VNU Journal of Science: Mathematics – Physics, Vol. 35, No. 3 (2019) 114-124

Original Article

Prediction of Weinberg Angle in Discretized

Kaluza-Klein Theory

Nguyen Van Dat*

Faculty of Physics, VNU University of Science, 334 Nguyen Trai, Hanoi, Vietnam

Received 09 May 2019

Accepted 28 May 2019

Abstract: In Discretized Kaluza-Klein theory (DKKT) the gauge fields emerge as components of

gravity with a single coupling constant. Therefore, it provides a new approach to fix the

parameters of the Standard Model, and in particular the Weinberg angle. We show that in our

approach using DKKT, the predicted value of Weinberg angle is exactly the one measured in the

electron-positron collider experiment at Q = 91.2 GeV/c. The result is compared with the one

predicted the group theoretic methods.

Keywords: Weinberg angle, Discretized Kaluza-Klein theory, DKKT.

1. Introduction

The Weinberg angel W [1-3] is the most important parameter of the Standard model, which

relates the two coupling constant g and g’ corresponding to the two underlying gauge groups SU 2 L

and U 1Y . It can be introduced by the following mixing of the gauge field W3 and B to form the

physical photon and Z 0 fields

A cosW

Z 0

sin W

sin W B

3

cosW W

In terms of the coupling constants g and g’, the Weinberg angle can be expressed as

________

Corresponding author.

Email address: dnvdat@gmail.com

https//doi.org/ 10.25073/2588-1124/vnumap.4350

114

(1)

N.V. Dat / VNU Journal of Science: Mathematics – Physics, Vol. 35, No. 3 (2019) 114-124

cosW

sin W

g

g g 2

2

g

g g 2

2

115

(2)

(3)

Therefore, it relates the masses of the W and Z 0 as follows

mW mZ cosW

It can also relates the coupling constants g and g’ to the electric charge

e g sin W g cosW

(4)

(5)

Originally, since the Weinberg angle is a free unfixed parameter of the Standard Model, its value

can only be determined by experimental measurement. It’s most precise measurements carried out in

electron-positron collider experimnets at a value of Q = 91.2 GeV/c, have given the value [4]

sin 2 W 0.231200 0.00015

(6)

Due to the radiative corrections and renormalization effects, sin 2 W is a running constant, giving

different values at different energies. For example, at Q = 0.16 GeV/c the value of sin 2 W is

0.2397 0.0013 and at Q = 7-8 TeV/c, it is 0.23142.

The theoretical prediction of the Weinberg angle is possible in unified theories when only one

coupling constant is used instead of g and g’. In this paper, we will refer to the works of Fairlie [5, 6]

as an example of such theories. Essentially, these works are based on the simple gauge group

SU 2 |1 having only one coupling constant leading to the prediction of the Weinberg angle

sin W 0.25

(7)

It is remarkable that the predicted value of the Weinberg angle is very far from the experimental

value. It means that it is not at the measured electroweak scale energy, but at another one. At this

energy scale the two gauge groups SU 2 L and U 1 are embedded into a single simple one, which

gives only one coupling constant. By choosing different simple unification gauge group like SU 5 ,

E8 , … we might have different predictions, but since their energy scales are far from the electroweak

one, those can not reproduce the above experimental value.

In this paper, we follow a different approach, using the discretized Kaluza-Klein theory developed

by Viet and Wali [7, 12] to predict the Weinberg angle. The new feature of this approach is that it is

not a group theoretic but a geometric one. Surprisingly, this approach have given the value sin 2 W as

0.23077, which is very close to the experimental value.

This paper organized as follows. In the section 2, we will briefly review the Discretized KaluzaKlein theory (DKKT). In the section 3, we will calculate the Weinberg angle in this framwork. In the

section 4, we will compare the obtained prediction with ones of Fairlie. In the section 5, we will

discuss the results and physical implications.

N.V. Dat / VNU Journal of Science: Mathematics – Physics, Vol. 35, No. 3 (2019) 114-124

116

2. Overview of Discretized Kaluza-Klein theory

In 1994, Landi, Viet and Wali [6] have revised the work of Felder, Frohlich and Chamseddine

[13], which extend the non commutative geometric model of Connes and Lott [14]. Surprisingly, the

zero modes of the original Kaluza-Klein theory [15, 16] have emerged. This inspired Viet and Wali to

develop the DKKT with a full physical content having a pair of gravity, vector and scalar fields. In

each pair one field is massless and the other is massive. Based on these results, the NCG can be used

to construct the theories in the space-time extended by discrete extra dimensions. The main advantage

of DKKT is that it can avoid the infinite tower of massive fields in the usual Kaluza-Klein one with

continuous extra dimensions. Discrete extra space-time means that one have different sheets of the

usual space-time. For instance, if the extra dimension consists of two points, one must have two copies

of the space-time.

