First Simultaneous Determination of Inclusive and Exclusive | V ub |

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In this letter we report the first simultaneous determination of the absolute value of the Cabibbo-Kobayashi-Maskawa (CKM) matrix element V ub using inclusive and exclusive decays.Exclusive determinations of |V ub | focus on reconstructing explicit final states such as B → πℓν ℓ [1], Λ b → pµν µ [2], or B s → K µν µ [3], whereas inclusive determinations study B meson decays undergoing b → uℓν ℓ transitions and are indiscriminate of the u → X u hadronization process.The world averages of either method are only marginally compatible [1], |V excl.ub | = (3.51± 0.12) |V incl.ub | = (4.19± 0.16) with a ratio of |V excl.ub |/|V incl.ub | = 0.84 ± 0.04, which deviates 3.7 standard deviations from unity.The underlying arXiv:2303.17309v2[hep-ex] 16 Aug 2023 reason for this tension is unknown.New physics explanations are challenging (see e.g.Refs.[4][5][6][7]), leading to some to speculate the existence of until now unaccounted systematic effects [8].This motivates the simultaneous determination in a single analysis, in which B → πℓν ℓ and the B → X u ℓ νℓ rates can be simultaneously extracted and systematic effects can be correlated.
The presented measurement of inclusive and exclusive b → uℓν ℓ decays uses the same collision events and a similar analysis strategy as Refs.[9,10].Charmless semileptonic decays are reconstructed by relying on the complete reconstruction of the second B meson in the e + e − → Υ(4S) → B B process.This approach allows for the direct reconstruction of the X u system of the B → X u ℓ νℓ process.Specifically, the four-momentum transfer squared, q 2 = (p B − p Xu ) 2 , and the number of charged pion candidates of the X u system, N π ± , can be reconstructed.This allows for the separation of B + → π 0 ℓ + ν ℓ and B 0 → π − ℓ + ν ℓ from other B → X u ℓ νℓ decays.The main background in the measurement stems from the much more abundant B → X c ℓ νℓ decays and a multivariate suppression strategy is used to reduce this and other background processes.Charge conjugation is implied throughout.The inclusive B → X u ℓ νℓ branching fraction is defined as the average branching fraction of B + and B 0 meson decays.Furthermore, we denote ℓ = e, µ, and use natural units: We analyze (772 ± 10) × 10 6 B meson pairs recorded at the Υ(4S) resonance energy and 79 fb −1 of collision events recorded 60 MeV below the Υ(4S) peak.Both data sets were recorded at the KEKB e + e − collider [11] by the Belle detector.Belle is a large-solid-angle magnetic spectrometer.A detailed description of its performance and subdetectors can be found in Ref. [12].The particle identification and selection criteria are the same as in Ref. [9].
Monte Carlo (MC) samples of B meson decays and continuum processes (e + e − → q q with q = u, d, s, c) are simulated using the EvtGen generator [13].The normalization of continuum events is calibrated with the measured off-resonance data.A detailed description of all samples and decay models is given in Ref. [9].The simulated samples are used for background subtraction and to correct for detector resolution, selection, and acceptance effects.The used sample sizes correspond to approximately ten and five times, respectively, the Belle collision data for the B meson production and continuum processes.
Semileptonic B → X u ℓ νℓ decays are simulated as a mixture of specific exclusive modes and nonresonant contributions using a "hybrid" approach [14][15][16]: the triple differential rate of inclusive and exclusive predictions are combined such that the partial rates of the inclusive prediction are recovered.This is achieved by assigning weights to the inclusive contribution as a function of the generator-level q 2 , E B ℓ , and M X .Here E B ℓ and M X denote the energy of the lepton in the signal B rest frame and the invariant mass of the X u system produced in the B → X u ℓ νℓ decay.For the inclusive contribution, we use two different calculations: the De Fazio and Neubert (DFN) model [17] (with m KN b = (4.66±0.04)GeV, a KN = 1.3±0.5)and the Bosch-Lange-Neubert-Paz (BLNP) model [18] (with m SF b = 4.61 GeV, µ 2 SF π = 0.20 GeV 2 ).The difference between the two models is treated as a systematic uncertainty.The simulated inclusive B → X u ℓ νℓ events are hadronized with the JETSET algorithm [19] into final states with two or more mesons.We study two different tunes of the fragmentation parameters and assign their difference as a systematic uncertainty.The exclusive contributions are modeled as follows: B → π ℓ νℓ decays are modeled using the Bourrely-Caprini-Lellouch (BCL) form factor parameterization [20]; B → ρ ℓ νℓ and B → ω ℓ νℓ decays are modeled using the Bharucha-Straub-Zwicky (BSZ) form factors [21] from the fit of Ref. [22] to light-cone sum rule (LCSR) predictions [21] and the measurements of Refs.[23][24][25]; B → η ℓ νℓ and B → η ′ ℓ νℓ are modeled using pole form factors obtained from fits to LCSR [26].For the branching fractions the world averages from Ref. [27] are used.
Semileptonic B → X c ℓ νℓ decays are dominated by B → D ℓ νℓ and B → D * ℓ νℓ decays.We simulate them with the form factors of Refs.[28][29][30] and values determined by the measurements of Refs.[31,32].Other B → X c ℓ νℓ decays are simulated as a mixture of resonant and nonresonant modes, using the parameterization of Ref. [33] for the modeling of B → D * * ℓ νℓ form factors.The known difference between inclusive and the sum of measured exclusive B → X c ℓ νℓ decays is simulated with B → D ( * ) η ℓ + ν ℓ decays.
