Three-nucleon force in chiral EFT with explicit $\Delta(1232)$ degrees of freedom: Longest-range contributions at fourth order

TL;DR

This paper investigates the long-range two-pion exchange three-nucleon forces within chiral effective field theory, explicitly including the $
232$ resonance, and compares results with $
232$-less models to understand the resonance's impact.

Contribution

It provides a detailed analysis of the three-nucleon force with explicit $
232$ degrees of freedom at leading-loop order, highlighting the resonance's role and comparing with $
232$-less approaches.

Findings

Resonance saturation effectively captures $
232$ effects in $
232$-less models.

Explicit $
232$ degrees of freedom influence the long-range three-nucleon force.

Comparison shows good agreement between $
232$-explicit and $
232$-less results at N4LO.

Abstract

We analyze the longest-range two-pion exchange contributions to the three-nucleon force at leading-loop order in the framework of heavy-baryon chiral effective field theory with explicit $\Delta(1232)$ degrees of freedom. All relevant low-energy constants which appear in the calculation are determined from pion-nucleon scattering. Comparing our results with the ones obtained in the $\Delta$-less theory at N4LO, we find effects of the $\Delta$ isobar for this particular topology to be rather well represented in terms of resonance saturation of various low-energy constants in the $\Delta$-less approach.