We propose a general renormalization method, which avoids completely the use
of lattice perturbation theory. We present the results from its numerical
applications to two-fermion operators on a $16^3 \times 32$ lattice, at
$\beta=6.0$.Comment: 3 pages postscript file. Contribution to Lattice '9

A. Vladikas

Bernard

C. Pittori

C.T. Sachrajda

G. Martinelli

Gavela

Lepage

M. Testa

Maiani

Martinelli

English

arXiv

MARTINELLI, Guido

PITTORI C

SACHRAJDA CT

VLADIKAS A.

TESTA, Massimo

Archivio della ricerca- Università di Roma La Sapienza

RESULTS FROM A NONPERTURBATIVE RENORMALIZATION OF LATTICE OPERATORS

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Nuclear Physics B - Proceedings Supplements

Results from a non-perturbative renormalization of lattice operators

We propose a general renormalization method, which avoids completely the use of lattice perturbation theory. We present the results from its numerical applications to two-fermion operators on a 16^3 \times 32 lattice, at \beta=6.0.We propose a general renormalization method, which avoids completely the use of lattice perturbation theory. We present the results from its numerical applications to two-fermion operators on a $16~3 \times 32$ lattice, at $\beta=6.0$

Martinelli, G.

Pittori, C.

Sachrajda, Christopher T.

Testa, M.

Vladikas, A.

CERN Document Server

1Results from a Non-Perturbative Renormalization of Lattice OperatorsG. Martinellia b, C. Pittoric , C.T. Sachrajdad, M. Testaaand A. VladikaseaDipartimento di Fisica, Universita di Roma `La Sapienza' and INFN, Sezione di Roma I,P.le A. Moro, I-00185 Rome, Italy.bTheory Division, CERN, CH-1211 Geneva 23, Switzerland.cL.P.T.H.E., Universite de Paris Sud, Centre d'Orsay, 91405 Orsay, France.dDepartment of Physics, The University, Southampton SO9 5NH, U.K.eDipartimento 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 propose a general renormalization method, which avoids completely the use of lattice perturbation theory.We present the results from its numerical applications to two-fermion operators on a 163 32 lattice, at  = 6:0.1. IntroductionFollowing the ideas presented last year at thisconference [1], we have pursued a theoreticaland numerical study, described in detail in [2],on the feasibility of a non-perturbative renor-malization of generic composite operators in lat-tice QCD. We propose a renormalization methodwhich avoids completely the use of lattice pertur-bation theory, known to be an important source ofuncertainty in the extraction of physical results.This method, applicable to any general compositeoperator, is particularly useful when other non-perturbative techniques, based onWard identities[3] or on renormalization conditions on hadronstates [4], are not viable. A similar, but lim-ited attempt, can be found in [5]. Here wepresent the main results of numerical applicationto two-fermion operators. Further applications tothe four-fermion operators of the eective weakhamiltonian, and to heavy-light currents in theheavy quark eective theory, are under way [6].2. Non-Perturbative RenormalizationOur renormalization method consists in mimi-cking on the lattice what is usually done in per-Talk presented by C. Pittori.turbation theory. One may x the renormaliza-tion conditions of a certain operator by impos-ing that suitable Green functions, computed be-tween o-shell quark and gluon states, in a xedgauge, at a scale , coincide with their tree levelvalue. For illustrative purposes, we consider spe-cic applications to matrix elements of two-quarkoperators. We dene the renormalized operatorO() = ZOO(a), by introducing the renormaliza-tion constant ZO, which is found, by imposingthe non-perturbative renormalization conditionZOa; g(a)Z 1 a; g(a) O(pa)jp2=2= 1: (1) Ois the forward amputated Green function2of the bare operator, computed between o-shell quark states with four-momentum p, withp2= 2, in the Landau gauge. Z is the quarkwave function renormalization, which can be non-perturbatively dened from the conserved vectorcurrent, or from the quark propagator (Z0 ).This scheme solves the problem of large correc-tions in lattice perturbation theory, which are au-tomatically included in the renormalization con-stants. The renormalized operator is independentof the regularization scheme; it depends, however,2Suitably projected on the tree-level, and traced overcolour and spin [2].2Figure 1. ZVas a function of 2a2.on the external states and on the chosen gauge.Matching with the corresponding operators, insome other standard continuum renormalizationscheme (e.g. MS), then requires a continuumperturbative calculation only, which is a commonstep in all approaches, standard or \boosted" per-turbation theory [7] and non-perturbative renor-malization. Our proposal is expected to work,whenever it is possible to x the virtuality of theexternal states  so that QCD  1=a, in or-der to keep under control both non-perturbativeand discretization eects. If such a \window"does not exist at current  values, an accuratematching becomes impossible, not only in our ap-proach, but also with any other method.3. Numerical ResultsIn order to study the -dependence of therenormalization constants, we have performed asimulation with 36 quenched congurations, ona 163 32 lattice, at  = 6:0, in the Landaugauge. To reduce discretization errors, we con-sider improved operators with the Clover fermionaction, at K = 0:1425 (amq 0:07). Our non-perturbative (NP) results are shown in gs.1-Figure 2. ZAas a function of 2a2,4. We compare them with \Boosted Perturba-tion Theory" (BPT), using an eective couplingVs' 1:68 LATTs. To monitor the distortionsdue to discretization, the perturbative computa-tion has also been performed on a lattice of thesame size as the non-perturbative one, with axed spacing. We call this procedure \BoostedDiscrete Perturbation Theory" (BDPT). In thegures the dashed line is from BPT, the curveis from BDPT, and the straight line is from theWard identities (WI) method.ZV: as expected, it is independent from the scale,within statistical errors, up to large values of2, where discretization errors become important.By using the points at 2a2 1, correspondingto   2 GeV, we get ZNPV= 0:84(1), to be com-pared with ZWIV= 0:824(2) and ZBPTV= 0:83.The three methods are in good agreement.ZA: it is also nite at all orders, and can be de-termined from WI. However, in the axial case(as in the pseudoscalar case) we expect to nda non-perturbative contribution from the pionpole, vanishing for large 2[2]. We interpretthe strong -dependence of ZA, at low 2, asthe non-perturbative eect of the pseudoscalarstate. There is no clear plateau between the non-3Figure 3. ZSas a function of 2a2.perturbative regime and the large  region. Nev-ertheless, it attens around 2a2 1, just beforelattice artefacts become large in DBPT, and weget ZNPA= 1:04(3), which agrees with the valueZWIA= 1:06(2). BPT gives ZAsmaller than one,indicating a  15% higher orders eect.ZS: this is an ideal quantity to check the valid-ity of our method, since it cannot be determinedusing WI, and it is not aected by the pion pole.It is logarithmically dependent on the scale, andgauge dependent. We do not expect agreementwith BPT, either at low 2, due to higher ordereects, or at high 2, where lattice distortions areimportant, as shown by BDPT. NP results fol-low the theoretical expectations and we nd goodagreement with BPT in the range 0:3  2a2 1.ZP: the pseudoscalar density (not shown here)is coupled to the pion, at low 2; moreover ithas large one-loop perturbative corrections, of or-der  35%. It is then not surprising that theNP result lies well below the BPT value, becauseof higher orders, not accounted for by one-loopBPT.With standard perturbation theory, the dis-crepancy would have been even worse.ZS=ZP: this ratio has a behaviour similar to ZA.At 2a2 1, the value obtained is in agreementFigure 4. ZS=ZPas a function of 2a2.with that determined by WI, whereas, given theresult for ZS, the signicant dierence with BPTshould be due to ZP. In conclusion, our numericalstudy suggests that there does exist a window, 1:1   2 GeV, where the method can be applied,already at  = 6:0, and we expect the useful rangeto get larger, as  increases.We thank S. Petrarca for an early partecipationto this work.REFERENCES1. G. Martinelli et al., Nucl. Phys. B (Proc.Suppl.) 34 (1994) 507.2. G. Martinelli et al., CERN-TH.7342/94, sub-mitted to Nucl. Phys. B.3. L. Maiani and G. Martinelli, Phys. Lett. B178(1986) 265; G. Martinelli et al., Phys. Lett.B311 (1993) 241.4. M.B. Gavela et al., Phys. Lett. B211 (1988)139.5. C. Bernard et al., Nucl. Phys. B (Proc.Suppl.) 17 (1990) 593; 20 (1991) 410.6. G. Martinelli et al., work in progress.7. G.P. Lepage and P.B. Mackenzie, Phys. Rev.D48 (1993) 2250.

Results from a Non-Perturbative Renormalization of Lattice Operators

Results from a Non-Perturbative Renormalization of Lattice Operators

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