# Hadron physics potential of future high-luminosity -factories at the and above

## Abstract

We point out the physics opportunities of future high-luminosity -factories at the resonance and above. Currently the two -factories, the SuperB factory in Tor Vergata, Italy and the Belle II factory in KEK, Japan, are under development and are expected to start operation in 2017 and 2016, respectively. In this paper we discuss numerous interesting investigations, which can be performed in the center-of-mass energy region from the and up to 11.5 GeV, where an efficient data taking operation should be possible with the planned -factories. These studies include abundant production and decay properties; independent confirmation and if found, exhaustive exploration of Belle’s claimed charged bottomonia; clarification of puzzles of interquarkonium dipion transitions; extraction of the light quark mass ratio from hadronic decays; analysis of quarkonium and exotic internal structure from open flavour decays, leading to severe symmetry violations; clarification of whether a hybrid state has similar mass to the bottomonium, making it a double state; searches for molecular/tetraquark states that should be more stable with heavy quarks; completion of the table of positive-parity mesons and study of their basic properties; production of heavy baryon pairs, that, following weak decay, open vistas on the charmed baryon spectrum and new channels to study CP violation; confirmation or refutation of the deviation from pQCD of the pion transition form factor, by extending the reach of current analysis; and possibly reaching the threshold for the production of triply-charmed baryons. If, in addition, the future colliders can be later upgraded to 12.5 GeV, then the possibility of copious production of pairs opens, entailing new studies of CP violation and improved, independent tests of the CKM picture (through determination of ), and of effective theories for heavy quarks.

###### pacs:

12.15.Hh and 12.38.Qk and 12.39.Hg and 13.30.Ce and 13.20.Gd and 14.20.Lq and 14.20.Mr and 14.40.Nd and 14.40.Pq and 14.40.Rt###### Contents:

## 1 Introduction

In the last years we have witnessed a renaissance of hadron spectroscopy caused by a series of discoveries made at -factories. Before -factories, most of the known states in the spectrum of charmonium and bottomonium lay below the open-flavour thresholds, a well-developed phenomenology based on potential quark models was quite successful in describing properties of these states, and, in particular, threshold effects could be cast into constant mass shifts. However the situation changed dramatically after -factories started operation and data collection. A lot of new states were observed, the majority of them lying above the open-flavour thresholds in the spectra of both charmonium and bottomonium. Many such states in fact are close to various thresholds, so that the threshold effects appear to be the dominant feature, and a drastic departure from the Breit–Wigner form in line shapes is expected, especially when the coupling to the corresponding open-flavour channels is in an wave. At the same time, a lot of theoretical ideas were put forward to explain the data. As a result, while some experimental observations remain not yet well understood theoretically, a large number of theoretical predictions do call for improvements in the experimental situation, which are expected to be achieved at the next-generation -factories, such as SuperB and Belle II.

This paper is mostly devoted to the heavy quarkonium studies at the centre-of-mass energy above the (4S). We will review available experimental results and different theoretical models applicable in the heavy quarkonium physics with a significant emphasis on unresolved theoretical issues. The potentially important experimental studies in this area, feasible for the future high-luminosity -factories, will also be discussed. In particular, the SuperB experiment plans to collect during several years of running about 75 ab at the (4S) and a few ab at the (5S) and, probably, at the (6S) and around. With expected integrated luminosity significantly larger than previously available at -factories, such experiment will become a valuable source of experimental data which not only will allow to improve statistics for previously studied reactions, but it will enable to study various rare decays and reactions, not achievable at present.

A large part of our discussion will be the isoscalar vector hidden-bottom state that resides in this region, with mass and width, according to recent measurements (1),

(1) |

In the framework of the quark
model, this state is a conventional meson, where
is the radial quantum number, while , , and denote the quark spin,
the quark–antiquark angular momentum, and the total spin, respectively^{1}

We will then extend the discussion to larger energies, particularly the resonance, but also several interesting thresholds expected above it. To keep the article within a manageable size we have limited ourselves to brief comments on most of the topics. We hope that the references given will be a reasonable starting point for the interested reader.

Measurements that can be mostly conducted at the , and much of the flavour discussion involving decays, have been the object of other studies and different theoretical approaches, so we have purposefully avoided dwelling on them here. We have included some discussion on quarkonium states, but the reader may want to refer to the quarkonium working group report (2) for extended coverage of many topics.

## 2 Closed-flavour analysis at

### 2.1 Dipion transitions from

The , well above open-bottom threshold, decays mainly to open-flavour channels. Closed-flavour decays are however also easily observable, and those with the largest branching fractions are dipion transitions into other bottomonium states. They are an important source of information on the nature of this state. Two types of dipion transitions can be identified: transitions without spin flip ( with ) and transitions with spin flip ( with ).

#### Dipion transitions without -quark spin flip

Dipion transitions with measured by Belle (3) peak at the same centre-of-mass energy (see Fig. 1),

(2) |

and the deviation of this quantity (more than ) from the maximum position in the hadronic cross section (vertical dashed line in Fig. 1) has been discussed theoretically assuming different interpretations. In particular, - rescattering was suggested in (6) as a possible mechanism responsible for the peak shift in the dipion invariant mass distribution.

Another interesting feature of the distribution shown in Fig. 1 is an almost zero cross section level outside the resonance region. It indicates that the states cannot be produced in the continuum, but only through the resonance formation mechanism. In contrast, the production cross section level is rather high at all energies below the mass threshold. This difference remains to be explained theoretically. Measurements of the exclusive channel cross section as a function of the centre-of-mass energy may shed some light on this difference as well as on the discussed above mass deviation.

Furthermore, Belle reported (7) observation of an unexpectedly large () production at the energy comparing with one at the . In particular, the measured values

(3) |

appear to be by about two orders of magnitude larger than
similar dipion decays of , with ^{2}

Another question related to the dipion transitions from states is the shape of the dipion invariant mass spectrum. Before the observation of the and dipion decays, there was a question about the peculiar dipion invariant mass spectrum of the transition (4); (17). In analogous heavy quarkonium transitions, such as and , the invariant mass distribution shows a single broad peak towards the higher end of the phase space (see, for example, the first plot in Fig. 2). In the transition, however, the structure is quite different, with two bumps showing up (the second plot in Fig. 2). Similar double-bump structures were observed later in higher dipion transitions (see Figs. 3 and 4). In particular, the invariant mass distribution for the transition was measured first by the Belle Collaboration (18), and later on by the BaBar Collaboration, together with the (9). Finally, measurements of the transitions, with , were reported by the Belle Collaboration (7); (15); (16).

Compounding this problem, one should notice that the kinematically allowed phase space is quite different in various transitions. For , the dipion invariant mass is limited up to about 560 MeV, similarly to the analogous charmonium transition . On the other hand, most of the transitions having a more complicated dipion invariant mass distribution have larger phase space, with the exception of the .

From the theory point of view, these double peak structures were studied by a number of authors, for reviews, see (19); (20); (21). The problem was further investigated in recent years (see, for example, (10); (5); (22)). Among various explanations, it is interesting to notice the proposal of an isovector exotic state in the bottomonium mass region (21); (23); (24).

A small bump close to the lower end of the mass distribution in the was noticed in (25). There has been no agreement whether the -wave final state interaction (in agreement with the strong (or ) signal in the reaction (26)), or rather a relativistic correction, is the key to understanding the structure in the region 400-600 MeV.

When the emitted pions are soft, an effective chiral Lagrangian can be used, with first steps taken already at the time of the charm discovery (27). At lowest order each channel requires four unknown parameters (28) that cannot be measured elsewhere, so that the predictive power is very limited.

Another issue is the importance of open-bottom coupled
channels (29); (30); (31). They cannot be simply
neglected in the effective field theory, because the gap between heavy quarkonium
and open-flavour heavy meson thresholds is often small compared with the hard scale
of 1 GeV. Assuming these thresholds are more important than the contact terms
described by the chiral Lagrangian^{3}

In fact, theorists have provided (10); (5) a parameter-free description of the dipion spectra stressing the role of the Adler Zero Requirement (AZR) imposed on the amplitude, as well as that of the Final State Interaction (FSI). Predictions of the approach (10); (5) are shown in Figs. 2 and 3 with the green dashed line, as well as in Fig. 5. In particular, dips in the spectrum are explained by the AZR which suppresses the corresponding amplitude at small pion momenta. In the meantime, the peak structure at the lower end of the dipion invariant mass spectrum (see Fig. 5) is explained in (10) to be due to an interplay of the FSI and AZR. Then the angular distributions ( is the angle between the initial and the ) are presented in (10); (5) for various transitions and are compared with the data from (7); (4) (see Fig. 6).

In addition, three-dimensional plots for the differential decay probability for various dipion decays were suggested and presented in (5) (see examples shown in Fig. 7), where the dimensionless variable is defined as

While general features of the data have been correctly captured, there remain details to be understood. In the second panel of Fig. 2, the dip predicted in the is deeper than experimental data. A similar result was also found in the effective Lagrangian formalism taking into account the FSI (21), where the dip was made consistent with the data with the help of an additional theorised isovector state. Indeed two exotic isovector structures and have been reported in decays, see Sec. 2.2, in all of the , and dipion modes. It is important therefore to revisit the lower transitions, including now the effect of possible states, and to perform a systematic study of all the data for transitions to understand the true mechanism of these decays.

Experimentally, it would be very helpful to measure the dipion and also the invariant mass distributions in the precisely, so that one can extract the information of coupling of the states to the . The information can then be fed back to the lower dipion transitions, eventually providing a deeper understanding of these puzzling decays.

The future high-luminosity -factories could contribute more precise data on the lower end of the mass distribution, where Belle data are somewhat insufficient, in order to observe these nontrivial structures in the dipion transitions. Moreover, this is also a region where the more “normal” dipion transitions such as the show striking differences with the “abnormal” ones like the : the lower end is rather suppressed for the former while it is enhanced for the latter.

Next, particular attention should be paid to the region around 1 GeV (in since the other final states do not have enough phase space). Detailed knowledge in this 1 GeV region would be helpful in understanding the nature of both the and the light scalar . The transition offers a rather unique possibility for studying light scalar mesons in the sense that both the initial and final particles (besides the pions) are flavour singlet states. In the limit, the production couplings should be equal for up, down, and strange flavours. Hence, this transition is able to provide information on the light scalar mesons complementary to those from the decays such as and .

Precise measurements of the transition in the region near the threshold would also be quite useful, because the couples strongly to two kaons. The rate for this reaction has been found to be about 9 times smaller than the dipion channel, with a partial width of about 0.067 MeV (7).

Additional important information on the dipion transitions can be obtained with a -factory running at the . The decay widths, dipion invariant mass spectra and angular distributions in the transitions, with , should be compared to ones obtained at the . Due to its larger mass, the open-beauty mass threshold effects should be smaller than for the and contributions from different processes can be somewhat separated.

#### Dipion transitions with -quark spin flip

Nominally, inverting the spin of the heavy quark in a decay is a process suppressed by a factor . Data however do not always make it obvious. There is a recent Belle Collaboration observation of the and states produced via in the region (34) (with the data sample of 121.4 fb collected near the peak).

It raises yet another question related to dipion transitions since the reported ratios

(4) |

are unexpectedly large. Indeed, while the pair is in the spin-triplet state in the vectors , it appears to be in spin-singlet state in the axials . This implies that, unlike the transition, proceeding without heavy-quark spin flip, naively such a flip must occur in , so that the amplitude for the latter process is expected to be suppressed by when compared with that for the former transition.

