Ab-initio Determination of Light Hadron Masses

TL;DR

This work presents a full ab initio calculation of the masses of protons, neutrons, and other light hadrons, using lattice quantum chromodynamics, and represents a quantitative confirmation of this aspect of the Standard Model with fully controlled uncertainties.

Abstract

More than 99% of the mass of the visible universe is made up of protons and neutrons. Both particles are much heavier than their quark and gluon constituents, and the Standard Model of particle physics should explain this difference. We present a full ab-initio calculation of the masses of protons, neutrons and other light hadrons, using lattice quantum chromodynamics. Pion masses down to 190 mega electronvolts are used to extrapolate to the physical point with lattice sizes of approximately four times the inverse pion mass. Three lattice spacings are used for a continuum extrapolation. Our results completely agree with experimental observations and represent a quantitative confirmation of this aspect of the Standard Model with fully controlled uncertainties.

Figures (8)

• Figure 1: Effective masses $aM$=$\log[C(t/a)/C(t/a+1)]$, where $C(t/a)$ is the correlator at time $t$, for $\pi$, $K$, $N$, $\Xi$ and $\Omega$ at our lightest simulation point with $M_\pi$$\approx$190 MeV ($a \approx 0.085$ fm with physical strage quark mass). For every 10th trajectory, the hadron correlators were computed with Gaussian sources and sinks whose radii are approximately 0.32 fm. The data points represent mean $\pm$ SEM. The horizontal lines indicate the masses $\pm$ SEM obtained by performing single mass correlated cosh/sinh fits to the individual hadron correlators with a method similar to that of Bernard:2002pc.
• Figure 2: Pion mass dependence of the nucleon ($N$) and $\Omega$ for all three values of the lattice spacing. (A): masses normalized by $M_\Xi$, evaluated at the corresponding simulation points. (B): masses in physical units. The scale in this case is set by $M_\Xi$ at the physical point. Triangles on dotted lines correspond to $a$$\approx$0.125 fm, squares on dashed lines to $a$$\approx$0.085 fm and circles on solid lines to $a$$\approx$0.065 fm. The points were obtained by interpolating the lattice results to the physical $m_s$ (defined by setting 2$M_K^2$-$M_\pi^2$ to its physical value). The curves are the corresponding fits. The crosses are the continuum extrapolated values in the physical pion mass limit. The lattice-spacing dependence of the results is barely significant statistically despite the factor of 3.7 separating the squares of the largest ($a{\approx}0.125$ fm) and smallest ($a{\approx}0.065$ fm) lattice spacings. The $\chi^2$/degrees of freedom values of the fits in (A) are 9.46/14 ($\Omega$) and 7.10/14 ($N$), whereas those of the fits in (B) are 10.6/14 ($\Omega$) and 9.33/14 ($N$). All data points represent mean $\pm$ SEM.
• Figure 3: The light hadron spectrum of QCD. Horizontal lines and bands are the experimental values with their decay widths. Our results are shown by solid circles. Vertical error bars represent our combined statistical (SEM) and systematic error estimates. $\pi$, $K$ and $\Xi$ have no error bars, because they are used to set the light quark mass, the strange quark mass and the overall scale, respectively.
• Figure S1: Effective masses for different source types in the pion (left panel) and nucleon (right panel) channels. Point sources have vanishing extents, whereas Gaussian sources, used on Coulomb gauge fixed configurations have radii of approximately 0.32 fm. Clearly, the extended sources/sinks result in much smaller excited state contamination.
• Figure S2: Forces in the molecular dynamics time history. We show here this history for a typical sample of trajectories after thermalization. Since the algorithm is more stable for large pion masses and spatial sizes, we present --as a worst case scenario-- the fermionic force for our smallest pion mass ($M_\pi{\approx}190$ MeV; $M_\pi L{\approx}4$). The gauge force is the smoothest curve. Then, from bottom to top there are pseudofermion 1, 2, the strange quark and pseudofermion 3 forces, in order of decreasing mass. No sign of instability is observed.