# Isospin analysis of charmless *B*-meson decays

## Abstract

We discuss the determination of the CKM angle \(\alpha \) using the non-leptonic two-body decays \(B\rightarrow \pi \pi \), \(B\rightarrow \rho \rho \) and \(B\rightarrow \rho \pi \) using the latest data available. We illustrate the methods used in each case and extract the corresponding value of \(\alpha \). Combining all these elements, we obtain the determination \(\alpha _\mathrm{dir}={({86.2}_{-4.0}^{+4.4} \cup {178.4}_{-5.1}^{+3.9})}^{\circ }\). We assess the uncertainties associated to the breakdown of the isospin hypothesis and the choice of the statistical framework in detail. We also determine the hadronic amplitudes (tree and penguin) describing the QCD dynamics involved in these decays, briefly comparing our results with theoretical expectations. For each observable of interest in the \(B\rightarrow \pi \pi \), \(B\rightarrow \rho \rho \) and \(B\rightarrow \rho \pi \) systems, we perform an indirect determination based on the constraints from all the other observables available and we discuss the compatibility between indirect and direct determinations. Finally, we review the impact of future improved measurements on the determination of \(\alpha \).

## 1 Introduction

Over the last few decades, our understanding of *CP* violation has made great progress, with many new constraints from BaBar, Belle and LHCb experiments among others [1, 2]. These constraints were shown to be in remarkable agreement with each other and to support the Kobayashi–Maskawa mechanism of *CP* violation at work within the Standard Model (SM) with three generations [3, 4]. This has led to an accurate determination of the Cabibbo–Kobayashi–Maskawa matrix (CKM) encoding the pattern of *CP* violation as well as the strength of the weak transitions among quarks of different generations [5, 6, 7, 8, 9, 10]. These constraints prove also essential in assessing the viability of New Physics models with well-motivated flavour structures [11, 12, 13, 14].

As the CKM matrix is related to quark-flavour transitions, most of these constraints are significantly affected by hadronic uncertainties due to QCD binding quarks into the observed hadrons. However, some of these constraints have the very interesting feature of being almost free from such uncertainties. This is in particular the case for the constraints on the CKM angle \(\alpha \) that are derived from the isospin analysis of the charmless decay modes \(B\rightarrow \pi \pi \), \(B\rightarrow \rho \rho \) and \(B\rightarrow \rho \pi \). Indeed, assuming the isospin symmetry and neglecting the electroweak penguin contributions, the amplitudes of the \({SU}(2)\)-conjugated modes are related. The measured branching fractions and asymmetries in the \(\mathcal{B} ^{\pm ,0}\rightarrow (\pi \pi )^{\pm ,0}\) and \(B^{\pm ,0}\rightarrow (\rho \rho )^{\pm ,0}\) modes and the bilinear form factors in the Dalitz analysis of the \(B^{0}\rightarrow (\rho \pi )^0\) decays provide enough observables to simultaneously extract the weak phase \(\beta +\gamma =\pi -\alpha \) together with the hadronic tree and penguin contributions to each mode [15, 16, 17]. Therefore, these modes probe two different corners of the SM: on one side, they yield information on \(\alpha \) that is a powerful constraint on the Kobayashi–Maskawa mechanism (and the CKM matrix) involved in weak interactions, and on the other side, they provide a glimpse on the strong interaction and especially the hadronic dynamics of charmless two-body *B*-decays.

*CP*asymmetry in the \(B^0\!\rightarrow \pi ^0\pi ^0\) decay is predicted at 68% CL:

The rest of this article goes as follows. In Sect. 2, we discuss the basics of isospin analysis for charmless *B*-meson decays. In Sect. 3, we provide details on the extraction of the \(\alpha \) angle, focusing on the \(B\rightarrow \pi \pi \), \(B\rightarrow \rho \rho \) and \(B\rightarrow \rho \pi \) modes in turn, before combining these extractions in a world average. In Sect. 4, we consider the uncertainties attached to the extraction of \(\alpha \). First, we discuss the \({SU}(2)\) isospin framework underlying these analyses, considering three sources of corrections: the presence of \(\varDelta I={3} / {2}\) electroweak penguins (isospin breaking due to different charges for the *u* and *d* quarks), \(\pi ^0\)–\(\eta \)–\(\eta '\) mixing (isospin breaking due to different masses for the *u* and *d* quarks), \(\rho \) width (additional amplitude to include in the isospin relations for \(B\rightarrow \rho \rho \)). In addition, we discuss the statistical issues related to our frequentist framework. In Sect. 5, we extract the hadronic (tree and penguin) amplitudes associated with each decay, comparing them briefly with theoretical expectations. We use our framework to perform indirect predictions of observables in each channel and discuss the compatibility with the available measurements in Sect. 6. We perform a prospective study to determine the impact of reducing the experimental uncertainties on specific observables in Sect. 7, before drawing our conclusions. Dedicated appendices gather additional numerical results for observables in three modes, separate analyses using either the BaBar or Belle inputs only, and a brief discussion of the quasi-two-body analysis of the charmless \(B^0\rightarrow a_1^\pm \pi ^{\mp }\) that may provide some further information on \(\alpha \).

## 2 Isospin decomposition of charmless two-body *B* decays

*h*=\(\pi ,\rho \) and \(i,j=-,0,+\)) is described by the weak transition \(\bar{b}\rightarrow \bar{u} u\bar{d}\), followed by the hadronisation of the \((\bar{u} u\bar{d},q)\) system, where \(q=d(u)\) is the spectator quark in the neutral (charged) component of the mesons isodoublet. The weak process receives dominant contributions from both the tree-level \(\bar{b}\rightarrow \bar{u}(u\bar{d})\) charged transition and the flavour-changing neutral current penguin transition, \(\bar{b}\rightarrow \bar{d} (u\bar{u})\), whose topologies are shown in Fig. 1.

### 2.1 Penguin pollution

*u*,

*c*,

*t*) occurring in the

*W*loop:

*u*,

*c*,

*t*)-loop mediated topologies, respectively.

*CP*-conjugate isodoublet (\(\bar{B}^0\), \(B^-\)) can be expressed as

*B*-meson that arises naturally in physical observables. For consistency between all the \({SU}(2)\)-related decay modes considered in the isospin analysis, the same phase convention has been applied to define the amplitudes for the charged

*B*meson. The

*CP*invariance of the strong interaction means that the same hadronic amplitudes \(T^{ij}\) and \(P^{ij}\) are involved in the

*CP*-conjugate processes, whereas a complex conjugation is applied to the weak phases.

*CP*-conjugate amplitudes:

*CP*-conjugate amplitudes describing the \(B^0\) and \(\bar{B}^0\) mesons decaying into the same final state \(h_1^ih_2^j\). In particular, the time-dependent analysis of the \(B^0/{\bar{B}^0}\rightarrow h_1^+h_2^-\) decay yields the

*CP*asymmetry:

*t*is either the decay time of the meson, or (in the case of

*B*-factories) the time difference between the

*CP*- and tag-side decays. The coefficients can be expressed as

*B*charmless decays modify this picture: if we introduce the effective angle corresponding to the phase of \(\lambda \), we have [19]

General decomposition of the amplitudes \(A^{ij}=\langle h_1^ih_2^j|\mathcal{H}_s|u\bar{u}\bar{d},q\rangle \) (\(q=u,d\); \(i,j=-,0,+\)) in terms of the reduced matrix elements \(A_{\varDelta I,I_f}\) for a pair of distinguishable isovector mesons \(h_1^i\) and \(h_2^j\)

\(A^{ij} = \langle h_1^ih_2^j|\mathcal{H}_s|u\bar{u}\bar{d}, q\rangle \) | \(A_{\frac{5}{2},2}\) | \(A_{\frac{3}{2},2}\) | \(A_{\frac{3}{2},1}\) | \(A_{\frac{1}{2},1}\) | \(A_{\frac{1}{2},0}\) |
---|---|---|---|---|---|

\(A^{+0} = \langle h_1^+h_2^0|\mathcal{H}_s|u\bar{u}\bar{d},u\rangle \) | \(-\sqrt{1/6}\) | \(+\sqrt{3/8}\) | \(-\sqrt{1/8}\) | \(+\sqrt{{1} / {2}}\) | 0 |

\(A^{0+} = \langle h_1^0h_2^+|\mathcal{H}_s|u\bar{u}\bar{d},u\rangle \) | \(-\sqrt{1/6}\) | \(+\sqrt{3/8}\) | \(+\sqrt{1/8}\) | \(-\sqrt{{1} / {2}}\) | 0 |

\(A^{+-} = \langle h_1^+h_2^-|\mathcal{H}_s|u\bar{u}\bar{d},d\rangle \) | \(+\sqrt{1/12}\) | \(+\sqrt{1/12}\) | \(+1/2\) | \(+1/2\) | \( +\sqrt{1/6}\) |

\(A^{-+} = \langle h_1^-h_2^+|\mathcal{H}_s|u\bar{u}\bar{d},d\rangle \) | \(+\sqrt{1/12}\) | \(+\sqrt{1/12}\) | \(-1/2\) | \(-1/2\) | \( +\sqrt{1/6}\) |

\(A^{00} = \langle h_1^0h_2^0|\mathcal{H}_s|u\bar{u}\bar{d},d\rangle \) | \(+\sqrt{1/3}\) | \(+\sqrt{1/3}\) | 0 | 0 | \( -\sqrt{1/6}\) |

### 2.2 General isospin decomposition and application to the \(\rho \pi \) final state

One can factorise the decay amplitudes in two parts. First, the weak decay \(\bar{b}\rightarrow u\bar{u} \bar{d}\) (common to all \(B_q\rightarrow h_1^ih_2^j\) decay processes) corresponds to a shift of isospin \(\varDelta I\). The hadronisation into two light mesons can then be described as \(\langle h_1^ih_2^j|\mathcal{H}_s|u\bar{u}\bar{d},q\rangle \) where \(\mathcal{H}_s\) represents the isospin-conserving strong interaction Hamiltonian. The Wigner–Eckart theorem can be used to express these amplitudes in terms of reduced matrix elements, \(A_{\varDelta I,I_f}\), identified by the shift \(\varDelta I\) and the final-state isospin \(I_f\) (\(I_f=0,1,2\)).^{1} Table 1 yields the decomposition of the \(A^{ij}\) amplitudes in the general case of two distinguishable isovector mesons (\(h_1\ne h_2\)).

*CP*-conjugate amplitudes. Moreover, the sum of the decay amplitudes of the charged modes \((A^{+0}+A^{0+})\) is a pure \(A_{\frac{3}{2},2}\) isospin amplitude.

*CP*-conjugate amplitudes of the charged modes, or equivalently as a function of the amplitudes of the neutral modes only:

### 2.3 Application to the \(\pi \pi \) and \(\rho \rho \) cases

*CP*-conjugate amplitudes.

^{2}In the case of the \(\rho \rho \) channel, one should consider a different set of independent amplitudes for each of the three possible polarisations (which is identical for the two \(\rho \) mesons).

*CP*-conjugate amplitudes using branching ratios and

*CP*asymmetries for all the modes. Figure 2 illustrates this construction, which translates the measurement of \(\alpha _\mathrm{eff}\) into a determination of the CKM angle \(\alpha \). This procedure is affected by discrete ambiguities, since there are several manners of reconstructing the two isospin triangles. This leads to a fourfold ambiguity for \(\sin (2\alpha )\), i.e. an eightfold ambiguity on the solutions of \(\alpha \) in \([0,180]^\circ \), in general. These additional solutions are called “mirror solutions”. If one or both triangles are flat, several mirror solutions become degenerate, decreasing the number of distinct solutions for \(\alpha \).

Decomposition of the amplitude \(\langle h^ih^j|\mathcal{H}_s|u\bar{u}\bar{d},q\rangle \) (\(q=u,d\); \(i,j=-,0,+\)) in terms of the isospin amplitudes \(A_{\varDelta I,I_f}\) for indistinguishable mesons in the final state (\(h_1=h_2=h\)). A global factor \(\sqrt{3}\) is applied to all the coefficients with respect to the general coefficients given in Table 1

\(A^{ij} = \langle h^ih^j|\mathcal{H}_s|u\bar{u}\bar{d},q\rangle \) | \(A_{\frac{5}{2},2}\) | \(A_{\frac{3}{2},2}\) | \(A_{\frac{1}{2},0}\) |
---|---|---|---|

\(A^{+0} = \langle \pi ^+\pi ^0|\mathcal{H}_s|u\bar{u}\bar{d},u\rangle \) | \(-1\) | \({3} / {2}\) | 0 |

\(A^{+-} = \langle \pi ^+\pi ^-|\mathcal{H}_s|u\bar{u}\bar{d},d\rangle \) | \(\sqrt{1/2}\) | \(+\sqrt{1/2}\) | \( +1\) |

\(A^{00} = \langle \pi ^0\pi ^0|\mathcal{H}_s|u\bar{u}\bar{d},d\rangle \) | \(+1\) | \(+1\) | \(-\sqrt{1/2}\) |

## 3 Determining the weak angle \(\alpha \)

### 3.1 Procedure

If we consider that each of the five amplitudes \(A^{ij}\)\((ij=+-,-+,+0,0+,00)\) receives two complex contributions, tree and penguin, the hadronic contributions to the generic decay system \(B^{i+j}\rightarrow h_1^ih_2^j\) can be parametrised with 20 real parameters, in addition to the weak phase \(\alpha \) (one overall phase being irrelevant). In the case of a Bose-symmetric final state (\(h_1=h_2\)), the dimension of the parameter space reduces down to 13. Assuming the isospin relations between the amplitudes discussed in Sect. 2, the three decay systems \(B\rightarrow \pi \pi \), \(B\rightarrow \rho \rho \) and \(B\rightarrow \rho \pi \) provide enough experimental measurements to fully constrain the parameter space, hereafter denoted \({\varvec{{p}}}=(\alpha ,\)\({\varvec{{\mu }}}\)), where \({{\varvec{\mu }}}\) represents the set of independent hadronic parameters (tree and penguin amplitudes). The actual dimension of the parameter space depends on the decay system and will be discussed in the following subsections.

*N*-dimensional parameters space through the frequentist statistical approach discussed in detail in Refs. [5, 6, 10, 22] and in Sect. 4.2, but we find it useful to briefly summarise its main features here. The set of experimental observables, denoted \({\varvec{\mathcal {O}}}_{\mathrm{exp}}\), is measured in terms of likelihoods that can be used to build a \(\chi ^2\)-like test statistic:

*N*parameters \(\varvec{p}\) free to vary. The absolute minimum value of the test statistic, \(\chi ^2_\mathrm{min}\), quantifies the agreement of the data with the theoretical model (assuming the validity of the SM and \({SU}(2)\) isospin symmetry in the present case). Converting \(\chi ^2_\mathrm{min}\) into a

*p*value is, however, not trivial a priori, as one has to interpret \(\chi ^2(\varvec{p})\) as a random variable distributed according to a \(\chi ^2\) law with a certain number of degrees of freedom. The actual number of degrees of freedom of the system can be ill defined in the case where the experimental observables are interdependent (see for instance the related discussion in Sect. 3.4). In the case of

*M*independent observables, the number of degrees of freedom of the system is defined as \(N_{\mathrm{dof}}=M-N\). This occurs in the Gaussian case, but it also can apply in non-Gaussian cases in the limit of large samples, under the conditions of Wilks’ theorem [23].

*p*value, which is computed assuming that \(\varDelta \chi ^2(\alpha )\) is \(\chi ^2\)-distributed with one degree of freedom:

*p*value larger than \(1-\)CL. The derivation, robustness and coverage of this definition for the

*p*value will be further discussed in Sect. 4.2.

Although the relevant information on the \(\alpha \) constraint is fully contained in the *p* value function \(p(\alpha )\), confidence intervals will be derived in the following subsections. It is worth noticing that the *p* value for \(\alpha \) usually presents a highly non-Gaussian profile and that the \({SU}(2)\) isospin analysis suffers from (pseudo-)mirror ambiguities. Therefore, the confidence intervals provided must be interpreted with particular care.

### 3.2 Isospin analysis of the \(B\rightarrow \pi \pi \) system

The three decays \(B^{0,+}\rightarrow (\pi \pi )^{0,+}\) depend on 12 hadronic parameters and the weak phase \(\alpha \). One can set one irrelevant global phase to zero, and one can eliminate further parameters using the complex isospin relation Eq. (20) and its *CP* conjugate (four real constraints) as well as the absence of the penguin contribution to the charged mode amplitude (two real constraints). The remaining system of amplitudes features six degrees of freedom. The isospin-related \(B\rightarrow \pi \pi \) decays (and similarly each of the helicity states of the \(B\rightarrow \rho \rho \) mode) can thus be described with 6 real independent parameters, including the common weak phase \(\alpha \).

^{3}in Table 3. Six independent observables are available, allowing us to constrain the six-dimensional parameter space of the \({SU}(2)\) isospin analysis. These branching fractions and

*CP*asymmetries are related to the decay amplitudes as

*B*meson.

World averages for the relevant experimental observables in the \(B\rightarrow \pi ^i\pi ^j\) modes: branching fraction \(\mathcal{B}^{ij}_{\pi \pi }\), time-integrated *CP* asymmetry \(\mathcal{C}^{ij}_{\pi \pi }\), time-dependent asymmetry \(\mathcal{S}^{ij}_{\pi \pi }\) and correlation (\(\rho \))

Observable | World average | References |
---|---|---|

\(\mathcal{B}^{+-}_{\pi \pi }\)\((\times 10^6)\) | \(5.10\pm 0.19\) | |

\(\mathcal{B}^{+0}_{\pi \pi }\)\((\times 10^6)\) | \(5.48\pm 0.34\) | |

\(\mathcal{B}^{00}_{\pi \pi }\)\((\times 10^6)\) | \(1.59\pm 0.18\) | |

\(\mathcal{C}^{00}_{\pi \pi }\) | \(-0.34\pm 0.22\) | |

\(\mathcal{C}^{+-}_{\pi \pi }\) | \(-0.284\pm 0.039\) | |

\(\mathcal{S}^{+-}_{\pi \pi }\) | \(-0.672\pm 0.043\) | |

\(\rho (C^{+-}_{\pi \pi },S^{+-}_{\pi \pi })\) | \(+0.013\) |

*P*are three complex parameters, among which one can be taken as real to set the global phase convention.

*a*, \({\bar{a}}\) and \(\mu \) are real positive parameters related to the modulus of the decay amplitudes and \({\bar{\alpha }}\) and \(\varDelta \) are relative phases. \(A^{+-}\) is chosen to be a real positive quantity, which sets the phase convention. The weak phase \(\alpha \) clearly appears as \(2\alpha =\arg ({\bar{A}}^{+0}/A^{+0})\). The parameter \(\bar{\alpha }\), satisfying \(\bar{\alpha }=2\arg ({\bar{A}}^{+-}/A^{+-})\), would coincide with \(\alpha \) in the limit of a vanishing penguin contribution.

^{4}

*c*, \(\bar{c}\) and \(\mathcal{S}^{+-}\) are left invariant by reflection,

*CP*asymmetry in the \(B^0\rightarrow \pi ^0\pi ^0\) decay, \(S^{00}\). This observable can be written as:

*CP*violation observed in the \(B^0\rightarrow \pi ^+\pi ^-\) decay [31, 32]. However, for any finite \(\alpha \) in the vicinity of zero, the isospin relation can be satisfied if arbitrary large penguin amplitudes are allowed (the limit case \(\alpha \rightarrow 0\) happens if \(|T^{+-}|\), \(|T^{00}|\), |

*P*| are sent to infinity; see Eq. (28) and Refs. [36, 42]). The limit \(\alpha \rightarrow 0\) was also discussed in the context of the Bayesian statistical approach in Ref. [41]. External data, e.g., based on \({SU}(3)\) consideration, could be used to set bounds on the penguin over tree ratio and thus further constrain \(\alpha \) around 0. These bounds stand beyond the pure \({SU}(2)\) approach adopted here and we will not attempt at including such additional information for the following reasons. On one hand, other charmless modes (such as \(\rho \pi \)) will provide further information on the small \(\alpha \) region, and on the other hand, we will ultimately use the determination of \(\alpha \) in a global fit together with other well-controlled constraints on the CKM parameters [18]: both types of constraints will rule out the region around \(\alpha =0\), so that additional theoretical hypotheses on the size of the \(B\rightarrow \pi \pi \) amplitudes are not necessary for our purposes.

### 3.3 Isospin analysis of the \(B\rightarrow \rho \rho \) system

*CP*asymmetry in the \(B^0\rightarrow \rho ^0\rho ^0\) decay, \(\mathcal{S}^{00}\), is experimentally accessible for the final state with four charged pions, potentially lifting some of the discrete ambiguities affecting the determination of \(\alpha \). However, the current measurement \(\mathcal{S}^{00}=0.3\pm 0.7\) suffers from large uncertainties, leaving pseudo-mirror solutions in \(\alpha \). The available experimental observables

^{5}and their current world averages are summarised in Table 4. Under the \({SU}(2)\) isospin hypothesis, the direct

*CP*asymmetry in \(B^+\rightarrow \rho ^+\rho ^0\) vanishes and we will not take into account the experimental measurement of this quantity (which is consistent with our hypothesis and will be used to test this assumption in Sect. 4.1). Seven independent observables are available for the longitudinal helicity state of the \(\rho \) mesons, allowing us to over-constrain the six-dimensional parameter space of the \({SU}(2)\) isospin analysis.

World averages for the relevant experimental observables in the \(B\rightarrow \rho ^i\rho ^j\) modes: branching fraction \(\mathcal{B}^{ij}_{\rho \rho }\), fraction of longitudinal polarisation \(f_L^{ij}\), time-integrated *CP* asymmetry \(\mathcal{C}^{ij}_{\rho \rho }\), time-dependent asymmetry \(\mathcal{S}^{ij}_{\rho \rho }\) and correlation (\(\rho \))

Observable | World average | References |
---|---|---|

\(\mathcal{B}^{+-}_{\rho \rho }~ \times ~ f_L^{+-}\)\((\times 10^6)\) | \((27.76\pm 1.84)\)\(\times \)\((0.990\pm 0.020)\) | |

\(\mathcal{B}^{+0}_{\rho \rho }\)\(\times \)\(f_L^{+0}\)\((\times 10^6)\) | \((24.9\pm 1.9)\)\(\times \)\((0.950\pm 0.016)\) | |

\(\mathcal{B}^{00}_{\rho \rho }\)\(\times \)\(f_L^{00}\)\((\times 10^6)\) | \((0.93\pm 0.14)\)\(\times \)\((0.71\pm 0.06)\) | |

\(\mathcal{C}^{+-}_{\rho _L\rho _L}\) | \(-0.00\pm 0.09\) | |

\(\mathcal{S}^{+-}_{\rho _L\rho _L}\) | \(-0.15\pm 0.13\) | |

\(\rho (C^{+-}_{\rho _L\rho _L},S^{+-}_{\rho _L\rho _L})\) | \(+0.0002\) | |

\(\mathcal{C}^{00}_{\rho _L\rho _L}\) | \(0.2\pm 0.9\) | |

\(\mathcal{S}^{00}_{\rho _L\rho _L}\) | \(0.3\pm 0.7\) |

The region \(\alpha = [14.0,76.0]^\circ \cup [112.0,158.0]^\circ \) is excluded at more than 3 standard deviations by the \({SU}(2)\) isospin analysis of the \(B\rightarrow \rho \rho \) system. On the bottom panel of Fig. 7, a zoom around \(\alpha =0\) exhibits a small discontinuity, corresponding to \(\chi ^2(\alpha =0)=1.61\) and indicating that this hypothesis is mildly disfavoured by the data, consistently with the absence of large direct CP asymmetries in \(B\rightarrow \rho \rho \) decays.

### 3.4 Isospin analysis of the \(B\rightarrow \rho \pi \) system

The three neutral and two charged \(B^{0,+}\rightarrow (\rho \pi )^{0,+}\) decays can be described with 10 complex (tree and penguin) amplitudes and one weak phase \(\alpha \), i.e., 21 real parameters. Assuming the pentagonal isospin relation Eq. (13) that leaves only two independent complex penguin contributions, the number of degrees of freedom of the system is reduced to 12 after setting an irrelevant global phase to zero. The dimension of the system gets down to 10 if we consider the three neutral decays only. The time-dependent Dalitz analysis of the neutral \(B^{0}\rightarrow (\rho ^\pm \pi ^{\mp },\rho ^0\pi ^0) \rightarrow \pi ^+\pi ^-\pi ^{0}\) transitions has been shown to carry enough information to fully constrain the isospin-related \(B\rightarrow \rho \pi \) system, thanks to the finite width of the intermediate \(\rho \)-mesons that yields a richer interference pattern of the three-body decay [15, 16]. For instance, 11 independent phenomenological observables can be defined from the flavour-tagged time-dependent Dalitz distribution of the three-pion decay, e.g., the branching fractions of the three intermediate \(\rho ^i\pi ^j\) transitions, the corresponding direct and time-dependent *CP* asymmetries and two relative phases between their amplitudes.

The extraction of the parameter \(\alpha \) through the \({SU}(2)\) analysis of the neutral modes will be referred to as the “Dalitz analysis” and will be considered first in Sect. 3.4.1. An extended analysis (referred to as the “pentagonal analysis”), including the information coming from the charged decays \(B^+\rightarrow \rho ^+\pi ^0\) and \(B^+\rightarrow \rho ^0\pi ^+\), will be discussed in Sect. 3.4.2.

#### 3.4.1 Dalitz analysis of the neutral \(B^0\rightarrow (\rho \pi )^0\) modes

^{6}and \(f_i\) (\(i=-,+,0\)) is the form factor accounting for the \(\rho ^i\) line-shape. Neglecting the tiny \(B^0\) width difference \(\varDelta \varGamma _d\), the time-dependent decay rate can be written as a function of three combinations of these amplitudes:

*CP*asymmetries in the \(B\rightarrow \rho ^i\pi ^j\) intermediate states while the \(\mathcal {I}\)coefficients parametrise the mixing-induced

*CP*asymmetries.

Relative value of the quasi-two-body related \(\mathcal U\) and \(\mathcal I\) coefficients extracted from the time-dependent Dalitz analysis of the \(B^0\rightarrow (\rho \pi )^0\) decay [48, 49]. The corresponding form-factor bilinear is indicated for each coefficient. The red vertical line indicates the overall normalisation, defined by \(U^+_+=1\)

Relative value of the interference-related \(\mathcal U\) and \(\mathcal I\) coefficients extracted from the time-dependent Dalitz analysis of the \(B^0\rightarrow (\rho \pi )^0\) decay [48, 49]. The corresponding form-factor bilinear is indicated for each coefficient. The red vertical line indicates the overall normalisation, defined by \(U^+_+=1\)

*CP*-conjugate modes are illustrated on Fig. 8. The resulting constraint on \(\alpha \) is shown on the top panel of Fig. 9.

#### 3.4.2 “Pentagonal” analysis of the \(B\rightarrow \rho \pi \) system

The amplitudes of the two charged decay modes \(B^+\rightarrow \rho ^+\pi ^0\) and \(B^+\rightarrow \rho ^0\pi ^+\) are related to the amplitudes of the three neutral decays \(B^0\rightarrow (\rho \pi )^0\) through the pentagonal relation Eq. (13). The measurement of the branching ratios and *CP*-asymmetries of the charged modes may provide additional constraints to the isospin system. Considering simultaneously the charged and the neutral modes, the \(\mathcal U\) and \(\mathcal I\) observables that describe the relative amplitudes of the neutral decays must be supplemented with an absolute normalisation. This is achieved by identifying the sum of branching ratios, \(\mathcal{B}^{+-}_{\rho {\pi }}+\mathcal{B}^{-+}_{\rho {\pi }}\), with the scaled amplitude \(\mu ^{+}(\mathcal{U}^+_+ + \mathcal{U}^+_-){\tau _{B^0}}/{2}\), where \(\mu ^{+}\) is the absolute normalisation of the relative \(\mathcal U\) and \(\mathcal I\) coefficients. The experimental measurements of the branching fractions for the charged and neutral \(B\rightarrow \rho \pi \) modes and the *CP* asymmetries for the charged modes are listed in Table 8.

World averages for the branching fractions \(\mathcal{B}^{ij}_{\rho {\pi }}\) and time-integrated *CP* asymmetries \(\mathcal{C}^{ij}_{\rho {\pi }}\) for the charged (top) and neutral (bottom) \(B\rightarrow \rho \pi \) decay modes

Observable | World average | References |
---|---|---|

\(\mathcal{B}^{+0}_{\rho {\pi }}\)\((\times 10^6)\) | \(10.9\pm 1.4\) | |

\(\mathcal{C}^{+0}_{\rho {\pi }}\) | \(-0.02\pm 0.11\) | |

\(\mathcal{B}^{0+}_{\rho {\pi }}\)\((\times 10^6)\) | \(8.3\pm 1.2\) | |

\(\mathcal{C}^{0+}_{\rho {\pi }}\) | \(-0.18^{+0.17}_{-0.09}\) | |

\(\mathcal{B}^{\pm {\mp }}_{\rho {\pi }}\)\((\times 10^6)\) | \(23.0\pm 2.3\) | |

\(\mathcal{B}^{00}_{\rho {\pi }}\)\((\times 10^6)\) | \(2.0\pm 0.5\) |

*CP*-conjugate amplitudes are defined as

### 3.5 Combined result

*B*-meson unitarity triangle [10, 18]. The following relation can be derived:

*B*-factory experiments, BaBar and Belle, have measured all these observables independently, it is also possible to perform separate \({SU}(2)\) isospin analyses for each of the three decay channels and each experiment. The corresponding constraints are discussed in Appendix A.

## 4 Additional uncertainties on the \(\alpha \) determination

In this section, we are going to test the limits of the assumptions made to extract \(\alpha \) from the data: the breakdown of isospin symmetry and the statistical approach used to build confidence intervals.

### 4.1 Testing the \({SU}(2)\) isospin framework

The charges of the

*u*and*d*quarks are taken as identical. The \(\varDelta I={3} / {2}\) contribution induced by the electroweak penguins topology to the \(B^+\rightarrow h^+h^0\) decay is considered negligible. As the gluonic penguins only yield a \(\varDelta I={1} / {2}\) isospin contribution, the pure \(\varDelta I={3} / {2}\)\(B^+\rightarrow h^+h^0\) decay only receives tree contributions in the absence of electroweak penguins. In this limit, the weak phase \(\alpha \) can be identified as half the phase difference between the amplitude of the charged mode and its*CP*conjugate.The masses of the

*u*and*d*quarks are taken as identical. Isospin symmetry is assumed to be exact in the strong hadronisation process following the weak transition \(\bar{b}(q)\rightarrow \bar{u} u\bar{d}(q)\) where*q*represents the light spectator quark*u*or*d*. This assumption allows one to relate the decay amplitudes of the charged (\(\bar{b}u\)) meson to the decay amplitudes of its isospin-related neutral state \(\bar{b}d\) according to the pentagonal (triangular) identity given in Eq. (13) (respectively Eq. (20)).The \(\rho \rho \) final state is supposed to obey Bose–Einstein statistics, and thus the analysis for the \(\pi \pi \) and the \(\rho \rho \) systems are supposed to follow the same isospin decomposition (for a given \(\rho \) polarisation). However, the \(\rho \) mesons cannot be distinguished only in the limit of a vanishing width. Once the finite \(\rho \) width is taken into account, additional terms (forbidden by Bose symmetry fin the limit \(\varGamma _\rho =0\)) must be taken into account in the isospin decomposition of the amplitudes.

#### 4.1.1 \(\varDelta I={3} / {2}\) electroweak penguins

*u*and

*c*as well as the electroweak operators \(\mathcal{O}_7\) and \(\mathcal{O}_8\) (suppressed by tiny Wilson coefficients), the amplitude ratio

*B*decay. Consequently, the impact of the electroweak penguin can be accounted for introducing the single real parameter \(r_{\textsc {ewp}}={P_\textsc {ew}^{+0}}/{T^{+0}}\) in the isospin analysis of the \(B\rightarrow hh\) systems.

^{7}The following 68% CL intervals are obtained:

#### 4.1.2 Isospin-breaking effects due to mixing in the \(\pi \pi \) system

We have worked up to now under the assumption that isospin symmetry was exact for the hadronic part of the *B*-meson decay. Even though isospin-breaking effects are known to be tiny, due to the very small mass difference between the *u* and *d* quarks, it is interesting to assess more precisely how it could affect our analysis.

*CP*-conjugate amplitudes.

World averages for the branching ratios and direct *CP* asymmetries of the \(B^{0,+}\rightarrow \pi ^{0,+}\eta ^{(')}\) modes [66]

Observable | World average |
---|---|

\(\mathcal{B}^{\pi ^0\eta }\ (\times 10^6)\) | \(\left( {0.41}_{+0.18}^{-0.17}\right) \) |

\(\mathcal{B}^{\pi ^{0}\eta {'}}\ (\times 10^6)\) | 1.2 ± 0.4 |

\(\mathcal{B}^{\pi ^+\eta }\)\((\times 10^6)\) | 4.02\({\pm }\)0.27 |

\(\mathcal{B}^{\pi ^+\eta {'}}\)\((\times 10^6)\) | \(\left( {2.7}_{+0.5}^{-0.4}\right) \) |

\(\mathcal{C}^{\pi ^+\eta }\) | \(0.14\pm 0.05\) |

\(\mathcal{C}^{\pi ^+\eta {'}}\) | \(-0.06\pm 0.15\) |

*B*meson decay. Consequently, the amplitude system is further constrained by introducing the measured

*CP*asymmetry in the \(B^+\rightarrow \pi ^+\pi ^0\) mode, see Table 10, which was not considered in the isospin-symmetric analysis.

World averages for the \(C^{+0}\) direct *CP* asymmetries for the \(B\rightarrow \pi ^+\pi ^0\) decay and for the longitudinally polarised state in \(B^+\rightarrow \rho ^+\rho ^0\) decay

*CP*conjugates, to the \(B\rightarrow \pi \pi \) system. We use Ref. [67] for the numerical estimates of the mixing parameters:

#### 4.1.3 Additional isospin-breaking effects for the \(\rho \rho \) and \(\rho \pi \) systems

The isospin breaking due to the \(\pi ^0\)–\(\eta \)–\(\eta '\) mixing may affect the \(\rho ^+\rightarrow \pi ^+\pi ^0\) decay but turns out negligible, as the leading term in \(\epsilon _\eta \) is suppressed by the small \(\rho ^+\rightarrow \pi ^+\eta \) decay rate: \({\varGamma (\rho ^+\rightarrow \pi ^+\eta )}/{\varGamma (\rho ^+\rightarrow \pi ^+\pi ^0)}<0.6\%\) [69].

Differences in the di-pion couplings for the neutral and charged \(\rho \) mesons are experimentally limited to less than 1%: indeed, \(1-({\varGamma _{\rho ^+}}/{\varGamma _{\rho ^0}})=(0.2\pm 0.9)\%\) [69].

Isospin breaking affecting the \(\rho ^0,\omega \rightarrow \pi ^-\pi ^+\) interference is restricted to a small window in the \(\pi \pi \) mass spectrum: this effect, integrated over the whole \(\pi \pi \) range, is estimated at the order of 2% [21].

The dominant source of isospin breaking is actually due to the large decay width \(\varGamma _\rho \), which makes the two final-state mesons distinguishable in the \(B\rightarrow \rho _1(\pi \pi )\rho _2(\pi \pi )\) decay. This results in a residual \(I_f=1\) amplitude contribution, forbidden by Bose–Einstein symmetry in the limit \(\varGamma _\rho =0\), but potentially as large as \(({\varGamma _\rho }/{m_\rho })^2\sim 4\%\). It is a slight abuse of language to call this effect an isospin-breaking contribution, as it does not vanish in the limit \(m_u=m_d\) (but it does in the limit of a vanishing decay width \(\varGamma _\rho \sim (m_{(\pi \pi )_{1}}-m_{(\pi \pi )_{2}})=0\)).

*CP*violation in the charged decays. The measured

*CP*asymmetry in the \(B^+\rightarrow \rho ^+\rho ^0\) mode given in Table 10 is thus included as an additional constraint in the presence of these isospin-breaking terms. Figure 16 illustrates the determination of the weak phase \(\alpha \) in the case of isospin-breaking contributions smaller than \(|r_T|,|r_P|<4\%\) and the very conservative bound \(|r_T|,|r_P|<10\%\). A small shift on the preferred \(\alpha \) value near \(90^\circ \), \(\delta \alpha =-0.6^\circ \)\((-1.2^\circ )\), results from isospin-breaking contributions limited to 4% (10%), respectively. The linear increase of the 68% interval, \(({}^{+0.2}_{-1.7})^\circ \) for \(|r_T|,|r_P|<4\%\), has to be understood as an upper limit of the impact of these contributions on the determination of \(\alpha \).

Isospin breaking arises similarly in the \(B\rightarrow \rho \pi \) system through mixing in the light pseudoscalar and vector sectors. However, as long as the weak phase determination is limited to the neutral \(B^0\rightarrow (\rho \pi )^0\) Dalitz analysis, the isospin constraints reduces to the triangular penguin relation Eq. (14). Any isospin breaking in the tree amplitudes can be absorbed by a redefinition of the unconstrained charged amplitudes \(T^{+0}\), \(T^{0+}\). Corrections to the penguin amplitude relation Eq. (14) in the \(B\rightarrow \rho \pi \) system will be suppressed by the small penguin-to-tree ratios. For instance, the impact of the \(\pi ^0\)–\(\eta \)–\(\eta '\) mixing on the weak phase determination in the \(B^0\rightarrow (\rho \pi )^0\) analysis is suppressed by factors like \(\epsilon _{\eta ^{(')}}\left| {P_{\rho \eta ^{(')}}}/{T^+}\right| \), where \(P_{\rho \eta ^{(')}}\) represents the penguin contribution to the \(B^+\rightarrow \rho ^+\eta ^{(')}\) transition and \(T^+\) is defined in Eq. (48). The corresponding deviation is found to be negligible, \(\delta \alpha <0.1^\circ \) [21]. Therefore, apart for the electroweak penguin contribution discussed in Sect. 4.1.1, no further isospin breaking will be considered for the \(B^0\rightarrow (\rho \pi )^0\) analysis.

### 4.2 Impact of the statistical treatment

#### 4.2.1 *p* values based on the bootstrap approach and Wilks’ theorem

In Sect. 3.1, we have outlined our framework to determine confidence intervals on \(\alpha \), starting from a test statistic \(\varDelta \chi ^2 (\alpha )\), which is converted into a *p* value following Eq. (24). This procedure can be proven to be exact in the simple case where the observables obey Gaussian probability distribution functions and they are linearly related to the parameters of interest. Following Wilks’ theorem, this can be extended to more general cases, at least asymptotically, if the data sample is large and the observables resolutions are small enough to consider the problem as locally linear [23]. In such case, the construction in Sect. 3.1 ensures exact coverage: if one repeated the determination of \(\alpha \) using independent data sets from many identical experiments, the 68% CL interval for \(\alpha \) would encompass the true value of \(\alpha \) in 68% of the cases.

However, many effects can alter this picture. Even in the exact Gaussian case, the *p* value can be distorted when the observables have a nonlinear dependence on the fundamental parameters of interest. An example consists in Eq. (12), where the reexpression of the *CP*-asymmetries in terms of \(\alpha _\mathrm{eff}\) is nonlinear and implements the trigonometric boundary on *CP*-asymmetries \(\sqrt{\mathcal{C}^2+\mathcal{S}^2}<1\) [6]. More generally, one may wonder if we stand close to the hypotheses of Wilks’ theorem with the current set of data. Otherwise, the construction given in Sect. 3.1 might suffer from under- or over-coverage. In this section, we will assess the finite-size errors associated with our statistical framework by considering a different construction of the *p* value that takes into account some of the effects deemed subleading in the Wilks-based approach.

