the medium modification of hadrons - theory...

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THE MEDIUM MODIFICATION OF HADRONS

J. WambachTU Darmstadt

Erice-2002, Sept 18, 2002

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Erice-2002, Sept 18, 2002

where do masses of light hadrons come from?

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Erice-2002, Sept 18, 2002


LQCD = iqD/q − 14Ga

µνGµνa − qm

qq mu,d ∼ 5 MeV

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Erice-2002, Sept 18, 2002



µνGµνa − qm

qq mu,d ∼ 5 MeV

’naively’ Mh ∼ 10− 20 MeV

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Erice-2002, Sept 18, 2002



µνGµνa − qm

qq mu,d ∼ 5 MeV

’naively’ Mh ∼ 10− 20 MeVapproximate chiral SU(2)L × SU(2)R symmetry

→ parity doublets

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Erice-2002, Sept 18, 2002



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

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QCD Vacuum

• in the physical vacuum quarks and gluons condense

〈qq〉 6= 0 〈G2µν〉 6= 0

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QCD Vacuum


〈qq〉 6= 0 〈G2µν〉 6= 0

• chiral symmetry breaking via instantons

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QCD Vacuum


〈qq〉 6= 0 〈G2µν〉 6= 0


D.B. Leinweber, hep-lat/0004025

〈qq〉 = πρ(0)

’Banks-Casher’ relation

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QCD Vacuum


〈qq〉 6= 0 〈G2µν〉 6= 0


D.B. Leinweber, hep-lat/0004025

〈qq〉 = πρ(0)

’Banks-Casher’ relation

J.W. Negele, NPPS 73 (1999) 92

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Chiral Symmetry Breaking

• Instantons mediate interaction between quarks

Leff = q(i∂/−mq)q + g[(qq)2 + (qiγ5~τq)2] + · · ·

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• BCS-like transition=⇒ quarks condense 〈qq〉 ∼ −2/fm3

=⇒ ’constituent mass’ Mq ∼ 0.3 GeV

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• 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

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QCD Phase Diagram

Hadron Gasπ, ρ, N,∆, . . .

quasiparticles?q, g, collective modes?

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QCD Phase Diagram

Hadron Gasπ, ρ, N,∆, . . .

quasiparticles?q, g, collective modes?

hadrochemical freeze out

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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

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Hadron Gas

〈qq〉 =∂Ω(0)∂m

q


= 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

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Hadron Gas

〈qq〉 =∂Ω(0)∂m

q


= 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!

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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

)

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.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)

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Chiral Symmetry and Light Hadrons

vanishing of Mq at large p leads to parity doubling!

mesons

pQCD

R. Rapp, nucl-th/0204003

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mesons

pQCD


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

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mesons

pQCD


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!

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Hadronic Correlators

how does chiral symmetry restoration

affect hadron masses?

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(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 . . .

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(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)

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.’Goldstone’ Bosons

Pions

|~k| = 0.3GeV

M. Urban et al. NPA 673 (2000) 357

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.’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

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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

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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

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χS =V

T(〈〈(qq)2〉〉 − 〈〈qq〉〉2) = V T lim

~q→0

∫ ∞

0

dω2

ω2ρS(ω, ~q) = T

∂2Ω(µ, T )

∂m2q


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


CAππ =

MAππ

σAT

/Mp

ππ

σp

T

F. Bonutti et al. NPA 677 (2000) 213

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χS =V

T(〈〈(qq)2〉〉 − 〈〈qq〉〉2) = V T lim

~q→0

∫ ∞

0

dω2

ω2ρS(ω, ~q) = T

∂2Ω(µ, T )

∂m2q


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


CAππ =

MAππ

σAT

/Mp

ππ

σp

T

F. Bonutti et al. NPA 677 (2000) 213

see also talks by Metag and Oset

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.Dileptons in URHIC’s

HADES (GSI) CERES and NA50 (CERN) PHENIX (BNL)

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Sources of DileptonsA A

+

-x

+

-h',q

h,q-

+

-h'

h

q

+

-

q-

+-

g

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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

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.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

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.Dilepton Rates

dNl+l−

d4xd4q= −Lµν(q)

( 1

π

)ImDelm

µν (ω, ~q)



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

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.Dilepton Rates

dNl+l−

d4xd4q= −Lµν(q)

( 1

π

)ImDelm

µν (ω, ~q)



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

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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

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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

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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



convolute over expanding fireball

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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



convolute over expanding fireball

chiral symmetry restoration?

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.Vector Mesons and Chiral Symmetry

R. Barate et al (ALEPH) EPJ C4 (1998) 409

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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

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’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]

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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

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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

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.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

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the medium modification of hadrons - theory...

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