  We consider a spin-$1$ resonance produced with an arbitrary spectrum of
velocities and decaying into a pair of massless leptons, and we study the
probability density function of the energy of the leptons in the laboratory
frame. A special case is represented by the production of $W$ bosons in
proton-proton collisions, for which the energy of the charged lepton from the
decaying $W$ can be measured with sufficient accuracy for a high-precision
measurement of $M_W$. We find that half of the resonance mass is a special
value of the lepton energy, since the probability density function at this
point is in general not analytic for a narrow-width resonance. In particular,
the higher-order derivatives of the density function are likely to develop
singularities, such as cusps or poles. A finite width of the resonance restores
the regularity, for example by smearing cusps and poles into local stationary
points. The quest for such points offers a handle to estimate the resonance
mass with much reduced dependence on the underlying production and decay
dynamics of the resonance.