We reconstruct e + e − collision events with the multivariate tagging algorithm of Ref. [34].The algorithm uses a hierarchical approach utilizing neural networks to fully reconstruct one of the two B mesons in hadronic final states (labeled as B tag ).The B tag reconstruction efficiency is calibrated using B → X c ℓ νℓ decays following the prescription outlined in [9].The identified final state particles forming the B tag are masked and b → uℓν ℓ signal candidates are reconstructed by identifying an electron or muon candidate in the events, requiring E B ℓ = |p B ℓ | > 1 GeV as measured in the signal B rest frame.To reject background from the much more abundant B → X c ℓ νℓ decays, eleven distinguishing features are combined into a single discriminant using boosted decision trees (BDTs) and utilizing the implementation of Ref. [35].The most discriminating training features are the reconstructed neutrino mass, M 2 miss , the vertex fit probability of the decay vertex between the hadronic system X and the signal lepton ℓ, and the number of identified K ± and K 0 S in the X system.Same as in [9], we select a working point that corresponds to a signal efficiency of 18.5%, which rejects 98.7% of B → X c ℓ νℓ decays, defined with respect to all events after the B tag selection.To test the modeling of B → X c ℓ νℓ and other backgrounds in the extraction variables, q 2 and N π ± , we also utilize the events failing the BDT selection and find good agreement [36].We further separate events by the reconstructed M X , categorizing M X < 1.7 GeV into five q 2 bins ranging in [0, 26.4] GeV 2 as a function of the N π ± multiplicity for the interval of [0, 1, 2, ≥ 3].Events with M X ≥ 1.7 GeV are analyzed only in bins of N π ± as they are dominated by background.To enhance the B → π ℓ νℓ purity in the low-M X N π ± = 0 and N π ± = 1 events, we apply a selection on the thrust of 0.92 and 0.85, respectively.It is defined by max |n|=1 ( i |p i • n|/ i |p i |), when summing over the neutral and charged constituents of the reconstructed X system in the center of mass frame.For B → π ℓ νℓ events, we expect a more collimated X u system than for B → X c ℓ νℓ and other B → X u ℓ νℓ processes, resulting in a higher thrust value.
The q 2 : N π ± bins and the M X ≥ 1.7 GeV N π ± distribution are analyzed using a simultaneous likelihood fit, which incorporates floating parameters for the modeling of the B → π ℓ νℓ form factor, the binned templates, and systematic uncertainties as nuisance parameters.Specifically, the shape of B → π ℓ νℓ template is linked to the form factors by correcting the efficiency and acceptance effects.The fit components we probe are the normalizations of B → π ℓ νℓ decays, other B → X u ℓ νℓ signal decays, and of background events dominated by B → X c ℓ νℓ decays.The f + and f 0 form factors describing the B → π ℓ νℓ decay dynamics are parameterized with expansion coefficients a + n and a 0 n using the BCL expansion, at expansion order [20,37], and a 0 2 is expressed by the remaining coefficients to keep the kinematical constraint f + (0) = f 0 (0).We constrain the expansion coefficients to the lattice QCD (LQCD) values of Ref. [37], combining LQCD calculations from several groups [38,39].Note that the measured distributions have no sensitivity for f 0 and we thus neglect its effects in the decay rate.The inclusion of the f 0 expansion coefficients, however, reduces uncertainties on the B → π ℓ νℓ rate through the correlation to the f + shape.In order to utilize the full experimental knowledge of the B → π form factors to date, we constrain its shape to the combined lattice QCD and experimental information of Refs.[40][41][42][43].The fit scenario with only lattice QCD constraints is studied for a standalone comparison with other experimental results.Events (M X < 1.7 GeV) Data/MC We consider additive and multiplicative systematic uncertainties in the likelihood fit by adding bin-wise nuisance parameters for each template.The parameters are constrained to a multinormal Gaussian distribution with a covariance reflecting the sum of all considered systematic effects, and the correlation structure between templates from common sources is taken into account.This includes detector and reconstruction related uncertainties, such as the tracking efficiency for low and high momentum tracks, particle identification efficiency uncertainties, and the calibration of the B tag reconstruction efficiency.We further consider uncertainties on the B → X u ℓ νℓ and B → X c ℓ νℓ shapes from the form factors, non-perturbative parameters, and their compositions.The u → X u fragmentation uncertainties are evaluated by changing the default Belle tune of fragmentation parameters to the values used in Ref. [44].We further vary the ss-production rate γ s = 0.30 ± 0.09, spanning the range of Refs.[45,46].The largest uncertainties on the exclusive branching fraction measurements are from the calibration of the tagging efficiency (±4.1%) and the B → X u ℓ νℓ modeling (±3.5%).The largest uncertainties on the inclusive branching fraction measurement are from the B → X u ℓ νℓ (±10.9%)modeling and the u → X u fragmentation (±5.3%).The uncertainties of the modeling of the B → X c ℓ νℓ background are ±1.2% and ±2.8% for the B → π ℓ νℓ and B → X u ℓ νℓ branching fractions, respectively.
Figure 1 shows the q 2 : N π ± distribution of the signal region after the fit and with only using LQCD information: B + → π 0 ℓ + ν ℓ and B 0 → π − ℓ + ν ℓ events are aggregated in the N π + = 0 and N π + = 1 bins, respectively, whereas contributions from other B → X u ℓ νℓ processes are in all multiplicity bins.The high M X bins constrain the B → X c ℓ νℓ and other background contributions.We use the isospin relation and B 0 /B + lifetime ratio to link the yields of B + → π 0 ℓ + ν ℓ and B 0 → π − ℓ + ν ℓ .The fit has a χ 2 of 13.8 with 21 degrees of freedom, corresponding to a p-value of 88%.The measured B + → π 0 ℓ + ν ℓ and B 0 → π − ℓ + ν ℓ yields are corrected for efficiency effects to determine the corresponding branching fractions B. The measured inclusive yield is calculated from the sum of B + → π 0 ℓ + ν ℓ , B 0 → π − ℓ + ν ℓ , and other B → X u ℓ νℓ events and unfolded to correspond to a partial branching fraction ∆B with E B ℓ > 1.0 GeV, also correcting for the effect of final state radiation photons.We find for exclusive and inclusive |V ub | with the uncertainties denoting the statistical error, systematic error, and error from theory (either from LQCD or the inclusive calculation).The correlation between the exclusive and inclusive |V ub | is ρ = 0.07.The determined value for inclusive |V ub | is compatible with the determination of Ref. [9].For the ratio of inclusive and exclusive V ub values, we find In the nominal result, we utilize the full theoretical and experimental knowledge of the B → π ℓ νℓ form factor, combining shape information from the measured q 2 spectrum with LQCD predictions, as provided by Ref. [37].The determined (partial) branching fractions in this scenario are ∆B(B → X u ℓν ℓ ) = (1.39 ± 0.14 ± 0.22) × 10 −3 , (10) with a correlation of ρ = 0.12 between inclusive and exclusive branching fractions and assuming isospin relation.This fit leads to a more precise value of |V ub | from B → π ℓ νℓ and we find with the same inclusive calculation V incl.

