AbstractThe Standard model of particle physics is based on nonabelian gauge theories. Since there are observed phenomena which cannot be explained with ordinary Standard model, this theory can be further generalized. This paper treats an extension of the Standard model by introducing a generalization of nonabelian gauge theories.
The most general model which is experimentally verified is the Standard model of particle Physics. A few years ago, the Higgs boson was observed at the Large Hadron Collider (Chatrchyan et al. 2012). Standard model of particle physics describes the electroweak and the strong interaction which are fundamental forces of nature. Both interactions are described by a nonabelian gauge theory based on the gauge group . The electromagnetic interaction which relies on the Lie group is obtained by symmetry breaking of the electroweak interaction. Gravitational interactions are not included in the Standard model.
However, the Standard model of particle physics cannot describe some experiments with sufficient accuracy. An example of a phenomenon where the Standard model fails is the asymmetry of matter and antimatter in the universe (Canetti et al. 2012). This asymmetry is also called “Baryon asymmetry” and is still an unsolved problem in physics. A model which explains the baryon asymmetry is an intrinsic electric dipole moment in elementary particles (The ACME Collaboration 2014). Such a dipole moment would lead to a difference in the decay rates of matter and antimatter. The validity of the electric dipole argument for explanation of baryon asymmetry is still an open question in physics.
Recently, an intrinsic dipole moment in elementary particles with electric charge is proposed as an additional degree of freedom in Topological Dipole Field Theory (Linker 2015). The intrinsic dipole moment has a topological nature, i.e. the physical system does not depend explicitly on the magnitudes of this dipole moment. Due to this fact it is constructed a topological quantum field theory for this intrinsic dipole moment. By adding a 2-form dipole field to the ordinary electromagnetic field strength tensor an extension of Quantum electrodynamics is obtained. This implies a modified dynamics of the force carriers of electromagnetism. Another generalization of electrodynamics is the Born-Infeld model (Goenner 2014). It is a nonlinear generalization of Maxwell’s field equations.
In this research paper it is showed how nonabelian gauge theories can also be further generalized. Generalizations of Yang-Mills theory were proposed in supersymmetric theories. There are existing several examinations about supersymmetric Yang-Mills theories (Beisert 2012). May be a quantum field which can be expressed as a matrix with dimension . In nonabelian field theory the field can be decomposed as
Here, is the constant Lie group generator and is the generator index which runs from to . For generator indices and spacetime indices Einstein’s summation convention is used. May be the 1-form gauge connection. Then the nonabelian field strength tensor is given by
With a coupling constant . This research paper shows how the field strength tensor (2) can be generalized. The generalization of the gauge field strength tensor is very similar to the generalization procedure performed in the original paper of Topological Dipole Field Theory (TDFT). It is respected the principle of gauge invariance during derivation of equations. After that, a simple computation with TDFT is shown.
A further generalization of TDFT can be obtained in similar manner to the derivation of TDFT in the original research paper. After the derivation of nonabelian TDFT, a calculation to extended Quantum chromodynamics is performed.
Formulation of nonabelian TDFT
A plausible generalization of the field strength tensor (2) has the following form:
Here, is the intrinsic dipole moment corresponding to the gauge interaction. This intrinsic dipole moment satisfies with the general Čech coboundary map and is an observable of the theory. This map is a gauge covariant map that satisfies for an arbitrary 2-form field . Since the field strength tensor transforms under a gauge group as it must hold the transformation rule
Also the general dipole field transforms by the rule (4) under gauge transformations. Because is a local group, the ordinary Čech coboundary map has to be modified. Due to the transformation property one can construct a gauge connection that satisfies the gauge transformation condition . It holds the relation
with the number of intersecting topological bases which surround a certain spacetime point, the spacetime point position vector pointing at the intersection of all topological bases and the vector which points from to the intersection of bases with the -th base excluded. The vector is an infinitesimal quantity. From (5) it follows that it must be:
It is easy to show that the following quantity is a differential operator:
Moreover, equation (7) is a proper gauge covariant derivative since the ordinary covariant derivative has the property . With above considerations, a gauge-invariant topological action can be constructed. For abelian gauge fields, the topological term of the TDFT has the form
with the Minkowski spacetime manifold and the Lagrange multiplier . If is a nonabelian field and after replacing the operator with , the Lagrangian density 4-form transforms as . Hence, the gauge-invariant topological term of TDFT in the nonabelian case reads:
Due to linearity of any differential operators and the trace, it can be shown in a similar way as in the original research paper of TDFT that (9) represents a Witten-type topological quantum field theory in the intrinsic curvature . Moreover, it holds the exactness condition since the operator evaluates the Čech coboundary map only in the base space of the fiber bundle that represents the nonabelian gauge theory. Equation (9) yields equation (8) if the dipole fields are abelian.
