On the Flavor Problem in Strongly Coupled Theories - THEP Mainz

128 Chapter 3. Solving the Flavor Problem in Strongly Coupled Theories
The covariant derivative in (3.73) reads
�
� �r (DµS) αa = δabδαβ∂µ − igD5δαβ GD µ T r �
� �r � � r ∗δab
ab + igS5 GS T µ αβ Sbβ , (3.74)
in which the color indices of the SU(3)D (SU(3)S) are denoted by greek (latin) letters
and the index r = 1 . . . , 8 runs over the number of generators of the respective group.
If the vev of S is supposed to break the bulk SU(3)D × SU(3)S down to its diagonal
subgroup, it must read
Inserting the linear combinations (3.54) and using
cos θ =
〈Saα〉 = vS
√ δaα . (3.75)
2NC
gS5
�
g 2 S5 + g2 D5
and (3.56), it follows from the kinetic term
��DµS Tr
� †� � µ
D S �
∋ 1
2
, sin θ =
g 2 s5 v2 S
2NC c 2 θ s2 θ
gD5
�
g 2 S5 + g2 D5
, (3.76)
A r µA r µ . (3.77)
Comparing this with the expressions (2.93) and (2.102), one can infer that for the
axigluon
vIR = L m2 A
M 2 KK
= L
g 2 s
v 2 S
2NC c2 θ s2 θ M 2 KK
, (3.78)
which we already anticipated in (3.64). In SM implementations of chiral color, vS is
directly proportional to the axigluon mass. This is also true for an axigluon implemented
in an RS model, if the IR brane provides the only source of SU(3)D × SU(3)S
breaking. However, we could rule out this choice of BCs for the model at hand in
Section 3.4, because it leads to a too light first KK mode. This is rooted in the fact
that the IR brane is connected to the electroweak breaking scale. The vev of every
scalar confined to the IR is naturally bounded from above by the cutoff vS ≤ MKK.
However, if the axigluon has Dirichlet BCs in the UV, the mass of its first excitation
will be in the TeV range even for vS well below MKK, as discussed in Section 2.3.
Already in the SM it is questionable to assume such a minimal extension of the Higgs
sector, even if the vev of S can in principle be chosen arbitrarily large. The problem
arises not from the axigluon mass, but from the top quark mass. If the Yukawa
interactions are dimension five operators as in (3.59), the ratio 〈S〉/Λ should be of
order one if the top mass is to be generated with a perturbative Yukawa coupling.
That makes it problematic to neglect higher dimensional operators with additional S
insertions.
In the extension of the RS model it brings about another drawback. If vS/MKK ∼ 1,
the contributions to the mixed chirality Wilson coefficients in (3.66) do not have an
additional suppression and become equally large as the original terms in (3.20) so that