Masakiyo Kitazawa(Osaka Univ.)

HQ2008, Aug. 19, 2008

Hot Quarks in Lattice QCD

Masakiyo Kitazawa(Osaka Univ.)

HQ2008, Aug. 19, 2008

Lattice QCD and Hot Quarks

1. Introduction to Lattice QCD 2. Hot quarks

in lattice QCD

3. Discussions

WHY we study Lattice QCD? WHY we study Lattice QCD? WHY we study Lattice QCD? WHY we study Lattice QCD?

Lattice QCD provides a “first principle” calculation of QCD.

•Lattice results justify QCD as well as lattice itself.•inputs for the physics beyond the standard model.

•hadron mass spectrum PACS-CS collab. 2007

•reproduces experiments quite well!

Will lattice QCD take over heavy-ion experiments?

Path Integral – Quantum Mechanics Path Integral – Quantum Mechanics Path Integral – Quantum Mechanics Path Integral – Quantum Mechanics

(

0

)1

0

, ,

e ( , )xp exp ( , )

F I

F

I

iH t tF F I

iH tn

i i ii

n

i ii

I F I

t

n tii

q t q t q e q

i t Dq i L q q dt

dq q

q

e

d L

q

q q

transition amplitude in Feynman’s path-integral

n-dimensional integral;With fixed n, this amplitude is numerically calculated in principle.

t

tI

t1

t2

t3

tn

tF

Path Integral – Field Theory Path Integral – Field Theory Path Integral – Field Theory Path Integral – Field Theory

4( ), ( ), exp ( , )F

I

t

F F I I tt t D i d xL x x

( ), t xinfinite degrees of freedom for each t :

•discretize space-time and sum up all field configurations

•Lattice QCD is formulated in the path integral formalism.Note:

t

xy

t

Systematic Errors Systematic Errors

Lattice action : discrete QCD action

•approaches QCD action in a0 limit•various choices

1( )

4b

bL iD m F F

different results for different actions

•quarks actions:

•Wilson•staggard (KS)

•Domain wall•Ginsparg-Wilson

in numerical simulations,

•a : lattice spacing•V : lattice volume•m : quark mass

a0 (continuum limit)Vinfinite mmphys (chiral extrapolation)in the real world

heavy numeciral cost Lattice2007, Karsch

critical temp.

Dynamical Quarks Dynamical Quarks

Example : Meson propagator

•neglect quark-antiquark loops•~103 times faster than full calc.

full QCD quenched QCD

•quenched (Nf=0)•Nf=2 (two light quarks)•Nf=2+1 (two-light + strange)

Simulation settings

heav

ier

calc

ulat

ion

M(x) M(y) M(y)M(x)

Lattice QCD at Lattice QCD at TT>0 >0 Lattice QCD at Lattice QCD at TT>0 >0

Tre H Hn n

n

eZ : Partition function

1Tr e H

ZO O : Expectation value of O

1

T

•Lattice is not the real-time simulation.•Lattice can deal with only the equilibrium system.

Statistical mechanics in equilibrium

Note:

•imaginary-time action0exp EDU d LZ

( )E ML L t i •periodicity at ==1/T

Hn n

n

Z e ( )F Ii t t HF Ie

( )F Ii t t exp ( , )

F

I

t

tD i dtL

Bulk Thermodynamics Bulk Thermodynamics Bulk Thermodynamics Bulk Thermodynamics

ZPartition function:

•thermodynamic quantities:2 ln

,T

V

Z

T

ln

,p TV

Z

actually, we calculate

ln /E ZS

s, susceptibilities, etc…

•energy density •pressure p

sudden increase of at T~190MeV

Cheng, et al., 2007

Correlation Function (Propagator) Correlation Function (Propagator) Correlation Function (Propagator) Correlation Function (Propagator)

1 2( )( ) (0)O OD

Imaginary-time propagator(Correlation function)

ni

observables on the lattice

Spectral function( , ) Im ( , )Rp D p

1Tr e H

ZO O Expectation values:

( , )nD i pF.T.

1 2( ) [ ( ), (0)] ( )RD t O t O t

Real-time propagator

dynamical propagation( , )RD pF.T.

discrete and noisy

continuous function

analytic continuation

Ill-posed problem

Note: •Only the Euclidean propagator is calculated on the Lattice.

Maximum Entropy Method (MEM) Maximum Entropy Method (MEM)

method to infer the most probable image with the lattice data and a set of prior information

Asakawa,Hatsuda,Nakahara, 2001

Charmonium spectral function above Tc

•charmonium survives even above Tc up to 1.5~2Tc.

Datta, et al. 2004

Summary for the First Part Summary for the First Part

•Lattice QCD at finite T is formulated based on the quantum statistical mechanics, with path integral in the Euclidean space.•It treats the equilibrium physics.

1Tr e H

ZO O

•The propagator calculated on the lattice is the imaginary-time function.•Analytic continuation is needed to extract dynamical information.

1 2( )( ) (0)O OD ( , )p

•We need ideas to measure observables on the lattice.

•topics not mentioned here: finite density / viscosities / Polyakov loop / etc.

Hot Quarks in Lattice QCD

Karsch, Kitazawa, PLB658,45 (2007); in preparation.

Hot Quarks in sQGP Hot Quarks in sQGP Hot Quarks in sQGP Hot Quarks in sQGP

Success of recombination model suggests theexistence of quark quasi-particles in sQGP.

Lattice simulations do not tell us physics under observables.

Quarks at Extremely High Quarks at Extremely High TT Quarks at Extremely High Quarks at Extremely High TT

•Hard Thermal Loop approx. ( p, , mq<<T )•1-loop (g<<1)

Klimov ’82, Weldon ’83Braaten, Pisarski ’89

( , ) p

“plasmino”

p / mT

/

mT

6T

gTm

0

1( , )

( , )S

p

p γ p

•Gauge invariant spectrum

•2 collective excitations having a “thermal mass” ~ gT

•The plasmino mode has a minimum at finite p.

• width ~g2T

p / m

/

m

Decomposition of Quark Propagator Decomposition of Quark Propagator Decomposition of Quark Propagator Decomposition of Quark Propagator

0

free

0

( ,)

)( ) (

SE E

p p

pp

p

0 ((

))

2

E m

E

p

p

pp

0

0

( )( , ) ( , )

(( ) ),

S S

S

p p

p

p

p

Free quark with mass mHTL ( high T limit )0

HTL

0

( , )( ) ( )

Sp p

p pp

p / mT

/

mT

2 2E m p p

0

0

( )

( , )

(

( )

, )

( , )

p

p

p

p

p

Quark Spectrum as a function of Quark Spectrum as a function of mm00 Quark Spectrum as a function of Quark Spectrum as a function of mm00

Quark propagator in hot medium at T >>Tc

- as a function of bare scalar mass m0

•How is the interpolating behavior?•How does the plasmino excitation emerge as m00 ?

m0 << gT

m0 >> gT

We know two gauge-independent limits:

m0mT-mT

+(,p=0) +(,p=0)

Fermion Spectrum in QED & Yukawa Model Fermion Spectrum in QED & Yukawa Model Fermion Spectrum in QED & Yukawa Model Fermion Spectrum in QED & Yukawa Model Baym, Blaizot, Svetisky, ‘92

0

1( )

2L i i m g

Yukawa model:

1-loop approx.:

m/T=0.01

0.80.450.3

0.1

+(

,p=

0)

Spectral Function for g =1 , T =1

0 / 1m T thermal mass mT=gT/4

0 / 1m T single peak at m0

Plasmino peak disappearsas m0 /T becomes larger.

cf.) massless fermion + massive bosonM.K., Kunihiro, Nemoto,’06

Simulation Setup Simulation Setup Simulation Setup Simulation Setup

•quenched approximation•clover improved Wilson•Landau gauge fixing

T size # of conf.

3Tc 7.45 643x16 51 (0)

483x16 51 (0)

7.19 483x12 51 (0)

1.5Tc 6.87 643x16 51 (7)

483x16 51 (0)

6.64 483x12 51 (0)

1.25Tc 6.72 643x16 71 (31)configurations generated

by Bielefeld collaboration

•vary bare quark mass m0

Correlator and Spectral Function Correlator and Spectral Function Correlator and Spectral Function Correlator and Spectral Function

( / 2 )

/ 2 / 2( ) ( )

eC d

e e

E1E2

Z1Z2

observablein lattice

dynamicalinformation

2-pole structure may be a goodassumption for +().

1 21 2( ) ( ) ( )E EZ Z

4-parameter fit E1, E2, Z1, Z2

Correlation Function Correlation Function Correlation Function Correlation Function 0

0

( , ) ( ) ( )

( ) ( )S

C C C

C C

0

( )C

•We neglect 4 points near the source from the fit.•2-pole ansatz works quite well!! ( 2/dof.~2 in correlated fit)

643x16, = 7.459, = 0.1337, 51confs.

/T

1 2 ( )1 2( ) e eE EC z z

Fitting result

0

1 1 1

2 c

m

Spectral Function Spectral Function Spectral Function Spectral Function

E1E2

Z1

Z2

E1E2

Z1Z2

T = 3Tc 643x16 (= 7.459)

E2

E1

2

1 2

Z

Z Z

= m0 pole of free quark

m0 / T

E /

TZ

2 / (

Z1+

Z 2)

T=3Tc

1

2

1

2

( ) ( )

( )

E

E

Z

Z

0

1 1 1

2 c

m

Spectral Function Spectral Function Spectral Function Spectral Function T = 3Tc 643x16 (= 7.459)

E2

E1

2

1 2

Z

Z Z

= m0 pole of free quark

m0 / T

E /

TZ

2 / (

Z1+

Z 2)

1

2

1

2

( ) ( )

( )

E

E

Z

Z

•Limiting behaviors for are as expected.•Quark propagator approaches the chiral symmetric one near m0=0.•E2>E1 : qualitatively different from the 1-loop result.

0 00,m m

T=3Tc

Temperature Dependence Temperature Dependence Temperature Dependence Temperature Dependence

•mT /T is insensitive to T.•The slope of E2 and minimum of E1 is much clearer at lower T.

T = 3Tc

T = 1.5Tc

minimum of E1

E2

E1

2

1 2

Z

Z Z

m0 / T

E /

TZ

2 / (

Z1+

Z 2)

1-loop result might be realized for high T.

643x16

T = 1.25Tc

Lattice Spacing Dependence Lattice Spacing Dependence Lattice Spacing Dependence Lattice Spacing Dependence

643x16 (= 7.459)

483x12 (= 7.192)

E /

T

E2

E1

m0 / T

same physical volumewith different a.

•No lattice spacing dependence within statistical error.

T=3Tc

Spatial Volume Dependence Spatial Volume Dependence Spatial Volume Dependence Spatial Volume Dependence

E2

E1

m0 / T

E /

T

T=3Tc

643x16 (= 7.459)

483x16(= 7.459)

same lattice spacingwith different aspect ratio.

•Excitation spectra have clear volume dependence even for N/N=4.

Extrapolation of Thermal Mass Extrapolation of Thermal Mass Extrapolation of Thermal Mass Extrapolation of Thermal Mass

Extrapolation of thermal mass to infinite spatial volume limit:

•Small T dependence of mT/T, •while it decreases slightly with increasing T.•Simulation with much larger volume is desirable.

mT /T

3 3/ ~ 1/N N V

T=1.5Tc

T=3Tc

mT /T = 0.800(15) mT = 322(6)MeV

mT /T = 0.771(18)mT = 625(15)MeV

483x16

643x16

T=1.25Tc

mT /T = 0.816(20) mT = 274(8)MeV

Pole Structure for p>0 Pole Structure for p>0

•E2<E1; consistent with the HTL result.•E1 approaches the light cone for large momentum.

HTL(1-loop)

•2-pole approx. works well again.

Discussions?

Charm Quark Charm Quark from Datta et al. PRD69,094507(2004).

•Charm quark is free-quark like, rather than HTL.•The J/ peak in MEM seems to exist above 2mc.

mcpreliminary

threshold 2mc

T=1.5Tc

Role of Thermal Mass Role of Thermal Mass

•Does chiral symmetry breaking take place even with mT?•Does thermal mass contribute to the stability of mesons?

2 25[( ) ( ) ]HTL SL D G i

+ +

1( , )HTL T HTLD p m

p

( , )HTL p

Interaction:

Hidaka, MK, PRD75, 011901(R) (2007)

YES

NO. Mesons are unstable even for <2mT.

Away Side Particle Distribution Away Side Particle Distribution

Quark mass ~TPartons have position dependent mass.

orbit of light in medium

slow

fast

orbit of quarks in sQGP

light

heavy

in very progress…

high

low T

Summary Summary Summary Summary

•Quarks seem to behave as a good quasi-particles.•Thermal gluon field gives rise to the thermal mass in the light quark spectra.•The plasmino mode disappears for heavy quarks. •The ratio mT/T is insensitive to T near Tc.

Lattice simulations provide us many information about the sructure of quark propagator successfully.

Information about the quark propagator will usedfor phenomenological studies of the QGP.

Future Work Future Work Future Work Future Work

full QCD / gauge dependence / volume dependence / …

Effect of Dynamical Quarks Effect of Dynamical Quarks Effect of Dynamical Quarks Effect of Dynamical Quarks

Quark propagatorin quench approximation:

screen gluon field suppress mT?

meson loop will have strong effect if mesonic excitations exist

In full QCD,

massless fermion + massive boson 3 peaks in quark spectrum! M.K., Kunihiro, Nemoto, ‘06