Instanton effects on the heavy-quark static potential
U. Yakhshiev, Hyun-Chul Kim, B. Turimov, M.M. Musakhanov, Emiko Hiyama
aa r X i v : . [ h e p - ph ] A p r INHA-NTG-02/2016
Instanton effects on the heavy-quark static potential
U.T. Yakhshiev,
1, 2, ∗ Hyun-Chul Kim,
1, 2, 3, † B. Turimov, ‡ M.M. Musakhanov, § and Emiko Hiyama ¶ Department of Physics, Inha University, Incheon 22212, Republic of Korea RIKEN Nishina Center, RIKEN, 2-1 Hirosawa, 351-0115 Saitama, Japan School of Physics, Korea Institute for Advanced Study (KIAS), Seoul 02455, Republic of Korea Theoretical Physics Department, National University of Uzbekistan, Tashkent-174, Uzbekistan (Dated: June, 2016)We investigate the instanton effects on the heavy-quark potential, including its spin-dependentpart, based on the instanton liquid model. Starting with the central potential derived from theinstanton vacuum, we obtain the spin-dependent part of the heavy-quark potential. We discuss theresults of the heavy-quark potential from the instanton vacuum. We finally solve the nonrelativistictwo-body problem, associating with the heavy-quark potential from the instanton vacuum. Theinstanton effects on the quarkonia spectra are marginal but are required for quantitative descriptionof the spectra.
PACS numbers: 12.38.Lg, 12.39.Pn, 14.40.PqKeywords: Instanton-induced interactions, heavy-quark potential, quarkonia ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] ¶ [email protected] I. INTRODUCTION
Heavy-quark physics has evolved into a new phase. Charmonium-like states, which are known as XYZ states [1–13]and quite possibly exotic ones, conventional bottomonia including the lowest-lying state η b [14–20], and heavy pen-taquark states [21] have been newly reported by various experimental collaborations (see also recent reviews [22–25]).These novel findings of heavy hadrons have renewed interest in heavy-quark spectra and have triggered subsequentlya great deal of experimental and theoretical work (see for example the following reviews [26–30]). Among these newlyobserved heavy hadrons, the conventional bottomonium η b (1 S ) is placed in a crucial position. Even though it is thelowest-lying bottomonium, it has been observed only very recently [14–18] and the precise measurement of its massprovides a subtle test for any theory about heavy quarkonia, based on quantum chromodynamics (QCD) [31–33].Various theoretical methods for the quarkonium spectra have been developed over decades (see recent reviews [27–29, 34]), among which the potential model has been widely used for describing properties of the quarkonia [35, 36]. Theform 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 forthis Coulomb-like attraction [37–40]. The running coupling constant for the Coulomb-like interaction was consideredwith higher order corrections in pQCD [41–45]. However, the distance of the quark and the anti-quark gets fartherapart, certain nonperturbative contributions should be taken into account in the potential. Quark confinement [46] isshown to be the most essential nonperturbative part obtained at least phenomenologically from the Wilson loop forthe heavy-quark potential, which rises linearly at large distances [35, 36]. This linearly rising potential was extensivelystudied in lattice QCD [47–54].There are yet another nonperturbative effects on the heavy-quark potential from instantons [55], which are known tobe one of the most important topological objects in describing the QCD vacuum. These instanton effects on the heavy-quark potential were already studied many years ago [56–58], spin-dependent aspects of the heavy-quark potentialbeing emphasized. The central part of the heavy-quark potential was first derived [59], based on the instanton liquidmodel for the QCD vacuum [60–62]. In Ref. [59], the Wilson loop was averaged in the instanton ensemble to getthe heavy-quark potential, which rises almost linearly as the relative distance between the quark and the antiquarkincreases, then it starts to get saturated. The results of Ref. [59] were also simulated in lattice QCD [63–65]. Thoughthe instanton vacuum does not