# Exploring \(B\rightarrow \pi \pi , \pi K\) decays at the high-precision frontier

## Abstract

The \(B\rightarrow \pi \pi ,\pi K\) system offers a powerful laboratory to probe strong and weak interactions. Using the isospin symmetry, we determine hadronic \(B\rightarrow \pi \pi \) parameters from data where new measurements of direct CP violation in \(B^0_d\rightarrow \pi ^0\pi ^0\) resolve a discrete ambiguity. With the help of the *SU*(3) flavour symmetry, the \(B\rightarrow \pi \pi \) parameters can be converted into their \(B\rightarrow \pi K\) counterparts, thereby allowing us to make predictions of observables. A particularly interesting decay is \(B^0_d\rightarrow \pi ^0 K_\mathrm{S}\) as it exhibits mixing-induced CP violation. Using an isospin relation, complemented with a robust *SU*(3) input, we calculate correlations between the direct and mixing-induced CP asymmetries of \(B^0_d\rightarrow \pi ^0 K_\mathrm{S}\), which are the theoretically cleanest \(B\rightarrow \pi K\) probes. Interestingly, they show tensions with respect to the Standard Model. Should this \(B\rightarrow \pi K\) puzzle originate from New Physics, electroweak penguins offer an attractive scenario for new particles to enter. We present a strategy to determine the parameters characterising these topologies and obtain the state-of-the-art picture from current data. In the future, this method will allow us to reveal the \(B\rightarrow \pi K\) dynamics and to obtain insights into the electroweak penguin sector with unprecedented precision.

## 1 Introduction

*B*-meson system has been an exciting playground for theorists and experimentalists to test the flavour- and CP-violating sector of the Standard Model (SM) [1], which is encoded in the Cabibbo–Kobayashi–Maskawa (CKM) matrix [2, 3]. After an era of pioneering measurements at the

*B*factories with the BaBar and Belle experiments as well as the Tevatron, the experimental stage is currently governed by the Large Hadron Collider (LHC) with its dedicated

*B*-decay experiment LHCb. In the near future, Belle II at the KEK Super

*B*Factory will join these explorations, allowing for exciting new opportunities [4, 5], which will be complemented by the LHCb upgrade [6].

In this endeavour, \(B\rightarrow \pi K\) channels are a particularly interesting decay class (for a selection of original references, see Refs. [7, 8, 9, 10, 11, 12, 13, 14, 15, 16]). These modes are dominated by QCD penguin topologies as the tree contributions are strongly suppressed by the tiny CKM matrix element \(|V_{ub}|\). In the case of \(B^+\rightarrow \pi ^0K^+\) and \(B^0_d\rightarrow \pi ^0K^0\), colour-allowed electroweak (EW) penguin topologies enter at the same level as colour-allowed tree amplitudes, contributing \(\mathcal{O}(10\%)\) to the decay amplitudes. As an illustration, we show the decay topologies that contribute to the \(B_d^0 \rightarrow \pi ^0 K^0\) channel in Fig. 1. Since New Physics (NP) may well enter through EW penguins [17, 18, 19, 20, 21, 22], these \(B\rightarrow \pi K\) modes are especially promising. Examples of specific models are given by NP scenarios with extra \(Z'\) bosons [19, 20, 21, 22], which are receiving a lot of attention in view of anomalies in rare *B*-decay data (see Ref. [23] and references therein).

In general, NP contributions are associated with new sources of CP violation that can be probed through CP-violating observables. In this respect, \(B^0_d\rightarrow \pi ^0 K_\mathrm{S}\) is a particularly interesting decay as it is the only \(B\rightarrow \pi K\) mode exhibiting mixing-induced CP violation [24, 25]. This phenomenon emerges from interference between \(B^0_d\)–\(\bar{B}^0_d\) mixing and decay processes of \(B^0_d\) and \(\bar{B}^0_d\) mesons into the \(\pi ^0K_\mathrm{S}\) final state. As we will demonstrate in this paper, the mixing-induced CP asymmetry of \(B^0_d\rightarrow \pi ^0 K_\mathrm{S}\) plays an outstanding role for testing the SM with the \(B\rightarrow \pi K\) system. This paper complements Refs. [26, 27], where we gave a compact presentation of the main results discussed in detail below.

Analyses of non-leptonic *B* decays are in general very challenging due to hadronic matrix elements of four-quark operators entering the corresponding low-energy effective Hamiltonians. In the case of the \(B\rightarrow \pi K\) decays, the flavour symmetries of strong interactions imply relations between the \(B\rightarrow \pi K\) amplitudes and those of the \(B\rightarrow \pi \pi , KK\) systems, which allow us to eliminate the hadronic amplitudes or to determine them from experimental data for the latter decays.

In our analysis, we aim at keeping the theoretical assumptions about strong interactions as minimal as possible, and shall use results from QCD factorization (QCDF) to include *SU*(3)-breaking corrections [13]. A central role is played by an isospin relation between amplitudes of neutral \(B\rightarrow \pi K\) decays. Complementing it with an *SU*(3) input to just fix a certain normalisation, this relation allows us to calculate a correlation between the direct and mixing-induced CP asymmetries of the \(B^0_d\rightarrow \pi ^0 K_\mathrm{S}\) mode [25]. We find an intriguing tension with the SM, implying either that the current central values of the relevant observables will change in the future or signals of NP contributions which involve in particular new sources of CP violation.

In order to clarify this situation and to reveal the dynamics underlying the EW penguin contributions of the \(B\rightarrow \pi K\) decays, we develop a new strategy to determine the corresponding parameters. It utilises again the isospin relation between the neutral \(B\rightarrow \pi K\) decays as well as its counterpart for the charged modes. As the experimental picture is sharper for the latter case, we perform a detailed analysis of these modes, resulting in the currently most stringent constraints on the EW penguin parameters. In the future, these quantities can be determined with the help of measurements of the mixing-induced CP asymmetry of \(B^0_d\rightarrow \pi ^0 K_\mathrm{S}\). We illustrate the promising potential of this method by discussing a variety of scenarios. The Belle II experiment offers exciting prospects for future measurements of the CP asymmetries in \(B^0_d\rightarrow \pi ^0 K_\mathrm{S}\) [4, 5], which will allow us to enter a new territory in terms of precision. Concerning \(B\rightarrow \pi K, \pi \pi \) modes with charged pions and kaons in the final states, the LHCb upgrade will also have an important impact for the implementation of the new strategy.

The outline of this paper is as follows: in Sect. 2, we discuss the hadronic parameters following from the current \(B\rightarrow \pi \pi \) data, where an important new ingredient is given by measurements of direct CP violation in \(B^0_d\rightarrow \pi ^0\pi ^0\). Having these parameters at hand, we apply the *SU*(3) flavour symmetry to calculate their \(B\rightarrow \pi K\) counterparts in Sect. 3, exploring also the impact of *SU*(3)-breaking corrections. In Sect. 4, we utilize the isospin symmetry to calculate correlations between the CP asymmetries of \(B^0_d\rightarrow \pi ^0K_\mathrm{S}\) and discuss the intriguing picture following from the current measurements. In Sect. 5, we present the details of the new method to determine the EW penguin parameters, apply it to the current data and demonstrate that we can match the expected experimental precision in the era of Belle II and the LHC upgrade(s) with the theoretical uncertainties. Finally, we summarize our conclusions in Sect. 6.

## 2 The \(\varvec{B\rightarrow \pi \pi }\) system

### 2.1 Amplitude structure

*t*and

*q*quarks, respectively, and introduce

### 2.2 Observables

*B*-meson decays, we introduce direct CP asymmetries as

Overview of the currently available \(B \rightarrow \pi \pi \) measurements. Note that the branching ratios are actually CP-averaged quantities

Mode | \({\mathcal Br}[10^{-6}]\) | \(A_\text {CP}^f\) | \(S_\text {CP}^f\) | Ref. |
---|---|---|---|---|

\(B^0_d\rightarrow \pi ^+ \pi ^-\) | \(5.12\pm 0.19 \) | \(0.31 \pm 0.05\) | \(-\,0.66 \pm 0.06\) | |

\(B^0_d\rightarrow \pi ^0 \pi ^0\) | \(1.59\pm 0.18\) | \(0.33\pm 0.22\) | – | |

\(B^+\rightarrow \pi ^+ \pi ^0\) | \( 5.5\pm 0.4 \) | \(0.03\pm 0.04 \) | – | [29] |

### 2.3 Hadronic parameters

*x*and \(\Delta \), we use the ratios \(R_{+-}\) and \(R_{00}\) defined in Eq. (21) and (22). Using the CP-averaged branching ratios in Table 1 and \(\tau _{B^+}/\tau _{B_d^0}=1.076 \pm 0.004\) [29], we obtain

*x*and \(\Delta \) can then be obtained analytically using Eqs. (24) and (25) [17]:

*x*and \(\Delta \) can be resolved through the direct CP asymmetry of the \(B^0_d \rightarrow \pi ^0\pi ^0\) channel, resulting in

*SU*(3) flavour symmetry and the \(B^\pm \rightarrow \pi ^0K^\pm \) channel, where the unphysical solution would result in a large direct CP asymmetry that is excluded by experimental data. The clean new constraint following from the \(A_\mathrm{CP}^{\pi ^0\pi ^0}\) is consistent with these considerations.

## 3 The \(\varvec{B \rightarrow \pi K}\) system

### 3.1 Amplitude structure

*SU*(3) flavour symmetry to the hadronic matrix elements, we obtain the following result [11, 24, 49]:

*SU*(3) limit. The smallness of this phase is actually a model-independent feature, as noted in Refs. [48, 49]. In the remainder of this paper, we use \(\omega =0^\circ \). Making numerical studies, we find that values of \(\omega \) up to \(10^\circ \) would not have an impact on our analysis. The parameter \(R_q\) describes

*SU*(3)-breaking effects. Following Ref. [25], we allow for corrections of \(30\%\) by taking \(R_q = 1.0\pm 0.3\). As a theory benchmark scenario, we assume

*q*and \(\phi \), are related by the isospin symmetry as

*U*-spin symmetry of strong interactions from data for the \(B^+\rightarrow K^+ \bar{K}^0\) decay [17, 45]. The most recent analysis gives the following result [42]:

### 3.2 Determination of the hadronic parameters

The \(B \rightarrow \pi K\) system is related to the \(B \rightarrow \pi \pi \) modes through the *SU*(3) flavour symmetry of strong interactions, which allows us to convert the \(B\rightarrow \pi \pi \) parameters determined in Sect. 2.3 into their \(B\rightarrow \pi K\) counterparts [17, 18]. As EW penguins play a negligible role in the \(B\rightarrow \pi \pi \) system, the resulting hadronic \(B\rightarrow \pi K\) parameters are essentially not affected by possible NP contributions to the EW penguin sector.

A complication arises from exchange (*E*) and penguin-annihilation (*PA*) topologies, which are present in the \(B \rightarrow \pi \pi \) system but do not contribute to the \(B \rightarrow \pi K\) modes. These contributions are dynamically suppressed and expected to play a minor role. Using data for \(B_s^0 \rightarrow \pi ^-\pi ^+\) and \(B_d^0\rightarrow K^- K^+\) decays [50], which exclusively emerge from such topologies, and the *SU*(3) flavour symmetry, the *E* and *PA* contributions can be constrained. As discussed in detail in Ref. [42], this results in effects at the few percent level of the overall \(B\rightarrow \pi K\) amplitudes. In the future, measurements of CP asymmetries in the \(B_s^0 \rightarrow \pi ^-\pi ^+\) and \(B_d^0\rightarrow K^- K^+\) decays will result in more precise determinations of these effects [42], allowing us to take these corrections into account.

*SU*(3) flavour symmetry, we have

*SU*(3)-breaking corrections if contributions from the colour-suppressed tree topology are neglected. The \(B^+\rightarrow \pi ^+ \pi ^0\) decay allows us to determine the \(|\mathcal {T}+\mathcal {C}|\) amplitude, which can be converted into its \(B\rightarrow \pi K\) counterpart using

*SU*(3)-breaking effects. We can write this quantity as

In Fig. 3a, we compare this determination with \(r_\mathrm{c}^{\pi \pi }\) and \(\delta _\mathrm{c}^{\pi \pi }\) in Eq. (38). Here the latter parameters give the red ellipse, whereas the blue circle follows from Eq. (58). In Fig. 3b we zoom in on the red ellipse and show the precision that can be obtained for \(r_\mathrm{c}^{\pi \pi }\) and \(\delta _\mathrm{c}^{\pi \pi }\) in the era of Belle II and the LHCb upgrade, using the expected uncertainty for the \(B\rightarrow \pi \pi \) observables [4, 5], as well as \(\gamma = (70 \pm 1)^\circ \) [4, 5, 6] and \(\phi _d = (43.2 \pm 0.6)^\circ \) [46]. We have shifted the ellipse to get agreement with the blue contour, and observe that both constraints have actually similar precision. In order to guide the eye, we have also added the dashed red ellipse which corresponds to the current data.

*SU*(3)-breaking corrections within the current experimental precision. In order to quantify this feature, we reverse Eq. (56) and use it to determine \(R_{T+C}\) from the value of \(r_\mathrm{c}\) in Eq. (38), yielding

*SU*(3)-breaking corrections as large as \(100\%\) of the factorizable effects in Eq. (55) were considered.

*SU*(3)-breaking effects. Considering non-factorizable corrections of up to \(20 \%\) through

*r*and \(\delta \) which enter the amplitude of the \(B_d^0\rightarrow \pi ^-K^+\) channel. They are related to their \({{B ^0_d \rightarrow \pi ^-\pi ^+}} \) counterparts through the

*SU*(3) relation

*SU*(3)-breaking corrections. Allowing again for such effects of \(20\%\) through

*U*-spin partner of the \(B^0_d\rightarrow \pi ^-K^+\) decay [42]:

*SU*(3)-breaking effects or contributions from exchange and penguin-annihilation topologies.

### 3.3 Observables and dynamics

#### 3.3.1 Branching ratios

*q*and \(\phi \), while

*R*only involves colour-suppressed EW penguins. Using the expressions in Eq. (42), we can express these ratios in terms of the hadronic parameters introduced above.

*r*and \(r_\mathrm{c}\) are small parameters of \(\mathcal {O}(0.1)\), and make expansions in terms of \(r_\mathrm{(c)}\), which yields

#### 3.3.2 Colour-suppressed electroweak penguins

*R*, we obtain

Overview of the current measurements in the \(B\rightarrow \pi K\) system [29]

Mode | \({\mathcal Br}[10^{-6}]\) | \(A_\text {CP}\) | \(S_\text {CP}\) |
---|---|---|---|

\(\bar{B}^0_d\rightarrow \pi ^+ K^-\) | \(19.6\pm 0.5\) | \( -\,0.082 \pm 0.006\) | – |

\(\bar{B}^0_d\rightarrow \pi ^0 \bar{K}^0\) | \(9.9\pm 0.5\) | \( 0.00 \pm 0.13\) | \(0.58 \pm 0.17\) |

\(B^+\rightarrow \pi ^+ K_S\) | \( 23.7\pm 0.8\) | \(-\,0.017\pm 0.016 \) | – |

\(B^+\rightarrow \pi ^0 K^+ \) | \( 12.9\pm 0.5\) | \(0.037\pm 0.021 \) | – |

*R*and \(A_{\text {CP}}^{\pi ^-K^+}\) allow the determination of the colour-suppressed EW penguin contributions \(\tilde{a}_C\) and \(\tilde{a}_S\). Neglecting sub-leading terms, we find

*q*in Eq. (40), we obtain

Input and hadronic \(B\rightarrow \pi K\) parameters obtained from the current \(B\rightarrow \pi \pi \) data, including uncertainties from *SU*(3)-breaking effects as discussed in Sect. 3.2

Parameter | Value |
---|---|

\(\gamma \) | \((70 \pm 7)^\circ \) |

\(\phi _d\) | \((43.2 \pm 1.8)^\circ \) |

| \(0.09 \pm 0.03 \) |

\(\delta \) | \((28.6\pm 21.4)^\circ \) |

\(r_\mathrm{c}\) | \(0.17 \pm 0.05\) |

\(\delta _\mathrm{c} \) | \((1.9 \pm 21.4)^\circ \) |

\(\rho _\mathrm{c}\) | \(0.03 \pm 0.01 \) |

\(\theta _\mathrm{c} \) | \((2.6\pm 4.6)^\circ \) |

#### 3.3.3 Direct CP asymmetries and sum rules

### 3.4 Vanishing CP violation in the electroweak penguin sector

*q*. In view of the discussion in Sect. 3.3.2, we neglect contributions from colour-suppressed EW penguin topologies. The observables take then the following forms [17]:

^{1}Interference between \(B_d^0\)–\(\bar{B}_d^0\) mixing and decays of \(B_d^0\) or \(\bar{B}_d^0\) mesons into the \(\pi ^0 K_\mathrm{S}\) final state gives rise to a mixing-induced CP asymmetry, which satisfies the following general relation [25, 58]:

*q*in Eq. (40) and the hadronic parameters in Table 3 yields

SM predictions of the \(B \rightarrow \pi K\) observables and comparison with the current experimental data

Observable | SM Prediction | Experiment |
---|---|---|

| \(0.93 \pm 0.03\) | \(0.89 \pm 0.04\) |

\(R_\mathrm{n}\) | \(1.13 \pm 0.10\) | \(0.99 \pm 0.06\) |

\(R_\mathrm{c}\) | \(1.11 \pm 0.08\) | \(1.09 \pm 0.06\) |

\(A_\text {CP}^{\pi ^\pm K^\mp }\) | \(-\,0.085 \pm 0.064\) | \(-\,0.082 \pm 0.006\) |

\(A_\text {CP}^{\pi ^\pm K^0}\) | \(0.003 \pm 0.005\) | \(-\,0.017 \pm 0.016\) |

\(A_\text {CP}^{\pi ^0K^\pm }\) | \(-\,0.007 \pm 0.11\) | \(0.037 \pm 0.021\) |

\(A_\text {CP}^{\pi ^0 K_S}\) | \(-\,0.07 \pm 0.15\) | \(0.00 \pm 0.13\) |

\(S_\text {CP}^{\pi ^0 K_S}\) | \(0.81 \pm 0.07\) | \(0.58 \pm 0.17\) |

\(\mathcal {B}r(B_d^0\rightarrow \pi ^- K^+)\times 10^6\) | \(20.6 \pm 0.7\) | \(19.6 \pm 0.5\) |

\({\mathcal {B}r(B^+ \rightarrow \pi ^+ K^0)\times 10^6}\) | Normalization | \(23.7 \pm 0.8\) |

\(\mathcal {B}r(B^+\rightarrow \pi ^0 K^+)\times 10^6\) | \(13.1 \pm 1.0\) | \(12.9 \pm 0.5\) |

\(\mathcal {B}r(B^0_d\rightarrow \pi ^0 K^0)\times 10^6\) | \(9.1 \pm 0.9\) | \( 9.9 \pm 0.5\) |

In Table 4, we summarize the SM predictions and experimental data for the various \(B \rightarrow \pi K\) observables. The errors are dominated by the currently large uncertainty of the *SU*(3)-breaking parameter \(R_q\). For the branching ratios, we use the measured branching ratio of \(B^+ \rightarrow \pi ^+ K^0\) to fix the normalization \(|P'|\). We observe that all predictions are well within the current experimental measurements. The excellent agreement of *R* and \(A_{\text {CP}}^{\pi ^-K^+}\) with the measurements reflects the smallness of the colour-suppressed EW penguin contributions found in Sect. 3.3.2, where these observables were used to determine the colour-suppressed EW penguin parameters. The largest deviation arises in the ratio \(R_\mathrm{n}\), where there is a tension of a bit more than \(1\sigma \) significance.

## 4 Correlations between CP asymmetries of \(\varvec{B^0_d\rightarrow \pi ^0 K_\mathrm{S}}\)

### 4.1 Preliminaries

The mixing-induced CP asymmetry of the \(B_d^0\rightarrow \pi ^0K_\mathrm{S}\) channel is a particularly interesting probe for testing the SM. In the previous section, we have used hadronic parameters which were determined from \(B\rightarrow \pi \pi \) data by means of the *SU*(3) flavour symmetry, resulting in the picture shown in Fig. 4. Interestingly, we can obtain a much more precise correlation in the \(A_\mathrm{CP}^{\pi ^0 K_\mathrm{S}}\)–\(S_\mathrm{CP}^{\pi ^0 K_\mathrm{S}}\) plane, as was pointed out first in [25]. In the following, we discuss this determination in more detail and update the analysis of [25]. In the next section, we also add a new element to the discussion, which strengthens the tension within the SM.

*SU*(3) flavour symmetry [9]:

*q*and \(\phi \), in particular also for the SM case as described by Eq. (40). Having \(\phi _{00}\) at hand, the expression in Eq. (101) allows us to calculate a contour in the \(A_\mathrm{CP}^{\pi ^0 K_\mathrm{S}}\)–\(S_\mathrm{CP}^{\pi ^0 K_\mathrm{S}}\) plane. The corresponding correlation relies only on the clean isospin relations in Eqs. (111) and (112) and the

*SU*(3) input given by \(R_{T+C}\) in Eq. (116), which is a very robust parameter as discussed in Sect. 3.2.

*q*and \(\phi \), such as in the SM which we consider in the following discussion, the amplitudes \(A_{3/2}\) and \(\bar{A}_{3/2}\) are fixed. Using the direct asymmetries \( A_{\text {CP}}^{\pi ^0 K_{\text {S}}} \) and \(A_{\text {CP}}^{\pi ^- K^+}\) taking the forms

Let us illustrate this method by taking \(A_{\text {CP}}^{\pi ^0 K_{\text {S}}}\) from the sum rule in Eq. (93) and central values of the measured observables. The four orientations of the resulting triangles are shown in Fig. 6, and correspond to the angles \(\phi _{00}\) and mixing-induced CP asymmetries \(S_{\text {CP}}^{\pi ^0 K_{\text {S}}}\) given in Table 5. The triangles are drawn in arbitrary units, since only the shape of the triangles is important for the determination of \(\phi _{00}\).

If we now vary the direct CP asymmetry of \(B_d^0 \rightarrow \pi ^0 K_S\), we obtain a correlation between \(S_{\text {CP}}^{\pi ^0 K_{\text {S}}}\) and \(A_{\text {CP}}^{\pi ^0 K_{\text {S}}}\) [17, 25]. This results in the four contours in the \(A_{\text {CP}}^{\pi ^0 K_{\text {S}}}\)–\(S_{\text {CP}}^{\pi ^0 K_{\text {S}}}\) plane shown in Fig. 7a, b, where we have also taken the experimental errors and the uncertainties of \(R_{T+C}\) and \(R_q\) into account. The four contours correspond to the configurations in Fig. 6 where \(\phi _{00}\) is labelled with the same colour. We have also included the current experimental data point for the CP asymmetries from Table 2, and the vertical band refers to the sum rule value of \(A_{\text {CP}}^{\pi ^0 K_{\text {S}}}\) in Eq. (93). In addition, the narrow bands illustrate a future scenario including only the expected theory uncertainties for \(R_q\) and \(R_{T+C}\) in Eqs. (41) and Eq. (62), respectively.

### 4.2 Discrete ambiguities

*r*, \(\delta \) and \(r_\mathrm{c}\) can be expressed in terms of the strong phase \(\delta _\mathrm{c}\) using the ratio \(R_\mathrm{n}\) and \(A_\mathrm{CP}^{\pi ^-K^+}\). In addition, we use \(\mathcal {B}r(B_d^0\rightarrow \pi ^-K^+)\) to fix the normalization \(|P'|\) which then enters \(r_\mathrm{c}\) via

*SU*(3) flavour symmetry.

The angles \(\phi _{00}\) and the corresponding mixing-induced CP asymmetries \(S_{\text {CP}}^{\pi ^0 K_{\text {S}}}\) following from the triangle construction for current data using \(A_{\text {CP}}^{\pi ^0 K_{\text {S}}}\) in Eq. (93)

\(\phi _{00}\) | \(S_{\text {CP}}^{\pi ^0 K_{\text {S}}}\) | \(\phi _{00}\) | \(S_{\text {CP}}^{\pi ^0 K_{\text {S}}}\) |
---|---|---|---|

\(-49.8^\circ \) | 0.989 | \(-\,22.9^\circ \) | 0.903 |

\(128.9^\circ \) | \(-\) 0.988 | \(145.5^\circ \) | \(-\) 0.967 |

### 4.3 How to resolve the \({B \rightarrow \pi K}\) puzzle?

In Fig. 8, we summarize the intriguing picture following from the isospin triangles, showing only the contour remaining once the constraints from Sect. 4.2 have been applied. In comparison with Fig. 4, we obtain a much cleaner picture, requiring only *SU*(3) input for the parameters \(R_{T+C}\) and \(R_q\), which are very robust as discussed in Sect. 3. On the other hand, Fig. 4 relies on the *SU*(3) flavour symmetry for the determination of the hadronic \(B\rightarrow \pi K\) parameters from their \(B\rightarrow \pi \pi \) counterparts.

The observed discrepancy arising within the SM between the isospin relation and the measured mixing-induced CP asymmetries is at the \(2\sigma \) level. However, as pointed out previously, also the isospin relation, which does not depend on the mixing-induced CP asymmetry of \(B_d^0\rightarrow \pi ^0 K_\mathrm{S}\), exhibits a discrepancy with the SM at the \(1\sigma \) level.

On the other hand, the puzzling situation may also be a signal of NP effects in the EW penguin sector, thereby affecting the values of *q* and \(\phi \). A particularly exciting aspect is the sensitivity to new sources of CP violation.

## 5 Extracting the electroweak penguin parameters

### 5.1 Preliminaries

In the previous section, we have used the isospin relations in Eqs. (111) and (112) to calculate a correlation between the direct and mixing-induced CP asymmetries of the \(B^0_d\rightarrow \pi ^0K_\mathrm{S}\) channel, resulting in an intriguing picture for the current experimental data that may be an indication of a modified EW penguin sector. In view of this result and to test the corresponding SM sector, it would be very interesting to determine the EW penguin parameters *q* and \(\phi \) from experimental data and to compare the corresponding results with the SM prediction (see Eq. (40)). The parameter \(R_q\) is then only needed for the SM prediction of *q* while the CP-violating phase \(\phi \), which vanishes in the SM, may give a “smoking-gun” signal of new sources of CP violation.

*q*plane. In order to convert the given value of \(|A_{3/2}|=|\bar{A}_{3/2}|\) into the parameter

*N*, we use again the

*SU*(3) relation in Eq. (116). For the current charged \(B\rightarrow \pi K\) decay data, we arrive at the contours shown in Fig. 11a. As was the case in Sect. 4, we have a four-fold ambiguity for \(\Delta \phi _{3/2}\) since the triangles can be flipped around the \(A_{3/2}\) and \(\bar{A}_{3/2}\) axes. This is represented by the four different colours for the contours in Fig. 11a. Moreover, for every value of \(\Delta \phi _{3/2}\), there are two contours in the \(\phi \)–

*q*plane due to solving a quadratic equation, giving two contours of every colour and eight contours in total. We find discontinuities of the contours around \(q\sim 1\), \(\phi \sim 70^\circ \), because \(|A_{3/2}|\) cannot become arbitrarily small as then the amplitudes in Eq. (123) cannot form triangles anymore.

*R*via Eq. (81). Contrary to the SM case discussed above, now the theoretically allowed \(\phi _{0+}\) depends on

*q*and \(\phi \). At the same time, the \(\phi _{0+}\) obtained from the triangle construction also depends on \(\phi \). In Fig. 11, we show this angle for each of the eight branches of the triangle determinations in the same colour. In addition, in grey we show the theoretically allowed values of \(\phi _{0+}\) as a function of \(\phi \), using the exact expression but neglecting colour-suppressed EW penguin contributions. For this theoretical prediction, we use the

*q*as a function of \(\phi \) from the associated triangle contour. This implies that each of the eight triangle contours has a different theoretical prediction for \(\phi _{0+}\) as function of \(\phi \). We observe that one of the contours in Fig. 11c and one in Fig. 11e is clearly excluded by the theoretical constraint on \(\phi _{0+}\). We have removed those curves in Fig. 11b.

*q*plane. In contrast to the analysis using the isospin relations, we require now also the strong phase \(\delta _\mathrm{c}\). In Fig. 11b, we have added the resulting contour, which is in excellent agreement with two branches of the isospin triangle construction. This curve is actually also consistent with the SM value of

*q*and \(\phi \).

We note that the allowed parameter space for *q* and \(\phi \) following from the current data of the charged \(B\rightarrow \pi K\) system is significantly reduced in comparison with the situation discussed in Ref. [25]. Moreover, we have presented a transparent way to calculate the contours in the \(\phi \)–*q* plane and do not have to make a fit to the data. The constraints on *q* and \(\phi \) have actually a highly non-trivial structure that follows from the isospin relation and can be understood in an analytic way. The only additional *SU*(3) input is the quantity \(R_{T+C}\) discussed in Sect. 3.2, which is required for the conversion of \(|A_{3/2}|\) into the parameter *N*.

In Fig. 12a, we discuss the uncertainties of the various input parameters, focusing on the contour in the \(\phi \)–*q* plane in Fig. 11b that is in agreement with the \(R_\mathrm{c}\) constraint. When adding the individual errors in quadrature, we obtain the uncertainty band in Fig. 11b. In Fig. 12b, we illustrate the error budget as a pie chart. We observe that \(\gamma \) and the branching ratios play the major roles, while \(R_{T+C}\) has a slightly smaller impact on the error budget.

