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Chiral Theory of Nuclear Matter and Nuclei

Chiral Theory of Nuclear Matter and Nuclei. strong interactions - ingredients, problems, partial solutions constructing a working hadronic model applications of a hadronic model. nuclear matter nuclear structure neutron stars, heavy-ion collisions. projects, outlook. D. Zschiesche

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Chiral Theory of Nuclear Matter and Nuclei

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  1. Chiral Theory of Nuclear Matter and Nuclei • strong interactions - ingredients, problems, partial solutions • constructing a working hadronic model • applications of a hadronic model nuclear matter nuclear structure neutron stars, heavy-ion collisions • projects, outlook D. Zschiesche G. Zeeb K. Balazs M. Reiter Ch. Beckmann P. Papazoglou

  2. Strong Interactions at low Energies QCD as theory of strong interactions well established radiative corrections generate running coupling constant α QCD quarks (asymptotic freedom) hadrons

  3. Strong interactions: drastic phenomenological consequences electrodynamics α ~ 1/137 , αn <<α coupling strength: QCD α ~ 1 , αn~ α QCD: q q e- e- g g  QED: (1-gluon exchange as important as 2-gluon exchange, …) proton (uud), neutron (ddu) : m ~ 20 MeV but total mass Mp , Mn ~ 1 GeV ! dynamical mass creation

  4. Chiral Symmetry: left- and right-handed particles decouple true for all vector interactions L/R = ½( 1 -/+ 5) L/R _ _ _ _ _  A = (L+R) A(L+R) = L AL + R A R e-.q , g R/L L/R e-.q m mass terms violate symmetry L/R _ _ _ _ _ m  = m (L+R)(L+R) = m (LR + R L) m << Etypical chiral symmetry useful, mu,d << Mn, ms < Mn

  5. QCD vacuum has a complex structure! _ Eqq ~ Ekin + Epot < 0 !condensation _ <0|q q|0>, <0| G G|0> = 0 / left-handed (k || s) right-handed (k || s) particles mass terms couple chirality _ qR qL _ qR qR qL qL _ qLqR mass generation! G G G G _ _ _ _ _ M  = M (L+R)(L+R) = M (LR + R L) _ _ _ _ _  A = (L+R) A(L+R) = L AL + R A R

  6. what about calculating larger systems - nuclei? currently not feasible within a quark picture hadronic description quarks, gluons hadrons nuclear matter, nuclei quark/gluon picture hadrons _ <0|q q|0> <0||0> <0| G G|0> <0||0> _ _ _ _ _ G N N N N G <N N> N N  MNN N dynamical mass

  7. construct a chirally symmetric interaction _ _ _ _ _ _ LI ~ (N N)2 + (N i 5 N)2 = (LR + RL)2 - (LR - RL)2 L R Original Nambu Jona-Lasinio _ _ bosonize: N NN i 5 N   _ R L ~ 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

  8. Degrees of Freedom SU(3) multiplets: n (ddu) p (uud) Baryons  - (sdd)  0  (sdu)  + (suu)  - (ssd)  0 (ssu) hyperons _ _  0 (sd) + (su) Scalar Mesons - (ud)  0 , ,   +(du)  - (us) 0 (ds) _ _ _ _ _ _ _ _ _ _  ~ <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) _ _ _ _ _ plus pseudoscalars, axial vectors and gluonic field 

  9. 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) _ _ _

  10. fit parameters to hadron masses  mesons * *         K* p,n ’    Model can reproduce hadron spectra via dynamical mass generation!  K 

  11. fields change in a dense and hot medium MN ~ g 0 (+ g0 + g0 ) e.o.m:  ~ - g /m2 s In the medium the vacuum condensate is reduced ( < 0) Inside of an atomic nucleus MN*/MN ~ 0.6 strong scalar attraction! ~ - 300 MeV plus vector repulsion from surrounding nucleons: VV ~ g  ~ - g /m 2 V ~ 240 MeV Vs- VV ~ - 540 MeVVLS ~ d/dR (VS- VV) large LS splitting

  12. 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

  13. important reality check nuclear matter (infinite matter, same number of p and n, no Coulomb) asymmetry energy E/A (p- n) equation of state E/A () binding energy E/A ~ -15.2 MeV saturation (B)0 ~ .16/fm3 compressibility  ~ 223 MeV asymmetry energy ~ 31.9 MeV phenomenology: 200 - 300 MeV 30 - 35 MeV

  14. 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 !

