IL NUOVO CIMENTO VOL. 107 A, N. 8 Agosto 1994

Scaling, Asymptotic Scaling and Improved Perturbation Theory(*).

A. PATRASCIOIU(1) and E. SEILER(2) (1) Physics Department, University of Arizona - Tucson, AZ 85721, USA

Division de Physique Thdorique, Institut de Physique Nuctdaire Universit~ de Paris-Sud - F-91406 Orsay Cedex, France

(2) Max-Planck-Institut fiir Physik, Werner-Heisenberg-Institut F6hringer Ring 6, 80805 Munich, Germany

(ricevuto il 10 Gennaio 1994; approvato l'l Febbraio 1994)

Summary. -- Contrary to recent claims, we show that lattice perturbation theory reproduces quite accurately Monte Carlo results of short-distance quantities. We argue that the present numerical situation strongly suggests the occurrence of a (zero temperature) deconfining transition in QCD at non-zero lattice coupling.

PACS 11.15.Ha - Lattice gauge theory. PACS ll.10.Lm - Nonlinear or nonlocal theories and models.

For several years now, the lattice community has been aware that in both two-dimensional (2D) O(N) ~ models and 4D gauge theories (with or without fermions) Monte Carlo (MC) results show good scaling, but poor asymptotic scaling [1]. For the non-expert, let us state that scaling means invariance-of-mass ratios under variation of the cut-off, while asymptotic scaling means the special relation between the lattice coupling constant g and the typical mass predicted by the perturbative renormalization group

(1) m(g) = CAL(g),

where the lattice scale parameter AL(g) is given by

(2) AL = gel e x p [ - c2/g2].

Here cl, c2 are determined by the universal first two terms of the Callan-Symanzik fl-function; C is a constant that cannot be computed perturbatively.

Another known fact is that the agreement with asymptotic scaling improves if one uses other definitions of the coupling constant [2]. The general idea behind these improved perturbation theory (PT) schemes is this: consider some short-distance

(*) The authors of this paper have agreed to not receive the proofs for correction.

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1348 A. PATRASCIOIU and E. SEILER

quantity, such as the energy density E ( g ) . Since in asymptotically free theories PT is applicable at short distances, one starts with the PT expansion of E ( g )

(3) E ( g ) = 1 - a l g 2 - a2g 4 - ...

and defines a new coupling constant

(4) ~2 _ 1 - E ( g )

a l

Here E ( g ) is the true (MC) value of the energy and the hope is that although for g - o 0 , one has ~/g----~ 1, ~ may already incorporate enough non-perturbative information to make the approach to the continuum faster. Obviously, one can rewrite eqs. (1) and (2) in terms of ~ and MC data show that the agreement with asymptotic scaling is improved [2].

Although this fact has been known for a few years, it received new attention and support when recently Lepage and Mackenzie [3] pointed out that in fact PT in the lattice coupling g fails even for short-distance quantities, like (tr Ulmk) in the Landau gauge. The purpose of this paper is to discuss the claim by Lepage and Mackenzie and present what we believe is the correct interpretation of the data.

We became intrigued by the Lepage and Mackenzie claim that PT in g fails at short distances because, in a recent paper [4] we verified that in the 0(3) model PT could reproduce quite accurately (< 3.5%) even large-distance quantities, like the magnetic susceptibility and the correlation length. Our study included lattices as large as 283 • 288 at modest values of fl - 1 / g 2 < 3. So it seemed strange that a short-distance quantity like the energy, known to approach its thermodynamic value on quite small lattices, would fail to be reproduced accurately by PT. In fact, as illustrated in table I, in both the 2D 0(3) model and the 4D SU(3) pure gauge theory, there is good agreement between the MC and PT values for (s(0). s(1)), respectively

TABLE I. - Comparison of the energy computed by second-order PT wi th its MC value.

a) 2D 0(3) spin model b) 4D SU(3) lattice gauge theory

/~ E (MC) E (PT) fl E (MC) E (PT)

