Dileptons from off-shell transport approach. E lena 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
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Dileptons from off-shell transport approach
Elena Bratkovskaya
5.07.2008 , HADES Collaboration Meeting XIX,
GSI, Darmstadt
Theory (status: last millenium < 2000) :
Implementation of in-medium vector mesons (r,w) scenarios (= ‚dropping‘ mass and ‚collisional broadening‘) in on-shelltransport models:
r meson spectral function
In-medium models:
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!)
<=>
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)
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
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
In-medium scenarios:
dropping mass collisional broadening dropping mass + coll. broad.
Collisional width GCB(M,r) = g r <u sVNtot>
r-meson spectral function:
E.L.B., NPA 686 (2001), E.L.B. &W. Cassing, NPA 807 (2008) 214
(s ½ < 2.2 GeV)
(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 :
e+
e+
g*
N
N
R
g*
R
N
R
N
N
N
N
e-
N
e-
=
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):
calculated before (and used in transport calculations before)!
2007 (HADES): The DLS puzzle is solved by accounting for a larger pn bremsstrahlung !!!
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
Ernst et al, PRC58 (1998) 447
Bremsstrahlung in UrQMD 1.3 (1998)
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
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:
+
Consistent microscopic calculations for e+e-
bremsstrahlung from NN and mN collisions!
E.L.B. &W. Cassing, NPA 807 (2008) 214
Original paper: H.F. Jones, M.D. Scadron, Ann. Phys. 81 (1973) 1
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 !
E.L.B. &W. Cassing, NPA 807 (2008) 214
PLUTO
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 )
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
Total deuteron energy:
I
II
E.B., W. Cassing and U. Mosel, NPA686 (2001) 568