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1
.
THE MEDIUM MODIFICATION OF HADRONS
J. WambachTU Darmstadt
Erice-2002, Sept 18, 2002
where do masses of light hadrons come from?
⇐ ⇒ •
1
.
THE MEDIUM MODIFICATION OF HADRONS
J. WambachTU Darmstadt
Erice-2002, Sept 18, 2002
where do masses of light hadrons come from?
LQCD = iqD/q − 14Ga
µνGµνa − qm
qq mu,d ∼ 5 MeV
⇐ ⇒ •
1
.
THE MEDIUM MODIFICATION OF HADRONS
J. WambachTU Darmstadt
Erice-2002, Sept 18, 2002
where do masses of light hadrons come from?
LQCD = iqD/q − 14Ga
µνGµνa − qm
qq mu,d ∼ 5 MeV
’naively’ Mh ∼ 10− 20 MeV
⇐ ⇒ •
1
.
THE MEDIUM MODIFICATION OF HADRONS
J. WambachTU Darmstadt
Erice-2002, Sept 18, 2002
where do masses of light hadrons come from?
LQCD = iqD/q − 14Ga
µνGµνa − qm
qq mu,d ∼ 5 MeV
’naively’ Mh ∼ 10− 20 MeVapproximate chiral SU(2)L × SU(2)R symmetry
→ parity doublets
⇐ ⇒ •
1
.
THE MEDIUM MODIFICATION OF HADRONS
J. WambachTU Darmstadt
Erice-2002, Sept 18, 2002
where do masses of light hadrons come from?
LQCD = iqD/q − 14Ga
µνGµνa − qm
qq mu,d ∼ 5 MeV
’naively’ Mh ∼ 10− 20 MeVapproximate chiral SU(2)L × SU(2)R symmetry
→ parity doublets
instead
MN ∼ 1 GeV Ma1 ∼ 1.5Mρ
⇐ ⇒ •
2
.
QCD Vacuum
• in the physical vacuum quarks and gluons condense
〈qq〉 6= 0 〈G2µν〉 6= 0
• chiral symmetry breaking via instantons
⇐ ⇒ •
2
.
QCD Vacuum
• in the physical vacuum quarks and gluons condense
〈qq〉 6= 0 〈G2µν〉 6= 0
• chiral symmetry breaking via instantons
D.B. Leinweber, hep-lat/0004025
〈qq〉 = πρ(0)
’Banks-Casher’ relation
⇐ ⇒ •
2
.
QCD Vacuum
• in the physical vacuum quarks and gluons condense
〈qq〉 6= 0 〈G2µν〉 6= 0
• chiral symmetry breaking via instantons
D.B. Leinweber, hep-lat/0004025
〈qq〉 = πρ(0)
’Banks-Casher’ relation
J.W. Negele, NPPS 73 (1999) 92
⇐ ⇒ •
3
.
Chiral Symmetry Breaking
• Instantons mediate interaction between quarks
Leff = q(i∂/−mq)q + g[(qq)2 + (qiγ5~τq)2] + · · ·
⇐ ⇒ •
3
.
Chiral Symmetry Breaking
• Instantons mediate interaction between quarks
Leff = q(i∂/−mq)q + g[(qq)2 + (qiγ5~τq)2] + · · ·
• BCS-like transition=⇒ quarks condense 〈qq〉 ∼ −2/fm3
=⇒ ’constituent mass’ Mq ∼ 0.3 GeV
⇐ ⇒ •
3
.
Chiral Symmetry Breaking
• Instantons mediate interaction between quarks
Leff = q(i∂/−mq)q + g[(qq)2 + (qiγ5~τq)2] + · · ·
• BCS-like transition=⇒ quarks condense 〈qq〉 ∼ −2/fm3
=⇒ ’constituent mass’ Mq ∼ 0.3 GeV
• NLO-order in 1/Nc
M. Oertel et al, NPA 676 (2000) 247
recoversVDM
⇐ ⇒ •
4
.
