Adding Particle Widths

Question

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?

Answer 1

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.