Application Of The Functional RG With Background Field Methods

J. Braun Heidelberg University

December 9, 2005 TU Darmstadt Outline

Motivation

Functional Renormalization Group - General Introduction

Gauge Theories And Functional Renormalization Group

Applications: QCD at finite temperature

Conclusions and outlook Motivation - challenging questions of QCD

◮ running QCD coupling −→ small momentum behavior? ◮ confinement −→ mechanisms? ◮ chiral symmetry breaking −→ Tχ = Td ? ◮ many-flavor QCD

◮ phase diagram of QCD (Bethke ’04) −→ order of the phase transition(s)? ◮ QCD at high temperature ◮ topological effects in QCD −→ e.g. role of instantons?

(Karsch et al. ’03) Motivation - Why Renormalization Group?

Lattice QCD

◮ implementation of fermions (staggered, Wilson, domain wall, ...) is difficult ◮ large current quark masses −→ extrapolation required (chiral perturbation theory) ◮ finite volume −→ IR-cutoff introduced −→ extrapolation required ◮ chemical potential −→ sign problem

Why Renormalization Group?

◮ allows to describe physics across different length scales (microscopic theory → macroscopic theory) ◮ allows for chiral fermions ◮ implementation of quarks with and without current quark masses ◮ critical behavior: importance of long-range fluctuations ◮ study QFT’s in infinite and finite volume ◮ non-perturbative method ◮ truncation is required

=⇒ complementary method to Lattice QCD Motivation - Why Renormalization Group?

Lattice QCD

◮ implementation of fermions (staggered, Wilson, domain wall, ...) is difficult ◮ large current quark masses −→ extrapolation required (chiral perturbation theory) ◮ finite volume −→ IR-cutoff introduced −→ extrapolation required ◮ chemical potential −→ sign problem

Why Renormalization Group?

◮ allows to describe physics across different length scales (microscopic theory → macroscopic theory) ◮ allows for chiral fermions ◮ implementation of quarks with and without current quark masses ◮ critical behavior: importance of long-range fluctuations ◮ study QFT’s in infinite and finite volume ◮ non-perturbative method ◮ truncation is required

=⇒ complementary method to Lattice QCD FUNCTIONAL RENORMALIZATION GROUP - GENERAL INTRODUCTION Functional Renormalization Group - Derivation I

◮ Goal: Flow Equation for Γk which interpolates between Sbare and Γ1PI

k 0 1PI k Λ bare Γk −→→ Γ Γk −→→ S

◮ Schwinger Functional:

S[φ]+R Jφ WΛ[J] ZΛ[J]= Dφ e− ≡ e ; ≡ Dφ(p) ZΛ ZΛ Zp.Λ ! ◮ IR-regularization → insertion of an IR-regulator

δ ∆Sk [ ] S[φ] ∆Sk [φ]+R Jφ Wk [J] Zk [J] := e− δJ ZΛ[J]= Dφ e− − ≡ e ZΛ ◮ 1 2 properties of the IR-regulator ∆Sk [φ]= 2 p φ(−p)Rk (p )φ(p) R

2 2 Rk p k 2lim2 ( ) ∼ p /k →0 2 Rk p 2lim2 ( ) = 0 k /p →0 2 lim Rk(p ) → ∞ k→Λ Functional Renormalization Group - Derivation II

δW [J] ◮ ¯ k Legendre-transformation → effective average action with φ = δJ

Γk [φ¯]= −Wk [J]+ Jφ¯ − ∆Sk [φ] Z ◮ flow equation for the effective action (Wetterich ’93)

(2) 1 1 − 1 k∂k Γk ≡ ∂t Γk = 2 STr ∂t Rk Γk + Rk = 2
◮ how to treat chiral fermions? Ψ Ψ ¯ → construct regulator which preserves symmetries, i. e. Rk = Rk (i∂/) ◮ how to treat gauge theories? →coming up soon ... Functional Renormalization Group - Derivation II

δW [J] ◮ ¯ k Legendre-transformation → effective average action with φ = δJ

Γk [φ¯]= −Wk [J]+ Jφ¯ − ∆Sk [φ] Z ◮ flow equation for the effective action (Wetterich ’93)

