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arXiv:hep-ph/0606086v2 24 Aug 2006
A Negative S parameter from Holographic Technicolor
Johannes Hirn1and Ver´onica Sanz2
1IFIC - Universitat de Val`encia, Edifici d’Instituts de Paterna, Apt. Correus 22085, 46071 Val`encia, Spain
2Departamento de F´ısica Te´orica y del Cosmos, Universidad de Granada, Campus de Fuentenueva, 18071 Granada, Spain
We present a new class of 5D models, Holographic Technicolor, which fulfills the basic requirements
for a candidate of comprehensible 4D strong dynamics at the electroweak scale. It is the first
Technicolor-like model able to provide a vanishing or even negative tree-level Sparameter, avoiding
any no-go theorem on its sign. The model is described in the large-Nregime. Sis therefore
computable: possible corrections coming from boundary terms follow the 1/N suppression, and
generation of fermion masses and the Sparameter issue do split up. We investigate the model’s 4D
dual, probably walking Technicolor-like with a large anomalous dimension.
Introduction: The idea that electroweak symmetry
breaking (EWSB) could be due to the onset of the strong-
coupling regime in an asymptotically-free gauge theory
was first put forward to solve the hierarchy problem in
[1]. Technicolor was based on the example of massless
QCD with two flavors, where the global SU(2) ×SU(2)
symmetry is spontaneously broken to the diagonal sub-
group. A similar theory with a mass scale of order 3000
larger would feed its three GBs to the SM SU(2)L×U(1)Y
gauge fields, yielding masses for the W±and Z, without
an associated Higgs boson. It was however shown that a
simple rescaled version of QCD fails, since it leads to the
famous Sparameter being too large and positive as com-
pared to the value extracted from experiments [2], unless
the number of techni-colors is small. This last possibility
is however undesirable, as it signifies the loss of our last
non-perturbative handle, namely the large-Nexpansion.
The recent developments in Holographic QCD [3, 4, 5]
give us a computable way of departing from rescaled
QCD. The models of Holographic QCD aim to describe
the dynamics of the QCD bound states in terms of a 5D
gauge theory: the input parameters in such a description
can be identified with the number of colors, the confine-
ment scale and the condensates. The present class of
models for dynamical EWSB works in a similar spirit.
For the first time, the tree-level Sparameter is negative.
This has further consequences in the gauge boson spec-
trum.
Holographic Technicolor: Our starting point is
a model in five-dimensions (5D) describing electroweak
symmetry breaking via boundary conditions (BCs). The
extra dimension we consider here is an interval. The
two ends of the space are located at l0(the UV brane)
and l1(the IR brane), with the names UV/IR implying
w(l0)>w(l1). We focus on metrics that can be recast as
ds2=w(z)2ηµν dxµdxν−dz2. We only consider the
dynamics of the bulk 5D symmetry SU(2)L×SU(2)R×
U(1)B−Lgauge symmetry. As in Higgsless models [6],
the BCs are chosen to break the LR symmetry to the
diagonal SU(2)Don the IR brane, while the breaking on
the UV brane reduces SU(2)R×U(1)B−Lto the hyper-
charge subgroup. The remaining 4D gauge symmetry is
thus U(1)Q.
An important ingredient of Holographic Technicolor
comes from the lessons learned in Holographic QCD:
breaking on the brane is too soft to account for all
phenomena found in QCD, in particular power correc-
tions at high energies due to condensates. Besides this
breaking by BCs, we therefore introduce breaking in
the bulk. In the following, the bulk source of breaking
will be a crossed kinetic term between L and R gauge
fields, just as in [7]. (The z-dependence of this term
could be obtained from the profile of a scalar.) At
the quadratic level, this well-defined procedure may
effectively be summarized as yielding different metrics,
wA(z)6=wV(z) [7]. This bulk breaking will play an
important role in our description of strong dynamics at
the TeV.
The spectrum: In terms of physical states, no
massless mode survives except for the photon. The
remainder will pick up masses via the compactification.
For the class of metrics that decrease away from the UV
as AdS or faster (gap metrics), the massive modes can
be separated into two groups: ultra-light excitations [6]
and KK-modes. If we interpret the ultra-light modes as
the Wand Z, the gap suppresses the KK contributions
to the electroweak observables [6]: this can be seen
clearly using Sum Rules (SRs).
For any gap metric, the KK modes are repelled from
the UV brane, and the massive modes approximately
split into separate towers of axial and vector fields (and
Bfields). Thus, W′, the first KK mode above the W
would a priori be a vector (the techni-rho), while the
next one, W′′ , would be an axial resonance (techni-a1),
etc... One can extract SRs involving KK-mode masses
(excluding ultra-light modes)
∞
X
n=1
1
M2
Xn≃Zl1
l0
dzwX(z)αX(z)Zz
l0
dz′
wX(z′),(1)
where X=V, A, B and αV,B (z) = 1 and αA(z) =
Rl1
z
dz′
wA(z′)/Rl1
l0
dz′′
wA(z′′). The SR in Eq.(1) is exact at or-
der OG0, where Gis the gap between the ultra-light
mode and the heavy modes: in AdS, w(z) = l0/z and
2
0
2
4
6
8
10
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4
FIG. 1: Masses at OG0divided by l1for the lightest vector
and axial KK modes of the W, as a function of the condensate
in their respective channel oV,A , for d= 2.
the gap is G= log(l1/l0). As in Holographic QCD, the
function αA(z) [5] is the wavefunction of the “would-be”
Goldstone boson matrix, DµU(x): it is monotonously
decreasing with BCs αA(l0) = 1 and αA(l1) = 0.
On the other hand, another exact SR can be obtained,
involving both heavy and ultra-light modes. For gap
metrics, it can be expanded to obtain the mass of the
ultra-light mode: at order O(1/G), we get
M2
W≃1/ Zl1
l0
dz(wV(z) + wA(z)) Zl1
l0
dz′
wA(z′)!,(2)
which can be shown to agree with the expression involv-
ing the 4D gauge coupling gand techni-pion decay con-
stant f
M2
W=g2f2
4+O1/G2=1
Gl2
1
+O1/G2,(3)
as expected from Technicolor. Eq.(3) shows that, at lead-
ing order O(1/G), M2
W,Z do not feel any breaking of con-
formality in the bulk: their mass is dominated by the UV
physics. On the other hand, Eq.(1) showed that the KK
masses do feel the effect of this bulk breaking at lead-
ing order OG0. Also, since their wave-functions are
repelled, the KK-modes have masses that are quite in-
sensitive to the UV brane position. Their appearance is
due to the fragmentation of the continuum of states due
to the IR breaking of conformal invariance. Therefore,
their mass is dictated by the position of the IR brane,
m∝1/l1and does in addition depend on the conden-
sates, as shown in FIG. 1 for the metrics of Eq.(6): the
ratio mA1/mV1tends to be lowered as negative conden-
sates are switched on.
Many other results can be shown in terms of SRs
[8], and we just outline them briefly. For example, as
is standard in 5D models, non-oblique corrections are
produced at low-energy: four-fermion interactions are
generated by the exchange of KK states. It can be shown
that the expression of the resulting Fermi constant in
terms of the techni-pion decay constant is obtained from
the SM by replacing v→f. Since the model is based on
an SU(2) ×SU(2) (gauge) symmetry in the bulk, broken
to the diagonal subgroup by the IR BCs, it possesses
custodial symmetry. This implies that the low-energy
rho parameter ρ∗(0) is strictly equal to one at tree
level, as was found in the deconstructed case [9] and
indicated by [10]. Also, the KK modes, being repelled
from the UV brane, are insensitive to the UV BCs. The
KK spectrum is therefore isospin symmetric up to 1/G
corrections: Wndegenerate with Zn. In addition, since
the KK contribution is small due to the large masses of
the KK modes, one concludes that the Tparameter is
suppressed in these models. Finally, one can also show
from two SRs that the E4and E2contributions to the
WLWLscattering vanish [6].
The SParameter: The tree-level contribution to the
Sparameter, being a low-energy effect due to strong dy-
namics responsible for spontaneous symmetry-breaking,
can be expressed [2] in terms of the L10 coupling of
chiral lagrangians Stree =−16πL10. The value extracted
from LEP physics is [11] S=−0.13(0.07) ±0.10 with
reference Higgs mass mH= 117(150) GeV, where the
value in parentheses is the most recent analysis of data
at the Zpole (2005). A sizeable negative L10 would
easily upset the experimental constraint (note that in
Nc= 3 QCD, −16πL10 ∼0.3). On the other hand,
large-Nmodels of strong dynamics predict the value
of L10 in terms of contribution of spin-1 resonances
L10 =−1/4P∞
n=1 f2
Vn−f2
An, via their decay constants
fXnaccording to [12], whereas other contributions are
down by 1/N . Higgsless models thus face a serious
challenge, a no-go theorem [13]: L10 is bounded to be
negative. This is readily understood by using a SR: one
can translate the sum over resonance contributions into
a purely geometric factor
L10 =−N
48π2Zl1
l0
dz
l0
w(z)1−α(z)2,(4)
where we have defined N/12π2≡l0/g2
5. The bound
α(z)61 implies that L10 is negative and proportional
to the loop expansion parameter, N. The most natural
value for L10 will thus drive a large positve Sparame-
ter, excluding the simplest realization of the model. For
example, pure AdS yields Stree =N/4π.
One possibility would be to consider these models in
the low-Nregime. This situation is most unwelcome, as
has been stressed by many authors [14] The main rea-
son is that the value of Nplays an important role: it
sets the range of computability of the model. Low N
implies strong coupling of the gauge KK modes. A way
3
of putting it is via the position-dependent cutoff [15, 16]:
a cutoff Λ at the position where w(z) is normalized to
unity will be redshifted for processes located near a po-
sition zas Λ(z) = Λ√g00 = Λw(z). For example, in
pure AdS, the 5D loop expansion breaks down when
Λ(z)z∼24π3l0/g2
5= 2πN . The other parameter playing
an important role is the gap G. Reproducing the Fermi
constant and the Wmass implies N G ∼500. Pushing
to low values of Nis thus asking for a bigger separation
between the Wand its KK modes, which would con-
flict with the premise of perturbativity: strong coupling
would set in before the resonances tame the high-energy
behavior of amplitudes.
Returning to the large-Nregime, one is then cornered
to hope for miraculous cancellations. Efficient possibili-
ties would be: introducing IR localized kinetic terms pro-
portional to SU(2)Dor hoping for cancellations against
fermion contributions [6]. Both possibilities face again
new challenges, difficult to resolve. Trying to add large
localized kinetic terms with the “wrong” sign, which are
of order 1/N directs again towards the low-Nproblem.
Besides it leads to a tachyon instability [13]. The way
out with bulk fermions poses a problem of naturalness
and dangerously ties the Sparameter problem with the
fermion mass hierarchy, and therefore with non-oblique
corrections [6, 17].
Here we propose a different point of view, which arises
naturally in Holographic QCD and should therefore ap-
pear in a Technicolor-like model. Local order parameters
of the symmetry-breaking imply a different behavior for
the Vand A combinations of bulk fields [3, 5]. In the sim-
plest realization of this IR behavior [7], L10 is modified
from Eq.(4) to read
L10 =−N
48π2Zl1
l0
dz
l0wV(z)−wA(z)αA(z)2,(5)
where wV,A are the metrics felt by the axial and vector
combinations of fields.
