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

Abstract

We revisit the $e^+e^- \to ϕπ^+π^-$ 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 $K_1 \toϕK$ decay and the final-state $K\bar{K} \to π^+π^-$ rescattering. With the same theoretical formalism and model parameters, the $e^+e^- \to ϕK^+K^-$ reaction is investigated, and it is found that the predicted total cross sections for the $e^+ e^- \to ϕK^+ K^-$ reaction are in good agreement with the existing BESIII measurements. Our study shows that the triangle singularity in the $ϕK^+K^-$ 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 $ϕ(2170)$ signal in the $e^+ e^- \to ϕK^+ K^-$ 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 $ϕ(2170)$ state is produced in the $e^+ e^-$ annihilation from the $K_1$-$\bar{K}$-$K$ triangle loop.