Dileptons from off-shell Dileptons from off-shell transport approachtransport approach

EElenalena Bratkovskaya Bratkovskaya

5.07.2008 , HADES Collaboration Meeting XIX,5.07.2008 , HADES Collaboration Meeting XIX,GSI, DarmstadtGSI, Darmstadt

OverviewOverview

• Study of Study of in-medium effectsin-medium effects in heavy-ion collisions require: in heavy-ion collisions require: off-shelloff-shell transport dynamics transport dynamics in-medium transition ratesin-medium transition rates time-integration methodstime-integration methods

•BremsstrahlungBremsstrahlung

• HSD results and HSD results and comparison of transport models comparison of transport models

• Elementary channels:Elementary channels:-Dalitz decay-Dalitz decay-Dalitz decay-Dalitz decay

•pp, pnpp, pn and and pdpd reactions vs. new HADES data reactions vs. new HADES data

Dileptons from transport modelsDileptons from transport models

Theory Theory (status: last millenium < 2000)(status: last millenium < 2000) : : Implementation of in-medium vector mesons (Implementation of in-medium vector mesons () scenarios (= ) scenarios (= ‚dropping‘ ‚dropping‘

mass and mass and ‚collisional broadening‘‚collisional broadening‘) in ) in on-shellon-shell transport models:transport models:

BUU/AMPT (Texas) ( > 1995)BUU/AMPT (Texas) ( > 1995)

HSD ( > 1995)HSD ( > 1995)

UrQMD v. 1.3 (1998)UrQMD v. 1.3 (1998)

RQMD (Tübingen) (2003), RQMD (Tübingen) (2003), but NO explicit propagation of vector mesonsbut NO explicit propagation of vector mesons

IQMD (Nantes) (2007), IQMD (Nantes) (2007), but NO explicit propagation of vector mesonsbut NO explicit propagation of vector mesons

Theory Theory (status: this millenium > 2000) :(status: this millenium > 2000) :

Implementation of in-medium vector mesons (Implementation of in-medium vector mesons () scenarios (= ) scenarios (= ‚dropping‘ ‚dropping‘

mass and mass and ‚collisional broadening‘‚collisional broadening‘) in ) in off-shelloff-shell transport models:transport models: HSD (>2000)HSD (>2000)

BRoBUU (Rossendorf) (2006) BRoBUU (Rossendorf) (2006)

0.00.5

1.01.5 0.0

0.5

1.0

1.5

1

3

5

7

9

M (GeV/c2)

B=0

q (G

eV/c

)

0.00.5

1.01.5 0.0

0.5

1.0

1.5

1

2

3

4

M (GeV/c2)

B=

0

q (G

eV/c

)

0.00.5

1.01.5 0.0

0.5

1.0

1.5

1

2

3

4

M (GeV/c2)

B=2

0

q (G

eV/c

)

0.00.5

1.01.5 0.0

0.5

1.0

1.5

1

2

3

4

-Im D (M,q,B,T) (GeV

-2)

T=150 MeV

M (GeV/c2)

B=3

0

q (G

eV/c

)

Changes of the particle properties in the hot and dense Changes of the particle properties in the hot and dense baryonic mediumbaryonic medium

How to treat in-medium effects in transport approaches?

In-medium models:

chiral perturbation theory chiral perturbation theory

chiral SU(3) model chiral SU(3) model

coupled-channel G-matrix approachcoupled-channel G-matrix approach

chiral coupled-channel effective field chiral coupled-channel effective field theory theory

predict changes of the particle predict changes of the particle properties in the hot and dense medium, properties in the hot and dense medium, e.g. e.g. broadening of the spectral function broadening of the spectral function

meson spectral functionmeson spectral function

From Kadanoff-Baym equations to transport equationsFrom Kadanoff-Baym equations to transport equations

drift termdrift term Vlasov termVlasov term

collision term =collision term = ‚loss‘ term -‚loss‘ term -‚gain‘ term‚gain‘ term

Operator <> - 4-dimentional generalizaton of Operator <> - 4-dimentional generalizaton of the Poisson-bracketthe Poisson-bracketbackflow termbackflow term

Generalized transport equations =Generalized transport equations = first order gradient expansion of the Wigner first order gradient expansion of the Wigner transformed Kadanoff-Baym equations:transformed Kadanoff-Baym equations:

The imaginary part of the retarded propagator is given by the normalized The imaginary part of the retarded propagator is given by the normalized spectral function:spectral function:

For bosons in first order gradient expansion:For bosons in first order gradient expansion:XPXP – width of spectral function = – width of spectral function = reaction rate of particle (at phase-reaction rate of particle (at phase-space position XP)space position XP)

W. Cassing et al., NPA 665 (2000) 377; 672 (2000) 417; 677 (2000) 445W. Cassing et al., NPA 665 (2000) 377; 672 (2000) 417; 677 (2000) 445

Backflow termBackflow term incorporates the incorporates the off-shell behavioroff-shell behavior in in the particle propagationthe particle propagation !! vanishes in the quasiparticle limit vanishes in the quasiparticle limit

Greens function SGreens function S< < characterizes the characterizes the number of particles (N) and their number of particles (N) and their properties (A – spectral function )properties (A – spectral function )

General testparticle off-shell equations of motion General testparticle off-shell equations of motion

Employ Employ testparticle Ansatztestparticle Ansatz for the real valued quantity for the real valued quantity ii S S<<XP XP --

insert in generalized transport equationsinsert in generalized transport equations and determine equations of motion ! and determine equations of motion !

General testparticle off-shell equations of motion:General testparticle off-shell equations of motion:

with

Note:Note: the common factor 1/(1-Cthe common factor 1/(1-C(i)(i)) can be absorbed in an ‚eigentime‘ of particle (i) !) can be absorbed in an ‚eigentime‘ of particle (i) !

W. Cassing , S. Juchem, NPA 665 (2000) 377; 672 (2000) 417; 677 (2000) 445W. Cassing , S. Juchem, NPA 665 (2000) 377; 672 (2000) 417; 677 (2000) 445

On-shell limitOn-shell limit

2) Γ(X,P) such that

E.g.: Γ = const

=Γ=Γvacuumvacuum (M)(M)

‚Vacuum‘ spectral function with constant or mass dependent width :spectral function AXP does NOT change the shape (and pole position) during propagation through the medium (backflow term vanishes also!)(backflow term vanishes also!)

1) Γ(X,P) 0

quasiparticle approximation quasiparticle approximation : : A(X,P) = 2 A(X,P) = 2 (P(P22-M-M22))

||||Hamiltons equation of motionHamiltons equation of motion - -

independent on Γ !independent on Γ !Backflow termBackflow term - - which which incorporates the off-shell incorporates the off-shell behavior in the particle behavior in the particle propagation - propagation - vanishes in vanishes in the quasiparticle limit !the quasiparticle limit !

Hamiltons equation of motion - Hamiltons equation of motion - independent on Γ !independent on Γ !

0Γand0Γ PX

• for each particle species for each particle species ii ( (i i = = NN, , RR, , YY, , , , , K, …) the phase-space density f, K, …) the phase-space density fi i followsfollows

the the transport equations transport equations

with collision termswith collision terms IIcoll coll describing elastic and inelastic describing elastic and inelastic hadronic reactions: hadronic reactions:

baryon-baryon, meson-baryon, meson-meson, formation and decay of baryon-baryon, meson-baryon, meson-meson, formation and decay of baryonic and mesonicbaryonic and mesonic resonances, stringresonances, string formation and decay formation and decay (for inclusive particle production:(for inclusive particle production:

BB BB X , mB X , mB X, X =many particles)X, X =many particles)

with with propagationpropagation of particles in self-generated of particles in self-generated mean-field potential mean-field potential U(p,U(p,)~Re()~Re(retret)/2p)/2p00

•Numerical realization – solution of classical equations of motion + Numerical realization – solution of classical equations of motion + Monte-Carlo Monte-Carlo simulationssimulations for test-particle interactions for test-particle interactions

‚‚On-shell‘ transport modelsOn-shell‘ transport models

)f,...,f,(fIt),p,r(fUUt M21colliprrp

Basic concept of the ‚on-shell‘ transport modelsBasic concept of the ‚on-shell‘ transport models (VUU, BUU, QMD etc. ): (VUU, BUU, QMD etc. ):

1)1) Transport equations Transport equations = first order gradient expansion of the Wigner = first order gradient expansion of the Wigner transformed Kadanoff-Baym equations transformed Kadanoff-Baym equations

2)2) Quasiparticle approximation Quasiparticle approximation or/andor/and vacuum spectral functions vacuum spectral functions : : A(X,P) = 2 A(X,P) = 2 (p(p22-M-M22) A) Avacuumvacuum(M) (M)

Short-lived resonances in semi-classical transport modelsShort-lived resonances in semi-classical transport models

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.610-3

10-2

10-1

100

101

102

103

/=5

M [GeV/c2]

spectral function

A(M

)

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.610-3

10-2

10-1

100

101

102

103

/=0

M [GeV/c2]

spectral function

A(M

)

In-mediumIn-medium

Vacuum Vacuum ((narrow narrow

statesstates

In-medium:In-medium:production of broad statesproduction of broad states

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.110-2

10-1

100

101

M [GeV]

bare mass coll. broadening

E=1.5 GeV

208Pb X e+e- Xd/

dM

[b

/GeV

]

BUU: M. Effenberger et al, PRC60 (1999)BUU: M. Effenberger et al, PRC60 (1999)

,

ρ)p,(M,MΓ)ReΣM(M

ρ)p,(M,ΓM

π

2ρ)p,A(M 2

totret2

02

tot2

, widthwidthIm Im retret

Spectral function:Spectral function:

Example : Example : -meson propagation through the medium -meson propagation through the medium within the on-shell BUU modelwithin the on-shell BUU model broad in-medium spectral function does not broad in-medium spectral function does not become on-shell in vacuum in ‚on-shell‘ become on-shell in vacuum in ‚on-shell‘ transport models!transport models!

Off-shell vs. on-shell transport dynamicsOff-shell vs. on-shell transport dynamics

010

2030

40

0.2

0.4

0.6

0.8

1.0

0.00.2

0.40.6

0.81.0

010

2030

40

0.2

0.4

0.6

0.8

1.0

0.00.2

0.40.6

0.81.0

010

2030

40

0.2

0.4

0.6

0.8

1.0

1.2

0.00.2

0.40.6

0.81.0

010

2030

40

0.2

0.4

0.6

0.8

1.0

1.2

0.00.2

0.40.6

0.81.0

C+C, 2.0 A GeV, b=1 fmdropp. mass + coll. broad.

dN/d

M [

a.u.

]

M [GeV/c2 ]

time [fm/c]

-meson off-shell

-meson on-shell

dN/d

M [

a.u.

]

M [GeV/c2 ]

time [fm/c]

-meson on-shell

dN/d

M [

a.u.

]

M [GeV/c2 ]

time [fm/c]

-meson off-shell

dN/d

M [

a.u.

]

M [GeV/c2 ]

time [fm/c]

The off-shell spectral function becomes on-shell in the vacuum dynamically The off-shell spectral function becomes on-shell in the vacuum dynamically by propagation through the medium!by propagation through the medium!

