Electromagnetic Probes in Heavy-Ion Collisions IIhees/dm12/dm12-2.pdf · Electromagnetic Probes in...

Electromagnetic Probesin Heavy-Ion Collisions II

Hendrik van Hees

Frankfurt Institute of Advanced Studies (FIAS)

September 5, 2012

Hendrik van Hees (FIAS Frankfurt) Em. Probes in HICs II September 5, 2012 1 / 33

Outline

1 Electromagnetic probes and vector mesonsRelation to chiral symmetry

2 Elementary vacuum cross sections: hadrons→ `+`−

chiral symmetry constraintsElectrodynamics of pions and ρ mesons (VMD model)Dalitz decays of hadron resonances

3 Dileptons in pp and pA collisions at SIS energiesThe Transport Model GiBUUBaryon-resonance model at SIS energiesDileptons in pp and pNb reactions at HADES

4 Conclusions and Outlook


Why Electromagnetic Probes?

γ, `±: only e. m.interactionswhole matter evolution

ρ/ω γ∗

a1

e+

e−π, . . .

0 1 2 3 4 5

mass [GeV/c2]

dN

ee / d

ydm

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

Fig. by A. Drees (from [RW00])


Vector Mesons and electromagnetic Probes

photon and dilepton thermal emission rates given by sameelectromagnetic-current-correlation function (Jµ = ∑f Qf ψf γµψf )

Π<µν(q) =

∫d4x exp(iq · x)

⟨Jµ(0)Jν(x)

⟩T = −2nB(q0) Im Π(ret)

µν (q)

q0dNγ

d4xd3~q= −αem

2π2 gµν Im Π(ret)µν (q)

∣∣∣q0=|~q|

fB(p0)

dNe+e−

d4xd4k= −gµν

α2

3q2π3 Im Π(ret)µν (q)

∣∣∣q2=M2

e+e−fB(p0)

Caveat: NOT manifestly Lorentz covariant⇔ heat-bath rest frame!

to lowest order in α: 4παΠµν ' Σ(γ)µν

derivable from underlying thermodynamic potential, Ω!


Vector Mesons and chiral symmetry

vector and axial-vector mesons↔ respective current correlators

ΠµνV/A(p) :=

∫d4x exp(ipx)

⟨JνV/A(0)J

µV/A(x)

⟩ret

Ward-Takahashi Identities of χ symmetry⇒Weinberg-sum rules

f 2π = −

∫ ∞

0

dp20

πp20[Im ΠV(p0, 0)− Im ΠA(p0, 0)]

spectral functions of vector (e.g. ρ) and axial vector (e.g. a1) directlyrelated to order parameter of chiral symmetry!


Vector Mesons and chiral symmetry

0 1 2 3

s [GeV2]

0

0.02

0.04

0.06

0.08

-Im

ΠV

,A/(

πs)

[d

im.-

less

] V [τ −> 2nπ ντ]

A [τ −> (2n+1)π ντ]

ρ(770) + cont.

a1(1260) + cont.

from [Rap03]

Mass

Spec

tral

F

unct

ion

"a1"

"ρ"

pert. QCD

Dropping Masses?

Mass

Sp

ectr

al

Fu

nct

ion

"a1"

"ρ"pert. QCD

Melting Resonances?

from [Rap05]


Chiral-symmetry constraints

different realizations of chiral symmetryequivalent only on shell (“low-energy theorems”)model-independent conclusions only in low-temperature/density limit(chiral perturbation theory) or from lattice-QCD calculationsQCD sum rules (see Lect. I):allow for dropping-mass or melting-resonance scenariouse phenomenological hadronic many-body theory (HMBT) to assessmedium modifications of vector mesons

build models with hadrons as effective degrees of freedombased on (chiral) symmetriesconstrained by data on cross sections, branching ratios,... in vacuumin-medium properties assessed by many-body (thermal) field theory


Example: vector-meson dominance model

early model for electromagnetic interaction of charged pions[Sak60, KLZ67, GS68, Her92, Hee00]

QED like U(1)-gauge model with massive vector meson for ρ0 and π±

Stuckelberg: introduce auxiliary scalar field for free vector mesons:

