26.2.2. 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

\[\mathrm{LS}(\vec{\theta}_\mathrm{pass},\vec{\theta}_\mathrm{fail}) = \sum_{i \in h_\mathrm{pass}}\left(\frac{n_i - n_i^\mathrm{pass}(\vec{\theta}_\mathrm{pass})}{\sigma_i}\right)^2 + \sum_{i \in h_\mathrm{fail}}\left(\frac{n_i - n_i^\mathrm{fail}(\vec{\theta}_\mathrm{fail})}{\sigma_i}\right)^2,\]

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

\[\mathrm{pdf}_\mathrm{sgn}(m|\mu,\sigma) \propto e^{-\frac{1}{2}\left(\frac{m-\mu}{\sigma^2}\right)^2},\]

and a background component, described by an exponential distribution

\[\mathrm{pdf}_\mathrm{bkg}(m|\lambda) \propto e^{-\lambda m}.\]

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

\[ \begin{align}\begin{aligned}n_i^\mathrm{pass}(N_\mathrm{sgn},\epsilon_\mathrm{sgn},\mu,\sigma_\mathrm{pass},N_\mathrm{bkg},\epsilon_\mathrm{bkg},\lambda_\mathrm{pass}) = N_\mathrm{sgn}\epsilon_\mathrm{sgn}\mathrm{pdf}_\mathrm{sgn}(m_i|\mu,\sigma_\mathrm{pass}) + N_\mathrm{bkg}\epsilon_\mathrm{bkg}\mathrm{pdf}_\mathrm{bkg}(m_i|\lambda_\mathrm{pass}),\\n_i^\mathrm{fail}(N_\mathrm{sgn},\epsilon_\mathrm{sgn},\mu,\sigma_\mathrm{fail},N_\mathrm{bkg},\epsilon_\mathrm{bkg},\lambda_\mathrm{fail}) = N_\mathrm{sgn}(1-\epsilon_\mathrm{sgn})\mathrm{pdf}_\mathrm{sgn}(m_i|\mu,\sigma_\mathrm{fail}) + N_\mathrm{bkg}(1-\epsilon_\mathrm{bkg})\mathrm{pdf}_\mathrm{bkg}(m_i|\lambda_\mathrm{fail}),\end{aligned}\end{align} \]

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:

• Minuit