Fermi-Liquid Theory for Strong Interactions

Transcription

1 Fermi-Liquid Theory for Strong Interactions H. Lenske Institut für Theoretische Physik, U. Giessen

2 The theorist s dream

3 from asymptotic freedom to confinement to the nuclear shell model Nucleus ~ cold, degenerate Fermi-Gas of Quasiparticles U=U 0 +U so l σ

4 NN-Interaction from the Lattice N. Ishii, S. Aoki, T. Hatsuda, Phys. Rev. Lett. 99, (2007). m π /m ρ = 0.595

5 Agenda: Density Functional & Fermi Liquid Theory Landau-Migdal Parameters and Nuclear Matter Landau-Migdal Parameters and Nuclear Dynamics Hypermatter and Neutron Stars Summary

6 II. Density Functional Theory for Fermi Liquids

7 Existence Theorems on the Dynamics of Interacting Quantum Many-Body Systems: Kohn-Sham (~1960) : QM many-body systems DFT of E[ρ] Kohn-Hohenberg (~1963) : DFT E[ρ,τ] Nuclei : E[ρ p,ρ n,τ p,τ n,κ p,κ n ] Nuclei: Energy Density Functional depending in proton (q=p) and neutron (q=n) densities, currents 1 E ρ τ κ T ρ τ κ T τ E ρ τ κ 2 00 q, q, q... = q, q, q... = ( q) + int ( q, q, q...)

8 in Infinite Nuclear Matter: dk k 1 dk dk E( ρ) = n ( k) + n ( k ) n ( k ) Γ ( k k ρ) qs 3 3 qs 1 q' s' 2 qs, q' s' 1 2 q pn, (2 π) 2m 2 q qq, ', ss, ' (2 π) = (2 π) s=± 1/2 Cold (T=0) Nuclear Matter: k n ( k) ( e ( k)) ( e ( k)) e U ( k ) 2 2 F qs = ΨΨ =Θλqs qs Θλq q λq = qf = + q F unpolarized 2mq Spin-Saturated Nuclear Matter Σ s =N s =2: dk k 1 dk dk E( ρ) = N n ( k) + N N n ( k ) n ( k ) Γ ( k k ρ) s 3 q s s' 3 3 q 1 q' 2 qq, ' 1 2 q pn, (2 π) 2m 2 = q qq, ' (2 π) (2 π) Isopin-Saturated (symmetric) Nuclear Matter Σ q =N q =2: dk k 1 2 dk1 dk2 E( ρ) = NN s q nk () + ( NN) ( 3 s q nk 3 3 1)( nk2) 00( k1 k2 ρ) (2 π) 2m 2 Γ (2 π) (2 π)

9 Elements of Density Functional Theory ( ( )) ( ) E( ρ, κ) E( ρ, κ ) + T + U ρ δ n + δκ 0 0 q q 0 q q q q= p,n ( ) ( ) + f ρ δn δ n + d ρ δκ δκ... qq' 0 q q' qq' 0 q q' q,q' = p,n q,q' = p,n The quasi-particle Self-Energy: U = δ E = Γ ( ρ )n + n n δ Γ ρ + κ κ δ Γ ρ (pair) ( ) ( ) q int qq' q' q' q'' q'q'' q' q'' q'q'' δρq 2 q' 2 q'q'' δnq 2 q'q'' δnq The residual interaction (restoring force): 2 δ 1 δ f =Γ ( ρ ) + 2 n Γ ρ + n n Γ ρ +... ( ) ( ) qq' qq' q'' q'q'' k' k'' k'k'' q '' δnq 2 k 'k '' δnqδnq'

10 the simplest case - E(ρ) ~ E HF (ρ): but:

11 two ways to solve the nuclear many-body problem: The ab initio shell model choice: simple ( bare ) interaction complicated wave function The DFT choice: complicated ( effective ) interaction simple wave function: T + V E χ = 0 T +Γ E φ = 0 ( ) ( ) M Γ= V + Vg Q Γ 12 = φ V χ φ Γ φ F Brueckner G-Matrix Theory (K-Matrix Theory)

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13 The Quasi-Particle Scattering Amplitude and Landau-Migdal Parameters A( k, k,cos ϑ) = N( k ) f( k, k ) 1 2 F 1 2 A= F + F' τ τ + Gσ σ + G' σ σ τ τ ( ) Ak (, k,cos ϑ) = (2 + 1) F+ F' τ τ + Gσ σ + G' σ σ τ τ P(cos ϑ) Density of States at the Fermi-surface: mkf 1 1 Nk ( F ) = ~ 2 2 eq π kf MeVfm (LM Parameters are constrained by the Landau Sum Rules)

14 Landau-Migdal Parameters Relation to Static (ground state) Properties Empirical density ρ sat =0.16/fm 3 : K~250MeV ; m*/m~0.7 ; E sym ~30 MeV (±10%)

15 The Giessen Approach to In-Medium Interactions K = V + VgNNQFK

16 The DDRH-DFT Lagrangian

17 Nuclear Matter DBHF Vertices Isoscalar Vertices Isovector Vertices

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19 Symmetric Nuclear Matter Pure Neutron Matter Region of Instability!

