Comments on Baryon Transition Form Factors

TL;DR

The paper addresses how space-like baryon transition form factors relate to resonance pole structure of the $S$-matrix, arguing that resonance properties must be defined via analytic continuation to the pole with complex residues $s_R$. It employs a dynamical coupled-channel framework (JBW) to compute multipoles $oldsymbol{\mathcal{M}}_{\\ell\pm}(W,Q^2)$ and to extract pole-residue based transition form factors $H(Q^2)$, highlighting how background interactions and gauge-invariant couplings influence the complex phase. The analysis of the Roper resonance shows nontrivial imaginary parts in $H(Q^2)$ and strong $Q^2$-dependent phases, challenging simple three-quark radial-excitation interpretations and suggesting a dynamically generated state in the JBW approach. The work underscores the necessity of analytic continuation to the resonance pole for universal resonance properties and calls for improved data and sophisticated methods to reliably separate pole and background contributions in TFF extractions.

Abstract

We discuss in rather general terms the properties of space-like baryon transition form factors. In particular, we argue why these are necessarily complex-valued, what can be deduced from the respective phase motion and why dealing with real valued transition form factors in general leads to misleading results. For illustration the transition form factors for the Roper resonance as derived in the Jülich-Bonn-Washington framework are discussed.

Figures (6)

• Figure 1: Diagrammatic representation of the vertex function $\pi N\to R$, denoting nucleon/pion/resonance states by a full/dashed/double line, respectively. The full vertex function ($\Gamma$) is given by the sum of the bare vertex ($\bullet$) plus a contribution where the external particles interact via the background interaction ($T_{\rm bg}$), shown as the gray ellipse.
• Figure 2: Diagrammatic representation of the transition form factor $H$, denoting nucleon/photon/resonance states by a full/wavy/double line, respectively. The full form factor is given by the sum of the bare vertex ($\bullet$) plus a contribution where the external particles interact via the background interaction $\tilde{T}_{\rm bg}$, shown as the gray ellipse.
• Figure 3: Top panel: Degrees of freedom of a one-meson electroproduction reaction. Bottom panel: Connection between reaction independent transition form factors $H_{N\to N^*}$ and observables. For more details and formulas see Refs. Mai:2021vswMai:2021auiMai:2023cbp.
• Figure 4: JBW multipole parametrization. For any fixed quantum number, the multiples $\{\mathcal{M}_{\ell\pm}(W,Q^2)| \mathcal{M}=E,M,L\}$ are determined through a set of coupled-channel integral equations with respect to the scattering potential $V$ and the electroproduction term $V^{\gamma^*}$.
• Figure 5: Left panel: Representative results of the recent dynamical coupled-channel (JBW) approach. Right panel, overview of other approaches (adapted from a recent review Ramalho:2023hqd), employing strictly real transition form factors. The black and red data points show the extractions from the CLAS collaboration for single CLAS:2009ces and double CLAS:2007bvsMokeev:2015lda pion production, respectively.