Setting the scale for the CLS 2+1 flavor ensembles
DDESY 16-162, WUB/16-05
Setting the scale for the CLS flavor ensembles
Mattia Bruno a , b , Tomasz Korzec c , Stefan Schaefer a a John von Neumann Institute for Computing (NIC), DESYPlatanenallee 6, D-15738 Zeuthen, Germany b Physics Department, Brookhaven National Laboratory, Upton, NY 11973, USA c Department of Physics, Bergische Universität WuppertalGaussstr. 20, D-42119, Wuppertal, Germany
Abstract
We present measurements of a combination of the decay constants of the light pseu-doscalar mesons and the gradient flow scale t , which allow to set the scale of thelattices generated by CLS with flavors of non-perturbatively improved Wil-son fermions. Mistunings of the quark masses are corrected for by measuring thederivatives of observables with respect to the bare quark masses. Keywords:
Lattice QCD, Scale setting
PACS: a r X i v : . [ h e p - l a t ] J un Introduction
A lattice scale is a dimensionful quantity which can be used to form dimensionlessratios of observables with a well-defined continuum limit. In principle, its choice isarbitrary, however, the precision to which we can extract the scale on the latticeand the accuracy, to which its experimental value is known, will affect the precisionof the final results.Here, we will determine two such scales for the setup chosen in the CLS simu-lations with N f = 2 + 1 flavors of O( a ) improved Wilson fermions and the tree-levelLüscher–Weisz gauge action, which has been described in detail in Ref. [1]. The twoscales are a combination of the pseudoscalar decay constants f π and f K as well as t , the gluonic dimension two quantity introduced by Lüscher in Ref. [2] using theWilson flow.Other observables commonly used in scale setting are the mass of the Ω − baryon [3], the Υ - Υ (cid:48) mass splitting [4] or the scale r [5] defined from the forcebetween two static quarks. For the latter, like for t , the physical value is not knownfrom experiment, but has to be computed on the lattice. Still such quantities canbe very useful as intermediate scales due to their high statistical accuracy and thefact that their definition does not include valence quarks. This makes them alsouseful in studies which connect results with different flavor content in the sea. Thestudy presented here is in some aspects similar to the one by QCDSF [6], wherecombinations of hadron masses are used to set the scale in the determination of t .The results of lattice QCD simulations are dimensionless ratios of observables.Since we restrict ourselves to three flavors, with the two light ones degenerate, theseratios will differ from those found in Nature. Therefore the choice of input observablewill affect the global normalization of the results. However, the scale also enters inthe definition of the physical quark mass point and therefore also directly affectsthe ratios of observables. Because of the latter, N f = 2 + 1 flavor results becomeunique only after specifying the lattice scale and the observables used to set thequark masses. Deviations from uniqueness, however, are very small effects as longas one remains in the low energy sector of the theory where decoupling holds [7].The paper is organized as follows: In Section 2 we first give a brief overview ofthe ensembles as well as the observables we consider. Section 3 discusses the issueswhich occur in the extraction of the masses and matrix elements in the presenceof open boundary conditions in time, as used in the CLS simulations. In Section 4the method to correct for mistunings by using mass derivatives of the observables isdetailed, before presenting the results in Section 5 and concluding. We want to set the scale for the ensembles generated by the CLS 2+1 effort whichuse the tree-level O( a ) improved Lüscher-Weisz gauge action and improved Wilsonfermions with a non-perturbative c sw [8]. Three values of the coupling have been2 d β N s N t κ u κ s m π [MeV] m K [MeV] m π L H101 3.40 32 96 0.13675962 0.13675962 420 420 5.8H102 3.40 32 96 0.136865 0.136549339 350 440 4.9H105 3.40 32 96 0.136970 0.13634079 280 460 3.9C101 3.40 48 96 0.137030 0.136222041 220 470 4.7H400 3.46 32 96 0.13688848 0.13688848 420 420 5.2H401 3.46 32 96 0.136725 0.136725 550 550 7.3H402 3.46 32 96 0.136855 0.136855 450 450 5.7H200 3.55 32 96 0.137000 0.137000 420 420 4.3N202 3.55 48 128 0.137000 0.137000 420 420 6.5N203 3.55 48 128 0.137080 0.136840284 340 440 5.4N200 3.55 48 128 0.137140 0.13672086 280 460 4.4D200 3.55 64 128 0.137200 0.136601748 200 480 4.2N300 3.70 48 128 0.137000 0.137000 420 420 5.1J303 3.70 64 192 0.137123 0.1367546608 260 470 4.1
Table 1 : List of the ensembles. In the id, the letter gives the geometry, the first digit thecoupling and the final two label the quark mass combination. employed β = 3 . , 3.55 and 3.7, which correspond roughly to a lattice spacing of a = 0 .
085 fm , .
065 fm and .
05 fm , respectively. Data limited to degenerate quarkmasses is also available at β = 3 . . An overview can be found in Tab. 1, withensemble N203 first described in Ref. [9].Apart from β = 3 . , these ensembles lie along lines of constant sum of the barequark masses am q = (1 /κ q − / with degenerate light quarks m ud ≡ m u = m d and an average quark mass m sym . κ q is the standard hopping parameter of theWilson quark action [10]. Using the quark mass matrix M q = diag ( m u , m d , m s ) , wetherefore have m sym = tr M q = 2 m ud + m s = const . (2.1)This line has been chosen, because it implies a constant O( a ) improved coupling [11] ˜ g = g (1 + 13 b g a tr M q ) , (2.2)irrespective of the knowledge of the improvement coefficient b g .To further specify the chiral trajectories, we have to define a point in the ( m ud , m s ) plane through which it is supposed to pass. To this end, we have used thedimensionless variables φ = 8 t m π and φ = 8 t ( m + 12 m π ) (2.3)with the requirement that the chiral trajectory intersects the symmetric line m ud = m s at φ = 1 . . Here, m π and m K are the masses of the pseudoscalars corresponding3o the pion and the kaon. The value of φ = 1 . comes from a preliminary analysisof the quark mass dependence of φ ; only the final analysis can tell in how far thischiral trajectory goes through the point of physical φ and φ .Three points in this strategy need special attention: First of all, eq. (2.1) doesnot imply a constant sum of renormalized quark masses, which is already violatedto O( a ) tr M R = Z m r m [(1 + a ¯ d m tr M q )tr M q + ad m tr M ] , (2.4)as worked out in Ref. [12], whose notation we are using. Secondly, the tuning in φ is correct only up to a certain degree and it is at the current stage by no means clearthat the thus defined trajectories also go through the physical quark mass points.The potential mistuning needs to be taken into account in the analysis.Furthermore, the definition of the point of physical quark masses depends atfinite lattice spacing on the scale. The tuning has been done using t for which theprecise experimental value is not known. It might therefore be preferable to use thedecay constants also at finite lattice spacing, but this will necessarily lead to chiraltrajectories which are no longer matched to the same level of accuracy. The physical quantity used here to convert the lattice measurements to physicalunits is the combination of the pseudoscalar decay constants of pion f π and kaon f K , which along with its next to leading order expansion in SU(3) chiral perturbationtheory [13] is given by f π K ≡
23 ( f K + 12 f π ) ≈ f (cid:20) − L π − L K − L η + 16 B tr M f ( L + 3 L ) (cid:21) . (2.5)The a priori unknown low energy constants L and L (defined at the scale µ = 4 πf )appear only in the tr M term and logarithms are given by L x = m x / (4 πf ) ln ( m x / (4 πf ) ) .A constant tr M therefore implies a constant f π K up to logarithmic corrections. Notethat due to Eq. (2.4) we also expect O( am ) effects to violate the constancy of thiscombination.With f π K as a scale, we can define a second set of dimensionless variables y π = m π (4 πf π K ) and y K = m (4 πf π K ) (2.6)for which the linear combination y π K = y π / y K is again constant in leading orderChPT along our chiral trajectory.Using the experimental values of the meson masses corrected for isospin break-ing effects [14] and the PDG values for the decay constants [15], we use as inputparameters m π = 134 . m K = 494 . f π = 130 . f K = 156 . . (2.7)4he value of f K comes from a direct experiment only up to the contributionof the CKM matrix element V us , which ultimately is extracted using theory input.At the current level of accuracy, the associated uncertainties are acceptable, how-ever, in the future a direct measurement from our simulations will be an interestingverification of this assumption. The finite spatial volume of the lattices can affect the quantities we are interested in.A detailed study of these effects is planned in the future, but a general requirementis that one has to ensure f π L (cid:29) and m π L (cid:29) for them to be small.For the lattices apart from H200 listed in Table 1, we have L ≥ . fm and m π L > throughout. The chiral perturbation theory prediction [16, 17] indicatesthat the systematic effects on f π and m π are below our statistical uncertaintieson the ensembles which enter our analysis, however, in some cases they are notcompletely negligible. The largest finite volume effect is on the N200 ensemble,where it amounts to 70% of the statistical error. We therefore apply the one-loopfinite volume corrections to all data. The remaining effect, not accounted for by thiscorrection, should be significantly below the statistical uncertainty and can thereforebe neglected.The H200 ensemble is excluded from the analysis, because the finite volumeeffect in the decay constant is too large. It is at the same physical parameters as theN202 ensemble, but with L/a = 32 instead of and is therefore the only latticewith L ≈ . The measured finite volume effect between the two volumes in thedecay constant is − . . , where ChPT predicts a − . correction. While theaccuracy of the data is not high enough for a detailed comparison, the correctionbeyond the ChPT prediction is to be significantly smaller for the larger volumes onwhich we base our computation. Three types of fermionic observables are required for the scale setting in this paper:the masses and decay constants of the pseudoscalar mesons, as well as the PCACquark masses. The open boundary conditions of the gauge field configurations do notpose a fundamental problem in the analysis due to the fact that the transfer matrixis not changed [18, 19]. Still some parts of the analysis have to be adapted becauseof the broken translational invariance at the boundaries at x = 0 and x = T . Byconstruction, the boundary states share the quantum numbers of the vacuum and,if source or sink of the two-point functions come close to the boundaries, the wholetower of these states contributes to correlation functions.As usual, pseudoscalar masses and decay constants are extracted from corre-lation functions of the pseudoscalar density P rs = ¯ ψ r γ ψ s and the improved axialvector current A µ = ¯ ψ r γ µ γ ψ s + ac A ˜ ∂ µ P rs with non-perturbatively tuned coefficient5 A [20]. The two-point functions f rs P ( x , y ) = − a L (cid:88) (cid:126)x,(cid:126)y (cid:104) P rs ( x , (cid:126)x ) P sr ( y , (cid:126)y ) (cid:105) ,f rs A ( x , y ) = − a L (cid:88) (cid:126)x,(cid:126)y (cid:104) A rs ( x , (cid:126)x ) P sr ( y , (cid:126)y ) (cid:105) , (3.1)where r and s are flavor indices, are estimated with stochastic sources located ontime slice y , for which we choose either y = a or y = T − a . This choice and thegeneral procedure are suggested by the comparison of various strategies in Ref. [21]. To obtain the vacuum expectation values, we have to define the plateaux regionsin which excited state contributions can be neglected. As in Ref. [22], the generalstrategy to define plateaux is divided in two steps. First, we perform preliminaryfits including the first excited state, where the fit interval is chosen such that thismodel describes the data well by using a χ test.In the second step, only the function describing the ground state contributionis used, with the fit range given by the region where the excited state contributionas determined by the first fit is negligible compared to the statistical errors of thedata.From our measurements of f P and f A we observe, at fixed lattice spacing, thatboundary effects increase as the up-down quark masses are lowered. This turns outto be particularly relevant for the quantity R PS ( x , y ) defined below in Eq. (3.4)where, according to our criterion for the definition of a plateau, boundary effects canbe neglected starting from x ≈ for pion masses around
200 MeV , as shown inFigure 1. Nevertheless, despite the fraction of the lattice which is discarded, we havebeen able to extract meson decay constants with one percent accuracy (and higher)on all ensembles, as reported in Table 2. Here, and all other cases presented, we usethe statistical analysis method of Ref. [23], taking into account the autocorrelationof the data including contributions from the slowest observed modes of the MarkovChain Monte Carlo as determined in Ref. [1].For the PCAC masses deviations from a flat behavior constitute a pure dis-cretization effect. We have observed the largest ones at β = 3 . , where a plateaucan be identified at distances of around . from the boundaries, while at β = 3 . this distance shrinks to . . In the presence of open boundary conditions, the pseudoscalar correlator f P has theasymptotic behavior f P ( x , y ) = A ( y ) e − m PS x + A ( y ) e − m (cid:48) x + B ( y ) e − ( E − m PS )( T − x ) + . . . , (3.2)6
20 40 60 80 100 120 x /a . . . . . . R
420 MeV 200 MeV
Figure 1 : The effective quantity R = R PS ( x , a ) defined in Eq. (3.4), from which thedecay constants are extracted, for ensembles N202 and D200. Both of them share thesame β = 3 . , but differ in the quark masses, the pion having a mass of ≈ MeV forthe former and
MeV for the latter. For the combined observable, the sources are at y = a and y = T − a , with the sink varying over the temporal extent of the lattice.The horizontal lines indicate the plateau average and its uncertainty, the vertical lines theplateau ranges. Smaller pion mass leads to boundary effects reaching farther into the bulk. T (cid:29) x (cid:29) y , where we included the contribution from the first excited state.The third exponential term originates from the first boundary excited state, a finitevolume two pion state. In large volume, E ≈ m PS , leading to the sinh -likefunctional form presented in Ref. [19].Taking into account the leading corrections from excited states for this formula,which are exponentially suppressed with the distance of the sink from the sourceand the boundaries, respectively, results in am eff ( x ) ≡ log f P ( x ) f P ( x + a ) = am PS (1 + c e − E x + c e − E ( T − x ) + . . . ) , (3.3)with E = m (cid:48) − m PS and only c and c depending on the source position y . Asdiscussed in the previous section, we determine the plateau range in x , where theexponential corrections can be safely neglected compared to the statistical uncer-tainties. The results of the plateau fits can be found in Table 2.To check for possible systematics, also direct fits of Eq. (3.2) including termsof excited states have been tried, without going through the effective mass. Thedifferences of the results are significantly below our statistical accuracy. The vacuum expectation values needed for the extraction of the decay constants areobtained from the plateaux in x of the ratio (where we drop the flavor indices rs ) R PS ( x , y ) = (cid:20) f A ( x , y ) f A ( x , T − y ) f P ( T − y , y ) (cid:21) / . (3.4)This ratio is formed such that matrix elements of operators close to the boundarydrop out. In this case, the plateaux are defined by fitting the ratios R PS with R PS ( x , y ) = R (1 + c ( y ) cosh[ − E ( T / − x )]) , (3.5)since it is invariant under time reversal transformations. Once the relevant matrixelement is known, the pseudoscalar decay constants are computed from f PS = Z A (˜ g ) (cid:2) b A a tr M q + ˜ b A am rs (cid:3) f barePS (3.6) f barePS = (cid:114) m PS R averPS , (3.7)where R averPS is the plateau average of the ratio previously introduced.The third observable we are interested in is the PCAC quark mass m rs ( x , y ) =˜ ∂ x f rs A ( x , y ) / (2 f rs P ( x , y )) where ˜ ∂ x is the symmetric derivative in time direction.With the same technique described for the effective mass in Sect. 3.2, plateaux in x are also found for this quantity, which is then multiplicatively renormalized (upto O ( a ) corrections) according to m rs, R = Z A Z P (cid:2) b A − ¯ b P ) a tr M q + (˜ b A − ˜ b P ) am rs (cid:3) m rs . (3.8)8 d t / a a m π a m K a m a m a f π a f K H . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) H . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) H . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) C . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) H . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) H . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) H . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) N . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) N . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) N . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) D . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) N . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) J . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) T a b l e : V a l u e s o f t , t h e p s e ud o s c a l a r q u a r k m a ss e s a ndd ec a y c o n s t a n t s a s w e ll a s t h e P C A C m a ss e s . F o r e a c h e n s e m b l e w e g i v e t h e m e a s u r e d v a l u e s i n t h e r o w s w i t h t h e l a b e l s a nd i n t h e r o w b e l o w t h e v a l u e s a f t e r a s h i f tt o φ = . . perturbation theory [25] ˜ b A − ˜ b P = − . g + O ( g ) , ˜ b A = 1 + 0 . g + O ( g ) , ¯ b A = ¯ b P = O ( g ) . (3.9)The finite renormalization factor Z A has been computed using the Schrödinger Func-tional [26] and its chirally rotated variant [27]. We use the latter result due to itshigher statistical accuracy, measured directly at the simulated values of g , thus ne-glecting terms of order b g g . The non-perturbative running of the scale-dependentfactor Z P is not yet completed [28]. Hence, in the following we will consider onlyratios of renormalized PCAC masses which do not depend on renormalization fac-tors.Starting from the leading matrix element of f P , a second possibility to obtain thedecay constants is based on the PCAC relation. The former can be obtained from aratio similar to R PS where the axial two point functions in the numerator are simplyreplaced by their corresponding f P . For this quantity, however, we observed muchstronger boundary contaminations and therefore we did not follow this strategy tocompute the pseudoscalar decay constants. As we have seen above, it is necessary to control small corrections in the quarkmasses from the ones at which the simulations have been performed. This couldbe done by reweighting [29, 30], but here we only consider the leading correctionsin a Taylor expansion. Since the required shifts are typically determined from thefit to the data, this has the advantage that we can include its effect easily in thefull data analysis without need of interpolation between the measured points of areweighting.For a general function f ( ¯ A ( m ) , . . . , ¯ A n ( m )) of expectation values of primaryobservables ¯ A i = (cid:104) A i (cid:105) , the derivative with respect to a parameter m of the theoryreads ddm f = (cid:88) i ∂f∂ ¯ A i (cid:20)(cid:10) ∂A i ∂m (cid:11) − (cid:10) ( A i − ¯ A i )( ∂S∂m − ∂S∂m ) (cid:11)(cid:21) (4.1)with S denoting the action of the theory. In the analysis we then use f ( m (cid:48) ) → f ( m ) + ( m (cid:48) − m ) ddm f ( m ) (4.2)neglecting higher order terms. Non-perturbatively determined values have become available while writing up the presentanalysis [24]. .
000 0 .
002 0 . a ∆ m q . . . . . . . t / a . . . a ∆ m q . . . . . . . a f π Figure 2 : Examples of the mass shifts for the t and f π along the symmetric line m ud = m s at β = 3 . . The data points correspond to the measurements on the ensembles H400 andH402 as well as H401 in the case of t . The shaded bands give the linear approximationto the mass dependence starting from the leftmost point. Deviations from the linearapproximation are smaller than the increase in the statistical uncertainties. For the measurement of the derivative we therefore need to compute the explicitderivative of the observable as well as the one of the action. If m is the bare quarkmass, the derivative of hadronic correlation functions is easily evaluated as in ∂ m tr (cid:2) D + m Γ 1 D + m (cid:48) Γ (cid:48) (cid:3) = − tr (cid:2) D + m ) Γ 1 D + m (cid:48) Γ (cid:48) (cid:3) . (4.3)The numerical effort is limited: for each propagator a second inversion on the solu-tion is necessary.The second term in Eq. (4.1) contains the derivative of the action − ∂ m log det( D + m ) = − tr ( D + m ) − , (4.4)which can be evaluated using stochastic estimates of the trace. For our ensembles, weused 16 sources and found the noise introduced by them to be significantly inferiorto the gauge noise of this observable. To test the method we use ensembles at β = 3 . which have been generated alongthe symmetric line H400, H401 and H402, where H402 has a sea quark mass whichis roughly 19% larger than the one of H400 and H401 has roughly twice this mass.The results are displayed in Figure 2. We give the direct measurements on the threeensembles as well as the prediction indicated by the shaded band obtained fromensemble H400.For t we can shift roughly 9% in the quark mass before doubling the statisticaluncertainty, for the decay constant this level is reached at a 5% shift. No deviations11rom the linear approximation beyond the statistical error are visible in the displayedregion. Since the shifts we apply in the following are smaller than those, we assumethat the systematic error from dropping higher orders in the Taylor expansion canbe neglected compared to the increase in statistical uncertainty. There is no unique choice of chiral trajectory in the m ud – m s plane along which onemoves as the pion mass is changed, as there is no unique choice of matching conditionbetween different lattice spacings. These choices, however, have an impact on theease with which the extrapolation to physical quark masses and to the continuumcan be performed. As already mentioned above, the chiral trajectories defined by tr M q = const lead todiscretization effects of O( am ) given in Eq. (2.4). To avoid them, an improved proxyfor the quark mass is needed. There are basically two options: the PCAC quarkmasses and the meson masses. The former has the advantage that a trajectorydefined through a constant sum of these quark masses automatically leads to aconstant coefficient of b g in Eq. (2.2). The disadvantage is that the improvementcoefficients b A − b P and ˜ b A − ˜ b P are known only perturbatively. However, our massesare small and so are the one-loop values of these combinations.We opt for using the sum of the mass squares of pseudoscalar mesons m + m π ,which in leading order of Chiral Perturbation Theory is proportional to the sum ofthe three light quark masses. While these do not introduce any discretization effectsat O( a ) , it might introduce a small variation of the improved coupling, since thesum of quark masses varies due to higher order effects in ChPT. As we will seebelow, on a chiral trajectory defined through a constant φ = 8 t ( m + m π ) alsothe sum of the renormalized quark masses is constant on the per-cent level. We cantherefore safely assume that the effect of a variation in the term coming with b g canbe neglected. The obvious extension of the strategy used in the planning of the simulation is tocontinue with t as a scale parameter, i.e. finding the physical value of φ alongwhich we move towards the chiral limit. Since the physical value of t is not knownbeforehand, we determine it implicitly from another dimensionful observable.The analysis therefore starts by assuming a certain physical value of t = ˜ t .Together with Eq. (2.7), this defines the target point ( ˜ φ , ˜ φ ) at which we can readoff physical results of the calculations. Starting from the simulated ensembles, shiftsalong the line ∆ m u = ∆ m d = ∆ m s are now performed to reach ˜ φ . This is the12irection in which tr M q changes fastest and therefore the effects due to the trun-cation of the Taylor expansion are expected to be smallest at the target ˜ φ . As anintermediate result we get values of √ t /a , af π K ( φ ) and their product at constant φ = ˜ φ , which now have to be extrapolated to ˜ φ .For this extrapolation we use two different functional forms, one given by NLOChPT, the other a Taylor expansion around the symmetric point. As noted inRef. [31], along the line adopted in our simulations, the linear term in the quark massdoes not contribute to the Taylor expansion and we can therefore use F contT ( φ ) = c + c ( φ − φ sym2 ) .The ChPT formula F χ ( φ ) can easily be derived from Eq. (2.5). Note that inNLO ChPT t is constant along our trajectory at this order [32] and we have astraightforward relation between φ , φ and the meson masses. At NLO, the ratiosare therefore unambiguously given by the logarithms predicted by ChPT √ t f π K ( √ t f π K ) sym = f π K ( f π K ) sym = 1 −
76 ( L π − L sym π ) −
43 ( L K − L symK ) −
12 ( L η − L sym η ) . (5.1)These continuum relations are augmented by a term to account for the leadingdiscretization effects. In general, we adopt √ t f π K = F contT /χ ( φ ) + c T/χ a t sym0 (5.2)and will see below that our data is well compatible with this ansatz.One result is a value of √ t f π K at ˜ φ and ˜ φ as defined by ˜ t . Using the physicalvalue of f π K , this gives a value of t in physical units. The final goal is to find thefixed point, at which this value agrees with the input ˜ t . This then defines thephysical value of t and in turn the physical value of φ .As we see from Figure 3, both the ChPT formula as well as the Taylor expan-sion, fitted to our data, hardly differ in the range of our points. Also at physicalquark masses, the difference amounts to roughly half the statistical uncertainty.However, such a difference might not be enough to properly quantify the systematicuncertainty associated to the chiral extrapolations. More specifically, in ChPT asensible way to estimate the size of the higher order terms is by changing the ex-pansion parameter. In SU (2) chiral perturbation this is done by using either theconstant f in the chiral limit or f π ( m π ) , which leads to the x and ξ expansions.To mimic this, we use either a constant scale proportinal to √ t or f π K ( m π ) , thusleading to φ and y π respectively.Taking into account the full propagation of the errors through the fixed pointcondition, we therefore arrive at physical values of φ phys4 = 1 . and (cid:113) t phys0 = 0 . (5.3)for the ChPT ansatz with y π . The fit has an excellent quality characterized by a χ = 8 at 8 degrees of freedom. The quadratic extrapolation gives φ phys4 = 1 . and (cid:113) t phys0 = 0 . (5.4)13 . . . . . . . . . φ . . . . √ t f π K β = 3 . β = 3 . β = 3 . β = 3 . . . . . . a /t . . . . .
00 0 .
01 0 .
02 0 .
03 0 .
04 0 .
05 0 . y π . . . . √ t f π K .
000 0 .
001 0 .
002 0 .
003 0 .
004 0 . a f π K . . . . Figure 3 : The dimensionless quantity √ t f π K along the line φ = 1 . in the top row,along the line of y π K = 0 . in the bottom row. In the left panels, we present allmeasurements as a function of φ together with the fit result of the quadratic function(solid) and the ChPT Eq. (5.1). The lattice spacing increases from bottom to top, withthe uppermost lines corresponding to the continuum limit. In the right panel the continuumextrapolation of the data at the symmetric point is shown. We observe discretization effectsup to for the coarsest lattice spacing.
