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\begin{document}
\newcommand{\abs}[1]{\lvert#1\rvert}
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%\title{{\bf Topology of space at sub-quark level \\ 
%and masses of quarks and leptons}}
\title{Topology of space at sub-quark level 
and masses of quarks and leptons\thanks{presented at 
{\it XXV International Workshop on the Fundamental Problems 
of High Energy Physics and Field Theory}, Protvino, 25-28 June 2002, arXiv:}
\author{{\bf V.N. Yershov} \\
\vspace{3.0\baselineskip}                                               
{\normalsize\it University College London, Mullard Space Science Laboratory,} \\
{\normalsize\it Holmbury St.Mary, Dorking RH5 6NT, United Kingdom\footnote{present address}} \\
%\vspace{3.0\baselineskip}    
{\normalsize\it Pulkovo Observatory, Pulkovskoye shosse 65-1, St.Petersburg 196140, Russia} \\
{\normalsize e-mail: {\it vny@mssl.ucl.ac.uk}}
}     

\date{}
\maketitle

%\begin{abstract}
{\small 
The global shape (topology) of the 
Universe is not derivable from General Relativity but should be 
determined by observations.  Here we  propose a method for estimation of 
this shape using patterns of fundamental physical parameters,
for example the spectrum of fermion masses. We suppose that this pattern
might appear because of specific topology at sub-quark level.
We restrict ourselves to analysis of a topological object
described by F.Klein in 1882 and show that 
its properties result in formation of structures  
reproducing leptons and quarks with all their quantum numbers 
and masses.} 
%\end{abstract}

\vspace{0.3cm}
\section{Introduction}

\vspace{0.1cm}
General relativity and the Big Bang theory assume that 
the universe is curved. But the character of this curvature 
(shape of the universe) cannot be deduced from
these theories without additional observations. Developments 
in theoretical and observational cosmic topology 
are progressing rapidly\cite{luminet}. 
Much observational effort has been directed towards the determination 
of the universe's curvature. Fewer observations have been aimed at   
topology. In the case of the universe with positive curvature there 
is a theoretical possibility to observe the same object from different
directions and to arrive to some conclusions. But these observations are
probably far beyond our technical capabilities. Some non-trivial
features, such as multi-connectivity or the hypertoroidal
character of space, can probably be revealed by observing patterns 
in the distribution of galaxies and quasars\cite{roukema}, but the result 
will never be definitive. More accurate results one may 
probably obtain by high resolution measurements
of the cosmic microwave background radiation\cite{bond,levin}.

\vspace{0.1cm}
Here we would like to propose another method for determination
of the universe's shape. If we suppose that some (or all)
of the fundamental physical parameters are topology-dependent,
then we can try to find a proper topological model that
matches these parameters. Many physical parameters 
are known with high accuracy, and our method should be accurate
and more definitive than observations of
distant objects at the limits of our possibilities.

\vspace{0.1cm}
There is one set of parameters, which forms an enigmatic 
pattern probably related to the topology of space.
Since the discovery of quarks, it was found that
there are only twelve elementary fermions 
grouped in three families (or generations) with  properties 
repeating from generation to generation. But masses of these
particles \cite{properties} are distributed in a rather odd way
 (Table \ref{t:masses}). 

\begin{table}[htb]
\caption{Experimental rest masses of quarks and leptons
in units of the proton mass, $m_p$ (first two 
charged leptons, $e$ and $\mu$, are given with sufficient number 
of digits for comparison purposes)}
\label{t:masses}
\begin{center}
\small
\begin{tabular} {|c|c|c|c|c|c|} \hline
\multicolumn{2}{|c|}{First generation} & \multicolumn{2}{|c|}{Second generation}
& \multicolumn{2}{|c|}{Third generation} \\ \hline
$\nu_e$ &  $\le 3\cdot 10^{-9}$  & $\nu_\mu$ & $\le 2\cdot 10^{-4}$ &
 $\nu_\tau$ & $\le 2\cdot 10^{-2}$ \\ %\hline
$e$ &  0.00544617  & $\mu$ & 0.1126095 & $\tau$ & $1.8939\pm0.0003$ \\
$u$ &  0.001 to 005  & $c$ &  1.2 to 1.4  &
$t$ & $185\pm 5$ \\ %\hline
$d$ & 0.003 to 0.009 & $s$ & 0.08 to 0.18 & $b$ & 4.2 to 4.7 \\ \hline 
\end{tabular}
\end{center}
\end{table}

The Standard Model of particle physics uses  
these masses as its input parameters and does not explain their origin.
Although the experimental masses of quarks are not known with high accuracy,
their distribution is wide and looks random. This distribution can be  
considered as a good pattern for testing topological models.
Masses of the charged leptons are known to higher
accuracy, and one can use them for estimation of quality of the model.
There are many models proposed for explanation of this
mass hierarchy \cite{pati,salam}
focusing basically on  unification\cite{georgi} or 
supersymmetric unification\cite{dimopoulos}, technicolor\cite{weinberg},
or strings and branes\cite{green}, but the problem is still there.
There are also theories investigating the possibility of randomness of
this pattern\cite{donoghue}.


