String-inspired special grand unification

Abstract

We discuss a grand unified theory (GUT) based on an |$SO(32)$| GUT gauge group broken to its subgroups including a special subgroup. In the |$SO(32)$| GUT on the six-dimensional (6D) orbifold space |$M^4\times T^2/\mathbb{Z}_2$|⁠, one generation of the standard model fermions can be embedded into a 6D bulk Weyl fermion in the |$SO(32)$| vector representation. We show that for a three-generation model, all the 6D and 4D gauge anomalies in the bulk and on the fixed points are canceled out without exotic chiral fermions at low energies.

1. Introduction

Grand unification [1] is one of the most attractive ideas for constructing unified theories beyond the standard model (SM). It is known—see, e.g., Refs. [2,3]—that the candidates for grand unified theory (GUT) gauge groups in four-dimensional (4D) theories are only |$SU(n)\ (n\geq 5)$|⁠, |$SO(4n+2)\ (n\geq 2)$|⁠, and |$E_6$| because of the rank and type of their representations, while the candidates for GUT gauge groups in higher-dimensional theories are |$SU(n)\ (n\geq 5)$|⁠, |$SO(n)\ (n\geq 9)$|⁠, |$USp(2n)\ (n\geq 4)$|⁠, |$E_n\ (n=6,7,8)$|⁠, and |$F_4$|⁠. Many GUTs have been discussed before, e.g. in Refs. [1,4–8] for 4D GUTs, Refs. [9–22] for 5D GUTs, and Ref. [23] for 6D GUTs.

Recently, a new type of GUT has been proposed by the author [24]. In other GUTs, the GUT gauge groups are broken to their regular subgroups, e.g., |$E_6\supset SO(10)\times U(1)\supset SU(5)\times U(1)\times U(1) \supset G_{\rm SM}\times U(1)\times U(1)$|⁠. In the new GUT, called a special GUT, the GUT gauge groups are broken down to special subgroups. (For Lie groups and their regular and special subgroups, see, e.g., Refs. [3,25–27]).

In Ref. [24], the author proposed an |$SU(16)$| special GUT whose gauge group |$SU(16)$| is broken to a special subgroup |$SO(10)$|⁠. The results are summarized as follows. In a 4D |$SU(16)$| special GUT, one generation of quarks and leptons can be embedded into a 4D |$SU(16)$||${\bf 16}$| Weyl fermion; the 4D |$SU(16)$| gauge anomaly restricts the minimal number of generations. Unfortunately, the minimal number is 12 in the 4D framework. In a 6D |$SU(16)$| special GUT on the 6D orbifold space |$M^4\times T^2/\mathbb{Z}_2$|⁠, one generation of quarks and leptons can be embedded into a 6D |$SU(16)$||${\bf 16}$| Weyl fermion; the 6D |$SU(16)$| gauge anomaly and the 4D |$SU(16)$| gauge anomaly on the fixed points restrict the minimal number of generations—three generation of quarks and leptons are allowed without 4D exotic chiral fermions.

Superstring theory [28,29] has been considered as a candidate for a unified theory to describe all the interactions, including gravity. There have been a lot of attempts to construct the SM from string theories. One such is based on |$E_8\times E_8$| and |$SO(32)$| heterotic string theories [30–41]. Usually, |$E_8\times E_8$| heterotic string model building is much more popular than |$SO(32)$| one. One of the biggest reasons seems to be that when we only consider regular embeddings, for the branching rules of |$SO(32)\supset SO(10)(\times U(1)^{11})$|⁠, the |$SO(32)$| vector and adjoint representations |${\bf 32}$| and |${\bf 496}$| do not contain the |$SO(10)$| spinor representations |${\bf 16}$| and |${\bf \overline{16}}$|⁠, while for the branching rules of |$E_8\supset SO(10)(\times U(1)^{3})$|⁠, the |$E_8$| adjoint representation |${\bf 248}$| contains the |$SO(10)$| spinor representations |${\bf 16}$| and |${\bf \overline{16}}$|⁠. However, for a special embedding, on the other hand, the branching rules of |$SO(32)(\supset SU(16)\times U(1)_Z)\supset SO(10)\times U(1)_Z$| for the |$SO(32)$| vector and adjoint representations |${\bf 32}$| and |${\bf 496}$| are given by  

\begin{align} {\bf 32}&= ({\bf 16})(2) \oplus({\bf \overline{16}})(-2), \label{branching-rule-32-SO10}\\ \end{align}

(1.1)

 

\begin{align} {\bf 496}&= ({\bf 210})(0) \oplus({\bf 45})(0) \oplus({\bf 120})(4) \oplus({\bf 120})(-4) \oplus({\bf 1})(0), \label{branching-rule-496-SO10} \end{align}

(1.2)

where we follow the convention of the |$U(1)$| normalization in Ref. [3]. Obviously, an |$SO(32)$| vector representation is decomposed into |$SO(10)$| spinor representations, and an |$SO(32)$| adjoint representation contains an |$SO(10)$| bi-spinor representation |${\bf 210}$|⁠. When we take into account the special embedding |$SU(16)\supset SO(10)$|⁠, |$SO(32)$| gauge theories easily contain |$SO(10)$| spinors. In the following discussion, we will not consider how to realize models from string theories.

There are several good features of special GUTs pointed out in Ref. [24]. First, almost all unnecessary |$U(1)$|s can be eliminated; e.g., |$SO(32)\supset G_{\rm SM}\times U(1)^{12}$| by using only regular embeddings, while |$SO(32)(\supset SU(16)\times U(1)_Z\supset SO(10)\times U(1)_Z) \supset G_{\rm SM}\times U(1)^2$| by using regular and special embeddings. Second, by using only regular embeddings, the SM fermions cannot be embedded into an |$SO(32)$| vector representation |${\bf 32}$|⁠, while by using regular and special embeddings |$SO(32)(\supset SU(16)\times U(1)_Z\supset SO(10)\times U(1)_Z) \supset G_{\rm SM}\times U(1)^2$|⁠, the SM fermions can be embedded into an |$SO(32)$| vector representation.

It is shown, e.g. in Refs. [2,3], that any 4D |$SO(32)$| gauge theory is a vectorlike theory since an |$SO(32)$| group has only real representations. To realize the SM, i.e., a 4D chiral gauge theory, we take orbifold space construction [42,43]. This allows us to realize 4D Weyl fermions from 5D Dirac fermions, 6D Weyl fermions, etc. In the 6D |$SU(16)$| special GUT [24], the nonvanishing vacuum expectation value (VEV) of a 5D |$SU(16)$||${\bf 5440}$| brane scalar is responsible for breaking the |$SU(16)$| GUT gauge group to its special subgroup |$SO(10)$| via the Higgs mechanism [44,45]. For |$SO(32)$| special GUTs, the |$SO(32)$| GUT gauge group can be broken to |$SO(10)$| by using the nonvanishing VEV of a scalar in an appropriate representation of |$SO(32)$|⁠; the lowest-dimensional representation is 86768. (The spontaneous symmetry breaking of |$SU(n)$| to its special subgroups has been discussed in, e.g., Refs. [46,47].)

In this paper, we will discuss an |$SO(32)$| special GUT on the 6D orbifold spacetime |$M^4\times T^2/\mathbb{Z}_2$|⁠. As in 6D |$SU(16)$| special GUTs, we need to take into account 6D and 4D gauge anomalies. In the same way as for the 6D |$SU(16)$| gauge anomaly in the 6D |$SU(16)$| special GUT [24], the 6D |$SO(32)$| gauge anomaly can be canceled out by introducing 6D positive and negative Weyl fermions in the same representation of the |$SO(32)$| gauge group. Unlike an |$SU(16)$| gauge group, an |$SO(32)$| gauge group itself has no 4D gauge anomaly for any fermion in any representation of |$SO(32)$|⁠, but there can be 4D gauge anomalies for its subgroups. We will see this in Sect. 3 in detail.

The main purpose of this paper is to show that in a 6D |$SO(32)$| special GUT on |$M^4\times T^2/\mathbb{Z}_2$| we can realize three generations of the 4D SM Weyl fermions from six 6D |$SO(32)$||${\bf 32}$| bulk Weyl fermions without 4D exotic chiral fermions at low energies, and without any 6D and 4D gauge anomaly.

This paper is organized as follows. In Sect. 2, before we discuss a special GUT based on an |$SO(32)$| gauge group, we quickly review the basic properties of |$SO(32)$| and its subgroups shown in Ref. [3]. In Sect. 3, we construct a 6D |$SO(32)$| special GUT on |$M^4\times T^2/\mathbb{Z}_2$|⁠. Section 4 is devoted to a summary and discussion.

2. Basics of SO(32) and its subgroups

First, we check how to embed the SM Weyl fermions into |$SO(32)$| vector multiplets. For regular and special embeddings |$SO(32)\supset SU(16)\times U(1)_Z\supset SO(10)\times U(1)_Z$|⁠, an |$SO(32)$| vector representation |${\bf 32}$| is decomposed into |$SO(10)$| spinor representations |${\bf 16}$| and |${\bf \overline{16}}$|⁠. Further, the |$SO(10)$| spinor representation |${\bf 16}$| is decomposed into |$G_{\rm SM}\times U(1)_X=SU(3)_C\times SU(2)_L\times U(1)_Y\times U(1)_X$| representations:  

\begin{align} {\bf 16}&= ({\bf 3,2})(-1)(1) \oplus({\bf \overline{3},1})(-2)(-3) \oplus({\bf \overline{3},1})(4)(1)\nonumber\\ &\quad\oplus({\bf 1,2})(3)(-3) \oplus({\bf 1,1})(-6)(1) \oplus({\bf 1,1})(0)(5). \end{align}

(2.1)

