outline:

Overview of the Brueckner theory of Nuclear Matter 3BF within the meson-exchange model of the interaction Skyrme-type parametrization of the BHF energy functional In medium Nucleon-Nucleon cross section Transport Parameters and dissipation of collective modes in NS Optical – Model Potential for nucleon-Nucleus collisions at E<200 MeV.

NN effective interaction from Brueckner Theory and Applications to Nuclear Systems

U. Lombardo Dipartimento di Fisica e Astronomia and INFN-LNS,Catania (Italy)

Dynamics and Correlations in Exotic Nuclei YITP, Kyoto 2011

Collaboration

M. Baldo, HJ Schulze, INFN CataniaI. Bombaci, Pisa UniversityU. Lombardo, Catania University and INFN-LNSA. Lejeune, Liege University J.F. Mathiot, Clermont-Ferrand,CEAW. Zuo, ZH Li . IMP-CAS, LanzhouH.Q. Song, Fudan University

strongly constrained by the most recent data (few thousands, up to 350 MeV) on NN phase shifts, energy scattering parameters, and deuteron binding energies.

The non-relativistic nuclear many-body problem

ijv

)(1

CoulombnovTHA

i

A

jiiji

Nuclear matter is an homogeneous system made of pointlike protons and neutrons interacting through the Hamiltonian

Nucleon-Nucleon phase shiftsAs an example, Argonne v18 potential :

Wiringa, Stoks, Schiavilla, PRC51, (1995) 38

Due to the short range repulsive core of the NN interaction, standard perturbation theory is not applicable.

Bonn B and Exper phase shifts

Many Body Approaches

Non-relativistic:

• Brueckner-Bethe-Goldstone expansion• Variational method• Chirl perturbation theory - > Schwenk

Relativistic :

• DBHF method

Ab initio approaches, realistic NN potentials, no free parameters

The Brueckner-Bethe-Goldstone theory of Nuclear Matter

101

)()(

: potential particle-singleauxiliary

HHUVUTvTHA

i

A

jiiji

In the BBG expansion:

• H1 expansion is recast in terms of G-matrix

• and then rearranged according to the order of correlations : twobody, three-body ,……(hole line expansion)

Fkkk

a

F kkGkkm

k

A

E',

2

';'2

1

25

3(

bakk ba

babaNNNN G

keke

kkQkkVVG

;

),(),(;

healing

1S0

12 12V GY = F

Two and three hole-line diagrams in terms of the Brueckner G-matrixs

The BBG expansion

Bethe-Faddeev

BHF approximation

The variational method in its practical form

Method used to calculate the upper bound to the ground state energy:

0EH

E

Trial w.f. is the antisymmetrized product of two-body correlation functions acting on an unperturbed ground state Φ:

Pandharipande & Wiringa, 1979; Lagaris & Pandharipande, 1981

,....),()(,...),( 2121 rrrfrr ijji The correlation function f represents the correlations induced by the two-body potential. It is expanded in the same spin-isospin, spin-orbit and tensor operators appearing in the NN interaction.The correlation function is obtained from

.

0

H

f

Nucleon-Nucleon Interaction: Argonne v18

(Wiringa, Stoks & Schiavilla, Phys. Rev. C51, 38 (1995))

Neutron Matter

Symmetric Matter?

Dependence on the Many-Body Scheme

Variational (Akmal, Pandharipande

& Ravenhall, PRC 58, 1804 (1998))

It makes us confident that these approximations do work!

same NN interaction

BBG:Catania group

Song,Baldo,Giansiracusa and Lombardo. PRL 81 (1998):3 hole lines: Bethe-Faddeev equation

Convergence of hole line expansion

UG =gap choiceUC=continuous choice

At the level of 3 hole line approximation the EoS is independen of auxiliary potential

2+3 hole

E0= <K> + Σi<V>i = - 16 .0 MeV

<V>2= - 40 MeV <V>3= 3 MeV <V>4= 0.5 MeV B.Day,1987

saturation point

3BF in Nuclei

10B

puzzle of analyzing power

Ishikawa Phys.Rev. C59 (1999)

