Complex energy method in four-body Faddeev-Yakubovsky equations

Abstract

The Complex Energy Method [Prog. Theor. Phys. 109, 869L (2003)] is applied to the four body Faddeev-Yakubovsky equations in the four nucleon system. We obtain a well converged solution in all energy regions below and above the four nucleon break-up threshold.PACS numbers: 25.10.+s, 11.80.Jy, 02.40.Xx Calculations for scattering systems in configuration space require boundary conditions which increase in complexity with growing particle numbers. These boundary conditions appear in the form of Green's functions in momentum space which carry singularities of increasing complexity. The Green's functions are expressed as G 0 = 1/(E + iε − H 0 ) where E and H 0 are the total and kinetic energy, respectively, and the limit ε → 0 has to be taken. In the two-body system there is one (relative) momentum variable p and G 0 has a pole in the complex p-plane. It is easy to handle it using the principal value prescription and (half) the residue theorem (PVR). In the three body case there arises already a difficulty in the form of so called moving singularities [1,2], however, PVR is still applicable [3], or one can use the contour deformation [4,5,6,7] (CD) technique. Summarizing these techniques, first one takes the limiting value ε → 0 and next the equation is solved avoiding the integration path on the complex plane. This is illustrated in Fig. 1.The situation is more complicated in the four body system. Employing a separable potential and a separable expansion technique for the three body, and [2+2] subamplitudes, the four body Faddeev-Yakubovsky (FY) equations [8] for four identical particles can be expressed as * Electronic address: j-uzu@ed.noda.tus.ac.jp

Calculation of Expectation Values of Operators in the Complex Scaling Method

“…Such boundary conditions are not appearing in momentum space, but then one deals with singularities which are treated by calculating the resolvent in the complex energy plane by going over the singularity by small finite radii [27,28]. This technique is widely known as complex energy method and was recently employed successfully for describing reactions above four-body breakup threshold with realistic interactions [29,30] and also for calculations on the lattice [31].…”

A Λnnthree-body resonance

“…In addition the Complex Energy Method (CEM) [9] is powerful and useful tool to solve the few-body system for the continuum energy regime. It has been used not only to the Faddeev calculation but to the four-body Yakubovsky calculation [10,11].…”

Four-Body Faddeev-Yakubovsky Calculation Using the Finite Range Expansion Method

“…In particular, above the four-nucleon break-up threshold, the boundary conditions become extremely complicated, and so is the analytic structure of the momentum plane when the equations are solved in momentum space, where the rigorous Faddeev-Yakubovsky (FY) equations 1) are applicable. Recently, we have shown that this hurdle can be cleared 2) with the complex energy method 3) (CEM).…”

“…The four-body energy is denoted by E, and we choose E = 0 at the four-body break-up threshold. The energies of the three-body, two-two, and two-body subsystems are denoted by [31] , [22] , and [2] , respectively. Their expressions are given in Eq.…”