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Toward the quantitative theory of the instanton liquid (I).
Nuclear Physics B302 (1988) 559-573 North-Holland, Amsterdam
TOWARD THE QUANTITATIVE THEORY OF THE I N S T A N T O N L I Q U I D (I). Phenomenology and the method of collective coordinates
E.V. SHURYAK Institute of Nuclear Physics, Novosibirsk 630090, USSR
Received 2 October 1987
This paper is the first in a series applying the method of collective coordinates to the description of an ensemble of interacting topological fluctuations, the so-called instanton liquid. We first present the results of the semiclassical theory of instantons and then proceed to a summary of known phenomenological facts, both coming from real experiments (for QCD) and from the lattice data. Then we describe the main ideas of the collective coordinate method.
1. Introduction N o n - a b e l i a n gauge fields are the main ingredient of the theories by means of which we h o p e to understand strong, weak and electromagnetic interactions, as well as their unification. Although asymptotic freedom allows rather simple calculations on the perturbative level, we are still far from an understanding of the nonperturbative p h e n o m e n a . T h e m o s t obvious source of information is, of course, the rich p h e n o m e n o l o g y available for Q C D . However, some time was needed in order to understand from which angle to look at the problem. Let me use a physical analogy which illustrates well what I mean by this. Consider some solid: its elementary excitations (such as p h o n o n s , electrons, holes, etc.) are analogous to elementary excitations of Q C D , the hadrons. One m a y know a lot about the spectroscopy of these excitations, measure their scattering cross sections etc., but it is impossible to understand what they are w i t h o u t realizing that solids are made of certain atoms arranged in a particular way, that their electrons are collectivized and form certain energy zones, etc. Quite similarly, it is impossible to understand what hadrons are without some insight into the structure of gauge and quark fields present in the Q C D vacuum. T h e most ambitious and straightforward approach towards this problem was suggested several years ago [1]; it involved computer experiments based on the 0550-3213/88/$03.50©Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
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lattice formulation of quantum gauge theories [2]. However, we turn to it somewhat later, and first summarize the very important conclusions reached in the studies of the QCD vacuum by means of the so-called QCD sum rules. It turns out that experimental data have shown rather nontrivial dependence of the behaviour of the correlation function on the quantum numbers of the probes used [3]. Moreover, for the spin-zero channels, it was found that the perturbative behaviour is violated already at very small distances, about 0.1 fm. These observations mean that the nonperturbative fields in the QCD vacuum are nontrivial and very inhomogeneous in space-time, possessing different intrinsic scales [4]. (If so, the lattice numerical experiments become even more difficult.) Large-scale computer experiments have been attempted during the last few years, and indeed a lot of interesting information on the vacuum properties have been obtained. However, the nonabelian gauge theories in four dimensions are in fact too difficult a problem, and it is nearly impossible to go beyond a certain accuracy level. Even for the simplest quantum mechanical examples (we consider one of them in detail in paper IV of this series), in order to obtain a really precise description of the ground-state wave function, etc., the number of lattice sites (per dimension) should be taken in the order of a few hundred, or even thousand. Nevertheless, any type of empirical information is but a basis for some deeper insight into the problem. As in all other branches of physics, it should be used for testing of some approximate (but simple and understandable) models of the phenomena. We have to develop some adequate concepts which can characterize the complicated distribution of the color and quark fields in the QCD vacuum, similar to the concepts of "atoms" and "crystals", explaining how the electromagnetic fields and electron wave functions are distributed in solids. We should also look for any small parameters in the problem, which usually lead to tremendous simplifications. Only if one simplifies the problem to the level at which it becomes treatable, one may try to "solve" it. The present series of papers is an attempt at this. Trying to explain our main idea, we consider the QED vacuum first. In zeroth order in the fine structure constant, the free electromagnetic field in vacuum is just a superposition of the small-amplitude zero-point oscillations of independent field modes, the gaussian noise as we will call it. For nonabelian theories, the high-virtuality harmonics (we use euclidean space-time) produce similar "noise", modulo small perturbative corrections. But such a description fails if one considers more slowly variable fields. We know that some fluctuations of the gauge field are qualitatively different from the small-amplitude "noise". In particular, we have learned about topologically nontrivial localized fluctuations of the gauge field called instantons (or pseudoparticle, or PPs). The phenomenological analysis of the data performed by the present author [4] has led to a very promising situation: a set of nontrivial numerical relations between phenomenological instanton parameters were found, suggesting a qualitative picture of the instanton liquid. The aim of this series of papers is to test
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whether these findings may or may not be explained from the microscopic theory of the instanton interactions. Comparing our approach to the lattice one from the more practical point of view, one may note that the former uses too many parameters to describe the fields, and most of them deal with the high-frequency perturbative "noise". Our aim is to select a minority of the most important coordinates from the majority of the noninteresting ones. Comparing with current lattice studies, one can see that the economy in the number of parameters may really be enormous: the number of parameters per fixed space-time volume is reduced by the factor 104-105. That is why, with a modest computer we are able to study phenomena which are definitely beyond the reach of modern lattice calculations. Of course, we should also warn the reader that it is a priori unclear whether the selected subset of configurations are really dominant for the true ensemble of vacuum fields. People are becoming somewhat convinced only if they see that such a model can explain some nontrivial observations. That is why we start with a summary of the phenomenological facts which we aim to explain in this series of works (a more detailed review can be found e.g. in ref. [5]). Then in this paper we turn to more methodical questions, related to introduction of the collective coordinates for the fields. It should probably be emphasized from the start that, generally speaking, it is by no means a new idea. For example, the local fields in magnetics are often described in terms of the domain wall positions; the deformation of solids can be described by prescribing positions and properties of dislocations; the flow velocity in liquids can be characterized by the positions of the vortex lines, etc. Our collective parameters describe properties of the complicated color-field configurations, being a kind of superposition of a number of instantons and anti-instantons. The present paper is an introductory one; we introduce notations and give the main references, and describe in condensed form the main findings related to instanton properties. We also show that the particular results of ref. [4] are now also supported by other observations, both coming from preliminary results of the variational approach [9] (" preliminary" means for the simplest trial functions taken at random) and new lattice studies. Also, we discuss the main points of the programme aiming to be a quantitative effective theory expressed in terms of collective coordinates. In paper II we consider the simplest nonabelian gauge t h e o r y - SU(2) gluodynamics. We compare various trial functions, derive the interaction law for the pseudoparticles, perform a numerical simulation of the resulting statistical mechanics of the instanton liquid, and end up with a set of physical results, ranging from the integral instanton density to rather delicate measurements of the correlation functions. The difficulty of the problem increases enormously when light quarks are involved, for they generale specific interaction between the PPs. Our studies of these
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phenomena are collected in paper III. The central question discussed in it is whether the instantons do or do not generate the so-called quark condensate, manifesting spontaneous chiral symmetry breaking, and our answer to this is definitely positive. We show, that in the presence of light quarks, the instanton liquid consists of essentially two components: the "polymer", creating the condensate all over the space, and the separate "molecules". Their relative role depends nontrivially on the number of light quark flavours. Paper IV is devoted to the much simpler "toy model", a quantum-mechanical motion in a double-well potential. In this case, it is possible to obtain both very accurate lattice data for the ensemble of the paths, and to extend our programme significantly further. The path containing tunnelling events moving back and forth can also be considered as a one-dimensional "liquid of kinks", and although in this case their mutual interaction is much less important, the studies turn out to be very instructive. We should not end this introduction with the impression that our instanton liquid explains all known features of the QCD vacuum. We claim that we understand the strongest colour fields causing deviations from asymptotic freedom behaviour of the correlation functions at small distances. We also claim that we quantitatively understand chiral symmetry breaking and related physics (e.g. properties of the pion). However, we have no explanation for colour confinement: in the instanton liquid, the quarks have simply a finite effective mass. Should this fact be so disturbing? I do not think so: it is too difficult to explain everything at once, and it is good enough if we have understood at least something. Let me again use the analogy to solid state physics here: the three decades following the development of quantum mechanics were extremely fruitful for it, in spite of the fact that e.g. the supercondutivity phenomenon had no explanation during these years.