FIG. 3.
The q 2 spectra of B 0 → π + ℓ − νℓ obtained from the fit of the combined LQCD and experimental information (orange, solid) and from the fit to LQCD only (green, dashed) are shown.The data points are the background subtracted post-fit distributions, corrected for resolution and efficiency effects and averaged over both isospin modes.In addition, the LQCD pre-fit prediction of [37] for the B 0 → π + ℓ − νℓ form factor is shown (grey).
with a correlation ρ = 0.11 and a ratio of compatible with the world average within 1.2 standard deviations.Fig. 2 (bottom) compares the obtained values and we also find good agreement between the isospin conjugate exclusive values of |V ub |. Figure 3 compares the fitted q 2 spectra of the differential rate of B 0 → π + ℓ − νℓ for both fit scenarios as well as for the LQCD input [37].
The inclusion of the full experimental and theoretical knowledge leads to a higher rate at low q 2 .In summary, we presented the first simultaneous determination of inclusive and exclusive |V ub | within a single analysis.In the ratio of both |V ub | values many systematic uncertainties such as the tagging calibration or the lepton identification uncertainties cancel and one can directly test the SM expectation of unity.We recover ratios that are compatible with this expectation, but 1.5 standard deviations higher than the ratio of the current world averages of inclusive and exclusive |V ub |.This tension is reduced to 1.2 standard deviations when including the constraint based on the full theoretical and experimental knowledge of the B → π ℓ νℓ form factor shape.We average our inclusive and exclusive values from both approaches using LQCD or LQCD and additional experimental information and find, |V ub | = (3.84± 0.26) × 10 −3 , (LQCD + exp.) (15) respectively.These values can be compared with the expectation from CKM unitarity of Ref. [48] of |V CKM ub | = (3.64 ± 0.07) × 10 −3 and are compatible within 1.2 and 0.8 standard deviations, respectively.The applied approach of simultaneously fitting q 2 and the number of charged pions in the X u system will benefit from the large anticipated data set of Belle II.Additional fit scenarios and inclusive |V ub | values from other theory calculations of the partial rate are provided in the supplemental material [36].
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-

