Zero-locus diagnostic of the direct contact term in $\phi\to\pi^+\pi^-\pi^0$

Abstract

We formulate, within a fixed FSI-improved amplitude convention, a zero-locus diagnostic for the direct contact term in $\phi\to\pi^+\pi^-\pi^0$. The resonant $\rho\pi$ contribution is dressed channel by channel with the complex Omn\`es function, the direct term is multiplied by the averaged Omn\`es factor, and a real constant background is retained as a smooth residual amplitude. In this convention, $F^{\rm FSI}=R+c e^{i\delta_{\rm dir}}D$ with $c=g_{\phi3\pi}^{\rm eff}$, and the difference of two constrained invariant-mass gradients of the reduced Dalitz density gives a local quadratic identity for $c$ on the zero-contour. The construction provides a closure and stability diagnostic complementary to a global Dalitz-plot fit. For the benchmark input $g_{\phi3\pi}^{\rm eff}=-24.0~{\rm GeV}^{-3}$, we identify a numerically stable zero-contour branch around $M_{+0}=0.460\pm0.025~{\rm GeV}$ and $M_{+-}\le0.560~{\rm GeV}$, where the tight-contour median estimator reproduces $g_{\phi3\pi}^{\rm est}\simeq -24.0~{\rm GeV}^{-3}$, a closure recovery of the injected coupling within the chosen FSI-improved convention rather than an independent extraction. We further vary the mass-window selection of the branch. The median remains close to the injected value under shifts of the $M_{+0}$ window and moderate changes of its width, while the mean and standard deviation are more sensitive to outlier points near unstable quadratic branches. The method therefore identifies Dalitz-plane regions where the direct-term interference is locally well-conditioned and provides useful guidance for control-region choices and systematic variations in future global fits.