  To explore the interplay of NLO matching and next-to-leading logarithmic
(NLL) parton showers, we consider the simplest case of $\gamma^*$ and
Higgs-boson decays to $q\bar q$ and $gg$ respectively. Not only should shower
NLL accuracy be retained across observables after matching, but for global
event-shape observables and the two-jet rate, matching can augment the shower
in such a way that it additionally achieves next-to-next-to-double-logarithmic
(NNDL) accuracy, a first step on the route towards general NNLL. As a
proof-of-concept exploration of this question, we consider direct application
of multiplicative matrix-element corrections, as well as simple implementations
of MC@NLO and POWHEG-style matching. We find that the first two
straightforwardly bring NNDL accuracy, and that this can also be achieved with
POWHEG, although particular care is needed in the handover between POWHEG and
the shower. Our study involves both analytic and numerical components and we
also touch on some phenomenological considerations.
