# Disentangling weak and strong interactions in \(B\rightarrow K^{*}(\rightarrow K\pi )\pi \) Dalitz-plot analyses

## Abstract

Dalitz-plot analyses of \(B\rightarrow K\pi \pi \) decays provide direct access to decay amplitudes, and thereby weak and strong phases can be disentangled by resolving the interference patterns in phase space between intermediate resonant states. A phenomenological isospin analysis of \(B\rightarrow K^*(\rightarrow K\pi )\pi \) decay amplitudes is presented exploiting available amplitude analyses performed at the BaBar, Belle and LHCb experiments. A first application consists in constraining the CKM parameters thanks to an external hadronic input. A method, proposed some time ago by two different groups and relying on a bound on the electroweak penguin contribution, is shown to lack the desired robustness and accuracy, and we propose a more alluring alternative using a bound on the annihilation contribution. A second application consists in extracting information on hadronic amplitudes assuming the values of the CKM parameters from a global fit to quark flavour data. The current data yields several solutions, which do not fully support the hierarchy of hadronic amplitudes usually expected from theoretical arguments (colour suppression, suppression of electroweak penguins), as illustrated from computations within QCD factorisation. Some prospects concerning the impact of future measurements at LHCb and Belle II are also presented. Results are obtained with the CKMfitter analysis package, featuring the frequentist statistical approach and using the Rfit scheme to handle theoretical uncertainties.

## 1 Introduction

Non-leptonic *B* decays have been extensively studied at the *B*-factories BaBar and Belle [1], as well at the LHCb experiment [2]. Within the Standard Model (SM) some of these modes provide valuable information on the Cabibbo–Kobayashi–Maskawa (CKM) matrix and the structure of CP violation [3, 4], entangled with hadronic amplitudes describing processes either at the tree level or the loop level (the so-called penguin contributions). Depending on the transition considered, one may or may not get rid of hadronic contributions which are notoriously difficult to assess. For instance, in \(b\rightarrow c\bar{c}s\) processes, the CKM phase in the dominant tree amplitude is the same as that of the Cabibbo-suppressed penguin one, so the only relevant weak phase is the \(B_d\)-mixing phase \(2\beta \) (up to a very high accuracy) and it can be extracted from a CP asymmetry out of which QCD contributions drop to a very high accuracy. For charmless *B* decays, the two leading amplitudes often carry different CKM and strong phases, and thus the extraction of CKM couplings can be more challenging. In some cases, for instance the determination of \(\alpha \) from \(B\rightarrow \pi \pi \) [5], one can use flavour symmetries such as isospin in order to extract all hadronic contributions from experimental measurements, while constraining CKM parameters. This has provided many useful constraints for the global analysis of the CKM matrix within the Standard Model and the accurate determination of its parameters [6, 7, 8, 9], as well as inputs for some models of New Physics [10, 11, 12, 13].

The constraints obtained from some of the non-leptonic two-body *B* decays can be contrasted with the unclear situation of the theoretical computations for these processes. Several methods (QCD factorisation [14, 15, 16, 17], perturbative QCD approach [18, 19, 20, 21, 22, 23], Soft-collinear effective theory [24, 25, 26, 27, 28]) were devised more than a decade ago to compute hadronic contributions for non-leptonic decays. However, some of their aspects remain debated at the conceptual level [29, 30, 31, 32, 33, 34, 35, 36, 37], and they struggle to reproduce some data on *B* decays into two mesons, especially \(\pi ^0\pi ^0\), \(\rho ^0\rho ^0\), \(K\pi \), \(\phi K^*\), \(\rho K^*\) [37]. Considering the progress performed meanwhile in the determination of the CKM matrix, it is clear that, by now, most of these non-leptonic modes provide more a test of our understanding of hadronic process rather than competitive constraints on the values of the CKM parameters, even though it can be interesting to consider them from one point of view or the other.

Our analysis is focused on the study of \(B\rightarrow K^*(\rightarrow K\pi )\pi \) decay amplitudes, with the help of isospin symmetry. Among the various \(b\rightarrow u\bar{u}s\) processes, the choice of \(B\rightarrow K^*\pi \) system is motivated by the fact that an amplitude (Dalitz-plot) analysis of the three-body final state \(K\pi \pi \) provides access to several interference phases among different intermediate \(K^*\pi \) states. The information provided by these physical observables highlights the potential of the \(B\rightarrow K^*\pi \) system (VP) compared with \(B\rightarrow K\pi \) (PP) where only branching ratios and CP asymmetries are accessible. Similarly, the \(B\rightarrow K^*\pi \) system leads to the final \(K\pi \pi \) state with a richer pattern of interferences and thus a larger set of observables than other pseudo-scalar–vector states, like, say, \(B\rightarrow K\rho \) (indeed, \(K\pi \pi \) exhibits \(K^*\) resonances from either of the two combinations of \(K\pi \) pairs, whereas the \(\rho \) meson comes from the only \(\pi \pi \) pair available). In addition, the study of these modes provides experimental information on the dynamics of pseudo-scalar–vector modes, which is less known and more challenging from the theoretical point of view. Finally, this system has been studied extensively at the BaBar [38, 39, 40, 41] and Belle [43, 44] experiments, and a large set of observables is readily available.

Let us mention that other approaches, going beyond isospin symmetry, have been proposed to study this system. For instance, one can use SU(3) symmetry and SU(3)-related channels in addition to the ones that we consider in this paper [45, 46]. Another proposal is the construction of the fully SU(3)-symmetric amplitude [47] to which the spin-one intermediate resonances that we consider here do not contribute.

The rest of this article is organised in the following way. In Sect. 2, we discuss the observables provided by the analysis of the \(K\pi \pi \) Dalitz-plot analysis. In Sect. 3, we recall how isospin symmetry is used to reduce the set of hadronic amplitudes and their connection with diagram topologies. In Sect. 4, we discuss two methods to exploit these decays in order to extract information on the CKM matrix, making some assumptions about the size of specific contributions (either electroweak penguins or annihilation). In Sect. 5, we take the opposite point of view. Taking into account our current knowledge of the CKM matrix from global analysis, we set constraints on the hadronic amplitudes used to describe these decays, and we make a brief comparison with theoretical estimates based on QCD factorisation. In Sect. 6, we perform a brief prospective study, determining how the improved measurements expected from LHCb and Belle II may modify the determination of the hadronic amplitudes before concluding. In the appendices, we discuss various technical aspects concerning the inputs and the fits presented in the paper.

