# The gluonic and decays into the meson

###### Abstract

The inclusive and exclusive decays into the meson plus others are investigated in a model based on the QCD anomaly. The invariant mass distribution is discussed. The constraint of the effective coupling is obtained from the data of the exclusive decay modes. The branching ratio of is predicted to be as large as , which can be tested in the forthcoming CLEO-c experiments.

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^{†}preprint: LMU 20-03

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^{†}preprint: hep-ph/0301038

1. Introduction.

A few years ago, unexpected large branching ratios of decaying
into final states with an meson such as and were observed
by the CLEO collaborationBrowder et al. (1998); Behrens et al. (1998) and recently confirmed by
BaBarAubert et al. (2001) and BelleBel . This stimulated many theoretical activities in
understanding the special role of the meson in B decays.
As the contribution of traditional four quark operators from the effective Hamiltonian in the
standard model (SM) is far below the dataDatta et al. (1998); Ali and Greub (1998), various exotic mechanisms
were introduced such as a large coupling between the gluon and through the QCD
anomalyAtwood and Soni (1997); Hou and Tseng (1998); Fritzsch (1997); Gronau and Rosner (1996); Dighe et al. (1997), intrinsic content inside
Halperin and Zhitnitsky (1998); Yuan and Chao (1997) and positive interference between several contributions in
the SMLipkin (1991); Beneke and Neubert (2002) et.al. The large contribution arising from new physics was
also discussedKagan and Petrov (1997); Hou and Tseng (1998).
Among those theoretical efforts, the possible enhancement from the QCD anomaly is of particular
interest since it is well known that the meson plays an very special role in the dynamics of
low energy QCD Fritzsch and Minkowski (1975).

The mass eigenstates and are related to flavor octet and singlet states , through a mixing matrix:

(1) |

where is the mixing angle and , have the flavor content: and . The associated axial currents are and respectively. Through the QCD anomaly, the divergence of the flavor singlet axial current is non-zero and is given by

(2) |

where is the dual of . This breaks the global symmetry for massless particles and makes the flavor singlet state evade to be a Goldstone boson of chiral symmetry. The QCD anomaly gives its main contribution to the large mass of which is much heavier than the other flavor octet states such as , s and suggests a large gluon content in .

The QCD anomaly indicates a strong coupling between and the gluon field. It is then natural to understand the large branching ratio of through the QCD anomaly. In the literatures there are two different ways to handle this problem. The one is through two-body decay process from some effective Hamiltonian due to QCD anomalyFritzsch (1997). The other one is through three-body process Atwood and Soni (1997); Hou and Tseng (1998). In the first step decay, the quark decays into the quark and a virtual gluon , then decays into and a on shell gluon . This model has some advantages in explaining the spectrum of invariant mass distribution of recoiling hadrons. However, the effective vertex seems to be too small from various approaches Hou and Tseng (1998); Muta and Yang (2000); Ali and Parkhomenko (2002); Artuso et al. (2002). In both of the approaches, the effective coupling between and gluon may contain complicate non-perturpative quark-gluon interactions. It is then better to treat it as a free phenomenological parameter rather than to evaluate it from perturbative calculationsFritzsch (1997).

In this note we focus on the phenomenological analysis of the first mechanism. The effective Lagrangian in this model is given by Fritzsch (1997)

(3) |

where and are the strong coupling constant and Fermi constant, is the effective coupling. From this effective Hamiltonian, the decay arises from the subprocess . The evaluation of matrix elements is straightforward:

(4) |

Applying relation to the divergences of both flavor singlet and octet axial currents and ignoring small quarks masses, the matrix elements can be rewritten as

(5) |

In the quark rest frame, the decay branching ratio is given by

(6) |

(7) |

where is the lifetime of meson.

2. Recoil mass distribution

For two-body like subprocess such as , the invariant mass is directly related to the energy of the meson through the relation: , where and are the masses of and meson. The small quark mass has been ignored. In the two-body decay of , the energy of is fixed from energy-momentum conservation. A typical value of the pole mass GeV will lead to a narrow peak with the invariant mass of GeV. This seems to be disfavored by the current data since the experiment reported a peak at about 2 GeV with a relative wide width in the recoil mass distribution Browder et al. (1998); Aubert et al. (2001).

However, the above estimation may be too naive. Note that in the two-body decay process, the exact distribution of the recoil mass strongly depends on the wave function of the meson which is theoretically hard to estimate. It is too early to draw the conclusion that the current data already disfavored all the two-body models.

To illustrate the non-perturbative bound state effects here we adopt a simple model proposed by Altarelli et.al. Altarelli et al. (1982) a number of years ago which is based on the Fermi motion of the quark inside meson. The basic idea of this model is that the Fermi motion of the quark and the spectator quark in the B meson make them have back-to-back relative three-momenta in the rest frame. The momentum is assumed to obey a Gaussian distribution as follows

(8) |

where is normalized as . The mean value of is . In this model the spectator quark is on always handled as on shell while the quark is treated as off-shell. Through energy-momentum conservation, the effective mass of the b quark is determined as

