Unification of functional renormalization group equations István Nándori MTA-DE Részecsefiziai Kutatócsoport, MTA-Atomi, Debrecen MTA-DE Részecsefiziai Kutatócsoport és a ATOMKI Rács-QCD Lendület Kutatócsoport özös szemináriuma, Debrecen, 2013
1 Kadanoff-Wilson Kadanoff s blocing Wilson s blocing RG flow 2 Wegner-Houghton RG harp momentum cutoff Derivative exapnsion 3 Polchinsi RG mooth momentum cutoff 4 Effective action RG Wetterich RG Regulator functions Litim-Pawlowsi optimization PM optimization 5 Unified regulator C regulator New optimization scenario
Quantum Field Theory Particle Physics Elementary particles = fermionic fields ψ(x) = gauge fields A µ (x) = scalar fields ϕ(x) Quantum Field Theory = d d xl (ϕ, µ ϕ, x), Z [J] = N D[ϕ]e [ϕ]+ d d xjϕ Γ[ϕ] + ln Z [J] dxjϕ = 0 Γ[ϕ] measurable quantities effective action (at 1-loop order) Γ[ϕ] = [ϕ] + 1 2 d d ( ) (2π) d ln (2) [ϕ] + O( 2 )
Renormalization High-energy (UV) symmetries define the model [ ] 1 = d d x 2 z( µϕ) 2 + u cos(ϕ) Low-energy (IR) hadronization measurements renormalization: UV IR, scale-dependence: u, z Γ [ϕ], Γ =Λ [ϕ], Γ 0 [ϕ] Γ
Kadanoff s blocing cale-invariance and Kadanoff blocing (1966) Changing the observation scale Kadanoff blocing construction lattice space: a H a = J a i j lattice space: 2a H 2a = J 2a i j cale-invariance Partition function is invariant Z Tr exp[ βj a i j ] = Tr exp[ βj 2a i j ] Partition function is invariant RG flow equation d J(a) = F(J, a) da
Wilson s blocing Wilson s generalization of blocing construction (1971) cale-invariant system (second order phase-transition) Kadanoff blocing construction partition function invariant If no scale-invariance new interactions are generated during the blocing functional form of the partition function is not invariant H a = J a i j H 2a = J 2a i j + G 2a i j Wilson s idea: starts with the general functional form H a = J a i j + G a i j, G a = 0 functional form of the partition function is preserved
RG flow Wilsonian RG H a = J a i j + G a i j H 2a = J 2a i j + G 2a i j H 3a = J 3a i j + G 3a i j RG flow equations d da J(a) = F 1(J, G, a), RG flow diagram d da G(a) = F 2(J, G, a), J G
harp momentum cutoff Wegner Houghton RG method (1973) blocing using a sharp momentum cutoff, ( δ) Z = D[ φ + ϕ]e [ φ+ϕ] = D[ φ]e δ [ φ], φ = φ + ϕ, = 1 blocing relation: e δ [ φ] = D[ϕ]e [ φ+ϕ] D[ϕ]e [ φ] (1) [ φ]ϕ (2) [ φ] ϕ2 2 Wegner Houghton RG equation, exact if δ 0 [φ] = 1 2 Tr ln [ (2) ] [φ], (2) [φ] = δ2 [φ] δφ 2
Derivative exapnsion Wegner Houghton RG method (1973) derivative (gradient) expansion [ [φ] = d d x V (φ) + 1 2 Z (φ)( µ φ) 2 + 1 ] 4! Y (φ)( µ φ) 4 +... WH RG + leading order of the derivative expansion (LPA) [ V = d α d ln 2 + V ], V = 2 φ V (φ), α d 1-loop perturbative result: V V Λ = V =0 = V Λ + 1 [ ] 2 2 + V Λ [φ] d d (2π) d ln Ω d 2(2π) d properties = advantage: exact in LPA = disadvantage: confront to the derivative expansion
mooth momentum cutoff Polchinsi RG method (1984) using a smooth momentum cutoff K (y), y = p 2 / 2 Polchinsi RG equation in LPA V = 2 [V ]2 K 0 + d 2 V α d 0 dy K (y) K = y K (y) and K 0 = yk (y) y=0 if 0 dy K (y) = 1 (= K 0 = 1) = Polchinsi RG: V = 2 2 V V d 2 α d V = WH RG: V = 2 V V d 2 α d V properties = advantage: in LPA independent of K(y), = advantage: compatible with the derivative expansion = disadvantage: beyond LPA depends on K(y) = disadvantage: some incorrectness beyond LPA
