outline : Overview of the Brueckner theory of Nuclear Matter

1. Dynamics and Correlations in Exotic Nuclei YITP, Kyoto 2011 NN effectiveinteractionfromBruecknerTheory and ApplicationstoNuclearSystems U. Lombardo Dipartimento di Fisica e Astronomia and INFN-LNS,Catania (Italy) • outline: • Overviewof the BruecknertheoryofNuclearMatter • 3BF within the meson-exchangemodelof the interaction • Skyrme-typeparametrizationof the BHF energyfunctional • In medium Nucleon-Nucleon cross section • TransportParameters and dissipationofcollectivemodes in NS • Optical – ModelPotentialfornucleon-Nucleuscollisions at E<200 MeV.

2. Collaboration M. Baldo, HJ Schulze, INFN Catania I. Bombaci, Pisa University U. Lombardo, Catania University and INFN-LNS A. Lejeune, LiegeUniversity J.F.Mathiot,Clermont-Ferrand,CEA W. Zuo, ZH Li . IMP-CAS, Lanzhou H.Q.Song, FudanUniversity

3. The non-relativistic nuclear many-body problem Nuclearmatterisanhomogeneous system madeofpointlikeprotons and neutrons interactingthroughthe Hamiltonian strongly constrained by the most recent data (few thousands, up to 350 MeV) on NN phase shifts, energy scattering parameters, and deuteron binding energies. Nucleon-Nucleon phase shifts As 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.

4. Bonn B and Experphaseshifts

5. 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

6. The Brueckner-Bethe-Goldstone theory of Nuclear Matter healing 1S0 • 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)

7. The BBG expansion  BHF approximation Two and three hole-line diagrams in terms of the Brueckner G-matrixs Bethe-Faddeev

8. The variational method in its practical form Pandharipande & Wiringa, 1979; Lagaris & Pandharipande, 1981 Methodusedtocalculate the upper boundto the ground state energy: Trial w.f.is the antisymmetrizedproductoftwo-bodycorrelation functionsactingon anunperturbedground state Φ: . The correlationfunction f represents the correlationsinducedby the two-bodypotential. Itisexpanded in the samespin-isospin, spin-orbit and tensoroperatorsappearing in the NN interaction. The correlationfunctionisobtainedfrom

9. Dependence on the Many-BodyScheme Nucleon-Nucleon Interaction: Argonne v18 (Wiringa, Stoks & Schiavilla, Phys. Rev. C51, 38 (1995)) same NN interaction BBG: Catania group Neutron Matter ? Symmetric Matter Variational (Akmal, Pandharipande & Ravenhall, PRC 58, 1804 (1998)) Itmakesusconfidentthattheseapproximationsdo work!

10. Convergenceofholelineexpansion At the levelof3 holelineapproximationthe EoSisindependenofauxiliarypotential 2+3 hole UG=gapchoice UC=continuouschoice E0= <K> + Σi<V>i = - 16 .0 MeV <V>2= - 40 MeV <V>3= 3 MeV <V>4= 0.5 MeVB.Day,1987 Song,Baldo,Giansiracusa and Lombardo. PRL 81 (1998): 3 holelines: Bethe-Faddeevequation

11. saturationpoint

12. 3BF in Nuclei puzzle ofanalyzingpower IshikawaPhys.Rev. C59 (1999) N-del.scatt. At E=3MeV 10B Caurier, P. Navrdtil, W.E.Ormand, and J.P.Vary, Phys. Rev. C64, 051301 (2001)

13. Meson-exchangeModelof2- and 3- body interaction P. Grangé et al., PRC 40, 1040 (1989): , ,σ,ωexchange N + + ,N* Fujita-Miyazawa model two-body Diracseapolarization Z-diagrams are introducedas the relativisticcorrectionton.r. BHF approach Brown,Weise,Baym,Speth, Nucl.Part.Phys.1987

14. NucleonExcitations Δ - resonance spin-flipofone quark S=1/2 Lπ=1- Jπ=1/2+ 1s 2s N* - resonance a radially excited three-quark state P11 (1440) couplingconstantsfromN* and Δdecaywiths + quark structure

15. N Input: mesonparametersfitting NN experimentphaseshifts Output: 2BF and 3BF based on the mesonparameters No adjustableparameters !

16. Double-selfconsistentapproachof BHF-3BF saturatingoneparticle Ф12 two-bodycorrelationfunction : V Ф12 = G Фo12

17. Esym (ρ) = BA (ρ)|SNM - BA (ρ)|PNM asystiff asysoft • saturationpoint: • =0.17 fm-3 , E= -16 MeV • symmetryenergy at saturation • Sv≈30 MeV • incompressibility at saturation • K ≈ 244 MeV Danielewicz Plot HIC collectiveflows ZH Li, U Lombardo and H-J Schulze, W,Zuo PRC(2008)

18. sub-saturationdensities 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 Esym (ρ)=Esym(ρo)+ L/(3 ρo) (ρ-ρo)+Ksym /(18ρo2)(ρ-ρo)2 From HIC (isoscaling, multigragmentation, n-p products,..) stillambiguous

19. EoSfrom BHF : individualcontributions and comparisonwith DBHF NN excitations (Z-diagrams) nucleonresonances (F-M) : Δ(1232),R*(1440) strong compensation !

20. BHF with 2+3 body force The G-matrixexpansion (fullBruecknertheory) isconverging The non relativisticBHFtheorywithtwo body forceisconsistent withothern.r.many body approaches (e.g., variationalmethod) Includingrelativisticeffects (Z-diagrams) the BHF EoSfullyoverlaps the DBHF The three-bodyforcepushes the saturation density to the empirical value, butitseffectissmall on the saturationenergy (1-2 MeV) and compressionmodulus (220->240 MeV) issmall

21. reconciling nuclei and nuclearmatter Ambitious task istobuild up a unified EDF for nuclei and nuclearmatter: The questionis: Towhatextent a Skyrme-typeforces can reproduce the BHF (numerical) EDF aswellasexperimentalobservables? Ourhope in suchstudyistolearnwhatismissing in the theoryofnuclearmatter and whichSkyrmeparametrizationscannot match nuclearmatter weperformed a ‘weightedfit’ ofnuclearmatter (theory) and nuclei (experiment) via the Skyrmefunctional: Gambacurta,Li,Colo’,Lombardo, Van Giai,Zuo, PRC 2011

22. Skyrme-like Energy Functionals (LNSx) splitting U into (S,T) • Despitelargedeviations in the individualspin-isospincomponentsofenergypotential U(S,T) • largecompensationgives rise tosimilarEoSspeciallyaroundsaturation • Since U(S,T) probe differentobservables (compressibility,symmetryenergy, GR’s,…) itlooks • convenienttofit U(S,T) insteadof total U • SimultaneousSkyrme-likefitofnuclearmatter (theory) and nuclei (exp) changing the • relative weights: th=BHF Oα = nuclearbindingenergies and radii. Changing the relative weightswe produce differentparametrizations: LNSn (n=1,2,3,…)

23. HF calculationsfrom the Skyrme-likeparametrizationsof BHF theor <- ( LNS1 – LNS5 ) -> exp closed-shellnuclei b. energy and radii SnIsotopicchain GiantResonances open problems: tensor and spin-orbit

24. Energy Density Functional on MicroscopicBasis Baldo,Robledo,Schuck,Vinas finite rangeterm (surface)

25. MicroscopicTransportEquation: n = 1 - n In BHF nuclearpotentialand cross sectionsare calculatedfromG-matrix j • Applications: • simulationsofheavyioncollisions (test-particlemethod) • fromlinearization : n = n0 + δn -> collectivemodes • nuclearfluidodynamics: transportparameters in dense matter • (high-energy HIC, neutronstars)

26. gradT heat flux A PhenomelogicalTransportEq.s HeatConductionEq (K thermalconductivity) Navier-StokesEq. bulk viscosity shearviscosity

27. TransportParameters Linearizing the Boltzmannequation the connection withphenomenological transportequationsisestablishedaswellas the transportparameters shearviscosity thermalconductivity is the in-medium NN cross section

28. In-medium NN cross sections Tocalculate the cross sections at high density the free scatteringamplitude mustbereplacedbyin-mediumscatteringamplitude (G-matrix) • medium effects in the cross section: • Pauliblocking (Q): nucleonsscatterintounoccupiedstates • Strong meanfield (H0) betweentwocollisions • Compressionof the leveldensities in entry and exitchennels

29. theoreticalframework (G-matrix) quasiparticle spectrum in-medium dσ/dΩ mean free path, viscosity, heatconductivity,…

30. G In medium effects on the scattering cross section dσ/dΩ(θ) =N2 |<p|G (θ)|p’ >|2 A) -------------FiniteRangeInteraction--------------------------- }N level density: }N effective mass vs ρ effective mass vs β empirical OMP data support the predictionm*n>m*pin ANM

31. Meanfield and EffectiveMass vs β empirical OMP data support the predictionm*n>m*pin ANM Weexpectthat in neutron-richmatter. σnnislesssuppressedthanσpp

32. B)--------------------Pauliblocking p>pF----------------------------------------- { =0 pf=0 free space >0 pf >0 in medium pF momentum transfer p ≈ p’ ≈ pF θ Δp = 2pF sin (θ/2) = Fermi sphere In CM frame backward and forwardscatteringssizablysuppressed byPauliblocking

33. transportparameters in NS structureof NS: chemicalcomposition,superfluidity, phase transitions,… transportphenomena rotation, glitches, cooling,collectivemodes,…

34. NSinternalstructure nucleons,hyperons, kaons, quarks in beta-equilibriumwithleptons • beta-equilibriumwithelectrons and muons: p + e¯ n +  • hyperonizedmatter: n + n n +  ( p + ¯) at  > 2o • kaoncondensation n p + K¯at > 2-3o • transitiontoquark matterHP QP (u,d,s) at  ~ 6o restrict on only the first item (p,n. e- ) transitionto QGP (in progress)

35. Viscositycontrols the rotationaldynamics and the dampingofcollectivemodes NS is a viscousfluidrapidlyrotating Glitches and post-glitchesrelaxationtime (superfluidity ?) couplingbetween rotation and collectivemodes ( r-modes) ChandrasekharInstability

36. Thermalconductivitycontrolsthermalevolution ofNeutronStars (Tsurfvs. Tcore) nn  (nn) +  (nn)  nn minimal cooling:

37. ANM withβ = β (ρ) p + e¯  n + e n  p + e¯ + e p + ¯  n +  n  p + ¯ + 

38. Cross Sections in β-stable matter p + e¯  n + e n  p + e¯ + e ANM withβ = β (ρ) nn collisions np collisions non linear behaviour of proton mean field and effective mass

39. shear viscosity Beta-stable SNM vs.PNM electrons Shternin & Yakovlev PRD(2008), APR Benhar & Carbone, arXiv09112.0129,CBF σo(Ω) → σ(Ω): η~10·η0 Flowers & Itoh, ApJ (1979) isospineffect: η(proton) → η(neutron) at higher density m →m* : ηn >> ηeShternin & Yakovlev PRD(2008) no r-mode damping

40. thermal conductivity NS Cooling Yakovlev et al, Phys.Reports 354 (2001) 1

41. Dissipationofr-modes A non radialcollectiveexcitationof a NS isdescribedby a velocityfield collectiveenergy: dissipationtimescale (viscosityonly): η(r )= η(ρ(r)) density profileof NS isrequired !

42. Tolman-Oppenheimer-Volkov(TOV) and nuclearEoS Input  Equation of State P=P(, p) Output Mass-Radius plot

43. Time scale ofnonradialmode damping fromshearviscosity Li,Lombardo,Peng, PRC (2008) integrationover the star (0≤r≤R): r → ρ(r) → η(ρ) → TOV constant mass approximation l=2 r-mode viscositytime scale thermal cond.

44. Transitiontodeconfinedphase in neutronstars Despite the contributionofquarkstoviscosityissmallerthanhadrons the phasetransition ispushing the hadronphasetohigher density allowingan extra contributiontoviscosity

45. ChandrasekharInstability (1970) Y22 -nonradial mode: ~ ω0 (Coriolisforce) Inertial frame J0 >> J22whichdecreases,being Y22 a sourse ofgravitationalradiation (GeneralRelativity) Corotatingframe J’22 =  (J-J22 ) |J’22|increases more and more accompaniedbylargerfrequency and amplitude and then more angularmomentum loss forradiation Sincerotating NS exist, GR instabilitymustbestoppedby Some dampingof Y22 the best candidate isviscosity interplay between GR drivinginstability and viscositydamping criticalvelocity: expectedtobedetected in terrestriallabs ( LIGO,VIRGO,…)

46. interplay between GR drivinginstability and viscositydamping criticalvelocity: Ω~1000 Hz → τGR ~ 100 sec(depending only on rotation) constant density approximationunderestimates the effectofviscosity

47. NuclearPotential

48. OpticalPotential vs. nucleon-NucleusScattering mass – shell relation BBG hole-lineexpansion in G-matrix Hughenholtz-VanHovetheorem: on-shellself.energy procedure: for a givenapproximationof∑ one solve k=k(E) and determinesselfconsistently the on-shellselfenergy

49. NuclearPotential corepolarization Diracsea pol. … + + +

50. n and p self-energies p,E EF p,E EF p,E EF DiracSea