Search for strongly interacting dark matter at Belle II

A small component of dark matter (DM) that is strongly interacting with the standard model sector is consistent with various experimental observations. Despite the small abundance, strongly-interacting DM can lead to pronounced signals in DM direct detection experiments. We study Belle II sensitivity on strongly-interacting DM that has a MeV-GeV mass and couples with electrons. By taking into account the substantial interactions between DM and electrons within detectors, we compute the ``ceiling'' of the mono-photon signature at Belle II, beyond which the mono-photon channel loses its sensitivity, and visible ECL clusters due to DM scatterings assume significance. We study two ECL signatures for strongly-interacting DM: the mono-cluster and the di-cluster channels. To carry out detailed calculations and to compare with other constraints, we consider DM models with light mediators, as they naturally lead to sizable interaction cross sections. We compute exclusion regions for the di-cluster, mono-cluster, and mono-photon channels. We find that Belle II (with currently accumulated data of 362 fb$^{-1}$) can rule out a significant portion of the parameter space above the ceilings of the constraints from various DM direct detection and neutrino experiments, for the vector mediator case with mass $\gtrsim 10$ MeV. Belle II also offers superior constraints on new light particles compared to PBH for the scalar mediator with mass $\gtrsim 10$ MeV.


I. INTRODUCTION
Dark matter (DM) is usually assumed to have a weak interaction cross section with standard model (SM) particles [1][2][3].Nonetheless, strongly-interacting DM that has a significant interaction cross section with the SM sector is allowed if it only constitutes a small DM abundance [4][5][6].One of the intriguing aspects of strongly-interacting DM lies in its potential to significantly increase the velocity of DM in astrophysical environments, thereby enhancing signals in DM direct detection (DMDD) experiments.Notable scenarios include up-scatterings induced by cosmic rays [7][8][9], diffuse supernova neutrinos [10], and blazars [11].
A small component of DM can be strongly interacting with the SM sector because various experimental constraints can be alleviated due to either the strong interaction cross section or the small DM abundance.For example, the flux of strongly-interacting DM at underground DMDD experiments is shielded by the overburden.The constraints from DM indirect detection are limited by the small DM abundance.The CMB constraints start to lose sensitivity when the DM fraction becomes less than 0.4% [12].
Particle colliders present a great opportunity for strongly-interacting DM searches, as they are not impeded by the strong absorption effects and the small DM abundance.The conventional DM signature at colliders is the missing momentum search [13][14][15][16], which, however, is not applicable to strongly-interacting DM.This is because strongly-interacting DM has a substantial interaction cross section with the detectors, and is therefore more likely to scatter with the detectors, leading to visible collider signatures, such as trackless jet signals at the LHC [17][18][19].Thus, in the missing momentum search, there exists a "ceiling" on the DM-SM interaction cross section, beyond which DM is no longer considered invisible at colliders [18].
As previous investigations have primarily focused on hadron collider signals arising from strongly-interacting DM, we study its signatures at electron-positron colliders in this paper.This is particularly relevant for sub-GeV DM that interacts with electrons.Sub-GeV DM is less likely to deposit significant nuclear recoil energy in many of the leading underground DMDD experiments that aim to detect weakly interacting massive particles (WIMPs).Consequently, electron recoils assume significance in DMDD experiments for sub-GeV DM, because of the small electron mass [20,21].The study of electron collider constraints on sub-GeV electro-philic DM becomes imperative for a comprehensive understanding.
Thus, we study constraints on strongly-interacting DM from the Belle II experiment, which is operated at √ s = 10.58GeV and is an ideal experiment to detect DM within the MeV-GeV mass range that interacts with electrons [22][23][24][25][26][27][28][29][30][31].As DM is neutral and has no track in the central drift chamber (CDC), our detection strategy relies on the electromagnetic calorimeter (ECL) detector at Belle II.With a substantial interaction with the ECL, arXiv:2312.08970v2[hep-ph] 19 May 2024 DM can scatter with atomic electrons in the ECL, resulting in recoiled electrons with significant energy.Subsequently, these recoiled electrons generate electromagnetic (EM) showers, giving rise to distinct "cluster" signatures in the ECL.We analyze two types of DM-induced cluster signals at Belle II: the mono-cluster and the dicluster.We then compute exclusion regions in the parameter space spanned by the DM-electron interaction cross section and the DM mass, for three DM signatures, including the mono-cluster, the di-cluster, and the monophoton channels.We find that the di-cluster channel typically probes the parameter space above the ceiling of the mono-photon channel.In contrast, the mono-cluster channel rarely extends into new parameter space beyond that probed by the mono-photon channel.
To compare Belle II sensitivity with various constraints from DMDD and neutrino experiments, we consider DM models with light mediators that couple to both DM and electrons.We find that Belle II can rule out a significant portion of the parameter space above the ceilings of the constraints from various DMDD [32][33][34][35] and neutrino experiments [36,37], for the vector mediator case with mass ≳ 10 MeV.For the ultralight mediator case, however, the parameter space probed by Belle II and that probed by DMDD experiments are well separated.We also find that Belle II constraints on beyond-the-SM (BSM) particles are better than those from primordial black holes (PBH), for the scalar mediator with mass ≳ 10 MeV.
The rest of the paper is organized as follows.In section II, we provide a brief discussion of the DM-induced signatures at Belle II.We then discuss in detail three different DM-induced signatures at Belle II: the di-cluster channel in section III, the mono-photon channel in section IV, and the mono-cluster channel in section V. To compute Belle II sensitivity on strongly-interacting DM, we consider DM models with four different types of light mediators in section VI.We compute experimental constraints on the light mediators in section VII, including electron g − 2, electron beam dump, BaBar, and Møller scattering.We compute Belle II sensitivities on the DM models with light mediator models in section VIII, and further compare them with constraints from DMDD and neutrino experiments in section IX.We also compare Belle II sensitivities on BSM particles with PBH constraints in section X.We summarize our findings in section XI.In appendix A, we provide the cross sections of the e + e − → χ χ, e + e − → χ χγ, and χe − → χe − processes, for the four different mediator models.

II. ECL RESPONSE OF DM INTERACTION
In this section we discuss the detector response to DM interactions with electrons.We focus on the ECL detector at Belle II, which is composed of CsI crystals.The electrons in the ECL can receive significant recoil energy when hit by the incident DM.The recoiled electron can subsequently generate EM showers in the ECL, leading to the so-called "cluster" signature if the total energy deposited across multiple cells exceeds 20 MeV [38].We note that this energy threshold is similar to the critical energy of the ECL, which is E c = 20.7 MeV [39].
To intuitively estimate the interaction rate with the ECL, we define the mean free path of DM in ECL as follows where n E is the electron number density of the ECL, and σ c is the DM-electron cross section with the electron recoil energy E r > E c .Thus, it is expected that one has to seriously take into account interactions between DM and electrons within detectors for DM searches at Belle II, when λ c ≲ L E , where L E is the length of the ECL detector.In the following, we investigate how these DM-detector interactions invalidate the missing momentum signature and give rise to new visible DM signatures within the ECL.By comparing with actual calculations, we find that the λ c ≲ L E condition serves as a fairly good criterion for determining the ceilings of the missing momentum signature.

III. DI-CLUSTER SIGNAL AT BELLE II
DM can be pair-produced at Belle II via the e + e − → χ χ process with χ denoting the DM particle, as illustrated in Fig. (1).Because DM is neutral, it leaves no track in the CDC.If the interaction between DM and the ECL is sufficiently strong, both χ and χ can result in clusters in the ECL.Thus, the observable signal for SIDM consists of two back-to-back trackless clusters in the center of mass (c.m.) frame; we denote this as the di-cluster signature.Note that the experimental reconstruction of two back-to-back trackless clusters in the c.m. frame may encounter challenges.This is attributed to the potential scenario where only partial energy of DM is deposited in the ECL.In this case, the two trackless clusters are back-to-back in the transverse plane.In our analysis, we only consider the final state DM particles in the barrel region of the ECL detector, which has better hermiticity compared to endcap regions [40].The barrel region of the ECL detector has a polar angle of 32.2 • < θ < 128.7 • in the lab frame.We compute the DM-induced di-cluster signal events in the barrel region of the ECL via where L is the integrated luminosity at Belle II, z * χ = cos θ * χ with θ * χ being the polar angle of χ in the c.m. frame, dσ χ χ/dz * χ is the differential cross section for the e + e − → χ χ process, and P χ (P χ ) is the interaction probability between DM χ (χ) and the ECL.Here, we have used the fact that z * χ ≡ cos θ * χ = −z * χ in the e + e − → χχ process.Hereafter, for a kinematic variable A in the lab frame, we use A * to denote its value in the c.m. frame.
Because Belle II collides a 7 GeV electron with a 4 GeV positron [41], the polar angle of χ (or χ) in the lab frame is related to its value in the c.m. frame via where z = cos θ with θ being the polar angle of χ (or χ) in the lab frame, z * = cos θ * denotes the z value in the c.m. frame, β = 3/11, β χ = 1 − 4m 2 χ /s with m χ being the mass of DM and s being the square of the centerof-mass energy of Belle II.The integration in Eq. ( 2) is performed such that both DM particles are within the barrel region.To do so, we multiply the integrand of Eq. (2) with a Boolean function which takes the value of unity when both f (z * χ ) and f (−z * χ ) take a value between cos(128.7 • ) and cos(32.2• ).In the massless limit, this leads to −0.745 < z * χ < 0.745.We compute the interaction probability between DM χ and the ECL via where dσ χe /dt χe is the differential cross section for the χe − → χe − process, with t χe being its t-channel Mandelstam variable.To compute P χ, one simply substitute χ with χ in Eq. ( 4).The cross sections of dσ χ χ/dz * χ and dσ χe /dt χe for different DM models are given in appendix A.
To analyze the di-cluster events, we employ the basic trigger of di-photons, since both processes lead to two clusters in the ECL that have no preceding tracks in the CDC.The basic selection condition of di-photons consists of two ECL clusters with E * > E * th = 2 GeV [41], where E * is the energy in the c.m. frame.For the χe − → χe − process, one has t χe = −2m e E r where E r is the electron recoil energy in the lab frame.Recoiled electrons then generate EM showers in the ECL, leading to clusters with energy E = E r in the lab frame.In the c.m. frame, the selection condition th leads to an upper bound on t χe : where γ = 1/ 1 − β 2 .The lower bound on t eχ is where The dominant background for the DM-induced dicluster signal arises from the SM di-photon process, e + e − → γγ, which in the c.m. frame has a production cross section [42] where z * γ = cos θ * γ with θ * γ being the photon polar angle in the c.m. frame.In order for both final state photons within the barrel region, we integrate z * γ in the range of −0.745 < z * γ < 0.745.This gives rise to a total diphoton cross section of ∼ 1.4 nb, leading to ≃ 5.1 × 10 8 (≃ 7.0 × 10 10 ) di-photon events in the SM with 362 fb −1 (50 ab −1 ) luminosity.
Unlike photons, which deposit nearly their entire energy in the ECL, DM only deposits a fraction of its energy in the ECL, if the DM mass is not small or the DM-electron cross section is not strong enough.In these regions of the parameter space, a detector cut on the energy deposited in the ECL could be instrumental in discriminating the signal process from the e + e − → γγ process.However, because our analysis spans a broad parameter space, where the DM mass ranges from MeV to several GeV and the DM-electron cross-section varies from moderate to very strong, we do not impose such a detector cut on the energy deposited in the ECL in our current study.We leave this to a future analysis where one can optimize the detector cuts for different regions of the parameter space.
As discussed in the beginning of this section, the experimental reconstruction of the back-to-back signature in the c.m. frame from the two DM clusters is affected by the partial energy deposition in the ECL.This is because one cannot obtain the true momentum of the DM particle in the c.m. frame without the precise measurement of its energy in the lab frame.Nevertheless, one can still use the back-to-back signature in the transverse plane for the two DM clusters.Therefore, the e + e − → γγγ events are also potential background events, if one of the final state photons escapes from the beam direction with small transverse momentum.
To assess the background from the e + e − → γγγ process, we generate 10 5 events by using MadGraph with a 1 MeV cut for each photon; the total cross section is ∼ 8.6 nb.We further select events with the following cuts: (1) There is one photon that either escapes in the beam direction or has an energy below 0.1 GeV.(2) The other two photons must both have E * > E * th and are located in the barrel region of the ECL with an opening angle larger than 150 • in the c.m. frame.We find that 5248 events satisfy this selection, leading to a di-photon cross section of ∼ 0.45 nb.Thus, in the e + e − → γγγ process, one expects ∼ 1.6 × 10 8 (∼ 2.25 × 10 10 ) di-photon events for the integrated luminosity of 362 fb −1 (50 ab −1 ).

