A method for model-independent measurement of the CKM angle $\beta$ via time-dependent analysis of the $B^0\to D\pi^+\pi^-$, $D\to K_S^0\pi^+\pi^-$ decays

A new method for model-independent measurement of the CKM angle $\beta$ is proposed, that employs time-dependent analysis of flavour-tagged $B^0\to D\pi^+\pi^-$ decays with $D$ meson decays into CP-specific and~$K_S^0\pi^+\pi^-$ final states. This method can be used to measure the angle $\beta$ with future data from the Belle II and LHCb experiments with the precision level of one degree.


Introduction
The B-factory experiments at SLAC [1] and KEK [2] have made impressive progress in studies of the CP symmetry breaking in B meson decays. The LHCb [3] experiment has been contributing significantly to this field since recently. The CP-violating phenomena observed so far are in agreement with the KM mechanism of the CP symmetry breaking proposed by Cabibbo, Kobayashi and Maskawa [4,5]. Nevertheless, theoretical estimates [6] claim that the KM mechanism cannot provide the value of CP violation large enough to generate the observed baryon asymmetry of the Universe [7]. Thus, searches for other mechanisms of CP violation and tests of the KM mechanism should be continued.
Comparison of the angle β values of the Unitarity Triangle (UT) [8] measured in different processes is a valuable test of the KM mechanism. The value of sin 2β obtained using the b → ccs transitions [9][10][11][12][13] is currently the most precisely measured parameter related to the UT angles [14]: sin 2β (b→ccs) = 0.691 ± 0.017. The value of sin 2β measured in the b → cud transitions [15] is consistent with the b → ccs result though it is statistically limited: sin 2β (b→cud) = 0.66 ± 0.10 ± 0.06. (1.2) Within the Standard Model, the angle β measurements in b → ccs and b → cud transitions should give the same result up to the hadronic corrections that are expected to be small. However, due to the difference of the b → ccs and b → cud structure (see Figure 1), the New Physics phenomena may manifest themselves differently in these transitions [16]. The doubly Cabibbo-suppressed loop contributions to the b → ccs transitions, limiting the interpretation of measurements, can be controlled using the SU (3) flavor symmetry, as it is shown by De Bruyn and Fleischer in Ref. [17]. Bias of the observable 2β value can be controlled at the level of 0.3 • assuming 20% accuracy in U -symmetry approximation.
The obtained value of sin 2β leaves the ambiguity β → π/2−β, which can be resolved by measuring cos 2β. Several approaches to measure cos 2β in the b → cud transitions using the time-dependent Dalitz plot analysis were discussed: (1) the analysis of B 0 → Dh 0 , D → K 0 S π + π − decays was proposed in Ref. [18], (2) the analysis of B 0 → D CP π + π − decays was mentioned in Ref. [19] and considered in detail in Ref. [20], (3) the analysis of B 0 → Dπ + π − , D → K 0 S π + π − decays was mentioned in Ref. [20]. Only the B 0 → Dh 0 , D → K 0 S π + π − decays analysis was implemented in practice providing the first [21] as well as the most precise at the moment measurements of cos 2β [22,23]. 1 These results indicate positiveness of the cos 2β as expected within the KM mechanism.
Measurements of cos 2β in B 0 → Dh 0 , D → K 0 S π + π − decays require knowledge of the phase difference ∆δ D between the amplitudes of D 0 → K 0 S π + π − and D 0 → K 0 S π + π − decays that varies over the phase space and cannot be measured directly. The common workaround is to build a phenomenological decay amplitude model and obtain the D meson decay amplitude phase from the model. A model uncertainty is inherent in this approach.
The LHCb and Belle II [24] experiments are expected to collect samples of B meson decays much larger than those available today. Precision of model-dependent measurements of the angle β with that statistics will probably be limited by the model uncertainty. Indeed, currently the model uncertainty is assessed mostly from the statistical error of 1 Results of the cos 2β measurement in B 0 → Dh 0 , D → K 0 S π + π − decays via joint analysis of the Belle and BaBar experiments data are being prepared for publication at the moment. It is expected to be the most precise measurement of cos 2β before the Belle II data is available. See the talk by M. Roehrken at the 52nd Rencontres de Moriond EW 2017 conference. model parameters, assuming that the obtained value exceeds the uncertainty related to justification of the model approach. There is no reason to rely on this assumption in a percent-precision-level measurement.
The idea of binned Dalitz plot analysis proposed in Ref. [25] was to overcome the limitations of model-dependent consideration of multibody decays. The initial idea is related to measuring the UT angle γ in B ± → DK ± , D → K 0 S π + π − decays. It was developed further and extended to several other applications in Refs. [26][27][28][29][30][31][32][33]. A measurement of cos 2β in Ref. [23] has been performed in a model-independent way using these ideas.
In this work, the model-independent approach is considered in a context of the angle β measurement in time-dependent analysis of B 0 → Dπ + π − decays with D meson decaying into CP-specific and K 0 S π + π − states. It is shown the angle β and necessary hadronic parameters of the B 0 → D 0 π + π − decay can be obtained in a single measurement. Formalism of the time-dependent analysis of the B 0 → Dπ + π − decays is described in Sec. 2. The method for model-independent measurement of the angle β with the B 0 → Dπ + π − decays is developed in Sec. 3. The statistical precision with future data of the Belle II and LHCb experiments is evaluated in Sec. 4. The measurement bias due to the neglect of b → cud transition and charm mixing is considered in appendices B, C, and D.
2 Time-dependent analysis of B 0 → Dπ + π − decays Phenomenology of time-dependent CP violation measurements at an asymmetric-energy e + e − B-factory is described elsewhere [34]. The decay probability density for a flavourtagged B meson is expressed by where ∆t ∈ (−∞, ∞) is the proper decay time of a tagged B meson counted from the moment of the tagging B meson decay, 2 q B = 1 (q B = −1) corresponds to B 0 (B 0 ) flavour at ∆t = 0, ∆m B is the mass difference between the B meson mass eigenstates, τ B is the B 0 lifetime, and where f denotes the B meson final state and where q and p are the parameters of B meson mixing and Hereafter, absence of direct CP symmetry breaking in B and D meson decays as well as absence of CP symmetry breaking in B meson mixing are assumed 3 which where f denotes the state obtained by CP conjugation of state f . The amplitude of B 0 → D 0 π + π − , D 0 → f D can be expressed as where A D is the D 0 meson decay amplitude and A B depends on the Dalitz plot variables µ 2 ± ≡ m 2 (Dπ ± ). The amplitude of the CP-conjugated process, The parameters D f and F f from Eq. (2.1) take the form If the D meson is reconstructed in a flavour-specific final state, then D flv = 1 and F flv = 0. 4 A CP-specific D meson final state with CP parity ξ D results in The final state K 0 S π + π − introduces the second Dalitz plot resulting in dependence of the D meson decay probability density and the phase difference between the D 0 and D 0 decay amplitudes on the Dalitz plot variables m 2 ± = m 2 K 0 S π ± : In this case, the B meson decay probability density from Eq. (2.1) depends on time and four Dalitz plot variables.
In principle, any multibody self-conjugated final state, such as K 0 S K + K − , π + π − π 0 or K + K − π + π − can be considered, but the K 0 S π + π − state is the most experimentally clean and has rich resonance structure leading to significant variation of the phase difference ∆δ D over the Dalitz plot and good sensitivity to the CP violation parameters. Similar formalism can be developed for other multibody hadronic D meson final states, such as K − π + π 0 . The D meson decay probability densities p D and p D would be independent in that case.

Binned Dalitz plot analysis
The decay probability densities derived in the previous section can be expressed in terms of the parameters of the binned Dalitz plot. We follow the notation introduced in Ref. [28], where the D 0 → K 0 S π + π − Dalitz plot is divided into 2N bins (we use N = 8). The partitioning is done so that the bin index i ranges from −N to N excluding zero and the sign inversion i → −i corresponds to the Dalitz plot reflection m 2 + ↔ m 2 − . The parameters K i , K i , C i and S i are defined for the i th bin: where integration is performed over the i th bin and The relation (2.10) and symmetry of the Dalitz plot partitioning lead to the relations In a similar way, we divide the B 0 → Dπ + π − decay Dalitz plot into 2M = 2 × 8 bins and define the parameters k j , c j and s j for that Dalitz plot, where the bin index j ranges from −M to M excluding zero. A time-dependent B 0 → Dπ + π − decay probability density is defined for the j th bin. In the case of double Dalitz decay B 0 → Dπ + π − , D → K 0 S π + π − , the decay probability density is defined for each combination of B 0 Dalitz plot bin j and D 0 Dalitz plot bin i: (3.4) The following substitutions are used to express the coefficients D and F in the form suitable for the binned analysis: The expression Eq. (2.9) for the CP-specific D meson decays transforms into The double Dalitz plot case with the D 0 → K 0 S π + π − decay results in We consider the parameters K i , C i and S i to be known because they can be measured in decays of coherent D 0 D 0 pairs [35]. The 2M parameters k j , M parameters c j , M parameters s j and the angle β constitute 4M + 1 unknown parameters.
The parameters k j can be measured precisely in the time-integrated analysis of B 0 → D 0 π + π − decays with D 0 meson decaying into hadronic state K − π + . The expected fraction of events in the j th Dalitz plot bin is The second term in Eq. (3.8) is negligible even at the Belle II precision level. The B 0 → Dπ + π − with CP-specific D meson decays provide 2M independent constraints (Eq. (3.6)) and do not allow one to resolve the system. It should be noted that the above statement does not depend on CP parity of the D meson final state, particularly, final states with the same CP parities can be used and inclusion of a final state of the opposite CP parity would not increase the number of constraints.
) allowing to measure the parameters c j and s j together with the angle β in the joint analysis of the B 0 → Dπ + π − with CP-specific and D → K 0 S π + π − decays for any N and M. 5 The B 0 → Dπ + π − , D → K 0 S π + π − decays alone provide enough constraints to measure the parameters c j and s j , and the angle β for 2M(N − 1) ≥ 1. 5 An important feature of the described setup is that the values of sin 2β and cos 2β cannot be considered as independent parameters. Indeed, the transformation with an arbitrary scale η = 0 does not change the expressions for decay probability densities and the scale η can not be determined.

Symmetrized
The number of parameters related to the B 0 → D 0 π + π − binned Dalitz plot can be reduced by a factor of 2 considering the j th and −j th bins as a single bin. For the symmetrized in this way B 0 → D 0 π + π − decay Dalitz plot binning, the expressions Eq. (3.6) and Eq. (3.7) should be modified as follows: and where the dilution factor is the single parameter for the j th symmetric bin. The analysis procedure is slightly different in the case of symmetrized binning of the B 0 → D 0 π + π − Dalitz plot. Flavour-specific D meson decays are not needed. A combined time-dependent fit of the B 0 → D 0 π + π − with D meson decays into CP-specific and K 0 S π + π − final states should be performed in order to measure the dilution factors d j together with the angle β. The K 0 S π + π − final state is still necessary since the CP-specific final states provide M constraints while there are M + 1 unknown parameters. 6 The symmetrization of binning leads to a certain loss of information. Particularly, the B 0 → Dπ + π − with CP-specific D meson decays are not sensitive to the cos 2β (Eq. (3.11)) in this case. A quantitative evaluation of the sensitivity decline related to the symmetrized B 0 → D 0 π + π − Dalitz plot partitioning is described in the next section.

