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Theory of Thermal Electromagnetic Radiation
Ralf Rapp Cyclotron Institute +
Dept. of Physics & Astronomy Texas A&M University College Station, Texas
USA
JET Summer School
The Ohio State University (Columbus, OH) June 12-14, 2013
1.) Intro-I: Probing Strongly Interacting Matter
• Bulk Properties: Equation of State
• Phase Transitions: (Pseudo-) Order Parameters
• Microscopic Properties: - Degrees of Freedom - Spectral Functions
⇒ Would like to extract from Observables: • temperature + transport properties of the matter • signatures of deconfinement + chiral symmetry restoration • in-medium modifications of excitations (spectral functions)
“Freeze-Out” QGP Au + Au
1.2 Dileptons in Heavy-Ion Collisions
Hadron Gas NN-coll.
Sources of Dilepton Emission: • “primordial” qq annihilation (Drell-Yan): NN→e+e-X -
e+
e-
ρ
• thermal radiation - Quark-Gluon Plasma: qq → e+e- , … - Hot+Dense Hadron Gas: π
+π - → e+e- , …
-
• final-state hadron decays: π0,η → γe+e- , D,D → e+e- X , … _
1.3 Schematic Dilepton Spectrum in HICs
Characteristic regimes in invariant e+e- mass, M2 = (pe++ pe- )2
• Drell-Yan: primordial, power law ~ M- n
• thermal radiation: - entire evolution - Boltzmann ~ exp(-M/T)
T/qeeFB
eeTdM
dRVdMddN 0
31 −∝≈ e
τ
q0≈ 0.5GeV ⇒ Tmax ≈ 0.17GeV , q0≈ 1.5GeV ⇒ Tmax ≈ 0.5GeV
Thermal rate: Im Πem(M,q;µB,T)
• Thermal e+e- emission rate from hot/dense matter (λem >> Rnucleus )
• Temperature? Degrees of freedom? • Deconfinement? Chiral Restoration?
1.4 EM Spectral Function + QCD Phase Structure
• Electromagn. spectral function - √s ≤ 1 GeV : non-perturbative - √s > 1.5 GeV : pertubative (“dual”)
• Modifications of resonances ↔ phase structure: hadronic matter → Quark-Gluon Plasma?
√s = M
e+e- → hadrons ~ Im Πem / M2
)T,q(fMqdxd
dN Bee023
2em
44 π
α−= Im Πem(M,q;µB,T)
1.5 Low-Mass Dileptons at CERN-SPS CERES/NA45 e+e- [2000]
Mee [GeV]
• strong excess around M ≈ 0.5GeV, M > 1GeV • quantitative description?
NA60 µ+µ- [2005]
1.6 Phase Transition(s) in Lattice QCD
• different “transition” temperatures?! • extended transition regions • partial chiral restoration in “hadronic phase”! (low-mass dileptons!) • leading-order hadron gas
Tpcconf ~170MeV
Tpcchiral ~150MeV
≈ 〈〈
qq〉〉
/ 〈q
q〉
-
-
[Fodor et al ’10]
2.) Chiral Symmetry in QCD - Nonperturbative QCD, Chiral Breaking + Hadron Spectrum 3.) Thermal Electromagnetic Emission Rates - EM Spectral Function: Hadronic vs. Partonic Regimes 4.) Vector Mesons in Medium - Many-Body Theory, Spectral Functions + Chiral Partners (ρ-a1) 5.) Quark-Gluon Plasma Emission - Perturbative vs. Lattice-QCD Rates, “Quark-Hadron-Duality” 6.) Dilepton + Photon Spectra in Heavy-Ion Collisions - Space-Time Evolution, Phenomenology + Interpretation 7.) Summary and Conclusions
