On the Flavor Problem in Strongly Coupled Theories - THEP Mainz

it is then a good approximation to consider VCKM ∼ U d L
1.2. Solutions to the Flavor Problem 29
to first order. If one further
assumes the Yukawa matrices to be symmetric, which implies � U d �∗ R = U d
L , one finds
Yd = U d RY diag
d
(U d L) † � V ∗ CKMY diag
d V †
CKM
⎛
� ⎝
(d + s)λ 4 sλ 3 A(ρ − iη)λ 3
sλ 3 sλ 2 Aλ 2
A(ρ − iη)λ 3 Aλ 2 1
⎞
√
2mb ⎠ . (1.61)
v
In the less restrictive case that Yd is not symmetric, but (U d R )∗ ij ∼ (U d L )ij, the coefficients
in front of the λ n s are undetermined, but the hierarchical structure remains.
From this analysis, no statement can be made about the structure of the Yukawa in
the up-sector Yu. However, as we will see, flavor symmetries that act in the same way
in the up- and down sector will relate the structures of Yu and Yd and also theories
with Yukawa unification at some GUT scale suggest such a relation.
It is safe to say that the approaches which try to explain this flavor structure are not
as inventive as the different solutions to the hierarchy problem. It basically comes
down to the question of what represents the source of the small parameter λ in (1.61).
Most of these models rely on abelian flavor symmetries, which have problems with
FCNCs from the irrelevant operators 1.1. On the other hand, non-abelian symmetry
groups, which lead to a successful suppression of these operators abandon an explanation
for the structure of the Yukawa matrices. The following sections will give a short
overview over the different models and highlight the role of partial compositeness as
a solution to the flavor problem.
Abelian Flavor Symmetries
Motivated by the fact, that the Yukawa matrices with the structure (1.61) (λ ↔ λu
for the up-sector) can reproduce the correct quark masses and a hierarchical CKM
matrix, Froggatt and Nielsen realized [70], that one can write down a simple ansatz
that parametrizes the small parameter λ, in writing the Yukawa couplings as
� �n φ
ΛFl
q L H q c R . (1.62)
Here, one assumes a local continuous U(1) flavor symmetry with different charges for
left- and right-handed quarks depending on their family. The SM Higgs is assumed to
be uncharged, but n insertions of a flavor charged scalar φ, the flavon, are necessary
in order to end up with an invariant operator, which in turn leads to a corresponding
power suppression by the flavor scale ΛFl. This scalar will take on a vev, so that
〈φ〉
∼ λ , (1.63)
ΛFl