## Running coupling and Yang-Mills theory

Searching casually with google I have come upon a paper by Acharya (see here). Acharya is Emeritus at Arizona State University and most of his papers appeared just on arxiv. It is a well-known matter that arxiv completely revolutionized the way scientific communication is diffused and the most impressive proof of this has been Grisha Perelman and his demonstration of the Poincare’ conjecture. This means that it is not more necessarily true that the right information is that published on archival journals but sometimes it happens to appear only on arxiv. While physicists may be skeptical about unpublished material, mathematicians are surely more open to the fact that a full community can act as a peer-review vehicle and agree on the rightness or wrongness of some results.

Acharya tries to show that the Cornwall’s view about QCD is right. Cornwall has been a pioneer on giving the proper view of Yang-Mills theory in the infrared (see here). His idea that the gluon mass is dynamically generated is in agreement with our view in this blog and more generally with the fact that the lattice gluon propagator does not go to zero in the infrared limit. As a by-product Acharya claims that the running coupling of QCD does not reach any fixed point but tends to be completely screened, that is, it is expected to go to zero! This is exactly what one presently sees on the lattice and several authors agree upon (see this post). He derives this assuming that the beta function remains negative for the all values of the coupling constant.

To see that this is the case, we consider our classical solution given here that is

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

that, as already said, it is true when the field satisfy the dispersion relation $p^2=\Lambda^2\left(\frac{Ng^2}{2}\right)^{1 \over 2}$ so that this is a massive solution. It is easy to see that the following Callan-Symanzick equation is true

$\Lambda\frac{\partial\phi}{\partial\Lambda}-4Ng^2\frac{\partial\phi}{\partial (Ng^2)}-2\phi=0$

that says us that the beta function is $\beta(g) = -4Ng^2$, always negative as expected by Acharya, and this means that at lower momenta the coupling goes to zero as the fourth power of momentum. This conclusion was also obtained on the lattice by Philippe Boucaud et al. (see here). So, the way Acharya describes the behavior of the running coupling is the same we were able to obtain from our exact classical solution and seems the right one as seen from lattice computations.

Do we have always to distrust unpublished preprints on arxiv?