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baryon, neutron, proton, auto-parametric, resonance, whispering, autonomous, gallery, modes, nonabelian, nonlinear, non-pertubative, supratransmission , supraconductivity, breather, nonpertubative, pseudosphere, phase, berry, Gordon, sine-Gordon, Baecklund, Thirring, Rayleigh, fine structure, iteration, iterative
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%keywords: baryon, neutron, proton, auto-parametric, resonance, whispering, autonomous, gallery, modes, nonabelian, nonlinear, non-pertubative, supratransmission , supraconductivity, breather, nonpertubative, pseudosphere, phase, berry, Gordon, sine-Gordon, Baecklund, Thirring, Rayleigh, fine structure, iteration, iterative
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\begin{document}
\title{Soliton Compton Mass from Auto--Parametric Wave--Soliton Coupling}
\author{B{\sc ernd} ~B{\sc inder}}

\date{20.11.2002, 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}
In this paper a self-excited Rayleigh-type system models the
auto-parametric wave-soliton coupling via phase fluctuations. The
parameter of dissipative terms determine not only the most likely
quantum coupling between solitons and linear waves but also the
most likely mass of the solitons. Phase fluctuations are mediated
by virtual photons coupling at light-velocity in a permanent
Compton scattering process. With a reference to the SI-units and
proper scaling relations in length and velocity, the final result
shows a highly interesting sequence: the likely soliton Compton
mass is about 1.00138 times the neutron and 1.00276 times the
proton mass.
\end{abstract}
\maketitle
%#############################################################

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

For nonlinear field theory models in 1+1--dimensional space--time
the equations of motion admit finite energy and finite width
solutions called solitons \cite{AbS}. Solitary waves were
discovered in the first half of the nineteenth century by Rusell,
the word soliton was invented by Kruskal, the sine--Gordon (SG)
model by Skyrme \cite{Skyrme}. Solitons retain their identity
after collisions, can annihilate with anti--solitons,
many--soliton solutions obey Pauli's exclusion principle. In
1+1--dimensional space--time there are two non--trivial minimal
quantum field theories which describe non--perturbative phenomena:
the SG model and the massive Thirring model \cite{Thirring} (a
self-coupled Dirac field, see the Lagrangians \cite{Remoissenet}),
both are intimately related \cite{Coleman}. It is an interesting
question how stationary solitons (like breather) get their
absolute mass/energy. In this paper a self-excited Rayleigh-type
system \cite{rayleigh} will model an auto-parametric wave-soliton
coupling via phase-fluctuations. This system can as a classical
example model simplified music instruments (in the original work
of Rayleigh a clarinet reed). In \cite{rayleigh07} it has recently
been shown, that Rayleigh-type self--excited auto-parametric
systems \cite{trvn} can stimulate "whispering gallery modes" and
model Coulomb interaction between sine-Gordon solitons. In this
paper exactly the same model will be applied to determine the most
likely Compton mass of the soliton.
\\
%---------------------------------------------------------------------------------------------------

%#############################################################
\paragraph*{Quantum field theory.}
%#############################################################

In quantum field theory, particles correspond to fluctuations of a
space-time function that minimizes the action for some Lagrangian
$L$ which is itself the space integral of a Lagrangian density
$\cal L$. A low-dimensional (bosonic) SG field of a single
hermitian scalar field $\theta$ in 1+1 dimensions with Lagrangian
density ${\cal L}= {\mu\over 2}
\partial_{\nu}\theta\partial^{\nu}\theta~-V(\theta)$ is a function
of one space dimension and time. The time independent field
equations reads $\mu {\partial_r^2 \theta} = {\partial_\theta V }$
which can also be written as
\begin{equation}
V(\theta) =  {\mu \over 2} (\partial_r \theta)^2 \,
\labeqn{potenergy},
\end{equation}
where $\mu$ is the soliton mass. In the quantum model, the
expectation of the square of the phase gradient contributes to the
soliton self-energy, where the phase gradient usually scales
proportional to the wave number that is
proportional to the frequency. \\

%#############################################################
\paragraph*{Rayleigh--type perturbed SG.}
%#############################################################

