12.2.3. Lifetime 2D fit#

Unbinned likelihood fit to the 2D joint distribution of decay time t and decay-time uncertainty σt of D0Kπ+π0 decay candidates reconstructed in simulation, to determine the lifetime of the D0 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:

2logL(θ)=2idatalogpdf(ti,σt,i|θ),

where the index i runs over the data candidates, ti is the observed decay time for candidate i, σt,i is the observed decay time for candidate i, which 2D PDF depends on some unknown parameters identified by the vector θ.

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

pdf(t,σt|τ,b,s,μ,λ,γ,δ)=pdf(t|σt,τ,b,s)pdf(σt|μ,λ,γ,δ).

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

pdf(t|σt,τ,b,s)0ettrue/τG(tttrue|b,sσt)dttrue

where τ is the lifetime, ttrue is the true decay time, and G is a Gaussian with mean b and width sσt describing the resolution model. The PDF is normalized in the range 2<t<4 ps for any value of σt.

The decay-time uncertainty PDF is parametrized by a Johnson’s SU distribution

pdf(σt|μ,λ,γ,δ)e12[γ+δsinh1(σtμλ)]21+(σtμλ)2,

and is normalized in the range 0<σt<0.5 ps.

The fit is developed using the following frameworks: