# RS1, Custodial Isospin and Precision Tests

###### Abstract

We study precision electroweak constraints within a RS1 model
with gauge fields and fermions in the bulk. The electroweak gauge symmetry
is enhanced to ,
thereby providing a custodial isospin symmetry
sufficient to suppress excessive contributions to the
parameter. We then construct complete models, complying with
all electroweak constraints,
for solving the hierarchy problem, without supersymmetry or large
hierarchies in the fundamental couplings. Using the
AdS/CFT correspondence our models can be interpreted as dual to a strongly
coupled conformal Higgs sector with *global* custodial
isospin symmetry, gauge
and fermionic matter
being fundamental fields external to the CFT. This scenario has interesting
collider signals, distinct from other RS models in the literature.

## 1 Introduction

There is a puzzle at the heart of particle physics which has become ever sharper in the last two decades of experimental and theoretical research. The minimal Standard Model (SM) is thusfar in superb agreement with experiment, not just in terms of the central functions for which it was designed, but remarkably, in every accidental detail following from its minimality, such as suppressed flavor-changing neutral currents (FCNC’s), proton stability, and a host of precision electroweak effects. Yet undeniably, the SM effective field theory suffers from the hierarchy problem and forces us to look beyond. Any approach for solving the hierarchy problem involves extending the SM at the weak scale and, in one way or another, threatens the economical and detailed agreement with experiment. Given this fundamental tension it is of considerable importance to identify within the different approaches to the hierarchy problem, robust effective field theory mechanisms which protect the key features of particle phenomenology, as well as the future experimental implications of these mechanisms.

The Randall-Sundrum I model (RS1) [1] [2] presented an exciting approach to the hierarchy problem based on geometrical hierarchies arising from warped higher-dimensional spacetime. However, most of the finer but interesting phenomenological issues are sensitive to the UV completion of the original RS1 effective field theory. The AdS/CFT correspondence [3] offers a great deal of insight into the RS1 proposal [4]. Via this correspondence, RS1 is dual to a purely 4D theory of particle physics and gravity, albeit one involving a strongly-coupled sector which is conformally invariant between the Planck and weak scales. The RS1 Kaluza-Klein excitations as well as fields localized on the “IR” brane are interpreted as TeV-scale composites of the strong sector. Fundamental fields coupled to strong CFT operators appear together as bulk RS fields. In the original RS1 model, all SM fields are localized on the IR brane, and therefore the model is dual to TeV-scale compositeness of the entire SM. The details of this compositeness determine the fate of the various phenomenological questions, but they are dual to details of the unknown UV completion of the RS1 effective theory.

There is another direction one can take. On the 4D side, to solve the hierarchy problem it is sufficient for just the Higgs to be a TeV-scale composite of a strongly interacting sector [5], the masses of higher-spin fundamental fields being protected by chiral or gauge symmetries. Of course for gauge boson and fermion masses to arise at the weak scale, the fundamental fields must couple to the Higgs sector. There is a simple way of studying this in the dual RS setting by continuing to take the Higgs to be localized on the IR brane, but taking gauge bosons and fermions to propagate in the higher dimensional bulk. The great advantage of doing this is that the key phenomenological issues become IR-dominated, and therefore addressable, in weakly-coupled RS effective field theory, rather than being sensitive to its UV completion. We find this approach very exciting and promising.

Let us briefly review the history of such studies. With bulk gauge fields, the calculability of weak scale effects at first seemed a liability, with large harmful effects for compositeness [6] [7] [8] [9] [10] and precision electroweak observables [7] [8] [9] [10]. Reference [8] presented their results in terms of the Peskin-Takeuchi and parameters [11], which facilitated a global fit to the data. It was later realized that placing fermion fields in the bulk allowed one to greatly soften some of these effects [12] [7] [13] [14]. There were further dividends in that bulk fermion masses provided a simple attractive mechanism for generating Yukawa structure without fundamental hierarchies in the RS1 action [15] [12] [16]. Furthermore the same mechanism automatically protects the theory from excessive FCNC’s [12] [16]. The issue of gauge coupling running and unification was discussed in Refs. [17] [18] [19] [20], with complete models with unification constructed in Ref. [20]. In particular a mechanism for protecting baryon stability was given in reference [20], adapting some key features of the mechanism of reference [21].

The last major phenomenological obstacle remaining in this program of research has been the problem of excessive contributions [7] [8] [9] to the Peskin-Takeuchi parameter [11]. The usual model-building rule for protecting this parameter is to ensure that the Higgs sector, when considered in isolation from gauge and fermion fields, should have a custodial isospin symmetry after electroweak symmetry breaking, under which the ’s form a triplet. However, the various RS1 models studied already appear to comply with this rule, since they make use of the minimal Higgs on the IR brane. However, the problem can be identified when one views these models from the dual 4D perspective. Bulk RS gauge fields are dual to both fundamental 4D gauge fields and to the CFT operators to which they might couple, namely conserved global symmetry currents of the CFT. This CFT represents the entire Higgs sector on the 4D side, of which the minimal Higgs is a light composite. The dual of the entire CFT Higgs sector enjoying a global custodial isospin symmetry is therefore to have a custodial isospin gauge symmetry in the RS bulk. Earlier RS models focused on just the SM gauge symmetry in the bulk. From the dual point of view, their difficulties with the parameter trace to the absence of custodial isospin symmetry in the CFT Higgs sector.

In this paper we study just such a bulk custodial isospin scenario and show that this extra gauge symmetry protects the parameter adequately. We are then able to construct fully realistic models satisfying all precision electroweak and other constraints. This is significant because we accomplish this in a non-supersymmetric approach to the hierarchy problem and without invoking any large fundamental hierarchies. In a composite Higgs model, the scale of compositeness can be made a free parameter. It can be raised at the cost of fine-tuning in the sense of the hierarchy problem, or lowered at the cost of strong interactions becoming more phenomenologically dangerous. The same is true in our RS model, where the scale of Kaluza-Klein resonances is dual to the compositeness scale. Our fit to the body of precision test data requires such resonances to be above about TeV.

An important consideration in this fit arises from the third generation quarks, especially the tension between the need to generate a large top quark mass while suppressing large corrections to bottom quark couplings to the . While some of the collider signals of our model are familiar expectations of strong interactions above the weak scale, some are more distinctively linked to the third generation constraints.

Our study illustrates the utility of RS effective field theory as a weakly coupled approach to a traditionally strongly-coupled and difficult subject: the possibility that the hierarchy problem is solved by non-supersymmetric physics above the weak scale. It allows us to calculate the signs and sizes of the leading effects on interesting observables in terms of model inputs, rather than just rough estimates. RS calculability is bought at a price. The dual strongly coupled theory must have special features [3] [4]: it must be approximately conformally invariant over the Planck-weak hierarchy, have a large- type expansion, and have a large gap in the spectrum of CFT scaling dimensions, with only a finite number being close to marginal (which automatically includes any symmetry CFT currents). Nevertheless, we have found that these special features do not pose any extra phenomenological liability, and are indeed an asset. Further, we expect many of our conclusions to survive even if some of the above theoretical control parameters are relaxed in Nature.

This paper is organized as follows. Section 2 describes the set-up of our model. Section 3 is a brief discussion of electroweak precision variables and the subtleties peculiar to our model. Sections 4 and 5 derive the tree level contributions to the Peskin-Takeuchi and parameters [11]. Section 6 derives the top loop contribution to in our model, bulk custodial isospin ensuring UV finiteness. Section 7 shows how our model can naturally fit the electroweak data. Section 8 describes the central new collider signals of our model. Section 9 describes the 4D dual CFT interpretation of our model and results. Section 10 provides further discussion and the outlook for future progress in this arena. Many of the more technical details have been relegated to appendices.

## 2 The Model

### 2.1 Overview

We are going to study a model with gauge symmetry in the bulk of a warped extra dimension. In order to recover the usual we will break with orbifold boundary conditions on the Planck brane to , keeping the IR brane symmetric. Then we will break spontaneously on the Planck brane. In one of the scenarios we consider, we will further break in the bulk by a small amount.

