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:

\[-2\log L(\vec{\theta}) = -2\sum_{i\in\mathrm{data}} \log\,\mathrm{pdf}(t_i,\sigma_{t,i}|\vec{\theta}),\]

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

\[\mathrm{pdf}(t,\sigma_t|\tau,b,s,\mu,\lambda,\gamma,\delta) = \mathrm{pdf}(t|\sigma_t,\tau,b,s)\,\mathrm{pdf}(\sigma_t|\mu,\lambda,\gamma,\delta).\]

The decay-time PDF is written as the convolution between the exponential decay and the experimental resolution function

\[\mathrm{pdf}(t|\sigma_t,\tau,b,s) \propto \int_0^\infty e^{-t_\mathrm{true}/\tau}G(t-t_\mathrm{true}|b,s\sigma_t) dt_\mathrm{true}\]

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

\[\mathrm{pdf}(\sigma_t|\mu,\lambda,\gamma,\delta) \propto \frac{e^{-\frac{1}{2}\left[\gamma+\delta\text{sinh}^{-1}\left(\frac{\sigma_t-\mu}{\lambda}\right)\right]^2}}{\sqrt{1+\left(\frac{\sigma_t-\mu}{\lambda}\right)^2}},\]

and is normalized in the range \(0 < \sigma_t < 0.5\) ps.

The fit is developed using the following frameworks: