arXiv:0901.2117v2 [hep-ph] 26 Mar 2009
Realistic Composite Higgs Models
Charalampos Anastasiou, Elisabetta Furlan, and Jos´ e Santiago
Institute for Theoretical Physics, ETH, CH-8093, Zurich, Switzerland
(Dated: March 26, 2009)
We study the role of fermionic resonances in realistic composite Higgs models. We consider the
low energy effective description of a model in which the Higgs arises as the pseudo-Goldstone boson
of an SO(5)/SO(4) global symmetry breaking pattern. Assuming that only fermionic resonances
are present below the cut-off of our effective theory, we perform a detailed analysis of the electroweak
constraints on such a model. This includes the exact one-loop calculation of the T parameter
and the anomalous ZbL¯bL coupling for arbitrary new fermions and couplings.Other relevant
observables, like b → sγ and ∆B = 2 processes have also been examined. We find that, while
minimal models are difficult to make compatible with electroweak precision tests, models with
several fermionic resonances, such as the ones that appear in the spectrum of viable composite
Higgs models from warped extra dimensions, are fully realistic in a large region of parameter
space. These fermionic resonances could be the first observable signature of the model at the LHC.
One main objective of the Large Hadron Collider (LHC) is to discover the precise mecha-
nism of electroweak symmetry breaking (EWSB). A well motivated hypothesis is that there
exists a Higgs boson which is not a fundamental scalar. Instead it could be a composite
state of a strongly coupled theory, the pseudo-Goldstone boson of a spontaneously broken
global symmetry . Compositeness can explain the insensitivity of EWSB to ultravio-
let physics, while the pseudo-Goldstone nature of the Higgs boson may explain the little
hierarchy between the scale of new physics and the scale of EWSB.
This mechanism has recently received increased attention, due to the realization that
calculable composite Higgs models can be constructed in five dimensions . The main idea
is an old one , but only recently realistic models in warped extra dimensions [2, 4, 5, 6, 7]
have been constructed. The experience with five-dimensional models indicates that custodial
symmetry  and a custodial protection of the ZbL¯bLcoupling  are likely ingredients of
realistic constructions. Little Higgs models  also use the idea of a Higgs boson which is
the pseudo-Goldstone boson of a spontaneously broken global symmetry. In order to solve
the little hierarchy problem, they employ the mechanism of collective symmetry breaking,
which ensures that the Higgs mass remains insensitive to ultraviolet physics at one loop.
The main phenomenological implications of a Higgs boson which is the pseudo-Goldstone
boson of an extended broken symmetry are largely independent of the particular details of
how the global symmetry is broken. They can therefore be conveniently described using an
effective Lagrangian approach [11, 12]. A reasonable starting point is a symmetry breaking
pattern that includes custodial symmetry. A minimal example of such a pattern is given
by a global SO(5) symmetry broken at a scale f to its custodially symmetric subgroup
SO(4) [4, 5].
Using this effective Lagrangian approach, it was argued in  that composite Higgs
models with an SO(5)/SO(4) symmetry breaking pattern are difficult to make compatible
with electroweak and flavor precision data without introducing a substantial fine-tuning.
The argument was based on very minimal models, in which fermionic resonances did not
span full representations of the SO(5) group. In addition, the estimation of electroweak
observables was made neglecting contributions which are formally subleading, but can be
relevant in specific situations. Two recent works extended the analysis of  to models in
which the fermionic composites span full representations of SO(5) [13, 14]. Although these
analyses differ in several aspects, like the degree of explicit SO(5) symmetry breaking and
the symmetry breaking patterns, their outcome is somewhat similar. Only a small region
of parameter space is allowed by electroweak precision data in models with no significant
fine-tuning and one set of fermionic composites spanning a vector representation of SO(5).
In this paper, we investigate thoroughly the viability of models with an SO(5)/SO(4)
symmetry breaking pattern. Our analysis extends previous works in two ways, by making
a careful computation of the effect of the new fermionic states on electroweak observables
and by considering the effect of an extended fermionic sector.
In our study of electroweak precision constraints we use an exact one-loop calculation of
the relevant electroweak observables. We do this in complete generality, and our analytic
formulae can be used in other models. In particular, our result for the anomalous ZbL¯bL
coupling is, to the best of our knowledge, the first complete calculation for an arbitrary
number of new quarks with generic couplings. We find that such an exact computation can
be important when formally subleading effects in commonly employed approximations are
We also examine the possibility of multiple sets of fermionic composites, departing from
minimal constructions. Our motivation is the model presented in , in which a five-
dimensional realization of a composite Higgs model with SO(5)/SO(4) symmetry breaking
pattern was shown to be fully compatible with electroweak precision tests, flavor observ-
ables, electroweak symmetry breaking and the observed dark matter relic abundance. The
non-minimal fermion sector of the model in  was indispensable in order to render its pre-
dictions compatible with experimental data. The goal of our article is to present an effective
four-dimensional description of composite Higgs models with a non-minimal fermionic con-
tent and discuss their electroweak and flavor constraints. We show that there are large
regions of parameter space in which composite Higgs models can provide a fully realistic
description of EWSB without fine tuning.
The paper is organized as follows. In section II we briefly review the effective description
of composite Higgs models with an SO(5)/SO(4) symmetry breaking pattern, including the
experimental constraints of minimal models. Section III is devoted to a description of the
relevant fermionic sector of the theory and its effects on electroweak precision observables
and flavor physics. In section IV we present exact one-loop expressions for electroweak
and flavor precision observables which are valid in general extensions of the SM. The main
phenomenological implications of our model are discussed in section V, where we describe two
options for realistic composite Higgs models, a very simple one and a slightly more involved
one which is closer to the realistic examples we know from extra dimensions. Finally, we
conclude in section VI.
