Theory Overview on Spectroscopy
A theoretical overview of the exotic spectroscopy in the charm and beauty quark sector is presented. These states are unexpected harvest from the and hadron colliders and a permanent abode for the majority of them has yet to be found. We argue that some of these states, in particular the and the recently discovered states and , discovered by the Belle collaboration are excellent candidates for tetraquark states , with light quarks. Theoretical analyes of the Belle data carried out in the tetraquark context is reviewed.
Theory Overview on Spectroscopy
Ahmed Ali††thanks: Speaker.
Deutsches-Elektronen Synchrotron DESY, Notkestrasse 85, D-22607 Hamburg
The title of my talk is both ambitious and pretentious! I hasten to state that the mandate given to me is rather limited, namely to review the phenomenology of hadronic states discovered recently in the mass region of the charmonia and the bottomonia. Spearheaded by the experiments at the B factories and the Tevatron, with the experiments at the LHC as welcome new-comers, an impressive number of new states have been reported. Generically called , and , these states defy a conventional quarkonia interpretation; this certainly holds for the majority of them. Their gross properties, such as the spin-parity assignments, masses, production mechanisms and decay modes, have been discussed in a number of comprehensive reviews [1, 2].
There have been a number of more recent developments in the field of quarkonium spectroscopy and I will confine myself just to their discussion. They involve the observation of the two charged bottomonium-like resonances by the Belle Collaboration  in the and mass spectra that are produced in association with a single charged pion in annihilation at energies near the resonance. Here are the P-wave spin-singlet bottomonia states. Calling the charged particles and , their masses and the decay widths averaged over the five final states are, respectively, MeV, MeV, and MeV, MeV. The favoured quantum number assignments for both are . This discovery was preceded by the observation of the and states, also by the Belle Collaboration  in the reaction , with the masses MeV and MeV. These measurements yield hyperfine splitting in the bottomonium sector, defined as the mass difference between the -wave spin-singlet state and the weighted average of the corresponding -wave triplet states, , , with MeV and MeV. They are consistent with theoretical expectations and also with the hyperfine splitting measured in the charmonium sector MeV , consistent with zero. Theoretically expected widths of and are of order 100 keV , which are too small to be measured by Belle.
Still on the subject of , the BaBar collaboration  has presented evidence of its production in the decay , followed by the decay , in the distribution of the recoil mass against the at the mass MeV, which is consistent with the Belle measurements . The width of is consistent with the experimental resolution, and the reported product branching ratio is . In this, and also in , the first error is statistical and the second systematic. The isospin-violating decay is expected to have a branching fraction of about [8, 9], and the branching fraction ; hence, the measured product branching ratio is as anticipated theoretically. It is noteworthy that the decay , which is suppressed by at least an order of magnitude compared to the decay , has not been observed. The observation of the singlet -state in the charmonium sector has also been reported this year by the CLEO collaboration  in the process at the center-of-mass energy MeV. In fact, CLEO pioneered the technique of searching for peaks in the mass spectrum recoiling against the , and the resulting mass MeV measured by this method is consistent with an earlier measurement of the mass from the decay . The product branching ratio is in agreement with theoretical expectations, and is also very similar to what has been reported by Babar for the corresponding product branching ratio, quoted above. However, there is an intriguing hint in the CLEO measurements of the cross section for , which rises at MeV. Since this is close to the mass of the hadron , which is a candidate for the hidden tetraquark state, it would suggest that the mechanism has something to do with the rise in the cross section. This remains to be confirmed in the next round of precise experiments.
2 Current experimental anomalies
There is a number of anomalous features in the Belle data taken in the center-of-mass energy region near the mass. The first of these was reported some three years ago [12, 13] in the processes , measured in the center-of-mass energy range between 10.83 GeV and 11.02 GeV. The enigmatic features of the Belle data are (i) the anomalously large decay widths (or cross sections) for the mentioned final states, and (ii) the dipion invariant mass distributions recoiling against the and states, which are at variance with similar spectra measured in the transitions involving lower mass bottomonium states (with ). To quantify the problem, the reported partial widths are MeV and MeV. Compared to the corresponding partial decay widths of the lower three states, keV, keV, and keV, the production of the in the energy region near the is larger by two to three orders of magnitude. The order keV partial widths are well-accounted for in the QCD multipole expansion [14, 15] based essentially on the Zweig-suppressed process shown in Fig. 1 (left-hand frame). The dipion invariant mass spectrum anticipated in the QCD multipole expansion is shown on the example of the decay in Fig. 1 (right-hand frame) and compared with the data taken from the Belle collaboration at . They are in excellent agreement with each other. Not so, for the dipionic transitions measured in the region, in which the dipionic mass spectra are dominated by the scalar meson and the tensor meson (for the mode) and by the and mesons (for the mode). This is illustrated in Fig. 2 for the process which shows the distributions in the (left-hand frame) and in the helicity angle ( distribution (right-hand frame). The dipion mass spectrum measured near the clearly shows peaks at and . An interpretation of the process in terms of the production and decay of a tetraquark state [17, 18] (histograms and the solid curves) accounts well the experimental distributions. We will return to discuss the underlying dynamical model later in section 4 of this report.
Not only are the cross sections for () near the anomalously large by at least two orders of magnitude, the same holds for the production of the P-wave spin-singlet bottomonia states (), for which the production cross sections for and are also anomalously large . The ratios of the production cross-sections in the indicated final states relative to that for the production are as follows :
We have already commented on the anomalous production cross sections in the modes near the region. The ratios given in the last two equations above for the and are found to be of order unity, a feature which violates theoretical expectations as the processes involve heavy quark spin-flip, which are suppressed by in the amplitude. It is obvious that the production mechanisms of all five processes involving () and () are exotic. In particular, the true mechanisms at work avoid the Zweig-suppression seen in similar dipionic transitions and evade power suppression due to the spin-flip transitions for the case. It is worth recalling that no excess of the kind seen in the Belle measurements near the [12, 13, 4] is seen by them or any other experiment either at energies below or above the region. Any plausible theoretical explanation must account for all these features.
These measurements have invoked a number of theoretical ideas. Particularly interesting is the suggestion by Bondar et al. , in which the resonances and are assumed mostly of a ’molecular’ type due to their respective proximity with the and thresholds. Thus, the internal dynamics of the states and is dominated by the coupling to meson pairs and , respectively. In particular, the pair within the and is an equal mixture of a spin-triplet and spin-singlet with the relative phase orthogonal between the two resonances, i.e.,
Here and stand for the para- and ortho-states with negative parity. The assignments (2.0) would predict that the mass difference should be equal to that between the and masses. The observed mass difference of 46 MeV  is in neat agreement with this argument. The spin-structure in (2.0) also suggests that the resonances and have the same decay width. This again is in agreement within measurement errors with the Belle data : MeV and MeV. The maximal ortho-para mixing of the heavy quarks in the and resonances described by Eq. (2.0) also implies couplings of comparable strengths to channels with states of ortho- and para-bottomonium, leading to the following couplings of these resonances to the channels and :
where , and denote the polarization vectors of the corresponding spin-1 states, and and are the pion energy and its three-momentum, respectively; and are a priori unknown coupling constants to be determined by data. The amplitudes described by Eq. (2.0) applied to the decays and yield the right pattern of destructive and constructive interferences seen in the Dalitz distributions of these processes . All of these arguments are plausible. Further variations on the molecular theme and predictions can be seen in [20, 21, 22, 23].
However, the structure suggested in Eq. (2.0) is a postulate not yet seen in decays other than those of the . A particular case in point are the decays of the , where the available phase space for the decays and are much larger. Hence, the implications of Eqs. (2.0) and (2.0) should be, at least qualitatively, very similar to those discussed in the context of the Belle data from the region. This remains to be tested. In addition, there are also some specific features of the Belle data which do not go hand-in-hand with the usual understanding of a hadronic molecule, the closest example of which is the Deuteron. The masses of the and are above the respective thresholds. The Deuteron mass, on the other hand, lies below the threshold by about 2.2 MeV. Also, the decay widths of the and are not particularly small, as one would expect for a hadron molecule. On the contrary, their decay widths are similar in order of magnitude as that of the . This is also curious as the other ’hadronic molecule’ discussed at length in a similar context, namely the , has a much smaller (by at least an order of magnitude) decay width, with the current 90% C.L. limit being MeV .
