The S-parameter in Holographic Technicolor Models
The S-parameter in
Holographic Technicolor Models
Kaustubh Agashea, Csaba Csákib, Christophe Grojeanc,d, and Matthew Reeceb
a Department of Physics, Syracuse University, Syracuse, NY 13244, USA
b Institute for High Energy Phenomenology
Newman Laboratory of Elementary Particle Physics,
Cornell University, Ithaca, NY 14853, USA
d CERN, Theory Division, CH 1211, Geneva 23, Switzerland
c Service de Physique Théorique, CEA Saclay, F91191, Gif-sur-Yvette, France
kagashe@phy.syr.edu, csaki@lepp.cornell.edu,
christophe.grojean@cern.ch, mreece@lepp.cornell.edu
Abstract
We study the S parameter, considering especially its sign, in models of electroweak
symmetry breaking (EWSB) in extra dimensions, with fermions localized near the
UV brane. Such models are conjectured to be dual to 4D strong dynamics triggering
EWSB. The motivation for such a study is that a negative value of S can significantly
ameliorate the constraints from electroweak precision data on these models, allowing
lower mass scales (TeV or below) for the new particles and leading to easier discovery at
the LHC. We first extend an earlier proof of S > 0 for EWSB by boundary conditions in
arbitrary metric to the case of general kinetic functions for the gauge fields or arbitrary
kinetic mixing. We then consider EWSB in the bulk by a Higgs VEV showing that
S is positive for arbitrary metric and Higgs profile, assuming that the effects from
higher-dimensional operators in the 5D theory are sub-leading and can therefore be
neglected. For the specific case of AdS5 with a power law Higgs profile, we also show
that S ∼ +O(1), including effects of possible kinetic mixing from higher-dimensional
operator (of NDA size) in the 5D theory. Therefore, our work strongly suggests that
S is positive in calculable models in extra dimensions.
http://arxiv.org/abs/0704.1821v2
1 Introduction
One of the outstanding problems in particle physics is to understand the mechanism of
electroweak symmetry breaking. Broadly speaking, models of natural electroweak symmetry
breaking rely either on supersymmetry or on new strong dynamics at some scale near the
electroweak scale. However, it has long been appreciated that if the new strong dynamics is
QCD-like, it is in conflict with precision tests of electroweak observables [1]. Of particular
concern is the S parameter. It does not violate custodial symmetry; rather, it is directly
sensitive to the breaking of SU(2). As such, it is difficult to construct models that have S
consistent with data, without fine-tuning.
The search for a technicolor model consistent with data, then, must turn to non-QCD-
like dynamics. An example is “walking” [2], that is, approximately conformal dynamics,
which can arise in theories with extra flavors. It has been argued that such nearly-conformal
dynamics can give rise to a suppressed or even negative contribution to the S parameter [3].
However, lacking nonperturbative calculational tools, it is difficult to estimate S in a given
technicolor theory.
In recent years, a different avenue of studying dynamical EWSB models has opened up
via the realization that extra dimensional models [4] may provide a weakly coupled dual
description to technicolor type theories [5]. The most studied of these higgsless models [6] is
based on an AdS5 background in which the Higgs is localized on the TeV brane and has a very
large VEV, effectively decoupling from the physics. Unitarization is accomplished by gauge
KK modes, but this leads to a tension: these KK modes cannot be too heavy or perturbative
unitarity is lost, but if they are too light then there are difficulties with electroweak precision:
in particular, S is large and positive [7]. In this argument the fermions are assumed to be
elementary in the 4D picture (dual to them being localized on the Planck brane). A possible
way out is to assume that the direct contribution of the EWSB dynamics to the S-parameter
are compensated by contributions to the fermion-gauge boson vertices [8, 9]. In particular,
there exists a scenario where the fermions are partially composite in which S ≈ 0 [10],
corresponding to almost flat wave functions for the fermions along the extra dimension. The
price of this cancellation is a percent level tuning in the Lagrangian parameter determining
the shape of the fermion wave functions. Aside from the tuning itself, this is also undesirable
because it gives the model-builder very little freedom in addressing flavor problems: the
fermion profiles are almost completely fixed by consistency with electroweak precision.
While Higgsless models are the closest extra-dimensional models to traditional technicolor
models, models with a light Higgs in the spectrum do not require light gauge KK modes for
unitarization and can be thought of as composite Higgs models. Particularly appealing are
those where the Higgs is a pseudo-Nambu-Goldstone boson [11, 12]. In these models, the
electroweak constraints are less strong, simply because most of the new particles are heavy.
They still have a positive S, but it can be small enough to be consistent with data. Unlike
the Higgsless models where one is forced to delocalize the fermions, in these models with
a higher scale the fermions can be peaked near the UV brane so that flavor issues can be
addressed.
Recently, an interesting alternative direction to eliminating the S-parameter constraint
has been proposed in [13]. There it was argued, that by considering holographic models
of EWSB in more general backgrounds with non-trivial profiles of a bulk Higgs field one
could achieve S < 0. The aim of this paper is to investigate the feasibility of this proposal.
We will focus on the direct contribution of the strong dynamics to S. In particular, we
imagine that the SM fermions can be almost completely elementary in the 4D dual picture,
corresponding to them being localized near the UV brane. In this case, a negative S would
offer appealing new prospects for model-building since such values of S are less constrained
by data than a positive value [14]. Unfortunately we find that the S > 0 quite generally,
and that backgrounds giving negative S appear to be pathological.
The outline of the paper is as follows. We first present a general plausibility argument
based purely on 4D considerations that one is unlikely to find models where S < 0. This
argument is independent from the rest of the paper, and the readers interested in the holo-
graphic considerations may skip directly to section 3. Here we first review the formalism
to calculate the S parameter in quite general models of EWSB using an extra dimension.
We also extend the proof of S > 0 for BC breaking [7] in arbitrary metric to the case of
arbitrary kinetic functions or localized kinetic mixing terms. These proofs quite clearly show
that no form of boundary condition breaking will result in S < 0. However, one may hope
that (as argued in [13]) one can significantly modify this result by using a bulk Higgs with a
profile peaked towards the IR brane to break the electroweak symmetry. Thus, in the crucial
section 4, we show that S > 0 for models with bulk breaking from a scalar VEV as well.
Since the gauge boson mass is the lowest dimensional operator sensitive to EWSB one would
expect that this is already sufficient to cover all interesting possibilities. However, since
the Higgs VEV can be very strongly peaked, one may wonder if other (higher dimensional)
operators could become important as well. In particular, the kinetic mixing operator of L,R
after Higgs VEV insertion would be a direct contribution to S. To study the effect of this
operator in section 5, it is shown that the bulk mass term for axial field can be converted to
kinetic functions as well, making a unified treatment of the effects of bulk mass terms and
the effects of the kinetic mixing from the higher-dimensional operator possible. Although we
do not have a general proof that S > 0 including the effects of the bulk kinetic mixing for
a general metric and Higgs profile, in section 5.2 we present a detailed scan for AdS metric
and for power-law Higgs vev profile using the technique of the previous section for arbitrary
kinetic mixings. We find S > 0 once we require that the higher-dimensional operator is of
NDA size, and that the theory is ghost-free. We summarize and conclude in section 6.
2 A plausibility argument for S > 0
In this section we define S and sketch a brief argument for its positivity in a general techni-
color model. The reader mainly interested in the extra-dimensional constructions can skip
this section since it is independent from the rest of the paper. However, we think it is worth-
while to try to understand why one might expect S > 0 on simple physical grounds. The
only assumptions we will make are that we have some strongly coupled theory that sponta-
neously breaks SU(2)L×SU(2)R down to SU(2)V , and that at high energies the symmetry is
restored. With these assumptions, S > 0 is plausible. S < 0 would require more complicated
dynamics, and might well be impossible, though we cannot prove it.1
Consider a strongly-interacting theory with SU(2) vector current V aµ and SU(2) axial
vector current Aaµ. We define (where J represents V or A):
d4x e−iq·x
Jaµ(x)J
= δab
qµqν − gµνq2
2). (2.1)
We further define the left-right correlator, denoted simply Π(q2), as ΠV (q
2)−ΠA(q2). In the
usual way, ΠV and ΠA are related to positive spectral functions ρV (s) and ρA(s). Namely,
the Π functions are analytic functions of q2 everywhere in the complex plane except for
Minkowskian momenta, where poles and branch points can appear corresponding to physical
particles and multi-particle thresholds. The discontinuity across the singularities on the
q2 > 0 axis is given by a spectral function. In particular, there is a dispersion relation
ΠV (q
ρV (s)
s− q2 + iǫ
, (2.2)
with ρV (s) > 0, and similarly for ΠA.
Chiral symmetry breaking establishes that ρA(s) contains a term πf
πδ(s). This is the
massless particle pole corresponding to the Goldstone of the spontaneously broken SU(2)
axial flavor symmetry. (The corresponding pions, of course, are eaten once we couple the
theory to the Standard Model, becoming the longitudinal components of the W± and Z
bosons. However, for now we consider the technicolor sector decoupled from the Standard
Model.) We define a subtracted correlator by Π̄(q2) = Π(q2) +
and a subtracted spectral
function by ρ̄A(s) = ρA(s)− πf 2πδ(s). Now, the S parameter is given by
S = 4πΠ̄(0) = 4
(ρV (s)− ρ̄A(s)) . (2.3)
Interestingly, there are multiple well-established nonperturbative facts about ΠV − ΠA, but
none are sufficient to prove that S > 0. There are the famous Weinberg sum rules [17]
ds (ρV (s)− ρ̄A(s)) = f 2π , (2.4)
ds s (ρV (s)− ρ̄A(s)) = 0. (2.5)
Further, Witten proved that Σ(Q2) = −Q2(ΠV (Q2)−ΠA(Q2)) > 0 for all Euclidean momenta
Q2 = −q2 > 0 [18]. However, the positivity of S seems to be more difficult to prove.
Our plausibility argument is based on the function Σ(Q2). In terms of this function,
S = −4πΣ′(0). (Note that in Σ(Q2) the 1/Q2 pole from ΠA is multiplied by Q2, yielding
a constant that does not contribute when we take the derivative. Thus when considering
1For a related discussion of the calculation of S in strongly coupled theories, see [15].
Σ we do not need to subtract the pion pole as we did in Π̄.) We also know that Σ(0) =
f 2π > 0. On the other hand, we know something else that is very general about theories that
spontaneously break chiral symmetry: at very large Euclidean Q2, we should see symmetry
restoration. More specifically, we expect behavior like
Σ(Q2) → O
, (2.6)
where k is associated with the dimension of some operator that serves as an order parameter
for the symmetry breaking. (In some 5D models the decrease of ΠA − ΠV will actually be
faster, e.g. in Higgsless models one has exponential decrease.) While we are most familiar
with this from the OPE of QCD, it should be very general. If a theory did not have this
property and ΠV and ΠA differed significantly in the UV, we would not view it as a sponta-
neously broken symmetry, but as an explicitly broken one. Now, in this context, positivity
of S is just the statement that, because Σ(Q2) begins at a positive value and eventually
becomes very small, the smoothest behavior one can imagine is that it simply decreases
monotonically, and in particular, that Σ′(0) < 0 so that S > 0.2 The alternative would be
that the chiral symmetry breaking effects push Σ(Q2) in different directions over different
ranges of Q2. We have not proved that this is impossible in arbitrary theories, but it seems
plausible that the simpler case is true, namely that chiral symmetry restoration always acts
to decrease Σ(Q2) as we move to larger Q2. Indeed, we will show below that in a wide variety
of perturbative holographic theories S is positive.
