  Infrared- and collinear-safe (IRC-safe) observables have finite cross
sections to each fixed-order in perturbative QCD. Generically, ratios of
IRC-safe observables are themselves not IRC safe and do not have a valid
fixed-order expansion. Nevertheless, in this paper we present an explicit
method to calculate the cross section for a ratio observable in perturbative
QCD with the help of resummation. We take the IRC-safe jet angularities as an
example and consider the ratio formed from two angularities with different
angular exponents. While the ratio observable is not IRC safe, it is "Sudakov
safe", meaning that the perturbative Sudakov factor exponentially suppresses
the singular region of phase space. At leading logarithmic (LL) order, the
distribution is finite but has a peculiar expansion in the square root of the
strong coupling constant, a consequence of IRC unsafety. The accuracy of the LL
distribution can be further improved with higher-order resummation and
fixed-order matching. Non-perturbative effects can sometimes give rise to order
one changes in the distribution, but at sufficiently high energies Q, Sudakov
safety leads to non-perturbative corrections that scale like a (fractional)
power of 1/Q, as is familiar for IRC-safe observables. We demonstrate that
Monte Carlo parton showers give reliable predictions for the ratio observable,
and we discuss the prospects for computing other ratio observables using our
method.
