  The analytic study of differential cross sections in QCD has typically
focused on individual observables, such as mass or thrust, to great success.
Here, we present a first study of double differential jet cross sections
considering two recoil-free angularities measured on a single jet. By analyzing
the phase space defined by the two angularities and using methods from
soft-collinear effective theory, we prove that the double differential cross
section factorizes at the boundaries of the phase space. We also show that the
cross section in the bulk of the phase space cannot be factorized using only
soft and collinear modes, excluding the possibility of a global factorization
theorem in soft-collinear effective theory. Nevertheless, we are able to define
a simple interpolation procedure that smoothly connects the factorization
theorem at one boundary to the other. We present an explicit example of this at
next-to-leading logarithmic accuracy and show that the interpolation is unique
up to $\alpha_s^4$ order in the exponent of the cross section, under reasonable
assumptions. This is evidence that the interpolation is sufficiently robust to
account for all logarithms in the bulk of phase space to the accuracy of the
boundary factorization theorem. We compare our analytic calculation of the
double differential cross section to Monte Carlo simulation and find
qualitative agreement. Because our arguments rely on general structures of the
phase space, we expect that much of our analysis would be relevant for the
study of phenomenologically well-motivated observables, such as
$N$-subjettiness, energy correlation functions, and planar flow.
