Introduction to Renormalization

with Applications

in Condensed-Matter and High-Energy Physics

Institute for Theoretical Physics, University of Cologne

Lecture course, winter term 2017/2018

Michael M. Scherer

February 3, 2018

Contents

Contents 1

1 Introduction 5

1.1 Aspects of the renormalization group . . . . . . . . . . . . . . . . . . . . . . . 6

1.2 Phase transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3 Critical phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4 Perturbative quantum field theory . . . . . . . . . . . . . . . . . . . . . . . . 9

1.5 Renormalization-group based definition of QFTs . . . . . . . . . . . . . . . . 10

1.6 QFTs in the high-energy limit . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 Phase transitions and critical phenomena 13

2.1 Ising model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.1.1 Remarks on solutions of the Ising model in d = 1 and d = 2 . . . . . . 14

2.1.2 Symmetries of the Ising model . . . . . . . . . . . . . . . . . . . . . . 15

2.2 XY model and Heisenberg model . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3 Universality and critical exponents . . . . . . . . . . . . . . . . . . . . . . . . 16

2.4 Scaling hypothesis for the free energy . . . . . . . . . . . . . . . . . . . . . . . 17

2.4.1 Derivation of power laws from scaling hypothesis . . . . . . . . . . . . 18

2.4.2 First attempt to calculate critical exponents . . . . . . . . . . . . . . . 19

2.5 Correlations and hyperscaling . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.6 Ginzburg-Landau-Wilson theory . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.6.1 Functional integral representation of the partition function . . . . . . 22

2.6.2 Saddle-point approximation (SPA) . . . . . . . . . . . . . . . . . . . . 27

2.6.3 Critical exponents in saddle-point approximation∗ . . . . . . . . . . . 28

2.7 Kosterlitz-Thouless phase transition . . . . . . . . . . . . . . . . . . . . . . . 29

2.7.1 XY model in two dimensions . . . . . . . . . . . . . . . . . . . . . . . 29

2.7.2 Limiting cases at low and high temperatures . . . . . . . . . . . . . . 30

3 Wilson’s renormalization group 31

3.1 General strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.2 Momentum-shell transformation . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.2.1 Perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.2.2 Combinations in eq. (3.23) that contribute in eq. (3.19) . . . . . . . . 36

1

Contents

3.3 Determination of fixed-point coupling below four dimensions . . . . . . . . . . 393.4 Epsilon expansion and Wilson-Fisher fixed point . . . . . . . . . . . . . . . . 413.5 Summary and comparison to experiment . . . . . . . . . . . . . . . . . . . . . 443.6 Excursion: Relations to QFTs in high-energy physics . . . . . . . . . . . . . . 45

3.6.1 Landau-pole singularity and the triviality problem . . . . . . . . . . . 473.6.2 Fine-tuning and the hierarchy problem . . . . . . . . . . . . . . . . . . 49

4 Functional methods in quantum field theory 514.1 Introductory remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.2 Generating functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.2.1 Correlation functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.2.2 Schwinger functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.3 Effective action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.4 Perturbative renormalization . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.4.1 Divergencies in perturbation theory . . . . . . . . . . . . . . . . . . . 574.4.2 Removal of divergencies . . . . . . . . . . . . . . . . . . . . . . . . . . 604.4.3 Divergencies in perturbation theory: general analysis . . . . . . . . . . 634.4.4 Classification of perturbatively renormalizable theories . . . . . . . . . 66

5 Functional renormalization group 695.1 Effective average action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.1.1 The regulator term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705.2 Wetterich equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.2.1 Some properties of the Wetterich equation . . . . . . . . . . . . . . . . 735.2.2 Method of truncations . . . . . . . . . . . . . . . . . . . . . . . . . . . 745.2.3 Example: Local potential approximation and for a scalar field . . . . . 75

5.3 Functional RG fixed points and the stability matrix . . . . . . . . . . . . . . 775.3.1 Example: critical exponents for scalar field theories . . . . . . . . . . . 79

5.4 Ultraviolet completion of quantum field theories . . . . . . . . . . . . . . . . . 815.4.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 815.4.2 Asymptotic safety scenario (Weinberg 1976, Gell-Mann-Low 1954) . . . . . . . . 835.4.3 Candidate theories for asymptotic safety . . . . . . . . . . . . . . . . . 86

A Remark on generating functional 89A.1 Connected and disconnected Green’s functions . . . . . . . . . . . . . . . . . 89A.2 Example: two-point function . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

B Divergencies beyond 1-loop 93B.1 Recap: Renormalized action and 1-loop divergencies . . . . . . . . . . . . . . 94B.2 Divergencies beyond 1-loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96B.3 Callan-Symanzik equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97B.4 Inductive proof of perturbative renormalizability (idea) . . . . . . . . . . . . 99

2

Recommended literature

I. Herbut, A Modern Approach to Critical Phenomena, Cambridge UniversityPress. No-frills introduction to critical phenomena and basics of the renormalization groupa la Wilson. Excellent first read to become acquainted with the physics and concepts. Thestart of this lecture follows this presentation, i.e. my chapters on the Ginzburg-Landau-Wilsontheory and Wilson’s RG.

J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, Oxford UniversityPress (2nd edition). 1000+ pages of quantum field theory and statistical mechanics forenthusiasts with a lot of details, background and also explicit calculations. Also contains alsosome more formal and mathematical aspects. Can also be considered as a general referencework. Does not contain functional renormalization. My chapter on the functional integralapproach to QFT is a compilation of Chaps. 5,6, 7 & 9 of this tome.

H. Gies, Introduction to the functional RG and applications to gauge theories, e-Print:hep-ph/0611146. https://arxiv.org/pdf/hep-ph/0611146.pdf. Sections 1 and2 give a smooth and straightforward introduction to the functional RG formalism and a verynice instructive example for its application (the euclidean anharmonic oscillator). Inspiredparts of my chapter on functional RG.

M.E. Peskin and D.V. Schroeder, An Introduction to Quantum Field Theory,Addison-Wesley. The introductory QFT textbook for particle physicists. After a canonicalintroduction to relativistic QFT in part I, the second part deals with functional integrals. Alsocontains a pretty nice chapter on critical phenomena which provides a complementary pointof view. No functional RG.

Further useful reads

• J. Cardy, Scaling and Renormalization in Statistical Physics, Cambridge Lecture Notesin Physics.

• P. Kopietz, L. Bartosch, F. Schutz, Introduction to the Functional RenormalizationGroup, Lecture Notes in Physics 798.

• M. Salmhofer, Renormalization. An Introduction. Springer, Berlin 1998, ISBN 3540646663

3

Chapter 1

Introduction

Quantum field theory provides a framework for the description of all fundamental interac-tions (strong, weak, electromagnetic, maybe gravity), phase transitions in particle physics,statistical mechanics and condensed-matter physics:

Quantum field theory is the framework for the discussion of systems with a large/infinitenumber of coupled degrees of freedom.

In quantum field theory and statistical mechanics renormalization is required to treat infini-ties which appear in calculated quantities, typically induced by effects of self-interactions.Moreover, even when no infinities occur in loop diagrams in QFT, renormalization of massesand fields appearing in the Lagrangian is needed.

Renormalization is a procedure to adjust the theoretical parameters describing a systemin such a way as not to change the measurable properties on the length of time scales ofinterest.

Generally speaking, in quantum or statistical field theory, a system at one scale is described bya set of different parameters (particle masses, couplings, interactions,...). The renormalizationgroup (RG) then describes how these parameters are changed when the system is consideredat a different scale, i.e. the couplings of a system are running couplings, running with thescale. A variation of the scale loosely corresponds to changing the magnifying power of anappropriate microscope for viewing the system.

The renormalization group refers to a mathematical procedure that facilitates the system-atic study of the changes of a physical system when viewed at different length or energyscales.

Upon changing the scale, it might turn out that a particular description of a system in termsof a chosen Lagrangian becomes inappropriate and new degrees of freedom arise. This canalso be assessed by the RG.

5

1. Introduction

In the following, I will first discuss some of the aspects of the RG to give a rough overview.These different aspects will then be worked out in more detail during the lecture.

1.1 Aspects of the renormalization group

• phase transitions and critical phenomena in:

– statistical mechanics (liquid-vapor transitions, ferromagnetic transitions,...)

– condensed-matter/solid-state physics (superfluid/superconducting transitions,...)

– quantum chormodynamics (chiral symmetry-breaking, confinement-defconfinementtransition,...)

– Standard-Model-Higgs sector

• perturbative quantum field theory

– divergencies and their removal

– predictivity of QFTs

– classification of (perturbatively) renormalizable QFTs

• quantum field theories in the high-energy limit

– triviality problem

– non-perturbatively renormalizable QFTs

– UV completion

– asymptotic freedom/safety

• renormalization-group based definitions of quantum field theories

6

1.2. Phase transitions

Renormalization-group methods are relevant to a large diversity of fields

⇒ many (apparently) different implementations

⇒ sometimes hard to access

→ Functional RG provides unified formulation.

1.2 Phase transitions

• Sketchy phase diagram of water :

theory space

S

+

h

TTc

m ≠ 0 m = 0m > 0

m < 0FM PM

temperature

orde

r par

amet

er

liquid-gas (1st order)

temperature

pres

sure

solid phaseliquid phase

vapour

Tc

pc critical point

• Definition:

A phase transition is a point in parameter space (T, p, µ, h, ...) where the thermo-dynamic potential becomes non-analytic.

– non-analyticity only possible in thermodynamic limit.

– in a finite system: partition function is sum of analytic functions of its parameters⇒ analytic

– phase transitions may be continuous (2nd order) or discontinuous (1st order).

– transition lines in phase diagram of water: 1st order transitions

– at critical point in phase diagram of water: 2nd order transition

• continuous phase transition: change of phase of a macroscopic system in equilibrium isnot accompanied by latent heat.

• discontinuous phase transition: change of phase of a macroscopic system in equilibriumis accompanied by latent heat.

• define physical quantity to distinguish phases: order parameter

7

1. Introduction

• examples for order parameters:

– average density (liquid-gas)

– resistance (liquid-solid)

– magnetization (Fe, AFM: La2Cu2)

– condensate (BEC: Li, Rb)

– superfluid density(Al, Pb, YBa2Cu3O6.97,4He, 3He) temperature

orde

r par

amet

er

liquid-gas (1st order)

• examples for phase transitions: Ising ferromagnet (“Ising model”), superfluid transition,Bose-Einstein condensation, superconducting transition, Higgs mechanism, confinement-deconfinement transition (QCD, lattice spin models)

1.3 Critical phenomena

• Critical phenomena occur in continuous/2nd order phase transitions

• definition:

Critical phenomena: non-analytic properties of systems near a continuous phasetransition.

• definition:

Critical point : point in the phase diagram where a continuous phase transition takesplace.

• observation 1:

– at phase transition, we see droplets in bubbles and bubbles in droplets in all sizes(strong fluctuations).

– through microscope: same picture at any magnification.⇒ system is scale invariant

– a nice video which exhibits this behavior with an Ising model can be found here:https://www.youtube.com/watch?v=MxRddFrEnPc

• observation 2:

– Completely different systems with many degrees of freedom and complex interac-tions show quantitatively identical behavior.

– this can be described with a small set of variables and scaling relations−→ notion of universality

8

1.4. Perturbative quantum field theory

• example: liquid-gas coexistence curves of fluids near the critical point:Here, we consider the condensation line, i.e. the first order transition line, where thegas coexists with the liquid, near the critical point. We show the temperature-densityplane (see figure below) with a coexistence region below the curve.

– for Ne, A, Kr, Xe, N2, O2,CO, CH4

– curves are identical when rescaled with ρc, Tc

– figure taken from E. A. Guggenheim,Journal of Chemical Physics 13, 253 (1945).

– ∆ρ ∼ (T − Tc)β with β ≈ 0.33

– in some paramagnetic-ferromagnetic transitionsm ∼ (Tc − T )β with β ≈ 0.33 (MnF2)

PRINCIPLE OF CORRESPONDING ST,\TES 257

.95

.75.

FIG. 2.

these formulae should be used for computing values of pg. There are however occasions when one requires relatively accurate values not of pg itself but of (Pl- pg) / pc; on such occasions formula (6.4) . in view of its extreme simplicity and surprisingly high accuracy has much to recom-mend it. An example of its use will occur in Section 16.

7. VAPOR PRESSURE

At temperatures considerably below the critical temperature, say T<0.65Tc, when formula (6.2) for Pu becomes inaccurate it is convenient to con-sider the equilibrium vapor pressure Prather than pg. According to the principle of corre-sponding states one should expect P fPc to be a universal function of T /Tc• In particular the temperatures T8 at which the equilibrium pres-

sure P is one-fiftieth of the critical pressure should be corresponding temperatures for differ-ent substances and the ratio of T. to Tc should have a universal value. On the other hand Tb the boiling points at a pressure of one atmosphere are not corresponding temperatures for different substances. In rows 9 and 10 of Table I are given Tb the boiling point at a pressure of one atmos-phere, and T. the boiling point at a pressure one-fiftieth the critical pressure. In rows 11 and 12 are given the ratios To/Te and T./Tc. It will be seen that the values of the latter are, as expected, more nearly the same than the values of the former.

8. ENTROPY OF EVAPORATION

According to Trouton's rule the molar entropies of evaporation for different substances have

Downloaded 15 Nov 2011 to 163.10.21.132. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions

• specific heat and susceptibility also follow simple power laws

• power laws: CV ∼ |t|−α and χ ∼ |t|γ with the reduced temperature: t = (T − Tc)/Tc– α, β, γ: critical exponents

– critical exponents can be non-rational, e.g., β ≈ 0.326419(3) (best theoreticalestimate)

– critical exponents often only depend on symmetries of a system and not on details

• questions:

– where does the emergence of universality in different systems come from?

– can we calculate the critical exponents?

1.4 Perturbative quantum field theory

• Remark on relations between euclidean QFT and statistical mechanics

– Euclidean QFT in d-dimensional spacetime ∼ classical statistical mechanics ind-dimensional space

– Euclidean QFT in (d+1)-dimensional spacetime, 0 ≤ τ ≤ β ∼ quantum statisticalmechanics in d-dimensional space

– Euclidean QFT in d-dimensional spacetime ∼ High-temperature quantum statis-tical mechanics in d-dimensional space

• observation: perturbatively calculated correlation functions 〈φ(x1)...φ(xn)〉 are gener-ally divergent, i.e. naıvely they are not well-defined.

• example: φ4 theory in four dimensions

– action: S[φ] =∫d4x

12φ(x)(−∂2

µ +m2)φ(x) + λ4φ(x)4

9

1. Introduction

– one-loop two-point function gives contribution to mass term ∼ m2:

〈φ(p)φ(−p)〉|1−loop =

∫d4x eipx〈φ(x)φ(x)〉 ∼ λ

∫R4

d4q

(2π)4

1

q2 +m2∼

temperature

orde

r par

amet

er

liquid-gas (1st order)

– introduce high-momentum cutoff Λ:

λ

∫R4

d4q

(2π)4

1

q2 +m2−→ λ

∫q2≤Λ2

d4q

(2π)4

1

q2 +m2= Λ

Ω4

2

(Λ2 −m2 log(1 + Λ2/m2)

)– cutoff regularization: quantitative control of divergence

• in summary:

– tree level: 〈φ(p)φ(−p)〉|p=0 ∼ m2

– with 1-loop correction: m2 + λ(c1Λ2 + c2m2 log(1 + Λ2/m2))

– interpretation:

∗ m2: parameter of theory

∗ m2 + λ(...): physical correlation function (should not be Λ-dependent!)

⇒ adjust m = m(Λ), λ = λ(Λ), φ = φ(Λ) such that 〈φ(x1)...φ(xn)〉 is Λ-independent and finite for Λ→∞.

→ perturbative renormalization

• questions:

– is this strategy always possible (classification of theories)?

– are the predictions of the theory independent from the regularization scheme?

– what is the role/interpretation of the cutoff (effective theories)?

– why are so many theories realized in nature (for example the standard model ofparticle physics) renormalizable within perturbative QFT?

1.5 Renormalization-group based definition of QFTs

• observation:

– systematic analysis of divergencies of quantum field theories is difficult beyondperturbation theory

• idea: integration of fluctuations in one narrow momentum shell (= “RG step”) is finite

⇒ construct integration over all fluctuations by (infinite) sequence of RG steps

⇒ Exact/functional renormalization group, flow equations

10

1.6. QFTs in the high-energy limit

– RG flow of average effective action Γk

– flow equation describes continuous trajectoryconnecting the microscopic actionand the full (quantum) effective action:

S = Γk=Λ −→ Γ = Γk=0

(see also QFT II)temperature

orde

r par

amet

er

liquid-gas (1st order) theory space

S

1.6 QFTs in the high-energy limit

• for perturbatively renormalizable QFTs, the high-energy conditions (parameters) atcutoff Λ can be chosen such that the low-energy predictions are independent of Λ.

• questions:

– how “natural” are these initial conditions for different systems?

– do we need to “fine tune” the conditions?

−→ hierarchy problem in the standard model of particle physics

– can cutoff of perturbatively renormalizable QFTs be removed (i.e. be sent to ∞)?

∗ if yes: QFT is valid on all scales (fundamental theory)

∗ if no: QFT predicts its own failure (e.g. triviality problem)

⇒ hint towards new physics

• example (for hint towards new physics):

– Fermi’s theory: 4-Fermion interaction amplitude λ for neutrino-neutrino scattering

– dimensional analysis:

M∼ λ+ λ2Λ2 ∼

temperature

orde

r par

amet

er

liquid-gas (1st order) theory space

S

+

⇒ Failure! Amplitude blows up

→ new physics can cure this problem (within perturbation theory):

introduce vector boson

temperature

orde

r par

amet

er

liquid-gas (1st order) theory space

S

+

– mechanism can also be occur towards low energies, indicating appearance of anordering transition, where the relevant degrees of freedom of the system change(e.g. in terms of condensation)

• are there QFTs that are perturbatively non-renormalizable and non-perturbativelyrenormalizable?

– Weinberg’s Asymptotic Safety scenario

– possibly applies to quantum gravity

– reminiscent of asymptotic freedom in QCD

11

Chapter 2

Phase transitions and critical phenomena

2.1 Ising model

Before, we discuss the theory of critical phenomena from a general point of view, we reconsidersome of the basic models which show (second order) phase transitions. The Ising model is abasic model to describe a simple ferromagnet or the disorder-order transition in binary alloys.Here, we will use it to introduce some basic notions, definitions and ideas.

• The partition function of the Ising model reads:

Z =∑

si=±1, i=1,...,N

e−βH (2.1)

where β = 1/(kBT ) is the inverse temperature multiplied by the Boltzmann constant.Further, we define the energy H of a magnetic dipole/spin configuration s1, s2, ..., sN:

H = −J∑〈i,j〉

sisj − h∑i

si (2.2)

• model shall be defined on the sites of a lattice (chain, quadratic, cubic, hypercubic)

• 〈i, j〉 indicates a sum over nearest neighbors

• J is the nearest-neighbor coupling, h is an external magnetic field

• for T J (also if h = 0): expect that all dipoles point in same direction

• for T J (also if h = 0): interaction between dipoles negligible→ random arrangement

• magnetization per dipole:

m = 〈sj〉 =1

Z

∑si=±1, i=1,...,N

sje−βH (2.3)

• magnetization m can serve as an order parameter for the phase transition ath = 0, T = Tc

• ferromagnetic (FM) phase: m 6= 0

13

2. Phase transitions and critical phenomena

• paramagnetic (PM) phase: m = 0

• for h 6= 0 : m 6= ∀ T

• equation (2.1) can be solved exactly in d ∈ 1, 2

temperature

orde

r par

amet

er

liquid-gas (1st order) theory space

S

+

h

TTc

m ≠ 0 m = 0m > 0

m < 0FM PM

2.1.1 Remarks on solutions of the Ising model in d = 1 and d = 2

In one dimension (d = 1, dipole chain of length N →∞) it is found that m = 0 ∀ T > 0⇒ no phase transition.

Reasoning:

• assume all dipoles are oriented in one direction (i.e. system is in FM state)

• flip half of the dipoles ⇒ energy cost ∆E = 2J (one pair points in opposite directions)→ domain wall

• pair with dipoles in opposite directions can be chosen in N different ways

⇒ increase of entropy: ∆S = kb logN (with N being the number of microstates realizingone domain wall)

⇒ free energy F = E − TS is always lowered in large system (N 1) by flipping halfof the dipoles , even at infinitesimal T

⇒ ordered state is unstable ⇒ m 6= 0 only at T = 0.

Competition between E and S is different in d > 1:

• state with spontaneous magnetization at T > 0 is possible in d = 2 (R. Peierls)

• exact solution of the 2d-Ising model by L. Onsager:

– system has continuous phase transition with kBTc/J = 2.269

– for T < Tc and near Tc: m ∼ (Tc − T )1/8 (magnetization) and C ∼ − log |Tc − T |(specific heat)

• no exact solution in d = 3. We know, however:

– 3d-Ising model has critical point

– characteristics known with great accuracy → we’ll calculate them in this lecture!

dimensionality is crucial for critical behavior.

14

2.2. XY model and Heisenberg model

2.1.2 Symmetries of the Ising model

• at h = 0: global symmetry under transformation si → −si ∀ i

• this is called Z2 (or Ising) symmetry, it is a discrete symmetry

• Z2 symmetry is present in PM phase (m = 0)

• Z2 symmetry is broken in FM phase:

– there is nothing in the Hamiltonian, however, that explicitly breaks this symmetry

– direction of magnetization depends on history of the system

−→ phenomenon of spontaneous symmetry breaking

• problem: if partition function is calculated by summing over all configurations

⇒ magnetization will vanish due to Z2 symmetry

• to describe ordered phase (FM with 6= 0):

– restrict space of configurations over which sum in Z is performed, e.g., at h 6= 0

– take limit h→ 0 after thermodynamic limit has been performed

⇒ m 6= 0 survives the h→ 0 limit

• actual physical process is different from mathematical procedure (as it is induced bythe history of the system)

2.2 XY model and Heisenberg model

Allow dipoles to point arbitrarily:

• in the plane: XY model

• in space: Heisenberg model

These models then have continuous symmetries.

• partition function:

Z =

∫ N∏i=1

(δ(|~Si| − 1)dd~Si

)eβ(J

∑〈i,j〉

~Si·~Sj+~h∑i~Si) (2.4)

symmetry of the model will be crucial for critical behavior.

15

2. Phase transitions and critical phenomena

2.3 Universality and critical exponents

• some properties of a system near critical points are the same for completely differentphysical systems

• example: specific heat near liquid-gas critical point = specific heat in PM-FM transition

⇒ some macroscopic properties of systems near continuous phase transitions are ind-pendent from the microscopic interactions between particles

→ these properties only depend on dimensionality & symmetry

→ different physical systems exhibiting the same behavior near critical point

→ universality or universal critical behavior

• universal critical behavior can be expressed in terms of power laws, e.g., in liquid-gastransition with reduced temperature t:

1. specific heat near Tc:

CV = C±|t|−α , (2.5)

where C± refers to t > 0 or t < 0.

2. compressibility near Tc:

κT =1

ρ

∂ρ

∂p= κ±|t|−γ . (2.6)

- the analogue in a magnetic system is the susceptibility : χ ∝ |t|−γ .

3. difference between gas and liquid densities along coexistence curve near the criticalpoint:

ρL − ρG = ρc(−t)β . (2.7)

- the analogue in a magnetic system is the magnetization: m ∝ (−t)β.

4. critical isotherm (T = Tc) near p = pc:

p− pcpc

=

∣∣∣∣ρL − ρGρc

∣∣∣∣δ . (2.8)

- analogue in a magnetic system: m ∝ h1/δ at t = 0 and h→ 0.

• α, β, γ, δ:

– non-trivial and (not completely independent) real numbers

– universal, i.e. they are the same for a whole class of various phase transitions

– they are called the critical exponents

– C+/C− and κ+/κ− are also universal (amplitude ratios)

Experimental evidence for universality of the critical exponents was found in 1930s. Micro-scopic understanding and calculation of critical exponents was achieved in 1970s with thehelp of the renormalization group.

16

2.4. Scaling hypothesis for the free energy

Table 2.1: Theoretical estimates for critical exponents of the Ising, XY and Heisenberguniversality classes. The term Ising specifies the symmetry group of the corresponding orderparameter to be Z2. In the case of the XY universality class the symmetry is O(2) and forHeisenberg it is O(3). The indices in the upper line denote the dimensionality

Exponent Ising2 Ising3 XY3 Heisenberg3

α 0 (log) 0.110(1) -0.015 -0.10β 1/8 0.3265(3) 0.35 0.36γ 7/4 1.2372(5) 1.32 1.39δ 15 4.789(2) 4.78 5.11

2.4 Scaling hypothesis for the free energy

The power laws near a critical point can be derived from the assumption of scaling :

• consider a magnetic system near paramagnet-ferromagnet transition

• magnetization shall have preferred axis, i.e. can be assumed to be a scalar → uniaxialferromagnet

→ same symmetry as Ising model → Ising universality class

• in vicinity of critical point (T = Tc, h = 0): thermodynamic potential (free energy)becomes non-analytic/singular

• decompose Gibbs free energy per unit volume fG(T, h) into singular and regular part:

fG(T, h) = f(T, h) + freg(T, h) . (2.9)

• sufficiently close to critical point the singular part f(T, h) is assumed to follow thescaling hypothesis/assumption:

f(T, h) = |t|1/wΨ±

(h

|t|u/w), (2.10)

where Ψ+(z) for t > 0 and Ψ−(z) for t < 0 is a function of a single variable.

• Note:

– the scaling assumption is very restrictive

– once we assume it ⇒ power laws for CV ,m, χ + two equations for α, β, γ, δ.

17

2. Phase transitions and critical phenomena

2.4.1 Derivation of power laws from scaling hypothesis

- magnetization m at h = 0: the magnetization is given as the negative first derivativeof the free energy with respect to the magnetic field h evaluated at h = 0, i.e.

m =− ∂f

∂h

∣∣∣h=0

= −|t| 1−uw Ψ′±(0) (2.11)

⇒ for t > 0 : m = 0 ⇒ Ψ′+(0) = 0 (2.12)

for t < 0 : m ∝ (−t)β ⇒ β =1− uw

(2.13)

- uniform magnetic susceptibility near Tc:

χ =− ∂2f

∂h2

∣∣∣h=0

= −|t| 1−2uw Ψ′′±(0) (2.14)

and with χ ∝ |t|−γ ⇒ γ =2u− 1

w(2.15)

- specific heat:

CV =− T ∂2f

∂T 2

∣∣∣h=0

= −Ψ±(0)

Tc

1

w

(1

w− 1

)t

1w−2 (2.16)

⇒ α = 2− 1

w(2.17)

- rewriting of the scaling assumption: provides critical exponent δ

f(T, h) =h1/uΨ±

(h

|t|u/w)

where Ψ±(z) = z−1/uΨ±(z) (2.18)

⇒ m = −1

uh

1u−1Ψ±(∞) at t = 0, h 1 (2.19)

⇒ δ =u

1− u (2.20)

From eqs. (2.13), (2.15), (2.17), (2.20), we can derive the scaling laws:

α+ 2β + γ = 2 (2.21)

α+ β(δ + 1) = 2 (2.22)

18

2.4. Scaling hypothesis for the free energy

- remarks:

• the most precisely measured value of any critical exponent is α in the superfluid tran-sition in 4He in d = 3: αexp ≈ −0.0127± 0.0003

– space shuttle experiment:

https://journals.aps.org/prb/abstract/10.1103/PhysRevB.68.174518

• the scaling form of the free energy, eq. (2.10), guarantees that the critical exponents arethe same below and above the critical point

• the concept of scaling rationalizes the appearance of power laws near the critical pointand yields experimentally correct relations – the scaling laws

- questions:

• why does the scaling assumption hold?

• how to compute scaling functions?

2.4.2 First attempt to calculate critical exponents

We take the Ising model in Curie-Weiss mean-field approximation (probably like in your stat-mech course) and see what we get...

In mean-field approximation each dipole only feels the average local magnetic field (see prob-lem set 1)

⇒ replace J∑〈i,j〉 sisj by J

∑〈i,j〉〈si〉sj in Ising model with an average magnetization per

spin m = 〈si〉

⇒ m =

∑s1=±1 s1 e

β(zmJ+h)s1∑s1=±1 e

β(zmJ+h)s1= tanh

(zmJ + h

kBT

), (2.23)

where z = 2d counts the number of nearest neighbors on a d-dimensional hypercubic lattice

• at h = 0:

– m 6= 0 for T < Tc = zJ/kB

– m = 0 for T > Tc

• expansion near m = 0 and at h = 0 ⇒ m = TTc

√3(1− T/Tc) ⇒ β = 1/2

• expansion at T = Tc for small h ⇒ m =(

3hkBTc

)1/3⇒ δ = 3

• Further, mean-field approach predicts γ = 1 and α = 0 (see problem set 1)

• no dependence on d within mean-field theory⇒ this deviates from experimental findings!

• experiment (d = 3): β ≈ 1/3 and α small but finite (e.g. with Xe)

– this cannot be attributed to material properties

– need better theory!

19

2. Phase transitions and critical phenomena

2.5 Correlations and hyperscaling

Two-point correlation function or order parameter correlation function:

G(~r, t) = 〈(m(~r)−m)(m(0)−m)〉 = 〈m(~r)m(0)〉 −m2 (2.24)

Here, m(~r) is the local value of the magnetization, m is the average magnetization and〈...〉 denotes the ensemble or thermal average. We have exploited translational invariance.Sometimes convenient to study the (spatial) Fourier transform of G(~r), i.e. work in wave-vector space

G(~q, t) =

∫ddre−i~q·~rG(~r, t) (2.25)

• assume scaling form (for large distances r):

G(r, t) =Φ±

(rξ(t)

)rd−2+η

where ξ(t) ∝ |t|−ν (2.26)

and ξ is called the correlation length, d is the dimensionality.

• the correlation length denotes the typical length scale of the regions where the degreesof freedom are strongly coupled

• ξ diverges at the critical point

• ν and η are two new critical exponents

– ν is called the correlation length exponent

– η is called the anomalous dimension

• relate ν and η to the exponent γ by means of the non-local magnetic susceptibility:

χ(~r − ~r′) :=∂m(~r)

∂h(~r′)

∣∣∣h=0

(2.27)

– general theory of linear response: χ(~q) = G(~q)

– uniform susceptibility, eq. (3.3): χ = χ(~q = 0) =∫dd~r G(~r, t)

– use scaling assumption, eq. (2.26): χ = C ·ξ2−η with the constant C =∫dd~z Φ±(z)

zd−2+η

– For T > Tc: values of magnetization at two distant points is uncorrelated

⇒ Φ+(z) should decay exponentially for large z

– For T < Tc: here m 6= 0. Deviations from finite magnetization also uncorrelatedat large distances

⇒ Φ−(z) should decay exponentially for large z

– From exponential decay of Φ± we conclude that integral C is finite in both cases

→ comparison with the definition of γ

⇒ γ = ν(2− η) (Fisher’s scaling law)

20

2.5. Correlations and hyperscaling

• further assumption: the only relevant length scale near Tc is provided by the correlationlength ξ:

free energy

unit volume= f ∝ ξ(t)−d (2.28)

– this is an additional scaling assumption → hyperscaling

– Differentiating f twice w.r.t. the temperature, we obtain Josephson’s scaling law :

C ∝ |t|νd−2 ⇒ α = 2− νd– Josephson’s scaling law involves the dimensionality d

SUMMARY:

• we have 6 critical exponents (α, β, γ, δ, ν, η) to describe singular behavior of ther-modynamic functions and of the correlation function near critical points

• we introduced scaling hypothesis for f and G⇒ 4 equations for the critical exponents⇒ reduce number of independent exponents to 2⇒ only have to compute ν and η to know all 6 exponents (if hyperscaling holds)

• Example: assume Fourier transform of the correlation function to be G−1(q) = ~q2 + tfor t > 0 in d = 3

⇒ G(~r) =∫ d3~q

(2π)3ei~q·~r

q2+t= 1

4πre−r√t ⇒ ν = 1/2, η = 0

⇒ scaling laws are satisfied (check!) together with exponents from mean-field theory,except Josephson’s law (which is only satisfied in d = 4)

21

2. Phase transitions and critical phenomena

2.6 Ginzburg-Landau-Wilson theory

In this section, we study interacting bosons as relevant to the liquid-superfluid transitionin 4He (λ transition, Tc ≈ 2K). The construction can be performed analogously for othertransitions. The goal of this section is to express the (grand-canonical) partition function Zin terms of the so-called path integral or functional integral. For functional integrals manyconvenient tools have been developed which facilitate systematic approximate evaluations ofthe partition function. So, the general strategy is the following:

• We want to evaluate the partition function of an interacting many-particle system (forexample 4He) to study thermodynamics, phase transitions, etc.

→ write down (grand-canonical) partition function Z = Tre−β(H−µN)

→ express Z in terms of a functional integral, i.e. Z =∫DΦe−S[Φ], with action S[Φ]

→ learn about various tools to systematically evaluate Z in its functional representation

• the Hamiltonian for interacting bosons reads in second quantization

H = H(a†, a) =∑α,β

εαβ a†αaβ +

∑α,β,γ,δ

Vαβγδ a†αa†β aδaγ (2.29)

• the particle number operator is

N =∑α

a†αaα (2.30)

2.6.1 Functional integral representation of the partition function

Coherent states (more details available in the book by Altland & Simons)

• standard orthogonal basis for quantum mechanical many-body states:

|nα1 , nα2 , ..., nαN 〉 =N∏i=1

(a†αi

)nαi√nαi !

|0〉 (2.31)

– αi label the states forming basis in single-particle Hilbert space of dimension N

– |0〉 is the vacuum

– a†αi , aαi are bosonic creation/annihilation operators → single-particle state |αi〉– standard commutation relations [aαi , a

†αj ] = δαiαj

– any many-particle state is a linear combination:

|Φ〉 =

∞∑nα1=0

· · ·∞∑

nαN=0

Φnα1 ,...,nαN|nα1 , nα2 , ..., nαN 〉 (2.32)

22

2.6. Ginzburg-Landau-Wilson theory

• Definition: a coherent state is a common eigenstate of the annihilation operators:

aαi |Φ〉 = Φαi |Φ〉 , (2.33)

with i = 1, 2, ..., N and complex eigenvalues Φαi

– for such a state we have (tutorial)

Φnα1 ,...,nαN=

N∏i=1

(Φαi)nαi√

nαi !⇒ |Φ〉 =

∞∑nα1=0

· · ·∞∑

nαN=0

∏i=1

(Φαi a

†αi

)nαinαi !

|0〉

= e∑αi

Φαi a†αi |0〉 (2.34)

– similarly: 〈Φ|a†αi = 〈Φ|Φ∗αi → bra version: 〈Φ| = 〈0|e∑αi

Φ∗αi aαi

⇒ a†αi |Φ〉 = ∂∂Φαi|Φ〉 which can be shown by Taylor expansion of eq. (2.34)

– overlap of two coherent states: 〈Φ|Φ′〉 = e∑αi

Φ∗αiΦ′αi (not orthogonal!)

– coherent state form overcomplete set with resolution of unity (without proof):∫ ∏α

dΦ∗αdΦα

2πie−∑α Φ∗αΦα |Φ〉〈Φ| = 1 (2.35)

where we omitted the index i for simplicity

– eq. (2.35) can be shown by proof for N = 1 and then directly generalized to N > 1

• we will use the coherent states to re-express the Hamiltonian operator (which in turnis expressed in terms of creation and annihilation operators) in terms of eigenstates

23

2. Phase transitions and critical phenomena

Path-integral representation of the grand-canonical partition function

• grand-canonical partition function Z = Tr e−β(H−µN) =∑

n〈n|e−β(H−µN)|n〉where the sum extends over a complete set of Fock states |n〉

• together with eq. (2.35), we get:

Z =

∫ ∏α

dΦ∗αdΦα

2πie−∑α Φ∗αΦα〈Φ|e−β(H−µN)|Φ〉 (2.36)

• divide imaginary “time interval” β into M pieces and insert unity operator (M − 1)times

⇒ Z =

∫ M−1∏k=0

∏α

dΦ∗α,kdΦα,k

2πie−∑M−1k=0

∑α Φ∗α,kΦα,k

M∏k=1

〈Φk−1|e−∆β(H−µN)|Φk〉 (2.37)

where we have the boundary condition Φ0 = ΦM = Φ and ∆β = β/M

• for ∆β 1:

〈Φk−1|e−∆β(H−µN)|Φk〉 = 〈Φk−1|Φk〉e−∆β〈Φk−1|(H−µN)|Φk〉/〈Φk−1|Φk〉 +O(∆β2) (2.38)

• in Hamiltonian and number operator all creation operators are to the left of annihilationoperators → normal ordering

• for abitrary normally ordered function A, the definition of coherent states implies:

〈Φ|A(a†α, aα)|Φ′〉 = A(φ∗α, φ′α)e

∑α Φ∗αΦ′α (2.39)

⇒ for M →∞ : (2.40)

Z = limM→∞

∫ M−1∏k=0

∏α

dΦ∗α,kdΦα,k

2πie−(

∑M−1k=0

∑α Φ∗α,k(Φα,k−Φα,k+1))

× e−∑M−1k=0 ∆β

(H(Φ∗α,k,Φα,k+1)−µ

∑α Φ∗α,kΦα,k+1

)(2.41)

where k labels moments in imaginary time.

24

2.6. Ginzburg-Landau-Wilson theory

Continuum limit

• the continuum limit of eq. (2.41) gives the functional integral :

Z =

∫Φα(0)=Φα(β)

DΦ∗α(τ)DΦα(τ)e−S[Φ∗α(τ),Φα(τ)] (2.42)

with the action

S =

∫ β

0dτ

[∑α

Φ∗α(τ)(−∂τ − µ)Φα(τ) +H[Φ∗α(τ),Φα(τ)]

](2.43)

• where α can be: momentum, position ~r, lattice site,...

• interacting bosons: choose continuous position coordinate α = ~r

• partition function of a system of interaction bosons =

“sum over all possible complex functions Φ(~r, τ) of space and imaginary time which areperiodic in imaginary time.”

• we can decompose the periodic function (periodicity due to boundary condition!):

Φ(~r, τ) =1

β

∫d~k

(2π)d

∑ωn

Φ(~k, ωn)ei~k·~r+iωnτ (2.44)

with the bosonic Matsubara frequencies ωn = 2πn/β = 2πnkBT , n ∈ Z

• remarks:

– in the case of fermions → antiperiodic, anticommuting Grassmann numbers,ωn,F = (2n+ 1)π/β

– example: non-interacting bosonic system

Z = Z0 =∏~k,ωn

∫dΦ∗(~k, ωn)dΦ(~k, ωn)

2πie−β−1

∑~k,ωn

(−iωn+ ~2~k2

2m−µ)|Φ(~k,ωn)|2

(2.45)

=∏~k,ωn

β

−iωn + ~2~k2

2m − µ(2.46)

→ recover Bose-Einstein condensation from F0 = −kBT logZ0 and N = −∂F0∂µ ...

– at T = 0, the Matsubara frequencies form a continuum

25

2. Phase transitions and critical phenomena

Path-integral representation at finite temperature

• at finite T : Fourier mode ω0 = 0 separated by finite amount ∝ T from the action forthe other modes (ωn 6= 0)

S = β−1

∫d~k

(2π)d

∑n

(−iωn − µ) |Φ(~k, ωn)|2 + ... (2.47)

• ωn modes (|n| ≥ 1) only yield analytic contribution to free energy

⇒ these modes are non-critical and do not affect universal properties

• for finite temperature transition, only retain the ω0-modes in the partition function

⇒ fields independent of imaginary time!

⇒ partition function

Z =

∫DΦ∗(~r)DΦ(~r)e−S (2.48)

– example: non-relativistic bosons of mass m

→ energy dispersion ε~k = ~2~k2

2m , i.e. non-interacting part of Hamiltonian, eq. (2.43):

H0[Φ∗(~r),Φ(~r)] = β

∫d~r

[Φ∗(~r)

(−~2∇2

2m

)Φ(~r)

](2.49)

→ contact interaction V (~r1 − ~r2) = λ δ(~r1 − ~r2), cf. eq. (2.29), gives the action:

S[Φ] =1

kBT

∫d~r

[~2

2m|∇Φ(~r)|2 − µ|Φ(~r)|2 + λ|Φ(~r)|4

](2.50)

– together with Z this defines Ginzburg-Landau-Wilson theory

– for superfluid transition (in 4He), Φ(~r) is complex: two real components (N = 2)

– contact interaction is sufficient to understand critical behavior

– more general: field Φ with N real components and same action as in eq. (2.50)

– for N = 1: symmetry is Z2 (Φ(~r)→ −Φ(~r))

⇒ same symmetry as Ising model

⇒ eq. (2.50) with N = 1 describes uniaxial FM transition

– for N = 3: same symmetry as Heisenberg model

Ginzburg-Landau-Wilson (GLW) partition function with general N serves as meta-modelto which specific models like Ising, XY, or Heisenberg model reduce in the critical region.

→ only universal critical behavior may be expected to be given correctly by GLW theory(not thermodynamics outside the critical region)

26

2.6. Ginzburg-Landau-Wilson theory

2.6.2 Saddle-point approximation (SPA)

SPA: evaluation of GLW partition function, eq. (4.1), with value of integrand at its maximum:

⇒ F = kBT · S[Φ0] (2.51)

• here Φ0 is the configuration that minimizes the action: δSδΦ

∣∣Φ0

= 0

• Φ0 will be independet of the coordinate and be (1) Φ0 = 0 , or (2) |Φ0|2 = µ/(2λ)

• take µ as a tuning parameter:

(a) for µ < 0: non-trivial solution (2) is impossible ⇒ Φ0 = 0

(b) for µ > 0: S[Φ0] = −µ2

4λVkBT

< S[0] = 0 ⇒ Φ0 is finite

→ Φ0 = 0 becomes a local maximum

→ action has mexican hat shape with a continuum of minima

• a finite value of Φ0 signals the ordered phase ⇒ take Φ0 as order parameter

• in general the order parameter is defined as 〈Φ(~r)〉 which in the SPA coincides with Φ0

In terms of temperature T :

• transition in SPA takes place at Tc where µ(T ) changes sign

⇒ −µ is coefficient of the quadratic term of S, which changes sign near the transition

µ ∼ (Tc − T ) +O((Tc − T )2

)(2.52)

Spontaneous symmetry breaking:

• only absolute value of Φ0 is fixed by |Φ0|2 = µ/(2λ), the phase is left arbitrary

• any choice of phase breaks global U(1) invariance (Φ(~r)→ eiφΦ(~r)) of the action (2.50)

⇒ U(1) symmetry is spontaneously broken in the ordered/superfluid phase!

• remarks:

– U(1) symmetry is analogue of rotational symmetry that is broken by the directionof magnetization, e.g., in FM transition

– consequence of the breaking of continuous symmetries→ Goldstone bosons appear

– spontaneous breaking of Z2: no Goldstone bosons

27

2. Phase transitions and critical phenomena

2.6.3 Critical exponents in saddle-point approximation∗

• since µ ∼ Tc − T in GLW theory → differentiating S[Φ0] twice w.r.t T :

→ specific heat only has a finite discontinuity at the transition

⇒ critical exponent α = 0

• define superfluid susceptibility:

χ(~r − ~r′) :=δ2 logZ[J ]

δJ∗(~r)δJ(~r′)

∣∣∣J=0

(2.53)

where

S → S +

∫d~r [Φ(~r)J(~r) + Φ∗(~r)J∗(~r)] (2.54)

defines Z[J ]

⇒ χ(~r − ~r′) = 〈Φ∗(~r)Φ(~r′)〉 − |〈Φ(~r)〉|2 (2.55)

– note the similarity to eq. (2.24)

– also note that the source field J(~r) directly couples to Φ in the action in thesame way as the real magnetic field would enter the partition function of magneticsystems, see eq. (2.27).

• to compute χ in saddle-point approximation consider small fluctuations of Φ(~r) aroundthe saddle point (for µ < 0, i.e. expand the action around trivial saddle point)

→ then neglect λ ⇒ action is quadratic in fluctuating fields

⇒ Z[J ] can be computed by completing the square:

logZ[J ] =

∫d~rd~r′J∗(~r)χ0(~r − ~r′)J(~r′) + const. (2.56)

where

χ0(~r) = kBT

∫d~k

(2π)dei~k·~r

~2~k2

2m − µ(2.57)

• near Tc where µ(Tc) = 0 rewrite χ0 as:

χ0(~r) =2mkBTc

~2· F (r/ξ)

rd−2(2.58)

with the scaling function F (z) = zd−2∫ d~q

(2π)dei~q·~r

q2+1,

and the correlation length ξ = ~/√

2m|µ|• this result for χ0 can be cast into the form in eq. (2.26)

– since µ ∼ Tc − T near Tc:correlation length exponent ν = 1/2 and anomalous dimension η = 0

⇒ saddle-point approximation leads to the mean-field values of the critical expo-nents and is independent of dimension d and number of field components N → notsatisfactory!

28

2.7. Kosterlitz-Thouless phase transition

2.7 Kosterlitz-Thouless phase transition

• no SSB of continuous symmetry in in 2 dimensions ⇒ no obvious order parameter

• can there still be a phase transition?

• Kosterlitz & Thouless (1973, Nobel prize 2016):

Yes, phase transition by unbinding of topological defects (vortices)

2.7.1 XY model in two dimensions

• classical statistical model on 2-dimensional square lattice with action:

S[φi] ≡ βH[φi] = −J∑〈i,j〉

cos (φi − φj) (2.59)

• φi ∈ [0, 2π] is an angle variable attached to each point

• the action has a global (continuous) U(1) symmetry: φi → φi + α

• physical picture/interpretation:

– spins with “easy plane” (free rotation in x− y-plane, frozen in z-direction)

– lattice regularization of complex boson theory at finite temperature T and at fre-quencies below T :

H =

∫d2x

[κ

2|~∇Ψ|2 +

t

2|Ψ|2 + u|Ψ|4

](2.60)

– where Ψ = Ψ1 + iΨ2 = reiφ

– more precisely:

∗ in-plane amplitude – phase only comes with (∇φ)2 (gapless mode).

∗ the above model, eq. (2.59) is a regularized (discrete) version of this part ofthe complex boson theory and reproduces it at long wavelength (see below).

∗ the amplitude fluctuations are gapped and do not contribute to the long-wavelength physics.

• agenda:

– we first argue that the low- and high-T phases of the XY omdel are qualitativelydistinct (algebraic vs. exponential decay of correlation functions)

– then we will explain this following Kosterlitz and Thouless.

29

2. Phase transitions and critical phenomena

2.7.2 Limiting cases at low and high temperatures

Low-temperature phase

• At T = 0, SSB can take place and we have an ordered phase profile with φi = φ0.

• At low T , we therefore expect an almost ordered phase profile φi ≈ const. = φ0, andonly small local fluctuations of the phase.

• thus we decompose φi = φ0 + δφi, where δφi are slowly varying small fluctuations

⇒ βH[φi] = −JT

∑〈i,j〉

cos (δφi − δφj) ≈ const.+J

2T

∑〈i,j〉

(δφi − δφj)2 (2.61)

continuum limit︷︸︸︷−→ const.+a2J

2T

∫d2r(~∇φ)2 =

J

2T

∫d2q~q2(φq)

2 (2.62)

• where J = a2J , a is the lattice constant and we have dropped the constant term.

• this reproduces the continuum model with the long wavelength limit.

• the asymptotic (long-wavelength) behavior of the correlation function is:

〈ei(φ(~r)−φ(~r′))〉 ≈ e− 12〈(φ(~r)−φ(~r′))2〉 = e−

T2πJ

log|~r−~r′|a =

( |~r − ~r′|a

)− T2πJ

(2.63)

• this is an algebraic decay, quasi long-range order replacing SSB due to smooth long-wavelength fluctuations.

High-temperature phae

High-temperature phase... .

30

Chapter 3

Wilson’s renormalization group

3.1 General strategy

Observation: In a finite system the correlation length is bound by the system size

⇒ singular thermodynamic behavior (critical behavior) comes from thermodynamic limit

→ i.e. from modes with arbitrarily low energies in “∞”-large system:

infrared singularity

Consider the action in Eq. (2.50):

S[Φ] =1

kBT

∫d~r

[~2

2m|∇Φ(~r)|2 − µ|Φ(~r)|2 + λ|Φ(~r)|4

](3.1)

• m provides energy scale ⇒ rescale into chemical potential and interaction:

2mµ/~2 −→ µ, 2mλ/~2 −→ λ (3.2)

• near critical point (T ≈ Tc): rescale action 2mkBTcS/~2 → S =∫~r |∇Φ|2−µ|Φ|2 +λ|Φ|4

⇒ superfluid susceptibility is function of wavevector, chemical potential and interaction only:

χ(~k) = F (k, µ, λ) (3.3)

This expression implicitly depends on an

ultraviolet cutoff Λ ∼ 1

a, (3.4)

restricting all wavevectors to |~k| < Λ.

• a defines the shortest length scale in the system (lattice spacing, atom size,...)

• we will find that its exact value does not affect the critical behavior

31

3. Wilson’s renormalization group

Strategy: low-energy modes (slow modes) induce singular behavior.

⇒ integrate only over shell of high-energy modes (large k, fast modes):

Λ

b< k < Λ , where b ≈ 1 (3.5)

→ recast partition function for remaining slow modes (k < Λb ) into form before integration

⇒ values of µ and λ have to be changed:

µ −→ µ(b), λ −→ λ(b) (3.6)

⇒ susceptibility for low k:

χ(~k) = bxF (bk, µ(b), λ(b)) (3.7)

• argument “bk” due to change of shortest length scale Λ→ Λ/b

• factor bx: in units of length scale, χ has dimension x, i.e. χ ∼ Λ−x

• Note: this strategy closely ressembles the real-space RG discussed on your first problemset http://www.thp.uni-koeln.de/~scherer/set01.pdf only in momentum space!

Question: What happens for b→∞ (more and more modes integrated out)?

• suppose limb→∞ λ(b) = λ∗

⇒ χ(~k) = bxF (bk, µ(b), λ∗) , (3.8)

for large b.

• if λ∗ is small ⇒ use perturbation theory in λ∗

• i.e. determine effective renormalized coupling for low-energy modes first

Critical behavior:

• assume λ∗ exists and is finite

• further, assume for b 1 and small µ < 0: µ(b) ≈ µ by

• now, choose b such that µ by = µ0 with constant µ0

⇒ χ(~k) =

∣∣∣∣µ0

µ

∣∣∣∣x/y F(k

∣∣∣∣µ0

µ

∣∣∣∣1/y , µ0, λ∗

)(3.9)

• then, the Fourier transform χ(~r) exhibits the desired scaling form

• the correlation length exponent and the susceptibility exponent are accordingly:

ν = 1/y, γ = x/y . (3.10)

• we still need the functions λ(b) and µ(b) to get x and y (next section!).

32

3.1. General strategy

Possible complications: Success of strategy is not guaranteed:

• as b increases: new terms could appear in action for slow modes

λ6(b)|Φ|6, λ2,2(b)|∇Φ|2|Φ|2 (3.11)

• generated terms must still be invariant under symmetries of original action

• as b→∞: a coupling is called

(1) irrelevant if limit is zero (i.e. it was correct to neglect it)

(2) relevant if limit is finite (i.e. coupling needs to be included)

• theory with finite number of relevant couplings is called renormalizable (next chapter!)

33

3. Wilson’s renormalization group

3.2 Momentum-shell transformation

Implement above strategy for the partition function of eq. (2.50)

→ therefore divide the field Φ into slow and fast modes:

Φ(~r) = Φ<(~r) + Φ>(~r) (3.12)

where

• Φ< are the modes with k < Λ/b

• Φ< are the modes with Λ/b < k < Λ

⇒ partition function:

Z =

∫ ∏k<Λ

dΦ∗(~k)dΦ(~k)

2πie−(S0<+S0>+Sint) (3.13)

=

∫ ∏k<Λ

b

dΦ∗(~k)dΦ(~k)

2πi

∫ ∏Λb<k<Λ

dΦ∗(~k)dΦ(~k)

2πie−(S0<+S0>+Sint) , (3.14)

with S0< =

∫ Λ/b

0

d~k

(2π)d(~k2 − µ)|Φ(~k)|2 , (3.15)

and S0> =

∫ Λ

Λ/b

d~k

(2π)d(~k2 − µ)|Φ(~k)|2 , (3.16)

and Sint = λ

∫ Λ

0

d~k1...d~k4

(2π)3dδ(~k1 + ~k2 − ~k3 − ~k4)Φ∗(~k4)Φ∗(~k3)Φ(~k2)Φ(~k1) . (3.17)

• the interaction term in Fourier space can be represented pictorially

• each field Φ is represented by a line with an arrow pointing into the vertex

• a field Φ∗ is represented by a line with an arrow pointing out of the vertex

• we also show the momentum labels of the fields:

theory space

S

+

h

TTc

m ≠ 0 m = 0m > 0

m < 0FM PM

temperature

orde

r par

amet

er

liquid-gas (1st order)

temperature

pres

sure

solid phaseliquid phase

vapour

Tc

pc critical point

fast modes

slow modes

(~k1) (~k2)

(~k3) (~k4)

34

3.2. Momentum-shell transformation

3.2.1 Perturbation theory

Zeroth order in λ:

Z = Z0<Z0> ⇒ Z factorizes

→ in Z0> all the modes have large k

⇒ Z0> only contributes to regular, analytic part of free energy

• now, bring Z0< into exact old form by first rescaling b~k → ~k:

S0< =1

bd

∫ Λ

0

d~k

(2π)d

(~k2

b2− µ

)|Φ(~k)|2 (3.18)

• define: µ(b) = µ b2 and Φ(~k)

b(d+2)/2 → Φ(~k)

⇒ S0< now precisely takes the old form in terms of rescaled Fourier components

• variables Φ(~k) are to be integrated over

⇒ critical behavior is not affected by their rescaling (also in the integration measure)

→ only adds regular part to free energy.

⇒ we find µ(b) = µ b2 +O(λ)

⇒ ν = 1/y = 1/2 → mean-field result!

First order in λ:

avoiding slow modes → can be done without encountering the singularity

Z =Z0>

∫ ∏k<Λ/b

dΦ∗(~k)dΦ(~k)

2πie−S0< [1

−λ∫ Λ

0

d~k1...d~k4

(2π)3dδ(~k1 + ~k2 − ~k3 − ~k4)〈Φ∗(~k4)Φ∗(~k3)Φ(~k2)Φ(~k1)〉0> +O(λ2)

](3.19)

where we have introduced the fast-mode average

〈A〉0> =1

Z0>

∫ ∏Λ/b<k<Λ

dΦ∗(~k)dΦ(~k)

2πie−S0>A . (3.20)

As S0 is quadratic in Φ and diagonal in ~k we find from Gaußian integration (problem set 3):

〈Φ∗(~k1)Φ(~k2)〉0 =(2π)d

k21 − µ

δ(~k1 − ~k2) , (3.21)

〈Φ∗(~k4)Φ∗(~k3)Φ(~k2)Φ(~k1)〉0 = 〈Φ∗(~k4)Φ(~k2)〉0〈Φ∗(~k3)Φ(~k2)〉0 + ( ~k1 ↔ ~k2) , (3.22)

where the second equation is a variant of Wick’s theorem:

- average factorizes into product of all possible averages of pairs of Φ∗ and Φ.

35

3. Wilson’s renormalization group

So, next we want to calculate the following part of eq. (3.19):

−λ∫ Λ

0

d~k1...d~k4

(2π)3dδ1234

1

Z0>

∫ ∏Λ/b<k<Λ

dΦ∗(~k)dΦ(~k)

2πiΦ∗(~k4)Φ∗(~k3)Φ(~k2)Φ(~k1)e−S0> (3.23)

where we have introduced the short-hand notation δ1234 = δ(~k1 + ~k2 − ~k3 − ~k4).

3.2.2 Combinations in eq. (3.23) that contribute in eq. (3.19)

...by giving a finite average:

1. ~k4 = ~k2 and Λ/b < k2 < Λ and ~k3 = ~k1 and k1 < Λ/b- then one part of eq. (3.23) reads

− λ∫ Λ

0

d~k1...d~k4

(2π)3dδ1234Φ∗(~k3)Φ(~k1)

1

Z0>

∫ ∏Λ/b<k<Λ

dΦ∗(~k)dΦ(~k)

2πiΦ∗(~k4)Φ(~k2)e−S0>

= −λ

∫ Λ

0

d~k1...d~k4

(2π)3dδ1234Φ∗(~k3)Φ(~k1)

((2π)d

~k22 − µ

δ(~k2 − ~k4)

)(3.24)

momenta ~ki as chosen above︷︸︸︷= −λ

∫ Λ/b

0

d~k1d~k3

(2π)dδ(~k1 − ~k3)Φ∗(~k3)Φ(~k1)

∫ Λ

Λ/b

d~q

(2π)d1

~q2 − µ(3.25)

= −λ∫ Λ/b

0

d~k

(2π)dΦ∗(~k)Φ(~k)

∫ Λ

Λ/b

d~q

(2π)d1

~q2 − µ (3.26)

or the same with ~k4 = ~k1 and Λ/b < k1 < Λ and ~k3 = ~k2 and k2 < Λ/b

2. same with ~k4 and ~k3 exchanged

→ combinations 1. and 2. give 4 different ways

theory space

S

+

h

TTc

m ≠ 0 m = 0m > 0

m < 0FM PM

temperature

orde

r par

amet

er

liquid-gas (1st order)

temperature

pres

sure

solid phaseliquid phase

vapour

Tc

pc critical point

fast modes

slow modes

(~k1) (~k2)

(~k3) (~k4)

3. when all k < Λ/b

4. when all Λ/b < k < Λ → only contributes to prefactor in eq. (3.19), i.e. to analyticpart of free energy!

→ omit factor from fast modes as it only contributes to analytic part.

36

3.2. Momentum-shell transformation

⇒ Z ∝ Z< with action in Z<:

S< =

∫ Λ/b

0

d~k

(2π)d

(~k2 − µ+ 4λ

∫ Λ

Λ/b

d~q

(2π)d1

q2 − µ

)|Φ(~k)|2

+ λ

∫ Λ/b

0

d~k1...d~k4

(2π)3dδ(~k1 + ~k2 − ~k3 − ~k4)Φ∗(~k4)Φ∗(~k3)Φ(~k2)Φ(~k1) +O(λ2) (3.27)

→ rescale to bring cutoff back to Λ and Φ(~k)/b(d+2)/2 → Φ(~k)

⇒ µ(b) = b2

(µ− 4λ

∫ Λ

Λ/b

d~q

(2π)d1

q2 − µ +O(λ2)

)(3.28)

λ(b) = b4−dλ+O(λ2) (3.29)

→ eq. (3.28): interaction shifts critical value of µ to

µc(b) = 4λ

∫ Λ

Λ/b

d~q

(2π)d1

q2+O(λ2) (3.30)

• here, we use a series expansion for small c: 1q2−c = 1

q2 + cq4 +O(c2)

• location of the critical point is not a universal quantity (depends on Λ)!

• consider deviation of chemical potential from its critical value at current value of b:

µ = µ− µc(b) ⇒ µ(b) = b2

[µ− 4λ

∫ Λ

Λ/b

d~q

(2π)d

(1

q2 − µ −1

q2

)+O(λ2)

](3.31)

• therefore, in the critical region where |µ| Λ2|:

µ(b) = b2µ

[1− 4λ

Λd−4Sd(2π)d

∫ 1

1/b

xd−1

x4dx+O(λ2)

](3.32)

• the integral in eq. (3.32) can be evaluated approximately close to four dimensions, i.e.we assume that |d− 4| 1:∫ 1

1/b

xd−1

x4dx =

1

d− 4

(1− 1

bd−4

)= log b+O(d− 4) (3.33)

• dimensionless interaction: λ = λΛd−4Sd/(2π)d with Sd = 2πd/2/Γ(d/2)

µ(b) = µ b2(

1− 4λ log b+O(λ(d− 4), λ2, λµ))≈ µ b2−4λ+O(λ2) (3.34)

37

3. Wilson’s renormalization group

• now, assume λ to be small. Consider b→∞:

1. for µ > 0, µ(b)→∞: superfluid phase (spontaneously broken symmetry)

2. for µ < 0, µ(b)→ −∞: normal phase (symmetric)

3. µ = 0, i.e. µ = µc(b), µ(b) ≡ 0: fixed point of the transformation

– fixed point is unstable, µ grows when perturbed from fixed-point value

– µ is a relevant coupling (= tuning parameter of the transition)

• from definition of y (µ by = µ0, µ0 fixed) and eq. (3.34)

⇒ y = 2(1− 2λ∗ +O(λ2∗)) (3.35)

where

λ∗ = limb→∞

λ ⇒ ν =1

2+ λ∗ +O(λ2

∗) (3.36)

• Note:

– in d > 4 and λ 1: λ∗ = 0, see eq. (3.29) ⇒ λ is irrelevant coupling

→ mean-field exponents are exact

→ λ∗ = 0 represents stable fixed point for d > 4

– in d < 4: weak interaction grows with b ⇒ next-order term must be included

→ µ and λ are relevant couplings at the non-interacting fixed point at λ∗ = 0.

38

3.3. Determination of fixed-point coupling below four dimensions

3.3 Determination of fixed-point coupling below four dimensions

We want to calculate λ∗ below four dimensions. Therefore, we calculate the O(λ2)-term ineq. (3.29). Explicitly, the O(λ2) term from eq. (3.19) reads

λ2

2

∫ Λ

0

d~p1...d~p4d~q1...d~q4

(2π)6δ(~p1 + ~p2 − ~p3 − ~p4)δ(~q1 + ~q2 − ~q3 − ~q4)

× 〈Φ∗(~p4)Φ∗(~p3)Φ(~p2)Φ(~p1)Φ∗(~q4)Φ∗(~q3)Φ(~q2)Φ(~q1)〉0> (3.37)

• To compute the average 〈...〉0>: pair up Φ∗ and Φ with equal wavevectors Λ/b < k < Λin all possible ways, similar to procedure in Sec. 3.2.2

• If two unpaired legs remain in the low-energy shell k < Λ/b ⇒ contribution to µ(b)

• If four unpaired legs remain in the low-energy shell k < Λ/b ⇒ contribution to λ(b)(to calculate the correction to the scaling of λ(b) and λ∗ we need these contributions!)

⇒ diagrams contributing to quadratic order in λ (joining legs of two vertices):

theory space

S

+

h

TTc

m ≠ 0 m = 0m > 0

m < 0FM PM

temperature

orde

r par

amet

er

liquid-gas (1st order)

temperature

pres

sure

solid phaseliquid phase

vapour

Tc

pc critical point

(~k1) (~k2)

(~k3) (~k4)

fast modes

slow modes

(a) (b)

• diagram (a) can be constructed in 16 different ways.

• diagram (b) can be constructed in 4 different ways.

• only connected diagrams need to be considered

• disconnected diagrams cancel out upon re-exponentiation into S<, cf. linked-clustertheorem (later in the lecture)

⇒ upon re-exponentiation the O(λ2) term becomes:

− λ2

2

∫ Λ/b

0

d~k1...d~k4

(2π)3dδ(~k1 + ~k2 − ~k3 − ~k4)Φ∗(~k4)Φ∗(~k3)Φ(~k2)Φ(~k1)

×[

16

∫ Λ

Λ/b

d~q

(2π)d1

(q2 − µ)((~q + ~k2 − ~k4)2 − µ)+ 4

∫ Λ

Λ/b

d~q

(2π)d1

(q2 − µ)((~k1 + ~k2 − ~q)2 − µ)

](3.38)

• note: we have replaced µ with µ as the difference is O(λ) ⇒ affects only next order!

• momentum-shell trafo induced wavevector dependence in the interaction ∼ Φ∗Φ∗ΦΦ.

39

3. Wilson’s renormalization group

• to define a single interaction coupling constant λ(b): choose value at ~ki = 0

⇒ λ(b) = b4−dλ

(1− 10λ

∫ Λ

Λ/b

d~q

(2π)2

1

(q2 − µ)2+O(λ2)

)(3.39)

→ at critical point µ = 0 (perform integral, introduce λ):

λ(b) = b4−dλ(

1− 10λ log b+O(λ2))

= b4−d−10λ+O(λ2)λ (3.40)

→ together with eq. (3.34):

µ(b) = b2µ(

1− 4λ log b+O(λ(d− 4), λ2, λµ))

= µb2−4λ+O(λ2) (3.41)

this completes the calculation to lowest non-trivial order.

40

3.4. Epsilon expansion and Wilson-Fisher fixed point

3.4 Epsilon expansion and Wilson-Fisher fixed point

Cast momentum-shell transformations µ(b) and λ(b), eqs. (3.34) and (3.40), into differentialform for |4− d| 1, 4− d = ε:

dµ(b)

d log b= βµ = µ(b)

(2− 4λ(b) +O(ελ, λ2, λµ)

)(3.42)

dλ(b)

d log b= βλ = ελ(b)− 10λ(b)2 +O(ελ2, λ3, λ2µ) (3.43)

• this defines the renormalization group β functions!

• momentum-shell trafo (=RG trafo) ⇒ flow in the space of coupling functions

Understand the flow in the λ− µ plane

- look for fixed points (βµ = βλ = 0) and analyze their stability

- we find two different fixed points:

1. the (trivial) Gaussian fixed point (GFP): µ = 0, λ = 0

2. the (non-trivial) Wilson-Fisher fixed point (WFFP): µ = 0, λ =ε

10+O(ε2)

- for ε < 0 the WFFP is unphysical as λ < 0 and further it is unstable in both directions:→ change of b (integrates out more modes)⇒ µ(b) and λ(b) are driven away from WFFP.

- for ε > 0: the WFFP is physical as λ > 0.

Analysis of stability

- For analysis of stability we look at β functions in λ− µ plane (see Fig. 3.1)

• the WFFP is stable in the λ direction and unstable in the µ direction: critical point

• the GFP is unstable in both directions

⇒ The critical behavior for ε > 0 is governed by the nontrivial Wilson-Fisher fixed point

• parameter ε controls the size of the corrections to MFT critical behavior

• consider ε as a continuous parameter

• in d = 4: slow flow to zero ⇒ logarithmic corrections to scaling

41

3. Wilson’s renormalization group

0.00 0.02 0.04 0.06 0.08 0.10 0.12-0.04

-0.02

0.00

0.02

0.04

λ

μ

Figure 3.1: The arrows in the plot are ~β(λ, µ) = (βλ, βµ)|λ,µ. The black square indicates the

GFP and the red square the WFFP. In the λ direction, the arrows point into the WFFPindicating its stabilty.

Flow near fixed points from stability matrix

Mij :=∂βxi∂xj

∣∣∣xi=x∗i

(3.44)

where xi = µ, λ.

• eigenvectors of M with positive eigenvalues: unstable directions at a given fixed point

→ relevant direction

→ tuning required to flow to CP (corresponds to tuning to critical temperature)

• eigenvectors of M with negative eigenvalues: stable directions at a given fixed point

→ irrelevant direction

• critical point : only one relevant direction with eigenvalue y, cf. eq. (3.36)

⇒ correlation length exponent ν = 1/y

– here Mij is already diagonal at WFFP with y = 2− 2ε/5

⇒ ν =1

2+

ε

10+O(ε2)

– for ε = 1(d = 3): ν ≈ 0.6

– the experimental value is νexp ≈ 0.670, i.e. we’re already better than MFT

• a single eigenvalue determines the flow for positive and negative chemical potential

→ ν is equal on both sides of the transition!

42

3.4. Epsilon expansion and Wilson-Fisher fixed point

Determination of anomalous dimension

• from rescaling of momenta/fields by momentum-shell trafo:

k2 → Zkk2, µ→ Zµµ, λ→ Zλλ (3.45)

• define η = dZkd ln b

∣∣∣λ=λ∗

⇒ η = O(ε2)

• ... calculation ... ⇒ Fisher’s scaling law!

Role of newly generated couplings

• for example a sixth order term ∼ g|Φ|6 is generated due to the (quartic) coupling λ:

theory space

S

+

h

TTc

m ≠ 0 m = 0m > 0

m < 0FM PM

temperature

orde

r par

amet

er

liquid-gas (1st order)

temperature

pres

sure

solid phaseliquid phase

vapour

Tc

pc critical point

(~k1) (~k2)

(~k3) (~k4)

fast modes

slow modes

(a) (b)

• standard (dimensional) rescaling:

βg =dg

d ln b= 2(3− d)g +O(gλ) (3.46)

⇒ g is irrelevant at the Wilson-Fisher fixed point near d = 4

– in d = 3: linear term vanishes → coupling becomes marginal

i.e. neither relevant nor irrelevant

– higher-order terms in βg make g irrelevant at Wilson-Fisher fixed point in d = 3

• dimensional analysis: all other couplings are also irrelevant near d = 4 (next chapter!)

43

3. Wilson’s renormalization group

3.5 Summary and comparison to experiment

• near and below d = 4:

– effect of the elimination of high-energy degrees of freedom in GLW theory is onlythe renormalization of already existing couplings

– all other couplings are ultimatively irrelevant

⇒ scaling forms of physical quantities (free energy, χ) near transition!

– corrections to scaling from irrelevant variables (i.e. quartic interaction)

• above d = 4:

– critical behavior is of mean-field type

– interaction is dangerously irrelevant → violation of hyperscaling!

• For a field with N real components and O(N) rotational symmetry one finds:

ν =1

2+

N + 2

4(N + 8)ε+

(N + 2)(N2 + 23N + 60)

8(N + 8)3ε2 +O(ε3) (3.47)

– series has been computed to 6th order: https://arxiv.org/abs/1705.06483

(May 2017)

– XY model (N = 2 in d = 3): critical exponent ν

ν = 0.5 + 0.1ε+ 0.055ε2 − 0.0126091ε3 + 0.0713876ε4 − 0.194548ε5 + 0.707019ε6 +O(ε7)

– direct substitution of ε = 1:

ε0 ε1 ε2 ε3 ε4 ε5 ε6

0.5 0.6 0.655 0.642 0.714 0.519 1.226

– ε expansion does not converge → it is a divergent series: asymptotic series!

– errors at any given order is bounded by next order term (see tutorial)

• One can extract very accurate results from the ε expansion series using resummationtechniques (e.g. Borel summation): educated guess about the function being expandedfrom the finite number of computed terms.

• correlation length exponent ν for N ∈ 1, 2, 3:

ν N = 1 N = 2 N = 3

ε6 expansion (resummed) 0.6292(5) 0.6690(10) 0.7059(20)experiment 0.64± 0.004 0.670± 0.007 0.72± 0.01

• critical exponents nowadays benchmark for different theoretical methods (RG approaches,CFT, Monte Carlo methods)

44

3.6. Excursion: Relations to QFTs in high-energy physics

3.6 Excursion: Relations to QFTs in high-energy physics

Disclaimer: This section cannot replace a full lecture on particle physics QFT, of course. Ihope to give you a reasonable account for the bare basics, here and refer to the textbook byPeskin & Schroeder for more details. The focus of the presentation is the application of theRG beta functions from the previous section which allow us to understand two problems inthe Standard Model of Particle Physics: the hierarchy problem and the triviality problem.

To describe non-relativistic interacting bosons in three spatial dimensions, we have em-ployed the action, eq. (2.50)

S[Φ] =1

kBT

∫d~r

[~2

2m|∇Φ(~r)|2 − µ|Φ(~r)|2 + λ|Φ(~r)|4

], (3.48)

which together with the partition function in functional representation

Z =

∫DΦ∗(~r)DΦ(~r)e−S , (3.49)

allows us to calculate thermodynamic properties from operator averages/correlation func-tions. This quantum field theoretical formalism that we have developed for the system ofmany interacting bosons works analoguously for other systems with a large/infinite numberof coupled degrees of freedom (fields).

• The functional integral approach is also applied in the context of particle physics/high-energy physics!

• in particle physics it is often used to calculate scattering amplitudes (e.g. for largeparticle collider experiments)

• Particle-physics (PP) is typically formulated as a field theory following the laws of thetheory of special relativity

→ (1+3) dimensional Minkowski spacetime with metric ηM =

1 0 0 00 −1 0 00 0 −1 00 0 0 −1

• Particle physics can also be studied in euclidean formulation which is often very conve-

nient

→ Euclidean space with metric ηE =

−1 0 0 00 −1 0 00 0 −1 00 0 0 −1

• Therefore, we need to introduce a Wick rotation (t = −itE) in the time coordinate

which then provides an evolution in imaginary time

• Euclidean functional integral for partice physics quantum field theory emphasizes deepconnection to statistical mechanics and condensed-matter physics.

• For the calculation of particle physics scattering amplitudes → analytic continuation

• In summary : Euclidean formulation of functional integral is natural for statistical me-chanics and convenient for particle physics!

45

3. Wilson’s renormalization group

Remarks on relations between particle physics QFT and statistical mechanics

1. euclidean QFT in d-dimensional spacetime ∼ classical stat-mech in d-dimensional space

2. euclidean QFT in d+ 1-dim spacetime, 0 ≤ τ ≤ β ∼ quantum stat-mech in d-dim space

3. euclidean QFT in d-dim spacetime ∼ high-T quantum stat-mech in d-dim space

Remarks on the Standard Model of Particle Physics

The Standard Model of Particle Physics describes the electromagnetic, weak and strong in-teractions and classifies all known elementary particles. Does not include gravity (for reasons,which we will learn later in the lecture course). Standard Model includes Higgs boson:

• Higgs boson is a massive scalar particle/field with self-interactions

• it provides a mechanism to give mass to leptons (electron, muon, tauon) and quarks

• “Higgs-like” particle has been measured at LHC in 2012 with mass mH ∼ 125 GeV

• generally couples to other fields in Standard Model (e.g. quarks... most importantlythe top quark and gauge fields)

• for our purposes it is sufficient to consider the Higgs field individually with a simplifiedaction for a complex scalar field in four-dimensional Euclidean space:

S[Φ] =

∫d4x

[|∂µΦ(x)|2 − µ|Φ(x)|2 + λ|Φ(x)|4

], (3.50)

• note the striking similarity to the interacting bosons from the previous sections!

• in fact, we have carried out the RG transformations in 4− ε euclidean dimensions

• to obtain the RG beta functions for this theory, we just have to set ε = 0 in eqs. (3.42)

• this provides us with the leading order beta functions for the Higgs field (without thecouplings to other Standard Model fields):

βµ = 2µ(b)− 4λ(b)µ(b) (3.51)

βλ = −10λ(b)2 (3.52)

• implicit to these equations is the choice of an ultraviolet cutoff Λ setting the largestmomentum scale of the problem.

• The cutoff Λ could be interpreted as the largest energy or momentum scale where theStandard Model of Particle Physics can be well defined.

• Ideally we fantasize that a Standard Model of Elementary Particles can be valid atarbitraily high energies and momentum and therefore we hope that we can somehowsend Λ→∞!

• The next section explores wether this is possible!

46

3.6. Excursion: Relations to QFTs in high-energy physics

3.6.1 Landau-pole singularity and the triviality problem

We can directly integrate eq. (3.52) by separation of variables. For simplicity of notation, wesuppress the tilde on top of the λ and just write λ. Before, we reformulate the derivative withrespect to log b:

d log b

db=

1

b⇒ d log b =

1

bdb ⇒ dλ(b)

d log b= b

dλ(b)

db. (3.53)

Therefore, we have to solve

bdλ(b)

db= −10λ(b)2 ⇒

∫ λIR

λUV

dλ

λ2= −10

∫ bIR

bUV

db

b⇒ − 1

λ

∣∣∣λIR

λUV

= −10 log b∣∣∣bIRbUV

(3.54)

⇒ 1

λUV− 1

λIR= −10 (log bIR − log bUV) with bUV = 1 and bIR =

ΛUV

ΛIR(3.55)

⇒ λ(ΛUV) =λ(ΛIR)

1− 10ΛIR log(

ΛUVΛIR

) (3.56)

• Here, ΛUV = Λ is the ultraviolet cutoff, eq. (3.4) which defines the highest momen-tum/energy scale in the system.

• ΛIR is infrared momentum scale down to which we integrated out the momentum modes.

• Now, suppose, we have a collider experiment at some momentum or energy scale ΛIR

where we measure the scattering of particles.

• Note that while the experiment operates at relatively high energies at the collider,these energies ∼ ΛIR can be considered low as compared to a ultraviolet scale ΛUV

which defines the highest energies/momenta where the model is assumed to be valid.

• The experiment carried out at energies ∼ ΛIR shall allow us to fix the interactionbetween particles λIR = λ(ΛIR) defined at that scale.

• this allows us to interpret eq. (3.56): It gives us the value of the interaction λ(ΛUV) atscale ΛUV if at scale ΛIR the value λIR was measured.

• eq. (3.56) has a singularity – the Landau pole singularity – at scale: ΛL = ΛIRe1

10λIR .

theory space

S

+

h

TTc

m ≠ 0 m = 0m > 0

m < 0FM PM

temperature

orde

r par

amet

er

liquid-gas (1st order)

temperature

pres

sure

solid phaseliquid phase

vapour

Tc

pc critical point

(~k1) (~k2)

(~k3) (~k4)

fast modes

slow modes

(a) (b)

scale

inte

ract

ion

IR

IR

UVL

47

3. Wilson’s renormalization group

• the presence of the Landau pole indicates that the interaction coupling λ grows strongat large momentum scales

⇒ as we approach the Landau pole our perturbative approach is not reliable anymore.

New physics

• if Landau pole exists beyond perturbation theory→ theory has limited range of validity

→ theory cannot be considered as being a fundamental theory

→ we cannot extend it beyond ΛL

→ new physics is expected to appear, e.g., new degrees of freedom, unkown particles,non-perturbative physics

Triviality problem

• the Landau-pole singularity can also be put in different terms:

– demand that our theory must be valid on all scales (i.e. also for ΛUV →∞)

– this means, we have to send the Landau pole to infinity ΛL →∞– this requires to have λIR → 0.

– once λIR = 0 on one scale it remains zero on all scales

– therefore the theory becomes non-interacting, i.e. trivial on all scales.

– therefore the Landau pole problem is also called the triviality problem!

Remarks

• A Landau pole also appears in another sector of the Standard Model: Quantum elec-trodynamics (QED) has a beta function de

d log b ∼ − e3

12π2 where e is the EM coupling.

• A more elaborate RG analysis predicts: QED-Landau pole is expected to appear at hugeultraviolet scales ΛL,QED mPlanck ∼ 1019 GeV, where the Planck mass is defined asmPlanck =

√~c/G with G being the gravitational constant. mPlanck defines the scale

where quantum gravity effects are expected to occur and a new theory beyond theStandard Model is needed which includes gravity. Therefore, QED-Landau pole is notconsidered as a problem of practical importance within the Standard Model.

• Further analysis with the Standard Model parameters suggests that also Landau-polein the Higgs sector only appears beyond mPlanck. Due to fluctuations of the top quark,however, another problem below mPlanck is sometimes discussed: The quartic couplingλ in the Higgs sector might drop below zero and render the Higgs potential unboundedfrom below or induce a second (deeper) minimmum with a possibly finite decay ratefrom the present metastable state. This is often referred to as vacuum instability.

• In case the sign of the beta function is different from the scenarios discussed in thissection (e.g. φ3 theory in d = 6 or QCD), the coupling drops to zero for large momentumscales. This behavior is typically referred to as asymptotic freedom (problem set 4). TheLandau pole then appears in the infrared and indicates, for example, a change of degreesof freedom, as the system becomes strongly coupled.

48

3.6. Excursion: Relations to QFTs in high-energy physics

3.6.2 Fine-tuning and the hierarchy problem

The second problem which we discuss here and which appears in the Standard Model is theso-called hierarchy problem or naturalness problem. It is related to the beta function for theterm which is bilinear in the bosonic fields, eq. (3.51):

βµ = 2µ(b)− 4λ(b)µ(b) . (3.57)

• for simplicity, we assume that we are in the perturbative regime, where λ 1 is small(sufficiently far away from the Landau pole).

→ the leading behavior of µ(b) is given by the simplified equation:

bdµ(b)

db= 2µ(b) (3.58)

• this equation is easily integrated by separation of variables and gives:

µ(ΛIR) = µ(ΛUV)

(ΛUV

ΛIR

)2

(3.59)

• Note that for a precise calculation the b-depencence of λ(b) should be taken into account.

• Generally, for the Standard Model, there will be further couplings contributing toeq. (3.51), most importantly the coupling to the top quarks (fermions).

• Still, the most important contribution to the beta function of µ is the dimensionalterm ∝ 2µ(b) which we already determined in Sec. 3.2.1 from zeroth order perturbationtheory (i.e. a purely dimensional argument).

• The relation in eq. (3.59) implies that an extreme fine tuning has to be performedon µ(ΛUV), when ΛUV ΛIR and at ΛIR, the parameter µ(ΛIR) has to match anexperimentally determined (physical) value.

• Let’s make a numerical example:

– suppose we make a particle collider experiment at energy scale ΛIR = 173GeV(which is at the scale of the mass of the top quark)

– suppose a measurement allows us to fix the parameter√µ(ΛIR) = 125GeV

(which suggests that this parameter can be connected to the Higgs mass!)

– So far, no direct evidence for physics beyond the Standard Model has been foundin particle colliders

→ assume that Standard Model is a valid theory also on much higher scales

→ some ideas suggest a grand unified theory (GUT) at ΛUV ∼ 1015GeV.

– With these numbers, we conclude that we have to choose a parameter

µ(ΛUV) = 4.67640625 · 10−22GeV2 (3.60)

• in the Standard Model this seems unnatural, because there is no “knob” which canbe used to adjust this value (hence naturalness problem). Still, for some reason theStandard Model seems to be a system which is tuned close to criticality.

• in the interacting boson system, where we observe critical behavior, this tuning seems tobe “natural” as we have to adjust the temperature T very close to its critical value Tc.

49

3. Wilson’s renormalization group

theory space

S

+

h

TTc

m ≠ 0 m = 0m > 0

m < 0FM PM

temperature

orde

r par

amet

er

liquid-gas (1st order)

temperature

pres

sure

solid phaseliquid phase

vapour

Tc

pc critical point

(~k1) (~k2)

(~k3) (~k4)

fast modes

slow modes

(a) (b)

scale

IR UV

scale

inte

ract

ion

IR

IR

UVL

µIRultra-fine adjusment

Figure 3.2: Sketch of the hierarchy problem.

50

Chapter 4

Functional methods in quantum field theory

4.1 Introductory remarks

In the previous chapter, we have seen that systems in statistical mechanics, condensed-matterphysics as well as in particle physics can be described in terms of a functional integral whichin short-hand notation can be written as

Z =

∫DΦe−S[Φ] , (4.1)

with the action S[Φ]. We have given an example for the action S[Φ] in terms of a complexscalar field which finds applications in the superfluid transition of non-relativistic interactingbosons as well as in the Higgs sector of the Standard Model of Particle Physics. More generally,there can be other action functionals S[Φ] with different numbers of boson components,fermion fields, gauge fields, other symmetries,... , i.e. the field variable Φ collects all relevantdegrees of freedom.

Therefore, a profound understanding of quantum and statistical field theories requiresa sophisticated toolbox of theoretical methods. Functional methods are suitable for thecomputation of generating functionals for correlation functions. The generating functionalscontain all relevant physical information about a theory. In this chapter and the next one,we will introduce some of these functional methods which are applicable to an extremelybroad range of physical systems. Further, theses functional methods also provide a deeperconceptual understanding of quantum and statistical field theories. In particular, we will

• introduce generating functionals for more efficient calculations

• analyse divergencies in perturbation theory

• give a classification of perturbatively renormalizable theories

• give an inductive proof of perturbative renormalizability

• introduce an exact renormalization group equation

• learn how to apply functional renormalization

51

4. Functional methods in quantum field theory

4.2 Generating functional

In quantum and statistical field theory all physical information is stored in correlation func-tions, e.g., the two-point correlation function from eq. (2.26) or the susceptibility eq. (2.27).In a particle collider experiment with two incident beams and (N − 2) scattering products,this process can be described by the N -point correlation function. In the following, we willrefer to quantum or statistical field theory just shortly as QFT.

• in Euclidean QFT the correlation functions are defined as

〈Φ(x1)...Φ(xn)〉 := N∫DΦ Φ(x1)...Φ(xn)e−S[Φ] , (4.2)

where xi are suitable coordinates in position or momentum space.

• we fix the normalization N such that 〈1〉 = 1.

• functional integral measure includes implicit regularization with ultraviolet cutoff Λ∫DΦ −→

∫ΛDΦ. (4.3)

• the regularized measure shall preserve the symmetries of the theory.

• in case we have to deal with Minkowski-valued correlators, we assume that these canbe defined from Euclidean ones by analytic continuation.

• with minor modifications, the following discussion also works for fermions.

All n-point correlation functions are summarized by the generating functional :

Z[J ] =

∫DΦe−S[Φ]+Φ·J , (4.4)

with the source term Φ · J =∫ddxΦ(x)J(x).

• Recall that at finite temperature, we had Z[0] = exp[−βF ], where F is the free energy.

4.2.1 Correlation functions

• the functional derivative acts as:

δΦ(x)

δΦ(y)= δ(d)(x− y) , or

δJ(x)

δJ(y)= δ(d)(x− y) . (4.5)

• therefore, we obtain

δ

δJ(x)e∫ddyJ(y)Φ(y) = Φ(x)e

∫ddyJ(y)Φ(y) (4.6)

⇒ 〈Φ(x1)...Φ(xn)〉 =

(1

Z[J ]

δ

δJ(x1)...

δ

δJ(xn)Z[J ]

) ∣∣∣J=0

(4.7)

⇒ Z[J ] contains full information about the theory.

• Definition: n-point correlator in the presence of a source J :

〈Φ(x1)...Φ(xn)〉J =1

Z[J ]

∫DΦ Φ(x1)...Φ(xn)e−S[Φ]+Φ·J . (4.8)

52

4.2. Generating functional

What does Z[J ] generate?

• Z[J ] is the generating functional and generates the correlation functions 〈Φ(x1)...Φ(xn)〉

• For example, the two-field Green’s function can be pictorially represented as

〈Φ(x1)Φ(x2)〉 = x2 x1 (4.9)

• Observation: Perturbative calculation of the four-field Green’s function

〈Φ(x1)Φ(x2)Φ(x3)Φ(x4)〉 (4.10)

(e.g. using the Wick theorem) contains disconnected contributions of the form:

〈Φ(x1)Φ(x2)〉〈Φ(x3)Φ(x4)〉 =x4 x3

x2 x1(4.11)

• these contributions can be expressed in terms of the exact two-field Green’s function

〈Φ(x1)Φ(x2)〉 = x2 x1 (4.12)

⇒ they do not exclusively contain information about correlations involving four fields.

⇒ we carry along redundant information!

• for more details: see App. A

Dyson-Schwinger equations

• the quantum equations of motion follow from the translational invariance of DΦ:∫DΦ

δ

δΦ(x)

(e−S[Φ]+Φ·J

)= 0 (4.13)

⇒∫DΦ

(J(x)− δS

δΦ(x)

)e−S[Φ]+Φ·J = 0 (4.14)

⇒ J(x)− 〈 δSδΦ

(x)〉J = 0 (4.15)

• These are the Dyson-Schwinger equations (DSE). They can also be written as(J(x)− δS

δΦ(x)

[δ

δJ

])Z[J ] = 0 (4.16)

• The DSEs provide an infinite hierarchy of equations relating the Φ-field correlationfunctions with each other

– for example, 2-point correlation function can be written in terms of 4-point corre-lation function (problem set 5)

53

4. Functional methods in quantum field theory

4.2.2 Schwinger functional

A more efficient way to store the information about the correlation functions is given by theSchwinger functional :

W [J ] = logZ[J ] = log

∫DΦe−S[Φ]+J ·Φ (4.17)

• Schwinger functional is generating functional for the connected Green’s functions.

• Observation: W [J ] contains diagrams of the form:

⇒ W [J ] still carries redundant information in particular cases.

• we would like to have a generating functional for the one-particle-irreducible (1PI)Green’s functions.

4.3 Effective action

We define the classical field:

ϕ(x) := 〈Φ〉J =1

Z[J ]

∫DΦ Φ(x)e−S[Φ]+J ·Φ =

δW

δJ(x)(4.18)

⇒ ϕ = ϕ[J ] (4.19)

• invert ϕ[J ] and compute J [ϕ].

We define the effective action Γ as the Legendre transform of W [J ]:

Γ[ϕ] = supJ

(J · ϕ−W [J ]) (4.20)

• for any given ϕ a special J = Jsup is singled out for which J · ϕ−W [J ] approaches itssupremum (J = J [ϕ]).

• this definition of Γ guarantees that Γ is convex.

• at J = Jsup:

δΓ[ϕ]

δϕ(x)= J(x) +

∫y

δJ [y]

δϕ(x)ϕ(y)−

∫y

δW [J ]

δJ [y]︸ ︷︷ ︸use eq. (4.20)

δJ [y]

δϕ(x)= J(x) (4.21)

⇒ δΓ[ϕ]

δϕ= J (4.22)

• eq. (4.21) is the quantum equation of motion (all quantum/statistical effects included).

• for comparison, the classical field equation is δSδΦ(x) = 0.

54

4.3. Effective action

Functional integro-differential equation for effective action

e−Γ[ϕ]+J ·ϕ = eW [J ] =

∫DΦe−S[Φ]+J ·Φ (4.23)

e−Γ[ϕ] =

∫DΦe

−S[Φ]+∫ δΓ[ϕ]

δϕ(Φ−ϕ)

(4.24)

e−Γ[ϕ]Φ→Φ+ϕ︷︸︸︷

=

∫DΦe

−S[Φ+ϕ]+∫ δΓ[ϕ]

δϕΦ

(4.25)

• functional integro-differential equation

• exact determination of Γ[ϕ] available only for rare special cases.

• a solution of eq. (4.25) can be attempted by a vertex expansion:

Γ[ϕ] =∞∑n=1

1

n!

∫x1,...,xn

Γ(n)(x1, ..., xn)ϕ(x1)...ϕ(xn)

F.T.︷︸︸︷=

∞∑n=1

1

n!

∫p1,...,pn

δ(p1 + p2 + ...+ pn)Γ(n)(p1, ..., pn)ϕ(p1)...ϕ(pn) (4.26)

• the expansion coefficients Γ(n) are the one-particle irreducible (1PI) proper vertices.

• inserting eq. (4.26) into eq. (4.25) and comparing coefficients of the field monomials inan infinite tower of coupled integro-differential equations for the Γ(n):

Dyson-Schwinger eqautions (DSE)

• functional method of constructing approximate solutions by truncated (= finite ansatzfor the series in eq. (4.26)) DSEs has highly developed applications in gauge theories.

• n-point correlation functions can be reconstructed from the Γ(n).

• in particular the analysis of divergencies is simpler using the Γ(n).

Inverse propagator

We consider the two times functional derivative of the effective action with respect to theclassical fields:

Γ(2)(x, y) =δ2Γ

δϕ(x)δϕ(y)(4.27)

we find Γ(2) = G−1 with G(x, y) = δ2WδJ(x)δJ(y) = W (2)(x, y)

⇒ Γ(2)W (2) = 1 (4.28)

or more explicitly ∫ddq′′

(2π)2Γ(2)(q, q′′)W (2)(q′′, q′) = (2π)dδ(d)(q − q′). (4.29)

55

4. Functional methods in quantum field theory

Perturbation theory for Γ[ϕ]

From eq. (4.25) we obtain:

Γ[ϕ] = − log

∫ΛDΦe

−S[Φ+ϕ]+∫xδΓ[ϕ]δϕ(x)

Φ(x)(4.30)

with

• classical/background field ϕ

• fluctuation field Φ

We can rewrite this as an implicit equation

Γ[ϕ] = S[ϕ] +

“loops′′︷ ︸︸ ︷Γl[ϕ] (4.31)

with

Γl[ϕ] = − log

∫ΛDΦe

−(S[Φ+ϕ]−S[ϕ])+∫xδΓ[ϕ]δϕ(x)

Φ(x)(4.32)

Now, we make a saddle-point approximation (expand around Φ = 0)

S[ϕ+ Φ] = S[ϕ] +

∫x

δS

δϕ(x)Φ(x) +

1

2

∫x,y

δ2S

δϕ(x)δϕ(y)︸ ︷︷ ︸=S(2)(x,y)

Φ(x)Φ(y) + ... (4.33)

⇒ Γl[ϕ] = − log

∫ΛDΦ exp

−1

2

∫x,yS(2)(x, y)Φ(x)Φ(y) +

∫x

δΓlδϕ(x)

Φ(x) +O(S(3))︸ ︷︷ ︸neglect→1−loop contribution

(4.34)

neglecting part marked in the above equation gives the 1-loop contribution to effective action:

Γ1l[ϕ] = − log

∫ΛDΦ exp

−1

2

∫x,yS(2)(x, y)Φ(x)Φ(y)

(4.35)

= − log

(detΛS(2)

)− 12

(4.36)

now use log detA = Tr logA

⇒ Γ1l[ϕ] =1

2Tr logS(2)[ϕ] (4.37)

This provides us with important expression for perturbation theory for Γ[ϕ] at one-loop order:

Γ[ϕ] = S[ϕ] +1

2Tr logS(2)[ϕ] (4.38)

In momentum space, the one-loop integral explicitly reads:

Γ1l =1

2

∫ddq

(2π)dddq′

(2π)d(2π)dδ(d)(q − q′)︸ ︷︷ ︸Tr

(logS(2)[ϕ]

)(q, q′) (4.39)

56

4.4. Perturbative renormalization

4.4 Perturbative renormalization

Here, we want to understand more systematically how renormalization in perturbation theoryworks. This will lead us to a classification of perturbatively renormalizable theories.

4.4.1 Divergencies in perturbation theory

Example: φ3 theory on the 1-loop level. The corresponding action can be written as:

S[ϕ] =

∫ddx

1

2(∂µϕ)2 +

1

2m2ϕ2 +

1

3!gϕ3

(4.40)

• Remark: this theory is unphysical as the potential is not bounded from below:

theory space

S

+

h

TTc

m ≠ 0 m = 0m > 0

m < 0FM PM

temperature

orde

r par

amet

er

liquid-gas (1st order)

temperature

pres

sure

solid phaseliquid phase

vapour

Tc

pc critical point

(~k1) (~k2)

(~k3) (~k4)

fast modes

slow modes

(a) (b)

scale

inte

ract

ion

IR

IR

UVL

scale

IR UV

µIRultra-fine adjusment

'

• However, there is a well-defined perturbative expansion and further the vertex structureappears in many physically relevant theories.

S(2)[ϕ] = −∂2+m2 + gϕ(x) (4.41)

⇒ Γ1l[ϕ] =1

2Tr log

(−∂2 +m2 + gϕ(x)

)(4.42)

=1

2Tr

[log

(1 +

1

−∂2 +m2gϕ(x)

)]+ const. (4.43)

= −1

2

∞∑n=1

1

nTr

[( −g−∂2 +m2

ϕ(x)

)n]+ const. (4.44)

• after Fourier transformation to momentum space, we find the proper vertices:

Γ(n)1−loop = −(n− 1)!

2(−g)n

∫ddq

(2π)d1

q2 +m2

1

(q + p1)2 +m2...

1

(q + p1 + ...+ pn−1)2 +m2

(4.45)

• Feynman diagram for eq. (4.45) (1-loop 1PI diagram):

57

4. Functional methods in quantum field theory

• ultraviolet (= large momentum) behavior of the loop-integration for q2 p21, ..., p

2n,m

2:

Γ(n)1−loop ∼

∫ddqq−2n (4.46)

therefore, it diverges for d ≥ 2n.

• more explicitly:

– d = 1: all correlation functions are finite (quantum mechanics)

– d = 2: 1-point function diverges (tadpole):

Γ(1)1−loop =

g

2

∫ddq

(2π)d1

q2 +m2(4.47)

– ...

– d = 6: 1-,2-, and 3-point functions diverge

• Regularization:

cut the momentum integral by integrating in a sphere |q| < Λ with Λ2 p21, ..., p

2n,m

2

→ cutoff regularization, see also Eq. (3.4)

Now, focus on the case d = 6: Determination of divergencies by expansion in p21, ..., p

2n,m

2:

Γ(1)1−loop =

g

2

∫ Λ d6q

(2π)6

1

q2 +m2=g

2

∫d6q

(2π)6

(1

q2− m2

q4+m4

q6+O(

1

q8)

)(4.48)

where we employed a Taylor series expansion of the integrand in the external momenta andthe mass p2

1, ..., p2n,m

2. Further, we use the spherical integral:∫ Λ d6q

(2π)6= VS5

∫ Λ dq

(2π)6q5 (4.49)

where VS5 = π3. Therefore, we get:

Γ(1)1−loop =

g

27π3

[Λ4

4− m2Λ2

2+m4 log

Λ

m+O(1)

](4.50)

• The log-term contains an infrared divergence

• therefore, we have introduced an infrared cutoff at m

• Here, we will focus on the analysis of the ultraviolet divergencies

• similarly, we obtain:

Γ(2)1−loop =

g2

27π3

[Λ2

2+ (2m2 +

p2

3) log

Λ

m+O(1)

](4.51)

Γ(3)1−loop =

g3

26π3log

Λ

m+O(1) (4.52)

58

4.4. Perturbative renormalization

• why d = 6?

– d = 6 is special as divergencies appear only in vertices that maximally are part ofthe original action:

S[ϕ] =

∫Γ

(1)0−loopϕ+

1

2

∫Γ

(2)0−loopϕ

2 +1

3!

∫Γ

(3)0−loopϕ

3 (4.53)

– here we have introduced the tree level or bare vertices:

Γ(1)0−loop = 0 (4.54)

Γ(2)0−loop = p2 +m2 (4.55)

Γ(3)0−loop(p1, p2,−(p1 + p2)) = g (4.56)

– note that Γ(1)0−loop can be generated by a shift ϕ→ ϕ+ const.

– furthermore, the dependence of divergencies on the external momenta equals the

momentum dependence of the bare vertices Γ(n)0−loop.

– on the contrary in d = 8: Γ(4)0−loop = 0 and Γ

(4)1−loop diverges

59

4. Functional methods in quantum field theory

4.4.2 Removal of divergencies

Divergent part of 1-loop 1PI functional in d = 6:

Γ1−loop,div =

∫d6x

[ga1(Λ)ϕ(x) +

1

2g2a2(Λ)(∂µϕ)2(x) +

1

2g2a3(Λ)ϕ2(x) +

1

3!g3a4(Λ)ϕ3(x)

](4.57)

where from eqs. (4.50), (4.51) and (4.52) we have:

a1(Λ) =1

27π3

[Λ4

4− m2Λ2

2+m4 log

Λ

m

](4.58)

a2(Λ) =1

3 · 27π3log

Λ

m(4.59)

a3(Λ) =1

27π3

[−Λ2

2+ 2m2 log

Λ

m

](4.60)

a4(Λ) =1

26π3log

Λ

m(4.61)

Note: the origin of these divergencies is that fluctuations on all scales contribute to the vertexfunctions.

• Physical problem: How do the low-energy observables (susceptibility, magnetization,coupling, scattering amplitude, particle masses), that are given by the full vertex func-tions, relate to the parameters in the bare action g,m2, ϕ that describe the high-energyparameters of a microscopic theory at scale Λ?

• Idea: Fix the coupling, mass, chemical potential, etc. at a renormalizartion point/scaleµ (scale of measurement) to the (physical) values mR, gR, ϕR by renormalization condi-tions for the full vertices:

The renormalized physical parameters are fixed by means of the effective action:

Γ(2)(p2 → 0) = m2R (fixes physical mass) (4.62)

∂

∂p2Γ(2)(p2 → 0) = 1 (fixes normalization of field) (4.63)

Γ(3)(p21 = p2

2 = p23 = 0) = gR (fixes physical coupling) (4.64)

• This makes sure that the renormalized parameters coincide with the measurements froma (low-energy) experiment.

• More general : fix parameters at p2 → µ2, where µ is the renormalization point/scale

⇒ Γ(2)(p2 = µ2) = m2 6= m2R(p = 0) (4.65)

• Note:

– mR and gR are real physical input

– eq. (4.63) is a normalization prescription. Finite rescaling is still permitted.

60

4.4. Perturbative renormalization

Renormalization in perturbation theory:

Expansion in the renormalized coupling gR:

S[ϕ] ≡ SΛ[ϕ] =

∫ [1

2ϕ(−∂2 +m2)ϕ+

1

3!gϕ3

](4.66)

≡ SR[ϕR] + δSR[ϕR] (4.67)

=

∫ [ finite︷ ︸︸ ︷1

2ϕR(−∂2 +m2

R)ϕR +1

3!gRϕ

3R (4.68)

+1

2(Zϕ − 1)(∂µϕR)2 +

1

2δm2ϕ2

R +1

3!gR(Zg − 1)ϕ3

R︸ ︷︷ ︸counterterms

](4.69)

with

ϕ = Z1/2ϕ ϕR (4.70)

g = gRZg

Z3/2ϕ

(4.71)

m2 = (m2R + δm2)

1

Zϕ(4.72)

and the parametrization of the relation between bare and renormalized quantities is given by

• Zϕ: wave function renormalization

• Zg: coupling renormalization

• mR, gR, ϕR: renormalized quantities

Now, assume that Zϕ, Zg, δm2 have an expansion in gR such that

(Zϕ − 1), δm2, (Zg − 1) = O(g2R) (4.73)

⇒ second line in eq. (4.69) is of the same order as 1-loop contribution, cf. Eq. (4.57)

→ 1-loop calculation has to be performed with SR[ϕR]

→ we then have:

Γ[ϕR] = SΛ[ϕ] +1

2Tr logS

(2)Λ [ϕ] +O(2− loop) (4.74)

= SR[ϕR] + δSR[ϕR] +1

2Tr logS

(2)R [ϕR] +O(2− loop) (4.75)

now the renormalization conditions, eqs. (4.62), fix the relations Zϕ, Zg, δm2:

In particular, for the divergent parts, we obtain:

Γ[ϕR] = O(Λ0) (4.76)

⇒ δSR[ϕR] +1

2Tr logS

(2)R [ϕR] = O(Λ0) (4.77)

61

4. Functional methods in quantum field theory

with eq. (4.57) for perturbative calculation with gR:

1

2Tr logS

(2)R [ϕR] =

∫d6x

[gRa1(Λ)ϕR(x) +

1

2g2Ra2(Λ)(∂µϕR)2(x)

+1

2g2Ra3(Λ)ϕ2

R(x) +1

3!g3Ra4(Λ)ϕ3

R(x)

](4.78)

and ai from eq. (4.58), we obtain:

Zϕ = 1− g2Ra2(Λ) (4.79)

Zg = 1− g2Ra4(Λ) (4.80)

δm2 = g2Ra3(Λ) (4.81)

⇒ the assumption eq. (4.73) is selfconsistent

⇒ Γ[ϕR] is finite (on a 1-loop level) and only depends on physical parameters!

Interpretation:

The original parameters in S[ϕ] (m2, g, ...) are not the ones measured in a (low-energy)experiment but they are connected to the latter ones by transformations.

This means the original (= bare) parameters are functions of the cutoff :

m = m(Λ), g = g(Λ) (4.82)

• This tuning of the bare parameters does not increase the number of parameters in theϕ3 theory for d ≤ 6.

• For example: in d = 8, the vertex Γ(4) is also divergent, so δS has to include a ϕ4 term(+ 1 parameter!). This term generates further divergencies in higher orders, such thatthe number of parameters increases without limit.

62

4.4. Perturbative renormalization

4.4.3 Divergencies in perturbation theory: general analysis

• Is it possible to choose the counter terms δm2, δZϕ = Zϕ − 1, etc. for any theory suchthat the low-energy physics is finite?

• under which conditions the theory remains predictive?

Canonical dimension of a field:

• consider the free propagator of a field:

∆(p) ≡(

Γ(2)0 (p)

)−1∼︸︷︷︸

p→∞

1

|p|σ (4.83)

where we have used the assumptions that the limit p→∞ is O(d) symmetric

• then the canonical dimension of a field is defined as

[ϕ] :=d− σ

2(4.84)

• Examples:

– scalar field:

∆(p) =1

p2 +m2∼︸︷︷︸

p→∞

1

p2⇒ [ϕ] =

d− 2

2(4.85)

– fermion field (Dirac):

∆(p) =1

pµγµ +m∼︸︷︷︸

p→∞

1

p⇒ [ϕ] =

d− 1

2(4.86)

– Note: In these examples, the canonical dimension can also be inferred from theaction (dimensionless). This is not possible for higher spin fields.

• in theories with polynomial interactions, the interaction can be written as a sum ofvertices:

V (ϕ) ∼∫ddx

(∂

∂x

)kϕ1(x)n1ϕ2(x)n2 ...ϕs(x)ns (4.87)

where the ϕi are different types of fields.

• pictorially, such a vertex can be represented as:

theory space

S

+

h

TTc

m ≠ 0 m = 0m > 0

m < 0FM PM

temperature

orde

r par

amet

er

liquid-gas (1st order)

temperature

pres

sure

solid phaseliquid phase

vapour

Tc

pc critical point

(~k1) (~k2)

(~k3) (~k4)

fast modes

slow modes

(a) (b)

scale

inte

ract

ion

IR

IR

UVL

scale

IR UV

µIRultra-fine adjusment

'

'1, n1

'2, n2

'3, n3

63

4. Functional methods in quantum field theory

• the dimension of a vertex is:

δ[V ] = −d+ k +s∑i=1

ni [ϕi] (4.88)

• vertex in momentum space

V (ϕ) ∼∫ddp1...d

dpn1+n2+...+nsδ(d)(p1 + ...+ pn1+...+ns)p

kϕ1(p1)...ϕs(pn1 + ...+ pns)

(4.89)

⇒ vertex contains momentum-conserving δ function

• divergencies in the n-point proper vertices Γ(n) appear in the integrations over thefluctuation momenta. Perturbation theory sorts the contributions to Γ(n) according tothe number of independent momentum integrations (= number of loops).

Superficial degree of divergence of a diagram γ:

If all integration momenta in a diagram γ are scaled by a factor λ 1

⇒ diagram is scaled by a factor λδ(γ) with:

δ(γ) = dL−∑i

Iiσi +∑α

vαkα (4.90)

where

• L: number of loops

• Ii: number of internal lines

• vα: number of vertices of type α

• kα: number of momenta/derivatives

Then we can distinguish three different cases:

1. if δ(γ) > 0 a regularized diagram diverges at least like Λδ(γ).

2. if δ(γ) = 0 a diagram diverges at least like log Λ.

3. if δ(γ) < 0 a diagram is superficially convergent (divergencies can only come fromsubdiagrams).

64

4.4. Perturbative renormalization

• Example (ϕ3 theory):

– here, we have σ = 2, k = 0 ⇒ δ(γ) = dL− 2I

– in d = 6 and L = 1 (1-loop level): δ(γ) = 6− 2I

⇒ for I = 1, 2, 3 −→ δ(γ) = 4, 2, 0

theory space

S

+

h

TTc

m ≠ 0 m = 0m > 0

m < 0FM PM

temperature

orde

r par

amet

erliquid-gas (1st order)

temperature

pres

sure

solid phaseliquid phase

vapour

Tc

pc critical point

(~k1) (~k2)

(~k3) (~k4)

fast modes

slow modes

(a) (b)

scale

inte

ract

ion

IR

IR

UVL

scale

IR UV

µIRultra-fine adjusment

'

'1, n1

'2, n2

'3, n3

I = 1 I = 2 I = 3

4 2 log

• Different expression of δ(γ) by topological relations:

– be Ei the number of external lines of a field ϕi

– we find

δ(γ) = d−∑i

[ϕi]Ei +∑α

vαδ(Vα) (4.91)

– with this equation, we can give a

classification of (perturbatively) renormalizable theories...

65

4. Functional methods in quantum field theory

4.4.4 Classification of perturbatively renormalizable theories

Suppose an analysis of the superficial degree of divergence is sufficient to get the informationabout all possible divergencies. This is the statement of the BPHZ theorem (Bogoliubov,Parasiuk, Hepp, Zimmermann):

All divergencies of a perturbatively renormalizable theory can be removed by counter termsthat correspond to superficially divergent amplitudes. (very technical proof!)

In the following: [ϕi] ≥ 0 then a theory is:

• Non-renormalizable, if at least one vertex has a positive dimension δ(Vα).

By considering diagrams with increasing number v of vertices (higher loops) of this type,we can make the degree of divergence arbitrarily large and this for any 1PI correlationfunction Γ(n) (any number of external legs Ei)

⇒ the removal of divergencies then requires infinitely many counter terms i.e. ∞-manyphysical parameters

⇒ Loss of predictivity!

• Super-renormalizable, if all δ(Vα) < 0

⇒ only a finite number of diagrams are superficially divergent.

Example (ϕ4 theory in d = 3):

δ(V ) = −3 + 0 + 4 · 1

2= −1 (4.92)

⇒ δ(γ) = 3− 1

2E − v (4.93)

which gives the divergent diagrams:

– E = 2:

theory space

S

+

h

TTc

m ≠ 0 m = 0m > 0

m < 0FM PM

temperature

orde

r par

amet

er

liquid-gas (1st order)

temperature

pres

sure

solid phaseliquid phase

vapour

Tc

pc critical point

(~k1) (~k2)

(~k3) (~k4)

fast modes

slow modes

(a) (b)

scale

inte

ract

ion

IR

IR

UVL

scale

IR UV

µIRultra-fine adjusment

'

'1, n1

'2, n2

'3, n3

I = 1 I = 2 I = 3

4 2 log

v = 1 v = 2 v = 2

– E = 4 has at least v = 2 ⇒ δ(γ)|E≥2 < 0.

• Renormalizable, if at least one δ(Vα) = 0 (and there is no δ(Vα) > 0).

⇒ ∞-many diagrams have δ(γ) > 0, however, at fixed Ei: δ(γ) independent from vα

⇒ only a finite number of Γ(n) is divergent

⇒ only a finite number of counter terms is required

⇒ finite number of physical parameters

⇒ Theory is predictive!

66

4.4. Perturbative renormalization

Examples:

δ(V )

(4.91)︷︸︸︷= −d+ k +

s∑i=1

ni[ϕi] (4.94)

1. scalar fields without derivative interactions (k = 0):

[ϕ] =d− 2

2⇒ δ(V ) = −d+ n

d− 2

2= d(

n

2− 1)− n (4.95)

• ϕ3 is renormalizable in d = 6

• ϕ4 is renormalizable in d = 4

• ϕ6 is renormalizable in d = 3

2. spin-1/2, [Ψ] = d−12 (Dirac field)

• (ΨΨ)2 in d = 2 (Gross-Neveu model)

• ϕΨΨ in d = 4 (Yukawa model)

• ΨΨϕ2 in d = 3

3. spin-1 (no gauge theories): [Vµ] = d2

• renormalizable only in d = 2, (Skin ∼∫∂V ∂V ), e.g., with ΨγµVµΨ interaction

4. spin ≥ 3/2: no perturbatively renormalizable theories.

5. gauge theories, spin 1: [Aµ] = d−22

• F aµνF aµν in d = 4 with interactions ΨγµAµΨ, ∂µϕAµϕ, ϕ2A2 renormalizable

6. No physically acceptable (from the point of view of particle physics) and renormalizabletheory exists for d > 4 (perturbatively)!

It is not known whether this property is logically connected with the fact that spacetimehas just four dimensions or is a mere coincidence. (from the book by J. Zinn-Justin)

67

Chapter 5

Functional renormalization group

The functional renormalization group provides a nonperturbative and practical theoreticalframework for quantum field theoretical calculations. More specifically, it represents theimplementation of Wilson’s idea for the renormalization group with the functional methodswhich we have learned in the previous chapter. The central object of the functional RG is aflow equation – the Wetterich equation, which describes the evolution of correlation functionsor the generating functional under successive application of momentum-shell integrations interms of a functional differential structure.

5.1 Effective average action

We will define the effective average action Γk as an action functional which depends on amomentum-shell parameter k (also called ”infrared cutoff scale” not to be confused with theultraviolet cutoff Λ). Γk will be defined such that it interpolates between two importantlimits:

1. For k → Λ: Γk corresponds to the bare action to be quantized, i.e. Γk→Λ ' Sbare.

2. For k → 0: Γk corresponds to the full effective action, i.e. Γk→0 = Γ.

To achieve this, we modify the generating functional Z[J ], cf. eq. (4.4) by adding a term∆Sk[Φ] to the action:

eWk[J ] ≡ Zk[J ] =

∫ΛDΦe−S[Φ]−∆Sk[Φ]+J ·Φ (5.1)

where the additional term reads

∆Sk[Φ] =1

2

∫ddq

(2π)dΦ(−q)Rk(q)Φ(q) . (5.2)

It is called a regulator term which is quadratic in the field Φ (like the derivative term or the“mass” term m2). Therefore, it can be interpreted as a momentum-dependent mass term.

69

5. Functional renormalization group

5.1.1 The regulator term

The regulator function Rk(q) has to satisfy three different conditions to qualify for the desiredregularization properties for the effective average action Γk:

1. It should implement infrared regularization which is realized by the condition:

limq2

k2→0

Rk(q) > 0 (5.3)

2. We want to recover the standard effective action Γ when k → 0, i.e. when we removethe infrared cutoff. The regulator should therefore vanish in this limit:

limk2

q2→0

Rk(q) = 0 (5.4)

3. The functional integral should be dominated by the stationary point of the action inthe limit of large k → Λ→∞. This can be achieved by the condition:

limk→Λ→∞

Rk(q)→∞ (5.5)

This filters out the classical field configuration of the action Γk→Λ → S + const.

q

Rk,∂tRk

Figure 5.1: Sketch of a representative regulator function Rk(q2) (black solid line). For later

purpose, we also sketch its derivative ∂tRk(q2) (orange dashed). For this plot, we chose

Rk(q2) = 2k2/(e(q2/k2)4

+1)−1. You can check that another suitable choice would be Rk(q2) =

(k2 − q2)Θ(k2 − q2).

70

5.2. Wetterich equation

5.2 Wetterich equation

We now want to study the trajectory of the interpolating functional Γk between the limitsk → 0 and k →∞ which we have constructed to correctly give Γ and Sbare, respectively.

Therefore it is convenient to use the short-hand notation:

t = logk

Λ∂t = k

∂

∂k. (5.6)

We keep the source term J independent from k and start with the derivative of the generatingfunctional

∂teWk = ∂t

(∫ΛDΦe−S[Φ]−∆Sk[Φ]+J ·Φ

)(5.7)

⇒ eWk︸︷︷︸=Zk

∂tWk =

∫ΛDΦ(−∂t∆Sk[Φ])e−S[Φ]−∆Sk[Φ]+J ·Φ (5.8)

⇒∂tWk =︸︷︷︸(5.2)

−1

2

∫q

∂tRk(q)[

1

Zk

∫ΛDΦ Φ(−q)Φ(q)e−S[Φ]−∆Sk[Φ]+J ·Φ

]︸ ︷︷ ︸

=〈Φ(−q)Φ(q)〉

(5.9)

We now recall that the (k-modified) connected propagator is

Gk(q) =δWk

δJδJ(q) = 〈Φ(−q)Φ(q)〉 − ϕ(−q)ϕ(q) (5.10)

where according to eq. (4.18), we have

ϕ(q) = 〈Φ(q)〉 =δWk[J ]

δJ(q)(5.11)

and for convenience we have suppressed the index J for the exepectation values 〈...〉J .

So, with eq. (5.9) we obtain

∂tWk = −1

2

∫q∂tRk(q) (Gk + ϕ(−q)ϕ(q)) (5.12)

= −1

2

∫q∂tRk(q)Gk − ∂t∆Sk[ϕ] (5.13)

Note the modified argument in ∆Sk in the above equation!

Before, we go on with the calculation, we can now define the average effective action as amodified Legendre transform reading

Γk[ϕ] = supJ

(J · ϕ−Wk[J ])−∆Sk[ϕ] (5.14)

cf. eq. (4.20).

71

5. Functional renormalization group

Accordingly, the corresponding quantum equation of motion, cf. eq. (4.21), is also modified:

J(x) =δΓk[ϕ]

δϕ(x)+ (Rkϕ)(x) . (5.15)

This can be used to obtain:

δJ(x)

δϕ(y)=

δ2Γk[ϕ]

δϕ(x)δϕ(y)+Rk(x, y) . (5.16)

Furthermore, we also have

δϕ(y)

δJ(x′)=

δ2Wk[J ]

δJ(x′)δJ(y)≡ Gk(y − x′) . (5.17)

From the previous two equations, we can now deduce the important equation

δ(x− x′) =δJ(x)

δJ(x′)=

∫y

δJ(x)

δϕ(y)

δϕ(y)

δJ(x′)(5.18)

=

∫y

(δ2Γk[ϕ]

δϕ(x)δϕ(y)+Rk(x, y)

)Gk(y − x′) (5.19)

=

∫y

(Γ

(2)k [ϕ] +Rk

)(x, y)Gk(y − x′) . (5.20)

In compact notation this reads:

1 = (Γ(2)k +Rk)Gk (5.21)

With all these preparations, we can now derive an expression for the scale derivative of theaverage effective action at fixed ϕ and at J = Jsup:

∂tΓk[ϕ] =︸︷︷︸(5.14)

−∂tWk[J ] + (∂tJ) · ϕ− ∂t∆Sk[ϕ] = −∂tWk[J ]− ∂t∆Sk[ϕ] (5.22)

=︸︷︷︸(5.13)

1

2

∫q∂tRk(q)Gk . (5.23)

Together with eq. (5.21), this gives the Wetterich equation

∂tΓk[ϕ] =1

2Tr

[(Γ

(2)k [ϕ] +Rk

)−1∂tRk

]. (5.24)

This equation will be the starting point for further investigations. Often this equation isalso referred to as Wetterich’s flow equation, functional RG flow equation or just the flowequation.

72

5.2. Wetterich equation

5.2.1 Some properties of the Wetterich equation

1. Eq. (5.24) is an exact equation and the propagator appearing in the trace operation onthe r.h.s. of the equation is the exact propagator. At this stage no approximations havebeen made.

2. Despite being exact, eq. (5.24) displays a one-loop structure as signaled by the traceoperation on the r.h.s.

3. Eq. (5.24) is a functional differential equation for the functional Γk. No functionalintegral has to be performed in contrast to eq. (4.4).

4. Eq. (5.24) was derived from a generating functional involving a functional integral. Asan alternative perspective, we may also define QFT based on eq. (5.24) without usingthe starting point of the functional integral.

5. The regulator term guarantees IR regularization by construction. Further, the derivativeof the regulator ∂tRk appears on the r.h.s. of the Wetterich equation. This also ensuresultraviolet regularization, cf. fig. 5.1. Moreover, the peaked structure of fig. 5.1 alsoimplements the Wilsonian idea of momentum-shell integration.

6. The solution of the Wetterich equation corresponds to a renormalization group trajec-tory in so-called theory space which is spanned by all possible symmetry-compatible fieldoperators in the space of action functionals. The RG trajectory interpolates betweenSbare and Γ = Γk→0.

7. The regulator term has to fulfill the basic conditions given in Sec. 5.1.1. Apart fromthese conditions, it can be chosen arbitrarily. The precise RG trajectory then dependson the choice of Rk and reflects the RG scheme dependence. The final point of the RGtrajectory, however, is independent of the regulator choice.

3.2. The Functional Renormalisation Group

Oi

O1

O2

O3k=

k=0

Figure 3.4.: The flow equation interpolates between the classical action at the renormal-isation group scale and the full quantum e↵ective action at k = 0. Theflow is a trajectory in the space of all possible theories which are spanned byorthogonal operators/couplings Oi, and serve as expansion coecients, e.g.O1 = (2)(p = 0). For di↵erent implementations of the regulation procedurethe flow varies at non-vanishing k, but the end-points of the flow are un-ambiguous. Furthermore, the trajectory depends on the chosen truncation,as may the quantum e↵ective action. By choosing satisfying truncations thelimit k → 0 must be rendered independent of the truncation.

Figure 3.5.: Graphical representation for the flow equation of a real scalar particle givenin eq. (3.15). The solid line with the shaded circle represents the full non-perturbative propagator of the scalar field. The crossed circle indicates theinsertion of the scale derivative of the regulator @tRk(p).

explicitly on higher n-point functions up to n + 2, which in turn can be derived from the

Wetterich equation eq. (3.15) by taking functional derivatives with respect to the fields

that are external to the particular n-point process. Note that herein all o↵-shell contri-

butions must be taken into account in the course of the derivation. Only after the last

functional derivative they can be dropped to get the physical contributions. However,

the dependence on higher n-point functions holds at arbitrary order. So accordingly, the

flow equation generates an infinite tower of coupled di↵ertial equations. Thus, for most

practical applications truncations are inevitable, which means that the set of contributing

44

8. The Wetterich equation can be used to derive perturbation theory: The loop-expansionin perturbation theory expands the action in powers of ~: Γk = S + ~Γ1−loop

k +O(~2).

Therefore, to one-loop order, we can use Γ(2)k = S(2) on the r.h.s. of the Wetterich

equation. This gives:

∂tΓ1−loopk =

1

2Tr log(S(2) +Rk) (5.25)

⇒ Γ1−loopk = S +

1

2Tr logS(2) + const. (5.26)

73

5. Functional renormalization group

5.2.2 Method of truncations

The functional RG provides an exact formulation of the renormalization group. For practicalcalculations, however, approximations for the action functionals have to be made, i.e. theaction functional that are plugged into eq. (5.24) have to be truncated.

Vertex expansion

One such truncation or approximation scheme is the vertex expansion as given before:

Γk[ϕ] =

∞∑n=1

1

n!

∫x1,...,xn

Γ(n)k (x1, ..., xn)ϕ(x1)...ϕ(xn) (5.27)

Inserting this expansion into the Wetterich equation provides RG flow equations for the vertex

functions Γ(n)k .

Derivative expansion

Another important example is the derivative expansion which is often used for the descriptionof scalar field theories:

Γk[ϕ] =

∫x

(1

2Zk(ϕ)(∂µϕ)2 +O(∂4) + Uk(ϕ)

)(5.28)

where Uk(ϕ) is the effective potential which again can be expanded in field monomials, e.g.,Uk = 1

2m2kϕ

2 + 14!λkϕ

4 + ... .

74

5.2. Wetterich equation

5.2.3 Example: Local potential approximation and for a scalar field

In this section, we study the local potential approximation (LPA) for scalar field theories.The LPA is a reasonable ansatz/truncation when the momentum dependence of the vertexfunctions is not important and the anomalous dimension is small. This is the case for thecritical behavior of O(N) symmetric scalar models.

The LPA for the Wetterich equation, eq. (5.24), is given by the following ansatz for the flowingeffective action

Γ[ϕ] =

∫~x

1

2[~∇ϕ]2 + Uk(ϕ) . (5.29)

• The only cutoff dependence is in the effective potential Uk(ϕ).

• Uk(ϕ) is an arbitrary function of the real field ϕ = ~ϕ = (ϕ1, ϕ2, ..., ϕN ).

• For a discussion of the ground state and, e.g., its preserved or spontaneously brokensymmetries the effective potential Uk(ϕ) is the most important quantity. It correspondsto the discussion of the free energy in Sec. 2.6.2.

• We consider a (d − 1)-dimensional system at T = 0 (or equivalently a d-dimensionalclassical system)

⇒ ϕ depends on the d dimensional vector ~x.

• We assume that the theory shall be an O(N) invariant scalar theory, i.e. the action isinvariant under the symmetry ϕa → Rabϕb with any N ×N orthogonal matrix R

⇒ effective potential only depends on inner product ρ = 12

∑Na=1 ϕaϕa = 1

2ϕ · ϕ.

We will use the following cutoff insertion

∆Sk[ϕ] =1

2

∫ddq

(2π)dϕa(−q)Rk(q)ϕa(q) , (5.30)

where summation over the index a is implied and with regulator function

Rk(p) = (k2 − p2)Θ(k2 − p2) . (5.31)

Θ(x) is the step function.

We can now insert the ansatz eq. (5.29) into the Wetterich equation, eq. (5.24). In a firststep, we obtain

∂tUk[ϕ] =1

2

∫ddq

(2π)dTr

[(q2δab +

δ2Uk(ϕ)

δϕaδϕb+Rk(q)δab

)−1

∂tRk(q)

](5.32)

where the Tr operation goes over internal indices of the field a, b.

We now use that the effective potential only depends on the field invariant ρ = 12ϕ · ϕ, i.e.

Uk(ϕ) = Uk(ρ). To that end, we rewrite the derivatives of Uk with respect to the fields ϕ asderivatives w.r.t. ρ

δUk(ρ)

δϕa= U ′k(ρ)ϕa ⇒ δ2Uk(ρ)

δϕaδϕb= U ′′k (ρ)ϕaϕb + U ′k(ρ)δab (5.33)

75

5. Functional renormalization group

• To study the ground state properties of the theory, we now use that a constant fieldconfiguration yields a minimum for the effective potential, i.e. ϕa(x) = const., cf.Sec. 2.6.2.

• Just as in section 2.6.2, the value of an O(N) symmetric potential Uk(ϕa) = −µρ +λρ2 + ... will only depend on the length of the vector ~ϕ0, the direction is arbitrary.

⇒ we can choose the coordinates so that ϕa,0 points in the first direction:

~ϕ0 = (v, 0, ..., 0) with v =√

2ρ (5.34)

Inserting all of this into eq. (5.32), we obtain

∂tUk(ρ) =1

2

∫ddq

(2π)d∂tRk(q)Tr

[(q2δab +Rk(q)δab + U ′k(ρ)δab + 2ρU ′′k (ρ)δa1

)−1]

(5.35)

=1

2

∫ddq

(2π)d∂tRk(q)

(1

q2 +Rk(q) + U ′k(ρ) + 2ρU ′′k (ρ)+

N − 1

q2 +Rk(q) + U ′k(ρ)

)(5.36)

Inserting now the specific form of the regulator function, we can simplify this expression

∂tUk(ρ) = kd+2Sd

(1

k2 + U ′k(ρ) + 2ρU ′′k (ρ)+

N − 1

k2 + U ′k(ρ)

), (5.37)

with Sd = πd/2/[(2π)dΓ(d/2 + 1)] (tutorial).

• The flow equation eq. (5.37) exhibits that the flow of the effective potential is driven bytwo essential contributions, i.e. (1) the contribution from a radial mode ∼ k2 +U ′k(ρ) +2ρU ′′k (ρ) and (2) the contribution from N−1 Goldstone modes ∼ k2 +U ′k(ρ) as expectedin a scenario with a spontaneously broken continuous symmetry.

• For vanishing external sources, the Goldstone modes are massless.

• A simple extension of this approximation scheme consists in multiplying a uniform wavefunction renormalization Zk to the kinetic term in the action ∝ q2ϕ(−q)ϕ(q). Then theanomalous dimension can be given in terms of the flow equation η = −∂t logZk.

• The set of flow equations for Uk and Zk is already well-suited to describe many propertiesof scalar field theories in 2 ≤ d ≤ 4 (and not only close to d = 4), e.g.:

– critical properties of scalar field theories in d = 3, see sec. 5.3.1!

– (multi-)critical properties of scalar field theories in d < 3

– aspects of the Kosterlitz-Thouless transition for d = 2, N = 2

– non-linear sigma model

– Mermin-Wagner-Hohenberg theorem in d = 2

– ... (for further reading, see e.g., https://arxiv.org/pdf/hep-ph/0005122.pdf)

76

5.3. Functional RG fixed points and the stability matrix

5.3 Functional RG fixed points and the stability matrix

Quite generally, we can expand the action functional for a quantum or statistical field theoryas before in terms of

• running couplings gi,k and field operators Oi:

Γk[ϕ] =∑i

gi,kOi[ϕ], e.g. Oi[ϕ] =ϕ2, ϕ4, (∂ϕ)2, . . .

. (5.38)

In the following, we will be interested in fixed-point solutions, as these allow us to evaluate thescaling behavior of physical quantities, for example, close to a second-order phase transition.

To that end we study the RG flow of the

• dimensionless couplings:

gi,k = gi,kk−dgi , (5.39)

where dgi is the canonical dimensionality of the coupling.

Note that the canonical dimension of the coupling is related to the canonical dimension ofthe vertex by the simple relation:

dgi = −δ(Vgi) . (5.40)

The reason for choosing dimensionless couplings is that a fixed point, which is a scale-freepoint, is hard to identify when dimensionful couplings are present, since each dimensionfulcoupling corresponds to a scale.

The RG flow of the couplings is given in terms of

• β functions:

βgi(gn) = ∂tgi,k = −dgigi,k + fi(gn), (5.41)

where fi(gn) are functions of the dimensionless couplings.

The first term reflects the scale-dependence due to the canonical dimensionality, whereas thesecond term carries the quantum/statistical corrections to the scaling.

Fixed point

A fixed point is defined by a set of values for the dimensionless couplings gn ∗ where the allthe beta functions vanish simultaneously:

βgi(gn ∗) = 0 ∀ i . (5.42)

77

5. Functional renormalization group

Linearized flow and stability matrix

To determine the critical exponents, which enter the scaling of observable quantities in thevicinity of a second-order phase transition, we linearize the flow around the fixed point gn ∗.We find for the linearized flow

∂tgi,k = βgi = Bij(gj,k − g∗j ) +O

((g − g∗)2

), (5.43)

where we have introduced the stability matrix

Bij =

∂βi∂gj

∣∣∣g∗. (5.44)

We diagonalize the stability matrix Bij , introducing the negative eigenvalues θI

Bij V I

j = −θIV Ii . (5.45)

Here, the V Ii are the eigenvectors enumerated by the index I which labels the order of the θI

according to their real part, starting with the largest one, θ1.

Solution of linearized flow

The RG eigenvalues allow for a classification of physical parameters by analyzing the solutionof the coupling flow in the fixed-point. The solution to this linearized equation is given by

gi,k = gi ∗ +∑I

CIVIi

(k

k0

)−θI, (5.46)

Herein, CI is a constant of integration and k0 is a reference scale.

• Eigendirections with ReθI > 0 are called relevant directions and drive the systemaway from the fixed point as we evolve the flow towards the infrared. Those infraredunstable directions determine the macroscopic physics.

• Eigendirections with ReθI < 0 die out and flow into the fixed point towards theinfrared. They are thus called the irrelevant (infrared stable) directions.

• For the marginal directions ReθI = 0, it depends on the higher-order terms in theexpansion about the fixed point.

For physics which takes place in the vicinity of the fixed point, we then have:

The number of relevant and marginally-relevant directions determines the number of physicalparameters to be fixed.

For the flow away from the fixed point, the linearized fixed-point flow (5.43) generally isinsufficient and the full nonlinear β functions have to be taken into account.

78

5.3. Functional RG fixed points and the stability matrix

5.3.1 Example: critical exponents for scalar field theories

Flow of dimensionless potential

To study the fixed point properties of O(N) models, we can use the flow equation for theeffective potential, which we derived previously, eq. (5.37). For the fixed-point analysis, weintroduce dimensionless quantities, i.e. we rescale the potential and the field with appropriatepowers of the cutoff k, i.e. ρ = kd−2ρ and uk(ρ) = k−dUk(ρ).

Then, we obtain from eq. (5.37),

∂tuk(ρ) = −duk(ρ)− (2− d)uk(ρ)′ρ+ Sd

(1

1 + uk(ρ)′ + 2uk(ρ)′′ρ+

N − 1

1 + uk(ρ)′

). (5.47)

Quartic potential and beta functions in d = 3

To evaluate this equation further, we expand the dimensionless effective potential uk in termsof a running non-vanishing minimum κk and a four-boson interaction λk:

uk =λk2

(ρ− κk)2 . (5.48)

Projection prescription for beta functions of the couplings

Instead of the flow of the function uk, we would like to have beta functions for the individualcouplings, i.e. in this case κk, λk. Therefore, we can use projection prescriptions, for example,the coupling λk can be obtained from the potential uk by the following projection:

λk =∂2uk(ρ)

∂ρ2

∣∣∣ρ=κk

. (5.49)

Accordingly, we obtain the flow equation for the quartic coupling from the prescription:

∂tλk =∂2(∂tuk(ρ))

∂ρ2

∣∣∣ρ=κk

. (5.50)

For the flow of κk, we use the fact that the first derivative of uk by definition of the potentialvanishes at the minimum, i.e. u′k(κk) = 0. This implies:

0 = ∂tu′k(κk) = ∂tu

′k(ρ)|ρ=κk + (∂tκk)u

′′(κk) (5.51)

⇒ ∂tκk = − 1

u′′(κk)∂tu′k(ρ)|ρ=κk . (5.52)

This procedure yields beta functions/flow equations for κk and λk in d = 3:

βκ = ∂tκk = −κk +1

6π2

((N − 1) +

3

(1 + 2κkλk)2

), (5.53)

βλ = ∂tλk = −λk +λ2k

3π2

((N − 1) +

9

(1 + 2κkλk)3

). (5.54)

79

5. Functional renormalization group

Non-Gaußian fixed points for N = 2

A numerical search for fixed points of the above beta functions βκ = βλ = 0, e.g., for thechoice N = 2 yields a solution:

κ∗ = 0.0337737... and λ∗ = 10.8375778... . (5.55)

Stability matrix and critical exponents

The matrix of first derivatives reads

B =

− 2λπ2(2κλ+1)3 − 1 − 2κ

π2(2κλ+1)3

− 18λ3

π2(2κλ+1)4

2λ(

9−9κλ

(2κλ+1)4+1)

3π2 − 1

. (5.56)

Evaluation at the fixed point values, provides the stability matrix B which the has negativeeigenvalues:

θ1 = 1.611, θ2 = −0.3842 . (5.57)

So, we have one relevant (θ1 > 0) and one irrelevant direction (θ2 < 0) at this non-Gaußianfixed point. This corresponds the Wilson-Fisher fixed point which we have studied withWilson’s RG approach.

The eigenvalue θ1 is related to the scaling of the field bilinear at the Wilson-Fisher fixedpoint: θ1 = y, cf. (3.35). Therefore, the functional RG prediction for the correlation lengthexponent at that level of truncation (LPA4) is:

νLPA4 =1

θ1≈ 0.62 . (5.58)

Higher orders in the potential and anomalous dimension

This estimate can easily be improved with the functional RG approach, by allowing for amore general expansion of the effective potential, i.e.:

uk =

nmax∑i=2

λi,ki!

(ρ− κk)i . (5.59)

and by amending the equation for the anomalous dimension η = −∂t logZk. For examplewith nmax = 6 and including η 6= 0, we find the following results:

Application in d = 3 νFRG νZJ ηFRG ηZJ

N = 0 Polymer chains 0.59 0.5882(11) 0.040 0.0284(25)N = 1 Liquid-vapor transition 0.64 0.6304(13) 0.044 0.0335(25)N = 2 Helium superfluid transition 0.68 0.6703(15) 0.044 0.0354(25)N = 3 Ferromagnetic phase transtion 0.73 0.7073(35) 0.041 0.0355(25)N = 4 0.77 0.741(6) 0.037 0.035(4)N = 10 0.89 0.859 0.021 0.024

Table 5.1: Critical exponents for O(N)-models in three dimensions.

Here, we compare our results to high accuracy computations by J. Zinn-Justin et al. us-ing resummed perturbation expansions at five-loop order. Given the simplicity of the FRGapproximation these results show a very reasonable agreement.

80

5.4. Ultraviolet completion of quantum field theories

5.4 Ultraviolet completion of quantum field theories

5.4.1 Preliminaries

Wilson’s RG construction as well as the functional RG generally work in the context of(euclidean) quantum and statistical field theories, as they are just a strategy to evaluate thefunctional integral.

→ Here, we will discuss its implications in the context of high-energy physics.

⇒ We consider the action and the functional integral:

S =

∫ddx

[1

2(∂µφ)2 +

1

2m2φ2 +

λ

4!φ4

](5.60)

Z =

∫ΛDφ exp(−S[φ]) (5.61)

with UV cutoff Λ.

• Momentum-shell transformation/integration ≡ transformation in space of couplings

• RG transformation ⇒ flow in the space of couplings!

RG transformations close to the Gaußian fixed point:

S∗G =

∫ddx

1

2(∂µφ)2 , (5.62)

m2∗|G = λ∗|G = λ∗6|G = 0 (5.63)

• In the Gaußian fixed point regime: neglect higher-order corrections to rescaling of cou-plings (a la Wilson):

m(b)2 = m2b2, λ(b) = λ b4−d, λ6(b) = λ6b6−2d , ... (5.64)

• Or a la FRG, cf. eq. (5.46):

m2k ∝ k−2, λk ∝ kd−4, λ6,k ∝ k2d−6 , ... (5.65)

• Generally:

g(b) = bθ g , b > 1 , or (FRG) : gk ∝ k−θ . (5.66)

with the classification:

– relevant if θ > 0 (driven away from Gaußian FP under RG trafos (k → 0))

– irrelevant if θ < 0 (vanish under RG trafos = IR stable)

– marginal if θ = 0 (higher orders determine stability)

81

5. Functional renormalization group

• Generally, the coefficient gl,ni of an operator with l derivatives and ni powers of the fieldφi (with canonical dimension [φi]) has the following behavior under RG transformationsclose to the Gaußian FP:

gl,ni(b) = b−(l+∑i ni[φi]−d)gl,ni = b−δ(V )gk,ni (5.67)

or gk;l,ni ∝ kl+∑i ni[φi]−d = kδ(V ) (5.68)

where δ(V ) is the canonical dimension of the vertex, cf. eq. (4.88).

⇒ The classification relevant, irrelevant, marginal corresponds to the classificationsuper-renormalizable, non-renormalizable, renormalizable.

Consequence in d = 4:

It is found in particle physics that all theories that play a role are perturbatively renormaliz-able theories:

→ Standard Model of Particle Physics!

Until now, it seemed to be a mere coincidence that all the (fundamental) particle physicstheories that appear in nature are (perturbatively) renormalizble.

Wilson’s interpretation:

• Suppose there is a more fundamental theory that has the symmetries of the StandardModel of Particle Physics (SU(3) × SU(2) × U(1)) at a ultraviolet cutoff scale Λ andhere it can be described by a quantum field theory.

• Generically, this QFT will have an action with all the allowed ∞-many couplings

• If the coupling coefficients are not too large, i.e. they are close to the Gaußian FP

⇒ infrared physics will be dominated solely by the relevant and the marginal couplings

⇒ this corresponds to a renormalizable QFT !

Example:

Potential of the φ4 sector of the standard model:

V (φ) =1

2m2φ2 +

λ

4!φ4 +

λ6

6!φ6 + ... (5.69)

with λi = O(1).

⇒ RG:

λk,i≥6 ∝ ki(d/2−1)−dd=4︷︸︸︷= ki−4 → 0 (for k → 0) . (5.70)

Accordingly, λk decreases slowly (logarithmically) and m2k generally grows large quickly (fine-

tuning/hierarchy problem) for k → 0.

82

5.4. Ultraviolet completion of quantum field theories

5.4.2 Asymptotic safety scenario (Weinberg 1976, Gell-Mann-Low 1954)

• Perturbatively renormalizable theories automatically arise in the infrared (IR), if theQFT system is dominated by the Gaußian fixed point.

• What happens if the system is dominated by a non-Gaußian (i.e. interacting) fixedpoint?

• Can theories be renormalizable if perturbation theory fails?

Why do we need renormalizability?

• IR physics should be separated from UV (cutoff Λ independent).

• number of physical parameters <∞, i.e. QFT should be predictive.

Assumption:

Suppose there is a (non-Gaußian/interacting) ultraviolet fixed point Γ∗ in the theory spaceof action functionals.

→ we can express fixed-point action in terms of suitable expansion of field operators Oi:

Γ∗[ϕ] =∑i

g∗iOi[ϕ], e.g. Oi[ϕ] =ϕ2, ϕ4, (∂ϕ)2, . . .

. (5.71)

• the corresponding beta functions of the dimensionless couplings gi all vanish:

βgi(gn ∗) = 0 . (5.72)

and at least one coupling g∗i 6= 0 (to make it non-Gaußian).

• Now, if we can find RG trajectory connecting Γ∗ with a meaningful physical theory ΓIR

at some infrared scale.

⇒ We have found a QFT, which can be extended to arbitrarily high scales, since forΛ→∞ we just run into the fixed point.

• Such a theory is called asymptotically safe, as it is free from pathological divergencies.

83

5. Functional renormalization group

Ultraviolet fixed-point behavior

• We already discussed the critical behaviour in the vicinity of a fixed point, however,with a perspective where we evolve the RG scale k towards the infrared.

• For the concept of asymptotic safety, where the fixed-point is supposed to govern thetheory in the ultraviolet we invert the discussion.

• For convenience we display again eq. (5.46) for linearized flow in fixed-point regime

gi = g∗i +∑I

CI V Ii

(k0

k

)θI.

• This gives us modified classification scheme when following flow towards higher scales:

– θI < 0: ultraviolet repulsive (RG irrelevant)

– θI > 0: ultraviolet attractive (RG relevant)

– θI = 0: ... higher orders ... (RG marginal)

Construction of asymptotic safety

• The directions with ReθI < 0 run away from the fixed point as we increase k (greenarrows in plot below).

• The directions with ReθI > 0 run into from the fixed point as we increase k (bluearrows in plot below).

• We define the critical hypersurface S:

S is the set of all points in theory space that run towards the fixed point as k →∞,i.e. the points lying on RG relevant trajectories.

84

5.4. Ultraviolet completion of quantum field theories

• Consequently, as we decrease k, irrelevant directions rapidly approach critical surface:

⇒ Observables in the IR are all dominated by the properties of the fixed point,independently of whether the flow has started exactly on or near the critical surface.

• The tangential space of S is spanned by the relevant RG eigenvectors V Ii .

• The number of linearly independent relevant directions at the fixed point correspondsto the dimension of S:

dimS ≡ ∆ = number of relevant directions with θI > 0 . (5.73)

• Renormalization: Set all CI = 0 for all irrelevant directions (θI < 0)

⇒ Limit k → Λ→∞ exists and the system runs into the FP

⇒ Everything is finite!

⇒ Cutoff independence

• How many physical parameters are there?

– (number of physical parameters) = (number of non-trivial initial conditions) = ∆= dimS <∞ if ∆ <∞

– This establishes the predictivity within the asymptotic safety scenario!

85

5. Functional renormalization group

5.4.3 Candidate theories for asymptotic safety

Gravity

Einstein’s theory of gravity is perturbatively non-renormalizable.

→ suggested from the dimensional/canonical analysis of the Einstein Hilbert action:

S =1

16πGN

∫d4x√−det(gµν) (R− 2λ) (5.74)

where

• gµν is the metric tensor

• we work in 4-dimensional spacetime

• R is the Ricci scalar

• λ is the cosmological constant

• GN is the gravitational constant

• we have set the speed of light c = 1

Variation of Einstein-Hilbert action w.r.t. metric field yields Einstein’s field equations.

Perturbative non-renormalizability of gravity

A simple way to see that the theory of gravity is perturbatively non-renormalizable withoutgoing into the details of the general theory of relativity is given as follows:

• We first recall Coulomb’s law, reading

V (r) = α1

r, (5.75)

where α = e2

4π is the fine-structure constant which is a dimensionless quantity α ≈ 1137 .

(Also e is dimensionless coupling, stating that QED is perturbatively renormalizable.)

• Now, look at gravity: In the non-relativistic limit and for low gravitational fields Ein-stein’s gravity is consistent with Newton’s gravity.

• In Newton’s gravity, we have Newton’s law, reading:

V (r) = GNM1M2

r. (5.76)

• Comparison with to Coulombs’s law, this implies that Newton’s gravitational constantGN has canonical dimension −2 (in units of mass or momentum).1

• Therefore, the theory is perturbatively non-renormalizable!

1Recall that if vertex has dimension δ[V ] = ∆, then the corresponding coupling g has dimension δ[g] = −∆.

86

5.4. Ultraviolet completion of quantum field theories

Asymptotically safe gravity

The above analysis is valid a the Gaußian fixed point, i.e. when an expansion about G∗N = 0is justified. The question is now, whether the Einstein-Hilbert action admits a non-Gaußianfixed point with a finite number of relevant parameters in the sense of the asymptotic safetyscenario. Evidence for this scenario has been collected with the FRG methods with increasingefforts in the last 15 years, see for example

→ see https://arxiv.org/pdf/1205.5431.pdf for lecture notes.11

−0.2 −0.1 0.1 0.2 0.3 0.4 0.5

−0.75

−0.5

−0.25

0.25

0.5

0.75

1

λ

g

Type IIIaType Ia

Type IIa

Type Ib

Type IIIb

Fig. 2 RG flow in the g-λ−plane. The arrows point in the direction of increasing coarse graining,i.e., of decreasing k. (From [11].)

In [11] this system has been analyzed in detail, using both analytical and numer-ical methods. In particular all RG trajectories have been classified, and exampleshave been computed numerically. The most important classes of trajectories in thephase portrait on the g-λ−plane are shown in Fig. 2.

The RG flow is found to be dominated by two fixed points (g∗,λ ∗): the GFPat g∗ = λ ∗ = 0, and a NGFP with g∗ > 0 and λ ∗ > 0. There are three classes oftrajectories emanating from the NGFP: trajectories of Type Ia and IIIa run towardsnegative and positive cosmological constants, respectively, and the single trajectoryof Type IIa (“separatrix”) hits the GFP for k → 0. The high momentum propertiesof QEG are governed by the NGFP; for k → ∞, in Fig. 2 all RG trajectories on thehalf–plane g> 0 run into this point. The two critical exponents are a complex con-jugate pair θ1,2 = θ ′ ± iθ ′′ with θ ′ > 0. The fact that at the NGFP the dimensionlesscoupling constants gk,λk approach constant, non-zero values then implies that thedimensionful quantities run according to

Gk = g∗k2−d , λk = λ ∗ k2 . (21)

Hence for k → ∞ and d > 2 the dimensionful Newton constant vanishes while thecosmological constant diverges.

So, the Einstein-Hilbert truncation does indeed predict the existence of a NGFPwith exactly the properties needed for the Asymptotic Safety construction. Clearlythe crucial question is whether the NGFP found is the projection of an exact fixedpoint in the full theory or merely the artifact of an insufficient approximation. Thisquestion has been analyzed during the past decade within truncations of ever in-creasing complexity. All investigations performed to date support the existence of a

Figure 5.2: Figure taken from https://arxiv.org/pdf/hep-th/0110054.pdf

Gross-Neveu models in 2 < d < 4

Gross-Neveu models are models of Dirac fermions with four-fermion interactions, e.g.:

SGN =

∫ddx

(ψ /∂ψ + λψ(ψψ)2

). (5.77)

These models are perturbatively non-renormalizable as the coupling carries a negative massdimension. This conclusion, however, is only an artifact of the perturbative quantizationprocedure. In fact they are non-perturbatively renormalizable in 2 < d < 4 at a non-Gaußianfixed point and hence can be extended to arbitrarily high scales, see, e.g.:

https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.55.363.

87

Appendix A

Remark on generating functional

A.1 Connected and disconnected Green’s functions

What does Z[J ] generate? – Extended answer.

• Example: Φ4 theory in 4 dimensions

S[Φ] = S0[Φ] + SI [Φ] =

∫d4x

(1

2Φ(−∂2 +m2)Φ +

1

4λΦ4

)(A.1)

– where Φ is a real scalar field

– and ∂2 = ∂µ∂µ and instead of −µ we now use m2 as parameter for the field bilinear

– and S0[Φ] =∫d4x

(12Φ(−∂2 +m2)Φ

)and SI [Φ] =

∫d4x

(14λΦ4

)• with eq. (4.6), we then obtain:

Z[J ] =

∫DΦe−S[Φ]+Φ·J (A.2)

= e−λ

4

∫d4x(

δδJ(x)

)4

Z0[J ] (A.3)

with

Z0[J ] :=

∫DΦe−S0[Φ]+Φ·J (A.4)

• we rewrite Z0[J ]

Z0[J ] =

∫DΦe−

12

∫ddxddyΦ(x)K(x,y)Φ(y)+

∫ddxJ(x)Φ(x) (A.5)

• where K(x, y) = (−∂2 +m2)δ(d)(x− y)

• and the inverse of K: ∫ddz G(x, z)K(z, y) = δ(d)(x− y) (A.6)

89

A. Remark on generating functional

• now, parametrize

Φ(x) = Φ(x) +

∫ddz G(x, z)J(z) (A.7)

⇒ Z0[J ] =

∫DΦe−

12

∫ddxddyΦ(x)K(x,y)Φ(y)e

12

∫ddxddy J(x)G(x,y)J(y) (A.8)

=

∫DΦe−S0[Φ]e

12

∫ddxddy J(x)G(x,y)J(y) (A.9)

• here, we have used DΦ(x) = DΦ(x) ⇐ translational invariance of DΦ (measure)

• so, we obtain for the generating functional

Z[J ] = e−SI [δδJ ]∫DΦeS0[Φ]e

12

∫ddxddy J(x)G(x,y)J(y) (A.10)

Z[J ] =1

Ne−SI [

δδJ ]e

12

∫ddxddy J(x)G(x,y)J(y) (A.11)

• where N = det1/2(−∂2 +m2)/√

2π∞

from Gaußian integration (cf. problem set 3).

• schematically Z[J ] can be represented by the following picture:

• derivatives of Z[J ] with respect to the source fields J

→ δnJZ[J ] generate the connected + disconnected Green’s functions

90

A.2. Example: two-point function

A.2 Example: two-point function

• for example the 2-point function is generated by

δ2Z[J ]

δJ(x)δJ(y)= e−SI [

δδJ ] δ2

δJ(x)δJ(y)e

12J ·G·J (A.12)

• with

J ·G · J =

∫ddxddy J(x)G(x, y)J(y) = J J (A.13)

• where the second equation provides a pictorical representation of the integral over thetwo sources J with propagator G.

• this yields

δ2Z[J ]

δJ(x)δJ(y)= e−SI [

δδJ ] G(x, y) + (G · J)(x)(G · J)(y) e 1

2J ·G·J (A.14)

with

(G · J)(x) =

∫ddyG(x, y)J(y) = (J ·G)(x) (A.15)

as G(x, y) = G(y, x). CHECK THE FOLLOWING EQS.

⇒ δ2Z[J ]

δJ(x)δJ(y)=[G(x, y)− λ

2

∫ddzG(x, z)G(y, z)

(δ

δJ(z)

)2

− λ

2

∫ddz(G · J)(x)G(y, z)

(δ

δJ(z)

)3

+7

4λ2

∫ddzddz′G(x, z)G(y, z′)

(δ

δJ(z)

)3( δ

δJ(z′)

)3 ]Z[J ] (A.16)

⇒ redundant information!

91

Appendix B

Divergencies beyond 1-loop

In this chapter, we analyse the appearing divergencies beyond 1-loop, dicuss the Callan-Symanzik equations and give an idea of the inductive proof of perturbative renormalizability.

93

B. Divergencies beyond 1-loop

B.1 Recap: Renormalized action and 1-loop divergencies

94

B.1. Recap: Renormalized action and 1-loop divergencies

95

B. Divergencies beyond 1-loop

B.2 Divergencies beyond 1-loop

96

B.3. Callan-Symanzik equations

B.3 Callan-Symanzik equations

97

B. Divergencies beyond 1-loop

98

B.4. Inductive proof of perturbative renormalizability (idea)

B.4 Inductive proof of perturbative renormalizability (idea)

99