# Unbinned model-independent measurements with coherent admixtures of multibody neutral *D* meson decays

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## Abstract

Various studies of Standard Model parameters involve measuring the properties of a coherent admixture of \({D} ^0\) and \({\overline{D}{}} {}^0\) states. A typical example is the determination of the Unitarity Triangle angle \(\gamma \) in the decays \(B\rightarrow DK\), \(D\rightarrow {{K} ^0_\mathrm{\scriptscriptstyle S}} {{\pi } ^+} {{\pi } ^-} \). A model-independent approach to perform this measurement is proposed that has superior statistical sensitivity than the well-established method involving binning of the \(D\rightarrow {{K} ^0_\mathrm{\scriptscriptstyle S}} {{\pi } ^+} {{\pi } ^-} \) decay phase space. The technique employs Fourier analysis of the complex phase difference between \({D} ^0\) and \({\overline{D}{}} {}^0\) decay amplitudes and can easily be generalised to other similar measurements, such as studies of charm mixing or determination of the angle \(\beta \) from \({{B} ^0} \rightarrow D h^0\) decays.

## 1 Introduction

Precise measurements of \(C\!P\) violation in decays of beauty hadrons is one of the key methods to search for effects of physics beyond the Standard Model. The phenomenon of \(C\!P\) violation is described in the Standard Model (SM) by the Cabibbo–Kobayashi–Maskawa (CKM) mechanism [1, 2], where \(C\!P\) violation enters as a complex phase in the unitary \(3\times 3\) matrix (CKM matrix) describing transitions between quarks of the three generations due to charged-current weak interactions. A common representation of the CKM matrix is the Unitarity Triangle (UT), the sides and angles of which are experimentally observable parameters. The fundamental \(C\!P\)-violating phase, the angle \(\gamma \) of the UT (also known in the literature as \(\phi _3\)), can be obtained with extremely low theoretical uncertainty [3] from tree-dominated \({b} \) hadron decays and thus serves as a “standard candle” for searches of effects beyond the Standard Model in other heavy flavour processes.

Various techniques have been proposed to measure \(\gamma \) experimentally in the decays of *B* mesons into final states with neutral *D* mesons [4, 5, 6, 7]. The \(C\!P\) violation in these decays is generated by interference of \({b} \rightarrow {c} \) and \({b} \rightarrow {u} \) quark level transitions once the neutral *D* meson is reconstructed in a final state accessible to both \({D} ^0\) and \({\overline{D}{}} {}^0\) decays. The neutral *D* meson in this case forms a coherent admixture of \({{D} ^0} \) and \({{\overline{D}{}} {}^0} \) states which is denoted here as *D*. One of the most sensitive techniques involves analysis of the Dalitz plot density of multibody *D* decays such as \(D\rightarrow {{K} ^0_\mathrm{\scriptscriptstyle S}} {{\pi } ^+} {{\pi } ^-} \) [8, 9].

Two different techniques have been developed and implemented experimentally to extract \(\gamma \) from \(B\rightarrow DK\) decays using multibody *D* meson final states. One is model-dependent, with the complex amplitude of the *D* decay obtained by fitting the flavour-specific \({{D} ^0} \) decay density to a model [10, 11, 12, 13, 14, 15, 16]. This technique offers optimal statistical precision since the fit can be performed in an unbinned fashion, however, it suffers from uncertainty, which is difficult to quantify, due to modelling of the \({{D} ^0} \) amplitude. Another method is a binned model-independent approach, where information as regards the behaviour of the strong phase across the phase space of the \({D} ^0\) decay is obtained from samples of quantum-correlated \({{D} ^0} {{\overline{D}{}} {}^0} \) decays produced near kinematic threshold [8, 17, 18, 19, 20].

In the model-independent technique, one needs to determine the relation between the decay densities of quantum-correlated \({{D} ^0} {{\overline{D}{}} {}^0} \) decays and *D* decays from \(B\rightarrow DK\). This necessarily requires estimation of the decay density from scattered data, which is achieved by binning both decay densities. Each bin is assigned a number of parameters that characterise the averaged behaviour of the amplitude (its magnitude and phase) over the bin; these parameters are obtained by solving a system of equations that also includes the value of \(\gamma \). In general, the binned approach reduces statistical sensitivity compared to the unbinned model-dependent technique, but the procedure is developed in such a way that it produces an unbiased measurement even in the case of a very rough binning.

