Eur. Phys. J. C (2019) 79:446 https://doi.org/10.1140/epjc/s10052-019-6967-1 Regular Article - Theoretical Physics Three fermion generations with two unbroken gauge symmetries from the complex sedenions Adam B. Gillard, Niels G. Gresnigta Department of Mathematical Sciences, Xi’an Jiaotong-Liverpool University, 111 Ren’ai Road, Suzhou HET, Jiangsu 215123, China Received: 24 April 2019 / Accepted: 17 May 2019 / Published online: 27 May 2019 © The Author(s) 2019 Abstract We show that three generations of leptons and constant. Octonions also describe geometry in 10 dimen- quarks with unbroken Standard Model gauge symmetry sions [5], which have made them useful in supergravity and SU(3)c × U(1)em can be described using the algebra of superstring theories [6–9]. Recently there has been a revived complexified sedenions C ⊗ S. A primitive idempotent is interest in using the division algebras to attempt to construct constructed by selecting a special direction, and the action of a theoretical basis for the observed Standard Model (SM) this projector on the basis of C ⊗ S can be used to uniquely gauge groups and observed particle spectrum [10–17]. split the algebra into three complex octonion subalgebras A Witt decomposition of the adjoint algebra of left actions ∼ C ⊗ O. These subalgebras all share a common quaternionic (C ⊗ O)L = C(6) of C ⊗ O, generated from all composed subalgebra. The left adjoint actions of the 8 C-dimensional left actions of the complex octonions on themselves decom- C ⊗ O subalgebras on themselves generates three copies of poses (C⊗O)L into minimal left ideals [18]. The basis states the Clifford algebra C(6). It was previously shown that the of these ideals were recently shown to transform as a single minimal left ideals of C(6) describe a single generation of generation of leptons and quarks under the unbroken uni- fermions with unbroken SU(3)c ×U(1)em gauge symmetry. tary symmetries SU(3)c and U(1)em [11]. In a complemen- Extending this construction from C ⊗ O to C ⊗ S naturally tary work by [19] it was shown that by considering the right leads to a description of exactly three generations. adjoint action of C(6) on the eight minimal left ideals of the algebra, one can also include the spinorial degrees of free- dom.1 One copy of C(6) = C(8) therefore provides the full 1 Introduction degrees of freedom for one generation of Dirac spinors. The main result presented here is that if we instead consider a The role of division algebras, particularly the octonion alge- decomposition of (C ⊗ S)L , where the sedenions S are the bra, in particle physics has a long history. Theorems by Hur- next Cayley-Dickson algebra after the octonions, then these witz [1] and Zorn [2] guarantee that there are only four earlier results are naturally extended from one single gener- normed division algebras, which are also the only four alter- ation to exactly three. native division algebras; the reals R, complex numbers C, Given that a considerable amount of the SM structure for quaternions H, and the octonions O. The first results relating a single generation of fermions can be realised in terms of the the octonions to the symmetries of (one generation of) quarks complex octonions C ⊗ O, and its associated adjoint algebra ∼ goes back to the 1970s [3,4]. The hypothesis that octonions of left actions (C⊗O)L = C(6), a natural question is how to might play a role in the description of quark symmetries fol- extend these encouraging results from a single generation to lows from the observation that Aut(O) = G2 contains the exactly three generations. Although it is possible to represent physically important subgroup SU(3), corresponding to the three generations of fermion states inside a single copy of subgroup that holds one of the octonionic imaginary units C(6) [20,21], in that case the physical states are no longer 1 [11] describes one generation of fermions in terms of two out of eight minimal left ideals of C(6), without any mention of spinorial degrees of freedom. These can be acocunted for by including a copy of the complex quaternions C ⊗ H.[19] on the other hand writes down all eight of the minimal left ideals, which explicitly includes the spinorial a e-mail: [email protected] degrees of freedom. 123 446 Page 2 of 11 Eur. Phys. J. C (2019) 79 :446 the basis states of minimal left ideals as for the case of a 2 One generation of fermions as the basis states of the ∼ single generation. A three generation representation in terms minimal left ideals of (C ⊗ O)L = C(6) of C(6) therefore comes at the cost of giving up the elegant construction of minimal left ideals giving rise to a single One of the main observations in [11]2 and [19] is that the generation. Instead, one might therefore look for a larger basis states of the minimal left ideals of C(6) transform mathematical structure to describe three generations. as a single generation of leptons and quarks with unbro- Since describing a single generation requires one copy of ken SU(3)c and U(1)em gauge symmetries. In [11] the alge- the octonions, it seems reasonable to expect a three gener- bra C(6) is identified as the adjoint algebra of left actions ation model to require three copies of the octonions. This (C ⊗ O)L whereas in [19] C(6) is taken without any ref- makes the exceptional Jordan algebra J3(O) a natural can- erence to division algebras. In this section the minimal left ∼ didate. This intriguing mathematical structure and its role ideals of (C ⊗ O)L = C(6) are constructed using the Witt in particle physics has recently been explored by various decomposition for C(6). The basis states of these ideals are authors [22–24]. then shown to transform as a single generation of leptons and A different approach is taken here. C and O are both divi- quarks with unbroken SU(3)c and U(1)em. More details of sion algebras, and starting from R, all the other division alge- can be found in sections 4.5 and 6.6 of [11]. bras can be generated via the Cayley-Dickson process. Given Leptons and quarks are Dirac spinors. One way of includ- that this process continues beyond the octonions, one may ing the spinorial degrees of freedom is to enlarge the algebra wonder if the next algebra in the sequence, the sedenions S, from C ⊗ O to C ⊗ H ⊗ O and to consider both right and may be a suitable mathematical structure to represent three left adjoint actions. The approach taken in [11] it to con- generations of fermions. This paper confirms that the answer sider the combined left and right adjoint action of C ⊗ H ∼ is yes. which is C(2) ⊗C C(2) = C(4), providing a representa- Starting with C⊗S we repeat the initial stages of the con- tion of the Dirac algebra. On the other hand, in [19] all the struction of [11]. We define a projection operator by choos- spinor degrees of freedom are contained in a single copy of ing a special direction, and use it to obtain a split basis of C(6). The left actions of C(6) on its eight minimal left 14 nilpotent ladder operators, defining two maximal totally ideals generates the unbroken gauge symmetries, whereas isotropic subspaces (MTIS). But instead of using these nilpo- the right actions contain the Dirac and isospin symmetries ∼ tents to construct two minimal left ideals of (C ⊗ S)L ,we C(6) = C(4)Dirac ⊗C C(2)isospin. Therefore, C(6) is instead observe that these 14 nilpotents uniquely divide into large enough to contain the full degrees of freedom for a three sets of ladder operators, in such a way that each set pro- single generation of leptons and quarks. vides a basis for a C ⊗ O subalgebra of C ⊗ S. Our approach The octonions O are spanned by the identity 1 = e0 and therefore gives rise to three copies of the octonions from the seven anti-commuting square roots of minus one ei satisfying sedenions. In this construction one raising operator and one =−δ + , lowering operator is common to all three sets. The shared ei e j ije0 ijkek (1) ladder operators generate a common quaternionic subalge- where bra C ⊗ H =∼ C(2). From the three sets of ladder operators = = , 2 = , we construct six ideals. These ideals transform as three gen- ei e0 e0ei ei e0 e0 (2) erations of fermions and antifermions under the unbroken and is a completely antisymmetric tensor with value +1 gauge symmetry SU(3)c × U(1)em. ijk = , , , The next section reviews the construction of one gener- when ijk 124 156, 137, 235, 267 346 457. The mul- tiplication of octonions is shown in Fig. 1. The multiplication ation of fermions with unbroken SU(3)c × U(1)em gauge symmetry starting from the complex octonions C ⊗ O. Then structure here follows that of [11], but is not unique. In par- in Sect. 3 it is shown that the same construction can be ticular, the Fano plane in the earlier work [3] is based on a used to describe three generations of leptons in terms of different multiplication structure. three C(2) subalgebras of C(6), generated from three Each pair of distinct points lies on a unique line of three C ⊗ H subalgebras of C ⊗ O.