Lattice QCD at the physical point: light quark masses

Abstract

Ordinary matter is described by six fundamental parameters: three couplings (gravitational, electromagnetic and strong) and three masses: the electron's (m_e) and those of the up (m_u) and down (m_d) quarks. An additional mass enters through quantum fluctuations: the strange quark mass (m_s). The three couplings and m_e are known with an accuracy of better than a few per mil. Despite their importance, $m_u$, $m_d$ (their average m_{ud}) and m_s are relatively poorly known: e.g. the Particle Data Group quotes them with conservative errors close to 25%. Here we determine these quantities with a precision below 2% by performing ab initio lattice quantum chromodynamics (QCD) calculations, in which all systematics are controlled. We use pion and quark masses down to (and even below) their physical values, lattice sizes of up to 6 fm, and five lattice spacings to extrapolate to continuum spacetime. All necessary renormalizations are performed nonperturbatively.

Figures (2)

• Figure 1: Summary of our simulation points. The pion masses and the spatial sizes of the lattices are shown for our five lattice spacings. The percentage labels indicate regions, in which the expected finite volume effect Colangelo:2003hf on $M_\pi$ is larger than 1%, 0.3% and 0.1%, respectively. This effect is smaller than about 0.5% for all of our runs and, as described, we corrected for it. Error bars are statistical.
• Figure 2: Continuum extrapolation of the average up/down quark mass, of the strange quark mass and of the ratio of the two. The errors of the individual points, which are statistical only here, are smaller than the symbols in most of the cases. The only exceptions are the light quark mass and its ratio to the strange quark mass at the two finest lattice spacings. These exceptions underline the importance of using physical quark masses to reach a high accuracy.