aa r X i v : . [ h e p - l a t ] J u l f (2010) in Lattice QCD Mushtaq Loan a ∗ , Zhi-Huan Luo b and Yu Yiu Lam c a International School, Jinan University, Huangpu Road West, Guangzhou 510632, P.R. China b Department of Applied Physics, South China Agricultural University, Wushan Road, Guangzhou, 510642, P.R. China c Department of Physics, Jinan University, Huangpu Road West, Guangzhou 510632, P.R. China (Dated: May 27, 2009)We present a search for the possible I ( J P ) = 0(2 + ) tetraquark state with ss ¯ s ¯ s quark content inquenched improved anisotropic lattice QCD. Using various local and non-local interpolating fields wedetermine the energies of ground-state and second ground state using variational method. The stateis found to be consistent with two-particle scattering state, which is checked to exhibit the expectedvolume dependence of the spectral weights. In the physical limit, we obtain for the ground state,a mass of 2123(33)(58) MeV which is higher than the mass of experimentally observed f (2010).The lattice resonance signal obtained in the physical region does not support a localized J P = 2 + tetraquark state in the pion mass region of 300 −
800 MeV. We conclude that the 4 q system inquestion appears as a two-particle scattering state in the quark mass region explored here. PACS numbers: 11.15.Ha, 11.30.Rd,12.38.Ge
I. INTRODUCTION
The concept of multi-quark hadrons has received re-vived interest due to the narrow resonances in the spec-trum of states. Recently, several new particles were ex-perimentally discovered and confirmed as the candidatesof multi-quark states. These discoveries are expectedto reveal new aspects of hadron physics. Among thesediscoveries, the tetra-quark systems are also interestingin terms of their rich phenomenology, in particular formesons which still remain a most fascinating subjectof research. The 4 q states are interesting in terms ofthe recent experimental discoveries of X (3872) [1, 2, 3], Y (4260) [4] and D s (2317) [5, 6], which are expected tobe tetra-quark candidates.The Particle Data Group lists 2 tensor mesons withmasses in the range 1 . − . /c and considers themas well-established. The 2 ++ candidates, f (1950) [7, 8,9] and f (2010) [10] are isosinglet. The relevant channelsof decay are K ¯ K and ηη for the f (1950) and φφ and K ¯ K for f (2010). Due to their K ¯ K decay, one would expect f (1950) and f (2010) are very likely one state; the massshift could be a measurement error or could be causedby the K ¯ K threshold. However, the results for f (2010)favour an intrinsically narrower state, strongly coupled to φφ and weakly coupled to the other channel for allowed s -wave decays. Following the recent re-analysis of theBNL data [11] we discuss the state f (2010) as a s ¯ s state.The multi-quark states have been investigated in lat-tice QCD studied with somewhat mixed results [12, 13,14, 15]. At the present status of approximations, lat-tice QCD seems to provide a trustworthy guide into un-known territory in tetra-quark hadron physics [16, 17,18, 19, 20, 21, 22, 23, 24, 25, 26]. Using the quenched ∗ Corresponding author approximation, and discarding quark-antiquark annihila-tion diagrams, we construct s ¯ s sources from multipleoperators. Note that we are working in quenched approx-imation which in principal is unphysical. However, previ-ous lattice results on masses and decay constants turn outto be in good agreement with experimental values [27].This seems to suggest that it is plausible to use quenchedlattice QCD to investigate the mass spectra. We excludethe processes that mix q ¯ q and q ¯ q and allow the quarkmasses to vary from small to large values. In the absenceof quark annihilation, we do not expect any mixing of q ¯ q with pure glue. Thus we can express the q ¯ q corre-lation functions in terms of a basis determined by quarkexchange diagrams only (ignoring the single, double andannihilation diagrams among Wick’s contractions). An-other important question is whether the interpolating op-erator one uses has a significant overlap with the state inquestion. To construct an interpolating field which hassignificant overlap with the 4 q system, we adopt the so-called variational method to compute 2 × f ( s ¯ ss ¯ s ) states in the spectrum of2 × II. LATTICE STUDY FOR THE f ( ss ¯ s ¯ s ) The simplest local interpolators can be written interms of colour-singlet configuration of a product ofcolour-neutral meson interpolation fields. We proposea non- φφ interpolating field to extract the f ( ss ¯ s ¯ s )tetraquark state. This choice is designed to maximizethe possibility to observe attraction between tetraquarkconstituents at relatively large quark masses. With a φφ operator it is possible that there is a small amount ofthe compact 4 q component in the two-body interpolatingfield since the interpolator may contain a large contam-ination of φφ scattering states. We adopt the simplestnon- φφ - type interpolator of the form O ( x ) = 12 (cid:20)(cid:18) ¯ q aα ( x )( γ i ) αβ q aβ ( x ) (cid:19)(cid:18) ¯ Q bλ ( x )( γ j ) λσ Q bσ ( x ) (cid:19) − (cid:18) q ↔ Q, ¯ q ↔ ¯ Q (cid:19)(cid:21) , (1)with spin I ( J P ) = 0(2 + ). In the nonrelativistic limit theabove non-two state particle can not be decomposed into φφ . Thus the 4 q state can be singled out as much as pos-sible and the results are less biased by the contaminationof two-state scattering states.The other type of interpolating field is one in whichquarks and anti-quarks are coupled into a set of diquarkand antidiquark, respectively and has the form O ( x ) = ǫ abc (cid:2) q Tb C Γ Q c (cid:3) ǫ ade (cid:2) ¯ q d C Γ ¯ Q Te (cid:3) . (2)Accounting for both colour and flavour antisymmetry,possible Γs are restricted within γ and γ i . For Γ = γ γ i ( i = 1 , , J P = 1 − . For concreteness, we simulate the flavourcombination [ ss ] and [¯ s ¯ s ].To extract energies E n of s ¯ s We compute the 2 × C ij ( t ) = h X ~x tr h h ( O i )( ~x, t ) ¯ O j )( ~ , i f i i U , (3)where the trace sums over the Dirac space, and the sub-scripts f and U denote fermionic average and gauge fieldensemble average, respectively. Following [12, 16, 28] wesolve the eigenvalue equation C ( t ) v k ( t ) = λ k ( t ) v k ( t ) (4)to determine the eigenvectors v k ( t ). We use these eigen-vectors to project the correlation matrices to the spacecorresponding to the n largest eigenvalues λ n ( t ) C nij ( t ) = ( v i , C ( t ) v j ) , i, j = 1 , · · · , n (5)and solve the generalised eigenvalue problem equationfor the projected correlation matrix C nij . The resultinglarge-time dependence of the eigenvalues λ n ( t ) allows adetermination of ground and excited-state energies. Themass can be extracted by a hyperbolic-cosine fit to λ n ( t )for the range of t in which effective mass M eff ( t ) = ln (cid:20) λ ( t ) λ ( t + 1) (cid:21) (6)attains a plateau. In order to show the existence or ab-sence of the signature of tetraquark resonance on lattice,we establish lowest and as well as the second-lowest en-ergy levels for our 4 q system.Using a tadpole-improved anisotropic gluon action[29], we generate quenched configurations on two lattice volumes 16 ×
64 and 16 ×
80 (with periodic boundaryconditions in all directions). After discarding the initialsweeps, a total of 200 configurations are accumulated formeasurements at β = 4 . M φ . At m q a t = 0 . κ t = 0 . φ of 1 . M φ /M N at the chiral limit is 1 . ± . . . k t which coverthe strange quark mass region of m s < m q < m s , i.e., a t m q = 0 . , . , . , . , . , . , .
12. Inspiredby the good agreement of the ratio with the experimentalvalue, the scale was set alternatively by M φ /M N . Usingthe experimental value 938 MeV for the nucleon mass,the spacing of our lattice is a s = 0 . III. RESULTS AND DISCUSSION
Figs. 1 illustrate the two lowest energy levels extractedby fitting the effective masses over appropriate t intervals.The ground state eigenvalues show a conventional time-dependence near t ≃ T / χ /N DF , chosen according to criteria that χ /N DF is preferably close to 1 . t in Eq. (6), which suggests that the groundstate in question is correctly projected. Suppressing anydata point which has error larger than its mean value,the possible plateau is seen in the region 5 ≤ t ≤ t = 6 −
11 is found to optimize the χ /N DF .To avoid the clutter in Fig. 1 we do not show the pointsat larger t values which have larger error bars, and havelittle or no influence on the fits. The best fit curve tothe 4 q data has χ /N DF = 0 .