II. EFFECTIVE DESCRIPTION OF COMPOSITE HIGGS MODELS
The low energy effective description of a composite Higgs model with SO(5)/SO(4) sym-
metry breaking pattern can be described by a scalar φ in the fundamental representation of
SO(5), subject to the constraint
φ2= f2, (1)
where f is the scale of the global symmetry breaking, assumed to be somewhat larger than
the EWSB scale v ≈ 174 GeV. The first four components of φ, which transform as the
fundamental representation of SO(4), are denoted by?φ. The SU(2)L× U(1)Y subgroup
of SO(4) = SU(2)L× SU(2)R, where the hypercharge corresponds to the T3
weakly gauged.1The vacuum expectation value of?φ breaks the EW symmetry,
From Eq. (1) we see that the ratio
sα≡ sinα ≡
measures the Higgs compositeness, i.e. how the vev of φ is split between?φ and φ5. Canonical
normalization of the different components in φ, expanded around its vev, requires a rescaling
of the physical Higgs
h → cosα h ≡ cαh, (4)
whereas the would be Goldstone bosons are not modified. This redefinition implies an
important feature of Higgs compositeness, namely that Higgs couplings to gauge bosons
are suppressed with respect to the couplings in the SM by the factor cα=
?1 − 2v2/f2.
1An extra U(1) group is required to generate the correct Weinberg angle, but it is irrelevant for the present
discussion and will be disregarded.
In the case of fermions, the suppression factor depends also on the embedding of the SM
fermions in SO(5) representations [6, 15]. This suppression of the Higgs couplings affects the
quantum corrections to electroweak precision observables, leading to some sensitivity to the
ultraviolet cut-off . The leading effect can be taken into account by replacing the Higgs
mass with an effective (heavier) Higgs in the SM expressions of the one-loop corrections to
the electroweak precision observables,
where Λ = 4πf/√NG= 2πf is the ultraviolet cut-off of our effective theory, NGbeing the
number of Goldstone bosons. This modification gives rise to an additional contribution to
the Peskin-Takeuchi  S and T parameters
,∆T = −
where mh,refis the reference Higgs mass used in the electroweak fit and cW is the cosine of
the Weinberg angle.
Furthermore, custodial symmetry can naturally account for a suppressed contribution of
ultraviolet physics to the T parameter, but there is no reason not to expect a contribution
to the S parameter from higher dimensional operators. A reasonable estimate, assuming
that new physics couples linearly to the SM (otherwise this estimate should have an extra
loop suppression), is
The combination of the two corrections, (6) and (7), results in a positive shift to the
S parameter and a negative shift to the T parameter. We show in Fig. 1 the current
constraints on the S and T parameters, assuming a Higgs mass mh= 120 GeV. We also
show the contributions from Higgs compositeness and UV physics for different values of sα.
These constraints are obtained as follows. For reasons that will become apparent in the next
section, we have performed a fit to all the relevant electroweak observables, allowing the S
and T parameters and an anomalous coupling of the bLquark to the Z, that we denote by
δgbL, to vary (note that we have set U = 0 in our fit, as it is expected to be vanishingly
small in our model).2In Fig. 1, we have fixed the optimal value of δgbL= −2.5 × 10−4,
2We use Ref.  for the fit updated to the most recent experimental data . αs(mZ) = 0.1183 and
FIG. 1: 68%, 95% and 99% C.L. limits on S and T for a fit to electroweak observables with three
independent parameters (S, T and δgbL), fixing the optimal value of δgbL= −2.5 × 10−4and a
reference Higgs mass mh,ref= 120 GeV. The dots at the start and the tip of the arrows show the
effects due to Higgs compositeness, Eq. (6), and to UV effects on S, Eq. (7), respectively. The
effect is shown (from top to bottom) for the values of sα= 0.25,0.33 and 0.5.
although the result of a proper projection over the S −T plane does not differ significantly.
As we see in the figure, the above corrections put the model in gross contradiction with
experimental observations for any sizable Higgs compositeness. As emphasized in , this
is a quite generic implication of Higgs compositeness and custodial symmetry, that seems to
disfavor composite Higgs models as a realistic description of electroweak symmetry breaking.
However, we have not included yet in our discussion the effect of other composite states that
might be lighter than the cut-off of our effective description. In the next section we discuss
the effect that new fermionic states can have on electroweak precision tests.3
mt= 172.4 GeV have been fixed to the best fit values from a 5 parameter (αs(mZ), mt, S, T and δgbL).
We would like to thank E. Pont´ on for help with the fit and with Fig. 1.
3New bosonic resonances have been recently shown to be able to make even Higgs–less models compatible
with experimental data .
III. THE FERMIONIC SECTOR
If the Higgs is a composite state of a new strongly interacting theory, it is natural to
assume that the large top mass is due to partial top compositeness. The partners under
SO(5) of the composite states with which the top mixes can then play an important role
in electroweak precision tests. In particular, if the top sector is partly composite, large
corrections to the ZbL¯bL coupling are typically expected, unless some symmetry forbids
them . Fermions, in the fundamental or adjoint representation of the SO(5) group, are
natural building blocks that incorporate the left-right symmetry guaranteeing the absence
of large tree-level corrections to the ZbL¯bL coupling. In this article we will consider the
former possibility and include new (composite) vector-like fermions that transform in the
five-dimensional vector representation of SO(5), which decomposes under SO(4) = SU(2)L×
Ψ = (Q,X,T) ⇒ (5) = (2,2) ⊕ (1). (8)
Q and X form a bidoublet under SU(2)L× SU(2)R, they are SU(2)L doublets with hy-
percharges 1/6 and 7/6, respectively (and TR
= −1/2 and 1/2, respectively). T is an
SU(2)L× SU(2)Rsinglet with hypercharge 2/3. The SM quarks qLand tRhave the same
quantum numbers under the SM gauge group as Q and T, respectively.
The question is, given the large corrections to the S and T parameters from the Higgs
sector, can the one-loop contribution of this new fermionic sector to T be such that the model
is compatible with experimental data? As shown in Fig. 1, a large positive contribution to
T is required, but this has to be obtained without spoiling the good agreement with other
experimental observables. In particular, δgbLis extremely constrained experimentally. The
situation is exemplified in Fig. 2, in which we show the constraints on the contribution to
the T parameter and δgbLfrom the new quarks, for a fixed sα≈ 0.5 (f = 500 GeV). In
the figure, we have shown both the actual value of the allowed T and δgbL(bottom and
left axes, respectively) together with the values relative to the one-loop SM corrections (top
and right axes). We see that, for this value of sα, even at the 99% C.L. the T parameter
must be positive and it is allowed to vary at most by about 20% in units of the SM one-loop
correction. We also observe that δgbLis constrained in the range −0.5 ? δgbL/δgSM
The effects on the T parameter and the ZbL¯bLcoupling of new vector-like quarks with
the above quantum numbers through their mixing with the top were discussed in detail in
0.10.20.3 0.40.5 0.6
0.1 0.20.3 0.40.5
FIG. 2: 95% and 99% C.L. limits on T and δgbLfor a fit to electroweak observables with three
independent parameters (S, T and δgbL) assuming sα≈ 0.5 (f = 500 GeV) and a reference Higgs
mass mh,ref= 120 GeV (no projection). The top and right axes show the value of the T parameter
and δgbLin terms of the SM one-loop contribution. For this value of sαthere is no allowed region
at 68% C.L..
Ref. . The SU(2)Lquantum numbers of the different multiplets determine, to a great
degree, the sign of the contribution to the T parameter. Through their mixing with the top,
the different SU(2)Lmultiplets typically contribute (assuming only one type of mixing at a
time) as follows:
• Singlets (T) contribute positively to the T parameter and the ZbL¯bLcoupling.
• Hypercharge 7/6 doublets (X) contribute negatively to the T parameter.
• Hypercharge 1/6 doublets (Q) contribute positively to the T parameter.
Furthermore, there is a strong correlation between the contribution of vector-like singlets to
the T parameter and the ZbL¯bLcoupling (they are both positive and governed by the same
parameters) which implies a large positive correction to the anomalous ZbL¯bL coupling
(making it less compatible with experimental observations) in the case that the singlet
contributes a large positive amount to T. Due to the particular chirality of the final state,
doublets (Q,X) do not contribute significantly to the ZbL¯bLcoupling (they can contribute
to the ZbR¯bRcoupling but that is a more model dependent issue that is not correlated with
the contribution to the T parameter).
Given the constraints shown in Fig. 2, it is clear that a large contribution from the
singlet is bound to give problems, as it will also give a large contribution to the ZbL¯bL
coupling, which is extremelly constrained experimentally. Ideally, one would want a large
contribution from Q, but this is also difficult, due to the constraints imposed by the global
SO(5) symmetry which usually mean a larger (negative) contribution from X. The general
situation is more complicated, and several modes can give large contributions to make the
model compatible with EWPT. The difficulties we have outlined, however, mean that the
model can be quite predictive, and only a few patterns can be realized. Our goal is to
describe these patterns and their implications at the LHC.
We restrict our discussion to the top sector4qL,tR, which couples to a set of new vector-
like quarks, transforming as the (5) representation of SO(5), ΨL,R= (Q,X,T)L,Ras
− Lint= mi
The new sector is SO(5) invariant, with a mass Lagrangian given by
− LSO(5)= mi
where the indices i,j allow for the possibility of more than one set of fermionic composites
and the brackets in the second term indicate the contraction of the SO(5) indices. µij is
a hermitian matrix. φ is our scalar 5-plet and the terms involving it can be written, in an
SU(2)L× U(1)Y invariant way, as
¯Ψφ =¯Q˜ ϕ +¯ Xϕ +¯Tφ5, (11)
where ϕ and ˜ ϕ = iσ2ϕ∗are the SM Higgs doublets with hypercharge 1/2 and −1/2, respec-
tively. Note that these have to incorporate the rescaling of the physical Higgs. For instance
v + cα
4For simplicity we will assume the bottom mass to come from a direct Yukawa coupling, ¯ qLφbR. This small
explicit breaking of the SO(5)/SO(4) pattern will not have appreciable effects.
with ?ϕ0? = ?ϕ+? = 0. We also have φ5= fcα−v
terms with two or more scalars. From these expressions and the Lagrangians in Eqs. (9)
0) + ..., where the dots denote
and (10) we can compute the mass matrix and the Yukawa couplings for the quarks in the
model (including the couplings to the would be Goldstone bosons that will be required for
the calculation of the anomalous ZbL¯bLcoupling). The mass terms can be written in matrix
− L =
where we have implicitly written the mass matrix in block form and Qu
charge 2/3 components of Q and X, respectively. The interaction Lagrangian, Eq. (9), gives
a mixing between the fundamental fields qL, tRand the composite states Q and T, which
makes the Lagrangian non-diagonal before EWSB. If we diagonalize the mass matrix before
EWSB (with v = 0), we will end up with a massless SU(2)Ldoublet and a massless singlet
that are now an admixture of fundamental and composite states. These new massless states
have Yukawa couplings, thanks to their composite components (since, assuming that the
only explicit breaking of SO(5) is through mi
L,R, the Higgs only couples to composites).
In order to better understand this mechanism, we consider the case that there is only one
set of composite fermionic states below the cut-off of our theory. We can then diagonalize
the mass matrix, for v = 0, by means of the following rotations
qL→ cosθLqL+ sinθLQL,QL→ −sinθLqL+ cosθLQL, (14)
tR→ cosθRtR+ sinθRTR,TR→ −sinθRtR+ cosθRTR. (15)
Note that in the case of qLand QLwe are rotating entire doublets; these have the same quan-
tum numbers and therefore no traceable physical footprint of the rotation is left. The mixing
angles determining the degree of compositeness of qLand tRare, respectively, tanθL=mL
mT, where we have defined mT≡ mΨ+ fµ (note that now µ is not a matrix
but a number and that for v = 0 we have cα= 1).
With these field rotations, the mass Lagrangian for the charge 2/3 quarks reads
where we have denoted sL,R≡ sinθL,R, cL,R≡ cosθL,R.
We already see in this mass Lagrangian some of the constraints imposed by the global
symmetry. First, the top quark acquires mass through its mixing with the composite states.
In order to have a large enough top mass, tL and tR cannot be simultaneously mostly
fundamental (small sLand sR). Second, the mixing of the hypercharge 1/6 doublet with the
top (one possible source of positive contribution to the T parameter) is always smaller, by
a factor cL, than the mixing of the hypercharge 7/6 multiplet. If the two are degenerate or
X is lighter than Q (as happens in minimal five-dimensional models), then the system Q,X
usually contributes negatively to the T parameter. This effect together with the additional
negative contribution from the fact that the Higgs boson is composite lead to the problem
discussed previously. Either the fermion contribution to the T parameter is not large enough
and therefore incompatible with electroweak precision data (given the positive contribution
to the S parameter from UV physics) or large corrections to flavor preserving and violating
b couplings are introduced, again in conflict with experimental data.
