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\begin{document}
\title{With Iterative and Bosonized Coupling towards Fundamental Particle Properties}
\author{B{\sc ernd} ~B{\sc inder}}

\date{31.12.2002, updates: \href{http://www.quanics.com}{http://www.quanics.com}}
\affiliation{\it 88682 Salem, Germany,
\href {mailto:binder@quanics.com}{binder@quanics.com} \copyright
2002}

%#############################################################
\begin{abstract}
Previous results have shown that the linear topological
potential-to-phase relationship (well known from Josephson
junctions) is the key to iterative coupling and non-perturbative
bosonization of the 2 two-spinor Dirac equation. In this paper
those results are combined to approach the nature of proton,
neutron, and electron via extrapolations from the Planck scale to
the System of Units (SI). The electron acts as a bosonizing bridge
between opposite parity topological currents. The resulting
potentials and masses are based on a fundamental soliton mass
limit and two iteratively obtained coupling constants, where one
is the fine structure constant. The simple non-perturbative and
relativistic results are within measurement uncertainty and show a
very high significance. The deviation for the proton and electron
masses are approximately 1 ppb ($10^{-9}$), for the neutron 4 ppb.
\end{abstract} \maketitle


%#############################################################
\paragraph*{\bf Introduction.}
%#############################################################

In the context of the sine-Gordon equation (SG) \cite{AbS} mass
and coupling constants constitute phenomenological dimensions
without any direct relation to the topological processes. If
topological phases evolve (stepwise) proportional to the potential
(as often observed) a possible iterative approach to coupling
constants is already included \cite{thirring11} and provides i.e.
for an iterative approach to the fine--structure constant.
Additionally, a simple bosonization is possible and based on a
Dirac Hamiltonian that carries pairs of standard vector and scalar
potentials \cite{thirring11} and requires the standard hydrogen
Hamiltonian to arrive at the SG that represent bosonized
spin-orbit coupling which requires to
compare at least two interacting states carrying spin and opposite currents.\\

%#############################################################
\paragraph*{\bf Bosonizing a radial symmetric Dirac equation.}
%#############################################################

We start with interaction described by massless vector particles
(photons) and massless scalar mesons within a relativistic local
quantum field theory. The Dirac equation $\hat H_D \Psi =
M_{\gamma}\Psi$ based on a Dirac - Hamiltonian $\hat H_D$ for a
mixed potential consisting of a scalar potential $V_{s}(r)$ and a
vector potential $V_{v}(r)$ is given by
\begin{equation} \hat H_D = c\hat \alpha
\cdot \hat p + \hat \gamma_{0}
\[ {V_{0} + V_{s} } \] + V_{v}\,\labeqn{H_D},
\end{equation} where $\hat \alpha$ and $\hat
\gamma_{0}$ are the Dirac matrices \cite{Dirac-28, greiner-we}.
The vector and scalar potentials will provide for spin--spin and
spin--orbit coupling and be helpful to bosonize the Dirac equation
with two opposite parity two--spinors. In the same way the vector
Coulomb potential part of $V_{v}$ corresponds to the exchange of
massless photons between a nucleus and leptons orbiting around it,
the scalar Lorentz potential potential part of $V_{s}$ with $-1/r$
characteristics corresponds to the exchange of massless scalar
mesons. The resulting energy eigenvalue $V_{\gamma}$ corresponds
to opposite parity two-spinors components with total mass $V_{0}$.
The two-spinor wave--function in spherical symmetry is given by
\begin{equation}
\Psi \propto {\psi_{R}(r) {y } {^{m} _{j l_{R}}} \choose i
\psi_{L}(r) {y } {^{m} _{j l_{L}}}}\,\labeqn{sgxx},
\end{equation}
where ${y } {^{m} _{j l}} $ are the normalized spin-angular
functions constructed with Pauli spinors and spherical harmonics
of order $l$. The spinors $\psi_{R}$ and $\psi_{L}$ are
eigenfunctions with eigenvalues $l_{R}(l_{R} + 1)$, and
$l_{L}(l_{L} + 1)$, respectively. A new parameter $\kappa$ can be
interpreted as orbital (spin) excitation between the two--spinor
components, where $2\kappa= l_{R}(l_{R} + 1)-l_{L}(l_{L} + 1)>0$
characterizes a left/right spin-asymmetry or difference. The
radial functions have to obey the asymmetry relation
\begin{eqnarray}
\frac{{d^2\theta_{R}}} {{dr^2}} = - \frac{{\kappa }}
{r}\frac{{d\theta_{R}}} {{dr}} + \left[{\frac{{V_{0} - V_{\gamma} + V_{s} - V_{v} }} {{\hbar c }}} \right]\frac{{d\theta_{L}}}{{dr}}, \\
\nonumber%
\frac{{d^2\theta_{L}}} {{dr^2}}  = + \frac{{\kappa }}
{r}\frac{{d\theta_{L}}} {{dr}} + \left[ {\frac{{V_{0} + V_{\gamma}
+ V_{s} + V_{v} }} {{\hbar c }}} \right]\frac{{d\theta_{R}}}{{dr}}
\labeqn{ap02}.
\end{eqnarray}\\

%#############################################################
\paragraph*{\bf Topological currents.}
%#############################################################

The $1/r$-terms carry dimensional information and can be
interpreted as fractional dimension shifts, $\kappa$ by
spin--asymmetry (a fractional parity property) and $\alpha$ by
vector potentials. For SG--solitons a radial topological current
can be introduced according to
\begin{eqnarray}
{q(r)} = {\partial_r \theta \over r}, \quad {2 {V} \over \mu} =
{(\partial_r \theta)^2}=q(r)^2{r^2}  \labeqn{mass01},
\end{eqnarray}
where $q(r)$ can be interpreted as a (fractional) radial dimension
shift induced by vector and/or scalar currents \cite{doubleSG05}.
$q(r)$ simply represents the electric charge density. Let the
Dirac equation describe topological charge and current density,
the 2-dim. case allows to introduce orthogonal topological
currents or phase gradients by
\begin{equation}
\psi_{L}(r) =  {\partial_r \theta_{L} \over r},\quad \psi_{R}(r) =
{\partial_r \theta_{R} \over r}\ \labeqn{ap01},
\end{equation}
where
\begin{eqnarray} |\Psi|^2 = q(r)^2 = \psi_{L}(r)^{2} + \psi_{R}(r)^{2}
 \labeqn{dimcharge}.
\end{eqnarray} Constant $q$ \eqn{mass01} immediately
provides for a harmonic oscillator coupling potential $V\propto
r^2$ with proportionality between potential and phase
\begin{equation} V(r)
 = {\mu \over 2} \( q r\)^2, \quad V(\theta) = {\mu \over 2}q \theta
 ,\quad E_\mu = \mu c^2  \,\labeqn{sg04},
\end{equation}
and unit condition
\begin{eqnarray}
V_0= V(r=1)= V(\theta=q)={\mu \over 2} q^2
 \,\labeqn{sg02x}.
\end{eqnarray}

%#############################################################
\paragraph*{\bf Stationary boson exchange.}
%#############################################################

The balancing condition is given by
\begin{equation}  {\psi_{R}(r) \over \psi_{L}(r)}= {\alpha \over
\kappa } > 1 \,\labeqn{hg3d},
\end{equation}
this couples opposite parity radial functions, and provides for a
wide variety of possible Lorentz scalar $V_{s}(r)$ and vector
Coulombic potential $V_{v}(r)$ functions
\cite{franklin-99,franklin-00}. With \eqn{ap01} both types of
$1/r$--coupling, coulombic coupling and spin--asymmetry correspond
to relative topological phase evolutions
\begin{equation}
\theta_{L}=\pi\kappa f(r),\quad \theta_{R}+\theta_0 = \pi\alpha
f(r) \labeqn{phasev01},
\end{equation}
where $f(r)$ will be a special function of $r$ and $\theta_0$ a
phase offset. This couples opposite parity radial functions, and
provides for a wide variety of possible Lorentz scalar $V_{s}(r)$
and vector Coulombic potential $V_{v}(r)$ functions that have to
obey the asymmetry relation
\begin{equation}
\frac{V_{\gamma}}{V_{0}} = {V_{v}(r)+ \frac{\hbar c \alpha }{r}
\over V_{s}(r) + \frac{\hbar c \alpha }{r}} = {\frac{{{\alpha}^{2}
- {\kappa}^{2}}} {{{\alpha}^{2} + {\kappa}
^{2}}}}\,\labeqn{miei2}.
\end{equation} To couple opposite
parity components both type of potentials have to include
Coulombic $-\hbar c \alpha/r$ terms. But finally those two
components will merge after bosonization to one $-\hbar c
\alpha/r$ potential and give the SG equation. The two first order
ODE can now be combined to a Schr\"odinger/Klein--Gordon type
relativistic wave equation
that have bosonic solutions.\\

%#############################################################
\paragraph*{\bf Coupling by sine--Gordon.}
%#############################################################

With \eqn{hg3d} it is sufficient to consider only one part,
therefore we omit the index $\theta_{R} \rightarrow \beta \phi$
\begin{eqnarray}
\frac{{d^2\phi_{}}} {{dr^2}} =  \frac{{\kappa }}
{r}\frac{{d\phi_{}}} {{dr}} + \left[{\frac{{V_{0} + V_{\gamma} +
V_{s} + V_{v}}} {{\hbar c }}} \right]{\kappa \over
\alpha}\frac{{d\phi_{}}}{{dr}} \, \labeqn{ap10}.
\end{eqnarray}
With the SG potential
\begin{equation}
V(\phi) = \frac {\mu} {2\beta^{2}} [1 - \cos (\beta\phi_{}]
\labeqn{cole9},
\end{equation}
and SG equation
\begin{eqnarray}
{\frac{{d^2\phi_{}}}{{dr}^2}} - \beta^{-1}\sin (\beta\phi_{}) =
0\, \labeqn{ap18},
\end{eqnarray}
the vector and scalar potentials necessary to bosonize the Dirac
equation with two opposite parity two--spinors merge to one
Coulomb-type potential
\begin{eqnarray} V_{s} + V_{v}=
 - {\hbar c \alpha \over r} \labeqn{ap32},
\end{eqnarray} where
\begin{eqnarray}
{(V_{0}+V_{\gamma}) \over \hbar c }{ \kappa \over \alpha}  = \cos
(\beta\phi_{}/2) = {(V_{0}-V_{\gamma}) \over \hbar c }{ \alpha
\over \kappa}  \, \labeqn{ap21}.
\end{eqnarray} \\

%#############################################################
\paragraph*{\bf Background-soliton and soliton-soliton coupling.}
%#############################################################

According to \cite{beta10} basic background fluctuations mediated
are by linear waves with mean potential $2 \overline{V } = \hbar
c/\lambda_{1}$ that couple to solitons in a Compton--type
permanent scattering process. The soliton Compton wavelength
$\lambda_{\mu}$ is related to the effective background (cut--off)
wavelength by $\lambda_{1}=q^2\lambda_{\mu}$, $q^2$ characterizes
localization or the reduction of fluctuation amplitude between
background and soliton. Another $q^2$ factor corresponds to the
fluctuation reduction due to soliton--soliton coupling. This
means, that soliton--soliton interaction and correspondent
topological currents scale with $q^4$ with respect to the linear
and massless background reference, where the coupling cascade is
given by
\begin{equation}
 2 V_0 = q^2 E_{\mu}  =  q^{2} {h c \over \lambda_{\mu}} = 2q^4\overline{V }
\labeqn{relationq1}.
\end{equation}
More can be found in \cite{thirring11}.\\
%#############################################################
\paragraph*{\bf Iterative coupling.}
%#############################################################

The linear relationship between potential and phase \eqn{sg02x}
provides for an iterative solution, where the optimum phase shift
$\theta_{M}$ is given by
\begin{eqnarray}
\theta_{M} = \pi \alpha = \beta\phi_{} \,\labeqn{ap26},
\end{eqnarray}
\begin{equation}
{M \theta_{M}}  = - 2 \pi J \cos \theta_{M}\,\labeqn{alfa04}.
\end{equation}
The Sommerfeld fine structure constant can be well approached by
${\alpha_{137}}=1/137.00360094...$ \cite{alpha137MN, online1} and
fits well to the interpretation, that the topological phase
gradient or charge/current $q$ provides for
a dimensional shift. \\
In the permanent radial symmetric background--soliton Compton
scattering process a radial phase gradient will be induced that
corresponds to a radial coupling potential $V_r(\theta_{})$ driven
by $\overline{V}$
\begin{eqnarray}
1 - \cos(\theta_{}) = {V_r(\theta_{}) \over \overline{V}}
\,\labeqn{beta03}.
\end{eqnarray}
And again, the linear relation between phase and potential leads
quickly with $E_{\mu}=V(2\pi J)$ to the iterative condition for
the optimum phase shift
\begin{eqnarray}
1 - \cos(\theta_{}) = {\theta_{} \over \pi} {q^2 \over 2\pi J}={1
\over N}, \quad \theta_{} = \pi \kappa
  \,\labeqn{beta05}.
\end{eqnarray}
\Eqn{beta05}, $1/q^2=12\pi^2$, and $J={1 \over 2}$ provides for
$\kappa = 1836.11766...$. The value of $\kappa$ controls
the topological phase gradient induced by radial spin--asymmetry. \\

% ----------------------------------------------
\begin{widetext}
\begin{table}[t]
\begin{caption}
\newline \small{The results of the $\alpha$-$\kappa$ model compared to values in \cite{FUNDDATA}.}
\end{caption}\\
\begin{tabular}{p{90pt}p{30pt}p{105pt}p{45pt}p{90pt}p{135pt}}\hline
\textit{Name}& \textit{Not.} & $\approx$ \textit{meas. value} & \textit{dev.} & \textit{calculation}& \textit{comment}  \\
\hline & & & & & \\
fine structure const.& $\alpha^{-1}$ & $137.03601144(498)$ \cite{KNW-99} & $\approx 1.5 \cdot 10^{-8}$   &$137.036009411..$&  \eqn{alfa04}, from iteration,  \cite{alpha137MN}\\
%\hline
$N=3805.5$ & $\alpha_{3805.5}^{-1}$ & $137.03599976(50)$ & $ \approx 6 \cdot 10^{-9}$ & $137.036000525..$&  modal shifts in \cite{alpha137MN}\\
%\hline
spin asymmetry & $\kappa^{-1}$ & {1836.118} & $< 10^{-7}$  &$1836.1176608..$& $\approx 6\pi^5$ \eqn{beta05}, from iteration\\
%\hline
$N=683174$ & $\kappa_{683174}^{-1}$ &  &  &$1836.11748207..$&  \eqn{modshift}, incl. modal shift\\
\hline
current w.n.& $k_{\mu}$, $q_{\mu}$ & $4.76804431085... \cdot 10^{15}$m$^{-1}$ & exact &$2\pi q^2\Xi^2$m$^{-1}$ &  \eqn{basenergy}, energy $940.8637..$MeV\\
\hline Vacuum permittivity constant & $\epsilon_0$
&$8.95419..\cdot 10^{-12}$(s/m)$^2$ & $\approx1.35\%$ &
$8.97413..\cdot 10^{-12}$(s/m)$^2$ &
\eqn{couple4}-\eqn{gfactoreps0}, the fixed SI value is $\epsilon_0
=
10^7/(4\pi c^2)$.\\
\hline
electron & $\xi_{e-}$ & $5.431168277 \cdot 10^{ - 4}$ & $-9.8 \cdot 10^{-8}$ & $5.4311688808 \cdot 10^{ - 4}$&  \eqn{x_e}\\
%\hline
electron + correction& $\xi_{e-}$ & $5.431168277 \cdot 10^{ - 4}$
& $ < 10^{-9}$ & $5.4311682794 \cdot 10^{ - 4}$&  \eqn{x_e},
\eqn{modshift}, including modal shift ($-9.7 \cdot 10^{-8}$)\\
%\hline
proton & $\xi_{p+}$ & $0.9972454102$ & $+9.6 \cdot 10^{-8}$
& $0.99724531409$&  \eqn{x_p} \\
%\hline
proton + correction & $\xi_{p+}$ & $0.9972454102$ & $ < 10^{-9}$
& $0.99724541114$&  \eqn{x_p}, \eqn{modshift}, including modal shift ($+9.7 \cdot 10^{-8}$)\\
%\hline
neutron & $\xi_{n}$ & $0.9986200355$ & $ 4 \cdot 10^{-9}$  &
$0.99862003154$&
\eqn{x_n}\\
\hline ? small neutral & $\xi_{m} $ &  $q_{m} \approx q_{\pi} $ ?
& ? & 0.2728144..&
 \eqn{x_m}, 256.68 MeV \\
\hline
\end{tabular}
\labtab{tablekappa}
\end{table}
%\clearpage
\end{widetext}
% ----------------------------------------------

