One GUT theory predicts that a proton will decay in about \(10^{31}\) years, which means if you have \(10^{\text {si }}\) protons, you should see one decay per year. The Super-Kamiokande observatory in Japan holds about 20 million kg of water in its main detector, and it did not see any decays in 5 years of continuous operation. What limit does this observation place on proton decay and on the GUT theory described here?

The Grand Unified Theory (GUT) is a theoretical framework in particle physics. The main goal of GUTs is to combine the three fundamental forces of the Standard Model—electromagnetic, weak nuclear, and strong nuclear forces—into a single, unified force. Theories like this are very complex and scientists are still working to prove them.
One important prediction made by some GUTs is that protons, which are stable according to current observations, can actually decay over a very long time. This hypothesis suggests that everything in the universe would eventually disintegrate, releasing energy in the process.
GUT is crucial because it tries to explain the unity of nature on an elementary level. This would bring us closer to understanding the Big Bang and even answer questions about dark matter and dark energy.

The Super-Kamiokande Observatory is a major research facility in Japan. It's famous for studying neutrinos and searching for proton decay. The observatory is located deep underground to protect it from background radiation from space. This makes it an ideal place for detecting extremely rare events like proton decay.
The observatory consists of 20 million kilograms of ultra-pure water held within a massive tank. Around this tank, there are thousands of ultra-sensitive detectors. These detectors capture the Cherenkov radiation—light emitted when a particle moves faster than the speed of light in water—produced by potential proton decays.
The Super-Kamiokande plays a key role in testing theories of physics that cannot be experimented with on a smaller scale. Its huge water volume increases the chances of catching rare phenomena, making it an invaluable tool in the quest to understand the universe at its most fundamental level.

Proton lifetime refers to the predicted duration over which a proton can exist before decaying, according to certain theoretical models. Current experimental observations have yet to see a proton decay, indicating that they have an extremely long lifetime.
Some GUTs predict that a proton’s lifetime could be around \(10^{31}\) years. However, observations from the Super-Kamiokande, which have not detected any proton decays over five years, suggest that the proton lifetime might be even longer.
In the example exercise, we calculated the number of expected proton decays based on the GUT prediction and the number of protons in the observatory. The lack of observed decays allowed us to set a new lower limit on the proton lifetime: at least \(6.7 \times 10^{33}\) years. This finding helps refine our theoretical models and pushes the boundaries of our understanding of particle physics.
Understanding proton lifetime can provide critical insights into the early universe and the fundamental makeup of matter itself. It's a cornerstone question for both particle physicists and cosmologists.