Simultaneous explanation of $K$ and $B$ anomalies in vectorlike compositeness

We address the presently reported significant flavor anomalies in the $K$ and $B$ meson systems such as the CP violating Kaon decay ($\epsilon'/\epsilon$) and lepton-flavor universality violation in $B$ meson decays ($R_{K^{(*)}},$ and also commenting ${R_{D^{(*)}}}$), by proposing flavorful and chiral vector bosons as the new physics constitution at $\sim 1\,\mathrm{TeV}$. Interestingly, if the new (composite) vector bosons are quite heavier than $\sim 1\,\mathrm{TeV}$, we face a difficulty in addressing the anomaly in $\epsilon'/\epsilon$ consistently with the constraint from the $K^0$-$\overline{K^0}$ mixing. Both of the anomalies can be addressed within $1\sigma$ confidence levels individually, where the relevant parameter space will be investigated by the NA62 and KOTO experiments, in addition to direct searches at the large hadron collider.