Topological charge, renormalization and cooling on the lattice

Nuclear Physics B329 (1990) 683-697 North-Holland

TOPOLOGICAL CHARGE, RENORMALIZATION AND COOLING ON T H E L A T ' H C E * Massimo CAMPOSTRINI, Adriano Di GIACOMO, Haralambos PANAGOPOULOS and Ettore VICARI

Dipartimento di Fisica dell'Universit?l and L N.F N., 1-56100 P&a, Italy Received 20 February 1989 (Revised 31 July 1989)

We show that a definition of the topological susceptibility X on the lattice in terms of local fields, with the proper additive and multiplicative renormalizations, is well defined, self-consistent and compatible with results we obtain by the cooling method. Moreover, we can follow the behaviour of the multiplicative renormalization during cooling, and check its dependence on the coupling constant.

1. Introduction T h e existence of a non-zero topological susceptibility of the Q C D v a c u u m was p r o p o s e d some years ago [1-3] to solve the so-called U(1) problem, i.e. the explicit breaking of the U(1) axial symmetry. T h e topological susceptibility is defined by

x = fd'x

(01T(Q(x)Q(O))IO),

(1.1)

where

~2 Q ( x ) = 64rr------~e"'°°F~,(x ) F~a ( x )

(1.2)

is the topological charge density. Q is the divergence of the topological current K , [1]:

Q = O , K ~,

K,=

~2 __Eltet[~YAa( a _ 1 fabcdbzt¢." ~ 16~r 2 ,,~ O~A:, ~ g s ~.p~.y / .

* Partially supported by M.P.I. (Italian Ministry for Public Education). 0550-3213/90/$03.50©Elsevier Science Publishers B.V. (North-Holland)

(1.3)

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684

The prescription defining the product of the operators in eq. (1.1) is [4] (01T(Q(x)Q(O))IO) - 0r(0 ]T(K~'(x)Q(O))IO).

(1.4)

Chiral Ward identities lead to the prediction [3]

2~ f2 X = m2n+ rn2' - 2m~

(1.5)

X = (180 MeV)*.

(1.6)

or, with three light flavours,

Eq. (1.5) is derived from an expansion in 1/N¢, and the susceptibility is intended to be that of a pure gauge system with no quarks [3]. The lattice is a unique tool to determine from first principles quantities like X which have nontrivial dimensions in mass, and therefore cannot be computed by perturbation theory. A few different methods have been used to extract X from Monte Carlo simulations, which we shall briefly review. All of them indicate that X is different from zero, but up to now there are discrepancies among the results obtained by different methods. This paper is a contribution to clarify the situation. The fact that X has to be computed on a pure gauge system with no quarks, makes the problem simpler from the numerical point of view. However, the real difficulty of the problem is to read a value of the topological charge out of a discrete configuration, like the ones produced by numerical simulations on a lattice. On the other hand, lattice configurations do approximate continuous configurations which can carry topological charge. The methods which have been used up to now to extract the topological susceptibility from the lattice are mainly three: (i) A "naive" method [5-7] which consists in the determination (by Monte Carlo simulation) of the vacuum two-point correlation at zero momentum of an operator QL(X) which coincides with the topological charge Q(x) of eq. (1.2) in the limit of zero lattice spacing (a ~ 0). A possible choice for QL is

QL(x)

1 32~ra E e.,poTr(HS'(x)I-l°°(x)},

(1.7)

/*~,po

where H ~ is the parallel transport along the 1 × 1 Wilson loop; in the usual notation

I I ~ ( x ) = U~(x)U,(x + #)U~*(x + v)Uv*(x ) .

(1.8)

M. Campostrini et al. / Topologicalcharge

685

By this method

X~) = ~ ( ~x QL( X)QL(O)I

(1.9)

is measured. After subtraction of an additive perturbative tail nice asymptotic scaling is observed. The value of X~) determined in this way is smaller than expected X(~) = (50 MeV) 4.

(1.10)

An intriguing fact is, moreover, that the value of the topological charge ~ x Q L ( X ) o n most configurations is not an integer, as expected, but a small fraction of unity [6]. (ii) A "geometrical" method [8-10] which consists in interpolating between the links of the lattice to produce a smooth field configuration on which the value of the topological charge can be read by geometrical methods. Early determinations by this procedure in the literature [11,12] refer both to SU(2) and SU(3). Here again good asymptotic scaling is observed. The resulting value of ~, ,,0i) is much higher ix., A L than X~): X~i) = (230 MeV) 4.

(1.11)

An argument is usually given that this method should work at large values of fl ( f l - - 2 N / g 2 for gauge group SU(N)) since small instantons are exponentially suppressed as fl--* o0. At which values of fl this happens is not known, and presently there is no way to estimate the errors involved at finite fiAt the values of fl explored in refs. [11,12], different methods of interpolation give different values for the charge configuration by configuration, even if they more or less agree on the average value. It is not easy to master the error involved in the procedure [13,14] (for a review, see ref. [15]). More recent results [16,17] tend to give a lower value than (1.11), by a reduction of the lattice artifacts (the so-called dislocations) effected either by a choice of the action density different from Wilson's action [16] or by a blocking procedure [17]. (iii) A third method consists in cooling [18,19] equilibrium configurations to eliminate quantum fluctuations, and in reading subsequently the topological charge, either by measuring the average value of ~,xQL(X),or by counting the number of zero modes of the inverse fermion propagator [20], or by measuring the value of the action in units of the action of one instanton. The principle on which this method is based is that the cooling process eliminates quantum fluctuations but does not destroy classical configurations, like instantons, which carry the topological charge and which correspond to a minimum of the action. Thus cooling eliminates lattice artifacts but leaves the topological charge unchanged.

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M. Campostriniet al. / Topologicalcharge

This is only approximately true, because classical configurations carrying topological charge, placed on the lattice, are only approximate solutions of the equations of motion; therefore they are only metastable and survive a number of cooling steps, which depends on the size of the lattice, and is higher for larger lattices [21, 22]. The result of cooling obeys nice asymptotic scaling, and gives a value X~ii) for the topological susceptibility X~ii) = (120 MeV) 4,

(1.12)

which is different from X~) of eq. (1.10) a n d X~i) of eq. (1.11). The problem with this method is to control metastability and to check that no topological charge gets lost in the cooling process. An interesting remark to make is that the topological charge computed from cooled configurations using the definition (1.7) takes integer values [19]. There are two main ways to go to the continuum. One of them is the geometrical method, by which each configuration is replaced with a continuum configuration obtained by interpolation. The other way is to consider the lattice as a regularization of a field theory, with a momentum cutoff going to infinity as /3 ~ ~ . Local operators like QL defined by eq. (1.7) should tend to their continuum counterpart in this limit. At finite/3, physical information can be extracted if renormalization is carried out properly. The "naive" method (i) operates additive subtraction, but neglects multiplicative renormalization (as was usual in all early lattice works). When renormalization is properly carried out, a consistent field theoretical treatment is obtained, in which corrections can be evaluated order by order (at least in principle) and a well defined continuum limit will exist if the field theory exists. What is measured is the vacuum expectation value of an operator which has additive renormalizations, corresponding to the quartic divergences of X and a multiplicative renormalization Z, QL = ZQ, and by eqs. (1.1) and (1.4) Q2) -'~ ---Z2xa4+ E ~ c, n,

(1.13)

with

zl

z~

Z = 1 + -fl- + ~-2 + . . . .

(1.14)

Eq. (1.13) differs from the "naive" method by the presence of the multiplicative factor Z 2 in front of X. The addition of this factor promotes the "naive" method into a "field theoretical" method, where all approximations are under control. Previous evidence for multiplicative renormalization of local operators has been

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687

produced in refs. [23, 24]. In ref. [25] we have computed the coefficient Z 1 and we have estimated Z 2. The result of ref. [25] was, for SU(2) and a 124 lattice: Z 1= -2.14,

Z 2 - 1.6.

