Lattice QCD at the physical point: Simulation and analysis details

TL;DR

This work delivers a high-precision, first-principles determination of the light ($m_{ud}$) and strange ($m_s$) quark masses in $N_f=2+1$ QCD at the physical point, employing five lattice spacings, large volumes, and pion masses down to 120 MeV. It combines a HEX-smeared clover fermion action with a sophisticated HMC algorithm, nonperturbative RI/MOM renormalization, and a novel ratio-difference method to control renormalization and discretization errors, while using dispersive input to disentangle $m_u$ and $m_d$ from $m_{ud}$ and $m_s$. The study demonstrates robust scaling, finite-volume control, and a careful systematic error analysis, yielding $m_{ud}$ and $m_s$ with sub-2% precision and $m_s/m_{ud}$ to sub-1% accuracy; it further derives $m_u$ and $m_d$ from dispersive input, strongly disfavouring a massless up quark and impacting discussions of the strong CP problem. The results are presented in RI/MOM at 4 GeV, with consistent conversions to RGI and $ar{ ext{MS}}$ schemes, and are reinforced by quenched benchmarks and comprehensive crosschecks.

Abstract

We give details of our precise determination of the light quark masses m_{ud}=(m_u+m_d)/2 and m_s in 2+1 flavor QCD, with simulated pion masses down to 120 MeV, at five lattice spacings, and in large volumes. The details concern the action and algorithm employed, the HMC force with HEX smeared clover fermions, the choice of the scale setting procedure and of the input masses. After an overview of the simulation parameters, extensive checks of algorithmic stability, autocorrelation and (practical) ergodicity are reported. To corroborate the good scaling properties of our action, explicit tests of the scaling of hadron masses in N_f=3 QCD are carried out. Details of how we control finite volume effects through dedicated finite volume scaling runs are reported. To check consistency with SU(2) Chiral Perturbation Theory the behavior of M_π^2/m_{ud} and F_πas a function of m_{ud} is investigated. Details of how we use the RI/MOM procedure with a separate continuum limit of the running of the scalar density R_S(μ,μ') are given. This procedure is shown to reproduce the known value of r_0m_s in quenched QCD. Input from dispersion theory is used to split our value of m_{ud} into separate values of m_u and m_d. Finally, our procedure to quantify both systematic and statistical uncertainties is discussed.

Figures (13)

• 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:2005gd on $M_\pi$ is larger than 1%, 0.3% and 0.1%, respectively. In our runs this effect is smaller than about 0.5%, but we still correct for this tiny effect.
• Figure 2: Histogram of the inverse iteration count ($N_\mathrm{CG}^{-1}$ of the lightest pseudofermion) in the ensemble with the lightest quark mass per $\beta$. There is no danger of a tail stretching out to zero.
• Figure 3: Evolution of the maximum of each MD force during the MD integration. 256 steps correspond to one $\tau\!=\!1$ trajectory. Shown is one production stream of our "physical pion mass" ensemble at $\beta\!=\!3.31$. The other streams with the same parameters give a similar picture.
• Figure 5: Topological charge history at our finest lattice spacing ($\beta\!=\!3.8$ corresponds to $a^{-1}\!\sim\!3.7\,\mathrm{GeV}$) using two vigorous smearings (10 or 30 HYP steps) in the gluonic charge definition.
• Figure 6: Scaling of the nucleon and delta mass, at fixed $M_\pi/M_\rho=0.67$, versus $\alpha a$ and $a^2$.