14t a χ = 2 . , again with 8 degrees of freedom.By repeating the two fit ansatz with φ as the extrapolation variable, we obtaina second pair of values for φ phys4 : in the case of the Taylor expansion the differenceis negligible, instead for the ChPT fits the difference is − . , i.e. below thestatistical accuracy of the final result. We take this number as our final systematicuncertainty since it covers also the discrepancy between the results of the Taylorand ChPT extrapolations quoted above. This leads to (cid:113) t phys0 = 0 . (5.5)as final result of this strategy. Note that the systematic error also includes possibleuncertainties on the validity range of the chiral extrapolations, which turn out to beextremely stable (with variations on the 0.5% level) under the exclusion, from ourfits, of the two most chiral points and the four symmetric ones. For convenience, wegive the values of our observables shifted to φ = 1 . in Table 2.With this result we have fixed the chiral trajectory φ = φ phys4 , such that nowwe are in a position to set the scale. One method to obtain the lattice spacing inphysical units would be to chirally extrapolate t /a to φ phys2 and divide the resultby t phys0 of Eq. (5.5). We prefer a slightly different method, which avoids this lastchiral extrapolation and is instead based on directly measured values of t /a at thesymmetric point. Eq. (5.2) is fitted to (cid:112) t sym0 f π K ( φ ) along the line of φ = φ phys4 .Dividing the continuum and chirally extrapolated result by the experimental valueof f π K yields (cid:112) t sym0 = 0 . . (5.6)The lattice spacings in physical units are obtained by dividing t sym0 /a by t sym0 [fm ] and their values are reported in Table 3. β t sym0 /a a [fm] Table 3 : Lattice spacings from strategy 1 set by t at the symmetric point and physicalvalue of φ as given in Eq. (5.5). Note the numbers in the second column are weakly corre-lated, whereas the values of the lattice spacings have strong correlations due to Eq. (5.6). In the second strategy, we use f π K to set the scale, shifting each simulated latticesuch that y πK equals its physical value y phys πK = 0 . . This strategy is simpler15ince its physical value is known, see Eq. (2.7). To set the lattice spacing one wouldcompute af π K along the line of constant y πK .The disadvantage of this approach is that the parameters of our ensembles arefarther away from this chiral trajectory and therefore require larger shifts. Thisincreases the statistical uncertainties and also potential higher order effects in theTaylor expansion, which we neglect. To show which accuracy can be reached withthe current data, √ t f π K is plotted after the shift to physical y πK in the bottomplots of Figure 3. As we can see, the statistical uncertainties are significantly largerthan the ones encountered in Strategy 1, such that the applicability of the linearcorrection terms alone is no longer clear. We therefore do not consider this strategyto be competitive on the current data.Employing the same analysis strategy as in the previous section, using a poly-nomial function and the one given by ChPT, we arrive at (cid:113) t phys0 = 0 . forthe former and (cid:113) t phys0 = 0 . for the latter.The advantage of the strategy for the scale setting is a direct value of thelattice spacing from f π K = 147 . at the physical point. This leads to a =0 . , . , . and . for β = 3 . , . , . and . , respectively. The difference to the results in the previous section is adiscretization effect, which is already visible in the plots on the right hand side ofFigure 3. Because of the large statistical error encountered in strategy 2, we will now restrictourselves to the data obtained with the first strategy. One assumption entering theanalysis presented above is that the data presented here can be described by theleading discretization effects of order a at the level of statistical accuracy. To geta handle on this, in Figure 3 the dimensionless product √ t f π K is displayed as afunction of a /t at the symmetric point given by φ = φ phys4 . As we can see, thedata exhibits no deviation from a linear behavior, supporting further the assumptionmade in the ansatz 5.2. The effect of the chiral extrapolation is best studied by forming ratios between thevalue of the observable at the symmetric point and the one at parameters closer tothe chiral limit but at the same lattice spacing. In these ratios, some of the latticesystematics cancels such that for the chiral effects a high sensitivity can be reached.The original ensembles are along trajectories of constant sum of bare quarkmasses, matched at the 1% level using φ at the symmetric point. The results forthe ratios can be found in the left column of Figure 4. As we can see from the lowerplot, the sum of renormalized quark masses is not constant. These masses have beenimproved with a non-perturbatively determined c A , effects of the b -terms have been16eglected. The fact that the renormalized sum is not constant is a discretizationeffect. At β = 3 . their size is so large that they cannot be attributed to contribu-tions linear in the quark masses alone; higher order contributions are noticeable atthis coarse lattice spacing.In the right column of this plot we see the effect of the shift to a constant φ = 1 . , which is close to the physical value. The renormalized quark mass isnow constant on the per-cent level even for the coarsest lattice spacing, with theremaining effects compatible with reasonable values of the b -terms. This also justifiesour choice of aiming for a constant φ versus a constant tr M R : the difference betweenthese two options cannot be resolved by the statistical accuracy of the data and inany case is limited to the per-cent level.The effect on the ratios of t and f π K is less dramatic. In agreement withthe expectation both based on the Taylor expansion of a flavor symmetric quantityaround the symmetric point [6, 31] and ChPT [32], the chiral corrections are tiny, inparticular for the finer lattices. At the coarsest lattice spacing, some deviation fromthe constant behavior is still observed, which is reduced by the shift to φ = const.The chiral effect in f π K is more noticeable, with a correction on the level of − to the physical light quark mass point. Notice, that our data agrees wellwith the logarithms predicted by ChPT in Eq. (2.5). Many observables can be used as a lattice scale, all agree up to effects which comefrom an incomplete description of Nature. Here we neglect for instance quarksheavier than the strange, electromagnetism and isospin breaking. The main strategypursued in the present study is to use t as an intermediate scale, with the approachto the chiral limit along lines of constant φ = 8 t ( m + m π / . Using f π K as physicalinput, this allows the determination of the physical value of t . This strategy ispreferred due to the currently available ensembles in the CLS effort, because theensembles have been tuned with t as a scale.Starting with statistical accuracies for the decay constants on the level of . ,we are able to determine t at the per-cent level √ t = 0 . . (6.1)This compares well to previous determinations using 2+1 flavors by the BMWcollaboration [33] that quotes √ t = 0 . and is also within σ of theQCDSF result [6] . as well as RBC-UKQCD’s value [34] of . .Using 2+1+1 dynamical flavors the MILC [35] and HPQCD [36] collaborations find Note that in Eq. (5.1) we have expanded the denominator to NLO in ChPT. Keeping the fullexpression amounts to higher order effects and produces approximately a 1% shift in the ratioat physical quark masses, which is well captured by the statistical uncertainty of the data in theextrapolations of √ t f π K or af π K . . . . . . . t / t s y m tr M q = const φ = 1 . β = 3 . β = 3 . β = 3 . . . . . . . f π K / f s y m π K NLO ChPT . . . . . . √ t f π K / ( √ t f π K ) s y m . . . . . φ . . . . . . tr M R / (tr M R ) s y m . . . . . φ Figure 4 : Effect of the chiral extrapolation on t , f π K and the sum of perturbativelyimproved PCAC masses. The data is always normalized by the symmetric point. In theleft column, the data as measured on the simulated ensembles, where tr ( M q ) is kept fixed.On the right after the shifts to a constant φ = 1 . has been applied. In particular forthe quark mass sum we observe a significant effect. While the violation of tr ( M Rq ) =constbefore the shift cannot be explained by effects linear in the lattice spacing, we observe thatconstant φ implies constant renormalized quark mass to high accuracy. t = 0 . (cid:0) (cid:1) fm and . , respectively, which might be an effect ofthe number of flavors in the sea as is the two-flavor result √ t = 0 . [37].With additional ensembles becoming available, the analysis presented here willimprove. However, even the accuracies of the current study will already allow toreach a good precision in many physics projects. Acknowledgements.
We are grateful to our CLS colleagues for sharing the gauge field configurations onwhich this work is based. We would like to thank Rainer Sommer for continuous encour-agement and many useful discussions.We acknowledge PRACE for awarding us access to resource FERMI based in Italyat CINECA, Bologna and to resource SuperMUC based in Germany at LRZ, Munich.Furthermore, this work was supported by a grant from the Swiss National SupercomputingCentre (CSCS) under project ID s384. We are grateful for the support received by thecomputer centers.The authors gratefully acknowledge the Gauss Centre for Supercomputing (GCS)for providing computing time through the John von Neumann Institute for Computing(NIC) on the GCS share of the supercomputer JUQUEEN at Jülich SupercomputingCentre (JSC). GCS is the alliance of the three national supercomputing centres HLRS(Universität Stuttgart), JSC (Forschungszentrum Jülich), and LRZ (Bayerische Akademieder Wissenschaften), funded by the German Federal Ministry of Education and Research(BMBF) and the German State Ministries for Research of Baden-Württemberg (MWK),Bayern (StMWFK) and Nordrhein-Westfalen (MIWF).This work was supported by the United States Department of Energy under GrantNo. DE-SC0012704.
References [1] M. Bruno et al.,
Simulation of QCD with N f = 2 + 1 flavors of non-perturbativelyimproved Wilson fermions , JHEP (2015) 043, [ ].[2] M. Lüscher, Properties and uses of the Wilson flow in lattice QCD , JHEP (2010) 071, [ ].[3] D. Toussaint and C. T. H. Davies,
The Omega- and the strange quark mass , Nucl.Phys. Proc. Suppl. (2005) 234–236, [ hep-lat/0409129 ].[4]
Fermilab Lattice, HPQCD, UKQCD, MILC collaboration, C. T. H. Davieset al.,
High precision lattice QCD confronts experiment , Phys. Rev. Lett. (2004)022001, [ hep-lat/0304004 ].[5] R. Sommer, A New way to set the energy scale in lattice gauge theories and itsapplications to the static force and alpha-s in SU(2) Yang-Mills theory , Nucl.Phys.
B411 (1994) 839–854, [ hep-lat/9310022 ].[6] V. G. Bornyakov et al.,
Wilson flow and scale setting from lattice QCD , . ALPHA collaboration, M. Bruno, J. Finkenrath, F. Knechtli, B. Leder andR. Sommer,
Effects of Heavy Sea Quarks at Low Energies , Phys. Rev. Lett. (2015) 102001, [ ].[8] J. Bulava and S. Schaefer,
Improvement of N f = 3 lattice QCD with Wilson fermionsand tree-level improved gauge action , Nucl.Phys.
B874 (2013) 188–197, [ ].[9] G. S. Bali, E. E. Scholz, J. Simeth and W. Söldner,
Lattice simulations with N f = 2 + 1 improved Wilson fermions at a fixed strange quark mass , .[10] K. G. Wilson, Confinement of Quarks , Phys.Rev.
