A Very General But Not Ultimate Pattern of CKM Matrix

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 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 an $\bf U^q$ 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 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. Though the result obtained here is not the ultimate one yet, however, we are already very close to it.Comment: 8 pages, no figure

BAU production in the SNS_N-breaking Standard Model

CP violation (CPV) and the baryon asymmetry of the universe (BAU) are two of the most significant unresolved problems in physics. This article presents further research on the CPV problem in the standard model (SM) with thirty-two candidate sets of the ten "natural" parameters that exhibit the same CKM performance. These parameters are considered "natural" because they consist solely of the Yukawa couplings and the vacuum expectation value (VEV) of the unique Higgs doublet in the SM. We then investigate the CPV problem and the BAU problem using the Jarlskog measure of CPV, $\Delta_{CP}=J(m^2_t -m^2_c)(m^2_t -m^2_u)(m^2_c -m^2_u)(m^2_b -m^2_s)(m^2_b -m^2_d)(m^2_s -m^2_d)/ T^{12}$, given that CP symmetry is violated following the breakdowns of $S_N$ symmetries. Subsequently, we perform numerical tests in a simplified scenario where eight of the ten parameters are fixed, and the remaining two parameters are allowed to vary from their $S_2$-symmetric values ${\bf (x,~y)}=(-1,~1)$ to their current values in all thirty-two parameter sets. To estimate the enhancement of CPV in such processes, we propose a ratio $R_{\Delta} \equiv \Delta_{CP} / \Delta^{(0)}_{CP}$ between the running $\Delta_{CP}$ and its current value, denoted by $\Delta^{(0)}_{CP}$, which is approximately $10^{-20}$. In all thirty-two cases, the three-dimensional (3D) plots of $R_{\Delta}$ exhibit many regions that stick out of the $R_{\Delta}=10^{10}$ plane, especially in regions very close to ${\bf (x,~y)}=(-1,~1)$. These results demonstrate that the $S_N$-breaking Standard Model is already sufficient to violate CP symmetry explicitly and generate a significant amount of BAU. Furthermore, it solves existing problems without creating new ones.Comment: 17pages, 36 figure

Feynman Rules in the Type III Natural Flavour-Conserving Two-Higgs Doublet Model

We consider a two Higgs-doublet model with $S_3$ symmetry, which implies a $\pi \over 2$ rather than 0 relative phase between the vacuum expectation values $$and$$. The corresponding Feynman rules are derived accordingly and the transformation of the Higgs fields from the weak to the mass eigenstates includes not only an angle rotation but also a phase transformation. In this model, both doublets couple to the same type of fermions and the flavour-changing neutral currents are naturally suppressed. We also demonstrate that the Type III natural flavour-conserving model is valid at tree-level even when an explicit $S_3$ symmetry breaking perturbation is introduced to get a reasonable CKM matrix. In the special case $\beta = \alpha$, as the ratio $\tan\beta = {v_2 \over v_1}$ runs from 0 to $\infty$, the dominant Yukawa coupling will change from the first two generations to the third generation. In the Feynman rules, we also find that the charged Higgs currents are explicitly left-right asymmetric. The ratios between the left- and right-handed currents for the quarks in the same generations are estimated.Comment: 16 pages (figures not included), NCKU-HEP/93-1