Summarising this subsection devoted to dipion transitions, it is important to notice that, with the data sample of about 1 ab collected by a future -factory near the , not only high-statistics data will be available for dipion decays with charged pions, but similar decays with two neutral particles (pions and/or ’s) in the final state should be readily accessible. Yet another piece of critical information can be obtained with a few hundred fb taken at the .

### 2.2 states

In 2011 the Belle Collaboration announced the first observation of two charged bottomonium-like states, and , in five different decay channels of (, and , ) (15); (16), with averaged masses and widths (16)

(5) | |||

(6) |

These structures are produced from, and reconstructed in, conventional
states, so it is natural to assume that they contain a quark pair.
However, for ’s being electrically charged, a pure assignment is discarded.
These states are still in want of confirmation by other experiments and their
nature is not well understood yet. It is clear however that they provide an
excellent test ground for QCD. In particular, these newly observed states may be
relevant for understanding the dipion , in particular
, transitions. Given the proximity of the observed states to
two-body thresholds ^{4}

If an exact heavy-quark symmetry is assumed (implying the limit ), a proper basis to consider -wave pairs consists of the direct product of the eigenstates of the spin operators for the heavy-quark pair and the light-quark pair, , with and . Then the wave functions of the states can be represented as (35)

(7) | |||||

where, in the first column, the quantum numbers are quoted in the form with and being the isospin and total spin of the state, and with the superindices denoting its - and -parity, respectively.

This provides a possible explanation for the anomalous ratios (4) as no suppression is expected for the processes proceeding through the intermediate states, as cascades (35):

because components with both heavy-quark spin orientations have equal weight in the ’s wave functions. Besides that, representation (7) predicts the interference pattern for the amplitudes of the processes proceeding through the and . In particular, for the reactions (), because of the different relative sign between the two components of the wave function, Eq. (7) predicts constructive interference between the peaks and destructive interference outside this region (35). In the meantime, the interference pattern is quite opposite for the reactions . This prediction is consistent with the recent Belle data (16) (small dip between peaks in Fig. 8 versus a well-pronounced dip between peaks in Fig. 9).

Furthermore, it is argued in (14) that results of model calculations both for the dipion invariant mass spectrum as well as for the angular distribution

can be reconciled with the data, if the ’s are included into the analysis as intermediate states.

According to the updated analysis (16) (in line with the original report (15)), the Breit-Wigner masses of the and states lie just above the and thresholds, respectively. Theoretical work (39), taking into account the coupling to the intermediate and states, shows that masses slightly below the corresponding thresholds are also consistent with the data in the and (see the comparison in Fig. 10).

In any case, given the proximity of the states to these two-body -wave thresholds, the latter are expected to have a strong impact on the properties of the ’s. In particular, the analysis in (39) is based on the assumption that the ’s are shallow bound states. An alternative interpretation of their resonant nature suggested in (40) is based on the observation that the Belle data are consistent with the ’s as threshold cusps.

A coupled-channel approach to hadron spectroscopy near the open-bottom thresholds was developed in (41). In particular, meson exchange potentials between mesons at threshold were derived and the corresponding Schrödinger equations were solved numerically. As a result, the masses of the twin resonances were reproduced and a number of other possible bound and/or resonant states in other channels were predicted. Decay modes suggested for the experimental searches of the predicted states can be found in (41).

An alternative explanation for the states is proposed in (42), which employs a coupled-channel scheme (see Fig. 11)

(8) |

with and standing for the heavy () and light quark, respectively, and with denoting a light hadron (pion). Such an approach is based on the concept of the QCD string breaking with pion emission (further details can be found in (45)). In particular, for the dipion decays this scheme implies multiple iterations of the basic building block . The calculated production rates for the transitions () reveal peaks at the and thresholds. This computation does not assume any direct interaction between mesons, so that the singularities appear solely due to the interchannel coupling (8). Similar calculations of the dipion transitions into final state constitute a challenge being addressed by theorists with these tools.

