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This maybe is a naive basic question but I want to be sure. If I want to calculate the cross section of the process $$p\bar{p} \rightarrow W^+HX \rightarrow e^+\nu_e b\bar{b} X$$ ($H$ is a Higgs particle, $p$ a proton), I'm going to need the branching ratio of $$W^+H \rightarrow e^+\nu_e b\bar{b}$$ defined as:

$$ BR(W^+H \rightarrow e^+\nu_e b\bar{b})= \frac{\sigma(W^+H \rightarrow e^+\nu_e b\bar{b})}{\sigma_{total}} \tag1 $$

where $\sigma$ is the cross section of $W^+$'s decay into $e^+ \nu_e$ and $H$ into $b\bar{b}$; and $\sigma_{total}$ is the sum of the cross sections of all possible decays of $W^+$ and $H$.

Then, for the reaction $$W^+H \rightarrow e^+\nu_e b\bar{b}$$ I have two Feynman diagrams.

My question is:

$$\sigma(W^+H \rightarrow e^+\nu_e b\bar{b}) ) = \sigma(W^+ \rightarrow e^+ \nu_e) + \sigma(H \rightarrow b\bar{b})\tag2$$

and use it in Eq. (1) since each cross section is an area or should I use $$BR(W^+H \rightarrow e^+\nu_e b\bar{b}) = BR(W^+ \rightarrow e^+ \nu_e)·BR(H \rightarrow b\bar{b})\tag3$$

since each BR is the probability of the corresponding process in brackets and these two decays are independent?

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