# New Charmonium States in QCD Sum Rules: a Concise Review

###### Abstract

In the past years there has been a revival of hadron spectroscopy. Many interesting new hadron states were discovered experimentally, some of which do not fit easily into the quark model. This situation motivated a vigorous theoretical activity. This is a rapidly evolving field with enormous amount of new experimental information. In the present report we include and discuss data which were released very recently. The present review is the first one written from the perspective of QCD sum rules (QCDSR), where we present the main steps of concrete calculations and compare the results with other approaches and with experimental data.

^{†}

^{†}journal: Physics Reports

###### Contents

## 1 Introduction

We are approaching the end of a decade which will be remembered as the “decade of the revival of hadron spectroscopy”. During these years several colliders started to operate and produce a large body of experimental information. At the same time new data came from the existing colliders and also from the accelerators. In Table 1 we give a list of the new charmonium states observed in these accelerators.

state | production mode | decay mode | ref. |
---|---|---|---|

belle1 | |||

bellegg | |||

bellez3930 | |||

belley3 | |||

bellecc | |||

yexp | |||

belle3 | |||

cdfy | |||

Abe:2007sya | |||

belle3 | |||

babar1 | |||

belleggs | |||

belle4 | |||

bellez | |||

belle4630 | |||

belle4 |

In what follows we will review and comment all this information belle1 ; bellegg ; bellez3930 ; belley3 ; bellecc ; yexp ; belle3 ; cdfy ; Abe:2007sya ; babar1 ; belleggs ; belle4 ; bellez ; belle4630 ; cdf ; d0 ; babarx ; babaree ; cleox ; cdfx2 ; pdg ; belleE ; babar2 ; cdf2 ; cdf3 ; babar09 ; belleD ; babar3 ; cleo ; belle0810 ; babary2 ; babary ; babar4 ; belle5 ; babar5 ; babar6 ; bellez2 ; babarz ; belleB0 ; babarB0 ; babary3 .

The study of spectroscopy and the decay properties of the heavy flavor mesonic states provides us with useful information about the dynamics of quarks and gluons at the hadronic scale. The remarkable progress on the experimental side, with various high energy machines has opened up new challenges in the theoretical understanding of heavy flavor hadrons.

### 1.1 New experiments

The -factories, the PEPII at SLAC in the U.S.A., and the KEKB at KEK in Japan, were constructed to test the Standard Model mechanism for CP violation. However, their most interesting achievement was to contribute to the field of hadron spectroscopy, in particular in the area of charmonium spectroscopy. They are colliders operating at a CM energy near 10,580 MeV. The pairs produced are measured by the BaBar (SLAC) and Belle (KEK) collaborations. Charmonium states are copiously produced at the -factories in a variety of processes. At the quark level, the quark decays weakly to a quark accompanied by the emission of a virtual boson. Approximately half of the time, the boson materializes as a pair. Therefore, half of the meson decays result in a final state that contains a pair. When these pairs are produced close to each other in phase space, they can coalesce to form a charmonium meson.

The simplest charmonium producing meson decay is: . Another interesting form to produce charmonium in -factories is directly from the collision, when the initial state or occasionally radiates a high energy -ray, and the subsequently annihilate at a corresponding reduced CM energy, as ilustrated in Fig. 1.

When the energy of the radiated -ray is between 4000 and 5000 MeV, the annhilation occurs at CM energies that correspond to the range of mass of the charmonium mesons. Thus, the initial state radiation (ISR) process can directly produce charmonium states with .

### 1.2 New states

Many states observed by BaBar and Belle collaborations, like the , , , , , , , , , , , and , remain controversial. A common feature of these states is that they are seen to decay to final states that contain charmed and anticharmed quarks. Since their masses and decay modes are not in agreement with the predictions from potential models, they are considered as candidates for exotic states. By exotic we mean a more complex structure than the simple quark-antiquark state, like hybrid, molecular or tetraquark states. The idea of unconventional quark structures is quite old and despite decades of progress, no exotic meson has been conclusively identified. In particular, those with quantum numbers should mix with ordinary mesons and are thus hard to understand. Therefore, the observation of these new states is a challenge to our understanding of QCD.

