1
$\begingroup$

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:

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

$\endgroup$
0
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ I'm sure I used to know this, but now it looks wrong :/ $\endgroup$ – innisfree May 22 '17 at 12:43

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.