Precise measurement of the $D^0$ and $D^+$ lifetimes at Belle II

We report a measurement of the $D^0$ and $D^+$ lifetimes using $D^0\to K^-\pi^+$ and $D^+\to K^-\pi^+\pi^+$ decays reconstructed in $e^+e^-\to c\bar{c}$ data recorded by the Belle II experiment at the SuperKEKB asymmetric-energy $e^+e^-$ collider. The data, collected at center-of-mass energies at or near the $\Upsilon(4S)$ resonance, correspond to an integrated luminosity of $72\,{\rm fb}^{-1}$. The results, $\tau(D^0) = 410.5\pm1.1\,{\rm(stat)}\pm0.8\,{\rm(syst.)}\,{\rm fs}$ and $\tau(D^+) = 1030.4\pm4.7\,{\rm(stat)}\pm3.1\,{\rm(syst.)}\,{\rm fs}$, are the most precise to date and are consistent with previous determinations.

Accurate predictions of lifetimes of weakly decaying charmed and bottom hadrons are challenging because they involve strong-interaction theory at low energy.Predictions must resort to effective models, such as the heavy-quark expansion [1][2][3][4][5][6], which also underpin stronginteraction calculations required for the determination of fundamental standard-model parameters from hadrondecay measurements (e.g., to extract the strength of quark-mixing couplings from decay widths).Precise lifetime measurements provide excellent tests of such effective models.Lifetimes are also important inputs for a wide variety of studies because they are needed to compare measured decay branching fractions to predictions for partial decay widths.
Weakly decaying charmed hadrons have lifetimes ranging from about 0.1 to 1 ps [7].The world averages of the D 0 and D + lifetimes, 410.1±1.5 fs and 1040±7 fs, are almost exclusively determined from systematically limited per-mille-precision measurements made by FOCUS two decades ago [7,8].Recently, the LHCb collaboration precisely measured the lifetimes of the D + s meson and charmed baryons relative to that of the D + meson [9][10][11][12].Such relative measurements minimize systematic uncertainties due to decay-time-biasing event-selection criteria that are particularly severe at hadron colliders.By contrast, experiments at e + e − colliders, owing to the reconstruction of large charmed hadron yields without decaytime-biasing selections, have a great potential for absolute lifetime measurements.With the first layer of its vertex detector only 1.4 cm away from the interaction region, the Belle II experiment at the SuperKEKB asymmetricenergy e + e − collider [13,14] obtains a decay-time resolution two times better than the Belle and BABAR experiments [15], enabling high precision for the measurement of charmed lifetimes with early data.To limit systematic uncertainties this potential must be complemented with an accurate vertex-detector alignment, a precise calibration of final-state particle momenta, and powerful background discrimination.
In this Letter, we report high-precision measurements of the D 0 and D + lifetimes using D * + → D 0 (→ K − π + )π + and D * + → D + (→ K − π + π + )π 0 decays reconstructed in the data collected by Belle II during 2019 and the first half of 2020 at center-of-mass energies at or near the Υ (4S) resonance.(Charge-conjugate decays are implied throughout.)The data correspond to an integrated luminosity of 72 fb −1 .At Belle II, D * + mesons from e + e − → cc events are produced with boosts that displace the D 0 and D + decay points from those of production by approximately 200 µm and 500 µm on average, respectively.The decay time is measured from this displacement, L, projected onto the direction of the momentum, p, as t = m D L • p/| p| 2 , where m D is the known mass of the relevant D meson [7].The decay-time uncertainty, σ t , is calculated by propagating the uncertainties in L and p, including their correlations.The lifetimes are measured using a fit to the (t, σ t ) distributions of the reconstructed decay candidates.The sample selection and fit strategy have been optimized and validated using simulation; however, no input from simulation is used in the fit to data.To avoid bias, we inspected the lifetimes measured with the full data sample only after the entire analysis procedure was finalized and all uncertainties were determined.However, we examined the results from the subset of data collected during 2019 (approximately 13% of the total) before the analysis was complete.
The Belle II detector [13] consists of several subsystems arranged in a cylindrical structure around the beam pipe.The tracking system consists of a two-layer siliconpixel detector (PXD), surrounded by a four-layer doublesided silicon-strip detector (SVD) and a 56-layer central drift chamber (CDC).Only two out of 12 ladders were installed in the second layer of the PXD for this data sample.The combined PXD and SVD system provides average decay-time resolutions of about 70 fs and 60 fs, respectively, for the D 0 and D + decays considered here.A time-of-propagation counter and an aerogel ring-imaging Cherenkov counter that cover the barrel and forward end cap regions of the detector, respectively, are essential for charged-particle identification.The electromagnetic calorimeter fills the remaining volume inside a 1.5 T superconducting solenoid and serves to reconstruct photons and electrons.A dedicated system to identify K 0 L mesons and muons is installed in the outermost part of the detector.The z axis of the laboratory frame is defined as the central axis of the solenoid, with its positive direction determined by the direction of the electron beam.