## 2 Dalitz-plot amplitudes

Charmless hadronic *B* decays are a particularly rich source of experimental information [1, 2]. For *B* decays into three light mesons (pions and kaons), the kinematics of the three-body final state can be completely determined experimentally, thus allowing for a complete characterisation of the Dalitz-plot (DP) phase space. In addition to quasi-two-body event-counting observables, the interference phases between pairs of resonances can also be accessed, and CP-odd (weak) phases can be disentangled from CP-even (strong) ones. Let us, however, stress that the extraction of the experimental information relies heavily on the so-called isobar approximation, widely used in experimental analyses because of its simplicity, and in spite of its known shortcomings [48].

The \(B\rightarrow K\pi \pi \) system is particularly interesting, as the decay amplitudes from intermediate \(B\rightarrow PV\) resonances (\(K^\star (892)\) and \(\rho (770)\)) receive sizeable contributions from both tree-level and loop diagrams, and they interfere directly in the common phase-space regions (namely the “corners” of the DP). The presence of additional resonant intermediate states further constrain the interference patterns and help resolving potential phase ambiguities. In the case of \(B^0\rightarrow K^+\pi ^-\pi ^0\) and \(B^+\rightarrow K^0_S\pi ^+\pi ^0\), two different \(K^\star (892)\) states contribute to the decay amplitude, and their interference phases can be directly measured. For \(B^0\rightarrow K^0_S\pi ^+\pi ^-\), the time-dependent evolution of the decay amplitudes for \(B^0\) and \(\overline{B^0}\) provides (indirect) access to the relative phase between the \(B^0\rightarrow K^{\star +}\pi ^-\) and \(\overline{B^0}\rightarrow K^{\star -}\pi ^+\) amplitudes.

In the isobar approximation [48], the total decay amplitude for a given mode is a sum of intermediate resonant contributions, and each of these is a complex function of phase space: \(\mathcal{A}(\mathrm{DP})= \sum _i A_iF_i(\mathrm{DP})\), where the sum rolls over all the intermediate resonances providing sizeable contributions, the \(F_i\) functions are the “lineshapes” of each resonance, and the isobar parameters \(A_i\) are complex coefficients indicating the strength of each intermediate amplitude. The corresponding relation is \(\overline{\mathcal{A}}(\mathrm{DP})=\sum _i \overline{A_i}~\overline{F_i}(\mathrm{DP})\) for CP-conjugate amplitudes.

*i*, its direct CP asymmetry \(A_{\mathrm{CP}}\) is expressed as

*i*and

*j*contributing to the same total decay amplitude (i.e., between resonances in the same DP) is

*i*is

*q*/

*p*is the \(B^0-\overline{B^0}\) oscillation parameter.

The CP-averaged \(\mathcal{B}^{+-}=\mathrm{BR}(B^0\rightarrow K^{\star +}\pi ^{-})\) branching fraction and its corresponding CP asymmetry \(A_\mathrm{CP}^{+-}\). These observables can be measured independently in the \(B^0\rightarrow K^0_S\pi ^+\pi ^-\) and \(B^0\rightarrow K^+\pi ^-\pi ^0\) Dalitz planes.

The CP-averaged \(\mathcal{B}^{00}=\mathrm{BR}(B^0\rightarrow K^{\star 0}\pi ^{0})\) branching fraction and its corresponding CP asymmetry \(A_\mathrm{CP}^{00}\). These observables can be accessed both in the \(B^0\rightarrow K^+\pi ^-\pi ^0\) and \(B^0\rightarrow K^0_S\pi ^0\pi ^0\) Dalitz planes.

The CP-averaged \(\mathcal{B}^{+0}=\mathrm{BR}(B^+\rightarrow K^{\star +}\pi ^{0})\) branching fraction and its corresponding CP asymmetry \(A_\mathrm{CP}^{+0}\). These observables can be measured both in the \(B^+\rightarrow K^0_S\pi ^+\pi ^0\) and \(B^+\rightarrow K^+\pi ^0\pi ^0\) Dalitz planes.

The CP-averaged \(\mathcal{B}^{0+}=\mathrm{BR}(B^+\rightarrow K^{\star 0}\pi ^{+})\) branching fraction and its corresponding CP asymmetry \(A_\mathrm{CP}^{0+}\). They can be measured both in the \(B^+\rightarrow K^+\pi ^+\pi ^-\) and \(B^+\rightarrow K^0_S\pi ^0\pi ^+\) Dalitz planes.

The phase difference \(\varphi ^{00,+-}\) between \(B^0\rightarrow K^{\star +}\pi ^{-}\) and \(B^0\rightarrow K^{\star 0}\pi ^{0}\), and its corresponding CP conjugate \(\overline{\varphi }^{{00},-+}\). They can be measured in the \(B^0\rightarrow K^+\pi ^-\pi ^0\) Dalitz plane and in its CP-conjugate DP \(\overline{B^0}\rightarrow K^-\pi ^+\pi ^0\), respectively.

The phase difference \(\varphi ^{+0,0+}\) between \(B^+\rightarrow K^{\star +}\pi ^{0}\) and \(B^+\rightarrow K^{\star 0}\pi ^{+}\), and its corresponding CP conjugate \(\overline{\varphi }^{{-0},0-}\). They can be measured in the \(B^+\rightarrow K^0_S\pi ^+\pi ^0\) Dalitz plane and in its CP-conjugate DP \(B^-\rightarrow K^0_S\pi ^-\pi ^0\), respectively.

The phase difference \(\Delta \varphi ^{+-}\) between \(B^0\rightarrow K^{\star +}\pi ^-\) and its CP conjugate \(\overline{B^0}\rightarrow K^{\star -}\pi ^+\). This phase difference can only be measured in a time-dependent analysis of the \(K^0_S\pi ^+\pi ^-\) DP. As \(K^{\star +}\pi ^-\) is only accessible for \(B^0\) and \(K^{\star -}\pi ^+\) to \(\overline{B^0}\) only, the \(B^0\rightarrow K^{\star +}\pi ^-\) and \(\overline{B^0}\rightarrow K^{\star -}\pi ^+\) amplitudes do not interfere directly (they contribute to different DPs). But they do interfere with intermediate resonant amplitudes that are accessible to both \(B^0\) and \(\overline{B^0}\), like \(\rho ^0(770)K^0_S\) or \(f_0(980)K^0_S\), and thus the time-dependent oscillation is sensitive to the combined phases from mixing and decay amplitudes.

