α D in the hidden sector.
Dark photons can be produced either in the bremsstrahlung process or in the annihilation process.In the bremsstrahlung process, because dark photons are on-shell produced, the cross section does not depend on α D .However, in the annihilation process, the cross section depends on α D through the Breit-Wigner form of the dark photon propagator, since the hidden decay width of the dark photon is proportional to α D .  is rescaled from the result with 20 fb −1 in Ref. [32].The NA64 constraint with 2.84 × 10 11 electrons on target (gray shaded region) is taken from Ref. [60].
constraints on the coupling parameter ϵ as a function of α D , in which m A ′ is fixed to be 66.66 MeV.The upper bound on ϵ decreases with α D until it saturates at ϵ ≲ 1.7×10 −5 for α D ≲ 0.001.Constraints from the missing energy signature at NA64 and from the monophoton channel at Belle II are also shown in Fig. (4), which are independent of α D .It is remarkable that the "disappearing positron track" channel for α D ≲ 0.5 leads to a better constraint than the mono-photon channel at Belle II and the missing energy signature at NA64.

C NA64 constraints
In this section, we compute the NA64 constraints.Note that Ref. [48] provided the NA64 constraints on dark photon with α D = 0.1 and α D = 0.5.In our calculation, we compute the NA64 constraints in the resonance region by taking into account the number of signal events both from the bremsstrahlung process (N bre ) and from the annihilation process (N ann ).The NA64 constraint on the coupling parameter ϵ taking into account only the bremsstrahlung process (denoted as ϵ bre ) has been obtained in Ref. [60,90], by requiring N s = 2.3 where N s is the number of signal events.Since the signal from the bremsstrahlung process is proportional to ϵ 2 and independent on α D , we compute the number of signal events from the bremsstrahlung process via where N s = 2.3.
We compute the number of signal events from the annihilation process via where E = 100 GeV is the energy of the incident electrons, N EOT = 2.84×10 11 is the number of electrons on target, n e is the electron number density of the lead target, ϵ d ≃ 0.5 is the detection efficiency [60], σ ann is the cross section of the annihilation process e − e + → χ χ, E A ′ is the energy of the dark photon, L T is the length of the target, and T e (E ′ = E A ′ −m e , E, L T ) is the positron differential track-length distribution.We obtain the 90% C.L. constraints by using the criterion N ann + N bre = 2.3.We recompute the NA64 limits in Fig. (5) and compare them with those presented in the NA64 analysis [48], where GEANT4-based simulations are employed.As shown in Fig. (5), our calculation yields consistent results with Ref. [48].In particular, in the vicinity of m A ′ ≃ 250 MeV, where the e + e − annihilation process dominates, our analytic method utilizing Eq. (C.2) successfully reproduces the NA64 results [48], for both α D = 0.5 and α D = 0.1.This demonstrates the accuracy of our method, which utilizes the analytic expressions of the positron track length distribution (with the first two generations), in predicting dark photon signal events in the region dominated by the e + e − annihilation process.