Lρ = −14

VµνVµν +12

m2VµVµ +12(∂µ ϕ)(∂µ ϕ) + mϕ∂µVµ

gauge invariant under local transformation

δVµ(x) = ∂µχ(x), δϕ = mχ(x)

usual way of gauge fixing using gauge condition

∂µVµ = −ξmϕ

effective Lagrangian of free ρ meson, Stuckelberg and FP ghosts

Lρ,gf = −14

VµνVµν +m2

VµVµ − 12ξ

(∂µVµ)2 +12(∂µ ϕ)(∂µ ϕ)− ξm2

2ϕ2

+ (∂µη∗)(∂µη)− ξm2η∗η



so far: free ρ meson and free ghostsghosts only relevant for ideal gas thermodynamics

Vµ: four bosonic field degrees(3 transverse with mass m, 1 longitudial with mass

√ξm)

ϕ: 1 bosonic Stuckelberg ghost with mass√

ξmη∗, η: 2 pseudofermionic Faddeev Popov fields with mass

√ξm

in partition sum: 3 bosons with mass m + 2 bosons with mass√

ξm − 2 FPghosts with mass

√ξm⇒ effectively three bosons with mass m

partition sum independent of gauge parameter, ξ!ξ → ∞: “unitary gauge”→ only three bosonic ρ-degrees of freedom!

Coupling to pions: obey gauge invariance! (like scalar QED)

Lπ = (Dµπ)∗(Dµπ)−m2π |π|2 −

λ

8|π|4

Dµ = ∂µ + igVµ; g: ρππ coupling



add photons: Dµ = ∂µ + igVµ + ieAµ

Lagrangian for photons: usual gauge fixed QEDadditional direct ργ mixing [KLZ67]

Lργ = − e2gργ

VµνAµν

classical field equations: ⇒ electromagnetic current

jνem = ∂µAµν = ie(

1− ggργ

)π←→D νπ∗ +

egργ

m2Vµ +e2

g2ργ

∂µAµν

for gργ = g: jνem = eg m2Vν +O(e2): ⇒ “vector-meson dominance”



calculate ρ selfenegy

transversality from gauge invariance:

Σµνρ (q) =

(q2gµν − qµqν

)Σ(q2)

electromagnetic form factor of pions



fit to observables: em. form factor of π

best fit: g = 5.683, gργ = 5.171, mρ = 765 MeV/c2

strict VMD: g = gργ = 5.38, mρ = 770 MeV/c2

data: [B+85]



ππ → ππ phase shift in I = 1 channel

δ11 = arccos

Re Gρ

|Gρ|

data: [FP77]



ππ → ππ total cross section

π+ π−

ρ0

π−π+

+

π+

π+ ρ0

π−

π−data: [FP77]


Dalitz decays

V

γ∗

π

ℓ− ℓ+

γ∗

ℓ− ℓ+

P, S

γ

γ∗

ℓ−

γ

R

ℓ+

Dalitz decay:1 particle→ 3 particlesV: ω → π + γ∗ → π + `+ + `−

P, S:π, η → γ + γ∗ → γ + `+ + `−

R: Baryon resonances∆, N∗ → N + V → N + γ∗ →N + `+ + `−

vector-meson dominance

≡

hadronic

γ∗

hadronic

γ∗

V


The GiBUU Model

Boltzmann-Uehling-Uhlenbeck (BUU) framework for hadronic transportreaction types: pA, πA, γA, eA, νA, AAopen-source modular Fortran 95/2003 codeversion control via Subversionpublicly available realeases: http://gibuu.physik.uni-giessen.deReview on hadronic transport (GiBUU): [BGG+12]


http://gibuu.physik.uni-giessen.de

The Boltzmann-Uehling-Uhlenbeck Equation

time evolution of phase-space distribution functions

[∂t + (~∇pHi) · ~∇x − (~∇xHi) · ~∇p]fi(t,~x,~p) = Icoll[f1, . . . , fi, . . . , fj]

Hamiltonian Hi

selfconsistent hadronic mean fields, Coulomb potential,“off-shell potential”

collision term Icoll

two- and three-body decays/collisionsmultiple coupled-channel problemresonances described with relativistic Breit-Wigner distribution

A(x, p) = − 1π

Im Π(p2 −M2 − Re Π)2 + (Im Π)2 ; Im Π = −

√p2Γ

off-shell propagation: test particles with off-shell potential


Resonance Model

reactions dominated by resonance scattering: ab→ R→ cdBreit-Wigner cross-section formula

