Ghost Propagators on the Lattice Faddeev-Popov Matrix in Linear - PowerPoint PPT Presentation

Ghost Propagators on the Lattice Faddeev-Popov Matrix in Linear Covariant Gauge: First Numerical Results, arXiv:1809.08224 Martin Roelfs † , Attilio Cucchieri, David Dudal † , Tereza Mendes, Orlando Oliveira, Paulo J. Silva † KU Leuven Kulak, Department of Physics October 18, 2018 . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . .. . . . . .

Overview Introduction 1 Gauge Fixing Gribov Problem Solution to the Gribov problem 2 Solution to the Gribov problem on the lattice 3 Extension to Linear Covariant Gauge 4 Conclusions & Outlook 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ghost Propagators on the Lattice

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ghost Propagators on the Lattice

Yang-Mills theory Dirac Lagrangian L Dirac = ¯ ( i / ψ ∂ − m ) ψ (1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ghost Propagators on the Lattice

Yang-Mills theory Dirac Lagrangian L Dirac = ¯ ( i / ψ ∂ − m ) ψ (1) Demand local gauge invariance: ψ → e − ig ω a ( x ) t a ψ, ψ → ¯ ¯ ψ e ig ω a ( x ) t a (2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ghost Propagators on the Lattice

Yang-Mills theory Dirac Lagrangian L Dirac = ¯ ( i / ψ ∂ − m ) ψ (1) Demand local gauge invariance: ψ → e − ig ω a ( x ) t a ψ, ψ → ¯ ¯ ψ e ig ω a ( x ) t a (2) ( e − ig ω a ( x ) t a ψ ) ∂ µ ψ → ∂ µ ≈ ∂ µ ψ − ig ∂ µ ω ( x ) (3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ghost Propagators on the Lattice

Yang-Mills theory Dirac Lagrangian L Dirac = ¯ ( i / ψ ∂ − m ) ψ (1) Demand local gauge invariance: ψ → e − ig ω a ( x ) t a ψ, ψ → ¯ ¯ ψ e ig ω a ( x ) t a (2) ( e − ig ω a ( x ) t a ψ ) ∂ µ ψ → ∂ µ ≈ ∂ µ ψ − ig ∂ µ ω ( x ) (3) We find δ L ∝ ¯ ψγ µ ∂ µ ω a ( x ) t a ψ (4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ghost Propagators on the Lattice

Yang-Mills theory Dirac Lagrangian L Dirac = ¯ ( i / ψ ∂ − m ) ψ (1) Demand local gauge invariance: ψ → e − ig ω a ( x ) t a ψ, ψ → ¯ ¯ ψ e ig ω a ( x ) t a (2) ( e − ig ω a ( x ) t a ψ ) ∂ µ ψ → ∂ µ ≈ ∂ µ ψ − ig ∂ µ ω ( x ) (3) We find δ L ∝ ¯ ψγ µ ∂ µ ω a ( x ) t a ψ (4) Local gauge invariance could be restored by introducing a new field A µ which transforms as µ ω c + 1 δ A a µ = − f abc A b g ∂ µ ω a (5) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ghost Propagators on the Lattice

Yang-Mills theory D µ = ∂ µ + igA a µ t a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ghost Propagators on the Lattice

Yang-Mills theory D µ = ∂ µ + igA a µ t a F µν = − i g [ D µ , D ν ] = F a µν t a F a µν = ∂ µ A a ν − ∂ ν A a µ + gf abc A b µ A c ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ghost Propagators on the Lattice

Yang-Mills theory D µ = ∂ µ + igA a µ t a F µν = − i g [ D µ , D ν ] = F a µν t a F a µν = ∂ µ A a ν − ∂ ν A a µ + gf abc A b µ A c ν Yang-Mills Lagrangian L YM = − 1 µν F a µν + ¯ 4 F a ( i / ψ D − m ) ψ (6) ∫ ψ D ψ e i ∫ d 4 x L YM D A µ D ¯ (7) Z = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ghost Propagators on the Lattice

(a) There are infinitely many (b) The trick is to intersect each identical gauge configurations orbit exactly once related by gauge transform Figure: Image credit 1 ∫ ∫ ∫ D ¯ D A µ ( x ) ∝ A µ ( x ) D ω ( x ) (8) 1 Antonio Duarte Pereira (2016). “Exploring new horizons of the Gribov problem in Yang-Mills theories”. PhD thesis. Niteroi, Fluminense U.. arXiv: . . . . . . . . . . . . . . . . . . . . 1607.00365 [hep-th] . . . . . . . . . . . . . . . . . . . . . Ghost Propagators on the Lattice

