Resurgent Analysis of Ward-Schwinger-Dyson Equations
Building on our recent derivation of the Ward-Schwinger-Dyson equations for the cubic interaction model, we present here the first steps of their resurgent analysis. In our derivation of the WSD equations, we ensured that they possessed the properties of compatibility with the renormalisation group...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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Інститут математики НАН України
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| Цитувати: | Resurgent Analysis of Ward-Schwinger-Dyson Equations. Marc P. Bellon and Enrico I. Russo. SIGMA 17 (2021), 075, 18 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860089716706115584 |
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| author | Bellon, Marc P. Russo, Enrico I. |
| author_facet | Bellon, Marc P. Russo, Enrico I. |
| citation_txt | Resurgent Analysis of Ward-Schwinger-Dyson Equations. Marc P. Bellon and Enrico I. Russo. SIGMA 17 (2021), 075, 18 pages |
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| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | Building on our recent derivation of the Ward-Schwinger-Dyson equations for the cubic interaction model, we present here the first steps of their resurgent analysis. In our derivation of the WSD equations, we ensured that they possessed the properties of compatibility with the renormalisation group equations and independence from a regularisation procedure known to allow for comparable studies in the Wess-Zumino model. The interactions between the transseries terms for the anomalous dimensions of the field and the vertex are at the origin of unexpected features, for which the effect of higher order corrections is not precisely known at this stage: we are only at the beginning of the journey to use resurgent methods to decipher non-perturbative effects in quantum field theory.
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| first_indexed | 2026-03-19T11:29:49Z |
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| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 17 (2021), 075, 18 pages
Resurgent Analysis of
Ward–Schwinger–Dyson Equations
Marc P. BELLON and Enrico I. RUSSO
Sorbonne Université, CNRS, Laboratoire de Physique Théorique et Hautes Energies,
Paris, France
E-mail: marc.bellon@upmc.fr, erusso@lpthe.jussieu.fr
Received February 10, 2021, in final form July 30, 2021; Published online August 11, 2021
https://doi.org/10.3842/SIGMA.2021.075
Abstract. Building on our recent derivation of the Ward–Schwinger–Dyson equations for
the cubic interaction model, we present here the first steps of their resurgent analysis. In our
derivation of the WSD equations, we made sure that they had the properties of compatibility
with the renormalisation group equations and independence from a regularisation procedure
which was known to allow for the comparable studies in the Wess–Zumino model. The inter-
actions between the transseries terms for the anomalous dimensions of the field and the
vertex is at the origin of unexpected features, for which the effect of higher order corrections
is not precisely known at this stage: we are only at the beginning of the journey to use
resurgent methods to decipher non-perturbative effects in quantum field theory.
Key words: renormalization; Schwinger–Dyson equation; resurgence
2020 Mathematics Subject Classification: 81Q40; 81T16; 40G10
1 Introduction
Quantum field theories do not produce convergent perturbative series, while these perturbative
series are often the only information accessible up to now through an analytic treatment. Con-
verting these series in numbers and properties of the theory therefore requires some non trivial
summation methods, especially in the large coupling regime. A very useful method goes through
the definition of a Borel transform that will give the solution through a Laplace integral. How-
ever, in many cases, the Borel transform has singularities on the real axis which make the naive
Borel–Laplace summation ambiguous: integration on rotated axis lose the reality properties
of the original series. In such situations, results with much reduced ambiguities can be obtained
through the use of real averages of the different analytic continuations.
It has been known for a long time that a successful resummation of a divergent series requires
the knowledge of its asymptotic properties and recent works have tried to quantify the gains
which can result from a deeper knowledge of the properties of the Borel transform. We would
single out the paper [14] that show how the use conformal maps of the Borel plane allows for
the most precise results. However, such gains are only possible if the precise structures of the
singularities of the Borel transform are known.
In quantum mechanics, singularities of the Borel transform stem from the presence of non
trivial saddle point of the action functional, dubbed instantons, but in renormalisable quantum
field theories, new singularities appear, related to the behaviour of diagrams with a maximal
number of subdivergences which are called renormalons. Much work has been devoted to the
finding of classical field configurations which could explain these singularities, but with limited
This paper is a contribution to the Special Issue on Algebraic Structures in Perturbative Quan-
tum Field Theory in honor of Dirk Kreimer for his 60th birthday. The full collection is available at
https://www.emis.de/journals/SIGMA/Kreimer.html
mailto:marc.bellon@upmc.fr
mailto:erusso@lpthe.jussieu.fr
https://doi.org/10.3842/SIGMA.2021.075
https://www.emis.de/journals/SIGMA/Kreimer.html
2 M.P. Bellon and E.I. Russo
success even if we must cite [3, 4]. This work will be based on a quite different approach, in the
spirit of [8, 9], which made use of the tools of resurgence theory and in particular the alien
derivatives, in the study of the solution of a Schwinger–Dyson equation. This work has been
prepared by a number of studies [5, 6, 7] which addressed only what was later recognized as
the singularities nearest the origin of the Borel transform. These previous studies were however
limited to the supersymmetric Wess–Zumino model, first solved perturbatively at high order
in [11], or a very special case of the ϕ3
6 model we study here, where the vertex gets no radiative
corrections at one loop [6].
It is therefore very interesting that our recent work [10] gives a system of equations for the
determination of the renormalisation group functions of the ϕ3
6 model, that we called Ward–
Schwinger–Dyson equations, most suitable for a resurgent analysis. Indeed, this scheme has the
properties which made the analysis in [8] possible, the absence of any explicit regularisation
parameter and the invariance of the solution under the renormalisation group. Indeed one may
say that our computations tend to transform a leading log approximation for the propagator and
the vertex in a computation of the leading terms for the high order terms of the perturbative
solution, but with the possibility to go beyond these leading behaviours through the systematic
inclusion of corrections.
It is a quite old observation by Giorgio Parisi that the renormalisation group equations written
for the Borel transform of the propagator imply that at a position ρ in the Borel plane, the
propagator has a leading correction proportional to
(
p2/µ2
)b1ρ, with b1 the leading coefficient
of the β function [22]. These power corrections give rise to new divergences which seemed
impossible to renormalize, in particular in the case of the infrared ones. We will instead show
that these divergences, linked to the poles of the Mellin transforms of the graphs, simply enable
us to compute the singularities of the Borel transforms of the anomalous dimensions.
2 The Borel–Laplace resummation method
2.1 General properties
We do not have the presumption here to introduce the topic of Borel–Laplace resummation
techniques; there are excellent introductions in the literature [13] or [21]. We report here though
some basic properties that we will need in the further development.
The formal Borel transform is defined on formal series as
B :
(
z−1C
[[
z−1
]]
, ·
)
−→ (C[[ξ]], ⋆),
f̃(z) =
1
z
+∞∑
n=0
cn
zn
−→ f̂(ξ) =
+∞∑
n=0
cn
n!
ξn.
Let f̃ , g̃ in z−1C
[[
z−1
]]
be two formal series and f̂ , ĝ in C[[ξ]] be their Borel transforms. The
following properties hold
B
(
f̃ .g̃
)
= f̂ ⋆ ĝ, B
(
∂f̃
)
= −ζf̂ , B
(
z−1f̃
)
=
∫
f̂ ,
f̃(z) ∈ z−2C
[[
z−1
]]
=⇒ B
(
zf̃
)
=
df̂
dζ
,
with the derivatives and the integral defined term by term and ⋆ denoting the convolution
product of formal series. If f̂ and ĝ are convergent,
f̂ ⋆ ĝ(ζ) =
∫ ζ
0
f̂(η)ĝ(ζ − η) dη
Resurgent Analysis of Ward–Schwinger–Dyson Equations 3
for ζ in the intersection of the convergence domains of f̂ and ĝ. The analytic continuation
of the convolution product on a given path can also be expressed through such an integral,
but the integration path is in general much more complex than the path on which the analytic
continuation is taken.
The definition of the Borel transform can be extended to series with constant terms through
the introduction of a unit δ for the convolution product. The constant function equal to the
constant a is mapped to aδ and then the whole space of formal series C[[z−1]] can be mapped
by linearity.
A formal series f̃(z) = 1
z
∑+∞
n=0
an
zn is 1-Gevrey if
∃A,B > 0: |an| ≤ ABnn! ∀n ∈ N.
In this case, we write f̃(z) ∈ z−1C
[[
z−1
]]
1
. In this case and only in this case, its Borel transform
has a finite radius of convergence and we denote by C{ζ} the space of such functions.
The Borel transform can be inverted through the Laplace transform. Let θ ∈ [0, 2π[ and
set Γθ :=
{
Reiθ, R ∈ [0,+∞[
}
. Let f̂ ∈ C{ζ} be a germ admitting an analytic continuation in
an open subset of C containing Γθ and such that
∃c ∈ R, K > 0:
∣∣f̂(ζ)∣∣ ≤ Kec|ζ| (2.1)
for any ζ in Γθ. Then the Laplace transform of f̂ in the direction θ is defined as
Lθ
[
f̂
]
(z) =
∫ eiθ∞
0
f̂(ζ)e−ζz dζ.
When the bound (2.1) is verified, this expression is finite in the half-plane Re
(
zeiθ
)
> c and
therefore defines an analytic function of z in this domain, which is called a Borel sum of f̃ .
For a formal series f̃(z) ∈ z−1C
[[
z−1
]]
with a non-zero radius of convergence, equation (2.1)
is true for all θ ∈ [0, 2π[ and its Borel sum in any direction coincide with the usual sum of the
series. For more general Borel summable series, many interesting phenomena can arise, such
as the Stokes phenomenon: the singularities of the Borel transform imply differences between
the Borel sums defined in directions separated by these singularities and even in cases, where
the condition (2.1) is satisfied for all directions, the Borel sum will differ from its analytic
continuation in a path around infinity, giving a non trivial monodromy. These problems are
at the heart of the renewed interest in summability techniques in particular in the physics
community [1, 2, 16, 20].
2.2 Resurgent functions and alien derivatives
Borel summation heavily relies on the possibility of analytically continuing the Borel trans-
form on a neighbourhood of the integration axis. In fact, the situation is usually better, with
a continuation possible in the whole complex plane minus some set of singularities. These singu-
larities can be studied through alien derivatives. The alien operator ∆ω extracts the singularity
around ω of the Borel transform and translates it to the origin. Some care must be taken when n
singularities lie on the segment [0, ω] since there is no longer a canonical analytic continuation of
the Borel transform to the neighborhood of ω: each singularity can be avoided in two different
ways resulting in 2n possible analytic continuations. It is easy to show that in the simple case
without singularities between 0 and ω, ∆ω is a derivation with respect to the convolution prod-
uct. With adequate weighting of the different paths to ω, this can be made true in the general
case with any number of singularities on the path.
4 M.P. Bellon and E.I. Russo
The alien operators corresponding to the singularities nearest the origin have special signi-
ficance: being the one which limit the convergence radius of the Taylor series, they give the
dominant behaviour in their high orders. A singularity at a point ω gives a contribution with
a ratio 1/ω between successive orders. The first correction is important, since it is related
to the kind of singularity. In the case of power laws singularities (ξ − ω)−α, the exponent α
translates into a nα−1 factor in the n-th coefficient of the Taylor series.1 Translating to the
original asymptotic series, this can be seen to mean that we have the following asymptotic ratio
of successive terms of the coefficients
cn+1
cn
=
1
ω
(
n+ α+O
(
n−1
))
. (2.2)
Beware that this formula is for the cn’s as defined in the previous subsection, with an index
shifted by 1 with respect to other conventions. Many studies have focused on obtaining this
type of relation, but the properties of the alien derivatives allows for a more efficient deriva-
tion.
Since the alien derivative involves a translation and the ordinary derivative becomes the
multiplication by −ζ for the Borel transform, these two derivatives do not commute: we have
[∂,∆ω] = ω∆ω.
Alien derivatives can be considered to act on the formal power series in z−1 and one generally
keeps the same notation, since confusion is not possible. In this case, we can multiply the
operator ∆ω by a transmonomial to obtain ∆̇ω ≡ e−ωz∆ω. This modified operator ∆̇ω now
commutes with the derivative and since it is a derivation, ∆̇ωf represent a possible deformation
of the solution f of a system of differential equations.
