Updated determination of $D^0$-$\overline{D}{}^0$ mixing and CP violation parameters with $D^0\to K^+\pi^-$ decays

We report measurements of charm-mixing parameters based on the decay-time-dependent ratio of $D^0\to K^+\pi^-$ to $D^0\to K^-\pi^+$ rates. The analysis uses a data sample of proton-proton collisions corresponding to an integrated luminosity of $5.0$ fb$^{-1}$ recorded by the LHCb experiment from 2011 through 2016. Assuming charge-parity (CP) symmetry, the mixing parameters are determined to be $x'^2=(3.9 \pm 2.7) \times10^{-5}$, $y'=(5.28 \pm 0.52) \times 10^{-3}$, and $R_D=(3.454 \pm 0.031)\times10^{-3}$. Without this assumption, the measurement is performed separately for $D^0$ and $\overline{D}{}^0$ mesons, yielding a direct CP-violating asymmetry $A_D =(-0.1\pm9.1)\times10^{-3}$, and magnitude of the ratio of mixing parameters $1.00<|q/p|<1.35$ at the $68.3\%$ confidence level. All results include statistical and systematic uncertainties and improve significantly upon previous single-measurement determinations. No evidence for CP violation in charm mixing is observed.


Introduction
The mass eigenstates of neutral charm mesons are linear combinations of the flavor eigenstates, |D 1,2 = p|D 0 ± q|D 0 , where p and q are complex-valued coefficients. This results in D 0 -D 0 oscillations. In the limit of charge-parity (CP ) symmetry, oscillations are characterized by the dimensionless differences in mass, x ≡ ∆m/Γ ≡ (m 2 − m 1 )/Γ, and decay width, y ≡ ∆Γ/2Γ ≡ (Γ 2 − Γ 1 )/2Γ, between the CP -even (D 2 ) and CP -odd (D 1 ) mass eigenstates, where Γ is the average decay width of neutral D mesons. If CP symmetry does not hold, the oscillation probabilities for mesons produced as D 0 and D 0 can differ, further enriching the phenomenology. Long-and short-distance amplitudes govern the oscillations of neutral D mesons [1][2][3]. Long-distance amplitudes depend on the exchange of low-energy gluons and are challenging to calculate. Short-distance amplitudes may include contributions from a broad class of particles not described in the standard model, which might affect the oscillation rate or introduce a difference between the D 0 and D 0 meson decay rates. The study of CP violation in D 0 oscillations therefore offers sensitivity to non-standard-model phenomena [4][5][6][7].
This paper reports measurements of CP -averaged and CP -violating mixing parameters in D 0 -D 0 oscillations based on the comparison of the decay-time-dependent ratio of D 0 → K + π − to D 0 → K − π + rates with the corresponding ratio for the charge-conjugate processes. The analysis uses data corresponding to an integrated luminosity of 5.0 fb −1 from proton-proton (pp) collisions at 7, 8, and 13 TeV center-of-mass energies, recorded with the LHCb experiment from 2011 through 2016. This analysis improves upon a previous measurement [12], owing to the tripling of the sample size and an improved treatment of systematic uncertainties. The inclusion of charge-conjugate processes is implicitly assumed unless stated otherwise.
The neutral D-meson flavor at production is determined from the charge of the lowmomentum pion (soft pion), π + s , produced in the flavor-conserving strong-interaction decay D * (2010) + → D 0 π + s . The shorthand notation D * + is used to indicate the D * (2010) + meson throughout. We denote as right-sign (RS) the D * + → D 0 (→ K − π + )π + s process, which is dominated by a Cabibbo-favored amplitude. Wrong-sign (WS) decays, D * + → D 0 (→ K + π − )π + s , arise from the doubly Cabibbo-suppressed D 0 → K + π − decay and the Cabibbo-favored D 0 → K + π − decay that follows D 0 -D 0 oscillation. Since the mixing parameters are small, |x|, |y| 1, the CP -averaged decay-time-dependent ratio of WS-to-RS rates is approximated as [1][2][3][4] where t is the proper decay time, τ is the average D 0 lifetime, and R D is the ratio of suppressed-to-favored decay rates. The parameters x and y depend on the mixing parameters, x ≡ x cos δ + y sin δ and y ≡ y cos δ − x sin δ, through the strong-phase difference δ between the suppressed and favored amplitudes, which was measured at the CLEO-c and BESIII experiments [17,18]. If CP violation occurs, the decay-rate ratios R + (t) and R − (t) of mesons produced as D 0 and D 0 , respectively, are functions of independent sets of mixing parameters, R ± D , (x ± ) 2 , and y ± . The parameters R + D and R − D differ if the ratio between the suppressed and favored decay amplitudes is not CP symmetric, indicating direct CP violation. Violation of CP symmetry either in mixing, |q/p| = 1, or in the interference between mixing and decay amplitudes, φ ≡ arg qA(D 0 → K + π − )/pA(D 0 → K + π − ) = δ, are referred to as manifestations of indirect CP violation and generate differences between ((x + ) 2 , y + ) and Experimental effects such as differing efficiencies for reconstructing WS and RS decays may bias the observed ratios of signal decays and, therefore, the mixing-parameter results. We assume that the efficiency for reconstructing and selecting the K ∓ π ± π + s final state approximates as the product of the efficiency for the K ∓ π ± pair from the D 0 decay and the efficiency for the soft pion. The observed WS-to-RS yield ratio then equals R(t) multiplied by the ratio of the efficiencies for reconstructing K + π − and K − π + pairs, which is the only relevant instrumental nuisance. The asymmetry in production rates between D * + and D * − mesons in the LHCb acceptance and asymmetries in detecting soft pions of different charges cancel in the WS-to-RS ratio.
Candidate D * + mesons produced directly in the collision (primary D * + ) are reconstructed while suppressing background contributions from charm mesons produced in the decay of bottom hadrons (secondary D * + ) and misreconstructed decays. Residual contaminations from such backgrounds are measured using control regions. The asymmetry in K ± π ∓ reconstruction efficiency is estimated using control samples of charged D-meson decays. The yields of RS and WS primary D * + candidates are determined, separately for each flavor, in intervals (bins) of decay time by fitting the D * + mass distribution of candidates consistent with being D 0 decays. We fit the resulting WS-to-RS yield ratios as a function of decay time to measure the mixing and CP -violation parameters, including the effects of instrumental asymmetries, residual background contamination, and all considered systematic contributions. To ensure unbiased results, the differences in the decay-time dependence of the WS D 0 and D 0 samples are not examined until the analysis procedure is finalized.

