The equation of motion inside some material can be written as:
By comparing eq. 53 with the eq. 2, we can immediately write down the solutions for the time evolution of a neutral kaon state inside some material:
The mass and lifetime eigenvalues of and
change equally in
first order of the forward scattering amplitudes.
And the time evolution of a or
in matter becomes:
For an arbitrary inital neutral kaon state we can write following the formalism in section 1:
Consider the standard case of an initial pure :
(
counts time after regenerator,
time inside
regenerator):
To get a more simple expression, assume conservation, i.e.
:
where ,
,
and
.
A similar calculation yields:
For evaluating and
decays in vacuum, it is useful to rotate to
the
basis using the transformation matrix:
with
The decay rates for initial pure and
into
after
the regenerator is then given by:
The decay rates for initial pure or
after a regenerator where
the neutral kaons travel first through vacuum before reaching the regenerator
can be calculated similarly:
Moreover, the decay rate after different regenerators can be easily calculated by multiplying the corresponding matrices: