# Dynamical electro-weak symmetry breaking from deformed AdS:
vector mesons and effective couplings.

Marco Fabbrichesi  INFN, Sezione di Trieste, Trieste, Italy.  Maurizio Piai
Department of Physics, University of Washington, Seattle, WA 98195  Swansea
University, School of Physical Sciences, Singleton Park, Swansea, Wales, UK
Luca Vecchi  INFN, Sezione di Trieste, Trieste, Italy.  Scuola Internazionale
Superiore di Studi Avanzati, Via Beirut 4, Trieste, Italy.

###### Abstract

We study a modification of the five-dimensional description of dynamical
electro-weak symmetry breaking inspired by the AdS/CFT correspondence.
Conformal symmetry is broken in the low-energy region near the IR brane by a
power-law departure from the pure AdS background. Such a modification—while
not spoiling the identification of the IR brane with the scale of confinement—
has a dramatic effect on both the coupling of the first composite states to
the standard model currents and their self-couplings. Chiral symmetry breaking
can take place at a scale larger than the IR cut-off. This study shows that
observables, such as the precision parameter $\hat{S}$, which depend on the
couplings of the lightest composite states to the currents are very sensitive
to the details of the dynamics in the low energy region where conformal
symmetry is lost and electro-weak symmetry is broken just above the scale of
confinement. Therefore results of calculations of these observables in AdS/CFT
inspired scenarios should be interpreted conservatively. The most important
phenomenological consequence for physics at the LHC is that the bound on the
mass scale of the heavy excitations (technirho mesons) in a realistic model is
in general lower than in the pure AdS background with a simple hard-wall cut-
off in the IR.

###### pacs:

11.10.Kk, 12.15.Lk, 12.60.Nz

##  I Introduction

The mechanism responsible for electro-weak symmetry breaking in the standard
model will be tested at the Large Hadron Collider (LHC). One important goal
for this experimental program is to understand whether the interactions
responsible for electro-weak symmetry breaking are strong or weak. It is
essential to identify theoretically clean, measurable quantities that can help
distinguish these two possibilities unambiguously.

One might think that this is an easy task: after all, a strongly coupled model
in the spirit of technicolor TC  predicts the existence of towers of broad,
strongly coupled composite resonances, with a rich spectroscopy at the TeV
scale, in analogy with what is known about QCD at the GeV scale. Yet, indirect
experimental data about electro-weak symmetry breaking PT ; Barbieri  cannot
be easily reconciled with this framework and already suggest that, if electro-
weak symmetry breaking is a dynamical effect, the low-energy effective field
theory description of the new strongly-coupled sector has to exhibit features
that are not generic. Walking technicolor walking  is a plausible candidate
for such a strongly coupled model, based on conformal behavior and large
anomalous dimensions in the IR, but calculability within this framework has
been a challenge AS .

In recent years, based on the ideas of the AdS/CFT correspondence AdS/CFT ,
and on the pioneering work of Randall and Sundrum RS1 ; pheno , many models
have been investigated which exhibit in the low energy region the basic
properties expected in a walking theory, while being calculable. Examples now
exist of models that are compatible (within the errors) with the precision
data and can be discovered at the LHC. The literature on the subject is
already extensive AdS/TC ; HS ; MP ; higgsless ; composite . Most of these
models assume a conformal behavior of the strongly coupled sector in the
energy region spanning few orders of magnitude above the electro-weak scale
and the existence of a weakly-coupled effective field theory description of
the low-energy dynamics of the resonances. The construction of the effective
field theory is derived by writing a weakly coupled extra-dimension model with
a non-trivial gravity background, and by using the dictionary of the AdS/CFT
correspondence to relate back to four dimensions. A generic phenomenological
feature of all these models is that, unless a clever mechanism arranging for
non-trivial (often fine-tuned) cancellations is implemented, a quite severe
lower bound on the mass
$M_{1}\mathrel{\raisebox{-2.58334pt}{$\stackrel{{\scriptstyle\textstyle>}}{{\sim}}$}}$
2.5–3 TeV of the lightest spin-1 resonance (techni-rho) results, in particular
from the bounds on the electro-weak parameter $\hat{S}$ PT ; Barbieri . This
result, together with the assumption that the effective field theory be weakly
coupled (and hence calculable), gives rise to a spectacular signature (a sharp
resonance peak) at the LHC MP2 . Unfortunately, it is very difficult to
distinguish it from the signature of a generic, weakly-coupled extension of
the standard model with an extended gauge group, predicting a new massive
$Z^{\prime}$ gauge boson.

Indisputable evidence proving that a strongly-coupled sector is responsible
for electro-weak symmetry breaking would be the discovery of at least the
first two spin-1 resonances, hence proving that these new particles are not
elementary, but higher energy excitations of a composite object. The major
obstacle against this scenario is the unfortunate numerology emerging from the
combination of precision data and LHC high-energy discovery reach. If
$M_{1}\mathrel{\raisebox{-2.58334pt}{$\stackrel{{\scriptstyle\textstyle>}}{{\sim}}$}}$
2.5–3 TeV, than it follows that the mass of the second resonance must be
$M_{2}\mathrel{\raisebox{-2.58334pt}{$\stackrel{{\scriptstyle\textstyle>}}{{\sim}}$}}$
5–6 TeV and just beyond the region where LHC data are expected to give
convincing evidence AtlasTDR . Yet, a pretty mild relaxation of the
experimental bounds would be enough to change this situation radically, since
$M_{1}\sim 1.5$ TeV would imply $M_{2}\approx$ 2.5–4 TeV, well within reach
even at moderate luminosity MP2 . It is hence timely, just before LHC starts
collecting data, to question how accurate the AdS/CFT description of realistic
dynamical electro-weak symmetry breaking is, and whether some of the
approximations implied by this description could account for the desired
softening of the bounds, without at the same time spoiling the calculability
of the effective field theory.

In analogy with RS1 , the five-dimensional picture usually contains two hard
boundaries representing the UV and IR cut-off between which the theory is
conformal. This is the weakest link with the idea that electro-weak symmetry
breaking be triggered by a non-abelian gauge theory with an approximate IR
fixed point. Taken literally, this picture means that, both in the UV and in
the IR, conformal symmetry is lost instantaneously, via a sharp transition. As
for the UV cut-off, this is not a real problem from the low-energy effective
field theory point of view. The details of how an asymptotically-free
fundamental theory in the far UV enters a quasi-conformal phase below the UV
cut-off, can always be reabsorbed (via holographic renormalization HR ; MP )
in the definition of otherwise divergent low-energy parameters of the
effective field theory, defined at a given order in the perturbative expansion
of the effective field theory itself.

Rather different is the case against using a hard-wall regulator in the IR.
There is no sense in which IR effects decouple and can be renormalized away,
and hence the low-energy effects we are interested in, when comparing the
effective field theory to the experimental data, are inherently sensitive to
the choice of the IR regulator. On the one hand, the very validity of the
effective field theory description based on the AdS/CFT dictionary requires
that the hard-wall cut-off be at least a reasonable leading order
approximation (otherwise the effective field theory itself would be strongly
coupled, and not admit a controllable expansion). On the other hand,
corrections are expected to be present, and estimating their size and
understanding their phenomenological consequences is crucial, at the very
least in order to know what to expect in experiments such as those at the LHC,
which is going to test precisely the energy range close to the IR cut-off.

To be more specific. In the IR, three different phase transitions are taking
place: electro-weak symmetry breaking, conformal symmetry breaking and
confinement. These cannot define three parametrically separate scales, since
they are all triggered by the same physical effect, namely the fact that the
underlying (unknown) theory possesses an approximate fixed point in the IR.
Hence the RG flow of the underlying dynamics is not going to reach the IR
fixed point (which is only approximate), but will drift away from it at low
energies, after spending some time (walking) in its proximity. Yet, there is
no reason to expect these three effects to arise precisely at the same energy
(temperature), and they might define three distinct critical scales
(temperatures) that differ by $O(1)$ coefficients.

An illustration of this point can be obtained by considering an ${\cal N}=1$
supersymmetric QCD model with $N_{c}$ colors and $N_{f}$ fermions. At least at
large-$N_{c}$, for $3N_{c}/2<N_{f}<3N_{c}$, the theory is asymptotically free,
but has a fixed point in the IR seiberg ; cascade  (for recent progress
towards the rigorous construction of the gravity dual see, for instance, Nunez
). If $N_{f}$ is not far from the lower bound, so that the theory is strongly
coupled at distances larger than a UV cut-off $1/L_{0}$, then the theory might
be approximately described by a large-$N_{c}$ conformal field theory at strong
’t Hooft coupling. Suppose now that at some smaller energy, characterized by a
length scale $\bar{L}\gg L_{0}$, for some reason (for example the existence of
a suppressed symmetry-breaking higher-order operator, which acquires a large
anomalous dimension in the IR turning it into a relevant deformation) a
symmetry-breaking condensate forms, reducing further $N_{f}$ to a value
$N_{f}^{\prime}$ closer to or below $3N_{c}/2$. Symmetry-breaking drives the
theory away from the original fixed point, and induces the loss of conformal
symmetry. The coupling now runs fast (because the coupling itself was already
big and large anomalous dimensions are present), and (depending on
$N_{f}^{\prime}$) the theory either enters a new conformal phase at stronger
coupling or confines. The breaking of the global $SU(N_{f})_{L}\times
SU(N_{f})_{R}$, conformal symmetry-breaking and confinement take all place
approximately at the same scale. Yet, the energy at which the coupling reaches
its upper bound defines a new scale $L_{1}$ which might well be some numerical
factor away from $\bar{L}$, the scale at which the RG-flow trajectory departed
away from the fixed point.

If this is the qualitative behavior of the UV-complete dynamical model that is
ultimately responsible for electro-weak symmetry breaking , describing it as a
slice of AdS space between two hard walls is a good leading-order
approximation. Nevertheless, we may wonder whether a factor of 3 or 4
separating the scales of conformal symmetry-breaking and confinement can be
completely ignored, in the light of the phenomenological consequences at the
LHC that a mere factor of two might have. In this paper, we study the effect
of such a factor. We consider the simplest possible effective field theory
description of dynamical electro-weak symmetry breaking as a 5D weakly-coupled
system (see also MP ), introduce (besides the UV brane at $L_{0}$ and the IR
brane at $L_{1}$) a new discontinuity at the scale $\bar{L}$, very close to
the IR scale $L_{1}$, and assume that the background deviates from the AdS
case for $\bar{L}<z<L_{1}$.

As for the origin and description of electro-weak symmetry-breaking, we will
treat it as a completely non-dynamical effect localized in the IR, somehow in
the spirit of Higgless models. The breaking could take place at $L_{1}$ as
well as at $\bar{L}$ (or anywhere in between), as suggested by the SQCD
example above. We compare the effects on the electro-weak precision parameter
$\hat{S}$ in these two cases, as illustrative of two extreme possibilities,
without committing ourselves to either of them. The idea that chiral symmetry
breaking might, for a generic model, take place at a scale higher than
confinement has been in the literature for a while chiral/confinement , has
been supported by lattice evidence in some special case lattice , and has
recently been discussed also in string-inspired models SS .

A realistic model should also implement a dynamical mechanism generating the
mass of the standard model fermions. This can be done either via extended
technicolor higher-order interactions between the standard model fermions and
the new strong sector ETC ; APS  (represented in the 5D picture by Yukawa
interactions localized at the UV, with the symmetry-breaking vacuum
expectation value not localized, but exhibiting a non-trivial power-law
profile in the bulk), or via the assumption that standard model fields are
themselves (partially) composite, in the spirit of topcolor and related models
topcolor  (which would imply the fermions be allowed to propagate in the bulk
of the 5D model). A detailed discussion of how the global family symmetry of
the standard model is broken would be required in order to study how the
phenomenology of flavor-changing transitions and the physics of the third
generation would be affected by the proposed modification of the background.
In this paper, we treat the standard model fermions as non-dynamical fields,
described by a set of external currents, and do not address the problem of
their mass generation.

##  II Preliminaries

A non-trivial departure of the dynamics of the spin-1 resonances, with respect
to that on pure AdS geometry, may be either due to a modification of the
gravity background or to the presence of a non-dynamical background (dilaton).
Since we consider an effective field theory where only spin-1 states are
dynamical, it is not possible to distinguish between these two effects at this
level. We choose to describe the model in terms of a deformation of the
gravity background, for simplicity.

Consider the five-dimensional space described by the metric

$\displaystyle\mbox{d}s^{2}$ $\displaystyle=$ $\displaystyle
a(z)^{2}\left(\eta_{\mu\nu}\mbox{d}x^{\mu}\mbox{d}x^{\nu}\,-\,\mbox{d}z^{2}\right)\,,$
(1)

where $L_{0}<z<L_{1}$. We will assume that the geometry approaches pure AdS in
the UV region, $a(z)\rightarrow L/z$ as $z\rightarrow L_{0}$, and departs from
it at a scale $z\sim\bar{L}$. In most of the calculations we take $L_{0}=L$
for simplicity.

We are interested in describing a model that at low-energy (below $1/L_{1}$)
can be matched to the electro-weak chiral Lagrangian EWCL . This requires to
introduce a 5-dimensional gauge group which is at least $SU(2)_{L}\times
U(1)_{Y}$, but may be enlarged to accommodate custodial symmetry.
Irrespectively of the details, the model contains a vectorial sector (the
neutral part of which consists of the photon and its excitations) and an axial
sector (containing the $Z$ boson and its excitations). In this paper we
describe only the phenomenology connected with the neutral gauge bosons, hence
we dispense with the details of the complete symmetry group. For concreteness,
we take the vectorial sector to be described by the pure Yang-Mills $SU(2)$
theory with the following action:

$\displaystyle{\cal S}$ $\displaystyle=$
$\displaystyle\int\mbox{d}^{4}x\int_{L_{0}}^{L_{1}}\mbox{d}z\sqrt{G}\left[G^{MR}G^{NS}\left(-\frac{1}{2}{\rm
Tr}F_{MN}F_{RS}\right)+2g\delta(z-L_{0})G^{MN}{\rm Tr}J_{M}A_{N}\right]\,,$
(2)

where $F_{MN}=\partial_{M}A_{N}-\partial_{N}A_{M}+ig[A_{M},A_{N}]$ is the
field strength tensor of $A_{M}=A_{M}^{a}T^{a}$ with $T^{a}=\tau^{a}/2$ the
generators of $SU(2)$, where $g$ is the (dimensionful) gauge coupling, and
where $J_{M}=(J_{\mu}(x),0)$ is the four-dimensional external current
localized on the UV-brane.

Quantization requires to add appropriate gauge-fixing terms, canceling the
mixing terms between spin-0 and spin-1 fields, which in unitary gauge implies
$A_{5}(x,z)=0$.

After Fourier transforming in 4D,
$A_{\mu}(x,z)\equiv\int\frac{\mbox{d}^{4}q}{(2\pi)^{2}}e^{iqx}A_{\mu}(q)v(q,z)$,
the free bulk equations read

$\displaystyle\partial_{5}\left[a(z)\partial_{5}v(q,z)\right]$
$\displaystyle=$ $\displaystyle-q^{2}a(z)v(q,z)\,.$ (3)

Substituting the solutions in the action, and canceling the boundary terms at
$z=L_{1}$, without breaking the gauge symmetry, requires to impose Neumann
boundary conditions:

$\displaystyle\partial_{5}v(q,L_{1})$ $\displaystyle=$ $\displaystyle 0\,.$
(4)

This set of equations admits always a constant, massless zero mode.

Finally, the action can be rewritten as a pure boundary term at the UV, from
which one can read the vector two-point correlator, that for $L_{0}\rightarrow
L$ is

$\displaystyle\Sigma_{V}(q^{2})$ $\displaystyle=$ $\displaystyle
g^{2}\frac{v(q,L_{0})}{\partial_{5}v(q,L_{0})}\,,$ (5)

which can be expanded as

$\displaystyle\Sigma_{V}(q^{2})$ $\displaystyle=$ $\displaystyle
e^{2}\left(\frac{1}{q^{2}}\,+\,\sum_{i}\frac{R_{i}}{q^{2}-M_{i}^{2}}\right)\,,$
(6)

where $M_{i}$ ($i=1,2,\dots$) are the masses of the excited states, and
$R_{i}$ define their effective couplings to the four-dimensional currents,
normalized to the coupling $e^{2}$ of the massless mode (to be identified with
the electro-magnetic coupling of the photon).

The (dimensionful) bulk coupling $g$ controls the perturbative expansion used
to extract this correlator. It is not directly related to the effective
coupling $e$ of the standard model gauge boson (photon), but rather is related
to the strength of the effective interactions among its heavy (composite)
excitations. The relation between these two effective couplings depends on how
the theory is regularized in the UV, and is not a calculable quantity, because
of the divergences in the $L_{0}\rightarrow 0$ limit. A rigorous treatment
requires to introduce appropriate counterterms and treat the ratio
$e^{2}L/g^{2}$ as a free parameter. For the purposes of this paper, which
primarily require comparing identical UV settings with different IR
deformations, we can simplify this procedure by assuming that $L_{0}\ll L_{1}$
be finite and fixed, and express this ratio as a function of the scales and
couplings in the model. We discuss later how good the perturbative expansion
is by estimating the size of the effective self-coupling of the composite
states.

In order to compute $\hat{S}$ one has to introduce also the axial-vector
excitations, and a symmetry-breaking mechanism. For the purposes of this
paper, we only consider the Higgsless limit, defined by the introduction of a
localized, infinitely massive Higgs scalar which assumes a non-trivial
symmetry-breaking vacuum expectation value.

The axial-vector modes $v_{A}(q,z)$ still satisfy Eq (3), but their boundary
conditions (and the gauge fixing action) are modified. We consider two cases
in the following. In the first, symmetry-breaking takes place on the boundary
$L_{1}$ so that the axial-vector profiles $v_{A}(q,z)$ obey generalized
Neumann boundary condition:

$\displaystyle\partial_{5}v_{A}(q,L_{1})\,+\,mv_{A}(q,L_{1})$ $\displaystyle=$
$\displaystyle 0\,.$ (7)

The effective symmetry-breaking parameter $m$ has dimension of a mass. In the
limit $m\rightarrow 0$ one recovers the symmetric case, while for
$m\rightarrow+\infty$ one recovers the Dirichelet boundary conditions. The
mass of the $Z$ boson depends on $m$ is such a way that it vanishes for
vanishing $m$, but is determined by $L_{1}$ for arbitrarily large $m$. In the
second case we consider a symmetry-breaking vacuum expectation value localized
at a different point $\bar{L}<L_{1}$ in the fifth dimension. The modifications
to be implemented in this case will be discussed in the next sections.

All of this allows to define the axial-vector correlator $\Sigma_{A}(q^{2})$
by replacing in Eq. (5) $v_{A}(q,z)$ and its derivative to $v(q,z)$. After
these manipulations, the precision parameter $\hat{S}$ is given by

$\displaystyle\hat{S}$ $\displaystyle=$
$\displaystyle\left.e^{2}\cos^{2}\theta_{W}\,\frac{\mbox{d}}{\mbox{d}q^{2}}\left(\frac{1}{\Sigma_{V}(q^{2})}-\frac{1}{\Sigma_{A}(q^{2})}\right)\right|_{q^{2}=0}\,,$
(8)

where $e$ has been defined before, and corresponds to the electro-magnetic
coupling, while $\theta_{W}$ is the effective weak-mixing angle. We recall
here that an approximate extrapolation to large Higgs masses yields the
experimental limit
$\hat{S}\mathrel{\raisebox{-2.58334pt}{$\stackrel{{\scriptstyle\textstyle<}}{{\sim}}$}}0.003$
at the $3\sigma$ level Barbieri .

##  III Pure AdS background

We summarize here the results of the case in which the background is purely
AdS with $a(z)=L/z$, and assume for simplicity that $L_{0}=L$. The vector
correlator is

$\displaystyle\Sigma^{(0)}_{V}(q^{2})$ $\displaystyle=$
$\displaystyle\frac{g^{2}\left(J_{0}(L_{1}q)Y_{1}(Lq)-J_{1}(Lq)Y_{0}(L_{1}q)\right)}{q\left(J_{0}(L_{1}q)Y_{0}(Lq)-J_{0}(Lq)Y_{0}(L_{1}q)\right)}\,.$
(9)

In order to discuss the spectrum and couplings, the following approximations
can be used:

$\displaystyle\Sigma^{(0)}_{V}(q^{2})$ $\displaystyle\simeq$
$\displaystyle\frac{g^{2}J_{0}(L_{1}q)}{Lq^{2}\left(\frac{\pi}{2}Y_{0}(L_{1}q)-J_{0}(qL_{1})\left(\gamma_{E}+\log\frac{Lq}{2}\right)\right)}$
(10) $\displaystyle\simeq$
$\displaystyle\frac{g^{2}}{Lq^{2}\left(\frac{\pi}{2}\tan(L_{1}q-\frac{\pi}{4})-\left(\gamma_{E}+\log\frac{Lq}{2}\right)\right)}\,,$
(11)

the first of which is valid for $L\ll L_{1}$, and the second for $qL_{1}>1$.

From (10) one can read the coupling of the zero mode:

$\displaystyle e^{2}$ $\displaystyle=$ $\displaystyle\frac{g^{2}}{L\log
L_{1}/L}\,.$ (12)

From (11) one can look for the poles and the residues $R_{i}$. The poles
(besides the pole at zero) are in the vicinity of those of
$\tan(L_{1}q-\pi/4)$:

$\displaystyle M_{i}$ $\displaystyle\simeq$
$\displaystyle\frac{\pi}{4L_{1}}\left((4i-1)-\frac{2}{\gamma_{E}+\log(4i-1)\pi
L/(8L_{1})}\right)\,,$ (13)

while the residues are approximately given by:

$\displaystyle R_{i}$ $\displaystyle\simeq$
$\displaystyle\frac{4\log(L_{1}/L)}{-2+\frac{2\pi
L_{1}M_{i}}{1+\sin(2L_{1}M_{i})}},$ (14)

with $i=1,2,\dots$. These approximations are acceptably accurate as long as
$L\ll L_{1}$. A numerical calculation will be performed later on, when
discussing the phenomenology for some relevant choice of parameters.

