Renormalons in the non-linear O(N) sigma model

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M

ASTER

T

HESIS

Renormalons in the non-linear O(N)

sigma model

Author:

Alexander van Spaendonck

Student ID: 12456527 Supervisor: dr. Marcel Vonk Second Examiner: dr. Diego Hofman

A thesis of 60EC submitted in fulfillment of the requirements for the degree MSc Physics & Astronomy: Theoretical Physics

at the

Institute for Theoretical Physics Amsterdam

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UNIVERSITY OF AMSTERDAM

Abstract

Faculty of Sciences

Institute for Theoretical Physics Amsterdam

MSc Physics & Astronomy: Theoretical Physics

Renormalons in the non-linear O(N) sigma model

by Alexander van Spaendonck

The two-dimensional non-linear O(N) sigma model can, in the presence of an exter-nal field coupled to a conserved charge, be solved perturbatively using the thermo-dynamic Bethe Ansatz. The ansatz results in an integral equation that when solved leads to a full perturbative series for the ground state energy density. This expres-sion diverges, which is a consequence of the presence of renormalons. In this thesis we first review the basics of resurgence and renormalons after which we reproduce the results of Mariño and Reis who demonstrated that the asymptotic behaviour of the perturbative solution is dominated by renormalons. We then take the analysis a step further by extending the perturbative solution to a trans-series by adding non-perturbative terms that are related to the leading IR-renormalon.

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Acknowledgements

While writing this thesis I have received help from many people to whom I owe many thanks. First and foremost my supervisor Marcel Vonk who guided my through the whole research process and has always shown an honest interest in the progress I made. Moreover, I would like to thank prof. E.L.M.P. Laenen for the fruitful dis-cussions we had on the topic of renormalons and prof. J.S. Caux for explaining the Bethe ansatz to me. I am also very grateful to Tomás Reis who did not hesitate to write me back and answer any questions I had about his paper with prof. M. Mariño. Finally I want to thank all my friends and family for their support and interest in my thesis.

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Introduction

Quantum field theory is the fundamental framework developed in the 20th century that helps us understand a wide range of topics: from tiny subatomic particles to large scale cosmological models and from ordinary materials to black holes that lie far away in the universe. Quantum field theory is everywhere and thus studying it in full detail will unquestionably lead to an even deeper understanding of the physical world around us. Besides covering such a broad range of subjects, QFT has also led to the most precise theoretical predictions known today in physics, with the magnetic moment of the electron as the most notable example. Fascinatingly enough, this has all been achieved while relying heavily on perturbation theory.

In perturbation theory a parameter is added that interpolates between a solv-able unperturbed problem, and a more difficult perturbed problem. Expressing the solution as a power series in that parameter lets us solve the problem for each or-der of that parameter and the sum of all these solutions then is the full solution. Unfortunately, perturbative calculations are flawed: many mathematical functions that we encounter in physical theories do not admit a (convergent) power expansion and are therefore invisible to our perturbative methods. Pursuing this perturbative approach anyhow thus leads to incomplete solutions that also generally diverge. Fortunately, this latter issue can be resolved by various summation techniques that mathematicians developed over the past few centuries. These functions that do not admit a perturbative expansion however, also known as non-perturbative functions, remain a problem. This is a difficulty that one also encounters when studying large N gauge theories. The early pioneers of QCD that studied these models stumbled upon such divergent series and immediately recognized that the source of this di-vergence was different from any other that they had seen before. This particular asymptotic behaviour had different characteristics and thus led to what we now call a renormalon.

Around the same time, the French mathematician Jean Écalle developed his the-ory of ’resurgent’ functions. In it he constructed a framework describing the re-lation between asymptotic perturbative series and non-perturbative functions. Al-though he applied his theory of resurgence to systems of differential equations, a small group of physicists soon discovered his theory and realised to what extent it could be used in physics. In the decades thereafter resurgence was further devel-oped within the physics community and thoroughly applied in for example matrix models, topological string theory, Painlevé equations and many more. Today, there is growing interest in resurgence and the theory is applied to a more broad variety of topics within theoretical physics. One of those topics is the appearance of renor-malons in two-dimensional integrable models.

This thesis reviews the basics of resurgence theory and renormalons, and ap-plies these ideas to a particular integrable QFT model. In a previous study of [21] it was shown that energy density of the non-linear O(N) sigma model coupled to a Noether charge could be computed perturbatively up to arbitrary order. In the more recent study [14] it was then shown that the perturbative result, which diverges, is

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2 Contents controlled by renormalons. Our goal here will first of all be to reproduce the com-putation and large order analysis of the ground state energy density of the O(N) model. Then we want to take the analysis further and compute the first coefficients of the first non-perturbative sector. This calculation then allows us to extend the per-turbative result to a partial trans-series which contains additional non-perper-turbative terms.

The outline of this thesis is as follows: we first review the theory of resurgence in chapter1. Then we discuss what renormalons are and how we expect them to appear in an asymptotically free QFT model in chapter2. Next, in chapter3the two-dimensional non-linear O(N) sigma model is introduced. Following both deriva-tions [21,14] we reproduce their perturbative result for the ground state energy den-sity. Then in chapter 4 we analyse the large order behaviour of the perturbative result by both reproducing and extending the computations done in [14] leading to a trans-series. Finally, we summarise our results and discuss possible leads for the future in the conclusion.

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3

Chapter 1

Resurgence

1.1 Perturbation theory

In physics one deals often with so-called hard problems: equations that are not exactly solvable. One of the most common methods of circumventing this issue is to use perturbation theory. In perturbation theory we introduce a (coupling) parameter that, when set to zero, reduces the problem to a solvable problem (also called the unperturbed problem). One then assumes that the solution we are looking for can be written as a power series of that particular parameter and the problem is solved for each order of that parameter. In this way a single hard problem is transformed into infinitely many easy problems. Each of these yields a solution and the sum of them is in the end the solution to our original equation. A familiar (physics) example of this method would be the calculation of energy levels in quantum mechanics:

E(λ) =

∞

∑

n=0

Enλn=E0+E1λ+E2λ2+E3λ3+. . . (1.1)

In this context E(0) = E0corresponds to the solution of the unperturbed problem,

which is exactly solvable, while any nonzero value of λ is associated with the per-turbed problem. In this way, the difficult perper-turbed problem is decomposed into a series of equations whose solutions are the coefficients En. What rests is picking a

value of λ and then summing all the terms.

This method suffers from one obvious and glaring issue: The sum might not converge. In a substantial amount of physics problems these series actually diverge, especially in quantum field theory. Even if one decides to keep the expansion pa-rameter very small to suppress the higher order terms, they might still eventually diverge due to the factorial growth of the individual coefficients. In this case we say that the sum or series has a zero radius of convergence, since it only converges for

λ = 0. The reason for such divergent coefficients stems from a mathematical

ar-gument: in our attempt to compute the energy E(λ)of the perturbed problem, the

function might not be analytic in the point λ=0. Complex analysis tells us that if a function is non-analytic in a point, then it does not admit a (convergent) Taylor ex-pansion around that point. Our series - the right-hand side of (1.1) - therefore fails to be a Taylor expansion of E(λ)and is just what is called an asymptotic expansion. As

was famously argued in [2] by Freeman Dyson, this is the case for most observables in QED and similar arguments can be made for many other quantum problems.

