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

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.

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 Möbius 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 (which is not a Clifford algebra, unlike ℝ , ℂ and ℍ , because non associative) 27,47 for a possible description of the SU(3) gauge field 45,48 , 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 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( ) = 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 e 7 invariant 23 . 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 ℂ and ℍ the 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 includes 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: These 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 in 20 . Furey has recently suggested the appealing possibility to reformulate the SM group G = SU(3) × SU(2) × U(1) in terms of a = ℝ ⊗ ℂ ⊗ ℍ ⊗ tensor product algebra, restarting from Dixon's work, using the concept of Ideals, i.e. using subspaces of proper Clifford Algebras as "particles" (see 24,[51][52][53][54] for details). Also string theory and supersymmetrical theories invoked division algebra to study particle interactions 32,[55][56][57][58][59] . Moreover, G(2) as automorphism of has important applications in terms of the so-called G 2 structure or G 2 manifolds 60 , in the context of M-theory 61 .
Indeed, one solid reason for studying division algebras in relation to particle symmetries is that, unlike Lie algebras and Clifford algebras 21 , there is a finite number of division algebras and corresponding automorphisms (see again the extensive works of Dixon 20 ). 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 benefits from this algebraic facts, summarizing the main features of sedenions algebra.
The sedenion algebra is the fifth Cayley-Dickson algebra 4 = , where 0,1,2,3 correspond to reals, complex numbers, quaternions and octonions. This 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. iff for all a, b ∈ A we have (aa)b = a(ab), (ba)a = b(aa) 62 ), hence it cannot be a composition algebra 29,63 (where a composition algebra is an algebra A over a field K with a non-degenerate quadratic form N, called norm, that satifies N(ab) = N(a)N(b) for all a, b in A 19 ). However sedenion algebra is power-associative and flexible (an algebra is power-associative if the subalgebra generated by any one element is associative: it is a sort of lowest level of associativity 27 ; the flexible property, for any a, b ∈ A , can be defined as a(ba) = (ab)a ), and satisfies the weak inversive properties for non-zero elements. Each Cayley-Dickson algebra satisfies the weak inversive property: a −1 (ab) = a(a −1 b), (ba −1 )a = (ba)a −1 , a −1 (ab) = (ba)a −1 -see 30,62 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 to be E 16 = {e i ∈ S|i = 0, 1, . . . , 15} where e 0 is the real unit and e 1 , … , e 15 are anticommuting imaginary units. In this basis, a general element A ∈ is written as Because the sedenion algebra is not a division algebra, it contains zero divisors: for these are elements of the form There are 84 such zero divisors in sedenion space and the subspace of zero divisors with unit norm is homeomorphic 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 ℝ, ℂ, ℍ and are by themselves division algebras, their tensor products, such as ℂ ⊗ ℍ , ℂ ⊗ and ℝ ⊗ ℂ ⊗ ℍ ⊗ , 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 24,51-53 . 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] : This 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 ( F 4 , E 6 , E 7 , E 8 ) 49 , except for the exceptional G(2) which represents octonions automorphism itself. The exceptional groups on the last line/row are not exactly automorphisms 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 ( ℂ ⊗ ), quateroctonions ( ℍ ⊗ ) and octooctonions ( ⊗ ) and find correspondence into Jordan's algebras 27 . Exceptional E i are also largely used in supergravity and string theory 35,57 . Therefore it seems reasonable to continue the Cayley-Dickson algebraic construction into the nondivision algebras, such as .
Interestingly, in 29,68 the authors put in evidence an important relation between sedenions and the exceptional group G(2), demonstrated by Brown in 69 : where we know that Aut( ) = G(2) and S 3 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. The only difference between octonions and sedenions automorphism groups is a factor of the permutation group S 3 : 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-see 21,27 for details). Eq. (6) suggests that the fundamental symmetries of are the same as those of , even if the factor S 3 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.
The previous formula can be generalized, for an arbitrary algebra constructed via Cayley-Dickson process (for n > 3 ), into 29,70 This tells us that the underlying symmetry is always G(2), the automorphism group of the octonions. 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 nondivision flexible algebras, if n > 3 Cayley-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,   www.nature.com/scientificreports/ 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 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 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 S 3 automorphism factor, as suggested by 29 . This is consistent with the proposal of a S 3 -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 mathematical correspondence into the division algebras ℂ, ℍ, 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 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 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 products between the corresponding algebras (see Freudenthal-Tits magic square) could be effective symmetries but not fundamental forces.
A new particle content 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 automorphism criterion, no more physics is needed nor predicted, except for the six additional degrees of freedom, i.e. boson fields, 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 definition 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 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 peculiar properties.
The 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 subgroup of the special orthogonal group SO(7) that preserves any chosen particular vector in its 8-dimensional real spinor representation 21,47,76 . The group G(2) is the simplest among the exceptional Lie groups 31 ; it is well known that the compact simple Lie groups are completely described by the following classes: 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 n g 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). The 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 difference is the S 3 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 or . 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 27,63 for proper descriptions of the algebraic properties and insights. www.nature.com/scientificreports/ A N (= SU(N + 1)), B N (= SO(N + 1)), C N (= Sp(N)), D N (= SO(2N)) and exceptional groups G 2 , F 4 , E 6 , E 7 , E 8 , with N = 1, 2, 3, . . .(for D N , 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 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 .
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 ℤ 2 as a center (and SU(3) has ℤ 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 problem 80 which afflicts 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 theories 81-83 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 × 7 real matrices U of the group SO(7) have determinant 1, orthogonal relation UU † = 1 and fulfill the constraint U ab U ac = bc . The G(2) subgroup is described by the matrices that also satisfy the cubic constraint where T is an anti-symmetric tensor defining the octonions multiplication rules, whose non-zero elements are To explicitly construct the matrices in the fundamental representation, one can choose the first eight generators of G(2) as 50,79 : where a (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 a b = Tr Λ a Λ b = 2 ab . Λ 3 and Λ 8 are 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) ≅ S 6 ≅ 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: where e i are the unit vectors In the chosen basis of the generators it is manifest that, under SU(3) subgroup transformations, the 7-dimensional representation decomposes into 50,79 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. This representation describes a SU(3) quark {3} ,  50,79 , which decomposes into 50,79,88 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. 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 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 flavor N f = 1 case that the U(1) L=R = U(1) B baryon number symmetry of SU(3) QCD is reduced to a Z 2B symmetry 50,91 : 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 sufficient 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 characterized by 14 . 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) confinement 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) w of the Higgs-like field, the other 8 SU(3) gluons remaining untouched and massless. The Lagrangian of such a G(2)-Higgs model can be written as 17,50,79,88 : the quadratic scalar potential, with > 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 T abc , such a term disappears. Following the product in Eq. (21), the four singlets corresponds to w 2 Φ 2 , Φ 4 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 www.nature.com/scientificreports/ break G(2) and re-obtain the familiar unbroken SU(3) symmetry: it is easy to notice from the diagonal and nondiagonal structure of Eqs. (11)(12)(13)(14) that Plugging this scalar field vev into the square of the Higgs covariant derivative, we get the usual quadratic term in the gauge fields that gives the diagonal mass matrix M ab 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. This new scalar , which acquires a typical mass of from the expansion of the potential about its minimum 1 , 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 through the Higgs mechanism 1 , 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 S 3 which 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 S 3 -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). Additionally, it has been shown that, in a phenomenologically viable electroweak S 3 extension of the SM, S 3 symmetry 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 simplicity, we could assume that this hypothetical process, involving S 3 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 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 N seem 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 SU (3) have been extensively studied in 16,90,[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 . It is interesting to mention that it has been shown 104-107 that first 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. 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 M G , 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 Λ QCD ), on the other hand, the additional G(2) "gluons" could be light and participate in the dynamics. As long as w remains finite, as we know it should be in the SM and in its extensions, the massive 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 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) QCD 108 , 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 hybrid states (and, in general, the quark confinement for one-quark-N-G(2) gluons, with N ≥ 3 ), diquarks, (qqqq) tetraquarks and (qqqqq) pentaquarks. States with baryon number 0 and 3 are in common with QCD whereas n B = 1, 2 , of J = 1∕2 hybrids www.nature.com/scientificreports/ and J = 0, 1 diquarks respectively, are G(2) specific. A tentative spectrum for the bosonic diquarks from lattice simulations has been proposed in 108 . 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. The complete explicit decompositions of the products are: 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 a SU(3) neutron gas star, as discussed in 109 .