Since the |$SO(32)$| vector representation |${\bf 32}$| is real, a 4D Weyl fermion in the |$SO(32)$||${\bf 32}$| representation includes not only 4D SM Weyl fermions but also their conjugate fermions. To realize chiral fermions, we take the orbifold symmetry-breaking mechanism [42,43]. After taking into account orbifold effects, we can regard the zero modes of an |$SO(32)$||${\bf 32}$| fermion as one generation of the SM fermions plus a right-handed neutrino. Note that there are no 4D pure |$SO(32)$| gauge anomalies of any representation of the |$SO(32)$| gauge group, while there can be 4D |$SU(16)$| and |$U(1)$| anomalies generated by 4D Weyl fermions in complex representations of |$SU(16)$| and |$U(1)$|⁠, respectively. Then, after orbifolding, a maximal regular subgroup |$SU(16)\times U(1)_Z$| of |$SO(32)$| may be anomalous. We will discuss how to cancel out 4D pure |$SU(16)$|⁠, pure |$U(1)_Z$|⁠, mixed |$SU(16)$|–|$SU(16)$|–|$U(1)_Z$|⁠, and mixed grav.–grav.-|$-U(1)_Z$| generated by 6D bulk fermions in the next section.

We consider a symmetry-breaking pattern from |$SO(32)$| to |$G_{\rm SM}$|⁠. One way of achieving this is to use orbifold symmetry-breaking boundary conditions (BCs) and several GUT-breaking Higgses. One example is to choose orbifold BCs breaking |$SO(32)$| to |$SU(16)\times U(1)$| and to introduce three |$SO(32)$||${\bf 86768}$|⁠, |${\bf 496}$|⁠, |${\bf 32}$| scalar fields, where we assume their proper components get non-vanishing VEVs. First, the following orbifold BC for the |$SO(32)$| vector representation |${\bf 32}$| breaks |$SO(32)$| to |$SU(16)\times U(1)_Z$|⁠:  

\begin{align} P_{\bf 32}= \sigma_2\otimes I_2\otimes I_2\otimes I_2\otimes I_2, \end{align}

(2.2)

where the projection matrix |$P_{\bf 32}$| is proportional to the |$U(1)_Z$| generator and satisfies |$(P_{\bf 32})^2=I_{32}$|⁠. (The matrix form of |$P_{\bf 32}$| depends on the basis.) Next, the non-vanishing VEV of the |$SO(32)$||${\bf 86768}$| scalar field is responsible for breaking |$SO(32)\supset SU(16)\times U(1)_Z$| to |$SO(10)$| or |$SO(10)\times U(1)_Z$|⁠, where its branching rule of |$SO(32)\supset SU(16)\times U(1)_Z$| is given by  

\begin{align} {\bf 86768}&= ({\bf 18240})(0) \oplus({\bf 14144})(0) \oplus({\bf 5440})(8) \oplus({\bf \overline{5440}})(-8) \oplus({\bf 255})(0) \oplus({\bf 1})(0) \nonumber\\ &\quad \oplus({\bf 21504})(4) \oplus({\bf \overline{21504}})(-4) \oplus({\bf 120})(4) \oplus({\bf \overline{120}})(-4). \label{branching-rule-86768} \end{align}

(2.3)

|$SU(16)$||${\bf 18240}$| and |${\bf 5440}$||$({\bf \overline{5440}})$| contain singlets under their |$SO(10)$| special subgroup. Their nonvanishing VEV can break |$SU(16)$| to its special subgroup |$SO(10)$| [24], where their |$SO(10)$| decompositions are given in Ref. [3] by  

\begin{align} {\bf 18240}&= ({\bf 8910}) \oplus({\bf 5940}) \oplus({\bf 770}) \oplus({\bf 1050}) \oplus({\bf \overline{1050}}) \oplus({\bf 54}) \oplus2({\bf 210}) \oplus({\bf 45}) \oplus({\bf 1}), \label{branching-rule-18240}\\ \end{align}

(2.4)

 

\begin{align} {\bf 5440}&= {\bf 4125} \oplus{\bf \overline{1050}} \oplus{\bf 210} \oplus{\bf 54} \oplus{\bf 1}\ \ \ ({\bf \overline{5440}}= {\bf 4125} \oplus{\bf 1050} \oplus{\bf 210} \oplus{\bf 54} \oplus{\bf 1}). \label{branching-rule-5440} \end{align}

(2.5)

The VEV of the |$SO(32)$||${\bf 32}$| scalar, then, breaks |$(SO(32)\supset)SO(10)\times U(1)_Z$| to |$SU(5)$|⁠, where its branching rule of |$SO(32)\supset SU(16)\times U(1)_Z$| is given by  

\begin{align} {\bf 32}&= ({\bf 16})(2) \oplus({\bf \overline{16}})(-2) . \label{branching-rule-32} \end{align}

(2.6)

The VEV of the |$SO(32)$||${\bf 496}$| scalar further reduces |$(SO(32)\supset SU(16)\supset SO(10)\supset)SU(5)$| to |$G_{\rm SM}$|⁠, where its branching rule of |$SO(32)\supset SU(16)\times U(1)_Z$| is given by  

\begin{align} {\bf 496}&= ({\bf 255})(0) \oplus({\bf 120})(4) \oplus({\bf \overline{120}})(-4) \oplus({\bf 1})(0), \label{branching-rule-496} \end{align}

(2.7)

and the |$SU(16)$| adjoint representation |${\bf 255}$| is decomposed into |$SO(10)$| representations  

\begin{align} {\bf 255}={\bf 210}\oplus{\bf 45}. \label{branching-rule-255} \end{align}

(2.8)

(For further information, see, e.g., Ref. [3].)