%#############################################################
\paragraph*{\bf Origin of the basic soliton energy.}
%#############################################################

With Planck units assigned to the background fluctuation level we
get $\lambda_{1}=q^2 \lambda_{\mu}=1$, $q^{-2} E_{\mu}=1$. To get
the correct baryon mass scale in our SI system, we have to shift
the Planck scale velocity units to human artificial velocity units
based on SI length and arbitrary mass units. Planck velocity units
demand that the light velocity equals the unit velocity $c=u=1$
such, that the mean background energy $\overline{V}$ scales with
the square of the wave velocity and the SI unit energy scales with
the square of the unit velocity $u$ (in SI $E_{u}=$1J = 1kg
m$^2$/s$^2$)
\begin{equation}
{2\overline{V} \over  E_{u}}= {c^2 \over u^2 }=  \Xi^2, \quad \Xi
= 299792458
 \,\labeqn{energy5}.
\end{equation}
Practical applicability of the SI system motivates to expect a
unit velocity $0<u \ll c$ with $\Xi = c/u \gg 1$. Particle and
photon energies can be compared via Compton and photon wavelengths
that refer to the light velocity. Planck length units demand that
the 1-dimensional quantum energy of waves coupling to particles
$E_{\mu}$ is inversely proportional to the wavelength, especially
to the Compton wavelength with
    \begin{equation}
{E_{\mu} \over E_{u}} = {\lambda_{u} \over \lambda_{\mu}}
\,\labeqn{energy3}.
\end{equation}\\
As a result, the characteristic soliton wavelength of
one-dimension coupling is with \eqn{relationq1}, \eqn{energy5},
and \eqn{energy3} exactly given by \cite{thirring11}
\begin{equation}
\lambda_{\mu} = { \lambda_{u}  \over q^2\Xi^2 }\approx 1,31777...
\cdot 10^{-15}\textup{m}
 \,\labeqn{basenergy}.
\end{equation}
\Eqn{basenergy} provides for the basic soliton mass $\mu$ via
Compton relation $ \mu = h / (c \lambda_{\mu})= q^2 \Xi^2 h / (c
\lambda_{u})$. Realized in SI units the value is
\begin{eqnarray}
\mu = {\hbar \over c}{\Xi^2 \over 6\pi \textup{m}} \approx
1.67724... \cdot 10^{-27}\textup{kg} \,\labeqn{xuf4b},
\end{eqnarray} where
\begin{equation}
\lambda_{1} = q^2 \lambda_{\mu}=\hbar c/(2\overline{V}) = q^2
\lambda_{\mu} = |c^{-2}|/(2\pi)\text{m}\,\labeqn{lambda1}.
\end{equation} \\

%#############################################################
\paragraph*{\bf Currents and bridges.}
%#############################################################

Supported by the Dirac equation, the following steps provide
probably for the most effective strategy to obtain very accurate
fundamental particle energies. The basic soliton energy will split
[see \eqn{mass01} and \eqn{dimcharge}] into left- and
right--handed parts $E_{L},\,E_{R}$
\begin{equation}
E_{\mu}  = E_{L} + E_{R} = 2 q^2 \overline{V} \,\labeqn{mass03},
\end{equation}
where Planck units provide for a common base. The correspondent
left, right, and basic mass currents $q_{L},\,q_{R},\,q_{\mu}$,
respectively, can be defined with \eqn{hg3d} according to
\begin{eqnarray}
E_{\mu}  = q^2 = q_{L}^2 +  q_{R}^2 = q_{\mu}^2(\alpha^2 + \kappa^2) ,\nonumber \\
 2 \overline{V} = 1\,\labeqn{qnormal}.
\end{eqnarray}
A (self-)bound state energy $V_{\gamma} < V_{0}$ will be assigned
to the left--right current asymmetry that scales like
\eqn{relationq1} and defines the ratio
\begin{equation}
{V_{\gamma} \over V_{0}} = \({q_{\gamma} \over q_{\mu}}\)^4 =
\frac{q_{L}^{2} - q_{R}^{2}} {q_{L}^{2} + q_{R} ^{2}}=
\frac{q_{L}^{4} - q_{R}^{4}} {q_{L}^{4} + q_{R}
^{4}+2q_{L}^{2}q_{R} ^{2}}\,\labeqn{gra02b}.
\end{equation}
According to the soliton-soliton coupling relation
\eqn{relationq1} coupling currents scale with the fourth power in
$q$. But not the extra term $2q_{L}^{2}q_{R} ^{2}$ in the
normalizing denominator: it will be identified as the
low-dimensional internal left/right bosonization bridge and leads
to the one and only possible definition of a symmetric bridge in
accordance with \eqn{hg3d} and \eqn{gra02b}
\begin{equation}
q_{e-}={q_{L} q_{R}\over q_{\mu}}
\end{equation}
 that can be normalized with \eqn{qnormal} to
\begin{eqnarray} \xi_{e-}={q_{e-} \over q}= \alpha{q_{R}
 \over q}= \kappa{q_{L} \over q},\nonumber \\
q_{L} =\alpha q_{\mu},\quad q_{R} =\kappa q_{\mu}
\,\labeqn{gra03},
\end{eqnarray}
see appendix. Acting as a kind of direct topological bridge
$q_{e-}$ connects and balances left-- and right--handed currents
in accordance with the bosonization relation \eqn{hg3d}. If the
smaller current $q_{R}$ becomes reciprocal to the bigger current
$q_{L}$ (i.e. by inversion) the coupling bridge
$q_{e-}=q_{L}q_{R}/q_{\mu}$ can be removed by setting
\begin{equation}
q_{L}^{4} = q_{n}^{4} - q_{m}^{4}, \quad q_{\pi} = {q_{m}\over
q_{n}} q_{L} = {q_{n}\over q_{m}} q_{R},\, \labeqn{gra04}
\end{equation} providing for a neutral and symmetric decoupling bridge
$q_{\pi}$ with big neutral current $q_{n}$ and small neutral
current $q_{m}$, see appendix. The transition generates a
decomposition without a bridge or mixed term given by
\begin{equation}
 \({q_{\gamma} \over q_{\mu}}\)^4 =
{\frac{{q_{n}^{4} - q_{m}^{4}}} {{q_{n}^{4} + q_{m}^{4}}}}
\,\labeqn{gra07},
\end{equation}
where the neutral currents $q_{n}$ and $q_{m}$ scale with the
fourth power like $q_{\mu}$ and split into two (``orthogonal'')
parts as desired
\begin{equation}
E_{\mu}^2 = E_{n}^2 + E_{m}^2 = 4 q^4 \overline{V}^2
\,\labeqn{mass03}.
\end{equation}

%#############################################################
\paragraph*{\bf Identification of particles}
%#############################################################

Any (self-)bound stable particle should be lighter than $\mu$ of
\eqn{xuf4b}. With $q$ as the current, $q_{e_{-}}$, $q_{\gamma}$,
and $q_{n}$ can now be identified as electron, proton, and neutron
currents, respectively. It is proper to assign the coupling
currents and correspondent energy eigenvalues to dimensionless
$\xi_{x}=q_{x}/q=k_{x}/k_{\mu}$ values that correspond to the
ratio of measured particle Compton wavenumber given by
\begin{equation}
E_{x}=\hbar \omega_{x}= \hbar c k_{x}=\hbar \xi_{x} \omega_{\mu} =
\hbar  \xi_{x} c k_{\mu}\,\labeqn{C_f0},
\end{equation}
with respect to the reference soliton Compton wavenumber
$k_{\mu}$, see table 1 and appendix. The bosonizing bridge
$q_{e-}$ based on the bridge in \eqn{gra03} is with \eqn{C_f0}
given by
\begin{equation} {q_{e-}} = {\frac{{q}}
{\sqrt{\kappa^{-2} + \alpha^{-2}}}}= \xi_{e-}q \labeqn{x_e}
\end{equation} and will represent the electron,
the proton coupling and stabilizing partner. In equilibrium the
coupling can be assumed to happen in rotary motion where the
orbital phase evolution given by $\alpha$ provides for a velocity
$v = \alpha c$. The correspondent relativistic factor
$\gamma={1/\sqrt {1 - \alpha^{2}}}$ increases the soliton
background current of the co-rotating body-fixed frame
$q_{\gamma}$ and provides for the effective current
\begin{equation}
q_{p+} = {\frac{{q_{\gamma}}}{\sqrt {1 - \alpha^{2}}}} =
\(\frac{\kappa^{-2} - \alpha^{-2}}{\kappa^{-2} +
\alpha^{-2}}\)^{1/4}\frac{q}{\sqrt{1-\alpha^{2}}}
 = \xi_{p+}q\\
\labeqn{x_p},
\end{equation}
that represents the (proton) current in the laboratory frame. The
big neutral current \eqn{gra04} can be assigned to the neutron,
\begin{equation} {q_{n}}  =  \({\frac{{\kappa^{-2}}}
{\kappa^{-2} + \alpha^{-2}}}\)^{3/4} \(\frac{\kappa^{-2} -
\alpha^{-2}}{\kappa^{-2} + \alpha^{-2}}\)^{-1/4}q=
\xi_{n}q\,\labeqn{x_n},
\end{equation}
the small neutral current to an unknown
\begin{equation} {q_{m}} = \sqrt{\kappa \over \alpha} {q_{n}} = \xi_{m}q
\,\labeqn{x_m}.
\end{equation}
Using the fine structure value $\alpha^{-1} \approx 137.036$
according to the Dirac equation with Coulomb coupling it turns out
that for a special $\kappa^{-1} \approx 1836.118$ the system above
can describe fundamental particle energies and mass ratios.
Possible numbers are shown in table 1, $\kappa$ is given similar
to $\alpha$ as an iterative orbital coupling condition.\\

%#############################################################
\paragraph*{\bf 3-dim. charge and permittivity.}
%#############################################################

The coupling currents or fluxes ``flow'' usually 3-dimensional.
The Gauss relation connects the 1-d coupling topology to a
spherical symmetry such, that the fine structure constant can be
defined by
\begin{equation}
\alpha = {q^2 \over  4 \pi \epsilon_{00} \hbar c }
\,\labeqn{couple3}.
\end{equation}
This provides for
\begin{equation} \epsilon_{00}= {\lambda_{1} q \over \hbar c}, \alpha_{q} =  {q \over 4\pi}, \, M =\[ 1 \over \alpha_{q} \]=137
\,\labeqn{couple4},
\end{equation}\\
where $[\,]$ means next higher integral value \cite{rayleigh07}.
Leaving Planck units and introducing the Coulomb charge unit by
replacing $q$ with the SI elemental charge $e$, $\epsilon_{00} =
{\lambda_{1} e /( \hbar c)} \approx 8.974129\cdot
10^{-12}$(s/m)$^2$ is slightly above ($\approx1.35\%$) the SI
vacuum permittivity constant $\epsilon_0 $, see \eqn{couple4} and
\cite{thirring11}. If we assume, that \eqn{couple4} is correct,
the unit of charge is with the fixed SI value $\epsilon_0 =
10^7/(4\pi c^2)$ directly coupled to the unit of mass part of
$\hbar$. As a consequence, the dimension of charge in
\eqn{couple4} becomes kilogram and we have determined the most
likely charge--to--mass ratio of a fundamental baryon, where the
reduction $\epsilon_0/\epsilon_{00}$ could be due to nuclear
effects and binding energies (the charge--to--mass ratio of
protons depends always on the context) \cite{thirring11}.\\

%#############################################################
\paragraph*{\bf Local screening?}
%#############################################################

But there is another interesting approach: if we take into account
that the ratio $e/\hbar$ has been determined with the reduced
$V_{\gamma}$ in the bound state and not with the ``free'' $V_{0}$
value, the reduction of $V_{\gamma}$ with respect to $V_{0}$ due
to the radial phase gradient depends only on $\kappa$ and $\alpha$
and is with \eqn{miei2} $\approx 1.12\%$. The electric field
surrounding an electron polarizes the QED vacuum. In QED this
phenomenon leads to charge screening for the electron $q_{e}<q$ or
$a=q/q_{e}-1>0$ that can be found as a g-factor anomaly
$g_{e}=2(1+a)>2$. The screening can be translated a reduction of
the topological phase gradient due to the bound state potential
reduction $V_{\gamma}<V_{0}$ and dimensional shift. Splitting the
shift in a $q$-dependent and $V$-dependent part (includes
$\lambda_{\mu},\, c,\,e/\hbar$) where
\begin{equation} \epsilon_{00}= q^2
{\lambda_{\mu} \over c}{e \over \hbar},\quad \epsilon_{e}=
{q_{e}^2V_{\gamma}\over q^2 V_{0}} \epsilon_{00} \,\labeqn{eps0},
\end{equation} it turns out that $\epsilon_{e}=\epsilon_{0}$
requires a first order correction
\begin{equation} 1+a=q/q_{e}\approx 1.00115810 .. \,\labeqn{gfactoreps0}
\end{equation}that is quite near to the electron g-factor $g_{e}/2=1.00115962..$ .
The overall reduction $V_{0}-V_{\gamma}$ by spin--asymmetry
without screening effect for a given $\lambda_{\mu}$ is about
$10.42$MeV and could be indeed a source for strong interaction as
a form of cooperative phase shift and dimensional reduction.

It should be noted, that the iterative method to determine the
fine-structure extended by a variable charge $Z$ in ${M
\theta_{M}} = - 2 \pi J \cos (Z \theta_{M})$ in \eqn{alfa04} is a
chaotic system. As an iterative system it shows asymptotic stable
and converging regimes but also bifurcations and unstable regimes
for special feedback coupling strengths $Z$, the chaotic dynamics
should be found in charged nuclei where $Z>114$ and $J=1$
reaches the critical value. A simulation can be found at \cite{online2}. \\

%#############################################################
\paragraph*{\bf The small difference.}
%#############################################################

