## Classical Yang-Mills theory and mass gap

One of the key ingredients to build up a quantum field theory is to have a set of solutions of classical equations of motion to start with. Then, given such solutions, we are able to perform perturbation theory and to extract results from the theory to be compared with experiment. I think that my readers are familiar with standard approach having free equations of the theory solved. When path integrals are used, we solve for the Green function of the free theory but we are talking about the same thing: we know how to solve our theory in some limit and then we build on it. So, to give an answer to the question of the mass gap for Yang-Mills theory, we have to know how to solve the theory in a limit we are not so familiar: strong coupling limit. So far, very few was known about this limit except knowledge acquired through lattice computations. Also in this latter case, for several years a lot of confusion pervaded the field: Does gluon propagator go to zero or not? Enlarging volumes produced an answer that is a reason for hot debate yet: Gluon propagator does not go to zero at very low energy but reaches a finite value. In literature this is known as the decoupling solution to be contrasted with the scaling solution having a propagator going to zero at very small momenta. If we know gluon propagator, we are able to compute the behavior of QCD at very low energies (see here) and this is a well-known fact since eighties.

The question of existence of a class of solutions for Yang-Mills theory to work with at low energies has been successfully answered quite recently. I have written a nice pair of papers that went published in respectful journals and permitted to solve all this matter (see here and here). Two papers were needed because Terry Tao showed that a proof in a key theorem (mapping theorem) was not correct. After this, I was able to give  an answer that both agreed. My aim in this post is to explain, with some simple mathematics, what is the content of this theorem that produces a set of classical solutions to build up a quantum field theory in the low-energy limit for Yang-Mills theory and so QCD.

The key element is a mapping theorem. We map two classical theories, one of this we are able to solve exactly. So, consider a massless scalar field theory

$\Box\phi+\lambda\phi^3=0.$

Contrarily to common wisdom, we are able to solve this exactly. Our solution can be written down as

$\phi(x)=\mu\left(\frac{2}{\lambda}\right)^\frac{1}{4}{\rm sn}(p\cdot x+\theta,i)$

provided that

$p^2=\left(\frac{\lambda}{2}\right)^\frac{1}{2}\mu^2.$

Hera $\mu$ is an integration constant with the dimension of energy, $\theta$ is another integration constant and ${\rm sn}$ is the snoidal Jacobi’s elliptic function. Why is this solution so interesting? The reason is that we started with a massless equation and the solution describes a wave with a massive dispersion solution of a free particle! This is the famous mass gap when we translate this result to quantum field theory. I have done this here. So, the classical theory already has the feature of a mass gap. Scalar theory proves to be trivial for the simple reason that we produce, in the low-energy limit, free massive excitations. This is a long awaited result that is going to get increasingly confirmed from other theoretical studies. I will discuss this issue in another post.

What is the relation, if any, between a massless scalar field theory and Yang-Mills theory? Indeed, there exists a deep relation in the low-energy limit, when the coupling becomes increasingly large, as the solutions of the two theories can be mapped. So, for SU(3), mapping theorem shows that

$A_\mu^a(x)=\eta^a_\mu\phi(x)+O\left(\frac{1}{\sqrt{3}g}\right)$

being $\phi(x)$ our solution above provided the substitution $\lambda\rightarrow\sqrt{3}g$. This is a very beautiful result as this gives at once the following conclusions:

• Strong coupling solutions of classical Yang-Mills theory are free massive waves.
• Yang-Mills theory displays massive solutions already at classical level.
• Quantum theory maintains such conclusions as I showed in my papers.

Lattice computations beautifully confirmed this mapping theorem in d=2+1 as showed by Rafael Frigori in a very nice paper (see here). Strong hints are also seen in d=3+1 by other authors and it would be very nice to see an extended computation in this case as the one Frigori did in d=2+1. For yourselves, you can check with Mathematica or Maple the equations given above. You will also see that gauge invariance is not hindered.

### 4 Responses to Classical Yang-Mills theory and mass gap

1. Daniel de França MTd2 says:

Hi Marco!

Long time no see you! really missed your posts. So, let me ask you this:

Can’t you use the classical solution as a propagator, and then try to show that there is a gap for any quantum solution?

2. BlackGriffen says:

Have you been able to show that you have the complete set of solutions and not just a solution?

• mfrasca says:

This is a solution much in the same way plane waves are solution of the wave equation. If you change boundary conditions, things may change a lot and I have not a complete solution in this sense as happens for Liouville equation. As you can see, this is enough for my aims.

Marco