Advanced parametrisations for hadronic form factors

TL;DR

This work advances hadronic form-factor parametrisations by developing two conformal-analytic frameworks that access resonance poles and left-hand cuts on the second Riemann sheet while maintaining a clear link to partial-wave amplitudes. The first approach, based on an improved zeta-map, unfolds the left-hand cut and portions of the second sheet to enable a $\zeta$-SSE that places low-lying resonances inside the convergence radius, yielding fast, accurate extraction of pole parameters for single-channel pion form factors. The second approach extends to inelastic regimes with a four-sheet conformal map and a final $\psi$-SSE that pushes the left-hand cut to the unit circle, allowing a controlled multi-channel description and direct access to resonance poles on higher sheets. Tests on pion form factors using IAM/Omnès inputs show improved convergence and the ability to extract pole positions and residues, suggesting practical use for experimental data (e.g., $e^+e^-\to\pi^+\pi^-$) and potential generalisations to unequal-mass channels and three-channel systems.

Abstract

The rich analytic structure of hadronic form factors makes a theoretically consistent yet easily applicable parametrisation cumbersome. Consequently, most parametrisations are limited to reproducing the simplest analytic features sufficient to describe form factors on their first Riemann sheet. Here, we introduce two novel form factor parametrisations that allow resonance poles and left-hand cuts on the second Riemann sheet to be studied, while also making the connection to partial-wave amplitudes manifest.

Figures (11)

• Figure 1: Analytic structure of a form factor on the first (I) and second (II) Riemann sheets. The singularities include a left-hand cut (red line), a unitarity cut opening at $s_+$ (dark blue line) with higher-energy branch cuts (light blue line) opening at the inelastic threshold $s_\text{in}$, as well as resonance poles $s_r$ (purple dots).
• Figure 2: The complex $z$-plane. The unfolded right-hand cut (white and orange) separates the physical Riemann sheet (I) from the unphysical Riemann sheet (II). For $s_0 = s_-$, the left-hand cut of the second sheet (red) extends from $z = 1$ to $\infty$.
• Figure 3: The complex $\zeta$-plane for $z_0 = -1$ (left) and $z_0 = 0$ (right). Four sheets are visible on the plane, sheets I and $\text{II} \equiv \text{II}^+$ are separated by the RHC, $\text{II}^+$ and $\text{II}^-$ by the LHC, and $\text{II}^-$ and $\text{III}^+$ by a RHC. Another LHC still spans $\zeta = 1$ to $\infty$, see \ref{['app:riemann_sheets']} for details. Higher-energy branch cuts due to other channels are not represented. The physical timelike region where the scattering data lies is highlighted in white.
• Figure 4: Comparison between the modelled form factors $\Omega_l^I$ and the SSE fits to pseudo data. The fits are performed in the regions $s\in[0,1]\,\mathrm{GeV}^2$ for the scalar form factor $I=0$, $l=0$ (left) and $s\in[0,1.2]\,\mathrm{GeV}^2$ for the vector form factor $I=1$, $l=1$ (right).
• Figure 5: Predictions for the partial-wave amplitudes from fits to the theoretical form factors $\Omega_0^0$ (left) and $\Omega_1^1$ (right). The form factors are fitted in the ranges $s \in [0,1]\,\mathrm{GeV}^2$ and $s \in [0,1.2]\,\mathrm{GeV}^2$ respectively. Fitting with $n = 3$ truncated $z$ (dashed) and $\zeta$ (plain) SSEs, we find that only the one based on $\zeta$ gives physical results.