CP violation in electroweak interactions

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Andrzej Buras (2015), Scholarpedia, 10(8):11418. doi:10.4249/scholarpedia.11418 revision #151973 [link to/cite this article]
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Curator: Andrzej Buras



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.

Figure 1: Box diagrams contributing to $K^0-\bar K^0$ mixing in the SM.

Our phase conventions are given by: \[ CP|K^0\rangle=-|\bar K^0\rangle, \qquad CP|\bar K^0\rangle=-|K^0\rangle. \]

In the absence of mixing the time evolution of $|K^0(t)\rangle$ is given by \[ |K^0(t)\rangle=|K^0(0)\rangle \exp(-iHt)~, \qquad H=M-i\frac{\Gamma}{2}~, \] 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 \[\tag{1} i\frac{d\psi(t)}{dt}=\hat H \psi(t) \qquad \psi(t)= \left(\begin{array}{c} |K^0(t)\rangle\\ |\bar K^0(t)\rangle \end{array}\right) \] where \[ \hat H=\hat M-i\frac{\hat\Gamma}{2} = \left(\begin{array}{cc} M_{11}-i\frac{\Gamma_{11}}{2} & M_{12}-i\frac{\Gamma_{12}}{2} \\ M_{21}-i\frac{\Gamma_{21}}{2} & M_{22}-i\frac{\Gamma_{22}}{2} \end{array}\right) \] 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 \[ M_{21}=M^*_{12}~, \qquad \Gamma_{21}=\Gamma_{12}^*~,\quad\quad {\rm (hermiticity)} \] \[ M_{11}=M_{22}\equiv M~, \qquad \Gamma_{11}=\Gamma_{22}\equiv\Gamma~, \quad {\rm (CPT)} \] we have \[\tag{2} \hat H= \left(\begin{array}{cc} M-i\frac{\Gamma}{2} & M_{12}-i\frac{\Gamma_{12}}{2} \\ M^*_{12}-i\frac{\Gamma^*_{12}}{2} & M-i\frac{\Gamma}{2} \end{array}\right)~. \]

Diagonalizing (1) one finds:

Eigenstates: \[\tag{3} K_{L,S}=\frac{(1+\bar\varepsilon)K^0\pm (1-\bar\varepsilon)\bar K^0} {\sqrt{2(1+\mid\bar\varepsilon\mid^2)}} \] where $\bar\varepsilon$ is a small complex parameter given by \[\tag{4} \frac{1-\bar\varepsilon}{1+\bar\varepsilon}= \sqrt{\frac{M^*_{12}-i\frac{1}{2}\Gamma^*_{12}} {M_{12}-i\frac{1}{2}\Gamma_{12}}}= \frac{2 M^*_{12}-i\Gamma^*_{12}}{\Delta M-i\frac{1}{2}\Delta\Gamma} \equiv r\exp(i\kappa)~. \] with $\Delta\Gamma$ and $\Delta M$ given below.

Eigenvalues: \[ M_{L,S}=M\pm {\rm Re} Q~, \qquad \Gamma_{L,S}=\Gamma\mp 2 {\rm Im} Q \] where \[ Q=\sqrt{(M_{12}-i\frac{1}{2}\Gamma_{12})(M^*_{12}-i\frac{1}{2}\Gamma^*_{12})}. \] Consequently we have \[\tag{5} \Delta M= M_L-M_S = 2{\rm Re} Q~, \quad\quad \Delta\Gamma=\Gamma_L-\Gamma_S=-4 {\rm Im} Q. \]

The mass eigenstates $K_S$ and $K_L$ differ from the CP eigenstates \[ K_1={1\over{\sqrt 2}}(K^0-\bar K^0), \qquad\qquad CP|K_1\rangle=|K_1\rangle~, \] \[ K_2={1\over{\sqrt 2}}(K^0+\bar K^0), \qquad\qquad CP|K_2\rangle=-|K_2\rangle~, \] by a small admixture of the other CP eigenstate: \[ K_{\rm S}={{K_1+\bar\varepsilon K_2} \over{\sqrt{1+\mid\bar\varepsilon\mid^2}}}, \qquad K_{\rm L}={{K_2+\bar\varepsilon K_1} \over{\sqrt{1+\mid\bar\varepsilon\mid^2}}}\,. \]

Since $\bar\varepsilon$ is ${\cal O}(10^{-3})$, one has to a very good approximation: \[\tag{6} \Delta M_K = 2 {\rm Re} M_{12}, \qquad \Delta\Gamma_K=2 {\rm Re} \Gamma_{12}~. \] 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) \[\tag{7} \Delta M_K=M(K_{\rm L})-M(K_{\rm S}) = \Delta M_K= 0.5292(9)\times 10^{-2} \,{\rm ps}^{-1}. \] 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 (4) are independent of phase conventions. In fact the departure of $r$ from 1 measures CP violation in the $K^0-\bar K^0$ mixing: \[ r=1+\frac{2 |\Gamma_{12}|^2}{4 |M_{12}|^2+|\Gamma_{12}|^2} {\rm Im}\left(\frac{M_{12}}{\Gamma_{12}}\right) \approx 1-{\rm Im}\left(\frac{\Gamma_{12}}{M_{12}}\right)~. \]

This type of CP violation can be best isolated in semi-leptonic decays of the $ K_L$ meson. The non-vanishing asymmetry $a_{\rm SL}(K_L)$: \[\tag{8} \frac{\Gamma(K_L\to \pi^-e^+\nu_e )- \Gamma( K_L\to \pi^+e^-\bar\nu_e )} {\Gamma(K_L\to \pi^-e^+\nu_e )+ \Gamma( K_L\to \pi^+e^-\bar\nu_e )} = \left({\rm Im}\frac{\Gamma_{12}}{M_{12}}\right)_K = 2 {\rm Re} \bar\varepsilon \] 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: \[ K_{\rm L}\to 3\pi^0 {\rm ~~(via~K_2),}\qquad K_{\rm S}\to 2 \pi {\rm ~~(via~K_1).} \] 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: \[ K_{\rm L}\to 2\pi {\rm ~~(via~K_1),}\qquad K_{\rm S}\to 3 \pi^0 {\rm ~~(via~K_2).} \] 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) \[\tag{9} \varepsilon \equiv {{A(K_{\rm L}\rightarrow(\pi\pi)_{I=0}})\over{A(K_{\rm S}\rightarrow(\pi\pi)_{I=0})}}. \] 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 \[ \varepsilon = \bar\varepsilon+i\xi= \frac{\exp(i \varphi_\varepsilon)}{\sqrt{2} \Delta M_K} \, \left( {\rm Im} M_{12} + 2 \xi {\rm Re} M_{12} \right), \quad\quad \xi = \frac{{\rm Im} A_0}{{\rm Re} A_0}, \tag{10} \] where $\varphi_\varepsilon = (43.51\pm0.05)^\circ$. The phase convention dependence of $\xi$ cancels the one of $\bar\varepsilon$ so that $\varepsilon$ is free from this dependence. The isospin amplitude $A_0$ is defined below.

The important point in the definition (9) is that only the transition to $(\pi\pi)_{I=0}$ enters. The transition to $(\pi\pi)_{I=2}$ is absent. This allows to remove a certain type of CP violation that originates in decays only. Yet as $\varepsilon\not=\bar\varepsilon$ and only ${\rm Re}\varepsilon={\rm Re}\bar\varepsilon$, it is clear that $\varepsilon$ includes a type of CP violation represented by ${\rm Im}\varepsilon$ which is absent in the semileptonic asymmetry (8). A systematic classification of different types of CP violation will be given below.

While indirect CP violation reflects the fact that the mass eigenstates are not CP eigenstates, so-called direct CP violation is realized via a direct transition of a CP odd to a CP even state: $K_2\to \pi\pi$. A measure of such a direct CP violation in $K_L\to \pi\pi$ is characterized by a complex parameter $\varepsilon'$ defined as \[\tag{11} \varepsilon'\equiv\frac{1}{\sqrt{2}}\left(\frac{A_{2,L}}{A_{0,S}}- \frac{A_{2,S}}{A_{0,S}}\frac{A_{0,L}}{A_{0,S}}\right) \] where $A_{I,L}\equiv A(K_L\to (\pi\pi)_I)$ and $A_{I,S}\equiv A(K_S\to (\pi\pi)_I)$.

This time the transitions to $(\pi\pi)_{I=0}$ and $(\pi\pi)_{I=2}$ are included which allows to study CP violation in the decay itself. We will discuss this issue in general terms below. It is useful to cast (11) into \[\tag{12} \varepsilon'=\frac{1}{\sqrt{2}}{\rm Im}\left(\frac{A_2}{A_0}\right) \exp(i\Phi_{\varepsilon'}), \qquad \Phi_{\varepsilon'}=\frac{\pi}{2}+\delta_2-\delta_0, \] where the isospin amplitudes $A_I$, neglecting the $\Delta I=5/2$ contributions, parametrize different $K\to\pi\pi$ amplitudes (Cirigliano et al., 2012) \[ A(K^+\rightarrow\pi^+\pi^0)=\frac{3}{2} A_2 e^{i\delta_2} \] \[ A(K^0\rightarrow\pi^+\pi^-)=A_0 e^{i\delta_0}+ \sqrt{\frac{1}{2}} A_2 e^{i\delta_2} \] \[ A(K^0\rightarrow\pi^0\pi^0)= A_0 e^{i\delta_0}-\sqrt{2} A_2 e^{i\delta_2}\,. \] Here the subscript $I=0,2$ denotes states with isospin $0,2$ equivalent to $\Delta I=1/2$ and $\Delta I = 3/2$ transitions, respectively, and $\delta_{0,2}$ are the corresponding strong phases. The weak CKM phases are contained in $A_0$ and $A_2$. The isospin amplitudes $A_I$ are complex quantities which depend on phase conventions. On the other hand, $\varepsilon'$ measures the difference between the phases of $A_2$ and $A_0$ and is a physical quantity. The strong phases $\delta_{0,2}$ can be extracted from $\pi\pi$ scattering (Beringer et al., 2012) \[ \delta_0-\delta_2=(47.5\pm0.9)^\circ~. \] Then $\Phi_{\varepsilon'}\approx \pi/4$.

The real parts of the amplitudes $A_I$ are measured to be (Beringer et al., 2012) \[\tag{13} {\rm Re}A_0= (2.704\pm 0.001)\times 10^{-7}~\, {\rm GeV}, \quad {\rm Re}A_2= (1.210\pm 0.002) \times 10^{-8}~\, {\rm GeV}, \] and express the so-called $\Delta I=1/2$ rule \[ \frac{{\rm Re}A_0}{{\rm Re}A_2}=22.35. \]

$\varepsilon$ and $\varepsilon'$ can be found by measuring the ratios \[ \eta_{00}={{A(K_{\rm L}\to\pi^0\pi^0)}\over{A(K_{\rm S}\to\pi^0\pi^0)}}, \qquad \eta_{+-}={{A(K_{\rm L}\to\pi^+\pi^-)}\over{A(K_{\rm S}\to\pi^+\pi^-)}}. \]

Indeed, assuming $\varepsilon$ and $\varepsilon'$ to be small numbers one finds \[ \eta_{00}=\varepsilon-{{2\varepsilon'}\over{1-\sqrt{2}\omega}} ,~~~~ \eta_{+-}=\varepsilon+{{\varepsilon'}\over{1+\omega/\sqrt{2}}}, \] where $\omega={\rm Re} A_2/{\rm Re} A_0=0.045$. In the absence of direct CP violation $\eta_{00}=\eta_{+-}$. The ratio ${\varepsilon'}/{\varepsilon}$ can then be measured through \[\tag{14} {\rm Re}(\varepsilon'/\varepsilon)=\frac{1}{6(1+\omega/\sqrt{2})} \left(1-\left|{{\eta_{00}}\over{\eta_{+-}}}\right|^2\right)~. \]

Experimentally we have (Beringer et al., 2012) \[ \varepsilon=2.228(11)\times 10^{-3}e^{i\phi_\varepsilon},\quad {\rm Re}(\varepsilon'/\varepsilon)=16.6(23)\times 10^{-4}, \] where $\phi_\varepsilon=43.51(5)^\circ$.

Express Review of $B_{d,s}^0$-$\bar B_{d,s}^0$ Mixing

The flavour eigenstates in this case are \[\tag{15} B^0_d=(\bar bd),\qquad \bar B^0_d=(b \bar d),\qquad B^0_s=(\bar bs),\qquad \bar B^0_s=( b \bar s)~. \] They mix via the box diagrams in Fig. 1 with $s$ replaced by $b$ in the case of $B_{d}^0$-$\bar B_{d}^0$ mixing. In the case of $B_{s}^0$-$\bar B_{s}^0$ mixing also $d$ has to be replaced by $s$.

Dropping the subscripts $(d,s)$ for a moment, it is customary to denote the mass eigenstates by \[\tag{16} B_H=p B^0+q \bar B^0, \qquad B_L=p B^0-q \bar B^0, \] \[\tag{17} p=\frac{1+\bar\varepsilon_B}{\sqrt{2(1+|\bar\varepsilon_B|^2)}}, \qquad q=\frac{1-\bar\varepsilon_B}{\sqrt{2(1+|\bar\varepsilon_B|^2)}}, \] with $\bar\varepsilon_B$ corresponding to $\bar\varepsilon$ in the $K^0-\bar K^0$ system. Here “H” and “L” denote Heavy and Light respectively. As in the $B_d^0-\bar B_d^0$ system one has $\Delta\Gamma\ll\Delta M$, it is more suitable to distinguish the mass eigenstates by their masses than by the corresponding life-times. In the case of $B_s^0-\bar B_s^0$ one cannot neglect $\Delta\Gamma\ll\Delta M$ which implies interesting effects covered in Nierste, 2013; Borissov et al., 2013.

The strength of the $B^0_{d,s}-\bar B^0_{d,s}$ mixings is described by the mass differences \[\tag{18} \Delta M_{d,s}= M_H^{d,s}-M_L^{d,s}~. \]

$\Delta M_{d,s}$ can be expressed in terms of the off-diagonal element in the neutral $B$-meson mass matrix by using the formulae developed previously for the $K$-meson system. One has \[ \Delta M_q= 2 |M_{12}^{(q)}|, \qquad \Delta \Gamma_q=2 \frac{{\rm Re}(M_{12}\Gamma_{12}^*)}{|M_{12}|} \ll\Delta M_q, \qquad q=d,s. \tag{19} \] These formulae differ from (6) because in the B-system $\bar\varepsilon_B$ cannot be neglected and $\Gamma_{12}\ll M_{12}$.

We also have \[\tag{20} \frac{q}{p}=\sqrt{\frac{M^*_{12}-i\frac{1}{2}\Gamma^*_{12}} {M_{12}-i\frac{1}{2}\Gamma_{12}}}= \frac{2 M^*_{12}-i\Gamma^*_{12}}{\Delta M-i\frac{1}{2}\Delta\Gamma} =\frac{M_{12}^*}{|M_{12}|} \left[1-\frac{1}{2}{\rm Im}\left(\frac{\Gamma_{12}}{M_{12}}\right)\right] \] where higher order terms in the small quantity $\Gamma_{12}/M_{12}$ have been neglected.

As ${\rm Im}(\Gamma_{12}/M_{12})< {\cal O}(10^{-3})$,

  • The semileptonic asymmetry $a_{\rm SL}(B)$ is even smaller than $a_{\rm SL}(K_L)$. Typically ${\cal O}(10^{-4})$ (Nierste, 2013; Borissov et al., 2013). These are bad news.
  • The ratio $q/p$ is a pure phase to an excellent approximation. In fact one finds within the SM

\[\tag{21} \left(\frac{q}{p}\right)_{d,s}= e^{i2\phi_M^{d,s}}, \qquad \phi^d_M=-\beta, \qquad \phi^s_M=\pi-\beta_s, \] with $\phi_M^{d,s}$ defined previously. These are good news as we will see below.

Classification of CP Violation

In the study of CP violation in $B$ decays it turned out to be useful to make a classification of CP-violating effects that is more transparent than the division into the indirect and direct CP violation usually made in the context of $K$ physics.

Generally complex phases may enter particle--antiparticle mixing and/or the decay process itself. It is then natural to consider three types of CP violation:

  • CP Violation in Mixing
  • CP Violation in Decay
  • CP Violation in the Interference of Mixing and Decay

As the phases in mixing and decay are convention dependent, the CP-violating effects depend only on the differences of these phases. This is clearly seen in the classification given below.

CP Violation in Mixing

This type of CP violation can be best isolated in semi-leptonic decays of neutral $B$ and $K$ mesons. We have discussed the asymmetry $a_{SL}(K_L)$ before. In the case of $B$ decays the non-vanishing asymmetry $a_{SL}(B)$ (we suppress the indices $(d,s)$), \[\tag{22} \frac{\Gamma(\bar B^0(t)\to l^+\nu X)- \Gamma( B^0(t)\to l^-\bar\nu X)} {\Gamma(\bar B^0(t)\to l^+\nu X)+ \Gamma( B^0(t)\to l^-\bar\nu X)} =\frac{1-|q/p|^4}{1+|q/p|^4} = \left({\rm Im}\frac{\Gamma_{12}}{M_{12}}\right)_B \] signals this type of CP violation. Here $\bar B^0(0)=\bar B^0$, $B^0(0)= B^0$. For $t\not=0$ the formulae analogous to (1) should be used. Note that the final states in (22) contain “wrong charge” leptons and can only be reached in the presence of $B^0-\bar B^0$ mixing. That is one studies effectively the difference between the rates for $\bar B^0\to B^0\to l^+\nu X$ and $ B^0 \to \bar B^0 \to l^-\bar\nu X$. As the phases in the transitions $B^0 \to \bar B^0$ and $\bar B^0 \to B^0$ differ from each other, a non-vanishing CP asymmetry follows. Specifically $a_{\rm SL}(B)$ measures the difference between the phases of $\Gamma_{12}$ and $M_{12}$.

