Charmonium prospects from anisotropic lattice study

International workshop on “Heavy Quarkonium 2006”June 27-30, 2006 @ Brookhaven National Lab.

Hideaki Iida (Yukawa Institute for Theoretical Physics, Kyoto univ.)

collaboration withN. Ishii (Tokyo univ.), T. Doi (RIKEN BNL),

H. Suganuma and K. Tsumura (Dept. of Phys., Kyoto univ.)

- Using spatial boundary condition -

Introduction

・ Based on effective model analyses:T.Hashimoto, O. Miyamura, K. Hirose and T. Kanki, Phys.Rev.Lett.57 (1986) 2123 …Calculation of the mass shift of J/ψ around Tc. T.Matsui and H.Satz, Phys.Lett.B178 (1986) 416 …There occurs J/ψ suppression above Tc.

・ Lattice results using Maximal Entropy Method:T. Umeda, K. Katayama, O. Miyamura and H. Matsufuru, Int. J. Mod. A16 (2001) 2115;H. Matsufuru, O. Miyamura, H. Suganuma and T. Umeda, AIP Conf. Proc. CP594 (2001) 258　 … J/ψ survives at T ～ 1.1TcS. Datta, F.Karsch, P. Petreczky and I. Wetzorke, Phys.Rev.D69 (2004) 094507 etc.　 … J/ψ survives above Tc. (There is a resonance peak still T ～ 2Tc, then disappears gradually.)M.Asakawa and T. Hatsuda, Phys.Rev.Lett.92 (2004) 012001 　… J/ψ survives until T ～ 1.6Tc, then disappears immediately.

・ Experiments: SPS (CERN)…observation of anomalies of dilepton spectra (NA50, Pb-Pb collision) RHIC…High luminosity, Au+Au, …

Study of charmonium at high temperatures (Review)

Boundary condition dependence Lattice studies suggest J/ψ may survive even above Tc.

Question: Is it a compact J/ψ? Isn’t it a cc scattering state? ・ Reproducibility of MEM (especially the width) ・ Narrow width = compact state ? ・ Continuum st. becomes discrete in finite box

Our goal is to study whether the cc quasi-bound states above Tc is a compact J/ψ or a cc scattering state.

How? → Using the dependence of the energy of the state on the spatial boundary condition

Boundary condition dependence

If a state is

Compact charmonium

cc scattering state

No boundary condition dependence

Boundary condition dependence due to the relative momentum of cc

We impose a periodic boundary condition or an anti-periodic boundary condition on c and c quark, respectively for x,y,z direction.

Ref) N. Ishii et al. Phys.Rev.D71 (2004) 034001…Using boundary condition for quarks to distinguish penta-quark and NK scattering state

Anisotropic lattice

ttaNT /1

st aa

ta

A technical problem in finite temperature QCD

0.4/ ts aaξ

We use the anisotropic lattice in this study, where the temporal lattice spacing is smaller than the spatial one . .In this work, we use the anisotropy parameter ξ:

Calculation of hadron masses at high temperatures are difficult.

At finite temperature, the temporal lattice points Nt becomes the smaller as the temperature becomes the higher.

space

Imag

inar

y tim

e

sa

anisotropic lattice

st aa

We define effective mass by the lattice data :

Correlator

Then, the effective mass is almost equal to the mass of ground state.

Temporal correlator with extended operator :

To enhance the ground state overlap, we use the spatially extended operator with Coulomb gauge.

where represents the mass of the ground state.

If is sufficiently large, the obtained correlator at a temperature is dominated by the ground state. In this time region, has the following form:

(Zero momentum projected)

effective mass

If is dominated by the ground st., is independent of t.

Suitable for S-wave In S-wave, is optimal.

Lattice setupGauge sector

:bare anisotropy

Parameter set (for gauge configuration)

Standard Wilson gauge action (anisotropic lattice)

(Quenched approx.)

　　　 clover term (O(a) improvement)

　　　　 Wilson parameter

(This parameter set reproduces the J/ψ mass at zero temperature ）

Parameter set (for quarks)

Quark sector O(a) improved Wilson action (anisotropic lattice)

Lattice setup

Spatial boundary conditionBy changing the spatial boundary condition of c and c, we can distinguish a compact resonance state from a scattering state.

Periodic Boundary Condition (PBC)

Anti-periodic Boundary Condition (APBC)

( Note: Temporal boundary condition for quarks and anti-quarks are anti-periodic.)

We impose periodic boundary condition or anti-periodic boundary condition for quarks and anti-quarks.

☆After zero momentum projection, the total momentum of the system vanishes. However, c and c can have non-vanishing momentum, respectively.

S-wave case

J/ψ(JP=1-), mJ/ψ ＝ 3100MeVηc (JP=0-), mJ/ψ ＝ 2980MeV

Spatial boundary conditionA compact J/ψ has zero momentum on PBC and APBC after zero momentum projection. Therefore the energy of the state is less sensitive to spatial boundary condition.

In contrast, if a state is a cc scattering one, c and c in lowest energy have momentum and , respectively on APBC after zero momentum projection.

APBC for cc scattering state

c

c

PBC for cc scattering state

c,c

Note 1: This expression is only for S-wave!!Note 2: How is the effect of Yukawa potential? 　　　　 → Less than 20MeV Negligible (Estimated by the potential-model with Yukawa pot. in the finite box on PBC and APBC.)

Spatial boundary condition

cc scattering stateA compact J/ψ

Effective mass plot of J/ψ

T=1.11Tc T=1.32Tc

T=1.61Tc T=2.07Tc

○ ： PBC, △ ： APBC

fit rangeBest fit of PBC and APBC

The fit is done by the cosh type function.

・ No boundary condition dependence is observed.

Effective mass plot of ηc

T=1.11Tc T=1.32Tc

T=1.61Tc T=2.07Tc

○ ： PBC, △ ： APBC

The fit is done by the cosh type function.

・ No boundary condition dependence is observed.

Behavior of J/ψ mass

…Almost no boundary condition dependence of the energy.

J/ψ is a compact state above Tc (～ 2Tc).

Compact J/ψ⇒ Scattering state ⇒

Behavior of ηc mass

…Almost no boundary condition dependence of the energy. ηc is a compact state above Tc (～ 2Tc).

Compact ηc⇒ Scattering state ⇒

P-wave case

χc1 (JP=1+), mχc1=3510MeV

Calculation in χc1 channel (JP=1+)

χc1…P-wave state

→due to the centrifugal potential, the wave function tends to spread.

⇒ It is sensitive to vanishing of linear potential and appearance of Debye screening effect.

Dissociation temperature of χc1 would be lower than that of J/ψ and ηc.

Threshold of P-wave state In the P-wave case:

PBC: (BCx,BCy,BCz)=(P,P,P)

APBC: (BCx,BCy,BCz)=(A,A,A)

→ Lowest quanta: (nx,ny,nz)=(0,0,1)

→ Lowest quanta: (0,0,0)Threshold is lower than that in PBC case

Hybrid boundary condition (HBC): (BCx,BCy,BCz)=(P,P,A) BC is different in the direction

→ Lowest quanta: (0,0,0) The lowest-threshold

Largest between PBC and HBC

The highest-threshold

Difficulty with the optimization of the operator

Gaussian type and spherical extension of the operator may not be suitable for P-wave state.

→ We examined the extension radius ρ=(0-0.5)fm Point-source, Point-sink Extended, Point Extended, Extended

Effective mass in χc1 channel

Extended-source, Point-sink (ρ=0.2fm)

HBC2: (BCx,BCy,BCz)=(A,A,P)

PBCAPBCHBCHBC2

No plateau region even at T=1.1Tc!

between PBC and HBC

T=1.11Tc

Analysis ofχc1 channel with maximally entropy method(K. Tsumura (Kyoto Univ.))

Maximally entropy method (MEM)

A method to solve an inverse problem:

B = K AObtained image Mapping function which “dirty” the information

Information we want to know

B K A

This method is applicable to the extraction of the spectral function from the temporal correlator obtained by lattice QCD.

Temporal correlator from lattice QCD Desired spectral

function→can be extracted !!

…　We can obtain B from A uniquely with MEM.

[M. Asakawa, Y. Nakahara and T. Hatsuda, Prog. In Part. And Nucl. Phys 46 (2001) 459.]

MEM results for χc1 channel

PBC

Lattice setup: Wilson quark action with β=7.0 (at-1=20.2GeV)

as/at=4.0, lattice size 203×46 (L=0.78fm, T=1.62Tc)

① No compact bound state of χc1 ( ～ 3.51GeV) is observed. ② In the high energy region, there emerges a sharp peak around 6GeV.

→ χc1 already dissolves at T=1.62Tc

ω=6GeV

ω=3.5GeV

APBC

Almost no difference between PBC and APBC

Comparison between PBC and APBC

This peak may be considered to be a lattice artifact of Wilson fermion.The bound state of doubler(s) ? (pointed out by other groups)

→The peak around 6GeV is considered as a compact bound state.

MEM results of J/ψPBC APBC

Comp.ω=3GeV

(a) Spectral function on PBC (b) SPF on APBC (c) Comparison between PBC and APBC

No BCD

MEM results of ηC

PBC APBC

Comp.ω=3GeV

Peak around 3GeV + No Boundary Condition dep. → Survival of J/ψ and ηc above Tc

Different from the P-wave channel

No BCD

Summary and Conclusion・ We have investigated J/ψ and ηc above Tc using lattice QCD. ・ We have used the O(a) improved Wilson action for quarks.・ For the accurate measurement, we have used anisotropic lattice QCD.・ Changing the spatial boundary condition, we have examined whether J/ψ and ηc above Tc are compact states or scattering states of c and c.・ We have observed almost no boundary condition 　　 dependence of cc state above Tc.

This suggests that J/ψ and ηc survive above Tc( ～２ Tc ） .( ・ The level inversion of J/ψ and ηc may occur.)

Summary and Conclusions・ We have also investigated in χc1 channel above Tc using lattice QCD, because the dissociation temperature of χc1 may differ from those of J/ψ and ηc.

・ Unfortunately, we cannot extract a low-lying state (due to the difficulty of optimization of operators).

→ MEM on PBC and APBC

・ We extract the spectral function in χc1 channel with maximally entropy method (MEM). 　 No peak structure corresponding to χc1 is observed at T ～ 1.6Tc. (Consistent with other work)

・ The spectral functions in J/ψ and ηc channel has the peaks corresponding to J/ψ and ηc and those are independent of BC. → Compact state

( ・ There may be a compact bound state in high energy region (doubler(s)).)

Perspectives・ Analysis of P-wave meson with effective mass・ Further analysis of Maximum Entropy Method (MEM) + Spatial boundary condition dependence

(By K. Tsumura (Kyoto Univ.))

・ Other charmonium and charmed mesons, D mesons…

( D meson…If D becomes lighter, J/ψ→DD channel open. The width of charmonium possibly change. ) →Ongoing

・ Mechanism of the formation of the bound state above Tc