zooo,. 9 r Physik C P a r t i c l e s

Z. Physik C, Particles and Fields 5, 233-238 (1980)

andFields

~) by Springer-Verlag 1980

Instantons and the

I[N Expansion

M.J. Teper Rutherford Laboratory, Chilton, Didcot, Oxon, OX11 0QX, UK Received 27 September 1979, in revised form 23 January 1980

Abstract. We show that a non-interacting gas of instantons, gives a contribution which vanishes, in QCD, exponentially with N (the number of colours). If we include some of the interactions between instantons, then the resulting dilute gas of instantons--which is now in the form that is used in practice to investigate the structure of h a d r o n s - gives a contribution that is essentially constant with N. Thus it is entirely consistent for both the N-~ limit to be useful and for the dilute gas of instantons to be relevant for the properties of hadrons.

Introduction Instantons [1], and in particular the dilute gas of instantons [2, 3], have been used extensively to illuminate the dynamics of hadron structure in an increasingly quantitative way [4]. Nonetheless their use for theSe purposes has remained controversial. Re,cently [5] it has been argued that in fact any such important role for instantons cannot be consistent with the usefulness of the large N ( = number of colours) limit [6] in QCD. This is basically because instanton contributions appear to fall off as exp - c N, and so would not appear at any finite order in the 1/N expansion. So if one expects to find all the characteristic properties of hadrons in the 1/N expansion, they cannot be the outcome of instanton effects 1. In this note we examine more carefully the premise that instanton contributions fall as e -r and show 1 Note that one is not here concerned with the question of confinement. It has long been understood that instantons by themselves cannot confine in 3 + 1 dimensions. (In 1 + 1 dimensions the situation is quite different because a Wilson loop encloses a region of space-time, so that the whole of the fluctuation of an interior instanton necessarily plays across the loop). It is quite possible that the bulk of the interesting properties of the usual hadrons are only marginally influenced by the strict confinement phenomena, and may be obtained in a non-perturbative approximation that is not strictly confining. This is the approach followed for example in [4], where the approximation used is based on dilute gas of instantons. It is this role ofinstantons that is subject to controversy

that things are not so simple. After performing the integration over scale sizes we find that whereas the non-interacting gas of instantons does indeed disappear exponentially with N, the inclusion of interactions, even to a first approximation, completely alters the situation, so that the dilute gas as used in practice [4] gives, in fact, a contribution that is essentially constant with N.

The 1IN Expansion and Instantons We obtain the 1IN expansion of QCD by extending the colour gauge group of Q C D to SU(N), taking N large, and expanding the theory in powers of 1/N. So that the obvious quantities should remain finite as N ~ oo, we also take gZN = constnant. All this is well-reviewed in [6]. Analogous expansions may be devised for many other field theories. We do not of course need to impose g 2 N = const as a separate condition. The usual equation for the running coupling involves the /~ function and to any order of coupling the dominant diagrams (i.e. those with the largest power of N) will be planar diagrams, so that one automatically obtains

g2(q2) = 1/Ng I (q2)

(1)

to leading order in 1IN. Although N = 3 is not obviously near N = oo, and although one cannot show confinement explicitly as N ~ co, the qualitative features of hadronic physics that one obtains for N - * ov (and assuming confinement in that limit) are so similar to what is observed at N - - 3, that one is strongly tempted to assume that indeed 3 is close to oo and also that confinement occurs in the 1/N expansion. All this would mean that the dynamics producing the dominant features of hadrons should come in as a power of 1/N, and not for example as e -cs. On the other hand instanton effects behave roughly as

e- 1/0~,= {e- 1/0;}u ~ e-ON

(2)

and hence would not appear to any finite order in

0170-9739/80/0005/0233/$01.00

MJ. Teper: Instantons and the i/N Expansion

234

1IN as N ~ ~ . This would imply [5] that if one takes the 1IN expansion seriously, then instantons have nothing significant to tell us about the dynamics of hadrons. In this paper we are going to investigate whether the naive argument embodied in (2) is indeed correct. We shall carry out this investigation within the context of the approach to instanton physics espoused by the Princeton group [4]; that is to say we shall work with the dilute gas of instantons and antiinstantons. Perhaps it might be appropriate at this point to explain why this is a sensible thing to do, and to put the instanton calculations in a broader perspective. In the semiclassical limit one approximates the path integral by a sum over the minima of the action (with perturbative fluctuations around each of these minima). Now the Euclidean equations of motion possess exact classical solutions that correspond to n instantons. They also possess approximate classical solutions that consist of wellseparated instantons and anti-instantons--the dilute instanton gas. Since these latter field configurations are in no sense small fluctuations around the former they should certainly be included separately. Moreover when one considers small enough scale-sizes that the gas is reasonably dilute, the contribution of the exact multi-instanton configurations relative to that of the instanton-anti-instanton dilute gas configurations is negligible, for obvious combinatoric reasons. As we consider larger scale sizes there is a transition region where the interactions between instantons and anti-instantons can be accurately estimated as a correction: here too the pure multiinstanton configurations are negligible. There will then be a region where we can't say anything, and finally the region of large scale sizes where instantons and anti-instantons overlap to such an extent that for all practical purposes they represent small fluctuations around the exact n instanton solutions, and so here these exact solutions dominate instanton physics. We can characterise all this by saying that for 0 < p < Pc the dilute gas dominates, while for P > Pc the exact n instanton solutions dominate. It is thus clear that if one wants to consider the contribution of instantons to strict confinement, then it is the exact instanton solutions that are relevant, and the dilute gas has no role to play. On the other hand the question of hadron structure (at least for the lowest levels) is a quite different matter, and indeed the Princeton group have shown that Pc is large enough that the question of hadron structure may be approached through the dilute gas, with some corrections for the interactions of instantons and anti-instantons. Of course once we go beyond the very smallest scale sizes we begin to leave the semiclassical limit, and we must look again at the relevance of the classical instanton solutions. In particular for large enough

scale sizes the coupling becomes "large" and we may expect the vacuum to contain all kinds of more complex field configurations (vortices etc.) that might undo the effects of instantons on these large scale-sizes. Since instantons are the most "efficient" non-perturbative fluctuations, in the sense of possessing least action and, due to their point-like character, maximum entropy, a natural criterion for what is a "small" coupling is that the characteristic configuration of the gas of instantons and anti-instantons should be reasonably dilute on the scale sizes corresponding to that coupling; i.e. that the path integral prefer field configurations where most of space-time is in a "classical vacuum". Using this criterion we see that the scale-size Pc not only separates the region of scale-sizes dominated by the dilute gas from that dominated by the exact n instanton solutions, but simultaneously acts as a boundary beyond which any instanton physics becomes suspect. This suggests that to evaluate [7] the contribution of these exact instanton solutions will not be enough to derive any physical consequences; one must simultaneously understand the effects of other nonperturbative fluctuations that are surely there on these larger size scales. Since the work of the Princeton group suggests that Pc is large enough for hadron structure to be amenable to a dilute gas approach, and since the attractive physics one derives in the N ~ oe limit largely involves this same region of scale sizes, it is natural to consider the consistency of the 1IN expansion with the dilute gas of instantons, and this is the question we now turn to. The Dilute Instanton Gas in the N ~ ~ Limit

The one instanton contribution to the path integral goes as [8]

I=S[

A]e_ S

const [ g4(p) e- s'~/~ }N

(3)

(where from now on g 2 (p)= gl2 (P), and we explicitly show the dependence on the instanton size p). Certainly for small p, { ... } < 1 and so this contribution falls exponentially with N. For large enough p it may be that {...} > 1 and then it blows up exponentially with N. The former would imply that the instanton contribution is irrelevant at large N, while the latter, if the case, would make a nonsense of the 1IN expansion. Of course in calculating I we must include all possible numbers of instantons, with all possible locations, group orientations and scale sizes. We can only do this to the extent that we can neglect (or treat as a correction) their mutual interactions (the interactions being a measure of the extent to which the configuration of instantons and antiinstantons deviates from being a true classical

235

M.J. Teper: Instantons and the 1/N Expansion

solution); and when we do so we have the dilute gas approximation for instantons. For a fixed scale size p the instantons appear in the gas with a density D(p) [4, 8] 1 [ b2

8~2] N

D(p) ~ p4 ]. 9-~) exp - ~ - j,

(4)

fcN = const then the dilute gas contribution to physical quantities will also be eventually constant in N, contrary to the naive argument of (2). Clearly for this to be the case there must be some size scale at which b2

g~ The quantity that will appear in calculations of quantities of interest will usually be

dp o(p)

(s)

o P

and so we are interested in the behaviour of (5) as N--+ oo; in particular whether it behaves exponentially with N or not. It' is clear we cannot include arbitrarily dense configurations of instantons 2. Indeed once instantons overlap significantly it is in no sense a good approximation to calculate in terms of noninteracting instantons; one must use some entirely different calculational scheme. So the obvious physical constraint is that one only includes in the dilute gas instantons up to a scak~ size Pc (N). (And this is why we have made the upper limit in (5) Pc rather than oe) such that the fraction of space-time covered by instantons, f (Pc, N) satisfies

f(Pc, N )=fc N ~<1

(6)

where

f (Pc, N) - P~dp TC2

o7-2/D(p)

o P (gr

gi(p) 3

(7)

87~2

exp - g2 (p) -

1

(8)

and this size scale will then be chosen as our Pc- In the N ~ oe limit any scale p < Pc does indeed give a contribution that falls exponentially with N; so the surviving contribution of the dilute gas consists of instantons with scale sizes narrowly centered on Pc (the range of values being O(1/N)). This would be in accord with what one [4] finds with N = 3; scale sizes close to Pc play the dominant dynamical role in producing the hadronic "bag". Whether (8) can be satisfied and hence whether the dilute gas disappears exponentially with N or not depends critically on the value of b z. However independently of this value we can already see why the dilute gas can never give a contribution that grows exponentially with N. Such a growth would be reflected in a fraction f ~ that would also grow as eN; so the piece of the "instanton gas" giving the growing exponential would be pure mathematical fiction, having no place in the dilute gas. Now if we consider the L.H.S. of (8) as a function of gZ(p), it attains its maximum value of b Z / ( 4 9 2 ) 2 e -2 at g2(p)= 4=2. The value of b 2 may be obtained from footnote 3 where the one instanton contribution is calculated for SU (N) and we find b 2 = 16rc4e2eU3[ln2+5/12+ 12~'(-1)1 ~ 1 6 n 4 e 2 e -0.292

This is also the practical constraint employed in (4)3. Optimistically one would choose f~N = 1, but for our arguments the precise value does not matter (although it is a source of anabiguity when one wishes to calculate actual numbers), as long as it tends to some constant non-zero value as N ~ ce. It is of course possible that we cannot impose f(p~, N)= const and that in fact this fraction vanishes exponentially with N for any pc(N). Since as remarked above the instanton contribution to the quantities of interest in the theory has a N dependence that is determined by the same function that determines the N dependence of f(Pc, N) (up to inessential powers of p) it follows that if one is able to impose

2 Overlapping instanton configurations also raise severe problems of double counting: such a configuration might already be included as an ordinary perturbation about a much less dense configuration of instantons. This is a problem we address further in M. Teper, An Introduction to Instantons--Lectures at Rutherford Laboratory and University of O r e g o n - - i n preparation 3 Formally at least it may be that Pc = 0% since the integral in (7) may well be convergent. Note that for large p, and hence large g(p),{...}~l/g4(p) which in general should be a sufficient convergence factor