  15. Task: self-consistent relativistic mean-field calculation coupled 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 %) Best relativistic nuclear structure models good charge radii rch~ 0.5 % (+ LS splittings) SWS, Phys. Rev. C66, 064310 (2002)

  16. Lagrangian (in mean-field approximation) L = LBS + LBV + LV + LS + LSB baryon-scalars: _ LBS= -  Bi (gi  + gi  + gi  ) Bi baryon-vectors: _ LBV= -  Bi (gi  + gi + gi  ) Bi 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) / (020)] LV = - k’0/2 2 (2 + 2 + 2 ) + g4 (4 + 4 + 4 + 6 22) explicit symmetry breaking: LSB = - (/0)2 (c1 + c2)

  17. nuclide chart - deformation superheavies? Z =116 Dubna magic numbers stick out as spherical shapes neutron drip line (preliminary) number of protons exotic nuclei large isospin (new GSI, ISAC, RIA) number of neutrons

  18. 2-nucleon gap energies < 2p > = < E(Z+2,N) - 2 E(Z,N) + E(Z-2,N) >N spherical deformed neutrons protons

  19. Form Factors, Charge Densities < rch > ~ 0.5 %

  20. linear realisation fails! charge distribution in 208Pb higher-order couplings generate fluctuations (nuclear matter ok!)

  21. 2D calculation of Mg Isotopes NL3

  22. heavy nuclei - deformation of Nobelium Isotopes axis ratio 3:2 (see neutron stars) heavy nucleus Z = 102, N=150,152,154 2 ~ 0.32  0.02 2 ~ 0.31  0.02 SWS, Phys. Rev. C66, 064310 (2002) exp.: Herzberg et al.PRC65 014303 (2001)

  23. deformation of S , Ar isotopes H. Scheit et al., PRL 77, 3967 (1996) S Ar N = Z = 34 S. M. Fischer et al., PRL 84, 4064 (2000) oblate groundstate ( ~ -0.3) excited prolate state

  24. superdeformed nuclei constraint 2d calculation energy SWS, Phys. Rev. C66, 064310 (2002) Exp: T. L. Khoo et al., PRL 76, 1583 (1996)

  25. breathing nucleus - B/A [MeV] 208Pb av [1/fm3] effective compressibility eff~ 117 MeV EGMR ~ (eff / m<r2> )1/2 ~ 12.3 MeV (exp: 13.7 MeV)

  26. superheavy nuclei - new valleys of stability? GSI, Dubna, Berkeley - fuse two heavy nuclei to new stable(?) superheavy elements (Z = 114, 120, 126 ?) 2-nucleon gap energy spherical peaks : strongly bound deformed protons neutrons signal for magic number of Z=120 vanishes in deformed calculation (deformed gap, metastable states? more detailed study needed )

  27.  (uds) single-particle energies 40Ca = (20 p, 20 n) 40Ca = (20 p, 19 n, 1 ) hypernucleus Nuclear matter Model and experiment agree very well

  28. The “Ultimate” Neutron-Rich Nucleus Neutron star: M ~ 1.4 Msolar R ~ 13 km MHT = (1.4411  0.0035) Msolar (Hulse-Taylor) collection of mass measurements Thorsett,Chakrabarty, APJ 512 288 (‘99) Neutron stars - constraint on hadronic models

  29. 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

  30. typical neutron star mass(r) regions relevant to nuclear physics density(r)

  31. equation of state of nuclear matter varying isospin Ratio of neutrons  = (n -p) / (n +p)

  32. static “neutron” star M. Hanauske, D. Zschiesche, S. Pal, SWS, H. St\öcker, W. Greiner, Ap. J. 537, 958 (2000).. no hyperons particle cocktail hyperstar Input: Equation of State (), p() 2 4 /0 max ~ 25% of “exotic” matter ( ,  -,  -) Not too exotic!

