The quenched hadron spectrum in the continuum obtained with the Wilson quark
action in recent simulations on the CP-PACS is presented. Results for the light
quark masses and the QCD scale parameter are reported.Comment: Talk presented by K. Kanaya at Lattice97, Edinburg

A. Ukawa

Butler

G. Boyd

Hamber

K. Kanaya

M. Okawa

N. Ishizuka

R. Burkhalter

S. Aoki

S. Hashimoto

T. Yoshié

Weingarten

Y. Iwasaki

Y. Kuramashi

English

arXiv

Aoki S.

Boyd G.

Burkhalter R.

Hashimoto S.

Ishizuka N.

IWASAKI Yoichi

Kanaya K.

Kuramashi Y.

Okawa M.

UKAWA Akira

Yoshie T.

Tsukuba Repository  (Tulips-R)

CP-PACS result for the quenched light hadronspectrum著者 Aoki S., Boyd G., Burkhalter R., Hashimoto S.,Ishizuka N., Iwasaki Y., Kanaya K., KuramashiY., Okawa M., Ukawa A., Yoshie T.journal orpublication titleNuclear physics. B, Proceedings supplementsvolume 63number 1-3page range 161-163year 1998-04権利 (C) 1998 Elsevier Science B.V.URL http://hdl.handle.net/2241/91495doi: 10.1016/S0920-5632(97)00708-1arXiv:hep-lat/9709139v1  29 Sep 19971UTCCP-P-26Sept. 1997CP-PACS Result for the Quenched Light Hadron Spectrum∗CP-PACS Collaboration: S. Aoki,a G. Boyd,b R. Burkhalter,b S. Hashimoto,c N. Ishizuka,a,b Y. Iwasaki,a,bK. Kanaya,a,b Y. Kuramashi,d M. Okawa,d A. Ukawa,a and T. Yoshie´a,baInstitute of Physics, University of Tsukuba, Tsukuba, Ibaraki 305, JapanbCenter for Computational Physics, University of Tsukuba, Tsukuba, Ibaraki 305, JapancComputing Research Center, High Energy Accelerator Research Organization (KEK), Tsukuba,Ibaraki 305, JapandInstitute of Particle and Nuclear Studies, High Energy Accelerator Research Organization (KEK),Tsukuba, Ibaraki 305, JapanThe quenched hadron spectrum in the continuum obtained with the Wilson quark action in recent simulationson the CP-PACS is presented. Results for the light quark masses and the QCD scale parameter are reported.1. IntroductionEver since the pioneering work of Hamber andParisi[1] and of Weingarten[2], the calculation ofthe light hadron spectrum has been regarded as afundamental subject in lattice QCD. The exten-sive simulations on the GF11 computer[3] have es-tablished that the quenched spectrum agrees withexperiment within 5–10%. The next, and final,step within quenched QCD would be to furtheradvance the calculational accuracy so that pos-sible deviations from the experimental spectrumare made manifest. In this article we report themain results recently obtained with the CP-PACScomputer towards this goal.2. SimulationOur calculation is made with the plaquettegluon action and the Wilson quark action. Weemploy a spatial size of Lsa ≈ 3 fm to avoid finite-size effects, and a−1>∼2 GeV to facilitate the con-tinuum extrapolation. These requirements leadto the selection of β and lattice sizes listed inTable 1. Gauge configurations are generatedwith the over-relaxation and heat bath algorithmsmixed in a 4:1 ratio. Hadron propagators arecalculated for the five quark masses correspond-∗Talk presented by K. Kanaya at Lattice97, Edinburgh,Scotland, 22–26 July 1997.Table 1Parameters of simulation.