  We argue that the evolution kernel for the scale-dependence of the $B$-meson
light-cone distribution amplitude (LCDA) can be written, to all orders in
perturbation theory, in terms of the generator of special conformal
transformations in a modified theory: QCD at critical coupling in non-integer
$d\neq 4$ dimensions. Explicit expression for the eigenfunctions of the
evolution kernel is derived that is valid, again, to all orders. From a
practitioner's point of view the utility of this representation is that it
allows to "save one loop" and obtain the evolution kernel to a given order of
perturbation theory, up to a constant term, from the calculation of the
conformal anomaly at one order less. This construction is verified by explicit
calculation at two-loop level.
