The small- Nseries in the zero-dimensional ON model constructive expansions and transseries Dario Benedetti1 Razvan Gurau12 Hannes Keppler2 and Davide Lettera2

2025-05-06 0 0 857.84KB 49 页 10玖币

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The small-Nseries in the zero-dimensional O(N) model:

constructive expansions and transseries

Dario Benedetti1, Razvan Gurau1,2, Hannes Keppler2, and Davide Lettera2

1CPHT, CNRS, Ecole Polytechnique, Institut Polytechnique de Paris, Route de Saclay,

91128 PALAISEAU, France

2Heidelberg University, Institut f¨ur Theoretische Physik, Philosophenweg 19, 69120 Heidelberg, Germany

emails: dario.benedetti@polytechnique.edu, gurau@thphys.uni-heidelberg.de,

keppler@thphys.uni-heidelberg.de, lettera@thphys.uni-heidelberg.de

Abstract

We consider the 0-dimensional quartic O(N) vector model and present a complete study of the

partition function Z(g, N) and its logarithm, the free energy W(g, N), seen as functions of the coupling

gon a Riemann surface. Using constructive ﬁeld theory techniques we prove that both Z(g, N) and

W(g, N) are Borel summable functions along all the rays in the cut complex plane Cπ=C\R−. We

recover the transseries expansion of Z(g, N) using the intermediate ﬁeld representation.

We furthermore study the small-Nexpansions of Z(g, N ) and W(g, N ). For any g=|g|eıϕ on the

sector of the Riemann surface with |ϕ|<3π/2, the small-Nexpansion of Z(g, N) has inﬁnite radius

of convergence in Nwhile the expansion of W(g, N) has a ﬁnite radius of convergence in Nfor gin a

subdomain of the same sector.

The Taylor coeﬃcients of these expansions, Zn(g) and Wn(g), exhibit analytic properties similar to

Z(g, N) and W(g, N) and have transseries expansions. The transseries expansion of Zn(g) is readily

accessible: much like Z(g, N), for any n,Zn(g) has a zero- and a one-instanton contribution. The

transseries of Wn(g) is obtained using M¨oebius inversion and summing these transseries yields the

transseries expansion of W(g, N). The transseries of Wn(g) and W(g, N) are markedly diﬀerent: while

W(g, N) displays contributions from arbitrarily many multi-instantons, Wn(g) exhibits contributions of

only up to n-instanton sectors.

Contents

1 Introduction 2

2 Borel summable series and Borel summable functions 5

3 The partition function Z(g, N)7

3.1 Analytic continuation and transseries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.2 Convergent small-Nseries of Z(g, N) and transseries of its coeﬃcients Zn(g) ........ 12

arXiv:2210.14776v2 [hep-th] 7 Nov 2022

4 The free energy W(g, N)14

4.1 Constructiveexpansion ...................................... 15

4.2 Transseriesexpansion ....................................... 19

4.3 Diﬀerentialequations........................................ 21

A Asymptotic expansions 22

A.1 The φrepresentation of the partition function . . . . . . . . . . . . . . . . . . . . . . . . . . 22

B A simple generalization of the Nevanlinna-Sokal theorem 24

C Proofs of Propositions 26

C.1 Properties of Z(g, N)........................................ 26

C.2 Properties of Zn(g)......................................... 32

C.3 The LVE expansion, analyticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

C.4 Borel summability of Wn(g) and W(g, N) in Cπ......................... 38

C.5 Transseries expansion of Wn(g) and W(g, N) .......................... 41

D The BKAR formula 44

D.1 Feynam graphs and W(g, N) ................................... 45

References 49

1 Introduction

The most famous problem of the perturbative expansion in quantum ﬁeld theory is the existence of

ultraviolet divergences in the amplitudes of Feynman diagrams. This is successfully dealt with using the

theory of perturbative renormalization. However, even in one and zero dimensions (quantum mechanics

and combinatorial models, respectively), where renormalization is not needed, perturbation theory poses

another notorious challenge: in most cases the perturbative series is only an asymptotic series, with zero

radius of convergence. Borel resummation is the standard strategy to address this problem, but this comes

with its own subtleties. From a practical standpoint, we are often only able to compute just the ﬁrst

few terms in the perturbative expansion. At a more fundamental level, singularities are present in the

Borel plane, associated to instantons (and renormalons in higher dimensions). The instanton singularities

are not accidental: they stem from the factorial growth of the number of Feynamn diagrams with the

perturbation order, which is also the origin of the divergence of the perturbation series.1

From the resummation point of view, the most inconvenient feature of perturbation theory is that

it does not naively capture contributions from non-analytical terms. For example, it is well known that

instanton contributions of the type e−1/g (g > 0 being the coupling constant) can be present in the

evaluation of some quantity of interest, but they are missed in the perturbative series as their Taylor

expansion at g= 0 vanishes identically.

Such exponentially suppressed terms are the archetypal example of nonperturbative eﬀects, and their

evaluation poses an interesting challenge. Aiming to include them, but still relying for practical reasons

on perturbative methods, one ends up with a more general form of asymptotic expansion, known as

transseries, which is roughly speaking a sum of perturbative and nonperturbative sectors, for example:

F(g)'X

n≥0

angn+X

ggγiX

n≥0

bi,n gn.(1.1)

1The renormalon singularities speciﬁc to higher dimensions, are diﬀerent. They are located on the positive real axis and

stem from the factorial growth of the renormalized amplitude of a family of diagrams consisting in essentially one diagram

per perturbation order.

Over time it became increasingly clear that, in many examples of interest, using the theory of Borel

summation for the perturbative sector it is possible to reconstruct some information about the nonpertur-

bative ones. This relation between the perturbative and nonperturbative sectors is known as resurgence,

and it was originally developed by ´

Ecalle in the context of ordinary diﬀerential equations [1] (see [2] for

a modern review) and later extensively used in quantum ﬁeld theory: for recent reviews with a quantum

ﬁeld theory scope, see [3–5], and in particular [6], which contains also a comprehensive list of references to

applications and other reviews.

Zero-dimensional quantum ﬁeld theoretical models, which are purely combinatorial models,2are useful

toy models for the study of transseries expansions. Most conveniently, they allow one to set aside all

the complications arising from the evaluation and renormalization of Feynman diagrams. Moreover, their

partition functions and correlations typically satisfy ordinary diﬀerential equations, thus ﬁtting naturally

in the framework of ´

Ecalle’s theory of resurgence. The zero dimensional φ4, or more generally φ2kwith

k≥2, models in zero dimensions have been exhaustively studied [6,10]. At the opposite end, the rigorous

study of the Borel summability in fully ﬂedged quantum ﬁeld theory is the object of constructive ﬁeld

theory [11–13]. It should come as no surprise that the generalization of results on resurgence in zero

dimensions to fully ﬂedged quantum ﬁeld theory is very much an open topic. Not only one has to deal

with renormalization, but also in higher dimensions the ordinary diﬀerential equations obeyed by the

correlations become partial diﬀerential (Schwinger-Dyson) equations on tempered distributions, and one

can not simply invoke ´

Ecalle’s theory.

From this perspective, revisiting the resurgence in zero dimensional models using techniques inspired by

constructive ﬁeld theory can be of great use. Here, following such route, we consider the zero-dimensional

O(N) model with quartic potential.3Denoting φ= (φa)a=1,...N ∈RNa vector in RNand φ2=PN

a=1 φaφa

the O(N) invariant, the partition function of the model is:4

Z(g, N) = Z+∞

−∞ N

a=1

dφa

√2π!e−S[φ], S[φ] = 1

2φ2+g

4!(φ2)2.(1.2)

The N= 1 case has been extensively studied in [6]. One can analytically continue Z(g, 1), regarded

as a function of the coupling constant g, to a maximal domain in the complex plane. Subsequently,

one discovers that Z(g, 1) displays a branch cut at the real negative axis and that the nonperturbative

contributions to Z(g, 1) are captured by its discontinuity at the branch cut. A resurgent transseries is

obtained when one considers gas a point on a Rienamm surface with a branch point at g= 0. From now

on we parameterize this Riemann surface as g=|g|eıϕ and we choose as principal sheet ϕ∈(−π, π).

An approach to the study of the partition function in Eq. (1.2) in the case N= 1 is to use the steepest-

descent method [15, 16]. We concisely review this in Appendix A. One notes that on the principal sheet

only one Lefschetz thimble contributes. As gsweeps through the princial sheet the thimble is smoothly

deformed, but not in the neighborhood of the saddle point: the asymptotic evaluation of the integral is

unchanged. When greaches the negative real axis there is a discontinuous jump in the relevant thimbles

and a pair of thimbles (passing through a pair of conjugated non trivial saddle points of the action) starts

contributing, giving rise to a one-instanton sector in the transseries of Z(g, 1).

Another approach to the transseries expansion of Z(g, 1) is to use the theory of ordinary diﬀerential

equations [6, 10]. It turns out that Z(g, 1) obeys a second-order homogenous linear ordinary diﬀerential

2These are for example of interest in the context of random geometry, see for example [7–9].

3The same model has been considered in a similar context in [14], where the problem of constructing Lefschetz thimbles

in the N-dimensional space have been studied. By using the intermediate ﬁeld formalism we will bypass such problem here.

4Note that we do not use the usual normalization g/4 of the interaction in the O(N) model, but stick to g/4! in order to

facilitate the comparison with the literature on transseries which deals mostly with the N= 1 case for which the normalization

g/4! is standard. Also, we do not use the ’t Hooft coupling λ=gN , which is needed for a well deﬁned 1/N expansion: in this

paper we keep Nsmall.

equation for which g= 0 is an irregular singular point (e.g. [15]), giving another perspective on why the

expansion one obtains is only asymptotic.

More interestingly, one can wonder what can be said about the nonperturbative contributions to the

free energy, that is, the logarithm of the partition function W(g, 1) = ln Z(g, 1), or to the connected

correlation functions. If we aim to study the free energy, the steepest-descent method does not generalize

straightforwardly as we lack a simple integral representation for W(g, 1). One can formally write Z(g, 1)

as a transseries and then expand the logarithm in powers of the transseries monomial ec

g, thus obtaining a

multi-instanton transseries. However this is very formal, as the transseries is only an asymptotic expansion,

and we would like to have a direct way to obtain the asymptotic expansion of W(g, 1). The closest one can

get to an integral formula for the free energy is to use the Loop Vertex Expansion (LVE) [17], a constructive

ﬁeld theory expansion which we present in detail below. However, this is by no means simple, and deriving

directly the transseries expansion of W(g, 1) using the steepest-descent method on this formula proved so

far impractical.

The best method available for the study of the transseries expansion of W(g, 1), before our work, is

to use again the theory of ordinary diﬀerential equations. One can show that W(g, 1) obeys a non-linear

ordinary diﬀerential equation whose transseries solution can be studied [6].

In this paper, we consider a general Nand we revisit both the partition function Z(g, N) and the free

energy W(g, N) from a diﬀerent angle. The paper is organized as follows.

In Section 2, we review the Borel summability of asymptotic series as well as the notion of Borel

summable functions, deriving in the process a slight extension of the Nevanlinna-Sokal theorem.

In Section 3, we study Z(g, N ) in the intermediate ﬁeld representation. This allows us to quickly prove

its Borel summability along all the rays in the cut complex plane Cπ=C\R−. More importantly, the

intermediate ﬁeld representation provides a new perspective on the origin of the instanton contributions: in

this representation, the steepest-descent contour never changes, but when greaches the negative real axis

a singularity traverses it and detaches a Hankel contour around a cut. We insist that this Hankel contour is

not a steepest-descent contour, but it does contribute to the asymptotic evaluation of the integral, because

the cut is an obstruction when deforming the contour of integration towards the steepest-descent path.

It is precisely the Hankel contour that yields the one-instanton contribution. We then build the analytic

continuation of Z(g, N) to the whole Riemann surface, identify a second Stokes line, compute the Stokes

data encoding the jumps in the analytic continuation at the Stokes lines and discuss the monodromy of

Z(g, N). Next we observe that, because in the intermediate ﬁeld representation Nappears only as a

parameter in the action, we can perform a small-Nexpansion:

Z(g, N) = X

n≥0

n!−N

2n

Zn(g).(1.3)

We thus study Zn(g) for all integer n, proving its Borel summability in Cπand computing its transseries

expansion in an extended sector of the Riemann surface, with arg(g)∈(−3π/2,3π/2), which we denote

C3π/2.