IV. MONO-PHOTON SIGNAL AT BELLE II
The mono-photon signature is employed as a powerful tool for investigating dark sector particles at Belle II [23-25, 27, 28, 30] through the e + e − → χ χγ process.In the DM-induced mono-photon studies, DM is assumed to be invisible due to its extremely weak interaction with detectors.However, as the interaction increases, the assumption of null-interaction with detectors will eventually fail, leading to an exclusion region with both an upper boundary (ceiling) and a lower boundary, in the σ − m χ plane, where σ is the interaction cross section between DM and electrons.Below the lower boundary, the interaction strength is too small to yield detectable events; above the upper boundary, DM can leave a visible signal in detectors, rendering the mono-photon constraints invalid.In this section, we compute the exclusion region of the mono-photon signature at Belle II, by taking into account the interaction between DM and detectors.
The mono-photon signature at Belle II is analyzed first with a set of basic event selection cuts [41], including an ECL cluster with E * > 1.8 GeV caused by the photon, and vetoes on activities in the following three detectors: • CDC veto: no tracks with p * T > 0.2 GeV; • ECL veto: no other clusters with E * > 0.1 GeV; • KLM veto: no KLM clusters outside of the 25 • (3D, COM) cone of the signal photon. 1 In addition to the basic event selection cuts, more advanced detector cuts are employed to further reduce the SM backgrounds.To compute the exclusion region of the mono-photon signature, we adopt the low-mass signal region [41].In our analysis, we use the fitting function for the low-mas region [25]: where x ≡ E * γ /GeV with E * γ being the photon energy in the c.m. frame, θ min and θ max are the minimum and maximum angles for the photon in the lab frame, namely θ min < θ γ < θ max .About 300 mono-photon events from the SM backgrounds are expected in the low-mass region with 20 fb −1 of data [41].We rescale this number and obtain ∼5430 (∼7.5×10 5 ) mono-photon background events with 362 fb −1 (50 ab −1 ) of data.
We next discuss the mono-photon events arising from the DM process.The vetoes on the activities in the detectors are normally satisfied if the interaction cross section between DM and electrons is small.However, as the cross section increases, DM can lead to activities in the detectors, which can then be vetoed.Thus, to compute the upper boundary of the exclusion region for the monophoton signature, one has to take into account the three vetoes in the detectors.
We compute the mono-photon events in the low-mass region by using where dσ χ χγ /dE * γ dz * γ is the mono-photon differential cross section for the e + e − → χ χγ process, and P 0 is the probability of DM not inducing significant detector activities that are vetoed by the mono-photon detector cuts.For the mono-photon signal events in the low-mass region, the two final-state DM particles are typically within the ECL coverage.We need to compute the interaction probability between the two DM particles and the detectors.
In our analysis, we assume that DM is neutral and does not lead to any activity in the CDC.Thus, we only need to consider the probability of DM-induced activities in ECL and KLM.To assess P 0 , we compute the probability for DM to traverse both ECL and KLM without significant activity as follows where n K is the electron number density of the KLM, L K is the length of the KLM, and σ v χe (E χ ) is the χ−e scattering cross section.In our analysis, we evaluate σ v χe (E χ ) at the average energy To approximate the production rate of clusters with energy > 0.1 GeV in ECL and KLM, we obtain σ v χe by integrating the differential cross section with the recoil energy larger than 0.1 GeV.

V. MONO-CLUSTER SIGNAL AT BELLE II
In this section, we discuss another possible signature due to DM interactions with detectors: the mono-cluster signature.The mono-cluster signature typically occurs with a moderately strong DM-electron interaction cross section so that in the e + e − → χ χ process, only one DM particle leads to an ECL cluster, while the other DM penetrates both ECL and KLM without any trace.
The DM-induced mono-cluster events share a detector response closely resembling that of the mono-photon events.In both cases, there is a trackless cluster with a substantial amount of energy deposition in the ECL.Thus, in the mono-cluster analysis, we adopt the monophoton trigger, the basic event selection criteria of the mono-photon channel, and the low-mass signal region.
We compute the number of mono-cluster events in the low-mass region via where P χ is the DM interaction probability with the ECL, as given in Eq. ( 4), and P 0 is the DM punchthrough probability, as given in Eq. (11).Here E χ is the energy of χ in the lab frame, which is related to The factor of 2 accounts for the fact that each DM can deposit energy leading to an ECL cluster.

