Two event correlations in local pile-up free analysis

For an incident muon flux which will achieve $10^{10}$ events in a couple of weeks, i.e. 35 kHz, we will have to deal with an average multiplicity of 2 muons at a time in the TPC. In order to reduce the losses inherent to the global pileup condition (rejecting any other muons in the TPC within $\pm15 \mu s$) we will alternatively use a ``Local Pileup'' protection for the same $\pm15 \mu s$. This local pileup condition consists primarily of guaranteeing that no other muon stops within a particular radius of the muon of interest, or crosses through this same volume. Additionally, when an electron track is associated with the muon of interest one needs to guarantee that this electron could not have come from the other muon, which is equivalent to requiring that the second muon not stop anywhere in a cylinder defined by the electron vector and the local pileup protection distance as radius.

We showed in Section 4.2.2 and Fig. 16 how we would model the accidental acceptance in the local pilup studies and be able to remove a $10^{-4}$ accidental rate and still achieve $10^{-5}$ accuracy. However, we do not want to rely on removing a large background to get our final result and the situation is actually much better. First, we have not made use of all possible cuts in Fig. 16 (left) and we reasonably expect at least a factor of 2 improvement when all cuts are refined. But more importantly, in our prototype test run, the electron acceptance was very tiny, something like 5%. With this small acceptance we were lucky already to get the electron from the primary muon track. To ask also to track the electron from the secondary muon would kill our statistics to the point that we could not even demonstrate the background study presented above. Therefore in the figures presented, we allowed secondary muons in the TPC but did not require that we find the secondary electron pointing at the correct muon vertex. This blindness allows diffused muons and mistracked electrons from the secondary muon to be accepted more readily as coming from the primary muon. With the 75% electron acceptance of the final setup we can put in the strict requirement that we must have found the decay electron for each secondary muon allowed in the TPC. This procedure should reduce the background in Fig. 16 (left) by an order of magnitude, making the high rate local pileup data as viable as the slower global pileup data, only the analysis will be slightly more complicated.

The other two event correlation that could cause a time distortion is that of the so-called "double kill", where the second muon affects the detection of the first muon and simultaneously the second electron affects the detection of the first electron. Due to the high granularity and high time resolution of the external detectors, the probability of the electron from a secondary muon hitting the same electron detectors withing the same time bin is already down at the $10^{-5}$ level. This however does not represent a $10^{-5}$ distortion but only a $10^{-5}$ change in overall efficiency. The distorting effect of this type of external pileup also contains the probability that the two muon tracks interfered with each other in the TPC. That reduces the possible distortion to below the $10^{-6}$ level. We will make a study of this effect in the final analysis but do not expect it to be significant.