PHYSICAL REVIEW D 99, 014017 (2019) Spin structure of heavy-quark hybrids Nora Brambilla,1,2 Wai Kin Lai,1 Jorge Segovia,3 Jaume Tarrús Castellà,3 and Antonio Vairo1 1 Physik-Department, Technische Universität München, James-Franck-Straße 1, 85748 Garching, Germany 2 Institute for Advanced Study, Technische Universität München, Lichtenbergstrasse 2a, 85748 Garching, Germany 3 Grup de Física Teòrica, Dept. Física and IFAE-BIST, Universitat Autònoma de Barcelona, E-08193 Bellaterra (Barcelona), Spain (Received 9 October 2018; published 14 January 2019) A unique feature of quantum chromodynamics (QCD), the theory of strong interactions, is the possibility for gluonic degrees of freedom to participate in the construction of physical hadrons, which are color singlets, in an analogous manner to valence quarks. Hadrons with no valence quarks are called glueballs, while hadrons where both gluons and valence quarks combine to form a color singlet are called hybrids. The unambiguous identification of such states among the experimental hadron spectrum has been thus far not possible. Glueballs are particularly difficult to establish experimentally since the lowest lying ones are expected to strongly mix with conventional mesons. On the other hand, hybrids should be easier to single out because the set of quantum numbers available to their lowest excitations may be exotic, i.e., not realized in conventional quark-antiquark systems. Particularly promising for discovery appear to be heavy hybrids, which are made of gluons and a heavy-quark-antiquark pair (charm or bottom). In the heavy-quark sector systematic tools can be used that are not available in the light-quark sector. In this paper we use a nonrelativistic effective field theory to uncover for the first time the full spin structure of heavy-quark hybrids up to 1=m2 -terms in the heavy-quark-mass expansion. We show that such terms display novel characteristics at variance with our consolidated experience on the fine and hyperfine splittings in atomic, molecular and nuclear physics. We determine the nonperturbative contributions to the matching coefficients of the effective field theory by fitting our results to lattice-QCD determinations of the charmonium hybrid spectrum and extrapolate the results to the bottomonium hybrid sector where lattice- QCD determinations are still challenging. DOI: 10.1103/PhysRevD.99.014017 I. INTRODUCTION quark and an antiquark, and baryons, which contain three quarks. Nevertheless, QCD also allows for the existence of Quantum chromodynamics (QCD), the strong-interac- states formed by more than three valence quarks or by tion part of the standard model of particle physics, presents a unique problem: the elementary degrees of freedom solely gluonic excitations (glueballs) or by gluonic exci- (d.o.f.) of the theory, quarks and gluons, are not the tations bound with quarks and antiquarks (hybrids). d.o.f. accessible via the experiments. These elementary The quark model received experimental verification in d.o.f. are confined inside color-singlet states generically the late 1960s and, despite extensive searches, a hadron named hadrons, which form the spectrum of the theory. not fitting the quark-model classification scheme was not A very successful classification scheme for hadrons was unambiguously identified until the last decade, the first independently proposed by Murray Gell-Mann [1] and being the Xð3872Þ discovered by the Belle Collaboration in George Zweig [2] in 1964. It is called the quark model and 2003 [3]. Nowadays, more than two dozen nontraditional classifies hadrons according to their (valence) quark con- charmonium- and bottomoniumlike states, the so-called tent. Two main families were identified: mesons, made of a XYZ mesons, have been observed at experiments at B-factories (BABAR, Belle and CLEO), τ-charm facilities (CLEO-c and BESIII) and also proton-(anti)proton col- liders (CDF, D0, LHCb, ATLAS and CMS). There is Published by the American Physical Society under the terms of growing evidence that at least some of the new states are the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to nonconventional hadrons such as glueballs, hybrids or the author(s) and the published article’s title, journal citation, multiquark systems (see, e.g., reviews [4–7] for more and DOI. Funded by SCOAP3. details on the experimental and theoretical status of the 2470-0010=2019=99(1)=014017(11) 014017-1 Published by the American Physical Society NORA BRAMBILLA et al. PHYS. REV. D 99, 014017 (2019) subject). While multiquark states may, under some circum- Moreover, the former may be computed in perturbation stances, have an electromagnetic analogue in molecular theory. Another hierarchy of scales is due to the existence states, the discovery of a hadron containing a gluonic of an energy gap between the gluonic excitations and the excitation would confirm one of the most unique features of excitations of the heavy-quark-antiquark pair as observed QCD and ultimately open a new window on how matter in lattice data [35–38]. The gluon dynamics is nonpertur- is made. bative and, therefore, it occurs at the scale ΛQCD , while the In addition to the experiments mentioned above, many nonrelativistic heavy-quark-antiquark pair binds in the of which will remain active in the next two decades, a background potential created by the gluonic excitations new experiment, PANDA@FAIR, has been designed to at an energy scale of order mv2, where v is the relative produce at the primary collision point a rich environment velocity