Local pile-up free spectra

To demonstrate that we can also achieve a sufficiently low accidental rate with multiple muons in the chamber we have done some preliminary studies with the limited data set of our prototype setup. Fig. 16 (left) is a result of one such study. In this plot, for each muon electron pair which gives rise to a decay time, there is a second muon in the TPC simultaneously. Previously we showed that with global pileup protection we could reduce the accidental backgrounds to a level near $10^{-5}$ where the flat background distribution could easily be removed to give us better than $10^{-5}$ precision. Here, with a simpler definition of local pileup protection than we will use in the final run, we already show an accidental reduction from nearly 1:1 to $2\times10^{-4}$ (for now look only in the time regions of -100:-50 and 100:150). Note that this accidental rate is achieved when, for every good electron displayed there is a second muon decaying within the displayed time region! The background is of course higher than in global pileup and has a more complicated shape which we will discuss next.

Figure 16: Background rejection when 2 muons are in the TPC. Time distribution and accidental background (left), One component of the accidental supression (right). 1 channel corresponds to 200 ns.

By rejecting secondary muons which are both spatially and temporally close to the primary muon we create a distortion in the time distribution of the accidental background. An example of such a reduction is shown in Fig. 16(right). Here we show the result of completely rejecting secondary muons which cross the primary muon track in the y-z plane of the TPC. We intend to resolve most crossed tracks but this cut gives a clear example of the effect and also shows how we can deal with the backgrounds in the case we decide to reject all crossed tracks. In the figure we can see the complete rejection for identical start times with the rejection tapering off as the angular range of tracks no longer allows them to touch. Note that this plot is from real data and we get an entry from every pair of muons giving a high precision plot of the rejection function. The actual background shape is a convolution of the rejection function for the specific cuts of the analysis and the normal muon decay exponential. An example of this rejection appears in Fig. 16 (left) in the time region -40:0. The events between -150:-110 are the exponential tail from events outside the cut region at -150 channels (-30 $\mu s$) and are a normal part of the background fit.

A complete fit of the background under Fig. 16 (left) would include: the main decay curve, a flat background multiplied by the convolved rejection function and the decay curve of muons arriving earlier than -150 channels. Look at Fig. 16 (right) only as an indication of how far the rejection correction extends under the main decay curve. Previously we demonstrated that for a flat background, we can easily fit a $10^{-4}$ accidental rate and still achieve $10^{-5}$ accuracy in the main decay rate. If we have the correct rejection function for the local pileup case the same applies, we could obtain a 10$^{-5}$ result from the present $2\times10^{-4}$ background shown in Fig. 16 (left). The only caution would be that the rejection function must be stored along with the primary spectrum to insure identical cuts on the secondary muons.