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

Abstract

We formulate, within a fixed FSI-improved amplitude convention, a zero-locus diagnostic for the direct contact term in $\varphi \to {\pi }^{+}{\pi }^{-}{\pi }^0$. The resonant $\rho \pi$ contribution is dressed channel by channel with the complex Omnès function, the direct term is multiplied by the averaged Omnès factor, and a real constant background is retained as a smooth residual amplitude. In this convention, $F^{\mathrm{FSI}}=R+c{e}^{i{\delta }_{\mathrm{dir}}}D$ with $c=g_{\varphi 3\pi }^{\mathrm{eff}}$, 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 $g_{\varphi 3\pi }^{\mathrm{eff}}=-24.0{\mathrm{GeV}}^{-3}$, we identify a numerically stable zero-contour branch around $M_{+0}=0.460\pm 0.025\mathrm{GeV}$ and $M_{+-}\le 0.560\mathrm{GeV}$, where the tight-contour median estimator reproduces $g_{\varphi 3\pi }^{\mathrm{est}}\simeq -24.0{\mathrm{GeV}}^{-3}$, 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.

Comments: 11 pages, 2 figures