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An Exceptional Dark Matter from Cayley–Dickson Algebras

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Abstract

In this article I propose a new criterion to individuate the origin and the properties of the dark matter particle sector. The emerging candidates come from a straightforward algebraic conjecture: the symmetries of physical microscopic forces originate from the automorphism groups of main Cayley–Dickson algebras, from complex numbers to octonions and sedenions. This correspondence leads to a natural enlargement of the Standard Model color sector, from a SU(3) gauge group to an exceptional Higgs-broken G(2) group, following the octonionic automorphism relation guideline. In this picture, dark matter is a relic heavy G(2)-gluons ensemble, separated from the particle dynamics of the Standard Model due to the high mass scale of its constituents.
An Exceptional Dark Matter from Cayley–Dickson
Algebras
Nicolo' Masi ( masi@bo.infn.it )
University of Bologna
Research Article
Keywords: Dickson algebras, Dark matter (DM), WIMPS, 100GeV, Standard Model (SM), WIMP
DOI: https://doi.org/10.21203/rs.3.rs-609636/v1
License: This work is licensed under a Creative Commons Attribution 4.0 International License. 
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An Exceptional Dark Matter from Cayley–Dickson Algebras
Nicol`
o Masi
INFN & Alma Mater Studiorum - Bologna University,
Via Irnerio 46, 40126 Bologna, Italy
masin@bo.infn.it
Abstract
In this article I propose a new criterion to individuate the origin and the properties of the
dark matter particle sector. The emerging candidates come from a straightforward algebraic
conjecture: the symmetries of physical microscopic forces originate from the automorphism
groups of main Cayley–Dickson algebras, from complex numbers to octonions and sedenions.
This correspondence leads to a natural enlargement of the Standard Model color sector, from
aSU (3) gauge group to an exceptional Higgs-broken G(2) group, following the octonionic
automorphism relation guideline. In this picture, dark matter is a relic heavy G(2)-gluons
ensemble, separated from the particle dynamics of the Standard Model due to the high mass
scale of its constituents.
1 Introduction
Dark matter (DM) is a very long-standing problem of modern physics with no evident nor univo-
cal solution: all the efforts made, from particle theory [1; 2] to modified gravities [3; 4; 5], have
not been successful in clarifying its nature. The most convincing particle candidates, the weakly
interacting massive particles (WIMPS), have not been discovered yet: direct, indirect and collider
searches show no evidence of new particles approximately up to the 1 TeV scale [6; 2; 7; 8; 9].
This is a strong hint that the Naturalness criterion [10] for the Higgs sector and the so-called WIMP
Miracle [11] in particular, which postulate the existence of a thermal particle relic of the Big Bang
at the electroweak scale O(100GeV )which interacts via weak force, could not be a prerogative of
Nature or, at any rate, not sufficient to individuate the origin of dark matter. Even the possibility
that the weak interaction between DM and Standard Model (SM) particles is disfavored must be
considered: so a DM candidate could hide at a higher energy scale and it could be not capable
of interacting with the visible world, at least at the experimentally explored energies. Therefore,
to proceed in the dark sector investigation, one has to fill up the lack of a theoretical guideline
and integrate some new simplicity criteria to select reliable candidates and explain the complex
astrophysical and cosmological observations [12; 13; 14]. Today physics seems to need some
extra inputs to go beyond current paradigms and reach a deepest understanding of the dark matter
conundrum: in this complex situation mathematics could provide fresh insights and conjectures to
overcome physical prejudices.
Here we propose an approach based on a division algebras conjecture capable of selecting a unique
branch of heavy dark matter particles from a simple and minimally high symmetry. The criterion
1
is to identify fundamental interactions with the automorphism groups1of Cayley-Dickson alge-
bras. Then, from the automorphism of octonions (and sedenions) algebra, the promising excep-
tional symmetry group G(2) can be pinpointed to solve the DM problem. We will demonstrate
that, once broken through a Higgs-like mechanism, G(2) represents the optimal gauge group to
describe strong interaction and dark matter at the same time, shedding light on DM origin and
present behavior. To the best of our knowledge, no existing work in literature is devoted to the
possibility that dark matter is formed by the heavy gluons from a broken-G(2) gauge group, which
naturally incorporates the standard SU (3) color Quantum Chromodynamics (QCD): even if G(2)
lattice models have been largely applied to simplify standard QCD computations [15; 16], the
implications for dark matter theory have not been explored. Hence, the present dissertation is
not intended as a mere review of the current status of Cayley–Dickson algebras applied to particle
physics, but as a phenomenological proposal to incorporate DM in the Standard Model framework.
2 Fundamental forces from division algebras automorphisms: a con-
jecture for dark matter
In the last decades many attempts to connect the Standard Model of elementary particles with di-
vision algebras have been made, showing it is worthwhile establishing relations between algebraic
structures and symmetry groups [17; 18; 19; 20; 21; 22; 23; 24; 25].
It is well-known that following the Cayley–Dickson construction process [19; 17], one can build
up a sequence of larger and larger algebras, adding new imaginary units. In detail, from Hurwitz
and Zorn theorem [18], one can identify the so-called division algebras R,C,H,O,i.e. the only
four alternative algebraic fields with no non-trivial zero divisors [26; 19], which are real numbers,
complex numbers, quaternions and octonions, respectively. During the construction process, the
algebras lose some peculiar properties, one at a time. For example, complex numbers are not or-
dered but commutative, quaternions are not commutative but associative, whereas octonions lose
all the familiar commutative and associative properties, but they are still an alternative algebra
[26]. The process does not terminate with octonions: applying the Cayley–Dickson construction,
greater 2n-dimensional algebras can be constructed, for any positive integer n. For n > 3, how-
ever, as anticipated, they all include non-trivial zero divisors2,i.e. they have problems in a general
definition of norm. This was considered an obstacle for the use of these extended algebras, such
as n= 4 sedenions, in science. But, as shown in [28; 29], sedenions should not be ruled out as
playing a role in particle physics on the basis that they do not constitute a division algebra. We
will return to this topic later.
The link between unitary groups and division algebras Anhas been diffusely studied [30; 31;
32]. Unitary groups are the fundamental bricks to build the particle Standard Model, because
each fundamental force can be described by a unitary or special unitary group [33; 34; 35; 36;
37], being G=SU(3) ×SU(2) ×U(1) the SM group of strong SU (3), weak SU(2) and
electromagnetic U(1) interactions [33]. Besides its symmetry, the SM includes three fermions
families: between these three generations, particles differ by their flavour quantum number and
1An automorphism is a bijective way of mapping a mathematical object to itself preserving its structure: the set of
all automorphisms form the automorphism group, i.e. the symmetry group of the object.
2In abstract algebra, a non-zero element aof a ring Ris called a zero divisor if there exists a non-zero xsuch that
ax = 0. For general properties of zero divisors see [27].
2
mass, but their interactions are identical.
In the following, we want to briefly highlight the relations between the automorphisms of Cayley-
Dickson algebras and these important physical gauge groups, including some considerations about
the tripartite structure of the Standard Model.
Starting from the most simple complex algebra and SM symmetry group, it is easy to find a
direct connection between the electromagnetism (or Quantum Electrodynamics) U(1) formalism
and the complex number field C: in fact the group U(1), the smallest compact real Lie group,
corresponds to the circle group S1, consisting of all complex numbers with absolute value 1 under
multiplication, which is isomorphic to the SO(2) group of rotation [38]. All the unitary groups
contain copies of this fundamental group. For n1, one can also consider for the comparison
the n-torus Tn, that is defined to be Rn/Zn
=U(n)
=SO(2)n
=(S1)n, where /denotes the
quotient group between reals and integers, which shows off the deep connection between U(1)
gauge symmetry and other representations strictly connected to complex numbers [38; 39]. It is
also true that the n×ncomplex matrices which leave the scalar product h,iinvariant form the
group U(n) = Aut(Cn,h,i),i.e. the group of automorphisms of Cnas a Hilbert space [40].
These links are not surprising because, from a mathematical point of view, the existence of infinite
distinct wild automorphisms of the complex numbers, beyond identity and complex conjugation,
is well-known [20; 41]. We find another noteworthy examination in [39], where the unitary group
U(1) is showed as defining binary complex relations C×C,i.e. the U(1) numbers effectively
operate as automorphisms of Cvia multiplication of a phase factor. As we know, the complex
numbers can be expressed in polar coordinates and this implies that the general linear multiplica-
tive group C=C\0 = eC
=GL(1,C)is uniquely decomposable (ez=ex·eiy) into the totally
ordered group with real exponential |C|=eRand into the phase group with imaginary expo-
nentials C/|C|=eiR
=U(1), which is approximately U(1) (see [39; 42] for details). This
is another way to underline the intimate connection between the unitary group and the complex
numbers.3
Even SU (2) weak isospin can be clearly represented with the algebraic quaternionic basis, i.e.
Pauli matrices [22]: SU (2) naturally embeds into Has the group of quaternion elements of norm
1, with a perfect analogy with respect to U(1) and complex numbers. More precisely, the group
SU (2) is isomorphic to the group of quaternions of norm 1, and it is thus diffeomorphic4to the
3-sphere S3. Indeed, since unit quaternions can be used to represent rotations in 3-dimensional
space (up to a sign), there is a surjective homomorphism5from SU (2) to the rotation group SO(3)
[22]: one can show that the local SU(2) spinors6are exactly the same two-component spinors
derived from the local quaternion matrix representation, i.e. the four Pauli matrices. In other
words, the correspondence between the automorphism of quaternion algebra and the Standard
Model symmetry group of weak force can be clearly shown: for quaternions Aut(H) = SO(3),
where SO(3) is homomorphic to SU (2) in turn, and the universal cover of SO(3) is the spin
group Spin(3), which is isomorphic to SU (2). So SU(2) and SO(3) algebraic structures are
3Furthermore, from a physical point of view, one can also think at the Riemann-Silberstein field reformulation of
the electromagnetism [43] in terms of a complex vector that combines the electric field E, as the real part, and the
magnetic field B, as the imaginary part, in order to put in evidence this essential relation.
4A diffeomorphism is an isomorphism of smooth manifolds, i.e. a map between manifolds which is differentiable
and has a differentiable inverse.
5An homomorphism is a structure-preserving map between two algebraic structures of the same type.
6Spinors are defined as vectors of a representation of the group of automorphisms of a Clifford algebra defined on
space–time.
3
equivalent. An interesting demonstration of the correspondence between the two groups using
M¨obius transformation is described in [44]. The quaternionic representation of (electro-) weak
isospin has been used by many authors [45; 46].
Hence, both in the U(1) electromagnetic case and in the SU(2) weak interaction, the solutions can
be expressed in terms of division algebras, respectively the complex and the quaternion algebras:
the division property is important to define the mathematical structure and in the determination of
solutions. This could be a coincidence, but the possibility that fundamental gauge interactions can
be described by the apparatus of division algebras should be explored.
It seems logical to revise the next division algebra, the octonion algebra O(which is not a
Clifford algebra, unlike R,Cand H, because non associative) [47; 26] for a possible description
of the SU (3) gauge field [48; 45], but the result is less clear than in quaternion case for the SU (2)
gauge field. The interesting fact to be considered is that the group of automorphisms of the oc-
tonion algebra, the largest of the normed division algebras, corresponds to the exceptional Lie
algebra G(2), the smallest among the known exceptional Lie algebras: Aut(O) = G(2) [49]. So
it is noteworthy to point out that the Standard Model gauge group SU(3) is not isomorphic to the
group of automorphisms of the octonions, which is G(2). Nonetheless, it is possible to fix one of
the octonion basis elements to obtain seven possible subalgebras, each of which has a subgroup of
automorphisms isomorphic to SU (3). For example, SU(3) itself may be defined as the subgroup
of G(2) which leaves the octonionic unit e7invariant [22]. Of course, alternative SU (3) subgroups
of G(2) may be found, corresponding to other imaginary units. In addition, recent works in the
framework of particle physics show the possibility to rewrite Gell-Mann matrices of SU (3) strong
force (the group generators) with octonions [45]. Also split-octonions representations have been
proposed as alternative formalism for SU (3) color gauge symmetry [48].
But here a crucial difference appears: it must be noted that for Cand Hthe direct automorphism
groups contain an equal, or comparable, amount of “mathematical information” than U(1) and
SU (2) themselves (through the approximate algebraic correspondences, via homomorphism in
SU (2) case), whereas the exceptional G(2) group is certainly bigger than SM SU(3), as it in-
cludes SU (3) and is equipped with six additional generators [50]. In other words, if we want to
study the application of the octonion automorphism in physics, it is mandatory to invoke a gauge
group which is not the strong color symmetry SU (3).
Summarizing, for non real division algebras it turns out that:
Aut(C)
=U(1),Aut(H)
=SU (2),Aut(O)G(2).(1)
These relations show an ordered correspondence between (approximate) automorphisms of alge-
bras and gauge groups useful for Standard Model description, where G(2) contains SU (3) color
force. We will see in the next section that, besides SU(2) Pauli matrices, also G(2) generators can
be written in terms of SU (3) Gell-Mann matrices as 14 unitary matrices.
Fundamental correlations between division algebras and symmetry groups, as anticipated, have
been already stressed in the last decades. Using division algebras, Dixon proposed an elegant
representation of particle physics in [19]. Furey has recently suggested the appealing possibility
to reformulate the SM group G=SU (3) ×SU (2) ×U(1) in terms of a A=RCHO
tensor product algebra, restarting from Dixon’s work, using the concept of Ideals,i.e. using sub-
spaces of proper Clifford Algebras as “particles” (see [23; 51; 52; 53; 54] for details). Also
string theory and supersymmetrical theories invoked division algebra to study particle interactions
[55; 56; 57; 31; 58; 59]. Moreover, G(2) as automorphism of Ohas important applications in
terms of the so-called G2structure or G2manifolds [60], in the context of M-theory [61]. Indeed,
4
one solid reason for studying division algebras in relation to particle symmetries is that, unlike Lie
algebras and Clifford algebras [20], there is a finite number of division algebras and corresponding
automorphisms (see again the extensive works of Dixon [19]). If we start with a division algebra,
the physical symmetries are dictated by the mathematical structure and the choice of a proper
symmetry group is constrained.
To proceed with the reasoning, we are going to see why 16-dimensional sedenions can be eas-
ily added to this picture and how dark matter description can benefits from this algebraic facts,
summarizing the main features of sedenions algebra.
The sedenion algebra is the fifth Cayley-Dickson algebra A4=S, where A0,1,2,3correspond
to reals, complex numbers, quaternions and octonions. This is not a division algebra, it is non-
commutative, non-associative, and non-alternative7, hence it cannot be a composition algebra8
[63; 28]. However sedenion algebra is power-associative and flexible9(and satisfies the weak
inversive properties for non-zero elements10 - see [62; 29] for details). In principle, the Cayley-
Dickson construction can be indefinitely carried on and, at each step, a new power-associative and
flexible algebra is produced, doubling in size. So, in first approximation, no new fundamental
properties and information are added nor lost enlarging the algebra beyond sedenions. One can
choose a canonical basis for Sto be E16 ={eiS|i= 0,1, ..., 15}where e0is the real unit and
e1, ..., e15 are anticommuting imaginary units. In this basis, a general element ASis written as
A=
15
X
i=0
aiei=a0+
15
X
i=1
aiei, aiR.(2)
The basis elements satisfy the multiplication rules
e0= 1, e0ei=eie0=ei,
e2
i=e0, i 6= 0,(3)
eiej=γk
ij eki6= 0, i 6=j,
with γk
ij the real structure constants, which are completely antisymmetric. For two sedenions A, B,
one has
AB = 15
X
i=0
aiei! 15
X
i=0
bjei!=
15
X
i,j=0
aibj(eiej) =
15
X
i,j,k=0
fij γk
ij ek,(4)
where fij aibj.
Because the sedenion algebra is not a division algebra, it contains zero divisors: for Sthese
are elements of the form
(ea+eb)(ec+eb) = 0, ea, eb, ec, edS.(5)
7An algebra Ais alternative if the subalgebra generated by any two elements is associative, i.e. iff for all a, b A
we have (aa)b=a(ab),(ba)a=b(aa)[62].
8It is an algebra Aover a field Kwith a non-degenerate quadratic form N, called norm, that satifies N(ab) =
N(a)N(b)for all a, b in A[18].
9An algebra is power-associative if the subalgebra generated by any one element is associative: it is a sort of lowest
level of associativity [26]. The flexible property, for any a, b A, can be defined as a(ba) = (ab)a.
10Each Cayley-Dickson algebra satisfies the weak inversive property: a1(ab) = a(a1b),(ba1)a=
(ba)a1, a1(ab) = (ba)a1.
5
There are 84 such zero divisors in sedenion space and the subspace of zero divisors with unit
norm is homeomorphic to G(2) [64; 27]. To understand the role and emergency of zero divisors,
one has to consider not only single algebras but also compositions of them. For example, whereas
R,C,Hand Oare by themselves division algebras, their tensor products, such as CH,COand
RCHO, largely applied in SM algebraic extensions, are not, and in fact the zero divisors of
these algebras play a crucial role in the construction of Furey’s Ideals [52; 23; 51; 53]. Moreover,
the two by two compositions of division algebras, which are not division algebras and contain zero
divisors, are the subjects of the well-known Freudenthal–Tits magic square [65; 66; 67]:
R C H O
RSO(3) SU (3) Sp(3) F4
CSU (3) SU (3)2SU (6) E6
HSp(3) SU (6) SO(12) E7
OF4E6E7E8
a symmetric square11 which exhibits the “unexpected” relation between octonions products
and exceptional groups (F4,E6,E7,E8) [49], except for the exceptional G(2) which represents
octonions automorphism itself. The exceptional groups on the last line/row are not exactly au-
tomorphisms of the octonions products, because of mathematical problems in the definition of
projective planes, due to the appearance of zero divisors: they are called bioctonions (CO),
quateroctonions (HO) and octooctonions (OO) and find correspondence into Jordan’s algebras
[26]. Exceptional Eiare also largely used in supergravity and string theory [57; 35]. Therefore
it seems reasonable to continue the Cayley-Dickson algebraic construction into the non-division
algebras, such as S.
Interestingly, in [68; 28] the authors put in evidence an important relation between sedenions and
the exceptional group G(2), demonstrated by Brown in [69]:
Aut(S) = Aut(O)×S3.(6)
where we know that Aut(O) = G(2) and S3is the permutation group of degree three. So the in-
ner symmetries of this non-division algebra can be again extracted from the automorphism group
of octonions and, in particular, from a proper product of the exceptional G(2) group with a sym-
metric group. The only difference between octonions and sedenions automorphism groups is a
factor of the permutation group S3(this permutation group can be constructed from the triality12
automorphism of the spin group Spin(8), see [20; 26] for details). Eq.(6) suggests that the fun-
damental symmetries of Sare the same as those of O, even if the factor S3introduces a three
copies scenario, that is exactly what we need in order to describe the observed three generations
of fermions in the Standard Model of particles.
The previous formula can be generalized, for an arbitrary algebra constructed via Cayley-Dickson
process (for n > 3), into [28; 70]
Aut(An)
=Aut(O)×(n3)S3.(7)
11SO(N)and SU(N)are the usual special orthogonal and unitary groups of order N,Sp(3) is the symplectic group
of order three.
12Triality is a trilinear map among three vector spaces, most commonly described as a special symmetry between
vectors and spinors in 8-dimensional euclidean space.
6
This tells us that the underlying symmetry is always G(2), the automorphism group of the oc-
tonions. The higher Cayley-Dickson algebras only add additional trialities,i.e. copies of G(2),
and reasonably no new physics beyond sedenions. Futhermore, sedenion algebra might represent
the archetype of all non-associative and non-division flexible algebras, if n > 3Cayley-Dickson
algebras do not differ from sedenions for what concerns the multilinear identities (or algebraic
properties) content, as suggested in [71].
In this picture, sedenion algebra could constitute the searched simplicity criterion to select the full
symmetry of a three generations Standard Model strong force and include a new particle physics
content, which might represent the unknown dark matter sector. This could be also read as a sort
of a naive indirect proof that fundamental forces should be a small number (only three), because
all algebras beyond octonions point towards the very same exceptional group, adding only copies
(particle generations). Finally, as it will be discussed in the next section, to recover the usual
SU (3) strong force the octonions-sedenions automorphism group must be broken at our energy
scales and new physics extracted: this enlarged algebraic content is going to be associated to dark
matter.
So, without the presumption of a rigorous and definitive mathematical definition of the problem,
we can reformulate and summarize the algebraic phenomenological conjecture for the dark matter
sector in a general way as follows.
The fundamental symmetry of the Standard Model of particle physics with three fermion families
might be the realization of some tensor products between the associative division algebras and the
most comprehensive non-division algebra obtained through the Cayley–Dickson construction, i.e.
the sedenion algebra. The sedenionic description, like the octonionic one, corresponds, via auto-
morphism, to the simplest exceptional group G(2), but tripled. It could provide an explanation to
the N= 3 fermion families of the Standard Model, which lie in the sedenions S3automorphism
factor, as suggested by [28]. This is consistent with the proposal of a S3–invariant extension of the
Standard Model, as discussed in [72; 73; 74; 75].
The gauge groups U(1), SU (2), SU (3), describing the three fundamental forces, find mathemat-
ical correspondence into the division algebras C,H,Orespectively: Table 1 summarizes this cor-
respondence. However, whereas U(1) and SU (2) are approximate isomorphisms of complex and
quaternion algebras automorphisms (see Eq.(1)), the octonion and sedenion automorphism rela-
tions point towards a different group, which is manifestly larger than the usual 8-dimensional
SU (3) color group of the Standard Model, i.e. the 14-dimensional G(2) group; SU (3) and G(2)
differ for 6 dimensions-generators. Therefore
Aut(C)×Aut(H)×Aut(S) = Aut(C)×Aut(H)×Aut(O)×S3=
U(1) ×SU (2) ×G(2) ×S3
(8)
could give the overall unbroken Standard Model symmetry. This is the first main statement of
the present dissertation. Here the automorphism selection is invoked to predict something beyond
current SM, and SU (3) in particular, and it works as a guideline to replace SU(3) color itself with
the smallest exceptional group: fundamental forces must be isomorphic to the automorphisms
groups of the division algebras built up through the Cayley–Dickson construction. Tensor prod-
ucts between the corresponding algebras (see Freudenthal–Tits magic square) could be effective
symmetries but not fundamental forces.
Dark matter constituents come from the aforementioned difference between G(2) and SU (3)
groups and lie in the spectrum gap between them. Following the Cayley–Dickson algebraic auto-
morphism criterion, no more physics is needed nor predicted, except for the six additional degrees
7
Charge (ng) Group Force Algebra Dim Commutative Associative Alternative Normed Flexible
Q(1) U(1) EM C2 Yes Yes Yes Yes Yes
T(3) SU (2) Weak H4 No Yes Yes Yes Yes
C(8) SU (3) Strong Oor S8/16 No No Yes Yes Yes
DC(6) broken-G(2) Dark Strong Oor S8/16 No No No No Yes
Table 1 Schematic correspondence between forces, groups and algebras. In the first column the charge of the physical
interaction is displayed along with the number ngof associated generators (bosons). Q, T , C are usual SM electric
charge, weak isospin and color charge, respectively; here DC stands for “dark-colored”, to indicate the six broken gen-
erators which originate the massive exceptional G(2) bosons which have quark and anti-quark color quantum numbers
(see next section). The second and third columns associate gauge groups and forces, highlighting the link between
G(2) and the 6 new dark-colored particles, separated from visible strong phenomena. G(2) algebraic automorphism
representation is valid for both octonions and sedenions (the only difference is the S3factor). In principle, strong force
and dark sector represent the same interaction but they are disconnected, coming from the broken exceptional sym-
metry. For this reason their algebras are displayed as Oor S. Algebraic dimensions are showed in the fifth column.
As shown in the subsequent columns, each division algebra loses inner properties hierarchically, from commutativity
to alternativity, as the dimensions increase. All algebras are flexible (and power-associative). See [63; 26] for proper
descriptions of the algebraic properties and insights.
of freedom, i.e. boson fields, which represent the discrepancy between G(2) and SU(3) gen-
erators. Hence, the automorphism selection rule extends the color sector and provides a rich
exceptional phenomenology.
The novelty is the definition of a new algebraic criterion to predict dark matter physics, which
substitutes Higgs Naturalness and the Wimp Miracle. In this scenario, the strong force acquires
a more complex structure, which includes the usual color sector and an enlarged strong dark dy-
namics, due to six residual generators of exceptional G(2), which might gain mass via a symmetry
breaking: to recover standard SU(3) color strong force description, the new G(2) color sector
should be broken by a Higgs-like mechanism and separated into two parts, one visible and the
other excluded from the dynamics due to its high mass.
In the next section the emergency of these massive exceptional dark bosons is discussed, starting
from a deep analysis of the exceptional G(2) group.
3 A G(2) gauge theory for dark matter
G(2) can be described as the automorphism group of the octonion algebra or, equivalently, as
the subgroup of the special orthogonal group SO(7) that preserves any chosen particular vec-
tor in its 8-dimensional real spinor representation [47; 76; 20]. The group G(2) is the simplest
among the exceptional Lie groups [30]; it is well known that the compact simple Lie groups are
completely described by the following classes: AN(= SU(N+ 1)), BN(= SO(N+ 1)), CN(=
Sp(N)), DN(= SO(2N)) and exceptional groups G2, F4, E6, E7, E8, with N= 1,2,3, ...(for
DN,N > 2) [77]. Among them, only SU (2), S U(3), S O(4) and symplectic Sp(1) have 3-
dimensional irreducible representations and only one, SU (3), has a complex triplet representation
(this was one of the historical criteria to associate SU (3) to the three color strong force, with quark
states different from antiquarks states [78]). There is only one non-Abelian simple compact Lie
algebra of rank 1, i.e. the one of SO(3) SU (2) = Sp(1), which describes the weak force,
whereas there are four of rank 2, which generate the groups G(2),SO(5) Sp(2),SU (3) and
SO(4) SU (2) SU (2), with 14, 10, 8 and 6 generators, respectively [50].
8
If we want to enlarge the QCD sector to include dark matter, it is straightforward we have to
choose G(2) or SO(5). The group G(2), beside its clear relation with division algebras described
in the previous section, is of particular interest because it has a trivial center, the identity, and it
is its own universal covering group, meanwhile SO(5) has Z2as a center (and SU (3) has Z3);
SO(N)in general are not simply connected and their universal covering groups for n > 2are spin
Spin(N)[79]. It is also well-know in literature that G(2), thanks to its aforementioned peculiar-
ities, can be used to mimic QCD in lattice simulations, avoiding the so-called sign problem [80]
which afflicts SU (3). Proposing to enlarge QCD above the T eV scale and have the SM as a low
energy theory is surely not an unprecedented nor odd idea: for example, modern composite Higgs
theories [81; 82; 83] try to introduce (cosets) gauge groups beyond SU(3), such as SU(6)/SO(6),
SO(7)/SO(6) or SO(5)/SO(4), dealing with multiple Higgs, strong composite states and dark
matter candidates.
Focusing on the present proposal, G(2) can be constructed as a subgroup of SO(7), which has
rank 3 and 21 generators [50; 79]. The 7×7real matrices Uof the group SO(7) have determinant
1, orthogonal relation U U = 1 and fulfill the constraint UabUac =δbc . The G(2) subgroup is
described by the matrices that also satisfy the cubic constraint
Tabc =Tdef UdaUeb Uf c (9)
where Tis an anti-symmetric tensor defining the octonions multiplication rules, whose non-zero
elements are
T127 =T154 =T163 =T235 =T264 =T374 =T576 = 1.(10)
To explicitly construct the matrices in the fundamental representation, one can choose the first
eight generators of G(2) as [50; 79]:
Λa=1
2
λa0 0
0λ
a0
0 0 0
.(11)
where λa(with a {1,2, ..., 8}) are the Gell-Mann generators of SU (3), which indeed is a
subgroup of G(2), with standard normalization Trλaλb=TrΛaΛb= 2δab.Λ3and Λ8are diagonal
and represent the Cartan generators w.r.t. SU (3). The G(2) coset space by its subgroup SU (3)
is a 6-sphere G(2)/SU (3)
=S6
=SO(7)/SO(6) [84], in analogy with the composite Higgs
proposal [82].
The remaining six generators can be found studying the root and weight diagrams of the group
[85; 86; 87], and can be written as:
Λ9=1
6
022e3
202e3
2eT
32eT
30
,Λ10 =1
6
0λ2i2e3
λ20i2e3
i2eT
3i2eT
30
,(12)
Λ11 =1
6
052e2
502e2
2eT
22eT
20
,Λ12 =1
6
0λ5i2e2
λ50i2e2
i2eT
2i2eT
20
,(13)
Λ13 =1
6
072e1
702e1
2eT
12eT
10
,Λ14 =1
6
0λ7i2e1
λ70i2e1
i2eT
1i2eT
10
,(14)
9
where eiare the unit vectors
e1=
1
0
0
, e2=
0
1
0
, e3=
0
0
1
.(15)
In the chosen basis of the generators it is manifest that, under SU(3) subgroup transformations,
the 7-dimensional representation decomposes into [50; 79]
{7}={3} {3} {1}.(16)
Since all G(2) representations are real, the {7}representation is identical to its complex conjugate,
so that G(2) “quarks” and “anti-quarks” are conceptually indistinguishable. This representation
describes a SU (3) quark {3}, a SU(3) anti-quark {3}and a SU (3) singlet {1}. The generators
transform under the 14-dimensional adjoint representation of G(2) [50; 79], which decomposes
into [50; 79; 88]
{14}={8} {3} {3}.(17)
So the G(2) “gluons” ensemble is made of SU (3) gluons {8}plus six additional “gluons” which
have SU (3) quark and anti-quark color quantum numbers. As mentioned before, the center of
G(2) is trivial, containing only the identity, and the universal covering group of G(2) is G(2)
itself. This has important consequences for confinement [79; 88; 89; 90]: we will see that the
color string between G(2) “quarks” is capable of breaking via the creation of dynamical gluons.
As discussed in [50], the product of two fundamental representations
{7} {7}={1} {7} {14} {27},(18)
shows a singlet {1}: as a noteworthy implication, two G(2) “quarks” can form a color-singlet,
or a “diquark”. Moreover, just as for SU (3) color, three G(2) “quarks” can form a color-singlet
“baryon”:
{7} {7} {7}={1} 4{7} 2{14} 3{27} 2{64} {77}.(19)
Due to the fact that “quarks” and “antiquarks” are indistinguishable, it is straightforward to show
for the one flavor Nf= 1 case that the U(1)L=R=U(1)Bbaryon number symmetry of SU(3)
QCD is reduced to a ZZ(2)Bsymmetry [50; 91]: one can only distinguish between states with an
even and odd number of “quark” constituents.
Another useful example is
{7} {14} {14} {14}={1} ... (20)
From this composition it is clear that three G(2) “gluons” are sufficient to screen a G(2) “quark”,
producing a color-singlet hybrid qGGG. It is also true that:
{7} {7} {7} {7}= 4{1} ... (21)
so that the product contains four singlets.
Summarizing: a G(2) gauge theory has colors, anticolors and color-singlet, and 14 generators.
So it is characterized by 14 gluons, 8 of them transforming as ordinary gluons (as an octuplet
of SU (3)), while the other 6 G(2) gauge bosons separates into {3}and {3}, keeping the color
10
quarks/antiquarks quantum numbers, but they are still vector bosons. A general Lagrangian for
G(2) Yang-Mills theory can be written as [33; 50; 79]:
LY M [A] = 1
2Tr[F2
µν ],(22)
with the field strength
Fµν =µAννAµigG[Aµ, Aν],(23)
obtained from the vector potential
Aµ(x) = Aa
µ(x)Λa
2.(24)
with gGa proper coupling constant for all the gauge bosons and Λathe G(2) generators. The
Lagrangian is invariant under non-Abelian gauge transformations A
µ=U(Aµ+µ)U, with
U(x)G(2).G(2) Yang-Mills theory is asymptotically free like all non-Abelian SU(N)gauge
theories and, on the other hand, we expect confinement at low energies [79]. The G(2) confine-
ment is surely peculiar with a different realization with respect to SU (3), where gluons cannot
screen quarks (and screening arises due to dynamical quark-antiquark pair creation). In particular,
as we have already seen in Eq.(20), G(2) admits a new form of exceptional confinement. It has
been showed that G(2) lattice Yang-Mills theory is indeed in the confined phase in the strong
coupling limit [50].
But we know that G(2) is not a proper gauge theory for a real Quantum Chromodynamics theory.
Therefore we must add a Higgs-like field in the fundamental {7}representation in order to break
G(2) down to SU (3). The consequence is simple and fundamental: 6 of the 14 G(2) “gluons”
gain a mass proportional to the vacuum expectation value (vev) wof the Higgs-like field, the other
8SU (3) gluons remaining untouched and massless. The Lagrangian of such a G(2)-Higgs model
can be written as [16; 50; 79; 88]:
LG2H[A, Φ] = LY M [A] + (DµΦ)2V(Φ) (25)
where Φ(x) = 1(x),Φ2(x), ..., Φ7(x)) is the real-valued Higgs-like field, DµΦ = (µ
igGAµ is the covariant derivative and
V(Φ) = λ2w2)2(26)
the quadratic scalar potential, with λ > 0. Because of the {7} {7} {7}={1} ... singlet
state seen before, in the fundamental representation a Higgs cubic term should be considered but,
according to the antisymmetric property of Tabc, such a term disappears. Following the product in
Eq.(21), the four singlets corresponds to w2Φ2,Φ4and two vanish due to antisymmetry, making
the aforementioned potential general and consistent with G(2) symmetry breaking and renormal-
izability. We can choose a simple vev like Φ0=1
2(0,0,0,0,0,0, w)to break G(2) and re-obtain
the familiar unbroken SU (3) symmetry: it is easy to notice from the diagonal and non-diagonal
structure of Eq. (11-14) that
Λ18Φ0= 0 (unbroken generators) (27)
Λ914Φ06= 0 (broken generators) (28)
Plugging this scalar field vev into the square of the Higgs covariant derivative, we get the usual
quadratic term in the gauge fields
g2
GΦ
0
Λa
2
Λb
2Φ0Aa
µ(x)Aµ,b(x) = 1
2MabAa
µ(x)Aµ,b(x)(29)
11
that gives the diagonal mass matrix Mab for the gauge bosons, of which we can use the aforemen-
tioned trace normalization relation, which Gell-Mann matrices and G(2) generators share, to put
the squared masses terms g2
Gw2in evidence. This new scalar Φ, which acquires a typical mass of
MH=2λw (30)
from the expansion of the potential about its minimum [33], should be a different Higgs field w.r.t.
the SM one, with a much higher vev, in order to disjoin massive gluons dynamics from SM one,
and a strong phenomenology. Such a strongly coupled massive field could be ruled out by future
LHC and Future Circular Collider searches [92] (it is enough to think of heavy scalars models
searches, such as the two-Higgs doublet model [93] or the composite Higgs models [94]). In this
picture, as anticipated, following the standard Higgs mechanism to build up the dark candidates,
6 massless Goldstone bosons are eaten and become the longitudinal components of G(2) vector
gluons corresponding to the broken generators, which acquire the eigenvalue mass
MG=gGw(31)
through the Higgs mechanism [33], according to Eg. (29), and exhibit the color quarks/antiquarks
quantum numbers. No additional Yukawa-like terms are needed for the purpose of the present
proposal, so that quarks remain massless at the scale of G(2) symmetry breaking, since the SM
Higgs has not yet acquired its vev. Then, if the sedenions description via automorphisms group is
invoked, the symmetry breaking process could in principle act on three different copies of G(2),
expressed by the permutation factor S3which keeps track of the three fermion families. In other
words, a Higgs sector (the SM one or an additional strong–coupled one for G(2)) of a S3–invariant
extension of the SM could also break the flavour symmetry in order to produce the correct patterns
of different masses and mixing angles for fermions families (see [72; 95] for insights). Addi-
tionally, it has been shown that, in a phenomenologically viable electroweak S3extension of the
SM, S3symmetry should be broken to prevent flavor changing neutral currents [72] and the Higgs
potential becomes more complicated due to the presence of three Higgs fields [74]. For simplic-
ity, we could assume that this hypothetical process, involving S3breaking and Yukawa fermion
masses generation, triggers at the electroweak scale, without interfering with the G(2) Higgs po-
tential.
Using the Higgs mechanism to smoothly interpolate between SU (3) and G(2) Yang-Mills theory,
we can study the deconfinement phase transition. In the SU (3) case this transition is weakly first
order. In fact, in (3 + 1) dimensions only SU (2) Yang-Mills theory manifest a second order phase
transition, whereas, in general, SU (N)Yang-Mills theories with any higher Nseem to have first
order deconfinement phase transitions [96; 97; 98; 99; 100], which are more markedly first order
for increasing N. The peculiarities of the phase transition from lattice G(2)-Higgs to S U (3) have
been extensively studied in [15; 90; 101; 102; 103], confirming that G(2) gauge theory has a
finite-temperature deconfining phase transition mainly of first order and a similar but discernable
behavior with respect to SU (N)[103].
Moving back to the G(2) color string, the breaking of this string between two static G(2)
“quarks” happens due to the production of two triplets of G(2) “gluons” which screen the quarks.
Hence, the string breaking scale is related to the mass of the six G(2) “gluons” popping out of the
vacuum. The resulting quark-gluons bound states (colorless qGGG states) coming from the string
breaking, must be both G(2)-singlets and SU(3)-singlets. When we switch on the interaction with
the Higgs field, six G(2) gluons acquire a mass thanks to the Higgs mechanism. The larger is MG,
12
the greater is the distance where string breaking occurs. When the expectation value of the Higgs-
like field is sent to infinity, so that the 6 massive G(2) “gluons” are completely removed from the
dynamics, also the string breaking scale is infinite. Thus the scenario of the usual SU (3) string
potential reappears. For small w(on the order of ΛQC D), on the other hand, the additional G(2)
“gluons” could be light and participate in the dynamics. As long as wremains finite, as we know
it should be in the SM and in its extensions, the heavy G(2) “gluons” can mediate weak baryon
number violating processes [50] (only in the w limit baryon number is an exact discrete
symmetry of the Lagrangian). Finally, for w= 0 the Higgs mechanism disappears and we come
back to G(2). As stressed before, hereafter only high wvalues (with wmuch greater that the SM
Higgs vev) are considered in order to realize a consistent dark matter scenario.
For what concerns the hadronic spectrum of a hypothetical G(2)-QCD, the physics appears to
be qualitatively similar to SU (3) QCD [104], but richer. This can be easily demonstrated from
the decomposition of representations products, like Eq.(18), (19), (20), (21). In the (massless)
spectrum of the unbroken G(2) phase there are many more states beyond standard mesons and
baryons: one-quark-three-G(2) gluons states (and, in general, the quark confinement for one-
quark-N-G(2) gluons, with N3), diquarks, (qqqq)tetraquarks and (qqqqq)pentaquarks. G(2)
and SU (3) also share glueballs states, for any numbers of G(2) gluons (2 and 3 in the ground
states) and hexaquarks 13. Even collective manifestations of G(2) exceptional matter could be
different: for example, a G(2)-QCD neutron star could display a distinct behavior with respect to
aSU (3) neutron gas star [105].
In addition, and even more fundamental for the present dissertation, we gain an exceptional par-
ticle sector: if we move away the six G(2) gluons from the dynamics according to a TeV-ish (or
beyond TeV) mass scale, these bosons must be secluded and separated from the visible SM sector
in first approximation, without experimentally accessible electroweak interactions, unlike WIMPs,
and extreme energies (and distances) should be mandatory to access the G(2) string breaking. This
could be due to the very high energy scale of the G(2)SU(3) phase transition, occurring at much
greater energies than electroweak breaking scale. This is the realization of a beyond Naturalness
and WIMP Miracle criterion for dark matter search. Indeed, G(2) gluons, as SU (3) ones, are
electrically neutral and immune to interactions with light and weak W, Z bosons at tree level. An-
other advantage of a G(2) broken theory is that no additional families are added to the Standard
Model, unlike SU (N)theories.
Overall, this seems to be a good scenario for a cold14 dark matter (CDM) theory if we find
a stable or long-lived candidate. The six dark gluons can form heavy dark glueballs consti-
tuted by two or three (or multiples) G(2) gluons, according to {14} {14}={1} ... and
{14} {14} {14}={1} ... representations [104], with integer total angular momentum
J= 0,2and J= 1,3for 2-gluons and 3-gluons balls respectively. In principle, dark-colored
broken-G(2) glueballs should not be stable, if these heavy composite states themselves have no
extra symmetries to prevent their decay; since the proposed G(2) theory includes QCD, unlike
13The further explicit decompositions of the products are:
{7} {14} {14} {14}={1} 10{7} 6{14} 15{27} 20{64} 13{77} 13{77} 10{182} 15{189}
9{286} 3{378} 6{448} 3{729} {896} 2{924}
{7} {7} {7} {7}= 4{1} 10{7} 9{14} 12{27} 8{64} 6{77} 2{77} {182} 3{189}
{7}{7}{7}{7}{7}= 10{1}35{7}30{14}45{27}40{64}30{77}⊕ 11{77} 10{182}20{189}
5{286} {378} 4{448}
{7} {7} {7} {7} {7} {7}= 35{1} 120{7} 120{14} 180{27} 176{64} 145{77} 65{77}
65{182} 120{189} 5{273} 40{286} 15{378} 40{448} {714} 9{729} 5{924}
{14} {14}={1} {14} {27} {77} {77}
{14} {14} {14}={1} {7} 5{14} 3{27} 2{64} 4{77} 3{77} {182} 3{189} {273} 2{448}
14Cold means non-relativistic and refers to the standard Lambda-CDM cosmological model.
13
hidden Yang-Mills theories with no direct connections with the SM [106; 107], there exist states
that couple to both the dark-colored glueballs and SU (3) particles (for example the G(2)-breaking
Higgs fields): hence, whether at tree-level or via loops, these heavy glueballs would not be stable.
To avoid this, first of all the new G(2) Higgs should be at least more massive than the lightest
2-gluons glueball, so that MH> MGG, which implies the qualitative constraint 2λ > 2gGfrom
Eq.(30), (31). Secondly, the decays into meson states should be also forbidden. In principle, a
lightest JP C = 0++ state, in analogy with standard QCD, could dominate the glueball spectrum
[108], but this could be unstable, like the lightest meson π0and the other known scalar particle,
i.e. the SM Higgs h0. The possibility of a conserved charge or a peculiar phenomenon which
guarantees stability to the lightest glueball states should not be ruled out a priori, considering that
analytical and topological properties of Yang-Mills theory solutions are still not completely un-
derstood: even the fundamental problem of color confinement has not a definitive answer nor an
analytical proof.
For example, in analogy with the baryon number conservation and the forbidden proton decay into
π0, one can introduce a conserved additive gluon number Γfor the glueball states, which counts
the number of massive G(2) gluons (and “antigluons”), preventing the glueball from decaying
into SM mesons, which are not made of G(2) gluons (one has to keep in mind that these peculiar
bosons do mantain the color quarks/antiquarks quantum number). Indeed also the U(1)Bglobal
symmetry of the Standard Model which prevents the proton decay is an accidental symmetry and
not a fundamental law, that can be broken by quantum effects.
We have the same lack of knowledge for the glueballs interactions with their own environments:
as for residual nuclear force between hadrons in nuclei, the possibility of a residual binding inter-
action between glueballs, which prevents the decay (like neutron decay), must be investigated.
Possibly, one can also postulate a suppression scale 1/fG(∆M)for the couplings with SM which
depends on the relative mass difference between the interacting particles, i.e. the dark glueball and
the quarks: if the masses of the glueballs are too high w.r.t. the QCD scale, their decay might be
highly suppressed.
Another interesting chance is to invoke the JP C = 0++,2++ dark G(2) glueball states as graviton
counterparts in a AdS/CFT correspondence framework, as discussed in [109] for QCD glueballs
(even if this does not exhaust the quest for stability). Moreover, the scalar glueball could be part of
a scalar-tensor gravity approach [110], whereas the tensor glueball could play the role of a massive
graviton-like particle, for example in the context of bimetric gravity theory [111; 112], where the
massive gravitational dark matter can non-trivially interact with gravity itself. The couplings to
SM quarks of such a tensor DM can be by far too weak [112], making it undetectable in collider
searches; furthermore, the requirement of a correct DM abundance and stability constrains a non-
thermal spin-2 mass to be >> 1T eV [113].
Furthermore, if we assume there exists a global symmetry, a peculiar feature or a collective phe-
nomenon capable of stabilizing the lightest G(2) exceptional glueballs states, it must be consid-
ered that bosonic ensembles could eventually clump together to form a Bose-Einstein conden-
sate (BEC): once the temperature of a cosmological boson gas is less than the critical temper-
ature, a Bose-Einstein condensation process can always take place during the cosmic history of
the Universe, even if the high mass of these candidates should disfavor this scenario (for exam-
ple, the occurence of glueballs condensates and glueball stars have been recently discussed in
[106; 114; 115; 116]).
Concluding, it is reasonable to assert that is no more pretentious to hypothesize a not yet clarified
stabilization mechanism for heavy G(2) glueballs than to postulate ad hoc a Standard Model–
disconnected gauge symmetry to avoid dark matter candidates decay.
14
Such a dark sector can naturally accommodate the fact that there is only gravitational evidence
for dark matter so far, certainly disfavoring direct and indirect searches, and it can also quali-
tatively account for the observation that dark matter and ordinary matter are in commensurable
quantities (approximately 5:1 from recent Planck experiment measurements [117]), as they come
from the same broken gauge group. Given the forbidden or extremely weak interactions between
the heavy glueball states and ordinary matter, the usual WIMP-like scenario in which the DM relic
abundance is built via the freeze-out mechanism cannot be achieved, since these bosons are not in
thermal equilibrium with the baryon-photon fluid in the early Universe: their production should be
abruptly triggered by a first-order cosmological phase transition, fixing their abundance one and
for all. It is well-known that several non freeze-out models has been proposed in literature, such
as the FIMP (Feebly Interacting Massive Particle) cosmology, via a freeze-in mechanism [118].
In analogy, a two-gluons G(2) glueball, with an extremely small coupling (O(107)or less) with
the visible sector, could be ab initio produced out of thermal equilibrium for instance by the heavy
visible mediator decay, such as the new Higgs (HDM DM ), if MHMGG and the coupling
between DM and the heavy Higgs is very weak, realizing a sort of exotic-Higgs portal DM. The
most general renormalizable scalar G(2)-glueball potential of this type reads
V, S) = V(Φ) + M2
S
2S2+λS
4S4+λHS
2Φ2S2(32)
where Sis the scalar G(2) glueball field and MSits mass, λSits quartic self-interaction strength
and λHS the heavy Higgs-scalar glueball coupling. If the portal coupling is sufficiently small
[118], one can recover a correct dark matter relic abundance h2
=0.12 [117]. The situa-
tion can change if λSis large, i.e. scalar self-interactions are active: this could be the case of a
SIMP (Strongly Interacting Massive Particle) scenario, with a possible dark freeze-out mechanism
[118; 119], which prescribes DM thermalisation within the dark sector itself (through generic
nDM n
DM processes), which was initially populated by previous freeze-in. A proper balance,
to be determined, between λHS and λScould produce the right abundance for the complex G(2)
phenomenology. This issue will certainly deserve a dedicated study.
Finally we are going to briefly discuss possible manifestations of G(2) dark gluons in the
present Universe. Many vector bosons composite states have been proposed as DM candidates
in the last two decades: light hidden glueballs [120; 107; 121; 122], gluon condensates [123],
exceptional dark matter referring to a composite Higgs model with SO(7) symmetry broken to
the exceptional G(2) [124], SU (N)vector gauge bosons [125], vector BECs [126] and, in gen-
eral, non-Abelian dark forces [127]. These studies demonstrate the growing interest in beyond SM
non-Abelian frameworks where to develop consistent DM theories, without invoking string theory
and keeping the theoretical apparatus sufficiently minimal.
