Exploring the Origin of CP Violation in the Standard Model

Abstract

In this article, we present a very general but not ultimate solution of CPV problem in the standard model. Our study starts from a naturally Hermitian ${\bf M^2}\equiv M^q \cdot M^{q\dagger}$ rather than the previously assumed Hermitian $M^q$. The only assumption employed here is that the real part and imaginary part of $\bf M^2$ can be, respectively, diagonalized by a common $\bf U^q$ matrix. Such an assumption leads to a $\bf M^2$ pattern which depends on only five parameters and can be diagonalized analytically by a $\bf U^q$ matrix which depends on only two of the parameters. Two of the derived mass eigenvalues are predicted degenerate if one of the parameters $\bf C ~(C')$ in up- (down-) quark sector is zero. As the $\bf U^q$ patterns are obtained, thirty-six $V_{CKM}$ candidates are yielded and only eight of them, classified into two groups, fit empirical data within the order of $O(\lambda)$. One of the groups is further excluded in a numerical test, and the surviving group predicts that the degenerate pair in a quark type are the lightest and the heaviest generations rather than the lighter two generations assumed in previous researches. However, there is still one unsatisfactory prediction in this research, a quadruple equality in which four CKM elements of very different values are predicted to be equal. It indicates the $\bf M^2$ pattern studied here is still oversimplified by that employed assumption and the ultimate solution can only be obtained by diagonalizing the unsimplified $\bf M^2$ matrix containing nine parameters directly. The $V_{CKM}$ presented here is already very close to such an ultimate CPV solution.