The metric of RS1 can be written as:

(2.1) |

Here,

(2.2) |

where represents the curvature, , and is the 4D Minkowski metric. We take

(2.3) |

to solve the hierarchy problem.

In that background the lagrangian for our model reads:

(2.4) |

is the bulk lagrangian. We focus on first, discussing in section 2.4.

(2.5) | |||||

where the indices are contracted with the bulk metric , and is field strength for the gauge group, for , for and is for the gluon. is a triplet of whose sole purpose is to spontaneously break to at a mass scale below . Therefore, henceforth, we will simply work with the gauge theory with a mass term for :

(2.6) | |||||

In fact, it will be interesting to consider separate cases, Scenario I, where is small, but non-zero and Scenario II, where , i.e., is unbroken in the bulk.

includes the necessary fields to spontaneously break to and contains the SM Higgs field, now a bidoublet of :

(2.7) |

where will generate Yukawa couplings for fermions which will be discussed in section 2.4 and

(2.8) |

is the *induced* flat space metric in the IR brane.
After the usual field redefinition of [1],
Eq. (2.8) takes its canonical form:

(2.9) |

with , GeV, and the ratio of the Higgs vev to the warped down curvature scale is taken as

(2.10) |

We assume that brane-localized (kinetic) terms for bulk fields are of order loop processes involving bulk couplings and are therefore neglected in our analysis (however, see references [22, 23, 14] for effects of larger brane-localized kinetic terms for gauge fields and reference [24] for effects of brane-localized kinetic terms for fermions.).

### 2.2 by orbifold boundary condition

### 2.3 on UV brane

The breaking of occurs via a vev on the UV brane. There are two linear combinations of and ,

(2.11) |

where

(2.12) |

is the electroweak covariant derivative with . is the hypercharge gauge boson. It is . We couple to a Planckian vev on the UV brane which mimics boundary condition to a good approximation.

In terms of and , the five dimensional electroweak covariant derivative is now

(2.13) |

We have defined the hypercharge coupling,

(2.14) |

the charge

(2.15) |

the coupling

(2.16) |

and the – mixing angle

(2.17) |

### 2.4 Fermions

Since we have an enhanced bulk gauge symmetry,
namely , we have to promote the usual right handed fermionic
fields
to doublets of this symmetry.
Moreover, since we are breaking that
symmetry through the UV orbifold, one component of doublet
must be even and therefore has a zero-mode
while the other component must be odd and therefore does not have a zero-mode.
Therefore,
we are forced to double the number
of right handed doublets in such a way that from one of them
the upper component, for example up type quark, is even
whereas from the other the lower component, down type, is even
– this is
similar to obtaining quarks and lepton zero-modes
from different bulk multiplets in orbifolded GUT scenarios
[26, 27].
This doubling
of right handed particles is only needed in the quark sector, since
in the lepton sector we only need the right handed charge leptons^{5}^{5}5
Although in any embedding of this theory in a GUT, the minimal group
would be , thus needing a and another doublet. to be
massless after we compactify.
Explicitly,
we have three types of doublets under
per generation in such a way that^{6}^{6}6Only one
chirality will be discussed since the
other chirality is projected out by symmetry.:

(2.18) |

where the unprimed particles are the ones to have zero modes, i.e. to be . The extra fields needed to complete all representations are (since breaking of is on the Planck brane).

The general bulk lagrangian for fermions is:

(2.19) |

where is the sign function. Even though it will seem that we are adding a mass term, is compatible with a massless zero mode. This parameter controls the localization of the zero mode, for () the wavefunction near the Planck (IR brane) [15, 12].

The Yukawa couplings to Higgs (prior to field redefinition of Eq. (2.8) Eq. (2.9)) are necessarily localized on the IR brane:

(2.20) |

Note that because and zero-modes arise from different doublets, we are able to give them separate Yukawa couplings without violating on the IR brane.

