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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