Introduction to Dileptons

and in-Medium Vector Mesons

Ralf RappCyclotron Institute

+ Physics Department Texas A&M UniversityCollege Station, Texas

USA

2 Lectures at ECT* EM-Probes WorkshopTrento, 04. + 06.06.05

1.) Introduction 1.1 Electromagnetic Probes in Strong Interactions

• -ray spectroscopy of atomic nuclei: collective phenomena, …

• DIS off the nucleon: - parton model, PDF’s (high q2< 0) - nonpert. structure of nucleon [JLAB]

• Drell-Yan: pp → eeX (q2> 0: symmetry, nucl. shadowing)

• thermal emission: - compact stars (GRBs?!) - heavy-ion collisions: [SPS, RHIC, LHC, FAIR]

(q2=0) , e+e(q2>0)

u/d

What is the electromagnetic spectrum of QCD matter?

“Freeze-Out”QGPAu + Au

Creating Strong-Interaction Matter in the Laboratory

Hadron GasNN-coll.

Sources of Dilepton Emission:

• “primordial” (Drell-Yan) qq annihilation: NN→e+eX -

e+

e-

• emission from equilibrated matter (thermal radiation) - Quark-Gluon Plasma: qq → e+e , … - Hot+Dense Hadron Gas:

→ e+e , …

-

• final-state hadron decays: ,→ e+e , D,D → e+eX, … _

1.2 Objective: Use Dileptons to Probe the Nature of Strongly Interacting Matter

• Bulk Properties: Equation of State• Microscopic Properties: - Degrees of Freedom - Spectral Functions • Phase Transitions: (Pseudo-) Order Parameters

(some) Key Questions: Can we • infer the temperature of the matter?• establish in-medium modifications of →ee ?• extract signatures of chiral symmetry restoration?

1.3 Intro-III: EoS and Hadronic Modes

All information encoded in free energy:

• EoS: , ,

• correlation functions:

V/logT)T,( B Z

Z/)](j),x(j[ˆTr)t(exdi)q( iqx 04

Ts

)(P

qqj : “hadronic” current

↔ iso/scalarpairs!

↔ dileptons, photons!

QCD Order Parameters and Hadronic Modes

• condensate: ‹qq› = ∂∂mq

suscept.: s=∂2∂mq2 =(q=0) = 2 D(q=0)

~ (m)-2

• decay const: f2= ∫ ds/s ( Im V Im A )

2 ImD

1.0 T/Tc

s‹qq›- lattice QCD

[Weinberg ’67,Kapusta+Shuryak ’94]

-

1.4.1 Schematic Dilepton Spectrum in HICs

qq

Characteristic regimes in invariant e+e mass, M2=(pe++ pe)2 :

• Drell-Yan: power law ~ Mn ↔ high mass• thermal ~ exp(-M/T): - QGP (highest T) ↔ intermediate mass - HG (moderate T) ↔ low mass

Central Pb-Pb 158 AGeV

opencharm (DD) Drell-

Yan

• factor ~2 excess• open charm? thermal? …

Intermediate Mass: NA50

M [GeV]• final-state hadron decays saturate yield in p-A collisions

1.4.2 Dilepton Data at CERN-SPS

Low Mass: CERES/NA45

Mee [GeV]• strong excess around M≈0.5GeV• little excess in region

2. Thermal Electromagnetic Emission Rates - Vacuum: Quarks vs. Hadrons, Vector Mesons

3. Chiral Symmetry in QCD - Spontaneous Breaking, Hadronic Spectrum, Restoration

4. (Light) Vector Mesons in Medium - Hadronic Many-Body Approach - Dropping Mass, Chiral Restoration?!

5. QGP Emission

6. Thermal Photons

7. Dilepton Spectra in Heavy-Ion Collisions - Space-Time Evolution; Comparison to SPS and RHIC Data

8. Summary and Conclusions

Outline

2.) Electromagnetic Emission Rates

Tiqx )(j)x(jexdi)q(Π 04

emememE.M. Correlation Function:

e+

e-

γ

)T(fMqd

dR Bee23

2

4

)T(fqd

dRq B

230

Im Πem(M,q)

Im Πem(q0=q)

= O(1)= O(1)

= O(= O(ααs s ))

also: e.m susceptibility (charge fluct.): χ = Πem(q0=0,q→0)

In URHICs:• source strength: dependence on T, B, , medium effects, …• system evolution: V(), T(), B(), transverse expansion, …• nonthermal sources: Drell-Yan, open-charm, hadron decays, … • consistency!

2.2 E.M. Correlator in Vacuum: (e+e→hadrons)

)s(Im em)s(DIm

gm

V,, V

V

22

s,d,u

Sqc

)s()e(N

s

1

122

e+

e-

h1

h2…

s ≥ sdual~(1.5GeV)2 : pQCD continuum

s < sdual :Vector-Meson Dominance

q

q_

qq_

24

KK

e+

e-

after freezeoutV ~ 1/V

tot

thermal emissionFB ~ 10fm/c

M

)M(Nqd

dRxdqd

dMdN eeV

VeeeeV 4

43

2.3 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:

3.1 Chiral Symmetry and its Breaking in Vacuum

3.2 Consequences for the Hadronic Spectrum

3.3 Vector-Axialvector Correlation Functions and Chiral Restoration

3.) Chiral Symmetry in QCD

3.1.1 Chiral Symmetry in QCD: Vacuum

2

41

aq Gq)m̂Agi(q QCDL

current quark masses: mu ≈ md ≈ 5-10MeV

Chiral SU(2)V × SU(2)A transformation:

Up to OO(mq ), LQCD invariant under

Rewrite LQCD using 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”

3.1.2 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 , … ]

-

• mass generation , not observables!

but: hadronic excitations reflect SBCS:

• “massless” Goldstone bosons 0,±

(explicit breaking: f2 m2= mq ‹qq› )

• “chiral partners” split: M ≈ 0.5GeV !

• vector mesons and :

JP=0± 1± 1/2±

qqG*Mqq 42

-

iiAiAiA )a()qR()qR()(R 121

)(RA

chiral singlet !

3.2.2 Hadron Spectra and SBCS in Vacuum

Axial-/Vector Correlators

)Im(Ims

dsf IA

IV

112

pQCD cont.

)ms(fs)s(DImg

mIm a

a

aIA

222

41

1

1

1

• entire spectral shape matters • Weinberg Sum Rule(s)

“Data”: lattice [Bowman etal ‘02] Curve: Instanton Model [Diakonov+Petrov ’85, Shuryak]

● chiral breaking: |q2| ≤ 1 GeV2

● quark condensate:

nvac≈ (2Nf ) fm-3 !

325000 )MeV(|qq|

Constituent Quark Mass

3.3.1 “Melting” the Chiral Condensate

How?

Excite vacuum (hot+dense matter)

• quarks “percolate” / liberated Deconfinement • ‹qq› condensate “melts”, iral Symm. chiral partners degenerate Restoration(-, -a1, … medium effects → precursor!)

0 0.05 0.3 0.75 [GeVfm-3] 120, 0.50 150-160, 20 175, 50 T[MeV], had

PT many-body degrees of freedom? QGP (2 ↔ 2) (3-body,...) (resonances?) consistent extrapolate pQCD

-

1.0 T/Tc

m‹qq›-lattice QCD

3.3.2 Low-Mass Dileptons + Chiral Symmetry

• How is the degeneration realized ?• “measure” vector with e+e, but axialvector?

VacuumAt Tc: Chiral Restoration

E.M. Emission Rates:● proportional to e.m. correlator (photon selfenergy)

● vacuum: separation in perturbative (qq) -- nonpert. (, , )

at “duality scale” sdual ~ (1.5GeV)2

● in-med radiation: low-mass ↔ -meson, high-mass ↔ QGP

Chiral Symmetry:

● spontaneously broken in the vacuum mass generation! Mq* ~ ‹qq› ≠ 0 (low q2)

● hadronic spectrum: chiral partners split (-, - a1, …)

● excite vacuum → condensate melts → chiral restoration → chiral partners degenerate

Upshot of Chapters 2 + 3

-

-

4.1 Hadronic Many-Body Theory for Vector Mesons - -Meson in Vacuum

- -Selfenergies and Spectral Functions - Constraints and Consistency: Photo-Absorption, QCD Sum Rules, Lattice QCD

4.2 Vector Meson in URHICs: Hot+Dense Matter

4.3 Dropping -Mass; Vector Manifestation of CS

4.4 Chiral Restoration?!

4.) Vector Mesons in Medium

4.1 Many-Body Approach: -Meson in VacuumIntroduce as gauge boson into free + Lagrangian

2

21 g)(gintL

)k(D)(

kdgg)qk(D)k(D)(

kdg)q(

4

424

422

22

2

1202 )]M()m(M[)M(D )(

e.m. formfactor

scattering phase shift

2402 |)M(D|)m(|)M(F| )(

)M(DRe)M(DIm

)M(

1-tan

|F|2

-propagator:

D

AIm2

0 q

4.1.2 -Selfenergies in Hot + Dense Matter

D(M,q;B,T)=[M2-m2--B-M ]-1

modifications due to interactions with hadronsfrom heat bath In-Medium -Propagator

[Chanfray etal, Herrmann etal, RR etal, Weise etal, Oset etal, …](1) Medium Modifications of Pion Cloud

)f(vD)f(DvD medmedmed

1212

D= [k02-k

2-(k0 ,k)]-1In-med -prop.

→ mostly affected by (anti-) baryons

a1

Ge.g. + → , a1(1260) → +

fix coupling G via decay width (a1→)

>

>

R

h

(2) Direct -Hadron Interactions: + h → R

(i) Meson Gas (h = , K, )

Generic features: real parts cancel, imaginary parts add

)](f)(f[v)qk(D)(

kdR

Rk

hhRRhR

23

3

2

(ii) -Baryon Interactions (h = N, , …)

• S-wave: + N → N(1520), (1700) → + N , etc.

e.g.: (N(1520)→N) ≈ 25MeV, ((1700)→N) ≈ 130MeV

• Coupling constant → free decay:222

2

2 )q(FQq)M(AdMMfC cm

mm

mBNNB

NB

Decay Phase Space

N(1520)

(1700)

4.1.3 Constraints I: Nuclear Photo-Absorption total nuclear -absorption in-medium–spectral cross section function at photon point

>

>

,N*,*

N-1

)q,M(DImg

mq

)qq(ImqA

)q( med

NN

absA 044

2

4

00

0

0

em

N → N ,

N →

meson exchange!

direct resonance!

Light-like -Spectral Function, D(q0=q), and Nuclear Photo-Absorption

NA

-ex

[Urban,Buballa,RR+Wambach ’98]

On the Nucleon On Nuclei

• 2.+3. resonance melt (parameter) (selfconsistent N(1520)→N)

[Post,Mosel etal ’98]

• fixes coupling constants and formfactor cutoffs for NB

4.1.4 (770) 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 etal ’98]

[Post etal ’02]

[Cabrera etal ’02]

4.1.5 QCD Sum Rules + (770) in Nuclear Matter

)ss()(s

)s(DImg

m)s(Im

sdual

18

2

4

General idea: dispersion relation for correlation function

nn

n

Tiqx

Q

c)(j)x(jexdi)q(Π

2

42 0

sQ

)s(Ims

dsQ/)qQ(Π

20

222

• lhs: operator product expansion for large spacelike Q2:

Nonpert.Wilson coeffs (condensates)

• rhs: model spectral function at timelike s>0:

Resonance + pQCD continuum

-Meson:

...Q

)qq(C

Q

GQ)(

s

ss

6

2

4

22

2

2

2 31

81

ln

[Shifman,Vainshtein+Zakharov ’79]

4-quark condensate!

