1. RESEARCH REVIEW PROJECT
Convergence of Perturbation Series in Quantum Field Theory
Supervisor: Prof. James Stirling; Candidate Number: 6941V
Abstract. Perturbation expansions are ubiquitous in quantum field
theory. They are a standard tool for determining field theory pre-
dictions to high accuracy. The purpose of this paper is to review
considerations of perturbation series as mathematical entities: their
radius of convergence; their large-order behaviour; what physical in-
terpretation can be assigned to divergent perturbation series; and
recent research on the renormalon divergence.
1. Introduction
We consider the perturbative expansion of a function f (g),
∞
f P T (g) ≡ f (0) + cn g n . (1)
n=1
Here g is the strength of an interaction, and f P T (g) − f (0) measures deviations to the state
of the noninteracting system within the radius of convergence of f P T . In quantum field
theory it is sometimes assumed that, for g small, f (g) = f P T (g). The justification is experi-
ence: in the framework of the standard model, perturbative predictions have had enormous
success in predicting experimental results. However, agreement between experiment and the
perturbation series truncated at the Nth power in g, where N is a small number, does not
imply f (g) = f P T (g). Indeed, it is well worth knowing about the divergence of perturbation
series in quantum field theories because the divergence may eventually limit the accuracy of
predictions, and because the limit of the perturbation series becomes ill defined. In Section
2, Dyson’s argument for the divergence of perturbation series in Quantum Electrodynamics
(QED) will be reviewed. In Section 3, large-order estimates of coefficients cn will be discussed
for the anharmonic oscillator in quantum mechanics and for various quantum field theories.
Section 4 focuses on attempts at resumming divergent perturbation series in the framework
of Borel summation. Recent research on the renormalon divergence is reviewed in Section 5.
Conclusions are drawn in Section 6.
2. The Dyson Instability
In 1952, Dyson published a paper [1] on the convergence of perturbation series in QED. He
e2
suggested that in a fictitious world with a negative QED coupling constant, α = 4π < 0;
i.e., imaginary charge e, the vacuum is a metastable state. This gedanken experiment is
motivated by the fact that any function, expanded perturbatively around the point at which
the coupling constant α is zero, is influenced by the negative coupling region.
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2. In such a fictitious world, the Coulomb force is reversed, such that like-charged particles
attract one another, and oppositely charged particles repel. Dyson argues that one can
then construct a state constituted of N electrons in one region of space and N positrons
in a separate region of space such that each particle’s rest- plus kinetic-energy is smaller
than the absolute value of its potential energy. This potential energy owes to the attractive
Coulomb force of the other N-1 like-charged particles in the vicinity. Dyson claims that
one can construct such a state without having to use very small distances or large charge
densities, so the classical Coulomb force is a valid approximation. In this world, the vacuum
is no longer the state of lowest energy. Furthermore, in a quantum mechanical system, such
a state is always accessible through quantum tunneling. Hence there would be a non-zero
chance for the vacuum to decay into such a state in any finite amount of time. Once the
vacuum has decayed, the electron and positron regions would build a potential between them
that would facilitate further decay, resulting in a rapid disintegration of the vacuum.
Following this reasoning it appears that a microscopic negative coupling introduced into
the system can have a macroscopic effect, which is an event that cannot be described per-
turbatively. Since there is a finite probability of quantum tunneling from any state into a
decayed-vacuum state in a finite amount of time, no function f (α) will be analytic when the
coupling constant is negative.
It is a general property of the Taylor expansion for complex-valued functions that: 1) the
region of convergence is a circle centered on the expansion point; and 2) the function is
analytic within the entire region of convergence. Suppose f (α) has non-zero radius of con-
vergence. Then for some range of imaginary α, f (α) is analytic, which contradicts the above
argument. This shows that any perturbation series in QED has zero radius of convergence.
Dyson claimed that the decay of the vacuum involves the interaction of many particles, and
hence does not influence very small orders of the perturbation expansion. His approach
of examining the properties of a field theory at small negative coupling is the foundation
on which most large-order estimates of perturbation theory coefficients in field theories are
based. The Dyson instability is not exclusive to quantum electrodynamics. It is in fact a
general feature of most quantum field theories (see Fischer [2] for details).
