The nuclear equation of state is soft

C. Hartnack and J. AichelinSubatech/University of Nantes

H. OeschlerTechnical University of Darmstadt SQM, march 2006

Why do we want to know it

in Astrophysics?in Nuclear physics?

How we can measure it?

Why it is soft?Is this a robust

statement?

The nuclear equation of state

E/A as a fct of the density(nuclear physics)

Pressure as a fct of density(astrophysics)

T=0

hard

soft

Not theoretically accessible yet (Brückner-Hartree-Fock limited to ρ < ρ0)

Importance for nuclear/hadron physics:

Modifies considerably the energy which is available for particle production

Changes the mass of hadrons considerably

Makes that hadrons are not a gas of noninteracting particles

(and questions therefore the statistical approach to the hadronization)

Is responsible for collective effects like v1 or v2

Life of a type II supernova

Protonneutron star has about the same density as nuclei

Nuclear equation of state influence many astrophysical processesDouble pulsar rotation (astro-ph 0506566)

Binary mergers (astro-ph 0512126)Neutron star formation:

““Neutron” Star Composition in Neutron” Star Composition in 20052005

Σ,Λ

,Ξ,Δ

CFL2SC, ...

strange quark matter K—

(F. Weber, Prog. Part. Nucl. Phys. 54 (2005) 193-288 )

The present simulations of the stellar core collapse modeling suffer from

The complexity of 2D and 3D simulationsBecause several 2D/3D phenomena influence the

shock expansion likeStellar core rotation

ConvectionShock instabilities

and

The incompletely known nuclear equation of stateBecause different EOS’s show significant differences

(Janka, astro-ph/0405289)

Source of information: heavy ion reactions at energies withEbeam< 2 AGeV

Three propositions:

Proposition I:Volume

oscillationsinduced by heavy

ion collisions

Giant Monopol ResonanceA particle with spin = 0 excites the heavy nucleus, the

nucleus vibrates and the vibration frequency is proportional to the compressibility modulus

Most recent result

(nucl-th 0312020)

Compressibility modulus

K= 248 +/- 6 MeV

Density change is tiny Δρ/ρ < 0.01

Proposition IIIn-plane (v1) and elliptic (v2) flow in heavy ion reactions

View perpendicular to the reaction plane

In plane flow v1 at beam/targ

y

Elliptic flow v2 at midrapidity

In the overlap zone the density increases, at its surface the density gradient increases and therefore the pressure.

This pressure creates the flow (Frankfurt)

How can one relate the flow to the compressibility modulus K?Only way:

Simulation of heavy ion reactions on the computerUsing different values of K

comparison with the experimental data should allow to determine K

The IQMD model● Semiclassical microscopic model on an event by event basis● Full time-evolution of each event allows for a view inside the reaction● Includes nucleons, deltas , pions with their isospin degrees of freedom● Nuclear eos, Coulomb, asymmetry, Yukawa and momentum dep. pot.● Virtual treatment of strange particles allows for high statistics

How is IQMD related with an EOS?IQMD uses potentials between nucleons

Therefore it is able to describe non equilibrium processes The potential uses 4 free parameters a,b,c,L

These parameters are adjusted by the following procedure:We calculate in infinite nuclear matter

the expectation value of the Hamiltonian for T=0:E = <H> = a(ρ/ρ0) + b(ρ/ρ0)c

a,b,c are adjusted to reproduce –16 AMeV at ρ=ρ0

and a compressibility modulus K (curvature) at ρ0

L, the width of the Gaussian wavefunctions is chosen to fit bestthe nuclear surface

Having adjusted a,b,c at nuclear matter properties we can use thepotential now also in out of equilibrium processes IQMD

Time-evolution: the basic scales:0 fm/c: start of the reaction

4 fm/c: raise of resonance prod.

8 fm/c: max. central (nuc.)10 fm/c max central 12 fm/c: max number of 14 fm/c: dominate over 16-20 fm/c: nucleon spectra become`thermal'

20 fm/c: number stabilizes

Different in

medium cross

sectionsK=210

Different rangesof VNN

same EOS

No conclusive results yet The flow is a tiny effect which depends very sensitive on many

thingswhich are not completely under control in the transport theories

Different EOS,same range of VNN

Andronic

nucl-ex041102

4

Proposition III : K+ production in heavy ion reactionsWhy do K+ may measure the nuclear equation of state ?

HI reactions around 1AGeV: - heavy nuclei get considerably compressed (ρ >> ρ0)

- sqrt(s) too low to produce a K+ in first chance NN collisions N’s which produce K+ had collisions before by these collisions they test the medium. In these collsions Δ’s are produced, the main source of K+ Higher densities shorter mean free path more Δ’s collide before they disintegrate

Different EOS different density profiles different K+ yield

Light nuclei may serve as benchmark.

INDEED Strong collectivity:Mult/A in AuAu >Mult/A in CCbecause ρ(CC) << ρ(AuAu)

But:Sufficient sensitive to determine the equation of state?

First observation: yield for CC too high (CC = superposition of pp)

Why?

RAA increases by a factor of 5

The K+ mass shift and its consequences for HI reactions

Increase of the K+ mass (nucl-th 0404088)

Selfconsistent (IQMD)

Scattering length

Mass shift around 8% for Au+Au 1.5 AGeV

Mass increases

higher threshold

yield decreases

CC: well reproducedResult independent of EOS

Au+Au:Small differences between soft and hard EOSBut: sufficient to determine EOS?

To enhance sensitivity: consider σ(AuAu)/σ(CC)

K+ production in central Au+Au as compared to CC collisions

shows a strong dependence on the Compressibility modulus

Only a soft equation of state (K around 210 MeV) is

compatible with the data

This result is robust:If one varies input parameters which are not

precisely known N-Delta cross sections

KN-potentialΔ lifetimes

the conclusion does not changeCompletely independent programs give the

same results

Conclusions confirmed by a completely independent

observable:

Number of K+ per Apart

This variable is robust as well

CONCLUSIONS

The simulations of heavy ion experiments agree quantitatively with experiments only if the K+ change their mass

by interactions with the hadronic environment (predicted theoretically).

The ratio σK+(AuAu)/σK+(CC) is sensitive to the compressibility modulus of the hadronic

equation of state

Data agree with simulations for a soft EOS (K = 240 MeV)

Result is robust with respect to little known input parametersand confirmed by impact parameter dependence of the K+ yield. Thus it seems that this long standing problem has been solved.

But remember: what we observe is a hadronic system (10-20% π,Δ) at

at high excitation energy out of equilibrium which we describe with a transport theory whose Hamiltonian gives this EOS in

infinite matterThis is the best we can ever do.

Where are the K+ produced ?

-- around R = 4fm (mid-central)Corresponds to a density of 1.5 ρ/ρ°

More central more K+ rescatting

Shortest way(Perp to the reac.Plane)

Longest way (in the reac.Plane)

Mean free path not really small

What tell heavy ion reactions about a possible K- condensate?

Pons et al (astro-ph/0008389):

« The effects of kaon condensation on metastable stars is dramatic »

« A unique signature for kaon condensation

will be difficult to identify »

Also here accelerator experiments may give interesting results

KN interactions are part of the chiral SU(3) Lagrangien

in the mean field approximation

But not at all for the K-

Reason :strong interaction with the baryons

We have forgotten that K’s interact with the mediumSimplest approach relativistic mean field calculation

KN potentials consequences for K+ in HI coll

Mass shift around 10% for Au+Au 1.5 AGeV

K+ are « heavier », K- « lighter » in the nucl. environment

Influence of the scalar and vector part of the interaction

Vector or scalar only changes the cross section dramaticallyChange of the total yield much easier to observe than the(small) change of the in-plane flow

These excitation functions are all but trivialThe importance of the different channels varies with- beam energy- size of the system

pA are normally a good test groundbut not here:

low momenta: Fermi motion and potentialcompensate: σ(pA) = Nσ(pp)High momenta: KN potential negligible

pA is not sensitive to-- KN potential-- KN cross section

Overall well described basic process understood butthis time we cannot learn much from pA

The ΔN K+ cross section is unknown

Two different approaches conjectured --Tsushima --Isospin scaled NN K+ cross section (here named Giessen)

The two give 60% difference in the K+ yield

How reliable are transport model predicitons?

Trento workshop + many homework assignments in order to clarify the discrepancies between the results.

BarzBratkovskaya (HSD)Cassing (HSD)Chen (C. M. Ko)DanielewiszFuchs (QMD)GaitanosHartnack (IQMD)Larinov (Mosel)Reiter (URQMD)

After several iterations: differences almost exclusively due to different inputs - different (unknown) cross sections (especially in the Δ chanel) - treatment of Δ resonance in medium

Stopping

and

transversemomentum

okFor large systems

Pions make more problems but agree fairly for the same input.

In the standard versions we see differences in the pion yield of more than 70%.Why?: all codes need a Δ lifetime as input

Why problem?: Δ has a width; in order to populate the Δwe need wavepackets with a large width in energy. But in simulation programs we have sharp energies Commonly used descriptions:

1) Kitazoe: 1/τ ~ Phasespace at the given energy 2) Wigner (phase shift) τ ~ 2 d δ(E)/dE Γrex >> Γwave fct

3) τ = 1/ 120 MeV

Dramatic consequences for π’s in HIC

Fortunatelly the different approaches for the Δ lifetime do not change the slope of the K+ spectra but they change the yield at low pt.

Nopotential

KNpotential

K+ : form of dn/dy very similar but differences in yielddue to different cross sections.

ΓΔ =const

Soft EOS

-2 -1 0 1 20.00

0.02

0.04

0.06

soft Eos; fixed width;b=1fm

BarzCassingReiterHartnakLarionovChenFuchsGaitanos

K+

Au+Au0.96 AGeV

dN/dy

y-yc .m.

no potential

-2 -1 0 1 20.0

0.2

0.4

soft E os; fixed width;b=1fm

BarzCassingReiterHartnakLarionovChenFuchs

K+Au+Au 1.48 AGeV

dN/dy

y-yc.m.

no potential

-2 -1 0 1 20.0

0.1

0.2

0.3

soft E os; fixed width;b=1fm

BarzCassingReiterHartnakLarionovChenFuchsGaitanos

K+Ni+Ni 1.93 AGeV

dN/dy

y-yc.m.

no potential

-2 -1 0 1 20.00

0.02

0.04

0.06

soft E os; fixed width;b=1fmBarzCassingHartnakLarionovChenFuchs

K+Au+Au0.96 AGeV

dN/dy

y-yc.m.

with potential

-2 -1 0 1 20.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

soft Eos; fixed width;b=1fm

BarzCassingHartnakLarionovChenFuchs

K+Au+Au 1.48 AGeV

dN/dy

y-yc.m.

with potential

-2 -1 0 1 20.00

0.05

0.10

0.15

0.20

soft E os; fixed width;b=1fm

BarzCassingHartnakLarionovChenFuchs

K+Ni+Ni 1.93 AGeV

dN/dy

y-yc .m.

with potential

1

pT slopes in good agreement. Not trivial : phase space of the NN(Δ) NΛK+ collisions. + Fermi motion + KN collisions

The calculations show a squeeze due to the KN potential in heavy systems

Good news: even if yields vary by 30% all calculations (even using different cross sections, τ

Δ etc) point towards

a soft nuclear equation of state

This is the first time that we have solid information on the nuclearequation of state.

The result of a soft equation of state is very robust

KN potential +/- π lifetime BB K+ cross section

What’s about the K- ? K- are much more complicated than K+

Resonances in the K- N channel - which may disappear in the medium (Λ(1405) Koch, Weise)

More cross sections Λ+π N+ K-

because the s quark can be transferedto baryons. These cross sections diverge close to the threshold. (not included in Chen)Complicated in medium propertiesDue to the coupling to the resonances(Lutz, Kolomeitsev, Tolos….)

Still quasi particles?

There remains work to be done

K- rapidity distributions

Present status (or on what all agree):- Large difference between pp and AA is due to the production channel Λ+π N+ K- absent in pp- This cross section dominates the final yield- It couples the K- to the K+ yield

- K- have a steady state equilibrium due to the hugh difference of Λ+π N+ K- and N+ K- Λ+π cross sec. close to threshold

Influence of the K- and K+

potential on the final K- yield:

K+ on/off : more or less Λ to create the K-

K- on/off : varies the threshold

Both yield a factor of two:K- on /K+ on yield the same result as K- off / K+ off

Can one see the K potentials and cross sections directly?

Yes,When leaving the nucleusK+ gain and K- loose energy

therefore spectra distortedin a specific way

σ(K+,pT ) / σ( K-,pT ) atsmall pT is sensitive to theK potentials

Collisions change the slope:Slope mesures the KN crosssections

Influence of the KN collisions

Changes the slope remarkably

Influence of the KN potential

Changes the yield at small Ecm

Slope of the K+ spectra « measures » the number of KN collisions

SPS and RHIC energies

Simulation programs still in development (EPOS) and URQMD Strategy on which most agree:

Start with pp pA to have a known environment AA

For RHIC: to early, even the elementary degrees of freedom are still discussed (parton casc, strings, CCC, hadronic rescatt.)

In the non strange sector (data available) reasonable agreement

The more strange the particles the more differ the predicitonsDoes this allow to find the right process or are there parameters to fix pp yields

differ by a factor of

4 for Ω3 for antiΛΣ3 for Ξ 1.5 for K, ΛΣ

Pb +Pb 160 AGeV:One sees quite different reaction scenarios

Hijing/ HSD much more transparent than EPOS/URQMD

Hugh difference in the energy deposit (but almost not visiblein the pion yield).

For the (multi) strange sector the differences become enormous due to the différent reaction mechanisms

Ω(AA)/Ω(pp)

Hijing 58Epos 1000Urqmd1.3 1190 Urqmd2.1 4222

Λ yields differby a factor 2

Ξ yield by afactor of 28

Λ/antiΛ ratiosby 4

Conclusions: Theory predict without doubt that hadrons change their properties in a hadronic environment. Models are reliable for ρ<ρ°

Baryons and Mesons react differently

Transport theories have seen a lot of progress K+ and K- production are coupled (NKπΛ) The present knowledge of elem.σ requires effective masses

Data are only compatible with a soft hadronic « EOS » this result seems to be robust.

to progress further: new transport theories for spectral functions more data on pp, np, pA to determine input σ

Conclusions:At 1 AGeV: lot of progress due to intense collaborations data are only compatible with a soft nuclear EOS KN potential and cross section may be seen in spectra to progress further: new transport theories with spectral functions more data on pp, np, pA to determine input σ

At 200 AGeV (SPS): Comparison of simulation codes have just started presently hugh differences in the strangeness sector extrapolation pp/AA yields very different results to progress further: pp,pA strangeness dn/dy(dx) data urgently needed determination of observables sensitive to the proposed mechanisms At RHIC: Ideas and manpower needed, but we have no choice

Thanks to

E. Kolomeitsev and C. Hartnack who made the figures

Barz Gaitanos Bleicher (URQMD) Hartnack(IQMD)Bratkovskaya (HSD) Larinov (Mosel) Cassing (HSD) Reiter(URQMD)Chen (C. M. Ko) Topor Pop (Hijing) Danielewisz Werner(Epos)Fuchs (QMD)

The present level of data theory comparison