PLB32621S03702693(17)30108910.1016/j.physletb.2017.02.018The Author(s)PhenomenologyFig. 1LH panel: Sketch of the path for the case of two mass degenerate quarks, mu=md≡ml, from a point on the SU(3) flavour symmetric line (m0,m0) to the physical point denoted with a ⁎: (ml⁎,ms⁎). RH panel: The pseudoscalar octet meson.Fig. 1Fig. 2LH panel: Xt02, Xw02, Xπ2, Xρ2, XN2≈XΛ2, Xfπ for (β,κ0)=(5.50,0.120900) along the m‾=const. line, together with constant fits. Open symbols have MπL≲4, where L is the spatial lattice size, and are not included in the fit. The vertical line is the physical point. RH panel: The same for (β,κ0)=(5.80,0.122810).Fig. 2Fig. 3LH panel: (2MK2−Mπ2)/XS2 versus Mπ2/XS2, S=N, ρ, t0, w0 for (β,κ0)=(5.50,0.120900). Stars represent the physical points, the dashed line is the SU(3) flavour symmetric line. RH panel: The same for (β,κ0)=(5.50,0.120950).Fig. 3Fig. 4LH panel: PQ (and unitary) pseudoscalar mass results for M˜2=M2/Xπ2 with (β,κ0)=(5.50,0.120950) against valence quarks δμa+δμb. The data is given by circles, while subtracting out the nonlinear pieces (using the fit) gives the squares, together with the linear fit. The vertical dashed line is the symmetric point, while the horizontal dashed line represents the physical M˜π2. RH panel: Similarly for (β,κ0)=(5.65,0.122005).Fig. 4Fig. 5Similarly for the decay constant, f˜=f/Xfπ.Fig. 5Fig. 6LH panel: Unitary results for M˜2=M2/Xπ2 versus δml for (β,κ0)=(5.80,0.122810). RH panel: Equivalent unitary results for f˜=f/Xfπ.Fig. 6Fig. 7LH panel: Estimate of the cA improvement coefficient using the Schrödinger Functional, [15] as a function of g02=10/β. The vertical dashed lines denote the β range 5.40–5.80. RH panel: The ratio f(1)/f versus δμa+δμb for (β,κ0)=(5.80,0.122810).Fig. 7Fig. 8The continuum extrapolation of f˜R⁎. The extrapolated values are again given as open circles. The converted FLAG3 values, [16], are given as stars.Fig. 8Table 1Results for δml⁎.Table 1β5.405.505.655.80
δml⁎−0.01041(11)−0.008493(33)−0.008348(33)−0.007094(11)
Table 2Results for f˜πR⁎, f˜KR⁎, f˜ηsR⁎, together with the extrapolated continuum value.Table 2βa [fm]f˜πR⁎f˜KR⁎f˜ηsR⁎
5.400.0818(9)0.8739(52)1.0631(26)1.2540(97)
5.500.0740(4)0.8859(34)1.0573(17)1.2328(63)
5.650.0684(4)0.8806(34)1.0599(17)1.2423(62)
5.800.0588(3)0.8827(14)1.0587(07)1.2359(28)

∞00.8862(52)1.0568(26)1.2263(99)
Flavour breaking effects in the pseudoscalar meson decay constantsQCDSF–UKQCD CollaborationsV.G.BornyakovabcR.Horsleydrhorsley@ph.ed.ac.ukY.NakamuraeH.PerltfD.PleiterghP.E.L.RakowiG.SchierholzjA.SchillerfH.StübenkJ.M.ZanottilaInstitute for High Energy Physics, 142281 Protvino, RussiaInstitute for High Energy PhysicsProtvino142281RussiabInstitute of Theoretical and Experimental Physics, 117259 Moscow, RussiaInstitute of Theoretical and Experimental PhysicsMoscow117259RussiacSchool of Biomedicine, Far Eastern Federal University, 690950 Vladivostok, RussiaSchool of BiomedicineFar Eastern Federal UniversityVladivostok690950RussiadSchool of Physics and Astronomy, University of Edinburgh, Edinburgh EH9 3FD, UKSchool of Physics and AstronomyUniversity of EdinburghEdinburghEH9 3FDUKeRIKEN Advanced Institute for Computational Science, Kobe, Hyogo 6500047, JapanRIKEN Advanced Institute for Computational ScienceKobeHyogo6500047JapanfInstitut für Theoretische Physik, Universität Leipzig, 04109 Leipzig, GermanyInstitut für Theoretische PhysikUniversität LeipzigLeipzig04109GermanygJülich Supercomputing Centre, Forschungszentrum Jülich, 52425 Jülich, GermanyJülich Supercomputing CentreForschungszentrum JülichJülich52425GermanyhInstitut für Theoretische Physik, Universität Regensburg, 93040 Regensburg, GermanyInstitut für Theoretische PhysikUniversität RegensburgRegensburg93040GermanyiTheoretical Physics Division, Department of Mathematical Sciences, University of Liverpool, Liverpool L69 3BX, UKTheoretical Physics DivisionDepartment of Mathematical SciencesUniversity of LiverpoolLiverpoolL69 3BXUKjDeutsches ElektronenSynchrotron DESY, 22603 Hamburg, GermanyDeutsches ElektronenSynchrotron DESYHamburg22603GermanykRegionales Rechenzentrum, Universität Hamburg, 20146 Hamburg, GermanyRegionales RechenzentrumUniversität HamburgHamburg20146GermanylCSSM, Department of Physics, University of Adelaide, Adelaide SA 5005, AustraliaCSSMDepartment of PhysicsUniversity of AdelaideAdelaideSA5005AustraliaEditor: A. RingwaldAbstractThe SU(3) flavour symmetry breaking expansion in up, down and strange quark masses is extended from hadron masses to meson decay constants. This allows a determination of the ratio of kaon to pion decay constants in QCD. Furthermore when using partially quenched valence quarks the expansion is such that SU(2) isospin breaking effects can also be determined. It is found that the lowest order SU(3) flavour symmetry breaking expansion (or GellMann–Okubo expansion) works very well. Simulations are performed for 2+1 flavours of clover fermions at four lattice spacings.1IntroductionOne approach to determine the ratio Vus/Vud of Cabibbo–Kobayashi–Maskawa (CKM) matrix elements, as suggested in [1], is by using the ratio of the experimentally determined pion and kaon leptonic decay rates(1)Γ(K+→μ+νμ)Γ(π+→μ+νμ)=VusVud2(fK+fπ+)2MK+Mπ+(1−mμ2/MK+21−mμ2/Mπ+2)2(1+δem) (where MK+, Mπ+ and mμ are the particle masses, and δem is an electromagnetic correction factor). This in turn requires the determination of the ratio of kaon to pion decays constants, fK+/fπ+, a nonperturbative task, where the lattice approach to QCD may be of help. For some recent work see, for example, [2–10].The QCD interaction is flavourblind and so when neglecting electromagnetic and weak interactions, the only difference between the quark flavours comes from the mass matrix. In this article we want to examine how this constrains meson decay matrix elements once full SU(3) flavour symmetry is broken, using the same methods as we used in [11,12] for hadron masses. In particular we shall consider pseudoscalar decay matrix elements and give an estimation for fK/fπ and fK+/fπ+ (ignoring electromagnetic contributions).2ApproachIn lattice simulations with three dynamical quarks there are many paths to approach the physical point where the quark masses take their physical values. The choice adopted here is to extrapolate from a point on the SU(3) flavour symmetry line keeping the singlet quark mass m‾ constant, as illustrated in the left panel of Fig. 1, for the case of two mass degenerate quarks mu=md≡ml. This allows the development of an SU(3) flavour symmetry breaking expansion for hadron masses and matrix elements, i.e. an expansion in(2)δmq=mq−m‾,withm‾=13(mu+md+ms) (where numerically m‾=m0). From this definition we have the trivial constraint(3)δmu+δmd+δms=0. The path to the physical quark masses is called the ‘unitary line’ as we expand in the same masses for the sea and valence quarks. Note also that the expansion coefficients are functions of m‾ only, which provided we keep m‾=const. reduces the number of allowed expansion coefficients considerably.As an example of an SU(3) flavour symmetry breaking expansion, [12], we consider the pseudoscalar masses, and find to nexttoleadingorder, NLO, (i.e. O((δmq)2))(4)M2(ab‾)=M02+α(δma+δmb)M02+β016(δmu2+δmd2+δms2)M02+β1(δma2+δmb2)+β2(δma−δmb)2M02+…, where ma, mb are quark masses with a,b=u,d,s. This describes the physical outer ring of the pseudoscalar meson octet (the right panel of Fig. 1). Numerically we can also in addition consider a fictitious particle, where a=b=s, which we call ηs. We have further extended the expansion to the nexttonexttoleading or NNLO case, [13]. As the expressions start to become unwieldy, they have been relegated to Appendix A. (Octet baryons also have equivalent expansions, [13].)The vacuum is a flavour singlet, so meson to vacuum matrix elements 〈0OˆM〉 are proportional to 1⊗8⊗8 tensors, i.e. 8⊗8 matrices, where Oˆ is an octet operator. So the allowed mass dependence of the outer ring octet decay constants is similar to the allowed dependence of the octet masses. Thus we have(5)f(ab‾)=F0+G(δma+δmb)F0+H016(δmu2+δmd2+δms2)+H1(δma2+δmb2)F0+H2(δma−δmb)2+…. The SU(3) flavour symmetric breaking expansion has the simple property that for any flavour singlet quantity, which we generically denote by XS≡XS(mu,md,ms) then(6)XS(m‾+δmu,m‾+δmd,m‾+δms)=XS(m‾,m‾,m‾)+O((δmq)2). This is already encoded in the above pseudoscalar SU(3) flavour symmetric breaking expansions, or more generally it can be shown, [11,12], that XS has a stationary point about the SU(3) flavour symmetric line.Here we shall consider(7)Xπ2=16(MK+2+MK02+Mπ+2+Mπ−2+MK‾02+MK−2),Xfπ=16(fK++fK0+fπ++fπ−+fK‾0+fK−). (The experimental value of Xπ is ∼410MeV, which sets the unitary range.) There are, of course, many other possibilities such as S=N, Λ, Σ⁎, Δ, ρ, r0, t0, w0, [11,12,14].As a further check, it can be shown that this property also holds using chiral perturbation theory. For example for mass degenerate u and d quark masses and assuming χPT is valid in the region of the SU(3) flavour symmetric quark mass we find(8)Xfπ=f0[1+8f02(3L4+L5)χ‾−3L(χ‾)]+O((δχl)2), where the expansion parameter is given by δχl=χ‾−χl with χ‾=13(2χl+χs), χl=B0ml, χs=B0ms, f0 is the pion decay constant in the chiral limit, Li are chiral constants and L(χ)=χ/(4πf0)2×ln(χ/Λχ2) is the chiral logarithm. In eq. (8), as expected, there is an absence of a linear term ∝δχl.The unitary range is rather small so we introduce PQ or partially quenching (i.e. the valence quark masses can be different to the sea quark masses). This does not increase the number of expansion coefficients. Let us denote the valence quark masses by μq and the expansion parameter as δμq=μq−m‾. Then we have(9)M˜2(ab‾)=1+α˜(δμa+δμb)−(23β˜1+β˜2)(δmu2+δmd2+δms2)+β˜1(δμa2+δμb2)+β˜2(δμa−δμb)2+…, and(10)f˜(ab‾)=1+G˜(δμa+δμb)−(23H˜1+H˜2)(δmu2+δmd2+δms2)+H˜1(δμa2+δμb2)+H˜2(δμa−δμb)2+…, where in addition to the PQ generalisation we have also formed the ratios M˜2=M2/Xπ2, α˜=α/M02, … and f˜=f/Xfπ, G˜=G/F0, … (see Appendix A for the NNLO expressions). This will later prove useful for the numerical results. We see that there are mixed sea/valence mass terms at NLO (and higher orders). The unitary limit is recovered by simply replacing δμq→δmq.3The latticeWe use an O(a) nonperturbatively improved clover action with tree level Symanzik glue and mildly stout smeared 2+1 clover fermions, [14,15], for β≡10/g02=5.40, 5.50, 5.65, 5.80 (four lattice spacings). We set(11)μq=12(1κqval−1κ0c), giving(12)δμq=μq−m‾=12(1κqval−1κ0). A κ value along the SU(3) symmetric line is denoted by κ0, while κ0c is the value in the chiral limit. Note that practically we do not have to determine κ0c, as it cancels in δμq. (For simplicity we have set the lattice spacing to unity.)We first investigate the constancy of XS in the unitary region. In Fig. 2 we show various choices for XS. It is apparent that over a large range, starting from the SU(3) flavour symmetric line, reaching down and approaching the physical point, XS appears constant, with very little evidence of curvature. (Although not included in the fits, the open symbols have MπL∼3–4 and also do not show curvature.) Presently our available pion masses reach down to ∼220MeV.Based on this observation, we determine the path in the quark mass plane by considering Mπ2/XS2 against (2MK2−Mπ2)/XS2. If there is little curvature then we expect that(13)2MK2−Mπ2XS2=3Xπ2XS2−2Mπ2XS2 holds for S=N,ρ,t0,w0,… . In Fig. 3 we show this for (β,κ0)=(5.50,0.120900), (5.50,0.120950). We see that this is indeed the case. In addition κ0 is adjusted so that the path goes through (or very close to) the physical value. For example we see that from the figure, β=5.50, κ0=0.120950 is very much closer to this path than κ0=0.120900, [14].The programme is thus first to determine κ0 and then find the expansion coefficients. Then use11Masses are taken from FLAG3, [16]. isospin symmetric ‘physical’ masses Mπ⁎, MK⁎ to determine δml⁎ and δms⁎. PQ results can help for the first task. As the range of PQ quark masses that can then be used is much larger than the unitary range, then the numerical determination of the relevant expansion coefficients is improved. PQ results were generated about κ0, a single sea background, so γ˜1 was not relevant. Also some coefficients (those ∝(δμa−δμb)2) often just contributed to noise, so were then ignored. In Fig. 4 we show M˜π2 against δμa+δμb. From the SU(3) flavour breaking expansions the leadingorder or LO expansions are just a function of δμa+δμb; at higher orders, NLO etc., this is not the case (see eq. (9)). We see that there is linear behaviour (coincidence of the PQ data with the linear piece) in the masses at least for M˜π2≲3 or Mπ≲3×410MeV∼700MeV. In Fig. 5 we show the corresponding results for f˜. Again we see similar results for f˜ as for M˜2; while our fit is describing the data well, the deviations from linearity occur earlier.Furthermore the use of PQ results allows for a possibly interesting method for fine tuning of κ0 to be developed. If we slightly miss the starting point on the SU(3) flavour symmetric line, we can also tune κ0 using PQ results so that we get the physical values of (say) Mπ⁎, XN⁎ and MK⁎ correct. This gives κ0, δμl⁎, δμs⁎. The philosophy is that most change is due to a change in valence quark mass, rather than sea quark mass. Note that then 2δμl⁎+δμs⁎≠0 necessarily (while 2δml+δms always vanishes). For our κ0 values used here, namely (β,κ0)=(5.40,0.119930), (5.50,0.120950), (5.65,0.122005), (5.80,0.122810), [14] (on 243×48, 323×64, 323×64 and 483×96 lattice volumes respectively) tests show this is a rather small correction and we shall use this as part of the systematic error, see Appendix C.Of course the unitary range is much smaller, as can be seen from the horizontal lines in Fig. 4. In the LH panel of Fig. 6 we show this range as a function of δml for M˜π2, M˜K2 and M˜ηs2, together with the previously found fits. The expressions are given from eq. (9), setting δμ→δmq and then a→u, b→d with mu=md≡ml for M˜π2 etc. . Here we clearly observe the typical ‘fan’ behaviour seen in the mass of other hadron mass multiplets [12]. As we have mass degeneracy at the symmetric point, the masses radiate out from this point to their physical values. For both M˜2 and f˜ the LO completely dominates.As can be seen from the LH panel of Fig. 6 when M˜π takes its physical value, M˜π⁎, this determines the physical value δml⁎. These are given in Table 1. Note that due to the constraint given in eq. (3) then δms⁎=−2δml⁎.4Decay constantsThe renormalised and O(a) improved axial current is given by [17](14)Aμab;R=ZAAμab;IMP, with(15)Aμab;IMP=(1+[b‾Am‾+12bA(ma+mb)])Aμab,Aμab=Aμab+cA∂μPab, and(16)Aμab=q‾aγμγ5qb,Pab=q‾aγ5qb. Using the axial current we first define matrix elements(17)〈0Aˆ4M〉=Mf,〈0∂4PˆM〉=Mf(1), giving for the renormalised pseudoscalar constants(18)fR=ZA(1+cAf(1)f)(1+[(b‾A+bA)m‾+12bA(δma+δmb)])f. As indicated in Fig. 7, we note that cA is small (compared to unity) and that f(1)/f is constant and ∼O(1) in the unitary region. So for constant m‾ we can absorb the cAf(1)/f and (b‾A+bA)m‾ terms to give a change in the first coefficient(19)f˜R≡fRXfπR=1+(G˜+12bA)(δma+δmb)+…. For bA (only defined up to terms of O(a)) we presently take the tree level value, bA=1+O(g02).5Results5.1fK/fπAs