explain quark confinement, it will play a certain role in describing the characteristicsof the quarkonia. The feature of the instanton vacuum will be recapitulated briefly in the present work in the contextof 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 ananalytic form of the instanton contributions to the spin-dependent potential but were not able to compute them dueto 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 expandthe 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 comesfrom the leading order in the heavy-quark expansion. The heavy-quark propagator or the Wilson loop being averagedover the instanton medium, the central part can be derived. The spin-dependent contributions arise from the orderof 1 /m Q . As we will show in this work, the heavy-quark propagator is given as an integral equation. Expanding itin powers of 1 /m Q , we are able to compute the spin-dependent part of the heavy-quark potential as was first shownin Ref. [58]. We will evaluate these spin-dependent potentials and examine their behaviour. Then we will proceedto compute the instanton effects on the hyperfine mass splittings of quarkonia. Assuming that the interaction rangebetween a heavy quark and a heavy anti-quark is smaller than the inter-instanton distance, we can easily deal withthe effects of the instantons on the hyperfine mass splittings of the quarkonia. We find at least qualitatively that theinstantons have definite effects on those of the charmonia, while those of the bottomonia acquire tiny effects from theinstanton 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 onthe 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 Q order. InSection III we discuss the results of the instanton effects on the heavy-quark potential in detail and present numericalmethod used to solve the Schr¨odinger equation. We also present the spectrum low laying charmonium states and theestimates of the hyperfine mass splittings of these states. Finally, in Section IV we summarise the results and give afuture outlook related to the present work. II. FORMALISMA. Heavy-quark propagator
We start with the matter part of the QCD Lagrangian for the heavy quark, given as L Ψ = ¯Ψ( x ) (cid:0) i /D − m Q (cid:1) Ψ( x ) , (1)where i /D = i /∂ + /A denotes the covariant derivative, m Q stands for the mass of the heavy quark, and Ψ( x ) representsthe 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 = 1). Then we can decompose the heavy-quarkfield into the large component h v ( x ) and the small one H v ( x ) as followsΨ( x ) = e − im Q v · x (cid:2) h v ( x ) + H v ( x ) (cid:3) , (2)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 h v ( x ) = e im Q v · x (cid:18) /v (cid:19) Ψ( x ) , (3) /vh v ( x ) = h v ( x ) ,H v ( x ) = e im Q v · x (cid:18) − /v (cid:19) Ψ( x ) , (4) /vH v ( x ) = − H v ( x ) . The velocity vector allows one also to split the covariant derivative into the longitudinal and transverse componentsas /D = /v ( v · D ) + /D ⊥ , (5)where /D ⊥ = γ µ ( g µν − v µ v ν ) D ν . The transverse component of the covariant derivative satisfies the relations( i /D ⊥ ) = − D + 12 σ · G = P + σ · B , i /D ⊥ ( iv · D ) i /D ⊥ = E · D + σ · ( E × D ) , (6)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 H v = 12 m Q + iv · D i /D ⊥ h v (7)or equivalently we can integrate out the H v fields [67]. Thus, we arrive at the effective action expressed only in termsof the h v fields S eff [ h v , A ] = Z d x ¯ h v (cid:20) iv · D − i /D ⊥ m Q + iv · D i /D ⊥ (cid:21) h v , (8)where the first term will provide the central contribution to the heavy-quark potential while the second term isresponsible for the spin-dependent part.Using the effective Lagrangian given in Eq. (8), we can define the heavy quark propagator as (cid:20) iv · D − i /D ⊥ m Q + iv · D i /D ⊥ (cid:21) S ( x, y ; A ) = δ (4) ( x − y ) . (9)If we assume that the heavy-quark mass is infinitely heavy, then the heavy-quark propagator in the leading ordersatisfies the following equation( iv · D ) S ( x, y ; A ) = δ (4) ( x − y ) (10)and its solution in the rest frame v = (1 , ) is found to be S ( x, y ; A ) = P exp (cid:18) i Z y x d z A (cid:19) δ (3) ( x − y ) , (11)where A is the time component of the gluon field in four-dimensional Euclidean space. Note that since we considerthe 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 ) isthen expressed as an integral equation as follows S ( x, y ; A ) = S ( x, y, A ) − Z d z S ( x, z ; A ) (cid:20) i /D ⊥ m Q + iv · D i /D ⊥ (cid:21) S ( z, y ; A ) . (12)Since m Q is rather heavy, we can expand iteratively the full propagator (12) in powers of 1 /m Q , when we derive thespin-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 gauge-invariant manner V ( r ) = − lim T →∞ T ln h | Tr { W C [ A ] }| i , (13)where W C [ A ] denotes the Wilson loop expressed as W C [ A ] = P exp (cid:18) i I C d z µ A µ ( z ) (cid:19) . (14)The path is usually taken to be a large rectangle ( T × r ) as drawn in Fig. 1 with r = | x − x | = | y − y | . We first ( x , − T /
2) ( y , T / x , − T /
2) ( y , T / r T FIG. 1. The rectangular Wilson loop. consider the central potential from the instanton vacuum, restating briefly the results from Ref. [59]. The leading-orderexpectation value of the Wilson loop in Euclidean space is defined as h W C [ A ] i = Z DA µ Tr P exp (cid:18) i I C d x µ A µ ( x ) (cid:19) e −S YM (15)where S YM is the Yang-Mills action for the gluon field. The Wilson loop in the instanton medium can be written as W C [ I, ¯ I ] = P exp i I C d t X I, ¯ I a I, ¯ I , (16)where a I, ¯ I = ˙ x µ A I, ¯ Iµ ( x ). I ( ¯ I ) denotes the instanton (anti-instanton). A I, ¯ Iµ represent the instanton (anti-instanton)solutions of which the explicit expressions can be found in Appendix. The sum P I, ¯ I a I, ¯ I stands for the superpositionof N + instantons and N − anti-instantons for the classical gluon background field ¯ A µ , which is written as˙ x µ ¯ A µ ( x, ξ ) = N + X I =1 a I ( x, ξ ) + N − X ¯ I =1 a ¯ I ( x, ξ ) , (17)where ξ represents the set of collective coordinates for the instanton, consisting of its center z Iµ , the size ρ I , andSU( N c ) orientation matrix with the number of colors N c . The integration over the gluon fields given in Eq. (15) isthen replaced with the integrations over the set of collective coordinates of the instantons (anti-instantons) [59–61]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 S ( i )0 ( x, y ; a I, ¯ I ) = h y | ddt − X I, ¯ I a ( i ) I, ¯ I + iǫ − | x i , (18)where a ( i ) I, ¯ I represents the gluon field projected onto the corresponding i th Wilson line. Since T → ∞ , we can neglectthe short sides of the rectangular path. The separation between the two long Wilson lines is given as r , as shown inFig. 1. Using Eqs.(11) and (18), we can write the Wilson loop along the rectangle shown in Fig. 1 asTr W C = DD Tr h S (1)0 ( x , − T / , y , T / a I, ¯ I ) S (2)0 ( x , − T / , y , T / a I, ¯ I ) i iEE , (19)The double angle bracket hh· · · ii emphasizes the average over the instanton ensemble. Each heavy-quark propagatorin Eq. (19) is expanded in powers of the instanton and anti-instanton fields a (1 , I, ¯ I . Then the sum of the planardiagrams is carried out, which is the leading order in the 1 /N c expansion [70]. Note that the instanton vacuumhas two parameters characterizing the dilute instanton liquid [60, 71]: the average size of the instanton ¯ ρ ≃ .
33 fmand the average separation between instantons ¯ R = ( N/V ) − / ≃ N/V ≃ (200 MeV) . It allows one to use N/V N c as a small perturbation parameter. We refer to Ref. [59] for furtherdetails of the calculation.Using Eq. (A2), we can obtain the explicit form of the central potential from the instanton vacuum as V C = N V N c Z d z I Tr c " − P exp i Z T/ − T/ dx A (1) I ! P exp − i Z T/ − T/ dx A (2) I ! z I =0 + ( I → ¯ I )= 2 NV N c Z d z " − cos π | ~z | p | ~z | + ¯ ρ ! cos π | ~z + ~r | p | ~z + ~r | + ¯ ρ ! − ~z ( ~z + ~r ) | ~z || ~z + ~r | sin π | ~z | p | ~z | + ¯ ρ ! sin π | ~z + ~r | p | ~z + ~r | + ¯ ρ ! , (20)where z denotes the position of the instanton, which is one of the collective coordinates for the instantons. The traceTr c runs over the colour space and r is a distance between quark and antiquark. Further introducing the dimentionlessvariables y = z/ ¯ ρ and x = r/ ¯ ρ , one can rewrite the potential in terms of the dimensionless integral I ( x ) V C ( r ) = 4 πN ¯ ρ V N c I (cid:18) r ¯ ρ (cid:19) , (21) I ( x ) = Z ∞ y dy Z − dt " − cos π y p y + 1 ! cos π s y + x + 2 xyty + x + 2 xyt + 1 ! − y + xt p y + x + 2 xyt sin π y p y + 1 ! sin π s y + x + 2 xyty + x + 2 xyt + 1 ! . (22)As r goes to infinity, the potential is saturated to be a constantlim r →∞ V C ( r ) = 2∆ M Q , (23)where ∆ M Q is the correction to the heavy-quark mass from the instanton vacuum [59]∆ M Q = N V N c Z d z Tr c − P exp i ∞ Z −∞ dx A I (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z =0 + ( I → ¯ I )= 8 πN ¯ ρ V N c ∞ Z dy y πy p y + 1 ! = − π N ¯ ρ V N c (cid:18) J ( π ) + 1 π J ( π ) (cid:19) (24)calculated using again Eq. (A2). The average size of the instanton is regarded as a renormalization scale of theinstanton vacuum [61, 72]. Keeping in mind the fact that the current quark mass is scale-dependent and its value isusually given at µ = m c , certain scaling effects arising from the renormalization group equation for the quark massshould 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 massgiven in Ref. [73].We are now in a position to consider the spin-dependent parts of the heavy-quark potential. The general procedureis very similar to what was done in Eq. (19). Since we consider now the finite heavy-quark mass, we need to use thefull propagator given in Eq. (12) instead of the leading one. That is, we calculate the two Wilson lines asTr W C = DD Tr h S (1) ( x , − T / , y , T / a I, ¯ I ) S (2) ( x , − T / , y , T / a I, ¯ I ) i iEE . (25)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 and S inpowers of 1 /m Q i /D ⊥ m Q + iv · D i /D ⊥ ≈ m Q (cid:0) − D + σ · B (cid:1) + 14 m Q [ E · D + σ · ( E × D )] . (26)Then, the heavy-quark propagator for the i th Wilson loop can be iteratively expressed in powers of 1 /m Q as S ( i ) ( x, y ; A ) ≈ S ( i )0 ( x, y ; A ) − m Q Z d η S ( i )0 ( x, η ; A )( − D + σ i · B ) S ( i )0 ( η, y ; A ) − m Q Z d η S ( i )0 ( x, η ; A )( D · D + σ i · ( E × D )) S ( i )0 ( η, y ; A )+ 14 m Q Z d ηd η ′ θ ( η ′ − η ) S ( i )0 ( x, η ; A )( − D + σ i · B ) S ( i )0 ( η, η ′ ; A ) × ( − D + σ i · B ) S ( i )0 ( η ′ , y ; A ) . (27)Replacing the full propagator in Eq. (25) with Eq. (27), we obtain the following expressionTr W C = DD Tr h S (1)0 ( x , − T / , y , T / a I, ¯ I ) S (2)0 ( x , − T / , y , T / a I, ¯ I ) i iEE − m Q Z d ηd η ′ DD Tr h S (1)0 ( x , − T / , η , η ; a I, ¯ I )( − D + σ · B ) S (1)0 ( η , η , y , T / a I, ¯ I ) × S (2)0 ( x , − T / , η , η ; a I, ¯ I )( − D + σ · B ) S (2)0 ( η , η , y , T / a I, ¯ I ) iEE − m Q (cid:28)(cid:28) Tr (cid:20) S (1)0 ( x , − T / , y , T / a I, ¯ I ) Z d η S (2)0 ( x , − T / , η , η ; a I, ¯ I )( E · D + σ · ( E × D )) × S (2)0 ( η , η , y , T / a I, ¯ I ) iEE − m Q (cid:28)(cid:28) Tr (cid:20)Z d η S (1)0 ( x − T / , η , η ; a I, ¯ I )( E · D + σ · ( E × D )) S (1)0 ( η , η , y , T / a I, ¯ I ) × S (2)0 ( x , − T / , y , T / a I, ¯ I ) iEE + 14 m Q (cid:28)(cid:28) Tr (cid:20) S (1)0 ( x − T / , y , T / a I, ¯ I ) Z d ηd η ′ S (2)0 ( x , − T / , η , η ; a I, ¯ I )( − D + σ · B ) × S (2)0 ( η , η , η ′ , η ′ ; a I, ¯ I )( − D + σ · B ) S (2)0 ( η ′ , η ′ , y , T / a I, ¯ I ) iEE + 14 m Q (cid:28)(cid:28) Tr (cid:20)Z d ηd η ′ S (1)0 ( x , − T / , η , η ; a I, ¯ I )( − D + σ · B ) S (1)0 ( η , η , η ′ , η ′ ; a I, ¯ I ) × ( − D + σ · B ) S (1)0 ( η ′ , η ′ , , y , T / a I, ¯ I ) S (2)0 ( x , − T / , y , T / a I, ¯ I ) iEE . (28)Note that here we consider only the spin-dependent parts. For example, we can exclude the spin-independent term D / m Q , which is just the kinetic energy, and that proportional to σ · B , which disappears because of parityinvariance [58]. We can further simplify Eq. (28), leaving all spin-independent parts out, which are just part ofrelativistic corrections to the potential. Taking only the spin-dependent parts into account, we obtainTr W /m Q C = − m Q Z d ηd η ′ DD Tr h S (1)0 ( x , − T / , η , η ; a I, ¯ I )( − D ) S (1)0 ( η , η , y , T / a I, ¯ I ) × S (2)0 ( x , − T / , η , η ; a I, ¯ I )( σ · B ) S (2)0 ( η , η , y , T / a I, ¯ I ) i + Tr h S (1)0 ( x , − T / , η , η ; a I, ¯ I )( σ · B ) S (1)0 ( η , η , y , T / a I, ¯ I ) × S (2)0 ( x , − T / , η , η ; a I, ¯ I )( − D ) S (2)0 ( η , η , y , T / a I, ¯ I ) iEE + Tr h S (1)0 ( x , − T / , η , η ; a I, ¯ I )( σ · B ) S (1)0 ( η , η , y , T / a I, ¯ I ) × S (2)0 ( x , − T / , η , η ; a I, ¯ I )( σ · B ) S (2)0 ( η , η , y , T / a I, ¯ I ) iEE − m Q DD Tr h S (1)0 ( x , − T / , y , T / a I, ¯ I ) × Z d η S (2)0 ( x , − T / , η , η ; a I, ¯ I )( σ · ( E × D )) S (2)0 ( η , η , y , T / a I, ¯ I ) (cid:21)(cid:29)(cid:29) − m Q (cid:28)(cid:28) Tr (cid:20)Z d η S (1)0 ( x − T / , η , η ; a I, ¯ I )( σ · ( E × D )) S (1)0 ( η , η , y , T / a I, ¯ I ) × S (2)0 ( x , − T / , y , T / a I, ¯ I ) iEE . (29)The final expression for W /m Q C contains 1 /m Q , so that we can expand the exponential of Eq. (13) in powers of 1 /m Q .Then, Eq. (29) will lead to the spin-dependent parts of the heavy-quark potential from the instanton vacuum. Thederivation of the potential from Eq. (29) is lengthy but straightforward. In Ref. [58], it was shown in very detailhow one can obtain the spin-dependent parts of the heavy-quark potential in QCD. Since the form of Eq. (29) isvery similar to the corresponding one in Ref. [58], we will closely follow the method of Ref. [58] and refer to it. Theleading-order propagator given in Eq. (11) is identified as the path-order exponential along the time direction apartfrom the Dirac delta function. Using the identities for the path-ordered exponentials given in Appendix, we canproceed to compute each term in Eq. (29). Note that the instanton satisfies the self-duality condition G aµν = ± ˜ G aµν ( B = ± E ), which plays an essential role in deriving the spin-dependent potential from the instanton vacuum. Itmakes it possible to relate several independent potentials to the central potential given in Eq.(21). As a result, allthe spin-dependent potentials are expressed in terms of the central potential V SD ( r ) = 14 m Q ( L · σ − L · σ ) 1 r dV C ( r ) dr + σ · σ m Q ∇ V C ( r )+ 13 m Q (3 σ · n σ · n − σ · σ ) (cid:18) r ddr − d dr (cid:19) V C ( r ) , (30)where L i and σ i represent respectively the orbital angular momentum and the Pauli spin operator of the correspondingheavy quark, n designates the unit radial vector. The potential V C ( r ) denotes the central part of the potential thatwe already have shown in Eq. (21). We want to mention that we have used m Q = m ¯ Q . If one considers two heavyquarks with different masses, we can simply replace m Q with m Q m ¯ Q 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 ): V Q ¯ Q ( r ) = V C ( r ) + V SS ( r )( S Q · S ¯ Q ) + V LS ( r )( L · S ) + V T ( r ) (cid:2) S Q · n )( S ¯ Q · n ) − S Q · S ¯ Q (cid:3) , (31)where S Q ( ¯ Q ) stands for the spin of a heavy quark (heavy anti-quark) S Q ( ¯ Q ) = σ / S does their total spin S = S + S , and L represents the relative orbital angular momentum L = L − L . Each potential of Eq. (31) isdefined respectively as V SS ( r ) = 13 m Q ∇ V C ( r ) , V LS ( r ) = 12 m Q r dV C ( r ) dr , V T ( r ) = 13 m Q (cid:18) r dV C ( r ) dr − d V C ( r ) dr (cid:19) . (32)Thus, all three components of the spin-dependent potential are expressed in terms of the central potential V C ( r ). III. NUMERICAL CALCULATIONS, RESULTS AND DISCUSSIONSA. Instanton potential
In the instanton liquid model for the QCD vacuum, we have two important parameters, i.e., the average size ofthe instanton ¯ ρ ≃ .
33 fm and the average distance ¯ R ≃ MS byDiakonov and Petrov [60]. Thus, it is also of great interest to look into the dependence of the heavy-quark potentialfrom the instanton vacuum on these parameters. Moreover, the values given above should not be considered as theexact ones. For example, Refs. [80–82] considered 1 /N c meson-loop contributions in the light-quark sector and foundit necessesary to readjust the values of parameters as ¯ ρ ≃ .
35 fm and ¯ R ≃ .
856 fm. Lattice simulations of theinstanton vacuum suggested ¯ ρ ≈ .
36 fm and ¯ R ≈ .
