Dileptons from offshell transport approach

1
Dileptons from off-shell transport approach
Elena Bratkovskaya 5.07.2008 , HADES
Collaboration Meeting XIX, GSI, Darmstadt
2
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

3
Dileptons from transport models

• Theory (status last millenium lt 2000)
• Implementation of in-medium vector mesons (r,w)
  scenarios ( dropping mass and collisional
  broadening) in on-shell transport models
• BUU/AMPT (Texas) ( gt 1995)
• HSD ( gt 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 gt 2000)
• Implementation of in-medium vector mesons (r,w,f)
  scenarios ( dropping mass and collisional
  broadening) in off-shell transport models
• HSD (gt2000)
• BRoBUU (Rossendorf) (2006)

4
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?
5
From Kadanoff-Baym equations to transport
equations
Generalized transport equations first order
gradient expansion of the Wigner transformed
Kadanoff-Baym equations
Operator ltgt - 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 Slt 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
6
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 i SltXP -
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) !
7
On-shell limit
1) G(X,P) ? 0
quasiparticle approximation A(X,P) 2 p
d(P2-M2) Hamiltons equation of motion
- independent on G !
Backflow term - which incorporates the off-shell
behavior in the particle propagation - vanishes
in the quasiparticle limit !
2) G(X,P) such that E.g. G const
GGvacuum (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!)
ltgt

• Hamiltons equation of motion - independent on
  G !

8
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 p d(p2-M2)
  Avacuum(M)

• for each particle species i (i N, R, Y, p, r,
  K, ) the phase-space density fi follows the
  transport equations
• with collision terms Icoll describing
  elastic and inelastic hadronic reactions
• baryon-baryon, meson-baryon, meson-meson,
  formation and decay of baryonic and mesonic
  resonances, string formation and decay (for
  inclusive particle production
• BB -gt X , mB -gtX, 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

9
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 gtgt 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)
10
Off-shell vs. on-shell transport dynamics
Time evolution of the mass distribution of r and
w mesons for central CC collisions (b1 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!
11
Collision term in off-shell transport models
Collision term for reaction 12-gt34
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!
12
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 GtotGvacGColl

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

• Assumptions used in transport model (to speed up
  calculations)
• Collisional width in low density approximation
  GColl(M,p,r) g r ltu sVNtotgt
• replace ltu sVNtotgt by averaged value Gconst
  GColl(M,p,r) g r G

13
Modelling of in-medium spectral functions for
vector mesons

• In-medium scenarios
• dropping mass collisional
  broadening dropping mass coll.
  broad.
• mm0(1-a r/r0)
  G(M,r)Gvac(M)GCB(M,r) m GCB(M,r)

Collisional width GCB(M,r) g r ltu sVNtotgt
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
14
Modelling of in-medium off-shell production cross
sections for vector mesons

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

• High energy BB and mB interactions
• (s ½ gt 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
15
Time integration method for dileptons
Cf. G.Q. Li C.M. Ko, NPA582 (1995) 731
Reality
e
only ONE ee- pair with probability
Br(r-gtee-)4.5 .10-5
r
r
w
e-
Virtual time integ. method
t0
tabs
e
r
time
e-
tF
Calculate probability P(t) to emit an ee- pair
at each time t and integrate P(t) over time! r
t0 lt t lt tabs w t0 ltt lt infinity
tF final time of computation in the code t0
production time tabs absorption (or hadronic
decay) time
16
The time integration method for dileptons in HSD
Dilepton emission rate
e
r
e-
t00
time
tF
Dilepton invariant mass spectra
tF lt t lt infinity
0 lt t lt tF
The time integration method allows to account for
the in-medium dynamics of vector mesons!
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Summary I

• Accounting of in-medium effects requires
• off-shell transport models
• time integration method

18
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.

19
NN bremsstrahlung - SPA
Phase-space corrected soft-photon cross section
Soft-Photon-Approximation (SPA)
N N -gt N N ee-
quasi- elastic N N -gt N N
elastic NN
off-shell correction factor
SPA implementation in HSD ee- production in
elastic NN collisions with probability
20
Bremsstrahlung a new view on an old story

• New OBE-model (KaptariKämpfer, 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 !!!
21
HSD Dileptons from pp and pd - DLS

• bremsstrahlung is the dominant contribution in
  pd for 0.15 lt M lt 0.55 GeV at 1-1.5 A GeV

22
HSD Dileptons from AA at 1 A GeV - DLS

• bremsstrahlung and D-Dalitz are the dominant
  contributions in AA for 0.15 lt M lt 0.55 GeV at 1
  A GeV !

