## Classical Yang-Mills theory and mass gap

Yesterday I posted a paper on arxiv (see here). In this paper I have given a solution to the classical equations of motion of Yang-Mills theory. Indeed, as already seen for the quartic scalar field (see here), one can show, with undergraduate level arguments, that Yang-Mills theory has a massive exact solution at least at a classical level. This can be seen as a simple proof that a massless theory can indeed produce a mass gap and this mass gap is simply a dynamical effect arising from the self-interaction term of the equations of motion.

As obtained from standard textbooks, the equations of motion for the Yang-Mills potential $A^a_\nu$ are

$\small \ddot A^a_\nu-\Delta_2A^a_\nu-\partial_\nu(\partial^\mu A^a_\mu)+gf^{abc}A^{b\mu}(\partial_\mu A^c_\nu-\partial_\nu A^c_\mu)+$$gf^{abc}\partial^\mu(A^b_\mu A^c_\nu)+g^2f^{abc}f^{cde}A^{b\mu}A^d_\mu A^e_\nu = 0.$

We use latin letters for group indeces and greek letters for space-time indeces. As it is already seen for Maxwell equations, we fix properly the gauge to obtain a solution and we choose the Lorentz gauge $\partial^\mu A^a_\mu=0$ following a standard textbook procedure already known for the electromagnetic field. The coupling constant $g$ is adimensional and no other constants enter into the theory. $f^{abc}$ are structure constants of the gauge group.

In order to find a solution we make the choice to take all the components of the field equal. So we write $A^a_\mu=(A_\mu,A_\mu,\ldots,A_\mu)$ with $N^2-1$ elements for SU(N) and $A_\mu=(\phi,\phi,\phi,\phi)$ and so we have a total of $4(N^2-1)$ degrees of freedom. With this substitution the above equation assumes a more familiar form

$\ddot\phi-\Delta_2\phi+Ng^2\phi^3=0$

that is just the equation of the quartic scalar field and we know its solution! Checking back here we can write down the solution

$\phi=\Lambda\left(\frac{2}{Ng^2}\right)^{\frac{1}{4}}{\rm sn}(px+\varphi,i)$

where we have changed the name to the integration constant calling it $\Lambda$ and introduced another one, a phase, $\varphi$. The incredible interesting new is that the above solution holds only if the following dispersion relation holds

$E^2=p^2+\Lambda^2\left(\frac{Ng^2}{2}\right)^{\frac{1}{2}}$

and we see that the classical Yang-Mills field has acquired a mass and this mass goes to zero as the coupling $g$ goes to zero. So stronger is the coupling in the self-interacting term and larger is the mass gap. But this says us something more. Indeed, as already seen for the quartic scalar field, we have a mass spectrum that we can write down as

$m_n=(2n+1)\frac{\pi}{2K(i)}\left(\frac{Ng^2}{2}\right)^{\frac{1}{4}}\Lambda$

that as we have seen here agrees fairly well with lattice computations but we cannot pursue further this comparison as we are working at a classical level while to compare with lattice and experimental data a quantum field theory is needed. But already at this level the agreement is striking and worthing further analysis.

The lesson to be learnt is that an exact classical solution can give a lot of information about the physics one should expect for a quantum field theory. This is also true for the existence of a mass gap of Yang-Mills theory.