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%% subsection 3.1 QCD$_{1+1}$ with Fundamental Matter [slac-pub-7152-0-0-3-1 in slac-pub-7152-0-0-3: ^slac-pub-7152-0-0-3 >slac-pub-7152-0-0-3-2]
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\subsection{\usemenu{slac-pub-7152::context::slac-pub-7152-0-0-3-1}{ QCD$_{1+1}$ with Fundamental Matter }}\label{subsection::slac-pub-7152-0-0-3-1}
This theory was originally considered by 't Hooft in the limit of
large $N_c$ \cite{28}. Later Burkardt \cite{3}, and
Hornbostel, Pauli and I \cite{4}, gave essentially complete
numerical solutions of the theory for finite $N_c$, obtaining the
spectra of baryons, mesons, and nucleons and their wavefunctions.
The results are consistent with the few other calculations available
for comparison, and are generally much more efficiently obtained.
In particular, the mass of the lowest meson agrees to within
numerical accuracy with lattice Hamiltonian results \cite{29}.
For $N_c=4$ this mass is close to that obtained by 't Hooft in the
$N_c\rightarrow\infty$ limit \cite{28}. Finally, the ratio of
baryon to meson mass as a function of $N_c$ agrees with the
strong-coupling results of Date, Frishman and Sonnenschein
\cite{30}.
In addition to the spectrum, of course, one obtains the
wavefunctions. These allow direct computation of, \eg, structure
functions. As an example, Fig. \docLink{slac-pub-7152-0-0-3.tcx}[fig1]{1} shows the valence
contribution to the structure function for an SU(3) baryon, for two
values of the dimensionless coupling $m/g$. As expected, for weak
coupling the distribution is peaked near $x=1/3$, reflecting that
the baryon momentum is shared essentially equally among its
constituents. For comparison, the contributions from Fock states
with one and two additional $q\bar{q}$ pairs are shown in Fig. 2.
Note that the amplitudes for these higher Fock components are quite
small relative to the valence configuration. The lightest hadrons
are nearly always dominated by the valence Fock state in these
super-renormalizable models; higher Fock wavefunctions are typically
suppressed by factors of 100 or more. Thus the light-cone quarks
are much more like constituent quarks in these theories than
equal-time quarks would be. As discussed above, in an equal-time
formulation even the vacuum state would be an infinite superposition
of Fock states. Identifying constituents in this case, three of
which could account for most of the structure of a baryon, would be
quite difficult.
\begin{figure}
\epsfxsize=3.0in
\centerline{\epsfbox{8084A05.eps}}
\caption[*]{Valence contribution to the baryon structure function in
QCD$_{1+1}$, as a function of the light-cone longitudinal momentum
fraction. The gauge group is SU(3), $m$ is the quark mass, and $g$
is the gauge coupling \cite{4}.}
\label{fig1}
\end{figure}
\begin{figure}
\centerline{\epsfbox{8084A01.eps}}
\caption[*]{Contributions to the baryon structure function from higher
Fock components: (a) valence plus one additional $q\bar{q}$ pair;
(b) valence plus two additional $q\bar{q}$ pairs \cite{4}.}
\label{fig2}
\end{figure}