Initial Tests Of The Pion Beta Decay Detector - 5.4.4 Determination of the gate timing correction f-1

5.4.4 Determination of the gate timing correction f-1

The timing correction f^-1 accounts for differences in the pion and muon lifetimes as well as the difference in the physical processes. The difference in the physical processes arises from the fact that the p -> µ n ( g ) events are only seen when the muon decays (not when the pion decays). The Michel background is thus created by a two component process: the p -> µ n ( g ) decay and the subsequent µ -> e n n decay. The p -> e[nu]( g ) process, however, has only a single component. The effect of the DPG (see section 3.4) was to veto events in which a positron was emitted from the target within the first 10 ns of the pion stopping. This means no p -> e n ( g ) events were seen where the pion decayed during this time. In contrast, pions which decayed into muons within this initial 10 ns period were recorded provided the muon lived until after the 10 ns boundary.

The different lifetimes of the pion and muon significantly affected f^-1 as well. This becomes intuitively obvious when one considers that most pions (=26 ns) will decay within the 85 ns between the pion stopping and the end of the DPG while only a fraction of muons (=2.197 ) will decay in this same time period. The value f^-1 thus represents the ratio of positrons from p -> e n ( g ) decays to positrons from the decay chain which are created during the DPG.

The number of positrons from p -> e n ( g ) decays can be written

where is the total number of pions stopped in the target, is the efficiency of the detector (including solid angle, computer dead time, etc...), is the p -> e n ( g ) branching ratio, is the amount of delay after the pion is stopped in the target before the DPG opens, and is the width of the DPG.

The number of positrons from p -> µ n ( g ) decays can be written

where ,,,, and are defined as above. Pion decay channels other than p -> µ n ( g ) and p -> e n ( g ) have been neglected because they a smaller by several orders of magnitude. Evaluating the integrals, reduces to

where

Taking the ratio of to yields

5.1

where

with . Solving equation 5.1 for gives

where the approximation follows from the condition <<1.

Values of and were recorded using a digital oscilloscope at the time the experiment was performed. Offline, a value for was extracted from the data by comparing the timing spectra for prompt trigger data and prompt events in the DPG. Figure 5.12 shows plots of these spectra. Gaussian functions were fit each of these histograms and the difference between the means was ns. The widths of the Gaussian functions are due primarily to detector resolution and are therefore used to determine the uncertainty in both and . The standard deviations of the fits to the prompt trigger and DPG trigger timing spectra are 0.72 ns and 0.81 ns respectively leading to an uncertainty of 1.1 ns which is used for both and .

The value of is sensitive to the values of and as demonstrated in Table 5.2. The current experiment used a DPG with ns and ns corresponding to .