# and Isospin Breaking Effects on Amplitudes

###### Abstract

Several modes of decays into three pseudoscalar octet mesons PPP have been measured. These decays have provided useful information for B decays in the standard model (SM). Some of powerful tools in analyzing B decays are flavor and isospin symmetries. Such analyses are usually hampered by breaking effects due to a relatively large strange quark mass which breaks SU(3) symmetry down to isospin symmetry. The isospin symmetry also breaks down when up and down quark mass difference is non-zero. It is therefore interesting to find relations which are not sensitive to and isospin breaking effects. We find that the relations among several fully-symmetric decay amplitudes are not affected by first order breaking effects due to a non-zero strange quark mass, and also some of them are not affected by first isospin breaking effects. These relations, therefore, hold to good precisions. Measurements for these relations can provide important information about B decays in the SM.

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## I Introduction

Several decay modes of decays into three pseudoscalar octet mesons PPP have been measured3p-data-babar ; 3p-data-lhcb . has been a subject of theoretical studieshe3 . The new data have raised new interests in related theoretical studiesyadong ; gronau ; xu-li-he1 ; he-li-xu2 ; hy-cheng ; gronau-full . With more data from LHCb, one can expect that the study of will provide more important information for B decays in the standard model (SM).

A powerful tool to analyze B decays is flavor symmetrysu31 . Some of the interesting features of using flavor are the predictions of relations among different decay modes which can be experimentally tested. The flavor symmetry is, however, expected to be only an approximate symmetry because , and quarks have different masses. Since the strange quark has a relative larger mass compared with those of up and down quarks, it is the larger source of symmetry breaking. If up and down quark masses are neglected, a non-zero strange quark mass breaks flavor symmetry down to the isospin symmetry. When up and down quark mass difference is kept, isospin symmetry is also broken. The breaking effect is at the level of 20 percent for the and decay constants and . For 2-body pseudoscalar octet meson decays, although there are some breakings su32 , it works reasonably well, such as rate differences between some of the and two-body pseudoscalar meson decays he1 ; he2 . Analysis has also been carried out for decays using flavor recently. It has been shown that the decay and CP asymmetry patterns for the charged decays into , , and do not follow predictions. To explain data, large breaking effects are neededxu-li-he1 ; he-li-xu2 . Usually isospin breaking effects are much smaller because up and down quark masses are much smaller than the strange quark mass and the QCD scale.

Because of possible large flavor breaking effects for , the predicted relations among different decay modes can only provide limited information. One wonders whether there exist relations which are immuned from or even isospin breaking effects due to , and quark mass differences. To this end we carried out an analysis for decays using flavor symmetry to identify possible relations, and then include breaking effects due to a strange quark mass, and also up and down quark masses to see whether some relations still remain to hold. We find that the relations between several fully-symmetric decay amplitudes studied in Ref. gronau-full are not affected by the flavor breaking effects due a non-zero strange quark mass, and some of them are not even affected by isospin breaking effects. These relations when measured experimentally can provide useful information about decays in the SM. In the following we provide some details.

## Ii conserving amplitudes

We start with the description of decays into three pseudoscalar octet mesons from flavor symmetry. The leading quark level effective Hamiltonian up to one loop level in electroweak interaction for hadronic charmless decays in the SM can be written as

(1) |

where can be or the coefficients and , with and indicate the internal quark, are the Wilson Coefficients (WC). The tree WCs are of order one with, , and . The penguin WCs are much smaller with the largest one to be . These WC’s have been evaluated by several groups heff . are the KM matrix elements. In the above the factor has been eliminated using the unitarity property of the KM matrix.

The operators are given by

(2) |

where , and are the field strengths of the gluon and photon, respectively.

At the hadron level, the decay amplitude can be generically written as

(3) |

where contains contributions from the as well as due to charm and up quark loop corrections to the matrix elements, while contains contributions purely from one loop contributions. indicates one of the , and . forms an triplet.

The flavor symmetry transformation properties for operators , , and are: , , and , respectively. We indicate these representations by matrices in flavor space by , and . For , the non-zero entries of the matrices are given by he1

(4) |

And for , the non-zero entries are

(5) |

These properties enable one to write the decay amplitudes for decays in only a few invariant amplitudes su31 . Here is one of the mesons in the pseudoscalar octet meson which is given by,

(6) |

Construction of decay amplitude can be done order by order by using three ’s, , and the Hamiltonian , and also derivatives on the mesons to form . The conserving momentum independent amplitudes can be constructed by the following.

For the amplitude, we havexu-li-he1

(7) | |||||

One can write similar amplitude for the penguin contributions.

The coefficients , , and are constants which contain the WCs and information about QCD dynamics. Expanding the above amplitude, one can extract the decay amplitudes for specific decays in terms of these coefficients.

In the above we have described how to obtain flavor amplitudes which are momentum independent. However, due to the three body decay nature, in general, there are momentum dependence in the decay amplitudes. The momentum dependence can in principle be determined by analysing Dalitz plots for the decays. The lowest order terms with derivatives lead to two powers of momentum dependence. One can obtain relevant terms by taking two times of derivatives on each of the terms in Eq.(7) and then collecting them together. It has been shownxu-li-he1 that there are six independent ways of taking derivatives for each of the terms listed in eq. (7). For example after taking derivatives for , we have the following independent terms

The full list of the possible terms have been obtained in Ref.xu-li-he1 in the Appendix B. We will not repeat them here.

Using the above decay amplitudes, one can find some interesting relations among different decaysxu-li-he1 . It has been recently pointed out that there are additional relations among the fully-symmetric final states B decay amplitudes gronau-full . Study of these relations can provide further information about flavor symmetry in decays.

The fully-symmetric amplitudes is related to the usual decay amplitudes for the final mesons carrying momenta , for all three final mesons are distinctive, by

For the cases that two of them or all three of them are identical particles, the identical particle factorial factors should be taken cared. In Ref.gronau-full , how the fully-symmetric amplitudes can be determined experimentally has been discussed in detail. We will not repeat the discussions here. We concentrate on how these amplitudes are derived in the framework of flavor SU(3) symmetry and how they are affected by SU(3) breaking effects due to finite quark masses for u, d and s quarks.

To understand that why there are new relations between the fully-symmetric amplitudes for different decay modes, let us consider and decays as examples.

As the decay amplitudes may have momentum dependence, we should also check if the equality of the above two amplitudes are equal when taking into account of momentum dependence in the amplitudes. Expanding terms in Appendix B of Ref.xu-li-he1 , we find

(12) |

The coefficients and are given by,

and

One can see from the above that is no longer equal to . However, one can readily see from the above equations, that

(15) |

This fact makes the fully-symmetric amplitudes to satisfy

(16) |

Similarly, the penguin amplitudes and have the same properties discussed above for the tree amplitudes, and .

The total fully-symmetric amplitudes then have the relation

(17) |

Enlarging the amplitudes to fully-symmetric ones, indeed produce more relations.

Expanding eq. (7) and equations in Appendix B of Ref. xu-li-he1 , we obtain the following relations confirming those obtained in Ref.gronau-full .
For induced amplitudes, we have

,

,

.

.

For induced amplitudes, we have

,

.