Coulomb gauge and Schwinger-Dyson equations Peter Watson Instit - PowerPoint PPT Presentation

Coulomb gauge and Schwinger-Dyson equations Peter Watson Instit¨ ut f¨ ur Theoretische Physik Universit¨ at T¨ ubingen (supported by DFG contracts no. Re856/6-1 & Re856/6-2) Collaborator: H. Reinhardt P. W. and H. Reinhardt, arXiv:0812.1989, 0808.2436, PRD75:045021 Coulomb gauge and DSes – p.1/14

Outline Coulomb gauge QCD is very physical and Gauß’ law rules (as I’ll try to show), but this comes at the expense of covariance: it is also very technical (as I’ll try not to show)... • Temporal zero modes, total charge and physical degrees of freedom • Slavnov–Taylor identities (STids) • Ghost Dyson–Schwinger equation (DSe) • Summary and outlook Coulomb gauge and DSes – p.2/14

Charge! Consider (continuum) Yang–Mills theory: � D Φ exp { ı S Y M } , S Y M = 1 � E 2 − B 2 � � Z = dx 2 in terms of spatial ( � A ) and temporal ( A 0 ) gauge fields. Importantly, E is linear in A 0 . Coulomb gauge and DSes – p.3/14

Charge! To fix to Coulomb gauge, � ∇· � A = 0 , use Faddeev–Popov: � � A a � � � ∇· � � − � ∇· � 1 = D θδ D . Det Since − � ∇· � D only involves spatial operators, we still have spatially independent (time-dependent) gauge transforms and there are temporal zero-modes (not to mention the usual Gribov copies)! Replace Det in the identity with Det, the determinant with the zero-modes removed. Coulomb gauge and DSes – p.4/14

Charge! Now we convert to first order formalism by introducing an auxiliary field � π : � ı � ı � �� � � � � π 2 − 2 � π · � E 2 exp → D π exp E , 2 2 π ⊥ a = 0 ) and longitudinal ( � π into transverse ( � then split � ∇ · � ∇ φ ) parts. The action is linear in E ( ∼ A 0 ) so integrate out to leave a δ -functional constraint (Gauß’ law): � � A a � � π ⊥ a � � � ∇· � � � − � ∇· � Z = D Φ δ δ ∇· � D × Det D ab φ b + g ˆ � ρ a � ∇· � � exp { ı S ′ } , δ ρ a = f abc � A b · � π ⊥ c → color charge. Now for φ . . . ˆ Coulomb gauge and DSes – p.5/14

Charge! Integrating φ , and noting the temporal zero modes �� � � − 1 � D ab φ b + g ˆ ρ a � � ∇· � � − � ∇· � ρ a δ → δ d� x ˆ D δ ( φ + . . . ) Det gives us then �� � � � A a � � π ⊥ a � ∇· � � � ρ a Z = D Φ δ δ ∇· � δ d� x ˆ exp { ı S} with the final effective action 2 − π ⊥ 2 A − B 2 � � π ⊥ · ∂ 0 � S = dx � 2 + g 2 � � − 1 � − 1 � ∇ 2 � − � ∇· � − � ∇· � 2 ˆ ρ D D ρ ˆ . Coulomb gauge and DSes – p.6/14

No charge! Thanks to Gauß’ law, our system has the following properties: • two (physical) transverse degrees of freedom ( � π ⊥ ) with A , � � d� x ˆ ρ = 0 ) a conserved and vanishing total charge ( • ghosts and temporal zero modes have gone away and there are no energy divergences — the zero modes of the Faddeev–Popov operator are an expression of the Gribov problem. The temporal zero modes give rise to the total charge, what about the spatial zero modes (genuine Gribov copies)? • the gauge is temporally fixed However, this is formal and non-local. . . Coulomb gauge and DSes – p.7/14

Slavnov–Taylor identities To solve Dyson–Schwinger equations, we need a truncation scheme for vertices. This is done via Slavnov–Taylor identities. Go back to the original (local) action and fix to Coulomb gauge: ∇· � � A = 0 . . . Ghost term looks like � � D ab c b � − c a � ∇· � S FP = dx Because − � ∇· � D only involves spatial operators, we have invariance under a Gauß-BRST transform – a time-dependent BRS transform (N.B. θ → θ ( t,� x ) ): θ a x = c a x δλ t . D. Zwanziger, Nucl. Phys. B518 (1998) 237. Coulomb gauge and DSes – p.8/14

STids: 2-point After some work, for the 2-point proper functions... ı k i k 0 Γ 00 ( k 0 ,� k 2 Γ 0 Ai ( k 0 ,� k )Γ cc ( q 0 + k 0 ,� k ) = k ) � ı k i k 0 Γ A 0 k ( k 0 ,� k 2 Γ AAki ( k 0 ,� k )Γ cc ( q 0 + k 0 ,� k ) = k ) � • analogue of Landau gauge transversality • gluon polarization is not transverse (even at tree-level) • inverse ghost propagator independent of energy • A 0 -leg Green’s functions known in terms of others – (local) elimination of A 0 -field (Gauß’ law)! Coulomb gauge and DSes – p.9/14

STids: n-point After lots more work, for the 3 & 4-point proper functions... • STids form closed sets from which A 0 -leg Green’s functions known in terms of others – (local) elimination of A 0 -field (Gauß’ law)! • ⇒ It is possible that the Gauß-BRST charge can be identified with the physical charge, à la Kugo–Ojima (unlike, as it appears, in Landau gauge) • g 2 W 00 is RG invariant K. -I. Kondo, arXiv:0907.3249... Coulomb gauge and DSes – p.10/14

Ghost DSe -1 -1 = - • tree-level vertex • lattice gluon input ( m ≈ 2 √ σ ≈ 0 . 88 GeV): p 2 1 � W AA ( p ) ∼ p 4 + m 4 ( p 2 0 − � p 2 ) � � • no gluon anomalous dimension, W AA (0) = 1 /m 2 finite! p 2 = 0 ⇒ Γ cc (0) is an • regularize by subtracting at � independent input. G. Burgio, M. Quandt, H. Reinhardt, PRL 102, 032002. Coulomb gauge and DSes – p.11/14

Ghost DSe 5 1 10 10 Γ (0)=1.e-5 4 0 Γ (0)=1.e-4 10 10 Γ (0)=1.e-3 Γ (0)=1.e-2 3 -1 10 Γ (0)=1.e-1 10 Γ (0)=1.e-5 Γ (0)=1 Γ (0)=1.e-4 2 -2 (2 α s N c x) -0.5 Γ (0)=1.e-3 10 10 Γ (0)=1.e-2 Γ (0)=1.e-1 1 -3 10 10 Γ (0)=1 (2 α s N c x) 0.5 0 -4 10 10 -1 10 -8 -4 -2 0 2 4 -8 -6 -4 -2 0 2 4 6 -6 6 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 x [GeV^2/m^2] x [GeV^2/m^2] Infrared powerlaw for Γ cc (0) = 0 √ α s N c p 2 → 0) ∼ √ Γ cc ( � 2 σ | � p | Coulomb gauge and DSes – p.12/14

Ghost DSe • DSes are (functional) differential equations – more than one solution is possible with different Γ cc (0) • IR finite gluon can still give divergent ghost dressing (unlike Landau gauge) • For the powerlaw solution 1 ∼ σ g 2 W 00 ∼ p 4 ? � p 2 Γ 2 � cc Γ AA Coulomb gauge and DSes – p.13/14

Summary and outlook • Coulomb gauge is a physical choice • Gauß’ law dominates • two transverse degrees of freedom • vanishing and conserved total charge • STids come from Gauß-BRST • ghost DSe (toy version) To do: • investigate the physical/Gauß-BRST charge further • solve the other DSes: is g 2 W 00 ∼ σ/� p 4 ? (in progress) • find the physical input for Γ cc (0) Coulomb gauge and DSes – p.14/14