The reconstruction [20][21][22] and selection of the signal candidates avoid any requirement that could bias the decay time or introduce a variation of the efficiency as a function of decay time, as checked in simulation.Events are first selected by vetoing events consistent with Bhabha scattering and by requiring at least three tracks with loose upper bounds on their impact parameters and with transverse momenta greater than 200 MeV/c.These three tracks are not necessarily associated with the decay modes being reconstructed.
Candidate D 0 → K − π + decays are formed using pairs of oppositely charged tracks.Each track must have a hit in the first layer of the PXD, at least one hit in the SVD, at least 20 hits in the CDC, and be identified as a kaon, if negative, or else a pion.Low-momentum pion candidates are tracks consistent with originating from the interaction region that are required to have hits both in the SVD and CDC.They are combined with D 0 candidates to form D * + → D 0 π + decays.A global decaychain vertex fit [23] constrains the tracks according to the decay topology and constrains the D * + candidate to originate from the measured position of the e + e − interaction region (IR).Only candidates with fit χ 2 probabilities larger than 0.01 are retained for further analysis.The IR has typical dimensions of 250 µm along the z axis and of 10 µm and 0.3 µm in the two directions of the transverse plane.Its position and size vary over data-taking and are regularly measured using e + e − → µ + µ − events.The mass of the D 0 candidate, m(K − π + ), must be in the range [1.75, 2.00] GeV/c 2 .The difference between the D * + -and D 0 -candidate masses, ∆m, must satisfy 144.94 < ∆m < 145.90 MeV/c 2 (±3 times the ∆m resolution around the signal peak).Since the D 0 is assumed to originate from the IR, charmed mesons originating from displaced decays of bottom mesons would bias the lifetime measurement.They are suppressed to a negligible rate by requiring that the momentum of the D * + in the e + e − center-of-mass system exceeds 2.5 GeV/c.After requiring 1.851 < m(K − π + ) < 1.878 GeV/c 2 (signal region), multiple D * + candidates occur in a few per mille of the selected events.In such events, one randomly selected candidate is retained for subsequent analysis.
The signal region contains approximately 171 × 10 3 candidates with a signal purity of about 99.8%, as determined from a binned least-squares fit to the m(K − π + ) distribution (Fig. 1).In the fit, the D 0 → K − π + signal is modeled with the sum of two Gaussian distributions and a Crystal Ball function [24]; misidentified decays of D 0 → π + π − and D 0 → K + K − , each modeled with a Johnson's S U distribution [25] with parameters determined from simulation, do not enter the signal region; the remaining background, modeled with an exponential distribution, is dominated by candidates formed by random combinations of particles.
The selection of the D * + → D + (→ K − π + π + )π 0 candidates follows similar criteria to those for the D 0 mode, but with more stringent requirements to suppress a larger background contamination.Tracks identified as kaons or pions are required to have a hit in the first layer of the PXD, at least one hit in the SVD, and at least 30 hits in the CDC.They are combined to form D + → K − π + π + candidates.To suppress backgrounds from misreconstructed charmed-hadron decays, such as four-body hadronic or semileptonic decays, the lower-momentum pion must have momentum exceeding 350 MeV/c and the higher-momentum pion must not be identified as a lepton.Candidate π 0 → γγ decays are Candidates per 1 MeV/ reconstructed using photon candidates from calorimetric energy clusters that are not associated with a track.Each photon energy must be larger than 80, 30, or 60 MeV if detected in the forward, central, or backward region, respectively, of the calorimeter.Neutral-pion candidates with masses in the range [120, 145] MeV/c 2 and momenta larger than 150 MeV/c are combined with D + candidates to form D * + → D + π 0 decays.The D * + decay chain is fit using IR and π 0 -mass constraints.Only candidates with fit χ 2 probabilities larger than 0.01 are retained.The mass of the D + candidate, m(K − π + π + ), must be in the range [1.75, 2.00] GeV/c 2 and the difference between the D * + and D + masses in the range [138,143] MeV/c 2 (±3 times the ∆m resolution around the signal peak).The momentum of the D * + in the e + e − center-of-mass system must exceed 2.6 GeV/c to suppress D * + candidates from bottom mesons.This requirement is tighter than that used for D 0 candidates because of the less-precise π 0 -momentum resolution.