between the heavy quark and antiquark [39]. of gluons in order to promote the formation of glueballs and The observed energy gap requires the hierarchy of scales heavy hybrids [8].1 ΛQCD ≫ mv2 to be fulfilled. This energy gap, not present in To support the experimental effort different theoretical light-quark hybrids, has led to the observation that quar- tools have been developed for studying this kind of bound konium hybrids can be treated in the framework of the state. In the early days of hadron phenomenology several Born-Oppenheimer approximation [35,40–43]. In recent models were proposed for glueballs and hybrids [10–15], papers, the Born-Oppenheimer approximation has been and more sophisticated versions have been developed in the incorporated into an effective field theory formulation, last years [16–21]. QCD sum rules have also been applied which is both rigorous and systematic, and it has been used to the study of heavy hybrids [22–27]. The interested reader to compute the quarkonium hybrid spectrum [44–46]. The is referred to [28,29] for reviews on hybrids. mixing of standard and hybrid quarkonia has also been Approaches fully rooted in QCD are lattice simulations studied in this framework in Ref. [45]. and effective field theories (EFTs). The implementation of A complete picture of the quarkonium hybrid spectrum these two techniques is, however, not devoid of difficulties. cannot be obtained until, at least, the leading nonvanishing For lattice calculations it is challenging to address excited spin-dependent contributions are calculated. These terms states, which require a large operator basis leaving never- break the degeneracy within the spin multiplets. The pattern theless sometimes ambiguities in the ultimate identification and values of the spin splittings distinguishes different of the states. Studies of hybrids in lattice QCD have tradi- theoretical pictures for XYZ mesons; therefore their deter- tionally focused on the charmonium sector. Calculations in mination in the chamonium and bottomonium hybrid the bottomonium sector require smaller lattice spacings, sectors will be an important step forward in the under- making them computationally more demanding. A pioneer- standing of exotic quarkonium. ing quenched calculation of the excited charmonium spec- In this paper, combining the EFT suited for hybrids with trum was presented in Ref. [30]. This study was extended by information gained from lattice QCD, we derive for the first the RQCD collaboration [31,32] and by the Hadron Spectrum time the full set of spin-dependent hybrid potentials up to Collaboration [33,34]. The common result in all these studies order 1=m2.2 These include the well-known spin-orbit, has been the identification of the lowest hybrid charmonium spin-spin and tensor potentials that govern the fine and spin multiplet at about 4.3 GeV containing a state with exotic hyperfine splittings in a broad span of problems from quantum numbers J PC ¼ 1−þ . atomic to nuclear and molecular physics, and a new set of On the other hand, EFTs require clear separations interactions unique to heavy hybrids. between the relevant scales of the system in order to be The spin-dependent potentials contain perturbative weakly useful, and necessitate the knowledge of matching coef- coupled contributions and nonperturbative ones. The for- ficients often of nonperturbative nature. In QCD, ΛQCD is mer can be computed analytically and are known up to the scale of a few hundred MeV at which nonperturbative some order in the strong coupling. The latter can be fitted effects dominate and a weak-coupling treatment is no on data or, in the absence of data, on the spectrum obtained longer valid. For heavy-quark or quarkonium hybrids, from lattice QCD. Owing to the factorization of the EFTs, there is however at least one scale, the heavy-quark mass the whole heavy-quark flavor dependence of the spin- m, which is much larger than any other scale of the s dependent potentials is encoded in the perturbative con- ystem and, in particular, larger than ΛQCD . The existence tributions, while nonperturbative contributions are flavor of this hierarchy of scales justifies the use of nonrela- independent. Hence, we can use lattice data for charmo- tivistic effective field theories to factorize high-energy nium hybrids to fit the nonperturbative contributions and effects happening at the scale m from low-energy ones. predict the fine and hyperfine splittings of bottomonium hybrids, where precise lattice-QCD calculations have 1 proved difficult. The experimental search of hybrid mesons in the light-quark sector has instead a long history mostly developed by the BNL- 2 E852 and COMPASS collaborations and will continue with the In Ref. [47], relations between the spin splittings of different new experiment GlueX@JLab12 [9]. hybrid states were presented up to 1=m-suppressed terms. 014017-2 SPIN STRUCTURE OF HEAVY-QUARK HYBRIDS PHYS. REV. D 99, 014017 (2019) II. FORMALISM stands only for hybrids coupling with light hadrons, which may be neglected. The potential V κλλ0 can be organized into Following [44] we classify quarkonium hybrids accord- an expansion in 1=m and a sum of spin-dependent (SD) and ing to their short-distance behavior in the static limit, i.e., spin-independent (SI) parts, at small heavy-quark-antiquark distances, r ≪ 1=ΛQCD . There, quarkonium hybrids reduce to gluelumps, which ð1Þ ð2Þ are bound states made of a color-octet heavy-quark- ð0Þ V κλλ0 ðrÞ V κλλ0 ðrÞ V κλλ0 ðrÞ ¼ V κλ ðrÞδλλ0 þ þ þ ; ð3Þ antiquark pair and some gluonic fields in a color-octet m m2 configuration localized at the center of mass of the heavy- ð1Þ ð1Þ ð1Þ quark-antiquark pair [44,48,49]. A basis of gluelump states V κλλ0 ðrÞ ¼ V κλλ0 SD ðrÞ þ V κλλ0 SI ðrÞ; ð4Þ can be written as ð2Þ ð2Þ ð2Þ V κλλ0 ðrÞ ¼ V κλλ0 SD ðrÞ þ V κλλ0 SI ðrÞ: ð5Þ jκ; λi ¼ Piκλ Oa† ðr; RÞGia κ ðRÞj0i; ð1Þ ð0Þ where R and r are the center-of-mass coordinate and the Besides the static potential, V κλ ðrÞ, the other potentials relative distance of the heavy-quark-antiquark pair, respec- have not been computed in lattice QCD. We gain some tively; Piκλ is an operator that projects the gluelump to an information by looking at them at short distances: eigenstate of K · r̂ with eigenvalue λ, where K is the angular rΛQCD ≪ 1. At short distances the potentials may be momentum operator of the gluons and r̂ the unit vector organized as a sum of a perturbative part, which is typically along the heavy-quark-antiquark axis; Oa is a color-octet nonanalytic in r, and a nonperturbative part, which is a heavy-quark-antiquark field and Gia κ are gluonic fields in a series in powers of r. The perturbative part comes from color-octet configuration; κ ¼ K with P and C parity and PC integrating out modes scaling with the inverse quark- charge conjugation of the gluonic fields, and i and a are antiquark distance 1=r, which at short distances is much spin and color indices, respectively. Summations over larger than ΛQCD . The nonperturbative part comes from the indices i and a are implied. The states jκ; λi live in integrating out modes scaling with ΛQCD . The short- representations, characterized by λ, of the cylindrical- distance expansion is consistent with our quantum number symmetry group D∞h (with P replaced by CP), which is attribution to the states. It is limited, however, to the lowest- the same symmetry group of diatomic molecules. The lying, most compact, hybrid states, the ones most likely to fields Ψκλ ðt; r; RÞ associated to the states jκ; λi are the be described in terms of gluelumps. natural d.o.f. of the low-energy EFT for hybrids [44,46]. For the lowest-lying hybrid excitations of κ ¼ 1þ− ,3 the We use the same notation of [46] and refer to this EFT as spin-dependent potentials take the form the Born-Oppenheimer EFT (BOEFT). ð1Þ ij j The BOEFT describes the low-energy excitations V 1þ− λλ0 SD ðrÞ ¼ V SK ðrÞðPi† 1λ K P1λ0 Þ · S (E ∼ mv2 ) of the hybrids. It is obtained by sequentially integrating out the modes at the higher energy scales m, þ V SKb ðrÞ½ðr · P†1λ Þðri K ij Pj1λ0 Þ · S mv and ΛQCD . Here we extend the BOEFT of [44,46] to þ ðri K ij Pj† 1λ Þ · Sðr · P1λ0 Þ þ    ; ð6Þ include spin-dependent terms of order 1=m and 1=m2 . The Lagrangian describing the fields Ψκλ ðt; r; RÞ reads ð2Þ V 1þ− λλ0 SD ðrÞ ¼ V SLa ðrÞðPi† 1λ LQQ̄ P1λ0 Þ · S i Z X  i j j 3 † 3 þ V SLb ðrÞPi† 1λ ðLQQ̄ S þ S LQQ̄ ÞP1λ0 i j LBOEFT ¼ d Rd r Ψκλ ðt; r; RÞ i∂ t − V κλλ0 ðrÞ λλ0 þ V S2 ðrÞS2 δλλ0 þ V S12 a ðrÞS12 δλλ0  ∇2r þ V S12 b ðrÞPi† j i j i j þ Pi† κλ Piκλ0 Ψκλ0 ðt; r; RÞ þ …; ð2Þ 1λ P1λ0 ðS1 S2 þ S2 S1 Þ þ    ; m ð7Þ where the ellipsis stands for possible terms mixing hybrids with different quantum numbers κ, and hybrids with where LQQ̄ is the orbital angular momentum of the heavy- quarkonia. It also stands for hybrids coupling with light quark-antiquark pair, S1 and S2 are the spin vectors of the hadrons. The hybrids that we examine in this work are the heavy quark and heavy antiquark, respectively, S ¼ S1 þ lowest-lying excitations of κ ¼ 1þ− , which are separated S2 and S12 ¼ 12ðS1 · r̂ÞðS2 · r̂Þ − 4S1 · S2 . ðK ij Þk ¼ iϵikj is by a gap of order ΛQCD from excitations of different κ and the angular momentum operator for the spin-1 gluons. The ordinary quarkonia. Hence, in our case, the contributions projectors Pi1λ read coming from other hybrid states and quarkonia are inte- grated out and included in the potential. They do not show ð0Þ According to lattice QCD the hybrid static potentials V 1þ− λ ðrÞ 3 up as explicit d.o.f. in the Lagrangian (2), and the ellipsis are the lowest lying at short distances [35,37]. 014017-3 NORA BRAMBILLA et al. PHYS. REV. D 99, 014017 (2019) Pi10 ¼ r̂i0 ¼ r̂i ; ð8Þ pffiffiffi Pi11 ¼ r̂i ¼∓ ðθ̂i  iϕ̂i Þ= 2; ð9Þ np ð0Þ where r̂ ¼ ðsin θ cos ϕ; sin θ sin ϕ; cos θÞ, θ̂ ¼ ðcos θ cos ϕ, FIG. 1. Feynman diagram contributing to V SK . The double cos θ sin ϕ; − sin θÞ and ϕ̂ ¼ ð− sin ϕ; cos ϕ; 0Þ. The ellip- and curly lines represent the heavy-quark-antiquark octet and the ses in Eqs. (6) and (7) stand for terms suppressed by powers gluon field, respectively. The black dot stands for a cF Si · B=m of rΛQCD . induced vertex from the weakly coupled pNRQCD Lagrangian, the circles with a cross for the gluelump operators and the shaded The matching coefficients of the spin-dependent poten- circle represents nonperturbative gluon exchanges. tials read V SK ¼ V np SK ; ð10Þ np ð0Þ np ð0Þ np ð0Þ np ð0Þ V SLa ; V SLb ; V S2 ; V S12 b ∼ Λ3QCD . Spin-independent V SKb ¼ V np ð11Þ potentials are irrelevant at our accuracy. Only the r- SKb ; dependent part of the spin-independent 1=m potential V SLa ¼ V o SL þ V np ð12Þ could, in principle, contribute to the spin splittings in SLa ; second order perturbation theory if it is of order Λ2QCD =m V SLb ¼ V np SLb ; ð13Þ or larger. However, the nonperturbative contribution is necessarily r2 Λ4QCD =m ∼ Λ2QCD =m × ðΛQCD rÞ2 ≪ Λ2QCD =m V S2 ¼ V o S2 þ V np S2 ; ð14Þ and the perturbative one shows up only at one loop and is therefore of order mv4 as in the quarkonium case [52]. V S12 a ¼ V o S12 ; ð15Þ The flavor