In our case of six dark heavy gluons from the broken exceptional G(2) group, mainly as stable
JP C = 0++ and/or JP C = 2++ glueballs, it must be considered they could clump and organize
into dark matter halos, in form of a heavy bosons gas or in some fluid systems [128; 129], for
sufficiently low temperatures and/or enough high densities. Indeed, the idea of a superfluid dark
matter has recently attracted attention in literature, from Khoury’s promising proposal of a uni-
fied superfluid dark sector [130; 131]. The fact that dark matter particles could assume different
“phases” according to the environment (gas, fluid or BECs) is very suitable to account for the
plethora of dark matter observations at all scales, from galactic to cosmological ones. The realiza-
tion of this scenario usually involves axions or ALPS (axion like particles) [132], which are light
or ultralight (with masses from 1024 to 1 eV in natural units) and capable of reproducing DM
15
halos properties: dark matter condensation and self-gravitating Bose-Einstein condensates have
been extensively studied in [133; 134; 135].
In our scenario we hypothesize the main constituents of a possible dark fluid are broken-G(2) mas-
sive gluons composite states. They are heavy, so that we cannot invoke the axion-like description,
rather a nuclear/atomic approach, like for Rubidium condensates [136]. But still G(2) glueballs
could aggregate into extended objects: one can explore the possibility that heavy G(2) gluon dark
matter is capable of producing stellar objects. Many models of exotic stars made of unknown
particles have been proposed, especially for sub-GeV masses, such as bosonic stars (where the
particle is a scalar or pseudoscalar [137; 138; 139], most likely for a quartic order self-interaction
[139]), Proca stars (for massive spin 1 bosons [140; 141; 142], if we invoke 3-gluons glueballs),
BEC stars [143], or glueballs stars [116]. In these cases, DM candidates could be also studied from
peculiar behaviors of compact astrophysical objects, characterizing physical observables useful to
disentangle standard scenarios from exotic phenomenology. The key ingredient to try to build up
stellar objects with non-light bosons of mass mBis mainly the magnitude of the quartic order self-
interaction λBof the constitutive boson, since Mstar λB/m2
B, as shown in [139; 143]. Even
if it is fair that glueballs are strongly self interacting, it is quite hard to make precise estimates for
their scattering process, given our general limited knowledge concerning strongly coupled theo-
ries.
Even more ambitious, one should consider the possibility to probe the SU (3) G(2) phase tran-
sition from black hole formation, as suggested in [144] for quark matter, if black holes are formed
through exotic condensed matter stages, exploiting the theoretical framework of gauge/gravity
dualities [145]. It could be worthwhile to speculate on the possibility that, in extremely high pres-
sure and temperature quark matter phases, quarks can be seized by G(2) gluons to form qGGG
screened states, rearranging QCD matter into color-singlet hybrids.
Concluding, for a few years we can take advantage of gravitational waves astronomy as a power-
ful probe to distinguish, in principle, a hypothetical binary dark-colored gluons star from a binary
black hole system, due to possible differences in the gravitational wave frequency and amplitude,
as demonstrated in [106].
So dark G(2) glueballs can be very versatile and exploitable within existing theoretical specula-
tions.
4 Conclusions
If Nature physical description is intrinsically mathematical, fundamental microscopic forces might
be manifestation of the algebras that can be built via the Cayley-Dickson construction process. In
other words, algebras can guide physics through the understanding of fundamental interactions.
To translate the mathematical meaning into a physical language, one has to move from abstract
algebras to groups of symmetry, through a correspondence here proposed as an automorphism
relation. This leads to the discovery of a mismatch between SU (3) strong force and octonions:
the octonions automorphism group is the exceptional group G(2), which contains SU (3), but it
is not exhausted by SU (3) itself. In the difference between the physical content of G(2) and
SU (3) new particles lie, in the form of six additional heavy bosons organized in composite states,
disconnected by Standard Model dynamics: the dark-colored G(2) gluons. These gluons cannot
interact with SM particles, at least at the explored energy scales, due to their high mass and QCD
string behavior. Mathematical realism is the guide and criterion to build a minimal extension of
16
the Standard Model.
Hence, for the first time in literature, G(2) was treated and developed as a realistic symmetry to
enlarge the Standard Model, and not only as a lattice QCD tool for computation. G(2) is a good
gauge group to describe a larger interaction, which operated in the Early Universe before the emer-
gence of visible matter; when the Universe cooled down, reaching a proper far beyond TeV energy
scale at which G(2) is broken, usual SU (3) QCD appeared, while an extra Higgs mechanism pro-
duced a secluded massive sector of cold dark-colored bosons. The SM is naturally embedded
in this framework with a minimal additional particle content, i.e. a heavy scalar Higgs particle,
responsible for a Higgs mechanism for the strong sector symmetrical w.r.t. the electroweak one,
and a bunch of massive gluons, in principle with the same mass, whose composite states play the
role of dark matter. Some accidental stability mechanisms for the dark glueballs have been pro-
posed. The presence of a new Higgs field represents the usual need of a scalar sector to induce
the symmetry breaking of a fundamental gauge symmetry. The extra Higgs might have visible de-
cay channels, but both this Higgs and the dark G(2) glueballs belong to a very high energy scale
which is certainly beyond current LHC searches, i.e. a multi-TeV or tens/hundreds of TeV scale
defined by the vaccum expectation value of the extra Higgs. Such a DM is certainly compatible
with direct, indirect, collider searches and astrophysical observations, as it is extremely massive
and almost collisionless.
In addition, if one tries to extend the correspondence between mathematical algebras and physical
symmetries further beyond octonions, sedenions show an intriguing property: they still have G(2)
as a fundamental automorphism, but “tripled” by an S3factor, which resembles the three fermion
families of the Standard Model and its S3-invariant extension. We know larger symmetries can
be constructed using the products of octonions (sedenions) and the other division algebras, point-
ing towards subsequent exceptional groups, as illustrated by the Freudenthal–Tits magic square,
which have been the subjects of string theories, but we did not want to push the dissertation in
this direction. Indeed the choice of G(2), as automorphism of octonions and minimally enlarged
non-Abelian compact Lie algebra of rank 2, is the minimal exceptional extension of the Standard
Model including a reliable dark sector, requiring no additional particle families nor extra funda-
mental forces. This fact reconciles the particle desert observed between the SM Higgs mass scale
and the TeV scale. G(2) could guarantee peculiar manifestations in extreme astrophysical com-
pact objects, such as neutron stars and black holes, which can be observed studying gravitational
waves and dynamics. In future, an interplay between condensed matter community and particle
physicists could be necessary to deeply investigate dark matter properties.
The development of a definitive theory is beyond the purpose of the present phenomenological
proposal, which is intended as a guideline for further speculations and future works.
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