The axial correlator can be computed exactly:

$\displaystyle\Sigma^{(0)}_{A}(q^{2})$ $\displaystyle=$
$\displaystyle\frac{g^{2}\left(\left(qJ_{0}(L_{1}q)+mJ_{1}(L_{1}q)\right)Y_{1}(Lq)-J_{1}(Lq)\left(qY_{0}(L_{1}q)+mY_{1}(L_{1}q)\right)\right)}{q\left(qJ_{0}(L_{1}q)Y_{0}(Lq)+mJ_{1}(L_{1}q)Y_{0}(Lq)-J_{0}(Lq)\left(qY_{0}(L_{1}q)+mY_{1}(L_{1}q)\right)\right)}\,,$
(15)

and, for $L_{0}\ll L_{1}$, yields

$\displaystyle\hat{S}$ $\displaystyle=$
$\displaystyle\frac{\cos^{2}\theta_{W}L_{1}m(3L_{1}m+8)}{4(L_{1}m+2)^{2}\log\left(\frac{L_{1}}{L}\right)}\,.$
(16)

In the limit $m\rightarrow+\infty$ we have

$\displaystyle\hat{S}$ $\displaystyle=$
$\displaystyle\frac{3\cos^{2}\theta_{W}}{4\log{L_{1}/L}}\,.$ (17)

Imposing the ($3\sigma$-level) experimental limit we find that

$\displaystyle\frac{g^{2}}{L}$ $\displaystyle=$ $\displaystyle
e^{2}\log\frac{L_{1}}{L}\,=\,e^{2}\frac{3\cos^{2}\theta_{W}}{4\hat{S}}\,\mathrel{\raisebox{-2.58334pt}{$\stackrel{{\scriptstyle\textstyle>}}{{\sim}}$}}\,20\,,$
(18)

where $e^{2}\simeq 0.1$ is the effective coupling of the electro-magnetic
$U(1)_{Q}$ in the standard model. Since, as discussed later, $g^{2}/L$ gives a
measure of the effective strength of the self interactions between resonances
(and the dimensionful coupling $g$ is the expansion parameter in the 5D
action) the experimental bounds are satisfied only at the price of loosing
calculability, as is the unfortunate case also when trying to build QCD-like
technicolor models in 4D, either using the large-$N$ expansion, hidden local
symmetry, or deconstruction (see for instance deconstructTC ). We do not
discuss further this limit.

In the more interesting and realistic case in which $mL_{1}\ll 1$, the axial-
vector spectrum and couplings are approximately the same as the vectorial
sector. In this framework $m$ is just a free parameter, and we treat it as
such. With finite $mL_{1}\ll 1$, the mass of the lightest axial-vector state
is approximately $M_{Z}^{2}\simeq m/(L_{1}\log(L_{1}/L))$, and hence

$\displaystyle\hat{S}\simeq\frac{\cos^{2}\theta_{W}}{2\log
L_{1}/L}\,mL_{1}\simeq\frac{\cos^{2}\theta_{W}}{2}M_{Z}^{2}L_{1}^{2}\,$ (19)

satisfies the bounds on $\hat{S}$ for
$1/L_{1}\mathrel{\raisebox{-2.58334pt}{$\stackrel{{\scriptstyle\textstyle>}}{{\sim}}$}}1$
TeV, which depending on the value of $L_{0}/L_{1}$ translates into a bound
$M_{1}\mathrel{\raisebox{-2.58334pt}{$\stackrel{{\scriptstyle\textstyle>}}{{\sim}}$}}2.5-4$
TeV. For instance, for $g^{2}/L<1/2$ it requires
$M_{1}\mathrel{\raisebox{-2.58334pt}{$\stackrel{{\scriptstyle\textstyle>}}{{\sim}}$}}2.8$
TeV, and consequently
$M_{2}\mathrel{\raisebox{-2.58334pt}{$\stackrel{{\scriptstyle\textstyle>}}{{\sim}}$}}6$
TeV, which is beyond the projected reach of the LHC searches.

##  IV Departure from AdS

We now consider the possibility that conformal invariance be violated at some
energy regime above the confinement scale and suppose there exists a hierarchy
of scales $L_{0}=L<\bar{L}<L_{1}$ such that the space is the usual AdS for
$L_{0}<z<\bar{L}$, but departs from it in the IR region $\bar{L}<z<L_{1}$. Our
aim is to model this behavior without affecting the approximate description of
confinement provided by the IR brane (different motivations lead the authors
of HS  to other parameterizations). The simplest form one can choose in order
to achieve this goal is a power-law warp factor

$\displaystyle a(z)$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{cc}\frac{L}{z}&z<\bar{L}\cr\frac{L}{z}\left(\frac{\bar{L}}{z}\right)^{n-1}&z>\bar{L}\end{array}\right.\,.$
(22)

This parameterization may be viewed as a leading order approximation of a
smooth background describing the appearance of some relevant deformation in
the conformal field theory before the underlying fundamental theory confines.

We will see later that a power-law avoids generating an explicit mass gap from
the bulk equations, so that the quantity $1/L_{1}$ can still be interpreted as
the scale of confinement. Moreover, with our parameterization we can solve the
equations exactly and in a very straightforward way, which is in itself a
welcome property when modeling an otherwise untreatable dynamical system.

Most of the algebraic manipulations can be performed for generic $n$. Yet, we
discuss explicitly only the $n>1$ case. A variety of arguments, all ultimately
descending from unitarity, suggest that we should limit ourselves to $n\geq
1$. An extra-dimensional argument can be derived along the lines of weakenergy
, in which it is shown how the weaker energy condition leads to a $c$-theorem
controlling the behavior of the curvature in crossing a phase transition
towards the IR. This is related to the fact that, in the context of strongly-
coupled four-dimensional models, in going through a phase transition it is
reasonable to expect the effective number of light degrees of freedom to
decrease Appelquist . Hence the effective coupling of the effective field
theory description, which is related to the $1/N$ expansion, is expected to
increase. We show later in the paper that the effective self-coupling of the
heavy resonances is enhanced for $n\geq 1$, in agreement with the four-
dimensional intuitive expectation, and that this enhancement is controlled by
a power of the ratio of relevant scales, in agreement with naive expectations
for a theory with a generic deformation due to a relevant operator. The fact
that all of our results agree with the intuitive interpretation gives an
indication in support both of the power-law parameterization chosen here and
of the $n\geq 1$ restriction.

The solutions to the bulk equations in the IR region $z>\bar{L}$ are of the
form

$\displaystyle v^{IR}(q,z)$ $\displaystyle=$ $\displaystyle
z^{\frac{n+1}{2}}\left(c_{1}^{IR}(q)J_{\frac{n+1}{2}}(qz)\,+\,c_{2}^{IR}(q)Y_{\frac{n+1}{2}}(qz)\right)\,,$
(23)

while in the UV region

$\displaystyle v^{UV}(q,z)$ $\displaystyle=$ $\displaystyle
z\left(c_{1}^{UV}(q)J_{1}(qz)\,+\,c_{2}^{UV}(q)Y_{1}(qz)\right)\,.$ (24)

The bulk profile is obtained by applying the IR boundary conditions to
$v^{IR}$, and then by requiring that the junction of the two solutions be
smooth, so that no boundary action localized at $\bar{L}$ is left:

$\displaystyle\partial_{5}v^{IR}(q,L_{1})$ $\displaystyle=$ $\displaystyle
0\,,$ (25) $\displaystyle v^{IR}(q,\bar{L})$ $\displaystyle=$ $\displaystyle
v^{UV}(q,\bar{L})\,,$ (26) $\displaystyle\partial_{5}v^{IR}(q,\bar{L})$
$\displaystyle=$ $\displaystyle\partial_{5}v^{UV}(q,\bar{L})\,.$ (27)

The correlator is then obtained from Eq. (5) by using $v^{UV}$. From all of
this, one can extract the masses and couplings of the resonances. In
particular, the coupling of the zero-mode (photon) is

$\displaystyle e^{2}$ $\displaystyle=$
$\displaystyle\frac{(n-1)\frac{g^{2}}{L}}{(n-1)\log(\frac{\bar{L}}{L_{0}})+\left(1-\left(\frac{\bar{L}}{L_{1}}\right)^{n-1}\right)}\,.$
(28)

For $n=1$, or for $\bar{L}=L_{1}$, one recovers the AdS result (12). For $n>1$
and $\bar{L}<L_{1}$ this estimate is enhanced (for fixed $g^{2}/L$). In order
to understand how significant this effect is, one needs to compare this
coupling to the effective self-coupling, which is discussed in the next
section.

