Measurement of the $D\to K^-\pi^+$ strong phase difference in $\psi(3770)\to D^0\overline{D}{}^0$

We study $D^0\overline{D}{}^0$ pairs produced in $e^+e^-$ collisions at $\sqrt{s}=3.773$ GeV using a data sample of 2.92 fb$^{-1}$ collected with the BESIII detector. We measure the asymmetry $\mathcal{A}^{CP}_{K\pi}$ of the branching fractions of $D \to K^-\pi^+$ in $CP$-odd and $CP$-even eigenstates to be $(12.7\pm1.3\pm0.7)\times10^{-2}$. $\mathcal{A}^{CP}_{K\pi}$ can be used to extract the strong phase difference $\delta_{K\pi}$ between the doubly Cabibbo-suppressed process $\overline{D}{}^{0}\to K^-\pi^+$ and the Cabibbo-favored process $D^0\to K^- \pi^+$. Using world-average values of external parameters, we obtain $\cos\delta_{K\pi} = 1.02\pm0.11\pm0.06\pm0.01$. Here, the first and second uncertainties are statistical and systematic, respectively, while the third uncertainty arises from the external parameters. This is the most precise measurement of $\delta_{K\pi}$ to date.


Introduction
Within the Standard Model, the short-distance contribution to D 0 -D 0 oscillations is highly suppressed by the GIM mechanism [1] and by the magnitude of the CKM matrix elements [2] involved. However, long distance effects, which cannot be reliably calculated, will also affect the size of mixing. Studies of D 0 -D 0 oscillation provide knowledge of the size of these long-distance effects and, given improved calculations, can contribute to searches for new physics [3]. In addition, improved constraints on charm mixing are important for studies of CP violation (CP V ) in charm physics.
Charm mixing is described by two dimensionless parameters where M 1,2 and Γ 1,2 are the masses and widths of the two mass eigenstates in the D 0 -D 0 system. The most precise determination of the mixing parameters comes from the measurement of the time-dependent decay rate of the wrong-sign process D 0 → K + π − . These analyses are sensitive to y ≡ y cos δ Kπ − x sin δ Kπ and x ≡ x cos δ Kπ + y sin δ Kπ [4], where δ Kπ is the strong phase difference between the doubly Cabibbosuppressed (DCS) amplitude for D 0 → K − π + and the corresponding Cabibbo-favored (CF) amplitude for D 0 → K − π + . In particular, where Knowledge of δ Kπ is important for extracting x and y from x and y . In addition, a more accurate δ Kπ contributes to precision determinations of the CKM unitarity angle φ 3 1 via the ADS method [5]. Using quantum-correlated techniques, δ Kπ can be accessed in the mass-threshold production process e + e − → D 0 D 0 [6]. In this process, D 0 and D 0 are in a C-odd quantum-coherent state where the two mesons necessarily have opposite CP eigenvalues [3]. Thus, threshold production provides a unique way to identify the CP of one neutral D by probing the decay of the partner D. Because CP V in D decays is very small compared with the mixing parameters, we will assume no CP V in our analysis. In this paper, we often refer to K − π + only for simplicity, but charge-conjugate modes are always implied when appropriate.
We denote the asymmetry of CP -tagged D decay rates to K − π + as where S+ (S−) denotes the CP -even (CP -odd) eigenstate. To lowest order in the mixing parameters, we have the relation [7,8] 2r cos δ Kπ where R WS is the decay rate ratio of the wrong sign process D 0 → K − π + (including the DCS decay and D mixing followed by the CF decay) and the right sign process D 0 → K − π + (i.e., the CF decay). Here, D 0 or D 0 refers to the state at production. Using external values for the parameters r, y, and R WS , we can extract δ Kπ from A CP →Kπ . We use the D-tagging method [9] to obtain the branching fractions B D S± →K − π + as Here, n S± and ε S± are yields and detection efficiencies of single tags (ST) of S± final states, while n K − π + ,S± and ε K − π + ,S± are yields and efficiencies of double tags (DT) of (S±, K − π + ) final states, respectively. Based on an 818 pb −1 data sample collected with the CLEO-c detector at √ s = 3.77 GeV and a more complex analysis technique, the CLEO collaboration obtained cos δ Kπ = 0.81 +0.22+0.07 −0.18−0.05 [8]. Using a global fit method including external inputs for mixing parameters, CLEO obtained cos δ Kπ = 1.15 +0.19+0.00 −0.17−0.08 [8]. In this paper, we present a measurement of δ Kπ , using the quantum correlated productions of D 0 -D 0 mesons at √ s = 3.773 GeV in e + e − collisions with an integrated luminosity of 2.92 fb −1 [10] collected with the BESIII detector [11].

The BESIII Detector
The Beijing Spectrometer (BESIII) views e + e − collisions in the double-ring collider BEPCII. BESIII is a general-purpose detector [11] with 93 % coverage of the full solid angle. From the interaction point (IP) to the outside, BESIII is equipped with a main drift chamber (MDC) consisting of 43 layers of drift cells, a timeof-flight (TOF) counter with double-layer scintillator in the barrel part and single-layer scintillator in the end-cap part, an electromagnetic calorimeter (EMC) composed of 6240 CsI(Tl) crystals, a superconducting solenoid magnet providing a magnetic field of 1.0 T along the beam direction, and a muon counter containing multi-layer resistive plate chambers installed in the steel flux-return yoke of the magnet. The MDC spatial resolution is about 135 µm and the momentum resolution is about 0.5 % for a charged track with transverse momentum of 1 GeV/c. The energy resolution for showers in the EMC is 2.5 % at 1 GeV. More details of the spectrometer can be found in Ref. [11].

MC Simulation
Monte Carlo (MC) simulation serves to estimate the detection efficiency and to understand background components. MC samples corresponding to about 10 times the luminosity of data are generated with a geant4-based [12] software package [13], which includes simulations of the geometry of the spectrometer and interactions of particles with the detector materials. kkmc is used to model the beam energy spread and the initial-state radiation (ISR) in the e + e − annihilations [14]. The inclusive MC samples consist of the production of DD pairs with consideration of quantum coherence for all modes relevant to this analysis, the non-DD decays of ψ(3770), the ISR production of low mass ψ states, and QED and qq continuum

Type
Mode processes. Known decays recorded in the Particle Data Group (PDG) [15] are simulated with evtgen [16] and the unknown decays with lundcharm [17]. The final-state radiation (FSR) off charged tracks is taken into account with the photos package [18]. MC samples of D → S±, D → X (X denotes inclusive decay products) processes are used to estimate the ST efficiencies, and MC samples of D → S±, D → Kπ processes are used to estimate the DT efficiencies.

Data Analysis
The decay modes used for tagging the CP eigenstates are listed in Table 1, where π 0 → γγ, η → γγ, K 0 S → π + π − and ω → π + π − π 0 . For each mode, D candidates are reconstructed from all possible combinations of final-state particles, according to the following selection criteria.
Momenta and impact parameters of charged tracks are measured by the MDC. Charged tracks are required to satisfy | cos θ| < 0.93, where θ is the polar angle with respect to the beam axis, and have a closest approach to the IP within ±10 cm along the beam direction and within ±1 cm in the plane perpendicular to the beam axis. Particle identification is implemented by combining the information of normalized energy deposition (dE/dx) in the MDC and the flight time measurements from the TOF. For a charged π(K) candidate, the probability of the π(K) hypothesis is required to be larger than that of the K(π) hypothesis.
Photons are reconstructed as energy deposition clusters in the EMC. The energies of photon candidates must be larger than 25 MeV for | cos θ| < 0.8 (barrel) and 50 MeV for 0.84 < | cos θ| < 0.92 (end-cap). To suppress fake photons due to electronic noise or beam backgrounds, the shower time must be less than 700 ns from the event start time [19]. However, in the case that no charged track is detected, the event start time is not reliable, and instead the shower time must be within ±500 ns from the time of the most energetic shower.
Our π 0 and η candidates are selected from pairs of photons with the requirement that at least one photon candidate reconstructed in the barrel is used. The mass windows imposed are 0.115 GeV/c 2 < m γγ < 0.150 GeV/c 2 for π 0 candidates and 0.505 GeV/c 2 < m γγ < 0.570 GeV/c 2 for η candidates. We further constrain the invariant mass of each photon pair to the nominal π 0 or η mass, and update the four momentum of the candidate according to the fit results.
The K 0 S candidates are reconstructed via K 0 S → π + π − using a vertex-constrained fit to all pairs of oppositely charged tracks, with no particle identification requirements. These tracks have a looser IP requirement: their closest approach to the IP is required to be less than 20 cm along the beam direction, with no requirement in the transverse plane. The χ 2 of the vertex fit is required to be less than 100. In addition, a second fit is performed, constraining the K 0 S momentum to point back to the IP. The flight length, L, obtained from this fit must satisfy L/σ L > 2, where σ L is the estimated error on L. Finally, the invariant mass of the π + π − pair is required to be within (0.487, 0.511) GeV/c 2 , which corresponds to three times the experimental mass resolution.

