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

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 $26.5\text{}\text{}{\mathrm{fb}}^{-1}$. The ratio of neutral-to-charged kaon form factors at large momentum transfers ($12<{Q}^{2}<25\text{}\text{}{\mathrm{GeV}}^{2}$) is determined to be $0.21±0.01$, 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 $|F\left({\pi }^{±}\right)|/|F\left({K}^{±}\right)|={f}_{\pi }^{2}/{f}_{K}^{2}$, where $F\left({\pi }^{±}\right)$ and $F\left({K}^{±}\right)$ 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 $\psi \left(3770\right)\to {K}_{S}^{0}{K}_{L}^{0}$ for the first time. Within a $1\sigma $ contour of the likelihood value, the branching fraction for $\psi \left(3770\right)\to {K}_{S}^{0}{K}_{L}^{0}$ is determined to be $B=\left(2.6{3}_{-1.59}^{+1.40}\right)×{10}^{-5}$, and the relative phase between the continuum and $\psi \left(3770\right)$ amplitudes is $\varphi =\left(-0.3{9}_{-0.10}^{+0.05}\right)\pi $. The branching fraction is in good agreement with the $S$- and $D$-wave charmonia mixing scheme proposed in the interpretation of the "$\rho \pi $ puzzle" between $J/\psi $ and $\psi \left(3686\right)$ decays.