52
G. Altarelli and S. Forte
Fig. 3.7 Box diagrams
describing K 0 − ¯
K 0 mixing
at the quark level at 1-loop
d
d
d
d
s
s
s
s
u, c, t
u, c, t
W
W
are in good agreement with the predictions from the SM unitary triangle [16, 17]
(see Chap. 10).
As we have discussed, due to the GIM mechanism, there are no flavour changing
neutral current (FCNC) transitions at tree level in the SM. Transitions with |F | =
1, 2 are induced at one loop level. In particular, meson mixing, i.e. M → ¯
M off
diagonal |F | = 2 mass matrix elements (with M = K, D or B neutral mesons),
are obtained from box diagrams. For example, in the case of K 0 − ¯
K 0 mixing the
relevant transition is ¯
sd → s ¯
d (see Fig. 3.7). In the internal quark lines all up-type
quarks are exchanged. In the amplitude, two vertices and the connecting propagator
(with virtual four momentum p μ ) at one side contribute a factor (u i = u, c, t):
F GI M =
i
V
∗
u i s
1
p / − m ui
V u i d ,
(3.75)
which, in the limit of equal m ui , is clearly vanishing due to the unitarity of the CKM
matrix V . Thus the result is proportional to mass differences. For K 0 − ¯
K 0 mixing
the contribution of virtual u quarks is negligible due to the small value of m u and the
contribution of the t quark is also small due to the mixing factors V ∗
ts V td ∼ o(A 2 λ 5 ).
The dominant c quark contribution to the real part of the box diagram quark-level
amplitude is approximately of the form (see, for example, [19]):
ReH box =
G 2
F
16π 2 m
2
c Re(V
∗
cs V cd )
2 η 1 O
,
(3.76)
where η 1 ∼ 0.85 is a QCD correction factor and O = ¯
d L γ μ s L ¯
s L γ μ d L is the
4-quark dimension six relevant operator. To obtain the K 0 − ¯
K 0 mixing its matrix
element between meson states must be taken which is parametrized in terms of a
“B K parameter” which is defined in such a way that B K = 1 for vacuum state
insertion between the two currents:
0
|O
| ¯
K
0
=
16
3
f K m
2
K B K ,
(3.77)
where f K ∼ 113MeV is the kaon pseudoscalar constant. Clearly to the charm
contribution in Eq. (3.76) non perturbative additional contributions must be added,
some of them of o(m 2
K /m 2
c ), because the smallness of m c makes a completely
partonic dominance inadequate. In particular, B K is best evaluated by QCD lattice
simulations. In Eq. (3.76) the factor o(m 2
c /m 2
W ) is the “GIM suppression” factor
G. Altarelli and S. Forte
Fig. 3.7 Box diagrams
describing K 0 − ¯
K 0 mixing
at the quark level at 1-loop
d
d
d
d
s
s
s
s
u, c, t
u, c, t
W
W
are in good agreement with the predictions from the SM unitary triangle [16, 17]
(see Chap. 10).
As we have discussed, due to the GIM mechanism, there are no flavour changing
neutral current (FCNC) transitions at tree level in the SM. Transitions with |F | =
1, 2 are induced at one loop level. In particular, meson mixing, i.e. M → ¯
M off
diagonal |F | = 2 mass matrix elements (with M = K, D or B neutral mesons),
are obtained from box diagrams. For example, in the case of K 0 − ¯
K 0 mixing the
relevant transition is ¯
sd → s ¯
d (see Fig. 3.7). In the internal quark lines all up-type
quarks are exchanged. In the amplitude, two vertices and the connecting propagator
(with virtual four momentum p μ ) at one side contribute a factor (u i = u, c, t):
F GI M =
i
V
∗
u i s
1
p / − m ui
V u i d ,
(3.75)
which, in the limit of equal m ui , is clearly vanishing due to the unitarity of the CKM
matrix V . Thus the result is proportional to mass differences. For K 0 − ¯
K 0 mixing
the contribution of virtual u quarks is negligible due to the small value of m u and the
contribution of the t quark is also small due to the mixing factors V ∗
ts V td ∼ o(A 2 λ 5 ).
The dominant c quark contribution to the real part of the box diagram quark-level
amplitude is approximately of the form (see, for example, [19]):
ReH box =
G 2
F
16π 2 m
2
c Re(V
∗
cs V cd )
2 η 1 O
,
(3.76)
where η 1 ∼ 0.85 is a QCD correction factor and O = ¯
d L γ μ s L ¯
s L γ μ d L is the
4-quark dimension six relevant operator. To obtain the K 0 − ¯
K 0 mixing its matrix
element between meson states must be taken which is parametrized in terms of a
“B K parameter” which is defined in such a way that B K = 1 for vacuum state
insertion between the two currents:
0
|O
| ¯
K
0
=
16
3
f K m
2
K B K ,
(3.77)
where f K ∼ 113MeV is the kaon pseudoscalar constant. Clearly to the charm
contribution in Eq. (3.76) non perturbative additional contributions must be added,
some of them of o(m 2
K /m 2
c ), because the smallness of m c makes a completely
partonic dominance inadequate. In particular, B K is best evaluated by QCD lattice
simulations. In Eq. (3.76) the factor o(m 2
c /m 2
W ) is the “GIM suppression” factor