Data-MC agreement in background dominated sideband
Figure 5 shows the analyzed categories in q 2 : N π ± for M X < 1.7 GeV and the four M X ≥ 1.7 GeV bins in the B → X c ℓ νℓ enriched BDT sideband.We observe fair agreement in the background shapes with a p-value of 87%.The uncertainties incorporate all systematic uncertainties discussed in the text.The bottom panel shows the ratio of observed events and the MC expectation.

Consistency cross-check
The prediction is fitted to the observed data by minimizing where the floating parameters η and a are the template normalization and the BCL form facotrs, respectively.The bin-and template-wise nuisance parameters θ are normalized to the relative bin errors ϵ, and the associated bin-to-bin correlations arising from systematics are accounted in the fit by a global correlation matrix ρ θ .The BCL form factors are constrained by the covariance matrix Cov FF provided by FLAG.In this measurement, an additional fit with separate normalizations of the B 0 → π − ℓ + ν ℓ and B + → π 0 ℓ + ν ℓ decays is applied to check the consistency of the nominal results based on combining the two modes.All of the fitter setups are summarised in the following: • Setup 1-a: fit q 2 : N π ± spectra with LQCD and external experimental constraint on the BCL form factor and shared B → π ℓ νℓ normalization based on the isospin relation.
• Setup 1-b: same as 1-a, but with only LQCD constraint for the form factor.
The nominal results are based on setup 1-a and 1-b.With different fit scenarios, the numerical results of the fitted yields are summarised in Table I as well as the signal efficiencies.After all selections, the total measured data are 7715 ± 88 events.Figure 6 and Fig. 7 illustrate the post-fit spectra and the pulls of the template-and bin-wise Nuisance parameters, respectively.The featured behaviors are found to be consistent in all setups.The obtained |V ub | and branching fractions are listed in Table IV and Table V for the setup 2-a and 2-b, where the weighted average of two pion modes is derived based on the total covariance matrix.The final results are found to be fairly compatible with the nominal results in Table II and Table III  (3.78 ± 0.

FIG. 1 .
FIG.1.The q 2 : N π ± spectrum after the 2D fit is shown for the scenario that only uses LQCD information.The uncertainties incorporate all postfit uncertainties discussed in the text.
FIG. 2. The |V ub | values obtained with the fits using (top) LQCD or (bottom) LQCD and experimental constraints for the B 0 → π + ℓ − νℓ form factor are shown.The inclusive |V ub | value is based on the decay rate from the GGOU calculation.The values obtained from the previous Belle measurement[9] (grey band) and the world averages from Ref.[1] (black marker) are also shown.The shown ellipses correspond to 39.3% confidence levels (∆χ 2 = 1).

FIG. 7 .
FIG. 7. The pulls of bin-wise nuisance parameters.From left to right, the results are shown the fit setup 1-a, 1-b, 2-a and 2-b.The uncertainty of each pull shows the post-fit error normalized to the pre-fit constraint.

1 1 Fit B 0 + Fit B + 0 FIG. 8 .
FIG.8.Top: the q 2 spectra of B → π ℓ νℓ obtained from the fit of the combined LQCD and experimental information (orange, solid) and from the fit to LQCD only (green, dashed) are shown.The data points are the post-fit signal distributions, corrected for resolution and efficiency effects and averaged over both isospin modes.The input LQCD constraints from FLAG are shown in grey.Bottom left: the q 2 spectra obtained with separated (blue, solid) π + mode and (red, dashed) π 0 using the LQCD only information from FLAG to constrain the B → π ℓ νℓ form factor (setup 2-b).Bottom right: the results obtained by using the LQCD and experimental constraint (setup 2-a).The combined fit (setup 1-a) result is shown for comparison (black, dotted).
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- Xc ℓ νℓ sideband from the events rejected by the BDT selection is shown in the binning of the 2D fit.

TABLE IV .
The determined |V ub | results and various ratios based on the setup 2-a and 2-b, respectively.

TABLE VI .
The measured B → πℓν form factor BCL parameters and exclusive |V ub | with full correlations.The shape of q 2 is constrained by the LQCD fit results from FLAG.

TABLE VII .
The measured B → πℓν form factor BCL parameters and exclusive |V ub | with full correlations.The shape of q 2 is constrained by the combined LQCD and experimental fit results from FLAG.