12.2.3. Lifetime 2D fit#
Unbinned likelihood fit to the 2D joint distribution of decay time \(t\) and decay-time uncertainty \(\sigma_t\) of \(D^{0}\to K^{-}\pi^+\pi^0\) decay candidates reconstructed in simulation, to determine the lifetime of the \(D^{0}\) meson.
The input data is provided by the ROOT ntuple ntp
contained in the file example-data/lifetime.root
. The branches of the tree corresponding to the decay time and decay-time uncertainty are Dz_t
and Dz_t_err
, respectively.
To find the maximum of the likelihood, the following quantity is minimized:
where the index \(i\) runs over the data candidates, \(t_i\) is the observed decay time for candidate \(i\), \(\sigma_{t,i}\) is the observed decay time for candidate \(i\), which 2D PDF depends on some unknown parameters identified by the vector \(\vec{\theta}\).
The fit assumes no background contamination. The 2D joint PDF is written as a conditional PDF of the decay time given the value of the decay-time uncertainty as
The decay-time PDF is written as the convolution between the exponential decay and the experimental resolution function
where \(\tau\) is the lifetime, \(t_\mathrm{true}\) is the true decay time, and \(G\) is a Gaussian with mean \(b\) and width \(s\sigma_t\) describing the resolution model. The PDF is normalized in the range \(-2 < t < 4\) ps for any value of \(\sigma_t\).
The decay-time uncertainty PDF is parametrized by a Johnson’s SU distribution
and is normalized in the range \(0 < \sigma_t < 0.5\) ps.
The fit is developed using the following frameworks: