Survival of charmonia above Tc in anisotropic lattice QCD
aa r X i v : . [ h e p - l a t ] J un Survival of charmonia above T c in anisotropic lattice QCD Hideaki
Iida , ∗ ) , Takumi Doi , Noriyoshi Ishii , Hideo Suganuma and Kyosuke Tsumura YITP, Kyoto University, Sakyo, Kyoto 606-8502, Japan Dept. of Phys. & Astr., University of Kentucky, Lexington KY 40506, USA CCS, University of Tsukuba, Tsukuba, Ibaraki 305-8577, Japan Department of Physics, Kyoto University, Sakyo, Kyoto 606-8502, Japan Analysis Technology Center, Fujifilm Corporation, Kanagawa 250-0193, Japan
We find a strong evidence for the survival of
J/Ψ and η c as spatially-localized c ¯ c (quasi-)bound states above the QCD critical temperature T c , by investigating the boundary-condition dependence of their energies and spectral functions. In a finite-volume box, therearises a boundary-condition dependence for spatially spread states, while no such depen-dence appears for spatially compact states. In lattice QCD, we find almost no spatialboundary-condition dependence for the energy of the c ¯ c system in J/Ψ and η c channelsfor T ≃ (1 . − . T c . We also investigate the spectral function of charmonia above T c inlattice QCD using the maximum entropy method (MEM) in terms of the boundary-conditiondependence. There is no spatial boundary-condition dependence for the low-lying peaks cor-responding to J/Ψ and η c around 3GeV at 1 . T c . These facts indicate the survival of J/Ψ and η c as compact c ¯ c (quasi-)bound states for T c < T < T c . §
1. Introduction
J/Ψ suppression was theoretically proposed as an important signal of quark-gluon plasma (QGP). The basic idea of
J/Ψ suppression is that
J/Ψ disappearsabove the QCD critical temperature T c due to vanishing of the confinement po-tential and appearance of the Debye screening effect. In contrast, recently, somelattice QCD calculations indicate an opposite result that J/Ψ and η c survive evenabove T c . Spectral functions of charmonia are extracted from tempo-ral correlators at high temperature using the maximum entropy method (MEM) inRefs. 4)-8). Although there are some quantitative differences, the peaks correspond-ing to
J/Ψ and η c seem to survive even above T c ( T c < T < T c ) in the c ¯ c spectralfunction.However, one may ask a question on the “survival of J/Ψ and η c ” observed inlattice QCD. Are the c ¯ c states above T c observed in lattice QCD really compact(quasi-)bound states? Since colored states are allowed in the QGP phase, thereis a possibility that the observed c ¯ c state in lattice QCD is just a c ¯ c scatteringstate, which is spatially spread. Particularly in lattice QCD, even scattering stateshave discretized spectrum, due to the finite-volume effect. Therefore, for the fairjudgment of the “survival of charmonia above T c ”, it is necessary to clarify whetherthe c ¯ c systems are spatially compact (quasi-)bound states or scattering states.To this end, we study the spatial boundary-condition dependence of the energy ∗ ) E-mail: [email protected] typeset using
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TEX.cls h Ver.0.9 i H. Iida, T. Doi, N. Ishii, H. Suganuma and K. Tsumura and the spectral function for the c ¯ c systems ( J/Ψ and η c ) above T c in lattice QCD. Actually, for c ¯ c scattering states, there occurs a significant energy difference betweenperiodic and anti-periodic boundary conditions on the finite-volume lattice. In con-trast, for spatially compact charmonia, there is almost no energy difference betweenthese boundary conditions. Using this fact, we investigate the c ¯ c system above T c in terms of their spatial extent. §
2. Boundary-Condition Dependence and Compactness of States
We show briefly the method to distinguish spatially-localized states from scat-tering states on the finite-volume lattice.
For a compact c ¯ c (quasi-)bound state,the wave-function of the c ¯ c state is spatially localized, and therefore its energy is in-sensitive to the spatial boundary condition. In contrast, for a c ¯ c scattering state, thewave-function of the c ¯ c system is spatially spread, so that there occurs a significantboundary-condition dependence of the energy for the low-lying c ¯ c scattering state.Let us estimate the boundary-condition dependence. • Under the periodic boundary condition (PBC) , the momentum of a quark oran anti-quark is discretized as p k = 2 n k π/L ( k = 1 , , , n k ∈ Z ) on the finitelattice with the spatial volume L , and the minimum momentum is ~p min = ~ • Under the anti-periodic boundary condition (APBC) , the momentum is dis-cretized as p k = (2 n k + 1) π/L ( k = 1 , , , n k ∈ Z ). In this case, the minimummomentum is | ~p min | = √ π/L .The energy difference of the low-lying c ¯ c scattering state is estimated as ∆E scatt ≡ E scattAPBC − E scattPBC ≃ p m c + 3 π /L − m c ≃ L ≃ . m c . In Ref. 7), we consider possiblecorrection of the energy difference from a short-range potential of Yukawa type be-tween a quark and an anti-quark, and find that the correction is negligible comparedto the energy difference ∆E scatt ≃ §
3. Pole-Mass Measurements above T c in Lattice QCD First, we perform the standard pole-mass measurement of low-lying c ¯ c systemsat finite temperature in anisotropic quenched lattice QCD with the standard pla-quette action at β ≡ N c /g = 6 .
