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The Breit-Wigner cross section for resonant particle decay takes the form: $$\sigma_{fi}=\frac{\pi\hbar^2}{q^2} \frac{2j+1}{(2S_1+1)(2S_2+1)} \frac{\Gamma_i \Gamma_f}{(E-Mc^2)^2+\Gamma^2/4}$$ I am looking to (non-relativistically) explain the origin of each of these terms. This question is concerned however with just the: $$\frac{1}{(E-Mc^2)^2+\Gamma^2/4}$$ factor. Here is a rough overview of it's origin:

The resonant state is of finite life time so we can write it's wave function as: $$\psi(t)=\psi(0) e^{iEt/\hbar}e^{-\Gamma t/2}$$ taking the Fourier transform of this we get: $$\chi(E)\propto \frac{1}{(E-Mc^2)+i\Gamma/2} $$ the cross section is then proportional to $\chi \chi^*$ meaning: $$\sigma \propto \frac{1}{(E-Mc^2)^2+\Gamma^2/4}$$

I am ok with this explanation apart from the part in bold. How do we know that $\sigma \propto \chi \chi^*$? should we not be integrating over the matrix element of this with the initial (final?) state?

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    $\begingroup$ you have taken the initial (final) states in your xx* and it is a probability density and the crossection is proportional to the probability density. If you integrate over E you will find the total production crossection for that resonance. $\endgroup$ – anna v Mar 21 '17 at 16:49
  • $\begingroup$ @annav Sorry what do you mean by 'you have taken the initial (final) states in your xx* '? That I have already taken the matrix elements or something else? $\endgroup$ – Quantum spaghettification Mar 21 '17 at 16:54
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    $\begingroup$ yes, what you have written is the matrix element squared which is the probabilty density. look at page6 of this cpp.edu/~pbsiegel/phy40413/lectures/lecture10.pdf $\endgroup$ – anna v Mar 21 '17 at 17:16
  • $\begingroup$ To further expound on @anna v's (virtual) answer, the E in your wave function should actually be M. You wish to compute the decay amplitude $\psi(t=0)^* ~\psi(t)$. This is what you are re-expressing as a function of E instead of t in your Fourier transform, so of course the square of the amplitude χ(E) yields the cross section. $\endgroup$ – Cosmas Zachos Sep 7 '17 at 16:01
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The numerator term gamma i is the input width to the resonant state through an incoming state i and gamma f is the decay width of resonant state to the final state f. Now the resonant state can decay not only back to state i or go to state f it could go to very many states. This is given by the spreading width in the denominator the gamma term. Mc2 is what may be called as the energy of resonance. Where the cross section is maximum.

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