The coupling between linear waves and non--linear solitons could
be important for the understanding for scale-dependent
electromagnetic coupling. The Rayleigh-type perturbed SG equation
is given by
\begin{equation}
      \(\partial_{rr} - \partial_{tt}\) \theta
    - \sin\theta =   \[b^2 (\partial_r \theta)^2 - a^2 \]\partial_t\theta
\, \labeqn{josephson0V},
\end{equation}
where $\partial_{rr} - \partial_{tt}$ is the D'Alembert wave
operator. On the left side are the usual SG terms, on the right
side are with \eqn{potenergy} the typical Rayleigh wave
dissipation and regeneration terms. These two terms balance wave
dissipation and regeneration and can be found in self--excited
auto-parametric systems \cite{trvn}. The term $b^2 (\partial_r
\theta)^2 - a^2$ controls $\theta$ by weighting the dissipative
first order term $\partial_t\theta$. The amplitude of
the coupling wave can be assigned to a phase difference between two solitons. \\

%#############################################################
\paragraph*{Rayleigh--type auto--parametric system.}
%#############################################################

Following \cite{trvn} we characterize auto--parametric systems as
a vibrating system which consists of at least two subsystems: an
oscillator that is generally in a vibrating state and the excited
system which is excited indirectly and coupled to the oscillator
in a nonlinear way. A linear wave will be defined as a disturbance
which transports energy from one location to another location
without transporting matter evolving at light speed $c$ with $c
\,\partial_{r} \theta =
\partial_{t}\theta$. In the auto--parametric system the
dissipation and excitement parameter will depend on the phase
statistics. The Rayleigh--type system
\begin{equation}
    \partial_{tt}\theta + \theta = a^2 \partial_{t} \theta -
    b^2 c^{-2} (\partial_{t}\theta)^3
\, \labeqn{josephson2}
\end{equation} is called self--excited auto--parametric, and can be
exactly treated by phase averaging methods providing for the
semi--trivial solution \cite{trvn}. The excitation is here also
balanced by the (soliton vortex) regeneration term $V(\theta)$ of
\eqn{potenergy} with parameter $b^2$ and (linear mode) damping
term $\partial_{t} \theta$ with parameter $a^2$. So $a$, $b$, and
$c$ control the amplitude $\theta$ of intermediating linear waves.\\

%#############################################################
\paragraph*{The coupling mechanism.}
%#############################################################

The vertex solitons are assumed to be identical and not
synchronized in phase, such that the excitement of coupling modes
is based on the (statistical) phase difference. Let the SG
solitons be coupled such, that the amplitude of the perturbed
linear wave will be the phase difference $\theta$ of the perturbed
nonlinear SG solitons controlled by the potential of the phase
gradient $V(\theta)$. Usually the non-trivial equilibrium
fluctuation amplitude $\theta_0$ can be found with given parameter
$a$ and $b$ or ratio $q=b/a$, but in our case the scattering phase
fluctuation amplitude $\theta_0$ is given by the relative phase
fluctuation range of solitons. Regarding the kink-antikink
solution there exists a characteristic time scale or oscillation
frequency that can be related via coupling wave velocity to a
characteristic length scale or wavelength $\lambda_{\mu}$. The
fermion--like behavior pointed out by Skyrme \cite{Skyrme}
suggests to define an additive phase--fluctuation range by $2
\times 2\pi = 4\pi$. Therefore, the coupling amplitude for the
mediating wave will be given by
\begin{equation}
\theta_0 = 4 \pi \lambda_{\mu} \, \labeqn{rayleigh2}.
\end{equation}
Later $\lambda_{\mu}$ will be reintroduced as the Compton
wavelength of wave--soliton coupling.
\\

%#############################################################
\paragraph*{Phase averaging.}
%#############################################################

Statistically, there are phase oscillations or fluctuations
randomly modulated in a special phase interval, so we can assume
in the most simple case a maximum entropy phase modulation and
fluctuation between two vertex solitons and find the most likely
energy flow and coupling ratio from phase averaged solutions
\cite{Sanders85}. The phase--amplitude modulation with random
$\tau$ as a solution to \eqn{josephson2} can be written as
\cite{trvn}
\begin{equation}
      \theta = \theta_0 \cos(\tau + \psi_0)
\, \labeqn{josephson3},
\end{equation}
averaging over $\tau$ yields \cite{Sanders85,trvn} with constant
$\psi_0$
\begin{eqnarray}
      \partial_{t}\theta_0 \propto {1 \over 2} \(a^2 \theta_0
       - {3 b^2 \over 4 \lambda_{\mu}^2} \theta_0^3\)
      , \quad \partial_{t} \psi_0=0
\, \labeqn{josephson4}.
\end{eqnarray}

%#############################################################
\paragraph*{1-dimensional coupling soliton energy.}
%#############################################################

With constant fluctuation strength $\partial_{t}\theta_0 = 0$, the
averaged equations provide with \eqn{rayleigh2} and
\eqn{josephson4} for a balanced non-trivial equilibrium
\begin{equation}
{q } ={b \over a} = {1 \over {4 \pi} \sqrt{3\over 4} }, \quad { 2
\overline{V} \over E_{1d}} = \overline{(\partial_r \theta)^2} =
q^{-2} \, \labeqn{josephson5}.
\end{equation}
Consequently, the 1-dimensional coupling strength $q$ will enter
the energy definitions via mean unit energy $E_{1d}$ of
1-dimensional coupling
\begin{equation}
E_{1d} = E_{\mu} = q^2\overline{(\partial_r \theta)^2}\,=1\mu
c^2\, \labeqn{josephson6}.
\end{equation}
To compare our theoretical soliton coupling model to real existing
couplings, mass/energy has to be quantified. Let the unit mass be
the 1d coupling mass--energy of a soliton with $E_{\mu}=1\mu c^2$.
The mutual 1-d coupling to a photon with amplitude/wavelenght
fluctuation $\lambda_{\mu}$ can be regarded as a permanent Compton
scattering process with mass--energy value related to
$\lambda_{\mu}$ via Compton relation
\begin{equation}
{E_{\mu} } =  2 q^2 \overline{V } = {h c \over \lambda_{\mu}} \,
\labeqn{couple1}.
\end{equation}
%-----------------------------------------------------------------------------------

%#############################################################
\paragraph*{Energy scaling.}
%#############################################################

According to \eqn{josephson5} and \eqn{josephson6}, both $E_{\mu}$
and $\overline{V}$ scale with $c^2$. $E_{\mu}$ can be assigned to
the soliton energy and mass. The absolute energy reference is
based on the human artificial energy unit $E_u$ referring to the
kinetic energy of a unit mass 1$\mu$ moving at unit velocity $u$
(in SI 1J = 1kg m$^2$/s$^2$). We will now find the scaling
relations of $E_{\mu}$ and $\overline{V}$ with respect to the
energy unit $E_u$ provided by the system of units (SI) on one end
and by Planck units ($\hbar=c=\mu=1$) on the other end of the
scale. $\lambda_u=1m$ will be the human artificial reference
wavelength. Planck units demand that
\begin{itemize}
    \item the light velocity equals the unit velocity $c=u=1$,
    \item $2\overline{V(c=1)} = E_u(u=1)$,
    \item the the Compton length equals the unit length $\lambda_{\mu} = \lambda_u = 1$,
    \item $E_{\mu}(\lambda_{\mu}= 1) = E_u(\lambda_u = 1)$.
\end{itemize}
The algebraic form of the two scaling relations is given by
\begin{itemize}
    \item the mean soliton coupling energy $\overline{V}$ scales with the square of the wave velocity
    \begin{equation}
{2\overline{V} \over  E_{u}}= {c^2 \over u^2 }=  \Xi^2, \quad \Xi
= 299792458
 \labeqn{energy5},
\end{equation}
    \item the 1-dimensional quantum energy of waves coupling to
    particles $E_{\mu}$ is inversely proportional to the wavelength, especially to the Compton wavelength
    \begin{equation}
{E_{\mu} \over E_{u}} = {\lambda_{u} \over \lambda_{\mu}}
\labeqn{energy3}.
\end{equation}
\end{itemize}
Practical necessity motivates to choose a unit velocity $0<u \ll
c$ with $\Xi \gg 1$ \cite{FUNDDATA}.\\

%#############################################################
\paragraph*{Results.}
%#############################################################

The main result could be written as a characteristic soliton
wavelength of one-dimension coupling exactly given with
\eqn{josephson5} - \eqn{energy3} by
\begin{equation}
\lambda_{\mu} = { \lambda_{u}  \over q^2\Xi^2 }
 \labeqn{energy7},
\end{equation}
and provides for the soliton mass $\mu$ via Compton relation $ \mu
= h / (c \lambda_{\mu})= q^2 \Xi^2 h / (c \lambda_{u})$. Realized
in SI units the values are
\begin{eqnarray}
\mu = {\hbar \over c}{\Xi^2 \over 6\pi \textup{m}} \approx
1.67724... \cdot 10^{-27}\textup{kg},\nonumber \\  \lambda_{\mu} =
{12 \pi^2 \textup{m} \over \Xi^2} \approx 1,31777... \cdot
10^{-15}\textup{m} \labeqn{xuf4b}.
\end{eqnarray}
The corresponding SI-values for the wave number $k_{\mu}$, angular
frequency $\omega_{\mu}$, and energy $E_{\mu}$ are given by
\begin{eqnarray}
k_{\mu}=2\pi/\lambda_{\mu}=4.7680443... \cdot 10^{15} \textup{rad m$^{-1}$} ,\nonumber \\
\omega_{\mu} = k_{\mu} c = 1.4294237... \cdot 10^{24}\textup{rad s$^{-1}$},\nonumber \\
E_{\mu} = \mu c^2= \hbar c k_{\mu}= 940.86369... \textup{MeV}.
\labeqn{E0}
\end{eqnarray}
It turns out, that $\mu$ shows a highly interesting and
significant sequence with the fundamental masses of the two
prominent baryons (Neutron with mass $M_{n}$ and proton with mass
$M_{p+}$) \cite{FUNDDATA}:
\begin{eqnarray}
{\frac{{\mu}}{{M_{n}}}} \approx 1.001382...,\nonumber \\
{\frac{{\mu}}{{M_{p+}}}} \approx 1.002762... \labeqn{xp}.
\end{eqnarray}
The system-invariant soliton mass scale is 1.001382 times the
neutron and 1.002762 times the proton scale and could be
interpreted as a stability limit. The neutron is energetically
lower than $\mu$, this could support semi--stability. In this
context it is interesting to note, that two neutrons provide for
approximately the same mass gain as one proton with
respect to $\mu$. This mass differences could be relevant for
approaching (nuclear) binding energies with many--soliton models. \\

%#############################################################
\paragraph*{Can fundamental baryons be solutions to the SG?}
%#############################################################

\eqn{potenergy} obtained from the SG--Lagrangian is the central
control term of auto-parametric resonance and has also a central
role in \cite{sepp,phasetospace,Mapping}, where generalized
fine-structure constants coupling between topological phase fields
have been proposed. Solitons keep their energy in a permanent
scattering process, retain their identity after collisions, can
annihilate with anti--solitons, many--soliton solutions obey
Pauli's exclusion principle. As pointed out by Skyrme
\cite{Skyrme} this can be interpreted as a fermion--like behavior.
The interest in such low-dimensional SG models arises from their
integrability, duality properties, non-perturbative aspects, the
electric-magnetic duality in gauge theories, and as an universal
concept in nonlinear science. Solitons appear in almost all
branches of physics, such as hydrodynamics, condensed matter
phenomena, particle physics, plasma physics, nonlinear optics, low
temperature physics, nuclear physics, biophysics and astrophysics.
Coleman's "Quantum SG equation as the massive Thirring model"
\cite{Coleman} establish an identity between the quantized SG
model and the Dirac spin-1/2 equation in 1+1 dimensions. But
probably because of the low dimensionality and the wide acceptance
of the standard model (including quark model) the interest was not
considerably extended to the fundamental particle field. The
central role approaching the baryon mass scale is the shift of
human artificial units to Planck units (at the Fermi scale!), this
extrapolates the soliton energy by a factor $\Xi^2$. The
deviations in \eqn{xp} of course suggest to play with the
self-coupling Dirac equation and massive Thirring model
\cite{Thirring} and include other particles as dual or partner
soliton states. This could eventually fill the gap and build the
bridge to a 4-dimensional integrable quantum field theory.\\

%#############################################################
\paragraph*{Relevance to the previous work.}
%#############################################################

Extremely interesting is the topological charge $q$ obtained from
the auto--parametric model regarding topological phase fields and
electromagnetic coupling. Via Gauss relation the 1-dimensional
coupling parameter $q$ can also be related to a 3-d coupling with
spherical ``whispering Gallery modes'' (WGM) coupling with a 3-d
potential $\phi_{3d} = {1 \over 137 r}$. Why integral? When the
round-trip path fits integer numbers of the wavelength
(single-valuedness), WGM are formed. The corresponding quantum
number of orbital degeneracy is given by $M = \[ 4\pi \over q\] =
137$ where $[\;]$ means next higher integral value. In
\cite{sepp,phasetospace,Mapping} generalized fine-structure
constants based on a SG type pseudospherical coupling of
topological phase fields have been defined. $M=137$ with $q=e$
corresponds to the Sommerfeld fine structure constant $\alpha =
e^2/(4\pi \epsilon_0\hbar c) $ according to the Dirac theory of
magnetic monopoles \cite{Dirac-31} (a generater of the topological
Berry phase \cite{berry}).\\

\begin{thebibliography}{aaa}

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Scattering Transform}, SIAM, Philadelphia, (1981).

\bibitem{Skyrme} T. H. R. Skyrme, Proc. Roy. Soc. A {\bf 247}, 260 (1958); {\it
ibid.} A {\bf 262}, 237 (1961).

\bibitem{Thirring} W. Thirring, Ann. Phys. (N.Y.) {\bf 3}, 91 (1958).

\bibitem{Remoissenet} M. Remoissenet, in {\it Waves Called Solitons, Concepts and
Experiments}, Springer--Verlag, Heidelberg, 1994.

\bibitem{Coleman} S. Coleman, Phys. Rev. D {\bf 11}, 2088 (1975).

\bibitem{rayleigh} L. Rayleigh, Philos. Mag. {\bf 27}, 100 (1914).

\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{trvn}  A. Tondl, T. Ruijgrok, F. Verhulst, R. Nabergoj, {\it Autoparametric Resonance in Mechanical
Systems} Cambridge University Press, New York, (2000).

\bibitem{Sanders85} J. A. Sanders, F. Verhulst, "Averaging Methods in Nonlinear Dynamical Systems", Appl. Math. Sci. {\bf 59}, Springer--Verlag, New
York (1985).

\bibitem{FUNDDATA} Groom et al., "The Review of Particle Physics", The European Physical Journal {\bf C15},1
(2000).

\bibitem{sepp} B. Binder,{\it Josephson Effect, B\"acklund Transformations, and Fine Structure Coupling};
\href{http://philsci-archive.pitt.edu/documents/disk0/00/00/08/61/index.html}
{PITT-PHIL-SCI00000861}.

\bibitem{phasetospace} B. Binder,{\it Topological Phase Fields, Baecklund Transformations, and Fine Structure};
\href{http://philsci-archive.pitt.edu/documents/disk0/00/00/08/41/index.html}
{PITT-PHIL-SCI00000841}.

\bibitem{Mapping} B. Binder,{\it Charge as the Stereographic Projection of Geometric Precession on
Pseudospheres};\\
\href{http://philsci-archive.pitt.edu/documents/disk0/00/00/08/18/index.html}
{PITT-PHIL-SCI00000818}.

\bibitem{Dirac-31} P. A. M.  Dirac, Proc. Roy. Soc. London A {\bf
133}, 60 (1931).
\bibitem{berry} M. V. Berry, Proc. Roy. Soc. Lond. A {\bf 392},
45 (1984).


\end{thebibliography}

\end{document}
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