Electrolyte Sensor Na K

Abstract

A measurement of the CKM angle γ and related strong-phase parameters is per- formed using a novel, model-independent approach in B± →D(→K0

Sh′+H′−)H±

decays, where h(′) ≡π, K. The analysis uses a joint data sample of electron-positron collisions collected by the BESIII experiment at the Beijing Electron-Positron Col- lider II during 2010–2011 and 2021–2022, corresponding to an integrated luminosity of 8 fb−1, and proton-proton collisions collected by the LHCb experiment at the Large Hadron Collider during 2011–2018, corresponding to an integrated luminosity of 9 fb−1. The two datasets are analyzed simultaneously by applying per-event weights based on the amplitude variation over the D-decay phase space to enhance the sensitivity to CP-violating observables. The CKM angle γ is determined to be γ = (71.3 ± 5.0)◦, which constitutes the most precise single measurement to date.

Submitted To Phys. Rev. D

© 2026 CERN for the benefit of the LHCb collaboration and IHEP for the benefit of the BESIII collaboration. CC BY 4.0 licence. †Authors are listed at the end of this paper.

Introduction

In the Standard Model of particle physics (SM), the single irreducible complex phase in the Cabibbo–Kobayashi–Maskawa (CKM) quark-mixing matrix is responsible for all the CP-violating phenomena in the quark sector. Assuming unitarity of the CKM matrix, the

Cb + Vtdv ∗

tb = 0 (Vxy represents the transition amplitude from quark x to quark y) forms a closed triangle in the complex plane, known as the Unitarity Triangle (UT), which has been extensively studied due to the high experimental accessibility of its sides and angles . Among its three internal angles, γ = ϕ3 ≡arg(−VudV ∗

Cb)

can be measured experimentally by exploiting interference between b→cus and b→ucs transitions at leading order within B →D(∗)K(∗) decays. The extraction of γ in these decays has negligible theoretical uncertainty, as small as O(10−7) . The angle γ can also be measured with charmless B-meson decays . Comparison of the results measured with the two methods can provide insights into potential new-physics contributions entering loop-mediated processes within charmless b-hadron decays. In addition to direct measurements from B →D(∗)K(∗) decays, indirect determinations of γ are obtained from global fits to other CKM parameters under the assumption of unitarity . Since some of these inputs arise from loop-level processes, discrepancies between direct and indirect determinations would indicate the existence of new physics.

The world-average values of γ are obtained from global fits to various measurements that constrain the CKM matrix elements. When direct measurements of γ are excluded, the

−1.86)◦From Ckmfitter And Γ = (64.9 ± 1.4)◦From

UTfit . A recent combination of direct measurements by the LHCb collaboration yields γ = (62.8 ± 2.6)◦, while a combination of BaBar measurements yields γ = (69+17

−16)◦,

and a combination of Belle and Belle II measurements yields γ = (75.2 ± 7.6)◦. These direct measurements are consistent with the indirect constraints but have larger uncertainties. Hence, larger data samples and optimized analysis techniques are required to improve the precision on γ.

The most precise direct measurement of γ to date is from the LHCb experiment

Through The Decays B± →D(→K0

Sh′+h′−)h±, where h(′) is either a π or K meson, and D represents a superposition of the D0 and D0 mesons . The analysis employed a binned phase-space approach as proposed in Refs. , benefiting from the local asymmetries across the phase space. The reported result of γ = (68.7+5.2

−5.1)◦Is Statistically Limited. Since

the binned phase-space approach only exploits about 85% of the sensitivity to γ , further optimization of the analysis method is desirable. A novel approach that applies per-event weights to data has been proposed as an alternative . This method for the determination of γ has been found to use the phase-space information more effectively than the binned approach.

The strong-phase differences between the decay amplitudes of D0 and D0 to the

K0

Sh′+h′−final states are essential inputs for determining γ. However, uncertainties due to model assumptions of the decay amplitudes are difficult to quantify. In practice, the strong-phase differences are measured directly from quantum-correlated charm decays typically produced at the ψ(3770) resonance. The strong-phase differences for the binned approach have been measured by the CLEO and BESIII experiments , and are statistically limited. These uncertainties introduce a non-negligible systematic uncertainty to γ. However, these inputs can only be used by the binned approach.

This paper, which expands upon the accompanying Letter , presents a more precise

1

determination of the CKM angle γ by applying the novel approach to the same LHCb dataset used in Ref. . The required strong-phase parameters for this method are obtained from a joint measurement using quantum-correlated DD decays produced at the BESIII experiment. The BESIII data were collected during 2010–2011 and 2021–2022 in e+e−collisions at the ψ(3770) threshold, with an integrated luminosity of 8 fb−1, where

S/Lh′+H′−Final States. The Lhcb Data

were collected during 2011–2018 in pp collisions at center-of-mass energies of √s = 7, 8 and 13 TeV, corresponding to an integrated luminosity of 9 fb−1. The decay channels are the same as those in Ref. , namely B± →DK± and B± →Dπ±, where the D meson is reconstructed via the self-conjugate decays D→K0

Sh′+H′−. In This Analysis, Both

the BESIII and LHCb datasets have also been analyzed using the binned phase-space

Approach Except For The D→K0

S/LK+K−signal in BESIII data. The outline of this paper is as follows: the formalism of the novel approach is explained in Sect. 2; the BESIII and LHCb detectors are introduced in Sect. 3; selection of candidates from data and extraction of signal yields are described in Sect. 4, while Sect. 5 presents the method to obtain key observables from data; the CP observables and strong-phase parameters are obtained using the fits described in Sect. 6; the corresponding systematic uncertainties are analyzed in Sect. 7; Section 8 presents the results of the measurements of CP observables and strong-phase parameters, together with the interpretation of CP observables determining the CKM angle γ; and finally, Sect. 9 summarizes the measurement.

