Three generations of colored fermions with $$S_3$$ family symmetry from Cayley–Dickson sedenions

Eur. Phys. J. C (2023) 83:747 https://doi.org/10.1140/epjc/s10052-023-11923-y Regular Article - Theoretical Physics Three generations of colored fermions with S3 family symmetry from Cayley–Dickson sedenions Niels Gresnigta , Liam Gourlay, Abhinav Varma Department of Physics, Xi’an Jiaotong-Liverpool University, 111 Ren’ai Rd., Dushu Lake Science and Education Innovation District, Suzhou Industrial Park, Suzhou 215123, People’s Republic of China Received: 19 June 2023 / Accepted: 9 August 2023 © The Author(s) 2023 Abstract An algebraic representation of three generations of fermions with SU (3)C color symmetry based on the Cayley–Dickson algebra of sedenions S is constructed. Recent constructions based on division algebras convincingly describe a single generation of leptons and quarks with Standard Model gauge symmetries. Nonetheless, an algebraic origin for the existence of exactly three generations has proven difficult to substantiate. We motivate S as a natural algebraic candidate to describe three generations with SU (3)C gauge symmetry. We initially represent one generation of leptons and quarks in terms of two minimal left ideals of C(6), generated from a subset of all left actions of the complex sedenions on themselves. Subsequently we employ the S3 automorphism of order three, which is an automorphism of S but not of O, to generate two additional generations. Given the relative obscurity of sedenions, efforts have been made to present the material in a self-contained manner. 1 Introduction Despite its great practical success in colliders and other experiments, there are several unexplained features of the Standard Model of particle physics (SM) which lack a deeper theoretical motivation. These include, among others, a derivation of the SM gauge group from first principles, an explanation for why some representations of the SM gauge group correspond to particle multiplets whereas others do not, and an account for why fermions come in three generations. These theoretical shortcomings may be suggestive that the SM ultimately emerges from a more fundamental physical principle or mathematical structure. In an attempt to establish the geometric and algebraic roots of the SM, several proposals have been put forth over the years which take as its essential mathematical ingredients (tensor products of) the only four normed division algebras over the reals: R, C, H, and O. Instead of unifying the internal symmetries into a single larger group, as is done in grand unified theories (GUTs) such as SU (5) and Spin(10), these division algebraic approaches attempt to unify the gauge groups together with the leptons and quarks that they act on into a single unified algebraic framework, in terms of an algebra acting on itself. The octonions O, the largest of the division algebras, were first considered in the 70s for their intriguing efficacy in describing quark color symmetry [1]. Dixon [2–4] considers the algebra R ⊗ C ⊗ H ⊗ O and its invariant subspaces in connection to the particles and charges of the SM. The algebra R ⊗ C ⊗ H ⊗ O has exactly the right dimensions (32 complex) to describe one generation of fermions. In a closely related approach, Furey studies the minimal ideals of the Clifford algebras C(4), and C(6), generated from C ⊗ H, and C ⊗ O respectively [5,6]. In her approach, the leptons and quarks correspond to elements of these minimal ideals, and the gauge symmetries are those unitary symmetries that preserve the ideals. In particular, the C ⊗ O part of Dixon’s algebra can be associated to the color and electric charge internal degrees of freedom, with the color gauge group SU (3) corresponding to the maximal subgroup of the exceptional group G 2 of automorphisms of O which fixes one of the octonion units. Many others have contributed to these, and related, algebraic approaches including those based on topology [7–11], exceptional Lie groups [12–17], Clifford algebras [18–26], and Jordan algebras [27–32]. Existing division algebraic models offer an elegant algebraic construction for the internal space of a single generation of leptons and quarks. Despite several attempts [4,33,34], a clear algebraic origin for the existence of three generations a e-mail: (corresponding author) 0123456789().: V,-vol 123 747 Page 2 of 13 is yet to be found. The Pati–Salam model, as well as both the SU (5) and Spin(10) grand unified theories likewise correspond to single generation models, lacking any theoretical basis for three generations, which ultimately has to be imposed by hand. Furey identifies three generations of color states directly from the algebra C(6) generated from the adjoint actions of C ⊗ O [33]. The algebra C(6) is 64 complex dimensional. Constructing two representations of the Lie algebra su(3) within this algebra, the remaining 48 degrees of freedom transform under the action of the SU (3) as three generations of leptons and quarks. The most obvious extension is to include U (1)em via the number operator, which works in the context of a one-generation model, but fails to assign the correct electric charges to states in a three-generation model. A generalized action that leads to a generator that produces the correct electric charges for all states is introduced in [35]. Dixon on the other hand considers the algebra T6 = C ⊗ 2 H ⊗ O3 , where T = R ⊗ C ⊗ H ⊗ O in order to represent three generations, with a single generation being described by T2 , a complexified (hyper) spinor in 1+9D spacetime [3]. However, the choice T6 , as opposed to any other T2n appears rather arbitrary, although can be motivated from the Leech lattice. These division algebraic models share many similarities with those based on the exceptional Jordan algebra J3 (O) consisting of three by three matrices over O, which has likewise been proposed to describe three generations [27–32]. In these models, each of the three octonions in J3 (O) is likewise associated with one generation via the three canonical J2 (O) subalgebras of J3 (O). In [24] it is argued that R ⊗ C ⊗ H ⊗ O-valued gravity can naturally describe a grand unified field theory of Einstein’s gravity with a Yang–Mills theory containing the SM, leading to a SU (4)4 symmetry group that potentially extends the SM with an extra fourth family of fermions. The existence of a fourth generation of fermions lacks experimental support however. In [36] it is shown how, by choosing a privileged C subalgebra of O, it is possible to reduce ten dimensional spacetime represented by S L(2, O) to four dimensional spacetime S L(2, C). This process of dimensional reduction naturally isolates three H subalgebras of O: those that contain the privileged C subalgebra. These three intersecting H subalgebras are subsequently interpreted as describing three generations of leptons. Starting with R, each of the remaining three division algebras can be generated via what is called the Cayley–Dickson (CD) process. This process does not termina (...truncated)