some aspects of instantons

IL NUOV0 CIMENTO VoL. 41 A, N. 2 21 Set tembre 1977

Some Aspects of Instantons ( ' ) ('*).

R . D. PEcc]~z a n d H. R. Q u I ~ (***)

Institute o/ Theoretical Physics, Department o] Physics Stan]ord University - Stan]ord, Cal. 94305

(ricevuto il 27 Apri le 1977)

Summary . - - We s tudy various effects associated with Euclidean instanton solutions. By means of an instruct ive example, we discuss how the probabi l i ty of tunnelling among vacuum states in Minkowski space is connected with s ta t ionary (instanton) solutions in Euclidean space. Next we examine the cluster decomposition propert ies of non- Abelian gauge theories and indicate how these are recovered in the presence of instanton solutions. The va l id i ty of per turbat ion theory in these field theories is discussed. We show tha t per turba t ive analyses are only tenable in the deep Euclidean region, unless there is an intrinsic scale. Final ly , we examine the role tha t instantons have in providing violations of otherwise conserved quantum numbers. We study, in part icular , how fermion number nonconservation results by a careful examination of the path integral for the fermion-generating functional.

Introduction.

The c e n t r a l ro le t h a t n o n - A b e l i a n g a u g e t heo r i e s p l a y in a d e s c r i p t i o n of

w e a k , e l e c t r o m a g n e t i c a n d s t r o n g i n t e r a c t i o n s has s p u r r e d an i n t e n s e inves t iga -

t i o n of t h e i r p r o p e r t i e s . R e c e n t l y , t h e r e has b e e n p a r t i c u l a r i n t e r e s t in t h e role

t h a t f i n i t e - a c t i o n E u c l i d e a n so lu t ions (1) h a v e for t h e ove ra l l s t r u c t u r e of non-

(*) To speed up publication, the authors of this paper have agreed to not receive the proofs for correction. (**) Work supported in par t by NSF grants P H Y 75-18444 and MPS 75-20427. (***) A. P. Sloan Foundat ion fellow. On leave from Physics Department , Harvard University, Cambridge, Mass. 02138. (1) A . A . BELAVIN, A. M. POLYAKOV, A. S. SCHWARTZ and Y r . S. TYUPKIN: Phys. .Left., 50 B, 85 (1975).

309

310 ~. D. PECCE~ and m R. QUINN

Abelian theories. PoLY:~I~OV (5) suggested t h a t these <~instanton ~> solutions

could be b rought to bear on the question of quark confinement. Al though

this hope has remained unrealized~ it was shown b y 'T HOOFT (~) t ha t the existence of these ins tan ton solutions has direct physical impor tance in as much

as t hey solve the U~ prob lem of QCD. JAOKIW and I~EB~I (4) and CAt~I~A~,

DASHE1N" and GRoss (~) provided a physical in te rpre ta t ion of these s t a t ionary

Eucl idean configurations in t e rms of tunnel l ing among degenerate classical

vacua in Minkowski space.

The existence of these nonpe r tu rba t ive solutions of non-Abelian gauge

theories has raised a num ber of interest ing and challenging questions. ]n this

pape r we will t r y to examine in some detail a number of physical effects and

theoret ical points t ha t are associated with these ins tan ton solutions. We will

concent ra te on three main questions:

1) the physical role of Eucl idean solutions in a Minkowski field theory ,

2) the bear ing t h a t ins tan ton solutions have on the val id i ty of pe r tu r -

bat ion theory for non-Abel ian gauge theories,

3) the connect ion of ins tan ton solutions to pe r tu rba t ion theory anomalies

and their re lat ion to violat ion of otherwise conserved q u a n t u m numbers .

P a r t of our work, of necessity, overlaps wi th what has been previous ly

done (~.6). However , for pedagogical reasons, we feel i t i m p o r t a n t to m a k e

our discussion of these complicated ideas as self-contained as possible. Below, we shall ampl i fy somewhat on what we have done and present a p lan of this

paper . I n sect. 1 we discuss the t ransi t ion f rom a Minkowski field theory to a

Eucl idean field theory. We p a y par t icu lar a t t en t ion here to the role t h a t Hermi t i c i ty plays, since this will be crucial to our unders tanding later on of

fermion nonconservat ion. I n sect. 2 we describe how Eucl idean solutions are

connected with the possibil i ty of tunnell ing be tween different background

field configurations in Minkowski space. We approach these considerations

f rom a physical poin t of view b y considering a very ins t ruc t ive example in

Minkowski space. This example makes it appa ren t t h a t the Eucl idean in-

s t an ton solution of Belavin st al. (~) is the one t h a t describes the mos t p robable

tunnel l ing solution. Section 3 deals wi th the role t h a t the ins tan ton solutions have in non-Abel ian gauge theories. We address ourselves here to how the

cluster decomposi t ion proper t ies of these gauge theories are recovered, even in the presence of the ins tan ton solutions. I n this section we also examine the

(2) A. M. POLYAKOV: Phys. Lctt., 59 B, 82 (1975). (3) G. '1" HOOFT" Phys. Rev. Lett., 37, 8 (1976). (4) ]~. JACKIW and C. R~BBI: Phys. l~ev. Lett., 37, 172 (1976). (s) C. G. CALLAN, R. F. DASHEN and D. J. G~oss" Phys. Y~ett., 63B, 334 (1976). (~) G. '~ HOOFT: Phys. Rev. (in press).

S O M E A S P E C T S O F I N S T A N T O N S 311

validity of perturbation theory in the presence of instantons and deduce that perturbative analyses are only tenable in the deep Euclidean region, unless there is a large intrinsic scale in the theory. In sect. 4 we discuss the con- nection of instanton solutions with perturbation theory anomalies and how violations of usually conserved quantum numbers can occur. We focus prin- cipally on the violation of the fermion number and examine this question by directly computing the path integral for the fermion-generating functional. Section 5 contains a summary of our results.

1. - Minkowski and Eucl idean field theory.