Analysis Strategy

The amplitude of the B−→Dh−decay is a superposition of the favored B−→D0h−and

Sh′+H′−Decay. The Amplitude Of The

B+ →Dh+ decay is obtained by substituting −γ with +γ and exchanging AD and AD.

Are The Ratio And Strong-Phase Difference Between

the suppressed and favored B−decay amplitudes, respectively. As the decay phase space has two degrees of freedom, often referred to as the Dalitz plot , the amplitudes are

±. Neglecting Cp Violation And

mixing in D decays, as their impact on the phase-space distributions is expected to be well below current experimental sensitivity , the amplitudes are defined as AD ≡AD(z)

−, M2

+). The rates of the B−and B+ decays, pB and pB, can be derived as

2 Is The Decay Rate Of The D0(D0)→K0

Sh′+h′−process.

B Sin(Δdh

B ± γ).

2

The functions C and S are related to the strong-phase difference between D0 and D0

Where Δd Is The Strong Phase Of The D0 →K0

Sh′+h′−decay amplitude.

±

are independent of the phase-space coordinates. Therefore, a phase-space-dependent weighting function can be applied to Eq. 2, while xDh

±

remain unchanged, and the weighted integrals of the decay rates can then be evaluated experimentally. The binned phase-space approach is a special case in that regard, using a uniform weighting function within different regions of the Dalitz plot.

A novel approach, referred to as the Fourier split method, has been proposed in Ref. . It applies Fourier expansions of the strong-phase difference ϕ(z) ≡δD(z) −δD(z) to the functions in Eq. 2, which is equivalent to applying per-event Fourier weights cos(kϕ) and sin(kϕ). The distributions of ϕ(z), shown in Fig. 1, are derived from amplitude models measured by the Belle and BaBar experiments . As in the case of the binned approach, the choice of model affects the sensitivity to γ but does not bias the results.

This method achieves a lower statistical uncertainty on γ than the binned approach by making use of the included intra-bin variation of the strong-phase difference across the phase space.

The sensitivity to γ also depends on the decay rate pD and the background level in data, which were not considered in Ref. . To improve the sensitivity to γ, an additional weighting function is introduced, which is denoted as the optimal weight, wopt. The optimal weights, as shown in Fig. 2, are obtained using amplitude models for the D decay, which originate from the Belle and BaBar measurements . The weights also take into account the effects of efficiency and background in the LHCb measurement . The structure of these weights across the Dalitz plot reflects the statistical uncertainty of signal events, showing enhancement and suppression around the K∗(892)± (ϕ(1020)) resonance

Sπ+Π−(K0

SK+K−) decay. The values of these optimal weights across the Dalitz plots are provided in a HEPData record . Analogous to the symmetric splitting

+ > (<)M2

−in the binned approach, the diametrically opposed optimal weight, wopt(z), is also applied to improve the sensitivity. In summary, the complete set of weighting functions combines the optimal weights

Wopt(Z) Sin [Kϕ(Z)] ,

n = 2k −1.

Wopt(Z) Sin [Kϕ(Z)] ,

n = 2k −1.

(5)

With a maximum Fourier order of M, k enumerates from 0 (1) to M for the cosine (sine) weights, and 2 × (2M + 1) weighting functions are defined. This novel approach is denoted as the optimal Fourier method.

Φ

Figure 1: Distributions of the strong-phase difference ϕ(m2

And (Right) D→K0

SK+K−Dalitz plots. They are derived from amplitude models measured by the Belle and BaBar experiments.

Wopt

Figure 2: Distributions of the optimal weight wopt across the (left) D→K0

D→K0

SK+K−Dalitz plots. They are obtained from amplitude models measured by the Belle and BaBar experiments, and also take into account the effects of efficiency and background in the LHCb measurement . The integrals of the weighting functions are normalized to unity.

The weighted integrals of the B± decay rates yield 4 × (2M + 1) parameters,

(6)

where separate normalization factors hB± for B± decays are applied, thereby accounting for any production and detection asymmetry in the B± decays. The parameters P +

P −

n , Cn and Sn are wn-weighted integrals of the pD, pD, C and S functions, respectively. In addition to hB± that are proportional to the signal yields, a normalization over the

(7)

where fX denotes the functions pD, C and S for X = P, C and S, respectively. Implicit

(8)

are applied, because the C function and cosine weights are symmetric, while the S function and sine weights are antisymmetric across the Dalitz plot. However, due to the asymmetric nature of wopt, weighted C and S functions are no longer symmetric or antisymmetric, respectively, which yields nonzero Cn and Sn terms at every sine and cosine order. This relationship is identical to that in the binned method.

By including both B± →DK± and B± →Dπ± decays, the above equations contain 8 × (2M + 1) observables when the Fourier order k enumerates from 0 to M. The number of free parameters is 2 × (2M + 1) + 12 when the strong-phase parameters Cn, Sn are constrained by the BESIII dataset. As a result, under the condition M ≥1, these observables are sufficient to determine all the parameters.

As the B± →DK± and B± →Dπ± processes have the same weak phase difference γ, only two additional observables are needed to describe the B± →Dπ± decay. They are

(9)

and the CP observables of the B± →Dπ± channel can be determined using

Ξ Xdk

± .

(10)

This reduces the number of free parameters to 2 × (2M + 1) + 10. In the quantum-correlated ψ(3770)→D0D0 system, the decay amplitude can be

(11)

where the superscripts (1) and (2) uniquely label the two D mesons, and A(i)

D ) Is

the decay amplitude of the corresponding D0 (D0) decay. The strong-phase parameters, Cn, Sn, are measured in this system by a double-tag (DT) method , in which one of the D mesons is reconstructed as the signal decay, and the other is reconstructed as a tag

Lh′+H′−Decays Are Also Included As

signal to provide further constraints on the strong-phase parameters. The tag mode can be either a (quasi-)flavor-specific state f like K∓π±, a CP eigenstate like K+K−, or another signal decay. In addition, the single-tag (ST) method, where only one of the D mesons is reconstructed as the tag decay, is adopted to determine ST yields for normalization as detailed in Sect. 6.

