Lepton-flavor-violating semileptonic $\tau$ decay and $K\to\pi\nu\bar\nu$

We consider lepton-flavor violation in strangeness changing ($|\Delta S|=1$) semileptonic $\tau$-lepton decays arising from new physics encoded in a standard model effective Lagrangian. Its invariance under the standard model gauge group entails the relevance of other processes which can serve as complementary probes of the new physics operators. We show in particular that for some of them the bounds implied by current data on the rare kaon decays involving a neutrino pair, $K\to \pi\nu\bar\nu$, are stronger than the existing limits from direct searches for lepton-flavor-violating semileptonic $\tau$ decays. We discuss additional processes affected by the same operators and find that certain leptonic charged-meson decays also provide stricter constraints on a few more of them. Upcoming results of ongoing experiments such as Belle II and NA62 will further test the new physics parameter space.