Neutral pion momentum in hypertriton mesonic decay through a root-finding method

A root-finding method is used to study two-body mesonic decay in the hypertriton. We validate this Newton--Raphson root-finding approach by applying it to the negative-pion decay channel ($^{3}_{\Lambda}\mathrm{H} \rightarrow {}^{3}\mathrm{He} + \pi^{-}$), for which the pion momentum and lambda binding energy were recently reported by MAMI A1 Collaboration as $p_{\pi^-} = 113.789 \pm 0.020_{\text{stat.}} \pm 0.112_{\text{syst.}} \text{ MeV}/c$ and $B_{\Lambda} = 0.523 \pm 0.013_{\text{stat.}} \pm 0.075_{\text{syst.}}$ MeV, respectively. Using their reported $\Lambda$ binding energy, the root-finding method and an exact kinematic formula both yield $p_{\pi^{-}} = 113.790$ MeV/$c$, agreeing with each other. We then apply both the Newton--Raphson method and the exact formula to the neutral-pion decay channel ($^{3}_{\Lambda}\mathrm{H} \rightarrow {}^{3}\mathrm{H} + \pi^{0}$), for which the neutral pion momentum cannot be directly measured due to difficulties in experimental setup. Both methods agree, yielding a predicted neutral-pion momentum of $p_{\pi^{0}} = 118.129$ MeV/$c$. This validates the root-finding algorithm as a robust equivalent for predicting pion momenta that may be experimentally inaccessible in some cases. Furthermore, it establishes the method as a reliable tool for extension to three-body mesonic decays, for which the pion momentum is a continuum and the exact kinematic formula can no longer be applied. In addition, the pion momentum computed allows for its 4-momentum to be completely determined, a useful input for investigating its two-photon decay ($\pi^0 \to \gamma \gamma$) in the rest frame of the hypertriton.