(2) 1 1 − 1 k∂k Γk ≡ ∂t Γk = 2 STr ∂t Rk Γk + Rk = 2
◮ how to treat chiral fermions? Ψ Ψ ¯ → construct regulator which preserves symmetries, i. e. Rk = Rk (i∂/) ◮ how to treat gauge theories? →coming up soon ... Functional Renormalization Group - Derivation II

δW [J] ◮ ¯ k Legendre-transformation → effective average action with φ = δJ

Γk [φ¯]= −Wk [J]+ Jφ¯ − ∆Sk [φ] Z ◮ flow equation for the effective action (Wetterich ’93)

(2) 1 1 − 1 k∂k Γk ≡ ∂t Γk = 2 STr ∂t Rk Γk + Rk = 2
◮ how to treat chiral fermions? Ψ Ψ ¯ → construct regulator which preserves symmetries, i. e. Rk = Rk (i∂/) ◮ how to treat gauge theories? →coming up soon ... Functional Renormalization Group - Truncation

◮ set of equations: → flow equation for the one-point function

(3) (1) 1 Γk

∂t Γk = 2 ¡ → flow equation for the two-point function

(3) (3) (2) Γk Γk 1 ∂t Γk = − 2

(4) Γk → flow equation for the three-point function (3) Γk (5) (3) (3) Γ (3) (4) Γk Γk 1 k

∂t Γk = −3Γk + 3 − 2

¡ ¡

(3) Γk

→ flow equation for the 1 four-point function ... (n) (n+1) (n+2) ◮ in order to calculate Γk , we need Γk and Γk → truncate the set of flow equations by“choosing”a maximum number n = nmax NOTE: potential problem for strongly interacting theories (operator expansion) GAUGE THEORIES AND FUNCTIONAL RENORMALIZATION GROUP Gauge Theories and Functional Renormalization Group

”PROBLEM”: We have to preserve gauge symmetry!

◮ Breaking of gauge invariance by ...... standard gauge fixing procedure → Ward identities ... momentum cut-off breaks gauge invariance necessarily

SOLUTIONS:

◮ construct a gauge invariant regulator: (T. R. Morris ’99, ’00) idea: constructing SU(N)-Yang-Mills theory from a spontaneously broken SU(N|N) super-gauge extension.

◮ construct gauge invariant regulator: geometrical effective action (J. Pawlowski ’03) ◮ constructing gauge-invariant flows by including Ward identies which are modified due to the presence of the cutoff (Bonnini et al. ’93; Reuter et al. ’94; Ellwanger ’94; D’Attanasio et al. ’96) → e.g. vertex expansion in Landau-gauge (Pawlowski et al. ’04; C. S. Fischer, H. Gies ’04)

◮ use the Background-Field method (Abbott ’81) will be discussed in the following ... Gauge Theories and Functional Renormalization Group

”PROBLEM”: We have to preserve gauge symmetry!

◮ Breaking of gauge invariance by ...... standard gauge fixing procedure → Ward identities ... momentum cut-off breaks gauge invariance necessarily

SOLUTIONS:

◮ construct a gauge invariant regulator: (T. R. Morris ’99, ’00) idea: constructing SU(N)-Yang-Mills theory from a spontaneously broken SU(N|N) super-gauge extension.

◮ construct gauge invariant regulator: geometrical effective action (J. Pawlowski ’03) ◮ constructing gauge-invariant flows by including Ward identies which are modified due to the presence of the cutoff (Bonnini et al. ’93; Reuter et al. ’94; Ellwanger ’94; D’Attanasio et al. ’96) → e.g. vertex expansion in Landau-gauge (Pawlowski et al. ’04; C. S. Fischer, H. Gies ’04)

◮ use the Background-Field method (Abbott ’81) will be discussed in the following ... Gauge Theories and Functional Renormalization Group

”PROBLEM”: We have to preserve gauge symmetry!

