Wednesday, March 08, 2006

Topology and masses

In this post I'd like to talk about some papers I stumbled accross recently which both have to do with topological quantities and masses in gauge theories, although in a completely unrelated way.

The first paper is this one by a group of Italian and Greek researchers, in which they study the dependence of the string tension and lowest glueball mass on the vacuum angle $$\theta$$. Unfortunately, it is not really possible to simulate the QCD action with a $$\theta$$-term included, since the topological structure of a lattice gauge configuration is necessarily trivial since the lattice is finite and discrete. They bypass this by considering small values for $$\theta$$ and studying the expansion around $$\theta=0$$ to order $$\mathcal{O}(\theta^2)$$. The coefficients in that expansion can then be expressed in terms of correlators involving the topological charge operator. Measuring that on lattice is still not an easy task, because it has essentially to be reconstructed from what its value would be in the continuum, but a number of methods based either on the concept of "cooling" or on the spectrum of the Dirac operator (via the Atiyah-Singer index theorem) exist. Using a cooling-based method, the authors find that the $$\theta$$-dependence of their observables is rather small (the $$\mathcal{O}(\theta^2)$$ coefficients are of order -0.01 to -0.1) and decreases with an increasing number of colours roughly like $$1/N^2$$, which is expected from the large-$$N$$ limit.

The other paper is this one by Dvali, Jackiw and Pi, who show a way to extend the topological mass generation mechanism of the Schwinger model from two to four dimensions. The photon part of the Lagrangian of the Schwinger model can be rewritten in terms of the square of the Pontryagin density and the Chern-Simons current, where the latter is coupled to the anomalously non-conserved axial vector current. The resulting equation of motion plus the anomaly equation then combine to give a mass to the Pontryagin density, which can be considered as a massive pseudoscalar field. It is this formulation which the authors lift to four dimensions (subject to some relatively unimportant technicalities, and with the caveat that the resulting 4D action is non-renormalisable) to find that a mass for the Pontryagin density is also created in four dimensions. Phenomenologically, this is identified as a possible part of an effective field theory for the $$\eta'$$ by the authors.

At first I wasn't sure whether calling this a topological mechanism is entirely correct, since the anomaly equation for the axial vector current is needed to make it work, but I finally realised that, as the integral of the axial anomaly is equal to the index of the Dirac operator, it is of course truly topological by the Atiyah-Singer index theorem. This is actually quite astonishing (at least to me), since the usual diagrammatic treatment of the axial anomaly completely obscures its topological connection.