The fourth-root trick for the staggered determinant has long been controversial. Most recently, the debate has been rekindled by a series of papers by Mike Creutz, in which he argues that the rooting procedure fails in specific ways. While some of the arguments have been refuted by members of the staggered community, criticisms related to the question whether the rooted staggered theory can describe the axial anomaly correctly remain important. A direct physical probe of the axial anomaly is given by the η'-η splitting. Unfortunately, the determination of this splitting requires the evaluation of disconnected contributions to the η' correlator, which are very noisy and cannot be measured with sufficient precision to make a clear statement at the current time. In his recent paper, Stephan Dürr approaches the question of the correctness of the rooting procedure from the angle of a theory in which sufficient statistics can be readily obtained, namely the Schwinger model.
The Schwinger model is simply QED in 1+1 spacetime dimensions, as far as its action is concerned. Its physics is, however, radically different from that of QED in 3+1 dimensions, since firstly there is neither spin nor a physical gauge boson degree of freedom in 1+1d, and secondly the 1-dimensional Coulomb potential is linear and hence confining. The Schwinger model therefore has a spectrum similar to that of QCD, with a mass gap and meson degrees of freedom (note that there are neither baryons nor "photoballs" due to the abelian nature of the interaction [although there aren't any glueballs in 1+1d QCD either due to the absence of the gauge boson as a degree of freedom]), and can therefore serve as a laboratory for ideas in QCD. The basic meson η of the Schwinger model, which Schwinger demonstrated to have a mass squared of m2=e2/π (where e is the dimensionful gauge coupling in 1+1d), in particular, is an analogue of the η' in QCD, since its mass is mainly due to the axial anomaly.
The Schwinger model is much easier to simulate than QCD both because two dimensions are easier than four, and also because it turns out that reweighting works very well in two dimensions where the fermionic determinant can be evaluated exactly due to its comparably small size, so that one can generate quenched ensembles and include the fermionic determinant via reweighting. In particular the latter feature allows the generation of huge statistics (80,000 configurations in this case). Dürr employs an algorithm incorporating the introduction of instantons and antiinstantons as well as parity transformations to optimise the sampling of topological sectors. The resulting ensembles are then used to simulate the Nf=1(2) Schwinger model via reweighting with the rooted (unrooted) staggered fermion determinant. The latter is correct by construction; testing the former is the motivation for the study.
Using all-to-all propagators and U(1)-projected triply APE-smeared gauge links, Dürr is able to show the validity of the staggered index theorem with impressive precision. Turning to the meson spectrum, he finds that the connected part of the η has the same mass as the Nf=2 π meson up to cut-off effects, so that the mass of the physical η in the chiral limit comes entirely from the disconnected part. The ratio of the disconnected to the connected Green's functions for the η approaches the correct limiting value expected if the rooting trick works correctly. After a continuum and chiral extrapolation, he finds that the mass of the Nf=1 η meson agrees with Schwinger's analytical result.
This paper provides a very interesting study that adds to the empirical support for the correctness of the rooting procedure for staggered quarks. Of course it remains to see if this result will carry over to QCD, but I'd be honestly surprised if it didn;t. An analytical construction demonstrating the correctness of the rooted staggered formalism would of course be very welcome. Perhaps some of the recent results regarding the connection between staggered and overlap fermions will point the way in that regard.