dependence of the BOEFT Lagrangian is in the mass m and, at the order we are working here, in the V S12 b ¼ V np S12 b ; ð16Þ one-loop expression of the quark chromomagnetic match- np ð0Þ ing coefficient cF that enters V SK ; see the Feynman where we have separated the perturbative parts, V o SL , V o S2 diagram in Fig. 1. The expression of cF at Oðαs Þ has been and V o S12 , from the nonperturbative ones, labeled with np. derived in Ref. [53]. The perturbative parts are the spin-dependent parts of the In Ref. [44] the coupled Schrödinger equations resulting color-octet quark-antiquark potential, computed at leading from the Lagrangian (2) with the mixing of the static order in [50]. The nonperturbative parts can be written as an ð0Þ potentials V 1þ− λ ðrÞ, λ ¼ 0; 1 have been solved and the expansion in r2 : V np ¼ V np ð0Þ þ V np ð1Þ r2 þ   . The coef- hybrid spectrum has been calculated. The mixing with ficients V np ðiÞ do not depend on r and can be expressed in other static states is suppressed by a large energy gap. The terms of gluonic correlators. They are made out of the ð0Þ static potentials V 1þ− λ ðrÞ have been obtained from match- gluon fields in the gluelump and in the interaction vertices of the weakly coupled potential nonrelativistic QCD ing it to the static energies computed on the lattice in (pNRQCD) Lagrangian that can be read off from Refs. [37,38]. There are two types of solution correspond- Njm ls Njm ls Ref. [51]. Eventually, gluonic correlators may be computed ing to states with opposite parity, Ψþ j and Ψ− j ,4 on the lattice. For the purpose of this paper we consider them just as constants independent of each other that 0 ðNÞ 1 ψ 0 ðrÞv0lml ðθ; ϕÞ respect, however, the power counting of the EFT. X B C B ðNÞ C We aim to include terms up to order Λ3QCD =m2 and mv4 Clmljsms B p1ffiffi2 ψ þ ðrÞvþ1 Njmj ls jm Ψþ ðrÞ ¼ ðθ; ϕÞ Cχ sms ; @ lml A to the spin splittings. The perturbative spin-dependent ml ms ðNÞ p1ffiffi ψ ðrÞv −1 ðθ; ϕÞ potentials V o SL , V o S2 and V o S12 scale like m3 v4 if we 2 þ lml count the Coulomb potential as mv2 . For the nonperturba- ð17Þ np ð0Þ tive potential V npSK we need to include V SK , which scales np ð1Þ 0 1 like Λ2QCD , and V SK , which scales like Λ4QCD . Higher 0 order terms in the expansion of V np X B p1ffiffi ψ ðNÞ C SK may or may not be Ψ− Njmj ls ðrÞ ¼ Clmljsms B jm 2 − ðrÞvþ1 lml ðθ; ϕÞ Cχ sm ; relevant in dependence of the relative size of ΛQCD =m @ A s − p1ffiffi2 ψ ðNÞ −1 ml ms with respect to v. In the present analysis we neglect − ðrÞvlml ðθ; ϕÞ them. The term proportional to V SKb contributes only ð18Þ to off-diagonal terms (λ ≠ λ0 ) and turns out to be therefore negligible for the present computation. The size of the other potentials follows from dimensional analysis: The P and C of Ψ are P ¼ ð−1Þl and C ¼ ð−1Þlþs . 4 014017-4 SPIN STRUCTURE OF HEAVY-QUARK HYBRIDS PHYS. REV. D 99, 014017 (2019) where lðl þ 1Þ is the eigenvalue of L2 , with L ¼ Uncertainties associated to discretization effects were ðLQQ̄ þ KÞ; jðj þ 1Þ and mj are the eigenvalues of J2 estimated in Ref. [33] to be ∼40 MeV. No analogous and J3 , respectively, with J ¼ L þ S; sðs þ 1Þ is the study was carried out in Ref. [34], but similar associated jm uncertainties are expected. Adopting the same theoretical eigenvalue of S2 ; the Clmljsms are Clebsch-Gordan coeffi- setting as in [44], we define the heavy-quark masses in the cients. The eigenfunctions vλl ml are generalizations of the RS0 scheme [55], which has the advantage to not affect associated Legendre polynomials for systems with cylin- the quark chromomagnetic matching coefficient, cF , at drical symmetry. Their derivation can be found in textbooks the order we are working. For the charm mass we take such as Ref. [54]. The χ s ms are the spin wave functions. c ð1 GeVÞ ¼ 1.477 GeV and for αs at four loops with mRS ðNÞ ðNÞ The radial wave functions ψ 0 ; ψ þ ; ψ ðNÞ − are obtained three massless flavors, αs ð2.6 GeVÞ ¼ 0.26. In the fit the numerically by solving the coupled Schrödinger equations, lattice data are weighted by ðΔ2lattice þ Δ2high−order Þ1=2, where with N labeling the radially excited states. Δlattice is the uncertainty of the lattice data and Δhigh−order ¼ ðmlattice − mlattice spin-average Þ × ΛQCD =m is the estimated III. RESULTS size of the theoretical uncertainty due to higher order terms in the potential, with ΛQCD taken to be 0.5 GeV. We use standard time-independent perturbation theory to The size of the V np ðiÞ ’s is introduced to the fit through a compute the mass shifts in the heavy hybrid spectrum prior. The outcome of the fit is consistent with the non- produced by the spin-dependent operators in Eqs. (6) and (7) for the states with wave functions given in Eqs. (17) perturbative potentials scaling naturally, i.e., V np ðiÞ ∼ and (18). We carry out perturbation theory to second order ðfew hundreds MeVÞd , where d is the mass dimension np ð0Þ of V np ðiÞ . for the 1=m-suppressed operator proportional to V SK in The results of the fit to the lattice data of Ref. [34] are Eq. (6), and to first order for the other operators in Eqs. (6) shown in Fig. 2. The values of the nonperturbative and (7). Since the orbital wave functions vλl ml ðθ; ϕÞ are matching coefficients obtained in the fit are shown in eigenfunctions of L2 ¼ ðLQQ̄ þ KÞ2 instead of L2QQ̄ , the Table II. Each panel in Fig. 2 corresponds to one of the computation of the expectation values of operators con- multiplets of Table I. The most right (purple) boxes for taining LQQ̄ requires some care. The details on