Wednesday, March 29, 2006

Twisted Mass Fermions

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

$$D_{tw} = D + i\mu\gamma^5\tau^3$$

where the $$\tau^3$$ 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:

$$det[D_{tw}] = det[D^\dag D +\mu^2]>0$$

and hence does not have any zero eigenvalues.

A flavour-dependent chiral rotation

$$\psi\mapsto\exp(i\alpha\gamma^5\tau^3/2)\psi$$

leaves the continuum action with an added twisted mass term invariant, but mixes the ordinary mass $$m$$ with the twisted mass $$\mu$$. Hence one can see the twisted mass action for a given $$\mu$$ as being the result of applying this chiral rotation to the ordinary continuum QCD action, and vice versa. The basis in which the $$\mu$$ 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 $$O(a)$$ lattice artifacts with a twisted mass.

The twisted mass theory has its own problems, though: The appearance of $$\tau^3$$ 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 $$\gamma^5$$ 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.
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