Family Unification in Special Grand Unification
Naoki Yamatsu







Family Unification in Special Grand Unification
    Naoki Yamatsu
[Electronic address: yamatsu@high.hokudai.ac.jp]


Institute for the Advancement of Higher Education,
Hokkaido University

Sapporo, Hokkaido 060-0817, Japan

    December 30, 2023
===============================================================================================================================================================================




We discuss family unification in grand unified theory (GUT) based on an
SU(19) GUT gauge group broken to its subgroups including a special
subgroup.
In the SU(19) GUT on the six-dimensional (6D) orbifold space
M^4× T^2/ℤ_2, 
three generations of the 4D SM Weyl fermions can be embedded into a 6D
bulk Weyl fermions in an SU(19) second-rank anti-symmetric tensor
representation. 6D and 4D gauge anomalies can be canceled out by
considering proper matter content without 4D exotic chiral fermions at
low energies.





§ INTRODUCTION


The existence of three chiral generations of quarks and leptons is one
of the most mysterious fact in particle physics.
In addition, their hierarchical mass structures generated by
the Higgs mechanism via their corresponding Yukawa couplings strongly
suggest the existence of a hidden structure of nature.
There are many attempts to understand the origin of chiral generations
and/or their hierarchical masses structures by considering e.g.,
so-called horizontal symmetry (or family symmetry) in four-dimensional
(4D) theories 
<cit.>,
geometrical structures in higher-dimensional theories
<cit.>
and string theories
<cit.>.

As is well-known, quarks and leptons for each generation in the Standard
Model (SM) can be unified into one multiplet (or two multiplets) in
grand unified theories (GUTs) <cit.>.
There are many GUTs in 4D framework
<cit.>
and higher-dimensional space framework
<cit.>
(For review, see Refs. <cit.>.)

There are some attempts to unify GUT and family groups into a larger GUT
group in 4D and higher-dimensional theories 
<cit.>.
However, such attempts are based on GUT groups and their limited
subgroups so-called regular subgroups: e.g.,
E_8⊃ E_7⊃ E_6⊃ SO(10)⊃ SU(5)⊃
G_ SM(:=SU(3)_C× SU(2)_L× U(1)_Y).
There are other subgroups called special subgroups
(or non-regular subgroups): e.g.,
SO(248)⊃ E_8, USp(56)⊃ E_7, SU(27)⊃ E_6,
and SU(16)⊃ SO(10).
(For Lie groups and their subgroups, see e.g.,
Refs. <cit.>.)

Recently, new-type GUTs called special GUT based on GUT groups
SO(32) and SU(16) and their special subgroup SO(10) have been
proposed in Refs. <cit.>.
The main results of SO(32) and SU(16) special GUTs are summarized as
below.
In an SU(16) special GUT based on its GUT group SU(16) broken
to its special subgroup SO(10), a 4D SU(16) 16 Weyl fermion
can be identified with one generation of quarks and leptons; 4D SU(16)
gauge anomaly cancellation does not work in 4D framework, while it works
in 6D framework without any exotic chiral fermions once we take into
account SU(16) symmetry breaking effects <cit.>.
Almost the same results are obtained in an SO(32) special GUT
<cit.>. 

In special GUT framework, family unification can be considered by using 
GUT groups and their “regular-type” and “product-type” subgroups;
the former example is
SU(19)⊃ SU(16)× SU(3)× U(1);
the latter example is 
SU(48)⊃ SU(16)× SU(3),
where SU(16) contains an ordinary GUT gauge group SO(10) and SU(3)
is a family gauge group. 
(Their branching rules of
SU(19)⊃ SU(16)× SU(3)× U(1),
SU(48)⊃ SU(16)× SU(3),
etc. can be calculated e.g., by
using a projection matrix method 
shown in Refs. <cit.>.)

First, for a regular-type case,
an example of GUT gauge groups and their subgroup pair is
SU(19)⊃ SU(16)× SU(3)× U(1)
⊃ SO(10)× SU(3)× U(1).
It is a simple extension of SU(16).
The branching rule of
SU(19)⊃ SU(16)× SU(3)× U(1)
⊃ SO(10)× SU(3)× U(1)
for the SU(19) defining representation is 

    19=   
    ( 16,1)(3)
    ⊕( 1,3)(-16).

In this case, e.g., an SU(19) second-rank anti-symmetric tensor
representation contains three generations of quarks and leptons.
Its branching rule is given by

    171=   
    ( 16,3)(-13)
    ⊕( 120,1)(6)
    ⊕( 1,3)(-32),

where an SU(16) 120 representation is complex while
an SO(10) 120 representation is real.
A 4D Weyl fermion in an SO(10) 120 representation is
vectorlike, so when we take into account symmetry breaking
effects for SU(19) to SO(10), only three 4D SO(10) 16 Weyl
fermions remain chiral.
Also, an SU(19) second-rank symmetric tensor representation 190
contains an SU(16)× SU(3) ( 16,3) representation.
Note that the SU(19) 190 contains unwilling
SU(16)× SU(3) complex representations.
The SU(19) adjoint representation 360
contains not only  an SU(16)× SU(3)
( 16,3) but also its conjugate representation
(16,3).
(In Ref. <cit.>, R.M. Fonseca has also pointed out that
a 4D Weyl fermion in an SU(19) 171 representation contains the
SM fermions plus vectorlike particles only, which was found by using the
Susyno program <cit.>.)

Next, for a product-type case, an example of GUT gauge groups and their
subgroup pair is
SU(48)⊃ SU(16)× SU(3)⊃ SO(10)× SU(3).
The branching rule for the SU(48) defining representation is 

    48=   ( 16,3).

The 4D Weyl fermion in the SU(48) defining representation can be
identified with three chiral generations of quarks and leptons.
As the best of my knowledge, there is no way to construct an SU(48)
gauge theory that contains only three chiral generations
without any gauge anomalies at least for 4D, 5D, 6D framework.
We will not discuss its possibility in this paper. 

In this paper, we discuss a 6D SU(19) special GUT on
M^4× T^2/ℤ_2, whose GUT group includes an SO(10) GUT
group and an SU(3)_F family group. The main purpose of this paper is
to show that three generations of the 4D SM Weyl fermions can be
embedded into a 6D bulk Weyl fermions in an SU(19) second-rank
anti-symmetric tensor representation.
6D and 4D gauge anomalies can be canceled out by considering
proper matter content without 4D exotic chiral fermions at low energies.

This paper is organized as follows. In Sec. <ref>, before we
discuss a special GUT based on an SU(19) gauge group, we discuss basic
properties of SU(19) and its subgroups mainly by using technique in
Ref. <cit.>.
In Sec. <ref>, 
we construct a 6D SU(19) special GUT on M^4× T^2/ℤ_2.
Section <ref> is devoted to a summary and
discussion.




§ BASICS FOR SU(19) AND ITS SUBGROUPS



First, we check how to embed three generations of the SM Weyl fermions
into 4D Weyl fermion in an SU(19) second-rank anti-symmetric tensor
representation 171.
For regular and special embeddings
SU(19)⊃ SU(16)× SU(3)_F× U(1)_Z⊃ 
SO(10)× SU(3)_F× U(1)_Z,
an SU(19) second-rank anti-symmetric tensor  representation 
171 is decomposed into an SU(16) second-rank anti-symmetric
tensor representation 120, 
an SU(3)_F defining representations 3, and 
an SU(16)× SU(3)_F bi-fundamental representations 
( 16,3) given in Eq. (<ref>).
Further, as is well-known, the SO(10) spinor representation 16
is decomposed into
G_ SM× U(1)_X=SU(3)_C× SU(2)_L× U(1)_Y× U(1)_X
representations:

    16=   
    ( 3,2)(-1)(1)
    ⊕(3,1)(-2)(-3)
    ⊕(3,1)(4)(1)
       ⊕( 1,2)(3)(-3)
    ⊕( 1,1)(-6)(1)
    ⊕( 1,1)(0)(5).

That is, three generations of the SM Weyl fermions are embedded into
an 4D SU(19) 171 Weyl fermion.
In addition, an SU(16) complex representation 120 is
identified with an SO(10) real representation 120.
An 4D SU(16) 120 Weyl fermion is chiral,
while an 4D SO(10) 120 Weyl fermion is vectorlike.
When SU(19) is broken to SO(10),
SU(3)_F× U(1)_Z is broken to nothing. 
An 4D Weyl fermion in the SO(10)× SU(3)
( 1,3) is chiral, 
while three 4D SO(10) 1 Weyl fermion is vectorlike.
Thus, once SU(19) is broken to SO(10), an 4D SU(19) 191
Weyl fermion is decomposed into  three 4D SO(16) 16 Weyl
fermions and vectorlike fermions.



