High precision renormalization of the flavour non-singlet Noether currents in lattice QCD with Wilson quarks
Mattia Dalla Brida, Tomasz Korzec, Stefan Sint, Pol Vilaseca
HHigh precision renormalization of the flavour non-singletNoether currents in lattice QCD with Wilson quarks
Mattia Dalla Brida a , Tomasz Korzec b , Stefan Sint c , Pol Vilaseca d a Dipartimento di Fisica, Università di Milano-Bicocca and INFN, sezione di Milano-Bicocca,Piazza della Scienza 3, I-20126 Milano, Italy b Department of Physics, Bergische Universität Wuppertal, Gaußstr. 20, 42119 Wuppertal, Germany c School of Mathematics, Trinity College Dublin, Dublin 2, Ireland d Dipartimento di Fisica, Università La Sapienza di Roma and INFN, sezione di Roma,Piazzale A. Moro 2, I-00185, Roma, Italy
Abstract
We determine the non-perturbatively renormalized axial current for O( a ) improvedlattice QCD with Wilson quarks. Our strategy is based on the chirally rotated Schrödingerfunctional and can be generalized to other finite (ratios of) renormalization constants whichare traditionally obtained by imposing continuum chiral Ward identities as normalizationconditions. Compared to the latter we achieve an error reduction by up to one order ofmagnitude. Our results have already enabled the setting of the scale for the N f = 2 + 1 CLS ensembles [1] and are thus an essential ingredient for the recent α s determination bythe ALPHA collaboration [2]. In this paper we shortly review the strategy and presentour results for both N f = 2 and N f = 3 lattice QCD, where we match the β -values ofthe CLS gauge configurations. In addition to the axial current renormalization, we alsopresent precise results for the renormalized local vector current. Keywords:
Non-perturbative renormalization, Lattice QCD, Chiral symmetry, Wilson fermions a r X i v : . [ h e p - l a t ] J a n Introduction
Lattice regularizations with Wilson type fermions [3] are widely used in current latticeQCD simulations [4–10]. The ultra-locality of the action enables numerical efficiency andthus access to a wide range of lattice spacings and spatial volumes. Furthermore, Wilsonfermions maintain the full flavour symmetry of the continuum action, as well as the discretesymmetries such as parity, charge conjugation and time reversal. Unitarity is either realizedexactly, or, in the case of Symanzik-improved actions, approximately up to cutoff effectswhich vanish in the continuum limit.The price to pay for these advantages consists in the explicit breaking of all chiralsymmetries by the Wilson term in the action. Well-known consequences include the ad-ditive renormalization of quark masses, the mixing under renormalization of compositeoperators in different chiral multiplets and discretization effects linear in a , the latticespacing. Furthermore, the Noether currents of chiral symmetry are no longer protectedagainst renormalization.The matrix elements of the axial Noether currents between pion or kaon states andthe vacuum, parametrized by the decay constants f π,K , e.g. (cid:104) | A udµ (0) | π − , p (cid:105) = ip µ f π , A udµ ( x ) = ψ u ( x ) γ µ γ ψ d ( x ) , (1.1)can be related to the measured life times of pions and kaons. The decay constants are finitein the chiral limit, can be precisely measured in numerical simulations and are ideally suitedto set the scale in physical units. In order to achieve this with Wilson quarks one needs todetermine the correctly renormalized axial currents, ( A R ) f f µ ( x ) = Z A A f f µ , (1.2)(with flavour indices f , = u, d, s ), which are to be inserted into the matrix elements.Of course it is desirable that the error of the matrix elements is not dominated by theuncertainty of the current normalization constant.Over the last 30 years many efforts have been made to control the consequences ofexplicit chiral symmetry breaking with Wilson quarks. The main strategy consists inimposing continuum chiral symmetry relations as normalization conditions at finite latticespacing [11, 12]. This is usually done using chiral Ward identities, which follow from aninfinitesimal chiral change of variables in the QCD path integral. An example is thePCAC relation which determines the additive quark mass renormalization constant, asthe “critical value” of the bare mass parameter, where the axial current is conserved. Thefact that chiral symmetry is fully recovered only in the continuum limit implies that thechoice of normalization condition matters at the cutoff level; at a fixed value of the latticespacing the numerical results may occasionally differ substantially between any two suchchoices. Rather than interpreting this scatter as a systematic error, the modern approachconsists in choosing a particular normalization condition and in fixing all dimensionfulparameters (such as momenta or distances or background fields) in terms of a physicalscale. This defines a so-called “line of constant physics” (LCP), along which the continuumlimit is taken. As the lattice spacing a (or, equivalently, the bare coupling, g = 6 /β ), isvaried, this defines a function Z A = Z A ( β ) . Obviously, another choice for the LCP willresult in a different function Z (cid:48) A ( β ) . However, their difference will be, within errors, a2mooth function of β which vanishes asymptotically ∝ a or ∝ a if O( a ) improvement isimplemented. Hence, following a LCP ensures that cutoff effects are smooth functions of β and the choice of LCP becomes irrelevant in the continuum limit. Adopting this viewpoint,the relevant systematic error is therefore determined by the precision to which a chosenLCP can be followed.In this paper we apply a recently developed method to lattice QCD with N f = 2 and N f = 3 flavours, matching the lattice actions chosen by the CLS initiative [8, 10].Our method is based on the chirally rotated Schödinger functional ( χ SF ) [13, 14]. Thetheoretical foundation of this framework has been explained in [14] and it has passeda number of perturbative and non-perturbative tests [15–20]. In contrast to the Wardidentity method the axial current renormalization conditions follow from a finite chiralrotation in the massless QCD path integral with Schrödinger functional (SF) boundaryconditions. The renormalization constants are then obtained from ratios of simple 2-pointfunctions. For the axial current, this represents a significant advantage over the Wardidentity method [12, 21, 22] which involves 3- and 4-point functions. Hence, we observea dramatic improvement in the attainable statistical precision for Z A and some care isrequired to ensure that systematic errors are under control at a similar level of precision.We also discuss the normalization procedure for the local vector current. While flavoursymmetry remains unbroken on the lattice with (mass-degenerate) Wilson quarks, thecorresponding Noether current lives on neighbouring lattice points connected by a gaugelink, so that the use of the local vector current is often more practical.This paper is organized as follows: after a short reminder of the χ SF correlationfunctions in the continuum and the normalization conditions derived from them in sect. 2,we define in sect. 3 a couple of different LCPs which we have followed. We then presentthe Z A and Z V determinations for lattice QCD with N f = 2 and N f = 3 quark flavoursin sects. 4 and 5, respectively, together with various tests we have carried out. Sect. 6contains a summary of the main results of this work and some concluding remarks. Finally,the paper ends with three technical appendices: appendix A collects the parameters andresults of the simulations, appendix B provides a detailed discussion on the systematic errorestimates for our determinations, and appendix C gathers our set of chosen fit functionswhich smoothly interpolate our Z A , V results in β .The main results for N f = 2 are collected in table 4, while those for N f = 3 are given intables 6 and 7. These results can be directly applied to data obtained from the CLS 2- and3-flavour configurations, respectively [8, 10]. The N f = 3 results have, in fact, already beenused, and enabled the precision CLS scale setting in ref. [1] and the accurate quark-massrenormalization of ref. [23]. We start by considering massless two-flavour continuum QCD. The Euclidean space-timeis taken to be a hyper-cylinder of volume L with Schrödinger functional boundary con-ditions [24, 25]. In particular, in the Euclidean time direction, the quark and anti-quarkfields satisfy, P + ψ ( x ) | x =0 = 0 = ψ ( x ) P − | x =0 , (2.1)3nd similarly at time x = L with the change P ± → P ∓ . The SU(2) × SU(2) chiral andflavour symmetry leads to conserved isovector Noether currents, given by A aµ ( x ) = ψ ( x ) γ µ γ τ a ψ ( x ) , V aµ ( x ) = ψ ( x ) γ µ τ a ψ ( x ) , (2.2)with Pauli matrices τ a and isospin index a = 1 , , . SF correlation functions of thesecurrents with isovector boundary sources O a and O ak have been defined in [26, 27] and aregiven by (cid:104) A a ( x ) O b (cid:105) = − δ ab f A ( x ) , (cid:88) k =1 (cid:104) V ak ( x ) O bk (cid:105) = − δ ab k V ( x ) . (2.3)Passing from the isospin notation to fields with definite flavour assignments, A f f µ ( x ) = ψ f ( x ) γ µ γ ψ f ( x ) , V f f µ ( x ) = ψ f ( x ) γ µ ψ f ( x ) , (2.4)and similarly for the boundary sources, the correlation functions for isospin indices a = 1 , ,can be written in terms of the flavour off-diagonal fields, f A ( x ) = − (cid:104) A ud ( x ) O du (cid:105) , k V ( x ) = − (cid:88) k =1 (cid:104) V udk ( x ) O duk (cid:105) . (2.5)For the flavour diagonal fields in the isospin a = 3 components, e.g. A µ = (cid:16) A uuµ − A ddµ (cid:17) , (2.6)one may use flavour symmetry to write f A ( x ) = − (cid:104) A uu (cid:48) ( x ) O u (cid:48) u (cid:105) , (2.7)and analogously for k V . Note that the additional up-type flavour u (cid:48) is merely a notationaldevice to indicate the fermionic contractions taken into account when applying Wick’stheorem. Indeed, the sum of all the disconnected contributions for the flavour diagonal a = 3 components of SF correlation functions cancels exactly due to flavour symmetry.We now apply a flavour diagonal chiral rotation to the fields, ψ → exp (cid:16) i α γ τ (cid:17) ψ, ψ → ψ exp (cid:16) i α γ τ (cid:17) . (2.8)Choosing the rotation angle α = π/ then leads to the chirally rotated SF boundaryconditions, ˜ Q + ψ ( x ) | x =0 = 0 = ψ ( x ) ˜ Q + | x =0 , (2.9)with projectors ˜ Q ± = (1 ± iγ γ τ ) . Analogous boundary conditions with reverted projec-tors are obtained at x = L . Applying the same chiral field rotation to the axial currents, A udµ ( x ) → − iV udµ ( x ) , A uuµ ( x ) → A uuµ ( x ) , (2.10)one obtains either a vector current or remains with an axial current, depending on theflavour assignments. If the chiral rotation of the field variables is performed as a changeof variables in the functional integral, one arrives at the formal continuum identities f A = g uu (cid:48) A = − ig ud V , k V = l uu (cid:48) V = − il ud A , (2.11)4here the g - and l -functions are defined with χ SF boundary conditions, eqs. (2.9), forinstance g f f A ( x ) = − (cid:104) A f f ( x ) Q f f (cid:105) ( ˜ Q + ) . (2.12)Here, the boundary operators Q f f denote the chirally rotated versions of their SF coun-terparts, O f f . For the complete expressions and further details we refer to ref. [20].Regarding the case of QCD with N f = 3 quark flavours we note that the very samesteps can be taken provided the massless third quark does not take part in the chiral rota-tion and thus remains with standard SF boundary conditions [14]. Correlation functionsare then considered for the doublet fields only, i.e. the third quark never appears as avalence quark. In the lattice regularized theory with Wilson type quarks, relations such as (2.11) can onlybe expected to hold after renormalization and up to cutoff effects. One first has to ensurethat massless QCD with χ SF boundary conditions has been correctly regularized. Thisis achieved by tuning the bare mass parameter m to its critical value, m cr , where theaxial current is conserved, and by tuning a boundary counterterm coefficient z f such thatphysical parity is restored (cf. [20] for more details). In terms of the bare χ SF correlationfunctions one may choose the two conditions, m = ˜ ∂ g ud A ( x )2 g ud P ( x ) (cid:12)(cid:12)(cid:12)(cid:12) x = L/ = 0 , g ud A ( L/