A precise assessment of the model viability requires a study after a complete diagonal-
ization of the matrix in Eq. (16), since several modes can simultaneously give relevant con-
tributions which are difficult to disentangle qualitatively. A detailed analysis of electroweak
constraints in our model is therefore required to see if there are regions of parameter space
compatible with current data. This detailed analysis includes a precise calculation of the
main electroweak observables, which in our case can be encoded in the T parameter and
δgbL, including formally subleading contributions not proportional to large Yukawa couplings
and a careful scan over parameter space. This is the subject of the next two sections.
IV.EVALUATION OF PRECISION ELECTROWEAK OBSERVABLES
In this section, we compute the one-loop contribution of the new fermionic sector to the
most relevant electroweak observables, which receive large corrections due to the large value
of the top mass. The most important observables are the T parameter and the anomalous
ZbL¯bLcoupling.5Other observables, Bd,s−¯Bd,smixing, Bd,s→ µ+µ−, and b → sγ, may also
receive large one-loop corrections which are however less generic, depending for example on
the details of how the bottom quark gets a mass. These observables provide typically weaker
constraints than the T parameter and the anomalous ZbL¯bLcoupling (see for instance 
for a discussion in the context of MFV scenarios). Nevertheless, we have explicitly checked
that the constraints from B −¯B mixing and b → sγ are indeed typically weaker in our
Given the stringent constraints on the new fermionic contributions to the T parameter
and δgbLwe have found it important to calculate these observables as precisely as possible.
In our results we do not discard any one-loop diagram. For δgbL, in particular, we compute
the full dependence of the corresponding amplitude on all mass parameters (including the
Z, W and Goldstone boson masses) except for the bottom or lighter quark masses. Our
calculation of δgbLis general and the result can be readily used in other models.6
We perform all our calculations in the ’t Hooft-Feynman gauge, in which the Goldstone
bosons and the corresponding gauge bosons have the same mass. We consider an arbitrary
number of new quarks ψi
Q, with electric charge Q (Q = −1/3,2/3 or 5/3 in our model) and
Q. We parametrize their couplings to the Z and W bosons in the mass eigenstate mass mi
basis as (an implicit sum over the particle charges Q is always understood)
µ+ h.c. ,(18)
where VQ= 0 for the minimum Q in the model and PL,R= (1∓γ5)/2 are chirality projectors.
5The one-loop contribution from the fermionic sector to S, which was computed in , is negligible
compared with the tree level contribution from UV physics.
6Our calculations are done assuming point particles. The observables computed here are however finite and
dominated by scales of the order of the masses of the particles involved. In the composite Higgs models
we are interested in, the relevant masses are much smaller than the strong coupling scale and therefore
our approximation is valid.
In the SM, X
= δij, V
and all the other couplings vanish at tree
level. The couplings to the Goldstone bosons are denoted by
(Q−1)G++ h.c. . (20)
Note that we have extracted a factor of g/2cWand g/√2 in the couplings of the Goldstone
bosons to simplify the equations of the observables. Finally, the trilinear gauge boson and
the gauge-Goldstone boson interactions are those of the SM ,
+1 − 2s2
cW[gµν(k1− k2)ρ+ gνρ(k2− k3)µ+ gρµ(k3− k1)ν]Zµ(k1)W+
where all momenta are taken to flow into the vertices.
A.Result for the T parameter
The T parameter measures the amount of custodial symmetry breaking and can be defined
in terms of the vacuum polarization amplitudes of the SU(2)LWi
µgauge bosons as 
?Π+−(0) − Π33(0)?, (22)
where Πij(0) denotes the transverse part of the vacuum polarization amplitude evaluated at
igµνΠij(p2) + (pµpνterms) ≡
The T parameter was computed in  for an arbitrary number of vector-like singlets and
doublets. We have extended their calculation to arbitrary couplings. The final result, which
we reproduce here for completeness, is almost unchanged,
αβ|2)θ+(yα,yβ) + 2Re(VQL
αβ|2)θ+(yα,yβ) + 2Re(XQL
Here again the sum over Q is left implicit; we have defined y ≡ m2/m2
the corresponding quark, and the functions θ±read
Z, with m the mass of
θ+(y1,y2) ≡ y1+ y2−
θ−(y1,y2) ≡ 2√y1y2
− 2(y1logy1+ y2logy2) +y1+ y2
− 2 + log(y1y2) −∆
where ∆ is a divergent term that comes from dimensional regularization. We have left it
explicit so that it is possible to check the cancellation of poles in the T parameter. Taking
the SM limit (only t and b quarks) we obtain
2(θ+(yt,yt) + θ+(yb,yb))
≈ 1.19 . (27)
B.Anomalous ZbL¯bLcoupling at one loop
The amplitude for the decay of a Z boson to massless left-handed b-quarks,
Z(q) →¯bL(p2)bL(p1), (28)
can be written
¯b(p2)?ǫ(q)1 − γ5
where the gLcoupling can be modified from its value in the SM
L + δgL
due to new heavy quarks with W and Z boson interactions described by the Lagrangian of
Eqs. (17-20). There are severe constraints on the value of gLfrom the LEP experiments.
In our model, a symmetry protection of this coupling has been implemented to forbid tree
level corrections. However, modifications can occur via radiative corrections. Taking into
account one-loop effects we have
where we denote with Fheavyone-loop contributions from all the heavy quarks in the general
BSM model, including the top quark with couplings as in the BSM model. The prediction
FIG. 3: Diagrams with heavy quarks contributing to the ZbL¯bLamplitude.
includes already contributions from the top quark with SM couplings, Ftop
have been removed explicitly in the above equation.
Summing over all the diagrams in Fig. 3 and carrying out the renormalization procedure,
7In order to make our result more compact, here we already substituted for the trilinear gauge boson
and for the gauge-Goldstone boson couplings. Therefore, when applying this formula one should be
careful in writing all the fermion-(gauge/Goldstone) boson couplings appearing in it consistently with the
conventions of Eq. (21).
4− 1) + |WQL
We have defined
2 = ∆ − 2 + yi+ yj− 2yW− I2(yi,yj)?yi+ yj− 2yW− 3?
+2I3(yi,yW,yj)?yi− yW− 1??yj− yW− 1?+ log(yi)
2+ yi− yW− I2(yW,yW)
?− I2(yW,yW)?2yi− 2yW− 1?
−2I3(yW,yi,yW)?(yi− yW)2+ 2yW
yi+ yj− 2yW
(yi− yW)(yj− yW)
∆ + 1 + yi+ yj− 2yW− I2(yi,yj)?yi+ yj− 2yW+ 1?
