3 The Standard Model of Electroweak Interactions
51
It has become customary to make the replacement ρ, η → ¯
ρ, ¯
η with:
ρ − iη =
¯
ρ − i ¯
η
√
1 − λ 2
∼ ( ¯
ρ − i ¯
η)(1 + λ
2 /2 + . . . ).
(3.72)
Present values of the CKM parameters as obtained from experiment are [16] [17] (a
survey of the current status of the CKM parameters can also be found in Ref. [5]):
λ = 0.2258 ± 0.0014
A = 0.818 ± 0.016
¯
ρ = 0.164 ± 0.029;
¯
η = 0.340 ± 0.017
(3.73)
A more detailed discussion of the experimental data is given in Chap. 10.
In the SM the non vanishing of the η parameter (related to the phase ϕ in
Eqs. 3.69 and 3.70) is the only source of CP violation. Unitarity of the CKM matrix
V implies relations of the form
a V ba V ∗
ca = δ bc . In most cases these relations
do not imply particularly instructive constraints on the Wolfenstein parameters. But
when the three terms in the sum are of comparable magnitude we get interesting
information. The three numbers which must add to zero form a closed triangle in the
complex plane, with sides of comparable length. This is the case for the t-u triangle
(unitarity triangle) shown in Fig. 3.6 (or, what is equivalent in first approximation,
for the d-b triangle):
V td V
∗
ud + V ts V
∗
us + V tb V
∗
ub = 0
(3.74)
All terms are of order λ 3 . For η = 0 the triangle would flatten down to vanishing
area. In fact the area of the triangle, J of order J ∼ ηA 2 λ 6 , is the Jarlskog invariant
[18] (its value is independent of the parametrization). In the SM all CP violating
observables must be proportional to J, hence to the area of the triangle or to η. A
direct and by now very solid evidence for J non vanishing is obtained from the
measurements of and in K decay. Additional direct evidence is being obtained
from the experiments on B decays at beauty factories and at the TeVatron where the
angles β (the most precisely measured), α and γ have been determined. Together
with the available information on the magnitude of the sides all the measurements
Fig. 3.6 The unitarity
triangle corresponding to
Eq. (3.74)
1V V
tb ub
*
V V
tb ud
*
V V
tb us
*
51
It has become customary to make the replacement ρ, η → ¯
ρ, ¯
η with:
ρ − iη =
¯
ρ − i ¯
η
√
1 − λ 2
∼ ( ¯
ρ − i ¯
η)(1 + λ
2 /2 + . . . ).
(3.72)
Present values of the CKM parameters as obtained from experiment are [16] [17] (a
survey of the current status of the CKM parameters can also be found in Ref. [5]):
λ = 0.2258 ± 0.0014
A = 0.818 ± 0.016
¯
ρ = 0.164 ± 0.029;
¯
η = 0.340 ± 0.017
(3.73)
A more detailed discussion of the experimental data is given in Chap. 10.
In the SM the non vanishing of the η parameter (related to the phase ϕ in
Eqs. 3.69 and 3.70) is the only source of CP violation. Unitarity of the CKM matrix
V implies relations of the form
a V ba V ∗
ca = δ bc . In most cases these relations
do not imply particularly instructive constraints on the Wolfenstein parameters. But
when the three terms in the sum are of comparable magnitude we get interesting
information. The three numbers which must add to zero form a closed triangle in the
complex plane, with sides of comparable length. This is the case for the t-u triangle
(unitarity triangle) shown in Fig. 3.6 (or, what is equivalent in first approximation,
for the d-b triangle):
V td V
∗
ud + V ts V
∗
us + V tb V
∗
ub = 0
(3.74)
All terms are of order λ 3 . For η = 0 the triangle would flatten down to vanishing
area. In fact the area of the triangle, J of order J ∼ ηA 2 λ 6 , is the Jarlskog invariant
[18] (its value is independent of the parametrization). In the SM all CP violating
observables must be proportional to J, hence to the area of the triangle or to η. A
direct and by now very solid evidence for J non vanishing is obtained from the
measurements of and in K decay. Additional direct evidence is being obtained
from the experiments on B decays at beauty factories and at the TeVatron where the
angles β (the most precisely measured), α and γ have been determined. Together
with the available information on the magnitude of the sides all the measurements
Fig. 3.6 The unitarity
triangle corresponding to
Eq. (3.74)
1V V
tb ub
*
V V
tb ud
*
V V
tb us
*