Measurement of the charm-mixing parameter $y_{CP}$

A measurement of the charm-mixing parameter $y_{CP}$ using $D^0 \to K^+ K^-$, $D^0 \to \pi^+ \pi^-$, and $D^0 \to K^- \pi^+$ decays is reported. The $D^0$ mesons are required to originate from semimuonic decays of $B^-$ and $\overline{B}^0$ mesons. These decays are partially reconstructed in a data set of proton-proton collisions at center-of-mass energies of 7 and 8 TeV collected with the LHCb experiment and corresponding to an integrated luminosity of 3 fb$^{-1}$. The $y_{CP}$ parameter is measured to be $(0.57 \pm 0.13(\rm{stat.}) \pm 0.09(\rm{syst.}))\%$, in agreement with, and as precise as, the current world-average value.

Neutral charm mesons can change their flavor and turn into antimesons, and vice versa, before they decay.This phenomenon, known as flavor oscillation or D 0 -D 0 mixing, occurs because the eigenstates of the Hamiltonian governing the time evolution of the neutral D system are superpositions of the flavor eigenstates, |D 1,2 = p|D 0 ± q|D 0 , where p and q are complex parameters satisfying |p| 2 + |q| 2 = 1.In the limit of charge-parity (CP ) symmetry, q equals p and the oscillations are characterized by only two dimensionless parameters, x ≡ (m 1 − m 2 )/Γ and y ≡ (Γ 1 − Γ 2 )/2Γ, where m 1(2) and Γ 1 (2) are the mass and decay width of the CP -even (odd) eigenstate D 1 (2) , respectively, and Γ ≡ (Γ 1 + Γ 2 )/2 is the average decay width [1].The values of x and y are of the order of 1% or smaller [2].In presence of CP violation, the mixing rates for mesons produced as D 0 and D 0 differ, further enriching the phenomenology.
Because of D 0 -D 0 mixing, the effective decay width Γ CP + of decays to CP -even final states, such as h + h − (h = K, π), differs from the average width Γ.The latter can be measured in decays that involve an equal mixture of CP -even and CP -odd states, such as D 0 → K − π + .The inclusion of charge-conjugate processes is implied unless stated otherwise.The quantity is equal to the mixing parameter y if CP symmetry is conserved.Otherwise, it is related to x, y, |q/p|, and φ ≡ arg(qA/pA), as 2y CP ≈ (|q/p| + |p/q|) y cos φ − (|q/p| − |p/q|) x sin φ, where A (A) is the D 0 (D 0 ) decay amplitude [3,4].The approximation holds for decays, such as D 0 → h + h − , that can be described by a single amplitude.Neglecting the O(10 −3 ) difference between the phases of the D 0 → K + K − and D 0 → π + π − decay amplitudes, φ is universal and y CP is independent of the h + h − final state.The current world average value of y CP , (0.84 ± 0.16)% [2], is dominated by measurements at the B factories [5,6] and is consistent with the known value of y, (0.62±0.07)% [2].The only measurement of y CP at a hadron collider, (0.55 ± 0.63 (stat) ± 0.41 (syst))%, has been made by the LHCb collaboration using a sample of proton-proton collisions corresponding to an integrated luminosity of 29 pb −1 [7].Improving the precision of both y CP and y might lead to evidence of CP violation in D 0 -D 0 mixing if they differ significantly.This would offer sensitivity to a broad class of non-standard-model processes that could contribute to the mixing amplitude by increasing the oscillation rate and/or introducing CP -violation effects that are highly suppressed in the standard model [8][9][10][11][12][13].
In this Letter, a measurement of y CP using D 0 → K + K − , D 0 → π + π − and D 0 → K − π + decays is reported.The D 0 mesons are required to originate from semimuonic decays of B − or B 0 mesons, collectively referred to as B → D 0 µ − ν µ X.The difference between the widths of D 0 decays to CP -even and CP -mixed final states, is measured from a fit to the ratio between signal yields as a function of the D 0 decay time.The parameter y CP is then calculated from the measured value of ∆ Γ and the precisely known value of Γ [1] as , where m is the known value of the D 0 mass [1], L is the vector connecting the B and the D 0 decay vertices, and p is the momentum of the D 0 meson.Biases on the measurement of the D 0 decay time, which are usually introduced by requiring the decay products of the charm meson to be detached from the pp interaction point, are largely suppressed by imposing displacement requirements mainly on the muon of B → D 0 µ − ν µ X decays.This yields a selection efficiency as a function of the D 0 decay time (decay-time acceptance) which is very similar for D 0 → h + h − and D 0 → K − π + decays.Residual differences, of the order of a few percent, are corrected for in the analysis.The correction is validated using control samples of data, which also include D + → K − π + π + and D + → K + K − π + decays with D + decays originating from semimuonic B decays (referred to as B → D + µ − ν µ X).To avoid potential experimenter's bias, the measured value of y CP remained unknown during the development of the analysis and was examined only after the analysis procedure and the evaluation of the systematic uncertainties were finalized.Semileptonic decays of B mesons are partially reconstructed in a data set collected with the LHCb experiment in pp collisions at center-of-mass energies of 7 and 8 TeV and corresponding to an integrated luminosity of 3 fb −1 .The LHCb detector is a singlearm forward spectrometer equipped with precise charged-particle vertexing and tracking detectors, hadron-identification detectors, calorimeters, and muon detectors, optimized for the study of bottom-and charm-hadron decays [14,15].Simulation [16][17][18] is used to model all relevant sources of decays, correct the data for the decay-time acceptance, study the decay-time resolution, and to evaluate systematic uncertainties on the measurement.
The online event selection is performed by a trigger [19], which consists of a hardware stage, based on information from the calorimeter and muon systems, followed by a twolevel software stage, which applies a full event reconstruction.To select semimuonic B decays, the hardware trigger requires a muon candidate with transverse momentum exceeding 1.5 to 1.8 GeV/c, depending on the data-taking period.In the first level of the software trigger, the selected muon is required to be displaced from any pp interaction point.These requirements do not bias the decay time of the D candidate.In the second level of the software trigger, the muon candidate is associated with one, two, or three charged particles, all displaced from the closest interaction point.This association can bias the decay time, favoring shorter D flight distances, as the muon and the D decay products satisfying the trigger criteria must be consistent with originating from a common displaced vertex.
In the offline reconstruction, the muon candidate is combined with charged particles according to the topology and kinematics of B → D 0 µ − ν µ X and B → D + µ − ν µ X decays.The requirements to select B → D 0 µ − ν µ X decays are inherited from the analysis reported in Ref. [20]; those for B → D + µ − ν µ X decays are taken from Ref. [21].In these selections, the requirement of displacement from the closest interaction point for all the D decay products are particularly relevant for the measurement of y CP as they bias the D decay-time distribution, being more efficient for decays with a larger flight distance.The requirement is χ 2 IP > 9, where χ 2 IP is defined as the difference in the vertex-fit χ 2 of a given interaction point reconstructed with and without the track being considered.The following additional requirements, not used in Refs.[20,21], are applied.The Dµ invariant mass, m(Dµ), must not exceed 5.2 GeV/c 2 , to suppress genuine charm decays accidentally combined with unrelated muon candidates.The mass of the D candidate must be in the range 1.825-1.920GeV/c 2 .Its decay time must be larger than 0.15 ps to minimize a bias observed in simulation at t ≈ 0 due to the reconstruction of the B vertex.A requirement on the component of the D momentum transverse to the B flight direction is applied as a function of the corrected B mass to suppress decays of b hadrons into final states with a pair of charm hadrons, of which one decays semileptonically, and background from semitauonic The corrected B mass is determined from the Dµ invariant mass using the momentum of the Dµ system transverse to the B flight direction, p ⊥ (Dµ), to partially compensate for the momentum of the unreconstructed decay products, as m 2 (Dµ) + p 2 ⊥ (Dµ) + p ⊥ (Dµ).After the selection, these background contributions total to at most 1.5% of the signal yield.A contamination of about 1% of D decays produced directly in the pp collision (prompt D) is also estimated to be present in the selected sample.All these background decays are checked to have negligible impact on the measurement of y CP .
Figure 1 shows the D 0 mass distributions of the selected candidates.Prominent signal peaks at the known D 0 mass values are visible on top of a smooth background made of random combinations of charged particles faking a D 0 candidate.The small contamination of prompt D 0 decays is included in the signal peak.Binned χ 2 fits to the mass distributions determine the signal yields reported in Table 1, together with the yields of the control samples of D + decays.The fits use a probability density function (pdf) consisting of a Johnson S U distribution [22] (or the sum of a Johnson S U and a Gaussian distribution in the case of D 0 → K − π + and D + → K − π + π + decays) to describe the signal peak and a linear distribution to describe the background.
The sample is split into 19 disjoint subsets (bins) of D decay time spanning the range 0.15-4 ps.The signal yields are determined in each decay-time bin with fits to the D mass distribution using the same pdf as described above.In these fits all signal-shape parameters are fixed to the values from the decay-time-integrated fits, with the exception of the mean and width of the Johnson function.The ratio between D 0 → K + K − (or D 0 → π + π − ) and D 0 → K − π + signal yields as a function of decay time is fitted to determine the value

Decay
Signal yield [10 3 ] The fit minimizes a χ 2 function where the signal-yield ratio in a decay-time bin is described by the ratio of the integrals of two decreasing exponential functions, one for the numerator with exponent Γ CP + = ∆ Γ + Γ, and the other for the denominator with exponent Γ.The value of Γ is fixed to its world average of 2.4384 ps −1 [1], while ∆ Γ and a decay-time-independent normalization factor of the ratio are free to vary in the fit.It should be noted that Γ can be fixed to any arbitrary value, since the distribution of the ratio is only sensitive to ∆ Γ .In the fit, the signal-yield ratio is corrected in each decay-time bin by a factor calculated as the ratio of the decay-time acceptances of the decays in the numerator and the denominator.This correction is determined from simulation and shows up to 6% variations around unity as a function of D 0 decay time (Figure 2).The correction is similar in magnitude, but with an opposite trend as a function of t, for the determination of ∆ Γ with D 0 → K + K − and D 0 → π + π − decays.
Several null tests are performed on data to prove that the estimates of the signal yields are unbiased, and that the corrections from simulation are reliable.The tests use samples of (i) D + → K + K − π + and D + → K − π + π + decays, (ii) D + → K − π + π + decays, (iii) D 0 → K − π + decays, and (iv) D 0 → K + K − decays.In test (i), the width difference is measured by fitting the yield ratio of D + → K + K − π + to D + → K − π + π + decays.The corrections for the ratio of decay-time acceptances are similar to those in the y CP measurement.In tests (ii)-(iv), the selected data are split randomly into two independent sets: one is used as the denominator sample, and the other, featuring a tighter requirement of χ 2 IP > 60 for the D decay products, is used as the numerator sample.The threshold on χ 2 IP is chosen such that the decay-time acceptance of the channels used in the numerator and the denominator differ by up to 40%, i.e. almost an order of magnitude larger than the variation present in the y CP measurement.In all tests, the measured decay-width difference is consistent with zero, with fit p-values ranging from 8% to 84%.The two most precise tests, (ii) and (iii), correspond to a validation of the measurement of y CP with an uncertainty of 0.14%, which includes the limited knowledge of the decay-time acceptance correction.Another test (v) consists in measuring the decay-width difference of D + and D 0 mesons, using the largest-yield samples of D + → K − π + π + and D 0 → K − π + decays.In this measurement, the ratio of the decay-time acceptances presents variations up to about 10%.However, the decays considered in the numerator and the denominator have sufficiently different topologies that potential biases on the measurement of the width difference are not suppressed in the ratio at the same level as in the y CP measurement.In addition, the very different lifetimes between D + and D 0 mesons lead to a signal-yield ratio spanning over a very broad interval, with a maximum approximately 25 times larger than its minimum.The ratio of D + to D 0 lifetimes is determined to be 2.5141 ± 0.0082, where the uncertainty is only statistical, in agreement with the known value of 2.536 ± 0.019 [1].Biases that scale with ∆ Γ are excluded by this test within a relative precision of about 1%.In summary, the five tests yield results consistent with the expectations with a χ 2 of 5.5, which corresponds to a p-value of 36%.The tests also confirm that background originating from prompt D decays, from b-hadron decays to double-charm final states, and from semitauonic B decays can be neglected.They contaminate all samples considered in the tests with fractions similar to those estimated in the y CP measurement.
Figure 3 shows the acceptance-corrected signal-yield ratio measured for the D 0 → K + K − and D 0 → π + π − decays with respect to D 0 → K − π + decays, with fit projections overlaid.The obtained values of ∆ Γ and y CP are reported in Table 2.The use of a common reference sample (D 0 → K − π + ) does not introduce any significant correlation between the statistical uncertainties of the D 0 → K + K − and D 0 → π + π − measurements.

Decay
D 0 → K + K − 0.0153 ± 0.0036 ± 0.0027 0.63 ± 0.15 ± 0.11 D 0 → π + π − 0.0093 ± 0.0067 ± 0.0038 0.38 ± 0.28 ± 0.15 (0.28%) on y CP , are assigned for the measurement done with D 0 → K + K − (D 0 → π + π − ) decays.The correlation between the systematic uncertainties is 5%.They are dominated by the knowledge of the correction for the ratio of decay-time acceptances, which is limited by the finite size of the simulated samples.This yields systematic uncertainties of 0.0026 ps −1 (0.0037 ps −1 ) on ∆ Γ and 0.11% (0.15%) on y CP , which are uncorrelated between the D 0 → K + K − and D 0 → π + π − measurements.Other systematic uncertainties, contributing less, are associated with: the assumed decay model and composition of the simulated samples of semileptonic B decays (0.0006 ps −1 on ∆ Γ , 0.02% on y CP ); possible biases introduced by the fit method as determined in large ensembles of pseudoexperiments (0.0004 ps −1 on ∆ Γ , 0.02% on y CP ); and the assumption of negligible decay-time resolution (0.0003 ps −1 on ∆ Γ , 0.01% on y CP ).These systematic uncertainties are fully correlated between the measurements with D 0 → K + K − and D 0 → π + π − decays.Asymmetric production of D 0 and D 0 mesons from semileptonic B − and B 0 decays produce biases on y CP that are smaller than 10 −5 .Uncertainties on the relative alignement of subdetectors to measure the decay length have negligible contribution, as well as the uncertainty of the input value of Γ, 2.4384 ± 0.0089 ps −1 [1], used to determine y CP from ∆ Γ .Finally, consistency checks based on repeating the y CP measurement on independent subsamples chosen according to data-taking periods, trigger-selection criteria and interaction-point multiplicity all yield compatible results within statistical fluctuations.In summary, the charm mixing-parameter y CP is measured using D 0 → K + K − , D 0 → π + π − and D 0 → K − π + decays originating from semileptonic B − and B 0 decays produced in pp collision data collected with the LHCb experiment at center-of-mass energies of 7 and 8 TeV, and corresponding to an integrated luminosity of 3 fb −1 .The results from D 0 → K + K − , y CP = (0.63 ± 0.15 (stat) ± 0.11 (syst))%, and D 0 → π + π − decays, y CP = (0.38 ± 0.28 (stat) ± 0.15 (syst))%, are consistent between each other and with determinations from other experiments [2].The value of y CP measured in the D 0 → K + K − mode is the most precise to date from a single experiment.The two measurements are combined and yield y CP = (0.57± 0.13 (stat) ± 0.09 (syst))%, which is consistent with and as precise as the current world average value, (0.84 ± 0.16)% [2].The result is also consistent with the known value of the mixing parameter y, (0.62 ± 0.07)% [2], showing no evidence for CP violation in D 0 -D 0 mixing.

Table 1 :
Signal yields of the selected candidates.

Table 2 :
Measured values of ∆ Γ and y CP .The first uncertainty is statistical, the second is systematic.