Upload
natane
View
47
Download
0
Tags:
Embed Size (px)
DESCRIPTION
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. - PowerPoint PPT Presentation
Citation preview
Theory of Thermal Electromagnetic Radiation
Ralf RappCyclotron Institute +
Dept. of Physics & Astronomy Texas A&M UniversityCollege 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”QGPAu + Au
1.2 Dileptons in Heavy-Ion Collisions
Hadron GasNN-coll.
Sources of Dilepton Emission:
• “primordial” qq annihilation (Drell-Yan): NN→e+eX -
e+
e-
• thermal radiation - Quark-Gluon Plasma: qq → e+e , … - Hot+Dense Hadron Gas: → e+e , …
-
• final-state hadron decays: ,→ e+e , D,D → e+eX, … _
1.3 Schematic Dilepton Spectrum in HICs
Characteristic regimes in invariant e+e mass, M2 = (pe++ pe)2
• Drell-Yan: primordial, power law ~ Mn
• thermal radiation: - entire evolution - Boltzmann ~ exp(-M/T)
T/qeeFB
ee
TdMdR
VdMddN
0
31 e
q0≈ 0.5GeV Tmax ≈ 0.17GeV , q0≈ 1.5GeV Tmax ≈ 0.5GeV
Thermal rate: Im Πem(M,q;B,T)
• Thermal e+e emission rate from hot/dense matter (em >> 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 em/ M2
)T,q(fMqdxd
dN Bee023
2em
44 Im Πem(M,q;B,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 [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
≈
/ 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 (-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 (s = g2/4π << 1)
• degrees of freedom = quarks + gluons
2.1 Nonperturbative QCD
2
41
aq 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
aq Gq)m̂Agi(q QCDL
(bare quark masses: mu ≈ md ≈ 5-10MeV )
Chiral SU(2)V × SU(2)A transformation:
Up to OO(mq ), LQCD invariant under
Rewrite LQCD using left- and right-handed quark fields qL,R=(1±5)/2 q :
q)/i(q)(Rq VVV 2 exp
q)/i(q)(Rq AAA 25 exp
2
4
1aq
R
RRR
L
LLL G)m(
du
Di)d,u(du
Di)d,u(
OQCDL
R,LR,LR,L q)/i(q 2 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 >
>
>
>qLqR
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|0 )qq(qq 0
G/)m*M(q*)M(Dq
G*M,Gq])Gm[i(q
q
q
4
22
21eff
0200eff
L
G*M*Mp
)(
pdNNlog
VT
fc 422
2223
3
Z
223
3
240
*Mp
*M
)(
pdGNN*M
*M fc
Gap
Equation
2.3.2 (Observable) Consequences of SBCS
• mass gap , not observable!
but: hadronic excitations reflect SBCS
• “massless” Goldstone bosons 0,±
(explicit breaking: f2 m2= mq ‹qq› )
• “chiral partners” split: M ≈ 0.5GeV ! chiral trafo: ,
• Vector mesons and :
JP=0± 1± 1/2±
0042 |qq|G*Mqq
AR )(NNRA 1535
q)()(q)(R jj
jjA 2
12
12
1505
chiral singlet
-
-iijkkii
jiijj
ij
jij
j
AiAiA
)a(
q)(qiq)i()i(q
)qR()qR()(R
1
5505 2221
21
21
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 () - Axialvector (a1) spectral splitting
)Im(ImsdsI AVn
n 00002
31
210
21
222
|q)q(|αcI,|qq|mI
,fI,FrfI
sq
πA
[Weinberg ’67, Das et al ’67]
→(2n+1)
pQCD
pQCD
• Key features of updated “fit”: [Hohler+RR ‘12]
+ a1 resonance, excited states (’+ a1’), universal continuum (pQCD!)
→(2n) [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
2.5 QCD Sum Rules: and a1 in Vacuum
• dispersion relation:
• lhs: hadronic spectral fct. • rhs: operator product expansion
[Shifman,Vainshtein+Zakharov ’79]2
2
20 Q
)Q(Π
sQ
)s(Ims
ds
• 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 (-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
em• EM Correlation Fct.:
e+
e
)T,q(fQQxdd
dN Bee023
2em
44
Im Πem(M,q;T,B)
*(q))j
Qj()f(fi|j|ff|j|i
)QPP(E)(
pd
E)(
pde
xd
dNV
eefi
fif
ff
i
iieeee
4emem
3
3
3
34
4
11
22222
• Quark basis:
• Hadron basis:
(T,B)
ssdduuj 31
31
32 em
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(emIm)s(D
gm
V,, V
V Im
22
s,d,u
sqc
)s()e(Ns
1
122
e+
e
h1
h2…
s ≥ sdual ~ (1.5GeV)2
pQCD continuum
s < sdual Vector-Meson Dominance
q
q_
qq_
46
KK
e+
e
3.3 Low-Mass Dileptons + Chiral Symmetry
• How is the degeneration realized?• “measure” vector with e+e , axialvector?
VacuumT > Tc: Chiral Restoration
Im Πem(M,q)
3.4 Versatility of EM Correlation Function• Photon Emission Rate
)T,q(fqd
dRq B
0230 em
Im Πem(q0=q)
e+
e-
γ ~ O(O(ααs s ))
*
• EM Susceptibility ( → charge fluctuations)
Q2 Q 2 = χem = Πem(q0=0,q→0)
~ O(1) O(1))T,q(fMqd
dR Bee023
2
4 em
same correlator!
• EM Conductivity
em = 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 (-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 a1 as gauge bosons into free + +a1 Lagrangian
2
21 g)(gintL
1202 )]M()m(M[)M(D )(
EM formfactor
scattering phase shift
2402 |)M(D|)m(|)M(F| )(
)M(DRe)M(DIm
)M(
1-tan
|F|2
-propagator:
• 3 parameters: m(0), g,
4.2 -Meson in Matter: Many-Body Theory
D(M,q;B,T) = [M2 – (m(0))2 - - B - M ]-1
interactions with hadrons from heat bath In-Medium -Propagator
[Chanfray et al, Herrmann et al, Urban et al, Weise et al, Koch et al, …]
• In-Medium Pion Cloud
>>
R=, N(1520), a1, K1...
h=N, , K …
=• Direct -Hadron
Scattering
= +
[Haglin, Friman et al, RR et al, Post et al, …]
• estimate coupling constants from R→ + h, more comprehensive constraints desirable
4.3 Constraints I: Nuclear Photo-Absorption total nuclear -absorption in-medium-spectral cross section function at photon point
>
>
,N*,*
N-1
)q,M(Dg
mq
)qq(qA
)q(
NN
A 044
2
4
00
0
0 medemem
abs
ImIm
N → N ,
N →
meson exchange
direct resonance
4.3.2 Spectral Function in Nucl. Photo-Absorption
NA
-ex
[Urban,Buballa,RR+Wambach ’98]
On the Nucleon On Nuclei
• 2.+3. resonances melt (selfconsistent N(1520)→N)
• fixes coupling constants and formfactor cutoffs for NB
4.4 -Meson Spectral Function in Nuclear Matter
In-med. -cloud ++N→B* resonances
+N→B* resonances (low-density approx.)
In-med. -cloud + +N → N(1520)
Constraints:N , A N →N PWA
• Consensus: strong broadening + slight upward mass-shift• Constraints from (vacuum) data important quantitatively
N=0
N=0
N=0.50
[Urban et al ’98]
[Post et al ’02]
[Cabrera et al ’02]
4.5-Meson Spectral Functions “at SPS”
• -meson “melts” in hot/dense matter• baryon densityB more important than temperature
B/0
0 0.1 0.7 2.6
Hot + Dense Matter
[RR+Gale ’99]
Hot Meson Gas
[RR+Wambach ’99]
B =330MeV
4.6 Light Vector Mesons “at RHIC + LHC”
• baryon effects remain important at B,net = 0:
sensitive to B,tot= + B (-N = -N, CP-invariant)
• also melts, more robust ↔ OZI
- [RR ’01]
4.7 Intermediate Mass: “Chiral Mixing”
)q()q()()q( ,A
,VV
00 1
)q()q()()q( ,V
,AA
001 =
=
• low-energy pion interactions fixed by chiral symmetry
• mixing parameter
2
2
3
3
2 6224
fT)(f
)(kd
fk
k
[Dey, Eletsky +Ioffe ’90]
0
0 0
0
• degeneracy with perturbative spectral fct. down to M~1GeV
• physical processes at M ≥ 1GeV: a1 → e+e etc. (“4 annihilation”)
4.8 Axialvector in Medium: Dynamical a1(1260)
+ + . . . =
Vacuum:
a1
resonance
InMedium: + + . . .
[Cabrera,Jido, Roca+RR ’09]
• in-medium + propagators• broadening of - 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 2
122
02
1 q)q(αcI,mfI,fI sπ
[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]
V,A/sVacuum T=140MeV T=170MeV
4.10 Chiral Condensate + -Meson Broadening
/ q
q 0
-
-
effective hadronic theory
• h = mq h|qq|h > 0 contains quark core + pion cloud
= hcore + h
cloud ~ + +
• matches spectral medium effects: resonances + pion cloud
• resonances + chiral mixing drive -SF toward chiral restoration
>>
-
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 (-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)
q
q
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 221
2 0
00
0
0sinh
cosh • Euclidean Correlation fct.
)T,(G
)T,(G
V
V
free
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
em
/(C
T q
0)
5.3 Summary of Dilepton Rates: HG vs. QGP
dRee /dM2 ~ ∫d3q f B(q0;T) Im em
• 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 (-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
dNq
qdM)(Vd
dMdN e
tM,B
thermee
FB
thermee
fo
44
0
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 FB = (6.5±1) fm/c
[CERN Courier Nov. 2009]
Emp. scatt. ampl. + T- approximation
Hadronic many-body
Chiral virial expansion
Thermometer
Spectrometer
Chronometer
6.3.3 Spectrometer
• in-med + 4 + QGP • invariant-mass spectrum directly reflects thermal emission rate!
