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The naive quark model describes hadrons as
qˉq mesons andqqq baryons. Since quarks and gluons are the fundamental degrees of freedom of QCD, if gluons can act as building blocks similar to quarks to build up hadrons, from a phenomenological perspective, there may exist glueballs that are purely made up of gluons and hybrids that are composed of quarks and gluons. Glueballs, hybrids, and multiquark states (tetraquarks, pentaquarks, etc.) are usually called exotic hadrons in contrast to conventionalqˉq mesons andqqq baryons. Exotic hadrons are long-standing topics of interest in theoretical and experimental studies of particle physics, especially with the discovery of numerousXYZ particles in the current era based on experimental investigations. These particles exhibit exotic properties in their production and decay processes and are potential candidates for exotic hadrons [1]. Given thatqˉqg hybrids are made up of a quark-antiquark pair and a gluon, theJPC=1−+ states are most interesting since this quantum number is prohibited in conventionalqˉq mesons. There have been many theoretical studies on hybrids from a phenomenological perspective and the lattice QCD approach. It was determined that the lowest1−+ hybrid usually has a mass of approximately 1 GeV higher than the ground state vector meson with the sameqˉq component. For example, the mass of the1−+ hybrid with light flavors was estimated to be approximately 1.9 GeV, whereas those of the strangeonium-like and the charmonium-like counterparts are approximately 2.1-2.3 GeV [2, 3] and 4.1-4.3 GeV [4], respectively. Reliable candidates for1−+ hybrids have not been experimentally determined to date. The vector charmonium-like stateY(4260) (orψ(4230) named by PDG 2018 [5]), due to its very different properties compared to the conventional charmonia and the closeness of its mass to that of the1−+ cˉcg hybrid, has a possible assignment of1−− hybrid [6].ϕ(2170) [5], also known asY(2175) , was first observed as a result of the BABAR Collaboration in the initial-state-radiation processe+e−→γISRϕf0(980) in 2006 [7] and was confirmed later by BES and Belle [8, 9]. The similarity of its property toY(4260) also suggests asˉsg hybrid interpretation ofϕ(2170) .Phenomenologically,
qˉqg hybrids are usually studied in the constituent gluon model [10], in which the gluon acts as an effective degree of freedom, similar to constituent quarks in the quark model, or the flux tube model in which the gluon is taken as a transverse vibration mode of the flux-tube that binds theqˉq pair [11]. For the hybrids with a heavy quark-anti-quark pairQˉQ , the gluonic excitations along the flux-tube are fast objects, such that in the Oppenheimer approximation [12-14], their distribution obeys cylinder symmetry along theQˉQ -axis, and their motion effects on theQˉQ can be taken as a centrifugal barrier, apart from the binding linear potential. Based on the hybrid potentials simulated from the lattice QCD, one can solve the Schrödinger equation for theQˉQ system to obtain predictions regarding the spectrum of hybrids with properly tuned parameters. There have also been phenomenological studies on heavy quarkonium-like hybrids in which gluonic excitations are treated in a mean field manner [15-17].Lattice QCD is an ab initio non-perturbative approach for the study of strong interactions in the low energy scale and is applied extensively to the investigation of hybrids [18-26]. The mass of the hybrids can be derived from the correlation functions of hybrid-like operators
ˉq→Γq∘→B , whereˉqq is the color octet,→B is the chromomagnetic field strength,Γ represents specific combinations ofγ matrices, and the symbol∘ represents any possible summation of the spatial indices of→Γ and→B . A recent lattice calculation [4] revealed that there exists a{1−−,(0,1,2)−+} charmonium-like supermultiplet with nearly degenerate masses of approximately 4.2-4.4 GeV, which overlaps strongly with the hybrid-like operators. This observation implies that these states may have similar internal dynamics, whereas the spin-spin coupling of theˉqq and→B yields the different quantum numbers. In our previous work [27], the internal structure of this supermultiplet was investigated by calculating their Bethe-Salpeter (BS) wave functions based on the lattice QCD in the quenched approximation, in which the spatially extended interpolating field operatorsˉq→Γq(→x,t)∘→B(→x+→r,t) are introduced in the Coulomb gauge, whose matrix element between the vacuum and a state is defined as the BS wave function. It was determined that the BS wave functions of the states in this multiplet are very similar and exhibit interesting nodal structures, which imply that the distance between thecˉc and the→B operator is a meaningful dynamic variable for hybrids.In this work, we extend the aforementioned study strategy to strangeonium-like hybrids and focus on the
{1−−,(0,1,2)−+} states, to determine if a situation similar to that of thecˉcg hybrids can also occur for thesˉsg states. In contrast, since the quantum numbers1−− and0−+ are permitted by theqˉq mesons, in these channels, we will also use the spatially extendedsˉs operators for the quark fields with spatial separations to extract the related BS wave functions, from which we can investigate the internal structure of these states. By comparison of these two kinds of BS wave functions, we may obtain useful information on the possible different formation pattern of hybrids from conventional mesons. Forϕ(2170) , since it can be33S1 ,23D1 sˉs , or a candidate for the vectorsˉsg hybrid, its properties will be discussed based on the results of this study.This work is organized as follows. Section II gives a detailed description of our lattice setup and the numerical strategy, including the construction of the spatially extended operators, the data analysis procedure, and the results of the spectrum and BS wave functions. The discussion and the comparison of our results with those of relevant phenomenological studies are presented in Section III. Section IV is an overall summary.
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The pure gauge configurations are generated via the tadpole-improved gauge action [28, 29] on anisotropic lattices with an aspect ratio of
ξ=as/at=5 , whereas andat are the spatial and temporal lattice spacing, respectively. Two latticesL3×T=163×160(β=2.4) and243×192(β=2.8) with different lattice spacings are used to check the discretization artifacts. The parameters of the gauge ensembles are listed in Table 1, where theas values are determined fromr−10=410(20) MeV. For the strange valence quark, we use the tadpole-improved clover action, whose parameters are carefully tuned by requiring the dispersion relations of the vector and pseudoscalar mesons to be reproduced [30]. As will be addressed in the following sections, we will use spatially extended operators to calculate the relevant correlation functions. Therefore, the configurations are first fixed to the Coulomb gauge based on the standard gauge fixing procedure used in lattice QCD studies before the quark propagators are computed. In this investigation, we work in the quenched approximation for which the effects of dynamical quarks are omitted. As such, two-particle scattering states will not appear in the spectrum.β ξ as /fmLas /fmL3×T Nconf 2.4 5 0.222(2) 3.55 163×160 500 2.8 5 0.138(1) 3.31 243×192 200 Table 1. The input parameters for the calculation. Values of the coupling
β , anisotropyξ , lattice spacingas , lattice size, and number of measurements are listed.as/r0 is determined by the static potential; the first error ofas is the statistical error, and the second one is due to the uncertainty of the scale parameterr−10=410(20) MeV. -
The major goal of this study is to investigate the inner structure of
sˉsg hybrids of the quantum numbersJPC=1−−,(0,1,2)−+ . We introduce two types of spatially extended operators as sink operators. The first type includes the followingsˉsg operators:Ok1−−(r,t)=∑→x,|→r|=rˉsa(→x,t)γ5sb(→x,t)Babk(→x+→r,t),O0−+(r,t)=∑→x,|→r|=rˉsa(→x,t)γisb(→x,t)Babi(→x+→r,t),Ok1−+(r,t)=∑→x,|→r|=rˉsa(→x,t)γisb(→x,t)Babj(→x+→r,t)εijk,Ok2−+(r,t)=∑→x,|→r|=rˉsa(→x,t)γisb(→x,t)Babj(→x+→r,t)|εijk|,
(1) where
i,j,k=1,2,3 are the spatial indices,a,b=1,2,3 are color indices, andBabi=12εijkFabjk is the chromomagnetic field strength. The summation over→r with the same distance r leads to the operators having correct quantum numbers. It should be noted that on a hypercubic spacetime latice, the spinJ=2 corresponds toT2⊕E whereT2 and E are the irreducible representations of the lattice symmetry group O, so theOk2−+(r,t) are the three components ofT2 . Generally speaking, the masses of theT2 state and the E state, which correspond to the sameJ=2 state, are different byO(ans) terms (here, the power n depends on the specific lattice setups). This difference vanishes in the continuum limitas→0 . In this work, we ignore the mass splitting due to the discretization effect and treat the mass of theT2 state as an approximation of theJ=2 state, so we do not give the explicit expression of the E operators in Eq. (1). The possible ground hybrids in these four quantum numbers form a supermultiplet as expected. Obviously, the two constituent quarks are localized at the same space-time point, and the gluon component is placed at another space point. The BS wave function that we attempt to extract reflects the dynamics of these two parts.Since the quantum numbers
JPC=0−+,1−− are conventional ones forqˉq mesons, we also introduce the second type of spatially extendedsˉs operators by splitting the strange quark field s and its conjugateˉs by the spatial separation→r . This is explicitly expressed asP0−+(r,t)=∑→x,|→r|=rˉs(→x,t)γ5s(→x+→r,t),Pk1−−(r,t)=∑→x,|→r|=rˉs(→x,t)γks(→x+→r,t),
(2) where the summation over
→r with|→r|=r is performed to guarantee the correctJPC .In practice, we calculate the wall-source correlation functions of these operators. For example, we use the following wall-source operators for
1−+ states:OW,k(τ)=∑→y,→zˉsa(→y,τ)γiBabj(→z,τ)sb(→z,τ)εijk,
(3) where
τ indicates the source time slice. The wall-source operators for otherJPC states vary accordingly. The wall source operators for thesˉs operators arePW(τ)= ∑→y,→zˉs(→y,τ)Γs(→z,τ) withΓ=γ5,γi for0−+ and1−− , respectively. Finally, we calculate the correlation functions (for simplicity, we setτ=0 and omit the subscripts and superscripts that refer to specific symmetry channels and different spatial components) asC(r,t)=⟨Ok(r,t)OW,k†(0)⟩.
