Transcript Charmonium prospects from anisotropic lattice study

Charmonium prospects from
anisotropic lattice study
- Using spatial boundary condition -
International workshop on “Heavy Quarkonium 2006”
June 27-30, 2006 @ Brookhaven National Lab.
Hideaki Iida
(Yukawa Institute for Theoretical Physics, Kyoto univ.)
collaboration with
N. Ishii (Tokyo univ.), T. Doi (RIKEN BNL),
H. Suganuma and K. Tsumura (Dept. of Phys., Kyoto univ.)
Introduction
Study of charmonium at high temperatures (Review)
・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～1.1Tc
S. 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, …
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
We impose a periodic boundary condition or an anti-periodic
boundary condition on c and c quark, respectively for x,y,z direction.
Compact charmonium
No boundary condition dependence
cc scattering state
Boundary condition dependence
due to the relative momentum of cc
If a state is
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
Imaginary time
at  as
A technical problem in finite temperature QCD
At finite temperature, the temporal lattice
points Nt becomes the smaller as the
temperature becomes the higher.
space
1 / T  Nt at
Calculation of hadron masses at high
temperatures are difficult.
at  as
We use the anisotropic lattice in this study,
where the temporal lattice spacing a t is smaller
than the spatial one as .
.
In this work, we use the anisotropy parameter ξ:
ξ as / at  4.0
anisotropic lattice
Correlator
To enhance the ground state overlap, we use the spatially extended operator with Coulomb gauge.
Temporal correlator with extended operator :
(Zero momentum projected)
Suitable for S-wave
In S-wave,
is optimal.
effective mass
If is sufficiently large, the obtained correlator
at a temperature is dominated by
the ground state. In this time region,
has the following form:
where
represents the mass of the ground state.
We define effective mass
by the lattice data
If
is dominated by the ground st.,
:
is independent of t.
Then, the effective mass is almost equal to the mass of ground state.
Lattice setup
Gauge sector
(Quenched approx.)
Standard Wilson gauge action (anisotropic lattice)
:bare anisotropy
Parameter set (for gauge configuration)
Lattice setup
Quark sector
O(a) improved Wilson action (anisotropic lattice)
clover term
(O(a) improvement)
Wilson parameter
Parameter set (for quarks)
(This parameter set reproduces the J/ψ mass at zero temperature ）
Spatial boundary condition
By changing the spatial boundary condition of c and c, we can distinguish
a compact resonance state from a scattering state.
( 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.
Periodic Boundary Condition (PBC)
Anti-periodic Boundary Condition (APBC)
☆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＝3100MeV
ηc (JP=0-), mJ/ψ＝2980MeV
Spatial boundary condition
A 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.
c
c,c
c
PBC for cc scattering state
APBC for cc scattering state
Spatial boundary condition
A compact J/ψ
cc scattering state
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.)
Effective mass plot of J/ψ
○：PBC, △：APBC
The fit is done by the cosh type function.
T=1.32Tc
T=1.11Tc
Best fit of PBC and APBC
fit range
T=1.61Tc
T=2.07Tc
・No boundary condition dependence is observed.
Effective mass plot of ηc
○：PBC, △：APBC
The fit is done by the cosh type function.
T=1.11Tc
T=1.61Tc
T=1.32Tc
T=2.07Tc
・No boundary condition dependence is observed.
Behavior of J/ψ mass
Compact J/ψ⇒
Scattering state ⇒
…Almost no boundary condition dependence of the energy.
J/ψ is a compact state above Tc (～2Tc).
Behavior of ηc mass
Compact ηc⇒
Scattering state ⇒
…Almost no boundary condition dependence of the energy.
ηc is a compact state above Tc (～2Tc).
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)
→ Lowest quanta: (nx,ny,nz)=(0,0,1)
The highest-threshold
APBC: (BCx,BCy,BCz)=(A,A,A)
→ Lowest quanta: (0,0,0)
Threshold is lower than that in PBC case
Hybrid boundary condition (HBC): (BCx,BCy,BCz)=(P,P,A)
→ Lowest quanta: (0,0,0)
Largest
between PBC and HBC
BC is different
in the direction
The lowest-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)
PBC
APBC
HBC
HBC2
T=1.11Tc
between PBC and HBC
HBC2: (BCx,BCy,BCz)=(A,A,P)
No plateau region even at T=1.1Tc!
Analysis ofχc1 channel with maximally
entropy method
(K. Tsumura (Kyoto Univ.))

Maximally entropy method (MEM)
[M. Asakawa, Y. Nakahara and T. Hatsuda, Prog. In Part. And Nucl. Phys 46 (2001) 459.]
A method to solve an inverse problem:
Information we want to know
B = KA
Obtained image
Mapping function which “dirty” the information
… We can obtain B from A uniquely with MEM.
This method is applicable to the extraction of the spectral function
temporal correlator
obtained by lattice QCD.
Temporal correlator
from lattice QCD
B
K
A
from the
Desired spectral
function
→can be extracted !!
MEM results for χc1 channel
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)
ω=6GeV
PBC
ω=3.5GeV
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
APBC
Comparison between PBC and APBC
Almost no difference
between PBC and APBC
→The peak around 6GeV is
considered as
a compact bound state.
This peak may be considered to be
a lattice artifact of Wilson fermion.
The bound state of doubler(s) ? (pointed out by other groups)
MEM results of J/ψ
PBC
APBC
Comp.
ω=3GeV
No BCD
(a) Spectral function on PBC (b) SPF on APBC (c) Comparison between PBC and APBC
MEM results of ηC
PBC
ω=3GeV
APBC
Comp.
No BCD
Peak around 3GeV + No Boundary Condition dep. → Survival of J/ψ and ηc above Tc
Different from the P-wave channel
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(～２Tc）.
(・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