Renormalization constants of vector ($Z_V$) and axial-vector ($Z_A$) currents
are determined non-perturbatively in quenched QCD for a renormalization group
improved gauge action and a tadpole improved clover quark action using the
Schr\"odinger functional method. Non-perturbative values of $Z_V$ and $Z_A$
turn out to be smaller than the one-loop perturbative values by $O(10%)$ at
$a^{-1}\approx 1$ GeV. A sizable scaling violation of meson decay constants
$f_\pi$ and $f_\rho$ observed with the one-loop renormalization factors remains
even with non-perturbative renormalization.Comment: Lattice2001(improvement), 3 pages, 7 figure

Aoki, S.

Burkhalter, R.

Collaboration, CP-PACS

Fukugita, M.

Hashimoto, S.

Ide, K.

Ishikawa, K. -I.

Ishikawa, T.

Ishizuka, N.

Iwasaki, Y.

Kanaya, K.

Kaneko, T.

Kuramashi, Y.

Lesk, V.

Okawa, M.

Taniguchi, Y.

Ukawa, A.

Umeda, T.

Yoshié, T.

English

arXiv

Renormalization constants of vector (ZV) and axial-vector (ZA) currents are determined non-perturbatively in quenched QCD for a renormalization group improved gauge action and a tadpole improved clover quark action using the Schrödinger functional method. Non-perturbative values of ZV and ZA turn out to be smaller than the one-loop perturbative values by O(10%) at a-1 ≈ 1 GeV. A sizable scaling violation of meson decay constants fπ and fρ observed with the one-loop renormalization factors remains even with non-perturbative renormalization

CP-PACS Collaboration,

Aoki, S

Burkhalter, R

Fukugita, M

Hashimoto, S

Ide, K

Ishizuka, N

Iwasaki, Y

Kanaya, K

Kaneko, T

Kuramashi, Y

Lesk, V

Okawa, Masanori

Taniguchi, Y

Ukawa, A

Yoshié, T

Name not available

Nuclear Physics B - Proceedings Supplements

Non-perturbative renormalization for a renormalization group improved gauge action

Renormalization constants of vector (Zv) and axial-vector (ZA) currents are determined non-perturbatively in quenched QCD for an RG-improved gauge action and a tadpole-improved clover quark action using the Schrödinger functional method. Meson decay constants p and fπ show much better scaling when Zv and ZpA estimated for infinite physical volume are used instead of Z-factors from tadpole-improved one-loop perturbation theory

