fulltext

Left-right symmetric gauge model in a noncommutative geometry

on M

4

3 Z

4

Y. OKUMURA(*)

Department of Natural Science, Chubu University - Kasugai 487, Japan (ricevuto il 27 Novembre 1996; approvato il 3 Aprile 1997)

Summary. — The left-right symmetric gauge model with the symmetry of SU( 3 )c3

SU( 2 )L3 SU( 2 )R3 U( 1 ) is reconstructed in a new scheme of the noncommutative

differential geometry (NCG) on the discrete space M43 Z4which is a product space

of Minkowski space and four points space. The characteristic point of this new scheme is to take the fermion field to be a vector in a 24-dimensional space which contains all leptons and quarks. Corresponding to this specification, all gauge and Higgs boson fields are represented in 24 324 matrix forms. We incorporate two Higgs doublet bosons h and SU( 2 )Radjoint Higgs jRwhich are as usual transformed

as ( 2 , 2* , 0 ) and ( 1 , 3 , 22) under SU(2)L3 SU( 2 )R3 U( 1 ), respectively. Owing to

the revision of the algebraic rules in a new NCG, we can obtain the necessary potential and interacting terms between these Higgs bosons which are responsible for giving masses to the particles included. Among the Higgs doublet bosons, one CP-even scalar boson survives in the weak energy scale and other four scalar bosons acquire the mass of the SU( 2 )R3 U( 1 ) breaking scale, which situation leads to that

in the standard model. jR is responsible for spontaneously break SU( 2 )R3 U( 1 )

down to U( 1 )Yand so well explains the seesaw mechanism. Up and down quarks have

the different masses through the vacuum expectation value of h. PACS 11.15.Ex – Spontaneous breaking of gauge symmetries. PACS 02.40.Hw – Classical differential geometry.

PACS 12.15 – Electroweak interactions.

1. – Introduction

The efforts made so far to understand the Higgs mechanism responsible for the spontaneous breakdown of the gauge symmetry have proposed many attempts such as technicolor model, Kaluza-Klein model, strong coupling model and the approach based on the noncommutative differential geometry (NCG) on the discrete space, among

(*) E-mail address: okumHisc.chubu.ac.jp

which the NCG approach does not demand any extra physical modes except for the usual modes, contrary to others and provides the unified picture of the gauge and Higgs boson fields as the generalized connection in NCG on the discrete space M43ZN.

Since Connes [1] proposed the original idea to understand the Higgs mechanism in NCG on the discrete space M43 Z2, many works have been done so far [2-10]. The

present author [11, 16] also developed a unique version of NCG approach which is a natural extension of the differential geometry on ordinary continuous space. We define the generalized gauge field on the principal bundle with the base space M43 ZN by

adopting the one-form bases dxm _{in M}

4 and xk (k 41, R , N) in ZN. The generalized

gauge field introduced in such a way contains both the ordinary gauge field and the Higgs field, which realizes a unified picture of the gauge and Higgs fields on the same footing. According to our formulation of NCG, we have constructed the standard model [13], the left-right symmetric gauge model (LRSM) [14], SU(5) and SO(10) grand unified theories [11, 12]. Generally speaking, the NCG approach imposes severe constraints on the Higgs potential and interacting terms. However, we could succeed in reconstructing the standard model [13] because there exists one kind of Higgs boson in this model. We could perform the renormalization group analysis for the Higgs boson mass with the initial condition mH4k2mWstrongly suggested to hold at the energy of

the unification of electromagnetic and weak interactions

(

g 4k_{5 /3 g 8 for the coupling}

constants of SU( 2 )Land U( 1 )Ygauge group

)

. The Higgs boson mass is predicted to be

164 GeV for the top quark mass 175 GeV [17]. On the other hand, we were not so successful to reconstruct the left-right symmetric gauge model [14] because we could not reproduce the sufficient Higgs potential and interacting terms owing to the severe constraints by the algebraic rules in our previous NCG, though the Higgs kinetic terms are nicely introduced as curvatures on the discrete space. However, LRSM is still alive as a model with the intermediate symmetry of the spontaneously broken SO( 10 ) grand unified theory (GUT) and are very important to contain the two-doublet fields f1, f2

and the SU( 2 )R adjoint field jR. These Higgs bosons play important roles in giving

masses to up and down quarks and inducing the seesaw mechanism for the left- and right-handed neutrinos [18]. In order to improve our previous reconstruction of LRSM [14] we relax in this article the algebraic rules to obtain the necessary terms for the Higgs potentials and interactions.

This paper consists of five sections. Section 2 presents our noncommutative differential geometry on the discrete space M43 ZN. The generalized gauge field is

defined there and a geometrical picture for the unification of the gauge and Higgs fields is realized. An algebraic rule will be modified so as to present the Higgs potential terms sufficient to explain the phenomenology of LRSM. The third section provides the reconstruction of the Yang-Mills-Higgs sector in LRSM. The characteristic point is to take the fermion field to be a vector in a 24-dimensional space which has the same placement of leptons and quarks as in SO( 10 ) GUT. Corresponding to this specification, all gauge and Higgs boson fields are represented in 24 324 matrix forms. As the Higgs bosons, we incorporate h and jRtransformed as ( 2 , 2* , 0 ) and ( 1 , 3 , 22)

for SU( 2 )L3 SU( 2 )R3 U( 1 ), respectively. After the construction of Yang-Mills-Higgs

sector, the gauge and Higgs boson masses are investigated. In sect. 4, the fermionic sector of LRSM is proposed after the presentation of the general formulation. It will be shown that the fermionic Lagrangian including lepton and quark sectors together is expressed in very compact form by use of the 24-dimensional left and right fermion fields. The Lagrangian for the leptonic sector is nicely constructed so that the seesaw mechanism [18] works well. Quark sector contains the generation mixing and up and

down quarks acquire the masses through the vacuum expectation value of h. The last section is devoted to concluding remarks.

2. – Noncommutative geometry on the discrete space M43 ZN

The NCG on the discrete space M43 ZNhas been formulated in ref. [16], which we

here briefly review by fitting it to the presentation in this article. We first define the generalized gauge field on the principal bundle with the base space M43 ZN and the

structure group of the corresponding internal symmetry. Let us denote the generalized gauge field A(x , n) which is represented in the differential form of degree one as A(x , n) 4

!

i a† i(x , n) dai(x , n) 1

!

j bj † (x , n) dbj(x , n) , (2.1)

where the ai(x , n) and bj(x , n) are the square-matrix–valued functions relating to the

internal symmetry and are assumed to be [ai(x , n), bj(x , n) ] 40. The subscript i and j

are variables of the extra internal space which we cannot yet identify. Now, we simply regard ai(x , n) as a more fundamental field to construct gauge and Higgs fields. The

reason why bj(x , n) is incorporated is that the color gauge field Gm(x) is composed of

bj(x), whereas the gauge fields of the flavor symmetry and the Higgs field are

composed of ai(x , n), as shown later. However, in order to retain the consistent

definition of gauge field, the following condition has to be imposed:

!

i a_i†_{(x , n) a} i(x , n) 41 , (2.2)

!

j bj † (x , n) bj(x , n) 4 1 gs (2.3)

with the constant gs relating to the coupling constant of the color gauge field. d in

eq. (2.1) is the generalized exterior derivative defined as follows: d 4d1dx4 d 1

!

k dxk, (2.4) dai(x , n) 4¯mai(x , n) dxm, (2.5) dbj(x , n) 4¯mbj(x , n) dxm, (2.6) dxkai(x , n) 4 [2ai(x , n) Mnkxk1 Mnkxkai(x , n) ] , (2.7) dxkbj(x , n) 40 , (2.8) where dxm _{and x}

k are the one-form bases, taken to be dimensionless, in Minkowski

space M4 and in the discrete space ZN, respectively. We have introduced the

x-independent matrix Mnm generally expressed in the rectangular matrix whose

hermitian conjugation is given by Mnk†4 Mkn. The matrix Mnk turns out to determine

symmetry-breaking mechanism is encoded in the dx operation. We determine several

important algebraic rules in our NCG. Detailed descriptions of our NCG are shown in ref. [16] and we here expound it concisely. Let us at first introduce the shifting rule described by

xkai(x , n) 4ai(x , k) xk,

(2.9)

which shows how to change the discrete variable when xkis shifted from left to right in

an equation and makes the product of matrix-valued functions consistently calculable. Inserting eqs. (2.4)-(2.8) into eq. (2.1) and using eq. (2.9), A(x , n) is rewritten as

