  We present the result of lattice QCD calculation of the scalar, vector and
tensor form factors for the $B\to K\ell^+\ell^-$ decay, across the full
physical range of momentum transfer. We use the highly improved staggered quark
(HISQ) formalism for all valence quarks on eight ensembles of gluon field
configurations generated by the MILC collaboration. These include four flavours
of HISQ quarks in the sea, with three ensembles having the light $u/d$ quarks
at physical masses. In the first fully relativistic calculation of these form
factors, we use the heavy-HISQ method. This allows us to determine the form
factors as a function of heavy quark mass from the $c$ to the $b$, and so we
also obtain new results for the $D\to K$ tensor form factor. The advantage of
the relativistic formalism is that we can match the lattice weak currents to
their continuum counterparts much more accurately than in previous
calculations; our scalar and vector currents are renormalised fully
nonperturbatively and we use a well-matched intermediate momentum-subtraction
scheme for our tensor current. Our scalar and vector $B\to K$ form factors have
uncertainties of less than 4% across the entire physical $q^2$ range and the
uncertainty in our tensor form factor is less than 7%. Our heavy-HISQ method
allows us to map out the dependence on heavy-quark mass of the form factors and
we can also see the impact of changing spectator quark mass by comparing to
earlier HPQCD results for the same quark weak transition but for heavier
mesons.
