Phenomenology of light baryon resonances

TL;DR

This review surveys how light baryon resonances emerge from QCD and are extracted from data using a spectrum of approaches, including lattice QCD in finite volume, chiral unitary dynamics, and comprehensive amplitude analyses across multiple channels. A central theme is the identification of resonances through complex pole positions W_0 and residues a_{-1}, with careful treatment of analytic structure (cuts, branch points, triangle singularities) and background contributions. The work emphasizes the role of electromagnetic transition form factors to probe resonance structure and spatial distributions, highlighting how pole-based quantities provide a universal language across reactions. Overall, the article clarifies practical methods for disentangling resonant and nonresonant effects to build a coherent, multi-channel picture of the light baryon spectrum and its QCD foundations.

Abstract

Protons and neutrons are the building blocks of matter, glued together in nuclei by strong interactions. They can be excited by pions, real and virtual photons, neutrinos and other probes. These excitations are referred to as light baryon resonances. A short, pedagogical overview of the field is presented including experimental progress, interpretation of light baryon resonances, and a focus on analysis methods to extract the resonance spectrum from data.

Figures (6)

• Figure 1: Left: Scattering and bound states of the finite spherical well. The screenshots a)-c) show the situation for increasing potential depth, see also an online animation Doering:blog. The left column shows the $S$-wave $T$-matrix, $|t_0|$, in the complex-momentum $k$-plane (arb. units), the right column shows the phase shift. (a) For a shallow potential, there is no bound state, but only virtual state 1 and resonances 2 and 3. In (b), the scattering length is much larger than the dimension of the potential (universality). In (c), pole 1 became a deeply bound state. Pole 2 and its mirror pole 2' have met on the imaginary $k$-axis and then separated again as virtual states $\bar{2}$ and $\bar{2}'$, with $\bar{2}$ on its way to become a bound state and $\bar{2}'$ a deeper-bound virtual state. Center: The $P_{33}$ partial wave and the $\Delta(1232)3/2^+$ resonance. The data for this and the right picture (taken from Refs. Doring:2009biDoring:2009yv) represent the Single-Energy-Solutions (SES) from the SAID data base SAID-web. Right: The $S_{11}$ partial wave. In contrast to the $P_{33}$ partial wave, there are the two overlapping resonances $N(1535)1/2^-$ and $N(1650)1/2^-$ on top of a substantial background (compare to the spherical well). In addition, there is a sharp threshold cusp at $W=1486$ MeV next to the $N(1535)1/2^-$. See text for further explanations.
• Figure 2: Meson ($M$) baryon ($B$) transitions through $s$-, $t$-, $u$-channel, and contact $(ct)$ processes. Also, the $2\to 3$ processes to populate the effective $\sigma N$, $\rho N$, and $\pi\Delta$ channels in the ANL-Osaka and the JBW approaches are shown. The figure also labels incoming (outgoing ) c.m. momenta $p$ ($p'$) and baryon (meson) masses $M_1,\,M_3$ ($m_2,\,m_4$). Figure from Ref. Doring:2025sgb.
• Figure 3: Summary of results for the baryon parameters obtained in the ANL-Osaka Kamano:2013iva and JBW Ronchen:2022hqk approaches. The lengths of the bars correspond to the resonance widths. For these cases, PDG information ParticleDataGroup:2024cfk is displayed in blue. Note that for the ANL-Osaka model, only resonances up to $2\,{\rm GeV}$ in mass and $400\,{\rm MeV}$ in width are quoted. Pictures from Ref. Doring:2025sgb.
• Figure 4: Analytic structure of the $P_{11}$ partial wave, for complex scattering energies $E=W$. The circular cut is labeled "CC", the left-hand cut "LHC". The nucleon is labeled as $N$. See text for further explanations. Picture from Ref. Doring:2025sgb.
• Figure 5: Kinematics of an electroproduction experiment with the final meson-baryon state i. The scattering plane is defined by the respective in/outgoing electron momenta $k_e/k'_e$ with the electron scattering angle $\theta_e$. The reaction plane is spanned by the virtual photon and the outgoing meson, scattered by an angle $\theta$. The momenta $q$ and $p$ correspond to the virtual photon and target nucleon while $k'_i$ and $p'_i$ correspond to the outgoing meson and baryon, respectively. Figure from Ref. Mai:2023cbp.