Single tags using CP modes
For the CP -even and CP -odd modes, the two variables beam-constrained mass M BC and energy difference ∆E are used to identify the signals, defined as follows: Mode Requirement (GeV)

ST Mode
Here p D and E D are the total momentum and energy of the D candidate, and E beam is the beam energy. Signals peak around the nominal D mass in M BC and around zero in ∆E. Boundaries of ∆E requirements are set at approximately ±3σ, except that those of modes containing a π 0 are set as (−4σ, +3.5σ) due to the asymmetric distributions. In each event, only the combination of D candidates with the least |∆E| is kept per mode. In the K + K − and π + π − modes, backgrounds of cosmic rays and Bhabha events are removed with the following requirements. First, the two charged tracks used as the CP tag must have a TOF time difference less than 5 ns and they must not be consistent with being a muon pair or an electron-positron pair. Second, there must be at least one EMC shower (other than those from the CP tag tracks) with an energy larger than 50 MeV or at least one additional charged track detected in the MDC. In the K 0 S π 0 mode, backgrounds due to D 0 → ρπ are negligible after restricting the decay length of K 0 S with L/σ L > 2. In the ρ 0 π 0 and K 0 S ω modes, mass ranges of 0.60 GeV/c 2 < m π + π − < 0.95 GeV/c 2 and 0.72 GeV/c 2 < m π + π − π 0 < 0.84 GeV/c  are required for identifying ρ and ω candidates, respectively.
After applying the criteria on ∆E in Table 2 in all the CP modes, we plot their M BC distributions in Fig. 1, where the peaks at the nominal D 0 mass are evident. Maximum likelihood fits to the events in Fig. 1 are performed, where in each mode the signals are modeled with the reconstructed signal shape in MC simulation convoluted with a smearing Gaussian function, and backgrounds are modeled with the ARGUS function [20]. The Gaussian functions are supposed to compensate for the resolution differences between data and MC simulation. Based on the fit results, the estimated yields of the CP modes are given in Table 3, along with their MC-determined detection efficiencies.

Double tags of the K − π + and CP modes
In the surviving ST CP modes, we reconstruct D → K − π + among the unused charged tracks. The D → K − π + candidate must pass the ∆E requirement listed in Table 2 Figure 2 shows an example of the results for one sample DT combination, (Kπ, K 0 S π 0 ). Table 3 lists the yields of the DT modes and their corresponding detection efficiencies as determined with MC simulations.

Purities of the CP Modes
It is necessary to determine the CP -purity of our ST modes. For the K 0 S π 0 (K 0 S η) mode, the issue is the background under the K 0 S peak. We use the sideband regions of the K 0 S mass, [0.470, 0.477] GeV/c 2 and [0.521, 0.528] GeV/c 2 , in the m π + π − distributions, to estimate the backgrounds from π + π − π 0 (π + π − η). The purity is estimated to be 98.5% (almost 100%) for the K 0 S π 0 (K 0 S η) mode. For the K 0 S ω, K 0 S π 0 π 0 and ρ 0 π 0 modes, due to the complexity of the involved non-resonant and resonant processes, we evaluate the CPpurity directly from our data. We use additional DT combinations, with a clean CP -tag in combination with the mode we wish to study. We look for signals where both D mesons decay with equal CP eigenvalue. If CP is conserved, the same-CP process is prohibited in the quantum-correlated DD production at threshold, unless our studied CP modes are not pure. If we take f S as the fraction of the right CP components in the CP tag mode, we have the yields of the same-CP process written as where mode S is chosen to be (nearly) pure in its CP eigenstate.
We take the modes K 0 S π 0 (S −) and K + K − (S +) as our clean CP tags to test the S− and S+ purities of our ST modes, respectively. We analyze our data to find (S , S) events using selection criteria similar to  Mode (S , S) n S ,S ε S ,S (%) f S (%) K + K − , K 0 S π 0 π 0 8 ± 3 11.80 ± 0.11 91.6 ± 16.7 (> 86.8) K + K − , ρ 0 π 0 13 ± 8 24.44 ± 0.16 84.0 ± 12.6 (> 70.6) K 0 S π 0 , K 0 S ω 7 ± 3 6.77 ± 0.08 94.6 ± 8.0 (> 90.6) Table 5: A summary of mode-dependent fractional systematic uncertainties, in percent. A "-" means the systematic uncertainty is negligible. However, a simplified procedure is used to obtain the yields. We implement a onedimensional fit to the M BC (S) distributions for the signal mode S of interest, while restricting the M BC (S ) distributions for the tagging modes S in the signal region 1.860 GeV/c 2 < M BC (K + K − ) < 1.875 GeV/c 2 and 1.855 GeV/c 2 < M BC (K 0 S π 0 ) < 1.880 GeV/c 2 . The DT signals are described with the signal MC shape convoluted with a Gaussian function, and backgrounds are modeled with the ARGUS function. Figure 3 shows the M BC (S) distributions in the DT events and the fits to the distributions. Table 4 lists the DT yields and the corresponding detection efficiencies. In the tested CP modes, the observed numbers of the same-CP events are quite small and nearly consistent with zero, which indicates that f S is close to 1. This one-dimensional fit may let certain peaking backgrounds survive; however, an over-estimated n S ,S leads to a more conservative evaluation of f S .