3 Boundary-effective-action approach to oblique cor-
rections. Simple cases with boundary breaking
In this section we review the existing results and calculational methods for the electroweak
precision observables (and in particular the S-parameter) in holographic models of elec-
troweak symmetry breaking. There are two equivalent formalisms for calculating these
parameters. One is using the on-shell wave function of the W/Z bosons [19], and the
electroweak observables are calculated from integrals over the extra dimension involving
these wave functions. The advantage of this method is that since it uses the physical wave
functions it is easier to find connections to the Z and the KK mass scales. The alternative
formalism proposed by Barbieri, Pomarol and Rattazzi [7] (and later extended in [20] to in-
clude observables off the Z-pole) uses the method of the boundary effective action [21], and
involves off-shell wave functions of the boundary fields extended into the bulk. This latter
method leads more directly to a general expression of the electroweak parameters, so we will
be applying this method throughout this paper. Below we will review the basic expressions
from [7].
A theory of electroweak symmetry breaking with custodial symmetry has an SU(2)L×
SU(2)R global symmetry, of which the SU(2)L×U(1)Y subgroup is gauged (since the S-
parameter is unaffected by the extra B − L factor we will ignore it in our discussion). At
2For a related discussion of the behaviour of Σ
in the case of large-Nc QCD, see [16].
low energies, the global symmetry is broken to SU(2)D. In the holographic picture of [7] the
elementary SU(2)×U(1) gauge fields are extended into the bulk of the extra dimension. The
bulk wave functions are determined by solving the bulk EOM’s as a function of the boundary
fields, and the effective action is just the bulk action in terms of the boundary fields.
In order to first keep the discussion as general as possible, we use an arbitrary background
metric over an extra dimension parametrized by 0 < y < 1, where y = 0 corresponds to the
UV boundary, and y = 1 to the IR boundary. In order to simplify the bulk equations of
motion it is preferential to use the coordinates in which the metric takes the form 1 [7]
ds2 = e2σdx2 + e4σdy2 . (3.1)
The bulk action for the gauge fields is given by
S = − 1
(FLMN)
2 + (FRMN)
. (3.2)
The bulk equations of motion are given by
µ − p2e2σAL,Rµ = 0, (3.3)
or equivalently the same equations for the combinations Vµ, Aµ = (AµL ± AµR)/
We assume that the (light) SM fermions are effectively localized on the Planck brane
and that they carry their usual quantum numbers under SU(2)L × U(1)Y that remains
unbroken on the UV brane. The values of these fields on the UV brane have therefore a
standard couplings to fermion and they are the 4D interpolating fields we want to compute
an effective action for. This dictates the boundary conditions we want to impose on the UV
brane
ALaµ (p
2, 0) = ĀLaµ (p
2), AR 3µ (p
2, 0) = ĀR 3µ (p
2), AR 1,2µ (p
2, 0) = 0. (3.4)
R are vanishing because they correspond to ungauged symmetry generators. The solutions
of the bulk equations of motion satisfying these UV BC’s take the form
2, y) = v(y, p2)V̄µ(p
2), Aµ(p
2, y) = a(y, p2)Āµ(p
2). (3.5)
where the interpolating functions v and a satisfy the bulk equations
∂2yf(y, p
2)− p2e2σf(y, p2) = 0 (3.6)
and the UV BC’s
v(0, p2) = 1, a(0, p2) = 1. (3.7)
The effective action for the boundary fields reduces to a pure boundary term since by
integrating by parts the bulk action vanishes by the EOM’s:
Seff =
d4x(Vµ∂yV
µ + Aµ∂yA
µ)|y=0 =
d4p(V̄ 2µ ∂yv + Ā
µ∂ya)|y=0 (3.8)
1In this paper, we use a (−+ . . .+) signature. 5D bulk indices are denoted by capital Latin indices while
we use Greek letters for 4D spacetime indices. 5D indices will be raised and lowered using the 5D metric
while the 4D Minkowski metric is used for 4D indices.
And we obtain the non-trivial vacuum polarizations for the boundary vector fields
ΣV (p
2) = −
∂yv(0, p
2), ΣA(p
2) = −
∂ya(0, p
2). (3.9)
The various oblique electroweak parameters are then obtained from the momentum ex-
pansion of the vacuum polarizations in the effective action,
Σ(p2) = Σ(0) + p2Σ′(0) +
Σ′′(0) + . . . (3.10)
For example the S-parameter is given by
S = 16πΣ′3B(0) = 8π(Σ
V (0)− Σ′A(0)). (3.11)
A similar momentum expansion can be performed on the interpolating functions v and a:
v(y, p2) = v(0)(y) + p2v(1)(y) + . . ., and similarly for a. The S-parameter is then simply
expressed as
S = −8π
(1) − ∂ya(1))|y=0. (3.12)
The first general theorem was proved in [7]: for the case of boundary condition breaking in
a general metric, S ≥ 0. The proof uses the explicit calculation of the functions v(n), a(n), n =
0, 1. First, the bulk equations (3.3) write
(0) = ∂2ya
(0) = 0, ∂2yv
(1) = e2σv(0), ∂2ya
(1) = e2σa(0). (3.13)
And the p2-expanded UV BC’s are
v(0) = a(0) = 1, v(1) = a(1) = 0 at y = 0 (3.14)
Finally, we need to specify the BC’s on the IR brane that correspond to the breaking
SU(2)L×SU(2)R → SU(2)D
∂yVµ = 0, Aµ = 0, (3.15)
which translates into simple BC’s for the interpolating functions
(n) = a(n) = 0, n = 0, 1. (3.16)
The solution of these equations are v(0) = 1, a(0) = 1 − y, v(1) =
dy′′e2σ(y
′′) −
dy′e2σ(y
′), a(1) =
dy′′e2σ(y
′′)(1−y′′)−y
dy′′e2σ(y
′′)(1−y′′). Consequently
dye2σ(y)dy −
dy′(1− y′)e2σ(y′)
(3.17)
which is manifestly positive.
