The FIT-pull Method: an experimental tool to monitor the track measurements and the B proper time.

In this note we describe a statistical tool, the \textit{FIT-pull method}, that can test the reliability of the measurements of the tracks and the vertices on real and Monte-Carlo data without knowledge of the truth information. The basic mathematical formalism is derived from the Lagrange Multipliers method and briefly described. Several tests are performed to prove its validity in different situations. %KDifferent useful examples are discussed. In particular, by using Monte-Carlo simulation, we demonstrate that the method can be applied to check if the measured tracks or vertices have biases or incorrect covariance matrices. For correct input measurements we obtain pull distributions with a normal Gaussian statistical form. In this case the B proper time value and its error, which is a function of the track and vertex measurements, are correctly calculated. However, in the case of incorrect measurements, for example due to a systematic error or to a scale factor of the covariance matrix, the pull distributions studied deviate from normal Gaussians and the B proper time measurement can be affected. In principle the method can, if necessary, be used to recover information from corrupted measurements. Its potential in this capacity is demonstrated for the particular case of the decay channel $B^0_d \to \pi^+ \pi^-$ with the reconstructed Monte-Carlo data produced in 2004.