L10 is still proportional to N, but the integrand in
Eq.(5) can reverse sign for zsuch that wA(z)α(z)2>
wV(z), and L10 may come out positive. The first conse-
quence is quite clear: a large-Nscenario is then preferred,
extending the pertubativity regime. In particular, the
bulk value of Swill not receive sizeable corrections from
the localized kinetic terms, since these are still suppressed
by 1/N. The Sparameter is therefore computable. An-
other important property of the bulk Sparameter is its
independence on the exact IR dynamics. Contrary to the
spectrum, contributions to Scome mainly from the bulk
far from the branes [8].
We now assume that the metrics behave as AdS near
the UV brane and deviate from conformality in the bulk
according to
wX(z) = l0
zexp νXz−l0
l1−l02d!.(6)
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
FIG. 2: Value of Stree/N —for d= 2 and for different values
of oV— as a function of the ratio of condensates in the two
channels oA/oV, and for the pure AdS case.
As explained at the beginning of the paper, this para-
metrically simple form encodes effects of couplings with
other background fields, whose dynamics we neglect here.
At order OG0, one can obtain an analytic expression
for Sin the case νA= 0 and νV<0
Stree =N
4π1−2
3d(Γ(−νV) + log(−νV) + γE).(7)
In νV≡12π3/2Γ(d+1/2)
d2Γ(d)3oV, NDA sets oV∼ O(1) [7].
In FIG.(2), we show the value of Stree/N for different
values of the ratio oA/oVfixing d= 2. A negative vector
condensate can lead to vanishing or negative Stree , even
more so if it is accompanied by an axial condensate of
the same sign: a direction not explored by the authors
of [4, 18]. Also, assuming oX∼ O (1), the effect dis-
appears if the dimension of the condensate is increased.
Our results thus extend those of [19], which indicated
that increasing oA−oV, preferably with a low d, could
decrease the Sparameter, in connection with a lowering
of the ratio mA1/mV1.
A refinement in the computation of the Sparameter
comes from taking into account the pion loop effects [2]
and subtracting the SM value with a reference Higgs mass
S=−16πL10 (µ) + 1
12πlog µ2
m2
H−1
6.(8)
From the understanding of the QCD case [12], one
expects the model to predict the value of L10 (µ) at the
matching scale of the model with a chiral lagrangian, i.e.
µ∼few/l1∼few TeV, the mass scale of the resonances.
The second term in Eq.(8) is then positive and of order
0.1, requiring a vanishing or slightly negative Stree, as
provided by the present model.
4
Four-dimensional dual: Holographic models
are inspired from the AdS/CFT correspondence [20].
The precise form of this conjecture relates two highly
symmetric theories and is, unfortunately, far from being
of direct phenomenological relevance. After a pioneering
work by Pomarol [21], authors in [22] explored the
audacious conjecture that more realistic models like
Randall-Sundrum [23] would somehow inherit properties
of the duality. Since then, more evidence has been
gathered towards a 5D/4D duality, the latest being
bottom-up models of Holographic QCD [3, 4, 5, 7, 24].
The success of these models in capturing the behavior
of a strongly-coupled theory like QCD provides an
incentive for applications to Technicolor. In this case,
one starts off on a firmer footing: in the presence of
condensates, the number of (techni)-colors can be made
large since it no longer in conflict with the Sparameter.
Let us show the effect in the 4D two-point correlator of
the current X=V, A, B of a metric of the form given by
Eq.(6). For large euclidean Q2, the two-point function
for this field Xreads [5]
ΠX−Q2≃ − N
12π2log Q2
µ2+λ(µ)+hO2dXi
Q2dX
(9)
where the parameter oX≡ hO2dXi/(N l−2dX
1)∼ O(1).
