Some Formulas for Invariant Phases of Unitary Matrices by Jarlskog

CP violation is expected in the standard model of particle physics with three or more families [1], [2]. Therefore it is an important problem that what is the measure of CP violation with such families which is invariant under the action of phase factors. To construct invariants for matrix action, the determinant is a useful tool. In the previous paper [3], C. Jarlskog succeeded to define invariants of CP violation by using the determinant for commutator of the quark mass matrices. For the case of 3 families, it is relatively easy to calculate it. Moreover, in that case, her determinant is propotional to an invariant phase of unitary matrices. Then she discussed invariant quantities for 4 families by using projection operators and the trace of some matrices, but she did not deal with her determinant itself for n = 4 [4]. Therefore the problem is still remained. An approach to this problem is to use a parametrization for