Truncation Effects in the FRG Method Istvn Nndori MTA-DE Particle - PowerPoint PPT Presentation

Functional Renormalization Group CSS regulator Optimization CSS in LPA CSS beyond LPA Summary Truncation Effects in the FRG Method István Nándori MTA-DE Particle Physics Research Group, University of Debrecen Non-perturbative Methods in Quantum Field Theory, Balatonfüred 2014

Functional Renormalization Group CSS regulator Optimization CSS in LPA CSS beyond LPA Summary Wetterich RG equation Functional Renormalization Group Wetterich RG equation ! R k ( p ) ≡ p 2 r ( y ) , y = p 2 k ∂ k Γ k = 1 k ∂ k R k , 2 Tr Γ ( 2 ) k 2 + R k k regulator: R k ! 0 ( p ) = 0 , R k ! Λ ( p ) = ∞ , R k ( p → 0 ) > 0 Approximations N cut Z  V k ( ϕ ) + Z k ( ϕ ) 1 � g n ( k ) 2 ( ∂ µ ϕ ) 2 + ... X ( 2 n )! ϕ 2 n Γ k [ ϕ ] = , V k = x n = 1 Problems = ⇒ approximated RG flow depends on R k = ⇒ phase structure depends on truncations

Functional Renormalization Group CSS regulator Optimization CSS in LPA CSS beyond LPA Summary Regulator functions Power-law regulator a r pow ( y ) = y b = ⇒ compatible with the derivative expansion, = ⇒ momentum integral analytic ( b = 1 , 2) but not UV safe Exponential regulator a r exp ( y ) = exp [ c y b ] − 1 = ⇒ compatible with the derivative expansion, = ⇒ UV safe but the momentum integral non-analytic Litim’s regulator ✓ 1 ◆ r opt ( y ) = a y b − 1 Θ ( 1 − y ) = ⇒ momentum integral analytic and UV safe, = ⇒ confront to the derivative expansion,

Functional Renormalization Group CSS regulator Optimization CSS in LPA CSS beyond LPA Summary Properties I. Compactly Supported Smooth (CSS) regulator general CSS regulator css ( y ) = exp [ cy b 0 / ( f − hy b 0 )] − 1 r gen exp [ cy b / ( f − hy b )] − 1 θ ( f − hy b ) I. N., JHEP 04 (2013) 150 normalised CSS regulator ( f ≡ 1 and y 0 fixed) 2 c − 1 exp [ ln ( 2 ) c ] − 1 θ ( 1 − hy b ) = θ ( 1 − hy b ) r norm ( y ) = css h i c yb ln ( 2 ) cy b exp − 1 1 − hyb − 1 2 1 � hy b I. N., I. G. Márián, V. Bacsó, PRD 89 (2014) 047701 = ⇒ smooth: compatible with the derivative expansion, = ⇒ compact support: UV safe, = ⇒ momentum integral cannot be performed analytically

Functional Renormalization Group CSS regulator Optimization CSS in LPA CSS beyond LPA Summary Properties II. Properties of the general CSS regulator ✓ 1 y b ◆ 0 c ! 0 r gen css = = Θ ( 1 − y ) , lim y b − 1 1 − y b 0 css = = y b f !1 r gen 0 lim y b , css ( y ) = = exp [ y b 0 ] − 1 h ! 0 , c = f r gen lim exp [ y b ] − 1 . Properties of the normalised CSS regulator ✓ 1 ◆ c ! 0 , h ! 1 r norm lim = y b − 1 θ ( 1 − y ) css 1 c ! 0 , h ! 0 r norm lim = css y b 1 c ! 1 , h ! 0 r norm lim = css exp [ ln ( 2 ) y b ] − 1

Functional Renormalization Group CSS regulator Optimization CSS in LPA CSS beyond LPA Summary Shortest Trajectory in Theory Space Optimization I. Amplitude ( ω = k � 2 V 00 k ) expansion in LPA Z 1 r 0 y 1 + d 1 2 m 2 k ∂ k V k = − α d k d X a 2 m � d ( − ω ) m � 1 ( 1 + r ) y + ω = d y d 0 m = 1 = ⇒ best convergence: Litim’s regulator (b=1, a=1) = ⇒ confront to the derivative expansion Shortest Trajectory (ST) in the Theory Space Jan M. Pawlowski, Annals of Physics 322 (2007) 2831 = ⇒ works in any order of the derivative expansion, = ⇒ it gives the Litim regulator in LPA, = ⇒ no explicit r(y) beyond LPA CSS is differentiable, recovers the Litim regulator in LPA BUT according to ST the Litim is NOT the best beyond LPA

Functional Renormalization Group CSS regulator Optimization CSS in LPA CSS beyond LPA Summary Principle of Minimal Sensitivity Optimization II. Principle of Minimal Sensitivity (PMS): L. Canet, B. Delamotte, D. Mouhanna and J. Vidal, PRD 67 (2003) 065004 0.664 1 r ( y ) = α u k ( ρ ) e y � 1 0.662 u 10 0.66 = ⇒ α opt = 6 0.658 ν 0.656 ν pms 0.654 0.652 0.65 1 2 3 4 5 6 7 8 9 10 α Léonie Canet, PRB 71 (2005) 012418 optimal choice for the parameters for a given regulator properties: = ⇒ works in any order of the derivative expansion, = ⇒ regulators cannot be compared directly CSS provides a framework to compare regulators directly.