◮ Breaking of gauge invariance by ...... standard gauge fixing procedure → Ward identities ... momentum cut-off breaks gauge invariance necessarily

SOLUTIONS:

◮ construct a gauge invariant regulator: (T. R. Morris ’99, ’00) idea: constructing SU(N)-Yang-Mills theory from a spontaneously broken SU(N|N) super-gauge extension.

◮ construct gauge invariant regulator: geometrical effective action (J. Pawlowski ’03) ◮ constructing gauge-invariant flows by including Ward identies which are modified due to the presence of the cutoff (Bonnini et al. ’93; Reuter et al. ’94; Ellwanger ’94; D’Attanasio et al. ’96) → e.g. vertex expansion in Landau-gauge (Pawlowski et al. ’04; C. S. Fischer, H. Gies ’04)

◮ use the Background-Field method (Abbott ’81) will be discussed in the following ... Gauge Theories and Functional Renormalization Group

”PROBLEM”: We have to preserve gauge symmetry!

◮ Breaking of gauge invariance by ...... standard gauge fixing procedure → Ward identities ... momentum cut-off breaks gauge invariance necessarily

SOLUTIONS:

◮ construct a gauge invariant regulator: (T. R. Morris ’99, ’00) idea: constructing SU(N)-Yang-Mills theory from a spontaneously broken SU(N|N) super-gauge extension.

◮ construct gauge invariant regulator: geometrical effective action (J. Pawlowski ’03) ◮ constructing gauge-invariant flows by including Ward identies which are modified due to the presence of the cutoff (Bonnini et al. ’93; Reuter et al. ’94; Ellwanger ’94; D’Attanasio et al. ’96) → e.g. vertex expansion in Landau-gauge (Pawlowski et al. ’04; C. S. Fischer, H. Gies ’04)

◮ use the Background-Field method (Abbott ’81) will be discussed in the following ... Background Field Method - General Introduction I

PROBLEM: the effective action under consideration should be gauge invariant, but already the definition of the functional integral requires gauge fixing. SOLUTION: split gauge field in fluctuation field and background field: A = A¯ + a

◮ generating functional with introduced Background Field: (Z[J, 0] = Z[J])

a δG (S[a+A¯] 1 R G aG a)+R Ja aa R Ja A¯a Z˜[J, A¯]= Da Det [a + A¯]e− − 2α x x µ µ = Z[J]e− x µ µ δΘb Z   with

=A 1 δG a(x) δ(a + A¯) ≡ δAa = Dab(A)Θb and ”Ghost”−action : [A] µ g µ δΘb(y) z }| { ◮ comparison of the“old” (Γ) and the Background Field (Γ)˜ effective action ˜ a ˜a ˜ ¯ ˜ ¯ a a Γ[A]= −W [J]+ x JµAµ Γ[˜a, A]= −W [J, A]+ x Jµ˜aµ with a δW R a a δW˜R A˜ = a ˜a = ˜a (J, A¯)= a µ δJµ µ µ δJµ Background Field Method - General Introduction I

PROBLEM: the effective action under consideration should be gauge invariant, but already the definition of the functional integral requires gauge fixing. SOLUTION: split gauge field in fluctuation field and background field: A = A¯ + a

◮ generating functional with introduced Background Field: (Z[J, 0] = Z[J])

a δG (S[a+A¯] 1 R G aG a)+R Ja aa R Ja A¯a Z˜[J, A¯]= Da Det [a + A¯]e− − 2α x x µ µ = Z[J]e− x µ µ δΘb Z   with

=A 1 δG a(x) δ(a + A¯) ≡ δAa = Dab(A)Θb and ”Ghost”−action : [A] µ g µ δΘb(y) z }| { ◮ comparison of the“old” (Γ) and the Background Field (Γ)˜ effective action ˜ a ˜a ˜ ¯ ˜ ¯ a a Γ[A]= −W [J]+ x JµAµ Γ[˜a, A]= −W [J, A]+ x Jµ˜aµ with a δW R a a δW˜R A˜ = a ˜a = ˜a (J, A¯)= a µ δJµ µ µ δJµ Background Field Method - General Introduction II

Is there a relation between Γ[0˜ , A¯] and Γ[A˜]?

