Asymptotic properties of $CP$ violation in the Standard Model

We present the analysis of the renormalization-based evolution of the $CP$-violation observables obtained from the $C$ matrix introduced by Jarlskog. We show that the observables $|\det C|$ and $\mathrm{Tr}{C}^2$ decrease very fast with the energy and their value is reduced at the Planck scale by 5 and 3 orders of magnitude, respectively, with respect to their low-energy values. On the other hand, the Jarlskog Cabibbo-Kobayashi-Maskawa matrix rephasing invariant $J$ increases with energy and at the Planck scale is 25% larger than at low energy. The absolute value of the coefficient $a_{CP}\sim \det C/(\mathrm{Tr}{C}^2{)}^{3/2}$ decreases with energy, and at the Planck scale it is 12% smaller than at low energy. We also find that the pattern of the eigenvalues of the $C$ matrix is such that two eigenvalues almost cancel each other and their absolute values are much bigger than the absolute value of the third eigenvalue. The low rate of $CP$ violation is a consequence of this pattern of the eigenvalues.

Figure 1

The dependence of $a_{CP}$ on $\xi$. The values of $a_{CP}$ are contained in the range $[-1,1]$, and $a_{CP}\to 0$ for $|\xi |\to \infty$.

Figure 2

The dependence of the Higgs quartic coupling $\lambda$ on energy. Only the region of energy where $\lambda$ is positive has stable vacuum and is physically acceptable. The top mass is equal to $m_t=173\mathrm{.}07$. The difference for one- and two-loop evolution is significant.

Figure 3

The dependence of the Higgs quartic coupling $\lambda$ on energy for two values of top quark mass: directly measured mass $m_t=173.07\mathrm{GeV}$ and $\overline{\mathrm{MS}}$ mass $m_t=160\mathrm{GeV}$. Here we include only two-loop evolution of $\lambda$.

Figure 4

RG evolution of $\det C$. In (a) we show the dependence of $\det C$ on the number of loops, and in (b) we show the two-loop evolution of $\det C$ for two values of the top quark mass: $m_t=173.07\mathrm{GeV}$ and $m_t=160\mathrm{GeV}$.

Figure 5

RG evolution of $\mathrm{Tr}{C}^2$. In (a) we show the dependence of $\mathrm{Tr}{C}^2$ on the number of loops, and in (b) we show the two-loop evolution of $\mathrm{Tr}{C}^2$ for two values of the top quark mass: $m_t=173.07\mathrm{GeV}$ and $m_t=160\mathrm{GeV}$.

Figure 6

RG evolution of $a_{CP}$. In (a) we show the dependence of $a_{CP}$ on the number of loops, and in (b) we show the two-loop evolution of $a_{CP}$ for two values of the top quark mass: $m_t=173.07\mathrm{GeV}$ and $m_t=160\mathrm{GeV}$.

Figure 7

RG evolution of the Jarlskog parameter $J$. In (a) we show the dependence of $J$ on the number of loops, and in (b) we show the two-loop evolution of $J$ for two values of the top quark mass: $m_t=173.07\mathrm{GeV}$ and $m_t=160\mathrm{GeV}$.

Figure 8

RG evolution of the two large eigenvalues of the matrix $C$. In (a) we show the dependence of the eigenvalues on the number of loops, and in (b) we show the two-loop evolution of the eigenvalues for two values of the top quark mass: $m_t=173.07\mathrm{GeV}$ and $m_t=160\mathrm{GeV}$.

Figure 9

RG evolution of the small eigenvalue of the matrix $C$. In (a) we show the dependence of the eigenvalue on the number of loops, and in (b) we show the two-loop evolution of the eigenvalue for two values of the top quark mass: $m_t=173.07\mathrm{GeV}$ and $m_t=160\mathrm{GeV}$.