### 2.1 Real-valued physical observables

*U*and

*I*observables defined in \(B\rightarrow \rho \pi \) [5], are expressed as the real and imaginary parts of ratios of amplitudes as follows:

*j*th resonance, i.e., we have the amplitude \(A_j=0\): the quantities \({\mathrm{Re}}(A_i/A_j)\) and \({\mathrm{Im}}(A_i/A_j)\) are not defined, but neither is the phase difference between \(A_i\) and \(A_j\). Therefore, in both parametrisations (real and imaginary part of ratios, or branching ratios, CP asymmetries and phase differences), the singular case \(A_\mathrm{CP}^j=\pm 1\) leads to some undefined observables. Let us add that this case does not occur in practice for our analysis.

In order to properly use the experimental information in the above format it will be necessary to use the full covariance matrix, both statistical and systematic, of the isobar amplitudes. This will allow us to properly propagate the uncertainties as well as the correlations of the experimental inputs to the ones exploited in the phenomenological fit.

## 3 Isospin analysis of \(B\rightarrow K^*\pi \) decays

The isospin formalism used in this work is described in detail in Ref. [51]. Only the main ingredients are summarised below.

*c*-quark penguin diagrams due to the re-expression of \(V_{cb}^*V_{cs}\) in Eq. (14). In the following, \(T^{ij}\) and \(P^{ij}\) will be called hadronic amplitudes.

One naively expects that colour-suppressed contributions will indeed be suppressed compared to their colour-allowed partner, and that electroweak penguins and annihilation contributions will be much smaller than tree and QCD penguins. These expectations can be expressed quantitatively using theoretical approaches like QCD factorisation [14, 15, 16, 17]. Some of these assumptions have been challenged by the experimental data gathered, in particular the mechanism of colour suppression in \(B\rightarrow \pi \pi \) and the smallness of the annihilation part for \(B\rightarrow K\pi \) [5, 22, 37, 55, 56, 57].

*B*meson into a final state can be written as

*m*are real magnitudes. Any additional term \(M_3e^{i\phi _3}e^{i\delta _3}\) can be expressed as a linear combination of \(e^{i\phi _1}\) and \(e^{i\phi _2}\) (with the appropriate properties under CP violation), leading to the fact that the decay amplitudes can be written in terms of any other pair of weak phases \(\{\varphi _1, \varphi _2\}\) as long as \(\varphi _1\ne \varphi _2\) (mod \(\pi \)):

We have thus to abandon the idea of an algorithm allowing one to extract both CKM and hadronic parameters from a set of physical observables. The weak phases in the parameterisation of the decay amplitudes cannot be extracted without additional hadronic hypothesis. This discussion holds if the two weak phases used to describe the decay amplitudes are different (modulo \(\phi \)). The argument does not apply when only one weak phase can be used to describe the decay amplitude: setting one of the amplitudes to zero, say \(m_2=0\), breaks reparametrisation invariance, as can be seen easily in Eqs. (23) and (24). In such cases, weak phases can be extracted from experiment, e.g., the extraction of \(\alpha \) from \(B\rightarrow \pi \pi \), the extraction of \(\beta \) from \(J/\psi K_S\) or \(\gamma \) from \(B\rightarrow DK\). In each case, an amplitude is assumed to vanish, either approximately (extraction of \(\alpha \) and \(\beta \)) or exactly (extraction of \(\gamma \)) [1, 2, 5].

In view of this limitation, two main strategies can be considered for the system considered here: either implementing additional constraints on some hadronic parameters in order to extract the CKM phases using the \(B \rightarrow K^*\pi \) observables, or fix the CKM parameters to their known values from a global fit and use the \(B \rightarrow K^*\pi \) observables to extract information on the hadronic contributions to the decay amplitudes. Both approaches are described below.

## 4 Constraints on CKM phases

We illustrate the first strategy using two specific examples. The first example is similar in spirit to the Gronau–London method for extracting the CKM angle \(\alpha \) [59], which relies on neglecting the contributions of electroweak penguins to the \(B\rightarrow \pi \pi \) decay amplitudes. The second example assumes that upper bounds on annihilation/exchange contributions can be estimated from external information.

### 4.1 The CPS/GPSZ method: setting a bound on electroweak penguins

In \(B\rightarrow \pi \pi \) decays, the electroweak penguin contribution can be related to the tree amplitude in a model-independent way using Fierz transformations of the relevant current–current operators in the effective Hamiltonian for \(B\rightarrow \pi \pi \) decays [6, 60, 61, 62]. One can predict the ratio \(R=P_\mathrm{EW}/T_{3/2}\simeq -3/2 (C_9+C_{10})/(C_1+C_2)=(1.35\pm 0.12) \%\) only in terms of short-distance Wilson Coefficients, since long-distance hadronic matrix elements drop from the ratio (neglecting the operators \(O_7\) and \(O_8\) due to their small Wilson coefficients compared to \(O_9\) and \(O_{10}\)). This leads to the prediction that there is no strong phase difference between \(P_\mathrm{EW}\) and \(T_{3/2}\) so that electroweak penguins do not generate a charge asymmetry in \(B^+\rightarrow \pi ^+\pi ^0\) if this picture holds: this prediction is in agreement with the present experimental average of the corresponding asymmetry. Moreover, this assumption is crucial to ensure the usefulness of the Gronau–London method to extract the CKM angle \(\alpha \) from an isospin analysis of \(B\rightarrow \pi \pi \) decay amplitudes [5, 6]: setting the electroweak penguin to zero in the Gronau–London breaks the reparametrisation invariance described in Sect. 3 and opens the possibility of extracting weak phases.

*SU*(2) symmetry guarantees that the matrix element with the combination of operators \(O_1-O_2\) vanishes, so that it does not enter tree amplitudes. A similar argument would hold for

*SU*(3) symmetry in the case of the \(K\pi \) system, but it does not for the vector–pseudo-scalar \(K^*\pi \) system. It is thus not possible to cancel hadronic matrix elements when considering \(P_\mathrm{EW}/T_{3/2}\), which becomes a complex quantity suffering from (potentially large) hadronic uncertainties [63, 64]. The size of the electroweak penguin (relative to the tree contributions), is parametrised as

*SU*(3) limit for \(B\rightarrow \pi K\) (and identical to the one obtained from \(B\rightarrow \pi \pi \) using the arguments in Refs. [60, 61, 62]), and \(r_\mathrm{VP}\) is a complex parameter measuring the deviation of \(P/T_{3/2}\) from this value corresponding to

*SU*(3) flavour relations suggest \(|r_\mathrm{VP}|\le 0.05\) [64, 65]. However, it is clear that both approximations can easily be broken, suggesting a more conservative upper bound \(|r_\mathrm{VP}|\le 0.30\).

