zotero/storage/LUZV4YBN/.zotero-ft-cache

PHYSICAL REVIEW D

VOLUME 57, NUMBER 2

15 JANUARY 1998

Divergence of perturbation theory: Steps towards a convergent series

Sergio A. Pernice Department of Physics and Astronomy, University of Rochester, Rochester, New York 14627

Gerardo Oleaga Departamento de Matematica Aplicada, Facultad de Matematicas, Universidad Complutense, 28040 Madrid, Spain
͑Received 14 August 1996; published 24 December 1997͒
The mechanism underlying the divergence of perturbation theory is exposed. This is done through a detailed study of the violation of the hypothesis of Lebesgue’s dominated convergence theorem using familiar techniques of quantum ﬁeld theory. That theorem governs the validity ͑or lack of it͒ of the formal manipulations done to generate the perturbative series in the functional integral formalism. The aspects of the perturbative series that need to be modiﬁed to obtain a convergent series are presented. Useful tools for a practical implementation of these modiﬁcations are developed. Some resummation methods are analyzed in the light of the above mentioned mechanism. ͓S0556-2821͑97͒04424-X͔
PACS number͑s͒: 11.10.Jj

I. INTRODUCTION

A typical quantity used to analyze the nature of the perturbative expansion in quantum ﬁeld theory is the partition function

͵1
Z͑␭͒ϭ Z0

͓d␾͔eϪS[␾],

͑1͒

with

͵ ͫ ͬ 1

1

␭

S͓␾͔ϭ ddx 2 ͑‫͒␾␮ץ‬2ϩ 2 m2␾2ϩ 4 ␾4 .

͑2͒

The normalization factor 1/Z0 is the partition function of the free ﬁeld (Z→1 when ␭→0). The analysis of the perturbative expansion of any Green’s function goes along similar
lines to that for Z. In the example above we consider a scalar ﬁeld theory for simplicity.
The traditional argument for understanding the divergent nature of the perturbative expansion can be traced back to Dyson ͓1͔. Although the form was different, the content of his argument is captured by the following statement: ‘‘If the perturbative series were to converge to the exact result, the function being expanded would be analytic in ␭ at ␭ϭ0. But the function (Z for example͒ is not analytic in ␭ at that value. Therefore, as a function of ␭, the perturbative series is either divergent or converges to the wrong answer.’’
Estimates of the large order behavior of the coefﬁcients of the perturbative series showed that the ﬁrst possibility is the one actually realized ͓2,3͔. That Z, as a function of ␭, is not analytic at ␭ϭ0 can be guessed by simply noting that if in its functional integral representation ͓Eq. ͑1͔͒ we make the real part of ␭ negative ͑though arbitrarily small͒, the integral diverges. In fact, there is a branch cut in the ﬁrst Riemann sheet that can be chosen to lie along the negative real axis, extending from ␭ϭϪϱ to ␭ϭ0 ͓4,5͔.
The above argument is very powerful and extends to the perturbative series of almost all other nontrivial ﬁeld theories. It has also motivated a series of very important calcu-

lations of the large order behavior of the perturbative coefﬁcients ͓2͔ and general analysis of the structure of ﬁeld theories ͓6͔, as well as improvements over perturbative computations of different physical quantities ͓2͔.
For all its power, it is fair to say that this argument, as is
typically the case with a reductio ad absurdum type of argument, fails to point towards a solution of the problem of divergence. It is only through the indirect formalism of Borel transforms that questions of the recovery of the full theory from its perturbative series can be discussed ͓5,7,8͔.
In this paper an alternative way of understanding the divergent nature of the perturbative series is presented. This way of understanding the problem complements the traditional argument brieﬂy described above, hopefully illuminating aspects that the traditional approach leaves obscure. In particular, as we will see, the arguments in this paper point directly towards the aspects of the perturbative series that need to be modiﬁed to achieve a convergent series. It is hoped that the way of understanding the problem presented here will help to provide new insights into the urgent problem of extracting nonperturbative information out of quantum ﬁeld theories.
In Sec. II we develop our analysis of the divergence of perturbation theory. In Sec. III we point out the ingredients that, according to the analysis of Sec. II, a modiﬁcation of perturbation theory would need to achieve convergence. We also present a remarkable formula ͑64͒ that allows us to implement such modiﬁcations in terms of Gaussian integrals, paving the way for the application of this convergent modiﬁed perturbative series to quantum ﬁeld theories. The proof of the properties of the function ͑64͒ is given in Appendix A. In Sec. IV we analyze recent work on the convergence of various optimized expansions ͓9–16͔ in terms of the ideas presented here. In Sec. V we summarize our results and mention directions of the work currently in preparation. Finally, in Appendix B, we apply the ideas of this paper in a simple but illuminating example for which we actually develop a convergent series by modifying the aspects of the perturbative series pointed out by our analysis as the source of divergence.

0556-2821/97/57͑2͒/1144͑15͒/$15.00

57 1144

© 1997 The American Physical Society

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DIVERGENCE OF PERTURBATION THEORY: STEPS . . .

1145

II. LEBESGUE’S DOMINATED CONVERGENCE THEOREM AND PERTURBATION THEORY

A. Wrong step in perturbation theory

Although the notation will not always be explicit, we will work in an Euclidean space of dimension smaller than 4 and in a ﬁnite volume.
Let us remember how the perturbative series is generated in the functional integral formalism for a quantity like Z:

͵ ͩ ͵ ͫ ͬ ͵ ͪ Z͑␭͒ϭ

͓d␾͔exp Ϫ

ddx

1 2

͑

‫ץ‬

␮

␾

͒

2ϩ

1 2

m

2␾

2

␭ Ϫ4

ddx␾4

͑3͒

͵ ͩ ͵ ͪ ͩ ͵ ͫ ͬͪ ͚ ϭ

ϱ ͑Ϫ1͒n ␭

͓d␾͔
nϭ0

n!

4

n
ddx␾4 exp Ϫ

ddx

1 2

͑

‫ץ‬

␮

␾

͒

2

ϩ

1 2

m

2␾

2

͑4͒

͵ ͩ ͵ ͪ ͩ ͵ ͫ ͬͪ ϱ
ϭ͚ nϭ0

͑Ϫ1͒n ␭ ͓d␾͔ n! 4

n
ddx␾4 exp Ϫ

ddx

1 2

͑

‫ץ‬

␮

␾

͒

2

ϩ

1 2

m

2␾

2

.

͑5͒

The ﬁnal sum is in practice truncated at some ﬁnite order N. The functional integrals that give the contribution of every order n are calculated using Wick’s theorem and Feynman’s diagram techniques with the corresponding renormalization.
We see then that the generation of the perturbative series in the functional integral formalism is a two-step process. First Eq. ͑4͒, the integrand, is expanded in powers of the coupling constant, and then Eq. ͑5͒, the sum, is interchanged with the integral.1
It will be convenient to have a simpler example in which the arguments of this paper become very transparent. Consider the simple integral

͵ͱ1
z͑␭͒ϭ

ϱ dxeϪ[x2ϩ͑␭/4͒x4]

␲ Ϫϱ

͑6͒

and its corresponding perturbative expansion

͵ ͚ ͩ ͪ 1
z͑␭͒ϭ ͱ␲

ϱ

ϱ ͑Ϫ1͒n

dx
Ϫϱ nϭ0

n!

␭ 4 x4

n
eϪx2

͑7͒

͚ ͵ ͩ ͪ 1 ϱ
ϭͱ␲ nϭ0

ϱ

͑Ϫ1͒n dx

␭ x4

n
eϪx2

Ϫϱ n! 4

͑8͒

ϱ

͚ ϵ ͑ Ϫ1 ͒ncn␭n.

͑9͒

nϭ0

This simple integral has been used many times in the past as a paradigmatic example of the divergence of perturbation

1In this paper we will often use the familiar word ‘‘integrand’’ to refer to eϪS or any functional inside the functional integration symbol. It would be more precise to preserve this word for eϪSint in the
measure deﬁned by the free ﬁeld. The terminology used here is,
however, common practice in the quantum ﬁeld theory literature
and also helps to emphasize the similarities with the intuitive ﬁnite
dimensional case presented below.

theory ͓8͔. It is therefore especially well suited for a comparison between the traditional arguments and the ones presented in this paper.
Again we use the two-step process to generate the perturbative series. First the integrand is expanded in powers of ␭, Eq. ͑7͒, and then the sum is interchanged with the integral ͑8͒. In this simple example the perturbative coefﬁcients can be calculated exactly for arbitrary n. In the large n limit they become

ͱ2
cnϳ 2␲͑ nϪ1 ͒! when n→ϱ.

͑10͒

With such factorial behavior, the series diverges for all ␭ different from zero as is well known. On the other hand, the function z(␭), as deﬁned in Eq. ͑6͒, gives a well-deﬁned positive real number for every positive real ␭. Therefore one or both of the two steps done to generate the perturbative series must be wrong.
Similarly, in the functional integral case normalized with respect to the free ﬁeld ͑1͒, Z is a well-deﬁned number while its perturbative series diverges. Again, one or both of the two steps must be wrong.
The ﬁrst step, the expansion of the integrand in powers of ␭, is clearly correct. As the integrand ͑not the integral͒ is analytic in ␭ for every ﬁnite ␭, the expansion merely corresponds to a Taylor series. The second step, the interchange of sum and integral, must therefore be the wrong one.
The next obvious step is then to recall the theorems that govern the interchange between sums and integrals in order to understand in detail why this step is wrong in our case. The most powerful theorem for this purpose is Lebesgue’s well-known theorem of dominated convergence. In a simpliﬁed version, sufﬁcient for our purposes, it says the following.
Let f N be a sequence of integrable functions that converge pointwisely to a function f ,

f N→ f as N→ϱ,

͑11͒

and bounded in absolute value by a positive integrable function h ͑dominated͒:

1146

SERGIO A. PERNICE AND GERARDO OLEAGA

57

FIG. 1. Exact integrand, zeroth, second, and fourth perturbative approximations. ␭ϭ1.