In order to incorporate the weak and strong interactions into this theory, one must extend it to the

case of nonabelian gauge vector fields. In 2015, Viet and Du have proven that in the case of the

discrete extra dimension of two points it is possible to extend DKKT to include the nonabelian gauge

fields in the two cases i) The gauge fields must be the same for both sheets of space-time or ii) The

gauge field on at least one sheet must be abelian. Viet, Dat, Han and Wali, have applied this result to

construct the Einstein-Yang-Mills-Dirac theories, having QCD coupled to gravity and electroweak

couple to gravity theories as special cases. Viet has also proposed to extend DKKT to include two

discrete extra dimensions, each having two points, leading to a unified theory where all four

interactions and the Higgs fields emerge as components of a generalized gravity.

In this article we will focus on the case, when the electroweak interaction emerges as a component

of gravity and couples to quarks and lepton as suggested in [12].

The construction is based on the spectral triplet. The first element of spectral triple is Hilbert space

L R . So, the generalization of spinor is direct sum of two chiral spinors,

L .

R

The second element is the Algebra using for function operations

I

4

(8)

L

R

, where

, I = L, R. The elements of this algebra is represented by 0-form diagonal matrix F,

f x 0

F x L

(9)

.

f R x

0

The third element is the Dirac operator, which that can be defined as an extension of the normal

Dirac operator as D d .e Θ , where d is usual Dirac operator in the four dimensions spacetime

4

im 5

D

,

5

im

0

im 5

Θ

.

5

0

im

(10)

With NCG space-time defined with the above spectral triplet, we can calculate the derivative of

the 0-forms by acting the Dirac operator on function F as follows

N.V. Dat / VNU Journal of Science: Mathematics – Physics, Vol. 35, No. 3 (2019) 114-124

117

DF D, F

f x

im f x f x

L

5

L

R

im

5

f x f x

f x

L

R

(11)

R

We can rewrite it in the following form

DF DX D , F DX 5 † D5 , F

DX F DX 5 † 5 F

(12)

where

0 m † 0 1

D5

, 1 0 .

m 0

(13)

If we use the representation of Dirac matrix, DX can be replaced by the generalized -matrices

0 5 0

i 5

Γ

,

Γ

(14)

5

0

i 0

Since the derivative of a 0-form is a 1-form, we can extend the module of 1-forms, which is the

generalization of the vector field in NCG to the following form

U M U M U 5U 5

u

5 L

i u5 L

i 5u5 R

u R

(15)

where U M is generalized functions (0-forms). The 1-form U contains two vectors and two scalars.

Now we can define the 2-form to be used as generalized field strength or curvature. The 2-forms

must extend from the derivatives of 1-forms. We have to define wedge product of two 1-forms as

follows

DX DX DX DX

DX DX 5 DX 5 DX

DX 5 DX 5 0,

U V

U V

U V 5 U V 5

U V 55 U 5V5

1

2

1

2

U V

(16)

U V

U V

U 5V

5

(17)

where tidle operation "~" on a generalized function is defined as follows

F f e f r, f

and

1

f1 f 2

2

(18)

118

N.V. Dat / VNU Journal of Science: Mathematics – Physics, Vol. 35, No. 3 (2019) 114-124

1 0

1 0

e

,r

.

0 1

0 1

Exterior derivative of 1-forms is given as

DU D,U DX M DX N DU MN

DU

DU

5

DU

55

1

2

1

2

U

U DU ,

U

5

m U U DU 5 ,

(19)

(20)

(21)

m U 5 L U 5 R .

3. The prediction of the Weinberg angle in DKKT

The gauge field A in DKKT takes the following form

a L i 5 a5†

A

i 5 a a

5

R

5 †

Γ A Γ A5

(22)

DX A DX A5 .

5

†

So, the gauge fields aL , aR and complex scalar field a5 are choosen as elements of the following

2 2 matrices

aL 0

A

,

aR

0

a 0

A5 5

.

†

0 a5

(23)

Now we can introduce the physical fields Wa x and B x by assigning

aL gWa x

a

14 1N F g'

YL

B x 12 1N F

2

2

YR

aR g' B x 1N F

2

m

h1*

h0 f

h

k

, H 0 .

a5 f k

m

h1

h0*

h1

fk

(24)

where g, g and f k are parameters, H is the usual Higgs doublet. YL and YR are hypercharge

opreators have the following forms

N.V. Dat / VNU Journal of Science: Mathematics – Physics, Vol. 35, No. 3 (2019) 114-124

1

1

YL 3 3

0

119

0

,

1

4

3 13

YR 0

0

0

0

0

0 0 .

2 0

0 0

0

2

13

3

0

0

(25)

The field strength is defined through wedge product and derivative of gauge fields as follows

(26)

F DA A A

From Eq. (17) and (21), we can calculate the components of F as

F DX DX

1

2

2 DX DX

5

1

2

DX DX

5

5

A A A , A

ma

A A A5 m

a5 a5 a5

†

5

(27)

DX DX F 2 DX DX F 5

5

DX DX F55

5

5

Let us calculate explicitly each component in terms of the physical boson and Higgs fields. The

first component is

1

F A A A , A

2

f L f R

1

a L a L a L , a L

2

1

a R a R a R , a R

2

Y 12 YR

g

W 14 g' L

B

4

4

(28)

where

W a Wa Wa

B B B .

The second component is

g

2

f abc Wb ,Wc ,

(29)

120

N.V. Dat / VNU Journal of Science: Mathematics – Physics, Vol. 35, No. 3 (2019) 114-124

F 5

1

2

A

A A5 m f 5 L f 5 R

(30)

where f 5L , f 5R is

f 5 L

1

2

2

2

fk

h

h

2

*

0

1

*

h0

a

5

*

*

h1

0

h0

*

a

a

gW x 2 14

g

YL 12 YR

2

B

x

B

x

(31)

m

h

h

h1

*

h1

1

h

h

m

0

a R a L

fk

*

h0

1

5

h

h

h1

0

1

a

a R a L

f 5 L

1

1

*

h0

1

a

a

gW

x

14

2

g

YL 12 YR

2

(32)

The third component is

F55 m a5† a5 a5† a5

m f k h0* h0 2m

f h m f k h0 m f k2 h1*h1

*

k 0

(32)

m2

f k2 h0* h0 h1* h1 2

fk

m2

f k2 HH 2 .

fk

The Lagrangian of the gauge sector now is calculated as

g

1

F, F

f k2

1

2 Trace F† F 2 F†5 F 5 F55† F 55 .

fk

(33)

Let calculate the first term in the Lagrangian, we have

F† F f † L f L f † R f R

g2

16

W W 14 g'2

YL2 12 YR2

16

B B

(34)

so we can get the trace of equation (35)

Trace F† F

g2

4

W W g'2

5

6

B B

(35)

N.V. Dat / VNU Journal of Science: Mathematics – Physics, Vol. 35, No. 3 (2019) 114-124

121

The second term can be derive similarly

F†5 F 5 f †5 L f †5 R f 5 L f 5 R

(36)

f †5 L f 5 L f †5 R f 5 R

1

TraceF†5 F 5 Trace f †5 L f 5 L f †5 R f 5 R

2

1

Trace a5† a5 ( a5† m)( a5 m)( a R a L )( aR aL )

4

1

4

h0* h0 h1* h1

f k2Trace

h h0 h h1

0

*

0

0

h0 h0* h1h1*

f k2Trace

*

1

h h hh

0

*

0 0

0

*

1 1

2

2

2 a b a b

2 YL 12 YR

g

W

W

1

g

B B .

4

4

4

So we have

TraceF†5 F 5

1 2

f H H .

2 k

(37)

where the covariant derivative of the Higgs field define as

g

g

i A Wi i .

2

2

The last term can be calculated as follow

(38)

2

m2

F F f HH 2 ,

fk

(39)

TraceF55† F 55 f k2V H , H .

(40)

†

55

55

4

k

Finally, replace equation (36), (38) and (41) into equation (34) we obtain

g

1

4

F

2

G

2

1

2

H H V H , H .

(41)

In order to have the right factors for kinetic terms of the gauge fields, we imply that

1 g2 1

fk g

f k2 4 4

(42)

1 25 1

3

g = g= g

.

2

g

6 4

10

Hence the Weinberg angle is calculated explicitly as follows

sin 2W

g'2

3

0.23077

2

g g' 13

2

(43)

N.V. Dat / VNU Journal of Science: Mathematics – Physics, Vol. 35, No. 3 (2019) 114-124

122

The deviation of this prediction is just 0.1% compared to the experimental value.