When a system of equations is given, alien derivatives can therefore be determined in two
stages. One first determines all possible deformations of a solution involving transmonomial
factors e−ωiz, extending the solution to a transseries. This transseries will depend on para-
meters ci and the possible alien derivatives are given by bridge equations, which express each
alien derivative through the action of some differential operator in the parameters ci. It is this
computation scheme that we will use throughout, rather than attempting to obtain directly the
asymptotic ratios in equation (2.2).
3 WSD equations for ϕ3
6
We introduced the Ward–Schwinger–Dyson scheme (often abbreviated to WSD in the following)
in the case of the model ϕ3
6 in [10]. We here recall only the equations which will be studied.
For their origin and possible extensions, the reader is invited to go back to this previous work.
They can be considered as variations on the Schwinger–Dyson equations written in terms of
derivatives of the effective action [15]. The lowest order primitive terms of the Ward–Schwinger–
Dyson equations for ϕ3
6 have the following diagrammatic form
ν = ν − 1
2
,
= + .
(3.1)
1If α is a nonpositive integer, (ξ−ω)−α is not singular at ω and the function has to be multiplied by log(ξ−ω)
to obtain the nα−1 behaviour.
Resurgent Analysis of Ward–Schwinger–Dyson Equations 5
The first equation allows us to determine the 2-point function, while the second one is for
the 3-point function. Dotted lines represent a vanishing incoming momentum, the decorations
represent the functions we compute
G(a, L) :=
∑
n≥0
1
n!
γn(a)L
n for ,
Y (a, L) :=
∑
n≥0
1
n!
υn(a)L
n for
and the square appears once and only once in the diagrams for the derivative of the propagator
Kν(a, L) := ∂ν
(
G(a, L)
p2
)
=
2pν
(p2)2
∑
n≥0
(γn+1 − γn)
Ln
n!
for .
In these equations, a := g2/(4π)3 is an equivalent of the fine structure constant which hides irre-
levant π factors, g is the coupling constant, ∂ν := ∂/∂pν , and L := log
(
p2/µ2
)
is the logarithmic
kinematic variable for a reference energy scale µ2. These decorations and the functions they
denote are always relative to the free propagator. They do not depend on any regularisation
parameter and satisfy the renormalisation group equations
∂LG = (γ + βa∂a)G, (3.2)
∂LY = (υ + βa∂a)Y, (3.3)
where γ(a) and υ(a) are the anomalous dimensions of the 2-point and 3-point function and β(a)
is the beta-function of the model. In [10], we established that β = 2υ + 3γ, but we must
point out that our convention for the function β differs from the usual ones. Since γ0 and υ0
are left undetermined by the equations, we fix them to 1 as a normalisation condition. Then
equations (3.2) and (3.3) impose the relations γ1 = γ and υ1 = υ which are used to determine
the renormalisation group functions γ and υ.
Decorations can be further composed as Cauchy products of the G and Y series. We can
assign to each internal line an operator W given as a product of G and Y , with its anomalous
dimension w given as
w = #Gγ +#Y υ,
and the renormalisation group equation
∂LW = (w + βa∂a)W.
Here is a non-exhaustive list of operators W that we could consider
W ∈
{
, , , , ,
}
.
In equations (3.1), two products appear, s (read “Samekh”) and q (read “Qof”), defined as
s(a, L) := Y GY (a, L) =
∑
n≥0
1
n!
sn(a)L
n for ,
q(a, L) := GY G(a, L) =
∑
n≥0
1
n!
qn(a)L
n for .
In particular, they satisfy
∂Ls = (γ + 2υ + βa∂a)s, ∂Lq = (2γ + υ + βa∂a)q,
6 M.P. Bellon and E.I. Russo
and we will write s = 2υ+ γ and q = 2γ+ υ. This formalism was tested in [10] by showing that
the renormalisation functions to order a2 computed with it matched with known results.
As is customary in resurgence literature, we work with a variable defined in the neighbourhood
of infinity, that is one which is proportional to 1/a. We use the variable r with a normalisation
such that:
rβ(r) = −1 + o
(
1
r
)
. (3.4)
This means that we define r = −1/(β1a) and the sign here is important because we want to keep
this trademark of asymptotic freedom. This normalisation ensures that the transseries terms
in the expansion of the anomalous dimensions will contain exp(nr) factors with integer n, and
the exponents we compute have simpler expressions.
Our computations make use of the Mellin transform representation of graphs. It is obtained
by replacing the ne propagators by full propagators with an additional exp(xL) factor. The eval-
uation of the resulting modified graph gives a function of ne complex parameters, meromorphic
with poles on linear subspaces.
The usefulness of such a function stems from the relation
Ln = ∂n
x e
xL
∣∣
x=0
,
which permits to obtain any function of L through the action of an infinite order differential
operator on exp(xL). The effect of the replacement of any propagator in a diagram can be
obtained from the action of this differential operator on (one of the parameters of) the Mellin
transform, followed by putting the parameter to 0. So, for example, the G series is described by
Gx := G(a, ∂x) =
∑
n≥0
γn(a)
∂n
x
n!
.
We assign also an operator O to the whole graph by multiplying the operators associated to all
of the internal lines Wi, with a factor 1/rl for a diagram with l loops, so that the evaluation of
the graph is just by applying this operator on the Mellin transform of the graph. For example,
for the graphs appearing in equations (3.1)
Oxy =
1
r
Kν
xsy for ,
1
r
qxsy for .
We do not keep Kν in our operators, since it can be expressed through G and its derivative
with respect to L: Kν
x can be traded for Gx, with only a multiplication of the Mellin transform
by x−1. In the one loop diagram we consider here, the propagator Kν is the only one in a path
between the exterior legs of the diagram, so that it could be obtained through the derivation
of the whole diagram with respect to the exterior momentum. In any case, we can just write G
instead of Kν . We can assign to O an anomalous dimension γO:
γO ≡ #Gγ +#Y υ − lβ, (3.5)
and write
∂LO = (γO − βr∂r)O.
Resurgent Analysis of Ward–Schwinger–Dyson Equations 7
The β function appears in equation (3.5) due to the 1/rl factor in the definition of O. It is
a combination of the anomalous dimensions γ and υ, in our model
β = 3γ + 2υ,
such that for the graphs in equation (3.1)
γO =
−γ for Oγ :=
1
r
Gxsy,
υ for Oυ :=
1
r
qxsy.
