Functional renormalization group approach to ultracold fermions

Functional renormalization groupapproach to ultracold fermions

冷却フェルミオンに対する汎関数繰り込み群による解析

Yuya Tanizaki

Department of Physics, The University of Tokyo

Theoretical Research Division, Nishina Center, RIKEN

January 2013

Master Thesis

Abstract

In this thesis, we study properties of many-body fermionic systems in a nonperturbative methodof quantum field theories. Especially we will concentrate on thermodynamic properties of theBCS-BEC crossover and those of dipolar fermionic systems. These systems are recently studiedfrom both the experimental and theoretical aspects, and thus we need to invent a theory todescribe them quantitatively. For this purpose, we use the functional renormalization group(FRG) method and develop it in fermionic systems. It is characteristic to this study discussingsuperfluid phase transitions without introducing Hubbard-Stratonovich fields, and it enables usto study those systems in a less biased and nonperturbative way. As a preparation for studyingthe BCS-BEC crossover, we discuss the vacuum physics in detail, especially the atom-dimerscattering problem. We also derive some formulae to estimate the number density of fermionicatoms, and rederive the Thouless criterion using the Ward-Takahashi identity. Applying FRGto the BCS-BEC crossover, we have reproduced the NSR theory in a systematic analysis anddiscussed possible corrections to it. In dipolar fermionic systems, we established the LandauFermi liquid theory with the dipole-dipole interaction using FRG and revealed that the systemshows 3P1 superfluid at sufficiently low temperatures even in weakly coupling regions.

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Table of Contents

Title . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

1 Introduction 1

2 Functional Renormalization Group 72.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Wetterich formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.1 Derivation of the Wetterich equation . . . . . . . . . . . . . . . . . . . . 112.2.2 The flow of correlation functions . . . . . . . . . . . . . . . . . . . . . . . 142.2.3 Renormalization group flows . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3 Polchinski’s formulation of FRG . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.3.1 Relation between the Wilsonian and 1PI effective actions . . . . . . . . . 202.3.2 The Polchinski equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.3.3 Wick ordered formulation of the FRG . . . . . . . . . . . . . . . . . . . . 24

2.4 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.4.1 Optimization criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.4.2 Properties of the optimized regulator . . . . . . . . . . . . . . . . . . . . 292.4.3 Example: the Litim regulator in LPA . . . . . . . . . . . . . . . . . . . . 30

2.5 Properties of flow equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.5.1 Rescaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.5.2 Fixed points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3 FRG for many-body fermions 373.1 Notations and setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.1.1 Possible forms of the effective action . . . . . . . . . . . . . . . . . . . . 393.2 RG flow equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.3 Fermi liquid theory from the RG viewpoint . . . . . . . . . . . . . . . . . . . . . 43

3.3.1 Tree-level RG analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.3.2 1-loop analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.4 Open problems to be considered . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4 BCS-BEC crossover 514.1 Introduction to the BCS-BEC crossover . . . . . . . . . . . . . . . . . . . . . . . 524.2 Scattering problems in the vacuum . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.2.1 Two-body scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.2.2 Three-body sector: atom-dimer scattering . . . . . . . . . . . . . . . . . 57

4.3 Number equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

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4.4 Thouless criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644.4.1 Ward-Takahashi identity . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.4.2 Spontaneous symmetry breaking . . . . . . . . . . . . . . . . . . . . . . . 66

4.5 Nozieres-Schmitt-Rink theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684.5.1 Deep BCS regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.5.2 Deep BEC regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5 Application of FRG to dipolar fermionic systems 795.1 Ultracold atomic systems with the DDI . . . . . . . . . . . . . . . . . . . . . . . 795.2 Basics of the DDI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 815.3 RG study within the RPA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5.3.1 Study of the Landau channel . . . . . . . . . . . . . . . . . . . . . . . . . 855.3.2 Study of the BCS channel . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

6 Summary and perspectives 93

A Quick derivation of the 1-loop RG expressions 95A.1 Feynman rules for the flow of vertex functions . . . . . . . . . . . . . . . . . . . 95A.2 Feynman rules for the flow of composite operators . . . . . . . . . . . . . . . . . 96

B Continuum limits 99B.1 Continuum limits of quantum field theories . . . . . . . . . . . . . . . . . . . . . 99B.2 Perturbative renormalizability of the φ4

4-theory . . . . . . . . . . . . . . . . . . . 103

C Angular momenta 107C.1 Clebsch-Gordan coefficients and spherical tensors . . . . . . . . . . . . . . . . . 107C.2 6j symbols and 9j symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110C.3 List of formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

D Properties of ΓSk in the vacuum 113D.1 Calculation of ΓSk (P 0, 0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113D.2 Spatial momentum dependence of ΓSk . . . . . . . . . . . . . . . . . . . . . . . . 115D.3 Numerical approximate solution of atom-dimer scattering . . . . . . . . . . . . . 118

E Channel decomposition of the dipole-dipole interaction 121E.1 DDI in the spherical coordinate . . . . . . . . . . . . . . . . . . . . . . . . . . . 121E.2 Particle-particle channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123E.3 Channel decomposition of the contact interaction . . . . . . . . . . . . . . . . . 128

Acknowledgments 129

References 131

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Chapter 1

Introduction

In this thesis, I will discuss many-body problems of non-relativistic fermionic theories usingthe functional renormalization group (FRG). Before going into the details, I would like to givebrief introductions in order to motivate us to have interests in these topics and to study themvia nonperturbative methods of quantum field theory (QFT). I hope that I can tell not onlyhow interesting physics these topics contain but how wide range of physics shares the similarkinds of problems.

About QFT

QFT is now a standard tool for broad areas of physics, especially for condensed matter physics,nuclear physics, and particle physics. In the case of condensed matter physics, suppose that wewould like to reveal thermodynamic properties of matter. With usual quantum and statisticalmechanics we must solve the Schrodinger equation with too many variables. In many interestingcases, we must cease to expect that we can solve such problems in this way. In the case ofparticle physics, we also encounter the problem which may seem to be more fundamental. Ifwe require the relativistic description of quantum mechanics, antiparticles should appear andthus processes with annihilation and with creation of particles appear from this fundamentalrequirement [1,2]. Historically this fact posed some paradoxes when one interpreted relativisticwavefunction as a single-particle one (e.g. see the Klein paradox [3, 4]).

Surprisingly, answers for these questions posed in different contexts are the same: We shoulduse the field description of quantum physics, i.e. QFT. In my understandings, we must relyon the renormalization group (RG) and its application to effective field theories in order tounderstand why the same method applies to such different situations. In QFT, every physicalquantity is written in terms of fields, and a particle is regarded just as a cluster of a given energyand momentum. Only when the perturbation theory in QFT works well, the elementary fieldis composed of creation and annihilation operators of the corresponding particle, which itself isnot required from any principles of QFT. Indeed, the ergodic or stable property of the vacuum,or the cluster decomposition property, allows us to introduce creation/annihilation operatorsfor particles [1, 2, 5], but they are not needed for the elementary ones.

As long as low-energy physics is concerned, short-distance behavior is washed out andphysics can be described with an effective theory related to the original one via RG transfor-mations [6]. That is, we can expect universality when physical scales such as the correlationlength are much larger than microscopic scales. At this stage, we already need not require thenaive renormalizability with power counting, and the only important thing is that the effectivecouplings are controlled under coarse-graining with the aid of the RG flow.

1

2 Chapter 1. Introduction

If we use this idea to the get an effective field theory, we at first have to write down anypossible terms allowed by the symmetry with given fields, and then calculate Feynman diagramswith a given cutoff. In this case, the bare Lagrangian does not take any simple forms and itcontains many non-renormalizable terms. This was pointed out by S. Weinberg [7] and it wasused to describe soft pions without current algebra. The methods and ideas of an effective fieldtheory are now widely used.

These experiences tell us an important lesson: Even if the high-energy theory may not besuitably described with QFT, QFT is still a powerful tool to describe low energy physics andprovides a systematic expansion in terms of the energy under the name of RG. To see this, letus explain some classic examples of effective field theories taken from condensed matter physicsand from particle physics.

At first, let us consider the theory of superconductivity. In many metals, it is well describedwith the Bardeen-Cooper-Schrieffer (BCS) theory [8], that is, the electrons are weakly attractedvia phonon exchange and it leads to the Cooper instability of the Fermi surface at sufficiently lowtemperatures. However, we must point out that the attractive interaction energy via phononexchange is typically of the order of 1meV and that it is much weaker than the Coulombrepulsion which energy scale is about the order of 10eV. At this point, the notion of effectivefield theories is very important. After coarse graining of the full theory containing the bareCoulomb repulsion, we can obtain the Landau Fermi liquid theory as its low-energy effectivetheory. Its typical cutoff scale is given by the Debye temperature, and for such low energyprocesses, the Coulomb repulsion is screened and weaken. As a result, the phonon attraction andthe effective Coulomb repulsion can compete. When the total interaction becomes attractive,the condensation of the Cooper pairs is formed.

In particle physics, theory of the Fermi weak interaction is regarded as a low energy effectivedescription of the Weinberg-Salam theory. Its typical cutoff scale is given, for example, by theW -boson mass mW ' 80GeV. Above that scale, the Fermi weak interaction theory breaksdown and many important properties such as the unitarity does not hold. As long as we onlyconsider low energy phenomena such as β-decay, then the Fermi four-point coupling gives aconvenient description.

Even for the standard model of particle physics, we believe that it should be a low-energyeffective description of some UV complete theory. If we adopt some grand unification scenario,the standard model with the gauge group SU(3) × SU(2) × U(1)Y is an effective theory witha cutoff scale about ΛGUT ' 1016GeV. Since our usual experiments search physics up to about1TeV, those two energy scales are well-separated. Therefore, as long as we believe that theperturbation theory works well in between those energy scales, the non-renormalizable couplingsare suppressed with a factor 1TeV/ΛGUT ' 10−13 1. It explains why we can describe particlephysics using a renormalizable theory, and as a result we can conclude that the baryon numberconservation, for example, is a good conservation law as a result of an accidental symmetrysince it can be violated only through higher-dimensional operators.

About FRG

In this way, the notion of RG pioneered by K. G. Wilson is applicable widely, and the descriptionwith effective field theories provides a systematic expansion in terms of energy scales. In thecontext of gauge theories, the inconsistency between the gauge symmetry and separation ofenergy scale poses a serious problem, and the nonperturbative and practical realization of theWilsonian RG is still open in this area. On the other hand, in condensed matter physics such

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problem associated with the non-linear symmetry transformation is often absent. The FRG isa realization of the Wilsonian RG in a suitable form for practical computations.

In 1984, J. Polchinski gave a version of the FRG in order to give a simple proof of perturba-tive renormalizability of the φ4

4-theory [9]. He separated the propagators into low-energy modesand high-energy modes, and derived a differential equation of RG transformations by graduallyintegrating out high-energy modes. At each step, the modes lying only in the infinitesimalshell of Euclidean momenta are integrated out, and then the differential equation consists onlyof tree and 1-loop diagrams. Of course, the diagrammatic expansion of correlation functionssolves the RG differential equation, but those Feynman diagrams has forest structures [10].Therefore, the subtraction of their subdivergences is necessary for the renormalization, and theproof of Dyson’s counting rule becomes awful, because we have to take care about symmetryfactors for correct cancellations of each subdivergence [10]. If we use the RG differential equa-tion given by J. Polchinski, we are free from those subdivergences and the proof of perturbativerenormalizability is given with a simple mathematical induction.

After about a decade, a suitable form of FRG for practical applications was established.This is called the Wetterich formalism, and it gives RG transformations for the 1PI effectiveaction [11–13]. If the tree structure appears in the vertex function, some drastic momentumdependence can appear, because if the sum of the external momenta exceeds the cutoff itsuddenly flows into the internal tree leg [14]. Since the resultant vertex function sticks to the1PI structure in the Wetterich formalism, such drastic change would not appear and it is easierto treat in practical applications. At the end of the RG flow, it indeed gives the 1PI effectiveaction of the original theory.

In a bosonic system, local potential approximation (LPA), which is the lowest order ap-proximation in the derivative expansion, already gives a powerful tool to study nonperturbativephysics. In this case, the notion of the optimized regulator is well established and also manyother techniques are developed [15,16]. On the other hand, in the case of fermionic field theoriesthere are many problems associated with the Fermi surface even in the weak coupling region.Technically, due to the anticommuting behavior of the fermionic fields the naive applicationof LPA is meaningless and it makes the nonperturbative problem of fermionic theories moredifficult [17]. In the case of strong coupling systems, the identification of correct low-energydegrees of freedom is a difficult problem and in most cases we need to put some ansatz to treatsuch systems. This is true also for the bosonic theories. Indeed, the formation of a bound state,for example, changes the scaling behavior drastically and the counting rule of the expansionwill be totally changed from that in the weak coupling region.

Problems tackled in this thesis

In this thesis we will attack problems of the Bardeen-Cooper-Schrieffer to Bose-Einstein conden-sate (BCS-BEC) crossover and of dipolar Fermi systems. Both of them now become importanttopics in the field of ultracold atoms. I will give an introductory review for each topic at thebeginning of the appropriate chapter. Here I would like to discuss why they are interesting andalso why they are difficult from a theoretical point of view.

At first, let us consider the BCS-BEC crossover. Imagine that we have a many-body two-component fermionic system with a short-range attractive interaction. If the coupling is veryweak, the system shows superfluidity described by the BCS theory. On the other hand, ifthe attraction is very strong, fermionic atoms form a bound state (often called a dimer) andthe ground state is a superfluid with the Bose-Einstein condensation (BEC) of those bosonic


dimers. If we change the coupling in between those two limits, the ground state changessmoothly without phase transitions and this is called the BCS-BEC crossover. This system isnow experimentally accessible thanks to developments of ultracold atomic experiments.

The BCS-BEC crossover gives a unified description of two different kinds of superfluidity,which provides important theoretical challenges. If we could give a clear and quantitatively goodprescription for this problem, we would acquire a new insight for the superfluid phase transitionespecially for the strongly coupled systems. We can expect that we will learn very importantlessons which may open ways for understanding of high-temperature superconductivity and forother strongly coupled systems associated with phase transitions such as the chiral symmetrybreaking in QCD. However, the quantitative description of the BCS-BEC crossover remains tobe intractable, although the system is defined with a very simple classical action.

We will analyze the BCS-BEC crossover applying FRG within a purely fermionic framework.That is, we do not introduce auxiliary bosonic fields using the Hubbard-Stratonovich transfor-mation [18, 19] in order to describe Cooper pairs in the BCS regime and bosonic molecules inthe BEC regime. Of course, the Hubbard-Stratonovich transformation provides a powerful toolto study nonperturbative physics when we already know the correct channel of instabilities,since it describes the condensates already in the mean field approximation. However, it alwayscontains some ambiguities associated with the Fierz transformation [20, 21] so that differentHubbard-Stratonovich transformations give different results within the mean field approxima-tion. and for more difficult systems the correct way to introduce Hubbard-Stratonovich fieldsis often nontrivial. We shall consider the BCS-BEC crossover with a less biased method andsee how instabilities emerge in the purely fermionic language using FRG.

Next we will consider a dipolar fermionic system. Compared with the BCS-BEC crossover,this system contains many open problems, and due to the complexities of interactions, wecan expect the occurrence of many unprecedented phenomena. The dipole-dipole interactionis anisotropic and long-ranged interaction, so scattering processes with higher harmonics willbecome dominant even if we consider low energy phenomena.

Furthermore, what is interesting is that those systems may be helpful to understand prop-erties of high density nuclear matter. In a normal nuclear matter, the interparticle distance isvery long and anisotropic natures of nucleon-nucleon interactions does not affect many-bodyproperties. However, nucleon-nucleon interactions contain anisotropy coming from the LS in-teraction, tensor force of pion exchanges, and so on. In high density nuclear matters, theseinteractions play an important role and the mean field calculation predict interesting physicssuch as neutron 3P2 superfluid, meson-condensation, etc. Those phenomena are considered tooccur inside neutron stars and to affect cooling mechanism. It is still difficult to observe suchphenomena but some counterparts in cold atoms may give some insights in a near future.

Composition of the thesis

Before closing the introduction, let us comment on composition of this thesis.In chap.2, we will review the method of FRG. At first, we introduce the language of field

theories using functional integrations, and we derive the Wetterich formalism, which will beused in this thesis for practical computations of 1PI effective actions. To get a physical insightof FRG, we compare it with another version of FRG proposed by J. Polchinski.

In chap.3, we introduce notations for fermionic systems and calculate some analytic expres-sions of the RG flow, which will be frequently used later. We also review the Landau Fermiliquid theory from the perspective of FRG, and understand it as an IR effective theory of many

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body fermionic system. The idea established here will play an important role, especially in theanalysis of the BCS regime in the BCS-BEC crossover and in the analysis of dipolar fermions.We also marshal problems to be considered in following chapters from lessons learned throughthe RG flow of Landau Fermi liquid theory.

In chap.4, we attack the BCS-BEC crossover using FRG with purely fermionic description.Starting from scattering problem of the model in the vacuum, we consider superfluid phasetransition using the RG flow in normal phases. We derive the number equation and the Thoulesscriterion from the RG point of view, and consider possible corrections to the Nozieres-Schmitt-Rink theory in a systematic way.

In chap.5, we discuss possible instabilities of dipolar fermionic systems using the scalingansatz of the Landau Fermi liquid theory. We will find that the ansatz is justified in weakcoupling regions and that 3P1 superfluid is predicted from our calculations.

We summarize results in chap.6, and discuss some future perspectives there.The rest of the thesis is composed of appendices. In Appendix A, we denote some computa-

tional techniques to derive diagrammatic expressions of the RG flow. We also discuss the wayto compute composite operators there. In Appendix B, we review the proof of perturbativerenormalizability using FRG, which was originally proposed by J. Polchinski. It will makethe connection between renormalizability of quantum field theories and fixed points clear. InAppendix C, we review the theory of angular momenta in order to fix its notations and conven-tions. It is used in calculations of the dipole-dipole interaction and we also list some convenientformulae. In Appendices D, E, we show some formula used in this thesis. In Appendix D, wecalculate some properties of the effective four point coupling in the vacuum, and they are usedfor the analysis in chap.4. In Appendix E, we can see details of calculations of the interactionmatrix of the dipole-dipole interaction which is used in chap.5.


Chapter 2

Functional Renormalization Group

In this chapter, we will review and explain the functional renormalization group (FRG), whichwill be applied to many-body fermionic systems in latter chapters. This is one of the concreterealizations of the renormalization group introduced by K. G. Wilson [6], which was originallyinvented for clear description of critical phenomena. After about a decade, J. Polchinski [9] gaveanother exact formulation of the renormalization group in simplifying the proof of perturbativerenormalizability of the scalar φ4 theory in 4-dimensional spacetime. The starting point ofFRG is based on exact relations among generating functionals of Green functions, and thereare several and equivalent versions of FRG depending on the choice of types of Green functions.In this chapter, we will see two types of flow equations of FRG: the Wetterich formalism forone-particle irreducible (1PI) vertices (sec.2.2), and Polchinski’s equation for the Wilsonianeffective action (sec.2.3). In latter chapters we will only use the Wetterich formalism of FRG,however seeing the relationship with other formulations will give us deeper insights for FRG.

In sec.2.4, we will give a discussion on the choice of regulators, which is an unphysicaldegree of freedom introduced in FRG in order to divide scales of the physics. In principle, theresults do not depend on the choice of regulators since we remove it at the end of calculations.However, in practice some approximations are necessary in solving the physical systems, andthose approximations can break down the independence of this unphysical degree of freedomand cause a delicate problem for the choice of regulators.

In sec.2.5, we discuss general properties of exact solutions of the RG flow equation. Thenotion of fixed points becomes important for this purpose, and the rescaling operation is intro-duced as a convenient tool.

2.1 Preliminaries

At first, we give the definition of notations and review basic properties of generating functionals.Let us denote the elementary fields by φn, where the label n represents the index (x, α) coveringthe space-time coordinate x and the flavor index α. For a functional F [φ] of elementary fieldsφn, F ,n represents the left functional derivative:

F ,n[φ] =δLδφn

F [φ], (2.1.1)

where the left and right derivatives in terms of fields φn are defined by

F [φ+ δφ]− F [φ] = δφnδLF [φ]

δφn=δRF [φ]

δφnδφn. (2.1.2)

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8 Chapter 2. Functional Renormalization Group

For two fields An and Bm, we use Einstein’s convention for the integral

AnBn =

∫dDx

∑α

Aα(x)Bα(x). (2.1.3)

Let S[φ] be an action which is a bosonic local functional of the elementary fields φn, andfor each φn we introduce the source Jn which has the same statistics with that of φn. Whenwe need to distinguish the statistics of fields, φok and φ1

l represent bosonic and fermionic fields,respectively, and corresponding sources are written as Jko and J l1. Therefore, Jnφn = Jko φ

ok+J

l1φ

1l

is a bosonic quantity and we define the generating functional W [J ] of connected Green functionsas follows:

Definition 2.1.1 (Schwinger functional). We define the generating functional for connectedGreen functions W [J ] by

exp (W [J ]) =

∫Dφ exp (−S[φ] + Jnφn) . (2.1.4)

W [J ] is called the Schwinger functional, and its moments are called Schwinger functions. Herewe are not interested in the normalization of the right hand side of Eq.(2.1.4), because thechange of the normalization only affects to add the field independent term in W [J ].

Connected Green functions are given by

〈φn1 · · ·φn`〉c = W,n1···n` [J ] =δLδJn1 · · ·

δLδJn`

W [J ]. (2.1.5)

Now we would like to introduce the one-particle-irreducible (1PI) effective action Γ[ϕ] as theLegendre transformation of W [J ]. However, in order that the definition of the 1PI effectiveaction makes sense, the convexity of the generating functional W [J ] is necessary and sufficient,and thus we need to check it at first.

Let us make a more precise and concrete statement. Let us define a functional ϕn[J ] by

ϕn[J ] = W,n[J ] = 〈φn〉c[J ]. (2.1.6)

According to a general theory of the superspace, the existence of the inverse function Jn[ϕ] is

ensured if the matrix of derivativesδLδJm

ϕn[J ] is invertible. Since

δLδJm

ϕn[J ] = W,mn[J ] = 〈φmφn〉c[J ], (2.1.7)

the existence of the inverse function Jn[ϕ] is nothing but the existence of the inverse Greenfunctions with any external fields J . Since any functions are expanded in a formal power seriesin terms of Grassmannian variables, what we need to do is to check existence of the inverse ofpropagators 〈φmφn〉c[Jo], where Jo is a bosonic external source.

Now let us see details about the matrix 〈φmφn〉c[Jo]. At first, we can immediately noticethat the matrix is block diagonal in the sense that 〈φonφ1

m〉c = 〈φ1nφ

om〉 = 0 since J1 = 0, and

then

〈φnφm〉c[Jo] =

(〈φonφom〉c[Jo] 0

0 〈φ1nφ

1m〉c[Jo]

). (2.1.8)

2.1. Preliminaries 9

Clearly, the boson sector 〈φonφom〉c is a symmetric matrix and the fermion sector 〈φ1nφ

1m〉c is a

skew-symmetric matrix.In the case of a theory only with bosons, we can prove the convexity of the effective action

[22] with a simple calculation.

Theorem 2.1.1 (Convexity of the Schwinger functional). Let us consider a theory onlywith bosons 1 described by the classical action S[φo]. Then the eigenvalues of the matrix〈φonφom〉c[Jo] are all positive, which may be infinite.

Proof. Let ρn be a real vector, then one can easily find that

ρnρm〈φonφom〉c[Jo] =

∫DφoDφ′o(ρn(φon − φ′

on))2e−S[φo]+Jo·φoe−S[φ′o]+Jo·φ′o∫

DφoDφ′oe−S[φo]+Jo·φoe−S[φ′o]+Jo·φ′o, (2.1.9)

and the right hand side of the equation is clearly positive. Since the matrix 〈φonφom〉c[Jo] issymmetric, we find that all the eigenvalues are positive.

Theorem 2.1.1 ensures that W [J ] is convex and then its Legendre transformation can beperformed. For the case of theories with fermions, we here simply assume 2 that the inversefunction of ϕn[J ] exists globally, which is denoted by Jn[ϕ]. Then the 1PI effective action isdefined:

Definition 2.1.2 (The 1PI effective action). The 1PI effective action Γ[ϕ] is defined as

Γ[ϕ] = Jn[ϕ]ϕn −W [J [ϕ]], (2.1.10)

where Jn[ϕ] is the inverse function of W,n[J ] = ϕn.

As a consequence of properties of the Legendre transformation, the following can be con-firmed immediately:

Theorem 2.1.2. We can get W [J ] as the inverse Legendre transformation of Γ[ϕ]:

W [J ] = Jnϕn[J ]− Γ[ϕ[J ]], (2.1.11)

whereδRΓ

δϕn[ϕ[J ]] = Jn. Therefore, we can find the relations

δ2LW [J ]

δJnδJm=δLϕm[J ]

δJn,δLδRΓ[ϕ]

δϕnδϕm=δLJ

m[ϕ]

δϕn. (2.1.12)

Therefore, using Eqs.(2.1.12) and the Leibniz rule, we get the two relations

δ2LW [J ]

δJnδJ lδLδRΓ[ϕ]

δϕlδϕm= δmn ,

δLδRΓ[ϕ]

δϕnδϕl

δ2LW [J ]

δJ lδJm= δnm. (2.1.13)

1The Yang-Mills theory is not in the scope of this theorem, since it essentially contains F.P. ghosts. Sincethe Faddeev-Popov determinant can change its sign, the proof given here does not hold. Therefore, in the gaugetheories, the more careful treatment of the path integral becomes important and this would be related to theGribov problem.

2To my best knowledge, the proof for general cases does not exit. Since the positivity of two-point Greenfunctions is related to well-defined field propagation, the Osterwalder-Schrader reflection positivity [5, 23] maysolve this problem.


Therefore, the quantity

Γ(2)nm[ϕ] =δLδϕn

δRδϕm

Γ[ϕ] (2.1.14)

is the inverse of full propagators 〈φnφm〉c with the source J [ϕ].In the perturbative expansion, Γ[ϕ] is the generating functional of 1PI vertices. This fact

can be seen via the tree expansion of connected Feynman diagrams with full propagators and1PI vertices. It can be proven in a direct way, however we here prove it following the discussionin chap.16 of Weinberg’s textbook [2].

Proof. Let us define the generating functional of connected Green functions W [J ] with 1PIvertices by

exp1

~

(W [J ]

)=

∫Dϕ exp

1

~(−Γ[ϕ] + Jnϕn) . (2.1.15)

Here we introduce the loop-counting parameter ~, and expand the generating functional W [J ]in the power series of ~:

W [J ] =∞∑n=0

~nW (n)[J ]. (2.1.16)

Now, let us take the limit ~→ 0, then the steepest decent method applied to the path integralin Eq.(2.1.15) becomes precise. Therefore, we find that

W (0)[J ] = Jnϕn[J ]− Γ[ϕ[J ]], (2.1.17)

withδRΓ

δϕn[ϕ[J ]] = Jn. This is the inverse Legendre transformation (2.1.11) so we find that

W [J ] = W (0)[J ]. Connected Green functions are given by the tree-level expansion with fullpropagators and 1PI vertices.

The following theorem turns out to be useful.

Theorem 2.1.3. Let us consider an infinitesimal change ∆W [J ] in the generating functional.Then the corresponding infinitesimal change of the 1PI effective action ∆Γ[ϕ] satisfies

∆Γ[ϕ] = −∆W [J [ϕ]]. (2.1.18)

Proof. We define the modified external field J ′[ϕ] by

δL(W + ∆W )

δJn

∣∣∣∣J ′[ϕ]

= ϕn.

Then the Legendre transformation is given by

(Γ + ∆Γ)[ϕ] = J ′n[ϕ]ϕn − (W + ∆W )[J ′[ϕ]].

Then, setting ∆J = J ′[ϕ]− J [ϕ], we get

∆Γ[ϕ] = ∆Jnϕn −

(∆Jn

δLW

δJn

∣∣∣∣J [ϕ]

+ ∆W [J [ϕ]]

)= −∆W [J [ϕ]],

which is the desired result.

2.2. Wetterich formalism 11

2.2 Wetterich formalism

Our purpose in this section is to derive the flow equation for the one-particle-irreducible (1PI)effective action [11–13], which is called Wetterich’s formulation of FRG. Here we use the pathintegral formalism developed in sec.2.1 to describe Euclidean QFT.

2.2.1 Derivation of the Wetterich equation

Let S[φ] be a classical action. Now let us consider the action with the infrared regulator Rk:

S[φ] +1

2φ ·Rk · φ ≡ S[φ] +

1

2Rnmk φnφm. (2.2.1)

Here the label k of the infrared regulator Rk describes scale of infrared regularization. Weassume that the regulator Rk be bosonic, so bosons and fermions do not mix through thisterm. In the boson sector Rnm

k = Rmnk , and in the fermion sector Rnm

k = −Rmnk .

Let us define the Schwinger functional Wk of the action S with the regulator Rk by

exp (Wk[J ]) =

∫Dφ exp

(−S[φ]− 1

2φ ·Rk · φ+ J · φ

). (2.2.2)

In the limit Rk → 0, we can find that Wk[J ] converges to the Schwinger functional W [J ] of theoriginal theory.

Definition 2.2.1 (The flowing 1PI effective action). We define the flowing 1PI effectiveaction Γk[ϕ] as the Legendre transform of Wk[J ] in the sense that

Γk[ϕ] +1

2ϕRkϕ = Jn[ϕ]ϕn −Wk[J [ϕ]], (2.2.3)

where J [ϕ] is determined by the condition

δLWk

δJn

∣∣∣∣J [ϕ]

= ϕn. (2.2.4)

In the definition of the 1PI effective action (2.2.3), we have subtracted the trivial contribu-tion for the two point vertex function from the regulator. In the limit of removing the regulatorRk → 0, this trivial contribution vanishes and Wk[J ]→ W [J ], and then Γk[ϕ] becomes the 1PIeffective action Γ[ϕ] of the classical action S[φ] in this limit. Furthermore, it should be noticedthat the quantity Γk[ϕ] has a nice property also in the limit Rk →∞:

Theorem 2.2.1. Let us consider the limit Rk → ∞. Then the 1PI effective action Γk[ϕ]converges to the classical action S[ϕ].

Proof. Notice that the inverse function Jn[ϕ] is given as Jn[ϕ] =δRΓkδϕn

+ ϕmRmnk . Using

definitions (2.2.2) and (2.2.3), we can find that

exp (−Γk[ϕ]) =

∫Dφ exp

(−S[φ]− 1

2φRkφ+ Jn[ϕ](φn − ϕn) +

1

2ϕRkϕ

)=

∫Dφ exp

(−S[φ+ ϕ] +

δRΓk[ϕ]

δϕnφn −

1

2φRkφ

).

In the limit Rk → ∞, the saddle point approximation becomes accurate and then Γk[ϕ] →S[ϕ].


Now let us derive the Wetterich equation which is the flow equation for the 1PI effectiveaction Γk[ϕ]. Let us consider infinitesimal change δRk of the regulator Rk, then it induceschange of the Schwinger functional Wk[J ] given by

δWk[J ] expWk[J ] =

∫Dφ(−1

2

δLδJ· δRk ·

δLδJ

)exp

(−S[φ]− 1

2φ ·Rk · φ+ J · φ

)= −1

2

δLδJ· δRk ·

δLδJ

exp(Wk[J ]). (2.2.5)

Therefore, we can find the expression for infinitesimal change δWk[J ] in terms of the Schwingerfunctional Wk[J ]:

δWk[J ] = −1

2δRnm

k

(δLWk

δJnδLWk

δJm+

δ2LWk

δJnδJm

). (2.2.6)

Theorem 2.2.2. Under infinitesimal change δRk of the regulator, the 1PI effective action Γk[ϕ]changes by

δΓk[ϕ] =1

2STr

[δRk

(Γ

(2)k [ϕ] +Rk

)−1]. (2.2.7)

The super-trace represents the ordinary trace of the boson sector subtracted by the ordinarytrace of the fermion sector. Here we have defined that

Γ(2)nmk [ϕ] =

δLδRΓk[ϕ]

δϕnδϕm. (2.2.8)

Proof. Using the property (2.1.18) of the Legendre transformation, we have

δΓk[ϕ] +1

2δRnm

k ϕnϕm = −δWk[J [ϕ]].

Then Eq.(2.2.4) and Eq.(2.2.6) give

δΓk[ϕ] =1

2δRnm

k

δ2LWk[J ]

δJnδJm

∣∣∣∣J [ϕ]

=1

2STr

[δRk(Γ

(2)k [ϕ] +Rk)

−1].

Here, we have used Eq.(2.1.13) and the fact that the factor (−1) appears in the fermion loopdue to statistics.

We express this equation diagrammatically as

δΓk =1

2

δRk

(Γ(2)k +Rk)

−1

, (2.2.9)

where the shaded blob represents the two-point vertex for infinitesimal change of the IRregulator δRk and the line represents the field dependent propagator (Γ

(2)k [ϕ] + Rk)

−1. If weassume that we can expand the 1PI effective action Γk[ϕ] in terms of the field variables ϕ, wecan obtain the coupled ordinary differential equation for the vertex functions.

From now on, we have discussed change of the 1PI effective action Γk[ϕ] or the Schwingerfunctional Wk[J ] under general infinitesimal change of the regulator Rk. Now, let us consider


the change of the regulator parametrized by infrared scale k ∈ [0,Λo], where Λo represents theultraviolet (UV) cutoff 3 of the theory described by the classical action S[φ]. This producesthe 1-parameter flow of the 1PI effective action Γk[ϕ]. In order for desirable properties of theflow, we require the following properties of the regulator: In the Fourier space, let us write theregulating term as

1

2φ ·Rk · φ =

1

2

∫q

φα,−qRαβk (q)φβ,q, (2.2.10)

where

∫q

=

∫dDq

(2π)Dand φα,q =

∫dDxφα(x)eiqx, then

1. Rk(q) > 0 in the limit q2/k2 → 0. (This ensures that Rkbehaves as an IR regulator.)

2. Rk(q)→ 0 in the limit q2/k2 →∞. (This avoids a new UV renormalization.)

2′. Rk is an L2-function, and satisfies Rk → 0 as k2 → 0.

3. Rk →∞ in the limit k → Λo. (The theory becomes classical at the UV scale.)

Let us see details of these requirements. The property 1 guarantees an infrared regularization:as an example, Rk(q) ∼ k2 in the region q2 k2 for the boson sector satisfies the property 1and it generates a mass k2 in boson propagators.

The property 2 is required to avoid introducing a new UV renormalization. For example,Rk(q) = k2 (1− k2/Λ2)

−1satisfies the properties 1 and 3, however it violates the property 2

and it causes new ultraviolet divergences to the theory because it behaves a new mass term.The property 2 ensures the existence of the well-defined generating functional Wk[J ] and 1PIeffective action Γk[ϕ] as long as the original theory, i.e., W [J ] or Γ[ϕ], is well-defined. Therefore,the rapidity of the convergence Rk(q)→ 0 as q2/k2 → 0 is important, and we can make a morerefined statement 2’.

The requirement Rk → 0 as k2 → 0 in the property 2 or 2’ is also important, because Γk[ϕ]becomes the ordinary 1PI effective action Γ[ϕ] at the endpoint of the flow k = 0. The property3 ensures that the flow starts from the classical action Γk=Λ[ϕ] = S[ϕ]. Thus, in principle, wecan evaluate the effective action Γ[ϕ] from the classical action S[ϕ] by solving the flow for Γk[ϕ]parametrized by k ∈ [0,Λ].

As a corollary of Theorem 2.2.2, we can derive the flow equation for Γk[ϕ]:

Theorem 2.2.3 (The Wetterich equation). The flowing 1PI effective action Γk[ϕ] satisfiesthe differential equation, called the Wetterich equation:

∂tΓk[ϕ] =1

2STr

[1

Γ(2)k [ϕ] +Rk

∂tRk

], (2.2.11)

where ∂t = k∂k. The flow starts from the classical action: Γk=Λ[ϕ] = S[ϕ] and reaches the 1PIeffective action of the full theory: Γk=0[ϕ] = Γ[ϕ].

Let us assume that we can expand the 1PI effective action Γk[ϕ] in terms of field variablesϕ and then we can draw Feynman diagrams for the Wetterich equation (2.2.11). For simplicity,

3For the renormalizable theory, the ultraviolet cutoff scale Λo is taken to be infinite.


we assume that the theory contains only bosonic fields, and that the Z2-symmetry ϕ 7→ −ϕholds. These assumptions allow us to expand the 1PI effective action as

Γk[ϕ] =1

2ϕn(G−1

o − Σk)nmϕm +

∑i≥2

1

(2i)!Γn1,n2,...,n2i−1,n2i

k,2i ϕn1ϕn2 · · ·ϕn2i−1ϕn2i

. (2.2.12)

Substituting (2.2.12) into the Wetterich equation (2.2.11), we can find its diagrammatic rep-resentations by comparing coefficients (See Appendix A for the detailed derivation). For 1PIvertices with some small number of legs, we obtain that(

∂tΓk,2 =)− ∂tΣk = , (2.2.13)

∂tΓk,4 = + , (2.2.14)

and so on. In these graphs, square vertices represent the negative of 1PI vertices−Γk,2i, internallines represent the full propagator (G−1

o −Σk +Rk)−1, and the shaded blob represents the two

point vertex ∂tRk. The negative signature of square vertices reflects the negative signature infront of the action in the Boltzmann factor exp(−S[φ]) of the path integral measure.

2.2.2 The flow of correlation functions

In terms of FRG, it is quite natural to define composite operators as infinitesimal changes of theWilsonian effective action [24,25]. In this review, we have focused on the Schwinger functionalW [J ] and the 1PI effective action Γ[ϕ] instead of the Wilsonian effective action, and thus wecan define the flow for correlation functions 4 in a systematic way [26].

Let I[J, φ] be a bosonic functional of fields φ and sources J , then the expectation valueI[J ] = 〈I[J, φ]〉 is defined as

I[J ] = e−W [J ]I

[J,δLδJ

]eW [J ] =

∫DφI[J, φ] exp (−S[φ] + J · φ)∫Dφ exp (−S[φ] + J · φ)

. (2.2.15)

This is the quantity defined only for the full theory, i.e., at the end of the flow. In order todiscuss the flow also for the functional I[J ], we extend this as

Ik[J ] = e−Wk[J ] exp

(−1

2

δLδJ·Rk ·

δLδJ

)(I

[J,δLδJ

]eW [J ]

). (2.2.16)

One should notice that the endpoint of the flow is clearly given by Ik=0[J ] = I[J ] and thatI[J ] = 0 means that Ik[J ] = 0 at any k ∈ [0,Λ]. This fact will also be derived in another waylater.

Now we can readily derive the flow equation. Taking the derivative of Eq.(2.2.16) withrespect to k directly will give the flow equation, however we here take another approach.

4If we assume that I[J, φ] is independent of J , Ik[φ] is a composite operator, and then this is a naturalextension of the flow of composite operators.


Theorem 2.2.4 (The flow of the functional Ik[J ]). Let us consider infinitesimal changesδRk of the regulator, then the corresponding infinitesimal change of the functional Ik[J ] is givenby

δIk[J ] = −δRnmk

(δLWk[J ]

δJnδLIk[J ]

δJm+

1

2

δ2LIk[J ]

δJnδJm

). (2.2.17)

Therefore, the right hand side is linear in Ik[J ] and it implies that the solution of the flow isunique 5. Especially, if Ik=0[J ] = 0 then Ik[J ] = 0 for any k.

Proof. Let us rewrite Eq.(2.2.16) as

exp (Wk[J ] + εIk[J ]) = exp

(−1

2

δLδJ·Rk ·

δLδJ

)exp (W [J ] + εI[J ]) ,

which is true up to the linear order of ε. Here ε can be regarded as an infinitesimal realparameter. Therefore, the flow equation (2.2.6) is true for Wk[J ] + εIk[J ] up to the order O(ε).The flow equation (2.2.6) for Wk[J ] is also true, so we find Eq.(2.2.17).

In the diagrammatic interpretation, the first term of Eq.(2.2.17) is coming from the treestructure, and thus even when the bare quantity I[J, φ] has a 1PI structure the quantity Ik[J ]

loses 1PI property. Next, we consider the correlation function Ik[ϕ] in the 1PI formalism. Letus define

Ik[ϕ] = Ik[J [ϕ]], (2.2.18)

where J [ϕ] is the inverse function of Wk,n[J ] = ϕn. We would like to derive the flow equation

for Ik[J ]. As we have discussed, the flow equation for Ik[J ] can be derived by recognizing thefunctional εIk[J ] as an infinitesimal change of the Schwinger functional. Since Γk[ϕ] and Wk[J ]is related by the Legendre transformation, this infinitesimal change induces the infinitesimalchange to the 1PI effective action so that it becomes Γk[ϕ]− εIk[ϕ].