D SM backgrounds due to charged particles
We have assumed in the analysis that charged particles lead to insubstantial backgrounds compared to neutral particles, such as photons and neutrons.Here, we provide a more detailed discussion of potential background sources due to charged particles.Backgrounds can arise from pairs of charged particles generated in e + e − collisions at the primary vertex, with notable examples including π + π − and µ + µ − .To be classified as a background contribution, the charged pair must satisfy the following criteria: The negatively charged particle within the pair must effectively mimic a fully reconstructed e − , exhibiting successful reconstructions in both CDC and ECL (with nearly all of its energy deposited in ECL).Additionally, the positively charged particle should emulate a poorly reconstructed e + , featuring a robust reconstruction in CDC, but with less than 5% of its energy detected by ECL and no significant energy deposition in KLM (to avoid KLM veto).
We first discuss backgrounds due to the π + π − final state generated in e + e − collisions at the primary vertex.First note that σ(e + e − → π + π − ) at Belle II is significantly suppressed by the pion form factor, making it substantially smaller than leptonic final states, such as µ + µ − .We estimate σ(e + e − → π + π − ) by utilizing the measurement of dσ(π + π − )/dz in e + e − collisions conducted in Ref. [91], where z = 2E π / √ s with √ s = 10.52 GeV.To mimic the e + e − final state, z has to be close to 1.However, due to limitations of the method used in Ref. [91], dσ(π + π − )/dz is provided only for z < 0.98.Since dσ(π + π − )/dz decreases with increasing z, we take a conservative estimate by considering the measurement in the range of 0.95 < z < 0.98; this gives σ(e + e − → π + π − ) ≃ 0.3 pb, thus leading to ≃ 1.5 × 10 7 events for an integrated luminosity of 50 ab −1 at Belle II.Since we only consider the barrel region of the ECL, we adopt the mis-identification rate for π − as e − in the high p T region, which is ∼ 3 × 10 −5 [92].To determine the probability for a π + to deposit < 5% of its energy in the ECL, we carry out GEANT4 simulations and find that this probability is ≲ 10 −5 .Taking into account these two factors, backgrounds due to the π + π − final state are ≃ 4.5 × 10 −3 .Thus, we expect that backgrounds due to a hadronic pair are negligible.
We next discuss backgrounds due to the µ + µ − final state generated in e + e − collisions at the primary vertex, which has a production cross section of ≃ 500 pb at √ s = 10.58GeV in the ECL barrel region, thus leading to ≃ 2.5 × 10 10 events for an integrated luminosity of 50 ab −1 at Belle II.To determine the background, we carry out GEANT4 simulations for both µ − and µ + in ECL.Because the energy resolution of ECL is δE/E ≃ 2% [32], we determine the probability for a µ − to mimic an e − in ECL by computing the probability for a µ − to deposit > 98% of its energy in ECL; we find the probability to be ≲ 10 −7 by using our GEANT4 simulations.We note that one of the processes in which a µ − can mimic an e − is one where µ − emits a very hard and collinear radiation.The probability for a µ + to deposit < 5% of its energy in ECL is found to be ≃ 90%.To estimate the KLM veto efficiency on the µ + , we adopt the KLM veto efficiency on photon, which is ∼ 4.5 × 10 −4 [61].The backgrounds are reduced to a single event, after taking into account the above three factors.Given that KLM is a muon detector, we expect its veto efficiency for muons to surpass that for photons.Consequently, we anticipate that backgrounds due to µ + µ − pairs will be negligible.Backgrounds can also arise from the e + e − e + e − final state, where a pair of e + and e − is outside the acceptance of detectors.Another background can arise from the τ + τ − final state, where τ + decays into e + /µ + /π + and τ − decays into e − .However, due to energy carried away from the other particles, these two types of backgrounds do not exhibit the expected electron and positron momenta characteristic of Bhabha scattering events and can be effectively vetoed via track measurements in the CDC.
Another potential background arises when a pair of muons is produced in collisions between e + and the ECL, via the bremsstrahlung process: e + + N → e + + N + µ + + µ − , which has a cross section of ∼300 nb.This leads to ∼3400 events with 6 × 10 11 positrons hitting the ECL target.Once again, we adopt the KLM veto efficiency on photon as the efficiency on muon, which is ∼ 4.5 × 10 −4 [61]; applying this veto efficiency to both muons makes this type of background negligible.

E Sensitivity on the scalar mediator model
In this section we investigate the Belle II sensitivity on an alternative DM model from the "disappearing positron track" channel.Thus, we consider the scalar mediator model with the following interaction Lagrangian where ϕ is the scalar mediator with mass m ϕ , and ℓ is the SM charged lepton.Similar to the invisible dark photon case, we assume m χ = m ϕ /3 and g χ ≫ eϵ such that ϕ decays dominantly into DM with the decay width We follow the same procedure in section 4 to compute the Belle II sensitivity on the scalar mediator model from the "disappearing positron track" channel, which is shown in Fig. (6).Similar to the invisible dark photon model, the sensitivity on the scalar mediator For the NA64 constraint, we adopt the limits from Ref. [90] due to the bremsstrahlung process, and use the method in appendix C for the resonance region.
model is also significantly enhanced in the resonance region near m ϕ = 66 MeV.The sensitivity in the resonance region is dominated by the annihilation process, e + e − A → ϕ → χ χ, which has the following cross section: The best limit on the coupling parameter in the resonance region is ϵ ≃ 2.4 × 10 −5 , which is slightly weakened as compared to the dark photon case.We note that the change of the ϵ limit can be explained by the fact that in the narrow width approximation, the new physics signal is proportional to (2J + 1)Γ(M → e + e − ), where J is the spin of the mediator M .