Applying the weighting functions to flavor-tagged decay rates yields 2 × (2M + 1)

5

where rD and δD are the amplitude ratio and strong-phase difference, respectively, between the doubly-Cabibbo-suppressed (DCS) and Cabibbo-favored (CF) processes, and RD is the coherence factor of the tag mode. Depending on the signal decay, sK is 1 for D→K0

And −1 For D→K0

Lh′+h′−, due to the different CP content of the neutral kaons. The parameters Nn (N n) refer to the case where the tag mode is a DCS (CF) decay like D0 →K+π−(D0 →K−π+).

(13)

where Nn (N n) corresponds to the wn(wn)-weighted integrals. The CP-even fraction F+ is 1 for CP-even tags, 0 for CP-odd tags, and F+ = 0.9406 ± 0.0032 (stat) ± 0.0021 (syst) for the π+π−π0 tag .

For the case where both signal and tag D mesons are reconstructed by the

N1N2, Can Be Computed By The Four

possible weighting-function combinations {wn1, wn1} × {wn2, wn2}. As a result, there are a total of 4 × (2Mh′ + 1) × (2Mh′′ + 1) parameters when the maximum Fourier order of

The D→K0

S/Lh′(′)+h′(′)−decay is set to Mh′(′). Letting s1 and s2 denote the signs of the first and second weights with the +(−) sign itself corresponding to wn(wn), and letting s1 and s2 denote the opposite signs, the parameters can be expressed as

(14)

where sK1(2) = 1 when the first (second) D meson decays into the K0

Sh′+H′−Final State,

and sK1(2) = −1 when the first (second) D meson decays into the K0 Lh′′+h′′−final state. In summary, the optimal Fourier method introduces an extra optimal weight in addition to the Fourier weights proposed in Ref. . Setting the maximum Fourier order

To Mh For D→K0

S/Lh′+h′−decays, applying the weighting functions to LHCb data yields 2 × (2Mh + 1) parameters for each B charge, B decay and D decay. The CP observables

And Flavor Parameters P ±

n can be determined from a simultaneous fit to B± →DK± and B± →Dπ± candidates when the input strong-phase parameters Cn, Sn are known. From BESIII data, 2 × (2Mh′ + 1) parameters of flavor tags, 2 × (2Mh′ + 1) parameters of CP tags, and 4 × (2Mh′ + 1) × (2Mh′′ + 1) parameters of K0

N In Besiii Data, And The Strong-Phase

parameters Cn, Sn, are obtained from fits to CP-tag and K0

Sh′+H′−-Tag Parameters. The

Cn and Sn parameters need to be shared between BESIII and LHCb data, while separate

P ±

n parameters are obtained independently from flavor tags in BESIII and B± →Dπ± decays in LHCb. Therefore, a joint fit to the BESIII and LHCb parameters allows for a simultaneous determination of the CP observables and strong-phase parameters.

Pseudoexperiments are conducted to validate the approach, where it is found that the smallest statistical uncertainty on γ is achieved with Mπ = 2 and MK = 1 when the dataset size is comparable to that analyzed in this paper. The increasing sensitivity with Mh is limited by sizes of both BESIII and LHCb datasets. The average uncertainty of γ from the optimal Fourier approach is also lower than that from the binned approach when both approaches are applied to the same ensemble. The study is performed under

6

ideal conditions where the generated amplitude models are the same as those used to compute the weighting functions. As the underlying model of the data is not perfectly known, imperfect description of the models may not lead to the most optimal results.

Nevertheless, the choice of model does not bias the central values, and the orders Mπ = 2 and MK = 1 are chosen to determine the CP observables xDK

And Ydπ

ξ .

Detectors And Simulation

The BESIII detector records symmetric e+e−collisions provided by the BEPCII storage ring in the center-of-mass energy range from 1.84 to 4.95 GeV with a peak luminosity of 1.1 × 1033 cm−2s−1 achieved at √s = 3.773 GeV. Large data samples in this energy region have been collected by the BESIII experiment . The cylindrical core of the BESIII detector covers 93% of the full solid angle and consists of a helium-based multilayer drift chamber (MDC), a plastic scintillator time-of-flight system (TOF), and a CsI(Tl) electromagnetic calorimeter (EMC), which are all enclosed in a superconducting solenoidal magnet providing a 1.0 T magnetic field, which was 0.9 T in 2012. The solenoid is surrounded by an octagonal flux-return yoke made of steel, interleaved with resistive-plate-counter muon-identification modules.

The charged-particle momentum resolution at 1 GeV/c is 0.5%, and the dE/dx resolu- tion is 6% for electrons from Bhabha scattering. The EMC measures photon energies with a resolution of 2.5% (5%) at 1 GeV in the barrel (end-cap) region. The time resolution in the TOF barrel region is 68 ps, while that in the end-cap region is 110 ps. The end-cap TOF system was upgraded in 2015 using multigap resistive-plate chamber technology, providing a time resolution of 60 ps . This upgrade benefits 63% of the data used in this analysis. More details can be found in Ref. .

The simulation samples produced with a Geant4-based software package , which includes the geometric description of the BESIII detector and the detector response, are used to determine detection efficiencies and estimate backgrounds. The simulation models the beam energy spread and initial-state radiation (ISR) in e+e−annihilations with the generator kkmc .

Inclusive simulation samples are produced including DD pairs corrected with quantum- correlation effects, non-DD decays of the ψ(3770), the ISR production of the J/ψ and ψ(3686) states, and continuum e+e−→uu, dd, ss processes incorporated in kkmc. All particle decays in the inclusive simulation samples are modeled with EvtGen using branching fractions either taken from the Particle Data Group , when available, or estimated with Lundcharm . Final-state radiation from charged final-state particles is incorporated using the Photos package .

Are Produced For Dt K0

S/Lh′+h′−vs. tag modes, where quantum-correlation effects are

Implemented. The D→K0

Sπ+π−decay is simulated with the amplitude model measured

Sπ+Π−

model implemented with U-spin breaking parameters . The D→K0

S/Lk+K−Decay

is simulated with the amplitude model developed by the BESIII experiment. Multibody tag-side decays are simulated with the aforementioned EvtGen according to the most recent models from experimental studies.

The LHCb detector [52, 53] is a single-arm forward spectrometer covering the pseudorapidity range 2 < η < 5, designed for the study of particles containing b or

7

c quarks. The detector used to collect the data analyzed in this paper includes a high- precision tracking system consisting of 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 T m, and three stations of silicon-strip detectors and straw drift tubes are placed downstream of the magnet. The tracking system provides a measurement of the momentum of charged particles with a relative uncertainty that varies from 0.5% at low momentum to 1.0% at 200 GeV/c. The minimum distance from a track to a primary pp collision vertex, the impact parameter, is measured with a resolution of (15 + 29/pT) µm, where pT is the component of the momentum transverse to the beam, in GeV/c. Different types of charged hadrons are distinguished using information from two ring-imaging Cherenkov detectors. Photons, electrons and hadrons are identified by a calorimeter system consisting of scintillating-pad and preshower detectors, an electro- magnetic and a hadronic calorimeter. Muons are identified by a system composed of alternating layers of iron and multiwire proportional chambers. The online event selection is performed by a trigger, which consists of a hardware stage, based on information from the calorimeter and muon systems, followed by a software stage, which applies a full event reconstruction. Triggered data further undergo a centralized, offline processing step to deliver physics-analysis-ready data across the entire LHCb physics program .

Simulation is required to correct for reconstruction and selection efficiencies in LHCb data. In the simulation, pp collisions are generated using Pythia with a specific LHCb configuration . Decays of unstable particles are described by EvtGen , in which final-state radiation is generated using Photos . The interaction of the generated particles with the detector and its response, are implemented using the Geant4 toolkit [42, 58] as described in Ref. . When generating simulation samples for background processes, the underlying pp interaction is reused multiple times, with an independently generated background decay for each . Some subdominant background processes are generated with RapidSim that mimics the LHCb detector acceptance, reconstruction efficiency and dynamics of the decay.

Event Selection And Signal Extraction

Quantum-correlated DD decays collected by the BESIII detector are reconstructed by requiring one of the D mesons to decay to a signal channel, K0

S/Lh′+H′−, And

the other to decay via a tag mode.

As Described In Sect. 2, Three Categories Of

tag modes are used in the analysis.

The Flavor And Cp-Tag Modes Are Listed In

Table 1. The decays where both D mesons are reconstructed in a signal chan- nel are selected according to the following self-conjugate DT modes:

K0

Sπ+π−vs.

L]Π+Π−}, Where Particles In Square Brackets

are not reconstructed. Final-state particles of charm-meson decays are reconstructed from candidates of charged tracks and neutral photons using selection criteria identical to those applied in

The Binned Analysis Of D→K0

Sπ+π−. In addition, the requirement of no extra π0

Lπ+Π−Final

states, as higher statistics are obtained without a decrease in signal purity. Meanwhile, to mitigate resolution effects in selected DD decay samples, D, K0

8

Table 1: Summary of tag modes selected against D→K0 S/Lh′+h′−.

Sω(→Πππ0)

kinematically constrained to their known masses . The charm-meson decays where all final-state particles are reconstructed are clas- sified as fully reconstructed modes. Following Ref. , a mode-specific requirement on the energy difference ∆E ≡ED −√s/2 is applied, and the beam-constrained mass

P

(√s/2)2 −|pD|2c2 is utilized as the fit variable. Here, √s/2 is the beam energy, ED is the sum of final-state-particle energies and pD represents the reconstructed mo- mentum of the D candidate both evaluated in the center-of-mass system. When there

L Meson Or Neutrino That Cannot Be Reconstructed

within the BESIII detector, a partial reconstruction method is adopted. The squared

Miss −|Pmiss|2C2, Where Emiss = √S/2 −Eother And

pmiss = −ptag −pother. Here, ptag represents the momentum of the fully reconstructed D candidate, while pother and Eother are the total momentum and energy of the other recon- structed particles in the partially reconstructed modes, all evaluated in the center-of-mass system. In the D→Keνe mode, Umiss = Emiss −|pmiss|c is employed as the fit variable as signal events form a peak around Umiss = 0 GeV.

Possible sources of background in D decays are estimated through inclusive simula- tion samples. Details on background studies, extraction of signal yields and efficiency

Determinations Of The D→K0

S/Lπ+π−signal channel can be found in Ref. . For the

Channel D→K0

S/LK+K−, the dominant peaking backgrounds are D→π+π−K+K−and

D→K0

S(→π0π0)K+K−with a contamination rate of 0.5% and 2.5%, respectively. The nonpeaking backgrounds are mainly from other DD decays and continuum processes. One-dimensional unbinned maximum-likelihood fits are performed to discriminate signal from background. In the fits, double-sided Crystal Ball functions are used as signal shapes, with all of their parameters fixed from simulated signal samples. The signal models are then convolved with Gaussian functions whose mean values and widths vary freely in order to take into account the different resolution effects between data and simulation.

The background shapes and contributions are directly derived from inclusive simulation

S →Π0Π0 Decay In D→K0

LK+K−vs. CP-tag modes. For those background components, the shapes are extracted from simulated signal samples and the background yields are estimated from data based on the signal yields of K0

Sk+K−

vs. CP-eigenstate modes. Example fit results for the D→K0

S/Lk+K−Channel Are Shown

in Fig. 3. Details on fits separated by individual tag modes along with their corresponding signal yields are given in Appendix A. In the datasets collected by the LHCb detector, decays of K0

S →Π+Π−Are Reconstructed

in two different categories. The first involves K0

Sππ Tag; And (Bottom Right) The D→[K0

L]KK vs. KK tag. pions to be reconstructed in the vertex detector, and the second involves K0

S Mesons That

decay later such that track segments of the pions cannot be formed in the vertex detector. These categories are referred to as long and downstream, respectively. The long category has better mass, momentum and vertex resolution than the downstream category, while the downstream category has higher reconstruction efficiency and contributes approximately two thirds of the dataset. The D-meson candidates are then formed by combining a

K0

S candidate with two tracks both with either the pion or kaon mass hypothesis. The B-meson candidate is subsequently built by combining a D candidate and another pion or kaon track. This constitutes eight categories in the B± →D(→K0 Sh′+h′−)h± data.

The selection criteria and fit model to the Dh± mass spectra are identical to those described in Ref. . Signal and background yields in data are extracted by a two- stage fit.

The first stage is the same as the global fit in Ref. , in which the parametrizations of signal and background components are determined. The global fit includes: the signal B± →Dh± decays; partially reconstructed B± →D∗(→Dγ/π0)h±,

S →Dk−Π+ Decays, Where A Pion Or Photon

is not reconstructed; the aforementioned background decays in B± →Dπ± with the com- panion pion misidentified as a kaon; and a combinatorial component in each category. All shape parameters are subsequently fixed in the second stage, in which another fit is performed in a tightened mass range, from 5150 to 5800 MeV/c2, to reject most of the partially reconstructed background events. The mass distributions with fit results also

10

included are shown in Fig. 4.

Observables From Data

The observables are obtained from weighted sums over data events, in which background contributions are subtracted and efficiency effects are corrected. The signal contribution in LHCb data is isolated by the sPlot technique from which per-event weights, denoted as fi, are obtained based on the mass fit results. However, sPlot is not applicable to BESIII data due to the known correlation between the fit variables and Dalitz plot coordinates.

The D-decay observables are computed from both BESIII data and simulated background samples, and the background observables are subtracted from the data observables. Efficiency corrections are applied to both datasets by applying inverse efficiency weights derived from relative efficiency distributions.

Relative efficiency distributions are described as functions of the Dalitz-plot coordinates and obtained from simulated signal samples. The D→K0

Sh′+H′−Decays In Both Besiii

and LHCb simulation samples are generated with a uniform phase-space distribution; thus, the Dalitz-plot distributions post selection are proportional to the efficiencies across the Dalitz plot, which include the effects of detector acceptance, reconstruction and selection.

Efficiency distributions are obtained by fitting each of the Dalitz-plot distributions of simulated signal samples with a two-dimensional polynomial function, with terms up to fourth order. In addition, extra per-event efficiency weights are introduced to correct the difference between data and simulation samples for D decays in BESIII. The main discrepancies exist in the tracking efficiency, the particle-identification (PID) efficiency of Kπ separation, and the reconstruction efficiency of K0

S Candidates. They Are Investigated

using a control sample of DD hadronic events . Subsequently, corrections are applied to

L]K+K−.

For tag modes, these differences largely cancel out in the ratio of ST to DT yields for fully reconstructed tag modes, and are found to be negligible for the partially reconstructed tag modes, such as Keνe, K0

Lπ0, And K0

Lπ0π0. Figure 5 shows example efficiency distributions from BESIII simulation. Figure 6 shows the efficiency distributions from B± →Dπ± simulation at LHCb, which are also used in the B± →DK± decays, as the Dalitz-plot distributions of simulated samples are found to be consistent between B± →Dπ± and B± →DK± decays.

The weighted signal yields from flavor tags are computed by

(15)

where the weights are obtained from the functions wn defined in Eq. 5, efficiency distribu- tions ϵ and the Dalitz-plot coordinates of candidate i. The computation of Nn (Nn) is

Sh+H−Data, Where The Tag Side Decays

into DCS (CF) D0 final states, e.g. K+π−(K−π+). These observables from the flavor tags are used to compute the flavored parameters, P ±

N . The Dcs Contributions In These

channels are corrected according to Eq. 12, where the strong-phase parameters Cn, Sn are fixed to model predictions and the hadronic parameters rD, δD and RD are fixed to values that are listed in Table 2. The corrected parameters P ±

Lπ+Π−Decays, Respectively, Are Found To Be

consistent.

9 Fb

Figure 4: Mass distributions and fit results of the (left) B± →DK± and (right) B± →Dπ± data, with the D meson decaying to either (top four) K0

Sk+K−Final

states. The first and third rows correspond to the long category, while the second and fourth rows correspond to the downstream category.

Simulation

Figure 5: Example efficiency distributions from BESIII simulation. For the Kπ tag mode,

Lk+K−Signal Modes. The Average Efficiencies

over the Dalitz plot are scaled to unity. Table 2: Parameters of the flavor tags that are used to correct flavor observables, taken from Ref. .

0.09

Similarly, weighted yields from CP tags are computed as

(16)

and four combinations of weighted yields of self-conjugated tags are computed by combining

Simulation

Figure 6: Efficiency distributions of B± →Dπ± decays with (top) D→K0

D→K0

SK+K−decays for the (left) long and (right) downstream categories of the K0 S candidate. The average efficiencies over the Dalitz plot are scaled to unity.

per-event weights from the phase space of both sides

I

({wn1,i1, wn1,i1} /ϵi1) · ({wn2,i2, wn2,i2} /ϵi2) ,

(17)

where the first (second) set of weights is based on the Dalitz-plot coordinates of the first (second) D meson. By summing over the same data samples, the observables are naturally correlated with each other. The covariance matrices between two weighted sums, P

2. Therefore, The Covariance

between observables of Eq. 16, e.g. Nn1 and Nn2, are

(18)

and the same computation is applied to other combinations. For the K0

Sπ+Π−Decay From

different tags. The total observables are obtained from combining those of all tags.

0 + P −

0 = 1.

Lπ+Π−Decay From

different tags. The total observables are obtained from combining those of all tags.

0 + P −

0 = 1. samples, covariances between CP-tag observables are computed by Eq. 18. Meanwhile, the correlations between self-conjugated tag observables are derived from pseudoexperiments with a much larger sample size, which are generated with the amplitude model as the correlations from data are found to bias the results under current statistics. The covariance estimates in both CP and self-conjugate tag observables with K0

S/Lk+K−

are also computed from an ensemble due to low statistics. The uncertainties of those observables are calculated from data as a special case of Eq. 18, in which the two weights

15

are identical, reducing the expression to the sum of squared weights. The weighted yields from LHCb data are computed by

(19)

which includes the extra signal weight, fi. The covariance between LHCb observables are

I∈B−→Dh−

wn1,iwn2,i (fi)2 / (ϵi)2 .

(20)

Figure 9 shows the weighted yields obtained from the B± →Dh±, D→K0

Sπ+Π−Data

in the downstream category. The model predictions are computed from the amplitude model measured in Refs. and physics parameters measured in Ref. . Sizable CP-violation effects are seen in the observables from the B± →DK± data, while much smaller CP asymmetries are present in the B± →Dπ± channel due to the small value of

Fit To Observables From Data

The parameters of interest are extracted by the least-squares approach, where a χ2 test statistic is constructed with the observables from data. This step is referred to as the CP fit. In Sect. 6.1, a joint fit to both BESIII and LHCb observables is performed to determine the CP observables, while in Sect. 6.2, a fit using only BESIII observables is carried out to determine the strong-phase parameters independently.

Joint Fit To Determine Cp Observables

For each CP tag in BESIII data, the 2 × (2Mh′ + 1) expected observables are computed

Defined As

efficiency-corrected ST yields of the CP tag and flavor tags, respectively. The ST yields and efficiencies are fixed to those measured in Ref. . The flavor parameters P ±

N Are

fixed to those computed from flavor tags, and the strong-phase parameters Cn are allowed to vary. The χ2 test statistic from CP tags is constructed as

Cp-Tag = [Nn −⟨Nn⟩, Nn −⟨Nn⟩]T V −1

CP [Nn −⟨Nn⟩, Nn −⟨Nn⟩] .

(22)

Here, the observed weighted yields Nn are computed from the weighted sums over the relevant data samples, in which the background observables, computed from the weighted sums over simulated background samples, are subtracted. The covariance matrices VCP are computed from the weighted sums as described in Sect. 5.

The 4 × (2Mh′ + 1) × (2Mh′′ + 1) expected observables of the self-conjugate tags are

N From Model

Figure 9: Observables from the (top) B± →DK± and (bottom) B± →Dπ± data sample with

D→K0

Sπ+π−decays in the downstream category. The CP-conjugated observable pairs, (left) N +

N Or N +

n , are plotted together to visualize CP asymmetries.

F,1N St

f,2) is the normalization factor.

The Factor Κ Ac-

counts for the different reconstruction methods between the two D mesons, with

Sπ+Π−Channel,

κ = 2 for the channels where the two D mesons decay to different final states, and κ = 1 for the channels where the two D mesons decay to the same final state. The branching

S →Π+Π−) Are Fixed To Their Known Values . The

number of DD pairs are measured to be NDD = (28.66 ± 0.25) × 106 , and N ST

F,1(2)

correspond to the efficiency-corrected ST yield of flavor tags of the first (second) D-meson decays. The χ2 test statistic of the self-conjugate tags is constructed in the same way as

The Cp Tags, Where 4 Sets Of Observables, N ±±

n1n2, are included. An additional weak constraint from the amplitude model is included to help determine the strong-phase parameters, following the same approach as in Ref. . Differences in

Sh′+H′−Decays, ∆Cn, ∆Sn,

are constrained to those from the amplitude models.

D0 →K0

Sh′+h′−decays are measured by the Belle and BaBar experiments , while

K0

Sh′+h′−models and the assumption of U-spin symmetry. Further U-spin breaking

Lπ+Π−Decays From The Besiii Measurement ,

and estimated with the approach described in Ref. for D0 →K0

Lk+K−Decays As

no measurement is available at the moment. Uncertainties of ∆Cn, ∆Sn, denoted as δ∆Cn, δ∆Sn, are estimated from pseudoexperiments, in which the amplitude model is altered by sampling input parameters within their measured uncertainties in Refs. , or by using alternative amplitude models that are measured independently . The χ2 test statistic of this constraint, e.g., for ∆Cn, is constructed as

Where Cn And C′

n are the measured strong-phase parameters of the D0 →K0

D0 →K0

Lh′+h′−decays, respectively. Furthermore, the expected values of ⟨∆Cn⟩are computed from the amplitude models, and Mh denotes the highest Fourier order in the fit with h = π, K.

The χ2 test statistic constructed from BESIII data is

K0

S/Lh′+h′−vs. CP-tag contains 12 categories: 4 signal D decays and 3 tag modes

S/Lh′′+H′′−Term Contains 9 Self-Conjugate Tags

as explained in Sect. 4. For LHCb data, the χ2 test statistic can be constructed by the same approach, using

)

N ± from Eq. 19, and their covariance matrices VB± from



.

(26)

This construction represents 16 categories of LHCb data. The normalization factors of each category, hB±, 2 × (2Mh + 1) flavor parameters of LHCb data in each D decay and

± ,

are free parameters in the fit. The CP observables of B± →Dπ± decays, as explained in Sect. 2, are described by four B± →DK± parameters and two B± →Dπ± parameters, which are also free parameters. Finally, the strong-phase parameters are shared between

C And Χ2

B, so that the CP observables can be extracted from a joint fit to both BESIII and LHCb data. Comparing with those fitted with only the BESIII data at the same orders Mπ = 2 and MK = 1, strong-phase parameters from the joint fit have slightly lower uncertainties in Sn, indicating extra sensitivity provided by LHCb data as presented in Appendix B. The joint fit is found to have a good quality, with the minimized χ2 corresponding to a p-value of 1%.

