Eur. Phys. J. C (2020) 80:169https://doi.org/10.1140/epjc/s10052-020-7698-z

Regular Article - Theoretical Physics

Light quark masses in Nf = 2 + 1 lattice QCD with Wilsonfermions

ALPHA Collaboration

M. Bruno1 , I. Campos2 , P. Fritzsch1 , J. Koponen3,a , C. Pena4,5 , D. Preti6 , A. Ramos7 , A. Vladikas8

1 Theoretical Physics Department, CERN, 1211 Geneva 23, Switzerland2 Instituto de Física de Cantabria IFCA-CSIC, Avda. de los Castros s/n, 39005 Santander, Spain3 High Energy Accelerator Research Organization (KEK), Tsukuba, Ibaraki 305-0801, Japan4 Departamento de Física Teórica, Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid, Spain5 Instituto de Física Teórica UAM-CSIC, Universidad Autónoma de Madrid, C/ Nicolás Cabrera 13-15, Cantoblanco, 28049 Madrid, Spain6 INFN, Sezione di Torino, Via Pietro Giuria 1, 10125 Turin, Italy7 School of Mathematics, Hamilton Mathematics Institute, Trinity College Dublin, Dublin 2, Ireland8 INFN, Sezione di Tor Vergata, c/o Dipartimento di Fisica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica 1, 00133 Rome, Italy

Received: 29 November 2019 / Accepted: 30 January 2020 / Published online: 22 February 2020© The Author(s) 2020

Abstract We present a lattice QCD determination of lightquark masses with three sea-quark flavours (Nf = 2 + 1).Bare quark masses are known from PCAC relations inthe framework of CLS lattice computations with a non-perturbatively improved Wilson-Clover action and a tree-level Symanzik improved gauge action. They are fullynon-perturbatively improved, including the recently com-puted Symanzik counter-term bA − bP. The mass renor-malisation at hadronic scales and the renormalisation grouprunning over a wide range of scales are known non-perturbatively in the Schrödinger functional scheme. Inthe present paper we perform detailed extrapolations tothe physical point, obtaining (for the four-flavour theory)mu/d(2 GeV) = 3.54(12)(9) MeV and ms(2 GeV) =95.7(2.5)(2.4) MeV in the MS scheme. For the mass ratiowe have ms/mu/d = 27.0(1.0)(0.4). The RGI values in thethree-flavour theory are Mu/d = 4.70(15)(12) MeV andMs = 127.0(3.1)(3.2) MeV.

1 Introduction

The lattice regularisation of QCD provides a well-definedprocedure for the determination of the fundamental param-eters of the theory (i.e. the gauge coupling and the quarkmasses) from first principles. The aim of the present workis the determination of the three lightest quark masses (i.e.those of the up, down and strange flavours), in a frameworkin which the up and down quarks are degenerate and all heav-

a e-mail: jonna.koponen@kek.jp (corresponding author)

ier flavours (i.e. charm and above), if present in the theory,would be quenched (valence) degrees of freedom. This isknown as Nf = 2 + 1 lattice QCD. Moreover, QED effectsare ignored.

The three-flavour theory adopted in this paper is presum-ably sufficient for determining light quark masses due to thedecoupling of heavier quarks [1–4]. Indeed, lattice worldaverages of light quark massesmu/d,ms do not show a signif-icant dependence on the number of flavours at low energiesfor Nf ≥ 2 within present-day errors [5]. This also holds forthe more accurately known renormalisation group indepen-dent ratio mu/d/ms. Recently, heavy-flavour decoupling hasbeen substantiated also non-perturbatively [6].

This paper is based on large-scale Nf = 2 + 1 flavourensembles produced by the Coordinated Lattice Simula-tion (CLS) effort [7,8]. The simulations employ a tree-levelSymanzik-improved gauge action and a non-perturbativelyimproved Wilson fermion action; see references [9–12]. Thesea quark content is made of a doublet of light degeneratequarks mq,1 = mq,2, plus a heavier one mq,3. At the physicalpoint mq,1,mq,2 = mu/d ≡ 12 (mu + md) and mq,3 = ms.

The bare quark masses produced by CLS [7,8] need tobe combined with renormalisation and improvement coeffi-cients in order to obtain renormalised quantities with O(a2)discretisation effects. We use ALPHA collaboration resultsfor the quark mass renormalisation and RenormalisationGroup (RG) running [13] in the Schrödinger functional(SF) scheme. Symanzik improvement is implemented forthe removal of discretisation effects from correlation func-tions, leaving us with O(a2) uncertainties in the bulk and

123

http://crossmark.crossref.org/dialog/?doi=10.1140/epjc/s10052-020-7698-z&domain=pdfhttp://orcid.org/0000-0002-5127-4461http://orcid.org/0000-0002-9350-0383http://orcid.org/0000-0001-8373-3215http://orcid.org/0000-0001-9773-5414http://orcid.org/0000-0003-4830-9245http://orcid.org/0000-0003-3874-3646http://orcid.org/0000-0003-1654-1816http://orcid.org/0000-0002-4214-0963mailto:jonna.koponen@kek.jp

169 Page 2 of 18 Eur. Phys. J. C (2020) 80 :169

O(g40a) ones at the time boundaries. We find that correla-tion functions extrapolations to the continuum limit are com-patible with an O(a2) overall behaviour. The counter-termsrequired for the improvement of the axial current are knownfrom refs. [14–16]. The present work has combined all theseelements, obtaining estimates of the up/down and strangequark masses, as well as their ratio. These are expressed asrenormalisation scheme-independent and scale-independentquantities, known as Renormalisation Group Invariant (RGI)quark masses. Of course we also give the same results in theMS scheme at scale μ = 2 GeV.

The bare dimensionless parameters of the lattice theoryare the strong coupling g20 ≡ 6/β and the quark massesexpressed in lattice units amq,1 = amq,2 and amq,3, witha the lattice spacing. They can be varied freely in simula-tions. Having chosen a specific discretisation of the QCDaction, its parameters must be calibrated so that three “input”hadronic quantities (one for each bare mass parameter andone for the lattice spacing) attain their physical values.Other physical quantities can subsequently be predicted.Such input quantities are typically very well-known fromexperiment, but they also need to be precisely computed onthe lattice; examples are ground state hadron masses anddecay constants (mπ , fπ ,mK, fK, . . .). Since the majorityof numerical large-scale simulations do not yet include thesmall strong-isospin breaking and electromagnetic effects,the physical input quantities have to be corrected accord-ingly. Following Ref. [8], we use the values of Ref. [17]

mphysπ = 134.8(3) MeV, mphysK = 494.2(3) MeV,f physπ = 130.4(2) MeV, f physK = 156.2(7) MeV. (1.1)

The calibration of the lattice spacing, referred to as scalesetting, usually singles out a dimensionful quantity as refer-ence scale fref [MeV]. Its dimensionless counterpart a frefis computed on the lattice for fixed values of the barecoupling at the point where the physical spectrum, suchas [amπ/(a fref)]g20 ≡ mπ/ fref , is reproduced in the bareparameter space (g20, amq,i ) of the lattice theory. In thisway, the lattice spacings a(g20) = [a fref ]g20 / fref , and con-sequently all computed observables, are obtained in physicalunits. When simulations approach the point of physical massparameters while the lattice spacing is lowered, computa-tional demands rapidly increase. In the present work resultsare obtained at non-zero lattice spacings and at quark masseswhich correspond to unphysical meson and decay constantvalues. Thus our data need to be extrapolated to the contin-uum limit and extra/interpolated to the physical quark massvalues. This is achieved with a joint chiral and continuumextrapolation. The present work pays particular attentionto these extrapolations and interpolations and the ensuingsources of systematic error.

So far we did not specify the reference scale fref . InRef. [8] the three-flavor symmetric combination f physπK =23 ( f

physK + 12 f physπ ) = 147.6(5) MeV, (obtained from the phys-

ical input of Eqs. (1.1)) was used for calibration, and for thedetermination of the hadronic gradient flow scale t0 [18]. 1

An artificial (theoretical) hadronic scale with mass dimen-sion −2, t0 is precisely computable with small systematiceffects [19,20], and thus well-suited as intermediate scale onthe lattice. Its physical value determined from CLS ensem-bles [8] reads

√8t phys0 = 0.415(4)(2) fm, (1.2)

at fixed

φ4 ≡[8t0(m

2K + 12m2π )

]phys = 1.119(21), (1.3)

where the first error of√

8t phys0 is statistical and the secondsystematic.

The theoretical framework of our work is explainedin Sect. 2. The definitions of bare current quark masses,their renormalisation parameters and the O(a)-improvementcounter-terms are provided in standard ALPHA-collaborationfashion. There is also a fairly detailed exposition of howthe so-called “chiral trajectory” (a line of constant physics –LCP) is traced by Nf = 2 + 1 CLS simulations. In Sect. 3we outline the computations leading to renormalised cur-rent quark masses as functions of the pion squared mass.These are computed in the SF renormalisation scheme at ahadronic (low energy) scale. In Sect. 4 we perform the com-bined chiral and continuum limit extrapolations in order toobtain estimates of the physical up/down and strange quarkmasses. Details of the ansätze we have used are provided inAppendix A and in Appendix B. Our final results are gatheredin Sect. 5. Preliminary results have been presented in [21].

2 Theoretical framework

We review our strategy for computing light quark masseswith improved Wilson fermions. In what follows equationsare often written for a general number of flavours Nf . Inpractice Nf = 2 + 1. Flavours 1 and 2 indicate the lighterfermion fields, which are degenerate; at the physical pointtheir mass is the average up/down quark mass. Flavour 3stands for the heavier fermion, corresponding to the strangequark at the physical point.

1 The gradient flow scale t0 is defined by the implicit relation{t2〈E(t)〉}t=t0 = 0.3, for the finite Yang–Mills energy density E(t)at flow time t ; see Section 6 of Ref. [7].

123

Eur. Phys. J. C (2020) 80 :169 Page 3 of 18 169

2.1 Quark masses, renormalisation, and improvement

The starting point is the definition of bare correlation func-tions on a lattice with spacing is a and physical extensionL3 × T :

f i jP (x0, y0) ≡ −a6

L3∑x,y

〈Pi j (x0, x)P ji (y0, y)〉,

f i jA (x0, y0) ≡ −a6

L3∑x,y

〈Ai j0 (x0, x)P ji (y0, y)〉, (2.1)

where the pseudoscalar density and axial current are

Pi j (x) ≡ ψ̄ i (x)γ5ψ j (x) , (2.2)Ai j0 (x) ≡ ψ̄ i (x)γ0γ5ψ j (x). (2.3)The indices i, j = 1, 2, 3 label quark flavours, which arealways distinct (i �= j).

The bare current (or PCAC) quark mass is defined via theaxial Ward identity at zero momentum and a plateau averagebetween suitable initial and final time-slices ti < tf ,

mi j ≡ atf − ti + a

×tf∑

x0=ti

[12 (∂0 + ∂∗0) f i jA0 + cAa∂0∂∗0 f

i jP

](x0, y0)

2 f i jP (x0, y0),

(2.4)

with the source P ji positioned either at y0 = a or y0 =T − a.2 The mass-independent improvement coefficient cAis determined non-perturbatively [14]. The average of tworenormalised quark masses is then expressed in terms of thePCAC mass mi j as follows:

miR + m jR2

≡ mi jR = ZA(g20)

ZP(g20, aμ)mi j

×[1 + (bA − bP)amq,i j + (b̄A − b̄P)aTr[Mq]

]

+ O(a2), (2.5)

where Mq ≡ diag(mq,1,mq,2, · · · ,mq,Nf ) is the matrix ofthe sea quark subtracted masses, characteristic of Wilsonfermions. Given the bare mass parameter m0,i ≡ (1/κi −8)/(2a), with κi the Wilson hopping parameter, these aredefined as

mq,i = 1/(2aκi ) − 1/(2aκcr) ≡ m0,i − mcr (2.6)

2 In our simulations we average correlation functions with the sourceat y0 = a and (time-reversed) correlation functions with the source aty0 = T − a. Since bare quantities are computed on lattices with openboundary conditions in time, we do not use translation invariance forthe source position.

where mcr ∼ 1/a is an additive mass renormalisation aris-ing from the loss of chiral symmetry by the regularisationand κcr is the critical (chiral) point. The average of two sub-tracted masses is then denoted by mq,i j ≡ 12 (mq,i +mq, j ) inEq. (2.5).

The axial current normalisation ZA(g20) is scale-indepen-dent, whereas the current quark mass renormalisation param-eter 1/ZP(g20, aμ) depends on the renormalisation scale μ.The renormalisation condition imposed on the pseudoscalardensity operator P ji defines the renormalisation scheme forthe quark masses. The schemes used in the present work (SFand MS) are mass-independent. Pertinent details will be dis-cussed in latter sections.

The improvement coefficients bA − bP and b̄A − b̄P ofEq. (2.5) cancel O(a) mass-dependent cutoff effects; theyare functions of the bare gauge coupling g20. The correspond-ing counter-terms of Eq. (2.5) contain the subtracted massesamq,i j and Tr[aMq], which require knowledge on the criticalmass mcr. This can be avoided by substituting these masseswith current quark masses and their sum. Their relationshipis [22],

mi j = Z[mq,i j + (rm − 1) Tr[Mq]

Nf

]+ O(a), (2.7)

where Z(g20) ≡ ZP/(ZSZA) and rm(g20) are finite normalisa-tions. ZS is the renormalisation parameter of the non-singletscalar density Si j ≡ ψ̄ iψ j and rm/ZS is the renormalisa-tion parameter of the singlet scalar density, which indirectlydefines rm; cf. Ref. [22]. In the above we neglect O(a) terms,as they only contribute to O(a2) in the b-counter-terms ofEq. (2.5). Substituting amq,i j → ami j in the latter expres-sion, we obtain

mi jR(μhad) = ZA(g20)

ZP(g20, aμ)mi j

[1 + (b̃A − b̃P)ami j

+{(b̃A − b̃P)1 − rm

rm+ (b̄A − b̄P) Nf

Zrm

}aMsumNf

]

+ O(a2), (2.8)

where we define

b̃A − b̃P ≡ bA − bPZ

,

Msum ≡ m12 + m23 + · · · + m(Nf−1)Nf + mNf 1= ZrmTr[Mq] + O(a). (2.9)

To leading order in perturbation theory the difference bA−bPis O(g20) and equals b̃A − b̃P. However, non-perturbativeestimates are likely to differ significantly, especially in therange of couplings g0 considered here (1.56 � g20 � 1.76).We will employ non-perturbative estimates of bA − bPand Z ; cf. Ref. [23]. The term multiplying Msum contains(1 − rm)/rm and (b̄A − b̄P). In perturbation theory rm =

123

169 Page 4 of 18 Eur. Phys. J. C (2020) 80 :169

1 + 0.001158CF Nf g40 [24,25], (1 − rm)/rm ∼ O(g40) and(b̄A − b̄P ) ∼ O(g40) [22]. A first non-perturbative study ofthe coefficients b̄A and b̄P produced noisy results with 100%errors [26]. Given the lack of robust non-perturbative resultsand the fact that the term in curly brackets is O(g40) in per-turbation theory, it will be dropped in what follows.

Once the quark mass averages m12R and m13R are com-puted say, in the SF scheme at a scale μhad, the three renor-malised quark masses can be determined. Since we are work-ing in the isospin limit (mq,1 = mq,2), the lighter quarkmass is given by m12R. Then one can isolate m13R from theratio m13R/m12R in which, as seen from Eq. (2.8), the Msumcounter-term cancels out.

The ALPHA Collaboration is devoting considerableresources to the determination of the non-perturbative evo-lution of the renormalised QCD parameters (strong couplingand quark masses) between a hadronic and a perturbativeenergy scale (μhad ≤ μ ≤ μpt). Quark masses are renor-malised at μhad ∼ O(ΛQCD) and evolved to μpt ∼ O(MW)[13,27–36] in the SF scheme [37,38]. Both renormalisationand RG-running are done non-perturbatively. At μpt pertur-bation theory is believed to be reliably controlled and we maysafely switch to the conventionally preferred, albeit inher-ently perturbative MS scheme.

We will be quoting results also for the scheme- and scale-independent renormalisation group invariant (RGI) quarkmasses M12 and M13 (corresponding to the current massesm12 and m13) as well as the physical RGI quark masses Mu/dand Ms derived from them. They are conventionally definedin massless schemes [39] by

Mi ≡miR(μ)[2b0g

2R(μ)

]− d02b0

× exp{

−∫ gR(μ)

0dg

[τ(g)

β(g)− d0

b0g

]}, (2.10)

for each quark flavour i . In our opinion, Mi is better suitedfor comparisons either to experimental results or other the-oretical determinations. Equation (2.10) is formally exactand independent of perturbation theory as long as the renor-malised parameters (gR,miR) and the continuum renormali-sation group functions (i.e. the Callan–Symanzik β-functionand the mass anomalous dimension τ ) are known non-perturbatively with satisfacory accuracy [13,33–36]. Theircomputation in the SF scheme with Nf = 3 massless quarkshas been carried out in Ref. [13].

Our determination of the renormalised quark masses isbased on the bare current mass averages mi jR; cf. Eqs. (2.5)and (2.8). The analogue of these expressions for the RGImass averages is given by

Mi j ≡ 12(Mi + Mj ) = M

mR(μhad)mi jR(μhad). (2.11)

Note that the ratio M/mR(μhad) is flavour-independent; cf.Eq. (2.10). In Ref. [13] it has been computed in the SF schemefor the Nf = 3 massless flavours at μhad = 233(8) MeV.

2.2 The chiral trajectory and scale setting

Our aim is to stay on a line of constant Physics within system-atic uncertainties of O(a2), as we vary the bare parametersof the theory (i.e. the gauge coupling g0 and the Nf = 2 + 1quark masses). In particular, if the improved bare gauge cou-pling

g̃20 ≡ g20(

1 + 1Nf

bg(g20)aTr[Mq]

)(2.12)

is kept fixed in the simulations, so does the lattice spacing,with any fluctuations being attributed to O(a2)-effects. Theproblem is that bg(g20) is only known to one-loop order in per-turbation theory [40,41]; bPTg = 0.012Nfg20. Thus, followingrefs. [42,43], we vary the quark masses at fixed g20, ensuringthat the trace of the quark mass matrix remains constant:

Tr[Mq] = 2mq,1 + mq,3 = const. (2.13)In this way the improved bare gauge coupling g̃20 is keptconstant at fixed β for any bg .3

This requirement leads to an unusual but unambiguousapproach to the physical point, shown in the (M12, M13)-plane in the left panel of Fig. 1. Initially, one starts at thesymmetric point (amq,1 = amq,2 = amq,3 = amsymq ) forsome fixed coupling β = 6/g20, and tunes the mass parameterof the simulation in such a way that Tr[Mq] = Tr[Mq]phys to agood approximation.4 This is achieved by varyingamsymq until(m2K + 12m2π )/ fref takes its physical value. Since it is propor-tional to Tr[Mq] at leading order in chiral perturbation theory(χPT), it suffices as tuning observable. In subsequent simu-lations, one successively lifts the mass-degeneracy towardsthe physical point by decreasing the light quark masses whilemaintaining the constant-trace condition. By doing so thephysical strange quark mass is approached from below as inFig. 1 (left panel). We call this procedure “the determinationof the chiral trajectory”.

Note that the improved renormalised quark mass matrixMR is given by [22]

Tr[MR] = Zmrm×

[(1 + ad̄mTr[Mq])Tr[Mq] + admTr[M2q ]

]+ O(a2).

(2.14)

3 In Ref. [44] (cf. Sect. 5.3.2) it was estimated that when, in someensembles, Tr[Mq] is not constant, the resulting effect on the shift ofthe lattice spacing is about 6 per mille. This estimate was based on the1-loop value of bg , bA − bP and Z .4 In practise one only tunes the bare quark mass am0, since amcr isunknown a priori, but constant at fixed β.

123

Eur. Phys. J. C (2020) 80 :169 Page 5 of 18 169

0

Mu/d

Msym

Mphys13

32Msym

0 Mu/d Msym

M13

M12

Mij

M12

0

Mu/d

Msym

Mphys13

32Msym

0 Mu/d Msym

φ∗2 φsym2

mij

√ 8t 0

φ2

β = 3.40 : m12√8t0[φ4]

β = 3.46 : m12√8t0[φ4]

β = 3.55 : m12√8t0[φ4]

β = 3.70 : m12√8t0[φ4]

β = 3.40 : m13√8t0[φ4]

β = 3.46 : m13√8t0[φ4]

β = 3.55 : m13√8t0[φ4]

β = 3.70 : m13√8t0[φ4]

β = 3.40 : m12√8t0[φ̃4]

β = 3.46 : m12√8t0[φ̃4]

β = 3.55 : m12√8t0[φ̃4]

β = 3.70 : m12√8t0[φ̃4]

β = 3.40 : m13√8t0[φ̃4]

β = 3.46 : m13√8t0[φ̃4]

β = 3.55 : m13√8t0[φ̃4]

β = 3.70 : m13√8t0[φ̃4]

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.0 0.2 0.4 0.6 0.8 1.0

Fig. 1 The left panel shows an idealisation of the chiral trajectory forrenormalised RGI current quark masses M12 and M13 in the continuum.The symmetric point (gray box) is defined by the trace of the renor-malised RGI quark mass matrix, M12 = M13 = Msym = 13 Tr[M], andthe physical point is indicated by red circles, where Mphys12 = Mu/d and

Mphys13 = 12 (Mu/d +Ms). The right panel shows our data φi j ≡√

8t0mi jversus φ2 ≡ 8t0m2π ∝ m12. Coloured (gray) points correspond to mass-shifted (-unshifted) points in parameter space, cf. the discussion in thetext

Since the dm-counter-term is proportional to squared baremasses, a constant Tr[Mq] does not correspond to a constantTr[MR]; the latter requirement is violated by O(a) effects.This is an undesirable feature, as it implies that the chiraltrajectory is not a line of constant-physics. In practice theseviolations have been monitored in Ref. [8] (Fig. 4, lowestlhs panel), where Tr[MR] has been computed, at constantTr[Mq ], from the current quark masses with 1-loop pertur-bative Symanzik b-coefficients. The violations appear to bebigger than what one would expect from O(a) effects.

These considerations have led the authors of Ref. [8] toredefine the chiral trajectory in terms of φ4 = const., whereφ4 ≡ 8 t0

(m2K +

1

2m2π

), (2.15)

and t0 is the gluonic quantity of the Wilson flow [18]; ithas mass dimension −2. Here mπ and mK are the lightestand strange pseudoscalar mesons respectively; at the phys-ical point these are the pion mphysπ and kaon m

physK . Keeping

φ4 constant is a Symanzik-improved constant physics con-dition. But φ4 is proportional to the sum of the three quarkmasses only in leading-order (LO) chiral perturbation theory(χPT). Thus, the improved bare coupling g̃20 now suffersfrom O(amqTr[Mq ]) discretisation effects due to higher-order χPT contributions. In practice, these turn out to besmall, as can be seen from Ref. [8] (Fig. 4, lowest rhs panel),where Tr[MR] has been computed, at constant φ4. The vio-lations appear to be at most 1% and thus the variation of theO(a) bg-term in g̃20 can be ignored.

Obviously, one must also ensure, through careful tuning,that the chosen φ4 = const. trajectory passes through thepoint corresponding to physical up/down and strange renor-malised masses (i.e. quark masses that correspond to thephysical pseudoscalar mesons mphysπ and m

physK ). This is done

by driving φ4 to its physical value φphys4 = 8t0[(mphysK )2 +

(mphysπ )2/2] through mass shifts [8]. The aim is to express thecomputed quantities of interest (in our case the quark masses)as functions of

φ2 ≡ 8 t0 m2π , (2.16)with φ4 held fixed at φ

phys4 , and eventually extrapolate them

to φphys2 = 8t0(mphysπ )2.The determination of the redefined chiral trajectory is not

straightforward. One needs to know the value ofφphys4 . The lat-ter is obtained from t0 and the pseudoscalar masses (correctedfor isospin-breaking effects) quoted in Eq. (1.1). But sincethe value of t0 is only approximately known, one starts withan initial guess t̃0, which provides an initial guess φ̃4. At eachβ, the symmetric point with degenerate masses (κ1 = κ3) istuned so that the computed t0/a2, amπ and amK combine asin Eq. (2.15) to give a value close to φ̃4. The other ensemblesat the same β have been obtained by decreasing the degener-ate (lightest) quark mass mq,1 = mq,2, while increasing theheavier mass mq,3 so as to keep Tr[Mq] constant. Thus theydo not correspond exactly to the same φ̃4. Small correctionsof the subtracted bare quark masses (or hopping parameters)are introduced, using a Taylor expansion discussed in Sect.IV of Ref. [8], in order to shift φ4 to the reference value φ̃4and correct analogously the measured PCAC quark masses

123

169 Page 6 of 18 Eur. Phys. J. C (2020) 80 :169

and other quantities of interest such as the decay constants.The procedure is repeated for each β and the same value φ̃4at the starting symmetric point.

All shifted quantities are now known at φ̃4 as functions ofφ2. Defining the combination of decay constants

fπK ≡ 23

(fK + fπ

2

), (2.17)

the dimensionless√

8t̃0 fπK is computed for all φ2 andextrapolated to φ̃2 = 8t̃0(mphysπ )2. The extrapolated

√t̃0 fπK ,

combined with the experimentally known f physπK , gives a bet-ter estimate of t̃0, and thus of φ̃4. As described in Sect. Vof Ref. [8], this procedure can be recursively repeated andeventually the physical value of t0 is fixed through f

physπK ; the

value in Eq. (1.2) from Ref. [8] leads to

φphys4 = 1.119(21), (2.18)

φphys2 = 0.0804(8). (2.19)

The main message is that once PCAC quark masses areshifted onto the chiral trajectory defined by the constant φphys4 ,they only depend on a single variable, namely φ2.

In analogy to the definitions (2.15) and (2.16), we alsodefine rescaled dimensionless bare current quark masses andtheir renormalised counterparts at scale μhad

φi j ≡√

8 t0 mi j , φi jR(μ) ≡√

8 t0 mi jR(μ). (2.20)

The redefined chiral trajectory is shown in Fig. 1 (rightpanel), where the light-light and heavy-light dimensionlessmass averages (φ12R and φ13R respectively) are plotted asfunctions of φ2. Extrapolating in φ2 to φ

phys2 amounts to the

simultaneous approach of the light and heavy quark masses tothe corresponding physical up/down and strange values. Allother physical quantities are then also at the physical point.Section 4 is dedicated to these extrapolations.

3 Quark mass computations

We base our determination of quark masses on the CLSensembles for Nf = 2 + 1 QCD, listed in Table 1. The baregauge action is the Lüscher–Weisz one, with tree-level coef-ficients [11]. The bare quark action is the Wilson, Symanzik-improved [10] one. The Clover term coefficient csw has beentuned non-perturbatively in Ref. [12]. Boundary conditionsare periodic in space and open in time, as detailed in Ref. [45].

For details on the generation of these ensembles see Ref.[7]. As seen in Table 1, results have been obtained at fourlattice spacings in the range 0.05 � a/fm � 0.086. For eachlattice coupling β = 6/g20, gauge field ensembles have beengenerated for a few5 values of the Wilson hopping param-

5 We note in passing that for the ensemble with β = 3.46 we only haveresults for degenerate quark masses.

eters κ1 = κ2 and κ3. The light pseudoscalar meson (pion)varies between 200 and 420 MeV. The heaviest value corre-sponds to the symmetric point where the three quark massesand the pseudoscalar mesons are degenerate. The strangemeson (kaon) varies between 420 and 470 MeV. Given thatour lightest pseudoscalars are relatively heavy (200 MeV),the chiral limit ought to be taken with care.

The bare correlation functions f i jP , fi jA of Eqs. (2.1) are

estimated with stochastic sources located on time slice y0,with either y0 = a or y0 = T − a. From them the cur-rent quark masses m12,m13 are computed as in Eq. (2.4),with the O(a)-improvement coefficient cA determined non-perturbatively in Ref. [14]. The exact procedure to select theplateaux range in the presence of open boundary conditionshas been explained in Refs. [7,44,46].

Having obtained the bare current quark massesm12,m13 atfour values of the coupling g20, we construct the renormaliseddimensionless quantities m12R(μhad) and m13R(μhad); cf.Eq. (2.8). For this we need the ratio ZA(g20)/ZP(g

20, μhad)

and the Symanzik b-counter-terms. Results for the axialcurrent normalisation ZA(g20) are available in Ref. [47],from a separate computation based on the chirally rotatedSchrödinger Functional setup of Refs. [48–50]. The com-putation of ZP(g20, μhad) in the SF scheme, for μhad =233(8) MeV, was carried out in Ref. [13] for a theory withNf = 3 massless quarks and the lattice action of the presentwork. The ZP results, shown in Eqs. (5.2) and (5.3) of Ref.[13], are in a range of inverse gauge couplings which coversthe β ∈ [3.40, 3.85] interval of the large volume simulationsof Ref. [8], from which our bare dimensionless PCAC massesare extracted.

Besides the ratio ZA/ZP, we also need the improvementcoefficient (b̃A−b̃P), which multiplies the O(a) counter-termproportional to ami j in Eq. (2.8). To leading order in pertur-bation theory b̃A − b̃P = −0.0012g20. Non-perturbative esti-mates based on a coordinate-space renormalisation schemehave been provided for Nf = 2 + 1 lattice QCD in Ref. [26].More accurate non-perturbative results have been subse-quently obtained by the ALPHA Collaboration, using suit-able combinations of valence current quark masses, mea-sured on ensembles with Nf = 3 nearly-chiral sea quarkmasses in small physical volumes [16,23]. Also these simu-lations have been carried out in an inverse coupling range thatspans the interval β ∈ [3.40, 3.85] of the large volume CLSresults of Ref. [8]. They are expressed in the form of ratiosRAP and RZ, from which (bA −bP) and Z are estimated; thus(b̃A − b̃P) = RAP/RZ. In Ref. [23], results are quoted fortwo values of constant Physics, dubbed LCP-0 and LPC-1. InLCP-0, RAP and RZ are obtained with all masses in the chirallimit. In LCP-1, one valence flavour is in the chiral limit (soit is equal to the sea quark mass), while a second one is heldfixed to a non-zero value. The physical volumes are alwayskept fixed. In Ref. [23], Eqs. (5.1), (5.2) and (5.3) refer to

123

Eur. Phys. J. C (2020) 80 :169 Page 7 of 18 169

Table 1 Details of CLSconfiguration ensembles,generated as described inRef. [7]. In the last column,ensembles are labelled by aletter, denoting the latticegeometry, a first digit for thecoupling and a further two digitsfor the quark mass combination

βa

fmL/a T/L κ1 κ3

mπMeV

mKMeV

mπ L Label

3.40 0.086 32 3 0.13675962 κ1 420 420 5.8 H101

32 3 0.136865 0.136549339 350 440 4.9 H102

32 3 0.136970 0.136340790 280 460 3.9 H105

48 2 0.137030 0.136222041 220 470 4.7 C101

3.46 0.076 32 3 0.13688848 κ1 420 420 5.2 H400

3.55 0.064 32 4 0.137000 κ1 420 420 4.3 H200

48 8/3 0.137000 κ1 420 420 6.5 N202

48 8/3 0.137080 0.136840284 340 440 5.4 N203

48 8/3 0.137140 0.136720860 280 460 4.4 N200

64 2 0.137200 0.136601748 200 480 4.2 D200

3.70 0.050 48 8/3 0.137000 κ1 420 420 5.1 N300

64 3 0.137123 0.1367546608 260 470 4.1 J303

LCP-0 results, while those in Eqs. (5.1), (5.4) and (5.5) referto LCP-1; differences are due to O(a) discretisation effects.

We have opted to use the LCP-0 values of b̃A − b̃P in thepresent work. The covariance matrices of the fit parametersof RAP as well as those of RZ are provided in Ref. [23]. Weassume that the covariance matrix between fit parameters ofRAP and RZ is nil. This is justified a posteriori, by repeat-ing the analysis with LCP-1 values, as a means to estimatethe magnitude of systematic errors arising from our choice.Moreover, we have also compared our LCP-0 results to thoseobtained from different fit functions, used in the preliminaryanalysis of Ref. [16], as well as from the perturbative esti-mate b̃A − b̃P. We find that the contribution arising from suchvariations is below ∼ 1% of the total error on renormalisedquark masses at the physical point.

As discussed in Sect. 2, the complicated Symanzikcounter-term in curly brackets, multiplyingaMsum in Eq. (2.8),is O(g40a) in perturbation theory. As there are no robust non-perturbative estimates of its magnitude at present, we willdrop this term, assuming that the O(g40a) effects it wouldremove are subdominant compared to O(a2) uncertainties.

As already explained in Sect. 2.2, our analysis is based onthe rescaled dimensionless quantities defined in Eqs. (2.15),(2.16), and (2.20). At each β value and for each gauge fieldconfiguration, we have results for t0/a2, am12 and am13 fromRefs. [7,8], from which φ12 and φ13 are obtained. The erroranalysis is carried out using the Gamma method approach[51–54] and automatic differentiation for error propagation,using the library described in Ref. [55]. This takes intoaccount all the existing errors and correlations in the dataand ancillary quantities (renormalisation constants, improve-ment coefficients, etc.), and estimates autocorrelation func-tions (including exponential tails) to rescale the uncertaintiescorrespondingly. Following [8], the estimate of the exponen-tial autocorrelation times τexp used in the analysis is the onequoted in [7], viz.,

τexp = 14(3) t0a2

. (3.1)

We have checked that without attaching exponential tails sta-tistical errors are 40–70% smaller in our final results. Thefull analysis has been crosschecked by an independent codebased on (appropriately) binned jackknife error estimation.Note that one of the strengths of data analysis based on theGamma-method is that each Monte Carlo ensemble is treatedindependently, and the final statistical uncertainty is deter-mined as a sum in quadratures of the statistical fluctuationsfor each ensemble. This allows to trace back which frac-tion of the statistical variance comes from each ensemble orancillary quantities, such as renormalisation constants (seeReferences 5–7 in [55]). This feature will be exploited in theerror budgets provided below.

The starting values for φ12R and φ13R on which the analy-sis is based are shown in Table 2, where renormalised quarkmasses are in the SF scheme at a scale μhad = 233(8) MeV.By suitably fitting these quantities as functions of φ2, andextrapolating to φphys2 , we obtain the results for physicalup/down and strange quarks at scale μhad, as detailed inSect. 4. Only then do we convert them to the RGI masses, bymultiplying them with the RG-running factor [13]

M

mR(μhad)= 0.9148(88), (3.2)

with the error added in quadrature; cf. Eq. (2.11).Before presenting our chiral fits in Sect. 4, we conclude

this section with a comment on finite-volume effects. Currentquark masses are not expected to be affected by finite-volumecorrections, since their values are fixed by Ward identities. Onthe other hand, meson masses, decay constants, and the ratiot0/a2 are expected to suffer from such effects. This can bedirectly checked in the ensembles H200 and N202, obtainedat β = 3.55 with degenerate masses and corresponding tovolumes of about 2 fm and 3 fm respectively. A glance at the

123

169 Page 8 of 18 Eur. Phys. J. C (2020) 80 :169

Table 2 Rescaleddimensionless current quarkmasses φ12 and φ13,renormalised in the SF schemeat μhad, for each CLS ensembleused in our analysis. Note thatfor simulation points H102,H105, C101 more than oneindependent ensembles exist,which have been run withdifferent algorithmic setups; wekeep those separate before fits.All points have been shifted tothe target chiral trajectory asdescribed in the text, and thequoted errors contain bothstatistical uncertainties and thecontribution fromrenormalisation and the massshift

β Ensemble t0/a2 φ2 φ12 φ13 φ12/φ13

3.40 H101 2.857(13) 0.747(18) 0.0917(26) 0.0917(26) 1

H102r001 2.877(19) 0.547(20) 0.0673(27) 0.1047(28) 0.643(10)

H102r002 2.883(18) 0.549(19) 0.0667(28) 0.1039(29) 0.642(10)

H105 2.886(11) 0.346(20) 0.0416(25) 0.1167(27) 0.357(15)

H105r005 2.896(38) 0.355(20) 0.0420(33) 0.1160(34) 0.362(19)

C101 2.900(19) 0.238(24) 0.0279(31) 0.1236(27) 0.226(21)

C101r014 2.899(14) 0.233(20) 0.0273(26) 0.1223(30) 0.222(17)

3.46 H400 3.656(20) 0.747(18) 0.0923(28) 0.0923(28) 1

3.55 N202 5.161(23) 0.747(18) 0.0978(26) 0.0978(26) 1

N203 5.138(16) 0.526(19) 0.0684(27) 0.1128(26) 0.606(10)

N200 5.155(16) 0.356(18) 0.0455(25) 0.1232(25) 0.369(13)

D200 5.171(16) 0.189(20) 0.0237(27) 0.1232(29) 0.176(17)

3.70 N300r002 8.592(41) 0.747(18) 0.0988(29) 0.0988(30) 1

J303 8.628(40) 0.278(20) 0.0364(31) 0.1326(39) 0.274(19)

relevant entries of Table II of Ref. [8] shows that quark massesdo not change as the volume is varied, while meson massesand decay constants vary by about 2.5%, which correspondsto differences of about 2−3.5σ .6 Standard SU(3) χPT NLOformulae are available for masses and decay constants [56];t0/a2 does not suffer from finite-volume effects up to NNLOcorrections [20]. In particular, the χPT-predicted effects formeson masses are below the percent level, since the latticespatial size in units of the inverse lightest pseudoscalar mesonmass is in the range [3.9, 5.8]. On the other hand, by directlycomparing the values in Table 2 obtained at the same lat-tice spacing and sea quark masses but different volumes (cf.Ref. [44]), it is seen that the finite-volume effects on t0/a2

and m2π are comparable and come with opposite signs. Asa result, they largely cancel in φ2, the variable in whichchiral fits are performed. Decay constants, which generallysuffer from larger finite-volume effects than meson masses,enter our computation indirectly only – firstly through NLOterms in chiral fits, where the finite-volume correction is sub-leading, and secondly through the physical value of

√8t0

determined in [8], where these corrections have already beentaken into account. We therefore expect that the quantitiesmost affected by finite-volume effects are the rescaled cur-rent quark masses φ12, φ13, due to the presence of

√8t0/a in

their definition. As mentioned above, these are much smallerthan our statistical uncertainty, cf. Table 2. In the rest ofour analysis we will therefore neglect this source of uncer-tainty.

6 The ensemble H200 is only used in this context in the present work.Since at β = 3.55 we have results at two larger volumes (N202/203/200and D200), we do not use H200 results in our analysis.

4 Extrapolations to physical quark masses

Having obtained the dimensionless renormalised currentmass combinations φ12R and φ13R at each β as functionsof φ2, we now proceed with the determination of the physi-cal values φud and φs. This is done in fairly standard fashionthrough fits and extrapolations. To begin with, we note thatthe two lighter degenerate quark masses are simply given byφ12, whereas the heavier strange one is obtained from thedifference7

φh = 2 φ13 − φ12. (4.1)It is then possible to perform simultaneous fits of φ12 andφh as functions of φ2 and the lattice spacing, subsequentlyextrapolating the results to φphys2 of Eq. (2.19) and the contin-uum limit, so as to obtain φud and φs. Variants of this methodconsist in simultaneous fits and extrapolations of either φ13or φ12 on one hand and their ratio φ12/φ13 on the other. Theseturn out to be advantageous, as does a certain combination ofratios involving φ12, φ13, φ2, and φ4, for reasons discussedbelow. We recall in passing that in the ratio φ12/φ13 all renor-malisation factors cancel.

We use fits based in chiral perturbation theory (χPT fits)which are expected to model the data well close to the chirallimit φ2 = 0. Recall that we have performed Nf = 2 + 1simulations on a chiral trajectory; starting from a symmet-ric point where all quark masses are degenerate, we increasethe mass of the heavy quark while decreasing that of the lightone, until the physical point is reached. Since both masses arevarying, it is natural to use SU(3)L ⊗SU(3)R chiral perturba-

7 Henceforth all quark masses will be renormalised. In order to simplifythe notation, we shall drop the subscript R from φ12R, φ13R in thissection, in Appendix A and in Appendix B.

123

Eur. Phys. J. C (2020) 80 :169 Page 9 of 18 169

tion theory, which bears explicit dependence on both masses.This works when all three quark masses in the simulationsare light enough for say, NLO χPT with three flavours to pro-vide reliable fits. In Ref. [57] it is stated that this is the casefor their data, obtained with domain wall fermions, as longas the average quark mass satisfies amavg < 0.01. As seen inTable 2 of Ref. [8], our PCAC dimensionless quark massesam12 and am13 also satisfy this empirical constraint. The realtest comes about a posteriori, when the SU(3)L ⊗ SU(3)RNLO ansätze are seen to fit our results well.

In Appendix A and Appendix B, ansätze for NLO χPT anddiscretisation effects are adapted to our specific parametri-sation in terms of φ2 and φ4. For the current quark massesthese are

φ12 = φ2[p1 + p2φ2 + p3K

(L2 − 1

3Lη

)]

+ a2

8t0[C0 + C1φ2] , (4.2)

φ13 = φK[p1 + p2φK + 2

3p3KLη

]

+ a2

8t0

[C̃0 + C̃1φ2

], (4.3)

where φK = (2φ4 − φ2)/2. The constants p1, p2, p3 andK are related to standard χPT parameters in Eqs. (A.8)-(A.11), whereas the chiral logarithms L2 and Lη are definedin Eq. (A.12). For justification of the ansatz used for the dis-cretisation effects, see comments after Eqs. (B.8) and (B.9).We stress again that φ12 and φ13 are functions of φ2 only, φ4being held constant. They have common fit parameters p1,p2 and p3.

Using the above expressions and consistently neglectinghigher orders in the continuum χPT terms, we obtain theratio of PCAC masses (cf. Eqs. (A.13) and (B.11))

φ12

φ13= 2φ2

2φ4 − φ2[

1 + p2p1

(3

2φ2 − φ4

)− K̃ (L2 − Lη

)]

+ a2

8t0(2φ4 − 3φ2)

[D0 + D1φ2

]. (4.4)

As discussed in Appendix B, the form of the cutoff effectsrespects the constraint φ12/φ13 = 1 at the symmetric pointmq,1 = mq,3, which is exact at all lattice spacings by con-struction.

For the combination defined in Eq. (A.14), we have

4φ132φ4 − φ2 +

φ12

φ2= 3p1 + 2p2φ4 + p3K

(L2 + Lη)

+ a2

8t0

[G0 + G1φ2

]. (4.5)

An alternative to NLO χPT fits is the use of power series,based simply on Taylor expansions around the symmetricpoint mq,1 = mq,2 = mq,3, for which φsym2 = 2φphys4 /3:φ12 = s0 + s1(φ2 − φsym2 ) + s2(φ2 − φsym2 )2

+a2

t0

[S0 + S1(φ2 − φsym2 )

], (4.6)

φ13 = s0 + s̃1(φ2 − φsym2 ) + s̃2(φ2 − φsym2 )2

+a2

t0

[S0 + S̃1(φ2 − φsym2 )

]. (4.7)

Note that imposing the constraint φ12 = φ13 at the symmetricpoint implies that s0 and S0 are common fit parameters. Theseexpansions are expected to give reliable results in the higherend of the φ2 range, underperforming close to the chiral limit.They are thus complementary to the chiral fits, which arebetter suited for the small-mass regime. In this sense the twoapproaches may provide a handle to estimate the systematicuncertainties due to these fits and extrapolations.

We explore various fit variants, in order to unravel thepresence of potentially significant systematic effects. Theyare encoded as follows:

• Fitted quantities and ansätze:[chi12] Fit of φ12 data only, using the χPT ansatz.[chi13] Fit of φ13 data only, using the χPT ansatz.[tay12] Fit of φ12 data only, using the Taylor expan-sion ansatz.[tay13] Fit of φ13 data only, using the Taylor expan-sion ansatz.[chipc] Combined fit to φ12 and φ13, using χPT.[chirc] Combined fit to φ13 and φ12/φ13, using χPT.[chirr]Combined fit to the ratio φ12/φ13 and the com-bination 2φ13/φK + φ12/φ2 using χPT.[tchir] Combined fit to φ13 and the ratio φ12/φ13,using the Taylor expansion for φ13 and χPT for φ12/φ13.

• Discretisation effects:[a1] Fits with terms ∝ a2/t0 only.[a2] Fits with terms ∝ a2/t0 and ∝ φ2a2/t0.

• Cuts on pseudoscalar meson masses:[420] Fit all available data, including the symmetricpoint; i.e. data satisfies mπ � 420 MeV.[360] Fit excluding the symmetric point; i.e. data sat-isfies mπ � 360 MeV.[300] Fit only points for which mπ ≤ 300 MeV.

Any given fit will thus be labelled as[xxxxx][yy][zzz],using the above tags.

The results obtained with the various fit methods at thephysical point (cf. Eq. (2.19)) are expressed in physical units

by dividing them out by√

8t phys0 . Multiplication by the fac-tor of Eq. (3.2) subsequently gives the RGI mass estimates

123

169 Page 10 of 18 Eur. Phys. J. C (2020) 80 :169

Fig. 2 Results for the RGI light(Mu/d ) and averaged(Mphys13 = (Mu/d + Ms)/2) quarkmasses from independent fits toeither M12 or M13. Results areconverted to MeV by dividing

out with√

8t phys0 . Dotted linesindicate the central value of thelatest FLAG average [5] forreference

shown in Figs. 2 and 3. We comment on the various fitansätze:Independent fits of φ12 and φ13: comparing light quarkmasses Mu/d (upper panel of Fig. 2) from [chi12][a1]and [chi12][a2] we find that they are sensitive to thepresence of a discretisation term ∝ a2/t0, albeit within∼ 1 − 2σ . This difference is attenuated when the morestringent mass cutoff [chi12][300] is enforced, mainlybecause the error increases as less points are fitted. The samequalitative conclusions are true for the Taylor expansion fits[tay12] of the light quark mass. On the other hand, thelower panel of Fig. 2 shows that the average quark massMphys13 is not sensitive to the details of the fit ansätze. Thisis not surprising, given that our simulations have been per-formed in a region of rather heavy pions 220 MeV ≤ mπ ≤420 MeV, with data covering the physical point Mphys13 , whileMu/d requires long extrapolations. The conclusion is thatindependent fits are reliable for φ13 but less so for φ12, andso we discard their results.Combined fits to φ12 and φ13: Fig. 3 shows that the fits[chipc][a1] and [chipc][a2] give results which aresensitive to the ansatz employed for the cutoff effects. This ismore pronounced for Mu/d and the ratio Ms/Mu/d, but per-sists also for Ms. Moreover, fits [chipc][a1][420] and[chipc][a1][360] display visible differences whencompared to fits of the [chipc][a2] variety; the latteragree with results obtained from different fit ansätze. For

these reason we have also discarded results from this analy-sis.Combined fits to φ13 and φ-ratios: As previously ex-plained, we have explored three ansätze, namely [chirc],[chirr], and [tchir]. In all cases Fig. 3 shows thatthere is no significant dependence of the results from thedetails of these fits, except for a very slight fluctuation of the[tchir][a1][420] results for Ms. Preferring to err onthe side of caution, we also discard [tchir] fits.

A few general points concerning the fit analysis deserveto be highlighted:

• In all our fits the χ2/dof is well below 1. This ispartly because our data are correlated – both from thefact that there are common renormalisation factors andimprovement coefficients, and because we are includingthe contribution to the χ2 from the fluctuations of themeson masses (horizontal errors). Therefore, while thegoodness-of-fit is in general satisfactory, we will refrainfrom quoting the corresponding p-values, since they arenot really meaningful.

• Unsurprisingly, the inclusion of a second discretisationterm ∝ φ2(a2/t0) in the fits contributes to an increaseof the error. This term is often compatible with zero,and almost always so within ∼ 2σ , suggesting that fits[a1] are safe. As stated previously, exceptions are fits[chi12] and [chipc], where inclusion of this termhas a strong effect.

123

Eur. Phys. J. C (2020) 80 :169 Page 11 of 18 169

Fig. 3 Results for the RGI light(Mu/d) and strange (Ms) quarkmasses, and their ratio, from thesimultaneous fits [chipc],[chirc], [chirr], and[tchir]. Results areconverted to MeV by dividing

out with√

8t phys0 . Dotted linesindicate the central value of thelatest FLAG average [5] forreference

• Within large uncertainties, the coefficients of the leadingcutoff effects (i.e. those ∝ a2/t0) depend on the fittedobservable, and are larger for φ13 than for φ12.

• The power-series fits [tay12] and [tay13] behaveremarkably well. Results from [tay12] vanish withinerrors in the chiral limit, except for fits going up to thesymmetric point, which are sometimes incompatible withnaught by 2–3 σ . This is evidence that our data are notprecise enough to capture the impact of chiral logs. Fits[tay13] to φlh are very stable, and impressively bet-ter than those obtained with the χPT ansatz. Indeed, ifone considers fits [texp1] and [texp2], which aresafest from the point of view of error estimation, all thefits considered provide compatible results for M13 withinone sigma. Notice, furthermore, that the constant termsof [tay12] and [tay13] are generally in good agree-

ment, signalling the consistency of the approach. It is alsointeresting to note that the coefficient of the quadraticterm is very small and always compatible with zero within1σ (save for two cases where it vanishes within 2σ ).

• Fits [chirc] and [chirr] appear to be the stablest.• NLO χPT appears to be suffering around and above

400 MeV.

5 Final results and discussion

Following the analysis of Sect. 4, we quote as final resultsthose obtained from the following procedure:

• The central values are those of a combined fit to theratio φ12/φ13 and the quantity 2φ13/φK +φ12/φ2, using

123

169 Page 12 of 18 Eur. Phys. J. C (2020) 80 :169

Fig. 4 Illustration of the chiral+continuum fit from which our central values are obtained. The grey band is the continuum limit of our fit, and thefull black point corresponds to our extrapolation to the physical point

NLO χPT, with pseudoscalar meson masses less than360 MeV and a discretisation term proportional to a2/t0(i.e., fit [chirr][a1][360]). The error from this fitwill appear as the first uncertainty in the results below.The fit is illustrated in Fig. 4.

• We estimate systematic errors from the spread of cen-tral values of all other [chirr] and [chirc] fits, forall pion mass cutoffs, and for both [a1] and [a2]. Thespread is intended to be the difference between the cen-tral value, obtained as described in the previous item,and the most distant central value of all other [chirr]and [chirc] fits. This is the second error of the resultsbelow. Recall that [chirc] are combined fits to φ13 andthe ratio φ12/φ13, using NLO χPT.

• Discard other fits, including [chipc], considered toounstable.

• All results have been obtained using the Symanzik b̃-parameters computed in the LCP-0 case (see discussion inSect. 3). Using LPC-1 results instead, has very marginaleffects on the error.

• In Sect. 3 we have also argued that for the quantities underconsideration finite volume effects are negligible.

The resulting RGI masses are

Ms = 127.0(3.1)(3.2) MeV,Mu/d = 4.70(15)(12) MeV. (5.1)

The quark mass ratio is obtained from

MsMu/d

= 2φll/φlh

− 1. (5.2)

Table 3 Contributions to the squared errors of our final quantities fromdifferent sources

Mu/d Ms Ms/Mu/d

Stat+chiral+cont 56% 40% 86%Fit systematics 39% 52% 14%

Renormalisation < 1% < 1% n/a

Running 5% 8% n/a

O(a) impr Negligible Negligible Negligible

Finite volume Negligible Negligible Negligible

Dependence on renormalisation is only implicit, from thejoint fit with φ13. The same procedures as above yield

MsMu/d

= 27.0(1.0)(0.4). (5.3)

The above results for RGI masses refer to the Nf = 2 + 1theory.

It is customary in phenomenological studies to report lightquark masses measured in the Nf = 2 + 1 lattice theory inthe MS scheme at 2 GeV, referred to the more physical QCDwith four flavours. This entails using Nf = 3 perturbativeRG-running from 2 GeV down to the charm threshold, fol-lowed by Nf = 4 perturbative RG-running back to 2 GeV;see for example Ref. [5]. We use 4-loop perturbative RG-running and the value for the ΛMSQCD parameter computed by

the ALPHA Collaboration in Ref. [34] to obtain8

msR(2 GeV) = 95.7(2.5)(2.4) MeV,8 In converting our results to MS we have taken into account the uncer-tainty in the matching factor coming from the error on ΛMSQCD, as well

as the covariance of the latter with our determination of Mi .

123

Eur. Phys. J. C (2020) 80 :169 Page 13 of 18 169

Fig. 5 Contributions to the statistical+chiral extrapolation+continuum limit uncertainties from each ensemble included in our analysis, for ourpreferred fit [chirr][a1][360]

mu/dR(2 GeV) = 3.54(12)(9) MeV. (5.4)

The mass ratio is obviously the same as in Eq. (5.3). We notein passing that switching to the four-flavour theory has a verysmall effect on MS results, since at 2 GeV the matching factoris mR(Nf = 4)/mR(Nf = 3) = 1.002.

The error budget for our computation is summarised inTable 3 and Fig. 5. Uncertainties are completely dominatedby our chiral fits. We have separated these errors into twocontributions; see first two lines of Table 3. The first erroris that of our best fit [chirr][a1][360], and includesthe statistical errors as well as the error from combined fitsin φ2 and a. The second uncertainty is the one arising uponvarying the fit ansätze and their φ2 range. All other errorsare clearly seen to be subdominant. It is worth noting that, asexpected, the largest contribution to the uncertainty comesfrom the ensembles with the lightest sea pion masses, espe-cially the one with the finest lattice spacing. It is then clearthat decreasing our errors would require more chiral ensem-bles, and more extensive simulations at light masses.

The current FLAG 2019 [5] world averages from Nf =2 + 1 simulations, in the MS scheme, reportedly quoted forthe Nf = 4 theory as explained above, are:

msR(2GeV) = 92.03(88)MeV,mu/dR(2GeV) = 3.364(41)MeV. (5.5)

The strange mass estimate is based on the results of Refs.[58–63], while the up/down one is based on Refs. [58–61,64].For the quark mass ratio, based on Refs. [58–60,63], FLAGquotes

msRmu/dR

= 27.42(12)MeV. (5.6)

Our results for the strange and light quark masses agree withthose of FLAG within 1.7σ and 1.2σ respectively and thusexhibit good compatibility albeit with bigger errors.

Acknowledgements We thank T. Blum, G. Herdoíza, L. Lellouch andA. Portelli for useful discussions. J.K., C.P., and A.V. thank CERN for

its hospitality. A.V. also thanks BNL for its hospitality. C.P. and D.P.thankfully acknowledge support through the Spanish projects FPA2015-68541-P (MINECO/FEDER) and PGC2018-094857-B-I00, the Cen-tro de Excelencia Severo Ochoa Programme SEV-2016-0597, and theEU H2020-MSCAITN-2018-813942 (EuroPLEx). We acknowledgePRACE for awarding us access to resource FERMI based in Italy atCINECA, Bologna, and to resource SuperMUC based in Germany atLRZ, Munich. Computer resources were also provided by the INFN(MARCONI cluster at CINECA, Bologna); a grant from the SwissNational Supercomputing Centre (CSCS) under project ID s38; theGauss Centre for Supercomputing (GCS) – through the John von Neu-mann Institute for Computing (NIC) – on the GCS share of the super-computer JUQUEEN at Jülich Supercomputing Centre (JSC); Altamira,provided by IFCA at the University of Cantabria; the FinisTerraeIImachine provided by CESGA (Galicia Supercomputing Centre); anda dedicated HPC cluster at CERN. We are grateful for the technicalsupport received by all the computer centers.

Data Availability Statement This manuscript has no associated dataor the data will not be deposited. [Authors’ comment: Several Terabytesof raw and processed data are stored in the several HPC systems usedin the project. Data can be made available by the authors on request, incompliance with the Open Data policies of the respective institutions.]

Open Access This article is licensed under a Creative Commons Attri-bution 4.0 International License, which permits use, sharing, adaptation,distribution and reproduction in any medium or format, as long as yougive appropriate credit to the original author(s) and the source, pro-vide a link to the Creative Commons licence, and indicate if changeswere made. The images or other third party material in this articleare included in the article’s Creative Commons licence, unless indi-cated otherwise in a credit line to the material. If material is notincluded in the article’s Creative Commons licence and your intendeduse is not permitted by statutory regulation or exceeds the permit-ted use, you will need to obtain permission directly from the copy-right holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.Funded by SCOAP3.

Appendix A Chiral perturbation theory expansions

We adapt standard χPT expressions to our specific parametri-sation of the data, stemming from our choice of chiral tra-jectory. We start, for example, from Eqs. (B5) and (B6) ofRef. [57], which are NLO chiral expansions of the light andstrange pseudoscalar mesons mπ and mK in terms of light

123

http://creativecommons.org/licenses/by/4.0/http://creativecommons.org/licenses/by/4.0/

169 Page 14 of 18 Eur. Phys. J. C (2020) 80 :169

and strange quark masses m1 = m2 and m3. These series areinverted, so that quark masses are functions of meson masses.The PCAC quark mass combinations m12 = (m1 + m2)/2and m13 = (m1 + m3)/2 are then formed and everythingis re-expressed in terms of the dimensionless quark massesφ12, φ13 and the dimensionless quantities φ2 and φ4, so thatwe arrive at

φ12 = φ22B0

√8t0

·{

1 − 168t0 f 20

(2L8 − L5)φ2

− 328t0 f 20

(2L6 − L4)φ4 − 124π28t0 f 20

×[

3

2φ2 ln

(φ2

8t0Λ2χ

)− 1

2φη ln

(φη

8t0Λ2χ

)] },

(A.1)

and

φ13 = 2φ4 − φ24B0

√8t0

·{

1 − 88t0 f 20

(−2L8 + L5)φ2

− 168t0 f 20

(4L6 + 2L8 − 2L4 − L5)φ4

− 124π28t0 f 20

φη ln

(φη

8t0Λ2χ

)}, (A.2)

where B0, f0, Lk(k = 4, 5, 6, 8) are standard χPT parame-ters and

φη ≡ 8t0 4m2K − m2π

3= 4φ4 − 3φ2

3. (A.3)

The NLO LECs Li and B0 are implicitly renormalised atscale Λχ . It is also useful to consider the ratio

φ12

φ13= 2φ2

2φ4 − φ2

{1 − 24

8t0 f 20(2L8 − L5)

[φ2 − 2

3φ4

]

− 116π2(8t0 f 20 )

[φ2 ln

(φ2

8t0Λ2χ

)

−φη ln(

φη

8t0Λ2χ

)]}, (A.4)

which does not require renormalisation. Note that at the sym-metric point (φ2 = 2φ4/3) the current quark masses ofEqs. (A.1) and (A.2) respect the constraint φ12 = φ13, whilethe ratio (A.4) is exactly 1. Note that the sum of ratios

4φ132φ4 − φ2 +

φ12

φ2= 3

2B0√

8t0

×{

1 − 168t0 f 20

(4

3L8 − 2

3L5 + 4L6 − 2L4

)φ4

− 148π2(8t0 f 20 )

[φ2 ln

(φ2

8t0Λ2χ

)

+φη ln(

φη

8t0Λ2χ

)]}, (A.5)

has the remarkable advantages of depending on just one com-bination of NLO LECs, and of being free of polynomialdependence on φ2.

We next rewrite Eqs. (A.1) and (A.2) in forms which aresuitable for combined fits, with common coefficients for φ12and φ13, obtaining

φ12 = φ2[p1 + p2φ2 + p3K

(L2 − 1

3Lη

)], (A.6)

φ13 = 2φ4 − φ22

[p1 + p2

(φ4 − φ2

2

)+ 2

3p3KLη

],

(A.7)

where the coefficients p1, p2, and p3 relate to LECs as fol-lows:

p1 = 12B0

√8t0

[1 − 32

8t0 f 20(2L6 − L4)φ4

]

≈ 12B0

√8t0

[1 − 32

8t0 f 2πK(2L6 − L4)φ4

], (A.8)

p2 = − 12B0

√8t0

16

8t0 f 20(2L8 − L5)

≈ − 12B0

√8t0

16

8t0 f 2πK(2L8 − L5), (A.9)

p3 = − 12B0

√8t0

. (A.10)

We also define

K ≡ (8t016π2 f 20 )−1 ≈ (8t016π2 f 2πK )−1, (A.11)with fπK given by Eq. (2.17). The chiral logarithms are

L2 ≡ φ2 ln φ2, Lη ≡ φη ln φη. (A.12)

The following points should be kept in mind:

• We are using only configurations along the φ4 = constantchiral trajectory. Terms proportional to φ4 are thus reab-sorbed into constant fit terms.

• Our expressions are linear in fit parameters, rather thannon-linear factors in which LECs appear explicitly.Determination of LECs is beyond the scope of the presentwork.

• By replacing f 20 by f 2πK in the above definition of K ,the coefficients of chiral logarithms are completely fixedrelative to the LO value; cf. Eqs. (A.1) and (A.2). Thiseliminates one fit parameter, pushing its effect to NNLO

123

Eur. Phys. J. C (2020) 80 :169 Page 15 of 18 169

LECs. In practice, the fact that terms with φ4 are reab-sorbed into the LO terms nullifies the effect in some fits,e.g., those for φ12 and φ13. A second advantage of thischoice is that the resulting ansätze are fully linear in thefit parameters. See also Eq. (2.5) in [8] and commentstherein on the reasons that f0 ≈ fπK and for preferringfπK to f0.

• We conveniently set the renormalisation scale to Λχ =1/

√8t0 � 476 MeV, simplifying the chiral logs. There

is no need to reabsorb ln(8t0Λ2χ ) terms in fit parame-ters. This is an unconventional choice, as common prac-tice consists in providing results for LECs at Λχ = mρor Λχ = 4π f0. Consequently, NLO LECs eventuallyobtained with our methodology may only be comparedto results in the literature after some extra work.

Using the above expressions and consistently neglectinghigher mass orders, we obtain for the ratio (A.4) of PCACmasses

φ12

φ13= 2φ2

2φ4 − φ2[

1 + p2p1

(3

2φ2 − φ4

)− K̃ (L2 − Lη

)].

(A.13)

For the combination (A.5) we have

4φ132φ4 − φ2 +

φ12

φ2= 3p1 + 2p2φ4 + p3K

(L2 + Lη).

(A.14)

With φ4 held constant, the quantities of Eqs. (A.6), (A.7),(A.13), and (A.14) are functions of φ2 only. We use theseexpressions to fit our data, after adding O(a2) terms whichmodel leading discretisation effects that have been neglectedthroughout this “Appendix”.

Appendix B Discretisation effects

In order to parametrise the discretisation effects of the quan-tities we fit, we first examine φi j ; cf. Eqs. (2.5) and (2.20). Itcan be written in the very general form

φi j = φconti j + f(a,

mi + m j2

,mi − m j

2, Tr[Mq]

), (B.1)

where φconti j is the continuum quantity and the function fcontains the discretisation effects which in general depend onthe lattice spacing a, the quark masses mi , m j , and the traceof the mass matrix Tr[Mq]. As we have discussed in Sect. 2,we will ignore O(g40Tr[Msum]) discretisation effects and onlyconsider the influence of O(a2) uncertainties. Also φi j has tobe symmetric with respect to the exchange of quarks, i ↔ j .We can thus parametrise f as follows:

f(a,

mi + m j2

,mi − m j

2, Tr[Mq]

)

= c0 a2

t0+ c1 a

2

t0

√8t0

(mi + m j

2

)+ c2 a

2

t0

√8t0Tr[Mq ]

+ c3 a2

t08t0

(mi + m j

2

)2+ c4 a

2

t08t0

(mi − m j

2

)2

+ c5 a2

t08t0Tr[M2q ] + c6

a2

t08t0(Tr[Mq])2

+ c7 a2

t0

√8t0

(mi + m j

2

)√8t0Tr[Mq] + O(a3). (B.2)

A further simplification is brought about by neglecting thedependence of c0, . . . , c7 on the bare coupling g20.

Next we write the function f in terms of φ2, recallingthat a constant φ4 constrains the relation between the heavier(strange) and light quark masses. This is done by first express-ing the current quark masses on the rhs of the above equationin terms of φ12 and φ13, followed by using their LO χPT rela-tions to φ2 and φ4. In particular, with β0 ≡ 1/(2B0√8t0),we see from Eqs. (A.6), (A.7) that to LO:

φ12LO= β0φ2, (B.3)

φ13LO= β0 1

2(2φ4 − φ2), (B.4)

√8t0

(m1 − m3

2

)φ12 − φ13 LO= β0

(3

2φ2 − φ4

), (B.5)

√8t0Tr[Mq] =

√8t0[2m1 + m3] LO= 2β0φ4, (B.6)

8t0Tr[M2q ] = 8t0[2m21 + m23] = 2φ212 + φ2hLO= 2β20 [3φ22 + 2φ24 − 4φ4φ2]. (B.7)

Inserting the above LO expressions in Eq. (B.2) we obtain,after some straightforward algebra, that for two light quarksthe discretisation function has the form

f12(a, φ2) ≡ f (a,m1, 0, Tr[Mq])

= a2

8t0

[C0 + C1φ2 + C2φ22

] + O(a3), (B.8)

where Ck (with k = 0, 1, . . . ) depend on the constants β0,φ4, and the coefficients cl , suitably rescaled by factors of 8t0(with l = 0, 1, . . . ). Similarly, for the heavier and a lightquark we obtain

f13(a, φ2) ≡ f (a,m1,m3, Tr[Mq])

= a2

8t0

[C̃0 + C̃1φ2 + C̃2φ22

] + O(a3). (B.9)

Note that, although in general coefficients Cn and C̃n arenot the same, in the case of m3 = m1 (symmetric point)f13 = f12 trivially.

The very fact that we have used LO χPT to obtain thelast two expressions (cf. Eqs. (B.3)–(B.7)) allows us to drop

123

169 Page 16 of 18 Eur. Phys. J. C (2020) 80 :169

O(a2φ22) contributions of f12 and f13. Moreover, standardpower-counting schemes in Wilson χPT [65,66] suggestthat terms of O(a2) enter at the same order as O(m2π ),which would imply that the terms of O(a2φ2) should alsobe dropped. We will nevertheless keep this term and exploreits impact.

For the ratio of φ12 and φ13 we have that

φ12

φ13= φ

cont12 + f12(a, φ2)

φcont13 + f13(a, φ2)= φ

cont12

φcont13+ f12

φcont13− f13φ

cont12

(φcont13 )2

+ · · · . (B.10)

We write the discretisation functions f12 and f13 as inEqs. (B.8) and (B.9) and then express coefficientsC0,C1, C̃0,C̃1, . . . in terms of the original coefficients ci of Eq. (B.2).After some algebra we end up with

φ12

φ13= φ

cont12

φcont13+ a

2

8t0

2φ4 − 3φ2(2φ4 − φ2)2

[D0 + D1φ2 + D2φ22

]

+ O(a3). (B.11)

The coefficients D0, D1, D2, . . . depend on the ci ’s. The fac-tor 2φ4 − φ2 in the discretisation term vanishes at the sym-metric point φ2 = 2φ4/3. This confirms that at the symmetricpoint the ratio φ12/φ13 is 1 by construction, for any latticespacing. In analogy to the arguments exposed above for f12and f13, we drop the D2φ22 term in our fits. Moreover, thevariation of the denominator (2φ4 − φ2)2 is relatively mild,ranging between ∼ 2 and ∼ 4.6 as φ2 varies between ∼ 0.1and ∼ 0.8 in our simulations. To simplify matters, we reab-sorb this O(1) term in re-definitions of D0 and D1.

Finally, for the combination of Eq. (A.14), we straightfor-wardly parametrise the discretisation errors in a way analo-gous to f12 and f13; see Eq. (4.5).

References

1. K. Symanzik, Infrared singularities and small distance behavioranalysis. Commun. Math. Phys. 34, 7 (1973). https://doi.org/10.1007/BF01646540

2. T. Appelquist, J. Carazzone, Infrared singularities and massivefields. Phys. Rev. D 11, 2856 (1975). https://doi.org/10.1103/PhysRevD.11.2856

3. B.A. Ovrut, H.J. Schnitzer, Gauge theories with minimal subtrac-tion and the decoupling theorem. Nucl. Phys. B 179, 381 (1981).https://doi.org/10.1016/0550-3213(81)90011-0

4. W. Bernreuther, W. Wetzel, Decoupling of heavy quarks in the min-imal subtraction scheme. Nucl. Phys. B 197, 228 (1982). https://doi.org/10.1016/0550-3213(82)90288-7, https://doi.org/10.1016/S0550-3213(97)00811-0

5. S. Aoki et al. (Flavour Lattice Averaging Group), FLAG review(2019). arXiv:1902.08191

6. ALPHA collaboration, F. Knechtli, T. Korzec, B. Leder, G. Moir,Power corrections from decoupling of the charm quark. Phys.

Lett. B 774, 649 (2017). https://doi.org/10.1016/j.physletb.2017.10.025. arXiv:1706.04982

7. M. Bruno et al., Simulation of QCD with N f = 2 + 1 flavors of non-perturbatively improved Wilson fermions. JHEP 02, 043 (2015).https://doi.org/10.1007/JHEP02(2015)043. arXiv:1411.3982

8. M. Bruno, T. Korzec, S. Schaefer, Setting the scale for the CLS2 + 1 flavor ensembles. Phys. Rev. D 95, 074504 (2017). https://doi.org/10.1103/PhysRevD.95.074504. arXiv:1608.08900

9. K.G. Wilson, Confinement of quarks. Phys. Rev. D 10, 2445 (1974)10. B. Sheikholeslami, R. Wohlert, Improved continuum limit lattice

action for QCD with Wilson fermions. Nucl. Phys. B 259, 572(1985). https://doi.org/10.1016/0550-3213(85)90002-1

11. M. Lüscher, P. Weisz, On-shell improved lattice gauge theories.Commun. Math. Phys. 97, 59 (1985). https://doi.org/10.1007/BF01206178

12. J. Bulava, S. Schaefer, Improvement of N f = 3 lattice QCD withWilson fermions and tree-level improved gauge action. Nucl. Phys.B 874, 188 (2013). https://doi.org/10.1016/j.nuclphysb.2013.05.019. arXiv:1304.7093

13. ALPHA collaboration, I. Campos, P. Fritzsch, C. Pena, D. Preti,A. Ramos, A. Vladikas, Non-perturbative quark mass renor-malisation and running in N f = 3 QCD. Eur. Phys. J. C78, 387 (2018). https://doi.org/10.1140/epjc/s10052-018-5870-5.arXiv:1802.05243

14. ALPHA collaboration, J. Bulava, M. Della Morte, J. Heitger,C. Wittemeier, Non-perturbative improvement of the axial cur-rent in N f = 3 lattice QCD with Wilson fermions and tree-levelimproved gauge action. Nucl. Phys. B 896, 555 (2015). https://doi.org/10.1016/j.nuclphysb.2015.05.003. arXiv:1502.04999

15. J. Bulava, M. Della Morte, J. Heitger, C. Wittemeier, Nonpertur-bative renormalization of the axial current in N f = 3 lattice QCDwith Wilson fermions and a tree-level improved gauge action. Phys.Rev. D 93, 114513 (2016). https://doi.org/10.1103/PhysRevD.93.114513. arXiv:1604.05827

16. ALPHA collaboration, G.M. de Divitiis, M. Firrotta, J. Heit-ger, C.C. Köster, A. Vladikas, Non-perturbative determination ofimprovement b-coefficients in N f = 3. EPJ Web Conf. 175,10008 (2018). https://doi.org/10.1051/epjconf/201817510008.arXiv:1710.07020

17. S. Aoki et al., Review of lattice results concerning low-energy par-ticle physics. Eur. Phys. J. C 77, 112 (2017). https://doi.org/10.1140/epjc/s10052-016-4509-7. arXiv:1607.00299

18. M. Lüscher, Properties and uses of the Wilson flow in lattice QCD.JHEP 08, 071 (2010). https://doi.org/10.1007/JHEP08(2010)071,https://doi.org/10.1007/JHEP03(2014)092. arXiv:1006.4518

19. R. Sommer, Scale setting in lattice QCD. PoS LATTICE2013, 015(2014). https://doi.org/10.22323/1.187.0015. arXiv:1401.3270

20. O. Bär, M. Golterman, Chiral perturbation theory for gradient flowobservables. Phys. Rev. D 89, 034505 (2014). https://doi.org/10.1103/PhysRevD.89.099905,https://doi.org/10.1103/PhysRevD.89.034505. arXiv:1312.4999

21. ALPHA collaboration, M. Bruno, I. Campos, J. Koponen, C. Pena,D. Preti, A. Ramos et al., Light and strange quark massesfrom N f = 2 + 1 simulations with Wilson fermions, PoS.LATTICE2018, 220 (2019). https://doi.org/10.22323/1.334.0220.arXiv:1903.04094

22. T. Bhattacharya, R. Gupta, W. Lee, S.R. Sharpe, J.M. Wu, Improvedbilinears in lattice QCD with non-degenerate quarks. Phys. Rev. D73, 034504 (2006). https://doi.org/10.1103/PhysRevD.73.034504.arXiv:hep-lat/0511014

23. ALPHA collaboration, G. M. Divitiis, P. Fritzsch, J. Heitger, C. C.Köster, S. Kuberski, A. Vladikas, Non-perturbative determinationof improvement coefficients bm and bA − bP and normalisationfactor ZmZP/ZA with Nf = 3 Wilson fermions. Eur. Phys. J. C79, 797 (2019). https://doi.org/10.1140/epjc/s10052-019-7287-1.arXiv:1906.03445

123

https://doi.org/10.1007/BF01646540https://doi.org/10.1007/BF01646540https://doi.org/10.1103/PhysRevD.11.2856https://doi.org/10.1103/PhysRevD.11.2856https://doi.org/10.1016/0550-3213(81)90011-0https://doi.org/10.1016/0550-3213(82)90288-7https://doi.org/10.1016/0550-3213(82)90288-7https://doi.org/10.1016/S0550-3213(97)00811-0https://doi.org/10.1016/S0550-3213(97)00811-0http://arxiv.org/abs/1902.08191https://doi.org/10.1016/j.physletb.2017.10.025https://doi.org/10.1016/j.physletb.2017.10.025http://arxiv.org/abs/1706.04982https://doi.org/10.1007/JHEP02(2015)043http://arxiv.org/abs/1411.3982https://doi.org/10.1103/PhysRevD.95.074504https://doi.org/10.1103/PhysRevD.95.074504http://arxiv.org/abs/1608.08900https://doi.org/10.1016/0550-3213(85)90002-1https://doi.org/10.1007/BF01206178https://doi.org/10.1007/BF01206178https://doi.org/10.1016/j.nuclphysb.2013.05.019https://doi.org/10.1016/j.nuclphysb.2013.05.019http://arxiv.org/abs/1304.7093https://doi.org/10.1140/epjc/s10052-018-5870-5http://arxiv.org/abs/1802.05243https://doi.org/10.1016/j.nuclphysb.2015.05.003https://doi.org/10.1016/j.nuclphysb.2015.05.003http://arxiv.org/abs/1502.04999https://doi.org/10.1103/PhysRevD.93.114513https://doi.org/10.1103/PhysRevD.93.114513http://arxiv.org/abs/1604.05827https://doi.org/10.1051/epjconf/201817510008http://arxiv.org/abs/1710.07020https://doi.org/10.1140/epjc/s10052-016-4509-7https://doi.org/10.1140/epjc/s10052-016-4509-7http://arxiv.org/abs/1607.00299https://doi.org/10.1007/JHEP08(2010)071https://doi.org/10.1007/JHEP03(2014)092http://arxiv.org/abs/1006.4518https://doi.org/10.22323/1.187.0015http://arxiv.org/abs/1401.3270https://doi.org/10.1103/PhysRevD.89.099905https://doi.org/10.1103/PhysRevD.89.099905https://doi.org/10.1103/PhysRevD.89.034505https://doi.org/10.1103/PhysRevD.89.034505http://arxiv.org/abs/1312.4999https://doi.org/10.22323/1.334.0220http://arxiv.org/abs/1903.04094https://doi.org/10.1103/PhysRevD.73.034504http://arxiv.org/abs/hep-lat/0511014https://doi.org/10.1140/epjc/s10052-019-7287-1http://arxiv.org/abs/1906.03445

Eur. Phys. J. C (2020) 80 :169 Page 17 of 18 169

24. M. Constantinou, M. Hadjiantonis, H. Panagopoulos, Renormal-ization of flavor singlet and nonsinglet fermion bilinear operators.PoS LATTICE2014, 298 (2014). https://doi.org/10.22323/1.214.0298. arXiv:1411.6990

25. RQCD collaboration, G. S. Bali, E. E. Scholz, J. Simeth, W. Södner,Lattice simulations with N f = 2 + 1 improved Wilson fermionsat a fixed strange quark mass. Phys. Rev. D 94, 074501 (2016).https://doi.org/10.1103/PhysRevD.94.074501. arXiv:1606.09039

26. P. Korcyl, G.S. Bali, Non-perturbative determination of improve-ment coefficients using coordinate space correlators in N f = 2+1lattice QCD. Phys. Rev. D 95, 014505 (2017). https://doi.org/10.1103/PhysRevD.95.014505. arXiv:1607.07090

27. M. Lüscher, R. Sommer, U. Wolff, P. Weisz, Computation of therunning coupling in the SU(2) Yang–Mills theory. Nucl. Phys. B389, 247 (1993). https://doi.org/10.1016/0550-3213(93)90292-W.arXiv:hep-lat/9207010

28. M. Lüscher, R. Sommer, P. Weisz, U. Wolff, A precise deter-mination of the running coupling in the SU(3) Yang–Mills the-ory. Nucl. Phys. B 413, 481 (1994). https://doi.org/10.1016/0550-3213(94)90629-7. arXiv:hep-lat/9309005

29. ALPHA collaboration, M. Della Morte, R. Frezzotti, J. Heit-ger, J. Rolf, R. Sommer, U. Wolff, Computation of the strongcoupling in QCD with two dynamical flavors. Nucl. Phys. B713, 378 (2005). https://doi.org/10.1016/j.nuclphysb.2005.02.013.arXiv:hep-lat/0411025

30. S. Capitani, M. Lüscher, R. Sommer, H. Wittig, Non-perturbative quark mass renormalization in quenched lat-tice QCD. Nucl. Phys. B 544, 669 (1999). https://doi.org/10.1016/S0550-3213(00)00163-2,https://doi.org/10.1016/S0550-3213(98)00857-8. arXiv:hep-lat/9810063

31. ALPHA collaboration, M. Della Morte, R. Hoffmann, F. Knechtli,J. Rolf, R. Sommer, I. Wetzorke et al., Non-perturbativequark mass renormalization in two-flavor QCD. Nucl. Phys. B729, 117 (2005). https://doi.org/10.1016/j.nuclphysb.2005.09.028.arXiv:hep-lat/0507035

32. P. Fritzsch, A. Ramos, The gradient flow coupling in theSchrödinger functional. JHEP 10, 008 (2013). https://doi.org/10.1007/JHEP10(2013)008. arXiv:1301.4388

33. ALPHA collaboration, M. Dalla Brida, P. Fritzsch, T. Korzec,A. Ramos, S. Sint, R. Sommer, Slow running of the gradient flowcoupling from 200 MeV to 4 GeV in Nf = 3 QCD. Phys. Rev. D95, 014507 (2017). https://doi.org/10.1103/PhysRevD.95.014507.arXiv:1607.06423

34. ALPHA collaboration, M. Bruno, M. Dalla Brida, P. Fritzsch,T. Korzec, A. Ramos, S. Schaefer et al., QCD coupling froma nonperturbative determination of the three-flavor Λ parame-ter. Phys. Rev. Lett. 119, 102001 (2017). https://doi.org/10.1103/PhysRevLett.119.102001. arXiv:1706.03821

35. ALPHA collaboration, M. Dalla Brida, P. Fritzsch, T. Korzec,A. Ramos, S. Sint, R. Sommer, Determination of the QCD Λ-parameter and the accuracy of perturbation theory at high ener-gies. Phys. Rev. Lett. 117, 182001 (2016). https://doi.org/10.1103/PhysRevLett.117.182001. arXiv:1604.06193

36. ALPHA collaboration, M. Dalla Brida, P. Fritzsch, T. Korzec,A. Ramos, S. Sint, R. Sommer, A non-perturbative explorationof the high energy regime in Nf = 3 QCD. Eur. Phys. J. C78, 372 (2018). https://doi.org/10.1140/epjc/s10052-018-5838-5.arXiv:1803.10230

37. M. Lüscher, R. Narayanan, P. Weisz, U. Wolff, The Schrödingerfunctional: a renormalizable probe for nonabelian gauge the-ories. Nucl. Phys. B 384, 168 (1992). https://doi.org/10.1016/0550-3213(92)90466-O. arXiv:hep-lat/9207009

38. S. Sint, On the Schrödinger functional in QCD. Nucl. Phys. B421, 135 (1994). https://doi.org/10.1016/0550-3213(94)90228-3.arXiv:hep-lat/9312079

39. J. Gasser, H. Leutwyler, Quark masses. Phys. Rep. 87, 77 (1982).https://doi.org/10.1016/0370-1573(82)90035-7

40. S. Sint, R. Sommer, The running coupling from the QCDSchrödinger functional: a one loop analysis. Nucl. Phys. B465, 71 (1996). https://doi.org/10.1016/0550-3213(96)00020-X.arXiv:hep-lat/9508012

41. S. Sint, P. Weisz, Further results on O(a) improved lattice QCDto one loop order of perturbation theory. Nucl. Phys. B 502,251 (1997). https://doi.org/10.1016/S0550-3213(97)00372-6.arXiv:hep-lat/9704001

42. W. Bietenholz et al., Tuning the strange quark mass in lattice sim-ulations. Phys. Lett. B 690, 436 (2010). https://doi.org/10.1016/j.physletb.2010.05.067. arXiv:1003.1114

43. W. Bietenholz et al., Flavour blindness and patterns of flavoursymmetry breaking in lattice simulations of up, down and strangequarks. Phys. Rev. D 84, 054509 (2011). https://doi.org/10.1103/PhysRevD.84.054509. arXiv:1102.5300

44. M. Bruno. The energy scale of the 3-flavour Λ parameter. Ph.D. the-sis, Humboldt University, Berlin (2015). https://doi.org/10.18452/17516

45. M. Lüscher, S. Schaefer, Lattice QCD with open boundaryconditions and twisted-mass reweighting. Comput. Phys. Com-mun. 184, 519 (2013). https://doi.org/10.1016/j.cpc.2012.10.003.arXiv:1206.2809

46. M. Bruno, P. Korcyl, T. Korzec, S. Lottini, S. Schaefer, On theextraction of spectral quantities with open boundary conditions.PoS LATTICE2014, 089 (2014). https://doi.org/10.22323/1.214.0089. arXiv:1411.5207

47. M. Dalla Brida, T. Korzec, S. Sint, P. Vilaseca, High precisionrenormalization of the flavour non-singlet Noether currents in lat-tice QCD with Wilson quarks. Eur. Phys. J. C79, 23 (2019). https://doi.org/10.1140/epjc/s10052-018-6514-5. arXiv:1808.09236

48. S. Sint, The chirally rotated Schrödinger functional with Wil-son fermions and automatic O(a) improvement. Nucl. Phys. B847, 491 (2011). https://doi.org/10.1016/j.nuclphysb.2011.02.002.arXiv:1008.4857

49. S. Sint, B. Leder, Testing universality and automatic O(a) improve-ment in massless lattice QCD with Wilson quarks. PoS LAT-TICE2010, 265 (2010). arXiv:1012.2500

50. M. Dalla Brida, S. Sint, A dynamical study of the chirally rotatedSchrödinger functional in QCD. PoS LATTICE2014, 280 (2014).arXiv:1412.8022

51. ALPHA collaboration, U. Wolff, Monte Carlo errors with lesserrors. Comput. Phys. Commun. 156, 143 (2004). https://doi.org/10.1016/S0010-4655(03)00467-3,https://doi.org/10.1016/j.cpc.2006.12.001. arXiv:hep-lat/0306017

52. ALPHA collaboration, S. Schaefer, R. Sommer, F. Virotta, Criticalslowing down and error analysis in lattice QCD simulations. Nucl.Phys. B 845, 93 (2011). https://doi.org/10.1016/j.nuclphysb.2010.11.020. arXiv:1009.5228

53. F. Virotta, Critical slowing down and error analysis of latticeQCD simulations. Ph.D. thesis, Humboldt-Universität zu Berlin,Mathematisch-Naturwissenschaftliche Fakultät I (2012)

54. ALPHA collaboration, H. Simma, R. Sommer, F. Virotta, Generalerror computation in lattice gauge theory. ALPHA Collaborationinternal notes (2012–2014)

55. A. Ramos, Automatic differentiation for error analysis of MonteCarlo data. Comput. Phys. Commun. 238, 19 (2019). https://doi.org/10.1016/j.cpc.2018.12.020. arXiv:1809.01289

56. G. Colangelo, S. Dürr, C. Haefeli, Finite volume effects for mesonmasses and decay constants. Nucl. Phys. B 721, 136 (2005). https://doi.org/10.1016/j.nuclphysb.2005.05.015. arXiv:hep-lat/0503014

57. RBC- UKQCD collaboration, C. Allton et al., Physical resultsfrom 2+1 flavor domain wall QCD and SU(2) chiral perturbationtheory. Phys. Rev. D 78, 114509 (2008). https://doi.org/10.1103/PhysRevD.78.114509. arXiv:0804.0473

123

https://doi.org/10.22323/1.214.0298https://doi.org/10.22323/1.214.0298http://arxiv.org/abs/1411.6990https://doi.org/10.1103/PhysRevD.94.074501http://arxiv.org/abs/1606.09039https://doi.org/10.1103/PhysRevD.95.014505https://doi.org/10.1103/PhysRevD.95.014505http://arxiv.org/abs/1607.07090https://doi.org/10.1016/0550-3213(93)90292-Whttp://arxiv.org/abs/hep-lat/9207010https://doi.org/10.1016/0550-3213(94)90629-7https://doi.org/10.1016/0550-3213(94)90629-7http://arxiv.org/abs/hep-lat/9309005https://doi.org/10.1016/j.nuclphysb.2005.02.013http://arxiv.org/abs/hep-lat/0411025https://doi.org/10.1016/S0550-3213(00)00163-2https://doi.org/10.1016/S0550-3213(00)00163-2https://doi.org/10.1016/S0550-3213(98)00857-8https://doi.org/10.1016/S0550-3213(98)00857-8http://arxiv.org/abs/hep-lat/9810063https://doi.org/10.1016/j.nuclphysb.2005.09.028http://arxiv.org/abs/hep-lat/0507035https://doi.org/10.1007/JHEP10(2013)008https://doi.org/10.1007/JHEP10(2013)008http://arxiv.org/abs/1301.4388https://doi.org/10.1103/PhysRevD.95.014507http://arxiv.org/abs/1607.06423https://doi.org/10.1103/PhysRevLett.119.102001https://doi.org/10.1103/PhysRevLett.119.102001http://arxiv.org/abs/1706.03821https://doi.org/10.1103/PhysRevLett.117.182001https://doi.org/10.1103/PhysRevLett.117.182001http://arxiv.org/abs/1604.06193https://doi.org/10.1140/epjc/s10052-018-5838-5http://arxiv.org/abs/1803.10230https://doi.org/10.1016/0550-3213(92)90466-Ohttps://doi.org/10.1016/0550-3213(92)90466-Ohttp://arxiv.org/abs/hep-lat/9207009https://doi.org/10.1016/0550-3213(94)90228-3http://arxiv.org/abs/hep-lat/9312079https://doi.org/10.1016/0370-1573(82)90035-7https://doi.org/10.1016/0550-3213(96)00020-Xhttp://arxiv.org/abs/hep-lat/9508012https://doi.org/10.1016/S0550-3213(97)00372-6http://arxiv.org/abs/hep-lat/9704001https://doi.org/10.1016/j.physletb.2010.05.067https://doi.org/10.1016/j.physletb.2010.05.067http://arxiv.org/abs/1003.1114https://doi.org/10.1103/PhysRevD.84.054509https://doi.org/10.1103/PhysRevD.84.054509http://arxiv.org/abs/1102.5300https://doi.org/10.18452/17516https://doi.org/10.18452/17516https://doi.org/10.1016/j.cpc.2012.10.003http://arxiv.org/abs/1206.2809https://doi.org/10.22323/1.214.0089https://doi.org/10.22323/1.214.0089http://arxiv.org/abs/1411.5207https://doi.org/10.1140/epjc/s10052-018-6514-5https://doi.org/10.1140/epjc/s10052-018-6514-5http://arxiv.org/abs/1808.09236https://doi.org/10.1016/j.nuclphysb.2011.02.002http://arxiv.org/abs/1008.4857http://arxiv.org/abs/1012.2500http://arxiv.org/abs/1412.8022https://doi.org/10.1016/S0010-4655(03)00467-3https://doi.org/10.1016/S0010-4655(03)00467-3https://doi.org/10.1016/j.cpc.2006.12.001https://doi.org/10.1016/j.cpc.2006.12.001http://arxiv.org/abs/hep-lat/0306017https://doi.org/10.1016/j.nuclphysb.2010.11.020https://doi.org/10.1016/j.nuclphysb.2010.11.020http://arxiv.org/abs/1009.5228https://doi.org/10.1016/j.cpc.2018.12.020https://doi.org/10.1016/j.cpc.2018.12.020http://arxiv.org/abs/1809.01289https://doi.org/10.1016/j.nuclphysb.2005.05.015https://doi.org/10.1016/j.nuclphysb.2005.05.015http://arxiv.org/abs/hep-lat/0503014https://doi.org/10.1103/PhysRevD.78.114509https://doi.org/10.1103/PhysRevD.78.114509http://arxiv.org/abs/0804.0473

169 Page 18 of 18 Eur. Phys. J. C (2020) 80 :169

58. RBC, UKQCD collaboration, T. Blum et al., Domain wall QCDwith physical quark masses. Phys. Rev. D 93, 074505 (2016).https://doi.org/10.1103/PhysRevD.93.074505. arXiv:1411.7017

59. S. Dürr, Z. Fodor, C. Hoelbling, S.D. Katz, S. Krieg, T. Kurth etal., Lattice QCD at the physical point: light quark masses. Phys.Lett. B 701, 265 (2011). https://doi.org/10.1016/j.physletb.2011.05.053. arXiv:1011.2403

60. S. Dürr, Z. Fodor, C. Hoelbling, S. Katz, S. Krieg et al.,Lattice QCD at the physical point: simulation and analy-sis details. JHEP 1108, 148 (2011). https://doi.org/10.1007/JHEP08(2011)148. arXiv:1011.2711

61. C. McNeile, C.T.H. Davies, E. Follana, K. Hornbostel, G.P. Lepage,High-precision c and b masses, and QCD coupling from current–current correlators in lattice and continuum QCD. Phys. Rev. D82, 034512 (2010). https://doi.org/10.1103/PhysRevD.82.034512.arXiv:1004.4285

62. Y. Maezawa, P. Petreczky, Quark masses and strong coupling con-stant in 2+1 flavor QCD. Phys. Rev. D 94, 034507 (2016). https://doi.org/10.1103/PhysRevD.94.034507. arXiv:1606.08798

63. MILC collaboration, A. Bazavov et al., MILC results for light pseu-doscalars. PoS CD09, 007 (2009). https://doi.org/10.22323/1.086.0007. arXiv:0910.2966

64. A. Bazavov et al., Staggered chiral perturbation theory in thetwo-flavor case and SU(2) analysis of the MILC data. PoS LAT-TICE2010, 083 (2010)

65. G. Rupak, N. Shoresh, Chiral perturbation theory for the Wilsonlattice action. Phys. Rev. D 66, 054503 (2002). https://doi.org/10.1103/PhysRevD.66.054503. arXiv:hep-lat/0201019

66. O. Bär, G. Rupak, N. Shoresh, Chiral perturbation theory at O(a2)for lattice QCD. Phys. Rev. D 70, 034508 (2004). https://doi.org/10.1103/PhysRevD.70.034508. arXiv:hep-lat/0306021

123

https://doi.org/10.1103/PhysRevD.93.074505http://arxiv.org/abs/1411.7017https://doi.org/10.1016/j.physletb.2011.05.053https://doi.org/10.1016/j.physletb.2011.05.053http://arxiv.org/abs/1011.2403https://doi.org/10.1007/JHEP08(2011)148https://doi.org/10.1007/JHEP08(2011)148http://arxiv.org/abs/1011.2711https://doi.org/10.1103/PhysRevD.82.034512http://arxiv.org/abs/1004.4285https://doi.org/10.1103/PhysRevD.94.034507https://doi.org/10.1103/PhysRevD.94.034507http://arxiv.org/abs/1606.08798https://doi.org/10.22323/1.086.0007https://doi.org/10.22323/1.086.0007http://arxiv.org/abs/0910.2966https://doi.org/10.1103/PhysRevD.66.054503https://doi.org/10.1103/PhysRevD.66.054503http://arxiv.org/abs/hep-lat/0201019https://doi.org/10.1103/PhysRevD.70.034508https://doi.org/10.1103/PhysRevD.70.034508http://arxiv.org/abs/hep-lat/0306021

Light quark masses in Nf=2+1 lattice QCD with Wilson fermionsALPHA CollaborationAbstract 1 Introduction2 Theoretical framework2.1 Quark masses, renormalisation, and improvement2.2 The chiral trajectory and scale setting

3 Quark mass computations4 Extrapolations to physical quark masses5 Final results and discussionAcknowledgementsAppendix A Chiral perturbation theory expansionsAppendix B Discretisation effectsReferences