In the OPAL experiment at LEP, the efficiencies for selecting W W W \(\rightarrow \ell v q_{1} \bar{q}_{2}\) and W W W \(^{-} \rightarrow q_{1} \bar{q}_{2} q_{3} \bar{q}_{4}\) events were \(83.8 \%\) and \(85.9 \%\) respectively. After correcting for background, the observed numbers of \(\ell v q_{1} \bar{q}_{2}\) and \(q_{1} \bar{q}_{2} q_{3} \bar{q}_{4}\) events were respectively 4192 and 4592 . Determine the measured value of the \(W\)-boson hadronic branching ratio \(B R\left(W \rightarrow q \bar{q}^{\prime}\right)\) and its statistical uncertainty.

Particle physics is the branch of physics that studies the fundamental particles of the universe and their interactions. The main particles include quarks, leptons, bosons, and their respective antiparticles. One such important particle is the W-boson, which mediates the weak nuclear force. The W-boson can decay into different particles, and understanding these decay processes allows physicists to probe the fundamental properties and interactions of particles. Additionally, in experimental particle physics, large detectors and colliders are used to observe and measure the decay products of particles like the W-boson.

The branching ratio (BR) measures the fraction of decays of a particle into a specific final state compared to the total number of decays of that particle. For the W-boson, which can decay both into quarks and leptons, the branching ratio into quarks (hadronic decay) is an important parameter. To calculate the branching ratio, you need:

• The number of observed hadronic and leptonic decay events
• The selection efficiencies for detecting these decays

This information is combined using the following formula for the branching ratio: \[ BR = \frac{N_{\text{had}}}{N_{\text{lep}} + N_{\text{had}}} \times \frac{e_{\text{lep}}}{e_{\text{had}}} \]Here, \(N_{\text{had}}\) and \(N_{\text{lep}}\) are the numbers of observed hadronic and leptonic events after background corrections. \(e_{\text{lep}}\) and \(e_{\text{had}}\) are the selection efficiencies for leptonic and hadronic events, respectively. The observed numbers and the selection efficiencies allow the accurate calculation of the branching ratio.

In measurements, statistical uncertainty quantifies the randomness in the observed data. In particle physics, where observations often follow Poisson statistics, the uncertainty in the number of events is modeled by the square root of the number of observed events. Here, to calculate the statistical uncertainty \(\Delta BR\) of the branching ratio, we use the propagation of uncertainties formula: \[ \Delta BR = BR \times \sqrt{ \left( \frac{\Delta N_{\text{had}}}{N_{\text{had}}} \right)^2 + \left( \frac{\Delta N_{\text{lep}}}{N_{\text{lep}}} \right)^2 } \]\(\Delta N_{\text{had}}\) and \(\Delta N_{\text{lep}}\) are the uncertainties in the observed hadronic and leptonic events, calculated as the square roots of \(N_{\text{had}}\) and \(N_{\text{lep}}\) respectively. This ensures that the reported branching ratio comes with an associated confidence level, reflecting the precision of the measurement.