2) = 0 (2.13)(with ˜ ∂ the symmetric lattice derivative). The division by the pseudo-scalar correlationfunction g ud P is not really necessary, however it is done for convenience, as it gives rise tothe definition of a (bare) PCAC quark mass m . Solutions to these equations define m cr and z ∗ f as functions of the bare coupling g , and the lattice size, L/a .Once the lattice regularization is correctly implemented, we expect e.g. Z A Z ζ g uu (cid:48) A ( x ) = − iZ V Z ζ g ud V ( x ) + O( a ) , (2.14)where Z ζ renormalizes a boundary quark or anti-quark field [14, 26, 28] and Z A , V arethe current normalization constants of interest. Requiring such identities to hold exactlyat finite lattice spacing thus fixes the relative normalization of axial and vector current.Replacing the latter by the exactly conserved lattice vector current (cid:101) V µ ( x ) (cf. ref. [20]), forwhich Z (cid:101) V = 1 , one may obtain Z A from either one of the ratios R g A = − ig ud (cid:101) V ( x ) g uu (cid:48) A ( x ) (cid:12)(cid:12)(cid:12)(cid:12) x = L/ or R l A = il uu (cid:48) (cid:101) V ( x ) l ud A ( x ) (cid:12)(cid:12)(cid:12)(cid:12) x = L/ . (2.15)Assuming that the parameters x (here set to L/ ), the boundary angle θ [29], the back-ground gauge field [24], and the precise definition for the zero mass and α = π/ point (2.13)are fixed, we define, on an ( L/a ) lattice and for a given bare coupling g = 6 /β , Z g,l A ( β, L/a ) = R g,l A . (2.16)5inally the choice of a line of constant physics (cf. section 3) defines a smooth function ( L/a )( β ) such that the normalization constants become functions of β alone, with thedifference between any two definitions vanishing smoothly with a rate ∝ a .We also comment on the appearance of a second up-type flavour u (cid:48) in (2.15). Whenapplying the chiral rotation (2.8) to the diagonal components of f A , the disconnecteddiagrams are mapped to disconnected diagrams on the χ SF side which can be shown toadd up to a pure cutoff effect. Their omission is thus perfectly legitimate, even if theformulation of the renormalization conditions then has an element of partial quenching toit. The situation is comparable with the Ward identity method in two-flavour QCD [12,21],where a fictitious s -quark can be introduced to eliminate the disconnected diagrams.Even though there exists a conserved vector current, in practice the local current isoften used and then requires renormalization, too. Its renormalization constant can beobtained from, R g V = g ud (cid:101) V ( x ) g ud V ( x ) (cid:12)(cid:12)(cid:12)(cid:12) x = L/ or R l V = l uu (cid:48) (cid:101) V ( x ) l uu (cid:48) V ( x ) (cid:12)(cid:12)(cid:12)(cid:12) x = L/ . (2.17)The same remarks as for the axial current normalization apply here, and with definitechoices for all parameters we set, Z g,l V ( β, L/a ) = R g,l V . (2.18)As in the case of the axial current normalization conditions, only 2-point functions arerequired, which connect the boundary quark bilinear sources with the currents in thebulk. This is a major advantage over the Ward identity method [12, 21] where 3- and4-point functions are required. Hence, one expects better statistical precision from thesimpler 2-point functions, and this will be confirmed below. Furthermore, as discussedin [20], the cutoff effects in the ratios are O( a ), due to the mechanism of automatic O( a )improvement [30], even if the PCAC mass and the axial current are not O( a ) improved bythe counterterm ∝ c A [26], or if the vector currents are not improved by the correspondingcounterterms ∝ c V , c (cid:101) V [27, 31].Finally, we emphasize that similar renormalization conditions can be devised for otherfinite renormalization constants. An interesting example is the ratio Z P /Z S , where Z P and Z S are the pseudo-scalar and scalar renormalization constant, respectively. We refer thereader to ref. [20] for more details. A line of constant physics requires to specify a physical (length) scale r which is keptfixed as the continuum limit is taken. A typical choice would be the pion decay constant, r = 1 /f π , either at the physical quark masses or in the chiral limit. Once calculatedfor a range of lattice spacings, this scale defines a function ( r/a )( β ) of the bare coupling β = 6 /g which fixes the lattice spacing a in units of the chosen physical scale. Choosingthe spatial lattice extent L/a , at a given beta, such that ( L/a )( β )( r/a )( β ) = L/r = C r (3.1)6with a numerical constant C r ) then fixes the spatial size of the finite volume system inunits of r . In practice we will choose C r such that the physical size of L will be somewhatlarger than half a femto metre. Note that this equation can be read in two ways: first,if one fixes C r and then chooses a set of β -values for which r/a is known, one obtains acorresponding set of values ( L/a )( β ) , which will not necessarily be integers. To evaluate thenormalization constants at these non-integer lattice sizes then requires some interpolationof results from neighbouring integer L/a -values at the same β . Alternatively, one couldchoose a set of integer L/a -values such that a choice for C r will imply a set of β -values.In general this means that the data for r/a may have to be interpolated in β . We willhere choose the first option, with the set of β -values taken over from the large volumesimulations by the CLS project [8, 10].Having set the scale one needs to ensure the correlation functions are calculated in thedesired situation of massless QCD and for the chosen chirally rotated boundary conditionsat α = π/ . This means one needs to tune the bare quark mass am and z f as functionsof β . We will discuss this in more detail below. Finally, the correlation functions dependon kinematic parameters, such as x or background field parameters such as θ . We havealready set x = L/ in eqs. (2.15,2.17) and we choose θ = 0 and work with vanishingSU(3) background field.With these parameter choices we will have, for a given r and C r in eq. (3.1), twodefinitions each for Z A and Z V , namely Z g,l A , V ( β ) = R g,l A , V ( β, a/L ) (cid:12)(cid:12) L/r = C r ; m =0; α = π/ , (3.2)either based on the g - or the l -ratios. We then expect e.g. that Z g A ( β ) = Z l A ( β ) + O( a ) , (3.3)where the a -effects are now expected to be smooth functions of the bare coupling. A possible refinement consists in using perturbation theory to reduce the cutoff effectsperturbatively. This requires to compute the R -ratios (2.15,2.17) perturbatively, with theexact same parameter choices as in the numerical simulations. We have performed thiscalculation to 1-loop order, R g,l A , V ( g , a/L ) = R g,l (0)A , V ( a/L ) + g R g,l (1)A , V ( a/L ) + O( g ) , (3.4)and for the chosen parameters we always find R g,l (0)A , V ( a/L ) = 1 , exactly. We may thendefine a 1-loop correction factor, r g,l A , V ( β, L/a ) = 1 + g R g,l (1)A , V (0)1 + g R g,l (1)A , V ( a/L ) , (3.5)and results for the coefficients R g,l (1)A , V are collected in table 1, for the relevant lattice resolu-tions L/a and the two lattice gauge actions used by CLS. Note that the 1-loop results are N f -independent and are thus obtained along the lines of ref. [20], the only difference being7he form of the free gluon propagator in the case of the Lüscher-Weisz gauge action [32].As an aside we note that our results converge to the known 1-loop results Z (1)A , V for aninfinitely extended lattice [33–35], i.e. for a/L = 0 . We also observe that the 1-loop cutoffeffects for the l -definitions are generally much smaller than for the g -definitions. Wilson gauge action
L/a R g (1)A ( a/L ) R l (1)A ( a/L ) R g (1)V ( a/L ) R l (1)V ( a/L ) − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . ∞ Z (1)A = − . Z (1)V = − . Lüscher-Weisz gauge action
L/a R g (1)A ( a/L ) R l (1)A ( a/L ) R g (1)V ( a/L ) R l (1)V ( a/L ) − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . ∞ Z (1)A = − . Z (1)V = − . Table 1 : Finite
L/a estimators for the current normalization constants at 1-loop order, and ourestimates for their asymptotic values; the latter agree with previous results in the literature [33–35].All results are given for
SU(3) . For given
L/a and β , the perturbatively improved current normalization constants arenow defined by Z g,l A , V , sub ( β, L/a ) = r g,l A , V ( β, L/a ) × Z g,l A , V ( β, L/a ) , (3.6)and, by construction, the O( a ) cutoff effects are subtracted to O( g ), reducing them toO( a g ). The subtracted data for the Z -factors are then treated as before: a choice ofa line of constant physics implies a set of β - and corresponding L/a -values to which thedata must be interpolated. We will see evidence for the effectiveness of this perturbativesubtraction of cutoff effects in Sect. 4 and 5. N f = 2 and N f = 3 In order to fix the physical scale r , we choose either the kaon decay constant r = 1 /f K ( N f = 2 ), or the gradient flow scale r = √ t ( N f = 3 ) [36]. In order to fix the respectiveconstants C r we proceed as follows. Given the set of values β i for i = 1 , , . . . (taken The choice of the scale from f K seems somewhat circular, as its measurement requires the correctlynormalized axial current. We use the results from ref. [8] which were obtained using Z A from a standardSF Ward identity determination. β ref either the largest or the smallest of the set.Choosing an integer lattice size L/a at the reference point β ref now fixes C r through C r = ( L/a )( β ref )( r/a )( β ref ) . (3.7)Having set the scale in this way, the L/a -values at the remaining β i follow from eq. (3.1).For all our choices the physical size of our space-time extent will be L ≈ . − . . Asmentioned before, except at the chosen reference value for β this requires interpolations ofsimulation results at integer L/a and our current simulation code, which is based on theopenQCD package [37, 38], requires that
L/a is also even.
