The equation of state from K

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The equation of state from K

• Christoph Hartnack and Jörg Aichelin
• Subatech, Nantes

Song of joy on a marriage of strangeness with the
properties of nuclear matter
Trento, May,2009
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Contents of the songbook

• I did it my way
• some words on (my) IQMD
• Those were the days my friend...
• Some recall on the nuclear eos
• Strangers in the night..
• Kaon production
• I will survive.
• How the strangeness signal remains robust
• My heart will go on..
• Summary and Outlook
• Total play timearound 40-50 minutes

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IQMD - some lyrics
And now, the end is near And so I face the final
curtain. My friend, I'll say it clear, I'll state
my case, of which I'm certain. I've lived a life
that's full. I travelled each and ev'ry
highway And more, much more than this, I did it
my way. Regrets, I've had a few But then again,
too few to mention. I did what I had to do And
saw it through without exemption. I planned each
charted course Each careful step along the
byway, And more, much more than this, I did it my
way. Yes, there were times, I'm sure you knew
When I bit off more than I could chew. But
through it all, when there was doubt, I ate it up
and spit it out I faced it all and I stood
tall And did it my way. I've loved, I've laughed
and cried. I've had my fill, my share of losing.
And now, as tears subside, I find it all so
amusing. To think I did all that And may I say -
not in a shy way ,"Oh no, oh no not me, I did it
my way". For what is a man, what has he got? If
not himself, then he has naught To say the things
he truly feels And not the words of one who
kneels. The record shows I took the blows And did
it my way

• - semiclassical model with
• quantum features
• - microscopic N-body description
• - calculation of heavy ion collisions on an
  event-by-event-basis
• -includes N, D, p with isospin d.o.f.
• - strange particles treated virtually
• 2 and 3 body potentials including nuclear
  equation of state, Coulomb...

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The nuclear equation of state (Skyrme)
3 parameters, 2 ground state condit. 1 remaining
d.o.f. compress. mod. K Artificial link between
curverture at ground state and high density
behaviour. Compression modulus Kgt170
MeV Problems of causality for high densities rgt
5-7 r0 Caution when extrapolating to high
densities
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Definition of the potentials
2 and 3 body interactions (no equilibrium
required)
Bethe Weizsaecker mass formula
symmetry term
Volume term (with eos)
Surface term
Coulomb term
(pairing term not included)
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The eos in IQMD
after the convolution of the Skyrme type
potentials supplemented by momentum dependent
interactions (mdi) for infinite saturated nuclear
matter at equilibrium
soft
hard
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The original idea of measuring the eos

• Eos describes the energy needed to compress
  nuclear matter
• A hard eos requires more energy for a given
  density than a soft one
• For a given density and a given available energy
  a soft eos leaves more thermal energy to the
  system than a hard eos
• R.Stock This thermal energy could be measured by
  regarding pion production

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At which density should we compare ?
Hard and soft eos reach different maximum
densities and the pion numbers are only slightly
different. For a small system the differences in
density vanish. The differences in pion yield as
well
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Different densities and different pressure
hard
soft
Next idea on eos do not use the compressional
energy but the repulsion of the potential
Nucleonic flow squeeze
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Out/in ratio of energy claims for a soft eos
Ratio of total energy out-of-plane to in-plane
raises linear with system size collective effect
Published 1994 Mod. Phys. Lett. A 9 (1994) 1151
By the way there was the influence of the eos to
some (pX2-pY2)/pT2 which lateron would be called
2v2 but I do not have the figure at hand
The neutron squeeze-out (LAND) has also been
described
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Systematics of 2v2
Influence of eos but also of cross
sections Protons and neutrons identical within
statistical errors (Easy32 MeV, linear ?
dependence)
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and now the directed flow
More visible difference between protons and
neutrons. Perhaps influence of asymmetry-eos, but
remember the cross-sections, force ranges,
density profiles etc
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and why not the pion flow ??
S.A. Bass stated in the mid 90ies that the flow
of pions in central events was depending on the
eos
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Pion flow sensitive on the eos, but the deltas
play the game

• Pion flow in central collisions is caused by the
  N-D potentials
• Importance of D lifetime (exposure time to
  potentials)
• May also apply to asymmetry-energy

?t
In central collision only if ND pot actif
Coulomb
eos
Hard Soft
N-?-pot
p
In peripheral coll rescattering yields negative
flow
p0
p0
p-
Bass et al.
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In-plane flow, Squeeze, pion flow
X
Z
Test of density gradient and geometry Transverse
flow dominated by cold  matter Dense matter
tends towards isotropy Pion flow test on
resonance matter Comparison of Plasticball
squeeze favors soft eosmdi
X
j
Y
For recent analysis on FOPI data see the
contribution of Willi Reisdorf
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Stopping depending on the eos
but much more on the cross sections

• QZZ 3pz2-p22pZ2-pT2 gt 0 longitudinal
  dominant
• lt 0 transversal dominant

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Studying stopping via vartl
Again the cross sections dominate the signal
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Stopping versus flow
hard
soft
Difference of eos but large fluctuations
pXdir
GSI-report 93-05
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Is there a signal directly related to the density?
The maximum density is increasing when softening
the eos. However there are large fluctuations
possible. Kaons are produced at high density and
may serve as signal
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Why kaons?
Strange particles that may not be reabsorbed
after production due to strangeness
conservation. Only weak decay possible
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Production of kaons in elementary processes

• Reactions governed by strangeness conservation
• Kaon-hyperon production energeticly preferable
  to kaon-antikaon production

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Subthreshold kaon production

