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An exceptional G(2) extension of the Standard Model from the correspondence with Cayley–Dickson algebras automorphism groups

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In this article I propose a new criterion to extend the Standard Model of particle physics 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, an additional ensemble of massive G(2)-gluons emerges, which is separated from the particle dynamics of the Standard Model.
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An exceptional G(2) extension
of the Standard Model
from the correspondence
with Cayley–Dickson algebras
automorphism groups
Nicolò Masi
In this article I propose a new criterion to extend the Standard Model of particle physics 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, an additional ensemble of massive G(2)-gluons
emerges, which is separated from the particle dynamics of the Standard Model.
Despite its great success and prediction capability, the Standard Model (SM) of particle physics is aicted
by internal and external problems, i.e. theoretical issues (such as hierarchy and strong CP problems) and not
explained phenomena, like dark matter (DM), dark energy or matter–antimatter asymmetry1. Above all, DM is
probably the most compelling and very long-standing problem of modern physics, with no evident nor univocal
solution: all the eorts made, from particle theory2,3 to modied gravities46, have not been successful in clarifying
its nature. e 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 scale3,710. is is a strong hint that the Naturalness criterion11 for the Higgs sector and the so-called
WIMP Miracle12, which postulate the existence of a thermal particle relic of the Big Bang at the electroweak scale
O(100
GeV) which interacts via weak force, could not be a prerogative of Nature or, at any rate, not sucient
to individuate the origin of dark matter and describe the physics beyond Standard Model. Even the possibility
that the weak interaction between DM and SM particles is disfavored must be considered: new particles could
hide at dierent energy scales and they could be not capable of interacting with the visible world, at least at the
experimentally explored energies.
erefore, to proceed in the investigation of beyond Standard Model phenomenology, one has to ll up the
lack of a theoretical guideline and integrate some new simplicity criteria to select reliable candidates and explain
the complex astrophysical and cosmological observations1315. 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 extension
of the SM, which introduces a branch of exceptional matter particles from a simple and minimally high symmetry.
e criterion is to identify fundamental interactions with the automorphism groups of Cayley-Dickson algebras
(an automorphism is a bijective way of mapping a mathematical object to itself preserving its structure: the set
of all automorphisms forms the automorphism group, i.e. the symmetry group of the object). en, from the
automorphism of octonions (and sedenions) algebra, the promising exceptional symmetry group G(2) can be pin-
pointed 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 a primordial high energy phase transition which generated the strong sector. is minimal extension of
the SM, via Cayley–Dickson algebras automorphism correspondence, uniquely x the content of particle physics.
To the best of our knowledge, no existing work in literature is devoted to an exceptional G(2) enlargement of
the strong sector nor to the possibility that dark matter is formed by massive gluons from a broken-G(2) gauge
OPEN
Physics Department, INFN & Bologna University, Via Irnerio 46, 40136 Bologna, Italy. email: masin@bo.infn.it
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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 computations16,17, the implications of such an
extension of the SM 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 build up an exceptional Standard Model framework and incorporate new particle physics.
Fundamental forces from division algebras automorphisms
In the last decades many attempts to connect the Standard Model of elementary particles with division algebras
have been made, showing it is worthwhile establishing relations between algebraic structures and symmetry
groups1826.
It is well-known that following the Cayley–Dickson construction process18,20, one can build up a sequence
of larger and larger algebras, adding new imaginary units. In detail, from Hurwitz and Zorn theorem19, one can
identify the so-called division algebras
,,,𝕆
, i.e. the only four alternative algebraic elds with no non-trivial
zero divisors20,27, 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 ordered but commutative, quaternions are not commutative but associative, whereas octonions lose all
the familiar commutative and associative properties, but they are still an alternative algebra27. e process does
not terminate with octonions: applying the Cayley–Dickson construction, greater 2 -dimensional algebras can
be constructed, for any positive integer n. For
n>3
, however, as anticipated, they all include non-trivial zero
divisors, i.e. they have problems in a general denition of norm (in abstract algebra, a non-zero element a of a
ring R is called a zero divisor if there exists a non-zero x such that
ax =0
; for general properties of zero divisors
see28). is was considered an obstacle for the use of these extended algebras, such as
n=4
sedenions, in science.
But, as shown in29,30, 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.
e link between unitary groups and division algebras
𝔸n
has been diusely studied3133. 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 group1,3437, being
G=SU (3SU (2U(1)
the SM group of strong SU(3),
weak SU(2) and hypercharge U(1) interactions1. Besides its symmetry, the SM includes three fermions families:
between these three generations, particles dier by their avour quantum number and mass, but their interac-
tions are identical.
In the following, we want to briey 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 nd a direct connection
between the electromagnetism (or Quantum Electrodynamics) U(1) formalism and the complex number eld
: 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
rotation38. 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 dened to be
n
nU(n)≅SO(2)n≅(S1)n
, where / denotes the quo-
tient group between reals and integers, which shows o the deep connection between U(1) gauge symmetry and
other representations strictly connected to complex numbers38,39. It is also true that the
n×n
complex matrices
which leave the scalar product
,
invariant form the group
U(n)=Aut(n,,)
, i.e. the group of automorphisms
of
n
as a Hilbert space40.
ese links are not surprising because, from a mathematical point of view, the existence of innite distinct
wild automorphisms of the complex numbers, beyond identity and complex conjugation, is well-known21,41.
We nd another noteworthy examination in39, where the unitary group U(1) is showed as dening binary
complex relations
×
, i.e. the U(1) numbers eectively operate as automorphisms of
via multiplication of
a phase factor. As we know, the complex numbers can be expressed in polar coordinates and this implies that
the general linear multiplicative group
=
0=eGL(1,
)
is uniquely decomposable (
ez=exeiy
) into
the totally ordered group with real exponential
|
|=e
and into the phase group with imaginary exponentials
|
|=eiU(1)
, which is approximately U(1) (see39,42 for details). is is another way to underline the
intimate connection between the unitary group and the complex numbers. Furthermore, from a physical point
of view, one can also think at the Riemann-Silberstein eld reformulation of the electromagnetism43 in terms of
a complex vector that combines the electric eld E, as the real part, and the magnetic eld B, as the imaginary
part, in order to put in evidence this essential relation.
