Is the confinement mechanism understood in 1+1 QCD ('t Hooft's model)?

Question

I was following the coleman lectures on "Aspects of symmetry" particularly the chapter about the 't Hooft's model. Then I have wandered into older papers like the 't Hooft's papers and many others. And concerning the reason why there are no free quarks in this model it seems to me that their reasoning is the following:

Calculate the dressed propagator and you will get something like :

$$\frac{ip_-}{2p_+p_- -M^2-\frac{g^2|p_-|}{\lambda\pi}+i\varepsilon}$$

This is a propagator depending on the cut-off $\lambda$.Then, because of the infra-red divergence, we have to restore gauge invariance taking the $\lambda\to 0$ limit. The pole of this propagator is shifted towards $p_-\to \infty$. We conclude that there is no physical single quark state.

But, Einhorn claims (in PhysRevD.14.3451) that the dependence on $\lambda$ has nothing whatever to do with the confinement mechanism. In order to prove that the 't Hooft's argument is wrong, Einhorn switch off the coulomb potential but retain a constant gauge-dependent term. He finds that the interaction between $\bar{q}q$ pairs cancels the term in the self energy, so that free quarks are produced. So the confinement must be obtained by other means.

By the way, It seems that Coleman is using the principal value method and not the original 't Hooft's regularization. So it is not obvious for me to realize whether Coleman agrees or not.

Is the underlying reason of confinememnt (in 't Hooft's model) clear currently? Are Einhorn's arguments wrong?

This post imported from StackExchange Physics at 2017-06-25 18:17 (UTC), posted by SE-user physics_teacher