To fulfill the requirements of single-valuedness, $\kappa$
coupling provides for both, orbital radial and standing waves, see
fig.\fig{kappa03}. The correspondent adjustment requires to
increase $\kappa' \rightarrow \kappa$ and angle $\theta'
\rightarrow \theta$ to obtain an integer $N$ in \eqn{beta05}. This
process stretches the wave and reduces the orbital wavenumber.
With integer $N=683174$ there is a very small increase in $\kappa'
\rightarrow \kappa$ given by
\begin{eqnarray}\delta_\kappa =
{1\over\kappa' N}{\pi \over q^2} - 1 \approx 9.7 \cdot 10^{-8},
\kappa N = {\pi \over q^2} ,\nonumber\\N=683174,\,
N'=683174.06648.. \, \labeqn{modshift}.
\end{eqnarray} In table 1 a similar shift can be found for both,
the proton (same direction) and electron (opposite direction),
while the neutron value is quite exact within measurement
uncertainty. Neglecting the orbital effect of $\alpha$, the
orbital wavelength stretch in fig.\fig{kappa03} and wavenumber
reduction can be assigned to the electron since $E_{e-} \approx
\overline{V}2\pi /N'$. In electron--to--proton mass ratio
measurements the proton mass now appears to be heavier. This could
be the reason why $E_{p+}=\xi_{p+}E_{\mu}$ is increased by the
relative difference $\delta_\kappa$.\\


%#############################################################
\paragraph*{\bf Conclusion.}
%#############################################################

% ------------------------------------------
\begin{figure} [t]
\center\includegraphics[scale=0.6] {kappa04.ps}
\caption{\small{Integer $N$ in \eqn{beta05} corresponds to both,
the orbital and radial electron wavenumber. Increasing the angle
$\kappa' \rightarrow \kappa$, $\theta' \rightarrow \theta$ to
adjust to the integer $N$, stretches the orbital standing wave and
reduces the orbital wavenumber, see also \eqn{modshift}.}}
\labfig{kappa03}
\end{figure}
% ------------------------------------------
Increasing the angle $\kappa' \rightarrow \kappa$, $\theta'
\rightarrow \theta$ fits to the shifts tabulated in table 1 and
provides for an overall deviation for the proton and electron mass
of $< 10^{-9}$. Regarding weak interaction, the small neutral
partner current should somehow be related to the neutrino, but
$256.7$ MeV would probably be too heavy for a neutrino. One could
ask if it is an unmeasurable massless fluctuation current related
to the massless background current. The local current and phase
reduction including phase gradient and dimensional reduction
(charge screening) provides for a possible understanding of
$\epsilon_{00}>\epsilon_{0}$ connected to strong interaction.
Regarding the currents, two neutrons provide approximately for the
same reduction like one proton, see table 1. This fits to the well
known proton--to--neutron number ratio found in heavy nuclei.
Generally, these mass differences could be relevant for
approaching (nuclear) binding energies with many--soliton models
\cite{beta10}. The corresponding energies are in the MeV--range.
The neutron is energetically lower than $\mu$, this fits with the
semi--stability of a free neutron.

Since the vector potential $V_{v}$ in \eqn{ap32} already includes
the negative coulomb term, $V_{v}=-V_{s}- {\hbar c \alpha / r}$
also includes the negative scalar potential. If we would make the
scalar component very small we would find simply the relativistic
hydrogen formalism providing for the famous result of Dirac
\cite{Dirac-28, greiner-we} including relativistic spin-orbit
coupling. The two central coupling constants can be determined
iteratively from the a linear SG potential--to--phase relationship
- a general key that also provides for balanced and bosonized
Dirac currents with a very high significance.

\clearpage

\begin{widetext}


\begin{thebibliography}{aaa}

\bibitem{AbS} M. J. Ablowitz and H. Segur,{\it Solitons and the Inverse
Scattering Transform}, SIAM, Philadelphia, (1981).

\bibitem{thirring11} B. Binder,{\it Bosonization and Iterative Relations Beyond Field Theories} (2002);
\href{http://philsci-archive.pitt.edu/documents/disk0/00/00/09/44/index.html}{PITT-PHIL-SCI00000944}.

\bibitem{Dirac-28} P. A. M. Dirac, Proc. R. Soc. Lond. A {\bf 117}, 610 (1928).

\bibitem{greiner-we} W. Greiner, {\it Relativistic Quantum Mechanics: Wave Equations}, Springer-Verlag
Berlin Heidelberg, 1990.

\bibitem{doubleSG05} B. Binder,{\it Higher-Dimensional Solitons Stabilized by Opposite Charge};
\href{http://philsci-archive.pitt.edu/documents/disk0/00/00/09/18/index.html}
{PITT-PHIL-SCI00000918},
\href{http://www.ma.utexas.edu/mp_arc-bin/mpa?yn=02-499}{UniTexas
mp-arc 02-499}.

\bibitem{franklin-99} J. Franklin, Mod. Phys. Lett. A {\bf 14}, 2409 (1999); \href{http://xxx.lanl.gov/abs/hep-ph/9812464}{hep-ph/9812464}.

\bibitem{franklin-00} A. S. De Castro and J. Franklin, Int. J. Mod. Phys. A {\bf 15}, 4355 (2000);\href{http://xxx.lanl.gov/abs/hep-ph/0002093}{hep-ph/0002093}.

\bibitem{beta10} B. Binder,{\it Soliton Compton Mass from Auto--parametric Wave--Soliton Coupling};
\href{http://www.ma.utexas.edu/mp_arc-bin/mpa?yn=02-475}{UniTexas
mp-arc 02-475}.

