Offshell Transport

1
Off-shell Transport
Frankfurt Institute for Advanced Studies
Elena Bratkovskaya 27.04.2007 , VI-SIM
Workshop GSI, Darmstadt
2
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?
3
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 Slt
(from 1962)
e.g. for bosons
Greens functions S / self-energies S
retarded (ret), advanced (adv) (anti-)causal (a,c
)

• do Wigner transformation

N -number distribution A - spectral function G -
reaction rate of particle (at phase-space
position XP)
Greens function Slt characterises the number of
particles (N) and their properties (A spectral
function )
4
From Kadanoff-Baym equations to transport
equations
Generalized transport equations the first order
gradient expansion of the Wigner transformed
Kadanoff-Baym equations
drift term
Vlasov term
Operator ltgt - 4-dimentional generalizaton of the
Poisson-bracket
backflow term
collision term gain term - loss term
The imaginary part of the retarded propagator is
given by normalized spectral function
For bosons in first order in gradient expansion
GXP width of spectral function reaction rate
of particle (at phase-space position XP)
To solve transport equation use a testparticle
ansatz for the real quantity FXPM which
represents the probability to find a particle in
a given phase-space cell
W. Cassing et al., NPA 665 (2000) 377 672 (2000)
417 677 (2000) 445
5
On-shell transport models
Basic concept of the on-shell transport models
(VUU, BUU, QMD etc. ) 1) Transport equations
first order gradient expansion of the Wigner
transformed Kadanoff-Baym equations 2)
quasiparticle approximation A(x,P) 2 p d(p2-M2)

• 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

6
Semi-classical transport models

• VUU, BUU, QMD etc. transport models are
  semi-classical !
• (first order gradient expansion of the Wigner
  transformed Kadanoff-Baym equations within
  quasiparticle approximation )
• Where are quantum effects ?
• - in Pauli blocking for baryons
• - in matrix elements for elementary
  processes
• Semi-classical on-shell transport models work
  very well in describing interactions of
  point-like particles and narrow resonances !
• In-medium effects gt strong broadening of
  spectral functions
• How to treat short-lived (broad) resonances in
  semi-classical
• transport models?

gt semi-...
7
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 thought the medium within
  on-shell BUU model
• broad in-medium spectral function does not
  become on-shell in vacuum in on-shell transport
  models!

GiBUU M. Effenberger et al, PRC60 (1999)
8
Problems in treatment of short-lived resonances
in the on-shell semi-classical transport models

• Problem
• dynamical changes of spectral function by
  propagation through the medium are NOT included
  in the on-shell semi-classical transport
  equations !
• the resonance spectral function can be changed
  only due to explicit collisions with other
  particles in the on-shell semi-classical
  transport models !
• Reason for the problem
• some term - backflow term - is missing in the
  explicit on-shell dynamical equations since
  this backflow term vanishes in the on-shell
  limit, however, NOT vanishes in the off-shell
  limit (i.e. becomes very important for the
  dynamics of broad resonances)!

Generalized transport equations
Operator ltgt - 4-dimentional generalizaton of the
Poisson-bracket
drift term
Vlasov term
backflow term
gain term loss term
W. Cassing et al., NPA 665 (2000) 377
9
From on-shell to off-shell transport dynamics
W. Cassing et al., NPA 665 (2000) 377 672 (2000)
417 677 (2000) 445

• Off-shell transport approach
• Generalized transport equations on the basis of
  the Kadanoff-Baym equations for
• Greens functions (accounting for the first order
  gradient expansion of the Wigner
• transformed Kadanoff-Baym equations beyond the
  quasiparticle approximation )
• Dynamical equations of motion
• for test-particle propagation in
• 8-dimensional (not 7 dimensional
• as for on-shell limit) phase space
• (r(t), p(t), E(t))

The off-shell spectral function becomes on-shell
in the vacuum dynamically by propagation through
the medium!
10
Spectral function in off-shell transport model
(1) Collision term for reaction 12-gt34
(2) Collisional width of the particle in the rest
frame (keep only loss term in eq.(1))
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

11
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
  in-medium off-shell cross sections s(s,r)

Application to dileptons In-medium transition
rates with momentum and density dependent
dynamical spectral functions of vector mesons
E.L..B., NPA 686
(2001), HSD predictions for HADES and CBM
(2006)
12
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

13
Dileptons from CC at 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

14
Dilepton pT and y spectra from CC at 2 A GeV
HADES
Mlt0.15 GeV/c2
Mgt0.55 GeV/c2
0.15ltMlt0.55 GeV/c2
Preliminary HADES data

• HSD predictions for pT and y spectra are in good
  agreement with HADES data for all M-bins!

Plots from Sudol Malgorzata
15
Dileptons from CC at 1 A GeV predictions for
HADES

• HSD shows a good agreement with TAPS data on p
  and h production

Enhancement factor

• Dominant contributions at 0.15ltMlt0.5 GeV
• 
  D-Dalitz decay and h-Dalitz decay
• In-medium effects broadening of r/w peak

16
10 years of DLS puzzle
HSD98

• DLS-puzzle (since 1997)
• good description by transport models of pp and
  pd data from 1 to 5 GeV
• missing yield at 0.15ltMlt0.5 GeV for CC and CaCa
  at 1.04 A GeV
• new (1997) DLS data for CC and CaCa at 1.04 A
  GeV are higher than the old (1995) DLS data by
  the same factor
• Accounting for in-medium effects does not provide
  enough enhancement at intermediate M!

HSD07
17
HADES_at_GSI and DLS_at_LBL
p0, ? acceptance
Detector design
Dilepton yield
p0?ee?
Hades DLS
??ee?
Hades DLS

• a direct comparison between HADES and DLS
  dielectron results not feasible
• very strong influence of the acceptance on the
  final shape of the dilepton spectra

18
DLS acceptance

• Not possible to reconstruct the DLS acceptance
  by only geometrical cuts (200ltqzlt600, qxlt70)
  due to the strong distortion of electron/positron
  trajectories by the magnetic fields

DLS acceptance (available from WEB v. 4.1,
August 1997)
Acc(M,ylab,pT) 0 or 1 has been build for
virtual photons, not single e, e-, using
Monte-Carlo simulation program (analogue to
PLUTO) gt some assumptions on dilepton sources
have been used

• Assumptions on anisotropic emission of e, e-
• give a factor of 2 in dilepton yield!

N 1 a cos2q
19
I. Consistency of the DLS acceptance M, pT

• Invariant mass spectra

• pT-distribution

Centrality selection

• HSD underestimates M and pT spectra for CC at 1
  A GeV

20
II. Consistency of the DLS acceptance - y

• y-distribution

• HSD overestimates y-spectra integrated over all
  M but is in a good agreement with the DLS y-data
  for Mgt0.25 GeV/c

gt what is going wrong?! Integrated DLS spectra
I(y) lt I(M)I(pT) by a factor of 3 ! Warning
DLS data for pT and y spectra for CC at 1 A GeV
are unpublished - taken from Ph.D. thesis by M.
Prunet (1995)
21
Exitation function of dilepton yields

• Dileptons are an ideal probe for vector meson
  spectroscopy in the nuclear medium and for the
  nuclear dynamics !

Dileptons excitation function for central AuAu
HSD NPA 674 (2000) 249

• In-medium effects can be observed at all
  energies from SIS to RHIC
• The shape of the theoretical dilepton yield
  depends on the actual model
• for the in-medium spectral function
• gt energy scan will allow to distinguish
  in-medium scenarios

22
Summary
Accounting of in-medium effects requires
off-shell transport models !

• Last millenium on-shell transport models (QMD,
  BUU)
• This millenium development of a new generation
  of
• transport
  theories/codes
• One of the first steps off-shell transport
  approach
• now default version of HSD
• also taken up by other groups, e.g. Rossendorf
  BUU
• Future/present developments modelling of phase
  transition based
• on QCD EoS and off-shell parton transport (PHSD)

23
----- END ------
24
Dilepton channels in HSD
all particles decaying to dileptons are firstly
produced in BB, mB or mm collisions
Factorization of diagramm
25
Dilepton channels in HSD
Note all decaying particles are firstly produced
in BB, mB or mm collisions
26
Dileptons in HSD

• The dilepton (ee- or mm-) spectra are
  calculated perturbatively with the time
  integration method.
• What is perturbative or the weighted method?
• Method for rare particle production (i.e.
  dileptons, strange particles at low energies,
  charm particles) in order to increase statistics
• In each NN or mN or mm collision the
  perturbative particles are produced with an
  individual weight.
• Weight probability to produce the particle of
  type i in NN collision
• wis(s)NN-gtiX /
  s(s)NNinelastic
• The perturbative particles scatter elastically
  and may participate in inelastic
• collisions, but do not change the momenta of
  nonperturbative particles (with
• weight1)!

27
Time integration method for dileptons
Reality
e
only ONE ee- pair with probability
Br(r-gtee-)4.5 .10-5
r
r
w
e-
Virtual time int. method
t0
tabs
e
r
time
e-
tF
Calculate probability P(t) to emit 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
28
The time integration method for dileptons in HSD
Dilepton emission rate
e
r
e-
t00
time
tF
Dilepton invariant mass spectra
0 lt t lt tF
0 lt t lt infinity
29
Dileptons HSD predictions for CBM
HSD predictions
In-medium modifications of ee- and mm- spectra
are very similar!