The presence of these hadronic uncertainties have important consequences for the method. Indeed, it turns out that including a non-vanishing \(P_\mathrm{EW}\) completely disturbs the extraction of \(\alpha \). The electroweak penguin can provide a \(\mathcal{O}(1)\) contribution to CP-violating effects in charmless \(b\rightarrow s\) processes, as its CKM coupling amplifies its contribution to the decay amplitude: \(P_\mathrm{EW}\) is multiplied by a large CKM factor \(V_{ts}V_{tb}^*=O(\lambda ^2)\) compared to the tree-level amplitudes multiplied by a CKM factor \(V_{us}V_{ub}^*=O(\lambda ^4)\). Therefore, unless \(P_\mathrm{EW}\) is particularly suppressed due to some specific hadronic dynamics, its presence modifies the CKM constraint obtained following this method in a very significant way.

It would be difficult to illustrate this point using the current data, due to the experimental uncertainties described in the next sections. We choose thus to discuss this problem using a reference scenario described in Table 11, where the hadronic amplitudes have been assigned arbitrary (but realistic) values and they are used to derive a complete set of experimental inputs with arbitrary (and much more precise than currently available) uncertainties. As shown in Appendix A (cf. Table 11), the current world averages for branching ratios and CP asymmetries in \(B^0\rightarrow K^{*+}\pi ^-\) and \(B^0\rightarrow K^{*0}\pi ^0\) agree broadly with these values, which also reproduce the expected hierarchies among hadronic amplitudes, if we set the CKM parameters to their current values from our global fit [6, 7, 8]. We choose a penguin parameter \(P^{+-}\) with a magnitude 28 times smaller than the tree parameter \(T^{+-}\), and a phase fixed at \(-7^\circ \). The electroweak \(P_\mathrm{EW}\) parameter has a value 66 times smaller in magnitude than the tree parameter \(T^{+-}\), and its phase is arbitrarily fixed to \(+15^\circ \) in order to get good agreement with the current central values. Our results do not depend significantly on this phase, and a similar outcome occurs if we choose sets with a vanishing phase for \(P_\mathrm{EW}\) (though the agreement with the current data will be less good).

*R*and, more significantly, \(r_{VP}\), have an important impact on the precision of the constraint on \((\bar{\rho },\bar{\eta })\).

This simple illustration with our reference scenario shows that the CPS/GPSZ method is limited both in robustness and accuracy due to the assumption on a negligible \(P_\mathrm{EW}\): a small non-vanishing value breaks the relation between the phase of \(R^0\) and the CKM angle \(\alpha \), and therefore, even a small uncertainty on the \(P_\mathrm{EW}\) value would translate into large biases on the CKM constraints. It shows that this method would require a very accurate understanding of hadronic amplitudes in order to extract a meaningful constraint on the unitarity triangle, and the presence of non-vanishing electroweak penguins dilutes the potential of this method significantly.

### 4.2 Setting bounds on annihilation/exchange contributions

As discussed in the previous paragraphs, the penguin contributions for \(B\rightarrow K^*\pi \) decays are strongly CKM-enhanced, impacting the CPS/GPSZ method based on neglecting a penguin amplitude \(P_\mathrm{EW}\). This method exhibits a strong sensitivity to small changes or uncertainties in values assigned to the electroweak penguin contribution. An alternative and safer approach consists in constraining a tree amplitude, with a CKM-suppressed contribution. Among the various hadronic amplitudes introduced, it seems appropriate to choose the annihilation amplitude \(N^{0+}\), which is expected to be smaller than \(T^{+-}\), and which could even be smaller than the colour-suppressed \(T^{00}_\mathrm{C}\). Unfortunately, no direct, clean constraints on \(N^{0+}\) can be extracted from data and from the theoretical point of view, \(N^{0+}\) is dominated by incalculable non-factorisable contributions in QCD factorisation [14, 15, 16, 17]. On the other hand, indirect upper bounds on \(N^{0+}\) may be inferred from either the \(B^+\rightarrow K^{*0} \pi ^+\) decay rate or from the *U*-spin related mode \(B^+\rightarrow K^{*0}K^+\).

*B*-meson state. This justifies the \(\beta \)-like shape of the constraint obtained when fixing the value of the annihilation parameter. The presence of the oscillation phase

*q*/

*p*here, starting from a decay of a charged

*B*, may seem surprising. However, one should keep in mind that the measurement of \(B^+\rightarrow K^{*0}\pi ^+\) and its CP-conjugate amplitude are not sufficient to determine the relative phase between \(A'\) and \(\bar{A}'\): this requires one to reconstruct the whole quadrilateral equation Eq. (15), where the phases are provided by interferences between mixing and decay amplitudes in \(B_0\) and \(\bar{B}_0\) decays. In other words, the phase observables obtained from the Dalitz plot are always of the form Eqs. (4), (5): their combination can only lead to a ratio of CP-conjugate amplitudes multiplied by the oscillation parameter

*q*/

*p*.

The lower plot of Fig. 2 describes how the constraint on \(\beta \) loosens around its true value when the range allowed for \(\left| N^{0+}/T^{+-}\right| \) is increased compared to its initial value (0.143). We see that the method is stable and keeps on including the true value for \(\beta \) even in the case of a mild constraint on \(\left| N^{0+}/T^{+-}\right| \).

## 5 Constraints on hadronic parameters using current data

As already anticipated in Sect. 3, a second strategy to exploit the data consists in assuming that the CKM matrix is already well determined from the CKM global fit [6, 7, 8]. The measurements of \(B\rightarrow K^\star \pi \) observables (isobar parameters) can then be used to extract constraints on the hadronic parameters in Eq. (16).

### 5.1 Experimental inputs

Two three-dimensional covariance matrices, cf. Eq. (10), from the BaBar time-dependent DP analysis of \(B^0\rightarrow K^0_S\pi ^+\pi ^-\) in Ref. [38], and two three-dimensional covariance matrices from the Belle time-dependent DP analysis of \(B^0\rightarrow K^0_S\pi ^+\pi ^-\) in Ref. [44]. Both the BaBar and Belle analyses found two quasi-degenerate solutions each, with very similar goodness-of-fit merits. The combination of these solutions is described in Appendix A.3, and is taken as input for this study.

A five-dimensional covariance matrix, cf. Eq. (11), from the BaBar\(B^0\rightarrow K^+\pi ^-\pi ^0\) DP analysis [40].

A two-dimensional covariance matrix, cf. Eq. (12), from the BaBar\(B^+\rightarrow K^+\pi ^+\pi ^-\) DP analysis [39], and a two-dimensional covariance matrix from the Belle \(B^+\rightarrow K^+\pi ^+\pi ^-\) DP analysis [43].

A simplified uncorrelated four-dimensional input, cf. Eq. (13), from the BaBar\(B^+\rightarrow K^0_S\pi ^+\pi ^0\) preliminary DP analysis [41].

These sets of experimental central values and covariance matrices are described in Appendix A, where the combinations of the results from BaBar and Belle are also described.

Finally, we notice that the time-dependent asymmetry in \(B\rightarrow K_S\pi ^0\pi ^0\) has been measured [49, 50]. As these are global analyses integrated over the whole DP, we cannot take these measurements into account. In principle a time-dependent isobar analysis of the \(K_S\pi ^0\pi ^0\) DP could be performed and it could bring about some independent information on \(B\rightarrow K^{*0}\pi ^0\) intermediate amplitudes. Since this more challenging analysis has not been done yet, we will not consider this channel for the time being.

### 5.2 Selected results for CP asymmetries and hadronic amplitudes

Using the experimental inputs described in Sect. 5.1, a fit to the complete set of hadronic parameters is performed. We discuss the fit results focusing on three aspects: the most significant direct CP asymmetries, the significance of electroweak penguins, and the relative hierarchies of hadronic contributions to the tree amplitudes. As will be seen in the following, the fit results can be interpreted in terms of two sets of local minima, out of which one yields constraints on the hadronic parameters in better agreement with the expectations from CPS/GPSZ, the measured direct CP asymmetries and the expected relative hierarchies of hadronic contributions.

#### 5.2.1 Direct CP violation in \(B^0\rightarrow K^{\star +}\pi ^-\)

Using the amplitude DP analysis results from these three measurements as inputs, the combined constraint on \(A_\mathrm{CP}(B^0\rightarrow K^{\star +}\pi ^-)\) is shown in Fig. 3. The combined value is 3.0 \(\sigma \) away from zero, and the 68% confidence interval on this CP asymmetry is \(0.21\pm 0.07\) approximately. This result is to be compared with the \(0.23\pm 0.06\) value provided by HFLAV [66]. The difference is likely to come from the fact that HFLAV performs an average of the CP asymmetries extracted from individual experiments, while this analysis uses isobar values as inputs which are averaged over the various experiments before being translated into values for the CP parameters: since the relationships between these two sets of quantities are non-linear, the two steps (averaging over experiments and translating from one type of observables to another) yield the same central values only in the case of very small uncertainties. In the current situation, where sizeable uncertainties affect the determinations from individual experiments, it is not surprising that minor discrepancies arise between our approach and the HFLAV result.

As can be readily seen from Eq. (14), a non-vanishing asymmetry in this mode requires a strong phase difference between the tree \(T^{+-}\) and penguin \(P^{+-}\) hadronic parameters that is strictly different from zero. Figure 4 shows the two-dimensional constraint on the modulus and phase of the \(P^{+-}/T^{+-}\) ratio. Two solutions with very similar \(\chi ^2\) are found, both incompatible with a vanishing phase difference. The first solution corresponds to a small (but non-vanishing) positive strong phase, with similar \(\left| V_{ts}V_{tb}^\star P^{+-}\right| \) and \(\left| V_{us}V_{ub}^\star T^{+-}\right| \) contributions to the total decay amplitude, and is called Solution I in the following. The other solution, denoted Solution II, corresponds to a larger, negative, strong phase, with a significantly larger penguin contribution. We notice that Solution I is closer to usual theoretical expectations concerning the relative size of penguin and tree contributions.

#### 5.2.2 Direct CP violation in \(B^+\rightarrow K^{\star +}\pi ^0\)

The \(B^+\rightarrow K^{\star +}\pi ^0\) amplitude can be accessed in a \(B^+\rightarrow K^0_\mathrm{S}\pi ^+\pi ^0\) Dalitz-plot analysis, for which only a preliminary result from BaBar is available [41]. A large, negative CP asymmetry \(A_\mathrm{CP}(B^+\rightarrow K^{\star +}\pi ^0) = -0.52\pm 0.14\pm 0.04 ^{+0.04}_{-0.02}\) is reported there with a 3.4 \(\sigma \) significance. This CP asymmetry has also been measured by BaBar through a quasi-two-body analysis of the \(B^+\rightarrow K^+\pi ^0\pi ^0\) final state [42], obtaining \(A_\mathrm{CP}(B^+\rightarrow K^{\star +}\pi ^0) = -0.06\pm 0.24\pm 0.04\). The combination of these two measurement yields \(A_\mathrm{CP}(B^+\rightarrow K^{\star +}\pi ^0) = -0.39\pm 0.12\pm 0.03\), with a 3.2 \(\sigma \) significance (Fig. 5).

#### 5.2.3 Hierarchy among penguins: electroweak penguins

#### 5.2.4 Hierarchy among tree amplitudes: colour suppression and annihilation

As already discussed, the hadronic parameter \(T^{00}_\mathrm{C}\) is expected to be suppressed with respect to the main tree parameter \(T^{+-}\). Also, the annihilation topology is expected to provide negligible contributions to the decay amplitudes. These expectations can be compared with the extraction of these hadronic parameters from data in Fig. 11.

For colour suppression, the current data provides no constraint on the relative phase between the \(T^{00}_\mathrm{C}\) and \(T^{+-}\) tree parameters, and only a mild upper bound on the modulus can be inferred; the tighter constraint is provided by Solution I that excludes values of \(|T^{00}_\mathrm{C}/T^{+-}|\) larger than 1.6 at \(95\%\) C.L. The constraint from Solution II is more than one order of magnitude looser.

Similarly, for annihilation, Solution I provides slightly tighter constraints on its contribution to the total tree amplitude with the bound \(|N^{0+}/T^{+-}|<2.5\) at \(95\%\) C.L., while the bound from Solution II is much looser.

### 5.3 Comparison with theoretical expectations

We have extracted the values of the hadronic amplitudes from the data currently available. It may prove interesting to compare these results with theoretical expectations. For this exercise, we use QCD factorisation [14, 15, 16, 17] as a benchmark point, keeping in mind that other approaches (discussed in the introduction) are available. In order to keep the comparison simple and meaningful, we consider the real and imaginary part of several ratios of hadronic amplitudes.

\(68\%\) confidence intervals for the real and imaginary parts of hadronic ratios according to our fit and the corresponding predictions in our implementation of QCD factorisation (QCDF). No prediction is given for the ratio \({P_\mathrm{EW}^\mathrm{C}}/{P^{+-}}\) due to numerical instabilities (see text)

Quantity | Fit result | QCDF |
---|---|---|

\(\displaystyle {\mathrm{Re}}\frac{N^{0+}}{T^{+-}}\) | \((-5.31, 4.73)\) | \(0.011 \pm 0.027\) |

\(\displaystyle {\mathrm{Im}}\frac{N^{0+}}{T^{+-}}\) | \((-9.59, 7.73)\) | \(0.003\pm 0.028\) |

\(\displaystyle {\mathrm{Re}}\frac{P_\mathrm{EW}^\mathrm{C}}{P_\mathrm{EW}}\) | (0.69, 1.14) | \(0.17\pm 0.19\) |

\(\displaystyle {\mathrm{Im}}\frac{P_\mathrm{EW}^\mathrm{C}}{P_\mathrm{EW}}\) | \((-0.48,-0.28)~\cup ~(-0.13,0.22)~\cup \) | \(-0.08\pm 0.14\) |

(0.34, 0.60) | ||

\(\displaystyle {\mathrm{Re}}\frac{P_\mathrm{EW}^\mathrm{C}}{P^{+-}}\) | (1.29, 2.08) | – |

\(\displaystyle {\mathrm{Im}}\frac{P_\mathrm{EW}^\mathrm{C}}{P^{+-}}\) | \((-1.09,-0.75)~\cup ~(-0.51,-0.10)~\cup \) | – |

\((-0.08,0.16)~\cup ~(0.47,0.83)\) | ||

\(\displaystyle {\mathrm{Re}}\frac{P_\mathrm{EW}^\mathrm{C}}{T^{+-}}\) | \((-0.12,0.34)\) | \(0.0027\pm 0.0031\) |

\(\displaystyle {\mathrm{Im}}\frac{P_\mathrm{EW}^\mathrm{C}}{T^{+-}}\) | \((-0.42,0.05)\) | \(-0.0015^{+0.0024}_{-0.0025}\) |

\(\displaystyle {\mathrm{Re}}\frac{P^{+-}}{P_\mathrm{EW}}\) | (0.49, 0.56) | \(3.9^{+3.2}_{-3.3}\) |

\(\displaystyle {\mathrm{Im}}\frac{P^{+-}}{P_\mathrm{EW}}\) | \((-0.03,0.16)\) | \(1.8\pm 3.3\) |

\(\displaystyle {\mathrm{Re}}\frac{P_\mathrm{EW}}{T^{+-}}\) | (0.0, 0.25) | \(0.0154^{+0.0059}_{-0.0060}\) |

\(\displaystyle {\mathrm{Im}}\frac{P_\mathrm{EW}}{T^{+-}}\) | \((-0.40,-0.09)~\cup ~(-0.02,0.02)\) | \(-0.0014^{+0.0023}_{-0.0022}\) |

\(\displaystyle {\mathrm{Re}}\frac{P^{+-}}{T^{+-}}\) | (0.023, 0.140) | \(0.053\pm 0.039\) |

\(\displaystyle {\mathrm{Im}}\frac{P^{+-}}{T^{+-}}\) | \((-0.20,-0.04)~\cup ~(0.0, 0.01)\) | \(0.016\pm 0.044\) |

\(\displaystyle {\mathrm{Re}}\frac{T^{00}_\mathrm{C}}{T^{+-}}\) | \((-0.26,2.24)\) | \(0.13\pm 0.17\) |

\(\displaystyle {\mathrm{Im}}\frac{T^{00}_\mathrm{C}}{T^{+-}}\) | \((-3.28,0.74)\) | \(-0.11\pm 0.15\) |

Our results for the ratios of hadronic amplitudes are shown in Fig. 12 and in Table 1. We notice that for most of the ratios good agreement is found. The global fit to the experimental data has often much larger uncertainties than theoretical predictions: with better data in the future, we may be able to perform very non-trivial tests of the non-leptonic dynamics and the isobar approximation. The situation for \(P_\mathrm{EW}^\mathrm{C}/P_\mathrm{EW}\) is slightly different, since the two determinations (experiment and theory) exhibit similar uncertainties and disagree with each other, providing an interesting test for QCD factorisation, which, however, goes beyond the scope of this study.

There are two cases where the theoretical output from QCD factorisation is significantly less precise than the constraints from the combined fit. For \(P_\mathrm{EW}^C/P^{+-}\), both numerator and denominator can be (independently) very small in QCD factorisation, and numerical instabilities in this ratio prevent us from having a precise prediction. For \(P^{+-}/P_\mathrm{EW}\), the impressively accurate experimental determination, as discussed in Sect. 5.2.3, is predominantly driven by the \(\varphi ^{00,+-}\) phase differences measured in the BaBar Dalitz-plot analysis of \(B^0\rightarrow K^+\pi ^+\pi ^0\) decays. Removing this input yields a much milder constraint on \(P^{+-}/P_\mathrm{EW}\). On the other hand in QCD factorisation, the formally leading contributions to the \(P^{+-}\) penguin amplitude are somewhat numerically suppressed, and compete with the model estimate of power corrections: due to the Rfit treatment used, the two contributions can either compensate each other almost exactly or add up coherently, leading to a \(\sim \)\(\pm \)100 relative uncertainty, which is only in marginal agreement with the fit output. Thus we conclude that the \(P^{+-}/P_\mathrm{EW}\) ratio is both particularly sensitive to the power corrections to QCD factorisation and experimentally well constrained, so that it can be used to provide insight on non-factorisable contributions, provided one assumes negligible effects from New Physics.