In this paper, a method to extract \(\gamma \) is proposed which does not involve binning and aims to combine the advantages of the model-dependent and model-independent approaches. Like the binned approach with optimal binning, it uses a construction inspired by a \({{D} ^0} \) amplitude model, but provides an unbiased measurement even if the wrong model is used. It is shown to offer better statistical sensitivity than the binned approach. The method employs Fourier analysis of the distribution of the complex phase difference between the \({D} ^0\) and \({\overline{D}{}} {}^0\) amplitudes. The method is illustrated using the “golden” channel \(B\rightarrow DK\) with subsequent \(D\rightarrow {{K} ^0_\mathrm{\scriptscriptstyle S}} {{\pi } ^+} {{\pi } ^-} \) decay, but can easily be generalised to other cases of \(\gamma \) determination where the binned model-independent technique is applicable: analyses using other three- or four-body \({{D} ^0} \) decays [21, 22, 23, 24, 25, 26], multibody *B* decays [27, 28] or analyses using correlated Dalitz plots of multibody *B*- and *D*-meson decays [29, 30].

Apart from measurements of \(\gamma \), similar model-independent techniques, which employ interference between \({{D} ^0} \) and \({{\overline{D}{}} {}^0} \) amplitudes, have been developed for other kinds of measurements: studies of \(C\!P\) violation and mixing parameters of \({{D} ^0} \) mesons [31, 32, 33], measurements of the UT angle \(\beta \) in \({{B} ^0} \rightarrow Dh^0\) (where \(h^0\) is a neutral light meson) and \({{B} ^0} \rightarrow D{{\pi } ^+} {{\pi } ^-} \) decays [34, 35]. In all these cases, the technique proposed can be applied instead of the binned methods.

## 2 Model-independent formalism with weight functions

In this section, the formalism for \(\gamma \) measurement is recalled to introduce the notation, and the established model-independent technique is reformulated in slightly different terms. This allows a demonstration that the binned approach is not the only possible method to perform such a measurement.

*D*mesons with opposite flavours. In the case of \({{{B} ^+}} \rightarrow D{{K} ^+} \) decays followed by \(D\rightarrow {{K} ^0_\mathrm{\scriptscriptstyle S}} {{\pi } ^+} {{\pi } ^-} \), the amplitude as a function of two variables of the

*D*decay Dalitz plot, the squared invariant masses \(m^2_+ \equiv m^2_{{{K} ^0_\mathrm{\scriptscriptstyle S}} {{\pi } ^+}}\) and \(m^2_-\equiv m^2_{{{K} ^0_\mathrm{\scriptscriptstyle S}} {{\pi } ^-}}\), is expressed as

*D*decay amplitudes: \(\overline{A}_D\leftrightarrow A_D\). A simultaneous analysis of the two amplitudes \(A_B\) and \(\overline{A}_B\) provides information on the unknown parameters \(\gamma \), \(r_B\), and \(\delta _B\).

*i.e.*the Dalitz plot distributions for \({{D} ^0} \) and \({{\overline{D}{}} {}^0} \) decays are symmetric under the exchange \(m^2_+\leftrightarrow m^2_-\) assuming \(C\!P\) conservation in \({{D} ^0} \) decays.

^{1}The functions \(C(\mathbf {z})\) and \(S(\mathbf {z})\) contain information as regards the motion of the complex strong phase over the Dalitz plot which cannot be obtained from flavour-specific

*D*meson decays:

*D*mesons produced at the kinematic threshold in the \(e^+e^-\rightarrow {{D} ^0} {{\overline{D}{}} {}^0} \) process to obtain this information. In this case, the two

*D*mesons are produced in a

*P*-wave such that their wave function is antisymmetric. As a result, if both

*D*mesons are reconstructed in the \({{K} ^0_\mathrm{\scriptscriptstyle S}} {{\pi } ^+} {{\pi } ^-} \) final state, the densities of two Dalitz plots will be correlated:

*D*mesons and \(h_{DD}\) is a normalisation factor. The necessary information as regards \(C(\mathbf {z})\) and \(S(\mathbf {z})\) is present in Eq. (8), but it is not straightforward to obtain the explicit expressions for the functions \(C(\mathbf {z})\) and \(S(\mathbf {z})\) from the observable distributions \(p_D(\mathbf {z})\), \(\bar{p}_{D}(\mathbf {z})\) and \(p_{DD}(\mathbf {z} _1, \mathbf {z} _2)\).

