- Email: [email protected]
Further studies on the smoothed SU(2) gauge configurations
3 August 2000
Physics Letters B 486 Ž2000. 448–453 www.elsevier.nlrlocaternpe
Further studies on the smoothed SU ž2/ gauge configurations Ying Chen b, Bing He b, He Lin b, Ji-min Wu a,b a
b
CCAST (World Laboratory), P.O.Box 8730, Beijing 100080, People’s Republic of China Institute of High Energy Physics, Academia Sinica, Beijing 100039, People’s Republic of China Received 24 April 2000; received in revised form 14 June 2000; accepted 20 June 2000 Editor: M. Cveticˇ
Abstract The field variables ŽFV., the strength tensor ŽST. and the topological charge density ŽTCD. of lattice SUŽ2. gauge field are studied in the momentum space. The evolution behaviors of their Fourier components are monitored during the smoothing procedure. The high frequency components of the FV can not be eliminated by smoothing, but all the Fourier amplitudes of the ST are uniformly suppressed. The distribution of the TCD in Euclidean spacetime is highly smoothed. Based on these facts, we give a new interpretation of the structure of the smoothed gauge configurations and argue that the ultraviolet and infrared cutoff introduced by lattice discretization account for the suppression of the instanton size distribution at both the large size end and the small size end. q 2000 Elsevier Science B.V. All rights reserved. PACS: 11.15.Ha; 11.38.Aw; 12.38.Gc Keywords: Smoothing; Instanton; Momentum space
1. Introduction The structures and the topological properties of the classical gauge field have been extensively explored on the lattice in the past several years. Based on the background field assumption, several smoothing schemes such as cooling w1x and APE smearing w2x are suggested to extract the classical components from the thermalized lattice gauge configurations, since it is argued that the quantum fluctuations can be eliminated through smoothing. By a proper defini-
E-mail address: [email protected] ŽY. Chen..
tion of the topological charge density on the lattice, the integer topological charges can be obtained up to a few percent, and the topological susceptibility shows good scaling properties and agrees well with the Witten–Veneziano relation w3x. As far as the local structure of the smoothed configuration is concerned, the density and the size distribution of instantons were investigated on the lattice. Instantons are self-dual solutions of the equation of motion in Euclidean spacetime. To extract the instanton information from the configurations, the action and the topological charge density are often approximated by the superposition of self-dual, non-interacting instantons and antiinstantons. The usual recognition scheme of instantons is carried out by looking for and filtering the peaks of the action density or the topological
0370-2693r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 0 0 . 0 0 7 6 5 - 6
Y. Chen et al.r Physics Letters B 486 (2000) 448–453
charge density. The instanton sizes are determined by the height of the peaks through the BPST parameterization w4x. However the results obtained by various authors are never in complete agreement w5–7x, e.g. the diluteness of the instanton ensemble are not unambiguously determined yet through the numerical simulations. Our work is to investigate how the smoothing procedure affects the lattice gauge configurations, especially how the different momentum components vary under the smoothing procedure. This is done directly in the momentum space. Since the finite lattice size and finite lattice spacing introduce a ultraviolet momentum cutoff and a minimal nonzero momentum, we conjecture that this has direct effects on the extraction of the structure of the smoothed configuration. This work is organized as follows. We give the relevant formulation in Section 2, where the expressions of smoothing transformation are given both in coordinate space and in momentum space. In Section 3 we study the evolutions of different Fourier components of the field variables ŽFV., the field strength tensor ŽST. and the topological charge density ŽTCD.. Section 4 is devoted to a new interpretation of the structure of the smoothed configurations. The final section gives the discussion and conclusion.
449
Dn FnmŽ x . s 0 is one of the fixed points of the cooling transformation Ž1.. To investigate the evolution of all the Fourier components during cooling, we modify the cooling iteration from the sequential upgrading of links to a parallel one which transforms all the links simultaneously from the uncooled configuration. Thus the transformation of Eq. Ž1. is performed equally to every link on the lattice. Unfortunately this modified transformation does not converge. Eq. Ž2. takes the following form in the momentum space w8x: AmŽ1. Ž q . s Ý hmn Ž q . AnŽ0. Ž q . q 16 a 2 Hm Ž q . , where
ž
hmn s Ž 1 y 16 a 2 qˆ 2 . dmn y
qˆm qˆn qˆ
2
/
Um
™
UmX s NSmq,
Am Ž x .
™A Ž x. sA Ž x. q a D Ž x. F 1 6
X
m
UmŽ1. Ž n . s N Ž 1 y c . U mŽ 0 . Ž n . q
ž
m
4
qOŽ a . .
2
n
nm
Ž 2.
If the O Ž a 4 . artifacts can be ignored, it is obvious that the classical solution of the equation of motion
qˆm qˆn qˆ 2
,
Ž 4.
c 6
SmŽ 0 . q Ž x . ,
/
Ž 5. and
Ž 1.
where N is the normalization factor which enables UmX to be an element of the SUŽ2. group and Sm is the sum of staples and depends on the form of the local action at each link. Eq. Ž1. can be expressed in terms of field variables Am ,
q
with qˆm s 2a sinŽ aqm ., HmŽ q . is the term including the 2 commutators. If HmŽ q . were taken small, the transverse component of AmŽ q . would be given a factor Ž1 y 16 a 2 qˆ 2 . n after n smoothing steps and would diverge for large momentum Žsuch as qm s Ž"pra, "pra," pra," pra.. when n becomes large. With this fact in consideration, the local transformation is further modified to a smearing version,
2. Formulations The normal cooling is a diffusion procedure and is realized by iteratively minimizing the local action of the lattice gauge field link by link. The link-upgrading of cooling follows the formula
Ž 3.
n
ž
hmn s 1 y
c 6
a 2 qˆ 2
/ žd
mn y
qˆm qˆn qˆ
2
/
q
qˆm qˆn qˆ 2
.
Ž 6.
The convergence of the modified cooling requires 0 F c F 0.75. It is remarkable that the term Hm that includes the commutators can not be ignored because of the non-Abelian property of Am . The transverse component of the large momentum Fourier coefficients AmŽ q . is not being suppressed as Ž1 y 6c a2 qˆ 2 . n after n smoothing steps, as is seen in our calculation.
Y. Chen et al.r Physics Letters B 486 (2000) 448–453
450
The lattice version of the topological charge density Q LŽ x . we used here is the naively discretized theoretical definition improved to O Ž a4 ., Q LŽ x . s y
1 512p 2
Ž 53 Q P Ž x . y 16 Q R Ž x . .
s a 4 Ž Q cont . Ž x . q O Ž a 4 . . ,
Ž 7.
where Q P Ž x . and Q R are constructed by plaquettes Pmn and rectangular operators Rmn , respectively. The lattice field strength tensor is defined from the clover average of Pmn and Rmn w12x, Fˆmn Ž x . s Im Ž 5² Pmn :clover y 3² Rmn :clover . .
Ž 8.
where q is the momentum qm s
2p Lm am
nm ,
nm s 0,1, . . . , Lm y 1,
Ž 12 .
It is remarkable that the momentum space vector potential AmŽ q . is not a gauge invariant and the longitudinal terms are pure gauge transformation and almost left unchanged during smoothing Žsee Eqs. Ž3. and Ž4.., so we will only check the transverse components. The amplitudes of the transverse components of each momentum mode are measured by < A Ž q . < 2 ' Ý Tr Ž ATm Ž q . ATm † Ž q . . m
3. Evolution of Am , Fm n and Q during smoothing In this work the lattice SUŽ2. theory is investigated by the smoothing scheme based on the tadpole-improved action S1 on an 8 3 = 24 anisotropic lattice with the anisotropy ratio j s a sra t s 3 w9x Ž a s and a t are spatial and temporal lattice spacings, respectively., S1 s b
Ý x , s)s
5 Ps sX 3 j u 4s
X
y
4 Pst j
qb Ý
3
x,s
u 2s u 2t
1 R s sX 12 j u 6s
y
y
1 R st j 12 u 4s u 2t
1 R sX s 12 j u 6s ,
Ž 9.
where b s 4rg 2 , g is the coupling, u s and u t are the tadpole parameters w10x. On the lattice the vector potential AmŽ x . of the SUŽ2. gauge field is defined as Am Ž x q am m ˆ r2 . s
1 2 igam
† m
žU Ž x . y U m
Ž x.
y 12 Tr Ž Um Ž x . y Um† Ž x . . .
/
Ž 10 . where am is the lattice spacing in the m direction. The Fourier transformation of AmŽ x . is defined as Am Ž q . s a4 Ý eyi qPŽ xqam mˆ r2.Am Ž x q am m ˆ r2 . , x
Ž 11 .
s
Ý Tr m ,n
ž
ž
Am Ž q . A†n Ž q . dmn y
qˆm qˆn qˆ 2
//
.
