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Application of the Functional RG with Background Field Methods

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Application of the Functional RG with Background Field Methods 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
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