method for borel-summing instanton singularities: introduction

P H Y S I C A L R E V I E W D V O L U M E 1 9 , N U M B E R 8 1 5 A P R I L 1 9 7 9

Method for Borel-summing instanton singularities: Introduction

William Y. Crutchfield I1 Institute for Theoretical Physics, State' University of New York, Stony Brook, New York 11794

(Received 5 December 1973)

The problem of Borel-summing the perturbation series of field theories possessing instanton solutions to the Euclidean equations of motion is studied. Such theories have a singularity on the positive real axis of the Borel plane along which the Borel integration is performed. A direct functional expression for the Borel transform of a functional integral is given, which yields a prescription for integration over such singular points. The problem of "double counting" perturbation expansions about multiple solutions to the Euclidean equations of motion is solved. In this first of two papers the method is introduced and applied to a trivial integral possessing an instanton. Also Borel singularities of nontrivial functional integrals not possessing instantons are found and compared to familiar large-order calculations.

I. INTRODUCTION AND RESULTS

In recent years , great advances have been made in Borel-summing the perturbative expansions of functional Borel summation converts the asymptotic ser ies derived from a functional integral to a convergent expansion which may be applied to the strong-coupling problem. However, there i s a c lass of very interesting theories whose perturbation se r i e s cannot be summed using these methods. These a r e theories possessing finite positive-action solutions to the Euclidean equations of motion: instant on^.^'^^^ One such theory is quantum chromodynamics, the leading candidate for a theory of the strong interactions. It has been shown that the Borel transform of the perturbation se r i e s in such theories has a singularity on the positive r ea l axis, along which the Borel-transformed function i s integrated.7s8 This singularity i s a symptom of the presence of "nonperturbative" contributions proportional to e x p ( - ~ , / g ~ ) . ~ . ~ Since such exponentials have a zero asymptotic expansion in gZ, i t i s often assumed that Borel-summing such theories i s impossible.

The principal result of this paper i s that the perturbative expansion for a functional integral possessing instantons may be Borel summed. (Un- fortunately, this need not imply that the infinite- volume limit of thermodynamic quantities always has a convergent Borel sum-see Ref. 7.) Given a Euclidean path integral in loop expansion form

a direct functional expression fo r i t s (modified) Borel transform may be written. The functional integral i s evaluated by integrating i t s modified Borel transform, B(z), on a contour encircling the positive rea l axis

When no instantons occur in the functional integral , this sum reduces to the usual Borel sum of the usual Borel transform. It i s easy, using the functional expression for the modified Borel transform (MBT), to calculate directly the character- i s t ics of singularities of the ordinary Borel t ransform. Thus the results of large orders of perturbation t h e ~ r y ' ~ ~ , * * ~ ~ . ' ~ calculations may be repro- duced.

When instantons do occur in the functional integral, the contour integration of the modified Bore!, sum encloses singular points of the function B(z). Such singular polnts represent, in general, contributions to the functional integral which a r e not apparent in the perturbation theory about the vacuum. These contributions may be calculated from perturbation theory about the instanton solution. In t e rms of the modified Borel t ransform, the nonperturbative contributions may be identified a s the discontinuity across the branch cut originating at the instanton singularity. This precise separation of influences from multiple solutions to the Euclidean equations of motion solves the "double-counting" problem of perturbation theory.

The double-counting problem occurs when a functional integral has more than one solution to the Euclidean equations of motion, e.g., a vacuum solution and a n instanton solution. The problem i s to combine properly the information contained in the perturbation expansion about each solution without "double counting" the contributions of some parts of configuration space, e.g., regions inter- mediate between two solutions. The solution given here i s that each perturbation se r i e s corresponds to a branch cut of the modified Borel transform.

A final claim for this method i s that i t i s a more

2370 @ 1979 The American Physical Society

19 - M E T H O D F O R B O R E L . S U M M I N G I N

delicate tool for examining the details of the analytic s tructure of the Borel transform of a functional integral than the large- order -expansion saddle-point r n e t h ~ d . ' ~ ~ ~ ~ ~ ' This i s due to the ability t o write a functional expression for the modified Borel transform. It i s not necessary to f i r s t estimate the large orders of perturbation theory in the coupling constant and then deduce what singularities of the Borel transform would cause them. The singularities and branch cuts may be calculated directly. This simplifies the calculational and conceptual problems of using the Borel transform.

This is the f i r s t of two papers on th is subject. In this paper, the modified Borel t ransform i s introduced and applied to simple functional integrals . Some of the examples do not have instanton solutions and have been studied elsewhere using the large-order saddle-point r n e t h ~ d . ' ~ ~ ~ ~ * ~ ~ ~ " This allows comparison of the modified Borel t ransform with more familiar techniques. Many of the examples studied, including the instanton example, a r e tr ivial functional integrals in the sense that they represent zero-dimensional space-time field theories, that is to say, ordinary integrals. These examples were chosen for pedagogical ease of presentation. Generalization to l e s s trivial field theories requires no more or no l e s s difficulty than generalizing a discussion of the saddle-point approximation from ordinary integrals to function- a l integrals , a s may be seen from the discussion of one-dimensional space-time field-theory examples, in the Appendix.

In the second paper of this se r i e s , the methods introduced in this paper a r e applied to the nontrivial example of a one-space-time-dimensional field-theory (quantum-mechanics) functional inte- g ra l which has the instanton problem. The results of the modified Borel sum a r e compared to numer- ica l calculations of the energy eigenvalues of the double-well anharmonic oscillator.

The outline of the present paper i s a s follows: In Sec. 11, a very brief review is made of ordinary Borel summation and how large-order saddle-point calculations make i t practical. In Sec. 111, the modified Borel sum i s introduced. Certain con- sequences of i t s definition a r e explored and i t s reiationship to the ordinary Borel transform made explicit. In Sec. IV, the MBT i s applied to tr ivial functional integrals. These examples demonstrate the results of Sec. 111 and introduce the techniques necessary for nontrivial field-theory applications. An integral with a n "instahton" i s studied numerically to prove that the predictions of the MBT a r e correct and that the proper prescription for the instanton singularity is given by the MBT. The Appendix examines the Borel singularities of sev-

S T A N T O N S I N G U L A R I T I E S : ... 2371

era1 one-dimensional field theories which have been reliably calculated using the usual large-order saddle-point methods.' The results of the Appendix a r e not new, but a r e included for com- pleteness and to allow comparison of the MBT with the usual methods in cases where no instanton i s present.

11. A REVIEW

In the following, we review the Borel summation method with emphasis on application of the large- order methods and the conformal-mapping methods.

It i s generally accepted that the formal power s e r i e s obtained by expanding a Euclidean path integral about a saddle point in powers of the coupling constant i s at best asymptotic, i.e., C E n ( g 2 ) " i s the formal perturbation se r i e s for E ( g 2 ) and for any N

N

E ( g 2 ) - C E n g 2 " li m n =O

g 2 N = o .

*2+0 (2.1)

Note the difference between this and a convergent power series. For fixed g2, an asymptotic s e r i e s can only approximate the t rue value of E ( g 2 ) even if a l l E n a r e known. The usual algorithm for the best approximation is to terminate the sum at the smallest n such that l ~ ~ ~ ~ ~ l s J E , + , ~ ~ ( ~ + ~ ) ~ . The last t e r m in the ser ies est imates the size of the e r ro r ,

For some asymptotic ser ies , the method of Borel summation allows E ( ~ ' ) to be calculated with arbitrari ly small e r r o r , even for large values of the coupling constant, provided sufficient t e rms in the perturbation se r i e s a r e available. Suppose E may be written a s a Laplace transform

E ( g 2 ) = jm dr e - z h 2 ~ (z) . 0

Let F be called the (ordinary) Borel t ransform of E. Denote the integral a s the Borel sum. What can the asymptotic perturbation se r i e s for E tell about ~ ( z ) ? If F has a power-series expansion (plus a 6 function to represent t e rms in E independent of g)

then i t is simple to show

The a, a r e clearly much more convergent than the E n a s a power series.'' In fact,, since pertur-

2372 W I L L I A M Y . C R U T C H F I E L D I 1

bation se r i e s in field theory tend to diverge like f a c t ~ r i a l s , ~ ~ ' ~ ~ ~ the power s e r i e s for F will be convergent in some circle about the origin and F will be an analytic function. Suppose also that the analyticity of ~ ( z ) allowed continuation from this circle of convergence to the entire positive z axis, a s i s necessary for evaluation of the integral (2.2). Given these assumptions, then E ( g 2 ) could be r e - covered from i t s asymptotic s e r i e s by Borel summation.

It is possible to show for some functions E ( g 2 ) that i t s Borel transform ~ ( z ) exists with the r e - quired properties, using Watson's theorem.13 The hypotheses of Watson's theorem require bounds on the asymptotic ser ies and on the size of the singularity-free region of E (g2 ) . In practice, rigorously fulfilling the hypothesis of Watson's theorem i s difficult, although i t has been done for some interesting quantum-mechanical prob- l e m ~ . ~ ~ ~ ~ ~

A practical problem in Borel summation i s how to proceed when only a few t e rms of the perturbation se r i e s for E ( g 2 ) have been calculated. In practice, a few t e rms can reasonably well represent ~ ( z ) near the origin, but the integral over z extends to infinity. The radius of convergence of the power s e r i e s is limited by the closest singularity. Outside this radius, a partial sum need not even approximate ~ ( z ) .

The generic solution to this problem i s to conformally map the z plane s o infinity i s brought near the origin while the singularities of F a r e equally distant.16 Then the f i r s t few t e r m s in the power s e r i e s in the conformally transformed variable approximate F uniformly along the contour of integration.

But this requires knowledge of the location of the singularities of F. Since the nearest singularity determines the large-order behavior of the power s e r i e s of F and hence E'S asymptotic power ser ies , the information i s contained in the large orders of perturbation theory. But clearly direct computation of the large-order behavior i s out of the ques- tion.

Recently, a method has become popular for approximately computing the large orders of perturbation theory for functional There a r e several variations on the basic method, but al l involve making a saddle-point integration on a Euclidean path integral. For example, the Kth order of perturbation theory for the partition function

i s evaluated a s a double saddle point in q ( t ) and g. I/K corrections to the leading behavior have been

FIG. 1. Singularity structure of Borel transform of the eigenvalues of the single-well anharmonic oscillator. Singularities occur a t integer multiples of "bounce" action. Branch cuts a r e present, but not shown.

calculated by perturbation theory about the saddle point. Agreement with available numerical and analytic computations has been e x ~ e l l e n t . ~ ~ ' ~ ~ * ~

As an example of the techniques above, consider the situation which exists for the eigenvalues of the single-well, anharmonic oscillator. The large- order saddle-point techniques show the singularity of F nearest the origin to be a t z =-So. (-So i s the action of the pseudoparticle o r "bounce" solution.) Further singularities a r e at integer multiples of -So (see Fig. 1). A convenient conformal transformation i s 3

The singularity structure of F(a) i s shown in Fig. 2. The Borel contour of integration now l ies within the radius of convergence of the power s e r - i e s in a. The t rue value of E ( ~ ' ) may be approximated arbitrari ly well using partial sums for F(a) in the integral (2,2). Numerical tes t s of this method have Deen extremely s~cces s fu l . ' ' ~

It may be possible to improve this method if

FIG. 2. Singularity structure of F(cu). Singular points a r e now located on a unit c i rc le , a s is the point z = *. There a r e no singularities inside the unit c i rc le . The singularities accumulate a t a = 1.

19 - M E T H O D F O R B O R E L - S U M M I N G I N S T A N T O N S I N G U L A R I T I E S : . . . 2373

more about the analytic structure of F were made available. Knowledge of the nonleading singulari- t ies could increase the accuracy of the approximation when a finite number of terms of perturbation theory a re available, by fitting the approximate ~ ( a ) to have these singularities. In any event, no amount of conformal transformation will help if a singularity exists on the positive real axis where F must be integrated. For this case, a generalization of the Borel summation technique i s required.

