Sep 30, 1998 - Jack-knife and [other] error analysis were all con- sistent with each other. So we could solidly draw various quantitative conclusions,...

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Chiral limit of light hadron mass in quenched staggered QCD

arXiv:hep-lat/9809184v1 30 Sep 1998

Seyong Kima∗ and Shigemi Ohtab† a

Department of Physics, Sejong University, Seoul, Korea

b

Institute for Particle and Nuclear Studies, KEK, Tsukuba, Ibaraki 305-0801, Japan

We discuss chiral limit of light hadron mass from our quenched staggered calculations with a high lattice cutoff of a−1 ∼3.7 GeV at β=6.5 and a large lattice volume of 483 × 64. We added six heavier quark mass values of mq a=0.0075, 0.015, 0.02, 0.03, 0.04 and 0.05 to the previously existing 0.01, 0.005, 0.0025, and 0.00125. An interesting curvature is observed in the m2π /mq to mq plot near mq a=0.01.

We have been reporting our quenched staggered light hadron mass calculations for the past few years [1]. Our inverse squared coupling is set at β=6.5 corresponding to a high cutoff of a−1 ∼3.7 GeV. The lattice volume of 483 × 64 covers about 2.6 fm across for each space dimension and hence is comfortable enough even for our lightest pion with mπ ∼0.06 a−1 =220 MeV. We calculated staggered quark propagators using “corner” and “even” wall sources of a few different wall sizes and point sink, with quark mass values set at mq a=0.01, 0.005, 0.0025, and 0.00125 for each of the 250 well-separated gauge configurations and then formed various light-hadron propagators. We did not see any significant autocorrelation among the hadron propagators, and our Jack-knife and [other] error analysis were all consistent with each other. So we could solidly draw various quantitative conclusions, of which most important are

Figure 1. Edinburgh plot comparing the ratios mN /mρ (vertical) and mπ /mρ (horizontal).

1. flavor symmetry breaking among different staggered definitions of pion and ρ meson are smaller than the statistical errors, 2. mπ /mρ is as small as 0.27±0.01, 3. mN /mρ is as small as 1.25±0.04. ∗ The

authors thank the Computer Center, Institute of Physical and Chemical Research (RIKEN), Japan, for the use of its VPP500 vector parallel super computing facility. † SO thanks the hospitality of the RIKEN BNL Research Center, Brookhaven National Laboratory, Upton, NY 11973-5000, USA, where he stayed for a year from September 1997. Most of the new research in this poster was done during this stay.

In addition we saw a possible sign of anomalous quenched chiral logarithm: the ratio m2π /mq seemed to increase toward lighter quark mass values. Unlike with Wilson-fermion quarks where no good way to accurately determine the critical value of the hopping parameter is known, with staggered-fermion quarks we have a good control of chiral symmetry and hence of quark mass. So our chance in either establishing or excluding the presence of this anomalous effect is better. It is

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Figure 2. ∆ (×), N (✸), ρ (✷) and π (+) mass (vertical) vs bare quark mass (horizontal).

Figure 3. ∆ (×), N (✸) and ρ (✷) mass (vertical) vs π mass (horizontal)

this issue that we want to address in this poster. Since it is prohibitively costly to push down the lightest quark mass value further, we decided to add several heavier quark mass values of mq a=0.0075, 0.015, 0.02, 0.03, 0.04 and 0.05. So far we have accumulated propagator calculations for 30 gauge configurations evenly distributed over the 250 available ones. We summarize the obtained light hadron mass spectrum in Table 1 and in Figure 1 and 2. The curvature in nucleon mass, as has been discussed since Lattice 97 [2] and is observed also in Figure 2, is probably not relevant for the anomalous quenched chiral log discussion because of renormalization in quark mass. More relevant is to compare the obtained pion mass with the other hadron mass estimates. In full QCD the correction for finite pion mass mπ should start with O(m2π ), but in quenched QCD it may start with O(mπ ) arising from the anomalous chiral log term. Actual behavior obtained from our lattice as shown in Figure 3 cannot yet distinguish these two cases: If we disregard chiral perturbation argument for the nucleon mass and try fitting to a naive form of mN = C0 + C1 mq + C2 m2q , where mq is the bare quark mass, we get C0 =0.248(2), C1 =9.3(2) and C2 =−31(4) with a confidence level of 91%. Similar naive fitting gives us C0 =0.354(2), C1 =5.4(2) and C2 =16(5)

