On the Flavor Problem in Strongly Coupled Theories - THEP Mainz

112 Chapter 3. Solving the Flavor Problem in Strongly Coupled Theories
absolute value of the Yukawa couplings from observables which are proportional to
positive powers of them, typically ∆F = 1 observables, which arise at the one loop
level with mass insertions. The authors of [180] find that direct CP violation from
Kaon decay, measured by ɛ ′ /ɛK gives the most severe upper bound. At Yd ≈ 6, the
derived bound on MKK becomes stronger than the one from |ɛK|, so that the best one
can arrange for would be a KK scale of MKK ≥ 3 TeV.
This problem can be avoided if instead of enhancing the Yukawa couplings, one lowers
the value of the volume of the extra dimension L = − ln ɛ, which appears linearly in
both the Wilson coefficients in ∆F = 1 and ∆F = 2 effective Hamiltonians [144].
This is the idea of the LRS models, which have been introduced in Section 3.1 as
a means to reduce the contributions to the oblique parameters. The reason for this
linearity is, that the coupling of the KK modes to a good approximation reads g √ L,
which becomes apparent if one computes the contributions mode by mode instead of
using the full 5D propagators. This coupling will therefore be smaller if the bulk is
truncated at scales ɛLRS = ΛEW/ΛLRS, with ΛLRS ≪ MPl. In the dual theory this
means, that the range of scales over which the theory is strongly coupled is smaller,
i.e. asymptotic freedom sets in already at the scale ΛLRS.
However, for the zero mode profiles of the light quarks holds F (c) ∼ ɛ −c−1/2 and if ɛ →
ɛLRS which is orders of magnitude smaller, the localization parameters must adjust,
because the values of the zero mode profiles are fixed by the mass relations (2.158).
Considering the original proposal [144], ɛLRS = 10 3 (LLRS ≈ 7), this means that he
c-parameters are by a factor of two larger in magnitude than in the original model.
However, the RS-GIM mechanism, illustrated by the naive picture 2.11 assumes, that
the integrand of the overlap integrals in (2.173) is proportional to a positive power
of he bulk coordinates t and t ′ . This will only be true as long as cqi + cqj + 2 > 0
(remember, all light flavors have cQ,q < −1/2). If the c parameters become smaller,
the overlap integrals (2.176) will not be dominated by the IR localization of the gluon
KK modes, but by the extreme UV localization of the fermion zero modes, so that the
main contributions arise from the region where t ≈ ɛ and the RS-GIM suppression is
undermined. This phenomenon is called UV dominance and its implications for flavor
changing as well as flavor diagonal neutral currents, for example the Zb¯b coupling,
have been analyzed in [179]. One can turn this argument around and formulate a
lower bound on the volume LLRS > 8.2, above which the bound from ɛK is at least
not stronger, but the enhancement in (3.41) cannot be balanced out by lowering L
alone.
Global Symmetries and MFV
The dangerous coefficients C4 and C5 appear only if flavor changing vertices with
both right- and left-handed down quarks are present. This can be prevented, either
by imposing a global symmetry which arranges for the singlet or doublet bulk mass
parameters to be equal or by aligning them with the down-type Yukawa matrix [184].
The latter is an implication of minimal flavor violation, which was introduced in Section
1.2. Several variants have been proposed, in which the alignment is assumed
only in the down sector, is extended to the up sector, or holds only for the first two