F Sensitivity of the radiative Bhabha scattering process
In this section we discuss the potential sensitivity of the "disappearing positron track" channel for positrons generated by the radiative Bhabha scattering (RBS) process, e + e − → e + e − γ.Although the RBS cross section is expected to be smaller by a factor of α ≃ 1/137 than that of the Bhabha scattering (BS) process, positrons from the RBS process exhibit a broader energy range, as shown in Fig. (7), which thus have the potential to probe different parameter space in the resonance region via the annihilation process.
As discussed in section 2, the missing energy in the positron track can be obtained by comparing the momentum of the positron inferred from the CDC track with the energy deposition in the ECL.In the BS events, the energy of the positron can be further crosschecked by its polar angle and the measurement of the electron.In contrast, there is no simple relation between the positron energy with its polar angle.But one can still reconstruct the energies and momenta of all the three final state particles and then use the momentum conservation to cross-check the measurement on the positron.Note that  ) pb is the total cross section for the RBS (BS) case.The BS case is given by Eq. (2.2).The RBS case is obtained from MadGraph simulations, where all the three final state particles are required to be within the ECL barrel region and have energy above 2 GeV; moreover, the photon is required to have a angular distance ∆R > 0.1 from both the electron and the positron.
the energy resolution of ECL is σ E /E = 1.6% (4%) at 8 GeV (100 MeV), and the angular resolution of ECL at high (low) energies is 3 (13) mrad [32].Thus the angular information of the photon can be well-measured by the ECL.To compute the cross section and the positron energy spectrum in the RBS process, we simulated 10 6 RBS (e + e − → e + e − γ) events by using Madgraph [93].In our simulation, we require each of the three final state particles to have E > 2 GeV and 32.2 • < θ < 128.7 • in the lab frame; we further require that the photon does not appear in the vicinity of both the electron and the positron such that the angular distance ∆R γe ± is greater than 0.1.Under these detector cuts, we find that the total cross section is ≃160 pb, corresponding to ≃ 8 × 10 9 events with a luminosity of 50 ab −1 .To compute the number of the signal events in the invisible dark photon model, we use Eq.(4.4) and Eq.(4.5), where dσ B /dE is obtained by the RBS events in our simulation, as shown in Fig. (7).In section 3, we have found that there are ≃ 94 backgrounds due to punch-through neutrons/photons for the ≃ 6 × 10 11 BS positrons in the ECL barrel region.Thus, one expect ≃ 1 background for the ≃ 8 × 10 9 RBS events.Fig. (8) shows the 90% CL upper bound on ϵ from the RBS process, where the criterion of N s = 3 is used.The resonance region for the RBS process expands to the range of 40 MeV ≲ m A ′ ≲ 80 MeV, due to the broader range of the positron energy.The limit improves within the mass range of 40 MeV ≲ m A ′ ≲ 60 MeV, reaching a level comparable to the NA64 constraints.

Figure 1 .
Figure 1.Schematic view of the signal event in the "disappearing positron track" channel, in the transverse plane of the Belle II detector.

Figure 2 .
Figure 2. Feynman diagrams of the annihilation process e + e −A → χχ (left), and the bremsstrahlung process e + N → e + N χχ (right).The process with χχ radiated from the initial state e + is included in the analysis, but not shown here.

4 N A 6 4 L 3 Figure 3 .
Figure 3. Belle II sensitivities on the coupling parameter ϵ between the DP and SM particles with 50 ab −1 integrated luminosity as a function of the DP mass m A ′ from the "disappearing positron track" with N b = 94 (blue solid) and with N b = 1000 (blue dashed).Here we fix α D = 0.001 and m χ = m A ′ /3.Also shown are the Belle II sensitivity from the mono-photon channel with 50 ab −1 (black dashed), and the NA64 constraints (gray shaded region); see main text for details.

Figure 4 .
Figure 4. Belle II sensitivity on the coupling parameter ϵ with 50 ab −1 integrated luminosity as a function of α D from the "disappearing positron track" (blue solid), where m A ′ = 66.66 MeV and m χ = m A ′ /3.The Belle II sensitivity from the mono-photon channel with 50 ab −1 (black dashed)is rescaled from the result with 20 fb −1 in Ref.[32].The NA64 constraint with 2.84 × 10 11 electrons on target (gray shaded region) is taken from Ref.[60].

3 d is a p p e a r in g e + Nb = 9 4 N A 6 4 L = 50 ab − 1 α D = 0.001 m χ = m φ / 3 Figure 6 .
Figure 6.Belle II sensitivity on the coupling parameter ϵ as a function of the scalar mediator mass m ϕ from the "disappearing positron track" with N b = 94 and L = 50 ab −1 .Here we fix α D = 0.001 and m χ = m ϕ /3.For the NA64 constraint, we adopt the limits from Ref.[90] due to the bremsstrahlung process, and use the method in appendix C for the resonance region.

Figure 7 .
Figure 7.The normalized positron energy spectra of both the BS (e + e − → e + e − ) and the RBS (e + e − → e + e − γ) processes, where σ tot ≃ 160 (1.2 × 10 4 ) pb is the total cross section for the RBS (BS) case.The BS case is given by Eq. (2.2).The RBS case is obtained from MadGraph simulations, where all the three final state particles are required to be within the ECL barrel region and have energy above 2 GeV; moreover, the photon is required to have a angular distance ∆R > 0.1 from both the electron and the positron.

N A 6 4 L 3 Figure 8 .
Figure 8. Belle II sensitivities on the coupling parameter ϵ with 50 ab −1 integrated luminosity as a function of the DP mass m A ′ from the "disappearing positron track" channel for the Bhabha scattering process with N b = 94 (blue solid) and for the radiative Bhabha scattering process with N b = 1 (blue dashed).Here we fix α D = 0.001 and m χ = m A ′ /3.Also shown are the Belle II sensitivity from the mono-photon channel with 50 ab −1 (black dashed), and the NA64 constraints (gray shaded region).