Introduction to Renormalization

Home , Correlation function, Effective action, Landau pole, Length scale, Quantum field theory, Quartic interaction

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

January 26, 2021

Contents

Contents 1

1 Introduction5 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...... 39 3.4 Epsilon expansion and Wilson-Fisher ﬁxed point ...... 41 3.5 Summary and comparison to experiment...... 44 3.6 Excursion: Relations to QFTs in high-energy physics...... 45 3.6.1 Landau-pole singularity and the triviality problem...... 47 3.6.2 Fine-tuning and the hierarchy problem...... 49

4 Functional methods in quantum field theory 51 4.1 Introductory remarks...... 51 4.2 Generating functional ...... 52 4.2.1 Correlation functions...... 52 4.2.2 Schwinger functional...... 54 4.3 Effective action...... 54 4.4 Perturbative renormalization ...... 57 4.4.1 Divergencies in perturbation theory ...... 57 4.4.2 Removal of divergencies...... 60 4.4.3 Divergencies in perturbation theory: general analysis...... 63 4.4.4 Classification of perturbatively renormalizable theories...... 66

5 Functional renormalization group 69 5.1 Effective average action ...... 69 5.1.1 The regulator term...... 70 5.2 Wetterich equation...... 71 5.2.1 Some properties of the Wetterich equation...... 73 5.2.2 Method of truncations...... 74 5.2.3 Example: Local potential approximation and for a scalar field . . . . . 75 5.3 Functional RG fixed points and the stability matrix ...... 77 5.3.1 Example: critical exponents for scalar field theories...... 79 5.4 Ultraviolet completion of quantum field theories...... 81 5.4.1 Preliminaries...... 81 5.4.2 Asymptotic safety scenario (Weinberg 1976, Gell-Mann-Low 1954) ...... 83 5.4.3 Candidate theories for asymptotic safety...... 86

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

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

2 Recommended literature

φ−1

I. Herbut, A Modern Approach to Critical Phenomena, Cambridge University Press. No-frills introduction to critical phenomena and basics of the renormalization group `ala Wilson. Excellent ﬁrst read to become acquainted with the physics and concepts. The start of this lecture follows this presentation, i.e. my chapters on the Ginzburg-Landau-Wilson theory and Wilson’s RG.

J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, Oxford University Press (2nd edition). 1000+ pages of quantum ﬁeld theory and statistical mechanics for enthusiasts with a lot of details, background and also explicit calculations. Also contains also some more formal and mathematical aspects. Can also be considered as a general reference work. Does not contain functional renormalization. My chapter on the functional integral approach 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 and 2 give a smooth and straightforward introduction to the functional RG formalism and a very nice instructive example for its application (the euclidean anharmonic oscillator). Inspired parts 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 canonical introduction to relativistic QFT in part I, the second part deals with functional integrals. Also contains a pretty nice chapter on critical phenomena which provides a complementary point of view. No functional RG.

Further useful reads

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

• P. Kopietz, L. Bartosch, F. Sch¨utz,Introduction to the Functional Renormalization Group, Lecture Notes in Physics 798.

3 Contents

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

4 Chapter 1

Introduction

Quantum ﬁeld theory provides a framework for the description of all fundamental interactions (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/inﬁnite number of coupled degrees of freedom.

In quantum field theory and statistical mechanics renormalization is required to treat infinities 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 masses and fields appearing in the Lagrangian is needed.

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

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

The renormalization group refers to a mathematical procedure that facilitates the systematic study of the changes of a physical system when viewed at diﬀerent length or energy scales.

Upon changing the scale, it might turn out that a particular description of a system in terms of a chosen Lagrangian becomes inappropriate and new degrees of freedom arise. This can also 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-defconfinement transition,...) – 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 liquid-gas (1st order) theory space 1.2. Phase transitions S + Renormalization-group methods are relevant to a large diversity of fields many (apparently) different implementations ⇒ orderparameter sometimes hard to access ⇒ Functional RG providestemperature unified formulation. → h m > 0 1.2 Phase transitions m ≠ 0 m = 0 T • Sketchy phase diagram of water: FM Tc PM m < 0

solid phase liquid phase pressure pc critical point

vapour

temperature Tc

• Deﬁnition:

A phase transition is a point in parameter space (T, p, µ, h, ...) where the thermodynamic 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 is not accompanied by latent heat.

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

• deﬁne physical quantity to distinguish phases: order parameter

7 1. Introduction

• examples for order parameters: – average density (liquid-gas) liquid-gas (1st order)

– resistance (liquid-solid)

– magnetization (Fe, AFM: La2Cu2) – condensate (BEC: Li, Rb) orderparameter – superﬂuid density 4 3 (Al, Pb, YBa2Cu3O6.97, He, He) temperature

• 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 phase transition.

• deﬁnition:

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

• 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 interactions 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, consider the condensation line, i.e. the ﬁrst order transition line, where the gas coexists with the liquid, near the critical point in the density-temperature plane (coexistence region below the curve).

– for Ne, A, Kr, Xe, N2,O2,CO, CH4

– curves are identical when rescaled with ρc,Tc – ∆ρ (T T )β with β 0.33 ∼ − c ≈ – in some paramagnetic-ferromagnetic transitions m (T T )β with β 0.33 (MnF ) ∼ c − ≈ 2 • speciﬁc heat and susceptibility also follow simple power laws

• power laws: C t −α and χ t γ with the reduced temperature: t = (T T )/T V ∼ | | ∼ | | − c c – α, β, γ: critical exponents – critical exponents can be non-rational, e.g., β 0.326419(3) (best theoretical ≈ estimate) – 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 in ∼ d-dimensional space – Euclidean QFT in (d + 1)-dimensional spacetime, 0 τ β quantum statistical ≤ ≤ ∼ mechanics in d-dimensional space liquid-gas (1st order) – Euclidean QFT in d-dimensional spacetime High-temperature quantum statis- ∼ tical mechanics in d-dimensional space

• observation: perturbatively calculated correlation functions φ(x )...φ(x ) are gener- h 1 n i ally divergent, i.e. na¨ıvely they are not well-deﬁned. orderparameter

4 • example: φ theory in four dimensions temperature

– action: S[φ] = R d4x 1 φ(x)( ∂2 + m2)φ(x) + λ φ(x)4 2 − µ 4 – one-loop two-point function gives contribution to mass term m2: ∼ Z Z 4 4 ipx d q 1 φ(p)φ( p) 1−loop = d x e φ(x)φ(x) λ 4 2 2 h − i| h i ∼ R4 (2π) q + m ∼

9 1. Introduction

– introduce high-momentum cutoﬀ Λ:

Z 4 Z 4 d q 1 d q 1 Ω4 2 2 2 2 λ 4 2 2 λ 4 2 2 = Λ Λ m log(1 + Λ /m ) R4 (2π) q + m −→ q2≤Λ2 (2π) q + m 2 −

– cutoﬀ regularization: quantitative control of divergence

• in summary: – tree level: φ(p)φ( p) m2 h − i|p=0 ∼ 2 2 2 2 2 – with 1-loop correction: m + λ(c1Λ + c2m log(1 + Λ /m )) – 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(x )...φ(x ) is Λ- ⇒ h 1 n i 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 of particle physics) renormalizable within perturbative QFT?

1.5 Renormalization-group based deﬁnition of QFTs

• observation: – systematic analysis of divergencies of quantum ﬁeld theories is diﬃcult beyond perturbation 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 ⇒ liquid-gas (1st order) theory space – RG flow of average effective action Γk – flow equation describes continuous trajectory S connecting the microscopic action and the full (quantum) effective action:

orderparameter S = Γ Γ = Γ k=Λ −→ k=0 (see also QFT II)temperature

10 1.6. QFTs in the high-energy limit

1.6 QFTs in the high-energy limit

• for perturbatively renormalizable QFTs, the high-energy conditions (parameters) at cutoﬀ 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 cutoffliquid-gas of perturbatively (1st order) renormalizable QFTs be removedtheory (i.e. space be sent to )? ∞ ∗ if yes: QFT is valid on all scales (fundamental theory) S ∗ if no: QFT predicts its own failure (e.g. triviality problem) liquid-gas (1st order)hint towards new physics theory space ⇒

orderparameter • example (for hint towards new physics): S temperature – Fermi’s theory: 4-Fermion interaction amplitude λ for neutrino-neutrino scattering

orderparameter – dimensional analysis:

temperature λ + λ2Λ2 + M ∼ ∼

Failure! Amplitude blows up ⇒ new physics can cure this problem+ (within perturbation theory): → introduce vector boson

– mechanism can also be occur towards low energies, indicating appearance of an ordering 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-perturbatively renormalizable?

– Weinberg’s Asymptotic Safety scenario – possibly applies to quantum gravity – reminiscent of asymptotic freedom in QCD

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 reconsider some of the basic models which show (second order) phase transitions. The Ising model is a basic 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: X Z = e−βH (2.1)

{si=±1, i=1,...,N}

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 s , s , ..., s : { 1 2 N } X X H = J s s h s (2.2) − i j − i hi,ji i

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

• i, j indicates a sum over nearest neighbors h i • 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: 1 X m = s = s e−βH (2.3) h ji Z j {si=±1, i=1,...,N}

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

• ferromagnetic (FM) phase: m = 0 6 13 liquid-gas (1st order) theory space S +

orderparameter 2. Phase transitions and critical phenomena temperature

• paramagnetic (PM) phase: m = 0

• for h = 0 : m = T 6 6 ∀ • equation (2.1) can be solved exactly in d 1, 2 ∈ { }

h m > 0 m ≠ 0 m = 0 T FM Tc PM m < 0

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 = k log N (with N being the number of microstates realizing ⇒ b one domain wall) free energy F = E TS is always lowered in large system (N 1) by flipping half ⇒ − of the dipoles , even at infinitesimal T ordered state is unstable m = 0 only at T = 0. ⇒ ⇒ 6 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 < T and near T : m (T T )1/8 (magnetization) and C log T T c c ∼ c − ∼ − | c − | (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 s s i i → − 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 Z symmetry ⇒ 2 • to describe ordered phase (FM with = 0): 6 – restrict space of configurations over which sum in Z is performed, e.g., at h = 0 6 – take limit h 0 after thermodynamic limit has been performed → m = 0 survives the h 0 limit ⇒ 6 → • actual physical process is different from mathematical procedure (as it is induced by the 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 Y P ~ ·~ ~ P ~ Z = δ( S~ 1)ddS~ eβ(J hi,ji Si Sj +h i Si) (2.4) | i| − i i=1

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 diﬀerent physical systems

• example: specific heat near liquid-gas critical point ˆ=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-gas transition with reduced temperature t:

1. speciﬁc heat near Tc: −α C = C± t , (2.5) V | | where C± refers to t > 0 or t < 0.

2. compressibility near Tc:

1 ∂ρ −γ κ = = κ± t . (2.6) T ρ ∂p | | - the analogue in a magnetic system is the susceptibility: χ t −γ. ∝ | | 3. diﬀerence between gas and liquid densities along coexistence curve near the critical point: ρ ρ = ρ ( t)β . (2.7) L − G c − - the analogue in a magnetic system is the magnetization: m ( t)β. ∝ − 4. critical isotherm (T = Tc) near p = pc: δ p pc ρL ρG − = − . (2.8) pc ρc - 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 the help 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 Heisenberg universality classes. The term Ising speciﬁes the symmetry group of the corresponding order parameter to be Z2. In the case of the XY universality class the symmetry is O(2) and for Heisenberg 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 -0.10 β 1/8 0.3265(3) 0.35 0.36 γ 7/4 1.2372(5) 1.32 1.39 δ 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 uniaxial → ferromagnet 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 the scaling hypothesis/assumption:

1/w h f(T, h) = t Ψ± , (2.10) | | t u/w | |

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 C , m, χ + two equations for α, β, γ, δ. ⇒ V 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 derivative of the free energy with respect to the magnetic ﬁeld h evaluated at h = 0, i.e.

∂f 1−u 0 m = = t w Ψ±(0) (2.11) − ∂h h=0 −| | for t > 0 : m = 0 Ψ0 (0) = 0 (2.12) ⇒ ⇒ + 1 u for t < 0 : m ( t)β β = − (2.13) ∝ − ⇒ w

- uniform magnetic susceptibility near Tc:

2 ∂ f 1−2u 00 χ = = t w Ψ±(0) (2.14) − ∂h2 h=0 −| |

2u 1 and with χ t −γ γ = − (2.15) ∝ | | ⇒ w

- speciﬁc heat:

2 ∂ f Ψ±(0) 1 1 1 − w 2 CV = T 2 = 1 t (2.16) − ∂T h=0 − Tc w w −

1 α = 2 (2.17) ⇒ − w

- rewriting of the scaling assumption: provides critical exponent δ 1/u h −1/u f(T, h) =h Ψ˜ ± where Ψ˜ ±(z) = z Ψ±(z) (2.18) t u/w | |

1 1 −1 m = h u Ψ˜ ±( ) at t = 0, h 1 (2.19) ⇒ −u ∞ u δ = (2.20) ⇒ 1 u − 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 transition in 4He in d = 3: α 0.0127 0.0003 exp ≈ − ± – 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 are the same below and above the critical point

• the concept of scaling rationalizes the appearance of power laws near the critical point and 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 problem set 1) replace J P s s by J P s s in Ising model with an average magnetization per ⇒ hi,ji i j hi,jih ii j spin m = s h ii P β(zmJ+h)s1 s1 e zmJ + h m = s1=±1 = tanh , (2.23) P β(zmJ+h)s1 k T ⇒ s1=±1 e B where z = 2d counts the number of nearest neighbors on a d-dimensional hypercubic lattice • at h = 0: – m = 0 for T < T = zJ/k 6 c B – m = 0 for T > Tc T p • expansion near m = 0 and at h = 0 m = 3(1 T/Tc) β = 1/2 ⇒ Tc − ⇒ 1/3 3h • expansion at T = Tc for small h m = δ = 3 ⇒ kB Tc ⇒ • 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) h − − i h i − 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. h i Sometimes convenient to study the (spatial) Fourier transform of G(~r), i.e. work in wavevector space Z G(~q, t) = ddre−i~q·~rG(~r, t) (2.25)

• assume scaling form (for large distances r):

r Φ± ξ(t) G(r, t) = where ξ(t) t −ν (2.26) rd−2+η ∝ | | and ξ is called the correlation length, d is the dimensionality.

• the correlation length denotes the typical length scale of the regions where the degrees of 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: ∂m(~r) χ(~r ~r0) := (2.27) − ∂h(~r0) h=0 – general theory of linear response: χ(~q) = G(~q) – uniform susceptibility, eq. (3.3): χ = χ(~q = 0) = R dd~rG(~r, t) – use scaling assumption, eq. (2.26): χ = C ξ2−η with the constant C = R 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 < T : here m = 0. Deviations from ﬁnite magnetization also uncorrelated c 6 at 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 correlation length ξ: free energy = f ξ(t)−d (2.28) unit volume ∝ – 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 thermodynamic 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 + t for t > 0 in d = 3

3 i~q·~r √ G(~r) = R d ~q e = 1 e−r t ν = 1/2, η = 0 ⇒ (2π)3 q2+t 4πr ⇒ 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 transition in 4He (λ transition, T 2K). The construction can be performed analogously for other c ≈ transitions. The goal of this section is to express the (grand-canonical) partition function Z in terms of the so-called path integral or functional integral. For functional integrals many convenient tools have been developed which facilitate systematic approximate evaluations of the partition function. So, the general strategy is the following:

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

write down (grand-canonical) partition function Z = Tre−β(Hˆ −µNˆ) → express Z in terms of a functional integral, i.e. Z = R Φe−S[Φ], with action S[Φ] → D learn about various tools to systematically evaluate Z in its functional representation → • the Hamiltonian for interacting bosons reads in second quantization

ˆ ˆ † X † X † † H = H(ˆa , aˆ) = αβaˆαaˆβ + Vαβγδ aˆαaˆβaˆδaˆγ (2.29) α,β α,β,γ,δ

• the particle number operator is

ˆ X † N = aˆαaˆα (2.30) α

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:

nαi N † Y aˆαi n , n , ..., n = 0 (2.31) | α1 α2 αN i p | i i=1 nαi !

– α label the states forming basis in single-particle Hilbert space of dimension N { i} – 0 is the vacuum | i † – aˆα , aˆ are bosonic creation/annihilation operators single-particle state α i αi → | ii † – standard commutation relations [ˆaαi , aˆαj ] = δαiαj – any many-particle state is a linear combination:

∞ ∞ X X Φ = Φn ,...,n nα , nα , ..., nα (2.32) | i ··· α1 αN | 1 2 N i nα1=0 nαN =0

22 2.6. Ginzburg-Landau-Wilson theory

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

aˆ Φ = Φ Φ , (2.33) αi | i αi | i

with i = 1, 2, ..., N and complex eigenvalues Φαi – for such a state we have (tutorial)

† nαi N nα ∞ ∞ Φ aˆ Y (Φ ) i X X Y αi αi Φ = αi Φ = 0 nα1 ,...,nαN p nαi ! ⇒ | i ··· nαi ! | i i=1 nα1=0 nαN =0 i=1 P † Φα aˆα = e αi i i 0 (2.34) | i P ∗ † ∗ α Φα aˆα – similarly: Φ aˆα = Φ Φ bra version: Φ = 0 e i i i h | i h | αi → h | h | † ∂ aˆαi Φ = ∂Φ Φ which can be shown by Taylor expansion of eq. (2.34) ⇒ | i αi | i P ∗ 0 0 Φα Φα – overlap of two coherent states: Φ Φ = e αi i i (not orthogonal!) h | i – coherent state form overcomplete set with resolution of unity (without proof): Z ∗ Y dΦ dΦα − P ∗ α e α ΦαΦα Φ Φ = 1 (2.35) 2πi | ih | α 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 turn is 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

ˆ ˆ ˆ ˆ • grand-canonical partition function Z = Tr e−β(H−µN) = P n e−β(H−µN) n nh | | i where the sum extends over a complete set of Fock states n {| i} • together with eq. (2.35), we get: Z ∗ Y dΦ dΦα − P ∗ − ˆ − ˆ Z = α e α ΦαΦα Φ e β(H µN) Φ (2.36) 2πi h | | i α

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

M−1 ∗ M Z dΦ dΦ PM−1 P ∗ Y Y α,k α,k − Φ Φα,k Y −∆β(Hˆ −µNˆ) Z = e k=0 α α,k Φ − e Φ (2.37) ⇒ 2πi h k 1| | ki k=0 α k=1

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

• for ∆β 1:

• in Hamiltonian and number operator all creation operators are to the left of annihilation operators normal ordering → • for abitrary normally ordered function A, the deﬁnition of coherent states implies:

† 0 ∗ 0 P ∗ 0 Φ A(ˆa , aˆ ) Φ = A(φ , φ )e α ΦαΦα (2.39) h | α α | i α α for M : (2.40) ⇒ → ∞ Z M−1 ∗ Y Y dΦα,kdΦα,k − PM−1 P Φ∗ (Φ −Φ ) Z = lim e ( k=0 α α,k α,k α,k+1 ) M→∞ 2πi k=0 α PM−1 ∗ P ∗ − ∆β H(Φ ,Φα,k+1)−µ Φ Φα,k+1 e k=0 α,k α α,k (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 ∗ ∗ −S[Φα(τ),Φα(τ)] Z = Φα(τ) Φα(τ)e (2.42) Φα(0)=Φα(β) D D

with the action " # Z β X S = dτ Φ∗ (τ)( ∂ µ)Φ (τ) + H[Φ∗ (τ), Φ (τ)] (2.43) α − τ − α α α 0 α

• 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 are periodic in imaginary time.”

• we can decompose the periodic function (periodicity due to boundary condition!): Z 1 d~k X ~ Φ(~r, τ) = Φ(~k, ω )eik·~r+iωnτ (2.44) β (2π)d n ωn

with the bosonic Matsubara frequencies ω = 2πn/β = 2πnk T , n Z n B ∈ • remarks: – in the case of fermions antiperiodic, anticommuting Grassmann numbers, → ωn,F = (2n + 1)π/β – example: non-interacting bosonic system

Z ∗ ~ ~ −1 P 2~k2 2 Y dΦ (k, ωn)dΦ(k, ωn) −β (−iωn+ ~ −µ)|Φ(~k,ωn)| Z = Z = e ~k,ωn 2m (2.45) 0 2πi ~k,ωn Y β = (2.46) ~2~k2 ~ iωn + µ k,ωn − 2m − recover Bose-Einstein condensation from F = k T log Z and N = ∂F0 ... → 0 − B 0 − ∂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 ω = 0 separated by ﬁnite amount T from the action for 0 ∝ the other modes (ω = 0) n 6 Z d~k X S = β−1 ( iω µ) Φ(~k, ω ) 2 + ... (2.47) (2π)d − n − | n | n

• ω modes ( n 1) only yield analytic contribution to free energy n | | ≥ 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 Z = Φ∗(~r) Φ(~r)e−S (2.48) D D – example: non-relativistic bosons of mass m 2~ 2 energy dispersion = ~ k , i.e. non-interacting part of Hamiltonian, eq. (2.43): → ~k 2m Z 2 2 ∗ ∗ ~ H [Φ (~r), Φ(~r)] = β d~r Φ (~r) ∇ Φ(~r) (2.49) 0 − 2m

contact interaction V (~r ~r ) = λ δ(~r ~r ), cf. eq. (2.29), gives the action: → 1 − 2 1 − 2 Z 2 1 ~ S[Φ] = d~r Φ(~r) 2 µ Φ(~r) 2 + λ Φ(~r) 4 (2.50) kBT 2m|∇ | − | | | |

– 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 Z (Φ(~r) Φ(~r)) 2 → − 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-model to 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 = k T S[Φ ] (2.51) ⇒ B · 0

• here Φ is the conﬁguration that minimizes the action: δS = 0 0 δΦ Φ0

• Φ will be independet of the coordinate and be (1) Φ = 0 , or (2) Φ 2 = µ/(2λ) 0 0 | 0| • take µ as a tuning parameter: (a) for µ < 0: non-trivial solution (2) is impossible Φ = 0 ⇒ 0 µ2 V (b) for µ > 0: S[Φ0] = < S[0] = 0 Φ0 is ﬁnite − 4λ kB T ⇒ Φ = 0 becomes a local maximum → 0 action has mexican hat shape with a continuum of minima →

• a ﬁnite value of Φ signals the ordered phase take Φ as order parameter 0 ⇒ 0 • in general the order parameter is deﬁned as Φ(~r) which in the SPA coincides with Φ h i 0 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 (T T ) + (T T )2 (2.52) ∼ c − O c − Spontaneous symmetry breaking:

• only absolute value of Φ is ﬁxed by Φ 2 = µ/(2λ), the phase is left arbitrary 0 | 0| • 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 direction of 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 T T in GLW theory differentiating S[Φ ] twice w.r.t T : ∼ c − → 0 specific heat only has a finite discontinuity at the transition → critical exponent α = 0 ⇒ • define superfluid susceptibility: δ2 log Z[J] χ(~r ~r0) := (2.53) − δJ ∗(~r)δJ(~r0) J=0 where Z S S + d~r [Φ(~r)J(~r) + Φ∗(~r)J ∗(~r)] (2.54) → defines Z[J] χ(~r ~r0) = Φ∗(~r)Φ(~r0) Φ(~r) 2 (2.55) ⇒ − h i − |h i| – note the similarity to eq. (2.24) – also note that the source field J(~r) directly couples to Φ in the action in the same way as the real magnetic field would enter the partition function of magnetic systems, see eq. (2.27). • to compute χ in saddle-point approximation consider small fluctuations of Φ(~r) around the 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: ⇒ Z log Z[J] = d~rd~r0J ∗(~r)χ (~r ~r0)J(~r0) + const. (2.56) 0 − where Z d~k ei~k·~r χ (~r) = k T (2.57) 0 B d 2~ 2 (2π) ~ k µ 2m −

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

2mkBTc F (r/ξ) χ0(~r) = (2.58) ~2 · rd−2 d−2 R d~q ei~q·~r with the scaling function F (z) = z (2π)d q2+1 , p and the correlation length ξ = ~/ 2m µ | | • this result for χ0 can be cast into the form in eq. (2.26) – since µ T T near T : ∼ c − c 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 not → satisfactory!

