We calculate lattice renormalisation constants of local and one-link quark
operators for overlap fermions and improved gauge actions in one-loop
perturbation theory. For the local operators we stout smear the SU(3) links in
the fermionic action. Using the popular tadpole improved L\"uscher-Weisz
actions at $\beta=8.45$ and $\beta=8.0$ we present numerical values for the Z
factors in the $\bar{MS}$ scheme (partly as function of the stout smearing
strength). We compare various levels of mean field (tadpole) improvement which
have been applied to our results.Comment: 7 page

Horsley, R

Perlt, H

QCDSF collaboration

Rakow, PEL

Schierholz, G

Schiller, A

English

arXiv

Holger Perlt

Roger Horsley

Paul E.L. Rakow

Gerrit Schierholz

Arwed Schiller

Crossref

Perturbative renormalisation of quark bilinear operators for overlap fermions with and without stout links and improved gauge action

We calculate lattice renormalisation constants of local and one-link quark
operators for overlap fermions and improved gauge actions in one-loop
perturbation theory. For the local operators we stout smear the SU(3) links in
the fermionic action. Using the popular tadpole improved L\"uscher-Weisz
actions at $\beta=8.45$ and $\beta=8.0$ we present numerical values for the Z
factors in the $\bar{MS}$ scheme (partly as function of the stout smearing
strength). We compare various levels of mean field (tadpole) improvement which
have been applied to our results

University of Liverpool Repository

PoS(LAT2005)238Perturbative renormalisation of quark bilinearoperators for overlap fermions with and withoutstout links and improved gauge actionR. HorsleySchool of Physics, University of Edinburgh,Edinburgh EH9 3JZ, UKE-mail: rhorsley@ph.ed.ac.ukH. Perlt∗†Institut für Theoretische Physik, Universität Regensburg,D-93040 Regensburg, GermanyE-mail: perlt@itp.uni-leipzig.deP.E.L. RakowTheoretical Physics Division, Department of Mathematical Sciences,University of Liverpool, Liverpool L69 3BX, UKE-mail: rakow@amtp.liv.ac.ukG. SchierholzJohn von Neumann Institut NIC / DESY Zeuthen,D-15738 Zeuthen, GermanyDeutsches Elektronen-Synchrotron DESY,D-22603 Hamburg, GermanyE-mail: Gerrit.Schierholz@desy.deA. SchillerInstitut für Theoretische Physik, Universität Leipzig,D-04109 Leipzig, GermanyE-mail: schiller@itp.uni-leipzig.de(QCDSF collaboration)We calculate lattice renormalisation constants of local and one-link quark operators for overlapfermions and improved gauge actions in one-loop perturbation theory. For the local operators westout smear the SU(3) links in the fermionic action. Using the popular tadpole improved Lüscher-Weisz actions at β = 8.45 and β = 8.0 we present numerical values for the Z factors in the MSscheme (partly as function of the stout smearing strength). We compare various levels of meanfield (tadpole) improvement which have been applied to our results.XXIIIrd International Symposium on Lattice Field Theory25-30 July 2005Trinity College, Dublin, Ireland∗Speaker.†Supported by DFG under contract FOR 465.c© Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. http://pos.sissa.it/PoS(LAT2005)238Perturbative renormalisation of quark bilinear operators H. Perlt1. IntroductionLattice calculations at small quark masses require an action with good chiral properties. Thesame is true for calculations of matrix elements of certain operators, which otherwise mix withoperators of opposite chirality. Ginsparg-Wilson fermions [1] have an exact chiral symmetry on thelattice [2], and thus are well suited for these tasks. A further advantage is that they are automaticallyO(a) improved [3]. Overlap fermions [4, 5, 6] provide a four-dimensional realisation of Ginsparg-Wilson fermions.It is well known that computer simulations with overlap fermions suffer from large computa-tional costs due to the large condition number. One possibility to reduce this number is by usingimproved gauge actions [7] and hence there is a need for the corresponding perturbative calcula-tions. In this paper we use the tadpole improved Lüscher-Weisz action [8, 9, 10]ST ILWG =6g2[c0 ∑plaquette13 ReTr(1−Uplaquette) + c1 ∑rectangle13 ReTr(1−Urectangle)+c3 ∑parallelogram13 ReTr(1−Uparallelogram)]. (1.1)The parameters c1 and c3 weight the contributions of the corresponding six-links loops. It is cus-tomary to impose the normalisation conditionc0 +8c1 +8c3 = 1. (1.2)Defining the lattice coupling β byβ = 6g2c0 (1.3)we choose in accordance with numerical simulations performed by the QCDSF collaboration [11,12] the following values [13]β c1 c38.45 −0.154846 −0.01340708.0 −0.169805 −0.0163414(1.4)A further step to reduce the computational costs consists of smearing the gauge link variablesin the fermionic action. The resulting fat links tend to reduce the density of eigenmodes speedingup the inversion of the fermion Dirac operator [14]. Morningstar and Peardon [15] have proposedan analytic smearing method (stout link method) which can be applied in Hybrid Monte Carlo(HMC) simulations. As by construction the stout links remain in the SU(3) group, this enables theforce term in the equations of motion for HMC to be easily determined.The calculations which we will present in the next sections have been performed for r = 1and the overlap parameter ρ = 1.4 (for the definition see [16]). The influence of stout links isinvestigated for local operators only.238 / 2PoS(LAT2005)238Perturbative renormalisation of quark bilinear operators H. Perlt2. RenormalisationTo obtain continuum results from lattice calculations of hadron matrix elements, the under-lying operators have to be renormalised. A non-perturbative determination of the correspondingrenormalisation factors would be preferable. However, often perturbative renormalisations are donefirst. Especially, if this calculation is performed analytically it provides useful information aboutthe intrinsic singular structure and possible complicated mixing properties. We use