Nuclear Physics B (Proc. Suppl.) 30 (1993) 373--376 North-Holland
i ~ Laitq q W~-~ ,'i n k'dl,"! [~1 ~a
Study of Spatial Size Effect in Quenched Wilson Hadron Spectroscopy at/3 = 6.3
K.Akemi a , Ph.deForcrand b , M.Fujisaki ~, T.Hashimoto c , H.C.Hege d , S.Hioki , O.Miyamura,, A.Nakamura f , M.Okuda a, I.O.Stamatescu s , Y.Tago ~ and T.Takaishi '
aComputational Science Research Laboratory, Fujitsu Limited, Ota-ku, Tokyo 144, Japan
bIPS,ETH-Zfirich, CH-8092 Zfirich, Switzerland
CDept. of Applied Physics, Fukui University, Fukui 910, Japan
aZIB, D-1000 Berlin 31, Germany
eDept, of Physics, Hiroshima University, Higashi-Hiroshima 724, Japan
fDept, of Physics, Waseda University, Tokyo 169, Japan
gFESt Heidelberg and Institut fiir Theoretische Physik, Universit~.t Heidelberg, D-6900 Heidelberg, Germany
Quenched Wilson hadron spectroscopy at fl = 6.3 has been studied by using parallel computer AP1000. Spectrum of hadrons composed of various combinations of hopping parameters are measured on 32 s 48 and 163 )
374 K. Akemi et aL / Quenched Wilson hadron spectroscopy at fl = 6.3
Table 1 Coefficients of mass formula
a b c d PS 0.000(3) 0.879(35) 0.316(129) 1.176(57) VC 0.068(12) 0.451(12) 1.138(430) 2.277(616) PR 0.033(14) 1.499(110) -0.909(364) 2,563(272) DL 0.140(11) 1.184(69) -0.043(218) 2.278(148)
0.0 i 6.0 8.0
323 48 z
anti periodic in ~ .-'"'" *
0.130 0.140 ,4 .-";" ,-.-" 0.143 x" .-'~," 0,148 ~*"
0.150 i i i
7.0 1/~ 2
Figure 1. Pseudoscalar mass for various combi- nations of hopping parameters.
requires a large amount of storage for the quark propagators. Abada et al.  measured spectrum of mesons composed of heavy and light quarks. We also measured spectrum of hadrons composed of several combinations of 5 different Wilson hop- ping parameters, 0.150, 0.148, 0.143, 0.140 and 0.130. AP1000 has 8Gbytes of main memory for 512 cells and three kinds of propagators can be kept on it.
We have measured 20 configurations for p.b. and a.p.b, on 163 x 48 lattice. On 323 x 48 lat- tice, 20 and 15 configurations have been measured for p.b. and a.p.b., respectively. The configura- tions are separated by 250 sweeps using the over- relaxed algorithm after 2000 sweeps of thermal- ization. The ratio of overrelaxation to heat bath is 9:1 . Propagators are measured for various channels. In this report, pseudo scalar (PS), vec- tor (VC), (1/2) + baryon (PR) and (3/2) + baryon (DL) are resented. We use standard local opera- tors as point sources.
3. HADRON SPECTRUM
First we report new results on 323 48 lattice with a.p.b. Results for p.b. are found in Ref.. Fig. 1 shows mass spectrum of pseudo scalar chan- nel for various combinations of hopping parame- ters ~1 and ~2. The horizontal line is *:2 and the dashed line is a fitted curve for fixed ~t- To fit the data, we take into account naive quark masses mi up to second order.
= a + b(ml + m2) + c(m + + amlm2,
where rn~ is defined as
!(2 i). rn~---- 2 ~ - - *:c
We use same type of expression for baryons,
m = + b(ml + m2 + m3) + c(m + +
+d(mlm2 + m2m3 + m3ml).
The data are fitted for *:1, *:2, *:3 _> 0.143. This simple formula for meson reproduces all the data within 5% error in the fitted region. The critical hopping parameter *:c determined from pseudo scalar channel is 0.15185(5). Several combina- tion of diagonal (al =- *:2) and off diagonal points (*;1 # *:2) gives consistent *:c within errors. The lattice spacing estimated from the value of vector mass at *:t = *;2 = ~c is a -1 = 3.0 GeV 1.0 GeV. The r - p ratio is 0.76 and p - p ratio is 1.52 for all ~s = 0.150.
3.1. Tempora l Boundary Cond i t ion The measurements denoted in section 2 were
performed on totally different configurations for p.b. and a.p.b. We had no data measured with
K. Akemi et al. I Quenched Wilson hadron spectroscopy at 13 = 6.3 375
Table 2 Comparison of temporal p.b. and a.p.b, on the same configurations t1 = t2 = (t3) = 0.150 for meson (baryon)
PS VC PR DL 16 ~ 4s p.b. 0.274(37) 0 .366(37) 0 .614(61) 0.652(45) 16 3 x 48 a.p.b. 0.271(38) 0.376(35) 0.602(57) 0.646(47) 323 x 48 p.b. 0.253(17) 0.340(10) 0.513(30) 0.562(14) 323 x 48 a.p.b. 0.264(20) 0.338(80) 0.516(28) 0.570(21)
Figure 2. size.
DL I PR
PS ~ i
10 . . . . ,10 20 30 Na
Hadron masses versus spatial lattice
i t i J o.14 lO :;o :;o 4~, .No-
Figure 3. Critical hopping parameter versus spa- tial lattice size.
p.b. and a.p.b, on the same configuration at the time of the conference.
After the conference, we measured additional 10 and 5 configurations on the 163 48 and 323 48 lattice, respectively, both for p.b. and a.p.b. to see the effects of temporal boundary condition. The results are summarized in Table 2 for the lightest combination of to's, i.e. all tc's are 0.150, as the boundary effects become most evident for these hadrons.
The fitting region of the data is r = 10-20 and 28-38 for mesons. We fitted this region with sin- gle cosh type fitting function using X 2 method. The errors are standard ones in X 2 fitting. For baryons, we choose the region r = 10 to 20 and fitted the data with single exponential using the same method.
The results are consistent between p.b. and a.p.b, at this hopping parameter. It is difficult to see statistically meaningful difference between p.b. and a.p.b, on the same configurations within
errors determined by X 2 fitting. Very high statis- tics will be required to find the difference even if it exists. In the following, we combine p.b. and a.p.b, data and make fitting for the combined data, as the difference is meaningless at present statistics.
3.2. Spatial Size Effects A single cosh or single exponential fit may suf-
fer from mixing of higher mass states. We fitted various intervals in the temporal direction and see that the differences are small and within errors.
For baryons, we make parity projection. We see a baryon corresponding to parity plus running forward in the temporal direction and parity mi- nus in the opposite direction when we make par- ity plus projection. We have inverse behavior for negative parity projection. They should be sym- metric, so we add one of them after r --* N~ - r to the other and fit the data. Sign change of one of parity partners can be observed between p.b.
376 K. Akemi et al. / Quenched B41son hadron spectroscopy at/3 = 6.3
and a.p.b. In Fig.2, the spatial size dependence of hadron
masses is shown for all hopping parameter ~s = 0.150. We find an evident finite spatial size effect for baryons between No = 16 and 32. Smaller spatial size gives larger values and the differ- ence between the central values are about two standard deviation both for PR and DL. As for mesons, we do not see a difference within statis- tical errors.
The values corresponding to No = 24 are calcu- lated using mass formula given by the Ape group . The error bars are propagated from errors in the critical hopping parameter and coefficients of the mass formula. Though there exist differ- ences between our method and Ape's, i.e. they use smeared sources and only p.b. for tempo- ral direction, their data are consistent with ours in the meson channels. Their baryon data gives closer value to our No = 32 data and same size of difference of about two standard deviation also can be found between No = 16 and 24.
The spatial size dependence of ~c is shown in Fig.3. We do not observe spatial size dependence within error bars.
We are indebted to M.Ikesaka, Y.Inada, K.Inoue, M.Ishii, T.Saito, T.Shimizu and H.Shiraishi at the Fujitsu parallel computing re- search facilities for their valuable comments on parallel computing.
1 QCD_TARO Collaboration, Nucl.Phys.B (Proc. Suppl.) 26 (1992) 644.
2 M.Guagnelli et al., Nucl.Phys.B378 (1992) 616.
3 F.Butler et al., Nucl.Phys.B (Proc. Suppl.) 26 (1992) 287.
4 M.Fukugita et al., phys.Rev.Lett.68 (1992) 761.
5 A.Abada et al., Nucl.Phys.B376 (1992) 172. 6 QCD_TARO Collaboration, these proceed-
ings. 7 QCD_TARO Collaboration, Nucl.Phys.B
(Proc. Suppl.) 26 (1992) 293.
Large spatial size effects are observed for quenched Wilson baryon spectrum between 163 48 and 32 a x 48 lattice at/3 = 6.3. The meson data show no difference between these lattices within errors. The data on the No = 24 lattice inter- polated from results by Ape group show closer values to ours on 323 x 48 lattice and spatial size effects seem small already for No = 24 at this /3. Butler et al. found no spatial size effects for physically larger spaces, which is consistent with our results.