In analogy to the discussion of the charged \(B\rightarrow \pi K\) system given above, we may also use the neutral \(B\rightarrow \pi K\) decays and their isospin amplitude relations to determine contours in the \(\phi \)–*q* plane. The key difference is that the measurement of the mixing-induced CP asymmetry of \(B^0_d\rightarrow \pi ^0 K_\mathrm{S}\) allows us to determine the angle \(\phi _{00}\) in a clean way through Eq. (104), thereby fixing the relative orientation of the neutral \(B\rightarrow \pi K\) isospin triangle and its CP conjugate. In contrast to using \(\phi _\mathrm{c}\) in Eq. (125) for the charged \(B\rightarrow \pi K\) decays, this determination is theoretically clean (although also \(\phi _\mathrm{c}\) is only affected by a small theoretical uncertainty). The charged and neutral \(B\rightarrow \pi K\) decays should result in constraints in the \(\phi \)–*q* plane that are consistent with each other.

### 5.2 Utilizing mixing-induced CP violation in \(B^0_d\rightarrow \pi ^0K_\mathrm{S}\)

*q*and \(\phi \) but to determine these parameters, further information is needed. It is provided by the mixing-induced CP asymmetry \(S_{\text {CP}}^{\pi ^0 K_{\text {S}}}\), which allows the extraction of the phase \(\phi _{00}\). If we use the values of the hadronic parameters \(r_\mathrm{c}\), \(\delta _\mathrm{c}\) and

*r*, \(\delta \) as determined in Sect. 3.2, we may convert this observable into a contour in the \(\phi \)–

*q*plane with the help of the following expression:

*R*and \(A_{\text {CP}}^{\pi ^-K^+}\), allowing us to take also these contributions into account. We first focus on the constraints from current data. Using the measurement of \(S_\mathrm{CP}^{\pi ^0 K_\mathrm{S}}\) in Table 2 gives

*q*plane resulting from \(S_{\text {CP}}^{\pi ^0 K_{\text {S}}}\) and the isospin determination separately for the three scenarios. We also give the SM point corresponding to the value of \(R_q\) in Eq. (41). For the constraints following from \(S_{\text {CP}}^{\pi ^0 K_{\text {S}}}\), we take into account the experimental uncertainties on \(A_{\text {CP}}^{\pi ^0 K_{\text {S}}}\) and \(S_{\text {CP}}^{\pi ^0 K_{\text {S}}}\) as given in Table 6. In addition, we take into account the theoretical

*SU*(3) uncertainties for the hadronic parameters that are required to determine

*q*from Eq. (139). We show these experimental and theoretical uncertainties separately in Fig. 14. This additional contour in the \(\phi \)–

*q*plane, combined with the isospin triangle contours, constrains the allowed values of

*q*and \(\phi \) significantly, as indicated by their small overlapping region in Fig. 14.

Scenarios for future measurements of \(S_\mathrm{CP}^{\pi ^0 K_\mathrm{S}}\)

Scenario | \(S_\mathrm{CP}^{\pi ^0 K_\mathrm{S}}\) | \(A_\mathrm{CP}^{\pi ^0 K_\mathrm{S}}\) | \(\phi _{00}\) |
---|---|---|---|

1 | \(0.67 \pm 0.042\) | \(-\,0.07 \pm 0.042\) | \((0.9\pm 3.3)^\circ \) |

2 | \(0.33 \pm 0.042 \) | \(-\,0.06 \pm 0.042\) | \((23.9\pm 2.6)^\circ \) |

3 | \(0.91 \pm 0.042\) | \(-\,0.07 \pm 0.042\) | \((-23.0 \pm 6.0 )^\circ \) |

For the isospin triangle constraints in Fig. 14, we only show the contours that remain after taking constraints from \(\phi _{0+}\) and \(R_\mathrm{c}\) into account. For the uncertainty, we only consider the uncertainty on \(R_{T+C}\) as in Eq. (62). The theory uncertainty (dashed line) matches the future experimental uncertainty (solid line), which is very promising.

*SU*(3)-breaking corrections can be reduced by a factor of four with respect to the current situation. Taking only these uncertainties into account, we obtain the constraints in Fig. 15. These considerations show the exciting potential of the new strategy, going even beyond the next generation of

*B*-decay experiments.

*q*for several values of \(\phi \), using the hadronic parameters in Table 3. Here the outer curves correspond to the maximum values that the sum rules can take. The behaviour of \(\Delta _\mathrm{SR}^{(\mathrm{I})}\) can be easily derived from Eq. (86), i.e. it is linear in

*q*with a slope proportional to \(\sin (\gamma -\phi )\). On the other hand, \(\Delta _\mathrm{SR}^{(\mathrm{II})}\) also depends on \(q^2\) as can be seen from Eq. (90). The grey horizontal bands show the sensitivity of the sum rules at Belle II, assuming an uncertainty for \(A_\mathrm{CP}^{\pi ^0 K_\mathrm{S}}\) of \(\pm 0.042\) and perfect measurements of the other observables entering Eqs. (85) and (89). The black data point corresponds to the SM values of

*q*and \(\phi \) using \(R_q\) in Eq. (41). Consequently, we observe that the experimental resolution would not be sufficient to reveal the NP effects in the EW penguin sector with the sum rules, in contrast to the new method presented above.

## 6 Conclusions

Employing information on the UT angle \(\gamma \) and the \(B^0_d\)–\(\bar{B}^0_d\) mixing phase \(\phi _d\), we use the currently available data for \(B\rightarrow \pi \pi \) decays to determine hadronic parameters which characterize these modes and describe the interplay between various tree-diagram-like and penguin topologies. We find agreement with previous studies although our results have higher precision. An important new element in this endeavour is given by measurements of direct CP violation in \(B^0_d\rightarrow \pi ^0\pi ^0\), allowing us to resolve a twofold ambiguity. The determination of the hadronic \(B\rightarrow \pi \pi \) parameters relies only on the isospin symmetry and is hence theoretically clean. Consequently, the corresponding results represent reference values for the comparison with QCD calculations. EW penguin topologies play a negligible role in the \(B\rightarrow \pi \pi \) system for the current experimental uncertainties but could be included in the future through more sophisticated analyses.

Utilizing the *SU*(3) flavour symmetry, we convert the hadronic \(B\rightarrow \pi \pi \) parameters into their counterparts in the \(B\rightarrow \pi K\) system. We test also the *SU*(3) flavour symmetry and obtain an impressive global picture which does not indicate any anomalously large non-factorizable *SU*(3)-breaking corrections. Correspondingly, we do not find indications of an enhancement of colour-suppressed EW penguin topologies when analysing the data. The cleanest SM prediction of the \(B\rightarrow \pi K\) observables is a correlation between the direct and mixing-induced CP asymmetries of the \(B^0_d\rightarrow \pi ^0K_\mathrm{S}\) decay. As we discussed in detail, it follows from an isospin relation between the neutral \(B\rightarrow \pi K\) decay amplitudes and uses the *SU*(3) flavour symmetry only to fix the magnitude of the \(\hat{T}'+\hat{C}'\) amplitude. In comparison with a previous study, a tension of the mixing-induced CP violation in \(B^0_d\rightarrow \pi ^0K_\mathrm{S}\) has become more pronounced due to a sharper determination of \(\gamma \). Moreover, we have considered the angle \(\phi _\pm \) as a new constraint, which also shows tension with respect to the SM. These discrepancies emerging from the current data suggest that either the values of the measured observables will change in the future or indicate NP effects with new sources of CP violation. In the former case, a reduction of the central value of the branching ratio of \(B^0_d\rightarrow \pi ^0K^0\) by about \(2.5\,\sigma \) with an increase of the mixing-induced CP asymmetry by about \(1 \,\sigma \) would give a situation in agreement with the SM. In the latter case, EW penguin topologies offer an attractive avenue for new particles to enter the \(B\rightarrow \pi K\) modes.

In view of this intriguing \(B\rightarrow \pi K\) puzzle and to test the corresponding sector of the SM, the EW penguin parameters *q* and \(\phi \) are in the spotlight. We have presented a new strategy to determine these quantities from the data for the neutral and charged \(B\rightarrow \pi K\) decays, employing again the corresponding isospin relations. Applying this method to the current data, we already obtain surprisingly stringent constraints in the \(\phi \)–*q* plane. They are consistent with the SM but leave also a lot of space for possible NP effects. In order to actually pin down \(\phi \) and *q* further information is needed, which is provided by the mixing-induced CP asymmetry of the \(B^0_d\rightarrow \pi ^0K_\mathrm{S}\) decay. Considering a variety of future scenarios, we have illustrated this determination and have shown that the theory uncertainties can match the expected experimental precision in the era of Belle II and the LHCb upgrade. Following these lines, we may determine \((q,\phi )\) and reveal the dynamics of the \(B\rightarrow \pi K\) system with unprecedented accuracy. The resulting picture will either confirm once again the SM or may eventually establish new flavour structures with possible new sources of CP violation.

## Footnotes

- 1.
As usual, we neglect tiny CP violation in the neutral kaon system.

## Notes

### Acknowledgements

This research has been supported by the Netherlands Organisation for Scientific Research (NWO) and by the Deutsche Forschungsgemeinschaft (DFG), research unit FOR 1873 (QFET).

## References

- 1.A.J. Buras, J. Girrbach, Rep. Prog. Phys.
**77**, 086201 (2014). arXiv:1306.3775 [hep-ph]ADSCrossRefGoogle Scholar - 2.N. Cabibbo, Phys. Rev. Lett.
**10**, 531 (1963)ADSCrossRefGoogle Scholar - 3.M. Kobayashi, T. Maskawa, Prog. Theor. Phys.
**49**, 652 (1973)ADSCrossRefGoogle Scholar - 4.T. Abe et al., [Belle-II Collaboration], arXiv:1011.0352 [physics.ins-det]
- 5.T. Aushev et al., arXiv:1002.5012 [hep-ex]
- 6.R. Aaij et al., [LHCb Collaboration], Eur. Phys. J. C
**73**, 2373 (2013). arXiv:1208.3355 [hep-ex] - 7.Y. Nir, H.R. Quinn, Phys. Rev. Lett.
**67**, 541 (1991)ADSCrossRefGoogle Scholar - 8.M. Gronau, O.F. Hernandez, D. London, J.L. Rosner, Phys. Rev. D
**52**, 6374 (1995). arXiv:hep-ph/9504327 ADSCrossRefGoogle Scholar - 9.M. Gronau, J.L. Rosner, D. London, Phys. Rev. Lett.
**73**, 21 (1994). arXiv:hep-ph/9404282 ADSCrossRefGoogle Scholar - 10.
- 11.
- 12.
- 13.M. Beneke, M. Neubert, Nucl. Phys. B
**675**, 333 (2003). arXiv:hep-ph/0308039 ADSCrossRefGoogle Scholar - 14.R. Fleischer, S. Recksiegel, F. Schwab, Eur. Phys. J. C
**51**, 55 (2007). arXiv:hep-ph/0702275 ADSCrossRefGoogle Scholar - 15.M. Gronau, J.L. Rosner, Phys. Lett. B
**666**, 467 (2008). arXiv:0807.3080 [hep-ph]ADSCrossRefGoogle Scholar - 16.C. Bobeth, M. Gorbahn, S. Vickers, Eur. Phys. J. C
**75**, 340 (2015). arXiv:1409.3252 [hep-ph]ADSCrossRefGoogle Scholar - 17.A.J. Buras, R. Fleischer, S. Recksiegel, F. Schwab, Nucl. Phys. B
**697**, 133 (2004). arXiv:hep-ph/0402112 ADSCrossRefGoogle Scholar - 18.A.J. Buras, R. Fleischer, S. Recksiegel, F. Schwab, Phys. Rev. Lett.
**92**, 101804 (2004). arXiv:hep-ph/0312259 ADSCrossRefGoogle Scholar - 19.V. Barger, L. Everett, J. Jiang, P. Langacker, T. Liu, C. Wagner, Phys. Rev. D
**80**, 055008 (2009). arXiv:0902.4507 [hep-ph]ADSCrossRefGoogle Scholar - 20.V. Barger, L.L. Everett, J. Jiang, P. Langacker, T. Liu, C.E.M. Wagner, JHEP
**0912**, 048 (2009). arXiv:0906.3745 [hep-ph]ADSCrossRefGoogle Scholar - 21.
- 22.N.B. Beaudry, A. Datta, D. London, A. Rashed, J.S. Roux, JHEP
**1801**, 074 (2018). arXiv:1709.07142 [hep-ph]ADSCrossRefGoogle Scholar - 23.W. Altmannshofer, C. Niehoff, P. Stangl, D.M. Straub, Eur. Phys. J. C
**77**, 377 (2017). arXiv:1703.09189 [hep-ph]ADSCrossRefGoogle Scholar - 24.
- 25.R. Fleischer, S. Jäger, D. Pirjol, J. Zupan, Phys. Rev. D
**78**, 111501 (2008). arXiv:0806.2900 [hep-ph]ADSCrossRefGoogle Scholar - 26.R. Fleischer, R. Jaarsma, K.K. Vos, arXiv:1712.02323 [hep-ph]
- 27.R. Fleischer, R. Jaarsma, E. Malami, K.K. Vos, Talk given at Rencontres de Moriond 2018, QCD and High Energy Interactions, La Thuile, Italy, 17–24 March 2018, to appear in the Proceedings. arXiv:1805.06705 [hep-ph]
- 28.M. Gronau, D. London, Phys. Rev. Lett.
**65**, 3381 (1990)ADSCrossRefGoogle Scholar - 29.C. Patrignani et al., [Particle Data Group], Chin. Phys. C
**40**, 100001 (2016)Google Scholar - 30.Y. Amhis et al., [Heavy Flavor Averaging Group (HFAG)], arXiv:1412.7515 [hep-ex]. For updates, see http://www.slac.stanford.edu/xorg/hfag/
- 31.J.P. Lees et al., [BaBar Collaboration], Phys. Rev. D
**87**, 052009 (2013). arXiv:1206.3525 [hep-ex] - 32.T. Julius et al., [Belle Collaboration], Phys. Rev. D
**96**, 032007 (2017). arXiv:1705.02083 [hep-ex] - 33.R. Aaij et al., [LHCb Collaboration], arXiv:1805.06759 [hep-ex]
- 34.L. Wolfenstein, Phys. Rev. Lett.
**51**, 1945 (1983)ADSCrossRefGoogle Scholar - 35.A.J. Buras, M.E. Lautenbacher, G. Ostermaier, Phys. Rev. D
**50**, 3433 (1994). arXiv:hep-ph/9403384 ADSCrossRefGoogle Scholar - 36.J. Charles et al., Phys. Rev. D
**91**, 073007 (2015). arXiv:1501.05013 [hep-ph]; for updates, see http://ckmfitter.in2p3.fr - 37.M. Gronau, D. Wyler, Phys. Lett. B
**265**, 172 (1991)ADSCrossRefGoogle Scholar - 38.D. Atwood, I. Dunietz, A. Soni, Phys. Rev. Lett.
**78**, 3257 (1997). arXiv:hep-ph/9612433 ADSCrossRefGoogle Scholar - 39.D. Atwood, I. Dunietz, A. Soni, Phys. Rev. D
**63**, 036005 (2001). arXiv:hep-ph/0008090 ADSCrossRefGoogle Scholar - 40.R. Fleischer, S. Ricciardi, Proceedings of the 6th International Workshop on the CKM Unitarity Triangle (CKM 2010). arXiv:1104.4029 [hep-ph]
- 41.A. Bevan et al., arXiv:1411.7233 [hep-ph]; for updates, see http://www.utfit.org
- 42.R. Fleischer, R. Jaarsma, K.K. Vos, JHEP
**1703**, 055 (2017). arXiv:1612.07342 [hep-ph]ADSCrossRefGoogle Scholar - 43.R. Fleischer, R. Jaarsma, K.K. Vos, Phys. Rev. D
**94**(11), 113014 (2016). arXiv:1608.00901 [hep-ph] - 44.M. Gronau, D. Pirjol, T.M. Yan, Phys. Rev. D
**60**, 034021 (1999) [Erratum: Phys. Rev. D**69**, 119901 (2004). arXiv:hep-ph/9810482 - 45.
- 46.
- 47.M. Beneke, G. Buchalla, M. Neubert, C.T. Sachrajda, Nucl. Phys. B
**606**, 245 (2001). arXiv:hep-ph/0104110 ADSCrossRefGoogle Scholar - 48.M. Neubert, J.L. Rosner, Phys. Rev. Lett.
**81**, 5076 (1998). arXiv:hep-ph/9809311 ADSCrossRefGoogle Scholar - 49.M. Neubert, J.L. Rosner, Phys. Lett. B
**441**, 403 (1998). arXiv:hep-ph/9808493 ADSCrossRefGoogle Scholar - 50.R. Aaij et al., [LHCb Collaboration], Phys. Rev. Lett.
**118**, 081801 (2017). arXiv:1610.08288 [hep-ex] - 51.J.L. Rosner, S. Stone, R.S. Van de Water, arXiv:1509.02220 [hep-ph]
- 52.A. Khodjamirian, T. Mannel, M. Melcher, Phys. Rev. D
**68**, 114007 (2003). arXiv:hep-ph/0308297 ADSCrossRefGoogle Scholar - 53.R. Fleischer, T. Mannel, Phys. Rev. D
**57**, 2752 (1998). arXiv:hep-ph/9704423 ADSCrossRefGoogle Scholar - 54.
- 55.M. Gronau, J.L. Rosner, Phys. Rev. D
**74**, 057503 (2006). arXiv:hep-ph/0608040 ADSCrossRefGoogle Scholar - 56.B. Aubert et al., [BaBar Collaboration], Phys. Rev. D
**79**, 052003 (2009). arXiv:0809.1174 [hep-ex] - 57.M. Fujikawa et al., [Belle Collaboration], Phys. Rev. D
**81**, 011101 (2010). arXiv:0809.4366 [hep-ex] - 58.S. Faller, R. Fleischer, T. Mannel, Phys. Rev. D
**79**, 014005 (2009). arXiv:0810.4248 [hep-ph]ADSCrossRefGoogle Scholar

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