Theorem 2.2.5 (The flow of correlation functions Ik[ϕ]). The flow equation for bosonic

correlation functions Ik[ϕ] is

∂tIk[ϕ] = −1

2STr

[1

Γ(2)k [ϕ] +Rk

I(2)k [ϕ]

1

Γ(2)k [ϕ] +Rk

∂tRk

]. (2.2.19)

Here we have defined that I(2)nmk [ϕ] =

δLδRIk[ϕ]

δϕnδϕm.

Proof. Since Γk[ϕ] − εIk[ϕ] is the Legendre transformation of Wk[J ] + εIk[J ], it satisfies theWetterich equation:

∂t

(Γk[ϕ]− εIk[ϕ]

)=

1

2STr

[1

Γ(2)k [ϕ]− εI(2)

k [ϕ] +Rk

∂tRk

]. (2.2.20)

By comparing linear terms in ε of the both sides, we reach the desired result.

5When expanding the both sides of the equation in the proof with respect to the parameter ε, we have toassume the absence of phase transitions associated with the perturbation εIk[φ]. This stability of the vacuum isrelated to the cluster-decomposition property (see chap. 4 of [1] and chap. 19 of [2]). For solving degeneraciesof the vacua, we need to take care about boundary conditions of the infinite volume limit or, equivalently, toadd the sufficiently many terms to the classical action (see chap. 5 and 16 of [5]).


The flow equation (2.2.19) for Ik[ϕ] has exact 1-loop property. Thus, especially when bare

quantities Ik=Λ[ϕ] are 1PI correlation functions, the 1PI structure of correlation functions Ik[ϕ]holds for arbitrary k ∈ [0,Λ]. Diagrammatically, we can express this as

∂tIk[ϕ] = −1

2∂tRk I

(2)k [ϕ], (2.2.21)

where the shaded blob represents the scale derivative of the regulator ∂tRk, the empty blobrepresents the vertex for I

(2)k [ϕ], and the line represents the field dependent propagator (Γ

(2)k [ϕ]+

Rk)−1.

From now on, we have restricted ourselves to the case where bare functionals I[J, φ] arebosonic in order to deal those as infinitesimal changes of the action. Now, let us allow for thecase where functionals I[J, φ] are fermionic quantities. This case, however, can be reduced tothe bosonic case. Let η be some fermionic variable, then ηI[J, φ] becomes a bosonic quantity.Therefore, all the discussions given above are valid for ηI[J, φ] when I[J, φ] is fermionic.

Recall that the regulator Rmnk is a bosonic quantity, so in Eq.(2.2.17) the fermionic parameter

η can be factored out since the labels m and n in the regulator can only run over the fieldswith the same statistics:

ηδIk[J ] = −δRnmk

(δLWk[J ]

δJnδLηIk[J ]

δJm+

1

2

δ2LηIk[J ]

δJnδJm

)= −ηδRnm

k

(δLWk[J ]

δJnδLIk[J ]

δJm+

1

2

δ2LIk[J ]

δJnδJm

).

Therefore, Eq.(2.2.17) is valid also for the fermionic quantity.

Next, let us check for Eq.(2.2.19) in the case of fermionic functionals. Denoting Gmn[ϕ] =Wk,mn[J [ϕ]], we get

η∂tIk[ϕ] = −1

2∂tR

mnk Gml[ϕ]

(δLδϕl

δRδϕ′l

ηIk[ϕ]

)Gl′n[ϕ].

Therefore, the extra minus sign appears only when m and n are fermionic labels in factoringout η from the left. Therefore, we get the flow equation for the fermionic functional Ik[ϕ]:

∂tIk[ϕ] = −1

2∂tR

nmk Gml[ϕ]I

(2)ll′

k [ϕ]Gl′n[ϕ] = −1

2Tr

[1

Γ(2)k [ϕ] +Rk

I(2)k [ϕ]

1

Γ(2)k [ϕ] +Rk

∂tRk

].

(2.2.22)In other words, since the vertex becomes fermionic, the extra minus sign for the fermionic loopdoes not appear, and thus the super-trace should be replaced by the standard trace operation.

In the following, we sometimes denote the flow equation as Eq.(2.2.19) instead of Eq.(2.2.22)

even for the fermionic functional Ik for the simplicity of the notation. In that case, we shouldinterpret the super-trace as the trace operation which gives the extra minus sign only whenthe fermionic loop is closed. In that interpretation, the fermionic loop is not closed in the flowequation of the fermionic vertex Ik and therefore the super-trace in this definition coincideswith the standard trace operation.


Dyson-Schwinger equation

As an example of applications of Eq.(2.2.19), we will find a relation of FRG to the Dyson-Schwinger (DS) equation. The Dyson-Schwinger equation [27, 28] is the equation of motionin quantum field theories. In the path integral formalism we can understand that the DSequation is nothing but the identity, which represents the invariance of the path integral. Themost standard example of the DS equation can be derived from the invariance of the flat pathintegral measure Dφ under translational shifts φ 7→ φ + εφ, where ε is an infinitesimal realparameter and φ’s are arbitrary. Then, we find that∫

Dφ(Jn − δRS

δφn[φ]

)exp (−S[φ] + J · φ) = 0, (2.2.23)

which is nothing but the DS equation.Let us rewrite the DS equation by using a correlation function.

Definition 2.2.2 (The DS operator). The Dyson-Schwinger (DS) operator IDS is defined as

InDS

[J,δLδJ

]= Jn − δRS

δφn

[δLδJ

]. (2.2.24)

With the DS operator, the DS equation can be written as InDS

[J, δL

δJ

]exp (W [J ]) = 0.

Since we have constructed the correlation function which denotes the DS equation, we canapply the flow equation by extending it with Eq.(2.2.16). Before going into this topic, let usbecome more familiar with the DS equation. Since we have obtained the DS equation for fullGreen functions, we can easily rewrite it as the DS equation for connected Green functions:

Jn −(δRS

δφn

[δLW [J ]

δJ+δLδJ

]1

)= 0. (2.2.25)

Following the notation given in Eq.(2.2.15), the left-hand side of Eq.(2.2.25) represents thenormalized expectation value InDS[J ]. Let us rewrite Eq.(2.2.25) as the DS equation for 1PIvertices, which can be achieved by the substitution of J = J [ϕ]. In this change of variables,the derivative with respect to Jn can be written as

δLδJn

=δLϕmδJn

∣∣∣∣J [ϕ]

δLδϕm

= W,nm[J [ϕ]]δLδϕm

.

Let us denote Gnm[ϕ] = W,nm[ϕ], then we can rewrite the DS equation as

δRΓ[ϕ]

δϕn−(δRS

δφn

[ϕ+G[ϕ] · δL

δϕ

]1

)= 0. (2.2.26)

This represents InDS[ϕ] = 0.Now let us apply the flow equation to the functional IDS. Since the subscript becomes too

messy if we simply use the notation given in Eq.(2.2.16), we define another notation as follows:

InDS[J ;Rk] = exp (−Wk[J ]) exp

(−1

2

δLδJ·Rk ·

δLδJ

)(InDS

[J,δLδJ

]exp (W [J ])

). (2.2.27)


As we have done in Eq.(2.2.18), we define the correlation function for the DS equation in the1PI formalism as

InDS[ϕ;Rk] = InDS[J [ϕ];Rk], (2.2.28)

where Wk,n[J [ϕ]] = ϕn. Since J-dependence of the functional InDS[J, φ] is very easy, we can

explicitly calculate InDS[J ;Rk] and InDS[ϕ;Rk]. By the definition of Eq.(2.2.27), we get

InDS[J ;Rk] =

∫Dφ(Jn − φmRmn

k −δRS[φ]δφn

)exp

(−S[φ]− 1

2φ ·Rk · φ+ J · φ

)∫Dφ exp

(−S[φ]− 1

2φ ·Rk · φ+ J · φ

)= Jn − ϕm[J ]Rmn

k −(δRS

δφn

[ϕ[J ] +

δLδJ

]1

). (2.2.29)

Substituting J = J [ϕ] into Eq.(2.2.29), we obtain that

InDS[ϕ;Rk] =δRΓk[ϕ]

δϕn−(δRS

δφn

[ϕm +Wk,ml

δLδϕl

]1

). (2.2.30)

Of course, in the limit k → 0, or Rk → 0, we get IDS[ϕ,Rk → 0]→ IDS[ϕ]. One can find that

IDS[ϕ;Rk] is a very natural extension of IDS[ϕ] by comparing Eqs.(2.2.26) and (2.2.30).In the context of FRG, the Dyson-Schwinger equation can be regarded as the result of the

flow equation for IDS[ϕ,Rk], as pointed out in the review [26] given by J. M. Pawlowski. In theclassical limit k → Λ, or Rk → ∞, we can find that Γk[ϕ] → S[ϕ] and that Wk,mn[J [ϕ]] → 0,

and thus IDS[ϕ,Rk] → 0 from the definition of Eq. (2.2.30). Since the functional IDS[ϕ;Rk]satisfies the flow equation (2.2.19), the existence and the uniqueness of the solution imply that

IDS[ϕ;Rk] = 0 (2.2.31)

at any k ∈ [0,Λ]. Especially, this means that the DS equation IDS[ϕ] = 0.

2.2.3 Renormalization group flows

So far, we have considered flows accompanying to infinitesimal changes of an IR scale k, whichis a parameter of the regulator Rk. Since the scale parameter k does not appear in the fulltheory (i.e. in the limit k → 0), the Schwinger functional W [J ] of the full theory is independentof k: k∂kW [J ] = 0. From this viewpoint, we can generalize situations in the following way [26].

Let us consider a theory with fundamental couplings gi. We would like to study the responseof the theory under infinitesimal changes of some scale s. For example, we can put s = k, whichis an IR scale of FRG, and we can also put s = µR, which is a renormalization scale of the fulltheory. Let us regard that the couplings gi and external sources Jn can be s-dependent 6. Weare interested in a differential operator

Ds = s∂

∂s+ γg

ijgi

∂

∂gj+ γJ

nmJ

m δlδJn

, (2.2.32)

which satisfiesDsW [J ] = 0. (2.2.33)

6For example, let the couplings gi be defined at some renormalization scale µR. Then, gi should depend ons if we set the scale s = µR. Furthermore, in this situation the external fields J should also depend on s if J ’scouple to renormalized fields.


In other words, we consider situations in which we can cancel the change of the Schwingerfunctional under the suitable rescaling of the couplings gi and the fields Jn. For simplicity, werestrict ourselves to the case where anomalous dimensions γJ do not changes the statistics andare independent of external sources J .

We would like to calculate the flow DsWk[J ]. By the assumption (2.2.33), W [J ] does notchange under the transformation s 7→ (1 + ε)s, gi 7→ gi + εγg

jigj, and Jn 7→ Jn + εγJ

nmJ

m. Weshould recall that

exp (Wk[J ]) = exp

(−1

2Rnmk

δLδJn

δLδJm

)exp (W [J ]) .

Under infinitesimal changes s 7→ (1 + ε)s, the regulator term changes as

1

2Rnmk

δLδJn

δLδJm

7→ 1

2(Rnm

k + εDsRnmk )

(δLδJn− εγJ ln

δLδJ l

)(δLδJm

− εγJ l′

m

δLδJ l′

)=

1

2(Rnm

k + ε(DsRnmk − 2γJ

nl R

lmk ))

δLδJn

δLδJm

. (2.2.34)

Let us denote as [(Ds − γJ)Rk]nm = DsR

nmk − 2γJ

nl R

lmk . Then, we can derive the flow equation

of the Schwinger functional Wk[J ] by replacing δRk by (Ds − γJ)Rk in Eq.(2.2.6):

DsWk[J ] = −1

2

(δ2LWk[J ]

δJnδJm+δLWk[J ]

δJnδLWk[J ]

δJm

)[(Ds − γJ)Rk]

nm . (2.2.35)

On the other hand, for the flow of correlation functions I[J ], the corresponding operator I[J, δLδJ

]may depend on s, which makes the equation a little bit complicated. Using the definition(2.2.16), we can easily find the RG flow of Ik[J ]:

DsIk[J ] +1

2

(δ2LIk[J ]

δJnδJm+ 2

δLWk[J ]

δJnδLIk[J ]

δJm

)[(Ds − γJ)Rk]

nm = ([Ds, I])k[J ]. (2.2.36)

Here, the left hand side of Eq.(2.2.36) is nothing but the linearization of Eq.(2.2.35) and theright hand side represents the contribution from the scale dependence of I[J, φ].

Now, let us go back to the 1PI formalism. We would like to define anomalous dimensionsγϕ of fields ϕ so that

DsΓ[ϕ] = 0, (2.2.37)

where

Ds = s∂

∂s+ γg

ijgi

∂

∂gj+ γϕ

nmϕn

δLδϕm

. (2.2.38)

We can achieve this by imposing γϕ = −γJ under the assumption of Eq.(2.2.33).

Proof. Let us calculate the left hand side of Eq.(2.2.37). Recall that Γ[ϕ] is given by Eq.(2.1.10),then we can get

DsΓk[ϕ] = (DsJn[ϕ])ϕn + γϕ

nmJ

mϕn − DsW [J [ϕ]].

The last term can be written in the following way:

DsW [J [ϕ]] = s∂W

∂s

∣∣∣∣J [ϕ]

+ s∂Jn[ϕ]

∂s

δLW

δJn

∣∣∣∣J [ϕ]

+ γgijgi

(∂W

∂gj+∂Jn

∂gi

δLW

δJn

)∣∣∣∣J [ϕ]

+ γϕnmϕn

δLJl

δϕm

δLW

δJ l

∣∣∣∣J [ϕ]

= (DsW )[J [ϕ]]− γJnmJm[ϕ]ϕn + (DsJn[ϕ])ϕn.

Therefore, we have found that DsΓ[ϕ] = (γJ + γϕ)nmJmϕn by using DsW [J ] = 0.


We can readily obtain the RG flow for Γk[ϕ] under the total scale change of s:

DsΓk[ϕ] =1

2STr

[1

Γ(2)k [ϕ] +Rk

[(Ds + γϕ)Rk]

]. (2.2.39)

We can see that this is a natural extension of Eq.(2.2.11). By substituting J = J [ϕ] intoEq.(2.2.36), we can find the RG flow of the correlation functions in the 1PI formalism:

DsIk[ϕ] +1

2STr

[Gk[ϕ] · [(Ds + γϕ)Rk] ·Gk[ϕ] · I(2)

k

]= ([Ds, I])k[ϕ], (2.2.40)

where Gk[ϕ] = (Γ(2)k [ϕ] +Rk)

−1.The start point of the above discussion is the invariance of the Schwinger functional or the

1PI effective action in the full theory. Therefore, we can straightforwardly lift Eqs.(2.2.39,2.2.40)to equations including general variations of the regulator Rk. We can achieve this simply byreplacing

Ds → DR =

∫dk δRnm

k

δ

δRnmk

∣∣∣∣s

+δs(R, δR)

s(R)Ds. (2.2.41)

The operator DR represents the total derivative with respect to the regulator Rk.

2.3 Polchinski’s formulation of FRG

In this section we shall explicitly give relationship between the Wetterich formalism and theanother effective action formalism given by J. Polchinski [9], which is essentially same with theWilsonian effective action. This is important not only because their relationship is interestingbut also because it helps us to interpret the physical meaning of the regulator in Wetterich’sapproach intuitively.

2.3.1 Relation between the Wilsonian and 1PI effective actions

Let S[φ] be a bare action of a continuum theory. Let us decompose it as

S[φ] =1

2

∫p

φ−p,αG−1o (p)

αβφp,β + Sint[φ], (2.3.1)

where

∫p

=

∫ddp

(2π)d, φp,α =

∫ddxφα(x)eipx, and Go(p) is the free propagator. The idea of

the Wilsonian effective action is very simple: by decomposing the fields into low-momentumparts and high-momentum parts and by integrating out high-momentum fields we would liketo obtain the low-energy effective action which is called the Wilsonian effective action. Thismotivates us to write the free propagator in the following way:

Go(p) = Go(p)K(p) +Go(p)[1−K(p)],

where K is some cut-off function which is monotonically decreasing and satisfies K(0) = 1,K(∞) = 0. Its outline is shown in Fig.2.1, and the equation itself does not depend on detailsof the choice of cutoff functions. If we set K(p) = 1 − Θ(p2/k2 − 1) with some cutoff scale k,

2.3. Polchinski’s formulation of FRG 21

the first term in the right-hand side represents the propagator of the fields with p2 < k2 andthe second term is the propagator of the fields with p2 > k2. Indeed we can easily show that∫

Dφ exp

(−1

2

∫p

φ−p,αG−1o (p)

αβφp,β − Sint[φ]

)(2.3.2)

=

∫Dφ`Dφh exp

(−1

2

∫p

φ`−p,αG−1o (p)

αβ

K(p)φ`p,β −

1

2

∫p

φh−p,αG−1o (p)

αβ

1−K(p)φhp,β − Sint[φ

` + φh]

)by performing the change of path integration variables. According to this fact, we define theinteraction part of the Wilsonian effective in the following way:

exp (−Sint,K [φ]) =

∫Dφh exp

(−1

2

∫p


αβ

1−K(p)φhp,β − Sint[φ+ φh]

), (2.3.3)

then we get the relation∫Dφ exp (−S[φ]) =

∫Dφ` exp

(−1

2

∫p

φ`−p,αG−1o (p)

αβ

K(p)φ`p,β − Sint,K [φ`]

). (2.3.4)

From Eq.(2.3.3), we can find that the Wilsonian effective action is essentially the Schwingerfunctional with a regulator. Let us see this explicitly. By shifting the path integral variableφh 7→ φh − φ in Eq.(2.3.3), we find that


∫Dφh exp

(−1

2

∫p


αβ

1−K(p)φhp,β − Sint[φ

h]

+

∫p

φ−p,αG−1o (p)

αβ

1−K(p)φhp,β −

1

2

∫p

φ−p,αG−1o (p)

αβ

1−K(p)φp,β

). (2.3.5)

Let us define the regulator R by

R = G−1o

(1

1−K− 1

), (2.3.6)

then we obtain that


∫Dφh exp

(−S[φh]− 1

2φhRφh + φ(G−1

o +R)φh − 1

2φ(G−1

o +R)φ

)= exp

(Wk[φ · (G−1

o +R)]− 1

2φ · (G−1

o +R) · φ). (2.3.7)

1.0

1.0

Figure 2.1: Outlines of the cutoff function KΛ and its derivative KΛ(p) = Λ∂ΛKΛ(p)


We have shown that the Wilsonian effective action is essentially the generating functional forconnected Green functions whose external legs are amputated.

By using Eq.(2.3.6), we can find the regulator corresponding to the sharp cut-off in theWilsonian effective action. In the sharp cut-off limit K(p) = 1−Θ(p2/k2 − 1) we obtain

Rsharp,k(p) = G−1o (p)

(1

Θ(p2/k2 − 1)− 1

). (2.3.8)

In this way, the regulator R in the 1PI formalism can be related to the corresponding cut-off function of the Wilsonian effective action formalism. Furthermore, the relation Eq.(2.3.6)between R and K shows that the appropriately chosen regulators Rk in Wetterich’s formulationdivide scales in physics and that it is closely related to coarse graining, or decimation of modes,in the Wilsonian renormalization group procedure.

Finally, let us calculate the explicit relation between Γk[ϕ] and Sint,K [φ]. Those two variablesφ and ϕ in the effective actions are related by

ϕ =δLWk[J ]

δJ

∣∣∣∣J=φ·(G−1

o +R)

= (G−1o +R)−1 · δLWk[φ · (G−1

o +R)]

δφ. (2.3.9)

Since Wk and Sint,K are related by Eq.(2.3.7), we obtain

ϕ = φ− (G−1o +R)−1 · δLSint,K [φ]

δφ. (2.3.10)

By using the definition (2.2.3), we can find that

Γk[ϕ] +1

2ϕ ·R · ϕ = φ · (G−1

o +R) · ϕ−Wk[φ · (G−1o +R)]. (2.3.11)

Hence, we obtain that

Γk[ϕ] = Sint,K [φ]− 1

2φ · (G−1

o +R) · φ+ φ · (G−1o +R) · ϕ− 1

2ϕ ·R · ϕ (2.3.12)

= Sint,K [φ] +1

2ϕ ·G−1

o · ϕ−1

2(φ− ϕ) · (G−1

o +R) · (φ− ϕ). (2.3.13)

In the classical limit K → 1, the Wilsonian effective action Sint,K=1 is trivially the classical oneSint[φ], which can be easily seen from Eq.(2.3.3) or Eq.(2.3.13).

2.3.2 The Polchinski equation

Now, let us derive the Polchinski equation [9], which is the flow equation for the Wilsonianeffective action Sint,K [φ]. Assume that the cut-off function K is a smooth function. Considerinfinitesimal changes δK, then they induce changes of the regulator R by

δR = G−1o

δK

(1−K)2= (G−1

o +R) · δKGo · (G−1o +R).

According to Eq.(2.3.7), we get

δSint,K [φ] = −δWk[J ]∣∣∣J=φ·(G−1

o +R)− φ · δR · δLWk[J ]

δJ

∣∣∣∣J=φ·(G−1

o +R)

+1

2φ · δR · φ. (2.3.14)


Here, the first term in the right hand side comes from the dependence of Wk[J ] on R, or on K,with fixed arguments J , and the second term comes from the change of the variable δJ = φ ·δR.Using Eq.(2.2.6), we find that

δSint,K [φ] =1

2δRnm

(δLWk

δJnδLWk

δJm+

δ2LWk

δJnδJm− 2φn

δLWk[J ]

δJm+ φnφm

)∣∣∣∣J=φ·(G−1

o +R)

. (2.3.15)

Here, we should notice that

δLWk[J ]

δJn

∣∣∣∣J=φ·(G−1

o +R)

= φn − (G−1o +R)−1

nm

δLSint,K [φ]

δφm. (2.3.16)

Then the second derivative of W [J ] can be represented as

δ2LWk[J ]

δJnδJm

∣∣∣∣J=φ·(G−1

o +R)

= (G−1o +R)−1

nm − (G−1o +R)−1

nl1(G−1

o +R)−1ml2

δ2LSint,K [φ]

δφl1δφl2. (2.3.17)

Up to a field independent term, we can obtain the following equation which represents the flowof Sint,K :

Theorem 2.3.1 (Polchinski equation). The interaction part of the Wilsonian effective actionSint,K [φ] satisfies the RG differential equation

δSint,K [φ] =1

2(δKGo)nm

(δLSint,K [φ]

δφn

δLSint,K [φ]

δφm− δ2

LSint,K [φ]

δφnδφm

)(2.3.18)

under infinitesimal changes of K. This is called the Polchinski equation.

We can see that in the diagrammatic expansion the first term in the right hand side comesfrom the tree contribution and the second term comes from the 1-loop contribution. Thanksto this structure of the Polchinski equation, the proof of perturbative renormalizability forrenormalizable theories becomes straightforward and does not require the analysis on detailedstructures of each Feynman diagram [9]. We need not care about subdivergences and thenthe complication due to Zimmermann forests is absent even in the discussion on all orderperturbation theory. For details of the proof, see Appendix B.

In order to obtain a diagrammatic expression for the Polchinski equation, let us assumethat the interaction part of the Wilsonian effective action Sint,K can be expanded in terms offield variables φ:

Sint,K [φ] =∞∑m=1

1

m!V n1,n2,...,nmK,m φn1φn2 · · ·φnm . (2.3.19)

Just for simplicity we assume that fields φn are bosonic, so each interaction vertex V n1,...,nmK is

totally symmetric in terms of its indices n1, . . . , nm. The Polchinski equation (2.3.18) can bewritten as

δV n1,...,nmK =

1

2(δKGo)l1l2

(∑σ∈Sm

m∑p=0

1

p!(m− p)!Vnσ(1),...,nσ(p),l1K,p+1 V

l2,nσ(p+1),...,nσ(m)

K,m−p+1 − V l1,l2,n1,··· ,nmK,m+2

),

(2.3.20)


and the diagrammatic representation is given by

δV n1,...,nmK =

∑partitions

ni1

ni2

...

nip

nip+1

nip+2

...

nim

−

n2 · · · nm−1n1 nm

, (2.3.21)

where the empty vertex with ` lines represents the interaction vertex V n1,...,n`K,` , and the line with

the blob represents the single scale propagator δKGo = (G−1o +R)−1 · δR · (G−1

o +R)−1. In thefirst diagram in the right hand side, the summation over partitions means that we should sumover all the partitions of the labels 1, . . . ,m into two subsets i1, . . . , ip and ip+1, . . . , im,which satisfies i1 < · · · < ip and ip+1 < · · · < im. In these graphs, all the external lines areamputated, which represents the fact that the Wilsonian effective action Sint,K represents thegenerating functional of the amputated, connected vertices.

2.3.3 Wick ordered formulation of the FRG

In this subsection, we shall consider the Wick ordering, which can remove some divergencesin the continuum limit. Although this procedure does not cancel all ultraviolet divergences,the Wick ordering of operators would be natural because renormalization procedures shouldderive not only the low-energy effective action but also observables in the effective theory.That is, we also need composite operator renormalization and the improvement with the Wickordering gives better convergence in the continuum limit. The renormalization group differentialequation in the Wick-ordered form was first introduced by Wieczerkowski [29].

At first, let us define the Wick ordering and consider its properties. For simplicity, we hereagain assume that fields φ are bosonic scalars. Suppose that a positive-definite covariance C isgiven. The free field theory with the covariance C is defined by the following path integration:

〈φ(x1) · · ·φ(xn)〉C =

∫Dφ exp

(−1

2φ C−1φ

)φ(x1) · · ·φ(xn). (2.3.22)

The Wick-ordered monomials of n degree are given by

ΩC [φ(x1) · · ·φ(xn)] = exp

(−1

2

δ

δφCδ

δφ

)φ(x1) · · ·φ(xn). (2.3.23)

Clearly, for any permutations σ ∈ Sn they satisfy

ΩC [φ(x1) · · ·φ(xn)] = ΩC

[φ(xσ(1)) · · ·φ(xσ(n))

], (2.3.24)

and the set of the Wick ordered monomials spans the algebra of totally symmetric polynomialssince the highest degree in ΩC(φ(x1) · · ·φ(xn)) is n. An important property of the Wick orderedmonomials is orthonormality in the sense that

〈ΩC [φ(x1) · · ·φ(xn)] ΩC [φ(y1) · · ·φ(ym)]〉C = δnm∏σ∈Sn

C(x(1), y(σ(1))). (2.3.25)

The easiest way to show this equality is to use the generating functional. Let us define

ΩC

[eJφ]

=∞∑n=0

1

n!

∫dx1 · · · dxnJ(x1) · · · J(xn)ΩC [φ(x1) · · ·φ(xn)] , (2.3.26)


then Eq.(2.3.25) can be written as⟨ΩC

[eφ(f)

]ΩC

[eφ(g)

]⟩C

= exp(fCg). (2.3.27)

Let us show Eq.(2.3.27). Since we can easily obtain that

ΩC [φ(J)] =∞∑n=0

1

n!

(−1

2

δ

δφCδ

δφ

)eφ(J) =

∞∑n=0

1

n!

(−1

2(JCJ)

)eφ(J) = e−(JCJ)/2+φ(J), (2.3.28)

we get⟨ΩC

[eφ(f)

]ΩC

[eφ(g)

]⟩C

= e−(fCf)/2−(gCg)/2〈eφ(f+g)〉C = e−(fCf)/2−(gCg)/2e(f+g)C(f+g), (2.3.29)

which implies the result. Especially, we have proven (2.3.25). Furthermore, we can readily get

ΩC [φ(x1) · · ·φ(xn)]ΩC [φ(y1) · · ·φ(ym)] = exp

(δ

δφxC

δ

δφy

)ΩC [φ(x1) · · ·φ(xn)φ(y1) · · ·φ(ym)],

(2.3.30)

whereδ

δφxis a functional derivative operator acting only on φ(x1) · · ·φ(xn), and δ

δφysimilarly

acts only on φ(y1) · · ·φ(ym).We can now obtain another diagrammatic representation for the Polchinski equation (2.3.18)

by expanding the Wilsonian effective action Sint,K [φ] in terms of the Wick ordered monomials.We consider the Wick ordering in terms of the low-energy free propagator CΛ, where

CΛ(x, y) = (KΛGo)(x, y) =

∫ddp

(2π)deip(x−y)

p2 +m2K(p2/Λ2). (2.3.31)

Under infinitesimal changes of the effective cutoff scale Λ 7→ Λe−δt, the effective action Sint,KΛ

changes as

∂tSint,KΛ[φ] =

1

2∂tCΛ,nm

(δLSint,K [φ]

δφn

δLSint,KΛ[φ]

δφm− δ2

LSint,KΛ[φ]

δφnδφm

), (2.3.32)

where ∂t = −Λ∂Λ. Let us expand the Wilsonian effective action Sint,KΛso that

Sint,KΛ[φ] =

∞∑j=1

1

j!Vn1,n2,...,nj

Λ,j ΩCΛ[φn1φn2 · · ·φnj ]. (2.3.33)

Let us calculate the left hand side of the Polchinski equation (2.3.32). Since

∂tΩCΛ[φn1 · · ·φnj ] = −1

2∂tCΛ,nm

δ2

δφnδφmΩC [φn1 · · ·φnj ], (2.3.34)

the Polchinski equation for Wick-ordered vertices VΛ becomes

∞∑j=1

1

j!∂tV

n1,n2,...,njΛ,j ΩCΛ

[φn1φn2 · · ·φnj ] =1

2∂tCΛ,nm

δLSint,KΛ[φ]

δφn

δLSint,KΛ[φ]

δφm. (2.3.35)

The expansion of the Wilsonian effective actions in the right hand side of Eq.(2.3.35) producesthe products of the Wick-ordered monomials, and then we would like to rewrite them in termsof the Wick-ordered monomials by using Eq.(2.3.30). Substitution and comparison of thecoefficients give us the following result:


Theorem 2.3.2. The Wick-ordered vertices VΛ satisfy the differential equations

∂tVn1,n2,...,nj

Λ,j =1

2

∞∑k=1

j∑m=0

∑σ∈Sj

∂t(CΛi1i′1· · ·CΛiki

′k)

(k + 1)!m!(j −m)!

×V nσ(1),...,nσ(m),i1,...,ikΛ,m+k V

nσ(m+1),...,nσ(j),i′1,...,i′k

Λ,(j−m)+k . (2.3.36)

Let us consider the case where Z2-symmetry φ 7→ −φ exists, then diagrammatic expressionsof Eq.(2.3.36) for small number of legs (j = 2 and 4) are given as

∂tVΛ,2 = + + + + · · · , (2.3.37)

∂tVΛ,4 = + + + + · · · . (2.3.38)

Here gray vertices represents Wick-ordered vertices, the usual lines represents the free propa-gator CΛ = KΛGo, and lines with a blob represent the single scale propagator ∂tKΛGo. Again,all the external lines are amputated.

2.4 Optimization

In principle, any formalisms of FRG give the same result and the result does not dependon a choice of regulators Rk, which suppress the fluctuation of low-energy modes in the flowequations. However, approximations which we will apply breaks this equivalence and the choiceof regulators Rk can become important. Let us discuss optimization of the RG flow, which is oneof the most important problem in practical applications of FRG to realistic systems. Usually,we have to adopt some approximation schemes to reveal interesting physics, and we will relyon the approximation if we have a systematic procedure of that approximation scheme. Here,we should notice that even if we have a systematic approximation, which can be applied atany order, we have to truncate the expansion at some finite order in practical calculations.Therefore, in approximate calculations physical quantities can depend on unphysical degrees offreedom, such as the unit of mass µR in the perturbation theory, or a choice of regulators Rk

in FRG, whose dependence should vanish for observables when approximations are absent.

Although it has not been formulated in a mathematically rigorous way, it is very importantto seek the optimal choice of the regulator Rk in a given truncation scheme. In the contextof FRG, such optimization procedures are proposed by D. Litim [30, 31], and he applied hisprocedure to the local potential approximation (LPA) and find the optimized regulator Rk(q) =

(k2 − q2)Θ(k2 − q2) for boson propagators and Rk(q) = q/(√

k2

q2 − 1)

Θ(k2 − q2) for fermion

propagators in a relativistic system. On the other hand, the principle of minimal sensitivity(PMS) is one of the acceptable criteria for optimization of an approximation scheme, whichwas first introduced by P. M. Stevenson in the context of the perturbation theory [32]. In thereview [26], J. M. Pawlowski indicates that the optimization criterion given by D. Litim can beunderstood in the PMS.

2.4. Optimization 27

2.4.1 Optimization criterion

We define the optimization criterion which tells us how to choose the regulator R which is calledthe optimized regulator. In the next section, we will see properties of the optimized regulator.

In this discussion, we assume that regulators Rk are monotonically decreasing as k decreases:Rk ≤ Rk′ for k < k′. At first, we define effective cut-off scales keff , which represent mass gapsof IR regularized theories.

Definition 2.4.1 (The effective cut-off scale). Choose and fix a regulator Rbase,k whichrepresents a specific flow. The full propagator with a regulator Rk is given by

G[ϕ,Rk] =1

Γ(2)k [ϕ] +Rk

, (2.4.1)

and we define the norm ||G[Rk]||sup by

||G[Rk]||sup = supϕ||G[ϕ,Rk]||L2 . (2.4.2)

Let dimG be the mass dimension of the full propagator, then for each k ∈ [0,Λ] we define theeffective cut-off scale keff(k) by 7

kdimGeff (k) = ||G[Rbase,k]||sup. (2.4.3)

Let us consider a concrete example and become familiar with it. Here we take a scalar fieldtheory and adopt the LPA, in which we put an ansatz for flowing effective actions so that

Γk[ϕ] =1

2

∫p

ϕ−pp2ϕp + Vk[ϕ]. (2.4.4)

In order to define keff(k), we have to fix a regulator Rbase,k. Here we use the sharp cut-offregulator (2.3.8), which gives us an explicit physical interpretation for cutoff scales k:

Rbase,k(p) = Rsharp,k(p) = p2

(1

Θ(p2/k2 − 1)− 1

). (2.4.5)

Then we can find that

G[ϕ,R] =(p2/Θ(p2/k2 − 1) + V

(2)k [ϕ]

)−1

≤(k2 + V

(2)k [ϕ]

)−1

.

Therefore, we get

k−2eff (k) = sup

ϕ

(k2 + V

(2)k [ϕ]

)−1

=(k2 + V

(2)k,min

)−1

. (2.4.6)

From Eq.(2.4.6), we can say that keff(k) gives the mass gap of the regularized theory with the

IR regulator Rsharp,k, and indeed at k = 0, keff(0) =√V

(2)0,min is the mass gap of the full theory.

7If we consider situations, where the renormalization of the full theory depends on the regulator R, we haveto divide the left hand side of Eq.(2.4.3) by the field renormalization Zφ. Here we fix the renormalization schemeof the full theory and use the same renormalization procedure for theories with IR regularizations.


Now that we have another parametrization of regulators Rk by the effective cut-off scalekeff , we can consider hyper-surfaces R⊥keff

whose elements regularizes the theory at the samephysical cut-off scale keff . Restricting variations of regulators into this hyper-surfaces, we canconsider the variation of regulators in the same keff . Then we can require that the regulator Rk

should behave as an IR regularization.

In order to define the optimization criterion as a stability condition, we need to studyfunctional dependence of correlation functions Ik[ϕ] on a set of regulators Rk′k′ . The flow

of Ik[ϕ] is given by Eq.(2.2.19), and since the operator is well under controlled at the startingpoint k = Λ we find that

Ik[ϕ] = IΛ[ϕ] +

∫ Λ

k

dk′

k′1

2(G[ϕ,Rk′ ]Rk′G[ϕ,Rk′ ])nmI

(2)nmk′ [ϕ]. (2.4.7)

Therefore, we can change the point of view: Instead of regarding Ik[ϕ] as a function of a

regulator R we can consider Ik[ϕ] as a functional depending on flows Rk′k′ and on the positionof the flow k ∈ [0,Λ]. To express this in an explicit way, we write the correlation function as

Ik[ϕ,R], where the argument R represents dependence on the set of regulators Rk′k′∈[0,Λ] and

the subscript k represents the position of the flow. At k = 0, the functional dependence I0[ϕ,R]on R disappears without any truncations.

Definition 2.4.2 (The optimization criterion). Let keff(k) be an effective cut-off scale. Thehyper-surface R⊥keff

with the same physical cut-off scale is defined as

R⊥keff= Rk|∀k ∈ [0,Λ] ||G[Rk]||sup = kdimG

eff (k). (2.4.8)

The optimization criterion in a closed form is given by

DR⊥ Ik[ϕ,R]∣∣∣R=Rstab

= 0, (2.4.9)

where Ik[ϕ,R] are correlation functions. Here DR⊥ =

∫dk′δRnm

⊥,k′δ

δRnmk′

represents the total

derivative in hyper-surface directions. Since the starting point and the end point of the flow isfixed, δR⊥,k′ = 0 at k′ = 0,Λ. This condition is already non-trivial even without any truncationsand Rstab is called the optimized regulator.

By changing the point of view via Eq.(2.4.7), we have reached the idea of stability undersmall variations of a “flow” but not of a regulator. In order to realize this idea, we have toconstruct the space of the flows which can be achieved by parametrization of regulators withrespect to the effective cutoff scale. This is done in Eq.(2.4.8) so that flows give the same massgap at the same scale parameter k. Thus, Eq.(2.4.9) is the requirement of the stability of theflow.

In this definition we require that δR0 = 0. However, we should notice that Ik[ϕ,R] onlydepends on Rk′ with k′ ≥ k according to Eq.(2.4.7). Therefore, we can remove the conditionδR0 = 0 from the optimization criterion. In other words, that condition is already includedwhen we have restricted regulators Rk as elements of R⊥keff

.


2.4.2 Properties of the optimized regulator

At first, let us rewrite definition (2.4.9) of the optimized regulator Rstab in a suitable form forpractical calculations to construct the optimized regulator. This can be easily done after thechange of viewpoint for the flow equation of correlation functions.

Theorem 2.4.1. The optimized regulator Rstab satisfies

δRnm⊥δ(G[ϕ,Rk]RkG[ϕ,Rk])ll′

δRnmI

(2)ll′

k [ϕ,R]

∣∣∣∣∣R=Rstab

= 0. (2.4.10)

Here we have defined the notation DR⊥ = δRnm⊥

δ

δRnmand by omitting the subscript k we

emphasize that it represents a variation of the flow.

Proof. We should notice that the derivative with the flowing parameter ∂t and DR⊥ commutewith each other. Therefore, by applying ∂t to Eq.(2.4.9) we can find that

DR⊥

(1

2STr

[(G[ϕ,Rk]RkG[ϕ,Rk])I

(2)k [ϕ,R]

])∣∣∣∣R=Rstab

= 0.

By taking functional derivatives of Eq.(2.4.9), we can get DR⊥ I(2)k [ϕ,R]

∣∣∣Rstab

= 0, and then we

can obtain Eq.(2.4.10).

Remark. This theorem indicates that if we apply the optimization criterion (2.4.9) for arbi-

trary correlation functions I the variation of GRG in the hyper-surface R⊥keffshould vanish

pointwise, i.e., for all k and for all indices. This would be an over-constraint and in practicalapplications we restrict the set of I used in optimization.

As we have pointed out in the remark, we have to restrict the set of the correlation functionsIk, to which is applied Eq.(2.4.10). Then, which Ik should be taken for the optimizationprocedure? Here we should realize that in any approximations we believe that there are finitenumber of the relevant operators Irel and that they plays the dominant role in physics.

Therefore, we should take the correlation functions Irel which is considered relevant in theapproximation scheme we use.

Theorem 2.4.2 (Principle of minimal sensitivity). Let us assume that a flow satisfyingEq.(2.4.10) exists. Then, Eq.(2.4.10) implies the PMS condition on the correlation functions

Irel used in the optimization criterion: for any I ∈ Irel we can find that

δRnm⊥δI0[ϕ,R]

δRnm

∣∣∣∣∣Rstab

= 0. (2.4.11)

Proof. We should notice that the starting point of the flow Irel,Λ is R-independent. There-fore, taking the variation of Eq.(2.4.7) with respect to the regulator R⊥ around the optimizedregulator we obtain

δRnm⊥δIk[ϕ,R]

δRnm


=1

2

∫ ∞k

dk′

k′(GRG)nm δR⊥

δI(2)nmk [ϕ,R]

δR


.

Since this is a linear equation for the correlation function Ik[ϕ,R] and the left-hand side is zeroat k = Λ, we can say that the left-hand side is identically zero at any k. Especially at k = 0,we have obtained the result.


2.4.3 Example: the Litim regulator in LPA

Let us consider a scalar field theory in the local potential approximation (LPA):

Γk[ϕ] =1

2

∫p

ϕ−pp2ϕp + Vk[ϕ]. (2.4.12)

In this approximation, D.Litim [30,31] suggests that Rk(p) = p2(k2/p2 − 1)Θ(k2/p2 − 1) is theoptimal choice, which is often called the Litim regulator. Here, following the review [26] givenby J.M.Pawlowski, we derive the Litim regulator from the optimization criterion (2.4.10). Inthe following, we should note that all the quantities in the LPA are evaluated with constantfields ϕ.

Due to the momentum conservation law, we can put

I(2)k [ϕ](p, q) = Ik(ϕ, p2)(2π)dδd(p− q), (2.4.13)

and in the LPA we can set

G[ϕ,R] =1

Γ(2)k [ϕ] +Rk

(p, q) =(2π)dδd(p− q)

p2 + V(2)k [ϕ] +Rk(p2)

. (2.4.14)

Then, Eq.(2.4.10) tells us that∫ddp

(2π)d

∫ddq

(2π)dδR⊥(p2)

δ

δR(p2)

(∂tRk(q

2)

(q2 +Rk(q2) + V(2)k [ϕ])2

)Ik(ϕ, q2)


= 0. (2.4.15)

By the dimensional argument, we can put Rk(q) = q2r(q2/k2), and then we introduce thevariable x = q2/k2. At first, we rewrite the q-integration into the x-integration.

k2

kd

∫ddq

(2π)dIk(ϕ, x)∂tRk(q

2)

(q2 +Rk(q2) + V(2)k (ϕ))2

= Ωd

∫ ∞0

xd/2−1dx

2

Ik(ϕ, x)(−2x2∂xr(x))

(x+ xr(x) + V(2)k [ϕ])2

= −Ωd

∫ ∞0

xd/2−1dxIk(ϕ, x)∂xr(x)

(1 + r(x) + V(2)k [ϕ]/x)2

.

Ωd is the solid angle of the d-dimensional space. For the integration by parts, let us consider∫ ∞0

dx∂xxd/2−1Ik(r + V (2)/x)

1 + r + V (2)/x=

∫ ∞0

dxxd/2−1Ik∂xr

(1 + r + V (2)/x)2(2.4.16)

+

∫ ∞0

dxxd/2−2Ik([(

d

2− 1

)+ x∂x ln Ik

]r + V (2)/x

1 + r + V (2)/x− V (2)/x

(1 + r + V (2)/x)2

).

The left hand side becomes −δd,2Ik(ϕ, 0) since r(x) and V (2)/x diverges as x→ +0. Therefore,we should search extrema of the quantity

δd,2Ik(ϕ, 0) +

∫ ∞0

dxxd/2−2Ik([(

d

2− 1

)+ x∂x ln Ik

]r + V (2)/x

1 + r + V (2)/x− V (2)/x

(1 + r + V (2)/x)2

)(2.4.17)

under variations of r in the r⊥ direction with Ik fixed.


Then, we have to calculate the r-derivative of the parenthesis in the integrand of Eq.(2.4.17)and we get

[(d/2− 1) + x∂x ln Ik](1 + r + V (2)/x) + V (2)/x

(1 + r + V (2)/x)3. (2.4.18)

Therefore, we can rewrite Eq.(2.4.15) as∫ ∞0

dxxd/2−1xd/2−2Ikδr⊥[(d/2− 1) + x∂x ln Ik](1 + r + V (2)/x) + V (2)/x

(1 + r + V (2)/x)3

∣∣∣∣rstab

= 0. (2.4.19)

Let as assume that the quantity of (2.4.18) is positive for the most relevant operator Ik such asIk=Λo [ϕ] = ϕ2. This is not a trivial assumption because Vk is not necessarily convex for k 6= 0and for a non-convex bare potential VΛo . Furthermore, we cannot ensure that the inequalityr+V (2)/x > 0 holds for any x and ϕ due to the approximation. Here we just expect such a subtleregion only has a minor contribution to the result for optimized r, and that this assumptioncan be justified in some way 8. Then, Eq.(2.4.19) says that we should minimize or maximizethe regulator r in the space r⊥keff

so that δr⊥ = 0 at the optimal regulator r = rstab. For thestability of the flow, we should minimize the regulator r since it indicates the minimum of thequantity (2.4.17) and then the minimal flows. On the other hand, maximizing the regulator rcorresponds to the most instable flow.

In above calculations we have not taken into account that r belongs to r⊥keff. From now

on, we have to use it in order to proceed the calculation further. By definition (2.4.8), we get

∀x ∈ [0,∞] x+ xr⊥(x) + V(2)

min ≥ 1 + V(2)

min, (2.4.20)

where we have used keff given by Eq.(2.4.6). By subtracting V(2)

min from the both sides ofEq.(2.4.20), we find that x + xr⊥(x) ≥ 1. Therefore, in the region x ≥ 1, we can set r⊥ = 0and this is consistent with the condition of Eq.(2.4.19), which indicates that the minimal valueof r is favored. And, we can saturate the inequality (2.4.20) with r(x) = 1/x− 1 when x ≤ 1,which implies that

rstab(x) = (1/x− 1)Θ(1/x− 1), (2.4.21)

which is nothing but the Litim regulator:

RLitim(q) = q2rstab(q2/k2) = (k2 − q2)Θ(k2/q2 − 1). (2.4.22)

Then we have obtained the optimized regulator for the scalar field theory in the local potentialapproximation.

Remark. It is very important for the practical applications that the Litim regulator (2.4.22)has a very simple form. Its derivative ∂tRk(q) = 2kΘ(k2/q2 − 1) also does not introduce anysingularities in the loop calculations, and then the low-momentum region would give a maincontribution which is a desirable property for the LPA. Furthermore, its simplicity often enablesus to calculate analytic expressions for the Wetterich equations, and then the numerical effortcan be reduced significantly.

8We can see that this statement is glossy. In the double well potential, V (2) is most negative at ϕ = 0,which defines the scale of the physics. Therefore, |V (2)| should be much smaller than the UV cutoff Λ2

o and thenr + V (2)/x is positive for x 1. For large momenta x 1, this may not be true but such regions should besuppressed in the flow equation due to the effective UV cutoff. In Eq.(2.4.19), this effect should be taken intoaccount through the factor δr⊥.


2.5 Properties of flow equations

In this section, we discuss general properties of the exact solutions of the RG flow. Thesediscussions were originally developed in the study of critical phenomena [6, 33–36], and now itbecomes an important tool also for quantum field theories.

The Wilsonian renormalization group consists of two steps: coarse graining (or decimationof modes) and a rescaling. In this thesis, we have only considered a coarse-graining in derivingflow equations since the rescaling is not necessary for calculations of Green functions. However,we will see that the rescaling is necessary to find fixed points of the RG flow and it revealsmany important properties of exact solutions.

For simplicity, the discussion given below is restricted to the d-dimensional scalar field theorywithin the Wetterich formalism; arguments in a more general context can be found in [35]. Thedescription given here is strongly influenced also by the papers [37–40].

2.5.1 Rescaling

With decimation of modes or coarse-graining, an effective cutoff k is reduced to another effectivecutoff k′ = e−tk. In order to consider fixed points of the RG flow, the lowered cutoff scale shouldbe brought up to the original one and then we need the rescaling. Let us consider the dilationtransformation

p 7→ p′ = eδtp, x 7→ x′ = e−δtx, φ(x) 7→ e−δtdφφ(x′). (2.5.1)

Here, dφ = 12(d − 2 + η) is the scaling dimension, which is to be determined in searching for

a fixed point of the RG flow. η represents the deviation of the scaling dimension from thecanonical dimension of the field, and it is called the anomalous dimension. In the momentumspace, the field transforms as

φp 7→ eδt(d−dφ)φp′ . (2.5.2)

Therefore, under the dilation, changes of fields φ are given by

δφp = eδt(d−dφ)φeδtp − φp = δt ((d− dφ) + pµ∂pµ)φp. (2.5.3)

We can find that the dilation changes the functional Γ[φ] by

δΓ = δtGdilΓ[φ], (2.5.4)

where Gdil is the operator given by

GdilΓ =

(∫p

pµ∂φp∂pµ

δ

δφp+ (d− dφ)

∫p

φpδ

δφp

)Γ[φ]. (2.5.5)

The first term in the parenthesis represents the effect of the scale transformation on momenta,and the second one represents the effect of the scale transformation of fields.

In some literatures, other conventions to represent the dilation transformation Gdil areadopted. Here I show two other forms which are frequently taken:

• By integrating by parts in the first term in Eq.(2.5.5), we can rewrite it as

GdilΓ = −(∫

p

φppµ∂pµ

δ

δφp+ dφ

∫p

φpδ

δφp

)Γ. (2.5.6)

2.5. Properties of flow equations 33

• Let us introduce the new derivative ∂′pµ , which acts only on coefficient vertices, i.e., doesnot act on the momentum-conserving delta-function. With this derivative, we can obtainthat

GdilΓ =

(d−

∫p

φppµ∂′pµ

δ

δφp− dφ

∫p

φpδ

δφp

)Γ. (2.5.7)

This follows due to p · ∂pδd(p) = −dδd(p).

Here we have denoted a traditional way of the rescaling. We can achieve the same conclusionin a neater way by making all the dimensional quantities dimensionless with the cutoff k.

2.5.2 Fixed points

We consider the renormalization group equation with rescaling. Let us denote the total deriva-tive with respect to t = − ln k/ko by d

dtin order to represent Wilson’s RG procedures, and we

denote the effect only from the first step by ∂∂t

. We have

dΓkdt

=∂Γk∂t

+ GdilΓk, (2.5.8)

in which the first term is given by the Wetterich equation (2.2.11), which represents the effectof coarse-graining, and the second term represents the effect of rescaling (2.5.1). Let us denotethe right hand side of Eq.(2.5.8) simply as G(η,Γk) to get

d

dtΓk = G(η,Γk). (2.5.9)

Here we make the dependence on a choice of the anomalous dimension η explicit. It is importantto notice that G acts on Γ in a nonlinear way.

A fixed point Γ∗ with the anomalous dimension η∗ is given by

G(η∗,Γ∗) = 0 (2.5.10)

so that it satisfiesd

dtΓ∗ = 0. That is, the fixed point represents the scale invariant theory. In

the vicinity of Γ∗ we have an expansion in terms of perturbations ∆Γ(t) so that

d

dt(Γ∗ + ∆Γ(t)) = L∆Γ(t) +Q∆Γ(t) + · · · , (2.5.11)

where we have separated the right hand side into a linear part L, a quadratic part Q, etc.

Definition 2.5.1 (Scaling exponents and scaling operators). Consider the eigenvalueequation:

LOi = λiOi. (2.5.12)

The eigenvalues λi are called scaling exponents and the eigenoperators Oi associated with λiare called scaling operators.

Let us assume that the set of the scaling operators Oi spans the full vector space. Underthis assumption, we can write any actions Γk near the fixed point Γ∗ as

Γk = Γ∗ +∑i

αi(t)Oi. (2.5.13)


αi(t) are nothing but effective coupling constants, sometimes called scaling fields [34]. In thelinear approximation, they satisfy

d

dtαi(t) = λiαi(t), (2.5.14)

and then we find thatΓk = Γ∗ +

∑i

αieλitOi. (2.5.15)

Recall that the limit k → 0 corresponds to t→∞. The following classification makes sense:

Definition 2.5.2 (Classification of the scaling operators). We classify scaling operatorsinto three different types as follows:

• If λi > 0, the associated scaling operator Oi is called relevant, which emanates Γk fromthe fixed point Γ∗ in the limit t→∞.

• If λi < 0, the associated scaling operator Oi is called irrelevant, which drives Γk into thefixed point Γ∗ in the limit t→∞.

• If λi = 0, the associated scaling operator Oi is called marginal and Γ∗ + αiOi is also afixed point for infinitesimal αi.

Remark. In order to describe critical phenomena, we should require that the effective action Γkapproaches to the fixed point Γ∗ asymptotically, and then we have to fine tune parameters sothat the relevant directions vanish. On the other hand, in order to define the continuum limitof the quantum field theory only the relevant operators dominate low-energy physics and thenthe irrelevant operators should be put to be zero (see Appendix B).

Especially when a marginal direction appears in leading order consideration, it is veryimportant to take into account higher order corrections. Let Omar be a marginal operator, andwe consider a perturbation Γk[ϕ] = Γ∗[ϕ] + Pt[ϕ] with

Pt[ϕ] = χ(t)Omar[ϕ] +∑i

µi(t)Oi[ϕ], (2.5.16)

where scaling fields µi(t) should be considered as quadratically small quantities. In this case,we consider up to the quadratic order of the perturbation Pt. One possibility is that at thequadratic order of the perturbation χ(t) shows the t-dependence, and in this case Omar is calleda marginally relevant/irrelevant operator. In this case, scaling fields µi(t) should be chosen sothat the perturbation Pt is an eigenperturbation in this order.

Let us suppose that the flow in the Omar direction vanishes even in the second order per-turbation. In this case, we should ask whether the eigenperturbation Pt itself vanishes in thequadratic order or not. Extracting quadratic order terms of the flow into the Oi direction, wefind that

d

dtµi(t) = λiµi(t) + biχ(t)2, (2.5.17)

where the first term comes from L(µi(t)Oi) and the second term comes from Q(χ(t)Omar). Ifthe feedback bi is absent, the flow is eliminated simply by taking µi = 0. Even if the feedbackexists, the flow of µi vanishes by choosing µi(t) = −biχ2(t)/λi as long as λi 6= 0. Therefore, if

2.5. Properties of flow equations 35

the mixing with another marginal operator does not exist, the flow of the perturbation can beeliminated up to the second order perturbation. If we can kill the flow along the Omar directionat any order in this way, the operator Omar is called exactly marginal. An exactly marginaloperator generates a line of fixed-points.

Remark. From now on, we have discussed the notion of fixed points in the Wetterich formalism,but all the statements given above is of course true for other versions of FRG. However, weshould make some comments on the relation between the Wetterich and Wilsonian effectiveactions.

Originally, F. J. Wegner discussed properties of critical fixed points based on the invarianceof the partition function under RG transformations [33–35]. Therefore, his arguments can bedirectly applicable to the Wilsonian effective action, and then fixed points in the Polchinskiversion of FRG have well-known properties. That is, if it is a critical fixed point then a lineof equivalent fixed points exists, and the line is generated by a marginal, redundant operator.This property is well studied by O. J. Rosten [40].

On the other hand, the connection between the Wetterich formalism and the Polchinskiformalism is not still clear when the rescaling operation accompanies coarse-graining in RGtransformations. Recently, a clear connection between them is given just above critical fixedpoints by O. J. Rosten [39], and it is explicitly shown that a line of equivalent fixed points alsoexists in the Wetterich formalism.


Chapter 3

FRG for many-body fermions

In this chapter, we will consider interacting many fermions using FRG. In deriving the lowenergy effective theory with the renormalization group method, identification of low-energyexcitations is most important but one of the most difficult problems. Indeed, there does notstill exist a systematic way to derive low energy excitations if it were composite particles,topological solitons, etc. In the case of fermionic systems, the identification is still difficult evenif we assume that low energy living particles are fermions given by elementary fermionic fields.In the Fermi liquid picture, such excitations are given by pseudo-particles in the vicinity of theFermi surface but the calculation of the Fermi energy from the microscopic model is not trivialat all.

In this chapter, we at first define the setup of physical systems to be considered in a generalway. In later chapters, we will treat specific models, e.g., BCS-BEC crossover and dipolarfermions, which are contained in the general discussion given in sec.3.1. In sec.3.2, we willderive the RG flow equation up to some lower orders in terms of the field expansion of the 1PIeffective action.

In order to get a physical intuitions for the RG flow in fermionic systems, we will reviewsome previous studies on the Landau Fermi liquid theory in terms of the FRG approach insec.3.3. After that, we consider the validity of such approach and pose some problems intreating the Fermi liquid from the microscopic theory in sec.3.4. Especially it will be turnedout that the self-energy correction is important for calculations of the Fermi sphere for theinteracting fermions, because we need to identify gapless excitations to separate high-energyand low-energy scales of physics quantitatively.

3.1 Notations and setup

In this section, we describe notations and the set up for many-body fermionic systems whichwe will consider.

Let ψα(x) be fermionic fields with the flavor index 1 α ∈ ↑, ↓ and with the spacetimelabel x = (τ,x). Here τ ∈ [−β/2, β/2], with the inverse temperature β = 1/T , represents theimaginary time and x ∈ Rd runs over the spatial directions. We require fermionic fields to beantiperiodic in the imaginary time direction:

ψα(β/2,x) = −ψα(−β/2,x). (3.1.1)

1In this section we treat this label as a spin index, but sometimes it represents another degrees of freedom.For example, in the case of the BCS-BEC crossover this label represents hyperfine states of an alkaline atom.

37

38 Chapter 3. FRG for many-body fermions

For the simplicity of the notation, we sometimes extend the fields ψ(x) as an antiperiodic fieldon Rd+1 with the period β in the imaginary time direction, that is, ψ(τ ± β,x) = −ψ(τ,x).The conjugate field ψ(x) is also introduced and satisfies the antiperiodic boundary conditionψ(τ ± β,x) = −ψ(τ,x).

We assume the homogeneity and thus the momentum space representation is quite usefulin calculations of physical quantities. For that purpose, let us define the Fourier transform offermionic fields as

ψp,α =

∫dxe−ipxψα(x), ψp,α =

∫dxeipxψα(x), (3.1.2)

where p = (ωn,p) and px = ωnτ +p ·x. Here the temporal component p0 is given by fermionicMatsubara frequencies ωn = π(2n+1)T with n ∈ Z. The difference of signatures e±ipx in (3.1.2)can be understood by regarding ψ as a linear functional and ψ as an antilinear functional:ψ(cf) = cψ(f) and ψ(cf) = c∗ψ(f) for a test function f and a complex number c. The inverseFourier transform of them is given by

ψ(x) =

∫ (T )

p

eipxψp, ψ(x) =

∫ (T )

p

e−ipxψp, (3.1.3)

where

∫ (T )

p

=1

β

∑ωn

∫ddp

(2π)dand the sum is the fermionic Matsubara sum.

Let us dictate transformation properties of fermionic fields ψ and ψ. We will require thesymmetry of the action under the following transformations:

• Global U(1) symmetry: Let ψ 7→ eiαψ and ψ 7→ e−iαψ with a constant α. The chemicalpotential µ defined below is the Lagrange multiplier corresponding of this symmetry.

• Rotational symmetry 2: Let us consider infinitesimal spatial rotations x 7→ x+ θ(n×x)around a unit vector n, then fields transform as

ψ 7→ ψ′ = exp(−iθn · S)ψ, ψ 7→ ψ′= ψ exp(iθn · S), (3.1.4)

where S = σ2

is the spin matrix and σ are the Pauli matrices. Under this transformation,ψ†Sψ transforms as a vector field.

• Euclidean time reflection 3: Let us consider the time reflection θ : (τ,x) 7→ (−τ,x), andwe define the antilinear operation on the field algebra defined as

θψα(τ,x) = ψα(−τ,x), θψα(τ,x) 7→ ψα(−τ,x), (3.1.5)

and θ(AB) = (θB)(θA) for functionals A,B.

• Parity: The discrete symmetry given by ψ(τ,x) 7→ ψ(τ,−x) and by ψ†(τ,x) 7→ ψ†(τ,−x)is called the parity.

2In the case of the BCS-BEC crossover, the labels α do not represent real spins, and we further require thatthe action is invariant under the spin SU(2) rotation.

3The invariance under the Euclidean time reflection is strongly related to the Hermitian property of theHamiltonian [23].

3.1. Notations and setup 39

To calculate the thermodynamic quantities, the imaginary time formalism will be used andthe grand partition function Z is given by

Z(β, µ) = Tr[exp(−β(H − µN))

]=

∫DψDψ exp(−S[ψ, ψ]), (3.1.6)

where H is the Hamiltonian, N is the number operator, β is the inverse temperature, and µ isthe chemical potential. S is the corresponding microscopic action, which is a functional of ψ, ψ,and the path integral in Eq.(3.1.6) is given by the Berezin integration. As usual, we decomposethe action S into the free part So and the interaction part Sint. The free quadratic part is

So[ψ, ψ] =

∫ (T )

k

ψα(k)[iωn + εo(k)− µ]ψα(k), (3.1.7)

where the kinetic energy is εo(p) = p2/2m. This free action satisfies the symmetries above,and furthermore it is also symmetric under the spin-SU(2) rotation.

3.1.1 Possible forms of the effective action

Let us write down the interaction part Sint. Although we require later that Sint is quartic atthe classical level, it is important for us to consider about the most general form of actionssince such terms will appear in the quantum effective action. The global U(1) symmetry andthe momentum conservation suggest that

Sint =∞∑n=1

1

(n!)2

∫ (T )

p1,...,pn;q1,...,qn

δd+1(p1 + · · ·+ pn − q1 − · · · − qn)

×Γα1...αnβn...β1

(p1, . . . , pn; qn, . . . , q1)ψp1,α1· · ·ψpn,αnψqn,βn · · ·ψq1,β1 , (3.1.8)

where δd+1(p− q) = βδp0,q0(2π)dδd(p− q). The Fermi statistics of fields gives totally antisym-metric properties to vertices: for any permutations σ, σ′ ∈ Sn

Γα1...αnβn...β1

(p1, . . . , pn; qn, . . . , q1) = sgn(σ · σ′)Γασ(1)...ασ(n)

βσ′(n)...βσ′(1)(pσ(1), . . . , pσ(n); qσ′(n), . . . , qσ′(1)). (3.1.9)

Using the parity invariance, we get the equality


(p1, . . . , pn; qn, . . . , q1) = Γα1...αnβn...β1

(Pp1, . . . , Ppn;Pqn, . . . , P q1), (3.1.10)

where P is the parity operator: P (p0,p) = (p0,−p). The invariance under the Euclidean timereflection implies that


(p1, . . . , pn; qn, . . . , q1) = [Γβ1...βnαn...α1

(θq1, . . . , θqn; θpn, . . . , θp1)]∗, (3.1.11)

where ∗ represents the complex conjugation and θ(p0,p) = (−p0,p).Especially, we shall pay attention to the self-energy correction and the four-point vertex.

Let us specify useful properties of them concluded from these symmetries. For the self-energyΣαβ(p) = −Γαβ(p; p), the rotational symmetry suggests that

Σαβ(p) = Σ(p0,p2)δαβ + Σs(po,p

2)p · Sαβ .


However, the second term does not satisfy the parity invariance and then it vanishes as long asthe parity symmetry is not broken. We conclude that the self-energy takes the form

Σαβ(p) = Σ(p0,p2)δαβ , (3.1.12)

where Σ(p0,p2) = Σ∗(−p0,p2) due to Eq.(3.1.11).Next, let us consider the properties of the four-point vertex. Let us write down the momen-

tum dependence of the four point vertex as

Γα1α2β2β1

(P ; q, q′) = Γα1α2β2β1

(P/2 + q, P/2− q;P/2− q′, P/2 + q′), (3.1.13)

where P represents the total momenta and q ± q′ represent the momentum transfer. Let usdecompose it with respect to the total spin S. Components with S = 0 consist of two terms

Γα1α2β2β1

(P ; q, q′)∣∣S=0

= Γdd(P ; q, q′)δα1β1δα2β2

+ Γss(P ; q, q′)(Sα1β1· Sα2

β2), (3.1.14)

where those coefficient functions are scalars under spatial rotations due to the parity invariance.Since those two terms are mixed under interchange of particles, the consequence of Fermistatistics is unclear with this expression. In order to make it obvious, we rewrite (3.1.14) usingFierz identities 4 so that

Γα1α2β2β1

(P ; q, q′)∣∣S=0

= ΓA(P ; q, q′)Psα1α2β2β1

+ ΓS(P ; q, q′)Paα1α2β2β1

(3.1.15)

where Psα1α2β2β1

=(δα1β1δα2β2

+ δα1β2δα2β1

)∝ (σiε

†)α1α2(εσi)β2β1 is the projection onto the spin-symmetric

component and Paα1α2β2β1

=(δα1β1δα2β2− δα1

β2δα2β1

)∝ (ε†)α1α2εβ2β1 is the projection onto the spin-

antisymmetric component. From the Fermi statistics, we can conclude that

ΓA(P ; q, q′) = −ΓA(P ;−q, q′) = −ΓA(P ; q,−q′), (3.1.16)

ΓS(P ; q, q′) = ΓS(P ;−q, q′) = ΓS(P ; q,−q′). (3.1.17)

The Euclidean time reflection requires that

ΓA/S(P ; q, q′) = [ΓA/S((−P 0,P ); (−q′0, q′), (−q0, q))]∗. (3.1.18)

When we can require the invariance under spin SU(2) rotations, the four point vertex onlycontains the S = 0 component and following terms do not appear.

The S = 1 component consists of three terms

Γα1α2β2β1

(P ; q, q′)∣∣S=1

= Γvec1i (P ; q, q′)Si

α1β1δα2β2

+ Γvec2j (P ; q, q′)δα1

β1Sj

α2

β2+ Γvec3

k (P ; q, q′)(Sα1β1× Sα2

β2)k,

(3.1.19)where the parity invariance requires that those coefficients are pseudovectors. If we would liketo find the consequence of the Fermi statistics, the Fierz transformation to the particle-particlechannel again gives clear descriptions:

Γα1α2β2β1

∣∣S=1

= Γsti (ε†)α1α2(εSi)β2β1 + Γts

j (Sjε†)α1α2εβ2β1 + Γtt

k ((Sε†)α1α2 × (εS)β2β1)k, (3.1.20)

4In order to make the rotational symmetry manifest in the particle-particle basis, we use ε, εσ as a basis of2× 2 matrices and ε†,σε† as a dual basis.

3.2. RG flow equation 41

where the momentum arguments are omitted. Now the Fermi statistics suggests that

Γst(P ; q, q′) = −Γst(P ;−q, q′) = Γst(P ; q,−q′), (3.1.21)

etc. The consequence from the Euclidean time reflection also becomes clear in this basis:

Γst(P ; q, q′) = [Γts(θP ; θq′, θq)]∗, Γtt(P ; q, q′) = −[Γtt(θP ; θq′, θq)]∗. (3.1.22)

The S = 2 component consists only of the tensor coupling

Γα1α2β2β1

(P ; q, q′)∣∣S=2

= Γtensij (P ; q, q′)(Tij)

α1α2β2β1

, (3.1.23)

where Tij is the tensor operator, given by

(Tij)α1α2β2β1

=1

2

(Siα1β1Sj

α2

β2+ Sj

α1

β1Siα2β2

)− δij

3(Sα1

β1· Sα2

β2). (3.1.24)

We will find in chap.5 that the dipole-dipole interaction contains this term at the classicallevel and then the spatial and spin rotations cannot be treated separately. We will postponestudying details of this term until chap.5.

Finally, let us comment on forms of possible interactions in the vacuum. In the nonrelativis-tic system, we can further require the Galilean invariance in the vacuum. In this case, verticesdepend on the center of mass momentum P only through the Galilean invariant combinationiP 0 + P 2/4m, and the phenomenological description of effective interactions becomes muchsimpler.

3.2 RG flow equation

In this section, we will write down the Wetterich equation of fermionic systems for the self-energy and the four-point vertex. We introduce diagrammatic expressions for convenience:

Σkαα′(p) = −Γk

αα′(p) =

p pα α′ , (3.2.1)

−Γkα1α2β2β1

(P ; q, q′) = −Γkα1α2β2β1

(p1, p2; p′2, p′1) =

β1 p′1 p′2β2

α1 p1 p2 α2

, (3.2.2)

where p1 = P/2 + q, p2 = P/2− q, p′1 = P/2 + q′, and p′2 = P/2− q′.Let us write down the Wetterich equation for these vertices. For derivation, we use the

diagrammatic method described in Appendix A. For the self-energy correction, the flow equationis

∂k

(p p

α α′)

= ∂k

(p p

α α′

l, β ), (3.2.3)

where we have introduced the new derivative ∂k, which acts only on the k-dependence of theinfrared regulator Rk in the loop propagators. The analytic expression of the flow equation forthe self-energy is given by

∂kΣkαα′(p) = −∂k

∫ (T )

l

e−il00−Γk

αββα′(p+ l; (p− l)/2, (p− l)/2)

G−1o (l)− Σk(l) +Rk(l)

(3.2.4)

= −∫ (T )

l

Γkαββα′(p+ l; (p− l)/2, (p− l)/2)

e−il00∂kRk(l)

(G−1o (l)− Σk(l) +Rk(l))2

. (3.2.5)


Therefore, we get

∂kΣk(p) = −∫ (T )

l

[3ΓAk + ΓSk ]

(p+ l;

p− l2

,p− l

2

)e−il

00∂kRk(l)

(G−1o (l)− Σk(l) +Rk(l))2

. (3.2.6)

The other terms in the four point coupling drop out since they do not satisfy the restrictionscoming from symmetries.

For the four-point vertex, the diagrammatic expression is

−∂kΓkα1α2

α′2α′1(P ; q, q′) = ∂k

1 2

1′ 2′+

1 2

1′ 2′

+

1 2

1′ 2′

+

1 2

1′ 2′ . (3.2.7)

The first and the second diagrams represent particle-hole fluctuations, and their sum satisfiesthe restriction coming from the Fermi statistics. We call them ZS and ZS’ diagrams, respec-tively. These two diagrams are important in the description of the Fermi liquid theory sincetheir singularities for small momentum transfers q±q′ ' 0 are related to the existence of “zerosound” as brought up by L. D. Landau [41,42]. The third diagram represents particle-particlefluctuations, which plays an essential role in the formation of Cooper pairs in the BCS theory,and then we will call the diagram BCS. The last one comes from feedback from the six pointvertex.

Let us calculate the analytic expression of these diagrams. In the following, we assume thespin SU(2) symmetry for simplicity. At first, we consider the ZS diagram. Using the Feynmanrule, we find that

1 2

1′ 2′= −

∫ (T )

l

Γkα1βγα′1

(l; P, q, q′)Γkγα2

α′2β(l; P, q, q′)

([G−1o − Σk +Rk](l))([G−1

o − Σk +Rk](q − q′ + l))

where

Γkα1βγα′1

(l; P, q, q′) = Γkα1βγα′1

(P

2+ q + l;

P2

+ q − l2

,P2

+ 2q′ − q − l2

), (3.2.8)

Γkγα2

α′2β(l; P, q, q′) = Γk

γα2

α′2β

(P

2− q′ + l;

−P2

+ 2q − q′ + l

2,−P

2+ q′ + l

2

). (3.2.9)

It satisfies relations Γkα1α2

α′2α′1(l − (q − q′); P, q, q′) = Γk

α2α1

α′1α′2(l; P,−q,−q′) and Γk

α1α2

α′2α′1(l − (q −

q′); P, q, q′) = Γkα2α1

α′1α′2(l; P,−q,−q′). We can calculate explicitly the summation over spin

indices. Then we get

Γkα1βγα′1

Γkγα2

α′2β(l; P, q, q′) =

1

2

(5ΓAk ΓAk + (ΓAk ΓSk + ΓSk ΓAk ) + ΓSk ΓSk

)(l; P, q, q′)Paα1α2

α′2α′1(3.2.10)

+1

2

(3ΓAk ΓAk + 3(ΓAk ΓSk + ΓSk ΓAk )− ΓSk ΓSk

)(l; P, q, q′)Psα1α2

α′2α′1

The contribution of the ZS diagram can be written as

−∂kΓAk (P ; q, q′)∣∣ZS

= −∂k∫ (T )

l


)(l; P, q, q′)

2([G−1o − Σk +Rk](l))([G−1

o − Σk +Rk](q − q′ + l)), (3.2.11)

3.3. Fermi liquid theory from the RG viewpoint 43

−∂kΓSk (P ; q, q′)∣∣ZS

= −∂k∫ (T )

l


)(l; P, q, q′)

2([G−1o − Σk +Rk](l))([G−1

o − Σk +Rk](q − q′ + l)). (3.2.12)

The calculation of the ZS’ diagram is very similar to that of the ZS diagram.

1 2

1′ 2′

=

∫ (T )

l

Γkα1βγα′2

(l; P, q,−q′)Γkγα2

α′1β(l; P, q,−q′)

([G−1o − Σk +Rk](l))([G−1

o − Σk +Rk](q + q′ + l)). (3.2.13)

The summation over spin indices also gives a similar result:

Γkα1βα′2γ

Γkγα2

βα′1(l; P, q,−q′) =

1

2

[5ΓAk ΓAk + (ΓAk ΓSk + ΓSk ΓAk ) + ΓSk ΓSk

](l; P, q,−q′)Paα1α2

α′2α′1

−1

2

[3ΓAk ΓAk + 3(ΓAk ΓSk + ΓSk ΓAk )− ΓSk ΓSk

](l; P, q,−q′)Psα1α2

α′2α′1

As a result, the contribution of the ZS’ diagram is given by

−∂kΓAk (P ; q, q′)∣∣ZS′

= ∂k

∫ (T )

l


)(l; P, q,−q′)

2([G−1o − Σk +Rk](l))([G−1

o − Σk +Rk](q + q′ + l)), (3.2.14)

−∂kΓSk (P ; q, q′)∣∣ZS′

= −∂k∫ (T )

l


)(l; P, q,−q′)

2([G−1o − Σk +Rk](l))([G−1

o − Σk +Rk](q + q′ + l)). (3.2.15)

Next, let us obtain the analytic expression for the BCS diagram.

1 2

1′ 2′

=1

2

∫ (T )

l

Γkα1α2β2β1

(P ; q, l − P2

)Γkβ1β2

α′2α′1(P ; l − P

2, q′)

([G−1o − Σk +Rk](l))([G−1

o − Σk +Rk](P − l))(3.2.16)

SinceΓk

α1α2β2β1

Γkββ2

α′2α′1

= 2ΓAk ΓAk Paα1α2

α′2α′1

+ 2ΓSk ΓSkPsα1α2

α′2α′1, (3.2.17)

the analytic expression of the BCS contribution is

−∂kΓAk (P ; q, q′)∣∣BCS

= ∂k

∫ (T )

l

ΓAk (P ; q, l − P2

)ΓAk (P ; l − P2, q′)

([G−1o − Σk +Rk](l))([G−1

o − Σk +Rk](P − l)), (3.2.18)

−∂kΓSk (P ; q, q′)∣∣BCS

= ∂k

∫ (T )

l

ΓSk (P ; q, l − P2

)ΓSk (P ; l − P2, q′)

([G−1o − Σk +Rk](l))([G−1

o − Σk +Rk](P − l)). (3.2.19)

3.3 Fermi liquid theory from the RG viewpoint

In this section, we will review the Landau Fermi theory [41,42] as a low energy effective theoryarising from the RG method. This point of view is developed in 1990s and reveals a universalnature of interacting fermionic systems [43–45]. We can find some mathematical rigoroustreatments of this problem in the textbook [46].

Let us consider the low energy effective action of interacting fermions. For this purposewe must identify the correct degrees of freedom in low energy physics, but this is a difficultproblem and depends on details of dynamics. Here we just guess that they are also a spin-1

2

fermion and also denoted by ψ. The quadratic part is again given by Eq.(3.1.7). The groundstate of this theory is the Fermi sea, with all states ε(k) < µ filled and all states ε(k) > µempty. Therefore, the low lying excitation are obtained by adding a particle just above theFermi surface, or removing one just below.


3.3.1 Tree-level RG analysis

Suppose that we already have integrated out fields with momenta outside the shell of thickness2Λo around the Fermi surface. For simplicity, we assume that the Fermi surface has rotationalinvariance5 and we also require the spin SU(2) symmetry. Let k be a momentum in the shellthen we can decompose it as

k = kF + `, (3.3.1)

where kF and ` are parallel to k and kF lies on the Fermi surface. Since Λo kF , the

momentum integration becomes

∫ddk ' kd−1

F

∫ Λo

−Λo

d`

∫dΩd, and then the free action in

Eq.(3.1.7) becomes

So =kd−1F

(2π)dβ

∑ω

∫ Λo

−Λo

d`

∫dΩdψα(k)[−iω + vF `]ψα(k), (3.3.2)

where vF is the Fermi velocity. Now we can readily find that Eq.(3.3.2) is invariant under theRG transformation:

1. Decimation: we integrate out modes with momenta lying between sΛo < |l| < Λo with0 < s < 1.

2. Rescaling: we recover the original momentum region by changing variables ω → sω(⇔ β → s−1β), `→ s`, associated with the rescaling ψ → s−3/2ψ, ψ → s−3/2ψ.

Now we have determined the scaling property of fields, then we can judge naturalness of thetheory by studying the scaling behavior of interaction terms.

Apart from the momentum delta function, we can easily find the scaling property of quarticterms. By multiplying scaling factors of each

∫ki

, of a factor βδω1+ω2,ω3+ω4 , and of fields (ψ†ψ)2,we find

(s2)4 × s−1 × (s−3/2)4 = s.

Therefore, if we have totally neglected the scaling behavior of the momentum delta functionδd(k1 + k2 − k′2 − k′1), the quartic interaction is irrelevant and then all the interactions areirrelevant. This result would be wrong, since we already know that gapless fermions are unstableunder, for example, the spontaneous symmetry breaking. Indeed we should take into accountan important subtlety in the kinematics. Momentum conservation requires that

k′2 = kF (n1 + n2 − n′1) + (`1 + `2 − `′1),

where ni is a unit vector in the direction of ki. Since the momentum k′2 must lie inside the thinshell around the Fermi sphere, possible configurations of directions ni are restricted. Especiallyin the limit Λo/KF → 0, the restriction becomes

|n1 + n2 − n′1| = 1. (3.3.3)

To make the point clearer, we restrict ourselves to consider 2-dimensional cases in thefollowing. The possible solutions of Eq.(3.3.3) can be classified into two types:

5This is only true for the case where the system is invariant under the spatial rotation. Therefore, the Fermisea of the electrons in conductors does not satisfy this condition because crystals break this symmetry.


1. If n1 6= −n2, the possible solutions are n1 = n′1 or n2 = n′1.

2. If n1 = −n2, the direction n3 is not restricted.

These are a restriction on directions and give (d − 1) constraints in the d-dimensional space.In this situation, there remains a delta function which only contains `i’s in its argument and itscales as s−1. In these special configurations, the interaction becomes marginal and producesvarious interesting physics such as superconductivity.

The interaction channel which corresponds to n1 = n′1 is called the Landau interactionchannel and the corresponding effective action is

Seffint

∣∣L

=1

4

k2(d−1)F

(2π)2dβ3

∑ω1,ω2,ω3

∏i=1,2

∫ Λo

−Λo

d`i

∫dΩi

d

∫ Λo ddQ

(2π)dΓΛ

αβγδ (k1

F ,k2F ;Q)

×ψα(k1, ω1)ψβ(k2, ω2)ψγ(k2 −Q, ω2 − Ω)ψδ(k1 +Q, ω1 + Ω). (3.3.4)

Here we have introduced the following notation for the nearly forward scattering vertex

ΓΛαβγδ (k1, k2;Q) = ΓΛ

αβγδ (k1, k2; k2 −Q, k1 +Q) (3.3.5)

with the transfer vertex Q = k′1 − k1 = (Ω,Q). Since the frequency dependence and thedependence on the deviations from the Fermi sphere ` of the vertex are irrelevant, we simplyput k1 = (πT,k1

F ) and k2 = (πT,k2F ). Nevertheless, we will retain the Q-dependence of the

vertex since we will find in the 1-loop analysis that the expansion of the vertex around Q = 0is not well defined when the cutoff Λ becomes of order of the temperature T/vF . Therefore, wecan define the two distinct limits of the vertex functions

ΓQΛαβ

γδ (k1F ,k

2F ) = lim

|Q|→0

[Γαβγδ (k1

F ,k2F ;Q, t)

∣∣∣Ω=0

], (3.3.6)

ΓΩΛ

αβ

γδ (k1F ,k

2F ) = lim

Ω→0

[Γαβγδ (K1

F ,K2F ;Q, t)

∣∣∣Q=0

]. (3.3.7)

In a rotationally invariant 2-dimensional system, ΓQ/Ω are functions only of the relative angle θbetween k1 and k2. The only remnant of the Fermi statistics is the constraint Γ

Q(A)Λ (θ = 0) = 0

and ΓΩ(A)Λ (θ = 0) = 0. For simplicity of the notation, we do not write the subscript Λ explicitly

in the following.

In a 2-dimensional case, functions of a relative angle can be expanded as a Fourier series 6

X(θ) =∞∑

l=−∞

Xle−ilθ, Xl =

∫dθ

2πX(θ)eilθ. (3.3.8)

The symmetry property X(θ) = X(−θ) becomes Xl = X−l and the consequence of the Pauli

principle, ΓQ/Ω(A)(θ) = 0, becomes∑

l ΓQ/Ω(A)l = 0.

6In the 3-dimensional system, this expansion should be replaced by the expansion with Legendre polynomialsX(θ) =

∑∞l=0XlPl(cos θ). Due to the difference of the normalization, we should replace Xl → Xl/(2l + 1) to

obtain formulae for the 3-dimensional case from ones for the 2-dimensional case.


3.3.2 1-loop analysis

Let us calculate 1-loop contributions to effective four-point vertices. There are three types of1-loop contributions, two of which mainly contribute to the Landau channel and are called ZSand ZS’ and the remaining one contributes to the BCS channel 7:

ZS =1 2

1′ 2′, ZS′ =

1 2

1′ 2′

, BCS =

1 2

1′ 2′

. (3.3.9)

The analytic expressions can be found in sec.3.2.Let us consider the Landau channel at first. For this purpose we set outgoing momenta

k′1 = k1 + Q and k′2 = k2 − Q, where Q is small enough, i.e. |Q| < Λ. In this channel wesimply neglect the contribution from the BCS graph because the configuration space of theloop momentum k5 is strongly restricted due to the center-of-mass momentum P (see Fig.3.1).The fermionic Matsubara sum over ω5 can be evaluated in a standard manner and gives

1

β

∑ω5

G(k5)G(k5 −Q) =nF (ε(k5))− nF (ε(k5 −Q))

iΩ + ε(k5)− ε(k5 −Q). (3.3.10)

Eq.(3.3.10) gives zero in the Ω-limit and then

−Λ∂

∂ΛΓΩ(θ12)

∣∣ZS

= 0. (3.3.11)

On the other hand, theQ-limit of the same product is not zero and gives a factor−β

4 cosh2(βvF `5/2).

With the sharp cutoff, the flow equations (3.2.11,3.2.12) of the vertex in this limit are given by

−Λ∂ΛΓQlA

= −β2vFΛ

cosh2(β2vFΛ

)N(εF )

2

(5

2

[ΓQl

A]2

+ ΓQlA

ΓQlS

+1

2

[ΓQl

S]2), (3.3.12)

−Λ∂ΛΓQlS

= −β2vFΛ

cosh2(β2vFΛ

)N(εF )

2

(3

2

[ΓQl

A]2

+ 3ΓQlA

ΓQlS − 1

2

[ΓQl

S]2). (3.3.13)

Here N(εF )/2 is the density of states at the Fermi surface, and here it is given by kF/2πvF .The similar analysis on the ZS’ graph shows that unless |k1 − k2|/kF < Λ/kF ( 1) the

loop integration is suppressed due to the restriction on possible configurations. Therefore, inthe following analysis we simply neglect that graph 8.

According to Eq.(3.3.11) and neglecting the ZS’ diagram, we find that ΓΩ is approximatelyinvariant under the renormalization group flow:

ΓΩ(A,S)(θ,Λ) ' ΓΩ(A,S)∗(θ). (3.3.14)

7In order to refer each diagram many times, we have assigned names to them. This nomenclature is basedon R. Shanker’s paper [44], which is also used in [45]. ZS stands for zero sound and we will find that ZS andZS’ diagrams are important for the Landau Fermi liquid. We will find later that the BCS diagram plays animportant role for the Cooper instability.

8Due to this approximation, the consequence on the Pauli exclusion principle on the vertex function Γq/Ω isviolated, i.e., even if we have put k1 = k2 or θ12 = 0, the vertex does not vanish. For the rigorous treatment ofthis problem, see [47].


Figure 3.1: In the BCS diagram, the possible configuration of the loop momentum is restrictedinto the gray circle. As long as the total momentum drawn with the blue arrow exist, thisrestriction suppresses the BCS diagram significantly. Here we have drawn the figure for the3-dimensional case, but the same is true for other dimensions.

We introduce the dimensionless Landau functions F s/a by

F s =N(εF )

2[3ΓΩ(A) + ΓΩ(S)], F a =

N(εF )

2[ΓΩ(A) − ΓΩ(S)]. (3.3.15)

They are suitable for the channel decomposition of particle-hole channels. Next let us try tosolve Eqs.(3.3.12,3.3.13), and for this purpose we introduce the running dimensionless couplingsA,B by

A =N(εF )

2[3ΓQ(A) + ΓQ(S)], B =

N(εF )

2[ΓQ(A) − ΓQ(S)]. (3.3.16)

For these couplings, the flow equations become

−Λ∂ΛAl = −β2vFΛ

cosh2[β2vFΛ

]A2l , (3.3.17)

−Λ∂ΛBl = −β2vFΛ

cosh2[β2vFΛ

]B2l . (3.3.18)

At the UV scale Λo, A(0)l = F s

l∗ and B

(0)l = F a

l∗. Then at IR scales we find the fixed point

A∗l =F sl∗

1 + F sl∗ , B

∗l =

F al∗

1 + F al∗ . (3.3.19)

From the flow equations, Eqs.(3.3.17,3.3.18), and the expression of the fixed point, Eq.(3.3.19),we find the stability condition for the Landau Fermi liquid, which is called the Pomeranchukstability condition [48]:

F sl∗ > −1, F a

l∗ > −1. (3.3.20)

If one of the conditions (3.3.20) does not hold, description with the Landau Fermi liquid be-comes inconsistent and there are many possibilities for physical origins of these instabilities.Pomeranchuk originally derived these conditions by studying energy gains under the deforma-tion of the Fermi surface in [48].

In the rest of this subsection, let us concentrate on the RG flow in the BCS channel. Forthis purpose, put k2 = −k1 and k′2 = −k′1. In this case, Q = k′1−k1 or Q′ = k′2−k1 need not be


small and then the ZS and ZS’ graphs are suppressed due to the restriction of configurations ofloop momenta. Therefore, we neglect these graphs and only take into consideration the effect ofthe BCS graph. This approximation respects the symmetry of the vertex, and thus it makes theanalysis much simpler than that of the Landau channel. Since we have set k1 +k2 = 0, analyticexpressions for the BCS diagram Eqs.(3.2.18,3.2.19) are simplified. The 1-loop contribution ofthe BCS graph to this channel is given by the flow equation

−Λ∂ΛΓAl = − tanh

(β

2vFΛ

)N(εF )

2(ΓAl )2, (l : odd) (3.3.21)

−Λ∂ΛΓSl = − tanh

(β

2vFΛ

)N(εF )

2(ΓSl )2. (l : even) (3.3.22)

By solving the flow equations (3.3.21, 3.3.22), we find the relation between the renormalized

couplings ΓA/Sl

∗and the bare couplings Γ

A/S(0)l :

ΓA/Sl

∗=

ΓA/S(0)l

1 + N(εF )2

ΓA/S(0)l

∫ Λo0

dΛΛ

tanh(β2vFΛ

) . (3.3.23)

The Landau Fermi liquid remains stable in the BCS channel at any temperature when theinteraction in the BCS channel is marginally irrelevant, i.e., bare couplings satisfy

ΓA/S(0)l > 0. (3.3.24)

On the other hand, if one of the conditions (3.3.24) is violated, the coupling becomes marginallyrelevant and a Landau pole appears at some low temperature. To evaluate the critical temper-ature, let us consider the low temperature limit. In the low temperature limit, we can calculatethe integration over Λ quite accurately:∫ Λo

0

dΛ

Λtanh

(β

2vFΛ

)' ln

(2eγEvFΛo

πT

). (3.3.25)

Therefore the critical temperature is

Tc =2eγE

πvFΛo exp

(1

N(εF )Ξl/2

), (3.3.26)

where Ξl is the maximally negative bare coupling constant which violates the condition givenin Eq.(3.3.24).

We should emphasize the difference between singularities in the BCS channel and Pomer-anchuk instabilities. In the case of Pomeranchuk instabilities, the Landau Fermi liquid theoryitself contains self-contradictions and some of the assumptions on low energy excitations mustbe wrong. However, in the case of singularities in the BCS channel, the Landau Fermi liquidtheory is consistent as long as temperatures are higher than the critical temperature given inEq.(3.3.26). When T > Tc, we can find no singularities from these analyses and then it isnatural to conclude that we can rely on the Fermi liquid description at T > Tc but some phasetransition occur at T = Tc so that it breaks down below Tc.

3.4. Open problems to be considered 49

3.4 Open problems to be considered

In this section, we pose some open problems based on the lessens obtained in the previoussections.

As we have seen in sec.3.3, as long as we assume that low energy excitations are given byfermionic quasiparticles and that the effective cutoff is sufficiently lowered to justify Eq.(3.3.2),the Landau Fermi liquid theory is justified and many important properties of many-body physicscan be described in a unified way. However, if we have satisfied at this stage, then we cannotpredict many-body physics quantitatively from data of few body physics. In order to use resultsin sec.3.3, we should already know parameters of the low energy effective action. We will seein sec.4.5 what happens when we calculate the critical temperature using the renormalizedcoupling of scattering physics.

In sec.3.3, we have also seen that the Pomeranchuk stability condition (3.3.20) is derivedfrom the RG flow of particle-hole channels. We would also like to ask what if the coupling isso strong that the Pomeranchuk instability occurs. The answer would depend on situations.In sec.4.5 we will see that weakly interacting bosonic bound states condense at sufficiently lowtemperatures if its density is very low but the Pomeranchuk stability condition is violated dueto the sufficiently strong attractive interaction. We should emphasize that deriving this resultwould become difficult if we have performed the Hubbard-Stratonovich transformation in theparticle-hole channel, although the bosonization in this channel may seem to be plausible fromobservation of the Pomeranchuk instability.

In order to clarify physical origins of these instability, it is much better to use computationalmethods which are less biased. From this viewpoint, we will use FRG for fermionic systemsand do not rely on a bosonization such as the Hubbard-Stratonovich transformation, althoughit is often a quite powerful tool when we have already identified the origin of instabilities.

In chap.4, we apply FRG to the BCS-BEC crossover, which is a familiar system and wealready know its properties to some extent. However, physics of the BCS-BEC crossover cannotbe described with the perturbation theory and we will encounter many nontrivial things as weshall see.

In chap.5, we will consider dipolar fermionic systems with FRG. This system is not yet wellstudied and it is fruitful to analyze what kinds of instabilities can occur even at qualitativelevels.


Chapter 4

BCS-BEC crossover

Let us discuss the BCS-BEC crossover in our framework. In the weakly attracting two-component fermionic system, the Cooper instability appears due to the fluctuations aroundthe Fermi surface at low temperatures. By A. Leggett [49], and by P. Nozieres and S. Schmitt-Rink [50], it was suggested in 1980s that the BCS superfluidity smoothly continues to theBEC of bosonic molecules as the attraction increases, which is called the BCS-BEC crossover.Although it had been considered as a theoretical toy model for a long time, the BCS-BECcrossover now becomes one of the interesting and realistic physical problems after successes ofthe superfluidity of fermionic pairs in the cold atom experiments in 2004.

In sec.4.1, we give a brief and qualitative description of the BCS-BEC crossover, and we alsoreview some experimental successes of this field. Since the experimental techniques in the coldatoms are developing, we can expect that this provides a severe testing ground for theoreticalphysicists.

We will use the FRG formalism developed in previous chapters to analyze the BCS-BECcrossover. For that purpose, we need to renormalize the bare coupling. In sec.4.2, we calculatescattering processes and renormalize the bare coupling gΛo using the s-wave scattering lengthaS, which provides the unique length scale in the vacuum. Furthermore, we also consider theatom-dimer scattering problem using FRG, and explicitly show the equivalence between ournew method and conventional ones.

For the BCS-BEC crossover, the relevant extensive quantities are the number density n andthe superfluid gap ∆. In sec.4.3, we will provide several ways to calculate the number densityn: one is a direct calculation from the full 1PI effective action Γk=0 and the another is the useof the flow equation of the scale dependent number density nk. Those ways are equivalent infull calculations, but in practical calculations the different methods may give different resultsdue to approximations.

In sec.4.4, we derive the condition for the appearance of the off-diagonal long range or-der (ODLRO), which gives the Thouless criterion [51]. At first, we consider the consequenceobtained by symmetries in the Wetterich formalism of the FRG in a general way using theWard-Takahashi identity. We will apply it to spontaneous symmetry breaking to derive theThouless criterion for superconductivity.

In sec.4.5, we reproduce the NSR theory in the FRG formalism from a purely fermionictheory. We discuss to what extent we can justify that theory from the RG viewpoint, and weestimate the possible correction which arises in a natural way. Especially, we discuss Gorkovand Melik-Barkhudarov correction in the BCS regime from the viewpoint of scale separationsand clarify its physical origin. We will see that similar ideas also give free bosonic picture inthe BEC regime without introducing any bosonic auxiliary fields.

51

52 Chapter 4. BCS-BEC crossover

4.1 Introduction to the BCS-BEC crossover

In this section, we give a brief and qualitative introduction to the BCS-BEC crossover, whichis a target of our applications of the FRG method.

We consider the two-component fermionic system with the contact interaction. That is, itis described by the classical action

S =

∫d4x

[ψ(x)

(∂τ −

∇2

2m− µ

)ψ(x) + gψ1(x)ψ2(x)ψ2(x)ψ1(x)

]. (4.1.1)

Here ψ, ψ describes the two-component fermionic field. In order to circumvent possible misun-derstandings, we use 1, 2 as the flavor labels, which should be replaced by ↑, ↓ in the comparisonwith notations in sec.3.1. Here we consider the attractive interaction (g < 0) between the dif-ferent fermionic species. We describe the nature of the BCS-BEC crossover before consideringhow we can realize such a system.

At first, let us suppose that there exists the attraction but it is weak enough. In this case,if the temperature is not so low the Fermi liquid should be a good picture for the model, as wehave seen in sec.3.3. However, the Fermi liquid picture is not valid at T = 0, since there is aphase transition to the BCS superfluid at the critical temperature T = Tc. This is suggestedby the existence of a Landau pole at temperatures lower than the critical temperature Tc givenin Eq.(3.3.26).

Next, let us consider what happens if the attraction is very strong. It is easy to imaginethat for such strong attractive interaction two fermions 1 and 2 form a bound state, which isoften called a dimer. Since the contact interaction only has the s-wave component in the partialwave expansion, that is, its orbital angular momentum L is zero, the bound state would alsobe an s-wave pairing. When the system is cooled down, the state will show the Bose-Einsteincondensation (BEC) of bosonic dimers

In the seminal paper [49] by A. Leggett, it was suggested that these two different superfluidstates connect to each other smoothly as the coupling changes. He pointed out that the BCSground state wave function

ΨBCS =∏k

(uk + vkc†k,↑c

†−k,↓)|vac〉 (4.1.2)

is applicable not only in the weak coupling region originally suggested by Bardeen, Cooper, andSchrieffer (BCS) but also in the strong coupling region if the chemical potential µ is treated

BECBCS0

Cooper pairs DimersUnitary gas

Figure 4.1: The schematic view of the BCS-BEC crossover. The horizontal axis represents theinverse of the scattering length aS, which controls the interaction between different fermionspecies.

4.1. Introduction to the BCS-BEC crossover 53

open channel

closed channelbound state

Ener

gy

Interparticle distance

Figure 4.2: A schematic of the Feshbach reso-nance for 6Li.

-2

-1

0

1

2

0 500 1000 1500 2000

Sca

tterin

g le

ngth

[100

nm]

Magnetic field [Gauss]

BCS-BEC crossover

BCS phase No bound state

BEC phase Bound state

Figure 4.3: Characteristic behavior of the scat-tering length aS of 6Li.

self-consistently. The theory was extended to finite-temperature treatments by Nozieres andSchmitt-Rink (NSR) [50], which is called the NSR theory. This smooth connection betweenthe BCS state of fermionic atoms and the BEC state of bosonic molecules is the BCS-BECcrossover (see Fig.4.1).

Now we discuss the experimental access of the BCS-BEC crossover. From now on, theunique possibility of the crossover is provided by ultracold atomic gas experiments of fermionicatoms. By cooling down the system (typically about 100nK), energies of particles becomesufficiently low so that the s-wave scattering becomes dominant. Effectively, such interactionscan be treated as contact interactions, and due to the Pauli exclusion principle the scatteringbetween the same species is forbidden. Remarkably, in these systems the interaction betweendifferent species can be tuned with the external magnetic field via the Feshbach resonances.We will see some details below.

Let us explain about the Feshbach resonance [52]. To be specific, we consider the caseof 6Li, which is an alkaline atom, and Fig.4.2 gives a schematic view of the inter-particlepotential. Roughly speaking, each curve is described by the Lennard-Jones potential. With themagnetic field in the broad Feshbach resonance region, we may concentrate on the two statescharacterized by |F,mF 〉. Here F represents the total spin and mF represents its projectionalong the magnetic field. The two states are given by |1〉 := |1

2, 1

2〉 and |2〉 := |1

2,−1

2〉. If

we alternatively use the electronic and nuclear spins as labels, the states |1〉 and |2〉 have theelectronic spin −1

2and have the nuclear spin 1 and 0, respectively.

S = 1 in Fig.4.2 represents an electronic spin-triplet state, and it is called an open channel.Similarly, S = 0 represents an electronic spin-singlet state. Each curve represent the interactionfor the corresponding channel. The electronic Zeeman coupling to the triplet state enables usto tune the relative position of the curves, described by δ(B) in Fig.4.2, with magnetic field. Asa result, the energy level of the singlet bound state can be tuned around that of the scatteringstate of the open channel continuously. When the detuning ν = 0, the scattering lengthaS diverges and this is the Feshbach resonance. Since the ultracold atomic clouds are diluteenough, the density, the temperature, and the scattering length are the only scale parameters.Therefore, after the normalization of the thermodynamic quantities with the density, they onlydepends on (kFaS)−1 and T/εF . At the resonance aS =∞, any scale parameters except for thetemperatures does not affect the thermodynamic properties, and this region is called unitarity.The characteristic behavior of the scattering length aS can be found in Fig.4.3.

Let us comment on experimental developments of this area. In 1999, JILA group [55]


Figure 4.4: 40K Cooper pair condensate [53].These images are taken after the projection ofthe fermionic system to a molecular gas withthe detuning ∆B = 0.12, 0.25, and 0.55G (leftto right) on BCS side.

Figure 4.5: 6Li Cooper pair condensate [54].The initial temperatures of the atomic cloudsare T/TF = 0.2, 0.1, and 0.05 (left to right) inthe BCS side (B = 900G).

succeeded to achieve quantum degeneracy of the Fermi gas (40K). In 2003, some groups (JILA,MIT, etc. ) succeeded to realize the BEC of dimers [56, 57] using different elements (6Li inMIT, 40K in JILA). The observations of the BCS superfluid of these gases are done in 2004 bythe same groups [53, 54]. In Figs.4.4 and 4.5, we show the density profile of these condensateof the Cooper pairs in the momentum space.

4.2 Scattering problems in the vacuum

Our final goal is to describe many-body physics from the knowledge of few body physics. Forthat purpose, we need to calculate the flow equation in the vacuum to calculate scatteringproperties in the formulation of FRG. In the followings, we often use the unit 2m = 1 forsimplicity of calculations.

In order to calculate the scattering physics in the vacuum, we should take the zero densityand the zero temperature limit. This limit should be taken with some care. We should taketwo limits n → 0 and T → 0 satisfying that the temperature T is always above the criticaltemperature Tc. For that purpose, recall that the partition function Z is given by

Z = Tr[exp−β

(H − µN

)], (4.2.1)

where H is the Hamiltonian and N is the number operator. As long as the ground state energyper particle is bounded from below, we can make it by taking sufficiently large negative valuesof the chemical potential µ and T → 0. If there is no bound state we can simply take µ = 0. Ifthe bound state appears, the binding energy Eb of the bound state can produce a pole in thevertex function to give a finite density and we should let 2µ < −Eb to circumvent this problem.

Let us consider the self-energy correction. We obtain that

∂kΣk(p) = ∂k

∫d3l

(2π)3

∫dl0

2π

Γ(p+ l)e−il00

Go(l)− Σk(l) +Rk(l). (4.2.2)

We can close the contour of the l0 integration encircling the upper-half plane, and then the l0

integration vanishes since the integrand has poles only in the lower-half plane. As a result wefind that

Σk(p) = 0 (4.2.3)

in the vacuum.

4.2. Scattering problems in the vacuum 55

4.2.1 Two-body scattering

Next, let us consider the renormalization of the four-point vertex. Since the feedback fromthe higher order vertex and the particle-hole loop vanish in a similar way, the only possiblequantum correction come from the particle-particle loop. The loop correction of four-pointvertices is given by

∂kΓA/Sk (P ; q, q′) = −

1 2

1′ 2′

= − limβ→∞

∂k

∫ (T )

l

ΓA/Sk (P ; q, l − P

2)Γ

A/Sk (P ; l − P

2, q′)

([G−1o +Rk](l))([G−1

o +Rk](P − l)). (4.2.4)

At the bare scale, ΓAΛo = 0 and ΓSΛo = gΛo . Therefore, at any scale the anti-symmetric partvanishes; ΓAk (P ; q, q′) = 0. Even for the symmetric part ΓSk , the relative momentum dependencedoes not appear, since at the bare scale ΓSk = gk is a constant. Therefore, we may put ΓSk (P ) =ΓSk (P ; q, q′). Now we have found that the vertex functions ΓSk in the right hand side of Eq.(4.2.4)can be put outside of the integration, we can readily get

∂k1

ΓSk (P )= lim

β→∞∂k

∫ (T )

l

1

([G−1o +Rk](l))([G−1

o +Rk](P − l)). (4.2.5)

Since the k-dependence does appear only through Rk in the right hand side, ∂k coincides with∂k. Then we obtain that

1

ΓSk (P )− 1

gΛo

=

∫d4l

(2π)4

(1

([G−1o +Rk](l))([G−1

o +Rk](P − l))(4.2.6)

− 1

([G−1o +RΛo ](l))([G

−1o +RΛo ](P − l))

)Let us choose the regulator given by Rk(l) = (k2 − l2)θ(k2 − l2), which is a nonrelativisticversion of the Litim regulator discussed in sec.2.4. Recall that we use the unit 2m = 1, thenthe integration over l0 gives

1

ΓSk (P )− 1

gΛo

=

∫d3l

(2π)3

(1

iP 0 − 2µ+ l2 +Rk(l) + (P − l)2 +Rk(P − l)(4.2.7)

− 1

iP 0 − 2µ+ l2 +RΛo(l) + (P − l)2 +RΛo(P − l)

).

In order to fix the renormalization condition, let us put P = 0 and iP 0 = 2µ in Eq.(4.2.7). Itis easy to calculate the integration analytically, and then we get the result

1

ΓSk (iP 0 = 2µ,P = 0)− 1

gΛo

=1

6π2(Λo − k). (4.2.8)

At k = 0, ΓSk=0(P 0 = i(2q2 − 2µ),P = 0) represents the T -matrix for the scattering betweentwo different fermion species in the center of mass system. Therefore, it should be related tothe scattering length aS in the case P 0 = −2iµ and P → 0.

The T -matrix and the scattering length as is related by lim|q|→0

T (q′, q) =2πaSmr

=4πaSm

for

the particles with different species 1. Therefore, we should fine tune the bare coupling gΛo of

1In general, we can show that in the center of mass system

f(q, q′) = −mr

2πT (q′, q)


the contact interaction so that1

gΛo

=1

8πaS− Λo

6π2. (4.2.9)

By the dimensional analysis, we can recover the formula in the original unit if necessary:1

gΛo

=m

4πaS− Λo

6π2.

Let us explicitly calculate the P -dependence of the 1PI four-point vertex ΓSk=0(P ) of thesinglet channel. In the limit Λo →∞ we get

1

ΓSk=0(P )− 1

8πaS=

∫d3l

(2π)3

[1

iP 0 − 2µ+ (P2

+ l)2 + (P2− l)2

− 1

2l2

]. (4.2.10)

We can explicitly calculate this integration without any divergence to find that

ΓS(P 0,P 2) =8π

1/aS −√

(iP 0 − 2µ+ P 2/2)/2. (4.2.11)

We should notice that this expression satisfies the Galilean invariance since the center of massmomentum appears only through the combination iP 0 +P 2/4m. We can find calculations withgeneral cutoff scales k in Appendix D. There we also discuss how to remove the UV cutoff scaleΛo.

This is obtained in the imaginary time formalism, and then we need to perform the Wickrotation P 0 +2iµ 7→ iE+0 in order to get a result in the real time formalism. In this scatteringprocess, we need to input the total energy E = 2 q

2

2m= q2/m in the center of mass system

P = 0, and then the scattering amplitude is given by

f(q′, q) = −m4π

8π

2m/aS −√−(2m)3(q2/m)/2

=1

i|q| − 1/aS. (4.2.12)

This is nothing but the effective range expansion of the contact interaction. Since we havetaken the limit Λo →∞, the effective range becomes zero.

When the scattering length is negative, aS < 0, then zeros do not appear in the real axis ofthe energy E. Then, there is no bound state and we can take the limit µ 0 as a result. Onthe other hand, if aS > 0, then the pole of ΓS appears with the dispersion relation

E =P 2

2− 2

a2S

=P 2

4m− 1

ma2S

. (4.2.13)

This implies that there exists a bound state in the spin singlet channel, and its binding energyEb is given by 1/ma2

S. This is a well-known result for the two-component fermionic systemwith the contact interaction. We should restrict the chemical potential µ < −Eb/2 = −1/a2

S

for the zero density condition.

for the non-relativistic elastic scattering. Here q′ = |q|r and mr is the reduced mass. We can find the proofof this relation in some textbooks on the scattering theory, e.g., in chap.10 of the book [58]. We can find theresult for the general situations in chap.3 of the textbook [1].


4.2.2 Three-body sector: atom-dimer scattering

From previous calculations, in the case aS → −0 the notion of weakly attracting fermions isa good starting point and the Landau Fermi liquid theory gives a good description of manybody physics. On the other hand, in the limit aS → +0 a bound state, or a dimer, appears andfermions are gapped by half of the binding energy Eb/2 = 1/a2

S. Therefore, we have to derivethe bosonic picture from our fermionic theory in order to describe low-energy properties in thisregime. For this purpose, the calculation for the dimer-dimer scattering is important.

As suggested by S. Weinberg [59–61] in his work on composite particles and on multi-particle scattering problems, the vertex for the dimer-dimer scattering should be representedwith a vertex for the atom-dimer scattering. We can easily see this fact from the viewpoint ofthe Wetterich equation, and we at first consider the scattering problem between an atom anda dimer.

In the case aS < 0, the effective action up to the quartic terms is given by

Γ =

∫p

ψp(ip0 + p2 − µ)ψp (4.2.14)

+

∫P

(∫q

ψ↑,P/2+qψ↓,P/2−q

)8π

1/aS −√

(iP 0 − 2µ+ P 2/2)/2

(∫q′ψ↓,P/2−q′ψ↑,P/2+q′

).

The pole of the four-point vertex is given by the dispersion relation −iP 0 = P 2/2− 2/a2S − 2µ,

and then we should take µ < −1/a2S in order for the zero density condition. We can take the

limit µ −1/a2S to get

Γ =

∫p

ψp

(ip0 + p2 +

1

a2S

)ψp −

∫P

ΦP

16π/aS

(1 +

√1 +

a2S

2(iP 0 + P 2/2)

)iP 0 + P 2/2

ΦP . (4.2.15)

Here we have introduced the following short-hand notation for the simplicity of the expressions:

ΦP =

∫q′ψ↓,P/2−q′ψ↑,P/2+q′ , ΦP =

∫q

ψ↑,P/2+qψ↓,P/2−q. (4.2.16)

We should emphasize that Eq.(4.2.16) does not represent any transformations such as theHubbard-Stratonovich transformation and that it is just a definition of the convenient notationfor the composite operators. From Eq.(4.2.15), we can identify the boson propagator by iden-tifying the pole which is necessary for the Lehmann-Symanzik-Zimmermann (LSZ) reductionformula [62] in calculating scattering processes. That is, we can identify the fermion four-pointvertex as

= , (4.2.17)

where in the left hand side the arrowed double line represents the dimer propagator D(P ) =

−(1 +

√1 +

a2S

2(iP 2 + P 2

2))/(2iP 0 + P 2) and the vertex represents

√32π/aS =

√8π/m2aS.

Now let us consider the scattering between an atom ↑ and a spin-singlet bound state. Wewould like to determine the scattering length aad of the atom-dimer scattering. For the atom-dimer scattering, contributing Feynman diagrams consist of tree and 1PI diagrams. The tree


diagram for the atom-dimer scattering T -matrix is represented as

T adtree = =(√

32π/aS)2

i(−i/a2S) + 1/a2

S

= 16πaS =8π

maS. (4.2.18)

Here we should notice that momenta of external legs are given by p = p′ = (i/a2S,0) for external

legs of a fermion and P = P ′ = 0 for external legs of a dimer. Thus, the transfer momentumof the internal fermion line is given by q = (−i/a2

S,0). Since the T -matrix is related to theatom-dimer scattering length aad through the formula T ad = 2πaad/mr = 3πaad/m, we get

atreead /aS =

8

3' 2.67 (4.2.19)

as a tree diagram contribution.Next, let us evaluate contribution of 1PI diagrams to the atom-dimer scattering. Diagram-

matically, the flow equation can be represented as follows:

∂k = ∂k

+ + +

. (4.2.20)

Here we should regard that external legs of each diagram are antisymmetrized. The firstdiagram may seem to vanish at first sight since it represents a feedback from higher vertices,but it can contribute due to non-locality of effective vertices 2. Analytic expression of Eq.(4.2.20)is given by

−∂kΓk↑↑↓↓↑↑(p1, p2, p3; p′3, p′2, p′1)

= ∂k

∫l

(Γk↑↑↑↓↓↑↑↑(p1, p2, l, p3; p′3, l, p

′2, p′1)

[G−1o +Rk](l)

+Γk↑↑↓↓↓↓↑↑(p1, p2, p3, l; l, p

′3, p′2, p′1)

[G−1o +Rk](l)

)(4.2.21)

+∂k

∫l

(ΓSk (p2+3)Γk

↑↑↓↓↑↑(p1, l, p2+3 − l; p′3, p′2, p′1)

[G−1o +Rk](l)[G−1

o +Rk](p2+3 − l)+

ΓSk (p1+3)Γk↑↑↓↓↑↑(l, p2, p1+3 − l; p′3, p′2, p′1)

[G−1o +Rk](l)[G−1

o +Rk](p1+3 − l)

+Γk↑↑↓↓↑↑(p1, p2, p3; p′2+3 − l, l, p′1)ΓSk (p′2+3)

[G−1o +Rk](l)[G−1

o +Rk](p′2+3 − l)+

Γk↑↑↓↓↑↑(l, p2, p3; p′1+3 − l, p′2, l)ΓSk (p′1+3)

[G−1o +Rk](l)[G−1

o +Rk](p1+3 − l)

)+∂k

∫l

1

[G−1o +Rk](l)

×(−

ΓSk (p2+3)ΓSk (p′2+3)ΓSk (p1 + l)

[G−1o +Rk](l + p1−1′)[G−1

o +Rk](p2+3 − l)+

ΓSk (p1+3)ΓSk (p′2+3)ΓSk (p2 + l)

[G−1o +Rk](l + p2−1′)[G−1

o +Rk](p1+3 − l)

+ΓSk (p2+3)ΓSk (p′1+3)ΓSk (p1 + l)

[G−1o +Rk](l + p1−2′)[G−1

o +Rk](p2+3 − l)−

ΓSk (p1+3)ΓSk (p′1+3)ΓSk (p2 + l)

[G−1o +Rk](l + p2−2′)[G−1

o +Rk](p1+3 − l)

),

where we have introduced the abbreviation p2+3 = p2 + p3, p1−1′ = p1 − p′1, etc.

2Let us emphasize that this fact is consistent with a general property in nonrelativistic scattering physics:We can solve problems for n-body scattering without knowing about (n+ 1)-body scattering problems. As wewill see later, the feedback from 1PI eight-point vertices can be written with vertices in three-body sector. Sincethis point is often missed, this feedback is neglected in many previous studies.


We would like to decompose it corresponding to the diagram

= . (4.2.22)

Therefore, we define a new vertex function gadk motivated by Eq.(4.2.22) so that

Γk↑↑↓↓↑↑(p1, p2, p3; p′3, p

′2, p′1) (4.2.23)

= −ΓSk (p2+3)gadk (p1, p2+3; p′2+3, p′1)ΓSk (p′2+3) + ΓSk (p1+3)gadk (p2, p1+3; p′2+3, p

′1)ΓSk (p′2+3)

+ΓSk (p2+3)gadk (p1, p2+3; p′1+3, p′2)ΓSk (p′1+3)− ΓSk (p1+3)gadk (p2, p1+3; p′1+3, p

′2)ΓSk (p′1+3).

Indeed, we can show that this decomposition is possible directly from a formal diagrammaticexpansion of six-point vertex in terms of the coupling aS. With this decomposition, we canrelatively easily represent the three-body sector in eight-point vertices, which can contributethe first diagram in the right hand side of Eq.(4.2.20). In eight-point vertices, the three-bodysector is composed of the following diagrams:

∣∣∣∣∣∣3-body

= + + +

+ + + + . (4.2.24)

Now we can immediately obtain an expression for the feedback term in the flow equation ofvertex functions gadk . Amputate the bosonic external lines in each graph, and close two fermionlines with the two-point function ∂kRk so as not to make particle-hole loops. This procedureis clearly possible in each graph in Eq.(4.2.24), and then we here explicitly show that feedbackfrom higher-order vertices is possible even in the vacuum, but we also find that higher-bodysectors do not come in for nonrelativistic scatterings.

In order to obtain other part of the flow equation of ∂kgadk , let us substitute Eq.(4.2.23) into

Eq.(4.2.21) and use of the flow equation for ΓSk . Besides the above eight terms coming from thefeedback, we have another three terms for the flow equation of gadk :

∂kgadk (p1, p2+3; p′2+3, p

′1) = 8 feedback terms

+∂k

∫l

(ΓSk (p1+2+3 − l)gadk (l, p1+2+3 − l; p′2+3, p

′1)

[G−1o +Rk](l)[G−1

o +Rk](p2+3 − l)+gadk (p1, p2+3; p′1+2+3 − l, l)ΓSk (p′1+2+3 − l)

[G−1o +Rk](l)[G−1

o +Rk](p′2+3 − l)

+ΓSk (p1 + l)

[G−1o +Rk](l)[G−1

o +Rk](l + p1−1′)[G−1o +Rk](p2+3 − l)

). (4.2.25)

Diagrammatically, Eq.(4.2.25) can be represented as

∂k = ∂k

+ +

+ 8 feedback terms, (4.2.26)


where “eight feedback terms” are immediately obtainable from Eq.(4.2.24) as we have men-tioned. Precisely speaking, we should multiply a factor −32π/aS to the both sides of Eq.(4.2.25)to get Eq.(4.2.26). The function gk is related to the 1PI contribution to the atom-dimer scat-tering T -matrix, and the LSZ reduction formula gives

T ad1PI =32π

aSgadk=0

((i/a2

S,0), 0; 0, (i/a2S,0)

). (4.2.27)

Let us see some properties of this new vertex function gadk . The restriction coming fromFermi statistics is automatically respected in Eq.(4.2.23), and then it does not restrict the formof gadk . Using the Euclidean time reflection invariance, we find that

gadk (p1, p2+3; p′2+3, p′1) = [gadk (θp′1, θp

′2+3; θp2+3, θp1)]∗. (4.2.28)

Next, let us reduce the number of variables in gadk . Due to the momentum conservation,P := p1 + p2+3 = p′1 + p′2+3. Furthermore, we can easily find that we may put P = (i/a2

S,0) inEq.(4.2.25) to extract the quantity in Eq.(4.2.27). Introducing

gadk (q, q′) := gadk (P + q,−q;−q′, P + q′), (4.2.29)

with P = (i/a2S,0), we can rewrite Eq.(4.2.25), or Eq.(4.2.26), as

∂kgadk (q, q′) = ∂k

∫l

ΓSk (−l)[G−1

o +Rk](l + P )

(gadk (l, q′)

[G−1o +Rk](−l − P − q)

+gadk (q, l)

[G−1o +Rk](−l − P − q′)

+1

[G−1o +Rk](−l − P − q)[G−1

o +Rk](−l − P − q′)

)+ feedback. (4.2.30)

In Appendix D, we solve this equation with an approximation in which we neglect the feedbackeffect from eight-point vertices. In the following, instead of solving this flow equation we discussits properties and find formal solutions using Dyson-Schwinger equations.

Formal solutions

Let us discuss a formal solution of Γk↑↑↓↓↑↑ , or gadk . As we will see below using Dyson-Schwinger

equations, the vertex function gadk (q, q′) satisfies the integral equation

gadk (q, q′) =

∫l

ΓSk (−l)[G−1

o +Rk](l + P )

(gadk (l, q′)

[G−1o +Rk](−l − P − q)

+1

[G−1o +Rk](−l − P − q)[G−1

o +Rk](−l − P − q′)

), (4.2.31)

and similarly satisfies

gadk (q, q′) =

∫l

ΓSk (−l)[G−1

o +Rk](l + P )

(gadk (q, l)

[G−1o +Rk](−l − P − q′)

+1

[G−1o +Rk](−l − P − q)[G−1

o +Rk](−l − P − q′)

). (4.2.32)

At k = 0, these integral equations reduce to the integral equation for the atom-dimer scatteringT -matrix, which was derived by G. Skornyakov and K. Ter-Martirosyan [63].


Using the Dyson-Schwinger equation (2.2.26), or Eqs.(2.2.30,2.2.31), we can derive inte-gral equations given above. Let us recall that Eq.(2.2.31) suggests that the Dyson-Schwingerequation must be true even for IR regularized theories. For the six-point vertex Γk

↑↑↓↓↑↑ in our

model, we have two Dyson-Schwinger equations in the vacuum. Diagrammatically, they can berepresented as

= + , (4.2.33)

where black dots represent the bare contact interaction, and as

= + + . (4.2.34)

In deriving the second Dyson-Schwinger equation, Eq.(4.2.34), for the six-point vertex, weuse the Dyson-Schwinger equation for the four-point vertex ΓSk and the first Dyson-Schwingerequation, Eq.(4.2.33), for the six-point vertex. This second expression (4.2.34) of the Dyson-Schwinger equation is equivalent to the integral equation for three-body scattering problemsderived by S. Weinberg in [61].

Combining these two Dyson-Schwinger equations and applying the decomposition (4.2.23)of the six-point vertex, we can obtain an integral equation. Its diagrammatic expression isgiven by

= + . (4.2.35)

This is nothing but the integral equation (4.2.31). One can find direct derivation of this integralequation by considering all possible Feynman diagrams in the paper [64]. We here inventeda new way to derive the integral equation (4.2.31) for the three-body scattering T -matrix byusing FRG, and explicitly proved that those two formalism can give the same answer.

Let us explicitly show that the solution of this integral equation satisfies the flow equation(4.2.30). For this purpose, it is better to write down the integral equation (4.2.35) symbolicallyas

gad = G ·K · gad +G ·K ·G, (4.2.36)

where

gad = , G = , K = . (4.2.37)

Here we made their k-dependence just for simplicity of notations. We can formally write downthe solution of Eq.(4.2.36):

gad = (1−G ·K)−1 ·G ·K ·G = G ·K ·G+G ·K ·G ·K ·G+ · · · . (4.2.38)

In order to derive the flow equation, let us take a derivative with respect to k of the both sidesof Eq.(4.2.38). Leibniz rule and formal resummation gives

∂kgad = ∂k(G ·K ·G) + ∂k(G ·K) · (G ·K ·G+ · · · ) + (G ·K ·G+ · · · ) · ∂k(K ·G) (4.2.39)

+(G+G ·K ·G+ · · · ) ·K · ∂kG ·K · (G+G ·K ·G+ · · · )+(G ·K ·G+G ·K ·G ·K ·G+ · · · ) · ∂kK · (G ·K ·G+G ·K ·G ·K ·G+ · · · ).


Combining Eqs.(4.2.38,4.2.39), we obtain

∂kgad = ∂k(G·K ·G)+∂k(G·K)·gad+gad·∂k(K ·G)+(G+gad)·K ·∂kG·K ·(G+gad)+gad·∂kK ·gad.

(4.2.40)Now it is clear that Eqs.(4.2.24,4.2.26) and Eq.(4.2.40) are equivalent as it should be sinceDyson-Schwinger equation must hold at any values of k.

It would be very important to comment that, in the flow equation of gadk , quadratic termsof the vertex gadk appears, and that it originates from the feedback from eight-point vertices inthe flow of six-point vertices. This structure must become important when we discuss Efimovphysics without introducing auxiliary composite fields, since limit cycle structure can occurunder existence of this quadratic term.

Remark. We can expect that a formulation for the dimer-dimer scattering problems is alsogiven in a similar way, but this task is still open in our formalism. Even though we can solvesuch problems by deriving integral equations directly from Dyson-Schwinger equation, as wehave done here for the atom-dimer scattering, formulating that problem in the context of FRGis still important. Using FRG, we can study many-body physics by reflecting knowledge offew-body physics thanks to ideas of scale separations. Therefore, formulating few-body physicsin FRG and solving it, we can immediately find some knowledge of many-body physics.

4.3 Number equation

In specifying thermodynamic states, it is natural to fix the number of fermions NF rather thanthe chemical potential µ especially in the strong coupling region. Of course, those two quantitiesare related via the Legendre transformation and then they have equivalent informations on asystem in principle. However, in the strong coupling region we often need a crude approximationfor calculations and lose the correct relation between them. In order to obtain sensible resultseven in such situations, we should treat relevant operators themselves, and in this section wewill explicitly show why the NSR theory gives a correct picture of the BCS-BEC crossover fromthe viewpoint of the composite operator flow in sec.2.2.

Let us define the number of fermions nk in IR regularized theories by

nk = − 1

βV

∂Γk∂µ

∣∣∣∣ψ,ψ=0

.

In stead of using this formula for the number density n = nk=0, we will try to use the compositeoperator flow equation. The number density operator is given by

n[ψ, ψ] = ψ(τ + 0,x)ψ(τ,x), (4.3.1)

and then we can extend it as a correlation function nk[ψ, ψ] according to Eq.(2.2.18). It obeysthe flow equation (2.2.19). Let us denote the expansion of the correlation function nk[ψ, ψ] interms of fields ψ, ψ as

nk[ψ, ψ] = nk + n(2)nmk ψnψm + · · · . (4.3.2)

At k = Λo, nΛo = 0 and n(≥4)Λo

= 0 since they are absent in the classical limit, and n(2)k =

δαβ δ4(x − y) in the coordinate basis. At k = 0, nk=0 should give the number density of the

4.3. Number equation 63

full theory n. Let us introduce a diagrammatic expression for the expansion of Eq.(4.3.2) asfollows:

nk[ψ, ψ] = + ψ ψ +

ψ ψ

ψ ψ

+ · · · . (4.3.3)

We can obtain the flow equation for each coefficient. Let us write it down using Eq.(2.2.19)and using the technique described in Appendix A. The flow equations for the constant andtwo-point parts are given by

∂k = ∂k = − , (4.3.4)

∂k

ψ ψ

= ∂k

+

. (4.3.5)

Triangle vertices represent operators in Eq.(4.3.3), square ones represent the negative of 1PIvertices, and the blob represent the derivative of the IR regulator ∂kRk. Lines with an arrowrepresent propagators (G−1

o − Σk + Rk)−1, and we should regard that all external lines are

amputated in the both sides of equations.On the other hand, we can calculate each coefficient in nk[ψ, ψ] directly using formulae

(A.2.3,A.2.5). Let us denote the insertion of the two-point vertex ψψ(x) with the crossedvertex. Then, we get

= =

∫ (T )

l

−2e−il00

[G−1o − Σk +Rk](l)

, (4.3.6)

ψ ψ = ψψ(x) + . (4.3.7)

For later convenience, we give the analytic expression of Eq.(4.3.6). In the numerator, thenegative sign appears due to the fermion loop, the factor 2 appears due to the number of spindegrees of freedom, and the factor e−il

00 appears due to the infinitesimal difference of imaginarytimes of field arguments in the definition of Eq.(4.3.1).

Substituting the result of Eq.(4.3.7) into the right hand side of Eq.(4.3.4), we can seeconsistency between the flow equation and its solution. For this purpose, let us substituteEq.(4.3.7) in to the flow equation of nk to get

∂knk = − = − − . (4.3.8)

Recall that the flow equation for the self-energy is given by Eq.(3.2.6), which can be diagram-matically represented as

∂kΣkαα′(p) = ∂k

(p p

α α′

l, β ). (4.3.9)


Combining Eqs.(4.3.8,4.3.9), we obtain that

∂knk =

∫ (T )

l

−2e−il00

([G−1o − Σk +Rk](l))2

(−∂kRk(l) + ∂kΣk(l)) = ∂k . (4.3.10)

As a result, we find the consistency between the flow equation of the correlation functionnk[ψ, ψ] and its analytical expression with 1PI vertices.

As a by-product of this consideration, we have now obtained two different ways in order forevaluations of the number density. One way is to use the formula (4.3.6) at k = 0, then we canobtain the number density directly:

n = nk=0 =

∫ (T )

l

−2e−il00

G−1o (l)− Σ(l)

, (4.3.11)

where Σ = Σk=0. As long as we have solved the flow equation of the self-energy, we can obtainthe number density using it.

Another way to get the number density is to use the flow equation (4.3.8). The analyticexpression is given by

∂knk =

∫ (T )

l

2e−il00∂kRk(l)

([G−1o − Σk +Rk](l))2

+

∫ (T )

l,l′

2e−i(l0+l′0)0

([3ΓAk + ΓSk ](l + l′; l−l

′

2, l−l

′

2))∂kRk(l

′)

([G−1o − Σk +Rk](l))2([G−1

o − Σk +Rk](l′))2.

(4.3.12)If we solve the flow equation containing up to four-point vertices, we can immediately evaluate∂knk. Therefore, we can also evaluate n = n0 using this flow equation, which does not requireto solve another renormalization group equation.

Of course, Eqs.(4.3.11,4.3.12) are equivalent, and they should give the same answer. How-ever, in practical calculations, we have to adopt some approximation schemes in solving theWetterich equation. Approximations break this equivalence and they will give different results.For example, in the NSR theory the self-energy correction is totally neglected and then if weuse Eq.(4.3.11) the number density is just that of free fermions. That must be totally wrong atleast in the BEC regime. In Eq.(4.3.12), on the other hand, the second term in the right handside contains the effect of the formation of dimers through the four-point vertex, and it wouldgive a better solution when Σk is put to be 0 due to approximations.

4.4 Thouless criterion

In order to realize good approximations to the Wetterich equation, we need to have a physicalinsight to the system which we are studying. Especially if the system has a symmetry then wecan use it as a guiding principle and we would like to respect the symmetry in the flow of theeffective action. In this section, we would like to study the effect of symmetries in the FRGformalism.

We also consider a second order phase transition, and especially we will derive the Thoulesscriterion for the superfluid phase transition, which plays an essential role in the determinationof the critical temperature.

4.4. Thouless criterion 65

4.4.1 Ward-Takahashi identity

At first, let us briefly derive the Ward-Takahashi (WT) identity [65,66] for the full theory andby finding the correlation functional IWT[J, φ]. Let us assume that the theory is invariant underan infinitesimal transformation of fields φn 7→ φn + εΦn[φ], i.e., the product of the measure Dφand the Boltzmann factor exp (−S[φ]) does not change under the transformation. Here Φn[φ]is a local functional of φ. Change path integral variables φn 7→ φn + εΦn in Eq.(2.1.4), then weobtain

0 =

∫Dφ [exp (−S[φ] + Jn(φn + εΦn[φ]))− exp (−S[φ] + Jnφn)]

= εJn∫Dφ Φn[φ] exp (−S[φ] + Jnφn) .

Therefore, this is the WT identity and we can rewrite it as

JnΦn

[δLδJ

]exp (W [J ]) = 0. (4.4.1)

Performing the Legendre transformation and substituting δLWδJn

= ϕn, we can find that

δRΓ[ϕ]

δϕn

(Φn

[ϕ+G[ϕ] · δL

δϕ

]1

)= 0, (4.4.2)

where G[ϕ] = W (2)[J [ϕ]]. Especially when we are considering a symmetry generated by a lineartransformation, we can simply put (Φn[ϕ+G · (δL/δϕ)]1) = Φn[ϕ], and then the 1PI effectiveaction turns out to have the same symmetry with the classical action.

An immediate application of the above consideration shows the following: Suppose that theclassical action S[φ] and the flat path integral measure Dφ are invariant under the symmetrytransformation independently. Then, as long as one uses the regulator Rk which preserves thesame global symmetries, the effective action Γk at all scales k also satisfies those symmetries.

It is usually a difficult problem to find a symmetric regulator for a nonlinear symmetry.Investigating the situation where the symmetry is explicitly broken by the regulator is thusimportant, and indeed we will encounter such situations when the Nambu-Goldstone bosons aretreated as elementary fields and when the non-Abelian gauge theories are considered. Accordingto Eq.(4.4.1), the correlation function representing the WT identity is

IWT[J, φ] = JnΦn[φ]. (4.4.3)

In order to apply Wetterich’s formulation, we define the flowing correlation function IWT[J ;Rk]by

eWk[J ]IWT[J ;Rk] = exp

(−1

2

δLδJ·Rk ·

δLδJ

)(IWT

[J,δLδJ

]exp(W [J ])

). (4.4.4)

Theorem 2.2.4 suggests that IWT[J ;Rk] = 0 at any k, which is already trivial from the expressionof IWT[J,Rk] in Eq.(4.4.4) and from Eq.(4.4.1). Explicit calculations show that

IWT[J ;Rk] =

∫Dφ (JnΦn[φ]− φnRnm

k Φm[φ]) exp(−S[φ]− 1

2φ ·Rk · φ+ J · φ

)∫Dφ exp

(−S[φ]− 1

2φ ·Rk · φ+ J · φ

)= e−Wk[J ]

(JnΦn

[δ

δJ

]− δLδJn

Rnmk Φm

[δLδJ

])eWk[J ] (4.4.5)

= (Jm − ϕn[J ]Rnmk )

(Φm

[ϕ[J ] +

δLδJ

]1

)−Rnm

k

(δLδJn

Φm

[ϕ[J ] +

δLδJ

]1

).


Let us perform the Legendre transformation in the sense of Eq.(2.2.3), and then we obtain theflowing correlation function

IWT[ϕ;Rk] = IWT[J [ϕ];Rk] (4.4.6)

=δRΓkδϕm

(Φm

[ϕ+Gk[ϕ] · δL

δϕ

]1

)−Rnm

k Gk[ϕ]nl

(δLδϕl

Φm

[ϕ+Gk[ϕ] · δL

δϕ

]1

).

As we have seen in the case of the Dyson-Schwinger equation, we can regard the WT identityIWT = 0 as a consequence of the integrated flow. Let us consider the classical limit k → Λo,then the regulator becomes heavy Rk →∞. Since Gk[ϕ] = (Γ

(2)k [ϕ] + Rk)

−1, this limit impliesthat Rk ·Gk → 1 and Gk → 0. We have obtained that

limk→Λo

IWT[ϕ;Rk] =δRS[ϕ]

δϕnΦn[ϕ]− STr

(δLδϕm

Φn[ϕ]

). (4.4.7)

In Eq.(4.4.7), the first term in the right hand side gives the infinitesimal change of the classicalaction S[φ], and the second term represents the Jacobian of the path integral measure Dφ.Indeed, under the transformation φn 7→ φn + εΦn[φ] the path integral measure becomes

Dφ =∏n

dφn 7→∏n

dφm

(δmn + ε

δLδφm

Φn[φ]

)= DφSDet

(δmn + ε

δLδφm

Φn[φ]

). (4.4.8)

Up to the first order of the infinitesimal parameter ε, the Jacobian becomes

SDet

(δmn + ε

δLδφm

Φn[φ]

)= exp εSTr

(δLδφm

Φn[φ]

), (4.4.9)

and the second term in Eq.(4.4.7) appears. Since the product Dφ exp(−S[φ]) is invariant underthe symmetry transformation, the right hand side of Eq.(4.4.7) should vanish by hypothesis.

Hence, we get the unique solution of the flow equation (2.2.19) so that IWT[ϕ;Rk] = 0 for anyk, especially for k = 0.

Remark. With the help of the Batalin-Vilkovisky formalism, or the anti-field formalism, onetries to control the explicit violation of gauge symmetries due to IR regulators. The studiesin this direction is reviewed in detail in the review [67], which is based on the Polchinskiequation not on the Wetterich formalism which we use here. One has also studied FRG in thebackground field gauge, and for example we can find the papers [26, 68, 69]. In these studies,the modified version of the Slavnov-Taylor identity, which extends Eq.(4.4.6) so that it containsthe BRST transformed fields, often plays an important role. On the other hand, there are otherapproaches to invent a manifestly gauge invariant formalism of FRG. For example, see [70–72]for the approach to construct the manifestly gauge invariant effective action using the Wilsonlines. Any programs in these studies are not completed, and further investigations are stillnecessary.

4.4.2 Spontaneous symmetry breaking

Let us discuss the spontaneous symmetry breaking (SSB) in the context of FRG. For ourpurpose, it is enough to discuss SSB associated with the second order phase transition.

4.4. Thouless criterion 67

Critical fixed point

Critical surface

Space of theories

Line of theories with different temperatures

Symmetric theories

Symmetry broken phase

Figure 4.6: A space of theories with a second order phase transition. The red curve representsa line of theories with different temperatures. We indicate two RG flows, one of which showsthe flow for a symmetric theory and another one represents the flow for a symmetry brokentheory.

To solve degeneracies of vacua, we add a symmetry breaking term O[φ] to the bare actionS[φ] at the UV scale k = Λo. In this case, we also define the flowing Schwinger functionalWk[J,∆Λo ] by

exp(Wk[J, g]) =

∫Dφ exp

(−S[φ]−∆ΛoO[φ]− 1

2φ ·Rk · φ+ J · φ

), (4.4.10)

where ∆Λo is a new coupling constant associated with the symmetry breaking termO. The flow-ing 1PI effective action Γk[ϕ,∆Λo ] is again defined by the Legendre transformation of Wk[J,∆Λo ]in the sense of Eq.(2.2.3). Consider the infinitesimal transformation φn 7→ φn + εΦn[φ] whichpreserves Dφ exp(−S[φ]), then we find that∫

Dφ [(J − φ ·Rk) · Φ[φ]] exp

(−S[φ]−∆ΛoO[φ]− 1

2φ ·Rk · φ+ J · φ

)= g

∫Dφ[δRO[φ]

δφ· Φ[φ]

]exp

(−S[φ]−∆ΛoO[φ]− 1

2φ ·Rk · φ+ J · φ

). (4.4.11)

The left hand side of Eq.(4.4.11) is nothing but the WT operator IWT[J,∆Λo ;Rk] defined in

Eq.(4.4.4). Let us denote that δO[φ] =δRO[φ]

δφ· Φ[φ], then the right hand side of Eq.(4.4.11)

becomes gδOk[J, g]. Therefore, the WT identity should be modified as

IWT[ϕ,∆Λo ;Rk] = gδOk[ϕ,∆Λo ]. (4.4.12)

Let k = 0, then the discussion reduces to the one for the original theory. If the symmetry isnot broken, all physical quantities are analytic in terms of the coupling g and the WT identity(4.4.2) holds in the limit ∆Λo → 0. On the other hand, if the theory is in the symmetry brokenphase, the right hand side of Eq.(4.4.12) does not vanish at k = 0 and in the limit ∆Λo → 0and the WT identity does not hold.

Since we assume the second order phase transition, the critical fixed point exists. To bespecific, let us consider a line of theories parametrized by temperatures T which is drawn as a


red curve in Fig.4.6. At T = Tc, the line crosses the critical surface and the second order phasetransition occurs. Let us consider the RG flow in a symmetry broken phase. Here we assumethat the symmetry is not explicitly broken through the regulating term. At k = Λo, the theoryis classical and the symmetry is not broken and then there exists a critical scale k = kc, atwhich the IR regularized theory lies on the critical surface 3. For k > kc the RG flow coincideswith the symmetric one in the limit ∆Λo → 0, and for k < kc flowing 1PI effective action Γkdoes not satisfy the symmetry.

In order to get the Thouless criterion for the BCS-BEC crossover, let us consider fermionictheories with the singlet pairing. Since our regulator ψRkψ manifestly satisfies the global U(1)symmetry, Eq.(4.4.12) becomes

δΓk[ψ, ψ] = gδOk[ψ, ψ]. (4.4.13)

Our purpose is not to derive the gap equation but to get the critical temperature, and thus wecan neglect anomalous vertices with legs ≥ 4 because they also vanish in the limit k → kc orT → Tc with ∆Λo → 0.

As a symmetry breaking term, we take a fermion bilinear

O =

∫dx

1

2

(ψ(x)εψ(x)− ψ(x)εψ(x)

). (4.4.14)

Within the approximation neglecting higher order anomalous vertices, we find that Eq.(4.4.13)becomes

∆k := = ∆Λo

+

. (4.4.15)

If the original theory shows the spontaneous symmetry breaking, scale dependent gaps ∆k doesnot vanish in the limit ∆Λo → 0 for k < kc. Therefore, at k = kc the four point vertex in thesinglet channel ΓSk (P ) diverges if we put P = 0. Especially at the critical temperature T = Tc,the critical scale kc = 0, and then we obtain the Thouless criterion for superconductivity [51]:ΓS(P )→∞ as P → 0.

Remark. We should emphasize that Eq.(4.4.15) is a result of approximations, which are validonly when ∆Λo → 0 and k kc. In order to calculate the correct gap in the symmetrybroken phase, we have to invent an approximation which is compatible with the WT identityEq.(4.4.12) or Eq.(4.4.13). Even though our model only has a simple symmetry, i.e., the globalU(1) symmetry, it is still nontrivial how to keep the constraint in the RG flow of symmetrybroken phases. In this thesis we do not attack this problem, but since we have derived theThouless criterion for the superfluid phase transition we can calculate the critical temperatureof our model.

4.5 Nozieres-Schmitt-Rink theory

In this section, we try to reproduce the NSR theory from the viewpoint of FRG. Since it isvalid only in the deep BCS and in the deep BEC regimes, we consider those two limits and

3Here we implicitly assume that any exotic behaviors of the RG flow is absent: the RG flow crosses thecritical surface only once. In our application of FRG to the BCS-BEC crossover, this assumption is plausible.

4.5. Nozieres-Schmitt-Rink theory 69

simply connect the results. At the same time, we also consider some possible corrections to theNSR theory using the RG flow equation. We will invent the better approximations motivatedby those analyses.

Again we use the unit 2m = 1 for simplicity of calculations.

4.5.1 Deep BCS regime

Let us consider the asymptotic behavior in the limit (kFaS)−1 → −∞. We call this the deepBCS regime. At least in the vacuum, fermions interact via the weak attraction. At first, assumethat an effective cutoff scale k2 is much larger than the temperature T and the chemical potentialµ. Then the flow equation does not change so much from that in the vacuum, and as long ask2 T, µ we may approximate the effective action Γk as

Γk[ψ, ψ] '∫ (T )

p

ψpG−1o (p)ψp + gk

∫ (T )

p,p′,q′,q

δp+p′,q+q′ψ↑,pψ↓,p′ψ↓,q′ψ↑,q. (4.5.1)

Here G−1o (p) = ip0 + p2 − µ and g−1

k =1

ΓSk (P = 0)=

1

8πaS− k

6π2. Clearly, this approximation

should break down if the cutoff scale becomes of the order of physical scales T, µ.At first, let us evaluate an amount of the self-energy correction. It would be natural to

input the external momentum of the order of√µ ∼ kF , then for evaluation of the self-energy

we should use the effective coupling at some scale k ∼ √µ ∼ kF since the external momentumbehaves as an IR cutoff for the coupling in the self-energy diagram effectively. At the 1-looplevel, we find that

Σ(p) 'p p

α α′

l, β

'∫ (T )

l

gkF e−il00

il0 + l2 − µ

= −gkF∫

d3l

(2π)3

1

eβ(l2−µ) + 1

= −µ4√µaS/π

1− 4kFaS/(3π)

∫ ∞0

dxx2

exp (µ(x− 1)/T ) + 1. (4.5.2)

Therefore, it roughly gives the correction of the Fermi momentum kF from a naive estimationµ = k2

F , since k2F = µ+ Σ in our approximations here.

µ = k2F

(1 +

4

3πkFaS

)=k2F

2m

(1 +

4

3πkFaS

). (4.5.3)

Since the chemical potential can be regarded as an energy which is necessary to add a particleto the system, this indicates that due to the attractive interaction µ is smaller than the Fermienergy εF = k2

F/2m. Here we have derived the leading term in terms of kFaS and the momentumintegration of l is evaluated at T = 0. Anyway, we have found that in the small coupling regionthe self-energy just shift the chemical potential µ in Eq.(4.5.3) to the Fermi energy εF = k2

F/2m.Since we have found that the Fermi surface is a good notion in the deep BCS regime

(kFaS)−1 −1, we can use the Landau Fermi liquid theory described in sec.3.3. A consequenceof those analyses is that if we have integrated out the modes except for those in the vicinityof the Fermi surface then the scaling ansatz works well. In this scaling regime, only the BCS


diagram can contribute to the superfluid phase transition and other diagrams, ZS and ZS’, dropout.

As a first step, let us totally neglect the effect of the ZS and ZS’ diagram since those diagramscannot contribute in the both high- and low-energy regions (Fig.4.7). In this approximation,we can evaluate the flow of the coupling. Again only the spin-singlet part of the four-pointvertex survives in this approximation, and the relative momentum dependence of it does notexist (see Sec.4.2). Then, the flow equation becomes

∂kΓSk (P ) ' −∂k

1 2

1′ 2′

= −∂k∫ (T )

l

ΓSk (P )2

(G−1o (l)− Σk(l) +Rk(l))2

. (4.5.4)

Since in the deep BCS regime we have already found that the self-energy correction is quitesmall, let us neglect Σk in the flow of the coupling. In this approximation, we get

1

Γk(P )− 1

gΛo

=

∫ (T )

l

[1

(G−1o (l) +Rk(l))2

− 1

(G−1o (l) +RΛo(l))

2

]. (4.5.5)

We find that in the limit Λo →∞ and at k = 0

1

Γk=0(P )=

1

8πaS+

∫d3l

(2π)3

1− nF((

P2− l)2 − µ

)− nF

((P2

+ l)2 − µ

)iP 0 − 2µ+

(P2− l)2

+(P2

+ l)2 − 1

2l2

. (4.5.6)

The Thouless criterion gives1

ΓS(P = 0)= 0, which leads us to the condition

− 1

aS=

2

π

∫ ∞0

√εdε

[tanh

(βc2

(ε− µ))

2(ε− µ)− 1

2ε

]. (4.5.7)

This is the gap equation at T = Tc used in the NSR theory, which just reproduces the result ofthe BCS theory as we have seen in sec.3.3. Indeed, in the low temperature limit and puttingµ = k2

F , we find that the critical temperature Tc is given by

Tc =8eγE

πe2εF exp

(π

2kFaS

)' 0.613 . . .× εF exp

(− π

2kF |aS|

), (4.5.8)

where εF = k2F/2m is the Fermi energy and γE = 0.5772 . . . is the Euler constant.

From now on, we have totally neglected the contributions of the ZS and ZS’ diagramsalthough we have no reason to neglect them in an intermediate energy region characterized byk2 ∼ T, µ. Let us evaluate the effect of those diagrams

1 2

1′ 2′+

1 2

1′ 2′

,

and then we will reproduce the leading correction to the BCS theory originally suggested byGorkov and Melik-Barkhudarov [73]. Similarly, the self-energy correction also emerges fromthe same energy scale as we have seen in Eq.(4.5.2), but we neglect its effect to the RG flow ofthe coupling constant for simplicity. Recall that low energy excitations occur around the Fermi


Energy scale

UV cutoff

Fermi energy

RG flow is described by the flow in vacuum.Only the BCS diagram renormalizes effective couplings between fermions.

Matter starts to affect the RG flow.Especially, particle-hole loops screen interactions between fermions.

Fermi liquid theory describes low energy physics. The scaling ansatz picks up only the BCS diagram in the RG flow.

Figure 4.7: Separation of energy scales in the deep BCS regime.

surface. Therefore, a typical scale of the center of mass momentum incoming to the effectivecoupling in the ZS or ZS’ diagrams is again of the order of

√µ ' kF . Furthermore, the transfer

momenta in the ZS or ZS’ diagram is also of the order of kF . Let us assume that we can use the1-loop approximation for these diagrams in the weak coupling region, then their contributiongZS+ZS′

kFto the effective coupling becomes

gZS+ZS′

kF= −

∫ (T )

l

g2kF

G−1o (l)G−1

o (q + l), (4.5.9)

where |q0| ∼ 2πT and |q| ∼ kF . Here we have not restricted the possible configurations of theloop momenta l, but this expression automatically satisfies desirable properties. To see this,let us perform the Matsubara sum, which gives

gZS+ZS′

kF= g2

kF

∫d3l

(2π)3

nF (l2)− nF ((l + q)2)

iq0 + (l + q)2 − l2. (4.5.10)

In the high-energy region specified by l2, (l + q)2 k2F , the Fermi distribution function nF

suppresses the integrand exponentially and this is consistent with the fact that the ZS andZS’ diagrams vanish in the vacuum. On the other hand, in the low-energy region specified byl2, (l + q)2 ' k2

F , the possible configurations of the loop momentum l is too restrictive andthe quantum correction from this region should be small. This is consistent with the analysisin sec.3.3, which implies that the ZS and ZS’ diagrams does not contribute to the superfluidinstability in the scaling regime. Therefore, we can crudely regard that Eq.(4.5.10) representsthe loop correction coming from the intermediate region and that it roughly gives the correctionto the effective coupling gkF used in the BCS diagram in the scaling region.

Now, let us evaluate the effect of the ZS and ZS’ diagrams more quantitatively. In Eq.(4.5.10),the result depends on the length of transfer momenta |q|. Since we plug the result into the BCSdiagram evaluated in the lower energy region, it would be natural that we apply the projectionto the s-wave scattering. Since the transfer momentum is represented as q = p ± p′ withincoming and outgoing momenta p,p′, we redefine the effective contribution gZS+ZS′

kFas

gZS+ZS′

kF= g2

kF

∫ 1

−1

d cos θ

2

∫d3l

(2π)3

nF (l2)− nF ((l + q(θ))2)

iq0 + (l + q(θ))2 − l2, (4.5.11)


where |q(θ)| = kF√

2(1− cos θ) and θ is the relative angle between p and p′. For simplicity,let us evaluate the loop integration in Eq.(4.5.11) at T = 0, then it becomes∫

d3l

(2π)3

nF ((l− q/2)2)− nF ((l + q/2)2)

(l + q/2)2 − (l− q/2)2

=

∫ ∞0

l2dl

4π2

∫ 1

−1

d cos θlθ(k2

F − (l2 − lq cos θl + q2/4))− θ(k2F − (l2 + lq cos θl + q2/4))

2lq cos θl

=1

4π2

∫ 1

−1

d cos θl2

√k2F −

q2

4(1− cos2 θl).

We now find that Eq.(4.5.11) becomes

gZS+ZS′

kF= g2

kF

kF4π2

∫ 1

−1

d cos θ

2

∫ 1

−1

d cos θl2

√1− 1

2(1− cos θ)(1− cos2 θl)

= N(εF )g2kF

1 + ln 4

3' gkF × 0.795 . . .× 2

π(kFaS). (4.5.12)

Here we have introduced the density of state N(εF ) = kF/4π2 = mkF/2π

2. Therefore, at thelower energy scales we should use the effective coupling constant replaced as

gkF 7→ gkF + gZS+ZS′

kF= gkF

(1 +N(εF )gkF

1 + ln 4

3

)' gkF

(1 +

1 + ln 4

3

2kFaSπ

). (4.5.13)

This replacement affects the exponent in Eq.(4.5.8) in the leading order to add a constant

−1 + ln 4

3. Therefore, the critical temperature is reduced by a factor (4e)−1/3 ' 0.45 and then

we should replace Eq.(4.5.8) by

Tc =8

πeγE−2−(1+ln 4)/3εF exp

(− π

2kF |aS|

)(4.5.14)

in the deep BCS regime (kFaS)−1 −1. Using the idea of the RG flow, we have found thatthe Gorkov and Melik-Barkhudarov correction is reproduced in a quite natural way. What isimportant is that, since we have a scale parameter k, we can flexibly change approximationmethods at each scale and find physical insights of important diagrams. We expect that theseinsights obtained here will help us in going beyond Gorkov and Melik-Barkhudarov corrections,especially when we approach to resonance regions from the BCS side.

4.5.2 Deep BEC regime

Next let us consider the opposite limit (kFaS)−1 1, which is called the deep BEC regime 4.In the vacuum a bosonic dimer is formed as a spin-singlet bound state in this region and itsbinding energy Eb is given by 1/ma2

S according to the analysis in sec.4.2, which is large in thedeep BEC regime.

In the vacuum, the effective action is given by Eq.(4.2.15) in which the chemical potential µis put −1/a2

S. In order to clarify the matter effect, let us introduce another chemical potentialµd defined by µ = −1/a2

S + µd/2.

4We use the same notation kF as a length scale given by the number density n via the formula n = k3F /3π

2

even not in the BCS regime. We also use the notation εF as a unit of energy scale: εF = k2F /2m = (3π2n)2/3/2m.


Again, we evaluate the self energy correction with the 1-loop approximation with the full fourpoint vertex in vacuum. We should emphasize that this is still a nonperturbative approximationmotivated from the RG analysis, and it is consistent with a Dyson-Schwinger equation.

Σ(p) 'p p

α α′

l, β

=

∫ (T )

l

e−il00

il0 + l2 + 1/a2S − µd/2

ΓSk=0(p+ l)

= −16π

aS

∫ (T )

l

e−il00

1 +

√1 +

a2S

2(il0 + ip0 + (p+ l)2/2− µd)

[il0 + l2 + 1/a2S − µd/2][il0 + ip0 + (p+ l)2/2− µd]

. (4.5.15)

Let us perform the Matsubara sum, which gives

1

β

∑l0=(2n+1)πT

e−il00

1 +

√1 +

a2S

2(il0 + ip0 + (p+ l)2/2− µd)


=

∮Co

dω

2πi

−e+ω0

eβω + 1

1 +

√1 +

a2S

2(−ω + ip0 + (p+ l)2/2− µd)

[−ω + l2 + 1/a2S − µd/2][−ω + ip0 + (p+ l)2/2− µd]

(4.5.16)

Here Co is a contour enclosing the imaginary axis in the ω plane. See Fig.4.8 and we canchange the contour of the integration as shown there. Since we assume that the binding energyis large, 1/a2

S > µd/2 and then the branch cut in Fig.4.8 does not intersect the imaginary axis.Therefore, we have found that

1

β

∑l0=(2n+1)πT

e−il00

1 +

√1 +

a2S

2(il0 + ip0 + (p+ l)2/2− µd)


= −nB((p+ l)2/2− µd

) 2

ip0 + (p+ l)2/2− l2 − 1/a2S − µd/2

(4.5.17)

−nF (l2 + 1/a2S − µd/2)

1 +

√1 +

a2S

2(ip0 + (p+ l)2/2− l2 − 1/a2

S − µd/2)

ip0 + (p+ l)2/2− l2 − 1/a2S − µd/2

+

∫C2

dω

2πi

−1

eβω + 1

1 +

√1 +

a2S

2(−ω + ip0 + (p+ l)2/2− µd)

[−ω + l2 + 1/a2S − µd/2][−ω + ip0 + (p+ l)2/2− µd]

,

where C2 is the contour wrapping around the branch cut (see Fig.4.8). In calculating thisresult, we should notice that p0 is also a fermionic Matsubara frequency. The second and thirdterms in the right hand of Eq.(4.5.17) are exponentially suppressed in the limit β/a2

S → ∞.Let us neglect these higher order corrections and simply approximate the self energy as

Σ(p) =16π

aS

∫d3l

(2π)3

2nB

((p+l)2

2− µd

)ip0 + (p+ l)2/2− l2 − 1/a2

S − µd/2. (4.5.18)

We can now evaluate the number density n using the formula of Eq.(4.3.11) with the self


Re

Im

Re

Im

Re

Im

Figure 4.8: Change of contours in the Matsubara sum. It also reveals the pole positions andthe branch cut of the integrand.

energy given by Eq.(4.5.18):

n =

∫ (T )

p

−2e−ip00

G−1o (p)− Σ(p)

'∫ (T )

p

(−2)e−ip00[Go(p) +Go(p)Σ(p)Go(p)]

=

∫ (T )

p

−2e−ip00

ip0 + p2 + 1a2S− µd

2

(4.5.19)

−2

∫d3p

(2π)3

d3l

(2π)3

1

β

∑p0

e−ip00

(32π/aS)nB

((p+l)2

2− µd

)[ip0 + p2 + 1

a2S− µd

2

]2 [ip0 + (p+l)2

2− l2 − 1

a2S− µd

2

] .We expand the full propagator in terms of the self-energy Σ(p) in this expression, and wewill discuss what happens when we do not expand it at last of this section. The first termin Eq.(4.5.19) gives the Fermi distribution function. Since fermions are gapped by a half ofthe binding energy, this term is exponentially small. Let us evaluate the second term. TheMatsubara sum gives

1

β

∑p0

e−ip00[

ip0 + p2 + 1a2S− µd

2

]2 [ip0 + (p+l)2

2− l2 − 1

a2S− µd

2

] ' −nF(

(p+l)2

2− l2 − 1

a2S− µd

2

)[

(p+l)2

2− l2 − p2 − 2

a2S

]2 ,

(4.5.20)where we again neglect exponentially small contributions in terms of β/a2

S. We obtain that

n = 2

∫d3p

(2π)3

d3l

(2π)3

(32π/aS)nF

((p+l)2

2− l2 − 1

a2S− µd

2

)nB

((p+l)2

2− µd

)[(p− l)2/2 + 2/a2

S]2

= 2

∫d3P

(2π)3nB

(P 2

2− µd

)∫d3q

(2π)3

32πaSnF

(P 2

2− (P

2− q)2 − 1

a2S− µd

2

)[(2q)2/2 + 2/a2

S]2 . (4.5.21)

Here we have changed integration variables so that P = p + l and q = (p − l)/2. Let usevaluate the integration over the relative momentum q. Since the center of mass momentum P

4.6. Summary 75

should be of the order of√T due to the Bose distribution function, we may put it to be zero in

the Fermi distribution function. As a result, since we may approximate the Fermi distributionfunction as nF (−q2 − 1/a2

S) ' 1, the relative momentum integration gives∫d3q

(2π)3

32π/aS

[2q2 + 2/a2S]

2 = 1.

Therefore, number density becomes

n = 2

∫d3P

(2π)3nB

(P 2

2− µd

). (4.5.22)

We have confirmed that the free bosonic picture naturally emerges in the deep BEC regime.In the deep BEC regime, the biding energy and the energy scale of thermal excitations are

well separated. Therefore, the change of the RG flow from the vacuum one can occur only inlow energy scales. As a first approximation, let us neglect its effect to the four point vertex.Then only the BCS diagram contributes and the Thouless criterion is again given by Eq.(4.5.7).Let us substitute µ = −1/a2

S + µd/2 into Eq.(4.5.7) to get

− 1

aS=

2

π

∫ ∞0

√εdε

[tanh

(βc2

(ε+ 1/a2S − µd/2)

)2(ε+ 1/a2

S − µd/2)− 1

2ε

]. (4.5.23)

Since β/a2S 1 in the deep BEC regime, we may put the hyperbolic tangent in the integrand

to be 1 neglecting exponentially small corrections. In this approximation, the right hand sidebecomes −

√1/a2

S − µd/2 and then the Thouless criterion gives µd = 0 at T = Tc.Now we can calculate the transition temperature Tc substituting µd = 0 into Eq.(4.5.22).

Since n = k3F/3π

2, we get

k3F

3π2= 2

∫d3P

(2π)3

1

eβcP 2/2 − 1= T 3/2

c

1

π2

√π

2ζ(3/2). (4.5.24)

Since εF = k2F in our unit 2m = 1, we find that

Tc/εF =

(2

9πζ(3/2)2

)1/3

= 0.218 . . . . (4.5.25)

This is the BEC transition temperature of the free bosonic system.Here again, ideas of scale separation were so powerful. Thanks to the fact that RG flows in

the vacuum and in matters coincide at high energy scales k n1/3, we can concentrate on low-energy dynamics even if we use purely fermionic formulations in the BEC regime. This enablesus to extract nonperturbative features in our calculations, and to discuss phase transitions ofBEC. In order to go beyond this approximations, it is very important to take into account theinteraction between dimers and for this purpose calculating dimer-dimer scattering in terms ofthe flow equation is necessary.

4.6 Summary

In this chapter, we consider the two-component fermionic system with attractive contact inter-actions. Our central goal of this chapter is to calculate critical temperatures of superfluid phase


transitions from the knowledge of scattering physics without putting artificial assumptions inthose calculations. The main idea of those analysis is the following: in the high energy regionimportant physical process should be approximately same with that in the vacuum, and effectsof many body physics mainly affects low energy physics only. In order to realize this idea it isquite natural to separate energy scales, and by employing FRG we can consider the BCS-BECcrossover in a systematic way.

At first, in order to find properties of physics in the vacuum, we calculate scattering processusing FRG. Due to the absence of antiparticles or holes, the FRG equations automaticallyclose at each sector and are free from infinite hierarchy. Thus, in principle we can calculatethe scattering process in a precise way. As a result we have seen that the two body sector inthe vacuum is precisely solved and we have obtained the four-point vertex in the vacuum. Wealso discussed the atom-dimer scattering by studying the three-body sector, and derived theflow equation for a new vertex whose dimer legs are amputated. Important lesson learned fromatom-dimer scattering is that, feedback from higher-order vertices exists in general cases butstill flow equations decouple from higher-body sectors. This is very important property whichis missed in previous studies.

We derived some formula to calculate number densities n using FRG, and also derivedthe Thouless criterion using the Ward-Takahashi identity. Especially, we discussed the Ward-Takahashi identity in a general way and showed that it can be understood as an integrated flowof the corresponding correlation function as long as phase transitions do not occur in the RGflow. We also proposed that for symmetry broken theories there exits a critical cutoff scale kcat which the second order phase transition occurs by crossing the critical surface, although itis not still a mathematically rigorous statement.

Finally, we reproduce the NSR theory by considering the deviation of the RG flow of manybody fermionic systems from that of the vacuum. In the BCS regime, high energy physicsis controlled by scatterings in vacuum and low energy physics is controlled by the LandauFermi liquid theory. Therefore, it is easy to control resummation of Feynman diagrams basedon scaling ansatz on those energy scales, and by considering the higher order effect in theintermediate regime we have reproduced the Gorkov and Melik-Barkhudarov correction in anatural way.

In the BEC regime elementary excitations are bosonic dimers, and in order to describe thephysics based on fermionic theories, nonperturbative description is required even at the lowestorder approximation. Based on knowledge of the RG flow in the vacuum and separation ofenergy scales of many-body physics from that in the vacuum (binding energy), we evaluatedthe fermion self energy correction so as to find a bosonic picture. As as result, we can obtainfree bosonic picture in our formalism, and then we can consider scattering between dimers asa future work.

As a concluding remark, let us comment on previous studies by other groups. Researches onthe BCS-BEC crossover using FRG is done in the paper [74, 75] by introducing the Hubbard-Stratonovich field in the particle-particle channel. In this case, one can apply the local potentialapproximation to the effective potential of the bosonic field and they gave quantitatively correctresults in the BCS and BEC regions. In order to circumvent ambiguities due to the Hubbard-Stratonovich transformations associated with the Fierz transformation to some extent, theyinvented the rebosonization technique to eliminate dynamically generated fermion quartic cou-plings [76]. Their numerical calculation gives almost correct dimer-dimer scattering length, firstcalculated by D. S. Petrov et al. [77, 78], and asymptotic formula of critical temperatures of

4.6. Summary 77

interacting Bose gas so called Lee-Yang-Huang correction [79–81].Without introducing the Hubbard-Stratonovich field, R. Haussmann calculated transition

temperatures of the BCS-BEC crossover [82, 83]. Later, his group established the variationalmethod with the 2PI formalism for the BCS-BEC crossover, which is called the self-consistentT -matrix method [84]. Since their approximation breaks the global U(1) symmetry the phasetransition becomes first order, but their analysis provides plausible numerical results in theBCS and BEC regions. Especially, by introducing the external fields for two point functionsthe description of the BEC region becomes transparent. These are not in the framework ofFRG, but combining their analysis to our FRG method would be an interesting problem.


Chapter 5

Application of FRG to dipolarfermionic systems

So far, most ultracold atomic experiments are performed with alkaline atoms, and in suchsystems the dominant interaction can be treated as a contact interaction (see sec.4.1). Inthis chapter, we consider a totally different system: ultracold atoms with strong dipole-dipoleinteractions. In such systems the interaction is long-ranged and anisotropic, and we can expectunprecedented phenomena absent in conventional systems with the contact interaction.

Experimental successes in realizations of ultracold dipolar gases reported in the papers[85, 86] encourage us to perform theoretical calculations in detail for such systems. These arevery recent progresses in this area. Indeed, those papers were published in May 2012. Thereforewe will explain them in sec.5.1, and especially we will see details about the fermionic dipolarsystem realized with dysprosium-161 [86].

Our original motivation to study dipolar systems is to reveal physical properties of densenuclear matter. Although such dense systems of nucleons have not yet been realized in experi-ments, it is believed to be realized inside of neutron stars. In the dense system, the strong andanisotropic nature of the nuclear force become important and it is theoretically predicted thatmany unobserved phenomena such as nucleon superfluid, meson condensation, etc. are real-ized [87]. Acquiring new insights for dense nuclear matter from ultracold atomic experimentsis our purpose of this study.

In sec.5.2, we will see properties of the dipole-dipole interaction briefly and it will providephysical insights for our system. In sec.5.3, we establish the Landau Fermi liquid theory with thedipole-dipole interaction using RG methods and find a new kind of superfluidity characteristicto a dipolar fermionic system.

5.1 Ultracold atomic systems with the DDI

Let us explain experimental current status of dipolar gases in this section. The dipole-dipoleinteraction (DDI) between two magnets separated by r is given by

V αββ′α′(r) =

γ2

4πr3

[S1

αα′ · S2

ββ′ − 3(S1

αα′ · r)(S2

ββ′ · r)

], (5.1.1)

where Si represents the spin vector of a particle i. Here γ is the coupling constant for theDDI, called the magnetic moment or the gyromagnetic ratio. It is often measured in the unitof the Bohr magneton µB = e~

2me. Since the spin and orbital angular momenta are coupled in

79

80 Chapter 5. Application of FRG to dipolar fermionic systems

Attractive

Repulsive

Figure 5.1: The dipole-dipole interaction (DDI) between two magnets.

Eq.(5.1.1) it reveals an anisotropic nature (see Fig.5.1). We should also notice that it decreasesas 1/r3 as the distance r increases, and its behavior is much slower than that of the van derWaals interaction which decreases as 1/r6.

The first experimental study of quantum dipolar gases was the realization of the dipolarBEC of chromium-52 (52

24Cr) by the Stuttgart group [88]. Chromium is a transition metalwith a magnetic moment of 6µB. In order to make the DDI between the chromium atomsdominant, the researchers used the Feshbach resonance to turn off the contact interaction andthey observed the d-wave collapse of dipolar condensates [89].

In the field of the dipolar BEC, the BEC of erbium-168 (16868Er) is reported in May 2012

by the Innsbruck group [85]. Erbium is a lanthanide atom with the magnetic moment 7µB. Aremarkable property of erbium-168 is that it has 6 Feshbach resonances in low magnetic fieldenvironments. Indeed, they observed the d-wave collapse of the dipolar BEC in erbium about1 Gauss, although very high magnetic field lager than 500 Gauss is necessary for the case ofchromium-52. This may open possibilities to study a purely dipolar bosonic gas thanks to theweakness of external fields.

Next, let us care about fermionic dipolar gases. In May 2012, the achievement of thequantum degenerate dipolar fermionic gas was first reported with dysprosium-161 (161

66Dy) bythe Illinois and Stanford group [86]. This had been one of the unrealized experimental challengesin ultracold atomic experiments 1. Therefore this step is qualitatively new, and the observationof the Fermi sea of the dipolar fermions itself is a very important result (see Fig.5.2).

Now we will explain the experiment of dysprosium-161. Dysprosium is a lanthanide atomand its isotopes 161Dy and 163Dy are the most magnetic fermionic atoms with the dipole moment10µB. Usually, achieving the quantum degeneracy of identical fermions is difficult, because theFermi statistics prohibit s-wave scattering with the short-range interaction and because therethermalization process from the p-wave collision ceases due to the centrifugal barrier belowsome threshold temperature about 10-100µK.

From this fact, they chose the sympathetically cooling of 161Dy with its bosonic isotope162Dy. Both are spin polarized into high-magnetic-field seeking states: |F = 21

2,mF = −21

2〉

for 161Dy and |J = 8,mJ = −8〉 for 162Dy, where F = I + J is the total spin, I the nuclearspin 2, and J = 8 the total electronic angular momentum. By switching off the optical dipoletrap gradually, spin-polarized 161Dy are evaporated to a quantum degenerate state. Fig.5.2shows the result, and the population shows the better coincidence to that of the Thomas-Fermidistribution, which reflects the Fermi-Dirac distribution, than that of the Gaussian fit, whichreflects the Boltzmann distribution. The estimated temperature is T/TF = 0.21(5).

1For other highly magnetic fermionic atoms 53Cr (6µB) and 167Er (7µB), the temperature is still not below10µK. For the fermionic polar molecule 40K87Rb (0.57D) the temperature is about the Fermi temperature(T/TF ' 1.4), but the dissociation process from chemical reactions prevents further evaporative cooling.

2I = 0 for 162Dy, and I = 5/2 for 161Dy

5.2. Basics of the DDI 81

Figure 5.2: Quantum degeneracy of the dipolar fermions [86]. (a) Single-shot image of thepopulation in the momentum space. (b) Average of six images. (c),(d) Density integration alongthe horizontal, vertical axis, respectively. Green curves represent the fit with the Gaussian,while the red curves are fit with the Thomas-Fermi distribution. This reveals that T/TF ' 0.2.

We should also emphasize that they have also achieved the quantum degeneracy with theforced evaporative cooling of spin-polarized 161Dy without the bosonic isotope 162Dy. Theiranalysis shows that T/TF = 0.7 with TF = 500nK. This would suggest the importance ofdipolar collisions and they might provide rethermalization processes.

In following sections, we will consider two-component fermionic system with the DDI. Hith-erto, such systems have not been realized but the recent progress is opening the new physicswith dipolar systems. We can expect the realization of similar systems with the model whichwe will consider.

5.2 Basics of the DDI

At first, let us remark on basic properties of the DDI. We consider the magnetic dipole inter-action (5.1.1) between spin-1

2fermions. We should be attentive to following properties of the

DDI:

• The interaction is invariant under the combined SU(2) spin rotation and SO(3) spatialrotation, however the orbital angular momentum L and the spin S are not conservedindependently.

• The DDI has a long-range feature, i.e. it damps as 1/r3.

Since we will frequently use the momentum-space representation, it is convenient to give theFourier transform of Eq.(5.1.1):

V αββ′α′(q) =

1

3γ2[3(Sαα′ · q)(Sββ′ · q)− Sαα′ · S

ββ′

]. (5.2.1)

The interaction term V in the Hamiltonian is given in the coordinate space as

V =1

2

∫d3xd3yψα(x)ψβ(y)V αβ

β′α′(x− y)ψβ′(y)ψα′(x). (5.2.2)


In the momentum space, Eq.(5.2.2) becomes

V =1

2

∫P

∫k,k′

ψα,P /2+kψβ,P /2−kVαββ′α′(k − k

′)ψβ′,P /2−k′ψα′,P /2+k′ . (5.2.3)

We decompose Eq.(5.2.3) into some channels with spherical harmonics. Here we regard thatthe total momentum P = 0 and that all the relative momenta k and k′ lie on the Fermi sphere.Therefore, the tensor interaction V αβ

β′α′(k − k′) is a function only of directions k and k′. Wecan find details of calculations in Appendix E to get an explicit form of the interaction matrix,but here let us discuss properties of the DDI to get a physical insight of the result of thosecalculations.

For later use, let us consider decomposition of more general interactions V αββ′,α′(k,k

′) andobtain expansion formulae. We define

V(SS′)SzS′z

(k,k′) =∑α,β

∑α′,β′

CSSz12α, 1

2βCS′S′z12α′, 1

2β′

[V αββ′α′(k − k

′)− V αβα′β′(k + k′)]. (5.2.4)

The inverse formula of Eq.(5.2.4) is given by

V αββ′α′(k,k

′) =∑SS′

∑SzS′z

CSSz12α, 1

2βCS′S′z12α′, 1

2β′V

(SS′)SzS′z

(k,k′). (5.2.5)

Since we assume here that the spin of particles is 1/2, the Clebsch-Gordan coefficients only takesthe form CSSz

12α, 1

2β, and S and S ′ only take the values 0, 1. In the case of higher spin species, we

can obtain a similar formula. Decomposition with respect to relative angular momenta gives

V(SS′)ss′ (k,k′) =

∑LM

∑L′M ′

V(SS′)LMs,L′M ′s′YLM(k)Y ∗L′M ′(k

′), (5.2.6)

where coefficients are given by

V(SS′)LMs,L′M ′s′ =

∫d2Ωk

∫d2Ωk′Y

∗LM(k)YL′M ′(k′)V

(SS′)ss′ (k,k′). (5.2.7)

We can interpret this quantity (5.2.7) as a tree-level transition amplitude from the channellabeled by (L′, S ′;M ′, s′) to another channel labeled by (L, S;M, s). Since this is the decom-position of the interaction for general cases, the above statements is also applicable to effectiveinteractions after a cutoff scale is lowered.

Let us now restrict possible terms in the effective actions using the symmetry arguments.

• The Fermi statistics, ψα,kψβ,k′ = −ψβ,k′ψα,k, implies that under the interchange of thespin variables α↔ s−α in Eq.(5.2.5) associated with k 7→ −k should give a factor (−1).

This suggests that the coefficients V(SS′)LMs,L′M ′s′ does not vanish only when L is odd and

S = 1, or when L is even and S = 0. This should be true for L′ and S ′, too.

• Next, let us consider the invariance under the parity transformation ψ(ω,k) 7→ ψ(ω,−k)and similar for ψ. This implies that the interaction in Eq.(5.2.6) is invariant under the

transformation k(′) 7→ −k(′) and then V(SS′)LMs,L′M ′s′ does not vanish only when (−)L+L′ =

+1. This means that L+L′ should be an even integer, i.e. L′ = L,L± 2, L± 4, · · · . Thecombining the above two arguments, S is a good quantum number for our model.

5.2. Basics of the DDI 83

• The rotational symmetry implies conservation of the total angular momentum J2 =(L + S)2 and its projection quantum number Jz = M + s. Since the total spin S isalso a good quantum number, we can consider the spin-triplet couplings and the spin-singlet couplings independently. For S = 1, we need to require J = J ′, which implies thatL = L′, L′±1, L′±2, and then the invariance under the parity suggests that L = L′, L′±2.For the spin singlet case, we can conclude that L = L′ and M = M ′.

Since L and S are not good quantum numbers in general, and furthermore M and s arenot good quantum numbers even in our model, we would like to use the representations whichdiagonalizes the total angular momenta J and its projection Jz. For this purpose, let us definethe coefficients

VJJz ,J ′J ′zLS;L′S′ =

∑M,Sz

∑M ′,S′z

CJJzLM,SSz

CJ ′J ′zL′M ′,S′S′z

V(SS′)LMs,L′M ′s′ , (5.2.8)

or more explicitly we have

VJJz ,J ′J ′zLS;L′S′ =

∑M,Sz

∑M ′,S′z

CJJzLM,SSz


∑αβ

∑α′β′

CSSzsα,sβC

S′S′zsα′,sβ′

×∫

dΩk

∫dΩk′Y

∗LM(k)YL′M ′(k

′)[V αββ′α′(k − k

′)− V αβα′β′(k + k′)]. (5.2.9)

This is explicitly calculated in Appendix E.The DDI in Eq.(5.2.2) is non-local, however we can obtain the local interaction by going

back to the original action with the magnetic field. This procedure is equivalent to eliminatethe quartic coupling (5.2.2) with the Hubbard-Stratonovich transformation:

S =

∫ β

0

dτ

∫d3x

(ψ(x)(∂τ −

∇2

2m− µ)ψ(x) +

1

2(∇Ai)2

)+

∫ β

0

dτ

∫d3x γ

(ψσ

2ψ)· (∇×A) (5.2.10)

To see this, let us integrate out the gauge field A from the action, then the effective interactionbecomes

exp

(−∫ β

0

dτV (τ)

)=

∫DA exp

[−∫

dτd3x

(1

2(∇Ai)2 + γεijkψ

σi2ψ∂jAk

)]. (5.2.11)

To see that the interaction V introduced here coincides with the one in Eq.(5.2.3), we completethe square with respect to A:∫

d3x

(1

2(∇Ai)2 + γεijkψ

σi2ψ∂jAk

)=

∫q

(q2

2Ai,−qAi,q + γεijk

(ψσj2ψ)−qiqjAk,q

)=

∫q

q2

2

(Ak,−q + γεijk

(ψσi2ψ)−q

iqjq2

)(Ak,q + γεi′j′k

(ψσi′

2ψ)q

−iqj′q2

)−γ

2

2

∫q

εijkεi′j′kqjqj′

q2

(ψσi2ψ)−q

(ψσi′

2ψ)q


Therefore, up to an additive constant, the interaction becomes

V =γ2

2

∫q

(qqi′ − δii′)(ψσi2ψ)−q

(ψσi′

2ψ)q. (5.2.12)

Let us evaluate the Fourier transform of the spin composite operator:(ψσi′

2ψ)k′−k

=

∫d3xe−i(k

′−k)·x∫p

e−ip·x∫p′eip′·xψpσiψp′

=

∫p

ψp+kσiψp+k′ (5.2.13)

Now we have found that the traceless part in Eq.(5.2.12) becomes

Vtraceless =γ2

2

∫p,p′,q

ψp+qSiψp

(qiqj −

δij3

)ψp′−qSjψp′

=1

2

∫p,p′,q

ψα,p+qψβ,p′−qVαβ,β′α′(q)ψβ′,p′ψα′,p, (5.2.14)

which coincides with the one in Eq.(5.2.3) after the change of the variables P = p + p′,k′ = (p − p′)/2 and k = q + k′. On the other hand, the trace part in Eq.(5.2.12) gives acontact term:

Vtrace = −γ2

3

∫q

(ψσi2ψ)−q

(ψσi2ψ)q

= −γ2

3

∫d3x

(ψσi2ψ)

(x)(ψσi2ψ)

(x). (5.2.15)

Since there usually exists a centrifugal barrier, and such a contact term does not play animportant role. The classical equation of motion implies that

Ai,q = γεijkiqjq2

(ψσk2ψ)q, (5.2.16)

or−∇2A = γ∇×

(ψSψ

). (5.2.17)

We should also notice that ∇·A = 0 from Eq.(5.2.16), and this is a consequence of the fact thatthe action in Eq.(5.2.10) describes the dynamics of the magnetic interaction with the Coulombgauge condition.

5.3 RG study within the RPA

We at first study the weak coupling regime using the random phase approximation (RPA). Notonly just applying the RPA, we also consider the justification of those analyses based on theRG flow. The following discussion will be performed in a parallel way with the analysis insec.3.3.

Basically we rely on the Landau Fermi liquid theory again. We assume the existence of theFermi sphere and suppose that low energy excitations live in the vicinity of the Fermi surface.

5.3. RG study within the RPA 85

Due to weakness of the interaction, we neglect renormalization of the coupling in vacuum.We also neglect the self-energy correction, since unless the parity invariance is broken the selfenergy correction just shifts the chemical potential µ to the Fermi energy εF .

Then, in order to discuss the stability of the Fermi liquid theory, we need to check theabsence of singularities in the four-point vertex. In the RG flow equation, there are threediagrams contributing to the flow of dipolar couplings:

ZS =1 2

1′ 2′, ZS′ =

1 2

1′ 2′

, BCS =

1 2

1′ 2′

.

We will confirm that ZS and ZS’ diagrams lead to generalizations of the Pomeranchuk insta-bility to dipolar systems, but in the weak coupling region such instability is absent. From theanalysis of particle-particle fluctuations coming from the BCS diagram, the superfluid instabil-ity is induced in the low temperature we will find that the new type of superfluidity (the 3P1

superfluid) appears.

5.3.1 Study of the Landau channel

Let us discuss whether the diagrams ZS or ZS’ can affect low energy physics. We discuss theZS diagram at first:

ZS =1 2

1′ 2′.

In order for the diagram not not to be suppressed in terms of the cutoff scale, it is requiredthat both of vertices denoted by gray squares be automatically marginal at any loop momenta.In other words, it is necessary and sufficient for the ZS diagram to become marginal that themomenta k2 (or equally k′2) is not restricted after we specify the momenta k1 and k′1. This isonly possible if k1 ' k′1, indeed if so it implies that k2 ' k′2 and we may freely choose a pointof the Fermi surface for k2 because k′2 automatically lie on the Fermi surface too. Similarly,the ZS’ diagram is not suppressed only when k2 ' k′1 and especially when k1 ' k2 ' k′2 ' k′1both of the ZS and ZS’ diagram contribute to the quantum effect, which respects the Pauliexclusion principle.

We call the channel such that k1 ' k′1 as the Landau channel and let us analyze thischannel. If k1 ' k2 further, not only the ZS graph but also the ZS’ graph contributes inthe same order of magnitude, however here we do not take into account the ZS’ diagram forsimplicity of calculations.

Let us perform the Matsubara sum in the ZS diagram. We neglect the dependence of thevertices on the Matsubara frequencies of the external legs, then we should calculate

1

β

∑ω5

G(k5)G(k5 −Q) =1

β

∑ω5

1

iω5 + ε(k5)− µ1

i(ω5 − Ω) + ε(k5 −Q)− µ. (5.3.1)

Here Q = (iΩ,Q) represents the transfer momentum. The summation over the fermionicMatsubara frequencies can be written as

1

β

∑ω5

G(k5)G(k5 −Q) =

∮Co

dpo

2πi

−1

eβpo + 1

1

po + ε(k5)− µ1

po − iΩ + ε(k5 −Q)− µ, (5.3.2)


where Co is the closed curve enclosing the imaginary axis in Fig.5.3. Performing this contourintegration, we immediately obtain that

1

β

∑ω5

G(k5)G(k5 −Q) =sinh β

2(ε(k5)− ε(k5 −Q))

iΩ + [ε(k5)− ε(k5 − q)]

−1/2

cosh β2

(ε(k5)− µ) cosh β2

(ε(k5 −Q)− µ).

(5.3.3)In the limit |Q| |k5|, we can approximate it as

1

β

∑ω5

G(k5)G(k5 −Q) =1

2

− sinh(βvF

2k5 ·Q

)iΩ + vF k5 ·Q

1

cosh2(βvF

2l5) , (5.3.4)

and the naive limit Q → 0 is ambiguous as usual. This singularity reflects to singularities ofeffective couplings and two different low-energy effective coupling constants are defined:

ΓQΛαβ

β′α′(k,k′) = lim

|Q|→0

[ΓΛ

αββ′α′(k, k

′; k′ −Q, k +Q)∣∣∣Ω=0

], (5.3.5)

ΓΩΛ

αβ

β′α′(k,k′) = lim

Ω→0

[ΓΛ

αββ′α′(k, k

′; k′ −Q, k +Q)∣∣∣Q=0

]. (5.3.6)

By neglecting the ZS’ diagram, we found that the Ω-limit V Ω is invariant under the RG flow,and that the Q-limit V Q obeys the flow equation

Λ∂ΛΓQΛαβ

β′α′(k,k′) = Λ∂Λ

(∫ Λo

Λ

+

∫ −Λ

−Λo

)k2Fdl52π2

(−β

4

)1

cosh2 β2vF l5

×∫

d2k5

4πΓQΛ

αγ

γ′α′(k,k5)ΓQΛγ′β

β′γ(k5,k′)

= N(εF )β2vFΛ

cosh2 β2vFΛ

∫d2k5

4πΓQΛ

αγ

γ′α′(k,k5)ΓQΛγ′β

β′γ(k5,k′) (5.3.7)

where N(εF ) is the density of states of a single spin component at the Fermi surface:

N(εF ) =1

2π2k2F

dk

dεk

∣∣∣∣εk=µ

=k2F

2π2vF. (5.3.8)

Re

Im

Figure 5.3: The position of the poles in the frequency plane po = iω5.


That is, d3k/(2π)3 ' N(εF )dξ. By defining the dimensionless coupling constants

V =N(εF )

4πΓ, (5.3.9)

we get

Λ∂ΛVQ

Λ

αβ

β′α′(k,k′) =

β2vFΛ

cosh2 β2vFΛ

∫d2k5V

QΛ

αγ

γ′α′(k,k5)V QΛ

γ′β

β′γ(k5,k′), (5.3.10)

The channel decomposition in the Landau channel is given by

V Qαβ

β′α′(k,k′) =

∑L,Lz

∑S,Sz

∑L′L′z

∑S′S′z

(−)s−α′CSSzsα;s−α′(−)s+βC

S′S′zsβ′;s−βYLLz(k)Y ∗L′L′z(k

′)V QA,A′ , (5.3.11)

where A = (L,Lz, S, Sz) and A′ = (L′, L′z, S′, S ′z). The angular integration in Eq.(5.3.10)

becomes ∫d2k′′

∑γγ′

V Qαγ

γ′α′(k,k′′)V Qγ

′ββ′γ(k

′′,k′)

=

∫d2k′′

∑γγ′

∑LLzSSz

∑L′′L′′zS

′′S′′z

(−)s−α′CSSzsα;s−α′(−)s+γC

S′′S′′zsγ′;s−γYLLz(k)Y ∗L′′L′′z (k′)V Q

A,A′′

×∑

L′′′L′′′z S′′′S′′′z

∑L′L′zS

′S′z

(−)s−γCS′′′S′′′zsγ′;s−γ(−)s+βC

S′S′zsβ′;s−βYL′′′L′′′z (k)Y ∗L′L′z(k

′)V QA′′′,A′

=∑

LLzSSz

∑L′L′zS

′S′z


S′S′zs−β;sβ′YLLz(k)Y ∗L′L′z(k

′)

×∑L′′L′′z

∑S′′S′′z

∑S′′′S′′′z

V QA;L′′L′′zS

′′S′′zV QL′′L′′zS

′′′S′′′z ;A′

∑γγ′

(−)s−γCS′′S′′zsγ′;s−γ(−)s+γC

S′′′S′′′zsγ′;s−γ

= (−)2s∑Λ,Λ′


S′S′zs−β;sβ′YLLz(k)Y ∗L′L′z(k

′)∑Λ′′

V QA,A′′V

QA′′,A′ . (5.3.12)

Let us introduce the matrix notation V Q = (V QA,A′), then it obeys the RG flow

Λ∂ΛVQ

Λ = (−)2sβ2vFΛ

cosh2 β2vFΛ

(V Q

Λ

)2

, (5.3.13)

and its solution is given by

V QΛ =

V Qo

1 + (−)2sV Qo

(tanh β

2vFΛo − tanh β

2vFΛ

) . (5.3.14)

Since Λo is considered to be sufficiently large, we may approximate that tanh (βvFΛo/2) ' 1,then we can readily identify the stability conditions for the Landau channels. Let the fermionspin s be 1/2. If and only if all the eigenvalues of V q

o is less than 1, singularities in therenormalization group flow (5.3.14) do not arise and the Fermi liquid picture is consistent.Especially when we restrict ourselves in the weak coupling region, the stability condition ismanifestly satisfied and the Fermi liquid picture should be valid.


5.3.2 Study of the BCS channel

Here let us study the BCS diagram to see the BCS instability, and let us concentrate on theRG flow in the BCS channel. For this purpose, put k2 = −k1 and k′2 = −k′1. In this case,Q = k′1 − k1 or Q′ = k′2 − k1 need not be small and then the ZS and ZS’ graphs are suppresseddue to the restriction of possible configurations of loop momenta. Therefore, we neglect thesegraphs and only take into consideration the effect of the BCS graph. In these approximations,the 1-loop contribution of the BCS graph to this channel is given by the flow equation

Λ∂ΛΓαββ′α′(k,k′) = Λ∂Λ

(−1

2

)(∫ Λo

Λ

+

∫ −Λ

−Λo

)k2Fd`5

2π2

tanh(β2vF `5

)2vF `5

∫d2k′′

4πΓαβδγ (k,k′′)Γγδβ′α′(k

′′,k′)

=N(εF )

2tanh

(β

2vFΛ

)∫d2k′′

4πΓαβδγ (k,k′′)Γγδβ′α′(k

′′,k′), (5.3.15)

where N(εF ) =k2F

2π2vFis the density of states for one spin projection. With respect to the

dimensionless couplings given in Eq.(5.3.9), we get

Λ∂Λ Vαββ′α′(k,k

′)∣∣∣BCS

=1

2tanh

(β

2vFΛ

)∫d2k′′V αβ

δγ (k,k′′)V γδβ′α′(k

′′,k′). (5.3.16)

Let us calculate the angular integration in Eq.(5.3.16) using the decomposition (5.2.5,5.2.6).Substituting the decomposition of the interaction

V αββ′α′(k,k

′) =∑LSM

∑L′S′M ′

CS,α+β12α; 1

2βCS′,α′+β′

12α′; 1

2β′V

(SS′)LM,α+β;L′M ′,α′+β′YLM(k)Y ∗L′M ′(k

′), (5.3.17)

we find that ∫d2k′′

∑γδ

V αβδγ (k,k′′)V γδ

β′α′(k′′,k′) (5.3.18)

=

∫d2k′′

∑γδ

∑LSM

∑L′′S′′M ′′

CS,α+β12α; 1

2βCS′′,γ+δ

12γ; 1

2δV

(SS′′)LM,α+β;L′′M ′′,γ+δYLM(k)Y ∗L′′M ′′(k

′′)

×∑

L′′′S′′′M ′′′

∑L′S′M ′

CS′′′,γ+δ12γ; 1

2δCS′,α′+β′

12α′; 1

2β′V

(S′′′S′)L′′′M ′′′,γ+δ;L′M ′,α′+β′YL′′′M ′′′(k

′′)Y ∗L′M ′(k′)

=∑LSM

∑L′S′M ′

CS,α+β12α; 1


12α′; 1

2β′YLM(k)Y ∗L′M ′(k

′)

×∑L′′M ′′

∑S′′S′′′

∑γδ

CS′′,γ+δ12γ; 1

2δCS′′′,γ+δ

12γ; 1

2δV

(SS′′)LM,α+β;L′′M ′′,γ+δV

(S′′′S′)L′′′M ′′′,γ+δ;L′M ′,α′+β′

=∑LSM

∑L′S′M ′

CS,α+β12α; 1


12α′; 1

2β′YLM(k)Y ∗L′M ′(k

′)

×∑L′′M ′′

∑S′′,s′′

V(SS′′)LM,α+β;L′′M ′′,s′′V

(S′′S′)L′′′M ′′′,s′′;L′M ′,α′+β′ .

Therefore, the expansion of Eq.(5.3.16) gives the set of flow equations:

−Λ∂ΛV(SS′)LMs,L′M ′s′ =

1

2tanh

(β

2vFΛ

) ∑L′′M ′′

∑S′′s′′

V(SS′′)LM,s;L′′M ′′,s′′V

(S′′S′)L′′′M ′′′,s′′;L′M ′,s′ . (5.3.19)


Formally, we can solve the differential equations given in Eq.(5.3.19). Notice that it takesthe form

Λ∂ΛVΛ =1

2tanh

(β

2vFΛ

)V 2

Λ , (5.3.20)

where VΛ is the matrix with coefficients V(SS′)LMs,L′M ′s′ at the scale Λ. The formal solution of this

matrix differential equation is given by

V (Λ) = Vo

(1 + Vo

∫ Λo

Λ

dΛ′

Λ′1

2tanh

(β

2vFΛ′

))−1

. (5.3.21)

Indeed, this solution satisfies at Λ = Λo V = Vo and Eq.(5.3.20), as desired. The Landau Fermiliquid remains stable in the BCS channel at any temperature when the interaction in the BCSchannel is marginally irrelevant, i.e. the bare couplings satisfy

Vo > 0. (5.3.22)

On the other hand, if a condition in Eq.(5.3.22) is violated, the coupling becomes marginallyrelevant and a Landau pole appears at some low temperature. To evaluate the critical temper-ature, let us consider the low temperature limit. Then, the critical temperature is

TSC =2eγE

πvFΛo exp

(− 1

|Vmax|/2

), (5.3.23)

where Vmax is the maximal negative eigenvalue of the matrix V .The interaction matrix elements is given by

V =N(εF )

4π(1 + (−)1+L′+2s−S′)ΓΛ,Λ′ (5.3.24)

where

ΓJJz ,J ′J ′zLS,L′S′ = −(−)S+JδJJ ′δJzJ ′z

3

2γ2s(s+ 1)(2s+ 1) (5.3.25)

×√

(2L+ 1)(2L′ + 1)(2S + 1)(2S ′ + 1)

S ′ S 2L L′ J

S s sS ′ s s2 1 1

×

[(HL +HL′)C

2010,10C

20L′0,L0 + (−)L10

∑l

HlCl0L0,10C

l0L′0,10

1 1 2L L′ l

].

The details of calculations to obtain this result can be found in Appendix E. In the case of spins = 1/2, only the spin triplet channel S = S ′ = 1 remains. We get

ΓJ=0 =

(3/4 00 0

), ΓJ=1 =

(−3/8 0

0 0

), ΓJ=2 '

(0.075 0.0610.061 0.050

), (5.3.26)

where we have picked up the values for L,L′ = 1, 3 in (5.3.26). The maximal negative eigen-

value of the matrix V is given by Vmax = −3

4N(εF )

γ2

4πand the corresponding channel is 3P1.

Therefore, we can expect the 3P1 superfluid below the critical temperature

Tc =2eγE

πvFΛ exp

(− 8

3N(εF )(γ2/4π)

).


Let us consider the situation in which we add the contact interaction

V c =g

2

∫d4xψ†α(x)ψ†β(x)ψβ(x)ψα(x). (5.3.27)

In this case, g > 0 means the repulsive contact interaction, and g < 0 the attractive contactinteraction as usual. The channel decomposition shows that this remains non-zero only if J = 0and L = S = L′ = S ′ = 0. Therefore, the dipole-dipole interaction and the contact interactiondo not mix at least in this approximation for the BCS channel, and the transition temperatureof this channel (1S0) is given by

T =2eγE

πvFΛ exp

(− 1

N(εF )|g|

), (5.3.28)

if g < 0. We can judge the relevant channel by comparing the relevant scale for those channels.The border of those two phases are given by

|g| = 3

8

γ2

4π. (5.3.29)

5.4 Summary

From the RPA analysis of a dipolar fermionic system with spin 1/2, we have found two stabilitycondition. The first one corresponds to a generalization of the Pomerunchuk stability condition(3.3.20) of the usual Fermi liquid to the dipolar system. This instability is absent when thecoupling is sufficiently weak. The second one is the superfluid instability, which appears if thetemperature is sufficiently low even when the coupling is very weak.

The phase diagram of our analysis is shown in Fig.5.4. The vertical axis shows the strengthof the dipole-dipole interaction and the horizontal axis shows the coupling of the contact inter-action. When the attractive contact interaction is dominant, the system shows 1S0 superfluid,

superfluid superfluid

Break down of Fermi

Liquid picture

Figure 5.4: Phase diagram of a dipolar fermionic system in sufficiently low temperatures. Thehorizontal axis is strength of dimensionless contact couplings, and the vertical axis shows thatof the dipole-dipole interaction. The green region shows emergence of 1S0 superfluid due tothe attractive contact interaction, and the blue region shows that of 3P1 superfluid due tothe dipole-dipole interaction. The white region shows break-down of the Landau Fermi liquidtheory due to the Pomeranchuk instability.

5.4. Summary 91

and it is shown as the green region in the figure. On the other hand, if the dipole-dipole inter-action becomes dominant it shows 3P1 superfluid as shown with the blue region in the figure.This type of superfluidity has not been found in other systems, and it reveals characteristicbehaviors of the dipole-dipole interaction.

Before closing this chapter, let us comment on the relation of our studies to previous worksof other groups. In Fig.5.4, the white region represents break-down of the Landau Fermi liquidpicture due to the extended Pomeranchuk instabilities of Eq.(3.3.20). These instabilities arealready shown in the paper [90] by T. Sogo et al. using the RPA analysis at T = 0. Theyintroduced bosonic auxiliary fields in particle-hole channels and identified those Pomeranchukinstabilities using mean field approximations. However, both of their analysis and ours canonly be justified in the weak bare coupling region, we need more refined calculations to revealthe correct physical origin of those instabilities.

In strong coupling regions, unbiased analyses have not been done and we can only accessvariational calculations at the zero temperature with some ansatz on ground state wave func-tions. In the paper [91] by B. M. Fregoso and E. Fradkin, they revealed that the spin-polarizedstate is energetically stable if the contact repulsion is so strong, and that between the param-agnetic and ferromagnetic phases the intermediate state called ferronematic states is a possibleground state. In the paper [92] by K. Maeda et al., they propose another ground state, calledantiferrosmectic-C phase, motivated by the alternating-layer-spin (ALS) structure [93] asso-ciated with meson condensations in high-density nuclear matters, and they showed that it isenergetically stable if the dipole-dipole interaction and the contact repulsion are strong enough.

It is important to confirm that whether those possible ground states indeed occur. In orderto give convincing explanations, we should provide more sophisticated calculations and discussthe competition of possible ground states from unbiased viewpoints. Since the competition ofpossible instabilities often appears in many body physics, establishing those analysis providesunderstandings not only on dipolar systems but also on many physical open problems. Espe-cially, it will give deeper insights on the high-density nuclear matter which are important tounderstand neutron stars.


Chapter 6

Summary and perspectives

Main purpose of this thesis is revealing properties of many-body fermionic systems based onnonperturbative and unbiased method of quantum field theories. In order to realize it, we usethe functional renormalization group and discuss its approximations in a systematic way toclarify physical origins of interesting phenomena.

Chapter 2 is devoted to review methods of the functional renormalization group, whichis our main tool to study many-body fermions. We introduced the Wetterich formalism anddiscuss its properties by comparing with other versions of FRG. We also consider optimizationof the renormalization group flow proposed by D. Litim.

In chap.3, we reviewed the Landau Fermi liquid theory based on FRG. We have learned thatthe Landau Fermi liquid theory is justified as long as we assume that low energy excitationsare fermionic quasiparticles. We also learn how possible instabilities of the Fermi liquid appearfrom the RG point of view. The idea established here is a foundation in analyzing the BCSregime in the BCS-BEC crossover and dipolar fermions.

We applied FRG to the BCS-BEC crossover without any bosonization in chap.4. Startingfrom scattering problem of the model in the vacuum, we consider superfluid phase transitionusing the RG flow in normal phases. In scattering problems, we considered the atom-dimerscattering in detail, and we obtained the atom-dimer scattering length without solving integralequations, which is not rigorous but improves the tree approximation significantly. In order tocalculate thermodynamic properties with our formalism, we derived some formula to calculatenumber densities n using FRG, and also derived the Thouless criterion. In that argument, wediscussed the Ward-Takahashi identity in a general way and showed that it can be understoodas an integrated flow of the corresponding correlation function.

We considered the BCS and BEC regimes separately, and as a possible connection betweenthose two in leading order approximations we reproduced the NSR theory. In the BCS regime,we have seen that high energy physics is controlled by vacuum physics and low energy physicsis described by the Landau Fermi liquid theory. By taking into account possible corrections inintermediate energy scales, we have reproduced the Gorkov and Melik-Barkhudarov correctionin a natural way. In the BEC regime elementary excitations are bosonic dimers, and in order todescribe the physics based on fermionic theories, nonperturbative description is required evenat the lowest order approximation. Based on knowledge of the RG flow in the vacuum andseparation of energy scales of many-body physics from that in the vacuum, we evaluated thefermion self energy correction so as to reproduce the bosonic picture.

In chap.5, we discuss possible instabilities of dipolar fermionic systems using the scalingansatz of the Landau Fermi liquid theory. We will find that the ansatz is justified in weakcoupling regions and that 3P1 superfluid is predicted from our calculations. We also reproduced

93

94 Chapter 6. Summary and perspectives

possible instabilities in particle-hole channels suggested by T. Sogo et al. [90] which correspondsto Pomeranchuk instabilities in the usual Landau Fermi liquid theory, and gave some discussionon relations of our result to previous studies.

Throughout this thesis, we have seen that FRG provides a very powerful tool to studynonperturbative quantum field theories, but also that it is not yet completed and requiresimprovement especially for fermionic field theories. Separation of energy scales provides veryimportant ideas to study physics, and establishing FRG is directly related to polishing uptechniques to realize it in practical computations. Indeed, it is applicable not only many-bodyphysics but also few body physics, and it may open a new way to analyze interesting physicsclearly and intuitively. I believe that FRG will provide more powerful means to study verywide range of physics in the future.

Appendix A

Quick derivation of the 1-loop RGexpressions

In this appendix, we derive the abridged way to derive 1-loop expressions of the Wetterichequation (2.2.11) and the composite operator flow equation (2.2.19). Of course, we can obtainthem by expanding the both sides of the flow equation in terms of field variable since wealready know their nonperturbative expressions, however it becomes very complicated as theorder of vertices increases. We here would like to obtain the quick derivation of them using theknowledge of Feynman diagrams in the perturbation theory.

A.1 Feynman rules for the flow of vertex functions

Let us rewrite the Wetterich equation (2.2.11) as

∂kΓk[ϕ] = ∂k1

2STr ln

[Γ

(2)k [ϕ] +Rk

], (A.1.1)

where ∂k acts only on the k-dependence of the IR regulator Rk. In order to use Feynmandiagrams, we decompose the effective action into the quadratic part and the others. Let usdenote the decomposition as

Γk[ϕ] =1

2ϕ ·G−1

k · ϕ+ Vk[ϕ], (A.1.2)

where Vk[ϕ] does not contain quadratic terms. The inverse propagator G−1k is often written as

G−1o − Σk, where Go is the free propagator and Σk is the self-energy correction at the scale k.

In order to derive Feynman rules, we rewrite

1

2STr ln

[Γ

(2)k +Rk

]=

1

2STr ln

[(G−1

k +Rk) + V(2)k [ϕ]

]=

1

2STr ln(G−1

k +Rk)−1

2STr

∑n≥1

(−)n

n

((G−1

k +Rk)−1V

(2)k [ϕ]

)n.

The first term is independent of field variables, and for the most purposes we may simplydiscard that part. Eq.(A.1.1) becomes

−∂kΓk[ϕ] = ∂k1

2STr

∑n≥1

(−)n

n

((G−1

k +Rk)−1V

(2)k [ϕ]

)n. (A.1.3)

95

96 Appendix A. Quick derivation of the 1-loop RG expressions

We can easily interpret the right hand side of Eq.(A.1.3) diagrammatically. To simplify the

interpretation, we at first neglect the derivative ∂k in the right hand side of Eq.(A.1.3) for amoment. Then it is just the summation over all the possible 1-loop diagrams with the followingrules:

1. Write down the all possible 1-loop diagrams with the interaction Vk[ϕ] with the givenexternal lines. Regard that all the external lines are amputated.

2. To each vertex, attach the corresponding coupling in −Vk[ϕ].

3. To each internal lines, attach the propagator (G−1k +Rk)

−1 = (G−1o − Σk +Rk)

−1.

4. Take the super-trace. In the momentum space, take the trace of the flavor indices and

attach the loop integration

∫ddl

(2π)d. For the fermion loop, attach the factor −1.

5. The symmetric factor appears as the same way in the usual Feynman rule.

After all these procedures, we get some analytic expressions and we get the result by takingthe derivative of the k-dependence of Rk. We should not forget the negative sign in the lefthand side of Eq.(A.1.3).

A.2 Feynman rules for the flow of composite operators

In the same way, we can derive the quick rule for the composite operator flow. Again, we denoteEq.(2.2.19) as

∂kIk[ϕ] = ∂k1

2STr

[1

Γ(2)k [ϕ] +Rk

I(2)k [ϕ]

]. (A.2.1)

Substituting the decomposition given in Eq.(A.1.2) into Eq.(A.2.1), we find that

∂kIk[ϕ] = ∂k1

2STr

[∑n≥0

((G−1

k +Rk)−1(−V (2)

k [ϕ]))n

(G−1k +Rk)

−1I(2)k [ϕ]

]. (A.2.2)

In this case, we should write down all the possible 1-loop diagrams which contains one of thecouplings in Ik[ϕ] only once. The other rules to get diagrammatic expressions are the samewith those of vertex functions.

In this case, the expectation value itself of the composite operator is important for manycases. Therefore, we usually should not discard the field independent terms. When the ul-traviolet divergence appears in the continuum limit, we have to apply the composite operatorrenormalization. In general, operator mixing appears, that is, not only the multiplicative renor-malization but also the subtraction of lower-dimensional operators is necessary to get a finiteresult [10].

Finally, let us derive useful formula to get explicit forms of correlation functions. We oftenhave to calculate the quantity Wk,nm using the 1PI effective action for obtaining the explicit

form of the correlation functions Ik[ϕ]. Of course, when we decide that we calculate it usingthe flow equation (2.2.19), the calculation for the explicit form is unnecessary. However, itsometimes provides the way to check the consistency of the calculations and also gives the

A.2. Feynman rules for the flow of composite operators 97

physical insights for the problems. Therefore, let us write down diagrammatic rules to obtaincorrelation functions.

For simplicity, we assume that the original correlation function I[J, φ] does not dependon sources J , that is, it is a usual composite operator I[φ]. For this case, the formulae(2.2.16,2.2.18) give

Ik[ϕ] = I

[ϕ+Gk[ϕ] · δL

δϕ

]1, (A.2.3)

where

Gk[ϕ]nm = Wk,nm[J [ϕ]] =δ2LWk[J ]

δJnδJm

∣∣∣∣J=

δRΓkδϕ

+ϕ·Rk. (A.2.4)

Therefore, it suffices to give the diagrammatic rule to calculate Gk[ϕ]. Generally, we can saythat this quantity is the functional inverse of the second derivative of the effective action. Againusing the decomposition (A.1.2), we get

Gk[ϕ]−1 =δLδRΓkδϕδϕ

+Rk = (G−1k +Rk) + V

(2)k [ϕ]. (A.2.5)

Formally, we can take the inverse of the quantity (A.2.5) to get

Gk[ϕ] = (G−1k +Rk)

−1 − (G−1k +Rk) · V (2)

k [ϕ] · (G−1k +Rk)

−1 (A.2.6)

+(G−1k +Rk) · V (2)

k [ϕ] · (G−1k +Rk) · V (2)

k [ϕ] · (G−1k +Rk)

−1 − · · · .

