Time for another post in our series about lattice fermions. So far we have talked about naive, Wilson, staggered and Ginsparg-Wilson fermions. In this post, we are going to take a look at a still fairly new approach to lattice fermions that is known under the name of twisted mass QCD (tmQCD).

What one does in this approach is to take the Dirac operator for a flavour doublet of fermions and add to it a chirally twisted mass term

where the acts in flavour space. This extra term together with the doublet structure has the consequence that the worrisome exceptional configurations that plague Wilson quarks (remember, those were the configurations where the additive mass renormalisation that is allowed for Wilson fermions because they violate chiral symmetry takes the renormalised mass through zero) no longer exist, since the twisted Dirac operator has positive determinant:

and hence does not have any zero eigenvalues.

A flavour-dependent chiral rotation

leaves the continuum action with an added twisted mass term invariant, but mixes the ordinary mass with the twisted mass . Hence one can see the twisted mass action for a given as being the result of applying this chiral rotation to the ordinary continuum QCD action, and vice versa. The basis in which the term vanishes is known as the physical basis.

On the lattice, the twisted mass is usually added to the Wilson Dirac operator (which needs it most, since it suffers from exceptional configurations). The resulting action can then be used to study quarks at small masses, where the Wilson action itself would fail. It also has the added benefit that certain observables are automatically free of lattice artifacts with a twisted mass.

The twisted mass theory has its own problems, though: The appearance of in the twisted mass term means that the up- and down-type quarks have opposite signs of the twisted mass, and hence isospin is no longer conserved. Also, the appearance of implies that parity is no longer a symmetry, although a generalised parity operation involving the twist angle can be defined as a symmetry of the twisted theory.

In closing, it should be stressed again that the exact meaning and properties of twisted mass are still a very active field of research, and some surprises may still be expected. I should also add that I am not really an expert on tmQCD (though other people here in Regina are), so corrections and additional remarks are particularly welcome on this post.

## Tuesday, March 28, 2006

## 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 . Unfortunately, it is not really possible to simulate the QCD action with a -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 and studying the expansion around to order . 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 -dependence of their observables is rather small (the coefficients are of order -0.01 to -0.1) and decreases with an increasing number of colours roughly like , which is expected from the large- 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 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.

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 . Unfortunately, it is not really possible to simulate the QCD action with a -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 and studying the expansion around to order . 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 -dependence of their observables is rather small (the coefficients are of order -0.01 to -0.1) and decreases with an increasing number of colours roughly like , which is expected from the large- 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 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.

Labels:
topology

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