The LHCb detector
The LHCb detector [19] is a single-arm forward spectrometer covering the pseudorapidity range 2 < η < 5, designed for the study of particles containing b or c quarks. The detector achieves high precision charged-particle tracking using a silicon-strip vertex detector surrounding the pp interaction region, a large-area silicon-strip detector located upstream of a dipole magnet with a bending power of about 4 Tm, and three layers of silicon-strip detectors and straw drift tubes placed downstream of the magnet. The tracking system provides a measurement of charged-particle momentum p with a relative uncertainty varying from 0.5% at low momentum to 1.0% at 200 GeV/c. The typical decay-time resolution for D 0 → K + π − decays is 10% of the D 0 lifetime. The polarity of the dipole magnet is reversed periodically throughout data-taking. The minimum distance of a charged-particle trajectory (track) to a proton-proton interaction space-point (primary vertex), the impact parameter, is measured with (15 + 29/p T ) µm resolution, where p T is the component of the momentum transverse to the beam, in GeV/c. Charged hadrons are identified using two ring-imaging Cherenkov detectors. Photons, electrons and hadrons are identified by scintillating-pad and preshower detectors, and an electromagnetic and a hadronic calorimeter. Muons are identified by alternating layers of iron and multiwire proportional chambers. The online event selection is performed by a hardware trigger, based on information from the calorimeter and muon detectors, followed by a software trigger, based on information on displaced charged particles reconstructed in the event.
Offline-like quality detector alignment and calibrations, performed between the hardware and software stages, are available to the software trigger for the 2015 and 2016 data [20,21]. Hence, for these data the analysis uses candidates reconstructed in the software trigger to reduce event size.

Event selection and candidate reconstruction
Events enriched in D * + candidates originating from the primary vertex are selected by the hardware trigger by imposing that either one or more D 0 decay products are consistent with depositing a large transverse energy in the calorimeter or that an accept decision is taken independently of the D 0 decay products and soft pion. In the software trigger, one or more D 0 decay products are required to be inconsistent with charged particles originating from the primary vertex and, for 2015 and 2016 data, loose particle-identification criteria are imposed on these final-state particles. Each D 0 candidate is then combined with a low-momentum positive-charge particle originating from the primary vertex to form a D * + candidate.
In the offline analysis, criteria on track and primary-vertex quality are imposed. To suppress the contamination from misidentified two-body D 0 decays, the pion and kaon candidates from the D 0 decay are subjected to stringent particle-identification criteria. An especially harmful background is generated by a 3% contribution of soft pions misreconstructed by combining their track segments in the vertex detector with unrelated segments in the downstream tracking detectors. The track segments in the vertex detector are genuine, resulting in properly measured opening angles in the D * + → D 0 π + s decay. Since the opening angle dominates over the π + s momentum in the determination of the D * + mass, such spurious soft pions tend to produce a signal-like peak in the D * + mass spectrum. In addition, they bias the WS-to-RS ratio because the mistaken association with downstream track-segments is prone to charge mismeasurements. We suppress such candidates with stringent requirements on a dedicated discriminant based on many low-level variables associated with track reconstruction [22]. Candidates consistent with the D * + decay topology are reconstructed by computing the two-body mass M (D 0 π + s ) using the known D 0 and π + masses [23] and the reconstructed momenta [24]. The mass resolution is improved by nearly a factor of two with a kinematic fit that constrains the D * + candidate to originate from a primary vertex [25]. If multiple primary vertices are reconstructed, the vertex resulting from the fit with the best χ 2 probability is chosen. The sample is further enriched in primary charm decays by restricting the impact-parameter chi-squared, χ 2 IP , of the D 0 and π + s candidates such that the candidates point to the primary vertex. The χ 2 IP variable is the difference between the χ 2 of the primary-vertex fit reconstructed including or excluding the considered particle, and offers a measure of consistency with the hypothesis that the particle originates from the primary vertex. Only opposite-charge particle pairs with K ∓ π ± mass within 24 MeV/c 2 (equivalent to approximately three times the mass resolution) of the known D 0 mass [23] and and π + π − masses more than 40 MeV/c 2 away from the D 0 mass are retained. Accidental combinations of a genuine D 0 with a random soft pion are first suppressed by removing the 13% of events where more than one D * + candidate is reconstructed. We then use an artificial neural-network discriminant that exploits the π + s pseudorapidity, transverse momentum, and particle-identification information, along with the track multiplicity of the event. The discriminant is trained on an independent RS sample to represent the WS signal features and on WS events containing multiple candidates to represent background. Finally, we remove from the WS sample events where the same D 0 candidate is also used to reconstruct a RS decay, which reduces the background by 16% with no significant loss of signal.

Yield determination
The RS and WS signal yields are determined by fitting the M (D 0 π + s ) distribution of D 0 signal candidates. The decay-time-integrated M (D 0 π + s ) distributions of the selected RS and WS candidates are shown in Fig. 1. The smooth background is dominated by favored D 0 → K − π + and D 0 → K + π − decays associated with random soft-pion candidates. The sample contains approximately 1.77×10 8 RS and 7.22×10 5 WS signal decays. Each sample is divided into 13 subsamples according to the decay time, and signal yields are determined for each subsample using an empirical shape [11]. We assume that the signal shapes are common to WS and RS decays for a given D * meson flavor whereas the descriptions of the backgrounds are independent. The decay-time-dependent WS-to-RS rate ratios R + and R − observed in the D 0 and D 0 samples, respectively, and their difference, are shown in Fig. 2. The ratios and difference include corrections for the relative efficiencies for reconstructing K − π + and K + π − final states.

Determination of oscillation parameters
The mixing parameters are determined by minimizing a χ 2 function that includes terms for the difference between the observed and predicted ratios and for systematic effects, The observed WS-to-RS yield ratio and its statistical uncertainty in the decay-time bin i are denoted by r ± i and σ ± i , respectively. The associated predicted value R ± i corresponds to the decay-time integral over bin i of Eq. (1), including bin-specific corrections. The parameters associated with these corrections are determined separately for data collected in different LHC and detector configurations and vary independently in the fit within their constraint χ 2 corr in Eq. (2). Such corrections account for small biases due to (i) the decay-time evolution of the 1%-10% fraction of signal candidates originating from b-hadron decays, (ii) the approximately 0.3% component of the background from misreconstructed charm decays that peak in the signal region, and (iii) the effect of instrumental asymmetries in the K ± π ∓ reconstruction efficiencies. The secondary-D * + fraction is determined by fitting, in each decay-time bin, the χ 2 IP distribution of RS D 0 signal decays. The peaking background, dominated by D 0 → K − π + decays in which both final-state particles are misidentified, is determined by extrapolating into the D 0 signal mass region the contributions from misreconstructed charm decays identified by reconstructing the two-body mass under various mass hypotheses for the decay products. The relative efficiency ± r accounts for the effects of instrumental asymmetries in the K ± π ∓ reconstruction efficiencies, mainly caused by K − mesons having a larger nuclear interaction cross-section with matter than K + mesons. These asymmetries are measured in data to be typically 0.01 with 0.001 precision, independent of decay time. They are derived from the efficiency ratio + r = 1/ − r = (K + π − )/ (K − π + ), obtained by comparing the ratio of D − → K + π − π − and D − → K 0 S (→ π + π − )π − yields with the ratio of the corresponding charge-conjugate decay yields. The asymmetry between D + and D − production rates [26] cancels in this ratio, provided that the kinematic distributions are consistent across samples. We therefore weight the D − → K + π − π − candidates so that their kinematic distributions match those in the D − → K 0 S π − sample. We then determine ± r as functions of kaon momentum to account for the known momentum-dependence of the asymmetry between K + and K − interaction rates with matter. In addition, a systematic uncertainty for possible residual contamination from spurious soft pions is included through a 1.05-1.35 scaling of the overall uncertainties. The scaling value is chosen such that a fit with a constant function of the time-integrated WS-to-RS ratio versus false-pion probability has unit reduced χ 2 .
The observed WS-to-RS yield ratios for the D 0 and D 0 samples are studied first with bin-by-bin arbitrary offsets designed to mimic the effect of significantly different mixing parameters in the two samples. To search for residual systematic uncertainties, the analysis is repeated on statistically independent data subsets chosen according to criteria likely to reveal biases from specific instrumental effects. These criteria include the data-taking year (2011-2012 or 2015-2016), the magnet field orientation, the number of primary vertices in the event, the candidate multiplicity per event, the trigger category, the D 0 momentum and χ 2 IP with respect to the primary vertex, and the per-candidate probability to reconstruct a spurious soft pion. The resulting variations of the measured CP -averaged and CP -violating parameters are consistent with statistical fluctuations, with p-values distributed uniformly in the 4%-85% range.

