A critical point in the distribution of lepton energies from the decay of a spin-1 resonance

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.