\section{Choise of topology}

\vspace{0.1cm}
We restrict ourselves to the (3+1)-dimentional manifold, where 
we shall be looking for the most symmetric topological object.
Intuitively, A.Einstein and A.Friedmann imagined the 
universe having spherical shape of positive, negative
or zero curvature. But a sphere, having its two 
surfaces (inner and outer), is not a good candidate for the shape
of the universe.
By definition, the universe is self-consistent: there is 
nothing ``outside'' of the universe, and this ``outside'' 
doesn't exist. The spherical shape, applied to 
the universe,  would mean the co-existence of two different universes.
The topological object corresponding to the fact of the 
universe's self-consistency and uniqueness is the very well known
 Klein bottle. Having a unique hyper-surface,  the universe with
 Klein bottle topology - similar to the sphere - 
can be of positive, negative or zero curvature.
This object, described by F.Klein in 1882 \cite{kastrup}, 
was a result of his work on the theory of invariants under group
 projective  transformations. 


\vspace{0.1cm}
The main feature of the Klein bottle is the unification of
its inner and outer surfaces. In the case of the universe, 
the unification might well occur at the sub-quark level, giving
 rise to structures of particles.
Due to the clear pattern in the properties of fermions
these particles cannot be considered as the fundamental constituents of
matter. It is logical to think that matter is structured 
further down to a simplest possible object, which we hereafter 
refer to as ``a preon''. We associate it with the unification 
area of Klein bottle topology.

\section{Preons} 

The area of unification between the inner and outer manifestations of space
(the preon),  might be considered as an area where
space is turned ``inside-out''. One can attribute the known 
fundamental interactions to the geometrical properties of this inversion. 
For example, distances $z$, measured from the preon's centre,
can be equivalently expressed in terms of their reciprocal values $z'=z^{-1}$,
if considered within the inverted manifestation of space. Thus, any
potential, which is proportional to $1/z$, in the inverted manifestation
of space will be proportional to $1/z'$ (or simply $z$).
The Coulomb potential, $\phi_e \propto 1/z$,  
in the inverted manifestation of space should manifest itself strong-likely,
$\phi'_e \rightarrow \phi_s \propto z$, and vice versa. 
The distance $z_0$, where $\abs{z}=\abs{z'}$ and $\abs{\phi_s}=\abs{\phi_e}$,
should characterize the scale, at which the preons stabilize when  
forming structures.


\vspace{0.1cm}
Let us suppose, the primitive constituent of matter (the preon $\Pi$) to
possess a specific symmetry $\mathbf{\Pi}, \overline{\mathbf{\Pi}} \in {\bf 3}_c$.
Then, via rotational transformations it can be translated into one 
or another state corresponding to three different colours and two 
different electric charges.
For the formal representation of the preons 
we can use  a triplet of three-component column vectors
$\mathbf{\Pi}_i=\{\Pi_j\}_i$
 ($i,j=0,1,2$): 

\begin{equation*}
\Pi_{ij}=
 \begin{cases}
 +1, & i=j \\
 -1, & i\neq j 
 \end{cases}
\text{\hspace{0.6cm}}   \text{\hspace{0.2cm}} 
\overline{\mathbf{\Pi}}_i=-\mathbf{\Pi}_i
%\label{eq:preonmatrix}
\end{equation*}
 
with their unit charges

\begin{equation*}
Z(\mathbf{\Pi}_i)=\sum_{j=0}^2{\Pi_{ij}}
%\label{eq:charge}
\end{equation*} 

and unit masses

\begin{equation*}
m(\mathbf{\Pi}_i)=\abs{\sum_{j=0}^2{\Pi_{ij}}}.
%\label{eq:mass}
\end{equation*} 


The charge of a system composed of various preons or preon groups will be

\begin{equation}
Z=\sum_{k=1}^N{\sum_{i=0}^2{\sum_{j=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 $m$ (as well as the  
reciprocal mass $m'$) of the system:

\begin{equation}
m= (1-\delta_{Z,0})
\sum_{k=1}^N\abs{{\sum_{i=0}^2{\sum_{j=1}^n{\Pi_{ij}^k}}}}, \hspace{0.7cm}
m'= (1-\delta_{Z,0})
\sum_{k=1}^N\abs{{\sum_{i=0}^2({\sum_{j=1}^n{\Pi_{ij}^k})^{-1}}}},
 \label{eq:mass}
\end{equation} 

where $\delta_{Z,0}$ is the Kronecker delta-function.
Assuming that the strong (coloured) and electric interactions are
opposite manifestations of the same property of space
and, taking into account the well-known pattern of the coloured
interaction (two like-charged but unlike-coloured particles are
attracted, otherwise they repel), we can characterise  
the pattern of the combined chromo-electric interaction between two
preons, $i$ and $k$, by the following force $F_{ik}$ (normalised to unity):

\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 force).
Expression (\ref{eq:mass})  reflects the possibility
that the preon acquires its mass because of its accelerated 
motion along its world line. The acceleration is caused 
by local curvature of space in the area of inversion.
Using a unit value for the radius of this local curvature, one finds that
this acceleration is proportional to $c^2$. The coefficient of
proportionality can be considered as the preon's inertial mass.
However, from the point of view of an observer placed in the preon's
coordinate frame, the preon is at rest, and the above mentioned
inertial mass will be perceived as the preon's gravitational
mass, $m_g$. If now the preon is additionally forced to move 
along spatial coordinates (with acceleration $a$), it would 
resist this motion exactly in the same way as when moving along
its world line (time-axis). A force $F=m_ia$, which is required 
in order to accelerate the preon, is proportional to $a$ with 
the coefficient of proportionality $m_i$, and the observer will
conclude that the preon also possesses an inertial mass,  $m_i$.
In fact, in our model the preon's inertial and gravitational  
masses are generated by the same mechanism of acceleration,
and they are essentially the same thing; $m_i\equiv m_g$.
It is also well known, that positive and negative  
electric charges can be equivalently expressed in terms of
reversibility of time (first noted by P.A.M. Dirac).
This fact, together with the above mentioned mechanism  of
 mass acquisition, is reflected in (\ref{eq:mass}) indicating
 the possibility of cancellation of local acceleration (mass)
 in a structure with two combined unlike-charged preons. 

\section{Combining preons}

The pattern of forces (\ref{eq:seforce})
necessarily results in combinative preon structures. The simplest 
structure is a charged  preon doublet

\begin{equation*}
\varrho^\pm_{ik}= \mathbf{\Pi}_i+\mathbf{\Pi}_k, \text{\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, \text{\hspace{0.6cm}} i,k=0,1,2
%\label{eq:gluon}
\end{equation*}
 
(nine combinations)  is also possible.
The preon doublets will be deficient in one or two colours
having, for example, the following colour charges for $g$:
$r\overline{r}$, $r\overline{g}$, $r\overline{b}$, 
$g\overline{g}$, $g\overline{b}$, $b\overline{b}$,
$\overline{r}g$, $\overline{r}b$, $\overline{g}b$.  
The preons in these doublets will be
separated by the distance $z_0$.
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*}

The null value for the mass of the neutral doublet is a result of  
the above mentioned correspondence between charges and directions along the 
$ict$ coordinate. When two unlike-charged preons combine 
(say red and antigreen), their 
oppositely directed velocities are mutually cancelled (resulting in a 
neutral system). The same thing happens to their accelerations, which 
is reflected in the formula (\ref{eq:mass}) for the mass of the system.
Then, if an additional charged preon is added to the neutral
doublet, the mass and the charge of the system are restored. 
For example, 

\begin{equation}
m(\mathbf{\Pi}_r,\overline{\mathbf{\Pi}}_g,\mathbf{\Pi}_b)=1
\label{eq:mgluon}
\end{equation}

but still 

\begin{equation}
m'(\mathbf{\Pi}_r,\overline{\mathbf{\Pi}}_g,\mathbf{\Pi}_b)=\infty.
\label{eq:mprimgluon}
\end{equation}


\vspace{0.1cm}
The charged doublets $\varrho$ ($2\Pi$ and  $2\overline{\Pi}$)
 will not be free for long because their colour potentials 
($\phi_s \propto z$) are infinite at infinite
distance. Any distant preon of the same charge but with a
complementary colour will be attracted to
the pair. In this way,  $3\Pi$ and  $3\overline{\Pi}$ particles will
be formed, which we call hereafter
{\sf Y}-particles. The mass of the {\sf Y}-particle
is 3 units (in the units of the preon's mass), and its charge (positive
or negative) is of the same magnitude. Its colour will be 
complete, but locally, the $rgb$ colours  of its three preons 
will be distributed in a plane forming a closed loop.

\vspace{0.1cm}
{\sf Y}-particles cannot be free because their strong potentials 
are only partially closed within loops. 
Thus, distant like-charged {\sf Y}-particles will combine  
and form {\sf Y} \DoT{\sf Y}  structures
(Fig.\ref{fig:twoy}{\bf a}). Here we use 
\DoT{\sf Y} just in order to indicate the fact that the second 
particle is turned through $180^\circ $ in respect with the first one.
This is the only possible mutual orientation of two
combined like-charged {\sf Y}-particles.  
The colour pattern of this structure can be written as  

\begin{equation}
\begin{vmatrix} r_1 & & & b_2 \\  & g_1 &
g_2 &   \\ b_1 & & & r_2 \end{vmatrix}.
\label{eq:antiparallel}
\end{equation}

\begin{figure}[htb]
\centering \epsfysize=9cm
\includegraphics[scale=0.8]{yerfig1.eps}
\caption{({\bf a}) Two combined like-charged
{\sf Y}-particles turned through $180^\circ$ with respect 
to one another; and ({\bf b}) two coupled unlike-charged {\sf Y}-particles
turned through $120^\circ$ with respect to one another.}
\label{fig:twoy}
\end{figure}


Two unlike-charged
{\sf Y}-particles combine by turning clockwise through $120^\circ$
with respect to one another ($\uparrow$), Fig.\ref{fig:twoy}{\bf b},
or anticlockwise ($\downarrow$), with two 
corresponding colour patterns:

\begin{equation}
\begin{vmatrix} \overline{g}_1 & & r_2 &  \\  & \overline{b}_1 &
& g_2   \\ \overline{r}_1 & & b_2 &  \end{vmatrix}
 \text{\hspace{0.5cm} or \hspace{0.5cm} }   
\begin{vmatrix} \overline{g}_1 & & b_2 &  \\  & \overline{b}_1 &
& r_2   \\ \overline{r}_1 & &
g_2 &  \end{vmatrix}. \label{eq:parallel}
\end{equation}

\vspace{0.1cm}
The three-colour completeness of {\sf Y} permits up to three of them
to combine if all of them are like-charged. These three {\sf Y}
will be joined in a closed loop with the following distribution of colours:

\begin{equation}
\begin{vmatrix}  & r_1 &  & b_2 & & g_3 \\
 b_1 &  & g_2  &  & r_3 &  \\
  & g_1 &  & r_2 & & b_3  \end{vmatrix}
 \text{\hspace{0.5cm} or \hspace{0.5cm} } 
\begin{vmatrix}  & r_1 &  & g_2 & & b_3 \\
 b_1 &  & r_2  &  & g_3 &  \\
  & g_1 &  & b_2 & & r_3  \end{vmatrix},
\label{eq:threey}
\end{equation}


corresponding to the clockwise and anticlockwise mutual orientation
of the components.

%%%%%%%

\begin{figure}[htb]
\centering \epsfysize=9cm
\includegraphics[scale=0.8]{yerfig2.eps}
\caption{ Three like-charged {\sf Y}-particles joined with their vertices 
directed ({\bf a}) towards the centre of the structure (left-handed state)
and ({\bf b}) away from the centre (right-handed state).  }
\label{fig:electron}
\end{figure}


These structures are shown in Fig.\ref{fig:electron}{\bf a}, where the
vertices of their {\sf Y}-components are directed towards their common centre.
A 3\DoT{\sf Y}-state with the vertices of {\sf Y}  directed away from their
common centre is also possible (Fig.\ref{fig:electron}{\bf b}).
One state can be obtained from the other 
by mirror-reflection of all its components about the 
circular axis of symmetry. 
We shall refer to these two states as the left- and right-handed ones
(3{\sf Y}$_L$ and 3{\sf Y}$_R$).
The mass of the 3{\sf Y}-particle is the sum of masses of its
nine preons (9 units). Similarly, its charge  is 9 preon
charge units.

\vspace{0.1cm}
Couples of the unlike-charged {\sf Y} can form chains
{\sf Y}$\overline{\sf Y}$ - \DoT{$\overline{\sf Y}$}\DoT{\sf Y} -
 {\sf Y}$\overline{\sf Y}  \ldots$
 with the following colour patterns:

\begin{multline}
\left|
\begin{smallmatrix} & r_1 & &  \overline{b}_2 \\ b_1 &
& \overline{g}_2 &   \\ & g_1 & & \overline{r}_2
\end{smallmatrix}
\right| +
\left|
\begin{smallmatrix} \overline{r}_3 & & b_4 &  \\  &
\overline{g}_3 &  & r_4  \\ \overline{b}_3 & & g_4 &
\end{smallmatrix}
\right| +
\left|
\begin{smallmatrix}  & g_5 &  & \overline{r}_6 \\ r_5
&  & \overline{b}_6 &  \\  & b_5 &  & \overline{g}_6
\end{smallmatrix}
\right| +
\left|
\begin{smallmatrix} \overline{g}_7 & & r_8 &  \\  &
\overline{b}_7 &  & g_8  \\ \overline{r}_7 & & b_8 &
\end{smallmatrix}
\right| +
\left|
\begin{smallmatrix}  & b_9 &  & \overline{r}_{10} \\
g_9 &  & \overline{r}_{10} &   \\ & r_9 &  & \overline{b}_{10}
\end{smallmatrix} \right| +
\left|
\begin{smallmatrix} \overline{b}_{11} & & g_{12} &  \\
& \overline{r}_{11} & & b_{12}  \\ \overline{g}_{11} & & r_{12} &
\end{smallmatrix} \right| + \dots
\label{eq:nuright}
\end{multline}

or

\begin{multline}
\left|
\begin{smallmatrix} & r_1 & &  \overline{g}_2 \\ b_1 &
& \overline{r}_2 &   \\ & g_1 & & \overline{b}_2
\end{smallmatrix}
\right| +
\left|
\begin{smallmatrix} \overline{b}_3 & & r_4 &  \\  &
\overline{r}_3 &  & g_4  \\ \overline{g}_3 & & b_4 &
\end{smallmatrix}
\right| +
\left|
\begin{smallmatrix}  & b_5 &  & \overline{r}_6 \\ g_5
&  & \overline{b}_6 &  \\  & r_5 &  & \overline{g}_6
\end{smallmatrix}
\right| +
\left|
\begin{smallmatrix} \overline{g}_7 & & b_8 &  \\  &
\overline{b}_7 &  & r_8  \\ \overline{r}_7 & & g_8 &
\end{smallmatrix}
\right| +
\left|
\begin{smallmatrix}  & g_9 &  & \overline{b}_{10} \\
r_9 &  & \overline{g}_{10} &   \\ & b_9 &  & \overline{r}_{10}
\end{smallmatrix} \right| +
\left|
\begin{smallmatrix} \overline{r}_{11} & & g_{12} &  \\
& \overline{g}_{11} & & b_{12}  \\ \overline{b}_{11} & & r_{12} &
\end{smallmatrix} \right| + \dots
\label{eq:nuleft}
\end{multline}

\vspace{0.1cm}
corresponding to two ($\uparrow$ and $\downarrow$) possible 
states (\ref{eq:parallel}).
The colour patterns repeat after each  six consecutive
${\sf Y} \overline{\sf Y}$ groups, producing  
12{\sf Y}-period chains closed in loops (their 12-th chain element
is compatible with the first one).

\begin{figure}[htb]
\centering \epsfysize=9cm
\includegraphics[scale=0.8]{yerfig3.eps}
\caption{12{\sf Y}-structure.
 Mirror-reflection of all the components in   
 the loop about the loop's circular axis translates
 the structure into itself because of the equal number of
 {\sf Y} directed with their vertices towards and outwards the 
 centre of the structure.}
\label{fig:twonu}
\end{figure}

\vspace{0.1cm}
The pattern (\ref{eq:nuright}) is  visualised in
Fig.\ref{fig:twonu}, where brighter colours are assigned 
to the negatively charged preons. 
According to (\ref{eq:charge}) and (\ref{eq:mass})
the {\sf 12Y}-structures are electrically neutral and massless. 
The spatial distribution of any particular colour (say, red positive)
appears as a clockwise ($\uparrow$) or anticlockwise
($\downarrow$) helix, each being a complete rotation 
around the closed loop axis.


\vspace{0.1cm}
It is interesting to note that mirror-reflection of all the 
 components of 12{\sf Y} about the circular axis of the loop
reproduces the  same 12{\sf Y}-structure,
 without translating it into the opposite 
(right-handed) state. Thus, it is topologically impossible to
differentiate 12{\sf Y}$_L$ from 12{\sf Y}$_R$ because the number of
their {\sf Y}-components oriented with their vertices 
towards the centre of the loop coincides with the number
of those directed outwards.

\vspace{0.1cm}
The 12{\sf Y}-particle, consisting of 36 preons, is massless
unless coupled to a charged particle, say {\sf Y} or 3{\sf Y},
which restores its mass. 
 For example, according to (\ref{eq:mass}),
 the mass of {\sf Y}$^*$={\sf Y}+12{\sf Y} 
 is 39 preon mass units (3+36).
The (3{\sf Y}+12{\sf Y})-particle is of 45 mass units (9+36), etc.
Coupling {\sf Y} with 12{\sf Y} and 3{\sf Y} with 12{\sf Y} 
is possible because of attractive forces arising due to 
the particles' local patterns of colours and charges
(analogous to the van der Waals forces between molecules).
The strength of these forces depends on the compatibility
of colour patterns (helices) of the interacting particles.
The colour pattern of {\sf 12Y$_\uparrow$}
does not match that of {\sf 3Y$_\downarrow$}.
Only {\sf 3Y$_\uparrow$} and {\sf 12Y$_\uparrow$} or 
{\sf 3Y$_\downarrow$} and {\sf 12Y$_\downarrow$} can combine.
 Unlike this case, if {\sf 3Y} combines with another
 {\sf 3Y}, or {\sf 12Y} with another {\sf 12Y}, their helices  
should be opposite.   

\section{First generation of quarks and leptons}

\vspace{0.1cm}
By their properties, 3{\sf Y} and 12{\sf Y} can be readily
associated with the electron and its neutrino (leptons of the first
generation).  The charge of
3{\sf Y}, divided by 9, gives us the
conventional unit charge of the electron. 
Then charges of {\sf Y}$^*$ or 2{\sf Y}$^*$-particles
correspond to the fractional charges of 1/3 and 2/3 
(hinting at possible quark's constituents).
The {\sf Y}$^*$-structure cannot be free because of its 
 strong potential increasing with distance. It will further
combine with other {\sf Y}$^*$. If two {\sf Y}$^*$-particles 
have likewise helix patterns, they will couple via an intermediate 
 12{\sf Y}-particle with an opposite-helical pattern. 
This {\sf Y}$^*_\uparrow$ {\sf 12Y}$_\downarrow$ 
\DoT{\sf Y}$^*_\uparrow $ link
can be associated with the $up$ quark. It will be of +6 charge units and 78
mass units ($2\times 3$ charge and $2\times 39$ mass units of the 
$2\times${\sf Y}$^*$-structures). 