We start by recalling some elements related to the construction of *p* values in our context, as discussed in Refs. [22, 70, 71, 72]. We want to assess how much the data is compatible with the hypothesis that the true value of the weak phase \(\alpha \), denoted \(\alpha _t\), is equal to some fixed value \(\alpha \), i.e., \({\mathcal H}_{\alpha }: \alpha _t = \alpha \). This hypothesis is composite, as it sets the value of some of the theoretical parameters, but not all of them. Indeed, the hadronic parameters (tree and penguin amplitudes) are also theoretical parameters required in our isospin analysis but are not set in \({\mathcal H}_{\alpha }\). These hadronic parameters are nuisance parameters, denoted collectively as \(\varvec{\mu }\).

*X*by using the notation \(\varDelta \chi ^2 (X;\alpha )\). A

*p*value is built by calculating the probability to obtain a value for the test statistic at least as large as the one that was actually observed, assuming that the hypothesis \({\mathcal H}_{\alpha }\) is true:

*h*of the test statistic is obtained from the PDF

*g*of the data as

*p*value thus provides evidence against the hypothesis \({\mathcal H}_{\alpha }\). We notice that in general the

*p*value Eq. (63) exhibits a dependence on the nuisance parameters through the PDF

*h*, even though the test statistic \(\varDelta \chi ^2 \) itself is independent of \(\varvec{\mu }\).

For linear models, in which the observables *X* depend linearly on the parameter \(\alpha _t\), \(\varDelta \chi ^2 (X;\alpha )\) is a sum of standard normal random variables, and is distributed as a \(\chi ^2 \) with \(N_{\mathrm{dof}}=1\). Under the conditions of Wilks’ theorem [23], this property can be extended to non-Gaussian cases, the distribution of \(\varDelta \chi ^2 (X;\alpha )\) will converge to a \(\chi ^2 \) law depending only on the number of parameters tested. The *p* value becomes independent of the nuisance parameters \(\varvec{\mu }\) and can be still interpreted as coming from a \(\chi ^2 \) law with \(N_{\mathrm{dof}}=1\). This is the rationale for the statistical framework presented in Sect. 3.1.

*p*value for a given value of \(\alpha \) by setting the nuisance parameters to the value minimising \(\chi ^2 (\alpha ,\varvec{\mu })\). This approach assesses the role played by nuisance parameters by replacing them with an estimator \(\hat{\varvec{\mu }}(\alpha )\) that depends on \(\alpha \). Other constructions could have been considered, with more conservative statistical properties (supremum, constrained supremum, etc.) [74], but this would go beyond the scope of our study. We focus here on the bootstrap approach: it is relatively simple to implement, it exhibits good coverage properties in the examples considered here and it provides a first glimpse of the role played by nuisance parameters in coverage, which is neglected in the Wilks-based approach used up to now.

#### 4.2.2 Comparison of *p* values for the extraction of \(\alpha \)

The comparison between the two approaches may be performed by examining their coverage properties. Once computed, a *p* value can be used to determine confidence-level intervals for the parameter of interest \(\alpha \). These intervals have a correct frequentist interpretation if the *p* value exhibits exact coverage, i.e., for any \(\beta \) between 0 and 1, \(P(p\le \beta )=\beta \). Over-coverage \(P>\beta \) corresponding to a conservative *p* value and too wide a CL interval, under-coverage \(P<\beta \) corresponding to a liberal *p* value and too narrow a CL interval. In general, an exact *p* value, or if not possible, a reasonably conservative *p* value, is desirable, at least for the confidence levels of interest. This conservative approach is generally adopted in high-energy physics in order to avoid rejecting a hypothesis (such as “the SM is true”) too hastily.

*p*value using the Wilks-based approach. In the case of the bootstrap approach, the statistical coverage of the \(\alpha \) intervals has been studied through a full frequentist exploration of the space of nuisance parameters. A complete analysis with toy Monte Carlo simulations was carried out in order to compute the PDF

*h*and thus the

*p*value for the angle \(\alpha \). The individual constraints on \(\alpha \) from the \(B\rightarrow \pi \pi \), \(B\rightarrow \rho \rho \) and \(B^0\rightarrow \pi ^+\pi ^-\pi ^0\) systems, are displayed in Fig. 17, comparing the bootstrap and Wilks-based approaches. Considering the 68% CL interval on \(\alpha \), the Wilks-based approach is slightly more conservative than the bootstrap one for both \(B\rightarrow \pi \pi \) and \(B\rightarrow \rho \rho \) systems, whereas the situation is reversed for the \(B^0\rightarrow (\rho \pi )^0\) analysis. The 68% CL intervals obtained with the bootstrap method for the three systems are

*p*values to check whether they are exact, conservative or liberal. Even though \(p^{\mathrm {bootstrap}}\) does not depend on the true value of the nuisance parameters explicitly, its distribution does in general. For illustrative purposes, we choose here to compute the distribution of both tests assuming as true values \(\alpha = 92.5^\circ \) and \(\varvec{\mu } = \hat{\varvec{\mu }}(\alpha = 92.5^\circ )\) (we do not attempt to investigate other values of \(\alpha \) or the nuisance parameters). The computation of the PDF of \(p^{\mathrm {bootstrap}}\) requires a twofold recursion of the bootstrap procedure (double bootstrap).

^{8}The

*p*value for the \(\varDelta \chi ^2\) test is obtained assuming that it obeys Wilks’ theorem while the bootstrap test is assumed to be a true

*p*value, i.e., uniformly distributed over [0, 1].

*p*values as a function of

*p*, we have expressed

*p*in units of \(\sigma \) (assumed test significance) and expressed the cumulative distribution function (integral of the PDF from 0 to

*p*) once again in units of \(\sigma \) (true test significance). A

*p*value with exact coverage corresponds to a diagonal straight line, over-coverage (under-coverage) happens if the curve is above (below) this diagonal. In our particular case, one can clearly see the improved coverage from the bootstrap test with respect to the \(\varDelta \chi ^2\) case, assuming Wilks’ theorem. The bootstrap distribution has a flat distribution at least up to the \(2\sigma \) level, as shown by the curve in Fig. 19, which is close to a diagonal straight line (this is expected since we set the true values of the nuisance parameters to their plug-in values). On the other hand, the \(\varDelta \chi ^2\) test combined with Wilks’ theorem is too aggressive (under-covers) for confidence levels above \(0.6\sigma \) for the global combination of the three \(B\rightarrow hh\) decays modes and above \(1.2\sigma \) when considering \(B\rightarrow \pi \pi \) and \(B\rightarrow \rho \rho \) only. Due to the highly CPU-consuming nature of this computation, we have not tried to perform the same computation for other true values of \(\alpha \) and the nuisance parameters, but one may expect that the results of these coverage tests should hold for values of \(\alpha \) in the vicinity of the solution compatible with its indirect determination.

68% CL intervals for the weak phase \(\alpha \) for different theoretical hypotheses, statistical approaches and channels. In the “Iso” column, Y indicates that the analysis is performed assuming isospin symmetry, whereas N denotes the inclusion of isospin breaking, namely, the effect of \(\pi ^0\)–\(\eta \)–\(\eta '\) mixing in \(B\rightarrow \pi \pi \) and the breakdown of isospin triangle relations up to \(|r_T|,|r_P|<4\%\) in \(B\rightarrow \rho \rho \), as discussed in Sect. 4.1.2 (isospin-breaking effects are neglected in the \(B^0\rightarrow (\rho \pi )^0\) Dalitz analysis). In the “\(\textsc {ewp}\)” column, Y indicates that a contamination from \(\varDelta I=3/2\) electroweak penguins is included as discussed in Sect. 4.1.1, whereas N corresponds to setting these penguin contributions to zero. In the “Stat” column, B corresponds to the bootstrap approach and W to the Wilks-based one. For each channel or combination of channels, the deviation with respect to the indirect determination \(\alpha _\mathrm{ind}\) is indicated within brackets

Iso | \(\textsc {ewp}\) | Stat | \(B\rightarrow \rho \rho \) | \(B\rightarrow \pi \pi \) | \(B\rightarrow \pi \pi \) + \(\rho \rho \) | \(B^0\rightarrow (\rho \pi )^0\) | All combined |
---|---|---|---|---|---|---|---|

Y | N | W | \({92.0}_{-4.8}^{+4.7}\)\((0.03\sigma )\) | 93.0 ± 14.0 \((0.6\sigma )\) | \({92.1}_{-5.5}^{+5.2}\)\((0.03\sigma )\) | \({54.1}_{-10.3}^{+7.7}\)\((3.0\sigma )\) | \({86.2}_{-4.0}^{+4.4}\)\((1.3\sigma )\) |

Y | N | B | \({92.0}_{-4.0}^{+4.2}\)\((0.03\sigma )\) | 92.5 ± 13.5 \((1.1\sigma )\) | \({92.2}_{-5.3}^{+4.9}\)\((0.03\sigma )\) | \({54.0}_{-17.0}^{+10.0}\)\((2.7\sigma )\) | \({86.1}_{-5.0}^{+5.3}\)\((1.1\sigma )\) |

Y | Y | W | \({90.1}_{-4.8}^{+4.7}\)\((0.4\sigma )\) | 91.2 ± 14.2 \((0.5\sigma )\) | \({90.1}_{-5.6}^{+5.1}\)\((0.3\sigma )\) | \({52.9}_{-11.1}^{+8.7}\)\((3.0\sigma )\) | \({85.6}_{-4.2}^{+4.1}\)\((1.5\sigma )\) |

N | N | W | \({91.4}_{-5.7}^{+4.9}\)\((0.2\sigma )\) | 92.5 ± 15.5 \((0.4\sigma )\) | \({91.4}_{-6.0}^{+5.4}\)\((0.2\sigma )\) | \({54.1}_{-10.3}^{+7.7}\)\((3.0\sigma )\) | \({84.4}_{-4.3}^{+5.2}\)\((1.5\sigma )\) |

N | Y | W | \({89.8}_{-4.9}^{+4.6}\)\((0.5\sigma )\) | 91.0 ± 15.0 \((0.3\sigma )\) | \({89.8}_{-5.3}^{+4.9}\)\((0.5\sigma )\) | \({52.9}_{-11.1}^{+8.7}\)\((3.0\sigma )\) | \({83.3}_{-3.1}^{+6.1}\)\((1.6\sigma )\) |

### 4.3 Summary for the direct determination of \(\alpha \)

Our results for the direct determination of the weak phase \(\alpha \) are summarized in Table 11 for the different model hypotheses and the different statistical approaches considered up to now. The 68% CL interval (only for the solution near \(90^\circ \)^{9}) is reported as well as the compatibility with the indirect \(\alpha _\mathrm{ind}\) determination Eq. (2). We see that depending on the approach, the central value for the combination shifts by \(2^\circ \) or less, remaining thus within the error quoted in Eq. (1). The uncertainty remains between \(4^\circ \) and \(5^\circ \) when isospin-breaking effects are allowed. On the other hand, as discussed in Sect. 4.2.2, the comparison between the bootstrap and Wilks-based approaches suggests a sizeable uncertainty attached to the statistical framework, but this uncertainty is attached to the direct extraction of \(\alpha \) from the three \(B\rightarrow \pi \pi \), \(B\rightarrow \rho \rho \) and \(B\rightarrow \rho \pi \) modes and cannot be used as such when it is combined with other constraints, e.g., within global fits. It is likely that one would get a smaller uncertainty if a similar analysis was performed with an extended set of observables leading to a more accurate determination of \(\alpha \).