Next, we consider a symmetry breaking pattern from SU(19) to
G_ SM.  
One way of achieving it is to use orbifold symmetry breaking boundary
conditions (BCs) and several GUT breaking Higgses. One example is 
to choose orbifold BCs breaking SU(19) to 
SU(16)× SU(3)_F× U(1) and 
to introduce SU(19) 10830, 360, 
19 scalar fields, where we assume their proper components acquire
non-vanishing VEVs.
First, the following orbifold BC for the SU(19) defining representation 
19 breaks SU(19) to SU(16)× SU(3)_F× U(1)_Z:

    P_ 19=(I_16,-I_3),

where P_ 19 stands for a projection matrix defined in 
Eq. (<ref>).
The non-vanishing VEV of the SU(19) 10830 scalar
field is responsible for breaking 
SU(19)⊃ SU(16)× SU(3)_F× U(1)_Z to
SO(10)× SU(3)_F, where its branching rule of 
SU(19)⊃ SU(16)× SU(3)_F× U(1)_Z is given by

    10830=   
    ( 5440,1)(12)
    ⊕( 1360,3)(-7)
    ⊕( 136,6)(-26)
    
       ⊕( 120,3)(-26)
    ⊕( 16,8)(-45)
    ⊕( 1,6)(-64).

An SU(16) 5440 contains singlet
under its SO(10) special subgroup. 
Its nonvanishing VEV can break SU(16) to its special subgroup
SO(10) <cit.>, where their SO(10)
decompositions are given in Ref. <cit.> by 

    5440=    4125⊕1050⊕ 210⊕ 54⊕ 1.

The VEV of an SU(19) 19 scalar breaks 
(SU(19)⊃)SO(10)× SU(3)_F×  U(1)_Z to
SU(5)× SU(3)_F or
(SU(19)⊃)SO(10)× SU(3)_F×  U(1)_Z to
SO(10)× SU(2)_F, where its breaking rule is
given in Eq. (<ref>).
If the three VEVs of the proper components of SU(19) 19
scalars can break (SU(19)⊃)SO(10)× SU(3)_F×  U(1)_Z to
SU(5).
The VEV of the SU(19) 360 scalar further
reduces (SU(19)⊃ SO(10)× SU(3)_F⊃ SO(10))⊃ SU(5) 
to G_ SM, 
where its branching rule of 
SU(19)⊃ SU(16)× SU(3)_F× U(1)_Z is given by

    360=   
    ( 255,1)(0)
    ⊕( 1,8)(0)
    ⊕( 1,1)(0)
    ⊕( 16,3)(19)
    ⊕(16,3)(-19),

where the SU(16) 255
is decomposed into SO(10) ( 210⊕ 45),
and the SO(10) 45 is decomposed into SU(5)
( 24⊕ 10⊕10⊕ 1).
(For further information, see e.g., Ref. <cit.>.)




§ SU(19) SPECIAL GRAND UNIFICATION



As in Refs. <cit.>, 
an SU(19) special GUT on 6D orbifold
spacetime M^4× T^2/ℤ_2 with 
the Randall-Sundrum (RS) type metric
<cit.>
given by 

    ds^2=e^-2σ(y)(η_μνdx^μdx^ν+dv^2)+dy^2,

where y is the coordinate of RS warped space,
v is the coordinate of S^1,
η_μν=(-1,+1,+1,+1), 
σ(y)=σ(-y)=σ(y+2π R_5),
σ(y)=k|y| for |y|≤π R_5, and
v∼ v+2π R_6.
There are four fixed points on T^2/ℤ_2 
at (y_0,v_0)=(0,0), (y_1,v_1)=(π R_5,0), 
(y_2,v_2)=(0,π R_6), and (y_3,v_3)=(π R_5,π R_6).
For each fixed point, the ℤ_2 parity reflection 
is described by 

    P_j: (x_μ,y_j+y,v_j+v) → (x_μ,y_j-y,v_j-v),

where j=0,1,2,3, and P_3=P_1P_0P_2=P_2P_0P_1.
5th and 6th dimensional translation 
U_5: (x_μ,y,v)→(x_μ,y+2π R_5,v) 
and U_6: (x_μ,y,v)→(x_μ,y,v+2π R_6)  
satisfy U_5=P_1P_0 and U_6=P_2P_0, respectively.