QCD Phase Diagram
Hadron Gasπ, ρ, N,∆, . . .
quasiparticles?q, g, collective modes?
hadrochemical freeze out
⇐ ⇒ •
5
.
Chiral Symmetry Restoration
Hadron Gas
〈qq〉 =∂Ω(0)∂m
q
〈〈qq〉〉〈qq〉
= 1−∑
h
Σh%sh(µ, T )
F 2πM2
π
Σh = mq
∂Mh
∂mq
= M2π
∂Mh
∂M2π
; %sh(µ, T ) =
∂Ω(µ, T )∂Mh
⇐ ⇒ •
5
.
Chiral Symmetry Restoration
Hadron Gas
〈qq〉 =∂Ω(0)∂m
q
〈〈qq〉〉〈qq〉
= 1−∑
h
Σh%sh(µ, T )
F 2πM2
π
Σh = mq
∂Mh
∂mq
= M2π
∂Mh
∂M2π
; %sh(µ, T ) =
∂Ω(µ, T )∂Mh
ideal gas (pions, nucleons)
〈〈qq〉〉〈qq〉 ≈ 1− T 2
8F 2π
− 0.3µ3
µ30
. . . chiral pert. theory
⇐ ⇒ •
5
.
Chiral Symmetry Restoration
Hadron Gas
〈qq〉 =∂Ω(0)∂m
q
〈〈qq〉〉〈qq〉
= 1−∑
h
Σh%sh(µ, T )
F 2πM2
π
Σh = mq
∂Mh
∂mq
= M2π
∂Mh
∂M2π
; %sh(µ, T ) =
∂Ω(µ, T )∂Mh
ideal gas (pions, nucleons)
〈〈qq〉〉〈qq〉 ≈ 1− T 2
8F 2π
− 0.3µ3
µ30
. . . chiral pert. theory
but near the phase boundary
• proliferation of states!
• hadronic interactions!
⇐ ⇒ •
6
.Chiral Symmetry and Light Hadrons
nucleon mass on the lattice
D.B. Leinweber et al. PRD 61 (2000) 074502
MN = aMq + fna(mq)
ΣN = mq
(a
∂Mq
∂mq
+∂fna
∂mq
)
⇐ ⇒ •
6
.Chiral Symmetry and Light Hadrons
nucleon mass on the lattice
D.B. Leinweber et al. PRD 61 (2000) 074502
MN = aMq + fna(mq)
ΣN = mq
(a
∂Mq
∂mq
+∂fna
∂mq
)
quark mass on the lattice
J. Skullerud et al. PRD 64 (2001) 074508
SEq (p) =
Zq(p)
iγµpµ + Mq(p)
⇐ ⇒ •
7
.
Chiral Symmetry and Light Hadrons
vanishing of Mq at large p leads to parity doubling!
mesons
pQCD
R. Rapp, nucl-th/0204003
⇐ ⇒ •
7
.
Chiral Symmetry and Light Hadrons
vanishing of Mq at large p leads to parity doubling!
mesons
pQCD
R. Rapp, nucl-th/0204003
baryons
1/2 3/2 5/2 7/2 9/2 1/2 3/2 5/2 7/2 9/2Spin
0.9
1.1
1.3
1.5
1.7
1.9
2.1
2.3
2.5
Nucleons Deltas
T.D. Cohen and L.Ya. Glozman, IJMP A16 (2001) 1327
⇐ ⇒ •
7
.
Chiral Symmetry and Light Hadrons
vanishing of Mq at large p leads to parity doubling!
mesons
pQCD
R. Rapp, nucl-th/0204003
baryons
1/2 3/2 5/2 7/2 9/2 1/2 3/2 5/2 7/2 9/2Spin
0.9
1.1
1.3
1.5
1.7
1.9
2.1
2.3
2.5
Nucleons Deltas
T.D. Cohen and L.Ya. Glozman, IJMP A16 (2001) 1327
• chiral breaking for Jp = 12
±and 3
2
±
• high-energy states decouple (Σh ≈ 0)
• limited number of degrees of freedom!
⇐ ⇒ •
8
.
Hadronic Correlators
how does chiral symmetry restoration
affect hadron masses?
(retarded) correlators: (T, µ)
Di(ω, ~q) = i
∫d4x eiqxθ(x0)〈〈[Ji(x), Ji(0)]〉〉
quark currents:Ji(x) = q(x)Γiq(x) Γi = 1, γµ, γ5, γµγ5 . . .
⇐ ⇒ •
8
.
Hadronic Correlators
how does chiral symmetry restoration
affect hadron masses?
(retarded) correlators: (T, µ)
Di(ω, ~q) = i
∫d4x eiqxθ(x0)〈〈[Ji(x), Ji(0)]〉〉
quark currents:Ji(x) = q(x)Γiq(x) Γi = 1, γµ, γ5, γµγ5 . . .
spectral functions:
ρi(ω, ~q) = − 1
πImDi(ω, ~q) Di(ω, ~q) =
∫ ∞
0
dω′2ρi(ω
′, ~q)
ω′2 − ω2 + iη
hadronic description:
q(x)Γiq(x) → φi(x) LQCD(q, Aµ) → Leff(φi, ∂µφi)
⇐ ⇒ •
9
.’Goldstone’ Bosons
Pions
|~k| = 0.3GeV
M. Urban et al. NPA 673 (2000) 357
Kaons (K−)
M. Lutz and C. Corpa, NPA 700 (2002) 309
⇐ ⇒ •
10
.
Fluctuations of the Chiral Condensate
scalar susceptibility
χS =V
T(〈〈(qq)2〉〉 − 〈〈qq〉〉2) = V T lim
~q→0
∫ ∞
0
dω2
ω2ρS(ω, ~q) = T
∂2Ω(µ, T )
∂m2q
⇐ ⇒ •
10
.
Fluctuations of the Chiral Condensate
scalar susceptibility
χS =V
T(〈〈(qq)2〉〉 − 〈〈qq〉〉2) = V T lim
~q→0
∫ ∞
0
dω2
ω2ρS(ω, ~q) = T
∂2Ω(µ, T )
∂m2q
scalar spectral function
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
x 10-4
0 1 2 3 4 5 6 7 8 9 10
Z. Aouissat et al. PRC 61 (2000) 12202
⇐ ⇒ •
10
.
Fluctuations of the Chiral Condensate
scalar susceptibility
χS =V
T(〈〈(qq)2〉〉 − 〈〈qq〉〉2) = V T lim
~q→0
∫ ∞
0
dω2
ω2ρS(ω, ~q) = T
∂2Ω(µ, T )
∂m2q
scalar spectral function
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
x 10-4
0 1 2 3 4 5 6 7 8 9 10
Z. Aouissat et al. PRC 61 (2000) 12202
CAππ =
MAππ
σAT
/Mp
ππ
σp
T
F. Bonutti et al. NPA 677 (2000) 213
⇐ ⇒ •
10
.
Fluctuations of the Chiral Condensate
scalar susceptibility
χS =V
T(〈〈(qq)2〉〉 − 〈〈qq〉〉2) = V T lim
~q→0
∫ ∞
0
dω2
ω2ρS(ω, ~q) = T
∂2Ω(µ, T )
∂m2q
scalar spectral function
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
x 10-4
0 1 2 3 4 5 6 7 8 9 10
Z. Aouissat et al. PRC 61 (2000) 12202
CAππ =
MAππ
σAT
/Mp
ππ
σp
T
F. Bonutti et al. NPA 677 (2000) 213
see also talks by Metag and Oset
⇐ ⇒ •
11
.Dileptons in URHIC’s
HADES (GSI) CERES and NA50 (CERN) PHENIX (BNL)
Sources of DileptonsA A
+
-x
+
-h',q
h,q-
+
-h'
h
q
+
-
q-
+-
g
⇐ ⇒ •
11
.Dileptons in URHIC’s
HADES (GSI) CERES and NA50 (CERN) PHENIX (BNL)
Sources of DileptonsA A
+
-x
+
-h',q
h,q-
+
-h'
h
q
+
-
q-
+-
g
Dilepton Spectrum (schematic):
10-12
10-11
10-10
10-9
10-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
1
10
0 1 2 3 4 5
mass [GeV/c2]
dNee
/ dy
dm
πo,η Dalitz-decays
ρ,ω
Φ
J/Ψ
Ψl
Drell-Yan
DD
Low- Intermediate- High-Mass Region> 10 fm > 1 fm < 0.1 fm
⇐ ⇒ •
12
.Dilepton Rates
dNl+l−
d4xd4q= −Lµν(q)
( 1
π
)ImDelm
µν (ω, ~q)
Delmµν (ω, ~q) = i
∫d4x eiqxθ(x0)〈〈[Jelm
µ (x), Jelmν (0)]〉〉
Jemµ =
2
3uγµu− 1
3dγµd− 1
3sγµs
⇐ ⇒ •
12
.Dilepton Rates
dNl+l−
d4xd4q= −Lµν(q)
( 1
π
)ImDelm
µν (ω, ~q)
Delmµν (ω, ~q) = i
∫d4x eiqxθ(x0)〈〈[Jelm
µ (x), Jelmν (0)]〉〉
Jemµ =
2
3uγµu− 1
3dγµd− 1
3sγµs
Vector Dominance
Jvµ(x) = −
M2ρ
gρµ(x)
Leff = Lπ + Lρ +∑
i
LN∗i
+ Lπρ +∑
i
LπN∗i
+∑
i
LρN∗i
⇐ ⇒ •
12
.Dilepton Rates
dNl+l−
d4xd4q= −Lµν(q)
( 1
π
)ImDelm
µν (ω, ~q)
Delmµν (ω, ~q) = i
∫d4x eiqxθ(x0)〈〈[Jelm
µ (x), Jelmν (0)]〉〉
Jemµ =
2
3uγµu− 1
3dγµd− 1
3sγµs
Vector Dominance
Jvµ(x) = −
M2ρ
gρµ(x)
Leff = Lπ + Lρ +∑
i
LN∗i
+ Lπρ +∑
i
LπN∗i
+∑
i
LρN∗i
⇐ ⇒ •
12
.Dilepton Rates
dNl+l−
d4xd4q= −Lµν(q)
( 1
π
)ImDelm
µν (ω, ~q)
Delmµν (ω, ~q) = i
∫d4x eiqxθ(x0)〈〈[Jelm
µ (x), Jelmν (0)]〉〉
Jemµ =
2
3uγµu− 1
3dγµd− 1
3sγµs
Vector Dominance
Jvµ(x) = −
M2ρ
gρµ(x)
Leff = Lπ + Lρ +∑
i
LN∗i
+ Lπρ +∑
i
LπN∗i
+∑
i
LρN∗i
M. Urban et al PRL 88 (2002) 042002
⇐ ⇒ •
13
.
Dilepton Rates
0 0.2 0.4 0.6 0.8 1
dRe+ e- /d
M2 [G
eV-2
fm-4
]
M [GeV]
10 -8
10 -7
10 -6
10 -5 T = 150 MeVρB = 0.5ρ0
M. Urban et al., NPA 673 (2000) 357
⇐ ⇒ •
13
.
Dilepton Rates
0 0.2 0.4 0.6 0.8 1
dRe+ e- /d
M2 [G
eV-2
fm-4
]
M [GeV]
10 -8
10 -7
10 -6
10 -5 T = 150 MeVρB = 0.5ρ0
M. Urban et al., NPA 673 (2000) 357
close to QGP rate with resummed HTL
⇐ ⇒ •
13
.
Dilepton Rates
0 0.2 0.4 0.6 0.8 1
dRe+ e- /d
M2 [G
eV-2
fm-4
]
M [GeV]
10 -8
10 -7
10 -6
10 -5 T = 150 MeVρB = 0.5ρ0
M. Urban et al., NPA 673 (2000) 357
close to QGP rate with resummed HTL
R. Rapp, nucl-th/0204003
convolute over expanding fireball
⇐ ⇒ •
13
.
Dilepton Rates
0 0.2 0.4 0.6 0.8 1
dRe+ e- /d
M2 [G
eV-2
fm-4
]
M [GeV]
10 -8
10 -7
10 -6
10 -5 T = 150 MeVρB = 0.5ρ0
M. Urban et al., NPA 673 (2000) 357
close to QGP rate with resummed HTL
R. Rapp, nucl-th/0204003
convolute over expanding fireball
chiral symmetry restoration?
⇐ ⇒ •
14
.Vector Mesons and Chiral Symmetry
R. Barate et al (ALEPH) EPJ C4 (1998) 409
Weinberg sum rules∫ ∞
0
ds
s(ρV (s)− ρA(s)) = F 2
π ;∫ ∞
0
ds (ρV (s)− ρA(s)) = 0
M2ρ = ag2
ρF 2π ; a =
(1−
M2ρ
M2a1
)−1
a ∼ 2.2M. Urban et al NPA 697 (2001) 338
⇐ ⇒ •
15
.Vector Mesons and Chiral Symmetry
’gauged’ O(4) model: φ = (φ0, ~φ)
M = φ0 + i~τ ~φ = Aei~τ~θ, Dµ = ∂µ − igY µ; Y µ = ~ρµ~τ + ~aµ1~τ5
Leff =1
4Tr[DµMDµM† − µ2MM† + · · · − 1
8FµνF µν +
M20
4YµY µ]
⇐ ⇒ •
15
.Vector Mesons and Chiral Symmetry
’gauged’ O(4) model: φ = (φ0, ~φ)
M = φ0 + i~τ ~φ = Aei~τ~θ, Dµ = ∂µ − igY µ; Y µ = ~ρµ~τ + ~aµ1~τ5
Leff =1
4Tr[DµMDµM† − µ2MM† + · · · − 1
8FµνF µν +
M20
4YµY µ]
tree level results:
M2ρ = M2
0 + h2〈〈φ0〉〉2; M2a1 = M2
0 + (h1 + h2)〈〈φ0〉〉2
=⇒ δMi ∼ O(T 2) for M0 = 0
⇐ ⇒ •
15
.Vector Mesons and Chiral Symmetry
’gauged’ O(4) model: φ = (φ0, ~φ)
M = φ0 + i~τ ~φ = Aei~τ~θ, Dµ = ∂µ − igY µ; Y µ = ~ρµ~τ + ~aµ1~τ5
Leff =1
4Tr[DµMDµM† − µ2MM† + · · · − 1
8FµνF µν +
M20
4YµY µ]
tree level results:
M2ρ = M2
0 + h2〈〈φ0〉〉2; M2a1 = M2
0 + (h1 + h2)〈〈φ0〉〉2
=⇒ δMi ∼ O(T 2) for M0 = 0
one-loop results:
• mixing theorem statisfied
• δMi ∼ O(T 4)
M. Urban et al PRL 88 (2002) 042002
⇐ ⇒ •
16
.Summary and Outlook
• spectral properties of light hadrons ↔ structure of the QCD vacuum
− spontaneous χSB via instantons → mass generation− lattice as ’benchmark’− role of confinement?
• strongly interacting matter under extreme conditions
− QCD phase diagram in the (µ, T )-plane− location of the ’endpoint’− is it accesible experimentally?− chiral symmetry restoration and deconfinement− ’hadronic description’ of the phase transition?
• in-medium spectral functions
− spectral functions broaden! (relation to chiral symmetry restoration?)− except for scalar channel ↔ fluctuations of the chiral condensate− ab-initio spectral functions
⇐ ⇒ •