This ensures that both sides of the WSD equations obey the same renormalisation group equa-
tions and is instrumental in the proof we have given in [10] that the solutions of the WSD
equations obey renormalisation group equations. The β function is then the logarithmic deriva-
tive of the effective charge r−1Y 2G3, which have important combinatorial properties, see for
example [25].
From Ward–Schwinger–Dyson equations (3.1) we can extract equations for the anomalous
dimensions:
γ = (γ + βr∂r)γ − T2
2β1
OγH
γ ,
υ =
T3
β1
OυH
υ,
where Hγ and Hυ are functions associated to the two graphs through Mellin transforms.
4 Singularity structure of the Mellin transform
4.1 General properties
The poles of the Mellin transforms Hγ and Hυ give the dominant contributions in the evaluation
of the anomalous dimensions. These two functions are given by
Hγ(x, y) =
Γ(1− x− y)Γ(2 + x)Γ(2 + y)
Γ(4 + x+ y)Γ(1− x)Γ(1− y)
,
Hυ(x, y) =
Γ(1− x− y)Γ(1 + x)Γ(2 + y)
Γ(3 + x+ y)Γ(2− x)Γ(1− y)
,
and they have poles when the argument of one of the Γ function in their numerators is a negative
integer or zero. In terms of natural integers n, n′, n′′, we have therefore poles on the line with
equations
2 + x = −n,
2 + y = −n′,
1− x− y = −n′′,
and
1 + x = −n,
2 + y = −n′,
1− x− y = −n′′,
respectively for Hγ and Hυ.
The singularity structures of these functions are represented in Figure 1. We can also infer
from these plots the properties of the residue along any pole line. Consider one of them, whenever
a zero line crosses it, the residue get a zero, unless another pole line goes through the intersection
point. One can see that a zero line is present at each intersections of two pole lines, so that the
8 M.P. Bellon and E.I. Russo
Figure 1. Singularity structure of the characteristic functions H: the continuous blue lines represent
the poles, the dot-dashed red ones the zeroes.
residues never get poles of their own. In both cases, the singularity represented by the blue line
1− x− y = 0 is the closest one to the origin. For any H function we write the decomposition
H(x, y) =
∑
k
hk(x, y)
k − x− y
+
h′k(y)
k + x
+
h′′k(x)
k + y
+
∑
n,m
h̃n,mxnym, (4.1)
where the index sum is intentionally left unspecified because it depends on the particular choice
of H and a priori there should be four different summations but we did not want to weight the
notation too much. This description separates the poles of different kind whose residues are hk,
h′k, h
′′
k and an analytic part described by h̃n,m. Obviously if H is symmetric for x ↔ y then
h′k = h′′k, but it is not the case for Hυ. We give here their values for small k:
k hk h′k h′′k
Hγ 1
xy(2 + xy)
4!
— —
2
xy(3 + xy)(1− xy)
5!
(y − 2)(y − 1)
2
(x− 2)(x− 1)
2
3
xy(xy − 2)2(xy + 4)
6!2
(
y2 − 1
)
(2− y)(3− y)
3!
(
x2 − 1
)
(2− x)(3− x)
3!
Hυ 1
x(1 + y)
3!
1− y
2
—
2
xy(1 + y)(1− x)
4!
(
1− y2
)
(y − 2)
3!
2− x
2
The decompositions (4.1) of the functions H(x, y) give corresponding decompositions of the OH
constructs. The poles for x+y = k give rise to functions Ek in this decompositions. For the poles
depending on a single variable, the corresponding term can be further decomposed as a product
of factors respectively associated to the variables x and y: this part of H(x, y) is the product
of two functions depending on a single variable and the action of O on it will give the product
of the action of Ox on the function of x and the action of Oy on the function of y. The factor
Resurgent Analysis of Ward–Schwinger–Dyson Equations 9
corresponding to a single pole in a single variable will be called Fk. Functions Ek and Fk, were
first considered in [6] and were fundamental tools in [12] and in [7].
4.2 Fk functions
These functions capture the contribution to anomalous dimensions due to k+x = 0 or k+y = 0
poles of the characteristic functions H. For W in {G,s,q} and the corresponding w in {γ, s, q},
we define
Fw
k := Wx
(
x
k + x
)
= Wx
(∑
n≥0
(−1)nxn+1
kn+1
)
= −
∑
m≥1
(−1)mwm
km
.
We could have defined Fw
k as the action of W on the rational function 1/(k + x), but this would
produce terms which have a constant part and this would bring some difficulties with the Borel
transforms.
It follows immediately from the definition that
∂LW
(
x
k + x
)
= −kFw
k + w. (4.2)
Indeed, we can write
∂LW
∑
n≥0
(−1)n
xn+1
kn+1
= (w − βr∂r)
∑
n≥0
(−1)n
wn+1
kn+1
=
∑
n≥0
(−1)n
wn+2
kn+1
= k
∑
n≥2
(−1)n
wn
kn
= −k
(
Fw
k − w
k
)
.
The generic equation (4.2) can be specialised to the three different kinds of Fk, associated to
the three kinds of propagator like operators appearing in the Ward–Schwinger–Dyson equations.
The resulting equations allow to compute the corresponding Fk:
(γ − βr∂r)F
γ
k = γ − kF γ
k ,
(s− βr∂r)F
s
k = s− kF s
k ,
(q − βr∂r)F
q
k = q − kF q
k .
There is potentially an infinity of Fk terms which can contribute to the WSD equations, but
in our computations, we will need only a finite number of them. The first one in importance
is F q
1 but we also have all the Fk with a constant term in the residue of the corresponding pole.
These cases can be read in Figure 1 since they correspond to the poles parallel to an axis which
are not crossed by zeroes at their intersection with the other axis. In the case of Hγ , since it
is symmetric by x ↔ y, it suffices to describe the poles along one direction. These poles come
from Γ(2 + x) and thus start for x = −2 and end when a zero is crossed for y = 0, produced
by the term 1/Γ(4 + x+ y), therefore for x = −4 and lower. In the end, we have contributions
from two values of k, 2 and 3. In the case of Hυ, we do not have this symmetry and we must
distinguish the two directions: along x, the poles start at x = −1, from Γ(1+ x), and hit a zero
at x = −3, from 1/Γ(3 + x + y) taken at y = 0, leaving the two poles at −1 and −2. Along
the y direction, the important poles are determined by Γ(2 + y) for the start of the poles and
get a zero in the residue from the same factor than for the x poles, leaving only one pole for
y = −2. In the equation for the anomalous dimension γ there will be contributions from F s
k
and F γ
k , while for the one for the anomalous dimension υ there will be contributions from F s
k
and F q
k .
10 M.P. Bellon and E.I. Russo
4.3 Ek functions
The Ek functions capture the contribution to anomalous dimensions due to k−x−y = 0 poles of
the characteristic functions H. Differently from F k functions, they are not associated to a single
propagator but to the whole diagram. We define them through the action of the operators Oxy:
Ek = Oxy
hk(x, y)
k − x− y
. (4.3)
In this article we will work with the functions Eγ
k and Eυ
k given by
Eγ
k =
Gxsy
r
hγk(x, y)
k − x− y
, (4.4)
Eυ
k =
qxsy
r
hυk(x, y)
k − x− y
. (4.5)
As remarked already in [6], the relation (4.3) can also be written with exchanged places of
derivatives and variables
Ek =
hk(∂1, ∂2)
k − ∂1 − ∂2
O(L1, L2),
where ∂i stands for ∂Li and as usual, the variables are set to zero after all differentiations are
evaluated. Inverting k−∂1−∂2 is challenging, but when acting on O(L1, L2), it can be brought
to a form that only refers to the r variable and therefore act similarly on the function Ek. We can
also see that an arbitrary power of ∂1 + ∂2 followed by the evaluation at L1 = L2 = 0 can be
replaced by first evaluating L1 = L2 = L, applying the same power of ∂L and finally put L = 0.
We can then write, with O′ = hk(∂1, ∂2)O,
1
k − ∂1 − ∂2
O′ =
1
k
∑
n
1
kn
∂n
LO′ =
1
k
∑
n
1
kn
(γO − βr∂r)
nO′ =
1
k − γO + βr∂r
O′,
which ultimately brings the equation
(k − γO + βr∂r)Ek = hk(∂1, ∂2)O(L1, L2).
The rather formal definitions (4.4) and (4.5) can be converted to the following equations
(k + γ + βr∂r)E
γ
k = hγk(∂1, ∂2)
G(L1)s(L2)
r
,
(k − υ + βr∂r)E
υ
k = hυk(∂1, ∂2)
q(L1)s(L2)
r
.
One must remark that it is the inclusion of the 1/r factor in the definition of these Ek that
brings the simplification of the anomalous dimensions γO to −γ and υ.
Using the functions Ek and Fk, the contribution of the diagrams can be written as
OH =
c
r
+
∑
k
Ek +
1
r
∑
w
∑
k
zkwF
w
k +R, (4.6)
with c the constant giving the leading term and R collecting all other possible terms. Notice
that Ek functions include a factor 1/r while the Fk do not. This explains that the two sums
do not come with the same 1/r factor. The Fw
k come with a factor zkw, which is just a number
which is given by
zkw = −1
k
hwk (0),
Resurgent Analysis of Ward–Schwinger–Dyson Equations 11
where hwk (0) denotes the constant term of either h′k(y) or h′′k(x) according to which Mellin
variable is associated to the line and the prefactor −1/k comes from the identity
1
k + x
=
1
k
(
1− x
k + x
)
.
We could have kept other terms of the residue, but they would not contribute to the exponents.
4.4 R function
Our computations presume that the dominant contributions come from the poles of the Mellin
transform and more specifically from the poles near the origin. We therefore need some way of
bounding the contributions coming from the remainder of the Mellin transform, once a finite
number of poles has been subtracted.
The solution is not easy, since for the terms with exponential factors, all derivatives with
respect to L of the propagator corrections are now of the same order. In a first attempt to
find the corrections to the asymptotic behaviour of the series for the Wess–Zumino model [7],
a solution could be devised by using the conjecturally exact expansion of the Mellin transform
as sum of the pole contributions, when using a particular extension of the residues to the
whole (x, y) plane. This approach is however limited by the appearance of multizeta values with
high depth, which rapidly go beyond the cases with known reductions.
In a following work [8], we could find a much easier solution. It used a transformation of the
propagator which can be written as
G(L) =
∑
n
1
n!
γnL
n −→ Ǧ(λ) =
∑
n
γnλ
−n−1.
The relation between G and its transform looks like G is the Borel transform of this new func-
tion Ǧ, but it is not a proper interpretation, since the natural product for G is not a convolution
product. Nevertheless, the derivation with respect to L becomes the multiplication by λ after
this transform, so that it is easy to convert the renormalisation group equation for G in an equa-
tion for Ǧ. In this equation, a term with an exp(kr) factor in Ǧ gets multiplied by λ − k, so
that Ǧ gets a singularity for λ = k.
On the other side, the pairing of G with the Mellin transform is straightforward in this form.
We just have to make the sum of the products of the xn terms in H and the λ−n−1 terms
in Ǧ which is just the residue of H(x, y)Ǧ(x) at x = 0. This can be conveniently expressed as
a contour integral around the origin. If Ǧ depends on exp(kr) and has therefore singularities
for x = k, the contour integral will involve also evaluation of H or its derivatives at x = k: this
certainly does not work out if H has a pole at this point, so that this computation can only be
done after subtracting a number of poles of H, but the upshot is that one can obtain in this
way the contribution from the remainder of H in a form which contains sufficiently many 1/r
factor to not have any influence on the exponents we compute.
This construction must also be applied for the s and q products, which must be directly
obtained from their respective renormalisation group equations, since the pointwise product
in the variable L has no easy equivalent in the variable λ. The full development of this formalism
is certainly complex, since the Mellin transform must be evaluated not only at the origin, but
also, after suitable subtractions, at integer points. Nevertheless, for the sake of our limited
ambition in this work, it is possible to consider that the rest function which accounts for all
but linear terms in h′k and h′′k and the regular part of the H function is controlled. We can
characterise it as being
R =
∮ ∮
HregǑ,
12 M.P. Bellon and E.I. Russo
where Hreg is the function H with subtracted polar parts. This subtraction unfortunately cannot
be put in the form of a canonical projection as would be possible for H a function of a single
variable.
5 Trans-series corrections
5.1 Results
From the resurgent point of view the situation is quite intricate, but there are things that we
can easily establish. First of all, from the dominant term in rβ(r) in equation (3.4), one sees
that there is a dominant (k+ ∂r)Fk term in the equations for the Fk and (k− ∂r)Ek in the ones
for the Ek. We therefore see that Fk can be modified by a term proportional to e−kr and Ek by
a term proportional to ekr. This is however not sufficient to characterise these terms. The next
term in an expansion in 1/r cannot be compensated by the derivation of a series in powers
of r−1, which have a derivative starting with r−2, so that one has to multiply such terms by
a power of r, generically with a non-integer exponent which will be the dominant one in the
exponential terms in order to cancel the terms of order 1/r.
The purpose of this section is to show how to compute the values of these dominant exponents
for the corrections proportional to er and e−r for the anomalous dimension γ and υ. The situation
appears more complex than in the Wess–Zumino model, where there is only one Ek and one Fk
for each positive integer k and all exponents have been computed in [8]. Talking about the first
trans-series corrections means to talk about the closest singularities in the Borel plane through
their relations to alien derivatives. In turn, these singularities control the asymptotic behaviour
of the perturbative series. Terms that are proportional to er are linked to the singularities at −1
in the Borel plane while the e−r terms are linked to the singularities at 1. We will use the
notation that [k] indicates the part of a function which has a factor ekr, so that [0] indicates
the classical part and we will compute the exponents for the [1] and [−1] parts, corresponding
respectively to the singularities in −1 and 1 of the Borel transform.
The results of this section are expressed in terms of three quantities g, u and b appearing in
the first orders of the renormalisation group functions as
γ[0] =
g
r
+O
(
1
r2
)
= − T2
12β1
1
r
+O
(
1
r2
)
,
υ[0] =
u
r
+O
(
1
r2
)
=
T3
2β1
1
r
+O
(
1
r2
)
,
β[0]r = (3γ[0] + 2υ[0])r = −1 +
b
r
+O
(
1
r2
)
,
with
β1 =
T2
4
− T3,
b =
β2
β2
1
=
1
β2
1
(
11
24
T2T3 −
11
144
T 2
2 − 3
4
T 2
3 − 1
2
T5
)
.
These expressions appear different from previously published results, notably [17], due to our
conventions: we write the renormalisation group equations with β(r) as the variation of log r.
This shifts the powers of g and introduces a factor of 2 in β.
The trans-series corrections will come with dominant powers of r. We will name their expo-
nents η, θ and the pair of conjugated numbers λ± as follows: γ[1] is proportional to rηer;
Resurgent Analysis of Ward–Schwinger–Dyson Equations 13
υ[1] to rθer; both γ[−1] and υ[−1] are dominated by the two terms rλ
±
e−r, with a definite
relation between the dominant terms in γ[−1] and υ[−1]. Our results are summarized by
η = g + b, θ = b− 2
3
u, λ± = −2g − b± |3g|
√
1 +
4u
3g
.
Remarkably the difference
η − θ =
1
3
does not depend on the Casimir operators Ti, apart in the case, where β1 vanishes and our
change of variable is ill-defined.
We were surprised to find that λ± is algebraic and even complex in general. Real asymptotic
behaviors can only be obtained by combining two conjugate terms involving λ+ and λ−.
Let us show how they are calculated.
5.2 Preparatory steps
We start again from the WSD equations
γ = (γ + βr∂r)γ − T2
2β1
OγH
γ ,
υ =
T3
β1
OυH
υ,
and rewrite them using equation (4.6) as
γ = (γ + βr∂r)γ − T2
2β1
(
1
6r
+
∑
k
Eγ
k +
1
r
(
−1
2
(
F γ
2 + F s
2
)
+
1
3
(
F γ
3 + F s
3
)))
, (5.1)
υ =
T3
β1
(
1
2r
+
∑
k
Eυ
k +
1
r
(
−1
2
F q
1 +
1
6
F q
2 − 1
2
F s
2
))
, (5.2)
while neglecting the R terms.
For both er and e−r trans-series order we can neglect Ek for k ≥ 2. This occurs because these
functions satisfy the equations
(k + γ + βr∂r)E
γ
k = hγk(∂1, ∂2)
G(L1)s(L2)
r
,
(k − υ + βr∂r)E
υ
k = hυk(∂1, ∂2)
q(L1)s(L2)
r
,
and if we denote the descending powers by
Nn = N(N − 1) · · · (N − n+ 1)
we have
hγk(x, y) =
(−1)k−1
(k − 1)!Γ(4 + k)
(2 + x)k+2(2 + y)k+2,
hυk(x, y) =
(−1)k−1
(k − 1)!Γ(3 + k)
(1 + y)kxk.
The lowest degree monomial will be xy for these hk except for hυ1 , where it is just x. Higher
orders in x and y for hk extract terms with higher order in L in the propagators and these terms
14 M.P. Bellon and E.I. Russo
also begin at higher order in 1/r. Indeed, since we have that wn+1 = (w − βr∂r)wn, wn+1 is
of one order higher than wn. Since the equations for the Ek with k larger than two are not
resonant, they are of the same order than the term produced by hk, which is at least two order
smaller than γ or υ, unable to modify the exponents. At this stage, we only have to consider Eγ
1
and Eυ
1 . This simplification might not be true at higher trans-series order. From now on let us
forget the subscript 1.
Furthermore, the Fk functions do not contribute to the dominant exponents for trans-series
term proportional to er, as can be seen from the computation of Fk[1]:(
(k + w − βr∂r)F
w
k
)
[1] = w[1]. (5.3)
The left hand side can be expanded as(
(k + w − βr∂r)F
w
k
)
[1] = (k + w − βr∂r)[0]F
w
k [1] + (k + w − βr∂r)[1]F
w
k [0].
If we parametrise Fw
k [1] as follows
Fw
k [1] = errf
w
k cwk (1 + · · · ),
we have
∂rF
w
k [1] =
(
1 +
fw
k
r
+O
(
r−2
))
Fw
k [1].
Using that Fw
k [0] is at least of order 1 in 1/r, the dominant term of equation (5.3) gives that
Fw
k [1] ∼ 1
k + 1
w[1],
which can be specialised to the following dominant behaviours
F γ
k [1] ∼
1
k + 1
γ[1],
F s
k [1] ∼
1
k + 1
s[1] =
1
k + 1
(γ[1] + 2υ[1]),
F q
k [1] ∼
1
k + 1
q[1] =
1
k + 1
(2γ[1] + υ[1]).
Since in equations (5.1) and (5.2) they appear with a prefactor of r−1 they will always be
subdominant and for our purpose negligible. The dominant contributions proportional to er
come from E[1].
The opposite situation occurs for the e−r terms. The E[−1] do not contribute. It suffices to
consider the equations(
(1 + γ + βr∂r)E
γ
)
[−1] =
1
12r
(γs)[−1],(
(1− υ + βr∂r)E
υ
)
[−1] =
1
6r
q[−1],
and observe that the left hand side is dominated by (k + 1)Ek[−1]. This happens because
Ek[0] ∈ r−2C[[1/r]], since O contains a factor of r−1 and hk do not contain constant terms; also
βr∂rE[−1] ∼ E[−1]
because the factor −1 coming from e−r compensates with β[0]r = −1 + · · · .
This means that Eγ [−1] will always be subdominant. Any contribution are suppressed by
two factors of r: one from the original equation and one from order [0] series. So even if there
was resonance, which in fact occur, the shift is sufficient to neglect these terms. The situation
is, a priori, different for Eυ[−1]. In fact due to the exceptional right hand side for Eυ, where no
order [0] series appear, Eυ might contribute. Nevertheless we will see in Section 5.4 that F q
1 [−1]
is resonant and will dominate over Eυ.
Resurgent Analysis of Ward–Schwinger–Dyson Equations 15
5.3 Exponents of the leading singularities at −1
We take the following form for the functions Eγ at the er level
Eγ [1] = errϵcϵ(1 + · · · ),
Eυ[1] = errϵ̄cϵ̄(1 + · · · ).
In equation (5.1), the first term on the right hand side has the following leading contribution(
(γ + βr∂r)γ
)
[1] ∼ −γ[1]
which generates a factor of 2 with the γ[1] of the left hand side. Putting this in equation (5.2)
we obtain the dominant terms
γ[1] ∼ 3gEγ
1 [1], (5.4)
υ[1] ∼ 2uEυ
1 [1]. (5.5)
Then ϵ and ϵ̄ can be obtained from the equations for Eγ and Eυ
(
(1 + γ + βr∂r)E
γ
)
[1] ∼ − 1
12r
(γs)[1],(
(1− υ + βr∂r)E
υ
)
[1] ∼ − 1
6r
q[1],
where we have neglected higher order terms on the right hand side. At this transseries order
we have resonance: the highest order terms in the left hand side cancel exactly due to β[0]r =
−1 + O(1/r). No contribution from Eγ [0] or Eυ[0] occurs because Eγ [0] ∈ r−3C[[1/r]] and
Eυ[0] ∈ r−2C[[1/r]]. We are left with
g + b− ϵ
r
Eγ [1] ∼ 0,
−u+ b− ϵ̄
r
Eυ[1] ∼ − 1
6r
q[1].
In the first one of these equations, the right hand side is negligible, because it contains either γ[0]
or s[0] which give an additional 1/r factor. In the last one, we use q[1] = 2γ[1] + υ[1] and the
relation (5.5) to end up with
ϵ = g + b,
ϵ̄ = −2
3
u+ b
in order to have non trivial solutions for Eγ and Eυ. Then, using the relations (5.4) and (5.5)
shows that the dominant exponents in γ[1] and υ[1] are the ones announced before.
5.4 Exponents of the leading singularities at 1
Here the situation is less straightforward, with different important terms in the equations for γ
and υ and interactions between them.
For γ[−1], the leading resonance comes from the term (γ + βr∂r)γ. The F2 and F3 terms,
even if they are not resonant, give contributions of the order γ[−1]/r which cannot be neglected.
The first subdominant terms come from β[−1] and γ[0]. In particular equation (5.1) becomes,
when regrouping all gγ[1] terms
(g + λ+ b)γ[−1] + 2gυ[−1] ∼ 6g
(
−1
2
(
F γ
2 + F s
2
)
[−1] +
1
3
(
F γ
3 + F s
3
)
[−1]
)
.
16 M.P. Bellon and E.I. Russo
The right hand side can be further simplified using
Fw
k [−1] ∼ 1
k − 1
w[−1].
We therefore obtain that
(g + λ+ b)γ[−1] + 2gυ[−1] ∼ −4g(γ[−1] + υ[−1]),
giving
(5g + λ+ b)γ[−1] ∼ −6gυ[−1]. (5.6)
If υ[1] had a smaller exponent than γ[1], this would give an equation for λ, but this is not the
case.
In the expression for υ[−1], F1 stands out because it is resonant, so it adumbrates the other Fk.
Indeed, in the equation for F q
1(
(1 + q − βr∂r)
)
F q
1 [−1] = q[−1]
the highest terms cancel, so if we call φ the dominant exponent for F q
1 , we have
1
r
(q1 + b+ φ)F q
1 [−1] ∼ 2γ[−1] + υ[−1], (5.7)
with q1 = u + 2g the first coefficient in q. The other Fk do not have these cancellations, so
that they give contributions smaller by a factor 1/r to υ[−1] in equation (5.2), which at the
approximation level we use, gives a further relation
υ[−1] ∼ −u
r
F q
1 [−1]. (5.8)
Putting together equations (5.6), (5.7) and (5.8), we get an equation for the dominant power λ:
q1 + b+ φ = −u(2χ+ 1),
where
χ =
−6g
5g + λ+ b
is the proportionality factor between γ[−1] and υ[−1] from equation (5.6). Equation (5.7) further
shows that φ = λ+ 1 and we will use that, with our rescaling, the first coefficient of β is −1 so
that 1 = −3g − 2u. Finally λ satisfies
λ2 + 2(2g + b)λ+ b2 + 4gb− 5g2 − 12ug = 0,
with solutions
λ± = −2g − b± |3g|
√
1 +
4u
3g
.