Vacuum:<(qq)2> = <qq>2- -

0.2% 1%

Comparison to hadronicmany-body models

• roughly consistent• sensitive to detailed shape

Nuclear Matter: <(qq)2> decreases softening of -propagator

• decreasing mass or increasing width

-

QCD Sum Rule Results: (770) in Nuclear Matter [Leupold etal ’98]

4.2 Vector-Meson Spectral Functions in

High-Energy Heavy-Ion Collisions:

Hot and Dense Matter

[RR+Gale ’99]

4.2.1 -Meson Spectral Functions at SPS

• -meson “melts” in hot and dense matter• baryon densityB more important than temperature • reasonable agreement between models

B/0 0 0.1 0.7 2.6

Hot+Dense Matter Hot Meson Gas

[RR+Wambach ’99]

[Eletsky etal ’01]

Model Comparison

[RR+Wambach ’99]

4.2.2 Light Vector Mesons at RHIC

• baryon effects important even at B,tot= 0 : sensitive to Btot= + B (-N and -N interactions identical)

• also melts, more robust ↔ OZI

-

4.2.3 Lattice Studies of Medium Effects

)2/sinh(

))2/1(cosh(),(Im),(

0

00

00 Tq

TqTqdqT

calculatedon lattice

MEM

1-

0-

extracted

[Laermann, Karsch ’04]

4.2.4 Comparison of Hadronic Models to LGT

)2/sinh(

))2/1(cosh(),(Im),(

0

00

00 Tq

TqTqdqT

calculate

integrate

More direct!

Proof of principle, not yet meaningful (need unquenched)

4.3 Scenarios for Dropping -Meson Mass

(3) Vector Manfestation of Chiral Symmetry

(a) Vacuum: effective L with L≡(“VM”)

(b) Finite Temperature:thermal - and -loop expansion → f(T) , m(T)

QCD-matching requires “intrinsic” T-dependence of bare m(0), g

dropping -mass

[Harada+Yamawaki, ’01]

(1) Naïve Quark Model: m≈ 2Mq* → 0 at chiral restoration (problem: kinetic energy of bound state)

(2) Scale Invariance of LQCD: implement into effective Lhad

universal scaling law mmmmqqqq *N

*N

//T 3131

vac[Brown+, Rho ’91]

4.4 Dilepton Rates and Chiral Restoration dRee /dM2 ~ f B Im em

• Hard-Thermal-Loop result much enhanced over Born rate

• “matching” of HG and QGP automatic!

• Quark-Hadron Duality at low mass ?!

• Degenerate axialvector correlator?

[Braaten,Pisarski+Yuan ’90][qq→ee][qq+O(s)-HTL]

--

4.4.2 Current Status of a1(1260)

>

> >

>

N(1520)…

,N(1900)…

a1 + + . . .

Exp: - HADES (A): a1→(+-) - URHICs (A-A) : a1→

)DImg

mDIm

g

m(

s

dsf a

a

a1

1

1

2

4

2

42

5.1 Pertubative vs. Lattice QCD

5.2 Emission from “Resonances”

5.) Dilepton Emission from the QGP

Im [ ] = + + + …

5.1 Perturbative vs. Lattice QCD

Baseline:q

q

_

e+

e

But: small M → resummationsfinite-T perturbation theory (HTL)

q

q

collinear enhancement: Dq,g=(t-mD

2)-1 ~ 1/αs

[Braaten,Pisarski+Yuan ‘91]

• large enhancement at low M

• not shared by lattice calculations: threshold + resonance structures (photon rate?!) [Bielefeld

Group ‘02]

5.2 QGP Dileptons from Bound States

→ based on finite-T lattice potentials approach to “zero-binding line” ~ stable-mass-resonance

[Shuryak+Zahed ‘04]

• double-peak structure due to zero-binding line + mixed phase• factor 1.5-2 enhancement at M≈1.5GeV; depends on quark width

Dilepton Spectrumratio to pert. qq rate

Mee/mq

_

[Casalderrey+ Shuryak ‘04]

6.) Thermal Photon Emission Rates

Quark-Gluon Plasma

q

gq

But: other contributions in OO(αs)

collinear enhanced Dg=(t-mD2)-1~1/αs

[Aurenche etal ’00, Arnold,Moore+Yaffe ’01]

Bremsstrahlung Pair-ann.+scatt. + ladder resummation (LPM)

“Naïve” LO: q + q (g) → g (q) + γ

[Kapusta,Lichard+Seibert ’91, … , Turbide,RR+Gale’04]

Hot and Dense Hadron Gas

γ

a1,

Im Πem(q0=q) ~ Im D (q0=q)

Low energy: vector dominance

High energy: meson exchange

Emission Rates

Total HG ≈ in-med QGP ! to be understood…

7.1 Space-Time Evolution of URHIC’s - Formation and Freezeouts - Trajectories in the QCD Phase Diagram

7.2 Comparison to Data - Dileptons at SPS (√s = 17, 8 GeV) - Photons at SPS (√s = 17 GeV) - RHIC (√s = 200 GeV)

7.) Dilepton Spectra in Relativistic Heavy-Ion Collisions

7.1.1 Hadron Production in Heavy-Ion Collisions→ well described by hadron gas in thermal+chemical equilibrium:

SPS / RHIC: “chemical freezeout” close to phase boundary need to construct “evolution”

• before: up to earliest “formation” time 0 ↔ T0 > Tchem

• after: down to thermal freezeout f ↔ Tfo < Tchem

)T,;E(f)(

kdd)T,(n BkiBi

3

3

2

[Braun-Munzinger etal ‘03]

Caveat: conserve hadron ratios after chem. f.o. chemical potentials for , K, N,…: ,K,N > 0 for T < Tchem

7.1.2 Trajectories in the Phase Diagram• Basic assumption: entropy (+baryon-number) conservation fixes T(B) in the phase diagram

• Time scale: hydrodynamics, e.g. VFB()=(z0+vz) (R┴0+ 0.5a┴2)2

N [GeV] [fm/c]

Acc)T/(),T;q,M(qxdd

dNq

qMd)(Vd

dMdN N

i

thermee

FB

thermee

fo

exp440

3

0

Thermal Dilepton Emission Spectrum

Lower SPS Energy

• enhancement increases!• confirms importance of baryonic effects (predicted)

7.2.1 Low-Mass Dileptons at SPS

Top SPS Energy

• QGP contribution small• medium effects! • drop. mass or broadening?!

Ti≈300MeV, more QGP contr.

Hydrodynamics (chem-eq)

[Kvasnikowa,Gale+Srivastava ’02]

7.2.2 Intermediate-Mass Dileptons at SPS: NA50

e.m. corr. continuum-like: Im Πem~ M2 (1 + s/+…)

Ti≈210MeV , HG-dominated

Thermal Fireball (chem-off-eq)

[RR+Shuryak ’99]

QGP + HG!

7.2.3 Photon Spectra at the SPS: WA98

Hydrodynamics: QGP + HG[Huovinen,Ruuskanen+Räsänen ’02]

• T0≈260MeV, QGP-dominated

• still true if pp→X included

[Turbide,RR+Gale’04]Expanding Fireball + pQCD

• pQCD+Cronin at qt >1.5GeV T0=205MeV suff., HG dom.

• low mass: thermal dominant• int. mass: cc e+X , rescatt.? e-X

[RR ’01]

-

[R. Averbeck, PHENIX]

7.3 Dilepton Spectrum at RHIC

MinBias Au-Au (200AGeV)

run-4 results eagerly awaited …

thermal

8.) Conclusions

• Thermal E.M. Radiation from QCD matter - Low-mass dileptons: in-med (, ) ↔ Chiral Restoration!? - Intermediate mass: qq annihilation ↔ QGP Radiation!? - similar for photons but M=0

• extrapolations into phase transition region in-med HG and QGP shine equally bright (“duality”) deeper reason? lattice calculations? axialvector mode?

• phenomenology for URHICs so far promising - importance of model constraints - precision data+theory needed for definite conclusions

• much excitement ahead: PHENIX, NA60, CERES, HADES, ALICE, … and theory!

-