1.50 0.6016 0.6389 5.70 0.5491 1.60 0.6357 0.6631 6.00 0.5937 1.70 0.6642 0.6843 6.10 0.6050 1.75 0.6766 0.6939 6.20 0.6136 1.80 0.6879 0.7029 6.30 0.6225 1.85 0.6983 0.7115 6.40 0.6307 1.90 0.7079 0.7195 9.00 0.7652 1.95 0.7168 0.7272 12.0 0.8225 2.00 0.7251 0.7344 18.0 0.8845 2.05 0.7329 0.7412

0.6114 0.6326 0.6392 0.6456 0.6517 0.6576 0.7627 0.8248 0.8851

SCALING~ ASYMPTOTIC SCALING AND IMPROVED PERTURBATION THEORY 1349

(1/3)(trUplaq). The PT values were calculated using the following second-order formulae:

(5) O(N): E(PT) - (s( 0) . s(1)) = 1

(fl = l / g 2 ; see [5], eq. (13)),

N - 1 N - 1

4/3 32/32

1 (tr gplaq ) = 1 2 1.2248 (6) SU(3): E(PT) - ~ fl f12

(fl = 6 / g 2 ; see [6], eqs. (8), (9)). The MC data were taken from [7] in the case of the 2D 0(3) model and from [3] in

the case of 4D SU(3) lattice gauge theory. We believe that Lepage and Mackenzie reached their erroneous conclusion

because they discussed the non-gauge-invariant observable (1/3)(trUlink) in the Landau gauge. On the lattice this gauge-fLxing condition corresponds to transforming every configuration into the gauge that minimizes the quantity

(7) G = ~ RetrU1ink. links

They state that at fl = 6 / g z = 6 the MC data yield 1 - (1/3)(tr U~nk) = 0.139, while the first-order PT result is only 0.97• alat=0.078 ( a l a t = g 2 / 4 r c ) . This latter statement is incorrect as can be seen by considering the special case alat = 0. The only configurations which contribute to the partition function are those having Upl~ = 1. On a periodic lattice, as employed in MC simulations, there are non-trivial configurations of this kind, namely the well-known ,,torons,. These are configurations not gauge equivalent to the trivial one (all Ulmk = 1) because the product of the U's over some loops being closed by the periodicity of the lattice (the so-called Polyakov loops) is not equal to 1. A simple example is obtained by putting U~,k = 1 for all links in the 1, 2, 3 directions and Ulink = M ~ 1 with some M �9 SU(3) for all links in the 4-direction.

Since for those configurations there is no gauge transformation that could lead to Uunk = 1 for all links on a finite periodic lattice, even at 5tla t : 0 , 1 -- (1/3)(tr U~nk) is not 0. To assure that this observable does go to 0 for a~at--) 0 one would have to (,freeze- the Polyakov loops to 1; according to Sharpe [8], who provided the MC data employed by Lepage and Mackenzie, in his study no contraints were applied to the Polyakov loops, hence 1 - (1/3)(tr Ulink) will not go to 0 for ~la t going to 0 on a finite periodic lattice.

Put differently, the torons are zero modes of the action, even after imposing the Landau gauge. To do PT requires eliminating all zero modes. In principle that can be achieved in two ways: either, for finite L, by constraining the Polyakov loops to 1, or by going to the infinite volume limit. The perturbative expression reported in [3], eq. (1), corresponds to letting L --) ~ without worrying about any zero modes (as we verified numerically). The same is true about the perturbative expression for the energy given in eq. (6). So if one compares those PT expressions to MC data, one should be sure that the latter have reached their thermodynamic limit. While for the energy this has been established, we have severe doubts that this is the case for (Ul ink) . The reason is that even at/~ = ~ the presence of the toron zero modes at finite

1350 A. PATRASCIOIU and E. SEILER

L influences the value of (U~k), but not of (Vplaq). The other possibility to test the validity of PT would be to compare the PT expressions to MC data obtained with the constraint of fixing all Polyakov loops to 1. This would clearly bring the expectation value of U~k closer to 1 and improve the agreement with PT.

Lepage and Mackenzie claimed that al~t is ,ca poor expansion parameter)~, producing large second-order corrections to the first-order expressions. In fact, as eq. (6) shows, for fl = 6 the second-order term in E(PT) is only about 10% of the first-order term and the agreement between PT and MC data is quite good.

We conclude therefore that there is no indication at the present time that ordinary lattice PT fails to reproduce accurately short-distance MC data. Some readers may be puzzled by this statement in view of our long-standing suggestion that in both 2D O(N) ~ models and 4D gauge theories PT may fail to produce the correct asymptotic expansion even for short-distance Green's functions [9, 10]. The resolution of this apparent paradox is the following: from a mathematical point of view, a series ~ aig ~

represents the correct asymptotic expansion to all orders of a function f(g), if the following inequality holds:

(8) f ( g ) - aig i < dk+lg k+l ,

with some constants dk for any k/> 0 and g sufficiently small. For a quantity such as the energy density, there is no doubt that at fixed lattice size L, ineq. (8) is obeyed by the PT series. The question, raised by us in the past, is whether (5) remains valid even when L goes to ~ and in the references quoted above we explained our reasons for doubting that. Our claim, however, does not imply that, for instance, in the SU(3) gauge model at ~ = 6 the second-order PT formula will not reproduce quite accurately the MC data for (tr Uplaq). Indeed the first two terms of the PT answer are the dominant ones and they are undoubtly correct; the trouble should start only at second order, where one encounters loops. For an infinite lattice that trouble could manifest itself by a small change of the coefficient of g4, which would not necessarily destroy the good approximate agreement between PT and MC.

Even though the original claim by Lepage and Mackenzie that PT in the lattice coupling g fails for short distances is unjustified, one may still wonder if the better agreement with asymptotic scaling observed in the ~ scheme does not imply that ~ is a better expansion parameter. In our opinion the better agreement with asymptotic scaling is accidental: it occurs because the second-order term in the expansion of the energy (eqs. (2) and (3)) is negative. Indeed, another way of stating asymptotic scaling (eqs. (1) and (2)) is to say that dfl/d log ~ should approach 1/c2. The actual numerical data produce a curve which crosses 1/c2 with a certain negative slope [ll , 2]. The numerical situation is similar in the ~ scheme, but the slope is smaller.

According to our statement above, this situation has nothing to do with asymptotic freedom; indeed it also occurs in the 0(2) model, known not to be asymptotically free--see table II, where we compare (d/dfl)(dfl/d In 5) in the g and schemes (numerical data from [12]).

Our final conclusion is that the most plausible interpretation of the numerical situation in 2D O(N) ~ models and 4D gauge theories is the following: since the data show very good scaling, a critical point must be nearby. Since the data do not show asymptotic scaling, that critical point is not as expected at g = 0, but at some get > 0.

SCALING, ASYMPTOTIC SCALING AND IMPROVED PERTURBATION THEORY 1351

TABLE II. - X - (d/dfl)(dfl/d In ~) in the g and ~ schemes in the 2D 0(2) spin model.

a) g scheme (fl = 1/g) b) ~ scheme (fl = l/H)

0.800 0.653 0.495 0.023 0.820 0.682 0.512 0.150 0.840 0.628 0.531 0.084 0.860 0.603 0.551 0.136 0.885 0.618 0.578 0.268 0.909 0.588 0.606 0.260 0.935 0.576 0.639 0.331 0.962 0.540 0.676 0.394 0.980 0.518 0.704 0.480 0.990 0.498 0.719 0.431 1.000 0.494 0.734 0.464 1.010 0.471 0.749 0.473

Around this critical point a massive continuum limit can be constructed, but it will definitely not enjoy the famous asymptotic freedom. Over one year ago [13, 14], we used published numerical data to estimate gcr in QCD. The remarkable consequence of that analysis was that deviations from the perturbat ive running of as(Q 2) should become detectable at quite moderate energies-- less than 1 TeV. The latest L E P data show precisely this type of deviations and in fact an alternative explanation (existence of light gluinos) has already been offered [15,16]. Whichever may turn out to be the t rue explanation, it is certainly most important for the lattice community to make a deliberate effort to prove or disprove the get > 0 scenario.

AP would like to thank the Max-Planck-Institut ftir Physik (Werner-Heisenberg- Institut) for its hospitality.

R E F E R E N C E S

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1352 A. PATRASCIOIU and E. SEILER

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