Outline
well tested at high energies, Q2 > 1 GeV2:
• perturbation theory (αs = g2/4π << 1) • degrees of freedom = quarks + gluons
2.1 Nonperturbative QCD
241
µνaq Gq)m̂Agi(q −−/+∂/=QCDL
(mu ≈ md ≈ 5 MeV )
Q2 ≤ 1 GeV2 → transition to “strong” QCD:
• effective d.o.f. = hadrons (Confinement) • massive “constituent quarks” mq* ≈ 350 MeV ≈ ⅓ Mp (Chiral Symmetry ~ ‹0|qq|0› condensate! Breaking)
↕ ⅔ fm _
2.2 Chiral Symmetry in QCD Lagrangian 2
41
µνaq Gq)m̂Agi(q −−/+∂/=QCDL
(bare quark masses: mu ≈ md ≈ 5-10MeV )
Chiral SU(2)V × SU(2)A transformation: Up to O(mq ), LQCD invariant under Rewrite LQCD using left- and right-handed quark fields qL,R=(1±γ5)/2 q :
q)/i(q)(Rq VVV 2ταα
⋅−= expq)/i(q)(Rq AAA 25 ταγα
⋅−= exp
241
µνaqRR
RRLL
LL G)m(duDi)d,u(d
uDi)d,u( −+⎟⎠⎞
⎜⎝⎛/+⎟
⎠⎞
⎜⎝⎛/= OQCDL
R,LR,LR,L q)/i(q 2τα
⋅−exp Invariance under isospin and “handedness”
qL
qL
g
2.3 Spontaneous Breaking of Chiral Symmetry strong qq attraction ⇒ Chiral Condensate fills QCD vacuum: 0≠〉+〈=〉〈 LRRL qqqqqq
qL qR
qL -qR
- [cf. Superconductor: ‹ee› ≠ 0 , Magnet ‹M› ≠ 0 , … ]
-
Simple Effective Model: 2)qq(Gq)mi(q q −−∂/=effL
• assume “mean-field” , expand: • linearize:
• free energy:
• ground state:
〉〈≡ 00 |qq|0χ )qq(qq δχ += 0
G/)m*M(q*)M(Dq
G*M,Gq])Gm[i(q
q
q
4
2221
eff
0200eff
−+=
−≡+−−∂/=−
χχχL
G*M*Mp
)(pdNNlogV
Tfc 42
2222
3
3−+=−= ∫
πΩ Z
223
3
240
*Mp
*M)(pdGNN*M
*M fc+
=⇒=∂
∂∫ π
Ω Gap Equation
2.3.2 (Observable) Consequences of SBCS
• mass gap , not observable! but: hadronic excitations reflect SBCS
• “massless” Goldstone bosons π 0,±
(explicit breaking: fπ2 mπ2= mq ‹qq› )
• “chiral partners” split: ΔM ≈ 0.5GeV ! chiral trafo: ,
• Vector mesons ρ and ω: JP=0± 1± 1/2±
〉〈−=≅ 0042 |qq|G*MqqΔ
σπ AR )(NNRA 1535
µµµ ωα
τγγγα
τγωα =−+≈ + q)()(q)(R jj
jjA 2121
21
505 chiral singlet
-
-iεijkτk ii
jiijj
ij
jij
j
AiAiA
)a(
q)(qiq)i()i(q
)qR()qR()(R
µµ
µµµ
µµ
αρ
ττττγγα
ρα
τγτγγα
τγ
τγρα
1
5505 22212121
21
×+=
−+≈−+≈
=
+
• quantify chiral breaking?
2.3.3 Manifestation of Chiral Symmetry Breaking
Axial-/Vector Correlators
pQCD cont.
“Data”: lattice [Bowman et al ‘02] Theory: Instanton Model [Diakonov+Petrov; Shuryak ‘85]
● chiral breaking: |q2| ≤ 1 GeV2
Constituent Quark Mass
2.4 Chiral (Weinberg) Sum Rules • Quantify chiral symmetry breaking via observable spectral functions • Vector (ρ) - Axialvector (a1) spectral splitting
)Im(ImsdsI AVn
n ΠΠπ
−−= ∫00002
31
210
21
222
|q)q(|αcI,|qq|mI,fI,FrfI
sq
πA
=−=
=−= −− ππ
[Weinberg ’67, Das et al ’67]
τ→(2n+1)π
pQCD
pQCD
• Key features of updated “fit”: [Hohler+RR ‘12]
ρ + a1 resonance, excited states (ρ’+ a1’), universal continuum (pQCD!)
τ→(2n)π [ALEPH ‘98, OPAL ‘99]
2.4.2 Evaluation of Chiral Sum Rules in Vacuum
• vector-axialvector splitting (one of the) cleanest observable of spontaneous chiral symmetry breaking • promising (best?) starting point to search for chiral restoration
• pion decay constants • chiral quark condensates 00002
31
210
21
222
|q)q(|αcI|qq|mI
fIFrfI
sq
πA
=−=
=−= −− ππ
2.5 QCD Sum Rules: ρ and a1 in Vacuum
• dispersion relation:
• lhs: hadronic spectral fct. • rhs: operator product expansion
[Shifman,Vainshtein+Zakharov ’79] 2
2
20 Q
)Q(ΠsQ)s(Im
sds ααΠ =
+∫∞
• 4-quark + gluon condensate dominant
2.) Chiral Symmetry in QCD - Nonperturbative QCD, Chiral Breaking + Hadron Spectrum 3.) Thermal Electromagnetic Emission Rates - EM Spectral Function: Hadronic vs. Partonic Regimes 4.) Vector Mesons in Medium - Many-Body Theory, Spectral Functions + Chiral Partners (ρ-a1) 5.) Quark-Gluon Plasma Emission - Perturbative vs. Lattice-QCD Rates, “Quark-Hadron-Duality” 6.) Dilepton + Photon Spectra in Heavy-Ion Collisions - Space-Time Evolution, Phenomenology + Interpretation 7.) Summary and Conclusions
Outline
3.1 EM Correlator + Thermal Dilepton Rate
〉〉〈〈−= ∫ )(j)x(jexdi)Q(Π iQx 0emem4
emνµµν• EM Correlation Fct.:
e+
e-
)T,q(fQQxdd
dN Bee023
2em
44 π
α−= Im Πem(M,q;T,µB)
γ*(q) )j
Qj()f(fi|j|ff|j|i
)QPP(E)(pd
E)(pde
xddN
V
eefi
fif
ff
i
iieeee
νµνµ
πδπ
Π
πΠΓ
4emem
3
3
3
34
4
11
22222
±〉〈〉〈×
−−== ∫
• Quark basis: • Hadron basis:
(T,µB)
ssdduuj µµµµ γγγ 31
31
32 −−=em
µφ
µω
µρ
µµµµµµ γγγγγ
jjj
ss)dduu()dduu(j
31
231
21
31
61
21
−+=
−++−=em
→ 9 : 1 : 2
[McLerran+Toimela ’85]
3.2 EM Correlator in Vacuum: e+e-→ hadrons
=)s(emImΠ)s(Dg
mV
,, VV Im
22
∑⎥⎥⎦
⎤
⎢⎢⎣
⎡
φωρ
∑ ⎥⎦
⎤⎢⎣
⎡++−
s,d,u
sqc
)s()e(Ns π
απ
1122
e+
e-
h1 h2
…
s ≥ sdual ~ (1.5GeV)2 pQCD continuum s < sdual Vector-Meson Dominance
q q _
ρ +ω +φ
ρ Ι =1
qq _
ππ 4π + 6π +...
KK
e+ e-
π - π +
ρ
3.3 Low-Mass Dileptons + Chiral Symmetry
• How is the degeneration realized? • “measure” vector with e+e- , axialvector?
Vacuum T > Tc: Chiral Restoration
Im Πem(M,q)
3.4 Versatility of EM Correlation Function • Photon Emission Rate
)T,q(fqd
dRq B
0230π
αγ em−= Im Πem(q0=q)
e+
e-
γ ~ O(αs )
γ*
• EM Susceptibility ( → charge fluctuations) 〈Q2〉 - 〈Q〉 2 = χem = Πem(q0=0,q→0)
~ O(1) )T,q(f
MqddR Bee
023
2
4 παem−
=
same correlator!
• EM Conductivity σem = lim(q0→0) [ -Im Πem(q0,q=0)/2q0 ]
2.) Chiral Symmetry in QCD - Nonperturbative QCD, Chiral Breaking + Hadron Spectrum 3.) Thermal Electromagnetic Emission Rates - EM Spectral Function: Hadronic vs. Partonic Regimes 4.) Vector Mesons in Medium - Many-Body Theory, Spectral Functions + Chiral Partners (ρ-a1) 5.) Quark-Gluon Plasma Emission - Perturbative vs. Lattice-QCD Rates, “Quark-Hadron-Duality” 6.) Dilepton + Photon Spectra in Heavy-Ion Collisions - Space-Time Evolution, Phenomenology + Interpretation 7.) Summary and Conclusions
Outline
4.1 Axial/Vector Mesons in Vacuum Introduce ρ, a1 as gauge bosons into free π +ρ +a1 Lagrangian ⇒
ππρρππρ µµ
µµπρ
⋅⋅−∂×⋅= 2
21 g)(gintL
1202 −−−= )]M()m(M[)M(D )(ρππρρ Σ
π EM formfactor
ππ scattering phase shift
ρ π π
2402 |)M(D|)m(|)M(F| )(ρρπ =
⎟⎟⎠
⎞⎜⎜⎝
⎛= )M(DRe
)M(DIm)M(
ρ
ρππδ 1-tan
|Fπ|2 δππ
ρ -propagator:
• 3 parameters: mρ(0), g, Λρ
4.2 ρ-Meson in Matter: Many-Body Theory
Dρ(M,q;µB,T) = [M2 – (mρ(0))2 - Σρππ - ΣρB - ΣρM ]-1
interactions with hadrons from heat bath ⇒ In-Medium ρ-Propagator ρ
Σπ
Σπ
ρ [Chanfray et al, Herrmann et al, Urban et al, Weise et al, Koch et al, …]
• In-Medium Pion Cloud
Σπ
> >
R=Δ, N(1520), a1, K1...
h=N, π, K … ΣρΒ,Μ =
ρ • Direct ρ-Hadron Scattering
Σρππ = +
[Haglin, Friman et al, RR et al, Post et al, …]
• estimate coupling constants from R→ ρ + h, more comprehensive constraints desirable
4.3 Constraints I: Nuclear Photo-Absorption total nuclear γ -absorption in-medium ρ -spectral cross section function at photon point
> >
Σπ
Σπ
ρ
Δ,N*,Δ*
N-1
)q,M(Dgm
q)qq(qA)q(
NN
A 0442
4
00
0
0 =−==−= medemem
absImIm ρ
ρ
ργ
ρπα
Πρπασ
γ N → π N ,Δ
γ N → Β*
meson exchange
direct resonance
ρ
4.3.2 ρ Spectral Function in Nucl. Photo-Absorption
γN γA
π-ex
[Urban,Buballa,RR+Wambach ’98]
On the Nucleon On Nuclei
• 2.+3. resonances melt (selfconsistent N(1520)→Nρ)
• fixes coupling constants and formfactor cutoffs for ρNB
4.4 ρ-Meson Spectral Function in Nuclear Matter
In-med. π-cloud + ρ+N→B* resonances
ρ+N→B* resonances (low-density approx.)
In-med. π-cloud + ρ+N → N(1520)
Constraints: γ N , γ A π N →ρ N PWA
• Consensus: strong broadening + slight upward mass-shift • Constraints from (vacuum) data important quantitatively
ρN=ρ0
ρN=ρ0
ρN=0.5ρ0
[Urban et al ’98]
[Post et al ’02]
[Cabrera et al ’02]
4.5 ρ-Meson Spectral Functions “at SPS”
• ρ-meson “melts” in hot/dense matter • baryon density ρB more important than temperature
ρB/ρ0 0 0.1 0.7 2.6
Hot + Dense Matter
[RR+Gale ’99]
Hot Meson Gas
[RR+Wambach ’99]
µB =330MeV
4.6 Light Vector Mesons “at RHIC + LHC”
• baryon effects remain important at ρB,net = 0: sensitive to ρB,tot= ρΒ + ρB (ρ-N = ρ-N, CP-invariant)
• ω also melts, φ more robust ↔ OZI
- - [RR ’01]
4.7 Intermediate Mass: “Chiral Mixing”
)q()q()()q( ,A
,VV
µνµνµν ΠεΠεΠ 00 +−= 1
)q()q()()q( ,V
,AA
µνµνµν ΠεΠεΠ 001 +−==
=
• low-energy pion interactions fixed by chiral symmetry
• mixing parameter
22
33
2 6224
π
π
πω
ωπε
fT)(f
)(kd
f kk
≈= ∫
[Dey, Eletsky +Ioffe ’90]
0
0 0
0
• degeneracy with perturbative spectral fct. down to M~1GeV
• physical processes at M ≥ 1GeV: πa1 → e+e- etc. (“4π annihilation”)
4.8 Axialvector in Medium: Dynamical a1(1260)
+ + . . . =
Σρ
Σπ
Σρ
Σπ
Σρ
Σπ π
ρ
π
ρ Vacuum: a1
resonance
In Medium: + + . . .
[Cabrera,Jido, Roca+RR ’09]
• in-medium π + ρ propagators • broadening of π-ρ scatt. Amplitude • pion decay constant in medium:
4.9 QCD + Weinberg Sum Rules in Medium
T [GeV]
[Hatsuda+Lee’91, Asakawa+Ko ’93, Klingl et al ’97, Leupold et al ’98, Kämpfer et al ‘03, Ruppert et al ’05]
)(sdsI AVn
n ρρπ
−= ∫2
122
02
1 q)q(αcI,mfI,fI sπ ===− ππ
[Weinberg ’67, Das et al ’67; Kapusta+Shuryak ‘94]
• melting scenario quantitatively compatible with chiral restoration • microscopic calculation of in-medium axialvector to be done
[Hohler et al ‘12] s [GeV2]
ρV,A/s Vacuum T=140MeV T=170MeV
4.10 Chiral Condensate + ρ-Meson Broadening 〈〈
qq〉〉
/ 〈q
q〉0
-
-
effective hadronic theory • Σh = mq 〈h|qq|h〉 > 0 contains quark core + pion cloud
= Σhcore + Σh
cloud ~ + + • matches spectral medium effects: resonances + pion cloud • resonances + chiral mixing drive ρ-SF toward chiral restoration
Σπ
Σπ
> >
-
ρ
2.) Chiral Symmetry in QCD - Nonperturbative QCD, Chiral Breaking + Hadron Spectrum 3.) Thermal Electromagnetic Emission Rates - EM Spectral Function: Hadronic vs. Partonic Regimes 4.) Vector Mesons in Medium - Many-Body Theory, Spectral Functions + Chiral Partners (ρ-a1) 5.) Quark-Gluon Plasma Emission - Perturbative vs. Lattice-QCD Rates, “Quark-Hadron-Duality” 6.) Dilepton + Photon Spectra in Heavy-Ion Collisions - Space-Time Evolution, Phenomenology + Interpretation 7.) Summary and Conclusions
Outline
• marked low-mass enhancement • comparable to recent lattice-QCD computations
Im [ ] = + + + …
5.1 QGP Emission: Perturbative vs. Lattice QCD
Baseline: q q
_ e+
e-
small M → resummations, finite-T perturbation theory (HTL)
Σq
Σq
collinear enhancement: Dq,g=(t-mD
2)-1 ~ 1/αs
[Braaten,Pisarski+Yuan ‘91]
[Ding et al ’10]
dRee/d4q 1.45Tc q=0
5.2 Euclidean Correlators: Lattice vs. Hadronic
]T/q[)]T/(q[)T;q,q(dq)T;q,( VV 2
212 0
00
0
0sinh
cosh −= ∫∞ τ
ρπ
τΠ• Euclidean Correlation fct.
)T,(G)T,(G
V
Vτ
τfree
Hadronic Many-Body [RR ‘02] Lattice (quenched) [Ding et al‘10]
• “Parton-Hadron Duality” of lattice and in-medium hadronic …
5.2.2 Back to Spectral Function
• suggestive for approach to chiral restoration and deconfinement
-Im
Πem
/(C
T q
0)
5.3 Summary of Dilepton Rates: HG vs. QGP dRee /dM2 ~ ∫d3q f B(q0;T) Im Πem
• Lattice-QCD rate somwhat below Hard-Thermal Loop • hadronic → QGP toward Tpc: resonance melting + chiral mixing • Quark-Hadron Duality at all Mee?! (QGP rates chirally restored!)
2.) Chiral Symmetry in QCD - Nonperturbative QCD, Chiral Breaking + Hadron Spectrum 3.) Thermal Electromagnetic Emission Rates - EM Spectral Function: Hadronic vs. Partonic Regimes 4.) Vector Mesons in Medium - Many-Body Theory, Spectral Functions + Chiral Partners (ρ-a1) 5.) Quark-Gluon Plasma Emission - Perturbative vs. Lattice-QCD Rates, “Quark-Hadron-Duality” 6.) Dilepton + Photon Spectra in Heavy-Ion Collisions - Space-Time Evolution, Phenomenology + Interpretation 7.) Summary and Conclusions
Outline
6.1 Space-Time Evolution + Equation of State
• 1.order → lattice EoS: - enhances temperature above Tc - increases “QGP” emission - decreases “hadronic” emission
• initial conditions affect lifetime
• simplified: parameterize space-time evolution by expanding fireball
• benchmark bulk-hadron observables
• Evolve rates over fireball expansion:
[He et al ’12]
Au-Au (200GeV)
)p(Acc),T;q,M(qxdd
dNqqdM)(VddM
dN etM,B
thermee
FB
thermee
fo±
∫∫= µτττ
τ440
3
0
6.1.2 Bulk Hadron Observables: Fireball Model
• Mulit-strange hadrons freeze-put at Tpc • Bulk-v2 saturates at ~Tpc
[van Hees et al ’11]
6.2 Di-Electron Spectra from SPS to RHIC
• consistent excess emission source • suggests “universal” medium effect around Tpc
• FAIR, LHC?
QM12
Pb-Au(17.3GeV)
Pb-Au(8.8GeV)
Au-Au (20-200GeV)
[cf. also Bratkovskaya et al, Alam et al, Bleicher et al, Wang et al …]
6.3 In-In at SPS: Dimuons from NA60 • excellent mass resolution and statistics • for the first time, dilepton excess spectra could be extracted!
+ full acceptance correction…
[Damjanovic et al ’06]
6.3.2 NA60 Multi-Meter: Accept.-Corrected Spectra
• Thermal source! • Low-mass: good sensitivity to medium effects, T~130-170MeV • Intermediate-mass: T ~ 170-200 MeV > Tpc • Fireball lifetime τFB = (6.5±1) fm/c
[CERN Courier Nov. 2009]
Emp. scatt. ampl. + T-ρ approximation Hadronic many-body Chiral virial expansion
Thermometer
Spectrometer
Chronometer
6.3.3 Spectrometer
• in-med ρ + 4π + QGP • invariant-mass spectrum directly reflects thermal emission rate!
µ+µ- Excess Spectra In-In(17.3AGeV) [NA60 ‘09]
Mµµ [GeV]
Thermal µ+µ- Emission Rate
[van Hees+RR ’08]
dMRd
Vqd
dRqqdM)(VddM
dNFB
foµµµµ
τ
τ
µµ ττ 440
3
0
≈= ∫∫thermtherm
Mµµ [GeV]
6.4 Conclusions from Dilepton “Excess” Spectra • thermal source (T~120-230MeV)
• in-medium ρ meson spectral function - avg. Γρ (T~150MeV) ~ 350-400 MeV ⇒ Γρ (T~Tpc) ≈ 600 MeV → mρ - “divergent” width ↔ Deconfinement?!
• M > 1.5 GeV: QGP radiation
• fireball lifetime “measurement”: τFB ~ (6.5±1) fm/c (In-In)
[van Hees+RR ‘06, Dusling et al ’06, Ruppert et al ’07, Bratkovskaya et al ’08, Santini et al ‘10]
6.5 The RHIC-200 Puzzle in Central Au-Au
• PHENIX, STAR and theory: - consistent in non-central collisions - tension in central collisions
6.6 Direct Photons at RHIC
• v2γ,dir as large as for pions!?
• underpredcited by QGP-dominated emission
← excess radiation
• Teffexcess = (220±30) MeV
• QGP radiation? • radial flow?
Spectra Elliptic Flow
[Holopainen et al ’11, Dion et al ‘11]
6.6.2 Thermal Photon Radiation
thermal + prim. γ
[van Hees, Gale+RR ’11]
• flow blue-shift: Teff ~ T √(1+β)/(1-β) , β~0.3: T ~ 220/1.35 ~ 160 MeV • “small” slope + large v2 suggest main emission around Tpc
• other explanations…? [Skokov et al ‘12; McLerran et al ‘12]
6.7 Direct Photons at LHC
• similar to RHIC (not quite enough v2) • non-perturbative photon emission rates around Tpc?
● ALICE
[van Hees et al in prep]
Spectra Elliptic Flow
• Spontaneous Chiral Symmetry Breaking in QCD Vacuum: - ‹qq› ~ mq* ≠ 0 , chiral partners split (π-σ, ρ-a1, …) - low-mass dileptons dominated by ρ
7.) Conclusions
• Hadronic Medium Effects: - “melting” of ρ (constraints!) ⇒ hadronic liquid!? - connection to chiral symmetry: QCD/chiral sum rules (a1)
• Extrapolate EM Emission Rates to Tpc ~ 170MeV ⇒ in-med HG and QGP shine equally bright (“duality”) !
• Phenomenology for URHICs: - versatile + precise tool (spectro, chrono, thermo + baro-meter) - SPS/RHIC (NA45,NA60,STAR): compatible w/ chiral restoration - tension STAR/PHENIX; LHC (ALICE) and CBM/NICA?
-
6.3.4 Sensitivity of Dimuons to Equation of State
• partition QGP/HG changes, low-mass spectral shape robust
[cf. also Ruppert et al, Dusling et al…]
6.3.5 NA60 Dimuons with Lattice EoS + Rate
• partition QGP/HG changes, low-mass spectral shape robust
First-Order EoS + HTL Rate Lattice EoS + Lat-QGP Rate
Mµµ [GeV] [van Hees+RR ’08]
In-In (17.3GeV) Tin =190MeV
Tc =Tch =175MeV
[cf. also Ruppert et al, Dusling et al…]
Tin =230MeV Tpc =Tch =175MeV
6.3.6 Chronometer
• direct measurement of fireball lifetime: τFB ≈ (6.5±1) fm/c • non-monotonous around critical point?
In-In Nch>30
6.4 Summary of EM Probes at SPS
• calculated with same EM spectral function!
Mµµ [GeV]
In(158AGeV)+In
6.1 Fireball Evolution in Heavy-Ion Collisions
• “chemical” freezeout Tchem~ Tc ~170MeV, “thermal” freezeout Tfo ~ 120MeV • conserve entropy + baryon no.: Ti → Tchem → Tfo • time scale: hydrodynamics, fireball VFB(τ ) = (z0+vz τ ) π (r0 + 0.5a┴ τ 2)2
µN [GeV] τ [fm/c]
)p(Acc),T;q,M(qxdd
dNqqdM)(VddM
dN etM,B
thermee
FB
thermee
fo±
∫∫= µτττ
τ440
3
0
Thermal Dilepton Spectrum:
Isentropic Trajectories in the Phase Diagram
6.3.2 NA60 Data Before Acceptance Correction
• Discrimination power of model calculations improved, but … • can compensate spectral “deficit” by larger flow: lift pairs into acceptance
hadr. many-body + fireball
emp. ampl. + fireball
schem. broad./drop. + HSD transport chiral virial
+ hydro
6.3.7 Dimuon pt-Spectra + Slopes: Barometer
• slopes originally too soft
• increase fireball acceleration, e.g. a┴ = 0.085/fm → 0.1/fm
• insensitive to Tc = 160-190 MeV
Effective Slopes Teff
6.2 Mass vs. Temperature Correlation • generic space-time argument:
⇒ ⇒ Tmax ≈ M / 5.5 (for Im Πem =const)
23em44
33
0e /T/M
FBeeee )MT(M
Im)T(Vqxdd
dNqxddqM
dMddN −∝= ∫
Πτ
55.T/Mee TdTdMdN !!" eT)(M,Im em#
• thermal photons: Tmax ≈ (q0/5) * (T/Teff)2
→ reduced by flow blue-shift! Teff ~ T * √(1+β)/(1-β)
2.4.4 Weinberg (Chiral) Sum Rules + Order Parameters
)Im(ImsdsI AVn
n ΠΠπ
−−= ∫
21
0
21
222
0
31
q)q(αcII
fI
FrfI
s
π
Aππ
=
==
−=
−
−
[Weinberg ’67, Das et al ’67]
• Moments Vector-Axialvector
• In Medium: energy sum rules at fixed q [Kapusta+Shuryak ‘93]
• correlators (rhs): effective models (+data) • order parameters (lhs): lattice QCD
promising synergy!
Lower SPS Energy (Ti ~ 170MeV→ Tfo~100MeV)
• enhancement increases!? • supports importance of baryonic effects
6.2 Low-Mass Di-Electrons: CERES/NA45 Top SPS Energy (Ti ~ 200MeV → Tfo~110 MeV)
• QGP contribution small • medium effects! • drop. mass or broadening?!
[RR+Wambach ‘99]
6.1.2 Emission Profile of Thermal EM Radiation • generic space-time argument:
⇒ ⇒ Tmax ≈ M / 5.5 (for Im Πem =const)
23em44
33
0e /T/M
FBeeee )MT(M
Im)T(Vqxdd
dNqxddqM
dMddN −∝= ∫
Πτ
55em eT)(M, .T/Mee TImdTdM
dN !!" #
• Additional T-dependence from EM spectral function
• Latent heat at Tc not included (penalizes T > Tc, i.e. QGP)
e.g. , A=a1,h1
fix G via Γ(a1→ρπ) ~ G2 v2 PS ≈ 0.4GeV, …
> >
R
h
4.6 ρ -Hadron Interactions in Hot Meson Gas resonance-dominated: ρ + h → R, selfenergy:
Effective Lagrangian (h = π, K, ρ)
ρ
Generic features: • cancellations in real parts • imaginary parts strictly add up
∫ ±+= )](f)(f[v)qk(D)(kd
RR
kh
hRRhR ωωπ
Σ ρρ2
3
3
2
µννµ
πρ ρπ )(AGA ∂=L
2.5 Dimuon pt-Spectra and Slopes: Barometer
• vary fireball evolution: e.g. a┴ = 0.085/fm → 0.1/fm • both large and small Tc compatible with excess dilepton slopes
pions: Tch=175MeV a┴ =0.085/fm
pions: Tch=160MeV a┴ =0.1/fm
4.3.3 Acceptance-Corrected NA60 Spectra
• rather involved at pT>1.5GeV: Drell-Yan, primordial/freezeout ρ , …
[van Hees + RR ‘08]
4.5 EM Probes in Central Pb-Au/Pb at SPS
• consistent description with updated fireball (aT=0.045→0.085/fm) • very low-mass di-electrons ↔ (low-energy) photons
[Liu+RR ‘06, Alam et al ‘01]
Di-Electrons [CERES/NA45] Photons [WA98]
[van Hees+RR ‘07]
Model Comparison of ρ-SF in Hot/Dense Matter
• first sight: reasonable agreement • second sight: differences! ρB vs. ρN • Implications for NA60 interpretation?!
[RR+Wambach ’99]
[Eletsky etal ’01]
• ImΣV ~ ImTVN ρN + ImTVπ ρπ ~ σVN,Vπ + dispersion relation for ReTV
[Eletsky,Belkacem,Ellis,Kapusta ’01]
3.8 Axialvector in Medium: Explicit a1(1260)
Σρ
Σπ Σπ
>
> > >
N(1520)…
Δ,N(1900)… a1 + + . . .
Exp: - HADES (πA): a1→(π+π-)π
- URHICs (A-A) : a1→ πγ )Im(Imsdsf AV ΠΠππ −−= ∫2
[RR ’02]
after freezeout τV ~ 1/ΓV
tot
thermal emission τFB ~ 10fm/c
τΔΓM)M(N
qddRxdqddM
dN eeVV
eeeeV →→ ∝= ∫ 443
3.3 The Role of Light Vector Mesons in HICs
Γee [keV] Γtot [MeV] (Nee )thermal (Nee )cocktail ratio
ρ (770) 6.7 150 (1.3fm/c) 1 0.13 7.7 ω(782) 0.6 8.6 (23fm/c) 0.09 0.21 0.43 φ(1020) 1.3 4.4 (44fm/c) 0.07 0.31 0.23
⇒ In-medium radiation dominated by ρ -meson! Connection to chiral symmetry restoration?!
Contribution to invariant mass-spectrum:
∑=n
nn
Qc)Q(Π 2
2α
Nonpert. Wilson coeffs. (condensates)
0.2% 1%
4.5 Constraints II: QCD Sum Rules in Medium dispersion relation
for correlator: • lhs: OPE (spacelike Q2): • rhs: hadronic model (s>0):
[Shifman,Vainshtein +Zakharov ’79]
sQ)s(Im
sds
Q)Q(Π
+= ∫
∞
20
2
2αα Π
[Hatsuda+Lee’91, Asakawa+Ko ’92, Klingl et al ’97, Leupold et al ’98, Kämpfer et al ‘03, Ruppert et al ’05]
)ss()(s
)s(DImgm
)s(Im
sdual−+
−=
Θπα
π
Π ρρ
ρρ
18
2
4