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Dileptons from off-shell transport approach

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Dileptons from off-shell transport approach

Elena Bratkovskaya

5.07.2008 , HADES Collaboration Meeting XIX,

GSI, Darmstadt

- 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

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)

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?

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

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) !

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 !

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

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)

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 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!

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

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

- 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

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

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!

Accounting of in-medium effects requires :

- off-shell transport models
- time integration method

e+

e+

g*

N

N

R

g*

R

N

R

N

N

N

N

e-

N

e-

=

- 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.

g*->e+e-

Phase-space corrected soft-photon cross section:

Soft-Photon-Approximation (SPA):

NN bremsstrahlung - 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:

Bremsstrahlung – a new view on an ‚old‘ story

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 !!!

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

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

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

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

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

- 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

Ernst et al, PRC58 (1998) 447

Bremsstrahlung in UrQMD 1.3 (1998)

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

Dileptons from A+A - UrQMD 2.2 (2007)

NO bremsstrahlung in UrQMD 2.2

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

HADES - RQMD‘07

Dileptons from A+A - RQMD (Tübingen)

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

Bremsstrahlung in IQMD (Nantes)

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

Bremsstrahlung in BRoBUU (Rossendorf)

SPA implementation in BRoBUU :

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

- similar to IQMD (Nantes)

Summary II

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!

- Elementary channels:
- h-Dalitz decay
- D-Dalitz decay

- pp, pn and pd reactions vs. new HADES data

- 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

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

- 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

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 !

- 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

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

"quasi-free" p+n 1.25 GeV

e+e->90

PLUTO

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

Add the following channels:

1) p + n d +

2) d + p pspec + d +

Now HSD agreeswith PLUTO on the h- Dalitz decay!

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

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 !

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

HADES collegues:

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

+ Wolfgang

+

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-

NN bremsstrahlung: OBE-model

‚pre‘

‚post‘

‚pre‘

‚post‘

+ gauge terms

The strategy to restore gauge invariance is model dependent!

charged meson exchange

contact terms (from formfactors)

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

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) !

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