Instanton effects on the heavy-quark static potential

We investigate the instanton effects on the heavy-quark potential,including its spin-dependent part, based on the instanton liquid model. Starting with the central potential derived from the instanton vacuum, we obtain the spin-dependent part of the heavy-quark potential. We discuss the results of the heavy-quark potential from the instanton vacuum. We finally solve the nonrelativistic two-body problem, associating with the heavy-quark potential from the instanton vacuum. The instanton effects on the quarkonia spectra are marginal but are required for quantitative description of the spectra.

Various theoretical methods for the quarkonium spectra have been developed over decades (see recent reviews [27][28][29]34]), among which the potential model has been widely used for describing properties of the quarkonia [35,36]. The form of the potential at short distances is governed by the Coulomb-like interaction arising from perturbative QCD (pQCD). In the lowest order, one-gluon exchange between a heavy quark and a heavy anti-quark is responsible for this Coulomb-like attraction [37][38][39][40]. The running coupling constant for the Coulomb-like interaction was considered with higher order corrections in pQCD [41][42][43][44][45]. However, the distance of the quark and the anti-quark gets farther apart, certain nonperturbative contributions should be taken into account in the potential. Quark confinement [46] is shown to be the most essential nonperturbative part obtained at least phenomenologically from the Wilson loop for the heavy-quark potential, which rises linearly at large distances [35,36]. This linearly rising potential was extensively studied in lattice QCD [47][48][49][50][51][52][53][54].
There are yet another nonperturbative effects on the heavy-quark potential from instantons [55], which are known to be one of the most important topological objects in describing the QCD vacuum. These instanton effects on the heavyquark potential were already studied many years ago [56][57][58], spin-dependent aspects of the heavy-quark potential being emphasized. The central part of the heavy-quark potential was first derived [59], based on the instanton liquid model for the QCD vacuum [60][61][62]. In Ref. [59], the Wilson loop was averaged in the instanton ensemble to get the heavy-quark potential, which rises almost linearly as the relative distance between the quark and the antiquark increases, then it starts to get saturated. The results of Ref. [59] were also simulated in lattice QCD [63][64][65]. Though the instanton vacuum does not explain quark confinement, it will play a certain role in describing the characteristics of the quarkonia. The feature of the instanton vacuum will be recapitulated briefly in the present work in the context of the quarkonium hyperfine mass splittings.
In this work, we will examine the instanton effects on the heavy-quark potential from the instanton vacuum, including the spin-dependent parts in addition to the central one. In fact, Eichten and Feinberg [58] derived an analytic form of the instanton contributions to the spin-dependent potential but were not able to compute them due to difficulties of deriving the static energy or the central static potential induced from instantons. Diakonov et al. [59] calculated this central part of the heavy-quark potential from the instanton vacuum, as mentioned previously. Thus, in the present work, we want to obtain the instanton-induced spin-dependent parts of the heavy-quark potential, following closely Refs. [58,59]. To derive the spin-dependent potential from the instanton vacuum, we first expand the matter part of the QCD Lagrangian for the heavy quark with respect to the inverse of a heavy-quark mass (1/m Q ), as usually was done in heavy-quark effective theory (HQET). As was obtained from Ref. [59], the central part comes from the leading order in the heavy-quark expansion. The heavy-quark propagator or the Wilson loop being averaged over the instanton medium, the central part can be derived. The spin-dependent contributions arise from the order of 1/m 2 Q . As we will show in this work, the heavy-quark propagator is given as an integral equation. Expanding it in powers of 1/m 2 Q , we are able to compute the spin-dependent part of the heavy-quark potential as was first shown in Ref. [58]. We will evaluate these spin-dependent potentials and examine their behaviour. Then we will proceed to compute the instanton effects on the hyperfine mass splittings of quarkonia. Assuming that the interaction range between a heavy quark and a heavy anti-quark is smaller than the inter-instanton distance, we can easily deal with the effects of the instantons on the hyperfine mass splittings of the quarkonia. We find at least qualitatively that the instantons have definite effects on those of the charmonia, while those of the bottomonia acquire tiny effects from the instanton vacuum because of the heavier mass of the bottom quark.
The paper is organized in the following way. In the next section II, we explain how to derive the instanton effects on the heavy-quark potential systematically. We first review the results of Ref. [59] within the heavy-quark expansion. Then we show the corrections to the spin-dependent heavy-quark potential, which come from the 1/m 2 Q order. In Section III we discuss the results of the instanton effects on the heavy-quark potential in detail and present numerical method used to solve the Schrödinger equation. We also present the spectrum low laying charmonium states and the estimates of the hyperfine mass splittings of these states. Finally, in Section IV we summarise the results and give a future outlook related to the present work.

A. Heavy-quark propagator
We start with the matter part of the QCD Lagrangian for the heavy quark, given as where i / D = i / ∂ + / A denotes the covariant derivative, m Q stands for the mass of the heavy quark, and Ψ(x) represents the field corresponding to the heavy quark. As was done in HQET [66,67], we assume that the heavy-quark mass m Q goes to infinity with the velocity v of the heavy quark fixed (v 2 = 1). Then we can decompose the heavy-quark field into the large component h v (x) and the small one H v (x) as follows which is just the Foldy-Wouthuysen transformation [68,69] used in the nonrelativistic expansion in QED. The h v (x) and H v (x) fields are defined respectively as The velocity vector allows one also to split the covariant derivative into the longitudinal and transverse components as where The transverse component of the covariant derivative satisfies the relations where G µν stands for the gluon field strength tensor. E and B denote the chromoelectric and chromomagnetic fields, respectively. Using the equations of motion, we can remove the small field H v (x) by the relation or equivalently we can integrate out the H v fields [67]. Thus, we arrive at the effective action expressed only in terms of the h v fields where the first term will provide the central contribution to the heavy-quark potential while the second term is responsible for the spin-dependent part.
Using the effective Lagrangian given in Eq. (8), we can define the heavy quark propagator as If we assume that the heavy-quark mass is infinitely heavy, then the heavy-quark propagator in the leading order satisfies the following equation and its solution in the rest frame v = (1, 0) is found to be where A 4 is the time component of the gluon field in four-dimensional Euclidean space. Note that since we consider the instanton field, which is the classical solution in Euclidean space, we work in Euclidean space from now on. Equation (11) implies that the heavy quark propagates along the time direction. The full propagator S(x, y; A) is then expressed as an integral equation as follows Since m Q is rather heavy, we can expand iteratively the full propagator (12) in powers of 1/m Q , when we derive the spin-dependent heavy-quark potential.
B. Heavy-quark potential from the instanton vacuum The static heavy-quark potential is defined as the expectation value of the Wilson loop in a manifestly gaugeinvariant manner where W C [A] denotes the Wilson loop expressed as The path is usually taken to be a large rectangle (T × r) as drawn in Fig. 1  consider the central potential from the instanton vacuum, restating briefly the results from Ref. [59]. The leading-order expectation value of the Wilson loop in Euclidean space is defined as where S YM is the Yang-Mills action for the gluon field. The Wilson loop in the instanton medium can be written as where a I,Ī =ẋ µ A I,Ī µ (x). I (Ī) denotes the instanton (anti-instanton). A I,Ī µ represent the instanton (anti-instanton) solutions of which the explicit expressions can be found in Appendix. The sum I,Ī a I,Ī stands for the superposition of N + instantons and N − anti-instantons for the classical gluon background fieldĀ µ , which is written aṡ where ξ represents the set of collective coordinates for the instanton, consisting of its center z Iµ , the size ρ I , and SU(N c ) orientation matrix with the number of colors N c . The integration over the gluon fields given in Eq. (15) is then replaced with the integrations over the set of collective coordinates of the instantons (anti-instantons) [59][60][61] such that Eq. (15) can be understood as an average over instanton ensemble. The leading-order heavy-quark propagator in the rest frame is written in terms of the superposition of the instantons where a (i) I,Ī represents the gluon field projected onto the corresponding ith Wilson line. Since T → ∞, we can neglect the short sides of the rectangular path. The separation between the two long Wilson lines is given as r, as shown in Fig. 1. Using Eqs. (11) and (18), we can write the Wilson loop along the rectangle shown in Fig. 1 as The double angle bracket · · · emphasizes the average over the instanton ensemble. Each heavy-quark propagator in Eq. (19) is expanded in powers of the instanton and anti-instanton fields a (1,2) I,Ī . Then the sum of the planar diagrams is carried out, which is the leading order in the 1/N c expansion [70]. Note that the instanton vacuum has two parameters characterizing the dilute instanton liquid [60,71]: the average size of the instantonρ ≃ 0.33 fm and the average separation between instantonsR = (N/V ) −1/4 ≃ 1 fm, where the instanton density is given as 4 . It allows one to use N/V N c as a small perturbation parameter. We refer to Ref. [59] for further details of the calculation.
Using Eq. (A2), we can obtain the explicit form of the central potential from the instanton vacuum as where z denotes the position of the instanton, which is one of the collective coordinates for the instantons. The trace Tr c runs over the colour space and r is a distance between quark and antiquark. Further introducing the dimentionless variables y = z/ρ and x = r/ρ, one can rewrite the potential in terms of the dimensionless integral I(x) y + xt y 2 + x 2 + 2xyt sin π y y 2 + 1 sin π y 2 + x 2 + 2xyt y 2 + x 2 + 2xyt + 1 .
As r goes to infinity, the potential is saturated to be a constant where ∆M Q is the correction to the heavy-quark mass from the instanton vacuum [59] dy y 2 1 + cos πy calculated using again Eq. (A2). The average size of the instanton is regarded as a renormalization scale of the instanton vacuum [61,72]. Keeping in mind the fact that the current quark mass is scale-dependent and its value is usually given at µ = m c , certain scaling effects arising from the renormalization group equation for the quark mass should be taken into account in order to estimate the effects on the heavy-quark mass from the instanton vacuum. The instanton effects would be slightly decreased, when one matches the scale of ∆M Q to the charmed quark mass given in Ref. [73].
We are now in a position to consider the spin-dependent parts of the heavy-quark potential. The general procedure is very similar to what was done in Eq. (19). Since we consider now the finite heavy-quark mass, we need to use the full propagator given in Eq. (12) instead of the leading one. That is, we calculate the two Wilson lines as Considering the fact that 1/m Q can be regarded as a small parameter, we can expand the full propagators in Eq. (25) iteratively in powers of 1/m Q . Using the relations given in Eq. (6), we first expand the term between S 0 and S in powers of Then, the heavy-quark propagator for the ith Wilson loop can be iteratively expressed in powers of 1/m Q as Replacing the full propagator in Eq. (25) with Eq. (27), we obtain the following expression Note that here we consider only the spin-dependent parts. For example, we can exclude the spin-independent term D 2 /2m Q , which is just the kinetic energy, and that proportional to σ · B, which disappears because of parity invariance [58]. We can further simplify Eq. (28), leaving all spin-independent parts out, which are just part of relativistic corrections to the potential. Taking only the spin-dependent parts into account, we obtain The final expression for W 1/m 2 Q C contains 1/m 2 Q , so that we can expand the exponential of Eq. (13) in powers of 1/m 2 Q . Then, Eq. (29) will lead to the spin-dependent parts of the heavy-quark potential from the instanton vacuum. The derivation of the potential from Eq. (29) is lengthy but straightforward. In Ref. [58], it was shown in very detail how one can obtain the spin-dependent parts of the heavy-quark potential in QCD. Since the form of Eq. (29) is very similar to the corresponding one in Ref. [58], we will closely follow the method of Ref. [58] and refer to it. The leading-order propagator given in Eq. (11) is identified as the path-order exponential along the time direction apart from the Dirac delta function. Using the identities for the path-ordered exponentials given in Appendix, we can proceed to compute each term in Eq. (29). Note that the instanton satisfies the self-duality condition G a µν = ±G a µν (B = ±E), which plays an essential role in deriving the spin-dependent potential from the instanton vacuum. It makes it possible to relate several independent potentials to the central potential given in Eq. (21). As a result, all the spin-dependent potentials are expressed in terms of the central potential where L i and σ i represent respectively the orbital angular momentum and the Pauli spin operator of the corresponding heavy quark, n designates the unit radial vector. The potential V C (r) denotes the central part of the potential that we already have shown in Eq. (21). We want to mention that we have used m Q = mQ. If one considers two heavy quarks with different masses, we can simply replace m 2 Q with m Q mQ in Eq. (30). The spin-dependent potential V SD can be now decomposed into three different parts, i.e., the spin-spin interaction V SS (r), the spin-orbit coupling term V LS (r), and the tensor part V T (r): where S Q(Q) stands for the spin of a heavy quark (heavy anti-quark) S Q(Q) = σ 1(2) /2, S does their total spin S = S 1 + S 2 , and L represents the relative orbital angular momentum L = L 1 − L 2 . Each potential of Eq. (31) is defined respectively as Thus, all three components of the spin-dependent potential are expressed in terms of the central potential V C (r).

