ELEMENTARY PARTICLE PHYSICS

Introduction: the spin of nucleons

The proton and the neutron form the nucleus of all atoms we have discovered so far. Our picture of the inner most structure of these two nucleons is that they are bound states of partons, the quarks and the gluons, see Figure 1. The proton takes a spin of ½, which must be composed from the individual spins of its constituents. Quark models predicted that almost all the proton’s spin is actually carried by the quarks. It came therefore as a very big surprise that an experiment, the European Muon Collaboration (EMC), found that only half of the spin of the proton originates from the quarks [2]. This finding was named the spin crisis and attempts to explain of this puzzle led to an enormous amount of work.

Results and Methods

The simulations of lattice QCD are based on Markov chain Monte Carlo methods. In particular, we have used a Hybrid Monte Carlo algorithm which combines a Metropolis accept/reject step with a molecular dynamics time evolution of the gluons fields. This algorithm is used to generate gluon field configurations which are stored and then used to compute physical observables. Typical numbers of so generated gluon field configurations are 2,000–5,000. The lattice size we employ are 64³ x 128 and 80³ x 160 lattice points and since the internal degrees of freedom of the quark and gluon field is 12, this amounts to a size of a single gluon field configuration of 19 and 44 GBytes.

A basic ingredient to compute a physical observable are so-called quark propagators from which e.g. the proton mass or, as in this project, the individual contributions of quarks and gluons to the proton spin can be calculated. The quark propagators are computed by solving a very large linear system of equations with a coefficient matrix of 40 million time 40 million for our smaller lattice size. On first sight, this sounds impractical, but what helps is that this coefficient matrix is sparse with only the diagonal and a few sub-diagonals being filled. This allows to hard code the required matrix time vector multiplications as required in linear solver algorithms, e.g. conjugate gradient. In fact, in the early days of lattice QCD, the conjugate gradient algorithm has been the method of choice. However, since then substantial improvements could be achieved and nowadays algorithms, partly developed by us [4], based on multi grid methods are used which led to orders of magnitude reduced computational cost. It is only through these most significant algorithmic improvements in combination with new, ever improved supercomputer architectures, which led to the success of this project.

The HMC simulations performed for the larger volume were performed on 125 nodes using a hybrid parallelization employing 16 MPI tasks with 3 OpenMP threads per node. By using the above mentioned highly optimized numerical algorithms like algebraic multigrid methods and 4^th order force gradient integrators for the molecular dynamics of the HMC algorithm, the time per new gauge configuration could be reduced to 3 hours and 15 mins, resulting in 19,500 core-hours per trajectory. This requires a total of 60 M core-hours for one physical point ensemble, producing 3,000 gauge configurations of 132 TB in two parallel streams written during 200 days on the WORK partition of SuperMUC-NG. For the scaling of our algorithm, see Figure 2.

In a dedicated effort, which is embedded in the research programme of the Extended Twisted Mass Collaboration (ETMC), we were able to disentangle the various contributions to the proton spin. Since nature has decided—for an unknown reason—that besides the light up and down quarks also heavier strange and charm quarks exist, we included also the contributions of these quarks for the proton spin in order to obtain a comprehensive picture. The results can be seen in Figure 3, where we show the absolute size of the individual contributions as well as the percentage the different quarks and gluons contribute. Our finding that the effect of the gluons is rather large and that all contributions add up to the expected value of ½ provides a nice confirmation of the experimental results mentioned above. The proton spin puzzle is going to be put together and we start to see the picture it reveals.

Besides the spin, the proton also has an average momentum with a value of 1. In the same setup as was used for the proton spin we could also disentangle the various contributions of the quarks and the gluons to the proton average momentum. The result is shown in Figure 3, and, again, we nicely see, how the different contributions add up to the value of 1, as observed in nature.