89 fm [83–86], 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 instantonvacuum on three different sets of parameters, that is, Set I [60, 71], Set IIa [80–82], and Set IIb [83–86]. The parameterdependence of the potential can be easily understood from the form of leading-order potential expressed in Eq. (21).While the prefactor ¯ ρ / ¯ R N c , which includes both the parameters, governs the overall strength of the potential, itsrange is dictated only by the instanton size ¯ ρ through the dimensionless integral I ( r/ ¯ ρ ).When the quark-antiquark distance is smaller than the instanton size, i.e., r ≪ ¯ ρ ( x ≪ I ( x ) with respect to xI ( x ) ≃ (cid:20) π − π J (2 π ) (cid:21) x + (cid:20) − π (438 + 7 π )30720 + J (2 π )80 (cid:21) x + O ( x ) , (33)which yields the central potential in the form of a polynomial V C ( r ) ≃ π ¯ ρ ¯ R N c (cid:18) . r ¯ ρ − . r ¯ ρ (cid:19) . (34)As the distance between the quark and the antiquark grows larger than the intstanton size, i.e. r ≫ ¯ ρ ( x ≫ I ( x ) ≃ − π h πJ ( π ) + J ( π ) i − π x + O ( x − ) . (35)Consequently, the central potential at large r can be approximately written as V ( r ) ≃ M Q − g NP r . (36)The second term behaves like the Coulomb-like potential. So, crudely speaking, this can be understood as a nonper-turbative contribution to the perturbative one gluon exchange potential from the instanton vauum at large r . Thecoupling constant g NP in Eq.(36), which is defined as g NP := 2 π ¯ ρ / ( N c ¯ R ), could be regarded as a nonperturbativecorrection to the strong coupling constant α s ( r ). When r goes to infinity r → ∞ , the potential is saturated atthe value of 2∆ M Q . As discussed already in Ref. [59], it implies that the instanton vacuum can not explain quarkconfinement. 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 ≃ . M Q ≃ .
06 MeV.The Set IIb produces ∆ M Q ≃ .
72 MeV. r [fm] V C ( r ) [ M e V ] Set ISet IIb ..................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... r [fm] V SS ( r ) [ M e V ] Set ISet IIb ......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... r [fm] V L S ( r ) [ M e V ] Set ISet IIb .................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................. r [fm] V T ( r ) [ M e V ] Set ISet IIb .....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
FIG. 2. Each contribution to the heavy-quark potential as a function of r for the two different sets of the instanton parameters¯ ρ and ¯ R . The upper-left panel depicts the central part of the potential, the upper-right panel draws the spin-spin interaction, thelower-left panel illustrates the spin-orbit part, and the lower-right one shows the tensor interaction. The solid curve correspondsto Set I with ρ = 0 .
33 fm and ¯ R = 1 fm from the phenomenolgy [60, 71], whereas the dashed one corresponds to Set IIb with ρ = 0 .
36 fm and ¯ R = 0 .
89 fm from the lattice simulations [83–86]. The mass of the charm quark is chosen to be m c = 1275 MeV. Figure 2 draws r dependence of each term of the heavy-quark potentials from the instanton vacuum. We takeinto account the charm quark sector as an example. We also show the dependence of each term of the potentialon two different sets of parameters, that is, Set I and Set IIb. One can see that the central part of the potentialincreases monotonically at small distances r ≪ ¯ ρ and later becomes almost linear at the distances comparable with0the 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 → ∞ ) ≃ . V C ( r → ∞ ) ≃ .
44 MeV with Set IIb. The spin-spin interaction part isof particular interest among these contributions to the spin-dependent potential. In pQCD, it is given as a point-likeinteraction [87] in the leading order. On the other hand, the spin-spin interaction from the instanton vacuum lookssimilar 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 ≈ . R is employed, since all terms turn out to be very sensitive to¯ R on account of the prefactor ¯ ρ / ¯ R N c . It implies that a less dilute instanton medium yields stronger interactionsbetween a heavy quark and a heavy antiquark. However, one has to keep in mind that the value of ¯ R should not becontinually decreased, because the whole framework of the instanton liquid model is based on the diluteness of theinstanton medium where the packing parameter proportional to (¯ ρ/ ¯ R ) must be kept as a small parameter.On the other hand, the change of the ¯ ρ value seems less effective in the spin-dependent parts of the potential. Thisis again due to the fact that all spin dependent parts have the prefactor ¯ ρ/ ¯ R N c where instanton size appears in thefirst order, after rewriting the spin-dependent parts in terms of the dimensionless integral I ( x ) (see Eq. (22)) and itsderivatives. As mentioned in the previous section, the average size of the instanton has a physical meaning of therenormalization scale [61, 72], which is a crucial virtue of the instanton liquid model. Thus, ¯ ρ = 0 .
33 fm indicatesthe renormalization scale µ = 600 MeV. Bearing in mind this meaning of ¯ ρ , we should not take the value of ¯ ρ freely.Note that the value of ¯ ρ − = 600 MeV implies the strong couipling constant frozen at ¯ ρ − . Thus, Fig. 2 shows thedependence of the heavy-quark potential on both ¯ ρ and ¯ R within a constraint range of their values. In the case of thebottom quarks and anti-quarks, the instanton effects are quite much suppressed because of the large bottom quarkmass.For completeness, we provide the expression for the matrix elements of Q ¯ Q potential in Eq. (31) h S +1 L J | V Q ¯ Q ( r ) | S +1 L J i = V ( r ) + (cid:20) S ( S + 1) − (cid:21) V SS ( r ) + 12 [ J ( J + 1) − L ( L + 1) − S ( S + 1)] V LS ( r ) (cid:26) − J ( J + 1) − L ( L + 1) − S ( S + 1)] [ J ( J + 1) − L ( L + 1) − S ( S + 1) + 1]2(2 L − L + 3)+ 2 S ( S + 1) L ( L + 1)(2 L − L + 3) (cid:27) V T ( r ) , (37)where we have used the conventional spectroscopic notation S +1 L J given in terms of the total spin S , the orbitalangular 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 Schr¨odinger equation withthe potential from the instanton vacuum given in Eq. (31) (cid:20) − ~ m Q ∇ + V Q ¯ Q ( r ) − E (cid:21) Ψ JM ( r ) = 0 , (38)where m Q arises from the doubled reduced mass of the quarkonium system and Ψ JM represents the wave functionof 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 L -integrable basis functions { Φ LSJM,n ; n = 1 − n max } Ψ JM ( r ) = n max X n =1 C ( J ) n,LS Φ LSJM,n ( r ) (39)and the Rayleigh-Ritz variational principle is employed. Thus, one can formulate a generalized eigenvalue problemgiven as n max X m =1 h Φ LSJM,n (cid:12)(cid:12)(cid:12)(cid:12) − ~ m Q ∇ + V Q ¯ Q ( r ) − E (cid:12)(cid:12)(cid:12)(cid:12) Φ LSJM,m i C ( J ) m,LS = 0 . (40)1The normalized radial part of the basis wave functions φ Ln ( r ) is expressed in terms of the Gaussian basis functions φ Ln ( r ) = L + r − L − n √ π (2 L + 1)!! ! / r L e − ( r/r n ) , (41)where r n , n = 1 , , ..., n max stand for variational parameters. In the case of a two-body problem, the total number ofthe variational parameters can be reduced by choosing the geometric progression in the form of r n = r a n − , whichproduces a good convergence of the results. Thus, we need only three variational parameters, i.e. r , 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 inthe present approach. The bound or quasibound charmonium states with the masses below or around the thresholdmass M Q ¯ Q ≃ m c + ∆ M Q ), where m c = 1275 is the charm quark mass [73], are listed in Table I with the twodifferent sets of the instanton parameters. Other states above threshold will appear as resonances in the presentapproach. One can see that the instanton effects are not small in reproducing the mass of quarkonia. For example, This work This workSet I Set IIb Experiment [73]¯ ρ = 1 / R = 1 fm [60, 71] ¯ ρ = 0 .
36 fm, R = 0 .
89 fm [83–86] [MeV][MeV] [MeV] M η c . ± . M J/ψ . ± . M χ c . ± . M χ c . ± . M χ c . ± . m c = 1275 MeV. in the case of the potential with parameter Set I, the contribution to the mass of a charmonium is determinedby ∆ M c ¯ c = M c ¯ c − m c . For example, the contribution of the instanton effects to the η c mass turns out to be118 .
81 MeV, which is approximately about 30 % in comparison with the experimental data 433.60 MeV. As discussedalready, the potential from the instanton vacuum is sensitive to the instanton parameters. Therefore, the changein the instanton parameters strongly affects the spectrum of Q ¯ Q states. For example, parameter Set IIb gives theresult ∆ M η c ≃ .
64 MeV, which is almost 50 %, compared to the data. Parameter Set IIa gives slightly largerresults than those with Set IIb. When it comes to the
J/ψ state, the instanton effects on the Q ¯ Q mass becomessmaller in comparison with the experimental data. However, it is still important to consider them, since ∆ M J/ψ is119.57 MeV (205.36 MeV) with Set I (Set IIb) used, compared with the data 540.92 MeV. On the other hand, weobtain ∆ M χ c ≃ .
43 MeV (Set I) and ∆ M χ c ≃ .
86 MeV (Set IIb). Parameter Set I reproduces χ c , χ c and χ c as quasibound states while parameter Set IIb yields them as the definite bound states.It is of also interest to discuss the effects of the hyperfine mass splitting from the instanton vacuum. The contributionto the hyperfine mass splitting of each low-lying charmonium state is listed in Table II. While the instanton effects This work This workSet I Set IIb Experiment [73]¯ ρ = 1 / R = 1 fm [60, 71] ¯ ρ = 0 .
36 fm, R = 0 .