͉fN͉рh , ᭙N.

͑12͒

Then, it is true that

͵ ͵ ͵ lim f Nϭ lim f Nϭ f .

͑13͒

N→ϱ

N→ϱ

As a special case, if the convergence ͑11͒ is uniform and the measure of integration is ﬁnite, then the interchange is also valid. It should be emphasized that Lebesgue’s theorem follows from the axioms of abstract measure theory. Therefore if the problem under consideration involves a well-deﬁned measure, as is the case for the quantum ﬁeld theories considered here ͓16͔, the theorem holds.
In our case we can write, formally,2

͚ ͩ ͵ ͪ 1 N ͑Ϫ1͒n ␭
fN͓␾͑x͔͒ϭ Z0 nϭ0 n! 4

n
ddx␾4

ͩ ͵ ͫ ͬͪ ϫexp Ϫ

ddx

1 2

͑

‫ץ‬

␮

␾

͒

2

ϩ

1 2

m

2␾

2

͑14͒

for the functional integral case and

͚ ͩ ͪ 1 N ͑Ϫ1͒n
f N͑ x ͒ϭ ͱ␲ nϭ0 n!

␭ 4 x4

n
eϪx2

͑15͒

for the simple integral example. One important aspect of the dominated convergence theo-
rem approach to analyze the divergence of perturbation theory is that it focuses on the integrands, objects that are relatively simple to analyze. On the contrary, the analyticity approach brieﬂy described in the Introduction focuses on the integrals which are much more difﬁcult to analyze. So before we try to understand the aspects of the dominated convergence theorem that fail in our case, it will be useful to study some properties of the integrand for the intuitive simple example. In Fig. 1, the exact integrand, together with some perturbative approximations, are displayed. We can appreci-

2See previous footnote.

ate the way in which the successive approximations behave. For small x, and up to some critical value that we call xc,N ͑where the subindex c stands for critical while the subindex N indicates that this value changes with the order͒ the perturbative integrands approximate the exact integrand very well. Even more, xc,N grows with N. But for x bigger than xc,N a ‘‘bump’’ begins to emerge. The height of these bumps, as we will see in detail shortly, grows factorially with the order, while the width remains approximately constant. So the larger the order in perturbation theory, the larger the region in which the perturbative integrands approximate the exact integrand very well, but the stronger the upcoming deviation. As we will see shortly, it is precisely this deviation that is responsible for the divergence of the perturbative series and the famous factorial growth. We will also see that an exactly analogous phenomenon happens in the functional integral case and is again responsible for the divergence of the perturbative series.
Returning to the problem of understanding the aspects of the dominated convergence theorem that fail in the perturbative series, we will now show that the sequence of integrands of Eq. ͑14͒ and Eq. ͑15͒ converges, respectively, to the exact integrands

ͩ ͵ ͫ ͬ ͵ ͪ 1
Fϭ Z0 exp Ϫ

ddx

1 2

͑

‫ץ‬

␮␾

͒

2ϩ

1 2

m

2

␾

2

␭ Ϫ4

ddx␾4

͑16͒

and

ͱ f ϭ 1 e Ϫ[x2ϩ͑␭/4͒x4],
␲

͑17͒

but not in a dominated way. That is, there is no positive integrable function h that satisﬁes Eq. ͑12͒.

B. Failure of domination in the simple example

That the sequence of integrands of Eq. ͑14͒ and Eq. ͑15͒ converges, respectively, to the exact integrands ͑16͒ and ͑17͒ is obvious, since, as mentioned before, for ﬁnite ␭ they are analytic functions of ␭ and so their Taylor expansions converge ͑at least for ﬁnite ﬁeld strength͒. To see the failure of the domination hypothesis it is convenient to analyze the
‘‘shape’’ of every term of f N . Namely, for the ﬁeld theory case,

ͩ ͵ ͪ 1 ͑Ϫ1͒n 1
cn͓␾͑x͔͒ϵ Z0 n! 4n ␭

n
ddx␾4

ͩ ͵ ͫ ͬͪ ϫexp Ϫ

ddx

1 2

͑

‫ץ‬

␮

␾

͒

2

ϩ

1 2

m

2

␾

2

,

͑18͒

while for the simple integrand

ͩ ͪ 1 ͑Ϫ1͒n
cn͑x ͒ϭ ͱ␲ n!

␭ 4

n
x4neϪx2.

͑19͒

In this section we analyze the failure of the domination hypothesis for the simple example ͑6͒ because it turns out to be

57

DIVERGENCE OF PERTURBATION THEORY: STEPS . . .

1147

FIG. 2. Exact integrand and fourth perturbative approximation together with the third and fourth terms. ␭ϭ1.

remarkably similar to the ﬁeld theory example analyzed in
the next section. In Fig. 2 we can examine the functions c3(x) and c4(x) for ␭ϭ1 corresponding to the simple integrand case that we analyze ﬁrst. The maximum of cn(x) is reached at

xmaxϭϮ͑ 2n ͒1/2.

͑20͒

There, for large n, the function takes the value

c

n͑

x

max͒ ϭ

2

1 ␲ 3/2

͑

Ϫ

1

͒

n͑

n

Ϫ

1

͒

!

␭

n

.

͑21͒

On the other hand, the width remains constant as n increases
as can be seen by a Gaussian approximation around the maximum xmaxϭ(2n)1/2:

1 cn͑ x ͒Ϸ 2␲3/2 ͑ Ϫ1 ͒n͑ nϪ1 ͒!exp͕Ϫ2͓xϪ͑ 2n ͒1/2͔2͖␭n.
͑22͒

The integration of this Gaussian approximation gives, for large n,

͵1 dxcn͑x͒Ϸ 2

& 2␲

͑

Ϫ

1

͒

n͑

n

Ϫ

1

͒!

␭

n

,

͑23͒

in accordance with Eq. ͑10͒ if we take into account the factor of 2 coming from the two maxima Ϯ(2n)1/2.
The mechanism of convergence of the f N’s to f now becomes clear. The f N’s are made out of a pure Gaussian ͑the ‘‘free’’ term͒ plus ‘‘bumps’’ ͑the perturbative corrections͒ that alternate in sign ͑see Fig. 2͒. The height of these bumps grows factorially with the order, while their width remains
approximately constant. More speciﬁcally, the Gaussian approximation around the maxima ͓Eq. ͑22͔͒, which becomes exact when the order goes to inﬁnity, has a variance inde-
pendent of the order. For ﬁxed N and for x smaller than a
certain value, the bumps exhibit a delicate near-cancellation,
leaving only a small remnant that modiﬁes the free integrand
into the interacting one. However, for x larger than that
value, the last bump begins to emerge and, being the last,

does not have a successor to cancel it ͑in the N→ϱ limit, there is no last bump and the convergence is achieved for
every x). Consequently, beyond a certain value xc,N , the function f N deviates strongly from f and is governed by the uncanceled Nth bump, with height proportional to (NϪ1)! and ﬁnite variance. This is so because, since the height of the
bump grows factorially with the order, for N large enough
the last bump is far greater than all the previous ones and
remains almost completely uncanceled. Furthermore, since
the variance of the bumps is independent of the order, this
means that for every ﬁnite order, there is a region of ﬁnite
measure in which the perturbative integrand is of the order
of the height of the last bump. In Fig. 2 we can see how the
function c4(x) is left almost completely uncanceled by c3(x) and dominates the deviation of f 4 from f .
That xc,N ͑the value of ͉x͉ up to which the perturbative integrand very accurately approximates the exact one͒ grows with N, going to inﬁnity when N→ϱ, is a simple consequence of Taylor’s theorem applied to the analytic function e Ϫ␭x4/4.
The above analysis makes clear the failure of the domination of the sequence of Eq. ͑15͒ towards f ͓Eq. ͑17͔͒. Indeed, any positive function h(x) with the property

͉fN͑x͉͒рh͑x͒, ᭙N,

͑24͒

fails to be integrable, since it has to ‘‘cover’’ the bump,
whose area grows factorially with N. Therefore, although the
sequence of f N(x)’s converges to f (x), the convergence is not dominated, as we wanted to show.
Equation ͑23͒, together with the above comments, indicates that the same reason for which the sequence of integrands ͑19͒ fails to be dominated is the one that produces the factorial growth in the perturbative series.
In the ﬁeld theory case, although we cannot rely on ﬁgures such as Eqs. ͑1͒ and ͑2͒ to guide our intuition, we will see that the analogy with the simple integral example is so close that the interpretation is equally transparent.

