Flavour breaking effects in the pseudoscalar meson decay constants
V.G. Bornyakov, R. Horsley, Y. Nakamura, H. Perlt, D. Pleiter, P.E.L. Rakow, G. Schierholz, A. Schiller, H. Stüben, J.M. Zanotti
aa r X i v : . [ h e p - l a t ] D ec ADP-16-46/T1002DESY 16-241Edinburgh 2016/19Liverpool LTH 1116December 14, 2016
Flavour breaking effects in the pseudoscalarmeson decay constants
V. G. Bornyakov a , R. Horsley b , Y. Nakamura c , H. Perlt d ,D. Pleiter e , P. E. L. Rakow f , G. Schierholz g , A. Schiller d ,H. St¨uben h and J. M. Zanotti i – QCDSF-UKQCD Collaborations – a Institute for High Energy Physics, Protvino,142281 Protvino, Russia,Institute of Theoretical and Experimental Physics, Moscow,117259 Moscow, Russia,School of Biomedicine, Far Eastern Federal University,690950 Vladivostok, Russia b School of Physics and Astronomy, University of Edinburgh,Edinburgh EH9 3FD, UK c RIKEN Advanced Institute for Computational Science,Kobe, Hyogo 650-0047, Japan d Institut f¨ur Theoretische Physik, Universit¨at Leipzig,04109 Leipzig, Germany e J¨ulich Supercomputing Centre, Forschungszentrum J¨ulich,52425 J¨ulich, Germany,Institut f¨ur Theoretische Physik, Universit¨at Regensburg,93040 Regensburg, Germany f Theoretical Physics Division, Department of Mathematical Sciences,University of Liverpool, Liverpool L69 3BX, UK g Deutsches Elektronen-Synchrotron DESY,22603 Hamburg, Germany h Regionales Rechenzentrum, Universit¨at Hamburg,20146 Hamburg, Germany i CSSM, Department of Physics, University of Adelaide,Adelaide SA 5005, Australia
Abstract
The SU(3) flavour symmetry breaking expansion in up, down andstrange quark masses is extended from hadron masses to meson decay onstants. This allows a determination of the ratio of kaon to pion decayconstants in QCD. Furthermore when using partially quenched valencequarks the expansion is such that SU(2) isospin breaking effects can alsobe determined. It is found that the lowest order SU(3) flavour symme-try breaking expansion (or Gell-Mann–Okubo expansion) works very well.Simulations are performed for 2+1 flavours of clover fermions at four latticespacings. One approach to determine the ratio | V us /V ud | of Cabibbo-Kobayashi-Maskawa(CKM) matrix elements, as suggested in [1], is by using the ratio of the experi-mentally determined pion and kaon leptonic decay ratesΓ( K + → µ + ν µ )Γ( π + → µ + ν µ ) = (cid:12)(cid:12)(cid:12)(cid:12) V us V ud (cid:12)(cid:12)(cid:12)(cid:12) f K + f π + ! M K + M π + − m µ /M K + − m µ /M π + ! (1 + δ em ) , (1)(where M K + , M π + and m µ are the particle masses, and δ em is an electromagneticcorrection factor). This in turn requires the determination of the ratio of kaonto pion decays constants, f K + /f π + , a non-perturbative task, where the latticeapproach to QCD may be of help. For some recent work see, for example, [2–10].The QCD interaction is flavour-blind and so when neglecting electromagneticand weak interactions, the only difference between the quark flavours comes fromthe mass matrix. In this article we want to examine how this constrains mesondecay matrix elements once full SU (3) flavour symmetry is broken, using thesame methods as we used in [11, 12] for hadron masses. In particular we shallconsider pseudoscalar decay matrix elements and give an estimation for f K /f π and f K + /f π + (ignoring electromagnetic contributions). In lattice simulations with three dynamical quarks there are many paths to ap-proach the physical point where the quark masses take their physical values. Thechoice adopted here is to extrapolate from a point on the SU (3) flavour symmetryline keeping the singlet quark mass m constant, as illustrated in the left panel ofFig. 1, for the case of two mass degenerate quarks m u = m d ≡ m l . This allowsthe development of an SU (3) flavour symmetry breaking expansion for hadronmasses and matrix elements, i.e. an expansion in δm q = m q − m , with m = ( m u + m d + m s ) , (2)(where numerically m = m ). From this definition we have the trivial constraint δm u + δm d + δm s = 0 . (3)2 m l* ,m s* ) m s =m l m=const m lR m sR m l m s (m ,m ) π ++0− π Y +1−1 (ud) I (du) − π K (ds) K (us)K (su) K (sd) η Figure 1:
LH panel: Sketch of the path for the case of two mass degenerate quarks, m u = m d ≡ m l , from a point on the SU (3) flavour symmetric line ( m , m ) to thephysical point denoted with a ∗ : ( m ∗ l , m ∗ s ). RH panel: The pseudoscalar octet meson. The path to the physical quark masses is called the ‘unitary line’ as we expandin the same masses for the sea and valence quarks. Note also that the expansioncoefficients are functions of m only, which provided we keep m = const. reducesthe number of allowed expansion coefficients considerably.As an example of an SU (3) flavour symmetry breaking expansion, [12], weconsider the pseudoscalar masses, and find to next-to-leading-order, NLO, (i.e. O (( δm q ) )). M ( ab ) = M + α ( δm a + δm b )+ β ( δm u + δm d + δm s ) + β ( δm a + δm b ) + β ( δm a − δm b ) + . . . , (4)where m a , m b are quark masses with a, b = u, d, s . This describes the physicalouter ring of the pseudoscalar meson octet (the right panel of Fig. 1). Numericallywe can also in addition consider a fictitious particle, where a = b = s , which wecall η s . We have further extended the expansion to the next-to-next-to-leading orNNLO case, [13]. As the expressions start to become unwieldy, they have beenrelegated to Appendix A. (Octet baryons also have equivalent expansions, [13].)The vacuum is a flavour singlet, so meson to vacuum matrix elements h | b O| M i are proportional to 1 ⊗ ⊗ ⊗ b O is an octetoperator. So the allowed mass dependence of the outer ring octet decay constantsis similar to the allowed dependence of the octet masses. Thus we have f ( ab ) = F + G ( δm a + δm b )+ H ( δm u + δm d + δm s ) + H ( δm a + δm b ) + H ( δm a − δm b ) + . . . . (5)The SU (3) flavour symmetric breaking expansion has the simple property that forany flavour singlet quantity, which we generically denote by X S ≡ X S ( m u , m d , m s )3hen X S ( m + δm u , m + δm d , m + δm s ) = X S ( m, m, m ) + O (( δm q ) ) . (6)This is already encoded in the above pseudoscalar SU (3) flavour symmetric break-ing expansions, or more generally it can be shown, [11, 12], that X S has a sta-tionary point about the SU (3) flavour symmetric line.Here we shall consider X π = ( M K + + M K + M π + + M π − + M K + M K − ) ,X f π = ( f K + + f K + f π + + f π − + f K + f K − ) . (7)(The experimental value of X π is ∼
410 MeV, which sets the unitary range.)There are, of course, many other possibilities such as S = N , Λ, Σ ∗ , ∆, ρ , r , t , w , [11, 12, 14].As a further check, it can be shown that this property also holds using chiralperturbation theory. For example for mass degenerate u and d quark masses andassuming χ PT is valid in the region of the SU (3) flavour symmetric quark masswe find X f π = f " f (3 L + L ) χ − L ( χ ) + O (( δχ l ) ) , (8)where the expansion parameter is given by δχ l = χ − χ l with χ = (2 χ l + χ s ), χ l = B m l , χ s = B m s , f is the pion decay constant in the chiral limit, L i arechiral constants and L ( χ ) = χ/ (4 πf ) × ln( χ/ Λ χ ) is the chiral logarithm. Ineq. (8), as expected, there is an absence of a linear term ∝ δχ l .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). Thisdoes not increase the number of expansion coefficients. Let us denote the valencequark masses by µ q and the expansion parameter as δµ q = µ q − m . Then we have˜ M ( ab ) = 1 + ˜ α ( δµ a + δµ b ) − ( ˜ β + ˜ β )( δm u + δm d + δm s ) + ˜ β ( δµ a + δµ b ) + ˜ β ( δµ a − δµ b ) + . . . , (9)and˜ f ( ab ) = 1 + ˜ G ( δµ a + δµ b ) − ( ˜ H + ˜ H )( δm u + δm d + δm s ) + ˜ H ( δµ a + δµ b ) + ˜ H ( δµ a − δµ b ) + . . . , (10)where in addition to the PQ generalisation we have also formed the ratios ˜ M = M /X π , ˜ α = α/M , . . . and ˜ f = f /X f π , ˜ G = G/F , . . . (see Appendix A for theNNLO expressions). This will later prove useful for the numerical results. We seethat there are mixed sea/valence mass terms at NLO (and higher orders). Theunitary limit is recovered by simply replacing δµ q → δm q .4 The Lattice
We use an O ( a ) non-perturbatively improved clover action with tree level Symanzikglue and mildly stout smeared 2 + 1 clover fermions, [15], for β ≡ /g = 5 . .
50, 5 .
65, 5 .
80 (four lattice spacings). We set µ q = 12 κ val q − κ c ! , (11)giving δµ q = µ q − m = 12 κ val q − κ ! . (12)A κ value along the SU (3) symmetric line is denoted by κ , while κ c is the valuein the chiral limit. Note that practically we do not have to determine κ c , as itcancels in δµ q . (For simplicity we have set the lattice spacing to unity.)We first investigate the constancy of X S in the unitary region. In Fig 2 weshow various choices for X S . It is apparent that over a large range, starting from π /X π ( X S l a t ) S = t S = NS = w S = ρ S = π S = f π ( β , κ )=(5.50,0.120900) X f π lat π /X π ( X S l a t ) S = t S = NS = w S = ρ S = π S = f π ( β , κ ) = (5,80,0.122810) X f π lat Figure 2:
LH panel: X t , X w , X π , X ρ , X N ≈ X , X f π for ( β, κ ) = (5 . , . m = const. line, together with constant fits. Open symbols have M π L ∼ < β, κ ) = (5 . , . the SU (3) flavour symmetric line, reaching down and approaching the physicalpoint, X S appears constant, with very little evidence of curvature. (Althoughnot included in the fits, the open symbols have M π L ∼ ∼
220 MeV.Based on this observation, we determine the path in the quark mass plane byconsidering M π /X S against (2 M K − M π ) /X S . If there is little curvature then we5xpect that 2 M K − M π X S = 3 X π X S − M π X S (13)holds for S = N, ρ, t , w , . . . . In Fig. 3 we show this for ( β, κ ) = (5 . , . π /X S2 ( M K − M π ) / X S S = NS = ρ S = t S = w ( β , κ )=(5.50,0.120900) 0.00 0.05 0.10 0.15 0.20 0.25 0.30M π /X S2 ( M K − M π ) / X S S = NS = ρ S = t S = w ( β , κ )=(5.50,0.120950) Figure 3:
LH panel: (2 M K − M π ) /X S versus M π /X S , S = N , ρ , t , w for ( β, κ ) =(5 . , . SU (3)flavour symmetric line. RH panel: The same for ( β, κ ) = (5 . , . (5 . , . κ is adjustedso that the path goes through (or very close to) the physical value. For examplewe see that from the figure, β = 5 . κ = 0 . κ = 0 . κ and then find the expansioncoefficients. Then use isospin symmetric ‘physical’ masses M ∗ π , M ∗ K to determine δm ∗ l and δm ∗ s . PQ results can help for the first task. As the range of PQ quarkmasses that can then be used is much larger than the unitary range, then thenumerical determination of the relevant expansion coefficients is improved. PQresults were generated about κ , a single sea background, so ˜ γ was not relevant.Also some coefficients (those ∝ ( δµ a − δµ b ) ) often just contributed to noise, sowere then ignored. In Fig. 4 we show ˜ M π against δµ a + δµ b . From the SU (3)flavour breaking expansions the leading-order or LO expansions are just a functionof δµ a + δµ b ; at higher orders, NLO etc. , this is not the case (see eq. (9)). We seethat there is linear behaviour (coincidence of the PQ data with the linear piece)in the masses at least for ˜ M π ∼ < M π ∼ < √ ×
410 MeV ∼
700 MeV. In Fig. 5we show the corresponding results for ˜ f . Again we see similar results for ˜ f as for˜ M ; while our fit is describing the data well, the deviations from linearity occurearlier. Masses are taken from FLAG3, [16]. δµ a + δµ b M ( ab ) / X π pq datapq data (linear piece) ( β , κ )=(5.50,0.120950) −0.02 0.00 0.02 0.04 0.06 0.08 0.10 δµ a + δµ b M ( ab ) / X π pq datapq data (linear piece) ( β , κ ) = (5.65,0.122005) Figure 4:
LH panel: PQ (and unitary) pseudoscalar mass results for ˜ M = M /X π with ( β, κ ) = (5 . , . δµ a + δµ b . The data is givenby circles, while subtracting out the non-linear pieces (using the fit) gives the squares,together with the linear fit. The vertical dashed line is the symmetric point, while thehorizontal dashed line represents the physical ˜ M π . RH panel: Similarly for ( β, κ ) =(5 . , . −0.02 0.00 0.02 0.04 0.06 0.08 0.10 δµ a + δµ b f ( ab ) / X f pq data pq data (linear piece) ( β , κ ) = (5.50,0.120950) −0.02 0.00 0.02 0.04 0.06 0.08 0.10 δµ a + δµ b f ( ab ) / X f pq datapq data (linear piece) ( β , κ ) = (5.65,0.122005) Figure 5:
Similarly for the decay constant, ˜ f = f /X f π . Furthermore the use of PQ results allows for a possibly interesting methodfor fine tuning of κ to be developed. If we slightly miss the starting point on the SU (3) flavour symmetric line, we can also tune κ using PQ results so that we getthe physical values of (say) M ∗ π , X ∗ N and M ∗ K correct. This gives κ , δµ ∗ l , δµ ∗ s . Thephilosophy is that most change is due to a change in valence quark mass, ratherthan sea quark mass. Note that then 2 δµ ∗ l + δµ ∗ s = 0 necessarily (while 2 δm l + δm s always vanishes). For our κ values used here, namely ( β, κ ) = (5 . , . . , . . , . . , . ×
48, 32 × ×
64 and 48 ×
96 lattice volumes respectively) tests show this is a rather smallcorrection 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 horizontal7 δ m l M / X π π K η s sym. pt. ( β , κ ) = (5.80,0.122810) −0.008 −0.006 −0.004 −0.002 0.000 0.002 δ m l f R / X R f π π K η s sym. pt. ( β , κ ) = (5.80,0.122810) Figure 6:
LH panel: Unitary results for ˜ M = M /X π versus δm l for ( β, κ ) =(5 . , . f = f /X f π . lines in Fig. 4. In the LH panel of Fig. 6 we show this range as a function of δm l for ˜ M π , ˜ M K and ˜ M η s , together with the previously found fits. The expressions aregiven from eq. (9), setting δµ → δm q and then a → u , b → d with m u = m d ≡ m l for ˜ M π etc. . Here we clearly observe the typical ‘fan’ behaviour seen in themass of other hadron mass multiplets [12]. As we have mass degeneracy at thesymmetric point, the masses radiate out from this point to their physical values.For both ˜ M and ˜ f the LO completely dominates.As can be seen from Fig. 6 when ˜ M π takes its physical value, ˜ M ∗ π , this deter-mines the physical value δm ∗ l . These are given in Table 1. Note that due to the β δm ∗ l -0.01041(11) -0.008493(33) -0.008348(33) -0.007094(11) Table 1:
Results for δm ∗ l . constraint given in eq. (3) then δm ∗ s = − δm ∗ l . The renormalised and O ( a ) improved axial current is given by [17] A ab ; R µ = Z A A ab ; IMP µ , (14)with A ab ; IMP µ = (cid:16) h b A m + b A ( m a + m b ) i(cid:17) A abµ , A abµ = A abµ + c A ∂ µ P ab , (15)and A abµ = q a γ µ γ q b , P ab = q a γ q b . (16)8sing the axial current we first define matrix elements h | b A | M i = M f , h | d ∂ P | M i = M f (1) , (17)giving for the renormalised pseudoscalar constants f R = Z A c A f (1) f ! (cid:16) h ( b A + b A ) m + b A ( δm a + δm b ) i(cid:17) f . (18)As indicated in Fig. 7, we note that c A is small (compared to unity) and that −0.050.000.05 c A −0.02 0.00 0.02 0.04 0.06 δµ a + δµ b f ( ) ( ab ) /f ( ab ) pq dataunitary data( β , κ )=(5.80,0.122810) Figure 7:
LH panel: Estimate of the c A improvement coefficient using the Schr¨odingerFunctional, [15] as a function of g = 10 /β . The vertical dashed lines denote the β range5 .
40 – 5 .
