Search for a dark leptophilic scalar produced in association with τ + τ − pair in e + e − annihilation at center-of-mass energies near 10.58 GeV

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The astrophysical observation of the dark matter in the universe [1], and measured excess over Standard Model (SM) expectations in the anomalous magnetic moment of the muon, (g −2) µ [2], could be signatures of new physics beyond the SM.Recently, models with a dark leptophilic scalar (ϕ L ), which couples directly only to leptons [3,4], have been introduced at mass scales substantially lighter than the weak scale.Models, in which a generic dark scalar (ϕ) couples to quarks as well, are strongly constrained by the existing limits on the decays through flavor-changing neutral current, such as, B → Kϕ and K → πϕ [5,6].The leptophilic models evade most of such existing bounds with a minimal scenario that includes a mixing between ϕ L and the SM particles [7,8].These models can explain the observed excess in measured (g − 2) µ [9][10][11], violation of lepton flavor universality [12,13], or recent hints of new physics in a modelindependent framework [14].
In this model, mixing between ϕ L and the Higgs boson produces couplings proportional to fermion masses, arXiv:2207.07476v3[hep-ex] 17 Feb 2024 described by the following term in the Lagrangian [10]: where ξ denotes the strength of flavor-independent coupling to leptons (ℓ) with mass m ℓ , and v = 246 GeV [15] is the vacuum expectation value of the Higgs field.
The data used in this analysis was recorded by the Belle experiment from the collision of 8 GeV electrons with 3.5 GeV positrons at the KEKB collider [16].The Belle detector, a large-solid-angle magnetic spectrometer, is described in detail elsewhere [17].Outward from 15 mm radius beam pipe [18], it consists of a four-layer silicon vertex detector (SVD), a 50-layer central drift chamber (CDC), an array of aerogel threshold Cherenkov counters (ACC), a barrel-like arrangement of time-of-flight scintillation counters, and an electromagnetic calorimeter comprised of CsI(Tl) crystals (ECL), all located inside a superconducting solenoid coil that provides a 1.5 T magnetic field.Clean electron identification is obtained by combining the responses of the ECL, CDC, and ACC detectors, while muons are identified by CDC and resistive plate chambers in the instrumented iron flux-return located outside the coil.
The data-set corresponds to a luminosity of 626 fb −1 collected after the upgrade of the SVD sub-detector in October 2003.Out of these, 562 fb −1 was collected at the Υ (4S) resonance and the remaining at a centerof-mass (c.m.) energy 60 MeV below the resonance.
An important aspect of this analysis, in which it differs from the previous search performed by the BABAR experiment [32], is background modeling using MC samples and data in control regions.We use the multivariate analysis technique to enhance the presence of the signal over the background, as well as to define control regions, corresponding to regions enriched with each background component.The normalizations of the backgrounds are obtained by fitting the different MC components to data in different control regions.Studies of e + e − and µ + µ − invariant masses as the discriminating variables are carried out by blinding the signal region until the optimization of the selection criteria is complete.In the final set of fits in the signal region, a uniform shape with Poisson fluctuations is added as an additional component to account for background from the unsimulated SM four-lepton processes e + e − → τ + τ − e + e − and e + e − → τ + τ − µ + µ − .
We look for events with four tracks, each selected with a systematic uncertainty on the tracking efficiency of 0.35% [19].To suppress mis-reconstructed and beaminduced tracks, we require the transverse (dr) and longitudinal (|dz|) projection of the distances of the closest approach to the interaction point (IP) be smaller than 10 mm and 50 mm, respectively.This selection criteria probes the parameter space with ξ ∼ 1, which corresponds to a decay length of ϕ L less than ∼ 10 mm.For the m ϕ L < 0.1 GeV region, decay lengths can be larger than 10 mm.In such cases, we require looser criteria of: dr < 50 mm, and |dz| < 50 mm.
The net charge of the event is required to be zero.In the ϕ L → e + e − (µ + µ − ) channel, we require at least one track to be identified as e + (µ + ) and one track to be identified as e − (µ − ) by our particle identification (PID) system.Correction factors for efficiency and the misidentification rates are obtained using control samples from data, and applied to MC.The precision of these correction factors is included as a systematic uncertainty.
We reconstruct ϕ L candidates by fitting each pair of e + e − or µ + µ − to a common vertex, while the remaining tracks in the event come from 1-prong decays of the two τ leptons.Between 25% and 50% of the signal events have more than one ϕ L candidate, with the average multiplicity decreasing from 1.7 to 1.3 at higher m ϕ L values.We choose the candidate with the smallest opening angle in the laboratory frame to ensure there is exactly one ϕ L per selected event.The efficiency to select the true ϕ L candidate per signal event is more than 98% (83%) for ϕ L → e + e − (µ + µ − ) channel.
The major background for ϕ L → e + e − search comes from e + e − → τ + τ − events, where one of the τ ± leptons decays into a ρ ± producing a π 0 , which decays into e + e − γ final state, thereby faking the event topology of the signal.The major background for ϕ L → µ + µ − search till m ϕ L = 1 GeV comes from e + e − → τ + τ − events, where one of the τ ± leptons decay contains 3 charged pions, some of which are misidentified as muons.Beyond m ϕ L = 1 GeV, the two muons mostly come from semileptonic decays of heavy quarks in e + e − → qq events.
To suppress most of the Bhabha, µ + µ − and twophoton backgrounds, we use rectangular selection criteria on the 2-dimensional plane: where the missing mass (M miss ) is evaluated using the four tracks and all neutrals detected in the final state, and θ CM miss is the polar angle of the missing momentum in the c.m. frame.
To suppress the remaining backgrounds, we train a multiclass boosted decision tree (BDT) for each channel, using the GradientBoostingClassifier model available in scikit-learn [33].We define five BDT scores to discriminate between the signal and four different types of backgrounds: τ + τ − , e + e − (µ + µ − ), q q and BB in The top four variables ranked according to their feature importance in the BDT for the ϕ L → e + e − channel are the thrust in the c.m. frame [34], the opening angle between the daughters of ϕ L candidate in the laboratory frame, M miss and the transverse distance of the vertex of the ϕ L candidate from the IP.The top four variables for the ϕ L → µ + µ − channel are the invariant mass of τ + and τ − daughter tracks, thrust, M miss and the total energy of the reconstructed ϕ L candidate in the laboratory frame.The other variables used in the BDT are: event shapes (ratios of Fox-Wolfram moments [35]), missing particles (visible energy, and direction of missing momentum in the c.m. frame), ϕ L candidate (transverse momentum of daughter particles), PID (number of leptons, pion-kaon discriminator for ϕ L daughters), neutral activity (number of π 0 , the sum of energy deposited in ECL not associated with a track), and invariant mass of the system formed by the ϕ L candidate and its nearest photon.
In order to understand the background processes, we define the general control region (GCR) with negligible signal contributions for each channel, by requiring the signal score to be less than 0.5.The di-lepton mass distributions in the GCRs are shown in Fig. 2. We obtain the scale factors for each background component via a simultaneous fit across both channels.In order to estimate the uncertainty of the scale factors, we define a special control region (SCR) for each of those four backgrounds by requiring the corresponding BDT score to be greater than 0.5.We take the difference between the scale factors obtained from GCR and SCR as the uncertainty of each background contribution, except for the two-photon background, where the uncertainty is purely statistical.For the dominant background processes of τ + τ − , the scale factor is consistent with unity, with 6% (11%) relative uncertainty in e + e − (µ + µ − ) channel.We define the signal region (SR) with signal score > 0.95 (0.65), as an optimum choice that maximizes the sensitivity for the e + e − (µ + µ − ) channels, where the signal efficiency varies between 0.5% to 7.5% (5% to 17%).The distributions of e + e − and µ + µ − invariant mass in SR are shown in Fig. 3, along with the MC backgrounds (stacked histograms) and signal distributions (red histograms).The ratio between the data and the sum of the MC backgrounds is shown at the bottom of each figure .No obvious narrow peak structure is observed, except for the J/ψ signal in the µ + µ − channel.A slight excess of data above the MC samples in both channels is expected due to the above-mentioned unsimulated processes.
We search for narrow peaks in e + e − (µ + µ − ) invariant mass distributions by performing binned maximum likelihood fits, where the likelihood is defined as a product of Possion distributions with expected events obtained from template histograms, and Gaussian distributions describing systematic uncertainties, as implemented in HistFactory [36].We use one bin from 2m e (2m µ ) to m ϕ L − 2σ ϕ L , 2 to 8 bins in m ϕ L ± 2σ ϕ L window, and one bin from m ϕ L + 2σ ϕ L to 250 MeV (7 GeV).Here σ ϕ L is the resolution of the ℓ + ℓ − mass distribution for the signal, and it varies in the [5,30] MeV range, increasing at larger values of m ϕ L .The mass of ϕ L is kept fixed in the fit and scanned from 40 MeV to 210 MeV at 10 MeV intervals for e + e − channel, and from 225 MeV to 6.5 GeV at 25 MeV intervals for the µ + µ − channel.We skip the ±50 MeV window around the nominal mass of J/ψ and ψ(2S), where we expect peaking backgrounds.The fit includes systematic uncertainties from luminosity, tracking efficiency, momentum scale and PID corrections of ϕ L daughter tracks, scale factors and selection efficiency of BDT.To account for the unsimulated processes, we include an additional uniform background component.
