Effects of RG running in breaking $\mu-\tau$ reflection symmetry, conventional versus minimal seesaw

Neutrinos are the most abundant massive particles in the universe, yet they remain the most elusive. These ghostly particles zip through everything, from the core of the sun to the center of the Earth, rarely interacting with ordinary matter. For decades, physicists have been trying to understand why they have mass and how they change their identity as they travel. This phenomenon, known as oscillation, occurs because neutrinos come in three distinct "flavors"—electron, muon, and tau—and can shift from one to another. The way these flavors mix is described by a mathematical map called the PMNS matrix. Recent experiments have revealed a striking pattern in this mixing: the muon and tau flavors seem to behave almost identically, and the angle describing their mixing appears to be exactly halfway between two extremes. This observation has led scientists to propose a specific rule governing nature, called mu-tau reflection symmetry. This rule predicts that the mixing should be perfect and that a specific property called the CP-violating phase, which distinguishes matter from antimatter, should take on a precise value. However, the real world is rarely perfect. Current measurements show small but significant deviations from these ideal predictions, suggesting that the symmetry is broken. The big question is: why is it broken? Is it because the symmetry was never perfect to begin with, or because something happened to it as the universe cooled down after the Big Bang?

A team of researchers at Dibrugarh University in India has taken a fresh look at this problem by simulating how these neutrino properties evolve over time. They focused on a scenario where the symmetry was perfect at the very high energy levels that existed in the early universe, and then broke down as the energy dropped to the levels we see today. To do this, they used a theoretical tool called the renormalization group, which acts like a time machine for physics equations, allowing scientists to calculate how particle properties change as the energy scale shifts from the birth of the universe to the present day. The researchers compared two different theories about how neutrinos get their mass. The first is the standard "Type-I seesaw" model, which assumes three heavy, invisible particles exist to generate the tiny masses of the light neutrinos we observe. The second is the "minimal seesaw" model, a more economical version that assumes only two of these heavy particles exist. In this simpler model, the math dictates that one of the three light neutrinos must have zero mass, a feature that drastically reduces the number of free variables the scientists have to work with.

The researchers set out to see if this stripped-down, minimal model could still reproduce the messy, imperfect data we measure in laboratories today. They started their simulation at a high-energy scale of 10 to the power of 14 GeV, a point where the mu-tau reflection symmetry was assumed to be exact. From there, they let the equations run down to the energy scale of the top quark, which is roughly 172 GeV, mimicking the cooling of the universe. They tested two different scenarios for the CP-violating phase and two different arrangements of neutrino masses, known as normal and inverted ordering. In every case, they found that the minimal model worked. Even with one neutrino forced to have zero mass and the symmetry starting out perfect, the natural evolution of the equations successfully generated the small deviations seen in real-world experiments. The simulation showed that as the energy scale dropped, quantum corrections gradually nudged the mixing angles and phases away from their perfect values, resulting in the specific numbers observed by experiments like T2K and NOvA. This confirmed that a simpler universe with fewer heavy particles is still capable of explaining the complex neutrino behavior we see today.

Having established that the minimal model works, the team then asked a deeper question: does this simpler universe behave differently from the standard one as it evolves? They compared the running of the mass differences between the two models to see if the reduction in heavy particles left a detectable fingerprint. They found that the answer depends heavily on which mass difference is being looked at. The difference in mass between the first two neutrino types, known as the solar mass-squared difference, showed a clear sensitivity to the underlying model. In many of their simulations, the path taken by this value in the minimal model diverged noticeably from the path taken in the standard model, especially when the energy was high. This suggests that if we could measure this value with extreme precision at very high energies, we might be able to tell whether nature chose the three-particle or two-particle version of the seesaw mechanism.

In contrast, the difference in mass involving the third neutrino, the atmospheric mass-squared difference, proved to be much more stubborn. In almost every scenario the researchers tested, this value evolved in nearly the same way regardless of whether the universe had two or three heavy particles. The two models produced almost identical results for this specific measurement, making it a poor tool for distinguishing between the two theories. The researchers also discovered that the choice of the CP-violating phase at the start of the simulation mattered. In one configuration, the differences between the two models were quite large, while in another, they were so small they were almost negligible. Furthermore, the size of the difference between the models often shrank as the assumed scale of supersymmetry breaking increased, meaning that at higher energy thresholds, the two theories began to look more alike.

The study concludes that while the minimal seesaw model is a perfectly viable explanation for neutrino data, it is not entirely indistinguishable from the standard model. The radiative breaking of the symmetry leaves a subtle but measurable trace, particularly in the solar mass difference. This finding is significant because it suggests that the history of how neutrino masses evolved carries information about the fundamental structure of the universe. Even though the minimal model uses fewer ingredients, it does not simply mimic the standard model; it follows a slightly different trajectory through the energy scales. This implies that future, more precise measurements of neutrino properties could potentially reveal whether nature prefers the economical two-particle solution or the more complex three-particle version, offering a rare glimpse into the high-energy physics that shaped the early universe.