# The perturbative QCD predictions for the rare decay

###### Abstract

In this work, we calculate the branching ratios and CP violations of the decay modes with both charged and neutral mesons for the first time in the pQCD approach. Considering the recent observation of the BESIII collaboration that report a direct evidence of the quark-antiquark structure about the scalar meson , we regard as the quark component in our present work, and then make predictions of this decay model. The branching ratios of our calculations are and . We also calculate the CP violation parameters of decay modes. The relatively large branching ratios make it easily to be tested by the running LHC-b experiments, and it can help us to understand both the inner properties and the QCD behavior of the scalar meson.

## I Introduction

It is well-known that, the rare decays, which only have pure annihilation contributions in Standard Model(SM) due to the totally different quark components between the initial and final state mesons, can provide rich information of CP violations and signals of possible new physics beyond the SM. The QCD factorization approach Beneke:2000ry (); Beneke:1999br (), where the non-factorizable spectator scattering contributions and the annihilation contributions are adjustable parameters, which make the prediction unreliable. However, in the perturbative QCD(pQCD) approach Li:1994cka (); Li:1995jr (); Li:1994iu (), many rare decay modes can been studied Xiao:2011tx (); Yu:2005rm (); Li:2004ep (); Ali:2007ff (), where the theoretical results were coincident well with the experimental data and it proved that the successful application of pQCD approach to mesons rare decays. Since the first scalar meson was observed by the Belle collaboration in the charged decay mode Abe:2002av (), and afterwards confirmed by BaBar Aubert:2003mi (), a lot of other scalar mesons have been discovered in the experiment successively, many researches have been done about light scalar mesons Liu:2013cvx (); Dou:2015mka (); Lu:2006fr (); Colangelo:2010bg (); Wang:2006ria (); Cheng:2005nb (); Liu:2010kq (); Zou:2017yxc (). However, as far as we know, there are very few works about the decays( denote the scalar mesons) to be studied in these general factorization approaches, besides the Dou:2015mka () and Liu:2013lka ().

For a long time, the scalar mesons, especially for the and , which are important for understanding the chiral symmetry and confinement in the low-energy region, are one of the key problems in the nonperturbative QCD Achasov:2017zhy (). However, the inner structure of scalar mesons is still a contradiction in both the theoretical and experimental side, and many works have been done about the scalar meson in order to solve this problem. In Ref. Achasov:2017zhy (), the authors list many evidences that sustain the four-quark model of the light scalar mesons based on a series of experimental data. In Ref. Shen:2006ms (), the predicted result of is times difference from the experimental result, and the author conclude that cannot be interpreted as . In Ref. Weinstein:1990gu (), the authors showed that the production of the and and of low-mass pairs have properties of the molecules. Moreover, the scalar meson are identified as the quark-antiquark gluon hybrid. Nevertheless, these interpretations of the scalar mesons make theoretical calculations difficult, apart from the ordinary model.

In theoretical side, there are two interpretations about light scalar mesons below GeV in Review of Particle Physics Tanabashi:2018oca (), the scalars below GeV, including , , and , form a flavor nonet, and , , and (or ) that above GeV form another flavor nonet. In order to describe the structure of these light scalar mesons , the authors of Ref. Cheng:2005nb () presented two Scenarios to clarify the scalar mesons (here, we only focus on the flavor wave function of the meson, which are given in Ref. Jaffe:1976ig ()):

(1) Scenario 1, the light scalar mesons, which involved in the first flavor nonet, are usually regarded as the lowest-lying states, and the other nonet as the relevant first excited states. In the ordinary diquark model, the quark components of are

(1) |

(2) Scenario 2, the scalar mesons in the second nonet are regarded as the ground states(), and the mass between GeV is first excited states. This Scenario indicate that the scalars below or near GeV are four-quark bound states, while other scalars consist of in Scenario 1. So the quark components of are

(2) |

Recently, BES III collaboration declare that the flavor wave function of and are two-quark component through the decays and (and the charge conjugated ones), the decay modes are direct probe of the quark components of and Achasov:2018grq (). And in Ref. Ablikim:2018ffp (), BES III declare the - mixing in the and decay modes, which is the first observation of - mixing in experiment. But in this work, we will let the mixing effect aside and want to make comprehensive research in the future work.

In this present work, motivated by the uncertain inner structure of the , we explore the branching ratios and CP-violating asymmetries of rare decay mode ^{1}^{1}1 will abbreviated as in the last part. in perturbative QCD approach within the traditional two-quark model for the first time. Because the LHC-b collaboration are collecting more and more B mesons decays data, so we believe that our results can be testified by the experiment in the near future time.

This article is organized roughly in this order: in Section II, we give a theoretical framework of the pQCD, list the wave functions that we need in the calculations, and also the perturbative calculations; in Section III, we make numerical calculations and some discussions for the results that we get; and at last, we summary our work in the final Section.

## Ii The Theoretical Framework And Perturbative Calculation

The pQCD approach have been widely applied to calculate the hadronic matrix elements in the B mesons decay modes, it is based on the factorization. The divergence of the end-point singularity can be safely avoided by preserving the transverse momenta in the valence quark, and the only input parameters are the wave functions of the involved mesons in this method. Then the transition form factors and the different contributions, whose may contain the spectator and annihilation diagrams, are all calculated in this framework.