We use the profile likelihood ratio as the test statistic [37] to compare data with signal-plus-background or background-only hypothesis.The fraction of each background component and the additional uniform component are allowed to vary within their uncertainties.The fit returns the signal yield as well as the normalization factor for each background component, along with the nuisance parameters describing systematic uncertainties.In order to obtain the signal significance, we first calculate the p-value, the probability that the data is explained as the statistical fluctuation of the background.We then calculate the signed significance, where the sign follows the convention elaborated in Ref. [38].The signal significances are shown in the bottom sub-plots in Fig. 6 for e + e − (top) and µ + µ − (bottom) channels.We find all the rithm [39,40] as implemented in RooStats [41].A toy Monte Carlo based numerical integration technique [41] is used to cross-check our Bayesian result, which agrees within a couple of percents across the whole mass range.
The UL of the signal cross-section at 90% confidence level (CL) [42] are shown in the top sub-plots in Fig. 6.We also cross-check our results using an alternate fitting method as used by BABAR experiment [32], where the background is modeled as a smooth polynomial from sideband data, and the signal is modeled as a Gaussian.The observed significance from these two methods agree within 0.35 σ on the average.Both the cross-section and the proper decay length (cτ ) of the dark leptophilic scalar depend on the coupling constant ξ.For m ϕ L > 0.1 GeV, the obtained UL on ξ is consistent with the assumption that cτ is short enough to have negligible influence on the signal detection efficiency.However, for m ϕ L < 0.1 GeV, cτ of ∼ 10 to 50 mm is expected for ξ ∼ 1.To take this dependence into account, we simulate the events with two additional values of cτ = 10 mm and cτ = 50 mm, and re-perform the entire analysis to determine the UL on the crosssection for these values.Using these UL and the known relation between cτ and ξ, we iteratively determine the UL on the ξ, as shown in the top sub-plot of Fig. 6.The exclusion region of the coupling constant ξ vs. m ϕ L is shown in Fig. 7, overlaid with previous results [32,[43][44][45].Our limits are tabulated in Table I.
A fit to the ratio of limits obtained by the BABAR experiment [32] and our limits show that our results are more constraining by 19% on the average.We exclude the parameter space with m ϕ L between [0.04,4] GeV favored by (g − 2) µ at 90% CL [10,11].
In conclusion, we search for a dark leptophilic scalar and set the UL on the cross-section of e + e − → τ + τ − ϕ L , ϕ L → e + e − process in the range [0.6, 7] fb and on the cross-section of e + e − → τ + τ − ϕ L , ϕ L → µ + µ − process in the range [0.1, 2] fb at 90% CL.There is no such leptophilic scalar with mass less than 4 GeV that can explain the observed excess in (g − 2) µ .
This work, based on data collected using the Belle detector, which was operated until June 2010, was supported by the Ministry of Education, Culture, Sports, Science, and Technology (MEXT) of Japan, the Japan Society for the Promotion of Science (JSPS), and the Tau-

FIG. 2 :
FIG. 2: Data and MC distributions of e + e − invariant mass for ϕL → e + e − channel (top) and µ + µ − invariant mass for ϕL → µ + µ − channel (bottom) in the GCR.All corrections and scale factors are applied to the MC distributions, after normalizing them to the integrated luminosity of data.

FIG. 5 :
FIG. 5: µ + µ − invariant mass distributions are shown in the top inset with 10 MeV bin-width in the signal region corresponding to m ϕ L = 0.825 GeV and 2.425 GeV, which have the second highest and highest observed significance of 2.1 and 2.2 standard deviations in this channel, respectively.The data are shown as black dots, while the signal, τ + τ − , other Monte Carlo components of the backgrounds and the additional uniform background component are shown by pink, cyan, yellow and green histograms, respectively.The statistical and systematic components of the uncertainty on total background have been added in quadrature and are shown by the shaded histogram.The bottom sub-plots compare the signal distribution with data minus the background contributions.

FIG. 6 :
FIG.6: Observed upper limits at 90% CL on the signal cross-section with mean proper decay lengths (cτ ) of 0 mm, 10 mm and 50 mm, respectively, are shown in the top subplots as a function of the dark leptophilic scalar mass for ϕL → e + e − channel (top) and ϕL → µ + µ − channel (bottom).The bottom sub-plots in both of the figures show the observed significance for each channel.See text for details.
tute, Foreign Large-size Research Facility Application Supporting project, the Global Science Experimental Data Hub Center of the Korea Institute of Science and Technology Information and KREONET/GLORIAD; the Polish Ministry of Science and Higher Education and the National Science Center; the Ministry of Science and Higher Education of the Russian Federation, Agreement 14.W03.31.0026, and the HSE University Basic Research Program, Moscow; University of Tabuk research grants S-1440-0321, S-0256-1438, and S-0280-1439 (Saudi Arabia); the Slovenian Research Agency Grant Nos.J1-9124 and P1-0135; Ikerbasque, Basque Foundation for Science, Spain; the Swiss National Science Foundation; the Ministry of Education and the Ministry of Science and Technology of Taiwan; and the United States Department of Energy and the National Science Foundation.These acknowledgements are not to be