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### Dileptons from off-shell transport approach

### NN bremsstrahlung - SPA

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

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

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

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

### Bremsstrahlung in UrQMD 1.3 (1998)

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

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

### Bremsstrahlung in IQMD (Nantes)

### Bremsstrahlung in BRoBUU (Rossendorf)

### Summary II

### NN bremsstrahlung: OBE-model

### Test in HSD:bremsstrahlung production in NN collisions (only elastic vs. all)

Elena Bratkovskaya

5.07.2008 , HADES Collaboration Meeting XIX,

GSI, Darmstadt

Overview

- Study of in-medium effects in heavy-ion collisions require:
- off-shell transport dynamics
- in-medium transition rates
- time-integration methods
- Bremsstrahlung
- HSD results and comparison of transport models
- Elementary channels:
- h-Dalitz decay
- D-Dalitz decay
- pp, pn and pd reactions vs. new HADES data

Dileptons from transport models

Theory (status: last millenium < 2000) :

Implementation of in-medium vector mesons (r,w) scenarios (= ‚dropping‘ mass and ‚collisional broadening‘) in on-shelltransport models:

- BUU/AMPT (Texas) ( > 1995)
- HSD ( > 1995)
- UrQMD v. 1.3 (1998)
- RQMD (Tübingen) (2003), but NO explicit propagation of vector mesons
- IQMD (Nantes) (2007), but NO explicit propagation of vector mesons
- Theory (status: this millenium > 2000) :
- Implementation of in-medium vector mesons (r,w,f) scenarios (= ‚dropping‘ mass and ‚collisional broadening‘) in off-shelltransport models:
- HSD (>2000)
- BRoBUU (Rossendorf) (2006)

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

r meson spectral function

In-medium models:

- chiral perturbation theory
- chiral SU(3) model
- coupled-channel G-matrix approach
- chiral coupled-channel effective field theory
- predict changes of the particle properties in the hot and dense medium, e.g. broadening of the spectral function

How to treat in-medium effects in transport approaches?

From Kadanoff-Baym equations to transport equations

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

Operator <> - 4-dimentional generalizaton of the Poisson-bracket

drift term

Vlasov term

backflow term

Backflow term incorporates the off-shell behavior in the particle propagation

! vanishes in the quasiparticle limit

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

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

For bosons in first order gradient expansion:

GXP – width of spectral function = reaction rate of particle (at phase-space position XP)

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

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

General testparticle off-shell equations of motion

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

Employ testparticle Ansatz for the real valued quantity iS<XP -

insert in generalized transport equations and determine equations of motion !

General testparticle off-shell equations of motion:

with

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

On-shell limit

1) Γ(X,P) 0

quasiparticle approximation :

A(X,P) = 2 pd(P2-M2)

||

Hamiltons equation of motion -

independent on Γ !

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

2) Γ(X,P) such that

E.g.: Γ = const

G=Γvacuum(M)

‚Vacuum‘ spectral function with constant or mass dependent width G:

spectral function AXP does NOT change the shape (and pole position) during propagation through the medium(backflow term vanishes also!)

<=>

- Hamiltons equation of motion - independent on Γ !

‚On-shell‘ transport models

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

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

2)Quasiparticle approximation or/and vacuum spectral functions :

A(X,P) = 2 pd(p2-M2) Avacuum(M)

- for each particle species i (i = N, R, Y, p, r, K, …) the phase-space density fi followsthe transport equations
- with collision termsIcoll describing elastic and inelastic hadronic reactions:
- baryon-baryon, meson-baryon, meson-meson, formation and decay of baryonic and mesonicresonances, string formation and decay (for inclusive particle production:
- BB -> X , mB ->X, X =many particles)
- with propagation of particles in self-generated mean-field potential U(p,r)~Re(Sret)/2p0
- Numerical realization – solution of classical equations of motion + Monte-Carlo simulations for test-particle interactions

Short-lived resonances in semi-classical transport models

Spectral function:

width G ~ -Im Sret /M

Vacuum (r =0)narrow states

In-medium:

production of broad states

In-medium

r >> r0

Example :

r-meson propagation through the medium within the on-shell BUU model

- broad in-medium spectral function does not become on-shell in vacuum in ‚on-shell‘ transport models!

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

Off-shell vs. on-shell transport dynamics

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

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

On-shell model:

low mass r and w mesons live forever and shine dileptons!

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

Collision term in off-shell transport models

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

with

The trace over particles 2,3,4 reads explicitly

for fermions

for bosons

additional integration

The transport approach and the particle spectral functions are fully determined once the in-medium transition amplitudes G are known in their off-shell dependence!

Spectral function in off-shell transport model

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

with

Spectral function:

total width: Gtot=Gvac+GColl

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

- Assumptions used in transport model (to speed up calculations):
- Collisional widthin low density approximation: GColl(M,p,r) = g r <u sVNtot>
- replace <u sVNtot> by averaged value G=const: GColl(M,p,r) = g r G

Modelling of in-medium spectral functions for vector mesons

In-medium scenarios:

dropping mass collisional broadening dropping mass + coll. broad.