The CP observables from the joint fit are shown in Fig. 10, including the two- dimensional likelihood contours for the pairs of observables (xDK

Ξ , Ydπ

ξ ). The contours are determined from the statistical uncertainties and correlations

Lhcb 9 Fb−1

Figure 10: Results of CP observables from the joint fit. The contours correspond to 68% and 95% confidence levels of observable pairs, (left) (xDK

Ξ

). from the joint fit, and those from systematic uncertainty studies as detailed in Sect. 7. The likelihood distributions are assumed to be Gaussian-like and were validated by the binned measurement . The lengths of the two vectors from the origin to (xDK

Individual Rdk

B± values from B± →DK± decays. The nonzero opening angle between these two vectors, which equals 2γ, shows a clear sign of CP violation. Several cross-checks have been performed to validate the fit procedure. The fits to data are performed with higher Fourier orders, (Mπ, MK) = (2, 2), (3, 1) and (3, 2), and the results of CP observables and strong-phase parameters are found to be consistent. Fits are also performed using only the LHCb data with both the binned and unbinned approaches, in which the unbinned strong-phase parameters are fixed to those measured in Fig. 11, and the binned strong-phase parameters are fixed to those measured in Refs. [24, 25].

Correlation between the resultant CP observables was studied with pseudoexperiments and was found to be around 0.8 for each of the observables. Using the same pseudoexperiments their consistency in six-dimensional parameter space was found to correspond to a p-value of 39%.

Searches for intrinsic bias are conducted by generating and fitting an ensemble of pseudoexperiments. The yields and mass distributions are based on the fits to data in Sect. 4, and the distributions across the phase space are generated from the baseline amplitude models . Standardized-residual distributions of the CP observables are found to be consistent with the standard normal distribution.

Determination Of Strong-Phase Parameters

The strong-phase parameters Cn, Sn serve as crucial inputs for future measurements of γ with the novel approach. To avoid potential correlations between future γ measurements and this measurement, results of these parameters determined by a fit to BESIII data

S/Lπ+Π−And K0

S/LK+K−channels combined are also reported in this analysis. The Fourier orders are chosen to be one order higher than in the baseline with Mπ = 3 and MK = 2, so that the truncation choice does not limit future higher-precision

S

Figure 11: Strong-phase parameters of the (top) D→K0

S/Lk+K−

decays from BESIII data. measurements. The strong-phase parameters are obtained with the χ2 test statistic as written in Eq. 25 and listed in Table 3, where the first uncertainties are statistical and the second systematic. The correlations between each source of uncertainty are included in the Supplemental Material . Figure 11 shows good agreement between the model predictions and fitted results of the strong-phase parameters, where the statistical and systematic uncertainties have been combined.

Based on a pseudoexperiment study described in Sect. 6.1, the average standardized residuals of three strong-phase parameters of D→K0

Phase Parameters Of D→K0

S/LK+K−decays are found to deviate from zero. Their biases are less than 30% of the statistical uncertainties and are due to low statistics in data, so are not used to correct central values of the strong-phase parameters, but assigned as a source of systematic uncertainty as described in Sect. 7. The statistical uncertainties of

20

Table 3: Results of the strong-phase parameters Cn, Sn from BESIII-only data under Fourier orders Mπ = 3, MK = 2.

S/Lk+K−Decays Are Found To Be Underestimated

under the current sample size, thus corrections of statistical uncertainties are applied to the fit results.

Systematic Uncertainties

The systematic uncertainties affecting this measurement are considered from two groups: those associated with the BESIII inputs and those related to LHCb. The uncertainties of the strong-phase parameters are summarized in Tables 4–7, while those for the CP observables are listed in Table 8.

The dominant sources of systematic uncertainties from BESIII are identified as fol-

21

lows: biases estimated from pseudoexperiments; statistical fluctuations in the flavor-tag observables and ST yields; uncertainties associated with charm mixing and the hadronic parameters used for flavor tags; limited statistics of the uniform phase-space simulation samples for efficiency profiles, and the finite statistics of background samples for modeling the background shapes and yields.

Most of these uncertainties can be addressed through dedicated pseudoexperiments. The sets of values for the flavor and charm-mixing parameters, ST yields, efficiency profiles, signal efficiencies, background models and yields are generated independently according to Gaussian distributions with means and standard deviations set to their baseline values and uncertainties. The CP fit is then repeated for each varied set, and the standard deviation of the resulting pull distribution for each strong-phase parameter is assigned as the relative systematic uncertainty.

Since the DCS contributions in flavor tags and some of the correlation matrices depend on the amplitude models, the imperfection of the models can bias the strong-phase parameters. To assess this, alternative amplitude models are employed, and the differences in the extracted strong-phase parameters are assigned as the corresponding systematic uncertainties.

Another source of uncertainty originates from the tracking efficiency of pions from

K0

S decays. This effect is due to the relatively low momentum and displaced topology of these secondary pions, making their tracking efficiencies less well constrained than those of prompt tracks. Although the overall uncertainties related to PID and tracking are found to be negligible after correcting the efficiency with the method described in Sect. 5,

Residual Mismodeling Of The Pions From K0

S decays can still affect the signal efficiencies. To quantify this, the signal efficiencies and efficiency profiles are varied according to the tracking-efficiency corrections derived from π± control samples. The resulting shifts in the refitted strong-phase parameters are taken as systematic uncertainties.

Possible mismodeling of the detector resolution between data and simulation is taken into consideration. The efficiency profiles are derived separately using truth-level and reconstruction-level information of reconstructed events in the simulation samples. The differences in the extracted strong-phase parameters with these two kinds of efficiency profiles are treated as related systematic uncertainties.