a Mathematicaprogram which has been developed for one-loop lattice perturbative calculations [17] and has beenextended to overlap fermions with improved gauge actions.We define renormalised operators O byOS (µ) = ZSO (a,µ)O(a) , (2.1)where S denotes the renormalisation scheme. ZSO(a,µ) is the renormalisation factor connectingthe lattice operator O(a) with the renormalised operator OS (µ) at scale µ . We use the MOMscheme by computing the amputated Green function ΛO of the operator O and define the ZO viaZMOMO(a,µ)ZMOMψ (a,µ)ΛO∣∣p2=µ2 = ΛtreeO + other Dirac structures , (2.2)where ZMOMψ (a,µ) is the quark wave function renormalisation factor. The renormalisation con-stants can be converted to the MS scheme,ZMSψ,O(a,µ) = ZMS,MOMψ,O ZMOMψ,O (a,µ) , (2.3)where the conversion factors ZMS,MOMψ,O are calculable in continuum perturbation theory.3. One-link operatorsLet us first consider the one-link operatorsOµν =i2ψ¯(x)γµ↔Dν ψ(x)− traces , (3.1)O5µν =i2ψ¯(x)γµγ5↔Dν ψ(x)− traces , (3.2)which are related to the first moments of unpolarised and polarised nucleon structure functions,where↔Dν is the left-right covariant lattice derivative. The chiral properties of overlap fermionsimply that matrix elements of Oµν and O5µν give identical results. Therefore, we restrict our calcu-lations to the unpolarised case, i.e. to Oµν . For a detailed discussion see [18].The amputated one-loop Green function Λµν obtained from (3.1) has the formΛµν(a, p) = γµ pν +g2CF16pi2{[(13 +ξ)log(a2 p2)−4.29201ξ +b1]γµ pν+[43 log(a2 p2)+12ξ +b2]γν pµ +[−23 log(a2 p2)−12ξ +b3]δµν 6p+b4 δµνγν pν +(−43 +ξ)pµ pνp26p}, (3.3)238 / 3PoS(LAT2005)238Perturbative renormalisation of quark bilinear operators H. Perltwith ξ as gauge parameter (Feynman gauge: ξ = 0) and CF = 4/3. The constants bi are linearcombinations of finite lattice integrals [17] and depend on the used lattice fermionic and gaugeactions. For the chosen actions and parameters we obtainAction b1 b2 b3 b4 bΣβ = 8.45 −5.6115 −3.8336 2.7793 0.3446 −16.180β = 8.0 −5.2883 −3.7636 2.7310 0.3331 −15.733(3.4)In (3.4) we have added the contribution of the quark self energy bΣ which is needed for the calcu-lation of the wave function renormalisation.From (2.2), (2.3) and (3.3) we determine the Z factors in the MS scheme for the commonlyused representations under the hypercubic groupτ(6)3 : Ov2a ≡12(O14 +O41) , τ(3)1 : Ov2b ≡O44−13 (O11 +O22 +O33) .They have the formZMSvi (a,µ) = 1−g2CF16pi2[163 log(aµ)+Bvi(ck,ρ)](3.5)withBvi(ck,ρ) =409 +bvi +bΣ , bv2a = b1 +b2 , bv2b = b1 +b2 +b4 . (3.6)It is well known that the naive perturbative results suffer from lattice artefacts. Therefore,mean field (tadpole) improvement [19] has been proposed to rearrange the perturbative series. Incase of overlap fermions the tadpole improved Z factor is given by [16]ZT IO = ZMFO(ZOZMFO)pert, (3.7)where ZMFOis the mean field approximation of ZO . For overlap fermions we haveZMFO =ρρ−4(1−u0), ZMFOpert = 1+g2T I CF16pi24ρ kT Iu . (3.8)Here u0 denotes the mean value of the link. The boosted parameters are chosen as [16]g2T I = g2/u40 , cTI0 = c0 , cTIi = u20 ci (i = 1,3), CT I0 = c0 +8cT I1 +8cT I3 . (3.9)kT Iu is the one-loop contribution of the perturbative expansion for u0 with cTIi inserted for the cor-responding gauge actions (kTIu = 5.3625/5.0835 for β = 8.45/8.0).In case of overlap fermions one needs to improve the quark propagator as well which leads toa mean field improved ρ parameterρT I = ρ−4(1−u0)u0. (3.10)Results obtained with ρT I are denoted as fully tadpole improved (FTI). The Z factors have the formZT I,MSvi = ZMFO{1−g2TICF16pi2[163CT I0log(aµ)+BTIvi (cTIk ,ρ)]}, (3.11)238 / 4PoS(LAT2005)238Perturbative renormalisation of quark bilinear operators H. PerltZFTI,MSvi = ZMFO{1−g2TICF16pi2[163CT I0log(aµ)+BTIvi (cTIk ,ρTI)]}. (3.12)The following table shows numerical results for the various levels of improvement at a = 1/µOperator β B ZMS BTI ZT I,MS BFT I ZFT I,MSv2a 8.45 −22.430 1.315 0.502 1.393 −0.077 1.411v2b 8.45 −22.085 1.311 0.793 1.384 0.230 1.401v2a 8.0 −22.036 1.310 0.603 1.390 −0.108 1.412v2b 8.0 −21.703 1.305 0.892 1.381 0.199 1.402(3.13)It can be read off from Table (3.13) that the one-loop corrections B for the improved perturbativeZ factors become smaller as expected. Thus the perturbative series is better behaved. For β =8.45 and representation v2b we can compare the perturbative Zs (bold faced numbers in (3.13))with a quenched Monte Carlo simulation [20] giving ZMC,MS = 1.98(3). Using the stout smearingprocedure (see Section 4) the resulting factors are ZMC,MS1−smear = 1.47(4) and ZMC,MS2−smear = 1.34(3) withthe value of the smearing parameter ω = 0.15.4. Local operators and stout smearingZ factors for local fermionic operators for overlap fermions and a set of improved gauge ac-tions have been determined in [16]. A recalculation has been performed by Ioannou and Panagopou-los [21]. In this section we show the influence of stout smearing on the perturbative Z factors.By construction a stout smearing step is performed on a gauge link variable Uµ(x) as [15]U (n+1)µ (x) = eiQ(n)µ (U,ωµν )U (n)µ (x), (4.1)where n denotes the step of smearing. The parameters ωµν characterise the strength of smearing:they are the weights of the perpendicular staples associated to the link (x,x + µˆ). For our pertur-bative calculation we have assumed the isotropic case ωµν = ω and a small value of ω . Variousinvestigations suggest values of 0.1 < ω < 0.3. Therefore, we have expanded (4.1) to first orderin ω . Furthermore, we have restricted ourselves to n = 1. The resulting stout link has been in-serted into the fermionic action modifying the corresponding Feynman rules for the quark-gluonvertices. The corresponding results are obtained in powers of ω . As a possible choice in the tadpoleimprovement we assumed that our approximate stout smearing has been done for the mean fieldrescaled links (this does not change ZMFOand does not rescale ω).As examples we have calculated the Z factors in MS-scheme for the scalar and axial vectoroperators for the TILW action at β = 8.45 and the perturbative improvement levels discussed inthe previous section. For the scalar operator we find (a = 1/µ)ZS = 1.168−0.248ω−0.154ω2 ,ZT IS = 1.309−0.488ω−0.239ω2 ,ZFT IS = 1.359−0.241ω−0.685ω2 .238 / 5PoS(LAT2005)238Perturbative renormalisation of quark bilinear operators H. PerltThese numbers can be compared with ZMC obtained from a quenched MC simulation at ω = 0.15and a single smearing [20]:ZS = 1.127 , ZTIS = 1.230 , ZFT IS = 1.307; ZMCS,0−smear = 1.36(1) , ZMCS,1−smear = 1.13 .The same has been done for the axial vector operator. We get in this caseZA = 1.156−0.475ω +0.092ω2 ,ZT IA = 1.268−0.860ω +0.179ω2 ,ZFT IA = 1.303−0.560ω−0.346ω2 .The comparison with MC gives for ω = 0.15ZA = 1.087 , ZT IA = 1.114 , ZFT IA = 1.211; ZMCA,0−smear = 1.42(1) , ZMCA,1−smear = 1.16 .Contrary to the non-perturbative case the perturbative stout smearing decreases only mildly therenormalisation factors.References[1] P. H. Ginsparg and K. G. Wilson, Phys. Rev. D25 (1982) 2649.[2] M. Lüscher, Phys. Lett. B428 (1998) 342.[3] S. Capitani, M. Göckeler, R. Horsley, P. E. L. Rakow and G. Schierholz, Phys. Lett. B468 (1999) 150.[4] R. Narayanan and H. Neuberger, Nucl. Phys. B443 (1995) 305; Phys. Lett. B302 (1993) 62.[5] H. Neuberger, Phys. Lett. B417 (1998) 141; Phys. Lett. B427 (1998) 353.[6] F. Niedermayer, Nucl. Phys. Proc. Suppl. 73 (1999) 105.[7] T. DeGrand, A. Hasenfratz and T. G. Kovacs, Phys. Rev. D67 (2003) 054501.[8] M. Lüscher and P. Weisz, Commun. Math. Phys. 97 (1985) 59 [Erratum-ibid. 98 (1985) 433].[9] M. Lüscher and P. Weisz, Phys. Lett. B158 (1985) 250.[10] M. G. Alford, W. Dimm, G. P. Lepage, G. Hockney and P. B. Mackenzie, Phys. Lett. B361 (1995) 87.[11] M. Gürtler [QCDSF Collaboration], PoS (LAT2005).[12] T. Streuer [QCDSF Collaboration], PoS (LAT2005).[13] C. Gattringer, R. Hoffmann and S. Schaefer, Phys. Rev. D65 (2002) 094503.[14] T. DeGrand and S. Schaefer, Phys. Rev. D71 (2005) 034507.[15] C. Morningstar and M. J. Peardon, Phys. Rev. D69 (2004) 054501.[16] R. Horsley, H. Perlt, P. E. L. Rakow, G. Schierholz and A. Schiller [QCDSF Collaboration], Nucl.Phys. B693 (2004) 3 [Erratum-ibid. B713 (2005) 601].[17] M. Göckeler, R. Horsley, E. M. Ilgenfritz, H. Perlt, P. Rakow, G. Schierholz and A. Schiller, Nucl.Phys. B472 (1996) 309.[18] R. Horsley, H. Perlt, P. E. L. Rakow, G. Schierholz and A. Schiller [QCDSF Collaboration],[hep-lat/0505015].[19] G. P. Lepage and P. B. Mackenzie, Phys. Rev. D48, 2250 (1993).[20] M. Gürtler [QCDSF Collaboration], private communication.[21] M. Ioannou and H. Panagopoulos, PoS (LAT2005).238 / 6

Perturbative renormalisation of quark bilinear operators for overlap
  fermions with and without stout links and improved gauge action

Perturbative renormalisation of quark bilinear operators for overlap fermions with and without stout links and improved gauge action

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