Time evolution of the mass distribution of Time evolution of the mass distribution of andandmesons for central C+C mesons for central C+C collisions (b=1 fm) at 2 A GeV for collisions (b=1 fm) at 2 A GeV for dropping mass + collisional broadening scenariodropping mass + collisional broadening scenario

E.L.B. &W. Cassing, NPA 807 (2008) 214E.L.B. &W. Cassing, NPA 807 (2008) 214

On-shell model:On-shell model:low mass low mass

andandmesons live mesons live forever and shine forever and shine

dileptons!dileptons!

Collision term in off-shell transport modelsCollision term in off-shell transport models

Collision termCollision term for reaction 1+2->3+4: for reaction 1+2->3+4:

withwith

The trace over particles 2,3,4 reads explicitly The trace over particles 2,3,4 reads explicitly for fermionsfor fermions for bosonsfor bosons

The transport approach and the particle spectral functions are fully The transport approach and the particle spectral functions are fully determined once the determined once the in-medium transition amplitudes Gin-medium transition amplitudes G are known are known

in their in their off-shell dependenceoff-shell dependence!!

additional integrationadditional integration

Spectral function in off-shell transport modelSpectral function in off-shell transport model

Assumptions used in transport model (to speed up calculations):Assumptions used in transport model (to speed up calculations):

• Collisional widthCollisional widthin low density approximation:in low density approximation:CollColl(M,p,(M,p,) = ) = VNVNtottot

• replace replace VNVNtottot by averaged value G=const: by averaged value G=const: CollColl(M,p,(M,p,) = ) = GG

Collisional widthCollisional width of the particle in the rest frame of the particle in the rest frame (keep only loss term in eq.(1)):(keep only loss term in eq.(1)):

,

MΓ)ReΣM(M

ΓM

π

2A(M) 2

totret2

02

tot2

Spectral function:Spectral function:

Collisional width is defined by all possible interactions in the local cellCollisional width is defined by all possible interactions in the local cell

total width:total width:tottot==vacvac++CollColl

withwith

Modelling of in-medium spectral functions for Modelling of in-medium spectral functions for vector mesonsvector mesons

In-medium scenarios:In-medium scenarios:

dropping mass collisional broadening dropping mass + coll. dropping mass collisional broadening dropping mass + coll. broad.broad.

m*=mm*=m00(1-(1-) ) (M,(M,)=)=vacvac(M)+(M)+CBCB(M,(M,) m* & ) m* & CBCB(M,(M,Collisional widthCollisional widthCBCB(M,(M,) = ) = VNVNtottot

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.010-2

10-1

100

101

102

/ 0 1 2 3 5

M [GeV/c2]

dropping mass

A(M

)

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.010-3

10-2

10-1

100

101

102

/ 0 1 2 3 5

M [GeV/c2]

collisional broadeningA

(M)

meson spectral function:meson spectral function:

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.010-3

10-2

10-1

100

101

102

M [GeV/c2]

/ 0 1 2 3 5

dropping mass + collisional broadening

A(M

)

• Note:Note: for a consistent off-shell transportfor a consistent off-shell transport one one needsneeds not only in-medium not only in-medium spectral functions but also spectral functions but also in-medium transition ratesin-medium transition rates for all channels with for all channels with vector mesons, i.e. the full knowledge of the vector mesons, i.e. the full knowledge of the in-medium off-shell cross in-medium off-shell cross sections sections (s,(s,))

E.L.B., NPA 686 (2001), E.L.B. &W. Cassing, NPA 807 (2008) 214E.L.B., NPA 686 (2001), E.L.B. &W. Cassing, NPA 807 (2008) 214

Modelling of in-medium off-shell production Modelling of in-medium off-shell production cross sections for vector mesonscross sections for vector mesons

• Low energy BB and mB interactions Low energy BB and mB interactions (s (s ½ ½ < 2.2 GeV)< 2.2 GeV)

11 2 3 4 5 6 7 8 9 101010-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

10011 2 3 4 5 6 7 8 9 1010

10-4

10-3

10-2

10-1

100

101

11 2 3 4 5 6 7 8 9 101010-4

10-3

10-2

10-1

100

101

mN+2m

N->N

s1/2 [GeV]

mN+2m

mN+3m

N->N

(s)

[m

b]

(s)

[m

b]

-p->0p

+p->+p

string threshold

N->N

(s)

[m

b]

free, =0 drop. mass, =3

0

col. broad., =30

drop.mass +col. broad., =30

ρ)A(M,

dM

Ms,dσdM~(s)σ

ρNπNM

M

ρNπNmax

min

• High energy BB and mB interactions High energy BB and mB interactions (s (s ½ ½ > 2.2 GeV)> 2.2 GeV)

New in HSD:New in HSD: implementation of the in-medium implementation of the in-medium spectral functions A(M,spectral functions A(M,) for broad ) for broad resonancesresonances inside FRITIOF inside FRITIOF

Originally in FRITIOF (PYTHIA/JETSET): Originally in FRITIOF (PYTHIA/JETSET): A(M) with constant width around the pole A(M) with constant width around the pole mass Mmass M00

E.L.B. &W. Cassing, NPA 807 (2008) 214E.L.B. &W. Cassing, NPA 807 (2008) 214

Time integration method for dileptons Time integration method for dileptons

tt00 ttabsabs

timetime FF

F – F – final time of computation in the codefinal time of computation in the codett00 – production time – production timettabsabs – absorption (or hadronic decay) time – absorption (or hadronic decay) time

ee--

e+e+

ee--

e+e+

‚‚Reality‘:Reality‘:

‚‚Virtual‘ – time integ. method:Virtual‘ – time integ. method:

only only ONE e+e- pairONE e+e- pair with probability ~with probability ~Br(Br(->e+e-)=4.5->e+e-)=4.5 . .1010-5-5

Calculate probability P(t) to Calculate probability P(t) to emitemit anan e+e- pair e+e- pair at at each time teach time t and integrate P(t) over time!and integrate P(t) over time!: t: t0 0 < t < t< t < tabsabs

: t: t00 <t < <t < infinityinfinity

Cf. G.Q. Li & C.M. Ko, NPA582 (1995) 731Cf. G.Q. Li & C.M. Ko, NPA582 (1995) 731

The time integration method for dileptons in HSD The time integration method for dileptons in HSD

ee--

e+e+Dilepton emission rate:Dilepton emission rate:

N(t)]Δt)[N(tΔt

1

dt

dNΔt,dt:ynumericall

dtN(t)dtN(t)N(t)dt

eN)(t)eeN(ρ

Γ

cτ),eeBr(ρN

γτ

1(t)

dt

)eedN(ρ

F

F

τ

τ

00

γτ

t

0

totρ

FF

ΔM

1e(M)Br

ΔM

1

c)γ(

Δt(M)Γ

dM

dN c)γ(

τΓτ

time

eeρ FtoteeρeeρF

Dilepton invariant mass spectra:Dilepton invariant mass spectra:

0 < t < 0 < t < FF

timetimett00=0=0

FF < t < < t < infinityinfinity

The time integration method allows to account for the in-medium The time integration method allows to account for the in-medium dynamics of vector mesons!dynamics of vector mesons!

Summary ISummary I

Accounting of in-medium effects requires :Accounting of in-medium effects requires :

1)1) off-shell transport models off-shell transport models 2)2) time integration methodtime integration method

Dilepton channels in HSDDilepton channels in HSD

•All particles decaying to dileptons are All particles decaying to dileptons are first produced in BB, mB or mm collisionsfirst produced in BB, mB or mm collisions

•‚‚Factorization‘ of diagrams Factorization‘ of diagrams in the in the transport approach:transport approach:

•The dilepton spectra are calculated The dilepton spectra are calculated perturbatively perturbatively with the with the time integration methodtime integration method..

N N

N

N

R

e+

*e-

=

e-

N N

N R

N

R

e+

*

NN bremsstrahlung - SPANN bremsstrahlung - SPA

Soft-Photon-ApproximationSoft-Photon-Approximation (SPA): (SPA):

N N -> N N eN N -> N N e++ee--

-->e>e++ee--

Phase-space corrected Phase-space corrected soft-photon cross section:soft-photon cross section:

elastNNσ

1

qdMd

qM,s,dσ

qdMd

)qM,dP(s,

SPA implementation in HSD: SPA implementation in HSD:

ee++ee- - production in production in elastic NN collisionselastic NN collisions with probability: with probability:

elasticelastic

NNNN

‚‚quasi- elastic‘ quasi- elastic‘ N N -> N NN N -> N N

‚‚off-shell‘ correction factoroff-shell‘ correction factor

Bremsstrahlung – a new view on an ‚old‘ storyBremsstrahlung – a new view on an ‚old‘ story

2007 (HADES):2007 (HADES): The DLS puzzle is solved by accounting The DLS puzzle is solved by accounting for a larger pn bremsstrahlung !!!for a larger pn bremsstrahlung !!!

0.0 0.1 0.2 0.3 0.4 0.510-4

10-3

10-2

10-1

100

101

0.0 0.1 0.2 0.3 0.4 0.510-4

10-3

10-2

10-1

100

101

Bremsstrahlung

SPA, in HSD'97 Schäfer et al.'94 Shyam&Mosel'03 Kaptari&Kämpfer'06

p+n, 1.04 GeV

d/d

M [b

/(G

eV/c

2 )]

Schäfer et al.'94 de Jong&Mosel'96 Shyam&Mosel'03 Kaptari&Kämpfer'06

M [GeV/c2]

p+p, 1.04 GeV

d/d

M [b

/(G

eV/c

2 )]

New OBE-model New OBE-model (Kaptari&Kämpfer, NPA 764 (2006) 338):(Kaptari&Kämpfer, NPA 764 (2006) 338):

• pn pn bremstrahlung is bremstrahlung is largerlarger by a factor of by a factor of 4 4 than it has been than it has been calculated before (and used in transport calculations before)!calculated before (and used in transport calculations before)!

• pp pp bremstrahlung is smaller than pn, however, bremstrahlung is smaller than pn, however, not zeronot zero; consistent ; consistent with the 1996 calculations from with the 1996 calculations from F.F. de Jong in a T-matrix approachde Jong in a T-matrix approach

HSD: Dileptons from p+p and p+d - DLSHSD: Dileptons from p+p and p+d - DLS

• bremsstrahlungbremsstrahlung is the dominant contribution in p+d is the dominant contribution in p+d for 0.15 < M < 0.55 GeV at ~1-1.5 A GeV for 0.15 < M < 0.55 GeV at ~1-1.5 A GeV

0.1 0.3 0.5 0.7 0.9 1.110

-4

10-3

10-2

10-1

100

101

0.1 0.3 0.5 0.7 0.9 1.110

-4

10-3

10-2

10-1

100

101

0.1 0.3 0.5 0.7 0.9 1.110

-4

10-3

10-2

10-1

100

101

0.1 0.3 0.5 0.7 0.9 1.110

-4

10-3

10-2

10-1

100

101

0.1 0.3 0.5 0.7 0.9 1.110

-4

10-3

10-2

10-1

100

101

0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.510

-4

10-3

10-2

10-1

100

101

Dalitz Dalitz Dalitz Dalitz Brems. NN Brems. N All

p+d, 1.04 GeV

d/d

M [b

/(G

eV/c

2 )]

Dalitz Dalitz Dalitz Dalitz Brems. NN Brems. N All

p+d, 1.27 GeV

p+d, 1.61 GeV

d/d

M [b

/(G

eV/c

2 )]

0

p+d, 1.85 GeV

p+d, 2.09 GeV

d/d

M [b

/(G

eV/c

2 )]

M [GeV/c2]

p+d, 4.88 GeV

M [GeV/c2]

0.1 0.3 0.5 0.7 0.9 1.110

-4

10-3

10-2

10-1

100

101

0.1 0.3 0.5 0.7 0.9 1.110

-4

10-3

10-2

10-1

100

101

0.1 0.3 0.5 0.7 0.9 1.110

-4

10-3

10-2

10-1

100

101

0.1 0.3 0.5 0.7 0.9 1.110

-4

10-3

10-2

10-1

100

101

0.1 0.3 0.5 0.7 0.9 1.110

-4

10-3

10-2

10-1

100

101

0.1 0.3 0.5 0.7 0.9 1.110

-4

10-3

10-2

10-1

100

101

Dalitz Dalitz Dalitz Dalitz Brems. pp All

p+p, 1.04 GeV

d/d

M [b

/(G

eV/c

2 )]

Dalitz Dalitz Dalitz Dalitz Brems. pp All

p+p, 1.27 GeV

p+p, 1.61 GeV

d/d

M [b

/(G

eV/c

2 )]

p+p, 1.85 GeV

p+p, 2.09 GeV

d/d

M [b

/(G

eV/c

2 )]

M [GeV/c2]

p+p, 4.88 GeV

M [GeV/c2]

HSD: Dileptons from A+A at 1 A GeV - DLSHSD: Dileptons from A+A at 1 A GeV - DLS

• bremsstrahlungbremsstrahlung and and -Dalitz are the dominant contributions -Dalitz are the dominant contributions in A+A for 0.15 < M < 0.55 GeV at 1 A GeV !in A+A for 0.15 < M < 0.55 GeV at 1 A GeV !

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

10-3

10-2

10-1

100

101

102

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

10-3

10-2

10-1

100

101

102

Brems. NN Brems. N All

DLS Dalitz Dalitz Dalitz Dalitz

C+C, 1.04 A GeVin-medium effects: CB+DM

d/d

M [b

/(G

eV c

2 )]

M [GeV/c2]

Brems. NN Brems. N All

DLS Dalitz Dalitz Dalitz Dalitz

C+C, 1.04 A GeVno medium effects

d/d

M [b

/(G

eV c

2 )]

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.010-3

10-2

10-1

100

101

102

103

104

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.010-3

10-2

10-1

100

101

102

103

104

Brems. NN Brems. N All

DLS Dalitz Dalitz Dalitz Dalitz

Ca+Ca, 1.04 A GeVin-medium effects: CB+DM

d/d

M [b

/(G

eV c

2 )]

M [GeV/c2]

Brems. NN Brems. N All

DLS Dalitz Dalitz Dalitz Dalitz

Ca+Ca, 1.04 A GeVno medium effects

d/d

M [b

/(G

eV c

2 )]

0.0 0.2 0.4 0.6 0.810-8

10-7

10-6

10-5

10-4

10-3

10-2

0.0 0.2 0.4 0.6 0.810-8

10-7

10-6

10-5

10-4

10-3

10-2

HADES

HSD: Dalitz Dalitz Dalitz Dalitz Brems. NN Brems. N All

C+C, 1.0 A GeVno medium effects

1/N

dN/d

M [

1/G

eV /c

2 ]

HSD: Dalitz Dalitz Dalitz Dalitz Brems. NN Brems. N All

M [GeV/c2]

HADES

C+C, 1.0 A GeVin-medium effects: CB+DM

1/N

dN/d

M [

1/G

eV /c

2 ]

HSD: Dileptons from C+C at 1 and 2 A GeV - HADESHSD: Dileptons from C+C at 1 and 2 A GeV - HADES

• HADES data show exponentially decreasing mass spectra HADES data show exponentially decreasing mass spectra • Data are Data are better described by in-medium scenarios with collisional broadening better described by in-medium scenarios with collisional broadening • In-medium effects are more pronounced for In-medium effects are more pronounced for heavy systemsheavy systems such as Au+Au such as Au+Au

0.0 0.2 0.4 0.6 0.8 1.010-8

10-7

10-6

10-5

10-4

10-3

10-2

0.0 0.2 0.4 0.6 0.8 1.010-8

10-7

10-6

10-5

10-4

10-3

10-2

HADES

HSD: Dalitz Dalitz Dalitz Dalitz Brems. NN Brems. N All

C+C, 2.0 A GeVno medium effects

1/N

dN/d

M [

1/G

eV /c

2 ]

HSD: Dalitz Dalitz Dalitz Dalitz Brems. NN Brems. N All

M [GeV/c2]

HADES

C+C, 2.0 A GeVin-medium effects: CB+DM

1/N

dN/d

M [

1/G

eV /c

2 ]

Bremsstrahlung in UrQMD 1.3 (1998)Bremsstrahlung in UrQMD 1.3 (1998)

Ernst et al, PRC58 (1998) 447Ernst et al, PRC58 (1998) 447

0.0 0.2 0.4 0.6 0.8 1.0 1.210-3

10-2

10-1

100

Kaptari'06 1.0 2.1 4.9

Bremsstrahlung pn

UrQMD'97 1.0 2.1 4.9

M [GeV/c2]

d/d

M [b

/(G

eV/c

2 )]

Bremsstrahlung-UrQMD’98 Bremsstrahlung-UrQMD’98 smallersmaller than bremsstrahlung from Kaptari’06 than bremsstrahlung from Kaptari’06 by a factor of 3-6by a factor of 3-6

