Lepton Flavor Universality at Belle and Belle II

. We report the test of lepton ﬂavor universality at Belle and Belle II experiments considering b → c and b → s transitions. We use 711 fb − 1 and 189 fb − 1 of data collected at Υ (4 S ) resonance from electron-positron collisions for Belle and Belle II, respectively. For the b → c transition, we present the measurement of lepton-ﬂavor-universality in B 0 → D ∗− ` + ν ` decay, exclusive measurement of R ( D ) & R ( D ∗ ) in Belle data, and inclusive R ( X e /µ ) in Belle II. For the b → s transitions, we provide measurements of R K and R K ∗ using Belle data, measurement of the branching fraction for the decay B → K ∗ `` , and test of lepton-ﬂavor-universality in B → J /ψ K using Belle II data. Additionally, we report the lepton-ﬂavor-universality in Ω 0 c → Ω − ` + ν ` decay using 922 fb − 1 of data collected by the Belle detector.

SM [9]. The LFU test can also be performed by measuring R(X) =

B(B → Xτν) B(B → X ν)
, which is the complimentary measurement to R(D ( * ) ) via inclusive reconstruction. The R(X) measurement is more challenging due to the background from a less constrained X system. The LFU can be tested using light leptons by measuring R(X e/µ ).

Measurement of LFU in
Belle [10] has performed the LFU measurement in exclusive semileptonic B decay using B 0 → D * − (D 0 (K + π − )π − ) ν decay with a data sample of 711 fb −1 . The analysis is done using an untagged approach, leading to high efficiency with a sizable background. The background from fake D * is suppressed by constraining D * momentum in the CM frame to be  and muons provides a test of the LFU, i.e., The result is consistent with unity within the uncertainty.

Measurement of R(D) & R(D * ) at Belle
Belle [5] has measured simultaneously R(D) and R(D * ) using the semileptonic tagging method. Here, the tag-side B is reconstructed via B 0/± → D ( * ) − ν decay and signal side

Measurement of R(X e/µ ) at Belle II
Belle II [11] has checked the LFU by inclusive measurement of R(X e/µ ) using 189 fb −1 of data sample with hadronic-B tagging approach. In this measurement, the signal side B flavor and kinematics in constrained by tagging the other B in its fully hadronic decays, which leads to good signal purity at a cost of lower signal reconstruction efficiency. The X system on the signal side contains a large variety of different charged and neutral final-state particles. As we reconstruct only the lepton in the signal side, the lepton momentum in the B signal rest frame, p * , is used to extract the signal yield. We required the lepton to have a high probability to be an electron or muon and p * > 1.3 GeV/c to suppress backgrounds from hadrons faking as leptons and secondary leptons from b → c → ( , s) cascades and B → Xτν. The signal yields for B → Xeν and B → Xµν channels are extracted simultaneously in 10 bins of p * , with one-dimensional binned ML fit. The fit results are shown in Fig. 3. There are 48034 ± 286, 58569 ± 429 signal events for B → Xeν and B → Xµν channels, respectively. The R(X e/µ ) measured to be 1.033 ± 0.010(stat) ± 0.020(sys), for p * > 1.3 GeV/c. This is the first inclusive test of (e, µ) lepton flavor universality in semileptonic B → X ν decays. The measurement is in agreement with unity within 1.5σ, with a world-leading precision of 2.2%. This measurement paved the path for R(X) = R(X τ/ ) measurement.

LFU in b → s transition
b → s is propagated through loop-level transition and is an important probe to test the LFU by measuring R K and R K * for B → K and B → K * decays, i.e., According to SM this ratio should be 1 [12], as the coupling of leptons to gauge boson is independent of flavor. LHCb [13] has reported a deviation of 3.1σ in R K + measurement for q 2 ∈ [1.1 − 6.0] GeV 2 /c 4 bin using 9 fb −1 data sample, q 2 is invariant mass square of lepton  pair. Similarly, R K * + and R K 0 S measurements from LHCb [14] have 1.4σ and 1.5σ deviations from SM, calculated in 3 fb −1 data sample.

