Emergence of ${}^4$H $J^{\pi }=1^{-}$ resonance in contact theories

Abstract

We obtain the $s$- and $p$-wave low-energy scattering parameters for n${}^3$H elastic scattering and the position of the ${}^4$H $J^{\pi }=1^{-}$ resonance using the pionless effective field theory at leading order. Results are extracted with three numerical techniques: confining the system in a harmonic oscillator trap, solving the Faddeev-Yakubovsky equations in configuration space, and using an effective two-body cluster approach. The renormalization of the theory for the relevant amplitudes is assessed in a cutoff-regulator range between $1{\text{fm}}^{-1}$ and $10{\text{fm}}^{-1}$. Most remarkably, we find a cutoff-stable/RG-invariant resonance in the ${}^4$H $J^{\pi }=1^{-}$ system. This $p$-wave resonance is a universal consequence of a shallow two-body state and the introduction of a three-body $s$-wave scale set by the triton binding energy. The stabilization of a resonant state in a few-fermion system through pure contact interactions has a significant consequence for the powercounting of the pionless theory. Specifically, it suggests the appearance of similar resonant states also in larger nuclei, like 16-oxygen, in which the theory's leading order does not predict stable states. Those resonances would provide a starting state to be moved to the correct physical position by the perturbative insertion of sub-leading orders, possibly resolving the discrepancy between data and contact EFT.

Comments: 11 pages, 6 figures