Additional measurements of the Belle Collaboration on the ’s have very recently been reported (46). Two additional observation modes of the states seem to have been found. They are and , and the and were found in the missing mass spectra of the pion in the and final states, respectively. Assuming that the , and modes saturate the decays of the states, the reported branching fractions are listed in Table 1.

Mode | ||
---|---|---|

Furthermore, evidence for the neutral partners of the charged ’s was observed in the . Clearly, the models proposed for the states have to be confronted with the new data.

Additional insight on the nature of near-threshold resonances can be obtained from an unbiased analysis of the data in the region below threshold. Indeed, in the appropriate mode, the bound state should reveal itself as a peak below the corresponding open-bottom thresholds in the line shape. In particular, data on radiative decays of the states could potentially confirm or rule out their bound-state interpretation. To this end, high-statistics and high-resolution data in the region below the nominal thresholds is needed to study the radiative transitions , , and . As mentioned above, bound states would reveal themselves as below-threshold peaks which, given the expected high-statistics data from a future -factory, can potentially be observed.

To summarise, a data sample expected to be collected by a high-luminosity -factory with the known production reaction should be sufficient to perform a high-statistics analysis, to exclude alternative interpretations, and to fix the parameters of the states to a high accuracy, in particular their position relative to the corresponding thresholds. A sophisticated line shape form, respecting unitarity and accounting for threshold vicinity, should be used for the data analysis. The production of the ’s can be further investigated at the energy, and the additional information would be helpful in identifying their nature.

### 2.3 states

The idea put forward in (35) that the and states can be explained as molecule-type structures residing at the corresponding thresholds was extended further in (35); (47); (48) and a possible existence of a few sibling states, denoted as with and defined by four orthogonal combinations,

(9) | |||||

was predicted and their properties were outlined. The mesonic components of the , , and are the , , and , respectively.

It has to be noticed that, since the binding mechanisms responsible for the formation of the ’s are still obscure, only model-dependent conclusions can be made concerning the existence or nonexistence of the ’s. In particular, if the binding mechanism operates in the channel (or in both and channels), then all four ’s are expected to exist. However if this mechanism operates only in the channel, then only two sibling states ( and ) are predicted. In what follows it is assumed that all four ’s exist and the corresponding predictions found in the literature are quoted and discussed.

To proceed, it is convenient to adopt the classification scheme suggested in (47) exploiting the heaviness of the quark, that implies that its spin is decoupled from the remaining degrees of freedom, so that a convenient basis can be built in terms of the states , where stands for the heavy pair spin (), while denotes all remaining degrees of freedom, like angular momentum, light quark spins, etc. In particular, this basis matches naturally the decompositions (7) and (9) above, if is associated with the light-quark total spin (47). Since the degree of freedom plays the role of a spectator in all decay processes, it is sufficient to just pick up the term with the appropriate structure in the final state in order to extract the corresponding decay amplitude. Although a detailed microscopic theory is needed to predict each individual decay width, relative coefficients between different decay widths can be evaluated solely using these simple symmetry-based considerations. Such predictions are (47)

(10) | |||

(11) |

and (48)

(12) |

Relation (10) can be verified with the help of the Belle measurement of the ’s widths (equations (5) and (6) above) and indeed it is approximately fulfilled.

Further predictions for specific decay channels follow from the structure of the wave functions (7) and (9), as explained above. In particular, (47)

(13) |

or, similarly (48)

(14) |

(15) |

where denotes a suitable configuration of light hadrons. It should be noted however that relations (13)-(15) rely only on the structure of the amplitudes in the strict heavy-quark symmetry limit. These predictions acquire corrections due to the hyperfine splitting, conveniently parameterised in terms of the difference (47); (48) as well as due to the difference in kinematic phase space. With these effects taken into account and within an effective Lagrangian technique with binding contact interaction, the following predictions were obtained in (48):

where is a certain combination of various coupling constants appearing in the effective Lagrangian (see (48)).

