  We clarify some aspects of the LSZ formalism and wave function
renormalisation for unstable particles in the presence of electroweak
interactions when mixing and CP violation are considered. We also analyse the
renormalisation of the CKM mixing matrix which is closely related to wave
function renormalisation. We critically review earlier attempts to define a set
of "on-shell" wave function renormalisation constants. With the aid of an
extensive use of the Nielsen identities complemented by explicit calculations
we corroborate that the counter term for the CKM mixing matrix must be
explicitly gauge independent and demonstrate that the commonly used
prescription for the wave function renormalisation constants leads to gauge
parameter dependent amplitudes, even if the CKM counter term is gauge invariant
as required. We show that a proper LSZ-compliant prescription leads to gauge
independent amplitudes. The resulting wave function renormalisation constants
necessarily possess absorptive parts, but we verify that they comply with the
expected requirements concerning CP and CPT. The results obtained using this
prescription are different (even at the level of the modulus squared of the
amplitude) from the ones neglecting the absorptive parts in the case of top
decay. The difference is numerically relevant.
