Modeling the Probability for Proton Decay

The experimental search for proton decay was undertaken because of the implications of the grand unification theories. The lower bound for the lifetime is now projected to be on the order of τ = 10³³ years. The probability for observing a proton decay can be estimated from the nature of particle decay and the application of Poisson statistics.

寻找质子衰变的实验研究是由于大统一理论的 implications。目前对质子寿命的下限估计约为 τ = 10³³ 年。观察到质子衰变的概率可以从粒子衰变的性质和泊松统计的应用来估计。

质子数N可以由衰变方程进行建模

where λ = 1/t = 10^-33/ year is the probability that any given proton will decay in a year. Since the decay constant λ is so small, the exponential can be represented by the first two terms of the exponential series.

其中 λ = 1/t = 10⁻³³ / 年是单个质子在一年内衰变的概率。由于衰变常数 λ 非常小，指数可以表示为指数级数的前两项。

For a small sample, the observation of a proton decay is infinitesmal, but suppose we consider the volume of protons represented by the Super Kameokande neutrino detector in Japan. The number of protons in the detector volume is reported by Ed Kearns of Boston University to be 7.5 x 10³³ protons. For one year of observation, the number of expected proton decays is then

对于一个小样本来说，观察到质子衰变是微不足道的，但假设我们考虑日本超神冈中微子探测器所代表的质子体积。据波士顿大学的埃德·凯恩斯报告，探测器体积中的质子数为7.5 x 10 33个。对于一年的观测，预期的质子衰变数目为

About 40% of the area around the detector tank is covered by photodetector tubes, and if we take that to be the nominal efficiency of detection, we expect about three observations of proton decay events per year based on a 10³³ year lifetime.

探测器水箱周围约40%的区域被光电探测器管覆盖，若将此视为探测效率，基于质子寿命为10³³年的数据，我们预计每年能观测到约三次质子衰变事件。

So far, no convincing proton decay events have been seen. Poisson statistics provides a convenient means for assessing the implications of the absence of these observations. If we presume that a = 3 observed decays per year is the mean, then the Poisson distribution function tells us that the probability for zero observations of a decay is

到目前为止，尚未观察到有说服力的质子衰变事件。泊松统计提供了一种方便的方法，用于评估这些观测结果的含义。如果我们假设每年观察到3次衰变是平均值，那么泊松分布函数告诉我们零次衰变观测的概率是

This low probability for a null result suggests that the proposed lifetime of 10³³ years is too short. While this is not a realistic assessment of the probability of observations because there are a number of possible pathways for decay, it serves to illustrate in principle how even a non-observation can be used to refine a proposed lifetime.

这种获得零结果的概率较低，表明所提出的寿命为10³³年可能过短。虽然这并不是对观测概率的现实评估，因为衰变存在多种可能的途径，但它说明了原则上即使没有观测到衰变，也可以用来修正所提出的寿命。