### ii.1 Wave Functions and Distribution Amplitudes

In kinematics aspects, we adopt the light-cone coordinate system in our calculation. Assuming the meson to be rest in the system, we can describe the momenta of the mesons in light-cone coordinate system, where the momenta are expressed in the form of with the definition and .

In our calculation, the wave function of the hadron can be found in Refs. Lu:2000em (); Keum:2000wi (); Keum:2000ph ()

(3) |

where the distribution amplitude(DA) of meson is written as mostly used form, which is

(4) |

the normalization factor can be calculated by the normalization relation with is the color number and decay constant MeV. Here, we choose shape parameter GeV Ali:2007ff ().

For the scalar meson , the wave function can be read as Cheng:2005ye (); Cheng:2005nb ():

(5) |

where denotes the momentum fraction of the meson, and , are light-like dimensionless vectors.

The is leading-twist distribution amplitude, the explicit form of which is expanded by the Gegenbauer polynomials Cheng:2005ye (); Cheng:2005nb ():

(6) |

and the twist-3 DAs and are adopted the asymptotic forms in our predictive calculation ,

(7) | |||

(8) |

where and are the vector and scalar decay constants of the meson respectively, is Gegenbauer moment and in DA of is Gegenbauer polynomials, these parameters are scale-dependent. A lot of calculations have been carried out about the light scalar mesons in various model Brito:2004tv (); Maltman:1999jn (); Shakin:2001sz (). In this article, we adopt the value for decay constants and Gegenbauer moments in the DAs of the as listed follow, which were calculated in QCD sum rules at the scale GeV Cheng:2005nb ():

(9) |

It’s noticeable that only the odd Gegenbauer moments are taken into account due to the conservation of vector current or charge conjugation invariance. And we also pay attention to only the Gegenbauer moments and because the higher order Gegenbauer moments make tiny contributions and can be ignored safely.

And the Gegenbauer polynomials are

(10) |

The vector and scalar decay constants satisfy the relationship

(11) |

with

(12) |

and is the mass of the scalar meson and and are the running current quark masses in the meson. From the above relationship, it is clear to see that the vector decay constant is proportional to the mass difference between the and quark, the mass difference is so small after considering the symmetry breaking that would heavily suppress the vector decay constant, which lead to the vector decay constants of the scalar mesons are very small and can be negligible. Likewise, for the same reason that only the odd Gegenbauer momentums are considered, the neutral scalar mesons can not be produced by the vector current, so in this work we adopt the vector constant .

And the normalization relationship of the twist-2 and twist-3 DAs are

(13) |

### ii.2 Perturbative Calculations

For decay mode, the relevant weak effective Hamiltonian can be written as Buchalla:1995vs ()

(14) |

where is Fermi constant, and and are Cabibbo-Kobayashi-Maskawa (CKM) factors, () is local four-quark operator, which will be listed as follows, and is corresponding Wilson coefficient.

(1) Current-Current Operators (Tree):

(15) |

(2) QCD Penguin Operators:

(16) |

(3) Electroweak Penguin Operators:

(17) |

with the color indices , and . The denotes the quark and quark, and is corresponding charge.

The momenta of the , , meson in the light-cone coordinate read as

(18) |

with the mass and the mass ratio .

And the corresponding light quark’s momenta in each meson read as

(19) |

Then based on the pQCD approach, we can write the decay amplitude as

(20) |

where is the conjugate momenta of , and is the largest energy scale in hard function . The suppress the soft dynamics Li:1997un () and make a reliable perturbative calculation of the hard function , which come from higher order radiative corrections to wave functions and hard amplitudes. represent universal and channel independent wave function, which describes the hadronization of mesons.

Fig. 1 display the typical Feynman diagrams of the decays at the lowest order, and this decay only have pure annihilation topologies. We can find that this decay is similar to the Xiao:2011tx (); Li:2004ep (), which have four diagrams contributing to the , (a),(b) are factorization annihilation diagrams, other two diagrams are nonfactorization annihilation diagrams. As depicted in Fig. 1, we calculate the factorizable and non-factorizable annihilation diagrams respectively. We use and denote the factorizable and non-factorizable annihilation contributions respectively, and the subscript () denote the contributions of the Feynman diagrams Fig. 1(a) and (b) (Fig. 1(c) and (d)) and the superscript , , is the , and vertex, respectively. The vertex is the Fierz transformation of the .

First, the total contribution of the Feynman diagrams Fig. 1 (a) and (b), which only involve the wave function of the final light scalar mesons, are

(1)

(21) |

the evolution function is defined by

(22) |

where the largest energy scales to eliminate the large logarithmic radiative corrections are chosen as:

(23) |

(2)

(24) |

Then the total non-factorizable annihilation decay amplitudes for the Fig. 1 (c) and (d) diagrams are

(3)

(25) |

(4)

(26) |

with the color factor .

The evolution function is

(27) |

with the hard scales

(28) |

The hard scattering kernels function involved in the above expression are written as: