Hendrik van Hees - University of Texas at Austin

Meson and Baryon Spectral Functionsand Dileptons

Hendrik van Hees

Johann Wolfgang Goethe University Frankfurt

March 07, 2012

Hendrik van Hees (GU Frankfurt) Hadron resonances March 07, 2012 1 / 27

Outline

1 Motivation for electromagnetic probes ↔ hadron resonances

2 Resonance model at SIS energies (with J. Weil, U. Mosel)The Transport Model GiBUUBaryon-resonance model at SIS energiesDileptons in pp and pNb reactions at HADES

3 Dilepton production at the SPS (with R. Rapp)Hadronic many-body theoryDilepton emission from thermal and nonthermal sourcesComparison to NA60 data

4 Medium modfications of the ∆ (with R. Rapp)

5 Conclusions and Outlook


Electromagnetic probes in heavy-ion collisionsγ, `±: no strong interactions

reflect whole “history” of collision:

from pre-equilibrium phasefrom thermalized mediumQGP and hot hadron gasfrom VM decays after thermalfreezeout

ρ/ω γ ∗ e−π, . . .

e+

0 1 2 3 4 5

M [GeV/c]

J/

Drell−Yan

DD

Low− Intermediate− High−Mass Region> 10 fm > 1 fm < 0.1 fm

ψ ’

π0, η Dalitz decays

dN

/(d

y d

m)

ρ/ω

Φ

Ψ

[Fig. by A. Drees]

vacuum and in-mediumhadron properties needed!

[R. Rapp, J. Wambach, HvH, Landoldt-Bornstein, I/23, 4-1 (2010), arXiv: 0901.3289 [hep-ph] ]


Electromagnetic Probes and Vector Mesons

`+`− thermal emission rates ⇔ em. current-correlation function, Πµν

[L. McLerran, T. Toimela 85, H. A. Weldon 90, C. Gale, J.I. Kapusta 91]

dNe+e−

d4xd4q= −gµν α2

3q2π3Im Π(ret)

µν (q)∣∣∣q2=M2

e+e−fB(q0)

vector-meson dominance model:

Πµν =Gρ

γ∗ γ∗

hadronic many-body theory for vector mesons

ρ ρ

π

π B , a , K ,...*1 1

π,...N, K,

ρ ρ

elementary processes ⇔ cut self-energy diagrams


The GiBUU Model

Boltzmann-Uehling-Uhlenbeck (BUU) framework for hadronictransport

reaction types: pA, πA, γA, eA, νA, AA

open-source modular Fortran 95/2003 code

version control via Subversion

publicly available realeases:http://gibuu.physik.uni-giessen.de

Review: [O. Buss et al, Phys. Rept. 512, 1 (2012)]


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) = − 1

π

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 → cd

Breit-Wigner cross-section formula

σab→R→cd =2sR + 1

(2sa + 1)(2sb + 1)

4π

p2lab

sΓab→RΓR→cd

(s −m2R)2 + sΓ2

tot

applicable for low-energy nuclear reactions Ekin . 1.1 GeV

example: 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

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 mesons


p p at HADES (Ekin = 3.5 GeV)


“ρ meson” in pp

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

“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 model

binding effects, Fermi smearing, Pauli blockingfinal-state interactionsproduction from secondary collisions

sensitivity on medium effects of vector-meson spectral functions?


Dileptons at the SPS: Hadronic many-body theory

radiation from thermal sources: Hadronic many-body theory

[R. Rapp, J. Wambach 99]

baryon effects importantnB+nB relevant quantity (not net-baryon density)!