Perturbative calculation with TDFT
To evaluate the integral
the factor is expanded into Taylor series. After the Taylor expansion the general 2-form field can be decomposed into an exact term and into a non-exact term , i.e.
with a coupling constant such that . Without loss of generality, the topological bases which generate the Čech coboundary are chosen such that they are absorbing local gauge transformations, i.e. it can be set . Since runs over the fields for all one can pick an arbitrary field for arbitrary that is set equal to . For matching the spacetime point where is defined, the field lies on the intersection of all topological bases. All other fields with can be obtained by considering all possible generalized Čech cocycles. From (11) it follows , hence:
The evaluation of the generalized coboundary map on yields also a term and by choosing a positive sign it follows from (12) the topological action:
May be the incoming gauge boson fields fixed and , i.e. it must not be integrated over all possible gauge connection states. Additionally it is set . It is easy to show that all nonzero powers less than the fourth power which can be formed with are vanishing when weighted with the factor . Since it can be set due to translational invariance of the integration measure one obtains:
When integrating over multinomials in one can use the basic property . After this integration, the integration over can be performed. Finally, the averaging of multinomials in with weight factor yields a number which is independent on the choice of the topological bases which generates the Čech complex. Perturbative evaluations of (10) show that 5-boson-scattering or higher order scattering can occur. However, quantum chromodynamics is a field theory which has to be treated non-perturbative in many cases. The additional coupling constant is unique for every kind of gauge boson and has to be determined by experiments with particle colliders.
Topological Dipole Field Theory offers a higher complexity in calculations than the ordinary Standard model does. Phenomena where the ordinary Standard model fails like the baryon asymmetry can be predicted more precisely by TDFT. The main advantage of TDFT is that except the topological dipole moments no supersymmetric partners of every Standard model particle or other hypothetical concepts which require a lot of rigorous experimental verifications is introduced. More insights in phenomena in particle physics and cosmology that are still undiscovered are possible by TDFT.
The ACME Collaboration; et al. "Order of Magnitude Smaller Limit on the Electric Dipole Moment of the Electron." Science, 2014, 343 (269): 269–72. doi: 10.1126/science.1248213
Linker, P. "Topological Dipole Field Theory." The Winnower, 2015, 2: e144311.19292.
Goenner, H. "On the History of Unified
Field Theories. Part II. (ca. 1930 – ca. 1965)."
Living Rev. Relativity, 2014, 17 (5): 1-241. doi:10.12942/lrr-2014-5
Showing 19 Reviews
- It should be B = bB'+W with coupling constant b and the intrinsic curvature B'. Only this choice satisfies \Delta B = b \Delta B' + \Delta W = \Delta W = W (positive sign is chosen here). Also this choice allows to obtain a perturbative expansion in terms of a coupling constant. The Lagrangian density has the correct form: L = 1/2 \epsilon^(\alpha \beta \mu \nu) (bB'_(\alpha \beta)^a W_(a \mu \nu) + \lambda W_(\alpha \beta)^a W_(a \mu \nu))
- After equation (10) it is written: "the factor exp(i \int_M F \wedge *F) is expanded...". It should be exp(i 1/4 \int_M tr (F \wedge *F)) instead of exp(i \int_M F \wedge *F).
Special emphasis while calculating multinomials shows the depth of ground work for the paper. Great work.
An excellent approach for coboundary mapping and co-Boson field mapping. Commendable effort. Looking forward to more such work in the near future.
The theory is a mathematically consistent theory (a recent Review comment of "Patrick Linker", the author of the paper, Shows this). The data does Support author's conclusions."
Very good work , I like it very much !! You used some mathematicals expressions with high level !
Here it is clarified that the TDFT theory does not depend on the mathematical Details of the underlying cohomology theory. Therefore one can obtain an interesting form of the topological term in the Lagrangian density.
The author has written an interesting Topic. Calculation steps are Logically consistent and on a high mathematical Level. At the end of the paper a more plausible equation (without using cohomology theory) is provided. An interesting fact is that the theory does not depend on mathematical Details of the underlying cohomology theory which makes TDFT a very interesting theory.
Topological Dipole Field Theory offers a higher complexity in calculations than the ordinary Standard model does. The ordinary Standard model of particle physics cannot describe some experiments with sufficient accuracy, for exemple, is the asymmetry of matter and antimatter in the universe. A nice model which explains the baryon asymmetry is an intrinsic electric dipole moment, and the validity of the argument, of this model, for explanation of baryon asymmetry is an open question in modern physics. The introduction of TDFT would be a great contribution to the explanation of this asymmetry. The topic is complex and calculations are in high level and logic, great work.
Its a wonderful topic. I like your idea on the argument on the standard model. It looks like you give a good explanation to your theory to offer higher calculations then the standard model. In this research , you give an advancement of the TDFT for the standard model.
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