# Understanding , , and in a Friedrichs-model-like scheme

###### Abstract

We developed a Friedrichs-model-like scheme in studying the hadron resonance phenomenology and present that the hadron resonances might be regarded as the Gamow states produced by a Hamiltonian in which the bare discrete state is described by the result of usual quark potential model and the interaction part is described by the quark pair creation model. In a one-parameter calculation, the , , and state could be simultaneously produced with a quite good accuracy by coupling the three P-wave states, , , predicted in the Godfrey-Isgur model to the , , continuum states. At the same time, we predict that the state is at about 3902 MeV with a pole width of about 54 MeV. In this calculation, the state has a large compositeness. This scheme may shed more light on the long-standing problem about the general discrepancy between the prediction of the quark model and the observed values, and it may also provide reference for future search for the hadron resonance state.

###### pacs:

12.39.Jh, 13.25.Gv, 13.75.Lb, 11.55.FvAs regards the charmonium spectrum above the open-flavor thresholds, general discrepancies between the predicted masses in the quark potential model and the observed values have been highlighted for several years. Typically, among the P-wave , , , and states, the , discovered by the Belle CollaborationUehara et al. (2006), is now assigned to charmonium state though its mass is about 50 MeV lower than the prediction in the quark potential model Eichten et al. (1978); Godfrey and Isgur (1985); Barnes et al. (2005). The properties of the other P-wave states have not been firmly determined yet. The was first observed in the by the Belle Collaboration in 2003 Choi et al. (2003). Although its quantum number is , the same as the , the pure charmonium interpretation was soon given up for the difficulties in explaining its decays. The pure molecular state explanation of also encounters difficulties in understanding its radiative decays. So its nature remains to be obscure up to now. As for the state, the is assigned to it several years ago, but this assignment is questioned for the mass splitting between and , and its dominant decay mode Guo and Meissner (2012); Olsen (2015). In Ref.Zhou et al. (2015), analyses of the angular distribution of to the final leptonic and pionic states also support the possibility of being a state, which means that it might be the same tensor state as the . Very recently, the Belle Collaboration announced a new result about the signal of which could be a candidate for the Chilikin et al. (2017). The state has not been discovered yet. These puzzles have been discussed exhaustively in the literatures (see refs. Chen et al. (2016); Esposito et al. (2017); Lebed et al. (2017) for example), but a consistent description is still missing.

In this paper, we adopt the idea of Gamow states and the solvable extended Friedrichs model developed recently Xiao and Zhou (2016, 2017, 2017), usually discussed in the pure mathematical physics literature, to study the resonance phenomena in the hadron physics, in particular the charmonium spectrum. Using the eigenvalues and wave functions for mesons in the Godfrey-Isgur (GI) model Godfrey and Isgur (1985) as input and modelling the interaction by the quark pair production (QPC) model, we found that the first excited , , and charmonium states could be reproduced with good accuracy in an one-parameter calculation, and the mass and width of state are also obtained as a prediction. These results are helpful in resolving the long-standing puzzle of identifying the observed P-wave state, and also shed more light on the interpretation of the enigmatic state. Furthermore, this method can also provide the explicit wave functions of resonances, “compositeness” and “elementariness” parameters for bound states, and scattering -matrix involving these resonances Xiao and Zhou (2016, 2017, 2017), which are rigorously obtained in the Friedrichs model and have important applications in further studies of the resonance properties. This scheme provides a general framework to incorporate the hadron interaction corrections to the spectra predicted by the quark model, and can be used in evaluating the other mass spectra above the open-flavor threshold to reconcile the gaps between the quark potential model predictions and the experimental results.

To introduce our theoretical framework, we begin by recalling some basic facts about the Friedrichs model. A resonance exhibiting a peak structure in the invariant mass spectrum of the final states could be understood as a Gamow state in the famous Friedrichs model in mathematical physics Friedrichs (1948). In the simplest version of the Friedrichs model, the full Hamiltonian is separated into the free part and the interaction part as

(1) |

and the free Hamiltonian

(2) |

has a discrete eigenstate with eigenvalue , and continuum eigenstates with eigenvalues , being the threshold for the continuum states, and they are normalized as

(3) |

The interaction part serves to couple the discrete state and the continuous state as

(4) |

where function denotes the coupling form factor between the discrete state and the continuum state and denotes the coupling strength. This eigenvalue problem for the Hamiltonian can be exactly solved. In the Rigged-Hilbert-Space (RHS) formulation of the quantum mechanics developed by A.Bohm and M.Gadella, the discrete state becomes generalized eigenstate with a complex eigenvalue, which corresponds to the resonance state called Gamow state Bohm and Gadella (1989); Civitarese and Gadella (2004). The relation of the Gamow state and the pole in the scattering amplitude in the -matrix theory is also straightforward Civitarese and Gadella (2004). By summing the perturbation series, I.Prigogine and his collaborators also obtained a similar mathematical structure Petrosky et al. (1991). Properties of Gamow states could be represented by the zero point of function on the unphysical sheet of the complex energy plane, where

(5) |

In general, when increases from 0, the zero point moves away from the real axis to the second Riemann sheet. The wave function of the Gamow state is expressed as

(6) |

and the conjuate for a pair of resonance poles on the second Rienmann sheet, where the means the analytical continuations of the integration Xiao and Zhou (2016). There could also be bound-state and virtual-state solutions both for and , with the wave function being

(7) |

for a bound (virtual) state () at () on the first (second) Riemann sheet below the threshold, where are the normalizations. For virtual states, the integral should be continued to on the second sheet. A generalization of the Friedrichs model to include multiple discrete states and multiple continuum states is also worked out and readers are referred to Refs.Xiao and Zhou (2016, 2017, 2017) for more detailed discussions.

Inspired from QCD one-gluon exchange interaction and the confinement, the Godfrey-Isgur model Godfrey and Isgur (1985), with partially relativized linear confinement, Coulomb-type, and color-hyperfine interactions, provides very successful predictions to the mass spectra of the conventional meson states composed of , , , , and quarks, but its predictions with regard to the states above the open-flavor thresholds are not as good as those below. These discrepancies might arise from the neglecting of the coupling between these “bare” meson states and their decay channels (both open and closed) as they mentioned Godfrey and Isgur (1985). In our scheme, the GI’s Hamiltonian which provides the discrete bare hadron eigenstates can be effectively viewed as the free Hamiltonian in the Friedrichs model, and the interactions between the bare states of and the continuum states is modeled by the QPC model Blundell and Godfrey (1996), which modify the spectrum above the open-flavor thresholds. The stronger the coupling is, the larger the influence is. The wavefunctions in the QPC model is chosen to be the same as the eigenstate solutions in the GI model which is approximated by a combination of a set of harmonic oscillator basis. Since the OZI-allowed channel will be more strongly coupled to the bare states than the OZI-suppressed channel, the pole shift is dominantly caused by the these channels. So, we include only the OZI allowed channels in our analysis.

In the spirit of the Friedrichs model, suppose a discrete state with spin , coupled to a continuum composed of two hadrons with a total angular momentum quantum numbers , orbital angular momentum quantum number , total spin , the center of mass (c.m.) momentum for the two particles, and the reduced mass . In the non-relativistic theory, the free Hamiltonian in the c.m. frame can be expressed as

(8) |

The interaction between the discrete states and the continuum states is rotationally invariant and we can confine ourselves to a fixed channel and omit the indices. The matrix elements of the interaction potentials can be expressed as Xiao and Zhou (2017)

(9) |

by absorbing a phase space factor in both and .

