ELEMENTARY PARTICLE PHYSICS

Results and Methods

We have analysed 4,800 gauge configurations of lattice size 48x24³ and lattice spacing a=0.066 fm. We employ a technique called distillation [2] which we have improved in this project [3]. The computationally most intensive part of the calculation requires solving the Dirac equation with the right-hand sides given by 200 eigenvectors of the 3D gauge Laplacian on each of the 48 time-slices of the lattice. The number of solutions of the Dirac equations (inversions) per gauge configuration equals 38,400. The quark mass in the Dirac operator corresponds to approximately half of the value for the charm quark in nature.

Figure 1 shows the effective masses of the ground state in the pseudoscalar channel. They plateau for large values of the time separation t/a between source and sink of the meson to the value of the mass of the particle, which in this case corresponds to the η_c meson in nature. The points marked by circles are obtained from the iso-scalar correlator, which includes the contribution of charm-annihilation and the points marked by stars are obtained from the iso-vector correlator, where charm-annihilation is absent. For the data points coloured in red we used our optimised distillation technique. The black data points are obtained with standard distillation and are shown for comparison. It is evident that with the optimal distillation method the effective masses reach their plateau at earlier times than they do with the standard distillation method. This is crucial to determine the mass of the iso-scalar meson. Due to the charm-annihilation effects the statistical errors of the data for the iso-scalar meson increase rapidlly with the time separation t/a. The  charm-annihilation effects lead to an upward shift in the iso-scalar meson mass compared to the iso-vector meson mass. We estimate this shift to be 99 MeV with an uncertainty of 15 MeV. In nature this shift has been indirectly estimated  to be about 7 MeV. We expect that the number and the mass of the charm quarks in our model calculation, which differ from their values in nature together with the absence of light quarks account for the difference in the mass shift. For the future we plan a calculation in a physical setting.

To achieve this precision, we developed a software package for the measurement of correlation functions using the improved distillation method. Our programs are based on “QCDlib”, a library written by us in C+MPI, that facilitates massively parallel QCD calculations. The inversions of the Dirac operator are performed by calling the package openQCD [4], namely a deflated SAP GCR solver with AVX instructions enabled for SuperMUC-NG. The computations for 4800 gauge configurations of lattice size 48x24³ ran on 384 cores (8 nodes) of SuperMUC-NG and required about 17 millions core-hours.

We have applied our method to 16 times larger gauge configurations of lattice size 96x48³. The setup is the same as described above but with a smaller lattice spacing a=0.049 fm and using 325 eigenvectors of the 3D Laplacian per time-slice. Figure 2 shows the timings for one inversion of the Dirac operator as a function of the number of cores used on SuperMUC-NG. We plot the ratio of the run-time on 768 cores to the run-times using more cores (speedup). The red dashed line corresponds to the ideal speedup. The performance of our code is close to the ideal line and shows its good scalability. Using different block sizes (BS) corresponds to changing this parameter in the deflated SAP GCR solver used. We have analysed 250 gauge configurations of lattice size 96x48³. These computations ran on 6912 cores (144 nodes) and required about 20 millions core-hours. The number of cores 6912 was chosen because, even though we begin to see deviations from the ideal speedup in the inversion times, it allows for more parallelisation in the temporal direction.

Ongoing Research / Outlook

We plan to compute the charmonium spectrum with hybrid candidates and the effects of charm-annihilation on close-to-physical gauge configurations of size 144x48³ which include the dynamics of three light degenerate quarks, whose mass is the same as the average light quark mass in nature and a physical charm quark [5]. This will provide information on mass and quark content dependence of the spectrum of charmonium. The Research Unit FOR 5269 [1] will develop new numerical algorithms for calculations of glueballs and charmonium including their mixing and decays. We will also apply some of the new techniques to compute the potential between a static (infinitely massive) quark and anti-quark pair. The latter is a first principle input to effective theories to study the spectrum and decays of charmonium including exotic XYZ candidates.