87. The results for themasses corresponding to the various values of the hoppingparameter κ t are tabulated in Table I.To interpret the ground state in terms of signaturesof a lattice resonance, we look at three possible sce-narios. First, we extract the mass splitting betweenthe tetraquark 0(2 + ) and the noninteracting φ + φ two- m e ff t FIG. 1: Effective mass of the I ( J ) = 0(2 + ) colour-singletlowest-lying ground state. The data correspond to m π ≃ q , Kaon and φ states, in thelattice units, for various values of κ t . κ t M q M ∗ q M K M φ particle state and compare our results to that derived inquenched chiral perturbation theory . Fig. 2 shows themass difference ∆ M = M q − M φ , together with thequenched one-loop energy shift in the finite box [31], as afunction of m π L for the lowest 4 q state from 16 ×
64 lat-tice in our calculation. We obtain the results for one-loopenergy shift by interpolating the coefficients A ( m π L )and B ( m π L ) listed in Ref. [31] for the range of m π L appropriate for our calculation on 16 ×
64 lattice for δ = 0 .
12 and 0 . m π L.The positive mass difference observed in this range ofpion mass suggests that the observed signal is unlikelyto be a tetraquark. We also notice that our data arereasonably consistent with one-loop quenched perturba-tion results [31] for m π L ≥ . δ = 0 .
12 and 0 . Since we are using the quenched approximation, the extractionof energy shift in a finite box using full QCD one-loop chiralperturbation theory is not applicable M ∆ ( G e V ) Lm π FIG. 2: The energy shift of the lowest I ( J P ) = 0(2 + ) stateas a function of m π L . The solid and dashed lines correspondquenched one-loop chiral perturbation results for δ = 0 . .
15, respectively.
It is interesting to note that our results are consistentwith quenched one-loop results despite the fact that thedisconnected contributions were not inserted in our cal-culation. This implies that disconnected correlator hasvery small or negligible contribution than the connectedone at several time separations.To confirm or discard the signature observed in Fig. 2,we examine the second scenario, i.e., the volume depen-dence of the spectral weight of these states. Theoreti-cally, if the state is a genuine resonance, then its spectralweight should be almost constant for any lattices with thesame lattice spacing. On the other hand, if it is a two-particle scattering state, then its spectral weight has anexplicit 1 /V dependence [32]. In the following, we shalluse the ratio of the spectral weights on two spatial vol-umes 16 and 20 to discriminate whether the hadronicstate in question is a resonance or a scattering state.Fig. 3 shows the ratio ( R = W /W ) of spectralweights of the lowest state and second-lowest state, ex-tracted from the time-correlation function of variationalmatrix as a function of m π . Since our two lattice sizes are16 ×
64 and 20 ×
80, the spectral weight ratio for a two-particle state should be W /W = V /V = 1 .
95. Wesee that the ratio R for the lowest state clusters around1 . m π ∈ [0 . , . + resonance with quark contents ( ss ¯ s ¯ s ).On the other hand, for smaller quark masses, R beginsto deviate from 1 . R being consistent with 1 . q state and the two- φ statemight be a flux -tube recombination between two φ atsome diquark and internal quark separations. This can W / W m π (GeV) FIG. 3: Spectral weight ratio W /W as a function of m π for the lowest state (solid circles) and next lowest state (solidtriangles). be verified by analyzing the 4 q potential of the tetraquarksystem. We do not intend to pursue such an analysis heresince this is not the focus of our present study. The spec-tral weight ratio of the first excited state turns out to beconsistent with 1 .
95, confirming our speculation that itis two-particle scattering state. The two states are rea-sonably well separated compared to the decay width of f (2010).Finally, the mass differences extracted can be extrapo-lated to the physical limit, which is the next important is-sue [33]. Since quenched spectroscopy is quite reliable formass ratio of stable particles, it is physically even moremotivated to extrapolate mass ratios rather than massesor mass differences. This allows for the cancellation ofsystematic errors since the hadron states are generatedfrom the same gauge field configurations and hence sys-tematic errors are strongly correlated. We use a set ofdata points with smallest m π to capture the chiral log be-haviour. Fig. 4 collects and displays the resulting massratios, illustrated in Table II, extrapolated to the physicallimit using linear and quadratic fits in m π . The differencebetween these two extrapolations gives some informationabout systematic uncertainties in the extrapolated quan-tities. Performing such extrapolations to mass ratios, weadopt the choice which shows the smoothest scaling ba-haviour for the final value, and use others to estimate thesystematic errors.The data at smallest five quark masses behave almostlinearly in m π and both the linear and quadratic fits es-sentially gave the identical results. The contributionsfrom the uncertainties due to chiral logarithms in thephysical limit are seen to be significantly less dominant.The mass difference ∆ M is ∼
100 MeV at the smallerquark masses, and weakly dependent on m π . The sig-nature of repulsion at quark masses near the physicalregime would imply no evidence of the resonance in the J = 2 channel. If this mass difference from two- φ thresh-old can be explained by the two- φ interaction, then the M a ss r a t i o ( R ) M m π FIG. 4: Extrapolation of the mass ratio ∆
M/M K for loweststate (open circle) and second lowest state (solid triangles)to the physical limit at a s = 0 .