σab→R→cd =2sR + 1

(2sa + 1)(2sb + 1)4π

p2lab

sΓab→RΓR→cd

(s−m2R)

2 + sΓ2tot

applicable for low-energy nuclear reactions Ekin . 1.1 GeVexample: σπ−p→π−p [Teis (PhD thesis 1996), data: Baldini et al, Landolt-Bornstein 12 (1987)]

0.0 0.2 0.4 0.6 0.8 1.0 1.20

10

20

30

40

50

60

70

80

Daten

total

∆(1232)

N(1440)

N(1535)

plab (GeV/c)

σ (

mb

)


Resonance Model

further cross sections


Extension to HADES energies

[WHM12]

keep same resonances (parameters from Manley analysis)

production channels in Teis: NN → N∆, NN → NN∗, N∆∗, NN → ∆∆extension to NN → ∆N∗, ∆∆∗, NN → NNπ,NN → NNρ, NNω, NNπω, NNφ,NN → BYK (B = N, ∆, Y = Λ, Σ)


Extension to HADES energies

good description of total pp, pn (inelastic) cross section

dilepton sourcesDalitz decays: π0, η → γ`+`−; ω → π0`+`−, ∆→ N`+`−

ρ, ω, φ→ `+`−: invariant mass `+`− spectra⇒spectral properties of vector mesonsfor details, see [WHM12]


p p at HADES (Ekin = 3.5 GeV)

10-3

10-2

10-1

100

101

0 0.2 0.4 0.6 0.8 1

dσ

/dm

ee [

µb/G

eV

]

dilepton mass mee [GeV]

p + p at 3.5 GeV

dataGiBUU total

ρ → e+e

-

ω → e+e

-

φ → e+e

-

ω → π0e

+e

-

π0 → e

+e

-γ

η → e+e

-γ

∆ → Ne+e

-


p p at HADES (Ekin = 3.5 GeV)

10-3

10-2

10-1

100

101

0 0.2 0.4 0.6 0.8 1

dσ

/dp

T [

µb/G

eV

]

0 0.2 0.4 0.6 0.8 1

transverse momentum pT [GeV]

0 0.2 0.4 0.6 0.8 1

10-3

10-2

10-1

100

101

0 0.2 0.4 0.6 0.8 1

dσ

/dp

T [

µb/G

eV

]

dataGiBUU total

ρ → e+e

-

ω → e+e

-

φ → e+e

-

ω → π0e

+e

-

π0 → e

+e

-γ

η → e+e

-γ

∆ → Ne+e

-

10-4

10-3

10-2

10-1

100

0 0.5 1 1.5 2

dσ

/dy [

µb]

m < 150 MeV

0 0.5 1 1.5 2

rapidity y

150 MeV < m < 470 MeV

0 0.5 1 1.5 2

470 MeV < m < 700 MeV

0 0.5 1 1.5 2

10-4

10-3

10-2

10-1

100

dσ

/dy [

µb]

700 MeV < m


“ρ meson” in pp

production through hadron resonancesNN → NR→ NNρ, NN → N∆→ NNπρ

0

200

400

600

800

1000

1200

1400

1600

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

dσ

/dm

[µ

b/G

eV

]

m [GeV]

ρ totalD13(1520)S11(1535)S11(1650)F15(1680)P13(1720)S31(1620)D33(1700)F35(1905)

Pythia

10-3

10-2

10-1

100

0 0.2 0.4 0.6 0.8 1

dσ

/dm

ee [

µb

/Ge

V]


ρ → e+e

-

D13(1520)S11(1535)S11(1650)F15(1680)P13(1720)

all Res.Pythia

S31(1620)D33(1700)F35(1905)

“ρ”-line shape “modified” already in elementary hadronic reactionsdue to production mechanism via resonances


p Nb at HADES (3.5 GeV)

medium effects built in transport modelbinding effects, Fermi smearing, Pauli blockingfinal-state interactionsproduction from secondary collisions

sensitivity on medium effects of vector-meson spectral functions?p + Nb at 3.5 GeV

