# Study - mixing from transition

###### Abstract

Various processes have been proposed previously to study - mixing through transition. Here we investigate in detail the difference between and transitions. It is found that the transition can provide additional constrains to the parameters of and mesons. Proposal is made to study - mixing from reaction at the upgraded Beijing electron positron collider with the BESIII detector.

###### pacs:

14.40.Cs, 13.25.Gv, 12.39.Mk## I Introduction

More than thirty years after their discovery, today the nature of light scalar mesons and is still in controversy. They have been described as quark-antiquark, four quarks, molecule, quark-antiquark-gluon hybrid, and so on. Now the study of their nature has become a central problem in the light hadron spectroscopy.

In the late 1970s, the mixing between the and resonances was first suggested theoretically in Ref.first (). Its mixing intensity is expected to shed important light on the nature of these two resonances, and has hence been studied extensively on its different aspects and possible manifestations in various reactions n1 (); n2 (); n3 (); n5 (); im (); uim (); n4 (); y2 (); y4 (); y5 (); y10 (); y11 (); eb (); we (); hanhart (). But unfortunately no firm experimental determination on this quantity is available yet. Obviously, more solid and precise measurements on this quantity are needed, such as by polarized target experiment on the reaction n3 (), decays im (); we (), and reactions from WASA at COSY y10 (). In Ref. we (), we pointed out that the - mixing intensity can be precisely measured through reaction at the upgraded Beijing electron positron collider with the BESIII detector.

In all these previous proposals, the is produced first, then transits to the by the - mixing , i.e., transition. In this article, we investigate in detail the difference between and transitions. We define two kinds of mixing intensities and for the and transitions, respectively. We find there are some differences between them. We find that has more dependence on the parameters of , especially the , while the has more dependence on the parameters of , especially . For this reason, using the reaction of to study only the mixing is not perfect enough. For better determination of all relevant parameters for the and mesons, it would be useful to find some reaction to study mixing in addition. Recently, CLEO Collaboration reported an experimental study of the reaction cleo (). The resonances are clearly showing up and dominant. ¿From isospin symmetry, the should be produced with the same rate as . This may provide a nice place for studying the - mixing from transition by reaction. From our estimation, more than 300 events can be reconstructed by the BESIII detector in the narrow peak with a width of about 8 MeV around the mass of 990 MeV in the invariant mass spectrum.

## Ii The - Mixing Amplitude

The basic theory for the - mixing was already pointed out by Achasov and collaborators first (). For the nearly degenerate (isospin 1) and (isospin 0), both can decay into . Due to isospin breaking effect, the charged and neutral kaon thresholds are different by about 8 MeV. Between the charged and neutral kaon thresholds the leading term to the - mixing amplitude is dominated by the unitary cuts of the intermediate two-kaon system and proportional to the difference of phase spaces for the charged and neutral kaon systems.

Considering the - mixing , the propagator of can be expressed as n1 () :

(1) |

where and are the denominators for the usual propagators of and , respectively :

(2) | |||||

(3) | |||||

(4) | |||||

(5) |

The is the mixing term. From first (); n3 (), the mixing due to loops gives

(6) | |||||

Since the mixing is mainly coming from the loops, we have , and this is the amplitude of - mixing . ¿From Eq.(6), the becomes large only when the is between the and , so it is a narrow peak with the width of about .

## Iii Two types of reaction and mixing intensity

### iii.1 Two types of reaction of - mixing

There are two types of reaction which can be used to study - mixing : and .

For the reaction as shown by the Feynman diagram in Fig.1, the influence of various X and Y on the - mixing can be removed by its comparison to the corresponding reaction as shown in Fig.2. We define the mixing intensity for the transition as the following

(7) |

where is the invariant mass squared of two mesons in final state. With Eqs.(1-6), one can get the as:

(9) | |||||

Similarly, for the reaction , we define the mixing intensity for the transition and get its formula as the following

(11) | |||||

We can redefine that:

(12) | |||||

(13) | |||||

(14) | |||||

(15) | |||||

(16) |

The becomes large when the is between the and . In this mass range, and are real, meanwhile and are imaginary; then and become:

(17) | |||||

(18) |

### iii.2 Predictions of and from various models and experiment information

¿From equations given above, one can see that the mixing intensity depends on , , , , and . Various models for the structures of and give different predictions for these coupling constants and mass eb (); m2 (); m3 (); m6 () as listed in Table 1 by No.A-D. There have also been some experimental measurements on these coupling and mass constants ge1 (); ge2 (); ge4 (); ge5 (); m5 (); zou1 (); zou2 (); bes1 () as listed by No.E-H. The corresponding predictions for the and from these various theoretical and experimental values of the coupling constants are calculated. In the calculation, the masses for , , and are taken from PDG2008 pdg06 () as MeV, MeV, MeV and MeV, respectively. We give the value of and at the MeV in Table 1 and the dependence of - mixing intensities and vs two-meson invariant mass in Fig.(3). There is obviously some difference between these two mixing intensities.

No. | model or experiment | ||||||||
---|---|---|---|---|---|---|---|---|---|

A | model eb () | 983 | 2.03 | 1.27 | 975 | 0.64 | 1.80 | 0.023 | 0.010 |

B | model eb () | 983 | 4.57 | 5.37 | 975 | 1.90 | 5.37 | 0.068 | 0.062 |

C | model m2 (); m6 (); kk () | 980 | 1.74 | 2.74 | 980 | 0.65 | 2.74 | 0.21 | 0.15 |

D | model m3 () | 980 | 2.52 | 1.97 | 975 | 1.54 | 1.70 | 0.005 | 0.006 |

E | SND ge1 (); ge2 () | 995 | 3.11 | 4.20 | 969.8 | 1.84 | 5.57 | 0.088 | 0.089 |

F | KLOE ge4 (); ge5 () | 984.8 | 3.02 | 2.24 | 973 | 2.09 | 5.92 | 0.034 | 0.025 |

G | BNL m5 () | 1001 | 2.47 | 1.67 | 953.5zou2 () | 1.36zou2 () | 3.26zou2 () | 0.019 | 0.014 |

H | CB zou1 () | 999 | 3.33 | 2.54 | 965bes1 () | 1.66bes1 () | 4.18 bes1 () | 0.027 | 0.023 |

### iii.3 Discussion on the difference of two mixing intensities and

We know from Eqs.(17,18), if the , and (i=a or f) are much smaller than 1, we will have . Then from these mixing intensities we only get the no matter which type of reactions one makes the measurement.

However, are these quantities really much smaller than 1 ? For between the and , , we have

(19) |

Here is about 100 MeV from many experiments, MeV, then one gets and , which are not so small.

For and with , and , since from Table 1 the smallest and are 0.5 and 5.7 respectively among various experimental determinations, the and are also larger than 0.1 and are not small enough to be neglected.

Then from Eqs.(17,18), one can see that besides the common numerator, the has additional dependence on the parameters of while the has additional dependence on the parameters of .

¿From above analysis, we understand why is different from as shown in Table 1 and Fig.3. Both mixing intensities depend on six parameters: , , , , , , which are all important for understanding the nature of the and mesons, but are not well determined yet. Therefore to measure in addition to will be very useful for pinning down these parameters.

To further demonstrate the importance of measuring the in addition to the , a typical example is given as the following.

No. | (MeV) | (GeV) | (GeV) | (MeV) | (GeV) | (GeV) |
---|---|---|---|---|---|---|

1 | 980 | 3.2 | 4.2 | 980 | 1.5 | 4.0 |

2 | 980 | 3.2 | 3.0 | 980 | 1.5 | 5.12 |

For two sets of parameters given in Table.2, we can see that the set No.1 is close to the SND values in Table.1. The set No.2 changes the not well measured and in their experimental uncertainties. We plot the corresponding diagrams of - mixing intensities and vs two-meson invariant mass as shown in the Fig.4. The two sets of parameters give almost identical but very different .

## Iv Possibility of measuring from

The Feynman diagram for the reaction is shown in Fig 5.

The invariant amplitude for this reaction is

(20) |

The coupling constant can be determined by the reaction .

(21) |

According to Ref. pdg06 (), MeV, MeV, and . The gives the dominant contribution of about 75.1% cleo (). So we have . From the formulae of Eqs.(1-6) and parameters of setting No.H listed in Table 1 , we can calculate the . It is and the invariant mass spectrum of for is shown in Fig.6.

The branching ratio of reaction is pdg06 (). At the upgraded Beijing electron positron collider with BESIII detector, about events and hence about events can be collected per year. From the branching ratio of , more than events are expected to be collected. Considering the reconstruction efficiency of 30%, more than 300 events can be reconstructed for this channel. Since all these events should concentrate in a narrow region of about 8 MeV around 990 MeV in the invariant mass spectrum, the narrow peak should be easily observed, hence the should be able to be measured by this reaction.

## V Summary

Various processes have been proposed previously to study - mixing through transition. In this article we investigate in detail the difference between and transitions. Two corresponding mixing intensities and are defined. It is found that besides the common numerator, the has additional dependence on the parameters of while the has additional dependence on the parameters of . Therefore to measure transition in addition to the transition will be very useful for pinning down these parameters. We examine the possibility of measuring the - mixing from for at the upgraded Beijing Electron Positron Collider with BESIII detector and find it is feasible.

Acknowledgements This work is partly supported by the National Natural Science Foundation of China (NSFC) and by the Chinese Academy of Sciences under project No. KJCX3-SYW-N2.

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