Efficiency fit

26.2.2. Efficiency fit#

Simultaneous least-squares fit to the $D^{0}$ -mass distributions of $D^{* +} \to D^{0} (\to K^{-} π^{+} π^{-} π^{+}) π^{+}$ 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 ( $h_{pass}$ ), the second with candidates failing the requirement ( $h_{fail}$ ).

The least-squares are computed as

LS ({\vec{θ}}_{pass}, {\vec{θ}}_{fail}) = \sum_{i \in h_{pass}} {(\frac{n_{i} - n_{i}^{pass} ({\vec{θ}}_{pass})}{σ_{i}})}^{2} + \sum_{i \in h_{fail}} {(\frac{n_{i} - n_{i}^{fail} ({\vec{θ}}_{fail})}{σ_{i}})}^{2},

where the index $i$ runs over the histogram bins, $n_{i}$ is the observed number of candidates in the bin, $σ_{i}$ is the uncertainty on $n_{i}$ , $n_{i}^{pass / fail}$ 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 ${\vec{θ}}_{pass / fail}$ .

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

{pdf}_{sgn} (m | μ, σ) \propto e^{- \frac{1}{2} {(\frac{m - μ}{σ^{2}})}^{2}},

and a background component, described by an exponential distribution

{pdf}_{bkg} (m | λ) \propto e^{- λ 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{array}{r} \begin{matrix} n_{i}^{pass} (N_{sgn}, ϵ_{sgn}, μ, σ_{pass}, N_{bkg}, ϵ_{bkg}, λ_{pass}) = N_{sgn} ϵ_{sgn} {pdf}_{sgn} (m_{i} | μ, σ_{pass}) + N_{bkg} ϵ_{bkg} {pdf}_{bkg} (m_{i} | λ_{pass}), \\ n_{i}^{fail} (N_{sgn}, ϵ_{sgn}, μ, σ_{fail}, N_{bkg}, ϵ_{bkg}, λ_{fail}) = N_{sgn} (1 - ϵ_{sgn}) {pdf}_{sgn} (m_{i} | μ, σ_{fail}) + N_{bkg} (1 - ϵ_{bkg}) {pdf}_{bkg} (m_{i} | λ_{fail}), \end{matrix} \end{array}

where $N_{sgn / bkg}$ is the total number of signal/background candidates and $ϵ_{sgn / bkg}$ 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