the each quantum number indicate the lattice results: the computation of these matrix elements can be found in middle line corresponds to the mass of the state obtained Appendix. from the lattice and the height of the box provides the Finally, we note that the expectation values of V o S2 ðrÞ uncertainty. The (red) dashed line indicates the spin- vanish for all our hybrid states. This follows from V o S2 ðrÞ ∼ average mass of the lattice results. The most left (green) δ3 ðrÞ and from the hybrid wave functions being 0 at the boxes for each quantum number correspond to the origin owing to the spin of the gluons in the gluelumps [44]. spin splittings from the perturbative contributions to We present results for the four lowest-lying spin multiplets Eqs. (10)–(16). The height of these boxes (Δp ) is an np ð0Þ estimate of the uncertainty, given by the parametric size shown in Table I. The six nonperturbative parameters V SK , np ð1Þ np ð0Þ np ð0Þ np ð0Þ np ð0Þ of the higher order corrections to the perturbative part V SK , V SLa , V SLb , V S2 , V S12 b that appear in the spin- of the potential, which is Oðmα5s Þ. The central (blue) dependent potentials in Eqs. (10)–(16) are obtained by fitting boxes for each quantum number are the full results the spin splittings resulting from our calculation to lattice including the nonperturbative contributions after fitting data of the charmonium hybrid spectrum. Two sets of lattice data from the Hadron Spectrum Collaboration are available, one set from Ref. [33] with TABLE II. Nonperturbative matching coefficients determined by a pion mass of mπ ≈ 400 MeV and a more recent set fitting the charmonium hybrid spectrum obtained from the hybrid from Ref. [34] with a pion mass of mπ ≈ 240 MeV. EFT to the lattice spectrum from the Hadron Spectrum Collabo- ration data of Ref. [34] with pion mass of mπ ≈ 240 MeV, respectively. The matching coefficients are normalized to their TABLE I. Lowest-lying quarkonium hybrid multiplets. The parametric natural size. We take the value ΛQCD ¼ 0.5 GeV. number labeling H reflects the order in which the state appears in np ð0Þ the spectrum from lower to higher masses. Note that the l ¼ 0 V SK =Λ2QCD 1.03 state is not the lowest mass state [44]. np ð1Þ V SK =Λ2QCD −0.51 np ð0Þ Multiplet l J PC ðs ¼ 0Þ J PC ðs ¼ 1Þ V SLa =Λ3QCD −1.32 np ð0Þ H1 1 1−− ð0; 1; 2Þ−þ V SLb =Λ3QCD 2.44 H2 1 1þþ ð0; 1; 2Þþ− np ð0Þ V S2 =Λ3QCD −0.33 H3 0 0þþ 1þ− np ð0Þ V S12 b =Λ3QCD −0.39 H4 2 2þþ ð1; 2; 3Þþ− 014017-5 NORA BRAMBILLA et al. PHYS. REV. D 99, 014017 (2019) the six nonperturbative parameters to the lattice data. Refs. [33,34], where only a mild light-quark mass depend- The height of these boxes provides the uncertainty of the ence is observed and the pattern and number of charmo- full result. This uncertainty is given by Δfull ¼ nium states at the two pion masses are the same. Moreover ðΔ2p þ Δ2np þ Δ2fit Þ1=2 , where the uncertainty of the non- the leading light-quark mass dependence of the hybrid perturbative contribution Δnp is estimated to be of the spectrum cancels out in the case of the spin splittings. The same parametric size as the next order contribution to the light-quark dependence of the nonperturbative matching matching coefficients. Δfit is the statistical error of the fit. coefficients of our EFT can be estimated by fitting the The light-quark mass dependence of the spin splittings in lattice data of Ref. [33] and comparing with the results hybrid charmonium can be studied comparing the results of presented in Fig. 2. The precision of the determination of 4.55 4.60 4.50 4.55 4.45 4.50 4.40 Mass (GeV) Mass (GeV) 4.35 4.45 4.30 4.40 4.25 H 1 multiplet H 2 multiplet spin average (4.393 GeV) 4.35 spin average (4.472 GeV) 4.20 perturbative perturbative total total lattice lattice 4.15 4.30 -- -+ -+ -+ ++ +- +- +- 1 0 1 2 1 0 1 2 4.70 4.75 4.65 4.70 4.60 4.65 Mass (GeV) Mass (GeV) 4.55 4.60 4.50 4.55 H 3 multiplet H 4 multiplet 4.45 spin average (4.575 GeV) 4.50 spin average (4.638 GeV) perturbative perturbative total total lattice lattice 4.40 4.45 0++ 1+- 2++ 1+- 2+- 3+- FIG. 2. Spectrum of the four lowest-lying charmonium hybrid multiplets. The lattice results from Ref. [34] with mπ ≈ 240 MeV are the most right (purple) boxes for each quantum number. The perturbative contributions to the spin-dependent operators in Eq. (7) added to the spin average of the lattice results (red dashed lines) are the most left (green) boxes. The central (blue) boxes for each quantum number are the full results from the spin-dependent operators of Eqs. (6) and (7) including perturbative and nonperturbative contributions. The unknown nonperturbative matching coefficients are fitted to reproduce the lattice data. The height of the boxes indicates the uncertainty as detailed in the text. 014017-6 SPIN STRUCTURE OF HEAVY-QUARK HYBRIDS PHYS. REV. D 99, 014017 (2019) the nonperturbative parameters is limited to the size of the between states of λ ¼ 0 and λ ¼ 1 induced by the kinetic- neglected higher order contribution in our EFT counting; energy operator [44], a phenomenon known in molecular we observe that the variation of the parameters is within or physics as Λ-doubling. The perturbative parts of the close to this theoretical uncertainty; therefore no significant matching coefficients produce a pattern of spin splittings light-quark dependence of the nonperturbative parameters which is opposite to the one predicted by lattice-QCD simulations and opposite to ordinary quarkonia due to can be inferred. the repulsive nature of the octet