+ Excess Spectra In-In(17.3AGeV) [NA60 ‘09]
M[GeV]
Thermal Emission Rate
[van Hees+RR ’08]
dMRd
Vqd
dRq
qdM)(Vd
dMdN
FB
fo
440
3
0
thermtherm
M[GeV]
6.4 Conclusions from Dilepton “Excess” Spectra
• thermal source (T~120-230MeV)
• in-medium meson spectral function
- avg. (T~150MeV) ~ 350-400 MeV
(T~Tpc) ≈ 600 MeV → m
- “divergent” width ↔ Deconfinement?!
• M > 1.5 GeV: QGP radiation
• fireball lifetime “measurement”: FB ~ (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
• v2,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.
[van Hees,Gale+RR ’11]
• flow blue-shift: Teff ~ T √(1+)/(1) , ~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]
SpectraElliptic Flow
• Spontaneous Chiral Symmetry Breaking in QCD Vacuum: - ‹qq› ~ mq* ≠ 0 , chiral partners split (-, -a1, …)
- low-mass dileptons dominated by
7.) Conclusions
• Hadronic Medium Effects: - “melting” of (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
M[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: FB ≈ (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!
M[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() = (z0+vz ) (r0 + 0.5a┴ 2)2
N [GeV] [fm/c]
)p(Acc),T;q,M(qxdd
dNq
qdM)(Vd
dMdN e
tM,B
thermee
FB
thermee
fo
44
0
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 transportchiral 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 em =const)
23em44
33
0e /T/M
FBeeee )MT(
MIm
)T(Vqxdd
dNqxdd
qM
dMddN
55.T/Mee TdTdM
dN eT)(M,Im em
• thermal photons:
Tmax ≈ (q0/5) * (T/Teff)2
→ reduced by flow blue-shift!
Teff ~ T * √(1+)/(1)
2.4.4 Weinberg (Chiral) Sum Rules + Order Parameters
)Im(ImsdsI AVn
n
21
0
21
222
0
31
q)q(αcI
IfI
FrfI
s
π
Aππ
[Weinberg ’67, Das et al ’67]
• Moments VectorAxialvector
• 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/NA45Top 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 em =const)
23em44
33
0e /T/M
FBeeee )MT(
MIm
)T(Vqxdd
dNqxdd
qM
dMddN
55em eT)(M, .T/Mee TIm
dTdMdN
• 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 (a1→) ~ G2 v2 PS ≈ 0.4GeV, …
>>
R
h
4.6 -Hadron Interactions in Hot Meson Gasresonance-dominated: + h → R, selfenergy:
Effective Lagrangian (h = , K, )
Generic features: • cancellations in real parts• imaginary parts strictly add up
)](f)(f[v)qk(D)(
kdR
Rk
hhRRhR
23
3
2
)(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 , …
[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-SF in Hot/Dense Matter
• first sight: reasonable agreement• second sight: differences! B vs. N
• Implications for NA60 interpretation?!
[RR+Wambach ’99]
[Eletsky etal ’01]
• ImV ~ ImTVN N + ImTV ~ VN,V + dispersion relation for ReTV
[Eletsky,Belkacem,Ellis,Kapusta ’01]
3.8 Axialvector in Medium: Explicit a1(1260)
>
> >
>
N(1520)…
,N(1900)…
a1 + + . . .
Exp: - HADES (A): a1→(+-) - URHICs (A-A) : a1→
)Im(Ims
dsf AV 2
[RR ’02]
after freezeoutV ~ 1/V
tot
thermal emissionFB ~ 10fm/c
M
)M(Nqd
dRxdqd
dMdN eeV
VeeeeV 4
43
3.3 The Role of Light Vector Mesons in HICs
ee [keV] tot [MeV] (Nee )thermal (Nee )cocktail ratio
(770) 6.7 150 (1.3fm/c) 1 0.13 7.7
0.6 8.6 (23fm/c) 0.09 0.21 0.43
1.3 4.4 (44fm/c) 0.07 0.31 0.23
In-medium radiation dominated by -meson!
Connection to chiral symmetry restoration?!
Contribution to invariant mass-spectrum:
n
nn
Q
c)Q(Π 2
2
Nonpert. Wilson coeffs. (condensates)
0.2% 1%
4.5 Constraints II: QCD Sum Rules in Mediumdispersion relation
for correlator:
• lhs: OPE (spacelike Q2): • rhs: hadronic model (s>0):
[Shifman,Vainshtein+Zakharov ’79]
sQ
)s(Ims
dsQ
)Q(Π
20
2
2
[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(DImg
m)s(Im
sdual
18
2
4