\documentclass{JHEP3}
\usepackage{amsmath}
\usepackage{amssymb}
\title{Self-consistent model of fermions}
\author{Vladimir Yershov \\
University College London, Mullard Space Science Laboratory \\ 
Holmbury St.Mary, Dorking RH5 6NT, United Kingdom \\ 
E-mail: \email{vny@mssl.ucl.ac.uk}} 
\abstract{
A composite model of fermions based on three-coloured preons
is discussed. It is shown that the chromoelectric interaction
between preons leads to formation of complex structures 
reproducing three generations of quarks and leptons with all
their quantum numbers and masses. The model is self-consistent
(it makes no use of input parameters). Masses of the generated structures
match the experimental values for the fundamental fermions.
}
%\keywords{bsm, pmo, gls}
\keywords{beyond Standard Model, phenomenological model, global symmetries}
\preprint{JHEP/036A/0702, 
\dedicated{ Couples are being joined together  \ \ \ \ \ \ \\
But they would like to be separated... \\
{\small Vladimir Vysotsky} \\
{\small \rm (on the gluons)}}

\begin{document}
\newcommand{\abs}[1]{\lvert#1\rvert}

\section{Introduction}
The Standard Model of particle physics does
not explain the hierarchical pattern observed in the masses
of quarks and leptons. This theory just uses them as a set of
its input parameters and cannot give answer to the question,
why the masses are distributed in the way they are. At a first
glance they seem to be random, except for the fact that the
masses generally increase with generation number.


%\vspace{0.2cm}
Numerous attempts to solve this puzzle have been made over
the last twenty years, but the problem is still there. The
observed pattern of the particle masses insistently points 
to structures beyond the quark scale. A number of models
for these structures have been proposed focusing basically 
on the ideas of unification \cite{georgi}, technicolor 
\cite{weinberg}, supersymmetric unification \cite{dimopoulos},
or strings and branes \cite{green}. The latter approach gives
some encouraging results, such as distributions resembling
the observed families of particles, but no more. 

%\vspace{0.2cm}
There are models based on hypothetical compositeness of the 
fundamental particles \cite{luty}, but they are 
also far from reality. Finally, there are theories declaring the
randomness of these masses \cite{donoghue}. This can easily be
disproved by building one case where the masses are shown to be
functionally dependent. For example, a model proposed by 
G.R.Filewood \cite{filewood}, which is based on the geometrical
approach to the structures of particles, predicts masses of 
some fermions to a fairly good accuracy. Thus, it demonstrates 
that at least some of the  particle masses have a common origin.
This case shows that success in finding fundamental building
blocks of nature can be achieved beyond the road marked by the
standard theory. Here we propose a different geometrical composite 
model based on primitive particles (called preons), which we 
believe is closer to reality. At least it correctly reproduces
masses of all three generations of fermions.    

%\vspace{0.2cm}
\section{Simple structures}
We consider a preon ($\Pi$) having three degrees of freedom,
$\Pi, \overline{\Pi} \in \bf{3}_c$ and represent it 
with a triplet of three-component column vectors 
$\mathbf{\Pi}_i=\{\Pi_j\}_i$
 ($i,j=0,1,2$, $i$ stands for the colour: $red$, $green$, or $blue$).
Positive charges (antipreons) are represented with 
$\mathbf{\overline{\Pi}}_i=- \mathbf{\Pi}_i$.
The components of these vectors are
\begin{equation*}
\Pi_{ij}=
 \begin{cases}
 +1, & i=j \\
 -1, & i\neq j 
 \end{cases}
\text{\hspace{0.6cm}} .
%\label{eq:preonmatrix}
\end{equation*}
We shall assign unit values for the preon's charge
\begin{equation*}
Z(\mathbf{\Pi}_i)=\sum_{j=0}^2{\Pi_{ij}}
\end{equation*} 
and mass
\begin{equation*}
m(\mathbf{\Pi}_i)=\abs{\sum_{j=0}^2{\Pi_{ij}}}.
\end{equation*} 
The charge of a system composed of
various preons or preon groups will be
\begin{equation}
Z=\sum_{k=1}^N{\sum_{j=0}^2{\sum_{i=1}^n{\Pi_{ij}^k}}},
\label{eq:charge}
\end{equation} 
where  $N$ is the number of preon groups and $n$ is the number of preons in
the given group. We can define the mass of the system, $m$, and its  
reciprocal mass, $m'$, as follows:
\begin{equation}
m= (1-\delta_{Z,0})
\sum_{k=1}^N\abs{{\sum_{j=0}^2{\sum_{i=1}^n{\Pi_{ij}^k}}}}, 
\hspace{0.3cm}
m'= (1-\delta_{Z,0})
\sum_{k=1}^N\abs{{\sum_{j=0}^2({\sum_{i=1}^n{\Pi_{ij}^k})^{-1}}}},
 \label{eq:mass}
\end{equation} 
where $\delta_{Z,0}$ is the Kronecker delta-function.
Taking into account the well-known properties of the coloured
interaction (two like-charged but unlike-coloured particles are
attracted, otherwise they repel), we can characterise 
the chromoelectric interaction between two preons, $i$ and $k$,
by the following force $F_{ik}$ (normalised to unity; $i$ and $k$
stand for the colour indices 0,1,2 or $red$, $gree$, and $blue$):
\begin{equation}
F_{ik}=\pm \frac{\mathbf{\Pi}_i \cdot \mathbf{\Pi}_k}{\abs{\mathbf{\Pi}_i
\cdot  \mathbf{\Pi}_k}}.
\label{eq:seforce}
\end{equation}
Positive and negative signs correspond respectively to the strong and 
electric manifestations of the chromoelectric interaction.
This force necessarily leads to formation of stable spatial preon
structures. The simplest preon structure is a charged  doublet
\begin{equation*}
\varrho^\pm_{ik}= \mathbf{\Pi}_i+\mathbf{\Pi}_k, \hspace{0.6cm} i,k=0,1,2
%\label{eq:doublon}
\end{equation*}
(six possible combinations for $\mathbf{\Pi}$ and six others for
$\overline{\mathbf{\Pi}}$). A neutral preon doublet  
\begin{equation*}
g^0_{ik}= \mathbf{\Pi}_i+\overline{\mathbf{\Pi}}_k, \hspace{0.6cm} i,k=0,1,2
%\label{eq:gluon}
\end{equation*}
(nine combinations) can also be formed.
The preon doublets will be deficient in one or two colours.
According to (\ref{eq:charge}) and 
(\ref{eq:mass}), 
\begin{equation*}
Z(\mathbf{\Pi}_i,\mathbf{\Pi}_k)=\pm 2, \hspace{0.4cm}
m(\mathbf{\Pi}_i,\mathbf{\Pi}_k)=2, \hspace{0.4cm}
m'(\mathbf{\Pi}_i,\mathbf{\Pi}_k)=\infty,
\end{equation*}
and 
\begin{equation*}
Z(\mathbf{\Pi}_i,\overline{\mathbf{\Pi}}_k)=0, \hspace{0.4cm}
m(\mathbf{\Pi}_i,\overline{\mathbf{\Pi}}_k)=0, \hspace{0.4cm}
m'(\mathbf{\Pi}_i,\overline{\mathbf{\Pi}}_k)=\infty.
\end{equation*}
If an additional charged preon is added to the neutral
doublet, the mass and the charge of the system are restored: 
\begin{equation}
Z(\mathbf{\Pi}_i,\overline{\mathbf{\Pi}}_k,\mathbf{\Pi}_l)=\pm 1, 
\hspace{0.3cm}
m(\mathbf{\Pi}_i,\overline{\mathbf{\Pi}}_k,\mathbf{\Pi}_l)=1,
\hspace{0.3cm}
\label{eq:mgluon}
\end{equation}
but still 
\begin{equation}
m'(\mathbf{\Pi}_i,\overline{\mathbf{\Pi}}_k,\mathbf{\Pi}_l)=\infty.
\label{eq:mprimgluon}
\end{equation}

%\vspace{0.2cm}
The charged doublets $\varrho$ ($2\Pi$ and  $2\overline{\Pi}$) 
will not be free for long because their strong potentials are 
infinite at infinite distance. Any distant preon of the same charge
but with a complementary colour will be attracted to the doublet.
In this way, triplets 
\begin{equation*}
y=\sum_{i=0}^2{\mathbf{\Pi}_i} \text{\hspace{0.4cm} or \hspace{0.4cm}}
\overline{y}=\sum_{i=0}^2{\overline{\mathbf{\Pi}}_i},
\text{\hspace{0.6cm}} i=0,1,2
%\label{eq:triplets}
\end{equation*}
will be formed with their charges
$Z_y=\pm 3$, and masses $m_y=m'_y=3$. 
Its colour will be complete (colourless state), but locally 
the $red$, $green$, and $blue$-colour charges of its three preon
components will be distributed in a plane forming 
a closed loop. Thus, a part of its preons' strong field will be closed
in this plane, with another part extended to infinity.

%\vspace{0.2cm}
Due to the force (\ref{eq:seforce}), pairs of $y$-particles will form 
secondary structures turning through $180^\circ$ or $\pm 120^\circ$
with respect to one another.
In order to formalise representation of these structures let
us introduce the following group of matrices:
\begin{equation*}
\alpha_0=
\begin{pmatrix} 1 & 0 & 0  \\ 
                0 & 1 & 0  \\
                0 & 0 & 1   
 \end{pmatrix}
 \text{,\hspace{0.5cm}}
\overline{\alpha}_0=-\frac{1}{2}   
\begin{pmatrix} 0 & 1 & 1  \\
                1 & 0 & 1  \\
                1 & 1 & 0 
\end{pmatrix},
%\label{eq:alpha0}
\end{equation*}
\begin{equation*}
\alpha_1=
\begin{pmatrix} 0 & 1 & 0  \\ 
                0 & 0 & 1  \\
                1 & 0 & 0   
 \end{pmatrix}
 \text{,\hspace{0.5cm}}
\overline{\alpha}_1=-\frac{1}{2}   
\begin{pmatrix} 1 & 0 & 1  \\
                1 & 1 & 0  \\
                0 & 1 & 1 
\end{pmatrix}, 
%\label{eq:alpha1}
\end{equation*}
\begin{equation*}
\alpha_2=
\begin{pmatrix} 0 & 0 & 1  \\ 
                1 & 0 & 0  \\
                0 & 1 & 0   
 \end{pmatrix}
 \text{,\hspace{0.5cm}}
\overline{\alpha}_2=-\frac{1}{2}   
\begin{pmatrix} 1 & 1 & 0  \\
                0 & 1 & 1  \\
                1 & 0 & 1 
\end{pmatrix}. 
%\label{eq:alpha2}
\end{equation*}
%\vspace{0.2cm}
Then a charged structure $\delta^\pm$ with two coupled $y$-particles,
mutually orientated through $180^\circ$-rotation, 
can be written as
\begin{equation*}
\delta^\pm = \alpha_i y + \overline{\alpha}_i y. 
%\label{eq:delta}
\end{equation*}
with $Z_\delta=\pm 6$,  $m_\delta=m'_\delta=6$.
States corresponding to the
different $i$ are equivalent. 
Unlike-charged $y$-particles will couple 
through their $\pm 120^\circ$ relative rotations:
\begin{equation}
\gamma^0 = \alpha_i y + \alpha_k \overline{y} 
\label{eq:gamma}
\end{equation}
with $Z_\gamma=0$, $m_\gamma=m'_\gamma=0$, $i \neq k$,
$\gamma^0 \equiv 3(g\overline{g})$.
Two states corresponding to the clockwise ($\uparrow$)
and anticlockwise ($\downarrow$) turns  in (\ref{eq:gamma})
are topologically different.

%\vspace{0.2cm}
The three-colour completeness of $y$ means that 
up to three of them can combine if they are like-charged.
Necessarily, they  combine into closed-loop chains (triplets).
There are eight possible states of the charged triplet, four of which 
correspond to the negatively charged $y$:
\begin{equation*}
e_{R\uparrow} = \sum_{i=0,1,2}{\alpha_i y}, \text{\hspace{0.8cm}} 
e_{R\downarrow} = \sum_{i=0,2,1}{\alpha_i y}, 
%\label{eq:electronR}
\end{equation*}
\begin{equation*}
e_{L\uparrow} = \sum_{i=0,1,2}{\overline{\alpha}_i y}, \text{\hspace{0.8cm}} 
e_{L\downarrow}= \sum_{i=0,2,1}{\overline{\alpha}_i y}, 
%\label{eq:electronL}
\end{equation*}
and four others correspond to $\overline{y}$.
The structure will have a charge $Z_e=\pm 9$ and 
a mass $m_e=m'_e=9$ (in the units of the preon's charge and mass).
$R$ and $L$ states correspond to two opposite spatial orientation 
of $y$ in the loop.  
The $R$-state can be obtained from $L$ by mirror-reflection 
of all $y$-components about the loop's circular axis. 


