Recent Progress in Lattice QCD

Recent progress in Lattice QCD is highlighted. After a brief introduction to the methodology of lattice computations the presentation focuses on three main topics: Hadron Spectroscopy, Hadron Structure and Lattice Flavor Physics. In each case a summary of recent computations of selected quantities is provided.Comment: Review talk given at Physics in Collision 2012, Strbske Pleso, Slovakia; 14 pages, 4 tables, 6 figure

Physics of eta-prime with rooted staggered quarks

The quark-mass dependence of the eta in the Schwinger model, which -- like the eta-prime in QCD -- becomes massive through the axial anomaly, is studied on the lattice with N_f=0,1,2. Staggered quarks are used, with a rooted determinant for N_f=1. In the chiral limit the Schwinger mass is reproduced, which suggests that the anomaly is being treated correctly.Comment: 11 pages, 7 figures, 2 tables; v2: presentation improved, 4 refs adde

Three Dirac operators on two architectures with one piece of code and no hassle

A simple minded approach to implement three discretizations of the Dirac operator (staggered, Wilson, Brillouin) on two architectures (KNL and core i7) is presented. The idea is to use a high-level compiler along with OpenMP parallelization and SIMD pragmas, but to stay away from cache-line optimization and/or assembly-tuning. The implementation is for N_v right-hand-sides, and this extra index is used to fill the SIMD pipeline. On one KNL node single precision performance figures for N_c=3, N_v=12 read 475 Gflop/s, 345 Gflop/s, and 790 Gflop/s for the three discretization schemes, respectively.Comment: 1+6 pages, 3 figures, proceedings of Lattice 2018; v2: typos in eqs. (3.1) and (4.2) corrected, results unchange

Validity of ChPT -- is M_\pi=135 MeV small enough ?

I discuss the practical convergence of the SU(2) ChPT series in the meson sector, based on 2+1 flavor lattice data by the Wuppertal-Budapest and Budapest-Marseille-Wuppertal collaborations. These studies employ staggered and clover-improved Wilson fermions, respectively. In both cases large box volumes and several lattice spacings are used, and the pion masses reach down to the physical mass point. We conclude that LO and NLO low-energy constants can be determined with controlled systematics, if there is sufficient data between the physical mass point and about 350 MeV pion mass. Exploratory LO+NLO+NNLO fits with a wider range reveal some distress of the chiral series near M_\pi ~ 400 MeV and suggest a complete breakdown beyond M_\pi ~ 500 MeV.Comment: 14 pages, 16 figures; written form of plenary talk at Lattice 201

Convergence issues in ChPT: a lattice perspective

This review addresses the practical convergence of the ChPT series in the p-regime. In the SU(2) framework there is a number of new results, and improved estimates of \bar\ell_3 and \bar\ell_4 are available. In the SU(3) framework few new lattice computations have appeared and the improvement in the precision of the low-energy constants L_i is comparatively slow. I sketch some of the convergence issues genuine to extensions of ChPT which include additional sources of chiral symmetry breaking (finite lattice spacing) and/or violations of unitarity (different sea and valence quark masses). Finally, it is pointed out that the quark mass ratios m_u/m_d, m_s/m_d happen to be such that no reordering of the chiral series is needed to accommodate the experimental pion and kaon masses.Comment: 10 pages, 11 figures, review talk given at "Kaon 13", Ann Arbo

Reconstructing Polyatomic Structures from Discrete X-Rays: NP-Completeness Proof for Three Atoms

We address a discrete tomography problem that arises in the study of the atomic structure of crystal lattices. A polyatomic structure T can be defined as an integer lattice in dimension D>=2, whose points may be occupied by $c$ distinct types of atoms. To ``analyze'' T, we conduct ell measurements that we call_discrete X-rays_. A discrete X-ray in direction xi determines the number of atoms of each type on each line parallel to xi. Given ell such non-parallel X-rays, we wish to reconstruct T. The complexity of the problem for c=1 (one atom type) has been completely determined by Gardner, Gritzmann and Prangenberg, who proved that the problem is NP-complete for any dimension D>=2 and ell>=3 non-parallel X-rays, and that it can be solved in polynomial time otherwise. The NP-completeness result above clearly extends to any c>=2, and therefore when studying the polyatomic case we can assume that ell=2. As shown in another article by the same authors, this problem is also NP-complete for c>=6 atoms, even for dimension D=2 and axis-parallel X-rays. They conjecture that the problem remains NP-complete for c=3,4,5, although, as they point out, the proof idea does not seem to extend to c<=5. We resolve the conjecture by proving that the problem is indeed NP-complete for c>=3 in 2D, even for axis-parallel X-rays. Our construction relies heavily on some structure results for the realizations of 0-1 matrices with given row and column sums

Lattice fermions with complex mass

We present evidence in the Schwinger model that rooted staggered fermions may correctly describe the m<0 sector of a theory with an odd number of flavors. We point out that in QCD-type theories with a complex-valued quark mass every non-chiral action essentially "borrows" knowledge about the theta-transformation properties from the overlap action.Comment: 8 pages, 4 figures. v2: eqn. (20) corrected, figs. 3,4 adjusted, conclusions unchanged. v3: text improved, 2 refs. added (version to appear in PRD