3. Large-Order Estimates of cn
In the early 1970s, Bender and Wu [3, 4] derived the exact asymptotic behavior of the
perturbation series coefficients for the Kth energy level of the one-dimensional anharmonic
oscillator in quantum mechanics (in units h = 1)
¯
d2 x2 x4
+ + λ − E K (λ) ψ(x) = 0. (2)
dx2 4 4
The coupling here is the strength λ of the quartic power perturbation to the harmonic
oscillator Hamiltonian.
Using carefully established analyticity properties of the energy of the oscillator, Bender and
Wu were able to link the nth -power perturbation coefficient cn to the imaginary part of the
2

3. Figure 1: Comparison of anharmonic oscillator potential at small positive and negative
coupling
energy at small negative coupling. This imaginary part of the energy arises because there
is a possibility of quantum tunneling out of the oscillator altogether; i.e., probability is not
conserved even in the ground state of the oscillator (see Figure 1).
Bender and Wu obtained the imaginary part of the energy for an arbitrary energy level via a
WKB analysis of transmission through the potential barrier. Using a perturbative expansion
around the energy for λ = 0
∞
1
E K (λ) = K + + cK λn , (3)
2 n=1 n
the result is √
12K 6 1
lim cK
= (−1)n+1 3n Γ(n + K + ), (4)
n→∞ n K!π 3/2 2
where the Γ-function is a generalization of the factorial function g(n) = n! to non-integer
values. With increasing order of expansion, the coefficients of the perturbation series for
the energy asymptotically approach a factorially divergent series with alternating sign. Ben-
der and Wu found agreement with results from a computer simulation to 150th order in
perturbation theory and with the results of an alternative approach for the lowest energy
levels.
The extension of this approach to quantum field theories is non-trivial. Lipatov [5] showed
that the problem could be simplified by approximating the paths of the particles at small
negative coupling by their corresponding classical paths with small quantum fluctuations.
He calculated the degree of divergence of massless renormalizable scalar field theories. His
approach was extended by many others: Brezin, Le Guillou and Zinn-Justin [6] were able
to rederive and extend the result obtained by Bender and Wu for the anharmonic oscillator;
Parisi [7] extended the method to fermion fields; Itzykson, Zuber, Parisi and Balian [8,9]
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4. as well as Bogomolny and Fateyev [10] extended the result to QED. A discussion of large-
order estimates in field theories, including Quantum Chromodynamics (QCD), is provided
in Fischer [2] and references therein.
The general feature is a factorial growth of the expansion coefficients at large order. Given
this fast divergence of the perturbation series, we will next describe the manner by which
one might formally extract the exact physical quantity f (g) from the divergent perturbation
series f P T (g) via the method of Borel summation.
4. Borel summation of the divergent perturbation series
Divergence of a perturbation series signals that the quantity being studied is not analytic at
the point about which the expansion is being performed. The possibility that there exists a
bijective correspondence between f P T and f is not excluded by the divergence of the former
series. In other words, as long as there is a unique mapping φ : f P T (g) → f (g), physical
results to arbitrary precision can be obtained from f P T .
The standard process of Borel summation is an attempt to obtain this information:
∞
φ : f P T (g) → h(g) = dt e−t B(gt), (5)
0
where we have defined
∞ ∞
cn
B(gt) = (gt)n using the cn of f P T (g) = cn g n . (6)
n=0 n! n=0
The motivation lies in the integral representation of the gamma function at integer points:
1 ∞
1= dte−t tn . (7)
n! 0
Inserting this index-n dependent identity at each term in the sum of f P T (g) and moving the
divergent sum inside the integral (thereby changing the function, which is what we aim to
achieve), we obtain the above mapping. Notice that the sum B(t) has improved convergence
by scaling cn down by a factor of n!. Under certain analyticity conditions on the function
f (g), this procedure yields a unique function h(g) whose perturbative expansion matches
f P T (g). We would then claim h(g) = f (g) and use non-perturbative results and experiment
to verify our claim.
Despite arguments that renormalizable theories are not in general Borel-summable, which
we will come to later in this section, in particular cases Borel-summation has been used
successfully. For example, Ogievetsky [11] in 1956 calculated the contribution of vacuum
polarization by a constant external magnetic field to the QED Lagrangian using Borel re-
summation of the perturbation series. His resummed series agrees with the nonperturbative
result found by Schwinger in 1951. Brezin, Le Guillou and Zinn-Justin [12, 13] used Borel
summation in condensed matter field theory to resum the divergent Wilson-Fisher expan-
sion for critical phenomena. At the time they obtained the most precise theoretical values
of the critical exponents for phase transitions in Ising-like systems.
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5. Figure 2: In order to ensure uniqueness of the Borel transform h(g), there are different
requirements on the minimum opening angle of the region of analyticity of f (g), depending
on the steepness of the divergence of f P T (g). Note that only the opening angle at the origin
matters, not the extent of the domain. (a) A minimum opening angle of π for factorial
divergence; (b) a minimum opening angle 0 < θ(ρ) < π for divergence as (n!)ρ for ρ < 1;
and (c) a minimum opening angle of 0 for divergence as (ln(n))n
While the Borel transform works well for some problems, in QED and QCD one often
finds that the Borel sum B(gt) has singularities, which have to be integrated over to get
h(g). Since the integration is along the positive real axis, these singularities introduce an
ambiguity through different possible choices of contour around these poles. In order to avoid
the poles and unambiguously define the Borel transform h(g), the function f (g) needs to
be analytic within a certain opening angle around the origin in the complex g plane. In
ensuring convergence, there is a competition between the magnitude of the opening angle
and the steepness of the divergence of the perturbation series. Nevanlinna[14] shows that for
a factorially divergent series, the analyticity region of f (g) must have an opening angle of π
at the origin. Weaker statements of Nevanlinna’s Theorem require for example a divergence
as slow as cn ∝ (ln(n))n for a wedge of zero opening angle at the origin (See Figure 2 for
details). For an overview of Borel summation, see the review by Fischer [2] and references
therein.
’t Hooft [15] showed that the region of analyticity of any Green function in QCD is limited to
a horn-shaped wedge of zero opening angle. He used the fact that analyticity in the coupling
constant is related to analyticity in momentum and, using a coupling parameter such that
the Gell-Mann Low function had only two terms, proved that the renormalization group
equations impose the above constraint on the analyticity region. His result was verified
by Khuri [16], who showed explicitly that the region of analyticity is independent of the
particular choice of coupling parameter. The horn-shaped analyticity region corresponds
to choice (c) in Figure 2 and is related to a maximum divergence cn ∝ (ln(n))n for Borel
summability, which is not satisfied by QCD (see references in Fischer [2]).
The zero opening angle is believed to be a general property of renormalizable theories. In
other words, the divergence of f (g) at the origin is too strong to allow for a reconstruction
of the series via Borel summation. There are two alternative interpretations. Alternative
A: the problem lies with the Borel resummation technique, and there exists a different map
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6. that formally recovers f (g) from f P T (g); viz.,
φ : f P T (g) → f (g) for g < gmax . (8)
In this case, the problem is merely of a technical nature. The function f (g) could be uniquely
defined for small values of the coupling g and there would be no ambiguity in the theory.
Alternative B: the perturbation series in QCD (or QED) could suffer from inherent ambigu-
ities and therefore cannot be resummed uniquely even at small coupling. There is no way of
formally defining the limit of the perturbation series without some non-perturbative input.
The Borel representation h(g) and its poles provide the formalism in which many contem-
porary authors (see for example Beneke [20] and Fischer [2]) discuss asymptotic estimates.
We will now turn to the renormalon divergence arising from the contribution of a particular
type of diagram in renormalizable theories.
5. Renormalon divergence
In perturbation theory, the number of Feynman diagrams usually grows with the order of
expansion, oftentimes as n!, where n is the order of expansion (see for example Jaffe [17]
or references in Lautrup [19]). Giving each Feynman diagram an amplitude of order 1 and
summing over all diagrams at every order with no cancellation, we can naively imagine a
way in which the large-order divergence of the perturbation series predicted in Section 3,
cn ∝ n!, could appear in the terms of the series.
This picture may not work in renormalizable theories: Gross and Neveu [18], Lautrup [19]
and ’t Hooft [15] found diagrams of a particular type whose contribution to the perturbation
series f P T at large order grows factorially with the order of expansion. Since this divergence
occurs only in renormalizable theories, it has been termed the renormalon divergence. In
some cases, for example QCD, the divergence due to these diagrams is claimed to be stronger
(see Beneke [20] and references therein) than that calculated by Lipatov’s method of using
the saddle-point technique around classical solutions at small negative coupling.
In diagrams contributing to the renormalon divergence in QED, photon lines are modified
by insertions of a large number of lepton loops (pair creation and annihilation of leptons),
see Figure 3. One may wonder how a small collection of all diagrams can produce a stronger
divergence than all diagrams taken together. There are two possible answers to this question:
1. The divergence of the perturbation series has been underestimated: Lipatov’s method
of finding the divergence of f P T , due to all Feynman diagrams at each order of the
perturbation series, relies on the saddle-point technique around the classical particle
path and hence does not take into account contributions to the integral from the tail,
far away from the classical path.
It is possible that the integral over the tail yields a result which is comparable or bigger
than the saddle-point approximation (see Figure 4). In this case, Lipatov’s method
would not be applicable and a stronger divergence than that calculated by the saddle-
point technique may be possible, allowing for a renormalon-type divergence of the total
series.
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7. (a) Figure 3: Renormalon diagrams in QED (b) Figure 4: An example of a function for
whose contribution to f P T at large order n which the saddle-point technique around x0
is proportional to n! is inapplicable
Dyson’s fundamental argument would still hold: the perturbation series is expanded
around a non-analytic point owing to the instability of the vacuum at negative coupling.
In this case one cannot, however, evaluate Green functions at quantum fluctuations
near the classical path for the purpose of calculating large-order terms of the pertur-
bation series.
For an overview of papers whose authors advocate the significance of renormalon di-
vergences, see Beneke’s review [20] and references therein. Beneke associates the renor-
malon divergence with non-perturbative power corrections to the perturbations series,
and claims that the magnitude of these non-perturbative corrections is comparable to
the magnitude of resummed renormalon terms.
2. Suslov [21] argues that cancellation among diagrams prevents the renormalon diver-
gence when all Feynman diagrams are taken into account and that Lipatov’s method
for calculating large-order terms in the perturbation series is valid despite the faster
divergence of a subclass of diagrams.
In either case, the general claim put forth by Dyson is not questioned: the perturbation
series is divergent in most quantum field theories. We will now summarize what we have
learned.
6. Conclusions
Fischer [2] has posed three questions to quantify the relationship between f (g) and f P T (g):
1. Convergence: in what domain of g is f (g) = f P T (g)? Dyson’s argument [1], that the
vacuum instability at negative coupling leads to non-analyticity of f (g) at the origin,
leads us to answer: f (g) = f P T (g) only at the origin.
2. Truncation: what are the properties of the remainder functions, RN (g), defined in
Eq. (9)?
N
RN (g) ≡ |f (g) − f (0) − cn g n |. (9)
n=1
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8. N
Since we want to use the truncated perturbation series f (0) + cn g n to approximate
n=1
f (g) at low coupling, RN (g) must be of order g N +1 or higher: RN (g) = O(g N +1 ) as
g → 0. If this is true for every N, then f P T (g) is an asymptotic expansion of f (g). By
definition, estimates of the remainder function have to be obtained from information
outside of the perturbation series. We did not have space to review estimates of RN (g)
here. See Fischer [2] for more details.
3. Uniqueness: how much physical information is contained in the set {cn } defined in
Eq. (10)?
∞
f P T (g) ≡ cn g n (10)
n=0
Can the function f (g) be unambiguously reconstructed from this set for small values
of g, or is there a fundamental limit to the precision of the perturbation series? In
other words, do we lose information by perturbing around the non-analytic origin?
We have reviewed Borel resummation, and some of its successes. In general for renor-
malizable theories, ’t Hooft [15] has argued that the method fails because of insufficient
analyticity properties of the function f (g) in the vicinity of the origin.
We have also learnt about Lipatov’s method of approximating large-order terms in the per-
turbation series by evaluating Green functions, at negative coupling, only at the classical
path and quantum fluctuations nearby. Applying this approach, we find that most quantum
field theories show a Γ(bn + c) divergence of the large-order coefficients with the order of
expansion n, where b,c are constants.
Finally, we have learned about a class of diagrams contributing to the particularly strong
renormalon divergence and we have encountered different interpretations of this divergence.
It may or may not be true that Lipatov’s method of approximating the divergence of the
perturbation series gives only a lower bound to the divergence of the series, yet this does not
contradict Dyson’s original argument for the divergence of the perturbation series.
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