Abstract
The Standard model of particle physics is successful in the description of many scattering processes in nature. There are also phenomena in nature like the Baryon asymmetry that are based on scattering processes which cannot be described by the Standard model. This research paper treats simple scattering processeses which are predicted by Topological Dipole Field Theory. These scattering processes have their origin in additional interactions between gauge bosons.
Introduction
Despite the
large progresses in theoretical and experimental physics there are still a
couple of phenomena which cannot be described by modern and well-established
theories. Such phenomena require an extension of well-known physical models. An
example of such an extension is the Supersymmetric Standard Model (Fayet 1975).
Since the Standard model doesn’t include gravitational interactions, also
Quantum Gravity models were proposed (Sundance 2007). A recent extension of the
Standard model is the Topological Dipole Field Theory (TDFT) which assumes that
gauge bosons couple to an intrinsic dipole moment (Linker 2015). This theory
adds a 2-form field to the ordinary 2-form gauge field strength
where
is an additional degree of freedom,
is the gauge connection and
is the gauge coupling constant. The field
is an observable of a Witten-type topological quantum field theory.
TDFT is originally based on Čech cohomology and can be formulated with a
Lagrangian density that is independent on the choice of the topological bases
that generate the Čech cohomology. The Lagrangian density of TDFT depends on
the 2-form fields
and
and also on a Lagrange multiplier
where it is integrated over all possible real-valued fields
and
. The 4-form topological Lagrangian density has the form:
Here, is the coupling constant for the coupling to the topological
dipole. The Lagrangian (1) generates a Lorentz-invariant and real-valued
action. If the action is dimensionless and lengths or times have dimension 1
the 2-form fields
and
must have dimension -2, while
and
are dimensionless.
An example of an unsolved problem in physics is the Baryon asymmetry (Sakharov 1991). Baryon asymmetry means that number of baryons is higher than the number of antibaryons in the universe. Another unsolved problem in physics is the origin of dark energy (Peebles et al. 2003). The fluctuations in dark energy are much stronger than the Standard model of particle physics is predicting. These phenomena can be described by TDFT more precisely since there are additional terms involved that modify the dynamics of some scattering processes. Moreover, phenomena related with dark energy and the creation of particles and antiparticles depends strongly on the dynamics of gauge bosons.
In this research paper it is shown how to calculate scattering processes that arise from these additional terms of TDFT. From the structure of the Lagrangian density Feynman rules for scattering with intrinsic topological dipoles are constructed. Also expectation values of topological dipole correlations are computed. After the treatment of basic Feynman rules, a transition probability per time is computed for a simple scattering process. This scattering process is a modified propagation of a gauge boson within a short time. After the calculation of the transition amplitude, the computational results are discussed with respect to baryon asymmetry and dark energy.
Theory
The theory is based on the topological
action (1) on a spacetime manifold which can be written in coordinate basis that is denoted by Greek
indices as follows:
Here, the Latin indices (here: ) are running over all Lie group generators
of the gauge theory which satisfy
.
Feynman rules for TDFT
For the Standard model of particle physics
Feynman rules are well-known. Feynman rules for Standard model propagators and
vertices are in TDFT exactly the same as in the ordinary Standard model. The
only difference between the TDFT and the Standard model is that there are occurring
the interaction terms and
when the generalized 2-form field strength
is expanded around
in the Lagrangian density. When these interaction terms are
expanded out in the path integral one obtains additional self-interactions in
the gauge bosons. More precisely, one obtains for all possible interactions
with topological dipoles the expression:
From (3) it is easy to see that there is a
product of fields
in the term indexed with
. If the gauge theory is abelian then the term indexed with
describes
self-interacting gauge bosons. In the nonabelian case, the factor
can be expanded in powers of
. May be
the power of
occurring in the Taylor series expansion. Then the term indexed
with
has
self-interacting gauge bosons. A Feynman diagram which describes a
-boson self-interaction is denoted as a rectangle with
legs (which illustrating incoming and outgoing bosons) that are
attached to this rectangle. Inside the rectangle two numbers
separated with comma are written. The number
is the power of
, while
is the power of the contribution
. Consequently, there has to be computed the expectation value of a
product of
tensor fields
. The following example shows a Feynman diagram for a 4-boson
interaction.
Figure 1: 4-boson interaction; Bosons are denoted by straight lines
For usual, there are written ingoing and
outgoing momenta and polarizations on the legs of the rectangle. Moreover, the
product of fields which has to be computed is written next to the rectangle. The
product
of
fields
is defined as
where the averaging denotes the weighted average with weighting factor
. An easy calculation shows that (4) can be written as:
If (5) is further evaluated one observes
that the exponent of in the integrand of (5) increases with
. Integration over
and normalizing (in this research paper it is assumed that the path
integral normalization factor is included in the functional integration measure)
yields powers of an infinite quantity
. In nonvanishing terms
this infinite quantity is of order
and can easily be absorbed in the bare coupling constant
by setting
. The coupling constant
is (when other UV divergences are absorbed) a physical constant.
