## What’s up with 2d Yang-Mills theory?

24/12/2010

Being near Christmas I send my wishes to all my friends around the World. Most of them are physicists like Attilio Cucchieri that I cited a lot in this blog with his wife Tereza Mendes for their beautiful works on Yang-Mills propagators. Attilio asked to me again about the behavior of 2d Yang-Mills propagators and my approach to the theory. As you know, lattice computations due to Axel Maas (see here) showed without doubt that here, differently from the higher dimensional case, the scaling solution appears. This is an important matter as ‘t Hooft showed in 1974 (see here) that, in the light cone gauge, Yang-Mills theory has no dynamics. Indeed, the Lagrangian takes a very simple form

${\cal L}=-\frac{1}{2}{\rm Tr}(\partial_- A_+)^2$

and the nonlinear part is zero due to the fact that the field has just two Lorentzian indexes. Please, note that here $A_\pm=\frac{1}{\sqrt{2}}(A_1\pm A_0)$ and  $x_\pm=\frac{1}{\sqrt{2}}(x_1\pm x_0)$. Now, you can choose the time variable as you like and if you take this to be $x_+$ you see that there are no dynamical equations left for the gluon field! You are not able to do this in higher dimensions where the dynamics is not trivial and the field develops a mass gap already classically. Two-dimensional QCD is not trivial as there remains a static (nonlocal) Coulomb interaction between quarks. You can read the details in ‘t Hooft’s paper.

Now, in my two key papers (here and here) I proved that, at the classical level the Yang-Mills field maps on a quartic massless scalar field in the limit of the coupling going to infinity. So, I am able to build a quantum field theory for the Yang-Mills equations in this limit thanks to this theorem. But what happens in two dimensions? The scalar field Lagrangian in the light-cone coordinate becomes

${\cal L}=-\partial_+\phi\partial_-\phi-\lambda\phi^4$

and you can see that, whatever is your choice for the time variable, this field has always a dynamics. In 2d I am not able to map the two fields and, if I choose a different gauge for the Yang-Mills field, I can find propagators that are no more bounded to be massive or Yukawa-like. Indeed, the scaling solution is obtained. This is a bad news for supporters of this solution but this is plain mathematics. It is interesting to note that the scalar field appears to have nontrivial massive solutions also in this case. No mass gap is expected for the Yang-Mills field instead.

Thank you a lot Attilio for pointing me out this question!

Merry Xmas to everybody!