R K * measurement at Belle
Belle [15] has measured R K * using a full data sample of 711 fb −1 . The decay modes reconstructed are B + → K * + (K + π 0 , K 0 S π + ) and B 0 → K * 0 (K + π − , K 0 S π 0 ) . The kinematic variables which distinguish the signal from the background are beam-energy-constrained mass, Here, p B and E B are the momentum and energy of the B candidate, and E beam is the beam energy. The background coming from the continuum and BB are suppressed using Neural Network (NN). The signal yield is obtained by performing a 1-dimensional unbinned extended ML fit in M bc , shown in Fig. 4. From the fit, the signal yields are 103 +13.4 −12.7 and 139.9 +16.0 −15.4 for electron and muon channels, respectively. From the fitted signal yield and signal MC efficiency, the R K * is calculated for charged B, neutral B, and the combined result from charged and neutral B, as shown in Fig. 5. The results for different bins are consistent with SM expectations within the uncertainty.

R K measurement at Belle
The analysis is performed in 711 fb −1 data sample of Belle [16]. The decay modes reconstructed are B + → K + and B 0 → K 0 S . The background from continuum and BB are suppressed using a NN which uses several event shapes, vertex quality, and kinematic variables. The signal yield is extracted by 3-dimensional unbinned extended ML fit in M bc , ∆E, and the translated NN output (O ). The fit results are shown in Fig. 6. There are 137 ± 14, 138 ± 15, 27.3 +6. 6 −5.8 , and 21.8 +7.0 −6.1 signal events for B + → K + µµ, B + → K + ee, B 0 → K 0 S µµ, and B 0 → K 0 S ee, respectively. The R K + , R K 0 , and R K are measured in different q 2 bins, as shown in Fig. 7. The results are in agreement with SM expectations within the uncertainty. In addition to this, we have the most precise measurement for R K in the J/ψ region and is consistent with the unity, i.e., R K + (J/ψ) = 0.994 ± 0.011(stat) ± 0.010(sys) and R K 0 (J/ψ) = 0.993 ± 0.015(stat) ± 0.010(sys).

Measurement of B(B → K * ) at Belle II
Belle II [17] has measured B(B → K * ) using 190 fb −1 of data. For this analysis, the decay modes reconstructed are B 0 → K * 0 (K + π − ) and B + → K * + (K + π 0 , K 0 S π + ) . The signal yield is extracted by 2-dimensional un-binned extended ML fit in M bc and ∆E for the events which pass tight requirement on the boosted decision tree, used to fight the background from continuum and BB. The fit results are shown in Fig. 8.

LFU prospects at Belle II
The R(X) measurement or in general inclusive processes are unique to Belle II. The estimated precision on R(X) for 189 fb −1 of data sample is ∼ 17%. The uncertainty (statistical + systematic) projections for R(D), R(D * ), and R(X) are shown in Fig. 11 [22]. A few ab −1 of data from Belle II will be sufficient to track the anomaly on R(D) − R(D * ) to be statistical or systematic origin. With the full data sample of Belle II, the total uncertainty for R(D)−R(D * ) will be 2 − 3% for different tagging approaches (semileptonic or hadronic). R(X) will also be of similar precision. Similarly, for b → s transition, the R K * projections for 1 ab −1 , 5 ab −1 , and 50 ab −1 are shown in Fig. 11. The R K + and R K * statistical sensitivity will be < 2% for the entire q 2 region and ∼ 3% for q 2 ∈ [1−6] GeV 2 /c 4 , with full data sample of 50 ab −1 . Contrary to LHCb, Belle II can check these observables both for low and high q 2 bins. Belle II will  provide an independent measurement to confirm the tension observed in R K − R K * with few ab −1 of data sample.

Summary
The flavor physics in e + e − collisions offers an extremely rich physics program with many opportunities to probe physics beyond the SM. At Belle II, we can access charged and neutral B with equal efficiency and also have equal sensitivity for electron and muon channels. Belle II can access inclusive decay modes in addition to exclusive decays, and analysis can be performed in tagged or un-tagged approaches. Belle II has collected ∼ 424 fb −1 of data which is comparable to the size of the BaBar data sample and can be combined with the Belle data sample to increase statistics. So far, Belle or Belle II have not observed any sign of LFU violation and a higher data sample will shed light on LFU anomalies. Some results show the potential of Belle II, and only a part of them are covered here. We will have more results from Belle II in near future.