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Anisotropic improved gauge actions: — perturbative and numerical studies —
ILIIWll WJ BB"d-"h'dHIh~!
PROCEEDINGS SUPPLEMENTS Nuclear Physics B (Proe. Suppl.) 73 (1999) 417-419
ELSEVIER
Anisotropic Improved Gauge Actions: - Perturbative and Numerical Studies S. Sakai, a, A. Nakamura, b, and T.Saito, c aFaculty of Education, Y a m a g a t a University bResearch Center for Information Science and Education, ttiroshima University CDepartment of Physics, Hiroshima University The A parameter on the anisotropic lattice, the spatial and temperature coupling constant g., gr and their derivative with respaect to the the anisotropy parameter ~ axe studied perturbatively for the class of improved actions, which cover tree level Symanzik's, Iwasaki's and QCDTARO's improved actions. The *7(= g.~/g.) becomes less than 1 for Iwasaki's and QCDTARO's action, which is confirmed nonperturbatively by numerical simulations. Derivatives of the coupling constants with respect to the anisotropy parameter , Og~./(9~ and [.gg,r/O~,change sign for those improved actions.
1. I N T R O D U C T I O N The basic parameters of anisotropic lattices with improved action are important information for numerical simulations. In this work, we mainly focus on perturbative calculations of these parameters, but some of t h e m are confirmed by numerical simulations. The improved actions we study in this work consist of terms S~,~ = a0 • L(1 x 1)~v + c~1 • L(1 x 2)~,~ where L(1 x 1) and L(1 x 2) represent plaquette and 6-1ink rectangular loops respectively, and c~0 and a l satisfies a0 + 8. a l = 1. The improved actions cover i) tree level Symanzik's action(al = -1/12)[1], ii) Iwasaki's action(al = -0.331)[2], and iii) Q C D T A R O ' s action(al = -1.409)[3] etc. For these classes of improved actions the anisotropic lattice is formulated in the same way as in the case of standard plaquette action[4]. We introduce the coupling constant and lattice spacing in space direction g,, aa and those in temperature direction g~, aT. With these parameters the action on the anisotropic lattice is written as, S--
~o" " S i j "~- flr " S4i
where/3~ = g~-2~ 1, fir = g~-2~R and ~R = ao/a~.. In this study, A parameter and the ratio 77 =
g¢/go are calculated and the derivatives of the coupling constants with respect to the anisotropy parameter, Og~/O~R, Og72/O~R will be discussed. 2. P E R T U R B A T I V E
CALCULATIONS
In weak coupling expansions, the relation between the couping constants on the anisotropic lattice and those on the isotropic lattice are obtained as follows[4]. We calculate the coefficients of F~ Fi~ and F~F~ in the action in one loop order.
s jI( R) 1
2
=
-~(g'~ - Ca(~R)). F~'~F~'~.aaaar
+
1 2 - Cr(~R))" ~(g~-
n n a3aar F~iF~i"
The same calculation is done on the isotropic lattice. We require that the effective action is independent of the method of regularization. Then the relations are derived. g;2(
n) = 9 ( 1 ) - 2 +
-
+
o ( g ( 0 2)
g~'2(~R) = g(1) -2 + (C~(~n) - Cr (1)) + O(g(1) 2) On an isotropic lattice(~R = 1), gauge invariance means Co(l) = C~(1).
0920-5632/99/$ - see front matter © 1999 ElsevierScience B.V. All rights reserved. PII S0920-5632(98)00616-1
S. Sakai et aL /Nuclear Physics B (Proc. Suppl.) 73 (1999) 417-419
418
For the calculation of C¢ and C~, we employ the background field method[5] used by Iwasaki and Sakal[6] in the calculation of the A parameters of improved actions on the isotropic lattice. The details of the method will be reported in the forthcoming publications. The C~(hR) and C~(hR) have infrared divergences, but in the difference they are cancelled. The cancellation of the divergence is a delicate problem. The following method is used.