## 6 Prospects for LHCb and Belle II

In this section, we study the impact of improved measurements of \(K\pi \pi \) modes from the LHCb and Belle II experiments. During the first run of the LHC, the LHCb experiment has collected large datasets of B-hadron decays, including charmless \(B^0,B^+,B_s\) meson decays into tree-body modes. LHCb is currently collecting additional data in Run-2. In particular, due to the excellent performances of the LHCb detector for identifying charged long-lived mesons, the experiment has the potential for producing the most accurate charmless three-body results in the \(B^+\rightarrow K^+\pi ^-\pi ^+\) mode, owing to high-purity event samples much larger than the ones collected by BaBar and Belle. Using \(3.0\ \mathrm{fb}^{-1}\) of data recorded during the LHC Run 1, first results on this mode are already available [68], and a complete amplitude analysis is expected to be produced in the short-term future. For the \(B^0\rightarrow K^0_S\pi ^+\pi ^-\) mode, the event-collection efficiency is challenged by the combined requirements on reconstructing the \(K^0_S\rightarrow \pi ^+\pi ^-\) decay and tagging the *B* meson flavour, but nonetheless the \(B^0\rightarrow K^0_S\pi ^+\pi ^-\) data samples collected by LHCb are already larger than the ones from BaBar and Belle. As it is more difficult to anticipate the reach of LHCb Dalitz-plot analyses for modes including \(\pi ^0\) mesons in the final state, the \(B^0\rightarrow K^+\pi ^+\pi ^0\), \(B^+\rightarrow K^0_S\pi ^+\pi ^0\)\(B^+\rightarrow K^+\pi ^0\pi ^0\) and \(B^0\rightarrow K_S^0\pi ^0\pi ^0\) channels are not considered here. In addition, LHCb has also the potential for studying \(B_s\) decay modes, and LHCb can reach \(B\rightarrow KK\pi \) modes with branching ratios out of reach for *B*-factories.

The Belle II experiment [69], currently in the stages of construction and commissioning, will operate in an experimental environment very similar to the one of the BaBar and Belle experiments. Therefore Belle II has the potential for studying all modes accessed by the *B*-factories, with expected sensitivities that should scale in proportion to its expected total luminosity (i.e., \(50\ \mathrm{ab}^{-1}\)). In addition, Belle II has the potential for accessing the \(B^+\rightarrow K^+\pi ^0\pi ^0\) and \(B^0\rightarrow K_S^0\pi ^0\pi ^0\) modes (for which the *B*-factories could not produce Dalitz-plot results) but these modes will provide low-accuracy information, redundant with some of the modes considered in this paper: therefore they are not included here.

Since both LHCb and Belle II have the potential for studying large, high-quality samples of \(B^+\rightarrow K^+\pi ^-\pi ^+\), it is realistic to expect that the experiments will be able to extract a consistent, data-driven signal model to be used in all Dalitz-plot analysis, yielding systematic uncertainties significantly decreased with respect to the results from *B*-factories.

Finally for LHCb, since this experiment cannot perform *B*-meson counting as in a *B*-factory environment, the branching fractions need to be normalised with respect to measurements performed at BaBar and Belle, until the advent of Belle II. This prospective study therefore is split into two periods: a first one based on the assumption of new results from LHCb Run1+Run2 only, and a second one using the complete set of LHCb and Belle II results. The corresponding inputs are gathered in Appendix C. We use the reference scenario described in Table 11 for the central values, so that we can guarantee the self-consistency of the inputs and we avoid reducing the uncertainties artificially because of barely compatible measurements (which would occur if we used the central values of the current data and rescaled the uncertainties). The expected uncertainties, obtained from the extrapolations discussed previously, are described in Table 12.

*B*-factories and LHCb Run1 + Run2). For the input values used in the prospective, the modulus of the \(P^{+-}/T^{+-}\) ratio will be constrained with a relative \(10\%\) accuracy, and its complex phase will be constrained within 3 degrees (we discuss 68% C.L. ranges in the following, whereas Fig. 13 shows 95% C.L. regions). Slightly tighter upper bounds on the \(|T^{00}_\mathrm{C}/T^{+-}|\) and \(|N^{0+}/T^{+-}|\) ratios may be set, albeit the relative phases of these rations will remain very poorly constrained. Assuming that the electroweak penguin is in agreement with the CPS/GPSZ prediction, its modulus will be constrained within \(45\%\) and its phase within 14 degrees.

The addition of results from the Belle II experiment corresponds to the second step of this prospective study. As illustrated by the green area in Fig. 13, the uncertainties on the modulus and phase of the \(P^{+-}/T^{+-}\) ratio will decrease by factors of 1.4 and 2.5, respectively. Owing to the addition of precision measurements by Belle II of the \(B^0\rightarrow K^{*0}\pi ^0\) Dalitz-plot parameters from the amplitude analysis of the \(B^0\rightarrow K^+\pi ^-\pi ^0\) modes, the \(T^{00}_\mathrm{C}/T^{+-}\) ratio can be constrained within a \(22\%\) uncertainty for its modulus, and within 10 degrees for its phase. Similarly, the uncertainties on the modulus and phase of the \(P_\mathrm{EW}/T_{3/2}\) ratio will decrease by factors 2.7 and 2.9, respectively. Concerning the colour-suppressed electroweak penguin, for which only a mild upper bound on its modulus was achievable within the first step of the prospective, can now be measured within a \(22\%\) uncertainty for its modulus, and within 8 degrees for its phase. Finally, the less stringent constraint will be achieved for the annihilation parameter. While its modulus can nevertheless be constrained between 0.3 and 1.5, the phase of this ratio may remain unconstrained in value, with just the sign of the phase being resolved. We add that one can also expect Belle II measurements for \(B^+\rightarrow K^+\pi ^0\pi ^0\) and \(B^0\rightarrow K_S\pi ^0\pi ^0\), however, with larger uncertainties, so that we have not taken into account these decays.

In total, precise constraints on almost all hadronic parameters in the \(B\rightarrow K^\star \pi \) system will be achieved using the Dalitz-plot results from the LHCb and Belle II experiments, with a resolution of the current phase ambiguities. These constraints can be compared with various theoretical predictions, proving an important tool for testing models of hadronic contributions to charmless *B* decays.

## 7 Conclusion

Non-leptonic B meson decays are very interesting processes both as probes of weak interaction and as tests of our understanding of QCD dynamics. They have been measured extensively at *B*-factories as well as at the LHCb experiment, but this wealth of data has not been fully exploited yet, especially for the pseudo-scalar–vector modes which are accessible through Dalitz-plot analyses of \(B\rightarrow K\pi \pi \) modes. We have focused on the \(B\rightarrow K^*\pi \) system which exhibits a large set of observables already measured. Isospin analysis allows us to express this decay in terms of CKM parameters and six complex hadronic amplitudes, but reparametrisation invariance prevents us from extracting simultaneously information on the weak phases and the hadronic amplitudes needed to describe these decays. We have followed two different approaches to exploit this data: either we extracted information on the CKM phase (after setting a condition on some of the hadronic amplitudes), or we determined of hadronic amplitudes (once we set the CKM parameters to their value from the CKM global fit [6, 7, 8]).

In the first case, we considered two different strategies. We first reconsidered the CPS/GPSZ strategy proposed in Refs. [64, 65], amounting to setting a bound on the electroweak penguin in order to extract an \(\alpha \)-like constraint. We used a reference scenario inspired by the current data but with consistent central values and much smaller uncertainties in order to probe the robustness of the CPS/GPSZ method: it turns out that the method is easily biased if the bound on the electroweak penguin is not correct, even by a small amount. Unfortunately, this bound is not very precise from the theoretical point of view, which casts some doubt on the potential of this method to constrain \(\alpha \). We have then considered a more promising alternative, consisting in setting a bound on the annihilation contribution. We observed that we could obtain an interesting stable \(\beta \)-like constraint and we discussed its potential to extract confidence intervals according to the accuracy of the bound used for the annihilation contribution.

In a second stage, we discussed how the data constrain the hadronic amplitudes, assuming the values of the CKM parameters. We performed an average of BaBar and Belle data in order to extract constraints on various ratios of hadronic amplitudes, with the issue that some of these data contain several solutions to be combined in order to obtain a single set of inputs for the Dalitz-plot observables. The ratio \(P^{+-}/T^{+-}\) is not very well constrained and exhibits two distinct preferred solutions, but it is not large and supports the expect penguin suppression. On the other hand, colour or electroweak suppression does not seem to hold, as illustrated by \(|P_\mathrm{EW}/P^{+-}|\) (around 2), \(|P_\mathrm{EW}^\mathrm{C}/P_\mathrm{EW}|\) (around 1) or \(|T^{00}_\mathrm{C}/T^{+-}|\) (mildly favouring values around 1). We, however, recall that some of these conclusions are very dependent on the BaBar measurement on \(\varphi ^{00,+-}\) phase differences measured in \(B^0\rightarrow K^+\pi ^+\pi ^0\): removing this input turns the ranges into mere upper bounds on these ratios of hadronic amplitudes.

Central values and total (statistical and systematic) correlation matrix for the global (top) and local (bottom, \(\Delta \mathrm{NLL} = 0.16\)) minimum solutions for the BaBar\(B^0\rightarrow K^0_S\pi ^+\pi ^-\) analysis

\(B^0\rightarrow K^0_S\pi ^+\pi ^-\) | Global min | \({\mathrm{Re}}\left[ \frac{q}{p} \frac{\overline{A}(K^{*-}\pi ^+)}{A(K^{*+}\pi ^-)} \right] \) | \({\mathrm{Im}}\left[ \frac{q}{p} \frac{\overline{A}(K^{*-}\pi ^+)}{A(K^{*+}\pi ^-)} \right] \) | \({\mathcal B}(K^{*+}\pi ^-)\) |
---|---|---|---|---|

\({\mathrm{Re}}\left[ \frac{q}{p} \frac{\overline{A}(K^{*-}\pi ^+)}{A(K^{*+}\pi ^-)} \right] \) | \(0.428 \pm 0.473\) | 1.00 | 0.90 | 0.02 |

\({\mathrm{Im}}\left[ \frac{q}{p} \frac{\overline{A}(K^{*-}\pi ^+)}{A(K^{*+}\pi ^-)} \right] \) | \(-0.690 \pm 0.302\) | 1.00 | \(-0.06\) | |

\({\mathcal B}(K^{*+}\pi ^-) (\times 10^{-6})\) | \(8.290 \pm 1.189\) | 1.00 | ||

\(B^0\rightarrow K^0_S\pi ^+\pi ^-\) | Local min (\(\Delta \mathrm{NLL} = 0.16\)) | \({\mathrm{Re}}\left[ \frac{q}{p} \frac{\overline{A}(K^{*-}\pi ^+)}{A(K^{*+}\pi ^-)} \right] \) | \({\mathrm{Im}}\left[ \frac{q}{p} \frac{\overline{A}(K^{*-}\pi ^+)}{A(K^{*+}\pi ^-)} \right] \) | \({\mathcal B}(K^{*+}\pi ^-)\) |

\({\mathrm{Re}}\left[ \frac{q}{p} \frac{\overline{A}(K^{*-}\pi ^+)}{A(K^{*+}\pi ^-)} \right] \) | \(-0.819 \pm 0.116\) | 1.00 | \(-0.19\) | \(-0.15 \) |

\({\mathrm{Im}}\left[ \frac{q}{p} \frac{\overline{A}(K^{*-}\pi ^+)}{A(K^{*+}\pi ^-)} \right] \) | \(-0.049 \pm 0.494\) | 1.00 | \(-0.01\) | |

\({\mathcal B}(K^{*+}\pi ^-) (\times 10^{-6})\) | \(8.290 \pm 1.189\) | 1.00 |

Finally, we performed prospective studies, considering two successive stages based first on LHCb data from Run1 and Run2, then on the additional input from Belle II. Using our reference scenario and extrapolating the uncertainties of the measurements at both stages, we determined the confidence regions for the moduli and phases of the ratios of hadronic amplitudes. The first stage (LHCb only) would correspond to a significant improvement for \(P^{+-}/T^{+-}\) and \(P_\mathrm{EW}/T_{3/2}\), whereas the second stage (LHCb+Belle II) would yield tight constraints on \(N^{0+}/T^{+-}\), \(P_\mathrm{EW}^C/T^{+-}\) and \(T^{00}_\mathrm{C}/T^{+-}\).

Non-leptonic *B*-meson decays remain an important theoretical challenge, and any contender should be able to explain not only the pseudo-scalar–pseudo-scalar modes but also the pseudo-scalar–vector modes. Unfortunately, the current data do not permit such extensive tests, even though they hint at potential discrepancies with theoretical expectations concerning the hierarchies of hadronic amplitudes. However, our study suggests that a more thorough analysis of \(B\rightarrow K\pi \pi \) Dalitz plots from LHCb and Belle II could allow for a precise determination of the hadronic amplitudes involved in \(B\rightarrow K^*\pi \) decays thanks to the isobar approximation for three-body amplitudes. This will definitely shed some light on the complicated dynamics of weak and strong interaction at work in pseudo-scalar-vector modes, and it will provide important tests of our understanding of non-leptonic *B*-meson decays.

## Notes

### Acknowledgements

We would like to thank all our collaborators from the CKMfitter group for useful discussions, and Reina Camacho Toro for her collaboration on this project at an early stage. This project has received funding from the European Union Horizon 2020 research and innovation programme under the Grant agreements No. 690575, No. 674896 and No. 692194. SDG acknowledges partial support from Contract FPA2014-61478-EXP.

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