^{3}

We investigate the mass spectra of ordinary heavy mesons, based on a nonrelativistic potential approach. The heavy–light quark potential contains the Coulomb-type potential arising from one-gluon exchange, the confining potential, and the instanton-induced nonperturbative local heavy–light quark potential. All parameters are theoretically constrained and fixed. We carefully examine the effects from the instanton vacuum. Within the present form of the local potential from the instanton vacuum, we conclude that the instanton effects are rather marginal on the charmed mesons.

The structure of hadrons containing a heavy quark is systematically understood when the mass of the heavy quark is taken to infinity. This is valid, since the heavy-quark mass

In the limit of

The potential models for heavy mesons are usually based on two important parts of physics: quark confinement and perturbative one-gluon exchange. While these two ingredients of the potentials successfully describe the properties of both quarkonia and heavy mesons, certain nonperturbative effects need to be considered. Diakonov et al. derived the central part of the heavy-quark potential from the instanton vacuum, using the Wilson loop [

In the present work, we aim at exploring carefully the heavy–light quark potentials, which were derived from the RIGM, examining their effects on the mass spectra of the heavy mesons. For simplicity and convenience, we will use the nonrelativistic framework in dealing with the heavy–light quark interactions from the RIGM. In any potential model for describing quarkonia and heavy mesons, there are two essential components: the quark confinement and the one-gluon exchange contribution, which we want to introduce in addition to the interaction from the instantons. Instead of a simple variational method used in Ref. [

This paper is organized as follows: In

The general structure of the heavy–light quark potentials is expressed as

The one-loop

In a practical calculation, the point-like spin–spin interaction needs to be smeared by using the exponential form

Since the main purpose of the present work is to consider the contribution of the nonperturbative heavy–light quark interaction from the instanton vacuum, we will introduce the effective instanton-induced heavy–light quark potential. For simplicity, we follow Ref. [

The density parameter

Yet more spin-dependent potentials [

Since the central and spin–spin potentials are given as the Dirac delta functions, we also need to introduce here a smearing function to remove any divergence that would be caused by them. So, we introduce a Gaussian type of smearing function,

The total potential can be constructed by combining the potentials from the instanton vacuum given in Eqs. (

The matrix element of the potential in the

Here we have taken the conventional spectroscopic notation

In Ref. [

A nonrelativistic potential approach for a heavy–light quark system is represented by the time-independent Schrödinger equation with the static potential

In the GEM the wavefunction is expanded in terms of a set of

The angular part of the basis function ^{1}

Since some of the remaining parameters cannot be determined theoretically, we construct several sets of parameters and call them Model I^{2}

Free parameters of the model:

Model | |||||||
---|---|---|---|---|---|---|---|

I |
0.450 | 0.169 | 1.43 | – | – | ||

I | 0.450 | 0.169 | 1.43 | 1.18 | 1.0 | ||

II | 0.490 | 0.165 | 0.95 | 1.19 | 1.0 | ||

III | 0.470 | 0.163 | 0.93 | 1.17 | 0.9 |

The results of the charmed meson masses corresponding to the different models are listed in

The results of the charmed

Model | I | II | III | Exp. | |
---|---|---|---|---|---|

1867.7 | 1787.0 | 1868.3 | 1868.0 | ||

2013.5 | 2006.4 | 2009.7 | 2010.2 | ||

2461.2 | 2461.5 | 2458.7 | 2456.7 | ||

2462.2 | 2461.2 | 2461.7 | 2460.1 | ||

2639.0 | 2593.4 | 2634.1 | 2630.4 | ||

2737.0 | 2732.6 | 2724.0 | 2719.8 |

Model I has the same parameter set as Model

Thus, we present the results of Model II, in which the free parameters are fitted to the experimental data. One can see that the results slightly change in comparison with Model I

The results of Model III are slightly better than those of Model II. As expected from the comparison of Model I with Model

The results of the charmed strange

Model | I | II | III | Exp. | |
---|---|---|---|---|---|

1969.1 | 1887.9 | 1969.0 | 1968.9 | ||

2108.3 | 2100.8 | 2113.2 | 2111.5 | ||

2538.3 | 2538.1 | 2543.1 | 2540.5 | ||

2546.2 | 2545.1 | 2555.2 | 2551.8 | ||

2703.7 | 2661.6 | 2697.4 | 2696.0 | ||

2792.6 | 2788.2 | 2780.5 | 2778.5 |

Finally, we would like to note that, although we have changed the density of the instanton medium

In

The results of the instaton effects on the low-lying charmed heavy mesons in units of MeV. The values of the relevant parameters are taken from those for Model I.

Heavy meson | Instanton contribution [MeV] | Exp. [MeV] |
---|---|---|

80.7 | ||

7.1 | ||

–0.3 | ||

0.1 | ||

45.6 | ||

4.4 | ||

81.2 | ||

7.5 | ||

0.2 | ||

1.1 | ||

42.1 | ||

4.4 |

In the present work, we have investigated the effects of the heavy–light quark potential from the instanton vacuum on the mass spectra of the conventional charmed mesons. First, we considered the confining potential that is proportional to the relative distance between the heavy and light quarks. The Coulomb-like potential, which arises from one-gluon exchange, has been included. The spin-dependent potentials were generated from the central part. Then we computed the mass spectra of the charmed mesons, employing the Gaussian expansion method to solve the nonrelativistic Schrödinger equation. The results are in good agreement with the experimental data even without the inclusion of the potential from the instanton vacuum. Then, we introduced the central and spin-dependent potentials from the instanton vacuum. The additional spin part of the potential was obtained from the central part of the instanton-induced potential. While the instanton effects are noticeable on the

Though the present form of the instanton-induced potential does not make any significant contribution to the heavy meson masses, there are some possible ways of elaborating the present analysis:

The present work is based on the nonrelativistic Schrödinger equation, since we aimed mainly at investigating the effects of the instanton-induced potential. However, once the light quark is involved, the inclusion of certain relativistic effects is inevitable.

The instanton-induced potentials used in the present work were derived from the random instanton gas model and are given as local ones. However, if one uses the instanton liquid model, the interaction between the heavy and light quarks turns out to be nonlocal [

Recently, Ref. [

Thus, one needs to study systematically nonperturbative effects on both heavy mesons and heavy baryons, arising from the instanton vacuum. The corresponding investigations are underway.

H.-Ch.K. is grateful to P. Gubler, A. Hosaka, T. Maruyama, and M. Oka for useful discussions. He wants to express his gratitude to the members of the Advanced Science Research Center at the Japan Atomic Energy Agency for their hospitality, where part of the present work was done. The work of Q.W. is supported by the National Natural Science Foundation of China (11475085, 11535005, 11690030) and National Major State Basic Research and Development of China (2016YFE0129300). The work of H.-Ch.K. and U.Y. is supported by the Basic Science Research Program through the National Research Foundation (NRF) of Korea funded by the Korean government (Ministry of Education, Science and Technology, MEST), Grant No. NRF2018R1A2B2001752 (H.-Ch.K.) and Grant No. 2016R1D1A1B03935053 (U.Y.).

Open Access funding: SCOAP

^{1}For more details, see Refs. [

^{2}A corresponding explanation of the model parameters will be given hereafter in the text.