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

## TL;DR

This paper analyzes the energy distribution of leptons from a spin-1 resonance decay, revealing a critical point at half the resonance mass that can aid in precise mass measurements, especially for W bosons.

## Contribution

It identifies a non-analytic point in the lepton energy distribution at half the resonance mass and shows how finite width effects smooth this feature, providing a new method for resonance mass estimation.

## Key findings

- Half the resonance mass is a special, non-analytic point in the lepton energy distribution.
- Finite resonance width smooths singularities, enabling more accurate mass measurements.
- The approach reduces dependence on production and decay dynamics for resonance mass determination.

## Abstract

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.