The fact that it is algebraic is already strange, but for generic ϕ3 models the argument of the
square root is even negative, thus giving imaginary numbers. For example for the one-component
case where T2 = T3 = T5 = 1, its value is λ± = 107
81 ± i
√
7
3 .
This result is quite surprising, since we expected that the exponents would remain ratio-
nal, since they only depend on the rational numbers b, g and u. In Section 2.2, we recalled
Resurgent Analysis of Ward–Schwinger–Dyson Equations 17
how, through the use of alien derivatives, a link can be established between deformations of
the solution involving exponential factors and the asymptotic behaviour of the perturbative se-
ries. The rλ
±
e−r terms in our transseries solutions therefore convert into nλ±−1n! asymptotic
contributions to the (n + 1)th term of the perturbative series. Terms with the two possible
values of λ must be combined to obtain real results and give in particular a slowly oscillating
factor cos(Im(λ+) ln(n)+ϕ). The dependence of this phase on the logarithm of n means that we
would have to reach quite high orders to have an unambiguous sign of the associated oscillations.
A more speculative point is that such singularities of the Borel transforms could be related to
singularities of the propagator at points proportional to the renormalisation group invariant
mass [9]. The complex values of λ± could imply complex values for the corresponding position
of the singularities of the propagator: such unusual analyticity properties of the propagator have
been proposed as signs of confinement of the corresponding field [19, 23, 24].
6 Conclusion
In this paper we have started the resurgent analysis of the Ward–Schwinger–Dyson equations
for the ϕ3
6 model. This is a clear illustration of the power of this new approach to quantum field
theory to address asymptotic properties of the perturbative series. The exponents we compute
are totally inaccessible to the simple minded considerations of specific graphs that allowed to
locate renormalon singularities of the Borel transform. Classical field configurations which could
reproduce such singularities in a semiclassical expansion would have serious difficulties to explain
such exponents.
It is clear that the situation is much more intricate than for the Wess–Zumino model studied
in [9], where infinite families of possible transseries deformations could be readily obtained.
Already at the level we consider here of the nearest singularities of the Borel transform, we have
a pair of complex conjugated exponents in the transseries expansion. At the following levels,
we would have three different objects of the type F2, which could mean that three different
exponents are possible at level [−2]. And we cannot neglect the possibility that higher order
corrections to the Ward–Schwinger–Dyson equations have an influence on this whole picture,
since some aspects of our computations depend on the precise way we have done the infrared
rearrangements. It is therefore our hope that new constraints can be obtained that would allow
us to tame this proliferation of new series and use them to study non-perturbative effects for
this model through these methods.
It has recently been remarked that in a quite interesting special case, where the field is in a bi-
adjoint representation, the first coefficient of the β function vanishes, while the theory remains
asymptotically free due to the sign of the second coefficient [18]. In this case, the transseries
solution would include powers of exp
(
r2
)
= exp
(
1/a2
)
, meaning that singularities appear only
for the Borel plane dual to a variable u = r2 or some equivalent one. Such a study could
reveal new aspects of resurgence studies, but needs a knowledge of the β function at higher
loop for us to be able to control the precise transmonomial appearing in the expansion of the
renormalisation group function.
This is but a first step in the study of non-perturbative effects in this particular quantum
field theory, but it nevertheless presents results not accessible by other methods.
Acknowledgements
The authors feel indebted to Dirk Kreimer for his many inspirational works and find befitting
that this work be published in a special issue in his honour.
18 M.P. Bellon and E.I. Russo
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1 Introduction
2 The Borel–Laplace resummation method
2.1 General properties
2.2 Resurgent functions and alien derivatives
3 WSD equations for phi 3 6
4 Singularity structure of the Mellin transform
4.1 General properties
4.2 Fk functions
4.3 Ek functions
4.4 R function
5 Trans-series corrections
5.1 Results
5.2 Preparatory steps
5.3 Exponents of the leading singularities at -1
5.4 Exponents of the leading singularities at 1
6 Conclusion
References
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| id | nasplib_isofts_kiev_ua-123456789-211348 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-19T11:29:49Z |
| publishDate | 2021 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Bellon, Marc P. Russo, Enrico I. 2025-12-30T15:53:29Z 2021 Resurgent Analysis of Ward-Schwinger-Dyson Equations. Marc P. Bellon and Enrico I. Russo. SIGMA 17 (2021), 075, 18 pages 1815-0659 2020 Mathematics Subject Classification: 81Q40; 81T16; 40G10 arXiv:2011.13822 https://nasplib.isofts.kiev.ua/handle/123456789/211348 https://doi.org/10.3842/SIGMA.2021.075 Building on our recent derivation of the Ward-Schwinger-Dyson equations for the cubic interaction model, we present here the first steps of their resurgent analysis. In our derivation of the WSD equations, we ensured that they possessed the properties of compatibility with the renormalisation group equations and independence from a regularisation procedure known to allow for comparable studies in the Wess-Zumino model. The interactions between the transseries terms for the anomalous dimensions of the field and the vertex are at the origin of unexpected features, for which the effect of higher order corrections is not precisely known at this stage: we are only at the beginning of the journey to use resurgent methods to decipher non-perturbative effects in quantum field theory. The authors feel indebted to Dirk Kreimer for his many inspirational works and find it befitting that this work be published in a special issue in his honour. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Resurgent Analysis of Ward-Schwinger-Dyson Equations Article published earlier |
| spellingShingle | Resurgent Analysis of Ward-Schwinger-Dyson Equations Bellon, Marc P. Russo, Enrico I. |
| title | Resurgent Analysis of Ward-Schwinger-Dyson Equations |
| title_full | Resurgent Analysis of Ward-Schwinger-Dyson Equations |
| title_fullStr | Resurgent Analysis of Ward-Schwinger-Dyson Equations |
| title_full_unstemmed | Resurgent Analysis of Ward-Schwinger-Dyson Equations |
| title_short | Resurgent Analysis of Ward-Schwinger-Dyson Equations |
| title_sort | resurgent analysis of ward-schwinger-dyson equations |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211348 |
| work_keys_str_mv | AT bellonmarcp resurgentanalysisofwardschwingerdysonequations AT russoenricoi resurgentanalysisofwardschwingerdysonequations |