A~j)/A(1)
1.2-
(c#'°"'(,,~)
+
(c#""'(1)
8
0.8,
1:01:2
1:4 l:s~l:s
2:0 2:2
_ cs,o,,ffl)) Figure 1. The ratio A(~R)/A(1) for improved and standard actions
- c~'.(1))
In the first and third terms, the infrared divergence is canceled exactly for improved and standard actions. The second term is already calculated by Karsch[4] where the regularization of the infrared divergence could be done by an analytic integration of the fourth component of the loop momentum. With these C~,(~R),Cr(~R) functions, A and q are expressed as follows. A(~R)/A(1) = exp(-ACa(~R) + A C t ( r ) ) 4b0 9r ------
~7
W
,',c~(6~) = c~"~(,,R) - c~"-~(1) = (C~m~(~R) - c s ' o " ~ ( ~ R ) )
+
I Standard Symanzlk1 [] Iwa~ld W QCDTARO
1+-lV'(Ca-Cr)/Z+O(g
4)
g¢ The ratios A(~R)/A are shown in Fig.1. Except for the QCDTARO's action, they have similar ~R dependences. The more interesting differences are seen in the ratio ~1. In Fig.2, we show Ca(~R)- C~(~R), which determines 7/. It is found that as cq decreases Ca(~R) - C ~ ( ~ R ) changes sign. Namely for Iwasaki's and QCDTARO's improved actions, 7/ < 1, and at cq ~ -0.160, Ca((R) - Cr(~R) ~ 0 for all ~R. This result is very interesting and is not what we have expected. Then in the next section, we calculate the parameter r/ nonperturbatively and confirm the perturbative results. 3. N O N P E R T U R B A T I V E
S T U D Y O F 7/
Nonperturbatively, the parameter 7/ is calculated by the relation ~ = ~R/~,[7][8][9] , where
0.4'
Standard Symanzi~
cj~.)-cj~,,): P,rt
0.2. 0.0.
a,:-0.160 \[~ QCDTARO
-.0.2.
-0.4.
o
Ii
-0.6. -0.8 -1.5
t~
-1'.0
a,
~.s
o:0
Figure 2. C~(~R) - C,(~R) as a function of ~1
~B is a bare the anisotropy parameter which appears in the action, 2N 1 S = --~-(-z-S,j + ~ S , , ) g"
qB
while the renormalized the anisotropy parameter is defined by ~R = he~at. For the probe of the scale in the space and temperature direction, we use the lattice potential in these directions which is defined by,
v.(~.,t,t) = tn( w.(l w"(l't)t + ) 1 ,i
S. Sakai et at~Nuclear Physics B (Prec. Suppl.) 73 (1999) 417--419 similarly for the potential in the space direction. Here we fix ~n = 2, and calculate the rate for a few ,~B
R(~B, t, '9 =
~=10.0:Improved
1.00]
[] ~ " I
a,=-0.16- =
0.95] 0.901
Iwasaki + QCDTARO
-i.o
-i.5
£I1 Z.5
oo] o.2/o,
VIP."' Vls'r"(~a=2) Standard 1.05J ~=12.0:Standard Symanzik~ I
dCJd% •
V.(~B, l, '9
Then we search for the point R = i by interpolating with respect to ~B[7][8]. The simulations are done on the 123 x 24 lattice, in the large/? region. The results are shown in Fig.3, together with those of perturbative calculations. It is found that the agreement of both results
'
0.,]
0:0
Figure 3. Perturbative and Nonperturbative results for r/(,~R = 2, fl)
419
,cj,l Syrr~nzik
0
dC4d~l Sum I
-1.6
\o%
O
QCDTARO
-0.2
\Standard
0 /
•
~ki
-i.2
-6.8
0.I
-d4
o'.o
Figure 4. Og'~2/O(R, and Og72/O(R from numerical estimation
The saturation of the sum rule is good. Notice that the derivatives change sign as al decreases. These coefficients play important role in the calculation of internal energy and pressure. As the sign of these coefficients changes for the improved actions, it may be interesting to calculate thermodynamic quantities with these improved actions. In conclusion, the study of anisotropic lattice with improved action shows very interesting properties as shown in Figs.l,2,3,4. The study of finite temperature physics with these improved action seems to be very exciting. REFERENCES
is good. So the cq dependences of the r/ratio is also confirmed by the numerical simulation, a very exciting result for improved actions. 4. E S T I M A T I O N OF ~O+R A N D ag7= O~. The exact calculation of these derivatives are not carried out yet. However we could estimate them from the slope of the Ca(~R) and Cr (~R) at ~n = 1 and ~R = 1.01. The results are shown in Fig.4. They should satisfy following sum rule[4],
OCa
OCt
11 x N
O~' + 0--~RI+"=1 = 48~'
1. K. Symanzik, Nucl. Phys. B226(1983), 187 2. Y. Iwasaki, Nucl. Phys. B258(1985), 141; Univ. ofTsukuba preprint UTHEP-118(1983) 3. QCD-TARO Collaboration, Ph. de Forcrand et el. hep-lat/9806008 4. F.Karsch, Nucl. Phys.B205[FS5](1982), 285 5. K.Dashen and D.Gross, Phys. l%ev. D23(1981),2340 6. Y. lwasaki and S. Sakai, Nucl. Phys. B248(1984),441 7. T.R.Klassen hep-lat/9803010 8. G.Burgers et el., Nucl. Phys. B204, 587, 1988 9. J. Engels, F. Karsch, T. Scheideler, Nucl.Phys.B(Proc SuppI)63A-C(1998),48
PROCEEDINGS SUPPLEMENTS Nuclear Physics B (Proe. Suppl.) 73 (1999) 417-419
ELSEVIER
Anisotropic Improved Gauge Actions: - Perturbative and Numerical Studies S. Sakai, a, A. Nakamura, b, and T.Saito, c aFaculty of Education, Y a m a g a t a University bResearch Center for Information Science and Education, ttiroshima University CDepartment of Physics, Hiroshima University The A parameter on the anisotropic lattice, the spatial and temperature coupling constant g., gr and their derivative with respaect to the the anisotropy parameter ~ axe studied perturbatively for the class of improved actions, which cover tree level Symanzik's, Iwasaki's and QCDTARO's improved actions. The *7(= g.~/g.) becomes less than 1 for Iwasaki's and QCDTARO's action, which is confirmed nonperturbatively by numerical simulations. Derivatives of the coupling constants with respect to the anisotropy parameter , Og~./(9~ and [.gg,r/O~,change sign for those improved actions.
