Dispersive analysis of the $J/ψ\toπ^0 γ^\ast$ transition form factor with $ρ$-$ω$ mixing effects

TL;DR

This work tackles discrepancies between theory and BESIII data for the J/ψ → π^0 γ* transition form factor by employing a Khuri–Treiman dispersive framework that enforces crossing symmetry and unitarity while incorporating ρ–ω mixing and multi-pion intermediate states. Light- and heavy-quark contributions are treated coherently: 2π and 3π dynamics via KT equations with Omnès rescattering, 4π effects through a dispersively improved ρ′(1450) model, and charmonium contributions via a constrained monopole term, all anchored by sum rules and high-energy behavior. The authors fit BESIII data with only two free parameters, revealing a strong ρπ0- dominated channel and an ωπ0 electromagnetic contribution, and extract a relative strong–electromagnetic phase of (62 ± 21)°. This dispersive treatment clarifies the role of isospin-violating mixing in shaping the TFF and provides a benchmark for interpreting the ρπ puzzle in J/ψ decays, with implications for future measurements of J/ψ → π^0 e^+ e^- and related processes.

Abstract

Motivated by the discrepancies noted recently between the theoretical predictions of the electromagnetic $J/ψ\to π^0 γ^*$ transition form factor and the BESIII data, we reanalyze this transition form factor using the dispersive Khuri-Treiman equations, with final-state interactions in both the direct channel and the crossed channels properly considered. This improved framework incorporates $ρ$-$ω$ mixing effects. The effect of four-pion states is evaluated through a dispersively improved vector-meson-dominance model. From this information, we propose a two-parameter fit that provides an excellent description of the BESIII data over the broad energy range from 0 to 2.8 GeV. We demonstrate that the $ρπ^0$ decay mode of the $J/ψ$ is dominated by strong interaction, while the $ωπ^0$ mode is dominated by one-photon exchange. From this, we extract the relative phase between the strong and the one-virtual-photon (electromagnetic) modes in hadronic decays of $J/ψ$ as $(62 \pm 21)^{\circ}$. This could provide useful information in understanding the long-standing $ρπ$ puzzle in $J/ψ$ decays.

Figures (9)

• Figure 1: Unitarity relation for the $J/\psi\to\pi^0\gamma^\ast$ TFF $f_{\psi\pi^0}$. The blue dashed lines denote pions, the wiggly lines represent photons, the double-solid lines represent the $J/\psi$, and the red dashed line indicates that the intermediate $\pi\pi$ states are to be taken on-shell.
• Figure 2: Solution 1 for the $P$-wave phase shift $\delta(s)$ from Ref. Schneider:2012ez, which is valid up to roughly 1.9 GeV. Solution 2 Pelaez:2024uav is valid up to roughly 1.3 GeV; see the main text for further details.
• Figure 3: The KT path of $t_{ \pm}(s)$ (cf. Eq. \ref{['eq:t+-']}). The trajectories start at the same point $A$ with $t_{\pm}(4 M_\pi^2)$ above the physical cut. At point $B$ with $t_{\pm}((M_\psi^2-M_\pi^2)/2)$, the $t_{-}$ contour reaches $4 M_\pi^2$ and moves below the cut. From point $C$ with $t_{ \pm}(M_-^2)$ onwards to point $D$ with $t_{ \pm}(M_+^2)$, $t_{-}$ and $t_{+}$ have the same real part but opposite imaginary parts. The trajectories finally meet at point $D$. From point $D$ onwards, both trajectories are real with $t_{-}$ moving towards zero and $t_{+}$ towards infinity.
• Figure 4: Real (left panel) and imaginary (right panel) parts of the $P$-wave basis solution $\mathcal{F}_a(s)$ for $J/\psi\to 3\pi$, compared with the Omnès function without any three-body crossed-channel rescattering effects. $\delta_{1,2}$ represent the phase shifts used from Solution 1 or 2. The results from the Omnès functions are close to the full $\mathcal{F}_a$ since the crossed-channel rescattering effects are minor.
• Figure 5: The discontinuities $\operatorname{disc}f^{(\rho^\prime)}_{\psi\pi^0}/(2\,i\,s)$ corresponding to two types of energy-dependent widths, cf. Eq. \ref{['eq:width_v1']} and Eq. \ref{['eq:width_v2']}.