Analytical expressions for the couplings and masses of the vector-like
resonances are rather involved, and numerical results will be plotted later
on. In order to gain a semi-quantitative understanding of how these quantities
are modified with respect to the pure AdS case, we discuss the (unrealistic)
extreme case in which $\bar{L}=L_{0}\ll L_{1}$. For $qz\gg 1$:

$\displaystyle J_{\frac{n-1}{2}}(qz)$ $\displaystyle\simeq$
$\displaystyle\sqrt{\frac{2}{\pi qz}}\cos\left(qz-\frac{n\pi}{4}\right)\,,$
(29) $\displaystyle Y_{\frac{n-1}{2}}(qz)$ $\displaystyle\simeq$
$\displaystyle\sqrt{\frac{2}{\pi qz}}\sin\left(qz-\frac{n\pi}{4}\right)\,$
(30)

and the masses of $i$-th resonances, for
$n\mathrel{\raisebox{-2.58334pt}{$\stackrel{{\scriptstyle\textstyle>}}{{\sim}}$}}1$,
are approximately given by the zeros of $J_{\frac{n-1}{2}}(qL_{1})$,

$\displaystyle M_{i}(n)$ $\displaystyle\simeq$
$\displaystyle\frac{2i-1}{2}\frac{\pi}{L_{1}}\,+\,\frac{n\pi}{4L_{1}}\,,$ (31)

with $i=1,2,\dots$. This agrees with the pure AdS case ($n=1$) at least for
$L/L_{1}\ll 1$. For the more realistic case in which $L_{0}\ll\bar{L}<L_{1}$,
the spectrum of heavy modes is going to be shifted, with masses heavier by
approximately $(n-1)\pi/(4L_{1})$ with respect to the AdS case, for those
resonances whose masses are comparable with the new scale $1/\bar{L}$. The
spectrum connects back to the pure AdS case for higher excitation number $i$.

As for the residues, the couplings to the currents of the heavy modes are
approximately going to be suppressed with a power-law dependence
$\approx\left(\frac{\bar{L}}{L_{1}}\right)^{(n-1)}$ with respect to the AdS
case. Again, this suppression applies only to the lightest resonances, those
for which the mass is shifted to higher values.

For the axial-vector case, if the symmetry breaking takes place at $L_{1}$ the
only change is the IR boundary condition:

$\displaystyle\partial_{5}v_{A}^{IR}(q,L_{1})\,+\,mv_{A}(q,L_{1})$
$\displaystyle=$ $\displaystyle 0\,,$ (32) $\displaystyle
v_{A}^{IR}(q,\bar{L})$ $\displaystyle=$ $\displaystyle
v_{A}^{UV}(q,\bar{L})\,,$ (33)
$\displaystyle\partial_{5}v_{A}^{IR}(q,\bar{L})$ $\displaystyle=$
$\displaystyle\partial_{5}v_{A}^{UV}(q,\bar{L})\,.$ (34)

For generic $n>1$, we find the following approximations for the mass of $Z$
boson and for $\hat{S}$:

$\displaystyle M_{Z}^{2}$ $\displaystyle\simeq$
$\displaystyle\left(\frac{\bar{L}}{L_{1}}\right)^{n-1}\frac{(n-1)m}{L_{1}\left(1-(\bar{L}/{L_{1}})^{n-1}+(n-1)\log\bar{L}/L\right)}\,,$
(35) $\displaystyle\hat{S}$ $\displaystyle\simeq$
$\displaystyle\cos^{2}\theta_{W}\left(\frac{1}{n+1}+\frac{1}{2}(\bar{L}/L_{1})^{2}-\frac{1}{n+1}(\bar{L}/L_{1})^{n+1}\right)L_{1}^{2}M_{Z}^{2}\,,$
(36)

where the last approximation is valid as long as $mL_{1}\ll 1$. For small
$\bar{L}/L_{1}$ this approximation would not hold, because of the dependence
of $M_{Z}$ on $m$ and on $\bar{L}/L_{1}$. We do not admit a parametric
separation between $\bar{L}$ and $L_{1}$, and hence the approximations are
acceptable. We also checked this numerically, using the exact bulk profiles
and correlators.

The other extreme possibility we are interested in is the one in which the
symmetry-breaking condensate is localized at $\bar{L}$, for which the boundary
conditions become

$\displaystyle\partial_{5}v_{A}^{IR}(q,L_{1})$ $\displaystyle=$ $\displaystyle
0\,,$ (37) $\displaystyle v_{A}^{IR}(q,\bar{L})$ $\displaystyle=$
$\displaystyle v_{A}^{UV}(q,\bar{L})\,,$ (38)
$\displaystyle\partial_{5}v_{A}^{IR}(q,\bar{L})$ $\displaystyle=$
$\displaystyle\partial_{5}v_{A}^{UV}(q,\bar{L})\,+\,\bar{m}v_{A}(q,\bar{L})\,.$
(39)

For generic $n$:

$\displaystyle M_{Z}^{2}$ $\displaystyle\simeq$
$\displaystyle\frac{(n-1)\bar{m}}{\bar{L}\left(1-(\bar{L}/L_{1})^{n-1}+(n-1)\log\bar{L}/L\right)}\,,$
(40) $\displaystyle\hat{S}$ $\displaystyle\simeq$
$\displaystyle\frac{\cos^{2}\theta_{W}\left(n+1-2\left(\bar{L}/L_{1}\right)^{n-1}\right)}{2(n-1)}\bar{L}^{2}M_{Z}^{2}\,$
(41)

in which in the last expression only the leading order of the expansion in
$M_{Z}$ has been kept, and all the expressions are valid as long as
$\bar{m}L_{1}\ll 1$.

Notice how the dependence of $M_{Z}$ on $\bar{m}$ is not suppressed by powers
of $\bar{L}/L_{1}$, as in the former case, where $m$ came from a localized
term at $L_{1}$. This result agrees with the intuitive notion that moving the
symmetry-breaking towards the UV enhances its effect for the zero-mode, while
suppressing the mass splitting of the heavy resonances. The result is well
illustrated by $\hat{S}$, which is proportional to $M_{Z}^{2}$ through the
position $\bar{L}$ or $L_{1}$ of the symmetry-breaking condensate in the fifth
dimension.

##  V Estimating the strength of the self-interactions

The departure from conformal invariance, explicitly added via a power-law
deviation from the AdS background in the IR region, might imply that the
dynamics of the effective field theory itself be strongly coupled, as is the
case for a QCD-like dynamical model. It has to be understood if the effective
field theory treatment still admits a power-counting allowing to use a cut-off
$L_{0}$ much larger than the electro-weak scale. A fully rigorous treatment of
this problem is not possible, because it requires to extend the effective
field theory Lagrangian beyond the leading order in $1/N_{c}$. Yet, a
reasonable estimate of the effective coupling can be extracted by looking at
the cubic and quartic self-couplings of the resonances, the structure of which
(at the leading order) is dictated by 5D gauge-invariance.

Consider first the pure AdS background and define

$\displaystyle g_{\rho}^{(i)\,2}$ $\displaystyle\equiv$
$\displaystyle\frac{g^{2}}{L}\frac{\int_{L_{0}}^{L_{1}}\frac{\mbox{d}z}{z}|v(M_{i},z)|^{4}}{\left(\int_{L_{0}}^{L_{1}}\frac{\mbox{d}z}{z}|v(M_{i},z)|^{2}\right)^{2}}\,.$
(42)

The expansion parameter is related to $g_{\rho}$, which we define as the
asymptotic limit of the effective self-coupling for large excitation number.
As long as $L_{0}\ll L_{1}$ and $M_{i}L_{1}\gg 1$, the bulk profiles of the
heavy modes can be approximated by

$\displaystyle v(M_{i},z)$ $\displaystyle\propto$
$\displaystyle\frac{z}{\sqrt{M_{i}}}\,J_{1}(M_{i}z)\,\propto\,\frac{\sqrt{z}}{M_{i}}\cos\left(M_{i}z-\frac{3\pi}{4}\right)\,$
(43)

yielding

$\displaystyle
g_{\rho}^{2}\,\equiv\,\lim_{i\rightarrow+\infty}g_{\rho}^{(i)\,2}$
$\displaystyle\simeq$ $\displaystyle\frac{3}{4}\frac{g^{2}}{L}\,.$ (44)

For the smallest values of $i=1,2$ this is a moderate underestimate. For
instance for $i=1$, from the exact solution one obtains $g_{\rho}^{(1)\,2}\sim
1.2g^{2}/L$. The meaning of this definition of $g_{\rho}$ is that it gives a
reasonable estimate of the strength of the self-coupling of the resonances,
and hence of the expansion parameter of the effective field theory (which is
related to the large-$N_{c}$ expansion). As expected, this turns out to be
controlled by $g^{2}/L$, up to $O(1)$ coefficients. The actual value of
$g^{2}$ is related (with the treatment of the UV cut-off used here) to the
coupling of the zero mode $e^{2}=g^{2}/(L\log L_{1}/L_{0})$, so that
$g_{\rho}^{2}\approx e^{2}\log(L_{1}/L_{0})$. This yields the relation between
strength of the effective coupling and the effective cut-off in the UV, which
as expected is logarithmic, ultimately because of conformal symmetry. The
requirement that this defines a perturbative coupling $g_{\rho}^{2}$ implies a
bound on $L_{1}/L_{0}$. Choosing for instance $L_{1}=100L_{0}$ (a value that
is not justifiable by applying naive dimensional analysis to the electro-weak
chiral Lagrangian), yields $g_{\rho}^{(i)\,2}\approx 0.3$, which means that
the effective field theory admits an acceptable expansion in powers of
$g_{\rho}^{2}/(4\pi)$ even with large choices of the UV cut-off $1/L_{0}$.

Generalizing this estimate in presence of the non-trivial background (22) is
somehow more difficult, largely because of the junction conditions at
$\bar{L}$. This can be done numerically, but for the present purposes a semi-
quantitative assessment of the size of the effective coupling suffices. We
again focus on large values of $M_{i}L_{1}$ and modify the definition of the
effective couplings to

$\displaystyle g_{\rho}^{(i)\,2}$ $\displaystyle\equiv$
$\displaystyle\frac{g^{2}}{L}\frac{\int_{L_{0}}^{L_{1}}\frac{\mbox{d}z}{z^{n}}|v(M_{i},z)|^{4}}{\bar{L}^{n-1}\left(\int_{L_{0}}^{L_{1}}\frac{\mbox{d}z}{z^{n}}|v(M_{i},z)|^{2}\right)^{2}}\,.$
(45)

The specific case we are interested in lies somewhere in between the pure AdS
and the pure power-law. In the latter case an acceptable approximation would
be:

$\displaystyle v(M_{i},z)$ $\displaystyle\propto$
$\displaystyle\frac{z^{\frac{n+1}{2}}}{\sqrt{M_{i}}}\,J_{\frac{n+1}{2}}(M_{i}z)\,\propto\,\frac{z^{\frac{n}{2}}}{M_{i}}\cos\left(M_{i}z-\frac{(n+2)\pi}{4}\right)\,.$
(46)

The effective coupling receives power-law contributions in $L_{1}/\bar{L}$,
plus terms that are logarithmic in $L_{0}/\bar{L}$ and hence subleading $O(1)$
corrections. The power-law is the most important effect and, for largish
choices of $L_{1}/\bar{L}$ and in the case $n>1$, we obtain:

$\displaystyle g_{\rho}^{2}$ $\displaystyle\simeq$
$\displaystyle\frac{3}{2(n+1)}\frac{g^{2}}{L}\left(\frac{L_{1}}{\bar{L}}\right)^{n-1}\,$
(47) $\displaystyle\simeq$
$\displaystyle\frac{3e^{2}}{2(n^{2}-1)}\left(\frac{L_{1}}{\bar{L}}\right)^{n-1}\,$
(48)

which, as in the pure AdS case, represents a defective approximation by
roughly a factor of 2 for the very first resonance. We see that, for
$g_{\rho}$ to be acceptably small as to define an expansion parameter,
$L_{1}/\bar{L}$ cannot be large.

The power-law dependence on $\bar{L}/L_{1}$ in Eq. (47) is expected in a non-
conformal effective theory, in presence of relevant operators, in which case
there cannot be a substantial scale separation between the UV cut-off and the
mass scale $L_{1}$ of the effective theory itself. This result agrees with
naive dimensional analysis counting. For instance, taking $\bar{L}=L_{0}$
implies that the model is strongly coupled, unless $(L_{1}/\bar{L})^{n-1}\ll
4\pi$, which implies a very low cut off, and the impossibility of describing
the resonances as weakly coupled.

Notice that this result depends smoothly on
$n\mathrel{\raisebox{-2.58334pt}{$\stackrel{{\scriptstyle\textstyle>}}{{\sim}}$}}1$.
But in trying to extend the analysis to $n<1$ one immediately faces a problem.
For instance, for $L_{0}\rightarrow\bar{L}\ll L_{1}$, $n<1$, and keeping
$g^{2}/L$ fixed, the effective coupling becomes vanishingly small. This
behavior would imply that, in the region of the parameters space in which the
theory admits an effective approach, the original conformal theory flows into
a new phase that is described by a new effective field theory which has
effectively a weaker coupling. This violates all possible logical
expectations, according to which such a phase transition always drives the
theory towards stronger coupling, such that the new effective field theory has
always a smaller number of light degrees of freedom, and hence a larger
expansion parameter. Though not rigorous, this argument seems to support the
hypothesis that only
$n\mathrel{\raisebox{-2.58334pt}{$\stackrel{{\scriptstyle\textstyle>}}{{\sim}}$}}1$
is an admissible choice.

From the phenomenological point of view, one way to assess how strongly
coupled is the first resonance, is to consider $\gamma_{1}$, the first excited
mode with the quantum numbers of a photon, and compare its partial width into
two standard model fermions $f$ to the partial width into two on-shell $W$
bosons, namely:

$\displaystyle\frac{\Gamma(\gamma_{1}\rightarrow
f\bar{f})}{\Gamma(\gamma_{1}\rightarrow W^{+}W^{-})}$ $\displaystyle\approx$
$\displaystyle\frac{8\alpha}{3}R_{1}\,\frac{48\pi}{g_{\rho}^{2}}\,\simeq\,\frac{\pi
R_{1}}{g_{\rho}^{2}}\,.$ (49)

For a weakly-coupled theory this approximate estimate should be $O(1)$ or
bigger. In other words, a rough estimate of the width of the first resonance
gives $\Gamma\approx g_{\rho}^{2}M_{1}/(48\pi)$, and hence the approximation
of treating this resonance as infinitely narrow (as expected at large-$N_{c}$)
makes sense only as long as $g_{\rho}^{2}$ is at most some $O(1)$ number. A
more detailed study of this quantities, and the phenomenological consequences
relevant at LHC energies, will be discussed in a subsequent study.