A(x , n) 4A(x, n)1G(x, n) (2.10)

with the definitions that

A(x , n) 4Am(x , n) dxm1

!

k Fnk(x) xk, (2.11) G(x , n) 4Gm(x , n) dxm, (2.12) where Am(x , n) 4

!

i a_i†_{(x , n) ¯} mai(x , n) , (2.13) Fnk(x) 4

!

i a_i†_{(x , n)[2a}i(x , n) Mnk1 Mnkai(x , k) ] , (2.14) Gm(x , n) 4

!

j bj † (x , n) ¯mbj(x , n) . (2.15)

Here, Am(x , n), Fnk(x) and Gm(x , n) are identified with the gauge field of the flavor

symmetry, the Higgs field and the color gauge field, respectively. We define the generalized field strength as

F(x , n) 4F(x, n)1 G(x, n) , (2.16)

where

F(x , n) 4dA(x, n)1A(x, n)RA(x, n) ,

(2.17)

G(x , n) 4dG(x, n)1gsG(x , n) R GA(x , n) .

(2.18)

In order to obtain the gauge-covariant generalized field strength, we address the nilpotency of d by arranging the following rules:

dxl(Mnkxk) 4 (Mnlxl) R (Mnkxk) 4MnlMlkxlR xk, (2.19)

and in addition, whenever the dxkoperation jumps over Mnlxl, a minus sign is attached. For example,

dxk

(

Mnlxla(x , n)

)

4 (dxkMnlxl) a(x , n) 2MnlxlR

(

dxka(x , n)

)

. (2.20)

With these considerations, we can easily calculate

(2.21) dxl

(

dxka(x , n)

)

4 dxl

(

2 a(x , n) Mnkxk1 Mnkxka(x , n)

)

4 4 2

(

dxla(x , n)

)

R Mnkxk2 a(x , n)(dxlMnkxk) 1 1(dxlMnkxk) a(x , n) 2MnkxkR

(

dxla(x , n)

)

4 4 2 (Mnlxl) a(x , n) R (Mnkxk) 1 (Mnlxl) R (Mnkxk) a(x , n) 1 1(Mnkxk) a(x , n) R (Mnlxl) 2 (Mnkxk) R (Mnlxl) a(x , n) , which yields (dxldxk1 dxkdxl) a(x , n) 40 . (2.22)

In eq. (2.21), we should note that xlR xk is independent of xkR xl for k c l which

characterizes the noncommutative property of the NCG in this article. It should be emphasized that eq. (2.21) follows without using the shifting rule in eq. (2.9). From eq. (2.22) we obtain the nilpotency of dx2a(x , n) 40. Then, the nilpotency of d

d2_{a(x , n) 4 (d}2_{1 dd}x1 dxd 1dx2) a(x , n) 40

(2.23)

follows with assumptions that dxm_{R x}

k4 2 xkR dxm. Then, we can find eq. (2.17)

written as

F(x , n) 4dai

†_{(x , n) R da}

i(x , n) 1A(x, n)RA(x, n)

(2.24)

using the definition of A(x , n) in eq. (2.1). da_i†(x , n) R dai(x , n) consists of three parts,

of which we investigate a two-form with xkR xl:

(2.25) dxai †_{(x , n) R d} xai(x , n) 4

!

k dxkai †_{(x , n) R}

!

l dxlai(x , n) 4 4

!

k

(

2 ai †_{(x , n) M} nkxk1 Mnkxkai†(x , n)

)

R

!

l

(

2 ai †_{(x , n) M} nlxl1 Mnlxlai†(x , n)

)

.

Here, we adopt new algebraic rule in order to move xk into the right-hand side in the

equation. This rule is an extension of the shifting rule in eq. (2.9) and is applied only in the equation with more than two-form differentials. It proceeds as follows:

(2.26) dxai †_{(x , n) R d} xai(x , n) 4

!

k

!

k 8

!

l

(

2 ai †_{(x , n) M} nk1 Mnkai † (x , k 8) e2iakk 8

₎

_Q Q eiakk 8

(

_{2 a} i(x , k 8) Mk 8 l1 Mk 8 lai † (x , l)

)

x_{k 8}R xl,

where the sum for k 8 is taken when ai(x , k) 4eiakk 8ai(x , k 8) is satisfied. Thus, k 8 may

be k itself, or it may represent another suffix and eiakk_{4 1. e}iakk 8 _{and e}2iakk 8 _emerge when xk jumps over ai(x , n) and ai

†_{(x , n), respectively and x}

k changes to xk 8. This

gauge field. The phase eia_{kk 8 corresponds to that U( 1 ) gauge field. This situation is}

explicitly explained in the reconstruction of LRSM in the following section. As a result, we obtain (2.27) dxai †_{(x , n) R d} xai(x , n) 4 4

!

k , k 8, l]FnkMk 8 l1 MnkFk 8 l1 MnkMk 8 l2 ai †_{(x , n) M} nkMk 8 leiakk 8ai(x , l)( xk 8R xl,

which combined with

!

k Fnk(x) xkR

!

l Fnl(x) xl4

!

k , k 8, l Fnk(x) Fk 8 l(x) xk 8R xl (2.28)

in A(x , n) R A(x , n) yields the equation

V_{n , k , k 8, l(x) 4}

(

Fnk(x) 1Mnk

)(

Fk 8 l(x) 1Mk 8 l

)

2 ai

†_{(x , n) M}

nkMk 8 leiakk 8ai(x , l) .

(2.29)

With these considerations we can find the explicit forms of the generalized field strength F(x , n) and G(x , n), (2.30) _{F(x , n) 4} 1 2 Fmn(x , n) dx m_{R dx}n 1

!

k c n DmFnk(x) dxmR xk1 1

!

k c n Vnk(x) xkR xn1

!

k c n

!

k 8l c k 8

!

V_{nkk 8 l}(x) x_{k 8}R xl, where (2.31) Fmn(x , n) 4¯mAn(x , n) 2¯nAm(x , n) 1 [Am(x , n), Am(x , n) ] , (2.32) DmFnk(x) 4¯mFnk(x) 2

(

Fnk(x) 1Mnk

)

Am(x , k) 1Am(x , n)(Mnk1 Fnk(x)

)

, (2.33) Vnk(x) 4

(

Fnk(x) 1Mnk

)

(Fkn(x) 1Mkn

)

2 Ynk(x) for k c n , (2.34) V_{nkk 8 l(x) 4}

(

Fnk(x) 1Mnk

)(

Fk 8 l(x) 1Mk 8 l

)

2 Ynkk 8 l(x) for k c n , l c k 8.

Ynk(x) and Ynkk 8 l(x) are the auxiliary fields written as

Ynk(x) 4

!

i a† i(x , n) MnkMknai(x , n) , (2.35) Y_{nkk 8 l(x) 4}

!

i a† i (x , n) MnkMk 8 leiakk 8ai(x , l) , (2.36)

which may be a constant or an independent fields. G(x , n) is expressed as G(x , n) 4 1 2 Gmn(x , n) dx m_{R dx}n_, (2.37) where Gmn(x , n) 4¯mGn(x , n) 2¯nGm(x , n) 1gs[Gn(x , n), Gm(x , n) ] . (2.38)

We address the gauge transformation of ai(x , n), which is defined as aig(x , n) 4ai(x , n) gf(x , n) , (2.39) bj g (x , n) 4bj(x , n) gc(x) , (2.40)

where gf(x , n) and gc(x) are the gauge functions with respect to the corresponding

flavor unitary group specified by the argument n and color group, respectively. It should be noted that gf(x , n) and gc(x) commute with each other. Let us define the dx

operation on gf(x , n) by (2.41) dgf(x , n) 4

g

d 1

!

k dxk

h

gf(x , n) 4 4 ¯mgf(x , n) dxm1

!

k [2gf (x , n) Mnk1 Mnkgf(x , k) ] xk.