1. I N T R O D U C T I O N The basic parameters of anisotropic lattices with improved action are important information for numerical simulations. In this work, we mainly focus on perturbative calculations of these parameters, but some of t h e m are confirmed by numerical simulations. The improved actions we study in this work consist of terms S~,~ = a0 • L(1 x 1)~v + c~1 • L(1 x 2)~,~ where L(1 x 1) and L(1 x 2) represent plaquette and 6-1ink rectangular loops respectively, and c~0 and a l satisfies a0 + 8. a l = 1. The improved actions cover i) tree level Symanzik's action(al = -1/12)[1], ii) Iwasaki's action(al = -0.331)[2], and iii) Q C D T A R O ' s action(al = -1.409)[3] etc. For these classes of improved actions the anisotropic lattice is formulated in the same way as in the case of standard plaquette action[4]. We introduce the coupling constant and lattice spacing in space direction g,, aa and those in temperature direction g~, aT. With these parameters the action on the anisotropic lattice is written as, S--
~o" " S i j "~- flr " S4i
where/3~ = g~-2~ 1, fir = g~-2~R and ~R = ao/a~.. In this study, A parameter and the ratio 77 =
g¢/go are calculated and the derivatives of the coupling constants with respect to the anisotropy parameter, Og~/O~R, Og72/O~R will be discussed. 2. P E R T U R B A T I V E
CALCULATIONS
In weak coupling expansions, the relation between the couping constants on the anisotropic lattice and those on the isotropic lattice are obtained as follows[4]. We calculate the coefficients of F~ Fi~ and F~F~ in the action in one loop order.
s jI( R) 1
2
=
-~(g'~ - Ca(~R)). F~'~F~'~.aaaar
+
1 2 - Cr(~R))" ~(g~-
n n a3aar F~iF~i"
The same calculation is done on the isotropic lattice. We require that the effective action is independent of the method of regularization. Then the relations are derived. g;2(
n) = 9 ( 1 ) - 2 +
-
+
o ( g ( 0 2)
g~'2(~R) = g(1) -2 + (C~(~n) - Cr (1)) + O(g(1) 2) On an isotropic lattice(~R = 1), gauge invariance means Co(l) = C~(1).
0920-5632/99/$ - see front matter © 1999 ElsevierScience B.V. All rights reserved. PII S0920-5632(98)00616-1
S. Sakai et aL /Nuclear Physics B (Proc. Suppl.) 73 (1999) 417-419
418
For the calculation of C¢ and C~, we employ the background field method[5] used by Iwasaki and Sakal[6] in the calculation of the A parameters of improved actions on the isotropic lattice. The details of the method will be reported in the forthcoming publications. The C~(hR) and C~(hR) have infrared divergences, but in the difference they are cancelled. The cancellation of the divergence is a delicate problem. The following method is used.