VI. DARK MATTER MODELS
The DM-induced di-cluster and mono-cluster events at Belle II depend on both the DM production process and the DM-electron scattering process, as illustrated by the two diagrams in Fig. (2).In contrast, DMDD experiments usually only probe the DM-electron scattering process.This prevents one from making a modelindependent comparison between the Belle II constraints on strongly-interacting DM with DMDD experiments.Instead, one has to consider concrete DM models to compare constraints from these experiments.Because Belle II is an electron-positron collider, we focus on electro-philic DM models, where DM only interacts with electrons in the SM sector.
To generate substantial DM-induced di-cluster and mono-cluster signal events at Belle II, a large DMelectron interaction cross section is needed.Since DM is neutral, we take the typical interaction cross section of a nucleus, ∼ 1 barn, as the benchmark for strongly-interesting DM.2Such a large DM-SM interaction cross section can be realized in models where there is a rather light mediator connecting DM and SM particles [17,18,[43][44][45][46][47].See also Ref. [48] where the mediator mass at present is significantly reduced compared to early universe by a phase transition in the dark sector. 3n our analysis we focus on DM models with light mediators.To gauge the required mass range of the mediators for a substantial cross section, we consider a simple model where a mediator couples to DM and electron.For nonrelativistic DM, the DM-electron cross section in DMDD experiments is where q (m) is the momentum (mass) of the mediator, µ χe = m χ m e /(m χ + m e ) with m χ (m e ) being DM (electron) mass, and g χ (g e ) is the coupling between the mediator and DM (electron).In the case where g e ≃ g χ ≃ 1, m χ ≳ m e and q 2 ≪ m 2 , we find that the mediator mass has to be as low as ∼ 1 MeV to yield σ χe ≃ 1 b.Thus, we focus on electro-philic DM models, where DM interacts with electrons via a mediator, which can be either spin-one or spin-zero.We consider the following interaction Lagrangian: where Z ′ µ and ϕ are the spin-one and spin-zero mediators, respectively.Here we consider four different cases: vector (V), axial-vector (A), scalar (S), and pseudo-scalar (P).Here, g i χ (g i e ) is the coupling to DM (electron), where the superscript i denotes the four different mediators.
The interpretation of electron g−2 data depends on the experimental determination of the fine structure constant α.By using the α value measured with rubidium (Rb) atoms [67] and cesium (Cs) atoms [68], it is shown in Ref. [69] that the new electron g − 2 measurement [70] has a 2.2 σ and -3.7 σ deviations from the SM prediction [71]: ( Given the intricate aspects of this measurement, we adopt a cautious approach in constraining new physics models: We add a 2σ to the central deviations in Eqs.(23)(24) and then use the largest deviation to constrain new physics contributions regardless the sign.Thus, the new physics contributions should satisfy Fig. (4) shows the constraints on the four types of mediators in Eqs.(14)(15)(16)(17).

B. electron beam dump experiments and BaBar
Light mediators that couple to electrons can be searched for both in electron beam dump experiments and in electron-positron colliders.Unlike the electron g − 2 constraint, which is insensitive to the interaction between DM and the mediator (at least in the leading order), the experimental constraints from electron beam dump experiments and electron colliders are very sensitive to the invisible decay width of the mediator.Thus, we discuss the constraints in two cases: (1) the invisible mode, where the mediator dominantly decays into DM; (2) the visible mode, where the mediator dominantly decays into SM particles.

Invisible mode
The invisible mode can occur in the case where m > 2m χ and g χ ≫ g e .For the mediator mass in the range of ∼ (1 − 100) MeV, the leading constraints come from the missing momentum search at NA64 [57], as shown in the left panel figure of Fig. (4).The NA64 constraints on the electron coupling g e are about one order of magnitude stronger than the electron g − 2 constraints.
We note that the NA64 constraints [57] are analyzed under the assumption that DM does not interact with detectors, thereby resulting in a missing momentum signature.However, this assumption becomes invalid in the case of strongly-interacting DM, where DM exhibits a substantial interaction cross section with SM particles and is likely to be absorbed by the NA64 detectors.Therefore, the NA64 constraints shown in the left panel of Fig. (4) are not applicable to strongly-interacting DM. 5

Visible mode
The visible mode can occur in the case where 2m χ > m.For the mediator mass in the range of ∼ (1 − 100) MeV, the primary decay channel is the e + e − final state.In this case, the leading constraints come from electron beam dump experiments and electron-positron colliders.
The right panel figure of Fig. (4) shows the constraints on the visible signals arising from the spin-1 mediator that has a vector coupling to electrons.Here, we rescale the dark photon constraints [58,61] that arise only from electron couplings; the re-scaling is done via g e = eϵ where ϵ is the mixing parameter in dark photon models [62,[73][74][75][76] [57].Four different mediators are shown: vector (solid), axialvector (dashed), scalar (dot-dashed), and pseudo-scalar (dotted).Right: Constraints on the spin-1 mediator that has a vector coupling to electrons g V e and decays dominantly into e + e − (visible mode).The electron g − 2 constraint is shown as the gray line.Other constraints are shown as shaded regions: NA64 [58], E774 [59], E141 [60], and BaBar [61]; these constraints are obtained by re-scaling the dark photon constraints [62] via ge = ϵe with e being the QED coupling and ϵ being the mixing parameter of the dark photon.Here we also show the Belle II sensitivity (dashed) with 50 ab −1 of data, which is obtained by re-scaling the dark photon constraints in Refs.[41,63].
E141 [60], and E774 [59]. 6 Remarkably, there is a large portion of parameter space consistent with various constraints: 5 MeV ≲ m ≲ 100 MeV and 10 −5 ≲ g e ≲ 10 −3 .The beam dump experiments lose sensitivity in this region primarily because of the short lifetime of the mediators, so that the mediators are likely to decay in the dump.The termination of the BaBar constraints at ∼ 20 MeV is largely due to the large SM background in the low invariant mass bins of the e + e − pair in the process of e + e − → γZ ′ → γe + e − .

