Dispersive analysis of the $\boldsymbol{φ\to γπ^0 π^0}$ process

Abstract

We present an analysis of the radiative decay $φ\to γπ^0 π^0$ in a dispersive framework, where the two-pion subsystem undergoes strong final-state interactions that cover the $f_0(500)$ and $f_0(980)$ regions. We employ a coupled-channel Muskhelishvili-Omnès framework that allows for a consistent treatment of two scalar resonances and crossed-channel singularities induced by the Born and vector-meson exchanges. We explicitly verify the equivalence between the modified and standard Muskhelishvili-Omnès representations for vector-meson pole contributions when the isoscalar Omnès matrix is chosen asymptotically bounded, and we adopt the standard representation in decay kinematics. This yields, for the first time, a parameter-free dispersive prediction for the kaon Born rescattering, which provides a dominant contribution. To obtain a good fit to the KLOE and SND data, we employ a once-subtracted coupled-channel dispersion relation with heavier left-hand cut contributions and two unknown subtraction constants. The results demonstrate the consistency among the data for $ππ$ scattering, $γγ$ fusion, and $φ$ radiative decay, thereby validating the underlying dispersive formalism and the input used for the hadronic Omnès matrix and left-hand cuts.

Figures (3)

• Figure 1: Momentum convention in the $\pi^0 \pi^0$ center-of-mass frame.
• Figure 2: The real and imaginary parts of the $\rho$-pole LHC (couplings set to unity) from the explicit expression and from the dispersive representation \ref{['eq:hv']}. The integrals are evaluated along the complex cuts (see e.g., Ref. Moussallam:2013una). Two evaluations fall on top of each other, while the sharp behavior at $s_\pm$ is due to the narrow width.
• Figure 3: The differential branching ratio for $\phi\to\gamma\pi^0\pi^0$ as a function of the two-pion invariant mass $\sqrt{s}$. The top panel shows the prediction for the Born rescattering contribution. The bottom panel shows the final fit to the data: the black band represents the fit uncertainty alone, and the grey band denotes the total uncertainty. The $S$-wave contribution is indicated by the dashed line. Data points excluded from the fit are shown as open circles.