Results
The efficiency-corrected WS-to-RS yield ratios are subjected to three fits. The first fit allows for direct and indirect CP violation; the second allows only for indirect CP violation by imposing R + D = R − D ; and the third is a fit under the CP -conservation hypothesis, in which all mixing parameters are common to the D 0 and D 0 samples. The fit results and their projections are presented in Table 1 and Fig. 2, respectively. Figure 3 shows the central values and confidence regions in the (x 2 , y ) plane. For each fit, 208 WS-to-RS ratio data points are used, corresponding to 13 ranges of decay time; distinguishing D * + from D * − decays; two magnetic-field orientations; and 2011, 2012, 2015, and 2016 data sets. The consistency of the data with the hypothesis of CP symmetry is determined from the change in χ 2 probability between the fit that assumes CP conservation and the fit in which CP violation is allowed. The resulting p-value is 0.57 (0.37) for the fit in which both direct and indirect (indirect only) CP violation is allowed, showing that the data are compatible with CP symmetry.
The fit uncertainties incorporate both statistical and systematic contributions. The statistical uncertainty, determined in a separate fit by fixing all nuisance parameters to their central values, dominates the total uncertainty. The systematic component is obtained by subtraction in quadrature. The leading systematic uncertainty is due to residual secondary-D * + contamination and does not exceed half of the statistical uncertainty. The second largest contribution is due to spurious soft pions. Smaller effects are due to peaking backgrounds for the CP -averaged results, and uncertainties in detector asymmetries for the CP -violating results. All reported results, p-values, and the contours shown in Fig. 3, include total uncertainties. Direct CP violation would produce a nonzero intercept at t = 0 in the efficiencycorrected difference of WS-to-RS yield ratios between D 0 and D 0 mesons shown in Fig. 2 (c). We parametrize this effect with the asymmetry measured in the fit that allows for direct CP violation, where the first uncertainty is statistical and the second systematic. Indirect CP violation would result in a time dependence of the efficiency-corrected difference of yield ratios, which is not observed in Fig. 2 (c). From the results of the fit allowing for direct and indirect CP violation, a likelihood for |q/p| is constructed using the relations x ± = |q/p| ±1 (x cos φ ± y sin φ) and y ± = |q/p| ±1 (y cos φ ∓ x sin φ). Confidence intervals are derived with a likelihood-ratio ordering [27], assuming that the parameter correlations are independent of the true values  of the mixing parameters. We determine 1.00 < |q/p| < 1.35 and 0.82 < |q/p| < 1.45 at the 68.3% and 95.5% confidence levels, respectively.
The R D result departs from the previous result based on a subset of the same data [12], which was biased by the then-undetected residual spurious-pion background. Since such background induces an apparent global shift toward higher WS-to-RS ratio values, the bias affects predominantly the R D measurement and less severely the mixing-parameter determination. The systematic uncertainties are significantly reduced because the dominant components are statistical in nature or sensitive to a generally improved understanding of the data quality.

Summary
We study D 0 -D 0 oscillations using D * + → D 0 (→ K + π − )π + decays reconstructed in a data sample of pp collisions collected by the LHCb experiment from 2011 through 2016, corresponding to an integrated luminosity of 5.0 fb −1 . Assuming CP conservation, the mixing parameters are measured to be x 2 = (3.9 ± 2.7) × 10 −5 , y = (5.28 ± 0.52) × 10 −3 , and R D = (3.454 ± 0.031) × 10 −3 . The results are twice as precise as previous LHCb results [12] that were based on a subset of the present data, and supersede them. Studying D 0 and D 0 decays separately shows no evidence for CP violation and provides the current most stringent bounds on the parameters A D and |q/p| from a single measurement, A D = (−0.1 ± 9.1) × 10 −3 and 1.00 < |q/p| < 1.35 at the 68.3% confidence level.