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

1

Introduction Gauge Fixing Gribov Problem

2

Solution to the Gribov problem

3

Solution to the Gribov problem on the lattice

4

Extension to Linear Covariant Gauge

5

Conclusions & Outlook

Ghost Propagators on the Lattice

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Introduction

Ghost Propagators on the Lattice

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Yang-Mills theory

Dirac Lagrangian LDirac = ¯ ψ

(i /

∂ − m

)ψ

(1)

Ghost Propagators on the Lattice

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Yang-Mills theory

Dirac Lagrangian LDirac = ¯ ψ

(i /

∂ − m

)ψ

(1) Demand local gauge invariance: ψ → e−igωa(x)taψ, ¯ ψ → ¯ ψeigωa(x)ta (2)

Ghost Propagators on the Lattice

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Yang-Mills theory

Dirac Lagrangian LDirac = ¯ ψ

(i /

∂ − m

)ψ

(1) Demand local gauge invariance: ψ → e−igωa(x)taψ, ¯ ψ → ¯ ψeigωa(x)ta (2) ∂µψ → ∂µ

(

e−igωa(x)taψ

)

≈ ∂µψ − ig∂µω(x) (3)

Ghost Propagators on the Lattice

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Yang-Mills theory

Dirac Lagrangian LDirac = ¯ ψ

(i /

∂ − m

)ψ

(1) Demand local gauge invariance: ψ → e−igωa(x)taψ, ¯ ψ → ¯ ψeigωa(x)ta (2) ∂µψ → ∂µ

(

e−igωa(x)taψ

)

≈ ∂µψ − ig∂µω(x) (3) We find δL ∝ ¯ ψγµ∂µωa(x)taψ (4)

Ghost Propagators on the Lattice

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Yang-Mills theory

Dirac Lagrangian LDirac = ¯ ψ

(i /

∂ − m

)ψ

(1) Demand local gauge invariance: ψ → e−igωa(x)taψ, ¯ ψ → ¯ ψeigωa(x)ta (2) ∂µψ → ∂µ

(

e−igωa(x)taψ

)

≈ ∂µψ − ig∂µω(x) (3) We find δL ∝ ¯ ψγµ∂µωa(x)taψ (4) Local gauge invariance could be restored by introducing a new field Aµ which transforms as δAa

µ = −f abcAb µωc + 1

g ∂µωa (5)

Ghost Propagators on the Lattice

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Yang-Mills theory

Dµ = ∂µ + igAa

µta

Ghost Propagators on the Lattice

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Yang-Mills theory

Dµ = ∂µ + igAa

µta

Fµν = − i g [Dµ, Dν] = F a

µνta

F a

µν = ∂µAa ν − ∂νAa µ + gf abcAb µAc ν

Ghost Propagators on the Lattice

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Yang-Mills theory

Dµ = ∂µ + igAa

µta

Fµν = − i g [Dµ, Dν] = F a

µνta

F a

µν = ∂µAa ν − ∂νAa µ + gf abcAb µAc ν

Yang-Mills Lagrangian LYM = −1 4F a

µνF aµν + ¯

ψ

(i /

D − m

)ψ

(6) Z=

∫

DAµD ¯ ψDψei ∫

d4xLYM

(7)

Ghost Propagators on the Lattice

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(a) There are infinitely many identical gauge configurations related by gauge transform (b) The trick is to intersect each

• rbit exactly once

Figure: Image credit1 ∫

DAµ(x) ∝

∫

D¯ Aµ(x)

∫

Dω(x) (8)

1Antonio 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 Z ∝

∫

dx dy eiS(x) (9)

Ghost Propagators on the Lattice

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Analogous to an integral Z ∝

∫

dx dy eiS(x) (9) The integral over y is redundant, so we define Z :=

∫

dx dy δ(y)eiS(x) (10)

Ghost Propagators on the Lattice

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Analogous to an integral Z ∝

∫

dx dy eiS(x) (9) The integral over y is redundant, so we define Z :=

∫

dx dy δ(y)eiS(x) →

∫

dx dy δ(y − f (x))eiS(x) (10)

Ghost Propagators on the Lattice

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Analogous to an integral Z ∝

∫

dx dy eiS(x) (9) The integral over y is redundant, so we define Z :=

∫

dx dy δ(y)eiS(x) →

∫

dx dy δ(y − f (x))eiS(x) (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/∂y | = ⇒

• ∂G

∂y

• ∫

δ(G) dy = 1 (11)

Ghost Propagators on the Lattice

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Analogous to an integral Z ∝

∫

dx dy eiS(x) (9) The integral over y is redundant, so we define Z :=

∫

dx dy δ(y)eiS(x) →

∫

dx dy δ(y − f (x))eiS(x) (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/∂y | = ⇒

• ∂G

∂y

• ∫

δ(G) dy = 1 (11) Substituting gives Z :=

∫

dx dy

• ∂G

∂y

• δ(G)eiS(x)

(12)

Ghost Propagators on the Lattice

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Z =

∫

DAµD ¯ ψDψei ∫

d4xLYM

All Aµ related by a gauge transform h = eiωa(x)ta are physically equivalent. Ah

µ = hAµh† − i

g h(∂µh†) ≈ Aµ + 1 g Dµω (13)

Ghost Propagators on the Lattice

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Z =

∫

DAµD ¯ ψDψei ∫

d4xLYM

All Aµ related by a gauge transform h = eiωa(x)ta are physically equivalent. Ah

µ = hAµh† − i

g h(∂µh†) ≈ Aµ + 1 g Dµω (13) Demand a gauge fixing condition Ga[Aµ] = 0.

Ghost Propagators on the Lattice

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Z =

∫

DAµD ¯ ψDψei ∫

d4xLYM

All Aµ related by a gauge transform h = eiωa(x)ta are physically equivalent. Ah

µ = hAµh† − i

g h(∂µh†) ≈ Aµ + 1 g Dµω (13) Demand a gauge fixing condition Ga[Aµ] = 0. E.g. ∂µAa

µ(x) = 0

Ghost Propagators on the Lattice

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Z =

∫

DAµD ¯ ψDψei ∫

d4xLYM

All Aµ related by a gauge transform h = eiωa(x)ta are physically equivalent. Ah

µ = hAµh† − i

g h(∂µh†) ≈ Aµ + 1 g Dµω (13) Demand a gauge fixing condition Ga[Aµ] = 0. E.g. ∂µAa

µ(x) = 0

1 =

• ∂G

∂y

• ∫

δ(G) dy → 1 = det

(

δG[Ah] δω

) ∫

Dω(x)δ(G[Ah])

Ghost Propagators on the Lattice

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Z =

∫

DAµD ¯ ψDψei ∫

d4xLYM

All Aµ related by a gauge transform h = eiωa(x)ta are physically equivalent. Ah

µ = hAµh† − i

g h(∂µh†) ≈ Aµ + 1 g Dµω (13) Demand a gauge fixing condition Ga[Aµ] = 0. E.g. ∂µAa

µ(x) = 0

1 =

• ∂G

∂y

• ∫

δ(G) dy → 1 = det

(

δG[Ah] δω

) ∫

Dω(x)δ(G[Ah]) δGa[Ah(x)] δωb(y) = δ∂µ(Aa

µ(x) + 1 g Dac µ ωc(x))

δωb(y) = 1 g ∂µDab

µ

(14)

Ghost Propagators on the Lattice

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Z =

∫

DAµD ¯ ψDψei ∫

d4xLYM

All Aµ related by a gauge transform h = eiωa(x)ta are physically equivalent. Ah

µ = hAµh† − i

g h(∂µh†) ≈ Aµ + 1 g Dµω (13) Demand a gauge fixing condition Ga[Aµ] = 0. E.g. ∂µAa

µ(x) = 0

1 =

• ∂G

∂y

• ∫

δ(G) dy → 1 = det

(

δG[Ah] δω

) ∫

Dω(x)δ(G[Ah]) δGa[Ah(x)] δωb(y) = δ∂µ(Aa

µ(x) + 1 g Dac µ ωc(x))

δωb(y) = 1 g ∂µDab

µ

(14) det

(δG[A]

δω

)

∝

∫

D¯ cDcei ∫

d4x¯ ca(−∂µDab

µ )cb

(15)

Ghost Propagators on the Lattice

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Gauge fixing gives L = −1 4

(

F a

µν

)2 −

1 2ξ (∂µAa

µ)2 + ¯

ψ

(i /

D − m

)ψ + ¯

ca( −∂µDab

µ

)

cb (16) Z =

∫

DAµD ¯ ψDψD¯ cDcei ∫

d4xL

Ghost Propagators on the Lattice

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Gauge fixing gives L = −1 4

(

F a

µν

)2 −

1 2ξ (∂µAa

µ)2 + ¯

ψ

(i /

D − m

)ψ + ¯

ca( −∂µDab

µ

)

cb (16) Z =

∫

DAµD ¯ ψDψD¯ cDcei ∫

d4xL

Apart from the familiar particles, we now have ghosts whose two-point function is given by

⟨

¯ ca(x)cb(0)

⟩

=

(

−∂µDab

µ (x)

)−1

(17)

Ghost Propagators on the Lattice

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Gauge fixing gives L = −1 4

(

F a

µν

)2 −

1 2ξ (∂µAa

µ)2 + ¯

ψ

(i /

D − m

)ψ + ¯

ca( −∂µDab

µ

)

cb (16) Z =

∫

DAµD ¯ ψDψD¯ cDcei ∫

d4xL

Apart from the familiar particles, we now have ghosts whose two-point function is given by

⟨

¯ ca(x)cb(0)

⟩

=

(

−∂µDab

µ (x)

)−1

(17) And all is well in the world... Or is it?