2. Instanton density in a dilute gas approximation
The topologically nontrivial classical configurations of the nonabelian gauge fields, "pseudoparticles" (below PPs) or instantons and anti-instantons, were discovered in the classical paper [6] by Belavin, Polyakov, Schwartz and Tyupkin in 1975. It was immediately recognized that this discovery has opened up completely new perspectives in gauge field theory, showing that the barriers separating the topologically distinct sets of gauge fields, are penetrable. The instanton is the path connecting the topologically distinct sectors, possessing the minimal possible action. Therefore, it is a solution of the Yang-Mills equation. Throughout this work we use it in the so-called singular gauge A ~ ( x ) = 2gl~x~p2/[x2(x 2 +
p2)],
(1)
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where ~ , is the so-called 't Hooft symbol. (For anti-instantons, one should put 7/ instead of ~.) Our field normalization is such that the field strength and the gauge action are as follows a_
S=
a
abc b c
( 4g1)fd4 ~ ae 2 -
-
-
(2)
The semiclassical theory of the instantons was developed in the classical paper by 't Hooft [7]. If one considers strong fluctuations of this kind, possessing a strong field G2~ - 1 / 0 2 in the region of small size O, one may be sure that it happens rarely, and therefore it is possible to ignore all other such fluctuations and to consider this fluctuation placed in the "perturbative" vacuum. Such an approach, known as the dilute gas approximation (DGA), leads to the following expression for the PP density (instantons and anti-instantons together): dnpv
dzdo
/ 2C N \
[T'~)
) b'/2b
[/~(O)]2N' h'/26( b
× e x p [ - / ~ ( 0 ) + (2No-
b'/2b)(b'/2b)lnfl(p)/fl(p)](1 + O ( 1 / f l ) ) ,
(3)
where 4.66 exp( - 1.68N~)
CN, = ~ 2 ( U c - 1 ) ! ( N o - Z)! ' The particular definition of the lambda parameter here is made via the Pauli-Villars regularization method (for other definitions, one should change the coefficient C). Note also that in eq. (3) we have included the two-loop effects in the effective charge (as it was suggested in ref. [8], following ref. [9]). The constants b, b' here are the well-known coefficients of the Gell-Mann-Low function b = ~ N ~ - ~3Ns, b,= ~"N~- ~N~Nj + N / U c ,
(4)
with N,., N/ being the number of colors and light quark flavors, respectively. Expression (3) is valid in the semiclassical domain, where the action per pseudoparticle fl(p)>> 1. One may observe that, extrapolating this expression literally to small fl, it leads to a peak in the distribution over radii and then even vanishes at fl ~ 0. Unfortunately, this peak is in the region where one cannot trust expression (3). Therefore, the integral density (and other integral effects) cannot be estimated
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564
in the DGA. Multiple attempts to use similar expressions for the evaluation of some observable effects (e.g. see ref. [10]) have failed, and it was finally realized that in order to do so one has to go beyond this approximation and to face the question about the physical nature of the effects which may cut off the integral over 0 at large p. Logically speaking, there are two candidates: (i) mutual repulsive interaction of the PPs; (ii) "melting" of PPs due to large quantum fluctuations. Below, we are going to show that in fact the former effect dominates, stabilizing the instanton density at the point where quantuum corrections are still small and the semiclassical approach is still quite reasonable. 3. Phenomenology of the instanton liquid
Soon after the discovery of the instantons, it was realized that they may be responsible for such important effects as the solution of the Weinberg U(1) problem [7], the spontaneous breaking of the chiral SU(Nf) symmetry [10], etc. (In other words, it was suspected that it was the instanton-induced forces between quarks which make the 7/' meson so heavy and the pion so light.) Unfortunately, due to the shortcomings of the DGA, it was not possible to make any theoretical estimates of the magnitude of these effects; thus these suspicions have remained a plausible although unproved hypothesis. A new page in the development of instanton physics was opened with the development of the so-called QCD sum rules [11], connecting the correlation function of various operators with hadronic phenomenology. The value of the so-called gluonic condensate has set the scale for the nonperturbative fields. As a result, one had the following phenomenological upper limit on the PP density
N++N npp
(G 2) V - < nmax-- 32~"2 - (197 MeV) 4.
(5)
where N+, N_ are the number of instantons and anti-instantons in the (space-time) volume V. Another important number came from the physics of the ~/' meson; it is the value of the so-called topological succeptibility [12] (in fact, in a world without light quarks): Xtop =
liInoe( ( N + -
N _ ) 2 ) / V - - ( 1 8 0 - 1 9 0 MeV) 4,
(6)
Generally speaking, this quantity depends both on the PP density in vacuum and on the correlations of their positions in space-time. In particular, in the simplest case of the independent Poisson distribution over N+ and N , one can see that this quantity is just equal to the PP density because ( ( N + - N_) 2) --- ( N + + N _ ) .
(7)
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565
Thus it is tempting to ascribe two independent experimental numbers (5,6) to instantons, taking only one free parameter, the PP density, of the order of 1 fm 4. (It was noticed by Novikov et al. [3] that the dilute instanton gas with its Poisson distribution over N+, N_ cannot be the true picture, as it contradicts the low-energy theorem. Indeed, ascribing all nonperturbative fields to instantons, one finds from it that the fluctuations in total PP number should have statistics different from the Poisson one because this theorem leads (under these assumptions) to ( ( N + + N )2) + <(N++ N _ ) ) = 4<(N++ N_))
(s)
and not just to {(N++ N_)). However, eq. (8) follows from very general arguments related to the correct renorm-invariant dependence of the calculated vacuum energy, so it should hold in any self-consistent model. In particular, as was shown by Dyakonov and Petrov [9], eq. (8) does take place for their approximate theory of the instanton liquid.) The most important conclusion drawn by Novikov et al. [3] from the analysis of the QCD sum rules was the following one: In the spin-zero channels, data indicate the appearance of some nonperturbative corrections already, at a momentum transfer of the order of Q2 = 20 GeV 2. Moreover, it seemed impossible to describe them by usual operator-product-expansion formulae. It means that the QCD vacuum does contain strong nonperturbative fields! In a series of papers [4], I have attempted to fit all these data by some simple instanton-based model. It has just two free parameters, the PP density and the typical instanton radius ~. Not going into details, I just summarize that a lot of phenomenological facts were found to agree with their values = ½ fm,
npp = 1 fm -4 .
(9)
If so, one may make four important observations: (i) The vacuum is reasonably dilute: nppp 4 << 1.
(10)
(ii) Instantons are nearly classical: fl(~)
- 1 0 >> 1 .
(11)
(iii) Interaction does not spoil instantons Aft int- 2 << f i ( p ) .
(12)
(iv) Interaction is not negligible, so we have some liquid rather than a dilute gas: explA/~int I 2>> 1.
(13)
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E. V. Shuryak / Instanton liquid (I)
It is clear that this instanton liquid picture of the QCD vacuum, being a phenomenological observation, cannot shed any light on the gauge theories other than QCD. However, later studies made by Dyakonov and Petrov [9] (their variational approach is discussed below) have lead to essentially the same vacuum properties for quarkless gauge theories as well. 4. Instantons on the lattice
The question of whether the ensemble of the gauge fields generated on the lattice does or does not contain an essential fraction of the topologically nontrivial fluctuations was raised from the first days of these studies, but it turns out not so easy to answer it. Only rather recent studies have resulted in more or less definite conclusions on this issue. We first say some words on the problems and then come to the results. Generally speaking, any studies of the topological effects on the lattice is a rather tricky business because in this case, the cornerstone of the topology is lost: all discretized configurations can be connected by a continuous transformation. As a result, all definitions of the topological charge on the lattice are conventions, counting only the fluctuations of a size much larger than the lattice spacing. (We are going to discuss this point in IV in detail, using as a toy model the quantummechanical double-well system.) As phenomenology suggests the typical instanton size in QCD to be around 0.3 fro, while the typical lattice spacing used is about 0.1-0.2 fro, such a cut-off may produce significant systematical errors. Another problem (common to all lattice studies) is related to matching of the units used. In principle, there is no question about the relation between the lattice parameter A L (defined in terms of the lattice spacing and the bare lattice coupling g) and A PV used throughout this work. In particular, for the simplest SU(2) gauge theory without fermions (for which most of the studies are made) the calculations of various physical effects can be compared for both regularizations, with the wellknown result Apv =
21.5A L .
(14)
However, it is not quite clear whether the bare lattice coupling g is indeed small enough, so that all diagrams except the two-loop ones (used in the derivation of (14)) can indeed be neglected. The lattice data themselves demonstrate, that the so-called asymptotic scaling is not yet very accurate. This means that so far the observables do depend not only on A L (as they asymptotically should do), but on other details of the numerical experiments as well. Therefore, one should use exact expressions like (14), whilst keeping in mind that some systematic corrections to them are possible. As was mentioned above, the upper bound on the PP density follows from the value of the nonperturbative gluonic condensate. For the SU(2) theory the best
E.V. Shuryak / lnstanton liquid (I)
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statistical accuracy was reported by Ishikawa et al. [13]:
( ( a ;a, ) )2 - - 8 × 108A
(15)
which leads to npp
< /'/max ~'~ 1.3
× 106A4L .
(16)
However, the lattice definition of the nonperturbative fields is based on the subtraction of large perturbative ones, and the systematic errors due to them are much larger than the statistical errors. Nevertheless, one may hope that (15) holds at least up to a factor of two. First attempts to measure the net topological charge Q of the gauge field configurations in the box [14] have lead to surprisingly small values of the topological susceptibilty Xtop ~-~ 2
× 103A4 .
(17)
However, later other definitions of the topological charge and essentially larger lattices have lead to much larger X top" In particular, the definition suggested by Woit was used in the experiment made by the Princeton group [15] with the result: Xtop -- 105A4 -
(18)
Using another definition of Q due to Liischer, the DESY group [16] has found a somewhat larger value Xtop •
2.6 x 105A4 .
(19)
Both results (18), (19) demonstrate reasonably accurate scaling, so their discrepancy in absolute magnitude is presumably due to a different cut-off implied by these two definitions of Q. Thus, it is a measure of the systematic errors involved, showing that we know the order of magnitude of Xtop, at best. A new chapter in the hunt for instantons was connected to the application of the cooling method [17]. Starting with a configuration from the ensemble, the programme minimizes the action till some minimum is reached. It may be either the trivial one (zero field), or that possessing the nonzero topological charge. These minima have passed a number of tests, showing that the objects observed are indeed instantons: they do have the needed action, exactly one-fermion zero mode, etc. Thus, now we have also some clear lower bound for the PP density: npp> nmin- 2 × 104A4 .
(20)
It is still much lower than the upper bound (16), but it clearly shows that the early estimates (17) were wrong.
E.V. Shuryak / Instanton liquid (I)
568
Moreover, recent data reported by the ITEP group [18] show clear evidence that the surviving instantons do have radii of about 1 of the interparticle spacing R, and even strong evidence for the PP repulsion, from the distribution over R. Of course, one may well criticize this method too. Say, these instantons are not really the action minima if the standard Wilson action is used; therefore they have finite lifetimes. Also, one may argue that close instanton-anti-instanton (and even instanton-instanton) pairs most probably annihilate during the minimization process, so the picture seen after "cooling" is strongly distorted. Summarizing this section, we may say that although the real accuracy of the particular numbers mentioned is not yet good enough (and even this is not yet well understood), steady progress is observed. The numbers for the topological susceptibility have changed by two orders of magnitude and now are definitely convergent. It was shown, that what is seen is, at least sometimes, the true instantons, possessing the continuous limit, and not just some topologically nontrivial lattice defects. What is the most important: due to these works the lattice-oriented people are becoming convinced, that the topological phenomena are indeed the important ingredients of the gauge field vacuum, providing a significant fraction (or maybe even the main part) of the nonperturbative fields in gauge vacuum.
5. The general idea of the collective coordinate method and the variational approach The main technical problems of any quantum field theory deal with evaluation of some statistical sum Z over all configurations of the system. Generally speaking, the collective coordinate method is just a selection of some subset of interesting variables a, (i = 1, N ) out of the (infinite) set of the remaining (noninteresting) ones. The standard formal trick used is the incorporation of the Faddeev-Popov unity factor into the statistical sum: 1=
fJ(A,a)
i=1
[8(l,(A,a))dai].
(21)
Here Ii(A(x ), ai) are N conditions and J(A(x), ai) is the corresponding jacobian. Taking the integration over a i apart and performing first integration over all other variables, one obtains the so-called effective theory expressed in terms of the collective variables ai: N Zeff=
fexp(- S~,~(a))i=1 1-I da,, N
exp( - S~ff( a ) ) =
fDA
exp( - S [ A ( x ) ]
}J(A, a) 1-I 8(T~(A, a)). i=1
(22)
E.V. Shuryak / Instanton liquid (I)
569
Applications of this method are well known in the simplest case, in which the definition of all collective variables is governed by some exact symmetry. For example, if one has found one single-instanton solution, one may well construct the whole set of them using translational, rotational and dilatational invariance of the classical Yang-Mills theory. (For the SU(2) colour group it leads to some " p l a n e " in the configuration space, parametrized by eight parameters.) For all directions (again, in the configuration space) transverse to this plane the action increases, thus the integrand in the statistical sum is peaked at this plane, which is the basis of 't Hooft's semiclassical theory. The eight-dimensional integral over this plane is left intact: the action is constant on it, so the sum is proportional to its volume. This lesson should teach us, that the basis for any similar calculation should be some N-dimensional manifold, to be called the ansatz plane below, which is able to absorb the non-gaussian part of the functional integral. However, generally speaking it is not necessary to hold the action constant on it. What we actually want to do is to integrate safely over the transverse coordinates, with integration over the collective coordinates remaining for separate investigation. Therefore, we have first to face the question whether the integrand is indeed peaked in the transverse directions. In the present series of papers we consider superpositions of instantons and anti-instantons, and these configurations do not possess any exact symmetries: the action in general depends nontrivially on all coordinates• In this case there are no classical configurations available: for any of them the first variation of the action, the current, is nonzero
~S j;(x) = 8A~
A=A . . . . .
