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

SLIDE 1

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

SLIDE 2

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

SLIDE 3

Charge!

Consider (continuum) Yang–Mills theory: Z =

DΦ exp {ıSY M}, SY M = 1

2

dx
E2 − B2

in terms of spatial ( A) and temporal (A0) gauge fields. Importantly, E is linear in A0.

Coulomb gauge and DSes – p.3/14

SLIDE 4

Charge!

To fix to Coulomb gauge, ∇· A = 0, use Faddeev–Popov: 1 =

Dθδ
∇·

Aa Det

−

∇· D

.

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

SLIDE 5

Charge!

Now we convert to first order formalism by introducing an auxiliary field π: exp ı 2

E2
→
Dπ exp

ı 2 π2 − 2 π · E

,

then split π into transverse ( ∇ · π⊥a = 0) and longitudinal ( ∇φ)

parts. The action is linear in E (∼ A0) so integrate out to leave a

δ-functional constraint (Gauß’ law): Z =

DΦδ
∇·

Aa δ

∇·

π⊥a Det

−

∇· D

×

δ

∇·

Dabφb + gˆ ρa exp {ıS′}, ˆ ρa = f abc Ab · π⊥c → color charge. Now for φ . . .

Coulomb gauge and DSes – p.5/14

SLIDE 6

Charge!

Integrating φ, and noting the temporal zero modes δ

∇·

Dabφb + gˆ ρa → δ

d

xˆ ρa

Det
−

∇· D −1 δ (φ + . . .) gives us then Z =

DΦδ
∇·

Aa δ

∇·

π⊥a δ

d

xˆ ρa

exp {ıS}

with the final effective action S =

dx
π⊥·∂0

A − B2 2 − π⊥2 2 +g2 2 ˆ ρ

−

∇· D −1 ∇2 − ∇· D −1 ˆ ρ

.

Coulomb gauge and DSes – p.6/14

SLIDE 7

No charge!

Thanks to Gauß’ law, our system has the following properties:

two (physical) transverse degrees of freedom (

A, π⊥) with a conserved and vanishing total charge (

d

xˆ ρ = 0)

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

SLIDE 8

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 SFP =

dx
−ca

∇· Dabcb 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)): θa

x = ca xδλt.

D. Zwanziger, Nucl. Phys. B518 (1998) 237.

Coulomb gauge and DSes – p.8/14

SLIDE 9

STids: 2-point

After some work, for the 2-point proper functions... k0Γ00(k0, k) = ı ki

k2Γ0Ai(k0,

k)Γcc(q0 + k0, k) k0ΓA0k(k0, k) = ı ki

k2ΓAAki(k0,

k)Γcc(q0 + k0, k)

analogue of Landau gauge transversality
gluon polarization is not transverse (even at tree-level)
inverse ghost propagator independent of energy
A0-leg Green’s functions known in terms of others

– (local) elimination of A0-field (Gauß’ law)!

Coulomb gauge and DSes – p.9/14

SLIDE 10

STids: n-point

After lots more work, for the 3 & 4-point proper functions...

STids form closed sets from which A0-leg Green’s

functions known in terms of others – (local) elimination of A0-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)

g2W00 is RG invariant
K. -I. Kondo, arXiv:0907.3249...

Coulomb gauge and DSes – p.10/14

SLIDE 11

Ghost DSe

1
1

=

tree-level vertex
lattice gluon input (m ≈ 2√σ ≈ 0.88GeV):

WAA(p) ∼ 1 (p2

0 −

p2)

p2
p4 + m4
no gluon anomalous dimension, WAA(0) = 1/m2 finite!
regularize by subtracting at

p2 = 0 ⇒ Γcc(0) is an independent input.

G. Burgio, M. Quandt, H. Reinhardt, PRL 102, 032002. Coulomb gauge and DSes – p.11/14

SLIDE 12

Ghost DSe

10

8

10

6

10

4

10

2

10 10

2

10

4

10

6

x [GeV^2/m^2]

10

4

10

3

10

2

10

1

10 10

1 Γ(0)=1.e-5 Γ(0)=1.e-4 Γ(0)=1.e-3 Γ(0)=1.e-2 Γ(0)=1.e-1 Γ(0)=1 (2αsNcx)

0.5

10

8

10

6

10

4

10

2

10 10

2

10

4

10

6

x [GeV^2/m^2]

10

1

10 10

1

10

2

10

3

10

4

10

5 Γ(0)=1.e-5 Γ(0)=1.e-4 Γ(0)=1.e-3 Γ(0)=1.e-2 Γ(0)=1.e-1 Γ(0)=1 (2αsNcx)

0.5

Infrared powerlaw for Γcc(0) = 0 Γcc( p2 → 0) ∼ √αsNc √ 2σ | p|

Coulomb gauge and DSes – p.12/14

SLIDE 13

Ghost DSe

DSes are (functional) differential equations

– more than one solution is possible with different Γcc(0)

IR finite gluon can still give divergent ghost dressing

(unlike Landau gauge)

For the powerlaw solution

g2W00 ∼ 1

p2Γ2

ccΓAA

∼ σ

p4 ?

Coulomb gauge and DSes – p.13/14

SLIDE 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 g2W00 ∼ σ/

p4? (in progress)

find the physical input for Γcc(0)

Coulomb gauge and DSes – p.14/14