%\vspace{0.2cm}
Longer chains will be formed of a mixture of the like- and unlike-charged
$y$-particles:
\begin{equation*}
\nu_{e\uparrow}= 
\begin{pmatrix}
 \alpha_0 & \alpha_1 & \overline{\alpha}_1 & \overline{\alpha}_2  \\
 \alpha_2 & \alpha_0 & \overline{\alpha}_0 & \overline{\alpha}_1  \\
 \alpha_1 & \alpha_2 & \overline{\alpha}_2 & \overline{\alpha}_0  
\end{pmatrix}
\begin{pmatrix}
 y \\
 \overline{y} \\
 \overline{y} \\
 y
\end{pmatrix}   
%\label{eq:neutrino}
\end{equation*}
or
\begin{equation*}
\nu_{e\downarrow}= 
\begin{pmatrix}
 \alpha_0 & \alpha_2 & \overline{\alpha}_2 & \overline{\alpha}_1  \\
 \alpha_1 & \alpha_0 & \overline{\alpha}_0 & \overline{\alpha}_2  \\
 \alpha_2 & \alpha_1 & \overline{\alpha}_1 & \overline{\alpha}_0  
\end{pmatrix}
\begin{pmatrix}
 y \\
 \overline{y} \\
 \overline{y} \\
 y
\end{pmatrix}.   
%\label{eq:neutrinop}
\end{equation*}
These ring-closed chains containing twelve $y$-particles  (36 preons) 
are neutral and massless, according to (\ref{eq:charge}) and
(\ref{eq:mass}), unless coupled to a charged particle,
say $y$ or $3y$, which restores its mass.
Unlike \ $e$, the left and
right-handed states of \  $\nu_e$ are equivalent
 ($\nu_{eR} \equiv \nu_{eL}$) because of specific rotational symmetry of  
the $\nu_e$-loop: the number of its negatively charged $y$
coincides with the number of the positively charged $\overline{y}$,
 and the mirror-reflection of all these components about the 
loop's circular axis translates $\nu_e$ into itself.   

\section{Combining $y$, $e$, and $\nu_e$}

Unlike $y$, with its partially (in one plane) closed 
strong field, the strong fields of $3y$ ($e$) and $12y$ 
($\nu_e$) are closed all over the sphere. Thus, these particles
can be found in free states, and their behaviour depends only on
their electric potentials. 
The particles $y$, $e$, and $\nu_e$ can combine because they possess 
their local patterns of colours. The structure $y^*=y+\nu_e$  will be massive
and charged, with its charge $Z_{y*}=\pm 3$, corresponding to the
charge of a single $y$-particle,  and its mass $m_{y*}=m'_{y*}=39$
preon mass units ($N_\nu+m_y=36+3$, $N_\nu$ is the number of preons
in $\nu_e$). The combination $e+\nu_e$ will have 
the charge of $e$ ($Z_{\nu e}=\pm 9$) and a mass of 45 mass units
 ($m_{\nu e}=m'_{\nu e}=N_\nu+m_e=36+9$).

%\vspace{0.2cm}
Similarly to $y$, the $y^*$-particle cannot be free 
because of its strong potential increasing with distance. 
It will combine further with other $y^*$-particles (via an intermediate 
$\nu_e$ of a complementary colour pattern) forming
 $y^*_\uparrow \nu_{e\downarrow}  y^*_\uparrow $-chains
and so on. A single $y^*\nu_e y^*$-chain (let's call it {\it up}-quark),
has the charge of two $y^*$-particles ($Z_u=\pm 6$) and
a mass $m_u=78$ mass units ($m_u=m'_u=2m_{y*}=2\times 39$).
 The positively charged $u^+$ can combine with 
the negatively charged $e^-\nu_e$, forming a 123-mass units
{\it down}-quark ($m_d=m'_d=m_u+m_{\nu e}=78+45$) with its
charge  $Z_d=Z_u+Z_e=+6-9=-3$
 charge units.

 %\vspace{0.2cm}
It is natural to suppose that particles of higher (heavier)
generations should be composed of simpler structures belonging to 
lower (lighter) generations.
For example, the muon neutrino  (a neutral particle) can be formed
of positively and negatively charged $y^*$ and
$\overline{y}^*$:
\begin{equation}
\nu_\mu=y\nu_{e\uparrow} \nu_{e\downarrow} \nu_{e\uparrow}\overline{y}
=y^* \nu_e \overline{y}^* , 
 \label{eq:numu}
\end{equation}
and the muon can be naturally structured as
\begin{equation}
\mu= (\nu_{e\downarrow} e^-_\downarrow \overline{y})
(\nu_{e\uparrow} \nu_{e\downarrow}  \nu_{e\uparrow}y)
=\overline{\nu}_e e^-\nu_\mu ,
\label{eq:muon}
\end{equation}
which corresponds to one of the muon's decay modes. 
Structures of other particles can be also derived from their 
known decay modes. Components of these structures should not be
considered as rigidly tied. Higher level structures are clusters,
or many-body systems, rather than single particles. 
In (\ref{eq:muon}) we enclose the clustered
components in parenthesis. Interactions between components should
modify the energy state (mass) of the system. We suppose that the
force acting on the components 
\begin{equation*}
F=-G\frac{M M'}{r r'} 
%\label{eq:force}
\end{equation*}
depends both on the sum $M$ of their masses $m_k$:
\begin{equation*}
M=\sum_k{m_k} , 
%\label{eq:summass}
\end{equation*}
and on their reduced mass $M'$:
\begin{equation*}
\frac{1}{M'}=\sum_k{\frac{1}{m'_k}} ,
%\label{eq:sumrecipr}
\end{equation*}
which corresponds to the central potential of the classical 
two-body problem with its reduced mass $M'$. Distancies $r$ and $r'$
are reciprocal to each other ($r'=1/r$).

%\vspace{0.2cm}
We also suppose that, according to the Newton's second law, the total
mass of the composite system, $m$, is proportional to this force:
$m\propto \abs{Fa^{-1}}$. 
 Using the unit values for acceleration ($a=1$), the universal gravitational
 constant ($G=1$) and, taking into account the unity of the product of two
reciprocal distances, $r r'\equiv 1$, one can estimate masses of the
 composite particles by
\begin{equation}
m =M M' =(m_1, m_2, \ldots , m_N)=\frac{m_1+m_2+\dots+m_N}
{1/m'_1+1/m'_2+\dots+1/m'_N} .
\label{eq:mtotal}
\end{equation}
For example, for the muon, $m_1=m'_1=48$, $m_2=m'_2=39$,  
its mass can be calculated as
\begin{equation*}
m_\mu  = (m_1, m_2)=\frac{m_1+m_2}{\dfrac{1}{m'_1}+\dfrac{1}{m'_2}}=
 \frac{48+39}{\dfrac{1}{48}+\dfrac{1}{39}}=1872.0023
 \text{\hspace{0.2cm} (preon mass units)}.
\end{equation*}
For the $\tau$-lepton $m_1=m'_1=156$, $m_2=m'_2=201$, and  
\begin{equation*}
m_\tau  = (156, 201)= 31356.232 \text{\hspace{0.2cm} (preon mass units)}.
\end{equation*}

For the proton, positively charged particle consisting of two $up$,
one $down$ quarks and a cloud of gluons $g$, masses of its components 
are $m_u=m'_u=78$, $m_d=m'_d=123$. As for the gluons, only those of them
should be taken into account, which are coupled to the quarks' preons.
The total number of the coupled gluons corresponding to the proton's
structure is $N_g=2N_u+N_d=2\times 78+123=279$.
Masses of these gluons, according to (\ref{eq:mgluon}) and
 (\ref{eq:mprimgluon}), are $m_g=1$, $m'_g=\infty$.
The resulting proton mass is 

\begin{equation}
m_p  = \frac{2m_u+m_d+N_gm_g}{2\frac{1}{m'_u}+
\frac{1}{m'_d}+N_g\frac{1}{m'_g}}=
16523.052 {\hspace{0.2cm} {\rm (preon \hspace{0.2cm} mass \hspace{0.2cm} units)}.}
\label{eq:mproton}
\end{equation}

This expression gives the proton-to-electron mass ratio, 
which is consistent with the very well-known experimental result:
\begin{equation*}
\frac{m_p}{m_e}=\frac{16523}{9}\approx 1836.
\end{equation*}

%\vspace{0.2cm}
Masses of the composite particles 
calculated with the use of (\ref{eq:mtotal}) and converted into
the $m_p$-units  with the use of (\ref{eq:mproton}) are summarised
in Table \ref{t:massresult}, where for comparison the experimental
masses \cite{properties} 
are listed in the last column (expressed also in units of the proton mass).
\begin{table}[htb]
%\TABLE[htb]{
\caption{Predicted and experimental rest masses of quarks and leptons}
\label{t:massresult}
\begin{center}
\small
\begin{tabular}
{|cc|c|c|c|c|} \hline
\multicolumn{2}{|c|}{Particle and} & {Number of } & Predicted & Mass & Experimental \\
\multicolumn{2}{|c|}{its structure} & {preons in each} & total mass &  converted & mass \cite{properties} \\
\multicolumn{2}{|c|}{} &  { component } & (preon units) & into $m_p$ & ($m_p$) \\ \hline
\multicolumn{6}{|c|} {First generation} \\ \hline
$e^-$ & $3y$ & 9 & 9 & 0.0005446 & 0.0005446 \\ 
$\nu_e$ & $12y$ & 36 & 0 & 0 & - \\
$u$ & $ y^* +  y^* $ & 78  & 78 & 0.00472 & 0.001 to 0.005 \\
$d$ & \ \ $u + \nu_e e^-$ & 123 & 123 & 0.00744 & 0.003 to 0.009 \\
\hline
\multicolumn{6}{|c|}{Second generation} \\
\hline
$\mu^-$ & $\nu_e e^- \nu_\mu$ & (48, 39) & 1872.6778 & 0.1133 & 0.1126 \\
$\nu_\mu $ & $y^* \nu_e \overline{y}^*$ & 114  & 0 & 0 & -  \\
$c$ & $ \stackrel{**}{y} + \stackrel{**}{y} $ & (165, 165) & 27225 & 1.648 & 1.2 to 1.4  \\
$s$ & \ \ $c+e^-$ & (165, 165, 9) & 2750.9 & 0.1665 & 0.08 to 0.18 \\
\hline
\multicolumn{6}{|c|}{Third generation} \\
\hline
$\tau^-$ & $\nu_\mu \mu^- \nu_\tau $ & (156, 201) & 31356.222 & 1.8977
 & 1.8939 \\
$\nu_\tau$ & $u \nu_e \overline{u}$ & 192  & 0 & 0 & - \\
$t$ & $ \stackrel{***}{y} +  \stackrel{***}{y}$ & (1767, 1767) & 3122289 & 188.96 
& $185\pm 5$ \\
$b$ & \ \ $t+\mu^-$ & (1767, 1767, 48, 39) & 76061.49 & 4.60 & 4.2 to 4.7  \\
\hline
\end{tabular}
\end{center}
\end{table}
%}
%\vspace{0.2cm} 
Table \ref{t:massresult} 
illustrates family-to-family similarities in the particle
structures. 
For instance, in
each family, the down-like quark appears as a combination of the
up-like quark, with a charged lepton belonging to the lighter family.
Charged leptons appear as a combination of the neutrino from the
same family with the neutrinos and charged leptons from the lighter
family. Ring structures similar to that of the the electron neutrino,
may also appear as ``heavy neutrinos'',
$\nu_h =12y^*$. They
can further form ``ultra-heavy'' neutrinos
$\nu_{uh} =3(\overline{y}^* \nu_h u)e^-$, and so on.
The components $\stackrel{**}{y}$ and $\stackrel{***}{y}$ (which, by their
properties, are similar to $y$) of $c$ and $t$ have the following structures: 
$\stackrel{**}{y}=u\nu_e u\nu_e e^-$ and $\stackrel{***}{y}= \nu_{uh}y$.

%\vspace{0.2cm}
The experimental masses of the charged leptons are known with 
high accuracy. They are given in Table \ref{t:massresult}
with sufficient number of digits to compare them with 
our results. In this way, we estimate the accuracy of our model to be 
of about 0.5\%. Much higher accuracy can be achieved by applying 
radiative corrections to the  masses of clustered components
in the structures of the charged particles.
One can calculate also the neutrino masses by detailed
evaluation of metric along the neutrino loop.
However, this is already another story, which will be discussed elsewhere.

\section{Conclusions}
Our model reproduces masses of three generations of the fundamental 
fermions without using input parameters. The accuracy of our model
is about 0.5\% (only the accurate experimental masses of the charged
leptons are used for comparison).
  
\acknowledgments{
The author thanks Dr. L.V. Morrison for his linguistic
support.}

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\end{document}