Γ[˜ ˜a, A¯] = Γ[A˜ = ˜a + A¯] =⇒ Γ[0˜ , A¯] = Γ[A˜ = A¯]

But why should we use a background field?

We can choose a gauge fixing term G a (Background Field gauge) in such a way that the background field effective action Γ[0˜ , A¯] is a gauge invariant functional of A¯ :

δG a(x) 1 G a[a]= Dac (A¯)ac =⇒ [a + A¯]= Dac (A¯)Dcb(a + A¯)δ(x − y) µ δΘb(y) g µ µ

◮ one can show that Γ[˜ ˜a, A¯] is invariant under the simultaneous transformations (Abbott ’81) 1 δA¯ a = Dab(A¯)Θb and δ˜aa = −f abc Θb˜ac µ g µ µ µ =⇒ ”physical limit”(˜a = 0): Γ[0˜ , A¯] is a gauge-invariant functional of A¯! Background Field Method - General Introduction II

Is there a relation between Γ[0˜ , A¯] and Γ[A˜]?

Γ[˜ ˜a, A¯] = Γ[A˜ = ˜a + A¯] =⇒ Γ[0˜ , A¯] = Γ[A˜ = A¯]

But why should we use a background field?

We can choose a gauge fixing term G a (Background Field gauge) in such a way that the background field effective action Γ[0˜ , A¯] is a gauge invariant functional of A¯ :

δG a(x) 1 G a[a]= Dac (A¯)ac =⇒ [a + A¯]= Dac (A¯)Dcb(a + A¯)δ(x − y) µ δΘb(y) g µ µ

◮ one can show that Γ[˜ ˜a, A¯] is invariant under the simultaneous transformations (Abbott ’81) 1 δA¯ a = Dab(A¯)Θb and δ˜aa = −f abc Θb˜ac µ g µ µ µ =⇒ ”physical limit”(˜a = 0): Γ[0˜ , A¯] is a gauge-invariant functional of A¯! Background Field Method - General Introduction III

◮ 1PI Green’s functions are generated by differentiating Γ[0˜ , A¯] with respect to A¯

What are the properties of the 1PI-diagrams generated by Γ[0˜ , A¯]?

◮ vertices involving a-fields are used inside Feynman-diagrams ◮ vertices involving A¯-fields are used for external lines

What kind of divergencies occur when calculating Γ[0˜ , A¯]?

◮ ghost fields and a-fields only appear inside loops → renormalization of these fields is not needed 1 ◮ ¯ 2 ¯ renormalization of the remaining fields: (Aµ)0 = ZA A, g0 = Zg g ◮ relation between Zg and ZA? 1 ˜ ¯ a a − 2 → gauge invariance of Γ[0, A] implies that (Fµν )0 = const. Fµν =⇒ Zg = ZA → determine β-function from a calculation of the background-field two-point function

Fermions can be included straightforwardly! Background Field Method - General Introduction III

◮ 1PI Green’s functions are generated by differentiating Γ[0˜ , A¯] with respect to A¯

What are the properties of the 1PI-diagrams generated by Γ[0˜ , A¯]?

◮ vertices involving a-fields are used inside Feynman-diagrams ◮ vertices involving A¯-fields are used for external lines

What kind of divergencies occur when calculating Γ[0˜ , A¯]?

◮ ghost fields and a-fields only appear inside loops → renormalization of these fields is not needed 1 ◮ ¯ 2 ¯ renormalization of the remaining fields: (Aµ)0 = ZA A, g0 = Zg g ◮ relation between Zg and ZA? 1 ˜ ¯ a a − 2 → gauge invariance of Γ[0, A] implies that (Fµν )0 = const. Fµν =⇒ Zg = ZA → determine β-function from a calculation of the background-field two-point function

Fermions can be included straightforwardly! Background Field Method - General Introduction III

◮ 1PI Green’s functions are generated by differentiating Γ[0˜ , A¯] with respect to A¯

What are the properties of the 1PI-diagrams generated by Γ[0˜ , A¯]?

◮ vertices involving a-fields are used inside Feynman-diagrams ◮ vertices involving A¯-fields are used for external lines

What kind of divergencies occur when calculating Γ[0˜ , A¯]?

◮ ghost fields and a-fields only appear inside loops → renormalization of these fields is not needed 1 ◮ ¯ 2 ¯ renormalization of the remaining fields: (Aµ)0 = ZA A, g0 = Zg g ◮ relation between Zg and ZA? 1 ˜ ¯ a a − 2 → gauge invariance of Γ[0, A] implies that (Fµν )0 = const. Fµν =⇒ Zg = ZA → determine β-function from a calculation of the background-field two-point function

Fermions can be included straightforwardly! Background Field Method And Functional RG

SO FAR, WE HAVE:

1 a ab bc c ,a ac cd d S¯ = Scl [A] − a D (A¯)D (A¯)a − c∗ D (A¯)D (A)c with A = A¯ + a 2α µ µ ν ν µ µ Z Z

◮ introduce regulator terms ∆Sk and perform RG improvement

1 a ab 2 b ,a ab 2 b S¯ = S + S + S −→ S = S¯ + a R (P )a + c∗ R (P )c cl gf gh k k 2 µ a,µν a ν c c Z Z 2 ◮ choose Pi in such a way that gauge invariance of Sk under simultaneous gauge transformation is maintained (Reuter et al. ’94, Freire et al. ’00)

2 ab ¯ 2 ab 2 ab ¯ 2 ab ¯ → e. g. (PA)µν = (−D(A) ) δµν and (Pc ) = (−D(A) ) (only A-dependent!) ◮ flow equation can be derived with the usual procedure

2 (2,0) 2 1 ˜ ¯ 1 ˜ ¯ − ∂t Γk [˜a, A]= 2 STr ∂t Rk (PA) Γk [˜a, A]+ Rk (PA) + ghosts
Spectrally Adjusted Cutoff I

◮ regulator improvement by choice of an appropriate argument (Litim et al ’02, Gies ’02)

2 ˜(2,0) ¯ Rk (Pi )= Rk (Γk [0,A])

◮ how does this improve the flow? ˜(2,0) → every eigenvalue of Γk obtains its“own optimized”regulator (spectral adjustment) → allows for different truncation schemes (see below) ◮ flow equation

1 ˜ ¯ 1 ˜(2,0) ¯ ˜(2,0) ¯ ˜(2,0) ¯ − ∂t Γk [˜a, A] = STr ∂t Rk (Γk [0,A]) Γk [0, A]+ Rk (Γk [0,A]) ˜a=0 2 h i

◮ from now on, we assume (define Γ[0˜ , A¯] = Γ[A¯]):

2 δ ˜ ¯ (2) ¯ ! ˜(2,0) ¯ Γ[0, A] ≡ Γk [A] = Γk [˜a, A] δA¯2 ˜a=0

Spectrally Adjusted Cutoff II

◮ the spectrally adjusted cutoff allows for different truncation schemes:

(n) (n) STANDARD SCHEME: Γ =0& ∂t Γ = 0 for (n > nmax , e.g. nmax = 3)

→ flow equation for the one-point function:

(3) (1) Γk (2) (3)

∂t Γk ∼ + B12∂t Γk + B13∂t Γk ¡ → flow equation for the two-point function:

(3) (3) (2) Γk Γk (2) (3) (4) ∂t Γk ∼ + + B22∂t Γk + B23∂t Γk + B24∂t Γk

(4) Γk → flow equation for the three-point function: (3) Γk (3) Γ(4) (3) (5) (3) Γk k Γk Γk (2) (3) (4) (5)

∂t Γk ∼ + + + B32∂t Γk +B33∂t Γk +B34∂t Γk +B45∂t Γk

¡ ¡

(3) Γk

→ flow equation for1 the four-point function ... Spectrally Adjusted Cutoff II

◮ the spectrally adjusted cutoff allows for different truncation schemes:

(n) (n) ”NEW” SCHEME: Γ = 0 for (n > nmax , e.g. nmax = 3) but keep track of all ∂t Γ

→ flow equation for the one-point function:

(3) (1) Γk (2) (3)

∂t Γk ∼ + B12∂t Γk + B13∂t Γk ¡ → flow equation for the two-point function:

(3) (3) (2) Γk Γk (2) (3) (4) ∂t Γk ∼ + + B22∂t Γk + B23∂t Γk + B24∂t Γk

(4) Γk → flow equation for the three-point function: (3) Γk (3) Γ(4) (3) (5) (3) Γk k Γk Γk (2) (3) (4) (5)

∂t Γk ∼ + + + B32∂t Γk +B33∂t Γk +B34∂t Γk +B45∂t Γk

¡ ¡

(3) Γk

1 (n) n+2 (i) → flow equation for the n-point function with (n > nmax ): ∂t Γ = i=2 Bni ∂t Γ P Spectrally Adjusted Cutoff III

final remarks on the spectrally adjusted cutoff:

(n) ◮ the“tower” of differential equations is a linear system of equations in ∂t Γ :

−→ −→ −→ 1 ∂t Γ= C + B · ∂t Γ =⇒ ∂t Γ = (1 − B)− C

◮ the spectrally adjusted cutoff provides an efficient reorganization of the flow equation

(Gies ’02) → already a small truncation can contain information from the flow of infinite many Γ(n) → applied to the calculation of the QCD β-function (H. Gies ’02, J.B. & H. Gies ’05) ◮ introducing a background field is a necessary condition for the use of the spectrally adjusted cutoff

◮ flow equation can be“translated”into a Proper-time representation: (Gies ’02)

1 s A¯ ∞ ds STr fˆ s (2) ∂t Γ[ ]= ( , η)exp − 2 Γ 2 0 k Z   with 1 fˆ(s, η)= fˆ (s)+ fˆ (s, η)+ fˆ (s, η) ∂ 1 2 3 s t Spectrally Adjusted Cutoff III

final remarks on the spectrally adjusted cutoff:

(n) ◮ the“tower” of differential equations is a linear system of equations in ∂t Γ :

−→ −→ −→ 1 ∂t Γ= C + B · ∂t Γ =⇒ ∂t Γ = (1 − B)− C

◮ the spectrally adjusted cutoff provides an efficient reorganization of the flow equation

(Gies ’02) → already a small truncation can contain information from the flow of infinite many Γ(n) → applied to the calculation of the QCD β-function (H. Gies ’02, J.B. & H. Gies ’05) ◮ introducing a background field is a necessary condition for the use of the spectrally adjusted cutoff

◮ flow equation can be“translated”into a Proper-time representation: (Gies ’02)

1 s A¯ ∞ ds STr fˆ s (2) ∂t Γ[ ]= ( , η)exp − 2 Γ 2 0 k Z   with 1 fˆ(s, η)= fˆ (s)+ fˆ (s, η)+ fˆ (s, η) ∂ 1 2 3 s t BACKGROUND FIELD METHOD APPLIED:

RUNNING QCD COUPLING AT FINITE TEMPERATURE AND CHIRAL SYMMETRY RESTORATION (J.B. & H. Gies, hep-ph/0512085) running coupling - truncation

◮ ansatz (operator expansion)

gluon gf gh 1 a a Γ = W (Θ) + Ψ(¯ iD/ + M ¯ )Ψ + Γ + Γ , Θ := F F k k ΨΨ 4 µν µν Zx with w2 2 w3 3 w4 4 2 1 2 W (Θ) = Z Θ+ Θ + Θ + Θ + ... , (g = Z − g¯ ) k A 2! 3! 4! A

◮ spectrally adjusted cutoff used

10 ◮ 2 2 kmax ∝ T : ”finite-size effect” (pµpµ = ωn + ~p ) one loop (T=0) T=0 ◮ 2 k 2 “3d-running” for T & k: g4D = T g3D ∗ 2 (g3D) ◮ IR-fixed point: α∗ ≡ α∗ = ≈ 2.4276 3D 4π e 1 T=100 MeV (k,T)

α T=300 MeV

0.1

0.1 1 k [GeV] critical value of αs - truncation

◮ definition of the“critical”value of the strong coupling:

The strong coupling must exceed the“critical”value αcr in order to have χSB

◮ ansatz: (SU(Nc ) gauge symmetry + chiral SU(Nf )L × SU(Nf )R flavor symmetry)