*D*mesons from correlated \({{D} ^0} {{\overline{D}{}} {}^0} \) pairs can offer additional information to resolve these ambiguities. For instance, decays where one of the

*D*mesons is reconstructed in a \(C\!P\) eigenstate and the other is reconstructed as \({{K} ^0_\mathrm{\scriptscriptstyle S}} {{\pi } ^+} {{\pi } ^-} \) constrain \(C(\mathbf {z})\), and they resolve the ambiguity (9), as well as fix the sign for \(C(\mathbf {z})\). The remaining ambiguity, the sign of \(S(\mathbf {z})\), can be resolved by a weak model assumption using isobar parametrisation of the

*D*decay amplitude [18]. In practice, several

*D*decay modes are combined to measure the same strong phase parameters [36], but the description below will concentrate only on \({{D} ^0} {{\overline{D}{}} {}^0} \) pairs where both

*D*mesons are decaying to \({{K} ^0_\mathrm{\scriptscriptstyle S}} {{\pi } ^+} {{\pi } ^-} \).

*D*decay, while for Eq. (10) double integral is performed over the Dalitz plots \(\mathcal {D} _1\) and \(\mathcal {D} _2\) of two decaying

*D*mesons. Unlike in the binned formalism described in Refs. [17, 18], here the terms proportional to \(|A_{D}(\mathbf {z})|\cdot |\overline{A}_{D}(\mathbf {z})|\) are not factored out, thus capital letters are used to distinguish the expressions of Eq. (13) from \(c_i\) and \(s_i\) coefficients commonly used in the binned formalism.

The values of weighted integrals for the flavour-specific *D* sample (\(p_n\) and \(\bar{p}_n\)), *B* sample (\(\bar{p}_{{B},n}\) and \(p_{{B},n}\)) and correlated \({{D} ^0} {{\overline{D}{}} {}^0} \) sample (\(p_{DD,mn}\)) can be obtained directly from each of the corresponding scattered data samples by replacing the integrals with sums over individual observed events. The values of the weighted integrals for the phase terms \(C_n\) and \(S_n\) are considered as free parameters constrained by Eq. (10). This allows the values of \(x_{\pm }\) and \(y_{\pm }\) to be obtained by solving the system of Eq. (10) and (11).

*D*decay amplitudes [18]. Specifically, if one defines the phase difference \(\Phi (m^2_+, m^2_-)\) as

The following section shows how to construct an unbinned model-independent formalism using a model-based phase-difference function \(\Phi (m^2_+, m^2_-)\) which will be a generalisation of the technique with phase-difference binning. For reasons which will become obvious, this approach will not be optimal from the point of view of statistical uncertainty, and it will serve solely as a demonstration. Subsequently, a more optimal approach based on a similar construction will be presented.

## 3 Unbinned technique using Fourier series expansion of phase difference

*D*decay becomes

*i.e.*use weight functions of the form \(\cos (n\phi )\) and \(\sin (n\phi )\), where

*n*is an integer number. The unknowns \(x_{\pm }\) and \(y_{\pm }\) will then enter the system of equations which relates the coefficients of the Fourier expansions of the \(p_D\), \(p_{DD}\), \(\bar{p}_{{B}}\) and \(p_{{B}}\) densities.

*M*.

*e.g.*the covariance between the \(a_n\) and \(b_m\) coefficients can be calculated as

## 4 Strategy with Fourier expansion on split Dalitz plot

The strategy outlined above is the simplest example of the approach using Fourier expansion of the phase-difference distribution to measure \(\gamma \). However, it is clear that this approach is not optimal from the point of view of statistical precision. The reason is that one integrates over all points of the phase space with the same expected phase difference, regardless of the magnitudes of the interfering \({{D} ^0} \rightarrow {{K} ^0_\mathrm{\scriptscriptstyle S}} {{\pi } ^+} {{\pi } ^-} \) and \({{\overline{D}{}} {}^0} \rightarrow {{K} ^0_\mathrm{\scriptscriptstyle S}} {{\pi } ^+} {{\pi } ^-} \) amplitudes. This effectively reduces the interference term and, as a consequence, the sensitivity to the relative phase between the two amplitudes. For similar reasons, the “optimal” binning scheme was introduced for the binned model-independent approach to improve the precision with the equal phase-difference binning [18].

