Factor a resonance only when the integral permits it

For an intermediate state of mass M and approximately constant total width $\Gamma$, a resonance propagator contributes the squared denominator $[(q^2-M^2)^2+M^2\Gamma^2]^{-1}$. Here $q^2$ is the intermediate invariant mass squared, and both M and width use energy units. If $\Gamma/M$ is small, the denominator is sharply concentrated near $q^2=M^2$. Integrating it against a factor that changes slowly across the peak permits a delta-function approximation with an explicitly normalized area.

This is a distributional approximation, not pointwise equality between an infinite spike and a smooth function. The integration range must include the resonant neighborhood, and phase space, production factors, partial widths, and selection weights must vary gently over that neighborhood. A nearby threshold can truncate the peak. A tight invariant-mass window or a rapidly changing acceptance can also invalidate the naive inclusive replacement, even when the width-to-mass ratio is numerically tiny. Check those scales rather than using narrowness as the only criterion.

Invariant phase space can be decomposed recursively using $dq^2/(2\pi)$ times production and decay measures in our convention. Combining that factor with the resonant area and the definition of partial width produces the familiar branching ratio $\mathcal B=\Gamma_f/\Gamma$. For a sufficiently inclusive isolated channel, the resulting rate is approximately production cross section times the desired branching fraction. This reasoning explains why the total width belongs in the propagator while a selected partial width belongs in the numerator of the channel rate.

For a spinning resonance, production prepares a polarization density matrix and decay analyzes it. Their helicity indices must be contracted before angular information is discarded. A spin-correlated factorization can remain valid while a product of separately spin-averaged angular distributions fails. Multiple overlapping resonances or a sizable nonresonant amplitude introduce coherent interference that need not be represented by separate positive event counts. The approximation should be applied to the specified observable and its amplitude structure, with a stated estimate or assessment of neglected terms.

A constant width gives an exponential survival law in proper time under the usual unstable-state approximation. A moving particle's laboratory mean lifetime is dilated by its Lorentz factor, and multiplying that time by its speed gives the mean flight distance. Different events can have different boosts, so the distance distribution of a mixed sample is not generally one exponential with a single averaged boost. A detector selection on displaced decays must integrate the survival probability over the accepted momentum and geometry distribution.

A width is an energy scale in a natural-unit amplitude; a decay rate in inverse seconds follows only after restoring hbar. Likewise, multiplying a lifetime by the speed of light alone gives the proper decay-length scale, not the flight distance of every moving particle. Short lifetime and narrow relative width are also separate comparisons: a width can be tiny compared with an enormous mass yet correspond to a short time. Keep the relevant reference scale and unit conversion explicit before assigning an experimental label such as prompt or displaced.