Wetterich RG Effective (average) action RG method (1993) Wetterich RG equation ( Γ = 1 2 Tr R + R Γ (2) ), R (p) p 2 r(y), y = p2 2 1-loop improved RG + IR regulator 1 2 R (p)ϕ 2, (R 0 (p) = 0, R Λ (p) =, R (p 0) 0) Γ = Γ Λ + 1 2 Λ d Γ = Γ = 1 2 d d p (2π) d ln[r (p) + Γ (2) Λ ], Γ 0 = Γ, Γ Λ = Γ Λ d 1 d d p 2 (2π) d ln[r (p) + Γ (2) Λ ] ( ) R + R, Γ (2) Λ Γ(2) Wetterich RG Λ d d p (2π) d Γ (2) Λ
Regulator functions Regulator functions and the sharp cutoff limit R (p 2 ) p 2 r(y) with y = p 2 / 2 r pow (y) = c 1 y b = Morris RG c 1 r exp (y) = exp (c 2 y b = Wetterich RG ) 1 = sharp cutoff limit (b ) = r sharp (y) = 1 Θ(y 1) 1 possible to recover the WH RG (will be shown for the Litim s regulator in LPA) properties = advantage: "unification" of RG equations? = advantage: compatible with the derivative expansion = disadvantage: regulator-dependence
Litim-Pawlowsi optimization Optimization I. amplitude expansion for the Wetterich RG in LPA a n = V = α d d dy r y 1+ d 2 0 P 2 + ω = m=1 ( ) 0 dy r d 2 (1 + r) 1+ d 2 2m d a 2m d ( ω) m 1, P n, ω = 2 V, P2 = (1 + r)y best convergence = Litim s optimised regulator ) r opt (y) = c 1 (y b 1 Θ(1 y), best choice: b = 1, c 1 = 1 = sharp cutoff limit (c 1 ) = WH RG properties = advantage: best critical exponents in LPA, (recovers the Polchinsi RG by a Legendre trans. in LPA) = disadvantage: confront to the derivative expansion, (Pawlowsi s generalization but no explicit r(y) beyond LPA)
PM optimization Optimization II. Principle of Minimal ensitivity (PM): r(y) = α 1 0.664 e y u 1 (ρ) u = α opt = 6 10 ν 0.662 0.66 0.658 0.656 0.654 0.652 0.65 ν pms 1 2 3 4 5 6 7 8 9 10 α optimal choice for the parameters properties: = advantage: any order of the derivative expansion = disadvantage: regulators cannot be compared
C regulator Compactly upported mooth (C) regulator Litim regulator (Litim-Pawlowsi scheme): differentiability? PM: comparability? C regulator "unification" of the regulators r gen css (y) = exp[cy b 0 /(f hy b 0 )] 1 exp[cy b /(f hy b )] 1 θ(f hy b ), r modif css (y) = exp[cy b 0 /(1 hy b 0 )] 1 exp[cy b /(1 hy b )] 1 θ(1 hy b ) lim r css gen = lim r css modif = y 0 b c 0 c 0,h=1 1 y0 b ( ) 1 y b 1 θ(1 y), lim r css gen = lim r css modif = y 0 b f h 0,c 0 y b, lim r css gen (y) = lim r css modif = exp[y 0 b] 1 h 0,c=f h 0,c=1 exp[y b ] 1.
- - - New optimization scenario C regulator and a new type of optimization d=1 dimension NO spontaneous symmetry breaing sine-gordon model in d = 1 dimension [ ] 1 Γ = dx 2 z ( µ ϕ x ) 2 + ū cos(ϕ x ) 1.0 power-law regulator, b = 3, d=1 1.0 < power-law regulator, b = 3, d=1 0.8 0.8 D u 0.6 u 0.6 0.4 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1/z- 1.0 1.2 1.4 optimalization: smallest D = 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1/z- 1.0 1.2 1.4 1.0 C, b=1, c=0.1, y 0=0.1, f=1 u 0.8 0.6 < < D 1.0 0.75 0.5 0.25 0.0 0.0 0.2 0.4 0.6 y 0 C, b=1, c=0.1, f=1 0.4 0.54 C, b=2, c=0.1, y 0=1 0.53 0.2 0.52 0.51 0 200 400 600 f 0.0 0.0 0.05 0.1 0.15 1/z- 0.2 0.25 0.3 D
ummary Wetterich RG "unification" of RG equations Γ = 1 2 Tr R Γ (2), R (p) p 2 r(y), y = p2 + R 2 C regulator "unification" of regulator functions r modif css (y) = exp[cy b 0 /(1 hy b 0 )] 1 exp[cy b /(1 hy b )] 1 θ(1 hy b ). single numerical code for all regulators no problem with the upper bound of the momentum integral similar smoothing problem in Nuclear Physics approximation to the Litim-Pawlowsi scheme beyond LPA regulators can be compared through the PM framewor for new type of optimization Outloo optimization of C, convergence of the derivative expansion