On the Flavor Problem in Strongly Coupled Theories - THEP Mainz

2.3. Profiles of Gauge Bosons 59
parametrization for the BCs,
∂tD ξ µν(q, t; t ′ �
�
) �
t=ɛ + = vUV D ξ µν(q, ɛ + ; t ′ ) , ∂tD ξ µν(q, t; t ′ �
�
) �
t=1− = −vIR D ξ µν(q, 1 − ; t ′ ) ,
(2.80)
and an equivalent set of equations for t ↔ t ′ . Here, vUV and vIR can either be
understood as functions of the vevs of some UV or IR brane localized scalars or as
to be taken in the limit where they are 0 or ∞ and represent Neumann or Dirichlet
BCs. The results derived in the following paragraphs will apply for both scenarios. It
follows for general bulk mass ∆, that
C > 1 = −q Y∆−3(q) + (−vIR + ∆ − 3)Y∆−2(q) ,
C > 2 = q J∆−3(q) − (−vIR + ∆ − 3)J∆−2(q) ,
C < 1 = −qɛ Y∆−3(qɛ) + (vUVɛ + ∆ − 3)Y∆−2(qɛ) ,
C < 2 = qɛ J∆−3(qɛ) − (vUVɛ + ∆ − 3)J∆−2(qɛ) . (2.81)
The limit ∆ → 3 for gauge bosons is straightforward.
We will study the gauge boson propagator in the effective theory, q ≪ 1, and will
elaborate an interpretation in the dual theory depending on different BCs. In general,
the small momentum limit yields
with
D ξ µν(q, t; t ′ ) = ηµν
L
4πrc M 2 KK
c1 = vUV(2 + vIR) + vIRɛ(2 − vUVɛ)
vUV(2 + vIR) + vIRɛ(2 − vUVɛ) = 1 , c2 =
c3 =
vIRɛ(vUVɛ − 2)
vUV(2 + vIR) + vIRɛ(2 − vUVɛ) , c4 =
�
c1t 2 < + c2 t 2 t ′2 + c3(t 2 + t ′2 � 2
) + c4 + O(q ) (2.82)
−vIRvUV
vUV(2 + vIR) + vIRɛ(2 − vUVɛ) ,
(2 + vIR)ɛ(2 − vUVɛ)
vUV(2 + vIR) + vIRɛ(2 − vUVɛ) .
(2.83)
The correct way to read these different contributions is to imagine a four quark diagram
with one vertex labeled by t and the other by t ′ , as shown in Figure 2.10.
Integration over t and t ′ will cause flavor violation at the corresponding vertex. If
a coefficient multiplies neither t nor t ′ , the corresponding term describes two flavor
diagonal vertices, which is the case for c4. 10 Coefficients which multiply either t or t ′
can refer to diagrams with a flavor change at either one vertex, as in the case for c3.
Finally, the coefficient of terms with both t and t ′ can induce flavor change at both
vertices This is the case for c1 and c2 in (2.82).
It makes not much sense to keep terms that are ɛ suppressed and neglect O(q 2 ) terms
in the expressions above, since ɛ ∼ 10 −16 . However, a vev on the UV brane is naturally
order Planck scale and therefore factors ∼ vUVɛ may become relevant and the ɛ
10 It is a � in flavor space and the integral over the profiles of the external fermions vanishes, due
to the orthonormality relation (2.125), derived in the next section.