the functional renormalization group method – an...

The Functional Renormalization Group Method –An Introduction

Rainer Stiele-Argüello

Laboratoire de Mathématiques et Physique Théorique (LMPT)Université François-Rabelais de Tours, France

&Institut de Physique Nucléaire de Lyon, France

[email protected]

[email protected]

Motivation FRG method QCD Application Summary

Outline

1 Motivation and basic idea

2 Functional Renormalization Group (FRG) method

3 Application to QCD

4 Summary

The Functional Renormalization Group Method – An Introduction 1Rainer Stiele-Argüello


FRG Method – What is it about?

http://vertixap.com/ptv_vision

Relate physics across different length scales. . . in continuum field theory


http://vertixap.com/ptv_vision


FRG Method – A variety of applications

The Functional Renormalization Group (FRG) method can beapplied to variety of physical systems

• strong interaction

• electroweak phase transition

• effective models in nuclear physics

• condensed matter systemse.g. Hubbard model, liquid He4, frustrated magnets,superconductivity . . .

• ultra-cold atoms

• quantum gravity

after A. Wipf: The Functional Renormalization Group Method – An Introduction,https://saalburg.aei.mpg.de/wp-content/uploads/sites/25/2017/03/wipf15.pdf


https://saalburg.aei.mpg.de/wp-content/uploads/sites/25/2017/03/wipf15.pdf


Motivation

• Cover physics across different length scales.

• Relate microscopic theory to macroscopic (effective) theory.⇒ effective/thermodynamic potential

• Loose the irrelevant details of the microscopic theory(→ How to decide what is relevant and what is not?).

• Cope with the important rôle of fluctuations (long-range in thevicinity of a critical point).

• Consider universality: certain behavior in the vicinity of a criticalpoint independent from the details of the theory (e.g. criticalexponents).

after B. Klein: Introduction to the Exact Renormalization Group,http://www.t39.ph.tum.de/T39_files/T39_people_files/klein_files/rgseminar.pdf


http://www.t39.ph.tum.de/T39_files/T39_people_files/klein_files/rgseminar.pdf


Key words and concepts

• for continuum field theory

• non-perturbative

• (known) microscopic laws → complex macroscopic phenomena

• flow from classical action S [ϕ] to effective action Γ [φ]

• scale dependent effective action Γk [φ]

• scale parameter k = adjustable screw of microscope• high resolution: large values of a momentum scale k• decreasing resolution of the microscope: lowering k

B.-J. Schaefer, http://physik.uni-graz.at/~dk-user/talks/Schaefer_October_6

after A. Wipf: The Functional Renormalization Group Method – An IntroductionThe Functional Renormalization Group Method – An Introduction 5Rainer Stiele-Argüello

http://physik.uni-graz.at/~dk-user/talks/Schaefer_October_6


Key words and concepts: flow & flow equation

• scale dependent effective action Γk interpolates betweenmicroscopical bare action Sclass in the UV and effective action Γ inthe IR

limk→∞

Γk = Sclass; limk→0

Γk = Γ

ability to follow k → 0 evolution ≡ ability to solve the theory

• flow from classical action S [φ] to effective action Γ [φ]:flow equation ∂kΓk

• Conceptually simple, technically demanding flow equations

• connects (any given) initial action (classical action) with fullquantum effective action

• exact RG flow: flow in ‘theory space’: trajectory is schemedependent but end point is not



Key words and concepts: flow & flow equation

• exact RG flow: flow in ‘theory space’: trajectory is schemedependent but end point is not

H. Gies: Introduction to the functional RG and applications to gauge theories; Lect. Notes Phys. 852 (2012)287-348, arXiv:hep-ph/0611146.



Literature

Introductory reviews and lecture notes:

• D. F. Litim, and J. M. Pawlowski: On gauge invariant Wilsonianflows; arXiv:hep-th/9901063 (1999).

• K. Aoki: Introduction to the nonperturbative renormalizationgroup and its recent applications;Int. J. Mod. Phys. B 14 (2000) 1249-1326.

• C. Bagnuls and C. Bervillier: Exact renormalization groupequations and the field theoretical approach to criticalphenomena;Int. J. Mod. Phys. A 16 (2001) 1825, arXiv:hep-th/0101110.

• J. Berges, N. Tetradis and C. Wetterich: Non-perturbativerenormalization flow in quantum field theory and statisticalphysics;Phys. Rept. 363 (2002) 223-386, arXiv:hep-ph/0005122.



Literature


• J. Polonyi: Lectures on the functional renormalization groupmethod;Central Eur. J. Phys. 1 (2003) 1-71, arXiv:hep-th/0110026.

• J. M. Pawlowski: Aspects of the functional renormalisationgroup; Annals Phys. 322 (2007) 2831-2915,arXiv:hep-th/0512261.

• B.-J., Schaefer and J. Wambach: Renormalization groupapproach towards the QCD phase diagram; Phys. Part. Nucl. 39(2008) 1025-1032, arXiv:hep-ph/0611191.

• R. Stiele: The Functional Renormalization Group and itsapplication to the Polyakov-Quark-Meson model; (2015).



Literature


• H. Gies: Introduction to the functional RG and applications togauge theories; Lect. Notes Phys. 852 (2012) 287-348,arXiv:hep-ph/0611146.

• P. Kopietz, L. Bartosch and F. Schütz: Introduction to thefunctional renormalization group;Lect. Notes Phys. 798 (2010) 1-380.

• A. Wipf: Statistical approach to quantum field theory;Lect. Notes Phys. 864 (2013) 1-390.



Basics of quantum field theory (QFT)

Action S[ϕ] → partition function Z[J]→ generating functional W[J] → effective action Γ[φ]

• generating functional of correlation functions: path integral

Z [J] =∫

Dϕ e−S[ϕ,J] , W[J] = lnZ[J]

• effective action = Legendre transform of W[J]

Γ [φ] =

∫J φ − W[J] with φ ≡ ⟨ϕ⟩ = δW

δJ

• Γ [φ] encodes properties of QFT

after A. Wipf: The Functional Renormalization Group Method – An Introduction,https://saalburg.aei.mpg.de/wp-content/uploads/sites/25/2017/03/wipf15.pdf


https://saalburg.aei.mpg.de/wp-content/uploads/sites/25/2017/03/wipf15.pdf


Mean-field approximation

Z [J] =∫

Dϕ e−S[ϕ,J] ill-defined

• mean-field approximation∫Dϕ e−

∫d4x L(ϕ) →

∫dφ e−S[φ] → e−S[φ]

replace with a spatially and temporally constant background fieldφ, ignoring fluctuations ϕ′

• → Grand canonical potential in mean field approximationΩ(φ) ∼ − lnZ



Idea of the FRG method

• idea of the renormalization group:path integral ⇐⇒ functional differential equation∫

d3k L(φ) ⇐⇒ ∂k Γk[φ]

• requires a scale (k) dependent average effective action Γk thatinterpolates between microscopical bare action Sclass andeffective action Γ

Sclass = Γk=ΛUV

Γk−→ limk→0

Γk = Γ



Average effective action Γk

Average effective action Γk

• effective action for fieldaverages over volume ∼1/ kd

• large k: close to microscopicaction

• contains only fluctuationswith q2 > k2

⇒ implement IR cutoff Rk (q)

• k = 0: IR cutoff absent →Γk→0 ≡ Γ






Average effective action and its flow equation

Average effective action Γk and its flow equation ∂k Γk

• large k: close to microscopic action

• solving the flow equation ∂k Γk while lowering k:successive inclusion of fluctuations with momenta2 > k2 to Γk

• Γk: effective action for field averages over volume ∼1/ kd

• k → 0: full effective action with all fluctuations included

L. M. HaasT. K. Herbst



Scale-dependent averaged effective action Γk

Construct a scale-dependent, averaged effective action Γk [φ]which is

• the effective action only including fluctuations with momenta2

> k2

• i.e. a ‘coarse-grained’ effective action, averaged over volumes∼1/ kd (i.e. quantum fluctuations on smaller scales areintegrated out!)

• for large k (= small length scales) very similar to the microscopicaction S [ϕ] (since long-range correlations do not yet play a rôle)

• for small k (= large length scales) includes long-range effects(long-range correlations, critical behavior, . . . )

→ look for a derivation of such an scale-dependent, average effectiveaction starting from the generating functional of correlation functionsafter B. Klein: Introduction to the Exact Renormalization Group,http://www.t39.ph.tum.de/T39_files/T39_people_files/klein_files/rgseminar.pdf


http://www.t39.ph.tum.de/T39_files/T39_people_files/klein_files/rgseminar.pdf


Constructing average effective action Γk

Recipe for scale-dependent, averaged effective action Γk [φ]

• add a scale dependent IR cutoff term to classical action

S [ϕ] → S [ϕ] + ∆Sk [ϕ]

• adds IR cutoff term to partition function

Zk [J] =∫

Dϕ exp(−S [ϕ] +

∫Jϕ−∆Sk [ϕ]

)→ scale-dependent generating functional Wk [J] = ln Zk[J]

• a modified Legendre transformation defines thescale-dependent effective action

Γk [φ] = −Wk [J] +∫

Jφ−∆Sk [φ]

Γk [φ] = − ln∫

Dϕ′ exp(−S [ϕ′ + φ]−∆Sk [ϕ

′] +

∫Jϕ′)



IR cutoff term & regulator function Rk

• Choice: quadratic functional with momentum dependent mass

∆Sk [ϕ] =12

∫pϕ†(p) Rk(p) ϕ(p)

Quadratic form ensures that one-loop equation can be exact.

• Conditions on cutoff function, regulator Rk(p):• for k → ∞ no modes integrated out yet: acts like δ(ϕ):

limk→∞

Rk(p2) → ∞

→ recover classical action: limk→∞ Γk [φ] = S [φ]

• recover effective action for k → 0: remove cutoff for k → 0:lim

k2/p2→0Rk(p2) → 0

• must be an IR regulator:lim

p2/k2→0Rk(p2) > 0

→ suppress dynamics of low momentum modes



Common regulator functions

• exponential regulator: Rk(p) = p2/[exp

(p2/k2

)− 1]

• optimized regulator: Rk(p) =(k2 − p2

)Θ(1 − p2/k2

)• quartic regulator: Rk(p) = k4/p2

• sharp regulator: Rk(p) = p2/Θ(k2 − p2

)− p2

• Callan-Symanzik regulator: Rk(p) = k2

B.-J. Schaefer: The Functional Renormalization Group and Phases of Strongly Interacting Matter. Habilitationthesis, Karl-Franzens-Universität Graz, June 2009



Flow equation

Γk [φ] = −Wk[Jk] +

∫Jk φ−∆Sk[φ] → ∂k Γk[φ]

Flow equation: Describes the change of the scale-dependentaveraged effective action at scale k with a change of this RGscale, and thus how the average effective actions on differentscales are connected.

•− ∂kWk[Jk] = −∂kWk[J]−

∫δWk

δJk∂kJk

•∂k

∫Jk φ =

∫φ ∂kJk

•− ∂k∆Sk[φ] = −1

2

∫ϕ† ∂kRk ϕ



Derivation of the flow equation

⇒ ∂k Γk[φ] = −∂kWk[J]− ∂k ∆Sk[φ]

•− ∂kWk[J] = · · · = 1

2

∫Gk ∂kRk + ∂k ∆Sk[φ]

with Gk =δ2Wk[J]δJ† δJ

connected two-point (Greens) function

⇒ ∂k Γk[φ] =12

∫Gk ∂kRk

•Gk =

δ2Wk[J]δJ† δJ

=δφ

δJ,

δ2Γk

δφ† δφ=

δJ†

δϕ† − Rk

⇒ Gk =

(δ2Γk

δφ† δφ+ Rk

)−1

≡(Γ(2)k + Rk

)−1



Derivation of the flow equation

⇒ ∂k Γk[φ] =12

∫∂k Rk

Γ(2)k [φ] + Rk

=12

Tr

(∂k Rk

Γ(2)k [φ] + Rk

)Wetterich equation

C. Wetterich: Exact evolution equation for the effective potential; Phys. Lett. B 301 (1993) 90-94

⇒ ∂k Γk[φ] =12

Tr ∂kRk

(1

Γ(2)k + Rk

)

B.-J. Schaefer,http://physik.uni-graz.at/~dk-user/talks/Schaefer_October_6

One-loop equation forthe full propagator withinsertion of ∂kRk





Flow equation


One-loop equation for the full propagator with insertion of ∂kRk.

∂k Γk[φ] =12

Tr

(∂k Rk

Γ(2)k [φ] + Rk

)= ∂k

12

Tr ln(Γ(2)k [φ] + Rk

)Unequal to derivative of one-loop effective action, since terms∂k Γ

(2)k [φ] not present.




Flow equation: regulator dependence

Average effective action: Γk [φ] = −Wk[Jk] +

∫Jk φ−∆Sk[φ]

Flow of average effective action: ∂k Γk[φ] =12

Tr

(∂k Rk

Γ(2)k [φ] + Rk

)

B.-J. Schaefer, http://physik.uni-graz.at/~dk-user/talks/Schaefer_October_6The Functional Renormalization Group Method – An Introduction 21Rainer Stiele-Argüello



Flow equation


Tr

(∂k Rk

Γ(2)k [φ] + Rk

)

• Exact (no approximations so far!) RG flow equation for effectiveaction

• Non-linear functional differential equation (involves the functionalderivatives Γ

(2)k [φ])

• . . . in its most general form completely unsolvable!



Flow equation


Tr

(∂k Rk

Γ(2)k [φ] + Rk

)

• In its most general form completely unsolvable:

δ

δφ∂kΓk[φ] = ∂k Γ

(1)k [φ] = −1

2Tr

Γ(3)k [φ] ∂kRk(

Γ(2)k [φ] + Rk

)2

∂k Γ(2)k [φ] = Tr

(Γ(3)k [φ]

)2∂kRk(

Γ(2)k [φ] + Rk

)3

− 12

Tr

Γ(4)k [φ] ∂kRk(

Γ(2)k [φ] + Rk

)2

⇒ flow of Γ(n)k [φ] needs Γ

(n+1)k [φ] & Γ

(n+2)k [φ]

⇒ hierarchy of flow equationsThe Functional Renormalization Group Method – An Introduction 23Rainer Stiele-Argüello


Flow equation

∂k Γk[φ] =12

Tr

(∂k Rk

Γ(2)k [φ] + Rk

)unsolvable

→ flow of Γ(n)k [φ] needs Γ

(n+1)k [φ] & Γ

(n+2)k [φ]

→ hierarchy of flow equations

⇒ need to truncate effective action Γk[φ] andrestrict it to correlators of nmax fields

Truncation: Projection onto finite-dimensional sub-theory space



Truncation

∂k Γk[φ]: hierarchy of flow equations







Truncation

∂k Γk[φ]: hierarchy of flow equations




H. Gies, Lect. Notes Phys. 852 (2012) 287-348,arXiv:hep-ph/0611146





Truncation

⇒ need to truncate effective action Γk[φ]: restrict it tocorrelators of nmax fields



H. Gies, Lect. Notes Phys. 852 (2012)287-348, arXiv:hep-ph/0611146

B.-J. Schaefer, . . .

improve truncation, optimize regulator, check stability, enlargesubspace






Truncation

⇒ need to truncate effective action Γk[φ]: restrict it tocorrelators of nmax fields


Examples of truncations:• Derivative expansion

Γk[φ] =

∫ddx(

Vk(φ) +12

Zk (∂µφ)2+O

((∂µφ)

4))

• Expansion in powers of the fields

Γk[φ] =∑

n

1n!