A. Instanton potential
In the instanton liquid model for the QCD vacuum, we have two important parameters, i.e., the average size of the instantonρ ≃ 0.33 fm and the average distanceR ≃ 1 fm between instantons, as we have already mentioned. These numbers were first proposed by Shuryak [71] within the instanton liquid model and were derived from Λ MS by Diakonov and Petrov [60]. Thus, it is also of great interest to look into the dependence of the heavy-quark potential from the instanton vacuum on these parameters. Moreover, the values given above should not be considered as the exact ones. For example, Refs. [80][81][82] considered 1/N c meson-loop contributions in the light-quark sector and found it necessesary to readjust the values of parameters asρ ≃ 0.35 fm andR ≃ 0.856 fm. Lattice simulations of the instanton vacuum suggestedρ ≈ 0.36 fm andR ≈ 0.89 fm [83][84][85][86], which is almost the same as those with the 1/N c meson-loop corrections. Thus, we want to examine the dependence of the heavy-quark potential from the instanton vacuum on three different sets of parameters, that is, Set I [60,71], Set IIa [80][81][82], and Set IIb [83][84][85][86]. The parameter dependence of the potential can be easily understood from the form of leading-order potential expressed in Eq. (21). While the prefactorρ 3 /R 4 N c , which includes both the parameters, governs the overall strength of the potential, its range is dictated only by the instanton sizeρ through the dimensionless integral I(r/ρ).
When the quark-antiquark distance is smaller than the instanton size, i.e., r ≪ρ (x ≪ 1), one can expand the dimensionless integral I(x) with respect to x which yields the central potential in the form of a polynomial As the distance between the quark and the antiquark grows larger than the intstanton size, i.e. r ≫ρ (x ≫ 1), we again get an analytic expression as follows Consequently, the central potential at large r can be approximately written as The second term behaves like the Coulomb-like potential. So, crudely speaking, this can be understood as a nonperturbative contribution to the perturbative one gluon exchange potential from the instanton vauum at large r. The coupling constant g NP in Eq. (36), which is defined as g NP := 2π 3ρ4 /(N cR 4 ), could be regarded as a nonperturbative correction to the strong coupling constant α s (r). When r goes to infinity r → ∞, the potential is saturated at the value of 2∆M Q . As discussed already in Ref. [59], it implies that the instanton vacuum can not explain quark confinement. In the case of parameter Set I, which is often considered in the light-quark sector, the value of ∆M Q is obtained to be ∆M Q ≃ 66.    2. Each contribution to the heavy-quark potential as a function of r for the two different sets of the instanton parameters ρ andR. The upper-left panel depicts the central part of the potential, the upper-right panel draws the spin-spin interaction, the lower-left panel illustrates the spin-orbit part, and the lower-right one shows the tensor interaction. The solid curve corresponds to Set I with ρ = 0.33 fm andR = 1 fm from the phenomenolgy [60,71], whereas the dashed one corresponds to Set IIb with ρ = 0.36 fm andR = 0.89 fm from the lattice simulations [83][84][85][86]. The mass of the charm quark is chosen to be mc = 1275 MeV. Figure 2 draws r dependence of each term of the heavy-quark potentials from the instanton vacuum. We take into account the charm quark sector as an example. We also show the dependence of each term of the potential on two different sets of parameters, that is, Set I and Set IIb. One can see that the central part of the potential increases monotonically at small