318
20. Introduction to the Phenomenology of Weak Interactions
written out the Hermitian conjugate terms explicitly, keeping the coupling
V ud complex for the sake of generality, and separating the vector from the
axial vector parts. Problem 20.11 shows that the different parts of (20.159)
transform under C as follows (normal ordering being understood in all cases):
¯
C : uγ ˆ
μ
¯
¯
ˆ
d → − ˆ
¯
dγ
μ u, ˆ
uγ ˆ
μ
ˆ
γ 5 d → ˆ
+dγ
μ γ 5 u, ˆ
(20.160)
ˆ
ˆ
and we also know that under C, W μ → −W μ
† (the dagger is as in the charged
ˆ
scalar field case, and the minus sign is as in the photon A μ case). Hence under
C, (20.159) transforms into
¯ ˆ
¯
V dγ
μ
ˆ
u ˆ W
† + V
∗ uγ ˆ
μ
¯
ˆ ˆ
ˆ
ud
μ
ud
d W μ + V ud dγ
μ
ˆ
γ 5 u ˆ W μ
†
¯
+ V ud
∗ uγ ˆ
μ
ˆ ˆ
γ 5 d W μ . (20.161)
ˆ
Under P, W μ behaves like an ordinary four-vector, so the ‘vector·vector’ products in (20.161) are even under P, while the ‘axial vector·vector’ products are
odd under P. Thus finally, under the combined CP transformation (20.159)
becomes
¯
¯
ˆ
ˆ
V dγ
μ
ud
u ˆ W μ
†
¯ ˆ ˆ
+ V ud
∗
ˆ
ˆ
¯
ˆ ˆ
uγ ˆ
μ d W
V ud dγ
μ γ 5 u ˆ W μ
†
− V ud
∗ uγ ˆ
μ
μ −
γ 5 d W μ . (20.162)
Comparing (20.159) with (20.162) we deduce the essential result that this
interaction conserves CP if and only if
V ud = V ud
∗ ,
(20.163)
that is, if the coupling is real. The same is true for all the other couplings V ij .
The couplings we have introduced in this chapter so far only involve the
real Fermi constant G F , and the elements of the Cabibbo-GIM matrix which
ˆ
enters into the relation between the weakly interacting fields (d
′ , s ˆ
′ ) and the
ˆ
fields with definite mass (d, s ˆ):
(
) (
) (
)
(
)
ˆ
d
′
cos θ C
sin θ
ˆ
ˆ
C
d
d
=
V
.
(20.164)
s ˆ
′
− sin θ
CGIM
C cos θ C
s ˆ
≡
s ˆ
All these couplings are plainly real. But could we perhaps parametrize the
ˆ
(d
′ , s ˆ
′ ) ↔ ˆ
(d, s ˆ) differently, so as to smuggle in a complex, CP-violating,
coupling?
This is the question that Kobayashi and Maskawa asked themselves in 1972
(Kobayashi 2009, Maskawa 2009). To answer it is not completely straightforward, because we can always change the phases of the quark fields by independent constant amounts. A rephasing of the quark fields in the transition
i ↔ j with coupling V ij changes V ij by the phase factor exp i(α i − α j ). We
need to know whether, after allowing for this rephasing of the quark fields, an
‘irreducible’ complex coupling can remain.
First of all, note that the matrix V CGIM appearing in (20.164) is orthogonal, and this property guaranteed the vanishing of tree-level neutral
20. Introduction to the Phenomenology of Weak Interactions
written out the Hermitian conjugate terms explicitly, keeping the coupling
V ud complex for the sake of generality, and separating the vector from the
axial vector parts. Problem 20.11 shows that the different parts of (20.159)
transform under C as follows (normal ordering being understood in all cases):
¯
C : uγ ˆ
μ
¯
¯
ˆ
d → − ˆ
¯
dγ
μ u, ˆ
uγ ˆ
μ
ˆ
γ 5 d → ˆ
+dγ
μ γ 5 u, ˆ
(20.160)
ˆ
ˆ
and we also know that under C, W μ → −W μ
† (the dagger is as in the charged
ˆ
scalar field case, and the minus sign is as in the photon A μ case). Hence under
C, (20.159) transforms into
¯ ˆ
¯
V dγ
μ
ˆ
u ˆ W
† + V
∗ uγ ˆ
μ
¯
ˆ ˆ
ˆ
ud
μ
ud
d W μ + V ud dγ
μ
ˆ
γ 5 u ˆ W μ
†
¯
+ V ud
∗ uγ ˆ
μ
ˆ ˆ
γ 5 d W μ . (20.161)
ˆ
Under P, W μ behaves like an ordinary four-vector, so the ‘vector·vector’ products in (20.161) are even under P, while the ‘axial vector·vector’ products are
odd under P. Thus finally, under the combined CP transformation (20.159)
becomes
¯
¯
ˆ
ˆ
V dγ
μ
ud
u ˆ W μ
†
¯ ˆ ˆ
+ V ud
∗
ˆ
ˆ
¯
ˆ ˆ
uγ ˆ
μ d W
V ud dγ
μ γ 5 u ˆ W μ
†
− V ud
∗ uγ ˆ
μ
μ −
γ 5 d W μ . (20.162)
Comparing (20.159) with (20.162) we deduce the essential result that this
interaction conserves CP if and only if
V ud = V ud
∗ ,
(20.163)
that is, if the coupling is real. The same is true for all the other couplings V ij .
The couplings we have introduced in this chapter so far only involve the
real Fermi constant G F , and the elements of the Cabibbo-GIM matrix which
ˆ
enters into the relation between the weakly interacting fields (d
′ , s ˆ
′ ) and the
ˆ
fields with definite mass (d, s ˆ):
(
) (
) (
)
(
)
ˆ
d
′
cos θ C
sin θ
ˆ
ˆ
C
d
d
=
V
.
(20.164)
s ˆ
′
− sin θ
CGIM
C cos θ C
s ˆ
≡
s ˆ
All these couplings are plainly real. But could we perhaps parametrize the
ˆ
(d
′ , s ˆ
′ ) ↔ ˆ
(d, s ˆ) differently, so as to smuggle in a complex, CP-violating,
coupling?
This is the question that Kobayashi and Maskawa asked themselves in 1972
(Kobayashi 2009, Maskawa 2009). To answer it is not completely straightforward, because we can always change the phases of the quark fields by independent constant amounts. A rephasing of the quark fields in the transition
i ↔ j with coupling V ij changes V ij by the phase factor exp i(α i − α j ). We
need to know whether, after allowing for this rephasing of the quark fields, an
‘irreducible’ complex coupling can remain.
First of all, note that the matrix V CGIM appearing in (20.164) is orthogonal, and this property guaranteed the vanishing of tree-level neutral