A striking clue and some more



My colleagues participating to “The many faces of QCD” in Ghent last year keep on publishing their contributions to the proceedings. This conference produced several outstanding talks and so, it is worthwhile to tell about that here. I have already said about this here, here and here and I have spent some words about the fine paper of Oliveira, Bicudo and Silva (see here). Today I would like to tell you about an interesting line of research due to Silvio Sorella and colleagues and a striking clue supporting my results on scalar field theory originating by Axel Maas (see his blog).

Silvio is an Italian physicist that lives and works in Brazil, Rio de Janeiro, since a long time. I met him at Ghent mistaking him with Daniele Binosi. Of course, I was aware of him through his works that are an important track followed to understand the situation of low-energy Yang-Mills theory. I have already cited him in my blog both for Ghent and the Gribov obsession. He, together with David Dudal, Marcelo Guimaraes and Nele Vandersickel (our photographer in Ghent), published on arxiv a couple of contributions (see here and here). Let me explain in a few words why I consider the work of these authors really interesting. As I have said in my short history (see here), Daniel Zwanzinger made some fundamental contributions to our understanding of gauge theories. For Yang-Mills, he concluded that the gluon propagator should go to zero at very low energies. This conclusion is at odds with current lattice results. The reason for this, as I have already explained, arises from the way Gribov copies are managed. Silvio and other colleagues have shown in a series of papers how Gribov copies and massive gluons can indeed be reconciled by accounting for condensates. A gluon condensate can explain a massive gluon while retaining  all the ideas about Gribov copies and this means that they have also find a way to refine the ideas of Gribov and Zwanzinger making them agree with lattice computations. This is a relevant achievement and a serious concurrent theory to our understanding of infrared non-Abelian theories. Last but not least, in these papers they are able to show a comparison with experiments obtaining the masses  of the lightest glueballs. This is the proper approach to be followed to whoever is aimed to understand what is going on in quantum field theory for QCD. I will keep on following the works of these authors being surely a relevant way to reach our common goal: to catch the way Yang-Mills theory behaves.

A real brilliant contribution is the one of Axel Maas. Axel has been a former student of Reinhard Alkofer and Attilio Cucchieri & Tereza Mendes. I would like to remember to my readers that Axel have had the brilliant idea to check Yang-Mills theory on a two-dimensional lattice arising a lot of fuss in our community that is yet on. On a similar line, his contribution to Ghent conference is again a striking one. Axel has thought to couple a scalar field to the gluon field and study the corresponding behavior on the lattice. In these first computations, he did not consider too large lattices (I would suggest him to use CUDA…) limiting the analysis to 14^4, 20^3 and 26^2. Anyhow, also for these small volumes, he is able to conclude that the propagator of the scalar field becomes a massive one deviating from the case of the tree-level approximation. The interesting point is that he sees a mass to appear also for the case of the massless scalar field producing a groundbreaking evidence of what I proved in 2006 in my PRD paper! Besides, he shows that the renormalized mass is greater than the bare mass, again an agreement with my work. But, as also stated by the author, these are only clues due to the small volumes he uses. Anyhow, this is a clever track to be pursued and further studies are needed. It would also be interesting to have a clear idea of the fact that this mass arises directly from the dynamics of the scalar field itself rather than from its interaction with the Yang-Mills field. I give below a figure for the four dimensional case in a quenched approximation

I am sure that this image will convey the right impression to my readers as mine. A shocking result that seems to match, at a first sight, the case of the gluon propagator on the lattice (mapping theorem!). At larger volumes it would be interesting to see also the gluon propagator. I expect a lot of interesting results to come out from this approach.



Silvio P. Sorella, David Dudal, Marcelo S. Guimaraes, & Nele Vandersickel (2011). Features of the Refined Gribov-Zwanziger theory: propagators, BRST soft symmetry breaking and glueball masses arxiv arXiv: 1102.0574v1

N. Vandersickel,, D. Dudal,, & S.P. Sorella (2011). More evidence for a refined Gribov-Zwanziger action based on an effective potential approach arxiv arXiv : 1102.0866

Axel Maas (2011). Scalar-matter-gluon interaction arxiv arXiv: 1102.0901v1

Frasca, M. (2006). Strongly coupled quantum field theory Physical Review D, 73 (2) DOI: 10.1103/PhysRevD.73.027701


Gluon condensate: The situation


Stephan Narison agreed to contribute to my blog with the following lines describing the current situation about the gluon condensate. It is a pleasure for me to put them here.

The gluon condensate \alpha_s G^2  introduced by Shifman-Vainshtein-Zakharov (SVZ) [see also Zakharov, contribution at Sakurai’s prize 1999: Int. J. Mod. Phys A14 (1999)4865 (here)] within the framework of QCD spectral sum rules  (QSSR) plays also an important role in gluodynamics. The original value of 0.04  GeV^4  obtained by SVZ from charmonium sum rules has been shown by Bell-Bertlmann (BB) to be underestimated by about a factor 2 from their analysis of the non-relativistec version of heavy quark sum rules. The BB result has been confirmed later on from QSSR analyzes of  different channels including  e^+e^-  into hadrons, tau-decay and charmonium by different groups, where the most recent value of  (0.07\pm 0.01) GeV^4 has been obtained [see for a review my 2 books (here and here) and the last paper on tau-decay: PLB673(2009)30 (here) ]. However, in order to extract reliably this (small) quantity one should work with sum rule which can properly disentangle its contribution where some possible competing contributions due to perturbative radiative corrections and to quark mass should not appear. This feature may explain some results in the literature. The non-vanishing of the gluon condensate and its positive sign has been seen in the lattice by  the Pisa group [A. Di Giacomo, G.C. Rossi, PLB100(1981)481 (here);  M. Campostrini, A. Di Giacomo, Y. Gunduc, PLB225(1989)393 (here)] and more recently by P.E. Rakow (here). Its positive sign is expected in a model with a magnetic confinement (H. Nambu), while phenomenologically, its eventual negative value would leave to serious inconsistencies in the QSSR approach. Its negative and non-universal values obtained from some tau-decays analysis can indicate the difficulty to extract its value among the competitive parameters present there where in the tau-decay width the gluon condensate contribution acquires an extra \alpha_s contribution compared to some other non-perturbative contributions. In fact, this peculiar properties have been the most important observation that non-perturbative contributions are small in this observable, then allowing an accurate determination of \alpha_s from tau-decays.

Gluon condensate


While I am coping with a revision of a paper of mine asked by a referee, I realized that these solutions of Yang-Mills equations implied by a Smilga’s choice give a proof of existence of a gluon condensate. This in turn means that a lot of phenomenological studies carried out since eighties of the last century are sound as are also their conclusions. E.g. you can check this paper where the authors find a close agreement with my findings about glueball spectrum. The ideas of these authors are founded on the concept of gluon and quark condensates. As they conclusions agree with mine, I have taken some time to think about this. My main conclusion is the following. If you have a gluon condensate, the theory should give \langle F\cdot F\rangle\ne 0 being F_{\mu\nu}^a the usual gluon field. So, let us work out this classically. Let us consider a scalar field mapped on the gluon field in such a way to have

A_\mu^a(t)=\eta_\mu^a \Lambda\left(\frac{2}{3g^2}\right)^\frac{1}{4}{\rm sn}\left[\Lambda\left(\frac{3g^2}{2}\right)^\frac{1}{4}t,i\right]

being sn a Jacobi snoidal function, and \eta_\mu^a a constant array of elements obtained by a Smilga’s choice. When you work out the product F\cdot F the main contribution will come from the quartic term producing a term \langle \phi(t)^4 \rangle. Classically, we substitute the average with \frac{1}{T}\int_0^T dt being the period T=4K(i)/[\Lambda\left(\frac{3g^2}{2}\right)^\frac{1}{4}]. The integration is quite straightforward and gives

\langle \phi(t)^4 \rangle=\frac{\Gamma(1/4)^2}{18K(i)\sqrt{2\pi}}\frac{\Lambda^4}{4\pi\alpha_s}

I will evaluate this average in order to see if the order of magnitude is the right one with respect to the computations carried out by Kisslinger and Johnson. But the fact that this average is indeed not equal zero is a proof of existence of the gluon condensate directly from Yang-Mills equations.

%d bloggers like this: