2005_10_01_archive

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