*B*meson flavours. Throughout this paper, the flavour (\({b} \) or \(\bar{{b}}\)) is consistently denoted by the absence or presence of a “bar” in the corresponding quantities, for example \(\bar{p}_{{B}}\) and \(p_B\), except for the subscript for \(C\!P\)-violating parameters \(x_{\pm }, y_{\pm }\), which is a commonly used notation.

*D*decay densities \(p^{\pm }_D(\phi )\) are defined as

*D*decay, following Eq. (37), they are related as

*D*decay densities in that case take the following form:

*D*decay from \(B^{\pm }\rightarrow DK^{\pm }\) take the following form in the split Dalitz plot case:

*D*phase parameters in the equations has now increased: there are \(4M+2\) independent coefficients \((a,b)^{C,S+}_n\) (\(0\le n\le M\) for

*a*and \(1\le n\le M\) for

*b*) plus a common normalisation factor \(h_{DD}\) in the system of equations (46). Nevertheless, the statistical precision in this approach appears to be better as will be seen in the feasibility study.

In principle, one can even consider splitting the Dalitz plot into more regions, but certainly the increase in the number of free parameters can diminish the possible gain in statistical precision. Any strategy involving splitting the Dalitz plot should be optimised taking into account the size of experimentally available samples of correlated \({{D} ^0} {{\overline{D}{}} {}^0} \) and \(B\rightarrow DK\) decays.

## 5 Simulation results

To test the feasibility of the proposed method, simulation studies using pseudoexperiments are performed. Samples of flavour-specific \(D\rightarrow {{K} ^0_\mathrm{\scriptscriptstyle S}} {{\pi } ^+} {{\pi } ^-} \) decays, correlated \({{D} ^0} {{\overline{D}{}} {}^0} \) pairs decaying to \({{K} ^0_\mathrm{\scriptscriptstyle S}} {{\pi } ^+} {{\pi } ^-} \), and \(D\rightarrow {{K} ^0_\mathrm{\scriptscriptstyle S}} {{\pi } ^+} {{\pi } ^-} \) decays from \(B\rightarrow DK\) are generated using the *D* decay amplitude measured by Belle collaboration [14]. Samples are simulated with \(r_B=0.1\), \(\gamma =60^{\circ }\) and \(\delta _B=130^{\circ }\), which is close to the results of the recent model-independent measurement of the \(B\rightarrow DK\), \(D\rightarrow {{K} ^0_\mathrm{\scriptscriptstyle S}} {{\pi } ^+} {{\pi } ^-} \) channel by the LHCb collaboration [19]. For each of those event samples, the Fourier series coefficients and their covariance matrices are calculated as described in Sect. 3. Systems of equations which contain relations between Fourier spectrum coefficients of flavour-specific *D*, \({{D} ^0} {{\overline{D}{}} {}^0} \) and \(B\rightarrow DK\) densities are then solved by maximising the combined likelihood to obtain the value of \(\gamma \).

The formalism in Sects. 3 and 4 involved Cartesian \(C\!P\)-violating parameters \(x_{\pm }\) and \(y_{\pm }\). This approach is likely more suitable when dealing with real data when one has to combine the results of different \(\gamma \)-sensitive analyses. In the simulation study presented here, the free parameters are chosen to be \(\gamma \), \(r_B\) and \(\delta _B\).

For the flavour-specific \(D\rightarrow {{K} ^0_\mathrm{\scriptscriptstyle S}} {{\pi } ^+} {{\pi } ^-} \) mode, a large sample of \(10^7\) generated events is used. This sample is not expected to contribute significantly to the uncertainty on \(\gamma \) since high-statistics data sets are available at both the *B* factories and the LHCb. The size of the \(B\rightarrow DK\), \(D\rightarrow {{K} ^0_\mathrm{\scriptscriptstyle S}} {{\pi } ^+} {{\pi } ^-} \) sample generated is \(10^4\) events for each *B* meson flavour, which corresponds roughly to 10 times the data sample from LHCb Run 1 [19]. Three scenarios with different correlated \({{D} ^0} {{\overline{D}{}} {}^0} \) sample sizes are considered, \(10^5\), \(10^4\) and \(10^3\) events. For comparison, the \(e^+e^-\rightarrow {{D} ^0} {{\overline{D}{}} {}^0} \) data sample collected by CLEO experiment where both *D* mesons decay into \({{K} ^0_\mathrm{\scriptscriptstyle S}} {{\pi } ^+} {{\pi } ^-} \) contains 473 events, however, many other *D* decay modes are used in the combined fit to obtain the phase coefficients (notably, the modes where one of the *D* mesons is reconstructed in a \(C\!P\) eigenstate or as \({{K} ^0_\mathrm{\scriptscriptstyle L}} {{\pi } ^+} {{\pi } ^-} \)) [36]. It is expected that the statistical uncertainty of the \({{D} ^0} {{\overline{D}{}} {}^0} \) sample of \(10^5\) events will contribute negligibly to the uncertainty on \(\gamma \), thus pseudoexperiments with this sample size probe uniquely how the approximation of the amplitude with a finite number of parameters (*i.e.* truncated Fourier series or limited number of bins) affects \(\gamma \) sensitivity. The low-statistics sample of \(10^3\) events, on the other hand, will demonstrate the contribution of a limited \({{D} ^0} {{\overline{D}{}} {}^0} \) sample to the sensitivity.