Then, we can find from eq. (2.1) the gauge transformation of A(x , n) to be Ag

(x , n) 4g21_{(x , n) A(x , n) g(x , n) 1g}21_{(x , n) dg(x , n) ,}

(2.42)

where g(x , n) 4gf(x , n) gc(x). From eq. (2.42), we can find the gauge transformations

for gauge and Higgs fields as

Amg(x , n) 4gf21(x , n) Am(x , y) gf(x , n) 1gf21(x , n) ¯mgf(x , n) , (2.43) Fgnm(x) 4gf21(x , n) Fnm(x) gf(x , m) 1gf21(x , n) ¯nmgf(x , n) , (2.44) Gmg(x , n) 4gc21(x) Gm(x , n) gc(x) 1 1 gs g21 c (x) ¯mgc(x) . (2.45)

In eq. (2.44), the operator ¯nmis defined as

¯nmgf(x , n) 42 gf(x , n) Mnm1 Mnmgf(x , m) ,

(2.46)

which seems to be regarded as a difference operator in ZN. Equation (2.44) is very

similar to other two equations and so it strongly indicates that the Higgs field is a kind of gauge field on the discrete space M43 ZN. From eq. (2.46), eq. (2.44) is rewritten

as

Fg

nm(x) 1Mnm4 gf21(x , n)

(

Fnm(x) 1Mnm

)

gf(x , m) ,

(2.47)

which makes it obvious that

Hnm(x) 4Fnm(x) 1Mnm

(2.48)

is an unshifted Higgs field, whereas Fnm(x) denotes the shifted one with the vanishing

transformation of the generalized field strength is found to be Fg

(x , n) 4g21_{(x , n) F(x , n) g(x , n) ,}

(2.49)

which ensures the introduction of the gauge-invariant Lagrangian.

In order to construct the Yang-Mills-Higgs Lagrangian, we define the metric structure of one-forms as

adxm_{, dx}n

b 4gmn_{, ax}

n, dxmb 40 , axn, xkb 42 dnk.

(2.50)

According to these metric structures and eq. (2.30) we can obtain the expression for gauge-invariant Yang-Mills-Higgs Lagrangian

(2.51) LYMH(x) 42

!

n 41 N ₁ gn2 a F(x , n), F(x , n)b 4 4 2 Tr

!

n 41 N ₁ 2 gn2 Fmn†(x , n) Fmn(x , n) 2Tr

!

n 41 N ₁ 2 gn2 Gmn†(x , n) Gmn(x , n) 1 1 Tr

!

n 41 N

!

k c n 1 gn2

(

DmFnk(x)

)

†DmFnk(x) 2 2 Tr

!

n 41 N ₁ gn2

!

k c n Vnk†(x) Vnk(x) 2Tr

!

n 41 N ₁ gn2

!

k c n

!

k 8 l c k 8

!

V_{nkk 8 l}† _{(x) V} nkk 8 l(x) ,

where gn are constants relating to the gauge coupling constant and Tr denotes the

trace over internal symmetry matrices. The last two terms are the potential and interaction terms of the Higgs particles.

3. – Reconstruction of the Yang-Mills-Higgs sector in LRSM

The purpose of this section is to reconstruct the left-right symmetric gauge model (LRSM) based on the formulation in the previous section. LRSM is the gauge theory with the symmetry SU( 3 )c3 SU( 2 )L3 SU( 2 )R3 U( 1 ) and is promising as an

alternative model of the standard model because LRSM exhibits the seesaw mechanism for the left- and right-handed neutrinos [18] and contains two doublet Higgs bosons to give masses to the up and down quarks separately. In addition, it is important because the symmetry of LRSM is the intermediate symmetry of the spontaneously broken SO( 10 ) GUT.

First, we must consider the formations of gauge fields and the Higgs fields which are incorporated in the generalized gauge field A(x , n). Generally speaking, some devices are needed to reproduce the necessary terms for the Higgs potential and interactions so as to explain the phenomenology of LRSM, whereas the kinetic terms of gauge and Higgs fields come out without any difficulty. In order to meet the consistent presentation of LRSM, we consider the discrete space M43 Z4on which we specify the

generalized gauge field A(x , n) (n 41, 2, 3, 4) and the fermion fields c(x, n) (n4 1 , 2 , 3 , 4 ). Let us begin with the specification of the fermion field c(x , n) on the

discrete space M43 Z4.

.

`

`

`

`

`

/

`

`

`

`

`

´

c(x , 1 ) 4 1 k2

.

`

`

`

`

`

´

ur L uLg ub L nL dLr dLg dLb eL

ˆ

`

`

`

`

`

˜

, _{c(x , 2 ) 4} 1 k2

.

`

`

`

`

`

´

ur R uRg ub R nR dRr dRg dRb eR

ˆ

`

`

`

`

`

˜

, c(x , 3 ) 4 1 k2

.

`

`

`

`

`

´

drc L dLgc dbc L eLc 2uLrc 2uLgc 2uLbc 2nLc

ˆ

`

`

`

`

`

˜

, _{c(x , 4 ) 4} 1 k2

.

`

`

`

`

`

´

drc R dRgc dbc R eRc 2uRrc 2uRgc 2uRbc 2nR c

ˆ

`

`

`

`

`

˜

, (3.1)

where the subscripts L and R denote the left-handed and right-handed fermions, respectively, and the superscripts r, g and b represent the color indices. The superscript c in c(x , 3 ) and c(x , 4 ) represents the antiparticle of the respective fermions. It should be noted that c(x , n) has the index for the three generation as do the explicit expressions for fermions in the right-hand sides of eq. (3.1). In more precise expressions, u , d , n and e in eq. (3.1) should be replaced by

u K

u

u c t

v

, _{d K}

u

d s b

v

, _{n K}

u

ne nm nt

v

, _{e K}

u

e m t

v

, (3.2)

respectively. Thus, c(x , n) (n 41, 2, 3, 4) is a vector in the 24-dimensional space. With these specifications of the fermion fields written as 24-dimensional vector, the generalized gauge fields A(x , n) must be written in 24 324 matrices to express the interactions between the fermions and the generalized gauge bosons. We begin with the construction of the Yang-Mills-Higgs sector of LRSM.

3.1. Yang-Mills-Higgs Lagrangian. – In order to get Yang-Mills-Higgs Lagrangian in LRSM we prepare the discrete space M43 Z4. First, we specify A(n)(n 41, 2, 3, 4)

in eqs. (2.10), (2.11) and (2.12) in the explicit forms of gauge and Higgs fields. Fnk4 Fkn†

is implied due to Mnk4 Mkn† as known from eq. (2.14). For simplicity, the argument x in

the Minkowski space M4is abbreviated in the expressions

A( 1 ) 4Am( 1 ) dxm1 Gm( 1 ) dxm1 F12x2,

where Am( 1 ) 4ALm( 1 ) 1Bm( 1 ) 42 i 2

m

!

k tk7 147 13ALm k 1 c 7 13Bm

n

, (3.4) Gm( 1 ) 42 i 2 a 41

!

8 12 7 l 8a7 13Gma(x) , (3.5) F124 f 7 147 13. (3.6) In eq. (3.4), ALm k_{, B}

m are SU( 2 )Land U( 1 ) gauge fields, respectively, tk(k 41, 2, 3) is

the Pauli matrix and c is the U( 1 ) hypercharge matrix corresponding to c(x , 1 ) and

c(x , 2 ) in eq. (3.1) expressed in c 4diag

g

1 3 , 1 3 , 1 3 , 21, 1 3 , 1 3 , 1 3 , 21

h

. (3.7)

In eq. (3.5), Gma is the color gauge field and l 8a is 4 34 matrix constructed from the

Gell-Mann matrix la _{by adding 0 components to fourth line and column}

l 8a 4

u

0 la 0 0 0 0 0 0

v

. (3.8)

In eq. (3.6), f is the field containing two doublet Higgs bosons with the expression that f 4

u

f2 0 f22 f11 f01

v

. (3.9)

Together with F12, we must write down M12which determines the symmetry-breaking

pattern of SU( 2 )L3 SU( 2 )R3 U( 1 ).

M124 M21†4 m127 147 134

u

m2 0 0 m1

v

7 147 13. (3.10)

h 4f1m12is transformed as ( 2 , 2* , 0 ) under SU( 2 )L3 SU( 2 )R3 U( 1 ).

A( 2 ) is specified as follows: A( 2 ) 4Am( 2 ) dxm1 Gm( 2 ) dxm1 F21x11 F24x4, (3.11) where Gm( 2 ) 4Gm( 1 ) and Am( 2 ) 4ALm( 2 ) 1Bm( 2 ) 42 i 2

m

!

k tk7 147 13ARkm1 c 7 1 3 Bm

n

, (3.12) F214 F12† 4 f†7 147 13, (3.13)

F244 j 7 b 7 13, b 4

u

0 0 0 1

v

. (3.14) In eq. (3.12), ARm k

(k 41, 2, 3) are the SU(2)Rgauge fields. j is denoted as j 4

u

j 2 j2 2 j0 2j2

v

. (3.15)

The matrix b 4diag (0, 0, 0, 1) in eq. (3.14) is needed in order to avoid the interaction between quarks and j, which leads to the hypercharge conservation. M24prescribes the

symmetry breaking of SU( 2 )R3 U( 1 ) down to U( 1 )Y. M244 m247 b 7 134

u

0 0 MR 0

v

7 b 7 1 3_. (3.16)

jR4 j 1 m24is transformed as ( 0 , 3 , 22) under SU(2)L3 SU( 2 )R3 U( 1 ).

A( 3 ) is described as

A( 3 ) 4Am( 3 ) dxm1 Gm( 3 ) dxm1 F34x4,

(3.17)

where Gm( 3 ) is given by replacing l 8a in eq. (3.5) by the complex conjugate 2l8a* and

so Am( 3 ) 4ALm( 3 ) 1Bm( 3 ) 42 i 2

m

!

k tk7 1 4 7 13ALm k 2 c 7 13Bm

n

, (3.18) Gm( 3 ) 4 i 2 a 41

!

8 12 7 l 8a* 7 13_G ma(x) , (3.19) F344 fA7147 13. (3.20) In eq. (3.20), f A 4 t2f * t2 4

u

f1 0_* 2f21 2f12 f02*

v

(3.21) and M344 M43†4

u

m1 0 0 m2

v

7 147 134 m127 147 13. (3.22)

M34is also responsible for the symmetry breaking of SU( 2 )L3 SU( 2 )R3 U( 1 ).

A( 4 ) is specified as

A( 4 ) 4Am( 4 ) dxm1 Gm( 4 ) dxm1 F42x21 F43x3,

where Gm( 4 ) 4Gm( 3 ) and Am( 4 ) 4ALm( 4 ) 1Bm( 4 ) 42 i 2

m

!

k tk7 147 13ARm k 2 c 7 13Bm

n

, (3.24) F434 F34† 4 fA†7 147 13, (3.25) F424 F24† 4 j†7 b 7 13. (3.26)

It should be noted that 2c represents the U(1) hypercharge of c(x, 3) and c(x, 4) in eq. (3.1) M424 M24†4

u

0 MR 0 0

v

7 b 7 1 3_. (3.27)

We assume M134 M144 M234 0 in the above specifications. If we would incorporate the

adjoint Higgs field jL for SU( 2 )L symmetry to completely maintain the left-right

symmetry, jLwould be assigned in F13. However, we do not consider the incorporation

of jL because the symmetry breaking of SU( 2 )L3 U( 1 ) down to U( 1 )e.m.is caused by

the Higgs doublet field f.

In the above presentations of the explicit form of the generalized gauge field A(n), the following should be remarked:

[ALm(n), Bm(n) ] 4 [ALm(n), Gm(n) ] 4 [Gm(n), Fnk] 40 , (3.28)

which enables us to consistently reconstruct the LRSM Lagrangian.

The gauge transformation functions g(x , n) (n 41, 2, 3, 4) in eq. (2.42) are given as

.

`

`

/

`

`

´

g(x , 1 ) 4e2iaa(x)_g L(x) gc(x) , e2iaa(x)_{U( 1 ) , g} L(x)  SU( 2 )L, gc(x)  SU( 3 )c, g(x , 2 ) 4e2iaa(x)_g R(x) gc(x) , e2iaa(x)_{U( 1 ) , g} R(x)  SU( 2 )R, gc(x)  SU( 3 )c, g(x , 3 ) 4eiaa(x)_g L(x) gc* (x) , eiaa(x) U( 1 ) , g_L(x)  SU( 2 )_L, g_c* (x)  SU( 3 )_c, g(x , 4 ) 4eica(x)_g R(x) gc* (x) , eiaa(x) U( 1 ) , gR(x)  SU( 2 )R, gc* (x)  SU( 3 )c, (3.29)

in which it should be remarked that these gauge functions are taken so as to satisfy [gL , R(x), gc(x) ] 4 [gL , R(x), e6iaa(x)] 4 [gc(x), e6iaa(x)] 40 .

(3.30)

With these preparations, we can find the Yang-Mills-Higgs Lagrangian as LYMH(x) 4 LGB1 LHK2 VHP,

(3.31)

where LGBand LHKare kinetic terms of the gauge and Higgs boson fields, respectively

expressed as (3.32) _LGB4 2 1 4

g

12 g12 1 12 g32

h

!

i 41 3 FLimnFL imn 2 1 4

g

12 g22 1 12 g43

h

!

i 41 3 FRimnFR imn 21 4

g

k 41

!

4 ₃ 2 gk2 Tr c2

h

BmnB mn 2 1 4

g

k 41

!

4 ₆ gk2

h

!

a 41 8 Gmn a Gamn.

By defining the gauge coupling constants as 1 gL2 4 12 g12 1 12 g32 , (3.33) 1 gR2 4 12 g22 1 12 g42 , (3.34) 1 gB2 4

!

k 41 4 ₃ 2 gk2 Tr c2, (3.35) 1 g_{s 8}2 4_{k 41}

!

4 ₆ gk2 , (3.36)

and carrying out the scale transformations of gauge fields, we find LGB4 2 1 4

!

i Fi LmnF_Limn2 1 4

!

i Fi RmnF_Rimn2 1 4 BmnB mn 2 1 4

!

a Gmn a _Gamn (3.37) where FL , Ri mn_{4 ¯m}A_{L , R}i n_{2 ¯n}A_{L , R}i m_{1 gL , R}eijkA_{L , R}j mA_{L , R}k n, (3.38) Bmn4 ¯mBn2 ¯nBm, (3.39) Gmna4 ¯mGna2 ¯nGma1 gc fabcGmbGnc, (3.40)

with the definition

gc4 gs 8gs4 gs

g

!

k 41 4 ₆ gk2

h

21 O2 . (3.41)

Higgs kinetic term LHK consists of two terms LhK and LjRK for the Higgs doublet

and SU( 2 )R adjoint bosons, respectively. Four kinetic terms of F12, F21, F34 and F43

contribute to LhKand as a result, LhKis written as

LhK4 12

g

1 g12 1 1 g22 1 1 g32 1 1 g42

h

Tr NDmhN2, (3.42) where Dmh 4¯mh 2 i 2gLk 41

!

3 tk_A Lmk h 2 i 2gRhk 41

!

3 tk_Ak Rm (3.43)

with the definition that h 4f1m124

u

f2 0 1 m2 f22 f11 f101 m1

v

. (3.44)

The rescaling ofkghh Kh with

gh4 12

g

1 g12 1 1 g22 1 1 g32 1 1 g42

h

(3.45)

leads eq. (3.42) to the standard form of the kinetic term:

LhK4 Tr NDmhN2

(3.46)

with the same expression for Dmh as in eq. (3.43). The kinetic term of the SU( 2 )R

adjoint Higgs boson jR is given as LjRK4 3

g

1 g22 1 1 g42

h

Tr NDmjRN2, (3.168) where DmjR4 ¯mjR2 i 2gRk 41

!

3 [tk_A Rk, jR] 1igBBmjR (3.48)

with the notation that

jR4 j 1 m244

u

j2 j2 2 j0 1 MR 2j2

v

. (3.49)

The rescaling of kgjjRK jRwith gj4 3

g

1 g22 1 1 g42

h

(3.50)

leads eq. (3.47) to the standard form:

LjRK4 Tr NDmjRN

2

(3.51)

with the same expression for DmjR as in eq. (3.48).

Then, we move to the Higgs potential and interacting terms which are explained somewhat in detail because the managements of the auxiliary fields in eqs. (2.35) and (2.36) are rather complicated. For simplicity, g14 g3 and g24 g4are assumed, which is

plausible from the explicit assignments of fields in eqs. (3.3), (3.11), (3.17) and (3.23).

VHPin eq. (3.31) consists of six terms as VHP4 Vhh†_{1 Vh h}A_{1 Vhj}

R1 VjRj†R1 V 8jRjR†1 VjRjR,

which are explicitly expressed hereafter. After the rescalingkghh Kh, Vhh†is written as (3.53) Vhh†₄ 12 g12gh2 Tr Nhh† 2 y12N21 12 g22gh2 Tr Nh† h 2y21N21 1 12 g12gh2 Tr NhA hA† 2 y34N21 12 g22gh2 Tr NhA†_hA 2y43N2,

where the auxiliary fields are written in 2 32 matrices by taking the trace of the common factor 14 7 13and expressed as y124

!

i ai†( 1 ) m12m21ai( 1 ) 4

!

i ai†( 1 )

u

m22 0 0 m12

v

ai( 1 ) , (3.54) y214

!

i ai†( 2 ) m21m12ai( 2 ) 4

!

i ai†( 2 )

u

m22 0 0 m12

v

ai( 2 ) , (3.55) y344

!

i ai†( 3 ) m34m43ai( 3 ) 4

!

i ai†( 3 )

u

m1 2 0 0 m22

v

ai( 3 ) , (3.56) y434

!

i ai †_{( 4 ) m} 43m34ai( 4 ) 4

!

i ai †_{( 4 )}

u

m21 0 0 m22

v

ai( 4 ) . (3.57)

In these equations, ai(n) (n 41, 2, 3, 4) are reduced to 232 matrices after taking the

trace of 14

7 13. In managing these auxiliary fields, we should remark that from the explicit assignments of gauge fields in eqs. (3.4), (3.12), (3.18) and (3.24), the only difference between Am( 1 )

(

Am( 2 )

)

and Am( 3 )

(

Am( 4 )

)

is the contribution of the U( 1 )

gauge field. Therefore, we can conjecture from the constituent form of the gauge field in eq. (2.13) that the relations

ai( 1 ) 4eiaiai( 3 ) , ai( 2 ) 4e iai_a

i( 4 )

(3.58)

hold for fundamental fields ai(n) (n 41, 2, 3, 4). Owing to eqs. (2.2) and (3.58), we can

find the relations

y121 y344 (m211 m22) Q 12, y211 y434 (m121 m22) Q 12.

(3.59)

Eliminating the independent auxiliary field y12 in eq. (3.53) by use of the equation of

motion, we obtain (3.60) Vhh†4

g

6 g12gh2 1 6 g22g2h

h

Tr Nhh† 1 hA hA†2 (m121 m22) Q 12N21 1 24 g22gh2 Tr

N

y212 h† h 2hA†_hA 1(m211 m22) Q 12 2

N

2 , where the last term is later combined with the term occurring in VjRjR

With the same rescaling ofkghh Kh, Vh hAis written as (3.61) Vh hA4 12 g12gh2 Tr NhhA† 2 y1243N21 12 g22gh2 Tr Nh†_hA 2y2134N21 1 12 g12gh2 Tr NhA h† 2 y3421N21 12 g22gh2 Tr NhA† h 2y4312N2, where y12434

!

i ai†( 1 ) m12m43eiaiai( 3 ) 4

!

i ai†( 1 )

u

m1m2 0 0 m1m2

v

eiai_a i( 3 ) , (3.62) y21344

!

i ai†( 2 ) m21m34eiaiai( 4 ) 4

!

i ai†( 2 )

u

m1m2 0 0 m1m2

v

eiai_a i( 4 ) , (3.63) y34214

!

i ai†( 3 ) m34m21e2iaiai( 1 ) 4

!

i ai †_{( 3 )}

u

m1m2 0 0 m1m2

v

e2iai_a i( 1 ) , (3.64) y43124

!

i ai†( 4 ) m43m12e2iaiai( 2 ) 4

!

i ai † ( 4 )

u

m1m2 0 0 m1m2

v

e2iai_a i( 2 ) . (3.65)

According to eq. (3.58) together with eq. (2.2), we find

y12434 y21344 y34214 y43124 m1m2Q 12.

(3.66)

Inserting eq. (3.66) into eq. (3.61), we obtain

Vh hA4

g

2 g12g2h 1 2 g22gh2

h

Tr NhhA† 2 m1m2Q 12N2 (3.67) because of Tr NhhA†2 m1m2Q 12N24 TrNh†hA2m1m2Q 12N2.

Then, let us move to VhjR. After the rescaling of kghh Kh and kgjjRK jR,

VhjRis written as (3.68) VhjR4 3 g12ghgj ]NhjR2 y1224N21 Nhj†R2 y1242N2( 1 1 3 g22ghgj ]NjRhA†2 y2443N21 NjRh†2 y2421N2( 1 1 3 g32ghgj ]N hA j†R2 y3442N21 N hA jR2 y3424N2( 1 1 3 g42ghgj ]Nj†Rh†2 y4221N21 Nj†RhA†2 y4243N2( ,

where y12244

!

i a†_{( 1 ) m} 12m24ai( 4 ) 4

!

i a†_{( 1 )}

u

0 0 m2MR 0

v

ai( 4 ) , (3.69) y12424

!

i a†_{( 1 ) m} 12m42ai( 2 ) 4

!

i a†_{( 1 )}

u

0 m1MR 0 0

v

e iai_a i( 2 ) , (3.70) y24434

!

i a†_{( 2 ) m} 24m43ai( 3 ) 4

!

i a†_{( 1 )}

u

0 0 m2MR 0

v

ai( 4 ) , (3.71) y24214

!

i a†_{( 2 ) m} 24m21ai( 1 ) 4

!

i a†_{( 2 )}

u

0 0 m1MR 0

v

e 2iai_a i( 1 ) , (3.72) y34424

!

i a†_{( 3 ) m} 34m42ai( 2 ) 4

!

i a†_{( 1 )}

u

0 m2MR 0 0

v

ai( 2 ) , (3.73) y34244

!

i a†_{( 3 ) m} 34m24ai( 4 ) 4

!

i a†_{( 3 )}

u

0 0 m1MR 0

v

e 2iai_a i( 4 ) , (3.74) y42214

!

i a†_{( 4 ) m} 42m21ai( 1 ) 4

!

i a†_{( 4 )}

u

0 m2MR 0 0

v

ai( 1 ) , (3.75) y42434

!

i a†_{( 4 ) m} 42m43ai( 3 ) 4

!

i a†_{( 4 )}

u

0 m1MR 0 0

v

e iai_a i( 1 ) . (3.76)

According to eq. (3.58) and by rewriting the above equations in terms of only ai( 1 ) and

ai( 2 ), VhjRis expressed as (3.77) VhjR4

g

1 g12 1 1 g22

h

3 ghgj ]NhjR2 y1224N21 N hA jR2 y3424N21 1NhjR†2 y1242N21 N hA j†R2 y3442N2( , where y12244

!

i a†_{( 1 )}

u

0 0 m2MR 0

v

e 2iai_a i( 2 ) , y34244

!

i a†_{( 1 )}

u

0 0 m1MR 0

v

e 2iai_a i( 2 ) , (3.78) y12424

!

i a†_{( 1 )}

u

0 m1MR 0 0

v

e iai_a i( 2 ) , y34424

!

i a†_{( 1 )}

u

0 m2MR 0 0

v

e iai_a i( 2 ) , (3.79)

yield with the aid of the equation of motion VhjR4

g

1 g12 1 1 g22

h

3 ghgj

y

Tr N(m1h 2m2hA) jRN21 Tr N(m2h 2m1hA) j†RN2 m211 m22

z

. (3.80) VjRj†Ris written as VjRjR†4 3 g22g2j ]Tr NjRjR†2 y24N21 Tr NjR†jR2 y42N2( , (3.81) where y244

!

i ai †_{( 2 ) m} 24m42ai( 2 ) 4

!

i ai †_{( 2 )}

u

MR2 0 0 0

v

ai( 2 ) , (3.82) y424

!

i ai†( 4 ) m42m24ai( 4 ) 4

!

i ai†( 4 )

u

0 0 0 MR2

v

ai( 4 ) . (3.83)

From eqs. (3.58) and (2.2), we readily know y241 y424 MR2Q 12 which yields the

equation (3.84) VjRjR†4 3 2 g22g2j Tr NjRjR†1 j†RjR2 MR2Q 12N21 1 6 g22g2j Tr

N

y241 jRjR†2 jR†jR2 MR2Q 12 2

N

2 , where the last term is combined with the last term in eq. (3.60). From eq. (2.2), we know the relation y212 m22 m212 m22 1 y24 MR2 4 12 (3.85)

which together with the equation of motion helps us introduce the equation (3.86) _{V 8}hjR4 6 g22gj2 1 114g2jOgh2

(

(m222 m21) OMR2

)

2 Tr

N

h† h 2hA†_hA 2(m222 m21)

u

1 0 0 21

v

2 2m2 2 2 m12 MR2

{

jRj†R2 j†RjR2 MR2

u

1 0 0 21

v

}

N

2 . In the limit of MRcm1, m2, this term is reduced to

V 8hjR4 6 g22g2j Tr

N

h†_{h 2h}A†Ah_2(m222 m12)

u

1 0 0 21

v

N

2 . (3.87)

It should be remarked that the invariance of eq. (3.86) for SU( 2 )R symmetry is

invariant for SU( 2 )L gauge transformation. Thus, we adopt this approximation

hereafter. As a result of the elimination of auxiliary fields y21 and y24, we find the

potential terms for h and jR written as Vhh†₄

g

6 g12g2h 1 6 g22gh2

h

Tr Nhh†1 hA hA†2 (m211 m22) Q 12N2, (3.88) VjRjR †₄ 3 2 g22gj2 Tr NjRj†R1 j†RjR2 MRQ 12N2. (3.89)

Then, the term VjRjRis written as

VjRjR4 3 g22gj2 ]Tr NjRjR2 y2424N21 Tr NjR†j†R2 y4242N2( , (3.90) where y24244

!

i ai†( 2 ) m24m24e2iaiai( 4 ) 40 , y42424

!

i ai†( 4 ) m42m42eiaiai( 2 ) 40 . (3.91)

These equations simplify eq. (3.90) to yield

VjRjR4

6

g22g2j

Tr NjRjRN2,

(3.92)

which is very important for the doubly charged Higgs boson j2 2 _{to be massive, as}

shown later.

3.2. Gauge boson mass. – Let us investigate the various aspects of our LRSM introduced until now. Gauge boson mass matrix can be extracted from kinetic terms of the Higgs bosons h and jR. According to eqs. (3.46) and (3.51), the following mass

matrix for the neutral gauge boson follows:

MNGB4

u

gL2m2 2gLgRm2 0 2gLgRm2 gR2(m21 MR2) 2gRgBMR2 0 2gRgBMR2 gB2MR2

v

, (3.93) where m2

4 (m121 m22) /2 and the rescaling ofk2 MRK MRis carried out. Eigenvalues of

this mass matrix are easily calculated with the approximation of MRcm to be Mg24 0 , (3.94) MZ24

g

gL21 gR2gB2 gR21 gB2

h

m2, (3.95) MZR 2 4 (gR21 gB2) MR2, (3.96)

where g and Z represent of course photon and neutral weak boson, respectively and ZR

denotes the extra neutral gauge boson expected in this model. As a result of the spontaneous breakdown of gauge symmetry in the flavor sector, only remaining symmetry is the U( 1 ) electromagnetic one. This is reflected on the vanishing mass of

photon field in eq. (3.94). If we define the coupling constant g 8 as g 82 4 gR 2 gB2 gR21 gB2 , (3.97)

the mass of the neutral gauge boson Zmis written as

MZ24 (gL21 g 82) m2,

(3.98)

which is the favorite form in accord with the standard model. In this sense, g 8 is a coupling constant of the gauge symmetry resulting from the spontaneous breakdown of

SU( 2 )R3 U( 1 ) down to U( 1 )Y. MZRis estimated to be so large that one cannot detect

ZR in the energy range of the accelerator available nowadays.

The unitary matrix U to transform (ALm3 , ARm3 , Bm) into (Am, Zm, ZRm) is given as

U 4

u

sin uW cos uW 0 cos uWsin uY 2sin uWsinuY cos uY cos uWcos uY 2sin uWcos uY 2sin uY

v

, (3.99) where sin uW4 g 8

k

gL21 g 82 , cos uW4 gL

k

gL21 g 82 , (3.100) sin uY4 gB

k

gR21 gB2 , cos uY4 gR

k

gR21 gB2 . (3.101)

uW is the Weinberg angle and uYis a mixing angle between the gauge bosons ARm3 and Bm. From eq. (3.99), we can denote the transformation relations for neutral gauge

bosons as

Am4 AL m3 sin uW1 AR m3 cos uWsinuY1 Bmcos uWcos uY,

(3.102)

Zm4 AL m3 cos uW2 AR m3 sin uWsin uY2 Bmsin uWcos uY,

(3.103)

ZR m4 AR m3 cos uY2 Bmsin uY.

(3.104)

If B 8m is defined as B 8m4 Bmcos uY1 AR m3 sin uY we find the well-known mixing relations

written as Am4 B 8mcos uW1 ALm3 sin uW, (3.105) Zm4 2 B 8msin uW1 ALm3 cos uW (3.106) and B 8m4 Bmcos uY1 ARm3 sin uY, (3.107) ZRm4 2 Bmsin uY1 ARm3 cos uY. (3.108)

breakdown of SU( 2 )R3 U( 1 ) down to U( 1 )Y. In this context, we investigate the U( 1 )Y

hypercharge assignment of two-Higgs doublet bosons in h expressed in eq. (3.44). From eq. (3.43), we find

(3.109) Dmh 4¯mh 2 i 2gLk 41

!

3 tkALk2 2i 2 h

u

2gRA 3 Rm 0 0 gRARm3

v

1 i 2h

u

0 gRk2WRm2 gRk2WRm1 0

v

.

If we denote h 4 (h2, h1), the U( 1 )Y hypercharges of h1 and h2 are 11 and 21 from

eq. (3.109), respectively, because

gRARm3 4 gR(B 8msin uY1 ZRmcos uY) 4g 8 B 8m1 gRZRmcos uY

(3.110)

due to eq. (3.101).

Then, the mass matrix for charged gauge bosons is given also from eqs. (3.46) and (3.51) as MCGB4

u

gL2m2 2gLgRm1m2 2gLgRm1m2 gR2

(

m21 ( 1 O2 ) MR2

)

v

. (3.111)

In the limit of MRcm, the eigenvalues are written as MW24 gL2m2, (3.112) MW2R4 1 2gR 2 MR 2 , (3.113)

which together with the investigation of eigenstates indicates that the mixing between

W6

Lmand WRm6does not take place. Wm64 WLm6is the charged weak boson and WR64 WRm6

is also the extra charged gauge boson expected in this model and its mass is so high that it cannot be detectable. Combining eqs. (3.112) with (3.98), the mass relation

MW4 MZcos uW

(3.114)

holds. This is the same relation as in the standard model.

Then, we address the numerical values of the coupling constants and mixing angles

uW and uY within tree level when g14 g24 g34 g4. From eqs. (3.33)-(3.36) and with the

remark of Tr c2_{4 8 /3, the following equations hold:}

gL24 g12 24 , gR 2 4 g1 2 24 , gB 2 4 g1 2 16 , gc 2 4 gs2 g12 24 , (3.115) from which g 8B24

g

1 gR2 1 1 gB2

h

21 4 g1 2 40 (3.116)

follows. Thus, we obtain sin2_u W4 3 8 , sin 2_u Y4 3 5 , (3.117)

which are values predicted by SO( 10 ) GUT. However, in our model the grand unification for coupling constants is not achieved because of the extra parameter gsin gc

though gL4k5 /3 g 8.

3.3. The Higgs boson mass. – The mass spectrum of the Higgs bosons in tree level is also extracted from the Higgs potential and interacting terms given in eqs. (3.67), (3.80), (3.87)-(3.89) and (3.92). According to eq. (3.109), we know the particle contents which are absorbed by the gauge bosons when the spontaneous breakdown of symmetry takes place. If f10and f02are denoted as

f014 1 k2 (f1 R 0 1 if1 I0 ) , f204 1 k2(f2 R 0 1 if02 I) , (3.118)

the following combination of f01 Iand f02 I:

f0I8 4 f1 I0 cos b 2f2 I0 sin b

(3.119)

is absorbed into the neutral gauge boson, whereas

f0I4 f1 I0 sin b 1f02 Icos b

(3.120)

remains physical. In these equations, use has been made of cos b 4 m1

k

m121 m22 4 m1 k2 m , sin b 4 m2

k

m121 m22 4 m2 k2 m . (3.121)

Similarly, if the combination of f11 and f12 is denoted as f1

8 4 f11cos b 2f21sin b ,

(3.122)

this field is absorbed into the charged gauge boson, whereas

f1

4 f11 sin b 1f12 cos b

(3.123)

remains physical. After all, five Higgs bosons out of eight components of f1 and f2

become massive. The masses of these Higgs bosons are obtained below. Then, the components of j absorbed into gauge bosons are easily extracted from eq. (3.48). j0_and j2_{are absorbed into Z}

R and WR2, respectively. The only surviving Higgs boson is the

doubly charged component j2 2_{whose mass is of the order of M}

R, as shown below.

For simplicity, g14 g24 g34 g4is assumed without loss of generality. In this limit, gh21 and g21j are given as g12/24 4g2 and g12/6 44g2, respectively. From eqs. (3.67),

(3.80), (3.87)-(3.89) and (3.92), we find the mass terms of above-mentioned surviving Higgs bosons. The masses of f0I, f6, and j2 2are explicitly written as

mf2I04 g

2

( cos2b) MR2,

m2 f64 1 2 g 2 ( cos b 1sin b)2_M R2, (3.125) mj22 24 g2(m2cos22 b 18M_R2) , (3.126)

respectively. However, f1 R0 and f2 R0 are mixed as

(3.127) M2 f1 R0 , f2 R0 4 g 2

m

_m2_(f 2 R 0 sin b 1f01 Rcos b)21 m2 2 (f2 R 0 sin b 2f0 1 Rcos b)21

1m2(f2 R0 cos b 1f1 R0 sin b)21 MR2(f2 R0 cos b 2f01 Rsin b)2

n

.

In the limit of MRcm, we can diagonalize this mass term as Mf201 R, f02 R4 mfR0 2 fR0 21 mf0R8 2 f0R82, (3.128) where

fR04 f02 Rsin b 1f1 R0 cos b , fR08 4 f02 Rcos b 2f01 Rsin b ,

(3.129)

and the masses of fR0 and fR08 are given as mfR0 2 4 g2

g

4 2 1 2 cos 2_{2 b}

h

_m2_, (3.130) mfR08 2 4 g2

m

MR2

g

1 1 1 2 cos 2_{2 b}

h

_m2

n

_. (3.131)

According to the mass values presented above, it is known that the only detectable Higgs boson at the weak energy scale is fR0, whereas other five Higgs bosons with

masses of the scale MRare far beyond the energy scale of the present and near future

accelerator. This is a characteristic feature of the model in the present article. The approximation m1`m2`m yields the mass relation

mf0R4 2 MW.

(3.132)

Above discussions hold only at tree level. However, we can perform the renormalization group analyses of these relations under the assumption that these relations hold at the point that the electromagnetic and weak interactions are unified as shown in eqs. (3.115)-(3.117). This has been in fact done in the case of the standard model in ref. [16].

4. – Reconstruction of the fermionic sector in LRSM

The brief summary about the general formulation to obtain the Dirac Lagrangian in NCG is presented with small modification fitting it to the reconstruction of LRSM in this article.

We at first define the covariant derivative acting on the spinor c(x , n) by Dc(x , n) 4

(

_{d 1A}f_{(x , n)}

₎

c(x , n) ,

where Af

(x , n) is the differential representation to make Dc(x , n) covariant against the gauge transformation. In the present formulation stated in sect. 3, Af_{(x , n)}

coincides with A(x , n) in eq. (2.10). Thus, the superscript f in eq. (4.1) is removed, hereafter

A(x , n) 4

(

Am(x , n) 1Gm(x , n)

)

dxm1

!

k

Fnk(x) xk,

(4.2)

where it should be noted that the scale transformations of gauge and Higgs fields are performed after getting the Dirac Lagrangian. By setting the algebraic rule

dxc(x , n) 4

!

k dxkc(x , n) 4

!

k Mnkxkc(x , n) 4

!

k Mnkc(x , k) xk, (4.3) Dc(x , n) is described as Dc(x , n) 4

m

(

¯m1 Am(x , n) 1Gm(x , n)

)

dxm1

!

k Hnk(x) xk

n

c(x , n) , (4.4) with Hnk(x) in eq. (2.48).

Here, we comment on the parallel transformation of the fermion field on the discrete space M43 ZN. If eq. (4.3) is denoted as

dxc(x , n) 4

!

k

¯nkc(x , k) xk,

(4.5)

the covariant derivative Dc(x , n) is rewritten as

(4.6) _{Dc(x , n) 4}

m

(

¯m1 Am(x , n) 1Gm(x , n)

)

dxmc(x , n) 1

1

(

¯nk1 Fnk(x)

)

c(x , k) xk

n

.

Equation (4.6) implies that Fnk(x) c(x , k) xk expresses the variation accompanying

the parallel transformation from k-th to n-th points on the discrete space just as the variation of parallel transformation on the Minkowski space M4 is Am(x , n) c(x , n) dxm. This insures that the shifted Higgs field Fnk(x) is the gauge field

on the discrete space.

The nilpotency of dx in this case is also important to obtain consistent explanations

of covariant derivative and parallel transformation. From eq. (4.3), we find (dxldxk1 dxkdxl) c(x , n) 40 ,

(4.7)

which implies that if the Higgs field Fnk(x) as the gauge field on the discrete space

vanishes, the curvature on the discrete space also vanishes. From eq. (4.7), the nilpotency of d is evident.

In order to obtain the Dirac Lagrangian, we introduce the associated spinor one-form by D A c(x, n) 4gmc(x , n) dxm1 i] OgY(n)(†c(x , n)

!

k xk, (4.8)

where gY(n) is also a 24 324 matrix relating to the Yukawa coupling constant. The

operator O is defined with an appropriate field A as

] OgY(n)( Ac(x, k) 4AgY(n) c(x , k)

if c(x , k) is the right-handed fermion field and

] OgY(n)( A4gY(n) Ac(x , k)

(4.10)

if c(x , k) is the left-handed fermion field. More precisely in the later construction, Yukawa coupling matrix gY(n) operates directly to the right-handed fermion cR4

k2 c(x , 2 ) in eq. (3.1). This prescription is needed to ensure the hermiticity of the Dirac Lagrangian. The Dirac Lagrangian is obtained by taking the inner product (4.11) LD(x , n) 4i Tr a DA c(x, n), Dc(x, n)b 4 4 i Tr

[

c(x , n) gm_(¯ m1 Am(x , n) 1Gm(x , n)

)

c(x , n) 1 1i c(x , n)

(

OgY(n)

)

!

k Hnk(x) c(x , k)

]

,

where we have used the definitions of the inner products for spinor one-forms considering eq. (2.50), aA(x , n) dxm_{, B(x , n) dx}n b 4 Tr A(x, n) B(x, n) gmn_, (4.12) aA(x , n) xk, B(x , n) xlb 42 dklTr A(x , n) B(x , n) (4.13)

with other inner products vanishing. The total Dirac Lagrangian is the sum over n 4 1 , 2 , R , N LD(x) 4

!

n 41 N LD(x , n) . (4.14)

Let us move to the explicit reconstruction of the Dirac Lagrangian corresponding to the specification of the fermion field c(x , n) in eq. (3.1). After the rescaling of the gauge and Higgs bosons, we can write Dc(x , n) and DA c(x, n) as follows:

(4.15) _{Dc(x , 1 ) 4}

k

¯m7 182 i 2

m

gLk 41

!

3 tk 7 14Ak Lm1 cgBBm1

!

a 41 8 12 7 l 8agcGma

n

l

7 713c(x , 1 ) dxm_{1 g}hh 7 147 13c(x , 2 ) x2, (4.16) A c(x, 1) 41D 24gmc(x , 1 ) dxm1 i] OgY( 1 )(†c(x , 1 )

!

k xk, (4.17) _{Dc(x , 2 ) 4}

k

¯m7 182 i 2

m

gRk 41

!

3 tk 7 14Ak Rm1 cgBBm1

!

a 41 8 12 7 l 8agcGma

n

l

7 713c(x , 2 ) dxm 1 ghh†7 147 13c(x , 1 ) x11 gjjR7 b 7 13c(x , 4 ) x4, (4.18) A c(x, 2) 41_D 24 gmc(x , 2 ) dxm1 i] OgY( 2 )(†c(x , 2 )

!

k xk , (4.19) _{Dc(x , 3 ) 4}

k

¯m7 182 i 2

m

gLk 41

!

3 tk 7 14Ak Lm2 cgBBm2

!

a 41 8 12 7 l 8a* gcGma

n

l

7 713c(x , 3 ) dxm 1 ghhA7147 13c(x , 4 ) x4,

(4.20) A c(x, 3) 41D 24gmc(x , 3 ) dxm1 i] OgY( 3 )(†c(x , 3 )

!

k xk, (4.21) _{Dc(x , 4 ) 4}

k

¯m7 182 i 2

m

gRk 41

!

3 tk 7 14Ak Rm2 cgBBm2

!

a 41 8 12 7 l 8a* gcGma

n

l

7 713c(x , 4 ) dxm_{1 g}hhA†7 147 13c(x , 3 ) x31 gjj†R7 b 7 13c(x , 2 ) x2, (4.22) A c(x, 4) 41D 24_g mc(x , 4 ) dxm1 i] OgY( 4 )(†c(x , 4 )

!

k xk.

From these equations together with eq. (4.11), we can find the Dirac Lagrangian LD

which consists of the kinetic term LKT and the Yukawa interaction term LYukawa

between fermions and Higgs bosons expressed as (4.23) LKT4 4 i cLgm

k

¯m7 182 i 2

m

gLk 41

!

3 tk 7 14Ak Lm1 cgBBm1 gc

!

a 41 8 12 7 l 8aGa m

n

l

7 13cL1 1i cRgm

k

¯m7 182 i 2

m

gRk 41

!

3 tk 7 14Ak Rm1 cgBBm1 gc

!

a 41 8 12_{7 l 8}a_Ga m

n

l

7 13cR, (4.24) LYukawa4 2 cLh 7 147 13gYcR2 cRh†gY†7 147 13cR2 21 2 cRgY †_j R7 b 7 13cRc2 1 2 cR c _j R † 7 b 7 13gYcR,

where cL4k2 c(x , 1 ) and cR4k2 c(x , 2 ) by use of eq. (3.1). In introducing LYukawa, gY( 1 ) 4gY, gY( 2 ) 4gY†, gY( 3 ) 4gY† and gY( 4 ) 4gYT are assumed to ensure the

hermiticity of the Yukawa interaction where gY is the Yukawa coupling constant

written in 24 324 matrix form as

gY4 diag (gu, gu, gu, gn, gd, gd, gd, ge) .

(4.25)

gu_{, g}d_{, g}n_{and g}e

in eq. (4.25) are complex Yukawa coupling constants denoted as 3 33 matrix in generation space.

We address the U( 1 )Y hypercharge matrix of left- and right-handed leptons and

quarks. Owing to the vacuum expectation value of jR, SU( 2 )R3 U( 1 ) gauge symmetry

breaks down to U( 1 )Y gauge symmetry. The resulting U( 1 ) gauge boson is B 8m in

eq. (3.107). Inserting the following equations

Bm4 B 8mcos uY2 ZRmsin uY,

(4.26)

ARm3 4 B 8msin uY1 ZRmcos uY

(4.27)

into eq. (4.23) and using eqs. (3.97) and (3.101), it is known that the U( 1 )Yhypercharge

matrix of cLis c itself and that of cR is c 1t3 7 144 diag

g

4 3 , 4 3 , 4 3 , 0 , 2 1 3 , 2 1 3 , 2 1 3 , 22

h

, (4.28)

It should be remarked that leptons do not interact with the color gauge boson owing to l 8a _{in eq. (3.8). The Yukawa interaction of lepton is expressed as}

LYukawal 4 (4.29) 4 2 lLh 7 13gYllR2 lRgYl†h†7 13lL2 1 2 lRgY l†_j R7 13lRc2 1 2 lR c _j R † 7 13gYllR,

where lL4 (nL, eL)t, lR4 (nR, eR)t and lRc4 (eRc, 2nRc)t. gYl is a lepton part of gY in

eq. (4.25). If the mass term is extracted from eq. (4.29), it is evident that the seesaw mechanism for the left- and right-handed neutrino [17] works well.

Quarks do not interact with j owing to the matrix b in eq. (3.14). The Yukawa interaction of quarks is written as

LYukawaq 4 2 qLh 7 19gYqqR2 qRgYq†h†7 19qL,

(4.30)

where qL4 (uL, dL)tand qR4 (uR, dR)t. gYq is a quark part of gY. Up and down quarks

obtain the masses through the vacuum expectation value of h and the Kobayashi-Maskawa matrix comes out by diagonalizing the generation mixing matrices gu_{and g}d

in gYin eq. (4.25).

5. – Concluding remarks

We have reconstructed the left-right symmetric gauge model based on the new scheme of our noncommutative differential geometry on the discrete space M43 Z4

which is a product space of the Minkowski space multiplied by four point space. The Higgs field is well explained as a part of the generalized connection of the principal bundle with the base space M43 ZN, as shown in eq. (2.11). One-form basis xk on the

discrete space is the fundamental ingredient in the present formulation to hold the noncommutative nature that xnR xk is independent of 2xkR xn. In general, though

the NCG approach can propose the Higgs potential which automatically yields the spontaneous breakdown of gauge symmetry, it imposes the severe constraints on Higgs potential and interacting terms. In ref. [14], we could not obtain the Higgs potential terms sufficient to explain the phenomenology of LRSM. Thus, we revised an algebraic rule so as to relax the severe constraint and meet the requirements of the phenomenology. This modification is shown in eq. (2.26). In addition, we adopt the fermion field to be a vector in a 24-dimensional space including the generation space together with the space of flavor and color symmetry. This fermion field has the same placement of lepton and quark as in SO( 10 ) GUT. As the Higgs bosons, we incorporate an adjoint jR and h containing two Higgs doublet bosons, which are transformed as

( 1 , 3 , 22) and (2, 2*, 0) under SU(2)L3 SU( 2 )R3 U( 1 ) symmetry, respectively. h

and jR are the standard ingredients in LRSM. In addition to h, hA is also considered.

Owing to these considerations, we could extend the discussions about the gauge and Higgs boson masses in sect. 3. In the case of g14 g24 g34 g4, we obtain the interesting

relations sin2_u

W4 3 /8 and sin2uY4 3 /5 which are equal to the predictions of GUT.

However, the grand unification of coupling constants is not achieved in this formulation. Judging from this, the case of g14 g24 g34 g4 indicates that the

electromagnetic and weak interactions are unified at this point. As for the Higgs bosons, the surviving boson at weak energy scale is only one Higgs component, others have masses at energy scale of MR. This is a very characteristic feature in the present

formulation. The mass of this surviving boson is 2 MWin the case of g14 g24 g34 g4and m1`m2`m. It is very interesting to carry out the renormalization group analysis of

this mass relation down to the weak energy scale. This type of analysis has already been done in [16]. According to this analysis, the Higgs boson mass is 164 GeV if the top quark mass is 175 GeV. A similar analysis will be done in near future.

* * *

The author would like to express his sincere thanks to Profs. J. IIZUKA, H. KASE,

K. MORITA and M. TANAKA for useful suggestion and invaluable discussions on the non-commutative geometry.

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