Insights into Nucleon Resonances via Continuum Schwinger Function Methods

TL;DR

This work surveys the application of continuum Schwinger function methods (CSMs) to baryon resonances, with a focus on the Roper $N(1440) 1/2^+$ and the Delta excitations $\Delta(1600) 3/2^+$ and $\Delta(1700) 3/2^-$. It argues that baryons are three-body bound states described by dressed valence quarks, often modeled as a $q(qq)$ quark-diquark system, and that Poincaré covariance and emergent hadron mass (EHM) in QCD are essential to connect spectrum and resonance electroproduction to fundamental theory. The authors present detailed predictions for resonance electroproduction form factors and helicity amplitudes that align with CLAS/JLab data, supporting a quark-core picture with meson-cloud corrections and highlighting consistency with direct $3$-body results in accessible kinematic regimes. They advocate advancing direct $3$-body calculations to validate diquark simplifications and strengthen the link between resonance structure and QCD dynamics.

Abstract

The first baryon resonance was discovered in the early 1950s. The Roper resonance joined the collection ten years later. Today, many baryon resonances are known and more are being discovered. As baryons, these states are the most fundamental three-body systems in Nature. They must all be understood, not just the isolated ground state nucleon. This contribution sketches applications of continuum Schwinger function methods to the baryon resonance problem. Whilst spectroscopy is of value, particular emphasis is placed on resonance electroproduction because transition form factors extracted from electroproduction data provide a keen tool for revealing resonance structure.

Figures (10)

• Figure 1: Linear, homogeneous integral equation for $\Psi$, the Poincaré-covariant matrix-valued function (Faddeev amplitude) for a baryon with total momentum $P=p_q+p_d=k_q+k_d$ constituted from three valence quarks, two of which are paired in a fully-interacting nonpointlike diquark correlation. $\Psi$ expresses the relative momentum correlation between the dressed-quarks and -diquarks. Legend. Shaded box -- Faddeev kernel, which explicitly shows the quark exchange binding mechanism; single line -- dressed-quark propagator; $\Gamma$ -- diquark correlation amplitude; and double line -- diquark propagator..
• Figure 2: Panel A. Proton. Diquark component breakdown of the canonical normalisation of the proton's Poincaré-covariant nucleon Faddeev wave function. The $[ud]_{0^+}$ isoscalar-scalar diquark (SC) is dominant, but material contributions also owe to the $\{uu\}_{1^+}$, $\{ud\}_{1^+}$ isovector-axialvector correlations (AV). SC$\,\otimes\,$SC -- 60%; SC$\,\otimes\,$AV -- 15%; AV$\,\otimes\,$AV -- 25%. Panel B. Analogous image for the first $1/2^+$ excitation of the ground-state nucleon. SC$\,\otimes\,$SC -- 67%; SC$\,\otimes\,$AV -- 15%; AV$\,\otimes\,$AV -- 18%.
• Figure 3: Legend for interpretation of $J=1/2$ baryon rest-frame quark + diquark angular momentum decompositions, which also identifies interference between the distinct orbital angular momentum basis components. The axes labels refer to distinct components of the $q(qq)$ wave function, which are explicitly detailed in Ref. Chen:2019fzn.
• Figure 4: Contributions of the various quark + diquark orbital angular momentum components to the canonical normalisation of the Poincaré-covariant wave function of a $J=1/2$ baryon after rest-frame projection: there are both positive (above plane) and negative (below plane) contributions to the overall positive normalisation. Panel A. Proton. Panel B. First $J=1/2^+$ excitation of the proton, identified as the Roper resonance. Panel C. First $J=1/2^-$ excitation of the proton, identified as the $N(1535)1/2^-$. Panel D. Second $J=1/2^-$ excitation of the proton, identified as the $N(1650)1/2^-$.
• Figure 5: Legend for interpretation of $J=3/2$ baryon rest-frame quark + diquark angular momentum decompositions, which also identifies interference between the distinct orbital angular momentum basis components. These states can possess ${\mathsf S}$, ${\mathsf P}$, ${\mathsf D}$, and ${\mathsf F}$-wave components. The axes labels refer to distinct components of the $q(qq)$ wave function, which are explicitly detailed in Ref. Liu:2022ndb.