In Section 4, we proceed to study W(g, N) = ln(Z(g, N )). We ﬁrst establish its Borel summability

along all the rays in Cπusing consturctive ﬁeld theory techniques. We then proceed to the small-N

expansion of this object:

W(g, N) = X

n≥1

n!−N

2n

Wn(g),(1.4)

and prove that this is an absolutely convergent series in a subdomain of C3π/2and that both W(g, N) and

Wn(g) are Borel summable along all the rays in Cπ. Finally, in order to obtain the transseries expansion

of Wn(g) and W(g, N) we note that Wn(g) can be written in terms of Zn(g) using the M¨oebius inversion

formula relating moments and cumulants. Because of the absolute convergence of the small-Nseries,

it makes sense to perform the asymptoitic expansion term by term, and thus we rigorously obtain the

transseries for W(g, N) in a subdomain of C3π/2. In the Appendices we gather some technical results, and

the proofs of our propositions.

Ultimately, we obtain less information on the Stokes data for W(g, N) than for Z(g, N). While for

Z(g, N) we are able to maintain analytic control in the whole Riemann surface of g, the constructive ﬁeld

theory techniques we employ here allow us to keep control over W(g, N) as an analytic function on the

Riemann surface only up to ϕ=±3π/2, that is past the ﬁrst Stokes line, but not up to the second one.

The reason for this is that close to ϕ=±3π/2 there is an accumulation of Lee-Yang zeros, that is zeros

of Z(g, N), which make the explicit analytic continuation of W(g, N ) past this sector highly non trivial.

New techniques are needed if one aims to recover the Stokes data for W(g, N ) farther on the Riemann

surface: an analysis of the diﬀerential equation obeyed by W(g, N) similar to the one of [18] could provide

an alternative way to access it directly.

One can naturally ask what is the interplay between our results at small Nand the large-Nnonpertur-

bative eﬀects, ﬁrst studied for the zero-dimensional O(N) model in [19] (see also [3] for a general review,

and [20] for a more recent point view). This is a very interesting question: indeed, the relation between the

two expansions is a bit more subtle than the relation between the small coupling and the large coupling

expansions for instance. The reason is that, when building the large Nseries, one needs to use the ’t Hooft

coupling, which is a rescaling of the coupling constant by a factor of N. This changes the N-dependence

of the partition function and free energy, making the relation between small-Nand large-Nexpansions

nontrivial. A good news on this front is that the analiticity domains in gbecomes uniform in N when

recast in terms of the ’t Hooft coupling [21]. But there is still quite some work to do in order to connect

the transseries analysis at small Nwith that at large N.

Main results. Our main results are the following:

•In Proposition 1, we study Z(g, N). While most of the results in this proposition are known for

N= 1, we recover them using the intermediate ﬁeld representation (which provides a new point

of view) and generalize them to arbitrary N∈R. In particular we uncover an interplay between

Z(g, N) and Z(−g, 2−N) which captures the resurgent nature of the transseries expansion of the

partition function for general N.

•Proposition 2 deals with the function Zn(g), notably its Borel summability, transseries, and associated

diﬀerential equation. To our knowledge, Zn(g) has not been studied before and all of the results

presented here are new.

•Proposition 3 and 4 generalize previous results in the literature [22] on the analyticity and Borel

summability of W(g, 1) to W(g, N ) and furthermore derive parallel results for Wn(g).

•Proposition 5 contains the transseries expansion of Wn(g), which has not been previously considered

in the literature. We also give a closed formula for the transseries expansion of W(g, N).

•Lastly, in Proposition 6, we derive the tower of recursive diﬀerential equations obeyed by Wn(g).

This serves as an invitation for future studies of the transseries of Wn(g) from an ordinary diﬀerential

equations perspective.

2 Borel summable series and Borel summable functions

When dealing with asymptotic series, a crucial notion is that of Borel summability. Less known, there

exists a notion of Borel summability of functions, intimately related to the Borel summability of series. In

this section we present a brief review of these notions, which will play a central role in the rest of the paper,

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Thesmall-Nseriesinthezero-dimensionalO(N)model:constructiveexpansionsandtransseriesDarioBenedetti1,RazvanGurau1,2,HannesKeppler2,andDavideLettera21CPHT,CNRS,EcolePolytechnique,InstitutPolytechniquedeParis,RoutedeSaclay,91128PALAISEAU,France2HeidelbergUniversity,InstitutfurTheoretischePhysik,Philoso...

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The small- Nseries in the zero-dimensional ON model constructive expansions and transseries Dario Benedetti1 Razvan Gurau12 Hannes Keppler2 and Davide Lettera2

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