Exploring the Origin of CP Violation in the Standard Model

  • Chilong Lin National Museum of Natural Science, No. 1, Guancian Rd, Taichung 40453, Taiwan
Keywords: CP violation (CPV), Cabibbo-Kobayashi-Maskawa (CKM) matrix, standard model


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 \(M^2 ≡ M^q \cdot M^{q^†}\) rather than the previously assumed Hermitian
Mq. The only assumption employed here is that the real part and imaginary part of \(M^2\) can be,
respectively, diagonalized by a common \(U^q\) matrix. Such an assumption leads to an \(M^2\) pattern which depends
on only five parameters and can be diagonalized analytically by a \(U^q\) matrix which depends on only
two of the parameters. Two of the derived mass eigenvalues are predicted to be degenerate if one of the
parameters \(C (C′)\) in up- (down-) quark sector is zero. As the \(U^q\) 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 that the \(M^2\) pattern studied here is still oversimplified by that employed assumption
and the ultimate solution can only be obtained by diagonalizing the unsimplified \(M^2\) matrix containing
nine parameters directly. The VCKM presented here is already very close to such an ultimate CPV solution.


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