# An essential singularity of the cotangent of the Coulomb-nuclear phase shift, and a finite limit of the nuclear part of the effective-range function derived at zero energy

## TL;DR

This paper investigates the behavior of the Coulomb-nuclear phase shift and the effective-range function at zero energy, revealing an essential singularity in the phase shift and a finite limit for the effective-range function, with implications for nuclear astrophysics.

## Contribution

It derives the finite limit of the nuclear part of the effective-range function at zero energy and analyzes the singularity of the Coulomb-nuclear phase shift, enabling analytical continuation of scattering amplitudes.

## Key findings

- Cotangent of the Coulomb-nuclear phase shift has an essential singularity at zero energy.
- The nuclear part of the effective-range function approaches a finite limit at zero energy.
- Results facilitate the extraction of asymptotic normalization coefficients from experimental data.

## Abstract

The Coulomb-nuclear phase shift $\delta^{(cs)}_l$, $\cot\delta^{(cs)}_l$ and a finite limit of the nuclear part $\Delta_l(k)$ of the effective-range function (ERF) are derived for an arbitrary orbital momentum $l$ when energy $E\rightarrow0$. It is proved that $\cot\delta^{(cs)}_l$ has an essential singularity at zero energy, but $\Delta_l(k)$ does not. The explicit finite limit of $\Delta_l(0)$ is found. The property of $\Delta_l(k)$ as a meromorphic function makes possible the analytical continuation of a re-normalized scattering amplitude from the physical energy region to a bound state pole. Then the asymptotic normalization coefficients (ANC) can be deduced from experimental phase-shift data and applied to radiative capture processes which are important in nuclear astrophysics for new elements creation. Our results are in agreement with the results published for $S$ wave scattering.