I have a particle (A) which undergoes a two-body, on-shell decay to particle (B). The associated width of the process is $\Gamma_{ABB}$.

Particle B under goes a two-body, on-shell decay to particle C and the associated width of the process is $\Gamma_{BCC}$.

Since the particles are all produced on shell I can use the narrow-width approximation. See below for roughly drawn diagram.

Is the width of the process $A\rightarrow B B \rightarrow CCCC$ approximately given by $\Gamma_{ABB}+\Gamma_{BCC}$? i.e can I simply sum the widths?

• 1. Is the internal line in the third diagram supposed to be on-shell/resonant or not? I.e. do you want to compute a Feynman diagram for the process "A decays into CCCC mediated by B" or the actual sequence of events "A decays into BB, then each B decays into CC"? 2. Since the width is essentially a decay rate (cf. physics.stackexchange.com/q/269852/50583), you could think yourself a bit about whether or not it makes sense to add the decay rate of two subsequent processes. Sep 4, 2017 at 10:57

No. The width of process $A\to BB\to CCCC$ is given by $\Gamma_{ABB}\times(\textrm{BR}_{BCC})^2$, where BR is the branching ratio of $B\to CC$.
Since the $B$ particles are on-shell, the width of the process $A\to BB\to CCCC$ does not depend on the $B\to CC$ width, but rather on the branching ratio.