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: X S[φ ] βH[φ ] = J cos (φ φ ) (2.59) i ≡ i − i − j hi,ji

• φ [0, 2π] is an angle variable attached to each point i ∈ • the action has a global (continuous) U(1) symmetry: φ φ + α i → i • 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 frequencies below T :

Z κ t H = d2x ~ Ψ 2 + Ψ 2 + u Ψ 4 (2.60) 2 |∇ | 2| | | |

iφ – where Ψ = Ψ1 + iΨ2 = re – 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 of the 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 qualitatively distinct (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 φ const. = φ , and i ≈ 0 only small local ﬂuctuations of the phase.

• thus we decompose φi = φ0 + δφi, where δφi are slowly varying small ﬂuctuations J X J X βH[φ ] = cos (δφ δφ ) const. + (δφ δφ )2 (2.61) ⇒ i −T i − j ≈ 2T i − j hi,ji hi,ji

continuum limit a2J Z J¯ Z z}|{ const. + d2r(~ φ)2 = d2q~q2(φ )2 (2.62) −→ 2T ∇ 2T q

• 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:

T 0 0 − 2πJ i(φ(~r)−φ(~r0)) − 1 h(φ(~r)−φ(~r0))2i − T log |~r−~r | ~r ~r e e 2 = e 2πJ a = | − | (2.63) h i ≈ a

• 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):

Z 2 1 ~ S[Φ] = d~r Φ(~r) 2 µ Φ(~r) 2 + λ Φ(~r) 4 (3.1) kBT 2m|∇ | − | | | |

• m provides energy scale rescale into chemical potential and interaction: ⇒ 2 2 2mµ/~ µ, 2mλ/~ λ (3.2) −→ −→

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

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 1 ultraviolet cutoﬀ Λ , (3.4) ∼ a

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 < k < Λ , where b 1 (3.5) b ≈ recast partition function for remaining slow modes (k < Λ ) into form before integration → b 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 problem set 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 ≈ y • now, choose b such that µ b = µ0 with constant µ0

x/y 1/y ! µ0 µ0 ∗ χ(~k) = F k , µ0, λ (3.9) ⇒ µ µ

• 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

λ (b) Φ 6, λ (b) Φ 2 Φ 2 (3.11) 6 | | 2,2 |∇ | | |

• 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 Y dΦ∗(~k)dΦ(~k) Z = e−(S0<+S0>+Sint) (3.13) 2πi k<Λ Z Y dΦ∗(~k)dΦ(~k) Z Y dΦ∗(~k)dΦ(~k) = e−(S0<+S0>+Sint) , (3.14) Λ 2πi Λ 2πi k< b b = d (k µ) Φ(k) , + (3.16) Λ/b (2π)S − | | Z Λ ~ ~ dk1...dk4 ~ ~ ~ ~ ∗ ~ ∗ ~ ~ ~ and Sint = λ 3d δ(k1 + k2 k3 k4)Φ (k4)Φ (k3)Φ(k2)Φ(k1) . (3.17) 0 (2π) − − • the interaction term in Fourier space can be represented pictorially orderparameter • each ﬁeld Φ is represented by a line with an arrow pointing into the vertex

∗ temperature • 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: solid phase liquid phase pressure ⇤(~k3) ⇤(~k4) pc critical point

vapour (~k1) (~k2) temperature Tc h m > 0 m ≠ 0 m = 0 T fast modes FM Tc PM 34 m < 0 slow modes 3.2. Momentum-shell transformation

3.2.1 Perturbation theory Zeroth order in λ: Z = Z Z Z factorizes 0< 0> ⇒ in Z all the modes have large k → 0> Z only contributes to regular, analytic part of free energy ⇒ 0> • now, bring Z into exact old form by ﬁrst rescaling b~k ~k: 0< → Z Λ ~ ~ 2 ! 1 dk k ~ 2 S0< = d d 2 µ Φ(k) (3.18) b 0 (2π) b − | |

~ • define: µ(b) = µ b2 and Φ(k) Φ(~k) b(d+2)/2 → S now precisely takes the old form in terms of rescaled Fourier components ⇒ 0< • 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 + (λ) ⇒ O ν = 1/y = 1/2 mean-field result! ⇒ → First order in λ: avoiding slow modes can be done without encountering the singularity → Z Y dΦ∗(~k)dΦ(~k) Z =Z e−S0< [1 0> 2πi k<Λ/b Z Λ ~ ~ # dk1...dk4 ~ ~ ~ ~ ∗ ~ ∗ ~ ~ ~ 2 λ 3d δ(k1 + k2 k3 k4) Φ (k4)Φ (k3)Φ(k2)Φ(k1) 0> + (λ ) (3.19) − 0 (2π) − − h i O

where we have introduced the fast-mode average

Z ∗ ~ ~ 1 Y dΦ (k)dΦ(k) −S0> A 0> = e A. (3.20) h i Z0> 2πi Λ/b

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

(2π)d Φ∗(~k )Φ(~k ) = δ(~k ~k ) , (3.21) h 1 2 i0 k2 µ 1 − 2 1 − Φ∗(~k )Φ∗(~k )Φ(~k )Φ(~k ) = Φ∗(~k )Φ(~k ) Φ∗(~k )Φ(~k ) + (k~ ~k ) , (3.22) h 4 3 2 1 i0 h 4 2 i0h 3 2 i0 1 ↔ 2 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 liquid-gas (1st order) theory space So, next we want to calculate the following part of eq. (3.19): Z Λ d~k ...d~k 1 Z dΦ∗(~k)dΦ(~k) 1 4 Y ∗ ~ ∗ ~ ~ ~ −S0> λ 3d δ1234 + Φ (k4)Φ (k3)Φ(k2)Φ(k1)e (3.23) − S0 (2π) Z0> 2πi Λ/b

where we have introduced the short-hand notation δ = δ(~k + ~k ~k ~k ). 1234 1 2 − 3 − 4

3.2.2 Combinations in eq. (3.23) that contribute in eq. (3.19) orderparameter ...by giving a ﬁnite average: 1. ~k = ~k and Λ/b < k < Λ and ~k = ~k and k < Λ/b { 4 2 2 3 1 1 } temperature - then one part of eq. (3.23) reads   Z Λ d~k ...d~k 1 Z dΦ∗(~k)dΦ(~k) 1 4 ∗ ~ ~ Y ∗ ~ ~ −S0> λ 3d δ1234Φ (k3)Φ(k1)  Φ (k4)Φ(k2)e  − (2π) Z0> 2πi 0 Λ/b 0 → m ≠ 0 m = 0 T fast modes FM Tc PM m < 0 slow modes

3. when all k < Λ/b 4. when all Λ/b < k < Λ only contributes to prefactor in eq. (3.19), i.e. to analytic → part 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 : ⇒ ∝ < < ! Z Λ/b d~k Z Λ d~q 1 S = ~k2 µ + 4λ Φ(~k) 2 < (2π)d − (2π)d q2 µ | | 0 Λ/b − Z Λ/b ~ ~ dk1...dk4 ~ ~ ~ ~ ∗ ~ ∗ ~ ~ ~ 2 + λ 3d δ(k1 + k2 k3 k4)Φ (k4)Φ (k3)Φ(k2)Φ(k1) + (λ ) (3.27) 0 (2π) − − O

rescale to bring cutoﬀ back to Λ and Φ(~k)/b(d+2)/2 Φ(~k) → → ! Z Λ d~q 1 µ(b) = b2 µ 4λ + (λ2) (3.28) ⇒ − (2π)d q2 µ O Λ/b − λ(b) = b4−dλ + (λ2) (3.29) O eq. (3.28): interaction shifts critical value of µ to → Z Λ d~q 1 2 µc(b) = 4λ d 2 + (λ ) (3.30) Λ/b (2π) q O

1 1 c 2 • here, we use a series expansion for small c: 2 = 2 + 4 + (c ) q −c q q O • 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: " # Z Λ d~q 1 1 µ˜ = µ µ (b) µ˜(b) = b2 µ˜ 4λ + (λ2) (3.31) − c ⇒ − (2π)d q2 µ˜ − q2 O Λ/b −

• therefore, in the critical region where µ˜ Λ2 : | | | " d−4 Z 1 d−1 # 2 Λ Sd x 2 µ˜(b) = b µ˜ 1 4λ d 4 dx + (λ ) (3.32) − (2π) 1/b x O

• the integral in eq. (3.32) can be evaluated approximately close to four dimensions, i.e. we assume that d 4 1: | − | Z 1 xd−1 1 1 dx = 1 = log b + (d 4) (3.33) x4 d 4 − bd−4 O − 1/b −

d−4 d d/2 • dimensionless interaction: λˆ = λΛ Sd/(2π) with Sd = 2π /Γ(d/2)

ˆ 2 µ˜(b) =µ ˜ b2 1 4λˆ log b + (λˆ(d 4), λˆ2, λˆµ˜) µ˜ b2−4λ+O(λ ) (3.34) − O − ≈

37 3. Wilson’s renormalization group

• now, assume λˆ to be small. Consider b : → ∞ 1. forµ ˜ > 0, µ˜(b) : superfluid phase (spontaneously broken symmetry) → ∞ 2. forµ ˜ < 0, µ˜(b) : normal phase (symmetric) → −∞ 3.˜µ = 0, i.e. µ = µ (b), µ˜(b) 0: fixed point of the transformation c ≡ – fixed point is unstable,µ ˜ grows when perturbed from fixed-point value – µ˜ is a relevant coupling (= tuning parameter of the transition)

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

2 y = 2(1 2λ∗ + (λ )) (3.35) ⇒ − O ∗ where

ˆ 1 2 λ∗ = lim λ ν = + λ∗ + (λ∗) (3.36) b→∞ ⇒ 2 O

• Note:

– in d > 4 and λˆ 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 liquid-gas (1st order) theory3.3. Determination space of ﬁxed-point coupling below four dimensions 3.3 Determination of ﬁxed-point couplingS below four dimensions+ 2 We want to calculate λ∗ below four dimensions. Therefore, we calculate the (λ )-term in O eq. (3.29). Explicitly, the (λ2) term from eq. (3.19) reads O fast modes 2 Z Λ λ d~p1...d~p4d~q1...d~q4 orderparameter 6 δ(~p1 + ~p2 ~p3 ~p4)δ(~q1 + ~q2 ~q3 ~q4) 2 0 (2π) − − − − Φ∗(~p )Φ∗(~p )Φ(~p )Φ(~p )Φ∗(~q )Φ∗(~q )Φ(~q )Φ(~q ) (3.37) × h 4 3 slow2 modes1 4 3 2 1 i0> temperature • To compute the average ... : pair up Φ∗ and Φ with equal wavevectors Λ/b < k < Λ h i0> in all possible ways, similar to procedure in Sec. 3.2.2 solid phase • If two unpaired legs remain in the low-energy shell k < Λ/b contribution to µ(b) liquid phase pressure ⇒ pc critical point• 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): ⇒ vapour temperature Tc h m > 0 m ≠ 0 m = 0 T FM Tc PM m < 0 (a) (b) ⇤(~k3) ⇤(~k4)

• 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 (~k1) (~k2) • disconnected diagrams cancel out upon re-exponentiation into S<, cf. linked-cluster theorem (later in the lecture)

upon re-exponentiation the (λ2) term becomes: ⇒ O 2 Z Λ/b ~ ~ λ dk1...dk4 ~ ~ ~ ~ ∗ ~ ∗ ~ ~ ~ 3d δ(k1 + k2 k3 k4)Φ (k4)Φ (k3)Φ(k2)Φ(k1) − 2 0 (2π) − − " # Z Λ d~q 1 Z Λ d~q 1 16 d + 4 d × Λ/b (2π) (q2 µ˜)((~q + ~k ~k )2 µ˜) Λ/b (2π) (q2 µ˜)((~k + ~k ~q)2 µ˜) − 2 − 4 − − 1 2 − − (3.38)

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

! Z Λ d~q 1 λ(b) = b4−dλ 1 10λ + (λ2) (3.39) ⇒ − (2π)2 (q2 µ˜)2 O Λ/b − at critical pointµ ˜ = 0 (perform integral, introduce λˆ): → ˆ ˆ2 λˆ(b) = b4−dλˆ 1 10λˆ log b + (λˆ2) = b4−d−10λ+O(λ )λˆ (3.40) − O together with eq. (3.34): → ˆ 2 µ˜(b) = b2µ˜ 1 4λˆ log b + (λˆ(d 4), λˆ2, λˆµ˜) =µb ˜ 2−4λ+O(λ ) (3.41) − O − 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 diﬀerential form for 4 d 1, 4 d = : | − | − dµ˜(b) = β =µ ˜(b) 2 4λˆ(b) + (λ,ˆ λˆ2, λˆµ˜) (3.42) d log b µ − O dλˆ(b) = β = λˆ(b) 10λˆ(b)2 + (λˆ2, λˆ3, λˆ2µ˜) (3.43) d log b λ − O

• this deﬁnes the renormalization group β functions!

• momentum-shell trafo (=RG trafo) flow in the space of coupling functions ⇒ Understand the flow in the λˆ 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, λˆ = 0 2. the (non-trivial) Wilson-Fisher ﬁxed point (WFFP):µ ˜ = 0, λˆ = + (2) 10 O

- for < 0 the WFFP is unphysical as λˆ < 0 and further it is unstable in both directions: change of b (integrates out more modes) µ˜(b) and λˆ(b) are driven away from WFFP. → ⇒ - for > 0: the WFFP is physical as λˆ > 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 λˆ direction and unstable in theµ ˜ direction: critical point

• the GFP is unstable in both directions The critical behavior for > 0 is governed by the nontrivial Wilson-Fisher ﬁxed point ⇒ • parameter controls the size of the corrections to MFT critical behavior

• consider as a continuous parameter

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

41 3. Wilson’s renormalization group

0.04

0.02

μ 0.00

-0.02

-0.04 0.00 0.02 0.04 0.06 0.08 0.10 0.12 λ

Figure 3.1: The arrows in the plot are β~(λ,ˆ µ˜) = (β , β ) . The black square indicates the λ µ |λ,ˆ µ˜ GFP and the red square the WFFP. In the λˆ direction, the arrows point into the WFFP indicating its stabilty.

Flow near ﬁxed points from stability matrix

∂β M := xi (3.44) ij ∗ ∂xj xi=xi where xi =µ, ˜ λˆ. • 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 M is already diagonal at WFFP with y = 2 2/5 ij − 1 ν = + + (2) ⇒ 2 10 O – for = 1(d = 3): ν 0.6 ≈ – the experimental value is ν 0.670, i.e. we’re already better than MFT exp ≈ • a single eigenvalue determines the flow for positive and negative chemical potential ν is equal on both sides of the transition! → 42 liquid-gas (1st order) theory space S +

fast modes orderparameter 3.4. Epsilon expansion and Wilson-Fisher ﬁxed point

slow modes temperature Determination of anomalous dimension • from rescaling of momenta/ﬁelds by momentum-shell trafo: solid phase 2 2 liquid phase k Zkk , µ Zµµ, λ Zλλ (3.45) pressure → → → pc critical point • deﬁne η = dZk η = (2) d ln b λˆ=λˆ∗ ⇒ O • ... calculation ... Fisher’s scaling law! vapour ⇒ (a) (b) Role of newly generated couplings temperature Tc h m > 0 • for example a sixth order term g Φ 6 is generated due to the (quartic) coupling λ: ∼ | | m ≠ 0 m = 0 T FM Tc PM m < 0 ⇤(~k3) ⇤(~k4)

(~k1) (~k2) • standard (dimensional) rescaling: dg β = = 2(3 d)g + (gλˆ) (3.46) g d ln b − O 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 only the 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:

1 N + 2 (N + 2)(N 2 + 23N + 60) ν = + + 2 + (3) (3.47) 2 4(N + 8) 8(N + 8)3 O

– 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 + 0.1 + 0.0552 0.01260913 + 0.07138764 0.1945485 + 0.7070196 + (7) − − O – direct substitution of = 1:

0 1 2 3 4 5 6 0.5 0.6 0.655 0.642 0.714 0.519 1.226

– 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 expansion series using resummation techniques (e.g. Borel summation): educated guess about the function being expanded from the ﬁnite number of computed terms.

• correlation length exponent ν for N 1, 2, 3 : ∈ { } ν N = 1 N = 2 N = 3 6 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. I hope to give you a reasonable account for the bare basics, here and refer to the textbook by Peskin & Schroeder for more details. The focus of the presentation is the application of the RG beta functions from the previous section which allow us to understand two problems in the Standard Model of Particle Physics: the hierarchy problem and the triviality problem. To describe non-relativistic interacting bosons in three spatial dimensions, we have employed the action, eq. (2.50) Z 2 1 ~ S[Φ] = d~r Φ(~r) 2 µ Φ(~r) 2 + λ Φ(~r) 4 , (3.48) kBT 2m|∇ | − | | | | which together with the partition function in functional representation Z Z = Φ∗(~r) Φ(~r)e−S , (3.49) D D allows us to calculate thermodynamic properties from operator averages/correlation functions. This quantum field theoretical formalism that we have developed for the system of many interacting bosons works analoguously for other systems with a large/infinite number of 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 large particle collider experiments) • Particle-physics (PP) is typically formulated as a field theory following the laws of the theory of special relativity 1 0 0 0  0 1 0 0  (1+3) dimensional Minkowski spacetime with metric ηM =  −  → 0 0 1 0  − 0 0 0 1 − • Particle physics can also be studied in euclidean formulation which is often very convenient  1 0 0 0  −  0 1 0 0  Euclidean space with metric ηE =  −  →  0 0 1 0  − 0 0 0 1 − • Therefore, we need to introduce a Wick rotation (t = it ) in the time coordinate − E which then provides an evolution in imaginary time • Euclidean functional integral for partice physics quantum field theory emphasizes deep connection 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 mechanics 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 interactions 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 m 125 GeV H ∼ • generally couples to other ﬁelds in Standard Model (e.g. quarks... most importantly the top quark and gauge ﬁelds)

• for our purposes it is sufficient to consider the Higgs field individually with a simplified action for a complex scalar field in four-dimensional Euclidean space: Z S[Φ] = d4x ∂ Φ(x) 2 µ Φ(x) 2 + λ Φ(x) 4 , (3.50) | µ | − | | | |

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

• in fact, we have carried out the RG transformations in 4 euclidean dimensions − • to obtain the RG beta functions for this theory, we just have to set = 0 in eqs. (3.42)

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

β = 2˜µ(b) 4λˆ(b)˜µ(b) (3.51) µ − β = 10λˆ(b)2 (3.52) λ − • implicit to these equations is the choice of an ultraviolet cutoﬀ Λ setting the largest momentum scale of the problem.

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

• Ideally we fantasize that a Standard Model of Elementary Particles can be valid at arbitraily high energies and momentum and therefore we hope that we can somehow send 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, we suppress the tilde on top of the λ˜ and just write λ. Before, we reformulate the derivative with respect to log b:

d log b 1 1 dλ(b) dλ(b) = d log b = db = b . (3.53) db b ⇒ b ⇒ d log b db Therefore, we have to solve

λ b Z IR Z IR λIR bIR dλ(b) 2 dλ db 1 b = 10λ(b) 2 = 10 = 10 log b (3.54) db − ⇒ λUV λ − bUV b ⇒ −λ λUV − bUV

1 1 ΛUV = 10 (log bIR log bUV) with bUV = 1 and bIR = (3.55) ⇒ λUV − λIR − − ΛIR

λ(ΛIR) λ(ΛUV) = (3.56) ⇒ ΛUV 1 10ΛIR log − ΛIR

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

• ΛIR is infrared momentum scale down to which we integrated out the momentum modes. liquid-gas (1st order) theory space • Now, suppose, we have a collider experiment at some momentum or energy scale ΛIR where we measure the scattering ofS particles. + • Note that while the experiment operates at relatively high energies at the collider, these energies Λ can be considered low as compared to a ultraviolet scale Λ ∼ IR fast modes UV which defines the highest energies/momenta where the model is assumed to be valid. orderparameter • The experiment carried out at energies Λ shall allow us to fix the interaction slow ∼modesIR temperature between 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) at solid phase scale Λ if at scale Λ the value λ was measured. liquid phase UV IR IR pressure 1 pc critical point (a) (b) 10λ • eq. (3.56) has a singularity – the Landau pole singularity – at scale: ΛL = ΛIRe IR .

vapour

temperature Tc h m > 0 m ≠ 0 m = 0 T interaction FM Tc PM m < 0 IR ⇤UV ⇤(~k ) ⇤(~k ) 3 4 ⇤IR scale ⇤L

47 (~k1) (~k2) 3. Wilson’s renormalization group

• the presence of the Landau pole indicates that the interaction coupling λ grows strong at 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 Λ → L 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 Λ ) UV → ∞ – this means, we have to send the Landau pole to inﬁnity Λ L → ∞ – this requires to have λ 0. IR → – 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- 3 trodynamics (QED) has a beta function de e where e is the EM coupling. d log b ∼ − 12π2 • A more elaborate RG analysis predicts: QED-Landau pole is expected to appear at huge 19 ultraviolet scales ΛL,QED mPlanck 10 GeV, where the Planck mass is defined as p ∼ mPlanck = ~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 the Standard Model is needed which includes gravity. Therefore, QED-Landau pole is not considered as a problem of practical importance within the Standard Model. • Further analysis with the Standard Model parameters suggests that also Landau-pole in the Higgs sector only appears beyond mPlanck. Due to fluctuations of the top quark, however, another problem below mPlanck is sometimes discussed: The quartic coupling λ in the Higgs sector might drop below zero and render the Higgs potential unbounded from below or induce a second (deeper) minimmum with a possibly finite decay rate from 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 this section (e.g. φ3 theory in d = 6 or QCD), the coupling drops to zero for large momentum scales. This behavior is typically referred to as asymptotic freedom (problem set 4). The Landau pole then appears in the infrared and indicates, for example, a change of degrees of 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 the so-called hierarchy problem or naturalness problem. It is related to the beta function for the term which is bilinear in the bosonic fields, eq. (3.51): β = 2µ(b) 4λ(b)µ(b) . (3.57) µ − • 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: → dµ(b) b = 2µ(b) (3.58) db • this equation is easily integrated by separation of variables and gives: 2 ΛUV µ(ΛIR) = µ(ΛUV) (3.59) ΛIR • 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 to eq. (3.51), most importantly the coupling to the top quarks (fermions). • Still, the most important contribution to the beta function of µ is the dimensional term 2µ(b) which we already determined in Sec. 3.2.1 from zeroth order perturbation ∝ theory (i.e. a purely dimensional argument). • The relation in eq. (3.59) implies that an extreme fine tuning has to be performed on µ(Λ ), when Λ Λ and at Λ , the parameter µ(Λ ) has to match an UV UV IR IR IR experimentally 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) p – 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 found in particle colliders assume that Standard Model is a valid theory also on much higher scales → some ideas suggest a grand uniﬁed theory (GUT) at Λ 1015GeV. → UV ∼ – With these numbers, we conclude that we have to choose a parameter µ(Λ ) = 4.67640625 10−22GeV2 (3.60) UV · • in the Standard Model this seems unnatural, because there is no “knob” which can be used to adjust this value (hence naturalness problem). Still, for some reason the Standard 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 to be “natural” as we have to adjust the temperature T very close to its critical value Tc.

49 liquid-gas (1st order) theory space S + 3. Wilson’s renormalization group

fast modes orderparameter interaction

slow modes IR temperature ⇤UV ⇤IR scale ⇤L solid phase liquid phase pressure pc critical point (a) (b)

vapour temperature Tc h m > 0 m ≠ 0 m = 0 ultra-ﬁne T µIR adjusment FM Tc PM scale m < 0 ⇤(~k ) ⇤(~k ) 3 4 ⇤IR ⇤UV

Figure 3.2: Sketch of the hierarchy problem.

