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{\small
January 16, 2003\\ }

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{\bf\large A Subcolor Quark Model for Baryonic, Leptonic  \\

and Dark Matter }

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Risto Raitio    \footnote{Retired. E-mail: risto@rai.compart.fi}

Dept. of Theoretical Physics,  \\

University of Helsinki, Helsinki, Finland	\\

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{\bf\large Abstract} 

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A subconstituent model for matter and its interactions is proposed. The model is based on a charge \small{1/3} and a neutral SU(3) of subcolor fermions, called trions, and a condensed matter-like vacuum, of which a simple scheme is outlined.  \end{center}

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\begin{center}PACS 12.60.Rc 
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\section{Introduction}
\label{sec:1}

A model for quark and lepton constituents, maxons, was suggested in [1]. The model was extended to describe dark matter in [2]. In this note we make an attempt to analyze in more detail the trion scheme for the basic constituents. 

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Unlike the present main stream high symmetry theories, our goal is to build a  low symmetry "urtheorie" for the immediately after Planck time universe. The vacuum of this model is suggested to be low temperature condensed matter-like. We discuss the possibility that this model makes at low energies a transition to the standard model and gravity. Proving that is, needless to say, no minor project.  Therefore, the nature of this note is humbly contemplative.

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The two basic constituents of the model are trions: 

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$m_{i}^{+}$ of charge $+1/3$ and color triplet and 

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$m^{0}$ of zero charge and color singlet. 

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\begin{flushleft}Both are subcolor triplets. We set \end{flushleft}

$\alpha_{s,subcolor} > \alpha_{s,color}$ and $T_{conf, subcolor} > T_{conf, color}$

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The confinement scale of subcolor is assumed to be at least several decades higher than for color.

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Let us form all possible three body states using the $m_{i}^{+}$ and $m^{0}$. We obtain eight states, the first generation quarks and leptons, together with their antiparticles as follows  

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$u_{k} = \epsilon_{ijk}m_{i}^{+}m_{j}^{+}m^{0}$  

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$\overline{d_{k}}$ = $m_{k}$ $m^{0}$ $m^{0}$ 

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$e^{-} = \epsilon_{ijk}m_{i}^{-}m_{j}^{-}m_{k}^{-} $ 	

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$\nu_{e} = m^{0}m^{0}m^{0} $

\begin{flushleft}where the subcolor indices are not shown for clarity.\end{flushleft}

These states are of light mass (or massless) like the nucleons are light compared to $\Lambda_{QCD}$. 

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Heretically, the trions do not carry the ordinary baryon, lepton, or weak i-spin quantum numbers. Therefore the character of the weak interaction may be different in principle from the standard model but phenomenologically it is required to explain the same experimental results. The success of this depends, of course, on the properties of the many body trion initial stage dynamics. As such, methods for computing low temperature condensed matter phenomena for electromagnetic interactions exist, for trionium atoms in this case. We can not exclude the possibility of having to introduce or seeing to appear as solutions objects of more complicated structure than the trions. In any case, the cautious reader should consider stop reading here.

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Weak interactions (and lepto-quark transitions) occur by trion intercange with the vacuum. These would require a careful discussion, beyond the scope of this note. 

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This setup of is based by the following arguments, or goals: 

\begin{enumerate}

\item the various forms of matter should be understood in a unified way with a minimal representations of fermions and gauge fields,  we choose $SU(3)_{subcolor}$ and $U(1)_{Q}$ to be the only primary gauge fields of the model,

\item the T=0 state should be a very low entropy system,

\item the vacuum must have the observed low vacuum energy density value,

\item the interactions of the model are required agree experimentally with the standard model \footnote{The standard model is analyzed from a fresh point of view in [3].} at energies  below  $\Lambda_{subcolor}$.

\end{enumerate}

We end up considering hadrons as "molecules" rather than "atoms" of the basic constituents. 

\section{The Trions and the Vacuum}
\label{sec:2}

We summarize the particles and their interactions at various distances or energy scales

\begin{enumerate}

\item trions; QED at all distances, $SU(3)_{subcolor}$ for $l_{lab} (= 10^{-17} cm) > l > l_{q} >> l_{p}$ where $l_{p}$ is the Planck length, $l_{q}$ = $1/\Lambda_{subcolor}$, and $l_{lab}$ is the current resolving accuracy of the present experiments (all numerical equalities are approximate only in this note),

\item quarks, leptons; effective	$SU(3)_{color}xSU(2)_{L}xSU(2)_{R}xU(1)$ at laboratory conditions above the electro-weak symmetry energy, about 200 GeV, and below $1/l_{q}$, 

\item atoms, molecules and condensed, as well as ordinary, matter; electromagnetic interaction (QED),

\item massive bodies; general relativity at distances $l_{universe} > l > 1$ cm. 

\end{enumerate}

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In regions devoid of matter the vacuum is the dominant physical substance. An interesting candidate model for vacuum is a condensed state of matter like superfluid or superconductor, eg. $^{3}$He-A. How to handle the standard theory vacuum at the same time as the "supervacuum"? May be the standard theory forces are emergent [4] from a microscopic theory, which in this case is the trion matter condensate. As a first step we consider joining the standard theory and "supervacuum" such that the former effectively disappears at $l << l_{q}$ cm and the latter dominates below that limit, which includes the conditions of the very early universe. 

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The vacuum must have specific properties as to produce something "emergent", the standard model, and be a vacuum with the near zero energy density. The standard approaches to the direction of Planck scales have been done with some "grand" symmetry, whether it is some GUT or superstring type of theory. Here we wish to take a look in the opposite direction, ie. to a lower symmetry small distance unified theory. Unification is understood here in terms of fermions, the trions.

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In the present scenario below the distances $l << l_{q}$ the subcolor force reaches at some scale the asymptotic freedom conditions leaving electromagnetism as the dominant force. This is good for two reasons. First, QED provides a "backbone" in the transitions from above to below the scale $l_{q}$. Secondly, QED may provide a mechanism to produce the equivalent of atoms for the condensed state vacuum in the ultra short distance world. Only in such a vacuum we may expect the analogy with the "helium droplet universe" to work. 

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With charged and neutral trions we now ask what kind of "atoms" can be expected to form the lowest energy state. The desired result coud be  $m^{+/-}-m^{0}$ pair(s), or two charged and one neutral m, resembling the $^{3}He$ nuclei, and orbiting $m^{-/+}$ "electrons" bound to orbit by the Coulomb potential.  This trionium atom would be of such size that the subcolor force is weaker then the Coulomb force. These "subatoms" are unobsevable with present or foreseeable future methods (unless we are extremely lucky). It can be considered as "black object", and one may have to look for other type of models for it.

\section{On Quark and Lepton Generations}
\label{sec:3}

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The masses of the generations are quite irregular: the first generation mass scale is roughly 0.3 GeV, the second roughly 1 Gev and the third between 1-100 GeV with the neutrino masses very small in the eV region telling that intrageneration differences are very large. We cannnot see any conventional interaction mechanism to generate the three quark and lepton generations. Therefore we suggest that vacuum excitations, pictorially trions on the excited trionium vacuum shell, make up the second and third quark and lepton generations. An interaction in this supervacuum may be naively modeled eg. as He-He atom potential. More generally, the supervacuum objects may be one or more of the following: trions (trionium, corresponding helium atoms) with suitable number of Fermi points and topological structures, some string theory objects which can vibrate or permit something else to vibrate with a finite number of states or even stable black holes. In [4] a possible mechanism with the standard model embedded in a soliton wall of higher dimension, as well as other mechanisms, are discussed. These are beyond the scope of this note (and the author's skills).

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At this point we compare our approach to older and deeper labored preon approaches [5], [6] and [7]. They use more internal symmetry in the form of GUT of some form, or supersymmetry. It is, of course, recognized that the mathematical generalizations of quantum theory to field theories of ever higher symmetry have, during the last eight decades, been very successful. The representations needed in the main stream theories are, however, often quite high, and include smaller or larger number of particles not yet observed. Leaving preons aside, the leading candidate theory, the string theory, is discussed in [8]. At present, no model or theory beats the standard theory in its region of validity. But its twenty parameters still call for an explanation in terms of a more profound theory, towards which we hope to make a modest step.  

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This trion model is more pragmatic than the above theories, or other theories of the last 30 years, though our goals are alike. The trion model is constructed bottom up, starting with a magnifying glass on fermions near the Planck length scale, trying to take as few as possible of them, and gauge fields. Higher symmetries do stabilize infinities of theories but are not (yet) required by experiments. Therefore, at this moment we are tentatively satisfied with the hypothesis the standard theory emerging from the trionium condensed matter world.

\section{Fermions in the Early Universe}
\label{sec:4}

We now analyze a short period of the very early universe where the trions were the only constituents. The initial state of our model universe consisted of a large number of fermion trions, most likely packed in a large "freezer". The entropy was very low, or zero. The system had a small pressure, only due to surface effects. When the box walls were "removed" physical processes began to evolve due to fluctuations of the supervacuum, and the entropy and temperature of the universe increased leading to the bang.

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At $T > T_{c, subcolor}$, the trions formed a dense gas. The charged trions had a chance of being in thermal equilibrium due to QED collisions. The neutral trions interacted with a much longer free mean path, assuming $\alpha > \alpha_{subcolor}$ and, by random fluctuations, they tended to gather together in lumps of various sizes, to be later observed as galaxy level structures. 

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At $T < T_{c, subcolor}$, formation of three trion bound states started, favoring neutral trions, producing quarks and leptons. It seems tempting that the "four fermi" $SU(3)_{subcolor}$ interaction of $d + u -> e + \nu$ forms a bound state, the W boson, and the Z is formed similarly. Especially the weak transition $e -> \nu$ looks pretty ugly in terms of the trions: three charged are emitted and three neutral absorbed. This can happen more naturally if we consider the trion popping in and out of vacuum. So we rather associate the transition as a rearrangement process. Very likely, there are other similar processes at this energy scale, associated with the standard theory symmetry breeakdown, to be discovered.  

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We suppose the top quark be the heaviest three trion bound state. The mass spectrum between below one eV and up to 200 GeV is an open question for all theories.  

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As concerns gravity, in a cold universe the dark matter still interacts with ordinary matter via gravitation. 	

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In relatively cold regions the trions may pair to form a condensed state of boson matter. The vev of such a condensate is an interesting quantity. It is tempting to speculate of its connection to dark energy and/or inflation. In an expanding universe, this energy density may go slowly to zero. Studies in certain He systems indicate that a zero or very small cosmological constant may be possible [4].  

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For authoritative general reviews of cold dark matter and energy, and the cosmological constant, see [9] and [10].  

\section{Conclusions}
\label{sec:5}

The purpose of the model discussed here is to approach a simple, unified picture of the known forms of matter using a minimal number of basic constituents and gauge fields. This model assumes that baryons, leptons and dark matter are described by the same subcolor level constituents, which have confining gauge interaction between them. In principle, the vacuum properties may be calculated from the trionium model. In spite of the "adventurous" (see the Foreword in [4], the monograph) nature of this note, it need not be too far fetched to use condensed matter calculation methods and analogies to build such a supervacuum in the future.


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\bibitem{ref:b} Risto Raitio, A Subcolor Quark Model for Baryonic, Leptonic and Dark Matter, arXiv: (v3, Dec 2, 2002).

\bibitem{ref:c} J. D. Bjorken, arXiv:. 

\bibitem{ref:d} G.E. Volovik, Physics Reports \textbf{351}, 195-348 (2001), arXiv:gr-gc/000509; "The Universe in a Helium Droplet", Oxford University Press (2003). 

\bibitem{ref:e} S. L. Adler, Frustrated SU(4) as the Preonic Precursor of the Standard Model, arXiv:hep­.

\bibitem{ref:f} H. Terazawa, High Energy  Physics in the 21st Century, in Proceedings of 22nd International Workshop on the Fundamental Problems of High Energy Physics and Field Theory;  KEK Preprint 99-46, July 1999.

\bibitem{ref:g}  H. Harari, Phys. Lett. \textbf{86B} (1979); M. A. Shupe, Phys. Lett. \textbf{86B} (1979). H. Harari and N. Seiberg, Phys. Lett. \textbf{98B} (1982); Nucl. Phys. \textbf{B204} (1982).

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\bibitem{ref:i} Joel R. Primack, Status of Cold Dark Matter, arXiv:. 

\bibitem{ref:j} P. J. E. Peebles, Bharat Ratra, The Cosmological Constant and Dark Energy, arXiv: (v2, 20 Nov 2002). 

\bibitem{ref:k} J. D. Bjorken, Phys. Rev. \textbf{D64}, 85008 (2001), arXiv:.   

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