## Low-energy effective Yang-Mills theory

As usual I read the daily from arxiv and often it happens to find very interesting papers. This is the case for a new paper from Kei-Ichi Kondo. Kondo was in Ghent last year (here his talk) and I have had the chance to meet him. His research is on very similar lines as mine. A relevant paper by him is about the derivation of the Nambu-Jona-Lasinio model from QCD (see here) with a similar hindsight I exposed in recent papers (see here and here). This new paper by Kondo presents a relevant attempt to derive a consistent low-energy effective Yang-Mills theory from the full Lagrangian. The idea is to decompose the gauge field into two components and integrate away the one that just contributes to the high-energy behavior of the theory. Kondo shows how a mass term could be introduced at the expenses of BRST symmetry breaking. This symmetry can be recovered at the cost of nilpotency. But this mass term is gauge invariant and gives rise to a meaningful propagator for the theory. Then, the computations show how Wilson’s area law is satisfied granting quark confinement and positivity reflection for the gluon propagator is violated granting gluon confinement too. The gluon propagator is then given in a Gribov-Stingl form

$D(p)=\frac{1+d_1p^2}{c_0+c_1p^2+c_2p^4}$

but this form is only recovered if the mass term is introduced in the original Yang-Mills Lagrangian as said above. It is interesting to note that, if this is the right propagator, a Nambu-Jona-Lasinio model could anyhow be derived taking the small momenta limit. A couple of observations are in order here. Firstly, Cucchieri and Mendes fits often recover this functional form (e.g. see here). Last but not least, this functional form is acausal but produces a confining potential increasing with the distance. But even if the mass term would be zero and no Gribov-Stingl form is obtained, Kondo shows that area law still holds and one has confinement yet. As a final conclusion, Kondo shows that his effective model describes confinement through a dual Meissner effect, a hypothesis that come out at the dawn of the studies in QCD.

This paper represents a fine piece of work. A point to be clarified is, given from other studies and lattice computations that gluon mass arises dynamically, how this approach should change and, most important, how the form of the propagator changes. I just suspect that my conclusions about this matter would be recovered.

Kei-Ichi Kondo (2011). A low-energy effective Yang-Mills theory for quark and gluon confinement arxiv arXiv: 1103.3829v1

Kei-Ichi Kondo (2010). Toward a first-principle derivation of confinement and
chiral-symmetry-breaking crossover transitions in QCD Phys. Rev. D 82, 065024 (2010) arXiv: 1005.0314v2

Marco Frasca (2010). Glueball spectrum and hadronic processes in low-energy QCD Nucl.Phys.Proc.Suppl.207-208:196-199,2010 arXiv: 1007.4479v2

Marco Frasca (2008). Infrared QCD International Journal of Modern Physics E 18, (2009) 693-703 arXiv: 0803.0319v5

Attilio Cucchieri, & Tereza Mendes (2009). Landau-gauge propagators in Yang-Mills theories at beta = 0: massive
solution versus conformal scaling Phys.Rev.D81:016005,2010 arXiv: 0904.4033v2