A new Δ method for bound state asymptotic normalization coefficients with a finite limit of the nuclear part of the effective-range function at zero energy

Abstract

A new bound state Δ method for finding an asymptotic normalization coefficient (ANC) is proposed. This is valid, while the standard effective-range function (ERF) Kl(k2) method does not work when the product of colliding particles charges increases. The denominator of the strict expression of the re-normalized scattering amplitude f˜l includes the factor dl(E)=Δl(E)+hr(E)−h(η), where η=1/aBk, aB is the Bohr radius. In the physical region, the Coulomb term hr=Reh(η). So an analytical continuation of hr(E) from the physical region to E≤0 can be found by fitting hr(E) for E≥0. The related analytical continuation of h(η) consists in a simple sign change, E→−E, using an explicit dependence of h(η) on E. This is important that for E≤0, abs(Δh)=abs(h−hr)=0 at E=0, and it increases when abs(E) does. Thus, a new real equation, dl(−ε)=0, is obtained for a binding energy ε. It is applied to find a residue W of f˜l at the bound state pole E=−ε, the nuclear vertex constant (NVC) and (ANC). The Coulomb-nuclear phase shift δlcs, cotδlcs and a finite limit of the nuclear part Δl(k2) of Kl(k2) are also derived for an arbitrary orbital momentum l when E→0. It is shown that cotδlcs has an essential singularity at zero energy, but Δl(k2) does not. The explicit finite limit of Δl(k2) when E→0 is found using the expression for Kl(k2). These results are in agreement with those for the S-wave scattering, which are widely accepted. The ANCs for the ground and first excited bound states for the vertex Be7↔3He4He are calculated using the proposed new method, and are compared with those for the approximate method when dl(E)=Δl(E) proposed by Ramírez Suárez and Sparenberg (2017). The fit of Δl(E) is found from the experimental phase-shift input data and the additional equation dl(−ε)=0.