In the rest of this writeup, I will take the point of view that all the five anomalous processes measured by Belle at energies near the mass [12, 13, 4] have very little to do with the decays. Following [17, 18, 25], I will argue here that the final states and are the decay products of the tetraquark , which lies in mass tantalizingly close to the mass. More precise experiments are needed to tell the two apart than is the case currently. In the context of the final states, this was suggested in [17, 18, 25] and the dynamical model was shown to be consistent with the observed cross sections. Also, the measured dipion invariant mass distributions show the predicted scalar-and tensor-meson resonant structure. Moreover, in the tetraquark context, it is easier to understand why the production cross sections for , which involves a transition, and for , which involves a transition, are comparable to each other. Detailed distributions, including the resonant and effects are still being worked out in the tetraquark picture.
3 Spectrum of bottom diquark-antidiquark states
Much of the discussion of the tetraquark states involves the concept of diquarks (and anti-diquarks) as effective degrees of freedom, which will be used here to calculate the mass spectra, production and decay of the tetraquark states. In particular, four-quark configurations in the tetraquarks are assumed not to play a dominant role. Following this, the mass spectrum of tetraquarks with , , and can be calculated using a Hamiltonian 
All diquarks, denoted here by are assumed to be in the color triplet , as the diquarks in the representation do not show binding . Here is the constituent mass of the diquark , is the spin-spin interaction between the quarks inside the diquarks, are the couplings ranging outside the diquark shells, is the spin-orbit coupling of diquark and corresponds to the contribution of the total angular momentum of the diquark-antidiquark system to its mass. The overall factor of is used customarily in the literature. As the isospin-breaking effects are estimated to be of order 5 - 8 MeV for the tetraquarks [25, 26], they are neglected in the mass estimates discussed below.
where the sum runs over the hadron constituents. The coefficient depends on the flavour of the constituents , and on the particular colour state of the pair. The constituent quark masses and the couplings for the colour singlet and anti-triplet states are given in . To calculate the spin-spin interaction of the states explicitly, one uses the non-relativistic notation  , where and are the spin of diquark and antidiquark, respectively, and is the total angular momentum. These states are then defined in terms of the direct product of the matrices in spinor space, , which can be written in terms of the Pauli matrices as:
which then lead to the definition such as . Others can be seen in .
The next step is the diagonalization of the Hamiltonian (3.0) using the basis of states with definite diquark and antidiquark spin and total angular momentum., There are two different possibilities : Lowest lying states and higher mass states . The states can be classified in terms of the six possible states involving the good (spin-0) and bad (spin-1) diquarks (here, is the parity and the charge conjugation)
i. Two states with :
ii. Three states with :
All these states have positive parity as both the good and bad diquarks have positive parity and . The difference is in the charge conjugation quantum number, the state is even under charge conjugation, whereas and are odd.
iii. One state with :
Keeping in view that for there is no spin-orbit and purely orbital term, the Hamiltonian (3.0) takes the form
The diagonalisation of the Hamiltonian (3.0) with the states defined above gives the eigenvalues which are needed to estimate the masses of these states. For the and states the Hamiltonian is diagonal with the eigenvalues 
Mass of the constituent diquark can be estimated in one of two ways: We take the Belle data  as input and identify the with the lightest of the states, , yielding a diquark mass . This procedure is analogous to what was done in , in which the mass of the diquark was fixed by using the mass of as input, yielding GeV. Instead, if we use this determination of and use the formula , which has the virtue that the mass difference is well determined, we get , yielding a difference of . This can be taken as an estimate of the theoretical error on , which then yields an uncertainty of about 30 MeV in the estimates of the tetraquark masses from this source alone. For the corresponding and tetraquark states, there are two states each, and hence the Hamiltonian is not diagonal. After diagonalising the matrices, the masses of these states are obtained.
