Symanzik improvement with dynamical charm: a 3+1 scheme for Wilson quarks

Journal of High Energy Physics, Jun 2018

Abstract We discuss the problem of lattice artefacts in QCD simulations enhanced by the introduction of dynamical charmed quarks. In particular, we advocate the use of a massive renormalization scheme with a close to realistic charm mass. To maintain O(a) improvement for Wilson type fermions in this case we define a finite size scheme and carry out a nonperturbative estimation of the clover coefficient csw. It is summarized in a fit formula csw(g 0 2 ) that defines an improved action suitable for future dynamical charm simulations. Open image in new window

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Symanzik improvement with dynamical charm: a 3+1 scheme for Wilson quarks

Received: May Symanzik improvement with dynamical charm: a 3+1 ALPHA 0 1 2 Patrick Fritzsch 0 1 2 5 Rainer Sommer 0 1 2 3 4 Felix Stollenwerk 0 1 2 4 Ulli Wol 0 1 2 4 0 Newtonstr. 15, 12489 Berlin , Germany 1 Platanenallee 6 , 15738 Zeuthen , Germany 2 1211 Geneva 23 , Switzerland 3 John von Neumann Institute for Computing (NIC) , DESY 4 Institut fur Physik, Humboldt-Universitat zu Berlin 5 CERN, Theoretical Physics Department We discuss the problem of lattice artefacts in QCD simulations enhanced by the introduction of dynamical charmed quarks. In particular, we advocate the use of a massive renormalization scheme with a close to realistic charm mass. To maintain O(a) improvement for Wilson type fermions in this case we de ne a nite size scheme and carry out a nonperturbative estimation of the clover coe cient csw. It is summarized in a t formula csw(g02) that de nes an improved action suitable for future dynamical charm simulations. Lattice Quantum Field Theory; Nonperturbative E ects - Collaboration 1 Introduction 2 3 5 6 1 3.1 3.2 3.3 4.1 4.2 4.3 Mass independent case Mass dependent case A 3+1 avor scheme 4 Scaling at constant nite volume physics Finite size de nition of the LCP Lattice realization of the LCP Determination of csw Conclusion Introduction Symanzik improvement Mass (in)dependent renormalization scheme and improvement Physicists have been rather fortunate that perturbation theory in the form of low order Feynman diagrams has been largely su cient to practically establish today's highly successful Standard Model of elementary particles from experiment. Due to asymptotic freedom even the strongly interacting sector of con ned quarks and gluons (QCD) could be pinned down in this way, based on high energy scattering of weakly interacting probes (e.g. e+ e and e p). The thus completely characterized theory is expected to also produce the hadronic bound states observed in nature. To extract these predictions | and thus ultimately validate the complete theory | nonperturbative techniques become indispensable. The only known nonperturbative de nition of QCD is the regularization on a lattice which in particular provides a systematic way to extract nonperturbative information about hadrons from the theory by Monte Carlo simulations of QCD on sequences of nite computational lattices. This in icts however several unavoidable distortions of the theory which need to be controlled. The lattice spacing a acts as an ultraviolet regulator that has to be extrapolated to zero up to tolerable errors. A nite system length L is introduced and has to be made e ectively in nite in a similar sense. Finally | as in experiment | all predictions come with statistical errors that have to be small enough for signi cant comparisons. In particular the symmetries of QCD get modi ed by the lattice regularization and some of them only re-emerge in the continuum limit a ! 0. Very obviously this holds true for in nitesimal translation invariance. Wilson [1] showed that gauge invariance can { 1 { parallel transporters around plaquettes or other small closed loops as we shall detail below. For the quark degrees of freedom more subtle questions arise. The naive gauge covariant discretization of the rst derivative in the Dirac action leads to fermion doubling, i.e., an unwanted proliferation of the degrees of freedom. It has turned out that this is in fact related to a certain principal incompatibility between the lattice discretization and chiral invariance [2], which is a very important approximate symmetry in QCD. There are several possibilities to coexist with the corresponding no-go theorem [2]. The present article refers to the choice proposed also by Wilson [3] which nowadays is called Wilson fermions. By quenched quarks or heavy quarks treated in a mixed action approach. References [10{20] aim at a reduction of the quadratic and higher, n 2, terms by tree-level and one-loop perturbation theory. We here focus just on the non-perturbative removal of the linear term in amc. This paper is a highly condensed summary of the PhD thesis [21] by Felix Stollenwerk. In section 2 we review on-shell Symanzik O(a) improvement followed by the comparisons { 2 { of mass dependent and independent renormalization schemes in section 3. In section 4 we de ne a line of constant physics by maintaining a set of nite size observables as a is changed. In section 5 this is employed for a nonperturbative determination of the improvement coe cient csw in a scheme of three light avors together with the close to physical dynamical charm quark. A t formula is presented that allows to implement this action in future large volume simulations at and close to the physical point. We end with some conclusions in section 6. 2 Symanzik improvement HJEP06(218)5 Symanzik has put forward a method [22{24] to accelerate the continuum limit in lattice eld theories by increasing the power of a characterizing the leading lattice artifacts. It is routinely applied to Wilson fermions to achieve an a2 continuum scaling. This so-called Symanzik improvement programme starts from an e ective continuum theory which rst only describes a given lattice theory including its leading order artefacts. In a renormalizable quantum eld theory we are at rst instructed to include in the action a combination of all possible terms up to (including) mass dimension four which possess the desired symmetry. They typically are small in number and are related to the number of free (renormalized) parameters. To reproduce the leading cuto e ects of O(a), the scope has to be widened to all such operators of dimension ve. They appear multiplied by dimensionless coe cient functions times an explicit factor of a. They could be determined by imposing a su cient number of conditions where observables of the lattice theory are matched. These are analogous to renormalization conditions to x the relevant terms up to dimension four. In a next step one then modi es the lattice action such that the dimension ve terms in the associated e ective continuum theory become zero. This can be achieved by including in the lattice theory a linear combination of discretized versions of the dimension ve operators with suitably tuned coe cients. Any ambiguity in this step only a ects the uncancelled higher order artefacts. What is exploited here is the structure of the e ective theory which allows to describe all O(a) artefacts by a nite set of parameters by organizing all possible terms according to their naive dimension. This could in principle be invalidated by large anomalous dimensions due to quantum corrections. For asymptotically free theories like QCD this does however not happen to any nite order of perturbation theory. In this sense the Symanzik programme is expected to work in a similar way as renormalizability and thus the existence of the continuum limit. In the following we shall restrict our focus on the simpler on-shell Symanzik improvement [25, 26]. This means that we only improve observables that are derived from correlation functions with operators separated by physical distances that remain nite in the continuum limit. Then the use of the quantum equations of motion is justi ed which allows to relate dimension ve operators to each other and thus to reduce the number of independent terms that have to be managed. As we shall report below we then need, apart from terms related to quark masses, only one single term, the Pauli or, on the lattice, the clover-leaf term [4] to cancel O(a) artefacts in QCD. { 3 { We now summarize the structure of the Symanzik e ective (continuum) Lagrangian that describes lattice QCD with Nf avors of Wilson quarks with a diagonal mass matrix M , including O(a) lattice artefacts, LSym = 1 2g02 tr[F and D= is the covariant Dirac operator. A subtlety here is, that due to the missing chiral symmetry of the lattice action, the adjustable constant is required to match the renormalization pattern of the lattice theory which di ers for the avor singlet and non-singlet components in M . The remaining lines show the dimension ve operators. We have already implemented simpli cations from using the equations of motion (on-shell improvement) here. In this way we could drop the terms a D2 , a D= M and atr[M ] D= . Similarly to the action also currents appearing in correlation functions have to be nontrivially matched by the Symanzik e ective theory. One has to allow for the mixing of terms of the same and the next higher dimension as far as they share all symmetries preserved by the lattice theory. We here restrict ourselves to the o -diagonal axial vector and pseudoscalar currents made of quarks of avor i 6= j, Aij = i 5 j ; P ij = i 5 j : Their mixing pattern is and (ASym)ij = f1 + !1 atr[M ] + !2 a(mi + mj )g Aij + !0 a@ P ij ; (PSym)ij = 1 + !10 atr[M ] + !20 a(mi + mj ) P ij : In continuum QCD chiral symmetry implies the PCAC relation for the renormalized currents. Formally, i.e., disregarding renormalization for the moment, it reads and it holds as an operator relation. This means that the above identity may be inserted in correlators with suitable operators O h Such relations are typically violated by O(a) artefacts in the lattice theory. They thus may serve as a source of relations which allow to determine coe cients of O(a) corrections in the Symanzik e ective theory and ultimately in the improved lattice theory. In mass independent renormalization schemes for QCD the conditions that de ne the renormalized theory are formulated at vanishing quark masses in terms of an external renormalization scale equations with only. This in particular leads to relatively simple renormalization group - and -functions depending on the dimensionless coupling only. In the technically convenient nite size schemes the system size L is often used as external scale. In this article we shall argue that in the presence of dynamical charmed quarks, mass dependent schemes, where renormalization factors are allowed to depend on quark masses, o er practical advantages. 3.1 Mass independent case HJEP06(218)5 In the case of mass independent renormalization schemes the relation between bare and renormalized gauge coupling reads (without improvement) gR2 = Zg(g02; a )g02; and Zg is determined in this way, once a suitable gR has been de ned. The relation between bare and renormalized masses has the structure mR;i = Zm(g02; a ) mq;i + (rm(g02) 1) tr[Mq]=Nf involving the bare subtracted quark masses of avor i mq;i = mi mcrit(g02) ; Mq = diag(mq;1; : : : ; mq;Nf ) : We remind here that both the necessity of the nonzero critical bare mass mcrit and the extra renormalization constant rm refer to our lattice regulator which breaks chiral symmetry. To implement Symanzik O(a) improvement the above relations have to be augmented by additional terms that are proportional to an explicit factor a [25]. In the case of the coupling this is of a relatively simple form. All we have to do is to replace g02 in (3.1) and (3.2) by g~02 = g02 1 + abg(g02) tr[Mq]=Nf ; and to add to our lattice action the `clover' contribution, rst introduced by Sheikholeslami and Wohlert (SW) [4], SSW = a5csw(g02) X x (x) i 4 F^ (x) (x) ; with F^ (x) being a lattice discretization of the (anti-Hermitian) eld strength tensor at site x. More details on the lattice action follow in the next section. This extra term corresponds to the contribution proportional to 0 in (2.1), while the bg modi cation in (3.4) represents the 5 part on the lattice. Sandwiching the matrix M + a f 1M 2 + 2tr[M ]M + 3tr[M 2] + 4(tr[M ])2g between and summarizes the remaining terms in (2.1). Following the notation of [27] the { 5 { (3.1) (3.2) (3.3) (3.4) (3.5) corresponding terms