89 fm [83–86] [MeV][MeV] [MeV]∆ M J/ψ − η c ± M χ c − χ c ± M χ c − χ c ± M χ c − χ c ± m c = 1275 MeV. come into play significantly on ∆ M c ¯ c , they turn out to be rather small in describing the hyperfine mass splittings of2the charmonia. This might be due to the fact that the spin-dependent part of the potential from the instanton vacuumis almost an order of magnitude smaller than the central part. The tensor interaction almost does not contributeto the results. As a result, the instanton effects on the hyperfine mass splittings are almost negligible. In order toobtain realistic results of the hyperfine mass splittings as well as of the charmonium masses, we need to include theCoulomb-like potential coming from the perturbative one gluon-exchange and the confining potential together withthat from the instanton vacuum. This workSet I Experiment [73]¯ ρ = 1 / R = 1 fm [60, 71] [MeV][MeV] M η b . ± . M Υ . ± . M χ b . ± . M χ b . ± . M χ b . ± . m b = 4180 MeV. IV. SUMMARY AND OUTLOOK
In the present work, we aimed at investigating the instanton effects on the heavy-quark potential, based on theinstanton liquid model. We first considered the heavy-quark propagator starting from the QCD Lagrangian, whichcomes into an essential play in deriving the heavy-quark potential. We showed briefly how to construct the heavy-quark potential from the instanton vacuum. Expanding the heavy-quark propagator in powers of the inverse mass ofthe heavy quark, we obtained the spin-dependent parts of the heavy-quark potential. We studied the dependence ofthe heavy-quark potential on the two essential parameters for the instanton vacuum, that is, the average size of theinstanton (¯ ρ ) and the inter-distance between the instantons ( ¯ R ). The results of the potential are very sensitive to theparameter ¯ R , while they are varied marginally with ¯ ρ changed. The spin-spin interaction shows r dependence similarto a Gaussian-type potential, which is distinguished from the point-like spin-spin interaction derived from perturbativeQCD. The spin-orbit potential behaves like the spin-spin interaction, whereas the tensor potential exhibits a differentcharacter. It increases until r reaches approximately 0 . N/V N c , we can obtain the corrections from the next-to-leading order ( N/V N c ) . In principle,it is not that difficult to compute them. Starting from the instanton operator corresponding to the Wilson line (seeEq.(17) in Ref. [59]), we can consider the next-to-leading order in the expansion with respect to the small packingparameter of the instanton medium. Though the corrections from the next-to-leading order might be very small, onecould use it for the fine-tuning of the mass spectrum of the quarkonia. The corresponding investigations are underway. Appendix A: Useful formulae
Using the instanton and anti-instanton fields A Iµ = x ν ¯ η aµν τ a ρ x ( x + ρ ) , A ¯ Iµ = x ν η aµν τ a ρ x ( x + ρ ) , (A1)3where ¯ η aµν and η aµν denote the ’t Hooft symbols [88], we can easily derive the path-ordered exponential as follows [59] P exp (cid:18) i Z ∞−∞ dx A I (cid:19) = − cos π | z | p ρ + z ! − i τ · z | z | sin π | z | p ρ + z ! , (A2)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 Diracdelta function. Thus, it is of great use to consider the identities derived in Ref. [58] for the path-order exponentialswhen we compute the spin-dependent parts of the heavy-quark potential. Defining the path-ordered exponential as P ( x , y ) := P exp (cid:18) i Z y x dz A ( z ) (cid:19) , (A3)we have the following identities P ( x , y ) P ( y , z ) = P ( x , z ) ,D i ( x ) P ( x , y ) − P ( x , y ) D i ( y ) = Z x y dzP ( x , z ) E i ( z ) P ( z, y ) ,P ( y , t ; x , t ) D i ( x , t ) P ( x , t ; y , t ) = D i ( y , t ) − ǫ ijk Z dα ( x − y ) j [ P ( y , t ; z , t ) B k ( z , t ) P ( z , t ; y , t )] , (A4)where z = α y + (1 − α ) x . D i denotes the spatial component of the covariant derivative. When time t goes to infinity,i.e. t = ± T → ∞ , the third identity is simplified to belim | t |→∞ P ( y , t ; x , t ) D ( x , t ) P ( x , t ; y , t ) = i ∇ y . (A5) ACKNOWLEDGMENTS
HChK wants to express his gratitude to A. Hosaka, M. Oka, and Q. Zhao for very useful comments and discussionsat “The 31st Reimei Workshop on Hadron Physics in Extreme Conditions at J-PARC”. HChK owes also debt ofthanks to the late D. Diakonov and V. Petrov for invaluable discussions and suggestions. This work is supported bythe Basic Science Research Program through the National Research Foundation (NRF) of Korea funded by the Koreangovernment (Ministry of Education, Science and Technology, MEST), Grant Numbers 2016R1D1A1B03935053 (UY)and 2015R1D1A1A01060707 (HChK). The work was also partly supported by RIKEN iTHES Project. [1] S. K. Choi et al. [Belle Collaboration], Phys. Rev. Lett., : 262001 (2003)[2] B. Aubert et al. [BaBar Collaboration], Phys. Rev. D, : 071103 (2005)[3] B. Aubert et al. [BaBar Collaboration], Phys. Rev. Lett., : 142001 (2005)[4] K. Abe et al. [Belle Collaboration], Phys. Rev. Lett., : 082001 (2007)[5] S. K. Choi et al. [Belle Collaboration], Phys. Rev. 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