## Physics Letters B at last!

16/10/2008

Yesterday, the Editor of Physics Letters B communicated to me that my paper (see here) was accepted for publication. This was great for at least one reason: Physics Letters B is one of the most important journals in our area of activity and the paper that was accepted gives a sensible mathematical proof of the form of the gluon and ghost propagators in the infrared and relative mass spectrum that implies the very existence of a mass gap for the Yang-Mills theory. The key theorem is what I called the “mapping theorem” where a SU(N) Yang-Mills theory is mapped on a $\lambda\phi^4$ theory whose solution in the low energy limit I presented here and Physical Review D (see here). This analysis is in perfect agreement with the scenario emerging from lattice computations but we have the nice situation of explicit formulas for the gluon propagator and the spectrum permitting explicit computations wherever needed.

Also in this case the peer-review system worked at best. Both Editor and referees permitted to correct what appeared a serious difficulty in the proof of the mapping theorem. Once I solved this the paper was straightforwardly approved for publication. I take this chance to thank them all publicly.

I give here the formulas for the gluon propagator in the Landau gauge (the ghost propagator is that of a free particle) and the spectrum:

$D_{\mu\nu}^{ab}(p^2)=\delta_{ab}\left(\eta_{\mu\nu}-\frac{p_\mu p_\nu}{p^2}\right)\sum_{n=0}^\infty\frac{B_n}{p^2-m_n^2+i\epsilon}$

being

$B_n=(2n+1)\frac{\pi^2}{K^2(i)}\frac{(-1)^{n+1}e^{-(n+\frac{1}{2})\pi}}{1+e^{-(2n+1)\pi}}$

and

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

being $\Lambda$ an integration constant of Yang-Mills theory, arising from conformal invariance, to be fixed experimentally and $K(i)$ an elliptic integral that is about 1.3111028777. From the mass spectrum is clearly seen the mass gap when $n=0$ is taken. Nature decides what $\Lambda$ is but an higher order theory should be able to derive it. We see that the spectrum of the theory is made of massive excitations that should not be called gluons. I think that glueballs is more appropriate.

So, this is a key moment for Yang-Mills theory. It implies a great understanding of the behavior of the theory in a regime not accessible before. Knowing the gluon propagator means that a Nambu-Jona-Lasinio model describes correctly the phenomenology at low energies. This we proved quite recently (see this post).