^{+}e

^{-}→J/ψ+η

_{c}in Bethe-Salpeter Framework

^{1}

^{2}

^{2}

^{*}

It is
shown that the virtual states and relative momentum of the constituents of the
bound states are very important for the cross section of the process e^{+}e^{-}→J/ψ+η_{c} with the help of the Bethe-Salpeter wave
function description of the bound states. The gauge invariance of the cross
section is demonstrated. The numerical results can explain the experimental
data.

Heavy quarkonium physics has been the traditional arena of Quantum Chromodynamics (QCD) since ^{1} for a certain process, the NR description of the bound states could not be straightforward applicable. In this case, more general framework, which is relativistic and robust to introduce enough number of parameters to describe the bound system, especially the off shell states of the heavy quarks, is needed. The Bethe-Salpeter (BS) wave function framework is one of the good choices. If various relativistic effects are found to be small or could be factorized from the static bound state, the BS framework will naturally leads to the NR descriptions.

In this paper, the cross section of the exclusive production process

The double charm pair production process in B factory energies is of special significance. These four (anti)charms can be respectively grouped into two colour-singlet pairs, hence the colour-octet mechanism never plays important role because of the relatively much smaller colour-octet matrix elements. One can then concentrate on the effects of relative movement and heavy quark off shell states. It has been found that the decay width, as well as the energy distribution of the

The ways of incorporating the relativistic corrections can be grouped into two: One is in the NRQCD framework, the other is to employ various relativistic wave functions of

On the other hand, the bound states can be described by BS wave functions [

Of course by the mention of this history we do not imply NRQCD cannot properly incorporate the spin effect of the bound state, but want to point out that it could be better to start from the completely relativistic framework, i.e., the full BS wave function without HQL, to investigate all the relativistic effects in the bound state, namely the off-shell states and the large relative movement of

The momentum of each line is explicitly written. are colour indices for the quark line. There is another diagram with all the arrows of three loop fermion propagators backward

propagators off-shell at the order of

Next section will be the introduction of the BS framework we employ, then followed by Section 3 for describing the calculation details. Here we do the loop integral directly in the momentum space rather than employing the Feynman parameters because of the complex form of the BS vertex. Such a method is quite general and can be applied to the calculation of much more complex production processes in the BS framework. We conclude and discuss at Section 4.

In quantum field theory, the BS wave function of a quarkonium is

where

for equal mass quark pair.

stand this point is that one can get the coordinate space BS wave function

ternal legs. Of course from such an observation we can only obtain the coupling vertex from the BS wave function in the special case that the B particle is on mass shell. However this is enough for our purpose since here we only study the case of the B particle production as the real physical particle. In the following concrete calculations, consistent with all the studies on the decay processes [

For incorporation of the inner information of the bound states obtained from their decay processes which we have investigated in the BS framework [

where

For pseudo-scalar meson

From comparing with the decay data, it has been found that the higher order Dirac structures do not contribute significantly to heavy quark bound states [

The ground state scalar wave function

which leave only one parameter

The invariant amplitude for the

with

and

The electromagnetic (U(1)) gauge invariance for this amplitude is explicit. When replacing the four-vector

because the totally antisymmetric tensor in

The key point in the calculation is the loop integral in Equation (12). In principle it is simple and straightforward, since there is no divergence so that no regularization and renormalization are needed. However, the complexity of the vertex function makes it difficult to employ the formal and conventional Feynman parameter formulations. it will be more difficult if one want to improve precision by employing a more general form of the vertex (more Dirac structures and more complex form of scalar function) rather than the simplified ones employed in this paper. Here we first integrate over the poles of the propagators and then do the numerical integral for the remaining 3-dimensional momentum. This method, though no help to give analytical results but gives a schematic way to do the calculation to obtain the numerical results for much more complex diagrams. As a matter of fact, one of the purposes for this paper is to explore a systematic and practical way to do calculations for production processes within the BS wave function framework, of which the complexity from the loop integrals is quite common.

Such a method is employed to calculate the high energy scattering amplitude with the help of the flow diagrams, which shows great advantage [

We denote the integral in Equation (12) as

Here

where

We first integrate

integral interval of

Here

with

and

with

The next step is the numerical integral of

The numerical results are listed in

. The center of mass energy is taken to be 10.58 GeV

(GeV) | |
---|---|

1 | 0.9 |

1.15 | 1.5 |

1.3 | 6.6 |

5.0 | |

1.5 | 1.6 |

1.6 | 2.2 |

1.7 | 3.1 |

1.8 | 4.7 |

1.9 | 8.1 |

2 | 12.2 |

2.1 | 74.6 |

2.25 | 8.2 |

2.5 | 1.5 |

3 | 0.4 |

4 | 0.05 |

1 to 1.9, the cross section varies very slowly. It goes up for larger

1. In the BS framework, when one takes into account the off shell states and large relative movement of the bound quarks, one can get a large contribution to the production cross section from the diagrams of

2. To get the best value of

The BS wave function framework can naturally take into account the off shell states of

A question left is how to combine with the calculations in [

There are two papers employing the diagrams like this paper. In [

The authors thank Z.-G. Si for discussion. D. Soper and J.-X. Wang is also thanked for comments on the U(1) gauge invariance. This work is partially supported by NSFC, NSF of Shandong Province with Grant Nos. JQ201101.