The definition of the meson state is different from the one in Ref. Hayne and Isgur (1982) by omitting the factor to ensure the correct normalizations. Then, the meson coupling can be defined as the transition matrix element

(10) |

where the transition operator is the one in the QPC model

(11) |

By the standard derivation one can obtain the amplitude and the partial wave amplitude as in Ref. Blundell and Godfrey (1996). Then the form factor which describes the interaction between and in the Friedrichs model can be obtained as

(12) |

where is the c.m. momentum, and being the masses of meson and respectively. Now, after including more continuum states, function can be expressed as

(13) |

where denotes the energy of the -th threshold. Notice that this function can be continued to an analytic function defined on a -sheet Riemann surface in the case of thresholds. The poles of a scattering amplitude are just the zeros of the function Xiao and Zhou (2017), and its real part and imaginary part represent the mass and half-width of the Gamow state. Only the Gamow states close to the physical region could significantly influence the observables such as cross section or invariant mass spectrum of the final states.

With the parameters in the GI model Godfrey and Isgur (1985), we first reproduced the results of GI by approximating the wave function of the P-wave charmonium states and the charmed mesons with 30 Harmonic Oscillator wave function basis. Using these wave functions of the meson states in the QPC model, one could then obtain the coupling form factor in the Friedrichs model. The only parameter of the QPC model is , which represents the quark pair production strength from the vacuum. In the literature, various values of are chosen in different situations, and a typical value is chosen as Barnes et al. (2005); Kokoski and Isgur (1987). However, since the wave functions used here are different from theirs, it is no need to choose the same value as there. We choose it to be a value around such that all the observed -wave first excited charmonium state spectrum can be reproduced well simultaneously in our scheme.

The coupled channels are chosen up to in the four cases. The state can couple to , , and in both and -wave. For the and states, the coupled channels are , and . In the case of the state, the coupled channels are , and .

The poles of scattering amplitude (zero of the function) could be extracted by analytically continuing to the closest Riemann sheet. To make this scheme more friendly to the experimentalists, one may approximate by a Breit-Wigner parametrization as , and the mass parameter is determined by solving

(14) |

on the real axis where means principal value integration and the Breit-Wigner partial width of the -th open channel is expressed as

(15) |

and the total width . It is worth mentioning that this approximation, Eq. (14) and (15) together, is only valid when it is used to represent a narrow resonance far away from the thresholds.

pole | GI | |||||

3920 | 10 | 3920-4i | ||||

3934-40i | ||||||

3871 | 0 | 3871-0i | ||||

3878 | 11 | 3878-5i | ||||

3895 | 37 | 3902-27i |

The numerical results of the extracted pole position and related Breit-Wigner parameters are shown in Table 1. If there is only one open channel, usually one Gamow state which originates from the bare state is expected, but sometimes there could also be an extra virtual state or bound state generated by the form factor when the coupling is strong, which exhibits the molecular nature of this state. Here, the is just of this nature. In Ref. Xiao and Zhou (2016), we discussed the general condition for this kind of virtual or bound state poles.

For the channel, and thresholds are open for the . This pole is shifted from the GI’s value down to about 7 MeV below the observed value. Its width is about 10 MeV, a little smaller than the observed one. The branching ratio between and is 2.4, which demonstrates that is its dominant decay channel. Its decay probability to is relatively small, but there is still some possibility that there is the contribution of in the mass distribution in experiments.

In the channel, one pole is shifted down from the bare state to about 3934 MeV with fairly large width, while another bound state pole emerges just below the threshold around MeV, which is consistent with the found in the experiment. If the coupling strength is tuned smaller, this bound-state pole will move across the threshold to the second sheet and becomes a virtual state pole. This pole is dynamically generated from the form factor which is an evidence of the molecular origin of the state. It is natural to assign this bound state pole to the , and the higher state generated from GI’s bare state might be related to the state.

In the channel, the state is found to be a narrow resonance at about 3878 MeV. We noticed that the mass of the newly observed candidate is at 3862 MeV with a width MeV Chilikin et al. (2017), having a large uncertainty. Although the channel is OZI-allowed for the state, in this calculation we find that the coupling between the channel and the bare is unexpectedly weak which causes the narrow width. This narrow width is roughly only the bin size of the data in Aubert et al. (2010); Uehara et al. (2006) and is also smaller than the one in the more recent Chilikin et al. (2017). So, in the future experiments, we propose a further exploration with a higher resolution in this energy region to see whether there is a narrow signal missing in the present data. An interesting observation is that there seems to be a simultaneous small excess at the vicinity of about 3860 MeV in experiment of both Belle Uehara et al. (2006) and BABAR Collaboration Aubert et al. (2010). Especially in BABAR’s data, the small structure extends to a dip around 3880MeV. Notice the data points in the region of in Fig. 1.

