# Going from width to cross section

Given the decay width of a process, $\Gamma(A\to B+C)$, is it possible to turn this around to find the production cross section, $\sigma(B+C\to A)$?

Edit: In particular I have been thinking of something like a Breit-Wigner resonance type example assuming that $\sigma(B+C)$ is dominated by $\sigma(B+C\to A)$ at relevant energies.

$$Im(\Sigma(m^2))=m\Gamma_{tot}$$ So the propagator becomes $$iG(p^2)=\frac{i}{p^2-m^2+im\Gamma_{tot}}$$ You can jump up to cross section then considering s channel $$\sigma = g^4{\mid\frac{i}{p^2-m^2+im\Gamma_{tot}} \mid}^2=g^4\frac{1}{(p^2-m^2)^2+(m\Gamma_{tot})^2}\sim g^4\frac{\pi}{m\Gamma_{tot}}\delta(p^2-m^2)$$ Now the resonant become on-shell particle. But the thing is you need to calculate all the decay channels such as $A\rightarrow B+C$ and $A\rightarrow B+X$ and $A\rightarrow Y+C$ etc... For a specific decay channel one should use branching ratio as well $$\sigma(B+C\rightarrow A \rightarrow f )= \sigma(B+C\rightarrow A )\frac{\Gamma_{f}}{\Gamma_{tot}}$$ As you see it is very tedious to calculate each decay process and their width(decay rate) but I believe one can succeed.