3.1 S > 0 for BC breaking with boundary kinetic mixing
The first simple generalization of the BC breaking model is to consider the same model but
with an additional localized kinetic mixing operator added on the TeV brane (the effect of
this operator has been studied in flat space in [7] and in AdS space in [19]). The localized
Lagrangian is
−gV 2µν . (3.18)
This contains only the kinetic term for the vector field since the axial gauge field is set to
zero by the BC breaking. In this case the BC at y = 1 for the vector field is modified
to ∂yVµ + τp
2Vµ = 0. In terms of the wave functions expanded in small momenta we get
(1)+τv(0) = 0. The only change in the solutions will be that now v(1)
= −τ−
e2σ(y
′)dy′,
resulting in
e2σ(y)dy −
(1− y′)e2σ(y′)dy′ + τ
(3.19)
Thus as long as the localized kinetic term has the proper sign, the shift in the S-parameter
will be positive. If the sign is negative, there will be an instability in the theory since fields
localized very close to the TeV brane will feel a wrong sign kinetic term. Thus we conclude
that for the physically relevant case S remains positive.
3.2 S > 0 for BC breaking with arbitrary kinetic functions
The next simple extension of the BPR result is to consider the case when there is an arbitrary
y-dependent function in front of the bulk gauge kinetic terms. These could be interpreted as
effects of gluon condensates modifying the kinetic terms in the IR. In this case the action is
S = −
φ2L(y)(F
2 + φ2R(y)(F
. (3.20)
φL,R(y) are arbitrary profiles for the gauge kinetic terms, which are assumed to be the
consequence of some bulk scalar field coupling to the gauge fields. Note that this case
also covers the situation when the gauge couplings are constant but g5L 6= g5R. The only
assumption we are making is that the gauge kinetic functions for L,R are strictly positive.
Otherwise one could create a wave packet localized around the region where the kinetic term
is negative which would have ghost-like behavior.
Due to the y-dependent kinetic terms it is not very useful to go into the V,A basis.
Instead we will directly solve the bulk equations in the original basis. The bulk equations of
motion for L,R are given by
L,R∂yA
µ )− p2e2σφ2L,RAL,Rµ = 0 (3.21)
To find the boundary effective action needed to evaluate the S-parameter we perform the
following decomposition:
ALµ(p
2, y) = L̄µ(p
2)LL(y, p
2) + R̄µ(p
2)LR(y, p
ARµ (p
2, y) = L̄µ(p
2)RL(y, p
2) + R̄µ(p
2)RR(y, p
2). (3.22)
Here L̄, R̄ are the boundary fields, and the fact that we have four wave functions expresses
the fact that these fields will be mixing due to the BC’s on the IR brane. The UV BC’s (3.4)
and the IR BC’s (3.15) can be written in terms of the interpolating functions as
(UV) LL(0, p
2) = 1, LR(0, p
2) = 0, RL(0, p
2) = 0, RR(0, p
2) = 1. (3.23)
LL(1, p
2) = RL(1, p
2), LR(1, p
2) = RR(1, p
∂y(LL(1, p
2) +RL(1, p
2)) = 0, ∂y(LR(1, p
2) +RR(1, p
2)) = 0.
(3.24)
The solution of these equations with the proper boundary conditions and for small values of
p2 is rather lengthy, so we have placed the details in Appendix A. The end result is that
S = −8π
φ2L∂yL
R + φ
|y=0 = −
(aLR + aRL), (3.25)
where the constants aRL are negative as their explicit expressions shows it. Therefore S is
positive.
4 S > 0 in models with bulk Higgs
Having shown than S > 0 for arbitrary metric and EWSB through BC’s, in this section,
we switch to considering breaking of electroweak symmetry by a bulk scalar (Higgs) vev.
We begin by neglecting the effects of kinetic mixing between SU(2)L and SU(2)R fields
coming from higher-dimensional operator in the 5D theory, expecting that their effect, being
suppressed by the 5D cut-off, is sub-leading. We will return to a consideration of such kinetic
mixing effects in the following sections.
We will again use the metric (3.1) and the bulk action (3.2). Instead of BC breaking we
assume that EWSB is caused by a bulk Higgs which results in a y-dependent profile for the
axial mass term
A2M . (4.1)
Here M2 is a positive function of y corresponding to the background Higgs VEV. The bulk
equations of motion are:
(∂2y − p2e2σ)Vµ = 0, (∂2y − p2e2σ −M2e4σ)Aµ = 0. (4.2)
On the IR brane, we want to impose regular Neumann BC’s that preserve the full SU(2)L×
SU(2)R gauge symmetry
(IR) ∂yVµ = 0, ∂yAµ = 0. (4.3)
As in the previous section, the BC’s on the UV brane just define the 4D interpolating fields
(UV ) Vµ(p
2, 0) = V̄µ(p
2), Aµ(p
2, 0) = Āµ(p
2). (4.4)
The solutions of the bulk equations of motion satisfying these BC’s take the form
2, y) = v(y, p2)V̄µ(p
2), Aµ(p
2, y) = a(y, p2)Āµ(p
2), (4.5)
where the interpolating functions v and a satisfy the bulk equations
∂2yv − p2e2σv = 0, ∂2ya− p2e2σa−M2e4σa = 0. (4.6)
As before, these interpolating functions are expanded in powers of the momentum: v(y, p2) =
v(0)(y) + p2v(1)(y) + . . ., and similarly for a. The S-parameter is again given by the same
expression
S = −8π
(1) − ∂ya(1))|y=0. (4.7)
We will not be able to find general solutions for a(1) and v(1) but we are going to prove that
(1) > ∂yv
(1) on the UV brane, which is exactly what is needed to conclude that S > 0.
First at the zeroth order in p2, the solution for v(0) is simply constant, v(0) = 1, as before.
And a(0) is the solution of
(0) = M2e4σa(0), a(0)|y=0 = 1, ∂ya(0)|y=1 = 0. (4.8)
In particular, since a(0) is positive at y = 0, this implies that a(0) remains positive: if a(0)
crosses through zero it must be decreasing, but then this equation shows that the derivative
will continue to decrease and can not become zero to satisfy the other boundary condition.
Now, since a(0) is positive, the equation of motion shows that it is always concave up, and
then the condition that its derivative is zero at y = 1 shows that it is a decreasing function
of y. In particular, we have for all y
a(0)(y) ≤ v(0)(y), (4.9)
with equality only at y = 0.
