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MCRG study and renormalized coupling constants
,..N~H ELSEVIER
IUCLEARPHYSIC~ PROCEEDINGS SUPPLEMENTS Nuclear Physics B (Proc. Suppl.) 42 (1995) 805-807
MCRG Study and Renormalized Coupling Constants QCD_TARO Collaboration M. Fujisaki, M. Okuda, Y. Tago Research Center for Coml)utational Science, Fujitsu Linlite(1, Mihama-ku, Chiba 261, .Japan Ph. de Forcrand IPS, ETH-Ziirich, CH-8092 Z(irich, Switzerland T. Hashimoto Del)m'tment of Ai)pli(,(l Phvsi(-s. Fa(uhx of Eugineering. Fukui University. Fukui 910, J a p a n S. Hioki, O. Miyannwa. Depm'tment of Physics. Hiroshima University. Higashi-Hiroshima 724..lapan A. Nakamura. F. Shoji Faculty of Education, Y a m a g a t a University. Yalnagata 990, .Japan I. O. Stamatescu, T. Takaishi. F E S T , Schmeilweg 5, 69118 Heidelberg and Inst. Theor. Physik, Universi~it Heidelberg, 69120 Heidelberg, G e r m a n y \Ve report our analysis of MCtRG study where we elnploy an iml)roved blocking scheme and present our first trial to determine effective coupling constants obtailmd on the blocked lattices.
1. I n t r o d u c t i o n
2. B l o c k i n g
The goal of the ,mm,'ri
in our early work we used Swendson factor 2 schen,' fin' the blocking transforlnation, while we enlploy a new blocking scheme described in Ref.[4], which we call "perfect blocking". It has tim form of
so(3) [sl. 0920-5632/95/$09.50© 1995 Elsevier Science B.V. All rights reserved. SSDI 0920-5632(95)00387-8
=
Scheme
(1
-
6c)U(~°(n,)
+ ~ 2_,
uT(.)
v,#,.
U;j)(~, +
,7)U,~il~(n +/.7)
(1)
where Ut(,°)(.) = U.(.). We c o n s m m t Q . ( n B ) on a blocked lattice fl'om U (~1 as
O,,(,,,,I = u.(~)(.)c22)(- + :)
(2)
The transforlnation kernel from original U's to blocked link variables UB is given by
:<"","> = c'exp(~n~ T,'U.,,(.~)Q~(,,.)) (3) Th(, transformation has an advant.age that a blocked link is lnade tYom more original link variabh,s than in other schemes and it is easy to be
806
QCD-TARO Collaboration~Nuclear Pt~vsics B (Proc. Suppl.) 42 (1995) 805 807
fitting kappa 1,6
- -
,
,
~
r
i
1
C 2 , ... 1.4
~
1.2
T
w3 w4 w5
-.-.-% 0.8
/
,
,-
2(=
~ . ,..&,
c'0) 0.6
C1
0.4
0.2
I
8
Figure 1. R e n o r m a l i z a t i o n G r o u p Transfornlation and R e n o r m a l i z e d T r a j e c t o r y in m a n y cou1)ling space
implemented in the 1)rogram. Moreover a "'(:lassical perfect action" (:orresl)olldillg to the transform a t i o n with , , = 0 . 0 7 7 and ~:=10.5 was foun(l[4].
3. O p t i m i z a t i o n tion
of Blocking
Transforma-
We u p d a t e a 324 lattice at fl=6.65 with 36690 sweeps after 44020 thermalization sweeps by the pseudo h e a t b a t h algorithm[6][7]. T h e blocking transformat.ion is p e r f o r m e d at every 10 sweeps. One m a y choose any blocking t r a n s f o r m a t i o n as fro" as it gives a coarse grained procedure, However it. depends on the scheme how fast we reach to a renormalized trajectory. In Fig.2 we show A/3 o b t a i n e d by nlatching a 324 lattice at 3 = 6 . 6 5 and 164 at /3 - A~i3 as a function of t," in Eq.(3) at the level 2 (i.e., on blocked 84 lattice). /,From this d a t a a" is set to be 18.8.
4. E v a l u a t i o n o f E f f e c t i v e C o u p l i n g s We f u r t h e r u p d a t e 10000 sweeps and perform the blocking tral~sformation with K= 18.8 at every 10 sweeps to 1)ro(ha:e 1000 blocked configurations of the size 4 :~. We present a preliminary results on effective COul)lings determinc(t by the delnon
I 10
12
I
J 14
J 26
Figure 2. A,,'3 o b t a i n e d on a 324 lattice at ,3=6.65 as a function of ~'
m e d m d on the first 100 configurations of these. An extensive analysis will be published elsewhere. T h e d e m o n m e t h o d was originally invented by Creutz[9] and recently a new idea to reduce syst:ematic errors revives it[10]. In the m e t h o d a dem o n visits links on a lattice and exchanges the energy. After m a n y nficrocanonical sweeps, the joint system of the denlon and the lattice is therrealized. \Ve m a y then know the action of the lattice by investigate the energy distribution of the demon. We let 100 demons visit 100 blocked configurations mtd carry out the microcanonical u p d a t e on each system. A preliminary result of the effective couplings are shown in Table.I. A l t h o u g h A/3 obtained at the level 2 show a reasonable matching, it is not clear t h a t we are near to the renormalized trajectory. We plan to continue our M C R G st, udy on lattices with the effective couplings obtained here. Table I llq 1 1.468(12) Sofa O.O57(4)
Effective Coupling Constants 1I'~2 1V22 0.(188(7)0.033(9) Twist O.232(8)
4D-Twist 0.140(5)
Chair
0.266(3)
QCD-TARO Collaboration~Nuclear Physics B (Proc. SuppL) 42 (1995) 805~07
ACKNOWLEDGEMENTS
We would like to express our thanks to P.Hasenfratz for usefltl and stinmlating discussions. Most of the calculations reported here were done on AP1000 at at the Fujitsu parallel computing research facilities, Kawasaki. We are indebted to M.Ikesaka, K.Inoue, M.Ishii, T.Saito, T.Shimizu and H.Shiraishi for their valuable con> ments on parallel computing. This work was sup1)orted in part by the Deutsch Fors(:hungsgenleinschaft, and by the Grand-in-Aid fi)r Scientific I/esearch of Minister of gducation.S('ien('e and Cult, u r e of Japan (C-06640371). REFERENCES
1. Y.Iwasaki, Report No.UTHEP-118, (Dec. 1983) (unpublished); Nucl.Phys. B25& 141 (1985); S.Sakai and H.Iso. Phys.Rev. D36 (1987), 2545-2552: Y.Iwasaki in these l)roceedings. 2. K.Symanzik, in Mathenmti(:al Prolflen~s in Theoretical Physics. edited by R.Schladcr et. al. (Lecture Note in Physics, \:o1.153), (Springer, Berlin, 1982), Nucl.Phys. B226 (1983) 2O5. 3. P.Hasenfl'atz and F.Niedermayer, Nucl.Phys. B414 (1993), 785. 4. A.H~senfi'atz,in these proceedings. 5. QCDTARO, Phys. Rev. Letters 71 (1993) 3068 3O66. 6. N.Cabibbo and E.Mal"inari, Phys.Letters 119B (1982) 38r. 7. A.D.Kennedy and B.J.Pendleton. Phys.Letters, 156B (1985) 393. 8. T.Takaishi, in ~hese proceedings and Heidelberg preprint, "Deterlnination of effective actions for SU(3) gauge theory submitted to Mod.Phys.Letters. 9. M.Crutz, Phys.Rev.Le~ters, 50 (1983) 1411. 10. M.Hasenbush, ICPinn and C.Wieezerkowski, Phys.Lett. B338 (1994) 308.
807
IUCLEARPHYSIC~ PROCEEDINGS SUPPLEMENTS Nuclear Physics B (Proc. Suppl.) 42 (1995) 805-807
MCRG Study and Renormalized Coupling Constants QCD_TARO Collaboration M. Fujisaki, M. Okuda, Y. Tago Research Center for Coml)utational Science, Fujitsu Linlite(1, Mihama-ku, Chiba 261, .Japan Ph. de Forcrand IPS, ETH-Ziirich, CH-8092 Z(irich, Switzerland T. Hashimoto Del)m'tment of Ai)pli(,(l Phvsi(-s. Fa(uhx of Eugineering. Fukui University. Fukui 910, J a p a n S. Hioki, O. Miyannwa. Depm'tment of Physics. Hiroshima University. Higashi-Hiroshima 724..lapan A. Nakamura. F. Shoji Faculty of Education, Y a m a g a t a University. Yalnagata 990, .Japan I. O. Stamatescu, T. Takaishi. F E S T , Schmeilweg 5, 69118 Heidelberg and Inst. Theor. Physik, Universi~it Heidelberg, 69120 Heidelberg, G e r m a n y \Ve report our analysis of MCtRG study where we elnploy an iml)roved blocking scheme and present our first trial to determine effective coupling constants obtailmd on the blocked lattices.
1. I n t r o d u c t i o n
2. B l o c k i n g
The goal of the ,mm,'ri
in our early work we used Swendson factor 2 schen,' fin' the blocking transforlnation, while we enlploy a new blocking scheme described in Ref.[4], which we call "perfect blocking". It has tim form of
so(3) [sl. 0920-5632/95/$09.50© 1995 Elsevier Science B.V. All rights reserved. SSDI 0920-5632(95)00387-8
=
Scheme
(1
-
6c)U(~°(n,)
+ ~ 2_,
uT(.)
v,#,.
U;j)(~, +
,7)U,~il~(n +/.7)
(1)
where Ut(,°)(.) = U.(.). We c o n s m m t Q . ( n B ) on a blocked lattice fl'om U (~1 as
O,,(,,,,I = u.(~)(.)c22)(- + :)
(2)
The transforlnation kernel from original U's to blocked link variables UB is given by
:<"","> = c'exp(~n~ T,'U.,,(.~)Q~(,,.)) (3) Th(, transformation has an advant.age that a blocked link is lnade tYom more original link variabh,s than in other schemes and it is easy to be
806
QCD-TARO Collaboration~Nuclear Pt~vsics B (Proc. Suppl.) 42 (1995) 805 807
fitting kappa 1,6
- -
,
,
~
r
i
1
C 2 , ... 1.4
~
1.2
T
w3 w4 w5
-.-.-% 0.8
/
,
,-
2(=
~ . ,..&,
c'0) 0.6
C1
0.4
0.2
I
8
Figure 1. R e n o r m a l i z a t i o n G r o u p Transfornlation and R e n o r m a l i z e d T r a j e c t o r y in m a n y cou1)ling space
implemented in the 1)rogram. Moreover a "'(:lassical perfect action" (:orresl)olldillg to the transform a t i o n with , , = 0 . 0 7 7 and ~:=10.5 was foun(l[4].
3. O p t i m i z a t i o n tion
of Blocking
Transforma-
We u p d a t e a 324 lattice at fl=6.65 with 36690 sweeps after 44020 thermalization sweeps by the pseudo h e a t b a t h algorithm[6][7]. T h e blocking transformat.ion is p e r f o r m e d at every 10 sweeps. One m a y choose any blocking t r a n s f o r m a t i o n as fro" as it gives a coarse grained procedure, However it. depends on the scheme how fast we reach to a renormalized trajectory. In Fig.2 we show A/3 o b t a i n e d by nlatching a 324 lattice at 3 = 6 . 6 5 and 164 at /3 - A~i3 as a function of t," in Eq.(3) at the level 2 (i.e., on blocked 84 lattice). /,From this d a t a a" is set to be 18.8.
4. E v a l u a t i o n o f E f f e c t i v e C o u p l i n g s We f u r t h e r u p d a t e 10000 sweeps and perform the blocking tral~sformation with K= 18.8 at every 10 sweeps to 1)ro(ha:e 1000 blocked configurations of the size 4 :~. We present a preliminary results on effective COul)lings determinc(t by the delnon
I 10
12
I
J 14
J 26
Figure 2. A,,'3 o b t a i n e d on a 324 lattice at ,3=6.65 as a function of ~'
m e d m d on the first 100 configurations of these. An extensive analysis will be published elsewhere. T h e d e m o n m e t h o d was originally invented by Creutz[9] and recently a new idea to reduce syst:ematic errors revives it[10]. In the m e t h o d a dem o n visits links on a lattice and exchanges the energy. After m a n y nficrocanonical sweeps, the joint system of the denlon and the lattice is therrealized. \Ve m a y then know the action of the lattice by investigate the energy distribution of the demon. We let 100 demons visit 100 blocked configurations mtd carry out the microcanonical u p d a t e on each system. A preliminary result of the effective couplings are shown in Table.I. A l t h o u g h A/3 obtained at the level 2 show a reasonable matching, it is not clear t h a t we are near to the renormalized trajectory. We plan to continue our M C R G st, udy on lattices with the effective couplings obtained here. Table I llq 1 1.468(12) Sofa O.O57(4)
Effective Coupling Constants 1I'~2 1V22 0.(188(7)0.033(9) Twist O.232(8)
4D-Twist 0.140(5)
Chair
0.266(3)
QCD-TARO Collaboration~Nuclear Physics B (Proc. SuppL) 42 (1995) 805~07
ACKNOWLEDGEMENTS
We would like to express our thanks to P.Hasenfratz for usefltl and stinmlating discussions. Most of the calculations reported here were done on AP1000 at at the Fujitsu parallel computing research facilities, Kawasaki. We are indebted to M.Ikesaka, K.Inoue, M.Ishii, T.Saito, T.Shimizu and H.Shiraishi for their valuable con> ments on parallel computing. This work was sup1)orted in part by the Deutsch Fors(:hungsgenleinschaft, and by the Grand-in-Aid fi)r Scientific I/esearch of Minister of gducation.S('ien('e and Cult, u r e of Japan (C-06640371). REFERENCES
1. Y.Iwasaki, Report No.UTHEP-118, (Dec. 1983) (unpublished); Nucl.Phys. B25& 141 (1985); S.Sakai and H.Iso. Phys.Rev. D36 (1987), 2545-2552: Y.Iwasaki in these l)roceedings. 2. K.Symanzik, in Mathenmti(:al Prolflen~s in Theoretical Physics. edited by R.Schladcr et. al. (Lecture Note in Physics, \:o1.153), (Springer, Berlin, 1982), Nucl.Phys. B226 (1983) 2O5. 3. P.Hasenfl'atz and F.Niedermayer, Nucl.Phys. B414 (1993), 785. 4. A.H~senfi'atz,in these proceedings. 5. QCDTARO, Phys. Rev. Letters 71 (1993) 3068 3O66. 6. N.Cabibbo and E.Mal"inari, Phys.Letters 119B (1982) 38r. 7. A.D.Kennedy and B.J.Pendleton. Phys.Letters, 156B (1985) 393. 8. T.Takaishi, in ~hese proceedings and Heidelberg preprint, "Deterlnination of effective actions for SU(3) gauge theory submitted to Mod.Phys.Letters. 9. M.Crutz, Phys.Rev.Le~ters, 50 (1983) 1411. 10. M.Hasenbush, ICPinn and C.Wieezerkowski, Phys.Lett. B338 (1994) 308.
807