We study the behaviour of the vector and axial current renormalisation
constants $Z_V$ and $Z_A$ as a function of the quark mass, $m_q$. We show that
sizeable $O(am_q)$ and $O(g_0^2 a m_q)$ systematic effects are present in the
Wilson and Clover cases respectively. We find that the prescription of
Kronfeld, Lepage and Mackenzie for correcting these artefacts is not always
successful.Comment: Contribution to Lattice'94, 3 pages PostScript, uuencoded compressed

A. Vladikas

Abada

Bernard

G. Martinelli

Kronfeld

Lepage

M. Crisafulli

Martinelli

V. Lubicz

English

arXiv

We study the behaviour of the vector and axial current renormalisation constants Z_V and Z_A as a function of the quark mass, m_q. We show that sizeable O(am_q) and O(g_0^2 a m_q) systematic effects are present in the Wilson and Clover cases respectively. We find that the prescription of Kronfeld, Lepage and Mackenzie for correcting these artefacts is not always successful.We study the behaviour of the vector and axial current renormalisation constants $Z_V$ and $Z_A$ as a function of the quark mass, $m_q$. We show that sizeable $O(am_q)$ and $O(g_0~2 a m_q)$ systematic effects are present in the Wilson and Clover cases respectively. We find that the prescription of Kronfeld, Lepage and Mackenzie for correcting these artefacts is not always successful

Crisafulli, M.

Lubicz, V.

Martinelli, G.

Vladikas, A.

CERN Document Server

hep-lat/9502012   16 Feb 19951Dependence of the current renormalisation constants on the quark massM. Crisafulli,aV. Lubicz,aG. Martinellia band A. Vladikasc aDipartimento di Fisica, Universita di Roma `La Sapienza' and INFN, Sezione di Roma,P.le A. Moro, I-00185 Rome, Italy.bTheory Division, CERN, CH-1211 Geneva 23, Switzerland.cDipartimento di Fisica, Universita di Roma `Tor Vergata' and INFN, Sezione di Roma II,Via della Ricerca Scientica 1, I-00133 Rome, Italy.We study the behaviour of the vector and axial current renormalisation constants ZVand ZAas a function ofthe quark mass, mq. We show that sizeable O(amq) and O(g20amq) systematic eects are present in the Wilsonand Clover cases respectively. We nd that the prescription of Kronfeld, Lepage and Mackenzie for correctingthese artefacts is not always successful.The numerical non perturbative estimate of thevector and axial current matrix elements is af-icted by systematic errors due to the nitenessof the lattice spacing a. These errors, which aremonitored by measuring the current renormali-sation constants ZVand ZA, are of O(amq) inlattice simulations based on the Wilson actionand O(g20amq) in those based on the Clover ac-tion (mqis the quark mass). For light quarks,at  = 6:0, the above eects were found to be 25% in the Wilson case. The main success ofClover improvement consists in reducing such sys-tematic eects to  5%. However, when we dealwith heavy quark masses mh, O(g20amh) eectsmay become relevant even in the Clover case. Inthis talk we present a preliminary study of thedependence of ZVand ZAon the quark mass.Recently, Lepage, Mackenzie [1] and Kronfeld[2] (abbreviated as KLM) attempted to absorbthese artefacts in modied normalisation factorswhich match the fermion elds to their continuumcounterparts. For clarity of presentation, we sep-arate their proposals into two parts:(1) KLM normalisation: This is the normali-sation factor between the free continuum propa-gator Pcont(t; ~p) and its Wilson discrete counter-part Platt(t; ~p) which are related by Pcont(t; ~p) =2K(1 + amq)Platt(t; ~p) at ~p =~0. This suggestsTalk given by A. Vladikas.that continuum fermion elds are given in termsof lattice ones by cont= a 3=2q2K(1 + amq) latt(1)where amq= 1=(2K)   1=(2Kcr) and the crit-ical hopping parameter Kcris obtained nonperturbatively. We call the above factor theKLM normalisation. The standard normalisationused previously in lattice simulations,  cont=a 3=2p2K latt, diers from the KLM one byterms of O(amq).(2) MFTI normalisation: Mean Field argu-ments of [1] suggest a further Mean Field Tad-pole Improved (MFTI) prescription for relatingthe lattice and continuum elds:U! U=u0; K !~K = Ku0amq! ~amq= 8Kcr[1=(2K)  1=(2Kcr)] (2)where u0is any reasonable MF estimate of the ex-pectation value of the link (we use u0= 1=(8Kcr)after [1]). This implies the MFTI normalisation cont= a 3=2q2~K(1 + a ~mq) latt(3)Bernard [3] took up these ideas and appliedthem to the non perturbative calculation of ZVfrom the ratio of the conserved (VC) to local (VL)vector current matrix elements. The spatial com-ponent of the conserved current VCkhas the stan-2Figure 1. Wilson action estimates of ZVobtainedfrom the following ratios: R(); RKLM();RMFTI(2). The errors are smaller than the sym-bols.dard KLM normalisation of eq.(1), but the tem-poral one, VC0, being point split in time, requiresan extra KLM factor [3]. With P5= 5 de-noting the pseudoscalar density, the nal KLMpredictions for ratios of the 3-point correlationfunctions are [3]R < P5jVC0jP5>< P5jVL0jP5>= ZV2 + am1+ am22(4)which means that an improved estimate of ZVmay be obtained fromRKLM< P5jVC0jP5>< P5jVL0jP5>22 + am1+ am2= ZV(5)and its MFTI version is given byRMFTI< P5jVC0jP5>< P5jVL0jP5>22 + a ~m1+ a ~m2(6)The spatial components, on the other hand,should behave like~R < P5jVCkjP5>< P5jVLkjP5>= ZV(7)Figure 2. Wilson action estimates of ZVobtainedfrom the following ratios:~R();^R(). The errorsare only shown when greater than the symbols.and the same is true for the ratio of 2-point cor-relation functions^R < 0jVCkj >< 0jVLkj >= ZV(8)(the above formulae dier from those of [3] be-cause the conserved current used in all our sim-ulations is symmetrised;VC(x)  1=2[VC(x) +VC(x )]). In Figs. 1 and 2 we show the resultsfor the above ratios, as obtained from some ELCdata of [4]. The relevant parameters of this simu-lation were: 20 confs.;  = 6:4; V = 24360. Thedata shown here are for xed light quark mass,Kl= 0:1485, and varying heavy quark massesKh= 0:1275; 0:1325; 0:1375;0:1425. The spatialmomenta are all set to zero. By comparing theresults for R, RKLMof Fig. 1 to~R of Fig. 2,we see that the KLM normalisation of [1{3] iscorrecting most of the systematic O(amq) eects,whereas the RMFTIestimate of Fig. 1 shows thatthe MFTI correction is of little importance. How-ever, all of these results were obtained from thesame matrix elements, < P5jV jP5>, as opposedto the ZVestimates of^R (Fig. 2), obtained from3< 0jV jV >. The latter estimate of ZVis incom-patible with all the others. This is a typical O(a)eect which the KLM normalisation fails to cor-rect. Recall that, for light quark masses, Cloverimprovement manages to correct these eects, byimplementing a conserved improved current VCIwhich diers from VCby a total divergence.We now pass to Clover fermions; here the lead-ing corrections are O(g20amh). We will show pre-liminary results from 40 confs. at  = 6:0 andV = 183 32. We have obtained ZVfrom ra-tios of 3-point functions (R of eq.(4)) at zerospatial momenta. The P5densities are locatedat t = 0 and 16. We have data from correla-tions obtained at degenerate quark masses Kh=0:1150; 0:1200;0:1250;0:1330; 0:1425; 0:1432. Wealso have results for non - degenerate masses withxed Kl= 0:1432 and Khvarying as above. Inthis case, ZVcan be checked from the Ward Iden-tity (W.I.) rVC(x) =12[1Kh 1Kl]S(x) whereS =  is the scalar density and ris the asym-metric lattice derivative. Although this W.I. suf-fers from O(amq) eects, the O(amq) improvedestimate of ZVcan also be derived from it [5].Here we only state the nal result:ZV=18[1Kh 1Kl]< P5j[2S(x) + S(x +^0) + S(x  ^0)]jP5>r0< P5jVLI0(x)jP5>(9)The above equation has a symmetric latticederivativerand the local improved currentVLI; thus it has no O(amq) terms. Note that alloperators (except for S) have the standard Clover- rotated fermion elds. The P5's are at rest. Forthe axial current renormalisation constant ZA, wehave used the estimate obtained from a gauge in-variant W.I.[6].The results are shown in Fig. 3; note thatthe degenerate mass data does not interpolatesmoothly the non degenerate points. This eectis related to the presence of a spectator quarkand is currently under investigation [5]. The nonzero slopes of the ZVand ZAcurves of Fig. 3show the presence of sizeable O(g2amq) linear ef-fects even in the case of the Clover improved ma-trix elements. A crucial observation from Fig. 3Figure 3. Clover action estimates of ZVand ZA.ZVis obtained from R at Kl= Kh() and atKl6= Kh(2). ZAis denoted by (). Curvessimply guide the eye. The errors are only shownwhen greater than the symbols.is that the slopes of the two Z's have oppositesigns. Thus, a universal KLM (or MFTI) massdependent factor cannot atten out both curvessimultaneously. This is another case in which theremedies of refs. [1{3] are shown to be inade-quate. More details, as well as more accurateresults, consolidating these points, will be shortlypresented in [5].REFERENCES1. G.P. Lepage and P.B. Mackenzie, Phys.Rev. D48 (1993) 2250, and G.P. Lepage,Nucl.Phys. B(Proc.Suppl.)26 (1992) 45.2. A.S. Kronfeld, Nucl.Phys. B(Proc.Suppl.)30(1993) 445.3. C.W. Bernard, Nucl.Phys. B(Proc.Suppl.)34(1994) 47.4. A. Abada et al. Nucl.Phys. B416(1994)675.5. M. Crisafulli et al., in preparation.6. G. Martinelli et al. Phys. Lett. B311 (1993)241.

Dependence of the current renormalisation constants on the quark mass

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DEPENDENCE OF THE CURRENT RENORMALISATION CONSTANTS ON THE QUARK MASS

DEPENDENCE OF THE CURRENT RENORMALISATION CONSTANTS ON THE QUARK MASS

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