CP-PACS Collaboration

Hiroshima University Institutional Repository

arXiv:hep-lat/0309185 v1   30 Sep 2003UTHEP-480UTCCP-P-144Non-perturbative renormalization of vector and axial vector currents inquenched QCD for a renormalization group improved gauge action ∗CP-PACS Collaboration: K. Ide,a S. Aoki,a M. Fukugita,b N. Ishizuka,a,c Y. Iwasaki,a,c K. Kanaya,aT. Kaneko,d Y. Kuramashi,d V. Lesk,c M. Okawa,e Y. Taniguchi,a A. Ukawaa,c and T. Yoshie´a,caInstitute of Physics, University of Tsukuba, Tsukuba, Ibaraki 305-8571, JapanbInstitute for Cosmic Ray Research, University of Tokyo, Kashiwa, Chiba 277-8582, JapancCenter for Computational Physics, University of Tsukuba, Tsukuba, Ibaraki 305-8577, JapandHigh Energy Accelerator Research Organization (KEK), Tsukuba, Ibaraki 305-0801, JapaneDepartment of Physics, Hiroshima University, Higashi-Hiroshima, Hiroshima 739-8526, JapanRenormalization constants of vector (ZV ) and axial-vector (ZA) currents are determined non-perturbatively inquenched QCD for an RG-improved gauge action and a tadpole-improved clover quark action using the Schro¨dingerfunctional method. Meson decay constants fρ and fpi show much better scaling when ZV and ZA estimated forinfinite physical volume are used instead of Z-factors from tadpole-improved one-loop perturbation theory.1. IntroductionIn a recent comprehensive study by the CP-PACS Collaboration ofNf=2 full QCD [1], mesondecay constants were found to exhibit a very largescaling violation over the range of lattice spacinga−1 ≈ 1 − 2GeV. This was disappointing sincean RG-improved gluon action and Sheikoleslami-Wohlert quark action with tadpole-improved cswwere used. In this calculation, however, one-loopperturbative Z-factors, albeit tadpole-improved,were used for currents. A natural questionwas whether scaling becomes improved if non-perturbative Z-factors are employed instead.At Lattice2001, we reported an initial studyof this problem using the Schro¨dinger functional(SF) method [2] within quenched QCD. We foundthe problem of anomalously large values appear-ing in the ensemble of hadron correlators towardstrong coupling where CP-PACS data of decayconstants had been taken. In this report, we haveanalyzed this problem in some detail. Here wepresent our final results on the Z-factors includ-ing these analysis.∗Talk presented by K. Ide.2. MethodWe follow the method developed by the AL-PHA collaboration [3], and work with a latticegeometry of L3 × T with T = 2L for ZV with avector operator at t = L, and for ZA with twoaxial vector operators at t = 3T/8 and t = 5T/8.Tree-level values are used for the coefficients ofboundary counter terms of the action. For im-proving the axial current, we adopt the one-loopperturbative value for the coefficient cA.Values of ZV and ZA are determined for β =2.2 – 8.0 which approximately covers the range ofthe CP-PACS quenched calculation [1], β = 2.187– 2.575. We have analyzed 200–20000 configura-tions depending on β value and lattice size.3. Exceptional ConfigurationsIt is straight-forward to calculate Z-factors forβ ≥ 2.6. For lower β values on large lattices suchas 83×16, however, anomalously large values ap-pear in the ensemble of hadron correlators. Thismakes it difficult to determine quark mass pre-cisely, and since this means uncertainties in κc,also that of Z-factors.0 200 400 600 800 1000Maximum of fP (cutoff)−0.004−0.00200.0020.004mq2x10−3Figure 1. Cutoff dependence of mq at β = 2.4 onan 83 × 16 lattice for three κ’s around κc.We suspect that these “exceptional” configura-tions are an artifact of quenched approximation;having very small or negative eigenvalues of theWilson-Dirac operator, they would be suppressedin full QCD. Since one cannot distinguish “ex-ceptional” configurations from “normal” ones onsome rigorous basis, we restrict the configurationsused for averaging to those having the value of arelevant hadron correlator below some cutoff. Wethen examine if uncertainties under variation ofthe cutoff are contained within some acceptablemagnitude.In Fig. 1 we illustrate this test formq for whicha cutoff is set for fP (see Ref. [3] for definition).We estimate κc from mq with the cutoff valueof 300, because mq is rather stable there. Theuncertainty in mq at the κc is ≈ ±2× 10−3, oncethe cutoff of fP is taken in the range 200 – 1000.In Fig. 2 we show how much the Z-factors de-pend on mq. ZV is insensitive to mq, and ZAis consistent within 10% or so, albeit apparentlyexhibiting a more pronounced dependence.We analyze the uncertainties in the statisticalaveraging of Z-factors themselves by applying acutoff in f1, as carried out in Ref. [2]. The conclu-sion is similar; ZV is very stable against variationof the cutoff, and ZA shows a more conspicuousvariation of 5% or so.Uncertainties of ZA on an 83 lattice of order15% in total lead to uncertainty of ZA normalizedat infinite volume of order 30%. The uncertainty,however, has little effect in a Pade´ fit of ZA andhence final results; ZA varies less than 3% at thelargest coupling β = 2.187, even if we artificiallyshift ZA at β = 2.4 by 30%.−0.004 −0.002 0 0.002 0.004mq0.720.740.760.780.800.820.840.86Z A−0.004 −0.002 0 0.002 0.0040.720.740.760.780.800.820.840.86Z Vﬀ -cut=200-1000 regionFigure 2. mq dependence of ZV and ZA at β =2.4 on an 83 × 16 lattice.0.0 0.1 0.2 0.3 0.4a/L0.60.70.80.91.0Z VL*=0.80 fmL*=infinity↓β=8.0↑β=2.20.0 0.1 0.2 0.3 0.4a/L0.60.70.80.91.0Z AL*=0.80 fmL*=infinity↓β=8.0↑β=2.2Figure 3. Size dependence of ZV and ZA.4. Results for Z-factorsWe determine the Z-factors for infinite volume(L∗ =∞) and also for a fixed finite physical vol-ume (L∗ = 0.8 fm corresponding to 83 lattice atβ = 2.6) for comparison. The lattice scale is setthrough the string tensionAs shown in Fig. 3, size dependence of Z-factors becomes sizable toward strong couplings.Since our quark action employs a tadpole-improved value of csw, we expect O(a) errors inthe Z-factors. Therefore we extrapolate or inter-0 1 2 3g20.40.50.60.70.80.91.01−looptadpoleL*=0.80 fmL*=infinityZVNPCZV0 1 2 3g20.60.70.80.91.01−looptadpoleL*=0.80 fmL*=infinityZAFigure 4. Results for ZV and ZA normalized atL∗ = 0.8 fm and L∗ =∞.polate results linearly in a/L to obtain estimatesat L∗ = 0.8 fm and at L∗ =∞.In Fig. 4 we show results of Z-factors as afunction of bare coupling g2, together with Pade´fits (solid curves in the figure) to them. Non-perturbative estimates give values smaller thanthe one-loop perturbative ones (dashed lines) byabout 20 % (15%) for ZV (ZA) at the largest cou-pling of the CP-PACS simulation, β = 2.187.ZNPCV determined from the ratio of the con-served vector current to the local one differs sig-nificantly from ZV from the SF method, becausethe local current is not O(a)-improved.5. Scaling Property of Decay ConstantsWe compare in Fig. 5 fpi and fρ determinedwith non-perturbative Z-factors normalized atL∗ =∞ (filled circles) with those using perturba-tive Z-factors (open up triangles). For compari-son, open squares are the results from the stan-dard plaquette and Wilson action[4].We observe a very encouraging result that withthe non-perturbative Z-factors scaling violationsare sizably reduced. Furthermore the continuumextrapolation yields values consistent with thosefrom the standard action.In the same figure, we overlay fpi and fρ deter-0.0 0.2 0.4 0.6 0.8 1.0 1.2a [GeV−1]100120140160180f pi  [MeV]L*=0.80fmL*=infinityImproved, PT ZStandard, PT Z0.0 0.2 0.4 0.6 0.8 1.0 1.2a [GeV−1]150200250300f ρ  [MeV]L*=0.80 fmL*=infinityImproved, ZVNPCf ρ  [MeV]Improved, PT  ZStandard, ZVNPCFigure 5. fpi and fρ vs. a for our improved actionwith non-perturbative and perturbative (PT) Z-factors together with results for the standard ac-tion [4].mined with Z-factors normalized at finite L∗ =0.8 fm (open circles). Scaling is best improvedwhen Z-factors are normalized at L∗ = ∞. Thisproperty is likely related to the fact that O(a/L)errors in Z-factors are removed in the limitingprocedure L∗ →∞.We also find that fρ determined from the con-served vector current (filled down triangle inFig. 5) exhibits a large scaling violation.This work is supported in part by Grants-in-Aid of the Ministry of Education (Nos. 12304011,12640253, 13135204, 13640260, 14046202,14740173, 15204015, 15540251, 15540279 ).REFERENCES1. CP-PACS collaboration: A. Ali Khan et al.,Phys. Rev. D65 (2002) 054505.2. CP-PACS collaboration: S. Aoki et al.,Nucl. Phys. B (Proc. Suppl.) 106 (2002) 8443. M. Lu¨scher et al., Nucl. Phys. B491 (1997)344.4. CP-PACS collaboration: S. Aoki et al.,Phys. Rev. Lett. 84 (2000) 238; Phys. Rev.D67 (2003) 034503.5. M. Lu¨scher et al., Nucl. Phys. B384 (1992)168.

Non-perturbative renormalization of vector and axial vector currents in quenched QCD for a renormalization group improved gauge action

Renormalization constants (Z-factors ) of vector and axial-vector currents are determined nonperturbatively in quenched QCD for a renormalization group improved gauge action and a tadpole-improved clover quark action using the Schrödinger functional method. Nonperturbative values of Z-factors turn out to be smaller than 1-loop perturbative values by O(15%) at a lattice spacing of a-1[approximate] 1 GeV. The pseudoscalar and vector meson decay constants calculated with the nonperturbative Z-factors show a much better scaling behavior compared to previous results obtained with tadpole-improved one-loop Z-factors. In particular, the nonperturbative Z-factors normalized at infinite physical volume show that the scaling violations of the decay constants are within about 100p to the lattice spacing a-1~1 GeV. The continuum estimates obtained from data in the range a-1~ 1–2 GeV agree with those determined from finer lattices (a-1~2-4 GeV) with the standard action

CP-PACS, Collaboration

Ishikawa, Kenichi

Ishikawa, T

Umeda, T

Nonperturbative renormalization of meson decay constants in quenched QCD for a renormalization group improved gauge action

arXiv:hep-lat/0110128 v1   16 Oct 2001UTHEP-452UTCCP-P-117Non-perturbative renormalization for a renormalization group improvedgauge action ∗CP-PACS Collaboration:S. Aoki,a R. Burkhalter,a,b M. Fukugita,c S. Hashimoto,d K. Ide,a N. Ishizuka,a,b Y. Iwasaki,a,b K. Kanaya,aT. Kaneko,d Y. Kuramashi,d V. Lesk,b M. Okawa,d Y. Taniguchi,a A. Ukawaa,b and T. Yoshie´a,baInstitute of Physics, University of Tsukuba, Tsukuba, Ibaraki 305-8571, JapanbCenter for Computational Physics, University of Tsukuba, Tsukuba, Ibaraki 305-8577, JapancInstitute for Cosmic Ray Research, University of Tokyo, Kashiwa, Chiba 277-8582, JapandHigh Energy Accelerator Research Organization (KEK), Tsukuba, Ibaraki 305-0801, JapanRenormalization constants of vector (ZV ) and axial-vector (ZA) currents are determined non-perturbatively inquenched QCD for a renormalization group improved gauge action and a tadpole improved clover quark actionusing the Schro¨dinger functional method. Non-perturbative values of ZV and ZA turn out to be smaller than theone-loop perturbative values by O(10%) at a−1 ≈ 1 GeV. A sizable scaling violation of meson decay constants fpiand fρ observed with the one-loop renormalization factors remains even with non-perturbative renormalization.1. IntroductionReliable lattice calculations of hadronic ma-trix elements and quark masses require bothhigh precision numerical simulations and non-perturbative determinations of renormalizationconstants (Z-factors). The CP-PACS collabo-ration recently carried out a sophisticated spec-trum calculation in Nf = 2 full QCD [1] usinga renormalization group (RG) improved gaugeaction and a tadpole improved clover quark ac-tion. However, non-perturbative Z-factors werenot available for this action combination. Henceanalyses had to rely on one-loop perturbative val-ues.As a first step toward a systematic study ofnon-perturbative renormalization for this action,we apply the Schro¨dinger functional method [2]to calculations of Z factors for vector (ZV ) andaxial-vector (ZA) currents in quenched QCD withthe same improved action. We examine in partic-ular whether a large scaling violation of mesondecay constants observed for this action [1] is im-proved with non-perturbative Z-factors. We re-port preliminary results in these proceedings.∗Talk presented by K. Ide.2. Calculational MethodWe follow the method developed by the AL-PHA collaboration [3]. Namely, we use a latticegeometry of L3 ·T with T = 2L for ZV with a vec-tor operator at t = L, and T = 3L for ZA withtwo axial vector operators at t = L and t = 2L,except at β = 2.2 and 2.4 for ZA (see sec. 3 for de-tails of this exception). Tree-level values are usedfor coefficients of boundary counter terms of theaction. For improving the axial current, we usethe one-loop perturbative value for the coefficientcA.Values of ZV and ZA are determined for β =2.2 – 8.0 which almost covers the range of theCP-PACS quenched spectrum calculation [1], β =2.187 – 2.575. Physical size is normalized at β =2.6 on an 83 lattice. For other β values, two lat-tice sizes are analyzed to match the physical sizeusing the string tension. Our action has O(a) er-rors since we employ a tadpole improved value ofcsw = (1− 0.8412/β)−3/4. Therefore we extrapo-late/interpolate results linearly in 1/L. We haveanalyzed 300–4000 configurations depending on βvalue and lattice size.0 50 100 150 200 250 3000246810f 10 50 100 150 200 250 300#iteration0246810f VFigure 1. Time history of f1 and fV at β = 2.4on an 83 × 16 lattice.3. Exceptional ConfigurationsIt is straight-forward to calculate Z-factors forβ > 2.4 for ZV and β > 2.8 for ZA. We find rea-sonable plateaux in the ratio of Green functionsfor the Z-factors in spite of the O(a) error of