We now discuss orbital excitations with having both good and bad diquarks. Concentrating on the multiplet, we recall that there are eight tetraquark states (), and the lightest isospin doublet is:
and the next in mass is: , and so on. Values of , and are estimated in . We identify the state with (in fact there are two of them, which differ in mass from each other by about 5 - 8 MeV, including isospin-breaking). This does not fix the quantity , which is the mass difference of the good and the bad diquarks, i.e. . Following Jaffe and Wilczek , the value of for diquark is estimated as MeV for , , and quarks. This is another source of potential uncertainty in estimating the tetraquark masses. The mass spectrum for the tetraquark states for with and states is plotted in Fig. 3 in the isospin-symmetry limit. It is difficult to quote a theoretical error on the masses shown, with MeV presumably a good guess. Other estimates of the tetraquark mass spectra in the charm and bottom quark sectors can be seen in [31, 32, 33].
3.1 Estimates of the charged tetraquark states
In the tetraquark picture, one also anticipates a large number of charged states whose mass spectrum can be calculated in an analogous fashion as for their neutral counterparts just discussed. We would like to propose that the two charged states and observed recently by the Belle Collaboration , and interpreted by them as the charged bottomonium states produced in the process and , are indeed charged tetraquark states with the quark content for the positively charged state (its charge conjugate being ). For the present discussion, they are produced in the decays of the tetraquark . According to this interpretation, the decay chains involve . A detailed dynamical model is under development with the aim of understanding the decay distributions in the kinematic variables available in these decays.
We have estimated the masses of the isospin partners of and , the two neutral tetraquark states, denoted as and . The non-diagonal mass matrix for the neutral states was, however, calculated numerically for . If we ignore the isospin-breaking effects in the tetraquark masses, which are small, then the charged counterparts have the masses GeV and GeV, given in Fig. 3. As involves one good and one bad diquark and involves two bad diquarks, including the -dependent term, the non-diagonal mass matrix gets modified to the following form
The two eigenvalues can be written as , with and , yielding
Using the default values of the parameters 
we have now the following predictions for the two charged tetraquark masses
These estimates are to be compared with the masses of the states and reported by the Belle Collaboration  MeV and MeV. They are in the right ball-park, but miss the measurements by approximately 30 MeV and 230 MeV, respectively. More importantly, the mass difference between the two states has been measured precisely  MeV. The expression for this mass difference using the Hamiltonian (3.0) is:
The smallest value for the mass difference (140 MeV) is obtained for , which goes up to 247 MeV for . Both are larger than the measurements. Thus, the Belle data suggests that the Hamiltonian used here has to be augmented with an additional contribution. As the masses of the observed states and are rather close to the thresholds and , respectively, this suggests that the threshold effects may impact on the masses and mass differences presented here.
4 Tetraquark-based analysis of the processes
The cross sections and final state distributions for the processes near the have been presented in the tetraquark picture in  improving the results on the process published earlier . The distributions for the process calculated in  had a computational error, which has been corrected in the meanwhile (see the Erratum in ). These analyses are briefly reviewed in this section. Concentrating on the processes , there are essentially three important parts of the amplitude to be calculated consisting of the following:
(i) Production mechanism of the vector tetraquarks in annihilation. To that end, we derive the equivalent of the Van-Royen-Weiskopf formula for the leptonic decay widths of the tetraquark states and made up of a diquark and antidiquark, based on the diagram shown in Fig. 4 (left-hand frame).
Here, and are the electric charges of the constituent diquarks of the and , is the fine-structure constant, the parameter takes into account differing sizes of the tetraquarks compared to the standard bottomonia, with anticipated, and GeV  is the square of the derivative of the radial wave function for taken at the origin. Hence, the leptonic widths of the tetraquark states are estimated as