Partners of and Productions in Decays
Recently, Belle Collaboration has reported a resonant state produced in , which is called . This state is charged, so it can not be interpreted as an ordinary charmonium state. In this paper, we analyze the octet to which this particle belongs and predict the masses of mesons in this octet. Utilizing flavor SU(3) symmetry, we study production rates in several kinds of decays. The and decay channels, favored by Cabibbo-Kobayashi-Maskawa matrix elements, can have branching ratios of . This large branching ratio could be observed at the running factories to detect particles containing a strange quark. We also predict large branching ratios of the and () particle production rates in non-leptonic decays and radiative decays. Measurements of these decays at the ongoing factories and the forthcoming Large Hadron Collider-b experiments are helpful to clarify the mysterious particles.
Recently, there are many exciting discoveries on new hadron states especially in the hidden-charm sector. Among these discoveries, the most intriguing one is the new relatively narrow peak named found by Belle Collaboration in the invariant mass spectrum of in the decay mode 2007wga (). There is a large branching fraction for the following decay chain:
Mass and width of this particle are measured as:
The most prominent characteristic is that it is electric charged, that’s to say, this new particle can not be described as an ordinary charmonium state or a charmonium-like state such as . On the other hand, this particle can decay to with a large rate through strong interactions, so it involves at least four quarks , though there is not any further detailed information on its inner dynamics at present.
In order to elucidate this particle, many theoretical studies Maiani:2007wz (); Rosner:2007mu (); Meng:2007fu (); Bugg:2007vp (); Qiao:2007ce (); Gershtein:2007vi (); Lee:2007gs () have been put forward. This meson could be viewed as a genuine tetraquark state with diquark anti-diquark content which has a large rate to Maiani:2007wz (). Moreover based on QCD-string, two different four-quark descriptions are proposed in Ref. Gershtein:2007vi (): one can be reduced to the ordinary diquark-diquark picture and the other one can not. Besides this kind of explanation, it has also been identified as the resonance of Rosner:2007mu (); Meng:2007fu () as its mass is close to the thresholds of and . Within this picture, the authors in Ref. Meng:2007fu () explored the production of and . Short distance contribution to is neglected and the main contribution is from long distance re-scattering effect via . With proper parameters, they can successfully explain the much larger production rate of than that of . In Ref. Bugg:2007vp (), Bugg took this meson as a threshold cusp. Recently, Qiao also tried to explain this meson with the baryonium picture Qiao:2007ce (). Using the technique of QCD sum rules, Lee calculated the masses of this particle and its strange partner in Ref. Lee:2007gs ().
Whether or not these scenarios describe the true dynamics of , this strange meson indeed plays an important role in the charmonium spectroscopy. In the present paper, we do not intend to give an explanation of this meson’s structure, but we want to analyze its partners within SU(3) symmetry: the octet to which the meson belongs and the corresponding singlet meson. Up to now, these is no experimental information on these mesons except . The decays of meson provide a firm potential in searching for these exotic mesons Rosner:2003ia (); Bigi:2005fr (), just like the observed decay channel . We will investigate the possibilities to detect these mesons in decays. In doing this, we will analyze decay amplitudes with the assumption of SU(3) flavor symmetry: to construct effective Hamiltonian using flavor SU(3) meson matrixes. The decay amplitudes can also be studied by using Feynmann diagrams. In the discussion of production with the graphic technique, we only consider short-distance contributions and neglect soft final state interactions. Specifically, the considered decays are divided into three categories: Cabibbo-Kobayashi-Maskawa (CKM) allowed non-leptonic decays; CKM suppressed non-leptonic decays; radiative decays. The first kind of decays have similar branching ratios with the observed , while the second type of decays is suppressed by about one order in magnitude and we will show that the running factories could hardly detect this kind of decays. Radiative decay is a natural filter to exclude the 0-spin mesons and furthermore this kind of process may go through with a sizable branching ratio.
In the next section, we will analyze the octet of meson within flavor SU(3) symmetry and try to estimate their masses. We will construct the effective Hamiltonian using meson matrices and then use them to study the production rates of mesons in decays. In Sec.III, we will introduce meson which consists of three charm quarks, together with a brief discussion on its production in decays. We will summarize this note in the last section.
