We compute lattice renormalisation constants of one-link quark operators
({\it i.e.} operators with one covariant derivative) for overlap fermions and
L\"uscher-Weisz gauge action in one-loop perturbation theory. Among others,
such operators enter the calculation of moments of polarised and unpolarised
hadron structure functions. Results are given for \beta=8.45, \beta=8.0 and
mass parameter \rho=1.4, which are commonly used in numerical simulations. We
apply mean field (tadpole) improvement to our results.Comment: 13 pages, 1 figur

A. Schiller

Alford

Capitani

DeGrand

G. Schierholz

Galletly

Gattringer

Göckeler

Gürtler

H. Perlt

Horsley

Ishibashi

Kikukawa

Lepage

Lüscher

Narayanan

Neuberger

Niedermayer

P.E.L. Rakow

R. Horsley

Physics Letters B

English

arXiv

Crossref

We compute lattice renormalisation constants of one-link quark operators (i.e., operators with one covariant derivative) for overlap fermions and Lüscher–Weisz gauge action in one-loop perturbation theory. Among others, such operators enter the calculation of moments of polarised and unpolarised hadron structure functions. Results are given for β=8.45 , β=8.0 and mass parameter ρ=1.4 , which are commonly used in numerical simulations. We apply mean field (tadpole) improvement to our results

Horsley, R.

Perlt, H.

Rakow, Paul E. L.

Schierholz, G.

Schiller, A.

QCDSF Collaboration

DESY

Renormalisation of one-link quark operators for overlap fermions with Luscher-Weisz gauge action

DESY Publication Database

We compute lattice renormalisation constants of one-link quark operators (i.e., operators with one covariant derivative) for overlap fermions and Luscher-Weisz gauge action in one-loop perturbation theory. Among others, such operators enter the calculation of moments of polarised and unpolarised hadron structure functions. Results are given for beta = 8.45, beta = 8.0 and mass parameter rho = 1.4, which are commonly used in numerical simulations. We apply mean field (tadpole) improvement to our results. (c) 2005 Elsevier B.V. All rights reserved

Rakow, P.E.L.

University of Regensburg Publication Server

Renormalisation of one-link quark operators for overlap fermions with Lüscher–Weisz gauge action

We compute lattice renormalisation constants of one-link quark operators
({\it i.e.} operators with one covariant derivative) for overlap fermions and
L\"uscher-Weisz gauge action in one-loop perturbation theory. Among others,
such operators enter the calculation of moments of polarised and unpolarised
hadron structure functions. Results are given for \beta=8.45, \beta=8.0 and
mass parameter \rho=1.4, which are commonly used in numerical simulations. We
apply mean field (tadpole) improvement to our results

