Special Grand Unification

We discuss new-type grand unified theories based on grand unified groups broken to their special subgroups as well as their regular subgroups. In the framework, when we construct four-dimensional (4D) chiral gauge theories, i.e., the Standard Model (SM), 4D gauge anomaly cancellation restricts the minimal number of generations of the 4D SM Weyl fermions. We show that in a six-dimensional (6D) $SU(16)$ gauge theory on $M^4\times T^2/\mathbb{Z}_2$ one generation of the SM fermions can be embedded into a 6D bulk Weyl fermion. For the model including three chiral generations of the SM fermions, the 6D and 4D gauge anomalies on the bulk and fixed points are canceled out without exotic 4D chiral fermions.

More than half a century ago, E. Dynkin showed in Refs. [23,24] that simple Lie algebras has not only regular subalgebras but also special (or irregular) subalgebras. All regular subalgebras are obtained by deleting dots from Dynkin diagrams, while special subalgebras are not. For example, for an embedding SU (3) ⊃ SU (2) × U (1), its SU (2) × U (1) is a maximal regular subalgebra of SU (3), while for an embedding SU (3) ⊃ SU (2), its SU (2) is a maximal special subalgebra. One of the branching rules of the regular subalgebra SU (3) ⊃ SU (2) × U (1) is 3 = (2)(1) ⊕ (1)(−2), where 3 in the left-hand side stands for an SU (3) representation, and (2)(1) and 1(−2) in the right-hand side show SU (2) × U (1) representations. (The same rule will be applied below.) One of the branching rules of the special subalgebra SU (3) ⊃ SU (2) is 3 = (3). For an embedding G ⊃ H, a maximal subalgebra H is a subalgebra of G if there is no larger subalgebra containing it except G itself. (See e.g., Refs. [3,25] in detail.) In principle, we can construct GUTs based on Lie groups broken to not only their regular subgroups but also their special subgroups. However, at present, there seem to be no GUTs by using special embeddings, which is referred as special grand unification or special GUTs. In the following, we consider large GUT gauge groups G broken to smaller GUT groups H, called small GUT gauge groups. The groups H is regarded as usual GUT groups e.g., SU (5), SO(10), E 6 . When we apply them for a special GUT model building, the rank of its large GUT gauge groups is usually much larger than small GUT gauge groups. Even a famous GUT review paper [2] written by R. Slansky does not contain enough information to construct special GUTs. Recently, simple Lie algebras with up to rank-15 and D 16 and their regular and special subalgebras are given in Ref. [3], where the information in its tables is not enough to survey special GUTs.
We can find candidates for large GUT gauge groups in special GUTs by using usual methods about branching rules of Lie algebras under their maximal subalgebras discussed in Refs. [3,25]. Under usual requirements for their small GUT gauge groups that have complex representations and contain the SM gauge group G SM := SU (3) C × SU (2) L × U (1) Y the candidates for large GUT gauge groups broken to maximal special subgroups in 4D theories are e.g., In five-or higher-dimensional theories, the requirement for the small GUT gauge groups that contain the SM gauge group G SM leads to additional candidates e.g., where almost the above pairs are explicitly shown by E. Dynkin in Ref. [23].
To construct unified theories beyond the SM based on a GUT gauge group G GUT ⊃ G SM , we must take into account for not only how to realize the SM gauge group at a vacuum but also how to embed the SM matter content into the GUT matter content. To do so, we need the information about the branching rules of Lie algebras under their subalgebras. It is known that their branching rules can be calculated by using their corresponding projection matrices introduced in Ref. [26]. Many examples of the branching rules of simple Lie algebras with up to rank 8 under their regular and special semi-simple subalgebras are listed in Ref. [27]. Recently, more examples up to rank-15 and SO(32) including non-semi-simple subalgebras are given in Ref. [3]. Other cases can be calculated by using appropriate computer programs, such as Susyno program [28] and LieART [29], and appropriate projection matrices, which can be calculated by using a usual weight diagram method discussed in Refs. [3,25].
It is also known that one generation of the five SM left-handed Weyl fermions, which consist of a quark doublet, up-and down-type quarks, a lepton doublet and a charged lepton, can be embedded into an SU (5) reducible representation 10 ⊕ 5 , where we will omit left-handed if it is not needed. One generation of the SM Weyl fermions and vectorlike fermions can be embedded into an SU (n) (n ≥ 6) reducible representation n(n−1) 2 ⊕ (n − 4)n . (Note that their 4D SU (n) gauge anomaly coefficients from the 4D Weyl fermions is zero. It can be checked by using tables in Ref. [3].) One generation of the SM fermions and a SM singlet fermion can be embedded into an SO(10) spinor representation 16. For the other candidates for rank-4 and -5 GUT gauge groups SO(9), U Sp(8), F 4 , SO (11), and U Sp(10), it is examined which representations of them can contain the SM fermions in Ref. [3].
There are several good features of special GUTs. First, we can eliminate almost all unnecessary U (1)s even if we consider very large GUT gauge groups. For example, the rank of a gauge group SO(32) is sixteen. We have the SM gauge group and twelve extra U (1)s if we use only regular embeddings, while we have only the SM gauge group and two extra U (1)s if we use a special embedding SU (16) ⊃ SO (10). Second, by using only regular embeddings, we cannot embed the representations of the SM fermions into an SO(32) vector representation 32, while by using regular and special embeddings SO(32) ⊃ SU (16) × U (1) ⊃ SO(10) × U (1), the representations of the SM fermions can be embedded into an SO(32) vector representation because an SO(32) vector representation 32 is decomposed into SO(10) spinor representations 16 and 16 [3]. Third, asymptotic freedom may be realized in special GUTs. This is because the larger contribution comes from large GUT gauge fields while the smaller contribution comes from the GUT fermions.
In this paper, we discuss special GUTs based on a large GUT gauge group G broken to its maximal special subgroup H(⊂ G), where H is regarded as a usual GUT gauge group. In 4D framework, to realize a 4D chiral gauge theory, the large GUT groups G and their maximal special subgroups H must contain complex representations. In the following discussion, we focus on G = SU (n) with less than rank-30 and H = SU (k), SO(4k + 2), E 6 . Also, the special groups H must have rank four or greater because the rank of the SM gauge group G SM is four. The conditions are satisfied by the following nine pairs of Lie groups and their maximal special subgroups: SU ( n(n−1) 2 ) ⊃ SU (n) (n = 5, 6, 7, 8); SU ( n(n+1) 2 ) ⊃ SU (n) (n = 5, 6, 7); SU (16) ⊃ SO(10); and SU (27) ⊃ E 6 .
We find the following facts from Tables in Ref. [3] and its calculating methods. We consider 4D SU (n) (n ≤ 30) special GUTs whose fermions are a 4D Weyl fermion in an SU (n) reducible representation (n − 4)n ⊕ n(n−1) 2 , where the representation is 4D anomaly-free. Only for the SU (16) ⊃ SO(10) and SU (27) ⊃ E 6 cases, their matter contents contain the SM chiral fermions. For the SU (16) ⊃ SO(10) case, the fermion is a 4D SU (16) 12 × 16 ⊕ 120 Weyl fermion, which stands for a 4D left-handed Weyl fermion in an SU (16) reducible representation 12 × 16 ⊕ 120 , where the similar notation is used below. As we will see later, for 4D framework there are twelve generations of the SM Weyl fermions but fortunately there is no exotic non-SM chiral fermions due to a special property of the SU (16) complex representation 120. For the SU (27) ⊃ E 6 case, its matter content contains non-SM chiral fermions.
The main purpose of this paper is to show that in a 6D SU (16) special GUT on M 4 × T 2 /Z 2 we can realize three generations of the 4D SM Weyl fermions from twelve 6D SU (16) 16 bulk Weyl fermions.
This paper is organized as follows. In Sec. 2, we first discuss basic properties of a special embedding SU (16) ⊃ SO(10) in 4D framework. After that, by using the special embedding, we construct a 6D SU (16) special GUT on M 4 × T 2 /Z 2 . Section 3 is devoted to a summary and discussion.

Special grand unification
Before we start to construct 6D special GUTs, we examine how to embed the SM Weyl fermions into SU (16)  (2.1) Further, the SO(10) spinor representation 16 is decomposed as usual into where the convention of their U (1) normalization is the same as one in Ref. [3]. Obviously, we can identify an SU (16) 16 fermion as one generation of the SM fermions plus a right-handed neutrino. The 4D anomaly coefficient of an SU (16) 16 fermion is non-zero, while the 4D anomaly coefficient of any SO(10) fermion is zero. For SU (16) models instead of SO(10) ones, we cannot construct 4D anomaly-free theories with a chiral matter content by using only 4D SU (16) 16 and/or 16 Weyl fermions. We need to introduce an additional Weyl fermion to cancel out 4D SU (16) gauge anomaly. A primary candidate is a 4D SU (16) 120 Weyl fermion, where 120 is the second lowest dimensional complex representation of SU (16) and has the second smallest value of the 4D anomaly coefficient in SU (16) complex representations with at least up to 10 8 dimension. Its anomaly coefficient of SU (16) is −12. By using SU (16) complex representations 16 and 120, we can realize a 4D anomaly-free chiral gauge theory whose fermion content is a 4D SU (16) 12 × 16 ⊕ 120 Weyl fermion.
From the above discussion, we find that in 4D framework the 4D anomaly-free SU (16) chiral gauge theories cannot realize the SM matter content because the 4D SU (16) anomaly cancellation does not allow three chiral generation of quarks and leptons. The 4D anomaly situation is exactly the same as each 4D fixed point in 6D theories with two dimensional orbifold extra dimension T 2 /Z 2 . However, as we will see below, even when effectively twelve 4D SU (16) 16 Weyl fermions from six 6D SU (16) 16 positive and negative Weyl fermions exist at an 4D fixed point, three 6D positive Weyl fermions have zero modes and their zero modes are only 4D SU (16) 16 Weyl fermion. When we take into account appropriate symmetry breaking effects, the zero modes become three generations of the SM fermions.
Here we check the contribution to 6D bulk and 4D brane anomalies from the above four 6D Weyl fermion set. Obviously, the fermion set does not contribute to 6D gauge anomaly because the number of 6D SU (16) 16 positive and negative Weyl fermions are the same. In other words, the fermion set forms 6D vectorlike fermions in the bulk. We calculate 4D gauge anomaly numbers at four fixed points (y j , v j )(j = 0, 1, 2, 3) by using 4D SU (16) anomaly coefficients listed in Ref. [3], where for our convention the anomaly coefficient of a 4D SU (16) 16 Weyl fermion is +1. The above first 6D positive Weyl fermion contributes +1 to the SU (16) anomaly number for all the four fixed points (y j , v j )(j = 0, 1, 2, 3); the second 6D positive Weyl fermion contributes +1 for the fixed points (y 0 , v 0 ) and (y 1 , v 1 ) and −1 for the fixed points (y 2 , v 2 ) and (y 3 , v 3 ); the third 6D negative Weyl fermion contributes +1 for the fixed points (y 0 , v 0 ) and (y 3 , v 3 ) and −1 for the fixed points (y 1 , v 1 ) and (y 2 , v 2 ); the last 6D negative Weyl fermion contributes +1 for the fixed points (y 0 , v 0 ) and (y 2 , v 2 ) and −1 for the fixed points (y 1 , v 1 ) and (y 3 , v 3 ). Therefore, one set of the four 6D SU (16) 16 Weyl fermions contributes +4 for the fixed point (y 0 , v 0 ) and 0 for the other three fixed points (y 1 , v 1 ), (y 2 , v 2 ), and (y 3 , v 3 ). Since the SM has three chiral generations of quarks and leptons, we need to introduce three sets of the four 6D Weyl fermions. Its total anomaly number from three set of the four 6D Weyl fermions are +12 for the fixed point (y 0 , v 0 ) and 0 for the other fixed points (y 1 , v 1 ), (y 2 , v 2 ), and (y 3 , v 3 ).
To cancel the 4D SU (16) anomaly on all the fixed points, we need to introduce at least one 4D Weyl fermion of an SU (16) non-anomaly-free complex representation on the fixed point (y 0 , v 0 ). As we saw before, a 4D SU (16) 120 Weyl fermion contributes −12 to the SU (16) anomaly number. Therefore, by introducing a 4D SU (16) 120 Weyl brane fermion at the fixed point (y 0 , v 0 ), all the 4D SU (16) anomalies at the fixed points are canceled out.

