Thermodynamics of the nonlinear sigma model

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Erik LivermoreThermodynamics of the nonlinear sigma model NTNU Norwegian University of Science and Technology Faculty of Natural Sciences Department of Physics

Master ’s thesis

Erik Livermore

Thermodynamics of the nonlinear sigma model

Perturbation theory in the large-N approximation

Master’s thesis in Applied Physics and Mathematics Supervisor: Jens Oluf Andersen

August 2020

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Erik Livermore

Thermodynamics of the nonlinear sigma model

Perturbation theory in the large-N approximation

Master’s thesis in Applied Physics and Mathematics Supervisor: Jens Oluf Andersen

August 2020

Norwegian University of Science and Technology Faculty of Natural Sciences

Department of Physics

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Abstract

In this thesis, we study the thermodynamics of the two-dimensional nonlinear sigma model, a toy model for the strong interaction. After presenting the fundamentals of thermal quantum field theory and a derivation of the nonlinear sigma model, we discuss the method of dimensional regularization, and use it to renormalize the model to one- loop order. We use this to calculate the running of the coupling, and confirm that the theory is asymptotically free. Next, we calculate vacuum integrals that arise from the sunset diagram and the basketball diagram, using a combination of dimensional regularization and numerics. Using perturbation theory, we calculate the renormalized free energy density to two-loop order. Further, we calculate the renormalized free energy density to three-loop order in the large-N approximation, where we let the number of fields go to infinity. Both the two-loop and three-loop corrections are well-behaved at low temperatures. At high temperatures, as the massless limit is approached, the two- loop contribution is a finite constant, and the three-loop contribution diverges. The infrared divergent behavior of the three-loop free energy density gets worse for stronger coupling.

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Sammendrag

I denne oppgava studerer vi termodynamikken til den ikke-lineære sigmamodellen i to dimensjoner, en leketøysmodell for den sterke vekselvirkningen. Vi presenterer grunn- leggende termisk kvantefeltteori og utleder modellen ved å fryse ut en av frihetsgradene i den O(N)-symmetriske lineære sigmamodellen. Dimensjonell regularisering, en metode for å regularisere divergente integraler, drøftes før den anvendes til å renormalisere den ikke-lineære sigmamodellen til første orden i koblingskonstanten. Vi bruker dette til å regne ut hvordan koblinga løper, og bekrefter at teorien har asymptotisk frihet. Videre regner vi ut vakuumintegraler som oppstår fra solnedgangdiagrammet og basketballdia- grammet ved å bruke en kombinasjon av dimensjonell regularisering og numerikk. Ved å bruke perturbasjonsteori regner vi ut den renormaliserte frie energien til første orden i koblingskonstanten. Deretter lar vi antall felt gå mot uendelig i store-N approksi- masjonen, og regner ut den renormaliserte frie energien til andre orden i den omskalerte koblingskonstanten. Ved lave temperaturer har både første- og andreordenskorreksjonene god oppførsel. Ved høye temperaturer går vi mot den masseløse grensa, der førsteor- denskorreksjonen er en endelig konstant, og andreordenskorreksjonen divergerer. Den infrarødt divergente oppførselen til andreordenskorreksjonen forverres ved sterkere vek- selvirkning.

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Preface

I would first like to thank my supervisor, professor Jens Oluf Andersen, for his continued support and guiding questions during my thesis work. With a subject matter that is this complex and abstract, and with dozens of conventions to navigate, his help has been truly indispensable.

Second, I am grateful for the many fruitful discussions and long evenings working alongside my good friend Martin Aria Mojahed, both on campus before the pandemic and online during.

For all things unrelated to physics during these five years in Trondheim, my friends have blown me away with their presence and warmth. Among them are the kindest, funniest and smartest people I’ve met. This includes my sisters, who (re-)appeared in the city in more recent years and made it feel more like home.

Finally, a big thank you to my parents for showing that you can change your trajec- tory at any moment, and granting me the privilege to do the same. The fact that they chose and choose to care for people while making the most of their time here continues to inspire me.

“It’s nice to be important, but it’s more important to be nice.”

-Hans P. Geerdes

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Notation and conventions

In this thesis, we employ the following notation and conventions:

• In Chapter 1 and Appendix C, [1] is used to draw several Feynman diagrams, with conventions adopted from the article. Notably, a solid line with an arrow

denotes a fermion, a dashed line denotes a scalar particle, a curly line denotes a gluon, and a wavy line denotes any other gauge boson.

• Bold typeface implies a spatial vector, e.g. p, and a greek upper index implies a d-vector, e.g. x^µ, where x⁰ is the timelike component and xⁱ, i= (1,2, . . . , d−1) are the spacelike components.

• The Minkowski metric, i.e. the metric of flat spacetime in 3+1 dimensions, is ηµν =η^µν = diag(1,−1,−1,−1). The generalization to ddimensions has η00 = 1 and d−1 entries of−1in the d×ddiagonal matrix.

• Unless otherwise stated, we use the Einstein summation convention. This means that two repeated indices imply a summation over those indices: T_aθ_a =P

a

T_aθ_a. Further:

– The Minkowski metric can be used to raise or lower indices: η_µνx^ν = x_µ, η^µνx_ν =x^µ.

– Repeated Greek indices imply a summation over the Minkowski metric: ∂µj^µ= η_µν∂^µj^ν.

• Natural units are used, i.e. ~ = c = kB = 1, where ~ is the reduced Planck constant,k_B is the Boltzmann constant, andcis the speed of light in the vacuum.

• The vacuum integral in ddimensions using dimensional regularization is denoted Z

p

≡

e^γ^Eµ² 4π

Z d^dp

(2π)^d, (1)

wherep^ν is the d−momentum,µis the renormalization scale and γE is the Euler- Mascheroni constant. We employ this shorthand both in Minkowski and Euclidean spacetime.

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• The sum-integral of imaginary time formalism ind−1spatial dimensions is defined by

P Z

p

=TX

n

Z d^d−1p

(2π)^d−1 , (2)

wherenrefers to the Matsubara frequencies ωn and p is the spatial momentum.

• For common special functionsf(x) such assin(x),lnx and Γ(x), we define

fⁿ(x)≡[f(x)]ⁿ . (3)

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Chapter 1

Introduction

1.1 Motivation

Quantum chromodynamics (QCD) is the quantum field theory that describes the strong interaction of quarks and gluons. It has been described as “our most perfect physical theory” by Nobel laureate Frank Wilczek, who identified three important research fron- tiers during a keynote talk [2]: high-temperature QCD, high-density QCD, and grand unification of QCD and the other theories of the Standard Model. While considerable progress has been made in all three areas in the two decades since the talk, it still rings true today. Therefore, in this thesis we study the thermodynamics of a model that has important similarities with QCD.

The gluons are massless particles that mediate the strong force, like photons mediate the electromagnetic force in quantum electrodynamics (QED). As the names suggest, QCD is the theory of color charge, and QED is the theory of electric charge. The quarks are fermions with spin ¹₂ and electric charge ²₃ or−¹₃. They are the fundamental constituents of hadrons, of which there are two types: mesons and baryons. Mesons consist of a quark and an antiquark or a superposition of several such pairs and are thus bosons. Baryons consist of three quarks¹ and are thus fermions. There are six quark flavors: up (u), down (d), strange (s), charm (c), bottom (b), and top (t), and we denote antiparticles with an overbar: q. Some examples of hadrons are the neutral pion ^√¹

2(uu−dd), a meson, the neutronudd and the protonuud, both baryons. There are eight gluons carrying color and anticolor, and three quarks carrying color and three antiquarks carrying anticolor for each quark flavor. This complexity, as well as the strength of the coupling, means that analytic results are difficult to obtain. One highly successful method in QCD is lattice QCD (LQCD), where spacetime is divided into a discrete lattice, and numerical simulations are run. This method is exact in the limit where the grid spacing goes to zero, though this limit is not always feasible.

QCD is a non-Abelian gauge theory, i.e. a theory where the Lagrangian is locally invariant under transformations within a given non-Abelian symmetry group called a

1There also exist exotic baryons that consist of five quarks and antiquarks. These are known as pentaquarks, and were unambiguously confirmed in 2019 [3].

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gauge group. The gauge group for QCD is SU(3)col, and the other gauge groups of the Standard Model are U(1)_Q for QED and SU(2)_L for the weak interaction, and SU(2)_L×U(1)_Y for the unified electroweak theory. The subscripts of the symmetry groups refer to the quantum numbers associated with the symmetry, i.e. color in QCD, electric charge in QED, weak isospin in the weak interaction, and weak isospin and weak hypercharge in electroweak theory. U(1) is an Abelian group, and all nontrivialSU(N) groups are non-Abelian, i.e. all groups withN >1. Theories with gauge group SU(N) are called Yang-Mills theories, and they have some shared properties, like asymptotic freedom.

1.1.1 Asymptotic freedom

Asymptotic freedom was discovered by Politzer [4], Gross and Wilczek [5], and is an essential property of QCD and other Yang-Mills theories. It means that the coupling gets weaker at higher energies or, correspondingly, shorter distances. The discovery of asymptotic freedom helped resolve two major issues in quantum field theory at the time:

the Landau pole and quark confinement. We briefly discuss the issues and their solutions here, but recommend Wilczek’s Nobel Prize lecture [6] for a more insightful and nuanced discussion.

Quark confinement refers to the fact that while quarks have been discovered to be the fundamental constituents of hadrons, it is impossible to isolate single quarks; no free quarks have been directly observed. How then could they be the most basic building block of hadrons, if hadrons cannot be disassembled to produce them?

The Landau pole occurs because of screening. In the vicinity of a charged source particle, virtual particles are induced so that opposite charged virtual particles are at- tracted to the source particle and like charged virtual particles are repelled from it. This polarization of the vacuum serves to diminish the observed charge of the source particle at a finite distance. Decreasing the distance, the screening effect is reduced, so that the interaction becomes stronger. Thus, as we shorten the distance, the interaction strength diverges to infinity, unless we choose to have no interaction at all.

The process by which asymptotic freedom occurs is calledantiscreening. With screening, the induced virtual particles diminish the observed charge at a finite distance. How- ever, if the virtual particlesenhancethe observed charge, we have antiscreening. Further, the virtual particles themselves carry charge, inducing more antiscreening particles, and more, and more, growing infinitely. Thus, an attempt to produce a single particle in a theory where antiscreening dominates would produce an infinite number of antiscreening particles, both requiring an infinite amount of energy and producing an infinitely strong interaction. The solution to this is that the single particle could never be produced at all, and confinement is a necessity of the theory. If we produce a quark with its antiquark in its near vicinity, the infinite growth of antiscreening does not happen, and only finite energy is required. Thus, we have mesons, such as pions. As long as the net charge is zero, we could also produce an odd number of quarks, and thus we have baryons.

QCD has three colors and three anticolors as the theory’s charge, and the sum of the three colors or three anticolors is zero, as well as the sum of a color and its anticolor.

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Because of this, the phenomenon of quark confinement is actually a consequence of the more fundamental phenomenon of color confinement. While quark confinement is the only definitively observed instance of color confinement, it is theorized that gluons could stick together to form colorlessglueballs, however the experimental status of glueballs is inconclusive [7].

In order to determine whether a theory exhibits asymptotic freedom, we calculate its beta function, which informs us about how the interaction strength varies with energy scale. If the beta function is negative, we have asymptotic freedom. For a gauge theory like QED or QCD, we can calculate the beta function by calculating the quantum corrections to the propagation of the gauge boson, i.e. the photon in QED and the gluons in QCD.

The photon is neutral, and hence there is no photon-photon vertex in QED. The first quantum correction to the photon propagator is given by the electron loop

.

This diagram contributes to charge screening, and the beta function is β(e) = e³

12π². (1.1)

Nevertheless, QED is safe from the Landau pole, as it occurs beyond the Planck scale, i.e. outside the scope of any current physics. Specifically, it occurs at an energy of order 10⁵⁶GeV [8], roughly a tenth of the mass of the Sun.