(~k1) (~k2)

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-matter physics as well as in particle physics can be described in terms of a functional integral which in short-hand notation can be written as Z Z = Φe−S[Φ] , (4.1) D

with the action S[Φ]. We have given an example for the action S[Φ] in terms of a complex scalar field which finds applications in the superfluid transition of non-relativistic interacting bosons 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 relevant degrees of freedom. Therefore, a profound understanding of quantum and statistical field theories requires a sophisticated toolbox of theoretical methods. Functional methods are suitable for the computation of generating functionals for correlation functions. The generating functionals contain 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 extremely broad range of physical systems. Further, theses functional methods also provide a deeper conceptual 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 functions, 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 will refer to quantum or statistical field theory just shortly as QFT. • in Euclidean QFT the correlation functions are defined as Z Φ(x )...Φ(x ) := Φ Φ(x )...Φ(x )e−S[Φ] , (4.2) h 1 n i N D 1 n

where xi are suitable coordinates in position or momentum space. • we fix the normalization such that 1 = 1. N h i • functional integral measure includes implicit regularization with ultraviolet cutoff Λ Z Z Φ Φ. (4.3) D −→ Λ D • the regularized measure shall preserve the symmetries of the theory. • in case we have to deal with Minkowski-valued correlators, we assume that these can be 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 Z[J] = Φe−S[Φ]+Φ·J , (4.4) D with the source term Φ J = R ddx Φ(x)J(x). · • Recall that at ﬁnite temperature, we had Z[0] = exp[ β ], where is the free energy. − F F 4.2.1 Correlation functions • the functional derivative acts as: δΦ(x) δJ(x) = δ(d)(x y) , or = δ(d)(x y) . (4.5) δΦ(y) − δJ(y) − • therefore, we obtain

δ R d R d e d yJ(y)Φ(y) = Φ(x)e d yJ(y)Φ(y) (4.6) δJ(x) 1 δ δ Φ(x1)...Φ(xn) = ... Z[J] (4.7) ⇒ h i Z[J] δJ(x1) δJ(xn) J=0 Z[J] contains full information about the theory. ⇒ • Deﬁnition: n-point correlator in the presence of a source J: 1 Z Φ(x )...Φ(x ) = Φ Φ(x )...Φ(x )e−S[Φ]+Φ·J . (4.8) h 1 n iJ Z[J] D 1 n

52 4.2. Generating functional

What does Z[J] generate? • Z[J] is the generating functional and generates the correlation functions Φ(x )...Φ(x ) h 1 n i • For example, the two-ﬁeld Green’s function can be pictorially represented as

Φ(x )Φ(x ) = x x (4.9) h 1 2 i 2 1

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

Φ(x )Φ(x )Φ(x )Φ(x ) (4.10) h 1 2 3 4 i (e.g. using the Wick theorem) contains disconnected contributions of the form:

x4 x3 Φ(x1)Φ(x2) Φ(x3)Φ(x4) = (4.11) h ih i x2 x1

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

Φ(x )Φ(x ) = x x (4.12) h 1 2 i 2 1 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 Φ: D Z δ Φ e−S[Φ]+Φ·J = 0 (4.13) D δΦ(x) Z δS Φ J(x) (x) e−S[Φ]+Φ·J = 0 (4.14) ⇒ D − δΦ

δS J(x) (x) = 0 (4.15) ⇒ − hδΦ iJ

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

δS δ J(x) Z[J] = 0 (4.16) − δΦ(x) δJ

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

– for example, 2-point correlation function can be written in terms of 4-point correlation 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 the Schwinger functional: Z W [J] = log Z[J] = log Φe−S[Φ]+J·Φ (4.17) D • 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: 1 Z δW ϕ(x) := Φ = Φ Φ(x)e−S[Φ]+J·Φ = (4.18) h iJ Z[J] D δJ(x) 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] = sup (J ϕ W [J]) (4.20) J · − • for any given ϕ a special J = J is singled out for which J ϕ W [J] approaches its sup · − supremum (J = J[ϕ]).

• this deﬁnition of Γ guarantees that Γ is convex.

• at J = Jsup: δΓ[ϕ] Z δJ[y] Z δW [J] δJ[y] = J(x) + ϕ(y) = J(x) (4.21) δϕ(x) y δϕ(x) − y δJ[y] δϕ(x) | {z } use eq. (4.20)

δΓ[ϕ] = J (4.22) ⇒ δϕ

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

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

54 4.3. Eﬀective action

Functional integro-diﬀerential equation for eﬀective action

Z e−Γ[ϕ]+J·ϕ = eW [J] = Φe−S[Φ]+J·Φ (4.23) D

Z δΓ[ϕ] −S[Φ]+R (Φ−ϕ) e−Γ[ϕ] = Φe δϕ (4.24) D

Φ→Φ+ϕ Z δΓ[ϕ] −S[Φ+ϕ]+R Φ e−Γ[ϕ] z}|{= Φe δϕ (4.25) D • 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: ∞ X 1 Z Γ[ϕ] = Γ(n)(x , ..., x )ϕ(x )...ϕ(x ) n! 1 n 1 n n=1 x1,...,xn

F.T. ∞ X 1 Z z}|{= δ(p + p + ... + p )Γ(n)(p , ..., p )ϕ(p )...ϕ(p ) (4.26) n! 1 2 n 1 n 1 n n=1 p1,...,pn

• 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 in an infinite tower of coupled integro-differential equations for the Γ(n): Dyson-Schwinger eqautions (DSE)

• functional method of constructing approximate solutions by truncated (= ﬁnite ansatz for 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 the classical ﬁelds: δ2Γ Γ(2)(x, y) = (4.27) δϕ(x)δϕ(y)

(2) −1 δ2W (2) we ﬁnd Γ = G with G(x, y) = δJ(x)δJ(y) = W (x, y)

Γ(2)W (2) = 1 (4.28) ⇒ or more explicitly Z ddq00 Γ(2)(q, q00)W (2)(q00, q0) = (2π)dδ(d)(q q0). (4.29) (2π)2 −

55 4. Functional methods in quantum field theory

Perturbation theory for Γ[ϕ] From eq. (4.25) we obtain:

Z δΓ[ϕ] −S[Φ+ϕ]+R Φ(x) Γ[ϕ] = log Φe x δϕ(x) (4.30) − Λ D with • classical/background field ϕ • fluctuation field Φ We can rewrite this as an implicit equation “loops00 z }| { Γ[ϕ] = S[ϕ] + Γl[ϕ] (4.31) with Z R δΓ[ϕ] −(S[Φ+ϕ]−S[ϕ])+ x δϕ(x) Φ(x) Γl[ϕ] = log Φe (4.32) − Λ D Now, we make a saddle-point approximation (expand around Φ = 0) Z δS 1 Z δ2S S[ϕ + Φ] = S[ϕ] + Φ(x) + Φ(x)Φ(y) + ... (4.33) x δϕ(x) 2 x,y δϕ(x)δϕ(y) | {z } =S(2)(x,y)     Z  Z Z   1 (2) δΓl (3)  Γl[ϕ] = log Φ exp S (x, y)Φ(x)Φ(y) + Φ(x) + (S ) ⇒ − Λ D −2 x,y x δϕ(x) O   | {z }  neglect→1−loop contribution  (4.34) neglecting part marked in the above equation gives the 1-loop contribution to effective action: Z Z 1 (2) Γ1l[ϕ] = log Φ exp S (x, y)Φ(x)Φ(y) (4.35) − Λ D −2 x,y

1 − 2 = log det S(2) (4.36) − Λ now use log det A = Tr log A 1 Γ [ϕ] = Tr log S(2)[ϕ] (4.37) ⇒ 1l 2 This provides us with important expression for perturbation theory for Γ[ϕ] at one-loop order: 1 Γ[ϕ] = S[ϕ] + Tr log S(2)[ϕ] (4.38) 2 In momentum space, the one-loop integral explicitly reads: 1 Z ddq ddq0 Γ = (2π)dδ(d)(q q0) log S(2)[ϕ] (q, q0) (4.39) 1l 2 (2π)d (2π)d − | {z } Tr

56 4.4. Perturbative renormalization

4.4 Perturbative renormalization

liquid-gas (1st order) theory space Here, we want to understand more systematically how renormalization in perturbation theory works. This will lead us to a classiﬁcation of perturbativelyS + renormalizable theories.

ultra-ﬁnefast modes µIR adjusment 4.4.1 Divergenciesorderparameter in perturbation theory interaction

scale slow modes Example:IR φ3 theorytemperature on the 1-loop level. The corresponding action can be written as: ⇤UV ⇤IR ⇤UV ⇤IR scale ⇤L solid phase liquid phase Z pressure d 1 2 1 2 2 1 3 pc criticalS[ϕ ]point = (a) d x (∂ ϕ(b)) + m ϕ + gϕ (4.40) 2 µ 2 3!

vapour • Remark: this theory is unphysical as the potential is not bounded from below: temperature Tc h m > 0 m ≠ 0 m = 0 T FM Tc PM m < 0 (~k ) (~k ) ⇤ 3 ⇤ 4 '

(~k1) (~k2) • However, there is a well-deﬁned perturbative expansion and further the vertex structure appears in many physically relevant theories.

S(2)[ϕ] = ∂2+m2 + gϕ(x) (4.41) − 1 Γ [ϕ] = Tr log ∂2 + m2 + gϕ(x) (4.42) ⇒ 1l 2 − 1 1 = Tr log 1 + gϕ(x) + const. (4.43) 2 ∂2 + m2 − ∞ 1 X 1 g n = Tr − ϕ(x) + const. (4.44) −2 n ∂2 + m2 n=1 −

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

Z d (n) (n 1)! n d q 1 1 1 Γ1−loop = − ( g) d 2 2 2 2 ... 2 2 − 2 − (2π) q + m (q + p1) + m (q + p1 + ... + pn−1) + m (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 p2, ..., p2 , m2: 1 n Z Γ(n) ddqq−2n (4.46) 1−loop ∼ therefore, it diverges for d 2n. ≥ • more explicitly:

– d = 1: all correlation functions are ﬁnite (quantum mechanics) – d = 2: 1-point function diverges (tadpole):

g Z ddq 1 Γ(1) = (4.47) 1−loop 2 (2π)d q2 + m2

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

• Regularization: cut the momentum integral by integrating in a sphere q < Λ with Λ2 p2, ..., p2 , m2 | | 1 n cutoﬀ regularization, see also Eq. (3.4) → 2 2 2 Now, focus on the case d = 6: Determination of divergencies by expansion in p1, ..., pn, m :

g Z Λ d6q 1 g Z d6q 1 m2 m4 1 Γ(1) = = + + ( ) (4.48) 1−loop 2 (2π)6 q2 + m2 2 (2π)6 q2 − q4 q6 O q8 where we employed a Taylor series expansion of the integrand in the external momenta and 2 2 2 the mass p1, ..., pn, m . Further, we use the spherical integral:

Z Λ 6 Z Λ d q dq 5 = V 5 q (4.49) (2π)6 S (2π)6

3 where VS5 = π . Therefore, we get:

g Λ4 m2Λ2 Λ Γ(1) = + m4 log + (1) (4.50) 1−loop 27π3 4 − 2 m O

• 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:

g2 Λ2 p2 Λ Γ(2) = + (2m2 + ) log + (1) (4.51) 1−loop 27π3 2 3 m O g3 Λ Γ(3) = log + (1) (4.52) 1−loop 26π3 m O 58 4.4. Perturbative renormalization

• why d = 6? – d = 6 is special as divergencies appear only in vertices that maximally are part of the original action: Z 1 Z 1 Z S[ϕ] = Γ(1) ϕ + Γ(2) ϕ2 + Γ(3) ϕ3 (4.53) 0−loop 2 0−loop 3! 0−loop

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

(1) Γ0−loop = 0 (4.54) (2) 2 2 Γ0−loop = p + m (4.55) Γ(3) (p , p , (p + p )) = g (4.56) 0−loop 1 2 − 1 2

– note that Γ(1) can be generated by a shift ϕ ϕ + const. 0−loop → – furthermore, the dependence of divergencies on the external momenta equals the (n) momentum dependence of the bare vertices Γ0−loop. (4) (4) – on the contrary in d = 8: Γ0−loop = 0 and Γ1−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: Z 1 1 1 Γ = d6x ga (Λ)ϕ(x) + g2a (Λ)(∂ ϕ)2(x) + g2a (Λ)ϕ2(x) + g3a (Λ)ϕ3(x) 1−loop,div 1 2 2 µ 2 3 3! 4 (4.57) where from eqs. (4.50), (4.51) and (4.52) we have: 1 Λ4 m2Λ2 Λ a (Λ) = + m4 log (4.58) 1 27π3 4 − 2 m 1 Λ a (Λ) = log (4.59) 2 3 27π3 m · 1 Λ2 Λ a (Λ) = + 2m2 log (4.60) 3 27π3 − 2 m 1 Λ a (Λ) = log (4.61) 4 26π3 m Note: the origin of these divergencies is that ﬂuctuations on all scales contribute to the vertex functions.

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

• Idea: Fix the coupling, mass, chemical potential, etc. at a renormalizartion point/scale µ (scale of measurement) to the (physical) values mR, gR, ϕR by renormalization conditions for the full vertices: The renormalized physical parameters are ﬁxed by means of the eﬀective action:

Γ(2)(p2 0) = m2 (fixes physical mass) (4.62) → R ∂ Γ(2)(p2 0) = 1 (fixes normalization of field) (4.63) ∂p2 →

(3) 2 2 2 Γ (p1 = p2 = p3 = 0) = gR (ﬁxes physical coupling) (4.64)

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

• More general: ﬁx parameters at p2 µ2, where µ is the renormalization point/scale → Γ(2)(p2 = µ2) =m ˜ 2 = m2 (p = 0) (4.65) ⇒ 6 R • 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: Z 1 1 S[ϕ] S [ϕ] = ϕ( ∂2 + m2)ϕ + gϕ3 (4.66) ≡ Λ 2 − 3! S [ϕ ] + δS [ϕ ] (4.67) ≡ R R R R ﬁnite " z }| { Z 1 1 = ϕ ( ∂2 + m2 )ϕ + g ϕ3 (4.68) 2 R − R R 3! R R # 1 1 1 + (Z 1)(∂ ϕ )2 + δm2ϕ2 + g (Z 1)ϕ3 (4.69) 2 ϕ − µ R 2 R 3! R g − R | {z } counterterms with

1/2 ϕ = Zϕ ϕR (4.70) Z g = g g (4.71) R 3/2 Zϕ 2 2 2 1 m = (mR + δm ) (4.72) Zϕ 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

2 Now, assume that Zϕ,Zg, δm have an expansion in gR such that (Z 1), δm2, (Z 1) = (g2 ) (4.73) ϕ − g − O R 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 S [ϕ ] → R R we then have: → 1 Γ[ϕ ] = S [ϕ] + Tr log S(2)[ϕ] + (2 loop) (4.74) R Λ 2 Λ O − 1 = S [ϕ ] + δS [ϕ ] + Tr log S(2)[ϕ ] + (2 loop) (4.75) R R R R 2 R R O − 2 now the renormalization conditions, eqs. (4.62), ﬁx the relations Zϕ,Zg, δm : In particular, for the divergent parts, we obtain:

Γ[ϕ ] = (Λ0) (4.76) R O 1 δS [ϕ ] + Tr log S(2)[ϕ ] = (Λ0) (4.77) ⇒ R R 2 R R O

61 4. Functional methods in quantum field theory

with eq. (4.57) for perturbative calculation with gR: " 1 Z 1 Tr log S(2)[ϕ ] = d6x g a (Λ)ϕ (x) + g2 a (Λ)(∂ ϕ )2(x) 2 R R R 1 R 2 R 2 µ R # 1 1 + g2 a (Λ)ϕ2 (x) + g3 a (Λ)ϕ3 (x) (4.78) 2 R 3 R 3! R 4 R

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

Z = 1 g2 a (Λ) (4.79) ϕ − R 2 Z = 1 g2 a (Λ) (4.80) g − R 4 2 2 δm = gRa3(Λ) (4.81)

the assumption eq. (4.73) is selfconsistent ⇒ Γ[ϕ ] is ﬁnite (on a 1-loop level) and only depends on physical parameters! ⇒ R 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 the ϕ3 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 that the 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, δZ = Z 1, etc. for any theory such ϕ ϕ − that 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: −1 1 ∆(p) Γ(2)(p) (4.83) ≡ 0 |{z}∼ p σ p→∞ | | where we have used the assumptions that the limit p is O(d) symmetric → ∞ • then the canonical dimension of a field is defined as d σ [ϕ] := − (4.84) 2

• Examples:

– scalar ﬁeld: 1 1 d 2 ∆(p) = [ϕ] = − (4.85) p2 + m2 |{z}∼ p2 ⇒ 2 p→∞ liquid-gas (1st order) theory space – fermion ﬁeld (Dirac):

1 1 + d 1 ∆(p) = S [ϕ] = − (4.86) pµγµ + m |{z}∼ p ⇒ 2 p→∞ ultra-ﬁnefast modes – Note: In theseµ examples,IR the canonical dimensionadjusment can also be inferred from the orderparameter interaction action (dimensionless). This is not possible for higher spin ﬁelds. scale slow modes ' IR • in theories with polynomial interactions, the interaction can be written as a sum of temperature ⇤ ⇤ vertices: ⇤UV IR UV ⇤IR scale ⇤L solid phase Z k liquid phase ∂ pressure d n1 n2 ns V (ϕ) d x ϕ1(x) ϕ2(x) ...ϕs(x) (4.87) pc critical point (a) ∼ ∂x(b)

where the ϕi are diﬀerent types of ﬁelds. vapour • pictorially, such a vertex can be represented as:

temperature Tc ' ,n h m > 0 2 2

m ≠ 0 m = 0 '1,n1 T FM Tc PM '3,n3 m < 0 ⇤(~k3) ⇤(~k4) 63

(~k1) (~k2) 4. Functional methods in quantum field theory

• the dimension of a vertex is:

s X δ[V ] = d + k + n [ϕ ] (4.88) − i i i=1

• vertex in momentum space Z V (ϕ) ddp ...ddp δ(d)(p + ... + p )pkϕ (p )...ϕ (p + ... + p ) ∼ 1 n1+n2+...+ns 1 n1+...+ns 1 1 s n1 ns (4.89)

vertex contains momentum-conserving δ function ⇒ • divergencies in the n-point proper vertices Γ(n) appear in the integrations over the ﬂuctuation momenta. Perturbation theory sorts the contributions to Γ(n) according to the 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: ⇒ X X δ(γ) = dL I σ + v k (4.90) − i i α α i α 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 from subdiagrams).

64 liquid-gas (1st order) theory space S +

ultra-ﬁnefast modes µIR adjusment orderparameter interaction

scale slow modes ' IR temperature ⇤UV ⇤IR ⇤UV '2,n2 4.4. Perturbative renormalization ⇤IR scale ⇤L solid phase liquid phase pressure '1,n1 • Example (ϕ3 theory): pc critical point (a) (b) – here, we have σ =' 23,,n k3= 0 δ(γ) = dL 2I ⇒ − – in d = 6 and L = 1 (1-loop level): δ(γ) = 6 2I vapour − for I = 1, 2, 3 δ(γ) = 4, 2, 0 ⇒ −→ temperature Tc h m > 0 I =1 I =2 I =3 m ≠ 0 m = 0 T ⇤4 ⇤2 log ⇤ FM Tc PM ⇠ ⇠ ⇠ m < 0 ⇤(~k3) ⇤(~k4) • Diﬀerent expression of δ(γ) by topological relations:

– be Ei the number of external lines of a ﬁeld ϕi – we ﬁnd X X (~k1) (~k2) δ(γ) = d [ϕ ] E + v δ(V ) (4.91) − i i α α i α

– 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 information about 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 terms that correspond to superficially divergent amplitudes. (very technical proof!)

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

• 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 correlation (n) function Γ (any number of external legs Ei) the removal of divergencies then requires infinitely many counter terms i.e. -many ⇒ ∞ liquid-gas (1st order) physical theoryparameters space Loss of predictivity! ⇒ S + • Super-renormalizable, if all δ(Vα) < 0 only a finite number of diagramsultra-finefast are modes superficially divergent. ⇒µIR adjusment orderparameter 4 interaction Example (ϕ theory in d = 3): scale slow modes ' IR temperature 1 ⇤UV ⇤IR δ(⇤VUV) = 3 + 0 + 4 = 1 (4.92) I =1 '2,n2 I =2 I =3 2 ⇤IR scale ⇤L − · − solid phase liquid phase 1 pressure '1,n1 δ(γ) = 3 E v (4.93) pc critical point (a) (b) ⇒ − 2 − ⇤4 ⇤2 log ⇤ '3,n3 ⇠ which gives⇠ the divergent⇠ diagrams:

vapour – E = 2: temperature Tc h m > 0 m ≠ 0 m = 0 T FM Tc PM v =1 v =2 v =2 m < 0 ⇤(~k3) ⇤(~k4)

– E = 4 has at least v = 2 δ(γ) ≥ < 0. ⇒ |E 2 • Renormalizable, if at least one δ(Vα) = 0 (and there is no δ(Vα) > 0). (~k ) (~k ) 1 2 -many diagrams have δ(γ) > 0, however, at fixed E : δ(γ) independent from v ⇒∞ i α 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:

(4.91) s X δ(V ) z}|{= d + k + n [ϕ ] (4.94) − i i i=1

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

d 2 d 2 n [ϕ] = − δ(V ) = d + n − = d( 1) n (4.95) 2 ⇒ − 2 2 − − • ϕ3 is renormalizable in d = 6 • ϕ4 is renormalizable in d = 4 • ϕ6 is renormalizable in d = 3

d−1 2. spin-1/2, [Ψ] = 2 (Dirac ﬁeld) • (ΨΨ)¯ 2 in d = 2 (Gross-Neveu model) • ϕΨΨ¯ in d = 4 (Yukawa model) • ΨΨ¯ ϕ2 in d = 3

d 3. spin-1 (no gauge theories): [Vµ] = 2 R • renormalizable only in d = 2, (S ∂V ∂V ), e.g., with Ψ¯ γ V Ψ interaction kin ∼ µ µ 4. spin 3/2: no perturbatively renormalizable theories. ≥ d−2 5. gauge theories, spin 1: [Aµ] = 2 a a ¯ 2 2 • FµνFµν in d = 4 with interactions ΨγµAµΨ, ∂µϕAµϕ, ϕ A renormalizable 6. No physically acceptable (from the point of view of particle physics) and renormalizable theory exists for d > 4 (perturbatively)! It is not known whether this property is logically connected with the fact that spacetime has just four dimensions or is a mere coincidence. (from the book by J. Zinn-Justin)

Chapter 5

Functional renormalization group

The functional renormalization group provides a nonperturbative and practical theoretical framework for quantum field theoretical calculations. More specifically, it represents the implementation of Wilson’s idea for the renormalization group with the functional methods which we have learned in the previous chapter. The central object of the functional RG is a flow equation – the Wetterich equation, which describes the evolution of correlation functions or the generating functional under successive application of momentum-shell integrations in terms 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 a momentum-shell parameter k (also called ”infrared cutoff scale” not to be confused with the ultraviolet cutoff Λ). Γk will be defined such that it interpolates between two important limits:

1. For k Λ: Γ corresponds to the bare action to be quantized, i.e. Γ → S . → k k Λ ' bare 2. For k 0: Γ corresponds to the full eﬀective action, i.e. Γ → = Γ. → k k 0 To achieve this, we modify the generating functional Z[J], cf. eq. (4.4) by adding a term ∆Sk[Φ] to the action: Z Wk[J] −S[Φ]−∆Sk[Φ]+J·Φ e Zk[J] = Φe (5.1) ≡ Λ D where the additional term reads 1 Z ddq ∆S [Φ] = Φ( q)R (q)Φ(q) . (5.2) k 2 (2π)d − k

