^{3}

We propose a method of extracting the Cabibbo–Kobayashi–Maskawa matrix element

The precise determination of Cabibbo–Kobayashi–Maskawa (CKM) matrix elements [

There are two main methods to extract the value of ^{1}^{2}

In this work we intend to provide another method to extract

We emphasize that this method becomes possible only with recent precise determination of all the form factors of semi-leptonic B decay. More precise experimental data of the branching fractions of two-body hadronic B decays give a more precise value of

In the next section we investigate the amplitudes of

Consider the hadronic two-body decay processes

(1) Tree amplitudes

(2) Color-suppressed amplitudes

(3) Exchange amplitudes

(4) W-annihilation amplitudes

Tree amplitude. For example, the

Color-suppressed amplitude. For example, the

Exchange amplitude where quark–antiquark pair creation occurs. For example, the

W-annihilation amplitude. For example, the

In this paper we do not consider the process which contains the contribution of

In

Two-body hadronic decays and their amplitudes. Note that the decay mode

Decay mode | Topologies | Penguin | Fraction |
|||
---|---|---|---|---|---|---|

T | E | |||||

C | E | |||||

C | E | |||||

T | ||||||

C | ||||||

T | E | yes | ||||

T | yes | |||||

T | C | |||||

T | C | |||||

T | C | yes | ||||

T | C | yes |

In general, several diagrams with different topologies contribute to the amplitudes for each decay process. We see that the amplitude of

Penguin diagram. The process of

We focus on the two-body decay

The

We obtain the value of ^{3}

In order to consider the effect of hadronic final state interactions, we introduce a relation between decay amplitudes which follows from isospin symmetry. The amplitudes of

We expect that this relation should be satisfied within 1% accuracy, because the isospin breaking effect should be proportional to

Isospin relation of three amplitudes in a complex plane. We choose the direction of the amplitude

There is no justification of this assumption, since it has been known that the inelastic final state interactions are important in B decays in general [

Once the formula of Eq. (

Now we are going to extract

Equations (

From Eq. (

For

For

Notice that this amplitude depends only on the form factor

For

Notice that this amplitude depends only on the form factor

In our analysis we use the experimental data, masses and branching fractions in Ref. [

From

Inputs for the determination from

Input | Value | Reference |
---|---|---|

[ |
||

[ |
||

[ |
||

[ |
||

[ |
||

[ |
||

[ |
||

[ |

Sources of uncertainty of

Error source | Uncertainty [%] |
---|---|

30.4 |

We find that the precise measurements of the branching fractions

From

Inputs for the determination from

Input | Value | Reference |
---|---|---|

[ |
||

[ |
||

[ |
||

see text | ||

[ |

This value of

From

Inputs for determination from

Input | Value | Reference |
---|---|---|

[ |
||

[ |
||

[ |
||

see text |

The value of

Summary of our results.

Mode | ||
---|---|---|

We have proposed a method of extracting the value of

Our final results are summarized in

We have also examined the effects of hadronic final state interactions in two-body hadronic decays. The extracted strong phase shifts are consistent with the previous works of Refs. [

We would like to thank H. Kakuno for helpful information about SuperKEKB and the Belle II experiment. K.M. and Y.S. were supported in part by a scholarship of Tokyo Metropolitan University for graduate students.

Open Access funding: SCOAP

^{1}It has been pointed out that this problem cannot be solved by New Physics [

^{2}The error with CNL parametrization comes from an excessive reduction of the number of parameters in form factors by using heavy quark symmetry. In fact, improvements are possible by including higher-order corrections (see, for example, Ref. [

^{3}We have also neglected the effect of non-factorizable spectator quark scattering, which violates the factorization [