Numerical simulations of the SF by means of standard Monte Carlo algorithms are knownto suffer from the topology freezing problem (see e.g. ref. [39] for a discussion). A possiblesolution is to follow the proposal of ref. [39] and simulate the theory with open-SF boundaryconditions. However, if for the given choice of parameters the problem is “mild”, one cancircumvent the issue in a straightforward manner by simply imposing the renormalizationconditions (2.18) and (2.13) within the trivial topological sector [40, 41]. In a continuumnotation, the correlation functions entering these definitions are modified as follows, g ud A ( x ) → g ud A , Q ( x ) = − (cid:104) A ud ( x ) Q du δ Q, (cid:105) ( ˜ Q + ) (cid:104) δ Q, (cid:105) ( ˜ Q + ) , (3.8)and analogously in all other cases. Here, the Kronecker δ in the functional integral selectsthe gauge field configurations with topological charge Q = 0 . Since relations based on chiralflavour symmetries should hold separately in each topological charge sector, this restrictionto the trivial sector is a legitimate modification of the current renormalization conditions.It provides a viable solution to the algorithmic problem of topology freezing in cases wherethis problem becomes marginally relevant; this means when the fraction of topologicallynon-trivial gauge field configurations in the relevant ensembles is not too large. For ourchoices of parameters, the percentage of gauge field configurations with Q (cid:54) = 0 is generallybelow 10%, and reaches approximately 30% only in a couple of cases (cf. tables 8 and 9).On the lattice the topological charge is not unambiguously defined. We follow refs. [40,41] and define the trivial topological sector as the set of gauge field configurations for which | Q | < . , where Q is discretized in terms of the Wilson flow and the clover definition of thefield strength tensor [36]. The flow time t is then kept fixed in physical units by requiring √ t = 0 . × L . am and z f The current normalization conditions require the χ SF correlation functions at zero quarkmass and with a chiral twist angle of π/ . In practice this is achieved by the simultaneoustuning of m and z f such that eqs. (2.13) are satisfied. In general a 2-parameter tuningcan be quite involved. However, here the non-perturbative O( a ) improvement of the action For ease of notation, in the following the subscript Q is implicitly understood, and we assume that allrelevant correlation functions are restricted to the Q = 0 sector. a ) uncertainty of the zero mass point is very much reduced. Sincea change in z f merely re-defines the matrix element used to define the PCAC mass, avariation of z f is expected to induce a small variation of m within this O( a ) uncertainty.The latter could in principle be reduced to O( a ) by including the c A -counterterm to theaxial current, but this will not be pursued here. Another important observation is that,once m and z f are within O( a ) of their target values, the sensitivity of the PCAC mass toa variation of z f is reduced to order a (cf. appendix B, discussion after eq. (B.8)). One istherefore led to conclude that the PCAC mass m is to a good approximation independentof z f , and the tuning of m and z f thus becomes straightforward; given a reasonable guessfor z f , one can first tune m , and then turn to z f . − . − . . . − . − . − . − . − . − . − . am cr a m am z f = 1 . z f = 1 . z f = 1 . z f = 1 . Figure 1 : Results for the PCAC mass as a function of the bare quark mass, for different values of z f . The dashed lines are linear fits to the data, while the solid vertical line indicates the locationof our final estimate for am cr ( g , a/L ) (s. main text). The results are for L/a = 8 and β = 5 . . As an illustration of this situation we discuss the N f = 2 case for L/a = 8 , β = 5 . .For the tuning we considered 3 values of κ = 1 / (2 am + 8) and 4 values of z f . We thengenerated around 2000 gauge field configurations separated by 10 MDUs for each of the 12ensembles, and measured the relevant correlation functions. Figure 1 collects the resultsfor the PCAC mass as a function of the bare quark mass, for the 4 different values of z f . Within statistical errors, the PCAC mass depends linearly on m and is essentiallyindependent of z f . A linear fit of m vs. m yields an estimate of m = m cr ( g , L/a ) forwhich m vanishes: these are collected in table 2. The results are perfectly compatible witheach other, and we take as our estimate for m cr the result of a weighted average of thesefour.Once the critical bare mass is fixed, a smooth interpolation of g ud A ( L/ in m gives theresults shown in figure 2. Over the chosen range, g ud A ( L/ so interpolated is perfectly linearin z f , and it is thus straightforward to determine the point z ∗ f where g ud A ( L/ vanishes i.e. z ∗ f = 1 . in this example. 10 f am cr κ cr . − . . . − . . . − . . . − . . average − . . Table 2 : Results for am cr ( g , L/a ) for four different values of z f , for L/a = 8 and β = 5 . . Theweighted average of the results is also given in the last row of the table. − . − . . .
02 1 .
28 1 . .
285 1 . .
29 1 . z ∗ f g u d A ( L / ) z f Figure 2 : Results for g ud A ( L/ as a function of z f . The dashed line is a linear fit to the data,while the solid vertical line indicates the location of our final estimate for z ∗ f (s. main text). Thevalues of g ud A ( L/ come from an interpolation to κ = 0 . , and are for L/a = 8 and β = 5 . . am cr and z ∗ f determined in this way turn out to be quiteaccurate in practice, cf. table 8. We remark that results for m cr could also be taken froma different source, for instance from standard SF simulations. In this case only z f needsto be tuned. The differences to the above procedure would be O( a ) both in m cr and in z ∗ f which, by the mechanism of automatic O( a ) improvement, induce O( a ) differences inobservables such as the current normalization constants [14, 20]. One also expects that aprecise tuning of m is less crucial in the χ SF than in the SF; the quark mass dependenceof physical observables around the chiral limit is quadratic rather than linear [42]. Besides statistical errors directly affecting the estimators for the current normalizationconstants, the other source of uncertainty originates from the precision to which a line ofconstant physics can be followed. In principle also this latter effect is of a statistical nature,however, some elements of modelling or estimates may be involved when propagating theseerrors to the normalization constants, so that it is partly justified labelling these effects assystematic.Our procedure consists of the following steps:1. The LCP together with the set of values β i translates to target values ( L/a )( β i ) .At each β i we choose lattices with even L/a straddling the target values. We hereanticipate that with our choices of LCPs the required lattice sizes are in the range
L/a = 8 to L/a = 16 . Note that all target values ( L/a )( β i ) come with statisticalerrors except for β = β ref , where, by definition, L/a is given as an (even) integer.2. For given β and L/a we determine the solutions am = am cr and z f = z ∗ f ofeqs. (2.13). In order to find their statistical errors which follow from the statisti-cal uncertainties on m and g ud A ( L/ , we use estimates for the relevant derivatives, ∂mL∂m L , ∂mL∂z f , ∂g ud A ∂m L , ∂g ud A ∂z f . (3.9)3. We then determine the induced error on the Z -factors by estimating their derivativeswith respect to the bare parameters, ∂Z A , V ∂z f , ∂Z A , V ∂m L . (3.10)It turns out that the derivatives (3.9) and (3.10) scale quite well with lattice size andlattice spacing, so that it is unnecessary to evaluate them for all parameter choices.Some cross checks are sufficient. The errors coming from the uncertainties in m and z f are then combined in quadrature and added, again in quadrature, to the statisticalerror.4. Where necessary, the results for Z A , V at the different L/a -values and fixed β i areinterpolated to the target ( L/a )( β i ) ; and the statistical error on ( L/a )( β i ) is propa-gated at this point. In the case where only one value of L/a has been simulated, an Note that the
L/a = 8 , β = 5 . , simulations listed in table 8, use slightly different values for am cr and z ∗ f from a previous, less precise determination. ∂Z A , V ∂ ( L/a ) (3.11)is used to assign a systematic error due to the difference ∆( L/a ) ≡ L/a − ( L/a )( β i ) ,also taking into account the statistical uncertainty on ( L/a )( β i ) . The resulting sys-tematic error is again added in quadrature.We emphasize that all systematic effects become essentially statistical errors providedenough data is produced to estimate the derivatives required to propagate the errors tothe normalization constants. In the following two sections we will present the lattice set-upand results for N f = 2 and N f = 3 lattice QCD. We will also come back to some of theabove points. N f = 2 flavours The CLS large volume simulations of 2-flavour QCD [8] were performed using non-pertur-batively O( a ) improved Wilson quarks and the Wilson gauge action. The matching toCLS data via the bare coupling requires that we use the same action in the χ SF. As forthe details of the action near the time boundaries we refer to ref. [20]. In particular thecounterterm coefficients c t ( g ) and d s ( g ) were set to their perturbative one-loop valuesusing the results of that reference. In general, the incomplete cancellation of boundary O( a )artefacts implies some remnant O( a ) effects in observables. However, for the estimatorsof the current normalization constants, eqs. (2.15,2.17), it can be shown that such O( a )effects only cause O( a ) differences [20].The CLS simulations were carried out for 3 values of the lattice spacing [8], corre-sponding to the β -values . , . and . . For future applications we have added a finerlattice spacing corresponding to β = 5 . . We choose the smallest CLS-value β = 5 . asreference value and set L/a = 8 at β = 5 . , (4.1)to define the starting point for the line of constant physics. We then fix the space-timevolume of the χ SF simulations in terms of the kaon decay constant, f K , evaluated atphysical quark masses. Taking af K from table 3 at β = 5 . yields f K L = 0 . , (4.2)and corresponds to L ≈ . . Imposing this condition at the other β -values then leadsto the (non-integer) L/a -values given in table 3. The quoted errors are a combination ofstatistical uncertainties, propagated from eq. (4.2) and the error on af K at the given β ’s.While the first 3 results for af K in table 3 have been directly measured [8] we haveestimated af K at the fourth value, β = 5 . , as follows: with af K at β = 5 . taken asstarting point we used the three-loop β -function for the bare coupling [43], in order todetermine the ratio of lattice spacings. The error is obtained by summing (in quadrature)the statistical error propagated from the result at β = 5 . , and a systematic error dueto the use of perturbation theory. The latter is estimated as the difference between thenon-perturbative result for af K at β = 5 . , and the same perturbative procedure, applied13 af K ( L/a )( β ) L/a † † This value is estimated using the perturbativerunning of the lattice spacing (s. main text).
