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Nucleon and Meson Matrix Elements Close to the Physical Point

The validity of Quantum Field Theory (QFT) is proven beyond any reasonable doubt, but at the same time it is clear that the Standard Model is incomplete in many respects (quantum gravity, dark matter, dark energy, CP violation). Also, there are many aspects of the Standard Model, in particular of the QCD (Quantum Chromodynamics) sector which are not yet understood, especially non-perturbative aspects. It is hoped that the combination of dedicated new experiments and Lattice QFT will allow us to improve our understanding of these aspects to the point that even subtle discrepancies between experimental data and theoretical calculations allow to unambiguously identify New Physics. The all important issue in this context is obviously the control of all uncertainties. Let us illustrate this statement with just one example: LHC experiments have shown that at LHC energies Multiple Hard Interactions, which are not yet well understood, give important, sometimes even dominant contributions. Deviations between Standard Model predictions and experimental findings are only relevant if the signal is large compared to the associated uncertainties which can only be judged after the corresponding correlation functions are quantitatively known. As these are non-perturbative objects the only known reliable method to determine their size is Lattice QCD.

Unfortunately, the reliable determination of systematic uncertainties for lattice calculations is very difficult, especially for calculations addressing hadron structure. While many people thought a decade ago that they can extrapolate their results simultaneously reliably to physical quark masses, infinite simulation volumes, arbitrarily fine lattices and completely suppressed admixtures of unwanted hadron states, it is now clear that these hopes where overly optimistic. Therefore, in recent years a large fraction of the community has adopted the strategy to improve lattice simulations first till some experimentally very well known hadron structure benchmark observables are reproduced before trusting any other prediction on hadron structure. The most prominent of these benchmark observables are the iso-vector quark momentum fraction and axial vector coupling constant g_A of the nucleon. The present project was part of such an effort undertaken by our SFB/TR-55 “Hadron Physics from Lattice QCD”.

To obtain reliable results we generated large ensembles of configurations on our home-built computer QPACE using improved Wilson fermions, two dynamical flavors, various masses down to pion masses of 150 MeV and different lattice spacings and volumes. We analyzed our analyses as carefully and completely as possible, performing many different fits, assuming, e.g., a mixture of contributions from different states. We optimized our smearing strategy and source/sink interpolators to get maximal overlap with the nucleon state etc. All of this combined lead to a dramatic improvement in the quality of our results, as is shown in figure 1. However, the resulting comparatively small error bars at nearly physical masses show also that there is still work to be done. While the larger mass points supported the hope that a downward shift of the results for decreasing mass could lead to agreement with experiment, our new results rule this possibility out convincingly.

Nucleon and Meson Matrix Elements Close to the Physical PointFigure 1: The difference in average momentum fraction of up and down quarks in a nucleon. The results of fits to experimental data are given in black, our new results are given in blue in comparison with earlier lattice results (green and red). The cost of lattice simulations increases dramatically with decreasing quark masses, which is why simulations are often done with heavier than physical quarks, resulting also in heavier than physical pions.
Source: Universität Regensburg/Germany, Institut für Theoretische Physik


The logical conclusion is that the only limit which we do not yet have under good control, namely the extrapolation to vanishing lattice spacing must be improved by simulating on finer lattices. This, however, is a very tall order because for lattice spacings below 0.05 fm the topological autocorrelation time becomes unacceptably long for standard algorithms. Consequently, we and several other groups in Europe, forming the CLS collaboration, have switched to simulations with open boundary conditions, which is the only known way to avoid the problem. We expect that the results of our LRZ project (run on HPC SuperMUC of LRZ Garching/Munich) will contribute to steer lattice simulations for hadron structure in general towards higher precision and will help to avoid in future over-optimistic extrapolations of lattice results obtained for unphysical large masses and/or for coarse lattices. In view of the rapid progress of lattice QCD observed in recent years (which is exemplified by our results) we expect that the lattice community will achieve a satisfactory agreement with all experimental data within a few years.

Prof. Dr. Andreas Schäfer
Institut für Theoretische Physik, Universität Regensburg
D-93040 Regensburg/Germany
e-mail: andreas.schaefer@physik.uni-r.de

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