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 desired regularization properties for the eﬀective average action Γk: 1. It should implement infrared regularization which is realized by the condition:

lim Rk(q) > 0 (5.3) 2 q →0 k2

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

lim Rk(q) = 0 (5.4) 2 k →0 q2

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

lim Rk(q) (5.5) k→Λ→∞ → ∞

This filters out the classical field configuration of the action Γ → S + const. k Λ → k R t , ∂ k R

2 Figure 5.1: Sketch of a representative regulator function Rk(q ) (black solid line). For later 2 purpose, we also sketch its derivative ∂tRk(q ) (orange dashed). For this plot, we chose 2 2 (q2/k2)4 2 Rk(q ) = 2k /(e + 1). You can check that another suitable choice would be Rk(q ) = (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 limits k 0 and k which we have constructed to correctly give Γ and S , respectively. → → ∞ bare Therefore it is convenient to use the short-hand notation: k ∂ t = log ∂ = k . (5.6) Λ t ∂k We keep the source term J independent from k and start with the derivative of the generating functional Z Wk −S[Φ]−∆Sk[Φ]+J·Φ ∂te = ∂t Φe (5.7) Λ D Z Wk −S[Φ]−∆Sk[Φ]+J·Φ e ∂tWk = Φ( ∂t∆Sk[Φ])e (5.8) ⇒ |{z} Λ D − =Zk   1 Z  1 Z   −S[Φ]−∆Sk[Φ]+J·Φ  ∂tWk = ∂tRk(q) Φ Φ( q)Φ(q)e  (5.9) ⇒ |{z} −2 q  Zk Λ D −  (5.2) | {z } =hΦ(−q)Φ(q)i

We now recall that the (k-modiﬁed) connected propagator is δW G (q) = k (q) = Φ( q)Φ(q) ϕ( q)ϕ(q) (5.10) k δJδJ h − i − − where according to eq. (4.18), we have δW [J] ϕ(q) = Φ(q) = k (5.11) h i δJ(q) and for convenience we have suppressed the index J for the exepectation values ... . h iJ

So, with eq. (5.9) we obtain 1 Z ∂tWk = ∂tRk(q)(Gk + ϕ( q)ϕ(q)) (5.12) −2 q − 1 Z = ∂tRk(q)Gk ∂t∆Sk[ϕ] (5.13) −2 q −

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 a modified Legendre transform reading

Γk[ϕ] = sup (J ϕ Wk[J]) ∆Sk[ϕ] (5.14) J · − − cf. eq. (4.20).

71 5. Functional renormalization group

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

δΓ [ϕ] J(x) = k + (R ϕ)(x) . (5.15) δϕ(x) k

This can be used to obtain: δJ(x) δ2Γ [ϕ] = k + R (x, y) . (5.16) δϕ(y) δϕ(x)δϕ(y) k

Furthermore, we also have

δϕ(y) δ2W [J] = k G (y x0) . (5.17) δJ(x0) δJ(x0)δJ(y) ≡ k −

From the previous two equations, we can now deduce the important equation Z 0 δJ(x) δJ(x) δϕ(y) δ(x x ) = 0 = 0 (5.18) − δJ(x ) y δϕ(y) δJ(x )

Z 2 δ Γk[ϕ] 0 = + Rk(x, y) Gk(y x ) (5.19) y δϕ(x)δϕ(y) − Z (2) 0 = Γk [ϕ] + Rk (x, y) Gk(y x ) . (5.20) y − In compact notation this reads:

1 (2) = (Γk + Rk)Gk (5.21)

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

∂tΓk[ϕ] = ∂tWk[J] + (∂tJ) ϕ ∂t∆Sk[ϕ] = ∂tWk[J] ∂t∆Sk[ϕ] (5.22) |{z} − · − − − (5.14)

1 Z = ∂tRk(q)Gk . (5.23) |{z} 2 q (5.13)

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

1 −1 ∂ Γ [ϕ] = Tr Γ(2)[ϕ] + R ∂ R . (5.24) t k 2 k k t k

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

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 on the r.h.s. of the equation is the exact propagator. At this stage no approximations have been made.

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

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

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

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

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

7. The regulator term has to fulfill the basic conditions given in Sec. 5.1.1. Apart from these conditions, it can be chosen arbitrarily. The precise RG trajectory then depends on the choice of Rk and reflects the RG scheme dependence. The final point of the RG 3.2. The Functional Renormalisation Group trajectory, however, is independent of the regulator choice.

k= O3

{Oi} k=0 2 O O1

Figure 3.4.: The flow equation interpolates between the classical action at the renormal- 8. The Wetterich equationisation can group bescale used⇤ toand derive the full perturbation quantum e↵ective theory: action at k The0. loop-expansion The 1−loop 2 in perturbation theoryflow expands is a trajectory the in action the space in of powersall possible of theories~:Γ whichk = S are+ spanned~ Γ by + (~ ). = k orthogonal operators/couplings (2)i, and serve as expansion coecients, e.g. O 2 (2) Therefore, to one-loop1 order, p we0 . Forcan di↵ useerent Γ implementationsk = S on of the the regulation r.h.s. procedure of the Wetterich equation. This gives: the flow( ) varies at non-vanishingOk, but the end-points of the flow are un- ambiguous.O = ( = Furthermore,) the trajectory depends on the chosen truncation, as may the quantum e↵ective1 action. By choosing satisfying truncations the limit k ∂ 0Γ must1−loop be rendered= ∂ independentTr log(S(2) of the+ R truncation.) (5.25) t k 2 t k → 1 Γ1−loop = S + Tr log S(2) + const. (5.26) ⇒ k 2 73

Figure 3.5.: Graphical representation for the ﬂow equation of a real scalar particle given in eq. (3.15). The solid line with the shaded circle represents the full nonperturbative propagator of the scalar ﬁeld. The crossed circle indicates the insertion 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 contributions 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 5. Functional renormalization group

5.2.2 Method of truncations The functional RG provides an exact formulation of the renormalization group. For practical calculations, however, approximations for the action functionals have to be made, i.e. the action 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:

∞ X 1 Z Γ [ϕ] = Γ(n)(x , ..., x )ϕ(x )...ϕ(x ) (5.27) k n! k 1 n 1 n n=1 x1,...,xn Inserting this expansion into the Wetterich equation provides RG ﬂow equations for the vertex (n) functions Γk .

Derivative expansion Another important example is the derivative expansion which is often used for the description of scalar ﬁeld theories: Z 1 2 4 Γk[ϕ] = Zk(ϕ)(∂µϕ) + (∂ ) + Uk(ϕ) (5.28) x 2 O

where Uk(ϕ) is the eﬀective potential which again can be expanded in ﬁeld monomials, e.g., 1 2 2 1 4 Uk = 2 mkϕ + 4! λkϕ + ... .

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 vertex functions is not important and the anomalous dimension is small. This is the case for the critical behavior of O(N) symmetric scalar models. The LPA for the Wetterich equation, eq. (5.24), is given by the following ansatz for the flowing effective action Z 1 2 Γ[ϕ] = [~ ϕ] + Uk(ϕ) . (5.29) ~x 2 ∇

• 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 broken symmetries the eﬀective potential Uk(ϕ) is the most important quantity. It corresponds to 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-dimensional − classical 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 is invariant under the symmetry ϕ R ϕ with any N N orthogonal matrix R a → ab b × eﬀective potential only depends on inner product ρ = 1 PN ϕ ϕ = 1 ϕ ϕ. ⇒ 2 a=1 a a 2 · We will use the following cutoﬀ insertion 1 Z ddq ∆S [ϕ] = ϕ ( q)R (q)ϕ (q) , (5.30) k 2 (2π)d a − k a where summation over the index a is implied and with regulator function

R (p) = (k2 p2)Θ(k2 p2) . (5.31) k − − Θ(x) is the step function. We can now insert the ansatz eq. (5.29) into the Wetterich equation, eq. (5.24). In a ﬁrst step, we obtain

Z d " 2 −1 # 1 d q 2 δ Uk(ϕ) ∂tUk[ϕ] = d Tr q δab + + Rk(q)δab ∂tRk(q) (5.32) 2 (2π) δϕaδϕb

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 = 1 ϕ ϕ, i.e. 2 · Uk(ϕ) = Uk(ρ). To that end, we rewrite the derivatives of Uk with respect to the fields ϕ as derivatives w.r.t. ρ

2 δUk(ρ) 0 δ Uk(ρ) 00 0 = Uk(ρ)ϕa = Uk (ρ)ϕaϕb + Uk(ρ)δab (5.33) δϕa ⇒ δϕaδϕb

75 5. Functional renormalization group

• To study the ground state properties of the theory, we now use that a constant field configuration 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 + 2 − λρ + ... will only depend on the length of the vector ~ϕ0, the direction is arbitrary. we can choose the coordinates so that ϕ points in the ﬁrst direction: ⇒ a,0 p ~ϕ0 = (v, 0, ..., 0) with v = 2ρ (5.34)

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

d 1 Z d q h −1i ∂ U (ρ) = ∂ R (q)Tr q2δ + R (q)δ + U 0 (ρ)δ + 2ρU 00(ρ)δ (5.35) t k 2 (2π)d t k ab k ab k ab k a1 1 Z ddq 1 N 1 = d ∂tRk(q) 2 0 00 + 2 − 0 2 (2π) q + Rk(q) + Uk(ρ) + 2ρUk (ρ) q + Rk(q) + Uk(ρ) (5.36)

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

d+2 1 N 1 ∂tUk(ρ) = k Sd 2 0 00 + 2 −0 , (5.37) k + Uk(ρ) + 2ρUk (ρ) k + Uk(ρ)

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

• The flow equation eq. (5.37) exhibits that the flow of the effective potential is driven by two essential contributions, i.e. (1) the contribution from a radial mode k2 + U 0 (ρ) + ∼ k 2ρU 00(ρ) and (2) the contribution from N 1 Goldstone modes k2 +U 0 (ρ) as expected k − ∼ k in 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 wave function renormalization Z to the kinetic term in the action q2ϕ( q)ϕ(q). Then the k ∝ − anomalous dimension can be given in terms of the ﬂow equation η = ∂ log Z . − t k

• The set of flow equations for Uk and Zk is already well-suited to describe many properties of 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 theory as before in terms of

• running couplings g¯ and ﬁeld operators : i,k Oi X Γ [ϕ] = g¯ [ϕ], e.g. [ϕ] = ϕ2, ϕ4, (∂ϕ)2,... . (5.38) k i,kOi Oi i

In the following, we will be interested in ﬁxed-point solutions, as these allow us to evaluate the scaling 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:

−dg gi,k =g ¯i,kk i , (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 of the vertex by the simple relation:

d = δ(V ) . (5.40) gi − gi The reason for choosing dimensionless couplings is that a ﬁxed point, which is a scale-free point, is hard to identify when dimensionful couplings are present, since each dimensionful coupling corresponds to a scale. The RG ﬂow of the couplings is given in terms of

• β functions:

β ( g ) = ∂ g = d g + f ( g ), (5.41) gi { n} t i,k − gi i,k i { n} where f ( g ) are functions of the dimensionless couplings. i { n} The ﬁrst term reﬂects the scale-dependence due to the canonical dimensionality, whereas the second 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 g ∗ where the all { n } the beta functions vanish simultaneously:

β ( g ∗ ) = 0 i . (5.42) gi { n } ∀

77 5. Functional renormalization group

Linearized flow and stability matrix To determine the critical exponents, which enter the scaling of observable quantities in the vicinity of a second-order phase transition, we linearize the flow around the fixed point g ∗ . { n } We find for the linearized flow

∂ g = β = B j(g g∗) + (g g∗)2 , (5.43) t i,k gi i j,k − j O − where we have introduced the stability matrix

j ∂βi Bi = . (5.44) ∗ ∂gj g

j We diagonalize the stability matrix Bi , introducing the negative eigenvalues θI

B j V I = θ V I . (5.45) i j − I i I Here, the Vi 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 solution of the coupling flow in the fixed-point. The solution to this linearized equation is given by

−θI X I k gi,k = gi ∗ + CI Vi , (5.46) k0 I

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

• Eigendirections with Re θ > 0 are called relevant directions and drive the system { I } away from the ﬁxed point as we evolve the ﬂow towards the infrared. Those infrared unstable directions determine the macroscopic physics.