So far, we have detailed the model, except for choice of ’s.
The parameters
give a simple mechanism for obtaining
hierarchical
Yukawa couplings without hierarchies
in Yukawa couplings. In short, light fermions
are localized near Planck brane () so that their
Yukawa couplings are
small
due to the small overlap with Higgs on TeV brane.
Left-handed top and bottom quarks are close to
(but )^{7}^{7}7As
we will show is necessary to
be consistent with
for KK masses few TeV., whereas right-handed top quark is localized
near TeV brane to get top
Yukawa.
With this set-up, FCNC’s
from exchange of
both gauge KK modes and “string states” (parameterized
by flavor-violating local operators in our effective field theory) are also
suppressed.
See references [12, 16] for details.

## 3 Electroweak precision observables

We begin with formalism for electroweak fit in the presence of new physics. It is convenient to discuss this in the framework of effective Lagrangian at the weak scale for SM with all the heavy non-standard physics integrated out [28] (as pioneered in references [29], but here retaining the Higgs field). This framework was used earlier in the RS model studied in reference [8]. The dimension- operators, obtained by integrating out heavy particles, which are important for the electroweak fit are:

(3.1) |

(3.2) |

where , and , in general, vary with the fermion.

Usually, the gauge-kinetic higher-dimensional operator in Eq. (3.1) and the (custodial-isospin violating) mass higher-dimensional operator in Eq. (3.2) translate into “oblique” parameters, and [11], respectively:

(3.4) |

while the fermionic operators in Eq. (LABEL:fermionoperator) are considered “non-oblique”. However, for a special form of fermion-Higgs higher-dimensional operators in Eq. (LABEL:fermionoperator), these can be field-redefined into purely oblique effects as we now discuss. In the particular RS models studied in references [8, 23], with all fermions on the IR brane, the equivalent of our field redefinition was achieved by setting the gauge boson wavefunction to be unity on the IR brane. Reference [14] studied bulk fermions and used analogous field redefinitions.

This special form is

(3.5) |

for all fermions,
where
denotes the hypercharge of the fermion
and is the hypercharge of the Higgs.
Setting Higgs to its vev in the fermion-Higgs operator
in Eq. (LABEL:fermionoperator)
induces non-canonical couplings of fermions to gauge bosons. However, doing
the following redefinition of gauge fields renders the fermionic couplings
to gauge bosons canonical^{8}^{8}8This can be extended (non-linearly)
into a manifestly -invariant redefinition.:

(3.6) |

This redefinition when substituted in SM gauge kinetic terms induces shifts in and i.e., purely oblique effects, so that we now have:

(3.7) |

As we will show, in our model, the fermion-Higgs operators have the special form only for the light fermions (and right-handed bottom), not for the top and left-handed bottom quark so that as far as the precision electroweak fit is concerned, couplings of have to be considered separately. Hence, we will focus on and parameters and in this paper.

In the following, we will integrate out Kaluza-Klein (KK) (heavy) modes of gauge/fermion fields in the RS1 model and compute the resulting dimension- operators for the (light) zero-modes (which correspond to the SM fields in the above Lagrangian). There are even higher-dimensional operators whose effect on electroweak precision observables can be considered. Naively, these are further suppressed by , where is the KK mass scale. In the present scenario, these are also -enhanced. Therefore, we are careful in what follows to consider KK mass scales such that even with this enhancement, these operators are suppressed relative to the dimension- operators. This justifies using a Higgs vev insertion approximation within gauge and fermions propagators in what follows.

## 4 Tree-level and contributions from gauge-Higgs sector

### 4.1 Contribution to

A powerful aspect of our model is that the bulk right-handed gauge symmetry enforces custodial isospin. We will see that the gauge sector does not make a logarithmically enhanced contribution to the parameter, and that the parameter is log enhanced, but suppressed by relative to .

In the effective theory (section 3), the operator contributes only to the mass at order . Consequently, the coefficient measures the amount of isospin breaking. In terms of vacuum polarizations, the effective theory contains a modified quadratic term,

(4.1) |

is polarization from integrating out tree level insertions of gauge KK modes. Thus, from Eqs. (3.2) and (4.1), we see that the coefficient of the operator is

(4.2) |

In the gauge sector, oblique corrections to the electroweak observables come from integrating out KK towers which couple to left handed zero modes (Fig. 1). We will find it convenient to convert sums over KK propagators (eigenfunctions) to five dimensional propagators (Green’s functions) (see appendix A), while leaving four dimensional fields on external lines (see appendix B for details).

We use Eq. (B.4) to calculate the contribution to from Fig1,

(4.3) |

where is the or zero-mode
gauge coupling.
The sum over includes all fields which couple to the external .
in
(4.3)
is the IR brane to IR brane,
five dimensional, mixed momentum-position space Green’s function in Feynman gauge.
We *subtract* the massless pole, , since the effective Lagrangian is obtained by integrating out only
heavy/KK modes at tree-level (graphs with internal zero-modes
are part of the Dyson resummation).

In the charged sector, the mixes with its own KK modes and those of the :

(4.4) |

In the neutral sector, mixes with its own KK modes, the , and those of the :

(4.5) |

The ’s contain IR brane to IR brane propagators and we have used the charge of Higgs (see Eq. (2.15)).

Using the propagators in the appendix B, we find

(4.6) |

and

(4.7) |

where , . In (4.6) and (4.1), we have also dropped terms at order that are suppressed relative to the leading terms above. Then,

(4.8) |

(4.9) |

Hence,
in scenario II,
the gauge sector *does not* contribute a log enhanced piece to custodial
breaking at this order
as would be the case in the absence of .

### 4.2 Contribution to S

At tree-level, there are no Feynman diagrams that contribute to as defined in Eq. (3.1). As discussed at the end of section 3, there may be even higher-dimensional operators which can contribute to precision variables which we argued on general grounds are small in our model. However, in the case of the parameter since the dimension- contribution, , vanishes, to be cautious, we will calculate dimension- contribution of the type depicted in Fig.1.

Since this is a tree-level calculation, there is no (kinetic) mixing involving the photon (of course, there cannot be any mass mixing even at loop level), so the quantity

(4.10) |

is related to , with

(4.11) |

Eq. (4.11) is equivalent to a shift in parameter, [11]. As we will see, this contribution can be neglected.

The ’s contribute to and , but we postpone a detailed discussion of the model’s prediction for electroweak precision measurements, since, as discussed in section 3 additional contributions to and arise from the rescaling of gauge bosons. A careful treatment of the fermionic sector is necessary.

## 5 Fermionic operators

The coefficients of the operators in Eq. (LABEL:fermionoperator) get contributions from integrating out KK modes of gauge fields (at tree-level) as shown in the Feynman diagrams of Figs. 2 and 3.

The Feynman diagram in Fig. 2
is evaluated by
integrating fermion zero-mode wavefunction with propagator from to TeV brane
(including metric/fünfbein factors). This
gives the fermion-Higgs higher-dimensional operator in
Eq. (LABEL:fermionoperator)
with
coefficients (up to ) as follows^{9}^{9}9See
also references [7, 9, 10, 14]
for discussion of this effect in a different
language.. From exchange of KK modes of , we get
(see appendix C for details)

From exchange of KK modes of hypercharge and

The factor of (inside integral) in first line is from fünfbein. and denotes the charge of fermion and Higgs (see Eq. (2.15).

A similar computation of the Feynman diagram in Fig. 3 gives coefficient of “compositeness” operator in Eq. (LABEL:fermionoperator), (from exchange of KK modes of ):

We obtain
for (as applicable
to light fermions): this coefficient is
negligible for TeV
and similarly for exchange of hypercharge or
KK modes^{10}^{10}10Coefficient of the operator
(light fermion) (top or left-handed bottom)
will be larger since
for top quark or left-handed bottom (see section
5.2), but it plays no role in
fit to precision electroweak data,
although it will affect, say, at high-energy colliders..
See also references [12, 13].

### 5.1 Light fermions

If light fermions are at (such that : suffices) in order to address flavor (as mentioned before [12, 16]), then second term proportional to in Eq. (LABEL:coeffy) can be neglected. Also, KK modes of couple very weakly to light fermions and so there are no operators proportional to charge (i.e., last term in Eq. (LABEL:coeffy)). Thus, coefficients and of these fermion-Higgs operators are of the special form discussed in section 3 with

(5.4) |

Hence, the parameter in our model is given by (see Eq. (3.7))

(5.5) | |||||

where is the mass of the lightest KK mode of gauge boson (see Eq. (D.2)). Here we have neglected from gauge-Higgs sector since it is of higher order in (see section 4.2).

There is an interesting possibility that we will not pursue here where contribution to parameter arising from the fermion-Higgs operators in Eq. (LABEL:fermionoperator) is suppressed completely for as can be seen from Eqs. (LABEL:coeffw) and (LABEL:coeffy). However, in order to fit observed light fermion masses with , we would have to introduce very small dimensionless numbers into our fundamental theory. While this is radiatively stable, it goes against the general philosophy adopted in this paper.

### 5.2 Top and bottom

We can obtain the Yukawa coupling in terms of the Yukawa coupling (see, for example, [12]):

(5.7) |

The quick argument for choosing for top quark is that, for (or ) , the Yukawa coupling is (exponentially) suppressed (see Eq. (5.7)) and hence we consider for top quark to obtain .

For , we get

(5.8) |

Since coefficient in Eq. (LABEL:coeffy) is different for than for light fermions, the effect on coupling of to arising from the operators in Eq. (LABEL:fermionoperator) cannot be redefined into (see discussion in section 3) and must be treated separately:

(5.9) | |||||

using the charges of Higgs and . Here, and are charges of and Higgs.

To obtain without hierarchy in Yukawa coupling ( is same for top and bottom), we choose for .

## 6 at loop level in Scenario II

An interesting case to consider is when is unbroken in the bulk (our Scenario II), . In that case, remarkably, bulk custodial isospin symmetry forces loop contributions to parameter to be UV finite (and hence calculable) and these are the dominant contribution to . This is because contribution to requires both electroweak symmetry breaking on IR brane and breaking which is localized on UV brane. For remainder of this section, we will consider this case.

Because custodial-isospin violation is due to breaking of by boundary condition on Planck brane, there is no zero-mode for and KK spectrum is different for and (see appendix D). Similarly, there is no zero-mode for and KK spectrum can be different for and (as can be seen from appendices E and F). Hence, loop diagrams will have to involve right-handed and/or (and other fermions) in order to give .

An example of a Feynman diagram with zero and KK modes, but without fermions is shown in Fig. 4. This diagram with zero-mode gives mass term

Let us estimate the Feynman diagram without Yukawa insertion (Fig. 5). The contribution of gives

As we will see, the Feynman diagram involving insertions of top Yukawa coupling, Fig. 6 dominates over the diagram without top Yukawa insertion (Fig. 5) (and diagrams with other fermions) and also over the diagram without fermions (Fig.4). So, we concentrate on the diagram in Fig.6.

### 6.1 from top quark KK modes

To calculate the diagram in Fig. 6, we need (a) spectrum of KK modes of left-handed top and bottom and also and its partner, (not the physical ) and (b) their couplings to Higgs (or the and “mass insertions”). The spectrum and couplings to Higgs are calculated in the appendices E and F to which the reader is referred for details. We find it not convenient to convert sum over KK modes into propagators (unlike before) and so we will work directly in terms of KK modes.

We begin with , where . In this case, we get a “very light” (much lighter than ) mode. This is ruled out experimentally.

For a slightly larger , namely, (with ), we can show that the lightest mode is not lighter than (and hence not ruled out experimentally unlike before), but its mass is smaller than that of the lightest and KK mode. Other modes of are almost degenerate with KK modes of . Also, because , mass insertions in Fig. 6 involving and KK modes are the same as those involving the zero-mode. One can show that this results in small .