distances r ≪ρ and later becomes almost linear at the distances comparable with the instanton size r ∼ρ as already discussed in Ref. [59]. At large r ≫ρ it starts to get saturated at the value V C (r → ∞) ≃ 133.2 MeV with Set I and V C (r → ∞) ≃ 271.44 MeV with Set IIb. The spin-spin interaction part is of particular interest among these contributions to the spin-dependent potential. In pQCD, it is given as a point-like interaction [87] in the leading order. On the other hand, the spin-spin interaction from the instanton vacuum looks similar to a Gaussian-type interaction. The spin-orbit potential behaves in a similar way to the spin-spin potential. The tensor interaction, however, shows a different r dependence. As r increases, the tensor potential vanishes at r = 0 and then starts to increase until r ≈ 0.4 fm, from which it begins to fall off. The strength of each part of the potential become stronger when smaller value ofR is employed, since all terms turn out to be very sensitive tō R on account of the prefactorρ 3 /R 4 N c . It implies that a less dilute instanton medium yields stronger interactions between a heavy quark and a heavy antiquark. However, one has to keep in mind that the value ofR should not be continually decreased, because the whole framework of the instanton liquid model is based on the diluteness of the instanton medium where the packing parameter proportional to (ρ/R) 4 must be kept as a small parameter.
On the other hand, the change of theρ value seems less effective in the spin-dependent parts of the potential. This is again due to the fact that all spin dependent parts have the prefactorρ/R 4 N c where instanton size appears in the first order, after rewriting the spin-dependent parts in terms of the dimensionless integral I(x) (see Eq. (22)) and its derivatives. As mentioned in the previous section, the average size of the instanton has a physical meaning of the renormalization scale [61,72], which is a crucial virtue of the instanton liquid model. Thus,ρ = 0.33 fm indicates the renormalization scale µ = 600 MeV. Bearing in mind this meaning ofρ, we should not take the value ofρ freely. Note that the value ofρ −1 = 600 MeV implies the strong couipling constant frozen atρ −1 . Thus, Fig. 2 shows the dependence of the heavy-quark potential on bothρ andR within a constraint range of their values. In the case of the bottom quarks and anti-quarks, the instanton effects are quite much suppressed because of the large bottom quark mass.
For completeness, we provide the expression for the matrix elements of QQ potential in Eq. (31) where we have used the conventional spectroscopic notation 2S+1 L J given in terms of the total spin S, the orbital angular momentum L, and the total angular momentum J satisfying the relation J = L + S.

B. Gaussian Expansion Method
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. (31) where m Q arises from the doubled reduced mass of the quarkonium system and Ψ JM represents the wave function of the state with the total angular momentum J and its third component M . We can solve Eq.(38) numerically, using the Gaussian expansion method (see review [89]) in which the wave function is expanded in terms of a set of The normalized radial part of the basis wave functions φ L n (r) is expressed in terms of the Gaussian basis functions where r n , n = 1, 2, ..., n max stand for variational parameters. In the case of a two-body problem, the total number of the variational parameters can be reduced by choosing the geometric progression in the form of r n = r 1 a n−1 , which produces a good convergence of the results. Thus, we need only three variational parameters, i.e. r 1 , a and n max .

C. Quarkonium states
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 the masses below or around the threshold mass M QQ ≃ 2(m c + ∆M Q ), where m c = 1275 is the charm quark mass [73], are listed in Table I with the two different sets of the instanton parameters. Other states above threshold will appear as resonances in the present approach. 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 ∆M cc = M cc − 2m c . For example, the contribution of the instanton effects to the η c mass turns out to be 118.81 MeV, which is approximately about 30 % in comparison with the experimental data 433.60 MeV. As discussed already, the potential from the instanton vacuum is sensitive to the instanton parameters. Therefore, the change in the instanton parameters strongly affects the spectrum of QQ states. come into play significantly on ∆M cc , they turn out to be rather small in describing the hyperfine mass splittings of the charmonia. This might be due to the fact that the spin-dependent part of the potential from the instanton vacuum is almost an order of magnitude smaller than the central part. The tensor interaction almost does not contribute to the results. As a result, the instanton effects on the hyperfine mass splittings are almost negligible. In order to obtain realistic results of the hyperfine mass splittings as well as of the charmonium masses, we need to include the Coulomb-like potential coming from the perturbative one gluon-exchange and the confining potential together with that from the instanton vacuum.

IV. SUMMARY AND OUTLOOK
In the present work, we aimed at investigating the instanton effects on the heavy-quark potential, based on the instanton liquid model. We first considered the heavy-quark propagator starting from the QCD Lagrangian, which comes into an essential play in deriving the heavy-quark potential. We showed briefly how to construct the heavyquark potential from the instanton vacuum. Expanding the heavy-quark propagator in powers of the inverse mass of the heavy quark, we obtained the spin-dependent parts of the heavy-quark potential. We studied the dependence of the heavy-quark potential on the two essential parameters for the instanton vacuum, that is, the average size of the instanton (ρ) and the inter-distance between the instantons (R). The results of the potential are very sensitive to the parameterR, while they are varied marginally withρ changed. The spin-spin interaction shows r dependence similar to a Gaussian-type potential, which is distinguished from the point-like spin-spin interaction derived from perturbative QCD. The spin-orbit potential behaves like the spin-spin interaction, whereas the tensor potential exhibits a different character. It increases until r reaches approximately 0.4 fm and then starts to fall off.
Having solved explicitly the Schrödinger equation with the heavy-quark potential purely induced by the instantons, we discussed the masses of the low-lying quakonia. The instanton contribution to the hyperfine mass splitting turns out to be tiny due to smallness of the spin-dependent 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 inter-distance between instantons.
It is of great importance to study carefully the mass spectra of the quarkonia and their decays by solving explicitly the Schrödinger equation, combining the heavy-quark 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, the nonperturbative gluon dynamics is expected to shed light on strong decays of the quarkonia involving pions. Since the central part of the heavy-quark potential was derived by using the small packing parameter N/V N c , we can obtain the corrections from the next-to-leading order (N/V N c ) 2 . In principle, it is not that difficult to compute them. Starting from the instanton operator corresponding to the Wilson line (see Eq. (17) in Ref. [59]), we can consider the next-to-leading order in the expansion with respect to the small packing parameter of the instanton medium. Though the corrections from the next-to-leading order might be very small, one could use it for the fine-tuning of the mass spectrum of the quarkonia. The corresponding investigations are under way.

Appendix A: Useful formulae
Using the instanton and anti-instanton fields x νη a µν τ a ρ 2 x 2 (x 2 + ρ 2 ) , AĪ µ = x ν η a µν τ a ρ 2 whereη a µν and η a µν denote the 't Hooft symbols [88], we can easily derive the path-ordered exponential as follows [59] P exp i ∞ −∞ dx 4 A I4 = − cos π|z| which was used for deriving the heavy-quark potential and the instanton corrections to the heavy-quark mass. The leading-order propagator given in Eq. (11) is the same as the path-ordered exponential apart from the Dirac delta function. Thus, it is of great use to consider the identities derived in Ref. [58] for the path-order exponentials when we compute the spin-dependent parts of the heavy-quark potential. Defining the path-ordered exponential as dα(x − y) j [P (y, t; z, t)B k (z, t)P (z, t; y, t)], where z = αy + (1 − α)x. D i denotes the spatial component of the covariant derivative. When time t goes to infinity, i.e. t = ± T 2 → ∞, the third identity is simplified to be lim |t|→∞ P (y, t; x, t)D(x, t)P (x, t; y, t) = i∇ y .