Feasibility study
Sensitivity of the described method is assessed with a series of toy Monte Carlo (MC) experiments. The equal-phase D 0 → K 0 S π + π − decay Dalitz plot binning deduced from the decay model published in Ref. [36] is used. The values of parameters K i , C i and S i for that binning are taken from measurement in Ref. [35].
A model-independent measurement of the angle β in B 0 → Dh 0 decays is considered as a reference procedure. The coefficients D and F from Eqs. (3.3) and (3.4) for the case of B 0 → Dh 0 decays can be obtained using the formal substitutions where ξ CP h 0 is the CP eigenvalue of h 0 meson and L is the angular moment of Dh 0 system. The MC events are generated with probability density functions (PDFs) of the form where the resolution function R, employed also as the background PDF, is a Gaussian with zero mean and f bkg is the background fraction. The function p w true is a PDF from Sec. 3 with the wrong B meson flavor tagging probability w factor The tagging power ε tag ≡ (1 − 2w) 2 characterizes effective reduction of data sample due to non-ideality of a B meson flavour tagging procedure. The tagging power ε tag = 0.3, typical for B factory experiments, is employed for the Belle and Belle II and ε tag = 0.08 is employed for the LHCb taking into account the recent progress in the flavour-tagging algorithms at hadronic machines [37]. 7 The values of PDF parameters for the Belle (II) and LHCb are chosen based on results from Refs. [15,23,38] and are shown in Table 1. Table 2 shows estimates of the signal yields for the Belle, Belle II and LHCb experiments. The estimates for Belle are obtained using the results from Refs. [15,23,39]. The estimates for Belle II are obtained by extrapolating the Belle yields assuming the same experimental conditions and 50 times larger integrated luminosity. The estimate signal yields corresponding to the data collected by LHCb in 2010 -2012 are based on the results from Refs. [38,40,41]. This period of data taking is referred to as Run I. The estimates for the LHCb signal yields corresponding to the end of current data taking period (Run II) and to the period of data taking after the planned upgrade (Upgr.) [42] are roughly estimated to be, respectively, 4 and 70 times larger than the Run I values, assuming the corresponding luminosity integrals equal 8 fb −1 and 50 fb −1 .
The signal yields for B 0 → Dπ + π − with flavour-specific D meson decays are relatively large for both Belle and LHCb. Thus, the uncertainties related to the parameters k j are neglected.

Parameters of the
Two models of the B 0 → D 0 π + π − decay amplitude are available in Refs. [38,39]. A simplified version of the model from Ref. [39] is used in this study (see Appendix A). The Dalitz Table 2. Estimates of the signal yields for the B 0 → D 0 {h 0 , π + π − }, D 0 → {f CP , K 0 S π + π − } (and C-conjugated) decays at the Belle, Belle II and LHCb experiments.

Mode
Belle Belle II LHCb Run I Run II Upgr. distribution and distributions of the D 0 π + and π + π − invariant masses obtained with this model are shown in Figure 2.
The equal-phase binning of the B 0 → D 0 π + π − decay Dalitz plot into 16 bins is performed using this model. The bin regions obtained and corresponding values of the parameters k j , c j and s j are shown in Figure 3.

Numerical experiments
Three approaches to measure the angle β are considered. Each approach implies the joint analysis of ∆t distributions for the B 0 → Dπ + π − with D meson decays into CP-specific and K 0 S π + π − final states. These approaches are: 1. The fit based on Eqs. (3.6) and (3.7) with 17 free parameters: eight (c j , s j ) pairs and the angle β.
3. Model-independent measurement of the angle β in the B 0 → Dh 0 decays as a reference. The angle β is the only free parameter in this case.

Measuring scheme
Belle Belle II LHCb Run I Run II Upgr.
The statistical precision of the angle β measurement for the initial value β = 22 • , obtained with each of the three approaches, is shown in Table 3. The analysis of B 0 → Dπ + π − decays provides precision about 1.5 times worse than the analysis of B 0 → Dh 0 decays. The prospects for the analysis of B 0 → Dh 0 decays at LHCb are not clear since there are neutral particles in the final state. The Belle II and upgraded LHCb have comparable potential to measure the angle β in B 0 → Dπ + π − decays. A combination of the results from B 0 → Dh 0 and B 0 → Dπ + π − analyses would yield the β precision in b → cud transitions below one degree. 8 Figure 4 illustrates prospects for the Belle II experiment: a fit result for the dilution factors d j (Figure 4a) and for the parameters c j and s j (Figure 4b) obtained with MC simulation for the input value β = 22 • .
The results presented are obtained with a simple method of the Dalitz plot binning 8 At the moment, the uncertainty related to the Ci and Si parameters measurement is about 1.1 • , as it is stated in Ref. [23]. The precision level below one degree can be achieved only if a more precise measurement of the parameters Ci and Si appears. Such a measurement can be provided by the BESIII collaboration and by a future Super c-τ factory experiment.  (the equal-phase binning). It is shown in Refs. [27,35] that the binning can be optimized to improve the statistical sensitivity by a factor of about 1.2.

Conclusions
A novel model-independent approach to measure the CKM angle β with time-dependent analysis of the B 0 → Dπ + π − decays dominated by the tree quark transition is proposed. It is shown that the angle β and the parameters of binned B 0 → D 0 π + π − decay Dalitz plot can be obtained from the single measurement. Statistical precision of the method is comparable to that of the model-independent angle β measurement in B 0 → Dh 0 decays. The fact that only charged particles compose the final states of B 0 → Dπ + π − , D → f CP and D → K 0 S π + π − decay chains for such f CP as K + K − , π + π − , and φK 0 S provides good experimental perspectives for LHCb.
The angle β can be measured with the one-degree precision level at the Belle II and LHCb experiments in b → cud transitions in a model-independent way, namely without the need to model neither the D 0 → K 0 S π + π − nor the B 0 → D 0 π + π − decay amplitudes. The combined time-dependent analysis of B 0 → Dh 0 and B 0 → Dπ + π − decays with D meson decaying into a f CP (f CP = K + K − , K 0 S π 0 etc.) and K 0 S π + π − states should be performed in order to achieve such precision.
The measurement bias inherent in the proposed method due to the neglect of the suppressed transition b → ucd and charm mixing is of order of 0.2 • (see Appendix D) and can be considered as a probably non-dominant systematic uncertainty. Table 4. List of resonances included in the B 0 → D 0 π + π − decay amplitude model. The resonance fit fraction is denoted by F and the resonance amplitude phase is denoted by ϕ A The B 0 → D 0 π + π − decay amplitude model A simple isobar model of the B 0 → D 0 π + π − decay amplitude, inspired by the result from Ref. [39], is used in numerical experiments. The resonances constituting the model are listed in Table 4. Each resonance is described by a relativistic Breit-Wigner function [43]. Energy-dependent resonance width and Blatt-Weisskopf barrier factors [44,45] are used.

B Formalism accounting for the b → ucd transition
A precise measurement of the angle β in the b → cud transitions requires understanding the bias due to the neglect of the suppressed decay B 0 → D 0 π + π − and charm mixing. Both processes produce additional interfering amplitudes for the B 0 → D 0 π + π − , D 0 → K 0 S π + π − decay shown on the scheme at Figure 5.
This appendix extends the formalism presented in sections 2 and 3 and accounts for the B 0 → D 0 π + π − decay. Corrections due to the charm mixing are considered in appendix C. Quantitative estimates of the bias due to the neglect of these processes are described in appendix D.
As discussed in Ref [46], the expressions (B.17), (B.18), and (B.19) describe also the time-dependent analysis of tagged B 0 → DK 0 S decays. The CKM angles β and γ, phase ∆δ B and parameter r B can be simultaneously measured in a such analysis. In contrast with the B 0 → Dh 0 decay, the r B value corresponding to the B 0 → DK 0 S decay can be as large as 0.2, improving sensitivity to the CP violation parameters. However, the expected number of reconstructed at a B factory B 0 → DK 0 S decays is about the order of magnitude less then the number of reconstructed B 0 → Dh 0 decays. Numerical experiments have been performed to estimate the statistical precision one may expect with the Belle II data. The results obtained with r B = 0.2 are These values are only marginally dependent on ∆δ B . The angle γ precision doesn't improve much if the β value is considered as known.

C Formalism accounting for the charm mixing
We assume conservation of CP symmetry in charm mixing. The B 0 → Dπ + π − , D → K 0 S π + π − decay amplitude taking into account charm mixing can be written as follows: where t D is the D meson proper decay time and functions describe the D meson time evolution. Here x and y are the charm mixing parameters and τ D is the D 0 lifetime. The corresponding amplitude of the Integrating Eqs. (C.4), (C.5) and (C.6) over i th bin of the D Dalitz plot and j th bin of the B Dalitz plot we obtain the expressions for the binned analysis: The expressions for CP specific D meson decays and B 0 → Dh 0 decay can be obtained as a particular cases of Eqs. (C.7) and (C.8): (C.11) D Estimate of the bias due to neglect of b → ucd transition and charm mixing The neglect of b → ucd transition and charm mixing leads to a bias of the observed value of the angle β. Numerical experiments have been performed to assess the bias value. Data samples for the numerical experiments are generated using the expressions from appendices B and C. The values of angle β and hadronic parameters c j and s j are extracted from the generated samples with the maximum likelihood method. The fit procedure uses equations from Sec. 3 (i.e. neglects the b → ucd transition and charm mixing). The results obtained are summarized in the Table 5.
A model of the suppressed B 0 → D 0 π + π − decay is needed to obtain the values of parameters k j , c j , s j , c j , s j , c j and s j defined in Eqs. (B.11) and (B.12). We use the Table 5. Estimates for the angle β measurement bias due to the neglect of b → ucd transition (3 rd column) and charm mixing (4 th column). The second column shows the D 0 decays combination used in the fit: n D , n + , n − are relative fractions of D 0 → K 0 S π + π − , D 0 → f CP+ and D 0 → f CP− decays yields, respectively.

Process
(n D , n + , n − ) δβ b→ucd δβ mix factorization assumption 9 to construct an ensemble of the B 0 → D 0 π + π − decay models. The B 0 → D 0 π + π − decay model described in appendix A is taken as a basis and the following modifications are applied: • The B 0 → D * − 2 (2460)π + transition amplitude is reduced by a factor of 10 since it cannot proceed through a tree weak diagram.
An ensemble of 100 B 0 → D 0 π + π − decay models is constructed with 100 random triples of phases ϕ(D * 2 ), ϕ(D v * ), ϕ(D * 0 ) corresponding to the D * 2 , D v * and D * 0 amplitudes, respectively. The values quoted in the third column of Table 5 for the B 0 → D 0 π + π − decay are the maximal biases over the ensemble of models.
The results of numerical experiments and formalism described in appendices B and C lead to the following conclusions: 1. The bias due to neglect of the charm mixing is 3 ÷ 4 times smaller than the bias due to neglect of the b → ucd transition.
2. The biases corresponding CP specific D meson final states with ξ D = +1 and ξ D = −1 have equal absolute values and opposite signs. This feature was previously pointed out in Ref. [47]. Eqs. (B.16), (B.19c), (C.9c) and (C.12c) show that the main terms are proportional to the ξ D while the first order corrections do not depend on ξ D .
3. Biases for the processes involving D → K 0 S π + π − decay are of the order of 0.1 • . Relative smallness of this value can be qualitatively explained by the pairwise reduction of bias in bins of the Dalitz plot. This effect generalizes the feature described in the previous item. The same reduction is takes place in the binned analysis of B 0 → D 0 π + π − decay.
4. The biases for the B 0 → D CP h 0 decays are large enough to be observed with the Belle II statistics. However, assuming the statistics ratio 2/1/1 of the K 0 S π + π − , ξ D = +1 and ξ D = −1 events, respectively (which is close to reality), the residual bias is about 0.1 • .
5. Most of the D 0 decays to CP eigenstates collected by LHCb have negative CP parity (D 0 → K + K − , π + π − ). This CP parity imbalance does not lead to a significant bias in the case of analysis of the B 0 → D 0 π + π − decays, in contrast with the B 0 → Dh 0 case, as it is shown in the third and fourth rows of the Table. 5. The resulting bias due to neglect of the b → ucd amplitude is at level of 0.2 • .