∫ ( n∏i

ddxiφ (xi)

)Γ(n)k (x1, . . . , xn)



Application to QCD

T.-K. Herbst,http://www.thphys.uni-heidelberg.de/~smp/Delta/Delta13/talks/Delta13_Herbst.pdf

J.-M. Pawlowski,http://www.congres.upmc.fr/erg2012/wp-content/PDF-Files/Talk_Pawlowski_ERG2012.pdf


http://www.thphys.uni-heidelberg.de/~smp/Delta/Delta13/talks/Delta13_Herbst.pdf

http://www.congres.upmc.fr/erg2012/wp-content/PDF-Files/Talk_Pawlowski_ERG2012.pdf


Quark-Meson truncation

• Ansatz for effective average action:

Γk[φ] =

∫d4x q

[i/∂ + g (σ + iγ5τ π)

]q+

12(∂µσ)

2+

12(∂µπ)

2−Vk(φ2)

with φ2 = (σ, π)2 and Vk=Λ

(φ2)= λ

4

(σ2 + π2 − v2

)2 − cσ

• Flow equation for grand canonical potential:

∂kΩk(φ2; T, µ

)=

k4

12π2

1

Eσcoth

(Eσ

2T

)+

3Eπ

coth(

Eπ

2T

)−

− 2NcNf

Eq

[tanh

(Eq − µq

2T

)+ tanh

(Eq + µq

2T

)]with

E2σ = k2 + 2Ω′

k + 4φ2Ω′′k , E2

π = k2 + 2Ω′k , E2

q = k2 + g2ϕ2

and Ω′k = ∂Ωk/∂φ

2



Flow in the Quark-Meson truncation

0

10

20

30

40

50

60

70

80

90

100

0 200 400 600 800 1000 1200

f π[M

eV]

k [MeV]

µ = 0

T = 0T = 175MeV

dynamical breaking ofchiral symmetry

⇒ generation of constituentquark masses



Temperature dependence of quark mass

0

10

20

30

40

50

60

70

80

90

100

0 50 100 150 200 250 300

f π[M

eV]

T [MeV]

µ = 0

only thermal flucs. of quarks+ quantum flucs. of quarks

+ flucs. of meson

the more fluctuations included⇒ the smoother the transition



Conclusions• Cover physics across different scales

→ relate microscopic theory to macroscopic (effective) theory⇒ important to have a systematic scheme of integrating out quantumfluctuations

• Functional differential equation that interpolatesclassical action at UV scale to effective action⇒ Scale-dependent averaged effective action→ IR cut-off -> IR regulator

• FRG flow equation: exact RG scheme: connectsclassical action at UV scale to the full quantum effectiveaction

Γ = S +

∞∑n=1

∆Γn in a loop expansionc⃝WDR

• Solution of flow equation relies on some truncation of the effectiveaction⇒ result of a calculation is not exact !fluctuations→ truncation: loose irrelevant details of microscopic theory


WDR


Thank You for your attention!• Cover physics across different scales

→ relate microscopic theory to macroscopic (effective) theory⇒ important to have a systematic scheme of integrating out quantumfluctuations

• Functional differential equation that interpolatesclassical action at UV scale to effective action⇒ Scale-dependent averaged effective action→ IR cut-off -> IR regulator

• FRG flow equation: exact RG scheme: connectsclassical action at UV scale to the full quantum effectiveaction

Γ = S +

∞∑n=1

∆Γn in a loop expansionc⃝WDR

• Solution of flow equation relies on some truncation of the effectiveaction⇒ result of a calculation is not exact !fluctuations→ truncation: loose irrelevant details of microscopic theory


WDR

the functional renormalization group method – an...

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