A~j)/A(1)
1.2-
(c#'°"'(,,~)
+
(c#""'(1)
8
0.8,
1:01:2
1:4 l:s~l:s
2:0 2:2
_ cs,o,,ffl)) Figure 1. The ratio A(~R)/A(1) for improved and standard actions
- c~'.(1))
In the first and third terms, the infrared divergence is canceled exactly for improved and standard actions. The second term is already calculated by Karsch[4] where the regularization of the infrared divergence could be done by an analytic integration of the fourth component of the loop momentum. With these C~,(~R),Cr(~R) functions, A and q are expressed as follows. A(~R)/A(1) = exp(-ACa(~R) + A C t ( r ) ) 4b0 9r ------
~7
W
,',c~(6~) = c~"~(,,R) - c~"-~(1) = (C~m~(~R) - c s ' o " ~ ( ~ R ) )
+
I Standard Symanzlk1 [] Iwa~ld W QCDTARO
1+-lV'(Ca-Cr)/Z+O(g
4)
g¢ The ratios A(~R)/A are shown in Fig.1. Except for the QCDTARO's action, they have similar ~R dependences. The more interesting differences are seen in the ratio ~1. In Fig.2, we show Ca(~R)- C~(~R), which determines 7/. It is found that as cq decreases Ca(~R) - C ~ ( ~ R ) changes sign. Namely for Iwasaki's and QCDTARO's improved actions, 7/ < 1, and at cq ~ -0.160, Ca((R) - Cr(~R) ~ 0 for all ~R. This result is very interesting and is not what we have expected. Then in the next section, we calculate the parameter r/ nonperturbatively and confirm the perturbative results. 3. N O N P E R T U R B A T I V E
S T U D Y O F 7/
Nonperturbatively, the parameter 7/ is calculated by the relation ~ = ~R/~,[7][8][9] , where
0.4'
Standard Symanzi~
cj~.)-cj~,,): P,rt
0.2. 0.0.
a,:-0.160 \[~ QCDTARO
-.0.2.
-0.4.
o
Ii
-0.6. -0.8 -1.5
t~
-1'.0
a,
~.s
o:0
Figure 2. C~(~R) - C,(~R) as a function of ~1
~B is a bare the anisotropy parameter which appears in the action, 2N 1 S = --~-(-z-S,j + ~ S , , ) g"
qB
while the renormalized the anisotropy parameter is defined by ~R = he~at. For the probe of the scale in the space and temperature direction, we use the lattice potential in these directions which is defined by,
v.(~.,t,t) = tn( w.(l w"(l't)t + ) 1 ,i
S. Sakai et at~Nuclear Physics B (Prec. Suppl.) 73 (1999) 417--419 similarly for the potential in the space direction. Here we fix ~n = 2, and calculate the rate for a few ,~B
R(~B, t, '9 =
~=10.0:Improved
1.00]
[] ~ " I
a,=-0.16- =
0.95] 0.901
Iwasaki + QCDTARO
-i.o
-i.5
£I1 Z.5
oo] o.2/o,
VIP."' Vls'r"(~a=2) Standard 1.05J ~=12.0:Standard Symanzik~ I
dCJd% •
V.(~B, l, '9
Then we search for the point R = i by interpolating with respect to ~B[7][8]. The simulations are done on the 123 x 24 lattice, in the large/? region. The results are shown in Fig.3, together with those of perturbative calculations. It is found that the agreement of both results
'
0.,]
0:0
Figure 3. Perturbative and Nonperturbative results for r/(,~R = 2, fl)
419
,cj,l Syrr~nzik
0
dC4d~l Sum I
-1.6
\o%
O
QCDTARO
-0.2
\Standard
0 /
•
~ki
-i.2
-6.8
0.I
-d4
o'.o
Figure 4. Og'~2/O(R, and Og72/O(R from numerical estimation
The saturation of the sum rule is good. Notice that the derivatives change sign as al decreases. These coefficients play important role in the calculation of internal energy and pressure. As the sign of these coefficients changes for the improved actions, it may be interesting to calculate thermodynamic quantities with these improved actions. In conclusion, the study of anisotropic lattice with improved action shows very interesting properties as shown in Figs.l,2,3,4. The study of finite temperature physics with these improved action seems to be very exciting. REFERENCES
is good. So the cq dependences of the r/ratio is also confirmed by the numerical simulation, a very exciting result for improved actions. 4. E S T I M A T I O N OF ~O+R A N D ag7= O~. The exact calculation of these derivatives are not carried out yet. However we could estimate them from the slope of the Ca(~R) and Cr (~R) at ~n = 1 and ~R = 1.01. The results are shown in Fig.4. They should satisfy following sum rule[4],
OCa
OCt
11 x N
O~' + 0--~RI+"=1 = 48~'
1. K. Symanzik, Nucl. Phys. B226(1983), 187 2. Y. Iwasaki, Nucl. Phys. B258(1985), 141; Univ. ofTsukuba preprint UTHEP-118(1983) 3. QCD-TARO Collaboration, Ph. de Forcrand et el. hep-lat/9806008 4. F.Karsch, Nucl. Phys.B205[FS5](1982), 285 5. K.Dashen and D.Gross, Phys. l%ev. D23(1981),2340 6. Y. lwasaki and S. Sakai, Nucl. Phys. B248(1984),441 7. T.R.Klassen hep-lat/9803010 8. G.Burgers et el., Nucl. Phys. B204, 587, 1988 9. J. Engels, F. Karsch, T. Scheideler, Nucl.Phys.B(Proc SuppI)63A-C(1998),48