Supported by 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 Numbers 2016R1D1A1B03935053 (UY) and 2015R1D1A1A01060707 (HChK) and The work was also partly Supported by RIKEN iTHES Project
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We investigate instanton effects on the heavyquark potential, including its spindependent part, based on the instanton liquid model. Starting with the central potential derived from the instanton vacuum, we obtain the spindependent part of the heavyquark potential. We discuss the results of the heavyquark potential from the instanton vacuum. Finally, we solve the nonrelativistic twobody problem, associated with the heavyquark potential from the instanton vacuum. The instanton effects on the quarkonia spectra are marginal but are required for quantitative description of the spectra.
Article funded by SCOAP^{3}
Heavyquark physics has evolved into a new phase. Charmoniumlike states, which are known as XYZ states [
Various theoretical methods for the quarkonium spectra have been developed over recent decades (see recent reviews [
There is yet another nonperturbative effect on the heavyquark potential from instantons [
In this work, we will examine the instanton effects on the heavyquark potential from the instanton vacuum, including the spindependent parts in addition to the central one. In fact, Eichten and Feinberg [
The paper is organized in the following way. In Section 2, we explain how to derive the instanton effects on the heavyquark potential systematically. We first review the results of Ref. [
We start with the matter part of the QCD Lagrangian for the heavy quark, given as
Using the effective Lagrangian given in Eq. (
The static heavyquark potential is defined as the expectation value of the Wilson loop in a manifestly gaugeinvariant manner
The rectangular Wilson loop.
We first consider the central potential from the instanton vacuum, restating briefly the results from Ref. [
The leadingorder heavyquark propagator in the rest frame is written in terms of the superposition of the instantons
Using Eq. (
We are now in a position to consider the spindependent parts of the heavyquark potential. The general procedure is very similar to what was done in Eq. (
The spindependent potential
In the instanton liquid model for the QCD vacuum, we have two important parameters, i.e., the average size of the instanton
When the quarkantiquark distance is smaller than the instanton size, i.e.,
Figure
The change of the
Each contribution to the heavyquark potential as a function of
For completeness, we provide the expression for the matrix elements of the QQ̄ potential in Eq. (
In order to evaluate the bound states in the spectrum of quarkonia, we need to solve the Schrödinger equation with the potential from the instanton vacuum given in Eq. (
We already mentioned that at large distance the instanton potential is saturated, so that there is no confinement in the present approach. The bound or quasibound charmonium states with masses below or around the threshold mass
One can see that the instanton effects are not small in reproducing the mass of quarkonia. For example, in the case of the potential with parameter Set I, the contribution to the mass of a charmonium is determined by Δ
It is of also interest to discuss the effects of the instanton vacuum on the hyperfine mass splitting. The contribution to the hyperfine mass splitting of each lowlying charmonium state is listed in Table
While the instanton effects come into play significantly in Δ
Lowlying charmonium states from the instanton potential. Charm quark mass is set to be
this work Set I 
this work Set IIb 
experiment [ 


2668.81  2753.64  2983.6±0.6  
2669.57  2755.36  3096.916±0.11  
2692.43  2800.86  3414.75±0.31  
2692.50  2801.11  3510.66±0.07  
2692.67  2801.70  3556.20±0.09 
Contributions to the hyperfine mass splittings of the lowlying charmonium states. Charm quark mass is set to be
this work Set I 
this work Set IIb 
experiment [ 


Δ 
0.72  1.72  113.32 ±0.70 
Δ 
0.07  0.25  95.91 ±0.32 
Δ 
0.24  0.84  141.45 ±0.32 
Δ 
0.16  0.59  45.54 ±0.11 
Lowlying bottomonium states from the instanton potential. Charm quark mass is set to be
this work Set I 
experiment [ 


8454.58  9399.0±2.3  
8454.76  9460.30±0.26  
8477.95  9859.44±0.52  
8477.97  9892.78±0.40  
8478.01  9912.21±0.40 
In the present work, we aimed at investigating the instanton effects on the heavyquark potential, based on the instanton liquid model. We first considered the heavyquark propagator starting from the QCD Lagrangian, which is essential in deriving the heavyquark potential. We showed briefly how to construct the heavyquark potential from the instanton vacuum. Expanding the heavyquark propagator in powers of the inverse mass of the heavy quark, we obtained the spindependent parts of the heavyquark potential. We studied the dependence of the heavyquark potential on the two essential parameters for the instanton vacuum, that is, the average size of the instanton
Having explicitly solved the Schrödinger equation with the heavyquark potential purely induced by the instantons, we discussed the masses of the lowlying quarkonia. The instanton contribution to the hyperfine mass splitting turns out to be tiny due to the smallness of the spindependent part of the potential. We also discussed the dependence of the results on the intrinsic parameters of the instanton vacuum, i.e. the average size of the instanton and the distance between instantons.
It is of great importance to study carefully the mass spectra of the quarkonia and their decays by explicitly solving the Schrödinger equation, combining the heavyquark potential derived in the present work with the confining and Coulomb potentials. Considering the fact that the instanton vacuum plays a key role in realizing chiral symmetry and its spontaneous breaking in QCD, nonperturbative gluon dynamics is expected to shed light on strong decays of quarkonia involving pions. Since the central part of the heavyquark potential was derived by using the small packing parameter
Using the instanton and antiinstanton fields
The leadingorder propagator given in Eq. (