We consider the matter content in the SU(19) special GUT that consists
of a 6D SU(19) bulk gauge boson A_M;
6D SU(19) 171 positive Weyl fermions with the orbifold BCs
(η_0,η_1,η_2,η_3)=(+,+,-,-) and 
(η_0,η_1,η_2,η_3)=(+,+,+,+) 
Ψ_ 171+ and Ψ_ 171+', and 
6D negative Weyl fermions with (-,+,-,+) and (-,+,+,-)
Ψ_ 171- and Ψ_ 171-', 
where η_j (j=0,1,2,3) stands for parity assignment for
each 6D fermion;
5D SU(19), 10830, 360 and 19 brane scalar
bosons at y=0 
Φ_ 10830, Φ_ 360, Φ_ 19,
Φ_ 19^'(α) (α=1,2);
a 4D SU(16)× SU(3)_F× U(1)_Z 
(120,1)(0) Weyl fermion,
four 4D SU(16)× SU(3)_F× U(1)_Z 
(120,1)(6) Weyl fermions,
sixty-four 4D SU(16)× SU(3)_F× U(1)_Z
( 1,3)(13) Weyl fermions,
four 4D SU(16)× SU(3)_F× U(1)_Z
( 1,3)(-32) Weyl fermions
at the fixed point (y_0,v_0)=(0,0)
ψ_120,
ψ_120^(a) (a=1,2,3,4),
ψ_3^(b) (b=1,2,⋯,64),
ψ_3^'(c) (c=1,2,3,4).
The matter content of the SU(19) special GUT is
summarized in Table <ref>.
Note that the 5D brane scalars is responsible for achieving
appropriate symmetry and the 4D brane fermions are necessary to
realize 4D gauge anomaly cancellation. Only their conditions do not
uniquely determine the matter content, so one may choose another
matter content; e.g., one may introduce 6D bulk scalars instead of 5D
brane scalars. 
We will see the roles of the bulk and brane fields in the following.



First, a 6D SU(19) bulk gauge boson A_M is decomposed into 
a 4D gauge field A_μ and 5th and 6th dimensional gauge fields A_y
and A_v. The orbifold BCs of the 6D SU(19) gauge field are given by

    (
    [ A_μ; A_y; A_v;     ])(x,y_j-y,v_j-v)
    =P_j 19(
    [  A_μ; -A_y; -A_v;      ])(x,y_j+y,v_j+v)
    P_j 19^-1,

where P_j 19 is a projection matrix satisfying 
(P_j 19)^2=I_19. 
We consider the orbifold BCs P_2 and P_3 preserving SU(19)
symmetry, while the orbifold BCs P_0 and P_1 reduce SU(19) to its
regular subgroup SU(16)× SU(3)_F× U(1)_Z.
We take P_j 19 as 

    P_j 19=
    {[        I_19       j=2,3; (I_16,-I_3)       j=0,1 ]..

In this case, the 4D SU(19) 360 gauge field A_μ have
Neumann BCs at the fixed points (y_2,v_2) and 
(y_3,v_3), while the 5th and 6th dimensional gauge
fields A_y and A_v have Dirichlet BCs because of the negative sign
in Eq. (<ref>).
On the other hand, since SU(19) symmetry is broken to 
SU(16)× SU(3)_F× U(1)_Z at the fixed points (y_0,v_0) and
(y_1,v_1), 
by using the branching rules of the SU(19) adjoint representation 
360 given in Eq. (<ref>)
as well as  Eqs. (<ref>) and (<ref>),
the SU(16)× SU(3)_F × U(1)_Z
(( 255,1)(0)⊕( 1,8)(0)⊕( 1,1)(0))
and (( 16,3)(19)
⊕(16,3)(-19)) 
components of the 4D gauge field A_μ 
have Neumann and Dirichlet BCs at
the fixed points (y_0,v_0) and (y_1,v_1), respectively;
the SU(16)× SU(3)_F× U(1)_Z 
(( 255,1)(0)⊕( 1,8)(0)⊕( 1,1)(0))
and (( 16,3)(19)
⊕(16,3)(-19)) 
components of the 5th and 6th dimensional gauge 
fields A_y and A_v have Dirichlet and Neumann BCs, respectively.
Thus, since the SU(16)× SU(3)_F× U(1)_Z 
(( 255,1)(0)⊕( 1,8)(0)⊕( 1,1)(0))
components of the 4D gauge field A_μ 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), SU(3)_F and U(1)_Z gauge fields;
since the other components of A_μ 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 SU(19) to SU(16)× SU(3)_F× U(1)_Z. 