=
(D~ G.~,) A=A...... =~0. a6
b
(23)
Therefore, writing the field potential as the sum of the ansatz and the fluctuating part
A~(x)
= A,~atZ(x) +
b,~(x),
(24)
one generally has the so-called linear term in the action expansion,
S(A)
=
S(A ansatz) + f dxj~(x)b~.(x) + "'"
(25)
which shifts the mean values of A away from our ansatz. D y a k o n o v and Petrov [9] have made quite a radical step, omitting this linear term (25). The physical idea behind it can be explained as follows. Suppose one has to add some external current Jo,~(x) .ext which exactly compensates the linear term, forcing the integrand to have its maximum just at our ansatz. However, this external current creates some work on the system, therefore its energy becomes greater than
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E.V. Shuryak / Instanton liquid(I)
that in the unperturbed vacuum:
E(jext) ~>E(jext=0) :Evac.
(26)
This inequality (which they have rigorously proved using the Feynman variational principle) is the essence of their work: they have suggested the variational approach to the problem. Indeed, taking a better ansatz, one finds a smaller current and, correspondingly, lower e(jext). However, this interesting work has left a lot of questions open. What ansatz is the best possible one, and how to find it? Is it possible, even in principle, to get rid of the current completely, so that the estimated quantity e ( j ext) is arbitrarily close to evac? 6. Which ansatz is the best?
Imagine the action distribution in the configuration space for a set of instantons and anti-instantons. The resulting landscape is similar to that in some mountain country, it possesses a series of peaks (extremely unprobable configurations) separated by a complicated system of deep valleys. If the PPs are well separated and nearly noninteracting, the action on the valley bottoms is nearly constant. If they approach each other, the slopes of the valleys increases, but it still may be much smaller on the bottoms than on the walls. Such a structure suggests the idea of natural separation of the longitudinal coordinates along the valleys from the transverse ones mentioned above. Indeed, everybody who once was in the mountains knows that there exist some outstanding line in each valley: it is the streamline on the bottom of the valley. Formally speaking, it is the line which is the minimum in the transverse directions. Another property: it is the line at which direction of the driving force (the first derivative) is tangent to it. In the gauge theory this driving force is in fact the current j. It can be decomposed into two parts, the longitudinal and the transverse one, according to the chosen ansatz plane. Thus, the best ansatz is that which has the properties of the streamline; for all points on this plane the current should be tangentential to it, in other words it should have
j~(x) = 0 .
(27)
This is the answer to the question posed at the end of the preceding section. Indeed, the current is nonzero, but its longitudinal part is not dangerous for us, as the integration over the collective variables is presumably made accurately. If the transverse part is absent, the linear term in (25) is zero and the integration over the transverse plane (leading to our effective action) can be made without any external currents etc.
E.V. Shuryak / Instanton liquid(l)
571
Unfortunately, even for the simplest toy models it is not easy to find such streamline configurations analytically (see e.g. ref [19]). There exists a simple numerical method suggested by myself [20] (see its discussion and application in paper IV of this series). The idea is quite simple: one should just start with two well-separated pseudoparticles and follow the direction of the driving force, which leads along the streamline downstream. However, for such a complicated system as the four-dimensional gauge field, the streamline configurations are not yet explicitly found and, as a first step, in the subsequent papers we approach this problem in a variational way, considering various trial functions which approximately describe this streamline. The quantity which we are going to minimize is now clear, it is j ~ . Thus, comparing various trial functions, we calculate the current and make (so far rather crude) estimates of the corrections induced by it.
7. The semielassical effective theory In this section we are going to outline the effective theory which corresponds to a gaussian (semiclassical) approximation. This is possible if the field fluctuates around our ansatz plane, and the quantum fluctuations, are not too strong. All steps are in fact quite standard. First, one has to expand the action up to the second order in the fluctuating field
S ( A ) = S ( A ansatz) q-
fdxjX'(x)G~(x) (28)
+ ½fba,(X)D a~' b"(x, Y)bb.(y ) dx dy + " .
and then integrate over the transverse coordinates in the gaussian approximation. If it is possible, the symbolic result for the effective action can be written as follows exp( - Sefr ) =
fDb± ( x ) e - s = exp( - s d
....
S current -
S quantum -
S jac°bian
) (29)
(where f Db± ... means f Db. J" I~N=lS(li)...). Let us discuss four terms of the resulting effective action subsequently. The first "classical" one is just the action distribution in the ansatz plane sclass(a) = S(Aansatz(a)).
(30)
Obviously it is the simplest one, and it is quite easy to deal with it, provided the ansatz is defined. However, it is the main part of the effective action only if its
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E. V. Shuryak / Instanton liquid (1)
current part S c~r~m =
½fj~(x)rn2,b~(X, y)jff~(y)dxdy
(31)
is reasonably small compared to it. Note, that this term is formally of the same magnitude as the classical term (in perturbative language it corresponds to the zero-loop order). The sum of (30), (31) is ansatz independent, and the relative role of each depends on how close the ansatz used is to the streamline. The third, quantum term in this action squantum(a) = a l n d e t ( n ±)
(32)
contains the functional determinant of the differential operator [] ±. Such a determinant was explicitly found for the instanton solution by 't Hooft [7], which can be used as a reference point for estimates of its magnitude. Finally, there is the fourth structure in the effective action, related to the transverse projectors and the corresponding jacobian sJ"c°bi~(a) = - l n J ( A a~satz, a ) .
(33)
The most natural way to write our conditions I, is to project them transversely. This means that the fluctuating part of the field is orthogonal to all tangent vectors Oa~ of the ansatz plane
ba~(X)
(i) I, = f dxba~(x) V,,~,(X).
(34)
For the nongauge systems, these vectors can be obtained by a differentiation of the ansatz, but for the gauge system, one has also to take care about the projector Pgauge, which guarantees that b is not just a pure gauge transformation
4
(x, a).
(x) =
In particular, it is most convenient to take both background gauge condition O~abK(x) ~b~ --- 0 ,
ba~(X )
~Flabl~(i)( V b ~ a"~j = 0 .
(35) and
V~a~(X)satisfying the (36)
E.V. Shuryak / Instanton liquid (I)
573
The jacobian in general can be written as
J=det
(fdxva~((i)x)PgaugeValz((i)x))1/2
•
(37)
In the semiclassical context it should be taken at the ansatz plane. Again, as for the quantum part of the effective action, in the first approximation it may be taken from the known expressions for the individual PPs. References [1] [2] [3] [4] [5] [6]
M. Creutz, Phys. Lett. 45B (1980) 313 K.G. Wilson, Phys. Rev. D10 (1974) 2445 V.A. Novikov, M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Nucl. Phys. B191 (1981) 301 E.V. Shuryak, Nucl. Phys. B203 (1982) 93; B214 (1983) 237 E.V. Shuryak, Phys. Reports Cl15 (1984) 151 A.M. Polyakov, Phys. Lett. 59B (1975) 82; A.A. Belavin, A.M. Polyakov, A.A. Schwartz and Yu.S. Tyupkin, Phys. Lett. 59B (1975) 85 [7] G. 't Hooft, Phys. Rev. 14D (1976) 3432, (E) 18D (1978) 2199; C. Bernard, Phys. Rev. D19 (1979) 3013 [8] V.A. Novikov, M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Usp. Fiz. Nauk (Sov. Phys. Usp. 136 (1982) 553) [9] D.I. Dyakonov and V.Yu. Petrov, Nucl. Phys. B245 (1984) 259 [10] D.G. Caldi, Phys. Rev. Lett. 39 (1977) 121; C.G. Callan, R. Dashen and D.J. Gross, Phys. Rev. D17 (1978) 2717, D19 (1979) 1826; R.D. Carlitz and D.B. Creamer, Ann. Phys. 116 (1979) 429; C.G. Callan, R. Dashen and D.J. Gross, Phys. Rev. D20 (1979) 3279 [11] M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Nucl. Phys. B163 (1980) 43, B165 (165) 45; Phys. Lett. 76b (1978) 971 [12] G. Veneziano, Nucl. Phys. B159 (1979) 213; E. Witten, Nucl. Phys. B156 (1979) 269 [13] K. Ishikawa, G. Shierholtz, H. Schneider and M. Teper, Nucl. Phys. B227 (1983) 221 [14] P. di Vecchia, K. Fabricius, C.G. Rossi and G. Veneziano, Nucl. Phys. B192 (1981) 392; Phys. Lett. 108B (1982) 323 [15] P. Woit, Phys. Rev. Lett. 51 (1983) 638; G. Bhanot et al. Nucl. Phys. B230 [FS10] (1984) 291 [16] I.A. Fox, J.P Gilchrist, M.L. Laursen and G. Shierholtz, Phys. Rev. Lett. 54 (1984) 749 [17] B. Berg, Phys. Lett. 104B (1981) 475; Y. Iwasaki and T. Yoshie, Phys. Lett. 131B (1983) 159; 131B (1984) 73; 143B (1984) 449; E.-M. Ilgenfritz et al., Nucl. Phys. B268 (1986) 693 [18] M.I. Polikarpov and A.I. Veselov, Moscow ITEP preprint 41 (1987) [19] I.I. Balitsky and A.V. Yung, Phys. Lett. 168B (1986) 113 [20] E.V. Shuryak, in Proc. Conference on numerical experiments in quantum field theories, Alma-Ata 1985, eds. A.A. Migdal (in Russian)
TOWARD THE QUANTITATIVE THEORY OF THE I N S T A N T O N L I Q U I D (I). Phenomenology and the method of collective coordinates
E.V. SHURYAK Institute of Nuclear Physics, Novosibirsk 630090, USSR
Received 2 October 1987
This paper is the first in a series applying the method of collective coordinates to the description of an ensemble of interacting topological fluctuations, the so-called instanton liquid. We first present the results of the semiclassical theory of instantons and then proceed to a summary of known phenomenological facts, both coming from real experiments (for QCD) and from the lattice data. Then we describe the main ideas of the collective coordinate method.
1. Introduction N o n - a b e l i a n gauge fields are the main ingredient of the theories by means of which we h o p e to understand strong, weak and electromagnetic interactions, as well as their unification. Although asymptotic freedom allows rather simple calculations on the perturbative level, we are still far from an understanding of the nonperturbative p h e n o m e n a . T h e m o s t obvious source of information is, of course, the rich p h e n o m e n o l o g y available for Q C D . However, some time was needed in order to understand from which angle to look at the problem. Let me use a physical analogy which illustrates well what I mean by this. Consider some solid: its elementary excitations (such as p h o n o n s , electrons, holes, etc.) are analogous to elementary excitations of Q C D , the hadrons. One m a y know a lot about the spectroscopy of these excitations, measure their scattering cross sections etc., but it is impossible to understand what they are w i t h o u t realizing that solids are made of certain atoms arranged in a particular way, that their electrons are collectivized and form certain energy zones, etc. Quite similarly, it is impossible to understand what hadrons are without some insight into the structure of gauge and quark fields present in the Q C D vacuum. T h e most ambitious and straightforward approach towards this problem was suggested several years ago [1]; it involved computer experiments based on the 0550-3213/88/$03.50©Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
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lattice formulation of quantum gauge theories [2]. However, we turn to it somewhat later, and first summarize the very important conclusions reached in the studies of the QCD vacuum by means of the so-called QCD sum rules. It turns out that experimental data have shown rather nontrivial dependence of the behaviour of the correlation function on the quantum numbers of the probes used [3]. Moreover, for the spin-zero channels, it was found that the perturbative behaviour is violated already at very small distances, about 0.1 fm. These observations mean that the nonperturbative fields in the QCD vacuum are nontrivial and very inhomogeneous in space-time, possessing different intrinsic scales [4]. (If so, the lattice numerical experiments become even more difficult.) Large-scale computer experiments have been attempted during the last few years, and indeed a lot of interesting information on the vacuum properties have been obtained. However, the nonabelian gauge theories in four dimensions are in fact too difficult a problem, and it is nearly impossible to go beyond a certain accuracy level. Even for the simplest quantum mechanical examples (we consider one of them in detail in paper IV of this series), in order to obtain a really precise description of the ground-state wave function, etc., the number of lattice sites (per dimension) should be taken in the order of a few hundred, or even thousand. Nevertheless, any type of empirical information is but a basis for some deeper insight into the problem. As in all other branches of physics, it should be used for testing of some approximate (but simple and understandable) models of the phenomena. We have to develop some adequate concepts which can characterize the complicated distribution of the color and quark fields in the QCD vacuum, similar to the concepts of "atoms" and "crystals", explaining how the electromagnetic fields and electron wave functions are distributed in solids. We should also look for any small parameters in the problem, which usually lead to tremendous simplifications. Only if one simplifies the problem to the level at which it becomes treatable, one may try to "solve" it. The present series of papers is an attempt at this. Trying to explain our main idea, we consider the QED vacuum first. In zeroth order in the fine structure constant, the free electromagnetic field in vacuum is just a superposition of the small-amplitude zero-point oscillations of independent field modes, the gaussian noise as we will call it. For nonabelian theories, the high-virtuality harmonics (we use euclidean space-time) produce similar "noise", modulo small perturbative corrections. But such a description fails if one considers more slowly variable fields. We know that some fluctuations of the gauge field are qualitatively different from the small-amplitude "noise". In particular, we have learned about topologically nontrivial localized fluctuations of the gauge field called instantons (or pseudoparticle, or PPs). The phenomenological analysis of the data performed by the present author [4] has led to a very promising situation: a set of nontrivial numerical relations between phenomenological instanton parameters were found, suggesting a qualitative picture of the instanton liquid. The aim of this series of papers is to test
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whether these findings may or may not be explained from the microscopic theory of the instanton interactions. Comparing our approach to the lattice one from the more practical point of view, one may note that the former uses too many parameters to describe the fields, and most of them deal with the high-frequency perturbative "noise". Our aim is to select a minority of the most important coordinates from the majority of the noninteresting ones. Comparing with current lattice studies, one can see that the economy in the number of parameters may really be enormous: the number of parameters per fixed space-time volume is reduced by the factor 104-105. That is why, with a modest computer we are able to study phenomena which are definitely beyond the reach of modern lattice calculations. Of course, we should also warn the reader that it is a priori unclear whether the selected subset of configurations are really dominant for the true ensemble of vacuum fields. People are becoming somewhat convinced only if they see that such a model can explain some nontrivial observations. That is why we start with a summary of the phenomenological facts which we aim to explain in this series of works (a more detailed review can be found e.g. in ref. [5]). Then in this paper we turn to more methodical questions, related to introduction of the collective coordinates for the fields. It should probably be emphasized from the start that, generally speaking, it is by no means a new idea. For example, the local fields in magnetics are often described in terms of the domain wall positions; the deformation of solids can be described by prescribing positions and properties of dislocations; the flow velocity in liquids can be characterized by the positions of the vortex lines, etc. Our collective parameters describe properties of the complicated color-field configurations, being a kind of superposition of a number of instantons and anti-instantons. The present paper is an introductory one; we introduce notations and give the main references, and describe in condensed form the main findings related to instanton properties. We also show that the particular results of ref. [4] are now also supported by other observations, both coming from preliminary results of the variational approach [9] (" preliminary" means for the simplest trial functions taken at random) and new lattice studies. Also, we discuss the main points of the programme aiming to be a quantitative effective theory expressed in terms of collective coordinates. In paper II we consider the simplest nonabelian gauge t h e o r y - SU(2) gluodynamics. We compare various trial functions, derive the interaction law for the pseudoparticles, perform a numerical simulation of the resulting statistical mechanics of the instanton liquid, and end up with a set of physical results, ranging from the integral instanton density to rather delicate measurements of the correlation functions. The difficulty of the problem increases enormously when light quarks are involved, for they generale specific interaction between the PPs. Our studies of these
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phenomena are collected in paper III. The central question discussed in it is whether the instantons do or do not generate the so-called quark condensate, manifesting spontaneous chiral symmetry breaking, and our answer to this is definitely positive. We show, that in the presence of light quarks, the instanton liquid consists of essentially two components: the "polymer", creating the condensate all over the space, and the separate "molecules". Their relative role depends nontrivially on the number of light quark flavours. Paper IV is devoted to the much simpler "toy model", a quantum-mechanical motion in a double-well potential. In this case, it is possible to obtain both very accurate lattice data for the ensemble of the paths, and to extend our programme significantly further. The path containing tunnelling events moving back and forth can also be considered as a one-dimensional "liquid of kinks", and although in this case their mutual interaction is much less important, the studies turn out to be very instructive. We should not end this introduction with the impression that our instanton liquid explains all known features of the QCD vacuum. We claim that we understand the strongest colour fields causing deviations from asymptotic freedom behaviour of the correlation functions at small distances. We also claim that we quantitatively understand chiral symmetry breaking and related physics (e.g. properties of the pion). However, we have no explanation for colour confinement: in the instanton liquid, the quarks have simply a finite effective mass. Should this fact be so disturbing? I do not think so: it is too difficult to explain everything at once, and it is good enough if we have understood at least something. Let me again use the analogy to solid state physics here: the three decades following the development of quantum mechanics were extremely fruitful for it, in spite of the fact that e.g. the supercondutivity phenomenon had no explanation during these years.
2. Instanton density in a dilute gas approximation
The topologically nontrivial classical configurations of the nonabelian gauge fields, "pseudoparticles" (below PPs) or instantons and anti-instantons, were discovered in the classical paper [6] by Belavin, Polyakov, Schwartz and Tyupkin in 1975. It was immediately recognized that this discovery has opened up completely new perspectives in gauge field theory, showing that the barriers separating the topologically distinct sets of gauge fields, are penetrable. The instanton is the path connecting the topologically distinct sectors, possessing the minimal possible action. Therefore, it is a solution of the Yang-Mills equation. Throughout this work we use it in the so-called singular gauge A ~ ( x ) = 2gl~x~p2/[x2(x 2 +
p2)],
(1)
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where ~ , is the so-called 't Hooft symbol. (For anti-instantons, one should put 7/ instead of ~.) Our field normalization is such that the field strength and the gauge action are as follows a_
S=
a
abc b c
( 4g1)fd4 ~ ae 2 -
-
-
(2)
The semiclassical theory of the instantons was developed in the classical paper by 't Hooft [7]. If one considers strong fluctuations of this kind, possessing a strong field G2~ - 1 / 0 2 in the region of small size O, one may be sure that it happens rarely, and therefore it is possible to ignore all other such fluctuations and to consider this fluctuation placed in the "perturbative" vacuum. Such an approach, known as the dilute gas approximation (DGA), leads to the following expression for the PP density (instantons and anti-instantons together): dnpv
dzdo
/ 2C N \
[T'~)
) b'/2b
[/~(O)]2N' h'/26( b
× e x p [ - / ~ ( 0 ) + (2No-
b'/2b)(b'/2b)lnfl(p)/fl(p)](1 + O ( 1 / f l ) ) ,
(3)
where 4.66 exp( - 1.68N~)
CN, = ~ 2 ( U c - 1 ) ! ( N o - Z)! ' The particular definition of the lambda parameter here is made via the Pauli-Villars regularization method (for other definitions, one should change the coefficient C). Note also that in eq. (3) we have included the two-loop effects in the effective charge (as it was suggested in ref. [8], following ref. [9]). The constants b, b' here are the well-known coefficients of the Gell-Mann-Low function b = ~ N ~ - ~3Ns, b,= ~"N~- ~N~Nj + N / U c ,
(4)
with N,., N/ being the number of colors and light quark flavors, respectively. Expression (3) is valid in the semiclassical domain, where the action per pseudoparticle fl(p)>> 1. One may observe that, extrapolating this expression literally to small fl, it leads to a peak in the distribution over radii and then even vanishes at fl ~ 0. Unfortunately, this peak is in the region where one cannot trust expression (3). Therefore, the integral density (and other integral effects) cannot be estimated
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in the DGA. Multiple attempts to use similar expressions for the evaluation of some observable effects (e.g. see ref. [10]) have failed, and it was finally realized that in order to do so one has to go beyond this approximation and to face the question about the physical nature of the effects which may cut off the integral over 0 at large p. Logically speaking, there are two candidates: (i) mutual repulsive interaction of the PPs; (ii) "melting" of PPs due to large quantum fluctuations. Below, we are going to show that in fact the former effect dominates, stabilizing the instanton density at the point where quantuum corrections are still small and the semiclassical approach is still quite reasonable. 3. Phenomenology of the instanton liquid
Soon after the discovery of the instantons, it was realized that they may be responsible for such important effects as the solution of the Weinberg U(1) problem [7], the spontaneous breaking of the chiral SU(Nf) symmetry [10], etc. (In other words, it was suspected that it was the instanton-induced forces between quarks which make the 7/' meson so heavy and the pion so light.) Unfortunately, due to the shortcomings of the DGA, it was not possible to make any theoretical estimates of the magnitude of these effects; thus these suspicions have remained a plausible although unproved hypothesis. A new page in the development of instanton physics was opened with the development of the so-called QCD sum rules [11], connecting the correlation function of various operators with hadronic phenomenology. The value of the so-called gluonic condensate has set the scale for the nonperturbative fields. As a result, one had the following phenomenological upper limit on the PP density
N++N npp
(G 2) V - < nmax-- 32~"2 - (197 MeV) 4.
(5)
where N+, N_ are the number of instantons and anti-instantons in the (space-time) volume V. Another important number came from the physics of the ~/' meson; it is the value of the so-called topological succeptibility [12] (in fact, in a world without light quarks): Xtop =
liInoe( ( N + -
N _ ) 2 ) / V - - ( 1 8 0 - 1 9 0 MeV) 4,
(6)
Generally speaking, this quantity depends both on the PP density in vacuum and on the correlations of their positions in space-time. In particular, in the simplest case of the independent Poisson distribution over N+ and N , one can see that this quantity is just equal to the PP density because ( ( N + - N_) 2) --- ( N + + N _ ) .
(7)
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Thus it is tempting to ascribe two independent experimental numbers (5,6) to instantons, taking only one free parameter, the PP density, of the order of 1 fm 4. (It was noticed by Novikov et al. [3] that the dilute instanton gas with its Poisson distribution over N+, N_ cannot be the true picture, as it contradicts the low-energy theorem. Indeed, ascribing all nonperturbative fields to instantons, one finds from it that the fluctuations in total PP number should have statistics different from the Poisson one because this theorem leads (under these assumptions) to ( ( N + + N )2) + <(N++ N _ ) ) = 4<(N++ N_))
(s)
and not just to {(N++ N_)). However, eq. (8) follows from very general arguments related to the correct renorm-invariant dependence of the calculated vacuum energy, so it should hold in any self-consistent model. In particular, as was shown by Dyakonov and Petrov [9], eq. (8) does take place for their approximate theory of the instanton liquid.) The most important conclusion drawn by Novikov et al. [3] from the analysis of the QCD sum rules was the following one: In the spin-zero channels, data indicate the appearance of some nonperturbative corrections already, at a momentum transfer of the order of Q2 = 20 GeV 2. Moreover, it seemed impossible to describe them by usual operator-product-expansion formulae. It means that the QCD vacuum does contain strong nonperturbative fields! In a series of papers [4], I have attempted to fit all these data by some simple instanton-based model. It has just two free parameters, the PP density and the typical instanton radius ~. Not going into details, I just summarize that a lot of phenomenological facts were found to agree with their values = ½ fm,
npp = 1 fm -4 .
(9)
If so, one may make four important observations: (i) The vacuum is reasonably dilute: nppp 4 << 1.
(10)
(ii) Instantons are nearly classical: fl(~)
- 1 0 >> 1 .
(11)
(iii) Interaction does not spoil instantons Aft int- 2 << f i ( p ) .
(12)
(iv) Interaction is not negligible, so we have some liquid rather than a dilute gas: explA/~int I 2>> 1.
(13)
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It is clear that this instanton liquid picture of the QCD vacuum, being a phenomenological observation, cannot shed any light on the gauge theories other than QCD. However, later studies made by Dyakonov and Petrov [9] (their variational approach is discussed below) have lead to essentially the same vacuum properties for quarkless gauge theories as well. 4. Instantons on the lattice
The question of whether the ensemble of the gauge fields generated on the lattice does or does not contain an essential fraction of the topologically nontrivial fluctuations was raised from the first days of these studies, but it turns out not so easy to answer it. Only rather recent studies have resulted in more or less definite conclusions on this issue. We first say some words on the problems and then come to the results. Generally speaking, any studies of the topological effects on the lattice is a rather tricky business because in this case, the cornerstone of the topology is lost: all discretized configurations can be connected by a continuous transformation. As a result, all definitions of the topological charge on the lattice are conventions, counting only the fluctuations of a size much larger than the lattice spacing. (We are going to discuss this point in IV in detail, using as a toy model the quantummechanical double-well system.) As phenomenology suggests the typical instanton size in QCD to be around 0.3 fro, while the typical lattice spacing used is about 0.1-0.2 fro, such a cut-off may produce significant systematical errors. Another problem (common to all lattice studies) is related to matching of the units used. In principle, there is no question about the relation between the lattice parameter A L (defined in terms of the lattice spacing and the bare lattice coupling g) and A PV used throughout this work. In particular, for the simplest SU(2) gauge theory without fermions (for which most of the studies are made) the calculations of various physical effects can be compared for both regularizations, with the wellknown result Apv =
21.5A L .
(14)
However, it is not quite clear whether the bare lattice coupling g is indeed small enough, so that all diagrams except the two-loop ones (used in the derivation of (14)) can indeed be neglected. The lattice data themselves demonstrate, that the so-called asymptotic scaling is not yet very accurate. This means that so far the observables do depend not only on A L (as they asymptotically should do), but on other details of the numerical experiments as well. Therefore, one should use exact expressions like (14), whilst keeping in mind that some systematic corrections to them are possible. As was mentioned above, the upper bound on the PP density follows from the value of the nonperturbative gluonic condensate. For the SU(2) theory the best
E.V. Shuryak / lnstanton liquid (I)
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statistical accuracy was reported by Ishikawa et al. [13]:
( ( a ;a, ) )2 - - 8 × 108A
(15)
which leads to npp
< /'/max ~'~ 1.3
× 106A4L .
(16)
However, the lattice definition of the nonperturbative fields is based on the subtraction of large perturbative ones, and the systematic errors due to them are much larger than the statistical errors. Nevertheless, one may hope that (15) holds at least up to a factor of two. First attempts to measure the net topological charge Q of the gauge field configurations in the box [14] have lead to surprisingly small values of the topological susceptibilty Xtop ~-~ 2
× 103A4 .
(17)
However, later other definitions of the topological charge and essentially larger lattices have lead to much larger X top" In particular, the definition suggested by Woit was used in the experiment made by the Princeton group [15] with the result: Xtop -- 105A4 -
(18)
Using another definition of Q due to Liischer, the DESY group [16] has found a somewhat larger value Xtop •
2.6 x 105A4 .
(19)
Both results (18), (19) demonstrate reasonably accurate scaling, so their discrepancy in absolute magnitude is presumably due to a different cut-off implied by these two definitions of Q. Thus, it is a measure of the systematic errors involved, showing that we know the order of magnitude of Xtop, at best. A new chapter in the hunt for instantons was connected to the application of the cooling method [17]. Starting with a configuration from the ensemble, the programme minimizes the action till some minimum is reached. It may be either the trivial one (zero field), or that possessing the nonzero topological charge. These minima have passed a number of tests, showing that the objects observed are indeed instantons: they do have the needed action, exactly one-fermion zero mode, etc. Thus, now we have also some clear lower bound for the PP density: npp> nmin- 2 × 104A4 .
(20)
It is still much lower than the upper bound (16), but it clearly shows that the early estimates (17) were wrong.
E.V. Shuryak / Instanton liquid (I)
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Moreover, recent data reported by the ITEP group [18] show clear evidence that the surviving instantons do have radii of about 1 of the interparticle spacing R, and even strong evidence for the PP repulsion, from the distribution over R. Of course, one may well criticize this method too. Say, these instantons are not really the action minima if the standard Wilson action is used; therefore they have finite lifetimes. Also, one may argue that close instanton-anti-instanton (and even instanton-instanton) pairs most probably annihilate during the minimization process, so the picture seen after "cooling" is strongly distorted. Summarizing this section, we may say that although the real accuracy of the particular numbers mentioned is not yet good enough (and even this is not yet well understood), steady progress is observed. The numbers for the topological susceptibility have changed by two orders of magnitude and now are definitely convergent. It was shown, that what is seen is, at least sometimes, the true instantons, possessing the continuous limit, and not just some topologically nontrivial lattice defects. What is the most important: due to these works the lattice-oriented people are becoming convinced, that the topological phenomena are indeed the important ingredients of the gauge field vacuum, providing a significant fraction (or maybe even the main part) of the nonperturbative fields in gauge vacuum.
5. The general idea of the collective coordinate method and the variational approach The main technical problems of any quantum field theory deal with evaluation of some statistical sum Z over all configurations of the system. Generally speaking, the collective coordinate method is just a selection of some subset of interesting variables a, (i = 1, N ) out of the (infinite) set of the remaining (noninteresting) ones. The standard formal trick used is the incorporation of the Faddeev-Popov unity factor into the statistical sum: 1=
fJ(A,a)
i=1
[8(l,(A,a))dai].
(21)
Here Ii(A(x ), ai) are N conditions and J(A(x), ai) is the corresponding jacobian. Taking the integration over a i apart and performing first integration over all other variables, one obtains the so-called effective theory expressed in terms of the collective variables ai: N Zeff=
fexp(- S~,~(a))i=1 1-I da,, N
exp( - S~ff( a ) ) =
fDA
exp( - S [ A ( x ) ]
}J(A, a) 1-I 8(T~(A, a)). i=1
(22)
E.V. Shuryak / Instanton liquid (I)
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Applications of this method are well known in the simplest case, in which the definition of all collective variables is governed by some exact symmetry. For example, if one has found one single-instanton solution, one may well construct the whole set of them using translational, rotational and dilatational invariance of the classical Yang-Mills theory. (For the SU(2) colour group it leads to some " p l a n e " in the configuration space, parametrized by eight parameters.) For all directions (again, in the configuration space) transverse to this plane the action increases, thus the integrand in the statistical sum is peaked at this plane, which is the basis of 't Hooft's semiclassical theory. The eight-dimensional integral over this plane is left intact: the action is constant on it, so the sum is proportional to its volume. This lesson should teach us, that the basis for any similar calculation should be some N-dimensional manifold, to be called the ansatz plane below, which is able to absorb the non-gaussian part of the functional integral. However, generally speaking it is not necessary to hold the action constant on it. What we actually want to do is to integrate safely over the transverse coordinates, with integration over the collective coordinates remaining for separate investigation. Therefore, we have first to face the question whether the integrand is indeed peaked in the transverse directions. In the present series of papers we consider superpositions of instantons and anti-instantons, and these configurations do not possess any exact symmetries: the action in general depends nontrivially on all coordinates• In this case there are no classical configurations available: for any of them the first variation of the action, the current, is nonzero
~S j;(x) = 8A~
A=A . . . . .
=
(D~ G.~,) A=A...... =~0. a6
b
(23)
Therefore, writing the field potential as the sum of the ansatz and the fluctuating part
A~(x)
= A,~atZ(x) +
b,~(x),
(24)
one generally has the so-called linear term in the action expansion,
S(A)
=
S(A ansatz) + f dxj~(x)b~.(x) + "'"
(25)
which shifts the mean values of A away from our ansatz. D y a k o n o v and Petrov [9] have made quite a radical step, omitting this linear term (25). The physical idea behind it can be explained as follows. Suppose one has to add some external current Jo,~(x) .ext which exactly compensates the linear term, forcing the integrand to have its maximum just at our ansatz. However, this external current creates some work on the system, therefore its energy becomes greater than
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E.V. Shuryak / Instanton liquid(I)
that in the unperturbed vacuum:
E(jext) ~>E(jext=0) :Evac.
(26)
This inequality (which they have rigorously proved using the Feynman variational principle) is the essence of their work: they have suggested the variational approach to the problem. Indeed, taking a better ansatz, one finds a smaller current and, correspondingly, lower e(jext). However, this interesting work has left a lot of questions open. What ansatz is the best possible one, and how to find it? Is it possible, even in principle, to get rid of the current completely, so that the estimated quantity e ( j ext) is arbitrarily close to evac? 6. Which ansatz is the best?
Imagine the action distribution in the configuration space for a set of instantons and anti-instantons. The resulting landscape is similar to that in some mountain country, it possesses a series of peaks (extremely unprobable configurations) separated by a complicated system of deep valleys. If the PPs are well separated and nearly noninteracting, the action on the valley bottoms is nearly constant. If they approach each other, the slopes of the valleys increases, but it still may be much smaller on the bottoms than on the walls. Such a structure suggests the idea of natural separation of the longitudinal coordinates along the valleys from the transverse ones mentioned above. Indeed, everybody who once was in the mountains knows that there exist some outstanding line in each valley: it is the streamline on the bottom of the valley. Formally speaking, it is the line which is the minimum in the transverse directions. Another property: it is the line at which direction of the driving force (the first derivative) is tangent to it. In the gauge theory this driving force is in fact the current j. It can be decomposed into two parts, the longitudinal and the transverse one, according to the chosen ansatz plane. Thus, the best ansatz is that which has the properties of the streamline; for all points on this plane the current should be tangentential to it, in other words it should have
j~(x) = 0 .
(27)
This is the answer to the question posed at the end of the preceding section. Indeed, the current is nonzero, but its longitudinal part is not dangerous for us, as the integration over the collective variables is presumably made accurately. If the transverse part is absent, the linear term in (25) is zero and the integration over the transverse plane (leading to our effective action) can be made without any external currents etc.
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Unfortunately, even for the simplest toy models it is not easy to find such streamline configurations analytically (see e.g. ref [19]). There exists a simple numerical method suggested by myself [20] (see its discussion and application in paper IV of this series). The idea is quite simple: one should just start with two well-separated pseudoparticles and follow the direction of the driving force, which leads along the streamline downstream. However, for such a complicated system as the four-dimensional gauge field, the streamline configurations are not yet explicitly found and, as a first step, in the subsequent papers we approach this problem in a variational way, considering various trial functions which approximately describe this streamline. The quantity which we are going to minimize is now clear, it is j ~ . Thus, comparing various trial functions, we calculate the current and make (so far rather crude) estimates of the corrections induced by it.
7. The semielassical effective theory In this section we are going to outline the effective theory which corresponds to a gaussian (semiclassical) approximation. This is possible if the field fluctuates around our ansatz plane, and the quantum fluctuations, are not too strong. All steps are in fact quite standard. First, one has to expand the action up to the second order in the fluctuating field
S ( A ) = S ( A ansatz) q-
fdxjX'(x)G~(x) (28)
+ ½fba,(X)D a~' b"(x, Y)bb.(y ) dx dy + " .
and then integrate over the transverse coordinates in the gaussian approximation. If it is possible, the symbolic result for the effective action can be written as follows exp( - Sefr ) =
fDb± ( x ) e - s = exp( - s d
....
S current -
S quantum -
S jac°bian
) (29)
(where f Db± ... means f Db. J" I~N=lS(li)...). Let us discuss four terms of the resulting effective action subsequently. The first "classical" one is just the action distribution in the ansatz plane sclass(a) = S(Aansatz(a)).
(30)
Obviously it is the simplest one, and it is quite easy to deal with it, provided the ansatz is defined. However, it is the main part of the effective action only if its
572
E. V. Shuryak / Instanton liquid (1)
current part S c~r~m =
½fj~(x)rn2,b~(X, y)jff~(y)dxdy
(31)
is reasonably small compared to it. Note, that this term is formally of the same magnitude as the classical term (in perturbative language it corresponds to the zero-loop order). The sum of (30), (31) is ansatz independent, and the relative role of each depends on how close the ansatz used is to the streamline. The third, quantum term in this action squantum(a) = a l n d e t ( n ±)
(32)
contains the functional determinant of the differential operator [] ±. Such a determinant was explicitly found for the instanton solution by 't Hooft [7], which can be used as a reference point for estimates of its magnitude. Finally, there is the fourth structure in the effective action, related to the transverse projectors and the corresponding jacobian sJ"c°bi~(a) = - l n J ( A a~satz, a ) .
(33)
The most natural way to write our conditions I, is to project them transversely. This means that the fluctuating part of the field is orthogonal to all tangent vectors Oa~ of the ansatz plane
ba~(X)
(i) I, = f dxba~(x) V,,~,(X).
(34)
For the nongauge systems, these vectors can be obtained by a differentiation of the ansatz, but for the gauge system, one has also to take care about the projector Pgauge, which guarantees that b is not just a pure gauge transformation
4
(x, a).
(x) =
In particular, it is most convenient to take both background gauge condition O~abK(x) ~b~ --- 0 ,
ba~(X )
~Flabl~(i)( V b ~ a"~j = 0 .
(35) and
V~a~(X)satisfying the (36)
E.V. Shuryak / Instanton liquid (I)
573
The jacobian in general can be written as
J=det
(fdxva~((i)x)PgaugeValz((i)x))1/2
•
(37)
In the semiclassical context it should be taken at the ansatz plane. Again, as for the quantum part of the effective action, in the first approximation it may be taken from the known expressions for the individual PPs. References [1] [2] [3] [4] [5] [6]
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