Structure of light hypernuclei in the framework of Fermionic Molecular Dynamics

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1 1 Structure of light hypernuclei in the framework of Fermionic Molecular Dynamics Martin Schäfer, Jiří Mareš Nuclear Physics Institute, Řež, Czech Republic H. Feldmeier, T. Neff GSI Helmholtzzentrum für Schwerionenforschung, Darmstadt, Germany 53rd International Winter Meeting on Nuclear Physics, Bormio

2 Hypernuclei Hypernuclei Hypernucleus nuclear system which contains besides nucleons and protons also one or more hyperons Why to study hypernuclei? test models of BB interactions (meson exchange models, quark models, chiral models,...) test nuclear models (RMF, RPA, NCSM, FMD,...) test models of hadrons (SU(3) symmetry, quark models,...) hypernuclear production (test reaction mechanisms) hypernuclear decays (study of weak interaction) no Pauli blocking for hyperons (probe the nuclear core) astrophysics (neutron stars,...) 2

3 Introduction 3 Introduction Main goal study of light hypernuclei (shell vs. cluster structure) information about the ΛN interaction modification of the nuclear core due to Λ Objectives : develop Fermion Molecular Dynamics for hypernuclei calculations of ground and excited states of s-shell (p-shell) hypernuclei (Λ separation energy B Λ, ρ N and ρ Λ densities, rms radii)

4 FMD model Fermionic Molecular Dynamics (H. Feldmeier, Nucl. Phys. A 515 (1990) 147 ) (T. Neff, H. Feldmeier, Nucl. Phys. A 738 (2004) 367 ) system of interacting fermions described by an antisymmetrized many-body function Q Antisymmetrization many-body wave function approximated by a Slater determinant spatial part of a single particle state represented by a Gaussian wave packet x q k = exp ( ( x b k ) 2 2a k ) χ k, χ k t complex width a k, complex b, complex χ and χ spin parameters (12 real parameters for each particle)

5 5 Minimization FMD model Time-independent variational calculation E min = Q ˆT k + ˆV NN + ˆV ΛN ˆT cm Q min q 1,...,q n Q Q gradient method (gradients evaluated analytically to ensure numerical stability) minimization with respect to single particle state parameters q k = {a k, b k, χ k, χ k } Result minimization yields an intrinsic state which is not parity and total angular momentum eigenstate J π broken symmetries have to be restored

6 6 Symmetries Symmetries (T. Neff, H. Feldmeier, Eur. Phys. J 156 (2008) 69 ) Parity projection parity projected state Q; π = ˆP π Q ˆP π = 1 2 (ˆ1 + π ˆΠ) Total angular momentum projection total angular momentum eigenstate is projected out of the minimized intrinsic state Q; J π MK = ˆP MK J ˆP π Q total angular momentum projector ˆP MK J ˆP MK J = 2J + 1 8π 2 dωdmk J (Ω)ˆR(Ω)

7 7 K-mixing Symmetries (T. Neff, H. Feldmeier, Eur. Phys. J 156 (2008) 69 ) Orthogonal eigenstates Q; J π Mκ = K Generalized eigenvalue problem Q; J π MK C Jπ κ K (Ĥ ˆT cm ) Q; J π Mκ = E Jπκ Q; J π Mκ K HK,K Jπ C Jπ κ K = E Jπ κ NK,K Jπ C Jπ κ K K HK,K Jπ = Q (Ĥ ˆT cm )ˆP KK J ˆP π Q = Q ˆP KK J ˆP π Q N Jπ K,K

8 V NN and V ΛN potentials Interactions NN two-body potentials V2-M0.0, V2-M0.6 (A. Volkov, Nucl. Phys. 74 (1965) 33 ) MTV (UCOM modified*) (R. Malfliet, J. Tjon, Nucl. Phys.A127 (1969) 161) ATS3M (UCOM modified*) (I. Afnan, Y. Tang, Phys. Rev. 175 (1968) 1337) * UCOM (H. Feldmeier, T. Neff, R. Roth, J.Schnack, Nucl. Phys. A632 (1998) 61) ΛN two-body potential G-matrix transformed YNG (Jülich, Nijmegen) k F dependence (Y.Yamamoto et. al, PTP Suppl. 117 (1994) 361) V ΛN (r) = 3 { (a i + b i k F + c i kf 2 )exp r 2 } i β 2 i 8

9 Results of s-shell hypernuclei 4 Λ He Parity projection of energy levels in 4 Λ He V variation without parity projection VAP π variation of the parity projected state Parity projection ˆP π increases B Λ

10 Results of s-shell hypernuclei 4 Λ He V NN dependence of energy levels in 4 Λ He Λ separation energy B Λ slightly changes with V NN B( 3 He; V2M0.6) = 7.18 MeV B( 3 He; MTV) = 6.45 MeV B( 3 He; ATS3M) = 5.40 MeV 10

11 Results of s-shell hypernuclei 4 Λ He V ΛN dependence of energy levels in 4 Λ He Substantial difference between Λ separation energies as well as B Λ (0 + ) B Λ (1 + ) for various V ΛN

12 Results of s-shell hypernuclei 4 Λ He k F dependence of energy levels in 4 Λ He Strong Fermi momentum dependence in the V ΛN part (k F acts as a scaling factor) k F = 0.8 fm 1 (Y.Yamamoto et al, PTP Suppl. 117 (1994) 361) k F = fm 1 ( 3 He rms radius approximation) k F = 0.72 fm 1 (test value)

13 Results of s-shell hypernuclei Mirror hypernuclei 4 Λ He and 4 Λ H 4 Λ He and 4 Λ H no difference in B Λ between 4 Λ He and 4 Λ H using YNG V ΛN B Λ exp.( 4 ΛHe; 0 + ) = MeV B Λ exp.( 4 ΛHe; 1 + ) = MeV B Λ exp.( 4 ΛH; 0 + ) = MeV B Λ exp.( 4 ΛH; 1 + ) = MeV

14 14 p-shell hypernuclei p-shell hypernucleus 7 Λ Li 7 Λ Li preliminary results extensive computational complexity k F = 0.95 fm 1 (Y.Yamamoto et al, PTP Suppl. 117 (1994) 361)

15 Conclusions 15 Conclusions FMD for hypernuclei developed calculations of s-shell hypernuclei 4 Λ H and 4 Λ He relevance of the symmetry restoration (B Λ, projected 0 + and 1 + state) weak V NN dependence of B Λ strong V ΛN dependence of B Λ strong k F dependence of B Λ (k F acts as a scaling parameter of YNG V ΛN potentials) preliminary results for 7 Λ Li Next steps : calculations of p-shell hypernuclei more sophisticated interactions (Argonne V18, V ΛN potentials with Λ Σ mixing, chiral V NN and V ΛN potentials) ΛΛ hypernuclei