Pileup

With our segmented detector, there will on average be much less than one muon decay recorded in any one channel per measuring period, but two or more may occasionally come close together in time and this will happen far more often at the beginning of the interval than at the end. This situation defines ``pileup.'' When unaccounted for, it introduces a time-dependence in the counting efficiency which directly affects the measured lifetime.

The probability p(t) that a single muon decays per unit of time after a time t decreases exponentially in time,

$$p(t) = \frac{1}{\tau}e^{-t/\tau}$$ (9)

Therefore, the probability to detect k events in a time span $\Delta t$ originating from a source containing N muons (with N small) is governed by the Binomial distribution

$$B_{p(t),\Delta t}(k;N) = \frac{N!}{k!(N-k)!} (p(t)\Delta t)^k (1-p(t)\Delta t)^{N-k}.$$ (10)

Here it is tacitly assumed that $\Delta t$ is small enough so that $\int\limits_{\Delta t} p(t)dt = p(t)\Delta t$. In the case that multiple events during $\Delta t$ cannot be distinguished by the experimental apparatus, only one event is counted instead of k. The average number of events counted in a time window $\Delta t$ at time t, $\mu(t)$, must be modified from $\mu(t) = \sum k B(k;N) = N p(t) \Delta t$ to

$$\begin{array}{rl} \mu(t) & = \sum\limits_{k=1}^{\infty} B(... ...(N-1)\Delta t}{2\tau} e^{-t/\tau} + \dots \right], \end{array}$$ (11)

i.e a deviation from the pure exponential shape is introduced. To reduce this deviation, the event separating power can be improved trivially by segmenting the detector. Additionally, event separating power can be obtained from information on the signal amplitudes and the pulse shapes from the detector elements. The magnitude of the deviation, i.e. the second term between square brackets, is then reduced by a factor F_segF_dh. The factor F_seg is equal to 180 and represents the number of tile segments. The factor F_dh quantifies the double-hit identification power and is based on pulse height discrimination in the two scintillator layers. We have performed GEANT simulations of the tile system using positrons with energies following the Michel spectrum. For positrons above the threshold of approximately 2 MeV, the energy deposition profile corresponds to a typical MIP from the cosmic ray tests. The simulation has been used to estimate the probability that a single particle yields a light level in both scintillator layers corresponding to 2 MIPs. This probability is found to 0.5%, corresponding to a value for F_dh in the above definition of 200. We have used the very conservative value of F_dh=10 instead to account for possible complications due to photostatistics, edge effects, and other resolution dampening processes. Figure 11 is a plot of the energy deposited in the inner versus the outer scintillators for one and two positrons passing through a tile during the same, unresolved, time period. It is very clear that a comfortable separation exists between one and two MIPs when using this two-dimensional representation as a basis for a cut.

Figure 11: Plot of the energy deposited in the inner versus the outer scintillators for one and two positrons passing through a tile during the same, unresolved, time period. This GEANT simulation shows that it is straightforward to use a two-dimensional cut to identify double hits.

To first order, the systematic shift in the measured lifetime is $\Delta\tau/\tau = \frac{(N-1)\Delta t}{2F_{seg}F_{dh}\tau}$, which is reduced by a factor $1+T/2\tau$ when counts are collected for a measuring period T. For the design parameters listed in table 4, a shift of 3 ppm is expected. However, the parameters determining this shift only need to be known with a combined accuracy of 30% in order to correct for the shift, which can be done with relative ease by modeling the level of unobserved pileup. Pileup is therefore not expected to limit the accuracy of the measured lifetime.

Apart from miscounting the number of events, pileup also introduces a time shift, of which the magnitude is determined by the procedure used to collect and analyze the data. In a setup in which the decay time is derived from the time at which the analog signal exceeds a certain threshold, the unavoidable walk can cause a substantial shift in time. By using waveform digitizers, this problem is largely eliminated. In 2, we faced and solved a similar problem, and at a much higher instantaneous rate on the detectors.

Assuming that the detection efficiency does not change during a small time interval $\Delta t$, the time associated to a pileup event which has individually unresolved hits is just the average time of both. As a result, the distribution of double hit events is shifted by about 15 ps with respect to the distribution of decay times. The effect of this shift is negligible compared to the effect introduced by the fact that this distribution has a different shape.