The pole mass of state is predicted at around 3902 MeV in this scheme. As we have mentioned, , , all couple to the channel, which has no definite -parity. This means that the enhancement above the threshold contains all the contributions from these states. To detect the signal, one needs to look for it in a negative -parity channel such as in this energy region.

Further remarks about the is in order. In our calculation, although it is found to be a bound state, we can not exclude the possibility of a virtual state nature Hanhart et al. (2007); Kang and Oller (2016), since only a small shift down of the parameter will move it to the second sheet. In Zhou and Xiao (2014), improving the approach adopted in Pennington and Wilson (2007), a dispersion relation method combined with the QPC model is also used in discussing the charmonium-like states, where can also be produced. However, since the wave function for the can not be obtained there and the wave function used in the QPC model there is inaccurate, further discussion on the nature of the may not be accurate. In the present scheme, the exactly solvable Friedrichs model provides a more solid theoretical setup and since the more accurate hadron wave function of the GI model is used in the QPC model, the result here would more accurately discribe the nature of the . Moreover, since the wave function for the can be rigoroursly solved in terms of the discrete state and the continuum states, one can also find out its compositeness and elementariness. The compositeness of a bound state, defined as the probability of finding the -th continuous states in the bound state, is expressed in the Friedrichs model as

(16) |

where the normalization factor is

(17) |

If the is a bound state, the relative ratio of finding and in the state is about if we tune the parameter such that the locates within , showing the dominance of the continuum part in this state, which also demonstrates its molecular dominant nature Tornqvist (1994); Gamermann et al. (2010); Guo et al. (2013); Meng et al. (2013). Since the position of the is very close to the threshold, the compositeness is sensitive to its position. In comparison, in Meng et al. (2013), by analyzing the production rate of CMS Chatrchyan et al. (2013) and CDF Acosta et al. (2004) data within the framework of NRQCD factorization, the component is estimated to be , which is consistent with our value. However, our result is different from Takizawa and Takeuchi (2013), in which the component is about . Nevertheless, both results favor a large component. Another QCD sum-rule analysis Matheus et al. (2009) predicts a larger component, about , but the mass of the state is too low, at around 3.77 GeV, compared to the observed one of . It is worth emphasizing that for a resonance, the compositeness and elementariness parameters will become complex numbers Xiao and Zhou (2017), so they have no rigourous definitions, but some definitions proposed in the literature Sekihara et al. (2015); Guo and Oller (2016) might be able to approximately describe these quantities.

In this paper, using the exactly solvable Friedrichs model, we propose a general framework to include the hadron interaction corrections to the quark model spectrum predictions, in particular to the generally accepted GI’s standard results. The explicit wave function for the resonances can be obtained and the compositeness and elementariness of the bound states can be calculated which are important for further study of the properties of the state. Using this scheme, we could reproduce the first excited P-wave charmonium-like states. In particular, we find that the could be dynamically generated in a natural way by the coupling of the bare state and continuum states, but its continuum components is larger. The is found unexpectedly to be a narrow one. We also predict the appearance of the state to be at about 3902 MeV with a pole width of about 54MeV. This scheme is promising in matching the predictions of GI model with the observed states. The acceptable consistency of our results and experiments means that the hadron interactions really give large corrections to the GI’s results for open flavor channels which can reconcile the descrepancy between the quark model prediction and the experiments.

###### Acknowledgements.

Helpful discussions with Dian-Yong Chen, Hai-Qing Zhou, Ce Meng, and Xiao-Hai Liu are appreciated. Z.X. is supported by China National Natural Science Foundation under contract No. 11105138, 11575177 and 11235010. Z.Z is supported by the Natural Science Foundation of Jiangsu Province under contract No. BK20171349.## References

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