Artifacts and ⟨A2⟩ power corrections: Reexamining Zψ(p2) and ZV in the momentum-subtraction scheme

Artifacts and hA2ipower corrections: Reexamining Z p2and ZVin the momentum-subtraction scheme Ph. Boucaud, 1 F. de Soto, 2 J. P. Leroy, 1 A. Le Yaouanc, 1 J. Micheli, 1 H. Moutarde, 3 O. Pe `ne, 1 and J. Rodrı ´guez-Quintero 4 1 Laboratoire de Physique The ´orique*(Ba ˆt.210), Universite ´de Paris XI, Centre d’Orsay, 91405 Orsay-Cedex, France 2 Dpto. de Fı ´sica Ato ´mica, Molecular y Nuclear, Universidad de Sevilla, Apdo. 1065, 41080 Sevilla, Spain 3 Centre de Physique The ´orique Ecole Polytechnique, 91128 Palaiseau Cedex, France 4 Dpto. de Fı ´sica Aplicada, Fac. Ciencias Experimentales, Universidad de Huelva, 21071 Huelva, Spain (Received 21 September 2005; published 9 August 2006) The next-to-leading-order (NLO) term in the operator product expansion (OPE) of the quark propagator vector part Z and the vertex function g1of the vector current in the Landau gauge should be dominated by the same hA2icondensate as in the gluon propagator. On the other hand, the perturbative part has been calculated to a very high precision thanks to Chetyrkin and collaborators. We test this on the lattice, with both clover and overlap fermion actions at 6:0, 6.4, 6.6, 6.8. Elucidation of discretization artifacts appears to be absolutely crucial. First hypercubic artifacts are eliminated by a powerful method, which gives results notably different from the standard democratic method. Then, the presence of unexpected, very large, nonperturbative, O4symmetric discretization artifacts, increasing towards small momenta, is demonstrated by considering ZMOM V, which should be constant in the absence of such artifacts. They impede in general the analysis of OPE. However, in two special cases with overlap action—(1) for Z ; (2) for g1, but only at large p2—we are able to identify the hA2icondensate; it agrees with the one resulting from gluonic Green functions. We conclude that the OPE analysis of quark and gluon Green function has reached a quite consistent status, and that the power corrections have been correctly identified. A practical consequence of the whole analysis is that the renormalization constant Z ( Z1 2of the momentum-subtraction (MOM) scheme) may differ sizably from the one given by democratic selection methods. More generally, the values of the renormalization constants may be seriously affected by the differences in the treatment of the various types of artifacts, and by the subtraction of power corrections. DOI: 10.1103/PhysRevD.74.034505 PACS numbers: 12.38.Gc, 11.15.Ha, 12.38.Aw, 12.38.Cy I. INTRODUCTION The study of the quark propagator and vertex functions in momentum space has been extensively pursued in the literature starting in the 70’s with analytical considerations [1–3], later completed by the discovery of the presence of a contribution of the A2operator, due to gauge fixing [4]. Numerical lattice QCD has more recently treated this issue [5–13]. The scalar part of the quark propagator is related via axial Ward identities to the pseudoscalar vertex function. The role of the Goldstone boson pole in the latter has been thoroughly discussed [14–17]. In our paper Ref. [18], we have initiated a new study of the propagator and of the vector vertex function, which we pursue here. This paper has been devoted to an improved treatment of hypercubic artifacts. Now we present the whole of the work, which also includes the analysis of the other types of artifacts and the extraction of the power corrections. We mainly study the vector part of the inverse quark propagator, the one which is proportional to 6p,Z , and the vector vertex function g1related to it by the Ward identity, or, equivalently, on the ratio ZMOM VpZ p2 g1p2. One of the reasons is to check for the effect of the hA2icondensate which has been discovered via power corrections to the gluon propagator and three point Green functions at large momenta [19–22] (let us recall that, in order to perform trustable perturbative calculations, we take always p> 2:6 GeV). OPE shows that the effect of the hA2icondensate should have almost the same magnitude in the Z p2of the quark as in the gluon propagator. Moreover, the perturbative-QCD corrections are known to be varying especially slowly (the anomalous dimension being zero at one loop in the Landau gauge), as we shall recall later. This should give a favorable situation to display the power corrections (recall that in the gluon case, it was difficult to disentangle the power and the logarithmic corrections, which were moreover very sensitive to the value of QCD). Now, for the clover action, the crude values plotted in literature (with only a selection of democratic points) for Z p2are extremely flat above 2 GeV [23]. This was formerly considered as being natural, as a consequence of the vanishing of the one-loop anomalous dimension. However, thinking more about it, it must be considered on the contrary as very worrying since it means that the decrease predicted both by the perturbative-QCD corrections and the hA2icondensate would not be seen. This leads us to start a very systematic study of the problem, with the following series of improvements on earlier works: *Laboratoire associe ´au CNRS, UMR 8627 PHYSICAL REVIEW D 74, 034505 (2006) 1550-7998=2006=74(3)=034505(21) 034505-1 ©2006 The American Physical Society (i) We reach an energy of 10 GeV by matching several lattice spacings so that we are in a better position to eliminate lattice artifacts and also to identify power corrections, which requires a large momentum range. (ii) We make use of a very efficient way of eliminating hypercubic artifacts. A first version was used to study gluon Green functions [24,25]. In Ref. [18], this method has been improved for the specific case of the quark Green functions, where such artifacts are huge, especially in Z and for the overlap action. Hypercubic artifacts have often been cured by the ‘‘democratic’’ method which considers only momenta with equilibrated values of the components. This method points in the right direction but, as we shall discuss in more details, is by far insufficient when the hypercubic artifacts reach such a level as illustrated in this article. In fact, we are then led to qualitatively different conclusions. (iii) We make systematic use of Ward-Takahashi identities relating the quark propagator and the vertex function. According to these Ward identities, ZMOM V should be independent of p2up to artifacts. Then, ZMOM Vis a very sensitive test of the presence of artifacts. The fact that we observe a strong dependence of the lattice ZMOM Von p2shows unambiguously the existence of large remaining discretization artifacts, this time respecting O4symmetry, and decreasing with negative power of momentum. We can also check the consistency of ZMOM V with other determinations of ZV. (iv) The above mentioned O4symmetric artifacts could constitute a very serious problem, since we do not know how to eliminate them, and they would hide the OPE power corrections. Fortunately, in the case of overlap fermions (with s0), we find that for Z , there remains only very small O4artifacts. Then, we can obtain a satisfactory estimate of the 1=p2correction due to the hA2icondensate. For this reason, we consider mainly overlap fermions (with s0), although they have some specific inconveniences (for example very large perturbative corrections). This work, as well as the preceding one, clearly shows that lattice artifacts are overwhelming at the start, both the hypercubic and O4ones. The hypercubic ones have been shown to be cleanly eliminated by our method. The remaining O4ones, on the other hand, cannot, and we do not foresee the possibility of a similarly efficient method, wherefrom we have to rely on situations where they are small for reasons which are not known, so that their smallness appears accidental. In that respect, it may seem worrying; but, on the other hand, we are very happy to have found that the OPE can be checked to a good accuracy and power corrections are found consistent with the gluon analysis. It must be noted that on the whole, the main body of the analysis relies on the overlap data, because we have found that they exhibit smaller O4 artifacts. Another obvious interest of the study is then to improve the determination of the momentum-subtraction (MOM) renormalization constants, by taking into account both continuum power corrections and artifacts; and, indeed it indicates that much care must be exerted in using the MOM renormalization approach when high precision will be required, a point which has already been illustrated by the Goldstone contribution to ZMOM P. In Sec. II, we will recall theoretical premises; in Sec. III we will indicate the lattice conventions and the simulations which we have performed, as well as recall the general problems of the improvement; the three following sections are devoted to artifacts, with a large development justified by the importance of the topic: in Sec. IV, we will recall the method to eliminate hypercubic lattice artifacts, and emphasize the difference with the democratic method; in Secs. V and VI, we discuss the other artifacts (Lorentz scalar artifacts, volume effects); in Sec. VII, we will give the results of the overlap action; in Sec. VIII, we perform several consistency checks: chiral symmetry of the overlap action VIII A; comparison with clover action VIII B; finally, we note the agreement with the previous gluon analysis, Subsection VIII C. In Sec. IX, we will give our conclusions and further discussions. II. THEORETICAL PREMISES We work in the Landau gauge. Let us first fix the notations that we will use in the continuum, the precise lattice definitions being given later. We will use all along the Euclidean metric. The continuum quark propagator is a 12 12 matrix Spfor 3-color and 4-spinor indices. The inverse propagator is expanded according to: S1pa;bZ p2i6pmp2;(1) where a,bare the color indices. Z p2being a standard lattice notation (for the precise lattice definition, see below, Sec. III). Obviously, one has in the continuum, with trace on spin and color: Z p21=12 TrSp=p2:(2) Sometimes one uses the alternative quantity: bp2Z p2mp2(3) to describe the scalar part of the propagator. Let us consider a colorless vector current  qq. The three point Green function Gis defined by Gp; qZd4xd4yeipyiqxhqy qxqx q0i: (4) PH. BOUCAUD et al. PHYSICAL REVIEW D 74, 034505 (2006) 034505-2 The vertex function is then defined by p; qS1pGp; qS1pq:(5) In the whole paper, we will restrict ourselves to the case where the vector current carries a vanishing momentum transfer q. In the following we will omit to write q0 and we will understand pas the bare vertex function computed on the lattice. From Lorentz covariance and discrete symmetries pa;bg1p2ig2p2pg3p2p6p ig4p2;6p (6) which should be obeyed approximately on the lattice, as we checked. A. Renormalization and Ward-Takahashi (WT) identities The renormalized vertex function is then Z2ZVp. Here we must say something of conventions for renormalization constants. The standard definition of renormalization constants has been to divide the bare quantity by the renormalization constant to obtain the renormalized quantity (except for photon or gluon vertex renormalizations Z1 which we do not use). Z2is the standard renormalization of fermions bare  Z2 p R,SpZ22SRp. In principle renormalization of composite operators, for instance ZV, should be defined similarly. We have followed this convention in our works on gluon fields. But, in the case of quark operators, an opposite convention has become standard in lattice calculations:  O bare Z1 O O R;we feel compelled to maintain this convention for the sake of comparison with parallel works on the lattice. This explains our writing of the renormalized vertex function. In the continuum ZV1(conserved current). We keep ZV since the local vector current on the lattice is not conserved, and the discrepancy, which is of course an artifact, generates however finite effects in graphs due to additional divergencies multiplying the aterms (which have higher dimension). The Ward identity in the renormalized form tells us that at infinite cutoff: Rpi@ @pS1 Rp:(7) After multiplying both sides by Z1 2to return to bare quantities ZVpi@ @pS1p;(8) which from (1)–(6) implies ZVg1p2Z p2;Z Vg3p22@ @p2Z p2; ZVg2p22@ @p2bp2;g 4p20: (9) We note that the first Eq. (9) implies that ZVis independent of the renormalization scheme up to artifacts (it is a ratio of bare quantities). Of course, this will hold up to terms vanishing as inverse powers of the cutoff at infinite cutoff, which are called artifacts in the lattice language. It must be recalled that, on the lattice, the Ward identity is not exact, but holds only up to artifacts, because we work at finite cutoff, and the deviation will be found very large in some cases. A very important consequence of the Ward identity for our study is that the ratio g1p2=Z p2is constant up to artifacts, or that deviations of this ratio from a constant are pure artifacts. Defining analogously the vertex function of the pseudoscalar (  q5q) density 5pg5p25;(10) the axial Ward identity implies ZP ZS mqg5p2bp2;(11) where mqis 1=2a1= 1=c. B. MOM renormalization; radiative corrections To perform renormalization on the lattice, we appeal as usual to the convenient MOM schemes, which does not refer to a specific regularization. To speak technically, the precise renormalization scheme that we use is the one called RI’ by Chetyrkin, Eq. (26) in Ref. [26]. This is in fact the most standard MOM scheme in the continuum, developed a long time ago by Georgi, Politzer, and Weinberg. It consists of setting the renormalized Green functions to their tree approximation at the renormalization point p2 i2, in the chiral limit. The inverse bare propagator S1pis normalized through: S1 Rpjp22a;bi6pmp2:(12) Making p22shows that ZMOM 2p2Z p21(13) up to artifacts. To renormalize the bare vertex function g1p2, we multiply it by the factor ZMOM 22ZMOM V2. The MOM renormalization for g1must be chosen so that the renormalized WT identity (Eq. (7)) gR 1p2ZR p2 holds, therefore gR 1p221; we deduce that ZMOM V2is the ratio Z 2=g12up to artifacts, but as remarked above, this must be nothing but the scheme independent ZV: therefore this ratio ZMOM V2is independent of 2, up to artifacts. From now on, we define ZMOM V as the ratio Z p2=g1p2(measured in fact on the lattice); we write ARTIFACTS AND hA2iPOWER CORRECTIONS: ... PHYSICAL REVIEW D 74, 034505 (2006) 034505-3 ZMOM Va1;p 2Z p2=g1p2(14) which recalls that ZMOM V, which should be in fact independent of p2in the limit where the cutoff a1is infinite, is not so at finite a1—i.e. there are artifacts. The Ward identity implies that Z pand g1phave, in particular, the same perturbative pscale dependence. From the calculations of Chetyrkin et al. [26,27], we may express, for example, the perturbative running of Z at large pas a function of the running MOMp. This is our main choice throughout this paper, although we discuss the effect of substituting an expansion in MSp. More precisely, we will always choose to use the definition of MOMpby the symmetric three-gluon vertex. The advantage of quarks is that we can reach an accuracy of four loops in the renormalization group (RG) expansion, because we do not need MOM 3(for symmetric MOMp), since the dimension of the fermion is 0 at lowest order in the Landau gauge. Since the expression is lengthy and not necessary for present understanding, we refer the reader to Appendix A. Of course, even with such accuracy, such an expression cannot be expected to hold for too small p;we esteem the lower bound to be pmin 2:6 GeV, from our experience in the case of gluons; indeed, we must avoid going down too close to the ‘‘bumps’’ which manifest clearly that the gluon Green functions become nonperturbative. The perturbative calculation requires a value of QCD;in order to compare with existing values from a different origin, it is convenient to reexpress all values in terms of the equivalent MS, whatever the scheme employed to make the perturbative expansion of the Green functions. One advantage of Z p2is that it is not as sensitive to the QCD value as the gluon quantities, because of the vanishing of the LO fermion anomalous dimension in the Landau gauge; we choose MS 0:237 GeV from a previous analysis [19], not far from the ALPHA estimate [28], MS 0:23819GeV; we will discuss in the end the sensitivity of our results to this choice. Z p2and g1p2should have also the same nonperturbative power corrections, up to a constant. We consider them now. C. Power correction from the hA2icondensate An OPE analysis as those performed in Refs. [19–22] leads to consider a hA2icondensate coupled to the quark propagator and vertex in Landau gauge. Let us recall that such a condensate could not contribute to gauge-invariant Green functions, and is present only in (gauge fixed) gauge-noninvariant Green functions. The meaning and magnitude of such a condensate has been extensively discussed in the recent literature. Our aim here is to detect its effect on the Green functions through OPE, which provides a way of testing theoretical ideas on its existence and magnitude. For the propagator, we can write S1pS1 pertpi6pdbarebare p2h:A2 bare :i 4N2 C1a;b ; (15) where we only keep the leading term in 6p. The calculation of the coefficients of the OPE has been performed in the chiral limit, and therefore one has as far as possible to stay near this limit. In the renormalization prescription denoted by ‘‘RI’’’ in Chetyrkin papers (which amounts to the standard MOM of Georgi and Politzer in the chiral limit), and expanding everywhere in terms of MOM s, we obtain from Eq. (15) (see Appendix B) Z p2 Zpert 2Zpert p2 Zpert 232 3pp 0 A20=0 hA2Ri 4N2 C1 1 p2;(16) where S1i6pa;bZ and S1 pert i6pa;bZpert up to Omq6p=p2-terms. The condensate hA2Ri is renormalized at the scale .Zpert is given in Eq. (2) and the coefficients (0being the fermion anomalous dimension to lowest order) are 011; 00; 0 A235Nc 12 35 4:(17) As we have noted, ZMOM Vp2should be constant in p2 from the Ward identity (9), up to artifacts; then it cannot receive any power correction from A2, and therefore g1 receives exactly the same contribution from the condensate as Z . We will use this as a very useful test. The essential step is then to fit this formula on the lattice data to extract hA2i. The renormalization constants at each ,Z 2;will enter in the fit as free parameters to be determined, although they would be expected a priori to be close to lattice perturbation theory predictions. Of course, in general, we have to add lattice artifacts to Eq. (16), and one of the main problems we will discuss is how to determine them accurately. An important warning must be made here, concerning the low accuracy in the perturbative calculation of the Wilson coefficient of A2written above, namely, it is only tree order with renormalization group improvement. Expanding in terms of MOM s, although it may seem natural, is completely arbitrary, and one would wish the results to be the same with MS s. While this is the case to a good precision for Zpert , this is obviously not the case here, due to the low order of the expansion: spis quite different in the two schemes. At p2:6 GeV, the ratio MOM=MS is around 2 and decreases slowly down to 1.4 at 10 GeV; taking into account the anomalous dimension amounts roughly to replace spby s, then the ratio of PH. BOUCAUD et al. PHYSICAL REVIEW D 74, 034505 (2006) 034505-4 coefficients in terms of the two coupling constants is only slightly closer to 1: it is 1.5 in average over the whole range. This means that the coefficient is reduced by 50% when using MS s. This is due to the fact that the ratio of coupling constants decreases only very slowly up to the largest available momenta. As a consequence, the determination of hA2iobtained by fitting the lattice data will be automatically affected by the same amount. We will give the results with the convention of using everywhere MOM s, as we have done for gluons. We shall first show that the power correction is indeed present and well determined, and then express it in terms of the condensate value, which suffers from the above uncertainty. We also note that the ratio of condensates fitted from gluons and quark Green functions, which should be 1 ideally, is not affected by this uncertainty, since the Wilson coefficients relative to the various Green functions differ mainly by purely algebraic numbers (the anomalous dimensions differ only slightly). 1 III. LATTICE CALCULATIONS A. Actions and parameters We have first used Sheikholeslam-Wohlert improved Wilson quarks (here called clover) with the cSW coefficients computed in [30]. 100 quenched gauge configurations have been computed at 6:0, 6.4, 6.6, 6.8 with volumes 244,164, and 84. We have performed the calculation for five quark masses but in practice, for what is our concern in this paper, the quark mass dependence has not surprisingly proven to be negligible; anyway, since the theoretical calculations are performed in the chiral limit, we have to work as close as possible to the chiral limit; then, we present only for simplicity the results for the lightest quark mass, about 30 MeV, i.e. 0:1346;0:135 38;0:135 15;0:134 89 for 6:0;6:4;6:6;6:8:(18) It should also be mentioned that all the results presented for clover action refer to the 244lattices unless stated otherwise. In addition to clover fermions, the use of overlap fermions [31] has been revealed to be necessary, and even crucial, to obtain a good determination of the power correction, because the most embarrassing type of artifacts, the O4invariant ones, have been found to be smaller in certain cases with the overlap action (see Sec. VII). We have used approximately the same physical masses, i.e. as in the clover case am00:03;0:016 67;0:0125;0:01 for 6:0;6:4;6:6;6:8(19) with s0and volumes of 164. The bare mass m0and sare defined from Dover 1sam0=21sam0=2 Dw1s  Dw1syDw1s q;(20) where Dw1s is the Wilson-Dirac operator with a (negative) mass term 1s, Dw1s1 2rr 1 2ar r1s: (21) We use s0.s0:4is considered preferable from locality requirements [32], however the difference is slight as soon as is larger than 6.0. The reason for using only the small lattice 164is well known; it is due to limitation in the special treatment needed for small eigenvalues of the Neuberger operator. In practice, as for clover action, we discuss only the lightest quark mass, roughly corresponding to the same 30 MeV. We express everywhere quantities in physical units. Throughout this paper we will use for the lattice spacings the values in the following Table Iwhich follow the  dependence found in Ref. [33], Appendix C, Formule C.1. B. Improvements The propagators Sx; 0from the origin to point xhave been computed and their Fourier transform is defined as: ~ SpX x eipxSx; 0:(22) They have been averaged among all configurations and all momenta pwithin one orbit of the hypercubic symmetry group of the lattice, exactly as for gluon Green functions in [24]. The two actions are Oaimproved; however, this is not sufficient to improve the propagator itself, which is an off shell quantity. In the case of overlap quarks, the propagator and other Green functions can be improved according to a standard and exact procedure [34] which should eliminate Oadiscretization errors in Green functions, at large p,in the perturbative regime 2 : TABLE I. Lattices spacings. 