We present a fully non-perturbative determination of the $O(a)$ improvement
coefficient $c_{\rm SW}$ in three-flavor dynamical QCD for the RG improved as
well as the plaquette gauge actions, using the Schr\"odinger functional scheme.
Results are compared with one-loop estimates at weak gauge coupling.Comment: 3 pages, 6 figures, Lattice2002(Improvement and Renormalization),
  Unnecessary files are remove

A. Ukawa

Ali Khan

Aoki

Jansen

K-I. Ishikawa

K. Kanaya

Lu¨shcer

M. Fukugita

M. Okawa

N. Ishizuka

N. Tsutsui

N. Yamada

S. Aoki

S. Hashimoto

T. Kaneko

T. Umeda

T. Yoshie´x

V. Lesk

Y. Iwasaki

Y. Kuramashi

Y. Taniguchi

English

arXiv

We have determined non-perturbative values of theO(a) improvement coefficient csw in three-flavor dynamical QCD for the RG improved as well as the plaquette gauge actions, using the Schro¨dinger functional scheme. Results are compared with 1-loop estimates at weak gauge coupling

CP-PACS Collaboration,

JLQCD Collaboration,

Aoki, S

Fukugita, M

Hashimoto, S

Ishikawa, Kenichi

Ishizuka, N

Iwasaki, Y

Kanaya, K

Kaneko, T

Kuramashi, Y

Okawa, Masanori

Lesk, V

Taniguchi, Y

Tsutsui, N

Ukawa, A

Umeda, T

Yamada, N

Yoshié, T

Name not available

arXiv:hep-lat/0211034 v2   3 Jul 20031Non-Perturbative Determination of cSW in Three-flavor Dynamical QCD∗CP-PACS and JLQCD Collaborations: S. Aoki,a M. Fukugita,b S. Hashimoto,c K-I. Ishikawa,a,dN. Ishizuka,a,d Y. Iwasaki,a K. Kanaya,a T. Kaneko,c Y. Kuramashi,c M. Okawa,e V. Lesk,d Y. Taniguchi,aN. Tsutsui,c A. Ukawa,a,d T. Umeda,d N. Yamadac and T. Yoshie´a,daInstitute of Physics, University of Tsukuba, Tsukuba, Ibaraki 305-8571, JapanbInstitute for Cosmic Ray Research, University of Tokyo, Kashiwa, Chiba 277-8582, JapancHigh Energy Accelerator Research Organization (KEK), Tsukuba, Ibaraki 305-0801, JapandCenter for Computational Physics, University of Tsukuba, Tsukuba, Ibaraki 305-8577, JapaneDepartment of Physics, Hiroshima University, Higashi-Hiroshima, Hiroshima 739-8526, JapanWe present a fully non-perturbative determination of the O(a) improvement coefficient cSW in three-flavordynamical QCD for the RG improved as well as the plaquette gauge actions, using the Schro¨dinger functionalscheme. Results are compared with one-loop estimates at weak gauge coupling.1. IntroductionRealistic simulation of QCD requires treatingthe light up, down and strange quarks dynami-cally. Incorporating a degenerate pair of up anddown quarks have become almost standard bynow, and a first attempt toward the continuumextrapolation has shown that the deviation of thequenched hadron mass spectrum from experiment[1] is sizably reduced[2]. Adding a dynamicalstrange quark is the next step, which has becomepossible with the recent algorithmic developmentfor odd number of fermions[3].The CP-PACS and JLQCD Collaborationshave jointly started a 2+1 flavor dynamical QCD,employing the polynomial HMC (PHMC) algo-rithm for strange quark and the HMC algo-rithm for up and down quarks. We choose therenormalization-group (RG) improved action forthe gauge fields, in order to avoid the lattice arti-fact present for the plaquette action[4]. We wishto use a fully O(a)-improved action for quarksto control lattice spacing errors. Here we reporton a non-perturbative determination of csw forthree-flavor QCD by the Schro¨dinger functionalscheme both for the plaquette and RG-improved∗Talk presented by S. Aoki0 0.01 0.02M−0.003−0.002−0.00100.0010.0020.003∆McSW=1.3cSW=1.5cSW=1.7Figure 1. ∆M as a function of M for the RGaction with Nf = 3 at β = 2.2.gauge actions.2. Method and SimulationsFor the determination of cSW, we basically fol-low the method of ref.[5], except for the choiceB for the boundary weight of the RG-improvedgauge action [6]. We refer to ref. [5] and refer-ences therein for notations in this report.We mainly use an 83 × 16 lattice in our de-termination of cSW for the RG-improved as wellas the plaquette action with Nf = 3 dynamicalquarks at several values of β. Simulations withNf = 4, 2, 0 are also made for comparison.2We measure the modified PCAC quark masses,M and M ′, and their difference ∆M =M −M ′,at several values of cSW and K. We have takenthese parameters to realize M = 0 by an inter-polation, except at strong couplings for the caseof Nf = 3, where an extrapolation to M = 0 isnecessary as shown in Fig. 1.From the linear fit of ∆M as a function of Mand cSW: δM = a0+a1M+a2cSW, we obtain theO(a) improvement coefficient cSW = (∆M(0)−a0)/a2, where ∆M(0) = 0.000277, marked by thehorizontal dotted line in Fig. 1, is the tree-levelvalue of ∆M on a 83 × 16 lattice. Note that thedependence of ∆M on cSW becomes weaker atstronger couplings, so that the determination ofcSW is more difficult, and hence the error is larger,at stronger couplings.3. ResultsIn the upper plot of Fig. 2 we show the non-perturbative value of cSW as a function of thebare gauge coupling g20 for the RG-improvedgauge action with Nf = 3(circles), 2(diamonds)and 0(squares), together with the one-loop es-timate(solid line) and the mean-field(MF) esti-mate(dashed line) used in ref. [2]. Similarly, re-sults for the plaquette action with Nf = 3(circles)and 0(squares) are given in the lower plot ofFig. 2, together with the one-loop estimate(solidline) and the non-perturbative values by the Al-pha collaboration with Nf = 2(dotted lines)[5]and 0(long-dashed line)[7].In both cases, the non-perturbative values ofcSW are almost Nf independent at weak couplingwhile they become larger for smaller Nf at strongcoupling. This tendency can be clearly seen inFig. 3, where cSW is plotted as a function ofNf forthe RG action(open symbols) and the plaquetteaction(solid circles).4. Comparison with perturbative esti-matesAt first sight, the non-perturbative cSW seemsto undershoot the one-loop estimate at weakcoupling for the RG action, while it convergessmoothly from above for the plaquette action.0 1 2 3g0211.21.41.61.8Nf=3Nf=2Nf=01−loopMF(CP−PACS)0 0.2 0.4 0.6 0.8 1 1.2g0211.21.41.61.822.2Nf=3Nf=01−loopNf=2(Alpha)Nf=0(Alpha)Figure 2. cSW as a function of g20 for the RG ac-tion (upper) and for the plaquette action(lower).We have found that the discrepancy seen for theRG action is caused by the one-loop O(a/L) er-ror in cSW2, which becomes leading after theO(a/L) error at tree level is removed by requir-ing ∆M = ∆M0. In Fig. 4, the non-perturbativecSW is compared with the one-loop estimate wehave calculated on the same lattice size employedin the simulation, 83 × 16. As seen from the fig-ure the non-perturbative value agrees with theone-loop estimate much better on the 83 × 16lattice than in the infinite box. Note that theO(g20a/L) contribution to cSW slightly depends onNf through the fermion tadpole in the presenceof the background gauge field of the Schro¨dingerfunctional scheme. Such an Nf dependence is in-2O(a) errors of cSW in general cause O(a2) errors inon-shell quantities, which are irrelevant in the O(a)improvement.30 1 2 3 4Nf1.01.11.21.3β=3.0 , RGβ=5.0β=8.85β=12.0β=9.6, PFigure 3. cSW as a function of Nf for RG andP(plaquette) actions.0 0.5 1g0211.021.041.061.08Nf=3Nf=01−loop,infinite box1−loop,83x16,Nf=31−loop,83x16,Nf=0Figure 4. cSW for the RG action at weak cou-pling, together with the one-loop estimate onthe 83 × 16 lattice for Nf = 3(dotted line) andNf = 0(dashed line).deed seen in the numerical data of Fig. 4.For the plaquette action we also confirm a pres-ence of the O(g20a/L) correction, which is smallon the 83 × 16 lattice, as shown in Fig. 5 wherecSW is plotted as a function of a/L at β = 24with Nf = 0. The one-loop estimates(solid cir-cles) reproduce the non-monotonic behaviour ofnon-perturbative values(open circles) well.5. DiscussionWe have determined the non-perturbativevalue of cSW for the RG action at several gaugecouplings with Nf = 3, 2, 0. In order to obtain0 0.05 0.1 0.15a/L1.0551.061.0651.071.075non−perturbative1−loopFigure 5. Non-perturbative cSW(open circles)and one-loop estimate(solid circles) as a functionof a/L at β = 24 for the plaquette action withNf = 0.an interpolation formula of cSW as a function ofg20 , we have to eliminate large O(g20a/L) correc-tions to cSW present for the RG action. We arecurrently investigating this problem.We are also measuring the hadron spectrum forthe RG action at β ≡ 2 with Nf = 3 using thepreliminary value of cSW, in order to determinethe corresponding lattice spacing.This work is supported in part by Grants-in-Aid of the Ministry of Education (Nos. 11640294,12304011, 12640253, 12740133, 13135204,13640259, 13640260, 14046202 14740173 ). N.Y.is supported by the JSPS Research Fellowship.REFERENCES1. CP-PACS Collaboration: S. Aoki, et al.,Phy. Rev. Lett. 84 (2000) 238.2. CP-PACS Collaboration: A. Ali Khan, et al.,Phy. Rev. Lett. 85 (2000) 46743. JLQCD Collaboration: S. Aoki, et al.,Phy. Rev. D65 (2002) 094507, and referencestherein.4. JLQCD Collaboration: S. Aoki, et al.,Nucl. Phys.(Proc.Suppl.) 106 (2002) 263.5. K. Jansen and R. Sommer, Nucl. Phys. B530(1998) 185.6. S. Aoki, R. Frezzotti, P. Weisz,Nucl. Phys. B540 (1998) 501.7. M. Lu¨shcer et al., Nucl. Phys. B491 (1997)323.

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Non-perturbative determination ofcsw in three-flavor dynamical QCD

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JLQCD Collaboration

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