# Up-, down-, strange-, charm-, and bottom-quark masses from four-flavor lattice QCD

###### Abstract

We calculate the up-, down-, strange-, charm-, and bottom-quark masses using the MILC highly improved staggered-quark ensembles with four flavors of dynamical quarks. We use ensembles at six lattice spacings ranging from fm to fm and with both physical and unphysical values of the two light and the strange sea-quark masses. We use a new method based on heavy-quark effective theory (HQET) to extract quark masses from heavy-light pseudoscalar meson masses. Combining our analysis with our separate determination of ratios of light-quark masses we present masses of the up, down, strange, charm, and bottom quarks. Our results for the -renormalized masses are MeV, MeV, MeV, MeV, and MeV, with four active flavors; and MeV with five active flavors. We also obtain ratios of quark masses , , and . The result for matches the precision of the most precise calculation to date, and the other masses and all quoted ratios are the most precise to date. Moreover, these results are the first with a perturbative accuracy of . As byproducts of our method, we obtain the matrix elements of HQET operators with dimension 4 and 5: MeV in the minimal renormalon-subtracted (MRS) scheme, , and . The MRS scheme [Phys. Rev. D97, 034503 (2018), arXiv:1712.04983 [hep-ph]] is the key new aspect of our method.

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^{†}preprint: FERMILAB-PUB-17/492-T

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^{†}preprint: TUM-EFT 107/18

Fermilab Lattice, MILC, and TUMQCD Collaborations

## I Introduction

Quark masses are fundamental parameters of QCD. They must be known accurately for precise theoretical calculations within the Standard Model, especially for testing whether quarks receive mass via Yukawa couplings to the Higgs field. Because of confinement, the quark masses can be defined only as renormalized parameters of the QCD Lagrangian. Thus, they must be determined by comparing theoretical calculations of an appropriate set of observables to experimental measurements of those observables. Lattice QCD makes it possible to calculate in a nonperturbative way simple observables, such as hadron masses. To determine the quark masses in lattice QCD, one needs to tune the bare lattice quark masses such that a suitable set of hadron masses coincide with their experimental values.

The resulting bare masses must be renormalized, preferably to a regularization independent scheme, such as the recently introduced minimal renormalon subtracted (MRS) mass Brambilla et al. (2018). One approach is to use lattice perturbation theory, but multiloop calculations are difficult so, in practice, nothing more than two-loop matching Mason et al. (2006); Skouroupathis and Panagopoulos (2009); Constantinou et al. (2016) is available in the literature. Another is to use nonperturbative renormalization to, for example, momentum-subtraction Martinelli et al. (1995); Lytle and Sharpe (2013) or finite-volume Capitani et al. (1999); Della Morte et al. (2005) schemes. Finally, one can use lattice gauge theory to obtain quantities in continuum QCD and apply multiloop continuum perturbative QCD to extract the quark masses. An example of the latter is the analysis of quarkonium correlators Allison et al. (2008). In practice, no regularization-independent scheme is in such common use as the modified minimal subtraction () scheme Bardeen et al. (1978) of dimensional regularization, so we shall use to quote results.

Our method studies how a heavy-light meson mass depends on the mass of its heavy valence (anti)quark Kronfeld and Simone (2000); Freeland et al. (2006); Gambino et al. (2017). Like the quarkonium correlators, our approach requires only continuum perturbation theory. On the other hand, the binding energy of a heavy-light meson is of order , so it is necessary to use heavy quark effective theory (HQET) to separate long- and short-distance scales. In this way, we can obtain the masses of the charm and bottom quarks and, at the same time, HQET matrix elements Kronfeld and Simone (2000). Because this analysis uses as inputs the bare masses of the up, down, and strange quarks—tuned to reproduce the pion and kaon masses Bazavov et al. (2017), it also yields the renormalized masses of these quarks.

Following Ref. Kronfeld and Simone (2000), our analysis is based on the HQET formula for the heavy-light meson mass Falk and Neubert (1993)

(1) |

where is the pseudoscalar (vector) meson mass; is the heavy-quark mass; and , , and are matrix elements of HQET operators with dimension 4 and 5. The last three correspond to the energy of the light quarks and gluons, the heavy quark’s kinetic energy, and the spin-dependent chromomagnetic energy, with coefficient for pseudoscalar mesons and for vector mesons. The chromomagnetic operator has an anomalous dimension, known to three loops Grozin et al. (2008), so depends logarithmically on the mass . The strategy is to use lattice QCD to compute as a function of and fit Eq. (1) to distinguish the terms on the right-hand side including, in principle, higher orders in Kronfeld and Simone (2000).