As $M_{12}$ and in particular $\Gamma_{12}$ suffer from large hadronic uncertainties, no precise extraction of CP-violating phases from this type of CP violation can be expected. Moreover as $q/p$ is almost a pure phase, see (20) and (21), the asymmetry is very small and very difficult to measure. But in the $B_s^0-\bar B_s^0$ system width effects are not negligible (Nierste, 2013).

CP Violation in Decay

This type of CP violation is best isolated in charged $B$ and charged $K$ decays as mixing effects do not enter here. However, it can also be measured in neutral $B$ and $K$ decays. The relevant asymmetry is given by \[\tag{23} {\cal A}_{\rm CP}^{\rm dir}(B^\pm\to f^\pm)=\frac{\Gamma(B^+\to f^+)-\Gamma(B^-\to f^-)} {\Gamma(B^+\to f^+)+\Gamma(B^-\to f^-)} =\frac{1-|\bar A_{f^-}/A_{f^+}|^2}{1+| \bar A_{f^-}/A_{f^+}|^2} \] where \[\tag{24} A_{f^+}=\langle f^+|{\cal H}^{\rm weak}| B^+\rangle, \qquad \bar A_{f^-}=\langle f^-|{\cal H}^{\rm weak}| B^-\rangle~. \] For this asymmetry to be non-zero one needs at least two different contributions to these amplitudes with different weak ($\phi_i$) and strong ($\delta_i$) phases. These could be for instance two tree diagrams, two penguin diagrams or one tree and one penguin. Indeed writing the decay amplitude $A_{f^+}$ and its CP conjugate $\bar A_{f^-}$ as \[\tag{25} A_{f^+}=\sum_{i=1,2} A_i e^{i(\delta_i+\phi_i)}, \qquad \bar A_{f^-}=\sum_{i=1,2} A_i e^{i(\delta_i-\phi_i)}, \] with $A_i$ being real, one finds \[\tag{26} {\cal A}_{\rm CP}^{\rm dir}(B^\pm\to f^\pm)=\frac{ -2 A_1 A_2 \sin(\delta_1-\delta_2) \sin(\phi_1-\phi_2)}{A_1^2+A_2^2+2 A_1 A_2 \cos(\delta_1-\delta_2) \cos(\phi_1-\phi_2)}~. \] The sign of the strong phases $\delta_i$ is the same for $A_{f^+}$ and $\bar A_{f^-}$ because CP is conserved by strong interactions. The weak phases have opposite signs.

The presence of hadronic uncertainties in $A_i$ and of strong phases $\delta_i$ complicates the extraction of the phases $\phi_i$ from data. An example of this type of CP violation in $K$ decays is $\varepsilon'$.

CP Violation in the Interference of Mixing and Decay

This type of CP violation is only possible in neutral $B$ and $K$ decays. We will use $B$ decays for illustration suppressing the subscripts $d$ and $s$. Moreover, we set $\Delta\Gamma=0$, even if in the case of $B_s^0-\bar B_s^0$ width effects can play some role. Formulae with $\Delta\Gamma\not =0$ can be found in Buras and Fleischer, 1998; Nierste, 2013.

Most interesting are the decays into final states which are CP-eigenstates. Then a time dependent asymmetry defined by \[\tag{27} {\cal A}_{\rm CP}(t,f)=\frac{\Gamma(B^0(t)\to f)- \Gamma( \bar B^0(t)\to f)} {\Gamma(B^0(t)\to f)+ \Gamma( \bar B^0(t)\to f)} \] is given by \[\tag{28} {\cal A}_{\rm CP}(t,f)= {\cal A}_{\rm CP}^{\rm dir}(f)\cos(\Delta M t)+{\cal A}_{\rm CP}^{\rm mix}(f)\sin(\Delta M t) \] where we have separated the decay CP-violating contributions (${\cal A}_{\rm CP}^{\rm dir}$) from those describing CP violation in the interference of mixing and decay $({\cal A}_{\rm CP}^{\rm mix}$). The latter type of CP violation is usually called the mixing-induced CP violation. One has \[\tag{29} {\cal A}_{\rm CP}^{\rm dir}(f)=\frac{1-\left\vert\xi_f\right\vert^2} {1+\left\vert\xi_f\right\vert^2}\equiv C_f, \quad {\cal A}_{\rm CP}^{\rm mix}(f)=\frac{2\mbox{Im}\xi_f}{1+ \left\vert\xi_f\right\vert^2}\equiv -S_f~, \] where $C_f$ and $S_f$ are popular notations found in the literature. Unfortunately, the signs in the literature differ from paper to paper and some papers interchange $B^0$ and $\bar B^0$ with respect to the one used by us in the definition of the asymmetry in (27). Therefore when comparing results from different papers one has to check these definitions.

