  We present a new strategy for the dispersive evaluation of the hadronic
light-by-light contribution to the anomalous magnetic moment of the muon
$a_\mu$. The new approach directly applies in the kinematic limit relevant for
$a_\mu$: one of the photons is treated as an external electromagnetic field
with vanishing momentum, so that the kinematics corresponds to a triangle. We
derive expressions for the relevant single-particle intermediate states, as
well as the tensor decompositions of the two-pion sub-processes that appear in
addition to those needed in the established dispersive approach. The existing
approach is based on a set of dispersion relations for the hadronic
light-by-light tensor in four-point kinematics. At present it is not known how
to consistently include in this framework resonant intermediate states of spin
2 or larger, due to the appearance of kinematic singularities that can be
traced back to the redundancy of the tensor decomposition. We show that our new
approach circumvents this problem and enables dispersion relations in the limit
of triangle kinematics that are manifestly free from kinematic singularities,
paving the way towards a data-driven evaluation of all relevant exclusive
hadronic intermediate states.
