On the Flavor Problem in Strongly Coupled Theories - THEP Mainz

114 Chapter 3. Solving the Flavor Problem in Strongly Coupled Theories
elements of ( � δD)ij take the form
( � δD)ij =
2(3 + cQi − cQj )
(3 + 2cQi )(3 − 2cQj )(2 + cQi − cQj )
F 2 (cQi )
F 2 . (3.48)
(cQj )
An analogous expression holds in the case of ( � δd)ij with cQi replaced by cdi . Since
all bulk mass parameters are close to −1/2 it is also a good approximation to replace
the rational function of cQi and cQj in (3.48) by the numerical factor 3/8. Using the
mass relations (2.158) as well as (2.159), it is straightforward to deduce from (3.47)
that (3.46) is replaced by
C al
4 = −Nc C al
5 ≈ − 4παs
M 4 KK
L mdms ≈ v2Y 2
d
2M 2 C4 , (3.49)
KK
in which the last expression gives the relation to the Wilson coefficient in the RS
model with anarchical bulk mass parameters. Depending on the size of the Yukawa
couplings, this allows to cancel the enhancement from renormalization group running
and the matrix elements for KK scales in the range of 1 − 2 TeV. Notice that the
O(v 2 /M 2 KK ) corrections from (� δd)ij have been neglected in (3.49), which is justified,
because the universality of ( � δd)ij ∼ 1 in combination with the unitarity of the U R d
matrices renders them negligibly small.
Alternative Solutions
One of the alternative strategies to ameliorate the RS flavor problem is to allow for
the Higgs to be shifted into the bulk [187]. In the case of a brane Higgs, the Yukawa
interactions are brane-localized operators, because the overlap with a brane Higgs and
the bulk fermions are given by the values of the fermion zero modes and the IR brane.
For a bulk Higgs, the overlap integral does include the Higgs profile, which is typically
exponentially peaked towards the IR brane, and will therefore be numerically larger
than the value of the fermion profiles at the brane (the “tail” of the Higgs profile will
always make for a larger overlap). In order for the mass relations to hold, this means
that the fermion profiles must be shifted towards the UV. One can imagine the effect
of a bulk Higgs by multiplying the zero mode profiles F (cQi ) and F (cqi ) in (2.158) by
the value of the overlap integral with the Higgs. This factor must be compensated by
the zero mode profiles, which, as in the case of larger Yukawas, results in a negative
shift in the bulk masses cQi
and cqi .
The Wilson coefficient C4 in (3.20) is generated by KK gluon exchange alone, and
since the gluon KK modes are not affected at all by the localization of the Higgs, they
are unchanged for a bulk Higgs compared to the brane Higgs scenario. Consequentially,
the UV shift of the quark profiles leads to smaller overlap integrals with the
KK tower of the gluon and therefore the more the Higgs is shifted into the bulk, the
more are FCNC constraints defused, compare [181, Table 1].
In the dual theory, promoting the Higgs to a bulk field makes it a mixed state of an
elementary and a composite scalar. One might object, that an elementary scalar in
the theory is contrary to the original idea of a composite Higgs as a solution to the
hierarchy problem. However, as long as the Higgs stays close to the IR brane the