Parametrizations of three-body hadronic $B$- and $D$-decay amplitudes in terms of analytic and unitary meson-meson form factors

Abstract

We introduce parametrizations of hadronic three-body $B$ and $D$ weak decay amplitudes that can be readily implemented in experimental analyses and are a sound alternative to the simplistic and widely used sum of Breit-Wigner type amplitudes, also known as the isobar model. These parametrizations can be particularly useful in the interpretation of CP asymmetries in the Dalitz plots. They are derived from previous calculations based on a quasi-two-body factorization approach in which two-body hadronic final state interactions are fully taken into account in terms of unitary $S$- and $P$-wave $\pi\pi$, $\pi K$ and $K \bar K$ form factors. These form factors can be determined rigorously, fulfilling fundamental properties of quantum field-theory amplitudes such as analyticity and unitarity, and are in agreement with the low-energy behaviour predicted by effective theories of QCD. They are derived from sets of coupled-channel equations using $T$-matrix elements constrained by experimental meson-meson phase shifts and inelasticities, chiral symmetry and asymptotic QCD. We provide explicit amplitude expressions for the decays $B^\pm \to \pi^+ \pi^- \pi^\pm$, $B \to K \ \pi^+ \pi^-$, $B^\pm \to K^+ K^-K^\pm$, $D^+ \to \pi^- \pi^+ \pi^+ $, $D^+ \to K^- \pi^+ \ \pi^+ $, $D^0 \to K^0_S \ \pi^+\ \pi^- $, for which we have shown in previous studies that this approach is phenomenologically successful, in addition, we provide expressions for the $D^0 \to K^0_S \ K^+ K^-$ decay. Other three-body hadronic channels can be parametrized likewise.