• Production of kaons at energies below the kinetic
  threshold for K production in elementary pp
  collisions
• Fermi momenta may contribute in energy
• Multistep processes can cumulate the energy
  needed for kaon production
• Importance of resonances for storing energy

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Subthreshold K production as a signal

• Dominance of ND channel (2 steps
  at least)
• Short mean free path required
• Need of high densities for producing kaons
• Eos yields differences in maximum densities
• This may yield differences in the kaon yield

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High density medium effects

• KN-Rescattering
• Optical potential (repulsive for K)

Several parametrizations are implemented Choose
Schaffner-Bielich RMF results as standard
Optical potential influences kaon propagation but
changes also the production threshold Influence
on total kaon yield reduction for K
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Kaons test high densities
Multistep processes require high densities, but
in medium effects penalize the high density
production
soft
hard
The penalty from KN pot reduces the effect but
the sensitivity to the eos still survives
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And we see an effect of the eos in the yields
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To be honest we should also talk about s
and this is not the only thing effecting the
yields
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Different transport models different ingredients
E.E.Kolomeitsev et al.
Need of more experimental input for elementary
reactions of BB MB

• Differences in unknown production cross sections
• Differences in delta lifetimes, potentials etc

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Many channels have to be implemented

• Each channel contains isospin subdivisions
• Only few channels (like pp?pLK) are measured by
  the experiment (even incomplete infos)
• Significant incertainties from parametrization of
  unknown channels or isospin subdivisions

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Eos cannot be deduced directly from kaon yields!
Incertainties of cross sections larger than eos
effect
AuAu
sNDTsushima sND.75 sNN
CC
However, the eos effect vanishes for small A
while the cross section effect persists up to
small A.
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The solution use ratios Au/C
KaoS data support soft eos
Data Ch.Sturm et al. RQMD Ch. Fuchs
IQMD supports this
Different models agree 10 for soft
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A observation which is robust
sNDTsushima sND.75 sNN
versus effects of production cross sections,
KN-potential, less stopping (reduced sNN) ,
lifetime of the ?,
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Analysis at lower beam energy
A soft equation of state is favoured.Acceptation
range for K.
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Au central versus peripheral
Different cross sections and potential parameters
may change the global yield. However, the
parameter a for the increase of the kaon yield N
with the number A of participating nucleons
(raising with centrality) N(K)N0 Aa depends
on the eos. A soft eos yields higher values than
a hard eos.
central
peripheral
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Determination of the eos from a
The relation between the compression modulus and
a is monotonously falling. KaoS data (Förster et
al.) favor a value below 200 MeV, i.e. a soft
eos.
KaoSFörster et al.
hard
soft
20 for soft
PRL 96 (2006) 012302
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System size dependence
Einc
A soft eos obtains higher kaon yields for heavy
systems
1.8
1.5
KaoS PRC in preparation
1.2
1.0
0.8
0.6
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Energy dependence of the system size systematics
Soft eos confirmed
System size Apart in AuAu agrees with that
Preliminary
30 for soft
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Nostalgic feelings density isomers

• K could reveal density isomers by a sudden rise
  in the excitation function of kaons - KaoS might
  measure it

A 2ndminimum would yield a sudden factor of 10 in
the kaon yield
Density isomers yield up to factors of 10 in K
production
800MeV
600
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KaoS DATA no isomer up to 3r0
A density isomer would have needed the strong
raise indicated by the arrows. IQMD calculations
using a KN optical potential and a soft eos are
consistent with KaoS data on AuAu and CC of
Sturm et al.
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Conclusions on Kaon data

• In a range up to 1.5 AGeV the kaon data are
  consistent with a soft equation of state
• Prediction by different models
• Robust against various modifications of KN-pot,
  s,etc
• Principle of scaling laws might be applied to
  other observables?
• The kaon excitation function of KaoS gives no
  rise for the existence of density isomers up to
  3r0 . The exitation function of E895
  seems to prolong this statement to even higher
  densities.

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Outlook
 Is now everything solved? 

• Assuming to know the static part of the volume
  term of equilibrated and saturated infinite
  nuclear matter for
• T0 we still have to take into account
• finite size and surface effects
• symmetry and isospin effects
• finite temperature effects
• resonance matter
• Still a lot of topics for those who are
  interested

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Thank you !!!
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Transport models different ingredients
E.E.Kolomeitsev et al.
Need of more experimental input for elementary
reactions of BB MB

• Differences in unknown production cross sections
• Differences in delta lifetimes, potentials etc

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but still allowing conclusions
Ch. Fuchs et al

• Hint on the existence of an optical KN potential

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Flow comes from low densities
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Time-evolution the basic scales
0 fm/c start of the reaction
4 fm/c raise of resonance prod.
8 fm/c max. central r(nuc.) 10 fm/c max central
r(p)
12 fm/c max number of D 14 fm/c p dominate
over D
16-20 fm/c nucleon spectra
become thermal'
20 fm/c p number stabilizes
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Time profile of kaons
max of K production at about 7-8fm/c max of
last K interaction at about 10fm/c K
production ending at about 15 fm/c K collide up
to about 30 fm/c
Importance of KN collisions for understanding
dynamics
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Going down in beam energy
A soft eos yields a ?1.4 at E0.8 AGeV, a hard
eos yields a ?1.2 Limits for lower E no
asymptotic yield for peripheral collisions
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A observation which is robust
sNDTsushima sND.75 sNN
versus effects of production cross sections,
KN-potential, D-lifetime even less stopping
(reduced sNN) does not change the effect
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Virtual propagation