##  VI Phenomenological implications

###  VI.1 Spectrum and couplings to the currents

We start with a numerical analysis of the spectrum and couplings of the
vectorial excited states. We perform the numerical analysis because the
results discussed in the previous section for these quantities give only semi-
quantitative approximate expressions. Since we always consider values of $m$
and $\bar{m}$ that are small compared to $1/L_{1}$, the results apply also to
the axial-vector modes, irrespectively of the choice of localizing the
symmetry-breaking effects at $L_{1}$ or at $\bar{L}$.

The masses $M_{i}$ depend in a complicated way on $L_{1}$, $\bar{L}$, $L_{0}$,
and $n$. In Figure 1 we plot the mass (in units of $1/L_{1}$) for the first
three excited states, as a function of $L_{1}/\bar{L}$. We compare four
choices of the relevant parameters, characterized by $n=2,3$ and by the choice
of the UV cut off $L_{0}=L_{1}/20$ and $L_{0}=L_{1}/100$.

Figure 1: Masses $M_{i}L_{1}$ of the first $i=1,2,3$ excited vector modes, as
a function of $L_{1}/\bar{L}$. The four curves are drawn for $n=2$ (green) and
$n=3$ (cyan), and for $L_{0}=L_{1}/20$ (thick line) and $L_{0}=L_{1}/100$
(thin line).

As anticipated, the masses are larger than in the $n=1$ case (pure AdS), which
is recovered when $\bar{L}=L_{1}$. The enhancement is only moderate, it
affects the heavier states only for large values of $L_{1}/\bar{L}$, and is
proportional to $n$.

The coupling $R_{i}$ is, in the pure AdS case, a monotonically decreasing
function of the excitation number $i$. In Figures 2 and 3 we plot the
numerical results obtained for this quantity, for the same choices of
parameters used for the masses. In going from $L_{1}/\bar{L}=1$ (pure AdS) to
larger values and/or to large $n$, a suppression of the coupling is obtained
for the lightest state. This suppression is a very big effect, and it becomes
relevant at large values of $L_{1}/\bar{L}$. As a result, for instance in the
case $n=3$, with
$L_{1}/\bar{L}\mathrel{\raisebox{-2.58334pt}{$\stackrel{{\scriptstyle\textstyle>}}{{\sim}}$}}4$
the third resonance has the strongest coupling, followed by the second and by
the first.

Figure 2: Relative coupling $R_{i}$ to the currents of the first $i=1,2,3$
excited vector modes, as a function of $L_{1}/\bar{L}$. The curves are drawn
for $n=2$, and for $L_{0}=L_{1}/20$ (thick line) and $L_{0}=L_{1}/100$ (thin
line).  Figure 3: Relative coupling $R_{i}$ to the currents of the first
$i=1,2,3$ excited vector modes, as a function of $L_{1}/\bar{L}$. The curves
are drawn for $n=3$, and for $L_{0}=L_{1}/20$ (thick line) and
$L_{0}=L_{1}/100$ (thin line).

###  VI.2 Self-couplings and symmetry-breaking

We want the 5D action to define a reasonable effective field theory treatment
of the strong dynamics and of the resulting electro-weak symmetry breaking
effects, with a well-behaved perturbative expansion. We implement this
requirement by imposing the bound
$g_{\rho}^{2}\mathrel{\raisebox{-2.58334pt}{$\stackrel{{\scriptstyle\textstyle<}}{{\sim}}$}}1/2$
(a reference value that we fix in such a way that for the choices of
parameters discussed here the ratio of partial width estimated in Eq. (49) is
$\mathrel{\raisebox{-2.58334pt}{$\stackrel{{\scriptstyle\textstyle>}}{{\sim}}$}}1$),
where $g_{\rho}$ has been defined in the body of the previous section. In the
pure AdS case we require that
$L_{1}/L_{0}\mathrel{\raisebox{-2.58334pt}{$\stackrel{{\scriptstyle\textstyle<}}{{\sim}}$}}200$,
which means that the model is very modestly sensitive to the position of the
UV cut-off and, unless extreme choices of $L_{0}\ll L_{1}$ are used, we can
neglect the effect of $L_{0}$ in driving the effective coupling strong. We can
therefore impose the bound directly on the modification due to the new non-
conformal energy regime:

$\displaystyle\left(\frac{L_{1}}{\bar{L}}\right)^{n-1}$
$\displaystyle\mathrel{\raisebox{-2.58334pt}{$\stackrel{{\scriptstyle\textstyle<}}{{\sim}}$}}$
$\displaystyle\frac{(n^{2}-1)}{3e^{2}}\,.$ (50)

For small values of $n\simeq 1$, the bound is not relevant, unless very large
values of $L_{1}/L_{0}$ are used. We do not discuss further this case. For
$n\mathrel{\raisebox{-2.58334pt}{$\stackrel{{\scriptstyle\textstyle>}}{{\sim}}$}}3$
the bound is very restrictive, and only $L_{1}/\bar{L}\sim O(1)$ is allowed.
This confirms the intuitive notion that if large power-law deviations are
allowed over a large energy window, the model is strongly coupled and does not
admit a perturbative and controllable effective field theory expansion. For
$n=$ 2 – 3, values of $L_{1}/\bar{L}\sim$ 3 – 8 are compatible with the
requirement that the effective field theory be weakly coupled, and offer an
interesting possibility from the phenomenological point of view. We focus on
this possibility.

The effects of symmetry breaking are encoded in the estimate of $\hat{S}$.
This is the quantity that ultimately sets a bound on $L_{1}$, and hence on the
mass of the excited resonances. If the symmetry-breaking effects are localized
at $L_{1}$, the analytical expression derived in Eq. (36) shows that, for all
practical purposes, the bounds are the same as those obtained in the pure AdS
case,
$L_{1}\mathrel{\raisebox{-2.58334pt}{$\stackrel{{\scriptstyle\textstyle<}}{{\sim}}$}}1$
TeV-1. This is the case because the only sizable suppression factors are the
$1/(n+1)$ and the $\bar{L}/L_{1}$ terms, but at large values of $n$ only
$\bar{L}/L_{1}\sim 1$ is allowed.

Let us discuss the case in which symmetry-breaking takes place at $\bar{L}$.
In order to assess how sizable the reduction in the experimental bounds is, we
require that $\hat{S}<0.003$, and calculate the minimum value of $1/L_{1}$
which is compatible with this bound, using the expression in Eq. (41). We show
the result in Figure 4 assuming various values of $L_{1}/\bar{L}$. We plot, as
a function of $n$, the lower bound for $\pi/(M_{Z}L_{1})$ – which, up to
boundary effects and model-dependent shifts, gives a reasonable estimate of
the ratio $M_{1}/M_{Z}$ (see Figure 1) – starting from the pure AdS case, but
without exceeding the ($n$-dependent) bound in Eq. (50).

Figure 4: Lower bound on $\pi/(M_{Z}L_{1})$ as a function of $n$ in the case
in which symmetry-breaking takes place at $\bar{L}$. The green curves are
obtained using $L_{1}/\bar{L}=1,2,3,4,5$ The cyan curve is obtained by using
the limiting value of $L_{1}/\bar{L}$ such that $g_{\rho}$ be perturbative. We
interrupt the (green) curves obtained at constant $L_{1}/\bar{L}$ at the value
of $n$ for which Eq. (50) would not be satisfied, which is at the intersection
with the cyan curve.

In the pure AdS case ($L_{1}/\bar{L}=1$), the lower bound in Figure 4 implies
(using the experimental value of $M_{Z}$)
$M_{1}\mathrel{\raisebox{-2.58334pt}{$\stackrel{{\scriptstyle\textstyle>}}{{\sim}}$}}3$
TeV, and
$M_{2}\mathrel{\raisebox{-2.58334pt}{$\stackrel{{\scriptstyle\textstyle>}}{{\sim}}$}}$
6-7 TeV. Going to larger values of $L_{1}/\bar{L}$ allows for a very
significant reduction of such bounds, even when this ratio is small enough to
be compatible with the requirement that the effective coupling $g_{\rho}^{2}$
be smaller than $1/2$. As a result, the value of the scale $1/L_{1}$ can be
greatly reduced. Values such as $M_{1}\sim 1.5$ TeV, $M_{2}\sim 3$ TeV and
$M_{3}\sim 4.5$ TeV are not excluded experimentally.

A detailed calculation of the coupling to the currents and of the partial
widths is necessary in order to draw firm quantitative conclusions, but these
preliminary estimates indicate that the first three resonances have $R_{i}\sim
0.15-0.35$, while
$g_{\rho}^{(i)\,2}\mathrel{\raisebox{-2.58334pt}{$\stackrel{{\scriptstyle\textstyle<}}{{\sim}}$}}0.5$.
These resonances should have a sizable branching fraction in standard-model
fermions, and a sizable production cross-section in Drell-Yan processes. In
particular, for this range of masses and couplings, LHC has a good chance of
detecting all of these states even at moderate integrated luminosity, by
combining data on $\mu^{+}\mu^{-}$ and $e^{+}e^{-}$ final states.

##  VII Discussion

The starting point for the construction of an effective field theory
description of dynamical electro-weak symmetry breaking is the assumption that
some fundamental, possibly asymptotically free, field theory, defined in the
far UV, flows towards an (approximate) strongly-coupled fixed-point in the IR.
Accordingly, there is a regime at intermediate-to-low energies in which the
(walking) theory can be described by a weakly-coupled five-dimensional model,
in the spirit of the AdS/CFT correspondence. The presence of a deformation
away from the AdS metric—in the form of some operator that becomes relevant
and dominates the dynamics at long distances—drives the model away from the
fixed point (inducing the loss of conformal behavior), produces non-trivial
condensates (which trigger spontaneous electro-weak symmetry breaking), and
ultimately leads the theory towards confinement (and hence introducing a mass
gap in the spectrum of bound states).

This paper proposes a toy-model that allows for a quantitative study of the
effects that such a relevant deformation might have on the low-energy
observable quantities, in the regime at and below the LHC relevant energies.
The basic idea is to parameterize the effects of such a deformation in terms
of a power-law departure from the AdS background over a limited energy window
just above the scale of confinement. This treatment proves to be useful thanks
to its intrinsic simplicity and the lack of any more systematic (calculable)
approach. It has its limitations as well. Hence we summarize and critically
analyze our results, in order to draw some important model-independent
conclusion and in order to highlight the areas where more work, and possibly
some guidance from the experimental data to come, are necessary.

First of all, the type of modification of the background we propose has a very
modest effect on the spectrum of composite resonances. The properties of such
spectrum are still determined by the presence of a hard-wall in the IR, that
acts both as a regulator and as a physical scale determining the mass gaps and
spacings. It is inappropriate to believe that this model can describe
accurately more than a handful of resonances, and one should be very careful
when talking about resonances with large excitation number $i$. Yet, the
model-independent message here is quite clear, and very important. While the
spectrum is substantially independent of the possible presence, and type, of
deformation that is driving the theory away from the fixed point in the IR,
the effective couplings of the resonances, both to other resonances and to the
standard model fermions, are very sensible to the departure from conformality
that this deformation is introducing.

The calculation of the coupling to the currents and the estimate of the self-
couplings show a large departure from the expectations based on the pure AdS
case, in presence of the same regulators in the IR and in the UV. The coupling
to the currents is suppressed, and the suppression in not a universal effect,
but rather it is different for different resonances. The self couplings are
enhanced with respect to the pure AdS case, following the four-dimensional
intuition. This poses some important limitation on how long it is admissible
to assume that it will take for the theory to flow from the region in
proximity of the IR fixed point, where it is walking, to the new phase
transition at which confinement takes place. It is very encouraging that our
estimates indicate that this regime, though limited, might be long enough to
allow for very sizable $O$(2-4) effects to result, without spoiling the
calculability of the effective field theory that the AdS/CFT language is
supposed to provide.

The deformation responsible for the loss of conformal symmetry might or might
not be related with electro-weak symmetry breaking. If not, then electro-weak
symmetry breaking is triggered at the same scale as confinement, as is the
case for QCD. In this case this model allows us to say that we do not expect
any significant modification of the precision electro-weak parameters and of
the coefficients of the electro-weak chiral Lagrangian with respect to the
results obtained in the pure AdS background. In this case, the couplings of
the excited states are the only observable quantities carrying information
about the existence of an energy regime above the scale of confinement where
the dynamics is not conformal.

At large-$N_{c}$ or in presence of a complicated fermionic field content in
the fundamental theory, the chiral symmetry breaking condensates may form at a
temperature larger than the scale of confinement. In this case, the formation
of such condensates might itself be the deformation that drives the theory
away from the fixed point, and that leads to confinement at some lower scale.
The phenomenological consequences of such a scenario are relevant not only for
the LHC, but even in analyzing LEP and TeVatron data. Our simple model allows
us to show that it is reasonable to expect that in this case the estimates of
the coefficients of the chiral Lagrangian (we focused on $\hat{S}$ because
best known and most model-independent) might be suppressed by large numerical
factors, without entering a strongly coupled regime for the effective field
theory, and with a resulting drastic reduction of the experimental bounds on
the masses of the lightest new spin-1 states (techni-$\rho$). This toy-model
highlights the fact that, whatever the fundamental theory is in the far UV, if
the dynamics contains a mechanism leading to a separation of the scales of
chiral symmetry breaking and confinement, then the expectations for $\hat{S}$,
and for other precision parameters related with isospin breaking, can be
changed drastically. At the LHC, this implies that, without requiring any
additional custodial symmetry, nor any fine-tuning, the dynamics itself might
be compatible with the detection of the first two or even three excited
states, which would provide unmistakable evidence for a strongly-coupled
origin of electro-weak symmetry breaking.

The techniques used here, and the choices of parameters we make, are affected
by systematic uncertainties. The numerical results we obtain are to be taken
as an indication of what is possible, rather than as robust predictions. Yet,
part of the results are completely general: for any admissible choice of
$L_{1}/\bar{L}$, of $n>1$ and of the position in the fifth dimension at which
we localize the symmetry-breaking terms, there is always a suppression of the
coupling of the vector mesons to the currents, an enhancement of their self-
couplings, and a suppression of $\hat{S}$. These are quantitative model-
independent results, indicating that for these quantities the pure AdS case
yields always a limiting, conservative estimate. And they all point in the
direction of making the experimental searches at the LHC easier.

###### Acknowledgements.

The work of MP is supported in part by the Department of Energy under the
grant DE-FG02-96ER40956, and by the Wales Institute of Mathematical and
Computational Sciences. The work of MF and LV is partially supported by MIUR
and the RTN European Program MRTN-CT-2004-503369. MF thanks SISSA for the kind
hospitality.

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