Download pdf - Study of spatial size effect in quenched Wilson Hadron spectroscopy at β = 6.3

Nuclear Physics B (Proc. Suppl.) 30 (1993) 373--376 North-Holland

i ~ Laitq q W~-~ ,'i n k'dl,"! [ ~ 1 ~a

P R O C E E D I N G S S U P P L E M E N T S

Study of Spatial Size Effect in Quenched Wilson Hadron Spectroscopy at/3 = 6.3

QCD_TARO Collaboration

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 '

aComputa t ional Science Research Laboratory, Fujitsu Limited, Ota-ku, Tokyo 144, J apan

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 Insti tut 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 )<48 lattices. We report results of spatial size effects based on various hadron masses. Dependence on boundary conditions along the temporal direction is also examined and discussed.

1. I N T R O D U C T I O N

We present results of hadron spectroscopy ob- tained from a Monte Carlo study a t /3 = 6.3 using the s tandard Wilson action in the quenched approximation. Special at tention is paid to spatial size effects. Boundary effects along the temporal direction are also discussed. The calculation is performed on a massively parallel computer AP1000 (512 cells at present) supported by Fujitsu [1].

The lattice sizes in our calculation are 163 x 48 and 323 × 48, which is below and above the size of Ape group [2], respectively. The spectrum is measured along the temporal (r) direction of 48 sites for periodic (p.b.) and anti periodic boundary conditions (a.p.b.). The spatial boundary conditions are all periodic.

The spatial size dependence was investigated by many groups. For recent work, see Refs. [3,4].

F.Butler et al. [3] measured the spect rum at /3 = 5.7 using Wilson fermions on 163 x 32 and 243 x 32 lattices with high statistics. They found no spatial size dependence between these lattices even at small quark masses. The a" - p ratio is about 0.5 at the smallest quark mass. The lattice spacing is naively est imated as a ~-, 0.13 fm a t /3 = 5.7. On the other hand, it is 0.071 fm at /3 = 6.3, i.e., about a factor of 2 finer. Our larger lattice roughly corresponds to their smaller one in physical units. We study spatial size effects in a smaller physical volume with a finer mesh and compare our results with Ape group's.

2. G E N E R A L F E A T U R E S OF T H E SIM- U L A T I O N

Recent development in hardware allows us to measure spectrum of hadrons composed of different quarks. Measurement of these quantities

0920-5632/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved.

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)

1.0

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 combinations of hopping parameters.

requires a large amount of storage for the quark propagators. Abada et al. [5] measured spectrum of mesons composed of heavy and light quarks. We also measured spect rum of hadrons composed of several combinations of 5 different Wilson hopping 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 lattice, 20 and 15 configurations have been measured for p.b. and a.p.b., respectively. The configurations are separated by 250 sweeps using the over- relaxed algori thm after 2000 sweeps of thermal- ization. The ratio of overrelaxation to heat bath is 9:1 [6]. Propagators are measured for various channels. In this report, pseudo scalar (PS), vector (VC), (1/2) + baryon (PR) and (3/2) + baryon (DL) are resented. We use standard local opera- tors as point sources.

3. H A D R O N S P E C T R U M

First we report new results on 323 × 48 lattice with a.p.b. Results for p.b. are found in Ref.[7]. Fig. 1 shows mass spectrum of pseudo scalar channel for various combinations of hopping parameters ~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 ( m l m 2 + m 2 m 3 + m3ml) .

The data are fitted for *:1, *:2, *:3 _> 0.143. This simple formula for meson reproduces all the da ta within 5% error in the fitted region. The critical hopping parameter *:c determined from pseudo scalar channel is 0.15185(5). Several combination 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. T e m p o r a l B o u n d a r y C o n d i t i o n The measurements denoted in section 2 were

performed on totally different configurations for p.b. and a.p.b. We had no da ta 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 t¢1 = t¢2 = (t¢3) = 0.150 for meson (baryon)

PS VC PR DL 16 ~ × 4s p.b. 0.274(37) 0 . 3 6 6 ( 3 7 ) 0 . 6 1 4 ( 6 1 ) 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)

0.5

0.0

Figure 2. size.

DL I PR

VC ~ •

PS ~ • i

i

10 . . . . ,10 20 30 Na

Hadron masses versus spatial lattice

0.16

0.15

i t i J o.14 lO :;o :;o 4~, .No-

Figure 3. Critical hopping parameter versus spatial lattice size.

p.b. and a.p.b, on the same configuration at the t ime 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 single cosh type fitting function using X 2 method. The errors are s tandard ones in X 2 fitting. For baryons, we choose the region r = 10 to 20 and fitted the da ta 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 statistics 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 pari ty projection. We see a baryon corresponding to pari ty plus running forward in the temporal direction and pari ty mi- nus in the opposite direction when we make parity plus projection. We have inverse behavior for negative pari ty 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 pa r tne r s 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 difference 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 [2]. The error bars are propagated from errors in the critical hopping parameter and coefficients of the mass formula. Though there exist differences between our method and Ape's, i.e. they use smeared sources and only p.b. for temporal 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.

A C K N O W L E D G E M E N T S

We are indebted to M.Ikesaka, Y.Inada, K.Inoue, M.Ishii, T.Saito, T.Shimizu and H.Shiraishi at the Fujitsu parallel computing research facilities for their valuable comments on parallel computing.

R E F E R E N C E S

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.

4. C O N C L U S I O N

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.