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 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 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 find a stable or long-lived candidate. Many vector bosons composite states have been proposed as DM candidates in the last two decades: light hidden glueballs [110][111][112][113] , gluon condensates 114 , exceptional dark matter referring to a composite Higgs model with SO(7) symmetry broken to the exceptional G(2) 115 , SU(N) vector gauge bosons 116 , vector Bose-Einstein condensates (BECs) 117 and, in general, non-Abelian dark forces 118 . 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, the six dark gluons can form dark glueballs constituted by two or three (or multiples) G(2) gluons, 108 , with integer total angular momentum J = 0, 2 and J = 1, 3 for 2-gluons and 3-gluons balls respectively. In principle, exceptionalcolored 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 SM 111,119 , there exist states that couple to both the exceptionalcolored glueballs and SU(3) particles (for example the G(2)-breaking Higgs field): 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 M H > M GG , which implies the qualitative constraint √ 2 > 2g G from Eq.(30), (31); secondly, the decays into meson states should be also forbidden. In principle, a lightest J PC = 0 ++ state, in analogy with standard QCD, could dominate the glueball spectrum 120 , but this could be unstable, like the lightest meson 0 and the other known scalar particle, the SM Higgs h 0 . 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 understood: 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 0 , 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 effects. A large class of models [121][122][123] imposes global discrete Z 2 (or generally Z n 124 ) or continuous U(1) symmetries to guarantee DM stability, in which DM is odd under the new symmetry while SM fields are assumed to be even: even in our case one can introduce such a Z 2 symmetry, inherited from the new G(2)-breaking Higgs field, 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 in 125 , 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 www.nature.com/scientificreports/ 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 1∕f G (ΔM) 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 opportunity is to invoke the J PC = 0 ++ , 2 ++ dark G(2) glueball states as graviton counterparts in a AdS/CFT correspondence framework, as discussed in 126 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 127 , whereas the tensor glueball could play the role of a massive graviton-like particle, for example in the context of bimetric gravity theory 128,129 , 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 129 , 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 TeV 130 .
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 condensation 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 in 119,[131][132][133] .
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 measurements 134 ), 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 fluid in the early Universe: their production should be abruptly triggered by a first-order cosmological phase transition, possibly fixing 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 mechanism 135,136 , 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. This requires an extremely small coupling ( O(10 −7 ) 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 ( H → DMDM ): this is viable if M H ≥ M GG and the coupling between DM and the heavy Higgs is sufficiently weak, realizing an exotic-Higgs portal DM 137 .
In principle, the most general scalar glueball effective potential, in the large N limit of a SU(N) gauge theory, may contain not only a quartic interaction, proper of a Higgs portal, but also the cubic and higher order terms 119,138,139 , in the form where S is the scalar G(2) glueball field, m the mass term and the coefficients are a i ≈ 1 , which could be obtained from lattice computations. The trilinear interaction might generate an attractive Yukawa-like potential 138 , whereas the quartic one may be repulsive, according to the sign of the coupling; the fifth and higher terms might be suppressed by the mass scale and the decreasing couplings. Choosing the simplest scalar case with a Z 2 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 S is the quartic self-interaction strength, HS the heavy Higgs-scalar glueball coupling and V (Φ) is described by Eq. (26); M S 2 = m 2 + HS w 2 ∕2 can be defined as the total mass after the G(2) symmetry breaking (which trivially implies HS ≤ 2M S 2 ∕w 2 ). If the portal coupling is sufficiently small 135 , one can recover a correct dark matter relic abundance Ωh 2 ≃ 0.12 134 . In fact, the approximate solution for this present-day DM abundance, assuming the initial number density of DM particles negligible, is 135 where g * s and g * are effective numbers of degrees of freedom for entropy and energy densities, g H is the intrinsic number of degrees of freedom of H (the expression is evaluated around T ≃ M H ) and where M Pl is the Planck www.nature.com/scientificreports/ mass. For g * s ≃ g * and Ŵ H→SS ≃ 2 HS M H /(8π) , Eq. (34) gives an estimate for the coupling intensity as a function of the DM abundance: It's easy to verify that, for Ωh 2 ≃ 0.12 , g * ≃ 100 and not M H ∕M S >> 1 ratios, HS << 1 . Furthermore, studying the exotic Higgs-portal 135  For w∕M H = O(1) , to reproduce the correct relic density the coupling should be very tiny, i.e. HS ∼ 10 −(10÷9) , as in most FIMP theories 135,140 .