Next consider the order p2 terms. What we wish to show is that ∂ya
(1) > ∂yv
(1) at the
UV brane. First, let’s examine the behavior of v(1): the boundary conditions are v(1)|y=0 = 0
and ∂yv
= 0. The equation of motion is:
(1) = e2σv(0) = e2σ > 0, (4.10)
so the derivative of v(1) must increase to reach zero at y = 1. Thus it is negative everywhere
except y = 1, and v(1) is a monotonically decreasing function of y. Since v(1)|y=0 = 0, v(1) is
strictly negative on (0, 1].
For the moment suppose that a(1) is also strictly negative; we will provide an argument
for this shortly. The equation of motion for a(1) is:
(1) = e2σa(0) +M2e4σa(1). (4.11)
Now, we know that a(0) < v(0), so under our assumption that a(1) < 0, this means that
(1) ≤ ∂2yv(1), (4.12)
with equality only at y = 0. But we also know that ∂yv
(1)∂ya
(1) at y = 1, since they both
satisfy Neumann boundary conditions there. Since the derivative of ∂ya
(1) is strictly smaller
over (0, 1], it must start out at a higher value in order to reach the same boundary condition.
Thus we have that
> ∂yv
. (4.13)
The assumption that we made is that a(1) is strictly negative over the interval (0, 1]. The
reason is the following: suppose that a(1) becomes positive at some value of y. Then as it
passes through zero it is increasing. But then we also have that ∂2ya
(1) = e2σa(0)+M2e4σa(1),
and we have argued above that a(0) > 0. Thus if a(1) is positive, ∂ya
(1) remains positive,
because ∂2ya
(1) cannot become negative. In particular, it becomes impossible to reach the
boundary condition ∂ya
(1) = 0 at y = 1. This fills the missing step in our argument and
shows that the S parameter must be positive.
In the rest of this section we show that the above proof for the positivity of S remains
essentially unchanged in the case when the bulk gauge couplings for the SU(2)L and SU(2)R
gauge groups are not equal. In this case (in order to get diagonal bulk equations of motion)
one needs to also introduce the canonically normalized gauge fields. We start with the generic
action (metric factors are understood when contracting indices)
4g25L
(FLMN)
4g25R
(FRMN)
h2(z)
(LM −RM )2
(4.14)
To get to a canonically normalized diagonal basis we redefine the fields as
g25L + g
(L−R) , Ṽ =
g25L + g
. (4.15)
To get the boundary effective action, we write the fields Ṽ , Ã as
Ã(p2, z) =
g25L + g
L̄(p2)− R̄(p2)
ã(p2, z) , (4.16)
Ṽ (p2, z) =
g25L + g
L̄(p2) +
R̄(p2)
ṽ(p2, z) . (4.17)
Here L̄, R̄ are the boundary effective fields (with non-canonical normalization exactly as
in [7]), while the profiles ã, ṽ satisfy the same bulk equations and boundary conditions as
a, v in (4.2)–(4.4) with an appropriate replacement for M2 = (g25L + g
2. In terms of the
canonically normalized fields, the boundary effective action takes its usual form
Seff =
Ṽ ∂yṼ + Ã∂yÃ
. (4.18)
And we deduce the vacuum polarization
ΣL3B(p
2) = −
g25L + g
(∂y ṽ(0, p
2)− ∂yã(0, p2)) (4.19)
And finally the S-parameter is equal to
S = − 16π
g25L + g
(∂y ṽ
(1) − ∂yã(1)) (4.20)
Since ã(n), ṽ(n), n = 0, 1 satisfy the same equations (4.2)–(4.4) as before, the proof goes
through unchanged and we conclude that S > 0.
5 Bulk Higgs and bulk kinetic mixing
Next, we wish to consider the effects of kinetic mixing from higher-dimensional operator in
the bulk involving the Higgs VEV – as mentioned earlier, this kinetic mixing is suppressed
by the 5D cut-off and hence expected to be a sub-leading effect. The reader might wonder
why we neglected it before, but consider it now? The point is that, although the leading
effect on S parameter is positive as shown above, it can be accidentally suppressed so that
the formally sub-leading effects from the bulk kinetic mixing can be important, in particular,
such effects could change the sign of S. Also, the Higgs VEV can be large, especially when
the Higgs profile is “narrow” such that it approximates BC breaking, and thus the large
VEV can (at least partially) compensate the suppression from the 5D cut-off. Of course, in
this limit of BC breaking (δ-function VEV), we know that kinetic mixing gives S < 0 only
if tachyons are present in the spectrum, but we would like to cover the cases intermediate
between BC breaking limit and a broad Higgs profile as well. In this section, we develop
a formalism, valid for arbitrary metric and Higgs profile, to treat the bulk mass term and
kinetic mixing on the same footing and then we apply this technique to models in AdS space
and with power-law profiles for Higgs VEV in the next section.
We first present a discussion of how a profile for the y-dependent kinetic term is equivalent
to a bulk mass term. This is equivalent to the result [13] that a bulk mass term can be
equivalent to an effective metric. However, we find the particular formulation that we present
here to be more useful when we deal with the case of a kinetic mixing. Assume we have a
Lagrangian for a gauge field that has a kinetic term
S = − 1
−gφ2(y)F 2MN (5.1)
We work in the axial gauge A5 = 0 and again the metric takes the form (3.1). We redefine
the field to absorb the function φ: Ã(y) = φ(y)A(y). The action in terms of the new field is
then written as
S = − 1
e2σF̃ 2µν + 2(∂yÃµ)
2 + 2
Ã2µ − 4(∂yÃµ)Ãµ
(5.2)
To see that the kinetic profile φ is equivalent to a mass term, we integrate by parts in the
second term
S = − 1
F̃ 2MN + 2e
(5.3)
Thus we find that a bulk kinetic profile is equivalent to a bulk mass plus a boundary mass.
The bulk equations of motion for the new variables will then be
∂2yÃµ − e2σp2Ãµ −
Ãµ = 0, (5.4)
and the boundary conditions become
∂yÃµ =
Ãµ. (5.5)
Note, that despite the bulk mass term, there is still a massless mode whose wavefunction is
simply φ(z). Now we can reverse the argument and say that a bulk mass must be equivalent
to a profile for the bulk kinetic term plus a boundary mass term.
