# 12.2.1. Simple 1D fit#

Unbinned likelihood fit to the beam-constrained mass distribution of $$B^{0}\to K^{*}\gamma$$ decay candidates reconstructed in simulation, to determine the fraction of signal decays.

The input data is provided by the ROOT ntuple BtoKstG contained in the file example-data/fitme.root. The branch of the tree corresponding to the beam-constrained mass is B0_mbc.

To find the maximum of the likelihood, the following quantity is minimized:

$-2\log L(\vec{\theta}) = -2\sum_{i\in\mathrm{data}} \log\,\mathrm{pdf}(m_i|\vec{\theta}),$

where the index $$i$$ runs over the data candidates, $$m_i$$ is the observed beam-constrained mass for candidate $$i$$, which PDF depends on some unknown parameters identified by the vector $$\vec{\theta}$$.

The fit model assumes a signal component peaking at the nominal $$B^{0}$$ mass, described by a Crystal Ball distribution with $$n=15$$

$\begin{split}\mathrm{pdf}_\mathrm{sgn}(m|\mu,\sigma,\alpha) \propto \left\{\begin{array}{lcccl} e^{-\frac{1}{2}\left(\frac{m-\mu}{\sigma}\right)^2} & & \mathrm{if} & & \frac{m-\mu}{\sigma} > \alpha\\ \left(\frac{n}{|\alpha|}-|\alpha|-\frac{m-\mu}{\sigma}\right)^{-n}& & \mathrm{if} & & \frac{m-\mu}{\sigma} \leq -\alpha\end{array}\right.,\end{split}$

and a background component, described by an Argus distribution

$\begin{split}\mathrm{pdf}_\mathrm{bkg}(m|m_0,c) \propto \left\{\begin{array}{lcccl} \frac{m}{m_0^3}\,\sqrt{1-\frac{m^2}{m_0^2}}\, e^{-\frac{1}{2}c^2\left(1-\frac{m^2}{m_0^2}\right)}& & \mathrm{if} & & m \leq m_0\\ 0& & \mathrm{if} & & m > m_0\end{array}\right..\end{split}$

Each PDF is normalized in the fit range that goes from 5.2 GeV/$$c^{2}$$ up to the threshold corresponding to $$m_0$$. The total PDF is sum of the signal and background PDFs weighted by the signal fraction $$f_\mathrm{sgn}$$

$\mathrm{pdf}(m|\mu,\sigma,\alpha,m_0,c) = f_\mathrm{sgn}\,\mathrm{pdf}_\mathrm{sgn}(m|\mu,\sigma,\alpha) + (1-f_\mathrm{sgn})\, \mathrm{pdf}_\mathrm{bkg}(m|m_0,c).$

The fit is developed using the following frameworks: