High-precision determination of the light-quark masses from realistic lattice QCD

TL;DR

This work addresses the precise determination of light-quark masses by combining three-flavor unquenched lattice QCD using the asqtad staggered action with NNLO perturbative matching. The authors perform a two-stage lattice-to-$\overline{MS}$ determination via the pole mass, organizing the perturbation theory with $\alpha_V(q^*)$ and applying Blended Largest Moment (BLM) scale setting to achieve NNLO accuracy. They compute the two-loop lattice self-energy, establish the $Z_m$ renormalization factor, and extract $m_s^{\overline{MS}}(2\,\mathrm{GeV})=(87\pm 4)\,\mathrm{MeV}$ and $\hat m^{\overline{MS}}(2\,\mathrm{GeV})=(3.2\pm 2)\,\mathrm{MeV}$, with $m_u$ and $m_d$ obtained from the MILC ratio $m_u/m_d=0.43$. The results substantially reduce perturbative uncertainties (by about a factor of two) and provide a robust, three-flavor framework for high-precision QCD parameters, offering improvement over prior one-loop determinations and guiding future NNLO analyses for heavy quarks and matrix elements.

Abstract

Three-flavor lattice QCD simulations and two-loop perturbation theory are used to make the most precise determination to date of the strange-, up-, and down-quark masses, $m_s$, $m_u$, and $m_d$, respectively. Perturbative matching is required in order to connect the lattice-regularized bare- quark masses to the masses as defined in the \msbar scheme, and this is done here for the first time at next-to-next-to leading (or two-loop) order. The bare-quark masses required as input come from simulations by the MILC collaboration of a highly-efficient formalism (using so-called ``staggered'' quarks), with three flavors of light quarks in the Dirac sea; these simulations were previously analyzed in a joint study by the HPQCD and MILC collaborations, using degenerate $u$ and $d$ quarks, with masses as low as $m_s/8$, and two values of the lattice spacing, with chiral extrapolation/interpolation to the physical masses. With the new perturbation theory presented here, the resulting \msbar\ masses are $m^\msbar_s(2 {GeV}) = 87(0)(4)(4)(0)$ MeV, and $\hat m^\msbar(2 {GeV}) = 3.2(0)(2)(2)(0)$ MeV, where $\hat m = \sfrac12 (m_u + m_d)$ is the average of the $u$ and $d$ masses. The respective uncertainties are from statistics, simulation systematics, perturbation theory, and electromagnetic/isospin effects. The perturbative errors are about a factor of two smaller than in an earlier study using only one-loop perturbation theory. Using a recent determination of the ratio $m_u/m_d = 0.43(0)(1)(0)(8)$ due to the MILC collaboration, these results also imply $m^\msbar_u(2 {GeV}) = 1.9(0)(1)(1)(2)$ MeV and $m^\msbar_d(2 {GeV}) = 4.4(0)(2)(2)(2)$ MeV. A technique for estimating the next order in the perturbative expansion is also presented, which uses input from simulations at more than one lattice spacing.

Figures (5)

• Figure 1: The two-loop diagrams that contribute to the pole-mass renormalization in the lattice scheme. The numbering is consistent with Ref. Panagop. The filled circles represent five one-loop sub-diagrams which dress the gluon propagator at that order (these contain gluon and ghost loops, and the measure term), in addition to the diagrams with internal-quark loops, which are shown explicitly. The quark-loop diagrams are understood to be summed over all flavors. The crosses represent vertices generated by the perturbative expansion of tadpole and other renormalization factors in the gluon and quark actions.
• Figure 2: Schematic representation of an IR subtraction for the two-loop pole mass; appropriate traces of the self-energy with an energy projector are implicit.
• Figure 3: The $n_f=0$ part of the two-loop pole mass $M^{(4)}$ for the "asqtad" action (in units of $\alpha_{\rm lat}^2$), for varying quark mass $m_0 a$ (upper panel). The squares and crosses distinguish two independent sets of calculations, each one by a different author. The lower panel shows the results after subtracting the known logarithms in $M^{(4)}$, which yields the term $A_{20}$ in in Eq. (\ref{['e:LatticeMpole']}).
• Figure 4: Comparison of the loop-quark mass dependence of pole-mass renormalization factors, on the continuum side of the matching, and on the lattice side. The lattice calculation used $am_{\rm valence}=0.001$. The continuum function was taken from Ref. Broadhurst; the solid lines are from analytic expressions for the associated limits, cf. Eq. (\ref{['e:Delta']})
• Figure 5: BLM scales $aq^*$ for the lattice-bare mass to $\overline{\text{MS}}$ mass matching factor $Z_m(\mu a)$, as functions of $\mu a$. The solid line shows the first-order scale $q_1^*$, while the dashed and dotted lines show the second-order scales $q_{2+}^*$ and $q_{2-}^*$, respectively, in regions where they are real. The first-order scale is the correct choice for $\mu a \alt 0.61$ and $\mu a \agt 3.74$, while the appropriate second-order scale applies to the region in between.