(1.15)

In the scaling region observed in Monte Carlo simulations, fl - 2.4-2.6, Z is a small number. What is measured is not X but X" Z2, which explains why the result is so small. The presence of Z also explains why the values of the topological charge observed before cooling (method (i)) are a fraction of unity, while, after cooling, i.e. after sending fl to infinity, the local operator QL gives the correct integer values for it. Our program, which was started in ref. [25], is to bring different methods to agree, trying to understand and to master the origin of the discrepancies. In this paper we develop a method which in a sense brings together the "field theoretical" procedure and the cooling procedure. The details of the method will be explained in sect. 2. Results and checks will be the subject of sect. 3. In sect. 4 we present our conclusions.

2. The method Eq. (1.13) can be rewritten in terms of the topological charge as follows ,

\,4/

(2.1)

Q is the topological charge contained in a lattice configuration, V is the volume, and both stay constant in the cooling procedure at least as long as no topological charge gets lost. What changes from one cooling step to the next is a flCff which varies from the fl of the initial configuration to infinity as cooling goes on. We implement a means of controlling the velocity of cooling, as will be described in sect. 3; with this we ensure (and subsequently verify) the thermalization of intermediate cooled configurations. fl~f can be determined using as a thermometer the density of action ( E ) , which is the density of internal energy in the language of statistical mechanics. We know from previous work that [23, 24]

dn ( E ) = ~.~ fl--T + A e - b a -

(2.2)

The coefficients d n, A, and b are known; the last term describes gluon condensation, and is a small fraction of ( E ) at fl = 2.3-2.6 and completely irrelevant at higher fl 's. Eq. (2.2) can be inverted to define a fl~ff as a function of ( E ) . ( E ) can

688

M. Campostriniet al. / Topologicalcharge

be read off the configurations after each cooling step and the corresponding flaf can be determined. As will be shown in sect. 3, a given number of cooling steps, starting from configurations with the same initial/3, will give a/3af that is almost the same for every configuration, with very small spread. Therefore, if we measure (Q2/L4) step after step, the cooling procedure corresponds to having eq. (2.1) at different values of/3af, but with the same value of (Q2/V) and, of course, of the coefficients of the perturbative expansions (2.1) and (1.14). The latter are also independent of the initial/3. The sum cn//3" is the same perturbative tail which is subtracted from the data in the "naive" procedure. Eq. (2.2) is only approximate, since cooled configurations are not exactly in thermal equilibrium. The accuracy of this approximation, with respect to the problem of topological charge renormalization, can be checked a posteriori; it will be discussed in detail in sect. 4. Eq. (2.1) is a bridge between the "naive" determination X(i) and the value X (iii) obtained by cooling. If it is satisfied the field theoretical method and the cooling procedure do agree. As we shall see, eq. (2.1) can also provide a control and a cross check of the losses of topological charge in the cooling process. A crucial test will be that Z(flefD extracted from data obtained by cooling configurations at different fl should be the same, regardless of the initial/3. In principle, a mixing of X with the gluon condensate f¢2 of dimension four is possible; this comes from a 8 (x) singularity of the correlation function ( TQ (x) Q (0)) which must be subtracted according to the prescription (1.4). This mixing M is zero at tree level. We have computed M to one-loop level, finding for our choice (1.7) of the operator summed over positive and negative directions:

g22 FaF,,] M = 2.174 X 1,,-4N3,~-2[ u P [ 4qr ~v ~ ] a ,,

(2.3)

(the computation was done using the background field method, as in ref. [25]). M should in principle be added to the right-hand sides of eq. (1.13) and (2.1). However, inserting the Monte Carlo value of f#2 [23] in eq. (2.3) gives a contribution

( M ) / ( a A L ) 4 = 2.4 x 104/3-2 .

(2.4)

Compared to the values for x / A 4 (see sect. 3) this mixing is of the same order of the error bars. Furthermore, due to its/3 -2 behaviour, it is expected to become even less significant during cooling. We have therefore neglected it in the present computation. We plan to determine ( M ) from more precise data. To this end it is important to know the behaviour of the gluon condensate f#2 during cooling; we are studying it by independent methods.

M. Campostrini et al. / Topological charge

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3. Results

We have tested the "field theoretical" method by a Monte Carlo simulation on a 12 4 lattice, with gauge group SU(2) and the usual Wilson action. Our Monte Carlo upgrading procedure was a standard heat bath. As in ref. [5], our topological charge density was the operator of eq. (1.7), averaged over positive and negative directions. We have first repeated the "naive" determination of refs. [5-7], obtaining the unrenormalized topological susceptibility Xu, defined as

c.

-~

---Xua 4 + E ~--k

(3.1)

n~3

(for a lattice of size L 4 ) . Xu is related by multiplicative renormalization to the physical topological susceptibility Xu = z2(

)x.

(3.2)

If asymptotic scaling holds, we have

a =

1 / 6

\51/121exp/- 3 2 -~-~r ~ )

I

(3.3)

The values of (Q~/L4) are listed in table 1. As in refs. [5-7], we know c 3 by computation: c 3 = 2.648 × 10-4. c4 is obtained by a best fit on fl >/2.7 data points, giving c4 = 2.12 x 10 -4. We checked that fits starting from higher fl give consistent results. As obtained in previous work, we find a reasonable asymptotic scaling (see fig. 1 and fig. 4) and a value of xJA4L = (2.63 + 0.13) X 104 (cf. refs. [5-7]). Starting from the last configuration, we performed longer runs for values of fl in the (tentative) scaling region. A sample of configurations, separated typically by 60 sweeps, was cooled, and the action density and topological charge were measured after each cooling step. Our cooling procedure is controlled by a speed parameter 3. Cooling of one link variable U~ is performed by replacing it with the new value U/ that minimizes the action (keeping all the other links fixed), with the constraint ~ Tr( (Ut* - U t ' t ) ( U l - Ut') } ~<82.

(3.4)

One cooling step consists in cooling sequentially all the links of the lattice. We checked that, for ~ ~<0.1, the order in which the links are cooled is immaterial and that further decreasing 6 changes only the speed of the process. We choose 8 = 0.05 for our production runs.

M. Campostrini et al. / Topological charge

690

TABLE I


10 5

2.400 2.425 2.450 2.475 2.500 2.525 2.550 2.600 2.650 2.700 2.800 2.900 3.000 3.250 3.500 3.750 4.000 4.500 5.000 6.000



sweep no,

3.501 ± 0.102 3.399 __+0.094 3.098 ± 0.035 2.968 +__0.082 2.710 __+_0.039 2.428 +_ 0.086 2.412 + 0.047 2.111 ____0.117 _ 1.868 +_ 0.031 1.927 ± 0.110 1.516 ± 0.078 1.406 ± 0.070 1.316 + 0.063 0.979 ± 0.047 0.724 ± 0.039 0.540 __ 0.031 0.470 ± 0.023 0.341 ± 0.020 0.239 ± 0.016 0.137 ± 0.008

'

I

2800 3000 10000 3000 10700 300 6000 300 300 300 300 300 300 300 300 300 300 300 300 300

I

I

I

'

7

0.035 I

0.030

i I

T

j

0.025

0.020

-

, I .... 2.4 Fig. 1.

I .... 2.45

/~

I .... 2.5

I

,~

2.55

xu/A4L as a function of ft. The solid lines show the average value ~(u/A4L and (Xu + AXu)/A4L •

Fig. 2 shows the values of fle~f as a function of the number of cooling steps, for two representative values of ft. The bars show the width of the distribution of fleff for our sample of configurations. We can neglect the width, which is relatively very small, and say that fletr has a well-defined value after a given number of cooling steps, starting from any thermalized configuration at a given value of/3. The values of fl~fr and (Q~/L 4> are given in table 2. The average is made over the sample of configurations, at a fixed value of fl and of the number of cooling steps.

691

M. Campostrini et al. / Topological charge ,

,

,

,

[

'

,

,

,

4O 20 /Tel i 10 E-

,7 2 0

5

10

cooling steps

Fig. 2. flerr as a function of the number of cooling steps, for an initial fl of 2.5 (upper curve) and 2.4 (lower curve). The bars show the width of the distribution. The solid lines are interpolations.

K n o w i n g the values of c 3 and c4, we write

(

zl Z2 ) Vtx-a 2 ` I+~+~

/Q~\

c3

C4

is-,'/

he,,

N,

) 1/2

"

(3.5)

T h e a p p r o x i m a t i o n of the perturbative tail by two terms already works at fl = 2.5, and will work better and better as flefe grows; the same should h a p p e n for the p e r t u r b a t i v e expansion of Z. If this is true, we should be able to m a k e a best square fit to our data by eq. (3.5), with v~-a2(fl) - ~Q2/V>(fl) and Z 2 as free parameters, using the s a m e Z 2 for all the different fl's. Indeed this " g l o b a l " fit gives a x2/n (d.o.f.)= 4 7 / 4 9 ; the resulting values of the parameters are Z 2 = 1.27 + 0.05,

%/~a2(fl =

2.4)

2.45) = (8.12 + 0.17) x 10 -3 ,

%/~a2(fl =

2.425) = (8.73 __+0.27) x 10 -3 ,

%/~aZ(fl=

vrxa2(fl =

2.475) = (7.27 + 0.15) × 10 -3,

vrxa2(fl --- 2.5)

= (10.24 + 0.32) x 10 -3,

= (6.30 + 0.13) x 10 -3. (3.6)

Z 2 c o m p a r e s satisfactorily with the estimate (1.15). A separate fit to Z 2 and grx-a2 for each value of fl gives consistent results. Fig. 3 shows X/A4L as a function of fl, and the average value y(/A4L= (2.39 + 0.05) X 10 s. X a4 as a function of fl is shown in fig. 4 together with the " n a i v e " result xua 4. Nice asymptotic scaling is observed in the range 2.4 ~< fl ~< 2.5.

M. Campostrini et al. / Topological charge

692

TABLE 2 flcfr and

(Q~/L4), as a function of the initial fl # = 2.4

step

]~eff

0 1 2 3 4 5 6 7 8 9 10

2.406 2.766 3.283 4.001 4.992 6.366 8,301 11.082 15.166 21.280 30.555

105

fl = 2.425

(Q~/L4>

3.50 ____0.10 _ 4.21 ____0.74 _ 4.11 + 0.73 4.12 + 0.78 4.32 + 0.88 4.84 + 1.05 5.74 + 1.25 6.99 + 1.49 8.50 + 1,76 10.16 + 2.07 11.95 + 2.42

fleff 2.433 2.809 3.349 4.102 5.146 6.606 8.684 11.708 16.218 23.103 33.805

fl = 2.475

step

flcrf

0 1 2 3 4 5 6 7 8 9 10

2.485 2.894 3.483 4.310 5.470 7.120 9.518 13.096 18.602 27.329 41.504

105

105

fl = 2.45



3.40 +__0.10 2.87 + 0.56 2.82 +__0.59 2.93 __+0.64 3,16 + 0.67 3.54 + 0.72 4.07 + 0.81 4.70 + 0.94 5.44 -+ 1.10 6.26 __+1.27 7.12 +__1.44

fleff 2.459 2.852 3.416 4.206 5.309 6.863 9.098 12.392 17.386 25.162 37.539

105

(Q2L/La)

3.10 +__0.04 2.87 __+0.32 2.87 + 0.30 2.98 + 0.30 3.16 + 0,31 3.43 ± 0.34 3.51 + 0.61 4.03 + 0.65 4.70 + 0.72 5.46 +__0.84 6.27 + 0.99

fl = 2.5

(Q~/L4)

2.97 2.18 2.18 2.26 2.41 2.67 3.07 3.57 4.13 4.70 5.22

and of the cooling step number.

+ 0.08 + 0.34 + 0.33 _+ 0.31 + 0.30 + 0.32 __+0.36 + 0.41 + 0.47 + 0.53 _+ 0.60

fleff 2.511 2.935 3.546 4.407 5.623 7.362 9.896 13.731 19.705 29.326 45.257

105


2.71 2.46 2.24 2.07 1.94 1.87 2.00 2.20 2.50 2.88 3.28

___0.04 + 0.25 + 0.21 + 0.19 + 0.18 + 0.19 + 0.38 + 0.45 _+ 0.53 + 0.62 + 0.72

We show in fig. 5 that Z(j~eff) as determined from different initial fl's does not depend on fl, thus verifying the field theoretical method. This figure also shows that Z(fleee) is well approximated by two perturbative terms. Fig. 6 gives a global view of the result; data points represent the r.h.s, of eq. (3.5), i.e. the measured (Q2L/L4) after subtraction of the known additive renormalization; the curves represent the 1.h.s. of eq. (3.5), using the central values (3.6) for Z 2 and v/x-a 2. At ~eff -----fl we reproduce the "naive" method, and for fl ~ oo we reproduce the result given by the cooling procedure. For smaller and smaller lattices the topological charge is lost more and more rapidly during the cooling procedure. This is demonstrated in fig. 7 by comparing data from an 84 and 64 lattice with our "reference" 124 lattice. At fl --- 2.55 a loss of topological charge of the same form is observed also for a 124 lattice (fig. 8). For higher fl's cooling works only on larger lattices, while the field theoretical method still gives correct results.

M. Campostrini et aL / Topological charge I

I

I

l

693

0.26

0.24

Io-%/A 0.22

0.20

,

,

I 2.4

. . . .

I 2.45

I 2.5

Fig. 3. x / A 4, determined from the "global" fit, as a function of/3. The solid lines show 2 / A 4 and

(2 + Ax)/A~.

l°°I "'"---.1 0.30 I 1

0

~'~ 4

X

a

4

~

0.10

0.03

, I .... 2.4

I .... 2.45

I , 2.5

t~ Fig. 4. xa 4 (crosses) and Xu a4 (squares) as a function of/3. The solid lines show 2 a 4, (X + AX) a4, 2u a4, and (2~ + AX) a4.

4. Conclusions We have demonstrated that it is possible to define on a lattice a local density of topological charge which tends to the continuum operator as the lattice spacing goes to zero. At finite values of the cutoff, additive and multiplicative renormalizations are present, which can be computed in perturbation theory, at least in the leading orders. We have shown that this procedure is well defined and effective by a cooling

694

M. Campostrini et aL / Topologicalcharge

1.00

.._..0.75 ",3 ~q 0 . 5 0

0.25

0.00

.... 0

I .... 0.1

I .... 0.2

I .... 0.3

I,, 0.4

1/fl e. Fig. 5. Z(flefr ), determined by dividing the 1.h.s. of eq. (3.5) by !/Fxa 2 obtained from the "global" fit. T h e errors induced by the uncertainty on i / ~ a 2 are negligible. Crosses: fl = 2.4; d i a m o n d s : /~ = 2.425; squares: fl = 2.45; circles: fl = 2.475; crossed squares: fl = 2.5.

Fig. 6. Global view of the r.h.s, of eq. (3.5); symbols as in fig. 5. Lines at constant fl show the l.h.s, as a function of Bert. The line at 1//~ef r = 0.41 is vr~ua2; the line at 1/~ef f = 0 is ~/~a 2.

M. C a m p o s t r i n i et a L /

'

'

0

695

Topological c h a r g e

0.000

0.1

0.2

0.3

0.4

Fig. 7. R.h.s. of eq. (3.5), for data taken at fl = 2.5 on a 6 4 (circles), 84 (squares) and 124 (crosses) lattice. The line is the result of the "global" fit to the 12'* data.

....

I ....

I ....

I ....

r'

%

v

0.004

!

0.002

v N

,,

[

0.1

,

, , L ,

,

0.2

,I .... 0.3

I, 0.4

0.000

1/~ee. Fig. 8. R.h.s. of eq. (3.5), for data taken at 13 = 2.55; symbols as in fig. 7. Data before cooling are in agreement with scaling predictions, while a loss of topological charge is observed for high l/elf.

procedure, which preserves the topological charge of each configuration while sending 18 to infinity. The multiplicative renormalization factor of the topological charge has been determined in this way from independent sets of data corresponding to different values of the initial fl, and is in agreement with the two-loop formula (1.14). By our procedure, the dependence of Z on flefe can be followed from fleff - 2.5 to fleff - 100. Z 1 is known by computation [25], and Z 2 is determined by a best square fit and agrees with the estimate given by a tadpole approximation, within the

696

M. Campostrini et al. / Topological charge

expected precision. We plan to compute Z 2 exactly. The additive renormalization already used in the "naive" method has also been checked, and proves to be well defined in our range of tiThe inclusion of the multii.licative renormalization constant Z transforms the "na'fve" method in a consistent field theoretical procedure, which also allows a cross-check of the losses of topological charge during the cooling process. We close with some remarks on our fleff, introduced in sect. 2. The description of quantum fluctuations in terms of a fleff is of course approximate, since configurations produced by cooling are not equilibrium configurations, and hence our fleff has not, strictly speaking, the meaning of a usual ft. Our procedure is, however, correct for the following reasons: (i) At each step of cooling we have measured not only the 1 × 1 plaquette, in terms of which our fl~ff is defined by eq. (2.2), but also the 1 × 2, the 2 x 2 plaquette and the 1 × 1 plaquette covered twice. A different fleff can be defined in terms of each of these quantities, by a formula analogous to eq. (2.2) [22]. The resulting fleff'S differ from each other, reflecting the fact that we are not at equilibrium. The differences are very small at the first cooling steps, increase as cooling goes on, and can be as large as 20-30% at 10-15 cooling steps. These numbers indicate the order of magnitude of the approximation involved in our description by a fleff of the fluctuations at the scale of 1-2 lattice spacings. We have repeated our determination of the topological susceptibility using all these different fl~ff's as parameters in eq. (2.1). The results are consistent with each other within 1-2 standard deviations. In fact, quantum corrections in eq. (2.1) are only important at the first cooling steps, when all the definitions coincide. When they start to disagree by more than, say 10%, additive renormalizations have already disappeared within the errors and the multiplicative renormalization is already almost equal to 1 (figs. 5, 6). (ii) No matter what definition of ~eff one assumes, fleff gets large as cooling proceeds, reflecting the fact that quantum fluctuations become smaller and smaller. After a few cooling steps, higher order terms in fldfl (c3/fl3ff, c4/fl~f ' Z2/fl~f ) become irrelevant within the errors, so that what one is really probing is the term Z1/fleff. A model independent way of testing this statement is to look at the way in which the topological charge tends to an integer value in a single configuration. Z 1 is known by explicit computation [25], and most of it (90%) comes from tadpole diagrams, in which a pair of quantum fields of the operator are contracted. Typically, Z 1 = co(Al,(O)A~,(O)> + c,

(4.1)

where c o and c 1 only depend on the form of the operator, and not on the explicit form of the lagrangian. The vacuum expectation values of the fields do depend on the form of the lagrangian, but are divergent quantities in the continuum limit, and

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a r e sensitive to the short-distance q u a n t u m fluctuations, which are those p r o b e d b y s m a l l W i l s o n loops. (iii) W h a t is i m p o r t a n t for o u r m e t h o d is n o t a p a r a m e t r i c f o r m which works exactly. By o u r m e t h o d we show, on the one hand, that difficulties of the " n a i v e " m e t h o d c o m e f r o m renormalization, a n d that r e n o r m a l i z a t i o n effects are correctly d e s c r i b e d b y a few terms of p e r t u r b a t i o n theory; on the other hand, we are able to c h e c k if t o p o l o g i c a l charge is preserved d u r i n g cooling. A s i m i l a r c o m p u t a t i o n for the SU(3) gauge g r o u p is on the way.

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