To have a chance of obtaining a positive value for L10,
we need hO2diV<hO2diA. This is in agreement with
Witten’s positivity condition for ΠA−ΠV[25], ensur-
ing the stability of the selected vacuum [26]. Holography
tells us that this bulk field Xis dual to some operator
Oon the 4D side with the same quantum numbers: the
correlators generated by Xand by Oare the same. In
this particular case we see that deviations from confor-
mality with a given power of z2din Eq.(6) mimick the
effects of a condensate of dimension 2din the 4D dual.
Generally speaking, non-perturbative effects in QCD-
like Technicolor models make them unreliable. The same
goes for the case of a flat extra-dimension, the cutoff of
the theory is quite low, Λ ∼2πN/l1and quantities like
the Sparameter are no longer computable. On the other
hand, extra-dimensional models in AdS behave in a sim-
ilar fashion to walking Technicolor. The warping sup-
presses convolutions of wave-functions, as walking kills
unwanted operators. But in pure AdS, one cannot choose
which operators will be suppressed: their scaling is dic-
tated by the warping, whereas gap metrics with viola-
tions of conformality like Eq.(6) do change the scaling.
The dual of Holographic Technicolor must be a
strongly-coupled theory, with the running in the UV
dictated by the one of a gap metric and with non-
perturbative dynamics affecting the vector and axial
channel in a similar way. If the 4D dual is going to
yield small or negative Sparameter, the net effect of
condensates in the vector and axial current must go
in the direction of wAα2> wV. For example, imagine
that strong dynamics generate a techni-condensate
hQ¯
Qiresponsible of breaking the Technicolor gauge
group SU(N): this condensate is represented in the 5D
dual as the rescaled vev of hΦi. Assume now that the
anomalous dimensions is large, for example, due to the
running mass in the 5D picture. Then, there will be
a difference between the canonical dimension of h¯
QQi
and the running dimension of the operator. A way of
modelling this anomalous dimension would be that the
vector and axial fields couple to a scalar representing the
techni-quark condensate, Φ, via a running mass , such
that mΦ(l0)2=−3/l2
0and mΦ(l1)2=d(d−4)/l2
0with
d < 3 (d= 2 for extreme walking).
Conclusions: In this paper we have shown quantita-
tively how technicolor models which depart from rescaled
QCD can exhibit a negative tree-level Sparameter. This
was done using a holographic model (i.e. using a 5D
gauge theory) for the resonances created by a strongly-
interacting theory such as technicolor. It is based on the
recent successes of similar 5D models for the resonances
of QCD. These successes themselves validated the idea
of the duality between 4D strongly-coupled theories and
5D weakly-coupled ones at the quantitative level.
We have presented the first Technicolor-like model able
to provide a small Sparameter, and to remain com-
putable since it is defined in the large-Nlimit. The 5D
picture shows generic features of this class of models: 1)
the metric has to fall off fast near the UV to generate a
gap, 2) deviations from conformality must be introduced
in the bulk, describing condensates, 3) a condensate of
natural size can produce the desired effect if it has di-
mension close to 4 (as would happen for αTC h¯
QQi2in
walking Technicolor), 4) W′and Z′(vector resonances)
then tend to become degenerate with the W′′ and Z′′
(axial) resonances.
In the present paper, the fermions were located for
simplicity on the UV brane. As soon as we let them live
in the bulk, much more interesting phenomena should
arise: one big advantage of the present models is that
the fermion profiles are not constrained by the require-
ment of cancelling the Sparameter contributions. The
issue of the Sparameter is therefore decoupled from that
of fermion mass generation or from Z→bb, which can
be addressed in a new view [8]. In particular, topcolor
assisted models would be implemented as in [27]. Ac-
knowledgments: We acknowledge hospitality from
Boston, Harvard and Yale Universities during the com-
pletion of this work. We also thank Tom Appelquist,
Tony Gherghetta, Ami Katz, Ken Lane, John March-
Russell, Toni Pich and Francesco Sannino for stimulat-
ing discussions. JH is supported by the EC RTN net-
work HPRN-CT-2002-00311 and by the Generalitat Va-
lenciana grant GV05/015.
5
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