ap-3-10.dvi


Acta Polytechnica Vol. 50 No. 3/2010

Asymptotic Power Series of Field Correlators

I. Caprini, J. Fischer, I. Vrkoč

Abstract

We address the problem of ambiguity of a function determined by an asymptotic perturbation expansion. Using a modified
formof theWatson lemma recently proved elsewhere, wediscuss a large class of functionsdeterminedby the sameasymptotic
power expansion and represented by various forms of integrals of the Laplace-Borel type along a general contour in theBorel
complex plane. Some remarks on possible applications in QCD are made.

1 Asymptotic perturbation
expansions

Perturbation expansions are known to be divergent
both in quantum electrodynamics and in quantum
chromodynamics, as well as in many other physically
interesting theories and models. In QED, divergence
was proved byF. J. Dyson in 1952 (see [1]). His result
has been revisited and reformulated by many authors
([2, 3], see alsoa review in [4]). Dysonproposed to give
the divergent series mathematical meaning by inter-
preting it as an asymptotic series to F(z), the sought
function:

F(z) ∼
∞∑

n=0

Fnz
n, z ∈ S, z → 0, (1)

where S is a point set having the origin as an accumu-
lation point, z being the perturbation parameter.
To see how dramatically the philosophy of pertur-

bation theory was changed by this step, let us first
recall the definition of an asymptotic series:
Definition: Let S be a region or point set having

the origin as an accumulation point. The power series
∞∑

n=0

Fnz
n is said to be asymptotic to the function F(z)

as z → 0 on S, and we write Eq. (1), if the set of
functions RN(z),

RN(z)= F(z)−
N∑

n=0

Fnz
n, (2)

satisfies the condition

RN(z)= o(z
N) (3)

for all N =0,1,2, . . ., z → 0 and z ∈ S.
Note that the asymptotic series is defined by a dif-

ferent limiting procedure than the Taylor one: taking
N fixed, one observes how RN(z) behaves for z → 0,
z ∈ S, the procedure being repeated for all N ≥ 0

integers. Convergencemaybe provablewithout know-
ing F(z), but asymptoticity can be tested only if one
knows both the Fn and F(z).
By (1), F(z) is not uniquely determined; there are

many different functions having the same asymptotic
series, (1) say. Theambiguityof a function givenbyan
asymptotic series is illustrated by the lemma of Wat-
son.

2 Watson lemma
Consider the following integral

Φ0,c(λ)=
∫ c
0

e−λx
α

xβ−1f(x)dx, (4)

where 0 < c < ∞ and α > 0, β > 0. Let f(x) ∈
C∞[0, c] and f(k)(0) defined as lim

x→0+
f(k)(x). Let ε be

any number from the interval 0 < ε < π/2.
Lemma1 (G.N.Watson): If the above conditions

are fulfilled, the asymptotic expansion

Φ0,c(λ) ∼
1
α

∞∑
k=0

λ−
k+β

α Γ

(
k + β

α

)
f(k)(0)

k!
(5)

holds for λ → ∞, λ ∈ Sε, where Sε is the angle

|argλ| ≤
π

2
− ε. (6)

The expansion (5) can be differentiated with respect to
λ any number of times.
For the proof, see for instance [5]. Let us add sev-

eral remarks:
1) The angle Sε of validity of (5), (6), is indepen-

dent of α, β and c.

2) Thanks to the factor Γ
(

k + β
α

)
, the expansion

coefficients in (5) grow faster with k than those of the
Taylor series for f(x).
3) The expansion coefficients in (5) are indepen-

dent of c. This illustrates the impossibility of a unique
determination of a function from its asymptotic ex-
pansion.

Presented at the International Conference “Selected Topics in Mathematical and Particle Physics” organized in honour of the 70th
anniversary of Professor Jiří Niederle at New York University, Prague, 5–7 May 2009.

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Acta Polytechnica Vol. 50 No. 3/2010

In the next section we shall give a modification to
the Watson lemma, which shows that under plausible
assumptions the straight integration contour can be
bent.

3 Modified Watson lemma
The modified Watson lemma we present below (and
call Lemma 2’) is a special case of Lemma 2, which
we publish and prove in Ref. [6]. The special form
given here is obtained from that given in [6] by setting
α = β =1.
Let G(r) be a continuous complex function of the

form G(r) = rexp(ig(r)), where g(r) is a real-valued
function given on 0 ≤ r < c, with 0 < c ≤ ∞. Assume
that the derivative G′(r) is continuous on the interval
0 ≤ r < c and a constant r0 > 0 exists such that

|G′(r)| ≤ K1rγ1, r0 ≤ r < c, (7)

for a nonnegative K1 and a real γ1.
Assume that the parameter ε > 0 exists such that

the quantities

A = inf
r0≤r<c

g(r), B = sup
r0≤r<c

g(r) (8)

satisfy the inequality

B − A < π −2ε. (9)

Let the function f(u) be defined along the curve
u = G(r) and on the disc |u| < ρ, where ρ > r0. Let
f(u) be holomorphic on the disc and measurable on
the curve. Assume that

|f(G(r))| ≤ K2rγ2, r0 ≤ r < c, (10)

hold for a nonnegative K2 and a real γ2.
Define the function Φ

(G)
b,c (λ) for 0 ≤ b < c by

1

Φ
(G)
b,c (λ)=

∫ c
r=b

e−λG(r)G(r)f(G(r))dG(r). (11)

Lemma 2’: If the above assumptions are fulfilled,
then the asymptotic expansion

Φ
(G)
0,c (λ) ∼

∞∑
k=0

λ−(k+1)Γ(k +1)
f(k)(0)

k!
(12)

holds for λ → ∞, λ ∈ Tε, where

Tε ={λ : λ = |λ|exp(iϕ),
−

π

2
− A + ε < ϕ <

π

2
− B − ε}.

(13)

We refer the reader to Ref. [6] for the proof of
Lemma 2 and its discussion. The above simplified ver-
sion, Lemma2’, is givenhere to illustrate some special

features of the general Lemma 2 and its possible ap-
plications.
Let us add several remarks to Lemma 2’:
1/Lemma2’ impliesWatson’s lemmawhen the in-

tegration contour is chosen to have the special form of
a segment of the real positive semiaxis, i.e. g(r) ≡ 0,
and f(r) ∈ C∞[0, c].
2/ Perturbation theory is obtained by setting λ =

1/z in (10), (11). Then, the function

F
(G)
0,c (z)=

∫ c
r=0

e−G(r)/z f(G(r))dG(r) (14)

has the asymptotic expansion

F
(G)
0,c (z) ∼

∞∑
k=0

zk+1f(k)(0) (15)

for z → 0 and z ∈ Zε, where

Zε ={z : z = |z|exp(iχ),
−

π

2
+ B + ε < χ <

π

2
+ A − ε}.

(16)

3/ The parameter ε in (9) is limited by 0 < ε <
π/2−(B−A)/2, but is otherwise arbitrary. Note how-
ever that the upper limit of ε depends on B − A and
may be considerably less than π/2. This happens, for
instance, if the integration contour is bent or mean-
dering.
4/ The parametrization G(r) = rexp(ig(r)) does

not include contours that cross a circle centred at
r = 0, either touching or doubly intersecting it, so
that the derivative G′(r) either does not exist or is
not bounded. In such cases, the parametrization has
to be modified.
5/ Let us remark that the proof of Lemma 2 in

Ref. [6] allows us to obtain remarkable correlations
between the strength of the bounds on the remainder
and the size of the angleswithinwhich the asymptotic
expansion is valid. It follows from [6] that the bounds
are proportional to

1
(|λ|−1)sinε

e−(|λ|−1)r0 sinε (17)

or to
CN(|λ|sinε)−(N+2), (18)

where N is the truncation order and the CN, N =
0,1,2, . . . are λ-independent positive numbers. The
bounds decrease with increasing ε, the parameter,
which determines the angles Tε and Zε, see (13) and
(16) respectively. As a consequence, the larger the an-
gle of validity, the looser the bound, and vice versa.

