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We present HPQCD's improved scalar, vector and tensor form factors for $B \toK$ semileptonic decays, using the heavy-HISQ formalism for more accuratenormalisation of the weak currents. Working with masses close to the physical$b$ on the finest ensemble and including three ensembles with physical lightquarks, we cover the full physical $q^2$ range with good precision. Ouruncertainties at $q^2=0$ are a factor of three better than earlier work.  We compare Standard Model observables using our form factors to experimentalmeasurements for the rare flavour changing neutral current processes $B \to K\ell^+\ell^-$ and $B \to K \nu\bar{\nu}$ and discuss the significance of thetensions that arise.

The search for new physics in $B \to K \ell^+\ell^-$ and $B \to K \nu\bar{\nu}$ using precise lattice QCD form factors