Horsley, R

Perlt, H

Rakow, PEL

Schierholz, G

Schiller, A

University of Liverpool Repository

arXiv:hep-lat/0505015v1  14 May 2005DESY 05-029Edinburgh 2005/03Leipzig LU-ITP 2005/014Liverpool LTH 648Renormalisation of one-link quark operators foroverlap fermions with Lu¨scher-Weisz gauge actionR. Horsley1, H. Perlt2,3, P.E.L. Rakow4, G. Schierholz5,6 and A. Schiller3– QCDSF Collaboration –1 School of Physics, University of Edinburgh,Edinburgh EH9 3JZ, UK2 Institut fu¨r Theoretische Physik, Universita¨t Regensburg,D-93040 Regensburg, Germany3 Institut fu¨r Theoretische Physik, Universita¨t Leipzig,D-04109 Leipzig, Germany4 Theoretical Physics Division, Department of Mathematical Sciences,University of Liverpool,Liverpool L69 3BX, UK5 John von Neumann-Institut fu¨r Computing NIC,Deutsches Elektronen-Synchrotron DESY,D-15738 Zeuthen, Germany6 Deutsches Elektronen-Synchrotron DESY,D-22603 Hamburg, GermanyAbstractWe compute lattice renormalisation constants of one-link quark operators (i.e.operators with one covariant derivative) for overlap fermions and Lu¨scher-Weiszgauge action in one-loop perturbation theory. Among others, such operators enterthe calculation of moments of polarised and unpolarised hadron structure functions.Results are given for β = 8.45, β = 8.0 and mass parameter ρ = 1.4, which are com-monly used in numerical simulations. We apply mean field (tadpole) improvementto our results.11 IntroductionIn a recent publication [1] we have computed lattice renormalisation constants of localbilinear quark operators for overlap fermions and improved gauge actions in one-loopperturbation theory. Among the actions we considered were the Symanzik, Lu¨scher-Weisz, Iwasaki and DBW2 gauge actions. The results were given for a variety of ρparameters. Furthermore, we showed how to apply mean field (tadpole) improvementto overlap fermions. In this letter we shall extend our work to one-link bilinear quarkoperators. Operators of this kind enter, for example, the calculation of moments ofpolarised and unpolarised hadron structure functions. The present calculations are muchmore involved than the previous ones, so that we shall restrict ourselves to the Lu¨scher-Weisz action, and to parameters actually being used in numerical calculations.The integral part of the overlap fermion action [2, 3, 4]SF = ψ¯[(1− am2)DN +m]ψ , (1)m being the mass of the quark, is the Neuberger-Dirac operatorDN =ρa(1 +X√X†X), X = DW − ρa, (2)where DW is the Wilson-Dirac operator, and ρ is a real parameter corresponding to anegative mass term. At tree level 0 < ρ < 2r, where r is the Wilson parameter. We taker = 1 and consider massless quarks.Numerical simulations of overlap fermions are significantly more costly than simula-tions of Wilson fermions. The cost of overlap fermions is largely determined by the condi-tion number of X†X. This number is greatly reduced for improved gauge field actions [5].For example, for the tadpole improved Lu¨scher-Weisz action we found a reduction factorof & 3 compared to the Wilson gauge field action [6]. The reason is that the Lu¨scher-Weiszaction suppresses unphysical zero modes, sometimes called dislocations [7]. A reductionof the number of small modes of X†X appears also to result in an improvement of thelocality of the overlap operator [5].We consider the tadpole improved Lu¨scher-Weisz action [8, 9, 10]SG =6g2[c0∑plaquette13ReTr (1− Uplaquette) + c1∑rectangle13ReTr (1− Urectangle)+ c3∑parallelogram13ReTr (1− Uparallelogram)],(3)2where Uplaquette is the standard plaquette, Urectangle denotes the loop of link matricesaround the 1×2 rectangle, and Uparalellogram denotes the loop along the edges of the three-dimensional cube. It is required that c0 + 8c1 + 8c3 = 1 in the limit g → 0, in order toensure the correct continuum limit. We defineβ =6g2c0 . (4)The remaining parameters are [10]:c1c0= −(1 + 0.4805α)20 u20,c3c0= −0.03325αu20,1c0= 1 + 8(c1c0+c3c0), (5)whereu0 =(13Tr 〈Uplaquette〉) 14, α = − log(u40)3.06839. (6)The final results cannot be expressed in analytic form (as a function of β and ρ)anymore. We therefore have to make a choice. Here we consider two couplings, β = 8.45and 8.0, at which we run Monte Carlo simulations at present [6, 11]. The correspondingvalues of c1 and c3 are [12]:β c1 c3 r0/a8.45 -0.154846 -0.0134070 5.29(7)8.0 -0.169805 -0.0163414 3.69(4)(7)In (7) we also quote the corresponding force parameters r0/a, as given in [12]. Assumingthat r0 = 0.5 fm, they translate into a lattice spacing of a = 0.095 fm at β = 8.45 anda = 0.136 fm at β = 8.0. The mass parameter was chosen to be ρ = 1.4. This appearedto be a fair compromise between optimising the condition number of X†X as well as thelocality properties of DN [13].The paper is organised as follows. In Section 2 we give a brief outline of our calculationsand present results for the renormalisation constants in one-loop perturbation theory. InSection 3 we tadpole improve our results, and in Section 4 we give our conclusions.2 Outline of the calculation and one-loop resultsThe Feynman rules specific for overlap fermions [14, 15] are collected in [1], while thegluon-operator and the gluon-gluon-operator vertices (needed for the cockscomb and op-erator tadpole diagrams) are independent of the fermion action and can be found in [16].3We consider general covariant gauges, specified by the gauge parameter ξ. The Landaugauge corresponds to ξ = 1, while the Feynman gauge corresponds to ξ = 0. In latticemomentum space the gluon propagator Dµν(k) is given by the set of linear equations∑ρ[Gµρ(k)− ξξ − 1 kˆµkˆρ]Dρν(k) = δµν , (8)whereGµν(k) = kˆµkˆν +∑ρ(kˆ2ρδµν − kˆµkˆρδρν)dµρ (9)anddµν = (1− δµν)[C0 − C1 a2kˆ2 − C2 a2(kˆ2µ + kˆ2ν)], kˆµ =2asinakµ2, kˆ2 =∑µkˆ2µ . (10)The coefficients {Ci} are related to the coefficients {ci} of the improved action byC0 = c0 + 8c1 + 8c3 , C1 = c3 , C2 = c1 − c3 . (11)The calculations are done analytically as far as this is possible using Mathematica. Partof the numerical results have been checked by an independent routine.The bare lattice operators O(a) are, in general, divergent as a → 0. We define finiterenormalised operators byOS(µ) = ZSO(a, µ)O(a) , (12)where S denotes the renormalisation scheme. We have assumed that the operators donot mix under renormalisation, which is the case for the operators considered in thisletter. The renormalisation constants ZO are often determined in the MOM scheme firstfrom the gauge fixed quark propagator SN and the amputated Green function ΛO of theoperator O:ZMOMψ (a, µ) SN∣∣p2=µ2= Stree , (13)ZMOMO (a, µ)ZMOMψ (a, µ)ΛO∣∣p2=µ2= ΛtreeO + other Dirac structures . (14)(Note that Zψ = 1/Z2.) The renormalisation constants can be converted to the MSscheme,ZMSψ (a, µ) = ZMS,MOMψ ZMOMψ (a, µ) ,ZMSO (a, µ) = ZMS,MOMO ZMOMO (a, µ) ,(15)4where ZMS,MOMψ , ZMS,MOMO are calculable in continuum perturbation theory, and there-fore are independent of the particular choice of lattice gauge and fermion actions.In [1] the wave function renormalisation constants where found to beZMOMψ (a, µ) = 1−g2CF16pi2[2(1− ξ) log(aµ) + 4.79201 ξ + bΣ] (16)in the MOM scheme, andZMSψ (a, µ) = 1−g2CF16pi2[2(1− ξ) log(aµ) + 3.79201 ξ + bΣ + 1] (17)in the MS scheme, with CF = 4/3 andβ bΣ8.45 -17.4298.0 -17.054(18)We consider the following one-link operatorsOµν = i2ψ¯(x)γµ↔Dν ψ(x) , (19)O5µν =i2ψ¯(x)γµγ5↔Dν ψ(x) , (20)where↔Dν=→Dν −←Dν is the (symmetric) lattice covariant derivative. While in our previouswork [1], which involved local bilinear quark operators, we only had to deal with the vertexdiagram shown on the left-hand side of Fig. 1, we now obtain contributions from additionaldiagrams: the operator tadpole and the cockscomb diagrams shown on the right-hand sideof Fig. 1.Figure 1: The one-loop lattice Feynman diagrams contributing to the amputated Greenfunction. From left to right: vertex, operator tadpole and cockscomb diagrams.5Action b1 b2 b3 b4β = 8.45 - 5.6115 -3.8336 2.7793 0.3446β = 8.0 - 5.2883 -3.7636 2.7310 0.3331Plaquette -10.6882 -4.7977 3.4612 0.5267Table 1: The coefficients {bi} for the tadpole improved Lu¨scher-Weisz action at β = 8.45and 8.0, as well as for plaquette action.The amputated Green function of the operator Oµν [eq. (19)] turns out to beΛµν(a, p) = γµpν +g2CF16pi2{[(13+ ξ)log(a2p2)− 4.29201 ξ + b1]γµpν+[43log(a2p2) +12ξ + b2]γνpµ +[−23log(a2p2)− 12ξ + b3]δµν 6p (21)+ b4 δµνγνpν +(−43+ ξ)pµpνp26p},where p is the external quark momentum, and the coefficients {bi} are given in Table 1for the tadpole improved Lu¨scher-Weisz action and, for comparison, for the plaquetteaction (with c1 = c3 = 0) as well. The latter numbers are independent of β. The Greenfunction Λ5µν(a, p) of the operator O5µν [eq. (20)] is obtained by multiplying the right-handside of (21) by γ5 from the right. The coefficients {b5i } turn out to be identical to {bi},as is expected for overlap fermions. Thus, Oµν and O5µν have the same renormalisationconstants. In the following we may therefore restrict ourselves to the operator Oµν .It has been checked numerically that the gauge dependent part of (21) is universal (i.e.independent of the lattice gauge and fermion action), in accordance with the argumentspresented in [1].Under the hypercubic group H(4) the 16 operators of type (19) fall into the followingfour irreducible representations [17]:τ(6)3 : Ov2a ≡12(O14 +O41) , (22)τ(3)1 : Ov2b ≡ O44 −13(O11 +O22 +O33) , (23)τ(1)1 : Ov2c ≡ O11 +O22 +O33 +O44 , (24)τ(6)1 : Ov2d ≡ O14 −O41 . (25)(We have given one example operator in each representation. A complete basis for eachrepresentation can be found in [17].) The operators (22) and (23) are widely used in6numerical simulations [18, 19, 6, 11]. They correspond to the first moment of the partondistribution. The operators (24) and (25) represent higher twist contributions in theoperator product expansion, and so are not used as much as operators in the first tworepresentations. For completeness we give results for all four representations, so that therenormalisation factors for all operators of the form (19) will be known. We denote thecorresponding amputated Green functions by Λv2a , Λv2b , Λv2c and Λv2d . From (21) we readoffΛv2a =12(γ1p4 + γ4p1) {1 +g2CF16pi2[(ξ +53) log(a2p2)− 3.79201 ξ + bv2a]}+g2CF16pi2(−43+ ξ)p1p4p26p , (26)Λv2b =(γ4p4 − 133∑i=1γipi) {1 +g2CF16pi2[(ξ +53) log(a2p2)− 3.79201 ξ + bv2b]}+g2CF16pi2(−43+ ξ)(p24 −133∑i=1p2i)6pp2, (27)Λv2c = 6p{1 +g2CF16pi2[(ξ − 1) log(a2p2)− 4.79201 ξ + bv2c]}, (28)Λv2d =(γ1p4 − γ4p1) {1 +g2CF16pi2[(ξ − 1) log(a2p2)− 4.79201 ξ + bv2d]}(29)withbv2a = b1+ b2 , bv2b = b1+ b2+ b4 , bv2c = b1+ b2+4 b3+ b4−43, bv2d = b1− b2 . (30)It is worth pointing out that with Wilson or clover fermions the Green functions Λv2cand Λv2d both show perturbative mixing of O(g2/a) with local operators. With overlapfermions these O(1/a) terms are completely absent, showing once again that overlapfermions behave much more like continuum fermions when mixing is a possibility.Using (14) and (16), we obtain the renormalisation constants in the MOM scheme:ZMOMv2a,v2b(a, µ) = 1−g2CF16pi2[163log(aµ) + ξ + bv2a,v2b + bΣ], (31)ZMOMv2c,v2d(a, µ) = 1−g2CF16pi2[ bv2c,v2d + bΣ ] . (32)As already mentioned, the conversion factors ZMS,MOMv2a,v2b,v2c,v2d are universal [16]. They are7given byZMS,MOMv2a,v2b = 1−g2CF16pi2(409− ξ), (33)ZMS,MOMv2c = 1−g2CF16pi2(−43), (34)ZMS,MOMv2d = 1 . (35)In the MS scheme we then findZMSv2a,v2b(a, µ) = 1−g2CF16pi2[163log(aµ) +409+ bv2a,v2b + bΣ], (36)ZMSv2c (a, µ) = 1−g2CF16pi2[−43+ bv2c + bΣ], (37)ZMSv2d (a, µ) = 1−g2CF16pi2[bv2d + bΣ] . (38)3 Tadpole improved resultsA detailed discussion of mean field – or tadpole – improvement for overlap fermions andextended gauge actions has been given in [1]. Here we will briefly recall the basic idea,before presenting our results.Tadpole improved renormalisation constants are defined byZTIO = ZMFO(ZOZMFO)pert, (39)where ZMFO is the mean field approximation of ZO, while the right-hand factor is computedin perturbation theory. For overlap fermions (with r = 1), and operators with nD covariantderivatives, we haveZMFO =ρ u1−nD0ρ− 4(1− u0) . (40)In our case nD = 1. It is required that ρ > 4(1− u0), which is fulfilled here (see Table 2).To compute the right-hand factor in (39), we have to remove the tadpole contribu-tions from the perturbative expressions of ZO first. This is achieved if we re-express theperturbative series in terms of tadpole improved coefficients:cTI0g2TI= u40c0g2,cTIig2TI= u60cig2, i = 1, 3 . (41)8β kTIu u408.45 0.543338pi2 0.651768.0 0.515069pi2 0.62107Table 2: The coefficient kTIu and the average plaquette u40 at β = 8.45 and 8.0.This does not fix all parameters, but leaves us with some freedom of choice. The simplestchoice is to defineg2TI =g2u40, cTI0 = c0 , cTIi = u20 ci , i = 1, 3 . (42)With this choiceCTI0 = c0 + 8cTI1 + 8cTI3 , CTI1 = u20C1 , CTI2 = u20C2 . (43)(Note that CTI0 6= 1. However, CTI0 → 1 in the continuum limit.) This means that wehave to replace every g2 by g2TI and every c1 and c3 by