for the ∆ resonance mass with confidence level of 8.3 × 10−10 , and C0 =0.200(1), C1 =4.6(1) and C2 =−3(2) with a confidence level of 9.5 × 10−4 for the ρ meson mass. On the other hand a fitting form of mN = C0 + C1 mπ + C2 m2π gives C0 =0.204(3), C1 =0.662(4) and C2 =1.21(8) with a confidence level of 1.1 × 10−4 for the nucleon mass. This form gives equally bad fitting as the naive form for ∆ resonance and the ρ meson. On the other hand Gell’Mann-Oakes-Renner mass ratio m2π /mq seems more suggestive. In full QCD we expect this ratio to behave like µ0 + µ1 mq a near the chiral limit mq a → 0 [3]. With the anomalous quenched chiral log present it would be modified by an additional µ′ ln mq a behavior. Our current result, shown in Figure 4, seems to suggest there is this anomalous logarithmic contribution, though the statistics is not good enough yet. Finite-volume effect in m2π may show up as a finite but non-zero intercept for m2π -mq curve [4]. Fitting to the form of m2π = C0 + C1 mq + C2 m2q gives C0 =0.00066(96), C1 =2.3(1) and C2 =12(2) with a confidence level of 99%. Note that since our bare quark mass mq in the current fitting range is small, higher order terms in mq is irrelevant. The small (consistent with zero within error) intercept C0 may suggest the absence of finite-volume effect. On the other hand, fitting to the form of ln m2π =

3 Table 1 Hadron mass estimates. Since heavier quarks are less sensitive to gauge field fluctuations, statistics for the additional heavier quark mass values are already good enough except for the two lighter ones of mq a=0.0075 and 0.015. We will add more samples for these two and other heavier quark mass values. mq a π ρ N ∆ 0.00125 0.0580(8) 0.212(4) 0.261(6) 0.374(5) 0.0025 0.0811(6) 0.214(2) 0.269(3) 0.378(3) 0.005 0.1131(5) 0.223(1) 0.293(2) 0.369(4) 0.0075 0.138(1) 0.241(2) 0.321(3) 0.399(5) 0.01 0.1582(5) 0.2434(8) 0.337(1) 0.401(2) 0.015 0.198(1) 0.269(2) 0.384(3) 0.445(4) 0.02 0.229(1) 0.291(1) 0.422(3) 0.468(3) 0.03 0.2857(8) 0.336(1) 0.498(2) 0.535(2) 0.04 0.3369(7) 0.379(1) 0.568(2) 0.598(2) 0.05 0.3850(6) 0.4216(9) 0.635(2) 0.662(3) c + ln mq − δm2η′ + dm2π + em4π [5] gives c=0.68(7), δ=0.005(2), d=2.8(6), e=−5(2) with a confidence level of 16%. Conclusions: Even with the current small statistics for the added heavier quark mass values our investigation of the quenched chiral logarithm is already showing an interesting sign. It may become conclusive when the statistics is improved from the current 30 configurations to the target 250 ones, especially at the quark mass values of mq = 0.0075 and 0.015 where the Gell’MannOakes-Renner mass ratio m2π /mq is showing interesting curvature. REFERENCES

Figure 4. Gell’Mann-Oakes-Renner mass ratio m2π /mq (vertical) vs bare quark mass mq (horizontal).

1. S. Kim and S. Ohta, Nucl. Phys. B (Proc. Suppl.) 63A-C (1998) 185 and ref. [2] therein. 2. T. Yoshie, Nucl. Phys. B (Proc. Suppl.) 63AC (1998) 3, and references therein; see also R. Burkhalter’s, R. Kenway’s, and T. Yoshie’s and others’ respective contributions in these proceedings. 3. S.R. Sharpe, Phys. Rev. D46 (1992) 3146; C.W. Bernard and M.F.L. Golterman, Phys. Rev. D46 (1992) 853. 4. R.D. Mawhinney, Nucl. Phys. B (Proc. Supp.) 47 (1996) 557. 5. S. Kim and D.K. Sinclair, Phy. Rev. D52 (1995) 2614.