In QCD however, the gluons have color charge, so they directly interact with each other. When fixing the gauge, we must also include a ghost, a fictional complex scalar field of fermionic nature, in our calculations so that they are gauge invariant. Hence, the nonzero diagrams contributing to the first quantum corrections are

, where the leftmost diagram is the quark loop, the middle diagram is the gluon loop, and the rightmost diagram is the ghost loop. The quark loop, like the electron loop in QED, provides screening. For the sake of gauge invariance, the ghost loop and gluon loop are calculated together, and they provide antiscreening. The resulting beta function is

β(α_s) =−α²_s 11

3 N_c−2 3n_f

, (1.2)

where αs is the QCD fine-structure constant, Nc is the number of colors, and n_f is the number of quark flavors. QCD has three colors and six quark flavors, which means

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that antiscreening prevails and we have asymptotic freedom. As a closing remark on asymptotic freedom, we note that the weak interaction also exhibits asymptotic freedom, but does not produce confinement. Due to the Higgs mechanism, the gauge bosons acquire large masses, rendering the interaction much too weak to have confinement. The Higgs mechanism occurs at a higher scale than the confinement scale, so the coupling never becomes strong. Furthermore, the Higgs phase and the confinement phase are continuously connected, i.e. there is no order parameter that determines the transition between the two phases, and there is no phase boundary [9].

1.1.2 States of quark matter

At ordinary pressures and temperatures, quarks are confined to hadrons. LQCD predicts that the potential between a quark and an antiquark in a meson is of the form [10]

V(r) =−A(r)

r +Kr , (1.3)

wherer is the separation distance between the quark and antiquark andA(r)andK are experimentally determined. As the distance increases, the second term dominates and the potential is effectively linear. If we try to separate the pair, we supply more and more energy to the system. We can model this as gluon flux tubes that hold the quarks together. The flux tubes are to be understood as narrow strings with high string tension, the constantK in Eq. (1.3). As energy is supplied, the flux tubes elongate until a new quark-antiquark pair is produced, see Figure 1.1. The original meson is net colorless, and the new quark-antiquark pair is produced in a color configuration that results in the two new mesons to be net colorless as well. Hence, we have color confinement, as it is impossible to separate a single quark. If we rather decrease the distance, the interaction strength decreases until the limitr→0where the quarks are considered non-interacting due to asymptotic freedom.

q q

g

g q q

g

g q q

g

g q q

g

(a) (b) (c) g

Figure 1.1: Confinement. q and qrepresent quarks and antiquarks, respectively, and g represents gluon flux tubes. (a) A quark-antiquark pair is held together by gluon flux tubes. (b) Energy is supplied, and the distance between the quark and the antiquark increases. (c) More energy is supplied, and a new quark-antiquark pair is produced.

The hadron masses are often far bigger than the sum of the current massesof their constituent quarks. The current mass is the mass term found in the Lagrangian, i.e. the one generated by the Higgs mechanism. The proton, for instance, has a mass of order 1 GeV, whereas the sum of the current masses of its quarks is of order10 MeV. That is because the quarks also have a dynamically generated mass, known as the constituent mass or condensate mass. This mass is generated through chiral symmetry breaking (χSB).

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A particle’s helicity is determined by whether its spin axis is in the same or opposite direction as the particle’s motion. These helicity states are called right-handed and left- handed, respectively, as they are mirror images of one another. As spin is a conserved quantum number in the strong interaction, helicity is conserved for massless particles.

For massive particles, however, there exist two inertial frames where the momentum has opposite direction and thus opposite helicity. χSB is a natural consequence of quark confinement [11]. Suppose for simplicity that the current mass is zero. Then, helicity is conserved, and we have chiral symmetry. If we then produce a bound quark-antiquark state, the two particles each have a wavefunction describing the spin and momentum distribution. The particles will have momenta directed towards each other or away from each other, they cannot be exactly the same due to Heisenberg’s uncertainty principle.

Thus, they will eventually travel away from each other, but the forces of confinement prevent this, and the particles must reverse direction. As the strong interaction leaves spin unchanged, the helicity is flipped, and the chiral symmetry is broken.

χSB occurs because the QCD vacuum has a nonzero chiral condensate, ψψ

= h0|ψ_LψR+ψ_RψL|0i 6= 0, (1.4) where ψR destroys a right-handed quark and ψ_L creates a left-handed quark, and vice versa. Thus, ψ_Lψ_R describes the rate at which helicity is changed from right-handed to left-handed, andψ_Rψ_L describes the rate at which helicity is changed the other way.

Sinceψdoes not destroy the vacuum ket|0iandψdoes not destroy the vacuum bra h0|, there must exist a sea of virtual quark-antiquark pairsqq in the condensate. The bound quarks will flip helicities at a rate proportional to

ψψ

, and hence they have acquired constituent or condensate mass.

ψψ

is so large that the constituent mass is roughly 100times bigger than the current mass for the up quark and the down quark, hence the proton mass is fully accounted for.

At high temperatures, hadronic matter undergoes a phase transition to a quark- gluon plasma(QGP). Here, the thermal energy of the system leads to a reduction in the coupling strength, so much so that chiral symmetry is restored and we no longer have color confinement. The plasma consists of free strongly interacting quarks and gluons, and we can model the quark-antiquark potential as

V_QGP =−C r exp

− r λ_D

, (1.5)

where λ_D ∝ _T¹ and C is a constant. Comparing this to Eq. (1.3), we see that this is equivalent to zero string tensionK, i.e. no gluon flux tubes and no confinement. QGP has been produced in heavy-ion collisions, for instance at CERN [12], but it is very short-lived as it rapidly turns into hadrons. The early universe is thought to have been dominated by the QGP phase during the quark epoch. As the universe expanded and cooled, color confinement andχSB occurred and the quarks formed hadrons, ending the quark epoch at an age of about 10⁻⁶s and a critical temperatureT_c∼200 MeV.

As such, high-temperature QCD is crucial in our understanding of the universe and QGP, but the complexity and number of strong interactions necessitate the use of approx- imations. LQCD predicts a T_c of about 175 MeV (2×10¹²K in SI units) [13], slightly

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higher than experimental results [12]. A drawback of lattice QCD is that is extremely computationally intensive, requiring extensive use of supercomputers [14].

For a more detailed introduction to QCD and QGP, see [10], from which this section has borrowed substantially.

1.2 Approach

One way of modeling the strong interaction is to abandon QCD altogether, and do analytic calculations in simpler models that have shared characteristics with QCD. These models would betoy modelsfor QCD, and while the use of a toy model obviously removes us somewhat from reality, the benefits of using a simpler model are twofold. First, it allows us to delve deeper into the theory, and we can perform calculations we are unable to do in QCD. This can reveal hidden aspects such as symmetries or cancellations, which we will then know to look for in QCD. Second, we can use the toy model as a testing ground of sorts, trying out various methods and calculational techniques, before deciding whether they are useful in QCD. Of course, these benefits are just in the paradigm of QCD, and work in the toy model has inherent value for the toy model itself, regardless of the applications.

In this thesis, we will study the two-dimensional nonlinear sigma model (NLSM) as a toy model for QCD. The NLSM has a dynamically generated mass gap [15], and we will show that it exhibits asymptotic freedom. The model describes the interactions of (N −1) identical scalar π fields, behaving as quasi-Goldstone bosons. In short, a Goldstone boson is a massless particle that appears when a continuous symmetry of the Lagrangian is broken by the vacuum state. Ourπfields are only quasi-Goldstone bosons, as they have a nonzero current mass. Similarly, there are three scalar mesons that are interpreted as the quasi-Goldstone bosons of the broken chiral symmetry in QCD, and only quasi because the quarks have a nonzero current mass. These mesons are the pions, the charged pionsπ⁺ =udand π⁻ =du, and the neutral pionπ⁰ = ^√¹

2(uu−dd). They are the lightest hadrons, with masses roughly an order of magnitude smaller than the masses of the nucleons, i.e. the neutron and the proton. We can think of the pions as the particles that mediate the nuclear strong force between nucleons in the core of an atom. In our study of the NLSM, we will often refer to theπ particles as pions.

We will study the NLSM using perturbation theory, where we treat all interactions as perturbations on the base case of a non-interacting theory. The NLSM is renormalizable, which means that all infinities arising from quantum corrections can be offset by a finite number of counterterms in the Lagrangian. Using renormalized perturbation theory, we will calculate the free energy density to gain insight into the thermodynamics of the NLSM. This involves calculating many Feynman diagrams, both in the vacuum and at finite temperature.

Somewhat tangential to the finite-temperature work contained in this thesis, we will also calculate some higher-order vacuum integrals and use Mathematica to evaluate the associated numerics.

The reader will notice that this thesis is calculation heavy, which is a necessity of

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this type of work in order to clarify the process.

1.3 Structure of the thesis

The structure of the thesis is as follows. We begin by deriving how to calculate the thermodynamic properties of a quantum field theory by using the functional integral formulation of the partition function in imaginary time formalism in Chapter 2. The partition function can be used to calculate any desired thermodynamic quantity, like pressure, entropy, and energy. In Chapter 3, we derive the two-dimensional NLSM, where spontaneous symmetry breaking and Goldstone bosons are essential aspects. Chapters 2 and 3 were written as part of the specialization project [16].

We then proceed to renormalize the theory to one-loop level in Chapter 4. As loop diagrams often produce divergent integrals, we discuss the method of dimensional regularization to isolate these divergences, and then use counterterm renormalization to renormalize the propagator and zero-point energy. After calculating the beta function and running of the coupling, we present techniques to calculate some integrals that arise from higher-order vacuum diagrams. In Chapter 5, we calculate the renormalized free energy density to two loops, before proceeding to the large-N approximation of the NLSM in Chapter 6. In the large-N approximation, we let the number of fields go to infinity, and this simplifies some calculations. This lets us calculate the free energy density to three loops. We discuss our results and provide an outlook for further study in Chapter 7.

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Chapter 2

Functional integral formulation of the partition function

This chapter, as well as the majority of Section 5.1 and Section 5.2, follows [17] closely, and will derive how to obtain the thermodynamic properties of a system described by a basic quantum field theory. They were written as part of the specialization project [16], but are included here for completeness.

In statistical mechanics, the grand canonical ensemble is a system with fixed temperature T, chemical potentialµ and volume V. The total number of particles N and the total energyE are both free to be exchanged with the surroundings. This system is fully described by its density matrix ρ, given by

ρ= exp [−β(H−µ_iN_i)] , (2.1) whereHis the system’s Hamiltonian,βis the inverse temperature, andN_iis a conserved number operator with an associated chemical potential µi. From this, we can calculate the grand canonical partition function Z:

Z =Z(β, µi, ...) = Trρ . (2.2) The expectation value of any observableO is given by

hOi= TrOρ Trρ = 1

Z TrOρ . (2.3)

The natural logarithm of the partition function can in turn be used to calculate any thermodynamic property of the system, such as:

P = ^ln_βV^Z, (2.4)

hN_ii= ¹_β_∂µ^∂

ilnZ , (2.5)

S= lnZ−β²^∂(lnZ)_∂β , (2.6)

hEi=µ_ihN_ii −_∂β^∂ lnZ , (2.7)

where P is the pressure, S is the entropy, andhN_ii and hEi are the expectation values of the particle number and energy, respectively.

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2.1 Quantum mechanical partition function

Before going to functional integrals, we will derive the partition function using creation and annihilation operators. We start by looking at a single-particle mode, where each particle in that mode has energy ω. The system is free, i.e. there are no interactions between the particles. Thus, we can consider it as a set of noninteracting quantum harmonic oscillators. If there arenparticles in the system, the state is|ni. The orthogonality and completeness relations are

n n⁰

=δ_nn⁰, (2.8)

X

n

|ni hn|= 1. (2.9) For bosons, any number of particles may occupy a single mode. We introduce the creation operator a^†, which creates a particle and inserts it into the mode it operates on,

a^†|ni=√

n+ 1|n+ 1i, (2.10)

and the annihilation operator, which annihilates a particle and removes it from the mode it operates on,

a|ni=√

n|n−1i . (2.11)

The annihilation operator destroys the vacuum, a|0i= 0, as there are no particles left to annihilate. Hence, the commutation relation between the operators is[a, a^†] = 1, and

|ni is an eigenstate of the number operator N = a^†a. We can construct any state by repeated application of the creation operator on the vacuum,

|ni= 1

n!(a^†)ⁿ|0i. (2.12)

For a bosonic harmonic oscillator, the Hamiltonian is H = 1

2ω(a^†a+aa^†) = 1

2ω(N + 1 +N) =ω N+ 1 2

, (2.13)

where ¹₂ω is the energy of the vacuum. We are now ready to calculate the partition function. Since there are no interactions present that can alter the particle number, we can assign a chemical potential µ, and the partition function becomes

Z = Tr e^{−β(H−µN)}=

∞

X

n=0

hn|e^{−β(w−µ)N−}¹²^βω|ni

=

∞

X

n=0

e^{−β(w−µ)n−}¹²^βω= e⁻¹²^βω

1−e^{−β(ω−µ)}, (2.14) where the sum over all states is a way of explicitly performing the trace operation. This leads to

lnZ =−1

βω−ln

1−e^{−β(ω−µ)}

, (2.15)

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which is our desired quantity.