1This integral exists since we assume that f(u) is measurable along the curve u = G(r) and bounded by (10).

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Acta Polytechnica Vol. 50 No. 3/2010

4 Some applications to
perturbative QCD

Todiscuss some applications of Lemma2’, we take the
Adler function [7],

D(s)= −s
dΠ(s)
ds

−1 . (19)

where Π(s) is the polarization amplitude defined in
terms of the vector current products for light quarks.
TheAdler functionD(s) is real analytic in the s-plane,
except for a cut along the timelike axis produced by
unitarity [7, 8]. In perturbativeQCD, any finite-order
approximant has cuts along the timelike axis, while
the renormalization-group improved expansion,

D(s)=D1 αs(s)/π + D2 (αs(s)/π)2 +
D3 (αs(s)/π)

3 + . . . ,
(20)

has, in addition, an unphysical singularity due to the
Landau pole in the running coupling αs(s). (20) is
known to be divergent, the Dn growing as n! at large
orders [9]–[12].

4.1 On the high ambiguity of
perturbative QCD

Todiscuss the implicationsofLemma2’,wefirstdefine
the Borel transform B(u) by [11],

B(u)=
∑
n≥0

bn u
n, bn =

Dn+1
βn0 n!

. (21)

It is usually assumed that the series (21) is convergent
on a disc of nonvanishing radius (this result was rig-
orously proved by David et al. [13] for the scalar ϕ4

theory in four dimensions). This is what is required in
Lemma2’ for the generalizedBorel transform f(G(r)).
If we assume that the series (20) is asymptotic,

Lemma 2’ implies a large freedom in recovering the
true function from its coefficients. All the functions
DG0,c(s) of the form

DG0,c(s)=
1
β0

∫ c
r=0

e
− G(r)

β0 a(s) B(G(r))dG(r) , (22)

where a(s) = αs(s)/π, admit the asymptotic expan-
sion

DG0,c(s) ∼
∞∑

n=1

Dn (a(s))
n, as(s) → 0, (23)

in a certain domain of the s-plane, which follows from
(13) and the expression of the running coupling a(s)
given by the renormalization group. No function of
the form DG0,c(s), (22), can be a priori preferred when
looking for the true Adler function.
Contributing only to the exponentially suppressed

remainder, neither the form or length of the contour,

nor the values of B(u) outside the convergence disc
can affect (23). The remainder to (23) is of the form

hexp(−d/β0a(s)) ∼ h
(
−Λ2/s

)d
. The quantities h

and d > 0 depend on the contour and on B(u) outside
the disc, which can be chosen rather freely. As a con-
sequence, (22) contains arbitrary power terms, to be
added to (23).

4.2 Analyticity and optimal conformal
mapping

In discussing the divergence of (20) and (21), the sin-
gularities of D(s) in the αs(s) plane and, respectively,
those of B(u) in theBorel plane are of importance. As
for B(u), some information about the location andna-
ture of the singularities can be obtained from certain
classes of Feynman diagrams (which can be summed,
see [10]–[12]), and from general arguments based on
renormalization theory, [9, 14]. It follows that B(u)
has branch points along the rays u ≥ 2 and u ≤ −1
(IR andUVrenormalons respectively). Other (though
nonperturbative) singularities, for u ≥ 4, areproduced
by instanton-antiinstantonpairs. (Due to the singular-
ities at u > 0, the series (20) is not Borel summable.)
No other singularities of B(u) in the Borel plane are
known, however. It is usually assumed that B(u) is
holomorphic elsewhere.
To make full use of the analyticity of B(u) in the

whole B, we shall use the method of optimal confor-
mal mapping [15]. Let K be the disc of convergence
of the series (21); clearly, K ⊂ B. Then, evidently, the
expansion (21) in powers of u can be replaced by that
in powers of w(u),

B(u)=
∑
n≥0

cn w
n, (24)

where the function w = w(u)with theproperty w(0)=
0 represents the conformalmapping of the region of B
onto the disc |w| < 1, on which (24) converges. It can
easily be seen that (24) has better convergence prop-
erties than (21) in this case: indeed, as was proved
in [15] by using the Schwarz lemma, the larger the
region mapped by w(u) onto |w| < 1, the faster the
large-order convergence rate of (24).
If w(u)maps thewholeB onto theunit disc |w| < 1

in the w plane, the mapping is called optimal. In this
case, (24) converges everywhere on B and the conver-
gence rate is the fastest [15]. The regionof convergence
of (24) coincides with B, the region of analyticity. In
this way, the optimal conformal mapping can express
analyticity in terms of convergence.
Inserting (24) into (22) we obtain an alternative

asymptotic expansion:

DG0,c(s)=
1
β0

∫ c
r=0

e
− G(r)

β0 a(s) ·∑
n≥0

cn [w(G(r))]
n dG(r) .

(25)

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Acta Polytechnica Vol. 50 No. 3/2010

Containing powers of the optimal conformal mapping
w(u) (which has the same location of singularities as
the expanded function B(u)), this representation im-
plements more information about the singularities of
B(u) than the series (21) in powers of u, even at finite
orders. Thus, it is to be expected that even the finite-
order approximants of (25)will provide amore precise
description of the function searched for [16, 17].

4.3 Analyticity may easily get lost

We shall briefly mention an intriguing situation show-
ing that careless manipulation with the integration
contour may have a fateful impact on analyticity. In
[18], two different integration contours in the u-plane
were chosen for the summation of the so-called renor-
malon chains [10]: for a(s) > 0 and a(s) < 0, a ray
parallel andclose to thepositiveand, respectively, neg-
ative semiaxis is chosen. As was expected and later
proved [19], analyticity is lost with this choice, the
summation being only piecewise analytic in s.
On the other hand, as shown in [20, 21], the Borel

summationwith thePrincipalValue (PV)prescription
of the same class of diagrams admits an analytic con-
tinuation in the s-plane, in agreementwith analyticity
except for a cut along a segment of the spacelike axis,
related to the Landau pole.

5 Conclusion
In this paper we have discussed some special conse-
quences of our general result published in [6], which is
based on a modification of the Watson lemma. It fol-
lows that a perturbation series, if regarded as asymp-
totic, implies a huge ambiguity of possible expanded
functions having the sameasymptotic expansionof the
type (1). This mathematical fact is often ignored or
overlooked in physical applications. Our contribution
consists in the fact that we have specified its special
subclass by Lemma 2 of Ref. [6]. Moreover, in the
present paper, we have considered a special subclass
of Lemma 2 (as defined by Lemma 2’ in section 3 of
this paper),whichwediscusshere inmoredetail due to
its direct applicability to perturbative QCD. To find
the true solution, additional information inputs are
unavoidable.
Applying the result to QCD, we conclude that the

contour of the integral representing the QCD correla-
tor can be chosen very freely. The same holds for the
Borel transform B(u) outside the convergence circle.
We have kept our discussion on a general level,

bearing in mind that little is known, in a rigorous
framework, about the analytic properties of the QCD
correlators in the Borel plane. If some specific prop-
erties are known or assumed, the integral representa-
tions will have additional analytic properties. Natu-
rally, the results obtained in [6] may also be useful in
other branchesof physicswhereperturbation series are
divergent.

Acknowledgement

One of us (I.C.) thanks Prof. J. Chýla and the Insti-
tute of Physics of the Czech Academy in Prague for
hospitality. J. F. thanksProf.P.Ra̧czkaand the Insti-
tute of Theoretical Physics of Warsaw University for
hospitality. SupportedbyCNCSIS in the frameworkof
ProgramIdei, ContractNo. 464/2009, andbyProjects
No. LA08015 of the Ministry of Education and AV0-
Z10100502 of the Academy of Sciences of the Czech
Republic.

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Irinel Caprini
National Institute of Physics andNuclear Engineering
Bucharest POB MG-6, R-077125 Romania

Jan Fischer
Institute of Physics
Academy of Sciences of the Czech Republic
182 21 Prague 8, Czech Republic

Ivo Vrkoč
Mathematical Institute
Academy of Sciences of the Czech Republic
115 67 Prague 1, Czech Republic

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