One of the important phenomena in the field of fundamental interactions presently described by the the Standard Model (SM), is the violation of CP symmetry, the combination of C-symmetry (charge conjugation symmetry) and P-symmetry (parity symmetry). If CP-symmetry was exact the outcome of physical processes, like decays of hadrons (composites of quarks), would not change if all particles taking part in the process would be interchanged with their anti-particles (C-symmetry) and left would be interchanged with right (P-symmetry). While the symmetries C and P are conserved in strong and electromagnetic interactions, not only are they violated in weak interactions but also their combination (CP) has been found in 1964 to be violated in the decays of neutral K-mesons (Christenson et al., 1964) (James Cronin and Val Fitch, Nobel Prize in Physics 1980). By now, CP-violation has been also discovered in flavour-violating decays of mesons containing heavy bottom (b) quarks. The present experimental results can be well described in the framework of the so-called CKM (Cabibbo-Kobayashi-Maskawa) model (Cabibbo, 1963; Kobayashi and Maskawa, 1973), for which Kobayashi and Maskawa received Nobel-Prize in 2008 and Cabibbo Dirac Medal of ICTP (2010). Yet, this type of description does not explain the origin of large hierarchies observed in the size of flavour and CP-violating processes nor the dominance of matter over antimatter present in the Universe. Moreover in certain processes, small deviations from the predictions of CKM theory indicate that there could be other sources of CP-violation beyond CKM framework. They could then manifest themselves not only in flavour violating processes, like particle-antiparticle mixing and rare decays of mesons and charged leptons but also in Electric Dipole Moments (EDMs) of neutral particles that are flavour conserving. The most recent review of all these topics can be found in Buras and Girrbach, 2014.
Particle-Antiparticle Mixing and Various Types of CP Violation
The formalism of particle--antiparticle mixing and CP violation is elaborated at length in two books (Branco et al., 1999; Bigi and Sanda, 2000). The essentials are given below concentrating on $K^0-\bar K^0$ mixing, $B_{d,s}^0-\bar B^0_{d,s}$ mixings and CP violation in $K$-meson and $B$-meson decays. These phenomena in $D^0-\bar D^0$ mixing are strongly polluted by long distance effects and subject to large uncertainties. For references see Buras and Girrbach, 2014.
Express Review of $K^0-\bar K^0$ Mixing
$K^0=(\bar s d)$ and $\bar K^0=(s\bar d)$ are flavour eigenstates which in the SM may mix via weak interactions through the box diagrams in Fig. 1.
Our phase conventions are given by:
\[
CP|K^0\rangle=-|\bar]
In the absence of mixing the time evolution of $|K^0(t)\rangle$ is given by
\[
|K^0(t)\rangle=|K^0(0)\rangle]
where $M$ is the mass and $\Gamma$ the width of $K^0$. A similar formula exists for $\bar K^0$.
On the other hand, in the presence of flavour mixing the time evolution of the $K^0-\bar K^0$ system is described by
\[\label{SCH}
i\frac{d\psi(t)}{dt}=\hat]
where
\[
\hat]
with $\hat M$ and $\hat\Gamma$ being hermitian matrices having positive (real) eigenvalues in analogy with $M$ and $\Gamma$. $M_{ij}$ and $\Gamma_{ij}$ are the transition matrix elements from virtual and physical intermediate states respectively. Using
\[
M_{21}=M^*_{12}~,]
\[
M_{11}=M_{22}\equiv]
we have
\[\label{MM12}
\hat]
Diagonalizing \ref{SCH} one finds:
Eigenstates:
\[\label{KLS}
K_{L,S}=\frac{(1+\bar\varepsilon)K^0\pm]
where $\bar\varepsilon$ is a small complex parameter given by
\[\label{bare3}
\frac{1-\bar\varepsilon}{1+\bar\varepsilon}=
\sqrt{\frac{M^*_{12}-i\frac{1}{2}\Gamma^*_{12}}
{M_{12}-i\frac{1}{2}\Gamma_{12}}}=
\frac{2]
with $\Delta\Gamma$ and $\Delta M$ given below.
Eigenvalues:
\[
M_{L,S}=M\pm]
where
\[
Q=\sqrt{(M_{12}-i\frac{1}{2}\Gamma_{12})(M^*_{12}-i\frac{1}{2}\Gamma^*_{12})}.