Thus be 3 (4/1;2)2 e - 2 ~_ e - 0 ' 2 9 2 ~ - - a < 1

(9)

and hence for all scale sizes b2

g~

8n 2

exp - ~

< 1

(10)

So we see that for any pc(N) 4, f(pc, N ) ~ e -oN and hence the dilute instanton gas disappears exponentially with N. Our more careful arguments have thus led to the same conclusion as the naive argument in (2). 4 This is not quite true. Ifg 2(p) should vary very weakly with large p, then e -oN dependence of the L.H.S. of (8) could be compensated for by say Pc ~ exp {exp aN} giving f(Pc, N) ~ const. However most physical quantities would have more powers of p in the denominator of the integrand than does f (Pc, N), and hence would vanish very fast with N. Moreover in this extreme example it is the contribution of ever greater size scales that allows f(Pc, N) to be constant. However we are only interested here in the usual hadronic size scales; the contribution to quantities dominated by such scales would fall exponentially with N in this case also

M.J. Teper: Instantons and the

236 T h e I n t e r a c t i n g l n s t a n t o n G a s in t h e N ~

~

Limit

Concluding Remarks

We have seen that the non-interacting instanton gas vanishes exponentially with N. However it very nearly does not do so in the sense that, in (9), 3/4 is after all very close to 1. It is quite possible therefore that the inclusion of some of the interaction between instantons may alter our conclusion. Moreover it is precisely the interaction between instantons and the consequent large "magnetic permeability" of the vacuum that in [4] leads to physical consequences. Treating instantons in 4 dimensional Euclidean space-time as 4 dimensional magnetic dipoles, Callan et al. find (see (4, 40, 41) of the last reference in [4]) that the interaction of an instanton of size p located in a cavity of size R which in turn is located in a medium of permeability #, alters its effective action by an amount 87C2 "~ 6 S : - 6 /L g-~) ) ~l't -- 1 ( R ) r

(11)

So far this is a standard magnetostatics calculation. Using #~> 1 (crudely /~-~ l q-4Ir2/g2(pc) ) and R = 2.2p, which numerical calculations show to be the point at which the dipole approximation to the instanton field begins to breakdown [4], Callan et al. determine 6 S to be 6 S = - 292/92 (p)

(12)

The fact that 6 S is negative is not really surprising: a neighbouring instanton and anti-instanton will tend to annihilate each other where they overlap. So to a first approximation the interaction of an instanton with all other instantons alters the density function to [4] _ _ l J" b 2

b(p)

6~2 ~N

p4 [g4(p) e X p - g ~ ) ) j

(13)

It is easy to see that now the maximum value of {... } is obtained for g2(p)= 3~2 and at this value ofg 2, { ... } = b2/(3~r2)2e -e ~ 4/3, So if we can choose a Pc such that g2(pc) ~ 27r2 we will obtain a fraction f(Pc, N) that is essentially constant9 Now this value of g2 corresponds to ~(pc N '

=

3) =

92(p~) 9 1 ~ -rc ~ -1 4~

3

6

1/N Expansion

(14)

2

which is surely a value that can be readily attained in Q C D s, corresponding to size scales on the order of the size of the ~bmeson, Thus the dilute gas of instantons modified to include interactions (to a first approximation) gives an essentially constant" contribution in the N -~ limit. 5 This is obviously an important step in the argument; if for example we found that we required a g2(pc)= 500, then we could not be sure that such a large couplingcould arise in the theory and hencethat such a Pcin fact exists 6 Throughout we have ignored mere powers of N

We have shown that if we define the dilute instanton gas by the obvious physical requirement that the fraction of space-time covered by instantons should be less than, but as close as possible to some well chosen constant (which is less than u n i t y ) - - t h u s ensuring that our field configurations do indeed consist of identifiable instantonsmthen the contribution of the non-interacting gas will vanish exponentially with N. This confirms the usual naive arguments, even though it is obvious that this is partly fortuitous. Indeed perhaps the point to emphasise is that the dilute gas is very close to not decreasing exponentially with N, as we can see in (9). Of course it is not really clear to what extent we can say that 3/4 is close to 1, without defining some scale. The physically relevant question is: are the corrections due to interactions in the dilute gas for p close to Pc, large compared to the difference between the numbers 3/4 and 1 in (9). To answer this question we borrowed the expressions (11) and (12) from [4] - - t h e y represent the leading corrections to the dilute g a s - - a n d confirmed that indeed these corrections were large enough that the modified dilute gas no longer decreased exponentially with N. Thus, contrary to the naive argument, the dilute instanton gas with first order corrections for interactions does not decrease exponentially with N. Of course it is not clear how reliable (I1) is as a measure of the leading approximation to the dilute gas. There is some ambiguity in the choice of R, and since it appears as R 4 this introduces a significant uncertainty. And of course the corrections to the Gaussian fluctuations, the large scale non-trivial background fields etc. probably also deserve serious consideration. However we do not wish to go here into the validity of the approximations used in [4]. We merely point out that it is interesting that everything seems to lie on the borderline between the two kinds of behaviour. It is also interesting that it is the same interaction piece that makes all the difference here that leads to the interesting physics [4] in detailed calculations of hadronic structure. In any case we conclude that the dilute instanton gas, when interactions are included, does survive the N ---, oo limit, so that there is no necessary contradiction between its usefulness and that of the 1/N expansion. The same arguments that have been applied here to QCD, could also be applied to other field theories 7

7 After completingthis work we learned of a similar investigation in the context of Cp"-I models: A. Din. P. Di Vecchia, W.J. Zakrzewski, Nucl. Phys. B155 (1979) 447. There the authors find (in our language theirviewpointdiffersslightly)the L.H.S of (8) to have a maximum value of 1/2x/e, so that the gas vanishes exponentially with N. They treat only the non-interactinggas; again it would be interesting to know what happens in the case of the interacting gas

M.J. Teper : Instantons and the 1/N Expansion (that are s c a l e - i n v a r i a n t at t h e classical level). I n a n y p a r t i c u l a r case o n e h a s to check w h e t h e r the c o n d i t i o n c o r r e s p o n d i n g to (8) c a n i n d e e d b e fulfilled. W e see n o g e n e r a l a r g u m e n t w h y t h a t s h o u l d o r s h o u l d n o t be t h e case.

References 1. A.M. Polyakov: Phys. Lett. 59B, 82 (1975) A. Belavin et al. : Phys. Lett. 59B, 85 (1975) S. Coleman: Uses of Instantons HUTP-78/A004 2. G. 't Hooft: Phys. Rev. Lett, 37, 8 (1976); Phys. Rev. D14, 3432 (1976)

237 3. A.M. Polyakov: Nucl. Phys. B120, 429 (1977) C. Callan, R. Dashen, D. Gross: Phys, Lett. 63B, 334 (1976) 4. C. Callan, R. Dashen, D. Gross: Phys. Rev. D17 2717 (1978); Phys. Rev. D19, 1826 (1979); Semiclassical methods in QCD: towards a theory of ha~ron structure Princeton preprint and AIP conference proceedings No. 55. 5. E. Witten: Nucl. Phys. B149, 285 (1979) 6. E. Witten : Baryons in the 1IN expansion HUTP-79/A007 7. B. Berg, M. Lfischer: DESY preprint 79/17 E. Corrigan, D. Fairlie, P. Goddard, S. Templeton: Nucl. Phys. B140 31 (78) E. Corrigan, P. Goddard, H. Osborn, S. Templeton: Nucl. Phys. B159, 469 (79) H. Osborn : Nuct. Phys. B159, 497 (79) 8. J. Koplik, A. Neveu, S. Nussinov: Nucl. Phys. B123, 109 (1977)