  We present the results of a lattice QCD calculation of the scalar and vector
form factors for the unphysical $B_s\to\eta_s$ decay, over the full physical
range of $q^2$. This is a useful testing ground both for lattice QCD and for
our wider understanding of the behaviour of form factors. Calculations were
performed using the highly improved staggered quark (HISQ) action on $N_f = 2 +
1 + 1$ gluon ensembles generated by the MILC Collaboration with an improved
gluon action and HISQ sea quarks. We use three lattice spacings and a range of
heavy quark masses from that of charm to bottom, all in the HISQ formalism.
This permits an extrapolation in the heavy quark mass and lattice spacing to
the physical point and nonperturbative renormalisation of the vector matrix
element on the lattice. We find results in good agreement with previous work
using nonrelativistic QCD $b$ quarks and with reduced errors at low $q^2$,
supporting the effectiveness of our heavy HISQ technique as a method for
calculating form factors involving heavy quarks. A comparison with results for
other decays related by SU(3) flavour symmetry shows that the impact of
changing the light daughter quark is substantial but changing the spectator
quark has very little effect. We also map out form factor shape parameters as a
function of heavy quark mass and compare to heavy quark effective theory
expectations for mass scaling at low and high recoil. This work represents an
important step in the progression from previous work on heavy-to-heavy decays
($b\to c$) to the numerically more challenging heavy-to-light decays.
