The Functional Renormalization Group Method – An Introduction · Callan-Symanzik regulator: Rk(p)...
Transcript of The Functional Renormalization Group Method – An Introduction · Callan-Symanzik regulator: Rk(p)...
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
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
Motivation FRG method QCD Application Summary
FRG Method – What is it about?
http://vertixap.com/ptv_vision
Relate physics across different length scales. . . in continuum field theory
The Functional Renormalization Group Method – An Introduction 2Rainer Stiele-Argüello
Motivation FRG method QCD Application Summary
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
The Functional Renormalization Group Method – An Introduction 3Rainer Stiele-Argüello
Motivation FRG method QCD Application Summary
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
The Functional Renormalization Group Method – An Introduction 4Rainer Stiele-Argüello
Motivation FRG method QCD Application Summary
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
Motivation FRG method QCD Application Summary
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
Motivation FRG method QCD Application Summary
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
The Functional Renormalization Group Method – An Introduction 6Rainer Stiele-Argüello
Motivation FRG method QCD Application Summary
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.
The Functional Renormalization Group Method – An Introduction 6Rainer Stiele-Argüello
Motivation FRG method QCD Application Summary
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.
The Functional Renormalization Group Method – An Introduction 6Rainer Stiele-Argüello
Motivation FRG method QCD Application Summary
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.
The Functional Renormalization Group Method – An Introduction 7Rainer Stiele-Argüello
Motivation FRG method QCD Application Summary
Literature
Introductory reviews and lecture notes:
• 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).
The Functional Renormalization Group Method – An Introduction 7Rainer Stiele-Argüello
Motivation FRG method QCD Application Summary
Literature
Introductory reviews and lecture notes:
• 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.
The Functional Renormalization Group Method – An Introduction 7Rainer Stiele-Argüello
Motivation FRG method QCD Application Summary
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
The Functional Renormalization Group Method – An Introduction 8Rainer Stiele-Argüello
Motivation FRG method QCD Application Summary
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
The Functional Renormalization Group Method – An Introduction 9Rainer Stiele-Argüello
Motivation FRG method QCD Application Summary
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
The Functional Renormalization Group Method – An Introduction 10Rainer Stiele-Argüello
Motivation FRG method QCD Application Summary
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
B.-J. Schaefer, http://physik.uni-graz.at/~dk-user/talks/Schaefer_October_6
The Functional Renormalization Group Method – An Introduction 11Rainer Stiele-Argüello
Motivation FRG method QCD Application Summary
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
The Functional Renormalization Group Method – An Introduction 12Rainer Stiele-Argüello
Motivation FRG method QCD Application Summary
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
The Functional Renormalization Group Method – An Introduction 13Rainer Stiele-Argüello
Motivation FRG method QCD Application Summary
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′)
The Functional Renormalization Group Method – An Introduction 14Rainer Stiele-Argüello
Motivation FRG method QCD Application Summary
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
The Functional Renormalization Group Method – An Introduction 15Rainer Stiele-Argüello
Motivation FRG method QCD Application Summary
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
The Functional Renormalization Group Method – An Introduction 16Rainer Stiele-Argüello
Motivation FRG method QCD Application Summary
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
The Functional Renormalization Group Method – An Introduction 17Rainer Stiele-Argüello
Motivation FRG method QCD Application Summary
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
The Functional Renormalization Group Method – An Introduction 18Rainer Stiele-Argüello
Motivation FRG method QCD Application Summary
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
The Functional Renormalization Group Method – An Introduction 19Rainer Stiele-Argüello
Motivation FRG method QCD Application Summary
Flow equation
B.-J. Schaefer, http://physik.uni-graz.at/~dk-user/talks/Schaefer_October_6
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.
The Functional Renormalization Group Method – An Introduction 20Rainer Stiele-Argüello
Motivation FRG method QCD Application Summary
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
Motivation FRG method QCD Application Summary
Flow equation
Flow of average effective action: ∂k Γk[φ] =12
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!
The Functional Renormalization Group Method – An Introduction 22Rainer Stiele-Argüello
Motivation FRG method QCD Application Summary
Flow equation
Flow of average effective action: ∂k Γk[φ] =12
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
Motivation FRG method QCD Application Summary
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
The Functional Renormalization Group Method – An Introduction 24Rainer Stiele-Argüello
Motivation FRG method QCD Application Summary
Truncation
∂k Γk[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
B.-J. Schaefer, http://physik.uni-graz.at/~dk-user/talks/Schaefer_October_6
The Functional Renormalization Group Method – An Introduction 25Rainer Stiele-Argüello
Motivation FRG method QCD Application Summary
Truncation
∂k Γk[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
B.-J. Schaefer, http://physik.uni-graz.at/~dk-user/talks/Schaefer_October_6
H. Gies, Lect. Notes Phys. 852 (2012) 287-348,arXiv:hep-ph/0611146
The Functional Renormalization Group Method – An Introduction 25Rainer Stiele-Argüello
Motivation FRG method QCD Application Summary
Truncation
⇒ need to truncate effective action Γk[φ]: restrict it tocorrelators of nmax fields
Truncation: Projection onto finite-dimensional sub-theory space
B.-J. Schaefer, http://physik.uni-graz.at/~dk-user/talks/Schaefer_October_6
H. Gies, Lect. Notes Phys. 852 (2012)287-348, arXiv:hep-ph/0611146
B.-J. Schaefer, . . .
improve truncation, optimize regulator, check stability, enlargesubspace
The Functional Renormalization Group Method – An Introduction 26Rainer Stiele-Argüello
Motivation FRG method QCD Application Summary
Truncation
⇒ need to truncate effective action Γk[φ]: restrict it tocorrelators of nmax fields
Truncation: Projection onto finite-dimensional sub-theory space
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)
The Functional Renormalization Group Method – An Introduction 27Rainer Stiele-Argüello
Motivation FRG method QCD Application Summary
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
The Functional Renormalization Group Method – An Introduction 28Rainer Stiele-Argüello
Motivation FRG method QCD Application Summary
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
The Functional Renormalization Group Method – An Introduction 28Rainer Stiele-Argüello
Motivation FRG method QCD Application Summary
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
The Functional Renormalization Group Method – An Introduction 28Rainer Stiele-Argüello
Motivation FRG method QCD Application Summary
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
The Functional Renormalization Group Method – An Introduction 29Rainer Stiele-Argüello
Motivation FRG method QCD Application Summary
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
The Functional Renormalization Group Method – An Introduction 30Rainer Stiele-Argüello
Motivation FRG method QCD Application Summary
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
The Functional Renormalization Group Method – An Introduction 31Rainer Stiele-Argüello
Motivation FRG method QCD Application Summary
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
The Functional Renormalization Group Method – An Introduction 32Rainer Stiele-Argüello
Motivation FRG method QCD Application Summary
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
The Functional Renormalization Group Method – An Introduction 32Rainer Stiele-Argüello