are included in the lattice theory by using an improved renormalized mass of avor i mR;i = Zm(g~02; a ) mq;i + rm(g~02) 1 tr[Mq] + a bm(g02)m2q;i + bm(g02)tr[Mq] mq;i Nf on the lattice. If now physical observables are parameterized by gR and mR;i then we expect them to converge to their continuum limits at a rate proportional to a2, provided that the prefactors of all improvement terms have the correct values to cancel all O(a) terms along 3, the mass dependent improvement terms have been found to lead to small e ects. Thus they could be set to their low order perturbative values without spoiling the targeted precision. Only the csw term (beside operator improvements to be discussed later) was found to be essential. For this reason the ALPHA collaboration and others have engaged in nonperturbative computations of csw(g02) for various numbers of light avors 0 Nf 4. They are found in refs. [28{34]. If we want to treat the physical charm quark in a mass independent scheme as discussed above, then there appear improvement terms multiplied by the charm mass amq;c. Also tr[Mq] will now be dominated by charm. A rough estimate puts these terms to around 0.5 and about an order of magnitude larger than the analogous strange contributions for lattice spacings on the order of 0:1 fm. Thus the respective b and other coe cients must be known much more precisely than for the light avors. This is even true for the virtual contributions in the absence of valence charm quarks. Even at the level a0 the term rm 1 is enhanced and very high precision for rm may be required. A simultaneous nonperturbative determination of all these improvement coe cients seems impractical and we hence rather give up the technical convenience of mass independent schemes and switch to massive ones. 3.2 Mass dependent case In massive renormalization schemes the Z factors are allowed to depend on the mass matrix M . We take and gR2 = Z~g(g02; a ; aM )g02 ; mR;i = Z~mi(g02; a ; aM ) mi m~ crit(g02; atr[Mq]) : (3.7) (3.8) Clearly all the correction terms proportional to amq;i in (3.4) and (3.6) can now be absorbed into the multiplicative Z~ factors. The term proportional to (rm 1) in (3.6) can be reproduced by the atr[Mq] dependent critical mass m~crit. It remains to keep the clover term (3.5) with a mass dependent coe cient function c~sw(g02; aM ). Note that the mass dependent O(a) modi cation of this improvement term formally only contributes at O(a2) and in this sense could be dropped. But in passing to mass dependent renormalization { 6 { we want to avoid these O(a2) e ects which are enhanced by the charm mass and keep the mass dependence of c~sw, too. Beside the de nition of renormalized parameters and the SW term the scheme has to include improved operators, and we focus on Aij and P ij as before. The de nitions (2.2) and (2.4) are now reinterpreted in terms of lattice elds all at the same site. By similar considerations as for the action we set and (AR)ij = Z~Aij (g02; aM ) Aij + ac~A(g02; aM )@ P ij (PR)ij = Z~Pij (g02; a ; aM )P ij (3.9) (3.10) HJEP06(218)5 for the renormalized and improved currents in our massive scheme.1 For the inclusion of an M dependence in c~A we refer to the earlier discussion for c~sw. 3.3 A 3+1 avor scheme So far we have assumed no degeneracies in our arbitrary quark mass matrix. For our later numerical work we shall nd it advantageous to reduce the number of relevant mass parameters by approximating nature by a simpli ed pattern. We shall be guided in principle by keeping the charm mass and the value of tr[Mq] close to their physical values. At the same time we deal with only two independent masses by setting Mq = diag(mq;l; mq;l; mq;l; mq;c) : (3.11) Beside the (subtracted) charm mass mq;c we have introduced a light quark mass mq;l which is imagined to equal approximately a third of the strange mass. We expect this scheme to be at the same time numerically manageable and close enough to the eventual target physics to avoid large mass dependent artefacts and renormalization e ects. In the following we shall be concerned with this kind of massive renormalization scheme and will omit again the tildes over the improvement coe cients cA; csw. Moreover we shall in particular embark on a nonperturbative calculation of csw(g02; aMq) with Mq chosen as in eq. (3.11). 4 Scaling at constant nite volume physics To prepare for a nonperturbative determination of csw we now envisage the construction of a set of lattice con gurations with bare parameters that correspond to a line of constant physics (LCP) with xed renormalized parameters in a nite volume scheme and a well 1We note that mass independent renormalization schemes have turned out to be very practical for QCD with Nf 3. While the large charm mass motivates to renormalize at nite mc, it is presumably of interest to keep mu = md = ms = 0, de ning all Z-factors at such a point where less parameters have to be tuned. In practice this means that renormalization factors and improvement coe cients are de ned at the point Mq = (0; 0; 0; mq;c) and are thus functions of amq;c. At this point they can be determined exactly as in an Nf = 3 theory, as long as the elds do not involve the charm quark. The small O(amq;l) change in csw amounts to a negligible overall O(a2) term. For elds containing charm, special renormalization conditions have to be imposed. { 7 { de ned discretization. This is done for a number of lattice spacings a and csw values in the relevant range. The quark masses for Nf = 3 + 1 are tuned to the vicinity of the values described at the end of the last section. In the next section we consider scaling violations in the PCAC relation (2.5) to single out `optimal' csw values for each of our lattice spacings which can then be represented and interpolated by a smooth t formula. We implement Schrodinger functional (SF) boundary conditions [35, 36] to set up our nite volume scheme. The 3+1 avors of Wilson quarks are coupled to gluons weighted with the tree level improved Luscher-Weisz gauge action [26]. Instead of compiling all details of these de nitions we refer here to [34] where almost the same setup has been used. The only small di erence is that for the fermionic boundary improvement coe cient ~ct, that is called cF in [34], we use the tree level approximation c~t = 1. We choose equal temporal and spatial extents T = L. The boundary gauge elds Ck; Ck0 will rst be set to zero in this section and to the nontrivial values given in [28, 34] in the following section. Our fermion elds are periodic in the space direction, i.e., the freedom to insert a periodicity angle into the SF is not used here. The bare quark masses will be speci ed in terms of the well known hopping parameters l = 1 2(aml + 4) ; c = 1 2(amc + 4) : 4.1 Finite size de nition of the LCP We start with the de nition of a renormalized coupling that we denote by 1. In recent years it has turned out that coupling constants derived from the gradient ow [37, 38] have an excellent signal quality for nite tori [39, 40] and for SF boundary conditions [41] at the range of system sizes that we shall want to simulate. We employ the coupling given by the expectation value of the action density 1 = t 2 N X 1 2 tr[G2 (x; t)] x0 = T=2; t = c2L2=8 : In this formula G is the (anti-Hermitian traceless) clover eld strength of the ` owed' eld and its ow time argument t is tied to the system size by c = 0:3 : with the lattice currents (2.2). The SF boundary operator [25] O ij = a6 X i(~x) 5 j (~y) ~x;~y { 8 { (4.1) (4.2) (4.3) (4.4) (4.5) 1 2 The lattice ow equation is based on the simple Wilson plaquette action, i.e., we work with the Wilson ow. The normalization factor N , which implies perturbation theory, has been derived in [41]. 1 = g02 + O(g04) in Apart from the coupling that runs with the system size T = L we need two more dimensionless renormalized quantities to x the light and charm masses. To that end we introduce the SF correlations f Aij (x0) = Ai0j (x0)Oji ; f Pij (x0) = P ij (x0)Oji ; 1 2 is built from boundary elds at x0 = 0 and yields a nonzero correlation. We next form the improved axial correlation involving the symmetric lattice derivative @~0. An e ective meson mass times system size is now coded into T : For very large T we would isolate the mass of the pseudoscalar non-singlet meson of two di erent light quark species. At nite T we instead have a combination of this mass and higher levels in the same channel with weights given by universal amplitude ratios [42]. In a third quantity we excite states with the quantum number of a singly charmed meson (fourth avor) and take 3 = T 1 2 2 : (4.6) (4.7) (4.8) The reason for the subtraction of 2 is as follows. We will have to solve the complicated tuning task to nd values of g02; l; c that produce certain soon to be prescribed values of 1;2;3. This task is facilitated if 3 is sensitive to c and shows only weak dependence on l . The subtraction cancels the latter dependence in the most naive constituent quark model. In Nf = 2 simulations it has been veri ed that this property holds in an approximate sense also beyond such a naive scenario. Another remark is in order to why we use a component of the axial vector rather than the pseudoscalar current to excite the desired quantum numbers. At large T the same meson mass would be isolated with the admixtures at nite T being di erent in the two cases. The PCAC relation implies that our 2 vanishes in the chiral limit, while this would not be the case for the analogous quantity derived with the P 12 correlator. Hence our choice is expected to optimize the tuning sensitivity for the light mass l . Finally also the inclusion of the improvement term (with perturbative 1-loop cA [43]) in fA;I is not a necessity here and, again at Nf = 2, it has been seen to make little e ect. Nevertheless we keep it as part of our choice. 4.2 nite size e ects where the bare lattice parameters are tuned such that pion, kaon, D-meson masses and a scale such as the pion decay constant equal their values compiled by the particle data group. From the resulting physical values of the renormalization group invariant (RGI) quark masses, Mu; Md; Ms; Mc we could then easily switch to a avor SU(3) symmetric point Ml = (Mu + Md + Ms)=3 for the three light species keeping Mc. Then a would be known in physical units and we may lower T =a = L=a such that we nd L 0:8 fm, for example, still having L=a 1. On the resulting lattice we read o universal continuum values for 1;2;3. The above strategy is, of course, impractical. Instead we have to somehow deduce the i with su cient precision, in order to then on a series of LCP lattices tune csw for { 9 { O(a) improvement and to later use this information to reach acceptably small cuto e ects in reasonably large volume simulations. The choice of the system size L together with the feasible L=a determine for which g02 we obtain nonperturbative information on csw. It should end up in a range useful for the nal simulations. Our strategy to determine i is to use here the approximation Nf = 2 with quenched strange and charmed quarks and con gurations from earlier simulations2 of the ALPHA collaboration [44, 45] together with additional measurements and some newly created congurations as well. For details on these well-de ned but tedious estimations we refer to [21]. We here just report the outcome in the form of values These numbers emerge from the Nf = 2 computation with errors at the percent level3 but are now taken to de ne our LCP which in the continuum limit yields a nite slab of continuum QCD with SF boundary conditions. An important question is now to which precision we have to tune our Nf = 3 + 1 lattices to the above target values i to then determine csw(g02) along the LCP. By a chain of heuristic considerations given in more detail in [21] we arrive at the requirement (4.9) HJEP06(218)5 1 1 . 4% ; 2 2 . 10% ; 3 3 . 4% : (4.10) This corresponds roughly to 5% precision in the scale T and the charm mass and 11% in the light mass. As discussed earlier the closeness to the physical charm mass is of particular relevance to avoid large cuto e ects proportional to the deviation values of the light quark masses, which are dominantly controlled by mq;c. The precise 2, on the other hand appear to be less critical. 