Exploring Real-time Functionson the Lattice with

Inverse Propagator and Self-Energy

Masakiyo Kitazawa(Osaka U.)

22/Sep./2011 Lunch Seminar @ BNL

QCD @ T>0 QCD @ T>0 T

RHIC LHCperturbation

Lattice QCD

How does the matterbehave in this

region?

pn

K

u

s

s

d

du

d

gg

g g

g

Tc*

Quantum Statistical Mechanics Quantum Statistical Mechanics

Static expectation value

Dynamical response

• EoS• chiral condensate• susceptibilities• screening mass• …

Spectral func:

Spectral Functions at T>0 Spectral Functions at T>0

quasi-particle excitationwidth ~ decay rate

transport coefficients

r(w

,p)

w

peaks

Kubo formulae

slope at the origin

0

( )~ lim

•shear viscosity : T12

•bulk viscosity : Tmm

•electric conductivity : Jii

Analytic Continuation Analytic Continuation

1 2 1 2ˆ ˆ( ) ( ) () ) )( exp( EE EG DUO O SO O

0( ) ( )n

nG i d e G

0( ) ( )i t

R RG d e G t

analytic continuationni i

Retarded (real-time) propagator

Spectral function1

( ) Im ( )RG

Imaginary-time (Matsubara) propagatorLattice

Dynamics

Analytic Continuation Analytic Continuation

MEM analysis of r (w)

most probable image estimated bylattice data + prior knowledge Asakawa, Hatsuda, Nakahara, 1999

qualitative structure of r (w)Asakawa, Hatsuda, 2004; Datta, et al., 2004;…

Lattice Dynamics

Analytic Continuation Analytic Continuation

Questions:

• Are there other useful formulas to relate real-time and Euclidean functions?

• Is r(w) only a real-time function worth analyzing?

We consider the inverse propagator.

cf.)sum rules Kharzeev, Tuchin; Karsch, Kharzeev, Tuchin, 2008Romatchke, Son, 2009; Meyer, 2010; …

A Difficulty in Analyzing Low Energy Spectrum A Difficulty in Analyzing Low Energy Spectrum

Moment Expansion Moment Expansion

cosh( ) ( )

co

( 1/

sh / 2

2 )S d

T

T

( )1 ( )( ) ( )

! cosh /( 1/ 2)

2

nn n

n n

TS d Sn T T

Taylor expandcoshw(t-1/2T)

moment of “thermal” spectrum

Cn n-th moment of r’(w).Aarts, et al., 2002Petreczky, Teany, 2006Ding, 2010

Low Energy Spectrum Low Energy Spectrum

L w

( ) ( ) 1

( ) ! 2

nnS

S n T

Contribution of higher order moments of low energy part is severely suppressed.

For L=T, the ratio is 2.1x10-5 for n=6.

Inverse Propagator and Analytic Continuation Inverse Propagator and Analytic Continuation

Inverse Propagator Inverse Propagator

Latticeobservable

Dynamicalinformation

analyticin

vers

e

analytic

inve

rseF.T.

F.T.

inve

rse

Numerical Procedure Numerical Procedure

Latticeobservable

Dynamicalinformation

inve

rse

inve

rse

Standard analysis

Present study

• Translational symmetry reduces costs for the inverse.• [S]-1 (t) is not the statistical average of fermion matrix.

F.T. Inv. F.T.

Quark Self-Energy Quark Self-Energy

1( , )

( , )S

p m

p

p

1( , ) Im ( , )S

p p

•decay rate of quasi-quark @pole

•Theoretically, ImS is useful to understand spectral properties.•experimental observable

Spectral function

Self-energy (ImS)Im. p

art

To Interpret Physics behind the SPC To Interpret Physics behind the SPC

1

0

1( , )

( , ) ( , )D

D

pp p

physical interpretation via optical theorem

1( ) ( )D

1Im ( )D

Disp. Rel.

w

w

1Re ( )

w

Self-Energy for Discrete Spectrum Self-Energy for Discrete Spectrum

Self-Energy for Discrete Spectrum Self-Energy for Discrete Spectrum

Poles of GR(w) and GR(w)-1

are staggered on real-axis.

Numerical Results for Inverse Quark Correlator Numerical Results for Inverse Quark Correlator

Quarks at Extremely High T Quarks at Extremely High T •Hard Thermal Loop approx. ( p, w, mq<<T )•1-loop (g<<1)

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

( , ) p

“plasmino”

p / mT

w

/ mT

6T

gTm

0

1( , )

( , )S

p

p γ p

•Gauge independent spectrum

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

•The plasmino mode has a minimum at finite p.

• width ~g2T

Simulation Setup Simulation Setup

•quenched approximation•Landau gauge fixing•clover improved Wilson

T/Tcb size Nconf

3 7.45 1283x16 28

643x16 51

7.19 483x12 51

1.5 6.87 1283x16 42

643x16 44

6.64 483x12 51

1.25 6.72 643x16 48

483x16 58

0.93 6.42 483x16 50

0.55 6.13 483x16 60

Karsch, MK, ’07; ’09MK, et al., in prep.

Ansatz for Spectral Function Ansatz for Spectral Function

2-pole structure for r+(w).

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

4-parameter fit E1, E2, Z1, Z2

Correlation Function Correlation Function 0

0

( , ) ( ) ( )

( ) ( )S

C C C

C C

0

( )C

•We neglect 7 points near the source from the fit.•2-pole ansatz works quite well!! ( c 2/dof. is of order 1 in correlated fit)

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

t /T

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

Fitting result

0 ( )C

( )sC

t /T

Quark Dispersion on 1283x16 Lattice Quark Dispersion on 1283x16 Lattice

T=3Tc HTL(1-loop)

E2 has a minimum at p>0

•Existence of the plasmino minimum is strongly indicated.•E2, however, is not the position of plasmino pole.

MK, et al.,in preparation

Quark Correlator at k=0, m=0 Quark Correlator at k=0, m=0

Quark Correlator

Inverse Correlator

Lines: fits by 2-pole ansatz Good agreement for 0.4< t <0.7

2-pole structure of quark spectrum supported

Quark Correlator at k>0, m>0 Quark Correlator at k>0, m>0

Quark Correlator

Inverse Correlator

Difference in correlators which behave similarly can become clear in terms of the inverse correlator.

2-pole inverse correlator is inconsistent with the lattice one.

1283x16, T=3Tc

Deviation near the Source Deviation near the Source

Free Wilson propagator:

Continuum:

Lattice:

Deviation near the Source 2 Deviation near the Source 2

Deviation from the 2-pole prediction scales in lattice unit

Summary Summary

A correlator which is consistent with a lattice result can be inconsistent with the inverse correlator.

Inverse correlator can further constrain real-time functions!

Analysis of inverse quark correlator supports the existence of normal and plasmino modes in quark spectrum near Tc.

Future Work Future Work

MEM analysis for the inverse correlator Analysis of meson propagator Comparison with analytic studies in terms of self-energy etc.

To Obtain a Better Image of r(w) To Obtain a Better Image of r(w)

Decay width might be evaluatedwith a reasonable statistical error.

For isolated peaks, ImD(w) at the peak represents the decay width of the quasi-particle excitation.

( , ) ~( )

ZD

E i

p

1( ) ( )D

1Im ( )D

typical errorbars in MEM analysis;insufficient to determine the width