cTI1 and cTI3 , respectively, whilekeeping c0 unchanged. The effect of introducing tadpole improved coefficients (42) is thatthe rescaled gluon propagator remains of the same form as we change u0, thus ensuringfast convergence.To compute ZMFO perturbatively, we need to know the perturbative expansion of u0to one-loop order [20, 10]. We writeu0 = 1− g2TICF16pi2kTIu . (44)In [1] we have computed kTIu for the Lu¨scher-Weisz action with coefficients CTI0 , CTI1and CTI2 . The numbers are given in Table 2 for our two values of β, together with the‘measured’ values of u40. Expanding (40) then givesZMFO pert = 1 +g2TICF16pi24ρkTIu . (45)Let us now rewrite the one-loop renormalisation constants of Section 2 asZv2a,v2b = 1−CF g216pi2[163C0log(aµ) +Bv2a,v2b(ρ, C)], (46)Zv2c,v2d = 1−CF g216pi2Bv2c,v2d(ρ, C) . (47)9Dividing (46) and (47) by (45) and inserting (40), we obtain mean field/tadpole improvedrenormalisation constants:ZTIv2a,v2b =ρρ− 4(1− u0){1− g2TICF16pi2[163CTI0log(aµ) +BTIv2a,v2b]}, (48)ZTIv2c,v2d =ρρ− 4(1− u0){1− g2TICF16pi2BTIv2c,v2d}, (49)where we have introduced the abbreviated notationBTI = B(ρ, CTI) +4ρkTIu . (50)The coefficients B(ρ, CTI) are the analogue of B(ρ, C), with C0, C1 and C2 being replacedby CTI0 , CTI1 and CTI2 , respectively. In (48) and (49) only the gluon propagator has beentadpole improved.To tadpole improved the fermion propagator as well, we must replace ρ by [1]ρTI =ρ− 4(1− u0)u0(51)in the right-hand perturbative factor of (39). This defines ‘fully tadpole improved’ renor-malisation constantsZFTIv2a,v2b =ρρ− 4(1− u0){1− g2TICF16pi2[163CTI0log(aµ) +BFTIv2a,v2b]}, (52)ZFTIv2c,v2d =ρρ− 4(1− u0){1− g2TICF16pi2BFTIv2c,v2d}(53)withBFTI = B(ρTI , CTI) +4ρTIkTIu . (54)In Table 3 we present our final results and compare tadpole improved and unimprovedrenormalisation constants. We see that the improved coefficients B are rather small in thecase of the operators v2a and v2b, much smaller than for Wilson and clover fermions [21],which raises hope that the perturbative series converges rapidly. This furthermore meansthat the dominant contribution to the renormalisation constants is given by the meanfield factor (40).4 SummaryWe have computed the renormalisation constants of one-link quark operators for overlapfermions and tadpole improved Lu¨scher-Weisz action for two values of the coupling, β =10Operator β B ZMS BTI ZTI,MS BFTI ZFTI,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 1.401v2c 8.45 −18.079 1.254 2.985 1.318 1.829 1.353v2d 8.45 −19.207 1.270 2.303 1.338 1.369 1.367v2a 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 1.402v2c 8.0 −17.890 1.252 3.038 1.316 1.643 1.358v2d 8.0 −18.954 1.266 2.371 1.336 1.239 1.371Table 3: The constants B and ZMS at a = 1/µ for various levels of improvement.8.45 and 8.0, being used in current simulations. The calculations have been performed ingeneral covariant gauge, using the symbolic language Mathematica. This gave us completecontrol over the Lorentz and spin structure, the cancellation of infrared divergences, aswell as the cancellation of 1/a singularities. However, the price is high. In intermediatesteps we had to deal with O(105) terms due to the complexity of the gauge field action.To improve the convergence of the perturbative series and to get rid of lattice artefacts,we have applied tadpole improvement to our results. This was done in two stages. In thefirst stage we improved the gluon propagator, while in the second stage we improved bothgluon and quark propagators.Results at other β values, ρ parameters (also including other gauge field actions withup to six links) can be provided on request.AcknowledgementsThis work is supported by DFG under contract FOR 465 (Forschergruppe Gitter-Hadronen-Pha¨nomenologie) and by the EU Integrated Infrastructure Initiative Hadron Physics(I3HP) under contract RII3-CT-2004-506078.11References[1] R. Horsley, H. Perlt, P. E. L. Rakow, G. 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Renormalisation of one-link quark operators for overlap fermions with
  Lüscher-Weisz gauge action

Renormalisation of one-link quark operators for overlap fermions with
  L\"uscher-Weisz gauge action

Renormalisation of one-link quark operators for overlap fermions with L\"uscher-Weisz gauge action

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