Check condition for choosing among the two startPos solutions

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Something seems to be inconsistent in the currently implemented choice among the two startPos solutions: Even when disabling all smearing by setting all applySmearing_XXX flags to False in smearing_cfi.py, I found a few events in the rho decay channel [1] in which the wrong startPos solution is chosen, because the transverse impact parameters have components perpendicular to the tau decay plane. This indicates that something is not consistent in the code that computes the component of the transverse impact parameter perpendicular to the tau decay plane.

My hypothesis right now is that there are some hidden assumptions in this code, which are valid for the pi, but not for the rho decay channel. This would explain why this problem is not seen in the pi decay channel.

I believe it is worth revisiting the choice among the two startPos solutions in general: for the a1 (three-prong) decay channel, one could base the choice of solution on the reconstructed tau decay vertex (e.g. by choosing the solution for which theta and phi of the tau four-vector are most compatible with the vector SV-PV).

[1] One such event is e.g. 1:1:37 in the file /local/karl/belle_eeToTauTau/aod/unwgt_pythia_extended/aodsim_1.root

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I have added a new feature to the code that alllows to determine how often we choose the wrong solution when resolving the two-fold sign ambiguity of the kinematic reconstruction. The result is that we get the sign wrong quite often! The following numbers refer how often we choose the correct sign. They depend on the tau decay channel:

Decay channel	Correct solution
pi_pi	84.9%
pi_rho	81.1%
pi_a1	69.6%
rho_rho	76.9%
rho_a1	69.2%
a1_a1	65.5%

The pi_pi decay channel exhibits the best performance.

The numbers were determined with the commit 5b8c9a9 for 100k events from the file /local/karl/belle_eeToTauTau/aod/unwgt_pythia_extended/aodsim_1.root with the setting startPosFinder.resolveSignAmbiguity = cms.string("TIP") and with smearing applied.

I found a problem in the code that chooses between solutions that affected rho and probably also a1 decays. This problem is already fixed in the numbers quoted above (the issue was that visTauP4 was used instead of leadTrackP4 when computing the plane on which to project the transverse impact parameters).

I asked Luca to read the paper arXiv:hep-ph/9307269 so that we can discuss potential pitfalls and improvements. Something in the code that chooses between solutions (now moved to a separate StartPosTIPCompatibility class) still does not quite seem to work as expected.

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I checked the numbers for the case that no smearing is applied:

Decay channel	Correct solution
pi_pi	100%
pi_rho	99.9%
pi_a1	99.3%
rho_rho	99.3%
rho_a1	99.9%
a1_a1	98.9%

Interestingly, even without smearing there is a similar dependence on the tau decay channel as in the case with smearing. The pi_pi decay channel exhibits the best performance in both cases.

Without smearing, the correct solution is taken with 99% probability when resolving the two-fold sign ambiguity of the kinematic reconstruction. This demonstrates that the wrong solutions result from the experimental resolution on the transverse impact parameters (for pi and who decays) and on the tau decay vertex (for a1 decays).

The correct solution is taken much more often with the "TIP" method than with the "kinFit_chi2" method (the following numbers are for the case that smearing is applied):

Decay channel	Correct solution
pi_pi	25.7%
pi_rho	41.6%
pi_a1	70.6%
rho_rho	62.5%
rho_a1	33.5%
a1_a1	61.9%