111. MODIFIED BOREL TRANSFORM

The proposed modified Borel transform i s analogous to the following suggestion made by ' t H ~ o f t . ~ Consider a Euclidean functional integral which has been rescaled a s in a loop expansion

We desire to express G ( ~ ' ) a s a Borel sum of some function F

It i s apparent that if F (z ) exists i t must be formally expressible a s

F i s proportional to the volume in path space of paths having action equal to z. Perhaps an even better analogy for F , considering that Euclidean functional integrals a r e equivalent to a statistical mechanics, i s the exponential of the entropy of a microcanonical ensemble. As i s usual in these statistical mechanical analogies, the action S ( A ) corresponds to classical energy, while the coupling constant g2 i s like the temperature.

A direct result of this expression for F i s that F (z ) i s singular for z equal to the action of a classical solution, This is in agreement with the predictions of the saddle-point method. However, i t is not immediately clear how to evaluate F at negative or complex z , a s would be necessary to examine singular points of F which were not positive or real , or how to compute F perturbatively. Nor does 't Hooft's suggestion tell how to integrate over the singularity on the positive z axis in the case of instantons.

To handle these problems, a generalization of the ordinary Borel transform is needed. Consider again the functional integral of (3.1). Functional integrals a r e the N- rn limit of integrals like

where X is the usual normalizing factor. By an application of Hankel's integral representation of the I? function17 this can be rewritten suggestively a s

where @ is a contour encircling the range of S, which is normalized to run from 0 to .o. An in- terchange of integrations i s allowable for finite M and reasonable actions yielding

A final step is to take the functional integral lim- i t , ~ ( z ) =lim,,B,(z). Note that since the denominator is raised to power N/2 + 1, naive power counting guarantees convergence even when S is only quadratic (as in free field theory).

It i s easy to deduce certain properties of the B,(z), and hence of B(z). The integral for B,(z) is very much like a Feynman parameter integral. Thus the Landau conditions may be applied to ask when BN(z) is singular.18 The Landau conditions have a direct interpretation in terms of solutions to the Euclidean equations of motion: B,(z) may be singular for z such that there exist xf for which

This means BN(z) may be singular for z equal to the action of a solution to the classical Euclidean equations of motion (discretized). This is true for complex a s well a s real-valued solutions. Such solutions a r e generally called pseudoparticles or instantons. I choose to use pseudoparticle a s the generic term for all such solutions, while real solutions (positive action) will be called instantons. We shall suppose, a s is usual in path-integral manipulations, that in the large-N limit, the solutions to the discretized equations approach a limit and identify that limit with the solution to the con- tinuum equations of motion.

In general, attached to each singularity will be a branch cut. Later explicit computations will be made describing these branch cuts. Because B(z) is integrated along a contour @ encircling the positive real axis, i t i s convenient for real , positive singular points to have their branch cuts running along the positive real axis.

B(z) is defined a s being evaluated on the f i rs t sheet when the branch cuts a r e a s described above. An equivalent way to describe B(z) on the first


sheet i s t o define i t by (3.6) when the xi a r e integrated over rea l values. Other sheets may be reached by allowing z to pass through one o r more cuts on the positive rea l axis. During such an analytic continuation, the contours of integration must be deformed away from the rea l axis to avoid the singular points.

The above implies that the only singularities on the f i r s t sheet a r e on the positive r ea l axis. These can only ar i se from rea l solutions with r ea l positive action. The contour integral for the modified Borel sum need not pass through these singulari- t ies since the contour passes above and below these singularities. Thus the positive-action pseudoparticle problem i s avoided.

Consider how this description reduces to the ordinary Borel sum when no instantons a r e present. The only singularity on the non-negative r ea l z axis i s at z = 0, the singularity arising from the vacuum solution. There i s a cut extending from th is singularity to positive infinity (see Fig. 3). The contour C may be shrunk onto the positive r ea l axis demonstrating the relationship between F and B,

This means that the discontinuity of ~ ( z ) may be calculated by applying the ordinary Borel t ransform to perturbation theory. This will be demonstrated by example in the following section.

B will still , in general, have singular points, but by the above assumption, they will not exist on the f i r s t sheet. The existence of such points is st i l l of interest , even when there a r e no instantons, because the singular points of ~ ( z ) will, in general, also be singular points of ~ ( z ) . These points may be reached by analytic continuation through the branch cut on the positive r ea l axis. The analytic continuation will require deformation of the hypercontour of integration.

When instantons occur, there will be other contributions to the Borel sum other than those visible in perturbations about the vacuum. The instanton singularities will have branch cuts attached to them. The total discontinuity of B will be a sum of discontinuities across branch cuts. In Fig. 4, the contour c has been deformed to encircle each branch cut separately. Note how at point A the total discontinuity i s a sum of two contributions. One i s the analytic continuation of the discontinuity a t z <S, in other words, the analytic continuation of F past i t s instanton singularity. The second i s the discontinuity across the new branch cut, a contribution which cannot appear in perturbations about the vacuum configuration. This discontinuity, a s shall be shown later , may be calculated

FIG. 3. Contour of integration C encircles pole and cut of B e ) . Singular points of B on second sheet a r e not shown.

from perturbation theory about the instanton solution. This separation of contributions from each perturbation theory i s my claimed solution to the "double-counting" problem.

The modified Borel transform i s seen to be more general than the ordinary Borel transform because ~ ( z ) i s defined to be an analytic function, while discB(z) will be a boundary value of an analytic function, allowing distributions a s well a s analytic functions. In the following sections B(z) will be shown to have singularities like (z -So)-*, for example. The discontinuity of such a singularity is proportional to 6"-'(2 - S o ) .

The following sections will show how the MBT may-actually be used in calculations. The conceptual advantages of the MBT may be summar- ized now:

(1) B(z) mky be a more general function. (2) B(z) has a clear meaning when z i s complex. (3) A clear prescription for handling singulari-

ties at rea l positive z can be provided. The only singularities on the f i r s t sheet a r e for rea l positive z .

(4) Contributions from various classical solutions a r e disentangled by the separate discontinuities.

(5) B(z) may be calculated directly. While in

FIG. 4. Contour C i s replaced with multiple contours, each of which encircle a single branch cut. At pointA the total discontinuity i s the sum of two discontinuities.

19 - M E T H O D F O R B O R E L - S U M M I N G I N S T A N T O N S I N G U L A R I T I E S : ... 2375

the following sections results predicted by the MBT will be verified by comparison with existing calculations of the large orders of perturbation theory, i t should be emphasized that the analytic structure of B is the interesting feature. The large orders of perturbation theory themselves a r e of little interest. Calculating the MBT allows direct visualization of the singularities, rather than the indirect information of the saddle-point method.

N. TRIVIAL EXAMPLES

The rest of this paper, and the succeeding paper, is devoted to illustrating the modified Borel transform with specific examples. The majority of this section i s devoted to analyzing path integrals in zero space-time dimensions, that i s , ordinary integrals. Despite the apparent triviality of this subject, i t i s a good introduction to the techniques used for nontrivial path integrals. In order to allow easier correspondence between these examples and real path integrals, the terminology of functional integrals will be used throughout. For example, the point at which the first derivative of the action is zero wi l l be called the classical solution to the equations of motion. There a re additional complications in real field theories arising from the near degeneracy of the action under translation of the instanton, but this will be introduced later. An additional advantage of beginning with ordinary integrals is that the integrals may be computed numerically and compared to the results of the modified Borel sum.

(A). In f ree field theory, the action i s purely quadratic in the path variable. For the first example, consider a discretized version of such an action

where the vector 5 i s the discretized path and A is a positive-definite matrix of dimension N. The modified Borel transform corresponding to this discretized free field path integral i s easily computed

In this example, the only N dependence resides in the determinant of the NX N matrix 2. This N dependence is conventionally removed by the path- integral normalization factor, which i s not explicitly included above. The factor of z-' yields a 6 function when i t s discontinuity i s taken, so the

modified Borel sum equals the value of the Gaus- sian integral, which i s independent of g 2 :

Because the action for this example is purely quadratic, there is no branch cut attached to the origin. In general, when S i s nontrivial, there will be a branch cut which will yield a complicated g 2 dependence from the integration over i t s discontinuity. As will be seen in less trivial examples, the singularity arising from a solution with no translational degeneracy is proportional to z-I

and the inverse square of the constant of propor- tionality i s the determinant of the matrix for small oscillations about the classical solution. In the Appendix, i t will be shown that the degree of the singularity is determined by the number of degenerate modes among the small oscillations,

(B). We now turn to the f i r s t of the promised pedagogic integrals. A favorite example is1'

This action does not possess instantons, that i s , real solutions to the Euclidean equations of motion with positive action. This complication is de- ferred until the next example. ~ ( g ~ ) has an asymptotic series expansion

which may be Borel transformed according to the prescription of Eq. (2.4)

It is easy to show that the coefficient of z n becomes asymptotically

B(z ) has the integral form

Clearly, this has no pinched singular points at real values of x. There are , however, pinched points at imaginary x which will cause B to be singular at z = -+ on the second sheet. This is an important fact because i t controls the radius of convergence of B expanded about the origin.

First let us find an expansion for B. One convenient method is to write B in an exponential form again using the definition of the I' function:

2376 W I L L I A M Y . C R U T C H F I E L D I 1 - 19

B(z) = / d x /"df t l / ' exp[- t ( ix2+ix4-z) ] These manipulations require z < 0, but after the

d% calculation i s over, z may be analytically continued -.. t o positive values. The integration over t may be

= 2 rixe-2i rim dt dt-x4/4t . (4.9) done using an integral definition of Kl

(4.10)

B(z) = l /z + ln(-z)Bl(z) + B,(z) , (4.11)

where

B, equals the power s e r i e s in (4.6), a s predicted by the relationship between F and B, (3.8). B2 may also be expressed in a convergent power ser ies , but since it does not contribute to the discontinuity, we shall not write i t down. As claimed earl- i e r , the 1/z singularity depends only on the second derivative of the "action" a t the "vacuum" (x= 0) solution. This decomposition of B around a singular point into singular t e rms , an analytic function multiplying an elementary function with a discontinuity, and a residual analytic function, i s a feature which will reappear in these examples. In general, the singular t e rms and the analytic function analogous to B, a r e calculable from ordinary perturbation theory (Feynman diagrams). In general, the function analogous to B2 is not calculable using ordinary functional techniques; the ability to calculate B, i s special to one-dimensional integrals.

There i s one final point to discuss for this mod- e l integral: the singularity which f ixes the radius of convergence of the power-series expansions for B. This study will give experience which shall be useful with the next model integral where the singularity is an instanton singularity. Also, i t is necessary to find this singularity to apply the program of (ordinary) Bore1 summation described in Sec. 11.

The singular points of F, that i s B,, a r e the same a s those of ~ ( z ) . The pinched singular points of ~ ( z ) a r e easily seen to be a t x=*i , and hence B is singular at z =-a. By the reasoning of Sec. 111, this singularity does not occur on the f i r s t sheet [on the f i r s t sheet, the denominator of (4.8) is clearly bounded away from zero]. The singular point will be reached by analytically continuing z

from the f i r s t sheet, through the positive r ea l axis , to z =-a. In this process, the points in x space which a r e solutions to x2/2 + x4/4 - z = 0 pass from a configuration which does not pinch the contour of integration, to one that does. This may be seen explicitly. The singular points in the integration variable a r e

There a r e four such points; the signs above a r e independent. When z equals -a on the f i r s t sheet, the points have pinched together, but without the contour between them. Consider z a s i t follows a path from -$, around zero, and back to -4. Fig- ure 5 shows how the singular points move when z is on this path. The contour of integration has deformed to avoid the points. When z reaches the end of the path, i t is convenient t o rotate the contour of integration, a s shown i n Fig. 6. The sense of the contour rotation i s determined by the direc- tion in which the path of z circled zero.

After rotation of the contour, ~ ( z ) may be expressed a s

In the above, the denominator i s positive definite for z < -2, and the previously used techniques for decomposing B may be applied, with the result

C, and C, a r e analytic functions with a convergent

M E T H O D F O R B O R E L - S U M M I N G I N S T A N T ' O N S I N G U L A R I T I E S : ...

FIG. 5. x contour of integration deformed to avoid poles a s z enters second sheet.

power s e r i e s in (z +a). C,(z) i s the Borel t ransform of perturbation theory about either minimum of the double-welled action in (4.14). The pole t e rm inside the parentheses i s the usual one for a minimum of a potential whose second derivative equals 2. The leading factor of 2 multiplying the singular t e r m s occurs because there a r e two independent pinched singular points, each of which contributes independently for z in a finite neighbor- hood of -$. The leading factor of i occurs because of the contour rotation. If z had circled zero with an opposite sense, -i would replace z. The radius of convergence of C, and C, i s se t by the nearest singular point, which i s the usual vacuum singularity a t z = 0.

There i s a final consistency check to make on this calculation. The asymptotic form of the perturbation-series expansion for ~ ( z ) i s given in (4.7). The singularity of B(z) a t z = -+ not only determines the radius of convergence of the power- s e r i e s expansion, but also the asymptotic form of the s e r i e s coefficients in B,(z), that i s , F(z) . It i s easy to check that the predicted asymptotic form agrees with the calculated asymptotic form. Note simply that on the second sheet at negative z

FIG. 6. x contour of integration rotated.

The leading singularity of B is the pole t e rm

The pole t e rm has a power-series expansion around z = 0. The coefficient of z n is

This agrees with (4.16) and (4.7). It i s easy to find a lso the nonleading corrections to the power- s e r i e s expansion of F (2) about z = 0, by computing the expansion of C,(z) ln(-z - i).

In summary, this example demonstrates how the modified Borel transform reduces to the ordinary Borel transform when no instantons a r e present. Around singular points, B may be decomposed in- to t e rms which may be computed from perturbation theory (Feynman diagrams of a trivial type) and which contribute to the discontinuity, and a n analytic function which does not contribute t o the discontinuity and, in general, i s not computable using functional methods. The singular points of P a r e identified a s the singular points of B. Their location i s given by solutions to the equations of motion. In this example in one dimension, the leading singularity i s a simple pole whose normalization is given by smal l oscillations about the "classical solution." The nonleading singular- i t ies may be computed by perturbing about the "classical solution." A rotation of the contour of integration is necessary to make the computation feasible.

(C). This finally brings us to the most interesting of these one-dimensional exampleslg

where

This potential has two minimums a t x=O, 1. There is another "classical" solution at x = $ , with positive "action" equaling kLi, This i s an instanton solution because the point in x space is r ea l and i t s action i s positive.

An asymptotic expansion for T (g2 ) may be calculated by perturbing about the two saddle points of v:

where

The perturbation ser ies about each saddle point have been added together in the above, since no

W I L L I A M Y . C R U T C H F I E L D I 1

"mass" equal to 1. B, may be computed by perturbing about that single well

+ 522 The total discontinuity of B i s twice that arising from a single well.

5 0 The important new feature of this model integral

14 12 10 z 8 z 6 z 4 is the instanton solution a t x=$. It i s necessary q2 to compute B for z near & in order to understand

F ~ G . ~ ( ~ 2 ) againstg2. ~~~i~~~ occurs for the correct prescription for integrating over the g2 approximately equal to So, the instanton action. singularity. Let V be expanded about y = x - g:

better resolution of the double-counting problem exists when using saddle-point methods. Figure 7 shows the results of numerically integrating ~ ( g ' ) . The value of T r i s e s with increasing g2 until about g 2 =&, at which point i t begins to de- crease. Since the t e rms in the asymptotic s e r i e s a r e a l l positive, the approximation must fail for g 2 of that order. Not surprisingly, this turnover point i s a t g2 approximately equal to the instanton action.

Using now familiar techniques, the expansion of ~ ( z ) around z = 0 i s easily computed,

where u =&-z . For z < 0, that i s o>&, the inte- g ra l over y i s convergent, a s i s easily seen using the large-argument approximation to K,. When z i s analytically continued to la rger values, the contour of integration over y must be rotated so that the integrand goes to zero a s (y ( - m. The sense of the contour rotation i s determined bv the r e - quirement

~ e [ ~ ~ ( $ - &)I - - - a s ~ Y I - ~ . (4.24) =2[J2s'z - ln(-z)B1(z"i "(') . (4b21) Let z pass below zero in the lower half plane so

B, and B, a r e analytic power-series expansions that the contour rotates counterclockwise. After about z = O . The pole te rm in brackets is the usual performing the rotation, K, may be expanded for singularity from a minimum of the "action" with smal l a ,

ylliuk + t e r m s analytic in a B(z) = -i /dy edy2/4(; + (1no)y c 2k11 ! (k + 2) ) 2 3 67 (;L (3 ((I) + t e r m s analytic in a . = 27ri - - - ( h u b i, (320)' k! (;+43 !

k -0 ) Note the factor of i multiplying the expression. If z had passed zero in the upper half plane, a factor of -i would occur instead. This i s a s required by the Schwartz reflection principle.

Before we use (4.25) to deduce the correct singularity prescription, (4.25) may be used to pre- dict the asymptotic form of T,. Just a s in the pre- vious section, the coefficients of the power s e r i e s l o r F approach an asymptotic form for large K which i s determined by the nearest singularity of B. The location of the singularity a t positive z , ra ther than negative z , does not change this fact, although i t does affect the particular asymptotic form. In particular, the leading singularity i s the pole at z equals &-. It causes the T, to have the same sign asymptotically, rather than alternate in

, sign. The reader may demonstrate the agreement between the asymptotic form and (4.20).

Let the contour (3 in the modified Bore1 sum which encircles the positive rea l axis be decomposed into two contours (3, and e 2 , a s in Fig. 4. el surrounds the singularity at z = 0 and the branch cut originating there. e, surrounds the singularity a t z =& and the branch cut beginning there. By shrinking the contours onto the cuts, the contour integrals become integrals of the discontinuity. (4.25) implies that the discontinuity on e , is purely rea l . Since the total discontinuity is imaginary by the Schwartz reflection principle, the integral over c?, need not be considered. It i s sufficient to take the imaginary part of the discontinuity across the branch cut beginning at the origin. In other words,

19 - M E T H O D F O R B O R E L - S U M R I I N G I N S T A N T O N S I N G U L A R I T I E S : . . . 2379

i t i s sufficient to integrate the real part of B, on a contour from zero to infinity that avoids the singularity

Such a prescription i s equivalent to a principal- value prescription for the singularity. It must be emphasized that this i s not the general prescription fo r integrating instanton singularities. This prescription is peculiar to one dimension. A prescription appropriate to instantons with translational invariance will be discussed in the following paper.

Now that a prescription for the integral is known, i t remains to show how to apply i t when only a finite number of terms in the power-series expansion of F a r e known, a s i s usually the case. Then the result of this Borel improved perturbative calculation may be compared to numerical values for ~ ( g ~ ) . As in Sec. 11, i t i s convenient to make a conformal change of variables

So is the instanton action, &. An expression of F in powers of a, may be derived, given the ser ies expansion in z :

Clearly, the radius of convergence extends to /a, / = 1. Thus a convergent expansion for T, which may be approximated by truncating the sum, is

T ( ~ ' ) = c fn [Re ~ d z e - ' " ~ a (z)"] n

The In a re universal numbers, not restricted to this example alone since the dependence on So has

1 1 ~ ~ ~ ~ '2 9 8 2-7 2-6 z ' ~

g2

FIG. 8. The difference between the true T ( g 2 ) and two approximations is plotted as a function of g2. The asymptotic series approximation uses the algorithm given in Sec. I. The MBT approximation uses eight terms in the sum (4.29).

been removed. In Fig. 8, the results of this approximation scheme a r e compared to the best possible approximation from an asymptotic series. E r ro r i s plotted against values of g2 for the asymptotic ser ies , and for the ser ies (4.29) truncated at eight terms. The MBT method i s at least 2.5 orders of magnitude better over the given range, which extends to g2 =So. Of course, this method may be made arbitrarily accurate by increasing the number of terms in the sum.

Note that only two pieces of information from the MBT calculation a r e used to calculate the above approximation: the location of the singularity and the proper singularity prescription. It is possible to improve the accuracy by including other information about the instanton singularity. The asymptotic form of the T, is determined by the leading singularity at So. If the leading singularity i s treated separately in the Borel sum, the quantity of interest becomes the deviation of TK from i t s asymptotic form. To be precise, in the Borel sum, write F(z) a s

Since the leading singularity has been subtracted away, the b, will be smaller than the fK (for large K), s o the partial sums will converge faster.

The second term may be explicitly integrated in the modified Borel sum. The results of using these improved convergence schemes a r e shown in Fig. 9. Treating the pole term in this manner

FIG. 9. Error is plotted against g2 for three approximations. A is the MBT approximation using 1 6 terms in the sum, with no explicit instanton correction. B is MBT with 8 terms, but with explicit instanton correction. C is 15 terms with explicit instanton correction.


yields an improvement roughly equal to including eight extra t e rms in the Borel sum. Further im- provements in accuracy follow from treating the log singularities similarly.

This model integral i s atypical of instanton integrals because the discontinuity across the instanton cut makes no contribution to the final answer. Keeping in mind this caveat, i t i s interesting to consider how the double counting of perturbation theory is resolved. There a r e three "classical solutions": two vacuum solutions and the instanton solution. Perturbation theory about the instanton solution i s not required, since the instanton discontinuity does not contribute in this case, but this does not mean that the instanton i s not important in the solution. F (z ) , arising from the perturbation se r i e s about the two vacuum solutions, diverges a t z =So precisely because of the jnstanton. This i s an example of a remark made ear l ie r , that in general an instanton makes two contributions: one which i s visible in the perturbation theory of solutions of lesser action, and the discontinuity contribution which has no effect on the solutions below.

One might wonder why the vacuum solution at x = 1 does not effect the perturbation se r i e s about the x=O vacuum and perhaps cause i t to diverge at x = 0. The answer i s that i t does, but not on the f i r s t sheet. Since the low orders of perturbation theory only probe the region near the saddle point, the discontinuity across the x = 0 branch cut will not diverge at x=O. But if the discontinuity is analytically continued around z =So and back to z = 0, i t willdivergebecause of the x = 1 vacuum solution. We might call such singularities hidden. It i s easy to s ee that if there exists a path in configuration space from one solutioi~ to another, along which the action is monotonic, then the singulari- t ies will not be hidden from each other.

Thus the instanton solution i s not hidden f rom either vacuum solution. The discontinuities from both vacuum solutions diverge a t z =So. Note that the total singularity at z =So calculated in (4.25) is that predicted by adding the perturbation theories about both vacuum solutions.

This model integral has illustrated the key steps necessary to Borel sum an instanton singularity: (1) locate the singularity, (2) determine an expansion of B about the singular z , (3) find the correct prescription fo r integrating over the singularity. If necessary, compute new discontinuity contributions for perturbation theory, (4) conformally map to a unit circle and calculate the expansion in the new variable, (5) integrate using whatever information about the singularity of F i s available. These a r e essentially the steps used in the sequel paper.

ACKNOWLEDGMENTS

I would like to thank C. Callan, J. Zinn-Justin, A. Sokal, and J. Collins for many useful conversa- tions and suggestions. I wish especially to thank S. Libby for his patient reading of the manuscript.

This work i s supported in part by NSF Grants No. PHY 76-15328 and No. MDS 75-22514.

APPENDIX: QUANTUM-MECHANICAL EXAMPLES WITHOUT INSTANTONS

In the preceding sections of this paper, the modified Borel transform was applied to trivial examples, principally ordinary integrals. In this Appendix, the MBT i s applied to functional integrals from quantum mechanics. The pedagogical purposes a r e multiple. First , i t will demonstrate the generalization of the MBT to true functional integrals. The main problem encountered i s the translational invariance of pseudoparticle solutions. The problem i s resolved by the collective coordinate method. The second object is to rede- rive a result already calculated using the large- order saddle-point method. The third example of the Appendix, the single-well anharmonic oscillator, has been considered in Ref. 1. This will allow the reader to compare the two methods a s applied to this example. The third purpose i s to demonstrate the simple relationship between the number of zero modes of a pseudoparticle and the degree of i t s singularity in the Borel plane. Al- though this result i s implicit in the results of the large-order saddle-point method, i t i s particularly simple and direct to show this using the MBT.

The f i r s t example will s ~ m p l y be to compute B(z) around z = 0 for a quantum-mechanical action. The second example will compute B(z) for an integral whose lowest-action real solution i s translationally invariant. This solution i s an instanton solution, but since there i s no lower-action solution, there i s no subtlety to integrating over i t s contribution. The only technique to be introduced here i s to remove the translational degeneracy through collective coordinates. This technique i s familiar from ordinary saddle-point evaluation of functional integrals. The translational degeneracy will be shown to affect the order of the singularity in a predictable fashion.

The final example serves a s a model for non- instanton applications of the RIBT, that is, applications in which a classical solution to the Euclidean equations of motion determines the radius of convergence of F ( z ) around z = 0, but the singularity does not lie on the Borel integration contour. The singularity caused by a time- translation-invariant pseudoparticle solution i s

19 - M E T H O D F O R B O R E L - S U M M I N G I N S T A N T O N S J N G U L A E I T I E S : . . . 2381

calculated. Again, the connection between degenerate modes and the degree of singularity i s noted.

(A). The action to be considered has te rms cubic and quartic, a s well a s quadratic, in the position variable x(t). When the discretized action i s expanded about the absolute minimum of the action and any trivial constant removed, the action, S, has the form

S = + $ . Z . ~ ; + S ( ~ ) + S ( ~ ) . (5.1)

The x , are coefficients of the eigenvectors of 2

e

in the path. A i s the s~nali-oscillations matrix, which i s positive definite because the minimum is absolute. S(3) and SC4) a r e cubic and quartic in the coordinates x i .

The functional integral will be normalized, a s i s usual, by division by the same integral with S( ' j and S(4) set to zero. Then B, may be writter.

where

In the exponential, the large-t behavior i s con- variance, only te rms with even powers of s ' ~ ' need trolled by z, while the sm all-t behavior i s con- be kept. The factors of t-' In the resulting ex- trolled by s '~ ' . Since the small-t region i s dom- pansion may be represented a s derivatives wi th inated by Sf4) , e x p ( - ~ ( ~ ) / G ) may be expanded a s respect to S'4'. Letting N - m, B ( z ) becomes a power se r i e s in s '~) . Because of n, - - x , in-

As usual, B, i s the Bore1 transform of ordinary perturbation theory. l/z i s the usual pole whose discontinuity yields a 6 function. B, demonstrates the assertion made ear l ie r that the residual analytic function i s not computable using ordinary functional techniques, because of the logarithm of Sc4' which occurs. Note that the simplicity of the analytic structure near a = 0, found in Sec. IV, persists for true functional integrals.

(B). In the preceding example, the small-oscillations matrix about the classical solution i s positive definite. Fo r solutions possessing a con- tinuous symmetry, for example, time translation invariance, the small-oscillations matrix will have a zero eigenvalue for each such symmetry. Collective coordinate techniques a re used to re-

move the degeneracy." As a concrete example, conslder the path in-

tegral for the trace of P exp(-TH), where P 1s the parity operator, x - - x , and H i s the Hamiltonian of the double-well anharmonic oscillator:

H=&2++{+-g%)2(++gx)2/g2. (5.4)

This path integral integrates over end coordinates x ( 0 ) = -x (T) ; the paths a re antiperiodic with period T . Such an integral would be used to find the splitting between the ground and f i rs t excited states of this Hamiltonian, w h ~ c h are degenerate when 2 = 0 (Refs. 5, 9).

The lowest-action classical solution of the path integral with i t s antiperiodic boundary conditions i s the one-instanton solution in time T. Since this

2382 W I L L I A M Y . C R U T C H F I E L D I 1 - 19

solution has the lowest action, there is no difficulty freedom of the parameter t, above. If the methods in integrating over i t s singularity. There will, of of the preceding section were applied to such a course, b e "instanton singularity problems" as - solution, the small-oscillations determinant would sociated with the n-instanton solution, f o r n = 3, be singular. The solution, a s i s usual, i t to in- 5 , . . . , but this does not interfere with the imme- troduce a constraint f o r the collective coordinates. diate calculation. F o r finite T , the instanton may Consider BN(z) with the action expanded about b e expressed in t e r m s of elliptic integrals. In the instanton solution f o r t ime T: infinite t ime, the rescaled instanton solution i s

r(ffN-1) Xinstanton(t) = & tanh(t - to) (5.5) sfidxi G .A. Z/2 + s4) + t ) a ,

and has action So=$. The t ime derivative of this (5.6)

f f = N / 2 + 1 , b = S , - z . solution i s an eigenvector of the small-oscillations matr ix with eigenvalue zero , reflecting the One is inser ted into the integral in the f o r m

. . T

J = J l + J 2 ( x i ) = d x ~ o ~ t ~ , n ~ t ~ t + t o ~ + ~ ~ l ~ o ~ ~ ~ ? l ~ ~ + ~ o ~ . l*O

i,bi a r e the normalized eigenvectors of A labelled in o rder of ascending eigenvalue. Note that J, i s independent of the xi. Because the action i s t ime translation invariant, the constraint can be integrated trivially

A t i s the restr ic t ion of A to vectors perpendicular to $,, and i s thus positive definite. The usual t rans - formations may now be applied,

(5.9)

The t e r m in square b racke ts differs f r o m the corresponding t e r m in (5.2) by >?. This occurs because xo was removed by collective coordinates. The constraint thus has an important effect on the analytic s t ruc- tu re of BN(z):

B ~ ( z ) = B , ( z ) + (So - Z ) - ~ " ~ B ~ ( Z ) ,

The r e a d e r may verify that the coefficient of (So -z ) ' / r ( l - + ) i n B, i s theusual t e r m f r o m p e r t u r - bation theory about the instanton solution a f te r translation invariance h a s been removed.

There a r e two points to be noted. F i r s t , the singularity associated with the solution has o r d e r - 2 . This a r i s e s direct ly f r o m the translational invariance. F o r a solution with n degenerate modes, the singularity would be (So - z ) ~ " ~ " ' ~ ' . Second, note that the perturbation s e r i e s h a s not been Bore l t ransformed using the prescr ipt ion given in Sec. I. The lth o r d e r i s divided by r ( l -a), not I'(l). The MBT mandates a different prescr ipt ion f o r this case, a s i s correct .

(C). As s tated e a r l i e r , this l a s t example will consider a functional integral whose Bore l t rans -

I - - - -

fo rm's radius of convergence is limited by a singularity at negative z . One technique which will be demonstrated i s the rotation of the contours of integration in a multidimensional integral during analytic continuation." Of pract ical in- t e r e s t i s the simple expression f o r the leading singularity in t e r m s of a functional determinant, a Jacobian, and the number of degenerate modes of the pseudoparticle solution.

The single-well anharmonic osci l la tor will b e considered f o r definiteness. It i s easy to general- i ze to other potentials. The Hamiltonian of the osci l la tor i s

H = L z p 2 + $2 + p x 4 . (5.11)

Let us compute the partition function, the t r a c e

19 M E T H O D F O R B O R E L - S U M M I N G I N S T A N T O N S I N G U L A R I T I E S : . . . 2383

of exp(-HT), with a functional integral. The trace i s enforced in the path integral by integrating over x(O)=x(T),

The partition function is not of direct physical interest, while the ground-state energy is. As is discussed in Ref. 11, the ground-state energy may be extracted from those parts of the partition function proportional to a single power of 1'.

B(z) for the partition function will be singular at z equal to the action of solutions to the Euclidean equations of motion. One solution is the trivial x(t) = O solution which causes the usual vacuum singularity at z =O. There are , however, two imaginary, translationally invariant pseudoparticle solutions which, in the infinite-T limit, a re

and have action

When z is on the first sheet, the integration hypercontour in B(z) is pinched at none of the pseudoparticle solutions. After continuation to z = - + on the second sheet, the hypercontour i s pinched at all of the set of translationally degenerate solu-

tions. The infinity of solutions again causes a zero-eigenvalue problem which must be removed by collective coordinates. Consider B (2 ) where the action has been expanded about the pseudoparticle solution:

The qi a re the normalized eigenvectors of the small-oscillations matrixA. $,>s the translation- mode eigenvector with eigenvalw zero. $, has a negative eigenvalue and will require rotation of the x, contour. A i s A restricted to the space perpendicular to $,. In rotating the x, contour, the integral acquires a factor of ;ti, the sign depending on the sense of the path z took to reach the second sheet. In terms of the new variable yo = i ix,, At is positive definite, and s ' ~ ' and s ' ~ ' must have explicit factors of *i inserted wherever yo appears in them. The modified cubic and quartic parts of the action are written with primes to emphasize this change:

P , i s the lth order of ordinary perturbation theory about the stationary point. A factor of 2 ar ises above because there a re positive and negative imaginary solutions a s shown in (5.13), both of which contribute.

The leading singularity of B ( z ) at z =So is

where

ldetAO 1 l I 2 & , P,=J -

detA J1 =Go.

A0 i s a normalizing functional determinant, usually chosen to be that of a purely harmonic oscillator. Note that the absolute value of detA is taken, so the determinant factor i s real (as opposed to the Jacobian factor, say). Thus the leading singularity of F is

2384 W I L L I A M Y . C R U T C I I F I E L D I 1 -- 19

IE. Brezin, J. C. LeGuillou, and J. Zinn-Justin, Phys. Rev. D 15, 1544 (1977).

'G. Par is i , Phys. Lett. E, 329 (1977). 3 ~ . Bervillier, J. IM. Drouffe, J. Zinn-Justin, and

C. Godreche, Phys. Rev. D 17, 2144 (1978). 4 ~ . Belavin, A. Polyakov, A. Schwartz, and Y. Tyupkin,

Phys. Lett. E, 85 (1975). 5 ~ . Gildener and A. Patrascioiu, Phys. Rev. D 2, 423

(1977). 6 ~ . G. Callan, R. Dashen, and D. J. Gross, Phys. Rev.

D 5, 2717 (1978). ?G. 't Hooft, lectures given a t the Ettore Majorana,

International School of Subnuclear Physics, Erice, Sicily, 1977 (unpublished).

8 ~ , Brezin, G. Par is i , and J. Zinn-Justin, Phys. Rev. D g , 408 (1977).

%. G. Callan, private communication. 'OL. N. Lipatov, Zh. Eksp. Teor. Fiz. Pls 'ma Red. 25,

116 (1977) [JETP Lett. 2 5 , 104 (1977)l. "J. Collins and D. Soper, Ann. Phys. (N.Y.) 112, 209

(1978). ''Experts will. be aware that there a r e infinitely Inany

ways to define the Borel transform in i ts more general

form, the Borel-LeRoy transform. This is a parti.- cularly useful choice made by G. ' t Hooft in Ref. 7.

1 3 ~ . H. Hardy, Divergent Ser ies (Oxford Univ. P re s s , London, 1949).

I4s. Graffi, V. Grecchi, and B. Simon, Phys. Lett. E B , 631 (1970).

155. D. Eckman, J. Magnen, and R. Senegor, Commun. Math. Phys. 2, 251 (1975).

"J. J, Loeffel, In Proceed~ngs of the 1976 Worltshop on Pad6 Approx~mants, e d ~ t e d by D. B e s s ~ s , J. G ~ l e - wtcz, and P. Mery, Report No. SACLAY-DPh-~/76- 20 (unpubl~shed).

'?I. Gradshteyn and I. Ryzhik, Table of Integrals , Ser ies and Products (Academic, New York, 1965).

Eden, P. Landshoff, D. Olive, and J. Polkinghorne, T h e Analyt ic S-Matrzx (Cambridge Univ. P re s s , Cam- bridge, England, 1966).

1 9 ~ . Fabricius and K. H. Miitter, Report No. WU B-78-5 (dnpubllshed).

2%A. Jeviclii, Nucl. Phys. u, 36 (1976), and refer- ences therein.

method for borel-summing instanton singularities: introduction

Documents