ρ ρ

π

π B , a , K ,...*1 1

π,...N, K,

ρ ρ


Sources of dilepton emission in heavy-ion collisions

1 initial hard processes: Drell Yan

2 “core” ⇔ emission from thermal source [McLerran, Toimela 1985]

1

qT

dN(thermal)

dMdqT=

∫d4x

∫dy

∫Mdϕ

dN(thermal)

d4xd4qAcc(M, qT , y)

use cylindrical thermal fireball with QGP, mixed and hadronic phase

3 “corona” ⇔ emission from “primordial” mesons (jet-quenching)

4 after thermal freeze-out ⇔ emission from “freeze-out” mesons[Cooper, Frye 1975]

N(fo) =

∫d3q

q0

∫qµdσ

µfB(uµqµ/T )

Γmeson→`+`−

ΓmesonAcc


M spectra (in pT slices)

norm corrected by ∼3% due to centrality correction(min-bias data: 〈Nch〉 = 120, calculation Nch = 140)

10-9

10-8

10-7

10-6

10-5

0.2 0.4 0.6 0.8 1 1.2 1.4

(1/N

ch)

d2N

µµ/(

dM

dη

) (2

0 M

eV)-1

M (GeV)

all qT

Tc=Tch=175 MeV, at=0.1 c2/fm

+ω-t extotal

ω-t exωφ

DY4π mix

FO ρprim ρ

QGPin-med ρ

NA60




10-9

10-8

10-7

10-6

10-5

0.2 0.4 0.6 0.8 1 1.2 1.4

(1/N

ch)

d2N

µµ/(

dM

dη

) (2

0 M

eV)-1

M (GeV)

0<qt<0.2 GeV


+ω-t extotal

ω-t exωφ

DY4π mix

FO ρprim ρ

QGPin-med ρ

NA60




10-9

10-8

10-7

10-6

10-5

0.2 0.4 0.6 0.8 1 1.2 1.4

(1/N

ch)

d2N

µµ/(

dM

dη

) (2

0 M

eV)-1

M (GeV)

0.2 GeV<qt<0.4 GeV


+ω-t extotal

ω-t exωφ

DY4π mix

FO ρprim ρ

QGPin-med ρ

NA60




10-9

10-8

10-7

10-6

10-5

0.2 0.4 0.6 0.8 1 1.2 1.4

(1/N

ch)

d2N

µµ/(

dM

dη

) (2

0 M

eV)-1

M (GeV)

0.4 GeV<qt<0.6 GeV


+ω-t extotal

ω-t exωφ

DY4π mix

FO ρprim ρ

QGPin-med ρ

NA60




10-9

10-8

10-7

10-6

10-5

0.2 0.4 0.6 0.8 1 1.2 1.4

(1/N

ch)

d2N

µµ/(

dM

dη

) (2

0 M

eV)-1

M (GeV)

0.6 GeV<qt<0.8 GeV


+ω-t extotal

ω-t exωφ

DY4π mix

FO ρprim ρ

QGPin-med ρ

NA60




10-9

10-8

10-7

10-6

10-5

0.2 0.4 0.6 0.8 1 1.2 1.4

(1/N

ch)

d2N

µµ/(

dM

dη

) (2

0 M

eV)-1

M (GeV)

0.8 GeV<qt<1.0 GeV


+ω-t extotal

ω-t exωφ

DY4π mix

FO ρprim ρ

QGPin-med ρ

NA60




10-9

10-8

10-7

10-6

10-5

0.2 0.4 0.6 0.8 1 1.2 1.4

(1/N

ch)

d2N

µµ/(

dM

dη

) (2

0 M

eV)-1

M (GeV)

1.0 GeV<qt<1.2 GeV


+ω-t extotal

ω-t exωφ

DY4π mix

FO ρprim ρ

QGPin-med ρ

NA60




10-9

10-8

10-7

10-6

10-5

0.2 0.4 0.6 0.8 1 1.2 1.4

(1/N

ch)

d2N

µµ/(

dM

dη

) (2

0 M

eV)-1

M (GeV)

2.0 GeV<qt<2.2 GeV


+ω-t extotal

ω-t exωφ

DY4π mix

FO ρprim ρ

QGPin-med ρ

NA60




10-9

10-8

10-7

10-6

10-5

0.2 0.4 0.6 0.8 1 1.2 1.4

(1/N

ch)

d2N

µµ/(

dM

dη

) (2

0 M

eV)-1

M (GeV)

2.2 GeV<qt<2.4 GeV


+ω-t extotal

ω-t exωφ

DY4π mix

FO ρprim ρ

QGPin-med ρ

NA60


Importance of baryon effects

baryonic interactions important!

in-medium broadening

low-mass tail!