Analogous to an integral ∫ d x d y e iS ( x ) Z ∝ (9) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ghost Propagators on the Lattice

Analogous to an integral ∫ d x d y e iS ( x ) Z ∝ (9) The integral over y is redundant, so we define ∫ d x d y δ ( y ) e iS ( x ) Z := (10) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ghost Propagators on the Lattice

Analogous to an integral ∫ d x d y e iS ( x ) Z ∝ (9) The integral over y is redundant, so we define ∫ ∫ d x d y δ ( y ) e iS ( x ) → d x d y δ ( y − f ( x )) e iS ( x ) Z := (10) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ghost Propagators on the Lattice

Analogous to an integral ∫ d x d y e iS ( x ) Z ∝ (9) The integral over y is redundant, so we define ∫ ∫ d x d y δ ( y ) e iS ( x ) → d x d y δ ( y − f ( x )) e iS ( x ) Z := (10) if y = f ( x ) is a unique solution of some function G ( x , y ) = 0, we can write δ ( G ( x , y )) = δ ( y − f ( x )) ∂ G � � ∫ = ⇒ � � δ ( G ) d y = 1 (11) � � | ∂ G / ∂ y | ∂ y � � . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ghost Propagators on the Lattice

Analogous to an integral ∫ d x d y e iS ( x ) Z ∝ (9) The integral over y is redundant, so we define ∫ ∫ d x d y δ ( y ) e iS ( x ) → d x d y δ ( y − f ( x )) e iS ( x ) Z := (10) if y = f ( x ) is a unique solution of some function G ( x , y ) = 0, we can write δ ( G ( x , y )) = δ ( y − f ( x )) ∂ G � � ∫ = ⇒ � � δ ( G ) d y = 1 (11) � � | ∂ G / ∂ y | ∂ y � � Substituting gives � ∂ G � ∫ � δ ( G ) e iS ( x ) � � Z := (12) d x d y � � ∂ y � . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ghost Propagators on the Lattice

∫ ψ D ψ e i ∫ D A µ D ¯ d 4 x L YM Z = All A µ related by a gauge transform h = e i ω a ( x ) t a are physically equivalent. µ = hA µ h † − i g h ( ∂ µ h † ) ≈ A µ + 1 A h g D µ ω (13) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ghost Propagators on the Lattice

∫ ψ D ψ e i ∫ D A µ D ¯ d 4 x L YM Z = All A µ related by a gauge transform h = e i ω a ( x ) t a are physically equivalent. µ = hA µ h † − i g h ( ∂ µ h † ) ≈ A µ + 1 A h g D µ ω (13) Demand a gauge fixing condition G a [ A µ ] = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ghost Propagators on the Lattice

∫ ψ D ψ e i ∫ D A µ D ¯ d 4 x L YM Z = All A µ related by a gauge transform h = e i ω a ( x ) t a are physically equivalent. µ = hA µ h † − i g h ( ∂ µ h † ) ≈ A µ + 1 A h g D µ ω (13) Demand a gauge fixing condition G a [ A µ ] = 0. E.g. ∂ µ A a µ ( x ) = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ghost Propagators on the Lattice

∫ ψ D ψ e i ∫ D A µ D ¯ d 4 x L YM Z = All A µ related by a gauge transform h = e i ω a ( x ) t a are physically equivalent. µ = hA µ h † − i g h ( ∂ µ h † ) ≈ A µ + 1 A h g D µ ω (13) Demand a gauge fixing condition G a [ A µ ] = 0. E.g. ∂ µ A a µ ( x ) = 0 ( ) ∫ δ G [ A h ] ∂ G � � ∫ D ω ( x ) δ ( G [ A h ]) � � 1 = δ ( G ) d y → 1 = det � � ∂ y δω � � . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ghost Propagators on the Lattice

∫ ψ D ψ e i ∫ D A µ D ¯ d 4 x L YM Z = All A µ related by a gauge transform h = e i ω a ( x ) t a are physically equivalent. µ = hA µ h † − i g h ( ∂ µ h † ) ≈ A µ + 1 A h g D µ ω (13) Demand a gauge fixing condition G a [ A µ ] = 0. E.g. ∂ µ A a µ ( x ) = 0 ( ) ∫ δ G [ A h ] ∂ G � � ∫ D ω ( x ) δ ( G [ A h ]) � � 1 = δ ( G ) d y → 1 = det � � ∂ y δω � � δ∂ µ ( A a µ ( x ) + 1 g D ac µ ω c ( x )) δ G a [ A h ( x )] = 1 g ∂ µ D ab = (14) µ δω b ( y ) δω b ( y ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ghost Propagators on the Lattice