  Several boosted jet techniques use jet shape variables to discriminate the
multi-pronged signal from Quantum Chromodynamics backgrounds. In this paper, we
provide a first-principles study of an important class of jet shapes all of
which put a constraint on the subjet mass: the mass-drop parameter ($\mu^2$),
the $N$-subjettiness ratio ($\tau_{21}^{(\beta=2)}$) and energy correlation
functions ($C_2^{(\beta=2)}$ or $D_2^{(\beta=2)}$). We provide analytic results
both for QCD background jets as well as for signal processes. We further study
the situation where cuts on these variables are applied recursively with
Cambridge-Aachen de-clustering of the original jet. We also explore the effect
of the choice of axis for $N$-subjettiness and jet de-clustering. Our results
bring substantial new insight into the nature, gain and relative performance of
each of these methods, which we expect will influence their future application
for boosted object searches.
