Application of correlated basis to a description of continuum states

1. Application of correlated basis to a description of continuum states 19th International IUPAP Conference on Few-Body Problems in Physics University of Bonn, Germany 31.08 – 05.09.2009 Wataru Horiuchi (Niigata, Japan) Yasuyuki Suzuki (Niigata, Japan)

2. Introduction • Accurate solution with realistic interactions • Nuclear interaction • Nuclear structure • Some difficulties • Realistic interaction (short-range repulsion, tensor) • Continuum description → much more difficult (boundary conditions etc.) • Contents • Our correlated basis • Method for describing continuum states from L2 basis • Examples (n-p, alpha-n scattering) • Summary and future works

3. Variational calculation for many-body systems Hamiltonian Realistic nucleon-nucleon interactions:central, tensor, spin-orbit Generalized eigenvalue problem Basis function

4. Correlated Gaussian and global vector Correlated Gaussian Global vector x2 x1 x3 Global Vector Representation (GVR) Parity (-1)L1+L2

5. Advantages of GVR Variational parameters A, u → Stochastically selected • No need to specify intermediate angular momenta. • Just specify total angular momentum L • Nice property of coordinate transformation • Antisymmetrization, rearrangement channels x2 y3 y1 y2 x1 x3

6. 4He spectrum Ground stateenergy Accuracy ～ 60 keV. H. Kamada et al., PRC64, 044001 (2001) 3H+p, 3He+n cluster structure appear W. H. and Y. Suzuki, PRC78, 034305(2008) P-wave S-wave 3He+n 3H+p Good agreement with experiment without any model assumption

7. For describing continuum states • Bound state approximation • Easy to handle (use of a square integrable (L2) basis) • Good for a state with narrow width • Ill behavior of the asymptotics • Continuum states • Can we construct them in the L2 basis? • Scattering phase shift

8. Formalism(1) The wave function of the system with E Key quantity: Spectroscopic amplitude (SA) A test wave function Inhomogeneous equation for y(r) U(r): arbitrary local potential (cf. Coulomb)

9. Formalism(2) The analytical solution G(r, r’): Green’s function v(r): regular solution h(r): irregular solution SA solved with the Green’s function (SAGF) Phase shift:

10. Test calculations Relative wave function • Neutron-alpha phase shift • Minnesota potential + spin-orbit • Alpha particle → four-body cal. • R-matrix • SAGF • Neutron-proton phase shift • Minnesota potential (Central) • Numerov • SAGF The SAGF method reproduces phase shifts calculated with the other methods.

11. Improvement of the asymptotics Ill behaviors of the asymptotics are improved

12. α+n scattering with realistic interactions Interactions:AV8’ (Central, Tensor, Spin-orbit) Alpha particle → four-body cal. Single channel calculation with α+n S. Quaglioni, P. Navratil, PRL101, 092501 (2008)NCSM/RGM K. M. Nollett et al. PRL99, 022502 (2007) Green’s function Monte Carlo • 1/2+　 → fair agreement • 1/2-, 3/2-→ fail to reproduce • distorted configurations of alpha • three-body force

13. Summary and future works • Global vector representation for few-body systems • A flexible basis (realistic interaction, cluster state) • Easy to transform a coordinate set • SA solved with the Green’s function (SAGF) method • Easy (Just need SA) • Good accuracy • Possible applications (in progress) • Coupled channel • Alpha+n scattering with distorted configurations (4He*+n, t+d, etc) • Extension of SAGF to three-body continuum states • E1 response function (cf. 6He in an alpha+n+n) • Complex scaling method (CSM) • Lorentz integral transform method (LIT) • Four-body continuum • Four-body calculation with the GVR • Electroweak response functions in 4He (LIT, CSM)

15. Decomposition of the phase shift Neutron-alpha scattering with 1/2+ Vc: central, tensor, spin-orbit