Application of correlated basis to a description of continuum

states19th International IUPAP Conference on Few-

Body Problems in Physics

University of Bonn, Germany31.08 – 05.09.2009

Wataru Horiuchi (Niigata, Japan)Yasuyuki Suzuki (Niigata, Japan)

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

Variational calculation for many-body systems

Hamiltonian

Basis function

Realistic nucleon-nucleon interactions: central, tensor, spin-orbit

Generalized eigenvalue problem

Correlated Gaussian and global vector

Global Vector Representation (GVR)

x1 x3

x2

Correlated Gaussian

Global vector

Parity (-1)L1+L2

Advantages of GVR

• No need to specify intermediate angular momenta.– Just specify total angular momentum L

• Nice property of coordinate transformation– Antisymmetrization, rearrangement channels

Variational parameters A, u → Stochastically selected

x1 x3

x2 y1 y2y3

4He spectrum

Good agreement with experiment without any model assumption

3H+p, 3He+n cluster structure appearW. H. and Y. Suzuki, PRC78, 034305(2008)

P-wave

S-wave3H+p

3He+n

Ground state energy Accuracy ～ 60 keV.H. Kamada et al., PRC64, 044001 (2001)

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

Formalism(1)

Key quantity: Spectroscopic amplitude (SA)

The wave function of the system with E

A test wave function

U(r): arbitrary local potential (cf. Coulomb)

Inhomogeneous equation for y(r)

Formalism(2)The analytical solution

G(r, r’): Green’s function

SA solved with the Green’s function (SAGF)

Phase shift:

v(r): regular solutionh(r): irregular solution

Test calculations

Neutron-proton phase shiftMinnesota potential (Central)• Numerov• SAGF

Neutron-alpha phase shiftMinnesota potential + spin-orbitAlpha particle → four-body cal.• R-matrix• SAGF

The SAGF method reproduces phase shifts calculated with the other methods.

Relative wave function

Improvement of the asymptotics

Ill behaviors of the asymptotics are improved

α+n scattering with realistic interactions

Interactions: AV8’ (Central, Tensor, Spin-orbit)Alpha particle → four-body cal. Single channel calculation with α+n 　

1/2+ 　 → fair agreement1/2-, 3/2- → fail to reproduce

• distorted configurations of alpha• three-body force

K. M. Nollett et al.PRL99, 022502 (2007)Green’s function Monte Carlo

S. Quaglioni, P. Navratil,PRL101, 092501 (2008)NCSM/RGM

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)

Decomposition of the phase shift

Vc: central, tensor, spin-orbit

Neutron-alpha scattering with 1/2+