The Koide formula is an unexplained empirical equation discovered by Yoshio Koide in 1981. In its original form, it is not fully empirical but a set of guesses for a model for masses of quarks and leptons, as well as CKM angles. From this model it survives the observation about the masses of the three charged leptons; later authors have extended the relation to neutrinos, quarks, and other families of particles. The Koide formula is where the masses of the electron, muon, and tau are measured respectively as m_e = 0.510998946MeV/c2, m_μ = 105.6583745MeV/c2, and m_τ = 1776.86MeV/c2; the digits in parentheses are the uncertainties in the last digits. This gives Q = 0.666661. No matter what masses are chosen to stand in place of the electron, muon, and tau, 1/3 ≤ Q < 1 . The upper bound follows from the fact that the square roots are necessarily positive, and the lower bound follows from the Cauchy–Bunyakovsky–Schwarz inequality. The experimentally determined value, 2/3, lies at the center of the mathematically allowed range. But note that removing the requirement of positive roots it is possible to fit an extra tuple in the quark sector (the one with strange, charm and bottom). The mystery is in the physical value. Not only is the result peculiar, in that three ostensibly arbitrary numbers give a simple fraction, but also in that in the case of electron, muon, and tau, Q is exactly halfway between the two extremes of all possible combinations: 1/3 (if the three masses were equal) and 1 (if one mass dominates). Robert Foot also interpreted the Koide formula as a geometrical relation, in which the value is the squared cosine of the angle between the vector and the vector (see dot product). That angle is almost exactly 45 degrees: When the formula is assumed to hold exactly (Q = 2/3), it may be used to predict the tau mass from the (more precisely known) electron and muon masses; that prediction is m_τ = 1776.969MeV/c2. Please note that solving the Koide formula can also predict the third particle mass to be around 3.
Jian Wang, Matthias Finger, Qian Wang, Yiming Li, Matthias Wolf, Varun Sharma, Yi Zhang, Konstantin Androsov, Jan Steggemann, Leonardo Cristella, Xin Chen, Davide Di Croce, Rakesh Chawla, Matteo Galli, Anna Mascellani, João Miguel das Neves Duarte, Tagir Aushev, Lei Zhang, Tian Cheng, Yixing Chen, Werner Lustermann, Andromachi Tsirou, Alexis Kalogeropoulos, Andrea Rizzi, Ioannis Papadopoulos, Paolo Ronchese, Hua Zhang, Siyuan Wang, Tao Huang, David Vannerom, Michele Bianco, Sebastiana Gianì, Sun Hee Kim, Kun Shi, Wei Shi, Abhisek Datta, Jian Zhao, Federica Legger, Gabriele Grosso, Ji Hyun Kim, Donghyun Kim, Zheng Wang, Sanjeev Kumar, Wei Li, Yong Yang, Ajay Kumar, Ashish Sharma, Georgios Anagnostou, Joao Varela, Csaba Hajdu, Muhammad Ahmad, Ekaterina Kuznetsova, Ioannis Evangelou, Muhammad Shoaib, Milos Dordevic, Meng Xiao, Sourav Sen, Xiao Wang, Kai Yi, Jing Li, Rajat Gupta, Muhammad Waqas, Hui Wang, Seungkyu Ha, Pratyush Das, Miao Hu, Anton Petrov, Xin Sun, Valérie Scheurer, Muhammad Ansar Iqbal, Lukas Layer
Jian Wang, Matthias Finger, Qian Wang, Yiming Li, Matthias Wolf, Varun Sharma, Yi Zhang, Konstantin Androsov, Jan Steggemann, Xin Chen, Rakesh Chawla, Matteo Galli, Anna Mascellani, João Miguel das Neves Duarte, Tagir Aushev, Tian Cheng, Yixing Chen, Werner Lustermann, Andromachi Tsirou, Alexis Kalogeropoulos, Andrea Rizzi, Ioannis Papadopoulos, Paolo Ronchese, Hua Zhang, Siyuan Wang, Tao Huang, David Vannerom, Michele Bianco, Sebastiana Gianì, Sun Hee Kim, Kun Shi, Abhisek Datta, Federica Legger, Gabriele Grosso, Ji Hyun Kim, Donghyun Kim, Zheng Wang, Sanjeev Kumar, Wei Li, Yong Yang, Ajay Kumar, Ashish Sharma, Georgios Anagnostou, Joao Varela, Csaba Hajdu, Muhammad Ahmad, Ioannis Evangelou, Milos Dordevic, Meng Xiao, Sourav Sen, Xiao Wang, Kai Yi, Jing Li, Rajat Gupta, Hui Wang, Seungkyu Ha, Pratyush Das, Anton Petrov, Xin Sun, Valérie Scheurer, Muhammad Ansar Iqbal, Lukas Layer, Marko Stamenkovic
Jian Wang, Matthias Finger, Qian Wang, Yiming Li, Matthias Wolf, Varun Sharma, Yi Zhang, Konstantin Androsov, Jan Steggemann, Leonardo Cristella, Xin Chen, Davide Di Croce, Rakesh Chawla, Matteo Galli, Anna Mascellani, João Miguel das Neves Duarte, Tagir Aushev, Lei Zhang, Tian Cheng, Yixing Chen, Werner Lustermann, Andromachi Tsirou, Alexis Kalogeropoulos, Andrea Rizzi, Ioannis Papadopoulos, Paolo Ronchese, Hua Zhang, Siyuan Wang, Tao Huang, David Vannerom, Michele Bianco, Sebastiana Gianì, Sun Hee Kim, Kun Shi, Wei Shi, Abhisek Datta, Jian Zhao, Federica Legger, Gabriele Grosso, Ji Hyun Kim, Donghyun Kim, Zheng Wang, Sanjeev Kumar, Wei Li, Yong Yang, Ajay Kumar, Ashish Sharma, Georgios Anagnostou, Joao Varela, Csaba Hajdu, Muhammad Ahmad, Ekaterina Kuznetsova, Ioannis Evangelou, Muhammad Shoaib, Milos Dordevic, Meng Xiao, Sourav Sen, Xiao Wang, Kai Yi, Jing Li, Rajat Gupta, Muhammad Waqas, Hui Wang, Seungkyu Ha, Pratyush Das, Miao Hu, Anton Petrov, Xin Sun, Valérie Scheurer, Muhammad Ansar Iqbal, Lukas Layer