*M*on the number of harmonics is set to \(M=1, 2, 4, 7, 11\), or 19. In addition, an unbinned model-dependent fit is performed to serve as a reference for the best possible statistical \(\gamma \) precision that can be reached.

*M*.

*M*for relatively large \({{D} ^0} {{\overline{D}{}} {}^0} \) samples sizes, while for a small \({{D} ^0} {{\overline{D}{}} {}^0} \) sample size of \(10^3\) the optimum is reached for \(M=1\) (

*i.e.*for the smallest possible number of free parameters, which is three for non-split and six for split Dalitz plot). It is possible that other multibody

*D*decays may require higher harmonics to reach optimal sensitivity. Another case when Fourier terms with \(n>1\) might be required is if the amplitude model \(A_D^\mathrm{(model)}(\mathbf {z})\) used to define \(\Phi (\mathbf {z})\) differs significantly from the true one.

Uncertainty of \(\gamma \) measurement with strategies using binned fit (with optimal binning) and using Fourier expansion (with non-split and split Dalitz plot)

Sample size | \(\gamma \) resolution (\({}^{\circ }\)) | ||
---|---|---|---|

Binned optimal | Fourier non-split | Fourier split | |

\(10^4\) \(B\rightarrow DK\), \(10^3\) \({{D} ^0} {{\overline{D}{}} {}^0} \) | \(4.33\pm 0.10\) | \(4.54\pm 0.10\) | \(3.73\pm 0.08\) |

\(10^4\) \(B\rightarrow DK\), \(10^4\) \({{D} ^0} {{\overline{D}{}} {}^0} \) | \(3.60\pm 0.08\) | \(4.51\pm 0.10\) | \(3.43\pm 0.08\) |

\(10^4\) \(B\rightarrow DK\), \(10^5\) \({{D} ^0} {{\overline{D}{}} {}^0} \) | \(3.49\pm 0.10\) | \(4.47\pm 0.10\) | \(3.32\pm 0.08\) |

The \(\gamma \) uncertainties for the optimal scenarios with the binned and unbinned techniques are compared in Table 1. The uncertainty of the approach with split Dalitz plot is significantly better than when the Dalitz plot is taken as a whole. It is also clear that the Fourier expansion technique with split Dalitz plot shows better sensitivity than the binned method using “optimal” binning, with the gain being the most significant for smaller \({{D} ^0} {{\overline{D}{}} {}^0} \) sample size. The technique, however, is still about 10% less sensitive than the unbinned model-dependent approach. The possibilities to further improve the sensitivity of the unbinned model-independent method are discussed in Sect. 7.

## 6 Practical considerations

To be applicable to real data, the technique should be able to deal with experimental effects such as backgrounds and non-uniform detection efficiency across the Dalitz plot. Since the background enters the decay density additively, it can be treated at the level of Fourier-transformed variables, by calculating the Fourier expansion of the background density and subtracting it from the coefficients calculated on data. On the other hand, the efficiency enters the density in a multiplicative way, thus Fourier expansion need to be applied to efficiency-corrected data. The correction can be applied on an event-by-event basis, by assigning each event a weight proportional to the inverse of efficiency while calculating the Fourier coefficients.

The studies presented above have been performed using a combined likelihood fit to both \(B\rightarrow DK \) and correlated \({{D} ^0} {{\overline{D}{}} {}^0} \) samples. It is also possible to perform the analysis in two stages, by first calculating the coefficients of Fourier transformation of the functions \(C(\phi )\) and \(S(\phi )\) from the \({{D} ^0} {{\overline{D}{}} {}^0} \) data, followed by a fit to \(B\rightarrow DK\) sample using the coefficients, their correlations and uncertainties from the first stage. This is likely to be more convenient in practice, since the data samples come from different experiments.