potential. This discrepancy The multiplets H1 and H2 correspond to states with can be reconciled thanks to the nonperturbative contribu- l ¼ 1, and negative and positive parity, respectively. The np ð0Þ mass splitting between H1 and H2 is a result of the mixing tions, in particular, due to the dominant one: V SK , which 10.84 10.86 10.82 10.80 10.84 10.78 Mass (GeV) Mass (GeV) 10.76 10.82 10.74 H1 multiplet H2 multiplet 10.72 spin average (10.790 GeV) spin average (10.840 GeV) perturbative perturbative total total 10.70 10.80 1-- 0-+ 1-+ 2-+ 1++ 0+- 1+- 2+- 11.10 10.94 11.08 10.92 11.06 10.90 Mass (GeV) Mass (GeV) 11.04 10.88 H3 multiplet H4 multiplet 11.02 10.86 spin average (11.060 GeV) spin average (10.900 GeV) perturbative perturbative total total 11.00 10.84 0++ 1+- 2++ 1+- 2+- 3+- FIG. 3. Spectrum of the four lowest-lying bottomonium hybrids computed by adding the spin-dependent contributions from Eqs. (10)–(16) to the spectrum obtained in Ref. [44]. The nonperturbative contribution to the matching coefficients is determined from the fit of the charmonium hybrids spectrum of Ref. [34] shown in Fig. 2. The average mass for each multiplet is shown as a red line. The results with only the perturbative contributions included and the full results with perturbative and nonperturbative contributions included are shown for each quantum number by the left (green) and the right (blue) boxes, respectively. The height of the boxes indicates the uncertainty as detailed in the text. 014017-7 NORA BRAMBILLA et al. PHYS. REV. D 99, 014017 (2019) is of order Λ2QCD =m, and, therefore, parametrically larger spectrum we have determined the nonperturbative contri- than the perturbative contributions, which are of order mv4. butions and obtained new predictions for the bottomonium A consequence of the countervail of the perturbative hybrid spin splittings where lattice-QCD calculations are contributions is a relatively large uncertainty on the full still very challenging and incomplete. result caused by the large nonperturbative contributions. We have found that the perturbative contributions, corre- Due to this uncertainty the mass hierarchies among the sponding to the spin-dependent color-octet quark-antiquark spin triplet states of some multiplets, in particular, for the potentials, generate a spin-splitting pattern opposite to the H4 charmonium hybrid multiplet, could not be firmly one observed in lattice QCD. In our framework, this established. deviation is compensated by the nonperturbative contribu- Having determined the values of the V np ðiÞ ’s by fitting to tions, in particular, by the order 1=m operator that is peculiar the charmonium hybrid spectrum computed on the lattice, of hybrid states and has no perturbative counterpart. we can predict the spin contributions in the bottomonium Finally, we note that, since this EFT setup can be hybrid sector, for which lattice determinations are yet very generalized to describe states with light d.o.f. other than gluons, such as heavy tetraquarks and pentaquarks [43,46], sparse. The only flavor dependence of the V np ðiÞ ’s relevant a similar analysis could be done to describe spin multiplets at the precision we aim for is in the mass and in the quark np ð0Þ of exotic quarkonia other than hybrids, eventually provid- chromomagnetic matching coefficient, cF , in V SK . ing an unified description of all heavy-quark-antiquark spin Both are known. We compute the bottomonium hybrid multiplets. spectrum by adding the spin-dependent contributions from Eqs. (10)–(16) to the spectrum computed in Ref. [44]. We ACKNOWLEDGMENTS show the results thus obtained in Fig. 3. For the bottom we have used the RS0 mass mRS b ð1 GeVÞ ¼ 4.863 GeV. This work has been supported by the DFG and the NSFC through funds provided to the Sino-German CRC 110 IV. CONCLUSIONS “Symmetries and the emergence of structure in QCD,” and by the DFG cluster of excellence “Origin and structure Using a recently developed nonrelativistic effective field of the Universe” ([56]). J. T. C. has been supported by theory [44–46] we have obtained for the first time the 1=m the Spanish MINECO’s Grants No. FPA2014-55613-P, and 1=m2 spin interactions characterizing the fine and No. FPA2017-86989-P and No. SEV-2016-0588. J. S. hyperfine splittings of heavy quarkonium hybrids, one type acknowledges the financial support from the Alexander of exotic hadron under intensive study at high-energy von Humboldt Foundation, from the European Union’s experiments at the B-factory in Japan, the τ-charm factory Horizon 2020 research and innovation program under the in China as well at the LHC at CERN and in perspective at Marie Skłodowska–Curie Grant No. 665919, and from FAIR in Germany. Spanish MINECO’s Juan de la Cierva-Incorporación pro- The spin interactions in quarkonium hybrids display gram, Grant Agreement No. IJCI-2016-30028. novel features. The most interesting one is the appearance of operators already at order 1=m that couple the total spin APPENDIX: MATRIX ELEMENTS OF of the heavy-quark-antiquark pair with the spin of the OPERATORS INVOLVING LQQ̄ gluons [45]. At order 1=m2, we have the spin-orbit, spin- spin and tensor operators familiar from the studies of The angular momentum operator in spherical coordi- standard quarkonia. In addition, at order 1=m2, two new nates reads relevant operators appear involving the projectors associ- i ated to the representations of D∞h . LQQ̄ ¼ −iϕ̂∂ θ þ θ̂∂ : ðA1Þ The spin interactions depend on some matching coef- sin θ ϕ ficients. Their contributions coming from integrating