On the Flavor Problem in Strongly Coupled Theories - THEP Mainz

28 Chapter 1. Introduction: Problems beyond the Standard Model
Observable Operator Bound on Λ in TeV
ɛK (ds c )(ds c ) 10 4 − 10 5
∆mK (ds c )(ds c ) 10 3 − 10 4
∆mD (cu c )(cu c ) 10 2 − 10 3
∆mBd (bd c )(bd c ) 10 2 − 10 3
Table 1.1: Rough bounds from Flavor observables on the suppression scale of the
corresponding four fermion operators.
full U(3) 5 flavor symmetry, which is only broken by the Yukawa interactions.
The functions F (mi) and ˜ F (mi, mj) however depend on the quark masses and break
the GIM protection. This dependence still leads in many cases to a powerful quadratic
suppression, e.g. ∼ (m2 u − m2 c)/M 2 W , but can be weaker in processes where it is only
logarithmic, e.g. ∼ log (mu/mc). 20 New Physics at the TeV scale will generically
introduce new particles with TeV masses, leading to large flavor changing neutral
currents (FCNCs) already at tree level, which are excluded by the good agreement of
various flavor observables with the SM. One might therefore find a viable solution for
the hierarchy problem, but the flavor sector will push the scale at which it is realized
orders of magnitude above the TeV scale.
In the following, the emphasis will be put on the quark sector again, even though
flavor violation is even more phenomenologically constrained in the lepton sector [157].
A solution to the flavor problem must then at least provide an explanation to the
question why the Wilson coefficients of the operators cited in Table 1.1 should be
small. Ideally, it would also explain the structure of the Yukawa matrices, which is
put in by hand in the SM. The exact structure of the Yukawa matrices cannot be
measured, because many parameters are not physical, but it can be approximated
under some assumptions [66]. Using the Wolfenstein parametrization [68], one can
write the CKM matrix and the Yukawa matrices in the form
⎛
⎞
⎛
⎞
Y diag
d
dλ
= ⎝
4 0
0
sλ
0
2 0 0
⎛
√
2mb
0⎠
v
1
VCKM = ⎝
, Y diag
u
1 − λ2 /2 λ Aλ3 (ρ − iη)
−λ 1 − λ2 /2 Aλ2 Aλ3 (ρ − iη) −Aλ2 1
uλ
= ⎝
4 u
0
0
cλ
0
2 0
u
0
⎞
√
2mt
0⎠
v
1
(1.59)
⎠ , (1.60)
with λ � 0.2 denoting the Cabbibo angle, λu � 0.06, and all other coefficients of O(1).
The CKM matrix is given by VCKM = (U u L )† U d L , where U u L and U d L denote the unitary
rotation matrices which rotate the left-handed up- and down-type quarks from the
mass to the interaction eigenbasis. One can choose the rotation matrices such that
U d L and U d R depend only on λ and U u L and U u R only on λu. Since the Cabibbo angle is
roughly equal to the parameter λ � � md/ms, which determines Y diag
d
, and λu < λ,
20 One might argue, that the top mass leads to a considerable breaking of the GIM, but it will
come with a very small CKM element, canceling this effect.