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\begin{CellGroup}
\Title{A structural model of quarks and leptons.}
\Author{{\rm {\rm Clemens HEUSON\\
Zugspitzstr. 4 , D-87493 Lauben , Germany\\
email: clemens.heuson@allgaeu.org}}}


\Abstract{A model is proposed in which quarks, leptons and perhaps gauge bosons are composites of magnetically charged  rishons \inlineTFinmath{T(q=1/3)}
and \inlineTFinmath{V(q=0)} with magnetic charges \inlineTFinmath{g=(1,2,-3)\multsp {g_0}}. Structural formulas of composite particles and their
interactions are given, with the binding lines representing magnetic fields. Higher generations are produced by adding rishon-antirishon pairs with
the three different  magnetic charges.}


\begin{CellGroup}
\Section*{1. Introduction}
It is unlikely that the experimentally extremely successful standard model of particle physics is a fundamental theory, since it contains a large
number of particles and unexplained parameters like masses and mixing angles. One way to reduce the number of states and parameters is the assumption
that quarks, leptons and perhaps some or all gauge bosons are composites of more fundamental particles called preons. Several composite models have
been proposed, see for example the review of Lyons [1].

The original rishonmodel [2] has been introduced as an attractive and economical counting scheme reproducing the first generation of quarks and leptons.
The main drawback was the lack of an dynamical theory and today presumable nobody believes, that the three quarkcolors can be obtained by ordering
of only two rishons. Later on a dynamical scheme was proposed, in which a \inlineTFinmath{{{(\Mvariable{SU}(3))}_H}} hypercolor-force binds rishons
with approriate \inlineTFinmath{{{(\Mvariable{SU}(3))}_C}} colors [3]. We find it not very attractive to assume an SU(3) like force with just another
strength of coupling, because on the way down from molecules to quarks and leptons nature never repeated itself in structure and interactions. On
the other hand one has then a large number of preons and (hyper-) gluons, so that the original goal of reducing the number of states cannot be acchieved.
As an alternative way for preon bindung magnetic forces have been suggested [4]. Wu [5] proposed a variation of the rishonmodel with electric- and
magnetic-like charges, a drawback was however that the simple addition of electric charges e/3 and 0 reproducing the first generation has been lost.
Koh, Pati, Rodrigues as mentioned in [5] suggested a rishonmodel with magnetic charges (1,3,-4)\inlineTFinmath{{g_0}} excluding exotic states, but
now the question arises, why nature should have omitted the magnetic charge 2\inlineTFinmath{{g_0}}. In this letter we propose a model of magnetically
charged rishons with the charge assignment g\(=\)(1,2,-3)\inlineTFinmath{{g_0}}. While this does not sound very new in the first moment, we shall
see that one gets a structural representation of composite quarks, leptons, gauge bosons and their interactions in analogy to chemical formula of
molecules with the binding lines representing magnetic fields instead of electron pairs. A simple picture of the generation structure of composite
fermions is provided.




\end{CellGroup}
\begin{CellGroup}
\Section*{2. The preons}
As in the original rishon model we assume that the quarks and leptons of the first generation are composed of two fundamental rishons T,V and the
corresponding antirishons \inlineTFinmath{\overvar{T}{\_}},\inlineTFinmath{\overvar{V}{\_}} with spin 1/2. We assume furthermore that rishons carry
magnetic charges. From electroweak interactions of leptons and the asymptotic freedom of quarks one knows, that composite fermions don't carry magnetic
charges. For this reason the nonzero magnetic charges  \inlineTFinmath{g={g_0}(i,j,k)\multsp }of the three rishons in a quark or lepton must cancel
to zero:



\dispTFNumberedEquationmath{i+j+k=0\multsp }
From the magnetic charge quantisation [6] with n an integer

\dispTFNumberedEquationmath{{q_1}\multsp {g_2}-{q_2}\multsp {g_1}=\frac{\hbar \multsp c\multsp n}{2}}
applied to the system \inlineTFinmath{{T_i}{V_j}} one obtains \inlineTFinmath{\frac{e}{3}{g_j}=\frac{\hbar \multsp c}{2}n}. Similarly for other
systems this gives