Ž 13 . Our goal of this paper is to show that any initial configuration will approach one of the fixed points of the smoothing transformation. According to our calculation, the evolution behaviors of AmŽ q . and Fmn Ž q . during smoothing are typical for any gauge configuration and insensitive to the lattice spacing, so the following discussions are all based on a randomly thermalized configuration at b s 1.1 which corresponds to the spatial lattice spacing a s s 0.233 fm w11x. The evolution of < AŽ q .< 2 of several momenta under the smoothing procedure is plotted in Fig. 1. After a period of oscillation, < AŽ q .< 2 tends to be stable, indicating that the configuration tends to the fixed point of the transformation. Because the contributions of all modes are significant, it is shown that the conjectured suppression of high momentum components is not the case. A similar analysis is performed for the field strength tensor and the charge density. Fig. 2 shows the evolution of < Fmn Ž q .< 2 ' † Ž . TrFmn Ž q . Fmn q along with the smoothing procedure. The damping of < Fmn Ž q .< 2 is very fast, and there is no sign that only the high momentum components are suppressed. Fig. 3 shows several Fourier components Q LŽ q . measured during the smoothing procedure. In contrast to what we see from Fig. 1 and Fig. 2, where most of the modes damp out to almost zero or tend to be stable, one can find that only the zero momentum component and some other
Y. Chen et al.r Physics Letters B 486 (2000) 448–453
451
Fig. 1. The evolution behavior of < AmŽ q .< 2 during the smoothing procedure, where qm s L2mpam nm . Six momentum modes, which are denoted by Ž n1 ,n 2 ,n 3 ,n 4 ., are plotted in the figure. Even after 300 smoothing steps, all curves tend to be stable. This indicates that the large momentum components of Am were not being smoothed out.
Fig. 3. The evolution behavior of < QŽ q .< 2 is plotted. The zero momentum component has several plateaus during the smoothing and is kept stable after 120 smearing steps. Most of the momentum components Žexcept for the zero momentum one which is kept stable after 120 smearing steps. are smoothed out even after 50 smoothing steps. The residual parts account for the structure of the smoothed configuration.
modes have some metastable plateaus. Thus we can conclude from the above facts that if a stable configuration, which corresponds to the fixed point of the smoothing transformation, is approached, it is some
of the Fourier components of the charge density surviving the smoothing procedure that give rise to the local structure of the smoothed configuration, because all the momentum modes of the gauge field still exist after so many steps of cooling.
4. The structure of the smoothed configuration The local structure of the smoothed configuration is explored by studying the distribution of the charge density in the whole spacetime, i.e, the whole lattice volume. With reference to the scheme introduced by Teper and Smith w6x, our first step is to recognize the instantons and antiinstantons. In doing so, it is assumed that the smoothed configuration is the superposition of instantons and antiinstantons and we employ the BPST parameterization of the single instanton. The peaks of the charge density can be taken as the candidates of the instantons. For a peak at x i , the raw estimation of the size of this instanton candidate Ždenoted by r i . is given by Fig. 2. The evolution behavior of < Fmn Ž q .< 2 during the smoothing procedure. The labels of the curves are the same as in Fig. 1. All the Fourier components of Fmn damp out rapidly.
Q LŽ x i . s
6
p 2r i4
.
Ž 14 .
452
Y. Chen et al.r Physics Letters B 486 (2000) 448–453
An iterative procedure is performed to correct the sizes, and several complicated schemes were developed to filter out the unwanted peaks, which are assumed not to be instantons and antiinstantons. For details see Ref. w6x. After correcting of the instanton radii and filtering out the unwanted peaks, one can get a set of candidates for the instanton and antiinstanton Ž x i , r i . < i s 1,2,3, . . . 4 . By the use of this formalism to each configuration during the smoothing procedure, we find the following facts: Ži. The number of peaks decreases rapidly as the smoothing proceeds. Žii. The position of the peaks are very stable and insensitive to the smoothing. Žiii. Some peaks undergo disappear–appear–disappear procedures. Živ. The radii of some peaks increase with the smoothing proceeding until they disappear, and some peaks are kept fixed. Žv. If n I denotes the number of positive peaks and n A negative peaks, n I y n A is not equal to Q L . However, the number of relatively stable peaks with radii 1.5a - r - 3.5a is consistent with Q L when the metastable configurations are reached. The wider peaks have no contribution to the measured charge. We interpret these facts as follows: Due to the finite size of the lattice volume and the finite lattice spacing, the number of momentum modes contributing to all functions of the spacetime coordinates are finite. Here the structure of the smoothed configuration is explored through the spacetime distribution of the TCD, with the local peaks and valleys the results of the superposition of the momentum modes of the TCD. So, many peaks may be the presence of the fluctuations of each momentum mode rather than true instantons, especially in the early period of the smoothing procedure. As the smoothing proceeds, the amplitude of each momentum mode varies and, accordingly, the superficial radius obtained from Eq. Ž14. varies too. The variation of the amplitude of some momentum modes may cause some measured peaks to vanish and reappear during the smoothing procedure. However, the position of each peak or valley is fixed, as is observed in our calculation. There must be some ambiguities if the physical structure was extracted from the superficial peaks, for example, < n I y n A < is inconsistent with the measured Q L , since the topological charge is equivalent
to the amplitude of the zero momentum mode of TCD Žthe spacetime integral of the nonzero momentum mode is zero.. Next we try to explain why the smallest size of ‘‘instantons’’ obtained on the lattice is about 2 a and why the large ‘‘instantons’’’’ are suppressed. We start from the continuous BPST instanton. Fourier transforming the field strength tensor of the single instanton one gets Fmn Ž k . s
1
4
Ž 2p .
2
yi kŽ xyx 0 .
Hd ke
Fmn Ž x .
a s y8p 2hmn l ar 2 K 0 Ž r < k < . .
Ž 15 .
With the self-dual condition Fmn s F˜mn , the topological charge is Qs
1 16p 2
Hd
4
x TrFmn Ž x . F˜mn Ž x .
A d 4 kK 0 Ž r < k < . K 0 Ž r < k < . .
H
Ž 16 .
On a L4 lattice with lattice spacing a, there is the maximal momentum cutoff pra and the minimal nonzero momentum is approximately 2Lap . We take the assumption that the instantons distribute well separately, so the charge contribution QŽ r . of each single instanton with radius r s rˆ a on the lattice is qualitatively given as pr
H2pˆrˆ drr K
Q Ž rˆ , L . s 3
3
0
Ž r . K0Ž r . .
Ž 17 .
L
QŽ rˆ ,8.,QŽ rˆ ,24. and QŽ rˆ ,`.. are plotted versus rˆ in Fig. 4. When L s 8, the charge gets significant contributions from the instantons with radii r in a narrow range 0.5a - r - 3a. The smallest instanton we measured is with radius r ; 2 a because still smaller instantons with radii r ; a cannot survive the smoothing procedure and cannot be measured on the lattice because the lattice artifacts for thus small instanton would be too large w12x. If we take the lower peaks Žwith larger superficial radius. as larger instantons, they do not contribute to the charge Q L and are difficult to be disentangled from the fluctuations of the small momentum modes. It should be noted that Eq. Ž17. is derived from the continuum BPST parameterization and the result is thus qualitative but not precise. Practically, Eq. Ž17. implies that
Y. Chen et al.r Physics Letters B 486 (2000) 448–453
453
it is difficult to separate the instanton information from the unphysical fluctuations due to the lattice artifacts, there may be ambiguities in extracting the topological structure from the spacetime distribution of the charge density on the lattice. This may give an explanation of the suppression at the large size end of the instanton size distribution and the discrepancy of the measured charge with the number of peaks of the charge density, which are taken as instanton candidates, in the smoothed lattice configuration.
Acknowledgements
Fig. 4. The figure shows the lattice artifacts in QŽ r .. The rigid value of Q is 1 and independent of r in the continuum.
the largest instanton which can be checked on the lattice is governed by the largest lattice side size, which actually gives the smallest lattice momentum. This is in agreement with the observation of Ref. w13x where the authors suggested that the largest instanton would be larger if one takes the twisted boundary condition or asymmetric lattice with larger size in one direction than the others, for example 12 3 = 36 in their work.
5. Conclusion We have investigated the evolution behavior of the gauge field under smoothing in momentum space. It is found that the UV fluctuations of the field variables cannot be smoothed out as expected, while the field strength tensor decreases rapidly as the smoothing proceeds. In fact it is the topological density that is smoothed. The very small instantons cannot survive on the lattice with finite lattice spacing, while the very large instantons are deformed to a large amount by the finite lattice size. Consequently
This work is supported by the Natural Science Foundation of China under the Grant No. 19677205 and No. 19991487, and by the National Research Center for Intelligent Computing System under the contract No. 99128.
References w1x J. Hoek, Phys. Lett. 166 BŽ1986. 199; J. Hoek, M. Teper, J. Waterhouse, Nucl. Phys. B 288 Ž1987. 589. w2x M. Albanese et al., Phys. Lett. 192B Ž1987. 163. w3x E. Witten, Nucl. Phys. B 156 Ž1979. 269; G. Veneziano, Nucl. Phys. B 159 Ž1979. 213. w4x A.A. Belavin, A.M. Polyakov, A.A. Schwartz, Y.S. Tyupkin, Phys. Lett. 59B Ž1975. 85. w5x Ph. de Forcrand, M. Garcia Perez, J. Hetrick, I.-O. Stamatescu, Nucl. Phys. B ŽProc. Suppl.. 63 Ž1998. 406. w6x D. Smith, M. Teper, Phys. Rev. D 58 Ž1998. 014505. w7x A. Hasenfratz, C. Nieter, hep-latr9806062. w8x C. Bernard, T. DeGrand, hep-latr9909083. w9x C.J. Morningstar, M. Peardon, Nucl. Phys. B ŽProc. Suppl.. 47 Ž1996. 258. w10x G.P. Lepage, P.B. Mackenzie, Phys. Rev. D 48 Ž1993. 2250. w11x Jian Bo Zhang, Min Jin, Da Ren Ji, Chin. Phys. Lett. 15 Ž1998. 865; Y. Chen, B. He, H. Lin, J. Wu, Commun. Theor. Phys., in press. w12x M.G. Perez, ´ A. Gonzalez-Arroyo, J. Snippe, P. van Baal, Nucl. Phys. B 413 Ž1994. 535. w13x P. de Forcrand, M.G. Perez, I.O. Stamatescu, Nucl. Phys. B ´ 499 Ž1997. 409.
Physics Letters B 486 Ž2000. 448–453 www.elsevier.nlrlocaternpe
Further studies on the smoothed SU ž2/ gauge configurations Ying Chen b, Bing He b, He Lin b, Ji-min Wu a,b a
b
CCAST (World Laboratory), P.O.Box 8730, Beijing 100080, People’s Republic of China Institute of High Energy Physics, Academia Sinica, Beijing 100039, People’s Republic of China Received 24 April 2000; received in revised form 14 June 2000; accepted 20 June 2000 Editor: M. Cveticˇ
Abstract The field variables ŽFV., the strength tensor ŽST. and the topological charge density ŽTCD. of lattice SUŽ2. gauge field are studied in the momentum space. The evolution behaviors of their Fourier components are monitored during the smoothing procedure. The high frequency components of the FV can not be eliminated by smoothing, but all the Fourier amplitudes of the ST are uniformly suppressed. The distribution of the TCD in Euclidean spacetime is highly smoothed. Based on these facts, we give a new interpretation of the structure of the smoothed gauge configurations and argue that the ultraviolet and infrared cutoff introduced by lattice discretization account for the suppression of the instanton size distribution at both the large size end and the small size end. q 2000 Elsevier Science B.V. All rights reserved. PACS: 11.15.Ha; 11.38.Aw; 12.38.Gc Keywords: Smoothing; Instanton; Momentum space
1. Introduction The structures and the topological properties of the classical gauge field have been extensively explored on the lattice in the past several years. Based on the background field assumption, several smoothing schemes such as cooling w1x and APE smearing w2x are suggested to extract the classical components from the thermalized lattice gauge configurations, since it is argued that the quantum fluctuations can be eliminated through smoothing. By a proper defini-
E-mail address: [email protected] ŽY. Chen..
tion of the topological charge density on the lattice, the integer topological charges can be obtained up to a few percent, and the topological susceptibility shows good scaling properties and agrees well with the Witten–Veneziano relation w3x. As far as the local structure of the smoothed configuration is concerned, the density and the size distribution of instantons were investigated on the lattice. Instantons are self-dual solutions of the equation of motion in Euclidean spacetime. To extract the instanton information from the configurations, the action and the topological charge density are often approximated by the superposition of self-dual, non-interacting instantons and antiinstantons. The usual recognition scheme of instantons is carried out by looking for and filtering the peaks of the action density or the topological
0370-2693r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 0 0 . 0 0 7 6 5 - 6
Y. Chen et al.r Physics Letters B 486 (2000) 448–453
charge density. The instanton sizes are determined by the height of the peaks through the BPST parameterization w4x. However the results obtained by various authors are never in complete agreement w5–7x, e.g. the diluteness of the instanton ensemble are not unambiguously determined yet through the numerical simulations. Our work is to investigate how the smoothing procedure affects the lattice gauge configurations, especially how the different momentum components vary under the smoothing procedure. This is done directly in the momentum space. Since the finite lattice size and finite lattice spacing introduce a ultraviolet momentum cutoff and a minimal nonzero momentum, we conjecture that this has direct effects on the extraction of the structure of the smoothed configuration. This work is organized as follows. We give the relevant formulation in Section 2, where the expressions of smoothing transformation are given both in coordinate space and in momentum space. In Section 3 we study the evolutions of different Fourier components of the field variables ŽFV., the field strength tensor ŽST. and the topological charge density ŽTCD.. Section 4 is devoted to a new interpretation of the structure of the smoothed configurations. The final section gives the discussion and conclusion.
449
Dn FnmŽ x . s 0 is one of the fixed points of the cooling transformation Ž1.. To investigate the evolution of all the Fourier components during cooling, we modify the cooling iteration from the sequential upgrading of links to a parallel one which transforms all the links simultaneously from the uncooled configuration. Thus the transformation of Eq. Ž1. is performed equally to every link on the lattice. Unfortunately this modified transformation does not converge. Eq. Ž2. takes the following form in the momentum space w8x: AmŽ1. Ž q . s Ý hmn Ž q . AnŽ0. Ž q . q 16 a 2 Hm Ž q . , where
ž
hmn s Ž 1 y 16 a 2 qˆ 2 . dmn y
qˆm qˆn qˆ
2
/
Um
™
UmX s NSmq,
Am Ž x .
™A Ž x. sA Ž x. q a D Ž x. F 1 6
X
m
UmŽ1. Ž n . s N Ž 1 y c . U mŽ 0 . Ž n . q
ž
m
4
qOŽ a . .
2
n
nm
Ž 2.
If the O Ž a 4 . artifacts can be ignored, it is obvious that the classical solution of the equation of motion
qˆm qˆn qˆ 2
,
Ž 4.
c 6
SmŽ 0 . q Ž x . ,
/
Ž 5. and
Ž 1.
where N is the normalization factor which enables UmX to be an element of the SUŽ2. group and Sm is the sum of staples and depends on the form of the local action at each link. Eq. Ž1. can be expressed in terms of field variables Am ,
q
with qˆm s 2a sinŽ aqm ., HmŽ q . is the term including the 2 commutators. If HmŽ q . were taken small, the transverse component of AmŽ q . would be given a factor Ž1 y 16 a 2 qˆ 2 . n after n smoothing steps and would diverge for large momentum Žsuch as qm s Ž"pra, "pra," pra," pra.. when n becomes large. With this fact in consideration, the local transformation is further modified to a smearing version,
2. Formulations The normal cooling is a diffusion procedure and is realized by iteratively minimizing the local action of the lattice gauge field link by link. The link-upgrading of cooling follows the formula
Ž 3.
n
ž
hmn s 1 y
c 6
a 2 qˆ 2
/ žd
mn y
qˆm qˆn qˆ
2
/
q
qˆm qˆn qˆ 2
.
Ž 6.
The convergence of the modified cooling requires 0 F c F 0.75. It is remarkable that the term Hm that includes the commutators can not be ignored because of the non-Abelian property of Am . The transverse component of the large momentum Fourier coefficients AmŽ q . is not being suppressed as Ž1 y 6c a2 qˆ 2 . n after n smoothing steps, as is seen in our calculation.
Y. Chen et al.r Physics Letters B 486 (2000) 448–453
450
The lattice version of the topological charge density Q LŽ x . we used here is the naively discretized theoretical definition improved to O Ž a4 ., Q LŽ x . s y
1 512p 2
Ž 53 Q P Ž x . y 16 Q R Ž x . .
s a 4 Ž Q cont . Ž x . q O Ž a 4 . . ,
Ž 7.
where Q P Ž x . and Q R are constructed by plaquettes Pmn and rectangular operators Rmn , respectively. The lattice field strength tensor is defined from the clover average of Pmn and Rmn w12x, Fˆmn Ž x . s Im Ž 5² Pmn :clover y 3² Rmn :clover . .
Ž 8.
where q is the momentum qm s
2p Lm am
nm ,
nm s 0,1, . . . , Lm y 1,
Ž 12 .
It is remarkable that the momentum space vector potential AmŽ q . is not a gauge invariant and the longitudinal terms are pure gauge transformation and almost left unchanged during smoothing Žsee Eqs. Ž3. and Ž4.., so we will only check the transverse components. The amplitudes of the transverse components of each momentum mode are measured by < A Ž q . < 2 ' Ý Tr Ž ATm Ž q . ATm † Ž q . . m
3. Evolution of Am , Fm n and Q during smoothing In this work the lattice SUŽ2. theory is investigated by the smoothing scheme based on the tadpole-improved action S1 on an 8 3 = 24 anisotropic lattice with the anisotropy ratio j s a sra t s 3 w9x Ž a s and a t are spatial and temporal lattice spacings, respectively., S1 s b
Ý x , s)s
5 Ps sX 3 j u 4s
X
y
4 Pst j
qb Ý
3
x,s
u 2s u 2t
1 R s sX 12 j u 6s
y
y
1 R st j 12 u 4s u 2t
1 R sX s 12 j u 6s ,
Ž 9.
where b s 4rg 2 , g is the coupling, u s and u t are the tadpole parameters w10x. On the lattice the vector potential AmŽ x . of the SUŽ2. gauge field is defined as Am Ž x q am m ˆ r2 . s
1 2 igam
† m
žU Ž x . y U m
Ž x.
y 12 Tr Ž Um Ž x . y Um† Ž x . . .
/
Ž 10 . where am is the lattice spacing in the m direction. The Fourier transformation of AmŽ x . is defined as Am Ž q . s a4 Ý eyi qPŽ xqam mˆ r2.Am Ž x q am m ˆ r2 . , x
Ž 11 .
s
Ý Tr m ,n
ž
ž
Am Ž q . A†n Ž q . dmn y
qˆm qˆn qˆ 2
//
.
Ž 13 . Our goal of this paper is to show that any initial configuration will approach one of the fixed points of the smoothing transformation. According to our calculation, the evolution behaviors of AmŽ q . and Fmn Ž q . during smoothing are typical for any gauge configuration and insensitive to the lattice spacing, so the following discussions are all based on a randomly thermalized configuration at b s 1.1 which corresponds to the spatial lattice spacing a s s 0.233 fm w11x. The evolution of < AŽ q .< 2 of several momenta under the smoothing procedure is plotted in Fig. 1. After a period of oscillation, < AŽ q .< 2 tends to be stable, indicating that the configuration tends to the fixed point of the transformation. Because the contributions of all modes are significant, it is shown that the conjectured suppression of high momentum components is not the case. A similar analysis is performed for the field strength tensor and the charge density. Fig. 2 shows the evolution of < Fmn Ž q .< 2 ' † Ž . TrFmn Ž q . Fmn q along with the smoothing procedure. The damping of < Fmn Ž q .< 2 is very fast, and there is no sign that only the high momentum components are suppressed. Fig. 3 shows several Fourier components Q LŽ q . measured during the smoothing procedure. In contrast to what we see from Fig. 1 and Fig. 2, where most of the modes damp out to almost zero or tend to be stable, one can find that only the zero momentum component and some other
Y. Chen et al.r Physics Letters B 486 (2000) 448–453
451
Fig. 1. The evolution behavior of < AmŽ q .< 2 during the smoothing procedure, where qm s L2mpam nm . Six momentum modes, which are denoted by Ž n1 ,n 2 ,n 3 ,n 4 ., are plotted in the figure. Even after 300 smoothing steps, all curves tend to be stable. This indicates that the large momentum components of Am were not being smoothed out.
Fig. 3. The evolution behavior of < QŽ q .< 2 is plotted. The zero momentum component has several plateaus during the smoothing and is kept stable after 120 smearing steps. Most of the momentum components Žexcept for the zero momentum one which is kept stable after 120 smearing steps. are smoothed out even after 50 smoothing steps. The residual parts account for the structure of the smoothed configuration.
modes have some metastable plateaus. Thus we can conclude from the above facts that if a stable configuration, which corresponds to the fixed point of the smoothing transformation, is approached, it is some
of the Fourier components of the charge density surviving the smoothing procedure that give rise to the local structure of the smoothed configuration, because all the momentum modes of the gauge field still exist after so many steps of cooling.
4. The structure of the smoothed configuration The local structure of the smoothed configuration is explored by studying the distribution of the charge density in the whole spacetime, i.e, the whole lattice volume. With reference to the scheme introduced by Teper and Smith w6x, our first step is to recognize the instantons and antiinstantons. In doing so, it is assumed that the smoothed configuration is the superposition of instantons and antiinstantons and we employ the BPST parameterization of the single instanton. The peaks of the charge density can be taken as the candidates of the instantons. For a peak at x i , the raw estimation of the size of this instanton candidate Ždenoted by r i . is given by Fig. 2. The evolution behavior of < Fmn Ž q .< 2 during the smoothing procedure. The labels of the curves are the same as in Fig. 1. All the Fourier components of Fmn damp out rapidly.
Q LŽ x i . s
6
p 2r i4
.
Ž 14 .
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An iterative procedure is performed to correct the sizes, and several complicated schemes were developed to filter out the unwanted peaks, which are assumed not to be instantons and antiinstantons. For details see Ref. w6x. After correcting of the instanton radii and filtering out the unwanted peaks, one can get a set of candidates for the instanton and antiinstanton Ž x i , r i . < i s 1,2,3, . . . 4 . By the use of this formalism to each configuration during the smoothing procedure, we find the following facts: Ži. The number of peaks decreases rapidly as the smoothing proceeds. Žii. The position of the peaks are very stable and insensitive to the smoothing. Žiii. Some peaks undergo disappear–appear–disappear procedures. Živ. The radii of some peaks increase with the smoothing proceeding until they disappear, and some peaks are kept fixed. Žv. If n I denotes the number of positive peaks and n A negative peaks, n I y n A is not equal to Q L . However, the number of relatively stable peaks with radii 1.5a - r - 3.5a is consistent with Q L when the metastable configurations are reached. The wider peaks have no contribution to the measured charge. We interpret these facts as follows: Due to the finite size of the lattice volume and the finite lattice spacing, the number of momentum modes contributing to all functions of the spacetime coordinates are finite. Here the structure of the smoothed configuration is explored through the spacetime distribution of the TCD, with the local peaks and valleys the results of the superposition of the momentum modes of the TCD. So, many peaks may be the presence of the fluctuations of each momentum mode rather than true instantons, especially in the early period of the smoothing procedure. As the smoothing proceeds, the amplitude of each momentum mode varies and, accordingly, the superficial radius obtained from Eq. Ž14. varies too. The variation of the amplitude of some momentum modes may cause some measured peaks to vanish and reappear during the smoothing procedure. However, the position of each peak or valley is fixed, as is observed in our calculation. There must be some ambiguities if the physical structure was extracted from the superficial peaks, for example, < n I y n A < is inconsistent with the measured Q L , since the topological charge is equivalent
to the amplitude of the zero momentum mode of TCD Žthe spacetime integral of the nonzero momentum mode is zero.. Next we try to explain why the smallest size of ‘‘instantons’’ obtained on the lattice is about 2 a and why the large ‘‘instantons’’’’ are suppressed. We start from the continuous BPST instanton. Fourier transforming the field strength tensor of the single instanton one gets Fmn Ž k . s
1
4
Ž 2p .
2
yi kŽ xyx 0 .
Hd ke
Fmn Ž x .
a s y8p 2hmn l ar 2 K 0 Ž r < k < . .
Ž 15 .
With the self-dual condition Fmn s F˜mn , the topological charge is Qs
1 16p 2
Hd
4
x TrFmn Ž x . F˜mn Ž x .
A d 4 kK 0 Ž r < k < . K 0 Ž r < k < . .
H
Ž 16 .
On a L4 lattice with lattice spacing a, there is the maximal momentum cutoff pra and the minimal nonzero momentum is approximately 2Lap . We take the assumption that the instantons distribute well separately, so the charge contribution QŽ r . of each single instanton with radius r s rˆ a on the lattice is qualitatively given as pr
H2pˆrˆ drr K
Q Ž rˆ , L . s 3
3
0
Ž r . K0Ž r . .
Ž 17 .
L
QŽ rˆ ,8.,QŽ rˆ ,24. and QŽ rˆ ,`.. are plotted versus rˆ in Fig. 4. When L s 8, the charge gets significant contributions from the instantons with radii r in a narrow range 0.5a - r - 3a. The smallest instanton we measured is with radius r ; 2 a because still smaller instantons with radii r ; a cannot survive the smoothing procedure and cannot be measured on the lattice because the lattice artifacts for thus small instanton would be too large w12x. If we take the lower peaks Žwith larger superficial radius. as larger instantons, they do not contribute to the charge Q L and are difficult to be disentangled from the fluctuations of the small momentum modes. It should be noted that Eq. Ž17. is derived from the continuum BPST parameterization and the result is thus qualitative but not precise. Practically, Eq. Ž17. implies that
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it is difficult to separate the instanton information from the unphysical fluctuations due to the lattice artifacts, there may be ambiguities in extracting the topological structure from the spacetime distribution of the charge density on the lattice. This may give an explanation of the suppression at the large size end of the instanton size distribution and the discrepancy of the measured charge with the number of peaks of the charge density, which are taken as instanton candidates, in the smoothed lattice configuration.
Acknowledgements
Fig. 4. The figure shows the lattice artifacts in QŽ r .. The rigid value of Q is 1 and independent of r in the continuum.
the largest instanton which can be checked on the lattice is governed by the largest lattice side size, which actually gives the smallest lattice momentum. This is in agreement with the observation of Ref. w13x where the authors suggested that the largest instanton would be larger if one takes the twisted boundary condition or asymmetric lattice with larger size in one direction than the others, for example 12 3 = 36 in their work.
5. Conclusion We have investigated the evolution behavior of the gauge field under smoothing in momentum space. It is found that the UV fluctuations of the field variables cannot be smoothed out as expected, while the field strength tensor decreases rapidly as the smoothing proceeds. In fact it is the topological density that is smoothed. The very small instantons cannot survive on the lattice with finite lattice spacing, while the very large instantons are deformed to a large amount by the finite lattice size. Consequently
This work is supported by the Natural Science Foundation of China under the Grant No. 19677205 and No. 19991487, and by the National Research Center for Intelligent Computing System under the contract No. 99128.
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