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. doi:10.1016/j.physletb.2012.08.021
Canetti, L. & Drewes, M. & Shaposhnikov, M. "Matter and Antimatter in the Universe." New J.Phys, 2012, 14: 095012. arXiv:1204.4186. Bibcode:2012NJPh...14i5012C.
doi:10.1088/1367-2630/14/9/095012
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.
doi: 10.15200/winn.144311.19292
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
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3Error 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).
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2Special emphasis while calculating multinomials shows the depth of ground work for the paper. Great work.
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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.
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2The 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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2Very good work , I like it very much !! You used some mathematicals expressions with high level !
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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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2Here 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.
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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.
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1Topological 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.
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1Its 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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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.
Use of Gaussian distribution before TDFT approach is great indeed. Explains in a practical base.
Very well deduced calculations.