A standard way to deal with this divergence is to simply truncate the series at some optimal point. To find this point consider some series f(z) =∑∞_n₌₀anznwhere

anare the coefficients. Let us assume that the large n behaviour of these coefficients

is an ∼ n!A−n, where A is some constant1. For sufficiently small values of z, the

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4 Chapter 1. Resurgence first few terms in the expansion will shrink until after some n= N the factorial part n! becomes dominant and the partial sum starts to grow again. To find this point we minimize|anzn|. One way of doing this is using Stirling’s approximation and

then finding the saddle point. Another way is to look for the value of n such that

|an+1zn+1|

|anzn| ' 1. Solving this yields(n+1)|z/A| = 1, which in the large n limit gives

us a truncation point N ' |A/z|. It shows us that a smaller value of z will let us compute more terms and therefore find a more accurate answer.

The aim of resurgence, however, is not to solve this problem and improve the convergence of these series. On the contrary, resurgence thrives on the existence of these divergent series in all of physics. What we will see is that in these perturbative solutions that diverge, a great amount of information about non-perturbative effects is hidden. If the series is asymptotic to particular class of function that we call a resur-gent functions - which in physics is generally the case - then the growth behaviour of the perturbative coefficients will contain all kinds information about instantons, solitons, D-branes and more. The theory of resurgence lets us extract this informa-tion and even improve our perturbative soluinforma-tion by adding non-perturbative terms to it. The beauty of all this is that all the information necessary to reconstruct the non-perturbative terms is contained in the perturbative coefficients. We can be com-pletely ignorant about the underlying physics (or mathematics) and just look for these non-perturbative effects in the series themselves.

1.2 Asymptotics and Borel summation

While discussing the divergent series that we often encounter in physics, it was just mentioned that resurgence does not attempt to sum these series. There exist, how-ever, procedures that exactly do that; they are called summation techniques. One of them will be very important in the context of resurgence and is named Borel summa-tion. After performing the Borel summation on a series f , one is left with what we call a Borel sum ˜f. This will be a function that is asymptotic to the original series. To understand resurgence it will be necessary to discuss Borel summation, but before doing that we first need to explain what this asymptotic relation really is.

1.2.1 Asymptotic expansion

The motivation for defining asymptotic relations is that we need to make a clear distinction between functions and infinite series. The former is an apparatus where for every number we insert it spits out a number. An infinite series is a sum of infinitely many terms (or functions), but it can never be a function itself. For every number we would like to insert, it is impossible to get a definite answer from the series, because it involves doing infinitely many calculations. An infinite series is therefore not really the same as a function. Nevertheless, we often write that particular Taylor series are equal to some analytic function. This is acceptable because the Taylor expansion is unique, but is remains a slight abuse of notation and can cause a lot of confusion when we also consider divergent series.

Fortunately, there is a relation that we actually can use to link infinite series and functions to one another. This is in fact the asymptotic relation that we have men-tioned before. If a function f(x)is asymptotic to some series we write

f(x) ∼g=

∞

∑

n=0

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1.2. Asymptotics and Borel summation 5 and say that g is the asymptotic expansion of f(x) around the point x = 0. Just like with a Taylor expansion one can also expand around any other (non-zero) point, but for convenience we will always adjust the coordinate x such that we expand around zero. Before giving the conditions that have to be met for such a relation there is one more subtility we need to adress. An asymptotic expansion of a function is definied around a point p (for which we conveniently choose coordinate x = 0) in a sectorial neighbourhood U of that point. Figure1.1 shows an example of such a sectorial neighbourhood. Without getting too much into mathematical details it can be stated that U is a subset ofC \{p}and is defined by two rays emanating from the point p. An example of such a set could be for instance U= {x=reiθ|r> 0 and

|θ| < π₃}. The necessity of this sectoral neighbourhood will be justified shortly, but

first, we are now ready to give a full definition of the asymptotic expansion.

FIGURE1.1: The sectorial neighbourhood U of the point p

We say that a function f(x)is asymptotic to the series g=_∑∞_n₌₀anxnin U around

the point p if for every sectorial subneighbourhood U0 ⊂ U and for every positive integer N, there exists a constant C>0 such that

f(x) − N

∑

n=0 anxn <C|xN+1|. (1.3)

It tells us that near the point p the asymptotic expansion is a good approximation of the function f . Every asymptotic series either diverges or converges. In the latter case this series is the familiar Taylor expansion and we will use an equality sign instead of the∼sign. One can easily check that the Taylor expansion truncated at N does in fact satisfy equation (1.3).

To get some more feeling for this definition one can consider as an example the function f(x) =e−1/x and its behaviour around the singular point x= 0. If we take the limit of this function as x goes to zero we get several different answers depending on the direction:

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6 Chapter 1. Resurgence lim x→0+ f(x) =0 lim x→0− f(x) =∞ lim x→0± f(ix) =undefined

This is already the first hint that the notion of a sectorial neighbourhood is necessary to define an asymptotic expansion. In the first limit we have a function that decays exponentialy fast to zero. This decay is more rapid than any polynomial decay and so the asymptotic expansion along the positive real axis that satisfies equation (1.3) is simply g=0. This expansion is actually also valid in the complex plane provided that Re(x) > 0. In the Re(x) ≤ 0 sector there is no asymptotic expansion at all: the second limit shows that the function blows up in the Re(x) < 0 region and in the third limit we see that along the imaginary axis the function endlessly oscillates.

This demonstrates that in general one always needs to work with these sectors. If, however, we manage to find an expansion that is valid in all directions, e.g. one single sector with opening angle 2π, then we have simply stumbled upon the Tay-lor expansion. This follows from Riemann’s theorem on removable singularities: If a function admits an asymptotic expansion on a punctured disk, then it must be bounded on that disk by (1.3). Riemann’s theorem then states that this function is holomorphically extendable over the point p. Then finally, when a function is holo-morphic at p, it’s asymptotic expansion must converge which therefore makes it a Taylor expansion.

A reasonable question one might pose is whether these asymptotic expansions are unique or not. To answer this: every continuous function has a unique asymp-totic expansion, but not every expansion is related to one particular function. To demonstrate the latter, we can take an arbitrary function f(x) ∼_∑∞_n₌₀anxnand add

our function e−1/xto it. Because its expansion is zero, we also get f(x) +e−1/x ∼

∑

∞

n=0

anxn. (1.4)

Actually any asymptotic series multiplied with this e−1/x can be added without changing the asymptotic behaviour in the Re(x) > 0 sector. We will call these non-perturbative terms, because they are "invisible" to the perturbative (power) se-ries. Later on we will be interested in adding non-perturbative terms to perturbative series which leads to what we call a trans-series.

In general, we will be confronted with a perturbative series that diverges. The actual function we seek will have this series as an asymptotic expansion. In order to find this function we will try to sum the perturbative series, but in doing so we will discover certain ambiguities. From these ambiguities we can, however, extract information that lets us extend our perturbative series to a trans-series. When we then sum this new expression the ambiguities disappear and we find a proper solu-tion. To perform the first step in this proces, we need to explain what a summation method is.

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1.3. Euler’s equation 7

1.2.2 Borel summation

A summation method is a procedure that attempts to assign a value to an infinite series. Because we are dealing with an infinite number of terms, simply adding them all together is not an option. In the case of a series that converges, these methods are not necessary and one can try to find the value to which the partial sums converge. However, in the case of an infinite series that diverges, this is no longer an option. One then has to resort to summation methods, and there is variety of techniques to pick from. One very common summation technique is the Borel summation. It is based on the following identity which holds for n≥0:

xn+1= 1

n!

