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