(9) |

and the energy of the quark is . Here one parameter is introduced which specifies both the distribution width and the mean value. As is linked to the average energy of quark inside the meson, in principle it can be calculated from theories based on non-perturbative methods or from some models. For example, the calculations from QCD sum rule Bagan et al. (1995) give GeV, and the value from relativistic quark model Hwang et al. (1995) is GeV. The value of can also be extracted directly from the data. A fit to photon energy spectrum give a value of about GeVAli and Greub (1995) while the fits to semi leptonic decays and give a value of 0.57 GeV Palmer et al. (1997); Hwang et al. (1996). Thus the value of is likely to lie in the range GeV. After including the Fermi motion, the differential decay width , should be replaced by

(10) |

where is the allowed maximum value of , is the differential decay rate in the meson rest frame, which is linked to the one in quark rest frame through a Lorentz boostPalmer et al. (1997); Du and Yang (1998) In Fig.1 the invariant mass distribution is generated in this model with different values of . Here we use . which is normalized to unity and independent on the value of

It can be clearly seen that the dependence is rather strong. The peak of the distributions significantly shifts from around GeV ( for GeV) to GeV (for GeV ). Considering the considerable uncertainties in both theory and experiment data, there is no significant disagreement in the recoil mass distribution of .

3. Bound on from inclusive and exclusive decays

The value of could be constrained from the exclusive decay modes
.
Note that although predictions of the standard effective Hamiltonian approach
are too low to account for the data of inclusive decay modes, the disagreement
in the exclusive decay modes are smallerAli and Greub (1998); Beneke and Neubert (2002).
Furthermore, the effective Hamiltonian approach can reproduce the correct patterns of
and which is observed in the experiments.
It implies that in exclusive decays modes, it may still play an important role, and
the interference between different contributions may also be importantChiang and Rosner (2002); Gronau and Rosner (2000).

Nevertheless, by saturating the current data of exclusive decays, the upper bound of the parameter can still be obtained. The decay amplitudes of decay modes and in this model read

(11) |

where . and are the form factors for and transition with momentum transfer . The value of is taken to be the effective one. i.e . In the calculations, we take GeV which corresponds to GeV and GeV.

The corresponding branching ratio can be evaluated through the following relation

(12) |

where is the momentum of one of the final state mesons in rest frame.

It is useful to define two kind of ratios which are independent of the parameter :

1) The ratio between and . This ratio is independent of the value of and only sensitive to the mixing. In this model one findsHe et al. (1998)

(13) |

In the following numerical calculations we take and Ball et al. (1996) as an illustration. This leads to a value of . Considering the CLEO data of Behrens et al. (1998), it follows that with the constraints from , this model can account for at most of the observed branching ratio. Note that the exact value of may vary with different sets of parameters , and ; the constraints from are only an order of magnitude estimate.

2) The ratio between and . In this model it is independent of the value of and details of mixing.

(14) |

The values of and in the BSW model Bauer et al. (1987) are which corresponds to , while from light cone QCD sum rule Ali et al. (1994); Ball (1998) and . Thus if this model gives the dominant contribution to these modes, the value of should be around 1. However, the current data gives a value of Behrens et al. (1998). This is a more clear and stronger constraint than the one from . With the observed small value of , this model can explain at most half of the branching ratio of and therefore is not the dominant mechanism of these processes. In Fig.2(c-f) the numerical results of branching ratios as a function of the effective coupling is given and compared with the data. As some inclusive decay modes have not yet been observed by the Babar and Belle collaborations, only the CLEO data are used in the numerical evaluations. It can be seen from the figure that the data of exclusive decay modes and impose strong constraints on the effective coupling. With these constraints, the maximum value of lies in the range:

(15) |

From Eq.(6) and (7), the branching ratio of inclusive decays as a function of is plotted in Fig.2(a-b) and compared with the data. In the decay the acceptance cut effects is taken into account, which leads to a 19 reduction from the calculation in Eq.(6). Given the upper bound of in Eq.(15) this model can still successfully reproduce the branching ratio within 1 range.

4. Prediction of radiative decay

From the effective Hamiltonian in Eq.(3), this model can
also contribute to the radiative decays into . Using relation
Eq.(The gluonic and decays into the meson), the ratio between and
can be predicted and found to be the same as in Ref.Ball et al. (1996)

(16) |

which is in good agreement with the data.

Furthermore, given the value of the effective coupling the decay rate of can be predicted. To this end let us first define the ratio

(17) |

which can be understood as the size of relative to . Taking and GeV which comes from the bounds from exclusive decays as an example, one finds

(18) |

Note that the strong coupling constant in the effective Hamiltonian has been separated from the effective coupling and absorbed in the matrix element of . It is expected that there is no significant running on the value of from energy scale to .

Since the radiative decay of is dominated by the process , the branching ratio of can be simply estimated as

(19) |

The observation of the process gives . Thus taking the maximum branching ratio of is estimated as

(20) |

The decay rate of can be estimated as follows

(21) |

Using the value of from Eq.(The gluonic and decays into the meson) one finds for the maximum branching ratio for

(22) |

Considering the detection efficiency of is about a few percent ( through ), it may be hard to find a signal of such a decay mode in BES due to limited statistics ( in BES samples are collected). But in the forthcoming CLEO-c project samples are planned to be produced. It is then promising to search for the signal and test the predictions from this model in the CLEO-c experiment.

###### Acknowledgements.

Y.F. Zhou acknowledges the support by the Alexander von Humboldt Foundation.## References

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