Caurier, P. Navrdtil, W.E. Ormand, and J.P. Vary, Phys. Rev. C64, 051301 (2001)

N-d el.scatt. At E=3MeV

Meson-exchange Model of 2- and 3- body interaction

, ,σ,ω exchange

P. Grangé et al., PRC 40, 1040 (1989):

+ + N

Fujita-Miyazawa model

,N*

two-body Dirac sea polarization

Z-diagrams are introduced as the relativistic correction to n.r. BHF approach

Brown,Weise,Baym,Speth, Nucl.Part.Phys.1987

8/30

4.2 ( )A

MeVE rr=D

spin-flip of one quarkΔ - resonance

a radially excited three-quark state 1s

2s

S=1/2 Lπ =1- Jπ=1/2+

P11 (1440)

N* - resonance

Nucleon Excitations

coupling constants from N* and Δ decay withs + quark structure

Input: meson parameters fittingNN experiment phase shifts

Output: 2BF and 3BF based on the meson parameters

N

No adjustable parameters !

Double-selfconsistent approach of BHF-3BF

* *13 23 1'2 ' 1'3 '

33 '

( ) 123 1'2 '3'NNNV V

( 0 ) ( 0 )

(1) ( 0 )

( 2 )

( )

( )NN NNN

NN NNN

matrixV V G V

V V V

V

r

r

-= Þ Þ Þ

= + Þ ×××

=×××

Ф12 two-body correlation function : V Ф12 = G Фo12

saturating one particle

ZH Li, U Lombardo and H-J Schulze, W,ZuoPRC(2008)

Esym (ρ) = BA (ρ)|SNM - BA (ρ)|PNM

• saturation point:

=0.17 fm-3 , E= -16 MeV

• symmetry energy at saturation

Sv≈ 30 MeV • incompressibility at saturation

K ≈ 244 MeV

Danielewicz Plot HIC collective flows

asysoft

asystiff

K∞(ρo) Esym (ρo) Esym (0.1) L

BHF 244 29.4 23. 74.4

Empirical 240±20 28-32 24.1±0.8 65.1 ± 15.5

source MGR B-W mass table DGR PGR NUSYM10, Riken 2010

sub-saturation densities

Esym (ρ)=Esym (ρo)+ L/(3 ρo) (ρ-ρo)+Ksym /(18ρo2)(ρ-ρo)2

From HIC (isoscaling, multigragmentation, n-p products,..) still ambiguous

EoS from BHF : individual contributions and comparison with DBHF

nucleon resonances (F-M) :

Δ(1232),R*(1440)

NN excitations (Z-diagrams)

strong compensation !

The G-matrix expansion (full Brueckner theory) is converging

The non relativistic BHF theory with two body force is consistent with other n.r. many body approaches (e.g., variational method) Including relativistic effects (Z-diagrams) the BHF EoS fully overlaps the DBHF

The three-body force pushes the saturation density to the empirical value, but its effect is small on the saturation energy (1-2 MeV) and compression modulus (220->240 MeV) is small

BHF with 2+3 body force

Ambitious task is to build up a unified EDF for nuclei and nuclear matter:The question is:To what extent a Skyrme-type forces can reproduce the BHF (numerical) EDF as well as experimental observables?

Our hope in such study is to learn what is missing in the theory of nuclear matter and which Skyrme parametrizations cannot match nuclear matter

we performed a ‘weighted fit’ of nuclear matter (theory) and nuclei (experiment) via the Skyrme functional:

reconciling nuclei and nuclear matter

Gambacurta,Li,Colo’,Lombardo, Van Giai,Zuo, PRC 2011

Skyrme-like Energy Functionals (LNSx)

Despite large deviations in the individual spin-isospin components of energy potential U(S,T) large compensation gives rise to similar EoS specially around saturationSince U(S,T) probe different observables (compressibility,symmetry energy, GR’s,…) it looks convenient to fit U(S,T) instead of total USimultaneous Skyrme-like fit of nuclear matter (theory) and nuclei (exp) changing the relative weights:

( ) ( ),

sky skyth thi jST ST i

i j j

w U U w O Oa aa

- + -å å th=BHF Oα = nuclear binding energies and radii.Changing the relative weights we produce different parametrizations: LNSn (n=1,2,3,…)

splitting U into (S,T)

HF calculations from the Skyrme-like parametrizations of BHF theor <- ( LNS1 – LNS5 ) -> exp

closed-shell nuclei b. energy and radii

Sn Isotopic chain

Giant Resonances open problems: tensor and spin-orbit

Energy Density Functional on Microscopic BasisBaldo,Robledo,Schuck,Vinas

finite range term (surface)

Microscopic Transport Equation:

( ) ( ' ' ' ')iji i j i j i j i j i j

j

dI n I n n n n n n n n

d

sæ ö÷ç ÷= = × -ç ÷ç ÷ç Wè øå å %% % %

2

2

pUp pm

e = +

In BHF nuclear potential and cross sections are calculated from G-matrix j

n = 1 - n

Applications:

simulations of heavy ion collisions (test-particle method)from linearization : n = n0 + δn -> collective modes nuclear fluidodynamics: transport parameters in dense matter (high-energy HIC, neutron stars)

Phenomelogical Transport Eq.s

3V

Tu u f p u ut

(K thermal conductivity)

shear viscosity

Navier-Stokes Eq.

Heat Conduction Eq

gradT

A

heat flux

bulk viscosity

xu

y

¶

¶xu

x

¶

¶

2( )

3u T u K T

t

2 21( ) ( )

20 FT v C Wh r l r=

21( ) ( )

12 F FT p v H Wk m r=

shear viscosity

thermal conductivity

Transport Parameters

[ ]4 2 1

1 2

0 0

1 / 4 ( , )2 2

F

FF

dE dW E E

e pJ

e s Je p

-- = -ò ò

Linearizing the Boltzmann equation the connection with phenomenologicaltransport equations is established as well as the transport parameters

( , )Es J is the in-medium NN cross section

medium effects in the cross section:

Pauli blocking (Q): nucleons scatter into unoccupied statesStrong mean field (H0) between two collisions Compression of the level densities in entry and exit chennels

To calculate the cross sections at high density the free scattering amplitude

1( ) ( )T E V V T E

E K= +

-

must be replaced by in-medium scattering amplitude (G-matrix)

( ) ( )Q

G E V V G EE Ho

= +-

In-medium NN cross sections

theoretical framework (G-matrix)

2 2'

*' '

' | | '2 2

f

p ap p

p ppp G pp U

m mt tt

tt t

e<

= + < > = +å å

'

'

*22

2 4

( )| | ( ) | ' |

4 z z

z z

np S

S SSS S

d mp G p

d

s JJ

pá ñ

W å:h

in-medium dσ/dΩ

quasiparticle spectrum

mean free path, viscosity, heat conductivity,…

dσ/dΩ(θ) =N2 |<p|G (θ)|p’ >|2

A) -------------Finite Range Interaction---------------------------

G

N

N

1 *p

FF

d mN

dp p

e-æ ö÷ç ÷= =ç ÷ç ÷çè ø

level density:

effective mass vs ρ

empirical OMP data support theprediction m*n>m*p in ANM

effective mass vs β

In medium effects on the scattering cross section

Mean field and Effective Mass vs β

empirical OMP data support theprediction m*n>m*p in ANM

We expect that in neutron-rich matter. σnn is less suppressed than σpp

θ

pF

Fermi sphereIn CM frame

Δp = 2pF sin (θ/2) =

backward and forward scatterings sizably suppressedby Pauli blocking

momentum transfer p ≈ p’ ≈ pF

B)--------------------Pauli blocking p>pF-----------------------------------------

=0 pf=0 free space>0 pf >0 in medium

transport parameters in NS

structure of NS: chemical composition,superfluidity, phase transitions,…

transport phenomena rotation, glitches, cooling,collective modes,…

nucleons,hyperons, kaons, quarks in beta-equilibrium with leptons

beta-equilibrium with electrons and muons : p + e¯ n + hyperonized matter: n + n n + ( p + ¯) at > 2o

kaon condensation n p + K¯ at > 2-3o

transition to quark matter HP QP (u,d,s) at ~ 6o

NS internal structure

restrict on only the first item (p,n. e- )transition to QGP (in progress)

Glitches and post-glitches relaxation time (superfluidity ?)