For each source of systematic uncertainty from BESIII, its impact on the strong-phase parameters is studied by analyzing data or pseudoexperiments at Mπ = 3 and MK = 2. All sources of systematic uncertainties can be found in Tables 4–7.

In the analysis of LHCb data, the major difference between this measurement and Ref. is the approach to extract observables. In this measurement, background subtrac- tion is performed with the sPlot technique, which assumes that the B-candidate mass and Dalitz-plot coordinates are independent variables. Potential correlations between these variables may introduce bias on the CP observables, and are evaluated by inspecting the Dalitz-plot-dependent B-candidate mass distributions in simulated samples. Pseudoexper- iments generated with such correlations are fitted with the baseline model. Mean biases of the CP observables are taken as systematic uncertainties.

Efficiency profiles from the simulated signal are needed in this measurement to correct B± →Dh± observables, thus imperfect modeling in simulation is considered as a source of systematic uncertainty. Correction profiles are obtained by comparing Dalitz-plot distributions between simulated signal and B± →Dπ± signal from data. This is possible as the B± →Dπ± decays have clean signal peaks in data and minimal CP violation. New

22

Table 4: Summary of all uncertainties on the strong-phase parameters of D0 →K0

Sπ+Π−Decays

from BESIII data. All uncertainties are quoted in units of 10−4.

15

15

Table 5: Summary of all uncertainties on the strong-phase parameters of D0 →K0

Lπ+Π−Decays

from BESIII data. All uncertainties are quoted in units of 10−4.

9

efficiency profiles are obtained by multiplying the baseline profiles with the correction profiles, and are used to generate pseudoexperiments. Weighted signal yields from the ensemble are computed with the baseline efficiency profiles, and mean biases of the CP observables are taken as systematic uncertainties. The sample size of simulated signal can affect the precision of the efficiency profiles, so its impact is assessed by sampling efficiency parameters from the fits to simulated samples and computing a number of new profiles. The subsequent computation of observables and fits to data indicate that the standard deviations of the CP observables are two orders of magnitude smaller than other systematic uncertainties. Therefore, no systematic uncertainty due to the simulation sample size is assigned.

23

Table 6: Summary of all uncertainties on the strong-phase parameters of D0 →K0

Sk+K−Decays

from BESIII data. All uncertainties are quoted in units of 10−4.

27

Table 7: Summary of all uncertainties on the strong-phase parameters of D0 →K0

Lk+K−Decays

from BESIII data. All uncertainties are quoted in units of 10−4.

24

The effect of detector resolution is taken into account by smearing the Dalitz-plot distributions in pseudoexperiments. The smearing is based on the m2

± Resolution As

determined in simulation and scaled to account for data-simulation differences. The baseline fit model is then applied to the ensemble, and mean biases of the CP observables are taken as systematic uncertainties. This source is equivalent to the bin-migration effect described in Ref. .

Systematic uncertainties related to the global fit are estimated by including extra effects when generating pseudoexperiments, and fitting the ensemble with the baseline model. Several background decays are not considered in the baseline fit due to their low yields, which include the semileptonic B →Dµνµ and B →D(→K0

Sπ± Decays Where The Final-State

tracks are wrongly assigned. Their yields relative to signal are estimated from either LHCb simulation or fast simulation , as well as the B-candidate mass and Dalitz-plot distributions. Alternative parameter sets describing the B-candidate mass are obtained by resampling data and simulated samples, and propagated to the fit in the tighter mass region and the CP fit. The relative yields of partially reconstructed background and PID efficiencies for misidentified background are fixed in the global fit. They are varied within their uncertainties in the mass fits, and standard deviations of the CP observables from the CP fit are assigned as systematic uncertainties. The partially reconstructed backgrounds are treated as a whole with the same Dalitz-plot distributions and uniform Dh mass distributions across the phase space, which ignores underlying physics effects like CP violation in B →D(∗)K(∗) decays. These effects are taken into account when generating pseudoexperiments, where the parameters in Ref. are used. Global fit- related systematic uncertainties are evaluated by adapting the same approach as in Ref. .

The Amount Of Cp Violation In K0

S decay and interactions between detector materials and neutral kaons are ignored in the measurement. These effects are included to generate a pseudosignal following the approach described in Ref. . The baseline fit model is used to fit the pseudosignal, and biases on the CP observables are assigned as systematic uncertainties. The impact of ignoring D mixing in LHCb data is assessed by the same approach, where the signal model including D mixing is described in Ref. .

From pseudoexperiments, increasing bias on the CP observables is found when the Fourier orders increase. This is due to sizeable correlations present within the covariance matrix determined from weighted sums of data in Eq. 20. Nevertheless, biases found in pseudoexperiments under baseline Fourier orders are small and assigned as a source of systematic uncertainty.

All uncertainties of CP observables are summarized in Table 8, and the systematic uncertainties are found to be one order of magnitude smaller than the statistical ones. For the sources that have been studied in both this measurement and Ref. , their impact is found to be at the same level. The extra sources due to the new method are not dominant, making the total systematic uncertainties comparable to those in Ref. . The overall systematic uncertainties on the CP observables from BESIII are at the same order as those from LHCb sources, which are much smaller than the statistical uncertainties.

25

Table 8: Summary of all uncertainties on the CP observables from BESIII and LHCb data. All uncertainties are quoted in units of 10−2.

Results Of Γ

From the joint analysis of BESIII and LHCb data, the CP observables, determined from

= (−1.12 ± 2.37 ± 0.20) × 10−2,

where the first uncertainties are statistical and the second systematic. Statistical and systematic uncertainties from BESIII and LHCb are also analyzed separately to study their individual impact. The values and correlations of each source of uncertainty are summarized in Tables 12–15 of Appendix C.

The measured CP observables can be reinterpreted by a frequentist treatment as described in Refs. , which is denoted as the Plugin method. The solution in the

= (311+17

−20)◦.

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