In 2003 a great deal of attention was given to pentaquark states, which may be exotic objects theory_penta ; mnnrl04 . After many negative results the study of these states was abandoned. However, in that same year the discovery by BELLE of the unusual charmonium state named X prompted a lively and deep discussion about the existence of exotic states in the charmonium sector (i.e. non pure states), exactly where, up to that moment, the calculations based on potential models had worked so well.

Establishing the existence of these states means already a remarkable progress in hadron physics. Moreover it poses the new question: what is the structure of these new states? The debate on exotic hadron structure was strongly revived during the short “pentaquark era”. Unfortunately, because of the uncertainties on the experimental side, it was very soon aborted. However, some interesting ideas were passed along to the subsequent discussion about the nature of the , and states, which is still in progress.

### 1.3 New quark configurations

Concerning the structure we can say that there are still attempts to interprete the new states as . One can pursue this approach introducing corrections in the potential, such as quark pair creation. This “screened potential” changes the previous results, obtained with the unscreened potential and allows to understand some of the new data in the approach lichao . Departing from the assignment, the next option is a system composed by four quarks, which can be uncorrelated, forming a kind of bag, or can be grouped in diquarks which then form a bound system maiani ; maiani2 ; maiani3 . These configurations are called tetraquarks. Alternatively, these four quarks can form two mesons which then interact and form a bound state. If the mesons contain only one charm quark or antiquark, this configuration is referred to as a molecule torn ; close ; swan1 . If one of the mesons is a charmonium, then the configuration is called hadro-charmonium volz12 . Another possible configuration is a hybrid charmonium manke ; luoliu . In this case, apart from the pair, the state contains excitations of the gluon field. In some implementations of the hybrid, the excited gluon field is represented by a “string” or flux tube, which can oscillate in normal modes swan1 .

The configurations mentioned in the previous paragraph are quite different and are governed by different dynamics. In quarkonia states the quarks have a short range interaction dominated by one gluon exchange and a long range non-perturbative confining interaction, which is often parametrized by a linear attractive potential. In tetraquarks besides these two types of interations, we may have a diquark-antidiquark interaction, which is not very well known. In molecules and hadro-charmonium the interaction occurs through meson exchange. Finally, in some models inspired by lattice QCD results, there is a flux tube formation between color charges and also string junctions. With these building blocks one can construct very complicated “stringy” combinations of quarks and antiquarks and their interactions follow the rules of string fusion and/or recombination suganuma ; rich_steiner . In principle, the knowledge of the interaction should be enough to determine the spatial configuration of the system. In practice, this is only feasible in simple cases, such as the charmonium in the non-relativistic approach, where having the potential one can solve the Schrödinger equation and determine the wave function. In other approaches the spatial configuration must be guessed and it may play a crucial role in the production and decay of these states.

### 1.4 New mechanisms of decay and production

The theoretical study of production and decays of the new quarkonium states is still in the beginning. Some of the states are quite narrow and this is difficult to understand in some of the theoretical treatments, as for example, in the molecular approach.

The narrowness of a state might be, among other possibilities, a consequence of its spatial configuration. Assuming, for example, that the is a set of quarks confined in a bag but with sufficient (or nearly sufficient) energy to decay into the two mesons and , why is it so difficult for them to do so? What mechanism hinders this seemingly simple coalescence process, which is observed in other hadronic reactions? In a not so distant past the same questions was asked in the case of the pentaquark. Now we know that the existence of this particle is, to say the least, not very likely. Nevertheless, as mentioned in a previous subsection, some of the ideas advanced in the pentaquark years might be retaken now in a different context to answer the question raised above. In fact, the pentaquark decay was energetically “superallowed”, since there was enough phase space available and no process of quark pair creation nor annihilation. Why was this quark rearrangement and hadronization difficult? It has been conjectured stech that the was in a diquark () - diquark () - antidiquark () configuration, such that, due to the not very well understood repulsive and/or attractive diquark interactions, the two diquarks were far apart from each other with the antiquark standing in the middle. In this configuration it was difficult for one diquark to capture the missing quark (to form a neutron), which was very far. These words were implemented in a quantitative model and this configuration was baptized “peanut”, since the three bodies were nearly aligned. Another pentaquark spatial configuration with small decay width was the equilateral tetrahedron with four quarks located in the corners and the antiquark in the center bgs . An other interesting conjecture based on lattice arguments was that, due to the dynamics of string rearrangement, the pentaquark constructed with three string junctions had, during the decay process, to pass through a very energetic configuration. Since its initial energy would be much less than that, this passage would have to proceed through tunneling and would therefore be strongly suppressed suganuma .