C. Failure of domination in quantum ﬁeld theory

For quantum ﬁeld theory, as for the simple example ana-
lyzed above, it is convenient to consider every term cn͓␾(x)͔ ͓Eq. ͑18͔͒ of the perturbative approximation f N ͓Eq. ͑14͔͒ to the exact integrand ͓Eq. ͑16͔͒,

ͩ ͵ ͫ ͬ 1 ͑Ϫ1͒n
cn͓␾͑ x ͔͒ϭ Z0 n! exp Ϫ

ddx

1 2

͑

‫ץ‬

␮

␾

͒

2

ϩ

1 2

m

2␾

2

ͫ ͵ ͬ ͪ ϩnln ͑␭/4͒ ddx␾4 ,

͑25͒

where we have written the nth power of the interaction in exponential form. The mathematical analysis below follows closely the discussions in Chap. 38 of Ref. ͓8͔. Although the problem treated there is different from the one treated here, many techniques used in ͓8͔ can be directly applied here.
For n large enough, the analysis of its ‘‘shape’’ reduces to the familiar procedure of ﬁnding its maxima, as in the case of the simple integrand. The equation determining the

1148

SERGIO A. PERNICE AND GERARDO OLEAGA

57

maxima of cn͓␾(x)͔ is the equation that minimizes the exponent, and can be thought of as the equation of motion of
the effective action

͵ ͫ ͬ ͫ ͵ ͬ S͓␾͔ϭ

ddx

1 2

͑

‫ץ‬

␮

␾

͒

2

ϩ

1 2

m

2

␾

2

Ϫ n ln

␭ 4

ddx␾4 ,

͑26͒

which is

4n

͵ Ϫ ٌ 2 ␾ ϩ m 2 ␾ Ϫ

␾ 3 ϭ 0.

ddx␾4

͑27͒

Making the change of variables

ͩ ͪ ͐ddx␾4 1/2

␾͑x͒ϭm 4n

␸͑mx͒

ͩ ͵ ͪ ϭmd/2Ϫ1

4n

1/2

␸͑mx͒,

ddu␸4͑u͒

͑28͒

we ﬁnd that ␸ satisﬁes the equation Ϫٌ2␸͑ u ͒ϩ␸͑ u ͒Ϫ␸3͑ u ͒ϭ0, uϵmx. ͑29͒

This equation corresponds to the instanton equation of the negative mass ␭␾4 theory. The analysis of its solutions can be found in many places. We are interested in solutions with
minimal, ﬁnite action. For these solutions, in the inﬁnite vol-
ume limit, scaling arguments provide very interesting infor-
mation. We mentioned at the beginning of Sec. II A that we
work in a ﬁnite volume. However, if the volume is large
enough, the inﬁnite volume arguments used below remain
valid up to errors that go to zero exponentially fast when the
volume goes to inﬁnity. Since the solution ␾max(x) „the subindex ‘‘max’’ indi-
cates that, in functional space, cn͓␾(x)͔ reaches its maximum at ␾max(x); this should not be confused with the fact that the the action ͑26͒ reaches its minimum there… is a minimum of the action ͑26͒, then given an arbitrary constant ␣, S͓␣␾max(x)͔ should have a minimum at ␣ϭ1 ͓8,17͔. This implies the equation

͵ ͵ ddx͑ ‫␾␮ץ‬max͒2ϩm2 ddx␾m2 axϪ4nϭ0.

͑30͒

Similarly, S͓␾max(␣x)͔ should also have a minimum at ␣ϭ1, implying

͵ ͵ ͑2Ϫd͒ d

ddx͑ ‫␾␮ץ‬max͒2Ϫm2

ddx␾m2 axϩ2nϭ0.

͑31͒

Solving the system of equations ͑30͒ and ͑31͒ we obtain

͵ ddx͑‫␾␮ץ‬max͒2ϭn d,

͑32͒

͵m2 ddx␾m2 axϭn͑ 4Ϫd ͒,

͑33͒

from which we conclude in particular that the integral

͵ ͫ ͬ ddx

1 2

͑

‫ץ‬

␮ ␾ max͒ 2 ϩ

1 2

m2␾

2 max

ϭ2n

͑34͒

is independent of the dimension. The relations ͑32͒ and ͑33͒ can be explicitly checked in the case dϭ1 ͑quantum mechanics͒, in which the solutions to Eq. ͑27͒ are known analytically. They are

ͩ ͪ ␾mdϭax1͑t͒ϭ

3n 2m

1/2

1

cosh͓m͑ tϪt0͔͒ ,

͑35͒

giving

͵d

t

͑

␾˙

dϭ1 max

͒2

ϭ

n

,

͑36͒

͵m2

d

t

͑

␾

dϭ1 max

͒

2

ϭ

3

n

.

͑37͒

Since ␸(u), introduced in Eq. ͑28͒ and satisfying Eq. ͑29͒, is dimensionless ͑as is uϭmx), and the corresponding ␾max(x) has ﬁnite action, the quantity

͵1
Aϵ 4 ddu␸4͑u͒

͑38͒

is a ﬁnite, pure number greater than zero ͓8͔. For the quantum-mechanical case mentioned above, Aϭ4/3. For the cases dϾ1, A is not explicitly known but, as noted, it must be a ﬁnite, positive, pure number. With the deﬁnition ͑38͒, Eq. ͑28͒ becomes

ͩ ͪ ␾max͑x͒ϭmd/2Ϫ1

n A

1/2
␸͑mx͒.

͑39͒

Since ␸(mx) satisﬁes the n-independent equation ͑29͒, we conclude that the ﬁeld strength of ␾max grows with the square root of the order n.
Equation ͑34͒, together with the deﬁnition ͑38͒ and the relation ͑39͒, allow us to write an expression for the action ͑26͒ at ␾ϭ␾max ,

ͫ ͬ S͓␾max͔ϭ2nϪnln

␭mdϪ4 n2 A

.

͑40͒

The value of cn͓␾(x)͔ at ␾ϭ␾max then becomes, for large n,

ͩ ͪ 1 ͑Ϫ1͒n

␭mdϪ4 n

cn͓␾max͑ x ͔͒Ϸ Z0 2␲ ͑ nϪ1 ͒! A

.

͑41͒

With the change of variables

␾͑ x ͒ϭ␾max͑ x ͒ϩmd/2Ϫ1␾q͑ mx ͒,

͑42͒

the Gaussian approximation of cn͓␾͔ around ␾max is

57

DIVERGENCE OF PERTURBATION THEORY: STEPS . . .

1149

ͩ ͪ 1 ͑Ϫ1͒n

␭mdϪ4 n

cn͓␸͑u͔͒Ϸ Z0 2␲ ͑nϪ1͒! A

ͩ ͵1
ϫexp Ϫ 2

d

d

u

1

d

d

u

2␾

q͑

u

1

͒

͕

͓

Ϫ

ٌ

2 u

ϩ
1

1

ͪ Ϫ 3 ␸ 2͑ u 1 ͒ ͔ ␦ ͑ u 1 Ϫ u 2 ͒ ͖ ␾ q͑ u 2 ͒

͑43͒

ͩ ͵1
ϫexp Ϫ 2

ddu1ddu2␾q͑u1͒

ͪ ϫ͓͑1/A ͒␸3͑u1͒␸3͑u2͔͒␾q͑u2͒ ,

͑44͒

Equations ͑50͒ and ͑51͒ suggest that the operator D, with the corresponding renormalization for dϾ1, generates a well-deﬁned Gaussian measure in a ﬁnite volume ͑remember dϽ4). In fact, the determinant of DlЈocal has been calculated many times in the past ͓8͔, and a generalization of a quantum-mechanical argument of Ref. ͓18͔ indicates that this is all we need to compute the determinant of DЈ. The argument goes as follows:

ͫ ͬ 1
Det͓DЈ͔ϭDet DlЈocalϩ A ͉v͗͘v͉

ͩ ͪ ϭDet͓DlЈocal͔

1

ϩ

1 A

͗

v

͉

D

lЈoϪca1l ͉

v

͘

.

͑52͒

where uϭmx and ␸(u), the solution of Eq. ͑29͒, is related to ␾max through Eq. ͑39͒. This Gaussian approximation becomes exact in the limit n→ϱ.
The second derivative operator, which we call D, is, then,

D ϭ D localϩ D nonlocal ,

͑45͒

with

D localϭ Ϫ ٌ 2 ϩ 1 Ϫ 3 ␸ 2

͑46͒

and

1 DnonlocalϭA ͉v͗͘v͉

with ͗u͉v͘ϭ␸3͑u ͒

͑47͒

and A given in Eq. ͑38͒.
The operator Dlocal is well known ͑see, for example, ͓8͔͒. It has d eigenvectors ͉0␮͘ with zero eigenvalues given by

‫ץ‬

͗u͉0␮͘ϭ ‫ץ‬u␮ ␸͑u͒.

͑48͒

These vectors are also zero eigenvectors of D, as can be seen by noting that ͉v͘ is orthogonal to them:

͗ v ͉ 0 ␮ ͘ ϭ 0.

͑49͒

They reﬂect the translation invariance of the action ͑26͒.
Dlocal is also known to have one and only one negative eigenvector. D, on the contrary, is a positive semideﬁnite
operator. We can prove this in a line-by-line analogy with
the corresponding proof for Dlocal , which uses Sobolev inequalities and is given in Appendix 38 of Ref. ͓8͔,

D у 0,

͑50͒

in the operator sense.
Projecting out the d-dimensional eigenspace of eigenvalue zero, the resulting operator, which we call DЈ, is positive deﬁnite:

D Ј ϭ D lЈocalϩ D nonlocalϾ 0.

͑51͒

This equation explicitly states that the projection over the
strictly positive eigenvectors modiﬁes only Dlocal . The nonlocal part, as we saw, is a projector orthogonal to the zero modes and is therefore not modiﬁed under that operation.

Since ␸(u) is orthogonal to ‫(␸␮ץ‬u) ͑the zero modes of D and Dlocal),

D lЈocal␸ ϭ D local␸ ϭ Ϫ 2 ␸ 3 .

͑53͒

The last equality follows from the deﬁnition of Dlocal in Eq. ͑46͒ and Eq. ͑29͒ satisﬁed by ␸. Inverting DlЈocal and remembering the deﬁnition of ͉v͘ and A in Eqs. ͑47͒ and ͑38͒, we
obtain

͗ v ͉ D lЈoϪca1l ͉ v ͘ ϭ Ϫ 2 A .

͑54͒

Replacing this result in Eq. ͑52͒, we arrive at the result

Det͓ D Ј ͔ ϭ Ϫ Det͓ D lЈocal͔ .

͑55͒

As already mentioned, DlЈocal has one and only one negative eigenvector; consequently its determinant is negative. Equation ͑55͒ indicates then that Det͓DЈ͔ is positive, as it should be according to Eq. ͑51͒. The effect of the nonlocal part is to change the sign of the determinant of the local part.
The preceding equations allow us to integrate the Gaussian approximation of cn͓␸(u)͔ given in Eqs. ͑43͒ and ͑44͒. Using the method of collective coordinates to project out the
zero modes, the Jacobian of the corresponding change of
variables is, at leading order in 1/n,

ͫ͵ ͬ d
Jϭ ͟

1/2
͑ ‫␾␮ץ‬max͒2ddx ,

͑56͒

␮ϭ1

where no sum over ␮ is implied. It can be shown that the solutions of Eq. ͑27͒ correspond-
ing to minimal action are spherically symmetric ͓8͔. Equation ͑56͒ can then be written as

ͫ ͵ ͬ 1
Jϭ d

d /2
͑ ‫␾␮ץ‬max͒2ddx ,

͑57͒

where now a sum over ␮ from 1 to d is implied. Using Eq. ͑32͒ we then ﬁnd

J ϭ n d /2.

͑58͒

With this expression, the functional integral of cn͓␸(u)͔ can be written as

1150

SERGIO A. PERNICE AND GERARDO OLEAGA

57

͵ ͩ ͪ 1

͑Ϫ1͒n

␭mdϪ4 n

Z0 ͓d␾͔cn͓␾͔ϭ 2␲ ͑nϪ1͒! A

͑59͒

ͩ ͫ ͬ ͪ ϫ͑Volmd͒nd/2

Ϫ Det

D lЈocal D0

Ϫ 1/2
,

͑60͒

where D0ϵϪٌ2ϩ1. The factors in the line ͑59͒ correspond to the value of cn͓␾͔ at ␾max up to the normalization 1/Z0 as can be seen in Eq. ͑41͒. The factor ‘‘Vol’’ arises after the
integration over the ﬂat coordinates corresponding to the center of ␾max . The nd/2 comes from the Jacobian of the change of variables as mentioned before. The factor md
arises after the rescaling of the ﬁelds that makes them dimensionless in both cn͓␾͔ and Z0. This happens because there are d more integration variables in Z0 due to the integration over the collective coordinates in the numerator. Finally, the factor (ϪDet͓DlЈocal͔)Ϫ1/2 is the result of the integration over the coordinates orthogonal to the zero modes of D, while (Det͓D0͔)1/2 is the dimensionless normalization factor ͑the mass dimension of both the numerator and the denominator was already taken care of in the term md). The minus sign is
due to the nonlocal part of D that, as proved above, simply
changes the sign of the determinant of the local part, making
it positive. Equations ͑59͒ and ͑60͒ should agree with the correspond-
ing result from Ref. ͓18͔ in the case dϭ1, where ϪDet͓DlЈocal/D0͔ϭ1/12 ͓8,18͔ and Aϭ4/3. We see that the results are identical provided we take into account the differ-
ent normalization here and a factor of 2 that accounts for the undetermined sign of the solution of Eq. ͑27͒, allowing both
positive and negative solutions that contribute equally to the
functional integral. For dϭ2 or 3, the formal expressions ͑59͒ and ͑60͒ need
of course to be renormalized. All the arguments in this sec-
tion remain valid for the theory with a Pauli-Villars regularization ͓8͔. The action ͑2͒ becomes

͵ ͫ ͩ ͪ S͓␾͔ϭ

ddx

1 2␾

Ϫ

ٌ

2

ϩ

ٌ4 ⌳2

ϩ

m

2

␾ϩ ␭ ␾4 4

ͬ ϩ

1 2

␦

m

2͑

⌳

͒

␾

2

.

͑61͒

tion about the analytic structure in ␭ ͓18͔. A completely
analogous procedure would give the large order behavior of any Green’s function.
Equations ͑39͒, ͑41͒, ͑43͒, ͑44͒, ͑59͒, and ͑60͒ allow us to draw an accurate picture of the mechanism underlying the lack of domination ͑in the sense of Lebesgue’s theorem͒ of the convergence of the sequence of perturbative integrands ͑14͒ towards Eq. ͑16͒, and consequently of the mechanism underlying the divergence of the perturbative series. In fact, this picture is very similar to the one described in the previous section for the simple integral example. This is perhaps not surprising given the similarity of their large order behavior.
In a ﬁnite volume, there is a region of ﬁnite measure in ﬁeld space in which the perturbative approximation f N͓␾(x)͔ of Eq. ͑14͒ approximates the exact integrand ͑16͒ with an error smaller than a given prescribed number. This
region grows with N, becoming the full ﬁeld space in the N→ϱ limit. As in the simple example, this is a consequence of Taylor’s theorem applied to the ͑analytic͒ integrand ͑16͒.
The problem is that, for any ﬁnite N, outside that region the approximate integrand f N͓␾(x)͔ strongly deviates from the exact one. This can be seen by noting that the maxima of
every term of f N grow factorially with the order. Therefore, for large enough N, the last term is far greater than the previous ones at its maxima. Furthermore, as shown above, the Gaussian approximation around that maxima ͑which becomes exact for N→ϱ) deﬁnes a measure that does not go to zero as N→ϱ ͓in fact, it is independent of N, Eqs. ͑43͒ and ͑44͔͒. This means that for every ﬁnite N, there is a region of ﬁnite measure in ﬁeld space ͑and this measure does not go to zero as N→ϱ) in which the deviation between the perturbative integrand and the exact one is of the order of the maxima of the last term of f N , i.e., of the order of (NϪ1)!. No integrable functional can therefore satisfy the property ͑12͒ of Lebesgue’s theorem.
This is the mechanism that makes the sequence of perturbative integrands, although convergent to the exact one, nondominated in the sense of Lebesgue’s theorem. It is therefore also the mechanism that makes the sequence of integrals ͑i.e., the perturbative series͒ divergent. In fact, as Eqs. ͑59͒ and ͑60͒ show, the famous factorial growth of the large order coefﬁcients of the perturbative series is a consequence, after integration, of exactly this behavior.

The modiﬁcation of the kinetic part of the action affects both Eq. ͑27͒ and the scaling arguments, but by an amount that decreases as ⌳Ϫ2 when the ultraviolet cutoff ⌳ becomes large.
As shown in Ref. ͓8͔, although the counterterm increases with the cutoff, it is also proportional to at least one power of ␭. Therefore if we take the small ␭ limit before the large cutoff limit, we are justiﬁed in ignoring the counterterm in Eq. ͑27͒ and in the scaling arguments. On the other hand, it contributes to the results ͑59͒ and ͑60͒ an amount that exactly cancels the divergence in the Det͓DlЈocal͔, making the ﬁnal expression ﬁnite as it should be.
In the large n limit, where the Gaussian approximations ͑43͒ and ͑44͒ become exact, the expressions ͑59͒ and ͑60͒ give the large order behavior of the perturbative series of Z ͑up to the factor of 2 mentioned above͒ without any assump-

III. STEPS TOWARDS A CONVERGENT SERIES
It was mentioned in the Introduction that the analysis of the divergence of perturbation theory presented in this paper would point directly towards the aspects of the perturbative series that need to be modiﬁed in order to generate a convergent series. This is the topic of the present section.
In the previous section we analyzed perturbation theory from the point of view of the dominated convergence theorem. We have detected the precise way in which the convergence of the sequence of perturbative integrands to the exact one takes place and the way this convergence fails to be dominated. We have learned that for any ﬁnite order N, the ﬁeld space naturally divides into two regions. In the ﬁrst one, which grows with the order, eventually becoming the full ﬁeld space ͑in the N→ϱ limit͒, the perturbative integrands

57

DIVERGENCE OF PERTURBATION THEORY: STEPS . . .

1151

very accurately approximate the exact one. In the other one, however, the deviation between the perturbative and exact integrands is so strong that the sequence of integrals diverges.
It is then clear that if we could somehow modify the integrands, order by order, in the region where they deviate from the exact one, while preserving them as they are in the other region, then, with a ‘‘proper’’ modiﬁcation, such a modiﬁed sequence of integrands would converge in a dominated way. According to the dominated convergence theorem, their integrals would then converge to the exact integral, achieving the desired goal of a convergent modiﬁed perturbation theory.
Let ⍀N be the region of ﬁeld space in which the Nth perturbative integrand approximates with a given prescribed error the exact integrand ͑16͒. The characteristic function Ch„⍀N ,͕␾(x)͖… of that region is equal to 1 for ﬁeld conﬁgurations belonging to it and 0 otherwise:

ͭ1
Ch„⍀N ,͕␾͑x ͖͒…ϵ 0

for ͕␾͑ x ͖͒෈⍀N, for ͕␾͑ x ͖͒ ⍀N.

͑62͒

One possible realization of the above strategy of modifying the integrands ͑14͒ in the ‘‘bad’’ region of ﬁeld space is to make them zero there. We would have

͚ ͩ ͵ ͪ f

NЈ

͓␾͑x͔͒ϭ

1 Z0

N nϭ0

͑Ϫ1͒n n!

eϪS0

␭ 4

n
ddx␾4

ϫCh„⍀N ,͕␾͑x ͖͒….

͑63͒

According to the analysis of the previous section, if we choose ⍀N appropriately, the sequence of f NЈ ͓␾(x)͔ will exhibit dominated convergence, and the corresponding interchange between sum and integral will now be allowed. A rigorous proof of this is left for a paper currently in preparation. For the purposes of the present argument, it is sufﬁcient to rely on the analysis of the previous section to assume its validity. Also, in the next section we will analyze, along the lines of the general ideas of this paper, some resummation schemes for which rigorous proofs of convergence have recently been given ͓9–16͔. As that analysis will show, these methods strongly rely on the general notions underlying Eq. ͑63͒. Their convergence supports, then, the validity of the dominated nature of the convergence of Eq. ͑63͒ towards Eq. ͑16͒.
An urgent issue, however, is the practical applicability of the above strategy. To implement it, we need a functional representation of the characteristic function ͑62͒ ͑or an approximation to it͒ that only involves Gaussian and polynomial functionals. In the same way in which a functional representation of the Dirac ␦ function allows us to perform functional integrals with constraints, the Faddeev-Popov quantization of gauge theories being the most famous example, a functional representation of the characteristic function ͑62͒ would allow us to functionally integrate only the desired region of functional space. Since, basically, the functionals we know how to integrate reduce to Gaussians multiplied by polynomials, the desired representation of the characteristic function should only involve those functionals. Conversely, if it only involves those functionals, all the so-

phisticated machinery developed for perturbation theory ͑including all the perturbative renormalization methods͒ would automatically be applicable. With this in mind, consider the function

͚M
W͑M,u͒ϵeϪMu
jϭ0

͑Mu͒j j!