Ii The octet and the singlet
Just as stated above, involves at least four quarks in constituent quark model, and there is an octet which belongs to in flavor SU(3) symmetry. Generally, we can deduce the particles in this octet using group theory: these particles, under the name , , , , and , are shown in Fig. 1. Besides, there exists one singlet meson called . In constituent quark model, quark contents of these mesons are listed by:
In reality, quark is slightly heavier than quark which is one of the origins for SU(3) symmetry breaking. Accordingly, the singlet can mix with eighth component of the octet , in analogy with and . Physical particles, named and , are mixtures of them and can be expressed as:
The mixing angle can be determined through measuring decays of these two particles in future. For simplicity, we will assume the mixing is ideal, i.e. . In this case, the quark contents are:
All together, one can use the following meson matrix to describe these mesons:
With the quark contents given in the above, we are ready to estimate masses of these particles. Isospin analysis predicts the equal masses for the four mesons with neither open nor hidden strangeness: . For the mesons with a strange quark, the mass differences between the lighter quarks and the heavier quark are required. One can compare masses of and to get some information: the mass of is MeV larger than that of . In heavy quark limit , the light system will not be affected by different heavy quark systems, thus we can simply assume a similar difference for mesons which predicts the mass of around MeV. Because the mass of newly observed meson is not far from the threshold of , meson is regarded as the resonance of Rosner:2007mu (). Under this mechanism, we could give more precise predictions on the masses for other mesons using experimental results for the and mesons. Our results are listed in Tab. 1 and uncertainties in this table are from that of masses of the charmed mesons. In the heavy quark limit, mesons with the same light system can be related to each other. But if the particles are viewed as tetra-quark states, the effective strange quark mass in could be different from that in the usual mesons as the light systems in the two kinds of particles are different. If mesons are described as molecules, probably they would not belong to a full SU(3) nonet and the predicted masses may not be suitable. Currently, there is no better solution and we will use this assumption in the present study. The recent QCD sum rule study predicts the mass by Lee:2007gs ():
which is above the and threshold by about 160 MeV. More experimental studies are required to test this description.
|Meson||Constituent Meson||Mass(||Decay Mode|
Experimentalists have observed the particle through the with . Assuming S-wave decay for meson, the quantum numbers can be determined as Maiani:2007wz (). In order to detect the other mesons, experimentalists will choose the proper final states to re-construct them, thus the predictions on ’s strong decays are required. Using the flavor SU(3) symmetry and , we also list the strong decays of other mesons in Tab. 1. With the assumption , another kind of possible decay modes is Maiani:2007wz (), where denotes a light vector meson.
In order to explore the production in decays, one can construct the effective Hamiltonian at hadron level using meson matrices Savage:1989ub (). In the following, to construct the related effective Hamiltonian, we will assume the flavor SU(3) symmetry. In decays, the initial state forms an SU(3) anti-triplet. The transition at quark level is either or 111If the quark pair is generated from the QCD vacuum rather than directly produced by the four-quark operator, this kind of contribution is expected to be suppressed by since there is at least one hard gluon required to produce the quark pair., which is described by the effective electro-weak Hamiltonian:
where . and are color indices. The transition is CKM favored: , while the transition is suppressed by . To construct the effective Hamiltonian at hadron level, only the flavor structures needs to be concerned. The effective electro-weak Hamiltonian given in Eq. (8) can also be written as an SU(3) triplet: (i=1 (u), 2 (d), 3(s)), where the only non-zero elements are for CKM favored decays , and for CKM suppressed channels . The final state mesons can be described by two nonet matrices: and . The effective Hamiltonian at hadron level could be constructed as:
where the upper index labels rows and the lower labels columns.
The above effective Hamiltonian can be related to Feynmann diagrams with the one-to-one correspondence and the lowest order diagrams are given in Fig. 2. The second term in eq.(9) corresponds to the second diagram in Fig. 2 (called -recoiling diagram) in which the spectator light quark in meson enters into the heavy meson. If the spectator quark goes to the light meson, we call this kind of diagram (the third one in Fig. 2) as the -emission diagram which corresponds to the third term in the effective Hamiltonian. In order to estimate relative sizes of these terms, we have to analyze diagrams at quark level. Final state mesons move very slowly and thus the gluon generating the quark pair is soft: . Thus after integrating out high energy scales, decay amplitudes can be expressed as matrix elements of a soft four-quark operator between initial and final states. The first term in Eq. (9) corresponds to the annihilation diagram (the first one in Fig. 2), as flavor indices of and in this term are contracted with each other. This kind of diagram is expected to be suppressed in two-body non-leptonic decays. But here since the gluons are soft, decay amplitudes can also be expressed as time-ordered products of a soft four-quark operator and the interaction Hamiltonian which contains only soft fields, thus this kind of contribution is comparable with contributions from the second and third terms in Eq. (9). For SU(3) flavor singlet mesons and , there are additional contributions which are given by the last two terms in Eq. (9). One kind of typical Feynmann diagram is also shown in Fig. 2 as the last two diagrams and it is the contribution from the higher Fock states of and . Even in charmless two-body decays Williamson:2006hb (), this kind of gluonic contribution is sizable. Here we do not have any implication and thus one can not neglect it with any a priori.
With the effective Hamiltonian given in Eq. (9), we give the decay amplitudes for the first kind of non-leptonic decay channels in Table 2. These decays are induced by the CKM allowed transition and go through with a large decay rate (typically the same order with the observed ). The flavor SU(3) symmetry implies the following relations for decays:
where mass differences and lifetime differences of mesons are neglected which can not produce large corrections. Although all of these decays are expected to go through with large branching fractions, decay rates may differ from each other for distinct coefficients. Two of the decays in the first line have been observed experimentally, while the possibility to observe the other two channels is a little smaller as the daughter meson from is relatively more difficult to measure. The decays in the second line is contributed from the third term of effective Hamiltonian given in Eq. (9), which should also have similar production rates. Among these four channels, and can have large branching ratios and the final states ( or ) are easily to be measured on the experimental side. Thus measurements of the invariant mass distribution in these two channels are helpful to detect the particles and determine relative sizes of and . The other decays are less possible to be measured in the running factories as either or is produced in the final state. The forthcoming LHC-b experiments and Super-B factories can measure these decays, together with the decays.
For , the heavy quark decays into , and is produced from vacuum. Subsequently, , and the spectator can be transferred into , and the quarks left form a kaon. The other states can also be produced by selecting a different quark pair or changing the quark by quark. We give the decay amplitudes for non-leptonic decay channels induced by transition in Table 3. These decays are suppressed by CKM matrix elements . is one example of this kind of decays and the product branching ratio is:
where and Yao:2006px (). The uncertainties are from the experimental results for . In the above calculation, mass differences and lifetime differences are neglected again. From the branching ratio for this decay chain, we can see that this kind of process receives strong suppression. Furthermore, the detection of is more difficult than , thus it could hardly be measured at the present two factories. The relations for the decay can be derived similarly using the effective Hamiltonian which are also useful in searching for the mesons:
As pointed out in ref.xyz (), the study of charmonium like states production in decays is easier. Here we also consider the Z(4430) particle production in decays. In this case, the spectator is a quark, thus the initial state is very simple: a singlet of flavor SU(3) group. But the effective electro-weak Hamiltonian can form an octet: . The effective Hamiltonian at hadron level can be written by:
where the non-zero elements of the transition Hamiltonian are for CKM allowed channels and for CKM suppressed channels with a factor . The corresponding Feynmann diagrams are given in Fig. 3. The coefficients for distinct contributions are given in Tab. 4. The CKM matrix element for the decay channels induced by is , which is in the same order with that of : . Thus without any other suppressions, these decays also have similar branching ratios () with . The decays in the second part of Tab. 4 are suppressed by , which are expected to have smaller decay rates (). Furthermore, the SU(3) symmetry implies the following relations: