We present an exploratory study of a gauge-invariant non-perturbative
renormalization technique. The renormalization conditions are imposed on
correlation functions of composite operators in coordinate space on the
lattice. Numerical results for bilinears obtained with overlap and
O(a)-improved Wilson fermions are presented. The measurement of the quark
condensate is also discussed.Comment: Lattice2003(improve), 3 page

B. Taglienti

Becirevic

E. Trevigne

G. Martinelli

Giusti

Guagnelli

J. Reyes

L. Giusti

Lüscher

Martinelli

S. Guerriero

S. Petrarca

V. Giménez

V. Lubicz

English

arXiv

We present an exploratory study of a gauge-invariant non-perturbative renormalization technique. The renormalization conditions are imposed on correlation functions of composite operators in coordinate space on the lattice. Numerical results for bilinears obtained with overlap and O(a)-improved Wilson fermions are presented. The measurement of the quark condensate is also discussed. © 2004 Published by Elsevier B.V

Gimemenez V.

Giusti L.

Guerriero S.

Lubicz V.

Martinelli G.

Petrarca S.

Reyes J.

Taglienti B.

Trevigne E.

Sissa Digital Library

Non perturbative renormalization in coordinate space

We present an exploratory study of a gauge-invariant non-perturbative renormalization technique. The renormalization conditions are imposed on correlation functions of composite operators in coordinate spare on the lattice. Numerical results for bilinears obtained with overlap and 0(a)-improved Wilson fermions are presented. The measurement of the quark condensate is also discussed

Gimenez V

Giusti L

Guerriero S

Martinelli G

Petrarca S

Reyes J

Taglienti B

LUBICZ, Vittorio

Archivio della Ricerca - Università di Roma 3

V. GIMENEZ

L. GIUSTI

S. GUERRIERO

V. LUBICZ

J. REYES

B. TAGLIENTI

E. TREVIGNE

MARTINELLI, Guido

PETRARCA, Silvano

Archivio della ricerca- Università di Roma La Sapienza

NONPERTURBATIVE RENORMALIZATION IN COORDINATE SPACE

Crossref

We present an exploratory study of a gauge-invariant non-perturbative renormalization technique. The renormalization conditions are imposed on correlation functions of composite operators in coordinate space on the lattice. Numerical results for bilinears obtained with overlap and O(a)-improved Wilson fermions are presented. The measurement of the quark condensate is also discussed.We present an exploratory study of a gauge-invariant non-perturbative renormalization technique. The renormalization conditions are imposed on correlation functions of composite operators in coordinate space on the lattice. Numerical results for bilinears obtained with overlap and O(a)-improved Wilson fermions are presented. The measurement of the quark condensate is also discussed

Gimenez, V.

Giusti, Leonardo

Guerriero, S.

Lubicz, V.

Martinelli, G.

Petrarca, S.

Reyes, J.

Taglienti, B.

Trevigne, E.

CERN Document Server

Giménez Gómez, Vicente

Giusti, L.

Martinelli, Guido

Reyes Gámez, Juan

Repositori d'Objectes Digitals per a l'Ensenyament la Recerca i la Cultura

arXiv:hep-lat/0309150v1  22 Sep 20031Non Perturbative Renormalization in Coordinate Space ∗ †V. Gime´nez a, L. Giusti b, S. Guerriero c, V. Lubiczc, G. Martinellid e, S. Petrarcad e, J. Reyesd e,B. Taglientie, E. Trevigne daDep. de F´ısica Teo`rica and IFIC, Univ. de Vale`ncia, Dr. Moliner 50, E-46100, Burjassot, Vale`ncia,Spain.bTheory Division, CERN, CH-1211 Geneva 23, SwitzerlandcDip. di Fisica, Univ. di Roma Tre and INFN-Sezione di Roma III, Via della Vasca Navale 84, I-00146Roma, Italy.dDip. di Fisica, Univ. di Roma “La Sapienza”, Piazzale A. Moro 2, I-00185 Roma, Italy.eINFN-Sezione di Roma I, Piazzale A. Moro 2, I-00185 Roma, Italy.We present an exploratory study of a gauge-invariant non-perturbative renormalization technique. The renormal-ization conditions are imposed on correlation functions of composite operators in coordinate space on the lattice.Numerical results for bilinears obtained with overlap and O(a)-improved Wilson fermions are presented. Themeasurement of the quark condensate is also discussed.Renormalization constants (RC) of lattice op-erators are crucial inputs to obtain physical quan-tities from the matrix elements of these operators.We have presented in this conference a numericalstudy of a non-perturbative (NP) technique pro-posed in [1]. This method has the advantages ofbeing gauge-invariant and free of contact terms,in addition with the very easy implementation ona simulation. The first two properties can be ofvital interest to renormalize the operators whichenter the K → pi pi decays.A very exploratory analysis was already pre-sented in the Lattice 2002 conference [2] for bilin-ears operators. The results presented there werenot conclusive and claimed for a more detailedstudy, which is the purpose of this contribution.We have considered only bilinears operators totest the method, since they are the simplest op-erators for which there exist results from othersNP techniques to compare with.The method consists on imposing renormal-ization conditions on the bare green functions,GΓ(x) ≡ 〈OΓ(x)O†Γ(0)〉, where OΓ(x) is a quark∗Presented by J. Reyes at “Lattice 2003”, Tsukuba.†This work has been supported in part by the EU IHPunder HPRN-CT-2000-00145 Hadrons/LatticeQCD.bilinear operator, OΓ(x) = ψ¯1(x) Γψ2(x). Onecan chooseZ2Γ(µ = 1/x)GΓ(x) = GΓ(x)(free cont.) , (1)in a window 1/a ≫ 1/x ≫ ΛQCD; we will callthis definition as the X-space scheme. One canalso obtain ZΓ(µ = 1/x) directly in another con-tinuum scheme, such MS, by imposing,ZMSΓ2(µ = 1/x)GΓ(x) = GΓ(µ, x)(MS) (2)We have analyzed results for O(a) improved Wil-son fermions at β = 6.45 (a−1 = 3.87(19) GeV)on a 323 × 70 lattice and overlap fermions atβ = 6.00 (a−1 = 2.00(10) GeV) on a 163 × 32volume. All the results presented here have beenextrapolated to the chiral limit (further informa-tion can be found in [3]).As an example of the general strategy adopted,we explain the analysis for the vector-vector cor-relator with Wilson fermions. In Fig. 1a) we showthe dependence of the correlator 〈OV (x)O†V (0)〉on x2. The na¨ıve dimensional behaviour (x2−3)is well satisfied but the data points are quite dis-persed (see also Fig. 1b)). To clarify the ori-gin of these effects we show in Fig. 1c) the ra-tio between the free correlator on the lattice and20 10 20 30 40 50 60x21e-071e-061e-050.00010.0010.010.11<Vµ(x)Vµ(0)>0.106 (x2)-3Free Cont. = 0.0616 (x2)-3a)0 10 20 30 40 50 60x200.511.52(<V(x) V(0)>Free Con. _____________<V(x) V(0)>Int. )1/2ZV=b)0 10 20 30 40 50 60x200.511.52(<V(x) V(0)>Free Con. _____________<V(x) V(0)>Free Lat. )1/2c)0 10 20 30 40 50 60x200.511.52(<V(x) V(0)>Free Con. _____________<V(x) V(0)>Int. )1/2(<V(x) V(0)>Free Con. _____________<V(x) V(0)>Free Lat. )1/2_______________________________∼ΖV =d)Figure 1.in the continuum, both in infinite volume (notethat in this case the expected value is 1 up toO(a) effects). Comparing both figures we observethat the tree level calculation manifests the samepattern that the interacting case. These consid-erations together with similar analysis (see [3]),brought us to conclude that we are observing dis-cretization effects, which we propose to reduceconsidering the following ratio:Z2Γ(µ = 1/x)=GΓ(x)(F. con.)GΓ(x)(I.)·GΓ(x)(F. lat.)GΓ(x)(F. con.), (3)where, I., F. mean interacting and free respec-tively. In this way we obtain the result plottedin Fig. 1d), which can be compared directly withFig. 1b). It is clear that in this way a great partof O(a) effects are canceled out in the ratio of eq.(3). The same procedure has been carried out forthe other bilinears and for the overlap data.As commented above one can also obtain ZOdirectly in another continuum scheme (MS), byimposing (2). Now one can evolve the RCs withthe MS Renormalization Group (RG) factors, upto a fixed scale, say 2 GeV.ZMS(2GeV) =Z(x)W (µ, 2GeV)R(µ, 1/x), (4)where R(µ, 1/x) is the matching between the X-space scheme and MS, which it has been calcu-lated at NLO [3]. W (µ, 2GeV) is the RGE factor.In principle µ is arbitrary as far as µ ∼ 1/x in or-der to avoid large logarithmic contributions. This5 10 15 20 25 30 35 40 45x20.50.60.70.80.91ZVZSMS___(2 GeV)ZP / ZSZSWilsona)0 5 10 15x20.60.811.21.41.61.8ZAb)ZVZV/ZAOverlapFigure 2.scale dependence should cancel in a RGI quan-tity up to truncation of the perturbative series.In practice we have chosen µ = 1/x and µ = 2/xand included the difference as a systematic errordue to NNLO effects.Our results are summarized in Fig. 2 for theWilson (2a) and overlap (2b) simulation, respec-tively. For the Wilson case we show, as an ex-ample, the RCs for the scalar density and vectorcurrent. We also show the ratio ZP /ZS . Thisratio (as well as ZV /ZA) allows to extract infor-mation on the renormalization window, since itshould be a plateau up to O(a) and NP effects.From Fig. 2a) we see that in the range x2 = 9÷21we observe a reasonable good plateau: For x2 ≤ 9large O(a) effects are present, and above x2 ∼ 21the presence of NP effects begins to be visible (seeZP /ZS). Therefore, we choose to obtain our finalresults for the RCs in the above range. We alsoplot the scalar and vector RCs, the first in MSat 2 GeV and in the X-space scheme. For ZV wesee a good plateau as one would expect. ZS(1/x)in the X-space exhibits an anomalous dimensionbehaviour as it should be. On the contrary in thevalue of ZMSS (2GeV) one expects that the anoma-lous dimension cancels (up to NNLO correction)and indeed a plateau is observed.For the overlap simulation, we show the cur-rents RCs, together with the ratio ZV /ZA. Sincechirality holds exactly, one expect ZV = ZA up to3Table 1Wilson results and comparison with other methods: RI-MOM [2] and Schro¨dinger Functional [5].ZΓ ZV ZA ZP /ZS ZS ZP ZTThis work 0.801(2)(18)(6) 0.833(2)(27)(6) 0.888(2)(8) 0.702(4)(27)(23) 0.624(3)(19)(21) 0.895(2)(21)RI-MOM 0.803(3) 0.833(3) 0.897(4) 0.679(8) 0.609(8) 0.898(6)SF 0.808(1) 0.825(8) 0.912(9) −− 0.61(1) −−NP effects. From Fig. 2b) one see that indeed thisis the case for x2 . 10, and we choose the range6÷ 10 to obtain the values of ZV , ZA. Contraryto the Wilson results, no renormalization windowhas been found for the other bilinears in the over-lap case. This is due to the larger lattice spacing(twice what we have for Wilson data) which con-sequently makes much narrower the renormaliza-tion window. In Table 1 we present our results forthe Wilson case and those with other NP meth-ods. All the RCs are in MS at 2 GeV. The firsterror in our results is statistic, the second one isan estimate of discretization errors and the lastone comes from NNLO effects due to the choiceof the scale (see above). For the overlap case wequote the results obtained for ZV and ZA:ZV = 1.471(2)(10)(20) ZA = 1.494(2)(9)(20), (5)the tiny difference between the results for ZV andZA is due to small NP effects. In any case they arecompatible within the errors. Our results for ZAis comparing with ZA = 1.55(4) obtained withthe WI method in [4]. From Table 1 and theabove results in the overlap case we conclude thatalthough the errors are, in most cases, larger thanthose obtained from other NP techniques, goodagreement is found for all the cases which we cancompare. We stress that one could reduce theerrors carring out a recursive volume matchingtechnique.We present a preliminary result for the chiralquark condensate, obtained in a novel way. Theidea is very simple and consists on studying thechiral limit of the trace of the quark propagatorin X-space together with its OPE:TrS(x) ∼ C1(x2)mx2− C2(x2)〈ψ¯ψ〉 + · · · (6)We stress that in (6), once the l.h.s. is renor-malized, the r.h.s. gets renormalized too, and no0 10 20 30 40 50 60x200.00050.001Tr [S MS___(µ, x)] / C 2(µ2x2)µ=2 GeVA + B x2Figure 3.other divergence (apart from the natural one atx = 0) is present, thus only the renormalizationconstant for the quark field is needed. We havecomputed the needed Wilson coefficients factorsin eq. (6). From eq. (6) and in the chiral limitone can obtain an estimate of the chiral conden-sate. This is shown in Fig. 3. From that figure wesee large discretization effects for x2 ≤ 15. Onecan see also that there is a contribution from thenext order term which goes as x2. From a linearfit we obtain our (preliminary) result:〈ψ¯ψ〉MS(2GeV) = −(276(5)(13)MeV)3 , (7)in good agreement with other lattice determina-tions. The first error is statistic and the secondis due to the uncertainty in the value of a−1.REFERENCES1. G. Martinelli et al., Phys. Lett. B 411 (1997)141.2. D. Becirevic, et al., [SPQcdR Collaboration],arXiv:hep-lat/0209168.3. V. Gime´nez et al., in preparation.4. L. Giusti, C. Hoelbling and C. Rebbi, Nucl.Phys. Proc. Suppl. 106 (2002) 739.5. M. Lu¨scher et al., Nucl. Phys. B491 (1997)344. M. Guagnelli et al., Nucl. Phys. B535(1998) 389. M. Guagnelli et al., Nucl. Phys.B595 (2001) 344.

Nonperturbative renormalization in coordinate space

http://dx.doi.org/10.1016/S0920-5632(03)02598-2

Non perturbative renormalization in coordinate space

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