To evaluate path integrals in (5[inkowski) field theory, one transforms the theory to Euclidean space-time. This transformation replaces the oscillating weight exp [ i S ~ ] by the damped weight exp [-- SE] , providing thus an unam- biguous definition of the integrals. Physical answers are obtained by transform- ing back the result to Minkowski space-time.

The passage from a Minkowski field theory to a Euclidean field theory is quite straightforward. Time is replaced by an ilnaginary parameter accord- ing to the rule

(1) x ° ~ t - > - - i x 4 .

All four-vector fields (or four-tensors) are similarly transformed. example,

(2) q ~ ~ (~0, q~) _+ (_ i~,, ~ ) ,

where % E = ( % , q ~ ) . For spinors, the transformation rules are equally simple.

nent spinor %t is transformed into a Euclidean spinor ~E:

Thus, for

A four-compo-

(3) ~p~ -~ ~/¥ .

5- o "]" Its Minkowski adjoint ~M-~ %~Y is transformed into the Euclidean adjoint 7pE:

(4) ~ , , ~ v+E.

The necessity of this last transformation rule is easily understood: one wants the Minkowski scalar ~M?/.,~t to go into the corresponding Euclidean scalar

5. FE FE" We find it convenient to preserve the form of the Dirae Lagrangian in Euclidean space. Thus we introduce Euclidean y-matrices ~ff which are related to the usual Minkowski y-matrices by

(5) r,Y - (~',, v) = ( i t °, v ) .

312 ~. D. PECOEI and m a. QUINN

We define the Euclidean action S~ by

(6)

where ~ is a function of the Euclidean fields. (The minus sign is conventional so that

Then, under the transformation described above, the oscillatory weight exp [iSM] with

(7) s~ > fdxod~x ~e~

is indeed replaced by the damping weight exp [--SE]. The Lagrangian density ~fM in Minkowski space is required physically

to be Hermitian. For a free Dirac particle the usual form of the Lagrangian density

(8) ~(~> = - - ~ / ~ }, ~-~, + m V'~

is not explicitly Hermitian. A Hermitian version can be easily constructed by writing

(9) ~ = - 5 ~ Y"~- ~ " - y" ~ - m ¢ ~ .

(This uses the fact that (yoy,)t__ yo7~.) The Lagrangian density (9) differs from the usual form by a total derivative term:

(10) = _ ~ + ~ o~(~r"~) •

Under ordinary circumstances, the extra total derivative has no physical consequences and £f(M 1) is equivalent to ~Lf~(~ ). However, when the theory admits instanton solutions, in Euclidean space, there are important differences. In these cases, it is clear that the correct physical starting point is provided by the explicitly Hermitian form ~f(M :).

Since the fermion fields ~ and v~ anticommute, we may write ~f~) (after normal ordering) equivalently as

(11)


The Eucl idean Lagrangian density, which corresponds to the above, follows

di rec t ly f rom our previous ly s ta ted rules:

1

I t is wor th r emark ing t ha t ~q~E has no explicit Hermi t i c i ty propert ies . The

kinet ic-energy t e r m is ant i -Hermit iun , while the mass t e r m is Hermi t ian .

(Our Eucl idean y-matr ices are an t i -Hermi t i an : (7~)r : - (y~)*.) This should

be of no concern, since Hermi t i c i ty is only required physically in Minkowski space.

2. - Instantons and Minkowski -space tunnell ing.

Le t us consider a gauge field theory in which the gauge group is S U~. I n order to discuss the physical in te rpre ta t ion of the Eucl idezn ins tan ton solu-

tions, i t is ex t r eme ly convenient to choose the gauge A~ = 0 (~,5). I n this gauge, as we shall see, there is a clear Minkowski-space equivalent of the topological

q u a n t u m n u m b e r of Bel~vin et al. (~). Wi th A o ~ 0, we still have a t our

disposal t ime- independent gauge t runsformutions ~2(x). I n par t icular , we consider gauge t rans format ions of the fo rm (*)

/ ; ~ RI J -= exp [iF(r)?.'¢].

The pa rame te r s R and 2 are a rb i t ra ry , as is the funct ion F(r). a a We can const ruct a field A,(~2) f rom the (( v a c u u m >> field A , ~ 0 every-

wher% by apply ing the gauge t rans format ion f2. Such fields will be called pure gauges. I n a ma t r ix notat ion, we have

(14:) A , ~-~- Au(.Q) --~ - - g (~u f2)-O -~ .

Physical ly, before we can pe r fo rm an integrat ion over the gauge fields Au,

which enter in the v a c u u m p a t h integral , we mus t specify the bounda ry con-

ditions on the fields A , . We shall require that~ as one goes to spat ia l or t empora l

infinity, the gauge fields Au approach fields t h a t are re la ted b y any gauge

t r ans fo rmat ion to the <~ v a c u u m i> field As z 0. This requi rement , however,

is not ye t s t rong enough. We require fu r ther t ha t line integrals of the t ype

~dxuAu give no physical effects. When Az approaches Au(~2) wi th f2 given

(*) Other gauge transformations besides those of eq. (33) are possible. They are, however, of no interest for our purposes.

21 - I1 N u o v o Gimengo A .

314 ~. D. P]~CC:EI and H. ~. QmN~

b y (13), this will only be so if we res t r ic t the class of pure gauge fields (14)

to t h a t in which, as x - + c~, ~2-~ 1. Defining, wi thout loss of generali ty,

/~(0) = 0, we shall require then t h a t

(15) ~(r) ~ 2~rn, n = 1, 2, . . . .

We shall label the functions F(r) b y their a sumpto t i c behavior , F~(r). The pure gauge fields character ized b y /V~(r) will be denoted b y A~(f2~) and only

this res t r ic ted class of pure gauge funct ions will be considered.