To achieve the SM gauge symmetry G_ SM at low energies, 
we consider the symmetry breaking sector via spontaneous symmetry
breaking. We introduce 5D SU(19) 10830, 360 and
19 brane scalar fields,
Ψ_ 10830, Φ_ 360, Φ_ 19, and
Φ_ 19^'(α) (α=1,2) on the 5D brane (y=0).
Their orbifold BCs are given by 

    Φ_ x^(')(x,v_ℓ-v)=   η_ℓ x^(')P_ℓ xΦ_ x^(')(x,v_ℓ+v),

where ℓ=0,2, x stands for
10830, 360 and 19,
η_ℓ x is a positive or negative sign, and
P_ℓ x is a projection matrix. 
We take η_ℓ 10830=-η_ℓ 360
=-η_ℓ 19=η_0 19'=-η_2 19'=-1.





















The branching rules of SU(19)⊃ SU(16)× SU(3)_F× U(1)_Z
for 10830, 360 and 19 are given in
Eqs. (<ref>), (<ref>), and
(<ref>), respectively.
For Φ_ 10830, the SU(16)× SU(3)_F× U(1)_Z
( 5440,1)(12)⊕( 120,3)(-26)
⊕( 1,6)(-64) components have zero modes;
for Φ_ 360, the SU(16)× SU(3)_F× U(1)_Z
( 255,1)(0)⊕( 1,8)(0)⊕( 1,1)(0)
components have zero modes;
for Φ_ 19, the SU(16)× SU(3)_F× U(1)_Z 
( 16,1)(3) components have zero modes,
and for Φ_ 19^'(α), the
SU(16)× SU(3)_F× U(1)_Z 
( 1,3)(-16) components have zero modes.
We assume that a scalar field Φ_ 10830 is responsible for
breaking  (SU(19)⊃)SU(16)× SU(3)_F× U(1)_Z to
SO(10)× SU(3)_F;
two scalar fields Φ_ 19^'(α) are responsible for
breaking  (SU(19)⊃ SU(16)× SU(3)_F× U(1)_Z)
⊃ SO(10)× SU(3)_F to SO(10);
the nonvanishing VEV of the scalar field 
Φ_ 19 breaks (SU(19)⊃)SO(10) to SU(5);
the nonvanishing VEV of Φ_ 360 breaks (SU(19)⊃)SU(5)
to G_ SM.

The SM Weyl fermions are identified with zero modes of a 6D SU(19) 
171 Weyl bulk fermion. The orbifold BCs of 6D SU(19) 
171 positive or negative Weyl bulk fermions can be written by 

    Ψ_ 171±^(')(x,y_j-y,v_j-v)   =
    η_j 171±^(')γ
    P_j 171Ψ_ 171±^(')
    (x,y_j+y,v_j+v),

where the subscript of Ψ ± stands for 6D chirality,
η_j 171±^(') is a positive or negative sign,
∏_j=0^3η_j 171±^(')=1,
6D gamma matrices γ^a (a=1,2,⋯,7) 
satisfy {γ^a,γ^b}=2η^ab
(η^ab=(-I_1,I_5)),
γ
:=-iγ^5γ^6
=γ_ 6D^7γ_ 4D^5,
γ_ 4D^5=I_2⊗σ^3⊗ I_2,
γ_ 6D^7=I_4 ⊗σ^3, 
and P_j 171±^(') is a projection matrix.
(The same notation is used in
Refs. <cit.>.)
In our notation, a 6D Dirac fermion Ψ_ D^ 6D
and 6D positive and negative Weyl fermions
Ψ_±^ 6D:=P_±^ 6DΨ_ D^ 6D
(P_±^ 6D:=(1±γ_ 6D^7)/2)
can be expressed by using 4D left- and right-handed Weyl fermions
ψ_L/R±^ 4D(=P_L/R^ 4Dψ_ D^ 4D)
(P_L/R^ 4D:=(1±γ_ 4D^5)/2),
where subscripts L/R and ± stand for
4D and 6D chiralities, respectively:

    Ψ_ D^ 6D
     :=
     (
     [ ψ_R+^ 4D; ψ_L+^ 4D; ψ_R-^ 4D; ψ_L-^ 4D;          ]), Ψ_+^ 6D
     =P_+^ 6DΨ_ D^ 6D
     =(
     [ ψ_R+^ 4D; ψ_L+^ 4D;        0;        0;          ]), Ψ_-^ 6D
     =P_-^ 6DΨ_ D^ 6D
     = (
     [        0;        0; ψ_R-^ 4D; ψ_L-^ 4D;          ]).

Note that to see the relation between P_j 171 and
P_j 19, we express the orbifold BCs of
Ψ_ 171±^(') by using a 19× 19
matrix form as the same as one of the gauge field in
Eq. (<ref>).
We can write the (αβ) component of 
Ψ_ 171±^(') as 
[Ψ_ 171±^(')]_αβ
=∑_a<bψ_ 171±^(')ab
[M_ 171ab]_αβ
(a,b,α,β=1,2,⋯,19),
where ψ_ 171±^(')abs are the component fields
of Ψ_ 171±^(') expanded by
M_ 171abs, and [M_ 171ab]_αβ=(1/√(2))
(δ_aαδ_bβ-δ_aβδ_bα).
In this notation, the orbifold BCs of the 6D SU(19) anti-symmetric
tensor fermion Ψ_ 171±^(') can be expressed by
using the projection matrix P_j 19
given in Eq. (<ref>)
instead of P_j 171:

    [Ψ_ 171±^(')(x,y_j-y,v_j-v)]_αβ   =
    η_j 171±^(')γ
     [P_j 19]_ακ
     [Ψ_ 171±^(')(x,y_j+y,v_j+v)]_κλ
     [P_j 19]_λβ,

where α,β,κ,λ=1,2,⋯,19;
[P_j 19]_αβ denotes the (αβ) element of
the projection matrix P_j 19 given in
Eq. (<ref>).
(The projection matrix of any SU(19) tensor product representation can
be expressed by the projection matrix of the SU(19) defining
representation P_j 19.)

Here, we check zero modes of, e.g., a 6D SU(19) 171 positive
Weyl fermion with orbifold BCs
(η_0,η_1,η_2,η_3)=(+,+,-,-) Ψ_ 171+.
At fixed points (y_2,v_2) and (y_3,v_3), 
the 4D SU(19) 171 left-handed Weyl fermion components have
Neumann BCs, while the 4D SU(19) 171 right-handed Weyl fermion
components have Dirichlet BCs.
At fixed points (y_0,v_0) and (y_1,v_1), 
the 4D SU(16)× SU(3)_F× U(1)_Z
( 16, 3)(-13) and
( 120, 1)(6)⊕( 1,3)(-32) 
left-handed Weyl fermion components have Neumann and Dirichlet BCs,
respectively, 
while the 4D SU(16)× SU(3)_F× U(1)_Z
( 16, 3)(-13) and
( 120, 1)(6)⊕( 1,3)(-32) 
right-handed Weyl fermion components have Dirichlet and Neumann BCs,
respectively. 
In this case, only the 4D SU(16)× SU(3)_F× U(1)_Z
( 16, 3)(-13) 
left-handed Weyl fermion has zero modes.
Also, for Ψ_ 171+^',
the 4D SU(16)× SU(3)_F× U(1)_Z
( 120, 1)(6)⊕( 1,3)(-32)
right-handed Weyl fermion has zero modes.
They are vectorlike once we take into account
the symmetry breaking effects of SU(16)× SU(3)_F× U(1)_Z
to SO(10).
Also, for Ψ_ 171- and Ψ_ 171-^',
there is no zero mode.
The parity assignments of Ψ_ 171±^(') are
summarized in Table <ref>.



