# Onset of $\eta$-meson binding in the He isotopes

## TL;DR

This paper investigates the conditions under which $ ext{eta}$ mesons bind to helium isotopes using few-body calculations with semi-realistic potentials, finding specific scattering length thresholds for binding.

## Contribution

It provides the first precise few-body calculations showing the minimal $ ext{eta}$-nucleon scattering length needed for binding in helium isotopes, incorporating energy dependence self-consistently.

## Key findings

- Binding in $^3$He requires Re$a_{ ext{eta}N} ext{ near } 1$ fm.
- Binding in $^4$He requires Re$a_{ ext{eta}N} ext{ exceeds } 0.7$ fm.
- Results align with recent effective field theory calculations.

## Abstract

The onset of binding $\eta$(548) mesons in nuclei is studied in the He isotopes by doing precise $\eta NNN$ and $\eta NNNN$ few-body stochastic variational method calculations for two semi-realistic $NN$ potentials and two energy dependent $\eta N$ potentials derived from coupled-channel models of the $N^{\ast}(1535)$ nucleon resonance. The energy dependence of the $\eta N$ subthreshold input is treated self consistently. It is found that a minimal value of the real part of the $\eta N$ scattering length $a_{\eta N}$ close to 1 fm is required to bind $\eta$ mesons in $^3$He, yielding then a few MeV $\eta$ binding in $^4$He. The onset of $\eta$-meson binding in $^4$He requires that Re$\,a_{\eta N}$ exceeds 0.7 fm approximately. These results compare well with results of recent $\eta NNN$ and $\eta NNNN$ pionless effective field theory calculations. Related optical-model calculations are also discussed.