10-1

100

0.55 0.6 0.65 0.7 0.75 0.8 0.85

dσ

/dm

ee [

µb

/Ge

V]

prelim. datavacCB

CB+shiftshift

0.55 0.6 0.65 0.7 0.75 0.8 0.85


ρ

0.55 0.6 0.65 0.7 0.75 0.8 0.85

ω


Comparison to old DLS data (pp)

HADES data consistent with DLS datachecked by comparing HADES data within DLS acceptance

DLS: p+p

10-3

10-2

10-1

100

dσ

/dm

ee [

µb

/Ge

V]

1.04 GeV

dataGiBUU total

ρ → e+e

-

ω → e+e

-

φ → e+e

-

ω → π0e

+e

-

π0 → e

+e

-γ

η → e+e

-γ

∆ → Ne+e

-

1.27 GeV

10-3

10-2

10-1

100

1.61 GeV

10-3

10-2

10-1

100

0 0.2 0.4 0.6 0.8 1 1.2

dσ

/dm

ee [

µb

/Ge

V]


1.85 GeV

0 0.2 0.4 0.6 0.8 1 1.2


2.09 GeV

0 0.2 0.4 0.6 0.8 1 1.2

10-3

10-2

10-1

100


4.88 GeV


Comparison to old DLS data (pd)

HADES data consistent with DLS datachecked by comparing HADES data within DLS acceptance

DLS: p+d

10-3

10-2

10-1

100

dσ

/dm

ee [

µb

/Ge

V]

1.04 GeV

dataGiBUU total

ρ → e+e

-

ω → e+e

-

φ → e+e

-

ω → π0e

+e

-

π0 → e

+e

-γ

η → e+e

-γ

∆ → Ne+e

-

pn Brems.1.27 GeV

10-3

10-2

10-1

100

1.61 GeV

10-3

10-2

10-1

100

0 0.2 0.4 0.6 0.8 1 1.2

dσ

/dm

ee [

µb

/Ge

V]


1.85 GeV

0 0.2 0.4 0.6 0.8 1 1.2


2.09 GeV

0 0.2 0.4 0.6 0.8 1 1.2

10-3

10-2

10-1

100


4.88 GeV


Conclusions and Outlook

dilepton spectra⇔ in-medium em. current correlatoreffective hadronic models for dilepton sources

vector-meson dominance model (VMD)low-mass region 0 ≤ M . 1 GeV: j(had)µ

em ∝ Vµ (V ∈ ρ, ω, φ)direct relation between dilepton signal and VM spectral functionsinteractions with mesons and baryonsmodels constrained by phenomenology in pp, pn, pAmedium modifications predicted by finite-temperature QFT

Elementary reactions at SIS energiesGiBUU for pp, pn with resonance model for all HADES energiespn still a problem?p Nb, AA work in progress


Bibliography I

[B+85] L. M. Barkov, et al., Electromagnetic Pion Form Factor in theTimelike Region, Nucl. Phys. B 256 (1985) 365.

[BGG+12] O. Buss, et al., Transport-theoretical Description of NuclearReactions, Phys. Rept. 512 (2012) 1.http://dx.doi.org/10.1016/j.physrep.2011.12.001

[BK84] M. Bando, T. Kugo, Is the ρ Meson a Dynamical Gauge Boson ofHidden Local Symmetry, Phys. Rev. Lett. 54 (1984) 1215.http://link.aps.org/abstract/PRL/v54/p1215

[FP77] C. D. Frogatt, J. L. Petersen, Phase-Shift Analysis of π+π−

Scattering between 1.0 and 1.8 GeV Based on Fixed TransferAnalyticity (II), Nucl. Phys. B 129 (1977) 89.

[GS68] G. J. Gounaris, J. J. Sakurai, Finite-Width Corrections to theVector-Meson-Dominance Prediction for ρ eˆ+ eˆ-, Phys. Rev.Lett. 21 (1968) 244.http://link.aps.org/doi/10.1103/PhysRevLett.21.244


http://dx.doi.org/10.1016/j.physrep.2011.12.001

http://link.aps.org/abstract/PRL/v54/p1215

http://link.aps.org/doi/10.1103/PhysRevLett.21.244

Bibliography II

[Hee00] H. van Hees, Renormierung selbstkonsistenter Naherungen in derQuantenfeldtheorie bei endlichen Temperaturen, Ph.D. thesis, TUDarmstadt (2000).http://fias.uni-frankfurt.de/~hees/publ/doc.pdf