• Eigendirections with Re θ < 0 die out and ﬂow into the ﬁxed point towards the { I } infrared. They are thus called the irrelevant (infrared stable) directions.

• For the marginal directions Re θ = 0, it depends on the higher-order terms in the { I } expansion 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 physical parameters to be fixed. For the flow away from the fixed point, the linearized fixed-point flow (5.43) generally is insufficient 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 the effective potential, which we derived previously, eq. (5.37). For the fixed-point analysis, we introduce dimensionless quantities, i.e. we rescale the potential and the field with appropriate 2−d −d powers of the cutoff k, i.e.ρ ˜ = k ρ and uk(˜ρ) = k Uk(ρ). Then, we obtain from eq. (5.37), 0 1 N 1 ∂tuk(˜ρ) = duk(˜ρ) (2 d)uk(˜ρ) ρ˜ + Sd 0 00 + − 0 . (5.47) − − − 1 + uk(˜ρ) + 2uk(˜ρ) ρ˜ 1 + uk(˜ρ)

Quartic potential and beta functions in d = 3

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

λ u = k (˜ρ κ )2 . (5.48) k 2 − k

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 individual couplings, 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:

2 ∂ uk(˜ρ) λk = 2 . (5.49) ∂ρ˜ ρ˜=κk

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

2 ∂ (∂tuk(˜ρ)) ∂tλk = 2 . (5.50) ∂ρ˜ ρ˜=κk

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

0 = ∂ u0 (κ ) = ∂ u0 (˜ρ) + (∂ κ )u00(κ ) (5.51) t k k t k |ρ˜=κk t k k 1 0 ∂tκk = 00 ∂tuk(˜ρ) ρ˜=κk . (5.52) ⇒ −u (κk) |

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

1 3 βκ = ∂tκk = κk + 2 (N 1) + 2 , (5.53) − 6π − (1 + 2κkλk) 2 λk 9 βλ = ∂tλk = λk + 2 (N 1) + 3 . (5.54) − 3π − (1 + 2κkλk)

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 the choice 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  2λ 1 2κ  − π2(2κλ+1)3 − − π2(2κλ+1)3 B = 9−9κλ . (5.56)  3 2λ 4 +1  18λ (2κλ+1) 1 − π2(2κλ+1)4 3π2 − Evaluation at the fixed point values, provides the stability matrix B which the has negative eigenvalues: θ = 1.611, θ = 0.3842 . (5.57) 1 2 − So, we have one relevant (θ1 > 0) and one irrelevant direction (θ2 < 0) at this non-Gaußian fixed point. This corresponds the Wilson-Fisher fixed point which we have studied with Wilson’s RG approach. The eigenvalue θ1 is related to the scaling of the field bilinear at the Wilson-Fisher fixed point: θ1 = y, cf. (3.35). Therefore, the functional RG prediction for the correlation length exponent at that level of truncation (LPA4) is: 1 νLPA4 = 0.62 . (5.58) θ1 ≈

Higher orders in the potential and anomalous dimension This estimate can easily be improved with the functional RG approach, by allowing for a more general expansion of the eﬀective potential, i.e.: n Xmax λ u = i,k (˜ρ κ )i . (5.59) k i! − k i=2 and by amending the equation for the anomalous dimension η = ∂ log Z . For example − t k with n = 6 and including η = 0, we ﬁnd the following results: max 6

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. using resummed perturbation expansions at ﬁve-loop order. Given the simplicity of the FRG approximation 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 the functional integral. Here, we will discuss its implications in the context of high-energy physics. → We consider the action and the functional integral: ⇒ Z 1 1 λ S = ddx (∂ φ)2 + m2φ2 + φ4 (5.60) 2 µ 2 4! Z Z = φ exp( S[φ]) (5.61) Λ D − 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:

Z 1 S∗ = ddx (∂ φ)2 , (5.62) G 2 µ 2 ∗ m = λ∗ = λ = 0 (5.63) ∗|G |G 6|G • In the Gaußian ﬁxed point regime: neglect higher-order corrections to rescaling of couplings (`ala Wilson):

2 2 2 4−d 6−2d m(b) = m b , λ(b) = λ b , λ6(b) = λ6b , ... (5.64)

• Or `ala FRG, cf. eq. (5.46):

m2 k−2, λ kd−4, λ k2d−6 , ... (5.65) k ∝ k ∝ 6,k ∝

• Generally:

g(b) = bθ g , b > 1 , or (FRG) : g k−θ . (5.66) k ∝ 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 ﬁeld φi (with canonical dimension [φi]) has the following behavior under RG transformations close to the Gaußian FP:

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

P − or g kl+ i ni[φi] d = kδ(V ) (5.68) k;l,ni ∝ where δ(V ) is the canonical dimension of the vertex, cf. eq. (4.88). The classiﬁcation relevant, irrelevant, marginal corresponds to the classiﬁcation ⇒ { } super-renormalizable, non-renormalizable, renormalizable . { } Consequence in d = 4: It is found in particle physics that all theories that play a role are perturbatively renormalizable theories: Standard Model of Particle Physics! → Until now, it seemed to be a mere coincidence that all the (fundamental) particle physics theories that appear in nature are (perturbatively) renormalizble.

Wilson’s interpretation:

• Suppose there is a more fundamental theory that has the symmetries of the Standard Model of Particle Physics (SU(3) SU(2) U(1)) at a ultraviolet cutoﬀ scale Λ and × × here 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: 1 λ λ V (φ) = m2φ2 + φ4 + 6 φ6 + ... (5.69) 2 4! 6! with λ = (1). i O RG: ⇒ d=4 i(d/2−1)−d z}|{ i−4 λ ≥ k = k 0 (for k 0) . (5.70) k,i 6 ∝ → → 2 Accordingly, λk decreases slowly (logarithmically) and mk 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 the QFT system is dominated by the Gaußian ﬁxed point.

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

• 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 space of action functionals. we can express fixed-point action in terms of suitable expansion of field operators : → Oi X Γ∗[ϕ] = g¯∗ [ϕ], e.g. [ϕ] = ϕ2, ϕ4, (∂ϕ)2,... . (5.71) i Oi Oi i

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

β ( g ∗ ) = 0 . (5.72) gi { n } and at least one coupling g∗ = 0 (to make it non-Gaußian). i 6 ∗ • 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 for ⇒ Λ 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 the theory in the ultraviolet we invert the discussion.

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

I X k θ g = g∗ + CI V I 0 . i i i k I

• 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 Re θ < 0 run away from the ﬁxed point as we increase k (green { I } arrows in plot below).

• The directions with Re θ > 0 run into from the ﬁxed point as we increase k (blue { I } arrows 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.

I • The tangential space of S is spanned by the relevant RG eigenvectors Vi . • The number of linearly independent relevant directions at the ﬁxed point corresponds to the dimension of S:

dim S ∆ = number of relevant directions with θ > 0 . (5.73) ≡ I

• 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) = ∆ = dim S < 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: → Z 1 4 q S = d x det(gµν)(R 2λ) (5.74) 16πGN − − 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 without going into the details of the general theory of relativity is given as follows:

• We ﬁrst recall Coulomb’s law, reading 1 V (r) = α , (5.75) r

2 where α = e is the ﬁne-structure constant which is a dimensionless quantity α 1 . 4π ≈ 137 (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: M M V (r) = G 1 2 . (5.76) N r

• Comparison with to Coulombs’s law, this implies that Newton’s gravitational constant G has canonical dimension 2 (in units of mass or momentum).1 N − • 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 GN = 0 is justified. The question is now, whether the Einstein-Hilbert action admits a non-Gaußian fixed point with a finite number of relevant parameters in the sense of the asymptotic safety scenario. Evidence for this scenario has been collected with the FRG methods with increasing efforts in the last 15 years, see for example see https://arxiv.org/pdf/1205.5431.pdf for lecture notes. →

Gross-Neveu models in 2 < d < 4 Gross-Neveu models are models of Dirac fermions with four-fermion interactions, e.g.: Z d 2 SGN = d x ψ¯∂ψ/ + λψ(ψψ¯ ) . (5.77)

These models are perturbatively non-renormalizable as the coupling carries a negative mass dimension. This conclusion, however, is only an artifact of the perturbative quantization procedure. In fact they are non-perturbatively renormalizable in 2 < d < 4 at a non-Gaußian ﬁxed point and hence can be extended to arbitrarily high scales, see, e.g.: https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.55.363.

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

Z 1 1 S[Φ] = S [Φ] + S [Φ] = d4x Φ( ∂2 + m2)Φ + λΦ4 (A.1) 0 I 2 − 4

– where Φ is a real scalar ﬁeld – and ∂2 = ∂ ∂ and instead of µ we now use m2 as parameter for the ﬁeld bilinear µ µ − – and S [Φ] = R d4x 1 Φ( ∂2 + m2)Φ and S [Φ] = R d4x 1 λΦ4 0 2 − I 4 • with eq. (4.6), we then obtain: Z Z[J] = Φe−S[Φ]+Φ·J (A.2) D

4 λ R 4 δ − 4 d x δJ(x) = e Z0[J] (A.3)

with Z Z [J] := Φe−S0[Φ]+Φ·J (A.4) 0 D

• we rewrite Z0[J] Z − 1 R ddxddyΦ(x)K(x,y)Φ(y)+R ddxJ(x)Φ(x) Z [J] = Φe 2 (A.5) 0 D

• where K(x, y) = ( ∂2 + m2)δ(d)(x y) − − • and the inverse of K: Z ddz G(x, z)K(z, y) = δ(d)(x y) (A.6) −

89 A. Remark on generating functional

• now, parametrize Z Φ(x) = Φ(˜ x) + ddz G(x, z)J(z) (A.7)

Z − 1 R ddxddyΦ(˜ x)K(x,y)Φ(˜ y) 1 R ddxddy J(x)G(x,y)J(y) Z [J] = Φ˜e 2 e 2 (A.8) ⇒ 0 D Z −S [Φ]˜ 1 R ddxddy J(x)G(x,y)J(y) = Φ˜e 0 e 2 (A.9) D

• here, we have used Φ(x) = Φ(˜ x) translational invariance of Φ (measure) D D ⇐ D • so, we obtain for the generating functional Z −S δ S [Φ]˜ 1 R ddxddy J(x)G(x,y)J(y) Z[J] = e I [ δJ ] Φ˜e 0 e 2 (A.10) D

1 −S δ 1 R ddxddy J(x)G(x,y)J(y) Z[J] = e I [ δJ ]e 2 (A.11) ˜ N ∞ • where ˜ = det1/2( ∂2 + m2)/√2π from Gaußian integration (cf. problem set 3). N − • schematically Z[J] can be represented by the following picture:

• derivatives of Z[J] with respect to the source ﬁelds J δnZ[J] generate the connected + disconnected Green’s functions → J

90 A.2. Example: two-point function

A.2 Example: two-point function

• for example the 2-point function is generated by

2 2 δ Z[J] −S δ δ 1 J·G·J = e I [ δJ ] e 2 (A.12) δJ(x)δJ(y) δJ(x)δJ(y)

• with Z 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 the two sources J with propagator G.

• this yields

2 δ Z[J] −S δ 1 J·G·J = e I [ δJ ] G(x, y) + (G J)(x)(G J)(y) e 2 (A.14) δJ(x)δJ(y) { · · }

with Z (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] h λ Z δ 2 = G(x, y) ddzG(x, z)G(y, z) ⇒ δJ(x)δJ(y) − 2 δJ(z) λ Z δ 3 ddz(G J)(x)G(y, z) − 2 · δJ(z) 7 Z δ 3 δ 3 i + λ2 ddzddz0G(x, z)G(y, z0) Z[J] (A.16) 4 δJ(z) δJ(z0)

redundant information! ⇒

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)