Viscosity controls the rotational dynamics and the damping of collective modes

NS is a viscous fluid rapidly rotating

coupling between rotation and collective modes( r-modes)

Chandrasekhar Instability

minimal cooling:

nn (nn) + (nn) nn

30.11 0.3thrZ

fmA

Thermal conductivity controls thermal evolution of Neutron Stars (Tsurf vs. Tcore)

: =

:

,

( - , )

4n p sym

F Fp e p e

n p e

A

charge conservation k k

chemical equilibrium

energy loss replacing a proton with a neutron

namely the symmetry energy a in B W mass formul N

E

a Z

p + e¯ n + e

n p + e¯ + e

p + ¯ n + n p + ¯ +

341

2sym

pF

E

ck

ANM with β = β (ρ)

Cross Sections in β-stable matter

p + e¯ n + e

n p + e¯ + eANM with β = β (ρ)

non linear behaviour of proton mean field and effective mass

nn collisions np collisions

shear viscosity

σo(Ω) → σ(Ω) : η~10·η0 Flowers & Itoh, ApJ (1979)isospin effect: η(proton) → η(neutron) at higher densitym → m* : ηn >> ηe Shternin & Yakovlev PRD(2008)

( )14 2 1

2 2 2

0 0

1( ) 1 / 4 ( , )

20 2 2

F

F FF

dE dT v C E E

e pJ

h r l e s Je p

-

-é ùê ú= -ê úê úë ûò ò

Shternin & Yakovlev PRD(2008), APRBenhar & Carbone, arXiv09112.0129,CBF

no r-mode damping

Beta-stable SNM vs.PNM electrons

thermal conductivity

NS Cooling

( )4 2 1

2 2

0 0

1( ) 1 / 4 ( , )

12 2 2

F

F F FF

dE dT p v H E E

e pJ

k m e s Je p

-= -ò ò

Yakovlev et al, Phys.Reports 354 (2001) 1

Dissipation of r-modes

2

( )l

i tlm

R rv i i e R Y

l Rwd wx we

æ ö÷ç= = Ñ ÷ç ÷çè ø

r rr

21| |

2E dV vr d= ò

r

collective energy:

dissipation time scale (viscosity only):

A non radial collective excitation of a NS is described by avelocity field

2

0

1 1( ) | |

2

R

i j

dEdV r v

E dth

t=- =- Ñò

η(r )= η(ρ(r)) density profile of NS is required !

Tolman-Oppenheimer-Volkov (TOV) and nuclear EoS

Input Equation of State P=P(, p)

Output Mass-Radius plot

Time scale of nonradial mode damping from shear viscosity

( )( ) 234

3

11 2 1

V

Ml l const

t R Rp

hr

r= - + = =e

Li,Lombardo,Peng, PRC (2008)

tk (s)

0.54 1016

1.5 1016

3.0 1015

6.0 1015

thermal cond.

constant mass approximation

integration over the star (0≤r≤R):r → ρ(r) → η(ρ) → TOV

viscosity time scale

l=2 r-mode

Transition to deconfined phase in neutron stars

Despite the contribution of quarks to viscosity is smaller than hadrons the phase transition is pushing the hadron phase to higher density allowing an extra contribution to viscosity

Chandrasekhar Instability (1970)Y22 - nonradial mode: ~ ω0 (Coriolis force)

Inertial frame

Corotating frame

J’22 = (J-J22 ) |J’22| increases more and more

accompanied by larger frequency and amplitude and then more angular momentum loss for radiation

Since rotating NS exist, GR instability must be stopped bySome damping of Y22 the best candidate is viscosity

1 10

( ) ( )GR c V ct w t w+ =interplay between GR driving instability and viscosity damping

critical velocity:

