Theory of Thermal Electromagnetic Radiationheinz/JET2013/lecturenotes/...Theory of Thermal Electromagnetic Radiation Ralf Rapp Cyclotron Institute + Dept. of Physics & Astronomy Texas

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

qq

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


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 ]


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


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!)


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

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Theory of Thermal Electromagnetic Radiationheinz/JET2013/lecturenotes/...Theory of Thermal Electromagnetic Radiation Ralf Rapp Cyclotron Institute + Dept. of Physics & Astronomy Texas