Ghost Propagators on the Lattice

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Gribov problem

Take an infinitesimal gauge transformation Aa ′

µ = Aa µ + 1

g Dab

µ ωb

(18) Then, due to the gauge condition ∂µAa

µ = 0, we see that

∂µAa ′

µ = ∂µAa µ

if ∂µDab

µ ωb = 0

(19) If the FP-operator −∂µDab

µ has zero-modes, it is singular, and the

gauge hasn’t been properly fixed at all. ∂µDab

µ ωb = ∂2ωa + gf abc∂µ(

Ac

µωb)

= 0 (20)

Ghost Propagators on the Lattice

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Gribov problem

(a) We want to intersect each orbit exactly once (b) But instead, we intersect each

• rbit an unknown number of times –
• r not at all

Image credit2

2Antonio 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

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Solution to the Gribov problem

Ghost Propagators on the Lattice

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

It is not enough just to demand ∂µAµ = 0. This ∂µAµ = 0 needs to correspond to a minimum.

Ghost Propagators on the Lattice

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

It is not enough just to demand ∂µAµ = 0. This ∂µAµ = 0 needs to correspond to a minimum. F[h] :=

∫

d4x tr

[

Ah

µAh µ

]

≈ 1 2

∫

d4x Aa

µAa µ +

∫

d4x

(

∂µAa

µ

)

ωa − 1 4

∫

d4x ωa[ ∂µDab

µ + Dab µ ∂µ]

ωb

Ghost Propagators on the Lattice

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

It is not enough just to demand ∂µAµ = 0. This ∂µAµ = 0 needs to correspond to a minimum. F[h] :=

∫

d4x tr

[

Ah

µAh µ

]

≈ 1 2

∫

d4x Aa

µAa µ +

∫

d4x

(

∂µAa

µ

)

ωa − 1 4

∫

d4x ωa[ ∂µDab

µ + Dab µ ∂µ]

ωb This guarantees that the second variation −1

2

[

∂µDab

µ + Dab µ ∂µ]

→ −∂µDab

µ > 0

Ghost Propagators on the Lattice

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Solution to the Gribov problem on the lattice

Ghost Propagators on the Lattice

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Lattice 101

On the lattice we work with gauge links, rather than the Aµ directly Uµ(x) = eiagtaAa

µ(x+aˆ

µ/2)

= ⇒ Aµ(x + aˆ µ/2) = 1 2agi

[

Uµ(x) − Uµ(x)†]

traceless

→ 2 ag tb Re tr

[

−itbUµ(x)

]

Ghost Propagators on the Lattice

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Gauge fixing

F[h] :=

∫ d4x tr [

Ah

µAh µ

]

is replaced by ELD: ELD := −

∑

µ

∑

x

tr

(

g(x)Uµ(x)g†(x + aˆ µ)

)

(21) Uµ(x) = eigataAa

µ(x+aˆ

µ/2), g(x) = eiτtaωa(x)

Ghost Propagators on the Lattice

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Gauge fixing

F[h] :=

∫ d4x tr [

Ah

µAh µ

]

is replaced by ELD: ELD := −

∑

µ

∑

x

tr

(

g(x)Uµ(x)g†(x + aˆ µ)

)

(21) Uµ(x) = eigataAa

µ(x+aˆ

µ/2), g(x) = eiτtaωa(x)

Starting from a gauge configuration Uµ(x), minimize Eq. (21)

Ghost Propagators on the Lattice

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Gauge fixing

F[h] :=

∫ d4x tr [

Ah

µAh µ

]

is replaced by ELD: ELD := −

∑

µ

∑

x

tr

(

g(x)Uµ(x)g†(x + aˆ µ)

)

(21) Uµ(x) = eigataAa

µ(x+aˆ

µ/2), g(x) = eiτtaωa(x)

Starting from a gauge configuration Uµ(x), minimize Eq. (21) At this minimum, Landau gauge is obtained

Ghost Propagators on the Lattice

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Gauge fixing

ELD := −

∑

µ

∑

x

Re tr

(

g(x)Uµ(x)g†(x + aˆ µ)

)

Uµ(x) = eigataAa

µ(x+aˆ

µ/2),

g(x) = eiτtaωa(x) ≈ 1 + iτtaωa(x)