(4) After the intermediate state insertion, the correlation function
C(r,t) can be parameterized asC(r,t)=1Nc∑n12mnL3⟨0|O(r,t)|n⟩⟨n|OW(0)|0⟩=1Nc∑n12mnL3⟨0|O(r,0)|n⟩⟨n|OW(0)|0⟩e−mnt≡∑nΦn(r)e−mnt,
(5) where
Nc is the degenerate degree ofr=|→r| ,mn is the mass of the n-th state, andΦn(r) is defined as the corresponding Bethe-Salpeter wave function up to an irrelevant constant factor. It should be noted thatmn is independent of r, so that if we simultaneously fitC(r,t) with different r using Eq. (5), we can obtainmn andΦn(r) . To be specific, ifnr differentC(r,t) values with different r values are considered, and N mass terms are involved in the fitting model, the number of parameters to be fitted isN⋅nr+N . Since we usually have 20-30 statistically meaningful data points for eachC(r,t) , the number of degrees of freedom is large enough in the fitting procedure. -
We start with the
1−+ channel, sinceJPC is a typical exotic quantum number which cannot be assigned to aqˉq meson in the quark model. After the correlation functionC(r,t) is calculated for theβ=2.4 andβ=2.8 lattices, Eq. (5) is used for data analysis wherein we useN=3 mass terms. For theβ=2.4 lattice, the r range is from 0 to0.9 fm (converted using the lattice spacingas in Table 1), and the upper limit of the fit window[tmin,tmax] is uniformly set to betmax=20 for all theC(r,t) , while thetmin varies from 6 to 3. For theβ=2.8 lattice, r has a value up to0.8 fm, andtmax is set totmax=30 , withtmin varying from 9 to 6. For each lattice, we perform a simultaneous correlated fit for all the differentC(r,t) values using the jackknife covariance matrix. Table 2 summarizes the fit results for the massmn withn=1,2,3 for different time windows[tmin,tmax] as well asχ2 per degree of freedom (χ2/dof ), which is approximately one and indicates that the fits are reasonable. The mass of the lowest three states is stable to some extent for differenttmin and as such is a reliable parameter. For the lowest two states, the mass values for theβ=2.4 lattice are slightly larger (approximately 100 MeV larger) than those forβ=2.8 lattice. This difference might be attributed to the finite lattice spacing effect, and the strange quark mass parameters on the two lattices are not tuned to be exactly the same in the sense of their physical meaning. Combining the results from the two lattices, we can determine that the mass of the lowest1−+ sˉsg state is approximately 2.2 GeV. However, the mass splitting of the lowest two states is approximately 1.4 GeV, which is almost the same as that of the1−+ charmonium-like hybrids, and thus exhibits quark mass independence to some extent.tmin χ2/dof m1 /GeVm2 /GeVm3 /GeVβ=2.4 tmax=20 ( sˉsg )6 1.12 2.232(22) 3.56(21) 7.5(2.7) 5 1.23 2.228(22) 3.61(26) 4.9(7) 4 1.36 2.248(13) 3.71(11) 5.7(5) 3 1.38 2.255(09) 3.65(07) 5.4(2) β=2.8 tmax=30 ( sˉsg )9 1.53 2.099(16) 3.55(07) 7.7(7) 8 1.41 2.168(15) 3.78(13) 5.1(3) 7 1.47 2.100(15) 3.40(07) 5.8(2) 6 1.26 2.110(13) 3.47(06) 5.5(1) Table 2. The fitted masses
mn of the1−+ states withn=1,2,3 from different time windows[tmin,tmax] andχ2 per degree of freedom (χ2/dof ) for theβ=2.4 andβ=2.8 lattices. All the masses are converted to values in physical units using the lattice spacingas in Table 1.Along with the masses, the BS wave functions of the states of
Φn(r) can be extracted from the joint fit toC(r,t) as shown in Fig. 1. The radial separation r is converted to the value in physical units, and the wave functions on the two lattices are compatible with each other. The BS wave functions exhibit a clear nodal structure along the r direction: the BS wave functionΦ1(r) of the ground state has no radial node, and that of the first excited state (Φ2(r) ere) has one node, whereas that of the third state has two nodes. These nodal structures are very similar to the non-relativistic two-body Schrödinger wave functions in a central potential. It should be noted that r is the spatial separation between thesˉs component and the color chromomagnetic field strength→B . The r behavior of the wave functions of the excitations may imply that within the1−+ sˉsg hybrid, the relative movement between thesˉs and the gluonic degrees of freedom can be viewed qualitatively as a two-body system in which r is a physically meaningful dynamic variable. The same data analysis strategy is applied to the2−+ channel. The fitted masses of the lowest three states are listed in Table 3, and the wave functions are shown in Fig. 2. In comparison with the case of1−+ , the masses of the2−+ states are slightly higher (100-200 MeV higher for the ground states) than their1−+ counterparts, but the pattern of the spectrum is similar. For the BS wave functions, the r behavior of the1−+ and2−+ states are similar. These observations support the assertion that the1−+ and2−+ states have almost the same inner structure and dynamics, whereas the small mass difference can be attributed to the different coupling between the spin of thesˉs subsystem and the gluonic degrees of freedom. This meets our expectation that1−+ and2−+ states with the nearly degenerate mass can be in the same supermultiplet. Of course, the possibility exists that these2−+ states can be the conventionalsˉs mesons sinceJPC=2−+ is permitted for aqˉq system. However, the masses we obtain are much higher than those of the1D2 sˉs states in the quark model. However, a previous lattice study [31] on charmonium states found that the2−+ cˉcg operator couples almost exclusively to a state with a mass of 4.4 GeV instead of the expected11D2 charmonium stateηc2 , with a mass of approximately 3.8 GeV. If this is also the case for thesˉs states, the hybrid assignment is favorable for the2−+ states we obtain in this work.Figure 1. (color online) The BS wave functions
Φn(r) (normalized asΦn(0)=1 ) of the lowest two1−+ states. r is the spatial separation between thesˉs component and the chromomagnetic operatorBi and is converted to the value in physical units. Open and filled data points are the results forβ=2.4 andβ=2.8 , respectively.Figure 2. (color online) The BS wave functions
Φn(r) (normalized asΦn(0)=1 ) of the lowest two2−+ states. r is the spatial separation between thesˉs component and the chromomagnetic operatorBi and is converted to the value in physical units. Open and filled data points are the results forβ=2.4 andβ=2.8 , respectively.tmin χ2/dof m1 /GeVm2 /GeVm3 /GeVβ=2.4 tmax=20 ( sˉsg )6 1.27 2.416(41) 3.65(23) 7.2(2.4) 5 1.37 2.406(41) 3.60(25) 5.5(8) 4 1.47 2.442(26) 3.69(19) 4.9(3) 3 1.56 2.426(19) 3.60(10) 5.0(1) β=2.8 tmax=30 ( sˉsg )9 1.47 2.361(23) 3.89(09) 12.2(2.4) 8 1.24 2.321(34) 3.64(18) 5.5(5) 7 1.37 2.341(27) 3.71(12) 6.0(3) 6 1.40 2.359(19) 4.00(08) 6.2(2) Table 3. The fitted masses
mn of the2−+ states withn=1,2,3 for different time windows[tmin,tmax] andχ2 per degree of freedom (χ2/dof ) for theβ=2.4 andβ=2.8 lattices. All the masses are converted to the values in physical units using the lattice spacingas in Table 1. -
The
0−+ and1−− states have conventional quantum numbers forqˉq mesons, and the mesons with these quantum numbers are usually assigned to then1S0 andn3S1 states in the quark model. Therefore, we start with the analysis of the wall-source correlation functionC(r,t) of thesˉs operators with the s and theˉs field separated by a spatial distance r. We also use the function form of Eq. (5) withN=3 mass terms. The upper bound of the fit window is fixed totmax=40 and 30, and the lower bound gradually decreases totmin=5 and 7 forβ=2.4 andβ=2.8 , respectively. The fitted masses of the1−− states are listed in Table 4 and those of the0−+ states are listed in Table 5. As previously indicated, we use the physical mass ofϕ(1020) to set the mass parameters of the strange quark in the fermion action on the two lattices with smaller gauge ensembles. However, the fitted mass of the ground state deviates from the physical mass slightly, which means that the strange quark masses are not tuned with sufficient precision. Therefore, one should consider this slight deviation when examining the data in the tables.tmin χ2/dof m1 /GeVm2 /GeVm3 /GeVβ=2.4 tmax=40 ( sˉs )8 0.84 1.013(1) 1.753(77) 2.16(22) 7 0.81 1.013(1) 1.787(93) 2.08(19) 6 0.83 1.014(1) 1.732(47) 2.13(12) 5 0.92 1.015(1) 1.709(36) 2.11(08) β=2.4 tmax=45 ( sˉsg )16 1.40 1.011(1) 1.72(12) — 15 1.42 1.011(1) 1.66(10) — 14 1.42 1.009(1) 1.77(08) — 13 1.67 1.007(1) 1.83(06) — β=2.8 tmax=30 ( sˉs )10 1.93 1.001(5) 1.634(114) 2.17(29) 9 1.97 1.003(4) 1.633(86) 2.02(18) 8 2.11 0.999(3) 1.665(81) 2.30(20) 7 2.39 0.998(3) 1.668(52) 2.34(17) β=2.8 tmax=45 ( sˉsg )19 1.17 1.006(4) 1.51(6) — 18 1.39 1.003(3) 1.55(6) — 17 1.50 1.005(3) 1.51(5) — 16 1.45 0.998(2) 1.58(4) — Table 4. The fitted masses
mn of the1−− states for two different types of operators and different time windows[tmin,tmax] andχ2 per degree of freedom (χ2/dof ) for theβ=2.4 andβ=2.8 lattices. All the masses are converted to values in physical units using the lattice spacingas in Table 1.tmin χ2/dof m1 /GeVm2 /GeVm3 /GeVβ=2.4 tmax=40 ( sˉs )8 0.70 0.7012(2) 1.690(34) 2.21(17) 7 0.69 0.7010(2) 1.698(39) 2.12(14) 6 0.73 0.7010(2) 1.699(30) 2.12(10) 5 0.89 0.7014(2) 1.669(22) 2.10(07) β=2.4 tmax=45 ( sˉsg )16 1.30 0.7008(3) 1.711(73) — 15 1.40 0.7007(3) 1.680(56) — 14 1.42 0.7007(3) 1.672(46) — 13 1.56 0.7009(3) 1.659(36) — β=2.8 tmax=30 ( sˉs )10 2.59 0.6483(8) 1.736(62) 2.84(34) 9 2.51 0.6505(8) 1.703(76) 2.33(25) 8 2.37 0.6512(8) 1.679(66) 2.23(20) 7 2.35 0.6516(7) 1.620(30) 2.46(11) β=2.8 tmax=45 ( sˉsg )19 1.09 0.6508(10) 1.621(85) — 18 1.03 0.6510(10) 1.557(67) — 17 1.04 0.6490(06) 1.711(50) — 16 1.06 0.6491(05) 1.740(39) — Table 5. The fitted masses
mn of the0−+ states for two different types of operators and different time windows[tmin,tmax] andχ2 per degree of freedom (χ2/dof ) for theβ=2.4 andβ=2.8 lattices. All the masses are converted to values in physical units using the lattice spacingas in Table 1.In the
1−− channel, the masses of the ground state and the first excited state are approximately 1 GeV and 1.7 GeV, respectively, which are compatible with those ofϕ(1020) andϕ(1680) . For theβ=2.4 lattice, the fitted massm3 of the third state is also stable with respect totmin , and the value is approximately 2.1 GeV, which is close to the expected mass of the33S1 state for the quark model. Them3 on theβ=2.8 lattice is also in this mass range but fluctuates more strongly withtmin .In the
0−+ channel, the ground state mass can be precisely determined withm1≈0.701 GeV atβ=2.4 and0.651 GeV atβ=2.8 . Since thesˉs pseudoscalar meson (labeled asηs ) is not a physical state, we cannot directly compare our result to the physical value . A previous calculation has been performed for theNf=2+1 full-QCD lattice formalism, which gives the predictionmηs=0.686(4) GeV [32], lying between our values for the two lattices. The deviation is small and can be attributed to our less precise tuning of the strange quark mass parameter and the other systematic uncertainties. The mass of the first excited state is approximately 1.6-1.7 GeV, which is almost degenerate with that of the first excited1−− state. There is no physical correspondence of this state at present, but it can be compared to the pseudoscalarη(1295)/η(1475) . However, it should be noted that this state is a puresˉs state, which results in a higher mass. As indicated in the previous section, the BS wave functionsΦn(r) of the0−+ and1−− states can be derived simultaneously with the masses, as shown in Figs. 3 and 4. It should be noted that r represents the separation between the s andˉs field. For the ground state and the first excited state in each channel, theΦn(r) values exhibit the expectation of the quark model in that the wave function of the ground state has no radial node whereas that of the first excited state has one node. Therefore, the two states can be assigned to be the1S and2S states of a non-relativisticsˉs system. The behavior of the wave function of the third state is strange in that it has no radial nodes even though it has two inflection points. Since we only use three mass terms to fit the correlation functions, the third state may have substantial contamination from higher states, which may result in this phenomenon. As such, the results of the third state are not seriously considered.Figure 3. (color online) The BS wave functions
Φn(r) (normalized asΦn(0)=1 ) of the lowest two0−+ states. r is the spatial separation between the quark fields s andˉs and is converted to the value in physical units. Open and filled data points are the results forβ=2.4 andβ=2.8 , respectively.Figure 4. (color online) BS wave functions
Φn(r) (normalized asΦn(0)=1 ) of the lowest two1−− states. r is the spatial separation between the quark fields s andˉs and is converted to the value in physical units. Open and filled data points are the results forβ=2.4 andβ=2.8 , respectively.We also use the
sˉsg -type operators (in Eq. (1)) to explore the properties of the0−+ and1−− states. We use Eq. (5) withN=2 mass terms to fit the correlation functions in large time ranges (tmax=45 for both lattices). The masses are listed in Table 4 and Table 5, in which it is apparent that they are consistent with those from thesˉs operators. Figures 5 and 6 show the BS wave functions with r as the spatial separation between thesˉs components and the chromomagnetic operator. The results for the two lattices are compatible with each other. It is interesting to see that for the0−+ and1−− channels, such wave functions of the ground state (1S state) and the first excited state (2S state) lie almost upon each other and there are no significant differences. In these two channels, given the masses that are compatible with the masses of the states obtained using thesˉs operators, the ground states and the first excited states can be assigned to the1S and2S sˉs mesons. As such the similarity of the wave functions of1S and2S states with respect to the distance r between the gluonic component and thesˉs component can be interpreted as follows: the s (orˉs ) field along with the chromomagnetic field can be viewed as a dresseds′ (orˉs′ ) field in the fundamental representation of the colorSU(3) group, which annihilates the s (anti)quark of thesˉs meson. In this sense, r represents the spatial size of the dressed quark field and the r fall-off does not have a dynamic significance. This is in contrast to the wave functions of the1−+ and2−+ states whose nodal structures imply that r is a dynamic variable for hybrids.Figure 5. (color online) The BS wave functions
Φn(r) (normalized asΦn(0)=1 ) of the lowest two1−− states. r is the spatial separation between thesˉs component and the chromomagnetic operatorBi and is converted to the value in physical units. Open and filled data points are the result forβ=2.4 andβ=2.8 , respectively. Since the two states can be assigned to be the1S and2S sˉs mesons, the similar r-behavior of their BS wave functions imply that this r is not a typical dynamical variable for thesˉs states.Figure 6. (color online) Similar to Fig. 5 but for
0−+ states. -
We discuss the obtained results in this section. The BS wave functions of the
1−+ and2−+ states show the typical behaviors of non-relativistic two-body Schrödinger wave functions with a central potential, in terms of the correspondence of the spectrum and the nodal structure of the wave functions. We emphasize that the variable r is the spatial distance between thesˉs component and the chromomagnetic field strength of the operators. Since1−+ is an exotic quantum number forqˉq mesons, the states with this quantum number must be a hybrid meson with additional gluonic degrees of freedom. The similarity of the spectrum and the wave functions of the2−+ states to those of the1−+ states indicates that they are also hybrid states. In this sense, the wave functions imply that r can be a meaningful dynamical variable forsˉsg hybrid mesons. In a previous lattice study oncˉcg hybrids [27], the same behaviors of the wave functions and the spectrum pattern were observed for the(0,1,2)−+ and1−− supermultiplets, based on which a "color halo" concept has been proposed, which states that a hybrid meson can be viewed as a relatively compact color octetqˉq pair surrounded by color octet gluonic degrees of freedom, such that the wave functions depict the relative motion between theqˉq pair and the gluonic excitation. In a non-relativistic model, the binding mechanism can be the potential between two effective color octet charges. Previous lattice studies [33] revealed that the potential of two static color charges has the feature of a Casimir scalingVD(r)=VD,0−CDαr+σDr,
(6) where D is the color
SU(3) representation of the charge withCD as the eigenvalue of the second order Casimir operator of the colorSU(3) ,VD,0 is the potential constant,α is the coefficient of the Coulomb part, andσD is the string tension, which is related to the conventional string tensionσ between a static quark and antiquark pair byσD=34CDσ . For the octet charge in this work, this relation isσD=9/4σ , which means that the interaction between the color octet objects is stronger than that between the color triplet ones. This explains the observation that the mass splitting (approximately 1.2 GeV) between the ground state and the first excited hybrid state is larger than the1S−2S mass splitting of theqˉq states (approximately 0.6 GeV).The "color halo" idea is conceptually different from the flux-tube framework of hybrids in the market [12-14, 34, 35], whereby the quark and anti-quark are bound by an effective potential induced by the excitation of gluonic degrees of freedom. In the leading Born-Oppenheimer approximation, the
QˉQ of a heavy quarkonium-like hybrid can be viewed as static color sources; the excited gluonic degrees of freedom are distributed along theQˉQ axis and obey the cylinder symmetry, whose effect can be treated as an excited static potential denoted byΛϵη , whereΛ=0,1,2,… is the projected total angular momentum of the gluons with respect to theQˉQ axis and is labeled asΣ,Π,Δ forΛ=0,1,2 , and so on,η represents the combined parity (P) and the charge conjugate (C) of gluon excitations withη=g,u forP⊗C=± , respectively, andϵ is the P parity of the glue state. Therefore, the quantum number of aQˉQ state with this potential isP=ϵ(−1)L+Λ+1,C=ηϵ(−1)L+S+Λ,
(7) where
ˆL=ˆLQˉQ+ˆJg withˆLQˉQ being the orbital angular momentum ofQˉQ with respect to the midpoint of theQˉQ axis andˆJg being the total angular momentum of the gluons. The groundΣ+g potential hasΛ=0,ϵ=+ andη=+ are the conventional static potential ofQˉQ of the Cornell type, and the P and C quantum numbers reproduce the conventional quantum number. The1−− and(0,1,2)−+ hybrid supermultiplet is associated with theΠ+u(L=1) potential, such that the radial Shrödinger equation isd2dr2u(r)+2μ[E−Veff(r)]u(r)=0,
(8) where r is the distance between Q and
ˉQ ,μ is the reduced mass of theQˉQ pair, andu(r) is related to the radial wave functionϕ(r) byu(r)=rϕ(r) . The effective potentialVeff isVeff=VQˉQ(r)+⟨ˆL2QˉQ⟩2μr2
(9) with
⟨ˆL2QˉQ⟩=L(L+1)−2Λ2+⟨ˆJ2g⟩ and⟨ˆJ2g⟩=2 . Obviously, the eigenvalues of E are independent of the total spin S of theQˉQ pair. One can use the lattice results to determineVQˉQ and then solve the preceding equation to obtain the masses of the hybrids. We do not wish to delve into such details in this work but mention thatϕn(r) behaves as a P-wave wave function in a central potential, and the mass splitting of the ground state and the first radial excited state is only a few hundred MeV for the1−− and(0,1,2)−+ hybrid states [14]. Even though the preceding deduction is based on the heavy quarkonium-like hybrids, this concept has been also applied to the phenomenological studies of strangeonium hybrids [34].In contrast to the flux-tube concept, we observe that for
(1,2)−+ strangeonium hybrids, the mass splitting of the ground and the first excited states is approximately 1.2-1.4 GeV, which is much larger than the prediction of the flux-tube model, and the nodal structure is present with respect to the spatial distance between thesˉs and the chromomagnetic field strength. A similar phenomenon also appears for charmonium-like hybrids without a clear quark mass dependence. It should be emphasized that even though the interpretation of the wave functions is debatable, the pattern of the spectrum should be reliable and model-independent since it is derived directly from the lattice QCD calculation.For the
0−+ and1−− channels, we obtain consistent results for the masses of the ground states and the first excited state using thesˉs andsˉsg type operators. Since we use the physical mass of theϕ(1020) meson to set the strange quark mass parameters, it is natural to almost reproduce the physical value of the mass of the vector ground state. The ground state mass of the pseudoscalar is approximately650−700 MeV, which is compatible with the previous lattice result ofηs . The masses of the first excited states in both channels are closely degenerate at1.7 GeV, and the mass of the first excitedsˉs vector meson is in agreement with that ofϕ(1680) . However, the BS wave functions in both channels, defined based on the dependence of spatial distance between the s andˉs quark field, show the expected radially nodal behavior of the non-relativisticsˉs two-body system. Therefore, the ground and the first excited states can be assigned to be the1S and2S sˉs mesons, respectively. We also obtain some information on the third state based on thesˉs type operator in each channel, whose mass is approximately 2.1-2.3 GeV. For the vector channel, this mass value is close to the mass ofϕ(2170) . However, since we only use three mass terms for the data fitting, the third state may have substantial contaminations from higher states, and the result is not reliable. When we use thesˉsg operator to study these two channels, we can only obtain information on the lowest two states. At present, there is no definitive conclusion to whether there is a1−− and(0,1,2)−+ supermultiplet of the strangeonium hybrids.Finally, we present some arguments related to
ϕ(2170) . Its mass is in the range of the33S1 and23D1 sˉs predicted by the quark model. If a1−− and(0,1,2)−+ sˉsg hybrid multiplet exists with nearly degenerate masses at approximately2.1−2.3 GeV,ϕ(2170) can also be a candidate for the1−− member. However, the assignment of its characteristic is still an open question. Till now,ϕ(2170) has been observed in many final states includingϕ(1020) , such asϕf0(980) ,ϕππ ,ϕη , andϕη′ . In theK+K−ππ andK+K−K+K− final states [36-38], there are also sizable components includingϕ(1020) . Ifϕ(2170) is a candidate for the1−− sˉsg hybrid, this decay pattern can be understood based on the color halo concept of the hybrids: the binding between the color octetsˉs and the gluonic degrees of freedom can easily break up such that thesˉs component is neutralized toϕ(1020) and the gluons are hadronized to light hadrons, which are in the flavor singlet. Furthermore, in contrast to the hadronic transition of conventional excited strangeonium states, these decays are less OZI suppressed due to the existing gluons within the strangeonium hybrids. Recently, the BESIII Collaboration reported the observation ofϕ(2170) in the processe+e−→η′ϕ with the resonance parametersMR=2177.5± 4.8(stat)±19.5(syst) MeV andΓR=149.0±15.6(stat)± 8.9(syst) MeV, andBr(ϕ(2170)→η′ϕ)Γe+e− is measured to be7.1±0.7(stat)±0.7(syst) eV [39]. Combining the result ofBr(ϕ(2170)→ηϕ)Γe+e−=1.7±0.7(stat)±1.3(syst) eV, one hasBr(ϕ(2170)→ηϕ)Γe+e−Br(ϕ(2170)→η′ϕ)Γe+e−=0.23±0.10(stat)±0.18(syst).