\bibitem{alpha137MN} B. Binder, `Berry's Phase and Fine Structure' (2002);
\href{http://philsci-archive.pitt.edu/documents/disk0/00/00/06/82/index.html}{PITT-PHIL-SCI00000682}.

\bibitem{online1}{B. Binder, alpha simulation; }
\href{http://www.quanics.com/spinorbit.html}%
{http://www.quanics.com/spinorbit.html}

\bibitem{rayleigh07} B. Binder,{\it Soliton Coupling Driven by Phase Fluctuations in Auto-Parametric Resonance};
\href{http://www.ma.utexas.edu/mp_arc-bin/mpa?yn=02-473}{UniTexas
mp-arc 02-473}.

\bibitem{FUNDDATA} Groom et al., "The Review of Particle Physics", The European Physical Journal {\bf C15},1
(2000).
\bibitem{KNW-99} E. Kr\"uger, W. Nistler, W. Weirauch, Metrologia {\bf 36}, 147 (1999).

\bibitem{online2}{B. Binder, Geometric Phase Bifurcation; }
\href{http://www.quanics.com/bifurcation137.html}%
{http://www.quanics.com/bifurcation137.html}


\end{thebibliography}
\section*{Appendix}
Some additional algebra regarding coupled and decoupled partner
currents:
\begin{eqnarray}
q^2 = q_{R}^2 + q_{L}^2 \\%
q^2(q_{R}^2 - q_{L}^2) = q_{R}^4 - q_{L}^4 \\%
\alpha q_{L} = \kappa q_{R} \\%
q_{R} =  q\kappa / \sqrt{\kappa ^ 2 + \alpha ^ 2} \\%
q_{L} =  q\alpha / \sqrt{\kappa ^ 2 + \alpha ^ 2} \\%
q_{\mu} = q / \sqrt{\kappa ^ 2 + \alpha ^ 2} \\%
q_{e-} = q_{R} q_{L} / q_{\mu} \\%
q_{p+}^4 = q^2 (q_{L} ^ 2 - q_{R} ^ 2) \gamma^4 \\%
q_{n}^4 =  q_{L}^4 / (1 - \kappa ^ 2 / \alpha ^ 2)\\%
q_{m}^4 =  q_{n}^4 (\kappa ^ 2 / \alpha ^ 2) \\\nonumber\\%
q_{n}^4 = (q_{L} ^ 4 - q_{R} ^ 4) (1 + \kappa ^ 2 / \alpha ^ 2) /
(1 - \kappa^ 4 / \alpha ^ 4) ^ 2 \\%
q_{n}^4 =  (q_{L} ^ 4 - q_{R} ^ 4) / (1 - \kappa ^ 2 / \alpha ^ 2)
/ (1 - \kappa ^ 4 / \alpha ^ 4)  \\%
q_{n}^4 = (q_{L} ^ 4 - q_{R} ^ 4) / (1 - \kappa ^ 2 / \alpha ^ 2)
^ 2 / (1 + \kappa ^ 2 / \alpha ^ 2)  \\%
%q_{n}^4 = (q_{L} ^ 4 - q_{R} ^ 4) / (1 - \kappa ^ 2 / \alpha ^ 2)
%^ 2 (1 + \kappa ^ 2 / \alpha ^ 2)^2  / (1 + \kappa ^ 2 / \alpha ^
%2) ^ 3 \\%
q_{n}^4 = q^4 (q_{L} ^ 4 - q_{R} ^ 4) / (q_{L} ^ 2 - q_{R} ^ 2) ^
2 / (1 + \kappa ^ 2 / \alpha ^ 2) ^ 3
\\%
q_{n}^4 =  q^4(q_{L} ^ 2 + q_{R} ^ 2) / (q_{L} ^ 2 - q_{R} ^ 2)/
(1 + \kappa ^ 2 / \alpha ^ 2) ^ 3  \\%
q_{n}^4 =  q^6 / (q_{L} ^ 2 - q_{R} ^ 2) / (1 + \kappa ^ 2 / \alpha ^ 2) ^ 3  \\%
q_{n}^4 = (q_{L} ^ 4 - q_{R} ^ 4) / (1 - \kappa ^ 4 / \alpha ^
4) ^ 2 (1 + \kappa ^ 2 / \alpha ^ 2) \\%
q_{n}^4 = q^2(q_{L} ^ 2 - q_{R} ^ 2) / (1 - \kappa ^ 4 / \alpha ^
4) ^ 2 (1 + \kappa ^ 2 / \alpha ^ 2) \\%
q_{n}^4 =  q^2(q_{L} ^ 2 - q_{R} ^ 2) / (1 - \kappa ^ 4 / \alpha ^
4) / (1 - \kappa ^ 2 / \alpha ^ 2) \\%
q_{n}^4 = q^4 / (1 - \kappa ^ 4 / \alpha ^ 4) / (1 + \kappa ^ 2 /
\alpha ^ 2) \\%
q_{n}^4 =  q^4 / (1 - \kappa ^ 2 / \alpha ^ 2) / (1 + \kappa ^ 2 /
\alpha ^ 2) ^ 2 %
\,\labeqn{appendix01}
\end{eqnarray}
\clearpage
\end{widetext}
\end{document}