23
HSD Dileptons from CC 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 AuAu

24
Bremsstrahlung in UrQMD 1.3 (1998)
Ernst et al, PRC58 (1998) 447
SPA

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

• Bremsstrahlung-UrQMD98 smaller than
  bremsstrahlung from Kaptari06
• by a factor of 3-6

• old bremsstrahlung missing yield for pd and
  AA at 0.15 lt M lt 0.55 GeV at 1 A GeV (consistent
  with HSD employing old SPA)

25
Dileptons from AA - UrQMD 2.2 (2007)
D. Schumacher, S. Vogel, M. Bleicher, Acta
Phys.Hung. A27 (2006) 451
NO bremsstrahlung in UrQMD 2.2
26
Dileptons from AA - RQMD (Tübingen)
C. Fuchs et al., Phys. Rev. C67 025202(2003)
HADES - RQMD07
DLS - RQMD03
1 A GeV

• NO bremsstrahlung in RQMD (missing yield for pd
  at 0.15 lt M lt 0.55 GeV at 1-1.5 A GeV)
• too strong D-Dalitz contribution (since no time
  integration?)

27
Bremsstrahlung in IQMD (Nantes)
M. Thomere, C. Hartnack, G. Wolf, J. Aichelin,
PRC75 (2007) 064902
HADES CC, 2 A GeV
SPA implementation in IQMD ee- bremsstrahlung
production in each NN collision (i.e. elastic and
inelastic) ! - differs from HSD and UrQMD98
(only elastic NN collisions are counted!)
28
Bremsstrahlung in BRoBUU (Rossendorf)
H.W. Barz, B. Kämpfer, Gy. Wolf, M. Zetenyi,
nucl-th/0605036
SPA implementation in BRoBUU ee- production
in each NN collision (i.e. elastic and inelastic)
! - similar to IQMD (Nantes)
29
Summary II

• Transport models give similar results ONLY with
  the same
• initial input !
• gt 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 ee-
• bremsstrahlung from NN and mN collisions!

30
Part II

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

31
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
• gt h-Dalitz decay contribution is under control !

E.L.B. W. Cassing, NPA 807 (2008) 214
32
D-Dalitz decay
Original paper H.F. Jones, M.D. Scadron, Ann.
Phys. 81 (1973) 1
33
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

34
D-spectral function
The main 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 !
35
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
• gt p (D), h dynamics under control !

E.L.B. W. Cassing, NPA 807 (2008) 214
36
pp _at_ 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 pp data!

E.L.B. W. Cassing, NPA 807 (2008) 214
37
Quasi-free pn (pd) reaction HADES data _at_ 1.25
GeV
?

• HSD predictions underestimates the HADES pn
  (quasi-free) data at 1.25 GeV
• 0.2ltMlt0.55 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 pd results at higher
  energies!
• In HSD pd p (pn)-with Fermi motion
  according to the Paris deuteron wave function

38
Quasi-free pn (pd) _at_ 1.25 GeV h-channel
Add the following channels
1) p n ?d ?
2) d p ?pspec d ?
Now HSD agrees with PLUTO on the h- Dalitz decay!
39
Quasi-free pn (pd) _at_ 1.25 GeV N(1520) ?!
N(1520)
2) M gt 0.45 GeV HSD preliminary result
for pd _at_1.25 GeV shows that the missing yield
might be attributed to subthreshold r-production
via N(1520) excitation and decay ?!
Similar to our NPA686 (2001) 568
N(1520)
Model for N(1520) according to Peters et al.,
NPA632 (1998) 109
40
Ratio pd/pp _at_ 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 !
41
Outlook
HADES succeeded the DLS puzzle is solved !
Outlook-1 need new pp, pd and pN data from HADES
for a final check!
Outlook-2 study in-medium effects with HADES
42
Thanks to
HADES collegues Yvonne, Gosia, Romain, Piotr,
Joachim, Tatyana, Volker Wolfgang


43
- More slides -
44
Dynamics of heavy-ion collisions gt complicated
many-body problem!
Correct way to solve the many-body problem
including all quantum mechanical features ?
Kadanoff-Baym equations for Green functions Slt
(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 to first order in
the gradients a standard approximation of
kinetic theory which is justified if the
gradients in the mean spacial coordinate X
are small
45
NN bremsstrahlung OBE-model
KaptariKämpfer, NPA 764 (2006) 338
OBE-model N N -gt N N ee-
post
pre
pre
post
gauge terms
The strategy to restore gauge invariance is model
dependent!
charged meson exchange
contact terms (from formfactors)
46
Test in HSDbremsstrahlung production in NN
collisions (only elastic vs. all)
In HSD assume ee- production from old SPA
bremsstrahlung in each NN collision (i.e. elastic
and inelastic reactions) gt can reproduce the
results by Gy. Wolf et al., i.e. IQMD (Nantes)
and BRoBUU (Rossendorf) !
47
Deuteron in HSD
In HSD pd p (pn) -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