Bound states and asymptotic limits for quantum chromodynamics in two dimensions

For quantum chromodynamics in two dimensions in the limit of an increasing number of colors ($N_C\to \infty$) with $\frac{g^2{N}_C}{\pi }\equiv m^2$ fixed, the meson bound-state problem requires the solution to 't Hooft's integral equation. For low masses, we express the 't Hooft equation as an explicit matrix problem, which enables us to calculate accurate masses ${m_n}^2$, ($n=0,1,\dots ,50\mathrm{}$) and wave functions ${\phi }_n^{a\overline{b}}\mathrm{}\left(x\right)\mathrm{}$ for the quark ($a$)-antiquark ($\overline{b}$) bound states. In the large-mass (or WKB) limit, the 't Hooft equation is solved analytically. We obtain the spectrum ${m_n}^2={\pi }^2{m}^2(n+\frac{3}{4})+({m_a}^2+{m_b}^2)\ln n+C\left({m_a}^2\right)+C\left({m_b}^2\right)+O\left(\frac{1}{n}\right)$, where $C\left({m_a}^2\right)$ is explicitly calculated. The corresponding WKB wave function, ${{\phi }^{a\overline{b}}}_n\mathrm{}\left(x\right)\mathrm{}\simeq \sqrt{2}sin\mathrm{}\left[\right(n+1\mathrm{})\mathrm{}\pi x+{\delta }_n^{a\overline{b}}\mathrm{}(x\left)\right]$, has a phase shift ${\delta }_n^{a\overline{b}}\mathrm{}\left(x\right)\mathrm{}$, which is crucial for obtaining the following new asymptotic results for meson bound-state amplitudes: (1) The normalization of the scaling term for $e^{+}{e}^{-}\to X$ and $e^{-}h\to e^{-}X$ is demonstrated to agree exactly with the parton model.(2) The inclusive cross section for $e^{+}{e}^{-}\to \mathrm{hX}$ is given by the sum over quark fragmentation ($a\to h+b$) functions $D_{\frac{h}{a}}\left(x_F\right)={\left|{\Phi }_n^{\mathrm{ab}}\right(\frac{1}{x_F}\left)\right|}^2$, where the complex amplitude ${\Phi }_n^{a\overline{b}}\mathrm{}\left(x\right)\mathrm{}$ is the analytic continuation of the wave function ${\phi }_n^{\mathrm{ab}}\mathrm{}\left(x\right)\mathrm{}$ to $x>\mathrm{}1\mathrm{}$. (3) Regge powers are shown to have the correct phase $A\sim s^{{\alpha }_{a\overline{b}}\mathrm{}\left(0\right)\mathrm{}}{e}^{-i\pi {\alpha }_{\mathrm{ab}}\mathrm{}\left(0\right)\mathrm{}}$. These calculations explicitly verify that hard (parton) and soft (dual-Regge) physics can indeed be combined and illustrate the new physics resulting from this unification.