These ideas are relevant for the new charmonium physics because if they are multiquark states, as it seems to be the case, their spatial configuration will play a more important role both in production and decay.

The production of some of these states (such as, for example, the ) has been observed both in and colliders, the latter being much more energetic. The produced in electron - positron collisions comes from decays, whereas in reactions it must come from a hard gluon splitting into a pair, which then undergoes some complicated fragmentation process. In theoretical simulations grinstein , it was shown that it is very difficult to produce if it is composed by two bound mesons.

Some reviews about these states can be found in the literature swanson ; bauer ; rosre ; kz ; zhure ; ss ; volre ; egmr ; olsen . There are at least two reasons for writting a report on the subject. The first one is because this is a rapidly evolving field with enormous amount of new experimental information coming from the analysis of BELLE and BABAR accumulated data and also from CLEO and BES which are still running and producing new data. New data are also expected to come from the LHCb, which will start to operate soon and will be generating data with very high statistics for the next several years. In the present report we include and discuss data which were not yet available to the previous reviewers.

The other reason which motivates us to write this report is that, on the theory side each review is biased and naturally emphasizes the approach followed by the authors. Thus, in Refs. swanson and volre the authors present the available data and then discuss their theoretical interpretations making a sketch of the existing theories. In swanson attention is given to the conventional quark antiquark potential model and to models of the interaction between mesons which form a molecular state. In volre a very nice overview of different theoretical approaches such as potential models, QCD sum rules and lattice QCD was presented. The author works out some pedagogical examples, using general considerations and simplified assumptions. The present review is the first one written from the perspective of QCD sum rules, where we present the main steps of concrete calculations and compare the results with other approaches and with experimental data.

In the next sections we discuss the experimental data and the possible interpretations for the recently observed , and mesons.

## 2 QCD sum rules

### 2.1 Correlation functions

QCD sum rules are discussed in many reviews svz ; rry ; SNB ; col emphasizing various aspects of the method. The basic idea of the QCD sum rule formalism is to approach the bound state problem in QCD from the asymptotic freedom side, i.e., to start at short distances, where the quark-gluon dynamics is essentially perturbative, and move to larger distances where the hadronic states are formed, including non-perturbative effects “step by step”, and using some approximate procedure to extract information on hadronic properties.

The QCD sum rule calculations are based on the correlator of two hadronic currents. A generic two-point correlation function is given by

(1) |

where is a current with the quantum numbers of the hadron we want to study.

The fundamental hypothesis of the sum rules approach is the assumption that there is an interval in over which the correlation function may be equivalently described at both the quark and the hadron levels. The former is called QCD or OPE side and the latter is called phenomenological side. By matching the two sides of the sum rule we obtain information on hadron properties.

### 2.2 The OPE side

At the quark level the complex structure of the QCD vacuum leads us to employ the Wilson’s operator product expansion (OPE) ope .

In QCD we only know how to work analytically in the perturbative regime. Therefore, the perturbative part of in Eq.(1) can be reliably calculated. However, this does not yet imply that all important contributions to the QCD side of the sum rule have been taken into account. The complete calculation has to include the effects due to the fields of soft gluons and quarks populating the QCD vacuum. A practical way to calculate the vacuum-field contributions to the correlation function is through a generalized Wilson OPE. To apply this method to the correlation function (1), one has to expand the product of two currents in a series of local operators:

(2) |

where the set includes all local gauge invariant operators expressible in terms of the gluon fields and the fields of light quarks. Eq. (2) is a concise form of the Wilson OPE. The coefficients , by construction, include only the short-distance domain and can, therefore, be evaluated perturbatively. Non-perturbative long-distance effects are contained only in the local operators. In this expasion, the operators are ordered according to their dimension . The lowest-dimension operator with is the unit operator associated with the perturbative contribution: , . The QCD vacuum fields are represented in (2) in the form of vacuum condensates. The lowest dimension condensates are the quark condensate of dimension three: , and the gluon condensate of dimension four: . For non exotic mesons, i.e. normal quark-antiquark states, the contributions of condensates with dimension higher than four are suppressed by large powers of , where is the typical long-distance scale. Therefore, even at intermediate values of , the expansion in Eq. (2) can be safely truncated after dimension four condensates. However, for molecular or tetraquark states, the mixed-condensate of dimension five: , the four-quark condensate of dimension six: and even the quark condensate times the mixed-condensate of dimension eight: , can play an important role. The three-gluon condensate of dimension-six: can be safely neglected, since it is suppressed by the loop factor .

In the case of the (dimension six) four-quark condensate and the (dimension eight) quark condensate times the mixed-condensate, in general factorization assumption is assumed and their vacuum saturation values are given by:

(3) |

Their precise evaluation requires more involved analysis including a non-trivial choice of factorization scheme BAGAN . In order to account for deviations of the factorization hypothesis, we will use the parametrization:

(4) |

where gives the vacuum saturation values and indicates the violation of the factorization assumption SNB ; LNT ; SNTAU .

### 2.3 The phenomenological side

The generic correlation function in Eq. (1) has a dispersion representation

(5) |

through its discontinuity, , on the physical cut. The dots in Eq. (5) represent subtraction terms.

The discontinuity can be written as the imaginary part of the correlation function:

(6) |

The evaluation of the spectral density () is simpler than the evaluation of the correlation function itself, and the knownledge of allows one to recover the whole function through the integral in Eq. (5).

The calculation of the phenomenological side proceeds by inserting intermediate states for the hadron, , of interest. The current is an operator that annihilates (creates) all hadronic states that have the same quantum numbers as . Consequently, contains information about all these hadronic states, including the low mass hadron of interest. In order for the QCD sum rule technique to be useful, one must parametrize with a small number of parameters. The lowest resonance is often fairly narrow, whereas higher-mass states are broader. Therefore, one can parameterize the spectral density as a single sharp pole representing the lowest resonance of mass , plus a smooth continuum representing higher mass states:

(7) |

where gives the coupling of the current with the low mass hadron, :

(8) |

For simplicity, one often assumes that the continuum contribution to the spectral density, in Eq. (7), vanishes bellow a certain continuum threshold . Above this threshold, it is assumed to be given by the result obtained with the OPE. Therefore, one uses the ansatz

(9) |

### 2.4 Choice of currents

Mesonic currents for open charm mesons are given in Table 2.

state | symbol | current | |
---|---|---|---|

scalar meson | |||

pseudoscalar meson | |||

vector meson | |||

axial-vector meson |

From these currents we can construct molecular currents which can be eigenstates of charge conjugation and -parity. Let us consider, as an example, a current with for the molecular system. It can be written as a combination of two currents liuliu ; stancu0 :

(10) |

and

(11) |

Since the charge conjugation transformation is defined as: and , we get

(12) |

(13) |

Therefore, the current

(14) |

has positive . However, this current is not a -parity eigenstate. The -parity transformation is an isospin rotation of the charge conjugated current:

(15) |

(16) |

In the case of a charged molecular current with , it can also be written as a combination of two currents:

(17) |

(18) |

The charge conjugation transformation in these currents leads to

(19) |

(20) |

and the isospin rotation gives

(21) |

(22) |

Therefore, the current

(23) |

has positive -parity.

In the case of tetraquark currents, they can be constructed in terms of color anti-symmetric diquark states: , where are color indices of the color group, is the charge conjugation matrix, and stands for Dirac matrices. The quantum numbers of the diquark states are given in Table 3.

state | current | |
---|---|---|

scalar diquark | ||

pseudoscalar diquark | ||

vector diquark | ||

axial-vector diquark |

From these diquarks we can construct tetraquark currents which can be eigenstates of charge conjugation and -parity. In the case of a current it can be written as a combination of scalar and axial-vector diquarks:

(24) |

and

(25) |

It is interesting to notice that the structure of the currents in Eqs. (24) and (25) relates the spin of the charm quark with the spin of the light quark. This is very different from the spin structure of the heavy quark effective theory isgur . Heavy quark effective theory, in leading order in 1/M, possesses a heavy-quark spin symmetry. Therefore, hadrons can be classified according to the angular momentum and parity of the light fields only. This gives and mesons with identical properties.

