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 (NC→∞) with g²NCπ≡m² 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 mₙ², (n=0,1,,50) and wave functions φₙab̄(x) for the quark (a)-antiquark (b̄) bound states. In the large-mass (or WKB) limit, the 't Hooft equation is solved analytically. We obtain the spectrum mₙ²=π²m²(n+3/4)+(mₐ²+mb²)lnn+C(mₐ²)+C(mb²)+O(1/n), where C(mₐ²) is explicitly calculated. The corresponding WKB wave function, φab̄ₙ(x)√2sin[(n+1)πx+δₙab̄(x)], has a phase shift δₙab̄(x), 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^-→X and e^-h→e^-X is demonstrated to agree exactly with the parton model.(2) The inclusive cross section for e⁺e^-→hX is given by the sum over quark fragmentation (a→h+b) functions Dh/a(xF)=|Φₙᵃᵇ(1xF)|², where the complex amplitude Φₙab̄(x) is the analytic continuation of the wave function φₙᵃᵇ(x) to $x>1$. (3) Regge powers are shown to have the correct phase A~s^αab̄(0)e^-iπαab(0). 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.