[Her92] M. Herrmann, Eigenschaften des ρ-Mesons in dichterKernmaterie, Dissertation, Technische Hochschule Darmstadt,Darmstadt (1992).http://www-lib.kek.jp/cgi-bin/img_index?200038480

[HY03] M. Harada, K. Yamawaki, Hidden local symmetry at loop: A newperspective of composite gauge boson and chiral phase transition,Phys. Rept. 381 (2003) 1.http://dx.doi.org/10.1016/S0370-1573(03)00139-X

[KLZ67] N. M. Kroll, T. D. Lee, B. Zumino, Neutral Vector Mesons and theHadronic Electromagnetic Current, Phys. Rev. 157 (1967) 1376.http://link.aps.org/abstract/PR/v157/i5/p1376


http://fias.uni-frankfurt.de/~hees/publ/doc.pdf

http://www-lib.kek.jp/cgi-bin/img_index?200038480

http://dx.doi.org/10.1016/S0370-1573(03)00139-X

http://link.aps.org/abstract/PR/v157/i5/p1376

Bibliography III

[LSY95] S. H. Lee, C. Song, H. Yabu, Photon - vector meson coupling andvector meson properties at low temperature pion gas, Phys. Lett. B341 (1995) 407.http:

//www.sciencedirect.com/science?_ob=GatewayURL&_origin=

SPIRES&_method=citationSearch&_volkey=03702693%23341%

23407&_version=1&md5=05053a52e85b02fde34213175c490b2a

[Mei88] U. G. Meissner, Low-Energy Hadron Physics from Effective ChiralLagrangians with Vector Mesons, Phys. Rept. 161 (1988) 213.http://dx.doi.org/10.1016/0370-1573(88)90090-7

[Pis95] R. D. Pisarski, Where does the ρ go? Chirally symmetric vectormesons in the quark - gluon plasma, Phys. Rev. D 52 (1995) 3773.http://dx.doi.org/10.1103/PhysRevD.52.R3773

[Rap03] R. Rapp, Dileptons in high-energy heavy-ion collisions, Pramana60 (2003) 675.http://dx.doi.org/10.1007/BF02705167


http://www.sciencedirect.com/science?_ob=GatewayURL&_origin=SPIRES&_method=citationSearch&_volkey=03702693%23341%23407&_version=1&md5=05053a52e85b02fde34213175c490b2a




http://dx.doi.org/10.1016/0370-1573(88)90090-7

http://dx.doi.org/10.1103/PhysRevD.52.R3773

http://dx.doi.org/10.1007/BF02705167

Bibliography IV

[Rap05] R. Rapp, The vector probe in heavy-ion reactions, J. Phys. G 31(2005) S217.http://arxiv.org/abs/nucl-th/0409054

[RW99] R. Rapp, J. Wambach, Low mass dileptons at the CERN-SPS:Evidence for chiral restoration?, Eur. Phys. J. A 6 (1999) 415.http://dx.doi.org/10.1007/s100500050364

[RW00] R. Rapp, J. Wambach, Chiral symmetry restoration and dileptonsin relativistic heavy-ion collisions, Adv. Nucl. Phys. 25 (2000) 1.http://arxiv.org/abs/hep-ph/9909229

[Sak60] J. J. Sakurai, Theory of strong interactions, Ann. Phys. (NY) 11(1960) 1.http://dx.doi.org/10.1016/0003-4916(60)90126-3


http://arxiv.org/abs/nucl-th/0409054

http://dx.doi.org/10.1007/s100500050364

http://arxiv.org/abs/hep-ph/9909229

http://dx.doi.org/10.1016/0003-4916(60)90126-3

Bibliography V

[UBW02] M. Urban, M. Buballa, J. Wambach, Temperature dependence of ρand a1 meson masses and mixing of vector and axial-vectorcorrelators, Phys. Rev. Lett. 88 (2002) 042002.http://dx.doi.org/10.1103/PhysRevLett.88.042002

[WHM12] J. Weil, H. van Hees, U. Mosel, Dilepton production inproton-induced reactions at SIS energies with the GiBUUtransport model (2012).http://arxiv.org/abs/arXiv:1203.3557


http://dx.doi.org/10.1103/PhysRevLett.88.042002

http://arxiv.org/abs/arXiv:1203.3557

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