Table 3 : Values of af K used to determine ( L/a )( β ) such as to satisfy the condition (4.2) for thegiven β . The χ SF simulations were performed at the neighbouring even integer L/a -values givenin the last column. between β = 5 . and β = 5 . . This systematic error is about 2.7 times larger than thestatistical one, and thus dominates the error on L/a at β = 5 . .Except for β = 5 . , the target values ( L/a )( β i ) resulting from condition (4.2), are veryclose to even integer values of L/a , so that interpolations between simulations at different
L/a can be avoided. At β = 5 . we simulated at the three L/a -values given in the lastcolumn of table 3 and interpolated to the target value (see appendix B.4 for more details).For each choice of β and L/a , along the lines of the discussion in Sect. 3.5, we have carriedout various tuning runs covering a range of am and z f , so as to determine the parameterssatisfying the conditions (2.13). The values of the tuned parameters and the results for m and g ud A ( L/ at these parameters are given in table 8. In table 4 we collect the results for Z A , V , both g and l definitions, at the four values ofthe lattice spacing. The statistics range from , to , measurements depending onthe ensemble, cf. table 8. The quoted uncertainties combine the statistical and systematicerrors. The statistical errors are at the level of . − . (cid:104) , depending on the Z -factor andensemble considered. Hence a significant contribution to the error comes from systematicuncertainties.As discussed in section 3.6, systematic errors result from uncertainties or deviations infollowing a chosen LCP, which correspond with statistical errors and deviations from zeroin m and g ud A ( L/a ) , as well as uncertainties in the target lattice extent L/a and systematicerrors arising from inter- or extrapolations from the simulated lattices sizes, if applicable.Tables 3 and 8 contain the relevant information for the case N f = 2 . The propagationof these uncertainties to the Z -factors is then performed following the steps outlined inSect. 3.6. We have carried out some additional simulations to estimate the derivativesin eqs. (3.9,3.10), and some perturbative calculation to check the expected scaling of thederivatives with the lattice size. We delegate a detailed discussion to appendix B. Herewe just note that with our statistics and our rather conservative approach, the propa-gated uncertainties are typically larger than the statistical errors for the R -estimatorseqs. (2.15,2.17) (cf. tables 10 and 11). 14 Z g A Z l A Z g V Z l V . . . . . . . . . . . . . . . . . . . . β Z g A , sub Z l A , sub Z g V , sub Z l V , sub . . . . . . . . . . . . . . . . . . . . Table 4 : Results for Z A , V , both g and l definitions, for N f = 2 non-perturbatively O( a ) improvedWilson fermions and Wilson gauge action. The lower part of the table contains the same resultsafter subtraction of the one-loop cutoff effects, cf. eq. (3.6). As discussed in section 3.2, we have also computed the relevant χ SF correlation functionsin perturbation theory to order g = 6 /β . Besides consistency checks and qualitative in-sight the main application consists in the perturbative subtraction of cutoff effects fromthe data. Note that this requires to emulate the non-perturbative procedure in all details,in particular the determination of am cr and z ∗ f according to eqs. (2.13). The lower partof table 4 contains the results for Z A , V after perturbative improvement. Comparing withthe unimproved results in the upper part of table 4, one can see that the g -definitions aremore affected, and are brought closer to the corresponding l -definitions by the perturbativeimprovement (cf. also figure 5). In any case, the perturbative corrections are at the level of1 per cent at most. In conclusion, our final results for Z A , V , either with or without pertur-bative improvement, turn out to be very precise and improve significantly on the standardSF determination based on chiral Ward identities (WIs) [8,21,44]. This is particularly truefor the case of Z A , which can be appreciated in figure 3 where the determinations of table4 are compared with those of refs. [8, 44]. In figure 4 we show instead a comparison for thecase of Z V , as obtained from the χ SF , cf. table 4, and from the standard SF (cf. ref. [21]).We note that a relevant contribution to the error of our results comes from propagatingthe uncertainties associated with maintaining the condition (4.2) i.e. keeping L constant(cf. table 10 and 11). We anticipate that due to the much more accurate knowledge of theLCP in terms of t (cf. table 5), and by using interpolations in L/a at all relevant β values,this source of error will be essentially eliminated in the case of N f = 3 (cf. section 5). a ) improvement The χ SF determinations (2.15) and (2.17) are expected to be automatically O( a ) improvedonce the bare parameters m and z f are properly tuned (cf. section 2.2). This means thatneither bulk nor boundary O( a ) counterterms are necessary to cancel O( a ) discretizationerrors in these quantities. This was confirmed to one-loop order in perturbation theory [20]and should hold generally. To this end we now look at the ratios between Z -factors coming15 .760.770.780.790.80.81 1.05 1.07 1.09 1.11 1.13 1.15 g Z g A Z l A Z SFA g Z g A,sub Z l A,sub Z SFA
Figure 3 : Comparison of different Z A determinations for N f = 2 , obtained from WIs in thestandard SF and from universality relations in the χ SF . The effect of the perturbative one-loopimprovement of the χ SF results is also shown (right panel). The χ SF results are those of table 4.The individual SF points are taken from refs. [8, 44], and are slightly displaced on the x -axis forbetter clarity. The solid black line corresponds to the SF results from the fit formula of ref. [8],and the dashed lines delimit the σ region of the fit. Note that the SF fit formula is obtainedby considering additional points with g < , here not shown, and by enforcing the perturbative1-loop behaviour for g → (see ref. [8] for the details). g Z g V Z l V Z SFV g Z g V,sub Z l V,sub Z SFV
Figure 4 : Comparison of different Z V determinations for N f = 2 , obtained from WIs in thestandard SF and from universality relations in the χ SF . The effect of the perturbative one-loopimprovement of the χ SF results is also shown (right panel). The χ SF results are those of table 4.The individual SF points are taken from refs. [21], and are slightly displaced on the x -axis forbetter clarity. The solid black line corresponds to the SF results from the fit formula of ref. [21],and the dashed lines delimit the σ region of the fit. Note that the SF fit formula is obtainedby considering additional points with g < , here not shown, and by enforcing the perturbative1-loop behaviour for g → (see ref. [21] for the details). .980.9850.990.99511.005 0 0.004 0.008 0.012 0.016 ( a/L ) Z l A /Z g A Z l A,sub /Z g A,sub ( a/L ) Z l V /Z g V Z l V,sub /Z g V,sub
Figure 5 : Continuum limit of the ratios between the l and g definitions of Z A (left panel) and Z V (right panel) for the case of N f = 2 quark-flavours; the effect of subtracting the lattice artefactsfrom the Z -factors to O( g ) is also shown. The dashed lines correspond to linear fits to the data,constrained to extrapolate to 1 for a/L = 0 . from the g - and l -definitions. The expectation that these ratios converge to 1 with O( a )corrections is indeed very well borne out by the data, cf. figure 5, where we also includefits to this expected behaviour. We emphasize that this is a non-trivial result: eventhough the bulk action is improved to match the CLS set-up, we did not O( a ) improve thecurrents entering the definitions (2.15,2.17) and (2.13). This result thus confirms automaticO( a ) improvement at the non-perturbative level, and, indirectly, the universality relationsbetween the χ SF and SF formulations. A direct way to test universality between the χ SF and SF formulations would be simply to study the continuum scaling of ratios of Z -factors as obtained from one and the other formulation. Provided the SF determinationsare properly improved, these should also approach 1 in the continuum limit with O( a )corrections. The large errors on the SF determinations do not allow us for a precise test ofthis expectation. However, the results in figure 3 and 4 clearly show that our determinationsare in fact compatible with the SF ones within errors. N f = 3 flavours The CLS simulations with N f = 2 + 1 flavours of non-perturbatively O( a ) improved Wilsonfermions [45] and Lüscher-Weisz (LW) gauge action, have been carried out for 5 values ofthe lattice spacing, with β -values between . and . [1, 2, 10]. For completeness we notethat CLS has also tried to simulate at a coarser lattice spacing corresponding to β = 3 . .However, these ensembles have been discarded for the scale determination in [1] due tovery large cutoff effects observed e.g. in t [10]. For this reason we will not consider this β -value in our study, however, we mention that it was adopted as starting point for theWard identity determination of Z A in ref. [22]. Given the relatively large set of latticespacings we here consider two different LCPs, with slightly different physical extent, L and L , which we define through the gradient flow time t [36]. The associated length scale r = √ t can be interpreted as a smoothing radius, and has been very precisely determined17or the CLS β -values ≥ . in [1, 2]. Using this scale we impose the conditions L / √ t = 1 . L / √ t = 1 . , (5.1)where the right hand sides were chosen in order to have exactly, L /a = 8 at β = 3 . L /a = 16 at β = 3 . , (5.2)respectively. Using the result for t in physical units [1], eqs. (5.1) translate to L ≈ . and L ≈ . . β t /a ( L /a )( β ) ( L /a )( β ) L/a .
40 2 . . , , , .
46 3 . . . , , , .
55 5 . . . , , , .
70 8 . . . , , , .
85 14 . −
16 16
Table 5 : CLS β -values and corresponding results for t /a in the SU(3) flavour symmetric limit [1,2]. The latter are used to determine the lattice sizes ( L , /a )( β i ) which satisfy the conditions(5.1). The χ SF simulations are performed at the neighbouring L/a ’s given in the last column ofthe table.
In table 5 we collect the relevant β values of the CLS simulations and the correspondingresults for t /a [2]. The latter are evaluated for equal up-, down-, and strange-quarkmasses, which are close to the physical average quark mass (see refs. [1, 2]). Table 5 alsogives the lattice sizes ( L , /a )( β ) which satisfy the conditions (5.1). Compared to the N f = 2 case (cf. table 3), it is obvious that these N f = 3 LCPs are much more accuratelydetermined. In order to exploit this higher precision, we performed simulations for several
L/a -values at each β (cf. table 5). This allowed us to accurately interpolate the Z -factorsto the target values (see appendix B.5 for more details). Table 9 contains a summary ofall simulations performed with the corresponding parameters. Due to both technical andhistorical reasons, we do not use the finest lattice spacing for the LCP defined in termsof L . Following this LCP up to β = 3 . would have required simulating lattices with L/a = 18 , , which are particularly inconvenient to parellelize with our current simulationprogram. Note also that CLS simulations at β = 3 . are ongoing and currently limitedto a single ensemble, so that the LCP with L may remain useful for a while. Moreimportantly, however, the comparison between both LCPs allows us to perform additionaltests on our results (cf. section 5.3).The lattice action we employ for the finite volume simulations matches the CLS actionin the bulk, i.e. the Lüscher-Weisz tree-level improved gauge action and 3 flavours of non-perturbatively improved Wilson quarks [45]. Close to the time boundaries of the latticethere is some freedom regarding the implementation of Schrödinger functional boundaryconditions. For the gauge fields we choose option B of ref. [32]; we refer the reader to thisreference for the details. Regarding the fermions, two quark flavours satisfy χ SF boundaryconditions (option τ = 1 of [14]), while the third one obeys the standard SF boundaryconditions [25]. In general, such a mixed set-up increases the number of O( a ) improvementcoefficients which need to be tuned in order to eliminate O( a ) discretization errors from18he time boundaries. As in the N f = 2 case, however, one can show that the correspondingcounterterms affect the renormalization constants Z A , V only at O( a ). For definiteness wehave used the one-loop estimate c t = 1 + g c (1)t , where the one-loop coefficient decomposesas follows, c (1)t = c (1 , + 2 × c (1 , ( χ SF ) + 1 × c (1 , ( SF ) . (5.3)The pure gauge contribution is taken from ref. [46], the fermionic χ SF contribution fromref. [20] and the SF contribution from ref. [29]. Furthermore, we use the tree-level values d s = 1 / [20] and ˜ c t = 1 [26]. In table 6 and 7 we collect the results for Z A , V , corresponding to the L - and L -LCP,respectively. The statistics we accumulated for the different ensembles ranges between3,200 and 31,000 measurements, with exact numbers given in table 9. The correspondingstatistical precision on the Z -factors is between . − . (cid:104) , depending on the exactquantity and ensemble. The errors quoted in the tables then combine the statistical errorswith the systematic errors originating from the uncertainties on the LPCs. β Z g A Z l A Z g V Z l V .
40 0 . . . . .
46 0 . . . . .
55 0 . . . . .
70 0 . . . . β Z g A , sub Z l A , sub Z g V , sub Z l V , sub .
40 0 . . . . .
46 0 . . . . .
55 0 . . . . .
70 0 . . . . Table 6 : N f = 3 results for Z A , V using the L -LCP, both for g and l definitions. The lower partof the table contains the results after subtraction of the one-loop cutoff effects, cf. eq. (3.6). Like in the N f = 2 case, the high statistical precision requires a careful assessmentof the systematic errors in order to arrive at reliable error estimates. Tables 5 and 9contain information on the accuracy with which the chosen LCPs are realized for oursimulation parameters. Our estimates for the systematic uncertainties due to deviationsfrom the chosen LCP were then obtained analogously to the case of N f = 2 ; we referthe reader to appendix B for the details. Here it is worth noting that, similarly to thiscase, the propagated uncertainties are typically larger than the statistical errors for the R -estimators, eqs. (2.15,2.17), cf. table 12. Even though the fermionic contributions were calculated with the Wilson gauge action, to this orderthe calculation only depends on the gauge background field, which is not modified when using the LWaction with option B of [32]. Z g A Z l A Z g V Z l V .
40 0 . . . . .
46 0 . . . . .
55 0 . . . . .
70 0 . . . . .
85 0 . . . . β Z g A , sub Z l A , sub Z g V , sub Z l V , sub .
40 0 . . . . .
46 0 . . . . .
55 0 . . . . .
70 0 . . . . .
85 0 . . . . Table 7 : Same as table 6 but for the L -LCP. In the lower halves of tables 6 and 7 we give the results for Z A , V after perturbativelysubtracting the lattice artefacts to one-loop order. The results have been obtained by firstimproving the Z A , V determinations for each L/a and g value, and then interpolating tothe proper ( L , /a )( β ) (see appendix B.5).Comparing the results for Z A , V before and after perturbative improvement, one seesthat the g -definitions are the most affected, and are brought closer to the corresponding l -definitions. All in all, the effect of the perturbative improvement is at most at the levelof a couple of percent (cf. figure 7). Hence, not too surprisingly perhaps, the situation isvery much the same as for the N f = 2 case. g Z g A Z l A Z SFA Z SFA,con g Z g A,sub Z l A,sub Z SFA Z SFA,con
Figure 6 : Comparison between different Z A determinations for N f = 3 , obtained either fromWIs in the standard SF or from universality relations in the χ SF . The χ SF results are takenfrom table 7 and the effect of the perturbative one-loop improvement is shown in the right panel.The individual SF points labelled Z SFA and Z SFA , con are taken from ref. [22] and correspond to thedefinitions Z A , and Z conA , , respectively, of that reference. The solid black line is the fit formulato Z SFA also given in [22] and the dashed lines delimit the σ region of the fit. Note that this fitfunction enforces the perturbative 1-loop behaviour for g → . In conclusion, our final results for Z A , V are very precise for both LCPs. Similarly to20he N f = 2 case, the results for Z A are significantly more accurate than the standard SFdetermination based on Ward identities [22]. This can be appreciated in figure 6, wherethe results from table 7 are displayed together with the 2 alternative definitions Z A , and Z conA , of ref. [22]. a ) improvement Given our estimates for Z A , V we can study the approach to the continuum limit of theratio between different definitions. We begin with figure 7 where the ratios between the g - and l -definitions are considered for the L - and L -LCPs; both the results before andafter perturbative improvement are shown. The conclusions are very much the same asfor the N f = 2 case. Considering the results before perturbative improvement, along bothLCPs, the g and l definitions deviate by at most a couple of per-cent. These differencesthen perfectly scale with a to zero as the continuum limit is approached. If perturbativeimprovement is implemented, these differences almost vanish even at the coarsest latticespacings. There is no significant deviation from a scaling, however, some small admixtureof higher order effects cannot be excluded either. It is also interesting to consider the ( a/L ) Z l A /Z g A Z l A,sub /Z g A,sub ( a/L ) Z l V /Z g V Z l V,sub /Z g V,sub ( a/L ) Z l A /Z g A Z l A,sub /Z g A,sub ( a/L ) Z l V /Z g V Z l V,sub /Z g V,sub
Figure 7 : Continuum limit of the ratios between the l and g definitions of Z A (left panels) and Z V (right panels) for the case of N f = 3 quark-flavours; the effect of subtracting the lattice artefactsfrom the Z -factors to O( g ) is also shown. The upper panels show the L -LCP results while thelower ones show those of the L -LCP. In all cases, the dashed lines correspond to linear fits to thedata constrained to extrapolate to 1 for a/L , = 0 . Note that the (tiny) effect of the statisticalcorrelation between numerator and denominator has been neglected in these ratios. continuum limit of the ratio between the definitions belonging to different LCPs i.e. the L - and L -LCP. An example of such a ratio is shown in figure 8. Also in this case, thecontinuum scaling of this ratio is the one expected, and the initial difference is at the 221er cent level. Apart from providing an important check of universality and automaticO( a ) improvement, these results show that considering one definition or the other for therenormalization of matrix elements of the axial and vector currents, will only introducesmall O( a ) differences over the whole range of lattice spacings covered. . . . . .
05 0 . .
15 0 . .
25 0 . . Z l X , L / Z g X , L a /t X=AX=V
Figure 8 : Continuum limit of the ratio between the Z l X , L definitions, X=A,V, corresponding tothe L -LCP, and the Z g X , L definitions corresponding to the L -LCP. The dashed lines correspondto linear fits to the data constrained to extrapolate to 1 for a /t = 0 . Finally, we look at ratios between χ SF and standard SF determinations. Towards thecontinuum limit these should also scale like a ) , if the SF determinations are O( a )improved. In figure 9 we show the continuum limit of the ratios between the standard SFdeterminations of ref. [22] and the χ SF results of table 7. We here consider both definitionsof this reference, and label them as Z SFA = Z A , and Z SFA , con = Z conA , , respectively (cf. [22]for the exact definitions).As one can see in figure 9, for their preferred definition, Z SFA , the expected scalingis only setting in around a /t < . , where the SF and χ SF determinations differ by acouple of per cent. At the coarsest lattice spacing, corresponding to β = 3 . , the deviationfrom the O( a ) scaling is significant. The results for Z g A show the largest deviation fromthe SF determination, which is about 6%. Considering the perturbatively improved χ SF results this difference is somewhat reduced to 4-5%, but O( a ) scaling is not observedeither. If we consider instead the alternative definition, Z SFA , con , the deviation is reducedto about 2 per cent at the coarsest lattice spacing for Z l A , while, remarkably, the resultsfor Z g A and Z conA , are compatible within errors. In particular, the difference between thisSF and both our χ SF determinations is perfectly compatible with an O( a ) effect over thewhole range of lattice spacings considered. While discretization effects can only be definedwith respect to some reference definition, we conclude that the alternative SF definition Z SFA , con is, within errors, perfectly scaling with a for β ≥ . relative to all χ SF definitions,whereas the preferred definition Z SFA of ref. [22] requires much finer lattices before this22xpected asymptotic behaviour sets in. With hindsight, Z SFA , con seems to be a better choicewithin the SF framework and also has been the preferred SF definition within the N f = 2 setup of refs. [8, 44]. a /t Z SFA /Z g A Z SFA /Z l A Z SFA,con /Z g A Z SFA,con /Z l A a /t Z SFA /Z g A,sub Z SFA /Z l A,sub Z SFA,con /Z g A,sub Z SFA,con /Z l A,sub
Figure 9 : Continuum limit of the ratios between the N f = 3 WI determinations of Z A of ref. [22],and the χ SF determinations Z g,l A (left panel) and Z g,l A , sub (right panel) of table 7. The Z SFA resultsare from the fit formula provided in ref. [22], and correspond to their preferred, Z A , , definition.The associated dashed lines (red and blue lines) are linear fits to the data with a /t < . ,constrained to extrapolate to 1 for a /t = 0 . The Z SFA , con results come instead from a fit of theresults for the alternative, Z conA , , definition considered in ref. [22]. The latter fit was obtained usingthe same fit ansatz used in ref. [22] for Z A , . The associated dashed lines (green and magentalines) are linear fits to all data, constrained to extrapolate to 1 for a /t = 0 . We have used a new method [20] based on the chirally rotated Schrödinger functional [14]to obtain high precision results for the normalization constants of the Noether currentscorresponding to non-singlet chiral and flavour symmetries. The matrix elements of theseaxial and vector currents play a crucial rôle in various contexts of hadronic physics. Ourmethod differs from the traditional Ward identity method [11, 12] in that it comparescorrelation functions which are related by finite chiral or flavour rotations, rather thaninfinitesimal ones. The major advantage compared to the Ward identity method consistsin the avoidance of 3- and 4-point functions in favour of simple 2-point functions. This verysignificantly improves on the precision achieved in previous determinations [8,21,22,44,47].In particular, for the case of Z A , we obtain a reduction of the error by up to an order ofmagnitude (cf. figure 3 and 6). The relatively poor precision obtained for Z A with thetraditional Ward identity methods [8, 21, 22, 44] (around the percent level at the coarsestlattice spacings of interest), has now become a limiting factor in several applications.For this reason, our results are in high demand and have already been used in severalworks [1,23,48]. In particular, the precise N f = 2+1 scale setting from a linear combinationof f K and f π in ref. [1] crucially relies on our values of Z l A in table 6 and the associateduncertainty is negligible compared to the statistical error of the bare hadronic matrixelements. In turn, the precise scale setting result of [1] is entering almost all studiesdone with CLS gauge configurations: in particular it has enabled the precise result forthe 3-flavour QCD Λ -parameter and thus α s ( m Z ) by the ALPHA-collaboration [2, 41,49, 50]. Further applications of our Z A -results include the non-perturbative quark mass23enormalization factor in [23] and the related determination of the light and strange quarkmasses [48]. Regarding the N f = 2 case, the potential improvement of the scale setting inref. [8] due to our Z A -results would be very significant, too. Tentative estimates anticipatea gain by a factor − in precision, when going from the finest to the coarsest latticespacing [51].In order to maximize the usefulness of our results we have chosen the same actions andthe same β -values for N f = 2 and N f = 3 lattice QCD as used by the CLS initiative [8, 10].Hence, anyone working with CLS gauge configurations will be able to directly use ourresults: for N f = 2 we recommend to use Z l A , V , sub from table 4, and for N f = 3 werecommend using Z l A , V , sub either of table 6 or 7. Although the results for Z l A , V , sub areslightly less precise than those for Z g A , V , sub , their L/a -interpolations turn out to be morerobust. Furthermore, the effect of the perturbative subtraction of cutoff effects is rathersmall and only marginally significant with current errors. While the precise choice of the χ SF results for the Z -factors is not crucial, it is however very important to be consistentand to not switch definitions when changing β . Only then cutoff effects are guaranteed tovanish smoothly at a rate ∝ a .Our determination of Z A , V ( β ) was carried out for each β -value independently, in orderto avoid adding statistical correlation between physics results at different lattice spacings.However, it is straightforward to fit our Z -factors to a smooth function of β (or g ), whichinterpolates to any intermediate β -value. We have included a few such fits in appendix Cto our preferred definitions Z l A , V , sub . We also include fits which incorporate the expectedperturbative behaviour to 1-loop order. However, the high precision obtained in the β -range covered by the data cannot be guaranteed outside this range. If a similar precision isrequired at higher β , an extension of our non-perturbative determination will be required.If t /a was known for higher β -values one could extend our chosen line of constant physicscovering another factor of 2 or so in the lattice spacing. The required simulations of the χ SF for lattice sizes up to
L/a = 32 would be feasible with current resources. Goingbeyond this range it may be advisable to choose a different line of constant physics froma finite volume observable, or at least estimate the errors incurred by deviating from theoriginal choice.In applications to hadronic physics one would also like to control the O( a ) effectscancelled by the counterterms to the currents. Close to the chiral limit, one essentiallyrequires the counterterm coefficients c A , V [26, 27]. We emphasize that our method ofdetermining the Z -factors does not rely on any assumptions about these counterterms andcan therefore be combined with results for c A , V from other studies, e.g. [47, 52]. The sameremark applies to the b -coefficients multiplying O( am ) counterterms, which have recentlybeen determined for the vector current in ref. [53].Looking beyond direct applications of our results in the CLS context, it is quite obviousthat the precision gains of this method are generic and could be implemented with anyother choice of Wilson type fermions. One would need to implement the χ SF boundaryconditions following ref. [14], as well as the χ SF correlation functions [20]. We also notethat the computer resources required are rather modest: in fact our largest lattice sizewas ; indeed, the main work for the present results went into painstakingly followinglines of constant physics and the determination of the corresponding uncertainties andtheir propagation to the Z -factors. We have reported many technical details in the hope24hat any further applications of the method will be able to benefit from our experience.One possible improvement we did not explore was to measure the derivatives (3.9,3.10) bycomputing the corresponding operator insertions into the correlation functions directly onthe tuned ensembles; this was done e.g. in refs. [1, 2] for the PCAC mass, t , and otherobservables, and this would certainly allow one to further improve on the precision, as noassumptions on the derivatives need to be made.Possible future applications of the χ SF include the determination of the ratio betweenpseudo-scalar and scalar renormalization constants, Z P /Z S . Advantages of the χ SF arealso expected for scale-dependent problems, such as the renormalization of 4-quark op-erators, where the contamination by O( a ) effects could be significantly reduced by themechanism of automatic O( a ) improvement [19]. Finally the χ SF offers new methods forthe determination of O( a ) improvement coefficients, which we hope to explore in the future. We would like to thank Rainer Sommer and Stefano Lottini for useful discussions, andPatrick Fritzsch, Tim Harris, and Alberto Ramos for helpful comments on the analysis ofthe data. We are moreover thankful to the computer centers at ICHEC, LRZ (project idpr84mi), and DESY-Zeuthen for the allocated computer resources and support. The codewe used for the simulations is based on the openQCD package developed at CERN [38].25
Simulation parameters and results
L/a β κ z f mL × g ud A ( L/ × P Q N ms . . . − . . − . .