Will these definitions the relat ionship of the ((winding n u m b e r ~ ) n (4,5)

to the q u a n t u m n u m b e r q of the Eucl idean solutions of Belavin et al. (~) is

now apparen t . Recall t h a t q was defined b y

(16)

where

(17)

and

(18)

q = _ _ _ fd 4 E E g~ x ~ F ~ , ~" ~" 327~ 2 3 2 7 t 2

K ~ = 2s ~ TrA~(O A s - 2 " ~,gA~, A~)

I f A t is a pure gauge, then K~ simplifies to

2i E ~ ~ (19) K~ ----- -~ g%~r Tr A~ A~ A t .

I n Minkowski space K~ has an analogous definition.

one finds

(20) K~ = 0,

f 32~2 (21) d S x K ° , = g~ n .

For A ° = O, A ~ = A~(~2.),

Consider the field configuration A~,(x, t), for any fixed t ime t. We can con-

s t ruc t a n u m b e r

(22) n(t) = 4 ~ dSxei~TrA~A~A~.

This number , in general, can t ake any value. However , /or pure gauges o] the class ff2n, where A~ = A~(Q.), th is num ber takes an integer value n. R e m e m b e r - ing t h a t K~ = 0, we find

(23) . d ~ x F , . / ~ , , = n ( c o ) - - n ( - - ~ ) --= q .

SOME ASPECTS OF INSTANTONS 315

Because the field configurations a t t = =~ co are pure gauges of the t y p e A(Y2~) for any physical configuration, the num ber q is an integer.

I t is i m p o r t a n t to note tha t , if A , is everywhere a pure gauge, since £2n is t ime independent , t hen n ( + co) = n(-- co) and hence q = 0 always. However ,

we m a y choose field configurations in the Minkowski space which are not every-

where pure gauges, bu t are such t h a t as t -+ -~ co they approach dilferent pure gauge fields A~(Sg~). I n such c i rcumstances we will have a field configura-

t ion in Minkowski space with q ¢ 0. We are free to choose field configura- t ion AF, such tha t , at t -+ - - co, A , -+ 0. This corresponds to the gauge choice

Q = 0. I n s tandard pe r tu rba t ive calculations, one includes only field configura- t ions which are obta ined b y per forming a g2 expansion abou t A = 0 for all

t imes. Clearly these configm'ations correspond to q ~- 0. The point raised b y

the existence of q va 0 Eucl idean s ta t ionary solutions is t ha t this procedure requires modification.

To i l lustrate the physical in te rpre ta t ion of a q =~ 0 configuration, i t is

convenient to consider a field in Minkowski space character ized b y

(24) A ° = O, A'(x , t) = ](t)A'(Q1),

where the funct ion ] is a rb i t r a ry apa r t f rom the boundary conditions

(25) / ( - c o ) = 0 , / ( + c o ) = 1 .

Such a field is obviously a q ~ I configuration. I t does not, of course, sat isfy

the field equat ions in Minkowski space-t ime, but , nevertheless, fields of this

t ype m a y give a significant contr ibut ion to the p a t h integral.

Le t us evaluate the contr ibut ion f rom configurations of the t ype (24) for a given choice of Q1 and the pa rame te r s R and ~ appear ing in s91. The simplest

way to do so is to in t roduce the a rb i t r a ry funct ion J(t) as a collective co- ordinate (7) b y insert ing into the p a t h integral

(26) = I I fD "(x, t)H fDf(t) 1-[ ~[A~(x , t) - - ](t)A?(~) -- ~7(x, t)]" a,t ,x , t t a , i ,x , t

The p a t h in tegral t hen becomes

(27)

(7) J . L . GERVAIS, A. JEVICKI and B. SAKITA: Phys. Rev., 12, 1038 (1975); E. TOM- ~OULIS: Phys. Bey. D, 12, 1678 (1975).

316 ~. D. PECCEI and H. ~. qVlNN

where

(28)

and

(29)

M = fd .x

K = d" xA~(9I)A~(Y2~)~ ° A d~(Y2~)A~(f2~)s~b~z~d~~ .

The first two t e rms of the exponent in (27) are just fd~x Ae(](t)A~(~)) and

are of order 1/g ~, since A ~ ( ~ ) is of order 1/g. The remaining t e r m is given by

(30) R(Z~, l(t)A~(f2~}) = ~ ( X + ](t)A~(f21)) -- ~(/(t)A~(f2~)) ,

where ~ q ~ ( A ) = - ~F,~(A)F,~(A) . Clearly this last t e r m vanishes as Z - - ~ 0

and, for small f luctuations 27, i t corresponds to corrections of order (1) to the t e r m of order 1/g 2. l~eglecting, for the momen t , the question of s tabil i ty, we

evalua te the cont r ibut ion to (27) for 2: = 0 configurations:

(31)

where the value of the numer ica l constants M and K depends on our choice

of the gauge funct ion f21. We recognize (31) as the p a t h integral formula t ion of

a one-dimensionM q u a n t u m mechanics problem, a part icle of mass M and co-ordinate / in the presence of a po ten t ia l barr ier V = K/S( / - - 1) 2. Fur the r -

more, the boun da ry conditions (25) tell us to evalua te the tunnell ing probabi l i ty at E = 0, since we are dealing with a v a c u u m - t o - v a c u u m transi t ion. The

s tandard semi-classical approx imat ion gives

/C+ ~ )

/C--co}

We have calculated M and K for some choices of the gauge ~1. We find

exp [(-- 8~2/g ~) el, where the cons tant c is typical ly of order 5--10. The exact

number is of course i r re levant , because our choice of q = I configurations is

a rb i t ra r i ly res t r ic ted (*). This is re la ted to the question of the X in tegra t ion

in (27). The answer we obtain is only re levan t if we have chosen t h a t set of

(*) In fact, to evaluate the contribution even of our restricted set of functions (24), we would have to allow all possible functions F. The question of whether or not these terms are significant in the integral then depends on whether the overall divergences which arise are of the same order as those of ~he q = 0 terms. We do not investigate this question here. I t has been discussed by 'T HOOFT (ref. (e)) for the instanton solutions.