D10 (1974) 2445–2459.[11] W. Bietenholz et al.,
Tuning the strange quark mass in lattice simulations , Phys.Lett.
B690 (2010) 436–441, [ ].[12] T. Bhattacharya, R. Gupta, W. Lee, S. R. Sharpe and J. M. Wu,
Improved bilinearsin lattice QCD with non-degenerate quarks , Phys.Rev.
D73 (2006) 034504,[ hep-lat/0511014 ].[13] J. Gasser and H. Leutwyler,
Chiral Perturbation Theory: Expansions in the Mass ofthe Strange Quark , Nucl. Phys.
B250 (1985) 465–516.[14] S. Aoki et al.,
Review of lattice results concerning low-energy particle physics , .[15] Particle Data Group collaboration, K. A. Olive et al.,
Review of ParticlePhysics , Chin. Phys.
C38 (2014) 090001.[16] J. Gasser and H. Leutwyler,
Light Quarks at Low Temperatures , Phys. Lett.
B184 (1987) 83.[17] G. Colangelo, S. Dürr and C. Haefeli,
Finite volume effects for meson masses anddecay constants , Nucl. Phys.
B721 (2005) 136–174, [ hep-lat/0503014 ].[18] M. Lüscher and S. Schaefer,
Lattice QCD without topology barriers , JHEP (2011) 036, [ ].[19] M. Lüscher and S. Schaefer,
Lattice QCD with open boundary conditions andtwisted-mass reweighting , Comput.Phys.Commun. (2013) 519–528, [ ].[20]
ALPHA collaboration, J. Bulava, M. Della Morte, J. Heitger and C. Wittemeier,
Non-perturbative improvement of the axial current in N f =3 lattice QCD with Wilsonfermions and tree-level improved gauge action , Nucl. Phys.
B896 (2015) 555–568,[ ].[21] M. Bruno, P. Korcyl, T. Korzec, S. Lottini and S. Schaefer,
On the extraction ofspectral quantities with open boundary conditions , PoS
LATTICE2014 (2014) 089,[ ].[22]
ALPHA collaboration, P. Fritzsch, F. Knechtli, B. Leder, M. Marinkovic,S. Schaefer et al.,
The strange quark mass and Lambda parameter of two flavorQCD , Nucl.Phys.
B865 (2012) 397–429, [ ]. ALPHA collaboration, S. Schaefer, R. Sommer and F. Virotta,
Critical slowingdown and error analysis in lattice QCD simulations , Nucl.Phys.
B845 (2011)93–119, [ ].[24] P. Korcyl and G. S. Bali,
Non-perturbative determination of improvement coefficientsusing coordinate space correlators in N f = 2 + 1 lattice QCD , .[25] Y. Taniguchi and A. Ukawa, Perturbative calculation of improvement coefficients toO(g**2a) for bilinear quark operators in lattice QCD , Phys.Rev.
D58 (1998) 114503,[ hep-lat/9806015 ].[26] J. Bulava, M. Della Morte, J. Heitger and C. Wittemeier,
Nonperturbativerenormalization of the axial current in N f = 3 lattice QCD with Wilson fermionsand a tree-level improved gauge action , Phys. Rev.
D93 (2016) 114513, [ ].[27] M. Dalla Brida, T. Korzec, S. Sint and P. Vilaseca,
High precision renormalization ofthe non-singlet axial current in lattice QCD with Wilson quarks , in preparation .[28] I. Campos, P. Fritzsch, C. Pena, D. Preti, A. Ramos and A. Vladikas,
Prospects andstatus of quark mass renormalization in three-flavour QCD , PoS
LATTICE2015 (2016) 249, [ ].[29] A. Hasenfratz, R. Hoffmann and S. Schaefer,
Reweighting towards the chiral limit , Phys.Rev.
D78 (2008) 014515, [ ].[30] J. Finkenrath, F. Knechtli and B. Leder,
One flavor mass reweighting in latticeQCD , Nucl.Phys.
B877 (2013) 441–456, [ ].[31] W. Bietenholz et al.,
Flavour blindness and patterns of flavour symmetry breaking inlattice simulations of up, down and strange quarks , Phys. Rev.
D84 (2011) 054509,[ ].[32] O. Bär and M. Golterman,
Chiral perturbation theory for gradient flow observables , Phys. Rev.
D89 (2014) 034505, Erratum: [Phys. Rev. D (2014) no.9, 099905],[ ].[33] S. Borsanyi et al., High-precision scale setting in lattice QCD , JHEP (2012)010, [ ].[34]
RBC, UKQCD collaboration, T. Blum et al.,
Domain wall QCD with physicalquark masses , Phys. Rev.
D93 (2016) 074505, [ ].[35]
MILC collaboration, A. Bazavov et al.,
Gradient flow and scale setting on MILCHISQ ensembles , Phys. Rev.
D93 (2016) 094510, [ ].[36] R. Dowdall, C. Davies, G. Lepage and C. McNeile, V us from π and K decayconstants in full lattice QCD with physical u, d, s and c quarks , Phys.Rev.
D88 (2013) 074504, [ ].[37] M. Bruno and R. Sommer,
On the N f -dependence of gluonic observables , PoS
LATTICE2013 (2013) 321, [ ].].