4.3 Lattice realization of the LCP We now report on a set of simulation results with lattices tuned to i within the required precision. They were found from a rst set of exploratory simulations in the right range followed by multidimensional interpolations. They serve to predict the `right' values of g0; l; c which are then con rmed or fed into re ned interpolations. The nal results from this somewhat demanding procedure (see [21] for more details) are collected in table 1. Note that a range of csw is covered for each value T =a. After some initial tests, all simulations have been performed in a (2+1+1)- avor setup, i.e., a degenerate doublet of light quarks is simulated by HMC [46] and the two remaining quark species are incorporated by the RHMC algorithm [47, 48]. In general, the simulations are conducted along the lines of ref. [49], appendix A, and more details and a cost gure will be provided [50]. Due to the enhanced spectral gap in the Dirac operator with SF boundary conditions and non-vanishing physical quark masses, the performed HMC simulations are very stable and unproblematic. 2Their availability was another argument to choose an extent around L 0:8 fm for the LCP. 3Not including partial quenching errors, of course. csw 2:1 2:2 2:3 2:4 1:8 1:9 2:0 2:1 2:2 1:5 1:6 1:7 1:8 1:9 2:0 1:5 1:6 1:7 1:8 1:9 1:5 1:6 1:7 1:8 g 1:7848 1:8067 0:12033 0:11960 0:59(3) 0:56( 2 ) 0:61(3) 0:60(3) 0:62( 2 ) 0:57(3) 0:58(3) 0:60(3) 0:62(3) 0:60(4) 0:60(4) 0:56( 2 ) 0:61( 2 ) 0:59( 2 ) 0:57( 2 ) 0:59( 2 ) 0:59( 2 ) 0:61( 2 ) 0:57( 2 ) 0:60( 2 ) 0:63( 2 ) 0:62( 2 ) 0:62( 2 ) 0:59( 1 ) 0:62( 2 ) 0:56( 1 ) 5:95( 1 ) 5:99( 1 ) 6:02( 1 ) 6:03( 1 ) 5:97( 1 ) 5:93( 1 ) 5:93( 2 ) 5:92( 2 ) 5:99(3) 5:93(3) 5:91(3) 5:93(3) 6:06(7) 5:84(5) 5:90(5) 5:94(3) 6:03(4) 5:91(8) 5:91(5) 5:85(5) 5:81(6) 5:81(8) 6:02(8) 6:02(6) 6:00(5) 5:87(6) For each of the groups of LCP con gurations at a given T =a we now want to nd the combinations of g02 and csw that lead to the cancellation of a certain lattice artefact. Equivalently we can say that we impose an improvement condition where we match a result to the continuum. As we indicated before the PCAC operator relation is a source of such conditions which has in fact been traditionally used [28, 29] to determine csw. To achieve a good sensitivity to csw a chromoelectric background eld is enforced in the SF by the nonvanishing The bare improved current (PCAC) quark mass is written in terms of the improved and then insert this value into Our condition to determine csw nally reads 0 = M ij = M ij M 0ij : M ij = rij (x0) + cAsij (x0) ; M 0ij = r0ij (x0) + cAs0ij (x0) ; at x0 = T : (5.5) 1 4 mij (x0) = rij (x0) + acAsij (x0) rij (x0) = 2f Pij (x0) sij (x0) = 2f Pij (x0) is, of course a lattice artefact, absent for a true operator relation in the continuum. A second possibility is given by inserting SF operators O0ij in (4.4) which are analogous to O ij but de ned [25] on the opposite boundary at x0 = T . We add a prime to the corresponding correlations and nally to m0ij (x0). We are now ready to state our improvement conditions mij (x0) = m0ij (x0) at x0 = T and x0 = T : 1 4 3 4 As we equate di erent versions of mij we never need renormalization Z factors. The two Euclidean time locations in x0 are chosen to stay away from the boundaries for smaller higher order cuto e ects but also from each other to have two reasonably independent conditions. Two conditions are imposed to determine csw and simultaneously cA. More precisely we rst determine cA = r0ij (y0) s0ij (y0) rij (y0) sij (y0) at y0 = T ; 3 4 (5.1) (5.2) (5.3) (5.4) (5.6) A closer inspection discloses that the above construction is actually symmetric under interchanging the insertion times x0 and y0. So far we left the quark species i 6= j general. As we shall at rst primarily be interested in light quarks, we shall in the following take ij = 12 = ud. csw 1:9 2:0 2:1 2:2 2:3 2:4 1:8 1:9 2:0 2:1 2:2 1:5 1:6 1:7 1:8 1:9 2:0 1:5 1:6 1:7 1:8 1:9 1:5 1:6 1:7 1:8 g 1:7848 1:8067 l c T mud T M ud c, we quote the corresponding current quark masses mud(x0) at mass di erence M ud from eq. (5.6). Errors from the relation g02 $ T =a are neglected. x0 = T =2, the improvement condition mass M ud(x0; y0) for x0 = 14 T and y0 = 34 T , and the individuai nldividual csw ccssww csw 2:211(25) 2:105(30) 1:828(19) 1:760(34) 1:686(20) g 2 0 1:8597 1:8163 1:7129 1:6600 1:6028 l 0:13450 0:13482 0:13640 0:13619 0:13605 0.14 0.138 0.136 0.134 0.132 T =a csw;I 8 12 16 20 24 2.2 2 1.8 1.4 1.2 1 csw,I data 0.11-lo4op PT Padé fit 0.138 w1.6 0.136 s c 0.134 0.132 24 2T0*/a =16 24 1220 186 12 using the global ansatz of eq. (5.7) (right). Results for csw;I are summarized in table 3. individual t results global t results 8 8 LCP Linear -t Impr.cond. 8 12 our line of constant physics. csw;I1.9 2:202(21) 2:1051.(824) 1:833(19) 1:7551.(728) 1:688(24) g 2 0 1:8575 1:8163 1:7144 1:6586 1:6035 l 0:13459 0:13482 0:13636 0:13623 0:13603 c 0:11841 0:12588 0:12993 0:13143 0:13204 T*/a = 24 We have determined the bare parameters for LCP lattices with T = L. To enhance the sensitivity to our improvement conditions we shall in the following however switch to the smaller spatial extent L = T =2 and asymmetric lattices. The T =a values are unchanged to accommodate the various insertion time slices. The results for M ud and M ud on these lattices are listed in table 2. A glance at the last column shows that we have managed to M udT for each value T =a. To precisely locate the zeroes we use the T M ud = O(a) = s [csw csw;I(g02)] : a T This t is global in T =a with one slope parameter s and the improvement values csw for each of our lattice sizes. It works very well with s = 1:26(8) as seen in the right panel 2 of gure 1. The corresponding bare parameters g0; l;I; c;I are determined by linearly interpolating the data in table 2. The resulting improvement values are compiled in table 3. In [21] it is discussed that ts of the type (5.7) with independent slopes for each T =a lead to almost the same results. They are shown in the left panel of gure 1. We now have a number of values csw;I for csw that obey an improvement condition for ve di erent values g02 corresponding to T =a = 8; 12; 16; 20; 24. In a last step we represent these data by a continuous t function that interpolates between our `sampling' points and matches the one-loop perturbative result [43] (5.7) (5.8) (5.9) i are csw = 1 + 0:196g02 + O(g04) ; for g02 ! 0. Our t is given by the Pade form csw;I(g02) = 1 + (A 1 + Ag02 + Bg04 ; 0:196)g02 A = The t parameters emerge with errors, of course, but at this point we neglect this and propose the t formula as a de nition of an O(a) improved action. In gure 2 we see that the data are represented well. An initially used g06 term in the numerator turned out not to be needed and receive a coe cient compatible with zero. In a few tests we nally have convinced [21] ourselves that the spreads that we have allowed in tuning the subdominant in the errors of the tted csw. 6 Conclusion We have emphasized that the large mass mc of charmed quarks leads to lattice artefacts proportional to amc 0:5 for presently attainable lattice spacings. In the usual setting of a mass independent renormalization scheme with O(a) Symanzik improvement amc appears in numerous improvement terms, which are an order more important than the analogous light quark terms. For unquenched charm even observables in the light quark sector are polluted by such cuto e ects. We nd it impractical to tune all corresponding coe cients precisely enough, but instead propose and design a scheme of renormalization and improvement conditions formulated at or close to the physical charm mass. With Schrodinger functional boundary conditions we introduce a renormalized coupling and e ective meson masses that de ne a nite size scheme. Held xed, they de ne a series of lattices where only the lattice spacing changes. We use them to compute the coe cient csw(g02) of the clover improvement term. Our result is wrapped up in the formula (5.9) that we would like to advocate for use in future dynamical charm quark simulations. Acknowledgments We thank O. Bar and T. Korzec for input and discussions at an early stage of this project and P. Weisz for a critical reading of the manuscript. 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Patrick Fritzsch, Rainer Sommer, Felix Stollenwerk, Ulli Wolff. Symanzik improvement with dynamical charm: a 3+1 scheme for Wilson quarks, Journal of High Energy Physics, 2018, 25, DOI: 10.1007/JHEP06(2018)025