(10) This ratio is much larger than the predictions of phenomenological studies based on the flux tube model or the constituent gluon model of hybrids, with the mechanism whereby the flux tube or the constituent gluon breaks up into a light
qˉq pair that reorganizes into two mesons with the original constituentsˉs . However, this ratio can be explained directly from the flavor octet-singlet mixing and the kinetics. Ifϕ(2170) is asˉsg hybrid in the 'color halo' picture, then the decayϕ(2170)→ϕη(η′) can take place as follows: a gluon is emitted by a constituent strange quark (or antiquark) and the original gluon(s) couple to the flavor singlet component of theη(η′) meson. Ifϕ(2170) is a higher excitedsˉs meson, then theη(η′) is generated by two gluons emitted by thesˉs pair. It should be noted that this process can be enhanced by the QCD axial anomaly. Since the decay dynamics is expected to be the same for theϕη andϕη′ decay modes, the ratio of the partial widths can be attributed to theη−η′ mixing and the kinetic factorsΓ(ϕ(2170)→ϕη)Γ(ϕ(2170)→ϕη′)=tan2θ(kηkη′)3,
(11) where
θ is the flavor octet-singlet mixing angle of theη−η′ system, andkη(′) is the magnitude of the decay momentum. If we take the physical massesmη=547 MeV andmη′=958 MeV and the mixing angleθ as varying between−10∘ and−20∘ , this ratio is estimated to be between0.14 and0.58 and compatible with the experimental value (the mixing angleθ is derived to be approximately|θ|≈13∘ using the central value 0.23). As such, forϕ(2170) , this ratio may not be an ideal criterion to distinguish a hybrid assignment from a conventionalsˉs meson. Apart from the decay modes involvingϕ(1020) , BESIII have reportedKˉK decay modes of resonances X observed in the procesesse+e−→X→KˉK with a peak of X at approximately2.2 GeV [38, 40]. For the characteristics ofϕ(2170) to be revealed, all the observed decay modes should be considered jointly in scrutinized theoretical discussions. -
The strangeonium-like hybrids were investigated based on lattice QCD in the quenched approximation. Two anisotropic lattices with different lattice spacings were used to examine finite
as effects. We constructed spatially extendedsˉsg operators with thesˉs component separated from the chromomagnetic field strength operator by a spatial distance r. We investigated the1−− and(0,1,2)−+ channels and calculated the corresponding correlation functions based on these operators in the Coulomb gauge. The ground state mass of the1−+ states was determined to be 2.1-2.2 GeV and that of the2−+ states was approximately 200 MeV higher. These results are consistent with previous lattice calculations and phenomenological studies. The masses of the first excited state were approximately 3.6 GeV in these two channels, such that the mass splitting of the first excited state and the ground state was approximately 1.2-1.4 GeV. This is much higher than the predictions obtained based on the flux-tube model, which is only a few hundred MeV. The BS wave functions of these states, defined by the matrix elements of the aforementioned operators between the vacuum and the states, were extracted and exhibit clear nodal structures in the r direction. This indicates that r is a meaningful dynamical variable reflecting the relative motion of the center-of-mass of thesˉs against the gluonic degrees of freedom. Both the spectrum and the wave functions of thesesˉsg states have features similar to those of theircˉcg counterparts and are consistent with the "color halo" concept of the hybrids in that the color octetqˉq pair is surrounded by gluons.In the
0−+ and1−− channels, we used both the spatially extendedsˉs andsˉsg operators to perform the calculations. The ground state mass of the vectorsˉs meson almost reproduced the mass ofϕ(1020) (it should be noted that we use the mass of theϕ(1020) to set the strange quark mass parameters), and the ground state mass of the pseudoscalar was 650-700 MeV, which is in agreement with theηs mass determined by previous lattice calculations. In both channels, the masses of the first excited states were almost degenerate at approximately 1.7 GeV and compatible with the mass ofϕ(1680) . The BS wave functions of these states with respect to the distance between s andˉs were qualitatively similar to the nonrelativistic wave function of a two-body system in that the BS wave function of the first excited state had a radial node. Therefore, the first excited state can be a2S sˉs meson. In contrast, in each channel, the BS wave functions with respect to the spatial distance of the octetsˉs and the gluonic degrees of freedom had similar profiles for the ground and the first excited state, which meant that this distance was less significant forsˉs mesons.We did not obtain reliable results for the
3S sˉs mesons and the possible vectorsˉsg hybrids. Therefore, we are unable to present a convincing explanation forϕ(2170) . Since the mass ofϕ(2170) is compatible with the quark model prediction of the3S sˉs meson and the predicted mass of the lowestsˉsg hybrids, both assignments ofϕ(2170) are possible. We argue that ifϕ(2170) is either the3S sˉs meson or a vectorsˉsg hybrid within the "color halo" picture discussed in the preceding section, the ratioΓ(ϕη)/Γ(ϕη′)=0.23±0.10(stat)± 0.18(syst) can be understood based on the hadronic transition of a strangeoium-like meson in addition toη−η′ mixing. Nevertheless, the nature ofϕ(2170) is still an open question to be investigated based on further experimental and theoretical studies.
Strangeonium-like hybrids on the lattice
- Received Date: 2020-07-30
- Available Online: 2021-01-15
Abstract: The strangeonium-like