The role of triangle singularity in the $B^0\to D^{-}{\pi }^{+}{a}_0(980)({\pi }^0\eta )$ decay

Abstract

The triangle singularity interpretation of the \replyhe{$a_1(1420)$} observed by the COMPASS Collaboration has been widely accepted. In this work, we investigate the triangle mechanism in the decay $B^0\to D^{-}{\pi }^{+}{a}_0(980)({\pi }^0\eta )$, where the $a_0(980)$ is treated as a dynamically generated state. The ${\bar{K}}^{\ast 0}$-$K^{+}$-$K^{-}$ triangle loop originates from the decay $B^0\to D^{-}{K}^{+}{\bar{K}}^{\ast 0}$, which has been observed by the Belle Collaboration, followed by the subsequent decay ${\bar{K}}^{\ast 0}\to K^{-}{\pi }^{+}$. The triangle amplitude develops a pronounced peak around 1420 MeV, which is reflected in the invariant mass spectrum of the $\pi a_0(980)$ system. The differential decay width is calculated and exhibits a narrow peak around $980\mathrm{MeV}$ in the ${\pi }^0\eta$ invariant mass distribution. Furthermore, the invariant mass distribution of the ${\pi }^{+}{\pi }^0\eta$ system shows a clear peak around $1420\mathrm{MeV}$, which further \replyhe{confirms} the triangle singularity explanation of the \replyhe{$a_1(1420)$}. We expect that the proposed $B^0$ decay mode could provide a potential platform for further exploring the triangle-singularity nature of the \replyhe{$a_1(1420)$} and could be tested in future experiments such as LHCb, BESIII, and Belle~II.