theaction, which implies viability of the Schro¨dingerfunctional method for our action.However, for lower β values on a large lattice,anomalously large values appear in the ensembleof f1, fV and fAA where ZV = f1/fV and ZA =√f1/fAA. This is illustrated with a time historyof f1 and fV at β = 2.4 in Fig. 1.In order to estimate Z-factors at low β values,we have investigated the properties of these “ex-ceptional configurations”. We find : i) Large val-ues of f1 and fV for ZV and f1 and fAA for ZA arestrongly correlated (see Fig. 1). ii) Histogramsof f ’s have a long tail toward very large valuesas shown in Fig. 2. We then impose a cutoff intaking the average of the f ’s, and find that Z-factors are stable against change of the cutoff aslong as anomalously large values are discarded,as the numerator and denominator for Z-factorsare correlated and effects mostly cancel out. SeeFig. 3.We then estimate Z-factors for low β valuestaking a certain value of the cutoff. For ZA atβ = 2.2 and 2.4, the lattice geometry is alsochanged from T = 3L to T = 2L because “ex-ceptional configurations” appear very frequently0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4f1050100150frequencyFigure 2. Histogram of f1 at β = 2.4 on an 83×16lattice.0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4cut−off value  of  f10.7600.7620.7640.7660.7680.770Z VFigure 3. Values of ZV estimated by discard-ing configurations with f1 larger than the cutoffvalue. Data are for an 83 × 16 lattice at β = 2.4.for the original geometry to the extent that thecutoff analysis above does not work. We havechecked at β = 2.6 that the change of geometrydoes not lead to any significant difference in ZA.4. Results for Z-factorsIn Fig. 4 we show results of Z-factors to-gether with Pade´ fits (solid curves in the figure)to them. Non-perturbative estimates give val-ues smaller than the one-loop perturbative ones(dashed lines) by about 10 % (6%) for ZV (ZA) atthe largest coupling of the CP-PACS simulation,β = 2.187.5. Scaling Property of Decay ConstantsWe compare in Fig. 5 fpi and fρ determinedwith non-perturbative (filled circles) and pertur-bative (open circles) Z-factors. Also shown arethe results from the standard plaquette and Wil-son action (squares)[4] using the perturbative Z-factors.We observe that, even with the non-pertur-bative Z-factors, large scaling violation of meson0 1 2 3g020.60.70.80.91.0non−perturbative ZPade fit1−looptadpole improved valueβ=2.187β=2.575{CP−PACSregionZV0 1 2 3g020.60.70.80.91.0non−perturbative ZPade fit1−looptadpole improved valueβ=2.187β=2.575{CP−PACSregionZAFigure 4. Results for ZV (top) and ZA (bottom).vestigated. A possible reason is the necessity ofnon-perturbatively fixing the O(a) and perhapshigher terms in the currents themselves. For theaxial vector current, it will be worth investigatingif non-perturbative estimates of the O(a) coeffi-cient cA yield a large value.6. ConclusionsWe have successfully applied the Schro¨dingerfunctional method to calculations of ZV and ZAfor the combination of a RG-improved gauge ac-tion and a tadpole improved clover quark actiondown to the lattice spacings a−1 = 1 − 2 GeVwhere the quenched CP-PACS data for decay con-stants were taken.While Z-factors estimated non-perturbativelyare smaller by O(10%) than perturbative ones forthis range, there still remain large scaling vio-lations of O(a) and higher in meson decay con-stants with non-perturbative Z-factors. Furtherwork is needed to examine if hadronic matrix ele-ments could be reliably extracted at lattice spac-ings much coarser than a−1 ≈ 2 GeV with op-0.0 0.2 0.4 0.6 0.8 1.0 1.2a [GeV−1]100120140160180f pi  [MeV]Improved, NP ZImproved, PT ZStandard, PT Z0.0 0.2 0.4 0.6 0.8 1.0 1.2a [GeV−1]150170190210230250270290f ρ [MeV]Improved, NP ZImproved, PT ZStandard, PT ZFigure 5. fpi (top) and fρ (bottom) vs. a for ourimproved action with non-perturbative (NP) andperturbative (PT) Z-factors together with resultsfor the standard action [4].erators improved non-perturbatively at O(a) andbeyond.This work is supported in part by Grants-in-Aid of the Ministry of Education (Nos. 10640246,10640248, 11640250, 11640294, 12014202,12304011, 12640253, 12740133, 13640260 ). VLis supported by the Research for Future Programof JSPS (No. JSPS-RFTF 97P01102).REFERENCES1. CP-PACS collaboration: A. Ali Khan et al.,hep-lat/0105015.2. M. Lu¨scher et al., Nucl. Phys. B384 (1992)168.3. M. Lu¨scher et al., Nucl. Phys. B491 (1997)344.4. CP-PACS collaboration: S. Aoki et al.,Phys. Rev. Lett. 84 (2000) 238.

S. Aoki

R. Burkhalter

M. Fukugita

S. Hashimoto

K. Ide

N. Ishizuka

Y. Iwasaki

K. Kanaya

T. Kaneko

Y. Kuramashi

V. Lesk

M. Okawa

Y. Taniguchi

A. Ukawa

T. Yoshié

Lüscher

Aoki

Crossref

http://ir.lib.hiroshima-u.ac.jp/files/public/1/14606/20141016121657107173/074502APS.pdf

Non-perturbative renormalization for a renormalization group improved
  gauge action

Non-perturbative renormalization for a renormalization group improved gauge action

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