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 6D SU(19) gauge anomaly because
of the same number of 6D SU(19) 171 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 4D anomaly coefficients listed
in Ref. <cit.>.
From Table <ref>,
at three fixed points (y_j,v_j) (j=1,2,3),
there are two 4D left- and right-handed Weyl fermions in
( 16,3)(-13), ( 120,1)(6), and
( 1,3)(-32) of SU(16)× SU(3)_F× U(1)
from the 6D SU(19) 171 positive and negative Weyl
fermions Ψ_ 171+, Ψ_ 171+',
Ψ_ 171-, and Ψ_ 171-'.
The vectorlike matter sets do not produce any 4D gauge anomalies.
At the other fixed point (y_0,v_0), there can be 4D pure SU(16),
pure SU(3)_F pure U(1)_Z, mixed SU(16)-SU(16)-U(1)_Z, mixed
SU(3)_F-SU(3)_F-U(1)_Z, and mixed --U(1)_Z
anomalies. In fact, the 6D SU(19) 
171 positive and negative Weyl fermions generate 
4D pure SU(16), pure SU(3)_F, pure U(1)_Z,
mixed SU(16)-SU(16)-U(1)_Z, mixed SU(3)_F-SU(3)_F-U(1)_Z,
and mixed --U(1)_Z
anomalies. We focus on how to cancel the 4D anomalies at the fixed point
(y_0,v_0) below.

To achieve 4D gauge anomaly cancellation at the fixed point (y_0,v_0),
we need to introduce 4D Weyl fermions in appropriate
representations of SU(16)× SU(3)_F× U(1).
First, we consider the pure SU(16) gauge anomaly cancellation.
The 4D SU(16) gauge anomaly of twelve 4D SU(16) 16
left-handed Weyl fermions
and four 4D SU(16) 120 right-handed Weyl fermions
is canceled out by the anomaly of
three 4D SU(16) 120 left-handed Weyl fermions.
From Table <ref>,
there are one 4D SU(16) 120 left-handed Weyl fermion 
ψ_120
and four 4D SU(16) 120 left-handed Weyl fermions
ψ_120^(a) (a=1,2,3,4) in the model.
The SU(16) gauge anomaly of ψ_120 is
canceled out by one of e.g., ψ_120^(1).
The other three 4D SU(16) 120 left-handed Weyl fermions
ψ_120^(a) (a=2,3,4) contribute the SU(16) gauge
anomaly on the fixed point (y_0,v_0).
Thus, the 4D brane fermions ψ_120
and ψ_120^(a) (a=1,2,3,4) cancel 
the 4D SU(16) gauge anomaly from the bulk fermions.
Second, for the pure SU(3)_F gauge anomaly, 
the 4D SU(3) gauge anomaly of sixty-four 4D SU(3) 3
left-handed Weyl fermions
and four 4D SU(3) 3 right-handed Weyl fermions
is canceled out by the anomaly of sixty-eight 4D SU(3)
3 left-handed Weyl fermions.
Third, for 4D pure U(1)_Z, mixed SU(16)-SU(16)-U(1)_Z,
mixed SU(3)_F-SU(3)_F-U(1)_Z, and mixed
--U(1)_Z anomalies are canceled out 
if the matter content is vectorlike from the view of the U(1)_Z
gauge theory.
The matter content shown in
Table <ref>
satisfies all the above requirements, so
any 6D and 4D gauge anomalies at the fixed points are canceled out.




§ SUMMARY AND DISCUSSION



In this paper, we have pointed out that in special GUT framework, family
unification may be achieved by using GUT groups and their
“regular-type” and “product-type” subgroups, such as 
SU(19)⊃ SU(16)× SU(3)× U(1) and 
SU(48)⊃ SU(16)× SU(3), respectively.

