Topological Mass Generation

So perhaps you thought the Higgs mechanism was the only mechanism for mass generation in four dimensional spacetime? Probably it is, but Roman Jackiw, co-discoverer of the axial chiral anomaly with Bell and Adler, spoke at Imperial College yesterday about an alternative and elegant method to generate a 4-dimensional mass term. Roman first described to us the three-dimensional model where a Chern-Simons term can be added to the Lagrangian to generate mass, but told us that his motivation would come from the Schwinger model in two-dimensions. In the Schwinger model massless Dirac fermions are added and then eliminated in order to generate a mass. With hindsight Topological Aspects of Gauge Theories by Jackiw, which is to appear in the Encylopaedia of Mathematical Physics would have been a good article to read before attending this seminar.

Roman went through the original model and then repeated the analysis using a number of dualised terms, he referred to this as going "towards the topological model". In particular he highlighted that the field acquires a mass due to the presence of a chiral anomaly in the axial vector current and leads to a massive pseudoscalar; the pseudoscalar being dual to the two-index field strength as well as being proportional to the divergence of the axial vector current.

Now Roman's aim was to take this two-dimensional model, made out of purely topological terms, and then write out the equivalent expression using the four dimensional topological objects. He said that he would call this topological mass generation since now he would refer to the terms we had before with their topological names.

In two dimensions, using the dualised terms, a pseudoscalar crops up that is the Chern-Pontryagin density, P, and the dual of the potential field, C^\mu=\epsilon^\mu\nu A_\nu, is the two-dimensional Chern-Simons current. These are suitable quantities to take across to four dimensions, however it turns out to be a requirement of the method that the dual of the axial current must be a conserved quantity, and this can be guaranteed to occur by adding two fields, added in the form of Lagrange multipliers to the dual Lagrangian (I omit the details here unfortunately because I still haven't opted for a way of putting TeX in these posts). Surprisingly when one does this in order to conserve the dual axial current, one obtains a gauge invariant dual Lagrangian - the two go hand in hand. The generalisation of the Schwinger model to four dimensions is now straightforward, and is carried out by using the four-dimensional topological terms. Roman finalised by mentioning two shortcomings of this approach, the first being that the anomaly producing dynamics has not been specified and as such this model presents a phenomenological model of mass generation. The second shortcoming was the resulting dual Lagrangian was a dimension eight operator, and this, I am told, presents difficulties for renormalization. However on the positive side the specific contribution needed for the anomaly appears in the expansion of the Born-Infeld action to quadratic terms. Furthermore Roman pondered whether it might not present a phenomenological description for the elusive \eta'. Roman reminded us that the \eta' is the ninth goldstone boson suspected to arise by promoting an SU(3)xSU(3) symmetry to a U(3)xU(3) symmetry. This topological mass generation if it were indeed applicable to the \eta' would give a numerical prediction of the \eta' mass. For some notes about the mysterious \eta' see here. Apologies for any mistakes, one day I will read the much-recommended book by Zee and learn some QCD. Now I've written it here I just have to do it...