9 Symmetry Violations and Quark Flavour Physics
607
The GIM mechanism [131], introduced to cancel the K L → μ + μ − amplitude,
required the orthogonal state
s c = −d sin θ + s cos θ ,
(9.215)
and an additional charge +2/3 quark, the charm quark, complementing the two
quark doublets.
We know today that three generations of up-type quarks u i (i = 1, 2, 3: u-, c-,
t-quark) with electrical charge +2/3 and three generations of down-type quarks
d i (i = 1, 2, 3: d-, s-, b-quark) with electrical charge −1/3 exist. In the Standard
Model, the masses and mixing of quarks arise from the Yukawa interaction with
the Higgs condensate which couples the left-handed quark fields u L and d L to the
right-handed quark fields u R and d R (for better readability the generation index is
suppressed). After spontaneous breaking of the electroweak symmetry the Yukawa
terms give rise to masses and mixing:
L Yukawa = −
v
√
2
¯
d L Y d d R + ¯
u L Y u u R
+ h.c.
The Yukawa matrices Y d and Y u are complex 3 × 3 matrices in generation or
flavor space and do not need to be diagonal. Indeed, in the Standard Model they are
not, and as a consequence the flavor states are not equal to the mass eigenstates of the
quarks. The mass eigenstates ˜
u L,R and ˜
d L,R are obtained by unitary transformations
of the above quark flavor states: ˜
u A = V A,u u A and ˜
d A = V A,d d A (with chirality
index A = L, R for left and right-handed quark fields, and suppressed generation
indices). The unitary matrices V A,u and V A,d diagonalize the Yukawa matrices and
one obtains the diagonal quark mass matrices for the up- and down-type quarks,
M u = diag(m u , m c , m t ) =
v
√
2
V L,u Y u V
†
R,u ,
M d = diag(m d , m s , m b ) =
v
√
2
V L,d Y d V
†
R,d .
The quark masses will appear as usual Dirac mass terms in the above Yukawa part
of the Lagrangian:
L Yukawa = − ¯ ˜
d L M d ˜
d R − ¯ ˜
u L M d ˜
u R + h.c.
If the up-type and and down-type Yuakawa matrices Y u and Y d cannot be
diagonalized simultaneously by the same transformations there is a net effect of
the change of the quark basis. The charged current terms of the Standard Model
Lagrangian combining left-handed up and down-type quarks therefore get a flavor
607
The GIM mechanism [131], introduced to cancel the K L → μ + μ − amplitude,
required the orthogonal state
s c = −d sin θ + s cos θ ,
(9.215)
and an additional charge +2/3 quark, the charm quark, complementing the two
quark doublets.
We know today that three generations of up-type quarks u i (i = 1, 2, 3: u-, c-,
t-quark) with electrical charge +2/3 and three generations of down-type quarks
d i (i = 1, 2, 3: d-, s-, b-quark) with electrical charge −1/3 exist. In the Standard
Model, the masses and mixing of quarks arise from the Yukawa interaction with
the Higgs condensate which couples the left-handed quark fields u L and d L to the
right-handed quark fields u R and d R (for better readability the generation index is
suppressed). After spontaneous breaking of the electroweak symmetry the Yukawa
terms give rise to masses and mixing:
L Yukawa = −
v
√
2
¯
d L Y d d R + ¯
u L Y u u R
+ h.c.
The Yukawa matrices Y d and Y u are complex 3 × 3 matrices in generation or
flavor space and do not need to be diagonal. Indeed, in the Standard Model they are
not, and as a consequence the flavor states are not equal to the mass eigenstates of the
quarks. The mass eigenstates ˜
u L,R and ˜
d L,R are obtained by unitary transformations
of the above quark flavor states: ˜
u A = V A,u u A and ˜
d A = V A,d d A (with chirality
index A = L, R for left and right-handed quark fields, and suppressed generation
indices). The unitary matrices V A,u and V A,d diagonalize the Yukawa matrices and
one obtains the diagonal quark mass matrices for the up- and down-type quarks,
M u = diag(m u , m c , m t ) =
v
√
2
V L,u Y u V
†
R,u ,
M d = diag(m d , m s , m b ) =
v
√
2
V L,d Y d V
†
R,d .
The quark masses will appear as usual Dirac mass terms in the above Yukawa part
of the Lagrangian:
L Yukawa = − ¯ ˜
d L M d ˜
d R − ¯ ˜
u L M d ˜
u R + h.c.
If the up-type and and down-type Yuakawa matrices Y u and Y d cannot be
diagonalized simultaneously by the same transformations there is a net effect of
the change of the quark basis. The charged current terms of the Standard Model
Lagrangian combining left-handed up and down-type quarks therefore get a flavor