Baryonic Spectral Functions at Finite Temperature - Physics

Baryonic Spectral Functions 

at Finite Temperature 

Masayuki Asakawa 

Department of Physics, Osaka University 

July 2008 

@ XQCD 2008

QCD Phase Diagram 

T 

LHC 

160-190 MeV 

100MeV ~ 10 12 K 

RHIC 

crossover 

CEP(critical end point) 

1st order 

QGP (quark-gluon plasma) 

Hadron Phase 

• chiral symmetry breaking 

• confinement 

order ? 

CSC (color superconductivity) 

5-10ρ 0 

μ B 

M. Asakawa (Osaka University)

Hadrons above Tc? 

Hadrons above T c 

• No a priori reason that no hadrons exist above T c 

• QGP looks like strongly interacting system (low viscosity...etc.) 

• Definition of Spectral Function (SPF) 

ρ 

J 

μν 

μ 

 

( k , k) 

e 

(2 π ) 

( ) 

0 : 

−( En−μ 

Nn)/ 

T 

0 

0 −Pmn 

/ T 4 

≡ 

3 ∑ 

μ ( ) ν ( ) ∓ δ 

nm , Z 

mn m n 

nJ 0 m mJ 0 n(1 e ) ( k−P 

) 

− ( + ) : Boson(Fermion) 

A Heisenberg Operator with some quantum # 

n : Eigenstate with 4-momentum 

P = P −P 

P 

μ 

n 

mn 

• Information on 

• Dilepton production 

• Photon Production 

• J/ψ suppression...etc. : 

encoded in SPF 


Microscopic Understanding of QGP 

Importance of Microscopic Properties of matter, 

in addition to Bulk Properties 

• In condensed matter physics, common to start from one particle states, 

then proceed to two, three, ... particle states (correlations) 

Spectral Functions: 

One Quark 

Two Quarks 

mesons 

color singlet 

• octet 

‣ diquarks 

Three Quarks 

baryons 

...... 

need to fix gauge 




Photon and Dilepton production rates 

• Photon production rate 

 

( , ) 

3 

dRγ α ρT 

ω = p p 

= 

3 

ω 

d p 

π exp( ω T) −1 

• Dilepton production rate 

4 2 

d R 

 

+ − 

l l 

α (2 ρT + ρL)( ω, p) 

= 

4 2 2 

d p 3π 

p exp( ω T) −1 

(for massless leptons) 

where ρ T and ρ L are given by 

 

ρ ( ω, p) = ρ ( ω, p)( P ) + ρ ( ω, p)( P ) 

μν T T μν L L μν 

ρ μν : QCD EM current spectral function 

 

ρ ( k , k) = ρ ( k , k) P + ρ ( k , k) 

P 

( ) ( ) 

μν 0 T 0 T μν L 0 L μν 

em 2 1 1 

Jμ = jμ = uγμu− dγμd − sγμs+ 

3 3 3 


Lattice calculation of spectral functions 

• No calculation yet for finite momentum light quark spectral functions 

 

( , ) 

3 

dRγ α ρT 

ω = p p 

= 

3 

ω 

d p 


LQGP collaboration, in progress 

So far, only pQCD spectral function has been used 

• Calculation for zero momentum light quark spectral functions by two groups 

4 2 

d R 

 

+ − 

l l 

α (2 ρT + ρL)( ω, p) 

= 

4 2 2 

d p 3π 


ρ T = ρ L @ zero momentum 

Since the spectral function is a real time quantity, 

necessary to use MEM (Maximum Entropy Method) 


QGP is strongly coupled, but... 

D’Enterria, LHC workshop @CERN 2007 


Lattice calculation of spectral functions 

• No calculation yet for finite momentum light quark spectral functions 

 

( , ) 

3 

dRγ α ρT 

ω = p p 

= 

3 

ω 

d p 


LQGP collaboration, in progress 

So far, only pQCD spectral function has been used 

• Calculation for zero momentum light quark spectral functions by two groups 

4 2 

d R 

 

+ − 

l l 

α (2 ρT + ρL)( ω, p) 

= 

4 2 2 

d p 3π 


ρ T = ρ L @ zero momentum 

Since the spectral function is a real time quantity, 

necessary to use MEM (Maximum Entropy Method) 


Spectral Functions above T c 

m~m s 

ss-channel at T/T c = 1.4 

spectral functions 

A(ω)/ω 2 

peak structure 

Lattice Artifact 

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


Another calculation 

log scale! 

dilepton production rate 

massless quarks 

smaller lattice 

~2 GeV @1.5T c 

Karsch et al., 2003 

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

How about for heavy quarks? 


J/ψ non-dissociation above T c 


J/ψ (p = 0) disappears 

between 1.62T c and 1.70T c 


Asakawa and Hatsuda, PRL 2004 


Result for PS channel (η c ) at Finite T 

η c (p = 0) also disappears 

between 1.62T c and 1.70T c 

Asakawa and Hatsuda, PRL 2004 


Importance of Understanding Baryons @Finite T 

Success of Recombination @RHIC 

v 

2 ⎛ p 

p 

⎜ ⎞ 3v 

⎛ ⎞ 

2 

⎟ ∼ ⎜ 

3 

⎟ 

⎝ ⎠ ⎝ ⎠ 

( p ) ∼ v and 

v ( p ) 

M p t B p t 

2 t 2 

2 t 2 

( x) 

n 

1+ ∼1+ 

nx 

• v 2M (p TM ) : v 2 B 

(p TB ) ~ 2 : 3 for p TM : p T 

B 

= 2 : 3 

φ 

dN ⎛ 

i 

dN ⎞ = N 

i2 i0 1 + 2v1cos( − 

0) + 2v2 cos 2( − 

0) 

+ 

dydϕ⎜dydϕd p 

⎜⎝ ⎠⎟ 

T 

( ϕ ϕ ϕ ϕ ) 


Constituent Quark Number Scaling 

• v 2M (p TM ) : v 2 B 

(p TB ) ~ 2 : 3 for p TM : p T 

B 

= 2 : 3 

• Partons are flowing and Partons recombine to make mesons and baryons 

Assumption 

Evidence of Deconfinement 

All hadrons are created at hadronization simultaneously 


Baryon Operators 

• Nucleon current 

( ) ( ) 

J 

N( x) = εabc ⎡⎣s ua( x) Cdb( x) γ5uc( x) + t ua( x) Cγ5db( x) uc( x) 

⎤⎦ 

s 

= − t = 1 Ioffe current 

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

J(x) 

• Euclidean correlation function at zero momentum 

 

3 

D(,0) τ = ∫ d x J(, τ x) J(0,0) 

∞ 

D(,0) τ = K(, τωρω ) ( ) dω 

∫ 

−∞ 

⎛1 

⎞ω 

exp⎜ 

−τT 

⎟ 2 T 

K(, τω) 

= 

⎝ ⎠ 

⎛ ω ⎞ ⎛ ω ⎞ 

exp⎜ ⎟+ exp⎜− 

⎟ 

⎝2T 

⎠ ⎝ 2T 

⎠ 


Spectral Functions for Fermionic Operators 

 

3 

D( τ,0) = ∫ d x J( τ, x) J(0,0) 

∞ 

D(,0) τ = K(, τωρω ) ( ) dω 

∫ 

−∞ 

ρ( ω) = ρ ( ω) γ + ρ ( ω): ρ ( ω), ρ ( ω) 

0 

0 s 

0 

= ρ ( ω) Λ γ + ρ ( ω) 

Λ γ 

0 0 

+ + − − 

s 

independent 

ρ ( ω) = ρ ( − ω), ρ ( ω) =−ρ ( −ω) 

0 0 

s 

ρ ( ω) = ρ ( − ω) = ρ ( ω) + ρ ( ω) ≥0 

semi-positivity 

+ − 

0 

s 

s 

ρ+ ( ω)( ρ− ( ω)) 

: neither even nor odd 

• For Meson currents, SPF is odd 

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

In the following, we analyze 

ρω ( ) 

≡ 

ρ ( ω) 

+ 

5 

ω 


Lattice Parameters 

1. Lattice Sizes 

32 3 ∗ 46 (T = 1.62T c ) 

54 (T = 1.38T c ) 

72 (T = 1.04T c ) 

80 (T = 0.93T c ) 

96 (T = 0.78T c ) 

2. β = 7.0, ξ 0 = 3.5 

ξ = a σ /a τ = 4.0 (anisotropic) 

3. a τ = 9.75 ∗ 10 -3 fm 

L σ = 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, 

ρ 1 5 

5 

( ω ) ρ ( ω ) sgn( ) 

4 

128 

ω ω 

+ 

= 

− 

= 

( 2π 

) 

Espriu, Pascual, Tarrach, 1983 

• Relation between lattice and continuum currents 

LAT ⎛ 

3/4 15/4 1 ⎞ 1 CON 

J (, τ x) = aτ 

aσ 

J (, τ x) 

⎜2 

κκ ⎟ 

⎝ 

Z 

τ σ ⎠ O 

3/2 

• ω max = 45 GeV ~ 3π/a σ (3 quarks) 

• In the following, lattice spectral functions are presented 

• Z 

O 

= 1 is assumed 


Stat. and Syst. Error Analyses in MEM 

Generally, 

The Larger the Number of Data Points 

and the Lower the Noise Level 

The closer the result is 

to the original image 

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 


Below T c : Light Baryon 

sss baryon 

parity - 

parity + 


Below T c : Charm Baryon 

ccc baryon 

parity - 

parity + 


Above T c : Light Baryon 

peak near zero + symmetric and equally separated peaks 

parity - 

parity + 


@Higher T 

only symmetric and equally separated peaks 

parity - 

parity + 


Above T c : Charm Baryon 

peak near zero + symmetric and equally separated peaks 

parity - 

parity + 


@Higher T 

only symmetric and equally separated peaks 

parity - 

parity + 


Statistical Analysis: Light Baryon @T=0 

Peaks are statistically significant 


Statistical Analysis: Charm Baryon @Finite T 

Peak near zero is statistically significant 


Origin of Near Zero Structure 

Scattering Term 

 

p = 

• Scattering term at 0 

a.k.a. Landau damping 

m 1 

( , p ) 

1 1 

ω 2 1 

J N 

⊗ 

 

p = 0 

 

m 2 

( ω , p 

) 

0 < ω = ω2 −ω1 ≤ m2 −m1 

• This term is non-vanishing only for 

• For J/ψ (m 1 =m 2 ), this condition becomes 

0 < ω = ω2 −ω1 

≤ε 

zero mode 

cf. QCD SR (Hatsuda and Lee, 1992) 


Scattering Term (two body case) 

m 1 

( , p ) 

1 1 

ω 2 1 

J N 

⊗ 

 

p = 0 

 

m 2 

( , p ) 

ω 2 1 2 1 

• This term is non-vanishing only for 

0 < ω = ω −ω 

≤ m −m 

T 

m , m 

1 2 

1 

0 

p ∼ 

0 

m 

− m 

2 1 

m 

− m 

2 1 

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


Scattering Term (three body case) 

m 1 

( , ) 

ω1 p 

1 

m 2 

ω2 p 

 

2 

⊗ m 

J 3 

N 

 

p = 0 

 

( , ) 

( ω , p 

) 

3 3 

T m1, m2, 

m3 

p 1 

∼ 0 

ω 


Negative parity: a possible interpretation 

anti-quark: parity - 

m 1 

( ω1, p 

1) 

L ∼ 0 

m 2 

⊗ 

J N ( ω2, p 

 

1) 

 

p = 0 

 

if 

+ + 

0 or 1 

then: 

parity − 

T 

m , m 

1 2 

1 

0 

p ∼ 


Origin of Symmetric Structure 

• Wilson Doublers 

Mass of Wilson Doublers with r = 1 in the continuum limit 

m 

2n 

π 

+ n π : number of momentum components equal to π (1,2,3) 

a 

• If quark mass can be neglected: 

• Masses of baryons with doublers 

• Scattering term peaks with quark-doubler, doubler-doubler pairs 

Approximately equally separated and symmetric in ω 


QCD Phase Diagram 

Quark-antiquark correlations 

T 

LHC 

Diquark correlations 

Baryon correlations 

RHIC 

QGP (quark-gluon plasma) 

160-190 MeV 

100MeV ~ 10 12 K 

crossover 

CEP(critical end point) 

1st order 

Hadron Phase 

• chiral symmetry breaking 

• confinement 

order ? 

CSC (color superconductivity) 

5-10ρ 0 

μ B 


Summary 

• Baryons disappear just above T c 

• A sharp peak with negative parity near ω=0 is 

observed in baryonic SPF above T c 

This can be due to diquark-quark scattering term and 

imply the existence of diquark correlation above T c 

• Diquarks disappear below meson disapperance temperature 

• Direct measurement of SPF of one and two quark operators 

with MEM is desired 

• To understand doubler contribution, calculation with finer lattice 

is desired

Baryonic Spectral Functions at Finite Temperature - Physics

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