Monday, April 24, 2006

A debate about staggered fermions

Recently, there have been a number of short papers on the arXiv that discussed some potential problems that the usual procedure of taking the fourth root of the staggered fermion determinant to obtain a single-flavour theory might bring with it.

As a little reminder, staggered fermions are obtained from naive fermions by redistributing the spinor degrees of freedom across different lattice sites. As a result, staggered fermions describe a theory with four (rather than the 16 naive) degenerate fermion flavours, usually called "tastes" to distinguish them from real flavours. In order to obtain a theory with a single physical flavour, one usually takes the fourth root of the fermionic determinant for staggered fermions; this is correct in the free theory and in perturbation theory, but nobody really knows whether it makes sense nonperturbatively.

In the paper starting this recent debate, Creutz claimed that this procedure leads to unphysical results. His argument is based on the observation that with an odd number of quark flavours, physics is not invariant under a change of sign of the quark mass term, and hence the chiral expansion must contain odd powers of the quark mass. Since the staggered theory is invariant under a change of sign of the quark mass, so will be its fourth-rooted descendant, and hence it can only pick up even terms in the chiral expansion. Thus, Creutz claims, staggered fermions describe incorrect physics.

Within a week, there was a reply from Bernard, Golterman, Shamir and Sharpe, who claim that Creutz's argument is flawed since the quark mass in the theory corresponding to the continuum limit of the rooted staggered theory is always positive, regardless of the sign of the original quark mass, and since moreover the nonanalyticity inherent in taking a root leads to the emergence of odd powers of the (positive) mass in the continuum limit.

This was followed by third paper by Dürr and Hoelbling, in which they show how one may define "smart" determinant for staggered fermions (by including a phase factor that depends on the topological index of the gauge field background) that allows to reach the regime of negative quark masses. I have to admit that I do not fully understand this work, and enlightenment from readers is appreciated.

The debate over the correctness of the fourth root trick for staggered fermions is likely to go on for a while, particularly given the fact that the choice of fermion discretisation has become an almost religious issue within the lattice community. Personally, I certainly hope that staggered fermions give the correct physics, but I am not sure whether I actually have enough evidence or understanding to have an opinion either way.

Update: The paper by Creutz has been updated with a reply to the objections raised by Bernard (leading to the rather strange situation of circular citations between papers bearing different date stamps). Creutz now argues that while the problems he mentions may go away in the continuum limit, observables that develop a divergent dependence on a regulator at isolated points (such as the chiral condensate at m=0) are an "absurd behaviour" for a regulator, and that Wilson fermions are preferable in this regard. I am not entirely sure in how far the existence of exceptional configurations is a less absurd behaviour, though. I suppose there may be another round in this debate (with yet more circular citations).

Tuesday, April 11, 2006

More on (2+1)d glueballs

In a new paper, Leigh, Minic and Yelnikov give a more detailed follow-up on their earlier paper (discussed on this blog here) about the analytical solution of (2+1)-dimensional pure Yang-Mills theory.

Their basic setup is as before, but they give a lot more details: They start with the functional Schrödinger picture analysis of (2+1)d pure Yang-Mills theory performed by Karabali, Kim and Nair to re-express the theory in terms of new variables, and then make a generalised Gaussian ansatz for the vacuum wave functional containing an undetermined kernel K(Δ/m2). The Schrödinger equation is then turned into an ordinary differential equation for K(L), which can be solved in terms of Bessel functions. It follows that the glueball masses can be written as products of a sum of Bessel function zeros and the Karabali-Kim-Nair mass. Leigh, Minic and Yelnikov compare their predictions to lattice results and get mostly good agreement (with some uncertainty about the correct identification of excited states in the lattice simulations in a few cases).

Finally, they note and discuss the almost degeneracy of the glueball spectrum that follows from the asymptotic form of the Bessel function zeros, as discussed here and here.

These are very interesting results and their work may be considered a major breakthrough, although I remain sceptical as to whether we are going to see anything similar in the (3+1)d case anytime soon (or ever).