On the Flavor Problem in Strongly Coupled Theories - THEP Mainz

46 Chapter 2. The Randall Sundrum Model and its Holographic Interpretation
Scalar (j1, j2) = (0, 0) ∆± = 2 ± √ 4 + c 2
Vector (j1, j2) = ( 1 1
2 , 2 ) ∆± = 2 ± √ 1 + c2 Spin 2 (j1, j2) = (1, 1) ∆± = 2 ± √ 4 + c2 Chiral Spinor s = 1
2
(j1, j2) = ( 1
1
3 1
2 , 0) or (0, 2 ) ∆ = 2 + |c − 2 |
Table 2.1: Relation between scaling dimension of the conformal fields and bulk mass
parameter (localization) in AdS5. Note, that for general BCs, left- and right-handed
chiral spinors will have different relations, see (e.g. [104, p.7&13]). This will be obsolete
for our definition of the c-parameter, compare section 2.4. This table was first given
in [93, p.2-3].
where the source fields A a µ are gauge fields, so that they must couple to conserved
currents O µ a in the CFT. Therefore, the existence of a bulk gauge group implies a
corresponding global symmetry of the dual CFT, where only a subgroup is gauged,
namely the one whose gauge bosons have Neumann boundary conditions on the UV
brane and thus possess a source field (which corresponds to the gauge boson in the
elementary sector). If only for a subgoup of a global symmetry gauge fields are introduced,
this global symmetry is explicitly broken scenario, which, if the gauge couplings
are small is in the literature called weakly gauging the global symmetry. Note however,
that for the composite sector alone LCFT, the global symmetry is preserved. The fact,
that one can impose a global symmetry in the composite sector by choosing the appropriate
bulk gauge group, was the original inspiration for the SU(2)L × SU(2)R bulk
group in the custodial extension of the RS model [105] and its at first sight arbitrary
breaking pattern. It is just the holographic implementation of the global custodial
symmetry of the SM Higgs sector with the SU(2)L × U(1)Y weakly gauged. Likewise,
the extensions proposed in this thesis will translate in this way to global symmetries
of the dual theory.
At this point we have still not introduced an IR brane, but applications of the duality
without one can already be found in the literature. The 5D formulation of a set-up
without IR brane, where gravity is closely localized at the UV brane (SM fields play
no role but would reside on the UV brane in a complete realistic model) is called
RS2 model, due to a follow-up paper by Randall and Sundrum in 1999 [56] and is
described by a dual strongly coupled CFT together with 4D gravity (first mentioned
in the Acknowledgements of [98]). If one assumes the SM confined to the UV brane
and adds non SM fields into the bulk, one describes the holographic dual of a class
of new physics models proposed by Georgi in 2007, called unparticles [101], which
proposes the existence of a new conformal sector coupled to the SM.
Analogue to the explicit breaking of the CFT by the introduction of a UV brane at
high energies, the introduction of an IR brane z = R ′ should lead to a breaking of
conformal invariance at low energy scales ΛIR ∼ 1/R ′ , below which scaling invariance
in the bulk is broken. As a first consequence, regarding our bulk scalar example above,
we observe that the introduction of an IR brane will replace the regularity condition
at z = ∞, which fixes C1 to zero in (2.27) by a boundary condition at z = R ′ and
thus leads to C1 �= 0. Since the regularity condition in the non-compactified theory is