?− yilog(yi) − yjlog(yj)
(yi− yW)2+ yWlog(yW)2yi− yW
(yi− yW)2+ yi
6= 3∆ − 4 + 2?yi− yW
−I3(yW,yi,yW)− yilog(yi) + yWlog(yW), (38)
The finite parts of the two-point and three-point master integrals can be easily evaluated
numerically from their integral representations:
I2(y1,y2) = −
dxlog[xy1+ (1 − x)y2− x(1 − x)],
x + y2− y3log
I3(y1,y2,y3) = −
xy1+ (1 − x)y2
xy1+ (1 − x)y3− x(1 − x)
The calculation of the anomalous ZbL¯bLcoupling required tensor one-loop Feynman inte-
grals with up to three propagators with different masses, and the external Z boson invariant
mass, which we computed using the scalar form factors decomposition in [24, 25] and the
program AIR . We note that our exact expressions above agree with the limits for the
ZbL¯bLamplitude that have been presented in Ref. .
The result for Ftop
SMcan be obtained from the above results in the special case of i = j = t
by substituting appropriately the SM couplings of the top quark with gauge and Goldstone
In the model presented here the only relevant contribution to the anomalous ZbL¯bL
coupling is given from charge 2/3 quarks in the loop. Yet, the result we give can be extended
to models with a different quark content. For example, if heavy charge −1/3 quarks were
relevant, their contribution to the anomalous ZbL¯bLcoupling can be obtained from the first
three lines of Eq. (32) with the substitutions
y → y(m2
C.Result for Bq−¯Bq
∆B = 2 processes can be conveniently parametrized in terms of the following dimension
6 effective Lagrangian,
where we have used standard notation
1 = ¯ qα
1= ¯ qα
2 = ¯ qα
2= ¯ qα
3 = ¯ qα
3= ¯ qα
4 = ¯ qα
5 = ¯ qα
We have computed the Wilson coefficients for the different operators Ci,˜Ci, due to the
exchange, in box diagrams, of charge 2/3 quarks with arbitrary couplings as parametrized
in Eqs. (18) and (20). The procedure requires Fierz rearragement but is otherwise standard.
We define the mass ratios
The final result reads:
Also˜Ci= Ci(L ↔ R).
In the above equations the functions g0(x,y) and g1(x,y) are
g0(x,y) = −J0(x) − J0(y)
g1(x,y) = −J1(x) − J1(y)
x − y
x − y
(1 − x)2+
(1 − x)2+
1 − x,
1 − x.
D.Results for b → sγ
B → Xsγ is an interesting observable, as it can be very sensitive to new physics. It probes
different combinations of top couplings than the other observables that we have considered so
far, and in principle it restricts further the allowed parameter space in our model. However,
because of the same reason, deviations in b → sγ are not necessarily correlated to those of T
and δgbL. This means that while arbitrary points in parameter space can be constrained by
b → sγ, small modifications of other sectors in the model outside the top (like for instance
details of how the b quark gets a mass) can easily render this observable compatible with
experimental measurements without modifying the values of T or δgbL.
We use the results of Ref. , in which the relevant matching conditions are computed
including the NLO QCD corrections in an arbitrary extension of the SM (however, we use
only LO QCD correction in our estimation). The relevant operators are
where e and gsare the electromagnetic and strong coupling constants, respectively, Fµνand
µνthe electromagnetic and gluonic field strength tensors and Taare the color matrices
normalized to TrTaTb= δab/2. Splitting the corresponding Wilson coefficients into a SM
part and a new physics part, at the matching scale,
7,8+ ∆C7,8, (59)
we can express the constraint on the new physics contribution as:
B(¯B → Xsγ) =
3.15 ± 0.23 − 8.03∆C7− 1.92∆C8
+ 4.96(∆C7)2+ 0.36(∆C8)2+ 2.33∆C7∆C8
where the SM contribution includes NNLO results . This result is to be compared with
the experimental average 
B(¯B → Xsγ)exp= (3.52 ± 0.23 ± 0.09) × 10−4. (61)
V. CONSTRAINTS ON THE FERMIONIC SECTOR AND COLLIDER IMPLI-
In this section we analyze the main electroweak and flavor experimental constraints on
our class of composite Higgs models. The fact that we require a very constrained but
non-negligible contribution from the fermionic sector to the T-parameter while the other
observables must not be disturbed significantly, renders the χ2quite sensitive to the model
parameters. This sensitivity could be reduced by choosing large values of f, since in the
infinite f limit we recover the SM. However, this possibility is not attractive because large
values of f do not help with addressing the usual SM fine-tuning problem. For small values,
EWSB is less fine-tuned but a larger contribution to T from the fermionic sector is required
to make the model compatible with precision data. From now on we will consider as our
benchmark scenario f = 500 GeV. This corresponds to the lowest point in Fig. 1 and the
model is subject to non-trivial constraints (see Fig. 2) while at the same time its naive
fine-tuning measure  is better than ∼ 10%.
From Figs. 1 and 2 we see that the region of parameter space allowed by precision ob-
servables is presumably not only very sensitive to the input parameters but also small.8
We have found that in such a situation, computing the electroweak observables exactly or
estimating them in the large Yukawa approximation can lead to important phenomenologi-
cal differences. In order to estimate this effect and to ensure probing all relevant regions in
parameter space, we have performed several scans, based on adaptive Monte-Carlo meth-
ods , which we require to search for phase-space regions with a small value of χ2. We
have implemented two types of scans: in the first type the χ2is obtained using the complete
one-loop calculation of the electroweak observables and in the second type the χ2is obtained
from the calculation of δgbLin the large Yukawa approximation.
In our scans we have restricted |µij| ? 4π. We have also checked that the mass parameters
Ψare typically below Λ in the case of two multiplets (but either mΨor mLare
close or above Λ in most of the parameter space for one multiplet) and the masses of the
8Whether this constitutes further fine-tuning is a debatable issue. The fermionic sector will also contribute,
in a full composite Higgs model, to the Higgs potential and a sensible measure of fine-tuning might be
the overlap between the regions with good EWSB and good compatibility with precision data. Ref. 
presented a 5D composite Higgs model in which the two regions overlap nicely.
fermions that affect the relevant observables are also typically below the cut-off of our
A. One multiplet
We now discuss the constraints imposed by electroweak and flavor precision data in the
case that there is only one fermionic (5) of SO(5) below the cut-off of our effective theory.
This case has been previously considered in the literature, in the same or similar models
(see [13, 14, 32]). The new result of our paper is an exact treatment at one loop of all the
important precision observables; we also elaborate further on the implications of this minimal
model for collider phenomenology (Ref.  also emphasized the collider implications of a
composite top). We view the case of one multiplet in this section as a preparation for the
more interesting case of two multiplets in the following section, where we will discuss in
detail the role of a non-minimal sector of fermionic composites below the cut-off.
We first show that it is often important to use an exact one-loop calculation for δgbL. We
start by performing a scan of the parameter space using the large Yukawa approximation.
As expected for the case of one multiplet, the allowed parameter space is quite small. For
f = 500 GeV, using the large Yukawa coupling approximation, we find two generic regions
in parameter space compatible with electroweak precision data at the 99% C.L.: one for
0.1 ? sL? 0.2 and one for sL∼ 1. These regions are shown with empty (green) squares in
the plots of Fig. 4. The region 0.1 ? sL? 0.2 (left plot in Fig. 4) is, however, a misleading
artifact of the large Yukawa coupling approximation. When we repeat the computation of
δgbLexactly at one loop, corresponding to the full (red) squares in Fig. 4, this region is
excluded at the 99% C.L. (the region survives only at the 99.9% C.L.). On the other hand,
the large sLregion (right plot in Fig. 4) survives when the full one-loop calculation is used,
but it turns out to be significantly smaller than what the analysis using the large Yukawa
coupling approximation indicates.
In Fig. 5, we show the fermionic spectrum below the cut-off of the theory Λ ∼ 3 TeV, in
the region of parameter space which is compatible with electroweak precision data. Above
the top, there is always a very light (m5
quark very close in mass to it (0 ≤ m(1)
heavier charge 2/3 quark (800 GeV ? m(2)
3? 500 GeV) charge 5/3 quark, then a charge 2/3
? 2 TeV). The other two quarks, with electric
3? 100 GeV for m5
3? 300 GeV) and finally a
0.1 0.12 0.14 0.16 0.18
0.2 0.22 0.24
0.97 0.975 0.98 0.985
0.99 0.995 1
FIG. 4: Contribution to δgbLas a function of sL.The other input parameters are left free
(keeping the total χ2using the estimation for δgbLwithin 99% C.L.). The full red (empty green)
squares correspond to the full contribution (large Yukawa estimation) of δgbL. The horizontal lines
correspond to the maximum allowed contribution to δgbL(assuming an optimal contribution to T).
0.992 0.994 0.996 0.998 1
FIG. 5: Spectrum of light (below Λ) fermionic states (including the top quark) for the region of
parameter space compatible with EWPT. The (green) crosses, (blue) dots and (red) empty squares
correspond to charge 5/3, 2/3 and −1/3 quarks, respectively. The latter, together with one charge
2/3 quark, are typically above Λ.
charges 2/3 and −1/3 respectively, are quite degenerate and typically heavier than Λ.
All these fields mix very strongly with the top and among themselves to provide the
required positive contribution to the T parameter without violating the bounds on δgbL.
This gives rise to large corrections to the top gauge couplings, VL
tbdepends on the details of how the bottom quark is embedded in the theory but they
are expected to be suppressed by Yukawas of the order of the bottom Yukawa). We show
in Fig. 6 the values of these couplings in the allowed region of our model. It is interesting
0.99 0.992 0.994 0.996 0.998 1
0.7 0.75 0.8
0.85 0.9 0.95
FIG. 6: XL
tt(empty green squares) and XR
tt(full red squares) as a function of sL(left panel) and
tb(right panel). In both cases all the points are compatible with precision data at the 99%
C.L. The SM values for these couplings are XL
tb= 1 and XR
that while these couplings receive large corrections, the indirect constraints on them from
radiative corrections to electroweak and flavor precision data are still satisfied thanks to
the extra contribution of the new quarks in the theory. The charged current top-bottom
coupling is only now starting to be constrained by single top production measurements at
the Tevatron  (see also )
tb| ? 0.66,(62)
where we have assumed (as it happens in our model) that the only sizable correction to the
Wtb vertex is in VL
tb. From Fig. 6 we see that this precision is not yet sufficient in order to
constrain our model. The situation will improve at the LHC, where this coupling can be
determined with an accuracy of about ∼ 10%  and therefore the model could be probed
not only through direct production of the new quarks but also in single top production.
The measurement of the Zt¯t coupling is much more difficult. It is currently unconstrained
and although measurements of Zt¯t production at the LHC can be in principle used to
measure it, we have checked that the achievable precision is likely insufficient to constrain
our model [36, 37].
Let us now turn to the collider implications of the new fermionic sector. Pair production
of new vector-like quarks with quantum numbers similar to ours has been discussed in detail
in the literature [6, 7, 20, 32, 38, 39, 40]. In particular, it was shown in [39, 40] that pair
production of charge 5/3 quarks as light as the ones in our model can be discovered in the
very early runs of the LHC. The lightest of the charge 2/3 quarks has sizable couplings to the
top, the bottom and the charge 5/3 quark. Thus, pair production of this quark will result
in many ZZt¯t and W+W−b¯b, which should allow for a relatively easy search. However, the
mass difference with the charge 5/3 quark will typically be too small to allow for a significant
fraction of cascade decays involving both quarks.
Similar properties are shared by the second charge 2/3 quark (order one coupling to all
the lighter modes) with the notable distinction that now the mass difference with the charge
5/3 quark is large enough to allow for a large branching ratio. Thus, we have a sizable set
of events with the spectacular signature
pp → q(2)
2/3→ q5/3¯ q5/3W−W+→ W+W+W+W−W−W−b¯b. (63)
Note however that the larger mass of this quark will significantly reduce the pair production
cross section. A detailed analysis of the signal and background is necessary to decide the
reach of the LHC on this channel.
The large couplings of all these fields to the top and bottom indicates that they can be
further tested through single production (for a discussion of single production of vector-like
quarks with these quantum numbers mixing with valence quarks at the Tevatron see ).
We have seen in the previous subsection that the allowed region of parameter space
compatible with experimental data is quite small if only one set of fermionic composites is
below the cut-off of our effective theory. Due to the explicit breaking of the SO(5) symmetry
induced by the mixing with the fundamental fields of the SM top sector, these composite
fermions also contribute to the Higgs potential. Naturalness suggests that, ideally, they
should provide the leading contribution to the effective potential, cutting-off the Higgs mass
not at Λ but rather at the mass of these fermionic resonances. However, given the severe
constraints that precision data impose on the model, it would be rather coincidental if the
permitted small region in parameter space generated also a viable pattern of electroweak
symmetry breaking with the correct vacuum expectation value. This problem has already
been observed in 5D models of composite Higgs, in which a non trivial fermionic spectrum
was required in order to obtain a successful realization of EWSB [6, 7].
As a first step in preparation of realistic composite Higgs models which fully incorporate
a satisfactory pattern of EWSB, we find it useful to study the effect on electroweak and
flavor precision data of a second set of light composite fermions. Note that here we do not
mean to simply include the effect of more Kaluza-Klein modes (in an analogous 5D picture)
or heavier resonances in a purely 4D picture. We are rather considering the possibility that
the spectrum of fermionic resonances at the scale of our strongly coupled theory is richer
than what we have considered so far.9
We go back to our original mass term, Eq. (13), in which now mL and mR are two-
dimensional vectors and mΨ and µ are 2 × 2 matrices (the former diagonal, the latter
hermitian). The generalization of sL,Rto measure the degree of top compositeness in the
presence of more than one composite is
1 − (Uq(0)
1 − (Ut(0)
where Uq(0)(Ut(0)) is the 3 × 3 unitary matrix that mixes qL, Q1
in order to make the mass matrix diagonal before EWSB. This definition generalizes to an
arbitrary number of extra composites.
The result of our scans in this case shows that the region of parameter space compatible
with electroweak and flavor precision data expands dramatically. This was to be expected,
due to the increase in the number of degrees of freedom. Nevertheless, it should be em-
phasized that the constraints from the pattern of SO(5)/SO(4) global symmetry breaking
are still imposed on our extended sector. Even more interesting is the fact that the allowed
parameter space not only is it larger, but we also find a plethora of patterns of phenomena,
some of which we discuss below.
The most important phenomenological feature that is allowed by experimental data when
two sets of composite fermions are below Λ is a much richer spectrum. The number and
couplings of light quarks are no longer fixed. We can have from just one single charge 2/3
9This is precisely what happens in realistic 5D composite Higgs models. For a 4D interpretation, along the
lines of the models discussed in this paper, see  and .
quark to a full, almost degenerate, set of quarks arranged in a (4) of SO(4). They can be
very strongly coupled to the top and bottom or almost not coupled at all. This has a number
of interesting consequences, that can be subjects for exploration at colliders:
-Complex collider signatures. The presence of a relatively large, sometimes quite de-
generate, set of new particles will require sophisticated analysis to disentangle the
contribution of different modes. Also, the possibility of long decay chains ending
in a large number of leptons, jets and missing energy can make it more difficult to
distinguish composite Higgs models from other new physics models in the early data.
-Importance of single heavy quark production. Single production, which tests di-
rectly the couplings of the new quarks to the top and bottom, becomes an important
tool, complementary to pair production and to indirect tests through electroweak pre-
cision observables, to fully reconstruct the fermionic sector of the model.
-Strongly composite tR. A strongly composite tR(as opposed to a strongly composite tL
as in the case of one multiplet) is now a quite common occurrence. This is a welcome
feature, as there are some UV completions for which extra sources of flavor violation
can be enhanced for a strongly composite tL.
We now discuss in a bit more detail some indicative spectrum patterns of new quarks
that we have found in our scans. One interesting possibility is the presence of a rather
light (m(4)∼ 500 GeV) full, almost degenerate, (4) of SO(4) as the only set of particles
below 1 TeV. This set of new quarks does not contribute significantly to the T parameter
or δgbL(as it barely breaks the custodial symmetry), a role that is mainly played by heavier
modes which are more difficult to produce at the LHC. The light quarks (two charge 2/3,
one 5/3 and one −1/3) all decay mainly to the top and therefore a large number of V V t¯t
events should be produced at the LHC (Refs. [39, 40] showed that the charge 5/3 quarks
should be easily discovered with very early data at the LHC). Once their masses are known
and some information on their couplings from single production is gathered, a reanalysis of
electroweak precision data should give a clear picture of the fermionic content of the model.
The possibility we have just mentioned, a light full (4) representation of SO(4), would
clearly point to the underlying symmetry structure of the theory. There are however other
regions of parameter space in which this structure is not so obvious. One example is the case
in which the only light mode, easily accessible at the LHC, is a charge 2/3 quark. Of course,
we know that in the context of composite Higgs models with little fine-tuning, a single
charge 2/3 singlet is not compatible with electroweak data. However, without additional
information, it will not be easy to disentangle the contribution of Higgs compositeness and
the new quark to electroweak precision data. Our study of electroweak constraints shows
however that the heavier modes must significantly contribute to the T-parameter and δgbL
and therefore should have large couplings to the top and/or bottom that would make them
accessible through single production.
A variety of viable spectra allows for the possibiliy in which we have several non-
degenerate modes below 1 TeV. For instance, there are regions in which the lightest new
quark is a charge 5/3 one, then a heavier charge 2/3 and then a heavier (but still relatively
light, m−1/3∼ 800 GeV) charge −1/3. The mass difference is commonly large enough to
allow for cascade decays that end in the top and up to three gauge bosons. This means we
can easily have up to eight gauge bosons and two b’s in the final state,
pp → q−1
→ W−W−W+tW+W+W−¯t → W−W−W+W+bW+W+W−W−¯b.
3→ W−q 2
3W+¯ q 2
3→ W−W−q 5
3W+W+¯ q 5
A fraction of the time q 2
or 4W + 2Z + b¯b. Thus, we have processes with a relatively large production cross section
3will decay into Zt so that the final state can also be 6W + Z + b¯b
(for quarks which are not too heavy) and a very complex final state with many jets, leptons
and missing energy. Although a detailed analysis is required to assess our capability of
understanding these complex processes, it is likely that they will be easy to discover but
very difficult to fully reconstruct and the detailed information on the quark masses may not
We note that the same features as we find here, a rich spectrum of light modes, some of
which do not contribute very strongly to electroweak observables, and a number of heavier
modes (but still lighter than the bosonic resonances) which contribute to render electroweak
observables compatible with experimental data has been recently observed in composite
Higgs models in five dimensions. In fact, this kind of spectrum proved crucial to make the
model in  simultaneously compatible with electroweak precision tests and with a realistic
pattern of electroweak symmetry breaking.