These various arguments lead us to keep Eq. (1) as our final answer for the direct extraction of \(\alpha \) from charmless *B* decays to be used in latter analyses.

## 5 Hadronic amplitudes

In addition to the CKM angle \(\alpha \), our study of the \(B\rightarrow \pi \pi \), \(B\rightarrow \rho \pi \) and \(B\rightarrow \rho \rho \) systems provides constraints on the hadronic amplitudes that cannot be computed in QCD directly. We can thus determine some features of hadronisation from the data, to be compared with the theoretical approaches proposed to describe these decays.

### 5.1 Penguin-to-tree ratios

*B*decays into light pseudoscalar mesons based on \({SU}(3)\) flavour symmetry yield smaller values for these ratios [79].

### 5.2 Testing colour suppression

^{10}:

*B*decays into light pseudoscalar mesons based on \({SU}(3)\) yield similar values for \(R_\mathcal{C}({\pi \pi })\), indicating that large phase and modulus of the colour-suppressed tree contribution \(T^{00}\) are required not only in the \(\pi \pi \) system, but also in other \(B\rightarrow PP\) decays [79].

## 6 Prediction of observables

### 6.1 \(B\!\rightarrow \pi \pi \) observables

*CP*asymmetries (\(\mathcal{C} _{\pi \pi },\mathcal{S} _{\pi \pi }\)) for the neutral modes \(B^0\!\rightarrow \pi ^+\pi ^-\) and \(B^0\!\rightarrow \pi ^0\pi ^0\). When excluding any of the experimental measurements, either for the branching ratios \(\mathcal{B} ^{ij}_{\pi \pi }\) or for the direct

*CP*asymmetries \(\mathcal{C} ^{ij}_{\pi \pi }\), the \(B\!\rightarrow \pi \pi \) amplitudes system is no longer over-constrained. As a consequence, the corresponding \(\chi ^2 _\mathrm{min}(\alpha _\mathrm{ind},\varvec{\mathcal{O}}_{! i})\) vanishes and the pull associated to these observables saturates its maximal value: \(\mathrm{Pull} (B^{ij}_{\pi \pi },C^{ij}_{\pi \pi })=\sqrt{\chi ^2 _\mathrm{min}(\alpha _\mathrm{ind},\varvec{\mathcal{O}})}\), which turns out to vanish thanks to the closure of both isospin triangles with the current world-average data. The yet unmeasured time-dependent asymmetry, \(\mathcal{S} ^{00}_{\pi \pi }\), is predicted to be

The experimental values, confidence intervals and pulls for the measured observables are reported in Table 14 in Appendix C.

### 6.2 \(B\!\rightarrow \rho \rho \) observables

*CP*asymmetries (\(\mathcal{C} _{\rho \rho },\mathcal{S} _{\rho \rho }\)) for both neutral modes \(B^0\!\rightarrow \rho ^+\rho ^-\) and \(B^0\!\rightarrow \rho ^0\rho ^0\) as shown in Fig. 30. Numerical values, confidence intervals and pulls for the measured observables are reported on Table 15 in Appendix C.

### 6.3 \(B\!\rightarrow \rho {\pi }\) observables

#### 6.3.1 \(B^0\!\rightarrow \pi ^{+}\pi ^{-}\pi ^{0}\) Dalitz analysis

The normalised \(\mathcal {U}\)and \(\mathcal {I}\)observables provide a complete description of the relative \(B^0\!\rightarrow \pi ^+\pi ^-\pi ^0\) decay amplitudes. As discussed in Sect. 3.4, the data from \(B^0\!\rightarrow (\rho \pi )^0\rightarrow \pi ^{+}\pi ^{-}\pi ^{0}\) Dalitz analyses disagrees with the indirect determination of \(\alpha \) at the level of 3.0 standard deviations. This disagreement is reflected through the indirect determination of several of the form-factor coefficients \(\mathcal {U}\)and \(\mathcal {I}\)as listed in Table 17 in Appendix C, in particular \({\mathcal U}_{+}^{-} \), \({\mathcal U}_{-}^{+} \), \(\mathcal {I}_{-}\), \({\mathcal U}_{+0}^{+\mathcal {R}e} \), \({\mathcal U}_{-0}^{+\mathcal {R}e} \) with pulls above \(2\sigma \), and \({\mathcal U}_{+-}^{+\mathcal {R}e} \), \({\mathcal U}_{+-}^{+\mathcal {I}m} \) with pulls near or above \(3\sigma \).

*CP*asymmetry given by Eq. (10) is written independently for the three neutral \(B^0\!\rightarrow \rho ^i\pi ^j\) intermediate states:

*CP*asymmetries, respectively, and the subscript

*i*refers to the charge of the emitted \(\rho ^i\) meson.

*CP*admixture, the parameters \(\mathcal{C} ^{\pm }\) are not the only measurements of direct

*CP*violation. Considering either the decay where the \(\rho ^\pm \) meson is emitted by the spectator interaction or by the

*W*exchange, we can use one of the following time-integrated flavour-independent asymmetries:

*CP*-mixed \(B^0\!\rightarrow \rho ^\pm \pi ^{\mp }\) intermediate state was first performed at

*B*-factories in 2005 [50, 82]. Introducing the flavour-integrated charge asymmetry:

*CP*-violating (

*CP*-conserving) parameters \(\mathcal{C} \) and \(\mathcal{S} \) (\(\Delta \mathcal{C} \) and \(\Delta \mathcal{S} \)) generate the flavour-dependent asymmetries:

*CP*asymmetries defined in Eq. (72) can be rewritten in terms of the experimental parameters as

*A*stands for \(A^+\), \(A^-\) or \(A_{\rho {\pi }}\)). The world averages of the Q2B parameters derived from the measured \(\mathcal {U}\)and \(\mathcal {I}\)coefficients are listed in Table 18 in Appendix C, together with their corresponding prediction in the \({SU}(2)\) isospin framework. These results are also shown in Figs. 33 and 34.

A reasonable agreement is observed for the parameters related to the \(B^0\rightarrow \rho ^0\pi ^0\) mode. The \({SU}(2)\) isospin fit favours large values for both direct and mixing-induced *CP* asymmetry parameters in the \(B^0\rightarrow \rho ^\pm \pi ^{\mp }\) decays, in contrast with experimental data. The overall \(3\sigma \) discrepancy is mostly reflected by the correlated flavour-independent *CP* asymmetries, \(A^\pm \), or equivalently the charge asymmetry, \(A_{\rho {\pi }}\). While the predicted *CP*-violating asymmetry averages, \(\mathcal{C}\) and \(\mathcal{S}\), are in a reasonable agreement with the experimental data, the *CP*-conserving terms, \(\varDelta \mathcal{C} \) and \(\varDelta \mathcal{S} \), deviate by more than \(2.5\sigma \).

#### 6.3.2 Role of the strong phases in the \(B^0\rightarrow (\rho \pi )^0\) analysis

*CP*asymmetry \(A_{\rho {\pi }}\) (see also the discussion of the Q2B analysis of the charmless \(B^0\rightarrow a_1^\pm \pi ^{\mp }\) decay in Appendix B).

#### 6.3.3 \(B\!\rightarrow \rho {\pi }\) pentagonal analysis

*CP*asymmetries in the charged modes can be determined. The corresponding pulls are given in Table 16 in Appendix C.

## 7 Prospective study

In this section, we discuss how improved measurements of some \(B\rightarrow \pi \pi \), \(B\rightarrow \rho \rho \), \(B\rightarrow \rho \pi \) observables can affect the accuracy on the CKM angle \(\alpha \). So far, the experimental data is statistically limited for all the \(B\rightarrow hh\) charmless modes considered here. Our prospective study aims at identifying the specific decay channels worth measuring more accurately to improve the resolution on \(\alpha \) significantly.

For simplicity, we adopt a systematic approach rather than relying on the expected performance of current or forthcoming flavour experiments such as LHCb or Belle II. We consider the subset of observables related to a specific \(B\rightarrow h^ih^j\) decay, and we determine how much the accuracy on \(\alpha \) is improved if we reduce the uncertainties for this subset of observables by the (arbitrary) factor \(\sqrt{2}\), or if we take the radical limit of setting these uncertainties to zero. The central value of the current world-average measurements and correlation coefficients are kept unchanged. For the neutral modes \(B^0\rightarrow h^+h^-\) and \(B^0\rightarrow h^0h^0\), we distinguish the case of a counting analysis (C) that gives access to the branching ratio only (e.g., the LHCb measurement of \(B^0\rightarrow \rho ^0\rho ^0\) [47]), and the case of flavour-tagged analyses: indeed, a time-integrated (TI) analysis extracts only the direct *CP* asymmetry whereas a time-dependent (TD) analysis yields both the direct and the mixing-induced *CP*-asymmetry parameters (e.g., the LHCb contribution to the study of \(B^0\rightarrow \pi ^+\pi ^-\) [33]).

The global combination of the decay-specific determinations of \(\alpha \) is so far dominated by the \(B\rightarrow \rho \rho \) data that provides a constraint on \(\alpha \) with a relative uncertainty at the level of 5%, i.e., \(\alpha _{\rho \rho }=({92.1}_{-4.9}^{+4.6}){^\circ }\) for the solution near \(90^\circ \) given in Eq. (35). As shown in Table 12, if all other observables remain unchanged, improving the accuracy of the branching ratio of the charged mode \(B^+\rightarrow \rho ^+\rho ^0\) would improve the resolution on \(\alpha \) only marginally, even in the case of a vanishing resolution (indicating that this observable has only a limited impact on the accuracy for \(\alpha \)). Improving the measurements for the neutral modes, in particular the colour-suppressed \(B^0\rightarrow \rho ^0\rho ^0\) decay, has a larger impact, essentially driven by the *CP*-asymmetries parameters. Improving the time-dependent asymmetries in the \(B^0\rightarrow \rho ^0\rho ^0\) is also worth investigating, e.g., in the second run of LHCb data taking. Reducing by \(\sqrt{2}\) all the \(B\rightarrow \rho \rho \) uncertainties would reduce the 68% CL interval for \(\alpha \) by more than 1\(^{\circ }\).

*CP*asymmetry in the colour-suppressed decay \(B^0\rightarrow \pi ^0\pi ^0\). A reduction by \(\sqrt{2}\) of the uncertainty of this single observable would reduce the 68% CL range for \(\alpha \) by \(1^\circ \). A similar improvement of all the measured \(B\rightarrow \pi \pi \) observables would reduce by about \(1.5^\circ \) the uncertainty on \(\alpha \).

68% CL interval for \(\alpha \) from the \({SU}(2)\) isospin analysis of the \(B\rightarrow \rho \rho \) data in the case that the uncertainty of some observables is reduced by a factor \(\sqrt{2}\) (third column) or set to zero (fourth column). The relative gain in resolution with respect to the current measurement is indicated within brackets. Only the preferred solution for \(\alpha \) close to \(90^\circ \) is reported

Decay | Analysis | Improved data | \(\sigma _\mathcal{O}\rightarrow \sigma ^{\scriptscriptstyle WA}_\mathcal{O}/\sqrt{2}\) | \(\sigma _\mathcal{O}\rightarrow 0\) |
---|---|---|---|---|

\({B}^0\!\rightarrow \rho ^+\rho ^- \) | C + TD | \(\mathcal{B} ^{+-},f_L^{+-},\mathcal{C} ^{+-},\mathcal{S} ^{+-}\) | \({{91.8}}_{-4.0}^{+4.4}\ {{(-12\%)}}\) | \({{92.2}}\pm 3.5\)\( {{(-26\%)}}\) |

C | \(\mathcal{B} ^{+-}, f_L^{+-}\) | \({91.8}_{-4.5}^{+4.9}\)\((-1\%)\) | \({92.1}_{-4.7}^{+4.6}\)\((-2\%)\) | |

TD | \(\mathcal{C} ^{+-}, S^{+-}\) | \({92.1}_{-4.3}^{+4.1}\)\((-12\%)\) | \({92.1}_{-3.7}^{+3.5}\)\((-24\%)\) | |

\({B}^0\!\rightarrow \rho ^0\rho ^0 \) | C + TD | \(\mathcal{B} ^{00}, f_L^{00}, \mathcal{C} ^{00}, \mathcal{S} ^{00}\) | \({{91.8}}_{-4.0}^{+4.1}\)\({{(-15\%)}}\) | \({{(91.7)}}\pm 3.1\)\( {{(-35\%)}}\) |

C | \(\mathcal{B} ^{00}, f_L^{00}\) | \({91.8}_{-4.5}^{+4.9}\)\((-1\%)\) | \({92.1}_{-4.7}^{+4.6}\)\((-2\%)\) | |

C + TD | \(\mathcal{C} ^{00}, \mathcal{S} ^{00}\) | \({91.8}_{-4.0}^{+4.1}\)\((-15\%)\) | 91.6 ± 3.1 \({{(-35\%)}}\) | |

\({B}^+\!\rightarrow \rho ^+\rho ^0 \) | C | \(\mathcal{B} ^{+0}\) | \({{92.2}}_{-4.7}^{+4.6}\ {{(-2\%)}}\) | \({{92.5}}_{-4.7}^{+4.3}\)\( {{(-5\%)}}\) |

\({B}^{\pm ,0}\!\rightarrow (\rho \rho )^{\pm ,0} \) | C + TI | \(\mathcal{B} ^{ij}, f_L^{ij}, \mathcal{C} ^{ij}, \mathcal{S} ^{ij}\) | \({{92.0}}_{-3.4}^{+3.3}\ {{(-29\%)}}\) | – |

68% CL interval for \(\alpha \) from the \({SU}(2)\) isospin analysis of the \(B\rightarrow \pi \pi \) data in the case that the uncertainty of some observables is reduced by a factor \(\sqrt{2}\) (third column) or set to zero (fourth column). The relative gain in resolution with respect to the current measurement is indicated within brackets. Only the preferred solution for \(\alpha \) close to \(90^\circ \) is reported

Decay | Analysis | Improved data | \(\sigma _\mathcal{O}\rightarrow \sigma ^{\scriptscriptstyle WA}_\mathcal{O}/\sqrt{2}\) | \(\sigma _\mathcal{O}\rightarrow 0\) |
---|---|---|---|---|

\({B}^0\!\rightarrow \pi ^+\pi ^- \) | C + TD | \(\mathcal{B} ^{+-},\mathcal{C} ^{+-},\mathcal{S} ^{+-}\) | \({{93.0}}\pm {14.0}\ {{(0\%)}}\) | \(93.0\pm 14.0\ {{(0\%)}}\) |

C | \(\mathcal{B} ^{+-}\) | 93.0 ± 14.0 \((0\%)\) | 93.0 ± 14.0 \((0\%)\) | |

TD | \(\mathcal{C} ^{+-}, S^{+-}\) | 93.0 ± 14.0 \((0\%)\) | 93.0 ± 14.0 \((0\%)\) | |

\({B}^0\!\rightarrow \pi ^0\pi ^0 \) | C + TI | \(\mathcal{B} ^{00},\mathcal{C} ^{00}\) | \(92.6\pm 13.0\ {{(-8\%)}}\) | \(\left( {{84.1}}_{+4.5}^{-3.5}\right) \ {{(-70\%})}\) |

C | \(\mathcal{B} ^{00}\) | 93.0 ± 14.0 \((0\%)\) | 93.5 ± 13.5 \((-3.6\%)\) | |

TI | \(\mathcal{C} ^{00}\) | 92.0 ± 13.0 \((-8\%)\) | \(\left( {84.1}_{+6.4}^{-5.6}\right) \)\((-57\%)\) | |

\({{B}^+}{\pi ^+\pi ^0}\) | C | \(\mathcal{B} ^{+0}\) | \(93.0\pm 14.0\ {{(0\%)}}\) | \(93.0\pm 14.0\ (0\%)\) |

\({B}^{\pm ,0}\!\rightarrow (\pi \pi )^{\pm ,0} \) | C + TD or TI | \(\mathcal{B} ^{ij},\mathcal{C} ^{+-},\mathcal{S} ^{+-},\mathcal{C} ^{00}\) | \({{92.5}}\pm {12.5}\ {{(-11\%)}}\) | – |

*p*value for \(\alpha \) when adding the \(S^{00}_{\pi \pi }\) observable under different hypotheses concerning the experimental resolution, and setting the central value as predicted in Eq. (70). In addition to a significant improvement on \(\alpha \), the measurement of \(\mathcal{S} ^{00}_{\pi \pi }\) would reduce the number of mirror solutions in the \(B\rightarrow \pi \pi \) isospin analysis (see the discussion around Eq. (33)).

The same prospective exercise is more delicate for the \(B^0\rightarrow \pi ^+\pi ^-\pi ^0\) system due to the discrepancy between the direct measurement \(\alpha _{\rho {\pi }}=({54.1}^{+7.7}_{-10.3}){^\circ }\cup ({141.8}^{+4.8}_{-5.4}){^\circ }\) (Eq. (40)) and the indirect determination from the global CKM fit (Eq. (2)). New measurements of this decay would certainly aim first at a better understanding of this discrepancy, rather than improving the accuracy on \(\alpha \) extracted from this channel. Assuming the central values given by the individual predictions for each \(\mathcal {U}\)and \(\mathcal {I}\)coefficient, as listed in Table 17, and keeping unchanged the current experimental resolution and correlations, we obtain a mild constraint on \(\alpha \), consistent with \(\alpha _\mathrm{ind}\), namely \(\alpha _{\rho {\pi }}^{\mathrm{fit}~\mathcal {U},\mathcal {I}}=({82}^{-33}_{-48}){^\circ }\). In this case, the reduction of the observable uncertainty by \(\sqrt{2}\) results in a reduction by approximately \(9^\circ \) for the 68% CL interval for \(\alpha \). On the other hand, keeping the current world-average central values and reducing the uncertainty on all the \(\mathcal {U}\)and \(\mathcal {I}\)observables by a factor \(\sqrt{2}\) leads to a more stringent 68% CL interval \(\alpha _{\rho {\pi }}^{\sigma \scriptscriptstyle /\sqrt{2}}=({54.1}^{+5.6}_{-6.9}){^\circ }\cup ({141.0}^{+1.9}_{-1.9}){^\circ }\), which remains difficult to interpret due to the discrepancies already discussed between the current data and our theoretical framework based on isospin symmetry. Additional measurements should hopefully provide a clearer and more consistent picture for the \(B\rightarrow \rho \pi \) sector before discussing any improvement in the extraction of \(\alpha \) from these modes.

## 8 Conclusion

Quark-flavour transitions provide particularly stringent tests of the Standard Model, both through rare decays and \(\mathcal{CP}\)-violating processes. An accurate knowledge of the Cabibbo–Kobayashi–Maskawa matrix is essential for these studies and it requires the combination of many precise constraints. We have focussed on the determination of the \(\alpha \) angle, which can be extracted with a high accuracy from two-body charmless *B*-meson decays extensively studied at *B*-factories and LHCb.

We recalled that this extraction can be done from \(B\rightarrow \pi \pi \), \(B\rightarrow \rho \pi \) and \(B\rightarrow \rho \rho \) decays but it is affected by the presence of penguin contributions. We explained how \({SU}(2)\) isospin symmetry can be used to constrain the structure of hadronic penguin and tree amplitudes, enabling the extraction of \(\alpha \) from branching ratios and \(\mathcal{CP}\) asymmetries. We gave details on the analyses of \(B\rightarrow \pi \pi \), \(B\rightarrow \rho \pi \) and \(B\rightarrow \rho \rho \) systems separately, before combining these results to reach an accuracy around 4\(^\circ \) on the direct determination of \(\alpha \). The \(B\rightarrow \pi \pi \) and \(B\rightarrow \rho \rho \) systems dominate the combination and they favour solutions in good agreement with the indirect determination of \(\alpha \) from a global CKM fit analysis, Eq. (2), it is not the case for the \(B\rightarrow \rho \pi \) system which favours different ranges of values for \(\alpha \) with a discrepancy at the level of \(3\sigma \) compared to the indirect determination. The combination of the three channels is dominated by \(B\rightarrow \rho \rho \), and to a lesser extent \(B\rightarrow \pi \pi \), resulting in the 68% CL confidence intervals given in Eq. (1).

We have then studied several uncertainties that may affect this extraction. We have tested the hypotheses underlying the \({SU}(2)\) isospin analysis: setting the \(\Delta I={3} / {2}\) electroweak penguins to zero, neglecting the difference of light-quark masses generating \(\pi ^0\)–\(\eta \)–\(\eta '\) mixing, setting the \(\rho \) width to zero to cancel \(\Delta I=1\) contributions thanks to Bose–Einstein symmetry. These effects may shift the central value of \(\alpha _\mathrm{dir}\) by around \(2^\circ \), while keeping the uncertainty around \(4^\circ \) to \(5^\circ \), thus remaining within the statistical uncertainty quoted in Eq. (1). In addition, we have discussed a few aspects concerning the statistical treatment used in order to extract the *p* value, comparing two statistical approaches to test the impact of hadronic nuisance parameters on coverage. The approach based on Wilks’ theorem and mainly used here proves to be more conservative than the bootstrap method for \(B\rightarrow \pi \pi \) and \(B\rightarrow \rho \rho \), but less conservative in the case of \(B\rightarrow \rho \pi \). The comparison of the coverage properties of the two approaches leads to a further uncertainty of around \(1^\circ \). We stress that this uncertainty is only attached to the combination of the three direct determinations of \(\alpha \) with the current data: it is likely to be reduced if one combines the direct determination of \(\alpha \) with other observables, leading to a more accurate determination of \(\alpha \) (which is in particular the typical case of global CKM fits).

Assuming the validity of the Standard Model and taking as an input the indirect determination of \(\alpha \) from the global CKM fit, we used the observables in \(B\rightarrow \pi \pi \), \(B\rightarrow \rho \pi \) and \(B\rightarrow \rho \rho \) decays to extract information on ratios of hadronic amplitudes (penguin-to-tree and colour-suppressed), finding results in broad agreement with expectations from QCD factorisation, apart from the ratio of the colour-suppressed to colour-allowed tree \(\pi \pi \) contributions: indeed both the phase and the modulus of this ratio do not agree well with theoretical expectations. It would be interesting to widen this discussion and see how various theoretical approaches to non-leptonic two-body decays can reproduce the patterns of hadronic amplitudes that we have extracted from the data.

Under the same hypotheses, we have also performed the indirect determinations of observables of interest (using all the other measurements available), comparing the pulls for those already observed and predicting the values of the remaining ones. The compatibility between direct measurements and indirect determinations is very good for the observables in the \(B\rightarrow \pi \pi \) and \(B\rightarrow \rho \rho \) systems, whereas we could identify a subset of \(B\rightarrow \rho \pi \) observables likely to be responsible for the discrepancies observed with respect to the Standard Model expectations in these modes. Among many other quantities, we have predicted the yet-to-be-measured mixing-induced \(\mathcal{CP}\) asymmetry in the \(B^0\!\rightarrow \pi ^0\pi ^0\) decay; see Eq. (33).

Finally, we have performed a prospective study to analyse how improved measurements for some subsets of observables can improve the uncertainty of \(\alpha \). In particular, we have noticed that an improved accuracy for the time-dependent asymmetries in \(B^0\rightarrow \rho ^0\rho ^0\) and the measurement of \(\mathcal{S} ^{00}_{\pi \pi }\) would reduce the uncertainty on \(\alpha \) in a noticeable way.

We have seen that the extraction of \(\alpha \) is now possible to a high accuracy, using many different channels and experimental sources. It would be very interesting to measure the remaining observables that we can accurately predict using the data already available. The current accuracy reached by \(\alpha \) makes it a particularly useful constraint both for the Standard Model and for searches of New Physics, although the theoretical computations for these transitions remain challenging. Some of these channels will be improved by the LHCb experiment. In addition, the advent of the Belle-II experiment will certainly lead to new and improved measurements for a large set of branching ratios and \(\mathcal{CP}\) asymmetries, providing an opportunity to understand better the results obtained for the \(B\rightarrow \rho \pi \) system and allowing us to extract hadronic parameters with a higher accuracy. Both avenues should be highly beneficial for the upcoming studies of flavour physics in the quark sector.

## Footnotes

- 1.
Considering only the valence quarks, the isospin shift \(\varDelta I\) can only take the values \({3} / {2}\) and \({1} / {2}\) in the \(\bar{b}\rightarrow (\bar{u} u\bar{d})\) transition. Tree and strong penguin topologies correspond to \(\varDelta I={1} / {2}\), whereas electroweak penguins contain both \(\varDelta I={1} / {2}\) and \(\varDelta I={3} / {2}\) contributions. For completeness, the possible \(\varDelta I={5} / {2}\) contribution due to long-distance rescattering effects [20, 21] is also reported in Table 1. This contribution, suppressed by a factor \(\alpha _{em}\sim 1/127\), will be neglected until Sect. 4.1.2.

- 2.
Unless otherwise stated, the subscript \(h_1=h_2\) is dropped and \(A^{ij}\) implicitly refers to the symmetrised amplitude in the case of a decay into two mesons of same type.

- 3.
The

*CP*asymmetry for the charged mode explicitly vanishes in the parametrisation of the amplitudes based on exact \({SU}(2)\) isospin symmetry. Therefore the corresponding observable is not included here. The current experimental measurement, fully consistent with the null hypothesis, would only affect the minimal value \(\chi ^2 _\mathrm{min}\) of the fit, but not the metrology of the parameters. This additional observable will be useful to study isospin-breaking effects in Sect. 4.1. - 4.
This model is a valid representation of the amplitude system for \(\alpha \ne 0\). When \(\alpha \) vanishes exactly, the constraints \(\bar{a}=a\) and \(\bar{\alpha }=0\) must be added to the system in order to satisfy the equality of the

*CP*-conjugate amplitudes \(A^{+-}={\bar{A}^{+-}}\). - 5.
As can be seen from Table 4, the longitudinal polarisation \(f_L\) is always used as an input in combination with branching ratios \(\mathcal{B}\), and both are defined as

*CP*-averaged quantities. - 6.
With this convention the \(A^i\) (\({\bar{A}}^i\)) amplitude carries a superscript referring to the electric charge of the \(\rho ^i\) meson in both \(B^0\) and \({\bar{B}}^0\) decays, i.e., \(A^{+}=A^{+-}\), \(A^{-}=A^{-+}\) and \({\bar{A}}^{+}={\bar{A}}^{-+}\), \({\bar{A}}^{-}={\bar{A}}^{+-}\) where the \(A^{ij}\) (\({\bar{A}}^{ij}\)) amplitude is defined in Sect. 2.1.

- 7.
While eliminating mirror solutions, this constraint does not distort the

*p*value for \(\alpha \) in the vicinity of the solution compatible with the indirect determination. We stress that we do not use such a constraint anywhere else in this article. - 8.
This computation is a CPU-consuming exercise which was carried on at the CC-IN2P3 computing farm (scoring \(49 \cdot 10^3\ \mathrm {HS06 \cdot hours}\), i.e. approximatively 200 CPUs over 1 day).

- 9.
For \(B\rightarrow \pi \pi \), the bootstrap approach (second row in Table 11) provides two peaks that are just separated at 68% CL, as can be seen in Fig. 17. However, in Table 11, a single range for \(\alpha \) is given for this method, which is obtained by merging the intervals corresponding to the two peaks. Even though this range is not the exact outcome of the bootstrap analysis for \(B\rightarrow \pi \pi \) at 68% CL, it allows us to perform more meaningful comparisons with the extractions of \(\alpha \) using the Wilks-based approach, for which the two peaks are not distinguished at this level of significance and a single range is obtained for \(\alpha \).

- 10.
We identify this ratio of “tree” amplitudes to a colour-suppressed ratio from the analysis of tree diagram topologies. In the \(\mathcal C\)-convention used here, this identification is correct only in the limit of vanishing

*u*- and*c*-penguin topological amplitudes, as discussed in Sect. 2.1. Moreover, the*W*-exchange topology that provides a colour-suppressed contribution that can be absorbed in the tree amplitudes for both \(B\rightarrow h^+h^-\) and \(B\rightarrow h^0h^0\) transitions are neglected here. A similar statement holds for the penguin-to-tree ratio discussed previously.

## Notes

### Acknowledgements

We would like to thank all our collaborators from the CKMfitter group for many useful discussions on the statistical issues covered in this article, and in particular K. Trabelsi and L. Vale Silva for a careful reading of the manuscript. SDG acknowledges partial support from Contract FPA2014-61478-EXP. This project has received funding from the European Union’s Horizon 2020 research and innovation programme under Grant agreements nos. 690575, 674896 and 692194.

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