The quantity $\xi_f$ containing all the information needed to evaluate the asymmetries (29) is given by \[\tag{30} \xi_f=\frac{q}{p}\frac{A(\bar B^0\to f)}{A(B^0 \to f)}= \exp(i2\phi_M)\frac{A(\bar B^0\to f)}{A(B^0 \to f)} \] with $\phi_M$, introduced in (21), denoting the weak phase in the $B^0-\bar B^0$ mixing. $A(B^0 \to f)$ and $A(\bar B^0 \to f)$ are decay amplitudes. The time dependence of ${\cal A}_{\rm CP}(t,f)$ allows to extract ${\cal A}_{\rm CP}^{\rm dir}$ and ${\cal A}_{\rm CP}^{\rm mix}$ as coefficients of $\cos(\Delta M t)$ and $\sin(\Delta M t)$, respectively.

Generally several decay mechanisms with different weak and strong phases can contribute to $A(B^0 \to f)$. These are tree diagram (current-current) contributions, QCD penguin contributions and electroweak penguin contributions. If they contribute with similar strength to a given decay amplitude the resulting CP asymmetries suffer from hadronic uncertainties related to matrix elements of the relevant operators $Q_i$. The situation is then analogous to the class just discussed. Indeed \[\tag{31} \frac{A(\bar B^0\to f)}{A(B^0 \to f)}=-\eta_f \left[\frac{A_T e^{i(\delta_T-\phi_T)}+A_P e^{i(\delta_P-\phi_P)}} {A_T e^{i(\delta_T+\phi_T)}+A_P e^{i(\delta_P+\phi_P)}}\right] \] with $\eta_f=\pm 1$ being the CP-parity of the final state, depends on the strong phases $\delta_{T,P}$ and the hadronic matrix elements present in $A_{T,P}$. Thus the measurement of the asymmetry does not allow a clean determination of the weak phases $\phi_{T,P}$. The minus sign in (31) follows from our CP phase convention $ CP |B^0\rangle= -|\bar B^0\rangle$, that has also been used in writing the phase factor in (30). Only $\xi$ is phase convention independent.

An interesting case arises when a single mechanism dominates the decay amplitude or the contributing mechanisms have the same weak phases. Then the hadronic matrix elements and strong phases drop out and \[\tag{32} \frac{A(\bar B^0\to f)}{A(B^0 \to f)}=-\eta_f e^{-i2\phi_D} \] is a pure phase with $\phi_D$ being the weak phase in $A(B^0 \to f)$. Consequently \[\tag{33} \xi_f=-\eta_f\exp(i2\phi_M) \exp(-i 2 \phi_D), \qquad \mid \xi_f \mid^2=1~. \] In this particular case ${\cal A}_{\rm CP}^{\rm dir}(f)=C_f$ vanishes and the CP asymmetry is given entirely in terms of the weak phases $\phi_M$ and $\phi_D$: \[\tag{34} {\cal A}_{\rm CP}(t,f)= {\cal A}_{\rm CP}^{\rm mix}(f) \sin(\Delta Mt) \qquad {\cal A}_{\rm CP}^{\rm mix}(f)={\rm Im}\xi_f=\eta_f \sin(2\phi_D-2\phi_M)=-S_f~. \] Thus the corresponding measurement of weak phases is free from hadronic uncertainties. A well known example is the decay $B_d\to \psi K_S$. Here $\phi_M=-\beta$ and $\phi_D=0$. As in this case $\eta_f=-1$, we find \[ {\cal A}_{\rm CP}(t,f)= -\sin(2\beta) \sin(\Delta Mt), \qquad S_f=\sin(2\beta) \] which allows a very clean measurement of the angle $\beta$ in the unitarity triangle. We will discuss other examples below.

We observe that the asymmetry ${\cal A}_{\rm CP}^{\rm mix}(f)$ measures directly the difference between the phases of the $B^0-\bar B^0$-mixing $(2\phi_M)$ and of the decay amplitude $(2\phi_D)$. This tells us immediately that we are dealing with the interference of mixing and decay. As $\phi_M$ and $\phi_D$ are phase convention dependent quantities, only their difference is physical, it is impossible to state on the basis of a single asymmetry whether CP violation takes place in the decay or in the mixing. To this end at least two asymmetries for $B^0 (\bar B^0)$ decays to different final states $f_i$ have to be measured. As $\phi_M$ does not depend on the final state, ${\rm Im}\xi_{f_1}\not={\rm Im}\xi_{f_2}$ is a signal of CP violation in the decay.

This ideal situation presented above does not always take place and two or more different mechanisms with different weak and strong phases contribute to the CP asymmetry making the formulae more involved and introducing hadronic uncertainties.

