Two dimensional QCD with matter in adjoint representation: What does it teach us?

Abstract

We analyse the highly excited states in $QCD_2 (N_{c}\rightarrow\infty) $ with adjoint matter by using such general methods as dispersion relations, duality and unitarity. We find the Hagedorn-like spectrum $ρ(m) \sim m^{-a}\exp(β_H m)$ where parameters $β_H$ and $a$ can be expressed in terms of asymptotics of the following matrix elements $f_{n_{\{k\}}} \sim \la 0|Tr(\barΨΨ)^{k}|n_{k}\ra$. We argue that the asymptotical values $f_{n_{\{k\}}}$ do not depend on $k$ (after appropriate normalization). Thus, we obtain $β_H= (2/π)\sqrt{π/g^2N_{c}}$ and $a = -3/2$ in case of Majorana fermions in the adjoint representation. The Hagedorn temperature is the limiting temperature in this case. We also argue that the chiral condensate $\la 0|Tr(\barΨΨ) |0\ra$ is not zero in the model. Contrary to the 't Hooft model, this condensate does not break down any continuous symmetries and can not be considered as an order parameter. Thus, no Goldstone boson appears as a consequence of the condensation. We also discuss a few apparently different but actually tightly related problems: master field, condensate, wee-partons and constituent quark model in the light cone framework.