SPA:SPA:

• SPA implementation in UrQMD (1998): SPA implementation in UrQMD (1998): ee++ee- - production in production in elastic NN collisionselastic NN collisions (similar to HSD)(similar to HSD)

• „„old“old“ bremsstrahlung: missing yield bremsstrahlung: missing yield for p+d and A+A at 0.15 < M < 0.55 for p+d and A+A at 0.15 < M < 0.55 GeV at 1 A GeV GeV at 1 A GeV (consistent with HSD (consistent with HSD employing „old SPA“)employing „old SPA“)

Dileptons from A+A - UrQMD 2.2 (2007)Dileptons from A+A - UrQMD 2.2 (2007)

D. Schumacher, S. Vogel, M. Bleicher, D. Schumacher, S. Vogel, M. Bleicher, Acta Phys.Hung.Acta Phys.Hung. A27A27 (2006) (2006) 451451

NO bremsstrahlung in UrQMD 2.2NO bremsstrahlung in UrQMD 2.2

Dileptons from A+A - RQMD (Tübingen)Dileptons from A+A - RQMD (Tübingen)

C. Fuchs et al.C. Fuchs et al., , Phys. Rev. C67 025202(2003)Phys. Rev. C67 025202(2003)

• NO bremsstrahlung in RQMD NO bremsstrahlung in RQMD ((missing yieldmissing yield for p+d at for p+d at 0.15 < M < 0.55 GeV at ~1-1.5 A GeV)0.15 < M < 0.55 GeV at ~1-1.5 A GeV)

• too strong too strong Dalitz contribution Dalitz contribution (since no time integration?) (since no time integration?)

HADES - RQMD‘07HADES - RQMD‘07

DLS - RQMD‘03DLS - RQMD‘03

1 A GeV1 A GeV

Bremsstrahlung in IQMD (Nantes)Bremsstrahlung in IQMD (Nantes)M. Thomere, C. Hartnack, G. Wolf, J. Aichelin, PRC75 (2007) 064902M. Thomere, C. Hartnack, G. Wolf, J. Aichelin, PRC75 (2007) 064902

SPA implementation in IQMD : SPA implementation in IQMD : ee++ee- - bremsstrahlung production in bremsstrahlung production in each NN collisioneach NN collision (i.e. elastic and inelastic) !(i.e. elastic and inelastic) !

- differs from HSD and UrQMD’98 (only elastic NN collisions are counted!)- differs from HSD and UrQMD’98 (only elastic NN collisions are counted!)

HADES: C+C, 2 A GeVHADES: C+C, 2 A GeV

Bremsstrahlung in BRoBUU (Rossendorf)Bremsstrahlung in BRoBUU (Rossendorf)H.W. Barz, B. Kämpfer, Gy. Wolf, M. Zetenyi, nucl-th/0605036H.W. Barz, B. Kämpfer, Gy. Wolf, M. Zetenyi, nucl-th/0605036

SPA implementation in BRoBUU : SPA implementation in BRoBUU :

ee++ee- - production in production in each NN collisioneach NN collision (i.e. elastic and inelastic) !(i.e. elastic and inelastic) !

- similar to IQMD (Nantes)- similar to IQMD (Nantes)

Summary IISummary II

Transport models give Transport models give similar resultssimilar results ONLYONLY with the same with the sameinitial input ! initial input !

=> => REQUESTS:REQUESTS:„„unification“ of the treatment of dilepton production in unification“ of the treatment of dilepton production in transport models:transport models: Similar cross sections for elementary channelsSimilar cross sections for elementary channels Time-integration methodTime-integration method for dilepton production for dilepton production Off-shellOff-shell treatment of broad resonances treatment of broad resonances

++Consistent microscopic calculations for eConsistent microscopic calculations for e++ee--

bremsstrahlung from NN and mN collisions! bremsstrahlung from NN and mN collisions!

Part IIPart II

• Elementary channels:Elementary channels:-Dalitz decay-Dalitz decay-Dalitz decay-Dalitz decay

•pp, pnpp, pn and and pdpd reactions vs. new HADES data reactions vs. new HADES data

-production cross section in pp and pn-production cross section in pp and pn

HSD:HSD: good description of the experimental datagood description of the experimental data (Celsius/WASA) (Celsius/WASA) on inclusive on inclusive production cross section in production cross section in pppp and and pnpn collisions collisions

=> => -Dalitz-Dalitzdecay contribution is under control ! decay contribution is under control !

10-3 10-2 10-1 10010-4

10-3

10-2

10-1

100

s1/2

th string

Parametrization (inclusive) pn->X pp->X

Exp. data (exclusive):pn->pnpp->pp

pN->X

[m

b]

s1/2-s0

1/2 [GeV]

E.L.B. &W. Cassing, NPA 807 (2008) 214E.L.B. &W. Cassing, NPA 807 (2008) 214

-Dalitz decay-Dalitz decay

Original paper:Original paper: H.F. Jones, M.D. Scadron, Ann. Phys. 81 (1973) 1 H.F. Jones, M.D. Scadron, Ann. Phys. 81 (1973) 1

-Dalitz decay-Dalitz decay

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35

10-7

10-6

10-5

10-4

10-3

0.0 0.1 0.2 0.3 0.4 0.5 0.6

10-7

10-6

10-5

10-4

10-3

Wolf Ernst (PLUTO) Krivoruchenko * 3/2

GM

=2.7

d(

->N

e+ e )/dM

M=1.232 GeV

M [GeV]

d(

->N

e+ e )/dM

Wolf Ernst (PLUTO) Krivoruchenko * 3/2

GM

=2.7M=1.5 GeV

M [GeV]

• similarsimilar results for the results for the -Dalitz -Dalitz electromagnetic decay electromagnetic decay fromfrom different models !different models !

