Vector Boson Mass Spectrum from Extra Dimension

doi:10.4236/jmp.2013.47133

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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: [email protected].vn, [email protected]

Received April 29, 2013; revised May 31, 2013; accepted June 28, 2013

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



. 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,,

xa d

 



, and express the

field function of coordinates as

with the periodicity condition supposed to be of the form:

xFxyFxy











xyLfF xy



(1)

where

is some parameter function depending on the

compactification length L(a).

The Condition (1) corresponds to the equation:



xygF xy



(2)







 

with the relations:



1ln 2π,

fninZ











 



 

(3)

In general we can put:

 



1ln i2π

aaa

FFF















(4)







where



and

are some functions of .









For neutral field,



is real and therefore



0, 0



.





We now consider the neutral vector field





satisfying the periodicity condition.







MVM

Vxy LfVxy (5)



MVM

aVxyg Vxy



(6) 



D. V. DUC, N. M. GIAO

992



The free vector field



MyVx is described by the

Lagrangian



LxyF F

aab



 



FFF



(7)

where































VV





(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:



aaA V

LxyF F

VgV





VgV

















(9)

where ab



is Minkowski metric for extra dimensions.

Now we define a set of new physical vector fields





by putting

 

 

 

 

aaa

GZZ

(10)

Note that the field strength tensor









 



(11)

remains unchanged and independent of a :



GZ F









 



(12)

Therefore Equation (9) can be transformed into the

form:

 

  

aaa

aG G

gZZ

Lxy



















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.





d,Sy xLxy

where



means the integration over the extradimen-

sions:



00 0

ddd,,d

LL Ld

yyyy









The principle of minimal action for S(y) then gives the

Euler-Lagrange equation







Lxy Lxy















(16)

which in turn leads to the Klein-Gordon equation







mZx



(17)







for the effective vector field x





defined as

xyZxy





(18)

In this way we obtain:

 For































(19)

And hence



 (20)

 For









The Lagrangian (13) has no kinetic term for the field





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):























 (21)

D. V. DUC, N. M. GIAO

993

2dFor illustration let us consider the case . In this

case a single vector field





Vxy can give two effec-



()



and

()x



. By putting, for tive vector fields

example,

 









and ,

 

12,2 1





Equation (20) gives:



aa V





and











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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