Even SU(2) weak isospin can be clearly represented with the algebraic quaternionic basis, i.e. Pauli matri-
ces23: SU(2) naturally embeds into
as 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 quaterni-
ons of norm 1, and it is thus dieomorphic to the 3-sphere
S3
(a dieomorphism is an isomorphism of smooth
manifolds, i.e. a map between manifolds which is dierentiable and has a dierentiable inverse). Indeed, since
unit quaternions can be used to represent rotations in 3-dimensional space (up to a sign), there is a surjective
homomorphism from SU(2) to the rotation group SO(3)23 (an homomorphism is a structure-preserving map
between two algebraic structures of the same type): one can show that the local SU(2) spinors are exactly the same
two-component spinors derived from the local quaternion matrix representation, i.e. the three Pauli matrices
along with the identity matrix I (spinors are dened as vectors of a representation of the group of automorphisms
of a Cliord algebra dened on space–time). 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 qua-
ternions
Aut()=SO(3)
, where SO(3) is homomorphic to SU(2) in turn, and the universal cover of SO(3) is
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the spin group Spin(3), which is isomorphic to SU(2). So SU(2) and SO(3) algebraic structures are equivalent.
An interesting demonstration of the correspondence between the two groups using
M̈obius
transformation is
described in44. e quaternionic representation of (electro-) weak isospin has been used by many authors45,46.
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 dene the mathematical structure and in the determination of solutions. is 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 Cliord algebra,
unlike
,
and
, because non associative)27,47 for a possible description of the SU(3) gauge eld45,48, but the
result is less clear than in quaternion case for the SU(2) gauge eld. e interesting fact to be considered is that
the group of automorphisms of the octonion 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 auto-
morphisms of the octonions, which is G(2). Nonetheless, it is possible to x 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 dened as the subgroup of G(2) which leaves the octonionic unit e
7
invariant23. 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 octonions45. Also split-octonions representations have been proposed
as alternative formalism for SU(3) color gauge symmetry48.
But here a crucial dierence appears: it must be noted that for
and
the direct automorphism groups con-
tain 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 includes SU(3) and is equipped with six additional generators50.
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:
ese relations show an ordered correspondence between (approximate) automorphisms of algebras 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 in20. Furey has recently suggested the appealing possibility to reformulate the SM group
G=SU (3SU (2U(1)
in terms of a
𝔸=𝕆
tensor product algebra, restarting from Dix-
on’s work, using the concept of Ideals, i.e. using subspaces of proper Cliord Algebras as “particles” (see24,5154
for details). Also string theory and supersymmetrical theories invoked division algebra to study particle
interactions32,5559. Moreover, G(2) as automorphism of
𝕆
has important applications in terms of the so-called
G2
structure or
G2
manifolds60, in the context of M-theory61.
Indeed, one solid reason for studying division algebras in relation to particle symmetries is that, unlike Lie
algebras and Cliord algebras21, there is a nite number of division algebras and corresponding automorphisms
(see again the extensive works of Dixon20). 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 easily added to this
picture and how dark matter description can benets from this algebraic facts, summarizing the main features
of sedenions algebra.
e sedenion algebra is the h Cayley-Dickson algebra
𝔸4=𝕊
, where
𝔸0,1,2,3
correspond to reals, complex
numbers, quaternions and octonions. is is not a division algebra, it is non-commutative, non-associative,
and non-alternative (an algebra A is alternative if the subalgebra generated by any two elements is associative,
i.e. i for all
a,bA
we have
(aa)b=a(ab),(ba)a=b(aa)
62), hence it cannot be a composition algebra29,63
(where a composition algebra is an algebra A over a eld K with a non-degenerate quadratic form N, called
norm, that saties
N(ab)=N(a)N(b)
for all a,b in A19). However sedenion algebra is power-associative and
exible (an algebra is power-associative if the subalgebra generated by any one element is associative: it is a sort
of lowest level of associativity27; the exible property, for any
a,bA
, can be dened as
a(ba)=(ab)a
), and
satises the weak inversive properties for non-zero elements. Each Cayley-Dickson algebra satises the weak
inversive property:
a1(ab)=a(a1b),(ba1)a=(ba)a1,a1(ab)=(ba)a1
—see30,62 for details. In principle,
the Cayley-Dickson construction can be indenitely carried on and, at each step, a new power-associative and
exible algebra is produced, doubling in size. So, in rst approximation, no new fundamental properties and
information are added nor lost enlarging the algebra beyond sedenions. One can choose a canonical basis for
𝕊
to be
E16 ={eiS|i=0, 1,
...
, 15}
where
e0
is the real unit and
e1,,e15
are anticommuting imaginary units.
In this basis, a general element
A𝕊
is written as
(1)
Aut()≅U(1), Aut()≅SU (2), Aut(𝕆)G(2).
(2)
A
=
15
i=0
aiei=a0+
15
i=1
aiei,ai
.
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e basis elements satisfy the multiplication rules
with
𝛾k
ij
the real structure constants, which are completely antisymmetric. For two sedenions A,B, one has
where
fij aibj
.
Because the sedenion algebra is not a division algebra, it contains zero divisors: for
𝕊
these are elements of
the form
ere are 84 such zero divisors in sedenion space and the subspace of zero divisors with unit norm is homeo-
morphic to G(2)28,64. 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
,,
and
𝕆
are by themselves division algebras,
their tensor products, such as
,
𝕆
and
𝕆
, 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 Ideals24,5153.
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 square6567:
𝕆
SO(3) SU(3) Sp(3)
F4
SU(3)
SU(
3
)2
SU(6)
E6
Sp(3) SU(6) SO(12)
E7
𝕆
F4
E6
E7
E8
is is a symmetric square (SO(N) and SU(N) are the usual special orthogonal and unitary groups of order
N, Sp(3) is the symplectic group of order three), 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. e exceptional groups on the last line/row are not exactly automorphisms of the octon-
ions products, because of mathematical problems in the denition of projective planes, due to the appearance
of zero divisors: they are called bioctonions (
𝕆
), quateroctonions (
𝕆
) and octooctonions (
𝕆𝕆
)
and nd correspondence into Jordans algebras27. Exceptional
Ei
are also largely used in supergravity and string
theory35,57. erefore it seems reasonable to continue the Cayley-Dickson algebraic construction into the non-
division algebras, such as
𝕊
.
Interestingly, in29,68 the authors put in evidence an important relation between sedenions and the exceptional
group G(2), demonstrated by Brown in69:
where we know that
Aut(𝕆)=G(2)
and
S3
is the permutation group of degree three. So the inner 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 symmetric group. e only dierence between octo-
nions and sedenions automorphism groups is a factor of the permutation group
S3
: this permutation group can
be constructed from the triality automorphism of the spin group Spin(8) (triality is a trilinear map among three
vector spaces, most commonly described as a special symmetry between vectors and spinors in 8-dimensional
euclidean space—see21,27 for details). Eq.(6) suggests that the fundamental symmetries of
𝕊
are the same as those
of
𝕆
, even if the factor
S3
introduces 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.
e previous formula can be generalized, for an arbitrary algebra constructed via Cayley-Dickson process
(for
n>3
), into29,70
is tells us that the underlying symmetry is always G(2), the automorphism group of the octonions. e
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 exible algebras, if
n>3
Cayley-Dickson algebras do not dier from sedenions for what concerns the
multilinear identities (or algebraic properties) content, as suggested in71.
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 rep-
resent the unknown dark matter sector. is 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,
(3)
e
0
=1, e
0
e
i
=e
i
e
0
=e
i
,
e2
i=−e0,i�= 0,
e
iej
=
γk
ij
eki
�=
0, i
�=
j
,
(4)
AB
=
(15
i=0
aiei
)(15
i=0
bjei
)
=
15
i,j=0
aibj(eiej)=
15
i,j,k
=
0
fij𝛾k
ij ek
,
(5)
(ea+eb)
(ec+eb)=0, ea,eb,ec,ed𝕊.
(6)
Aut(𝕊)=Aut(𝕆S3.
(7)
Aut(𝔸n)≅Aut(𝕆)×(n3)S3.
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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 denitive mathematical denition of the problem, we can
reformulate and summarize the algebraic phenomenological conjecture in a general way as follows.
e 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. e seden-
ionic description, like the octonionic one, corresponds, via automorphism, 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
S3
automorphism factor, as suggested by29. is is consistent with the proposal of a
S3
-invariant
extension of the Standard Model, as discussed in7275.
e gauge groups U(1),SU(2),SU(3), describing the three fundamental forces, nd mathematical correspond-
ence into the division algebras
,,𝕆
respectively: Table1 summarizes this correspondence. However, whereas
U(1) and SU(2) are approximate isomorphisms of complex and quaternion algebras automorphism groups (see
Eq.(1)), the octonion and sedenion automorphism relations point towards a dierent 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) dier for 6 dimensions/generators. erefore
could give the overall unbroken Standard Model symmetry. is is the rst main statement of the present disser-
tation. Here the automorphism selection is invoked to predict something beyond current SM, and SU(3) in par-
ticular, 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 products between the corresponding algebras (see Freudenthal–Tits magic square) could
be eective symmetries but not fundamental forces.
A new particle content come from the aforementioned dierence between G(2) and SU(3) groups and lie
in the spectrum gap between them. Following the Cayley–Dickson algebraic automorphism criterion, no more
physics is needed nor predicted, except for the six additional degrees of freedom, i.e. boson elds, which represent
the discrepancy between G(2) and SU(3) generators. Hence, the automorphism selection rule extends the strong
color sector and provides a rich exceptional phenomenology.
A further novelty is the denition of an original algebraic criterion to predict physics beyond the Standard
Model, 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 exceptional dynamics, due
to six residual generators of exceptional G(2), which might gain mass via a symmetry breaking: to recover stand-
ard 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 peculiar properties.
e next section is devoted to a deep analysis of the exceptional G(2) group and to the emergency of these
massive exceptional bosons.
A
G(
2
)
gauge theory for the strong sector
G(2) can be described as the automorphism group of the octonion algebra or, equivalently, as the sub-
group of the special orthogonal group SO(7) that preserves any chosen particular vector in its 8-dimen-
sional real spinor representation21,47,76. e group G(2) is the simplest among the exceptional Lie groups31;
it is well known that the compact simple Lie groups are completely described by the following classes:
(8)
Aut(Aut(Aut(𝕊)=Aut(Aut(Aut(𝕆S3=U(1SU(2G(2S3
Table 1. Schematic correspondence between forces, groups and algebras. In the rst column the charge of
the physical interaction is displayed along with the number
ng
of associated generators (bosons). Q,T,C are
usual SM electric charge, weak isospin and color charge, respectively; here EC stands for “exceptional-colored,
to indicate the six broken generators which originate the massive exceptional G(2) bosons which have quark
and anti-quark color quantum numbers (see next section). e second and third columns associate gauge
groups and forces, highlighting the link between G(2) and the 6 new exceptional-colored particles, separated
from visible strong phenomena. G(2) algebraic automorphism representation is valid for both octonions and
sedenions (the only dierence is the
S3
factor). In principle, strong force and exceptional sector represent the
same interaction but they are disconnected, coming from the broken exceptional symmetry. For this reason
their algebras are both displayed as
𝕆
or
𝕊
. Algebraic dimensions are showed in the h 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 exible (and power-associative). See27,63 for proper
descriptions of the algebraic properties and insights.
Charge (
ng
)Group Force Algebra Dim Commutative Associative Alternative Normed Flexible
Q(1) U(1) EM
2Yes Yes Yes Yes Ye s
T(3) SU(2) Weak
4No Yes Yes Yes Ye s
C(8) SU(3) Strong
𝕆
or
𝕊
8/16 No No Ye s Ye s Yes
EC(6) Broken-G(2) Exceptional
strong
𝕆
or
𝕊
8/16 No No No No Ye s
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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),SU(3),SO(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 histori-
cal criteria to associate SU(3) to the three color strong force, with quark states dierent from antiquarks states78).
ere 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, respectively50.
If we want to enlarge the QCD sector to include dark matter, it is straightforward we have to choose G(2)
or SO(5). e 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
2
as a center (and SU(3) has
3
); SO(N) in general are not simply connected and their
universal covering groups for
n>2
are spin Spin(N)79. It is also well-know in literature that G(2), thanks to its
aforementioned peculiarities, can be used to mimic QCD in lattice simulations, avoiding the so-called sign
problem80 which aicts SU(3). Proposing to enlarge QCD above the TeV scale and have the SM as a low energy
theory is surely not an unprecedented nor odd idea: for example, modern composite Higgs theories8183 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
generators50,79. e
7×7
real matrices U of the group SO(7) have determinant 1, orthogonal relation
UU =1
and fulll the constraint
UabUac =𝛿bc
. e G(2) subgroup is described by the matrices that also satisfy the cubic
constraint
where T is an anti-symmetric tensor dening the octonions multiplication rules, whose non-zero elements are
To explicitly construct the matrices in the fundamental representation, one can choose the rst eight generators
of G(2) as50,79:
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
Tr 𝜆a𝜆b=Tr ΛaΛb=2𝛿ab
.