The signal region in m(K − π + π + ) is defined as [1.855, 1.883] GeV/c 2 (Fig. 1).It contains approximately 59 × 10 3 candidates after randomly selecting one D * + candidate for the percent-level fraction of events where more than one is found.A binned least-squares fit to the m(K − π + π + ) distribution identifies about 9% of candidates in the signal region as background.Simulation shows that such background is composed of misreconstructed charmed decays and random track combinations.In the fit, the D + → K − π + π + signal is modeled with the sum of two Gaussian distributions and a Crystal Ball function; the background is modeled with an exponential distribution.
The lifetimes are determined with unbinned maximumlikelihood fits to the (t, σ t ) distributions of the candidates populating the signal regions.Each signal probabilitydensity function (PDF) is the convolution of an exponential distribution in t with a resolution function that depends on σ t , multiplied by the PDF of σ t .In the D + case, simulation shows that a Gaussian distribution is sufficient to model the resolution function.The mean of the resolution function is allowed to float in the fit to account for a possible bias in the determination of the decay time; the width is the per-candidate σ t scaled by a free parameter s to account for a possible misestimation of the decay-time uncertainty.The fit returns s ≈ 1.12 (1.29) for the D 0 (D + ) sample.In the D 0 case, an additional Gaussian distribution is needed to describe the 3% of candidates with poorer resolution.This second component shares its mean with the principal component but has its own free scaling parameter (s ≈ 2.5) for the broader width.
In the D 0 case, the signal region contains a 0.2% fraction of background candidates.Sensitivity to the background contamination and its effects on the decay-time distribution is very limited.For the sake of simplicity, the background is neglected in the fit and a systematic uncertainty is later assigned.In the D + case, the signal region contains a non-negligible amount of background, which is accounted for in the fit.The background is modeled using data with m(K − π + π + ) in the sideband [1.758, 1.814] ∪ [1.936, 1.992] GeV/c 2 (Fig. 1), which is assumed to contain exclusively background candidates and be representative of the background in the signal region, as verified in simulation.The background PDF consists of a zero-lifetime component and two exponential components, all convolved with a Gaussian resolution function having a free mean and a width corresponding to sσ t .To better constrain the background parameters, a simultaneous fit to the candidates in the signal region and sideband is performed.The background fraction is Gaussian constrained in the fit to (8.78 ± 0.05)%, as measured in the m(K − π + π + ) fit.
The PDF of σ t is a histogram template derived directly from the data.In the fit to the D 0 sample, the template is derived assuming that all candidates in the signal region are signal decays.In the fit to the D + sample, the template is derived from the candidates in the signal region by subtracting the scaled distribution of the sideband data.The PDF of σ t for the background is obtained directly from the sideband data.
The lifetime fits are tested on fully simulated data and on sets of data generated by randomly sampling the PDF with parameters fixed to the values found in the fits to the data.All tests yield unbiased results and expected parameter uncertainties, independent of the assumed values of the D 0 and D + lifetimes.The decay-time distributions of the data, with fit projections overlaid, are shown in Fig. 2. The measured D 0 and D + lifetimes 410.5 ± 1.1 (stat) ± 0.8 (syst) fs and 1030.4 ± 4.7 (stat) ± 3.1 (syst) fs, respectively, are consistent with their world averages [7].The systematic uncertainties arise from the sources listed in Table I and described below.The total systematic uncertainty is the sum in quadrature of the individual components.
The decay time and decay-time uncertainty are observed to be correlated in data and simulation reproduces these effects well.The dominant effect is that small σ t values correspond to larger true decay times (and vice versa).These correlations, when neglected in the fits, result in an imperfect description of the t distribution as a function of σ t .To quantify the impact on the results, our model that neglects the correlations is fit to 1000 samples of signal-only simulated decays, each the same size as the data.The samples are obtained by resampling, with repetition, a set of simulated e + e − collisions corresponding to an integrated luminosity of 500 fb −1 .Upper bounds of 0.16 fs and 0.39 fs on the average absolute deviations of the measured lifetimes from their true values are de-rived and assigned as the systematic uncertainty due to the imperfect resolution model for the D 0 → K − π + and D + → K − π + π + cases, respectively.For signal decays, the bias of the decay-time resolution function depends nearly linearly on the candidate mass and may not average out when the mass range is restricted.Varying the boundaries of the signal region shows that such a correlation has a negligible effect upon the measured lifetimes.
The background neglected in the D 0 → K − π + fit could result in a systematic bias on the measured lifetime.To estimate the size of the bias, we fit our model that neglects the background to 500 resampled sets of simulated e + e − collisions, each having the same size and signal-to-background proportion as the data.The measured lifetimes are corrected by subtracting the bias due to the neglected t vs. σ t correlations.The average absolute difference between the resulting value and the simulated lifetime, 0.24 fs, is assigned as a systematic uncertainty due to the neglected background contamination in the D 0 → K − π + fit.