,

͑64͒

where M is a positive integer. Note that W(M ,u) arises from 1ϭeϪMueϩMu by expanding the second exponential up to
order M . Here W(M ,u) has the following remarkable prop-
erties. ͑1͒ W(M ,u)→1 when M →ϱ for 0ϽuϽ1. The conver-
gence is uniform, with the error going to zero as

R

͑

M

,

u

͒

р

e

M

[

lnu

Ϫ͑

u

Ϫ1

͒

]

1
ͱ2 ␲

M

u 1 Ϫ u ϩ 1/M

.

͑65͒

͑2͒ W(M ,u)→0 when M →ϱ for 1Ͻu. The convergence is also uniform, with an error of the form

W͑ M ,u ͒рe M[lnuϪ͑uϪ1͒].

͑66͒

As we see, the exponent corresponds to the same function in both cases. For uϾ0, this function is always negative except at its maximum, at uϭ1, where it is 0. Therefore the convergence is in both cases exponentially fast in M , with
the exponent becoming more and more negative, for a ﬁxed
M , when u differs more and more from 1. The proof of properties ͑1͒ and ͑2͒ is in Appendix A.
If we replace u by a positive deﬁnite quadratic form ͗␾͉D͉␾͘/CN , then the insertion of Eq. ͑64͒ into the functional integral would effectively cut off the region of integration ͗␾͉D͉␾͘ϾCN :

͵ ͚ 1
ZNЈ ͓␾͑ x ͔͒ϭ Z0

N
͓d␾͔
nϭ0

͑ ϪSint͒n n!

ͩ ͪ ϫ eϪS0 lim W

͗␾͉D͉␾͘ M,

͑67͒

M→ϱ

CN

͚ ͵ 1 N ͑Ϫ1͒n

ϭ Z0 nϭ0

lim n! M→ϱ

͓d␾͔

ͩ ͪ ϫeϪS0͑Sint͒nW

͗␾͉D͉␾͘ M, CN

.

͑68͒

CN is a constant that changes with the order N of the expansion in ␭, increasing with N but in such a way that in the region where ͗␾͉D͉␾͘ϽCN , the difference between the perturbative and the exact integrands is smaller than a given
prescribed error. Since the convergence of W is uniform according to properties ͑1͒ and ͑2͒, with errors given in Eqs. ͑65͒ and ͑66͒, the corresponding interchange between the sum in Eq. ͑68͒ and the functional integral is justiﬁed. The
fact that u becomes a quadratic form implies that the result-
ing integrands are Gaussians multiplied by monomials.
Therefore the familiar Feynman diagram techniques can be
used to integrate them. It also implies that no new loops

1152

SERGIO A. PERNICE AND GERARDO OLEAGA

57

appear and the sum in j from Eq. ͑64͒ becomes an algebraic problem. A typical functional integral to compute has the form

͵ ͩ ͵ ͫ ͬͪ ͓d␾͔exp Ϫ

ddx

1 2͑

‫ץ‬

␮

␾

͒

2ϩ

1 2

m

2

␾

2

ϩ

͑

␾

D

␾

/

C

N

͒

ͩ ͵ ͪ ͩ ͵ ͪ n

m

ϫ ddx␾4

ddx␾D␾ ,

͑69͒

as can be seen by replacing the deﬁnition ͑64͒ into Eq. ͑68͒ with uϭ͗␾͉D͉␾͘/CN .
Note that at any given order in ␭ it is not necessary in principle to go to inﬁnity in M . That would amount to re-
placing the perturbative integrands by zero in the region ͗␾͉D͉␾͘ϾCN , realizing the strategy mentioned before. But since the convergence in W is uniform, a ﬁnite, large enough M ͑depending on the order in the expansion in the coupling constant͒ would sufﬁce to tame the behavior of the perturbative integrands and transform them into a dominated conver-
gent sequence. In fact, as we will see, many methods of
improvement of perturbation theory use effectively formula ͑64͒ without sending M →ϱ for any given ﬁnite order in perturbation theory. In any case, as already mentioned, that
limit is in principle computable, since it does not involve
new loops. Work in this direction is in progress. The convergence of the sequence ͑68͒ towards Z(␭) may
be thought, at ﬁrst sight, to be in conﬂict with our well-
established knowledge about the nonanalyticity of this function at ␭ϭ0. In fact, Eq. ͑68͒ seems to be a power series in ␭ ͑the powers of ␭ coming from the powers of Sint); therefore, if convergent, that power series would deﬁne a function of ␭ analytic at ␭ϭ0. It must be recognized, however, that the validity of Lebesgue’s dominated convergence theorem
is completely independent of any analyticity consideration.
Therefore, if its hypothesis is satisﬁed, its conclusions must
be valid. This being said, the question of how to reconcile the convergence of Eq. ͑68͒ with the nonanalyticity of Z(␭) deserves an answer. To begin with, even at ﬁnite order in ␭, the function ͑68͒ is not necessarily analytic at ␭ϭ0 despite its analytic appearance. This is because the constant CN may have an implicit nonanalytic dependence on ␭. In Appendix B this is actually the case in the context of a simple example
to which the present ideas are applied. But the mechanism that ultimately introduces the proper nonanalyticity in ␭ is the limit process N→ϱ. Given a nonanalytic function such as Z(␭) one can always construct a sequence of analytic functions that converge to it. Satisfying the hypothesis of the
dominated convergence theorem is a way of achieving that,
avoiding all the complicated and model-dependent issues of
nonanalyticity. Note that the validity of this hypothesis for a
given sequence of integrands can be checked independently
of any analyticity consideration.
In Appendix B we prove the convergence of the general
strategy discussed here for the simple integral example analyzed in Sec. ͑II B͒. For that case, making uϭ(x/xc,N)2, the function W(M ,u) becomes in the limit the characteristic function of the interval ͉x͉Ͻxc,N . We use this to explicitly compute the nonanalytic function z(␭) ͓Eq. ͑6͔͒, calculating only Gaussian integrals. We also show explicitly how a nonanalytic dependence of xc,N on ␭ naturally arises just by

FIG. 3. Function W(M ,x,xc,N) with xc,Nϭ1 for M ϭ3 ͑dashed line͒ and M ϭ60 ͑solid line͒. The convergence towards the characteristic function of the interval ͉x͉Ͻxc,N is apparent.
demanding the validity of Lebesgue’s hypothesis and how the N→ϱ limit process captures the full nonanalyticity of z(␭). The same method also works for the ‘‘negative mass case,’’ where the Borel resummation method fails. In Fig. 3, we can appreciate the convergence of W towards the characteristic function of the interval ͉x͉Ͻxc,N for xc,Nϭ1 for two different values of M .
IV. IMPROVEMENT METHODS OF PERTURBATION THEORY
The analysis of the mechanism of divergence of the perturbative series presented in this paper, together with the formula ͑64͒ and its properties, offers a large range of possibilities to construct a convergent series. In the previous section we have shown how that formula can be used to effectively cut off the region of ﬁeld space where the strong deviation between perturbative and exact integrands takes place. But as we will see, this is only one possible way, among many, to use Eq. ͑64͒ to transform the sequence of perturbative integrands into a dominated one.
Another example of its possible use is the so-called ‘‘optimized ␦ expansion’’ ͓19,20͔. In a series of papers ͓9– 11,21͔, it was proved that such an expansion converges for the partition function of the anharmonic oscillator in ﬁnite Euclidean time. The problem of convergence in the inﬁnite Euclidean time ͑or zero temperature͒ limit for the free energy or any connected Green’s function is still under investigation, as well as its extension to quantum ﬁeld theories ͓10,12͔. The method was proved to generate a convergent series for the energy eigenvalues ͓13,14͔, although such studies make heavy use of analyticity properties valid speciﬁcally in the models studied. In these works, it was realized that many methods of improvement of perturbation theory, such as the order-dependent mappings of Refs. ͓22,23͔, possess the same general structure as the linear ␦ expansion. A considerable amount of work has been dedicated to investigating the virtues and limitations of the method and extensions of it ͓11,15͔.
Although it is not appropriate to give a detailed analysis of these methods here, we would like to brieﬂy indicate how they can be understood in terms of the ideas presented in this

57

DIVERGENCE OF PERTURBATION THEORY: STEPS . . .

1153

paper. In what follows, our analysis is restricted to dϭ1 ͑quantum mechanics͒ where rigorous results about the convergence of the methods considered here are available.
Let us consider the case of the anharmonic oscillator. Its action is given in Eq. ͑2͒ for dϭ1. The idea of the method is to replace it by an interpolating action

͵ ͫ ͩ ͪ S␦ϭ

dt

1 2

͑

d

t␾

͒

2ϩ

1 2

m2ϩ ␭ ␣ 2m

␾2

ͩ ͪͬ ␭
ϩ␦

␾4Ϫ ␣ ␾2

.

