So far, it is more than twenty years that I publish in refereed journals and, notwithstanding a lot of exchange with my colleagues, I have never had the chance to work in a collaboration. The opportunity come thanks to Marco Ruggieri (see here). Me and Marco met in Gent at the Conference “The Many faces of QCD” (see here, here and here). We have had a lot of good time and discussed a lot about physics. I remember a very nice moment discussing with Attilio Cucchieri and Tereza Mendes in a pub, with a good beer, about the history that was going to form on the question of the propagators for Yang-Mills theory in a Landau gauge. About a month ago, Marco wrote to me about his new work in progress. He was managing to analyze the behavior of QCD condensates in a magnetic field through a couple of models: The linear sigma model and the Nambu-Jona-Lasinio model. The formalism for doing this was already known in literature due to the works of Ritus, Leung and Wang (see below) that analyzed the solutions of the Dirac equations in a constant magnetic field giving also the propagator. In our paper we introduce the constant magnetic field into the given phenomenological models through a minimal coupling. It is interesting to note that, while the sigma model is renormalizable, Nambu-Jona-Lasinio model is not and displays explicitly a dependence on a cut-off. This is not a concern here as this cut-off in QCD has a physical meaning as one can already see in asymptotic freedom studies. The motivation for this study was mainly a lattice analysis of this kind of physical situation (see here). The point is that some kind of condensates can form only with the presence of the external magnetic field. We were able to recover the values of magnetic susceptibility and the dependence of the chiral condensate on the magnetic field in the limit of small and large fields. Besides, we obtained an evaluation of the magnetic moment. The agreement with lattice computations is fairly good.
What I have learned from this work is that the use of phenomenological models, particularly their choice, can entail some difficulties with the expected behavior of QCD. First of all, the sigma model and the Nambu-Jona-Lasinio model are not so different: One can be obtained from the other through bosonization techniques. But while the latter cannot be renormalized, implying a contact interaction and a dimensional coupling , the former can. A curious result I obtained working on this paper with Marco is that the Yukawa model, written down as a non self-interacting scalar field interacting with a massless Dirac field, can be easily transformed into a Nambu-Jona-Lasinio model giving rise to chiral symmetry breaking! If Hideki Yukawa would have had this known, his breakthrough would have been enormous. On the other side, a sigma model is always renormalizable and this implies that any final result of a computation from it is independent on any cut-off used to regularize the theory. This is not what is seen in QCD where a physical scale depending on energy emerges naturally by integrating the equations of motion as already said above. Besides, condensates do depend explicitly on such a cut-off and this means that to regularize a sigma model to describe QCD at very low-energies implies a deviation from physical results. Last but not least, scalar models are trivial at low energies but we know that this is not the case for QCD that has the running coupling reaching a non-trivial fixed point in the infrared limit. For a Nambu-Jona-Lasinio model this is not a concern as it holds when the infrared limit is already reached with a fixed value of the strong force coupling. My personal view is that one should always use a Nambu-Jona-Lasinio model and reduces to a sigma model after a bosonization procedure so to fix all the parameters of the theory with the physical ones. In this sense, the renormalizability of the sigma model will be helpful to correctly represent mesons and all the low-energy phenomenology. The reason for this is quite simple: Nambu-Jona-Lasinio model is the right low-energy limit of QCD.
Marco Frasca, & Marco Ruggieri (2011). Magnetic Susceptibility of the Quark Condensate and Polarization from
Chiral Models arxiv arXiv: 1103.1194v1
RITUS, V. (1972). Radiative corrections in quantum electrodynamics with intense field and their analytical properties Annals of Physics, 69 (2), 555-582 DOI: 10.1016/0003-4916(72)90191-1
C. N. Leung, & S. -Y. Wang (2005). Gauge independent approach to chiral symmetry breaking in a strong
magnetic field Nucl.Phys.B747:266-293,2006 arXiv: hep-ph/0510066v3
P. V. Buividovich, M. N. Chernodub, E. V. Luschevskaya, & M. I. Polikarpov (2009). Chiral magnetization of non-Abelian vacuum: a lattice study Nucl.Phys.B826:313-327,2010 arXiv: 0906.0488v2