80. RH panel: The ratio f (1) /f versus δµ a + δµ b for ( β, κ ) = (5 . , . f (1) /f is constant and ∼ O (1) in the unitary region. So for constant m we canabsorb the c A f (1) /f and ( b A + b A ) m terms to give a change in the first coefficient˜ f R ≡ f R X R f π = 1 + (cid:16) ˜ G + b A (cid:17) ( δm a + δm b ) + . . . . (19)For b A (only defined up to terms of O ( a )) we presently take the tree level value, b A = 1 + O ( g ). f K /f π As 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. Usingthe coefficients for the SU (3) flavour breaking expansion for ˜ f as previously de-termined, and then extrapolating to the physical quark masses gives the resultsin Table 2. Finally using these results, we perform the final continuum extrapo-9 a [fm] ˜ f R ∗ π ˜ f R ∗ K ˜ f R ∗ η s ∞ Table 2:
Results for ˜ f R ∗ π , ˜ f R ∗ K , ˜ f R ∗ η s , together with the extrapolated continuum value. [fm ]0.80.91.01.11.21.3 f R / X R f π FLAG3f π /X f π f K /X f π f η s /X f π Figure 8:
The continuum extrapolation of ˜ f R ∗ . The extrapolated values are againgiven as open circles. The converted FLAG3 values, [16], are given as stars. lation, using the lattice spacings given in [14], as shown in Fig. 8. (The fits have χ / dof ∼ . / ∼ . f η s 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 R ∗ K gives a result of f K f π = 1 . , (20)(for simplicity now dropping the superscripts). The first error is statistical; thesecond is an estimate of the combined systematic error due to b A , SU (3) flavourbreaking expansion, finite volume and our chosen path to the physical point asdiscussed in Appendix C. Finally we briefly discuss SU (2) isospin breaking effects. Provided m is kept con-stant, then the SU (3) flavour breaking expansion coefficients ( ˜ α , ˜ G , . . . ) remain10naltered whether we consider 1 + 1 + 1 or 2 + 1 flavours. So although our numer-ical results are for mass degenerate u and d quarks we can use them to discussisospin breaking effects (ignoring electromagnetic corrections). We parameterisethese effects by f K + f π + = f K f π (cid:16) δ SU (2) (cid:17) , and expanding in ∆ m = ( δm d − δm u ) / δm l = ( δm u + δm d ) / δ SU (2) = 23 − f K f π ! − ∆ mδm l , (21)with ∆ mδm l = 32 M K − M K + M π + − ( M K + M K + ) . (22)At the physical point, using the FLAG3, [16], mass values gives ∆ m ∗ /δm ∗ l andhence using our determined value for f K + /f π + , we find δ SU (2) = − . . (23)Alternatively, this gives f K + f π + = 1 . . We have extended our programme of tuning the strange and light quark masses totheir physical values simultaneously by keeping the average quark mass constantfrom pseudoscalar meson masses to pseudoscalar decay constants. As for masseswe find that the SU (3) flavour symmetry breaking expansion, or Gell-Mann–Okubo expansion, works well even at leading order.Further developments to reduce error bars could include another finer latticespacing, as the extrapolation lever arm in a is rather large and presently con-tributes substantially to the errors, and PQ results with sea quark masses notjust at the symmetric point ( κ ) but at other points on the m = const . line. An alternative, but equivalent method is to first determine δm ∗ u , δm ∗ d directly. cknowledgements The numerical configuration generation (using the BQCD lattice QCD program[18]) and data analysis (using the Chroma software library [19]) was carried outon the IBM BlueGene/Qs using DIRAC 2 resources (EPCC, Edinburgh, UK),and at NIC (J¨ulich, Germany), the Lomonosov at MSU (Moscow, Russia) andthe SGI ICE 8200 and Cray XC30 at HLRN (The North-German SupercomputerAlliance) and on the NCI National Facility in Canberra, Australia (supported bythe Australian Commonwealth Government). HP was supported by DFG GrantNo. SCHI 422/10-1 and GS was supported by DFG Grant No. SCHI 179/8-1.PELR was supported in part by the STFC under contract ST/G00062X/1 andJMZ was supported by the Australian Research Council Grant No. FT100100005and DP140103067. We thank all funding agencies.
