Two-Pion Exchange Contributions to the Nucleon-Nucleon Interaction from the Roper Resonance

TL;DR

This work extends heavy-baryon chiral perturbation theory by treating the Roper resonance as an explicit degree of freedom, introducing a mass splitting $\rho = m_R - m_N$ and a transition coupling $g_A'$, and computes the leading long-range NN potential from two-pion exchange with intermediate Ropers. The potential, obtained via one- and two-Roper loop diagrams and regularized in dimensional regularization, is decomposed into isoscalar/isovector central, tensor, and spin-spin components and analyzed in momentum space. In first-order perturbation theory, the Roper contribution is sizeable in $D$-waves and slightly improves phase shifts across high partial waves compared to the nucleon-only TPE, though some channels remain challenging. The results illuminate the role of explicit resonances in EFT descriptions of nuclear forces, highlight the contrast with Delta saturation, and motivate future work incorporating Delta–Roper effects for a more complete NN interaction.

Abstract

We derive the long-range components of the nucleon-nucleon (NN) two-pion-exchange potential with an intermediate Roper resonance. Leading-order interactions in heavy-baryon chiral perturbation theory are considered. NN phase shifts with orbital angular momentum $L\geq 2$ are calculated in first-order perturbation theory and compared to those obtained without the Roper resonance. We show that the Roper contribution is sizeable for $D$ waves and improves the description of phase shifts for all the partial waves slightly. We also discuss the role of the Roper resonance in the NN interaction in the framework of resonance saturation.

Figures (5)

• Figure 1: Leading two-pion-exchange contributions to the two-nucleon potential with an intermediate Roper resonance. Solid, double, and dashed lines represent nucleons, Ropers, and pions, respectively. Propagators and vertices are from the Lagrangian $\mathcal{L}^{(0)}$, Eq. \ref{['Lag']}.
• Figure 2: Isoscalar (left panel) and isovector (right panel) components of the two-nucleon potential in units of GeV$^{-2}$, as a function of the transferred momentum $q$ in GeV: central (top), tensor multiplied by $q^2$ (middle), and spin-spin (bottom). The blue (red) dashed lines denote the contributions from two-pion exchanges without (solely from) intermediate Roper contributions. The red dotted lines represent contributions from two-pion exchanges via a single intermediate Roper state. The Green dashed lines refer to the contributions from subleading two-pion exchanges with $c_1 = 0, c_3 = -\frac{1}{2}c_4 =-\frac{g_A'^2}{4 \rho}$.
• Figure 3: $D$-wave phase shifts $^{2S+1}$D$_J$ and mixing angle $\varepsilon_2$ (in degrees) as a function of the laboratory energy $T_{\text{lab}}$ (in units of MeV). The (green) dashed curves correspond to the OPE contributions, the (blue) dash-dotted curves represent the TPE contributions, and the (red) solid/dash-dotted/dashed curves represent the sum of OPE and TPE contributions, including those from the intermediate Roper resonance/intermediate one-Roper resonance/ subleading TPE with $c_{1,3,4}$ determined as explained in the text.
• Figure 4: Same as \ref{['fig:Dwave']}, but for the $F$-wave phase shifts $^{2S+1}$F$_{J}$ and mixing angle $\varepsilon_3$.
• Figure 5: Same as \ref{['fig:Dwave']}, but for the $G$-wave phase shifts $^{2S+1}$G$_{J}$ and mixing angle $\varepsilon_4$.