10 and the renormalized anisotropy a s /a t = 4 . i.e. , a t = a s / ≃ (8 . − ≃ . × (14 − L ≃ (1 . and the temperature as (1.11 − T c . We use 999 gaugeconfigurations, picked up every 500 sweeps after the thermalization of 20,000 sweeps.For quarks, we use O ( a )-improved Wilson (clover) action on the anisotropic lattice. We adopt the hopping parameter κ = 0 . ρ = 0 . which is found to maximize the ground-state overlap. The energy of thelow-lying c ¯ c state is extracted from the temporal correlator of the spatially-extendedoperators, where the total momentum of the system is projected to be zero. urvival of charmonia above T c in anisotropic lattice QCD Table I. The energy of the c ¯ c system in J/Ψ ( J P = 1 − ) and η c ( J P = 0 − ) channels on PBCand APBC at β = 6 .
10 at each temperature. The superscripts
J/Ψ and η c denote quantitiesof J/Ψ and η c , respectively. All the statistical errors are smaller than 0.01GeV. The energydifference E APBC − E PBC observed in lattice QCD is also added. The observed energy differenceis very small, compared to the estimated energy difference ∆E scatt ≃ c ¯ c scattering states.Temperature E J/Ψ
PBC E J/Ψ
APBC E J/Ψ
APBC − E J/Ψ
PBC E η c PBC E η c APBC E η c APBC − E η c PBC . T c − . T c − . T c − . T c − Table I shows the boundary-condition dependence of the low-lying c ¯ c state en-ergy in J/Ψ ( J P = 1 − ) and η c ( J P = 0 − ) channels at finite temperatures. Bothin J/Ψ and η c channels, the energy difference between PBC and APBC is less than40MeV, which is much smaller than the energy difference ∆E scatt in the case of thelow-lying c ¯ c scattering states, i.e. , | E APBC − E PBC | ≪ ∆E scatt ≃ c ¯ c states are spatially-localized (quasi-)bound states as charmonia of J/Ψ and η c for 1 . T c < T < . T c .Here, we comment on a “constant contribution” to the meson correlator above T c , provided by “wrap-around quark propagation”, which is pointed out in Ref. 11).The η c channel is free from this extra contribution, but the J/Ψ channel suffersfrom it. Actually, in contrast to almost no temperature dependence of the η c mass,there is a temperature dependence of the J/Ψ mass, which may be an artifact dueto the constant contribution to the meson correlator.
However, the difference ofthe
J/Ψ mass between 1 . T c and 2 . T c is only about 140MeV, and this value israther small compared to the estimated energy difference ∆E scatt ≃ c ¯ c scattering states. Therefore, even including this extra effect, our main conclusion isunchanged also for J/Ψ above T c . In fact, J/Ψ and η c survive as spatially-localized(quasi-)bound states for 1 . T c < T < . T c . §
4. MEM Analyses for Spectral Functions above T c in Lattice QCD Next, we investigate the boundary-condition dependence of the spectral function A ( ω ) of the c ¯ c system above T c using the maximum entropy method (MEM) inlattice QCD. Using MEM, we extract the spectral function A ( ω ) from the temporalcorrelator G ( t ) of the point source and the point sink, where the total momentum isprojected to be zero. In this calculation, we use the Wilson quark action on a finelattice at β = 7 . i.e. , a t = a s / ≃ (20 . − ≃ . × − fm. The adoptedlattice size is 20 ×
46, which corresponds to the spatial volume L ≃ (0 . andthe temperature T ≃ . T c .Figure 2 shows the spectral functions in J/Ψ and η c channels on PBC (dottedline) and APBC (solid line). There appear low-lying peaks around 3GeV, whichcorrespond to the charmonia of J/Ψ and η c . We find no spatial boundary-condition H. Iida, T. Doi, N. Ishii, H. Suganuma and K. Tsumura (cid:67) (cid:68)
Fig. 1. The spectral function A ( ω ) of the c ¯ c system in (a) J/Ψ channel and (b) η c channel at1 . T c on PBC (dotted line) and APBC (solid line), normalized by the default function m ( ω ). Almost no boundary-condition dependence is found for the low-lying peaks around 3GeV, whichcorrespond to the charmonia of
J/Ψ and η c . In the η c channel, the dotted and the solid linescoincide. For the figures of the spectral functions with errorbars, see Figs. 9 and 10 in Ref. 7). dependence for the low-lying peaks, which indicate that the c ¯ c states correspondingto J/Ψ and η c appear as spatially-localized (quasi-)bound states even above T c . §
5. Summary and Conclusion
We have studied
J/Ψ and η c above T c in anisotropic lattice QCD to clarifywhether these states are spatially-localized (quasi-)bound states or c ¯ c scatteringstates. As a result, both in J/Ψ and η c channels, we have found almost no spatialboundary-condition dependence of the energy of the low-lying c ¯ c system even on afinite-volume lattice for (1 . − . T c . Also in the MEM analysis, we find no spatialboundary-condition dependence of the low-lying peaks corresponding to J/Ψ and η c in the spectral function at T ≃ . T c . These facts indicate that J/Ψ and η c survivein QGP as spatially-localized (quasi-)bound states for T c < T < T c . Acknowledgement
H. I. and H. S. thank the Yukawa Institute for Theoretical Physics, for fruitfuldiscussions at “New Frontiers in QCD 2008”.
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