The background contamination under the D + → K − π + π + peak is already accounted for in the fit of the D + lifetime using sideband data.In simulation, the sideband (t, σ t ) distribution describes the background (t, σ t ) distribution in the signal region well.The same might not hold in data given that some disagreement is observed between data and simulation in the t distribution of the candidates populating the sideband.We fit to one thousand samples of simulated data obtained by sampling the fit PDF for the signal region and by resampling from the simulated e + e − collisions for the sideband.The resulting samples feature sideband data that differ from the background in the signal region with the same level of disagreement as observed between data and simulation.The absolute average difference between the measured and simulated lifetimes, 2.52 fs, is assigned as a systematic uncertainty due to the modeling of the background (t, σ t ) distribution.In the lifetime fit, the fraction of background candidates in the signal region is constrained from the fit to the m(K − π + π + ) distribution.When we change this background fraction to values obtained from fitting to the m(K − π + π + ) distribution with alternative signal and background PDFs, the change in the measured lifetime is negligible.
During data-taking, a periodic calibration determines the alignments and surface deformations of the internal components of the PXD and SVD and the relative alignments of the PXD, SVD, and CDC using e + e −collision, beam-background, and cosmic-ray events [26].Unaccounted-for misalignment can bias the measurement of the charmed decay lengths and hence their decay times.Two sources of uncertainties associated with the alignment procedure are considered: the statistical precision and a possible systematic bias.Their effects are evaluated using simulated signal-only decays reconstructed with a misaligned detector.For the statistical contribu-tion, we consider configurations derived from comparison of alignment parameters determined from data acquired on two consecutive days.These configurations have magnitudes of misalignment comparable to the alignment precision as observed in data averaged over a typical alignment period.For the systematic contribution, we consider configurations derived from simulation studies in which coherent global deformations of the vertex detectors (e.g., radial expansion) are introduced [27].These deformations have magnitudes, determined by the most misaligned sensors, ranging from about 50 µm to 700 µm.The alignment procedure determines the magnitude of these deformations within 4 µm accuracy.We consider configurations in which the CDC is perfectly aligned and configurations in which it is misaligned.Possible effects on the determination of the IR are also introduced by using parameters measured on misaligned samples of simulated e + e − → µ + µ − events, to fully mimic the procedure used for real data.For each misalignment configuration, we fit to the reconstructed signal candidates and estimate the lifetime bias.We estimate the systematic uncertainty due to imperfect detector alignment as the sum in quadrature of the largest biases observed in each of the statistical and systematic contributions.The resulting uncertainties are 0.72 fs and 1.70 fs for D 0 → K − π + and D + → K − π + π + decays, respectively.The absolute length scale of the vertex detector is determined with a precision significantly better than 0.01% and contributes negligibly to the systematic uncertainty.
The measurement of momenta is calibrated with the peak positions of abundant charmed-, strange-, and bottom-hadron decays.Uncertainty in the scaling of the momenta results in a systematic uncertainty in the lifetimes of 0.19 fs for D 0 and 0.48 fs for D + .Uncertainties in the D 0 and D + masses [7] contribute negligibly to the systematic uncertainty.
As a cross-check, a statistically independent measurement of the D 0 lifetime is performed using approximately 146 × 10 3 D * + → D 0 (→ K − π + π − π + )π + decays reconstructed in data with criteria similar to those used for the D 0 → K − π + mode and a signal purity greater than 99%.The resulting lifetime, 408.8 ± 1.2 (stat) fs, agrees with the value determined from the D 0 → K − π + mode.
Finally, the internal consistency of the measurement is tested by repeating the full analysis on various subsets of the data, i.e., running periods and running conditions, charmed-meson momentum and flight direction, and D * + or D * − candidates.We have also studied different selection criteria and sideband definitions.In all cases, the resulting changes in the lifetimes are insignificant.
In conclusion, the D 0 and D + lifetimes are measured using e + e − → cc data collected by the Belle II experiment corresponding to an integrated luminosity of are the world's most precise to date and are consistent with previous measurements [7].Assuming that all systematic uncertainties are fully correlated between the two measurements, except those due to the background contamination (assumed uncorrelated), the total correlation coefficient is 18%.The ratio of lifetimes is τ (D + )/τ (D 0 ) = 2.510±0.013(stat)±0.007(syst).These results demonstrate the vertexing capabilities of the Belle II detector and confirm our understanding of systematic effects that impact future decay-time-dependent analyses of neutral-meson mixing and mixing-induced CP violation.

Figure 1 :
Figure 1: Mass distributions of (top) D 0 → K − π + and (bottom) D + → K − π + π + candidates with fit projections overlaid.The vertical dashed and (for the bottom plot) dotted lines indicate the signal regions and the sideband, respectively.

Table I :
Systematic uncertainties.