10-9

10-8

10-7

10-6

0.2 0.4 0.6 0.8 1 1.2 1.4

(1/N

ch)

d2N

µµ/(

dM

dη

) (2

0 M

eV)-1

M (GeV)

all qT


full modelno baryons

NA60


Dilepton rates: Hadron gas ↔ QGP

in-medium hadron gas matcheswith QGP

similar results also for γ rates

“quark-hadron duality”!?

consistent with chiral-symmetryrestoration

“resonance melting” rather than“dropping masses”


Medium modfications of the ∆

[HvH, R. Rapp, Phys. Lett. B 606, 59 (2005); J. Phys. G 31, S203 (2005)]

∆

Ν

π

1 1.1 1.2 1.3 1.4 1.5

MπN

[GeV]

0

20

40

60

80

100

120

140

160

180

δ33[d

eg]

Model fitfπN∆

=3.3, Λπ=290 MeV

Data (Arndt et al)

π N scattering phase shift


Medium modifications of pions and nucleons

pions: nucleon and ∆-hole excitations

+++

short-range correlations: Migdal resummation

...+

...+

g’11

g’12

g’22

g’11

= + +

= + +

1

2

nucleons: πN and πB,B=∆(1232), N∗(1440), N∗(1535), ∆∗(1600), ∆∗(1620)coupling constants fitted to partial decay widths B → πN

+

BHendrik van Hees (GU Frankfurt) Hadron resonances March 07, 2012 21 / 27

Medium modifications of pions

0 0.1 0.2 0.3 0.4 0.5k

0 [GeV]

0

10

20

30

40

50

60

70-I

m G

π

[GeV

-2]

T=0

T=180 MeV

T=180, ρN

=0

In-Medium π Spectral Function

ρN

=0.68ρ0

k=0.3GeV


Medium modifications of nucleons

0.6 0.8 1 1.2

MN

[GeV]

0

5

10

15

20

25

30

-Im

GN

[G

eV-1

]

mN

T=100 MeVT=180 MeV

In Medium N-spectral function

p=0.3GeV

T=100 MeV, %N=0.12%0, µπ=96 MeVT=180 MeV, %N=0.68%0, µπ=0Hendrik van Hees (GU Frankfurt) Hadron resonances March 07, 2012 23 / 27

Medium Modifications of the ∆

same diagram as in vacuum with dressed pion- and nucleonpropagators

vertex corrections: same resummed Migdal loops as for the pion

4-fermion vertices: same Migdal parameters as for the pion

+ +

B’

+ +

B ′=∆(1232), N∗(1440), N∗(1520), ∆∗(1600), ∆∗(1620), N∗(1700),∆∗(1700)



1 1.1 1.2 1.3 1.4 1.5 1.6

M∆ [GeV]

0

5

10

15

20

25-I

m G

∆ [G

eV-1

]vacuumT=100, ρ

Ν=0.12, µ

π=96

T=180, ρΝ

=0.68, µπ=0

∆(1232) Spectral Function at RHIC

p=0.3GeV



1 1.1 1.2 1.3 1.4 1.5 1.6

M∆ [GeV]

0

5

10

15

20

25-I

m G

∆

[GeV

-1]

vacuumT=70, ρ

N=0.19, µ

π=105

T=160, ρN

=1.8, µπ=0

∆(1232) Spectral Function at SIS-06

p=0.3GeV


Conclusions and Outlook

dilepton spectra ⇔ in-medium em. current correlator

SIS energies

GiBUU for pp, pn with resonance model for all HADES energiesnp still a problem?p Nb, AA work in progresssimilar study within UrQMD in progress (with S. Endres, M. Bleicher)

SPS and RHIC energies

excess yield dominated by radiation from thermal sourcesbaryons essential for in-medium properties of vector mesonsmelting vector mesons with little mass shift“quark-hadron duality” of `+`− rates around Tc

compatible with chiral symmetry restoration!studies in UrQMD(+hydro hybrid) model (with S. Endres, M. Bleicher)⇒ see talk by Marcus Bleicher

Medium modifications of the ∆


Hendrik van Hees - University of Texas at Austin

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