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
Hendrik van Hees (FIAS Frankfurt) Em. Probes in HICs II September 5, 2012 2 / 33
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])
Hendrik van Hees (FIAS Frankfurt) Em. Probes in HICs II September 5, 2012 3 / 33
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!
Hendrik van Hees (FIAS Frankfurt) Em. Probes in HICs II September 5, 2012 4 / 33
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!
Hendrik van Hees (FIAS Frankfurt) Em. Probes in HICs II September 5, 2012 5 / 33
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]
Hendrik van Hees (FIAS Frankfurt) Em. Probes in HICs II September 5, 2012 6 / 33
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
Hendrik van Hees (FIAS Frankfurt) Em. Probes in HICs II September 5, 2012 7 / 33
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
Hendrik van Hees (FIAS Frankfurt) Em. Probes in HICs II September 5, 2012 8 / 33
Example: vector-meson dominance model
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
Hendrik van Hees (FIAS Frankfurt) Em. Probes in HICs II September 5, 2012 9 / 33
Example: vector-meson dominance model
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”
Hendrik van Hees (FIAS Frankfurt) Em. Probes in HICs II September 5, 2012 10 / 33
Example: vector-meson dominance model
calculate ρ selfenegy
transversality from gauge invariance:
Σµνρ (q) =
(q2gµν − qµqν
)Σ(q2)
electromagnetic form factor of pions
Hendrik van Hees (FIAS Frankfurt) Em. Probes in HICs II September 5, 2012 11 / 33
Example: vector-meson dominance model
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]
Hendrik van Hees (FIAS Frankfurt) Em. Probes in HICs II September 5, 2012 12 / 33
Example: vector-meson dominance model
ππ → ππ phase shift in I = 1 channel
δ11 = arccos
Re Gρ
|Gρ|
data: [FP77]
Hendrik van Hees (FIAS Frankfurt) Em. Probes in HICs II September 5, 2012 13 / 33
Example: vector-meson dominance model
ππ → ππ total cross section
π+ π−
ρ0
π−π+
+
π+
π+ ρ0
π−
π−data: [FP77]
Hendrik van Hees (FIAS Frankfurt) Em. Probes in HICs II September 5, 2012 14 / 33
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
Hendrik van Hees (FIAS Frankfurt) Em. Probes in HICs II September 5, 2012 15 / 33
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]
Hendrik van Hees (FIAS Frankfurt) Em. Probes in HICs II September 5, 2012 16 / 33
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
Hendrik van Hees (FIAS Frankfurt) Em. Probes in HICs II September 5, 2012 17 / 33
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
)
Hendrik van Hees (FIAS Frankfurt) Em. Probes in HICs II September 5, 2012 18 / 33
Resonance Model
further cross sections
Hendrik van Hees (FIAS Frankfurt) Em. Probes in HICs II September 5, 2012 19 / 33
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)
Hendrik van Hees (FIAS Frankfurt) Em. Probes in HICs II September 5, 2012 20 / 33
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]
Hendrik van Hees (FIAS Frankfurt) Em. Probes in HICs II September 5, 2012 21 / 33
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
-
Hendrik van Hees (FIAS Frankfurt) Em. Probes in HICs II September 5, 2012 22 / 33
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
Hendrik van Hees (FIAS Frankfurt) Em. Probes in HICs II September 5, 2012 23 / 33
“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]
dilepton mass mee [GeV]
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
Hendrik van Hees (FIAS Frankfurt) Em. Probes in HICs II September 5, 2012 24 / 33
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
dilepton mass mee [GeV]
ρ
0.55 0.6 0.65 0.7 0.75 0.8 0.85
ω
Hendrik van Hees (FIAS Frankfurt) Em. Probes in HICs II September 5, 2012 25 / 33
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]
dilepton mass mee [GeV]
1.85 GeV
0 0.2 0.4 0.6 0.8 1 1.2
dilepton mass mee [GeV]
2.09 GeV
0 0.2 0.4 0.6 0.8 1 1.2
10-3
10-2
10-1
100
dilepton mass mee [GeV]
4.88 GeV
Hendrik van Hees (FIAS Frankfurt) Em. Probes in HICs II September 5, 2012 26 / 33
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]
dilepton mass mee [GeV]
1.85 GeV
0 0.2 0.4 0.6 0.8 1 1.2
dilepton mass mee [GeV]
2.09 GeV
0 0.2 0.4 0.6 0.8 1 1.2
10-3
10-2
10-1
100
dilepton mass mee [GeV]
4.88 GeV
Hendrik van Hees (FIAS Frankfurt) Em. Probes in HICs II September 5, 2012 27 / 33
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
Hendrik van Hees (FIAS Frankfurt) Em. Probes in HICs II September 5, 2012 28 / 33
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
Hendrik van Hees (FIAS Frankfurt) Em. Probes in HICs II September 5, 2012 29 / 33
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
Hendrik van Hees (FIAS Frankfurt) Em. Probes in HICs II September 5, 2012 30 / 33
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
Hendrik van Hees (FIAS Frankfurt) Em. Probes in HICs II September 5, 2012 31 / 33
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
Hendrik van Hees (FIAS Frankfurt) Em. Probes in HICs II September 5, 2012 32 / 33
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
Hendrik van Hees (FIAS Frankfurt) Em. Probes in HICs II September 5, 2012 33 / 33