@ brookhaven national laboratory april 2008 spectral functions of one, two, and three quark...

@ Brookhaven National Laboratory@ Brookhaven National LaboratoryApril 2008April 2008

Spectral Functions of

One, Two, and Three Quark Operators

in the Quark-Gluon Plasma

Spectral Functions of

One, Two, and Three Quark Operators

in the Quark-Gluon Plasma

Masayuki ASAKAWAMasayuki ASAKAWA

Department of Physics, Osaka UniversityDepartment of Physics, Osaka University

M. Asakawa (Osaka University)

QCD Phase Diagram

CSC (color superconductivity)

QGPQGP ((quark-gluon plasmaquark-gluon plasma))

Hadron Phase

T

B

• chiral symmetry breaking• confinement

order ?

1st order

crossover

CEP(critical end point)160-190 MeV

5-100

100MeV ~ 1012 K

RHIC

LHC


Microscopic Understanding of QGP

In condensed matter physics, common to start from one particle states, then proceed to two, three, ... particle states (correlations)

Importance of Microscopic Properties of matter,in addition to Bulk Properties

Spectral Functions:

One Quark

Two Quarks

mesons

color singlet

• octet

diquarks

Three Quarks

baryons

......

need to fix gauge

need to fix gauge

need to fix gauge

Kitazawa’s talk


Spectral Function Definition of Spectral Function (SPF)

0( ) /

0 / 43

,

( , )0 0 (1 ) ( )

(2 )

( )

0 :

n n

mn

E N TP T

mnn m

k k en J m m J n e k P

Z

J

: Boson(Fermion)

A Heisenberg Operator with some qua

n

mn m n

n P

P P P

ntum #

: Eigenstate with 4-momentum

• SPF is peaked at particle mass and takes a broad form for a resonance

Information on• Dilepton production• Photon Production• J/ suppression...etc. : encoded in SPF


Photon and Dilepton production rates

Dilepton production rate

Photon production rate

3

3

( , )

exp( ) 1Td R p p

d p T

4 2

4 2 2

(2 )( , )

3 exp( ) 1l l T L

d R p

d p p T

(for massless leptons)

where T and L are given by

: QCD EM current spectral function

( , ) ( , )( ) ( , )( )T T L Lp p P p P

2 1 1

3 3 3emJ j u u d d s s

0 0 0( , ) ( , ) ( , )T T L Lk k k k P k k P


Lattice calculation of spectral functions

3

3

( , )

exp( ) 1Td R p p

d p T

No calculation yet for finite momentum light quark spectral functions

LQGP collaboration, in progress

Calculation for zero momentum light quark spectral functions by two groups

4 2

4 2 2

(2 )( , )

3 exp( ) 1l l T L

d R p

d p p T

T = L @ zero momentum

Since the spectral function is a real time quantity, necessary to use MEM (Maximum Entropy Method)

So far, only pQCD spectral function has been usedSo far, only pQCD spectral function has been used


QGP is strongly coupled, but...

D’Enterria, LHC workshop @CERN 2007


Lattice calculation of spectral functions

3

3

( , )

exp( ) 1Td R p p

d p T

No calculation yet for finite momentum light quark spectral functions

LQGP collaboration, in progress

Calculation for zero momentum light quark spectral functions by two groups

4 2

4 2 2

(2 )( , )

3 exp( ) 1l l T L

d R p

d p p T

T = L @ zero momentum

Since the spectral function is a real time quantity, necessary to use MEM (Maximum Entropy Method)

So far, only pQCD spectral function has been usedSo far, only pQCD spectral function has been used


Spectral Functions above Tc

Asakawa, Nakahara & Hatsuda [hep-lat/0208059]

m~ms

at T/Tc= 1.4ss-channel

(ω

）/ω

2

Lattice Artifact

peak structure

spe

ctra

l fu

nct

ion

s


Another calculation

dile

pto

n p

rodu

ctio

n ra

te

Karsch et al., 2003

~2 GeV @1.5Tc

massless quarks

How about for heavy quarks?

smaller lattice

log scale!

In almost all dilepton calculations from QGP, pQCD expression has been used


J/non-dissociation above Tc

J/ (p 0) disappearsbetween 1.62Tc and 1.70Tc

Lattice Artifact

Lattice Artifact

Asakawa and Hatsuda, PRL 2004

2( ) /


Result for PS channel (c) at Finite T

c (p 0) also disappears

between 1.62Tc and 1.70Tc

A() 2()

Asakawa and Hatsuda, PRL 2004


Statistical and Systematic Error Analyses in MEM

The Larger the Number of Data Pointsand the Lower the Noise Level

The closer the result isto the original image

Generally,

Need to do the following:

• Put Error Bars and Make Sure Observed Structures are Statistically Significant

• Change the Number of Data Points and Make Sure the Result does not Change

Statistical

Systematic

in any MEM analysis


Statistical Significance Analysis for J/

Statistical Significance Analysis = Statistical Error Putting

T = 1.62Tc

T = 1.70Tc

Ave.

±1

Both Persistence and Disappearanceof the peak are Statistically Significant


Dependence on Data Point Number

N= 46 (T = 1.62Tc)V channel (J/)

Data Point # Dependence Analysis = Systematic Error Estimate


Baryon Operators Nucleon current

5 5( ) ( ) ( ) ( ) ( ) ( ) ( )N abc a b c a b cJ x s u x Cd x u x t u x C d x u x

1s t Ioffe current

Euclidean correlation function at zero momentum

3( ,0) ( , ) (0,0)

( ,0) ( , ) ( )

D d x J x J

D K d

• On the lattice, used s = 0, t = 1, u(x) = d(x) = q(x), JN(x) → J(x)

1exp

2( , )

exp exp2 2

TT

K

T T


Spectral Functions for Fermionic Operators3( ,0) ( , ) (0,0)

( ,0) ( , ) ( )

D d x J x J

D K d

( ) ( ( )) : neither even nor odd

Thus, need to and can carry out MEM analysis in [-max, max]

0 0

0

( ) ( ), ( ) ( )

( ) ( ) ( ) ( ) 0

s s

s

semi-positivity

00 0

0 0

( ) ( ) ( ) : ( ), ( )

( ) ( )

s s

independent

• For Meson currents, SPF is odd

5

( )( )

In the following, we analyze


Lattice Parameters

1. Lattice Sizes

323 32 (T = 2.33Tc)

40 (T = 1.87Tc)

42 (T = 1.78Tc)

44 (T = 1.70Tc)

46 (T = 1.62Tc)

54 (T = 1.38Tc)

72 (T = 1.04Tc)

80 (T = 0.93Tc)

96 (T = 0.78Tc)

2. = 7.0, 0 = 3.5

= a/a = 4.0 (anisotropic)

3. a = 9.75 10-3 fm

L = 1.25 fm

4. Standard Plaquette Action

5. Wilson Fermion

6. Heatbath : Overrelaxation= 1 : 4

1000 sweeps between measurements

7. Quenched Approximation

8. Gauge Unfixed

9. p = 0 Projection　　　

10. Machine: CP-PACS


Analysis Details Default Model

At zero momentum,

5

4

1 5( ) ( ) sgn( )

1282

Espriu, Pascual, Tarrach, 1983

Relation between lattice and continuum currents

3/ 2

3/ 4 15/ 4 1 1( , ) ( , )

2LAT CON

O

J x a a J xZ

• In the following, lattice spectral functions are presented

• 1OZ is assumed

max = 45 GeV ~ 3/a3 quarks)


Just In• No statistical or systematic analysis has been carried out, yet

parity +parity -

ccc baryon


ccc baryon channel

parity +parity -

ccc baryon


Scattering Term Scattering term at 0p

1m

2m

NJ

0p

1 1( , )p

2 1( , )p

2 1 2 10 m m

• This term is non-vanishing only for

• For J/ (m1=m2), this condition becomes

2 10 zero mode

a.k.a. Landau damping

cf. QCD SR (Hatsuda and Lee, 1992)


Scattering Term (two body case)1m

2m

NJ

0p

1 1( , )p

2 1( , )p

2 1 2 10 m m • This term is non-vanishing only for

2 1m m0

1 2,T m m

1 0p

(Boson-Fermion case, e.g. Kitazawa et al., 2008)

2 1m m


Scattering Term (three body case)

2 3 1m m m 0

1 2 3, ,T m m m

1 0p

1m

2m

NJ

0p

1 1( , )p

2 2( , )p

3 3( , )p 3m

2 3 1m m m


Negative parity: a possible interpretation

1m

2m

NJ

0p

1 1( , )p

2 1( , )p

0L

then: parity

0if

1 2,T m m

1 0p

anti-quark: parity -


sss baryon channel

parity +parity -


QCD Phase Diagram

CSC (color superconductivity)

QGPQGP ((quark-gluon plasmaquark-gluon plasma))

Hadron Phase

T

B

• chiral symmetry breaking• confinement

order ?

1st order

crossover

CEP(critical end point)160-190 MeV

5-100

100MeV ~ 1012 K

RHIC

LHC

Diquark, Quark-antiquark correlations


Summary It is important to know one, two, three...quark spectral functions for the understanding of QGP

Mesons and Baryons (new!) exist well above Tc

A sharp peak with negative parity is observed in baryonic SPF above Tc

This can be due to diquark-quark scattering term and imply the existence of diquark correlation above Tc

Direct measurement of SPF of one and two quark operators with MEM is desired

@ brookhaven national laboratory april 2008 spectral functions of one, two, and three quark...

Documents

osaka university slide

t c asakawa

asakawa osaka university

t c lattice artifact

pqcd spectral function

spf slide

groups t

t c massless quarks