ELEMENTARY PARTICLE PHYSICS

Introduction

With the advent of high luminosity spin polarized particle accelerators exploring the spin structure of hadrons, i.e. bound states from quarks and gluons, became a major research topic. In the beginning experimental results contradicted naïve expectations to such an extend that this physics was often summarized under the title “spin crisis”. To be precise, this term summarized the fact that the distribution of the total proton or neutron (collectively called nucleon) spin of ½ on quark and gluon spin as well as quark and gluon orbital angular momentum was hotly disputed.

Over the years, however, to a large extent thanks to lattice calculations, a quite detailed understanding was developed, such that today more subtle questions have moved into the focus of attention. Quantum Chromodynamics (QCD), the theory of quarks and gluons, has strong similarities with Quantum Electrodynamics (QED), the quantum theory of the electromagnetic interaction, but there exist also fundamental differences. As in QED, spin couples to the (color) magnetic fields and (color) charged quarks experience a (color) Lorentz force in such a (color) magnetic field. In contrast to QED, however, all quarks and gluons in a nucleon are “confined”, i.e. they cannot be extracted in isolation from a nucleon making access to the color analogs of well-known electromagnetic effects far more difficult. One can only observe certain color neutral correlations of quarks and gluon fields which are classified by abbreviations like d₂ the correlation we are concerned with in this project. This correlation is judged so important that it motivated several large scale experiments and many theory investigations.

Figure 1 shows an up to date compilation, with various model predictions to the right and various experimental data as well as our old lattice results [2] to the left. Solid symbols denote results for the proton and open ones those for the neutron. The most important fact presented in this figure is that d₂ is small. (Naively one could have expected numbers anywhere in the range from -0.02 to +0.02, as is illustrated by the scattering of model predictions.) Because d₂ can be related to the color Lorentz force on quarks in a color magnetic field, which is in turn oriented along the spin vector direction of the studied hadron, this implies that this force is weaker than one could have expected.

Another remarkable feature of this plot is that our old lattice results [3] (plotted in green) are displayed together with experimental data, illustrating the status lattice QCD calculations have gained meanwhile in hadron physics. In contrast model predictions of various sorts are depicted by the symbols on the right. They scatter widely over the phenomenologically plausible region.

While this kind of plotting is kind of flattering for us it also made us feel a bit uneasy because at the time (2005) lattice QCD techniques were far less developed and the error analysis leading to the displayed error bars does not fulfill our present standards. In particular the most problematic uncertainty, caused by the required extrapolation to zero lattice constant, i.e. the physical space-time continuum, could not be determined at the time. (In lattice QCD continuous space-time is substituted by a four-dimensional hypercubic lattice of constant lattice spacing.) This we have now improved.

Results and Methods

The problem of controlling the continuum limit is not only one of required computer resources (simulations with smaller lattice constants require far more lattice points. Actually, their number scales like the fourth power of the inverse lattice spacing) but also a fundamental theoretical one. QCD has different topological sectors and different ones have to be sampled to get correct results. For lattice constants below 0.05 fm this becomes unfeasible, a problem known as “topological freezing”. On the other hand, for lattice spacings above 0.1 fm discretization errors become large such that the typical lever arm for a continuum extrapolation is only about a factor of two. (The radius of a proton is roughly 0.7 fm, so one needs fine lattices to resolve its structure.)  Within the very large CLS collaboration of collaborations we have implemented an algorithmic formulation which circumvents this problem. Figure 2 and Figure 3 show some of the preliminary results we obtained.

Such calculations proceed in two steps. First one generates large ensembles of field configurations (typically each ensemble contains several thousand configurations) which contain information on the structure of all hadrons (e.g. bound states of quarks and gluons). From these one then extracts information on specific aspects of specific hadrons, like d₂ of the proton.

The main results of this project are updates of the green lattice points in Figure 1. Looking at our results in Figure 2 and 3 the most important observation is that the lattice constant dependence is significant for the proton, making this a showcase example for how important control of the continuum limit is and thus adding to the justification of the whole CLS effort.

Although, apart from all other systematic uncertainties, our results also contain the uncertainty of the continuum extrapolation the resulting error bars are significantly smaller than for the old simulations. In particular the result for the proton is now distinctly non-zero. (Its size actually fits now better to phenomenological expectations.)