Proton decay is a hypothetical form of particle decay in which the proton decays into lighter subatomic particles. One possible decay mode, common to many models (including Grand unified Theories and String Theories), is into a positron and a neutral pion: p → e⁺ + π⁰. The masses of these particles, expressed in units of MeV/c² where c is the speed of light in vacuo, are: mp=938 MeV/c², me=0.511 MeV/c² and mπ=135 MeV/c². As with radioactive decay the rate of decay is presumed to be proportional to the number of undecayed protons in a given sample.
(a) Show that N=N0e−λt, where N is the number of undecayed protons at time t, N0 is the number at time t = 0 and λ is a constant called the decay constant. [2]
(b) By considering the number of protons, dN, that decay in time dt:
(i) Show that the probability, dP for a proton to decay in this time can be expressed as dP=λe−λtdt. [2]
(ii) Show that all the probabilities (from t = 0 to t = ∞) sum to 1 as expected. [1]
(iii) Show that the probability that a proton remains undecayed after time t is given by p(t)=e−λt. [2]
(iv) Show that the mean time taken for a proton to decay, τ, known as the mean lifetime of the proton, is given by τ = 1/λ.
Hint: The mean of a quantity, f(t), is given by
⟨f(t)⟩=∫0∞p(t)dt∫0∞f(t)p(t)dt
You may assume, without proof, the standard integral
∫0∞xe−xdx=1
[2]
(c) The average heat outflux through the surface of the Earth is Q = 92 mWm⁻².
(i) Assuming that this outflux is produced entirely by proton decay, estimate a lower bound for the proton's mean lifetime. [4]
(ii) Why is this a considerable underestimate? [1]
(d) The Super-Kamiokande detector in Japan is currently the largest detector for the observation of proton decay. It consists of a tank containing 50,000 tons of ultrapure water with about 13,000 photomultiplier tubes, covering about 40% of the area around the tank that detect Cherenkov radiation produced by the movement of charged particles through the water.
(i) Assuming the detectors are 100% efficient, how long would you have to wait, on average, to observe one proton decay. [4]
(ii) How many decays would you expect to observe per year, on average? [2]
(iii) The detection process for such decays can be modelled using Poisson statistics where the probability of observing k decays per year, p(k), is given by
p(k)=k!μke−μ
with μ the average number of decays per year. So far, no convincing proton decays have been observed. What would be the probability of not observing any decays in a year and what would this mean for the lifetime you calculated in (c)(i)? [3]
(e) Assuming that any proton decay occurs via the process p → e⁺ + π⁰, with the proton decaying while at rest, determine the energies and speeds of the positron and neutral pion produced. Hint: The energy, E, of a particle of rest mass m and momentum p (velocity v) is given by E2=p2c2+m2c4 with p=γmv and γ=1−v2/c21. [3]