6.0 6.4 6.6 6.8 a1(GeV) 1.966 3.66 4.744 6.1 a(fm) 0.101 0.055 0.042 0.033 1 While finishing the article, we have become aware, thanks to D. Becirevic, of the calculation of the two-loop anomalous dimension of A2by [29], in the MS scheme. 2 Indeed, in our opinion, the argument on the vanishing of artifacts uses chiral symmetry of vacuum matrix elements, which holds only when spontaneous symmetry breaking can be neglected. Therefore, it should hold only at large palthough this is not stated explicitly. ARTIFACTS AND hA2iPOWER CORRECTIONS: ... PHYSICAL REVIEW D 74, 034505 (2006) 034505-5 ~ Sp ~ Sp1 2 1am0=2:(23) On the other hand, such a well-defined improvement recipe for the propagator with the clover action does not exist: we would have to fix the counterterms necessary to eliminate the Oaterms from the numerical study of the propagator itself. Note that the standard contact term with a perturbative evaluation of the coefficient leads to subtract from Z pa term of the type a=p2. Indeed, it subtracts a term of the type amp2, where mp2is the continuum mass term, vanishing at large p, with a rather small absolute value less than 30 MeV for p>2:6 GeV, i.e. in the region where we are working. But this term happens to be much smaller than the observed artifacts. We indeed find a decreasing artifact: a1:9 GeV3=p2Eq. (40), which amounts to a280 MeV at p2:6 GeV. This suggests that the standard perturbative improvement is rather ineffective. Anyway, the analysis of the artifacts is included in our discussion of remaining O4invariant artifacts, and in fact, as we shall see, this is not feasible with accuracy. From now on, the notation Spwill represent the improved quark propagator in the case of overlap quarks and the standard one in the case of clover quarks. Now there are still various ways to define the Z pand mpof Eq. (1) on the lattice. In both cases we fit the inverse quark propagator by ~ S1pa;bZ pi 6pmp2 (24) according to Eq. (1) and where  p1 asinap. We write Z pand mpbecause of the loss of the Lorentz invariance. Z pcan then be written as: Z p1 12 Tr pS1p= p2:(25) We could have chosen instead of  pany other definition equivalent to pup to terms of order Oa2, for example por 2=a sinap=2. It is important to stress that after eliminating the hypercubic artifacts as explained in Sec. IV, the different definitions give similar results; however, the definition we choose here permits to reach the final result with smaller hypercubic corrections. Note that in our first gluon papers, we did not extract the hypercubic artifacts in this manner, and therefore the choice of an appropriate definition for the lattice gluon propagator was very important, as well as the one of the gluon vertex function. The three point Green functions with vanishing momentum transfer are computed by averaging analogously over the thermalized configurations and the points in each orbit Gp; q 0h5~ Spy5~ Spi;(26) where the identity S0;x5Syx; 05has been used. The vertex function is then computed according to Eq. (5) and we choose for the lattice form factor g1 g1p21 36 Trp; q 0 p  6p  p2;(27) where the trace is understood over both color and Dirac indices. Finally, according to the Ward identity (9) we compute ZMOM Vsimply from Eq. (14) where any effective p2depen- dence of ZVshould come only from lattice artifacts. C. Classification of remaining artifacts The question of eliminating lattice artifacts has been perhaps our main difficulty in this work. Let us stress that the discretization artifacts we will discuss are all due to the QCD interaction. Indeed, our definition of Z p2is such that it is equal to 1, as in the continuum limit, when interaction is switched off, and we have also g11in that case. This illustrates that, in general, it may not be sufficient, by far, to extract the free case artifacts, or to use such prescriptions as replacing pby  p. Note that we have used Oaimproved actions. But of course, there still remains artifacts of higher order in the lattice spacing, and furthermore, the Green functions are not thereby improved. In the overlap case, a simple definite recipe exists, as we have explained above, to improve exactly the propagator and the other Green functions at large p, but obviously Oaartifacts at small pand Oa2 remain. Since we will have a detailed discussion, it is useful first to remind of the main species of artifacts which are expected. First, we have discretization artifacts, 3 which themselves split into two: (i) hypercubic artifacts, treated in Sec. IV which are the most visible because they break the elementary O4symmetry; they are seen, as we plot invariants of Green functions as a function of p2, as a large discrepancy between the value for different orbits at the same p2. With some simple democratic treatment, it is easy to get a relatively regular function of p2. Nevertheless, in general, there remain nonanalytic oscillations, and to eliminate them will be shown to be very important to obtain the final physical result, and requires sophisticated methods. (ii) O4invariant discretization artifacts, which remain after elimination of the hypercubic ones, and which will be discussed in Sec. V. Let us say that this is the weakest point, because we do not have theoretical principles to determine their form, neither is there a systematical empirical method to determine them. Second, there may be finite volume artifacts which will be discussed in Sec. VI, and very shortly since they do not seem sizable. 3 We will use the term ‘‘discretization artifact’’ preferably to the other common one, ‘‘ultraviolet artifact’’ because, as we shall find, these artifacts may show up at small pas well, due to nonperturbative effects. PH. BOUCAUD et al. PHYSICAL REVIEW D 74, 034505 (2006) 034505-6 IV. ELIMINATION OF HYPERCUBIC LATTICE ARTIFACTS A. Generalities Since we use hypercubic lattices our results are invariant under a discrete symmetry group H4, a subgroup of the continuum Euclidean O4, but not under O4itself. This implies that lattice data for momenta which are not related by an H4transformation but are by a O4rotation will in principle differ. Of course this difference must vanish in the continuum limit, i.e. when a!0. It must be considered as one species of the so-called discretization effects, or ultraviolet artifacts. They also may be termed as ‘‘anisotropy artifacts.’’ The hypercubic artifacts turned out to be particularly striking for Z . The results, including overlap-computed quantities, have already been presented in [18]. The raw lattice data for Z and ZMOM V exhibit dramatically the ‘‘half-fishbone’’ structure which is a symptom of strong hypercubic artifacts, and we recall that these effects are especially strong in the overlap case. Altogether we would like to recall the following hierarchy: first, the hypercubic artifacts are 1 order of magnitude larger for overlap quarks than for clover ones. Second, for both types of quarks the hypercubic artifacts for Z are 1 order of magnitude larger than those for g1. These hypercubic artifacts have been of course a long standing problem in lattice calculations, and methods have been devised for a long time to handle them. A general idea has been the so-called democratic one: the hypercubic effects are minimal when the four components of the momentum for a given p2do not differ too much (this is democracy between components; ideal democracy is for diagonal p/1;1;1;1). Then the question is how to make this criterion quantitative, the rationale being to find a compromise between two contradictory requirements: (1) to be as democratic as possible, which tends to reduce the number of points and (2) to retain enough points to have a real curve. The precise criterion is often something of a secret recipe, not communicated in papers. On studying the gluon propagator, the authors of Ref. [35] have made explicit a selection method, keeping only the orbits having a point within a cylinder around the diagonal. Several other similar criteria have been written. As it can be seen, the compromise is never very good, leaving rather large oscillations in Green functions. Moreover, it is not suitable for a precision treatment. The alternative idea which we propose, on the contrary, relies on the use of all the orbits, and a method to extract the physical point from an extrapolation of the different orbits. 4 A first successful application of this idea was for the gluon propagator [36]. B. Our method: the p2nextrapolation method We use here the final refined form of the method, the socalled p2nextrapolation method, presented in [18], and which has been shown to be necessary to obtain satisfactory results for Z . In order to perform a global fit, we start from the remark that we are dealing with dimensionless quantities g1and Z . It is thus natural to expect that hypercubic artifacts contribute via dimensionless quantities times a constant. 5 Next we assume that there is a regular continuum limit. We denote the generic Green function as Q; it depends a priori on p2, but also a2p4,a4p6:Qp2;a 2p4;a 4p6; a4p4;, where p2nP1;4p2n ; we Taylor expand it around the O4symmetric limit Qp2;0;0;0;.Of course we must truncate this Taylor expansion of Qin a and we choose to expand it up to a4. Note that at this stage, the function Qp2;0;0;0; may still depend on a through terms of the form a2p2, etc., but we do not consider presently this further dependence; it will be discussed in the next sections. As a conclusion, we have fitted our results over the whole range of p2according to the following formula, where the ci’s are constants independent of p2: Qp2;a 2p4;a 4p6;  Qp2;0;0c1 a2p4 p2 c2a2p4 p22c3 a4p6 p2 c4a4p4;(28) with indeed small 2’s. We have also checked the validity of this expansion for the free propagator. The functional form used for Qp2;0;0does not influence significantly the resulting artifact coefficients. We can even avoid using any assumption about this functional form by taking the value for Qp2;0;0at each p2as an independent parameter to be fitted. 6 C. Quantitative comparison with the democratic method We would like now to recall the quantitative comparison of our ‘‘ p2nextrapolation method’’ Eq. (28) with the more common ‘‘democratic selection’’ methods; the latter method is carefully defined in [35]. This comparison is important, since almost all works up to now are using some variant of the democratic method, and since the difference with this method is crucial, as we show, to extract power corrections. Let us consider the overlap case. If we try to select [35] the orbits which are in a cylinder around the diagonal with a radius 2=L, this is too restrictive anyway for our 164 4 The initial idea is due to Claude Roiesnel. 5 We neglect a possible logarithmic dependence on p2. 6 We have enough data for that. ARTIFACTS AND hA2iPOWER CORRECTIONS: ... PHYSICAL REVIEW D 74, 034505 (2006) 034505-7 overlap case, where we have only 22 orbits at the start. In order to have a less restrictive democratic criterion and to make a bridge with our own method, we will use the p2n’s defined above. In our language, democracy can be translated as having a small enough ratio p4=p22. Momenta proportional to (1, 1, 1, 1) and (1, 0, 0, 0) have ratios 1=4 (minimum ratio, maximally democratic) and 1 (maximum, totally undemocratic), respectively. We then retain the democratic orbits defined by an intermediate p4=p22 :5. This leaves already only 7 orbits out of 22 for every . In Fig. 1we plot for Z the result of this selection, as compared with our own method. Figure 1clearly shows visible oscillations in the democratic curve demonstrating that the hypercubic artifacts have not been totally eliminated, while after applying the p2nextrapolation method, Eq. (28), the curves are perfectly smooth; they do not either exhibit the oscillations which remain in previous methods. We prefer our own method for this reason and also because of the loss of information due to the rejection of ‘‘undemocratic’’ points, which leaves one with very few points. This appears crucial in the overlap case, where one is constrained to use small lattices. Then another very important aspect appears: while completely eliminating the hypercubic oscillations, we also considerably modify the mean value of the curve, as defined by an analytic fit; for the present case, as seen in the figure, our curve is considerably lower, with a quite steeper descent than the democratic one. We will return later to the fact that ZMOM Vremains not at all constant in pafter this treatment. D. The importance of optimizing the elimination of hypercubic artifacts Let us stress then that, as is particularly visible from this comparison, the difference of the a2p2nmethod with the standard democratic method is not at all academic in the context of the study of power corrections and renormalization constants. The difference with the previous versions of our method is also not negligible, as we have found. (1) It is obvious that with the standard democratic method, we would obtain quite different results for power corrections and O4symmetric artifacts, and therefore for the resulting perturbative contribution. In fact, it has not even been really considered that Z could be affected by such power corrections. (2) Moreover, we observe that the power corrections, as well as the residual O4symmetric discretization artifacts, extracted by the previous variants of our method for treating the hypercubic artifacts are not the same; indeed, when using a previous cruder treatment for overlap action, we have found an important a2p2 artifact, which disappears with the more refined treatment, and we were also finding different power corrections (with a weaker condensate). The clover case shows similar spurious issues. This means that for a too crude treatment, some hypercubic artifacts can be spuriously mimicked as part of O4symmetric discretization artifacts or continuum power corrections. This does not imply that the determination of power corrections is uncertain in this respect, but rather that it is very important to push hypercubic artifact elimination to the best to obtain the genuine continuum power corrections. Let us finally mention an interesting consequence; as is well known, the values of Green functions at different momentum points in a Monte Carlo lattice calculation are highly correlated, which should lead to very small 2=d:o:f:for fits describing the pdependence by smooth analytic expressions (by small, we mean well below one). It is not found so with too crude treatments of the hypercubic artifacts, because of the erratic oscillations which always remain in the latter methods mimicking statistical deviations. We observe however an impressive decrease of the 2down to its expected small value when we improve the treatment of the data, showing that we are now obtaining indeed very smooth functions as physically expected. Another encouraging fact is that, after having eliminated consistently hypercubic artifacts, the different definitions of Z on the lattice converge to the same final result (see the comments around Eq. (24)). V. PROOF OF THE PRESENCE OF O4 SYMMETRIC ‘‘NONCANONICAL’’ DISCRETIZATION ARTIFACTS We now start, in the rest of the discussion, from the data obtained through the above treatment of the hypercubic artifacts. They still differ from the continuum by renor- 0123456 p 1.5 2 2.5 3 3.5 Zψ β = 6.0 without hypercubic artefacts β = 6.4 without hypercubic artefacts β = 6.0 democratic selection β = 6.4 democratic selection FIG. 1 (color online). Comparison between the democratic method and our refined treatment for the overlap Z p2at 6.0 and 6.4. Above 2 GeV, precisely where one must study the power corrections, the difference is striking. PH. BOUCAUD et al. PHYSICAL REVIEW D 74, 034505 (2006) 034505-8 malization and by O4symmetric discretization artifacts. It happens that the determination of these artifacts is still harder in general than for hypercubic ones (as for finite volume artifacts, which we estimate to be weak, see the corresponding short section below). Indeed, in practice, there is no similar unambiguous method to determine the O4symmetric discretization artifacts. True, they manifest themselves by a residual variation with , and, in principle, we could study the variation with at each momentum, and then extrapolate to the continuum. However, this requires too many momenta and too much accuracy, if we want really to extract the power correction from the extrapolation to the continuum. (a) A possible general method for treating O4scalar artifacts. Then, a more practical method consists in assuming a prescribed analytical form for both the continuum and the artifacts, with some unknown parameters to be determined by the 2adjustment. One has therefore to appeal to our a priori knowledge (a) of the continuum, as a function of p; (b) of the structure of O4symmetric artifacts, as a function of pand a, so that we could make fits with prescribed functions depending on a limited number of free parameters. For the continuum, this is exactly what is provided by the OPE, with the renormalization constants zband hA2ias free parameters. For O4symmetric artifacts, a standard idea is to recourse to lattice perturbation theory, in which the structure of artifacts is easily explicited. After all, this is what we have invoked for the hypercubic artifacts. As for O4symmetric artifacts, the result is quite simple: in the case of a scalar function, and in the chiral limit, where there is no other dimensioned parameters than aand p, there can be no other artifacts than anpn, with n>0. Then we could work with a few parameters only. (b) Why it fails. But now, there appears a very unfortunate circumstance. It is seen that this usual assumption of a perturbative structure of artifacts does not work at all, at least in general. We observe undoubtedly O4 symmetric artifacts decreasing with p, i.e. for instance of the form 1=pnand n>0times some positive power of a. To show it, the best way is to consider the pdependence of ZMOM Vg1=Z , which should be momentum independent close to the continuum limit. Any pdependence is therefore to be attributed to artifacts. Now, the lattice data show quite clearly a very strong pdependence of ZMOM Vexcept at large p. This is true for clover action, and for overlap action as well, see Fig. 2 for overlap fermions. The artifacts are similar; they decrease monotonously with increasing pfor 6:4, 6.6, 6.8; they could be fitted by negative powers of the momentum squared; for the overlap case, this is also true for 6:0. (c) Consequences for determination of power corrections. The necessity of taking into account such negative powers of pis very embarrassing because, as we explained, we cannot distinguish the O4symmetric discretization artifacts by the sole dependence; we have to rely on the pdependence. Now it is clear that we will have much difficulty in distinguishing the power corrections and the artifacts decreasing with p, and therefore to determinate the condensate. One of the problems is that when increasing the number of negative power terms in the description of the artifact, the tendency is to get alternating signs, and therefore rather unstable results. There is no rationale as to where we should stop. In addition, we must consider that we have also as parameters in the fit the renormalization constants zb, which are practically chosen as an independent parameter at each . To add to the uncertainty, we observe that within the precision of the data, we can obtain equivalently good solutions by modifying the O4 artifacts with a correlative change in the zb’s. Then, in spite of many efforts, we have in fact not been able to extract stable and accurate values of hA2iby this method, although we have gotten a clear signal that it is positive and sizable. Then we have been able to fix the continuum power corrections only by 0246810 p 1.7 1.95 2.2 2.45 2.7 ZVβ = 6.0 β = 6.4 β = 6.6 β = 6.8 FIG. 2 (color online). Overlap ZMOM Vas a function of momentum, respectively, for 6:0and 6.4, and 6:6and 6.8, for the overlap action. The huge momentum dependence which is displayed must be entirely attributed to discretization artifacts. ARTIFACTS AND hA2iPOWER CORRECTIONS: ... PHYSICAL REVIEW D 74, 034505 (2006) 034505-9 tive estimates made by various authors within analytical approaches. One will find abundant references in the paper of Dudal et al., Ref. [42]. It is clear that this comparison must take care of the precise definition of the condensate, as regarding instance renormalization. B. Systematic errors on hA2i Of course, we are making many assumptions which introduce uncertainty in the value of hA2i. Recall that we do not claim to determinate it only from the present study, since we have several previous determinations from the gluonic sector. So we can also appreciate systematic errors from the consistency we obtain with these previous estimates (see above, Sec. VIII C). In fact, as we have observed in Sec. VIII C, the gluon propagator determination and the one from sare notably different, but the values are affected by very large errors, and are compatible with the present ones. From all the determinations, we could conclude that the systematic errors do no seem to exceed 1 GeV2. But there is in fact an important source of the systematic error, which is explained below, and which cannot be estimated by comparison with gluons because it is present in both: it is the fact that the Wilson coefficient is calculated only at low order. Then, it remains useful to discuss the sources of errors inside the quark sector itself, for which anyway the conditions are intrinsically very favorable. Since we do not claim to do phenomenology, but rather an exercise in quenched QCD, we have not bothered with the quenched approximation, which is also supposed for gluons. Chiral limit is assumed on the theoretical side, for instance to calculate the Wilson coefficients. Now, of course, we do not work at zero quark mass on the lattice. We have not tried to do a systematic chiral extrapolation on the lattice data, which would only lead to increase the statistical errors. We observe that Z seems very weakly dependent on our set of masses, which means that this limit is not a priori a problem at the smallest mass (at which we have made all our OPE analysis). Anyway, we can discard any catastrophic effect at very low quark masses through the consistency with the quenched gluon data. Let us now pass to more relevant effects. Some come from the treatment of artifacts, for which we lack theoretical basis, some are relative to our description of the continuum, which, although based on a much stronger theoretical basis, involves necessarily approximations. We do not return to finite volume artifacts, since we have nothing quantitative to say about their magnitude in the overlap case. For what concerns hypercubic artifacts, we may have an idea on the error remaining in their treatment by the variation observed with two variants, with a slightly different description of the continuum limit: hA2i3:23 3:0GeV2:(45) Whenever large O4symmetric artifacts are present, we are compelled, as we have seen, to rather arbitrary assumptions on their structure, since we have concluded that we cannot rely on lattice perturbation theory. The correlated errors in our determination of the ‘‘nonperturbative’’ artifacts and of the condensate seem very large, as judged from the range of values obtained in various fits, to such a point that we have renounced to extract any number in this case. Therefore, we consider only the case where we have strong hints that the artifacts are small, in which case we have mainly to consider uncertainties in the continuum description. They are themselves of two origins: the perturbative calculation of Wilson coefficients; the nonperturbative aspect, i.e. the relevance of the OPE expansion, the enumeration of operators, etc. As to uncertainties in perturbative calculations: (i) We have checked that computing the perturbative contribution to third or fourth order in perturbation does not change significantly the estimated condensate (only 7% of change). Another test is to reexpress the series in terms of MS instead of MOM.Z changes by less than 1% with various prescriptions. We can thus assume that the perturbative contribution has been expanded far enough. (ii) As we have explained in Subsection II C, the problem is much more important for the Wilson coefficient of the A2operator which has, on the contrary, only been computed to leading logarithm. A sign of this problem is seen by changing MOM into MS.A change of MOM into MS reduces it by 40% s10 GeV, and more for the smaller momenta; whence a reduction of the Wilson coefficient by 50% in average. Through a conspiracy with the smaller change in the perturbative contribution, this change amounts to an increase of the resulting condensate by 70%. Of course, similar effects are present for gluons, and, as we have explained, the ratio of condensates obtained from quarks and gluons will remain the same. More importantly, one must be aware that the power correction by itself remains well determined; what is not well determined is the translation of the power correction into ahA2icondensate value. Indeed, this translation depends on the theoretical evaluation of the Wilson coefficient, which is not accurate at present. As to the properly nonperturbative aspect, we may think of two sources of uncertainties in determining hA2i. The one stems from other operators which could enter with the same power in the OPE expansion. However, we do not find any such operator contribution in Z in the chiral limit. We could have some contamination since we are not exactly in the chiral limit, but it must be small, since we observe a very weak quark mass dependence. The other could be the possibility that pis not sufficiently large for PH. BOUCAUD et al. PHYSICAL REVIEW D 74, 034505 (2006) 034505-16 the leading correction 1=p2to completely dominate over the next ones. This possibility is represented by the 1=p4 term and the fact that it is small but nonzero in the fits shows indeed that we are not completely asymptotic at such large momenta; although it is very small, around 1% at 3 GeV, it leads to a change of around 15% in hA2i. On the other hand, with this term included, we find a very good stability of hA2iover the large range 2:6 GeV <p< 10 GeV, when varying the fitting window, which suggests that we have correctly accounted for the small subasymptotic effects. Actually, this term could also mimic a neglected logarithmic dependence; indeed, passing from MOM to MS as expansion parameter as explained above, the 1=p4term passes from 1:1=p4GeV4to 0:65  0:34GeV4=p4, therefore there is an appreciable variation, although the sign and order of magnitude are encouragingly stable. On the whole, the dependence of the Wilson coefficient on the scheme for sseems to be the most worrying source of uncertainty, yet it can be solved soon. Another concern is the value we use for QCD, conventionally converted to the MS scheme. Let us vary by 10% our MS 0:237 GeV. We find: MS 0:215–0:260 GeV !hA2i3:45–3:02 GeV2; (46) a quite moderate change indeed. There is naturally an increase for decreasing QCD because the larger power correction compensates for the slower falloff of the perturbative part. C. General consequences for the nonperturbative MOM renormalization approach True, the direct object of this study has been to verify the consistency of our OPE analysis of lattice data by extending it to the quark sector, and then comparing with the previous analysis of the gluon sector; and thereby to assess the soundness of our statement of large power corrections in ‘‘elementary’’ Green functions as well as of their interpretation in terms of the nongauge-invariant condensate hA2i. Now, one must also be aware of the strong consequences of this study, as well as of the preceding ones, on other problems, especially in precision studies of physical quantities. These consequences should be then reexamined in unquenched calculations. A first example is the determination of QCD in the gluonic sector; a purely perturbative fit leads to a first determination of QCD; the necessary inclusion of a 1=p2 power correction induces a striking modification in the value of QCD, as was found some years ago [19]. In general, important consequences are expected on quantities requiring a renormalization procedure. Very commonly, the renormalization is performed in the MOM method, which uses elementary Green functions in momentum space to calculate the renormalization factors. The naive MOM method would just start from raw lattice data, with minimal improvements such as a democratic selection and choice of an optimal momentum window. Then, our study makes clear the difficulties of this naive method, which render impossible to get accurate results at the precision of a few percents. The present study allows some improvements, but fails in extracting the O4artifacts. Let us enumerate the obstacles in the naive method and those who remain in spite of our attempts to improve it: (1) Power corrections.—First, we have a problem independent of the discretization of the action: the presence of the power corrections has the effect of modifying the lattice estimate of the renormalization constants defined in the perturbative regime. Indeed, only perturbative renormalization schemes can be connected between one another by analytical calculations (e.g. the MS quark mass), and also connected to Wilson coefficients in order to produce physical quantities (e.g. weak four-fermion interaction). Now, the lattice measurement can be performed only at rather small momenta, where power corrections are present. These corrections must be subtracted, especially if one requires a good precision. This extraction is not performed in the naive method. Such power corrections affect Z and various vertex functions. From the Ward identity, it happens that for quantities like ZV, the A2power corrections cancel between the vertex function and Z . But this is not true in general: in ZSand ZPfor instance, this cancellation does not occur. In such cases, the power corrections must be subtracted, in principle. Of course, one may wonder whether this is practically important. It depends on the accuracy we want to obtain. If we aim at a precision of a few percent, certainly we do require to take them into account, since they reach several percent around 4 GeV, 5% on Z in the clover case. Now, one often claims to go below 10% with dynamical quarks; then, such effects are deserving of consideration. What simplifies somewhat the problem raised by power corrections is that not only are they independent of the chosen discretization of the action, but they are often related to the same hA2icondensate, at least as regards the dominant power. Once hA2ihas been confidently determined by one analysis, it can be used for other Green functions. (2) the discretization artifacts that we have found set a more general difficulty, and one which is more embarrassing, in view of our lack of theoretical control. We have no reason to suppose that the special case we have studied is especially cata- ARTIFACTS AND hA2iPOWER CORRECTIONS: ... PHYSICAL REVIEW D 74, 034505 (2006) 034505-17 strophic. Yet, it shows already an embarrassing situation. —The first step is to have control on hypercubic artifacts to a good accuracy. It is deserving of mention