Using the charge conjugation transformations one gets:

(26) |

(27) |

Therefore, the current

(28) |

has positive . The was used in Eq. (28) to insure that . As in the case of the molecular current, the current in Eq. (28) is not a -parity eigenstate. However, other combinations of tetraquark currents can be constructed to be -parity eigenstates.

In general, there is no one to one correspondence between the current and the state, since the current in Eq. (28) can be rewritten in terms of a sum over molecular type currents through the Fierz transformation. In the appendix, we provide the general expressions for the Fierz transformation of the tetraquark currents into molecular type of currents as given in Eq. (14). However, as shown in the appendix, in the Fierz transformation of a tetraquark current, each molecular component contributes with suppression factors that originate from picking up the correct Dirac and color indices. This means that if the physical state is a molecular state, it would be best to choose a molecular type of current so that it has a large overlap with the physical state. Similarly for a tetraquark state it would be best to choose a tetraquark current. If the current is found to have a large overlap with the physical state, the range of Borel parameters where the pole dominates over the continuum would be larger, and the OPE for the mass would have a better convergence. These conditions will lead to a better sum rule; this means that the Borel curve has an extremum or is flat, and the calculated mass is close to the physical value. Therefore, if the sum rule gives a mass and width consistent with the physical values, we can infer that the physical state has a structure well represented by the chosen current. In this way, we can indirectly discriminate between the tetraquark and the molecular structures of the recently observed states. However, it is very important to notice that since the molecular currents, as the one in Eq. (14), are local, they do not represent extended objects, with two mesons separated in space, but rather a very compact object with two singlet quark-antiquark pairs.

When the current is fixed, we proceed by inserting it into Eq. (1). Contracting all the quark anti-quark pairs, we can rewrite the correlation function in terms of the quark propagators, and then we can perform the OPE expansion of these propagators. For the light quarks, keeping terms which are linear in the light quark mass , this expansion reads:

(29) | |||||

where we have used the fixed-point gauge. For heavy quarks, it is more convenient to work in the momentum space. In this case the expansion is given by:

(30) | |||||

### 2.5 The mass sum rule

Now one might attempt to match the two descriptions of the correlator:

(31) |

However, such a matching is not yet practical. The OPE side is only valid at sufficiently large spacelike . On the other hand, the phenomenological description is significantly dominated by the lowest pole only for sufficiently small , or better yet, timelike near the pole. To improve the overlap between the two sides of the sum rule, one applies the Borel transformation

(32) |

Two important examples are:

(33) |

and

(34) |

for . From these two results, (33) and (34), one can see that the Borel transformation removes the subtraction terms in the dispersion relation, and exponentially suppresses the contribution from excited resonances and continuum states in the phenomenological side. In the OPE side the Borel transformation suppresses the contribution from higher dimension condensates by a factorial term.

After making a Borel transform on both sides of the sum rule, and transferring the continuum contribution to the OPE side, the sum rule can be written as

(35) |

If both sides of the sum rule were calculated to arbitrary high accuracy, the matching would be independent of . In practice, however, both sides are represented imperfectly. The hope is that there exists a range of , called Borel window, in which the two sides have a good overlap and information on the lowest resonance can be extracted. In general, to determine the allowed Borel window, one analyses the OPE convergence and the pole contribution: the minimum value of the Borel mass is fixed by considering the convergence of the OPE, and the maximum value of the Borel mass is determined by imposing the condition that the pole contribution should be bigger than the continuum contribution.

In order to extract the mass without worrying about the value of the coupling , it is possible to take the derivative of Eq. (64) with respect to , and divide the result by Eq. (64). This gives:

(36) |

This quantity has the advantage to be less sensitive to the perturbative radiative corrections than the individual sum rules. Therefore, we expect that our results obtained to leading order in will be quite accurate.

### 2.6 Numerical inputs

In the following sections we will present numerical results. In the quantitative aspect, QCDSR is not like a model which contains free parameters to be adjusted by fitting data. The inputs for numerical evaluations are the following: i) the vacuum matrix elements of composite operators involving quarks and gluons which appear in the operator product expansion. These numbers, known as condensates, contain all the non-perturbative component of the approach. They could in principle, be calculated in lattice QCD. In practice they are estimated phenomenologically. They are universal and, once adjusted to fit, for example, the mass of a particle, they must have always that same value. They are the analogue for spectroscopy of the parton distribution functions in deep inelastic scattering; ii) quark masses, which are extracted from many different phenomenological analyses and used in our calculation; iii) the threshold parametr is the energy (squared) which characterizes the beginning of the continuum. Typically the quantity (where is the mass of the ground state particle) is the energy needed to excite the particle to its first excited state with the same quantum numbers. This number is not well known, but should lie between and GeV. If larger deviations from this interval are needed, the calculation becomes less reliable.