2) 99 .
8% 80028 5 . . . . .
0) 1 . .
5) 99 .
9% 1200210 5 . . . − . .
7) 6 . .
3) 98 .
6% 400412 5 . . . . .
3) 7 . .
8) 93 .
8% 240312 5 . . . . . − . .
9) 99 .
4% 240316 5 . . . − . .
4) 1 . .
6) 100% 1803
Table 8 : Parameters of the N f = 2 ensembles and corresponding results for mL and g ud A ( L/ .The total number of measurements we collected is given by N ms ; these are spaced by 10 MDUs.In the table we also give the percentage P Q of gauge fields with Q = 0 . L/a β κ z f mL × g ud A ( L/ × P Q N ms .
40 0 . . − . . . .
40 0 . . − . . − . .
10 3 .
40 0 . . − . . − . .
12 3 .
40 0 . . . . − . . .
46 0 . . . − . . .
46 0 . . . .
5) 0 . .
10 3 .
46 0 . . − . .
5) 3 . .
12 3 .
46 0 . . . . − . . .
55 0 . . − . . − . .
10 3 .
55 0 . . − . . .
12 3 .
55 0 . . − . .
6) 2 . .
16 3 .
55 0 . . − . . − . . .
70 0 . . − . . .
10 3 .
70 0 . . . . − . .
12 3 .
70 0 . . − . .
3) 0 . .
16 3 .
70 0 . . − . .
2) 0 . .
16 3 .
85 0 . . . .
0) 1 . . Table 9 : Parameters of the N f = 3 ensembles and corresponding results for mL and g ud A ( L/ .The total number of measurements we collected is given by N ms ; these are spaced by 10 MDUs.In the table we also give the percentage P Q of gauge fields with Q = 0 . / a β κ z f Z g A Z l A Z g V Z l V . . . . ( )( )( ) . ( )( )( ) . ( )( )( ) . ( )( )( ) . . . . ( )( )( ) . ( )( )( ) . ( )( )( ) . ( )( )( ) . . . . ( )( )( ) . ( )( )( ) . ( )( )( ) . ( )( )( ) . . . . ( )( )( ) . ( )( )( ) . ( )( )( ) . ( )( )( ) . . . . ( )( )( )( ) . ( )( )( )( ) . ( )( )( )( ) . ( )( )( )( ) . . . . ( )( )( )( ) . ( )( )( )( ) . ( )( )( )( ) . ( )( )( )( ) T a b l e : R e n o r m a li z a t i o n c o n s t a n t s Z g , l A , V f o r t h e d i ff e r e n t N f = e n s e m b l e s . T h e f o u r e rr o r s r e f e r t o ( c f . a pp e nd i x B ) : s t a t i s t i c a l, s y s t e m a t i cc o m i n g f r o m t h e un ce r t a i n t y o n a m c r ( ∆ m Z A , V ) , s y s t e m a t i cc o m i n g f r o m t h e un ce r t a i n t y o n z ∗ f ( ∆ z f Z A , V ) , a nd s y s t e m a t i cc o m i n g f r o mm a i n t a i n i n g t h e c o nd i t i o n ( . )( ∆ x Z A , V ) ; t h e l a tt e r i s o n l y r e l e v a n t f o r t h e l a s tt w o e n s e m b l e s . L / a β κ z f Z g A , s ub Z l A , s ub Z g V , s ub Z l V , s ub . . . . ( )( )( ) . ( )( )( ) . ( )( )( ) . ( )( )( ) . . . . ( )( )( ) . ( )( )( ) . ( )( )( ) . ( )( )( ) . . . . ( )( )( ) . ( )( )( ) . ( )( )( ) . ( )( )( ) . . . . ( )( )( ) . ( )( )( ) . ( )( )( ) . ( )( )( ) . . . . ( )( )( )( ) . ( )( )( )( ) . ( )( )( )( ) . ( )( )( )( ) . . . . ( )( )( )( ) . ( )( )( )( ) . ( )( )( )( ) . ( )( )( )( ) T a b l e : R e n o r m a li z a t i o n c o n s t a n t s Z g , l A , V , s ub f o r t h e d i ff e r e n t N f = e n s e m b l e s . L a tt i ce a r t e f a c t s h a v e b ee n s ub t r a c t e d a t O ( g ) i np e r t u r b a t i o n t h e o r y . T h e f o u r e rr o r s r e f e r t o ( c f . a pp e nd i x B ) : s t a t i s t i c a l, s y s t e m a t i cc o m i n g f r o m t h e un ce r t a i n t y o n a m c r ( ∆ m Z A , V ) , s y s t e m a t i cc o m i n g f r o m t h e un ce r t a i n t y o n z ∗ f ( ∆ z f Z A , V ) , a nd s y s t e m a t i cc o m i n g f r o mm a i n t a i n i n g t h ec o nd i t i o n ( . )( ∆ x Z A , V ) ; t h e l a tt e r i s o n l y r e l e v a n t f o r t h e l a s tt w o e n s e m b l e s . C o m p a r i n g w i t h t h e r e s u l t s o f t a b l e , o n l y t h e m e a n v a l u e s , s t a t i s t i c a l e rr o r s , a nd e rr o r s a ss o c i a t e d w i t h m a i n t a i n i n g t h ec o nd i t i o n ( . ) , a r e d i ff e r e n t . T h e m e a n v a l u e s a nd c o rr e s p o nd i n g s t a t i s t i c a l e rr o r s c a nb e o b t a i n e d f r o m t h o s e o f t a b l e t h r o u g h e q . ( . ) . F o r d e t e r m i n i n g t h e s y s t e m a t i ce rr o r s a ss o c i a t e d w i t h t h e un ce r t a i n t y o n a m c r a nd z ∗ f w e u s e t h e s a m ee s t i m a t e s f o r t h e r e l e v a n t d e r i v a t i v e s a s f o r t h e r e s u l t s i n t a b l e : w e t hu s o b t a i n t h e s a m e v a l u e s . T h e s y s t e m a t i ce rr o r s a ss o c i a t e d w i t h m a i n t a i n i n g t h ec o nd i t i o n ( . ) ,i n s t e a d ,i n v o l v e d i ff e r e n t d e r i v a t i v e s f o r t h e d a t a i n t a b l e nd t h e p e r t u r b a t i v e l y i m p r o v e d o n e s ( c f . a pp e nd i x B ) . / a β κ z f Z g A Z l A Z g V Z l V . . . . ( )( )( ) . ( )( )( ) . ( )( )( ) . ( )( )( ) . . . . ( )( )( ) . ( )( )( ) . ( )( )( ) . ( )( )( ) . . . . ( )( )( ) . ( )( )( ) . ( )( )( ) . ( )( )( ) . . . . ( )( )( ) . ( )( )( ) . ( )( )( ) . ( )( )( ) . . . . ( )( )( ) . ( )( )( ) . ( )( )( ) . ( )( )( ) . . . . ( )( )( ) . ( )( )( ) . ( )( )( ) . ( )( )( ) . . . . ( )( )( ) . ( )( )( ) . ( )( )( ) . ( )( )( ) . . . . ( )( )( ) . ( )( )( ) . ( )( )( ) . ( )( )( ) . . . . ( )( )( ) . ( )( )( ) . ( )( )( ) . ( )( )( ) . . . . ( )( )( ) . ( )( )( ) . ( )( )( ) . ( )( )( ) . . . . ( )( )( ) . ( )( )( ) . ( )( )( ) . ( )( )( ) . . . . ( )( )( ) . ( )( )( ) . ( )( )( ) . ( )( )( ) . . . . ( )( )( ) . ( )( )( ) . ( )( )( ) . ( )( )( ) . . . . ( )( )( ) . ( )( )( ) . ( )( )( ) . ( )( )( ) . . . . ( )( )( ) . ( )( )( ) . ( )( )( ) . ( )( )( ) . . . . ( )( )( ) . ( )( )( ) . ( )( )( ) . ( )( )( ) . . . . ( )( )( ) . ( )( )( ) . ( )( )( ) . ( )( )( ) T a b l e : R e n o r m a li z a t i o n c o n s t a n t s Z g , l A , V f o r t h e d i ff e r e n t N f = e n s e m b l e s . T h e t h r eee rr o r s r e f e r t o ( c f . a pp e nd i x B ) : s t a t i s t i c a l, s y s t e m a t i cc o m i n g f r o m t h e un ce r t a i n t y o n a m c r ( ∆ m Z A , V ) , a nd s y s t e m a t i cc o m i n g f r o m t h e un ce r t a i n t y o n z ∗ f ( ∆ z f Z A , V ) . T h ec o rr e s p o nd i n g r e s u l t s w i t h t h e O ( g ) l a tt i ce a r t e f a c t ss ub t r a c t e d a r e o b t a i n e d f r o m t h o s e a b o v e b y a pp l y i n g e q . ( . )t o t h e m e a n v a l u e s a nd c o rr e s p o nd i n g s t a t i s t i c a l e rr o r s . T h e s y s t e m a t i c un ce r t a i n t i e s a r e i n s t e a d l e f t un c h a n g e d ( c f . t a b l e ) . Error propagation, systematic error estimates and comparison withperturbation theory
In this appendix we describe in some detail the elements required to carry out the steps 2,3and 4 sketched in subsection 3.6, for the propagation of uncertainties. We first report onthe numerical estimates of the various derivatives (steps 2,3) for both N f = 2 and N f = 3 .These estimates are obtained on smaller lattices L/a = 8 and
L/a = 6 , respectively, andthen used for all lattice sizes. We therefore also summarize the expected scaling with L/a ofthese derivatives and confirm this to first non-trivial order in perturbation theory. Finally,the interpolation of the Z -factors in L/a (step 4, where necessary) is discussed, first for N f = 2 , where this step is almost avoidable, and then for N f = 3 , where interpolations arenecessary at most β -values, thus requiring a more thorough analysis. B.1 Estimating the derivatives: N f = 2 As described in section 3.6, to estimate the uncertainty in Z A , V originating from thoseof am cr and z ∗ f , we require the derivatives (3.9) and (3.10). We estimated these throughdedicated simulations at L/a = 8 and β = 5 . , measuring the relevant quantities for severaldifferent values of κ ( z f ) at fixed z f ( κ ), straddling the tuned values given in table 8. Theresults we obtained for the derivatives (3.9) are: ∂mL∂m L = 1 . , ∂mL∂z f = 0 . ,∂g ud A ∂m L = − . , ∂g ud A ∂z f = − . . (B.1)We observe that the z f -derivative of mL vanishes within an uncertainty much smaller thanthe values of the other derivatives, so that we may safely set it to zero. These results werethen used for all other ensembles and lattice sizes listed in table 8. The uncertainty in thetuning of mL and g ud A ( L/ is thus converted to one in m cr L and z ∗ f . Specifically, one hasthat, (cid:32) ∆( m cr L )∆ z ∗ f (cid:33) = A − (cid:32) ∆( mL )∆ g ud A (cid:33) where A = (cid:32) ∂mL∂m L ∂mL∂z f ∂g ud A ∂m L ∂g ud A ∂z f (cid:33) , (B.2)and ∆( mL ) , ∆ g ud A , ∆( m cr L ) and ∆ z ∗ f , are the uncertainties in mL , g ud A ( L/ , m cr L and z ∗ f , respectively. For the uncertainties ∆( mL ) and ∆ g ud A we took the (absolute) values of mL and g ud A ( L/ measured on the given ensemble (cf. table 8), plus 2 times the corre-sponding statistical errors. The corresponding systematic errors on the Z -factors can thenbe estimated as, (∆ m Z X ) + (∆ z f Z X ) = (cid:18) ∂Z X ∂m L (cid:19) (∆ m cr L ) + (cid:18) ∂Z X ∂z f (cid:19) (∆ z ∗ f ) , X = A , V . (B.3)Through the very same simulations used to determine the derivatives (B.1) we obtained, ∂Z g A ∂m L = 0 . , ∂Z l A ∂m L = − . ,∂Z g V ∂m L = 0 . , ∂Z l V ∂m L = − . , (B.4)29nd ∂Z g A ∂z f = − . , ∂Z l A ∂z f = − . ,∂Z g V ∂z f = − . , ∂Z l V ∂z f = − . . (B.5)We then used these values at all other L/a - and β -values of table 8. More precisely, wepropagated the errors according to eq. (B.3) using the measured absolute mean values towhich we added twice the statistical errors. Taking these results at the coarsest availablelattice spacing is a conservative choice which should be safe, also given that some favourableexpected scaling of the derivative towards larger L/a (s. below) is not taken advantage of.Complementary information obtained during the tuning runs for finding m cr and z ∗ f , furthercorroborates this assumption. Looking at eqs. (B.4,B.5) it is clear that the l -definitionshave in general larger systematic uncertainties due to their larger sensitivity to z f . Thisis then reflected in the Z -factors in tables 10 and 11 where the uncertainties are listedseparately. B.2 Estimating the derivatives: N f = 3 For N f = 3 we proceeded in very much the same way, except that we carried out a morecomplete study of L/a = 8 lattices at all β -values, except β = 3 . , and also includedadditional L/a = 6 lattices at β = 3 . and . . As before for the derivatives (3.9) we usedtheir mean values and set d mL/ d z f = 0 , while for the derivatives (3.10) we consideredtheir (absolute) mean values plus twice their statistical errors. However, here we didthis separately for all β -values, with β = 3 . results also applied at β = 3 . . Table 12contains the results for Z g,l A , V for all ensembles of table 9, including the different estimateduncertainties. As with N f = 2 , the soundness of our assumption is supported by experiencegained during the tuning runs to find m cr and z ∗ f . B.3 Expected scaling with
L/a and perturbative calculations
B.3.1 Expected
L/a -scaling
Using general arguments based on the Symanzik expansion and P parity [20] one mayobtain the expected scaling with the lattice spacing a of the derivatives of the PCAC massand g ud A with respect to m and z f , ∂g ud A ∂z f = O(1) , ∂g ud A ∂m L = O(1) , (B.6)and ∂mL∂m L = O(1) , ∂mL∂z f = O( a ) . (B.7)To explain how one arrives at these scaling properties we go through the example of thePCAC mass: ∂mL∂z f = L ∂∂z f (cid:32) ˜ ∂ g ud A g ud P (cid:33) = L (cid:0) m (cid:48) − m (cid:1) × g ud P; z f g ud P , (B.8)30here we have introduced the modified PCAC mass m (cid:48) through the equation ˜ ∂ g ud A; z f = 2 m (cid:48) g ud P; z f , (B.9)and the notation ; z f indicates differentiation with respect to z f [20]. The crucial point tonote is that this differentiation merely modifies the fields at the time boundaries and thusproduces a different matrix element for the PCAC relation and hence the modified PCACmass m (cid:48) . Since the difference m (cid:48) − m between two PCAC masses is of O( a ) in general,and the z f -derivative of the P -even correlation function g ud P is P -odd and thus of O( a ),we arrive at O( a ) for the complete expression. It is re-assuring to see that this derivativeis indeed found to be small in the simulations.Similar arguments lead to ∂Z g,l A , V ∂m L = O( a ) , ∂Z g,l A , V ∂z f = O( a ) , (B.10)where the derivatives are taken at fixed β, z f and β, am , respectively. B.3.2 Comparison with perturbation theory
Perturbation theory confirms all of these expected scaling properties. The derivatives(3.9) have only been considered to tree-level, which gives, ∂mL∂m L = 1 + O( g ) , ∂mL∂z f = O( g ) ,∂g ud A ∂m L = − g ) , ∂g ud A ∂z f = − g ) . (B.11)For the mass sensitivity of the axial current we have, to one-loop order, ∂Z g A ∂m L ≈ aL × (cid:40) − . × g + . . . (Wilson action) , − . × g + . . . (LW action) , (B.12) ∂Z l A ∂m L ≈ aL × (cid:0) − . × g + . . . (cid:1) (LW & Wilson action) , (B.13)and, for the vector current, ∂Z g V ∂m L ≈ aL × (cid:40) − . × g + . . . (Wilson action) , − . × g + . . . (LW action) , (B.14) ∂Z l V ∂m L ≈ aL × (cid:40) − . × g + . . . (Wilson action) , − . × g + . . . (LW action) . (B.15)Here, the one-loop coefficients are the values at L/a = 12 and are stable within 3-10 percent for the range
L/a from 8 to 16. Incidentally, this result resolves qualitatively a puzzleposed by the non-perturbative results, eq. (B.4), where the derivatives of the g - and l -definitions are almost the same in magnitude and opposite in sign, whereas the tree levelresults are 1 and 0, respectively, as first noticed in [20]. Note that perturbative results without explicit group factors assume gauge group SU(3) and fermionsin the fundamental representation. z f -sensitivity of the Z -factors is easily described: all z f -derivatives vanish at treelevel and the one-loop coefficients for the g -definitions are very small and vanish with arate roughly proportional to a for Z g A , V and both gauge actions. On the other hand, the l -definitions behave as expected (s. above): very similar numbers are obtained which are,for both gauge actions, within a few percent given by ∂Z l A , V ∂z f = − . × g aL + O( g ) . (B.16)For completeness we report the perturbative results for am cr and z ∗ f we obtain from thesame computation. For the critical mass to order g , the known values [32, 35, 54, 55] (with C F = 4 / for gauge group SU(3)), am cr = g C F × (cid:40) − . , (Wilson action), − . , (LW action) , (B.17)are already reproduced to 4-5 digits on lattices with L/a in the range from 8 to 16. As for z ∗ f , we have, to order g and for a/L → , z ∗ f = 1 + g C F × (cid:40) . , (Wilson action) , . , (LW action) , (B.18)where the Wilson action result is from ref. [20], whereas the LW action value is the one of L/a = 16 with a generous guess for the error. Also in this case the values at finite
L/a from 8 and 16 coincide with these numbers to 4-5 digits precision.We observe that the quantitative comparison of non-perturbative data with bareperturbation theory to order g works quite well in certain cases. For instance, for N f = 2 at β = 5 . , we compare the non-perturbative value am cr = − . to am cr = − . g ) , and similarly for z ∗ f = 1 . we need to compare to z ∗ f = 1 .
258 + O( g ) .Also the non-perturbative current normalization constants themselves are reproduced byone-loop perturbation theory at the 5-10 percent level (compare table 1 with tables 4 and 6,7). On the other hand, the majority of the derivatives differ very significantly, for instance, ∂Z g A ∂m L (cid:12)(cid:12)(cid:12)(cid:12) β =5 . ,L/a =8 = 0 . vs. .
057 + O( g ) , (B.19)is off by a factor 2, and for the l -definition the comparison is between − . and − . , which differs by a factor 5. For the z f -derivatives we note that perturbation theorycorrectly predicts the smallness of the sensitivity in the g -definition. Quantitatively, theperturbative z f -derivatives of Z l A , V at β = 5 . and L/a = 8 are − .
046 + O( g ) , to becompared with eq. (B.5), so again we observe a difference by a factor 2.Finally, for the interpolations in L/a of the data (s. below) we are also interested inthe derivatives of the Z -factors with respect to x = ( a/L ) at fixed bare coupling. Inperturbation theory we can obtain approximate results from table 1. Expanding R g,l A , V = 1 + g (cid:18) Z (1)A , V + k g,l A , V × a L + O( a ) (cid:19) + O( g ) , (B.20)32he x -derivatives to order g are approximately given by ∂Z g,l A , V ∂x ≈ g × k g,l A , V , (B.21)provided higher order cutoff effects are small. We find that this is quite well satisfied, withvery similar coefficients for both Wilson and LW actions, given approximately by k g A ≈ . , k g V ≈ . , (B.22)whereas k l A , V are ca. 20 to 30 times smaller in magnitude, for axial and vector cases,respectively, and come with the opposite sign. Comparing this with the β = 5 . non-perturbative data in eqs. (B.23) we see again that for the g -definitions these derivatives arereproduced by perturbation theory up to a factor 2, while for the l -definitions, perturbationtheory to O( g ) is clearly missing the bulk of the effect. While, as expected, the non-perturbative derivatives are smaller than for the g -definitions, it seems that the smallnessof the O( g ) term is an accident and higher orders are dominating at these values of β .To conclude this comparison, perturbation theory often gives valuable qualitative in-formation and may provide reasonable starting values for the tuning of am and z f . How-ever, quantitatively, the agreement with non-perturbative data at lattice spacings of inter-est for hadronic physics hugely varies for different observables. Hence, the main practicaluse of perturbation theory consists in the perturbative subtraction of cutoff effects. Here,even a qualitative agreement, which may be quantitatively off by a factor 2, still means awelcome reduction of cutoff effects by 50 percent, and our data analysis does indeed pointto such benefits.Given this situation, we have refrained from using perturbative data in our estimatesof the derivatives, and we have decided to ignore the favourable O( a ) scaling, eq. (B.10),when applying the results obtained at L/a = 8 at all other
L/a -values, too. In this respect,the tuning runs for am and z f , both for N f = 2 and N f = 3 , provided some consistencychecks which make us confident that the chosen procedure is indeed sound and ratherconservative. B.4 Interpolation in
L/a for N f = 2 Once the uncertainties associated with the conditions (2.13) have been propagated to Z A , V at given β and L/a , one needs to keep the LCP condition (4.2) and also take into accountthe corresponding uncertainties. The choice (4.2) is made such that the
L/a = 8 resultsat β = 5 . satisfy this condition by definition, while at β = 5 . we needed to interpolatethe results for L/a = 8 , , to the target value ( L/a )(5 .
