Comment on Effective-range function methods for charged particle collisions

TL;DR

This paper critiques recent methods for calculating asymptotic normalization coefficients in charged particle collisions, emphasizing the correct analytical properties of effective-range functions and clarifying misconceptions about their behavior at zero energy.

Contribution

The paper clarifies the analytical properties of the elta_l function and effective-range functions, correcting misconceptions in recent literature regarding their behavior at zero energy.

Findings

elta_l function is analytical at zero energy.

Effective-range function K_l(k^2) is meromorphic with bound state poles.

Re-normalized scattering amplitude form is valid for bound state analysis.

Abstract

The authors of a recent paper [Phys. Rev. C 97(2018) 044003] (Ref. [1]), D. Gaspard and J.-M. Sparenberg, attempt to consider an alternative method for the asymptotic normalization coefficients (ANC) calculating which differs from so-called approximate \Delta method proposed earlier [Phys. Rev. C 96(2017) 034601] (Ref. [2]) by O.L. Ram\'irez Su\'arez, and J. M. Sparenberg. The abstract in Ref. [1], in essence, declares that the approximation used in Ref. [2], where a \Delta_l function is introduced to fit the Coulomb-nuclear phase shifts {\delta}_l^{(cs)} at low energies, is not correct, since this \Delta_l function is not analytical at zero energy. In my view, this statement is erroneous. I believe that the origin of this mistake in Ref. [1] is due to adopting the re-normalized scattering amplitude form designed for resonant states, which is not valid for a bound state. It is shown in the fundamental work [Nucl. Phys. B 60 (1973) 443] Ref.[3] by Hamilton et al. that an effective-range function (ERF) K_l(k^2) is a meromorphic in the physical sheet except for the poles of bound states on the imaginary positive axis of the momentum complex plane. In my paper [Nucl. Phys A, 1010 (2021) 122174] Ref. [4], a strict form of the re-normalized scattering amplitude is derived, which can be used for the analytical continuation to a bound state pole. It is found in Ref. 4 that the \Delta_l function has analytical properties similar to those of the K_l(k^2) function. Thus, both functions have no singularities at zero energy.