Z ∞

0 t

n_e−t/x_dt _(1.5)

In our power series we would like to replace every xn+1_{with the right-hand side of}

(1.5). We do this in two steps, that we know by the names of Borel transformation (B) and Laplace transformation (L). Given a power series f(x) =∑_n∞₌₀anxn+1these

are B[f](t) = ∞

∑

n=0 antn n! L[f](x) = Z ∞ 0 f (t)e−t/xdt (1.6) The Borel summation then is simply applying these two transformations after one another:S [f](x) = L[B[f]](x).

With equation (1.5) in mind this might seem like a pointless exercise, as it sug-gests that performing the Borel summation on a series f spits out the same series again, but this is incorrect. Actually it produces a function - denoted by ˜f in Figure

1.2- whose asymptotic expansion is the original series f . If the series is convergent, then these two will be ’equivalent’, but in the case of a divergent series we obtain a new function. This comes from the fact that we interchange the summation and integration leading to a new expression.

FIGURE1.2: The Borel summation.

1.3 Euler’s equation

To see all of this in practice, we will discuss an example based on [1]: Euler’s equa-tion

x2d f

dx = x− f . (1.7)

This is a first order ordinary differential equation (ODE). Using a power series ansatz the following solution is obtained

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8 Chapter 1. Resurgence f(p)= ∞

∑

k=0 (−1)kk!xk+1, (1.8)

where the superscript(p)denotes that this is the power series solution. There are two major problems with it:

1. A first order ODE should have one parameter that is set by initial conditions of the problem. However, this parameter is clearly missing here and so this must be one of many possible solutions. If we solve the homogeneous differential equation x2 dg_dx = x−g, then we get a solution g(x) = ae1/x. We conclude that our power series somehow missed this term.

2. The series is divergent for any nonzero value of x. Regardless of how small we pick x, eventually the terms will diverge and so this solution seems unusable. The goal now is to resolve these two issues using Borel summation, and along the way we will get a better idea about what resurgence really is. We first apply the Borel transformation to our series to obtain

B[f(p)](t) = ∞

∑

k=0 (−t)k = 1 1+t. (1.9)

In the last step we analytically continue the function to also include values of t>1. The next step is applying the Laplace transformation which yields

f(x) = S [f(p)](x) = L[B[f(p)]](x) =

Z ∞

0

e−t/x

1+tdt. (1.10)

One can check that this expression indeed satisfies the Euler equation. Wheras f(p) had a zero radius of convergence, this result is well defined for all x > 0. Subse-quently, we would like to also include negative values of x. To achieve this it will be necessary to extend x to the complex plane. Moreover, we change our integra-tion contour: Let γx be the contour that goes from zero to infinity in a straight line

through the point x. By making this slight adjustment we keep the integral and thus also the function real. Figure1.3shows what the contour looks like for an arbitrary x in the complex plane. The solution itself then looks like

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1.3. Euler’s equation 9

FIGURE1.4: The two possible contours when integrating along the negative real axis

f(x) =

Z

γx

e−t/x

1+tdt, (1.11)

which is well defined for all x, except those lying on the negative real axis. For x ∈ _R<0 the integration contour stumbles upon a pole. For these values one could

try to adjust the contour slightly by taking a semi-circle above or below the pole, as shown in Figure1.4, but this gives us an ambiguity. Which contour should we pick? Before answering this question let us first examine the difference between both contours Z γ+x−γ−x e−t/x 1+tdt= Z γpole e−t/x 1+tdt=2πi·e 1/x_, _(1.12)

where the contour γpoleis just a closed contour around the pole at−1 in the

counter-clockwise direction. The result is an exponential term as seen in the solution of the homogeneous equation. What this ambiguity reveals is that our solution is part of a larger class of solutions. These are all solutions of the form

fa(x) = Z γ±x e−t/x 1+tdt+ae 1/x _(1.13)

where a is a free parameter set by initial conditions. This is now well defined as switching from integration prescription is equivalent to changing a→a±2πi. This result solves all previous issues we had with the power series solution: It is well defined for all values of x and has a free parameter to be set by initial conditions.

Previously we have stated that a function has a unique asymptotic expansion in a particular direction, but every expansion can relate to multiple functions, and that is exactly what this example confirms. All functions of the form (1.13) have the same asymptotic expansion, namely (1.8). However, the series is related to multiple func-tions, all of which might be the correct solution depending on boundary conditions. To recap: We started with a perturbative solution that was divergent and incom-plete. Borel summation gave us the actual function that solves the Euler equation, and analysing the integrand in the Borel plane led us to the exponential or non-perturbative term that was missing. One could say that the exponential term resurged, which is exactly where the name resurgence stems from.

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10 Chapter 1. Resurgence

1.4 Trans-series

In general we would like to extend our perturbative series by adding non-perturbative terms that we find with resurgence. This will lead to a new expression fp(x) +fnp(x)

that we will call a trans-series. As our example demonstrated, this extended series gives a well-defined and more accurate solution to the problem. Moreover, in many physics problems the Borel singularity might lie on the positive real axis, which ren-ders a Borel summation impossible unless deforms the contour and deals with this ambiguity. The expansion into a trans-series allows for summation by removing this ambiguity that arises while integrating around the pole.

To see how this all works, let us first state the definition of the most common type of one-parameter trans-series:

N

∑

n=0 σn ∞

∑

k=0 a(_kn)xke−nA/x_. _(1.14)

The σ is a free parameter that might be fixed by boundary conditions and the coef-ficients nA are determined by poles of the Borel transform. In our example we had only one pole (so N= 1) at−1, but in general there might be an infinite number of poles. A good example would be the case of instantons in a double well potential. The model allows for so-called (anti)kink solutions with an action A and multikink solutions with action nA. If we construct a perturbative solution of the ground state energy around one of the minima of the potential, we can analyse the series and we will find multiple poles in the Borel plane that correspond to the actions of these multikink solutions. This lets us construct a trans-series that will look2like this:

∞

∑

n=0 σn ∞

∑

k=0 a(_kn)xkenA/x ₌ a₀(0) + a₁(0)x + a(₂0)x2 + a(₃0)x3 + a₄(0)x4 + ... +σe−A/x a(₀1) + a₁(1)x + a₂(1)x2 + a(₃1)x3 + a₄(1)x4 + ... ) +σ2e−2A/x a(₀2) + a₁(2)x + a₂(2)x2 + a(₃2)x3 + a₄(2)x4 + ... ) ... ... ... ... ... ... ...

The first row is the perturbative series that we start with. The second row is called the 1-instanton sector and comes from a single kink solution. The third row is the 2-instanton sector and comes from the double kink solution, and so on. Thus, the particle’s ability to tunnel through the potential barrier leads to corrections of the ground state energy that one can find using conventional methods, but also by resur-gent analysis. As can be seen here, the higher instanton sector can carry their own perturbative series. When contructing a trans-series it is therefore also necessary to find the higher instanton coefficients. How this works will be discussed in the next section.

The theory of resurgence tells us that there is a deep connection between the perturbative series and the higher instanton sectors. We already saw that the per-turbative series allows us to compute the constants nA from examining the Borel transform of the series. Resurgence however goes even further teaches us that one can obtain the coefficients a(_kn) with n ≥ 1 from the original perturbative series. In 2_{Actually, the correct trans-series will also involve log}₍_A/λ₎_{factors, but we omit them here so we} can stick to the most basic, uncomplicated form of a trans-series.

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1.5. Asymptotic behaviour of perturbative series 11 multiple known problems, exact relations between these coefficients have been es-tablished3 and they suggests that the perturbative series actually contains an enor-mous amount of information that is hidden in plain sight. In fact, knowing all per-turbative coefficients allows one to compute all higher order instanton sectors, as we will discuss in the next section. This is an amazing fact, because it means that non-perturbative effects can actually be found via perturbation theory. Furthermore, this relation is a two-way street; from the higher instanton sectors one can fully repro-duce the perturbative series. The study of these intricate relations is the main subject of resurgence theory and requires knowledge of more advanced tools like alien cal-culus and Stokes automorphisms. An introduction to these concepts is beyond the scope of this thesis and so we refer the interested reader to [4] for more information on these topics.

In this study, we will be concerned with applying resurgence within the realm of physics. The goal will be to study the large order behaviour of the perturbative solu-tion to some physical problem and constructing a trans-series. We will also see that in general we do not necessarily have a closed form of the perturbative coefficients, like in the Euler equation. This means that we will only have a limited number of coefficients and so we can only determine A and a(_kn)with some uncertainties. How this is all done is the subject of the next section.

1.5 Asymptotic behaviour of perturbative series

In order to explain how one computes these new coefficients we assume - for sim-plicity - a series that has only one non-perturbative sector4. Looking at the behaviour of the perturbative coefficients as we go to higher orders, an interesting pattern emerges: the coefficients seem to grow approximately like

an∼n!A−n, (1.15)

for large values of n. This behaviour can be compared to that of the series f(z) =

∑∞n=0n!A−nzn+1, which has a Borel transform B[f](s) = ∑∞n(s/A)n = 1−1s/A. It

has a pole at s = A and thus we expect A to also be the instanton action of our original sequence an. To find this value A one can make use of (1.15) and argue that

A≈ an(n+1)

an+1 in the large n limit. For an accurate result one can calculate this quantity

for subsequent coefficients anwhich leads to a sequence of values that converges to

A.

The second objective is to compute the non-perturbative sector coefficients a(_k1). For that, a more detailed description of the growth is required. What follows is one of the most important relations from resurgence theory which demonstrates that the non-perturbative sector coefficients are hidden in the large order behaviour of the perturbative coefficients. This relation looks like this [13]:

a(n0)∼ A−n−bΓ(n+b)

∑

k≥0 a(_k1)Ak ∏k i=1(n+b−i) (1.16)

where a(_k1) are the coefficients of the non-perturbative sector and b is a constant. Working in inverse powers of n this relation can be rewritten to

3_{See for example section 5.2 of [}₃_{] in the context of Painlevé I.}

4_{Henceforth we will speak of the more general non-perturbative sector instead of instanton sector, as} these sectors can be related to other effects than instantons.

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12 Chapter 1. Resurgence a(n0) ∼ A−n−bΓ(n+b) a(₀1)+a (1) 1 A n + a₁(1)(1−b)A+a(₂1)A2 n2 +O[n −3_] (1.17) With this result, the non-perturbative coefficients can be calculated recursively up to any order we like, provided we have enough perturbative coefficients at our dis-posal. This can be seen from the fact that there will always be subleading correc-tions that scale with inverse powers of n. To suppress their contribucorrec-tions we need to take n as large as possible, hence we are limited by our perturbative calculation. Fortunately, we can apply a convergence accelerating method like the Richardson transform - which is explained in appendixA.4- to aid us. These non-perturbative coefficients that we eventually find will be directly related to the non-perturbative sector

σe−A/x(x)−b a(₀1)+a₁(1)x+a(₂1)x2+... (1.18)

Thus to recover this sector we will want to compute A, b and as many coefficients a(_k1). In the next chapter we will be interested in what these coefficients look like in the context of renormalons. As mentioned before, we saw that A is the action of the instanton field configuration in the case of instanton related divergences. Do such physical interpretations of the coefficients also exist for renormalon divergences?

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13

Chapter 2

Renormalons

There are many possible sources of divergence in perturbative solutions of physics problems. In the context of quantum field theory this most commonly is the instan-ton. These are non-trivial saddlepoints of the action, or equivalently solutions to the equations of motion with a finite action. In the case of instanton related diver-gences we observe a perturbative solution expressed in some coupling parameter g where the number of Feynman diagrams grows factorially with the power of g. Another characteristic property of the instanton induced divergence is that the non-perturbative terms that we need to add - in order to cancel the ambiguity from the Borel singularity - carry an exponentially small factor e−A/gwhere A is the action of the instanton solution.

There seems to be, however, another source of divergence for perturbative so-lutions in quantum field theory. These are related to special types of diagrams that appear at each power of the coupling and that, after integrating of loop momenta, grow factorially themselves. These diagrams are called renormalons and they were discovered first in renormalizable field theories in [5,7] by considering bubble dia-grams in Yang-Mills theories. According to M. Beneke in [8] "these diagrams still feature so prominent in discussions of renormalons that sometimes they are identi-fied with them".

The name renormalon was first suggested by ’t Hooft in [5]: At the time instan-tons were the only known source of divergences and as this phenomenon seemed to be characteristic for renormalizable QFT models, the name renormalon was adopted. To what part of this divergent behaviour we can identify the term ’renormalon’ re-mains imprecise. As mentioned, some identify the corresponding diagrams with renormalons, but as the divergence leads to one (or more) poles in the Borel plane, one can also identify these Borel singularities with renormalons. In this study the latter definition will be adopted, as will become clear shortly. There exists yet an-other definition, common among those that study applications of QCD, where renor-malons are identified with power corrections that these divergences result in. A brief explanation of these corrections will be presented later on, but we will not adopt this renormalon definition.

This chapter will be concerned with explaining what renormalons are and how we expect them to appear in an asymptotically free QFT model, like the non-linear O(N) sigma model that we discuss in the next chapter. The first section will re-view a classic example of renormalons in QCD in which one can explicitly relate the renormalon to a particular type of Feynman diagram. It will also illustrate in a straightforward way the distinction between UV-renormalons and IR-renormalons. In the second section we will get a bit more technical and use the concept of operator product expansion and renormalization group analysis to derive non-perturbative terms that are related to IR-renormalons. These terms will, by resurgence arguments, give us predictions on the asymptotic behaviour of the perturbative coefficients of

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14 Chapter 2. Renormalons our model. Lastly, in the third section we will go beyond leading order and derive the subleading terms in the non-perturbative sector that we can relate to 1-instanton sector coefficients that we know from resurgence.

2.1 Bubble diagrams

To illustrate what renormalons are and how they result in diverging perturbative coefficients, we will review the textbook example [8] of a renormalon (also discussed in [6]). This will be in the context of a gauge theory coupled to some Dirac fermions, like QED or QCD. In the latter case we could consider a theory with massless quarks and the following Lagrangian:

L =ψ(iγµDµ)ψ− 1 4G A µνG µν,A_, _(2.1)

with ψ and ψ quark fields and where G_µνA is the gluon field strength tensor. Let us now consider the time-ordered two-point function of two vector currents of massless quarks jµ=ψγµψin its momentum representation:

i

Z

d4xhTjµ(x)jν(0)ieiqx = (qµqν−q2gµν)Π(Q2), (2.2)

where Q2= −q2. Next, we will be interested in the Adler function that is defined as D(Q2) =4π2dΠ(Q

2₎

dQ2 . (2.3)

Thus, it is the derivative of the vacuum polarization function with respect to the external momentum Q2. The Adler function can be computed peturbatively and has many contributions that can be written in Feynman diagrams. One particular type of Feynman diagrams that contributes to this function is a fermionic bubble diagram. It is a sequence of fermion loops connected by gluon lines in one long chain, as shown in Figure2.1. A single fermion loop has, after renormalization, the

FIGURE2.1: Two contributions to Adler function that we will con-sider here (a). At each order of n we can insert n fermion loops into

the gluon line (b).

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2.1. Bubble diagrams 15

−β0 fαs ln(−k2/µ2) +C, (2.4)

where αsis the coupling between the quarks and gluons and the constant C depends

on the renormalization scheme. It has in the case of modified minimal substraction (MS) a value of−5

3. Also there is the variable µ, which denotes the the

renormaliza-tion scale. The beta funcrenormaliza-tion

β(αs) =µ2∂αs

∂µ2 =β0α

2

s+ O(α3s), (2.5)

has a leading order coefficient β0 =βN A+β_{0 f} of which the latter is the contribution

to the beta function from the fermions.

If we account for these fermion loops and compute their contribution to the Adler function for all orders n, corresponding to n insertions of a fermion loop, then we get the following expression:

D= ∞

∑

n=0 αs Z ∞ 0 dˆk2 ˆk2 F(ˆk 2₎ β_{0 f}αslnh ˆk2 Q2_e−5/3 µ2 in . (2.6)

Here we have introduced the dimensionless variable ˆk2 = −k2/Q2where k2 is the gluon momentum going through the chain. Moreover, there is the function F(ˆk2)

whose exact expression can be found in [10]1_{. If we now keep the}

renormaliza-tion scale µ fixed at all orders of n and perform the integrarenormaliza-tion over momenta, one can see that the dominant contributions to the integrand come from the small and large momenta regions, especially when n becomes very large. This can be inferred - heuristically - from the log-term; when n is non-zero, then the integrand is sup-pressed near ˆk2 = µ2

Q2e5/3 by a power of degree n2. Additionally, when ˆk2

µ2

Q2e5/3

and ˆk2 µ2

Q2e5/3 the contributions are enlarged by the logarithm. Hence it is

appro-priate to cut the integral in two and derive a small and large momenta expression for F(k2₎_[₈_]:

F(ˆk2) = 3CF

2π ˆk4+ O(ˆk6) for small momenta

F(ˆk2) = CF

3πˆk12 log ˆk

2₊5 6

+ O(ˆk−4) for large momenta.

(2.7)

These expressions inserted in equation (2.6) also suggest that the integrand decays powerlike to zero as ˆk2 goes to 0 or infinity. Thus, splitting the low and high mo-menta regions of the integral at the point ˆk2 = µ2

Q2e5/3 with equations (2.7) inserted

into (2.6) yields D(αs) ' CF π ∞

∑

n=0 αn_s+1 3 4 Q2 µ2e −5/3 −2 − β0 f 2 n n!+ Q 2 3µ2e −5/3 βn_{0 f}n!(n+ 11 6 ) . (2.8) It must be noted that despite the fact that the sum over all n of these terms is only approximately equal to D, the individual terms for n 1 approximate their coun-terparts in (2.6) arbitrarely well. Therefore, this new expression does fully capture the large order behaviour of (2.6). Furthermore, it is clearly factorially divergent:

1_F₍_ˆk2₎_{is called w}

D(τ)in [10]

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16 Chapter 2. Renormalons the fermion bubble contribution grows factorially with the number of fermion loops inserted. Contrary to instanton divergences where we spot a factorial growth of Feynman diagrams, here with we have a single type of diagram that appears at each order and grows factorially with that order.

As was done in the previous chapter, a Borel transformation can be applied to learn more about the divergent behaviour of the series. In this setting this leads to

B[D](t) = 3CF 4π Q2 µ2e −5/3 −2 1 1+ β0 f 2 t + CF 3π Q2 µ2e −5/3 1 (1−β_{0 f}t)2 + 5/6 1−β_{0 f}t . (2.9) The first term now explicitly shows a Borel singularity at t= −2/β_{0 f} coming from the integration over the IR and the second term a singularity at t= +1/β0 f from the

UV. Hence, we call these singularities IR- and UV-renormalons respectively. The value of the leading beta function coefficient clearly dictates the position of the renormalons: In the case that this leading coefficient is positive (as it is in QED) we have UV-renormalons on the positive real axis and IR-renormalons on the neg-ative real axis. In an asymptotically free theory, like QCD where we replace β_{0 f} with β0 = β0N A+β0 f < 0, we have the opposite. Thus, interestingly enough, we

see that renormalons are very closely related to the running of the coupling. In a UV-divergent theory the UV-renormalon sits on the positive axis preventing us from Borel summing the series along that axis and telling us that non-perturbative correc-tions need to be added. This does not come as a surprise as the coupling becomes strong in the UV which impairs perturbation theory. Moreover, in the context of in-stantons we mentioned that the Borel singularity is located at A which is the action of the instanton solution. In this renormalon example however, we see a direct link between this location of the pole and the beta function of the theory. This is not in-cidental; we will demonstrate in the next section that the leading coefficient of the beta function will always show up in A and that therefore we can predict its location a priori.

Another interesting consequence of these renormalon singularities is that they imply the existence of ’power corrections’. To see this recall that in the previous chapter we learned that the ambiguity that arises from integrating either above or below the Borel singularity needs to be canceled by adding non-perturbative terms that scale with e−A/α. The leading order solution to the running of the coupling (2.5) is

α(µ) = 1

−β0log(k2/Λ2). (2.10)

So for the IR-renormalon3at t = −2/β0we add terms that scale with

∼e2/(β0α(Q))_∼

Λ Q

4

. (2.11)

In general, a renormalon at t = −n/2β_{0 f} corresponds to a correction that scales with(Λ/Q)n. These power corrections can be quite interesting for those studying phenomenological applications of renormalons, but that subject is beyond the scope of this study and so we would like to refer to section 5 of [8] for more on the topic.

3_{Here we have replaced β}

0 fby β0(QCD). This can be justified diagrammatically as stated in section 3 of [8]

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2.2. Operator product expansion 17

2.2 Operator product expansion

In this section we will demonstrate how non-perturbative contributions emerge from considering the operator product expansion (OPE) of an arbitrary observable K. These non-perturbative terms will be related to IR-renormalons and the reasoning behind this derivation comes from RG-arguments made in [8,11,12,14]. We will consider an asymptotically free QFT model that contains some coupling parameter g2 with running of the coupling4

β(g) =µdg

dµ = −β0g

3₋

β1g5+ O(g7). (2.12)

where β0and β1are positive. In our model we might be interested in the

computa-tion of a generic observable K(g). This observable can be expressed in the form of a trans-series which consists of a perturbative and non-perturbative part:

K(g) =Kp(g) +Knp(g). (2.13)

The perturbative part will be a powers series Kp(g) =∑nang2nand any contribution

to the observable K that can not be written in this perturbative form will be contained in Knp(g). Under certain circumstances, it might however be possible to express the

observable in an OPE. An OPE is a tool to decompose an operator into a sum of local operators weighted by coefficient functions. A general OPE looks like

K(g) =

∑

i

1

qdiCi(q/µ, g(µ))hOii, (2.14)

where Ci(g)are scalar functions that can be computed perturbatively. Furthermore,

we have mass dimensions diof operatorsOiand some external scale q. If a particular

operator contribution is completely non-perturbative, then we expect it to be located in the Knp(g)part of the trans-series. Thus, the OPE can determine the form of

non-perturbative sectors of the trans-series.

Let us consider the contribution of a single operator of dimension d which is non-perturbative. Then we get the following contribution to Knp:

Knp(g) =

1

qdC(q/µ, g)hOi. (2.15)

The coefficient function C(q/µ, g)should obey the homogeneous Callan Symanzik equation [12] : µ ∂ ∂µ+β(g) ∂ ∂g −γ(g)C(q/µ, g(µ)) =0. (2.16)

This equation has the solution C(q/µ, g) =exp − Z g(q) g0 γ(g) β(g)dg C(1, g(q)), (2.17) 4_{The reason for this convention is that in the next chapter the non-linear O(N) sigma model will be} studied whose beta function has exactly this form.

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18 Chapter 2. Renormalons where g0 =g(µ)is a reference coupling and γ(g)is the anomalous dimension of the

operatorO. Similarly, we get for the external scale 1/qd: qd=µdexp −d Z g(q) g0 dg β(g) =µd g 2₍_q₎ g2 0 −dβ1/(2β20)_exp − d 2β0g2(q) + d 2β0g02 exp −d Z g(q) g0 1 β(g) + 1 β0g3 − β1 β2₀gdg ≡Λd g2(q)−dβ1/(2β20)_exp − d 2β0g2(q) exp −d Z g(q) 0 1 β(g)+ 1 β0g3 − β1 β2₀gdg , (2.18)

whereΛ is the dynamically generated scale

Λ=µ g2₀β1/(2β 2 0)_exp 1 2β0g20 exp Z g₀ 0 1 β(g)+ 1 β0g3 − β1 β2₀gdg . (2.19)

In equation (2.18), the final integral in the exponent is the beta function with its lead-ing and subleadlead-ing terms subtracted and one can see that this term is completely per-turbative in g2. A similar rewritting can be done in equation (2.17) when we know the anomalous dimension of the operatorO. In our model, that we will discuss in the next chapter, we will be concerned with operators that have γ(g) = γ0g2+ O(g3).

If we now put all the pieces together we get the following expression:

Knp(g) =C(1, g(q)) g2(q)−dβ1/ (2β2 0)+γ0/(2β0) exp − d 2β0g2(q) 1+ O(g2(q)), (2.20) where we have absorbed all constants in C(1, g(q)). Finally, we can assume that the coefficient function will have the following form [14]:

C(1, g(q)) =c(g2(q))n0₍₁_{+ O(}_g2₍_q₎₎₎_, _(2.21)

where c is a constant and n0a non-negative integer. This leads to our final expression

for the non-perturbative contribution Knp(g) =c g2 −b e−A/g2(1+ O(g2)), (2.22) Where A= d 2β0 and b= dβ1 2β2 0 − γ0 2β0 −n0. (2.23)

These are the coefficients that determine5the leading order asymptotic behaviour of the perturbative series Kp = ∑ ang2n. From resurgence we know in fact that these

coefficients will have the following large order behaviour [4,13]:

an∼ A−n−bΓ(n+b). (2.24)

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2.2. Operator product expansion 19 Thus, when analysing the asymptotic behaviour of perturbative coefficients it is pos-sible to check whether the dominant behaviour comes from renormalons, in which case we expect the relations (2.23) to hold. Now that we have seen how the OPE is related to the divergences of perturbative series, we would like to make several observations.

First of all, this whole line of reasoning started at the operator product expan-sion of the observable that we wanted to compute. The presence of a single operator in that OPE led to a nonperturbative sector described by coefficients A and b -that will affect the asymptotic behaviour of the perturbative series of our observ-able. Hence in the large order behaviour of our perturabtive series we can check the presence and absence of operators in the OPE. For example, the absence of a Borel singularity at some particular point on the real axis indicates the absence of operators of a certain dimension in the OPE.

To see this in practice, let us revisit our example from the previous section where we derived the Borel transform of the Adler function related to the current-current correlation function. There we saw that equation (2.7) determines the locations of the Borel singularities in (2.9). The fact that the leading (UV) singularity - which is the singularity closest to the origin - is located at t=1/β0 f and not at t=1/(2β0 f)

tells us that the OPE of the Adler function contains no dimension two operator. This implication is correct; the lowest dimensional (non-trivial) operator contributing to the Adler functions OPE is the gluon condensateh0|G_µνAGA,µν|0iwhich has a mass dimension of four6. The whole OPE of the Adler function is [8]:

D(Q) =C0(Q2/µ2) + 1

Q4C1(Q

2_/µ2_)h_GG_{i +} 1

Q4C2(Q

2_/µ2_)h_qq_{i + O(}_Q−6₎_{. (2.25)}

In the previous section we also learned that renormalons lead to power corrections. Now we can see that a d-dimensional operator is related to a Borel singularity at A= d/(2β0)and power corrections that scale as(Λ/Q)d.

In general, adding all the contributions to the large order behaviour of perturba-tive coefficients will lead to the following asymptotic relation:

an∼

∑

i

CiA−i n−biΓ(n+bi) 1+ O(

1

n), (2.26)

where every value of i is associated to a single operator contribution and Ci are

constants. From equations (2.23) and (2.24) we can also deduce that the most dom-inant asymptotic behaviour will come from the operators of the lowest dimension di, which correspond to the Borel singularity closest to the origin. In the case of

multiple dimension dioperators, we then can see that the operator corresponding to

the largest value of b will dominate among them. Thus, for the Adler function from our example we expect two leading renormalon contributions from the GG and qq condensates with the same Borel singularity location A in (2.26).

Besides IR-renormalons, there are also the UV-renormalons that affect the asymp-totic behaviour of our perturbative solutions. Their singularities, that are closely related to the familiar UV-divergences known from renormalization, are related to operators of dimensions larger than the spacetime [8,11]. In general, these D+d dimensional operators7lead to Borel singularities with

6_{This can be inferred from the lagrangian if our spacetime has four dimensions, which is the case} for QCD.

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20 Chapter 2. Renormalons A= − d 2β0 and b= −dβ1 2β2₀ + γ0 2β0 −n0, (2.27)

where again n0is a non-negative integer.

2.3 Beyond leading order

The next step is to go beyond leading order of the IR-renormalons. As was discussed in the previous chapter, resurgence allows one to fully compute the non-perturbative sector

e−A/g2 a₀(1)+a(₁1)g2+a(₂1)g4+ O(g6)

(2.28) provided one possesses enough perturbative coefficients. These coefficients a(n1)are

commonly called 1-instanton sector coefficients, as they are associated with instan-tons, but in the context of renormalons we will name them non-perturbative coeffi-cients.

In our OPE derivation we also found corrections to the leading order coming from coefficient functions C(1, g2₍_q₎₎_{and from the RG flow. The latter can be seen}

when we inspect the exponents exp −d Z g(q) 0 1 β(g)+ 1 β0g3 − β1 β2₀gdg (2.29) and exp − Z g(q) g0 γ(g) β(g)− γ0 β0g dg . (2.30)

These ’residual’ quantities yield perturbative corrections in powers of g2(q)that we ignored so far. When we include higher orders of the beta and gamma function we can compute the first order correction: from (2.29) we get

− dβ2 2β0 g2+dβ 2 1 2β3₀g 2_{+ O(}_g4₎_, _(2.31) and from (2.30) − γ0β1 2β2 0 g2+ γ1 2β0 g2+ O(g4), (2.32)

where our convention for the anomalous dimension is γ(g) =γ0g2+γ1g4+ O(g6).

Lastly, we have the coefficient function C(1, g2) = c0+c1g2+ O(g4)that we need

to account for. Adding this all together we get the following expression for the non-perturbative sector Knp(g) =c g2 −b e−A/g2 1+s1g2+ O(g4) (2.33) where s1= c1 c0 −γ0β1 2β2 0 + γ1 2β0 −dβ2 2β0 +dβ 2 1 2β3 0 (2.34) This result should match the first non-perturbative coefficient a(₁1) in (2.28)8 which

8_{Provided that the zeroth non-perturbative coefficient a}(1)

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2.3. Beyond leading order 21 can be obtained via resurgence. In a similar fashion one can obtain more non-perturbative coefficients; for the coefficient skone needs the first k+2 coefficients of

the beta function and the first k+1 coefficients of the anomalous dimension. More-over, one needs to know the coefficients ci appearing in the coefficient function of

the OPE.

In deriving a perturbative series of some observable we might want to switch to a different coupling parameter α. How do we then relate the asymptotic behaviour of (2.24) for a perturbative series in g2to the large order behaviour of coefficients bn

in the series∑_nbnαn? We expect

bn ∼ (A)−n−bΓ(n+b). (2.35)

For a new coupling α related to g2by

g2=d1α+d2α2+d3α3+ O[α4] (2.36)

we can substitute this relation into (2.33) and expand the whole expression in α to obtain

Knp(α) =c α)−be−A/α 1+s1α+ O(α2) (2.37)

with new coefficients

A= A d1 b= b s1= d1s1− A d1 (d 2 2 d2₁ − d3 d1 ) − d2 d1 b (2.38)

One can easily check this by inserting relation (2.36) into the non-perturbative ex-pression (2.33) and expanding in small α. We see that the locations of Borel sin-gularity is rescaled and that the b-coefficient remains unchanged. The sublead-ing powercorrections require more complicated adjustments. The transformation (2.36) provides us with enough tools to compute the subleading behaviour in non-perturbative sectors and compare them with results from Borel analysis.

In conclusion we can state that the renormalon divergence is closely related to the RG properties of the QFT model and that it shows a distinct behaviour different from instantons. With this derivation in mind we have now two possible paths towards the nonperturbative extension of our perturbative solution:

1. By analyzing the asymptotic behaviour of the perturbative coefficients

2. By computing the beta function, studying the operator product expansion and computing the corresponding anomalous dimensions of the contributing (rel-evant) operators

The latter method might seem tedious, and it probably will be in more complicated models, but it offers us a tool to confirm the existence of renormalons. If these two methods truly lead to the exact same non-perturbative sector, then this confirms that the complete solution belongs to the class of resurgent functions. Moreover, this then poses a highly interesting application of resurgence. Our derivation sug-gests that the non-perturbative extension is related to various operator condensates (through the OPE) and their anomalous dimension. Thus, resurgence can recover

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22 Chapter 2. Renormalons all those properties of the model via these renormalon divergences. Additionally, as renormalons are also relevant from a phenomenological perspective - remember the power corrections -, this allow one to possibly connect resurgence more closely to the experimental side of physics.

For this study however, we will mostly be interested in testing the existence of renormalons in an asymptotically free QFT model. Armed with relations (2.23) and (2.34), we are now ready to apply our knowledge from resurgence and renormalons to the non-linear O(N) sigma model in the next chapter.

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23

Chapter 3

The non-linear O(N) sigma model

The non-linear O(N)sigma model is a well-known toy model that has been studied extensively in the past. The reason for this interest is because in two dimensions the model is both exactly solvable and shares many similarities with QCD; most notably it is asymptotically free and has a dynamically generated mass gap. Given its apparent simplicity it has proven itself to be a useful laboratory for studying different phenomena seen in QCD, like for instance renormalon divergences. These renormalons were studied in the large N limit in [9] (and are reviewed in [6,8]). The model can also be coupled to a conserved current which allows one to perturbatively compute the free energy of the model [17] and the mass gap [18]. Using the Bethe Ansatz it was then shown by Volin in [21] that it is also possible to compute the full perturbative series for the ground state energy density. This computation was repeated in [14] using a slightly different route. We will review both derivations, which for the largest part will be along the same line. While doing this, we will try to spot where non-perturbative terms might be neglected with the prospect of reproducing those terms in the next chapter.

In this study the two-dimensional model will be considered. The action of the theory is

S=

Z

d2x 1

2g2∂µS·∂µS. (3.1)

The S field here is a vector field consisting of N components S = {S1, S2, ..., SN}

such that they satisfy the constraint S·S= 1. The constraint leads to a global O(N) symmetry: If the field S is rotated within the n-sphere by the same amount for all spacetime coordinates, then the Lagrangian is clearly invariant. Thus, this symmetry can be associated to a Noether current, which is

Jµij =Si∂µSj−Sj∂µSi. (3.2)

If Qij _{denotes the corresponding charge, then in the Hamiltonian formalism, the}

system is described in following manner:

H−hQ12, (3.3)

where we have chosen to couple the system to the charge Q12 _{via een external field}

h. From a condensed matter perspective this parameter h can be seen as a chemical potential that lowers the energy of particles charged under Q12. When the parameter h is now turned on beyond some threshold, numerous particles charged under Q will emerge in the system. This will lead to a particle density ρ and an energy density e.

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24 Chapter 3. The non-linear O(N) sigma model

FIGURE3.1: One particle on a circle (left) and N particles on a cir-cle (right). Beneath them the periodic boundary condition that one

imposes on the wavefunctions.

These two quantities can be computed via an integral equation which stems from the thermodynamic Bethe Ansatz. This ansatz is a well-known technique from con-densed matter physics, used to solve models with one spatial dimension. In the next section we will give a brief background on this method (based on [15,16]), and de-rive the integral equation, albeit in a slightly handwaving manner. In the subsequent section we will solve the integral equation following both the derivations of [21] and [14]. Then finally, we briefly review the approximations made in this perturbative derivation and provide an argument for what the non-perturbative sectors will look like.

3.1 Bethe ansatz

In a multiparticle state the behaviour of a single particle is governed by many-body dynamics. With the Bethe ansatz we impose periodic boundary conditions on indi-vidual particles that comprises only two-body interactions. To get a feeling for this, consider a single bosonic particle placed on a circle S1with length L. We must have the following boundary condition for this setup

eip1L₌_1, _(3.4)

which implies that the momentum p1 must be 2πn_L where n ∈ Z. When a second

particle is added, then the boundary condition changes: the particle gets from go-ing once around the circle not only a phase shift from its motion, but also from the interaction with the second particle. This interaction is described by the S-matrix element S12. When placing N (interacting) particles on our circle S1, then this leads

to the following boundary condition for a particle labeled by j: eipjL₌

N

∏

j6=i

Sij. (3.5)

When taking the logarithm of this expression we get pjL=2πnj−i

N

∑

i6=j

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3.1. Bethe ansatz 25 where nj ∈ Z comes from the branch of the logarithm we pick. In fact, the values

of nj will be unique for each j, that is ni = nj if and only if i = j. This is not a

consequence of (3.6) but a requirement. The numbers nj can thus be regarded as

quantum numbers (this is more thoroughly explained in [15,16]). An easy way to parametrize all the different particles momenta is by their rapidity θ. This is a single variable that parametrizes a particle’s ’two-momentum’ through the relations

p=m sinh θ and E= m cosh θ. (3.7)

One can easily see that these relations yield the familiar energy-momentum relation E2−p2 = m2. The S-matrices are a function of the difference of rapidities: Sij =

S(θi−θj). Thus we can rewrite our expression to

m sinh(θj) = 2πnj L − i L N

∑

i6=j log(S(θi−θj)), (3.8)

where on both sides we divided by L. Next, we will be interested in the thermody-namic limit of the theory. This means taking the L → ∞ and N → ∞ limit, while

keeping N/L fixed. If we define xj = nj/L to parametrize the rapidities and

quan-tum numbers nj on a real line we notice that these numbers become dense in the

thermodynamic limit. In this way we can rewrite our expression to m sinh(θ(x)) =2πx−i

Z

log(S(θ0(y) −θ(x)))dy, (3.9)

where in the limit the summation becomes an integral over y. The next step is dif-ferentiating the expression once with respect to θ(x):

m cosh(θ(x)) =2π dx d(θ(x))−i Z _d d(θ(x))log(S(θ 0₍_y_{) −} θ(x))dy. (3.10)

Finally, by a change of variables under the integral we can integrate over rapidities

θ0(y)instead of y. This is done by introducing the density of Bethe roots χ(θ) =2πdx_dθ .

Then we arrive at:

m cosh(θ) =χ(θ) − Z ₁ 2πi d d(θ0−θ)log(S(θ 0₋ θ)) 2πdy dθ0 dθ0 ≡χ(θ) − Z K(θ0−θ)χ(θ0)dθ0. (3.11)

The function K(θ)is called the kernel and is symmetric in its argument θ. The

bound-aries of the integral still need to be defined. This is a variable B defined by [18]

χ(θ) =0 for |θ| ≥ B, (3.12)

as individual particles prefer to have low energies and low momenenta correspond-ing to a small rapidity1. Thus we get

m cosh θ=χ(θ) −

Z B

−Bdθ

0

K(θ−θ0)χ(θ0), (3.13)

1_{Another interpretation if B is as the fermi rapidity, analogue to the fermi momentum in condensed} matter models.

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26 Chapter 3. The non-linear O(N) sigma model with the kernel

K(θ) = 1

2πi d

dθlog S(θ). (3.14)

Equation (3.13) is the integral equation and it will be the starting point of the com-putations in the next section.

3.2 Solving the integral equation

The integral equation tells us that after turning on the external field h, numerous particles with rapidities between −B and B will emerge. The density of rapidities

χ(θ), also known as the density of Bethe roots, is the solution to this integral equation

and is zero outside the support[−B, B]. From there, one can compute the energy-and particle densities

e = m 2π Z B −Bdθχ (θ)cosh θ and ρ= 1 2π Z B −Bdθχ (θ). (3.15)

A large value of h corresponds to a large value of B, and so it is possible to express the energy density in terms of B. In order to do that, the following function will be very helpful: R(θ) = Z B −B χ(θ0) θ−θ0dθ 0_. _(3.16)

It is called the resolvent function of Bethe roots and it has a discontinuity on the interval θ∈ [−B, B]. Moreover, outside this interval it is an odd function. This defi-nition lets us write the density of Bethe roots then also as a function of the resolvent:

χ(θ) = − 1

2πi(R(θ+ie) −R(θ−ie)), (3.17) where e is an arbitrarily small parameter (positive). One can easily check this equal-ity by inserting the expression back into equation (3.16) as a consistency check. The need for this resolvent function will become apparent shortly. Moreover, the func-tion can be transformed to its inverse Laplace transform ( ˆR) and back using the fol-lowing two transformations:

R(z) = Z ∞ 0 ˆ R(s)e−szds, Rˆ(s) = Z i∞+_e −i∞+e R(z)esz dz 2πi. (3.18)

The latter of these two is the Bromwich inversion formula.

The goal now will be to relate the energy density e to the inverse Laplace trans-form of the resolvent function. From there we will attempt to derive an expression for this quantity in terms of B. This procedure will be done perturbatively in a dou-ble scaling limit:

θ, B→∞ while keeping z =2(θ−B) fixed. (3.19)

In our expressions we will substitute any θ with B+z/2 such that in this large B limit we get a solution in a 1/B perturbative expansion. From a resurgence standpoint we can anticipate that exponentially small terms of the form e−nBwith n∈N>0will be

invisible to such an expansion and therefore an interesting non-perturbative sector to explore later on.

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3.2. Solving the integral equation 27 Using the fact that χ(θ)is symmetric one can rewrite the energy density to

e= m 4π Z B −Bχ (θ)(eθ+e−θ)dθ = m 2π Z B −Bχ (θ)eθdθ = me B 4π Z 0 −4Bχ (z)ez/2dz (3.20)

where χ(z) = χ(θ(z)). So far we have an exact expression, but in the next step we

consider this expression in the large B limit. it can then be written in a perturbative 1/B expansion of which the leading order (B0) is

e ' me B 4π Z 0 −∞χ(z)e z/2_dz = −me B 4π Z 0 −∞(R(θ+ie) −R(θ−ie))e z/2 dz 2πi = me B 4π Z +i∞+e −i∞+e R(z)ez/2 dz 2πi = me B 4π Rˆ(1/2). (3.21)

A valid question that the reader might have is why the first line in (3.21) is an ap-proximation if χ(z)has no support outside[−4B, 0]. The subtlety lies in the fact that this support is only finite at finite B. As χ is a solution of the integral equation that is determined by the value of B, it would be more appropriate to, at finite B, write

χ(z, B). Then in the large B expansion the support runs all the way up to−∞ and

we get subleading corrections to this expression from the growth of χ. These correc-tions are of a non-perturbative nature, a point to which we will come back in the last section of this chapter.

Up to this point both derivations of [21] and [14] are alike. The next step is finding an expression for the inverse Laplace transform of the resolvent function. If one manages to obtain such an expression, it is possible to express the energy density e as a function of B. To derive such an expression there are two possble pathways: The first one is by constructing a difference equation for the inverse Laplace transform of the resolvent. Then imposing the correct analytical properties as conditions on the solution of that equation one can guess the most general solution for ˆR. In the second method the Wiener Hopf decomposition of the kernel is used. This, in combination with some general knowledge from complex analysis, allows one to also solve for ˆR to leading order in 1/B. The former of these two methods was pursued in [21] and the latter in [14]. We will review both derivations, starting with the latter.

3.2.1 Wiener Hopf method

In this procedure we will start with the integral equation and try to solve it by re-working the equation into a particular expression that will be fourier transformed. This new transformed expression will contain several functions in ω (the conju-gate variable), of which one is directly related to ˆR. This is where the Wiener Hopf method comes in: the various functions will either be analytic in the upper or lower