20 The Nuclear Equation of State (F 0,F 1,F 0 )

21 Density Dependence of the Symmetry-Energy (F 0 )

22 B = B theo B B exp exp DDRH Results: B(A) and Charge Radii (F 0,F 0 ) Hartree Vertices (DME-Method: F. Hofmann, HL, PRC 1998) r = < r > < r > 2 2 theo 2 < r > exp exp

23 Neutron Skins in Ni and Sn Isotopes Neutron Skin and Symmetry Energy: Bonn A : a 4 = 32 MeV Groningen : a 4 = 26 MeV Sn Data: Krasnahorkay et al. PRL 82 (1999) 3216 (from Charge Exchange Spin-Dipole sum rules) F. Hofmann et al., PR C64 (2001) N. Tsoneva. H.L., PLB586 (2004), PRC77 (2008)

24 III. Nuclear Dynamics

25 Relation to dynamical Properties Dyson Equation for the 4-point Function Π =Π + Π Γ Π (0) (0) αβ αβ αγ γ γβ γ Π = 0 T G T 0 (0) + (0) αβ α αβ β 1 R λ ω,q = IΠλλ ω,q π ( ) ( ) ρ (1) ρ (2) ρ (1) ρ (2) + + (0) αn βn αe βe Gαβ ( 1, 2 ω ) = + de +... n En ω iη E ω iη

26 how to probe LM-Dynamics: F l surface vibrations, isoscalar giant resonances F l pygmy resonances, isovector giant resonances G l magnetic moments, isoscalar spin excitations G l pionic correlations, Gamov-Teller resonances

27 Sn Isotopes: DFT-HFB Results (N. Tsoneva, HL, PRC77 (2008), PRL 2010, PRL 2011)

28 Electric Dipole Response of Exotic Nuclei N. Tsoneva, H.L.

29 QRPA-Response 128 Sn Microscopic DD-QRPA (F 0,F 0 ) accumulated, normalized EWSR 1 R (E;E λ ) = E B (E λ) λ 1 c c S 1(E ) E E c Phys. Lett. B695 (2011) 174

30 Response Functions for 11 B 11 Be Nucl.Phys. A 739 (2004) 30 Probing G 0, H 0 Response Functions for 56 Fe 56 Mn Nucl. Phys. A 744 (2004)108. A. Ataie, H.L. 2011

31 (e,e ) Response Functions: 48 Ca

32 IV. Hypermatter and Neutron Stars

33 The BEST experimental Proof of Single Particle Motion in Nuclei: S. Bender, R. Shyam, HL, Nucl. Phys. A 839 (2010); P. Konrad, H.L.; Th. Gaitanos, U. Mosel, H.L.

34 DDRH Hypermatter Equation of State (Binding Energy per Baryon) Minimum at 10% Λ-content: B 0 =-18MeV at ρ 0 =0.21fm -3 (R σ =0.49, R ω =0.55)

35 PSR J Neutron Star Mass- Radius Relation DB-Interactions baryon octet leptons: e,µ beta equilibrium TOV equation

36 Summary and Outlook Elements of DFT and Fermi-Liquid Theory Quasi-Particle Interaction and Landau-Migdal Parameters Nuclear Matter, Hypermatter, Neutron Stars Challenges: Baryon Interactions from (L)QCD?! Linking Many-body Dynamics to In-Medium Interactions Flavour Dynamics in Matter Reaction Theory for weakly Bound Systems Credits to: Nadia Tsoneva, Urnaa Badarch, A. Ataie, A. Fedoseew, P. Konrad, Anika Obermann, Th. Gaitanos

37 Other Applications to Strong Interaction Physics: Dense Nuclear Matter: Landau Fermi-Liquid Theory and Chiral Lagrangian with Scaling, Phys.Rept. 347, C. Song Landau Theory of Relativistic Fermi Liquids, G. Baym, S.A. Chin, NPA 262 (1976) Symmetric and anti-symmetric Landau parameters and magnetic properties of dense quark matter K. Pal and A. K. Dutt-Mazumder, hep-ph: v2

38 Composition of Neutron Star Matter