Cosmic Rays and Beam Related Backgrounds

A potential source of systematic error in the present design is an unaccounted for flat background which causes a shift in the measured lifetime. Representative flat backgrounds originate from cosmic radiation and from muons which sneak into the detector after the measurement time has begun. Both will cause a constant rate $\rho$ in our detector. If unaccounted for in the analysis, this modifies the lifetime by a factor $1+\rho\tau/N_{0}$ (to first order for small times). The relative importance of a flat background grows exponentially with the measurement period T.

We consider the following example for a detector with a 28 cm radius and using the parameters defined in table 4. Fitting the spectrum to the simple function $N(t)=N_{0}e^{-t/\tau}$ then has a $\chi^2$ of 1.11 and a decrease of $\tau$ by 46 ppm. When a constant background is added to the fitting function, the $\chi^2$ reduces to an acceptable value of 0.998. The shift in $\tau$ is now reduced to 2.6 ppm, of which 1.5 ppm is due to the presence of cosmic ray events. Furthermore, the statistical error in the fitted lifetime is increased from 1 ppm to 1.7 ppm because of the correlation between the fitting parameters.

A similar effect may occur if the extinction in the muon channel is incomplete coupled with any inefficiency in the beam-defining muon counters upstream of the apparatus. With the muon-defining counter at the end of the channel, we will measure the flux of unwanted muons to a very high accuracy. When such a muon is registered by the counter, the entire fill will be discarded. For the case when a muon goes through the muon-defining counter and does not fire the counter, then we have the source of flat background being considered. For practical reasons, it is highly desirable to have the extinction factor of the channel and the efficiency of the muon-defining counter very high. Nevertheless, either of these flat background sources can be eliminated by making measurements of the background to a precision of 1%. This can be made by simply doubling the muon measuring period to nearly 10 lifetimes for part of the running time. At a rate of 20 Hz of such backgrounds, such a measurement would take less than 10 minutes.