The minimum number of factors
where (5) is nonvanishing is
since it holds
and
.
Modified propagation of a boson
Considering a vector boson with initial
energy-momentum state and initial polarization
, the vector boson goes to the final energy-momentum state
and final polarization
. Then the probability density that this boson is involved in this
reaction within a time
and within a volume
denoted by
in the Lorentz-invariant phase space volume
has the form:
Here, is the kinetic energy of the incoming boson,
the kinetic energy of the outgoing boson and
the expectation value of the T-Matrix associated with the 2-boson
self-interaction. From (3) it can be observed that the term
is the only term with nonvanishing topological expectation value
and without boson loop contributions which describes a 2-boson interaction.
The UV divergent Loop contributions which arise in higher exponents
can be absorbed by renormalization. For simplicity, only the term
is considered. Transforming this term into
-space one obtains:
In equation (7) the expression denotes the observation interval in spacetime direction
(with
and
) and a contraction in
due to the factor with
was performed. Also fluctuations in energy and momentum are included
in equation (7). Therefore the transition probability per unit time (6) also
allows processes with nonconservation of energy and nonconservation of
momentum. For large observation time the sinc-function can be approximated as a
delta distribution and energy and momentum is conserved in transitions. The
transition rate is proportional to
and is assumed to be small; hence, a small deviation from ordinary
Standard model is included by adding TDFT transition probabilities. If the boson
polarizations in (6) does not matter it can be averaged over all possible
polarizations. In the limit of very large momentum absolute values
the transition probability (6) decreases as
. It makes sense that the transition probability decreases rapidly
with momentum excess
.
Conclusions
The transition amplitudes given by the equations (6) and (7) are a model for boson density fluctuations. Bosons that undergo stochastic noise in small time scales can explain why dark energy fluctuations are much higher than the Standard model of particle physics predict. It makes sense that the highest amount of dark energy was present a few Planck times after the Big Bang since fluctuations predicted by TDFT were very significant. Baryon asymmetry arises from the high fluctuations in energy and momentum during this time period. The high-energy regions tend more to particle-antiparticle creation than the low-energy regions.
References
Fayet, P. "Supergauge invariant extension of the Higgs mechanism and a model for the electron and its neutrino." Nuclear Physics B, 2014, 90: 104. doi: 10.1016/0550-3213(75)90636-7
Sundance, O. et al. "Quantum gravity and the standard model." Class.Quant.Grav., 2007, 24: 3975-3994. doi: 10.1088/0264-9381/24/16/002
Linker, P. "Topological Dipole Field Theory." The Winnower, 2015, 2: e144311.19292. doi: 10.15200/winn.144311.19292
Linker, P. "Nonabelian Generalization of Topological Dipole Field Theory." The Winnower, 2015, 2: e144564.43935. doi: 10.15200/winn.144564.43935
Sakharov, D. "Violation of CP invariance, C asymmetry, and baryon asymmetry of the universe." Soviet Physics Uspekhi, 1991, 34 (5): 392–393. doi: 10.1070/PU1991v034n05ABEH002497
Peebles, P. J. E. & Ratra, B. "The cosmological constant and dark energy." Reviews of Modern Physics, 2003, 75 (2): 559–606. doi:10.1103/RevModPhys.75.559. arXiv:astro-ph/0207347
Showing 5 Reviews
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As far as I can see, mathematical calculations are correct. They are proved rigorously. The result concerning energy-momentum fluctuations is also very interesting, especially that it is a candidate explanation for Baryon Symmetry.
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Excellent work with rigorous calculations. This paper treats simple scattering processes which are predicted by dipole topological field theory. These scattering processes have their origin in additional interactions between gauge bosons. It nice and makes sense that the highest amount of dark energy was present a few Planck times after the big bang since fluctuations predicted by TDFT were very significant. It's very nice paper. Well written and anyone can understand. It only the rigorous calculus are very hard to understand it! But it is a good work.
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Your conclusion was interesting and was very good. I liked when refer to Planck time after the Big Bang which predicted by TDFT. I understand the TDFT and the standard model much better now. I like your explanation of the scattering process with the standard model by using TDFT as model and theory to nature.
"The transition amplitudes given by the equations (6) and (7) are a model for boson density fluctuations. Bosons that undergo stochastic noise in small time scales can explain why dark energy fluctuations are much higher than the Standard model of particle physics predict. It makes sense that the highest amount of dark energy was present a few Planck times after the Big Bang since fluctuations predicted by TDFT were very significant. Baryon asymmetry arises from the high fluctuations in energy and momentum during this time period. The high-energy regions tend more to particle-antiparticle creation than the low-energy regions."
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It's very nice paper ! Well written and anyone can understand it only the mathematicals formula are very hard to understand it ! It's good work :)
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I was expecting a Boson-Feynmann quadrature, however theory has been explained in a far more lucid manner. Very well deduced calculations.
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