# Narrow Width Approximation (NWA)

I understand how the NWA is used, setting the intermediate particles on-shell and allowing us to drop off-shell contributions, ultimately writing the NWA cross-section as a product of the production cross-section and relevant branching ratios only, and replacing a Bret-Wigner distribution with a single delta-function (which makes our lives that much easier!).

However, I'm less clear about how the approximation itself is derived, i.e. how we get:

$$\dfrac{1}{(s-M^2)^2+M^2\Gamma^2}\overset{\Gamma/M\to 0}{\longrightarrow}\dfrac{\pi}{M\Gamma}\delta(s-M^2).$$

There is a known limit $$\lim_{\epsilon \to 0} \frac {\epsilon}{\epsilon^2 + x^2} = \pi \delta (x)$$
Your expresaion may be rewritten as $$\frac1 {\Gamma M^3}\frac {\Gamma/M}{(s / M ^2 -1)^2 + (\Gamma/M)^2} \to \frac1 {\Gamma M^3} \pi\delta (s/M^2-1) = \frac1 {\Gamma M} \pi\delta (s-M^2)$$ This is slightly odd as I assume $\Gamma M^3$ remains constant whilst the limit is taken.
• In dimensionless quantities, the Poisson kernel goes to $\dfrac{{M^3\Gamma}}{(s-M^2)^2+M^2\Gamma^2}\overset{\Gamma/M\to 0}{\longrightarrow} {\pi}M^2 \delta(s-M^2)$. Oct 11, 2021 at 20:23
• @cosmas indeed, but the OP does not have the denominator $M^3 \Gamma$ Dec 8, 2021 at 15:26