Why can't we measure branching ratios of the Higgs at LHC? There are theoretical predictions for SM Higgs decay branching ratios. What about the experimental results on BRs?
All that I found was measurements of signal-strengths $\mu =  \frac{(\sigma \cdot BR)_{\text{meas}}}{(\sigma \cdot BR)_{\text{SM}}} $ and coupling strengths $\kappa$. 
Why are there no results on branching fractions up to now? And how do you actually measure the signal strength $\mu$ without measuring the branching ratio $BR$?
I would like to have a little detailed information about the actual differences between measuring $BR$, signal strength $\mu$ and coupling strength $\kappa$.
 A: The Higgs is inferred from its decays - no Higgs bosons themselves are actually detected. As such, an experiment can count the number of Higgs boson decays to a particular final state, if it can measure the particles in that final state.
For that reason, it is impossible to measure $\sigma$ or the branching ratios independently, you can only measure $\sigma \times BR$. 
To extract a branching ratio from the measurements, one would have to make a model-dependent assumption about the production cross-section, $\sigma$, or assume that there were no "invisible" decay channels with final states that we could not see.
Instead, what typically happens, is that the coupling strengths, $\kappa$, related to Lagrangian couplings in an effective field theory approach, are fitted to all of the data, simultaneously. That allows us to extract the Lagrangian couplings for the Higgs boson.
This needn't be true at an electron collider, at which the dominant production cross section would be $ee\to hZ$. This production cross section could be measured independently of the branching ratios by analysing the $Z$, because the electron collider environment is cleaner than that of the LHC.