• starting pointstarting point: the same : the same Lagrangian for the Lagrangian for the N-vertexN-vertex

• small differencessmall differences are related to a are related to a different treatment of the 3/2 spin different treatment of the 3/2 spin statesstates

-spectral function-spectral function

1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8

10-1

100

101

=Const (M) (e.g. Monitz width)

spectral function

M [GeV]

A(M

) [G

eV-1]

1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7

10-1

100

101

1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7

10-1

100

101

HSD RQMD UrQMD

M [GeV]

C+C, 2 A GeV, b=0.5 fm

dN/d

M

HSD RQMD UrQMD

M [GeV]

C+C, 2 A GeV, b=0.5 fm

dN/d

M

TheThe mainmain differences differences in the dilepton in the dilepton yield from the yield from the -Dalitz decay are -Dalitz decay are related not to the electromagnetic related not to the electromagnetic decay but to the decay but to the treatment oftreatment of--dynamics in the transport models !dynamics in the transport models !

– – dynamics vs. TAPS datadynamics vs. TAPS data

0.5 1.0 1.5 2.0

10-4

10-3

10-2

10-1

100

101

TAPS HSD

Ebeam

[A GeV]

C+C

Mul

t inc

Constraints on Constraints on , , by TAPS data: by TAPS data:

HSD:HSD: good description of TAPS data on good description of TAPS data on , , multiplicities and m multiplicities and mTT-spectra -spectra

=>=>, , dynamics under control !dynamics under control !

E.L.B. &W. Cassing, NPA 807 (2008) 214E.L.B. &W. Cassing, NPA 807 (2008) 214

pp @ 1.25GeV :pp @ 1.25GeV : new HADES data new HADES data

-Dalitz decay is the dominant channel (HSD consistent with -Dalitz decay is the dominant channel (HSD consistent with PLUTO)PLUTO)

HSD predictions:HSD predictions: good description of new HADES data p+p data!good description of new HADES data p+p data!

PLUTO

E.L.B. &W. Cassing, NPA 807 (2008) 214E.L.B. &W. Cassing, NPA 807 (2008) 214

0.0 0.2 0.4 0.6 0.8

10-7

10-6

10-5

10-4

10-3

10-2

M [GeV/c2]

HSD

Dalitz Dalitz Dalitz Dalitz Brems. pp All

p+p, 1.25 GeV

1/N

dN/d

M [

(GeV

/c2 )-1

]

Quasi-free pnQuasi-free pn (pd) reaction: HADES data @ 1.25 GeV (pd) reaction: HADES data @ 1.25 GeV

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

10-11

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2 p+d, 1.25 GeV1/

N

dN/d

M [

(GeV

/c2 )-1

] HSD

Dalitz Dalitz Dalitz Dalitz Brems. NN Brems. N All

M [GeV/c2]

"quasi-free" p+n 1.25 GeV

e+e->90

PLUTO

HSD predictions: HSD predictions: underestimates the HADES p+n (quasi-free) data at 1.25 GeV:underestimates the HADES p+n (quasi-free) data at 1.25 GeV:

1)1) 0.2<M<0.55 GeV:0.2<M<0.55 GeV:

-Dalitz decay-Dalitz decay by a factor of ~10 is larger in PLUTO than in HSD since the c by a factor of ~10 is larger in PLUTO than in HSD since the channels hannels

d + p d + p ppspecspec + d + + d + (‘quasi-free’ (‘quasi-free’ -production --production - dominant at 1.25GeV!)dominant at 1.25GeV!) and and p p + n + n d + d + were NOT taken into account ! were NOT taken into account !

Note: Note: these channels have these channels have NO impactNO impact for heavy-ion reactions and even for p+d results at for heavy-ion reactions and even for p+d results at higher energies!higher energies!

*In HSD:*In HSD: p+d = p + (p&n)-with Fermi motionp+d = p + (p&n)-with Fermi motion according to the Paris deuteron wave function according to the Paris deuteron wave function

10-3 10-2 10-1 10010-4

10-3

10-2

10-1

100

s1/2

th string

Parametrization: pn->X pp->X pn->d

Exp. data (exclusive):pn->pnpp->pppn->d

pN->X

[m

b]

s1/2-s0

1/2 [GeV]

Quasi-free pnQuasi-free pn (pd) @ 1.25 GeV: (pd) @ 1.25 GeV: -channel-channel

10-3 10-2 10-1 10010-6

10-5

10-4

10-3

10-2

10-1

HADES pd@1.25GeV

CELSIUS

pd->pd

[m

b]

s1/2-s0

1/2 [GeV]

1)1) p + n p + n d + d +

2)2) d + p d + p ppspecspec + d + + d +

Add the following channels:Add the following channels:

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

10-11

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2 p+d, 1.25 GeV

1/N

dN/d

M [

(GeV

/c2 )-1

]

HSD

Dalitz Dalitz Dalitz Dalitz Brems. NN Brems. N All

M [GeV/c2]

Now HSD agreesNow HSD agreeswith PLUTO with PLUTO on the on the - Dalitz decay!- Dalitz decay!

Quasi-free pnQuasi-free pn (pd) @ 1.25 GeV: (pd) @ 1.25 GeV: N(1520) ?!N(1520) ?!

1.01.0 1.3 1.51.5 1.8 2.02.0 2.3 2.52.5 2.8 3.03.010-4

10-3

10-2

10-1

100

101

102

Exp. data: N N

s1/2 [GeV]

N -> X

[m

b]

Free meson spectral function N->N(1520)->N non-resonant N->X

2 3 4 5 6 7 8 9 101010-4

10-3

10-2

10-1

100

101

Free meson spectral function NN->N(1520)->NN non-resonant NN->X

s1/2

th string

Exp. data:pp pp pp X

(s)

[m

b]

s1/2 [GeV]

pp X

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

10-11

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2 p+d, 1.25 GeV

1/N

dN

/dM

[(G

eV/c

2 )-1]

HSD

Dalitz Dalitz Dalitz Dalitz Brems. NN Brems. N All

M [GeV/c2] 2)2) M > 0.45 GeV:M > 0.45 GeV:

HSD HSD preliminarypreliminary result for p+d @1.25 GeV shows result for p+d @1.25 GeV shows that the missing yield might be attributed to that the missing yield might be attributed to subthreshold subthreshold production via N(1520) production via N(1520) excitation and decay ?! excitation and decay ?!

Similar to our Similar to our NNPPA686A686 (2001) (2001) 568568

0.1 0.3 0.5 0.7 0.9 1.110-4

10-3

10-2

10-1

100

101

M [GeV/c2]

d/d

M [b

/(G

eV/c

2 )]

pn

all

0

p+d, 1.27 GeVDLS

N(1520)N(1520)

N(1520)N(1520)

Model for N(1520):Model for N(1520): according to according to Peters et al., Peters et al., NPA632NPA632 (1998) (1998) 109109

Ratio pd/ppRatio pd/pp @ 1.25 GeV @ 1.25 GeV

HSD shows a qualitative agreement with the HADES data on the ratio:HSD shows a qualitative agreement with the HADES data on the ratio:

accounting for the accounting for the subthreshold subthreshold production via N(1520) decay should production via N(1520) decay should improve the agreement !improve the agreement !

0.0 0.1 0.2 0.3 0.4 0.5 0.61

10

HADES HSD

M [GeV/c2]

d+p/p+p, 1.25 GeV

Rat

io

OutlookOutlook

HADES succeeded:HADES succeeded:the DLS puzzle is solved !the DLS puzzle is solved !

Outlook-1: need new Outlook-1: need new pp,pp, pd pd andand NN data data from HADES for a final check!from HADES for a final check!

Outlook-2:Outlook-2: study in-medium study in-medium effects with HADESeffects with HADES

ThanksThanks to to

HADES collegues: HADES collegues:

Yvonne, Gosia, Romain, Piotr, Yvonne, Gosia, Romain, Piotr, Joachim, Tatyana, Volker …Joachim, Tatyana, Volker …

+ + WolfgangWolfgang

++

- More slides -- More slides -

Dynamics of heavy-ion collisions –> Dynamics of heavy-ion collisions –> complicated many-body problem!complicated many-body problem!

Correct wayCorrect way to solve the many-body problem including all quantum to solve the many-body problem including all quantum mechanical features mechanical features

Kadanoff-Baym equationsKadanoff-Baym equations for Green functions S for Green functions S<< (from 1962)(from 1962)

Greens functions S / self-energies Greens functions S / self-energies : :

e.g. for bosons

do Wigner transformation do Wigner transformation

retarded (ret), advanced (adv) (anti-)causal (a,c )

consider only contribution up toconsider only contribution up to first order in the gradientsfirst order in the gradients = a= a standard approximationstandard approximation of kinetic theory which is justified if the gradients in of kinetic theory which is justified if the gradients in the mean spacial coordinate X are smallthe mean spacial coordinate X are small

NN bremsstrahlung: OBE-modelNN bremsstrahlung: OBE-model

OBE-model:OBE-model: N N -> N N e N N -> N N e++ee--

‚‚pre‘pre‘ ‚‚post‘post‘

‚‚post‘post‘‚‚pre‘pre‘

+ gauge terms+ gauge terms

charged meson charged meson

exchangeexchange contact terms (from formfactors)contact terms (from formfactors)

The strategy to The strategy to restore gauge restore gauge invarianceinvariance is is

model dependent!model dependent!

Kaptari&Kämpfer, NPA 764 (2006) 338Kaptari&Kämpfer, NPA 764 (2006) 338

Test in HSD:Test in HSD:bremsstrahlung production in NN collisions bremsstrahlung production in NN collisions

(only elastic vs. all)(only elastic vs. all)

In HSD assume:In HSD assume: ee++ee- - productionproduction from „old“ SPA bremsstrahlung from „old“ SPA bremsstrahlung in each NN collisionin each NN collision (i.e. elastic and inelastic reactions)(i.e. elastic and inelastic reactions)=> => can reproduce the results by Gy. Wolfcan reproduce the results by Gy. Wolf et al., i.e. IQMD (Nantes) et al., i.e. IQMD (Nantes) and BRoBUU (Rossendorf) !and BRoBUU (Rossendorf) !

0.0 0.2 0.4 0.6 0.8 1.010-8

10-7

10-6

10-5

10-4

10-3

10-2

Bremsstrahlung: HSD: elast. NN HSD-test: all NN Wolf: Nantes Wolf: Rossendorf

M [GeV/c2]

C+C, 2.0 A GeV, HADES

1/N

dN/d

M [

1/G

eV /c

2 ]

Deuteron in HSDDeuteron in HSD

In HSD:In HSD: p+d = p + (p&n) -with Fermi motionp+d = p + (p&n) -with Fermi motion according according to the momentum distribution f(p) with Paris deuteron to the momentum distribution f(p) with Paris deuteron wave functionwave function

E.B., W. Cassing and U. Mosel, E.B., W. Cassing and U. Mosel, NNPPA686A686 (2001) (2001) 568568

Total deuteron energy: Total deuteron energy:

Dispersion relation I. (used):Dispersion relation I. (used):

(fulfill the binding energy constraint)(fulfill the binding energy constraint)

Dispersion relation II.:Dispersion relation II.:

II

IIII