# Nonabelian generalization of Topological Dipole Field Theory

1. 1.  Technische Universität Darmstadt

### Abstract

The 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.

### Introduction

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.

### Theory

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.

### Conclusions

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.

## References

Chatrchyan, S. et al. " Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC." Physics letters B, 2012, 716 (1): 30-61.

Canetti, L. & Drewes, M. & Shaposhnikov, M. "Matter and Antimatter in the Universe." New J.Phys, 2012, 14: 095012. arXiv:1204.4186. Bibcode:2012NJPh...14i5012C.

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:

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

Beisert, N. "Review of AdS/CFT Integrability: An Overview." Letters In Mathematical Physics , 2012, 99: 425. arXiv:1012.4000. Bibcode:2012LMaPh..99..425K.

### Showing 19 Reviews

3

Error fixing:

- 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).

This review has 2 comments. Click to view.

Mathematically, the TDFT is based on generalized fiber bundles. More precisely, generalized means that the fiber bundles have a non-smooth curvature but smooth connection. The 2-form curvature field of These fiber bundles consist of a smooth curvature F_0 which can be expressed in Terms of a smooth 1-form Connection A, i.e. F_0 = dA+gA \wedge A and an additional non-smooth curvature 2-form field B'. The field B' is assumed to be a non-continuous stochastic noise (while A is a smooth function). On a base manifold point x the curvature B'(x) is random and Independent on the field A. For this field the following axioms hold:

1) B' satisfies the Čech cocycle condition.

2) The average value of B' is zero because the fiber bundle is smooth and is determined only by the Connection field A if the stochastic noise is filtered out.

3) The B' field is distributed by a multivariate Gaussian distribution in terms of arbitrary non-smooth 2-form fields B (also fields that don't satisfy condition 1)) which is Lorentz invariant ( This means that any functions in terms of the field B' are averaged with a probability density function exp(iS_(top)) ).

4) The moment-generating function corresponding to the field B' is unique (This means that a topological Quantum field theory with observables B' has to be constructed).

From These 4 conditions TDFT can be formulated axiomatically. The Axiom 4) ensures that the stochastic noise remains the same over the whole fiber bundle regardless of the physical system and regardless of the processes taking place in the physical system. In more mathematical Terms, there exists an unique functor \Phi from the category of fiber bundles with smooth connections to the category of fiber bundles with non-smooth connections by attaching a unique non-smooth stochastic noise to the fiber bundle with smooth connection.

"The moment-generating function corresponding to the field B'" means the moment-generating function of the statistic noise (which has the probability density function exp(iS_(top)) ).

The 4 axioms are plausible since there is no dependence of the stochastic noise structure on any physical processes (this is ensured by assuming a topological Quantum field theory). Since the Dipole field B' was not observed yet it is assumed as a field that fluctuates around the value 0. A Gaussian distribution (the simplest form of a probability density which is Lorentz invariant) ensures that the stochastic noise is weakly correlated (and therefore a topological Dipole at base manifold Point x is not strongly correlated on another topological dipole at base manifold Point x').

The observables of the field theory are well-defined even if the B and B' fields are non-smooth because it is governed by a topological Quantum field theory in Terms of the fields B' and B where computations of expectation values yield a finite number.

The topological invariant that are computed with TDFT are called "Dipole correlation value". It is a value that does not depend on energy-momentum states; it is simply a number.

• Somak Chaudhuri

Use of Gaussian distribution before TDFT approach is great indeed. Explains in a practical base.

• Somak Chaudhuri
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2

Special emphasis while calculating multinomials shows the depth of ground work for the paper. Great work.

• Somak Chaudhuri
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2

An excellent approach for coboundary mapping and co-Boson field mapping. Commendable effort. Looking forward to more such work in the near future.

• Azizur Rahman Madhu
2

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."

• Ritahi Oussama
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Very good work , I like it very much !! You used some mathematicals expressions with high level !

This review has 1 comments. Click to view.
• Azizur Rahman Madhu

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."

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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.

2

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.

• José Roberto Mahon
1

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.

• Ritchie Yan
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1

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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