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Family Gauge Symmetry as an Origin of Koide's Mass Formula and Charged Lepton Spectrum

Recently we have proposed mechanisms to explain origins of the charged lepton spectrum as well as Koide's mass formula, on the basis of U(3)\times O(3) family gauge symmetry. In this note, we review the basic ideas of these mechanisms. Without technical details, and adding some speculations, we give a sketch of the mechanisms, what the important points are and what assumptions are involved. We adopt a known scenario, in which the charged lepton spectrum is determined by the vacuum expectation value of a scalar field that takes values on 3-by-3 matrix. Within this scenario, we propose a mechanism, in which the radiative correction induced by U(3) family gauge interaction cancels the QED radiative correction to Koide's mass formula. We consider SU(9)\times U(1) symmetry broken down to U(3)\times O(3) symmetry. This leads to a potential model which predicts Koide's mass formula and the charged lepton spectrum consistent with the experimental values, by largely avoiding fine tuning of parameters. These are discussed within an effective theory, and we argue for its validity and usefulness.