͑70͒

4

m

Clearly, the dependence on the parameter ␣ in S␦ is lost when ␦ϭ1. For that value, the action ͑70͒ reduces to Eq. ͑2͒. However, if we expand up to a ﬁnite order in ␦ and then make ␦ϭ1, the result still depends on ␣. The idea is to tune ␣, order by order in the expansion in ␦, so that the result is a convergent series. It was shown in the references mentioned above that the method works if ␣ is tuned properly. For example, in Ref. ͓9͔, the asymptotic scaling ␣ӍN2/3 was
used to prove the convergence of the method for the partition
function at ﬁnite Euclidean time. It is interesting to note that, originally ͓9,21͔, ␣ was tuned
according to heuristic prescriptions such as the ‘‘principle of minimal sensitivity’’ ͓20͔ ͑at any given order in ␦, choose ␣ so that the result is insensitive to small changes in it͒ or the criterion of ‘‘fastest apparent convergence’’ ͑the value of ␣ at which the next order in ␦ vanishes͒. But later ͓10,12͔, it was realized that the best strategy was simply to leave ␣ undetermined, ﬁnd an expression for the error ͑that obviously depends on ␣), and then choose ␣ so that the error goes to zero when the order in ␦ goes to inﬁnity. It is clear that a structural understanding of the convergence of the
method can help to construct the generalizations necessary to
overcome the difﬁculties associated with the convergence in
the inﬁnite volume limit for connected Green’s functions, as
well as the extensions to general quantum ﬁeld theories. To understand the ‘‘optimized ␦ expansion’’ in terms of
the ideas presented in this paper, let us expand the functional integral corresponding to the action ͑70͒ in powers of ␦ up to a ﬁnite order N, and make ␦ϭ1 as the method indicates,

͵ ͭ ͵ ͫ 1

1

Z͑ m,␭,␣,N ͒ϭ Z0 ͓d␾͔exp Ϫ dt 2 ͑ dt␾ ͒2

ͩ ͪ ͬͮͫ 1

␭␣

ϩ 2 m2ϩ 2m ␾2

͚N ͑Ϫ1 ͒n
nϭ0 n!

ͩ ͵ ͵ ͪ ͬ ␭
ϫ4

␾

4

Ϫ

␭␣ 4m

n
␾2 .

͑71͒

The general analysis of the mechanism of divergence of perturbation theory of Sec. II indicates that if the function ͑71͒ generates a convergent series with ␣ scaling properly with N, then, barring miraculous coincidences, the corresponding integrands should converge in a dominated way ͑or, even better, uniformly͒ towards the exact integrand ͑14͒. We want to obtain a qualitative understanding of how this method achieves that.

Expanding the binomial and making some elementary changes of variables in the indices of summation, we obtain the expression

͵ ͭ ͵ ͫ 1
Z͑m,␭,␣,N͒ϭ Z0

͓d␾͔exp Ϫ

dt

1 2

͑

d

t␾

͒

2

ͩ ͪ ͬͮͭ 1
ϩ

m2ϩ ␭␣

␾2

2

2m

͚N ͑Ϫ1 ͒i
iϭ0 i!

ͩ ͵ ͪ ͫ ͩ ͵ ͪ ͬͮ ͚ ␭
ϫ4

␾4

i

NϪi 1 kϭ0 k!

␭␣ 4m

k
␾2

.

͑72͒

This equation already shows some of the distinctive charac-
teristics of the method. As we see, the ith power of the interacting action in the expansion of eϪSint up to order N is
multiplied by

ͩ ͵ ͪͫ ͩ ͵ ͪ ͬ ͚ W͑NϪi͒ϵexp Ϫ͑␭␣/4m͒

␾2

NϪi 1 ␭␣ kϭ0 k! 4m

k
␾2 .

͑73͒

Note that W(N) corresponds to the function W(M ,u) with M ϭN (N is the order in the expansion of eϪSint) and the
variable u replaced by the quadratic form ͓(␭/4m)͐␾2͔/CN , where CNϭN/␣. Taking, for example, ␣ӍN2/3 as in Ref. ͓9͔ ͑where it was proved that with such a scaling the method generates a convergent series͒, we see that, according to the previous section, W(N) is an approximation of the ␪ function in the region of ﬁeld space charac-
terized by

͵␭␣
4m

d x ␾ 2 р N 1/3.

͑74͒

Equation ͑72͒, however, shows that the mechanism used to achieve dominated convergence cannot be reduced to a simple insertion of the function W(M ,u) with M ϭN and uϭ͓(␭/4m)͐␾2͔/CN . That would be the case if all the powers of the expansion of eϪSint up to order N were multiplied by W(N). But Eq. ͑72͒ shows that the ith power of the interacting action is in fact multiplied by W(NϪi).
At this point it is convenient to pause for a moment in our study of the ‘‘optimized ␦ expansion’’ to give some useful deﬁnitions.
Let us call passive mechanisms ͑to achieve dominated or uniform convergence of a sequence of integrands to the exact one͒ those that can be reduced to the product of the Nth perturbative integrand and the characteristic function of a region ⍀N of ﬁeld space for some sequence ͕⍀N͖.
Passive methods use only information that is already available in the perturbative integrands; they just get rid of the ‘‘noise’’ inherent to perturbation theory. Because of that, in addition to deﬁning a convergent series, they can also be very useful for studying perturbation theory itself. The func-
tion W(N,u), with u replaced by a properly selected quadratic operator, was specially designed to make passive methods practical. In a sense, Sec. III is a discussion of passive methods.

1154

SERGIO A. PERNICE AND GERARDO OLEAGA

57

Active mechanisms are those that are not passive, as de-
ﬁned above.
What kind of mechanism is the one underlying the ‘‘optimized ␦ expansion’’ method?
A trivial generalization of the proof, in the previous section, of the convergence of W(M ,u) towards the ␪ function for uϾ0 shows that the function

͚M Ϫ i
W¯ ͑ M ,u,i ͒ϵeϪMu
nϭ0

͑Mu͒i i!

͑75͒

also converges towards the ␪ function for uϾ0 in the limit

M →ϱ, i ﬁxed.

͑76͒

In this sense, the ‘‘optimized ␦ expansion’’ method does have passive aspects. As Eqs. ͑72͒ and ͑73͒ show, it amounts to multiplying the ith power of the expansion up to order N
of eSint by W¯ with uϭ͓(␭/4m)͐␾2͔/CN and CNϭN/␣. Since this function converges to the characteristic function of the region characterized by Eq. ͑74͒, this means that the ﬁrst i terms of the expansion up to order N of eSint are effectively multiplied by the same function ͑an approximate characteristic function͒ for iӶN. Therefore, the ﬁrst i terms, with iӶN, use only the information available in the perturbative series to converge to the exact integrand.
What about the other terms, i.e., the ones characterized by iՇN? Surprisingly, these terms produce a convergence of the corresponding integrands towards the exact one that is
faster than possible with only passive components.
It is not the place here to study this aspect in detail, and so
let us simply show this ‘‘faster than passive’’ convergence
for the simple integral example. Applied to the ‘‘massless’’ version of the integral ͑6͒, the
optimized ␦ expansion method was proved to generate a rapidly convergent sequence in Ref. ͓21͔. That is, the sequence given by

͚ ͵ ͩ ͪ N ͑Ϫ1͒n

INϵ
nϭ0

n!

ϱ dxeϪ␣͑N͒ x2
Ϫϱ

␭ 4

x

4

Ϫ

␣

͑

N

͒x

2

n

͑77͒

was proved to converge to

͵Iϵ ϱ dxeϪ␭x4/4

͑78͒

Ϫϱ

when ␣(N)ӍͱN with an error that goes to zero at the very
fast rate of RNϽCN1/4eϪ0.663N when N→ϱ. C is a numerical constant.
We are interested in understanding whether the corresponding convergence of the integrands is faster than passive. For our qualitative purposes, it is enough to observe, in Fig. 4, the convergence towards the exact integrand

I exa͑ x ͒ϭ e Ϫ͑␭/4͒x4

͑79͒

of both the perturbative integrand

͚N ͑ Ϫ␭x4/4 ͒n

I pertϭ
nϭ0

n!

͑80͒

FIG. 4. In the main plot, the superiority of the convergence of the fourth order optimized ␦ expansion ͑‘‘ode’’͒ with respect to the same order perturbative approximation is evident. In the subgraph, the difference between the ‘‘ode’’ and the exact integrand is plotted. Note the difference in the scales of the y axis of the main and subgraph.
and the optimized ␦ expansion integrand

͚ ͩ ͪ N
I odeϭ
nϭ0

͑

Ϫ1 n!

͒

n

e

Ϫ

␣

͑

N

͒

x

2

␭ 4 x4Ϫ␣͑N͒x2

n
,

͑81͒

with ␣(N)ӍͱN, for Nϭ4.
We can see how accurate the convergence of Iode(x) is, even at this low order. In particular, when the perturbative
integrand begins to diverge, Iode(x) continues to approximate the exact integrand remarkably well. In the inset, we can
appreciate the difference between Iexa(x) and Iode(x). Note the difference in the y axis scale of the main graph and the inset.
It is then clear that the optimized ␦ expansion method, with its subtle combination of passive and active components, manages to generate a sequence of integrands that ͑uniformly͒ converges towards the exact one at a rate that far exceeds the possibilities within a purely passive method.
From this qualitative discussion of the optimized ␦ expansion method we can deduce two general lessons: ͑1͒ Any method of improvement of the perturbative series in a given quantum theory, where a functional integral representation of the quantity under study exists, must rely, at the level of the integrands, on an improvement over the pointwise convergence of the Taylor series in the coupling constants of eϪS; ͑2͒ the problem of ﬁnding a convergent series reduces to the problem of ﬁnding a dominated convergent sequence of integrands towards eϪS. This second simple statement not only provides a guide to the construction of convergent schemes, but also emphasizes the fact that, in principle, a dominated convergent sequence of integrands does not have to have any relation whatsoever to the corresponding Taylor expansion. In order to be able to use the usual techniques of quantum ﬁeld theory, it is reasonable to restrict the search for a convergent scheme to a sequence of integrands of the general form

57

DIVERGENCE OF PERTURBATION THEORY: STEPS . . .

1155

N

͚ f NϭeϪS0

a

nS

n Int

f

n͑

͗

␾

͉

D

n͉

␾

͘

͒

,

͑82͒

nϭ0

where the functional f n of the quadratic form ͗␾͉Dn͉␾͘ should take care of the nondominated convergence that is bound to appear with only powers of the interacting action. The function W of Sec. III, with its possible generalizations, is an ideal candidate for this purpose. But the selection of the coefﬁcients an amounts to a pure problem in optimization of the convergence of the integrands — no a priori connection with any Taylor series is necessary.

V. CONCLUSIONS
In this paper we have exposed the mechanism, at the level of the integrands, that makes the perturbative expansion of a functional integral divergent. We have seen in detail how the sequence of integrands violates the domination hypothesis of Lebesgue’s dominated convergence theorem. That theorem, as is well known, establishes the conditions under which one is allowed to interchange an integration and a limit, in particular the interchange that takes place in the generation of perturbation series.
It was shown that at any ﬁnite order in perturbation theory, the ﬁeld space divides into two regions. In one region, whose measure grows with the order, the perturbative integrands very accurately approximate the exact integrand. In the other region, however, a strong deviation takes place. It was shown that the behavior in this second region violates the hypothesis of Lebesgue’s theorem and, consequently, generates the divergence of perturbation theory. The famous factorial growth of the large order coefﬁcients of the perturbative series was shown to be an effect, after integration, of the very mechanism that violates the hypothesis of the theorem.
All of the above was done explicitly without relying on the particular analytic properties of the models studied. It is therefore natural to assume that similar mechanisms of the violation of Lebesgue’s hypothesis are present in any other quantum ﬁeld theory, although for just renormalizable theories other mechanisms are responsible for renormalons. Studies in this direction are in progress.
The mechanism of divergence presented here points towards a simple way to achieve a convergent series: Integrate only in the ‘‘good’’ region of ﬁeld space. Since this region grows with the order, becoming in the limit the whole ﬁeld space, integrating in a correspondingly increasing region we would obtain a convergent series. A step towards a practical implementation of this program was made with the construction of the function W, Eq. ͑64͒. This function allows us to introduce a Gaussian representation of the characteristic function of regions of ﬁeld space, in much the same way that the imposition of constraints in the functional integral was allowed by a functional representation of the Dirac ␦ function. A rigorous proof of the convergence of this practical implementation of the above-mentioned strategy is in progress. In Appendix B it was applied to a simple integral example.
Finally, a qualitative analysis of the optimized ␦ expansion method of improvement of perturbation theory in terms of the ideas of this paper was presented. Some general prop-

erties of improvement methods, useful to generate new schemes, as well as to understand and improve old ones, have been established.

ACKNOWLEDGMENTS
S.P. was supported in part by U.S. Department of Energy Grant No. DE-FG 02-91ER40685. He would like to thank A. Duncan for numerous very useful discussions, and also A. Das, L. Orr, and S. Rajeev.

APPENDIX A

In this appendix we will prove the two properties of for-

mula ͑64͒.

Because for uϾ0 all the terms of the sum deﬁning

W(M ,u) are positive, we have trivially W(M ,u)Ͼ0. On the

other hand, since in the Taylor expansion of eMu all the

terms

are

positive,

we

have

͚

M nϭ0

(

M

u

)

n

/

n

!

р

e

M

u

.

Therefore

W(M ,u)р1. So for every M and positive or zero u we have

0рW͑ M ,u ͒р1.

͑A1͒

Consider ﬁrst the case 0ϽuϽ1:

͚ϱ
1ϪW͑M,u͒ϭeϪMu
nϭMϩ1

͑Mu͒n n! ϵR͑M,u͒.

͑A2͒

We will prove that R(M ,u)→0 when M →ϱ. Changing variables to jϭnϪM , we get

͚ R͑ M ,u ͒ϭeϪMu

͑Mu͒M ϱ M! jϭ1

͑Mu͒j

M! ͑ jϩM ͒!

͑A3͒

͚ р

e

Ϫ

M

u

͑

Mu͒ M!

M

j

ϱ ϭ1

͑Mu͒j ͑Mϩ1͒j

͑A4͒

р

e

Ϫ

M

u

͑

Mu͒ M!

M

u 1 Ϫ u ϩ 1/M

.

͑A5͒

But M M/M !→eM/ͱ2␲M for large M , and so

R

͑

M

,

u

͒

р

e

M

[

lnu

Ϫ͑

u

Ϫ1

͒

]

1
ͱ2 ␲

M

u 1 Ϫ u ϩ 1/M

.

͑A6͒

The exponent is negative in the region 0ϽuϽ1 since both lnu and (uϪ1) are negative there and ͉lnu͉Ͼ͉uϪ1͉. Therefore

R͑ M ,u ͒→0 when M →ϱ

͑A7͒

in the region 0ϽuϽ1 and property ͑1͒ is proved with an exponentially fast convergence.
In the region uϾ1, we have

͚ ͚ M
W͑M,u͒ϭeϪMu
nϭ0

͑Mu͒n рeϪMuuM M

n!

nϭ0

Mn n!

͑A8͒

р e M [lnuϪ͑uϪ1͒].

͑A9͒

1156

SERGIO A. PERNICE AND GERARDO OLEAGA

57

The ﬁrst inequality is valid because uϾ1 and the second

because

e

M

Ͼ

͚

M nϭ

0

M

n/

n

!

.

The

exponent

is

again

negative.

For uϾ1, both lnu and (uϪ1) are positive, but now

͉lnu͉Ͻ͉uϪ1͉. So property ͑2͒ is also valid with an exponen-

tially fast convergence.

For uϭ1 all we know is that W is bounded by Eq. ͑A1͒.

That is all we need. Numerics suggest W(M ,1)→1/2 when

M→ϱ.

This ﬁnishes our proof.

APPENDIX B

In this appendix we apply the strategy discussed in Sec.

III to generate a series convergent to the function z(␭) ͓Eq.

͑6͔͒. This is done using the function W of Eq. ͑64͒ and com-

puting exclusively Gaussian integrals; therefore, we restrict

ourselves to using only those techniques that are also avail-

able in quantum ﬁeld theory.

As mentioned in Sec. III, the simplest possible modiﬁca-

tion of the perturbative integrand ͑15͒ that would transform

the corresponding sequence into a dominated one amounts to

keeping them as they are for ͉x͉Ͻxc,N and replacing them by zero for ͉x͉Ͼxc,N . That is,

ͭ ͩ ͪ ͚ fNЈϭ

N
␲ Ϫ1/2
nϭ0

͑Ϫ1͒n n!

␭ 4 x4

n
eϪx2

for ͉x͉Ͻxc,N ,

0

for ͉x͉Ͼxc,N.

͑B1͒

In fact, choosing xc,N so as to properly avoid the region in which the deviation takes place, the sequence of f NЈ converges uniformly towards the exact integrand ͑17͒ as we will
show shortly. Consequently, the corresponding sequence of
integrals

͵ ͵ ͚ ͩ ͪ ϱ dx f NЈ ϭ␲Ϫ1/2 Ϫϱ

xc,N

N ͑Ϫ1͒n

dx

Ϫxc,N nϭ0 n!

␭ x4 4

n
eϪx2

͑B2͒

͚ ͵ ͩ ͪ N
ϭ ␲Ϫ1/2
nϭ0

xc,N ͑Ϫ1͒n

dx
Ϫxc,N

n!

␭ 4 x4

n
eϪx2

͑B3͒

will converge to the desired integral

͵ z͑ ␭ ͒ϭ␲Ϫ1/2 ϱ dxeϪ[x2ϩ͑␭/4͒x4]. Ϫϱ

͑B4͒

In Eq. ͑B2͒ the change in the limits of integration from Ϯϱ

to Ϯxc,N is just due to the deﬁnition of f NЈ in Eq. ͑B1͒. The interchange between sum and integral in Eq. ͑B3͒ is now

allowed because in the region ͓Ϫxc,N ,xc,N͔ we have uniform convergence ͑this is a stronger condition than domi-

nated convergence͒. The resulting integrals are not Gaussian

due to the ﬁnite limits of integration. We will show how they

can be calculated using only Gaussian integrals.

A trivial way to achieve convergence of the sequence of

integrals of the f NЈ of Eq. ͑B1͒ towards Eq. ͑B4͒ amounts to

keeping xc,N equal to a ﬁnite constant ‘‘a’’ independent of

N, while taking the limit N→ϱ. In this limit, Eq. ͑B3͒ be-

comes

identical

to

␲

Ϫ

1/2͐

a Ϫ

a

d

x

e

Ϫ[

x

2

ϩ

(

␭

/

4

)

x4

]

,

since

for

ﬁnite

a the Taylor series of the integrands converges uniformly.
Therefore, as already said, the interchange between sum and integral is legal. Finally, taking the limit a→ϱ, we would obtain the desired convergence towards z(␭).
However, better use can be made of the information available in f NЈ for ﬁnite N. For example, for every ﬁnite N, we can choose xc,N so that

͉

f

NЈ ͑

x

͒

Ϫ

f

͑

x

͉͒

р

⑀T,N 2xc,N

for ͉x͉Ͻxc,N ,

͑B5͒

with ⑀T,N going to zero as N→ϱ. Then, since we have

͉

f

NЈ

͑

x

͒

Ϫ

f

͑

x

͒

͉

р

e

Ϫ

[x

2 c,

N

ϩ

͑

␭/

4

͒x

c4, N ]

ϵ

⑀

c,
2

N

for ͉x͉Ͼxc,N , ͑B6͒

the f NЈ (x) will uniformly converge towards the exact integrand f (x) if Eq. ͑B5͒ is consistent with xc,N→ϱ when N→ϱ. Indeed, if this happens, we would have

ͯ͵ ͯ ϱ ͓ f͑x͒Ϫ f N͑x͔͒dx р⑀T,Nϩ⑀c,N→0 Ϫϱ

when N→ϱ. ͑B7͒

The term ⑀T,N comes trivially from Eq. ͑B5͒, while ⑀c,N comes from Eq. ͑B6͒ and the inequality

͵ϱ

eϪ[x2ϩ͑␭/4͒x4]dx

рe

Ϫ

[

x

2 c,

N

ϩ

͑

␭

/

4

͒ x c4,

N]

ϭ

⑀

c,N

,

͑B8͒

xc,N

valid for xc,NϾ1. Applying Taylor’s theorem to the function eϪ␭x4/4 one
can easily show that the condition ͑B5͒ is satisﬁed if

ͫ ͩ ͪ ͬ xc,Nϭ

͑

N

ϩ

1

͒!