C. Møller scattering
Because the contributions to the electron g−2 from the vector and axial-vector couplings have opposite signs 7 , the spin-1 mediator can have a vanishing contribution to the electron g−2 if both vector and axial-vector couplings are present.However, such a scenario is strongly constrained by the Møller scattering, e − e − → e − e − , from the SLAC E158 experiment [83].For the Z ′ mass ≲ 100 MeV, the E158 constraint is roughly mass-independent: 6 We note that many accelerator constraints on the dark photon rely on its production in meson decays: for example, π 0 → γA ′ , η → γA ′ , and ∆ → N A ′ in HADES [78], π 0 , η → γA ′ decays in PHENIX [79], π 0 decays in NA48 [80], and ϕ → ηA ′ decays in KLOE [81].Therefore, these constraints are not applicable to light mediators that only couple to electrons. 7In the leading order, there is no interference term between g V e and g A e in the electron g − 2 calculation; see e.g., [65] [82].
|g V e g A e | ≲ 10 −8 [82,84].Thus, it is difficult to have both vector and axial-vector couplings sizable simultaneously.In fact, in the democratic setting, namely g V e ∼ g A e , the SLAC E158 experiment imposes a stronger constraint than the electron g − 2 for the mediator mass ≳ 10 MeV; for the mediator mass ≲ 3 MeV, however, the electron g − 2 constraint can be somewhat alleviated with g V e ∼ g A e .Note that for a spin-one mediator that couples with electrons, parity-violating effects in Møller scattering arise when both vector and axial-vector couplings are present.Thus, in a new physics model with two electrophilic spin-one mediators, where one mediator only has a vector coupling with electrons and the other mediator only has an axial-vector coupling with electrons, constraints from the parity-violating asymmetry measurement in Møller scattering are absent.Moreover, cancellations to the electron g − 2 between the two mediators also render the electron g − 2 constraint unimportant, if the vector and axial-vector couplings are comparable.Therefore, in this case, the couplings between the two mediators and electrons can be both sizable.

VIII. BELLE II SENSITIVITY
In this section we analyze constraints from three different channels at Belle II: the di-cluster, mono-cluster, and mono-photon channels.We compute the 90% C.L. limits for all the three channels via the criterion of χ 2 = N 2 s /σ 2 b = 2.71 by assuming the Gaussian distribution, where N s is the number of signal events and  3) scalar (lower-left), (4) pseudoscalar (lower-right).The black dot-dashed curves denote λc = LE where LE = 37 cm.For the di-cluster channel, the dashed and solid lines merge due to the dominance of the systematic uncertainty over the statistical uncertainty in this channel.σ b is the uncertainty of the background, which includes both the statistical uncertainty and the systematic uncertainty.For the statistical uncertainty, we use σ b = √ N b , where N b is the number of the background events.For the systematic uncertainty in the di-cluster channel, we adopt σ b = 1%N b , which is the systematic uncertainty of the di-photon events [86].For the mono-cluster and mono-photon channels, we have neglected the systematic uncertainty associated with mono-photon events in the SM, due to the scarcity of literature on such systematic uncertainties and the relatively small number of background events.86), for both the mono-photon and mono-cluster channels, the exclusion regions typically exhibit both a lower boundary and an upper boundary; the parameter space below the lower boundary or above the upper boundary (ceiling) is not constrained.In contrast, the constraints from the dicluster channel manifest as exclusion regions with only lower boundaries, namely, without ceilings.
The ceilings in the mono-photon and mono-cluster channels arise when DM starts to have a substantial interaction cross section with detectors.The locations and shapes of these ceilings can also be estimated via λ c = L E with the energy of the incident DM being 5 GeV, where L E = 37 cm is the length of the ECL, as shown in Fig. (5) and Fig. (6).Significantly below the λ c = L E curve, DM rarely interacts with the ECL, manifesting itself as a missing signature; significantly above the λ c = L E curve, the multiple-scatterings become important.
The di-cluster constraints with the current data of L = 362 fb −1 [85] can already probe a significant portion of the parameter above the ceilings of both the monophoton channel and the mono-cluster channel.We find that increasing the current Belle II data to the total expected data of L = 50 ab −1 only improves the lower boundaries of the di-cluster constraints, which, however, have already been ruled out by either the mono-photon channel or the mono-cluster channel.
For the pseudo-scalar mediator case, the di-cluster constraints can exclude the entire parameter space above the ceilings of the mono-photon and mono-cluster channels.However, for the other three mediators, there is a significant portion of parameter space unconstrained by the di-cluster channel, above these two ceilings.We note that the parameter space that is probed by the mono-cluster channel predominantly falls within the mono-photon exclusion region; in instances where it extends beyond the mono-photon ceiling, it is concurrently constrained by the di-cluster channel.
We note that Belle II constraints are insensitive to the mediator mass m, if m ≪ √ s where √ s ≃ 10.5 GeV is the colliding energy of Belle II.This explains that the Belle II constraints given in Fig. (5) and Fig. ( 6) are nearly the same, except the lower boundary of the monophoton exclusion region, in the parameter space where the mediator can decay into a pair of the DM.As shown in Fig. (5) where the mediator mass is 10 MeV, the lower boundary of the mono-photon exclusion region for m χ < 5 MeV is shifted downward significantly compared to that for m χ > 5 MeV.This is because when m χ < 5 MeV, DM can be produced via the resonance of the mediator, thus boosting its production cross section.
The mono-photon ceilings only exist for small DM mass and start to disappear when m χ ≳ 0.2 GeV, as shown in Fig. (5) and Fig. (6).However, this is primarily due to the single-scattering assumption in our calcu-lation, in which the maximum value of the electron recoil energy becomes less than 0.1 GeV, if the DM mass exceeds 0.2 GeV.As the interaction cross section increases, multiple scatterings become important.Even if the energy deposited in the ECL in one scattering is low, the sum of the recoil energies in all DM scatterings with the ECL can still be significant, leading to an ECL cluster with energy larger than 0.1 GeV.Therefore, the ceiling for the case of m χ ≳ 0.2 GeV should appear with a sufficiently large interaction cross section. 9In our analysis, we have neglected multiple scatterings due to the complexity.We leave that to a future study.
Both di-cluster and mono-cluster channels lose sensitivity when m χ ≳ 0.1 GeV.This is also largely due to the single-scattering assumption used in our analysis.We require the cluster energy to be larger than 2 GeV (1.8 GeV) for the di-cluster (mono-cluster) channel, which cannot be realized in a single elastic scattering for a DM with m χ ≳ 0.1 GeV and E χ ≃ 5 GeV.
While the Belle II channels analyzed in this section constrain the product of g e and g χ , the experimental constraints analyzed in section VII only limit g e .As shown in Fig. ( 4), the electron g − 2 constraints apply to both the invisible and visible modes of the mediators and lead to a limit of g e ≲ (10 −5 − 10 −4 ), for the mediator mass ∼ (10 − 100) MeV.Combining these two types of constraints, we find that the mono-photon ceilings or the lower boundaries of the di-cluster exclusion regions in Fig. (5) and Fig. (6) are already in the parameter region of g χ ∼ (10 3 − 10 4 ) where perturbative calculations start to fail.However, as discussed in section VII, couplings to electrons can be quite substantial in a NP model with two spin-one mediators, where one mediator only has vector couplings and the other only has axial-vector couplings; in such a model, electron g − 2 constraints can be mitigated by cancellations between the two mediators, and NA64 constraints can be alleviated by the strong DMelectron interaction cross section.To summarize, we find that although it is challenging to perturbatively analyze the strongly-interacting DM signals at Belle II in a simple mediator model, there are viable models with more than one mediator.