  33. SWS, D. Zschiesche, J. Phys. G 29, 531 (2003) include stellar rotation expand g() in multipole moments M, R change, deformation excentricity() axis ratio of 3:2 Kepler period PK > 0.8 ms fastest known pulsar PK ~ 1.5 ms Mmax(max) = 1.94 Msolar masses change < 20 % Experimental numbers for frequency, radius, mass needed

  34. neutron star results - summary static star: TOV equations (P) low hyperon content fs ~ 1/3 (, -, -) Mmax ~ 1.54 Msolar1.82 Msolar (no hyperons) Rmin ~ 11.3 km 11.2 km “ rotating star: excentricity < 3:2 Kepler frequency 1/ 0.8 ms mass increases to ~ 1.9 Msolar during slow-down non-strange  strange star no backbending, phase transition in this model

  35. Influence of resonances r =g / gN not well determined!

  36. parity violation at Jefferson Lab Polarized e- scattering from 208Pb (850 MeV) e- e- +  Z0 p p,n Polarized cross section  interference term R-L GF Q2 Fn(Q2) neutron form factor  R+L 4  2 Fp(Q2) axial charge: QAP ~ 1 - 4 sin2w ~ 0 parity violation from neutrons!

  37. modify isovector interactions 1) LV = -a (Tr VV)2 - b Tr (VV)2  LV = … - g ( 4 + 4 + 6 2 2 ) 2) LVS = c 2 Tr VV + d Tr ( VV)  LVS ~ [ (1 - r ) 2 + r(2 + 2)] ( 2 + 2 ) vary  , r and look at neutron skins + star radii Horowitz,Piekarewicz, PRL 85, 5647 (‘01)

  38. neutron star Radius R and neutron skin rnp of lead Parity violating electron-nucleus scattering (Jefferson Lab) neutrons r=0.4 208Pb protons density (1/fm3) r=0 radius (fm) dial isospin interaction  (vector) and r (scalar) readjust parameters in every step (!!) SWS, PLB560, 164 (2003)

  39. 208Pb neutron skin as function of  rnp rn - rp ~ 0.26 fm no refit refitted “refit” = fit to BPb and rPb

  40. proton skin in Ar isotopes Reasonable agreement with data independent of  A. Ozawa et al., RIKEN preprint ‘02

  41. ultrarelativistic heavy-ion collisions Au Au E/A ~ 200 GeV (RHIC) p quarks and gluons not confined anymore (T ~ 2 * 1012 K)   high T Several 1000 produced particles Quark-gluon plasma n  _ u s g g measure particle numbers, determine T,  u d

  42. fitting particle ratios measured at RHIC (b) (a) No resonances resonances T [MeV] B [MeV] (a)170.8 48.3 (b) 153.3 51.0 (c) 174.0 46.0 Braun-Munzinger et al., PLB 518, 41 (‘01)

  43. Thermodynamical analysis agrees with the QGP picture D. Zschiesche et al, PLB547, 7 (2002). _ _ • measure p, p, , , …. • fit ratios of particle numbers • within model • determines temperature of fireball (a) T [MeV] B+B / 0 _ (a) 153.3 ~ 0.5 (b) 174.0 ~ 1.5 Braun-Munzinger et al., PLB 518, 41 (‘01)

  44. nucleon mass as function of T and µ

  45. 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

  46. 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

  47. include stellar rotation expand g() in multipole moments M, R change, deformation excentricity() Kepler period(M) axis ratio of 3/2 PK > 0.8 ms M() Mmax(max) = 1.94 Msolar masses change < 15 - 20 %

  48. neutron star Radius R and Pb neutron skin rnp vary  and r “reasonable values” 0   1.3 0 r  0.3

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