Exceptional composite dark matter
Eur. Phys. J. C
Exceptional composite dark matter
Guillermo Ballesteros 2 3
Adrián Carmona 1 3
0 Departament de Física Tèorica, Universitat de València and IFIC, Universitat de ValènciaCSIC , Dr. Moliner 50, 46100 Burjassot, València , Spain
1 Theoretical Physics Department, CERN , Geneva , Switzerland
2 Institut de Physique Théorique, Université Paris Saclay, CEA, CNRS , 91191 GifsurYvette , France
3 , Mikael Chala
We study the dark matter phenomenology of nonminimal composite Higgs models with S O (7) broken to the exceptional group G2. In addition to the Higgs, three pseudoNambuGoldstone bosons arise, one of which is electrically neutral. A parity symmetry is enough to ensure this resonance is stable. In fact, if the breaking of the Goldstone symmetry is driven by the fermion sector, this Z2 symmetry is automatically unbroken in the electroweak phase. In this case, the relic density, as well as the expected indirect, direct and collider signals are then uniquely determined by the value of the compositeness scale, f . Current experimental bounds allow one to account for a large fraction of the dark matter of the Universe if the dark matter particle is part of an electroweak triplet. The totality of the relic abundance can be accommodated if instead this particle is a composite singlet. In both cases, the scale f and the dark matter mass are of the order of a few TeV.

Contents
1 Introduction
Composite Higgs Models (CHMs) [
1–3
] are among the best
motivated extensions of the Standard Model (SM) of
particle physics. First of all, the hierarchy problem can be
solved by assuming the Higgs to be a bound state of a new
strongly interacting sector. This sector is supposed to respect
an approximate global symmetry G, which in turn is
spontaneously broken to H ⊂ G at a scale f ∼ TeV. The Higgs is
then expected to be naturally lighter than the scale of
compositeness by further assuming that it is a pseudo Nambu–
Goldstone Boson (pNGB) of this symmetry breaking pattern.
Moreoever, this approach could also help to understand the
puzzling hierarchy of fermion masses in the SM. Indeed, the
explicit breaking of G by linear interactions between the SM
fermions and composite operators [
4, 5
] translates into
mixings between elementary fields and fermionic resonances at
the confinement scale f . The Yukawa couplings emerge in
the physical basis before electroweak (EW) symmetry
breaking, being very much dependent on the dimension of the
composite operators [6]. Therefore, making the mixing of
different flavors depend on the dimension of their
respective operators, their masses at the EW scale could be very
different. In particular, the top quark Yukawa coupling can
be much larger than the other ones without any previous
enhancement in the UV. In such a case, the explicit breaking
of the global symmetry is triggered by the top quark linear
mixing.
The requirement that one Higgs doublet is part of the
pNGB spectrum restricts the amount of possible cosets.
In light of this background and the ongoing tests of the
scalar sector at the LHC, a systematic study of nonminimal
CHMs is a timely target that is worth aiming for. 1 Thus,
in this paper we consider a model based on the
symmetry breaking pattern S O(7)/G2 [
12
], which gives rise to
seven pNGBs transforming as 7 = (2, 2) + (3, 1) under
SU (2) × SU (2) ⊂ G2. Depending on which of the two
SU (2) groups is weakly gauged (and therefore identified
with the SM SU (2)L ) the three additional pNGBs
transform as a scalar real triplet or as three singlets of the EW
group. The former constitutes a version of the inert triplet
model [
13
] free of the hierarchy problem, in which we
concentrate throughout most of the paper. It is worth noticing
that the number of free parameters in the scalar potential of
the inert triplet model is smaller than in any other extended
Higgs sector (and on an equal footing with e.g. the
singlet Higgs portal). In this sense, the coset we study is also
minimal. Moreover, the collection of constraints obtained in
this work will also be of relevance for triplets in other
contexts.
We will highlight the main differences between the triplet
and the singlet cases in the last section of the paper. In both
cases, the neutral scalar can be forced to be odd under a Z2
symmetry which is shown to be compatible with the strong
sector dynamics. Thus, this sector respects the symmetry
O(7) ∼= S O(7) × Z2. 2 The neutral extra pNGB can then
account for part or all of the observed dark matter (DM) relic
abundance, depending on its SU (2)L quantum numbers.
Both models present several advantages in contrast to their
respective elementary counterparts. Indeed, the larger
symmetry on the strong sector constrains the number of
independent free parameters. If the breaking of the Goldstone
symmetry is mainly driven by the fermion sector, the scalar
potential depends only on three quantities, two of which can
be traded by the measured values of the Higgs mass and
quartic coupling. The remaining parameter is just the
compositeness scale f . The Z2 symmetry is automatically unbroken in
the EW phase. On another front, the symmetry breaking
pattern we consider provides a more interesting phenomenology
than the minimal CHM of Ref. [
7
]. First of all, it contains a
DM candidate. Since direct and indirect DM searches bound
the compositeness scale f from above, they also set a robust
upper limit on the mass of the new fermionic resonances,
1 The adjective nonminimal refers to CHMs that present an extended
scalar sector, contrary to the minimal model based on S O(5)/S O(4)
[
7
]. An exhaustive list of small groups that can be used to build
composite Higgs models can be found, for instance, in Ref. [
8
].
According to the dimension of the global symmetry group, the smallest
cosets are S O(6)/S O(5), S O(6)/[S O(4) × U (1)], S O(7)/S O(6) and
S O(7)/G2. From the point of view of the number of pNGBs, the
minimal choices are S O(6)/S O(5), S O(7)/S O(6), S O(8)/S O(7) and
S O(7)/G2. The first three have been already studied in the context of
DM [
9–11
], while S O(8)/S O(7) provides a very similar
phenomenology leading (only) to extra singlets.
2 Note that in what concerns the composite sector alone, the scalars are
exact Goldstones and hence their interactions are shift invariant.
(2.1)
(2.2)
(2.3)
which otherwise could not be estimated by other means than
finetuning arguments. Moreover, since these new resonances
decay into the extra scalars (for which there are no dedicated
searches), constraints on vectorlike fermions in light of
current LHC data could therefore be weakened.
The structure of the paper is as follows. In Sect. 2 we
demonstrate that the Z2 symmetry mentioned above can be
respected by the strong sector; and we compute the pNGB
sigma model for the triplet case. There, we also discuss
the representation theory for fermions and derive the scalar
potential. The possible collider signatures are described in
Sect. 3. The computation of the relic density, as well as the
analysis of potential direct and indirect detection signals are
presented in Sect. 4. Finally, in Sect. 5 we consider the
possibility that the three additional scalars transform as SU (2)L
singlets rather than as a triplet, highlighting the
phenomenological differences. We conclude in Sect. 6. Further details on
the algebra of S O(7) and G2 are provided in Appendices A
and B, while in Appendix C we stress the main
phenomenological consequences of sizable explicit symmetry breaking
in the lepton sector.
2 Viability of the Z2 symmetry
For a generic symmetry breaking pattern, let T i and X a
represent the unbroken and coset generators, respectively. Let
us also define = a X a , where a runs over all pNGBs.
The dμ = dμa X a symbol from the Maurer–Cartan one form
where we have denoted ad A(B) = [ A, B] and the subindex
X means the projection into the broken generators. It is
well known that the pNGB interactions in symmetric spaces
contain only even powers of 1/ f . Indeed, for symmetric
cosets, [X a , X b] = i f abi T i , and hence all even powers in
the expression above vanish. Consequently, the leadingorder
Lagrangian in derivatives describing the pNGB fields,
Lσ = 21 f 2Tr(dμdμ),
dμ =
∞
k=0
1
= f ∂μ
ωμ = −iU −1∂μU = dμa X a + EμiT i ,
with U = exp i
f
,
entering the nonlinear sigma model, reads
(−i )k
f k+1(k + 1)!
adk (∂μ )X
1
+ 24 f 4 [ , [ , [ , ∂μ ]]]X + · · ·
i 1
− 2 f 2 [ , ∂μ ]X − 6 f 3 [ , [ , ∂μ ]]X
1
dμ = f ∂μ
with
i
g1 = a1
1
g2 = a2
ˆ =
√
a a .
−1 + cos
1
− f + √a2 sin
√a1 ˆ
f
,
√a2 ˆ
f
.
In the equations above, a1 = 5/6, a2 = 11/18, and
constructed out of the trace of two dμ symbols, contains only
terms with even number of fields.
This concerns models like S O(6)/S O(5) [
9,14
], S O(7)/
S O(6) [11], S O(6)/S O(4) × S O(2) [
10
] or SU (4) ×
SU (4)/SU (4) [
15
] for example. We are instead interested
on the coset S O(7)/G2. The corresponding generators can
be found in Appendix A. They are normalized in such a
way that Tr(T a T b) = δab and Tr(X a X b) = δab. The
condition Tr(T i X a ) = 0 also holds. A straightforward
computation shows that this space is not symmetric. For example,
[N1, N2] = −i (M3+N3/√2)/√3. Nevertheles, the
leadingorder sigma model still contains only even powers of f . This
result relies on two properties. First, all commutators with
odd powers of f in Eq. (2.2) are parallel; likewise for all
even powers. More concretely,
+ g1 ˆ 2[ , ∂μ ]X + g2 ˆ 3[ , [ , ∂μ ]]X ,
Clearly, g1 consists of only even powers of 1/ f , while g2
contains only odd terms. And second, one can easily check
that both Tr(∂μ [ , ∂μ ]X ) and Tr([ , ∂μ ]X [ , [ ,
∂μ ]]X ) vanish, and so no odd powers of 1/ f appear in
the Lagrangian at leading order in derivatives. Given this,
SU (2)L invariance implies that only terms with an even
number of new multiplets are allowed, irrespectively of whether
they are singlets or triplets.
We can then impose a Z2 symmetry under which the
multiplet containing the new neutral scalar changes sign,
regardless of whether it is a singlet or a triplet. Clearly, in light of
the discussion above, this does not spoil the twoderivative
Lagrangian containing the kinetic term of the propagating
fields. Higherorder terms, instead, might be forbidden by
this symmetry without observable phenomenological
consequences. As in the rest of nonminimal CHMs with DM
scalars, as well as in their renormalizable counterparts, the
origin of this symmetry is not specified and it has to be
enforced by hand. It is nonetheless interesting to prove that
this is compatible with the shift symmetry even in a
nonsymmetric coset, as is the case here.
2.1 Gauge bosons
Let us focus now on the triplet case. We can compute the
leadingorder covariant derivative Lagrangian for the pNGBs
by promoting the derivatives in ωμ and dμ to covariant
derivatives, i.e., ∂μ → ∂μ − i g√3W μi Mi − i g Bμ F3 (see
Appendix A for the expressions of Mi and F3). At lowest
order in derivatives, this leads to
1
Lσ = Dμ H 2 1 − 3 f 2  
2
1 2
+ 2 Dμ 2 1 − 3 f 2 H 2
1
− 6 f 2
†ti (Dμ ) (Dμ )†ti
1 2
+ 3 f 2 ∂μ(H † H )( † Dμ ) − 3 f 2 H 2Dμ H 2
1
+ 6 f 2 ∂μ(H † H )
2
+ O
1
f 4
,
where we have defined the following SU (2)L × U (1)Y ⊂
S O(4) multiplets:
1
H = √ (h1 − i h2, h3 + i h4)T ∼ 21/2,
2
= (κ1 + i κ2, −η, −κ1 − i κ2)T ∼ 30,
and κ1, κ2, η, h1, h2, h3, h4 are the pNGBs associated to the
broken generators N1, N2, . . . , N7. H is identified with the
SMlike Higgs doublet living in the 7 representation of G2
and with the remaining real triplet. Besides, ti , with i =
1, 2, 3, read
(2.7)
(2.8)
(2.9)
(2.4)
(2.5)
(2.6)
1 ⎛ 0 1 0 ⎞ 1 ⎛ 0 −i 0 ⎞
t 1 = √2 ⎝ 1 0 1 ⎠ , t 2 = √2 ⎝ i 0 −i ⎠ ,
0 1 0 0 i 0
We have also redefined f → − f /(2√2/3). We will keep
this convention henceforth. We also identify h = h3 with
the physical Higgs boson. The part of the above Lagrangian
involving only h can easily be summed to all orders in 1/ f 2,
resulting in
1
Lσ ⊃ 2 (∂μh)2 +
1 g2 f 2 sin2
4
h
f
Wμ+W μ−
1
+ 8
(g2 + g 2) f 2 sin2
Zμ+ Z μ− ,
h
f
(2.10)
and tan θW = g /g as usual. In particular, we can see that
after EW symmetry breaking (EWSB), the W and Z bosons
get masses
1 g2 f 2 sin2
m2W = 4
h
f
m2Z = 41 (g2 + g 2) f 2 sin2
,
h
f
,
with h the Higgs vacuum expectation value (VEV), which
differs from the SM EW VEV v = f sin ( h / f ) ∼ 246
GeV. It is also clear that ρ = m2W /m2Z cW2 = 1, as expected
due to the custodial symmetry S O(4) ⊂ G2. The ratio of the
tree level coupling between the Higgs and the massive gauge
bosons to the corresponding SM coupling differs from unity
by the amount:
RhV V =
1 − ξ ,
v2
ξ = f 2 .
Clearly, given that f ∼ TeV, the ratio ξ 1. If the
SM group SU (2)L × U (1)Y of S O(7) is the only gauged
group in the EW sector, the global symmetry S O(7) is
broken explicitly. This becomes manifest in the (nonvanishing)
scalar potential. In order to compute it, we promote the
SU (2)L × U (1)Y gauge bosons to be in the adjoint of
the global S O(7) with the help of spurion fields. For
this aim, let us order the generators of S O(7) as T aˆ =
{F 1, . . . , F 7, M 1, . . . , M 7, N 1, . . . , N 7}. We can then write
Aaμˆ = W μi iaˆ + Bμϒ aˆ , aˆ = 1, . . . , 21, i = 1, 2, 3.
(2.14)
The spurions iaˆ and ϒ aˆ are given explicitly by the
expressions:
iaˆ =
√3 δ(i+7)aˆ ,
ϒ ˆ = δaˆ3.
a
(2.15)
Formally, they also transform in the 21 of S O(7). The dressed
μ = U −1 Aaˆ T aˆ U transforms under g ∈ S O(7) as
field AD μ
h( , g) AμD h−1( , g), with h ∈ G2, and decomposes as
a sum of irreps of G2. The same happens to the dressed
spurions iD = U −1 iaˆ T aˆ U and ϒD = U −1ϒ aˆ T aˆ U , with
the difference that the index i spans an SU (2)L triplet.
The gauge contribution to the scalar potential consists
therefore of the different invariants that can be built out of
the G2 irreps within iD and ϒD and can be expressed as an
where we have defined
Zμ = cos θW Wμ3 − sin θW Bμ,
Aμ = sin θW Wμ3 + cos θW Bμ
(2.11)
(2.12)
(2.13)
expansion in powers of g/gρ and g /gρ , with gρ the
characteristic coupling of the strong sector vector resonances.
Taking into account that iD and ϒD decompose as 7 ⊕ 14
under G2, we obtain only one independent invariant at
leading order:
, (2.16)
where c˜1,2 are 1 dimensionless numbers, mρ ∼ gρ f is
the typical mass of the vector resonances, and we have used
naive dimensional analysis [
16–18
] to account for the h¯ and
mass dependence of the radiative potential.
2.2 Fermions
The mixing between the elementary fermions and the
composite sector breaks explicitly the global symmetry S O(7),
because the former transform in complete representations of
the EW subgroup only. Let us first focus on the quark sector.
The mixing Lagrangian can be written as
Lmix ∼ λiqj q¯αiL ( qα)I (Oqj )I + λiuj u¯iR ( u )I (Ouj )I
+λidj d¯Ri ( d )I (Odj )I + h.c..
(2.17)
The indices i, j = 1, 2, 3 run over the three quak
generations. α = 1, 2 and I are instead SU (2)L and S O(7)
indices, respectively. The couplings λq33 and λu33 are supposed
to be order one and much larger than all other couplings.
This is expected from the dependence of the quark Yukawas
on these numbers, namely yuij,d ∼ λq†ik λku,jd /g∗ where g∗
is the typical coupling of the strong sector fermionic
resonances. The spurion fields are incomplete multiplets of
S O(7) × U (1)X .3 Formally, they transform in the same
representations as the corresponding composite operators, O.
We assume that the third generation right and left quarks
mix with composite operators transforming in the 12/3 and
the 352/3 of S O(7) × U (1)X , respectively. This is motivated
by the following branching rules under G2 × U (1)X and the
EW gauge group:
35 = 12/3 ⊕ 72/3 ⊕ 272/3
= 12/3 ⊕ 2±1/2 ⊕ 30 ⊕ 12/3 ⊕ 2±1/2 ⊕ · · ·
(2.18)
where the ellipsis stands for higherdimensional
representations in the branching rule of the 27.4 In order not to
break the EW symmetry, the spurions qα can only have
3 The addition of the extra (unbroken) U (1)X is necessary to
accommodate the SM fermion hypercharges.
4 Under S O(4) ∼= SU (2)L × SU (2)R ⊂ G2 : 7 = (2, 2) ⊕ (3, 1) and
27 = (1, 1) ⊕ (2, 2) ⊕ (3, 3) ⊕ (4, 2) ⊕ (5, 1).
nonzero entries in the doublets. However, the Z2
symmetry requires the components along the second one to
vanish.5 Its explicit expression can be found in Appendix A.
Similarly to the gauge boson case, the dressed spurion
qαD = U −1 qαU transforms under g ∈ S O(7) as qαD →
h( , g) qαD h−1( , g) with h ∈ G2 and decomposes as a
sum of irreps: qαD = m qαm . The fermion contribution
to the scalar potential can be written as
Vfermion( ) ≈ m4∗ (4Nπc)2
× ⎣
⎡
λq  2
g∗
j
c j Vj ( ) +
λq  4
g∗
k
⎤
ck Vk ( ) + · · · ⎦ ,
(2.19)
where m∗ ∼ g∗ f is the typical mass scale of the fermionic
resonances and again we have used naive dimensional
analysis to estimate the parametric dependence of the potential,
with Nc = 3 and ci and ci order one dimensionless
numbers. V j ( ), Vk ( ) and the terms indicated by the ellipsis
are the different invariants that can be built out of two, four
and higher number of insertions of qmI , respectively. For
simplicity, we have also defined λq = λq33. Note that no
terms proportional to λu appear, because the righttop
mixing, being a full singlet, does not break the global symmetry.
The scalar potential consists then of the lefthanded
topinduced and the gaugedinduced potentials. However, the
latter can be neglected if
c˜1g2g2
ρ
2π 2λH ⇒ c˜1gρ2
8,
(2.20)
where λH ∼ 0.13 is the SM Higgs quartic coupling and we
have disregarded the hypercharge contribution for simplicity.
Indeed, if this inequality holds, all observables computed
taking into account only the topinduced potential are (almost)
unaffected when the gauge potential is also included. This
occurs, in particular, for c˜1 ∼ 0.1 and moderately large
values of gρ ; and also if c˜1 ∼ 1 and gρ ∼ 1. Whereas the
former possibility may involve some additional tuning, the
latter arises naturally at large values of f , which, as we will
see, are the ones preferred to account for the observed DM
relic abundance. We consider this scenario hereafter. Then
V ( ) ≈ m2 f 2 16Nπc 2 yt2 [c1V1( ) + c2V2( )] ,
∗
(2.21)
where λq  has been traded by the top Yukawa coupling yt
(see below) and we have defined
5 Let (i, j) run over the nonvanishing entries of the spurions q1 and
q2 (see Appendix A). Then, note that under the Z2, the elements (i, j)
of the U matrix do not change sign. Therefore, the spurions are even
eigenstates: Z2( q1 ) = q1 and Z2( q2 ) = q2 . On the contrary, the
spurion accommodating the second doublet includes, for example, a
nonvanishing entry in (1, 4), while U14 ∼ h1κ1, that changes sign.
α
7
∼ V2( ) = H 2 cos
3H 2 + 2 2 cos
.
2 ˆ
f
4 ˆ
f
(2.23)
The scalar potential above depends only on two independent
unknowns, c1 and c2. They parametrize the two invariants
constructed out of 1 × 1 and 7 × 7 in Eq. (2.18),
respectively. Note that the potential features only an even number
of powers of . This is actually true at any order in λq /g∗,
because the spurions are Z2even and the Z2 invariance of the
potential requires to appear always squared. Let us further
keep the leadingorder potential in the expansion in powers
of 1/ f 2. This can be matched to the renormalizable piece
1 2
Vrenorm(H, ) = μ2H H 2 + λH H 4 + 2 μ  
2
+ 41 λ  4 + λH H 2 2.
(2.24)
The five parameters in Eq. (2.24) can be expressed in terms of
the parameters c1 and c2. These can be traded by the measured
values of the SM EW VEV and the Higgs quartic coupling,
λH . Up to the scale f , all parameters of phenomenological
relevance are then predictions. These are given in Table 1. It
can be checked that = 0 in the EW phase, since μ2 > 0
and λH > 0. And so, as anticipated, the Z2 symmetry is
not spontaneously broken. We would like to point out that the
negative sign of λ does not necessarily imply a (potentially
dangerous) runaway behavior at high energies, where the
effective description we use fails. The existence or not of
such a behavior, and of a possible minimum at higher energies
would depend on the specific way that the model is completed
in the UV.
It is also worth stressing that, after EWSB, the masses of
the charged and neutral components of are both equal to
m2 = μ2 + v2λH
2 2 9 v2
= 3 f λH 1 − 4 f 2
+ · · · , (2.25)
where the ellipsis stands for terms that are further suppressed
by powers of v2/ f 2. The splitting between the masses of
the charged components and that of the neutral one comes
only from (subdominant) radiative EW corrections. It can be
estimated to be M ∼ 166 MeV [
19
].
Finally, we can also compute the top Yukawa Lagrangian:
α
qα
¯L
= ct λq q¯L H˜ tR
α †
q D 88tR + h.c. ∼ Lyuk
f
ˆ
sin
2 ˆ
f
+ h.c.
2
= −yt (q L H˜ tR ) 1 − 3 f 2
2 + · · · + h.c.,
(2.26)
1 − 2ξ
Rhtt = √1 − ξ
where ct is an order one dimensionless parameter encoding
the UV dynamics and the product −2ct λq has been traded
by the top Yukawa, yt . This Lagrangian is explicitly
Z2invariant. If we add all terms involving only the Higgs boson,
the ratio of the tree level coupling of the Higgs to the massive
top quark to the corresponding SM coupling is
v2
, ξ = f 2 . (2.27)
3 Collider signatures
Different collider searches bound f from below. Among the
more constraining ones, we find monojet analyses, searches
for disappearing tracks, measurements of the Higgs to
diphoton rate and EW precision tests. The small probability for
an emission of a hard jet in association with two invisible
particles makes monojet searches less efficient than the other
two.
The Higgs decay width into photons is modified by order ξ
due to the nonlinearities of the Higgs couplings, as stated in
Eqs. (2.13) and (2.27), and, to a smaller extent, due to the new
charged scalars that can run in this loopinduced process. The
width (taking into account both effects) is given by [
20,21
]
α2v2m3h g2
(h → γ γ ) = 1024π 3 2m2W
1 − ξ A1(τW )
4yt2 1 − 2ξ
+ 3mt2 √1 − ξ
A1/2(τt ) +
λH
m2 A0(τκ )
κ
2
(3.1)
where τi = 4mi2/m2h , A0(x ) = −x 2(x −1 − F (x −1)),
A1/2(x ) = 2x 2(x −1 + (x −1 − 1)F (x −1)) and A1(x ) =
−x 2(2x −2 + 3x −1 + 3(2x −1 − 1)F (x −1)), while the
function F is given by F (x ) = arcsin2 √x . The Higgs
production cross section via gluon fusion is also modified by order
ξ effects:
σ (gg → h) =
(1 − 2ξ )2
1 − ξ
σ SM(gg → h),
with σ SM the SM production cross section. Given that ξ > 0
and λH > 0 (see Table 1), the production cross section
times branching ratio is always smaller than in the SM. A
(3.2)
combination of 7 and 8 TeV data from both ATLAS and
CMS [
22
] sets a lower bound of 0.66 on σ (gg → h →
γ γ )/σSM(gg → h → γ γ ) at 95 % C.L. This translates into
a bound on f 800 GeV. EW precision tests [
23
] push this
bound to f 900 GeV. Searches at future colliders (see
for example Ref. [
24
]) would determine the Higgs to
diphoton cross section with a much better accuracy. In particular,
the region f 1.5 TeV is expected to be probed in Higgs
searches at future facilities.
Finally, searches for disappearing tracks are sensitive to
pairproduction and the subsequent decay of the new scalars.
Indeed, the small splitting between the charged and the
neutral components of implies that the former has a decay
length exceeding a few centimeters. This produces tracks in
the tracking system that have no more than a few associated
hits in the outer region, in contrast with most of the SM
processes. To our knowledge, the most constraining search of
this kind was performed by the ATLAS collaboration in [
25
]
(similar results were found in the CMS analysis of Ref. [
26
]).
Searches of this type using 13 TeV data are not yet published.
The ATLAS search is optimized for a Wino (i.e. a generic
triplet fermion, χ ) with a width of ∼160 MeV, corresponding
to a lifetime of ∼0.2 ns, whose charged components
therefore decay predominantly into the neutral one and a soft pion.
In this respect, the search applies equally well to our scalar
triplet. The search rules out any mass below ∼270 GeV,
corresponding to a production cross section of ∼0.25 pb. The
latter takes into account the production of all χ +χ −, χ +χ 0
and χ −χ 0. The corresponding bound on f is therefore given
by the value at which the production cross section in the
scalar case equals the previous number (note that, for the
same mass, the scalar and triplet cross sections can be very
different).
In order to compute this cross section at the same level
of accuracy as the one considered in the experimental
reference, i.e. at NLO in QCD, we first implement the
renormalizable part of our model in Feynrules v2 [
27
]. UV
and R2 terms [
28
] are subsequently computed by means of
NLCOT [
29
]. The interactions are then exported to an UFO
model that is finally imported in MadGraph v5 [
30
] to
generate partonlevel events from which the total cross section is
computed for all values of f in the range 500, 600, . . . , 1500
GeV. The bound on f turns out to be only f 650 GeV.
However, future facilities could easily exceed the reach of
Higgs searches [
31–33
]. For example, a naive
reinterpretation of the results in Ref. [
31
] suggests that values of f as
large as ∼3.5 TeV could be tested in a future 100 TeV pp
collider.
4 Searches for dark matter
In this section of the paper we analyze the extent to which η,
the neutral component of the scalar triplet of our model,
can contribute to the DM of the Universe, given the current
experimental constraints. As we anticipated in the
Introduction, the compatibility of a global Z2 symmetry with the
breaking pattern S O (7)/G2 allows one to forbid η decays.
This neutral particle, which couples to the SM through weak
interactions and does not couple directly to the photon is, a
priori, a good weakly interacting massive particle (WIMP)
DM candidate with the adequate mass scale. Remarkably, the
mass of η and its relic abundance are, in our case, entirely
determined by the scale f , which makes the model extremely
predictive. In this last respect, the model is on pair with other
simple implementations of the WIMP idea, such as the
minimal DM model [
19
].
We recall that the total annihilation rate, σ v , and the relic
abundance, h2, of any thermal relic satisfy the approximate
relation
h2 ∼
3 × 10−27
σ v
cm s−1,
(4.1)
where the brackets indicate the average over the thermal
velocity distribution. Thus, if a thermal relic explains the
totality of the DM abundance ([ h2]DM ∼ 0.11 [
34
]), it
must have an annihilation rate of the order of σ v ∼
3 × 10−26 cm s−1.
Given the expression 4.1, the relic abundance turns to be
roughly proportional to the mass squared of the thermal relic.
The relic abundance for the neutral component of a scalar
triplet as a function of its mass was computed in [
35
].
Including nonperturbative effects, it was found that a mass of ∼2.5
TeV is required to obtain the measured DM abundance. In
Fig. 1, we recast this result as a function of the
compositeness scale f of our model, which is related to the mass of the
neutral component of the triplet, η, through Equation 2.25.6
As shown in Fig. 1, a scale f ∼ 8.6 TeV is required in this
case to account for the totality of the DM in the Universe. In
the remaining of this section we explore whether this scale is
compatible with the current bounds from the LHC and direct
and indirect detection experiments; and we determine how
much DM can be accounted for by η.
6 The result shown in Fig. 1 assumes that the portal coupling λH is
negligible. This is indeed the case in our model, since λH is roughly
an order of magnitude smaller than the gauge couplings.
2h 0.10
0.20
(4.2)
which scales with the inverse of the DM mass squared and
arises from the tree level exchange of a Higgs boson on t
channel (see Fig. 2a), where
f N =
fq =
q
q
mq
m N
N q¯ qN
= 0.30 ± 0.03,
(4.3)
and m N = 21 (mn + m p) ∼ 1 GeV is the nucleon mass.
However, due to the presence of derivative interactions
∼i ←∂→μ W μ in the Lagrangian, there is also a loopinduced
contribution independent of m . It comes from the virtual
exchange of W bosons (which is insensitive to λH ), because
they bring down a p2 ≈ m2 term which precisely cancels
the 1/m2 factor coming from the phase space integral; see
e.g. the diagrams in Fig. 2b–d. Such a cross section has been
computed in the heavy WIMP effective theory (HWET) [
37–
40
]. The leading term in the 1/m expansion (valid therefore
for m m W mq ) reads
σ (η N
→ η N )HWET = 1.3+0.4+0.4
−0.5−0.3 × 10−2 zb.
(4.4)
This value includes contributions from twoloop diagrams
and is universal, in the sense that it only depends on the
SU (2)L quantum numbers of the heavy particle, while
further details of the model (such as the spin of the WIMP or
its possible interaction with the Higgs) enter only through
1/m corrections.
In order to provide a conservative estimate of the
sensitivity of current and projected direct detection experiments
to this model, we show in Fig. 3 the sum of both
contributions to the spinindependent cross section as a function of
the compositeness scale (purple) versus the latest limits from
LUX [
41
] (dashed orange) together with the projected
sensitivities for LZ (dashed green) [
42
] and XENON1T (dashed
red) [
43
]. The latter are properly rescaled by [ h2]DM/ h2,
which takes into account that η could be just a subcomponent
of the whole relic abundance. In order to be more
conservative, we have used the 1σ upper values for both contributions
in σ (η N → η N ). It should be noted that this is only an
estimate of the DM–nucleon cross section, since the validity of
the HWET breaks down for low values of f (and hence low
masses). We also neglect possible interference effects. The
low sensitivity of current experiments ensures that making
more accurate predictions is not needed. Interestingly, the
order of magnitude of the estimated cross section is in the
ballpark of the aimed sensitivity for LZ, making the model
accessible via direct detection in the near future.
4.2 Indirect detection
Indirect DM searches use astrophysical and cosmological
observations to look for the effects of SM particles into which
DM is assumed to decay or annihilate. Concretely, they focus
on the cosmic microwave background (CMB) and the
detection of gamma and cosmic rays originating from decays or
annihilations of DM.
We consider the effects due to the W boson pairs that are
produced in the annihilation of η particles, which is the main
relevant channel in our case. We restrict our attention to three
different kinds of indirect probes, which provide the current
most constraining bounds on the annihilation rate of WIMPs:
the CMB, gamma rays coming from dwarf spheroidal
galaxies and gamma rays from the center of the Milky Way. The
last of these two observables have specific intrinsic
uncertainties due to our limited knowledge about the DM distribution
2
4
6
8
inside galaxy halos; and therefore the CMB leads to a more
robust bound. It is worth stressing that quite generically DM
constraints coming from indirect detection experiments are
less robust than direct detection ones and even less than those
coming from colliders, such as the LHC. Again, the reason is
the required modeling of astrophysical phenomena in
indirect detection experiments. Nevertheless, when taken at face
value these constraints are the most stringent ones on our
model and therefore they deserve to be considered in depth.
However, we warn the reader that the percentages of the DM
relic density that we derive in what follows should be taken
as an indicative approximation.
In Fig. 4 we compare the theoretical prediction for the
annihilation rate σ v of η particles from [
35
],7 as a
function of the scale f , with the current bounds from Planck,
H.E.S.S. and FERMI+MAGIC. The shape of all the curves,
peaking around ∼8.2 TeV is due to the use of Fig. 1 to
rescale the bounds on σ v from the various
collaborations. This is needed to account for the fact that the event
rate of any annihilation process scales as the square of the
local density of annihilating particles, which in the case
7 See Fig. 3 in Ref. [35] for a scalar triplet.
10–22
10–23
Fig. 4 The thermally averaged annihilation rate of ηη into W +W − as
a function of the compositeness scale, f . The black continuous line
is the theoretical prediction of the model. From top to bottom we also
show the upper bounds (rescaled with the square of the DM abundance)
from the following observations: H.E.S.S. dwarf spheroidal galaxies
[Burkert (dashed) and NFW (dotdashed) profiles, in orange], CMB
from Planck (blue, dotted), the combination of FERMI and MAGIC
dwarf spheroidal galaxies (red, dashed), and the Milky Way center as
seen by H.E.S.S. [NFW (dashed) and Einasto (dotdashed) profiles, in
brown]. See the main text for references. The green dashed line is the
expected sensitivity of CTA for observations of the Milky Way center
assuming an Einasto profile [
44
]. The vertical band at f 8.5 TeV
locates the scale that gives the total DM abundance with a 95% C.L.
from the prediction of the model; see also Fig. 1
of WIMPs can be assumed to be approximately
proportional to the relic density h2. The usual indirect
detection upper bounds on the DM annihilation rate assume that
all the DM in the Universe corresponds to a single WIMP
species of a given mass. In order to include the possibility
that η explains only a fraction of the total DM abundance,
the experimental bounds have thus to be multiplied by a
factor ([ h2]DM/[ h2])2 0.012 [ h2]−2. Obviously, this
takes into account the dependence of the abundance h2 of
η on its mass (or, equivalently, on the scale f ), as shown in
Fig. 1.
The weakest indirect detection bounds that we consider
come from the observation by the Cherenkov radiation
telescope H.E.S.S. of dwarf spheroidal galaxies, which are
strongly DM dominated systems and supposed to be free
from other gamma ray emission. These bounds correspond
to the two upper lines of Fig. 4; see [
45
]. The distance
between them comes from their different assumptions for
the radial distribution of DM in those galaxies. The upper
curve assumes a Burkert profile [
46
], which features a
constant inner density core, whereas the lower one is for an NFW
profile [
47, 48
], which peaks at the center. A more stringent
limit from dwarf spheroidals is reported by the
collaborations of the FERMI satellite and the Cherenkov telescope
MAGIC. The combination of their respective data leads to the
red dashed curve [49], assuming an NFW profile. Although
these data are more constraining, a direct comparison to the
results of H.E.S.S. is not straightforward, since the details of
the assumed profiles are different.
Notwithstanding the importance of dwarf spheroidal
galaxies for indirect DM detection, the center of the Milky
Way is thought to be the strongest gamma ray emitter and
therefore an important candidate for a potential indirect DM
detection. Clearly, the choice of DM profile is critical for
the interpretation of these observations, but unfortunately
the DM distribution in the center of our galaxy is
uncertain. Moreover, there is a number of baryonic
astrophysical sources of gamma rays which need to be accounted for
when considering the possible emission from the Galactic
center. These backgrounds are not well known either and
this implies a large source of uncertainty in addition to the
choice of DM profile. The pair of brown lines at the
bottom of Fig. 4 represent the current constrains from the Milky
Way galactic center obtained by H.E.S.S. [
50
] for two
profile choices: NFW (dashed) and Einasto [
51
] (dotdashed).
Although these observations appear to be the most stringent,
it has to be emphasized that they are also the ones whose
interpretation carries a larger uncertainty. It is interesting to
point out that modifying the parameters of the NFW
profile these limits can be weakened slightly above the Einasto
curve; see [
50
].
As we mentioned earlier, the most robust bounds come
from the CMB, and in particular from the Planck satellite.
The blue dashed line of Fig. 4 represents the bound obtained
in the analysis of [
52
]. According to the CMB upper bound on
the annihilation rate, our exceptional DM model can account
for as much as 80% of the DM abundance of the Universe at
95% C.L., as can be read from Figs. 1 and 4. This requires a
value of the composite scale f 7.5 TeV, which corresponds
to a triplet mass of ∼2.2 TeV. If instead we take the strongest
limits from the Galactic center from H.E.S.S. the maximum
percentage of the DM abundance that can be explained by η
is at most 36%, corresponding to f 4.25 TeV and a mass
of ∼1.25 TeV. Given that indirect detection sets the strongest
upper bounds in our model, we can conclude that a significant
amount of the DM of the Universe might be in the form of
the neutral component of our triplet.
Future indirect detection data could in principle test the
model with improved sensitivity in the range of f that is
relevant for DM. The Cherenkov Telescope Array CTA [
53
],
which should start taking data by 2021, may currently be
the best proposal that could contribute to that goal. Several
CTA sensitivity estimates exist in the literature, in particular,
for DM annihilation in the Galactic center into a pair of W
bosons [
44, 54, 55
]. These estimates vary depending on the
assumptions made about the final configuration design of
the telescope array, the observational strategy (including its
timespan) and several other factors. In Fig. 4 we report the
forecast of reference [
44
] for DM annihilation into W bosons,
appropriately rescaled with the DM abundance. According
to [
44
], it appears that once systematics effects are accounted
for, the upper bound that will be reachable with CTA for this
specific channel might not be too dissimilar from the most
stringent current limits obtained by H.E.S.S. [
50
]. However,
the value of the cross section that will be attainable with CTA
for the range of masses that interests us is estimated to be a
factor ∼4.5 lower in [
55
]; but this number accounts only for
statistical errors. It is clear that a proper comparison between
current bounds and different forecasts would require, at the
very least, the use of the same DM profile.
In principle, the model could also be constrained from
searches of monochromatic gamma lines due to the
annihilation of DM into two photons in the central regions of the
Milky Way. To the best of our knowledge, the latest and most
stringent upper bounds on the cross section for this process in
the relevant range of mass have been obtained by the H.E.S.S.
collaboration [
56,57
]. A DM mass of ∼1.2 TeV
approximately corresponds to f ∼ 4 TeV, which is the scale at which
the H.E.S.S. limits on DM annihilation into W +W − intersect
the theoretical prediction; see Fig. 4. The current strongest
bound for ηη → γ γ and DM masses around that value is
σ v ∼ 10−27 cm3/s at 95% C.L., assuming an Einasto
profile. For a scalar triplet with zero hypercharge, this cross
section has been computed (including the Sommerfeld effect) in
[35]. After the adequate rescaling with the DM abundance,
the theoretical prediction is σ v ∼ 5 × 10−28 cm3/s, which
is an order of magnitude lower than the aforementioned upper
bound. Although, once more, the DM profile dependence is
an important source of uncertainty,8 this channel is not more
constraining in our case than ηη → W +W −.
A CTA sensitivity estimate applicable for ηη → γ γ
was produced in [
60
] under the assumption of an NFW
profile. Translating this estimate to the relevant range of f
and after rescaling by the DM abundance, it gives σ v ∼
3 × 10−29 cm3/s, which is an order of magnitude lower
than the theoretical estimate. This means that CTA
observations of monochromatic gamma lines should allow one to
probe the model beyond the current H.E.S.S. bound from
ηη → W +W −. This type of search may in fact be able to
test all the range of f for which the limits on the annihilation
cross section into W bosons still allows one to account for a
significant fraction of the DM relic abundance.
5 Singlet case
The model we have explored so far provides a
hyperchargeless scalar triplet as DM candidate. This is a consequence of
weakly gauging one particular SU (2) of the two global ones
respected by the strong sector, which makes the fundamental
representation of S O(7) decompose as 7 = 21/2 ⊕ 30 under
8 See e.g. Refs. [
58,59
].
the EW subgroup SU (2)L . However, as was done in [12],
one can also weakly gauge the other SU (2) within S O(4) ∼=
SU (2)L × SU (2)R , under which 7 = 21/2 ⊕10 ⊕1±1,
obtaining an isospin singlet as potential DM candidate. We follow
this path in this section, highlighting the specific differences
between the two cases.
Gauge contribution to the scalar potential Contrary to the
triplet case, the potential of the new charged (and
hypercharged) scalars receives contributions proportional to g .
Equation 2.16 has to be modified by
This term modifies the mass splitting between κ± and η,
with respect to the case of the triplet. It gives a contribution
(mκ± − mη)/mη ∼ g 2/(Nc yt2) ∼ 0.05.
Fermion contribution to the scalar potential In the
singlet case, the charged and neutral scalars do not exchange
gauge bosons, and hence the first cannot decay into the
second. In order to avoid an overabundance of stable charged
particles, for which stringent constraints exist [
61–63
] new
sources of sizable explicit symmetry breaking have to be
considered. Being the second heaviest fermion, we assume that
this effect is driven by the bottom quark. There are many
different possible embeddings of the righthanded bottom
quark, bR . However, not all of them respect the Z2 symmetry
η ↔ −η, which makes the singlet scalar stable or generate
a bottom Yukawa coupling at leading order in λq λbR , with
λbR = λ3d3. We consider the case where bR mixes with the
72/3 within the 212/3, since it fulfills both conditions. Then
Eq. (2.21) still holds, but it has to be supplemented by the
(subleading) bottom contribution to the scalar potential:
Nc
Vbottom( ) = (4π )2 m4∗cˆ1
× 2H 2 + η2 + κ+κ− sin2
λbR  2 1
g∗
ˆ 2
ˆ
f
.
.
(5.1)
(5.2)
This term of the potential also contributes to breaking
the mass degeneracy between κ± and η, giving (mκ± −
mη)/mη ∼ (g∗ yb)2/yt4 ∼ 6 × 10−4 g∗ 10−2. In
addition, the following Yukawa couplings are generated:
7
qα
¯L
α †
q D i8
α i=1
cb λq∗ λbR f
= 2√6 ˆ
g∗
b i8bR + h.c. ∼ Lyuk,b
ˆ
f
sin
q¯L H cos
ˆ
f
2 0.08
h
2.5
f (TeV)
Fig. 5 The dependence of the relic abundance h2 in the singlet case
as a function of the compositeness scale f . The horizontal lines show the
measured central value and a 95% C.L. interval around it as determined
by Planck [
34
]
− i H˜ √
3 κ +
2 ˆ
sin
ˆ
f
3 κ +
bR + h.c.
= −ybq¯L
H − i H˜ √
2 f
. . . bR + h.c.,
(5.3)
where cb is a dimensionless order one parameter and we
have traded cb λq∗ λbR by −yb in the second expression. This
2√6 g
provides a vertex i √∗32 mfb t¯L bR κ +, which makes κ ± decay into
t b.
Collider implications Searches for disappearing tracks do
not constrain the singlet case. So, measurements of the Higgs
couplings dominate the reach of current and future facilities.
On another hand, analyses of invisible Higgs decays [
64, 65
]
forbid only f 300 GeV. Monojet searches are further
suppressed by the small coupling of the Higgs boson to η.
This can also be produced in gluon fusion via loops of top
quarks, but its coupling to the latter is suppressed with respect
to the top Yukawa by order ξ . The charged scalar can be
instead produced via gauge interactions. However, the small
rate together with the unclean final state containing tops and
bottoms, make its discovery challenging at the LHC. Future
facilities could probe this channel, though.
Relic density Given the small splitting between the masses
of the charged and the neutral components (which is driven
by the small gauge induced potential), the DM particles are
not expected to annihilate into κ +κ − final states. As a
consequence, the main annihilation channels are t t as well as
W + W −, Z Z and hh. The first channel dominates for small
f 1.7 TeV; see Fig. 7, right panel. The main reason is that
the annihilation into tops proceeds also via contact
interactions (analogous to the ones coming from Eq. (2.26)),
suppressed by 1/ f 2. In the unitary gauge, other DM interactions,
instead, are driven by the Higgs portal. This receives
contributions from both the scalar quartic coupling in the potential
100
LUX 2016
XENON1T (2t·y, projected)
LZ (goal, projected)
tree
1.0
1.5
3.0
3.5
2.0
Fig. 6 Spinindependent direct detection cross section as a function of
the compositeness scale, f , in the singlet DM case. We show the
theoretical estimate as a purple continuous line. We also show the current
limits (linearly rescaled with the DM abundance) from LUX (dashed
orange) [
41
] and the projected exclusion limits at 95% C.L. for LZ
(dashed green) [
42
] and XENON1T of 2 years in 1 ton (dashed red)
[
43
]
λH and from derivative operators like  H 2(∂ μη)2,
appearing in the sigmamodel Lagrangian. The ratio between these
two is given by (see Table 1)
1
2 λH
and therefore the derivative interactions dominate. The main
annihilation channel for large values of f is ηη → W + W −,
as shown in Fig. 7.9
Nonperturbative effects, like the Sommerfeld
enhancement of the formation of bound states are not relevant. For
each value of f we have computed the relic density by just
using micrOMEGAs v3 [
69
]. The result is shown in Fig. 5,
alongside the current observational band (as in Fig. 1). It
turns out that the whole relic abundance can be explained by
this model with f ∼ 3 TeV, for which mη ∼ 900 GeV. As we
will see, current direct and indirect searches do not exclude
this possibility. However, future experiments will have the
required sensitivity to test this prediction.
Direct searches Contrary to the triplet case, the DM–nucleon
interaction proceeds only via the Higgs exchange. As can be
seen in Fig. 6, current searches are not constraining enough
for this model, but future experiments will definitely probe
the whole parameter space.
Indirect searches The total thermally averaged cross section
for DM annihilation as a function of f is shown as a black
continuous line in the left panel of Fig. 7. As we already
mentioned, DM particles annihilate mostly into W + W − for
suf9 For an exhaustive discussion of the effects of higherdimensional
operators in related models see for example Refs. [
66–68
].
2. × 10–25
)
3/s1. × 10–25
m
c
(
>5. × 10–26
v
<
2.5
f (TeV)
Fig. 7 Left panel the theoretical prediction for the total thermally
averaged annihilation rate of DM particles as a function of the compositeness
scale, f , for the singlet DM case (black continuous line). We also show
the current upper bound (rescaled with the square of the DM
abundance) from observations by H.E.S.S. of the Galactic center (assuming
an Einasto profile, brown dotdashed line) [
50
] and the expected
sensitivity of CTA (also with an Einasto profile, green dashed) [
44
], for
ficiently large values of the compositeness scale; see Fig. 7,
right panel. For this reason, we also show in the left panel the
current upper constraints on ηη → W + W − from
observations of the Galactic center by H.E.S.S. (brown dotdashed
line) [
50
]. This bound assumes that DM particles annihilate
exclusively into W + W − (and, as we already discussed, it is
the most stringent one for this kind of process). We show as
well an estimate of the future sensitivity of CTA for the same
process (green dashed line) [
44
]. The remarks we made in
the triplet case concerning this estimate and its comparison
to the results of [
50
] also apply now. Clearly, the prediction
of the singlet model for the total cross section appears to be
well below the current bound and the future sensitivity for
the dominant channel. We conclude that the singlet variant
of exceptional composite DM is viable for all the interesting
values of f . In particular, for f ∼ 3.25 TeV, all the DM
abundance in the Universe can be accounted for. A
substantial improvement in the sensitivity of the next generation of
indirect probes will be needed to test this result.
6 Conclusions
The amount of evidence for the existence of DM, which
comes from astrophysics and cosmology, is overwhelming.
Today, the nature and origin of DM are regarded as one of
the biggest problems of contemporary physics. At the same
time, a large theoretical effort has been made directed towards
solving the gauge hierarchy problem. Therefore, the
possibility of establishing a link between the two is a tantalizing
idea. This is further supported by the socalled WIMP
miracle. As is well known, a WIMP of roughly TeV mass scale
can help to explain the inferred DM abundance through the
simple freezeout mechanism.
0.0
hh
tt
ZZ
1.5
WW
2.0
2.5
f (TeV)
3.0
3.5
ηη → W +W −. Right panel annihilation fraction of the main channels
for twobody final states; t t¯ (blue continuous line), W +W − (red dotted),
hh (green dotdashed) and Z Z (black dashed). Other (subdominant)
channels are not shown. As in previous plots, the vertical gray lines
indicate the range of values of f corresponding to the observed DM
abundance at 95% C.L
In spite of the current lack of definite new physics signals
at energies of the order of a few TeV, the aforementioned
ideas are still widely acknowledged to be excellent reasons to
expect a discovery at the LHC in the coming years. Moreover,
ongoing direct and indirect detection experiments also show
promising windows for the detection of DM particles at the
TeV scale, and the sensitivity of these techniques will keep
increasing in the near future.
We have worked out a nonminimal CHM containing
a Higgs doublet and three additional scalars: two
electrically charged and a neutral one. Depending on how the
SM gauge interactions break the global symmetry, they can
either transform as a whole SU (2)L triplet or as three
singlets. Contrary to the minimal CHM, this setup can explain a
large fraction of the observed DM relic abundance.
Moreover, significant improvements with respect to the
corresponding elementary extensions of the scalar sector are
also present. Indeed, if the global symmetries are broken
mainly in the fermion sector, our setup depends on a
single parameter ( f ) and the external Z2 symmetry
stabilizing the DM candidate is predicted to be exact also after
EW symmetry breaking. Were this not the case, the
potential would receive sizable contributions from the gauge
sector. These would affect this phenomenological study in
several ways: the relevant observables would not only depend
on f , but also on c˜1gρ . In order for the neutral scalar not
to break the Z2 symmetry by taking a VEV, the condition
c˜1gρ2 −2π 2λH /g2 ∼ −7 should hold. In any case, for
c˜1 ∼ gρ ∼ 1, the bounds on f would be modified only by a
small amount.
Assuming that the breaking of S O (7) is driven mainly by
the fermion sector, the fraction of the DM abundance that
can be accounted for in this framework depends on how the
three additional scalars are arranged. In the case in which
they form a triplet, the scale f is constrained to be below
∼4.25 TeV by H.E.S.S. observations of gamma rays from
the Galactic center. These would imply that at most ∼36–
46% of the DM abundance can be explained with this model,
depending on the shape of the DM radial distribution in the
Galactic center. This bound is relatively uncertain, precisely
due to our lack of detailed knowledge about the DM profile
in the innermost regions of the Galaxy and the modeling
of the gamma ray background in that region. Conversely,
CMB limits on the DM annihilation cross section are less
stringent (though more robust) and allow one to account for
∼80% of the DM abundance with the triplet model. Since
the relic density grows (approximately quadratically) with
the mass of the DM particle, and the tuning of the EW scale
needed to reproduce the correct Higgs mass grows also like
f 2, there is a linear dependence between tuning and relic
density. Therefore, it is clear that the values of f that would be
needed to account for the totality of the DM could be regarded
as less natural than those suggested by the current indirect
detection constrains. We would like to stress that the mild
tuning corresponding to values f giving m h2 ∼ 0.12 is
still acceptable since the Higgs mass is stable under radiative
corrections by construction.
We stress that these results assume a standard thermal
history of the Universe. A different thermal history, which in
principle is compatible with the model, could help to allow
one to account for a higher percentage of the DM relic
abundance in the triplet case, and would change the upper bound
on f .
In the case in which the three additional scalars are
arranged as three singlets, the neutral one can currently
explain the totality of the DM relic abundance. This is
simply because in this case the theoretical prediction for the DM
annihilation cross section is well below all the current
indirect detection upper bounds. In this case, the tuning required
to explain the totality of the DM is only increased by a mild
factor ∼3–10 with respect to the expectation from
naturalness arguments (see for instance Ref. [
17
]).
Future observations of the Galactic center from the
Cherenkov telescope CTA are expected to improve the
sensitivity on the cross section for several DM annihilations
channels. However, for DM annihilating into W + W − – which is
the common channel of interest for both of our scenarios –
the analysis of [
44
] indicates that the sensitivity that will be
achieved with CTA is not expected to increase significantly
beyond the current H.E.S.S. upper bounds in the range of
possible DM masses that are relevant for us.
However, a forecast of the CTA sensitivity to
monochromatic gamma ray lines (produced by DM annihilation into
two photons) [
60
] indicates that testing most of the relevant
range of f for DM in the triplet case should be possible with
this channel. Instead, this kind of search is not that useful in
the singlet case, since the cross section is much smaller than
any current or expected future bound.
Searches for disappearing tracks performed at the LHC
require f to be larger than 650 GeV in the triplet case, while
Higgs measurements rise this bound up to f ∼ 800 GeV
in either scenario. Future facilities could improve this bound
by almost a factor of 2. Likewise, current direct searches are
not constraining, while future experiments would be able to
probe all allowed values of f .
Clearly, the different searches are rather complementary.
Also, we have set a robust upper limit on the compositeness
scale. In generic CHMs, the latter can be obtained only if (less
definite) fine tuning arguments are advocated. Note also that
this bound translates into an upper limit on the mass, M , of
the fermionic resonances (roughly speaking, M f ).
Consequently, a comment on the implications of our findings for
the phenomenology of heavy vectorlike fermions is
necessary. In particular, let us focus on toplike resonances, for
these are the ones whose interaction with the SM sector is
stronger. These states can be produced in pairs in proton–
proton collisions. The production cross section is mainly
driven by QCD interactions, and hence model independent.
Experimental limits on the mass of these resonances rely only
on their branching ratio into the different lighter particles.
Searches performed in the LHC Run I constrain their masses
to be smaller than ∼900 GeV (see for example Ref. [
70
]).
More recent analyses [
71
] have pushed this limit just above
the TeV. The reach of current analyses is still far from the
largest mass allowed by DM experiments. In this respect,
our model – and also generic nonminimal CHMs with
EWcharged DM candidates – , favors a hadronic highenergy
collider as physics case for a future facility. On top of that,
all current studies consider that the new fermions decay only
into SM particles, not into other light scalars expected in
nonminimal CHMs. So, if these setups are to be considered
seriously, and they should, new dedicated searches need to
be developed straight away (see Refs. [
72–77
] for work in
this direction).
Acknowledgements We would like to thank Marco Cirelli, Richard
Ruiz, Javi Serra and Marco Taoso for useful discussions. We also thank
Javi Serra and Marco Taoso for comments and suggestions on a draft
version of this paper. The work of G.B. and A.C. is funded by the
European Union’s Horizon 2020 research and innovation programme under
the Marie SkłodowskaCurie grant agreements number 656794 (DEFT)
and 659239 (NP4theLHC14), respectively. The work of MC is partially
supported by the Spanish MINECO under Grant FPA2014 54459P and
by the Severo Ochoa Excellence Program under Grant SEV20140398.
G.B. thanks the CERN Theoretical Physics Department for hospitality
while part of this work was developed.
Open Access This article is distributed under the terms of the Creative
Commons Attribution 4.0 International License (http://creativecomm
ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit
to the original author(s) and the source, provide a link to the Creative
i
i
i
{F1, F2, F3} and √3 {M1, M2, M3} span two separate copies
of SU (2). In this particular basis, the vacuum (i.e. the vector
that it is annihilated only by the generators of G2) adopts the
form 0 = (0, 0, 0, 0, 0, 0, 0, f )T . The Z2even spurion qα
for the triplet case is given by
Commons license, and indicate if changes were made.
Funded by SCOAP3.
Appendix A: Representation theory of S O(7) and G2
Let us define the following 8 × 8 matrices [
12,78
]:
γ1 = i σ2 ⊗ i σ2 ⊗ i σ2, γ2 = σ1 ⊗ i σ2 ⊗ 1,
In this paper we consider instead an equivalent
representation obtained by rotating Jmn (i.e. Jmn → S† Jmn S) with the
following S matrix:
The Lie algebra of G2 ⊂ S O(7) and the coset space are
expanded, respectively, by the 14 generators, Fi , Mi , and the
7 generators, Ni [
12,79
]:
i
F1 = − 2 ( J24 − J51),
i
N1 = √ ( J24 + J51 + J73),
6
i
F2 = + 2 ( J54 − J12),
i
N2 = √ ( J54 + J12 + J67),
6
i
F3 = − 2 ( J14 − J25),
i
N3 = √ ( J14 + J25 + J36),
6
i
F4 = − 2 ( J16 − J43),
i
N4 = √ ( J16 + J43 + J72),
6
M3 = + √12
( J14 + J25 − 2 J36),
M4 = + √12
( J16 + J43 − 2 J72),
(A.1)
(A.3)
M2 = − √12
( J54 + J12 − 2 J67),
In the singlet case, q1 is changed by q1∗ whereas q2 remains
unchanged. In this case, the spurion for b reads
(A.4)
(A.5)
(A.6)
To recognize which combination of pNGBs spans the 21/2 or
the 30 of the EW group it is useful to remember that, if the
broken generators X a transform as
exp(−αi Yi ) X a exp(α j Y j ) = Rab X a
under an element h = exp(αi Y i ) of the unbroken group G2,
the pNGBs accompanying them inside U = exp(i a N a / f )
transform with the transposed matrix, i.e.,
For simplicity, let us focus first on the triplet case. If we define
N 0 ≡ −N 3, and
a
−N −
3[Mi , N ] = −t i N ,
and use their commutations relations, we get
√
[F 3, T ] = −0N ,
where t i , i = 1, 2, 3, are the threedimensional SU (2)
representation given in (2.9). Therefore,
e−iα j √3M j N eiαk √3Mk = N
√
− i α j 3[M j , N ] + · · ·
= (1 + i α j t j )N
+ · · · = eiα j t j N
and
∗ →
where we have defined
eiα j t j T
∗ ⇒
→ e−iα j t j ,
= ⎝
⎛ κ+ ⎞
−η ⎠ , and κ± =
−κ−
κ1√±2κ2 .
1
NH = √2
N 3 − i N 4
N 6 + i N 7
and use the commutation relations, we get
√
3[Mi , NH ] = − 21 σ i NH ,
which implies that
1
H = √2
h1 − i h2
h3 + i h4
This means that transforms properly as a hyperchargeless
SU (2) triplet. Analogously, if we define
1
[F 3, NH ] = − 2 NH , (B.9)
(B.8)
V ( ) ≈ m2 f 2
∗
3
j=1
+
with
λ  j j
g∗
Appendix B: SU (2)L × U (1)Y quantum numbers of the
pNGBs
transforms as an SU (2) doublet with Y = 1/2 hypercharge.
In the singlet case, this combination can be taken
(B.10)
α
( qαD)8
∼ V1( ) = H 2 sin2
ˆ 2
ˆ
f
,
(C.2)
(B.1)
(B.2)
(B.3)
(B.4)
(B.5)
(B.6)
(B.7)
An interesting possibility that has been explored recently is
that leptons could play a role in EWSB when they
transform in nonminimal irreps of the global group, see e.g.
Refs. [
80–82
]. (By nonminimal irreps we mean that they
can provide more than one independent invariant under the
unbroken group at leading order in the spurion expansion,
like e.g. the 14 in S O(5)/S O(4) or the 35 in S O(7)/G2.)
The rationale is that, when the quark sector transforms in
smaller representations of the Goldstone symmetry (like the
spinorial, the fundamental, the adjoint, ...), even a
moderate degree of compositeness in one of the lepton chiralities
can have a sizable impact in the Higgs potential. This is due
to the fact that the leading lepton contribution to the Higgs
quartic coupling scales in this case with λ 2/g2, whereas
∗
the top one goes with λq 4/g∗4, λq 2λt 2/g4 or λt 4/g∗4.
∗
Therefore, a relatively smaller value of λ /g∗ arising from
the charged lepton sector can provide a comparable effect to
the one coming from the top quark.
Moreover, the fact that all different lepton generations
could be partially composite, could enhance the lepton
contribution by a factor Ngen ∼ 3, compensating the color factor
Nc = 3 present in the top case. Indeed, the recent hints of
violation of lepton flavor universality observed by LHCb and
CMS in RK and R∗K [
83,84
] seem to provide a further
motivation to these scenarios, as discussed e.g. in [81].
In what follows, we will briefly discuss how a similar setup
works in the case of S O(7)/G2 and its impact on DM. We
assume that Oqj and Ouj transform in the 82/3 and the 12/3 of
S O(7) × U (1)X , respectively, whereas the composite
operators mixing with the lefthanded lepton doublets and the
righthanded charged singlets, OLj and O j , transform
respectively in the 1−1 and 35−1 of the same group. Then the scalar
potential can be written as
where we have defined the dressed spurions
The parameters c1, c2, j and c3, j , with j = 1, 2, 3, running
over the three lepton generations, which appear in the scalar
potential are order one dimensionless numbers. Note,
however, that c2, j and c3, j always enter in the same linear
combination. So, effectively, we are left with only three
independent unknowns (the coefficients of V1( ), V2( ) and V3( ))
which can be traded at the renormalizable level by the Higgs
VEV v, the Higgs quartic λH and the mass parameter of the
scalar triplet μ2 ; see Table 2 and Eq. (2.24). The mass of the
triplet in the EW phase is given by
m2 = μ
2
+ λH v
2
= μ
2
v2
1 − 3 f 2 + O
v4
10 We are thinking of the triplet case. In the singlet case one has to
change q1 by q1∗, with the rest of the spurions remaining the same.
Since μ2 ∼ f 2 λH v2, the triplet does not take a VEV
provided the underlying UV dynamics allows for a positive
μ2 (the same holds for the singlet if we weakly gauge the
other SU (2) as discussed in Sect. 5, since the main
contribution to the potential is still given by Eq. (2.24)). The main
difference with respect to the scenarios explored before is
that the relationship between μ and f is in principle not
known. However, the same phenomenological study could
be done having as an extra variable the ratio μ / f , which
we leave for future work.
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