Efficiency fit
25.2.2. Efficiency fit#
Simultaneous least-squares fit to the \(D^{0}\)-mass distributions of \(D^{*+}\to D^{0}(\to K^{-}\pi^+\pi^-\pi^+)\pi^+\) decay candidates passing and failing the kaon-momentum requirement \(p(K^{-})>1\) GeV/\(c\), to determine the efficiency of the requirement on both signal and background decays.
The input data is provided by the ROOT ntuple ntp
contained in the file example-data/momentum-scale.root
. The tree contains branches corresponding to the 4-momenta of the final state particles (K_P
, pi1_p
, pi2_p
, pi3_p
), a branch with the \(D^{0}\) mass (Dz_M
), a branch with the difference between the \(D^{*+}\) and \(D^{0}\) masses (DM
).
Two histograms of the \(D^{0}\) mass, each consisting of 150 bins in the range 1.8-1.95 GeV/\(c^{2}\), are filled from the data. The first with candidates passing the kaon-momentum requirement (\(h_\mathrm{pass}\)), the second with candidates failing the requirement (\(h_\mathrm{fail}\)).
The least-squares are computed as
where the index \(i\) runs over the histogram bins, \(n_i\) is the observed number of candidates in the bin, \(\sigma_i\) is the uncertainty on \(n_i\), \(n_i^\mathrm{pass/fail}\) is the predicted number of candidates in the bin passing/failing the kaon-momentum requirement, which depends on some unknown parameters identified by the vector \(\vec{\theta}_\mathrm{pass/fail}\).
To compute the predicted number of candidates, the fit model assumes that the \(D^{0}\)-mass distributions of the candidates passing/failing the selection can be described by a signal component peaking around the nominal \(D^{0}\) mass, described by a Gaussian distribution
and a background component, described by an exponential distribution
Each PDF is normalized in the fit range 1.8-1.95 GeV/\(c^{2}\). The predicted numbers of candidates passing/failing the selection in bin \(i\) are estimated by evaluating the above PDFs at the center of the bin \(m_i\) as
where \(N_\mathrm{sgn/bkg}\) is the total number of signal/background candidates and \(\epsilon_\mathrm{sgn/bkg}\) is the signal/background efficiency, and we have assumed that the signal/background shapes may have different widths/slopes depending on whether the candidates pass/fail the kaon-momentum requirement.
The fit is developed using the following frameworks: