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arXiv:nucl-th/0607035v1 17 Jul 2006

Isospin Dynamics in Heavy Ion Collisions

M.Di Toro

Laboratori Nazionali del Sud INFN,

Via S.Sofia 62, 95123 Catania, Italy,

and Physics-Astronomy Dept., University of Catania,

E-mail: ditoro@lns.infn.it

Some isospin dynamics results in heavy ion collisions from low to relativistic energies obtained

through transport approaches, largely inspired by David M.Brink, are rewieved. At very low ener-

gies, just above the Coulomb barrier, the stimulating implications of the prompt dipole radiation in

dissipative collisions of ions with large isospin asymmetries are discussed. We pass then to the very

rich phenomenology of isospin effects on heavy ion reactions at intermediate energies (few A GeV

range). We show that it can allow a “direct” study of the covariant structure of the isovector in-

teraction in the hadron medium. We work within a relativistic transport frame, beyond a cascade

picture, consistently derived from effective Lagrangians, where isospin effects are accounted for in

the mean field and collision terms. Rather sensitive observables are proposed from collective flows

(“differential” flows) and from pion/kaon production (π−/π+, K0/K+yields). For the latter point

relevant non-equilibrium effects are stressed. The possibility of the transition to a mixed hadron-

quark phase, at high baryon and isospin density, is finally suggested. Some signatures could come

from an expected “neutron trapping” effect. The importance of violent collision experiments with

radioactive beams, from few AMeV to few AGeV , is stressed.

I.INTRODUCTION: THE DAVID LEGACY

It is a great pleasure and honor to contribute to this

special day devoted to celebrate the 75th anniversary of

David Brink. After many years of acquaintance and col-

laboration I would like to state a few points I mostly

got from him: 1.) The Mean Field “moves” the world,

particularly true in nuclear dynamics, as I will show in

examples discussed here; 2.)

nuclear structure and reactions; 3.) Play always a deep

attention to the suggestions of young people; 4.) Fol-

low a “sensible” behaviour towards the political choices,

i.e. try to change something only when you can count

on something better. Here of course I will focus on the

physics part showing a series of results broadly inspired

by the David ideas.

The role of relativity in

In the last years the isospin dynamics has gained a lot

of interest, as we can see from the development of new

heavy ion facilities (radioactive beams), for the unique

possibilities of probing the isovector in medium interac-

tion far from saturation, relevant for the structure of un-

stable elements as well as for nuclear astrophysics see the

recent reviews [1, 2].

Here I will show some selected results of the mean

field transport approach in a wide energy range, from

few AMeV to few AGeV , in non-relativistic and rela-

tivistic frames. At low energies I will discuss the isospin

equilibration in dissipative collisions, fusion and deep-

inelastic, through a related observable, the Prompt Col-

lective Dipole Radiation. At high energies I will shortly

present isospin effects on collective flows, on particle pro-

duction and finally on the transition to a mixed hadron-

quark phase at high baryon density.

II.THE PROMPT DIPOLE γ-RAY EMISSION

The possibility of an entrance channel bremsstrahlung

dipole radiation due to an initial different N/Z distri-

bution was suggested at the beginning of the nineties

[3, 4], largely inspired by David discussions.

time a large debate was present on the disappearing of

Hot Giant Dipole Resonances in fusion reactions. David

was suggesting the simple argument that a GDR needs

time to be built in a hot compound nucleus, meanwhile

the system will cool down by neutron emission and the

GDR photons will show up at lower temperature. The

natural consequence suggested in [3] was that we would

expect a new dipole emission, in addition to the statisti-

cal one, if some pre-compound collective dipole mode is

present. After several experimental evidences, in fusion

as well as in deep-inelastic reactions [5, 6, 7, 8, 9] we have

now a good understanding of the process and stimulating

new perspectives from the use of radioactive beams.

During the charge equilibration process taking place in

the first stages of dissipative reactions between colliding

ions with different N/Z ratios, a large amplitude dipole

collective motion develops in the composite dinuclear sys-

tem, the so-called dynamical dipole mode. This collec-

tive dipole gives rise to a prompt γ-ray emission which

depends: i) on the absolute value of the intial dipole mo-

ment

At that

D(t = 0) =NZ

A

|RZ(t = 0) − RN(t = 0)| =

Rp+ Rt

A

ZpZt

????(N

Z)t− (N

Z)p

????,(1)

being RZ and RN the center of mass of protons and of

neutrons respectively, while Rpand Rtare the projectile

and target radii; ii) on the fusion/deep-inelastic dynam-

ics; iii) on the symmetry term, below saturation, that is

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2

FIG. 1:

D(t) (solid lines) and p−space DK(t) (dashed lines, in fm−1)

and the correlation DK(t) − D(t) at incident energy of

6AMeV and 9AMeV for b = 2fm.

The time evolution of the dipole mode in r−space

acting as a restoring force.

A detailed description is obtained in a microscopic

approach based on semiclassical transport equations, of

Vlasov type, introduced in the nuclear dynamics in col-

laboration with David [10], where mean field and two-

body collisions are treated in a self-consistent way, see

details in [11]. Realistic effective interactions of Skyrme

type are used. The numerical accuracy of the transport

code has been largely improved in order to have reliable

results also at low energies, just above the threshold for

fusion reactions [12, 13]. The resulting physical picture

is in good agreement with quantum Time-Dependent-

Hartree-Fock calculation [14]. In particular we can study

in detail how a collective dipole oscillation develops in

the entrance channel [13].

First, during the approaching phase, the two partners,

overcoming the Coulomb barrier, still keep their own re-

sponse. Then follows a dinuclear phase where the relative

motion energy, due to the nucleon exchange, is converted

in thermal motion and in the collective energy of the

dinuclear mean field. In fact the composite system is not

fully equilibrated and manifests, as a whole, a large am-

plitude dipole collective motion. Finally thermally equi-

librated reaction products are formed, with consequent

statistical particle/radiation emissions.

We present here some results for the

(N/Z asymmetric) reaction at 6 and 9 AMeV , recently

studied vs. the “symmetric”36S +96Mo counterpart in

ref.[9]. In Fig.1 (left columns) we draw the time evolu-

tion for b = 2fm of the dipole moment in the r-space

(solid lines), D(t) =

AX(t) and in p−space (dashed

lines), DK(t) = Π/?, where Π =

Pp(Pn) center of mass in momentum space for protons

(neutrons), is just the canonically conjugate momentum

32S +100Mo

NZ

NZ

A(Pp

Z−Pn

N), with

FIG. 2: Density plots of the neck dynamics for the32S +100Mo

system at incident energy of 6AMeV and 9AMeV .

of the X coordinate, see [13, 14, 15].

hand side columns we show the corresponding correla-

tion DK(t) − D(t) in the phase space. We choose the

origin of time at the beginning of the dinuclear phase.

The nice ”spiral-correlation” clearly denotes the collec-

tive nature of the mode. From Fig.1 we note that the

”spiral-correlation”starts when the initial dipole moment

D(t = 0), the geometrical value at the touching point, is

already largely quenched. This is the reason why the

dinucleus dipole yield is not simply given by the ”static”

estimation but the reaction dynamics has a large influ-

ence on it.

A clear energy dependence of the dynamical dipole

mode is evidenced with a net increase when we pass from

6AMeV to 9AMeV . A possible explanation of this ef-

fect is due to the fact that at lower energy, just above

the Coulomb barrier, a longer transition to a dinuclear

configuration is required which hinders the isovector col-

lective response. From Fig.2 a slower dynamics of the

neck during the first 40fm/c − 60fm/c from the touch-

ing configuration is observed at 6 AMeV .

collective dipole response sets in the charge is already

partially equilibrated via random nucleon exchange.

The bremsstrahlung spectra shown in Fig.3 support

this interpretation.

In fact from the dipole evolution given from the

Vlasov transport we can directly apply a bremsstrahlung

(”bremss”) approach [15] to estimate the “direct” photon

emission probability (Eγ= ?ω):

On the right

When the

dP

dEγ

=

2e2

3π?c3Eγ|D′′(ω)|2, (2)

where D′′(ω) is the Fourier transform of the dipole accel-

eration D′′(t). We remark that in this way it is possible

to evaluate, in absolute values, the corresponding pre-

equilibrium photon emission. In the same Fig.3 we show

statistical GDR emissions from the final excited residue.

We see that at the higher energy the prompt emission

represents a large fraction of the total dipole radiation.

In the Table we report the present status of the Dy-

namical Dipole data, obtained from fusion reactions.. We

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3

TABLE I: The percent increase of the intensity in the linearized γ-ray spectra at the compound nucleus GDR energy region

(the energy integration limits are given in the parenthesis), the compound nucleus excitation energy, the initial dipole moment

D(t = 0) and the initial mass asymmetry ∆ for each reaction.

Reaction

40Ca+100Mo

36S+104Pd

16O+98Mo

48Ti +64Ni

32S+100Mo

36S+96Mo

32S+100Mo

36S+96Mo

Increase (%)

16 (8,18)

E∗(MeV)

71

71

110

110

117

117

173.5

173.5

D(t = 0) (fm)

22.1

0.5

8.4

5.2

18.2

1.7

18.2

1.7

∆Ref

[5]0.15

0.17

0.29

0.05

0.19

0.16

0.19

0.16

36 (8,20)[6]

1.6 ±2.0 (8,21)[9]

25 (8,21)[7]

FIG. 3: The bremsstrahlung spectra for the32S +100Mo sys-

tem at incident energy of 6AMeV and 9AMeV (solid line)

and the first step statistical spectrum (dashed line) for three

impact parameters.

note the dependence of the extra strength on the in-

terplay between initial dipole moment and initial mass

asymmetry: this clearly indicates the relevance of the

fusion dynamics.

We must add a couple of comments of interest for the

experimental selection of the Dynamical Dipole: i) The

centroid is always shifted to lower energies (large defor-

mation of the dinucleus); ii) A clear angular anisotropy

should be present since the prompt mode has a definite

axis of oscillation (on the reaction plane) at variance with

the statistical GDR. At higher beam energies we expect

a further decrease of the direct dipole radiation for two

main reasons both due to the increasing importance of

hard NN collisions: i) a larger fast nucleon emission that

will equilibrate the isospin before the collective dipole

starts up; ii) a larger damping of the collective mode.

This has been observed in ref.[8] and more exps. are

planned [16].

Before closing I would like to note two interesting

developments for future experiments with radioactive

beams:

• The prompt dipole radiation represents a nice cool-

ing mechanism on the fusion path. It could be a

way to pass from a warm to a cold fusion in the

synthesis of heavy elements with a noticeable in-

crease of the survival probability, [17].

• The use of unstable neutron rich projectiles would

largely increase the effect allowing a detailed study

of the symmetry potential, below saturation, re-

sponsible of the restoring force of the dipole oscil-

lation [18]

III. ISOSPIN PHYSICS IN A COVARIANT

APPROACH

We move now to a relativistic framework for the de-

scription of the isovector part of the effective interac-

tion. I will focus then the attention on relativistic heavy

ion collisions, that provide a unique terrestrial opportu-

nity to probe the in-medium nuclear interaction in high

density and high momentum regions. An effective La-

grangian approach to the hadron interacting system is ex-

tended to the isospin degree of freedom: within the same

frame equilibrium properties (EoS, [19]) and transport

dynamics ([20, 21]) can be consistently derived. Within

a covariant picture of the nuclear mean field, for the de-

scription of the symmetry energy at saturation (a4 pa-

rameter of the Weizs¨ aecker mass formula) (in a sense

equivalent to the a1 parameter for the iso-scalar part),

extracted in the range from 28 to 36 MeV, there are dif-

ferent possibilities: (a) considering only the Lorentz vec-

tor ρ mesonic field, and (b) both, the vector ρ (repulsive)

and scalar δ (attractive) effective fields [22, 23, 24]. The

latter corresponds to the two strong effective ω (repul-

sive) and σ (attractive) mesonic fields of the iso-scalar

sector. We get a transparent form [2, 23]:

Esym=1

6

k2

E∗

F

F

+1

2

?

fρ− fδ

?m∗

E∗

F

?2?

ρB,(3)

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4

with E∗

Once the a4 empirical value is fixed from the ρ − δ

balance, important effects at supra-normal densities ap-

pear due to the introduction of the effective δ field. In

fact the presence of an isovector scalar field is increasing

the repulsive ρ-meson contribution at high baryon densi-

ties [2, 23] via a pure relativistic mechanism, due to the

different Lorentz properties of these fields (the vector ρ

field grows with baryon density whereas the scalar δ field

is suppressed by the scalar density). Dynamical, non-

equilibrium, effects can be more sensitive to such “fine

structure” of the isovector interaction. We will see the

vector couplings give γ-boosted Lorentz forces, and so

we expect a larger isospin dependence of the high energy

nucleon propagation. Moreover, the scalar δ field natu-

rally leads to an effective (Dirac) mass splitting between

protons and neutrons [2, 22, 23, 24, 25, 26], with influ-

ence on nucleon emissions and flows [27, 28]. In order

to explore the symmetry energy at supra-normal densi-

ties one has to select signals directly emitted from the

early non-equilibrium high density stage of the heavy ion

collision. A transverse momentum analysis is important

in order to select the high density source [29, 30]. The

description of the mean field is important, since nucleons

and resonances are dressed by the self-energies. This will

directly affect the energy balance (threshold and phase

space) of the inelastic channels.

F≡

?

k2

F+ m∗2.

Relativistic Transport

The starting point is a simple phenomenological ver-

sion of the Non-Linear (with respect to the iso-scalar,

Lorentz scalar σ field) Walecka model which corresponds

to the Hartree or Relativistic Mean Field (RMF) approx-

imation within the Quantum-Hadro-Dynamics [19]. Ac-

cording to this model the baryons (protons and neutrons)

are described by an effective Dirac equation (γµk∗µ−

M∗)Ψ(x) = 0, whereas the mesons, which generate the

classical mean field, are characterized by correspond-

ing covariant equations of motion. The presence of the

hadronic medium modifies the masses and momenta of

the hadrons, i.e.M∗= M + Σs (effective masses),

k∗µ= kµ− Σµ(kinetic momenta), where we have in-

troduced the scalar and vector self-energies Σs, Σµ.

For asymmetric matter the self-energies are different for

protons and neutrons, depending on the isovector me-

son contributions. We will call the corresponding models

as NLρ and NLρδ, respectively, and just NL the case

without isovector interactions. We will show also some

results with Density Dependent couplings, in order to

probe effects which go beyond the RMF picture. For

the more general NLρδ case the self-energies of protons

and neutrons read:

Σs(p,n) = −fσσ(ρs) ± fδρs3,

Σµ(p,n) = fωjµ∓ fρjµ

3, (4)

(upper signs for neutrons), where ρs= ρsp+ ρsn, jα=

jα

isospin scalar densities and currents and fσ,ω,ρ,δare the

coupling constants of the various mesonic fields. σ(ρs)

is the solution of the non linear equation for the σ field

[2, 23].

For the description of heavy ion collisions we solve

the covariant transport equation of the Boltzmann type

[20, 21] within the Relativistic Landau Vlasov (RLV )

method, phase-space Gaussian test particles [31], and

applying a Monte-Carlo procedure for the hard hadron

collisions. The collision term includes elastic and inelas-

tic processes involving the production/absorption of the

∆(1232MeV) and N∗(1440MeV) resonances as well as

their decays into pion channels, [32, 33].

A relativistic kinetic equation can be obtained from

nucleon Wigner Function dynamics derived from the ef-

fective Dirac equation [21]. The neutron/proton Wigner

functions are expanded in terms of components with def-

inite transformation properties. Consistently with the

effective fields included in our minimal model one can

limit the expansion to scalar and vector partsˆF(i)(x,p) =

F(i)

algebra a transport equation of Vlasov type for the

scalar part fi(x,p∗µ) ≡ Fi

M∗

Fµν

i

≡ ∂µp∗ν

i

+ ∂νp∗µ

obey to the following equation of motion:

p+ jα

n,ρs3= ρsp− ρsn, jα

3= jα

p− jα

nare the total and

S(x,p) + γµF(i)µ(x,p),i = n,p. We get after some

S/M∗

i; {p∗

µi∂µ+ [p∗

νiFµν

i

+

i(∂µM∗

i)]∂µp∗}fi(x,p∗µ) = 0, with the field tensors

i. The trajectories of test particles

d

dτxµ

d

dτp∗µ

i=

p∗

M∗

i(τ)

i(x),

iν(τ)

M∗

i

=p∗

i(x)Fµν

i

(xi(τ)) + ∂µM∗

i(x) .(5)

In order to have an idea of the dynamical effects of

the covariant nature of the interacting fields, we write

down, with some approximations, the “force” acting on

a particle. Since we are interested in isospin contributions

we will take into account only the isovector part of the

interaction [34]:

d? p∗

dτ

i

= ±fρpiν

M∗

i

??∇Jν

?∇ρ3∓ fδ?∇ρS3, (p/n)

3− ∂ν?J3

?

∓ fδ∇ρS3

≈ ±fρE∗

i

M∗

i

(6)

The Lorentz force (first term of Eq.(6) shows a γ =

boosting of the vector coupling, while from the second

term we expect a γ-quenched δ contribution. We remark

that the Lorentz-like force is absent in the non-relativistic

Vlasov transport equation discussed before. This nicely

shows the qualitative different dynamics of a fully rela-

tivistic approach. You cannot get it just inserting a rel-

ativistic kinematics in the classical transport equations.

E∗

M∗

i

i

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5

-1

-0,5

0

0,5

1

y/yproj

-40

-20

0

20

40

Fpn (y)

0 0,20,4

pT/pproj

0,6

0,81

-0,06

-0,04

-0,02

0

0,02

<V2

p-n>

FIG. 4: Differential neutron-proton flows for the132Sn +124

Sn reaction at 1.5 AGeV (b=6fm) from the three different

models for the isovector mean fields. Top: Transverse Flows.

Bottom: Elliptic Flows. Full circles and solid line: NLρδ.

Open circles and dashed line: NLρ.

IV.COLLECTIVE FLOWS

The flow observables can be seen respectively as the

first and second coefficients of a Fourier expansion of the

azimuthal distribution [35]:

?

mentum and y the rapidity along beam direction. The

transverse flow can be also expressed as: V1(y,pt) = ?px

The sideward (transverse) flow is a deflection of forwards

and backwards moving particles, within the reaction

plane. The second coefficient of the expansion defines

the elliptic flow given by V2(y,pt) = ?p2

the competition between in-plane and out-of-plane emis-

sions. The sign of V2indicates the azimuthal anisotropy

of emission: particles can be preferentially emitted ei-

ther in the reaction plane (V2 > 0) or out-of-plane

(squeeze − out, V2 < 0) [35, 36]. The pt-dependence

of V2is very sensitive to the high density behavior of the

EoS since highly energetic particles (pt≥ 0.5) originate

from the initial compressed and out-of-equilibrium phase

of the collision. For the isospin effects the neutron-proton

differential flows V(n−p)

1,2

(y,pt) ≡ Vn

have been suggested as very useful probes of the isovec-

tor part of the EoS since they appear rather insensitive

to the isoscalar potential and to the in medium nuclear

cross sections, [37, 38].

In heavy-ion collisions around 1AGeV with radioac-

tive beams, due to the large counterstreaming nuclear

currents, one may exploit the different Lorentz nature of

a scalar and a vector field, see the different γ-boosting

in the local force, Eq.(6). In Fig.4 transverse and el-

liptic differential flows are shown for the132Sn +124Sn

reaction at 1.5 AGeV (b = 6fm), that likely could be

studied with the new planned radioactive beam facilities

at intermediate energies, [34]. The effect of the different

structure of the isovector channel is clear. Particularly

dN

dφ(y,pt) ≈ 1+ 2V1cos(φ) +

p2

yis the transverse mo-

2V2cos(2φ) where pt =

x+ p2

pt?.

x−p2

p2

t

y

?. It measures

1,2(y,pt)−Vp

1,2(y,pt)

evident is the splitting in the high ptregion of the elliptic

flow. In the (ρ + δ) dynamics the high-ptneutrons show

a much larger squeeze−out. This is fully consistent with

an early emission (more spectator shadowing) due to the

larger repulsive ρ-field. The V2 observable, which is a

good chronometer of the reaction dynamics, appears to

be particularly sensitive to the Lorentz structure of the

effective interaction. We expect similar effects, even en-

hanced, from the measurements of differential flows for

light isobars, like3H vs.3He.

V.ISOSPIN EFFECTS ON SUB-THRESHOLD

KAON PRODUCTION AT INTERMEDIATE

ENERGIES

Particle production represent a useful tool to constrain

the poorly known high density behaviour of the nuclear

equation of state (EoS) [39, 40]. In particular pion and

(subthreshold) kaon productions have been extensively

investigated leading to the conclusion of a soft behaviour

of the EoS at high densities, [41, 42]. Kaons (K0,K+)

appear as particularly sensitive probes since they are pro-

duced in the high density phase almost without subse-

quent reabsorption effects [41]. At variance, antikaons

(¯K0,¯K−) are strongly coupled to the hadronic medium

through strangeness exchange reactions [41, 43]. Here we

show that the isospin dependence of the K0,+production

can be also used to probe the isovector part of the EoS:

we propose the K0/K+yield ratio as a good observable

to constrain the high density behavior of the symmetry

energy, Esym, [1, 2].

Using our RMF transport model we analyze pion and

kaon production in central197Au +197Au collisions in

the 0.8 − 1.8 AGeV beam energy range, with different

effective field choices for Esym. We will compare results

of three Lagrangians with constant nucleon-meson cou-

plings (NL... type, see before) and one with density de-

pendent couplings (DDF, see [24]), recently suggested

for better nucleonic properties of neutron stars [45]. In

the DDF model the fρis exponentially decreasing with

density, resulting in a rather ”soft” symmetry term at

high density. In order to isolate the sensitivity to the

isovector components we use models showing the same

”soft” EoS for symmetric matter.

Pions are produced via the decay of the ∆(1232) reso-

nance and can contribute to the kaon yield through colli-

sions with baryons: πB −→ Y K. All these processes are

treated within a relativistic transport model including an

hadron mean field propagation. The latter point, which

goes beyond the “collision cascade” picture, is essential

for particle production yields since it directly affects the

energy balance of the inelastic channels.

Fig. 5 reports the temporal evolution of ∆±,0,++res-

onances and pions (π±,0) and kaons (K+,0) for central

Au+Au collisions at 1 AGeV . It is clear that, while the

pion yield freezes out at times of the order of 50fm/c,

i.e. at the final stage of the reaction (and at low densi-

Page 6

6

010 203040

50 60

time (fm/c)

0

5

10

15

20

25

multiplicity

0 10203040

5060

time (fm/c)

0

0,02

0,04

0,06

0,08

NL

NLρ

NLρδ

DDF

π-

π0

π+

∆-

∆0

∆+

∆++

K0

K+

FIG. 5: Time evolution of the ∆±,0,++resonances and pi-

ons π±,0(left), and kaons K0,+(right) for a central (b = 0

fm impact parameter). Au+Au collision at 1 AGeV incident

energy. Transport calculation using the NL,NLρ,NLρδ and

DDF models for the iso-vector part of the nuclear EoS are

shown.

ties), kaon production occur within the very early stage of

the reaction, and the yield saturates at around 20fm/c.

Kaons are then suitable to probe the high density phase

of nuclear matter. From Fig. 5 we see that the pion re-

sults are weakly dependent on the isospin part of the nu-

clear mean field. However, a slight increase (decrease) in

the π−(π+) multiplicity is observed when going from the

NL (or DDF) to the NLρ and then to the NLρδ model,

i.e. increasing the vector contribution fρin the isovector

channel. This trend is more pronounced for kaons, see

the right panel, due to the high density selection of the

source and the proximity to the production threshold.

The results for the DDF model, density dependent cou-

plings with a large fρdecrease at high density, are fully

consistent. They are always closer to the NL case (with-

out isovector interactions) but the difference, still seen

for π+,−, is completely disappearing for K0,+, selectively

produced at high densities.

When isovector fields are included the symmetry po-

tential energy in neutron-rich matter is repulsive for neu-

trons and attractive for protons. In a HIC this leads

to a fast, pre-equilibrium, emission of neutrons. Such a

mean field mechanism, often referred to as isospin frac-

tionation [1, 2], is responsible for a reduction of the neu-

tron to proton ratio during the high density phase, with

direct consequences on particle production in inelastic

NN collisions.

Threshold effects represent a more subtle point. The

energy conservation in a hadron collision in general has

to be formulated in terms of the canonical momenta, i.e.

for a reaction 1 + 2 → 3 + 4 as sin = (kµ

(kµ

effective (kinetic) momenta and masses, an equivalent

relation should be formulated starting from the effective

1+ kµ

2)2=

3+ kµ

4)2= sout. Since hadrons are propagating with

1,6

1,6

1,8

2

2,2

2,4

2,6

2,8

3

π-/π+

0,6

0,811,2

Ebeam (AGeV)

1,4

1,6

1,82

1,1

1,2

1,3

1,4

1,5

K0/K+

DDF

NL

NLρ

NLρδ

NLDDρ

FIG. 6: π−/π+(upper) and K+/K0(lower) ratios as a func-

tion of the incident energy for the same reaction and models

as in Fig. 5. In addition we present, for Ebeam = 1 AGeV ,

NLρ results with a density dependent ρ-coupling (triangles),

see text. The open symbols at 1.2 AGeV show the corre-

sponding results for a132Sn +124Sn collision, more neutron

rich.

in-medium quantities k∗µ= kµ− Σµand m∗= m + Σs,

where Σsand Σµare the scalar and vector self-energies,

Eqs.(4).In reactions where nucleon resonances, espe-

cially the different isospin states of the ∆ resonance, and

hyperons enter, also their self energies are relevant for

energy conservation. We specify them in the usual way

according to the light quark content and with appro-

priate Clebsch-Gordon coefficients [33]. The self-energy

contribution to the energy conservation in inelastic chan-

nels will influence the particle production in two differ-

ent ways. On one hand it will directly determine the

thresholds and thus the multiplicities of a certain type of

particles, in particular of the sub-threshold ones, as here

for the kaons. Secondly it will modify the phase space

available in the final channel.

In fact in neutron-rich systems mean field and thresh-

old effects are acting in opposite directions on particle

production and might compensate each other. As an ex-

ample, nn collisions excite ∆−,0resonances which decay

mainly to π−. In a neutron-rich matter the mean field

effect pushes out neutrons making the matter more sym-

metric and thus decreasing the π−yield. The threshold

effect on the other hand is increasing the rate of π−’s

due to the enhanced production of the ∆−resonances:

now the nn → p∆−process is favored (with respect to

pp → n∆++) since more effectively a neutron is con-

verted into a proton. Such interplay between the two

mechanisms cannot be fully included in a non-relativistic

dynamics, in particular in calculations where the baryon

symmetry potential is treated classically [46, 47].

In the 0.8 − 1.8 AGeV range the sensitivity is larger

for the K0/K+compared to the π−/π+ratio, as we can

Page 7

7

see from Fig.6. This is due to the subthreshold produc-

tion and to the fact that isospin effects enter twice in

the two-steps production of kaons, [48]. Between the two

extreme DDF and NLρδ isovector interaction models,

the variations in the ratios are of the order of 14 − 16%

for kaons, while they reduce to about 8 − 10% for pions.

Interestingly the Iso-EoS effect for pions is increasing at

lower energies, when approaching the production thresh-

old. We have to note that in a previous study of kaon

production in excited nuclear matter the dependence of

the K0/K+yield ratio on the effective isovector interac-

tion appears much larger (see Fig.8 of ref.[33]). The point

is that in the non-equilibrium case of a heavy ion collision

the asymmetry of the source where kaons are produced

is in fact reduced by the n → p “transformation”, due

to the favored nn → p∆−processes. This effect is al-

most absent at equilibrium due to the inverse transitions,

see Fig.3 of ref.[33]. Moreover in infinite nuclear matter

even the fast neutron emission is not present. This result

clearly shows that chemical equilibrium models can lead

to uncorrect results when used for transient states of an

open system. In the same Fig. 6 we also report results

at 1.2 AGeV for the132Sn +124Sn reaction, induced

by a radioactive beam, with an overall larger asymmetry

(open symbols). The isospin effects are clearly enhanced.

VI.TESTING DECONFINEMENT AT HIGH

ISOSPIN DENSITY

The hadronic matter is expected to undergo a phase

transition into a deconfined phase of quarks and gluons

at large densities and/or high temperatures. On very

general grounds, the transition’s critical densities are ex-

pected to depend on the isospin of the system, but no ex-

perimental tests of this dependence have been performed

so far. Moreover, up to now, data on the phase transi-

tion have been extracted from ultrarelativistic collisions,

when large temperatures but low baryon densities are

reached. In order to check the possibility of observing

some precursor signals of some new physics even in col-

lisions of stable nuclei at intermediate energies we have

performed some event simulations for the collision of very

heavy, neutron-rich, elements. We have chosen the reac-

tion238U +238U (average proton fraction Z/A = 0.39)

at 1 AGeV and semicentral impact parameter b = 7 fm

just to increase the neutron excess in the interacting re-

gion. To evaluate the degree of local equilibration and

the corresponding temperature we have followed the mo-

mentum distribution in a space cell located in the c.m.

of the system; in the same cell we report the maximum

mass density evolution. Results are shown in Fig. 7. We

see that after about 10 fm/c a nice local equilibration is

achieved. We have a unique Fermi distribution and from

a simple fit we can evaluate the local temperature. At

this beam energy the maximum density coincides with

the thermalization, then the system is quickly cooling

while expanding. In Fig.7, lower panel, we report the

010 20 30 40 50 60

0

1

2

3

4

ρB,res [ρsat] , Q

0 10203040

500

20

40

60

80

T [MeV]

010 20 30 40 50 60

time [fm/c]

0

0,1

0,2

0,3

0,4

0,5

ε [GeVfm-3]

010 20 30 40 50 60

time [fm/c]

0,2

0,3

0,4

0,5

0,6

0,7

0,8

Z/A

FIG. 7: Uranium-Uranium 1 AGeV semicentral. Correlation

between density, temperature, momentum thermalization in-

side a cubic cell 2.5 fm wide, located in the center of mass of

the system. Lower panel: density, temperature, energy den-

sity, momentum, proton fraction. Curves in the upper-left:

black dots - baryon density in ρ0 units; grey dots - quadrupole

moment in momentum space; squares - resonance density.

time evolution of all physics parameters inside the c.m.

cell in the interaction region.. We note that a rather ex-

otic nuclear matter is formed in a transient time of the or-

der of 10 fm/c, with baryon density around 3−4ρ0, tem-

perature 50 − 60 MeV , energy density 500 MeV fm−3

and proton fraction between 0.35 and 0.40, well inside

the estimated mixed phase region, see the following..

A study of the isospin dependence of the transition

densities has been performed up to now, to our knowl-

edge, only by Mueller [49]. The conclusion is that, mov-

ing from symmetric nuclei to nuclei having Z/A ∼ 0.3,

the critical density is reduced by roughly 10%. Here we

explore in a more systematic way the model parameters

and we estimate the possibility of forming a mixed-phase

of quarks and hadrons in experiments at energies of the

order of a few GeV per nucleon.

Concerning the hadronic phase, we use the rela-

Page 8

8

tivistic non-linear Walecka-type model of Glendenning-

Moszkowski (GM...) [50], where the isovector part is

treated just with ρ-meson couplings, and the iso-stiffer

NLρδ interaction [51].

For the quark phase we consider the MIT bag model

[52] with various bag pressure constants. In particular we

are interested in those parameter sets which would allow

the existence of quark stars [53, 54], i.e. parameters sets

for which the so-called Witten-Bodmer hypothesis is sat-

isfied [55, 56]. One of the aim of our work it to show that

if quark stars are indeed possible, it is then very likely

to find signals of the formation of a mixed quark-hadron

phase in intermediate-energy heavy-ion experiments [51].

The scenario we would like to explore corresponds to

the situation realized in experiments at moderate energy,

in which the temperature of the system is at maximum of

the order of a few ten MeV . In this situation, only a tiny

amount of strangeness can be produced and therefore we

can only study the deconfinement transition from hadron

matter into up and down quark matter. Since there no

time for weak decays the environment is rather different

from the neutron star case.

Mixed phase

The structure of the mixed phase is obtained by impos-

ing the Gibbs conditions [59, 60] for chemical potentials

and pressure and by requiring the conservation of the

total baryon and isospin densities

µ(H)

B

P(H)(T,µ(H)

= µ(Q)

B

, µ(H)

3

= µ(Q)

3

,

B,3) = P(Q)(T,µ(Q)

ρB= (1 − χ)ρH

ρ3= (1 − χ)ρH

B,3),

B+ χρQ

3+ χρQ

B,

3, (7)

where χ is the fraction of quark matter in the mixed

phase. In this way we get the binodal surface which

gives the phase coexistence region in the (T,ρB,ρ3) space

[49, 60]. For a fixed value of the conserved charge ρ3,

related to the proton fraction Z/A ≡ (1 + ρ3/ρB)/2, we

will study the boundaries of the mixed phase region in the

(T,ρB) plane. We are particularly interested in the lower

baryon density border, i.e. the critical/transition den-

sity ρcr, in order to check the possibility of reaching such

(T,ρcr,ρ3) conditions in a transient state during an HIC

at relativistic energies. In the hadronic phase the charge

chemical potential is given by µ3= 2Esym(ρB)ρ3

we expect critical densities rather sensitive to the isovec-

tor channel in the hadronic EoS.

We compare the predictions on the transition to a de-

confined phase of the two effective Lagrangians GM3 and

NLρδ. The isoscalar part is very similar while the isovec-

tor EoS is different, because in GM3 we only have the

coupling to the vector ρ-field. In Fig. 8 we show the

crossing density ρcr separating nuclear matter from the

quark-nucleon mixed phase, as a function of the proton

ρB. Thus,

FIG. 8: Variation of the transition density with proton frac-

tion for various hadronic EoS parameterizations. Dotted line:

GM3 parametrization; dashed line: NLρ parametrization;

solid line: NLρδ parametrization. For the quark EoS, the

MIT bag model with B1/4=150 MeV . The points represent

the path followed in the interaction zone during a semi-central

132Sn+132Sn collision at 1 AGeV (circles) and at 300 AMeV

(crosses), see text.

fraction Z/A. We can see the effect of the δ-coupling

towards an earlier crossing due to the larger symmetry

repulsion at high baryon densities. The δ-exchange po-

tential provides an extra isospin repulsion of the hadron

EoS, and its effect shows up in a further reduction of the

critical density.

In the same figure we report the paths in the (ρ,Z/A)

plane followed in the c.m. region during the collision of

the n-rich132Sn+132Sn system, at different energies. We

see that already at 300 AMeV we are reaching the border

of the mixed phase, and we are well inside it at 1 AGeV .

Deconfinement Precursors

Statistical fluctuations could help in reducing the den-

sity at which drops of quark matter form. The reason is

that a small bubble can be energetically favored if it con-

tains quarks whose Z/A ratio is smaller than the average

value of the surrounding region. This is again due to the

strong Z/A dependence of the free energy, which favors

clusters having a small electric charge. These configura-

tions can easily transform into a bubble of quarks having

the same flavor content of the original hadrons, even if

the density of the system is not large enough to allow

deconfinement in the absence of statistical fluctuations.

Moreover, since fluctuations favor the formation of bub-

bles having a smaller Z/A, neutron emission from the cen-

tral collision area should be suppressed, what could give

origin to specific signatures of the mechanism described

in this paper. This corresponds to a neutron trapping ef-

fect, supported also by a symmetry energy difference in

the two phases. In fact while in the hadron phase we have

a large neutron potential repulsion (in particular in the

NLρδ case), in the quark phase we only have the much

Page 9

9

smaller kinetic contribution. If in a pure hadronic phase

neutrons are quickly emitted or “transformed” in pro-

tons by inelastic collisions, when the mixed phase starts

forming, neutrons are kept in the interacting system up to

the subsequent hadronization in the expansion stage [51].

Observables related to such neutron “trapping” could be

an inversion in the trend of the formation of neutron rich

fragments and/or of the π−/π+, K0/K+yield ratios for

reaction products coming from high density regions, i.e.

with large transverse momenta. In general we would ex-

pect a modification of the rapidity distribution of the

emitted “isospin”, with an enhancement at mid-rapidity

joint to large event by event fluctuations. A more de-

tailed analysis is clearly needed.

VII.PERSPECTIVES

We have shown in few examples the richness of the

physics we can describe using mean field transport equa-

tions, inspired by the pioneering works of David M.

Brink. We have seen that collisions of n-rich heavy ions

from low to intermediate energies can bring new infor-

mation on the isovector part of the in-medium interac-

tion in different regions of high baryon densities, quali-

tatively different from equilibrium EoS properties. We

have presented quantitative results for charge equilibra-

tion in fusion/deep inelastic reactions, differential col-

lective flows and yields of charged pion and kaon ratios.

Important non-equilibrium effects for particle production

are stressed. Finally our study supports the possibility

of observing precursor signals of the phase transition to a

mixed hadron-quark matter at high baryon density in the

collision, central or semi-central, of neutron-rich heavy

ions in the energy range of a few GeV per nucleon. As

signatures we suggest to look at observables particularly

sensitive to the expected different isospin content of the

two phases, which leads to a neutron trapping in the

quark clusters.The isospin structure of hadrons pro-

duced at high transverse momentum should be a good

indicator of the effect.

Many new ideas for fundamental experiments with ra-

dioactive beams are emerging. My picture of David Brink

like a tree always starting new braches is more and more

confirmed......

Acknowledgements

I would like to warmly mention the great experience

of collaborating with exceptional people on the top-

ics and projects shortly discussed here.

to list some of them: V.Baran, M.Colonna, A.Drago,

T.Gaitanos,V.Greco, A.Lavagno,

H.H.Wolter, S.Yildirim and the essential young con-

tributors L.Bonanno, G.Ferini, R.Lionti, N.Pellegriti,

V.Prassa, J.Rizzo and E.Santini.

I will try

B.Liu,M.Pfabe,

The interaction with experimental groups has been es-

sential, in particular I like to thank the DIPOLE Col-

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