The Branching fraction formula for a particular decay is ${\cal {B}}(B \rightarrow X)=\Gamma_{B\rightarrow X/\Gamma_B}$. But experimentally formula is written as ${\cal {B}}(B \rightarrow X) = N_{obs}/(N_{BB}\times \epsilon)$. Where $N_{obs}=$no. of $B\rightarrow X$ events observed, $N_{BB}=$no. of BB events created, $\epsilon=$detection efficiency. I am trying to relate these two formulas. Can anyone explain why the experimental BF formula is formulated like this?
Thanks
Here, I change the variable names for more pedagogic ones : $X$ for the resonance produced, $f$ for final state (the particle to which $X$ decays). The couple {$X$, f} represents what we call the "channel". But sometimes, we use channel as meaning the final state : decay channel. There is some ambiguity in the language. But it is not important here.
The resonance $X$ could decay to many final states. The branching ratio is formulated theoretically as $BR(X\rightarrow f)=\Gamma_f/\Gamma_X$. This expresses the ratio of events in the specific final state of the study, that is the fraction of events that decays to the final state you are looking for (numerator) with respect to all possible final states (denominator).
When dealing with an experimental measurement with a single channel, that is meaning, when you deal with the production and decay of a resonance, the analysis alone (without combination with other analyses) does not allow itself to separate the contribution from the production cross-section from the branching ratio. All you measure is the product of the two : there is a degeneracy of the couple "{production cross-section, branching ratio}" : there are infinite possible values.
To raise this degeneracy, you need an assumption.
If you make an assumption on the branching ratio, then you could make a measurement of the production cross-section only, that is the cross-section to produce the resonance $X$ that could decay to all possible final states. If you make an assumption on the production cross-section, then you could measure the branching ratio.
In your example, you wish to measure the branching ratio. In this case, you need to make an assumption on the production cross-section. A first way would be to fix the cross-section to the value predicted of the Standard Model (or a given model). Another option would be to fix the cross-section to the value predicted by the combination of several analyses.
In analysis, the resonance that is produced is made after a selection (with an efficiency eff selectoin) whose goal is to reduce drastically the background (processes that mimic your signal). So when you measure that you have found "Nobs" signal, this means that "Nobs" is the number that passes your selection. The selection is restrictive : you select less events that there are produced. So if you observe after the selection $Nobs$*, this means that $N_{obs}/efficiency_{selection}$ were produced, but your criteria of selection kept only a fraction $efficiency_{selection}$.
So if you measure $Nobs$ : this represent the number that has been :
-selected
-produced
-decayed to the final state you are interested in.
So $Nobs=Nproduction\times BR(X\rightarrow f)\times efficiency_{selection}$. If you invert this equation, this gives your formula.
Im not working in this fields, however, I would use a weighted average to define the observed branching ratio into the $i^{th}$ decay branch, $$ BR_i^{\textrm{(obs)}} := \frac{N_i^{\textrm{(obs)}} }{\sum_i N_i^{\textrm{(obs)}}} = \frac{N_i^{\textrm{(theo)}} \cdot \epsilon_i}{\sum_i N_i^{\textrm{(theo)}} \cdot \epsilon_i} $$ where $\epsilon_i$ is the detection efficiency for the $i^{th}$ decay branch, and $N_i^{\textrm{(theo)}} = N_i^{\textrm{(obs)}}/\epsilon_i$ is the (true, theoretical) number of decays in the $i^{th}$ branch. Since I'm actually interested in the "true" (theoretical) branching ratios, I would use $$ BR_i^{\textrm{(theo)}} := \frac{N_i^{\textrm{(theo)}} }{\sum_i N_i^{\textrm{(theo)}}} = \frac{N_i^{\textrm{(obs)}}/\epsilon_i }{\sum_i N_i^{\textrm{(theo)}}} $$ This is your formula.