out One can compute the following commutators of the the heavy-quark mass and the heavy-quark-antiquark dis- angular momentum operator and the unit vectors in tance are of perturbative nature, and, therefore, can be spherical coordinates calculated in an expansion in the strong coupling. The contributions coming from integrating out the scale ΛQCD ½LiQQ̄ ; r̂j0  ¼ r̂iþ r̂j− − r̂i− r̂jþ ; ðA2Þ are nonperturbative. Due to the separation of scales under- lying the construction of the EFT, the heavy quark- antiquark flavor dependence of the matching coefficients ½LiQQ̄ ; θ̂j  ¼ iϕ̂i r̂j0 þ i cotðθÞθ̂i ϕ̂j ; ðA3Þ can be factorized from the gluon correlators encoding the nonperturbative dynamics. Thus nonperturbative contribu- ½LiQQ̄ ; ϕ̂j  ¼ −iθ̂i ðr̂j0 þ cotðθÞθ̂j Þ; ðA4Þ tions to the matching coefficients can be determined in the charm sector and then used in the bottom sector. Using the from which we obtain the commutators with the projection latest lattice-QCD calculations of the charmonium hybrid vectors 014017-8 SPIN STRUCTURE OF HEAVY-QUARK HYBRIDS PHYS. REV. D 99, 014017 (2019) ½LiQQ̄ ; r̂j  ¼ ðr̂0j r̂i þ cotðθÞθ̂i r̂j Þ: ðA5Þ This expression vanishes for λ ¼ 0. In the case λ ¼ 1, For any λ, we have   cotðθÞ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r̂†i λ 0 Si l L r̂ QQ̄  l ¼ ðr̂† λ 0 · SÞ r̂  · LQQ̄ − p ffiffi ffi h i λðλ − 1Þ j i 2 j j i LQQ̄ ; r̂λ ¼ λ cotðθÞr̂λ θ̂ þ 1 − i r̂λ−1 r̂þ †   2 ðr̂ 0 · SÞ i rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼∓ λpffiffiffi ∂ θ − ∂ ϕ þ λ cotðθÞ λðλ þ 1Þ j i 2 sinðθÞ − 1− r̂λþ1 r̂− : ðA6Þ 2 ðr̂†λ0 · SÞ ¼∓ pffiffiffi K∓ : ðA12Þ 2 To compute the matrix elements of V SLa we rewrite the operator in the following way, The operators K act as the λ-raising and -lowering ðr̂i† λ LQQ̄ r̂λ0 Þ · S ¼ ðLQQ̄ δλλ0 þ r̂λ ½LQQ̄ ; r̂λ0 Þ · S i i† i operators for the angular wave functions vλlml , 0 1 LQQ̄ r̂†þ −r̂†− B C pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ @ r̂þ LQQ̄ þ cot θθ̂ 0 A·S K vλlml ðθ; ϕÞ ¼ lðl þ 1Þ − λðλ  1Þvλ1 lml ðθ; ϕÞ: ðA13Þ −r̂− 0 LQQ̄ − cot θθ̂ ¼ δλλ0 ½LQQ̄ þ λðcot θθ̂ þ r̂0 Þ · S þ iðr̂†λ × r̂λ0 Þ · S The second piece of the operator in Eq. (A10) can be ij j written in a similar way, ¼ δλλ0 L · S − ðr̂i† λ ðK 1 Þ r̂λ0 Þ · S δλλ0 2 ¼ ðJ − L2 − S2 Þ − ðr̂i† ij j λ ðK 1 Þ r̂λ0 Þ · S; ðA7Þ r̂†i †i † λ0 LQQ̄ S r̂λ ¼ ð½r̂λ0 ; L  þ LQQ̄ · r̂λ0 ÞS · r̂λ i l l i 2   cotðθÞ where we have used that ¼ LQQ̄ · r̂†λ0 − ðλ0 Þ2 pffiffiffi r̂λ · S: ðA14Þ 2 ½LQQ̄ þ λðcot θθ̂ þ r̂0 Þ2 λ2 cos θ In this case the operator vanishes for λ0 ¼ 0. For λ0 ¼ 1 ¼ L2QQ̄ þ 2 þ 2iλ 2 ∂ θ ≡ L2 ; ðA8Þ sin θ sin θ we have which is the operator whose eigenfunctions are our angular   wave functions, cotðθÞ r̂†i  LQQ̄ S r̂λ ¼ i l l ðr̂ · LQQ̄ Þ† − pffiffiffi r̂λ · S   2 λ2 cos θ  † LQQ̄ þ 2 þ 2iλ 2 ∂ θ vλlm ðθ; ϕÞ 2 i 0 r̂λ · S sin θ sin θ ¼∓ ∂ θ − ∂ ϕ þ λ cotðθÞ pffiffiffi sinðθÞ 2 ¼ lðl þ 1Þvλlm ðθ; ϕÞ: ðA9Þ ðr̂λ · SÞ ¼∓ K0†∓ pffiffiffi : ðA15Þ Next we show a detailed computation of the matrix 2 elements of the operator V SLb, The prime in K0∓ indicates that the operator depends on λ0 r̂†i λ0 ðLQQ̄ S i l þ Si LlQQ̄ Þr̂lλ : ðA10Þ instead of λ. Adding up both contributions we arrive at The first term in Eq. (A10) can be manipulated as follows: r̂†i λ ðLQQ̄ S þ S LQQ̄ Þr̂λ0 i l i l l r̂†i i l l λ0 S LQQ̄ r̂λ ¼ ðr̂†λ0 · SÞðr̂λ · LQQ̄ þ ½LlQQ̄ ; r̂lλ Þ   ðr̂λ0 · SÞ ðr̂†λ · SÞ 0 ¼∓ K†∓ pffiffiffi δλ1 ∓ p ffiffiffi K∓ δλ0 1 : ðA16Þ † 2 cotðθÞ p ffiffi ffi 2 2 ¼ ðr̂λ0 · SÞ r̂λ · LQQ̄ − λ : ðA11Þ 2 014017-9 NORA BRAMBILLA et al. PHYS. REV. D 99, 014017 (2019) [1] M. Gell-Mann, A schematic model of baryons and mesons, mass predictions with QCD sum-rules, J. Phys. G 39, Phys. Lett. 8, 214 (1964). 125003 (2012). [2] G. Zweig, Developments in the quark theory of hadrons, [26] R. Berg, D. Harnett, R. T. Kleiv, and T. G. Steele, Mass CERN Reports No. 8182/TH.401 and No. 8419/TH.412, predictions for pseudoscalar J PC ¼ 0−þ charmonium and 1964. bottomonium hybrids in QCD sum-rules, Phys. Rev. D 86, [3] S. K. Choi et al., Observation of a Narrow Charmoniumlike 034002 (2012). State in Exclusive B → K  π þ π − J=ψ Decays, Phys. Rev. [27] W. Chen, R. T. Kleiv, T. G. Steele, B. Bulthuis, D. Harnett, J. Lett. 91, 262001 (2003). Ho, T. Richards, and S.-L. Zhu, Mass spectrum of heavy [4] N. Brambilla et al., Heavy Quarkonium Physics (CERN, quarkonium hybrids, J. High Energy Phys. 09 (2013) 019. Geneva, 2005). [28] R. F. Lebed, R. E. Mitchell, and E. S. Swanson, Heavy- [5] N. Brambilla et al., Heavy quarkonium: Progress, puzzles, quark QCD exotica, Prog. Part. Nucl. Phys. 93, 143 (2017). and opportunities, Eur. Phys. J. C 71, 1534 (2011). [29] R. A. Briceno et al., Issues and opportunities in exotic [6] N. Brambilla et al., QCD and strongly coupled gauge theories: hadrons, Chin. Phys. C 40, 042001 (2016). Challenges and perspectives, Eur. Phys. J. C 74, 2981 (2014). [30] J. J. Dudek, R. G. Edwards, N. Mathur, and D. G. Richards, [7] S. L. Olsen, A new hadron spectroscopy, Front. Phys. 10, Charmonium excited state spectrum in lattice QCD, Phys. 121 (2015). Rev. D 77, 034501 (2008). [8] M. F. M. Lutz et al., Physics performance report for [31] G. Bali et al., Spectra of heavy-light and heavy-heavy PANDA: Strong interaction studies with antiprotons, arXiv: mesons containing charm quarks, including higher spin 0903.3905. states for N f ¼ 2 þ 1, Proc. Sci. LATTICE2011 (2011) 135. [9] J. Dudek et al., Physics opportunities with the 12 GeV [32] G. S. Bali, S. Collins, and C. Ehmann, Charmonium upgrade at Jefferson lab, Eur. Phys. J. A 48, 187 (2012). spectroscopy and