The utility of Eq. (1) rests on the definition of the quark mass .
In HQET, the natural definition is the pole mass (also known as the on-shell mass).
Although the pole mass is infrared finite Kronfeld (1998) and gauge independent Kronfeld (1998); Breckenridge et al. (1995) at
every order in perturbation theory, its value is ambiguous when all orders are considered Bigi et al. (1994); Beneke and Braun (1994).
At large orders, the coefficients of the self energy grow factorially, and a possible interpretation via Borel summation is
obstructed by a series of renormalon singularities Bigi et al. (1994); Beneke and Braun (1994).
This behavior is a manifestation of the strongly-coupled long-range gluon field that, remarkably, appears in perturbation theory.
Note that because is unambiguous, the ambiguity in must be cancelled by those in , , and
higher-dimension terms.^{1}^{1}1By forming the spin average, , and spin difference, , it is easy to see that
spin-independent and spin-dependent ambiguities are distinct.

To address this problem, some of us introduced the minimal renormalon-subtracted (MRS) mass in a companion paper Brambilla et al. (2018). It is defined by Eqs. (2.24) of Ref. Brambilla et al. (2018),

(2) |

where , the are the coefficients relating the mass to the pole mass, denote their asymptotic behavior, and , which is defined in Eqs. (2.25) and (2.26) of Ref. Brambilla et al. (2018), is the unambiguous part of the Borel sum of . To compute one uses the known behavior of the Beneke (1995); Pineda (2001); Komijani (2017), including their overall normalization Komijani (2017). In deriving Eq. (2), Ref. Brambilla et al. (2018) puts the leading renormalon ambiguity into a specific quantity of order , denoted , and transfers it from to . Below we write and to denote the unambiguous definitions of and in the MRS scheme.

A second feature of our technique may seem almost trivial. In Eq. (4.7), Ref. Brambilla et al. (2018) rewrites as

(3a) | ||||

(3b) |

where is the bare mass (in lattice units) of staggered fermions, and the subscript labels a reference mass; see Sec. III. Owing to the remnant chiral symmetry of staggered fermions, the first factor in Eq. (3a) is . In Eq. (3b), the factors are, respectively, a convenient fit parameter, the factor to run from scale to , the quantity in the big parentheses in Eq. (2), and the ratio of the freely chosen heavy-quark lattice mass to the reference mass. Equation (3) plays a key role: with for in Eq. (2), the first factor in Eq. (3b) is in the scheme; with removing the leading renormalon ambiguity, the product on the right-hand side of Eq. (3b) is indeed the MRS mass. By taking (the so-called approach), the analysis yields as well as the heavy-quark masses and .

The third important feature of our work is a data set with a wide range of lattice spacing, heavy-quark mass, and light valence and sea masses. These data, which were generated in a companion project to compute the - and -meson decay constants Bazavov et al. (2017), are very precise, with statistical errors of 0.005–0.12%. It is very challenging to take advantage of the statistical power and parameter range of the data set. In this paper, we use heavy-meson rooted all-staggered chiral perturbation theory Bernard and Komijani (2013) (HMrASPT) to describe the dependence of the heavy-light pseudoscalar meson masses on the light mesons. To make possible a fit to lattice data, Ref. Brambilla et al. (2018) combined the next-to-leading-order HMrASPT with the MRS mass to write heavy-light meson masses as a function of lattice spacing and heavy- and light-quark masses. The fit function, by construction, has the correct nonanalytic form in the chiral and HQET limits. Here, it is extended with enough analytic terms to mimic higher-order corrections and obtain a good fit.

We use twenty ensembles generated by the MILC Collaboration Bazavov et al. (2010a, 2013, 2008) with four flavors of sea quarks using the highly-improved staggered quark (HISQ) action Follana et al. (2007) and a one-loop Hart et al. (2009) tadpole-improved Alford et al. (1995) Symanzik-improved gauge action Weisz (1983); Weisz and Wohlert (1984); Curci et al. (1983); Lüscher and Weisz (1985a, b). The algorithm for the quark determinant uses the fourth-root procedure to remove the unwanted taste degrees of freedom Marinari et al. (1981); Follana et al. (2004); Dürr et al. (2004); Dürr and Hoelbling (2005); Wong and Woloshyn (2005); Shamir (2005); Prelovsek (2006); Bernard (2006); Dürr and Hoelbling (2006); Bernard et al. (2006); Shamir (2007); Bernard et al. (2007); Kronfeld (2007); Donald et al. (2011). A thorough description of the simulation program can be found in Ref. Bazavov et al. (2013). Since then, the simulations have been extended to smaller lattice spacings; up-to-date details are in Ref. Bazavov et al. (2017). Our procedures for calculating pseudoscalar meson correlators and for finding masses and amplitudes from these correlators are described in Refs. Bazavov et al. (2014a); *Bazavov:2014lja; Bazavov et al. (2017). The amplitudes are used in Ref. Bazavov et al. (2017) to calculate the decay constants of and mesons, and the corresponding meson masses are used here.

A preliminary report of this analysis can be found in Ref. Komijani et al. (2016). Instead of the MRS mass, at that time we used the renormalon-subtracted (RS) mass Pineda (2001), which also subtracts the leading renormalon ambiguity but at the same time introduces a factorization scale . In principle, the masses emerging from Eqs. (3) and (1) should not depend on , but we found more dependence than one would like. Moreover, it turns out to be necessary to introduce three scales in all, , with being used for Komijani et al. (2016). For that reason, we prefer the MRS over the RS mass.

This paper is organized as follows. Section II contains a description of the lattice-QCD simulations, focusing on the way we eliminate the lattice scale in favor of physical units. In Sec. III, we present our function of quark masses and lattice spacing that describes masses of heavy-light pseudoscalar mesons. In Sec. IV, we perform a combined-correlated fit to the meson masses; the fit is then extrapolated to the continuum and interpolated to physical values of the light quark masses. In Sec. V, we present our final results for the masses of the strange, charm and bottom quarks as well as quark mass ratios , , and . Combining our results with our separate determination of the quark-mass ratios and , where , we also report the up- and down-quark masses. In addition, we present our lattice-QCD determinations of , , and as well as flavor splittings and low-energy constants of heavy-meson chiral perturbation theory. Section VI compares our main results with work in the literature and offers some remarks on further work. An appendix gives the correlation matrices of the MRS masses of the charm and bottom quarks with HQET matrix elements, and of the charm-quark mass and quark-mass ratios.

## Ii Simulations summarized

The lattice data used in this work come from the same correlation functions used to determine leptonic decay constants of charmed and -flavored mesons in a companion paper Bazavov et al. (2017). For a full description of the simulation, the reader should consult Ref. Bazavov et al. (2017). Here we provide a brief summary.

We employ a data set that includes ensembles with five values of lattice spacings ranging from approximately 0.12 fm to 0.03 fm, enabling good control over the continuum extrapolation. Ensembles at a sixth lattice spacing, approximately 0.15 fm, are used only to estimate the continuum extrapolation error. The data set includes ensembles with the light (up-down), strange, and charm sea-masses close to their physical values (“physical-mass ensembles”) at all but the smallest lattice spacing, 0.03 fm. The data set also includes ensembles where either the mass of light sea quarks is heavier than in nature, or the mass of the strange sea quark is lighter than in nature, or both. As in Ref. Bazavov et al. (2017), we set the scale of the lattice spacing with a two-step procedure that uses the value of from the Particle Data Group (PDG), MeV Patrignani et al. (2016); *Rosner:2015wva, combined with the so-called method.

The first step in the scale-setting procedure takes to set the overall scale on each physical-mass ensemble. On these ensembles, we tune the valence light, strange, and charmed quark masses to reproduce the pion, kaon, and -meson masses. Then we calculate and , which are the mass and decay constant of a pseudoscalar meson with both valence-quark masses set equal to . We then form the ratio and take the continuum limit of and . These values and those of the quark mass ratios are then used as inputs to the second step of the procedure, which we call the method. In the method, the values of and are calculated on a given physical-mass ensemble, with , by adjusting the valence-quark mass until equals the physical-mass continuum limit of :

(4) |

In the method, all ensembles at the same bare gauge coupling, , as a given physical-mass ensemble are chosen to have the same lattice spacing and the same . This choice is known as a mass-independent scale-setting scheme.

At fm, we have only a 0.2 ensemble, so this procedure cannot be carried out. In this case, we rely on the derivatives with respect to , which are given in Ref. Bazavov et al. (2014a); *Bazavov:2014lja.

## Iii Construction of the fit function

In this section, we discuss in detail how to construct a function of quark masses and lattice spacing that describes masses of heavy-light pseudoscalar mesons. To this end, we use three effective field theories (EFTs), HQET, and HMrASPT, as mentioned already, and the Symanzik effective theory of cutoff effects Symanzik (1980, 1983); Lüscher and Weisz (1985a). We start with the merger of HQET and HMrASPT Brambilla et al. (2018) and incorporate generic lattice-spacing dependence, as well as higher-order terms in HQET and HMrASPT. Putting everything together, we obtain an EFT fit function for masses of heavy-light pseudoscalar mesons.

### iii.1 Leading-order Pt

Let us start with fixing our notation for quark masses associated with lattice ensembles with flavors of quarks. We use , , and to denote the simulation masses of the light (up-down), strange, and charm quarks, respectively; without the primes, we use , , and to denote the correctly tuned masses of the corresponding quarks; last, we use to denote a generic light quark mass. Further, we use to denote a generic heavy-light pseudoscalar (vector) meson composed of a light valence quark and a heavy valence antiquark . We also use , , and to denote the MRS, , and bare masses of antiquark , respectively. The relations between , , and are discussed in Sec. III.3.

In HMrASPT, the mass of meson is described by Eq. (4.2) of Ref. Brambilla et al. (2018)

(5) | ||||

where is the low energy constant (LEC) in the relation between the pion mass and the quark mass; () for pseudoscalar (vector) mesons; and are LECs that appear in (continuum) heavy-meson chiral perturbation theory (HMPT) Boyd and Grinstein (1995); and is the one-loop corrections to the mass of the meson in HMrASPT Brambilla et al. (2018). The arguments of and in Eq. (5) correspond to the light valence-quark mass; the set of three light sea-quark masses, which are not necessarily tuned to their physical values; and the lattice spacing . As usual for a one-loop PT result, contains a term nonanalytic as (a “chiral log”). For the pseudoscalar mesons with light flavors in the sea, we have.

(6) | ||||

where the indices and run over light sea-quark flavors and meson tastes, respectively; is the mass of the pseudoscalar meson with taste and flavors and ; is the lowest-order hyperfine splitting; is the flavor splitting between a heavy-light meson with light quark of flavor and one of flavor ; is the -- coupling; and are the taste-breaking hairpin parameters; is the mean-squared pion taste splitting; and and are parameters in SPT related to taste breaking in meson masses. Definitions of the residue functions , the sets of masses in the residues, and the chiral-log function at infinite and finite volumes are given in Ref. Brambilla et al. (2018) and references therein. The expression for is also given in Ref. Brambilla et al. (2018), but because we have lattice data only for pseudoscalar mesons, it is not needed here.

In Eq. (5), we set

(7) |

so that in the continuum limit the usual expression

(8) |

is recovered for physical values of sea-quark masses and .
With this choice for , the values that we obtain for , and are readily applicable
for calculations in HQET.^{2}^{2}2Note that in the context of Eq. (8), the matrix elements , and
depend on the light-quark masses.
In this work, we set , and we report , and for this choice.

At this stage, the fit parameters are via Eq. (3), , the kinetic energy , the chromomagnetic energy from which we obtain as in Eq. (10) below, and the LECs , , , and . Ideally, one would have data for both pseudoscalar- and vector-meson masses, and then one could set up separate fits for spin-independent and spin-dependent terms. In this work, however, only the pseudoscalar masses are available. The experimental masses of the and meson can be used to estimate

(9) |

which neglects contributions to the hyperfine splitting suppressed by a power of . The chromomagnetic operator has an anomalous dimension, however, so we obtain in Eq. (5) with

(10) |

using the three-loop relation Grozin et al. (2008) for the Wilson coefficient . For four active flavors,

(11) |

where .

As discussed in Sec. I and Ref. Brambilla et al. (2018), the matrix elements of HQET suffer in general from ambiguities related to renormalon singularities, although the ambiguities cancel in observables such as the meson mass. For instance, the ambiguity in cancels the leading-renormalon ambiguity in the pole mass. By construction, only the leading renormalon is removed to define the MRS mass. In principle, renormalon ambiguities in and remain. In practice, numerical investigation indicates that the subleading infrared renormalon of the pole mass is small Brambilla et al. (2018), which implies that the corresponding renormalon ambiguity in is not large. Moreover, the leading spin-dependent renormalon in is suppressed by a further power of .

### iii.2 Higher-order terms in Pt

Because we have very precise data with statistical errors of 0.005–0.12%, we can anticipate that NLO PT is not enough to describe fully the quark-mass dependence, especially for data with near . We therefore extend the function given in Eq. (5) by adding higher-order analytic corrections in powers of light quark masses and in inverse powers of the heavy quark mass. For the expansion in inverse powers of the heavy-quark mass, we introduce the dimensionless variable

(12) |

with . Then the natural size of coefficients of the corrections is of order 1. For expansion in light quark masses, following Refs. Bazavov et al. (2014a); *Bazavov:2014lja; Bazavov et al. (2017), we define dimensionless quark masses, which are natural expansion parameters in PT:

(13) |

where can be either the valence or sea light quarks. For simplicity, we drop the primes on the simulation s. The quark masses in the formula for can also be expressed in terms of .

We include all mass-dependent analytic terms at order by adding

(14) |

to the expression for in Eq. (5). With to set the overall scale of these higher-order terms, the coefficients become of order 1 or less. We also include all mass-dependent analytic terms at order , namely

(15) |

In practice, one can expect the terms without to be less important, but we keep all of them for consistent power counting.

To improve the expansion in inverse powers of the heavy quark, we add

(16) |

with three fit parameters to the right-hand side of Eq. (5). We also add and corrections to the LECs , and ; and corrections to the fit parameters in Eq. (14).

The heavy quark mass also affects the hyperfine splitting and the flavor splitting in Eq. (6). Although we could express these quantities in terms of and , we exploit the experimental values for the hyperfine splittings and flavor splittings in the and systems to calculate and for different quark masses. See our companion paper on decay constants Bazavov et al. (2017) for details.

We now discuss the effects of mistuning in the sea charm-quark mass . The effects can be divided into two parts: the effects on the pole mass (and, hence, the MRS mass) and the effects on the effective theory after the charm quark is integrated out. The former effects are taken into account in calculating the MRS mass from the mass; cf. Eq. (28). We treat the latter effects as in Ref. Bazavov et al. (2017). We use to denote the effective value of when the charm quark with mass is integrated out. At leading order in weak-coupling perturbation theory, one obtains (Manohar and Wise, 2000, Eq. (1.114))

(17) |

where is the correctly tuned value of charm-quark mass. Assuming , we take the effects of the mistuned mass into account by multiplying with

(18) |

where the extra fit parameter describes higher-order corrections to Eq. (17).

We must also include generic lattice artifacts in our analysis. Taste-breaking discretization errors from staggered fermions are already included in Eq. (6). In addition to these effects, various discretization errors, from gluons for example, must be taken into account. We include the leading lattice artifacts for by replacing.

(19) |

where is the scale of generic discretization effects, set to 600 MeV in this analysis. The factor of in the second-order term arises because the HISQ action is tree-level improved to order . Note that is not affected by heavy-quark discretization errors. As discussed in the Appendices of Ref. Bazavov et al. (2017), at leading order (LO) in HQET, heavy-quark discretization errors only affect the normalization of the heavy-quark state. Thus, and also , and at leading order in are free of heavy-quark discretization errors. For we replace

(20) |

where the term is added to incorporate effects of heavy-quark discretization errors. We incorporate similar corrections for and . Finally, we add and corrections to and ; and corrections to the parameters in Eq. (14).

### iii.3 Heavy-quark mass

Although the MRS mass is the key to our interpretation of the HQET mass formula, as indicated in Eq. (3) we arrange the fit to yield the mass. For , the relation between the and bare masses is

(21) |

where in the denominator is set from the scale setting quantity (here , as described in Sec. II).
With staggered fermions, there is no additive mass renormalization,
and to eliminate tree-level discretization errors from Eq. (21), we take the mass to be the tree-level pole mass.^{3}^{3}3The exact relation between and can be found in Appendix A of Ref. Bazavov et al. (2017).
Taking the ratio between two masses^{4}^{4}4For Wilson fermions with order- improvement, the following arguments hold for the mass defined through the axial
Ward identity, apart from details about the lattice artifacts.

(22) |

where the dots stand for higher-order terms in and . In fact, each higher order in is also multiplied by a quantity of order , as stated in the Introduction. In this analysis, we set the reference-quark mass to and the scale of the scheme to GeV. Thus, is a free parameter left to be determined in the fit to lattice data; cf. Eq. (3b).

To incorporate further heavy-quark discretization effects into Eq. (22), we multiply the right-hand side of Eq. (3b) by

(23) |

where the dimensionless coefficients are free fit parameters, and

(24) |

We multiply by a factor of so that the parameters become of order 1, based on the radius of convergence of various tree-level formulas for the HISQ action; see Appendix A of Ref. Bazavov et al. (2017). Because , the effects of a nonzero value of are negligible compared with the heavy-quark discretization effects. To incorporate generic lattice-spacing dependence into our analysis, we additionally multiply the right-hand side of Eq. (3b)

(25) |

To complete our approach to introducing via , we must describe the calculation of the second and third factors in Eq. (3b). The second factor simply uses the anomalous dimension to run from to the self-consistent scale

(26) |

where with four active flavors Baikov et al. (2014)

(27) |

The coefficient of is obtained from the five-loop results for the quark-mass anomalous dimension Baikov et al. (2014) and beta function Baikov et al. (2017). Finally, the third factor in Eq. (3b) is simply the relation derived in Ref. Brambilla et al. (2018), which at the four-loop level reads

(28) |

where the are known through order Marquard et al. (2015, 2016); the depend only on the coefficients of the beta function Beneke (1995); Komijani (2017) up to an overall normalization, which is given in Ref. Komijani (2017); the function appears in the definition of the MRS mass Brambilla et al. (2018); and contains the contribution from the charm sea quark. Because the nonzero mass of the charmed sea quark cuts off the infrared region that is the origin of factorial growth in the Ball et al. (1995), we subtract the renormalon with three massless active quarks and lump the charmed loops’ contributions into Ayala et al. (2014).

The detailed formulas for and can be found in Ref. Brambilla et al. (2018). The crucial aspects of Eq. (28) for the fits of the next section is that the renormalon-subtracted perturbative coefficients are small: for and three active flavors. The Borel resummed renormalon is computed from a function with a convergent expansion in . (In fact, our implementation of one of the factors in uses the convergent expansion until it saturates to numerical precision.)

### iii.4 Summary formulas

In summary, we fit our data for to

(29) |

where is the fit function and is in the continuum limit. From the preceding subsections [with free fit parameters in blue (arXiv)]:

(30) | ||||

where and . The HMrASPT self energy depends on , , , , , and , as well as and taste-independent . The breved quantities are

(31a) | ||||

(31b) | ||||

(31c) | ||||

(31d) | ||||

(31e) | ||||

(31f) |

where ; further

(32) | ||||

Thus, there are free parameters, 4 parameters [, , , and, in , ] with external priors, and hairpin parameters ( and ) from light-meson PT. and introduce 2 parameters each that are, however, frozen to reproduce PDG hyperfine and flavor splittings. The total number of fit parameters is 67 (compared with 60 for the decay-constant fit Bazavov et al. (2017)).

## Iv EFT fit to determine the quark masses

In Sec. III, we have constructed a function with 67 fit parameters that is motivated by EFTs. Here, we use this function to perform a correlated fit to partially-quenched data at five lattice spacings, from fm to fm, and at several values of the light sea-quark masses. A sixth lattice spacing, fm, is used to check discretization errors but is not included in the base fit. At the coarsest lattice spacings, we only have data with two different values for valence heavy-quark mass: and , where is the simulation value of sea charm-quark mass in each ensemble. It is close to but not precisely equal to the physical charm mass because of tuning errors. We include data with subject to condition , which is chosen to avoid large lattice artifacts. For every valence heavy quark, we use several light valence quarks with masses ; on ensembles with the mass of the strange sea quark close to its physical value, takes values in a subset of (in several cases the whole set). In the base fit, we obtain the meson masses from fits to two-point correlators with three pseudoscalar states and two opposite-parity states, which we denote “3+2” below. To investigate the error arising from excited state contamination, we also use meson-mass data from (2+1)-state fits.

The values of the bare masses corresponding to the light and strange quarks are taken from combinations of the physical pion and kaon masses, as discussed in Refs. Bazavov et al. (2017) and Bazavov et al. (2014a). Similarly, the physical charmed and bottom quarks are defined so that the - and -meson masses take their physical values. Because the gauge-field ensembles omit electromagnetism, we need to subtract electromagnetic effects from the experimentally measured masses, which means introducing a specific scheme to do so. We identify and adjust accordingly. Then, is tuned to obtain