Rephasing invariance and permutation symmetry in flavor physics
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Regular Article - Theoretical Physics
Rephasing invariance and permutation symmetry in flavor physics T. K. Kuo1,a , S. H. Chiu2,b 1 2
Department of Physics, Purdue University, West Lafayette, IN 47907, USA Physics Group, CGE, Chang Gung University, Taoyuan 33302, Taiwan
Received: 14 December 2019 / Accepted: 12 February 2020 © The Author(s) 2020
Abstract With some modifications, the arguments for rephasing invariance can be used to establish permutation symmetry for the standard model. The laws of evolution of physical variables, which transform as tensors under permutation, are found to obey the symmetry, explicitly. We also propose to use a set of four mixing parameters, with unique properties, which may serve to characterize flavor mixing.
1 Introduction One of the long-standing puzzles in the Standard Model (SM) is the existence of three families of fundamental fermions. (Throughout this paper, SM refers to a modified version with the addition of three massive Dirac neutrinos, so that one may treat the lepton sector on a par with the quark sector). These fermions are endowed with properties (masses and mixing parameters) which seem to be arbitrary. At the same time, it is also well-established that these properties are not static, but that they do evolve as the physical environment changes. They may thus be regarded as dynamical variables in the system. E.g., the masses and mixing parameters are dependent on the energy scale, and evolve according to the renormalization group equations (RGE). Similarly, as a coherent neutrino beam propagates, the neutrino mixing parameters change along its path. A third physical example concerns neutrino propagation in matter, where one finds that their masses and mixing become functions of the medium density. While it may be difficult to find order in the observed fermion properties, one can look for regularity in the laws of their evolution. In a previous paper [1], it was pointed out that these equations obey the symmetry [S3 ]4 = (S3 (u), S3 (d), S3 (), S3 (ν)), where S3 is the permutation group which operates on the three members in each of the a e-mail:
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four fermion sectors. Examples of calculated formulas of transition (neutrino oscillation in vacuum), as well as evolution equations (RGE and neutrino oscillation in matter) were examined. It was found that they obey the permutation symmetry, explicitly. In this paper, we will incorporate, systematically, rephasing into our study of permutation. It turns out that the two operations are closely related. Let Vαi denote the elements of either the CKM or the PMNS matrix. While rephasing attaches phases to Vαi according to their indices, permutation exchanges these indices. We will demonstrate that the SM Lagrangian, L S M , is invariant in form under either operation. Furthermore, rephasing invariance implies that Vαi must be grouped in specific combinations according to their indices. Applying a permutation to any such pro
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