3

The optical theorem links the imaginary part of the forward scattering amplitude to the total decay width of a particle: $\mathrm{Im}\,M_{i\to i} = m\Gamma_{tot}$. Here $\Gamma_{tot} = \frac{1}{2m} \sum_x |M_{i\to x}|^2$. My QFT lecturer told us that the Breit-Wigner propagator $\frac{i}{p^2-m^2+im\Gamma_{tot}}$ follows from that. How?

I tried to use the LSZ formula to replace the forward amplitude to the propagator. However I don't even know if the LSZ formula is applicable here (due to the complex mass). If one can use it I find something like (for a scalar particle) $iM_{i\to i} \sim (p^2 - m^2)^2 \cdot \frac{i}{p^2-m^2+i\varepsilon}$ which should simply vanish for on-shell particles, whether or not $m$ has an imaginary part.

So how does the Breit-Wigner propagator follows from the optical theorem? In particular why is the $\Gamma$ in the Breit-Wigner propagator the same as in the optical theorem?

toaster
  • 721
  • WP useful? – Cosmas Zachos Feb 05 '19 at 20:53
  • Sorry, but how does anything on this Wikipedia page answer this question? :D – toaster Feb 06 '19 at 21:43
  • It sends you to L Brown's friendly derivation to the generic propagator of an unstable particle, and whence the relativistic Breit-Wigner formula. I did not stick to the somewhat off-mainstream twists and turns of your proposed path to it. – Cosmas Zachos Feb 06 '19 at 21:46
  • This just explains that a complex pole of propagator is related to a exponential decay of the propagator. This does not show that the $\Gamma$ in the propagator is actually the total decay width as calculated in scattering theory: $\Gamma_{tot} = \frac{1}{2m} \sum_x |M_{i \to x}|^2$. – toaster Feb 07 '19 at 12:24
  • I took you to Lowell Brown's (6.314-6.3.23), but I can't make you drink... Perhaps if you took that primary section into consideration in restructuring your question you might have better luck. – Cosmas Zachos Feb 07 '19 at 16:34
  • @CosmasZachos Ehm... your answer is really not useful at all. SE is not about thinking that you take someone to the source; it's about explaining and giving concrete answers. – Helen Apr 06 '22 at 14:42
  • Answer? We hardly got to the question! Thanks for the advice. I have so far failed to understand SE as a bypass of standard material detailed in texts and reading them to the student who should have gone through them in their course. This is a clear-cut non-question that should have not been asked, or asked with more detail about where the mental block resides... – Cosmas Zachos Apr 06 '22 at 14:50
  • Linked. – Cosmas Zachos Apr 06 '22 at 14:57

0 Answers0