- m*=m0(1-a r/r0) G(M,r)=Gvac(M)+GCB(M,r) m* & GCB(M,r)

Collisional width GCB(M,r) = g r <u sVNtot>

r-meson spectral function:

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

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

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

- Low energy BB and mB interactions

(s ½ < 2.2 GeV)

- High energy BB and mB interactions

(s ½ > 2.2 GeV)

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

Originally in FRITIOF (PYTHIA/JETSET): A(M) with constant width around the pole mass M0

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

Time integration method for dileptons

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

‚Reality‘:

e+

only ONE e+e- pair with probability ~

Br(r->e+e-)=4.5 .10-5

r

r

w

e-

‚Virtual‘ – time integ. method:

t0

tabs

e+

r

time

e-

tF

Calculate probability P(t) to emitan e+e- pair at each time t and integrate P(t) over time!

r: t0 < t < tabs

w: t0 <t < infinity

tF – final time of computation in the code

t0 – production time

tabs – absorption (or hadronic decay) time

The time integration method for dileptons in HSD

Dilepton emission rate:

e+

r

e-

t0=0

time

tF

Dilepton invariant mass spectra:

tF < t < infinity

0 < t < tF

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

Summary I

Accounting of in-medium effects requires :

- off-shell transport models
- time integration method

e+

g*

N

N

R

g*

R

N

R

N

N

N

N

e-

N

e-

=

Dilepton channels in HSD- All particles decaying to dileptons are first produced in BB, mB or mm collisions

- ‚Factorization‘ of diagrams in the transport approach:

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

Phase-space corrected soft-photon cross section:

Soft-Photon-Approximation (SPA):

N N -> N N e+e-

‚quasi- elastic‘

N N -> N N

elastic

NN

‚off-shell‘ correction factor

SPA implementation in HSD:

e+e- production in elastic NN collisions with probability:

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

- pn bremstrahlung is larger by a factor of 4 than it has been

calculated before (and used in transport calculations before)!

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

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

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

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

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

SPA:

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

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

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

D. Schumacher, S. Vogel, M. Bleicher, Acta Phys.Hung.A27 (2006) 451

NO bremsstrahlung in UrQMD 2.2

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

HADES - RQMD‘07

DLS - RQMD‘03

1 A GeV

- NO bremsstrahlung in RQMD (missing yieldfor p+d at 0.15 < M < 0.55 GeV at ~1-1.5 A GeV)
- too strong D-Dalitz contribution (since no time integration?)

M. Thomere, C. Hartnack, G. Wolf, J. Aichelin, PRC75 (2007) 064902

HADES: C+C, 2 A GeV

SPA implementation in IQMD :

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

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

H.W. Barz, B. Kämpfer, Gy. Wolf, M. Zetenyi, nucl-th/0605036

SPA implementation in BRoBUU :

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

- similar to IQMD (Nantes)

Transport models give similar resultsONLY with the same

initial input !

=> REQUESTS:

„unification“ of the treatment of dilepton production in transport models:

- Similar cross sections for elementary channels
- Time-integration method for dilepton production
- Off-shell treatment of broad resonances

+

Consistent microscopic calculations for e+e-

bremsstrahlung from NN and mN collisions!

Part II

- Elementary channels:
- h-Dalitz decay
- D-Dalitz decay
- pp, pn and pd reactions vs. new HADES data

h-production cross section in pp and pn

- HSD:good description of the experimental data (Celsius/WASA) on inclusive h production cross section in pp and pn collisions
- => h-Dalitzdecay contribution is under control !

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

D-Dalitz decay

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

D-Dalitz decay

- similar results for the D-Dalitz electromagnetic decay from different models !
- starting point: the same Lagrangian for the gDN-vertex
- small differences are related to a different treatment of the 3/2 spin states

D-spectral function

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

p0, h, D – dynamics vs. TAPS data

- Constraints on p, h by TAPS data:
- HSD: good description of TAPS data on p, h multiplicities and mT-spectra
- => p (D), h dynamics under control !

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

pp @ 1.25GeV : new HADES data

PLUTO

- D-Dalitz decay is the dominant channel (HSD consistent with PLUTO)
- HSD predictions:good description of new HADES data p+p data!

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

e+e->90

PLUTO

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

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

0.2<M<0.55 GeV:

h-Dalitz decay by a factor of ~10 is larger in PLUTO than in HSD since the channels d + p pspec + d + (‘quasi-free’ h-production -dominant at 1.25GeV!) and p + n d + were NOT taken into account !

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

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

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

Add the following channels:

1) p + n d +

2) d + p pspec + d +

Now HSD agreeswith PLUTO on the h- Dalitz decay!

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

N(1520)

2)M > 0.45 GeV:

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

N(1520)

Similar to our NPA686 (2001) 568

Model for N(1520): according to

Peters et al., NPA632 (1998) 109

Ratio pd/pp @ 1.25 GeV

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

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

Outlook

HADES succeeded:

the DLS puzzle is solved !

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

Outlook-2: study in-medium effects with HADES

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

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

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

e.g. for bosons

Greens functions S / self-energies S :

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

- do Wigner transformation

consider only contribution up tofirst order in the gradients

= astandard approximationof kinetic theory which is justified if the gradients in

the mean spacial coordinate X are small

Kaptari&Kämpfer, NPA 764 (2006) 338

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

‚pre‘

‚post‘

‚pre‘

‚post‘

+ gauge terms

The strategy to restore gauge invariance is model dependent!

charged meson exchange

contact terms (from formfactors)

In HSD assume:

e+e- productionfrom „old“ SPA bremsstrahlung in each NN collision (i.e. elastic and inelastic reactions)

=> can reproduce the results by Gy. Wolf et al., i.e. IQMD (Nantes) and BRoBUU (Rossendorf) !

Deuteron in HSD

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

- Dispersion relation I. (used):
- (fulfill the binding energy constraint)

Total deuteron energy:

- Dispersion relation II.:

I

II

E.B., W. Cassing and U. Mosel, NPA686 (2001) 568

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