Observation of Significant Flavor-SU(3) Breaking in the Kaon Wave Function at $12<Q^2<25{\mathrm{GeV}}^2$ and Discovery of the Charmless Decay $\psi (3770)\to K_S^0{K}_L^0$

We present cross sections for the reaction $e^{+}{e}^{-}\to K_S^0{K}_L^0$ at center-of-mass energies ranging from 3.51 to 4.95 GeV using data samples collected in the BESIII experiment, corresponding to a total integrated luminosity of $26.5{\mathrm{fb}}^{-1}$. The ratio of neutral-to-charged kaon form factors at large momentum transfers ($12<Q^2<25{\mathrm{GeV}}^2$) is determined to be $0.21\pm 0.01$, which indicates a small but significant effect of flavor-SU(3) breaking in the kaon wave function, and, consequently, excludes the possibility that flavor-SU(3) breaking is the primary reason for the strong experimental violation of the pQCD prediction $|F({\pi }^{\pm })|/|F(K^{\pm })|=f_{\pi }^2/f_K^2$, where $F({\pi }^{\pm })$ and $F(K^{\pm })$ are the form factors, and $f_{\pi }$ and $f_K$ are the decay constants of charged pions and kaons, respectively. We also observe a significant signal for the charmless decay $\psi (3770)\to K_S^0{K}_L^0$ for the first time. Within a $1\sigma$ contour of the likelihood value, the branching fraction for $\psi (3770)\to K_S^0{K}_L^0$ is determined to be $\mathcal{B}=(2.63_{-1.59}^{+1.40})\times {10}^{-5}$, and the relative phase between the continuum and $\psi (3770)$ amplitudes is $\varphi =(-0.39_{-0.10}^{+0.05})\pi$. The branching fraction is in good agreement with the $\mathcal{S}$- and $\mathcal{D}$-wave charmonia mixing scheme proposed in the interpretation of the “$\rho \pi$ puzzle” between $J/\psi$ and $\psi (3686)$ decays.

Figure 1

Distributions of $X\equiv E_{K_S^0}/E_{\mathrm{beam}}$ at $\sqrt{s}=3.774$ (top) and 4.226 GeV (bottom), where the corresponding ${\chi }^2/\mathrm{n}.\mathrm{d}.\mathrm{f}$ of the fits are 1.06 and 0.50, respectively. Dots with error bars are data. The blue solid lines are the total fit results, the blue dashed lines are the signal component, the red dashed lines represent the $e^{+}{e}^{-}\to {\gamma }^{\mathrm{ISR}}\psi (3686)$ component, the orange dashed lines represent the $e^{+}{e}^{-}\to K^{*0}(892){\overline{K}}^0+\mathrm{c}.\mathrm{c}.$ component, and the violet dashed lines represent exponential functions describing the remaining background.

Figure 2

(a) Dressed cross sections of $e^{+}{e}^{-}\to K_S^0{K}_L^0$ and a fit with the coherent sum of a continuum and a $\psi (3770)$ resonance amplitude, using the local minimum in (c), where $\mathcal{B}=2.6\times {10}^{-5}$ and $\varphi =-0.39\pi$. Dots with error bars are data. Red solid, green dashed, and blue dashed lines are the fit results, the continuum production, and the $\psi (3770)$ production, respectively. (b) The ratio of neutral-to-charged kaon form factors. The blue solid, red dashed, and green dashed lines result from fits with different order polynomials, the corresponding ${\chi }^2/\mathrm{n}.\mathrm{d}.\mathrm{f}$ are 1.26, 1.28, 1.19, respectively. (c) The likelihood contours in the $\mathcal{B}[\psi (3770)\to K_S^0{K}_L^0]$ and the relative phase $\varphi$ plane. The filled areas are up to $3\sigma$ likelihood contours. The red cross shows the local minimum.