mixing with light quark and open charm [10] D. Horn and J. Mandula, A model of mesons with con- states from nF ¼ 2 lattice QCD, Phys. Rev. D 84, 094506 stituent gluons, Phys. Rev. D 17, 898 (1978). (2011). [11] T. Barnes, A transverse Gluonium potential model with [33] L. Liu, G. Moir, M. Peardon, S. M. Ryan, C. E. Thomas, P. Breit-Fermi hyperfine effects, Z. Phys. C 10, 275 (1981). Vilaseca, J. J. Dudek, R. G. Edwards, B. Joo, and D. G. [12] M. S. Chanowitz and S. R. Sharpe, Hybrids: Mixed states of Richards, Excited and exotic charmonium spectroscopy quarks and gluons, Nucl. Phys. B222, 211 (1983); Erratum, from lattice QCD, J. High Energy Phys. 07 (2012) 126. Nucl. Phys. B228, 588(E) (1983). [34] G. K. C. Cheung, C. O’Hara, G. Moir, M. Peardon, S. M. [13] T. Barnes, F. E. Close, F. de Viron, and J. Weyers, Q anti-Q Ryan, C. E. Thomas, and D. Tims, Excited and exotic G hybrid mesons in the MIT bag model, Nucl. Phys. B224, charmonium, Ds and D meson spectra for two light quark 241 (1983). masses from lattice QCD, J. High Energy Phys. 12 (2016) [14] J. M. Cornwall and A. Soni, Glueballs as bound states of 089. massive gluons, Phys. Lett. 120B, 431 (1983). [35] K. J. Juge, J. Kuti, and C. J. Morningstar, Gluon excitations [15] N. Isgur and J. E. Paton, A flux tube model for hadrons in of the static quark potential and the hybrid quarkonium QCD, Phys. Rev. D 31, 2910 (1985). spectrum, Nucl. Phys. B, Proc. Suppl. 63, 326 (1998). [16] Yu. S. Kalashnikova, Exotic hybrids and their nonexotic [36] G. S. Bali, B. Bolder, N. Eicker, T. Lippert, B. Orth, P. counterparts, Z. Phys. C 62, 323 (1994). Ueberholz, K. Schilling, and T. Struckmann, Static poten- [17] E. S. Swanson and A. P. Szczepaniak, Heavy hybrids with tials and glueball masses from QCD simulations with constituent gluons, Phys. Rev. D 59, 014035 (1998). Wilson sea quarks, Phys. Rev. D 62, 054503 (2000). [18] F. Brau and C. Semay, Semirelativistic potential model for [37] K. J. Juge, J. Kuti, and C. Morningstar, Fine Structure of the glueball states, Phys. Rev. D 70, 014017 (2004). QCD String Spectrum, Phys. Rev. Lett. 90, 161601 (2003). [19] A. P. Szczepaniak and P. Krupinski, Coulomb energy and [38] G. S. Bali and A. Pineda, QCD phenomenology of static gluon distribution in the presence of static sources, Phys. sources and gluonic excitations at short distances, Phys. Rev. D 73, 034022 (2006). Rev. D 69, 094001 (2004). [20] F. Buisseret and C. Semay, On two- and three-body descrip- [39] N. Brambilla, A. Pineda, J. Soto, and A. Vairo, Effective tions of hybrid mesons, Phys. Rev. D 74, 114018 (2006). field theories for heavy quarkonium, Rev. Mod. Phys. 77, [21] P. Guo, A. P. Szczepaniak, G. Galata, A. Vassallo, and E. 1423 (2005). Santopinto, Heavy quarkonium hybrids from coulomb [40] L. A. Griffiths, C. Michael, and P. E. L. Rakow, Mesons gauge QCD, Phys. Rev. D 78, 056003 (2008). with excited glue, Phys. Lett. 129B, 351 (1983). [22] J. Govaerts, L. J. Reinders, H. R. Rubinstein, and J. Weyers, [41] K. J. Juge, J. Kuti, and C. J. Morningstar, Ab Initio Study of Hybrid quarkonia from QCD sum rules, Nucl. Phys. B258, Hybrid b̄ gb Mesons, Phys. Rev. Lett. 82, 4400 (1999). 215 (1985). [42] E. Braaten, C. Langmack, and D. H. Smith, Born-oppenheimer [23] J. Govaerts, L. J. Reinders, and J. Weyers, Radial excitations approximation for the XYZ mesons, Phys. Rev. D 90, 014044 and exotic mesons via QCD sum rules, Nucl. Phys. B262, (2014). 575 (1985). [43] E. Braaten, C. Langmack, and D. H. Smith, Selection Rules [24] J. Govaerts, L. J. Reinders, P. Francken, X. Gonze, and for Hadronic Transitions of XYZ Mesons, Phys. Rev. Lett. J. Weyers, Coupled QCD sum rules for hybrid mesons, 112, 222001 (2014). Nucl. Phys. B284, 674 (1987). [44] M. Berwein, N. Brambilla, J. Tarrús Castellà, and A. Vairo, [25] D. Harnett, R. T. Kleiv, T. G. Steele, and H.-y. Jin, Axial Quarkonium Hybrids with nonrelativistic effective field vector J PC ¼ 1þþ charmonium and bottomonium hybrid theories, Phys. Rev. D 92, 114019 (2015). 014017-10 SPIN STRUCTURE OF HEAVY-QUARK HYBRIDS PHYS. REV. D 99, 014017 (2019) [45] R. Oncala and J. Soto, Heavy quarkonium hybrids: D 55, 407 (1997); Erratum, Phys. Rev. D 59, 099901(E) Spectrum, decay and mixing, Phys. Rev. D 96, 014004 (1999). (2017). [51] N. Brambilla, D. Gromes, and A. Vairo, Poincare invariance [46] N. Brambilla, G. Krein, J. Tarrús Castellà, and A. Vairo, constraints on NRQCD and potential NRQCD, Phys. Lett. Born-oppenheimer approximation in an effective field B 576, 314 (2003). theory language, Phys. Rev. D 97, 016016 (2018). [52] N. Brambilla, A. Pineda, J. Soto, and A. Vairo, The QCD [47] J. Soto, Heavy Quarkonium Hybrids, Nucl. Part. Phys. Proc. potential at Oð1=mÞ, Phys. Rev. D 63, 014023 (2000). 294, 87 (2018). [53] E. Eichten and B. R. Hill, Static effective field theory: 1=m [48] M. Foster and C. Michael, Hadrons with a heavy color corrections, Phys. Lett. B 243, 427 (1990). adjoint particle, Phys. Rev. D 59, 094509 (1999). [54] L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non- [49] N. Brambilla, A. Pineda, J. Soto, and A. Vairo, Potential Relativistic Theory, Vol. 3 of Course of Theoretical Physics, NRQCD: An effective theory for heavy quarkonium, Nucl. 3rd ed. (Pergamon Press, New York, 1977). Phys. B566, 275 (2000). [55] A. Pineda, Determination of the bottom quark mass from the [50] A. Pineda, More nonperturbative corrections to the fine Upsilon(1S) system, J. High Energy Phys. 06 (2001) 022. and hyperfine splitting in the heavy quarkonium, Phys. Rev. [56] www.universe-cluster.de. 014017-11