¯ ZA a a 1 ¯ ¯ Γk = ψ(iZψ∂/ + Z1g¯A/)ψ + Fµν Fµν + Z λ (V − A)+ Z+λ+(V + A) x 4 2 − − Z adj gf gh + Zσλ¯σ(S − P)+ ZVAλ¯VA[2(V − A) + (1/Nc )(V − A)] + Γ + Γ  ◮ four-fermion interactions ( λ¯i → 0 for k → Λ )

2 2 (V−A) = (ψγ¯ µψ) +(ψγ¯ µγ5ψ) 2 2 (V+A) = (ψγ¯ µψ) −(ψγ¯ µγ5ψ) ¯a b 2 ¯a b 2 (S−P) = (ψi ψi ) −(ψi γ5ψi ) adj z 2 z 2 (V−A) = (ψγ¯ µT ψ) +(ψγ¯ µγ5T ψ)

◮ ansatz for the four-fermion vertices: λ¯i (p1, p2, p3, p4) → λ¯i (pi = 0); Zψ(p) → Zψ(p = 0)

◮ mesonic operators and UA(1)-symmetry violating terms are neglected critical value of αs - flow equations ∂t λ √α √α α = 0

λi λj

T k ր √α √α λ α > 0 √α

λ i α > αcrit.

√α

◮ an example: flow equation for λ (zero temperature: H. Gies, J. Jaeckel, C. Wetterich ’03) − 2 1 FB 4 3 1 FB 4 12 + 9N l ( ), T g 2 g 2 l ( ), T c g 4 ∂t λ = 2λ − 2 1,1 ( k ) λ − 3 λVA − 2 1,2 ( k ) 2 − − 8π Nc − 256π Nc h i h i F 4 1 ( ), T 2 2 2 2 − l1 ( k ) 2λVA − Nf Nc (λ + λ+)+ λ − 2(Nc + Nf )λ λVA + Nf λ+λσ 4π2 − − −  V g 2 ◮ α = 4π > αcr −→ no fixed points −→ χSB (...),4 (...),4 T ◮ T T T li ( =0) > li ( k ) =⇒ αcr( k ) > αcr( =0) 1 λ ∼ m2

<σ>=6 0 σ chiral phase transition temperature & many-flavor QCD

α 4 s at T=130 MeV α 200 cr at T=130 MeV 2

1 150

α [Mev] 4 s at T=220 MeV 100

α cr cr at T=220 MeV 2 T

1 50

0 0.2 0.4 0.6 2 4 6 8 10 12 k [GeV] N f

◮ results for Tcr compared to lattice calculations:

Nf Tcr (bound) Tcr (lattice) (Karsch et al. ’03) 2 186 MeV 175 ± 8 MeV 3 161 MeV 155 ± 8 MeV

◮ approximately linear decrease of Tcr with increasing number of (massless) quark flavors ◮ critical number of (massless) quark flavors

12 (T. Appelquist et. al. ′96) cr N ≈ 10 RG for T=0 (H. Gies & J. Jaeckel ′05) f   12 RG for T>0 (J. B. & H. Gies ′05)  Conclusions and outlook

CONCLUSIONS ◮ Background Field Method allows for studies of gauge theories with the Functional RG ◮ Functional RG allows for systematic and consistent expansions of QFT ◮ operator expansion seems to be a promising tool for studing QCD

OUTLOOK ◮ study effect(s) of finite quark masses ◮ extension to finite density ◮ “penetrate”phase boundary (apply rebosonization technique)

◮ study Polyakov-Loop dynamics (extend previous work: J.B., H. Gies, H.-J. Pirner ’05) COLLABORATORS ◮ Polyakov-Loop: H. Gies (Heidelberg U.), H.-J. Pirner (Heidelberg U.) ◮ Running QCD coupling & four-fermion interactions at finite temperature: H. Gies ◮ Finite-Volume studies: B. Klein (GSI Darmstadt), H.-J. Pirner, A. H. Rezaeian

Thanks to GSI Darmstadt for financial support!