[Paper Review] The pion form factor at timelike momentum transfers in a dispersion approach

Dmitri Melikhov, Otto Nachtmann+1 more|ArXiv.org|Sep 13, 2002

Particle physics theoretical and experimental studies3 citations

TL;DR

This paper develops a dispersion-theoretic model for the pion form factor at timelike momentum transfers by resumming pion and kaon loops with $ hou\pi\pi$, $\rho KK$, and gauge-invariant $\rho\gamma$ couplings. The key result is a vector meson dominance-like form factor with an $s$-dependent effective $\rho$-meson decay constant $f^{\rm eff}_{\rho}(s)$, which successfully describes $e^+e^-$ annihilation and $\tau$ decay data, yielding precise $\rho$, $\omega$, and $\rho$--$\omega$ mixing parameters with $g_{\rho\to\pi\pi} = 11.6 \pm 0.3$ in agreement with chiral perturbation theory.

ABSTRACT

We consider a model with $ρππ$, $ρKK$, and gauge-invariant $ργ$ couplings and obtain the pion form factor $F_π$ at timelike momentum transfers by resummation of pion and kaon loops. We use a dispersion representation for the loop diagrams and analyse ambiguities related to subtraction constants. The resulting representation for $F_π$ is shown to have the form of the conventional vector meson dominance formula with one important distinction - the effective $ρ$-meson decay constant $f^{ m eff}_ρ$ turns out to depend on the momentum transfer. For the electromagnetic pion form factor we include in addition the $ρ-ω$ mixing effects. We apply the representations obtained to the analysis of the data on the pion form factors from $e^+e^-$ annihilation and $τ$ decay and extract the $ρ^-$, $ρ^0$ and $ω$ masses and coupling constants.

Motivation & Objective

• To develop a theoretically consistent dispersion-based description of the pion electromagnetic and charged current form factors in the timelike region $0 \leq q^2 \leq 1.5$ GeV$^2$.
• To address the limitations of standard vector meson dominance and the Gounaris-Sakurai formula by incorporating finite-width effects via loop resummation with proper subtraction constants.
• To extract precise values for the $\rho$ and $\omega$ meson masses, coupling constants, and $\rho$--$\omega$ mixing parameter from experimental data on $e^+e^-$ annihilation and $\tau \to \pi\pi\nu_\tau$ decay.
• To test the consistency of the model with chiral perturbation theory predictions, particularly for the $\rho\to\pi\pi$ coupling.

Proposed method

• A model with $\rho\pi\pi$, $\rho KK$, $\omega\pi\pi$, $\omega KK$, and gauge-invariant $\rho\gamma$, $\omega\gamma$ couplings is constructed to describe the pion form factor at timelike $q^2$.
• Pion and kaon loop diagrams are resummed using a dispersion representation with two subtractions to reconstruct the real parts from the imaginary parts.
• The subtraction constants are fixed by requiring agreement with the observed $\rho$-meson mass, decay constant, and $\rho\to\pi\pi$ coupling at $s = M_\rho^2$, leading to an $s$-dependent effective $f^{\rm eff}_{\rho}(s)$.
• The $\rho$--$\omega$ mixing effects are included in the electromagnetic form factor via a mixing parameter $\Delta$, which is fitted to data.
• The form factor is expressed in a modified VMD form: $F_\pi(s) = \frac{M_\rho^2 - B_{\rho\rho}(0)}{M_\rho^2 - s - B_{\rho\rho}(s)}$, where $B_{\rho\rho}(s)$ is the two-pion loop with $s$-dependent real part.
• Fits to CLEO and ALEPH data on $e^+e^- \to \pi^+\pi^-$ and $\tau \to \pi^-\pi^0\nu_\tau$ are performed to extract $M_\rho$, $f_\rho$, $g_{\rho\to\pi\pi}$, and $\Delta$.

Experimental results

Research questions

• RQ1 How can a dispersion-theoretic approach with loop resummation improve the description of the pion form factor in the timelike region compared to standard VMD or Gounaris-Sakurai models?
• RQ2 What is the impact of $\rho$--$\omega$ mixing on the electromagnetic pion form factor, particularly near $s = M_\omega^2$?
• RQ3 To what extent does the effective $\rho$-meson decay constant $f^{\rm eff}_{\rho}(s)$ depend on the momentum transfer $s$, and how does this affect the form factor shape?
• RQ4 Are the extracted values of the $\rho\to\pi\pi$ coupling constant and $\rho$--$\omega$ mixing parameter consistent with chiral perturbation theory predictions?
• RQ5 Can a unified description of both electromagnetic and charged current pion form factors be achieved within a single dispersion-based framework?

Key findings

• The effective $\rho$-meson decay constant is found to depend linearly on $s$, leading to a modified VMD form that improves agreement with data compared to standard models.
• The $\rho$--$\omega$ mixing contributes significantly to the electromagnetic form factor, increasing $|F_\pi|^2$ by 10% at $s = M_\rho^2$ and nearly 30% at $s = M_\omega^2$, which is crucial for fitting the data.
• The fitted value of the $\rho^{-}\to\pi^{0}\pi^{-}$ coupling constant is $11.80 \pm 0.05$ for $Q_{\rm upper} = 1025$ MeV, consistent with the ChPT prediction $g_{\rho\to 2\pi} = 2M_\rho/f_\pi = 11.7$, validating the model's consistency with low-energy QCD.
• The central values of the $\rho^{-}$ and $\rho^{0}$ masses are $775$ MeV and $774$ MeV, respectively, with $f_\rho = 149 \pm 1$ MeV, and the $\rho$--$\omega$ mixing parameter is $\Delta = 0.17 \pm 0.02$, both in good agreement with PDG values.
• The model successfully describes both $e^+e^- \to \pi^+\pi^-$ and $\tau \to \pi^-\pi^0\nu_\tau$ data, with fits showing excellent agreement in the $0 \leq s \leq 1$ GeV$^2$ range.
• The extracted $\omega$ mass is $782.0 \pm 0.5$ MeV, and the $\rho^{-}$ mass is $775 \pm 2$ MeV, with all parameters consistent within errors for neutral and charged $\rho$ mesons.