538
K. Kleinknecht and U. Uwer
The proposed models can be grouped into the following four categories:
1. Millistrong CP violation models [29–31] postulate the existence of C- and T -
violating terms of order 10 −3 in the strong interaction. The process K L →
π + π − is supposed to occur by the interference of two amplitudes: first, the K L
decays via the normal CP-conserving weak interaction, with = 1, into an
intermediate state X, and then this state decays into π + π − by a T -violating
strong interaction. The amplitude of the process is of order G F a, where G F
is the Fermi coupling constant and a is the coupling of this CP-violating
strong interaction. From the experimental value of |η +− |, one can conclude that
a ≈ 10 −3 .
2. Electromagnetic CP violation models [32–35] require large parts of the electromagnetic interaction of hadrons to be C- and T -violating, but P-conserving. A
two-step process K L → X → 2π could then occur through the interference of a
weak and an electromagnetic CP-violating amplitude. The product of G F with
the fine structure constant α is not too far from G F × 10 −3 , as required by the
magnitude of |η +− |.
3. Milliweak models assume that a part, of the order of 10 −3 , of the weak interaction
is CP-violating and is responsible for the observed effects. The decay K L → 2π
is then a one-step process, hence the name “direct CP violation”, and CP or T
violations of the order of 10 −3 should show up in other weak processes [36–44].
In these models, based on two doublets of quarks, CP violation is introduced in
different ways. In one example [44], CP violation is due to the Higgs couplings,
with flavor-changing neutral currents allowed; in another one [42], it is due
to right-handed weak currents. A bold alternative was considered in 1973 by
Kobayashi and Maskawa [48]: they saw that if there are three doublets of quarks,
there is a possibility of CP violation in the 3 × 3 weak quark mixing. Today, with
six quarks observed, this seems the most natural modelas discussed below.
4. The superweak model [45] postulates a new = 2 CP-violating interaction
that has a coupling (coupling constant g) smaller than the second-order weak
interaction. This interaction could induce a transition K L → K S , with a subsequent decay K S → 2π. More precisely, this interaction would cause a first-order
transition matrix element
M SW = =K|H SW |K ∼ gG F .
(9.58)
The mass difference itself is related to the second-order weak matrix element
M KK =
n
K|H W |n W |K
E K − E n + iε
.
(9.59)
where n is an intermediate state with energy E n and H W is the weak
Hamiltonian. In order that the CP-violating amplitude for K L → 2π relative
to the CP-conserving amplitude should be of the observed magnitude, the
ratio M SW /M KK must be of the order of 10 −3 . Since M SW ≈ gG F and
K. Kleinknecht and U. Uwer
The proposed models can be grouped into the following four categories:
1. Millistrong CP violation models [29–31] postulate the existence of C- and T -
violating terms of order 10 −3 in the strong interaction. The process K L →
π + π − is supposed to occur by the interference of two amplitudes: first, the K L
decays via the normal CP-conserving weak interaction, with = 1, into an
intermediate state X, and then this state decays into π + π − by a T -violating
strong interaction. The amplitude of the process is of order G F a, where G F
is the Fermi coupling constant and a is the coupling of this CP-violating
strong interaction. From the experimental value of |η +− |, one can conclude that
a ≈ 10 −3 .
2. Electromagnetic CP violation models [32–35] require large parts of the electromagnetic interaction of hadrons to be C- and T -violating, but P-conserving. A
two-step process K L → X → 2π could then occur through the interference of a
weak and an electromagnetic CP-violating amplitude. The product of G F with
the fine structure constant α is not too far from G F × 10 −3 , as required by the
magnitude of |η +− |.
3. Milliweak models assume that a part, of the order of 10 −3 , of the weak interaction
is CP-violating and is responsible for the observed effects. The decay K L → 2π
is then a one-step process, hence the name “direct CP violation”, and CP or T
violations of the order of 10 −3 should show up in other weak processes [36–44].
In these models, based on two doublets of quarks, CP violation is introduced in
different ways. In one example [44], CP violation is due to the Higgs couplings,
with flavor-changing neutral currents allowed; in another one [42], it is due
to right-handed weak currents. A bold alternative was considered in 1973 by
Kobayashi and Maskawa [48]: they saw that if there are three doublets of quarks,
there is a possibility of CP violation in the 3 × 3 weak quark mixing. Today, with
six quarks observed, this seems the most natural modelas discussed below.
4. The superweak model [45] postulates a new = 2 CP-violating interaction
that has a coupling (coupling constant g) smaller than the second-order weak
interaction. This interaction could induce a transition K L → K S , with a subsequent decay K S → 2π. More precisely, this interaction would cause a first-order
transition matrix element
M SW = =K|H SW |K ∼ gG F .
(9.58)
The mass difference itself is related to the second-order weak matrix element
M KK =
n
K|H W |n W |K
E K − E n + iε
.
(9.59)
where n is an intermediate state with energy E n and H W is the weak
Hamiltonian. In order that the CP-violating amplitude for K L → 2π relative
to the CP-conserving amplitude should be of the observed magnitude, the
ratio M SW /M KK must be of the order of 10 −3 . Since M SW ≈ gG F and