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We discuss matrix elements of the strangeness changing vector current usingtheir relation, due to analyticity, with $\pi K$ scattering in the $P$-wave. Wetake into account experimental phase-shift measurements in the elastic channelas well as results, obtained by the LASS collaboration, on the details ofinelastic scattering, which show the dominance of two quasi two-body channelsat medium energies.The associated form factors are shown to be completelydetermined, up to one flavour symmetry breaking parameter, imposing boundaryconditions at $t=0$ from chiral and flavour symmetries and at $t\to\infty$ fromQCD. We apply the results to the $\tau\to K\pi \nu_\tau$ and $\tau\to K\pi\pi\nu_\tau$ amplitudes and compare the former to recent high statistics resultsfrom B factories.

Analyticity constraints on the strangeness changing vector current and applications to $\tau\to K\pi \nu_\tau$, $\tau\to K\pi\pi \nu_\tau$