Λ3
and
Λ8
are diagonal and represent the Cartan genera-
tors w.r.t. SU(3). e G(2) coset space by its subgroup SU(3) is a 6-sphere
G(2)∕SU (3)≅S6SO(7)∕SO(6)
84, in
analogy with the composite Higgs proposal82.
e remaining six generators can be found studying the root and weight diagrams of the group8587, and can
be written as:
where
ei
are the unit vectors
In the chosen basis of the generators it is manifest that, under SU(3) subgroup transformations, the 7-dimen-
sional representation decomposes into50,79
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. is representation describes a SU(3) quark
{3}
,
(9)
Tabc =Tdef UdaUeb Ufc
(10)
T127 =T154 =T163 =T235 =T264 =T374 =T576 =1.
(11)
Λ
a=1
2
𝜆a
00
0𝜆
a0
0 00
.
(12)
Λ
9=1
6
0i𝜆2
2e3
i𝜆20
2e3
2eT
3
2eT
3
0
,Λ10 =1
6
0𝜆2i
2e3
𝜆20i
2e3
i
2eT
3
i
2eT
3
0
,
(13)
Λ
11 =1
6
0i𝜆5
2e2
i𝜆50
2e2
2eT
2
2eT
20
,Λ12 =1
6
0𝜆5i
2e2
𝜆50i
2e2
i
2eT
2i
2eT
20
,
(14)
Λ
13 =1
6
0i𝜆7
2e1
i𝜆70
2e1
2eT
1
2eT
10
,Λ14 =1
6
0𝜆7i
2e1
𝜆70i
2e1
i
2eT
1i
2eT
10
,
(15)
e
1=
(1
0
0)
,e2=
(0
1
0)
,e3=
(0
0
1).
(16)
{7}={3}{3}{1}.
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a SU(3) anti-quark
{3}
and a SU(3) singlet
{1}
. e generators transform under the 14-dimensional adjoint
representation of G(2)50,79, which decomposes into50,79,88
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. is has important consequences for
connement79,8890: we will see that the color string between G(2) “quarks” is capable of breaking via the creation
of dynamical gluons. As discussed in50, the product of two fundamental representations
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”:
Due to the fact that “quarks” and “antiquarks” are indistinguishable, it is straightforward to show for the
one avor
Nf=1
case that the
U(1)L=R=U(1)B
baryon number symmetry of SU(3) QCD is reduced to a
Z2B
symmetry50,91: one can only distinguish between states with an even and odd number of “quark” constituents.
Another useful example is
From this composition it is clear that three G(2) “gluons” are sucient to screen a G(2) “quark”, producing a
color-singlet hybrid qGGG . It is also true that:
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 char-
acterized 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 quarks/antiquarks quantum numbers, but they
are still vector bosons. A general Lagrangian for G(2) Yang–Mills theory can be written as1,50,79:
with the eld strength
obtained from the vector potential
with
gG
a proper coupling constant for all the gauge bosons and
Λa
the G(2) generators. e Lagrangian is invari-
ant 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 connement
at low energies79. e G(2) connement is surely peculiar with a dierent 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 connement. It has been showed that
G(2) lattice Yang–Mills theory is indeed in the conned phase in the strong coupling limit50.
But we know that G(2) is not a proper gauge theory for a real Quantum Chromodynamics theory. erefore
we must add a Higgs-like eld in the fundamental
{7}
representation in order to break G(2) down to SU(3).
e consequence is simple and fundamental: 6 of the 14 G(2) “gluons” gain a mass proportional to the vacuum
expectation value (vev) w of the Higgs-like eld, the other 8 SU(3) gluons remaining untouched and massless.
e Lagrangian of such a G(2)-Higgs model can be written as17,50,79,88:
where
Φ(x) = (Φ1(x),Φ2(x),,Φ7(x))
is the real-valued Higgs-like eld,
Dµ=(∂µigGAµ)�
is the covari-
ant derivative and
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
,
Φ4
and two vanishing due to antisymmetry, making the aforementioned potential general and consistent with
G(2) symmetry breaking and renormalizability. We can choose a simple vev like
Φ
0=
1
2
(0, 0, 0, 0, 0, 0, w
)
to
(17)
{14}={8}{3}{3}.
(18)
{7}{7}={1}{7}{14}{27},
(19)
{7}{7}{7}={1}4{7}2{14}3{27}2{64}{77}.
(20)
{7}{14}{14}{14}={1}
(21)
{7}{7}{7}{7}=4{1}
(22)
L
YM [A]=−
1
2
Tr [F2
𝜇𝜈]
,
(23)
F𝜇𝜈 =𝜕𝜇A𝜈𝜕𝜈A𝜇igG[A𝜇,A𝜈],
(24)
A
𝜇(x)=Aa
𝜇(x)
Λ
a
2.
(25)
LG2H
[A,Φ] = L
YM
[A]+(D𝜇Φ)
2
V
(Φ)
(26)
V(Φ) = 𝜆2w2)2
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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 Eqs. (1114) that
Plugging this scalar eld vev into the square of the Higgs covariant derivative, we get the usual quadratic term
in the gauge elds
that gives the diagonal mass matrix
Mab
for the gauge bosons, of which we can use the aforementioned trace
normalization relation, which Gell-Mann matrices and G(2) generators share, to put the squared masses terms
g
2
G
w
2
in evidence. is new scalar
, which acquires a typical mass of
from the expansion of the potential about its minimum1, should be a dierent Higgs eld 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 eld could be ruled out by future LHC and Future Circular Collider searches92
(it is enough to think of heavy scalars models searches, such as the two-Higgs doublet model93 or the composite
Higgs models94). 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
through the Higgs mechanism1, 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 mass-
less at the scale of G(2) symmetry breaking, since the SM Higgs has not yet acquired its vev. en, if the sedenions
description via automorphisms group is invoked, the symmetry breaking process could in principle act on three
dierent copies of G(2), expressed by the permutation factor
S3
which keeps track of the three fermion fami-
lies. 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 avour symmetry in order to produce the correct patterns of dierent
masses and mixing angles for fermions families (see72,95 for insights). Additionally, it has been shown that, in a
phenomenologically viable electroweak
S3
extension of the SM,
S3
symmetry should be broken to prevent avor
changing neutral currents72 and the Higgs potential becomes more complicated due to the presence of three
Higgs elds74. For simplicity, we could assume that this hypothetical process, involving
S3
breaking and Yukawa
fermion masses generation, triggers at the electroweak scale, without interfering with the G(2) Higgs potential.
Using the Higgs mechanism to smoothly interpolate between SU(3) and G(2) Yang–Mills theory, we can
study the deconnement phase transition. In the SU(3) case this transition is weakly rst 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 N seem to have rst order deconnement phase transitions96100, which are
more markedly rst order for increasing N. e peculiarities of the phase transition from lattice G(2)-Higgs to
SU(3) have been extensively studied in16,90,101103, conrming that G(2) gauge theory has a nite-temperature
deconning phase transition mainly of rst order and a similar but discernable behavior with respect to SU(N)103.
It is interesting to mention that it has been shown104107 that rst order phase transitions in the early Universe
could produce gravitational waves detectable by future space-based gravitational observatories such as LISA.
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. e 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 eld, six G(2) gluons acquire a mass thanks to the Higgs mechanism.
e larger is
MG
, the greater is the distance where string breaking occurs. When the expectation value of the
Higgs-like eld is sent to innity, so that the 6 massive G(2) “gluons” are completely removed from the dynamics,
also the string breaking scale is innite. us the scenario of the usual SU(3) string potential reappears. For small
w (on the order of
ΛQCD
), on the other hand, the additional G(2) “gluons” could be light and participate in the
dynamics. As long as w remains nite, as we know it should be in the SM and in its extensions, the massive G(2)
“gluons” can mediate weak baryon number violating processes50 (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, hereaer only high w values (with w much 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) QCD108, but richer. is 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 hybrid states (and, in general,
the quark connement for one-quark-N-G(2) gluons, with
N3
), diquarks, (qqqq) tetraquarks and (qqqqq)
pentaquarks. States with baryon number 0 and 3 are in common with QCD whereas
nB=1, 2
, of
J=12
hybrids
(27)
Λ18Φ0=0(unbroken generators)
(28)
Λ914Φ00(broken generators)
(29)
g
2
GΦ
0
Λ
a
2
Λ
b
2
Φ0Aa
𝜇(x)A𝜇,b(x)= 1
2
MabAa
𝜇(x)A𝜇,b(x
)
(30)
MH=2𝜆w
(31)
MG=gGw
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and
J=0, 1
diquarks respectively, are G(2) specic. A tentative spectrum for the bosonic diquarks from lattice
simulations has been proposed in108. 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. e complete 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
}.
Even collective manifestations of G(2) exceptional matter could be dierent: for example, a G(2)-QCD neu-
tron star could display a distinct behavior with respect to a SU(3) neutron gas star, as discussed in109.
In the next section the focus will be on the phenomenology of the massive G(2) glueball states.
The exceptional gluonic content of the theory: a possible dark matter
phenomenology
As discussed before, the G(2) extension of the SM produces an exceptional particle sector: if we move away the six
G(2) gluons from the dynamics, these bosons must be secluded and separated from the visible SM sector in rst
approximation, without experimentally accessible electroweak interactions, unlike WIMPs, and extreme ener-
gies (and distances) should be mandatory to access the G(2) string breaking. is 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. is could be the realization of a beyond Naturalness criterion. 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. Another 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 cold dark matter (CDM) theory (cold means non-relativistic
and refers to the standard Lambda-CDM cosmological model), if we nd a stable or long-lived candidate. Many
vector bosons composite states have been proposed as DM candidates in the last two decades: light hidden
glueballs110113, gluon condensates114, exceptional dark matter referring to a composite Higgs model with SO(7)
symmetry broken to the exceptional G(2)115, SU(N) vector gauge bosons116, vector Bose-Einstein condensates
(BECs)117 and, in general, non-Abelian dark forces118. ese 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 suciently minimal.
In our case, the six dark gluons can form dark glueballs constituted by two or three (or multiples) G(2) gluons,
according to
{14}{14}={1}
and
{14}{14}{14}={1}
representations108, with integer total
angular momentum
J=0, 2
and
J=1, 3
for 2-gluons and 3-gluons balls respectively. In principle, exceptional-
colored broken-G(2) glueballs should not be stable, if these massive composite states themselves have no extra
symmetries to prevent their decay; since the proposed G(2) theory includes QCD, unlike hidden Yang–Mills
theories with no direct connections with the SM111,119, there exist states that couple to both the exceptional-
colored glueballs and SU(3) particles (for example the G(2)-breaking Higgs eld): hence, whether at tree-level
or via loops, these heavy glueballs would not be stable. To avoid this, rst 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
2
𝜆>
2gG
from Eq.(30), (31); secondly, the decays into meson states should be also forbidden. In principle, a
lightest
JPC =0++
state, in analogy with standard QCD, could dominate the glueball spectrum120, but this could
be unstable, like the lightest meson
𝜋0
and the other known scalar particle, the SM Higgs
h0
. e 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 understood: even the fundamental problem of color connement has not a denitive
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
Ŵ
for 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)B
global 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 eects. A large class
of models121123 imposes global discrete
Z2
(or generally
Zn
124) or continuous U(1) symmetries to guarantee DM
stability, in which DM is odd under the new symmetry while SM elds are assumed to be even: even in our case
one can introduce such a
Z2
symmetry, inherited from the new G(2)-breaking Higgs eld, preventing the lightest
G(2) two-gluons glueball decay. A complementary choice is to invoke another multiplicative quantum number,
i.e. a G-parity conservation for a generic Yang–Mills theory as suggested in125, to generalize the C-parity and
apply it to meson-like multiplets; unlike the lightest QCD mesons
π
which possess electroweak interactions, the
lightest G(2) glueball could represent a sort of stable dark meson (due to the quark–antiquark quantum numbers
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carried by the two G(2) gluons), whose dynamics is constrained inside the G(2) broken sector itself, leading to
an exactly conserved G-parity and the impossibility to decay into G-even SM particles.
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 interaction between glueballs,
preventing the decay (like the neutron case in the nucleus), must be investigated.
Possibly, one can also postulate a suppression scale
1fGM)
for the couplings with SM which depends on
the relative mass dierence 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 opportunity is to invoke the
JPC =0++,2
++
dark G(2) glueball states as graviton coun-
terparts in a AdS/CFT correspondence framework, as discussed in126 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 approach127,
whereas the tensor glueball could play the role of a massive graviton-like particle, for example in the context of
bimetric gravity theory128,129, where the massive gravitational dark matter can non-trivially interact with grav-
ity itself. e couplings to SM quarks of such a tensor DM can be by far too weak129, making it undetectable in
collider searches; besides, the requirement of a correct DM abundance and stability constrains a non-thermal
spin-2 mass to be
>> 1
TeV130.
Furthermore, it must be considered that bosonic ensembles could eventually clump together to form a BEC:
once the temperature of a cosmological boson gas is less than the critical temperature, a Bose-Einstein condensa-
tion process can always take place during the cosmic history of the Universe, even if the not low mass of these
candidates should disfavor this scenario. For example, the occurence of glueballs condensates and glueball stars
have been recently discussed in119,131133.
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 qualitatively account for the observation
that dark matter and ordinary matter are in commensurable quantities (approximately 5:1 from recent Planck
experiment measurements134), as they come from the same broken gauge group. Given the forbidden or extremely
weak interactions between the G(2) 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 uid in the early Universe: their production should be abruptly
triggered by a rst-order cosmological phase transition, possibly xing their initial abundance. 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 mechanism135,136, in which the comoving DM abundance freezes when the
number densities of the visible sector, generating DM by decays or annihilations, become Boltzmann-suppressed,
ending the yield. is requires an extremely small coupling (
O(107)
or less) with the visible sector. Another
intriguing alternative is represented by the so-called Dark freeze-out, for which DM reaches an equilibrium heat
bath within the dark sector itself, never interacting with SM particles: in this case, the dark ensemble was initially
populated by a freeze-in-type yield from part of the visible sector.
In analogy, trying to construct a particle cosmology for the exceptional gluonic content of the present theory,
let’s suppose that a two-gluons G(2) glueball could be abinitio produced out of thermal equilibrium, for instance
by a heavy mediator decay in the visible sector heat bath, such as the new G(2)–Higgs (
HDMDM
): this is
viable if
MHMGG
and the coupling between DM and the heavy Higgs is suciently weak, realizing an exotic-
Higgs portal DM137.
In principle, the most general scalar glueball eective potential, in the large N limit of a SU(N) gauge the-
ory, may contain not only a quartic interaction, proper of a Higgs portal, but also the cubic and higher order
terms119,138,139, in the form
where S is the scalar G(2) glueball eld, m the mass term and the coecients are
ai1
, which could be obtained
from lattice computations. e trilinear interaction might generate an attractive Yukawa-like potential138, whereas
the quartic one may be repulsive, according to the sign of the coupling; the h and higher terms might be sup-
pressed by the mass scale and the decreasing couplings. Choosing the simplest scalar case with a
Z2
symmetry
with a negligible cubic interaction (according to the minimal Higgs-portal paradigm), from now on we adopt a
renormalizable scalar G(2)-glueball potential, coupled to the exotic heavy Higgs sector, of the type
where
𝜆S
is the quartic self-interaction strength,
𝜆HS
the heavy Higgs-scalar glueball coupling and
V(Φ)
is
described by Eq.(26);
MS
2=m2+𝜆HSw22
can be dened as the total mass aer the G(2) symmetry breaking
(which trivially implies
𝜆HS 2MS
2w2
). If the portal coupling is suciently small135, one can recover a correct
dark matter relic abundance
Ωh20.12
134. In fact, the approximate solution for this present-day DM abundance,
assuming the initial number density of DM particles negligible, is135
where
gs
and
g
are eective numbers of degrees of freedom for entropy and energy densities,
gH
is the intrinsic
number of degrees of freedom of H (the expression is evaluated around
TMH
) and where
MPl
is the Planck
(32)
V
(S)=
i=2
ai
i!
(
4𝜋
N
)
i2
m4iS
i
(33)
V
,S)=V(Φ) + m
2
2
S2+
𝜆
S
4
S4+
𝜆
HS
2
Φ2S
2
(34)
Sh24.48 ×108
g
H
g
s
g
M
S
GeV
M
Pl
Ŵ
HSS
M2
H
,
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mass. For
gsg
and
ŴHSS
2
HS MH
/(8
π)
, Eq.(34) gives an estimate for the coupling intensity as a func-
tion of the DM abundance:
It’s easy to verify that, for
Ωh20.12
,
g100
and not
MHMS
>>
1
ratios,
𝜆HS << 1
. Furthermore, stud-
ying the exotic Higgs-portal135 in the context of the G(2)-breaking mechanism, one can approximate again
Eqs.(34)–(35) as
Sh2/0.12
10242
HS
M
S
MH
and insert the scalar glueball mass for
m2
<< 𝜆
HSw2
, i.e. a scalar
mass integrally generated by the Higgs expectation value
as an explicit function of the Higgs-scalar coupling, to obtain:
For
wMH=O(1)
, to reproduce the correct relic density the coupling should be very tiny, i.e.
𝜆HS 10−(10÷9)
,
as in most FIMP theories135,140.
e situation can change if
𝜆S
is large enough (
𝜆S
>
103
122), i.e. if scalar self-interactions are active: the DM
particles, initially produced by the previous mechanism, may thermalize among themselves even if the dark
sector consists of only one particle species, due to number-changing processes (generic
nDM
n
DM
processes),
which reduce the average temperature of DM particles and increase the number density until equilibrium is
reached. e resulting relic abundance could therefore change even though the coupling between the visible
and dark sector is absent. is could be the case of a SIMP (Strongly Interacting Massive Particle) scenario with
a dark freeze-out mechanism122,135,137. For example, in case of a quartic self-interaction, thermalization of the
dark sector within itself through
24
scatterings is active if
𝜆S
exceeds a critical value, so that DM reaches a
thermal equilibrium. When DM is no longer relativistic,
42
processes dominate the dynamics in the so-called
cannibalization era, which ends when its rate drops below the Hubble rate, xing the DM number density to a
modied yield through the dark freeze-out. Odd processes, such as
23
, could be neglected for DM potential
without odd powers terms and are forbidden for multiplicative
Zn
symmetry conservation for even n122, like in
most well-established DM setups141. As shown in122,142, if DM relic abundance is solely computed via a Higgs
decay in a freeze-in framework, without dark thermalization for
42
interactions, it could be appreciably
underestimated (over an order of magnitude) and, consequently, the previous bounds for the parameters space
could be altered. Quartic self-interactions should also fulll additional constraints from cosmology140,142, such
as the isocurvature bound, for which the scalar mass is bounded from above:
with
H
the Hubble parameter at the inationary scale.
e resolution of Boltzmann equation for SIMP DM usually leads to scenarios where the dark freeze-out
temperature is less than the visible ensemble one, making DM naturally colder than SM particles. See141,143147
for insights regarding
32
dark thermalization and explicit formulas for the relic density for a SIMP DM with
32
annihilations, which do not have immediate analytic representations.