Systematic Uncertainties
In calculating A CP Kπ , uncertainties of most of efficiencies cancel out, such as those for tracking, particle identification and π 0 /η/K 0 S reconstruction. The efficiency differences ∆ S± = ∆( ε S± ε Kπ,S± ) of K − π + between data and MC simulation are studied for the modes S±. We use control samples to study ∆ S± . The K − π + final state is used for studying ∆ S± in the K + K − and π + π − modes; K − π + π 0 is used for the π 0 π 0 , ρπ 0 , 9 13.8 ± 1.8 14.5 ± 2.4 10.0 ± 2.3 8.0 ± 3.7 12.2 ± 2.0 K 0 S η 15.5 ± 3.5 16.3 ± 3.9 11.8 ± 3.8 9.7 ± 4.8 14.0 ± 3.6 K 0 S ω 13.5 ± 2.2 14.2 ± 2.8 9.7 ± 2.7 7.7 ± 3.9 11.9 ± 2.4 K 0 S π 0 and K 0 S η modes; K − π + π 0 π 0 is used for the K 0 S π 0 π 0 mode; and K + π − π − π + is used for the K 0 S ω mode. We determine ∆ S± in different CP -tag modes by comparing the ratio of the DT yields to the ST yields between data and MC. We find that ∆ S± are at 1% level for different CP -tag modes. In the formula of A CP Kπ , the dependence of ∆ S± on the CP mode is not canceled out. The resulting systematic uncertainty on A CP Kπ is 0.2 × 10 −2 . Some systematics arise from effects which act among several CP modes simultaneously. The efficiency of the cosmic and Bhabha veto (only for the KK and ππ modes) is studied based on the inclusive MC sample. We compare the obtained A CP Kπ with and without this requirement and take the difference of 0.6 × 10 −3 as a systematic uncertainty. For the CP modes involving K 0 S , CP -violating K 0 L → π + π − decays are also considered. Using the known branching fraction, we find this causes the change on A CP Kπ to be 0.8 × 10 −3 . Other systematic uncertainties, relevant to A CP Kπ , are listed in Table 5, which are uncorrelated among different CP modes.
The ∆E requirements are mode-dependent. We study possible biases of our requirements by changing their values; we take the maximum variations of the resultant B D S± →Kπ as systematic uncertainties.
Fitting the M BC distributions involves knowledge of detector smearing and the effects of initial-state and final-state radiation. In the case of ST fits, we scan the smearing parameters within the errors determined in our nominal fits. The maximum changes to n S± are taken as a systematic uncertainty. For the DT fits, we obtain checks on n Kπ,S± with one-dimensional fits to M BC (S) with inclusion of floating smearing functions. The outcomes of B D S± →Kπ are consistent with those determined from the two-dimensional fits, and any small differences are treated as systematic uncertainties.
Systematic effects due to the CP purities are checked, as stated in Sec. 5. We introduce the CP purities f S in calculating the B D S± →Kπ under different CP tagging modes and obtain the corrected B D S± →Kπ . We set the lower limits of f S and take the corresponding maximum changes as part of systematic uncertainties.

Results
We combine the branching fractions B D S+ →K − π + and B D S− →K − π + in Eq. (4) from two kinds of the CP modes based on the standard weighted least-square method [15]. Following Eq. (2), we obtain A CP Kπ = (12.7 ± 1.3 ± 0.7) × 10 −2 , where the first uncertainty is statistical and the second is systematic. The modedependent systematics are propagated to A CP Kπ and combined with the mode-correlated systematics. The values of A CP Kπ obtained for the 15 different CP mode combinations are also checked as listed in Table 6. Within statistical uncertainties, they are consistent with each other.

Summary
We employ a CP tagging technique to analyze a sample of 2.92 fb −1 quantum-correlated data of e + e − → D 0 D 0 at the ψ(3770) peak. We measure the asymmetry A CP Kπ = (12.7 ± 1.3 ± 0.7) × 10 −2 . Using the inputs of r 2 and y from HFAG [21] and R WS from PDG [15], we obtain cos δ Kπ = 1.02 ± 0.11 ± 0.06 ± 0.01. The first uncertainty is statistical, the second is systematic, and the third is due to the external inputs. Our result is consistent with previous results from CLEO [8]. Our result is the most precise to date, and helps to constrain the D 0 -D 0 mixing parameters and the angle φ 3 in the unitarity triangle of the CKM matrix.