Ideas from 23 June 2008 presentation:

• Aim:
  • test data/MC agreement on longitudinal profile characteristics
  • Examine parametrisation of ECAL showers (see e.g. "Fluctuations in Calorimetry Measurements",U.Amaldi 1981 Phys. Scr. 23 409-424, doi:10.1088/0031-8949/23/4A/012)

Need points in data/MC distributions of longitudinal profiles to be at depths corresponding to material upstream. To extract more precise integrated number of radiation lengths in front of each Si layer from Mokka and use these.

• Plot of longitudinal profiles, data (points/no horizontal error bars) and MC for representative set of energies in range 6 ... 45 GeV
  • Quantities of interest to compare data/MC (without fits), e.g.
    • median depth of shower (depth at which 50% of incident energy is deposited) - can also be related to shower max Comment: median is not needed any more because event by event analysis is incorrect due to sampling fraction fluctuations. Errors on event by event basis are shown in the attachment samplingfraction.gif for 30GeV. A MC sample which has both the energy depositions in silicon and in tungsten is used to quantify the sampling fraction fluctuations. These fluctuations are high. However there is still a bug in the MOKKA extension as can be seen in the total energy reconstructed. I am currently discussing with Gabriel Musat about the problem with the Mokka extension. For the moment the error estimate with the help of event by event analysis is abandoned. Instead the errors are estimated by dividing the sample into subsamples and fitting every subsample.

• • • ...
  • Also include on plot an example of fit to data, using parametrisation of form E(X0) = c (X0 - f )^ alpha . exp(- beta (X0 - f )); fix beta to 0.5 based on Amaldi paper (and fitted values all ~0.5 when free). Comments: problem when comparing data and MC in shower profile is reoccuring, why? Need to investigate.

• Study leakage energy, using parametrisation above, to give the fraction of shower energy which escapes beyond last ECAL layer:
  • (integral of parametrised fit to infinite depth)/(integral to max. calorimeter depth) vs. incident beam energy

• Three tests to demonstrate that parametrisation is well-behaved/stable before it can be used for leakage energy.
  • Peak position
    • Plot (pg 4 of 23 June 2008 presentation) of shower max. vs. beam energy.

• • • Include data and MC on same plot. Check data result for 30 GeV runs. Comment: done, MC is in red, data in black

• • Upstream material
    • Use fit to data of to extract f, the apparent material upstream of ECAL, should be constant for all beam energies. May not want to show this in paper, but useful as a control of the stability of the fit, gives confidence that the leakages inferred from the fit is reliable.

• • • Plot this for data and MC, compare results. As material upstream of ECAL is known in MC (value to obtain), include this on plot for comparison with f_MC.

• • Accuracy of integral in measured region
    • Verify (but not include in paper) that integral of fit to max. calorimeter layer == sum of energy in all layers
    • Plot (as control, not for paper) difference between these two as function of beam energy (done already)

• Verifying errors on fits: where there is concern about the fitted parameter uncertainties (e.g. f) suggest dividing the data into subsamples, e.g. 100 events, performing fits to these samples, and examining the distribution of a parameter, the rms of the distribution will be consistent with the original fit uncertainty from the entire sample if the fit uncertainties are correct. Done: examples shown for 45GeV run

• Plot of leakage energy vs. incident beam energy, for data, MC (using same fit/parametrisation method as data) and for MC (using MC truth to determine energy escaping last ECAL layer). Comment: need to figure out bug in Mokka extension in order to plot the MC truth

• Conclusions about data/MC agreement, and on how applicable the parametrisation is for our ECAL. Comment: looks good, slight difference in shower max