28-30 January 2004 KEK Workshop

“ Hadron physics and Lattice QCD ”

Takashi Umeda ( YITP, Kyoto-Univ. )

Koji Nomura ( Dep.of Physics, Hiroshima-Univ. )

Hideo Matsufuru ( CRC, KEK )

KEK Workshop / 28-30 Jan. 2004

created by

Meson spectral functions

near the deconfining transition

Contents

18

1) Introduction

2) Our approach

Maximum entropy method

Constrained curve fitting

3) Lattice setup

4) Numerical results

5) Discussion & Conclusion

KEK Workshop / 28-30 Jan. 2004

Charmonium properties below & above Tc

Introduction

18

Charmonium is an important signal for formation of QGP

heavy ion experiment at CERN(SPS), BNL(RHIC)

Theoretical understanding

from potential models

Mass shift near Tc

T.Hashimoto et al., Phys. Rev. Lett. 57 (1986) 2123.

suppression above Tc

T.Matsui and H.Satz, Phys. Lett. B178 (1986) 416.

from Lattice QCD

Meson correlators

T.Umeda et al., Int. J. Mod. phys. A16 (2000) 2215.

strong spatial correlation at T~1.5Tc

KEK Workshop / 28-30 Jan. 2004

J

/



Spatial correlation

18

KEK Workshop / 28-30 Jan. 2004

c



c

“Wave function” on the Coulomb gauge





r

,

t



=







r

,

t



/







0,

t



,







r

,

t



=

∑



x

〈



c





x





r

,

t





c





x

,

t



O



0



〉

†

strongly

correlate even

at

c



c

T

≃

1.5

T

c

T.Umeda et al., Int. J. Mod. Phys.A16 (2001) 2215

Our approach

18

KEK Workshop / 28-30 Jan. 2004

Temporal correlators in lattice simulations

C



t



=

∑



x

〈

O





x

,

t



O



0



〉

O





x

,

t



†

[ : meson operators ]

Spectral function

A







A.Abrikosov et al., Sov. Phys. J. 36 (1959) 636.

C



t



=

∫

d



K



t

,





A







,

K



t

,





=

e

−



t



e

−





N

t

−

t



1

−

e

−

N

t



Our approach

18

KEK Workshop / 28-30 Jan. 2004

Anisotropic lattice

Our analysis procedure

Maximum entropy method (MEM)

Nakahara et al., Phys. Rev. D60 (1999) 091503.

--- no assumption for the form of

Fit with ansatz for a spectral function

--- need information on the form of

--- with given form , more quantitative

Constrained curve fitting

Lepage et al., Nucl. Phys. B(PS)106 (2002) 12.

We use these two methods in complementary manner

smearing operators

enhancement of low frequency modes

A







A







A







Maximum entropy method (MEM)

18

KEK Workshop / 28-30 Jan. 2004

Reconstruction of a spectral function with MEM

Y.Nakahara et al., Phys. Rev. D60 (1999) 091503.

Standard least square fit ill-posed problem

MEM ( based on Bayes' theorem )

Maximization of

Q

=



S

−

L

S

=

∫

d



[

A







−

m







−

A







ln



A







m









]

m







=

m

DM



2

: model function

: constant to be integrated out

: Likelihood function



L

using function form from the Singular value decomposition

Constrained curve fitting (CCF)

18

KEK Workshop / 28-30 Jan. 2004

A simple modification of standrd least square fitting

based on the Bayesian statistics

G.Lepage et al., Nucl. Phys. B(PS)106 (2002) 12.



2





arg

2

≡



2





prior

2

,



prior

2

≡

∑

i



c

i

−



c

i



2





c

i

2

c

i



c

i

,





c

i

: fit parameters : input parameters as prior nowledge

many parameters fitting become stable

bias from the input parameters (prior knowledge)

MEM results are suitable for prior knowledge

Function form ansatz

18

KEK Workshop / 28-30 Jan. 2004

We suppose the shape of spectral function.