A. Search for mediators via visible channels
In addition to the DM processes, the light mediators can also be searched for at Belle II via the "visible" channels.Unlike the DM processes, which probe both g χ and g e , the visible channels only probe g e .Here we estimate the Belle II sensitivities on new light spin-one mediators with vector couplings, from the e + e − final state via the following two processes: (1) the radiative return process, e + e − → γZ ′ → γe + e − , in which the mediator Z ′ is produced nearly on-shell and subsequently decays into an electron-positron pair; (2) the t-channel process mediated by Z ′ that contributes to the Bhabha scattering.
The mediator in the radiative return process can be probed via the resonance search where the invariant mass of the final state electron-positron pair is reconstructed.The right panel figure of Fig. (4) shows the Belle II sensitivity in this channel (with 50 ab −1 of data), which is obtained by properly re-scaling the dark photon constraints in Refs.[41,63].We note that the expected Belle II limits (as well as the BaBar constraints in this channel, as also shown in the right panel figure of Fig. ( 4)) only probe mediators with mass ≳ 20 MeV, and are not relevant to our current study which focuses on mediators with mass ≲ 10 MeV.
Measurement of Bhabha scattering data can impose constraints on the new physics process of e + e − → e + e − mediated by a t-channel Z ′ .The leading new physics contribution arises from the interference term between the SM process and the Z ′ process: N s ≃ 2(g V e /e) 2 N SM , where N SM is the number of SM background events.To compute the 90% CL limits, we use the criterion of χ 2 = N 2 s /σ 2 b = 2.71, which is the same as the DM channels.We adopt 0.6%N SM [86] for the systematic uncertainty; the statistical uncertainty is negligible due to the large number of Bhabha scattering events.We find that the Belle II sensitivity at 90% CL from Bhabha scattering is g V e ≲ 0.02.We note that the constraints from Bhabha scattering are significantly weaker than those shown in the right panel of Fig. (4).

IX. COMPARISON WITH DMDD
In this section we compare the Belle II constraints with DMDD constraints (also neutrino experiments) that are due to electron recoils.The Belle II constraints consist of the limits from three channels: di-cluster, monocluster, and mono-photon.We consider the vector mediator model, given in Eq. ( 14), as the benchmark model in this section.To compare constraints with DMDD, we compute the reference cross section [20,88] σe where m Z ′ is the mediator mass, and |q| = αm e is the magnitude of the mediator's 3-momentum evaluated at the typical atomic scale.
Although Belle II constraints on the product of the couplings, g χ g e , do not exhibit a significant dependence on the mediator mass across most of the parameter space analyzed in this study, the DMDD constraints are very sensitive to the mediator mass.This is evident in the reference cross section defined in Eq. (26).To properly interpret the Belle II constraints in the context of DMDD experiments and other experimental constraints that rely  2) mono-photon (red), (3) monocluster (green).The gray shaded regions indicate the DMDD constraints: the lower boundary on the left panel is obtained by combining constraints from Xenon10 [32], Xenon1T [33], and SENSEI [87]; the upper boundary (ceiling) on the left panel and the exclusion region on the right panel are adopted from Ref. [35] (with electronic stopping only).Other constraints are shown as upper limits only (black solid curves): Xenon1T limits on boosted DM from solar reflection (denoted as "SRDM" here) [34]; and Super-K limits on cosmic ray boosted DM (denoted as "CRDM" here) [37] for mediator masses at 10 MeV (left panel) and eV (right panel).
on the DM-electron scattering process, we consider two distinct scenarios for the mediator mass: the ultralight mediator case, namely m Z ′ ≪ αm e , and the m Z ′ = 10 MeV case.In the m Z ′ = 10 MeV case, one has σe ≃ In the ultralight mediator case, one has σe ≃ The q-dependence of the DM-electron cross section can be encoded in the DM form factor [20,88].These two types of mediators have distinct DM from factors, which are F DM (q) = (αm e /|q|) 2 and F DM (q) = 1 for the m Z ′ ≪ αm e and m Z ′ = 10 MeV cases, respectively. 10e compare the Belle II constraints (using the current data of L = 362 fb −1 ) with the constraints from DMDD and neutrino experiments, for the m Z ′ = 10 MeV and m Z ′ ≪ αm e cases in the left and right panels of Fig. (7), respectively.
In the m Z ′ = 10 MeV case (the left panel figure of Fig. (7)), the lower boundary of the DMDD exclusion region consists of constraints from Xenon10 [32], Xenon1T [33], and SENSEI [87].There are also constraints on boosted-DM, including Super-K limits on cosmic rays boosted DM [37], and Xenon1T limits on DM due to solar reflection [34].We find that for the m Z ′ = 10 MeV case, the lower boundary of the Belle II mono-photon constraints is higher than the combined constraints from DMDD/neutrino experiments.We note that, however, the mono-photon constraints can easily surpass the DMDD constraints for a mediator with a mass larger than 10 MeV [28].This is because the Belle II constraints on σe are proportional to 1/m 4 Z ′ in the mass range of αm e ≪ m Z ′ ≪ √ s = 10.58GeV.Ceilings also exist in DMDD constraints.This is because DM has to penetrate the overburden (including both rock and the atmosphere) of the underground DMDD labs.If DM has a sufficiently large interaction cross section with SM particles, it gets absorbed before reaching the underground detectors. 11This also applies to constraints from neutrino experiments such as Super-K, MiniBooNE, and DUNE [37,104].Ref. [35] has analyzed the DMDD ceilings for DM-electron interactions, and found that the ceilings occur at σe ≃ 11 Experiments conducted at the top of the atmosphere or on Earth's surface, such as XQC [89,90], CSR [91], and RRS [92], provide a more promising avenue for investigating stronglyinteracting DM.See also Ref. [93] for a proposed experiment in the upper stratosphere aimed at mitigating the overburden.Recent analyses of constraints from these experiments on the interaction cross section between nuclei and strongly-interacting DM can be found in Refs.[94][95][96][97][98][99][100][101][102][103].
10 −22 (10 −25 ) cm 2 in the case where the DM form factor is F DM (q) = 1 (F DM (q) = (αm e /|q|) 2 ).As shown in the left panel figure of Fig. (7), for the m = 10 MeV case, Belle II is capable of ruling out almost the entire parameter space above the DMDD ceiling, except the narrow mass window of ∼(100-200) MeV. 12n the m Z ′ ≪ αm e case, the Belle II constraints probe a completely different parameter space compared to DMDD, as shown in the right panel figure of Fig. (7).The lower boundary of the mono-photon exclusion region is approximately four orders of magnitude larger than the ceiling of the DMDD exclusion region.Notably, the parameter region of 10 −25 cm 2 ≲ σe ≲ 10 −20 cm 2 for MeV-GeV DM is currently allowed, in the ultralight mediator case.FIG. 8. Belle II 90% C.L. sensitivity (combining constraints from mono-photon, mono-cluster, and di-cluster channels) on the reference cross section of the χ-e interaction in the scalar mediator case: m ϕ = 100 MeV (brown shaded region) and m ϕ = 10 MeV (green curve).We use 362 fb −1 of data for Belle II and FDM = 1 for the DM form factor. Super-K limits on χ from PBH evaporation are also shown, for the scalar EFT operator χχēe/Λ 2 with MPBH = 7.9 × 1014 g [36].
Light BSM particles with a mass below the Hawking temperature can be produced in PBH, leading to detectable signals at terrestrial experiments, if they have particle interactions with the SM sector [36,105,106].Ref. [36] analyzed the XENON1T and Super-K constraints on PBH-evaporated BSM particles that interact with electrons, for an EFT interaction of χχ lℓ/Λ 2 , where ℓ denotes the SM leptons.Such an EFT operator can be obtained by integrating out a scalar mediator with a mass above the typical energy of PBH, which is at most ∼ 10 MeV [36].To compare Belle II constraints with PBH constraints, we thus consider scalar mediator models with mediator masses of 10 MeV and 100 MeV, as shown in Fig. (8). 14The Belle II constraints consist of those from the mono-photon, mono-cluster, and dicluster channels.We find that the Belle II sensitivity with the current data (362 fb −1 ) surpasses the PBH constraint by more than four orders of magnitude, for the mediator mass of m ϕ = 100 MeV.If the mediator mass is lowered to 10 MeV, both the floor and the ceiling of the Belle II constraints move upward (as indicated by the black arrows in Fig. ( 8)); the resulting Belle II sensitivity is at the same level of the PBH constraints. 15

