When a substance is fragmented into smaller and smaller segments, eventually, you reach a point where division is no longer possible. At this point, you are left with a building block. To date, 12 distinct elementary particles have been identified, composed of a mixture of quarks and leptons, each available in six unique variants.

These variants are classified into three generations. Each generation contains one lepton that carries a charge and one that is neutral, forming various particles such as electrons, muons, and tau neutrinos. In the Standard Model, the masses of the three neutrino generations are represented by a three by three matrix.

A research team led by Professor Naoyuki Haba of the Graduate School of Science at Osaka Metropolitan University analyzed the collection of leptons that make up the neutrino mass matrix. Neutrinos are known to have less difference in mass between generations than other elementary particles, so the research team assumed that neutrinos have roughly the same mass between generations. They analyzed the neutrino mass matrix by randomly assigning each element of the matrix. They demonstrated theoretically, using the random mass matrix model, that leptonic aroma mixtures are large.

Clarifying the properties of elementary particles leads to the exploration of the universe and ultimately to the great topic of our origins! explained Professor Haba. Beyond the remaining mysteries of the Standard Model is a whole new world of physics.

After studying neutrino mass anarchy in the Dirac neutrino, seesaw, double seesaw models, the researchers found that the anarchy approach requires that the matrix measurement obey the Gaussian distribution. After considering several light neutrino mass models in which the matrix is composed of the product of several random matrices, the research team was able to demonstrate, as best it could at this stage, why the calculation of the squared difference of the masses of the neutrino is the closest with the experimental results in the case of the see-saw model with the Dirac and Majorana random matrices.

In this study, we have shown that the neutrino mass hierarchy can be explained mathematically using random matrix theory. However, this proof is not mathematically complete and is expected to be rigorously proved as random matrix theory continues to develop, said Professor Haba. In the future, we will continue with our challenge of elucidating the copy structure of three generations of elementary particles, whose essential nature is still completely unknown both theoretically and experimentally.

Reference: Neutrino mass square ratio and neutrinoless double beta decay in random neutrino mass matrices by Naoyuki Haba, Yasuhiro Shimizu, and Toshifumi Yamada, January 19, 2023, *Progress of theoretical and experimental physics*.

DOI: 10.1093/ptep/ptad010

The study was funded by the Ministry of Education, Culture, Sports, Science and Technology of Japan.

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