\begin{table}[htb]
\caption{Simple structures generated  by the  Klein bottle topology of space 
}
\label{t:preonsummary}
\begin{center}
%\small
\begin{tabular}
{|c|c|c|c|c|} \hline
Structure     & Constituents       & Number of            & Charge         & Mass            \\
              & of the             & possible states      & (in the preon  & (in the preon      \\
              & structure          &                      & charge units) & mass units)  \\ \hline
\multicolumn{5}{|c|} {First level structures consisting of single preons} \\ \hline
$\Pi^\pm$  & $1\Pi$       & 6 ($3^+$ and $3^-$)  & $\pm 1 $       &   1           \\ 
$\varrho^\pm$       & $2\Pi$       & 12 ($6^+$ and $6^-$) & $\pm 2 $       &   2           \\ 
$g^0$         & $2\Pi$       & 9                    &  0             &   0           \\
{\sf Y}$^\pm$ & $3\Pi$       & 2 ($1^+$ and $1^-$)  & $\pm 3 $       &   3           \\  
\hline
\multicolumn{5}{|c|}{Second level structures consisting of triplets ({\sf Y}-particles)} \\
\hline
{\sf Y}$^\pm$ & 1{\sf Y}        & 2 ($1^+$ and $1^-$)  & $\pm 3 $       &   3            \\ 
$\delta^\pm$  & 2{\sf Y}        & 2 ($1^+$ and $1^-$)  & $\pm 6 $       &   6            \\ 
$\gamma^0$    & 2{\sf Y}        & 2                    & 0              &   0             \\ 
$e^\pm$       & 3{\sf Y}        & 8 ($4^+$ and $4^-$)  & $\pm 9 $       &   9             \\ \hline
\multicolumn{5}{|c|}{Structures consisting of the second levels particles}  \\
\hline
$2e^\pm$      & $e^-+e^-$ or $e^++e^+$   & 6 ($3^+$ and $3^-$)  & $\pm 18$       &  18 \\
$2e^0$        & $e^-+e^+$ (or $3\gamma$)   & 4                    & 0              &  0 \\
$\nu^0$       & 12{\sf Y} (or $6\gamma$)   & 2                    & 0              &  0  \\
{\sf Y}$^*$ & $\nu$ \ \   + \ \  {\sf Y} & 4 ($2^+$ and $2^-$) & $\pm 3$   &  39 \\
$W^\pm$   & $\nu$ \ \ + \ \ $e$     & 8 ($4^+$ and $4^-$)  & $\pm 9$        &  45 \\
$u^\pm$       & {\sf Y}$^*$ \ \ $\nu$ \ \ {\sf Y}$^*$ & 4 ($2^+$ and $2^-$)  & $\pm 6$        &  78 \\
$\nu^0_\mu$   & {\sf Y}$^*$ \ \ $\nu$ \ \ $\overline{{\sf Y}}^*$ & 2                    &  0             &  0 \\
$d^\pm$       & $u$ \ \ + \ \ $W$ & 8 ($4^+$ and $4^-$) & $\pm 3$ & 123 \\
$\mu^\pm$     & \ $\nu_{\mu}$ \  + \ \ $W$ & 8 ($4^+$ and $4^-$) & $\pm 9$ & $(48,39)^*$ \\
\multicolumn{5}{|c|}{and so on ...}  \\     
\hline
\end{tabular}
\end{center}
{\footnotesize \hspace{2cm}*)two-component system }
\end{table}

\vspace{0.1cm}
Being positively charged, the $up$ quark will be able to couple to a
negative particle, such as $3\overline{\sf Y} 12{\sf Y}^0$ 
(with its 45 mass units and its -9 charge units).
The total mass of the resulting 
$3\overline{\sf Y}12{\sf Y}^0$ + 
{\sf Y}$^*$ 12{\sf Y}$^0$ \DoT{\sf Y}$^*$
structure (the $down$ quark) is $123= 45+78$, and its charge is
$-3=-9+6$. The preon structures are summarised in
Table \ref{t:preonsummary}.

 

\section{The second and third generations of particles}

It is natural to suppose  that particles of the second and third
generations should be composed of simpler structures belonging to the 
first generation.
For example, the muon neutrino  (a neutral particle) can be formed
of unlike-charged {\sf Y}$^*$={\sf Y$ \nu_e$} and
$\overline{\sf Y}^*=\overline{\sf Y} \nu_e$:


\begin{equation}
\nu_\mu= {\sf Y} 12{\sf Y}^0_\downarrow 
 \ 12{\sf Y}^0_\uparrow \  12{\sf Y}^0_\downarrow  \overline{\sf Y} 
={\sf Y}^*\nu_e\overline{\sf Y}^*, 
 \label{eq:numu}
\end{equation}

and the muon could be structured naturally as

\begin{equation}
\mu= (12{\sf Y}^0+3\overline{\sf Y}{\sf Y})
(12{\sf Y}^0_\downarrow 12{\sf
Y}^0_\uparrow  12{\sf Y}^0_\downarrow \overline{\sf Y})
=\overline{\nu}_e e^-\nu_\mu ,
\label{eq:muon}
\end{equation}


\vspace{0.1cm}
which corresponds to one of the muon's decay modes. 
Structures of other particles can also be derived from their 
known decay modes. These particles should not
be considered as having rigid structures. They
are clusters, or many-body systems,
rather than single particles. In
(\ref{eq:muon}) we enclose  the clustered components in parenthesis.
Interactions between the clustered components
modify the energy (mass) of the system. We suppose that, because 
of inversion of space, the force 

\begin{equation*}
F=-G\frac{M M'}{r r'} 
%\label{eq:force}
\end{equation*}

acting on the components depends both on the sum $M$ of their
masses $m_k$:

\begin{equation*}
M=\sum_k{m_k} , 
%\label{eq:summass}
\end{equation*}

and the sum $M'$ of their inverted reciprocal masses $1/m'_k$ 
(the reduced mass):

\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 $M'$.  
Let us also suppose that, according to the Newton's second law,
the resulting energy (mass) of the composite system,
is proportional to this force:
$M_{system}\propto \abs{F}a^{-1}$. 
Using unit values for the acceleration ($a=1$), the universal gravitational
constant ($G=1$), taking into account the unity of the product of two
reciprocal distances ($r r'=1$), and assuming all unit conversion 
coefficients to have unit values, one can estimate masses of the
composite particles by

\begin{equation}
M_{system}=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}

\vspace{0.2cm}
Using (\ref{eq:mtotal}) and calculating masses of fermions, 
one can notice that the particle masses
for the second and third generations are reproduced with a systematic
error of about 0.5\% with respect to their experimental values
(we don't present these values here).
The systematic differences between the experimental
and calculated masses linearly depend on the number of preons in the 
structural elements containing 3{\sf Y}. These differences can be 
easily taken into account by radiative corrections to the masses
as following:


\begin{equation}
m^{rc}=m+s  \eta,
\label{eq:corrmass}
\end{equation}

where $m$ is the original mass of the clustered component (in the 
units of the preon's mass), $m^{rc}$ is the corrected mass of the component,
and $\eta$ is the radiative correction factor:

\begin{equation*}
\eta=\frac{3}{2}(\pi q_e^{rc})^{-2}.
%\label{eq:mcorr}
\end{equation*}

 $q_e^{rc}$ is the radiatively 
corrected electron charge
calculated from the recursive expression

\begin{equation*}
q_e^{rc}=q_e-q_e^{rc}\eta,
\end{equation*}


and $q_e$ is the original electron charge, expressed in the 
preon's charge units. For $q_e=9$ the radiative correction 
factor $\eta=0.0018771$. $s$ is the preon factor for the  
given clustered component: 

\begin{equation*}
s= \left\lbrace
 \begin{array}{ll}
 \frac{N^+}{N^-} q_e, & {\rm for \hspace{0.2cm} the \hspace{0.2cm} 
positively \hspace{0.2cm} charged \hspace{0.2cm} components, \hspace{0.2cm}
 if \hspace{0.3cm}}  Z_{system}>0 \\
 -N^-, & {\rm for \hspace{0.2cm} the \hspace{0.2cm} negatively \hspace{0.2cm}
charged \hspace{0.2cm} components, \hspace{0.2cm}
 if \hspace{0.3cm}}  Z_{system}<0.  
 \end{array}
 \right.
\end{equation*}

$N^+$ and $N^-$ are respectively the preon numbers in the positively and negatively
 charged components of the cluster.

%%%%

\vspace{0.1cm}
Fermion masses 
calculated with the use of (\ref{eq:mtotal}) and (\ref{eq:corrmass})
 are summarised in Table \ref{t:massresult}. 
As an  example, we can calculate the muon's mass. The masses of the 
muon's  components, according to its structure (\ref{eq:muon}), are: 
$m_1=m'_1=48$, $m_2=m'_2=39$, $s=-123$, 
$m_2^{rc}=38.769$ (all in preon mass units). And the muon's mass is

\begin{equation*}
m_\mu =(m_1,m_2^{rc}) = \frac{m_1+m_2^{rc}}{\frac{1}{m'_1}+
\frac{1}{m_2^{'rc}}}=1860.917594
 {\hspace{0.2cm} {\rm (preon \hspace{0.2cm} mass \hspace{0.2cm} units)}.}
\end{equation*}

For the $\tau$-lepton:
 $m_1=m'_1=156$, $m_2=m'_2=201$,
$s=-201$, $m_2^{rc}=200.6227$,   

\begin{equation*}
m_\tau  = (m_1,m_2^{rc})=31297.1416 {\hspace{0.2cm} {\rm (preon 
\hspace{0.2cm}  mass \hspace{0.2cm} units)}.}
\end{equation*}


\vspace{0.2cm}
For the proton, positively charged particle consisting of two $up$,
one $down$ quarks and a cloud of gluons $g$, masses of its components 
are $m_u=m'_u=78$,  $N^+=78$,  $m_d=m'_d=123$, $N^-=123$ 
($m_u^{rc}=78.0107132$). As for the gluons, only those of them
should be taken into account, which are coupled to the quarks' preons.
The total number of these coupled gluons, in accordance with the proton's
structure ($2u+d$), is $N_g=2N^++N^-=279$.
The mass of each gluon coupled to a preon, according to (\ref{eq:mgluon}) and
 (\ref{eq:mprimgluon}), is $m_g=1$, $m'_g=\infty$.
The resulting proton mass is 

\begin{equation}
m_p  = \frac{2m_u^{rc}+m_d+N_gm_g}{2\frac{1}{m_u^{'rc}}+
\frac{1}{m'_d}+N_g\frac{1}{m'_g}}=
16525.3575 {\hspace{0.2cm} {\rm (preon \hspace{0.2cm} mass \hspace{0.2cm} units)}.}
\label{eq:pmass}
\end{equation}

Using  (\ref{eq:pmass}), we can convert $m_\mu$ and masses of other
particles from the preon mass units into the proton
mass units, $m_p$. These values are given  in the fourth column 
of Table \ref{t:massresult}. 
The experimental masses of the particles 
(also expressed in units of $m_p$) are listed in the last
column for comparison.

\begin{table}[htb]
\caption{Predicted and experimental rest masses of quarks and leptons.
Values given in the third column can be  converted into the proton
mass units dividing them by $m_p=16525.3575$}
\label{t:massresult}
\begin{center}
\small
\begin{tabular}
{|cc|c|c|c|c|} \hline
\multicolumn{2}{|c|}{Particle and}  & {Number of }  &  Predicted    & Predicted   & Experimental \\
\multicolumn{2}{|c|}{its structure} & {preons in the}  & masses (preon & masses   & masses (in $m_p$)   \\
\multicolumn{2}{|c|}{ (components)} & {components} & mass units)   & (in $m_p$) &  Ref.\cite{properties} \\ \hline
\multicolumn{6}{|c|} {First generation} \\ \hline
$\nu_e$ & 12{\sf Y}$^0$            & 36   & 0    & 0           & - \\
$e^-$ & $3\overline{{\sf Y}}$               & 9    & 9    & 0.000544618 & 0.000544617 \\ 
$u$ & {\sf Y}$^*\nu_e${\sf Y}$^*$ & 78   & 78   & 0.00472     & 0.001 to 0.005 \\
$d$ & $u$ $ \ \nu_e e^-$                 & 123  & 123  & 0.00743     & 0.003 to 0.009 \\
\hline
\multicolumn{6}{|c|}{Second generation} \\
\hline
$\nu_\mu $ & {\sf Y}$^*$ \ \ $\nu_e$  \ \  $\overline{{\sf Y}}^*$ & 114  & 0 & 0 & -  \\
$\mu^-$ & \ \ $\nu_\mu$ \ \ + \   $\nu_e e^-$    & (48, 39)   & 1860.9176 & 0.1126098 & 0.1126095 \\
$c$ & {\sf Y}$^{**}$ \ \ + \ \ {\sf Y}$^{**}$ & (165, 165) & 27122.89     & 1.6415  & 1.2 to 1.4 \\
$s$ &  \ $c$ \ \ \ + \ \ $e^-$ & (165, 165, 9) & 2745.37 & 0.16615 & 0.08 to 0.18 \\
\hline
\multicolumn{6}{|c|}{Third generation} \\
\hline
$\nu_\tau$ & $u$ \ \ \  $\nu_e$ \ \ \  $\overline{u}$ & 192  & 0 & 0 & - \\
$\tau^-$ & \ \ $\nu_\tau$ \ \  + \    $\nu_\mu \mu^-$ & (156, 201) & 31297.142 & 1.89388 & $1.8939\pm 0.0003$ \\
$t$ & {\sf Y}$^{***}$ \  + \ \ {\sf Y}$^{***}$ & (1767, 1767) & 3122289 & 188.96 & $185 \pm 5$ \\
$b$ & \  $t$ \ \ + \ \ $\mu^-$ & (1767, 1767, 48, 39) & 75813.46 & 4.588 & 4.2 to 4.7  \\
\hline
\end{tabular}
\end{center}
\end{table}


\vspace{0.1cm}
Table \ref{t:massresult} 
illustrates family-to-family similarities in the particle
structures. For example, 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'',
$\nu_h =12{\sf Y}^*$. They
can further form ``ultra-heavy'' neutrinos
$\nu_{uh} =3(\overline{\sf Y}^* \nu_h u)e^-$, and so on.
The {\sf Y}$^{**}$ and {\sf Y}$^{***}$ components of  
$charm$ and $top$ have the following structures: 
{\sf Y}$^{**}=u\nu_e u\nu_e e^-$ and {\sf Y}$^{***}= \nu_{uh}${\sf Y}.



%%%%%%%%%%%%%%%%%

\section{Conclusions}

Our model, based on the hypothesis of the universe having the Klein bottle
 topology, has generated fermion masses without 
using any experimental input parameters.
Predicted quark masses match the experimental 
tolerance intervals. Predicted masses of the charged leptons
($e, \mu, \tau$) agree with experiment to an accuracy of
about 2 parts in $10^6$.       
Thus, we can conclude that  
the Klein topology of the universe fits the experimental data fairly well
and it seems to account for the enigmatic pattern of the 
fermion masses.

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\end{document}