3. SO(32) special grand unification

We start by discussing an |$SO(32)$| special GUT on the 6D orbifold spacetime |$M^4\times T^2/\mathbb{Z}_2$| with the Randall–Sundrum (RS) type metric [23,48] given by  

\begin{align} ds^2=e^{-2\sigma(\,y)}\left(\eta_{\mu\nu}dx^{\mu}dx^{\nu}+dv^2\right)+dy^2, \end{align}

(3.1)

where |$y$| is the coordinate of RS warped space, |$v$| is the coordinate of |$S^1$|⁠, |$\eta_{\mu\nu}=\mbox{diag}(-1,+1,+1,+1)$|⁠, |$\sigma(\,y)=\sigma(-y)=\sigma(\,y+2\pi R_5)$|⁠, |$\sigma(\,y)=k|y|$| for |$|y|\leq \pi R_5$|⁠, and |$v\sim v+2\pi R_6$|⁠. There are four fixed points on |$T^2/\mathbb{Z}_2$|⁠, at |$(\,y_0,v_0)=(0,0)$|⁠, |$(\,y_1,v_1)=(\pi R_5,0)$|⁠, |$(\,y_2,v_2)=(0,\pi R_6)$|⁠, and |$(\,y_3,v_3)=(\pi R_5,\pi R_6)$|⁠. For each fixed point, the |$\mathbb{Z}_2$| parity reflection is described by  

\begin{align} P_j:\ (x_\mu,y_j+y,v_j+v)\ \to\ (x_\mu,y_j-y,v_j-v), \end{align}

(3.2)

where |$j=0,1,2,3$| and |$P_3=P_1P_0P_2=P_2P_0P_1$|⁠. The fifth- and sixth-dimensional translations |$U_5: (x_\mu,y,v)\to(x_\mu,y+2\pi R_5,v)$| and |$U_6: (x_\mu,y,v)\to(x_\mu,y,v+2\pi R_6)$| satisfy |$U_5=P_1P_0$| and |$U_6=P_2P_0$|⁠, respectively.

We consider the matter content in the |$SO(32)$| special GUT that consists of a 6D |$SO(32)$| bulk gauge boson |$A_{M}$|⁠; three 6D |$SO(32)$||${\bf 32}$| positive Weyl fermions |$\Psi_{{\bf 32}+}^{(a)}$||$(a=1,2,3)$| and three 6D negative ones |$\Psi_{{\bf 32}-}^{(b)}$||$(b=1,2,3)$|⁠; 5D |$SO(32)$||${\bf 86768}$|⁠, |${\bf 496}$|⁠, and |${\bf 32}$| brane scalar bosons at |$y=0$|⁠, |$\Phi_{\bf 86768}$|⁠, |$\Phi_{\bf 496}$|⁠, |$\Phi_{\bf 32}$|⁠; and a 4D |$SU(16)\times U(1)$||$({\bf \overline{120}})(0)$| Weyl brane fermion and twelve 4D |$SU(16)\times U(1)$||$({\bf {16}})(0)\oplus({\bf \overline{16}})(-2)$| Weyl brane fermions at the fixed point |$(\,y_0,v_0)=(0,0)$|⁠, |$\psi_{\bf \overline{120}}$|⁠, |$\psi_{\bf 16}^{(c)}$|⁠, and |$\psi_{\bf \overline{16}}^{(d)}$||$(c,d=1,2, \ldots,12)$|⁠. The matter content of the |$SO(32)$| special GUT is summarized in Table 1. We will see what kind of roles each field has in detail in the following.

Table 1.

The matter content in the |$SO(32)$| special GUT on |$M^4\times T^2/\mathbb{Z}_2$|⁠. The representations of |$SO(32)$| and the 6D, 5D, 4D Lorentz groups, the orbifold BCs of the 6D bulk fields and 5D brane fields, and the spacetime locations of the 5D and 4D brane fields are shown. The orbifold BCs stand for parity assignment

$\left(\begin{array}{@{}cc@{}}\eta_2&\eta_3\\ \eta_0&\eta_1\end{array}\right)$

for 6D fields and

$\left(\begin{array}{@{}c@{}}\eta_2\\ \eta_0\end{array}\right)$

for 5D fields, where the orbifold BCs for the 6D and 5D fields are given in Eqs. (3.8) and (3.5).

6D bulk field	|$A_M$|	|$\Psi_{{\bf 32}+}^{(a)}$|	|$\Psi_{{\bf 32}-}^{(b)}$|
|$SO(32)$|	|${\bf 496}$|	|${\bf 32}$|	|${\bf 32}$|
|$SO(5,1)$|	|${\bf 6}$|	|${\bf 4}_+$|	|${\bf 4}_-$|
Orbifold BC		$\left(\begin{array}{@{}cc@{}} -&-\\ -&-\\ \end{array} \right)$	$\left(\begin{array}{@{}cc@{}} -&+\\ -&+\\ \end{array} \right)$
5D brane field	|$\Phi_{\bf 86768}$|	|$\Phi_{\bf 496}$|	|$\Phi_{\bf 32}$|
|$SO(32)$|	|${\bf 86768}$|	|${\bf 496}$|	|${\bf 32}$|
|$SO(4,1)$|	4	1	1
Orbifold BC	$\left(\begin{array}{@{}c@{}} +\\ -\\ \end{array} \right)$	$\left(\begin{array}{@{}c@{}} +\\ +\\ \end{array} \right)$	$\left(\begin{array}{@{}c@{}} +\\ +\\ \end{array} \right)$
Spacetime	|$y=0$|	|$y=0$|	|$y=0$|
4D Brane field	|$\psi_{\overline{\bf 120}}$|	|$\psi_{{\bf 16}}^{(c)}$|	|$\psi_{\overline{\bf 16}}^{(d)}$|
|$SU(16)$|	|$\overline{\bf 120}$|	|${\bf 16}$|	|$\overline{\bf 16}$|
|$U(1)_Z$|	|$0$|	|$0$|	|$-2$|
|$SL(2,\mathbb{C})$|	|$(1/2,0)$|	|$(1/2,0)$|	|$(1/2,0)$|
Spacetime |$(\,y,v)$|	|$(0,0)$|	|$(0,0)$|	|$(0,0)$|

First, a 6D |$SO(32)$| bulk gauge boson |$A_{M}$| is decomposed into a 4D gauge field |$A_\mu$| and the fifth- and sixth-dimensional gauge fields |$A_y$| and |$A_v$|⁠. The orbifold BCs of the 6D |$SO(32)$| gauge field are given by  

\begin{align} \left( \begin{array}{@{}c@{}} A_\mu\\ A_y\\ A_v\\ \end{array} \right)(x,y_j-y,v_j-v) =P_{j{\bf 32}} \left( \begin{array}{@{}c@{}} A_\mu\\ -A_y\\ -A_v\\ \end{array} \right)(x,y_j+y,v_j+v) P_{j{\bf 32}}^{-1}, \label{Eq:gauge-field-BCs} \end{align}

(3.3)

where |$P_{j{\bf 32}}$| is a projection matrix satisfying |$(P_{j{\bf 32}})^2=I_{32}$|⁠. We consider the orbifold BCs |$P_0$| and |$P_1$| preserving |$SO(32)$| symmetry, while the orbifold BCs |$P_2$| and |$P_3$| reduce |$SO(32)$| to its regular subgroup |$SU(16)\times U(1)_Z$|⁠. We take |$P_{j{\bf 32}}$| as  

\begin{align} P_{j{\bf 32}}= \left\{ \begin{array}{@{}ll@{}} \sigma_2\otimes I_2\otimes I_2\otimes I_2\otimes I_2& \mbox{for}\ j=2,3,\\ I_{32}\ &\mbox{for}\ j=0,1 . \end{array} \right. \label{Eq:SO32-BCs-32} \end{align}

(3.4)

In this case, the 4D |$SO(32)$||${\bf 496}$| gauge field |$A_\mu$| has Neumann BCs at the fixed points |$(\,y_0,v_0)$| and |$(\,y_1,v_1)$|⁠, while the fifth- and sixth-dimensional gauge fields |$A_y$| and |$A_v$| have Dirichlet BCs because of the negative sign in Eq. (3.3). On the other hand, since |$SO(32)$| symmetry is broken to |$SU(16)\times U(1)_Z$| at the fixed points |$(\,y_2,v_2)$| and |$(\,y_3,v_3)$|⁠, by using the branching rules of the |$SO(32)$| adjoint representation |${\bf 496}$| given in Eq. (2.7), the |$SU(16)\times U(1)_Z$||$\left(({\bf 255})(0)\oplus({\bf 1})(0)\right)$| and |$\left(({\bf 120})(4)\oplus({\bf \overline{120}})(-4)\right)$| components of the 4D gauge field |$A_\mu$| have Neumann and Dirichlet BCs at the fixed points |$(\,y_2,v_2)$| and |$(\,y_3,v_3)$|⁠, respectively; the |$SU(16)\times U(1)_Z$||$\left(({\bf 255})(0)\oplus({\bf 1})(0)\right)$| and |$\left(({\bf 120})(4)\oplus({\bf \overline{120}})(-4)\right)$| components of the fifth- and sixth-dimensional gauge fields |$A_y$| and |$A_v$| have Dirichlet and Neumann BCs, respectively. Thus, since the |$SU(16)\times U(1)_Z$||$\left(({\bf 255})(0)\oplus({\bf 1})(0)\right)$| components of the 4D gauge field |$A_\mu$| have four Neumann BCs at the four fixed points |$(\,y_j,v_j)\ (j=0,1,2,3)$|⁠, they have zero modes corresponding to 4D |$SU(16)$| and |$U(1)_Z$| gauge fields; since the other components of |$A_\mu$| and any component of |$A_y$| and |$A_v$| have four Dirichlet BCs or two Neumann and two Dirichlet BCs at the four fixed points, they do not have zero modes. The orbifold BCs reduce |$SO(32)$| to |$SU(16)\times U(1)_Z$|⁠. (Since there are no zero modes of the extra-dimensional gauge fields |$A_y$| and |$A_v$|⁠, we cannot rely on symmetry breaking known as the Hosotani mechanism [49,50] in this setup).

To achieve the SM gauge symmetry |$G_{\rm SM}$| at low energies, we consider the symmetry-breaking sector via spontaneous symmetry breaking. We introduce 5D |$SO(32)$||${\bf 86768}$|⁠, |${\bf 496}$| and |${\bf 32}$| brane scalar fields |$\Phi_{\bf 86768}$|⁠, |$\Phi_{\bf 496}$|⁠, and |$\Phi_{\bf 32}$| on the 5D brane (⁠|$y=0$|⁠). Their orbifold BCs are given by  

\begin{align} \Phi_{\bf x}(x,v_\ell-v)&= \eta_{\ell{\bf x}}P_{\ell{\bf x}}\Phi_{\bf x}(x,v_\ell+v), \label{Eq:BC-SO(32)-scalar-x} \end{align}

(3.5)

where |$\ell=0,2$|⁠; |${\bf x}$| stands for |${\bf 86768}$|⁠, |${\bf 496}$|⁠, and |${\bf 32}$|⁠; |$\eta_{\ell{\bf x}}$| is a positive or negative sign; and |$P_{\ell{\bf x}}$| is a projection matrix. We take |$\eta_{0{\bf 86768}}=-\eta_{2{\bf 86768}}=-1$| and |$\eta_{\ell{\bf 496}}=\eta_{\ell{\bf 32}}=1$|⁠. The tensor products of |$SO(32)$| for |${\bf 496}$| and |${\bf 32}$| are given in Ref. [3] by  

\begin{align} {\bf 496}\otimes{\bf 496}&= ({\bf 86768})_{\rm S} \oplus({\bf 35960})_{\rm S} \oplus({\bf 527})_{\rm S} \oplus({\bf 1})_{\rm S} \oplus({\bf 122264})_{\rm A} \oplus({\bf 496})_{\rm A},\\ \end{align}

(3.6)

 

\begin{align} {\bf 32}\otimes{\bf 32}&= ({\bf 527})_{\rm S} \oplus({\bf 1})_{\rm S} \oplus({\bf 496})_{\rm A}. \end{align}

(3.7)

The branching rules of |$SO(32)\supset SU(16)\times U(1)$| for |${\bf 496}$| and |${\bf 86768}$| are given in Eqs. (2.7) and (2.3), respectively; for |${\bf 527}$|⁠, |${\bf 35960}$|⁠, and |${\bf 122264}$| they are listed in Ref. [3]. For |$\Phi_{\bf 86768}$|⁠, the |$SU(16)\times U(1)$||$\left(({\bf 18240})(0)\oplus({\bf 14144})(0)\oplus({\bf 5440})(8) \oplus({\bf \overline{5440}})(-8)\oplus({\bf 255})(0)\oplus({\bf 1})(0) \right)$| components have zero modes; for |$\Phi_{\bf 496}$|⁠, the |$SU(16)\times U(1)$||$\left(({\bf 255})(0)\oplus({\bf 1})(0)\right)$| components have zero modes; and for |$\Phi_{\bf 32}$|⁠, the |$SU(16)\times U(1)$||$({\bf 16})(2)$| components have zero modes. We assume that the nonvanishing VEV of the scalar field |$\Phi_{\bf 86768}$| is responsible for breaking |$(SO(32)\supset)SU(16)\times U(1)_Z$| to |$SO(10)$|⁠; the nonvanishing VEV of the scalar field |$\Phi_{\bf 32}$| breaks |$(SO(32)\supset)SO(10)$| to |$SU(5)$|⁠; the nonvanishing VEV of |$\Phi_{\bf 496}$| breaks |$(SO(32)\supset)SU(5)$| to |$G_{\rm SM}$|⁠.

The SM Weyl fermions are identified with zero modes of 6D |$SO(32)$||${\bf 32}$| Weyl bulk fermions. The orbifold BCs of a 6D |$SO(32)$||${\bf 32}$| positive or negative Weyl bulk fermion can be written by  

\begin{align} \Psi_{{\bf 32}\pm}(x,y_j-y,v_j-v)&= \eta_{j}(-i\Gamma^5\Gamma^6) P_{j{\bf 32}} \Psi_{{\bf 32}\pm} (x,y_j+y,v_j+v), \label{Eq:BC-SO(32)-fermion-32} \end{align}

(3.8)

where the subscript of |$\Psi$||$\pm$| stands for 6D chirality, |$\eta_{j}$| is a positive or negative sign, |$\prod_{j=0}^{3}\eta_j=1$|⁠, |$\Gamma^M$||$(M=1,2,\ldots,7)$| is a 6D gamma matrix, and |$P_{j{\bf 32}}$| is give in Eq. (3.4). Here, we check zero modes of, e.g., a 6D |$SO(32)$||${\bf 32}$| positive Weyl fermion with orbifold BCs |$(\eta_{0},\eta_{1},\eta_{2},\eta_{3})=(-,-,-,-)$|⁠. At fixed points |$(\,y_0,v_0)$| and |$(\,y_1,v_1)$|⁠, the 4D |$SO(32)$||${\bf 32}$| left-handed Weyl fermion components have Neumann BCs, and the 4D |$SO(32)$||${\bf 32}$| right-handed Weyl fermion components also have Neumann BCs. At the fixed points |$(\,y_0,v_0)$| and |$(\,y_1,v_1)$|⁠, the 4D |$SU(16)\times U(1)_Z$||$({\bf 16})(2)$| and |$({\bf \overline{16}})(-2)$| left-handed Weyl fermion components have Neumann and Dirichlet BCs, respectively, while the 4D |$SU(16)\times U(1)_Z$||$({\bf 16})(2)$| and |$({\bf \overline{16}})(-2)$| right-handed Weyl fermion components have Dirichlet and Neumann BCs, respectively. In this case, only the 4D |$SU(16)\times U(1)_Z$||$({\bf 16})(2)$| left-handed Weyl fermion has zero modes. Note that only 6D |$SO(32)$||${\bf 32}$| positive Weyl fermions suffer from 6D gauge anomalies.

From the above, to realize three generations of the SM chiral fermions, we introduce three sets of pairs of 6D |$SO(32)$||${\bf 32}$| positive and negative Weyl fermions to cancel out the 6D gauge anomalies. More explicitly, each set of 6D Weyl fermions consists of a 6D |$SO(32)$||${\bf 32}$| positive Weyl fermion with orbifold BCs |$(\eta_{0},\eta_{1},\eta_{2},\eta_{3})=(-,-,-,-)$| and a 6D negative one with orbifold BCs |$(-,+,-,+)$|⁠. Only the |$SU(16)\times U(1)_Z$||$({\bf 16})(2)$| components of the positive Weyl fermion have zero modes for its 4D left-handed Weyl fermion components because its 4D left-handed Weyl fermion components have Neumann BCs at all the fixed points. The corresponding 4D right-handed Weyl fermion components have Dirichlet BCs at all the fixed points. The other components of the positive Weyl fermion and all the components of the negative Weyl fermion have two Neumann and two Dirichlet BCs at four fixed points |$(\,y_j,v_j)$|⁠.

Here, we check the contribution to 6D bulk and 4D brane anomalies from the above 6D Weyl fermion sets. The fermion set does not contribute to the 6D |$SO(32)$| gauge anomaly because of the same number of 6D |$SO(32)$||${\bf 32}$| positive and negative Weyl fermions. We need to check 4D gauge anomaly cancellation at four fixed points |$(\,y_j,v_j)\ (j=0,1,2,3)$| by using the 4D anomaly coefficients listed in Ref. [3]. At two fixed points |$(\,y_j,v_j)\ (j=0,1)$|⁠, there is no 4D pure |$SO(32)$| gauge anomaly because any 4D anomaly coefficient of |$SO(32)$| is zero. At the other two fixed points |$(\,y_j,v_j)\ (j=2,3)$|⁠, there can be 4D pure |$SU(16)$|⁠, pure |$U(1)_Z$|⁠, mixed |$SU(16)$|–|$SU(16)$|–|$U(1)_Z$| and mixed grav.–grav.–|$U(1)_Z$| anomalies. At the fixed point |$(\,y_3,v_3)$|⁠, the anomalies generated from the 6D |$SO(32)$||${\bf 32}$| positive and negative Weyl fermions cancel each other; at the other fixed point |$(\,y_2,v_2)$|⁠, the 6D |$SO(32)$||${\bf 32}$| positive and negative Weyl fermions generate 4D pure |$SU(16)$|⁠, pure |$U(1)_Z$|⁠, mixed |$SU(16)$|–|$SU(16)$|–|$U(1)_Z$| and mixed grav.–grav.–|$U(1)_Z$| anomalies. We focus on how to cancel the 4D anomalies at the fixed point |$(\,y_2,v_2)$| below.

