  We present a systematic formalism based on a factorization theorem in
soft-collinear effective theory to describe non-global observables at hadron
colliders, such as gap-between-jets cross sections. The cross sections are
factorized into convolutions of hard functions, capturing the dependence on the
partonic center-of-mass energy $\sqrt{\hat s}$, and low-energy matrix elements,
which are sensitive to the low scale $Q_0\ll\sqrt{\hat s}$ characteristic of
the veto imposed on energetic emissions into the gap between the jets. The
scale evolution of both objects is governed by a renormalization-group
equation, which we derive at one-loop order. By solving the evolution equation
for the hard functions for arbitrary $2\to M$ jet processes in the leading
logarithmic approximation, we accomplish for the first time the all-order
resummation of the so-called "super-leading logarithms" discovered in 2006,
thereby solving an old problem of quantum field theory. We study the numerical
size of the corresponding effects for different partonic scattering processes
and explain why they are sizable for $pp\to 2\,\text{jets}$ processes, but
suppressed in $H/Z$ and $H/Z$+jet production. The super-leading logarithms are
given by an alternating series, whose individual terms can be much larger than
the resummed result, even in very high orders of the loop expansion.
Resummation is therefore essential to control these effects. We find that the
asymptotic fall-off of the resummed series is much weaker than for standard
Sudakov form factors.