In the case of $K$ decays, this type of CP violation can be cleanly measured in the rare decay $K_L\to\pi^0\nu\bar\nu$. Here the difference between the weak phase in the $K^0-\bar K^0$ mixing and in the decay $\bar s \to \bar d \nu\bar\nu$ matters.

We can now compare the two classifications of different types of CP violation. CP violation in mixing is a manifestation of indirect CP violation. CP violation in decay is a manifestation of direct CP violation. The third type contains elements of both the indirect and direct CP violation.

It is clear from this discussion that only in the case of the third type of CP violation there are possibilities to measure directly weak phases without hadronic uncertainties and moreover without invoking sophisticated methods. This takes place provided a single mechanism (diagram) is responsible for the decay or the contributing decay mechanisms have the same weak phases. However, there are other strategies, involving also decays to CP non-eigenstates, that provide clean measurements of the weak phases. They are discussed in detail in Buras and Fleischer, 1998 and the present situation of these efforts is summarized in Borissov et al., 2013.

CKM Matrix and the Unitarity Triangle

The central for discussion of CP violation in the Standard Model is the unitary $3\times3$ CKM matrix (Cabibbo, 1963; Kobayashi and Maskawa, 1973). It connects the weak eigenstates $(d^\prime,s^\prime,b^\prime)$ and the corresponding mass eigenstates $d,s,b$ through \[\tag{35} \left(\begin{array}{c} d^\prime \\ s^\prime \\ b^\prime \end{array}\right)= \left(\begin{array}{ccc} V_{ud}&V_{us}&V_{ub}\\ V_{cd}&V_{cs}&V_{cb}\\ V_{td}&V_{ts}&V_{tb} \end{array}\right) \left(\begin{array}{c} d \\ s \\ b \end{array}\right)\equiv\hat V_{\rm CKM}\left(\begin{array}{c} d \\ s \\ b \end{array}\right). \]

Many parameterizations of the CKM matrix have been proposed in the literature. Presently the following parametrization is commonly used \[\tag{36} \hat V_{\rm CKM}= \left(\begin{array}{ccc} c_{12}c_{13}&s_{12}c_{13}&s_{13}e^{-i\delta}\\ -s_{12}c_{23} -c_{12}s_{23}s_{13}e^{i\delta}&c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i\delta}& s_{23}c_{13}\\ s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\delta}&-s_{23}c_{12} -s_{12}c_{23}s_{13}e^{i\delta}&c_{23}c_{13} \end{array}\right)\,, \] with $c_{ij}=\cos\theta_{ij}$ and $s_{ij}=\sin\theta_{ij}$ ($i,j=1,2,3$) and the complex phase $\delta$ necessary for {\rm CP} violation. Thus CKM matrix depends on three angles $\theta_{12}$, $\theta_{13}$, $\theta_{23}$ and one phase that are not predicted by the theory but have to be determined from experiment. Presently, all four parameters are rather precisely determined and rather accurately $c_{13}=c_{23}=1$. One has then \[ s_{12}=|V_{us}|=0.2252(9), \quad s_{13}=|V_{ub}|=3.7(5)\cdot 10^{-3}, \] \[ s_{23}=|V_{cb}|=40.9(13)\cdot 10^{-3}, \quad \delta=68(4)^\circ, \] where the numbers in parentheses give the errors that should be significantly decreased in this decade.

In view of the smallness of $s_{ij}$ for analytic calculations it is useful to expand the elements of the CKM matrix in terms of a parameter $\lambda=s_{12}$ and use the Wolfenstein parameters (Wolfenstein, 1945), \[\tag{37} \lambda, \qquad A, \qquad \varrho, \qquad \eta \, . \] This can be done efficiently by making the following change of variables in the standard parametrization (36) (Buras et al., 1994) \[\tag{38} s_{12}=\lambda\,, \qquad s_{23}=A \lambda^2\,, \qquad s_{13} e^{-i\delta}=A \lambda^3 (\varrho-i \eta)~. \]

One finds then in particular \[\tag{39} V_{ud}=1-\frac{1}{2}\lambda^2-\frac{1}{8}\lambda^4, \qquad V_{ub}=A \lambda^3 (\varrho-i \eta), \] \[ V_{cd}=-\lambda, \qquad V_{cb}=A\lambda^2 \] \[ V_{td}=A\lambda^3(1-\bar\varrho-i\bar\eta),\qquad V_{tb}=1-\frac{1}{2} A^2\lambda^4, \] where \[\tag{40} \bar\varrho=\varrho (1-\frac{\lambda^2}{2}), \qquad \bar\eta=\eta (1-\frac{\lambda^2}{2}) \] and terms ${\cal O}(\lambda^5)$ and higher order terms have been neglected. A non-vanishing $\eta$ is responsible for CP violation in the SM. It plays the role of $\delta$ in the standard parametrization.

