On the Flavor Problem in Strongly Coupled Theories - THEP Mainz

2.4. Profiles of Fermions 71
which simultaneously give the mass eigenvalues of the KK modes xn. It follows also
from (2.118), that the BCs of fL and fR must be opposite, so that it is enough to give
one of them and in the following, the notion “BCs of Q” will always be understood
to be the BCs of its fL component. The following choices of BCs are possible,
fL
fR
(NN) (DD)
(DD) (NN)
(ND) (DN)
(DN) (ND)
But only the first two lines will give rise to a zero mode for either the fL or fR
component. The zero modes are massless, but introducing Yukawa couplings with a
brane localized Higgs will result in zero mode masses which depend on the overlap
of the corresponding Bessel function with the IR brane, and thus on the bulk mass
parameter c. In this context of model building, the third and last line give still
interesting solutions, because the (DN) component mimics a (NN) solution with
a Higgs localized on the UV brane, which implies that a IR localization c > −1/2
results in a very light first KK mode. These ultralight KK fermions have first been
recognized in [119, App A.2] and can arise in theories in which the SM top is in an
extended multiplet with additional fermions, which do not have a zero mode due to
(DN) BCs. Candidates are GUTs as in the original reference or higgsless models as
examined in [120].
An interpretation of the different BCs in the dual theory can be given in accordance
with the interpretation of a bulk gauge boson. If the BC in the UV is Neumann,
there exists an elementary fermion which mixes with a composite state of the same
quantum number. Only for (NN) BCs will this elementary state remain massless,
Dirichlet BC in the IR result in a mass depending on the scale of the IR brane and
the localization along the bulk. In the case of (DN) or (DD) BCs the dual theory
has no elementary fermion and the Dirichlet BC in the IR make only for a correction
of the masses of the composite meson tower.
In order to model SM fermions, we will rely on the first two solutions, so that the
masses and the integration constant is fixed by
αn = Jc∓ 1 (xnɛ)
2
Y 1
c∓ (xnɛ)
2
, (2.121)
in which the − sign holds if fL has a zero mode and and the + for the case that fR
has a zero mode.
In order to model the SM field content with two chiral zero modes for each quark
generation, we will have to introduce three copies of Q = (u, d) transforming as a
doublet under SU(2)L and a set of three downtype and three up-type SU(2)L singlets
q c = u c , d c , as already hinted at in Section 2.1. The Yukawa interactions couple these
bulk fermions to the Higgs, which should result in a Yukawa coupling for the zero
modes and therefore implies, that the zero mode of Q corresponds to an fL solution,
while the zero mode of q c is the fR solution of the bulk EOM. The ansatz (2.16) gives