Deformed Gauge Invariance with Massive Gauge Vector Bosons

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DOI: 10.4236/jmp.2017.81007    942 Downloads   1,328 Views  


We consider the concept of deformed gauge invariance. The described formalism allows the vector gauge bosons to be massive independently of Higgs mechanism. It also allows the possibility for the variability of gauge coupling constants in space-time.

Cite this paper

Duc, D. , Giao, N. and Dung, T. (2017) Deformed Gauge Invariance with Massive Gauge Vector Bosons. Journal of Modern Physics, 8, 82-86. doi: 10.4236/jmp.2017.81007.

1. Introduction

The mass problem for elementary particles in general, and for gauge vector bosons in particular, has been of actual character in many aspects, mostly in the construction of various unification models based on gauge invariance principle [1] - [7] where the Higgs mechanism for mass creation plays a crucial role.

On the other hand, in our recent works [8] - [12] , a mechanism for mass creation in space-time with extradimensions has been proposed. For vector bosons, this problem has been treated in more detail in [8] .

In this work, an alternative approach is proposed to give the possibility for gauge vector bosons to acquire mass independently of Higgs mechanism. It is based on a modified gauge principle and referred to as deformed gauge invariance [12] .

As a consequence, the proposed mechanism also allows the possibility for gauge coupling constants to be variable in space-time. This would be meaningful for the study of both micro and macro world [13] [14] .

2. Deformed U(1) Gauge Invariance

Let be some matter field with charge q and the transformation law


under gauge transformation with parameter

The covariant derivative is constructed by the formula


with the gauge field obeying the transformation law:


being some scalar function parameter.

The conventional field strength defined as


is no more imvariant under the transformation (3) but its deformed version:


Hence the corresponding invariant Lagrangian should be taken of the form:


The Euler?Lagrange equation


then gives:


Let us put the constraint on the gauge field:


This coincides with the ordinary Lorentz gauge condition when is constant.

Equation (8) now reads:


Now we restrict the consideration to a special form of, namely


where, and c being some scalar parameters, -some vector parameter.

With Equation (11) inserted Equation (10) becomes:


which corresponds to the expression


for mass of gauge boson A.

Equation (13) shows that in general mA can vary in value in space-time, except for the special case, where. It takes the value


at the origin.

3. Deformed Non-Abelian Gauge Invariance

We now proceed to the case of non-abelian gauge. Let be some matter field multiplet obeying the transformation law under gauge transformation


being representation matrices of the symmetry algebra


―structure constants.

The covariant derivative is introduced by the formula:


with the gauge fields transforming according to the rule:


G being gauge coupling constant.

The deformed field strength is constructed from the conventional one


in a similar way as Equation (5), namely:


with the transformation law:


Hence, the invariant Lagrangian for gauge fields should be:


where is proportional to and.

By performing further calculations in a similar way as for the -gauge with the deformed Lorentz gauge condition


taken into account the same expression (13) for mass mA will be obtained.

4. Variable Coupling Constants

From the Equations (2) and (17) of covariant derivatives it follows immediately that instead of the gauge coupling constants and G one should use


in the corresponding gauge interaction Lagrangians instead of and G.

For example, the -gauge interaction Lagrangians for charged scalar field and spinor field should be:



Hence, according to the formalism presented here the fine structure constant a can change the value in space-time. In this connection it is worth mentioning that the problem concerning the variability of is significant for the study of both macro and micro world. In fact, is has been realized that if so, many phenomena of the Nature related to the time evolution of the Universe might be theoretically explained. Take for example the Red Shift in cosmology traditionally treated as Doppler effect. Within our proposed mechanism, it might be explained in an alternative way more compatible with static Universe in General Relativity. Another example would be the Oklo problem [13] [14] which might be theoretically explained if the value of some milliards years ago was far different from that at present time.

The variability of coupling constants in space-time might also have the relation to the renormalization problem in quantum field theory, this topic is the subject of our further consideration.

5. Conclusion

In this work, the concept of deformed gauge is considered. The key idea is the introduction of some parameter function in the transformation law for gauge fields. The proposed formalism might be considered as the generalization of the traditional gauge invariance which corresponds to the special case. The formalism allows the gauge vector bosons to acquire mass with the value expressed in terms of. It also allows the possibility for the gauge coupling constant to be variable in space-time.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] Furlan, G., Jengo, R., Pati, J.C., Sciama, D.W. and Shafi, O. (1997) Superstrings, Supergravity and Unified Theories. World Scientific, Singapore City.
[2] Konuma, M. and Maskawa, T. (1981) Grand Unified Theories and Related Topics. World Scientific, Singapore City.
[3] Zee, A. (1985) Unity of Forces in the Universe. World Scientific, Singapore City.
[4] Becker, K., Becker, M. and Schwarz, J.H. (2007) String Theory and M-Theory.
[5] Green, M.B. and Gross, D.J. (1981) Unified Field Theory. World Scientific, Singapore City.
[6] O’Raifeartaigh, L. and Straumann, N. (1997) Earky History of Gauge Theories and Kaluza-Klein Theories. arXiv-helth /98/05.
[7] Moriyasu, K. (1985) An Elementary Primer for Gauge Theory. World Scientific, Singapore City.
[8] Duc, D.V. and Giao, N.M. (2013) Journal of Modern Physics, 4, 991.
[9] Duc, D.V. and Giao, N.M. (2014) Journal of Modern Physics, 5, 477.
[10] Duc, D.V., Giao, N.M. and Dung, T.T. (2015) International Journal of Theoretical Physics, 54, 1071.
[11] Duc, D.V. and Giao, N.M. (2016) International Journal of Theoretical Physics, 55, 959.
[12] Duc, D.V. (2011) Communications in Physics (Vietnam), 21, 259.
[13] Yasumori, F. (2004) Oklo Constraint on the Time Variability of Fine Structure Constant, Astrophysics, Clocks and Fundamental Constants. Lecture Notes in Physics. Springer Berlin, 167-185.
[14] Lamoreaux, S.K. and Torgeson, J.R. (2004) Physical Review D, 69, 1217016.

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