This formulation is very useful for a graphical representation of the most popular relation between the elements of the CKM-matrix implied by its unitarity \[\tag{41} V_{ud}^{}V_{ub}^* + V_{cd}^{}V_{cb}^* + V_{td}^{}V_{tb}^* =0. \]

Indeed the relation (41) can be represented as a “unitarity” triangle in the complex $(\bar\varrho,\bar\eta)$ plane shown in Fig. 2.
Figure 2: Unitarity Triangle.
One can demonstrate that the sides and angles of this triangles are invariant under any phase-transformations common in field theories and are physical observables. Consequently they can be measured directly in suitable experiments.

In particular we have \[ V_{ub}=|V_{ub}| e^{-i\gamma}, \quad V_{td}=|V_{td}| e^{-i\beta}, \quad V_{ts}=-|V_{ts}| e^{-i\beta_s} \] with \[ \gamma\approx 68^\circ, \quad \beta\approx 22^\circ, \quad \beta_s \approx -1^\circ\, . \] The area of the unitarity triangle is related to the measure of CP violation $J_{\rm CP}$ (Jarlskog, 1985): \[ \mid J_{\rm CP} \mid = 2\cdot A_{\Delta}, \] where $A_{\Delta}$ denotes the area of the unitarity triangle.

The lengths $CA$ and $BA$ in the unitarity triangle are given respectively by \[\tag{42} R_b \equiv \frac{| V_{ud}^{}V^*_{ub}|}{| V_{cd}^{}V^*_{cb}|} = \sqrt{\bar\varrho^2 +\bar\eta^2} = (1-\frac{\lambda^2}{2})\frac{1}{\lambda} \left| \frac{V_{ub}}{V_{cb}} \right|, \] \[\tag{43} R_t \equiv \frac{| V_{td}^{}V^*_{tb}|}{| V_{cd}^{}V^*_{cb}|} = \sqrt{(1-\bar\varrho)^2 +\bar\eta^2} =\frac{1}{\lambda} \left| \frac{V_{td}}{V_{cb}} \right|. \]

The unitarity relation (41) can be rewritten as \[\tag{44} R_b e^{i\gamma} +R_t e^{-i\beta}=1~. \] Therefore the knowledge of $(R_t,\beta)$ allows to determine $(R_b,\gamma)$ through \[\tag{45} R_b=\sqrt{1+R_t^2-2 R_t\cos\beta},\qquad \cot\gamma=\frac{1-R_t\cos\beta}{R_t\sin\beta}. \] Similarly, $(R_t,\beta)$ can be expressed through $(R_b,\gamma)$: \[\tag{46} R_t=\sqrt{1+R_b^2-2 R_b\cos\gamma},\qquad \cot\beta=\frac{1-R_b\cos\gamma}{R_b\sin\gamma}. \] These relations are remarkable. They imply that the knowledge of the coupling $V_{td}$ between $t$ and $d$ quarks allows to deduce the strength of the corresponding coupling $V_{ub}$ between $u$ and $b$ quarks and vice versa.

The triangle depicted in Fig. 2, $|V_{us}|$ and $|V_{cb}|$ give the full description of the CKM matrix. Looking at the expressions for $R_b$ and $R_t$, we observe that within the SM the measurements of four CP conserving decays sensitive to $|V_{us}|$, $|V_{ub}|$, $|V_{cb}|$ and $|V_{td}|$ can tell us whether CP violation ($\bar\eta \not= 0$ or $\gamma \not=0,\pi$) is predicted in the SM. This fact is often used to determine the angles of the unitarity triangle without the study of CP-violating quantities.

The determination of the unitarity triangle from various observables constitutes simultaneously a test of the Standard Model as all measured observables should imply the same values of $\bar\varrho$ and $\bar\eta$. The most sophisticated analyses of unitarity triangle these days are performed by the UTfitter (Bevan et al., 2014) and CKMfitter (Charles et al., 2014) groups. The most important observables in these analyses are $\varepsilon$, $\Delta M_s$, $\Delta M_d$ and the mixing induced CP asymmetry $S_{\psi K_S}$. Details of these analyses and the relevant references can be found in these papers and in the review Buras and Girrbach, 2014. While there are few tensions between various determinations of $\bar\varrho$ and $\bar\eta$, the Standard Model describes the data well, implying that the dominant part of the presently measured CP violation and related flavour violation is consistent with the CKM framework. Yet, the oncoming flavour precision era could still reveal some presence of new physics at work.


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