demonstrated in the RH panel of Fig. 6, we again expect LO behaviour for SU(3) flavour symmetry breaking for f˜ to dominate in the unitary region. Using the coefficients for the SU(3) flavour breaking expansion for f˜ as previously determined, and then extrapolating to the physical quark masses gives the results in Table 2. Finally using these results, we perform the final continuum extrapolation, using the lattice spacings given in [14], as shown in Fig. 8. (The fits have χ2/dof∼3.3/2∼1.6.) For comparison, the FLAG3 values, [16], are shown as stars. (Note that although fηs helps in determining the expansion coefficients, there is no further information to be found from the various extrapolated values.) Continuum values are also given in Table 2. Converting f˜KR⁎ gives a result of(20)fKfπ=1.192(10)(13) (for simplicity now dropping the superscripts). The first error is statistical; the second is an estimate of the combined systematic error due to bA, SU(3) flavour breaking expansion, finite volume and our chosen path to the physical point as discussed in Appendix C.5.2Isospin breaking effectsFinally we briefly discuss SU(2) isospin breaking effects. Provided m‾ is kept constant, then the SU(3) flavour breaking expansion coefficients (α˜, G˜,…) remain unaltered whether we consider 1+1+1 or 2+1 flavours. So although our numerical results are for mass degenerate u and d quarks we can use them to discuss isospin breaking effects (ignoring electromagnetic corrections). We parameterise these22An alternative, but equivalent method is to first determine δmu⁎, δmd⁎ directly. effects byfK+fπ+=fKfπ(1+12δSU(2)), and expanding in Δm=(δmd−δmu)/2 about the average light quark mass δml=(δmu+δmd)/2 gives, using the LO expansions (which from Figs. 4, 5 or more particularly Fig. 6, have been shown to work well)(21)δSU(2)=23(1−(fKfπ)−1)Δmδml, with(22)Δmδml=32MK02−MK+2Mπ+2−12(MK02+MK+2). At the physical point, using the FLAG3, [16], mass values gives Δm⁎/δml⁎ and hence using our determined value for fK+/fπ+, we find(23)δSU(2)=−0.0042(2)(2). Alternatively, this givesfK+fπ+=1.190(10)(13).6ConclusionsWe have extended our programme of tuning the strange and light quark masses to their physical values simultaneously by keeping the average quark mass constant from pseudoscalar meson masses to pseudoscalar decay constants. As for masses we find that the SU(3) flavour symmetry breaking expansion, or GellMann–Okubo expansion, works well even at leading order.Further developments to reduce error bars could include another finer lattice spacing, as the extrapolation lever arm in a2 is rather large and presently contributes substantially to the errors, and PQ results with sea quark masses not just at the symmetric point (κ0) but at other points on the m‾=const. line.AcknowledgementsThe numerical configuration generation (using the BQCD lattice QCD program [18]) and data analysis (using the Chroma software library [19]) was carried out on the IBM BlueGene/Qs using DIRAC 2 resources (EPCC, Edinburgh, UK), and at NIC (Jülich, Germany), the Lomonosov at MSU (Moscow, Russia) and the SGI ICE 8200 and Cray XC30 at HLRN (The NorthGerman Supercomputer Alliance) and on the NCI National Facility in Canberra, Australia (supported by the Australian Commonwealth Government). HP was supported by DFG Grant No. SCHI 422/101 and GS was supported by DFG Grant No. SCHI 179/81. PELR was supported in part by the STFC under contract ST/G00062X/1 and JMZ was supported by the Australian Research Council Grant No. FT100100005 and DP140103067. We thank all funding agencies.Appendix ANexttonextto leading order expansionWe give here the nexttonextto leading order PQ expansion or NNLO PQ expansion for the octet pseudoscalars and decay constants, which generalise the results of eqs. (4), (9) and eqs. (5), (10). For the pseudoscalar