# Exploring the Quark Mass Plane with Open Boundaries

**Principal Investigator:** Andreas Schäfer, Institut für Theoretische Physik, Universität Regensburg (Germany)

**HPC Platform:** JUQUEEN of JSC

This project, which ran on high performance computing (HPC) system JUQUEEN of JSC, is part of a very large international effort in Lattice quantum chromodynamics (QCD). It addressed important aspects of hadron physics for the first time or with unprecedented accuracy. None of this would have been possible without superior HPC power. Unfortunately, many crucial elements are of rather technical nature, like the use of open instead of periodic boundary conditions. Therefore, rather than trying to give a comprehensive overview of this multi-faceted research program, this report focuses on just one aspect, which is *comparatively* easy to explain.

Lattice QCD explores the properties of hadrons like the proton. Their structure is completely dominated by quantum effects. This can already be seen from the fact that the masses of the three valence quarks in a proton add up to just roughly 1 percent of the total proton mass. The rest is due to interactions. For all hadrons, i.e. bound states of interacting quarks, the multi-particle wave-functions which describe their structure are consequently very complicated. It is one of the central tasks of hadron physics to determine them with ever higher accuracy. To do so, several complementary approaches have been developed.

One of these is the expansion in so-called Fock states, i.e. components with ever larger numbers of fields. The leading Fock state of a proton consists, e.g., of a three-quark-wave function. The higher Fock states contain additional gluons or quark-antiquark pairs. The nice thing about the leading Fock state of any hadron is that it gives the dominant contribution to some experimentally observable reactions. These belong to a class of reactions in which all participating hadrons in the initial and final state and their momenta are detected (referred to as “exclusive reactions”). This requires very large statistics which are achieved thanks to the advent of high luminosity accelerators in the last decade. In this field models are in the process of being replaced by ab initio calculations.

Copyright: Institut für Theoretische Physik, Universität Regensburg

In the following the researchers will focus on the results shown in Fig.1 and will explain in detail how these plots have to be interpreted. Protons and neutrons have very similar QCD properties and are, therefore, collectively referred to as nucleons. Other similar particle families are the Sigma and Xi hadrons as well as the isolated Lambda hadron. For all of these states the leading Fock state is a three quark state.

The next point is a bit tricky: In quantum mechanics the measuring process affects the result. As quantum effects in hadrons are so strong, so is this effect. Therefore, one must specify exactly in which coordinate frame and for which measurement results like those shown in the figure apply.

The situation is simplest if a so-called infinite momentum frame is chosen in which the momentum of each of the three quarks is given as a fraction, x, of the total (infinite) hadron momentum. These three fractions have obviously to sum up to 1 such that two of them determine the third. This is why the wave function is plotted in the form chosen in the figure.

Another quantity which specifies the measurement process is the momentum transfer taking place during the measurement. This is common for all wave phenomena: For example, the better spatial resolution of an electron microscope compared to an optical one is due to the larger momentum transfer possible. One can show that all Fock state wave functions discussed converge for infinitely large momentum transfer to the same symmetric functional form, the asymptotic wave function phi*. Therefore, it is customary to plot the difference between the actual wave function (or “Distribution Amplitude”, DA) at a given momentum transfer and the asymptotic one.

After all these explanations the messages contained in figure 1 can be formulated as follows:

• The nucleon DA is very close to its asymptotic form, the Lambda DA is quite different, while the Sigma and Xi are inbetween.

• Often, two quarks have a similar momentum fraction, while the third one has a significantly larger one. This agrees with the leading quark-diquark picture used by many simple quark models.

The detailed information contained in these plots should help to better understand many exclusive reactions of these and similar hadrons.

**References**

[1] G.S. Bali et al., ArXiv: 1512.02050

**Scientific Contact:**

Prof. Dr. Andreas Schäfer (for the Regenburg QCD collaboration)

Institut für Theoretische Physik, Universitaet Regensburg

D-93040 Regensburg/Germany

andreas.schaefer [@] physik.uni-r.de

*January 2016*