Study of the $e^{+}{e}^{-}\to \varphi K^{+}{K}^{-}$ reaction within triangle dynamics and its implications for the $\varphi (2170)$

Abstract

We revisit the $e^{+}{e}^{-}\to \varphi {\pi }^{+}{\pi }^{-}$ reaction within the $K_1$-$\bar{K}$-$K$ triangle dynamics framework, in which the $e^{+}{e}^{-}$ pair annihilates through one-photon exchange approximation to produce a $K_1\bar{K}$ pair, followed by the $K_1\to \varphi K$ decay and the final-state $K\bar{K}\to {\pi }^{+}{\pi }^{-}$ rescattering. With the same theoretical formalism and model parameters, the $e^{+}{e}^{-}\to \varphi K^{+}{K}^{-}$ reaction is investigated, and it is found that the predicted total cross sections for the $e^{+}{e}^{-}\to \varphi K^{+}{K}^{-}$ reaction are in good agreement with the existing BESIII measurements. Our study shows that the triangle singularity in the $\varphi K^{+}{K}^{-}$ channel is strongly suppressed, because the higher $K^{+}{K}^{-}$ mass threshold shifts the kinematics away from the triangle singularity condition and the phase space near the $K_1\bar{K}$ threshold is very limited. Moreover, the interference between the tree-level and loop amplitudes eliminates the remaining signal. These combined effects naturally explain the absence of a distinct $\varphi (2170)$ signal in the $e^{+}{e}^{-}\to \varphi K^{+}{K}^{-}$ reaction, provide a strong test of the model, and reinforce the picture that both reactions are governed by the same underlying mechanism, in which the $\varphi (2170)$ state is produced in the $e^{+}{e}^{-}$ annihilation from the $K_1$-$\bar{K}$-$K$ triangle loop.

7 pages, 4 figures