Summary and discussion
In this paper, we constructed SU (16) special GUTs by using a special embedding SU (16) ⊃ SO (10). In this framework, we found that the 4D SU (16)  Since an SO(10) tensor product 16 ⊗ 16 ⊗ 120 contains singlet, the 6D SU (16) 16 bulk and 4D SU (16) 120 brane fermions can be mixed via the VEV of the 5D SU (16) 16 brane scalar once its corresponding brane interaction term is generated. The effective mass term divides up-type quark mass with down-type quark and charged lepton masses. Next, we introduced a 5D SU (16) 255 brane scalar field on the brane y = 0 and its VEV is responsible for breaking SU (5)(⊂ SU (16)) to G SM . The VEV of the brane scalar field can mix three 6D SU (16) 16 bulk Weyl fermions containing zero modes with the other nine 6D SU (16) 16 ones. Therefore, the SU (16) model seems to realize the SM fermion masses, but seems to give us no prediction about quark and lepton masses. We will leave the detail analysis in future studies. In the 6D SU (16) special GUT, we use the fact that an SU (16) complex representation 120 is identified with an SO(10) real representation 120 for the special embedding SU (16) ⊃ SO (10). It is important to eliminate non-SM chiral fermions at low-energy physics. This type of representations, which is complex under a Lie group but is real under its maximal special subgroup, cannot be found in the other special embeddings G ⊃ H with up to the rank-30 of G Many branching rules of SU (10) ⊃ SU (5), SU (15) ⊃ SU (5), and SU (15) ⊃ SU (6) are listed in Ref. [3]; for branching rules of SU (21) ⊃ SU (6), SU (21) ⊃ SU (7), SU (28) ⊃ SU (7), SU (28) ⊃ SU (8), SU (27) ⊃ E 6 , there are no references, but their branching rules can be calculated by using methods in Ref. [3]. We consider a special GUT based on a small GUT gauge group E 6 . As the same as SO(10) gauge theories, there is no 4D E 6 gauge anomaly, while there can be 4D SU (27) gauge anomaly in 4D chiral theories. We have to check how to cancel out the 4D SU (27) gauge anomaly. When we take the 4D anomaly coefficient of a 4D SU (27) 27 Weyl fermion as +1, the 4D anomaly coefficient of a 4D SU (27) 2925 Weyl fermion is −252, which can be calculated by using the general formula of 4D anomaly coefficients of SU (n) representations discussed in Refs. [3,36]. Thus, if we consider a model whose fermion matter content is a 4D SU (27) 252 × 27 ⊕ 2925 Weyl fermion, its matter content does not contain exotic non-SM chiral fermions. However, there are 252 generations of the SM Weyl fermions. It seems to be impossible to realize three chiral generations of quarks and leptons.
In our 6D SU (16) special GUT, we assumed that 5D brane scalar fields on the 5D brane on 5th dimensional space y = 0 are responsible for breaking the large GUT gauge group SU (16) to G SM . However, a part of its symmetry breaking may be replaced by orbifold BCs and/or the Hosotani mechanism breaking. In the above discussion, we took the projection matrix P j16 (j = 0, 1, 2, 3) for the SU (16) representation 16 as an identity matrix, but we can choose different ways. For example, when we take P 016 = P 116 = I and P 216 = P 316 = diag(I 15 , −1), the orbifold BCs break SU (16) to SU (15) × U (1), and we assume that the non-vanishing VEV of the SU (16) 5440 GUT Higgs breaks the large GUT gauge group SU (16) to the small GUT gauge group SO(10). The two symmetry breaking combination leaves only SU (5), where it can be recognized by using the correspondence of the root vectors between SU (16) and its special subgroup SO (10). Also, the SU (16) adjoint representation 255 is decomposed into SO(10) representations 210 and 45. The SO(10) representation 210 includes not only the SU (5) adjoint representation 24 and but also all the SU (5) fundamental representations 5, 10, 10, and 5. It can be used as GUT or the electro-weak symmetry breaking Higgses in gauge-Higgs GUT scenarios.
The use of the special embedding SU (16) ⊃ SO(10) may be interesting for string inspired GUT model builders because one of the regular embeddings of SO(32) is SU (16) × U (1). The special embedding may be incorporated with model buildings in an SO(32) heterotic string theory. For one of the amazing features, SO(10) spinor representations 16 and 16 are embedded into an SO(32) vector representation 32.
Another motivation may come from string GUT model buildings to realize higher-level gauge groups by lower "costs" compared with e.g., so-called diagonal embeddings [9,37,38]. For example, by using a diagonal embedding, we need four SO(10) 1 s to obtain SO(10) 4 , while by using an special embedding SU (16) 1 ⊃ SO(10) 4 , we need one SU (16) 1 , where the subscript is the index of embedding, which stands for the ratio of the second order Dynkin indices of corresponding representations between a Lie algebra and its subalgebra. The rank in the former is twenty, while one in the latter is fifteen [37].