Journal of Modern Physics, 2013, 4, 991-993
http://dx.doi.org/10.4236/jmp.2013.47133 Published Online July 2013 (http://www.scirp.org/journal/jmp)
Vector Boson Mass Spectrum from Extra Dimension
Dao Vong Duc1, Nguyen Mong Giao2
1Institute of Physics, Hanoi, Vietnam
2Institute of Physics, Ho Chi Minh City, Vietnam
Email: dvduc@iop.vast.ac.vn, nmgiao2011@yahoo.com.vn
Received April 29, 2013; revised May 31, 2013; accepted June 28, 2013
Copyright © 2013 Dao Vong Duc, Nguyen Mong Giao. This is an open access article distributed under the Creative Commons At-
tribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is prop-
erly cited.
ABSTRACT
Based on the mechanism for mass creation in the space-time with extradimensions proposed in our previous work,
(arXiv: 1301.1405 [hep-th] 2013) we consider now the mass spectrum of vector bosons in extradimensions. It is shown
that this spectrum is completely determined by some function of compactification length and closely related to the met-
ric of extradimensions.
Keywords: Boson Mass Spectrum; Extra Dimensions
1. Introduction
The topology of space-time extradimensions has been a
subject of intensive study in various aspects during the
last time [1-8]. On the other side, the problem of particle
masses, especially for gauge vector bosons, remains to be
of actual character.
In our previous work [9] we have proposed a mecha-
nism for the creation of particle masses based on the pe-
riodicity condition dictated from the compactification of
extradimensions. In this approach the original field func-
tions depend on both ordinary space-time coordinates
and those for extradimensions, the ordinary field func-
tions for ordinary 4-dimensional space-time are consid-
ered as effective field functions obtained by integration
of the original ones over extraspace-time.
The mass of vector bosons has been treated in [9] for
the simplest case with one extradimension. Along this
line, in this work we extend the results to the general
case for arbitrary number of extradimensions.
2. Effective 4-Dimensional Vector Fields
As in [9], we denote the 4 + d dimensional coordinate
vector by xM with ,5,6, ,4
M
d
. The Greek in-
dices will be used for conventional 4-dimensional Lor-
entz indices, 0,1, 2, 3
4aa
. For convenience we will use
the notations ,1,2,,
y
xa d
 

,,
, and express the
field function of coordinates as
Ma
with the periodicity condition supposed to be of the form:
xFxyFxy



;,
aa
aF
F
xyLfF xy

a
(1)
where
F
f
is some parameter function depending on the
compactification length L(a).
The Condition (1) corresponds to the equation:
F



,,
a
F
a
xygF xy
y
(2)


 
 
with the relations:

.
e,
1ln 2π,
a
aF
Lg
a
F
aa
FF
a
f
fninZ
L

g


 

 
(3)
In general we can put:
 

i
e
1ln i2π
a
F
aa
FF
aaa
FFF
a
f
n
L


g

(4)

a

a
where
F
and
F
are some functions of .

a
L

a
F
For neutral field,
F
,
F
f
is real and therefore

0, 0
a
Fn
.
We now consider the neutral vector field
,
My

Vx
satisfying the periodicity condition.


,,
M
aa
a
MVM
Vxy LfVxy (5)



,,
M
a
MVM
aVxyg Vxy
y
(6) 
C
opyright © 2013 SciRes. JMP
D. V. DUC, N. M. GIAO
992
The free vector field
,
MyVx is described by the
Lagrangian

1
,4
11
42
MN
MN
LxyF F
1
4
aa
b
aab
F
FF

 

FFF
(7)
where
N
M
MN
M
N
Aa
BA
ab
VV
a
ab
F
x
x
V
V
F
x
x
V
V
F
x
y
VV
yy










VV
0F
F

(8)
Now let us consider the case when the field component
4Aa under treatement does not depend on yb,
unless b = a. In this case ab
and the Lagrangian (7)
with Equation (6) taken into account becomes:



1
1
,4
1
2A
d
aaA V
a
LxyF F
VgV




A
aa
AV
VgV




(9)
where ab
is Minkowski metric for extra dimensions.
Now we define a set of new physical vector fields

a
Z
by putting
 
1
A
A
a
V
a
Z
VV
g
 
 
 
aaa
GZZ
(10)
Note that the field strength tensor



 

V
(11)
remains unchanged and independent of a :

a
GZ F



 

(12)
Therefore Equation (9) can be transformed into the
form:
 
 
  
2
A
aa
aaa
V
aG G
gZZ
1
1
,4
1
2
d
a
aa
a
Lxy


1
1
d
a
a

dSySy

(13)
With the constraint:
(14)
The Formula (13) tells that a set of d vector fields in
ordinary 4-dimensional space-time can be unified in a
single vector field in the whole (4 + d)-dimensional
space-time with the distribution characterized by the co-
efficients obeying the constraint (14). This constraint is
dictated by Equations (9)-(13). This result could be use-
ful in the construction of Unifying models for gauge in-
teractions.
3. Equation of Motion and Mass Spectrum
We start from the Lorentz invariant Lagrangian (13) and
the effective action defined as:
(15)
Here, as in Section 2, for convenience we use the nota-
tion y instead of x for extradimension coordinate.

4
d,Sy xLxy
dy
where
means the integration over the extradimen-
sions:

12
12
00 0
ddd,,d
d
LL Ld
yyyy



The principle of minimal action for S(y) then gives the
Euler-Lagrange equation




,,
0
aa
Lxy Lxy
ZZ




(16)
which in turn leads to the Klein-Gordon equation


20
a
a
Z
mZx
(17)


a
Z
for the effective vector field x




d,
aa
defined as
xyZxy

Z
(18)
In this way we obtain:
For
0a
:



2
0
A
a
Va
aa
gZ
a







(19)
And hence
2
2A
a
a
V
aa
Z
g
ma
 (20)
For
0a
:
The Lagrangian (13) has no kinetic term for the field

a
Z
which now can be considered as auxiliary field.
So, the conclusion is that a single neutral vector field in
space-time with extradimensions leads to a set of (not
more than d) effective neutral vector fields in ordinary
4-dimensional space-time with the masses obeying the
sum rule followed from (14) and (20):


2
21
A
a
a
V
aa
aZ
g
m

(21)
Copyright © 2013 SciRes. JMP
D. V. DUC, N. M. GIAO
Copyright © 2013 SciRes. JMP
993
2dFor illustration let us consider the case . In this
case a single vector field
,
M
Vxy can give two effec-

1

1
()
Z
x
and
Z
()x
. By putting, for tive vector fields
example,
 
1
1

2
2



and ,
 
12,2 1


Equation (20) gives:

22
2
aa
aa V
mg

Z
and

22
22
22
2
V
mg
11
22
11
1
V
mg

Z
Z
respectively
It is worth noting that by choosing appropriate values of
(a) and Minkowski metric of extradimensions the de-
scribed formalism could be useful in giving the relation
between the metric of extradimensions and tachyons
having negative squared mass.
4. Conclusion
In this work we extend the results obtained in our previous
work [9] to consider the mass spectrum for vector bosons.
It has been shown that a single neutral vector field in
space-time with d extradimensions corresponds to a set of
s (d) effective neutral vector fields in ordinary 4-di-
mensional space-time with masses obeying the sum rule
expressed in terms of metric of extradimensions and pa-
rameter function dictated from the periodicity conditions.
This would give a deeper understanding of the relation
between extradimensions and gauge bosons in the con-
struction of the Unified Gauge Theory of interactions.
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