Diagrammatically, Eq.(A.2.6) tells us that we should sum up the unbranched tree diagramswith the full propagator (G−1

k +Rk)−1 and with effective couplings −Vk[ϕ] at the scale k.

98 Appendix A. Quick derivation of the 1-loop RG expressions

Appendix B

Continuum limits

In sec.2.3, we have derived the Polchinski equation, which represents the renormalization groupflow of the Wilsonian effective action. By introducing the explicit ultraviolet cutoff Λo in theoriginal theory, we are free from UV divergences in all calculations. In the relativistic system, itis important to control short distance physics without knowing its details and for that purposewe require renormalizability for relativistic quantum field theories. In this appendix, we shalltake the continuum limit Λo → ∞ and explicitly see how perturbative renormalizability isensured in this formalism. For simplicity, we restrict ourselves to consider the φ4-theory in the4dimensional Euclidean spacetime (the φ4

4-theory) and prove its perturbative renormalizability.The proof with FRG was first given by Polchinski [9] and it was greatly simplified by Keller,Kopper, and Salmhofer [94].

B.1 Continuum limits of quantum field theories

The nonperturbative realization of the renormalization of the quantum field theory is sketchilysummarized in Fig.B.1. The critical manifold containing the fixed point is the set of the pointswhich flow into the fixed point in the RG flow; around the fixed point, the critical manifold isspanned by the infinite set of irrelevant and marginally irrelevant operators.

Let us consider the flow starting from the bare action which is sufficiently close to but notin the critical manifold. As shown by the dotted line in Fig.B.1, the effective action flows intothe vicinity of the fixed point and then runs away along one of the relevant directions. As aresult, it will reach a trivial theory which represents an infinitely massive theory.

We need a trick in order to get the continuum limit of the theory with finite masses. Atfirst, we should tune the bare action back towards the critical manifold in order to prevent thetheory from becoming infinitely massive. Simultaneously, we should rewrite physical quantities

Critical manifold

Renormalized trajectoryUV fixed point Infinitely massive theory

Figure B.1: Schematic view of the RG flow in the space of bare couplings.

99

100 Appendix B. Continuum limits

in terms of renormalized quantities. After these renormalization procedures, we take the limitthat the bare action touches the critical manifold, and then the RG flow splits into two parts.The first one goes into the fixed point, and the another one emanates from the fixed point alonga relevant direction, which path is called a renormalized trajectory. In terms of renormalizedquantities, the far end of the renormalized trajectory obtains a finite limit. In this sense, thecontinuum limit is said to be defined at a UV fixed point.

Using this idea, we will explain what is expected as a consequence of renormalizability ofthe φ4

4-theory following the discussion given by Polchinski [9]. We give the bare action, or theWilsonian effective action at the UV cutoff scale Λo, in the next form:

Sbare[φ] =1

2

∫p

p2 +m2

K(p2/Λ2o)φ−pφp + Sint,KΛo

[φ]

=1

2

∫p

p2 +m2

K(p2/Λ2o)φ−pφp +

∫d4x

(∆m2

2φ2 +

∆Z

2(∂µφ)2 +

λo4!φ4

). (B.1.1)

We would like to integrate out high-momentum modes of a field, so that we reduce the cutoffΛ = Λo in Eq.(B.1.1) to a much lower scale Λ = ΛR. Now we have known that, by changing theeffective action Sint,KΛ

[φ] according to the Polchinski equation (2.3.18), correlation functions oflow-momentum modes are unchanged.

Even though the action might start with a simple form Sbare, at lower scales Sint,KΛcontains

the nonrenormalizable interactions and becomes very complicated. However, in the infraredregion Λ2

R Λ2o, the effective action Sint,KΛR

will be strongly attracted to the set of renormalizedtrajectories R, which is totally parametrized only by the relevant and marginal couplings,i.e., the renormalizable couplings 1. In our case, by keeping the Z2 symmetry φ 7→ −φ, thesubmanifold R is three-dimensional, where the dimension 3 is determined by the number of therenormalizable operators φ2, φ∂2φ, and φ4.

We parametrize the effective theory by effective vertices V2n corresponding to the expansion(2.3.19):

Sint,KΛ[φ] =

∞∑n=1

1

(2n)!

∫p1,··· ,p2n

φp1 · · ·φp2n · (2π)4δ4(p1 + · · ·+ p2n)V2n(Λ; p1, · · · , p2n).

In order to define the perturbation theory, we introduce a formal variable g. First of all, Sint,KΛis

taken to be a formal power series in g, and hence

Sint,KΛ[φ] =

∞∑r=1

grS(r)int,KΛ

[φ]. (B.1.2)

At Λ = Λo, we set for r ≥ 1

S(r)int,KΛo

=

∫d4x

(∆m2

(r)

2φ2 +

∆Z(r)

2(∂µφ)2 +

λo(r)4!

φ4

). (B.1.3)

We should notice that the formal power series (B.1.2) starts from r = 1 at any scale Λ, and

furthermore S(r)int,KΛ

is an even polynomial of degree ≤ 2r + 2. The RG flow is given by the

1In the perturbative argument, distinction between relevant and marginal is unnecessary. But the distinctionwill be important if we consider the continuum limit in nonperturbative arguments [6]. See also sec.2.5.

B.1. Continuum limits of quantum field theories 101

Polchinski equation (2.3.18). The boundary conditions for the RG flow are mixed type: atΛ = Λo the theory is required to become φ4

4-theory and at Λ = 0 the renormalization conditionis given for the relevant parameters: From (B.1.3), we have

b.c. at Λ = Λo

(a) V(r)

2n (Λo) = 0 (for 2n ≥ 6),

(b) V(r)4 (Λo)(p1, p2, p3) = λo(r),

(c) V(r)2 (Λo)(p) = ∆Z(r)(Λo)p

2 + ∆m2(r)(Λo),

(B.1.4)

which imply that ∂wp V(r)2n (Λo) = 0 for 2n+w ≥ 5. Here we have omitted the last argument of the

vertices since it is determined from the other arguments due to the momentum conservation.We should remind that ∆m2

(r)(Λo), ∆Z(r)(Λo), and λo(r) have not been prescribed yet and theyshould be determined by the boundary conditions at Λ = 0, which are given by

b.c. at Λ = 0

(a) V(r)

4 (Λ = 0)(pR1 , pR2 , p

R3 ) = λR(r),

(b) ∂pµ∂pνV(r)2 (Λ = 0)(pR4 )|δµν = 2δµν∆Z

R(r),

(c) V(r)2 (Λ = 0)(pR5 ) = ∆mR

(r)

2,

(B.1.5)

where pRi (i = 1, . . . , 5) specify the renormalization points. The usual Bogoliubov-Parasyuk-Hepp-Zimmermann (BPHZ) renormalization condition can be obtained by setting pR1 = · · · =pR5 = 0 and λR(r) = δr1 and ∆ZR

(r) = ∆mR(r)

2= 0.

In the low energy scale Λ < m, the mass term changes the scaling of the fields drastically. Byusing another renormalization scale ΛR > m, we need not consider such complicated situationsince we can stop the RG flow at Λ = ΛR. For that purpose, we rewrite the condition (B.1.5)into the boundary conditions at Λ = ΛR:

b.c. at Λ = ΛR

(a) V(r)

4 (ΛR)(pR1 , pR2 , p

R3 ) = λR(r) +O(r),

(b) ∂pµ∂pνV(r)2 (ΛR)(pR4 )|δµν = 2δµν(∆Z

R(r) +O(r)),

(c) V(r)2 (ΛR)(pR5 ) = ∆mR

(r)

2+O(r),

(B.1.6)

where O(r) stands for the contribution of a finite number of connected, amputated Feynmangraphs of order r whose external momenta are pR1 , p

R2 , p

R3 or pR4 or pR5 and whose propagators

are K(p2/Λ2)p2+m2 .

In order to estimate effective couplings, we introduce a semi-norm on vertex functions. Letf(Λ, pl) = f(Λ; p1, · · · , pn) be a sufficiently regular function of momenta pl defined in thetheory with an effective cutoff Λ. We require the regularity of the function f(Λ; pl) so thatthe following definition of semi-norms makes sense: for any η > 0

||∂wf(Λ)||η = maxi1,··· ,iw

maxµ1,··· ,µw

sup|pi|<max2Λ,η

∣∣∣∂pµ1i1· · · ∂pµwiw f(Λ; p1, · · · , pn)

∣∣∣ . (B.1.7)

This definition is useful. Let us take the cutoff function KΛ(p2) = K(p2/Λ2) so that it im-mediately damps above p2 = Λ2 so as to behave like the step function. Then K(p2/Λ2) =−Λ∂ΛK(p2/Λ2) behaves as the delta function which has a peak around p2 = Λ2, so that∫

d4p

(2πΛ)4

Λ2K(p2/Λ2)

p2 +m2< C, and

∣∣∣∣∣∣∣∣∣∣∂wΛ2K(p2/Λ2)

p2 +m2

∣∣∣∣∣∣∣∣∣∣∞

< DwΛ−w,


that is, the scaling behavior expected from the mass dimension is correct.We shall derive the inequalities which are necessary for the proof of perturbative renor-

malizability. At first, let us rewrite the Polchinski equation (2.3.18) in terms of dimensionlessquantities A2n = Λ2n−4V2n (see Eq.(2.3.20)):(−Λ

∂

∂Λ+ 2n− 4

)A2n(Λ; pl) = −

∑partition

A2k(Λ; pil, q)Λ2K(q2/Λ2)

q2 +m2A2(n−k+1)(Λ;−q, pil′)

+1

2

∫d4q

(2πΛ)4

Λ2K(q2/Λ2)

q2 +m2A2n+2(Λ; q,−q, pl). (B.1.8)

We can readily get the following inequality for arbitrary η > 0∣∣∣∣∣∣∣∣(−Λ∂

∂Λ+ 2n− 4

)∂wA

(r)2n (Λ)

∣∣∣∣∣∣∣∣η

(B.1.9)

≤ cw,n,r

||∂wA(r)2n+2(Λ)||η +

(r,n,w)∑r′,k,wi

Λ−w1||∂w2A(r′)2k (Λ)||η||∂w3A

(r−r′)2(n−k+1)(Λ)||η

,

where the summation is taken over all the combinatorics for tree diagrams, i.e., r′ runs over1, . . . , r− 1, k runs over 1, . . . , n, and w1, w2, w3 runs over all the combinations for w1 +w2 +w3 = w. Here, cw,n,r is independent of Λ and η. We should notice that this is a very suitableform in terms of the perturbation theory. The first term in the right hand involves only lowerorders with respect to the coupling constant and the second term only involves the term withlarger n. Since 2n is the number of the legs, at each order of the perturbation A2n(Λ) vanishesfor large n. Thus we can expect that the induction will work.

As we will see in the next section, Eq.(B.1.9) is sufficient to prove the boundedness of the

norms ||A(r)n (ΛR)||(2Λ,η) as Λo → ∞. However, in order to prove the convergence we need the

following inequalities involving ∂Λo∂wp A

(r)n (Λ).

For irrelevant vertices, boundary conditions at Λ = Λo is well-known for us. Therefore, forcases 2n+ w ≥ 5, apply ∂Λo∂

wp to the equation:∫ Λo

Λ

ds

s

(−s ∂

∂s

)s4−2nA2n(s; pl) =

∫ Λo

Λ

dss3−2n

(1

2

∫d4q

(2πΛ)4

s2K(q2/s2)

q2 +m2A2n+2(s; q,−q, pl)

−∑

partition

A2k(s; pil, q)s2K(q2/s2)

q2 +m2A2(n−k+1)(s;−q, pil′)

), (B.1.10)

which can be easily derived from Eq.(B.1.8). Taking the norm, we get for 2n+ w ≥ 5∣∣∣∣∣∣Λ4−2n∂Λo∂wA

(r)2n (Λ)

∣∣∣∣∣∣η

(B.1.11)

≤ c′w,n,rΛ3−2no

||∂wA(r)2n+2(Λo)||η +

(r,n,w)∑r′,k,wi

Λ−w1o ||∂w2A

(r′)2k (Λo)||η||∂w3A

(r−r′)2(n−k+1)(Λo)||η

+c′′w,n,r

∫ Λo

Λ

dss3−2n

||∂Λo∂wA

(r)2n+2(s)||η +

(r,n,w)∑r′,k,wi

s−w1||∂Λo∂w2A

(r′)2k (s)||η||∂w3A

(r−r′)2(n−k+1)(s)||η

.

B.2. Perturbative renormalizability of the φ44-theory 103

Again, constants c′w,n,r and c′′w,n,r are independent of Λo, Λ, and η. On the other hand, one maytake the integration interval in Eq.(B.1.10) as [ΛR,Λ], which will be more appropriate choicefor the relevant vertices. Applying ∂wp and ∂Λo , we obtain that∣∣∣Λ4−2n∂Λo∂

wA(r)2n (Λ; p1, · · · , p2n)

∣∣∣− ∣∣∣Λ4−2nR ∂Λo∂

wA(r)2n (ΛR; p1, · · · , p2n)

∣∣∣ (B.1.12)

≤c′′w,n,r∫ Λ

ΛR

dss3−2n

||∂Λo∂wA

(r)2n+2(s)||M +

(r,n,w)∑r′,k,wi

s−w1||∂Λo∂w2A

(r′)2k (s)||M ||∂w3A

(r−r′)2(n−k+1)(s)||M

,

where M = max|pi|; i = 1, · · · , n − 1. Now we have finished preliminaries and then we canprove perturbative renormalizability of the Euclidean massive φ4

4-theory.

B.2 Perturbative renormalizability of the φ44-theory

Here we follow the proof of the perturbative renormalizability of the φ44-theory given by Keller,

Kopper, and Salmhofer [94]. The rigorous statement of the theorem is as follows:

Theorem B.2.1 (Perturbative renormalizability). Consider the field theory defined bythe action (B.1.1) and define the effective action Sint,KΛ

(Λo) as the solution of the Polchinskiequation (2.3.20) with the boundary conditions (B.1.4) and (B.1.5). Then, order by order withrespect to a formal variable g in the perturbation theory, the limit lim

Λo→∞V(r)n (Λ = 0; p1, · · · , pn−1)

exists for all r and n. Again, we have omitted the last argument of the vertices thanks to themomentum conservation.

Moreover, limΛo→∞

V(r)n (Λ = 0; p1, · · · , pn−1) is a smooth and polynomially bounded function of

p1, · · · , pn−1.

Let us add a convenient notation. The symbol P ln(z), where z = ΛΛR, Λo

ΛRwith Λ ∈ [ΛR,Λo]

and hence z ≥ 1, represents some possibly polynomial in ln(z). Also Pη represents somepolynomials in η. Each time it appears, the coefficients of the polynomial can be different aslong as they does not depend on η nor on Λ nor on Λo.

Lemma B.2.1 (Boundedness). For any η > 0 and for Λ ∈ [ΛR,Λo], we have

||∂wp A(r)2n (Λ)||η ≤ Λ−w · Pη ·

(P ′ ln Λ

ΛR

+Λ

Λo

P ′ ln Λo

ΛR

). (B.2.1)

Proof. We will prove it by induction. The induction hypothesis is that Eq.(B.2.1) is true forr < ro or n > no for any w ≥ 0. The induction starts since Eq.(B.2.1) holds trivially true forr = 0 or 2n > 2r + 2.

For the irrelevant vertices, i.e., for ∂wA(ro)2no (Λ) with w + 2no ≥ 5, we employ the boundary

condition at Λ = Λo to get

||Λ4−2no∂wp A(ro)2no (Λ)||η =

∣∣∣∣∣∣∣∣∫ Λo

Λ

ds∂s

(s4−2no∂wA

(ro)2no (s)

)∣∣∣∣∣∣∣∣η

≤∫ Λo

Λ

ds

s

∣∣∣∣∣∣s∂s (s4−2no∂wA(ro)2no (s)

)∣∣∣∣∣∣η.

Apply Eq.(B.1.9) to the integrand, and then the induction hypothesis gives

||Λ4−2no∂wp A(ro)2no (Λ)||η ≤

∫ Λo

Λ

ds

ss4−2no−w

(P ln

s

ΛR

+s

Λo

P lnΛo

ΛR

).


The integration by parts gives

∫ Λo

Λ

dss−aP lns

ΛR

= Λ−a+1

(P ln

Λ

ΛR

+a−1∑b=1

(Λ

Λo

)bPb ln

Λo

ΛR

)for a ≥ 2, and then we can obtain the result and arrive at Eq.(B.2.1).

Now let us consider the relevant vertices 2no + w ≤ 4. The boundedness of these verticeswill be derived from the finiteness condition at Λ = ΛR. Let us first discuss 2no = 4 and w = 0.Then

|(A(ro)4 (Λ)− A(ro)

4 (ΛR))(pR1 , pR2 , p

R3 )| ≤

∫ Λ

ΛR

ds|∂sA(ro)4 (s; pR1 , p

R2 , p

R3 )| ≤

∫ Λ

ΛR

ds||∂sA(ro)4 (s)||M ,

with M ≥ max|pRi | : i = 1, 2, 3. According to the induction hypothesis, Eq.(B.2.1) holds for

A(r)2n (Λ) with r = ro and 2n ≥ 6 or with r < ro and n ∈ N.

Now we can discuss the detail for the term O(ro) in part (a) of the boundary conditions(B.1.6). Since it is constructed with the finite number of the diagrams of order ro with lower

order vertices A(r)2n (ΛR), which can be already bounded by Eq.(B.2.1), and therefore we can

obtain that

|A(ro)4 (ΛR; pR1 , p

R2 , p

R3 )| ≤ c,

where c is a constant which is independent of Λ and Λo. Therefore,

|A(ro)4 (Λ; pR1 , p

R2 , p

R3 )| ≤ c+

∫ Λ

ΛR

ds||∂sA(ro)4 (s)||M ≤ c+

∫ Λ

ΛR

dss−1

(P ln

s

ΛR

+s

Λo

P lnΛo

ΛR

)≤(P ln

Λ

ΛR

+Λ

Λo

P lnΛo

ΛR

).

Therefore, we have obtained the bound for the renormalization point (p1, p2, p3). For the generalmomenta, the Taylor theorem gives

A(ro)4 (Λ; q1, q2, q3) = A

(ro)4 (Λ; pR1 , p

R2 , p

R3 ) +

3∑i=1

4∑µ=1

(pRi,µ − qi,µ)

∫ 1

0

dx∂qi,µA(ro)4 (Λ; q + x(pR − q)),

and then taking the semi-norm || · ||M we can obtain that∣∣∣∣∣∣A(ro)4 (Λ)

∣∣∣∣∣∣M≤ P ln

Λ

ΛR

+Λ

Λo

P lnΛo

ΛR

+

(Λ +

M

ΛR

ΛR

)Λ−1

(P ′ ln Λ

ΛR

+Λ

Λo

P ′ ln Λo

ΛR

)≤ P ln

Λ

ΛR

+Λ

Λo

P lnΛo

ΛR

.

This is nothing but Eq.(B.2.1) for η = M , and hence it is true for any η > 0.Next, let us evaluate the case with 2no = 2 and w = 2, i.e., estimate the bound for

∂2pA

(ro)2 (Λ). The assessment is analogous to that for the case 2no = 4. By estimating the

finite number of the Feynman diagrams contributing to O(r) in the part (b) of the boundary

conditions (B.1.6), we get∣∣∣∂2pA

(ro)2 (ΛR; pR4 )|δµν

∣∣∣ ≤ c′. The Taylor theorem gives

∂2pA

(ro)2 (ΛR : pR4 ) = ∂2

pA(ro)2 (ΛR : p = 0) +

∑µ

pR4,µ

∫ 1

0

dx∂pµ∂2pA

(ro)2 (ΛR;xpR4 ).

B.2. Perturbative renormalizability of the φ44-theory 105

We should notice that ∂2pA

(ro)2 (ΛR; p = 0) only has the term proportional to δµν due to the

Lorentz covariance, so we can find the boundedness for the first term. Applying the bound

||∂3pA

(ro)2 (ΛR)|| we can obtain that

∣∣∣∂2pA

(ro)2 (ΛR; p4)

∣∣∣ ≤ c′′ for some constant c′′ which is indepen-

dent of Λo. Thus,∣∣∣Λ2∂2pA

(ro)2 (Λ; pR4 )

∣∣∣ ≤ Λ2Rc′′ +

∫ Λ

ΛR

ds∣∣∣∣∣∣∂s (s2∂2

pA(ro)2 (s)

)∣∣∣∣∣∣M,

where M ≥ |pR4 |. Using Eq.(B.1.9), we can find the P ln-type bound for∣∣∣Λ2∂2

pA(ro)2 (Λ; pR4 )

∣∣∣.Again the Taylor theorem gives the bound for any momentum and then we can arrive at thebound (B.2.1).

For the case 2no = 2 and w = 1, the Taylor theorem gives

∂1pA

(ro)2 (Λ; p) = ∂1

pA(ro)2 (Λ; p = 0) +

∑µ

pµ

∫ 1

0

dx∂pµ∂1pA

(ro)2 (Λ;xp)

and the first term in the right hand side vanishes due to the Lorentz covariance. Therefore, wecan find the desired bound for ||∂1

pA(ro)2 (Λ)||η.

Finally, we consider the case with 2no = 2 and w = 0. By the renormalization condition,|A(ro)

2 (ΛR; pR5 )| < c′′′ and then

|Λ2A(ro)2 (Λ; p)| ≤ Λ2

Rc′′′ +

∫ Λ

ΛR

ds∣∣∣∣∣∣∂s (s2A

(ro)2 (s)

)∣∣∣∣∣∣M,

where M ≥ |pR5 |. We again obtain the P ln-type bound for∣∣∣Λ2A

(ro)2 (Λ; pR5 )

∣∣∣. The Taylor theorem

gives the desired result.

Now we can show the convergence of the effective actions in the limit Λo →∞ in the similarway.

Lemma B.2.2 (Convergence). Assume that for η ≥ 0 and Λ ∈ [ΛR,Λo]∣∣∣∣∣∣∂wp A(r)2n (Λ)

∣∣∣∣∣∣η≤ Λ−wP ln

Λo

ΛR

. (B.2.2)

Then we can find that ∣∣∣∣∣∣∂Λo∂wp A

(r)2n (Λ)

∣∣∣∣∣∣η≤ Λ−2

o Λ−w+1P lnΛo

ΛR

. (B.2.3)

Proof. The statement will be inductively proved, and the induction scheme is precisely whatis done in the proof of the previous lemma. Again the induction begins since the statement istrivial for r = 0 or 2n > 2r + 2 for any w ≥ 0.

Given no, consider separately the cases w + 2no ≥ 6 and w + 2no = 5. For w + 2no ≥ 6,Eq.(B.1.11) and application of Eqs.(B.2.2,B.2.3) to the right hand side of Eq.(B.1.11) give∣∣∣∣∣∣Λ4−2no∂Λo∂

wA(r)2no(Λ)

∣∣∣∣∣∣η≤ Λ3−2no−w

o P lnΛo

ΛR

+

∫ Λo

Λ

dss3−2noΛ−2o s−w+1P ′ ln Λo

ΛR

= Λ3−2no−wo P ln

Λo

ΛR

+ Λ−2o

(Λ5−2no−wo − Λ5−2no−w

)P ′′ ln Λo

ΛR

≤ Λ−2o Λ5−2no−wP ′′′ ln Λo

ΛR

.


Therefore, we have obtained the result for 2no + w ≥ 6. For 2no + w = 5, the first inequalityabove holds true but the following evaluation is different. So we get∣∣∣∣∣∣Λw−1∂Λo∂

wA(r)2no(Λ)

∣∣∣∣∣∣η≤ Λ3−5

o P lnΛo

ΛR

+

∫ Λo

Λ

dsΛ−2o s3−5+1P ′ ln Λo

ΛR

≤ Λ−2o P ′′ ln

Λo

ΛR

.

Thus, we have obtained the desired bound for the irrelevant vertices. Notice that this proof isvalid only for Λ < Λo but due to the continuity the limit Λ→ Λo justifies that the results holdtrue for any Λ ∈ [ΛR,Λo].

Now let us consider the effective vertices with 2no + w ≤ 4. We should notice that|∂ΛoA

(ro)4 (ΛR; pR1 , p

R2 , p

R3 )|, |∂Λo∂

2pA

(ro)2 (ΛR; pR4 )| and |∂ΛoA

(ro)2 (ΛR; pR5 )| do not exceed Λ−2

o P ln ΛoΛR

,since the contribution comes from the finite number of connected amputated diagrams contain-ing the vertex differentiated by ∂Λo and because the tree diagram vanishes due to ∂Λo∆Z

R(r) =

∂Λo∆mR(r)

2= ∂Λoλ

R(r) = 0.

We can find that Eq.(B.1.12) gives the appropriate bounds for |∂ΛoA(ro)4 (ΛR; pR1 , p

R2 , p

R3 )|,

|∂Λo∂2pA

(ro)2 (ΛR; pR4 )| and |∂ΛoA

(ro)2 (ΛR; pR5 )|. The applications of the Taylor theorem gives

Eq.(B.2.3) for these vertices.

Since the assumptions in the second lemma are ensured from the first one, we get the proofof the theorem of perturbative renormalizability. Let us give some concluding remarks on thisappendix.

In the proof, the notion of fixed points seems to be hidden although we have pointed outthat the continuum limit of the quantum field theory is to control bare actions so that theyapproach to a fixed point along a renormalized trajectory. This is a trick of the perturbativerenormalizability and it is related to the triviality problem of the φ4

4 theory.In these proofs, we have implicitly assumed that we can use the Gaussian fixed point (GFP)

which describes the massless free field theory as a UV fixed point of the φ44 theory. Indeed,

the theorem suggests that effective couplings obey the scaling law predicted from canonicaldimensions up to some small logarithmic corrections, which give small anomalous dimensions.At each order of the perturbation theory the GFP can thus play a role of a UV fixed point andthe irrelevant couplings are controlled by the renormalization condition.

However, when we discuss the (Borel) summability of the perturbation theory the situationbecomes totally different. The 1-loop RG flow of the φ4-coupling suggests that it is marginallyirrelevant around the GFP, and then the theory is not asymptotically free. The appearanceof the Landau pole in the UV regions prevents us from taking the continuum limit in a non-perturbative way. This means that if we would like to use the GFP as a UV fixed point thenthe theory should be free and can contain no interaction terms. In the case of the φ4

4 theory, itis believed that there exists no other UV fixed points and we should regard it as a low-energyeffective theory with some UV cutoff.

In the case of non-Abelian gauge theories, we know that they have asymptotic freedom.Therefore, it is widely believed that non-trivial fixed points of non-Abelian gauge theories canbe realized with appropriate fine-tuning of the renormalizable couplings.

Appendix C

Angular momenta

In this section, we give a brief review of the theory of angular momenta according to [95, 96].The purpose of this appendix is to specify the convention on the theory of angular momenta,since there exists several ones used in common. Let J = (Jx, Jy, Jz) be an operator of angularmomenta, satisfying the commutation relation

[Ji, Jj] = iεijkJk. (C.0.1)

We denote the simultaneous eigenvector of J2 and of Jz by |j,m〉, with m = −j,−j + 1, · · · , j:

J2|j,m〉 = j(j + 1)|j,m〉, jz|j,m〉 = m|j,m〉. (C.0.2)

Here we have normalized the eigenvector |j,m〉 as

〈j1,m2|j2,m2〉 = δj1,j2δm1,m2 , (C.0.3)

and the phase convention of states |j,m〉 with different m is chosen so that they satisfyJ±|j,m〉 =

√(j ∓m)(j ±m+ 1)|j,m± 1〉 for raising and lowering operators J± = Jx ± iJy.

C.1 Clebsch-Gordan coefficients and spherical tensors

Let us consider the coupling of two different angular momenta J1 and J2, and denote the totalangular momentum J = J1 + J2. We define the Clebsch-Gordan coefficients Cjm

j1m1;j2m2by

|j,m〉 =∑m1,m2

Cjmj1m1;j2m2

|j1,m1〉 ⊗ |j2,m2〉, (C.1.1)

or, denoting |j1, j2;m1,m2〉 = |j1,m1〉 ⊗ |j2,m2〉, we have Cjmj1m1;j2,m2

= 〈j1, j2;m1,m2|j,m〉.There still exists ambiguity of sign, since the normalization (C.0.3) does not tell relative phasesbetween states with different j’s. This ambiguity is resolved by requiring that (j||J1||j − 1) =−(j||J2||j − 1) are real and positive, where (j||Ji||j′) are reduced matrix elements defined byEq.(C.1.18).

It should be noticed that the Clebsch-Gordan coefficients Cjmj1,m1;j2,m−m1

j,m1 are elementsof a unitary transformation from the tensor product basis |j1, j2;m1,m−m1〉m1 to the irre-ducible basis |j,m〉j 1: ∑

m1

Cjmj1,m1;j2,m−m1

Cj′mj1,m1;j2,m−m1

= δjj′ . (C.1.2)

1Regard these are the basis for the Hilbert subspace specified by the conditions J21 = j1(j1+1), J2

2 = j2(j2+1)and Jz = J1z + J2z = m.

107

108 Appendix C. Angular momenta

α

β

γ

Figure C.1: The Euler angles

Therefore, we can invert the formula (C.1.1):

|j1,m1〉 ⊗ |j2,m−m1〉 =∑j

Cjmj1,m1;j2,m−m1

|j,m〉. (C.1.3)

In this phase convention, the Clebsch-Gordan coefficients satisfy following symmetries:

Cj3m3

j1m1;j2m2= (−)j1+j2−j3Cj3,−m3

j1,−m1;j2,−m2= (−)j1+j2−j3Cj3m3

j2m2;j1m1= (−)j1−m1

√2j3 + 1

2j2 + 1Cj2−m2

j1m1;j3−m3.

(C.1.4)Before introducing spherical tensors, let us consider the rotational matrix and its transfor-

mation properties. We denote the Euler angle αβγ as given in Fig.C.1. The correspondingrotational operator exp(−iθn · J) is given by

R(αβγ) = exp(−iαJz) exp(−iβJy) exp(−iγJz). (C.1.5)

We denote matrix elements of the operator R(αβγ) by Dj(αβγ):

Djm′m(αβγ) = 〈jm′|R(αβγ)|jm〉. (C.1.6)

The unitarity of the representation suggests that Djm′m(−γ,−β,−α) = Dj∗

mm′(α, β, γ). Let usapply the rotational matrix to the both sides of Eq.(C.1.1), then we get the expansion

Djm′m(αβγ) =

∑m1,m2

∑m′1,m

′2

Cjmj1m1;j2m2

Cjm′

j1m′1;j2m′2Dj1m′1,m1

(αβγ)Dj2m′2,m2

(αβγ). (C.1.7)

Inverting this relation, or using Eq.(C.1.3) instead of Eq.(C.1.1), we obtain the relation

Dj1m′1m1

(αβγ)Dj2m′2m2

(αβγ) =∑j

Cjm′

j1m′1;j2m′2Cjmj1m1;j2m2

Djm′1+m′2,m1+m2

(αβγ). (C.1.8)

Schur’s orthogonal relation suggests that∫dΩ

4πDj1∗m′1m1

(αβγ)Dj2m′2m2

(αβγ) =δj1j2

2j1 + 1δm′1m′2δm1m2 , (C.1.9)

Here we denote the integration over the solid angle by dΩ, expressed with the Euler angles as∫dΩ = 1

2π

∫ 2π

0dα∫ β

0sin βdβ

∫ 2π

0dγ. As a special case, we obtain∫ 2π

0

dϕ

∫ π

0

sin θdθY ∗lm(θ, ϕ)Yl′m′(θϕ) = δll′δmm′ , (C.1.10)

C.1. Clebsch-Gordan coefficients and spherical tensors 109

since Dlm0(αβ0) =

√4π

2l+1Y ∗lm(θϕ). From (C.1.8), we get the composition rule:

Yl1,m1(θ, ϕ)Yl2,m2(θ, ϕ) =∑l,m

√(2l1 + 1)(2l2 + 1)

4π(2l + 1)C lml1m1,l2m2

C l0l10,l20Ylm(θ, ϕ). (C.1.11)

The irreducible spherical tensor TLMM of rank L is the set of 2L+ 1 operators TLM withM = −L,−L+ 1, · · · , L satisfying

RTLMR−1 =

L∑M ′=−L

DLM ′M(αβγ)TLM ′ , (C.1.12)

where R = R(α, β, γ) is the rotational operator. Equivalently, we can characterize the irre-ducible spherical tensor using the commutation relations:

[J±, TLM ] =√

(L∓M)(L±M + 1)TL,M±1, (C.1.13)

[Jz, TLM ] = MTL,M . (C.1.14)

Due to the linearity of the condition (C.1.12), or of (C.1.13,C.1.14), the set of the irreduciblespherical tensors of the same rank forms a vector space.

We can form another irreducible tensor from two irreducible tensors TL1,M1(A1) and TL2,M2(A2)as follows:

TLM(A1,A2) =∑M1,M2

CLML1M1,L2M2

TL1M1(A1)TL2M2(A2). (C.1.15)

Here we have denoted all the other arguments of the irreducible tensors by Ai. Especially whenL2 = L1, we can form the rotational invariants:

(−)L1√

2L1 + 1T00(A1A2) =∑M1

(−)M1TL1M1(A1)TL1,−M1(A2). (C.1.16)

As a special case of this formula, we can obtain the addition theorem of the spherical harmonics:

Pl(cos θ) =4π

2l + 1

∑m

Y ∗lm(θ1, ϕ1)Ylm(θ2, ϕ2). (C.1.17)

The Wigner-Eckart theorem gives the projection quantum number dependence of matrixelements of an irreducible tensor:

〈j′m′|TLM |jm〉 = Cj′m′

jm,LM(j′||TL||j), (C.1.18)

where the second factor (j′||TL||j) is called the reduced matrix element, which is independentof projection quantum numbers.

Proof. Using the commutation relation (C.1.14), we find that (m′−m−M)(j′m′|TLM |jm) = 0.Therefore, the matrix elements (j′m′|TLM |jm) vanish unless m′ = m+M . Another commuta-tion relation (C.1.13) gives√

(j′ ±m′)(j′ ∓m′ + 1)(j′,m′ ∓ 1|TLM |jm) −√

(j ∓m)(j ±m+ 1)(j′m′|TLM |j,m± 1)

=√

(L∓M)(L±M + 1)(j′m′|TL,M±1|jm).

The same recursion relation holds for the Clebsch-Gordan coefficients Cj′m′

jm,LM , which satisfies

the same constraint (m′ −m −M)Cj′m′

jm,LM = 0. Therefore, the dependence of (j′m′|TLM |jm)

on projection quantum numbers is the same as that of Cj′m′

jm,LM .


C.2 6j symbols and 9j symbols

Here we consider the composition of three angular momenta J1,J2 and J3. For the uncoupledrepresentation, we can diagonalize the 6 operators J2

1 , J22 , J2

3 and J1z, J2z, J3z. There are threedifferent ways to couple those angular momenta with a resultant angular momentum J and itsprojection Jz:

(1) J1 + J2 = J12, J12 + J3 = J ,

(2) J2 + J3 = J23, J23 + J1 = J ,

(3) J1 + J3 = J13, J13 + J2 = J .

Let |j1j2(j12)j3; jm〉 be the state vector corresponding to the composition (1) above. Us-ing the Clebsch-Gordan coefficients, we can relate these vectors with the uncoupled basis|j1,m1; j2,m2; j3,m3〉:

|j1j2(j12)j3; jm〉 =∑

m12,m3

Cjmj12m12,j3m3

∑m1,m2

Cj12m12

j1m1,j2m2|j1,m1; j2,m2; j3,m3〉. (C.2.1)

Introducing similar notations, we can readily find the following:

|j2j3(j23)j1; jm〉 =∑

m1,m23

Cjmj1m1,j23m23

∑m2,m3

Cj23m23

j2m2,j3m3|j1,m1; j2,m2; j3,m3〉, (C.2.2)

|j1j3(j13)j2; jm〉 =∑

m13,m2

Cjmj13m13,j2m2

∑m1,m3

Cj13m13

j1m1,j3m3|j1,m1; j2,m2; j3,m3〉. (C.2.3)

Since states in each coupling choice form a complete set, there is a unitary transformation fromone scheme to another one. We define the Wigner 6j symbols, or simply called the 6j symbols,j1 j2 j12

j3 j j23

by

〈j1j2(j12)j3; jm|j2j3(j23)j1; j′m′〉 = δjj′δmm′(−)j1+j2+j3+j√

(2j12 + 1)(2j23 + 1)

j1 j2 j12

j3 j j23

.

(C.2.4)The previous factor of the symbol is introduced so that the 6j symbols are invariant under anypermutation of its columns or under interchange of the upper and lower arguments in each ofany two columns. For examples,

a b cd e f

=

a c bd f e

=

b a ce d f

=

d e ca b f

=

a e fd b c

= · · · .

According to the definition (C.2.4), the 6j symbols may be expressed in terms of the Clebsch-Gordan coefficients: ∑

m1,m2,m3,m12,m23

Cjmj12m12,j3m3

Cj12m12

j1m1,j2m2Cj′m′

j1m1,j23m23Cj23m23

j2m2,j3m3

= δjj′δmm′(−)j1+j2+j3+j√

(2j12 + 1)(2j23 + 1)

j1 j2 j12

j3 j j23

. (C.2.5)

Since the phase convention are given in the definition of the Clebsch-Gordan coefficients, thedefinition of the 6j symbols or the Racah coefficients is complete.

C.3. List of formulae 111

Let us next consider an addition of four angular momenta J1,J2,J3 and J4. There are threedifferent ways to couple those angular momenta which are not directly related to the productof the two 6j symbols:

(1) J1 + J2 = J12, J3 + J4 = J34, J12 + J34 = J ,

(2) J1 + J3 = J13, J2 + J4 = J24, J13 + J24 = J ,

(3) J1 + J4 = J14, J2 + J3 = J23, J14 + J23 = J .

The following argument is similar to that was done in the discussion for the 6j symbols.For each scheme of the composition, we introduce the state vectors |j1j2(j12)j3j4(j34); jm〉,|j1j3(j13)j2j4(j24); jm〉, and |j1j4(j14)j2j3(j23); jm〉, respectively. Using the Clebsch-Gordan co-efficients, we can explicitly write down those states as the tensor products. For instance,

|j1j2(j12)j3j4(j34); jm〉 =∑

m1,...,m4

∑m12,m34

Cjmj12m12;j34m34

Cj12m12

j1m1;j2m2Cj34m34

j3m3;j4m4|j1m1; j2m2; j3m3; j4m4〉.

(C.2.6)In order to realize the unitary transformation in the different composition schemes, we introducethe Wigner 9j symbols by

〈j1j2(j12)j3j4(j34); jm|j1j3(j13)j2j4(j24); j′m′〉 = δjj′δmm′Πj12j13j24j34

j1 j2 j12

j3 j4 j34

j13 j24 j

(C.2.7)

with Πab...c =√

(2a+ 1)(2b+ 1) · · · (2c+ 1). The 9j symbol does not change its value under thetransposition and under the even permutations of its elements. Under the odd permutations,the phase factor (−)R appears, where R is the sum of all arguments of the 9j symbol.

C.3 List of formulae

Here we list up the useful formulae to calculate the summation of the products of the Clebsch-Gordan coefficients. We pick up them from the textbook [96].

First we should recall the symmetries of the Clebsch-Gordan coefficients (C.1.4):

Ccγaα,bβ = (−)a+b−cCc,−γ

a,−α;b,−β = (−)a+b−cCcγbβ,aα, , (C.3.1)

and

Ccγaα,bβ = (−)a−α

√2c+ 1

2b+ 1Cb,−βa,α;c,−γ = (−)b+β

√2c+ 1

2a+ 1Ca,−αc,−γ;b,β. (C.3.2)

Sums Involving One Clebsch-Gordan Coefficients∑α

Caαaα,b0 =

√2a+ 1

∑α

(−)a−αCb0aα,a−α = (2a+ 1)δb0. (C.3.3)

Sums Involving Two Clebsch-Gordan Coefficients∑αβ

Ccγaα,bβC

c′γ′

aα,bβ = δcc′δγγ′ , (C.3.4)


∑cγ

Ccγaα,bβC

cγaα′,bβ′ = δαα′δββ′ , (C.3.5)

∑cγ

(2c+ 1)Cbβaα,cγC

bβ′

aα′,cγ = (2b+ 1)δαα′δββ′ , (C.3.6)

∑aα

(−)a−α(2a+ 1)Ccγaα,bβC

bβ′

aα′,cγ′ =√

(2b+ 1)(2c+ 1)δγ,−γ′δβ,−β′ . (C.3.7)

Sums Involving Three Clebsch-Gordan Coefficients∑αβδ

Ccγaα,bβC

eεbβ,dδC

dδaα,fϕ = (−)a+b+e+f

√(2c+ 1)(2d+ 1)Ceε

cγ,fϕ

a b ce f d

, (C.3.8)

∑αβδ

Caαbβ,cγC

dδbβ,eεC

dδaα,fϕ = (−)b+c+d+f (2d+ 1)

√2a+ 1√

2e+ 1Ceεcγ,fϕ

a b ce f d

, (C.3.9)

∑αβδ

(−)a−αCcγaα,bβC

eεdδ,bβC

fϕdδ,a−α = (−)b+c+d+f

√(2c+ 1)(2f + 1)Ceε

cγ,fϕ

a b ce f d

. (C.3.10)

Sums Involving Four Clebsch-Gordan Coefficients

For the simplicity of expressions, let us put Πab...c =√

(2a+ 1)(2b+ 1) · · · (2c+ 1).