To achieve 4D gauge anomaly cancellation at the fixed point |$(\,y_2,v_2)$|⁠, we introduce 4D brane Weyl fermions in appropriate representations of |$SU(16)\times U(1)$|⁠. First, we consider the pure |$SU(16)$| gauge anomaly cancellation. The 4D |$SU(16)$| gauge anomaly of twelve 4D |$SU(16)$||${\bf 16}$| left-handed Weyl fermions is canceled out by the anomaly of a 4D |$SU(16)$||${\bf \overline{120}}$| Weyl fermion [24]. (This can be checked by using the 4D |$SU(16)$| anomaly coefficients listed in Ref. [3].) When we introduce a 4D |$SU(16)\times U(1)$||$({\bf \overline{120}})(0)$| Weyl fermion at |$(\,y_2,v_2)$|⁠, its 4D |$SU(16)$| anomaly cancels one generated from the 6D bulk |$SO(32)$||${\bf 32}$| Weyl fermions. Next, 4D pure |$U(1)_Z$|⁠, mixed |$SU(16)$|–|$SU(16)$|–|$U(1)_Z$|⁠, and mixed grav.–grav.–|$U(1)_Z$| anomalies can be canceled out by introducing twelve 4D |$SU(16)\times U(1)$||$\left(({\bf 16})(0)\oplus({\bf \overline{16}})(-2)\right)$| left-handed Weyl fermions. This is because the matter content is vectorlike from the point of view of the |$U(1)_Z$| gauge theory. More explicitly, at the fixed point |$(\,y_2,v_2)$|⁠, there are twelve |$SU(16)\times U(1)_Z$||$({\bf 16})(2)$| left-handed Weyl fermions from the 6D |$SO(32)$||${\bf 32}$| Weyl fermions and we introduced twelve |$SU(16)\times U(1)_Z$||$({\bf \overline{16}})(-2)$| left-handed Weyl brane fermions. Also, the 4D |$SU(16)\times U(1)$||$\left(({\bf 16})(0)\oplus({\bf \overline{16}})(-2)\right)$| Weyl fermions do not generate a 4D pure |$SU(16)$| anomaly. Note that the above brane fermions become vectorlike when |$SU(16)\times U(1)$| symmetry is broken to |$SO(10)$|⁠, so there are no exotic chiral fermions at low energies, where the branching rules of |$SU(16)\supset SO(10)$| for an |$SU(16)$| complex representation |${\bf \overline{120}}$| (⁠|${\bf 120}$|⁠) are identified with an |$SO(10)$| real representation |${\bf 120}$|⁠:  

\begin{align} {\bf \overline{120}}={\bf 120}\ \ \ ({\bf 120}={\bf 120}). \end{align}

(3.9)

4. Summary and discussion

In this paper, we constructed an |$SO(32)$| special GUT by using a special breaking |$SU(16)$| to |$SO(10)$|⁠. In this framework, the zero modes of the 6D |$SO(32)$||${\bf 32}$| Weyl fermion can be identified with one generation of quarks and leptons; the 6D |$SO(32)$| and the 4D |$SU(16)\times U(1)$| gauge anomalies on the fixed points allow a three-generation model of quarks and leptons in the 6D framework; as in the |$SU(16)$| special GUT [24], exotic chiral fermions do not exist due to a special feature of the |$SU(16)$| complex representation |${\bf \overline{120}}$| once we take into account the symmetry breaking of |$SO(32)$| to |$SO(10)$|⁠.

In this paper, we simply assumed that the nonvanishing VEV of a scalar field |$\Phi_{\bf 86768}$| breaks |$(SO(32)\supset)SU(16)\times U(1)_Z$| to |$SO(10)$|⁠. Instead, we may consider a dynamical symmetry-breaking scenario [51–59] to realize the special breaking |$SU(16)$| to |$SO(10)$|⁠. Its breaking can be realized by using the pair condensation of a fermion in the |$SO(32)$| adjoint representation |${\bf 496}$| or the |$SU(16)$| second-rank antisymmetric tensor |${\bf 120}$||$({\bf \overline{120}})$|⁠. The analysis will be reported in a separate paper (T. Kugo and N. Yamatsu, in preparation). (The dynamical symmetry breaking of |$SU(16)$| to its special subgroup |$SO(10)$| is essentially the same as that of |$E_6$| to its special subgroups |$F_4$| and |$USp(8)$| or |$G_2$| or |$SU(3)$| as discussed in Ref. [58].)

To cancel 4D pure |$SU(16)$|⁠, pure |$U(1)_Z$|⁠, mixed |$SU(16)$|–|$SU(16)$|–|$U(1)_Z$|⁠, and mixed grav.–grav.–|$U(1)_Z$| anomalies on a fixed point, we introduced several brane Weyl fermions. For the mixed anomalies, one may rely on the Green–Schwarz (GS) anomaly cancellation mechanism [30] for the 4D version [22,60] by introducing a pseudo-scalar field that transforms nonlinearly under the anomalous |$U(1)$| symmetry.

Acknowledgements

The author would like to thank Yutaka Hosotani, Kentaro Kojima, Taichiro Kugo, Shogo Kuwakino, Kenji Nishiwaki, and Shohei Uemura for valuable comments.

Funding

Open Access funding: SCOAP|$^3$|⁠.