Rare decay modes of theKmesons in gauge theories

Rare decay modes of the kaons such as $K\to \mu \overline{\mu }$, $K\to \pi \nu \overline{\nu }$, $K\to \gamma \gamma$, $K\to \pi \gamma \gamma$, and $K\to \pi e\overline{e}$ are of theoretical interest since here we are observing higher-order weak and electromagnetic interactions. Recent advances in unified gauge theories of weak and electromagnetic interactions allow in principle unambiguous and finite predictions for these processes. The above processes, which are "induced" $\left|\Delta S\right|=1\mathrm{}$ transitions, are a good testing ground for the cancellation mechanism first invented by Glashow, Iliopoulos, and Maiani (GIM) in order to banish $\left|\Delta S\right|=1\mathrm{}$ neutral currents. The experimental suppression of $K_L\to \mu \overline{\mu }$ and nonsuppression of $K_L\to \gamma \gamma$ must find a natural explanation in the GIM mechanism which makes use of extra quark(s). The procedure we follow is the following: We deduce the effective interaction Lagrangian for $\lambda +\mathfrak{N}\to l+\overline{l}$ and $\lambda +\overline{\mathfrak{N}}\to \gamma +\gamma$ in the free-quark model; then the appropriate matrix elements of these operators between hadronic states are evaluated with the aid of the principles of conserved vector current and partially conserved axial-vector current. We focus our attention on the Weinberg-Salam model. In this model, $K\to \mu \overline{\mu }$ is suppressed due to a fortuitous cancellation. To explain the small $K_L-K_S$ mass difference and nonsuppression of $K_L\to \gamma \gamma$, it is found necessary to assume $\frac{m_{\mathcal{P}}}{m_{{\mathcal{P}}^{\prime }}}<<1\mathrm{}$, where $m_{\mathcal{P}}$ is the mass of the proton quark and $m_{{\mathcal{P}}^{\prime }}$, the mass of the charmed quark, and $m_{{\mathcal{P}}^{\prime }}<5\mathrm{}$ GeV. We present a phenomenological argument which indicates that the average mass of charmed pseudoscalar states lies below 10 GeV. The effective interactions so constructed are then used to estimate the rates of other processes. Some of the results are the following: $K_S\to \gamma \gamma$ is suppressed; $K_S\to \pi \gamma \gamma$ proceeds at a normal rate, but $K_L\to \pi \gamma \gamma$ is suppressed; $K_L\to \pi \nu \overline{\nu }$ is very much forbidden, and $K^{+}\to {\pi }^{+}\nu \overline{\nu }$ occurs with the branching ratio of ${\mathrm{\sim }10}^{-10\mathrm{}}$; $K^{+}\to {\pi }^{+}e\overline{e}$ has the branching ratio of ${\mathrm{\sim }10}^{-6\mathrm{}}$, which is comparable to the presently available experimental upper bound. The predictions of other models are briefly discussed. Relevant renormalization procedures and computational details are discussed in appendixes.