On the Flavor Problem in Strongly Coupled Theories - THEP Mainz

A List of Solutions to the Flavor Problem
3.3. Limitations of the RS GIM Mechanism 111
The situation described in the last section has triggered many new ideas in order to
make the bound from ɛK compatible with a KK scale in the ballpark of a TeV. This
section will contain a comprehensive review of these ideas. They will be separated
in three categories. First, parametric suppressions of the relevant Wilson coefficients.
Second, global symmetries which ensure that the coefficients of the dangerous mixed
chirality operators are either zero or extremely small and finally, alternative solutions,
which rely on mechanisms that can not be categorized in one of the above.
Parametric Suppressions
A key observation is, that the enhancement of ɛK is almost solely due to the Wilson
coefficient C4 and, to a lesser extent (especially with a custodial protection), C5 in
(3.20) (which is always understood with the corresponding replacements if cited in
the context of K − ¯ K mixing and the RS superscript will be dropped if no confusion is
possible). The parametric dependence of them becomes clearest using the ZMA and
(2.158), so that
C4 = −Nc C5 ≈ − 4παs
M 2 L
KK
2mdms
v2 |(Y (5D)
d )|
| det Y (5D)
d | ≈ − 4παs
M 2 KK
L 2mdms
v 2 Y 2
d
, (3.46)
where Yd denotes an order one parameter sensitive to the 5D Yukawa couplings in
the down sector. One possibility, which was first mentioned in [178], is therefore
to increase the down sector Yukawas, which corresponds to the application of the
reparametrization invariance (2.170). In terms of the 5D language, the overlap between
zero modes and the gauge boson KK tower becomes smaller, because the down
type quarks or the 5D electroweak doublets are more UV localized. In the dual theory
this is to be interpreted as decreasing the mixing angle and thus making the downtype
quarks or the doublets more elementary and less composite. As a consequence,
given the relation between zero mode profiles induced by the masses of the quarks
(2.158), either the Yukawas in the up-sector have to be larger as well or the electroweak
singlet up-type quarks must be localized closer to the IR brane, i.e. must be
more composite. Both seems like a good trade-off, because bounds on FCNCs in the
up-sector are considerably less constraining. In the numerical analysis, the Yukawa
couplings for both sectors are randomized, so that the absolute values of all entries are
in the range |Y | ∈ [0.3, 3], see Appendix A for details. However, one cannot arbitrarily
increase the value of the fundamental Yukawa couplings. One reason is that the
Yukawa couplings should not become non-perturbative, because the above analysis
cannot be made if one looses perturbative control. The point at which this happens
depends on the normalization chosen in defining the dimensionless Yukawa couplings
in (2.127) and therefore there is some freedom, see the discussion in [115, Sec 3.4] and
[181, App E.4]. However, in order to balance out the enhancement in (3.41), one would
need to enhance the Yukawas by a factor of ten. Larger Yukawas might therefore
ameliorate the bound from ɛK, but at some point one simply shifts the tuning over
to the Yukawa sector. Even if this would be accepted, there are upper bounds on the