J0 >> J22 which decreases, being Y22 a sourseof gravitational radiation (General Relativity)

expected to be detected in terrestrial labs ( LIGO,VIRGO,…)

Ω~1000 Hz → τGR ~ 100 sec(depending only on rotation)

1 10

( ) ( )GR c V ct w t w+ =

interplay between GR driving instability and viscosity damping critical velocity:

constant density approximation underestimates the effect of viscosity

Nuclear Potential

Optical Potential vs. nucleon-Nucleus Scattering

( ) ( ( ), ) ( ( ), )U E k E E i k E Eopt R I

1 2

2 2 2 2( , ) ( , ) ( , )

2 2

k kE k E k E k E

m m

mass – shell relation

procedure: for a given approximation of ∑ one solve k=k(E) and determines selfconsistently the on-shell selfenergy

on-shell self.energy

Hughenholtz-Van Hove theorem:0( )AF BE r r= =

BBG hole-line expansion in G-matrix

1 1' | | ' ' | | ' ' | | '

' ' ''' ' ''2 2

U GAU n n pp G pp n pp G pp n n pp ppp p p np A p A p Appp p ppn np p

d d ddd d

= = < > = < > + < >å å åé ùê úê úë û

Nuclear Potential

311 ( )1

p p p

VG G nG G G G G G

n n E n

d rd dd d d

-- -=- =- -

core polarization Dirac sea pol.

3123 2

( )NN bare bare

V V V V V r= + = +å

++ + …

n and p self-energies( , ) ( , ) ( , ) ( , )k E k E k E k Ebhf cpol tbf

p,E

EF

p,E

p,E

EF

EF

Dirac Sea

Evidence for “large” mass ?

Nice et al. ApJ 634, 1242 (2005) PSR J0751+1807 M = 2.1 +/- 0.2

Demorest et al. Nature 467, (2010) J1614-2230 M =1.97 +/- 0.04

Ozel, astro-ph /0605106 EXO 0748 – 676 M > 1.8

Quaintrell et al. A&A 401, 313 (2003) NS in VelaX-1 1.8 < M < 2

ρ ≈ 10 ρ0

OMP Imaginary part:absorption from BHF

from a virtual collisions with the Fermi sea

Im ∑bhf

ε > εF

ε < εF

Im ∑cpol

Im ∑3bf = 0At E>0 Im ∑bhf ≠ 0 only

Which information can be extracted from N-ion collisions? We must first single out the volume contribution from others (surface,spin-orbit,Coulomb)

from NM only the strengthsVv(E,ρn,ρp),Wv(E,)ρn,ρp)

on energy (or momentum ) ,density and isospin

n-A total cross sections

From Koning & Delaroche (NP 2003) parametrization of OMP fitting nucleon-A experimental cross sections one can extract the strength g(E) of the volume component

p-A react.cross sections

( )/

( )( , )

1 e r R avol

g Er E

-=

+V

The strength is energy ,density,and isospin dependent

56Fe + n

--Volume term: real part-- vs. energy

no Coulomb correction

with Coulomb correction

Comparison between BHF and empirical OMP

--Volume term: real part-- vs. asymmetry

28Si

208Pb

--Volume term: real part-- vs. asymmetry

absorption

Volume term (absorption)Vs. asymmetry

Volume term (absorption)vs. energy

theory underestimates empirical dataespecially at high energy

ConclusionsThe Brueckner theory with 3bf makes predictions is consistent with the empirical properties of nuclear matter around the saturation density

The meson-exchange model of the interaction allows to build up the two and threebody force with the same meson parameters adjusted on the experimantal phaseshifts on NN scattering

The NN virtual excitations (Z-diagrams) embody most relativistic effects , making non relativistic BHF comparable with DBHF

Preliminary Skyrme parametrization of the BHF potential energy

The estimate of the time-scale of viscosity dissipation of collective r-modes (with and without deconfined quark phase) is comparable with time-scale ofgravitational waves emission, confirming that the viscosity is the mechanismthat guarantees the stability of rotating NS

The BHF self-energy gives a microscopic basis to the optical-model potential, (volume part) describing nucleon-Nucleus collisions