Impact on lifetime measurement

In general terms the effect of $\mu d$ diffusion on the observed muon disappearance rate can be estimated as follows. We define

$\begin{displaymath} n(t)=e^{-(\lambda _{0}+\Lambda _{d})t}(1+\int ^{t}_{0}\Lambda _{d} \: p(r,t')e^{\Lambda_d t'}dt')\end{displaymath}$

The time distribution is thus described by an analytical part and a small correction simulated by Monte Carlo. The data points of this time distribution were Monte Carlo scattered with the statistics corrsponding to 10 $^{10}$ events and then fitted with a pure exponential in the interval 0.-15 $\mu s$ . The dependence of the extracted decay constants are presented in table 9. The two extremes of the corrections are easy to understand. At r $\ll$ against the diffusion scale, all transferred muons are lost and the correction is simply $\Lambda _{d}$ . At r $\gg$ compared to the diffusion scale the losses approach zero. Obviously, the correction can be significant compared to our planned experimental precision of $\pm 6 s^{-1}$ .

Table 9: Estimated correction to the observed decay rate $\lambda$ due to diffusion of $\mu d$ atoms, which leave a sphere of radius r around the primary muon stop point at C

1 ppm

r (mm)	correction to $\lambda$ ( $s^{-1}$ )
250	0.06
150	0.2
100	4.
75	36
50	74
25	125
10	139

The real value of this correction depends on the experimental cuts used. For the analysis of local-PU free events, muon and electron must lie within a cylinder around the electron vector. As this is only a 2-dimensional cut, the corrections will be somewhat reduced, say to a 45 s $^{-1}$ correction for a 5 cm cut radius.

**Figure:** $\mu d$ diffusion Monte Carlo, $\mu d$ 's hitting materials (left), $\mu$ 's stopping in hydrogen gas (right). The rectangle of points in the middle of the left image are the initial positions before diffusion.
$\resizebox*{0.9\textwidth}{0.4\textheight}{\includegraphics{udwall.eps} }$

For the analysis of the global-PU free events one can allow a much larger diffusion radius. In this case the correction will be small and depends to which material the $\mu d$ atoms diffuse. This question was investigated with a Monte Carlo model, where $\mu d$ atoms were started at the muon stop position and their path followed to study whether it intersects with chamber materials. Fig. 23 (left) shows the final position of those $\mu d$ atoms which hit chamber materials before they decay (1.4%). The TPC frames, TPC wires and chamber walls show up clearly. The square of points in the middle is the initial position of each muon in the fiducial volume before diffusion. Note that this distribution is rather flat, having been accentuated near the edges which are closer to materials (the normal stopping distribution falls off steeply near the edges). Fig. 23 (right) shows the decay positions of the other 98.6% of $\mu d$ atoms which decay harmlessly in the chamber gas.

Relative to the total number of $\mu d$ atoms existing at a particular time, Fig. 24 shows the fraction of muons decaying within each material vs. time (this simulation did not include captures on each material, but simply whether the $\mu d$ reached the material). In those structures that we can cover with Kapton the lifetime would only be reduced by about 10% due to capture on atoms with Z=6-7. For muon stops in the wires the lifetime would be reduced to $\sim$ 70 nanoseconds on the gold or tungsten. In summary, for a data sample with c $_{d}=1 ppm$ and the global PU-free condition, a $100 ppm$ fraction of muons is transferred to $\mu d$ , out of those $100 ppm\times 0.014=1.4 ppm$ stop on chamber material, but only about $1.4 ppm\times 0.25=0.35 ppm$ reach dangerous high Z material, where the capture rate dominates over muon decay. Though we plan further simulation on this question, diffusion appears to introduce a negligible correction for the global-PU free data sample.

**Figure:** $\mu d$ diffusion Monte Carlo, $\mu d$ 's hitting various materials versus time.
$\resizebox*{0.6\textwidth}{0.5\textheight}{\includegraphics{udgraph.eps} }$