5.1 The general case
We have seen above how to go between a bulk mass terms and a kinetic function. We
will now use this method to discuss the general case, when there is electroweak symmetry
breaking due to a bulk higgs with a sharply peaked profile toward the IR brane, and the same
Higgs introduces kinetic mixing between L and R fields corresponding to a higher dimensional
operator from the bulk. For now we assume that the Higgs fields that breaks the electroweak
symmetry is in a (2,2) of SU(2)L×SU(2)R, with a VEV 〈H〉 = diag(h(z), h(z))/
2.1 This
Higgs profile h has dimension 3/2. The 5D action is given by
(FLMN)
2 + (FRMN)
− (DMH)†(DMH) +
Tr(FLMNH
†HFMN R)
(5.6)
Here α is a coefficient of O(1) and Λ is the 5D cutoff scale, given approximately by Λ ∼
24π3/g25. The kinetic mixing term just generates a shift in the kinetic terms of the vector
and axial vector field, and we will write the bulk mass term also as a shift in the kinetic
term for the axial vector field. The exact form of the translation between the two forms is
given by answering the question of how to redefine the field with an action (note that m2
has a mass dimension 3)
wF 2MN +m
22g25AµA
(5.7)
to a theory with only a modified kinetic term. The appropriate field redefinition A = ρÃ
will be canceling the mass term if ρ satisfies
∂y(w∂yρ) = m
2g25e
4σρ, (5.8)
1An alternative possibility would be to consider a Higgs in the (3,3) representation of SU(2)L×SU(2)R.
together with the boundary conditions ρ′|y=1 = 0, ρ|y=0 = 1. The relation between the new
and the old expression for w will be w̃ = ρ2w. The action in this case is given by
−gw̃F̃ 2MN +
w̃(0)
(∂yρ)Ã
2|y=0 (5.9)
This last boundary term is actually irrelevant for the S-parameter: since it does not contain
a derivative on the field it can not get an explicit p-dependence so it will not contribute to
S, so for practical purposes this boundary term can be neglected.
With this expression we now can calculate S. For this we need the modified version of
the formula from [13], where the breaking is not by boundary conditions but by a bulk Higgs.
The expression is
dye2σ(wV − w̃A). (5.10)
In our case wV = 1− αh
2(y)2g2
while w̃A = wAρ
2 = (1 +
αh2(y)2g2
This formula also gives another way to see that S > 0 in the absence of kinetic mixing,
without analyzing the functions v(1) and a(1) from Section 4 in detail. Without kinetic
mixing, wV = 1 and w̃A = ρ
2, and the equation of motion for ρ is simply ∂2yρ = m
2g25e
In that case ρ is just the function we called a(0) in Section 4. Since we showed there that
a(0) ≤ 1, we see that our expression 5.10 gives an alternative argument that S > 0 without
kinetic mixing, because it is simply an integral of e2σ(1− ρ2) ≥ 0.
5.2 Scan of the parameter space for AdS backgrounds
Having developed the formalism for a unified treatment of bulk mass terms and bulk kinetic
mixing, we then apply it to the AdS case with a power-law profile for the Higgs vev. Requiring
(i) calculability of the 5D theory, i.e., NDA size of the higher-dimensional operator, (ii) that
excited W/Z’s are heavier than a few 100 GeV, and (iii) a ghost-free theory, i.e., positive
kinetic terms for both V and A fields, we find that S is always positive in our scan for
this model. We do not have a general proof that S > 0 for an arbitrary background with
arbitrary Higgs profiles, if we include the effects of the bulk kinetic mixing, but we feel that
such a possibility is quite unlikely based on our exhaustive scan. For this scan we will take
the parametrization of the Higgs profile from [22]. Here the metric is taken as AdS space
ds2 =
ηµνdx
µdxν − dz2
, (5.11)
where as usual R < z < R′. The bulk Higgs VEV is assumed to be a pure monomial in
z (rather than a combination of an increasing and a decreasing function). The reason for
this is that we are only interested in the effect of the strong dynamics on the electroweak
precision parameters. A term in the Higgs VEV growing toward the UV brane would mean
that the value of bulk Higgs field evaluated on the UV brane gets a VEV, implying that
there is EWSB also by a elementary Higgs (in addition to the strong dynamics) in the 4D
dual. We do not want to consider such a case. The form of the Higgs VEV is then assumed
to be
v(z) =
2(1 + β) logR′/R
(1− (R/R′)2+2β)
, (5.12)
where the parameter β characterizes how peaked the Higgs profile is toward the TeV brane
(β → −1 corresponds to a flat profile, β → ∞ to an infinitely peaked one). The other
parameter V corresponds to an “effective Higgs VEV”, and is normalized such that for
V → 246 GeV we recover the SM and the KK modes decouple (R′ → ∞ irrespective of β).
For more details about the definitions of these parameters see [22].2
We first numerically fix the R′ parameter for every given V, β and kinetic mixing pa-
rameter α by requiring that the W -mass is reproduced. We do this approximately, since we
assume the simple matching relation 1/g2 = R log(R′/R)/g25 to numerically fix the value of
g5, which is only true to leading order, but due to wave function distortions and the extra
kinetic term will get corrected. Then, ρ can be numerically calculated by solving (5.8), and
from this S can be obtained via (5.10).
We see that S decreases as we increase α. On the the hand, the kinetic function for
vector field (wV ) also decreases in this limit. So, in order to find the minimal value of S
consistent with the absence of ghosts in the theory, we find numerically the maximal value
of α for every value of V, β for which the kinetic function of the vectorlike gauge field is still
strictly positive. We then show contour plots for the minimal value of S taking this optimal
value of α as a function of V, β in Fig. 5.2. In the first figure we fix R′ = 10−8 GeV−1, which
is the usual choice for higgsless models with light KK W’ and Z’ states capable of rendering
the model perturbative. In the second plot we choose the more conventional value R = 10−18
GeV−1. We can see the S is positive in both cases over all of the physical parameter space.
We can estimate the corrections to the above matching relation from the wavefunction
distortion and kinetic mixing as follows. The effect from wavefunction distortion is expected
to be ∼ g2S/(16π) which is <∼ 10% if we restrict to regions of parameter space with S <∼ 10.
Similarly, we estimate the effect due to kinetic mixing by simply integrating the operator
over the extra dimension to find a deviation ∼ g6(V R′)2 log2 (R′/R) / (24π3)2. So, if restrict
<∼ 1 TeV and 1/R′ >∼ 100 GeV, then this deviation is also small enough. We see that
both effects are small due to the deviation being non-zero only near IR brane – even though
it is O(1) in that region, whereas the zero-mode profile used in the matching relation is
spread throughout the extra dimension.
In order to be able to make a more general statement (and to check that the neglected
additional contributions to the gauge coupling matching from the wave function distortions
and the kinetic mixing indeed do not significantly our results) we have performed an addi-
tional scan over AdS space models where we do not require the correct physical value of MW
to be reproduced. In this scan we then treat R′ as an independent free parameter. In this
case the correct matching between g and g5 is no longer important for the sign of S, since at
2Refs. [23] also considered similar models.
400 500 600 700 800 900 1000
400 500 600 700 800 900 1000
Figure 1: The contours of models with fixed values of the S-parameter due to the electroweak
breaking sector. In the left panel we fix 1/R = 108 GeV, while in the right 1/R = 1018 GeV.
The gauge kinetic mixing parameter α is fixed to be the maximal value corresponding to the
given V, β (and R′ chosen such that the W mass is approximately reproduced). In the left
panel the contours are S = 1, 2, 3, 4, 5, 6, while in the right S = 1, 1.5, 2.
every place where g5 appears it is multiplied by a parameter we are scanning over anyway
(V or α).
We performed the scan again for two values of the AdS curvature, 1/R = 108 and 1018
GeV. For the first case we find that if we restrict α < 10, 1/R′ < 1 TeV there is no case with
S < 0. However, there are some cases with S < 0 for α > 10, although in these cases the
theory is likely not predictive. For 1/R = 1018 GeV we find that S < 0 only for V ∼ 250
GeV and β ∼ 0, 1/R′ ∼ 1 TeV. In this case α is of order one (for example α ∼ 5). This case
corresponds to the composite Higgs model of [11] and it is quite plausible that at tree-level
S < 0 if a large kinetic mixing is added in the bulk. However in this case EWSB is mostly
due to a Higgs, albeit a composite particle of the strong dynamics, rather than directly by
the strong dynamics, so it does not contradict the expectation that when EWSB is triggered
directly via strong dynamics, then S is always large and positive. However, it shows that
any general proof for S > 0 purely based on analyzing the properties of Eqs. (5.8)-(5.10) is
doomed to failure, since these equations contain physical situations where EWSB is not due
to the strong dynamics but due to a light Higgs in the spectrum. Thus any general proof
likely needs to include more physical requirements on the decoupling of the physical Higgs.
6 Conclusions
In this paper, we have studied the S parameter in holographic technicolor models, focusing
especially on its sign. The motivation for our study was as follows. An alternative (to SUSY)
solution to the Planck-weak hierarchy involves a strongly interacting 4D sector spontaneously
breaking the EW symmetry. One possibility for such a strong sector is a scaled-up version
of QCD as in the traditional technicolor models. In such models, we can use the QCD
data to “calculate” S finding S ∼ +O(1) which is ruled out by the electroweak precision
data. Faced by this constraint, the idea of a “walking” dynamics was proposed and it can
be then argued that S < 0 is possible which is much less constrained by the data, but the
S parameter cannot be calculated in such models. In short, there is a dearth of calculable
models of (non-supersymmetric) strong dynamics in 4D.
Based on the AdS/CFT duality, the conjecture is that certain kinds of theories of strong
dynamics in 4D are dual to physics of extra dimensions. The idea then is to construct
models of EWSB in an extra dimension. Such constructions allow more possibilities for
model-building, at the same time maintaining calculability if the 5D strong coupling scale is
larger than the compactification scale, corresponding to large number of technicolors in the
4D dual.
It was already shown that S > 0 for boundary condition breaking for arbitrary metric
(a proof for S > 0 for the case of breaking by a localized Higgs vev was recently studied in
reference [24]). In this paper, we have extended the proof for boundary condition breaking
to the case of arbitrary bulk kinetic functions for gauge fields or gauge kinetic mixing.
Throughout this paper, we have assumed that the (light) SM fermions are effectively
localized near the UV brane so that flavor violation due to higher-dimensional operators
in the 5D theory can be suppressed, at the same time allowing for a solution to the flavor
hierarchy. Such a localization of the light SM fermions in the extra dimension is dual to SM
fermions being “elementary”, i.e., not mixing with composites from the 4D strong sector.
It is known that the S parameter can be suppressed (or even switch sign) for a flat profile
for SM fermions (or near the TeV brane) – corresponding to mixing of elementary fermions
with composites in the 4D dual, but in such a scenario flavor issues could be a problem.
We also considered the case of bulk breaking of the EW symmetry motivated by recent
arguments that S < 0 is possible with different effective metrics for vector and axial fields.
For arbitrary metric and Higgs profile, we showed that S > 0 at leading order, i.e., neglect-
ing effects from all higher-dimensional operators in the 5D theory (especially bulk kinetic
mixing), which are expected to be sub-leading effects being suppressed by the cut-off of the
5D theory. We also note that boundary mass terms can generally be mimicked to arbitrary
precision by localized contributions to the bulk scalar profile, so we do not expect a more
general analysis of boundary plus bulk breaking to find new features. Obtaining S < 0 must
then require either an unphysical Higgs profile or higher-dimensional operators to contribute
effects larger than NDA size, in which case we lose calculability of the 5D theory.
To make our case for S > 0 stronger, we then explored effects of the bulk kinetic mixing
between SU(2)L,R gauge fields due to Higgs vev coming from a higher-dimensional operator
in the 5D theory. Even though, as mentioned above, this effect is expected to be sub-leading,
it can nevertheless be important (especially for the sign of S) if the leading contribution to
S is accidentally suppressed. Also, the large Higgs VEV, allowed for narrow profiles in the
extra dimension (approaching the BC breaking limit), can compensate the suppression due to
the cut-off in this operator. For this analysis, we found it convenient to convert bulk (mass)2
for gauge fields also to kinetic functions. Although a general proof for S > 0 is lacking in
such a scenario, using the above method of treating the bulk mass for axial fields, we found
that S ∼ +O(1) for AdS5 model with power-law Higgs profile in the viable (ghost-free) and
calculable regions of the parameter space.
In summary, our results combined with the previous literature strongly suggests that S is
positive for calculable models of technicolor in 4D and 5D. We also presented a plausibility
argument for S > 0 which is valid in general, i.e., even for non-calculable models.
7 Acknowledgments
We thank Giacomo Cacciapaglia, Cédric Delaunay, Antonio Delgado, Guido Marandella,
Riccardo Rattazzi, Matthew Schwartz and Raman Sundrum for discussions. We also thank
Johannes Hirn and Veronica Sanz for comments on the manuscript. As we were finishing the
paper, we learned that Raman Sundrum and Tom Kramer have also obtained results similar
to ours [25]. C.C. thanks the theory group members at CERN for their hospitality during
his visit. K.A. is supported in part by the U. S. DOE under Contract no. DE-FG-02-85ER
40231. The research of C.C. is supported in part by the DOE OJI grant DE-FG02-01ER41206
and in part by the NSF grant PHY-0355005. The research of M.R. is supported in part by
an NSF graduate student fellowship and in part by the NSF grant PHY-0355005. C.G.
is supported in part by the RTN European Program MRTN-CT-2004-503369 and by the
CNRS/USA exchange grant 3503.
Note added
After submitting our paper to the arXiv, we learned of [26] which gives a proof for S > 0
for an arbitrary bulk Higgs profile in AdS background that is similar to our proof in Section
4. However, our proof of S > 0 is valid for a general metric and we have also included the
effect of kinetic mixing between SU(2)L and SU(2)R fields via higher-dimensional operator
(with Higgs vev) for the calculation of S in AdS background. We thank Deog-Ki Hong and
Ho-Ung Yee for pointing out their paper to us.
A Details of BC breaking with arbitrary kinetic func-
tions
Here we present the detailed calculation of S in the case with boundary breaking and arbi-
trary kinetic functions described in Section 3.2. Recall that we had the following decompo-
sition:
ALµ(p
2, y) = L̄µ(p
2)LL(y, p
2) + R̄µ(p
2)LR(y, p
ARµ (p
2, y) = L̄µ(p
2)RL(y, p
2) + R̄µ(p
2)RR(y, p
2), (A.1)
with boundary conditions
(UV) LL(0, p
2) = 1, LR(0, p
2) = 0, RL(0, p
2) = 0, RR(0, p
2) = 1. (A.2)
LL(1, p
2) = RL(1, p
2), LR(1, p
2) = RR(1, p
∂y(LL(1, p
2) +RL(1, p
2)) = 0, ∂y(LR(1, p
2) +RR(1, p
2)) = 0.
(A.3)
The action again reduces to a boundary term
Seff =
φ2L(0)Lµ∂L
µ + φ2R(0)Rµ∂R
, (A.4)
so we find that
S = −8π
φ2L∂yL
R + φ
|y=0 (A.5)
where we have done an expansion in terms of the momentum for all the wave functions as
usual as LL(y, p
2) = L
L (y) + p
L (y) + . . .. The lowest order wave functions satisfy the
following bulk equations:
J ) = 0, (A.6)
where I and J can refer to L or R. Imposing the BC’s these equations can be simply solved
in terms of the integrals
fL(y) =
, fR(y) =
(A.7)
L = 1− fL(y), L
R = fL(y), R
L = fR(y), R
R = 1− fR(y). (A.8)
In order to actually find S we need to go one step further, that is calculate the next order
terms in the wave functions I
J . These will satisfy the equations
J ) = e
2σφ2II
J , (A.9)
where for the I
J we use the solutions in (A.8). The form of the solutions will be given by
J (y) =
φ2I(y
due2σφ2I(u)I
J (u) + aIJ
, (A.10)
where aIJ are constants. In terms of these quantities the S-parameter is just given by
S = −8π
(aLR + aRL) (A.11)
One can again solve the boundary conditions to find the constants aRL , aLR . These turn out
to be
aRL = −
φ2L(y)
dy′e2σ(y
′)φ2L(y
′)(1− fL(y′))
φ2L(1)
φ2R(1)
dye2σ(y)φ2R(y)fR(y)
φ2L(y)
φ2R(y)
dy′e2σ(y
′)φ2R(y
′)fR(y
(A.12)
where
φ2R(y)
φ2L(1)
φ2R(1)
φ2L(y)
. (A.13)
A similar expressions applies for aLR with L ↔ R everywhere. Since 0 < fL,R < 1, we can
see that every term in the expression is manifestly positive, so S is definitely positive.
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http://arxiv.org/abs/hep-ph/0405040
http://arxiv.org/abs/hep-th/9802109
http://arxiv.org/abs/hep-th/9802150
http://arxiv.org/abs/hep-ph/0611358
http://arxiv.org/abs/hep-ph/0508279
http://arxiv.org/abs/hep-ph/0612242
http://arxiv.org/abs/hep-ph/0609104
http://arxiv.org/abs/hep-ph/0608241
http://arxiv.org/abs/hep-ph/0702234
http://arxiv.org/abs/hep-ph/0602177
	Introduction
	A plausibility argument for S > 0
	Boundary-effective-action approach to oblique corrections. Simple cases with boundary breaking
	S>0 for BC breaking with boundary kinetic mixing
	S>0 for BC breaking with arbitrary kinetic functions
	S>0 in models with bulk Higgs
	Bulk Higgs and bulk kinetic mixing
	The general case
	Scan of the parameter space for AdS backgrounds
	Conclusions
	Acknowledgments
	Details of BC breaking with arbitrary kinetic functions