Functional Renormalization Group CSS regulator Optimization CSS in LPA CSS beyond LPA Summary O(1), d=3 Model: O(1), d=3, LPA N cut " # Z 1 g n ( k ) 2 ( ∂ µ ϕ ) 2 + d 3 x X ϕ n Γ k [ ϕ ] = n ! n = 1 Critical exponent ν for the 3D O(1) model (regulator in exponential normalization, b parameter fixed) ! ν ν ν ν ν I. G. Márián, U. D. Jentscura, I.N., JPG 41 (2014) 055001 Best: c = 0 . 001, h = 1, b = 1 = ⇒ Litim limit

Functional Renormalization Group CSS regulator Optimization CSS in LPA CSS beyond LPA Summary QED 2 , d=2 Model: QED 2 , d=2, LPA  1 √ � Z 2 ( ∂ µ ϕ ) 2 + 1 k ϕ 2 + u k cos ( d 2 x 2 M 2 Γ k [ ϕ ] = 4 πϕ ) Critical ratio χ c for bosonized QED 2 (regulator in exponential normalization, b parameter fixed) ! χ c χ c χ c χ c χ c I. G. Márián, U. D. Jentscura, I.N., JPG 41 (2014) 055001 Best: c = 0 . 001, h = 1, b = 1 = ⇒ Litim limit

Functional Renormalization Group CSS regulator Optimization CSS in LPA CSS beyond LPA Summary Wavefunction renormalization Beyond LPA? → wavefunction renormalization O(N) model, d = 3, wavefunction renormalization  1 � Z 2 Z k ( ϕ )( ∂ µ ϕ ) 2 + g 1 ( k ) ϕ 2 + g 2 ( k ) ϕ 4 + ... d 3 x Γ k [ ϕ ] = 2 4 ! = ⇒ field-dependent Z k ( ϕ ) sine-Gordon model, d = 2, wavefunction renormalization Z  1 � 2 z k ( ∂ µ ϕ ) 2 + u k cos ( ϕ ) + ... d 2 x Γ k [ ϕ ] = 1.0 ⇒ field-independent z k = 1 / β 2 = 0.8 ⇒ β 2 0.6 = c = 8 π does not depend on R k < - u < > > > > 0.4 < > > 0.2 < < < < > 0.0 0 5 10 15 20 25 30 35 40 ~ 1/ z

Functional Renormalization Group CSS regulator Optimization CSS in LPA CSS beyond LPA Summary Truncations and phase structure Spontaneous Symmetry Breaking O(N=1) model in d = 1, truncated RG flow, LPA Z  1 2 ( ∂ µ ϕ ) 2 + g 1 ( k ) ϕ 2 + g 2 ( k ) � ϕ 4 + ... Γ k [ ϕ ] = dx 2 4 ! SG model in d = 1, truncated RG flow, beyond LPA  1 � Z 2 z k ( ∂ µ ϕ ) 2 + u k cos ( ϕ ) + ... Γ k [ ϕ ] = dx 1.0 < power-law regulator, b = 3, d=1 > 0.8 > D > > 0.6 > > > - u > Ê Gaussian g 2 > > 0.4 Ê Wilson Fisher > > Ê Infrared > 0.2 > 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 - 1/z - 1.0 - 0.8 - 0.6 - 0.4 - 0.2 0.0 g 1 N. Defenu, P . Mati, I.G. Márián, I. N., A. Trombettoni, in progress

Functional Renormalization Group CSS regulator Optimization CSS in LPA CSS beyond LPA Summary SG, d=1 Model: sine-Gordon, d=1, LPA + z Z  1 � 2 z k ( ∂ µ ϕ ) 2 + u k cos ( ϕ ) Γ k [ ϕ ] = dx 0.75 CSS regulator, d=1 dimension 0.7 0.65 0.6 D 0.55 0.5 0.45 Best values 0.4 Exponential limit Power-law limit 0.35 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 b I. N., I. G. Márián, V. Bacsó, PRD 89 (2014) 047701 Best: c → 0 (small but non-zero), b → 1 = ⇒ Litim limit

Functional Renormalization Group CSS regulator Optimization CSS in LPA CSS beyond LPA Summary Summary Wetterich RG equation R k ( p ) ≡ p 2 r ( y ) , y = p 2 k ∂ k Γ k = 1 k ∂ k R k , 2 Tr Γ ( 2 ) k 2 + R k k CSS regulator – "unification" of regulator functions 2 c − 1 exp [ ln ( 2 ) c ] − 1 θ ( 1 − hy b ) = θ ( 1 − hy b ) r norm ( y ) = css h i c yb ln ( 2 ) cy b exp − 1 1 − hyb − 1 2 1 � hy b single numerical code for all regulators no problem with the upper bound of the momentum integral regulators can be compared through the PMS Outlook optimization of CSS (best: Litim limit?)