Today I was reading a PhD thesis about matters we frequently discuss in this blog (see here). This is a very good work. But when I have come to the question of the running coupling I was somewhat perplexed. Indeed, there is a recurring wishful thinking about running coupling in a Yang-Mills. This prejudice claims that coupling in the low momenta limit should reach a non-trivial fixed point for the theory to be meaningful. Then, if you read the literature since the inception of the success of gauge theories you will read a myriad of papers claiming this “fact” that is not a fact having been never proved.

In this case we have two kind of evidences: lattice and experimental. These evidences show that the coupling at low momenta goes to zero, that is the theory is free also in the infrared! This is a kind of counterintuitive result as are all the results that are coming out from lattice computations. The reason for this relies on the fact that Yang-Mills theory is a scalar theory in disguise and so shares the same fate. But maybe, the most interesting result comes from Giovanni Prosperi and his group at University of Milan. They studied the meson spectrum and showed how the running coupling derived from measurements bends clearly toward zero. Their work has been published on Physical Review Letters (see here and here). They do this studying quarkonium spectra, a matter we discussed extensively in this blog. Their paper has been enlarged and published on Physical Review D (see here and here).

On the lattice the question is linked to the behavior of the gluon and ghost propagators. We have seen that the gluon propagator reach a non-null constant as momentum goes to zero and the ghost propagator behaves as that of a free particle. This means that if we write

for the gluon propagator and

for the ghost propagator, being and the dressing function, following Alkofer and von Smekal we can define a running coupling as

but the gluon propagator reaches a non-null value for and so and the ghost propagator goes like that of a free particle and so . This means nothing else that at low momenta. This is lattice response.

So, why with all this cumulating evidence people does not yet believe it? The reason relies on the fact that is very difficult to remove prejudices and truth takes some time to emerge. We have to live with them for some time to come yet.

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2 Responses to The question of the running coupling

Indeed, there is a mapping theorem that, given a solution of the classical equation of the scalar theory, this is also a classical solution of the Yang-Mills equations provided that all the components of the Yang-Mills field are taken to be equal. This mapping holds in the infrared limit for a quantum field theory but not for ultraviolet as, at high energies, quantum corrections are overwhelming and asymptotic freedom sets in.

Professor Frasca:

I find your comment that QCD is a “scalar theory” in disguise quite interesting, could you please elaborate this.

I enjoy your postings a lot

Thanks,

RR

Indeed, there is a mapping theorem that, given a solution of the classical equation of the scalar theory, this is also a classical solution of the Yang-Mills equations provided that all the components of the Yang-Mills field are taken to be equal. This mapping holds in the infrared limit for a quantum field theory but not for ultraviolet as, at high energies, quantum corrections are overwhelming and asymptotic freedom sets in.

I think you can look at my post

https://marcofrasca.wordpress.com/2008/07/15/classical-yang-mills-theory-and-mass-gap/

and the following papers of mine

http://arxiv.org/abs/0807.4299

http://arxiv.org/abs/0709.2042

Marco