473 fm. Also are shown themass ratio M K /M φ (solid diamonds). The dashed lines arethe linear fits in m π to the data. s ¯ s state can be regarded as a two- φ scattering state. TABLE II: Hadron mass ratios at various pion masses at a s = 0 .
473 fm. M π ( GeV ) M q − M φ M K (cid:0) M q − M φ M K (cid:1) ∗ M K M φ To verify whether analysis at relatively large quarkmasses would affect the manifestation of the J = 2 stateand aid to confirm the indication of a resonance, we allowthe quark mass to be m q > m s so that the thresholdfor the decay q ¯ q → ( q ¯ q )( q ¯ q ) is elevated. The heavyquark mass suppresses relativistic effects, which compli-cates the interpretation of light-quark states. The re-sulting extracted mass ratios are shown in Fig. 5 andtabulated in Table III.The behaviour observed for the mass differences be-tween the J = 2 and the two-particle states, at largequark masses, implies that at larger quark masses, thedata appear above the two- φ threshold by ∼
95 MeV andremains constant in magnitude as the physical regime isapproached. This trend continues in the physical limitwhere the masses exhibit the opposite behaviour to thatwhich would be expected in the presence of binding.Again, the positive mass difference could be a signatureof repulsion in this channel. This suggests that instead ofa bound state, we appear to be seeing a scattering state m π M a ss r a t i o ( R ) M FIG. 5: As in Fig. 4 but for larger quark mass. in J = 2 channel. Since the mass difference between thereported experimental s ¯ s mass and the physical 2 m φ continuum is ∼
20 MeV, the observed signal is too heavyto be identified with the empirical f (1950) or f (2010). TABLE III: Hadron mass ratios at larger quark masses. M π ( GeV ) M q − M φ M K (cid:0) M q − M φ M K (cid:1) ∗ M K M φ Using the physical kaon mass, M K = 503(5) MeV,we obtain a mass estimates of 2123(33)(58) MeV and2137(39)(64) for the s ¯ s tetraquark ground state andthe second ground-state, respectively. In each case, thefirst error is statistical, and second one is our estimate ofcombined systematic uncertainty including those comingfrom chiral extrapolation and quenching effects. Notethat we cannot estimate the discretization error since wehave only one lattice spacing to work with. Given the factthat the ratio does not show any scaling violations, wecould also quote the value of this quantity on our finestlattice, which has the smallest error. Nevertheless, order2% errors on the finally quoted values are mostly due tothe chiral extrapolations. The quenching errors mightbe the largest source of uncertainty. Note however, that in the case of mass ratios of stable hadrons, this is notexpected to be very important. It has been shown [34]that with an appropriate definition of scale, the massratios of stable hadrons are described correctly by thequenched approximation on the 1 −
2% level. To thisend we also calculated the pseudoscalar to vector mesonratio R SP and pseudoscalar to nucleon mass ratio R SN and found that in the physical limit these ratios differabout 1% from their corresponding experimental values.So we quote our quenching errors to be less than twopercent. IV. SUMMARY AND CONCLUSION
We presented the results of our investigation on thetetraquark systems in improved anisotropic lattice QCDin the quenched approximation. The mass of J = 2 statewas computed using field operators, which are motivatedby the non- ππ and diquark structure. In the quenchedapproximation, our results suggest that our interpola-tors have sufficient overlap with f ( ss ¯ s ¯ s ) to allow a suc-cessful correlation matrix analysis and produced the evi-dence that the mass of the lowest-lying state only agreesmarginally with the mass of f (2010). In the region ofpion mass which we are able to access, we saw no ev-idence of attraction that could be associated with theexistence of a resonance in J = 2 channel. Since ourestimated value for the mass of f ( s ¯ s ) is marginallyclose to its experimental value, we suspect that might bethe f (2010) resonance captured by our optimized corre-lator. However, on the other hand, our spectral weightratio for two different lattice volumes deviates from one(the essential criterion for resonance) with large errorsfor small quark masses, observed state exhibits the ex-pected volume dependence in the spectral weight for twoparticles in a box. The ground-state is found to be con-sistent with scattering state. Our estimated values serveas predictions of lattice QCD in quenched approxima-tion. Indeed, our simulation does not include dynamicalquarks, the final conclusions will have to wait till bothdisconnected correlators and annihilation contributionsare incorporated. Acknowledgments
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