5.4 Null Experiments. Setting Confidence Limits When No Counts Are Observed

Many experiments in physics test the validity of certain theoretical conservation laws by searching for the presence of specific reactions or decays forbidden by these laws. In such measurements, an observation is made for a certain amount of time T. Obviously, if one or more events are observed, the theoretical law is disproven. However, if no events are observed, the converse cannot be said to be true. Instead a limit on the life-time of the reaction or decay is set.

Let us assume therefore that the process has some mean reaction rate . Then the probability for observing no counts in a time period T is

(57)

This, now, can also be interpreted as the probability distribution for when no counts are observed in a period T. We can now ask the question: What is the probability that is less ₀? From (1),

(58)

where we have normalized (57) with the extra factor T. This probability is known as the confidence level for the interval between 0 to ₀. To make a strong statement we can choose a high confidence level (CL), for example, 90%. Setting (58) equal to this probability then gives us the value of ₀,

(59)

For a given confidence level, the corresponding interval is, in general, not unique and one can find other intervals which yield the same integral probability. For example, it might be possible to integrate (57) from some lower limit ' to infinity and still obtain the same area under the curve. The probability that the true is greater than ' is then also 90%. As a general rule, however, one should take those limits which cover the smallest range in .

Example 4. A 50 g sample of ⁸²Se is observed for 100 days for neutrinoless double beta decay, a reaction normally forbidden by lepton conservation. However, current theories suggest that this might occur. The apparatus has a detection efficiency of 20%. No events with the correct signature for this decay are observed. Set an upper limit on the lifetime for this decay mode.

Choosing a confidence limit of 90%, (59) yields

₀ = -1 / (100 x 0.2) ln (1 - 0.9) = 0.115 day^-1,

where we have corrected for the 20% efficiency of the detector. This limit must now be translated into a lifetime per nucleus. For 50 g, the total number of nuclei is

N = (N_a / 82) x 50 = 3.67 x 10²³,

which implies a limit on the decay rate per nucleus of

0.115 / (3.67 x 10²³) = 3.13 x 10^-25 day^-1.

The lifetime is just the inverse of which yields

8.75 x 10²¹ years   90% CL,

where we have converted the units to years. Thus, neutrinoless double beta decay may exist but it is certainly a rare process!