On the Flavor Problem in Strongly Coupled Theories - THEP Mainz

2.2. AdS/CFT 45
value of the operator O) comes trivially out to be zero, because (2.31) is quadratic
in the sources due to the regularity condition fixing C1 to zero in (2.29), which is
analogous to the statement, that the conformal symmetry is not spontaneously broken.
Higher order correlation functions can be computed by including self-interactions in
the bulk Lagrangian with an elegant technique introduced by Witten [99].
In the RS model, the UV brane is however not at z = 0, but at some non-zero position
z = R. We should expect that this affects the dual theory, so that it is not conformally
invariant anymore above some scale ΛUV ∼ 1/R, because the 5D theory is not scale
invariant in this region. In this situation, the analytic terms in (2.33) gain importance.
The first term, independent of the momentum q, can be interpreted as a mass term for
the source field in the dual Lagrangian, while the second expression can be obtained
by a kinetic term for the source field. They correspond to the usual propagator of a
4D scalar field with
m 2 eff
= 2(∆ − 4)(∆ − 3)
R 2 , (2.39)
and not explicitly given higher order analytic terms correspond to terms with additional
derivatives. Thus, the source field becomes dynamical and the dual theory
describes a conformal sector, probed by some field ϕ(x), called elementary for reasons
to become clear soon, which explicitly breaks conformal invariance at the scale at
which we placed the UV brane. Accordingly, the two-point function on the left-hand
side of (2.33) does not only describe the propagator of the conformal operators, but
of the mixed conformal-elementary states, with the elementary parts generating the
analytic contributions. In the 5D picture, the source fields can be understood as degrees
of freedom confined to the UV brane, while the conformal states probe the bulk.
In this case, the dual Lagrangian reads
L = ∂µϕ(x)∂ µ ϕ(x) + m 2 eff ϕ(x)2 + ω
Λ ∆−3 ϕ(x)O(x) + LCFT , (2.40)
with m ∼ k and ω some dimensionless parameter. Note, that in order for the mixing
term to form a singlet under all symmetry groups, φ and O need to have the same
conformal and gauge quantum numbers. The amount of mixing of elementary and
conformal sector in the mass eigenstates depends on the scaling dimension of O, which
in the 5D theory is equivalent to its localization along the extra dimension. The generalization
for more general bulk fields is straightforward (although the construction of
the dual CFT for chiral fermions requires some care [104]). From Equation (2.39) it is
evident, that a scaling dimension/localization of ∆ = 3 and ∆ = 4 gives rise to purely
massless eigenstates and it turns out, that these are the exact scaling dimensions for
gauge fields and gravitons, respectively, compare Section (2.3). The computation is
actually really similar to the case of a bulk scalar, only the relation between scaling
dimension and the bulk mass (2.28) differ from the spin 0 case due to the different
EOM for different spin bulk fields, see Table 2.1. For a bulk gauge field however, we
can deduce additional information about the dual CFT. The dual Lagrangian in this
case reads
L = − 1
4 F a µν(x)F µν
a (x) + ω A a µ(x)O µ a (x) + LCFT , (2.41)