The CKM matrix is unitary ie which leads to 9 unitarity conditions expressed in terms of the matrix elements. There are several (approximate) parameterisations of the CKM matrix, one of the more popular approaches is that of Wolfenstein , eqn(2), where the matrix elements are expressed in terms of powers of where is the Cabibbo angle. As can be seen from this parameterisation, the 9 complex elements of the matrix can be expressed in terms of 4 independent variables; three real parameters and an imaginary part of a complex number, The 18 parameters of the CKM matrix can be reduced to 4 because of the unitarity constraints and the arbitrary nature of the relative quark phases . It is the complex phase in the VCKM that leads to CP violation in the standard model.
The unitarity condition
VudVub* + VcdVcb* + VtdVtb* = 0is of particular interest since and This allows us to depict this condition as a triangle in the complex plane, as shown in fig 1. The angles of the triangle are related to the phase and can be measured in CP violating B-decays.
The non-closure of this triangle ie would suggest that our understanding of CP violation within the Standard Model was incomplete. Physics beyond the Standard Model can be further investigated, for example, by measuring CP asymmetries in several B decays that depend on the same unitarity angle or studying decays where zero asymmetries are expected in the Standard Model.
CP violation in the B system should be observable through the phenomenon of mixing, see for example . This mixing is dominated by box-diagrams with virtual fig 2.
The following decays
are into a CP eigenstate. If this is coupled with only a single diagram contributing to the decay, CP asymmetries can be constructed which are directly related to the angles of the unitarity triangle. For example, these conditions occur for the decay mode Here the number of B0d which decay at time t (where t is expressed in units of lifetime) is proportional to
By integrating eqns. (4) and (5) over time a similar asymmetry can be constructed which is proportional to . (Although for coherent B production ie the pair is produced in a definite CP state, this time integrated asymmetry is zero.) In addition this channel is experimentally very promising because of the dilepton decay of the Unfortunately additional decay diagrams contribute to the other channels listed in eqn(3) so there is no longer a complete cancellation of the hadronic matrix elements in the CP asymmetry. The channel, which is dependent on the angle is predicted to have large hadronic corrections from `penguin' diagrams, fig. 3. The (dependent on the angle ) also has additional hadronic contributions but, in addition, suffers from a very low branching fraction. Fortunately there are, of course, many other channels which can be used to measure CP violation, eg see ref. .
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