Chiral Theory of Nuclear Matter and Nuclei

• strong interactions - ingredients, problems, partial solutions• constructing a working hadronic model• applications of a hadronic model

D. Zschiesche G. Zeeb K. Balazs M. Reiter Ch. Beckmann P. Papazoglou

• projects, outlook

nuclear matternuclear structureneutron stars, heavy-ion collisions

Strong Interactions at low Energies

QCD as theory of strong interactions well established

radiative corrections generate running coupling constant αQCD

0

0.4

0.8

1.2

1.6

2

0 0.5 1 1.5 2 2.5

Q [Gev]

alp

ha

quarks (asymptotic freedom)

hadrons

Strong interactions: drastic phenomenological consequences

(1-gluon exchange as important as 2-gluon exchange, …)

electrodynamics α ~ 1/137 , αn << α

QCD α ~ 1 , αn ~ α

proton (uud), neutron (ddu) : m ~ 20 MeV

dynamical mass creation

coupling strength:

e- e-

q qggQCD:

QED:

but total mass Mp , Mn ~ 1 GeV !

Chiral Symmetry: left- and right-handed particles decouple

true for all vector interactions L/R = ½( 1 -/+ 5)

m = m (L+R)(L+R) = m (LR + R L)_ ____

A = (L+R) A (L+R) = L A L + R A R _ __ _ _

e-.q , g

L/R

mass terms violate symmetrye-.q

L/R

L/R

R/L

m

m << Etypical chiral symmetry useful, mu,d << Mn, ms < Mn

QCD vacuum has a complex structure!

<0|q q|0>, <0| G G |0> = 0

qL qR

_

qR qL

_

G G

qR qL

_qLqR

_

Eqq ~ Ekin + Epot < 0 ! condensation

_

mass generation!

left-handed (k || s) right-handed (k || s) particlesmass terms couple chirality

G G

/

M = M (L+R)(L+R) = M (LR + R L)_ ____

A = (L+R) A (L+R) = L A L + R A R _ __ _ _

hadronic description

quarks, gluons hadrons nuclear matter, nuclei

G N N N N G <N N> N N MN N N_ ___ _

dynamical mass

what about calculating larger systems - nuclei?currently not feasible within a quark picture

quark/gluon picture hadrons

<0|q q|0> <0||0>

_

<0| G G |0> <0||0>

construct a chirally symmetric interaction

LI ~ (N N)2 + (N i 5 N)2 = (LR + RL)2 - (LR - RL)2__ _ _ __

L

R

R

LOriginal Nambu Jona-Lasinio

bosonize: N N N i 5 N _ _

LI ~ N ( + i 5 ) N linear model_

L’I ~ 2 + 2 only mesons

~

total Lagrangian L ~ Lkin + g N ( + i 5 ) N - V( 2 + 2 ) _

non-linear: + i 5 = exp ( i 5 / f ) N = exp ( i 5 / 2f ) N~ ~ ~~

Degrees of Freedom

SU(3) multiplets:

n (ddu) p (uud)Baryons - (sdd) 0 (sdu) + (suu) - (ssd) 0 (ssu)

0 (sd) + (su) Scalar Mesons - (ud) 0 , , +(du)

- (us) 0 (ds) _

_

plus pseudoscalars, axial vectors and gluonic field

~ <u u + d d> ~ <s s> 0 ~ < u u - d d>_ __ __

_ _

_ _

_ _

K*0 (sd) K*+ (su) Vector Mesons - (ud) 0 , , +(du)

K*- (us) K*0 (ds)

_ _

_ _

_ _

hyperons

construction of the model

A) chirally symmetric SU(3) interaction

~ Tr [ B, M ] B , ( Tr B B ) Tr M

B) meson interactions ~ V(M) <> = 0 0 <> = 0 0

C) chiral symmetry m = mK = 0 explicit breaking ~ Tr [ c ] ( mq q q )

light pseudoscalars, breaking of SU(3)

_ _

_

fit parameters to hadron masses

’

mesons

Model can reproduce hadron spectra via dynamical mass generation!

p,n

K

K*

*

*

fields change in a dense and hot medium

MN ~ g 0 (+ g 0 + g 0 ) e.o.m: ~ - g /m2 s

~ - 300 MeVstrong scalar attraction!

VV ~ g ~ - g /m 2 V ~ 240 MeV

Vs - VV ~ - 540 MeV VLS ~ d/dR (VS - VV) large LS splitting

In the medium the vacuum condensate is reduced ( < 0)

Inside of an atomic nucleus MN*/MN ~ 0.6

plus vector repulsionfrom surrounding nucleons:

Nuclear Matter

vector fields non-zero B = j0 0 0 , 0 , 0 0

VMD: in n,p matter < 0 > ~ 0 symmetric matter < 0 > = 0

need to reproduce:

• binding E/A ~ -16 MeV• saturation (B)0 ~ .17/fm3

• compressibility ~ 200 -300 MeV

away from symmetric matter: asymmetry a4 ~ 30 MeV

important reality check

compressibility ~ 223 MeV asymmetry energy ~ 31.9 MeV

equation of state E/A () asymmetry energyE/A (p- n)

nuclear matter (infinite matter, same number of p and n, no Coulomb)

binding energy E/A ~ -15.2 MeV saturation (B)0 ~ .16/fm3

phenomenology: 200 - 300 MeV 30 - 35 MeV

Finite Nuclei (mean field, spherical)

16O 40Ca 208Pb

E/A [MeV] -7.30 (-7.98) -7.96 (-8.55) -7.56 (-7.86)

rch [fm] 2.65 (2.73) 3.42 (3.48) 5.49 (5.50)

LS [MeV] 6.1 (5.5-6.6) 6.2 (5.4-8.0) 1.59 (0.9-1.9) (p3/2- p1/2) (d5/2-d3/2) (2d5/2 - 2d3/2)

no nuclear fit, reasonable agreement magic numbers ok !

Task: self-consistent relativistic mean-field calculationcoupled 7 meson/photon fields + equations for nucleons in 1 to 3 dimensions

fit to known nuclear binding energies and hadron masses important step in the process - complicated structure of fitting surface, many minima

2d calculation of all measured (~ 800) even-even nuclei

error in energy (A 50) ~ 0.21 % (NL3: 0.25 %) (A 100) ~ 0.14 % (NL3: 0.16 %)

good charge radii rch ~ 0.5 % (+ LS splittings)

SWS, Phys. Rev. C66, 064310 (2002)

Best relativistic nuclear structure models

Lagrangian (in mean-field approximation)

L = LBS + LBV + LV + LS + LSB

baryon-scalars:

LBS = - Bi (gi + gi

+ gi ) Bi

LBV = - Bi (gi + gi

+ gi ) Bi

baryon-vectors:

meson interactions:

LBS = - k0/2 2 (2 + 2 + 2 ) + k1 (2 + 2 + 2 )2

+ k2/2 (4 + 2 4 + 4 + 6 2 2 ) + k3 2 - k4 4 - 4 ln /0 + 4 ln [(2 - 2) / (0

20)]

explicit symmetry breaking: LSB = - (/0)2 (c1 + c2 )

_

_

LV = - k’0/2 2 (2 + 2 + 2 ) + g4 (4 + 4 + 4 + 6 22)

nuclide chart - deformation

superheavies?Z =116 Dubnamagic numbers stick out

as spherical shapes

number of neutrons

num

ber

of p

roto

ns

exotic nuclei large isospin(new GSI, ISAC, RIA)

neutron drip line(preliminary)

2-nucleon gap energies

< 2p > = < E(Z+2,N) - 2 E(Z,N) + E(Z-2,N) >N

spherical

deformed

protons neutrons

Form Factors, Charge Densities

< rch > ~ 0.5 %

linear realisation fails!

charge distribution in 208Pb

higher-order couplings generate fluctuations(nuclear matter ok!)

2D calculation of Mg Isotopes

NL3

heavy nuclei - deformation of Nobelium Isotopes

exp.: Herzberg et al.PRC65

014303 (2001)

2 ~ 0.32 0.02

2 ~ 0.31 0.02

axis ratio 3:2(see neutron stars)

heavy nucleusZ = 102, N=150,152,154

SWS, Phys. Rev. C66, 064310 (2002)

deformation of S , Ar isotopes

H. Scheit et al., PRL 77, 3967 (1996)

SAr

S. M. Fischer et al., PRL 84, 4064 (2000)

N = Z = 34

oblate groundstate ( ~ -0.3)excited prolate state

superdeformed nuclei

Exp: T. L. Khoo et al., PRL 76, 1583 (1996)

constraint 2d calculation

SWS, Phys. Rev. C66, 064310 (2002)

ener

gy

breathing nucleus

208Pb

- B/A [MeV]

av [1/fm3]

effective compressibility eff ~ 117 MeV

EGMR ~ (eff / m<r2> )1/2 ~ 12.3 MeV (exp: 13.7 MeV)

spherical

deformed

protons neutrons

signal for magic number of Z=120 vanishes in deformed calculation(deformed gap, metastable states? more detailed study needed )

superheavy nuclei - new valleys of stability?

GSI, Dubna, Berkeley - fuse two heavy nuclei to new stable(?) superheavy elements

peaks : strongly bound

2-nucleon gap energy(Z = 114, 120, 126 ?)