## 7 Further directions of development

The proposed technique could be especially useful in the cases where a binned approach will limit precision due to small sample sizes of decays which determine the phase information. Examples are the \({{D} ^0} \rightarrow {{K} ^0_\mathrm{\scriptscriptstyle S}} {{K} ^+} {{K} ^-} \) mode, where the sample of quantum-correlated decays is small and currently only two bins are used in the \(\gamma \) measurement [19]. Another example is \(B\rightarrow DK\pi \) decays, where the phase coefficients corresponding to the three-body *B* decay are free parameters together with \(\gamma \) [29, 30]. Having an amplitude model which describes the strong phase variation across the *B* decay Dalitz plot with a small number of parameters should improve the statistical sensitivity.

Other analyses, where the coherent \({{D} ^0} \)–\({{\overline{D}{}} {}^0} \) admixtures are involved, are measurements of charm mixing and \(C\!P\) violation in mixing and measurement of the UT angle \(\beta \) in \(B\rightarrow D h^0\) decays. These classes of measurements utilise oscillations of \({{D} ^0} \) and \({{B} ^0} \) mesons, respectively, and thus the parameters of the \({{D} ^0} \)–\({{\overline{D}{}} {}^0} \) admixture are functions of decay time. In the proposed formalism, the coefficients of the Fourier series will be functions of decay time as well. While such analyses will certainly be more complicated than the case with constant coefficients, they are conceptually similar to the measurements using the binned technique which have already been carried out [33, 35].

## 8 Conclusion

A technique to perform unbinned model-independent analysis of a coherent admixture of \({D} ^0\) and \({\overline{D}{}} {}^0\) states decaying to a multibody final state is proposed. It is illustrated in detail using the measurement of the UT angle \(\gamma \) from \(B\rightarrow DK\) decays. Unlike the well-known technique with Dalitz plot binning, the proposed method employs Fourier analysis of the spectrum of the strong phase difference between the \({D} ^0\) and \({\overline{D}{}} {}^0\) amplitudes. While the method relies on an amplitude model to reach optimal statistical precision, it is unbiased by construction even if the wrong model is used.

A study of the feasibility of the proposed method has been performed with simulated pseudoexperiments. The precision of the method does not depend strongly on the number of Fourier expansion terms used, and even with only the single leading term yields sensitivity comparable to that of the binned model-independent approach. A modification of the procedure, where Fourier expansion is performed in two regions of the Dalitz plot separated according to the ratio of the suppressed and favoured amplitudes, provides \(\gamma \) sensitivity better than the most optimal binned strategy. The gain compared to the binned approach is especially significant if the size of the correlated \({{D} ^0} {{\overline{D}{}} {}^0} \) sample, which determines the strong phase in *D* meson decay, is small. Possible ways of improving the sensitivity of the proposed technique even further are identified and need further study.

The method is not limited to \(\gamma \) measurements with three-body *D* decays and can be generalised to any analysis where the parameters of a coherent admixture of \({{D} ^0} \) and \({{\overline{D}{}} {}^0} \) in a multibody final state need to be determined, such as measurements of charm mixing and \(C\!P\) violation, and measurements of the UT angle \(\beta \) in \(B\rightarrow D h^0\) decays. The technique could also be useful in \(\gamma \) measurements with a double Dalitz plot analysis of \(B\rightarrow DK\pi \), \(D\rightarrow {{K} ^0_\mathrm{\scriptscriptstyle S}} {{\pi } ^+} {{\pi } ^-} \) decay; in that case the Fourier expansion can be applied to both the *B* and the *D* Dalitz plots.

## Footnotes

- 1.
The same assumption is made throughout this paper.

## Notes

### Acknowledgements

The author is grateful to his colleagues from the LHCb collaboration for stimulating discussions and help in preparing the paper: Timothy Gershon, Mark Whitehead, Guy Wilkinson, Wenbin Qian, Susan Haines, Matthew Kenzie, and other members of “beauty decays to open charm” analysis working group. The author would like to thank Alex Bondar for inspiring the search for model-independent approaches beyond the well-developed binned technique. The work is supported by the Science and Technology Facilities Council (United Kingdom).

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