Least Squares fits to K+ ->  mu+ nu
-----------------------------------

These are least squares fits  to the beam and spectrometer tracks subject to the constraints 
that the  nu mass and the cda  are zero.
The fits were made using Blobel's APLCON for 6 parameters
subject to  2 constraints.  See   APLCON
and   talk with examples of use .

The fit parameters were momentum,  dx/dz,dy/dz for the K+ and mu+.
The covariance matrix was taken to be diagonal with error estimates  from
arXiv:1703.0850.

The object was to examine the chi**2 and pulls of the fit, and to determine  
the influence of the fit on measurement accuracy.

Fig. 1 ( 12 plots ) Plot 1 shows the missing mass distributions before and after fitting. Note the non-gaussian tails. Plot 2 shows the cda before and after fitting. Plot 3 shows the chi**2 p-distribution together with the chi**2 distribution. There is a long tail to the chi**2 distribution that results in the the low probability peak; the size of the peak suggests that ~ 8% of events are badly measured. The probabilty distribution has been flattened by multiplying the first estimate of the covariance matrix by 0.7 . Plots 4,5 explore the effect of cuts on probability, cda and chisq on the missing mass distribution. The probability and chisq cuts remove the non-gaussian tails. The CDA cut has no effect on the tails. Plot 6: This shows the ratio of fitted error to measurement error for the six fitted variables. The two-constraint fit improves the track momentum measurement accuracy by ~ 10% and the angle measurement by about a factor two. The beam variables are less influenced by the fit. Plots 7-12 show the pulls of the 6 variables. After the covariance matrix correction, all pulls have an rms close to unity; there is no evidence for systematic errors in any of the measured variables. Non-gaussian tails are present for all variables. Notice that the pulls in the momentum are identical for the track and beam; this is not understood. The pulls of the angles differ however, as can be seen on an event-by-event basis. To check the effect of systematic errors in momentum and angles, the input track momentum has been increased by 0.5 standard deviations and dx/dz (dy/dz) have been increased (decreased) by 0.5 standard deviations.

Fig. 2 ( 12 plots ) shows the fit results. Plots 7,8 ,9 (10) show that the pulls of the changed variables are displaced by -0.29, -0.62, +0.51,( -0.29 ) from zero, respectively, so demonstrating the sensitivity of the fit to systematic errors. ~/aplcon ./clg6mmcda.sh > clg6mmcda.out3 readfile6mmcda.F readfile6mmcda.kumac