∑βγεϕ

Caαbβ,cγC

dδeε,fϕC

gηeε,bβC

jµfϕ,cγ = Πadgj

∑kκ

Ckκgη,jµC

kκdδ,aα

c b af e dj g k

, (C.3.11)

∑βγεϕ

Caαbβ,cγC

dδeε,fϕC

bβeε,gηC

cγfϕ,jµ = Πbcd

∑kκ

√2k + 1Ckκ

gη,jµCaαdδ,kκ

a b cd e fk g j

. (C.3.12)

Sums Involving Clebsch-Gordan Coefficients and a 6j Symbol

Again, we put Πab...c =√

(2a+ 1)(2b+ 1) · · · (2c+ 1).

∑eε

(−)2eΠcdCeεbβ,dδC

eεfϕ,cγ

a b ce f d

= Ccγ

aα,bβCdδaα,fϕ, (C.3.13)

∑fϕ

(−)c+d+fΠceCfϕeε,aαC

fϕdδ,cγ

b a cf d e

= Ccγ

aα,bβCeεdδ,bβ. (C.3.14)

Appendix D

Properties of ΓSk in the vacuum

In this appendix, we calculate some properties of the symmetric part of the four point vertexfunction ΓSk in the vacuum defined by Eqs.(4.2.7) and (4.2.9). Although we can calculateΓSk (P ) for general momenta P and for general cutoff scales k analytically, we do not here givefull calculations to obtain the formula for general cases. Instead, we will give calculation forspecial but important cases to find properties of the vertex.

We use the unit 2m = 1 for simplicity of calculations.

D.1 Calculation of ΓSk (P 0, 0)

Let us calculate the vertex function ΓSk (P ) when the spatial momentum P = 0. Before startingcalculations, let us find an expression of ΓSk (P ) for general momenta and for general cutoffscales. Using the formula (4.2.7) with Eq.(4.2.9), we find that

1

ΓSk (P 0,P 2)− 1

8πaS=

∫d3l

(2π)3

[1

iP 0 − 2µ+ P 2

2+ 2l2 +Rk(l + P

2) +Rk(l− P

2)− 1

2l2

]

+

∫d3l

(2π)3

[1

2l2 + 2RΛo(l)− 1

iP 0 − 2µ+ P 2

2+ 2l2 +RΛo(l + P

2) +RΛo(l− P

2)

],

where Λo is an ultraviolet cutoff. Since the second line of this equation vanishes in the limitΛo →∞, we can eliminate the UV cutoff from this formula. As a result, we find that

1

ΓSk (P 0,P 2)=

1

8πaS−∫

d3l

(2π)3

[1

2l2− 1

iP 0 − 2µ+ P 2

2+ 2l2 +Rk(l + P

2) +Rk(l− P

2)

].

(D.1.1)By putting k = 0, we recover the formula (4.2.10).

Now, let us substitute P = 0 into Eq.(D.1.1), then we get

1

ΓSk (P 0, 0)=

1

8πaS−∫

d3l

(2π)3

[1

2l2− 1

iP 0 − 2µ+ 2l2 + 2Rk(l)

]. (D.1.2)

Since we use the regulator Rk(l) = (k2 − l2)θ(k2 − l2), we find that

1

ΓSk (P 0, 0)=

1

8πaS−∫ k

0

d3l

(2π)3

[1

2l2− 1

iP 0 − 2µ+ 2k2

]−∫ ∞k

d3l

(2π)3

[1

2l2− 1

iP 0 − 2µ+ 2l2

].

(D.1.3)

113

114 Appendix D. Properties of ΓSk in the vacuum

Integrations in Eq.(D.1.3) are quite simple, and we immediately obtain

1

ΓSk (P 0, 0)=

1

8πaS− 1

2π2

(k

2− k3/3

iP 0 − 2µ+ 2k2

)−

√iP 0

2− µ

8π

1− 2

πtan−1

k√iP 0

2− µ

.

(D.1.4)If aS < 0, there exists no bound states and we can put µ = 0. On the other hand, if aS > 0,then there exists a bound state and for the zero density condition we should put µ = −1/a2

S inthe vacuum, as we have already remarked in sec.4.2.

If IR cutoff scales k are much larger than other physical scales P 0, µ, then we find that

1

ΓSk (P 0, 0)' 1

8πaS− k

6π2= gk, (k →∞) (D.1.5)

as desired. Let us consider the opposite limit k → 0, then we get

1

ΓSk (P 0, 0)' 1

8πaS− k

4π2−

√iP 0

2− µ

8π

1− 2

π

k√iP 0

2− µ

=1

ΓSk=0(P 0, 0). (k → 0) (D.1.6)

That is, linear terms in k cancel in the Taylor expansion around k = 0. We can also easily findthat cubic terms with respect to k also cancel, and then the convergence of the limit k → 0 isvery fast. Let us consider the BEC case aS > 0, and we plot aS/Γ

Sk (P = 0) in Fig.D.1 using

Eq.(D.1.6).Later, we will find that it is useful to introduce the quantity ∂ω(1/ΓSk ) = i−1∂P 0(1/ΓSk )(P 0, 0).

Indeed, this quantity is necessary in Eq.(D.3.3).

∂ω1

ΓSk (P 0, 0)= − k3/6π2

(iP 0 − 2µ+ 2k2)2− 1

32π√

iP 0

2− µ

1− 2

πtan−1

k√iP 0

2− µ

− 1

8π2

k

iP 0 − 2µ+ 2k2. (D.1.7)

In the next section, we will use this to consider effects of explicit breaking of the Galileansymmetry.

0.5 1.0 1.5 2.0

0.008

0.006

0.004

0.002

Figure D.1: Graph of aS/ΓSk (P = 0) when aS > 0.

D.2. Spatial momentum dependence of ΓSk 115

D.2 Spatial momentum dependence of ΓSk

Now, let us discuss the dependence of the vertex function ΓSk (P ) on spatial momenta P . If thecutoff is removed so that k = 0, then the Galilean symmetry is satisfied and the dependence ofP can appear only through the combination iP 0 + P 2/2. However, the Galilean symmetry isexplicitly broken and this restriction does not hold in general cases.

Instead of calculating the full dependence of ΓSk on P , we try to find the leading orderexpansion of 1/ΓSk (P ) in terms of P . Due to the rotational invariance, the expansion startsfrom P 2 and then we will calculate its coefficient.

We introduce an expansion parameter s and then we consider the expansion of 1/ΓSk (P 0, s2P 2)around s = 0. Since the linear term in s must vanish due to the rotational symmetry, we haveto calculate the second derivative of 1/ΓSk (P 0, s2P 2) at s = 0. Using Eq.(D.1.1), we obtain that

∂2

∂s2

1

ΓSk (P 0, s2P 2)

∣∣∣∣s=0

=

∫d3l

(2π)3

∂2

∂s2

1

iP 0 − 2µ+ s2P 2

2+ 2l2 +Rk(l + sP

2) +Rk(l− sP2 )

∣∣∣∣∣s=0

= −∫

d3l

(2π)3

∂

∂s

sP 2 − P · (l + sP2

)θ(k2 − (l + sP2

)2) + P · (l− sP2

)θ(k2 − (l− sP2

)2)[iP 0 − 2µ+ s2P 2

2+ 2l2 +Rk(l + sP

2) +Rk(l− sP2 )

]2∣∣∣∣∣s=0

= −∫

d3l

(2π)3

P 2(1− θ(k2 − l2)) + 2(P · l)2δ(k2 − l2)

[iP 0 − 2µ+ 2l2 + 2Rk(l)]2. (D.2.1)

Therefore we should calculate the following integration

∂2

∂s2

1

ΓSk (P 0, s2P 2)

∣∣∣∣s=0

= −P 2

(∫ ∞k

d3l

(2π)3

1

(iP 0 − 2µ+ 2l2)2+

1

6π2

k3

(iP 0 − 2µ+ 2k2)2

).

(D.2.2)The integration in Eq.(D.2.2) can be evaluated as follows:∫ ∞

k

d3l

(2π)3

1

(iP 0 − 2µ+ 2l2)2=

1

8π2

∫ ∞k

dll2(

iP 0

2− µ+ l2

)2

=1

8π2

√iP 0

2− µ

∫ ∞k√iP0

2 −µ

dxx2

(1 + x2)2

=1

32π√

iP 0

2− µ

1− 2

πtan−1 k√

iP 0

2− µ

+

1

8π2

k

iP 0 − 2µ+ 2k2. (D.2.3)

Combining Eqs.(D.2.2,D.2.3) and Eq.(D.1.7), we find that

∂2

∂s2

1

ΓSk (P 0, s2P 2)

∣∣∣∣s=0

= P 2∂ω1

ΓSk (P 0, 0). (D.2.4)


Therefore,this means that up to the quadratic order the Galilean symmetry is respected andwe get

1

ΓSk (P 0,P 2)=

1

ΓSk (P 0, 0)+P 2

2!∂ω

1

ΓSk (P 0, 0)+ · · · = 1

ΓSk(P 0 − iP 2

2, 0) + · · · , (D.2.5)

where ellipses represent terms smaller than quadratic order of P 2. The effect of the Galileaninvariance breaking appears only in higher order terms. Neglecting these higher order termsand forcing the Galilean symmetry, we obtain the result:

1

ΓSk (P 0,P 2)=

1

8πaS− 1

2π2

(k

2− k3/3

iP 0 + P 2/2− 2µ+ 2k2

)

−

√iP 0

2+ P 2

4− µ

8π

1− 2

πtan−1

k√iP 0

2+ P 2

4− µ

. (D.2.6)

Before closing this appendix, let us evaluate spatial integrations of ΓSk , which are necessaryfor calculating the atom-dimer scattering. At first let us consider quantities in Eq.(D.3.3). Thatis, we consider the case aS > 0 and calculate two following quantities:∫ k d3l

(2π)3ΓSk (k2, l2),

∫ k d3l

(2π)3∂ωΓSk (k2, l2), (D.2.7)

where the notations are defined in Eq.(D.3.4). From Eq.(D.2.6), we find that

ΓS,−1k :=

1

ΓSk (k2, 0)=

1

8πaS− 1

2π2

(k

2− k3/3

3k2 + 2a2S

)−

√k2

2+ 1

a2S

8π

1− 2

πtan−1

k√k2

2+ 1

a2S

,

(D.2.8)

∂ωΓS,−1k := ∂ω

1

ΓSk (k2, 0)= − k3/6π2

(3k2 + 2a2S

)2−

1− 2π

tan−1 k√k2/2+1/a2

S

32π√

k2

2+ 1

a2S

− k/8π2

3k2 + 2a2S

. (D.2.9)

We should emphasize that ΓS,−1k vanishes quadratically in the limit k → 0 but ∂ωΓS,−1

k does notvanish. We now employ the leading order approximation in the expansion (D.2.5). Therefore,we get ∫ k d3l

(2π)3ΓSk (k2, l2) '

∫ k d3l

(2π)3[ΓS,−1k + l2∂ωΓS,−1

k /2]−1

=1

π2∂ωΓS,−1k

(k −

√2ΓS,−1

k

∂ωΓS,−1k

tan−1

[k

√∂ωΓS,−1

k

2ΓS,−1k

]). (D.2.10)

Using ∂ωΓSk = −(ΓSk )2∂ωΓS,−1k , we can similarly obtain that∫ k d3l

(2π)3∂ωΓSk (k2, l2) ' −

∫ k d3l

(2π)3[ΓS,−1k + l2∂ωΓS,−1

k /2]−2∂ωΓS,−1k (D.2.11)

=1

2π2ΓS,−1k

(2kΓS,−1

k

2ΓS,−1k + k2∂ωΓS,−1

k

−

√2ΓS,−1

k

∂ωΓS,−1k

tan−1

[k

√∂ωΓS,−1

k

2ΓS,−1k

]).

D.2. Spatial momentum dependence of ΓSk 117

These are the formula to be used in the first and second attempts at calculations of the atom-dimer scattering.

Next, let us consider quantities which may become necessary for the more complicatedmomentum dependence of gadk . Let us evaluate∫ k d3l

(2π)3

ΓSk (k2, l2)

[k2 + l2 + 1/a2S]n

,

∫ k d3l

(2π)3

∂ωΓSk (k2, l2)

[k2 + l2 + 1/a2S]n

(D.2.12)

for n = 0, 1, 2. The case of n = 0 is already calculated. Again using the same approximation,we find that∫ k d3l

(2π)3

ΓSk (k2, l2)

[k2 + l2 + 1/a2S]

=1

π2k∂ωΓSk

k√∂ωΓS,−1

k /2ΓS,−1k

√(ka2

S)/[(kaS)2 + 1]

k2(∂ωΓS,−1k /2ΓS,−1

k )− (ka2S)/[(kaS)2 + 1]

(D.2.13)

×

[k

√∂ωΓS,−1

k

2ΓS,−1k

tan−1

(√(kaS)2

(kaS)2 + 1

)−

√(kaS)2

(kaS)2 + 1tan−1

(k

√∂ωΓS,−1

k

2ΓS,−1k

)],

∫ k d3l

(2π)3

ΓSk (k2, l2)

[k2 + l2 + 1/a2S]2

=(kaS)2/[(kaS)2 + 1]

4π2kΓS,−1k

(D.2.14)

×

(kaS)2/(2(kaS)2 + 1)[k2(∂ωΓS,−1

k /2ΓS,−1k )− (ka2

S)/ ((kaS)2 + 1)]

+

√(kaS)2/[(kaS)2 + 1]

[k2(∂ωΓS,−1k /2ΓS,−1

k )− (ka2S)/ ((kaS)2 + 1)]2

×

(k2∂ωΓS,−1

k

2ΓS,−1k

+(kaS)2

(kaS)2 + 1

)tan−1

√(kaS)2

(kaS)2 + 1

−2k

√∂ωΓS,−1

k

2ΓS,−1k

√(kaS)2

(kaS)2 + 1tan−1

(k

√∂ωΓS,−1

k

2ΓS,−1k

)].

∫ k d3l

(2π)3

∂ωΓSk (k2, l2)

[k2 + l2 + 1/a2S]

=(kaS)2/[(kaS)2 + 1]

π2kΓS,−1k

(D.2.15)

×

k2∂ωΓS,−1k /(k2∂ωΓS,−1

k + 2ΓS,−1k )[

k2(∂ωΓS,−1k /2ΓS,−1

k )− (ka2S)/ ((kaS)2 + 1)

]+

k√∂ωΓS,−1

k /2ΓS,−1k

[k2(∂ωΓS,−1k /2ΓS,−1

k )− (ka2S)/ ((kaS)2 + 1)]2

×

2k

√∂ωΓS,−1

k

2ΓS,−1k

√(kaS)2

(kaS)2 + 1tan−1

√(kaS)2

(kaS)2 + 1

−

(k2∂ωΓS,−1

k

2ΓS,−1k

+(kaS)2

(kaS)2 + 1

)tan−1

(k

√∂ωΓS,−1

k

2ΓS,−1k

)].


∫ k d3l

(2π)3

∂ωΓSk (k2, l2)

[k2 + l2 + 1/a2S]

=−(k2∂ωΓS,−1

k /(2ΓS,−1k )

)2

((kaS)2/[(kaS)2 + 1])2

π2k5∂ωΓS,−1k

[k2(∂ωΓS,−1

k /2ΓS,−1k )− (ka2

S)/ ((kaS)2 + 1)]2 (D.2.16)

×

[−(3(kaS)2 + 1)k2∂ωΓS,−1

k + (kaS)22ΓS,−1k

(k2∂ωΓS,−1k + 2ΓS,−1

k )(2(kaS)2 + 1)

+1[

k2(∂ωΓS,−1k /2ΓS,−1

k )− (ka2S)/ ((kaS)2 + 1)

]×

(k2∂ωΓS,−1

k

2ΓS,−1k

+ 3(kaS)2

(kaS)2 + 1

)k

√∂ωΓS,−1

k

2ΓS,−1k

tan−1

(k

√∂ωΓS,−1

k

2ΓS,−1k

)

−

((kaS)2

(kaS)2 + 1+ 3k2∂ωΓS,−1

k

2ΓS,−1k

)√(kaS)2

(kaS)2 + 1tan−1

√(kaS)2

(kaS)2 + 1

]

D.3 Numerical approximate solution of atom-dimer scat-

tering

In this section, let us calculate atom-dimer scattering within an approximation neglecting feed-back effect from higher-order vertices.

In order to perform a further simple analysis, let us make a crude assumption to neglectthe momentum dependence of the vertex function gadk . In the integrand of the right hand sideof Eq.(4.2.25), we put p1 = p′1 = (i/a2

S,0) and p2+3 = p′2+3 = 0. Then we find an ordinarydifferential equation

∂kgadk = ∂k

∫l

(2gadk

ΓSk (−l + p1)

[G−1o +Rk](l)[G−1

o +Rk](−l)+

ΓSk (p1 + l)

[G−1o +Rk]2(l)[G−1

o +Rk](−l)

). (D.3.1)

As we have mentioned, we drop the feedback terms by hand. Performing the l0 integration, wefind that

∂kgadk = ∂k

∫d3l

(2π)3

(2gadk

ΓSk (iP 0 = l2 +Rk(l2), l2)

2[l2 + 1/a2S +Rk(l2)]

+ΓSk (iP 0 = l2 +Rk(l

2), l2)

4[l2 + 1/a2S +Rk(l2)]2

). (D.3.2)

Recall that we adopt the regulator Rk(l2) = (k2 − l2)θ(k2 − l2), and then its derivative takes

the form k∂kRk(l2) = 2k2θ(k2 − l2). Then Eq.(D.3.2) becomes

k∂kgadk = 2gadk

∫ k d3l

(2π)3k2

(− ΓSk (k2, l2)

(k2 + 1/aS)2+∂ωΓSk (k2, l2)

k2 + 1/a2S

)+

∫ k d3l

(2π)3k2

(− ΓSk (k2, l2)

(k2 + 1/a2S)3

+∂ωΓSk (k2, l2)

2(k2 + 1/a2S)2

), (D.3.3)

where

ΓSk (k2, l2) = ΓSk (iP 0 = k2, l2), ∂ωΓSk (k2, l2) =∂ΓSki∂P 0

(iP 0 = k2, l2). (D.3.4)

These quantities are calculated in Appendix D. Especially, see Eqs.(D.2.10,D.2.11).

D.3. Numerical approximate solution of atom-dimer scattering 119

1 2 3 4

0.20

0.15

0.10

0.05

Figure D.2: k-dependence of the vertex function gadk in the first attempt.

By solving the ordinary differential equation (D.3.3) using Eqs.(D.2.10,D.2.11), we find that

gadk=0/a

2S ' −0.224. (D.3.5)

The k-dependence of gadk in our approximation can be found in Fig.D.2. As a result, the 1PIcontribution to the atom-dimer scattering T -matrix at zero energy is given by

T ad1PI/aS ' −22.5. (D.3.6)

Therefore, the atom-dimer scattering length aad becomes

aad/aS ' 1.47. (D.3.7)

The correct value of the atom-dimer scattering length was derived by G. Skornyakov and K.Ter-Martirosyan [63] using the three-body Schrodinger equation, and it is given by aad = 1.18aS.Our result (D.3.7) is different from this correct value, and it would come from lack of thefeedback from higher vertex functions. However, it still improve the result (4.2.19) of the treediagram contribution significantly.


Appendix E

Channel decomposition of thedipole-dipole interaction

In this appendix, we explicitly calculate the decomposition of the dipole-dipole interaction(DDI) into channels labeled by angular momenta (J, Jz, L, S).

E.1 DDI in the spherical coordinate

At first, we rewrite the DDI in Eq.(5.2.3) in the spherical tensor basis. Since it has the quantumnumber L = 2 and J = 0 it should be proportional to

V α,ββ′α′(q) ∝

∑m

C002,−m;2,m

∑m1,m2

C2,−m1,m1;1,m2

(Sm1)αα′(Sm2)ββ′∑µ1,µ2

C2,m1,µ1;1,µ2

qµ1qµ2 . (E.1.1)

Here we have used the spherical basis for the vectors q = k − k′ and for the spin vector S:q±1 = ∓(qx± iqy)/

√2, q0 = qz and S±1 = ∓(Sx± iSy)/

√2, S0 = Sz. Indeed, we shall now show

that

(q · Sαα′)(q · Sββ′)−

q2

3Sαα′ · S

ββ′ =

∑m

(−)2−m∑m1,m2

C2,−m1,m1;1,m2


C2,m1,µ1;1,µ2

qµ1qµ2 .

(E.1.2)

Proof. Using the formula (C.3.7) and symmetries (C.3.3) of the Clebsch-Gordan coefficients,we get

(−)b+β(−)c+γ′∑aα

(−)a−αCaαb,−β;cγC

a,−αb,−β′;cγ′ = δγ,−γ′δβ,−β′ . (E.1.3)

Here we should notice that since C1m1µ1,1µ2

= (−)1+1−1C1m1µ2,1µ1

we need not consider a = 1 in the

sum in Eq.(E.1.3) for our purpose. Now we easily find that (C00lm:l,−m = (−)l−m√

2l+1)

∑m

(−)2−mC2,−m1m1,1m2

C2m1µ1,1µ2

= (−)1+µ1(−)1−m2δm2,−µ2δm1,−µ1 −1

3(−)1−m1(−)1−µ1δm1,−m2δµ1,−µ2

+(anti-symmetric µ1 ↔ µ2).

Since∑

m(−)mxmy−m = x · y for the vectors, we get the result.

121

122 Appendix E. Channel decomposition of the dipole-dipole interaction

Therefore, we have got the formula

V αββ′α′(q) =

γ2

q2

∑m

(−)2−m∑m1,m2

C2,−m1,m1;1,m2


C2,m1,µ1;1,µ2

qµ1qµ2 . (E.1.4)

Substitute q = k − k′ with k = k′. Notice that kµ = kY1µ(k) and for k′ < k

1

|k − k′|2= −2π

k2

∑l,lz

HlYl,lz(k)Y ∗l,lz(k′) + const.δ(k − k′), (E.1.5)

where Hl =l∑

n=1

1

nis the harmonic number. The coefficient of the spherical delta function is

constant in terms of the label l, and in the limit k′/k → 1 the coefficient diverges.

Proof. Assume that k′ < k and take the limit k′ → k later. We can now use the sum rule

1

|k − k′|=∑l,m

k′l

kl+1

4π

2l + 1Ylm(k)Y ∗lm(k′).

Therefore, we get

1

|k − k′|2=∑l1,m1

∑l2,m2

k′l1+l2

kl1+l2+2

(4π)2

(2l1 + 1)(2l2 + 1)Yl1m1(k)Y ∗l1m1

(k′)Yl2m2(k)Y ∗l2m2(k′).

The formula (C.1.11) and the orthogonal relation (C.3.4) give

1

|k − k′|2=∑lm

(∑l1,l2

k′l1+l2

kl1+l2+2C l0l10,l20C

l0l10,l20

)4π

2l + 1Ylm(k)Y ∗lm(k′)

Let us perform the summation over l1, l2 in order to simplify the formula and to extract thesingularity in the limit k′ → k.

This limit should be treated with some care. The infinite series 4π2l+1

∑l1,l2

(C l0l10,l20)2 diverges,

since C l0l10,l20 ' δl1,l2

√2l+12l1+1

. In order to circumvent this artificial divergence, we write this seriesas

4π

2l + 1

∑l1,l2

(C l0l10,l20)2 = 4π

∑l1,l2

[1

2l + 1(C l0

l10,l20)2 − (C00l10,l20)2

]+ 4π

∑l1,l2

(C00l10,l20)2 (E.1.6)

and the first term becomes −2πHl and the second term is a constant independent of l.Let us justify the above result. First we now use the formula (C.1.11):

2

2l + 1(C l0

l10,l20)2 =

∫ 1

−1

dxPl1(x)Pl2(x)Pl(x). (E.1.7)

In order to make the infinite series in Eq.(E.1.6) finite, we again multiply the convergence factorr = k′/k < 1 so that

2

2l + 1

∞∑l1,l2=0

rl1+l2(C l0l10,l20)2 =

∫ 1

−1

dx∞∑l1=0

rl1Pl1(x)∞∑l2=0

rl2Pl2(x)Pl(x) =

∫ 1

−1

dxPl(x)

1− 2rx+ r2.

E.2. Particle-particle channel 123

We would like to take the limit r → 1, but that procedure gives the divergence. The origin ofthe divergence is independent of l, so we here evaluate the difference of them:

xl := limr→1

∫ 1

−1

dxPl+1(x)− Pl(x)

1− 2rx+ r2=

1

2

∫ 1

−1

dxPl+1(x)− Pl(x)

1− x. (E.1.8)

In order to evaluate xl, we consider its generating function∞∑l=0

tl+1xl =1

2

∫ 1

−1

dx

1− x

(1− t√

1− 2tx+ t2− 1

)= ln(1− t). (E.1.9)

Therefore, we find that xl = −1/l for l ≥ 1, and then we obtain the result.

Since kµ = kY1µ(k), we can rewrite Eq.(E.1.4) as

V αββ′α′(k − k

′) = γ2∑l,lz

(−2πHlYl,lz(k)Y ∗l,lz(k

′) + const.δ(k − k′))

×∑m

(−)2−m∑m1,m2

C2,−m1,m1;1,m2

(Sm1)αα′(Sm2)ββ′

×∑µ1,µ2

C2,m1,µ1;1,µ2

(Y1µ1(k)− Y1µ1(k′))(Y1µ2(k)− Y1µ2(k′)). (E.1.10)

The spherical delta function gives zero, and we can now take the limit k′ = k. The spin matrixelements are given as

(Sm)αα′ = (sα|Sm|sα′) =√s(s+ 1)Csα

sα′,1m,

which gives

V αββ′α′(k − k

′) = −2πγ2s(s+ 1)∑l,lz

HlYl,lz(k)Y ∗l,lz(k′) (E.1.11)

×∑m

(−)2−m∑m1,m2

C2,−m1,m1;1,m2

Csαsα′,1m1

Csβsβ′,1m2

×∑µ1,µ2

C2,m1,µ1;1,µ2

(Y1µ1(k)− Y1µ1(k′))(Y1µ2(k)− Y1µ2(k′))

= −2πγ2s(s+ 1)∑l,lz

HlYl,lz(k)Y ∗l,lz(k′) (E.1.12)

×∑m

(−)2−m∑m1,m2

C2,−m1,m1;1,m2

Csαsα′,1m1

Csβsβ′,1m2

∑µ1,µ2

C2,m1,µ1;1,µ2

×

∑l′,l′z

3√4π(2l′ + 1)

Cl′l′z1µ1,1µ2

C l′010,10(Yl′l′z(k) + Yl′l′z(k

′))− 2Y1µ1(k)Y1µ2(k′)

.E.2 Particle-particle channel

Apply the channel decomposition (5.2.9) to the expression (E.1.12):

VJJz ,J ′J ′zLS,L′S′ =

∑M,Sz

∑M ′,S′z

CJJzLM,SSz


∑αβ

∑α′β′

CSSzsα,sβC

S′S′zsα′,sβ′ (E.2.1)

×∫

dΩk

∫dΩk′Y


′)[V αββ′α′(k − k

′)− V αβα′β′(k + k′)].


There are two terms which we should compute, however the second term can be represented interms of the first term. Let us denote the first term as V given in Eq.(E.2.5) and the secondterm as V given in Eq.(E.2.2) so that V = V + V . Then

VJJz ,J ′J ′zLS,L′S′ = −

∑M,Sz

∑M ′,S′z

CJJzLM,SSz


∑αβ

∑α′β′

CSSzsα,sβC


×∫

dΩk

∫dΩk′Y


′)V αβα′β′(k + k′)

= −∑M,Sz

∑M ′,S′z

CJJzLM,SSz


∑αβ

∑α′β′

CSSzsα,sβC

S′S′zsβ′,sα′ (E.2.3)

×∫

dΩk

∫dΩk′Y

∗LM(k)YL′M ′(−k′)V αβ

β′α′(k − k′)

= (−)1+L′+2s−S′VJJz ,J ′J ′zLS,L′S′ . (E.2.4)

Therefore, it suffices to calculate V because VJJz ,J ′J ′zLS,L′S′ = [1 + (−)1+L′+2s−S]V

JJz ,J ′J ′zLS,L′S′ .

Now we find that


∑M,Sz

∑M ′,S′z

CJJzLM,SSz


∑αβ

∑α′β′

CSSzsα,sβC


×∫

dΩk

∫dΩk′Y


′)

×− 2πγ2s(s+ 1)∑l,lz

HlYl,lz(k)Y ∗l,lz(k′)

×∑m

(−)2−m∑m1,m2

C2,−m1,m1;1,m2

Csαsα′,1m1

Csβsβ′,1m2

∑µ1,µ2

C2,m1,µ1;1,µ2

×

∑l′,l′z

3√4π(2l′ + 1)

Cl′l′z1µ1,1µ2

C l′010,10(Yl′l′z(k) + Yl′l′z(k

′))− 2Y1µ1(k)Y1µ2(k′)

,= −2πγ2s(s+ 1)

∑l

Hl

∑M,Sz

∑M ′,S′z

CJJzLM,SSz


(E.2.6)

×∑m

(−)2−m︷︸︸︷∑m1,m2

C2,−m1,m1;1,m2

∑αβ

∑α′β′

CSSzsα,sβC

S′S′zsα′,sβ′C

sαsα′,1m1

Csβsβ′,1m2

×∑

lz ,µ1,µ2

C2,m1,µ1;1,µ2

∫dΩk

∫dΩk′Y


′)Yl,lz(k)Y ∗l,lz(k′)

×

∑l′,l′z

3√4π(2l′ + 1)

Cl′l′z1µ1,1µ2

C l′010,10(Yl′l′z(k) + Yl′l′z(k

′))− 2Y1µ1(k)Y1µ2(k′)

.Here we should notice that the summation over lz is done at the underlined region in Eq.(E.2.6).

Let us perform the summation over the spin indices in the braced part in Eq.(E.2.6). Using


the formula (C.3.12) containing the 9j-symbols, we can find that

∑αβα′β′

CSSzsα,sβC

S′S′zsα′,sβ′C

sαsα′,1m1

Csβsβ′,1m2

= ΠssS′

∑k,κ

√2k + 1Ckκ

1m1,1m2CSSzS′S′z ,kκ

S s sS ′ s sk 1 1

.

(E.2.7)

Performing the summation overm1,m2 gives∑m1,m2

C2,−m1m1,1m2

Ckκ1m1,1m2

= δ2,kδ−m,κ from Eq.(C.3.4),

and then the over-braced part becomes

√5(2s+ 1)2(2S ′ + 1)CSSz

S′S′z ;2,−m

S s sS ′ s s2 1 1

. (E.2.8)

Next, let us perform the angular integration in the underlined part of Eq.(E.2.6). The key isthe formula ∫

dΩY ∗l3m3Yl1m1Yl2m2 =

√(2l1 + 1)(2l2 + 1)

4π(2l3 + 1)C l3m3l1m1,l2m2

C l30l10,l20, (E.2.9)

which is an immediate result of Eq.(C.1.11). Then the integration in the underlined partbecomes

∑l′l′z

3√4π(2l′ + 1)

Cl′l′z1µ1,1µ2

C l′010,10

×

(√(2l + 1)(2l′ + 1)

4π(2L+ 1)CLMllz ,l′l′z

CL0l0,l′0δL′,lδM ′,lz + δL,lδM,lz

√(2L′ + 1)(2l′ + 1)

4π(2l + 1)C llzL′M ′,l′l′z

C l0L′0,l′0

)

−2

√3(2l + 1)

4π(2L+ 1)CLMllz ,1µ1

CL0l0,10

√3(2L′ + 1)

4π(2l + 1)C llzL′M ′,1µ2

C l0L′0,10. (E.2.10)

Again the summation over µ1, µ2 gives∑

µ1,µ2C2,m

1µ1,1µ2Cl′l′z1µ1,1µ2

= δ2,l′δm,l′z in the first term, andin the second term the summation over lz, µ1, and µ2 gives

∑lz ,µ1,µ2

C2m1µ1,1µ2

CLMllz ,1µ1

C llzL′M ′,1µ2

=√

5(2l + 1)CLM2m,L′M ′

1 1 2L L′ l

(E.2.11)

from Eq.(C.3.8), then the underlined part becomes

3

4πC20

10,10 (δL′,l + δL,l)

√(2L′ + 1)

(2L+ 1)CLML′M ′,2mC

L0L′0,20

− 3

4π2

√2L′ + 1

2L+ 1CL0l0,10C

l0L′0,10

√5(2l + 1)CLM

2m,L′M ′

1 1 2L L′ l

. (E.2.12)


Substituting Eqs.(E.2.8,E.2.12) into Eq.(E.2.6), we get

VJJz ,J ′J ′zLS,L′S′ = −2πγ2s(s+ 1)

∑l

Hl

∑M,Sz

∑M ′,S′z

CJJzLM,SSz


(E.2.13)

×∑m

(−)2−m√

5(2s+ 1)2(2S ′ + 1)CSSzS′S′z ;2,−m

S s sS ′ s s2 1 1

×

[3

4πC20

10,10 (δL′,l + δL,l)

√(2L′ + 1)

(2L+ 1)CLML′M ′,2mC

L0L′0,20

− 3

4π2

√2L′ + 1

2L+ 1CL0l0,10C

l0L′0,10

√5(2l + 1)CLM

2m,L′M ′

1 1 2L L′ l

].

Again, we calculate the summation over the projection quantum numbers M,M ′, Sz, and S ′z.There are two terms in the square bracket in Eq.(E.2.13), however their dependence on theprojection quantum numbers are similar. The first term becomes (from the formula (C.3.12))

∑M,M ′,Sz ,S′z

CJJzLM,SSz


CLML′M ′,2mC

SSzS′S′z ;2,−m

=∑kκ

√(2L+ 1)(2S + 1)(2J ′ + 1)(2k + 1)Ckκ

2,m;2,−mCJJzJ ′J ′z ,kκ

J L SJ ′ L′ S ′

k 2 2

=∑k

√(2L+ 1)(2S + 1)(2J ′ + 1)(2k + 1)Ck0

2,m;2,−mCJJzJ ′J ′z ,k0


k 2 2

.(E.2.14)

Performing the summation over m as in Eq.(E.2.14), we find that

∑m

(−)2−m∑

M,M ′,Sz ,S′z

CJJzLM,SSz


CLML′M ′,2mC

SSzS′S′z ;2,−m

= δJJ ′δJzJ ′z√

(2L+ 1)(2S + 1)(2J ′ + 1)


0 2 2

. (E.2.15)

This 9j symbol can be reduced into the 6j symbol:

J L SJ L′ S ′

0 2 2

=(−)S+L′+2+J√

(2 · 2 + 1)(2J + 1)

S ′ S 2L L′ J

.


Therefore, substitution of Eq.(E.2.15) into Eq.(E.2.13) gives

VJJz ,J ′J ′zLS,L′S′ = −(−)S+L′+2+JδJJ ′δJzJ ′z

3

2γ2s(s+ 1)(2s+ 1)

∑l

Hl (E.2.16)

×

√5(2S ′ + 1)(2L′ + 1)

(2L+ 1)

√(2L+ 1)(2S + 1)(2J ′ + 1)√

(2 · 2 + 1)(2J + 1)

S ′ S 2L L′ J

S s sS ′ s s2 1 1

×[C20

10,10 (δL′,l + δL,l)CL0L′0,20 − (−)2+L′−L2

√5(2l + 1)CL0

l0,10Cl0L′0,10

1 1 2L L′ l

],

= (−)S+J+1δJJ ′δJzJ ′z3

2γ2s(s+ 1)(2s+ 1)

∑l

Hl (E.2.17)

×√

(2L+ 1)(2L′ + 1)(2S + 1)(2S ′ + 1)

S ′ S 2L L′ J

S s sS ′ s s2 1 1

×[(δL′,l + δL,l)C

2010,10C

20L′0,L0 + (−)L10C l0

L0,10Cl0L′0,10

1 1 2L L′ l

].

We have now obtained that

VJJz ,J ′J ′zLS,L′S′ = −(−)S+JδJJ ′δJzJ ′z

3

2γ2s(s+ 1)(2s+ 1) (E.2.18)

×√

(2L+ 1)(2L′ + 1)(2S + 1)(2S ′ + 1)

S ′ S 2L L′ J

S s sS ′ s s2 1 1

×

[(HL +HL′)C

2010,10C

20L′0,L0 + (−)L10

∑l

HlCl0L0,10C

l0L′0,10

1 1 2L L′ l

].

As a consistency check, let us confirm that a constant independent of l can be added to Hl

without any effect. Indeed, an additive constant term in Hl is canceled in Eq.(E.2.17). To seethis, we should notice that

∑l

C l0L0,10C

l0L′0,10

1 1 2L L′ l

=∑lm

(−)L+1−lC lmL0,10C

lmL′0,10

1 1 2L L′ l

=

1√(2 + 1)(2L′ + 1)

C1010,20C

L′0L0,20

=−1√

5C20

10,10

(−)L−0(−)L+L′−2

√5

C20L′0,L0. (E.2.19)

Using the parity selection rule, we find that Eq.(E.2.19) becomes (−)L+1C2010,10C

20L′0,L0/5, which

implies cancellation of the constant part in Hl.

As another consistency check, we can easily confirm that the expression in Eq.(E.2.18) sat-isfied restrictions coming from symmetry discussed in sec.5.2. Of course, it manifestly satisfiesthe rotational symmetry, and the parity is respected due to the third line of Eq.(E.2.18).


E.3 Channel decomposition of the contact interaction

In the case of the contact interaction V cαββ′α′(r) = gδαα′δ

ββ′δ

3(r), we can easily find the channeldecomposition. For the immediate reference of the formula, we derive it here. In the momentumrepresentation, the potential becomes

V cαββ′α′(k1 − k2) = gδαα′δ

ββ′ . (E.3.1)

Then, the BCS channel decomposition is


∑M,Sz

∑M ′,S′z

CJJzLM,SSz


∑αβ

∑α′β′

CSSzsα,sβC


×∫

dΩk

∫dΩk′Y


′)[V cαβ,β′α′(k − k′)− V c

αβ;α′β′(k + k′)]

= 4πgδL,0δL′,0∑Sz ,S′z

CJJz00,SSz

CJ ′J ′z00,S′S′z

∑αβ

∑α′β′

CSSzsα,sβC

S′S′zsα′,sβ′ [δαα′δββ′ − δαβ′δβα′ ]

= 4πg(1− (−)2s−S)δJJ ′δJzJ ′zδJSδSS′δL0δL′0. (E.3.3)

Acknowledgments

I am very grad to give great thanks to many people who advised and encouraged me in thisresearch. They affected me strongly in these two years of my master course, and it relies ontheir help to be able to submit this thesis.

At first, I would like to thank my supervisor Tetsuo Hatsuda 1. He spent his countless hourswith my discussion and often gave insightful advises in researches. Especially, he motivatedme to learn the functional renormalization group method and advised me to study dipolarfermionic systems with functional methods in connection with his strong interests on neutronstars. His wide vision of physics enlarges my interests and I am very happy to have a chanceto study with him.

I also thank Gergely Fejos1. He has also spent his precious long time in discussing withme my questions, and gave some advises and suggested related papers for those problems.Those discussions make my questions in a specific form and often clarifies ways to tackle thoseproblems.

I appreciate Tomoya Hayata12 estimating computational costs of numerical computationsand discussing possible improvements. He also suggested another applications of FRG to othersystems.

I thank Takahiko Miyakawa3 and Takaaki Sogo4 for their advises in calculating the interac-tion matrix of the dipole-dipole interaction. They showed their detailed calculations of otherchannels to me and it was very helpful in my calculations.

I would also like to appreciate Yoshimasa Hidaka1 talking about his research and providinginteresting topics. Although they do not connect to my research in a direct way, his discussionalways stimulated me and was very insightful, and I have learned a lot of things.

I also thank Arata Yamamoto1, Yasufumi Araki12, Yusuke Hama12, and Yuji Hirono12 forencouraging and advising me in leading master’s course and writing this thesis.

1Theoretical Research Division, Nishina Center, RIKEN2Department of Physics, The University of Tokyo3Faculty of Education, Aichi University of Education4Institut fur Physik, Universitt Rostock

129

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