The situation can change if S is large enough ( S > 10 −3122 ), 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 n DM → n ′ DM processes), which reduce the average temperature of DM particles and increase the number density until equilibrium is reached. The resulting relic abundance could therefore change even though the coupling between the visible and dark sector is absent. This could be the case of a SIMP (Strongly Interacting Massive Particle) scenario with a dark freeze-out mechanism 122,135,137 . For example, in case of a quartic self-interaction, thermalization of the dark sector within itself through 2 ↔ 4 scatterings is active if S exceeds a critical value, so that DM reaches a thermal equilibrium. When DM is no longer relativistic, 4 → 2 processes dominate the dynamics in the so-called cannibalization era, which ends when its rate drops below the Hubble rate, fixing the DM number density to a modified yield through the dark freeze-out. Odd processes, such as 2 ↔ 3 , could be neglected for DM potential without odd powers terms and are forbidden for multiplicative Z n symmetry conservation for even n 122 , like in most well-established DM setups 141 . As shown in 122,142 , if DM relic abundance is solely computed via a Higgs decay in a freeze-in framework, without dark thermalization for 4 → 2 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 fulfill additional constraints from cosmology 140,142 , such as the isocurvature bound, for which the scalar mass is bounded from above: with H * the Hubble parameter at the inflationary scale.
The 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. See 141,143-147 for insights regarding 3 → 2 dark thermalization and explicit formulas for the relic density for a SIMP DM with 3 ↔ 2 annihilations, which do not have immediate analytic representations.
For smaller interactions with the visible sector, the thermal production of DM particles is insignificant and DM must come from a non-thermal mechanism, leading to a Super-WIMP (SWIMP)-like scenario, for example through a direct DM-producing inflaton decay [148][149][150] , if the heavy Higgs scalar responsible for the G(2) → SU(3) transition is identified as the inflaton field. In this inflaton-portal case 151 , the glueball abundance is basically fixed by few parameters, i.e. the reheating temperature T rh , the inflaton mass and branching ratio into G(2) gluons/ scalar glueball B S 148 : For a cosmological reheating above the GeV scale, the B S (M S ∕M H ) factor should be quite small, i.e. < 10 −9 . The scenario works well especially for large DM masses, between the weak scale and the PeV scale 149 , and for extremely decoupled EeV candidates 151 .
Finally we are going to briefly discuss possible manifestations of G(2) gluons in the present Universe. The six massive gluons from the broken exceptional G(2) group, mainly as stable J PC = 0 ++ (and possibly J PC = 2 ++ ) glueballs, could clump and organize into dark matter halos, in form of a heavy bosons gas or in some fluid systems 152,153 , for sufficiently low temperatures and/or enough high densities. Indeed, the idea of a (super) fluid dark matter has recently attracted attention in literature, from Khoury's promising proposal of a unified superfluid dark sector 154,155 . 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 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 in [157][158][159] .
In our scenario we hypothesize the main constituents of a possible dark fluid are broken-G(2) massive gluons composite states. They are certainly massive, so that we cannot invoke the axion-like description. But still G(2)  www.nature.com/scientificreports/ 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 pseudoscalar [160][161][162] , most likely for a quartic order repulsive self-interaction 162 ), Proca stars (for massive spin 1 bosons 163-165 -one could invoke a correspondence with 3-gluons glueballs), BEC stars 166 , or QCD glueballs stars 133 . The key ingredients to try to build up stellar objects with non-light bosons are mainly the magnitude of the quartic order self-interaction 4 of the constitutive boson and its mass. Our glueball scalar field 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, g is the metric of the space-time, g its determinant and G is Newton's gravitational costant. The variation of the action with respect to the metric leads to the related Einstein equations 162,167 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 + 4 4 |S| 4 , where S ≡ 4 > 0 ( 4 < 0 ) signifies a repulsive (attractive) interaction: repulsive self-interactions can give rise to very dense boson stars. The maximum mass for such an object is 162,166,168 where M Pl ≡ 1∕  161 . 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 behaviour 103 , we could qualitatively approximate it to a SU(4) theory and obtain 4 ∼ 2 , which almost saturates the perturbativity condition 4 < 4π . For example, for such a 4 , both the aforementioned M max ranges, i.e. 10 −8 M ⊙ and few M ⊙ , fall into mass windows with weak constraints for MACHOs from up-to-date microlensing analysis 169,170 . Obviously, the feasibility of a star made of self-interacting scalars also depends on a correct estimate of the possible 3 → 1 , 3 → 2 and 4 → 2 annihilation processes inside the star 168 , which in turn depend on the symmetries (e.g. Z n ) of the exotic sector; for a non-conserved glueballs number, i.e. for a real scalar with no additional symmetries, 3∕4 → 2 processes could trigger the decay of massive bosons stars,leading to severe bounds on their mass: M max < 10 −11 M ⊙ .
For the so-called mini boson stars (with no scalar interaction potential) and Proca stars, the maximum mass is quite smaller and scales with ∼ . Furthermore, unlike complex scalar fields, 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 achieved 167 . These "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 observations 2,171,172 , mainly from colliding galaxy clusters dynamics, like the the Bullet Cluster 173 . The DM-DM scattering cross section must qualitatively satisfy which is consistent with the one obtained in 137 . In principle the intensity of the self-interaction is not effectively bounded by the scalar mass scale of the present theory: the more the DM particle is massive the more the constraint for 4 is relaxed. Exploiting Eq. (36) with a FIMP-like HS ∼ 10 −10 , Eq. (43) becomes For the aforementioned 4 ∼ 2 self-interaction assumption, this formula implies an expectation value for the heavy Higgs satisfying w ≥ 10 4 GeV, which is completely consistent with a beyond SM phenomenology. Such a www.nature.com/scientificreports/ FIMP/SIMP-like singlet scalar model, constrained by astrophysical observations and the perturbativity condition, naturally leads to a DM mass in the GeV/sub-GeV region 122 (Eq. (43)). This is also valid from Eq. 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 phenomenology 167 . 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 emission 167 . For example, a hypothetical binary SU(N) gluons star could be disentangled from a binary black hole system, due to possible differences in the gravitational wave frequency and amplitude, as demonstrated in 119 : this comes from the fact that the mass-radius relation of the glueball dark star, as in our case, is different from that of a black hole, as discussed before. In addition, in 119 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 in 174 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 dualities 175 . 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 by 109 demonstrates the possibility to distinguish a simplified 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 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 fundamental particles lie, in the form of six additional massive bosons organized in composite states, disconnected by Standard Model dynamics: the exceptional-colored G(2) gluons. These 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 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 emergence 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 produced a secluded sector of cold exceptional-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 masses, whose composite states might play the role of dark matter. Some accidental stability mechanisms for the dark glueballs have been proposed. 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 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 defined by the vaccum expectation value of the extra Higgs. The resulting exceptional glueball DM is certainly compatible with direct, indirect, collider searches and astrophysical observations, as it is almost collisionless. Several cosmological scenarios 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 symmetries further beyond octonions, sedenions show an intriguing property: they still have G(2) as a fundamental automorphism, but "tripled" by an S 3 factor, which resembles the three fermion families of the Standard Model and its S 3 -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. 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 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.
The development of a definitive theory is beyond the purpose of the present phenomenological proposal, which is intended as a guideline for further speculations. www.nature.com/scientificreports/ Publisher's note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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