(uds) single-particle energies

Model and experiment agree very well

Nuclear matter

40Ca = (20 p, 20 n) 40Ca = (20 p, 19 n, 1 ) hypernucleus

The “Ultimate” Neutron-Rich Nucleus

Neutron star: M ~ 1.4 Msolar R ~ 13 km

MHT = (1.4411 0.0035) Msolar (Hulse-Taylor)

Thorsett,Chakrabarty,APJ 512 288 (‘99)

collection of mass measurements

Neutron stars - constraint onhadronic models

static star easy to calculate (Tolman-Oppenheimer-Volkov)

dP(r)/dr = F((r), P(r), M(r) )

dM(r)/dr = 4 r2 (r)

start with (0), P(0) - integrate up to P =0 (surface of star)

NUCLEAR PHYSICS: equation of state (P)

“realistic” star include hyperons

typical neutron star

mass(r)

density(r)

regions relevant tonuclear physics

Ratio of neutrons = (n - p) / (n + p)

equation of state of nuclear matter varying isospin

static “neutron” star

particle cocktail

/02 4

max

Input: Equation of State (), p()

M. Hanauske, D. Zschiesche, S. Pal, SWS, H. St\öcker, W. Greiner, Ap. J. 537, 958 (2000)..

~ 25% of “exotic” matter ( , -,

-) Not too exotic!

hyperstar

no hyperons

include stellar rotation

expand g() in multipole moments M, R change, deformation

excentricity()

Kepler period PK > 0.8 ms

Mmax(max) = 1.94 Msolar masses change < 20 %

axis ratioof 3:2

Experimental numbers for frequency, radius, mass needed

SWS, D. Zschiesche, J. Phys. G 29, 531 (2003)

fastest known pulsar PK ~ 1.5 ms

neutron star results - summary

static star:

TOV equations (P)low hyperon content fs ~ 1/3 (, -, -)

Mmax ~ 1.54 Msolar 1.82 Msolar (no hyperons)Rmin ~ 11.3 km 11.2 km “

rotating star:

excentricity < 3:2 Kepler frequency 1/ 0.8 msmass increases to ~ 1.9 Msolar

during slow-down non-strange strange star

no backbending, phase transition in this model

Influence of resonances

r = g / gN

not well determined!

parity violation at Jefferson Lab

Polarized e- scattering from 208Pb (850 MeV)

e-

p

e-

p,n

Z0+

Polarized cross section interference term

R-L

R+L

GF Q2

4 2

Fn(Q2)

Fp(Q2)neutron form factor

axial charge: QAP ~ 1 - 4 sin2w ~ 0

parity violation from neutrons!

modify isovector interactions

1) LV = -a (Tr VV)2 - b Tr (VV)2

LV = … - g ( 4 + 4 + 6 2 2 )

2) LVS = c 2 Tr VV + d Tr ( VV)

LVS ~ [ (1 - r ) 2 + r(2 + 2)] ( 2 + 2 )

vary , r and look at neutron skins + star radii

Horowitz,Piekarewicz, PRL 85, 5647 (‘01)

neutron star Radius R and neutron skin rnp of lead

SWS, PLB560, 164 (2003)dial isospin interaction (vector) and r (scalar) readjust parameters in every step (!!)

208Pb

protons

neutrons

Parity violating electron-nucleus scattering (Jefferson Lab)

radius (fm)

dens

ity

(1/f

m3 )

r=0

r=0.4

208Pb neutron skin as function of

rnp rn - rp ~ 0.26 fm

no refit

refitted

“refit” = fit to BPb and rPb

proton skin in Ar isotopes

Reasonable agreement with data independent of

A. Ozawa et al., RIKEN preprint ‘02

ultrarelativistic heavy-ion collisions

E/A ~ 200 GeV (RHIC)

p

n

u

u

s

d

_

gg

quarks and gluons not confined anymore

(T ~ 2 * 1012 K)Several 1000 produced particles

high T

Au Au

measure particle numbers, determine T,

Quark-gluon plasma

fitting particle ratios measured at RHIC

No resonances resonances

(a) (b)

(a) 170.8 48.3(b) 153.3 51.0(c) 174.0 46.0

T [MeV] B [MeV]

Braun-Munzinger et al., PLB 518, 41 (‘01)

Thermodynamical analysis agrees with the QGP picture

(a)

(a) 153.3 ~ 0.5(b) 174.0 ~ 1.5

T [MeV] B+B / 0

Braun-Munzinger et al., PLB 518, 41 (‘01)

D. Zschiesche et al, PLB547, 7 (2002).

_

_ _• measure p, p, , , ….• fit ratios of particle numbers within model • determines temperature of fireball

nucleon mass as function of T and µ

Conclusions and Outlook

• working hadronic model• good description of masses and nuclear matter• competitive model for relativistic nuclear structure• reasonable neutron stars• PV e- scattering experiment not accurate enough• very good particle ratio fits for SPS/RHIC• first attempts in low-energy heavy-ion simulations

Conclusions and Outlook, continued

explore parameter space nuclear code: add rotation beyond mean field configuration mixing

relativistic Hartree in nuclear code heavy ions

initialization, vector fields, fragmentation

include stellar rotation

expand g() in multipole moments M, R change, deformation

excentricity()

axis ratioof 3/2

Kepler period(M)

PK > 0.8 ms

Mmax(max) = 1.94 Msolar

masses change < 15 - 20 %

M()

neutron star Radius R and Pb neutron skin rnp

vary and r “reasonable values” 0 1.3 0 r 0.3