CP violation

CP is a symmetry that relates particles and antiparticles. If CP would be an exact symmetry than it would be hard to explain why our universe consist almost exclusively of matter and hardly any antimatter can be found.

Charge conjugation (C, particles are replaced by their antiparticles) and Parity (P, Space Inversion) are discrete operators. They are conserved symmetries in strong interactions but they are maximally violated in weak interactions. However, it was thought for some time that the combination of the two is conserved in weak interactions. But in 1964 a small CP violation was observed in the Kaon system.

The Standard Model offers the following explanation: the eigenstates of the weak interaction of quarks do not coincide with the mass eigenstates, the transition between the two bases is described by the Cabbibo-Kobayashi-Maskawa (CKM) matrix $V_{CKM}$:

$$\mathbf{V_{CKM}} = \left( \begin{array}{ccc} V_{ud} & V_{us} & V... ...\ V_{cd} & V_{cs} & V_{cb}\\ V_{td} & V_{ts} & V_{tb}\\ \end{array} \right)$$

This matrix is unitary, so its parameters are bound by unitarity relations. Four parameters of the matrix remain free. In the Wolfenstein [#!wolf!#] parametrisation, the parameters are expanded in powers of the Cabbibo angle $\theta_C$.

$$\mathbf{V_{CKM}} = \left( \begin{array}{ccc} 1-\frac{1}{2}\lambd... ...lambda^2)\\ A\lambda^3(1-\rho-i\eta) & -A\lambda^2 & 1\\ \end{array} \right)$$

where is $\lambda$ = sin$\theta_C$. One of these parameters is complex ($\eta$) and it is the one that is responsible for CP violation. The unitarity constraints can be displayed as triangles in the complex plane where the sides are products of the CKM matrix parameters. One of such triangles is shown below:

$$V_{ud}*V_{td}^* + V_{us}*V_{ts}^* + V_{ub}*V_{tb}^* = 0$$

, with angles $\alpha, \beta, \gamma$.

Figure 1.1: CKM unitarity triangle.

The area of the triangle indicates the strength of the CP violation.

In order to measure CP violation in the B system, $B^0$ or $\bar B^0$ mesons have to be produced, their flavour tagged, and their decays reconstructed. The design of HERA-B was optimized for the measurement of the ``Golden Decay'':

$$B^0 or \bar B^0 \rightarrow J/\psi K^{0}_{S}$$

which through the asymmetry in the rates of the decays of $B^0$ and $\bar B^0$ mesons to the same final state will give sin2$\beta$, one of the angles of the unitarity triangle, with very little theoretical uncertainties:

$$sin2\beta \approx \frac{N_{\bar B^{0}} - N_{B^{0}}}{N_{\bar B^{0}} + N_{B^{0}}}$$

Other decays such as:

$$B^0 or \bar B^0 \rightarrow \pi^+ \pi^-$$

$$B^0 or \bar B^0 \rightarrow \rho^0 K^{0}_{S}$$

provide access to the angles $\alpha$ and $\gamma$. However, these decays are theoretically not as clean as the ``Golden Decay''. Due to contributing ``penguin'' amplitudes which are not easy to calculate, this affects the correspondence between the asymmetries and the angles of the unitarity triangle.