Simulations and Systematic Errors

We have made a preliminary study of a number of potential sources of systematic error in a measurement of the muon lifetime. We want to reduce the uncertainty on the muon lifetime to the 1 ppm level, which implies an absolute precision of about 2.2 ps per lifetime. The term absolute precision implies that the average measured times must be stable to a precision of 2.2 ps.

Any systematic problem with the experimental apparatus which causes a deviation from a pure exponential shape in the data can lead to an error in the measured muon lifetime. Deviations from one decay period to the next are generally much less dangerous than deviations within a given period. In particular, we must be vigilant for shifts in timing or pulse heights at early times (at the beginning of the measuring period, just after muon injection) compared to late times (at the end of the measuring period, many lifetimes after the muon injection) in a given cycle. Some of the more important error sources are:

1.

Gain and threshold instability;

2.

Pileup;

3.

Counting asymmetries due to spin precession;

4.

Rare decay modes;

5.

Cosmic rays and beam related backgrounds.

In the sections which follow, we examine each of these items in turn and make estimates based on simulations which indicate that these systematic errors are less than the proposed statistical uncertainty. The parameters given in table 4 are typical input values to the simulations. An overarching aspect of these studies is the distribution of positron energies and angles according to the Michel spectrum:

$$\frac{dP(y,\alpha)}{dy\,d\Omega} = n(y)\left[1+D(y)\cos\alpha \right]$$ (3)

with

n(y) = 2y²(3-2y), (4)

$$D(y) = \frac{2y-1}{3-2y}$$ (5)

for 100% polarized muons and

y = p_e / p_e,max, (6)

where p_e,max is the maximum momentum of the decay positron in the center of mass frame of the muon. The angle $\alpha$ is the angle between the momentum of the decay positron and the muon spin.

The rate of events at a given time is

$$N(t) = \frac{N_0}{\tau}e^{-t/\tau}$$ (7)

Table 4: Parameters used in the design of the experiment and their typical values. Symbols for selected variables as discussed in the next section.

Quantity	Symbol	Typical value
experimental goal ppm	$\delta\tau_{\mu}$	1 ppm
required number of events		1012
muon lifetime	$\tau$	2197.03 ns
accumulation period	Tacc	1 $\mu$s
measuring period	T	11 $\mu$s
$\mu$'s at T0	N0	12
geometrical acceptance		75%
detector segmentation	Fseg	180
average rate per tile		4.1 kHz
maximum rate per tile		22.7 kHz
time resolving power	$\Delta t$	10 ns
double-hit identification	Fdh	10
external magnetic field	B	75 G
spin precession frequency	$\omega$	1 MHz
fractional energy	y	0 - 1
detection threshold	$\xi$	0.04
spin modulation amplitude	D	$\frac{1}{3}$ - 1