The realization of electroweak symmetry breaking is a long-standing mystery that will
be soon tested in detail at the LHC. Composite Higgs models, in which the Higgs boson
arises as the (composite) pseudo-Goldstone boson of a spontaneously broken global symme-
try in a strongly coupled gauge theory, is an appealing candidate. It naturally protects the
electroweak scale from short distance physics and can even explain the suppression of the
electroweak scale with respect to the scale of new physics. Compatibility with current exper-
imental data seem to point to a relatively large scale of new bosonic resonances Λ ? 3 TeV
and to a custodially preserving symmetry breaking pattern to protect the T parameter.
Similarly, a left-right symmetry within the custodial symmetry naturally protects the ZbL¯bL
coupling from large corrections. A minimal symmetry breaking pattern that contains the
Higgs as a pseudo-Goldstone boson, is custodially symmetric and can protect the ZbL¯bL
coupling is a global SO(5) symmetry spontaneously broken to SO(4) and fermions in the
fundamental representation of SO(5) [4, 5, 9].
The low energy implications of this set-up can be simply analyzed with the aid of an
effective description based on an SO(5)/SO(4) non-linear sigma model. If the Higgs is quite
composite, i.e. if the scale of the global symmetry breaking is not far from the electroweak
scale, as one would expect in a natural (non fine-tuned) model, its couplings to the Standard
Model fields are significantly reduced. This results in a sensitivity to the cut-off of the theory
whose effect can be taken into account by defining an effective (heavier) Higgs mass. This
effective Higgs mass is the one that should be included in the calculation of electroweak
precision observables, giving a positive contribution to the S parameter and a negative
contribution to the T parameter. These contributions simply come from the fact that the
Higgs is composite and are therefore quite generic. UV physics, being custodially symmetric,
is not expected to contribute to the T parameter but it can give a tree level contribution to
the S parameter that, together with the contributions to S and T from Higgs compositeness,
put the model in gross contradiction with current experimental tests.
The large mass of the top quark, however, makes it natural to assume that it is partially
composite. In that case it will strongly couple to the fermionic resonances of the strongly
coupled theory which, if lighter than the cut-off of the theory, can have an important impact
on electroweak observables (due to their mixing with the top, which breaks explicitly the
custodial symmetry). We have considered the presence of one or more sets of fermionic
resonances, spanning full fundamental representations of SO(5). The SO(5) symmetry is
explicitly broken by the SM quarks, qLand tR(the former also breaks the custodial SO(4)
symmetry), which couple linearly to the strongly coupled theory. This coupling makes qL
and tRpartly composite and, through their composite components, couple to the Higgs and
get a mass. We have computed the exact contribution of this new sector to the relevant
electroweak precision observables, T and δgbL, and also to some of the flavor observables
that are strongly correlated to this new fermionic sector, Bq−¯Bqmixing and b → sγ. We
have performed an exact one-loop calculation which does not rely on any approximation and
presented the results in a general enough way that can be easily extended to models beyond
the one we have considered. Our exact calculation, that goes beyond the large Yukawa
approximation which is commonly used to estimate the contribution to the anomalous ZbL¯bL
coupling, can have an important impact if the model is strongly constrained, as happens in
our case in some regions of parameter space, or if one wants to do precision analysis of new
physics, as it will become necessary if new quarks are discovered at the LHC.
Armed with these detailed calculations of the most relevant electroweak observables, we
have used adaptive Monte-Carlo methods to scan the parameter space in search of the
regions allowed by experimental data. The result depends dramatically on whether there
exist just one or more fermionic multiplets below the cut-off of our theory.
The case of only one multiplet below Λ is quite constrained. When all the experimental
constraints are taken into account, only a small region with a very composite LH top sur-
vives. In this region the spectrum of new quarks and their couplings are almost univocally
determined by electroweak and flavor precision data. Above the top, there are typically two
quite light (? 500 GeV) quarks of charge 5/3 and 2/3, respectively (the former typically
slightly lighter) followed by another charge 2/3 quark with mass 800 GeV ? m(2)
? 2 TeV.
The lighter two quarks should be easily produced at the LHC and have large enough cou-
plings to the top to make single production an interesting channel to study. Furthermore,
they typically induce large enough corrections to the Vtbcoupling to be detectable at the
LHC. The strong constraints on this possibility and the fact that we have simply assumed a
realistic pattern of electroweak symmetry breaking - which is fully calculable and therefore
imposes further constraints in a UV completion of the model - have motivated us to consider
the possibility that more than one set of fermionic resonances contribute to electroweak ob-
servables. This motivation is reinforced by the experience with 5D UV completions of the
model, in which electroweak symmetry breaking imposes non-trivial constraints on the low
energy spectrum, usually requiring a more complex spectrum of light fermionic resonances
than the one we have found in the case of one multiplet.
The situation dramatically changes when two fermionic multiplets are allowed to con-
tribute to electroweak precision observables. The spectrum of light resonances is no longer
constrained; we have found from one single quark of charge 2/3 to four quarks in a full degen-
erate multiplet (4) of SO(4) below 1 TeV. Their couplings to the top can also vastly change,
which makes the study of single production even more interesting. When pair-produced,
these new quarks can produce long decay chains that contain a large number (up to eight)
of SM electroweak vector bosons and a b¯b pair. Thus, final states with many jets, leptons
and missing energy would be a common signature in these models.
We conclude that composite Higgs models with no bosonic resonances (apart from the
Higgs itself) below the cut-off of the low energy effective theory can be fully compatible with
current experimental constraints provided a quite rich spectrum of light fermionic resonances
is present in the model. These new fermionic resonances should be easily produced at the
LHC and would most likely be the first signal of new physics beyond the SM. Establishing
the symmetry pattern from the fermionic spectrum can however prove more difficult and will
require a detailed analysis of the full experimental information, including pair production,
single production and a detailed analysis of electroweak and flavor observables. This is really
important as other signatures that would definitely pin down the model as a composite Higgs
model, like the measurement of the Higgs couplings, longitudinal gauge boson scattering 
and the production of new bosonic resonances of the strongly coupled theory can take much
longer at the LHC .
We would like to thank J.A. Aguilar-Saavedra, R. Contino, M. Gillioz, U. Haisch , A.
Pomarol, E. Pont´ on and Z. Kunszt for useful discussions. This work was supported by the
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