\dispTFNumberedEquationmath{g=n\multsp {g_0}\multsp \multsp \Mvariable{with}\multsp \multsp {g_0}=\frac{3\multsp \Mvariable{\hbar c}}{2\multsp e}}
Since  the magnetic charges must cancel to zero, several charge assignments are possible. The most simple ones are: \inlineTFinmath{g/{g_0}} \(=\)
(1,1,-2), (1,2,-3), (1,3,-4). The first one cannot explain the three quark colors, and the third rises the question why nature should have omitted
the simple magnetic charge of size g\(=\)2\inlineTFinmath{{g_0}}. So we choose the second assignment providing us with the following fundamental
spin 1/2 preons and antipreons:\\




\dispTFNumberedEquationmath{{T_1},{T_2},{T_{-3}},{V_1},{V_2},{V_{-3}},{{ \overvar{T}{\_} }_{-1}},{{\overvar{T}{\_} }_{-2}},{{
\overvar{T}{\_} }_3},{{\overvar{V}{\_} }_{-1}},{{ \overvar{V}{\_} }_{-2}},{{ \overvar{V}{\_} }_3}}

T-rishons are electrically and magnetically charged dyons, while V-rishons are magnetically charged monopoles. The states carry the magnetic charge
in multiples of \inlineTFinmath{{g_0}} as index. The electric charges are \inlineTFinmath{q(T)=-q(\overvar{T}{\_})=e/3} and \inlineTFinmath{q(V)=q(\overvar{V}{\_})=0}
. Now one would like to know the masses of these fundamental preons, which cannot be answered in the affirmative. One possibility is that they are
all massless, avoiding the introduction of further massparameters. Alternatively one may assume that rishons with different magnetic and electric
charges carry different masses, as would be suggested by the classical dyon mass formula \inlineTFinmath{m=a{\sqrt{{q^2}+{g^2}}}} [7] .




\end{CellGroup}
\begin{CellGroup}
\Section*{3. Composite fermions}
From the above magnetic charge assignment g\(=\)(1,2,-3)\inlineTFinmath{{g_0}} it is clear that composite quarks and leptons of the first generation
have the structures  \inlineTFinmath{{R_1}-{R_{-3}}={R_2}} or \inlineTFinmath{\multsp {{\left( \overvar{R}{\_} \right) }_{-1}}-{{\left( \overvar{R}{\_}
\right) }_3}={{\left( \overvar{R}{\_} \right) }_{-2}}} where R\(=\)T,V. The single (double) line represents an attractive magnetic bindung between
a monopole/dyon and an antimonople/antidyon of strength one (two). One may also imagine that the lines resemble magnetic strings connecting the magnetically
charged rishons. Of course one has to consider composite states as dynamical rather than rigid structures. Due to quantum field theory one expects
a sea of virtual particles within any composite state. We now write down all composite quarks and leptons of the first generation. 


\mathGraphic{clemens.1.eps}

One sees that one has three quarks states arbitrarily assigned to the three quark colors R,G,B and exactly one state for each lepton. The mysterious
ordering of  rishons in a quark [2] now has a simple explanation. The three quark colors are distinguished by the magnetic charge of the rishon differing
from the other two. At first sight it seems natural that all these states have comparable masses because of their similar structure. We can't however
provide any dynamical calculations, since the three body problem is notoriously difficult. Therefore here and in the following we cannot answer questions
concerning the masses of composite states. For similar reasons we cannot say why spin 3/2 fermions are not observed, they may be excluded on dynamical
reasons or have higher masses. Another problem mentioned in [5] also arises here: the three color states are not absolutely degenerate because of
uneven electric charge interactions. This effect should however be small because of the large magnetic binding. Finally because of the present magnetic
charge assignment, contrary to the (1,3,-4) case, exotic composite fermions may be constructed as \inlineTFinmath{\multsp {R_1}-{{\left( \overvar{R}{\_}
\right) }_{-2}}-{R_1}} and \inlineTFinmath{\multsp {{\left( \overvar{R}{\_} \right) }_{-1}}-{R_2}-{{\left( \overvar{R}{\_} \right) }_{-1}}} where
R\(=\)T,V and we can say nothing about their existence and masses. 



The existence of three generations \inlineTFinmath{\multsp \multsp \imag :\multsp {{\nu }_e},{e^-},u,d\multsp \multsp } \inlineTFinmath{\Mvariable{II}:{{\nu
}_{\mu }},{{\mu }^-},c,s\multsp } \inlineTFinmath{\multsp \Mvariable{III}:\multsp {{\nu }_{\tau }},\multsp {{\tau }^-},t,b}   with the same quantum
numbers as the first one but different, approximately exponentially increasing masses is an unsolved puzzle resembling the old electron-muon puzzle
(remember Rabi's comment to the muon: who ordered that?). It is well known that radiative transitions between different generations are not observed,
which beside the wrong massdifferences compared to the compositeness scale presumably exclude radial excitations. The simplest solution is obtained
by adding rishon-antirishon-pairs to the first generation. Since we have rishons of magnetic charges \(\VerticalSeparator \)g\(\VerticalSeparator
\)\(=\)(1,2,3)\inlineTFinmath{{g_0}} at our disposal, we can construct four generations by adding \inlineTFinmath{{R_i}{{\left( \overvar{R}{\_} \right)
}_i}} - pairs (i\(=\)1,2,-3). Because of the unknown rishon masses and the lack of dynamical calculations one cannot say which pair corresponds to
which generation. Two effects seem to work against each other. If rishons with higher \(\VerticalSeparator \)g\(\VerticalSeparator \) have higher
masses they should correspond to higher generations. One the other side rishons with greater magnetic charge should exhibit a stronger magnetic binding
and therefore the resulting states should have lower mass. It is impossible to say without dynamical calculation which effect dominates the other
one. Due to the different structures it is conceivable that the generations differ in mass. We write down one possibility, bearing in mind, that
the reverse order of generations II-IV may be true. 

\mathGraphic{clemens.2.eps}

The problem arising here is that it is not clear a priori if the additional pairs consist of \inlineTFinmath{{T_i}{{\left( \overvar{T}{\_} \right)
}_i}} or \inlineTFinmath{{V_i}{{\left( \overvar{V}{\_} \right) }_i}} pairs or eventually mixtures of them.


\end{CellGroup}
\begin{CellGroup}
\Section*{ 4. Composite gauge bosons}
Here we consider the possibility that some or perhaps all gauge bosons of the standard model are composites of the fundamental rishons in (4). Beginning
with the weak bosons \inlineTFinmath{{W^+}} and \inlineTFinmath{{Z^0}} it seems likely that they are six-rishon-states. The charged weak bosons are
\inlineTFinmath{{W^+}=T\multsp T\multsp T\multsp V\multsp V\multsp V}  and \inlineTFinmath{{W^-}=\overvar{T}{\_}\overvar{T}{\_}\overvar{T}{\_}\overvar{V}{\_}\overvar{V}{\_}\overvar{V}{\_}}
in order to enable processes like \inlineTFinmath{{e^-}+{W^+}\rightarrow {{\nu }_e}} or \inlineTFinmath{\overvar{u}{\_}+{W^+}\rightarrow \overvar{d}{\_}}
. The neutral weak boson is \inlineTFinmath{{Z^0}=\Mvariable{RRR}\overvar{R}{\_}\overvar{R}{\_}\overvar{R}{\_}} to enable \inlineTFinmath{f+{Z^0}\rightarrow
f} , where f is any quark or lepton and \inlineTFinmath{R=T,V}. The structure formulas are : 

\mathGraphic{clemens.3.eps}

Other isomeric states of the W-boson may be obtained by interchanging the positions of \inlineTFinmath{{T_i}} and \inlineTFinmath{{V_i}} for i\(=\)1,2,-3.
Interestingly these three states can mediate the processes \inlineTFinmath{{{\left( \overvar{u}{\_} \right) }_C}+{W^+}\rightarrow {{\left( \overvar{d}{\_}
\right) }_C}} where C\(=\)R,G,B are the three quark-colors, while the state above gives \inlineTFinmath{{e^-}+{W^+}\rightarrow {{\nu }_e}}. One understands
that W-bosons can decay  in weakly interacting leptons as well as in quarks with color and how 6-rishon states can form, which is not clear in the
models in [2], [3] or [5].


Next we discuss the possibility that the gluons \inlineTFinmath{{G_{\alpha }}} (\(\alpha \)\(=\)1..8) are composites mediating processes like \inlineTFinmath{{u_R}\rightarrow
{u_B}}, \inlineTFinmath{{u_B}\rightarrow {u_G}}, \inlineTFinmath{{u_G}\rightarrow {u_R}} and similar ones for the d-quarks. The gluons if composite
must have the following structure:\\

\mathGraphic{clemens.4.eps}

The analogous gluons for the d-quarks are obtained by replacing \inlineTFinmath{{T_i}{{\left( \overvar{T}{\_} \right) }_i}\leftrightarrow {V_i}{{\left(
\overvar{V}{\_} \right) }_{i\multsp }}} where i\(=\)2,1,-3 in the above order. This structural representation could give an explanation for the box-structure
of gluons introduced by Shupe [2] and how 'ordering' of rishons in a quark is changed by gluons. Because of the 6-rishon-structure of the gluon similar
to W and Z, we face immediately the problem, why is the mass of the gluons zero as predicted by QCD ? Also unanswered remains the question of quarkconfinement.



While it is entirely possible that the photon \(\gamma \) is elementary, we consider the option of it's compositeness. Since the photon couples only
to charged particles it should consist of a mixture of \inlineTFinmath{{T_i}{{\left( \overvar{T}{\_} \right) }_i}}-pairs. One could envisage the
photon as six-rishon-state more in agreement with unification or as two rishon-state [2]. There are arguments in favour of the last case: a photon
must be able to couple to a d-quark containing one charged rishon, so only one \inlineTFinmath{{T_i}{{\left( \overvar{T}{\_} \right) }_i}}-pair can
be involved in this interaction and because we assume that T-rishons carry electric charge e/3 the electric interaction should also exist at the
level of preons. The structure formulas of the photon are:

\dispTFNumberedEquationmath{\gamma :\hspace{1em}{T_1}-{{\overvar{T}{\_} }_{-1}},{T_2}={{ \overvar{T}{\_}  }_{-2}},{T_{-3}}\equiv
{{\overvar{T}{\_} }_3}}

The above picture of a composite photon of course immediately faces several problems like the zero photon mass, photon selfinteractions or the possible
existence of \inlineTFinmath{V\overvar{V}{\_}} - states.

Finally we consider a possible compositeness [8] of the graviton g with spin 2 interacting with all particles without changing flavor ot color. If
composite it would in the simplest case be a mixture of \inlineTFinmath{R\multsp R\overvar{R}{\_}\overvar{R}{\_}} states (R\(=\)T,V) with spins (\(\uparrow
\)\(\uparrow \)\(\uparrow \)\(\uparrow \)). As in the case of the photon, the question why the graviton should be massless cannot be answered. Several
structures for the graviton are possible and presumable the graviton is a mixture of them. Both rishon-antirishon-pairs should be involved in interactions
giving the following formulas:

\mathGraphic{clemens.5.eps}


\end{CellGroup}


\begin{CellGroup}
\Section*{5. Composite particle interactions}
Weak interactions with charged bosons \inlineTFinmath{{W^{\pm }}} change the flavor of the involved particles, while those involving the neutral
\inlineTFinmath{{Z^0}} preserve flavor. As examples we display the following interactions with structural formulas:

\mathGraphic{clemens.6.eps}

Three incoming rishons are annihilated by three antirishons of the weak boson \inlineTFinmath{{W^-}}, while the remaining three form the outcoming
ones. From the first reaction one sees that \inlineTFinmath{{W^-}} changes flavor but not the color of quarks, which in the original rishon model
was mysteriously ascribed to an amorphous structure. It is clear that the weak bosons can couple in an equal manner to fermions of higher generations.

Strong interactions are described by gluons changing the color of quarks. We display the interaction of some gluons with an u and a d-quark:	

\mathGraphic{clemens.7.eps}

The gluon coupling \inlineTFinmath{{d_R}} and \inlineTFinmath{{d_B}} is obtained from the gluon coupling \inlineTFinmath{{u_R}} and \inlineTFinmath{{u_B}}
by replacing \inlineTFinmath{{T_2}{{\left( \overvar{T}{\_} \right) }_{-2}}\leftrightarrow {V_2}{{\left( \overvar{V}{\_} \right) }_{-2}}}. The coupling
of a gluon to an electron is not allowed, since only two of the six rishons constituting an gluon could annihilate with the corresponding antirishons
of the electron. So the present picture could explain why gluons couple only to quarks but not to leptons and change color but not flavor. In a similar
manner gluons can couple to quarks of higher generations.

The coupling of a photon to a charged quark or lepton is shown in the cases

\mathGraphic{clemens.8.eps}

One rishon is annihilated by the corresponding antirishon and flavor or color are not changed. It is clear that the photon can couple to all charged
particles.

Concerning the possible compositeness of the graviton [8] it is clear that it can couple to all composite particles without changing flavor or color.
Two rishons of the fermion are annihilated by the corresponding two antirishons of the graviton and the fermion remains unchanged in structure. One
example is:

\mathGraphic{clemens.9.eps}

With the exception of the photon, if constructed as two-rishon-state, all composite gauge bosons can couple only to composite fermions. The pleasant
feature of this approach is that we have an explanation, why the composite fermions have the observed interactions. The unpleasant feature is that
we cannot explain the masses of the gauge bosons and the different coupling constants. 

Several exotic bosons can be constructed and we show as examples: a leptoquark X, as needed for protondecay, changing a lepton in a quark and a colored
W-boson changing flavor and color of a quark.

\mathGraphic{clemens.10.eps}

CP-violation requires the transition \inlineTFinmath{\overvar{{K^0}}{\_}\big(s\multsp \overvar{d}{\_}\big)\rightarrow {{\Mfunction{K}}^0}(d\multsp
\multsp \overvar{s}{\_})}, which in the standard model  is described through a box diagram involving W-bosons and up-like quarks. In our structural
model this process could occur via exchange scattering, where the \inlineTFinmath{\overvar{R}{\_}R} - pair goes from  the \inlineTFinmath{s}-quark
to the \inlineTFinmath{\overvar{d}{\_}}-quark giving a d-quark and a \inlineTFinmath{\overvar{s}{\_}}-quark. A similar thing would happen in the
transition  \inlineTFinmath{\overvar{{B^0}}{\_}\big(b\multsp \overvar{d}{\_}\big)\rightarrow {B^0}\big(d\multsp \multsp \overvar{b}{\_}\big)}. Neutrino
oscillations are transitions \inlineTFinmath{{{\nu }_i}\leftrightarrow {{\nu }_j}} in vacuum or matter, which in our model can be described as transitions
\inlineTFinmath{{R_i}\multsp {{\left( \overvar{R}{\_} \right) }_i}\rightarrow \multsp {R_j}\multsp {{\left( \overvar{R}{\_} \right) }_j}} or \inlineTFinmath{{R_i}\multsp
{{\left( \overvar{R}{\_} \right) }_i}\rightarrow } energy.




\end{CellGroup}
\begin{CellGroup}
\Section*{6. Summary}
In summary we have proposed a structural model of composite quarks and leptons built by two rishons T(q\(=\)1/3) and V(q\(=\)0) each with magnetic
charges g\(=\)(1,2,-3)\inlineTFinmath{{g_0}} and their antiparticles. The model preserves much of the simplicity of the original rishon model. One
understands why composite fermions have the observed interactions with their properties and has a simple explanation of higher generations by adding
rishon-antirishon-pairs with different magnetic charges. The main problem is the lack of dynamical calculations, so that many questions concerning
particle masses, mixing parameters and relative coupling strengths cannot be answered. A severe theoretical problem is: can nonabelian quantum field
theories like QCD be obtained from a model of this kind ?




\end{CellGroup}
\begin{CellGroup}
\Section*{References:}
\Reference{[1] L. Lyons, Prog. Part. Nucl. Phys. 10, 227 (1983)}
\Reference{[2] H. Harari, Phys. Lett. 86B, 83 (1979),  M.A. Shupe, Phys. Lett. 86B, 87 (1979)}
\Reference{[3] H. Harari, N. Seiberg, Phys. Lett. 98B, 269 (1981)}
\Reference{[4] J.C. Pati, Phys. Lett. 98B, 40 (1981)}
\Reference{[5] D.D. Wu, Phys. Rev. D34, 280 (1986), Phys.Lett. 122B, 54 (1983)}
\Reference{[6] D. Zwanziger, Phys. Rev. D3, 880 (1970)}
\Reference{[7]  C. Montonen, D. Olive, Phys. Lett. 72B, 117 (1977)}
\Reference{[8]  H. Terazawa, K. Akama, Phys. Lett. 96B, 276 (1980),  C. Heuson, Phys. Lett. 105B, 381 (1981)}

\end{CellGroup}

\end{CellGroup}

\end{document}