### 2.4 and states

As argued in (35); (49), isovector and
states may possess isoscalar C-odd and C-even partners, also residing at the
thresholds. There is no general consensus in the literature
concerning the naming scheme for such states, so, for the sake of definiteness, we
refer to the C-even partners as ’s (in analogy with the C-even
charmonium state) while we use the notation ’s for C-odd states^{5}

Finally, it is argued in (49) that, while the C-odd isoscalar states are not directly accessible in the decays (neither in radiative decays nor in hadronic transitions), studies of the ratio (of its deviation from unity) of the yields for pairs of charged and neutral mesons in the decays may give an insight on the interaction of mesons in the isoscalar channel. In particular, in presence of a near-threshold isoscalar resonance, the isospin breaking Coulomb effect provides a specific behaviour of this ratio which can potentially be studied by the future -factories.

The search for the partners of the putative exotic states could also be conducted with any data taken at the resonance, which would facilitate additional phase space for the search. Spin splittings for states involving light quarks (as the ’s have due to their electric charge) are of order one or two hundred MeV, as for example the - splitting in the light meson spectrum. Since the candidates have masses in the 10605-10650 MeV range, the at 10876 MeV may be too light as a starting point to reconstruct decay chains containing the , , and partners. The state at 11020 MeV, on the other hand, offers the safety of the additional phase space, that will allow setting of more extensive exclusion limits and offer larger discovery potential.

### 2.5 Radiative decays of

A recent BaBar analysis using and events (50) allowed one to observe decay and to make precise measurements of the branching fractions for the and decays. A search for the and states was performed in the decays , however the significance of the result obtained was insufficient to draw conclusions regarding the masses, so that more data are needed.

Data expected with future high-luminosity -factories at the energy will allow precise studies of the radiative decays and which meets challenges provided by the lattice and model theoretical calculations of various and meson properties (hyperfine splittings with the states, total widths, various decay branching fractions, and so on).

Meanwhile the Belle Collaboration announced (51) a successful identification (with the significance of 15) of the state in the decay chain . The measured hyperfine splitting appeared to be lower than the world average by about 10 MeV, and this improves the agreement of the data with lattice simulations as well as with pNRQCD theoretical predictions. Remarkable progress was achieved by using the large sample of data for the state collected at the energy of the resonance. Related searches for radiative decays are expected soon.

Generally, a missing mass method for , , and can be applied with the higher statistics expected at the future -factories, however it will require longer chains with additional requirements, like the decay with .

The conjecture of the existence of ’s, negative -parity isovector partners of the ’s, opens extra opportunities for the future -factories in what the radiative decays are concerned. Indeed, because of the -parity, a natural way these states could be produced from the are radiative decays of the latter, as shown in Fig. 12 (adapted from (47)). Although such radiative decays are suppressed by the fine structure constant as compared, for example, to decays, the statistics expected at the future -factories should be sufficient to observe such decays.

is produced in the annihilation, so that there are two possible configurations, - and -wave, that when combined with spin yield the production quantum numbers . In the nonrelativistic approximation for quarks only the -wave contribution is retained. Then only the component of the wave function is considered which has the following decomposition in terms of the states: , where the component comes from the total spin while the component accounts for the angular momentum (47). Such identification leads, in particular, to the following prediction (47):

(16) |

where MeV, MeV, and MeV are the photon energies in the corresponding transitions.

In Fig. 12 various cascade reactions for the decays are shown graphically, specifically its radiative transitions with ’s created in the intermediate state. Measurements of such decay chains may become one of the most promising tasks for the future -factories.

### 2.6 Extraction of the light quark mass ratio

It was noticed recently that the light-quark mass ratio can be extracted to high accuracy from data for bottomonium decays (52). Specifically, the decays and are useful because the bottom meson loops are highly suppressed in these reactions (52). Similarly, decays of into should provide the same information on the ratio. Having a high-statistics measurement from a -factory would be useful to estimate the systematics.