For smaller interactions with the visible sector, the thermal production of DM particles is insignicant and
DM must come from a non-thermal mechanism, leading to a Super-WIMP (SWIMP)-like scenario, for example
through a direct DM-producing inaton decay148150, if the heavy Higgs scalar responsible for the
G(2)
SU (3)
transition is identied as the inaton eld. In this inaton-portal case151, the glueball abundance is basically xed
by few parameters, i.e. the reheating temperature
Trh
, the inaton mass and branching ratio into G(2) gluons/
scalar glueball
BS
148:
For a cosmological reheating above the GeV scale, the
BS(MSMH)
factor should be quite small, i.e.
<109
.
e scenario works well especially for large DM masses, between the weak scale and the PeV scale149, and for
extremely decoupled EeV candidates151.
Finally we are going to briey discuss possible manifestations of G(2) gluons in the present Universe. e six
massive gluons from the broken exceptional G(2) group, mainly as stable
JPC =0++
(and possibly
JPC =2++
)
glueballs, could clump and organize into dark matter halos, in form of a heavy bosons gas or in some uid
systems152,153, for suciently low temperatures and/or enough high densities. Indeed, the idea of a (super)
uid dark matter has recently attracted attention in literature, from Khoury’s promising proposal of a unied
superuid dark sector154,155. e fact that dark matter particles could assume dierent “phases” according to the
environment (gas, uid or BECs) is very suitable to account for the plethora of dark matter observations at all
scales, from galactic to cosmological ones. e realization of this scenario usually involves axions or ALPS (axion
like particles)156, which are light or ultralight (with masses from 10
24
to 1
eV
in natural units) and capable of
reproducing DM halos properties: dark matter condensation and self-gravitating Bose-Einstein condensates
have been extensively studied in157159.
In our scenario we hypothesize the main constituents of a possible dark uid are broken-G(2) massive gluons
composite states. ey are certainly massive, so that we cannot invoke the axion-like description. But still G(2)
(35)
𝜆
HS 1012
(
ΩSh
2
0.12 )
12
(
g
100 )
34
(
MH
MS)
12
.
(36)
MS
=
MGG
2gGw
𝜆
HS
2w,
(37)
S
h2/0.12
1024
5/2
HS
(w/M
H
)
.
(38)
M
S
GeV
<63/8
S
H
10
11
GeV
3/2
(39)
Sh2
2
×
108BS(MS/MH)
T
rh
GeV
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glueballs could aggregate into extended objects: one can explore the possibility that heavy G(2) gluon dark matter
is capable of producing stellar objects, which could populate the dark halos. 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 pseudoscalar160162, most likely for a quartic order repulsive self-interaction162), Proca stars
(for massive spin 1 bosons163165—one could invoke a correspondence with 3-gluons glueballs), BEC stars166, or
QCD glueballs stars133. e key ingredients to try to build up stellar objects with non-light bosons are mainly the
magnitude of the quartic order self-interaction
𝜆4
of the constitutive boson and its mass. Our glueball scalar eld
S evolving in the General Relativity framework can be described by the Einstein-Klein-Gordon (EKG) action,
where V(S) is the bosonic potential,
̄
S
the complex conjugate for a complex scalar, R is the Ricci scalar,
g𝜇𝜈
is
the metric of the space–time, g its determinant and G is Newton’s gravitational costant. e variation of the
action with respect to the metric leads to the related Einstein equations162,167 for static, spherically–symmetric
geometries. To build up a massive boson star, a quartic self-interaction potential is needed to balance the inward
gravitational force, in the form of the previously described (Eq.33) potential, i.e.
V
(
|
S
|
2)= M
2
S
2|
S
|
2+𝜆4
4|
S
|4
, where
𝜆S𝜆4>0
(
𝜆4<0
) signies a repulsive (attractive) interaction: repulsive self-interactions can give rise to very
dense boson stars. e maximum mass for such an object is162,166,168
where
MPl
1
G1.2 ×1019GeV M
13
is the Planck mass and
M
the solar mass, with a consequent
boson star radius119
For example, for a heavy
MS10
TeV scalar, the maximum values, as a function of the self-interaction, are
Mmax
10
8
𝜆
4M
and
RBS
10
4
𝜆
4
m: the resulting object carries a small mass w.r.t. usual stars (about
one hundredth of the Earth mass) and a sub-millimeter radius for not two high
𝜆4
, making it a tiny ball of glue-
balls. Instead, for a 1 GeV scalar,
Mmax
𝜆
4M
and
RBS
𝜆
4
×
10 km
stand, producing a solar mass object
with a neutron star-like radius. As stressed in168, at the maximum mass the radius is slightly larger than the
corresponding Schwarzschild radius. In addition, for
𝜆4=O(1)
, the relation is similar to the Chandrasekhar
mass for white dwarf stars
M
wd M
3
Pl
m2
p
: this could resemble the case of SU(N) glueballs, estimating the quartic
coupling as
𝜆4∼(4
𝜋
)2N2
168 (see Eq.(32)), being
4<4π
for perturbativity161. Even if it is fair that G(2) 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 theories. Nevertheless, considering the dimension of
the exceptional G(2) group and its lattice behaviour103, we could qualitatively approximate it to a SU(4) theory
and obtain
𝜆4
𝜋
2
, which almost saturates the perturbativity condition
4<4π
. For example, for such a
𝜆4
,
both the aforementioned
Mmax
ranges, i.e.
108M
and few
M
, fall into mass windows with weak constraints
for MACHOs from up-to-date microlensing analysis169,170. Obviously, the feasibility of a star made of self-inter-
acting scalars also depends on a correct estimate of the possible
31
,
32
and
42
annihilation processes
inside the star168, which in turn depend on the symmetries (e.g.
Zn
) of the exotic sector; for a non-conserved
glueballs number, i.e. for a real scalar with no additional symmetries,
34
2
processes could trigger the decay
of massive bosons stars,leading to severe bounds on their mass:
Mmax
<
1011M
.
For the so-called mini boson stars (with no scalar interaction potential) and Proca stars, the maximum mass
is quite smaller and scales with
M
2
Pl
MS
. Furthermore, unlike complex scalar elds, real scalars stars could not
possess the required stability: however, with a non-trivial time-dependent stress-energy tensor, long-term stable
oscillating geometries can be achieved167. ese “oscillatons” mass-radius relations are indeed very similar to the
boson stars ones.
It is well-known that the inclusion of self-interactions in the DM sector should be in agreement with the
upper limits which come from several astrophysical observations2,171,172, mainly from colliding galaxy clusters
dynamics, like the the Bullet Cluster173. The DM-DM scattering cross section must qualitatively satisfy
𝜎
MS
<1cm
2
g
1024 cm
2
GeV
. Assuming
𝜎
=
9
𝜆4
2
32
𝜋
M
2
S
at tree level135, the bound161 results in
which is consistent with the one obtained in137. In principle the intensity of the self-interaction is not eectively
bounded by the scalar mass scale of the present theory: the more the DM particle is massive the more the con-
straint for
𝜆4
is relaxed. Exploiting Eq.(36) with a FIMP-like
𝜆HS 1010
, Eq.(43) becomes
For the aforementioned
𝜆4𝜋2
self-interaction assumption, this formula implies an expectation value for the
heavy Higgs satisfying
w104
GeV, which is completely consistent with a beyond SM phenomenology. Such a
(40)
S
=
d4x
gR
16
𝜋G
g𝜇𝜈𝜕𝜇̄
S𝜕𝜈SV(S)
,
(41)
M
max ∼(0.1 ÷1)
𝜆4M
3
Pl
M2
S
GeV
2
(42)
R
BS ∼(0.1 ÷1)
𝜆4×10 km
M2
S
GeV2
.
(43)
|
𝜆4
|
<4102
(M
S
GeV )32
(44)
|
𝜆4
|
<105
(
w
GeV )32
.
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FIMP/SIMP-like singlet scalar model, constrained by astrophysical observations and the perturbativity condi-
tion, naturally leads to a DM mass in the GeV/sub-GeV region122 (Eq.(43)). is is also valid from Eq.(38) for
an ination scale
H
1012
GeV. e empirical Eq.(43) relation for the glueball scalar mass is quite equivalent
to the theoretical ones in Eqs.(35)–(36), for which
𝜆HS 1010
and
wMH=O(1)
imply
MS10−(5÷4)MH
.
DM candidates could be certainly studied from peculiar behaviors of the compact astrophysical objects they
form, characterizing physical observables useful to disentangle standard scenarios from exotic phenomenology167.
In fact, for a few years we can take advantage of both electromagnetic and gravitational waves astronomy as
powerful probes to discriminate compact objects as a function of their “compactness” (or “closeness” to a black
hole—a quantity related to the mass to radius ratio M/R is a possibility), “shadow” and gravitational waves
emission167. For example, a hypothetical binary SU(N) gluons star could be disentangled from a binary black
hole system, due to possible dierences in the gravitational wave frequency and amplitude, as demonstrated
in119: this comes from the fact that the mass-radius relation of the glueball dark star, as in our case, is dierent
from that of a black hole, as discussed before. In addition, in119 the authors show that, adopting a large N glueball
potential like the one in Eq.(32), the dark star is allowed to be more massive and larger at the same time, w.r.t.
the quartic potential case in Eq.(33). No accompanying luminous signal is obviously expected for G(2) glueballs
astrophysical objects, unlike generic beyond SM theories equipped with electro-weak interactions.
Even more ambitious, one should consider the possibility to probe the
SU (3)−G(2)
phase transition from
black hole formation, as suggested in174 for quark matter: this is possible if black holes are formed through
exotic condensed matter stages beyond degenerate neutron matter, maybe exploiting the theoretical framework
of gauge/gravity dualities175. It could be also worthwhile to speculate on the possibility that, in extremely high
pressure and temperature quark matter phases, unbroken G(2) quarks can combine into multiquarks particles or
can be seized by G(2) gluons to form qGGG screened states, rearranging QCD matter into a phase of color-singlet
hybrids and exotic hadrons. For example, the preliminary work by109 demonstrates the possibility to distinguish
a simplied G(2)–QCD neutron star using the mass-radius relation.
So dark G(2) glueballs can be very versatile and exploitable within theoretical speculations, especially for
exotic scalars coupled to a Higgs sector.
Conclusions
If Nature physical description is intrinsically mathematical, fundamental microscopic forces might be manifesta-
tion 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 correspond-
ence here proposed as an automorphism relation. is 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 dierence between the physical content of G(2) and SU(3) new
fundamental particles lie, in the form of six additional massive bosons organized in composite states, discon-
nected by Standard Model dynamics: the exceptional-colored G(2) gluons. ese gluons cannot interact with SM
particles, at least at the explored energy scales, due to their mass and QCD string behavior. Mathematical realism
has been the guide and criterion to build this minimal extension of the Standard Model.
Hence, for the rst 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 emergence of visible matter; when the Uni-
verse 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 produced a secluded sector of cold exceptional-colored bosons. e
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 masses, whose composite states might play the role of dark
matter. Some accidental stability mechanisms for the dark glueballs have been proposed. e presence of a new
Higgs eld represents the usual need of a scalar sector to induce the symmetry breaking of a fundamental gauge
symmetry. e extra Higgs might have visible decay channels, but it belongs to a very high energy scale which
is certainly beyond current LHC searches, i.e. a multi-TeV or tens/hundreds of TeV scale dened by the vaccum
expectation value of the extra Higgs. e resulting exceptional glueball DM is certainly compatible with direct,
indirect, collider searches and astrophysical observations, as it is almost collisionless. Several cosmological sce-
narios for these candidates were discussed, constraining the parameters space of the theory.
In addition, if one tries to extend the correspondence between mathematical algebras and physical symme-
tries further beyond octonions, sedenions show an intriguing property: they still have G(2) as a fundamental
automorphism, but “tripled” by an
S3
factor, 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, pointing 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 group 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 exotic sector, requiring no additional particle families nor extra fundamental forces. is 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 compact objects, such as boson stars made of G(2) glueballs,
which can populate the dark halos and be observed in the future studying their gravitational waves and dynamics.
e development of a denitive theory is beyond the purpose of the present phenomenological proposal,
which is intended as a guideline for further speculations.
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Received: 10 June 2021; Accepted: 3 November 2021
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Author contributions
I, N.M., am the only and corresponding author of the manuscript, which comes from a long and deep study and
chain of reasoning that lasted many years. is is a personal and thoughtful product of my ten-year activity in
the dark matter indirect search and phenomenology eld.
Competing interests
e author declares no competing interests.
Additional information
Correspondence and requests for materials should be addressed to N.M.
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... therein), has a long history, dating back to Gürsey, Günaydin, Ramond, Sikivie [34,35,36,37] and Morita [38,39,40] between mid '70s and early '80s, and more recently Bisht, Chanyal et al. [41,42,43] and Wolk [44]. In a broader framework, the algebraic formulation of the particle content and the gauge symmetries of the SM and beyond, crucially involving the octonions and related (exceptional) algebraic and geometric structures, has been the object of a number of works along the years; without any claim of completeness of our list, here we confine ourselves to