XI. SUMMARY
This study explores the potential of Belle II experiment to detect strongly-interacting DM that couples to electrons.As the interaction strength between DM and electrons increases, DM can no longer evade detection but instead deposit significant energy in the calorimeter, leading to ECL clusters.Thus, for strongly-interacting DM, the mono-photon channel becomes invalid.Instead, the visible signatures in the ECL, including di-cluster and mono-cluster channels, assume significance.
To illustrate the sensitivity of Belle II to stronglyinteracting DM, we consider several light mediator models with mass less than 10 MeV, including the vector mediator, the axial-vector mediator, the scalar mediator, and the pseudo-scalar mediator.We analyze constraints on such mediators from various experiments, including electron g − 2, electron beam dump experiments, electron-positron colliders, and Møller scattering.We find that these experiments impose a strong constraint on the coupling between the light mediator and electrons.Nonetheless, there is significant parameter space allowed: g e ≲ 10 −4 for mediator mass ∼ 10 MeV.We note that couplings to electrons can be substantial if there is more than one mediator.
We calculate the exclusion regions for the three different channels at Belle II: di-cluster, mono-cluster, and mono-photon.While the mono-cluster and mono-photon channels exhibit both a lower boundary and an upper boundary (ceiling), the di-cluster channel is characterized solely by a lower boundary.Remarkably, we find that the lower boundaries of the di-cluster channel typically reside below the upper boundaries of the monophoton channel.Therefore, the integration of these two channels proves effective in exploring the entire parameter space above the lower boundaries of the mono-photon exclusion region.
We further compare Belle II constraints with constraints from DMDD/neutrino experiments, as well as PBH constraints.We find that the di-cluster channel at Belle II can probe the parameter space above the ceilings of DMDD constraints for the vector mediator case with m Z ′ = 10 MeV; similar conclusions can be extended to mediator models with 10 MeV < m Z ′ ≪ √ s.For the m Z ′ ≪ αm e case, however, the parameter space probed by Belle II is well seperated from that probed by DMDD experiments.We also find that Belle II constraints on light BSM particles are stronger than PBH constraints, for the scalar mediator in the mass range of ∼ (10 − 100) MeV.
The differential cross sections of the e + e − → χ χ process for the four mediators are given by where z * χ = cos θ * χ with θ * χ being the polar angle of χ in the c.m. frame, s is the square of the center-of-mass energy, m Z ′ (Γ Z ′ ) is the mass (decay width) of the spin-1 mediator, m ϕ (Γ ϕ ) is the mass (decay width) of the spin-0 mediator, χ /s with m χ being the DM mass, and β e = 1 − 4m 2 e /s with m e being the electron mass.Note that one can take β e = 1 at Belle II.

Mono-photon cross section
The differential cross sections of the mono-photon signal from the e + e − → χ χγ process (with the phase space of DM particles integrated out) are given by,

Decay width
At the tree level, the mediator (Z ′ or ϕ) can decay into either χ χ or e + e − , when kinematically allowed.Thus, we compute the total decay width via where the subscript denotes the final state.The invisible decay widths for the four mediator models are given by Γ χ χ = Γ (Z ′ → χ χ) = g V

FIG. 1 .
FIG.1.Schematic view of the signal event in the di-cluster channel at the Belle II detector.

FIG. 4 .
FIG.4.Left: Constraints on mediators that have an electron coupling ge and decay dominantly into DM (invisible mode): electron g − 2 constraints (red) and NA64 constraints (blue)[57].Four different mediators are shown: vector (solid), axialvector (dashed), scalar (dot-dashed), and pseudo-scalar (dotted).Right: Constraints on the spin-1 mediator that has a vector coupling to electrons g V e and decays dominantly into e + e − (visible mode).The electron g − 2 constraint is shown as the gray line.Other constraints are shown as shaded regions: NA64[58], E774[59], E141[60], and BaBar[61]; these constraints are obtained by re-scaling the dark photon constraints[62] via ge = ϵe with e being the QED coupling and ϵ being the mixing parameter of the dark photon.Here we also show the Belle II sensitivity (dashed) with 50 ab −1 of data, which is obtained by re-scaling the dark photon constraints in Refs.[41,63].

Fig. ( 5 )
shows the Belle II constraints for mediators with a mass of 10 MeV; Fig.(6) shows the Belle II constraints for mediators with a mass of 1 MeV.Because the Belle II constraints presented here have a rather weak dependence on the mediator mass in the mass range of ≲ 1 MeV, the constraints given in Fig.(6) are applicable to mediators with mass < 1 MeV.We compute the constraints both with the current data of L = 362 fb −1[85], and with the total expected data of L = 50 ab −1 .As shown in both Fig. (5) and Fig. (

2 = 2 |q|=αme|F DM (q)| 2 . 2 χeπ(m 2 +
µ χe = m χ m e /(m χ + m e ) is the reduced mass.Here, we have assumed that both DM and electron are nonrelativistic, and the q-dependence is only in the matrix element M χe , which can be factorized as |M χe (q)| |M χe (q)| We provide the reference cross sections for the four mediator models as follows σe ≃ C(g e g χ ) 2 µ |q| 2 ) 2 e E 2 χ + 2s χe t χe + t 2 − 2t χe (s χe − 4m e E χ ) + t 2 χe and s χe are the Mandelstam variables of the DM-electron scattering process, which are given bys χe = 2m e E χ + m 2 e + m 2 χ , (A13) t χe = −2m e E r ,(A14)where E χ and E r are the energy of the initial state DM and the recoil energy of the electron, respectively, in the lab frame (the rest frame of the initial state electron).If one is interested in electron recoil events above the energy threshold of E th r , the total cross section can be obtained by χe − (m χ + m e ) 2 s χe − (m χ − m e ) 2 .(A16) 4. Reference cross section in DMDD