On the Flavor Problem in Strongly Coupled Theories - THEP Mainz

2.3. Profiles of Gauge Bosons 61
pure composite exchange, which can be found in the third line of Figure 2.7. Keep in
mind, that these diagrams are flavor-diagonal in Figure 2.7, but not flavor universal,
which will lead to the described flavor violation after rotation into the basis in which
the Yukawa matrices are diagonal. Also, the diagrams in the second and third line
of Figure 2.7 can give flavor conserving and ∆F = 1 contributions as well, the right
hand side only denotes the maximal number of vertices at which flavor violation can
occur simultaneously. Further diagrams can be constructed through additional mixing
from composites into the elementary states and back. However, each additional Oµ
propagator carries another factor M −2
KK and will the corresponding terms will thus be
suppressed.
An interesting result of this thesis is, that we are able to assign diagrams in the dual
theory to the different contributions of (2.82), by systematically considering different
BCs. In the following, the respective BCs and the contributing diagrams are
shown on the left hand side. Also, after performing the limits for the respective BCs,
ɛ-suppressed factors will be omitted and the results are given in Feynman-’t Hooft
gauge, ξ = 1. Let us start with the “cleanest” scenario.
(DN) Dirichlet BCs in the UV imply, that there is no elementary gauge boson
in the theory. Consequentially, there is no mixing as in (2.41). If there
were no composite fermions, the elementary and composite sector would
not talk to each other at all. There is also no symmetry breaking in the
composite sector, because all fields (for all possible gauge indices) have
Neumann BCs in the IR. Referring to Figure (2.7), that means, that we
do only expect the exchange of composite states, i.e. only the diagram
in the third line. Interestingly, for these BCs, (2.82) reduces to
D ξ=1
µν (q, t; t ′ ) = ηµν L
4πrc M 2 t
KK
2 < , (2.84)
which represents the term which we found to appear in (2.82) with
always the same coefficient regardless of the BCs. In the dual theory,
this corresponds to the exchange of exclusively composites, which is
only controlled by the bulk. That means, one can check that a different
localization, ∆ �= 3, will in general change this factor. This term will
however always be present as long as ∆ = 3 and a change of BCs will
only generate additional diagrams.