\]
Consequently we have
\[\label{deltak}
\Delta]
The mass eigenstates $K_S$ and $K_L$ differ from the CP eigenstates
\[
K_1={1\over{\sqrt]
\[
K_2={1\over{\sqrt]
by a small admixture of the other CP eigenstate:
\[
K_{\rm]
Since $\bar\varepsilon$ is ${\cal O}(10^{-3})$, one has to a very good approximation:
\[\label{deltak1}
\Delta]
The subscript $K$ stresses that these formulae apply only to the $K^0-\bar K^0$ system. They will be different in the $B_{d,s}^0-\bar B_{d,s}^0$ systems.
The $K_{\rm L}-K_{\rm S}$ mass difference is experimentally measured to be (Beringer et al., 2012)
\[\label{DMEXP}
\Delta]
Experimentally one has $\Delta\Gamma_K\approx-2 \Delta M_K$.
Generally to observe CP violation one needs an interference between various amplitudes that carry complex phases. As these phases are obviously convention dependent, the CP-violating effects depend only on the differences of these phases. In particular the parameter $\bar\varepsilon$ depends on the phase convention chosen for $K^0$ and $\bar K^0$. Therefore it may not be taken as a physical measure of CP violation. On the other hand ${\rm Re}~\bar\varepsilon$ and $r$, defined in \ref{bare3} are independent of phase conventions. In fact the departure of $r$ from 1 measures CP violation in the $K^0-\bar K^0$ mixing:
\[
r=1+\frac{2]
This type of CP violation can be best isolated in semi-leptonic decays of the $ K_L$ meson. The non-vanishing asymmetry $a_{\rm SL}(K_L)$:
\[\label{ASLK}
\frac{\Gamma(K_L\to]
signals this type of CP violation. Note that $a_{\rm SL}(K_L)$ is determined purely by the quantities related to $K^0-\bar K^0$ mixing. Specifically, it measures the difference between the phases of $\Gamma_{12}$ and $M_{12}$.
That a non--vanishing $a_{\rm SL}(K_L)$ is indeed a signal of CP violation can also be understood in the following manner. $K_L$, that should be a CP eigenstate $K_2$ in the case
of CP conservation, decays into CP conjugate final states with different rates. As ${\rm Re} \bar\varepsilon>0$, $K_L$ prefers slightly to decay into $\pi^-e^+\nu_e$ than $\pi^+e^-\bar\nu_e$. This would not be possible in a CP-conserving world.
$\varepsilon$ and $\varepsilon'$
Since two pion final states, $\pi^+\pi^-$ and $\pi^0\pi^0$, are CP even while the three pion final state $3\pi^0$ is CP odd, $K_{\rm S}$ and $K_{\rm L}$ decay to $2\pi$ and $3\pi^0$, respectively via the following CP-conserving decay modes:
\[
K_{\rm]
Moreover, $K_{\rm L}\to \pi^+\pi^-\pi^0$ is also CP conserving provided the orbital angular momentum of $\pi^+\pi^-$ is even. This difference between $K_L$ and $K_S$ decays is responsible for the large disparity in their life-times. A factor of 579. However, $K_{\rm L}$ and $K_{\rm S}$ are not CP eigenstates and may decay with small branching fractions as follows:
\[
K_{\rm]
This violation of CP is called indirect as it proceeds not via explicit breaking of the CP symmetry in the decay itself but via the admixture of the CP state with opposite CP parity to the dominant one. The measure for this indirect CP violation is defined as (I=isospin)
\[\label{ek}
\varepsilon
\equiv]
Note that the decay $K_{\rm S}\to \pi^+\pi^-\pi^0$ is CP violating (conserving) if the orbital angular momentum of $\pi^+\pi^-$ is even (odd).
The parameter $\varepsilon$ is a physical observable and is given as follows
\[
\varepsilon]= \bar\varepsilon+i\xi= \frac{\exp(i \varphi_\varepsilon)}{\sqrt{2} \Delta M_K} \,
\left( {\rm Im} M_{12} + 2 \xi {\rm Re} M_{12} \right),
\quad\quad