4. Comparison with the unified model based on the gauge symmetry SU(2|1)

In this section we discuss about six dimensonal Yang-Mill theory introduced by Fairlie [5, 6]. The

gauge field Am , m 1,...,6 has the form [5]

gA g B I

A

0

1,..., 4.

0

,

g B

(44)

Where g and g are the usual coupling constants. In this framework, the gauge fields A and B

transforms under SU(2) and U(1) respectively, where I is the 2 2 unit matrix,

parameter. The components in extra directions are specified as

i

0

A5 g

,

k

i

0

A6 g

.

k

is an arbitrary

(45)

Where is the Higgs doublet and k is an ad hoc parameter. The field strength Fmn obtained by

commutation of covariant derivative [5, 6]

F

1 gF A g F B I

0

g

0

F 6

D

0

0

, F 5

i D

g F B

D

, F56 2 g

1 i k

0

1 i k ,

iD

,

0

(46)

where

D gA g 1 B I .

(47)

The Lagrangian can be calculated as follows

1

1

Fmn2 F2 A

4

4

1

2

g / g 2 2 F2 B

4

2

2

1

D g 2 2k 2 g 2 .

2

The Weinberg angle is specified as follows

(48)

N.V. Dat / VNU Journal of Science: Mathematics – Physics, Vol. 35, No. 3 (2019) 114-124

1

tan w g / g 1 2

2

123

1/2

.

(49)

We have two alternatives for the right kinetic terms of the gauge fields in this theory with 2

leading to the same value of tan w .Therefore, we obtain

g / g 1 / 3

sin 2W

g'2

1

0.25.

2

2

g g'

4

(50)

This value of is significantly greater than the experiment value. Therefore, this theory will be

effective at an energy much higher than the electroweak one with the assumption that no new physics

emerges in the TeV range.

5. Conclusions

The geometric approach discussed in this paper is based on the DKKT, with the internal space

having only two points. It is straightforward to generalize into N points and arbitray number of the

internal dimensions. DKKT is a good way to unify all the interactions and Higgs field without

resorting to infinite tower of massive modes following Einstein's General Relativity. Surprisingly, the

geometric approach give the result with excellence agreement with the experiments at the electroweak

energy scale. So DKKT is valid at the currently accessible energy (unlike other unified theories). In

the geometric approach, we don't need to assume higher gauge symmetries to predict the Weinberg

angle. More phenomenological predictions of DKKT are in progress.

Acknowledgement

Thanks are due to Nguyen Ai Viet, Tran Minh Hieu and Pham Tien Du for their collaborations and

discussions. The research is funded by Vietnam National Foundation for Science and Technology

Development (NAFOSTED) under grant number 103.01-2017.319.

References

[1] T.P. Cheng, L.F. Li, Gauge theory of elementary particle physics, Clarendon Press, Oxford, 1984.

[2] L.B. Okun, Leptons and quarks, Elsevier, North-Holland, 2013.

[3] T.D. Lee, Particle physics and introduction to field theory, Contemporary Concepts in Physics Vol. 1, Harwood

Academic, New York, 1981.

[4] C. Amsler, Review of Particle Physics—Electroweak model and constraints on new physics, Physics Letters

(2008) B-667.

[5] D. Fairlie, Two consistent calculations of the Weinberg angle, Journal of Physics G: Nuclear Physics 5 (1979)

L55.

[6] D. Fairlie, Higgs fields and the determination of the Weinberg angle, Physics Letters B 82 (1979) 97–100.

[7] G. Landi, N.A. Viet, K.C. Wali, Gravity and electromagnetism in noncommutative geometry, Physics Letters

B 326 (1-2) (1994) 45-50.

[8] N.A. Viet, K.C. Wali, A discretized version of Kaluza–klein theory with torsion and massive fields, International

Journal of Modern Physics A 11 (13) (1996) 2403-2418.

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[9] N.A. Viet, K.C. Wali, Noncommutative geometry and a discretized version of Kaluza-Klein theory with a finite

field content, International Journal of Modern Physics A 11 (03) (1996) 533-551.

[10] N.A. Viet, K.C. Wali, Chiral spinors and gauge fields in noncommutative curved space-time, Physical Review D,

67(12) (2003) 124029.

[11] A.V. Nguyen, T.D. Pham, Non-Abelian gauge fields as components of gravity in the discretized Kaluza–Klein

theory. Modern Physics Letters A, 32(18) (2017) 1750095.