## 3 The meson

Since its first observation in August 2003 by Belle Collaboration belle1 , the represents a puzzle and, up to now, there is no consensus about its structure. The has been confirmed by CDF, D0 and BaBar cdf ; d0 ; babarx . Besides the discovery production mode , the has been observed in pronpt production cdf ; d0 . However, searches in prompt production babaree and in or formation cleox have given negative results so far. The current world average mass is

(38) |

and the most precise measurement to date is , as can be seen by Fig.2 cdfx2 .

The mass of the is at the threshold for the production of the charmed meson pair pdg , and this state is extremely narrow: its width is less than 2.3 MeV at 90% confidence level.

### 3.1 Experiment versus theory

Both Belle and Babar collaborations reported the radiative decay mode belleE ; babar2 , which determines . Belle Collaboration reported the branching ratio:

(39) |

Recent studies from Belle and CDF that combine angular information and kinematic properties of the pair, strongly favor the quantum numbers belleE ; cdf2 ; cdf3 . In particular, in ref. cdf3 it was shown that only the hypotheses and are compatible with data. All other possible quantum numbers are ruled out by more than three standard deviations. The possibility is disfavored by the observation of the decay into babar09 and also by the observation of the decays into by Belle and BaBar Collaborations belleD ; babar3 . On the other hand, the possibility is disfavored by the observation of the decay into by BaBar Collaborations babarate . In the following we will asume the quantum numbers of the to be .

From constituent quark models bg the masses of the possible charmonium states with quantum numbers are: and , and lattice QCD calculations give lat1 and lat2 . In all cases the predictons for the mass of the charmonium state are much bigger than the observed mass. However, a more recent lattice QCD calculation gives lat3 and, therefore, this interpretation can not be totally discarded meng1 . In any case, the strongest fact against the assignment for bg ; elq , is the fact that from the study of the dipion mass distribution in the decay, Belle belle1 and CDF cdf2 concluded that it proceeds through the decay. Since a charmonium state has isospin zero, it can not decay easily into a final state.

The proximity between the mass and the threshold inspired the proposal that the could be a molecular bound state with small binding energy torn ; close ; swan1 ; vol ; wong ; bira ; faes . As a matter of fact, Tornqvist, using a meson potential model torn2 , essentially predicted the in 1994, since he found that there should be molecules near the threshold in the and channels. The only other molecular state that is predicted in the potential model updated by Swanson swanson is a molecule at 4013 MeV.

In ref. swan1 Swanson proposed that the could be mainly a molecule with a small but important admixture of and components. With this picture the decay mode was predicted at a rate comparable to the mode. Soon after this prediction, Belle Coll. belleE reported the observation of these two decay modes at a rate:

(40) |

This observation establishes strong isospin and G parity violation and strongly favors a molecular assignment for . However, it still does not completely exclude a interpretation for since the isospin and G parity non-conservation in Eq. (40) could be of dynamical origin due to mixing tera or even due to final state interactions (FSI) containing loops, such as . Although all the ingredients (specially the charm form factors mnnr02 ; bcnn05 ) for the relevant effective field theory are available, there are no quantitative results in the FSI approach yet.

The decay was also observed by BaBar Collaboration babarate at a rate:

(41) |

which is consistent with the result in Eq. (40).

It is also important to notice that, although a molecule is not an isospin eingenstate, the ratio in Eq. (40) can not be reproduced by a pure molecule. In ref. x24 it was shown that for a pure molecule

(42) |

In refs. belleD ; babar3 Belle and BaBar Collaborations reported the observation of a near threshold enhancement in the system. The peak mass values for the two observations are in good agreement with each other: MeV for Belle and MeV for BaBar, and are higher than in the mass of the observed in the channel by MeV. Since this peak lies about 3 MeV above the threshold, it is very awkward to treat it as a bound state. According to Braaten braaten , the larger mass of the measured in the decay channel could be explained by the difference between the line shapes of the into the two decays: and . In the decay of a narrow molecular state into its constituents , the width of distorts the decay line shape of the bralu . Therefore, the peak observed in the decay channel could be a combination of a resonance below the threshold from the decay and a threshold enhancement above the threshold. In this case, fitting the invariant mass to a Breit-Wigner does not give reliable values for the mass and width. However, in a new measurement belle0810 Belle has obtained a mass MeV in the invariant mass spectrum, which is consistent with the current world average mass for . Using this new data and taking into account the universal features of the -wave threshold resonance Braaten and Stapleton concluded that the is a extremely weakly-bound charm meson molecule brasta .

Other interesting possible interpration of the , first proposed by Maiani et al. maiani , is that it could be a tetraquark state resulting from the binding of a diquark and an antidiquark tera . This construction is based in the idea that diquarks can form bound-states, which can be treated as confined particles, and used as degrees of freedom in parallel with quarks thenselves clo2 ; wilc ; fried . Therefore, the tetraquark interpretation differs from the molecular interpretation in the way that the quarks are organized in the state, as shown in Fig. 1.

The drawback of the tetraquark picture is the proliferation of the predicted states maiani and the lack of selection rules that could explain why many of these states are not seen dfp .

The authors of ref. maiani have considered diquark-antidiquark states with and symmetric spin distribution:

(43) |

The most general states that can decay into and are:

(44) |

Imposing the rate in Eq.(40), they get . It is important to notice that a similar mixture between and molecular states with the same mixing angle x24 , would also reproduce the decay rate in Eq.(40).

The authors of ref. maiani also argue that if dominates decays, then dominates the decays and vice-versa. They have also predicted that the mass difference between the particle in and decays should be maiani ; polosa

(45) |

There are two reports from Belle belleB0 and Babar babarB0 Collaborations for the observation of the and decays that we show in Figs. 4 and 5.

Although the identification of the from the decay in these two figures is very clear, this is not the case for the decay, where the evidence for the existence of such a state is not so clear.

In any case, if one accepts the existence of the in the decay , these reports are not consistent with each other. While Belle measures belleB0 :

(46) |

and

(47) |

BaBar measures babarB0 :

(48) |

and

(49) |

In both measurements, the mass difference between the two states is much smaller than the prediciton in Eq.(45).

If is a loosely bound molecular state, the branching ratio for the decay is suppresed by more than one order of magnitude compared to the decay . The prediction in refs. swanson ; braaten2 gives:

(50) |

which is, considering the errors, still consistent with the data from BaBar.

Recentely BaBar has reported the observation of the decay babar09 at a rate:

(51) |

while the prediction from ref. swan1 gives

(52) |

While this difference could be interpreted as a strong point against the molecular model and as a point in favor of a conventional charmonium interpretation lichao , it can also be interpreted as an indication that there is a significant mixing of the component with the molecule. As a matter of fact, the necessity of mixing a component with the molecule was already pointed out in some works elq ; mechao ; braaten3 ; suzuki ; dong ; li1 . In particular, in ref. suzuki it was phenomenologically shown that, because of the very loose binding of the molecule, the production rates of a pure molecule should be at least one order of magnitude smaller than what is seen experimentally.

In ref. grinstein , a Monte Carlo simulation of the production of a bound state with binding energy as small as 0.25 MeV, obtained a cross section of about two orders of magnitude smaller than the prompt production cross section for the observed by the CDF Collaboration. The authors of ref. grinstein concluded that -wave resonant scattering is unlikely to allow the formation of a loosely bound molecule, thus calling for alternative (tetraquark) explanation of CDF data. However, it was pointed out in ref. Artoisenet:2009wk that a consistent analysis of molecule production requires taking into account the effect of final state interactions of the and mesons. This observation changes the results of the Monte Carlo calculations bringing the theoretical value of the cross section very close to the observed one. Thus, the question of interpretation of production at hadronic machines is not yet settled.

### 3.2 QCDSR studies of

Considering the as a state we can construct a current based on diquarks in the color triplet configuration, with symmetric spin distribution: , as proposed in ref. maiani . Therefore, the corresponding lowest-dimension interpolating operator for describing as a tetraquark state is given by:

(53) | |||||

where denotes a or quark.

On the other hand, we can construct a current describing as a molecular state: