# Quark structure of the and resonances

###### Abstract

We discuss the nature of the and mesons: Are the ’s standard mesons or states with a significative continuum component? Is the a state with continuum coupling effects or a meson-meson molecule? To do that, we compare quark model and unquenched quark model results for the mass barycenter and splittings of the multiplet. Future and more precise experimental results will discriminate between the two interpretations. In the case of the , we interpret it a core plus higher Fock components due to the coupling to the meson-meson continuum, thus we think that it is compatible with the meson , with . The quantum numbers are in agreement with the experimental results found by the LHCb collaboration. In our view, the ’s mass is lower than the quark model’s predictions because of self energy shifts. We also provide an estimation of the open charm/bottom strong decay modes of the and mesons, such as and , and radiative transitions.

###### pacs:

12.35.Ki, 12.39.Pn, 13.25.Gv, 24.85.+p^{†}

^{†}thanks: Corresponding author: santopinto@ge.infn.it

## I Introduction

The nature of the meson, discovered by Belle in meson decay Choi:2003ue () and then confirmed by CDF Acosta:2003zx (), D0 Abazov:2004kp () and BABAR Aubert:2004ns (), is still unknown. There are two possible interpretations for this resonance: a weakly bound meson-meson molecule Hanhart:2007yq (); Baru:2011rs (); Swanson:2003tb (); Fulsom:2007ve (); Aceti:2012cb () or a charmonium state Suzuki:2005ha (); Meng:2007cx (); Pennington:2007xr (); Li:2009zu (); Li:2009ad (); Danilkin:2010cc (); charmonium (), both with quantum numbers. Indeed, recently the new results from the LHCb collaboration Aaij:2013zoa () have ruled out the hypothesis Burns:2010qq () and therefore only the one remains. For a summary of theoretical interpretations of the , see Ref. Swanson:2006st (). Thus, one has to make an assumption regarding its quark structure, compatible with quantum numbers, to study the properties of the , such as the decay modes.

The first possibility is to consider the as a charmonium state Suzuki:2005ha (); Meng:2007cx (); Pennington:2007xr (); Li:2009zu (); Li:2009ad (); Danilkin:2010cc (); charmonium (), with . Indeed, QM’s predictions show that the is the only state with quantum numbers and lying approximately in the same energy region as the . The relativized QM Godfrey:1985xj () predicts this state to have a mass of 3.95 GeV. Nevertheless, in Ref. charmonium () we have shown that the uncommon properties of the are due to its proximity to the decay threshold and cannot be easily explained within a standard quark-antiquark picture for mesons. Specifically, we have demonstated that ’s mass is lower than quark model’s predictions because of self energy shifts (see also Refs. Pennington:2007xr (); Li:2009ad ()). Thus, in our picture the is compatible with the meson , with , and is thus interpreted as a core plus higher Fock components due to the coupling to the meson-meson continuum charmonium ().

The second possibility is to interpret the as a meson-meson molecule with quantum numbers Hanhart:2007yq (); Baru:2011rs (); Swanson:2003tb (); Fulsom:2007ve (); Aceti:2012cb (). According to Refs. Voloshin:1976ap (), the system with quantum numbers can be found by pion exchange and forms a meson-meson molecule. More recent molecular model calculations Tornqvist:2004qy (), including quark exchange kernels for the transitions , in order to predict the decay mode of the Swanson:2003tb (), predict large isospin mixing due to the mass difference between and . Nevertheless, in Ref. Hanhart:2007yq () the authors observe that the one-pion exchange binding mechanism should be taken with greater caution in the case than in the one (see also Refs. Suzuki:2005ha (); Braaten:2007ct (); Fleming:2007rp ()).

Other studies proposed a diquark-antidiquark interpretation for the Burns:2010qq (); Maiani:2004vq () or different quantum number assignments within a charmonium picture Eichten:2004uh ().

Another important test for the properties of the consists of estimating its strong and radiative decay rates Suzuki:2005ha (); Meng:2007cx (); Swanson:2003tb (). In Ref. Meng:2007cx (), the authors re-examine the re-scattering mechanism for the X(3872), which decays to through the exchange of mesons between intermediate states and . Their results for the ratio , between the decay modes and , and for the rate , favor a charmonium interpretation for the . In Ref. Suzuki:2005ha (), the author uses semi-quantitative methods to study some properties of the ; he points out that the binding mechanism and the production rates are incompatible with the molecule interpretation. However these results have been criticized in several works Zanetti:2011ju (); Nielsen:2010ij (); Artoisenet:2009wk (); Thomas:2008ja (); Braaten:2007ct (); Swanson:2006st (). In particular in Ref. Artoisenet:2009wk (), the authors point out that the production rates in the molecular interpretation are compatible with Tevatron data once the charm-meson re-scattering effects are taken into account. In Refs. Bignamini:2009sk (); Bauer:2004bc () the authors observe also prompt production from the CDF collaboration and discuss whether a meson-meson molecule, with a dimension of a few fm and intrinsic fragility, can be promptly produced. By contrast, Refs. Swanson:2003tb (); Hanhart:2007yq (); Baru:2011rs () suggest a molecular interpretation for the .

The system was observed for the first time by ATLAS in 2012 Aad:2011ih () and later confirmed by D0 Abazov:2012gh (). The two collaborations gave an estimation for the mass barycenter of the multiplet, GeV Aad:2011ih () and GeV Abazov:2012gh (). Nevertheless, they did not provide results for the mass splittings between the members of the system.

It is worthwhile noting that in Ref. Abazov:2012gh () the authors state ”Further analysis is underway to determine whether this structure is due to the system or some exotic bottom-quark state”. Indeed, resonances are close to , and decay thresholds, thus their wave functions may include significative continuum components. We think that the present experimental data Aad:2011ih (); Abazov:2012gh () cannot exclude this possibility. Something similar also happens in the case of the , since its properties are influenced by the channel charmonium (); Danilkin:2010cc (); Suzuki:2005ha (); Meng:2007cx (); Pennington:2007xr ().

In this paper, we analyze the interpretation of the as a state with . To do that, we give results for the strong and radiative decays of the resonance. Our results are compared to those of the meson-meson molecular picture.

We also provide unquenched quark model (UQM) and relativized quark model (QM) results Ferretti:2013vua () for the mass barycenter of states, and, especially, for the splittings between them, that can be used to discuss this particular problem. Indeed, the magnitude of the splittings between multiplet’s states is still unknown. As discussed in the following, our UQM bottomonium (); Ferretti:2013vua (); charmonium (); Bijker:2009up (); Santopinto:2010zza (); Bijker:2012zza () and relativized QM Ferretti:2013vua (); Godfrey:1985xj () results for the mass barycenter of the system are in accordance with the present experimental data Aad:2011ih (); Abazov:2012gh (), the difference between our theoretical and the experimental results being within the error of a quark model calculation, of the order of MeV. Thus, only more precise experimental data for the masses of mesons will make it possible to distinguish between the two different interpretations of these resonances. See also Ref. Dib:2012vw (), where the authors calculate the mass barycenter and mass splittings between the members of the multiplet within different potential models.

As in the case of the , we also give results for the open bottom strong decays and radiative transitions of the states. These results, along with those for the masses of the resonances, can be a useful tool to investigate the properties of these mesons, that lay quite close to the first open bottom decay thresholds.

## Ii Results

### ii.1 Calculation of the spectrum with self energy corrections

In this section, we show our results for the spectrum of mesons with self energy corrections of Ref. charmonium (). In our calculation we fitted

(1) |

to the experimental data Nakamura:2010zzi (). Here, the physical mass of a meson, , results from the sum of a bare energy term, , computed within Godfrey and Isgur’s relativized QM of Ref. Godfrey:1985xj (), and a self energy correction,

(2) |

computed within the unquenched quark model formalism of Refs. charmonium (); bottomonium (); Santopinto:2010zza (). The values of Godfrey and Isgur’s model parameters are taken from Ref. charmonium ().

According to our result for the mass of the state (), i.e. 3.908 GeV, the is compatible with the meson and includes an extra component due to the coupling to the meson-meson continuum, which is responsible for the downward energy shift. The mass difference between our theoretical prediction and the experimental data is due to the intrinsic error of a QM calculation, of the order of MeV (see Fig. 1).

### ii.2 Mass barycenter and splittings of the system

Here, we show our UQM result for the masses of mesons with self energy corrections of Ref. Ferretti:2013vua (). As in the case, in our calculation we fitted (1) to the existing experimental data for mesons Nakamura:2010zzi (); Mizuk:2012pb () and the physical mass of a meson, is given by Eq. (2). The values of Godfrey and Isgur’s model parameters, used to calculate the ’s, are taken from Ref. Ferretti:2013vua (). More details can be found in Refs. charmonium (); Ferretti:2013vua (). Our results are reported in Table 1 and Fig. 2.

Then, we show our QM results for mesons, obtained by re-fitting the relativized QM mass formula Godfrey:1985xj () to the most recent experimental data Nakamura:2010zzi (); Mizuk:2012pb (). In this case, the Fock space is limited to the component, while in the UQM case, one also has higher Fock components such as . These results Ferretti:2013vua () are reported in Table 2.

9876 | 21 | 30 |

10242 | 13 | 18 |

10551 | -2 | 85 |

9894 | 21 | 30 |

10241 | 21 | 14 |

10510 | 17 | 13 |

### ii.3 Calculation of and of the open bottom strong decays of states in the pair-creation model

Here, we discuss our results for the open charm strong decay and the open bottom strong decay widths of states Ferretti:2013vua (). The decay widths are computed as charmonium (); bottomonium (); Ferretti:2013vua (); 3P0 (); Ackleh:1996yt (); Barnes:2005pb ()

(3a) | |||

where is the standard relativistic phase space factor charmonium (); bottomonium (); Ferretti:2013vua (); 3P0 (); Ackleh:1996yt (); Barnes:2005pb (), | |||

(3b) |

that depends on the relative momentum between and and on the energies of the two intermediate state mesons, and .

The decay is calculated as

(4) |

where the factor of 2 is introduced because the decays into or . The masses of the mesons, involved in the calculation of Eq. (4), are chosen as: MeV Nakamura:2010zzi (), MeV Nakamura:2010zzi (). Since the mass of the is very close to the threshold, even very small differences can substantially modify the results. Thus, we decided to do the calculations using the two latest measures reported by the PDG Nakamura:2010zzi (): MeV Choi:2011fc () and Aaij:2011sn (). The operator inside the amplitude is that of Refs. charmonium (); bottomonium (), which contains also a quark form factor and an effective pair-creation strength .

Finally, the result of our calculation, obtained with the values of the model’s parameters of Ref. charmonium (), are:

(5) | |||||

(6) |

respectively for the choice of and . These results must be compared to the existing experimental data: Nakamura:2010zzi (); Adachi:2008sua () and Nakamura:2010zzi (); Aubert:2007rva (). On the contrary, the open charm decays into and are forbidden by selection rules.

The open bottom strong decay widths of states are calculated via Eqs. (3). For the masses of the states, we use our UQM predictions of Ref. Ferretti:2013vua (). Indeed, our relativized QM predictions for ’s masses are of the order of MeV lower and, most important, below decay threshold (10.558 MeV) Nakamura:2010zzi (). See Tables 1 and 2. Thus, the presence/absence of these decay channels in the experimental data can be used to validate the QM or the UQM interpretation for these states. Our results are reported in Table 3.

State | Mass [MeV] Ferretti:2013vua () | [MeV] |

10495 | – | |

10580 | – | |

10578 | 13 |

### ii.4 QM calculation of ’s and ’s radiative transitions

In this section, we show our QM results for ’s and ’s radiative transitions. The radiative transitions of the are calculated according to Barnes:2005pb (); Eichten:1978tg (); Gupta (); Kwong:1988ae ():

(7a) | |||

Here, is the -quark charge in units of , is the fine structure constant, is the final photon energy, is the total energy of the final state, is the mass of the initial state, the matrix element | |||

(7b) | |||

involves the initial and final radial wave functions, and the angular matrix element is given by | |||

(7c) |

We calculate the matrix elements of Eqs. (7) assuming, for the initial and final states, the wave functions of Godfrey and Isgur’s relativized QM charmonium (); Godfrey:1985xj (). Finally, our results, obtained with the following values of the masses MeV Nakamura:2010zzi (); Choi:2011fc (), MeV Nakamura:2010zzi (), MeV Nakamura:2010zzi (), MeV Nakamura:2010zzi () and MeV Swanson:2003tb () are reported in Table 4.

Transition | ||||
---|---|---|---|---|

[MeV] | [KeV] | [KeV] | [KeV] | |

697 | 11 | 8 | ||

181 | 70 | 0.03 | ||

101 | 4.0 | 0 | ||

34 | 0.35 | 0 |

The calculation of radiative and strong decays of the , including also loop corrections, will be the subject of a subsequent paper.

The radiative transitions of states can be calculated analogously. In this case, Eq. (7a) can be re-written as Barnes:2005pb (); Eichten:1978tg (); Gupta (); Kwong:1988ae ():

(8) |

where, is the -quark charge in units of , is the total energy of the final state and is the mass of the initial state. We calculate the matrix elements of Eqs. (7) assuming, for the initial and final states, the wave functions of Godfrey and Isgur’s relativized QM charmonium (); Godfrey:1985xj (). Finally, our results, obtained with the values of the masses of Table 5, are reported in Table 6.

State | Mass(QM) [MeV] | Mass(UQM) [MeV] | Mass(Exp) [MeV] |
---|---|---|---|

9490 | 9489 | 9460 | |

10012 | 10022 | 10023 | |

10144 | 10112 | – | |

10154 | 10121 | 10164 | |

10161 | 10127 | – | |

10338 | 10358 | 10355 | |

10425 | – | – | |

10432 | – | – | |

10438 | – | – | |

10494 | 10495 | – | |

10511 | 10580 | – | |

10524 | 10578 | – |

Transition | ||||

[MeV] | [KeV] | [MeV] | [KeV] | |

983 | 0.6 | 984 | 0.2 | |

460 | 1.2 | 461 | 0.7 | |

138 | 6.1 | 139 | 6.4 | |

344 | 0.2 | 376 | 0.2 | |

69 | 0.9 | – | – | |

971 | 2.1 | 1061 | 1.7 | |

477 | 2.5 | 542 | 2.7 | |

155 | 7.4 | 223 | 22.8 | |

361 | 0 | 458 | 0 | |

86 | 0.4 | – | – | |

341 | 0 | 408 | 0.1 | |

79 | 1.0 | – | – | |

1010 | 3.9 | 1059 | 3.6 | |

489 | 3.8 | 540 | 4.0 | |

168 | 8.2 | 221 | 19.6 | |

373 | 0 | 456 | 0 | |

99 | 0 | – | – | |

354 | 0 | 406 | 0 | |

92 | 0.3 | – | – | |

357 | 0 | 441 | 0 | |

86 | 1.4 | – | – |

## Iii Conclusion

In this paper, we analysed some properties of the , to understand its nature and quark structure: state (plus higher Fock components) or molecule?

The starting point of our analysis was obviously the spectrum. Specifically, we calculated the spectrum of mesons with self energy corrections and argued that, according to our results, the is compatible with the meson and includes an extra component due to the coupling to the meson-meson continuum, which is responsible for the downward energy shift charmonium ().

The following step was to study the decay modes of the and, in particular, the strong and electromagnetic ones. Indeed, the radiative and strong decays of the offer a promising method to distinguish the charmonium and molecular assignments for this state Swanson:2003tb (). Our prediction for the decay width, resulting from a model Ackleh:1996yt (); charmonium (); Barnes:2005pb (); 3P0 () calculation, is compatible with the present (preliminary) experimental data Nakamura:2010zzi (); Adachi:2008sua (); Aubert:2007rva () and seems to favour a charmonium interpretation for the .

The results for radiative transitions of the seem to be quite sensitive to different model details and interpretation for the meson, i.e. charmonium state or meson meson molecule. Specifically, when one compares QM to molecular model’s calculations, as it is done in Table 4, it is clear that the two different pictures provide completely different results. In particular, the absence of the two decay modes and from molecular model’s predictions could be an important clue to distinguish between the two theoretical interpretation for the .

We also discussed the nature of the mesons: standard mesons or mesons with strong threshold effects? Our QM and UQM results for the mass barycenters and mass splittings for these states, when compared to the future experimental data, will allow to distinguish between the two interpretations. In particular, it will be shown if our UQM predictions are compatible with the experimental data or if the difference between them and the QM’s are only due the difficulties of fitting a large quantity of states in the sector within the UQM. This also holds for our calculations of the open bottom strong decays and radiative transitions of the resonances. Indeed, it is worthwhile noting that the decay into mesons is only compatible with the UQM prediction for the masses of the ’s.

###### Acknowledgements.

We acknowledge useful discussions with P. Colangelo. This work was supported in part by INFN and in part by CONACYT (Grant No. 78833) and by PAPIIT-DGAPA (Grant No. IN113711), Mexico.## References

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