that this accuracy cannot be obtained by the naive method of selecting democratic points; since the determination of renormalization constants relies on measures at large p, the error is particularly large. Moreover, the simplest version of our alternative method of ‘‘restoration of O4symmetry,’’ the old ‘‘p4’’ method, has not allowed one to get a good accuracy. We have had to go further with the p2n method. This requires already a good deal of work. But, finally, it seems that we are able to produce a systematic and accurate procedure. —On the other hand, we have a clear proof from ZV that there are large O4invariant artifacts at small p. This type of large artifacts seems to be a problem specific to the quark sector: we had not found similar evidence for gluonic Green functions. We do not have any theoretical mastering about the magnitude of these artifacts. It is also surprising that ZVbecomes relatively flat at large p. This suggests, but does not prove, that these artifacts are small at large p. All this is counter to the expectation of perturbation theory which would predict artifacts increasing with p. This lack of theoretical understanding will remain a problem for all the MOM determinations of renormalization constants. Indeed, we have not obtained on the other hand an empirical method which would be safe and systematic, to eliminate this new type of artifacts. This is disastrous. It also renders difficult the extraction of the power corrections, as we have observed in the clover case. The choice of the overlap action may help somewhat, but does not offer a general solution. Indeed, for Z , the overlap action seems to present small O4invariant artifacts. It introduces larger hypercubic artifacts, but this is not a decisive obstacle, as we have explained how to eliminate them accurately. Then one would conclude that all in all, the overlap action has an important advantage. Yet, it is weakened by our lack of theoretical understanding of the underlying reasons why it is so, which precludes any estimation of accuracy (calculations with other values of 1smay reveal instructive in this respect). Moreover, the problem remains entire for ZVand other vertex renormalization constants. Taking into account the large uncertainties coming from O4invariant artifacts, the MOM method for renormalization, even with the improvements we have proposed, may reveal not very practicable for precision calculations, although it is appealing by its simplicity, and quite efficient for ordinary purposes. Of course, at this point, one may think of the method of the ALPHA collaboration as a complementary one, technically difficult, but which allows a very clean treatment of discretization artifacts by using on shell quantities, and also allows to work at a very high energy scale, therefore eliminating power corrections (see, for example, for ZV,[43]). 8 ACKNOWLEDGMENTS We thank Michele Pepe, Damir Becirevic, and Claude Roiesnel for precious discussions and comments, and M. Pepe for help when initiating this work. Alain Le Yaouanc would like to thank Jean-Rene ´Cudell and Carlotta Pittori for many very useful discussions in a previous collaboration on the quark propagator. This work has been supported in part by the European Network ‘‘Hadron Phenomenology from Lattice QCD,’’ No. HPRN-CT-2000-00145. We have used for this work the APE1000 hosted by the Centre de Ressources Informatiques (Paris-Sud, Orsay) and purchased thanks to funding from the Ministe `re de l’Education Nationale and the CNRS. APPENDIX A: PERTURBATIVE EXPANSION TO FOUR LOOPS IN THE FULL MOM SCHEME Our aim is to express Z p2, in terms of MOM s, defined by the triple gluon vertex at symmetric momenta. More precisely, we are looking for the renormalization group improved expression, which resums the large logs logp2=2at large p. The expression then takes the form of a series in MOM spwith pure number coefficients. This series can be obtained from the knowledge of the anomalous dimension of Z in terms of MOM s, and of the function of the MOM scheme. This is possible in the Landau gauge to the order of four loops included, thanks to the papers of Chetyrkin and collaborators, Refs. [26,27]. First, one has in Sec. IV B of the first paper the expansion of @lnZ 2 @ln2as a series in sof the MS scheme to four loops (in fact the authors consider the inverse Z22 Z 21, so we have to invert their formula). Since there is no sterm in the Landau gauge, to reexpress the series in terms of the sof the MOM scheme at four loops requires the expansion of MS sin terms of MOM sonly to order three included. This expansion is provided by Sec. V of the second paper, by inverting the first equation of this section. On the other hand, we need the MOM function. It is given in the second paper, Sec. VI, at three loops, which is also sufficient to calculate the renormalization improved series for Z p2at four loops included; indeed, the four-loop coefficient of the function enters only as a factor of the one-loop anomalous dimension of the fermion 0, which is zero in the Landau gauge. We then obtain the following expansions: 8 After having issued the arXiv version, we have been advised of the paper [44] which has studied previously the same problem of extracting the A2condensate from the quark propagator. PH. BOUCAUD et al. PHYSICAL REVIEW D 74, 034505 (2006) 034505-18 Z is expressed in general as: Zpert 0=0101210 42 0 0:52012102 164 02 82 10020401122 02 323 02 3 7686 03163 13 02402 12 021 122 010162 13202411 62 124 023021412 3123 0162 11320123 1 4122032 10245 03: (A1) For nf0in the Landau gauge, we get: 00:0167 3; 294:7943 314 503:7:(A2) , which represents here MOMp(defined through the three-gluon vertex at symmetric momenta), is given, under the same conditions, by: p4 0t81 0 lnt 0t21 0t3 22 0162 1 2 0lnt2lnt1 1 0t4323 1 3 0lnt3803 1 3 0lnt2 12012643 1 3 0 lnt 22 03163 1 3 0;(A3) where tlnp2=2 MOM(in the MOM case, one stops at the 1=t3terms, and where 011; 151; 23072 10%:(A4) APPENDIX B: A2WILSON COEFFICIENT IN THE QUARK PROPAGATOR In order to renormalize the bare quark propagator in Eq. (15) we will define the following two renormalization constants, both in the momentum-subtraction (MOM) general scheme: Z 2 Zpert 2 ! a;b i6pmp2 p2mp2 S1p S1 pertp ! p22 : (B1) The constant in the top of the left-hand side of Eq. (B1) includes the nonperturbative contributions to the quark propagator to let it take the tree-level value all over the energy range (not only in the perturbative regime). We renormalize Eq. (15) by multiplying by that of the bottom (this purely perturbative MOM renormalization constant, computed to four loops in Ref. [26], is presented in Appendix A); as to the Wilson coefficient of A2,we calculate it at leading RG order. Zpert 21S1pi6pa;b Z p2 Zpert 2 Zpert 21S1 pertp i6pdp2 2; p2hA2Ri 4N2 C1a;b  i6pa;bZpert p2 Zpert 2dp2 2; p2 hA2Ri 4N2 C1;(B2) where A2R2Z1 A22:A2 bare :, 9 and where we only write the leading terms in 6p. Concerning dp2 2;, the same procedure used for gluon Green functions in Refs. [20–22] is in order here. Then, from Eq. (B2), multiplying by Zpert 2, and taking logarithm derivatives on  in both sides, we obtain the following RG equation: 0  40 A2  4d dln2dp2 2;0 (B3) that is identically satisfied with dp2 2;d1;p p00 A2=0;(B4) where d1;p is in fact beginning with pand 0 A2 is defined through d dln2lnZA20 A2  4O2;(B5) 9 the:  fixes the normal order subtracting the additive divergencies in A2 bare (see discussion in Ref. [45]). ARTIFACTS AND hA2iPOWER CORRECTIONS: ... PHYSICAL REVIEW D 74, 034505 (2006) 034505-19 and where all the involved one-loop coefficients are well known, 0 A235Nc 12 ; 00; 011:(B6) On the other hand, d1;p can be obtained by computing the only diagram involved in the ‘‘OPE business’’ for this case, → d(1, α (p)) = (N 2 C −1) 3g 2 (p). 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