q = 1 configurations which are stable against small fluctuations. (The (5-func-

t ion in (27) mere ly res t r ic ts these f luctuations to be or thogonal to the t ime

t rans la t ions of the original configuration, since these contr ibut ions are a l ready

included in fD]) . Our exercise, however , is ins t ruc t ive in t h a t i t shows clearly t h a t the

q ¢ 0 solutions which we seek mus t correspond to tunnel l ing be tween a pure

gauge configuration character ized b y winding number n a t t = - - co and a different pure gauge of winding num ber n d- q at t ---- -4- c~. For the simple one-

dimensional p rob lem of (3]) i t is known t h a t the semi-classical approxima-

t ion (32) is obta inable b y the s ta t ionary point evaluat ion of the p a t h inte-

gral in Eucl idean space (i.e. with t ---> -- ix4). This immedia te ly suggests t ha t

the correct procedure to obta in the t rue q = ] contr ibut ion (or a t least the

semi-classical app rox ima te solution for it) is to seek solutions of the field

theory in Eucl idean space which minimize the act ion there. These are, of

course, exac t ly the ins tan ton solutions, as shown b y BELAW~ et al. (1) ('). The physical in te rpre ta t ion of these solutions as a manifes ta t ion of tun-

nelling in Minkowski space is, of course, not new. We present this discussion

mere ly as a concrete example of the k ind of Minkowski-space configura-

tions involved in the hope t h a t i t provides a be t t e r in tui t ive unders tanding

of the physics involved. We would like to stress one point which we believe

is the subject of some confusion in the l i terature. This is the distinction between the winding num ber n which characterizes t ime- independent gauge choices

in the Ao =-0 gauge in Minkowski space and the number q - - - - n ( - } - c o ) -

- - n(--co) which is re la ted to the Belavin et al. (1) topological q u a n t u m number .

The t ime- independent semi-classieM states of the sys tem in Minkowski space

can be labelled b y In>. The num ber q characterizes distinct te rms in the p a t h integral <n d- q]n>.

3. - Green's functions, cluster decomposition and perturbation theory.

Now we consider the generat ing funct ional for the gauge field theory. As ment ioned in sect. 1 this generat ing functional , when expressed as a p a t h

integral , is eva lua ted in Eucl idean space-t ime. The resul t is then cont inued

back to Minkowski space-t ime. Since the bounda ry conditions of gauge field

configurations with different q's specify dist inct topologies, we can divide the

(*) In a recent preprint, GERVAIS and SAKITA (ref. (s)) discuss this point further. They attempt to write in Minkowski space the field configuration of the Euclidean instanton solution and examine the relevant tunnelling problem. Their procedure is extremely convoluted, but eventually arrives at the expected answer. Our example, hopefully, emphasizes the simpler physical aspects of the problem. (s) J. L. GERVAIS and B. SAKITA: CCNY preprint (CCNY-HEP-76 11).

3 1 ~ R . D . P:ECCEI ~ n d I t . R. QI3INN

path integral into a sum of terms:

(33) Z(J) -~ ~ fDAED~, exp [-- S~(A~., ~b, J)] ~(/(A)) .

Here b(f(A)) represents some gauge-fixing term and ?~ stands collectively for all other fields in the theory including any ghost fields introduced by the usual Fadeev-Popov prescription. We remark that this prescription remains valid for each term of the ~ . In terms of the Minkowski-space discussion given

q

previously, it represents our freedom to pick a gauge at t = - oo; in par- ticular, we can always choose n(--oo) ---- 0. In (33) J is a generic source term, coupled to any of the fields in the theory.

We remark that, in the q V= 0 sectors, the usual arguments which eliminute the inclusion of total derivatives in the Lagrangian density must be reviewed. In purticular, although ~/~ can be rewritten as ~,K#, it is no longer true that fd4x F.F vanishes. The addition of a term fd~xO.PF in the Minkowski action corresponds to an arbitrary phase exp [iOq] in the path integral. Hence, more precisely, one must admit a more general form than (33):

(34)

where

(35)

Z(J) -~ ~ exp [iOq] Z~(J), q

Z~(J) = f DA~DqS~ exp [-- S~(A~, q~E, J)]5[](A )] . q

The physical origin of the phase exp [iOq] is clearly discussed by CALL~, DASHEN and G~oss (5).

A further technical point must be discussed. As defined in eqs. (34) and (35), Z(J) suffers from the usual incalculable divergences and hence must be properly normalized. Let us define, for each q,

(36) Z (J) = z . (J) /Zo(O) .

Then the properly normalized Z is given by

2(J) = z ( J ) / ( z o ) = 2o(o).

If there are any massless fermion multiplets, coupled to non-Abelian fields, then this quantity simplifies to a single infinite sum, since then 2~(0), for q :/: 0, vanish identically (3,6).


Because the path integral (37) is now an infinite sum of terms, rather than the familiar single term, the calculation of connected Green's functions is somewhat involved. Each individual q-term does not possess the physically required property of cluster decomposition. However, it is clearly necessary that, for all v~lues of O, the Green's functions c~lculated from the complete sum be cluster decomponible. In order to obtain a useful set of caleulational rules, which we call a modified perturbation theory, one must rely on this fact. Furthermore, to conceive of any perturbative approach, we must ensure, as usual, that the expansion parameter be small whenever it enters the calculation. We will examine, in what follows, these two important points.

To deduce the modified perturbation theory discussed above, we proceed as follows.

i) Assume the full Z(J) has the property of cluster decomposition, in which case it can be written as

(38) 2(J) --~ exp [W(J)].

ii) Identify the coefficients of J~ in W(J) ~s the full connected n-particle Green's functions of the theory (not one-particle irreducible). I t is clear that, by means of (38), W(J) can, in principle, be computed to any desired order in g~ and exp [(--8~)/g ~] by evaluating the Zq(J) to that order. Since the relationship between Z and W is nonlinear, it is also obvious that the J~ term in W receives contributions from all terms proportional to J~, for all m < n, and for all Z~.

In practice, for the q # 0 terms, our ability to calculate is limited to a few leading contributions. However, all q # 0 terms are damped by a factor exp [(-- 8Jr2)/g~lql]. In order to perform any calculation accurately, we must require that this factor is sufficiently small that we can reasonably neglect the infinite number of incalculable terms. Thus it becomes crucially important to understand wh~t is the relevant coupling strength g~ defining the d~mping factor for the q V= 0 contributions to any Green's function.

Let us consider the case q ---- l . 'T HOOFT (e) has shown by explicit perturbative calculation that ZdJ) is of the form

(39) 21(J)

Here z and ~ are position and scale parameters appearing in the instanton solution and the function f depends explicitly on that solution. As it stands, both the z and ~ integrations may introduce divergences, but in the calculation of

aZdJ) I ~J(x~) ... ~J(x,,) I.,-o

320 n.D. PECCEI and m R. QI~INN

t hey operate to restore the explicit t ranslat ion and scale invarianee of the expression. However, the calculation is meaningless~ unless in these quantities,

or ra ther in the physically re levant Fourier t rnnsforms thereof, the integrat ion

is effectively cut off, so tha t the re levant contr ibutions are all f rom the region g~()~) << 1. Le t us examine a typical t e rm:

(40) fl(k~.., k~) = 1-I,=~ d'x, exp [ik,x,] %J(x~) ~.. ~J(x.) "

This is one of the many terms contr ibut ing to the n- and higher-point Green's functions, bu t i t suffices for our purposes.

To leading order in g one finds in general a resul t of the form (6)

(41)

_-- (2~), 5, ( 5 k,) 25_a. gS(22) ~=~

The functions ]~ and the dimensional factor d,, depend on the part icle types, bu t the separable nature of the expression is independent of the part icle type.

(I t is in fact a consequence of the original t ranslat ion invariance of the pa th integral .) The individual x integrations can be separately performed. Defining

(42)

we have finally

k r d~ exp [-- 8=2/g2(~)] h Ei(k~2). (43) / t (k, ... k . ) = (2~)~5s(~ ~)J~F-~: gS(22) ,=1

Equa t ion (43) displays clearly the fact t ha t the quantit ies which regulate the

integral are the individual four-momenta squared. This tells us tha t our modified per turba t ion theory is only tenable in a region where all the particles involved

have four-momenta large enough such tha t g~(1/k ~) is small. Denot ing by A 2

the intrinsic scale of the non-Abelian theory (9), we must require k~>> A 2 for ~ll i. This of course means tha t , wherever they are calculable, the inst~nton

effects are also small compared to the q = 0 contributions. Their importance

is in situations in which there are effects ~vhich are forbidden by symmet ry to occur in the q = 0 sector. An example is fermion number nonconservation, which we will discuss in sect. 4. The avoidance of unwanted symmet ry (the U~

problem) is another impor tan t effect.

(9) A. DE RUJULA and H. GEOttGI: _Phys. Rev. D, 13, 1296 (1976); E. C. POGGIO, H. R. Q u I ~ and S. W]~I~B]~RG: _Phys. l~ev. D, 13, 1958 (1976).

8 0 M ~ ASPECTS OF INSTANTONS 321

We remark tha t the existence of ins tanton solutions invalidates any a t t empt to s tudy physical on-mass-shell Green's function in unbroken non-Abelian

gauge theories by means of per turba t ion theory. In the q----0 sector, the usual Kinoshi ta (10) arguments on mass singularities allow one to indent i fy certain physical quantit ies which are free of logarithmic dependence on the

part icle masses. This proper ty , in ordinary circumstances, suffices to prove tha t when all the remaining invariants are large, usual per turbat ion theory is an expansion in a small parameter with finite coefficients. The situation is now changed, since for every ampli tude there will be additional terms arising from the q ve 0 sector. These contributions are explicitly proport ional to exp [--[8~2Iqf/g~(l/m2)]] and can only be neglected if g2(1/m2) is small. This, however, is not the case if, for example, the particles involved include massless gluons ! The upshot of these arguments is t ha t we cannot t rus t any per turba t ive arguments in unbroken-gauge theories except s tr ict ly in the deep Euclidean region.

In a theory where the non-Abelian gauge symmet ry is spontaneously broken, for example by the usual Higgs mechanism, the particles obtain masses propor-

t ional to the scalar vacuum expectat ion value (~} ~ F. '~ HooFT (6) has shown tha t the scalar corrections to the q----i sector also introduce factors which cut off the ~-integrations at 2 ~ :I/F. Thus, in such a theory, the modified per turba t ion theory can be used to calculate even light-particle on-mass-shell Green's functions, providing only tha t g~-(:l/F 2) is small.

A final point perhaps is worth making. The quan t i ty W(J) defined in eq. (38) is not an effective Lagrangian; it is the generating functional of the full connected

Green's functions. 'T HOO17T (6) in his paper defines an (( effective Lagran- gian ~). We find this language misleading, since the usual t ree graph i terat ion of ' t Hooft ' s (( effective Lagrangian )) would not give the dominant higher n-point contributions. Fur thermore , in writing this effective Lagrangian, 'T HOOFT makes the approximat ion Ix, -- z] ~ ~ and in so doing obscures the role of the external momenta in cut t ing off the ~-integrations. However, when one is only concerned with the contr ibution to the lowest nonvanishing n-point funct ion appearing in the q ---- 1 sector, as was the case in ' t Hooft ' s work, the difference between the connected Green's function and an effective Lagran- gian is nil.

4. - Instantons and anomal ies: violations of the fermion number.

The role of the ins tanton solutions in resolving the U1 problem of QCD 5 is by now well unders tood (~). The axial vector current J~ has an Adler-Bell-

(lO) T. KINOSI~ITA: Journ. Math. Phys., 3, 650 (1962); T. KINOSttlTA and A. UKA~VA: Phys. t~ev. D, 13, 1573 (1976).

322 ~. D. PECCE* and H. R. Q~IN~

Jack iw anomaly (~)

(44) - - 1 6 ~ ~ F ~ F ~ .

(45)

H e r e

I f we construct the difference of two axial charges, this difference will be

propor t iona l to the integral over the space- t ime of F_P. I f the theory admi ts ins t an ton solutions, this difference will not vanish and thus the axial charge

is not a good q u a n t u m number . I n weak- in terac t ion theories, like the Weinberg-

Salam SU2 x U1 t heory (n), there are also anomalies. These anomalies can be avoided in the gauge cur rent sector, bu t appear in currents which do not couple

to the gauge fields. B y a similar reasoning, 'T HOOFT (6) has shown t h a t in

these theories the fermion n u m b e r is not conserved.

I n this section, we would like to examine the question of fermion non-

conservat ion b y looking direct ly a t the generat ing funct ional of the fermion

Green 's functions. Our t r e a tmen t , thus, makes no direct appeal to the existence

of anomalies in the fermionic current . We shall focus in what follows in the

q = 4 - 1 sector of the theory , ~nd we will need to know cer ta in proper t ies ,

which now we discuss. The Lagrangian densi ty t h a t describes, in Eucl idean space, massless iso-

spinor Fe rmi fields in the presence of a classical ins tan ton (or ant i - ins tanton)

solution can be wr i t t en in the no ta t ion of (12) as

gr . E __ ~a#~ XEV (46) -2- A~ x ~ + ~ ~

is the ins tan ton solution (~,~) which we t ake a t z = 0 for convenience. For tE E an an t i - ins tan ton solution we just replace ~ b y ~ (*). Since y , = - - 7 ,

and the fields ~E, V~ an t i commute , we m a y write the Lagrangian (after normal

ordering) in a more compac t fo rm b y defining a (( ma t r ix ~)

(47)

Then

M = 7

, t (4s) zG = - MVE + VE M v , } .

(11) See, for example, S. L. ADLER: Lectures at the 1970 Brandeis Summer Institute, edited by S. D~SER, M. GRISARU and H. PENDLETON (Cambridge, Mass., 1970). (1~) S. WEINBE]CG: Phys. Rev. Lett., 19, 1264 (1967); A. SALAM: in Elementary Particle Physics: Relativistic Groups and Analyt iei ty (Nobel Symposium No. 8), edited by N. SVA~THOLM (Stockholm, 1968). (*) The T-symbols are defined by ~]a~= eal,~ (/~, v = 1, 2, 3), ~/a4~ ~ - - ~a~, ~a,4= ~a,,


'T ]:[OOFT (s) has shown tha t the mat r ix M has a zero mode %. Tha t is there exists a solution

(49) MFo = 0 .

This eigensolution is purely right handed and is real. For an ant i- instanton the zero mode is pure ly left handed. These solutions are readily constructed and the above propert ies easily follow. Le t us do so for the instanton case. The zero mode has the form

1 (50) ~o - (x, + 2~)t u~,

where the spinor u~ is purely right handed (*):

(51) (1 -- 7~) uR = 0 .

Applying M on the above ansatz yields an equation for u s for all indices v:

(52) z ~ , y t , z ~ ) u ~ = O .

which must hold

If we define a,n = 7°?,~ = (i, a), t ions for u~:

(53)

the above gives the following two equa-

(3 + v a . , ~ ) u ~ = 0 ,

( 3 ~ + " ~ a ~ % % ) u R = 0 .

These equations can be rewri t ten by using the propert ies of the ~-symbols as

(54)

Since

(55)

{ ( 3 ÷ ~ . x ) u ~ = 0 ,

(3a + i s , k T~ o:k -{- T~) u ~ = 0 .

the second equat ion above becomes

(56) [ ~ ( 3 + 7~a"~) - vi(1 - 75)]us = o .

(*) We define Y5 by y5=i7°~172~ 3 and adopt a conveninet representation where

' /5=(~ : ) . We shou ldno te tha t our ys is opposite in sign to 't Hooft 'seonvention

(ref. (e)).

324 R.D. ~ECCV, I and H. It. QUINN

Clearly (56) is only compatible with the first equation of (54), if

(57) (1 -- 75)ua ~ 0 ,

as claimed. The remaining equation for uR, in a representa t ion where

:) is pure ly real. Hence u R is a real r ight -handed spinor. This last point demon- strates tha t ~o is also a zero mode of M*. Finally, for completeness, we write

down the solution for anti- instantons. We have in this case

1 (5s) ~° - (x~ ÷ ~)~ u , ,

where

(59) (l÷y,)u L=O, (l--a"~)u L=O.

The fermion-number-nonconserving Green's functions arise in theories which are not left-right symmetr ic in their weak-isospin assignments, such as the usual Weinberg-Salam S U~× U~ theory (~9- They are associated with the existence of this purely r ight-handed mode. To see tha t this is so, we first consider a single-flavor S U~ theory. In S U2 x U1 theories we will require more flavors in order to avoid anomalies in the electromagnetic current . However , the generalization involved is s t raightforward and the additional flavors com- plicate the nota t ion unnecessarily. We also begin by discussing a fully massless theory.

Le t us consider first the left-right symmetr ic case. We wish to evaluate the leading contributions in g2 to the fermion Green's functions in any q :/= 0

sector. This is the problem of evaluat ing the fermion pa th integral, with sources ~ + y~.~ in the presence of a classical gauge field given b y the ins tanton

or mult i - instanton solution. We will examine the q = i sector, since this gives

the leading (in exp [-- 8z2/g2]) t e rm and we can use some of the results above.

The re levant quan t i ty is thus

where £f~ is given by (48). Since in the symmetr ic theory there is no anomaly in the fermion current , the mat r ix M is ant i - I te rmi t ian in the Hi lber t space of this theory and we can expand it in terms of the eigenstates F~ defined b y

SOME ASPECTS O:F INSTANTONS ~

The complex conjugate s ta tes ~* sat isfy the equat ion

M ~ ~ F~ - (62) * *------+~ *

The Fi are 8-component objects~ Dirac 4-spinors and doublets in isospin

space. They are funct ions of the ins tan ton scale and position pa ramete r s 2 and

z as well as of x. Because of the fo rm (47) of the ma t r ix M, it does not commute with isospin project ions and its eigenstates are specific combinat ions of isospin.

The eigenmode for ,t ---- 0 discussed previously gives an example of such a state.

We can define project ion operators P_+ ---- (1:1:~5)/2 and normalize the ~0~s b y

f d x F~ ~ 5~j. (63) 4 *

We write ~ = P ~ , and notice tha t , since L

(64) M~#~ ~-- ~.~ y ~ and MY'i~, = )'~ ~iL,

every ~i m u s t have nonvanishing left and r ight project ions except possibly a mode associate wi th +~---- 0. I n faet~ as discussed above, we know tha t a q ---- 1 ins tan ton has a zero cigenmode which is pure ly r ight handed~ whereas

for q z -- 1 there is a pure ly lef t -handed mode. We shall show how these modes

are ins t rumenta l in causing fermion number nonconservat ion in left-r ight

a symmet r i c theories. Once we have identified the set of wave funct ions F~R, F~, the fermion

in tegra t ion can be expressed in t e rms of these quanti t ies. We introduce a single

an t i commut ing operator for each mode of F and y*. Thus we can write, in the q : 1 sector,

i

(65)

and

(66) v*= Z +/,v,*) + gv:,,, t

where ~o~ denotes the single r ight -handed zero eigenmode of M. The coefficients

ai, bi, c, ei, ]~ and g are mutua l ly an t i commut ing and the Vi's represent c-number

wave functions. I t is convent ional to make the identification e~ : a~ ]~ ---- b~,

g----c* to stress the fact t ha t each coefficient involves only a single degree

of f reedom. However , this identification is not necessary. We argue t h a t i t

is sufficient to comple te ly define the fermion integrals to introduce one real

coefficient per mode in any fashion and to require the in tegrat ion rules

(67) a~:O, fdal=O, fdaa=-i for each coefficient.

326 R . D . P ~ c e ~ and i~. R. q C I ~

One finds, using (65) and (66),

(68) ~f~ = -- ~ ).i{]ia~gi -~- e, bi(1 -- ~,)}, i

where

( 4 , ~ i = J d V i ~ R ' X (69) = j d x 1 4 *

The fermion integral ZI(~, ~) can thus formally be wri t ten as (*)

(70) ZI(~, ~) = J ~ J d z exp [-- [8=~/g~(2)] + i0] .

"2(@oE)(~V) IX [~,~' + (@,~)(~kV)][~i( ~ -- 6,) + (@i*~)(V,%@, t

where

(@,) = f d ' x ~W,(x, z, (71) 2) , e t c .

This quan t i ty is of course infinite. 'T H o o ~ (~) has shown th a t i t can be rendered finite by dividing by Z0(0, 0). The fact t ha t the fermion number is conserved to all orders is demonst ra ted b y the appearance of only products of the form ( ~ ) ( ~ * ~ ) in Z~--never any t e rm which involves a single ~ or ~. This form depends on two facts:

i) ~f~ is bilinear in fermion fields,

ii) for every fermion mode there is a corresponding ant ifermion mode.

I t is this last p roper ty ~h ieh ceases to be t rue when we examine a theory in which the lef t -handed fermions are a doublet under the gauge group, while the r ight-handed fermions are a singlet. For such a theory one has, in the

one-instanton sector,

(72) ~e E = - ½{w'M(1 - - 7~)W + WM*(1 + 7~)W * +

~- ~+D(1 + Ys) V + vD*(1 -- Ys) ~ } , where

(73) D -= 1 / i~ ,y~ .

Jus t as we could define the states ~ a = P+ ~ from the eigenstates of M in L

a symmetr ic theory, so we can define a set of states ~ b y

(74) Dffi ~ ~ ~ , e~R =/)_+ ff~ • L

(*) The integrals over d~ and d4z re-establish scale and translation invariance for the fermion Green's function (see, e.g., ref. (s)).

SOME AS P ECTS OF I N S T A N T O N S ~9.7

We remark tha t all eigenmodes of D, satisfying the boundary condition tha t

the fermion fields vanish at infinity, have bo th left and r ight projections. In fact, we must also notice tha t there is a degeneracy in the spectrum of D corresponding to the isospin symmet ry of 39. We can define or thonormal Q~, using isospin projections ~±, and we assume in the following tha t this has been done in defining the @.

I t is clear thu t the Hi lber t space of the theory defined by (72) contains only the modes

P-~Pi, P+Ft ~nd .P+~, .P_~o*,

$ and tha t these contain one odd antifermion mode _P+%. If we write

(75) t

~* ~ * * •

i

there are as many ]~'s as there are a / s and similarly as m an y e's as b's. However, there is no t e rm in ~ to pair with the t e rm g% in F', because of the r ight-handed nature of the zero eigenmode %. If we evaluate the pa th integral, this odd mode results in fermion-number-violating Green's functions:

~" d2 (d~z (76) Z @ , ~]) = j y j exp [-[s~/g~(~)] + iO](~;v).

• IV[ [ L ( 1 - ~ ) + @~,~)(~7~v)][~A + (@,~)(e*~v)], i

where

(77) fl, = f d ,x e* o, .

This analysis can be readily generalized to a S U~ × U1 theory containing multiple flavors. Here, following "r H o o ~ (9, we use the t e rm flavor generically, to mean separate mult iplet of the weak interactions S U2L. In a more complicated

theory such as S U3×SU2x U1 each color is also a flavor in this sense. (Although this is not the common usage of the t e rm flavor, it is convenient for this purpose.) As long as we consider only the SU~ x UI interactions of fully massless quarks, the generalization is completely t r iv ia l - -one simply obtains one factor of the form (76) for each flavor of the theory. Thus the minimum fermion number violation is N~--the number of flavors in the theory. Since the ant i - ins tanton has a purely lef t -handed zero mode, the q = I sector will produce a t e rm which is the Hermi t ian conjugate of (76). These terms are all of the order exp [--8n2/g~], where g~ is of order a and so are experimental ly completely insignificant, as has been discussed by 'T HOOFT (la).