AppendixA Next-to next-to leading order expansion
We give here the next-to next-to leading order expansion or NNLO expansionfor the octet pseudoscalars and decay constants, which generalise the results ofeqs. (4), (9) and eqs. (5), (10). For the pseudoscalar mesons we have M ( ab ) = M + α ( δµ a + δµ b )+ β ( δm u + δm d + δm s ) + β ( δµ a + δµ b ) + β ( δµ a − δµ b ) + γ δm u δm d δm s + γ ( δµ a + δµ b )( δm u + δm d + δm s )+ γ ( δµ a + δµ b ) + γ ( δµ a + δµ b )( δµ a − δµ b ) , (24)and˜ M ( ab ) = 1 + ˜ α ( δµ a + δµ b ) − ( ˜ β + ˜ β )( δm u + δm d + δm s ) + ˜ β ( δµ a + δµ b ) + ˜ β ( δµ a − δµ b ) +(2˜ γ − γ ) δm u δm d δm s + ˜ γ ( δµ a + δµ b )( δm u + δm d + δm s )+˜ γ ( δµ a + δµ b ) + ˜ γ ( δµ a + δµ b )( δµ a − δµ b ) . (25)where ˜ M ( ab ) = M ( ab ) /X π and for an expansion coefficient ˜ α = α/M , ˜ β i = β i /M , i = 1, 2, and ˜ γ i = γ i /M , i = 1, 2, 3 and we have then redefined ˜ γ by˜ γ − ˜ α ( ˜ β + ˜ β + ˜ β ) → ˜ γ .The SU (3) flavour breaking expansion is identical for the decay constants, wejust replace M → F , α → G , β i → H i , γ i → I i in eq. (24) and ˜ α → ˜ G , ˜ β i → ˜ H i ,˜ γ i → ˜ I i in eq. (25). 12 Correlation functions
On the lattice we extract the pseudoscalar decay constant from two-point corre-lation functions. For large times we expect that C A P ( t ) = 1 V S h X ~x A ( ~x, t ) X ~y P ( ~y, t ) i = 12 M h h | b A | M ih | b P | M i ∗ e − Mt + h | b A † | M i ∗ h | b P † | M i e − M ( T − t ) i = − A A P h e − Mt − e − M ( T − t ) i , (26)and C P P ( t ) = 1 V S h X ~x P ( ~x, t ) X ~y P ( ~y, t ) i = 12 M h h | b P | M ih | b P | M i ∗ e − Mt + h | b P † | M i ∗ h | b P † | M i e − M ( T − t ) i = A P P h e − Mt + e − M ( T − t ) i , (27)where A 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 andkaon. V S is the spatial volume and T is the temporal extent of the lattice. Toincrease the overlap of the operator with the state (where possible) the pseu-doscalar operator has been smeared using Jacobi smearing, and denoted herewith a superscript, S for Smeared. We now set h | b A | M i = M f h | d ∂ P | M i = − sinh M h | b P | M i = M f (1) , (28)where f , f (1) are real and positive. By computing C A P S and C P S P S we find forthe matrix element of b A , M f = √ M × A A P S A P S P S × q A P S P S , (29)and for the matrix element of d ∂ P we obtain from the ratio of the C P P S and C A P S correlation functions f (1) f = sinh M × A P P S A A P S . (30)Some further details and formulae for other decay constants are given in [20, 21]. C Systematic errors
We now consider in this Appendix possible sources of systematic errors.13 ncertainty in b A Presently the improvement coefficient b A is only known perturbatively to leadingorder. We have estimated the uncertainty here by repeating the analysis with b A = 0 and b A = 2. This leads to a systematic error on f K /f π of ∼ . SU (3) flavour breaking expansion We first note that for the unitary range as illustrated in Fig. 6, the ‘ruler test’ in-dicates there is very little curvature. This shows that the SU (3) flavour breakingexpansion is highly convergent. (Each order in the expansion is multiplied by afurther power of | δm l | ∼ . ∼
220 MeV. Such expansions are very good compared to mostapproaches available to QCD. Comparing the LO (linear) approximation withthe non-linear fit gives an estimation of the systematic error. The comparisonyields the estimate to be ∼ .
004 for f K /f π . Finite lattice volume
All the results used in the analysis here have M π L ∼ >
4. We also have generatedsome PQ data for ( β, κ ) = (5 . , . × M π L > f .Making a continuum extrapolation (which is most sensitive to just the β = 5 . ∼ . Path to physical point
As discussed in section 3, we can further tune κ using PQ results to get thephysical values M ∗ π , X ∗ N and M ∗ K correct, to give κ , δµ ∗ l , δµ ∗ s . Setting δµ ∗ ≡ (2 δµ ∗ l + δµ ∗ s ) / δµ ∗ = 12 ˜ α X lat 2 π X lat 2 N , X ∗ π X ∗ N ! − − , (31)(while 2 δm l + δm s is always = 0). This gives for example for β = 5 . δµ ∗ ∼− . δm ∗ l (or δm ∗ s ) by this and making a continuum extrapolation(which is again most sensitive to this point) and comparing the result with thatof eq. (20) results in a systematic error of ∼ . Total systematic error
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