( MEM gives rough estimate of shape of spectral function. )

Breit-Wigner form:

A







=



2









,









=

C



m





2

−

m

2



2

−



2

m

2

C

m



: overlap, : mass, : width

We apply multi-Breit-Wigner fit

using the Constrained curve fitting

Quark action

18

KEK Workshop / 28-30 Jan. 2004

S

F

=

∑

x

,

y







x



K



x

,

y







y



K



x

,

y



=



x

,

y

−



t

[



1

−



4



U

4



x





x





4,

y





1





4



U

4



x

−



4





x

−



4,

y

]

−



s

∑

i

[



r

−



i



U

i



x





x





i

,

y





r





i



U

i



x

−



i





x

−



i

,

y

]

−



s

c

E

∑

i



4

i

F

4

i



x





x

,

y



r



s

c

B

∑

i



j



i

,

j

F

i

,

j



x





x

,

y

T.Umeda et al., Int. J. Mod. Phys. A16 (2001) 2215.

J.Harada et al., Phys. Rev. D64 (2001) 074501.

Constructed following the Fermilab approach

El-Khadra et al., Phys. Rev. D55 (1997) 3933.

(action retains explicit Lorentz invariant form)

Tadpole improved tree-level

r

=

1

/



Lattice setup

18

Nt T/Tc #conf. x Nsrc smearing

160 ~0 500 x 16 point, smear, (half-smear)

32 0.88 1000 x 16 smear

30 0.93 1000 x 16 smear

29 0.97 680 x 16 smear

27 1.04 300 x 16 smear

26 1.08 1000 x 16 smear, half-smear

24 1.17 1000 x 16 smear

20 1.40 1000 x 16 smear, half-smear

16 1.75 1000 x 16 smear, half-smear

KEK Workshop / 28-30 Jan. 2004

Anisotropic plaquette action in quenched approximation

T.Klassen, Nucl. Phys. B533 (1998) 557.

GeV

Scale set by Sommer scale



=

a

s

/

a

t

=

4

a

s

−

1

=

2.030



13



r

0

=

0.5

fm

Point correlator

18

KEK Workshop / 28-30 Jan. 2004

Type-1 t = 1 ~ 48 = 48 Type-2 t = 1 ~ 48 = 48

1 ~ 24 24 2 ~ 48 24

1 ~ 12 12 4 ~ 48 12

Type-2 reproduces T=0 result with =12, but Type-1 does not.

physical length is important we use smeared operators

N

df

N

df

N

df

Effective mass

18

KEK Workshop / 28-30 Jan. 2004

No change of correlators below Tc

Effective masses decrease as temperature increase

Change of vector meson is larger than that of pseudoscalar

Model function dependences

18

KEK Workshop / 28-30 Jan. 2004

MEM needs a model function as prior knowledge

we do not know definite model function

check model function dependences

MEM has an uncertainty of this size

Result below Tc

18

KEK Workshop / 28-30 Jan. 2004

correlators does not change below Tc

pole like structure almost no width

no mass shift from T=0

Result above Tc

18

KEK Workshop / 28-30 Jan. 2004

correlators change gradually as temperature increases

slightly wider structure

small or no mass shift from T=0

vector channel shows large change at high T

Result above Tc

18

KEK Workshop / 28-30 Jan. 2004

comparison between smeared and half-smeared results

at T/Tc=1.4 and 1.75 ( Nt=20 and 16 )

in PS channel at Nt=20

smeared & half-smeared results has similar peak

in PS channel at Nt=16, in V channel at Nt=20&16

similar behavior with free quark case

Temperature dependence

18

KEK Workshop / 28-30 Jan. 2004

CCF results has large systematic uncertainties

from input parameters as prior knowledges

small or no mass shift above and below Tc

wider peak structure above Tc

preliminary

preliminary

Discussion (potential model)

18

KEK Workshop / 28-30 Jan. 2004

Our results show no mass shift ( and no width ) below Tc

this is inconsistent with some potential model predictions

static quark potential

from Polyakov loop

or

from Wilson loop

Which potential should be

used for the potential model ?

Summary

18

KEK Workshop / 28-30 Jan. 2004

We study spectrul functions

from temporal charmonium correlators

We propose the analysis methods

MEM and fit analysis

below Tc

no mass shift and no width for PS and V channels

above Tc

peak structure at not so large T

small or no mass shift

finite width and grow as temperature increases

inconsistent with some potential models

Takashi Umeda

2003-09-04T17:47:02

Takashi Umeda