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Prediction of weinberg angle in discretized Kaluza-Klein theory

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 1Y . It can be introduced by the following mixing of the gauge field W3 and B to form the
physical photon and Z 0 fields

 A   cosW
Z 0   
     sin W

sin W   B 
 3
cosW  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

cosW 

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 cosW

It can also relates the coupling constants g and g’ to the electric charge
e  g sin W  g  cosW

(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.


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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 Wa  x  and B  x  by assigning
aL  gWa  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





ma



  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   Wa   Wa 
B    B   B .

The second component is

g
2




f abc Wb ,Wc   ,

(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  Wi 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 2W 

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   F2  A 
4
4
1
2
  g  / g  2   2 F2  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 2W 

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.
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