[12] N.A. Viet, N.V. Dat, N.S. Han, K.C. Wali, Einstein-Yang-Mills-Dirac systems from the discretized Kaluza-Klein

theory. Physical Review D, 95(3) (2017) 035030.

[13] A.H. Chamseddine, G. Felder, J. Fröhlich, Gravity in non-commutative geometry. Communications in

Mathematical Physics, 155(1) (1993) 205-217.

[14] A. Connes, J. Lott, Particle models and noncommutative geometry. Nucl. Phys. B, 18 (1991) 29-47.

[15] T. Kaluza, Zum unitätsproblem der physik, Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys.), (1921) (arXiv:

1803.08616) 966-972.

[16] O. Klein, Quantum theory and five-dimensional theory of relativity, 1926 Z. Phys, 37, (1987) 895.

Original Article

Prediction of Weinberg Angle in Discretized

Kaluza-Klein Theory

Nguyen Van Dat*

Faculty of Physics, VNU University of Science, 334 Nguyen Trai, Hanoi, Vietnam

Received 09 May 2019

Accepted 28 May 2019

Abstract: In Discretized Kaluza-Klein theory (DKKT) the gauge fields emerge as components of

gravity with a single coupling constant. Therefore, it provides a new approach to fix the

parameters of the Standard Model, and in particular the Weinberg angle. We show that in our

approach using DKKT, the predicted value of Weinberg angle is exactly the one measured in the

electron-positron collider experiment at Q = 91.2 GeV/c. The result is compared with the one

predicted the group theoretic methods.

Keywords: Weinberg angle, Discretized Kaluza-Klein theory, DKKT.

1. Introduction

The Weinberg angel W [1-3] is the most important parameter of the Standard model, which

relates the two coupling constant g and g’ corresponding to the two underlying gauge groups SU 2 L

and U 1Y . It can be introduced by the following mixing of the gauge field W3 and B to form the

physical photon and Z 0 fields

A cosW

Z 0

sin W

sin W B

3

cosW W

In terms of the coupling constants g and g’, the Weinberg angle can be expressed as

________

Corresponding author.

Email address: dnvdat@gmail.com

https//doi.org/ 10.25073/2588-1124/vnumap.4350

114

(1)

N.V. Dat / VNU Journal of Science: Mathematics – Physics, Vol. 35, No. 3 (2019) 114-124

cosW

sin W

g

g g 2

2

g

g g 2

2

115

(2)

(3)

Therefore, it relates the masses of the W and Z 0 as follows

mW mZ cosW

It can also relates the coupling constants g and g’ to the electric charge

e g sin W g cosW

(4)

(5)

Originally, since the Weinberg angle is a free unfixed parameter of the Standard Model, its value

can only be determined by experimental measurement. It’s most precise measurements carried out in

electron-positron collider experimnets at a value of Q = 91.2 GeV/c, have given the value [4]

sin 2 W 0.231200 0.00015

(6)

Due to the radiative corrections and renormalization effects, sin 2 W is a running constant, giving

different values at different energies. For example, at Q = 0.16 GeV/c the value of sin 2 W is

0.2397 0.0013 and at Q = 7-8 TeV/c, it is 0.23142.

The theoretical prediction of the Weinberg angle is possible in unified theories when only one

coupling constant is used instead of g and g’. In this paper, we will refer to the works of Fairlie [5, 6]

as an example of such theories. Essentially, these works are based on the simple gauge group

SU 2 |1 having only one coupling constant leading to the prediction of the Weinberg angle

sin W 0.25

(7)

It is remarkable that the predicted value of the Weinberg angle is very far from the experimental

value. It means that it is not at the measured electroweak scale energy, but at another one. At this

energy scale the two gauge groups SU 2 L and U 1 are embedded into a single simple one, which

gives only one coupling constant. By choosing different simple unification gauge group like SU 5 ,

E8 , … we might have different predictions, but since their energy scales are far from the electroweak

one, those can not reproduce the above experimental value.

In this paper, we follow a different approach, using the discretized Kaluza-Klein theory developed

by Viet and Wali [7, 12] to predict the Weinberg angle. The new feature of this approach is that it is

not a group theoretic but a geometric one. Surprisingly, this approach have given the value sin 2 W as

0.23077, which is very close to the experimental value.

This paper organized as follows. In the section 2, we will briefly review the Discretized KaluzaKlein theory (DKKT). In the section 3, we will calculate the Weinberg angle in this framwork. In the

section 4, we will compare the obtained prediction with ones of Fairlie. In the section 5, we will

discuss the results and physical implications.

N.V. Dat / VNU Journal of Science: Mathematics – Physics, Vol. 35, No. 3 (2019) 114-124

116

2. Overview of Discretized Kaluza-Klein theory

In 1994, Landi, Viet and Wali [6] have revised the work of Felder, Frohlich and Chamseddine

[13], which extend the non commutative geometric model of Connes and Lott [14]. Surprisingly, the

zero modes of the original Kaluza-Klein theory [15, 16] have emerged. This inspired Viet and Wali to

develop the DKKT with a full physical content having a pair of gravity, vector and scalar fields. In

each pair one field is massless and the other is massive. Based on these results, the NCG can be used

to construct the theories in the space-time extended by discrete extra dimensions. The main advantage

of DKKT is that it can avoid the infinite tower of massive fields in the usual Kaluza-Klein one with

continuous extra dimensions. Discrete extra space-time means that one have different sheets of the

usual space-time. For instance, if the extra dimension consists of two points, one must have two copies

of the space-time.

In order to incorporate the weak and strong interactions into this theory, one must extend it to the

case of nonabelian gauge vector fields. In 2015, Viet and Du have proven that in the case of the

discrete extra dimension of two points it is possible to extend DKKT to include the nonabelian gauge

fields in the two cases i) The gauge fields must be the same for both sheets of space-time or ii) The

gauge field on at least one sheet must be abelian. Viet, Dat, Han and Wali, have applied this result to

construct the Einstein-Yang-Mills-Dirac theories, having QCD coupled to gravity and electroweak

couple to gravity theories as special cases. Viet has also proposed to extend DKKT to include two

discrete extra dimensions, each having two points, leading to a unified theory where all four

interactions and the Higgs fields emerge as components of a generalized gravity.

In this article we will focus on the case, when the electroweak interaction emerges as a component

of gravity and couples to quarks and lepton as suggested in [12].

The construction is based on the spectral triplet. The first element of spectral triple is Hilbert space

L R . So, the generalization of spinor is direct sum of two chiral spinors,

L .

R

The second element is the Algebra using for function operations

I

4

(8)

L

R

, where

, I = L, R. The elements of this algebra is represented by 0-form diagonal matrix F,

f x 0

F x L

(9)

.

f R x

0

The third element is the Dirac operator, which that can be defined as an extension of the normal

Dirac operator as D d .e Θ , where d is usual Dirac operator in the four dimensions spacetime

4

im 5

D

,

5

im

0

im 5

Θ

.

5

0

im

(10)

With NCG space-time defined with the above spectral triplet, we can calculate the derivative of

the 0-forms by acting the Dirac operator on function F as follows

N.V. Dat / VNU Journal of Science: Mathematics – Physics, Vol. 35, No. 3 (2019) 114-124

117

DF D, F

f x

im f x f x

L

5

L

R

im

5

f x f x

f x

L

R

(11)

R

We can rewrite it in the following form

DF DX D , F DX 5 † D5 , F

DX F DX 5 † 5 F

(12)

where

0 m † 0 1

D5

, 1 0 .

m 0

(13)

If we use the representation of Dirac matrix, DX can be replaced by the generalized -matrices

0 5 0

i 5

Γ

,

Γ

(14)

5

0

i 0

Since the derivative of a 0-form is a 1-form, we can extend the module of 1-forms, which is the

generalization of the vector field in NCG to the following form

U M U M U 5U 5

u

5 L

i u5 L

i 5u5 R

u R

(15)

where U M is generalized functions (0-forms). The 1-form U contains two vectors and two scalars.

Now we can define the 2-form to be used as generalized field strength or curvature. The 2-forms

must extend from the derivatives of 1-forms. We have to define wedge product of two 1-forms as

follows

DX DX DX DX

DX DX 5 DX 5 DX

DX 5 DX 5 0,

U V

U V

U V 5 U V 5

U V 55 U 5V5

1

2

1

2

U V

(16)

U V

U V

U 5V

5

(17)

where tidle operation "~" on a generalized function is defined as follows

F f e f r, f

and

1

f1 f 2

2

(18)

118

N.V. Dat / VNU Journal of Science: Mathematics – Physics, Vol. 35, No. 3 (2019) 114-124

1 0

1 0

e

,r

.

0 1

0 1

Exterior derivative of 1-forms is given as

DU D,U DX M DX N DU MN

DU

DU

5

DU

55

1

2

1

2

U

U DU ,

U

5

m U U DU 5 ,

(19)

(20)

(21)

m U 5 L U 5 R .

3. The prediction of the Weinberg angle in DKKT

The gauge field A in DKKT takes the following form

a L i 5 a5†

A

i 5 a a

5

R

5 †

Γ A Γ A5

(22)

DX A DX A5 .

5

†

So, the gauge fields aL , aR and complex scalar field a5 are choosen as elements of the following

2 2 matrices

aL 0

A

,

aR

0

a 0

A5 5

.

†

0 a5

(23)

Now we can introduce the physical fields Wa x and B x by assigning

aL gWa x

a

14 1N F g'

YL

B x 12 1N F

2

2

YR

aR g' B x 1N F

2

m

h1*

h0 f

h

k

, H 0 .

a5 f k

m

h1

h0*

h1

fk

(24)

where g, g and f k are parameters, H is the usual Higgs doublet. YL and YR are hypercharge

opreators have the following forms

N.V. Dat / VNU Journal of Science: Mathematics – Physics, Vol. 35, No. 3 (2019) 114-124

1

1

YL 3 3

0

119

0

,

1

4

3 13

YR 0

0

0

0

0

0 0 .

2 0

0 0

0

2

13

3

0

0

(25)

The field strength is defined through wedge product and derivative of gauge fields as follows

(26)

F DA A A

From Eq. (17) and (21), we can calculate the components of F as

F DX DX

1

2

2 DX DX

5

1

2

DX DX

5

5

A A A , A

ma

A A A5 m

a5 a5 a5

†

5

(27)

DX DX F 2 DX DX F 5

5

DX DX F55

5

5

Let us calculate explicitly each component in terms of the physical boson and Higgs fields. The

first component is

1

F A A A , A

2

f L f R

1

a L a L a L , a L

2

1

a R a R a R , a R

2

Y 12 YR

g

W 14 g' L

B

4

4

(28)

where

W a Wa Wa

B B B .

The second component is

g

2

f abc Wb ,Wc ,

(29)

120

N.V. Dat / VNU Journal of Science: Mathematics – Physics, Vol. 35, No. 3 (2019) 114-124

F 5

1

2

A

A A5 m f 5 L f 5 R

(30)

where f 5L , f 5R is

f 5 L

1

2

2

2

fk

h

h

2

*

0

1

*

h0

a

5

*

*

h1

0

h0

*

a

a

gW x 2 14

g

YL 12 YR

2

B

x

B

x

(31)

m

h

h

h1

*

h1

1

h

h

m

0

a R a L

fk

*

h0

1

5

h

h

h1

0

1

a

a R a L

f 5 L

1

1

*

h0

1

a

a

gW

x

14

2

g

YL 12 YR

2

(32)

The third component is

F55 m a5† a5 a5† a5

m f k h0* h0 2m

f h m f k h0 m f k2 h1*h1

*

k 0

(32)

m2

f k2 h0* h0 h1* h1 2

fk

m2

f k2 HH 2 .

fk

The Lagrangian of the gauge sector now is calculated as

g

1

F, F

f k2

1

2 Trace F† F 2 F†5 F 5 F55† F 55 .

fk

(33)

Let calculate the first term in the Lagrangian, we have

F† F f † L f L f † R f R

g2

16

W W 14 g'2

YL2 12 YR2

16

B B

(34)

so we can get the trace of equation (35)

Trace F† F

g2

4

W W g'2

5

6

B B

(35)

N.V. Dat / VNU Journal of Science: Mathematics – Physics, Vol. 35, No. 3 (2019) 114-124

121

The second term can be derive similarly

F†5 F 5 f †5 L f †5 R f 5 L f 5 R

(36)

f †5 L f 5 L f †5 R f 5 R

1

TraceF†5 F 5 Trace f †5 L f 5 L f †5 R f 5 R

2

1

Trace a5† a5 ( a5† m)( a5 m)( a R a L )( aR aL )

4

1

4

h0* h0 h1* h1

f k2Trace

h h0 h h1

0

*

0

0

h0 h0* h1h1*

f k2Trace

*

1

h h hh

0

*

0 0

0

*

1 1

2

2

2 a b a b

2 YL 12 YR

g

W

W

1

g

B B .

4

4

4

So we have

TraceF†5 F 5

1 2

f H H .

2 k

(37)

where the covariant derivative of the Higgs field define as

g

g

i A Wi i .

2

2

The last term can be calculated as follow

(38)

2

m2

F F f HH 2 ,

fk

(39)

TraceF55† F 55 f k2V H , H .

(40)

†

55

55

4

k

Finally, replace equation (36), (38) and (41) into equation (34) we obtain

g

1

4

F

2

G

2

1

2

H H V H , H .

(41)

In order to have the right factors for kinetic terms of the gauge fields, we imply that

1 g2 1

fk g

f k2 4 4

(42)

1 25 1

3

g = g= g

.

2

g

6 4

10

Hence the Weinberg angle is calculated explicitly as follows

sin 2W

g'2

3

0.23077

2

g g' 13

2

(43)

N.V. Dat / VNU Journal of Science: Mathematics – Physics, Vol. 35, No. 3 (2019) 114-124

122

The deviation of this prediction is just 0.1% compared to the experimental value.

4. Comparison with the unified model based on the gauge symmetry SU(2|1)

In this section we discuss about six dimensonal Yang-Mill theory introduced by Fairlie [5, 6]. The

gauge field Am , m 1,...,6 has the form [5]

gA g B I

A

0

1,..., 4.

0

,

g B

(44)

Where g and g are the usual coupling constants. In this framework, the gauge fields A and B

transforms under SU(2) and U(1) respectively, where I is the 2 2 unit matrix,

parameter. The components in extra directions are specified as

i

0

A5 g

,

k

i

0

A6 g

.

k

is an arbitrary

(45)

Where is the Higgs doublet and k is an ad hoc parameter. The field strength Fmn obtained by

commutation of covariant derivative [5, 6]

F

1 gF A g F B I

0

g

0

F 6

D

0

0

, F 5

i D

g F B

D

, F56 2 g

1 i k

0

1 i k ,

iD

,

0

(46)

where

D gA g 1 B I .

(47)

The Lagrangian can be calculated as follows

1

1

Fmn2 F2 A

4

4

1

2

g / g 2 2 F2 B

4

2

2

1

D g 2 2k 2 g 2 .

2

The Weinberg angle is specified as follows

(48)

N.V. Dat / VNU Journal of Science: Mathematics – Physics, Vol. 35, No. 3 (2019) 114-124

1

tan w g / g 1 2

2

123

1/2

.

(49)

We have two alternatives for the right kinetic terms of the gauge fields in this theory with 2

leading to the same value of tan w .Therefore, we obtain

g / g 1 / 3

sin 2W

g'2

1

0.25.

2

2

g g'

4

(50)

This value of is significantly greater than the experiment value. Therefore, this theory will be

effective at an energy much higher than the electroweak one with the assumption that no new physics

emerges in the TeV range.

5. Conclusions

The geometric approach discussed in this paper is based on the DKKT, with the internal space

having only two points. It is straightforward to generalize into N points and arbitray number of the

internal dimensions. DKKT is a good way to unify all the interactions and Higgs field without

resorting to infinite tower of massive modes following Einstein's General Relativity. Surprisingly, the

geometric approach give the result with excellence agreement with the experiments at the electroweak

energy scale. So DKKT is valid at the currently accessible energy (unlike other unified theories). In

the geometric approach, we don't need to assume higher gauge symmetries to predict the Weinberg

angle. More phenomenological predictions of DKKT are in progress.

Acknowledgement

Thanks are due to Nguyen Ai Viet, Tran Minh Hieu and Pham Tien Du for their collaborations and

discussions. The research is funded by Vietnam National Foundation for Science and Technology

Development (NAFOSTED) under grant number 103.01-2017.319.

References

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[3] T.D. Lee, Particle physics and introduction to field theory, Contemporary Concepts in Physics Vol. 1, Harwood

Academic, New York, 1981.

[4] C. Amsler, Review of Particle Physics—Electroweak model and constraints on new physics, Physics Letters

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[5] D. Fairlie, Two consistent calculations of the Weinberg angle, Journal of Physics G: Nuclear Physics 5 (1979)

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[6] D. Fairlie, Higgs fields and the determination of the Weinberg angle, Physics Letters B 82 (1979) 97–100.

[7] G. Landi, N.A. Viet, K.C. Wali, Gravity and electromagnetism in noncommutative geometry, Physics Letters

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[11] A.V. Nguyen, T.D. Pham, Non-Abelian gauge fields as components of gravity in the discretized Kaluza–Klein

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[12] N.A. Viet, N.V. Dat, N.S. Han, K.C. Wali, Einstein-Yang-Mills-Dirac systems from the discretized Kaluza-Klein

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