β lattice a−1[GeV] Lsa[fm] Nconf5.90 323 × 56 1.97(1) 3.21(2) 8006.10 403 × 70 2.60(1) 3.04(2) 6006.25 483 × 84 3.12(2) 3.03(2) 4206.47 643 × 112 4.16(4) 3.03(3) 91ing to mpi/mρ ≈ 0.75, 0.7, 0.6, 0.5 and 0.4,the last point being a step closer to the chirallimit than hitherto attempted. All unequal quarkmass combinations allowed for degenerate u andd quarks are taken. We smear the quark sourcewith the function Ψ(r) = A exp[−B(mρ,mq) r],with B ≈ 0.33 fm. Hadron masses are extractedby uncorrelated fits together with a single elimi-nation jackknife procedure for estimating errors.3. Meson spectrumWe find m2PS for pseudoscalar and mV for vec-tor mesons to be well described by a linear func-tion of 1/K, which we adopt for our chiral fits.The physical point for the degenerate u and dquarks is fixed by mpi(135) and mρ(769), and weuse mK(498) or mφ(1019) for the s quark.In Fig. 1 we show the continuum extrapolationof strange vector meson masses with the mK asinput. A linear fit in a yields the continuum value20.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8a [GeV−1]0.800.850.900.951.001.051.10m [GeV]CPPACSExperimentGF11K*φmK inputFigure 1. Continuum extrapolation of K∗ and φmasses with the mK-input. GF11 estimate forinfinite volume[3] is plotted left at a= 0.0.0 1.0 2.0 3.0mPS2/mK21.61.82.02.22.42.6(mV2−mPS2 )/mK2experimentβ=5.90  323x56β=6.10  403x70β=6.25  483x84β=6.47  643x112Figure 2. Meson hyperfine splitting.which is smaller than the experimental value by3% for K∗ and by 4% for φ, to be compared withthe statistical error of 0.5%. If we take mφ as theinput for the s quark, mK∗ agrees with experi-ment to 0.6%, while mK is higher by 9%.The origin of the discrepancy can be traced tothe hyperfine splitting shown in Fig. 2. Our re-sults, which scale well, decrease too fast as mesonmass increases. The slope b of the curve is relatedto the J parameter through J = (1+ b)/2. A fastdecrease translates into a small value of J ; weobtain J = 0.377(11) in the continuum limit.4. Baryon spectrumWe show our typical chiral extrapolation forbaryon masses in Fig. 3. We find all baryonmasses to be linear in 1/K except for mN and6.44 6.46 6.48 6.50 6.52 6.541/K0.350.400.450.500.550.600.65mbaryon aNΞΛΣΞ∗Σ∗∆,Ωβ=6.10Figure 3. Baryon masses as a function of 1/Kfor the light quark at β = 6.1. For non-degenerate baryons, results with the heavier squark (mpi/mρ≈0.75) are plotted.0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8a [GeV−1]0.81.01.21.41.61.8m [GeV]Experiment∆NΩΛFigure 4. Continuum extrapolation of baryonmasses. Open symbols show GF11 results.mΛ, which show a clear negative curvature. Wechoose a fit cubic in 1/K for mN , quadratic formΛ, and linear for the other baryon masses.The negative curvature significantly lowers thenucleon mass compared to previous results at afinite lattice spacing, as shown in Fig. 4. Thecontinuum value obtained from a linear extrapo-lation is smaller than experiment by 2.3%, albeitconsistent within the statistical error of 3%. Themass of ∆ is 6% higher, with a statistical signifi-cance of 1.7 standard deviations.The nucleon mass at each β depends on theform adopted for the chiral extrapolation. In-cluding the form from quenched chiral perturba-tion theory, the continuum values, however, agreewithin 3%.30.40.60.81.01.21.41.61.8m  [GeV]GF11K−inputφ−inputExperimentKK*φN ∆ΛΣΣ*ΞΞ*ΩFigure 5. Quenched hadron spectrum in the con-tinuum limit.The extraction of the continuum value is mostuncertain for the nucleon, ∆, and Σ∗ masses, forwhich the errors in the continuum limit are about3%. For all the other channels a linear chiral ex-trapolation (quadratic for mΛ) followed by a lin-ear continuum extrapolation fits our data verywell, resulting in errors of 1–2%.Our final result for the baryon spectrum isshown in Fig. 5, together with the meson spec-trum for completeness.In the octet the strange baryon masses arelower than experiment, by 5–8% using mK as in-put, and by 3–5% for mφ as input. The Gell-Mann-Okubo (GMO) relation, however, is wellsatisfied, at 1% for both cases.For the decuplet the GMO relation takes theform of an equal spacing rule. This is also wellsatisfied with our results, the three spacings mu-tually agreeing within 5–10%. The average spac-ings, however, are too small by 30% with mK asinput and by 20% for mφ as input.The agreement of the baryon spectrum withexperiment is generally better if we use mφ asinput, particularly for the decuplet. However, wehave to remember that the K meson mass differsfrom experiment by 9% in this case, which is a 7standard deviation effect.5. Light quark massesWe calculate light quark masses with the per-turbative definition mqa = Zm(1/K − 1/Kc)/2and with those based on the axial Ward iden-0.0 0.1 0.2 0.3 0.4mρa3.03.54.04.55.05.5mud(µ=2GeV)  [MeV]PerturbativeWI local mpiWI local exp(mpi)−1WI local 1−exp(−mpi)WI extendedFigure 6. m=(mu +md)/2 at 2 GeV in the MSscheme calculated by various definitions.tity ∇µAµ = 2mqaP . The two methods give re-sults considerably different at finite lattice spac-ings. However, extrapolated linearly in the lat-tice spacing, they converge to a common valuewithin 5% as shown in Fig. 6. For average u,dquark mass at 2 GeV in the MS scheme, we ob-tain m= 4.1(2) MeV. For the s quark mass, wefind ms = 135(7)MeV and 111(4)MeV using mφand mK . The discrepancy reflects the quenchingerror in the strange meson spectrum.6. QCD scale parameterWe estimate ΛMSusing αP (3.40/a) andαMS(pi/a) including the 3-loop correction in theΛ parameter and the scale determined from mρ.The couplings are estimated from the average pla-quette with 2-loop perturbation theory. A linearcontinuum extrapolation of the two results con-verge and yields ΛMS= 230(5) MeV where theerror includes extrapolation uncertainties.This work is supported in part by theGrants-in-Aid of Ministry of Education (Nos.08NP0101, 08640349, 08640350, 08640404, 092462-06, 09304029, 09740226). GB and RB are sup-ported by JSPS.REFERENCES1. H. Hamber and G. Parisi, Phys. Rev. Lett. 47(1981) 1792.2. D. Weingarten, Phys. Lett. 109B (1982) 57.3. F. Butler et al., Nucl. Phys. B430 (1994) 179.

Nuclear Physics B - Proceedings Supplements

CP-PACS result for the quenched light hadron spectrum

Crossref

The quenched hadron spectrum in the continuum obtained with the Wilson quark action in recent simulations on the CP-PACS is presented. Results for the light quark masses and the QCD scale parameter are reported

Aoki, S

Boyd, G

Burkhalter, R

Hashimoto, S

Ishizuka, N

Iwasaki, Y

Kanaya, K

Kuramashi, Y

Okawa, Masanori

Ukawa, A

Yoshié, T

Hiroshima University Institutional Repository

arXiv:hep-lat/9709139 v1   29 Sep 19971UTCCP-P-26Sept. 1997CP-PACS Result for the Quenched Light Hadron Spectrum∗CP-PACS Collaboration: S. Aoki,a G. Boyd,b R. Burkhalter,b S. Hashimoto,c N. Ishizuka,a,b Y. Iwasaki,a,bK. Kanaya,a,b Y. Kuramashi,d M. Okawa,d A. Ukawa,a and T. Yoshie´a,baInstitute of Physics, University of Tsukuba, Tsukuba, Ibaraki 305, JapanbCenter for Computational Physics, University of Tsukuba, Tsukuba, Ibaraki 305, JapancComputing Research Center, High Energy Accelerator Research Organization (KEK), Tsukuba,Ibaraki 305, JapandInstitute of Particle and Nuclear Studies, High Energy Accelerator Research Organization (KEK),Tsukuba, Ibaraki 305, JapanThe quenched hadron spectrum in the continuum obtained with the Wilson quark action in recent simulationson the CP-PACS is presented. Results for the light quark masses and the QCD scale parameter are reported.1. IntroductionEver since the pioneering work of Hamber andParisi[1] and of Weingarten[2], the calculation ofthe light hadron spectrum has been regarded as afundamental subject in lattice QCD. The exten-sive simulations on the GF11 computer[3] have es-tablished that the quenched spectrum agrees withexperiment within 5–10%. The next, and final,step within quenched QCD would be to furtheradvance the calculational accuracy so that pos-sible deviations from the experimental spectrumare made manifest. In this article we report themain results recently obtained with the CP-PACScomputer towards this goal.2. SimulationOur calculation is made with the plaquettegluon action and the Wilson quark action. Weemploy a spatial size of Lsa ≈ 3 fm to avoid finite-size effects, and a−1>∼2 GeV to facilitate the con-tinuum extrapolation. These requirements leadto the selection of β and lattice sizes listed inTable 1. Gauge configurations are generatedwith the over-relaxation and heat bath algorithmsmixed in a 4:1 ratio. Hadron propagators arecalculated for the five quark masses correspond-∗Talk presented by K. Kanaya at Lattice97, Edinburgh,Scotland, 22–26 July 1997.Table 1Parameters of simulation.β lattice a−1[GeV] Lsa[fm] Nconf5.90 323 × 56 1.97(1) 3.21(2) 8006.10 403 × 70 2.60(1) 3.04(2) 6006.25 483 × 84 3.12(2) 3.03(2) 4206.47 643 × 112 4.16(4) 3.03(3) 91ing to mpi/mρ ≈ 0.75, 0.7, 0.6, 0.5 and 0.4,the last point being a step closer to the chirallimit than hitherto attempted. All unequal quarkmass combinations allowed for degenerate u andd quarks are taken. We smear the quark sourcewith the function Ψ(r) = A exp[−B(mρ,mq) r],with B ≈ 0.33 fm. Hadron masses are extractedby uncorrelated fits together with a single elimi-nation jackknife procedure for estimating errors.3. Meson spectrumWe find m2PS for pseudoscalar and mV for vec-tor mesons to be well described by a linear func-tion of 1/K, which we adopt for our chiral fits.The physical point for the degenerate u and dquarks is fixed by mpi(135) and mρ(769), and weuse mK(498) or mφ(1019) for the s quark.In Fig. 1 we show the continuum extrapolationof strange vector meson masses with the mK asinput. A linear fit in a yields the continuum value20.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8a [GeV−1]0.800.850.900.951.001.051.10m [GeV]CPPACSExperimentGF11K*φmK inputFigure 1. Continuum extrapolation of K∗ and φmasses with the mK-input. GF11 estimate forinfinite volume[3] is plotted left at a= 0.0.0 1.0 2.0 3.0mPS2/mK21.61.82.02.22.42.6(mV2−mPS2 )/mK2experimentβ=5.90  323x56β=6.10  403x70β=6.25  483x84β=6.47  643x112Figure 2. Meson hyperfine splitting.which is smaller than the experimental value by3% for K∗ and by 4% for φ, to be compared withthe statistical error of 0.5%. If we take mφ as theinput for the s quark, mK∗ agrees with experi-ment to 0.6%, while mK is higher by 9%.The origin of the discrepancy can be traced tothe hyperfine splitting shown in Fig. 2. Our re-sults, which scale well, decrease too fast as mesonmass increases. The slope b of the curve is relatedto the J parameter through J = (1+ b)/2. A fastdecrease translates into a small value of J ; weobtain J = 0.377(11) in the continuum limit.4. Baryon spectrumWe show our typical chiral extrapolation forbaryon masses in Fig. 3. We find all baryonmasses to be linear in 1/K except for mN and6.44 6.46 6.48 6.50 6.52 6.541/K0.350.400.450.500.550.600.65mbaryon aNΞΛΣΞ∗Σ∗∆,Ωβ=6.10Figure 3. Baryon masses as a function of 1/Kfor the light quark at β = 6.1. For non-degenerate baryons, results with the heavier squark (mpi/mρ≈0.75) are plotted.0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8a [GeV−1]0.81.01.21.41.61.8m [GeV]Experiment∆NΩΛFigure 4. Continuum extrapolation of baryonmasses. Open symbols show GF11 results.mΛ, which show a clear negative curvature. Wechoose a fit cubic in 1/K for mN , quadratic formΛ, and linear for the other baryon masses.The negative curvature significantly lowers thenucleon mass compared to previous results at afinite lattice spacing, as shown in Fig. 4. Thecontinuum value obtained from a linear extrapo-lation is smaller than experiment by 2.3%, albeitconsistent within the statistical error of 3%. Themass of ∆ is 6% higher, with a statistical signifi-cance of 1.7 standard deviations.The nucleon mass at each β depends on theform adopted for the chiral extrapolation. In-cluding the form from quenched chiral perturba-tion theory, the continuum values, however, agreewithin 3%.30.40.60.81.01.21.41.61.8m  [GeV]GF11K−inputφ−inputExperimentKK*φN ∆ΛΣΣ*ΞΞ*ΩFigure 5. Quenched hadron spectrum in the con-tinuum limit.The extraction of the continuum value is mostuncertain for the nucleon, ∆, and Σ∗ masses, forwhich the errors in the continuum limit are about3%. For all the other channels a linear chiral ex-trapolation (quadratic for mΛ) followed by a lin-ear continuum extrapolation fits our data verywell, resulting in errors of 1–2%.Our final result for the baryon spectrum isshown in Fig. 5, together with the meson spec-trum for completeness.In the octet the strange baryon masses arelower than experiment, by 5–8% using mK as in-put, and by 3–5% for mφ as input. The Gell-Mann-Okubo (GMO) relation, however, is wellsatisfied, at 1% for both cases.For the decuplet the GMO relation takes theform of an equal spacing rule. This is also wellsatisfied with our results, the three spacings mu-tually agreeing within 5–10%. The average spac-ings, however, are too small by 30% with mK asinput and by 20% for mφ as input.The agreement of the baryon spectrum withexperiment is generally better if we use mφ asinput, particularly for the decuplet. However, wehave to remember that the K meson mass differsfrom experiment by 9% in this case, which is a 7standard deviation effect.5. Light quark massesWe calculate light quark masses with the per-turbative definition mqa = Zm(1/K − 1/Kc)/2and with those based on the axial Ward iden-0.0 0.1 0.2 0.3 0.4mρa3.03.54.04.55.05.5mud(µ=2GeV)  [MeV]PerturbativeWI local mpiWI local exp(mpi)−1WI local 1−exp(−mpi)WI extendedFigure 6. m=(mu +md)/2 at 2 GeV in the MSscheme calculated by various definitions.tity ∇µAµ = 2mqaP . The two methods give re-sults considerably different at finite lattice spac-ings. However, extrapolated linearly in the lat-tice spacing, they converge to a common valuewithin 5% as shown in Fig. 6. For average u,dquark mass at 2 GeV in the MS scheme, we ob-tain m= 4.1(2) MeV. For the s quark mass, wefind ms = 135(7)MeV and 111(4)MeV using mφand mK . The discrepancy reflects the quenchingerror in the strange meson spectrum.6. QCD scale parameterWe estimate ΛMSusing αP (3.40/a) andαMS(pi/a) including the 3-loop correction in theΛ parameter and the scale determined from mρ.The couplings are estimated from the average pla-quette with 2-loop perturbation theory. A linearcontinuum extrapolation of the two results con-verge and yields ΛMS= 230(5) MeV where theerror includes extrapolation uncertainties.This work is supported in part by theGrants-in-Aid of Ministry of Education (Nos.08NP0101, 08640349, 08640350, 08640404, 092462-06, 09304029, 09740226). GB and RB are sup-ported by JSPS.REFERENCES1. H. Hamber and G. Parisi, Phys. Rev. Lett. 47(1981) 1792.2. D. Weingarten, Phys. Lett. 109B (1982) 57.3. F. Butler et al., Nucl. Phys. B430 (1994) 179.

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CP-PACS Result for the Quenched Light Hadron Spectrum

CP-PACS Result for the Quenched Light Hadron Spectrum

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