BOLOGNA SECTION. Paolo Finelli. University of Bologna.

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1 MB31 BOLOGNA SECTION Paolo Finelli University of Bologna INFN-MB31 Collaboration Meeting in Otranto - May 31, 2013

2 Density Functional Theory Pairing Particle-Vibration Coupling Optical Potentials Electron Scattering Collective modes Parity Violation Hypernuclei

3 Electron scattering & Parity violation PRL 108, (2012) P H Y S I C A L R E V I E W L E T T E R S week ending 16 MARCH 2012 Measurement of the Neutron Radius of 208 Pb through Parity Violation in Electron Scattering PHYSICAL REVIEW C 85,032501(R)(2012) Weak charge form factor and radius of 208 Pb through parity violation in electron scattering ρ w (fm -3 ) ρ ch r (fm) A pv = dσ/dω + dσ/dω A pv = G F Q 2 dσ/dω + + dσ/dω 2πα 2 In collaboration with Pavia (see Vorabbi) F W (Q 2 )= F W (Q 2 ) F ch (Q 2 ) d 3 r sin(qr) Qr ρ W (r)

4 Density functional: ground & collective states Meson-exchange Non linear Density dependent Point coupling Non linear Density dependent -phenomenological -chiral dynamics inspired

5 Particle-vibration coupling k k + k Particles Holes (a) Phonons (c) p h Particle-vibration coupling is still an open issue: so far no theoretical approach is self-consistent p,, p, Polarization h,, h, Corrections are always on top of the mean-field calculation p, (b) p Correlation h (d) h,

6 Particle-vibration coupling Total energy Starting potentials (Woods-Saxon?) Kohn-Sham equations apple r 2 2m + V KS (r) k(r) = k k (r) Iterative cycle Convergence old new 1 (in order to stabilize the numerics) Kinetic term External potential Exchange correlation term Hartree potential Particle-vibration coupling with RPA modes: O +,1 -,2 +,3 -,4 +,... V KS = E KS Energy dependent potentials Explicit dependence on KS orbitals and energies Extended Kohn-Sham equations apple r 2 2m + V KS (r)+v KS(r) k (r) = k k (r) Convergence old new 2 Iterative cycle Orthogonalization In collaboration with...

7 Pairing (from realistic forces) L L 0 (k) = 1 Z 1 0 dk 0 k 02 1 E(k 0 ) E(k) 2 = [ (k) (k F )] 2 + D(k) 2 D(k) 2 = L(k) 2 + L 0(k) 2 V LL V LL 0 (k, k 0 ) V L 0 L V L 0 L 0 L L 0 (k 0 ) R. Machleidt, D.R. Entem / Physics Reports 503 (2011) N 3 LO Idaho CD-Bonn AV-18 AV k -1 D 3P 2-3 F 2 In collaboration with S. Maurizio

8 Optical Potentials with V lowk T (p, q; q 2 )=V (p, q)+ 2 R dkk 2 V (p, k)f 2 (k) 1 q 2 k 2 T (k, q; q 2 ) d d V (p, q) = 2 d d T (p, q; q 2 )=0 R dkk 2 V (p, k)t (k, q; q 2 ) d d f 2 (k) 1 k 2 q 2 V Low k (p, q) =f (p)v (p, q)f (q)

9 Optical Potentials with V lowk NN potential G-matrix Phenomenological densities V lowk ρ mf U(E,r) = λ R V (E,r) + iλ I W(E,r) + λ R SO V SO(E,r) + iλ I SO W SO(E,r). In collaboration with...

10 Hypernuclei EΛ p1/2 p3/2 s Table 4 P -shell spin orbit splittings ɛ Λ (p) for six hypernuclei ( 13 Λ C, 16 Λ O, 40 Λ Ca, 89 Λ Y, 139 Λ La, 208 ΛPb). Experimental values [44], or empirical estimates [1,47,48], are shown in comparison with our theoretical predictions (FKVW), using a broad range of ζ parameters (see Eq. (12)), and other relativistic calculations with (RMFI [11]) or without (RMFII [14]) tensor coupling. All energies are given in kev. The asterisk means that a local fit has been necessary. Nucleus Exp. [kev] FKVW (0.4 ζ 0.66) RMFI [11] RMFII [14] 13 Λ C 152± 54 ± 36 [44] Λ O [47] [1] 40 Λ Ca Λ Y 90 [48] Λ La Λ Pb O

11 Hypernuclei Spherical (0,0) Oblate (β,γ=60 o ) γ L ωλ = g Λ ω ψ Λ γ µ ψ Λ ω µ + f Λ ω 2M Λ ψ Λ σ µν ψ Λ ν ω µ. This additional term modifies the effective Λ spin orbit potential as follows: V so,λ 1 [ (( 1 2MΛ 2 2 f Λ ) )] ω r r gω Λ + 1 ΣV Λ ΣΛ S l s. = = + Because of tensor forces, a Λ hyperon in a p state could induce different deformations In collaboration with... Prolate (β,γ=0 o )