3) = 9 . (cf. table 3). Wehave performed a simple linear interpolation in ( a/L ) which describes the data very well,similarly to the case N f = 3 which will be discussed in more detail below. For β = 5 . and . , the simulated L/a values are, within errors, compatible with the target valuesin table 3. Systematic errors related to the condition (4.2) were then estimated by usingthe slope of the interpolation in x = ( a/L ) at β = 5 . also for the higher β -values. Notethat we interpolate results with bare parameters am cr and z ∗ f tuned at the given β - and L/a -values. Hence the derivative defines the sensitivity to a change of the physical size ofthe system. This is a pure cutoff effect of O( a ) on the Z -factors. Since, by the choice ofthe variable x , a factor a is also divided out, the x -derivative is expected to be of O(1)33nd we expect a smooth dependence of this derivative on β ; this expectation is in factconfirmed by the results for N f = 3 where the slope shows a very mild β -dependence overthe whole range (cf. table 13). As we are looking at an O( a ) effect in disguise, it is nosurprise that the results depend on whether or not the cutoff effects have been subtractedperturbatively.Without perturbative subtraction we obtained the results, ∂Z g A ∂x = 0 . , ∂Z l A ∂x = 0 . ,∂Z g V ∂x = 1 . , ∂Z l V ∂x = 0 . , (B.23)while for the perturbatively improved ones we obtain, ∂Z g A ∂x = 0 . , ∂Z l A ∂x = 0 . ,∂Z g V ∂x = 0 . , ∂Z l V ∂x = 0 . . (B.24)The corresponding systematic error is then simply taken to be, (∆ x Z X ) = (cid:18) ∂Z X ∂x (cid:19) (∆ x ) , X = A , V , (B.25)which is summed in quadrature to (B.3) and the statistical error from the Monte Carlosimulations. The uncertainly ∆ x was estimated as: ∆ x = | x − x ( β ) | + 2 σ ( x ( β )) , (B.26)where x ( β ) = (( a/L )( β )) and σ ( x ( β )) is the associated error. As a further safeguardwe took for the derivatives (B.23) and (B.24) the (absolute) mean value plus twice theirstatistical error. We observe that the l -definitions have a milder L/a -dependence than the g -based ones, unless perturbative improvement is implemented. B.5 Interpolation in
L/a for N f = 3 Once the systematic errors deriving from the tuning of am and z f have been taken intoaccount, the results for Z A , V at different L/a and fixed β must be interpolated to either ( L /a )( β ) or ( L /a )( β ) , depending on the LCP; for completeness the values of Z A , V priorto interpolation are given in table 12. We have considered three types of interpolation in x = ( a/L ) , these are: linear using all 4 available values of L/a (cf. table 5), linear usingonly the 3 closest
L/a -values to the target ( L , /a )( β ) , and quadratic using all 4 L/a -values. Given this choice, the interpolations needed for the L - and L -LCPs only differfor β = 3 . , where, by definition, L /a = 8 is exact, and in the case of linear interpolationswith 3 points at β = 3 . . Recall that for β = 3 . no interpolation is required, as L /a = 16 is exact and this β -value has been excluded for the L -LCP.Starting with the L -LCP, the different interpolations describe the data quite well ingeneral, particularly so for the results at the two smallest lattice spacings and for definitions We recall that at leading order in PT the x -derivatives of the Z -factors are of O( g ) (cf. eq. (B.21)).They are thus expected to diminish, and eventually vanish, as g → . . . . . .
005 0 .
01 0 .
015 0 .
02 0 .
025 0 . a/L ) Z g A Z l A Z g V Z l V Figure 10 : L/a -interpolations for β = 3 . and the L -LCP. The upper two sets of points corre-spond to the Z g,l A results while the lower two sets are the Z g,l V results. The dashed lines are ourpreferred, quadratic, fits to the data, and the interpolation points are marked by a black verticalline. . . . . .
005 0 .
01 0 .
015 0 .
02 0 .
025 0 . a/L ) Z g A, sub Z l A, sub Z g V, sub Z l V, sub
Figure 11 : Same as figure 10, for the Z -factors with perturbative subtraction of the cutoff effects. l -correlators. The most relevant exception is given indeed by the linearinterpolation of Z g V at β = 3 . using all 4 values of L/a , for which we find a χ / d . o . f ≈ . . It should be noted, however, that the χ -criterion does not come with the usualprobability interpretation due to the errors being dominated by systematics. In any case,the interpolated values are generally compatible at the 1 σ level.Considering the perturbatively improved data, the quality of the interpolations is gen-erally improved, and all fits have excellent χ . The beneficial effect of the perturbativeimprovement can be appreciated by comparing figure 10 and 11, where the Z A , V interpo-lations at β = 3 . are shown for the cases before and after perturbative improvement,respectively. This example also illustrates the general feature that, before perturbativeimprovement, the l -definitions have a significantly milder L/a - and hence x -dependence.In addition, it is interesting to note that the x -dependence of the Z -factors does not changesignificantly over the range of β considered, but seems in general to diminish, as expected,as β → ∞ (cf. table 13). Based on these observations, we take as our final estimates forthe Z -factors the results of the quadratic fits, which have the largest errors. β ∂Z g A /∂x ∂Z l A /∂x ∂Z g V /∂x ∂Z l V /∂x .
46 0 . . . . .
55 0 . . . . .
70 0 . . . . β ∂Z g A , sub /∂x ∂Z l A , sub /∂x ∂Z g V , sub /∂x ∂Z l V , sub /∂x .
46 0 . . . . . − . . . . .
70 0 . . . . Table 13 : Results for ∂Z g,l A , V /∂x , where x = ( a/L ) , as a function of β for N f = 3 quark-flavours.The derivatives are estimated along the L -LCP from the linear fits using the 3 closest L/a -valuesto the target ( L /a )( β ) . Regarding the L -LCP, the situation is more complicated due to the fact that weneed to interpolate the data at the coarsest lattice spacing, β = 3 . . The quality of theinterpolations is still good in general, but there are a few significant exceptions. We notethat all these cases involve g -definitions: indeed, we obtain a pretty large χ / d.o.f. for thelinear fits of Z g V at β = 3 . and . , around . and respectively. Also the quadratic fitfor Z g A at β = 3 . has a large χ / d . o . f ≈ . . While in this case, however, the results ofthe interpolation are compatible with those of the linear fits within less than one standarddeviation, in the case of Z g V at β = 3 . the discrepancy between the linear and quadraticinterpolations is close to 3 standard deviations. The situation definitely improves whenthe perturbatively improved data are considered. In this case, with the exception of thelinear interpolations with 4 points of Z g V , A , sub at β = 3 . , all fits have very good χ , andgive compatible results within one standard deviation or so. As in the case of the L -LCP,we take as our final estimates for Z A , V the results of the quadratic fits, which have thebest χ -values and the largest errors. 36 Fit formulas for Z A , V ( g ) In this appendix we collect some useful fit formulas for the Z A , V results, both for N f = 2 and N f = 3 . We will focus on the data for Z l A , V , sub , cf. the discussion in sect. 6. C.1 N f = 2 For N f = 2 the final Z A , V results are given in table 4. Over the whole range of β ∈ [5 . , . ,the data for Z l A , sub is well described by a simple linear fit function, Z l A , sub = c + c g ,c , = (cid:18) . − . (cid:19) Cov = 10 − × (cid:18) . − . − . . (cid:19) , (C.1)which has a χ / d . o . f . = 0 . / .Similarly, for the vector current renormalization, Z l V , sub , a good description of thedata is given by, Z l V , sub = c + c g ,c , = (cid:18) . − . (cid:19) Cov = 10 − × (cid:18) . − . − . . (cid:19) , (C.2)which has a χ / d . o . f . = 0 . / . C.1.1 Matching with perturbation theory
It is also interesting to consider fit functions with the correct perturbative 1-loop behaviourfor g → (cf. sect. 3.2). In the case of Z l A , sub this is possible using a 2-parameterpolynomial fit, Z l A , sub = 1 − . g + c g + c g ,c , = (cid:18) − . − . (cid:19) Cov = 10 − × (cid:18) . − . − . . (cid:19) , (C.3)which gives a χ / d . o . f . = 1 . / . We note that the same fit ansatz was used to fit thestandard SF results of ref. [8]. Similarly, for the vector current data, Z l V , sub , we have, Z l V , sub = 1 − . g + c g + c g ,c , = (cid:18) − . − . (cid:19) Cov = 10 − × (cid:18) . − . − . . (cid:19) , (C.4)which gives a χ / d . o . f . = 1 . / . We stress that although the latter fit functions encodethe expected asymptotic behaviour far outside the β -range covered by the data, it is notrecommended to use them for β values much outside this range. For β ∈ [5 . , . , the twosets of fit functions agree within less than σ deviations.37 .2 N f = 3 For the case N f = 3 , our final Z A , V results are given in table 6 and 7. Having one additional β -value, it is natural to prefer an interpolation of the L -LCP data of table 7. The higherprecision of the data compared to N f = 2 , and the availability of a fifth data point suggeststo use 3-parameter fits in this case. We find that, for the whole range of β ∈ [3 . , . , Z l A , sub , is well described by the quadratic fit: Z l A , sub = c + c g + c g , (C.5)with coefficients and covariance given by c , , = . − . . Cov = 10 − × . − . . − . . − . . − . . , and χ / d . o . f . = 0 . / .For the vector current data, Z l V , sub , we use the same fit function, Z l V , sub = c + c g + c g , (C.6)and obtain c , , = . − . . Cov = 10 − × . − . . − . . − . . − . . , which gives a χ / d . o . f . = 1 . / . C.2.1 Matching with perturbation theory
Also in this case we consider fit functions with the correct perturbative 1-loop behaviourfor g → (cf. sect. 3.2). Applying a 3-parameter polynomial fit of the form Z l A , sub = 1 − . g + c g + c g + c g , (C.7)we obtain c , , = . − . . Cov = 10 − × . − . . − . . − . . − . . , which gives a χ / d . o . f . = 0 . / . We have also tried various Padé fits e.g. of the typeused in [22]. With these fits we experienced some technical problems with the bootstraptechnique, when trying to determine the covariance matrix for the fit parameters. Wetherefore also tried the the automatic differentiation procedure of ref. [56] which completelysolved this technical problem. It turns out, however, that the best fit function of this typedevelops a singularity at a β -value slightly above 4, and therefore does not provide a smoothinterpolation to the perturbative region. Given the good quality of the linear fits we didnot pursue any further non-linear options. 38egarding the vector current data, Z l V , sub , we have, Z l V , sub = 1 − . g + c g + c g + c g , (C.8)with c , , = . − . . Cov = 10 − × . − . . − . . − . . − . . , which gives a χ / d . o . f . = 1 . / .To conclude this appendix, we emphasize again that all given fits to the data arevery good if used as interpolations in the range of the non-perturbative data. Using themoutside this range is at the user’s own risk, even where perturbative information is used asa constraint. 39 eferences [1] M. Bruno, T. Korzec, and S. Schaefer, Setting the scale for the CLS flavor ensembles , Phys. Rev.
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