Data/MC Scale Factor

The scale factor for a set of cuts quantizes the difference between data and MC simulation. It should be multiplied on MC to get the scaled efficiency consistent with the efficiency in data. Fig. 1 shows lepton flow in data and in MC, and lepton data/MC scale factors.

Figure 1: Lepton flow in data and in MC, and lepton data/MC scale factors.

Ratio of Efficiencies

Scale factor is usually defined as the ratio of efficiencies,

where e_MC is the efficiency in MC which is straight forward to get because MC simulation has the truth of particle identity, and e_data is the efficiency in data which can be quite a challenge.

In the electron or muon case, we can use the electron or muon pair at Z boson peak, which is a reliable information that tells us we get a clean sample with negligible background in data, so reliable that we can use it as "standard candle" to calibrate detector and even measure luminosity. After that, select one leg to satisfy the trigger requirements in data, ask the second leg if it passes the set of cuts, and get the efficiency in data.

Ratio of Numbers

Due to the missing energy from the neutrinoes in tau's decays, the tau pair mass at Z boson peak is severely smeared. Instead, we will use W -> tau nu to select a relatively clean tau sample. There is no second leg to get efficiency data/MC. We will use the method of absolute number data/MC.

where n_MC is the absolute number of W -> tau nu events in MC normalized to the luminosity of data, and n_data is the number of W -> tau nu events observed in data after subtracting backgrounds.

Since it is an absolute thing, the uncertainties from luminosity in data and other cuts (not the cuts for the scale factor under study) which are not relevant in the ratio of the relative efficiencies, are now relevant in the ratio of the absolute numbers.