For a “regular-type” subgroup
SU(19)⊃ SU(16)× SU(3)× U(1), 
we have constructed an SU(19) special GUT by using a
special breaking SU(16) to SO(10). In this framework, the zero modes
of a 6D SU(19) 171 Weyl fermion can be identified with
three generations of quarks and leptons;
the 6D SU(19) gauge anomaly on the bulk and the 4D SU(19) or
SU(16)× SU(3)× U(1) gauge anomalies at each fixed point can
be canceled out; as in the SU(16) special GUT
<cit.>,  exotic 
chiral 
fermions do not exist due to a special feature of 
the SU(16) complex representation 120 once we take
into account the symmetry breaking of SU(19) to SO(10).

To cancel 4D pure SU(16), pure SU(3)_F, pure U(1)_Z, and mixed
anomalies on a fixed point, we introduced a lot of 4D Weyl fermions.
For the mixed anomalies, one may rely on Green-Schwarz (GS) anomaly
cancellation mechanism <cit.> for 4D version
<cit.>.
It may be achieved by introducing a pseudo-scalar field that
transforms non-linearly under the anomalous U(1)_Z symmetry.
In this case, the number of 4D Weyl fermions can be reduced, drastically.

We comment on the SM fermion masses in the SU(19) special GUT. Since
three generations of the SM fermions are unified into a 6D SU(19)
171 Weyl fermion, the masses of all quarks and leptons are
degenerate without 
SU(19)(⊃ SU(16)× SU(3)_F⊃ SO(10)× SU(3)_F)
breaking effects. 
We assumed that since the nonvanishing VEVs of 5D brane scalars
Ψ_ 19^'(α) break the SU(3)_F symmetry,
there is no reason to expect the unified masses of first, second and
third generations of up-type and down-type quarks, charged leptons, and
neutrinos, respectively; in addition, since the nonvanishing VEVs of 5D
brane scalars Ψ_ 10830, Ψ_ 19, and Ψ_ 360
break SU(16)(⊃ SO(10)) to G_ SM,
there is no reason to expect the degenerate mass of quarks and leptons
for each generation.
As discussed in e.g.,
Refs. <cit.>, on the UV brane y=0,
we can introduce SU(19)-invariant brane interaction terms among the 6D
bulk fermions and the 5D brane scalars because SU(19) tensor products
e.g., 171⊗ 171⊗10830,
171⊗171⊗ 360,
171⊗171⊗ 19⊗19,
etc. contain singlet.
The SU(16)× SU(3)_F× U(1)_Z ( 16,3)(-13)
and ( 120,1)(6) components of the SU(19) 171 bulk
fermions can be mixed via the VEVs of the 5D brane scalars once their
corresponding brane interaction terms or effective brane mass terms are 
generated, where SO(10)(⊂ SU(19)) 120 contains
( 3,2)(-1), (3,1)(4), 
(3,1)(-2), ( 1,2)(3), 
and ( 1,1)(0) of G_ SM.
We expect that the effective mass terms divide the degenerate mass for
each generation into up-type quark, down-type quark, charged lepton, and 
neutrino masses, where some VEVs or coupling constants must be
hierarchical to realize mass hierarchies for up-type and down-type
quarks and charged leptons. To realize tiny neutrino masses, it seems to
be better to introduce 5D symplectic Majorana fermions
<cit.> on the UV brane y=0. SU(19) brane 
interaction terms among the 6D bulk fermions, the 5D brane scalars, and
the 5D symplectic Majorana fermions lead to tiny neutrino masses via a
seesaw mechanism discussed in
Refs. <cit.>. 
The above brane interaction terms are essential to realize not only
quark and lepton masses but also their mixing matrices, i.e., 
the Cabibbo-Kobayashi-Maskawa (CKM)
<cit.> and
Maki-Nakagawa-Sakata (MNS) <cit.> matrices.
Since we can introduce a lot of 5D brane interaction terms in the
SU(19) special GUT, the model seems to realize the SM fermion
masses, but seems to give us no prediction about quark and lepton
masses and mixings. We will leave the detail analysis in future studies.


We have discussed how to embed three chiral generations of quarks
and leptons in triplet (a finite-dimensional representation) of a
non-Abelian compact group SU(3)_F.
Another direction for unifying generations may be considered by using
non-Abelian noncompact groups (e.g., SU(1,1)) and their
infinite-dimensional representation
<cit.>.




§ ACKNOWLEDGMENTS


The author would like to thank Kenji Nishiwaki
for critical reading of the manuscript and many valuable comments.

utphys 


