In this work we consider a mechanism for mass creation based on the periodicity condition dictated from the compactification of extradimensions. It is shown that the existence and the compactification of extradimensions are the origin for creating particle mass in ordinary 4-dimensional space-time. Mass of Higgs particles themselves would be also originated from the geometric topology of extradimensions.
Origin of Mass Mass of Higg Particles Extra Dimensions1. Introduction
The existence of space-time extradimensions has been a subject of intensive research study during the last decades [1] -[3] .
The topology of extradimensions, especially their compactification, plays a crucial role in many physical aspects, mostly in the construction of various models of unified theory of interactions, such as superstring theory, extended general relativity, and so on [4] -[7] .
It is worth noting, on the other hand, that it is in such approaches the particle mass always remains a problem of actual characters.
In this work we propose a mechanism for mass creation through the compactification of spacetime extradimensions. The crucial argument is the proposed periodicity condition dictated from the compactification of extra dimensions.
The original field functions depend on all space-time coordinate components including those for extradimensions, the ordinary field functions, ordinary 4-dimensional space-time considered as effective field functions obtained by integration of the original ones over extra space-time.
In Section 2, we present some general principles related to the compactification of extradimensions.
In Section 3, a mechanism for mass creation is treated.
2. Periodicity Compactification Condition
For simplicity let us begin with the case of one extra dimension. Denote the 5-dimensional coordinate vector by xM with M =, 5. The Greek indices, ,··· will be used as conventional 4-dimensional Lorentz indices (0, 1, 2, and 3). We do not directly care from the extra dimensions is topologically compactified, but instead a specific periodicity condition is put on the field functions depending on extra dimensions, namely
where is some parameter function depending on the compactification length L.
The condition (1) corresponds to the equation:
With the relations:
In general we can put
For neutral field, , is to be real and therefore
The periodicity condition (1) can be generalized for the case of arbitrary number of extra dimensions in the following manner.
For convenience we denote the extra dimension coordinates by and write
The periodicity condition (1) is now generalized to be:
and the corresponding Equation (2) becomes:
with the relations:
3. Effective Field Equation and Mass
The general procedure of our treatment is as follows. We start from the (4 + d) dimensional Lorentz invariant Kinetic Lagrangian L(x, y) and the action for the field F(x, y) defined as
where and the integral is performed over the whole extra space time.
The principle of minimal action for S(y) then gives the Euler-Lagrange equation
which in turn leads to the equation of Klein-Gordon type:
For the effective field defined as
For illustration let us consider in more details the cases of scalar, spinor and vector fields.
3.1. Scalar Field
The free neutral scalar field is described by the Lagrangian
where, is a Minkonski metric for extra dimensions:
By inverting (7) into (12), we obtain:
And from here the equation
For the effective field
With
It is worth nothing that the squared mass is positive if all the extra dimensions are space-live, and can be negative if there exist time-live extra dimensions.
For change scalar field instead of (12) we take
And instead of (13) we have:
And from here the equation the same Equation as (14) with:
3.2. Spinor Field
In (4 + d) dimensional space-time, the spinor field is described by a component function with the live Lagrangian
where denote (4 + d) Dirac matrices obeying the ant commutation relations:
By inverting
Into (19) we obtain:
And from here the equation
By acting from the left both sides of this equation by
And taking into account the relations (20) we have:
And hence
We note that if all the extra dimensions are space-like, if all are real, and can be negative if there exists time-like extra dimension
3.3. Vector Field
We restrict ourselves to the case d = l and consider the neutral vector field satisfying the periodicity condition
And in correspondence
The free vector field is described by the Lagrangian
where
By inverting (27) into (28) we have:
Now we define a new physical vector field by putting
Expressed in terms of, the Lagrangian (29) has the form:
The Lagrangian (31) leads to the equation:
which means that the effective vector field
Has squared mass
It’s positive or negative depending upon whether the extra dimension is space-like or time-like.
4. Conclusion and Discussion
In this work we have proposed a mechanism for the creation of particle mass. The key idea is that the mass is originated from the compactification of extra dimensions followed by the periodicity condition for the particle fields.
It is worth noting that according to the mechanism the existence of tachyon having negative squared mass is closely related to the existence of time-like extra dimensions.
In this work we have considered originality for mass creation, which is originated from the compactification of extradimensions. It is shown that the mass spectrum is completely determined by some functions of compactification length and closely related to the metric of extradimensions.
The key idea is that the existence and the compactification of extradimensions are the origin for creating particle mass in ordinary 4-dimensional space-time. In this connection one might think that the mass of Higgs particles themselves would be also originated from the geometric topology of extradimensions. The problem of whether there exists some mechanism allowing the extradimensions to create Higgs particles would be a problem of significant meaning. It might be also that the particles could acquire mass through different mechanisms, including those related to extradimensions. This could lead to thinking that the probability for experimentally finding Higgs particles is much smaller than the value theoretically obtained when Higgs mechanism is considered as the only one for mass creation.
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