~ 2 ~ 1~. D. PECCEI and ~ . R. Q u I ~

Our discussion shows how, in a massless theory, fermion-number-violating Green's functions arise as a consequence of the left-right asymmetry of the theory. In any left-right symmetric theory the matrix M in Minkowski space is explicitly Hermitian in the space of states of the theory. That being so, for every eigenmode ~ of ~ there exists a corresponding eigenmode ~* of ~+. The pairing of these eigenmodes guarantees fermion number conservation. The difference between the explicitly Hermitian Lagrangian (48) and the usual form F* M~ is just the total divergence of the fermion number current and this current must vanish when M is Hermitian in the space of states of the theory.

The non-Hermiticity of M(1 -- yh) + D(1 + ~5) makes it essential to use the explicitly Hermitian form (48) for the Lagrangian. The total fermion number current now cannot be conserved, since there are explicit fermion-number- violating Green's functions in the theory. This was already indicated by the argument of ' t Hooft (5) based on the existence of an anomaly coupled to this current and proportional to the operator FF. We remark, however, that the usual derivation of this result (11) is based on a perturbative calculation, using standard (q = 0) propagators. The q va 0 contribution to the propagators could in principle modify that argument. Because the anomaly actually comes from the short-distance part of perturbation theory graphs, one expects, however, that it should be relatively unaffected by q ve 0 contributions.

In all of this discussion, we have ~ri t ten formal expressions involving the eigenmodes F~ of M. In fact, we only know explicitly the form of Fo. Thus, although (70) and (76) present neat closed-form expressions for Z(~, ~), we cannot calculate anything beyond the lowest nonvanishing derivative

~ Z ... ~ I ,7=~=o ~ I "'" C~T]I ~]1

where ] represents the number of independent flavors of the theory, or for the asymmetric theory

In other words, we can calculate explicitly only the contribution to the lowest nonvanishing n-point function, all higher n-point functions depending on the unknown ~i's. The situation becomes even worse when we introduce a fermion mass. Even when the mass is common to all fermions of a given flavor, it introduces additional terms in Z in such a way that for a symmetric theory only the zero-fermion Green's functions are in fact calculable. Thus, we must rely on the exp [-- 8~2/g2fq[] suppression of the q ¢ 0 sectors, in order to use the theory for any calculation. The addition of masses does not alter the existence of fermion-number-violating Green's functions in the left-right asymmetric


theory, bu t again it produces a considerably more complicated Z. In a multi- flavor theory with quark mass terms, e.g. ml(~a q- bb) and m~(~v -q-dd), the theory can be left-right asymmetr ic while still vectorlike. This simply means

we make the weak-isospin assignments b L d L' bu t d ~ b ~" Such a theory

will be fermion number conserving, because there exist, in toto, as m an y fer-

mionic as ant i fermionie modes.

5. - S u m m a r y .

In this paper we have addressed ourselves to some issues in ins tanton physics. We have discussed, by means of an instruct ive example, how the ins tanton solutions yield the dominant contr ibut ion to tunnell ing among different vacuum configurations in Minkowski space. The fact t ha t there are topologically dist inct configurations reflects itself in an expression for the generat ing functional Z(J) which is a sum of terms characterized b y the topology.

This result has profound implications for a per tu rba t ive approach to non- Abelian gauge theories. To regain cluster decomposition, it is necessary to consider the contr ibut ions of all q-configurations in the generat ing functional. Since the q ~ 0 terms are in general incalculable, we require their contr ibut ion

to be insignificant. This occurs in two circumstances:

i) in any asymptot ical ly free theory, for the Green's functions in the deep Eucl idean region;

ii) in theories where the spontaneous symmet ry breaking acts to cut off the ins tan ton size.

Clearly this implies tha t the use of per turba t ive calculations to s tudy the infra-red region of non-Abelian gauge theories is suspect.

A final point t ha t we examined is the relat ion of ins tanton solutions to the violation of, otherwise conserved, quan tum numbers. We showed in the case

of fermion noneonservat ion how the violation occurred by direct ly examining the fermion-generat ing functional.

We are indebted to G. 'T HOOFT for some very helpful conversations on instantons. We are par t icular ly grateful to him for focusing our a t ten t ion on the Hermi t ic i ty propert ies of the fermion matr ix .

22 - I I :Vuovo Cimento A.

330 R. D. PECCEI a n d H. R. QUINN

R I A S S U N T O (*)

Si s t u d i a n o i v a r i ef fe t t i associa t i con le soluzioni i s t ~ n t o n i c h e eucl idce. P e r mezzo di u n esempio i s~rut t ivo , si d i scu te in ehe modo la p r o b a b i l i t h di t u n n e l t r a s t a t i vuo t i hello spazio di Minkowsk i ~ eo l lega ta con soluzioni s t az ionur i e ( i s t an ton i ) nel lo spazio eucl ideo. Quind i si e s a m i n a n o le p rop r i e t~ di deeompos iz ione degli a m m a s s i di tcor ie di gauge n o n abe l i ane e si i nd i ea come esse si~no r i p r i s t i n ~ t e in p r e s e n z a di soluzioni i s t a n t o n i c h e . Si d i scu te la v a l i d i t h del la t e o r i a delle p e r t u r b a z i o n i in ques te t eor ie di campo. Si m o s t r a che anul is i p e r t u r b a t i v e sono sos tenib i l i solo ne l la reg ione eucl idea p ro fonda , a m e n o che n o n ci sia u n a scala in t r in scca . Inf ine si esamin,~ il ruolo che h a n n o gli i s t a n t o n i ne l fo rn i r e v io laz ion i degl i a t t r i m e n t i c o n s e r v a t i n u m e r i quan t i c i . Si s tud ia , in pa r t i co la re , come r i su l t i la n o n c o n s e r v a z i o n c del n u m e r o di f e rmion i pe r mezzo di u n a t t e n t o e same de l l ' i n t eg ra l e del percorso pe r il funz iona le di gene raz ione dei f e rmion i .

(~) Traduzione a eu~'a della l?edazione.

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(*) HepeeeOeno peOar, ttueft.

some aspects of instantons

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