For fermions, the Pauli exclusion principle states that no more than one particle can occupy a single state, so the only states allowed are |0i and |1i. This means that the creation operator must destroy the state that already contains a particle, a^†|1i= 0, in order to have a consistent set of tools. Thus, the square of each of the operators is zero, aa=a^†a^†= 0, and we get the anticommutation relation

{a, a^†}= 1. (2.16) The Hamiltonian is

H= 1

2ω(a^†a−aa^†) = 1

2ω(N−1 +N) =ω N −1 2

, (2.17)

with the vacuum energy being−¹₂ω this time. We calculate the partition function in the same manner as before,

Z = Tr e^−β(H^−µN)=

1

X

n=0

hn|e^{−β(w−µ)N+}¹²^βω|ni

= e¹²^βω

1 + e^{−β(ω−µ)}

, (2.18)

and finally get

lnZ = 1

2βω+ ln

1 + e^{−β(ω−µ)}

. (2.19)

Now, we want to create a noninteracting gas by inserting particles of several modes into a cubical box with volume V =L³, which we will later take the macroscopic limit of. The particles will be either just fermions or just bosons, but we can deal with the two situations simultaneously. We impose the boundary condition that the wavefunction must disappear at the surface of the box. This necessitates that the length of the box L must contain an integer number of half wavelengths, in each of the three spatial directions:

λ_i = 2L/j_i, i= (x, y, z), j_i ∈Z⁺, (2.20) whereλ_i is the wavelength in directioniwith a corresponding quantum number j_i. The relation between particle momentum and wavelength is given by

|p_i|= 2π λi

. (2.21)

Hence, we can refer to a mode asj, with a single quantized three-dimensional momentum.

As the particles do not interact, the full Hamiltonian H is the sum of the Hamiltonians for each modeH_j, and likewise for the number operatorN. Thus, the partition function becomes

Z = Tr e^{−β(H−µN)}= Tr e^−β^P^j^(H^j^−µN^j⁾=Y

j

Tr e^−β(H^j^−µN^j⁾ =Y

j

Z_j, (2.22)

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i.e. the product of the partition functions for each modeZj. In turn, lnZ =X

j

lnZ_j =

∞

X

jx=1

∞

X

jy=1

∞

X

jz=1

lnZjx,jy,jz. (2.23) We take the macroscopic limit L → ∞, and let the sum over j_i become an integral.

Eqs. (2.20) and (2.21) implydji= ^L_πd|p_i|. We get lnZ= V

π³ Z ∞

0

d|p_x| Z ∞

0

d|p_y| Z ∞

0

d|p_z|lnZ(p). (2.24) Here, the partition function will only depend on the magnitude of the momentum, so we letR∞

0 → ¹₂R∞

−∞to finally obtain lnZ =V

Z d³p

2π³ lnZ(p), (2.25)

which becomes

lnZ =−V

Z d³p 2π³

1

2βω+ ln h

1−e^{−β(ω−µ)} i

(2.26) for bosons, and

lnZ =V

Z d³p 2π³

1

2βω+ lnh

1 + e^{−β(ω−µ)}i

(2.27) for fermions.

2.2 Transition amplitude in functional integral formulation

Let us now look at this from a field theory point of view. We consider a field operator φ(x, t) and its conjugate momentum operatorπ(x, t) that satisfy:

φ(x,0)|φi=φ(x)|φi , (2.28)

π(x,0)|πi=π(x)|πi, (2.29)

and have completeness relations Z

dφ(x)|φi hφ|= 1, (2.30)

Z

dπ(x)|πi hπ|= 1, (2.31)

and orthogonality relations

hφ_a|φ_bi=Y

x

δ(φa(x)−φb(x)), (2.32)

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hπ_a|π_bi=Y

x

δ(π_a(x)−π_b(x)), (2.33)

where |φ_ai and |φ_bi refer to two arbitrary states. The overlap between the field and its conjugate momentum is

hφ|πi= exp

i Z

d³x π(x)φ(x)

, (2.34)

and the Hamiltonian H is a functional of them through the Hamiltonian density H= H(π(x), φ(x)):

H= Z

d³xH(π(x), φ(x)). (2.35) We now assume the system is in state |φ_ai at a time t= 0. Then, at a later time t_f, it has evolved into e^−iHt^f |φ_ai. Thus, the transition amplitude for returning to the same state is hφ_a|e^−iHt^f |φ_ai. Let the time interval (0, t_f) be divided into N equal steps of size∆t. At each step, we insert the completeness relations of Eqs. (2.30) and (2.31) to get

hφ_a|e^−iHt^f |φ_ai= lim

N→∞

Z ^N Y

i=1

dπidφi

!

× hφ_a|π_Ni hπ_N|e^−iH∆t|φ_Ni hφ_N|π_N₋₁i

× hπ_N₋₁|e^−iH∆t|φ_N−1i. . .

× hφ₂|π₁i hπ₁|e^−iH∆t|φ₁i hφ₁|φ_ai . (2.36)

t t

∆t

φ_a φ₁ φ₂ φ_a φ_a φ_a

t_f 0

0 t_f

φ₄ φ₃

Figure 2.1: A visualisation of the path integral. On the left-hand side, the system starts in a state |φ_ai before splitting into several possible pathways to an intermediate state

|φ₁i. The systems propagate to new intermediate states for each timestep∆tuntil they once more reach |φ_ai. On the right-hand side, the limit ∆t→ 0 is taken, and a few of the infinite possible pathways are shown.

A good way to visualize this division is to imagine that the path emerging from |φ_ai

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splits intoN different pathways, each one propagating to a new intermediate state|φ_ii during each time interval∆t. This is shown conceptually in Figure 2.1. As we take the limit N → ∞, we have included every possible field configuration that leads back to

|φ_ai. In this limit, we also let∆t→0, so we can make the following expansion:

hπ_i|e^−iHⁱ^∆t|φ_ii ≈ hπ_i|(1−iHi∆t)|φ_ii

=hπ_i|φ_ii(1−iH_i∆t) (2.37) with

Hi = Z

d³xH(π_i(x), φi(x)). (2.38) The inner products in Eqs. (2.36) and (2.37) are given by Eqs. (2.32) and (2.34), and we obtain

hφ_a|e^−iHt^f |φ_ai= lim

N→∞

Z N

Y

i=1

dπidφi

!

δ(φ1−φa)

×exp







−i∆t

N

X

j=1

Z d³x

H(π_j, φj)−πj(φj+1−φj)/∆t







. (2.39)

Finally, taking the continuum limit of (2.39) yields hφ_a|e^−iHt^f |φ_ai=

Z [dπ]

Z φ(x,t_f)=φa(x) φ(x,0)=φa(x)

[dφ]

×exp

i Z tf

0

dt Z

d³x

π(x, t)∂φ(x, t)

∂t − H(π(x, t), φ(x, t))

, (2.40) with the bracket notation[dπ]and[dφ]denoting functional integration as defined in (2.39).

2.3 Partition function for bosons

The partition function for bosons is Z = Trh

e^−β(H−µⁱ^Nⁱ⁾i

=X

a

hφ_a|e^−β(H−µⁱ^Nⁱ⁾|φ_ai . (2.41) Eq. (2.41) is quite similar to the expression for the transition amplitude, and we want to express Z in a form similar to Eq. (2.40). Noether’s theorem, explained in Chapter 2 of [18], states that if the Lagrangian exhibits a continuous symmetry, there exists a conserved current j^µ such that ∂µj^µ = 0 and hence also a conserved charge. If the system has such a conserved charge with conserved charge densityN(π, φ), we make the replacement

H(π, φ)→ H(π, φ)−µN(π, φ), (2.42)

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with µ as an associated chemical potential like before. Making this replacement and introducing the imaginary time variableτ =it, we arrive at the expression

Z= Z

[dπ]

Z

periodic

[dφ]

×exp Z β

0

dτ Z

d³x

π∂φ

∂τ − H(π, φ) +µN(π, φ)

, (2.43)

where periodic means that the integral over φ is constrained so thatφ(x,0) =φ(x, β).

Let us calculate this for a free real scalar field. The Lagrangian density is L= 1

2∂_µφ∂^µφ− 1

2m²φ², (2.44)

wherem is the particle mass. The conjugate field is π = ∂L

∂(∂₀φ) = ∂φ

∂t , (2.45)

and hence the Hamiltonian density is H=π∂φ

∂t − L= 1 2π²+ 1

2(∇φ)²+ 1

2m²φ². (2.46)

As the Lagrangian has no continuous symmetries, there is no conserved charge and the field is neutral. Inserting this into (2.43) and carrying out the momentum integral yields

Z =N⁰ Z

periodic

[dφ] exp Z β

0

dτ Z

d³xL

, (2.47)

where N⁰ is a normalization constant which is irrelevant to any thermodynamics. The exponential in Eq. (2.47) is very similar to the action integral, except that one integrates overτ instead oft. We define

S =− Z β

0

dτ Z

d³xL= 1 2

Z β 0

dτ Z

d³x

"

∂φ

∂τ 2

+ (∇φ)²+m²φ²

#

. (2.48)

Integration by parts, together with the periodicity of φ, gives S= 1

2 Z β

0

dτ Z

d³xφ

− ∂²

∂τ² − ∇²+m²

φ . (2.49)

We Fourier expand the field, φ(x, τ) =

rβ V

∞

X

n=−∞

X

p

e^i(p·x+ωⁿ^τ)φn(p), (2.50)

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again keeping in mind the periodicity condition φ(x,0) = φ(x, β) which results in ωn= 2πnT. ωn are known as the Matsubara frequencies. We define the particle en- ergyω=p

p²+m² and substitute (2.50) into (2.49) to get S = 1

2β²X

n

X

p

(ω_n²+ω²)φ_n(p)φ^∗_n(p). (2.51) Sinceφ is a real field, we integrate out the phases and get the Gaussian integral

Z =N⁰Y

n

Y

p

Z ∞

−∞

d|φ_n(p)|exp

−1

2β²(ω²_n+ω²)|φ_n(p)|²

=N⁰Y

n

Y

p

(2π)^1/2

β²(ω_n²+ω²)−1/2

. (2.52)

Now, we can ignoreN⁰ and (2π)^−1/2, and the quantity of interest islnZ: lnZ =−1

2 X

n

X

p

ln

β²(ω_n²+ω²)

. (2.53)

Using Eq. (B.1) to evaluate the sum overn, we get lnZ =−X

p

1

2βω+ ln

1−e^−βω

=−V

Z d³p (2π)³

1

2βω+ ln

1−e^−βω

, (2.54)

which is the same result as we got earlier in Eq. (2.26), but with no chemical potential µ.

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Chapter 3

The nonlinear sigma model

This chapter was written as part of [16], but is included here for completeness. Here, we derive the nonlinear sigma model in two dimensions. In order to do this, we must first understand spontaneous symmetry breaking and Goldstone bosons.

3.1 Spontaneous symmetry breaking

Spontaneous symmetry breaking (SSB) takes place when the lowest energy state of a system has a smaller symmetry group than that of the full Lagrangian of the system. In field theory, such a state is called the vacuum state, and SSB occurs when the vacuum expectation value (VEV) for a field is nonzero. For sufficiently low energies, one typi- cally only observes fluctuations around the vacuum state, so the spontaneously broken symmetry is often called a hidden symmetry. To gain some intuition, we consider an everyday example:

A cylindrical thin metallic rod is subjected to a force on each of its ends, where the force is directed along the axis of the rod towards its center. As long as the force is small compared to the rigidity of the rod, the rod has rotational symmetry around its axis, an SO(2)symmetry. The force is directed along the rod’s axis, so the Lagrangian isSO(2)- symmetric. If we increase the force, the rod will eventually bend, and the symmetry is broken. The direction of the bend is completely arbitrary, and the Lagrangian is still SO(2)-symmetric, but the state of minimum energy is not.

For an example of SSB in a quantum field theory, we consider a complex Klein- Gordon (KG) fieldφwith the Lagrangian

L=∂µφ∂^µφ^∗−m²φφ^∗−1

4λφ²φ^∗2, (3.1)

where λ is a coupling constant. This Lagrangian has a global U(1) symmetry, it is invariant under the transformationφ→e^iαφ. In order for the potential to go to infinity when φ goes to infinity, we demand λ > 0. We assume m² < 0, which means that we cannot interpret −m²|φ|² as a mass term. To find the state of lowest energy, we must

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minimize the potential

V =m²|φ|²+λ

4|φ|⁴. (3.2)

By differentiating with respect to|φ|², we get

V⁰=m²+ 2λ|φ|², (3.3)

which means thatV has a minimum at

|φ|² =−m²

2λ ≡v². (3.4)

The phase ofφat minimum potential is arbitrary, and we see a direct analogy to the rod example: there is a continuum of states with minimum energy. One cannot guess the direction of the VEV beforehand, Nature simply realizes one. Without loss of generality, we assume the vacuum state to be entirely real, i.e. φ = v. We parametrize φ as fluctuations around the VEV with two real KG fieldsaand b,

φ=v+a+ib

√2 . (3.5)

Inserting Eq. (3.5) into Eq. (3.1), we get the Lagrangian L=−1

2m²v²+1

2∂_µa∂^µa+1

2∂_µb∂^µb+m²a²+ m²

√2va(a²+b²) + m²

8v²(a²+b²)². (3.6) Since m² <0, we see that the field ahas mass ma2 =−2m², while b is massless. This is becauseb describes vibrations along the line of constant potential going through the VEV. We call this a Goldstone boson, and this phenomenon was first described in [19].

Generally speaking, for any continuous symmetry that is spontaneously broken, there is a massless scalar for each of the broken generators of that symmetry. This is known as Goldstone’s theorem, and the idea is that each scalar represents a vibration which meets no resistance as there is no increase in the system’s potential energy.

3.2 Linear sigma model

We consider anO(N)-symmetric scalar φ⁴ theory, which has the Lagrangian L= 1

2∂_µφ∂^µφ−1

2µ²φ²−λ

4φ⁴ =L_kin−V , (3.7)

where µ is a mass parameter and λ is a dimensionless coupling constant. This is the linear sigma model. Like before,λ >0. φ has the parametrization

φ=σe^iT^j^φ^j^/v, (3.8)

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where v is the vacuum expectation value of φ1 ≡ σ, φj are the remaining fields with VEV= 0, and Tj are the broken generators of the symmetry, withj = 2, . . . , N. Using the same procedure as in the previous section, we find the nonzero VEV

v= r

−µ²

λ . (3.9)

We parametrizeσ as a fluctuation about the VEV,σ =v+ ˜σ. This yields the potential V =−µ²σ˜²−µ²

v σ˜³− 1 4

µ² v²σ˜⁴+1

4µ²v², (3.10)

where the last term is an additive constant we will henceforth ignore. The first term in Eq. (3.10) is the mass term of the fluctuation fieldσ, which shows that˜ m²_σ_˜ =−2µ². There are no other mass terms, so φ_j are all Goldstone bosons.

3.3 Nonlinear sigma model

We turn our attention to the kinetic term of the Lagrangian, reverting to the parametrization in Eq. (3.8),

1

2∂µφ∂^µφ= 1 2

∂µσ+iσT_j∂_µφ_j

v ∂^µσ−iσT_k∂_µφ_k v

= 1

2∂µσ∂^µσ+1 2σ²

T_j∂_µφ_j v

2

, (3.11) which leads us to our resulting Lagrangian

L= 1

2∂_µσ∂^µσ+ 1 2σ²

T_j∂_µφ_j v

2

+1

2λv²σ²−1

4λσ⁴, (3.12)

where we have used Eq. (3.9) and the relation m²_˜_σ =−2µ² to rewrite it in terms of λ.

Applying the Euler-Lagrange equations to the fieldσ returns its equation of motion,

∂µ

∂L

∂(∂µσ)

−∂L

∂σ = 0, (3.13)

∂_µ∂^µσ+

T_j∂_µφ_j v

2

−λv²σ+λσ³ = 0. (3.14)

We divide Eq. (3.14) by λand take the limitλ→ ∞ while keeping v constant to get

σ(σ²−v²) = 0, (3.15)

which has the solutions σ = 0 and σ =±v. This means that the field σ is in fact not dynamic, and σ˜ drops out of the Lagrangian. When we keep v constant while taking λ→ ∞, we also takem²_σ_˜ → ∞, so the physical meaning of this is that the system resides below the energy needed to create the σ˜ particle. Thus, since the entire potential V depends onσ˜ as shown in Eq. (3.10), we are left with only the kinetic term. In addition

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to this, ∂µσ = ∂µ˜σ = 0, so the Lagrangian will depend solely on the kinetics of the Goldstone bosons. This is the nonlinear sigma model. We reparametrizeφ,

φ= (σ,#»π), (3.16)

where #»π is a column vector of all N −1 Goldstone bosons previously parametrized as φj. In order to keep V = 0 in this new parametrization, π² = #»π · #»π must be constant.

Since we also know that σ is a constant, we can see from the ground state that σ =p

v²−π². (3.17)

In this representation,

∂µσ=− #»π ·∂µ#»π

√v²−π² , (3.18)

so our final Lagrangian is L= 1

2∂µφ∂^µφ= 1 2

|∂_µ#»π|²+ (#»π ·∂_µ#»π)² v²−π²

. (3.19)

Demandingπ²v², we expand the last term as a geometric series, L= 1

2

|∂_µ#»π|²+ 1

v²(#»π ·∂µ#»π)²+. . .

, (3.20)

from which we can derive our Feynman rules. We obtain p

πⁱ π^j = i

p²δ^ij, (3.21)

for the propagator, and

p₂ p₄

p3 p₁

π^j π^`

π^k πⁱ

=− i

v²[δîjδ^k`(p1+p2)·(p₃+p4)+δîkδ^j`(p1+p3)·(p₂+p4)+δî`δ^jk(p1+p4)·(p₂+p3)], (3.22) for the 4-vertex, where all momentap₁, . . . , p₄ are incoming.

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3.4 Nonlinear sigma model in 1+1 dimensions

Coleman showed that spontaneous breaking of continuous symmetries cannot occur in two dimensions, and hence there can be no Goldstone bosons [20]. Thus, we add a term c0σ =c0(v²−π²)^1/2 to the Lagrangian of Eq. (3.19) so that it isO(N −1)-symmetric from the outset. Expanding our new Lagrangian in powers of π², we get

L= 1 2

|∂_µ#»π|²−m²π²+g(#»π ·∂µ#»π)²−1 4gm²π⁴ +g²π²(#»π ·∂_µ#»π)²−1

8g²m²π⁶+. . .

+m²

g , (3.23)

where we recognize m²_π =m² ≡c₀/v as the mass of the #»π particles, and introduce the coupling constant g ≡ _v¹2. In order to derive our new Feynman rules, we must keep in mind that π⁴ = (π²)² = (#»π · #»π)(#»π · #»π). This yields 8 permutations of #»π through the interchange of parentheses or within each of the parentheses, cancelling the denominator of the π⁴ term. Thus, we get the propagator

p

πⁱ π^j = i

p²−m²+iδ^ij, (3.24)

and the 4-vertex

p₂ p4

p₃ p₁

π^j π^`

π^k πⁱ

=−ig

δîjδ^k`[−(p₁+p2)²+m²] +δîkδ^j`[−(p₁+p3)²+m²] +δî`δ^jk[−(p₁+p4)²+m²] , (3.25) where we have used conservation of momentum at the vertex to simplify the expression.

Now, the vertex we have calculated is actually the four-point function of four incoming particles, including every possible permutation that connects the particles. There are 4! = 24 such permutations, so the "bare" vertex to be used in other diagrams includes a factor 1/24 that will be multiplied with new combinatorial factors, depending on the diagram.

These Feynman rules, as well as the newly calculated six-point function, are summa- rized in Appendix C, for convenience.

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Chapter 4

Renormalization and vacuum integrals

4.1 Dimensional regularization

When calculating loop diagrams, we regularly come across divergent integrals. The most common one in this thesis is

I₁=

Z d²p (2π)²

i

p²−m²+i, (4.1)

where the integral is over two-dimensional Minkowski spacetime, and the Feynman pre- scription i is to avoid the poles at p² = m². We evaluate this type of integral by transitioning to Euclidean space using what is called a Wick rotation. Here, we let p⁰ →ip⁰, so that p² = (p⁰)²−p² → −(p⁰)²−p² =−p²_E, wherep²_E is the square of the Euclidean momentum. We see in Figure 4.1 that this lets us avoid the poles altogether, and hence we drop the term i in the propagator. As we will always perform a Wick rotation before evaluating these integrals, we drop the term i even when expressing them in Minkowski spacetime. Additionally, we will introduce another parameterlater in the chapter, and keeping this one would only add confusion. Thus, we have

I₁ =

Z d²p (2π)²

i p²−m² =

Z d²p_E (2π)²

1

p²_E+m² , (4.2)

where the right-hand side is over Euclidean space. We consider the Euclidean integral and drop the subscript E. Using Eq. (B.7), we get

I₁ =

Z d²p (2π)²

1

p²+m² = 2 4π

Z ∞ 0

p dp

p²+m² , (4.3)

This integral obviously diverges at large momenta, this is called an ultraviolet (UV) divergence. We need a way to regularize this divergence. The simplest way is to introduce

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Im(p0)

Re(p0) pp²+m²−i

−p

p²+m²+i

Figure 4.1: A Wick rotation, with the location of the poles marked by crosses. We see that the integral over the contour is zero, as it does not contain any poles. Thus, the integral along the imaginary axis and the integral along the real axis are equal and opposite.

a cutoff momentum Λ, where we integrate up to the cutoff rather than infinity. The calculations will then hold up to this energy. Using cutoff regularization, we obtain

I₁ = 1 4π ln

Λ² m² + 1

, (4.4)

and we see that the integral goes to infinity asΛ→ ∞. Evaluating the same integral in d= 4 dimensions, we have

I1=

Z d⁴p (2π)⁴

1

p²+m² = 2 (4π)²

Z ∞ 0

p³dp

p²+m², (4.5)

and we get

I₁(d= 4) = 1 (4π)²

Λ²−ln Λ²

m² + 1

, (4.6)

by using cutoff regularization. Looking at Eqs. (4.4) and (4.6), we see that power counting is a useful tool to estimate the degree of divergence. Integrals that are ∼ ^dp_p in the large momentum limit pm are logarithmically divergent, integrals ∼dp p are quadradically divergent, and so on. This nomenclature is used even in regularization methods that do not use a cutoff, and will be useful for a later proof in dimensional regularization.

Thermodynamics of the nonlinear sigma model

Master ’s thesis

Erik Livermore

Thermodynamics of the nonlinear sigma model

Perturbation theory in the large-N approximation

Erik Livermore

Thermodynamics of the nonlinear sigma model

Perturbation theory in the large-N approximation

Preface

Contents

Notation and conventions

Chapter 1

Introduction

1.1 Motivation

1.2 Approach

1.3 Structure of the thesis

Chapter 2

Functional integral formulation of the partition function

2.1 Quantum mechanical partition function

2.2 Transition amplitude in functional integral formulation

2.3 Partition function for bosons

Chapter 3

The nonlinear sigma model

3.1 Spontaneous symmetry breaking

3.2 Linear sigma model

3.3 Nonlinear sigma model

3.4 Nonlinear sigma model in 1+1 dimensions

Chapter 4

Renormalization and vacuum integrals

4.1 Dimensional regularization