mesons we have(24)M2(ab‾)=M02+α(δμa+δμb)+β016(δmu2+δmd2+δms2)+β1(δμa2+δμb2)+β2(δμa−δμb)2+γ0δmuδmdδms+γ1(δμa+δμb)(δmu2+δmd2+δms2)+γ2(δμa+δμb)3+γ3(δμa+δμb)(δμa−δμb)2, and(25)M˜2(ab‾)=1+α˜(δμa+δμb)−(23β˜1+β˜2)(δmu2+δmd2+δms2)+β˜1(δμa2+δμb2)+β˜2(δμa−δμb)2+(2γ˜2−6γ˜3)δmuδmdδms+γ˜1(δμa+δμb)(δmu2+δmd2+δms2)+γ˜2(δμa+δμb)3+γ˜3(δμa+δμb)(δμa−δμb)2, where M˜2(ab‾)=M2(ab‾)/Xπ2 and for an expansion coefficient α˜=α/M02, β˜i=βi/M02, i=1, 2, and γ˜i=γi/M02, i=1, 2, 3 and we have then redefined γ˜1 by γ˜1−α˜(16β˜0+23β˜1+β˜2)→γ˜1.The SU(3) flavour breaking expansion is identical for the decay constants, we just replace M02→F0, α→G, βi→Hi, γi→Ii in eq. (24) and α˜→G˜, β˜i→H˜i, γ˜i→I˜i in eq. (25).Appendix BCorrelation functionsOn the lattice we extract the pseudoscalar decay constant from twopoint correlation functions. For large times we expect that(26)CA4P(t)=1VS〈∑x→A4(x→,t)∑y→P(y→,t)〉=12M[〈0Aˆ4M〉〈0PˆM〉⁎e−Mt+〈0Aˆ4†M〉⁎〈0Pˆ†M〉e−M(T−t)]=−AA4P[e−Mt−e−M(T−t)], and(27)CPP(t)=1VS〈∑x→P(x→,t)∑y→P(y→,t)〉=12M[〈0PˆM〉〈0PˆM〉⁎e−Mt+〈0Pˆ†M〉⁎〈0Pˆ†M〉e−M(T−t)]=APP[e−Mt+e−M(T−t)], where A4 and P are given in eq. (16). We have suppressed the quark indices, so the equations with appropriate modification are valid for both the pion and kaon. VS is the spatial volume and T is the temporal extent of the lattice. To increase the overlap of the operator with the state (where possible) the pseudoscalar operator has been smeared using Jacobi smearing, and denoted here with a superscript, S for Smeared. We now set(28)〈0Aˆ4M〉=Mf〈0∂4PˆM〉=−sinhM〈0PˆM〉=Mf(1), where f, f(1) are real and positive. By computing CA4PS and CPSPS we find for the matrix element of Aˆ4,(29)Mf=2M×AA4PSAPSPS×APSPS, and for the matrix element of ∂4Pˆ we obtain from the ratio of the CPPS and CA4PS correlation functions(30)f(1)f=sinhM×APPSAA4PS. Some further details and formulae for other decay constants are given in [20,21].Appendix CSystematic errorsWe now consider in this Appendix possible sources of systematic errors.Uncertainty in bAPresently the improvement coefficient bA is only known perturbatively to leading order. We have estimated the uncertainty here by repeating the analysis with bA=0 and bA=2. This leads to a systematic error on fK/fπ of ∼0.008.SU(3) flavour breaking expansionWe first note that for the unitary range as illustrated in Fig. 6, the ‘ruler test’ indicates there is very little curvature. This shows that the SU(3) flavour breaking expansion is highly convergent. (Each order in the expansion is multiplied by a further power of δml∼0.01.) This is also indicated in Fig. 2, where our lowest pion mass there is ∼220MeV. Such expansions are very good compared to most approaches available to QCD. Comparing the LO (linear) approximation with the nonlinear fit gives an estimation of the systematic error. The comparison yields the estimate to be ∼0.004 for fK/fπ.Finite lattice volumeAll the results used in the analysis here have MπL≳4. We also have generated some PQ data for (β,κ0)=(5.80,0.122810) on a smaller lattice volume – 323×64. (This still has MπL>4.) Performing the analysis leads to small changes in f˜. Making a continuum extrapolation (which is most sensitive to just the β=5.80 point) and comparing the result with that of eq. (20) results in a systematic error of ∼0.005.Path to physical pointAs discussed in section 3, we can further tune κ0 using PQ results to get the physical values Mπ⁎, XN⁎ and MK⁎ correct, to give κ0, δμl⁎, δμs⁎. Setting δμ‾⁎≡(2δμl⁎+δμs⁎)/3 then at LO this average is given by(31)δμ‾⁎=12α˜((Xπlat2XNlat2/Xπ⁎2XN⁎2)−1−1) (while 2δml+δms is always =0). This gives for example for β=5.80, δμ‾⁎∼−0.0001. Changing δml⁎ (or δms⁎) by this and making a continuum extrapolation (which is again most sensitive to this point) and comparing the result with that of eq. 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