On the Flavor Problem in Strongly Coupled Theories - THEP Mainz

2.4. Profiles of Fermions 77
on the more general basis. The orthonormality relation (2.125) gives rise to
2
� 1
ɛ
dt t � f ± n (t, c) �2 1
=
N 2 n(c) ± f + n (1− , c) f − n (1− , c)
, (2.147)
xn
so that the normalization factor in (2.145) is fixed by
N −2
n (c) = � f + n (1 − , c) �2 � −
+ fn (1 − , c) �2 2c
− f
xn
+ n (1 − , c) f − n (1 − , c) − ɛ 2 f + n (ɛ, c) 2 .
(2.148)
The IR BCs determine the mass eigenvalues xn through (2.141), which for the zero
modes are well approximated by xn ≪ 1, or equivalently v ≪ MKK. It is therefore
convenient to expand (2.145) in this limit,
C Q,q
n (t) ≈
S Q,q
n (t) ≈ ±
� Lɛ
π F (cQ,q) t cQ,q ,
�
Lɛ
π xnF (cQ,q) t1+cQ,q − ɛ1+2cQ,q t−cQ,q , (2.149)
1 + 2cQ,q
which is suggestively called zero mode approximation (ZMA). Accordingly, the function
�
1 + 2c
F (c) ≡ sgn[cos(πc)]
, (2.150)
1 − ɛ1+2c which is proportional to the value of the zero mode C-profiles on the IR brane will be
called zero-mode profile. Note, that the S-profiles on the IR brane are proportional
to the inverse of the zero-mode profile.
We finish the discussion of the fermion profiles by giving the approximate behavior of
the zero-mode profile for different regions of the localization parameters
⎧
⎪⎨ −
F (c) ≈
⎪⎩
√ 1
−c− −1 − 2c ɛ 2 , −3/2 < c < −1/2 ,
. (2.151)
√
1 + 2c , −1/2 < c < 1/2 .
Since S−profiles are suppressed by xn compared to the C-profiles, the coupling between
fermions and the Higgs is controlled by (2.151), if the Yukawas are anarchic
matrices. As a consequence of the curvature of the extra dimension, the KK modes of
gauge bosons are also peaked towards the IR, indicated by the t dependence of (2.106).
Therefore, the behavior of the zero-mode profile shows that a UV localized fermion
zero mode will have exponentially small mass compared to the electroweak scale as
well as exponentially suppressed couplings to KK gauge bosons. In other words, the
same mechanism responsible for light masses ensures that flavor changing couplings,
which are generically mediated by the KK excitations of gauge bosons. This is the
RS-GIM mechanism and will be the subject of the next section. Note, that for the IR
localized top quark however, large overlaps and therefore large effects are expected,
which will be further examined in Chapter 4.