⑀

T,
2

N

4 ͑Nϩ1 ͒ 1/͓4͑Nϩ5/4͒]

␭

.

͑B9͒

Note that the nonanalytic dependence of xc,N on ␭ arises automatically from the imposition of Eq. ͑B5͒ to satisfy the
hypothesis of Lebesgue’s theorem.
Remember that the only condition on ⑀T,N ͑in order to achieve convergence of the sequence of integrals͒ is to go to
zero when N→ϱ consistently with xc,N→ϱ in that limit. Choosing, for example,

⑀

T

,

N

ϭ

e

Ϫ

4

N

1/4
,

we obtain, asymptotically,

͑B10͒

xc,N→͑ 4N/e␭ ͒1/4. This implies ͓through Eq. ͑B6͔͒

͑B11͒

⑀ c,N→ e Ϫ͑4N/e␭͒1/2ϪN/e.

͑B12͒

Equations ͑B10͒ and ͑B12͒ show the exponential rate at which the convergence of the sequence of integrals takes
place. Clearly the form ͑B10͒ for ⑀T,N is not unique, nor even
the most efﬁcient one, but enough to achieve convergence.

57

DIVERGENCE OF PERTURBATION THEORY: STEPS . . .

1157

TABLE I. Integration over the small ﬁeld conﬁgurations only produces a convergent series. In the last column the improvement over the perturbative values can be appreciated.

͚ ͩ ͪ ͵ M 1

lim
M→ϱ

nϭ0

n!

M

x

2 c,

N

n

ϱ

e

Ϫ͑

1

ϩ

M

/

x

2 c,N

͒

x

2

x

2͑

n

ϩ

t

͒d

x

Ϫϱ

͑B17͒

Order
2 4 6 8 20

Exact value ( ␭ ϭ 4/10)
0.837043 0.837043 0.837043 0.837043 0.837043

Convergent series
0.803160 0.830264 0.835516 0.836667 0.837044

Perturbative series
0.848839 0.854087 0.901897 1.316407 2.33755ϫ108

In Table I one can appreciate the numerical convergence for ␭ϭ4/10.
Up to now we have proved that the general strategy of
Sec. III does, in fact, generate a convergent sequence towards z(␭). However, the resulting integrals in Eq. ͑B3͒ are not Gaussians, making the applicability of the method in
quantum ﬁeld theory dubious, to say the least. We will show now that the integrals of Eq. ͑B3͒ can be computed, using Eq. ͑64͒ with uϭ(x/xc,N)2, calculating only Gaussian integrals. The steps involved are

͵ ͵ xc,N xreϪx2dxϭ ϱ xreϪx2 lim W͑ M ,x,xc,N͒dx

Ϫxc,N

Ϫϱ

M→ϱ

͑B13͒

͵ ϭ lim

ϱ

x

r

e

Ϫ

x

2
W

͑

M

,

x

,

x

c

,

N

͒

d

x

M→ϱ Ϫϱ

͚ ͩ ͪ ͵ M 1 M n ϱ

ϭ lim
M→ϱ

nϭ0

n!

x

2 c,N

Ϫϱ

ϫ

e

Ϫ

͑

1

ϩ

M

/x

2 c,

N

͒x

2
x

2

n

ϩ

r

d

x

.

͑B14͒ ͑B15͒

The two properties of W validate both equalities ͑B13͒ and ͑because of the uniformity of the convergence in W) ͑B14͒. In the last line, Eq. ͑B15͒, we just make explicit the meaning of Eq. ͑B14͒. So it is clear that these two properties are enough to prove the validity of Eq. ͑B15͒, where only Gaussian integrals are present. But it is a good exercise to ﬁnd a direct proof of it in the case at hand, where everything can be computed exactly. We do this next.
For r odd the integrals vanish, and so let us consider the case where r is even, that is, rϭ2t, for any integer t.
On the one hand, we have

͵ ͚ xc,N x2teϪx2dxϭ͑ xc,N͒2tϩ1 ϱ

Ϫxc,N

kϭ0

͑Ϫ1͒k k!

͑

͑xc,N͒2k k ϩ t ϩ 1/2͒

,

͑B16͒

where the necessary interchange between sum and integral to
arrive at the result is allowed due to the uniform convergence of the Taylor series of eϪx2 in the ﬁnite segment
͓Ϫxc,N ,xc,N͔. On the other hand,

͚M 1

ϭ lim
M→ϱ

nϭ0

n!

⌫ ͑ n ϩ t ϩ 1/2͒

ͩ ͪ ͩ ͪ ϫ

x

2 c,

N

t ϩ 1/2

1

ϩ

x

2 c,N

Ϫ ͑ n ϩ t ϩ 1/2͒

M

M

͚ ͩ ͪ ͚ ϭ lim

M

1

x

2 c,N

tϩ1/2 ϱ

͑Ϫ1͒k

M→ϱ nϭ0 n! M

kϭ0 k!

ͩ ͪ ϫ⌫͑nϩtϩkϩ1/2͒

x

2 c,

N

M

k

͑B18͒ ͑B19͒

͚ϱ
ϭ͑xc,N͒2tϩ1
kϭ0

͑Ϫ1͒k k!

͑xc,N͒2k ͑ k ϩ t ϩ 1/2͒

ͫ ͚ ͬ ͑kϩtϩ1/2͒ M ⌫͑nϩtϩkϩ1/2͒

ϫ

lim
M→ϱ

M ͑kϩtϩ1/2͒

nϭ0

n!

.

͑B20͒

In Eq. ͑B18͒ we have used the equation

͵ϱ Ϫϱ

x2ne

Ϫpx2dx

ϭ

⌫

͑ n ϩ 1/2͒ p nϩ1/2

,

͑B21͒

in Eq. ͑B19͒ we have expanded the last term of Eq. ͑B18͒ in

powers

of

x

2 c,

N

/

M

and

carried

out

some

cancellations,

and

ﬁnally in Eq. ͑B20͒ we have interchanged the M →ϱ limit

with the inﬁnite sum in k.

Comparing Eqs. ͑B16͒ and ͑B20͒, we see that the validity

of Eq. ͑B15͒ depends on the validity of the equation

͚ ͑kϩtϩ1/2͒ M ⌫͑nϩkϩtϩ1/2͒

lim
M→ϱ

M ͑kϩtϩ1/2͒

nϭ0

n!

ϭ1 ᭙ integers k,tϾ0.

͑B22͒

That this identity holds for every integer t and k can be seen by considering the following analytic function of the complex variable z:

͚ ͑1/z ͒ M ⌫͑nϩ1/z ͒

O

͑

z

͒ϵ

lim
M→ϱ

M

͑

1/z

͒n

ϭ

0

⌫͑nϩ1͒!

.

͑B23͒

If the identities ͑B22͒ hold, this function must be identically 1, since for 1/z jϭ jϩ1/2 with j integer it reduces to them, and for ever increasing j, we obtain a sequence accumulating at zϭ0 on which the function should be 1.
Conversely we will prove that O(z) is indeed identically
1 as an analytic function of z, proving in consequence the identities ͑B22͒ for arbitrary t and k. Consider the sequence 1/z jϭ jϩ1 for j integer. This sequence also accumulates at zϭ0, and for all its points we have

1158

SERGIO A. PERNICE AND GERARDO OLEAGA

57

͚ ͑ jϩ1 ͒ M ⌫͑nϩ jϩ1 ͒

O„1/͑ jϩ1 ͒…ϭ lim
M→ϱ

M ͑ jϩ1͒

nϭ0

⌫͑nϩ1͒

͚ ϭ lim M→ϱ

͑jϩ1͒ M M ͑ jϩ1͒nϭ0

⌸

j iϭ

1͑

i

ϩ

n

͒

ͫ ͚ ͬ ϭ lim M→ϱ

͑ jϩ1͒ M ͑ jϩ1͒

M nϭ0

n jϩO͑ n jϪ1͒

͑B24͒ ͑B25͒ ͑B26͒

ͫ ͬ ϭ lim M→ϱ

͑ jϩ1͒ M ͑ jϩ1͒

M ͑ jϩ1͒ ͑ jϩ1 ͒ ϩO͑ M j͒

M→ϱ
→ 1.

͑B27͒

Therefore O(z)ϭ1 for all z. This ﬁnishes the direct proof of Eq. ͑B15͒.

As was mentioned before, Eq. ͑B9͒, derived indepen-

dently of any analyticity consideration, and only with the

purpose of satisfying the hypothesis of Lebesgue’s theorem,

introduces a nonanalyticity in the sequence of integrals of f NЈ
even for ﬁnite N. But even for the case where xc,N is ﬁxed to a constant a, discussed before, in which the limit N→ϱ is

taken ﬁrst, and then a is sent to inﬁnity, and therefore the

sequence is made out of truly analytic functions, the conver-

gence towards z(␭) is perfectly compatible with analyticity

considerations.

The

functions

␲

Ϫ

1/2͐

a Ϫ

a

d

x

e

Ϫ

[

x

2

ϩ

(

␭

/

4)

x

4

]

͑the

result of the N→ϱ limit͒ are clearly analytic in ␭. But they

converge to ͑in fact they deﬁne͒ the nonanalytic function

z(␭) when a→ϱ. The limit of an inﬁnite sequence of ana-

lytic functions does not have to be analytic.

Another important issue is that the same method also

works for the ‘‘negative mass case,’’ where the Borel resum-

mation method fails. Indeed, from the discussion of this ap-

pendix it must be obvious that, with a proper scaling of xc,N , the f NЈ ’s with a negative quadratic part of the exponent also converge uniformly towards e[x2Ϫ(␭/4)x4] for x in

͓Ϫxc,N ,xc,N͔. Therefore, the sequence of integrals is also convergent.

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