On the Flavor Problem in Strongly Coupled Theories - THEP Mainz

124 Chapter 3. Solving the Flavor Problem in Strongly Coupled Theories
the composite sector. A four quark operator
1
Λ2 �
ψ γµ T a ψ �� ψ γ µ T a ψ � , (3.69)
will consequentially only have a non-zero coefficient, if the composite sector is invariant
under the corresponding global symmetry. Note, that here the composite scale Λ is
to be identified with the KK scale.
If now as an example, the 5D gauge boson of some bulk gauge symmetry only couples
to left-handed zero modes, this will translate to composites which will only couple
to left-handed quarks in the dual theory and from the list (3.20) only C1 would
receive contributions. One might object, that in the minimal RS model the bulk
SU(2)L gauge bosons seem to contribute to all Wilson coefficients in (3.20), except
C4 because of the color structure, but this does not hold true. The electroweak neutral
current is described by the exchange of the KK excitations of the photon and the Z,
which in return are mixtures of the hypercharge U(1)Y and the neutral SU(2)L gauge
bosons, just as in the SM. The hypercharge U(1)Y gives rise to vector couplings and
its admixture is the sole reason for the Z to couple to right-handed fermions. In the
limit of vanishing hypercharge, in which c2 w → 1, s2 w → 0 and Qd → T D 3 , one can check
that all electroweak contributions vanish apart from the one for C1, for which
Q 2 d α + (T D 3 − s2 w Qd) 2 α
s 2 wc 2 w
→ � T D 3
�2 g2 , (3.70)
4π
as one would expect. In the dual theory, there are only composite mesons with
couplings to left-handed quarks and thus a four quark interaction involving righthanded
quarks will not occur.
For the same reason purely left-handed, purely right-handed and mixed chirality four
fermion interactions get contributions from color-charged composite mesons, because
the gluon couples vectorially. If this group is now extended to SU(3)L × SU(3)R, the
composite sector will again only allow for four quark operators with only quarks of
the same chirality as external legs, because
1
Λ2 �
ψL γµ T a ��
LψL ψR γ µ T a �
RψR , (3.71)
vanishes if T a L and T a R belong to different Lie groups. This is independent of the gauge
couplings for the two bulk SU(3)s and therefore of the mixing angle θ in (3.56), because
in this basis the mixed chirality operators can simply not be constructed from
the meson fields available in the composite sector. Extreme values of tan θ will only
affect the contributions to C1 and � C1.
This is hidden by the fact that we choose the linear combination with vector and the
one with axial vector couplings in (3.54) to have different BCs. In terms of these combinations,
which correspond to the equivalent linear combinations of vector mesons
in the dual theory, it looks like their contributions to C4 and C5 cancel. In other
words, considering only the bulk symmetry, the composite sector has vector mesons
which couple either to left- or to righthanded currents and a change of basis may not
alter the fact, that the Wilson coefficients of four-quark operators which include both
chiralities are zero.
One can also understand the remaining terms in C4 in (3.66). The first two terms