Ghost Propagators on the Lattice

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Gauge fixing

ELD := −

∑

µ

∑

x

Re tr

(

g(x)Uµ(x)g†(x + aˆ µ)

)

Uµ(x) = eigataAa

µ(x+aˆ

µ/2),

g(x) = eiτtaωa(x) ≈ 1 + iτtaωa(x) Then, at O(1) : ELD =

∑

µ

∑

x

[

−N + a2g2 4 Aa

µ(x + aˆ

µ/2)Aa

µ(x + aˆ

µ/2) + O

(

a4)

Ghost Propagators on the Lattice

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Gauge fixing

ELD := −

∑

µ

∑

x

Re tr

(

g(x)Uµ(x)g†(x + aˆ µ)

)

Uµ(x) = eigataAa

µ(x+aˆ

µ/2),

g(x) = eiτtaωa(x) ≈ 1 + iτtaωa(x) Then, at O(1) : ELD =

∑

µ

∑

x

[

−N + a2g2 4 Aa

µ(x + aˆ

µ/2)Aa

µ(x + aˆ

µ/2) + O

(

a4) O(τ) : δELD δωa(x) = aτ

∑

µ

Re tr [ita∂µUµ(x)] ≈ −1 2a2gτ

∑

µ

∂+

µ Aa µ(x − aˆ

µ/2) + O

(

a4)

Ghost Propagators on the Lattice

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Gauge fixing

O(τ) : δELD δωa(x) = aτ

∑

µ

Re tr [ita∂µUµ(x)] ≈ −1 2a2gτ

∑

µ

∂µAa

µ(x + aˆ

µ/2) + O

(

a4) We see that at a minimum, Landau gauge is satisfied up to O

(a2)

corrections.

Ghost Propagators on the Lattice

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Gauge fixing

ELD := −

∑

µ

∑

x

Re tr

(

g(x)Uµ(x)g†(x + aˆ µ)

)

O

(

τ 2) : δ2ELD δωa(x)δωb(y) =

∑

µ

Re tr

[{

ta, tb} (Uµ(x) + Uµ(x − aˆ µ))

]

δxy − 2 Re tr

[

tbtaUµ(x)

]

δx+aˆ

µ,y − 2 Re tr

[

tatbUµ(x − aˆ µ)

]

δx−aˆ

µ,y

:= Mab

xy → −1

2

[

∂µDab

µ + Dab µ ∂µ]

+ O

(

a2) The second variation of ELD corresponds to the Faddeev-Popov

• perator, up to O

(a2) corrections, and can be expressed directly in

terms of Uµ.

Ghost Propagators on the Lattice

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Gauge fixing

δ2ELD δωa(x)δωb(y) = Mab

xy → −1

2

[

∂µDab

µ + Dab µ ∂µ]

+ O

(

a2) When in a (local) minimum, ωa

xMab xy ωb y > 0:

no zero-modes no infinitesimal Gribov-copies

Ghost Propagators on the Lattice

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Gauge fixing

δ2ELD δωa(x)δωb(y) = Mab

xy → −1

2

[

∂µDab

µ + Dab µ ∂µ]

+ O

(

a2) When in a (local) minimum, ωa

xMab xy ωb y > 0:

no zero-modes no infinitesimal Gribov-copies

Figure: Ghost dressing function p2G(p) in Landau gauge

Ghost Propagators on the Lattice

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Extension to Linear Covariant Gauge

Ghost Propagators on the Lattice

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Motivation

Study the Gribov problem in LCG

Ghost Propagators on the Lattice

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Motivation

Study the Gribov problem in LCG Lattice data in LCG, to compare with continuum results

Ghost Propagators on the Lattice

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Motivation

Study the Gribov problem in LCG Lattice data in LCG, to compare with continuum results Observables should be gauge invariant, but this has yet to be tested numerically on the lattice with gauge fixed configurations

Ghost Propagators on the Lattice

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Motivation

Study the Gribov problem in LCG Lattice data in LCG, to compare with continuum results Observables should be gauge invariant, but this has yet to be tested numerically on the lattice with gauge fixed configurations Using Nielsen identities one can transform Greens function from Landau to LCG. To test if these hold in the non-pertubative regime, lattice results are needed

Ghost Propagators on the Lattice

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Gauge Fixing LCG

LCG ∂µAa

µ = Λa

(22) Recall: L = −1 4

(

F a

µν

)2 −

1 2ξ (∂µAa

µ)2 + ¯

ψ

(i /

D − m

)ψ + ¯

ca( −∂µDab

µ

)

cb

Ghost Propagators on the Lattice

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Gauge Fixing LCG

LCG ∂µAa

µ = Λa

(22) Recall: L = −1 4

(

F a

µν

)2 −

1 2ξ (∂µAa

µ)2 + ¯

ψ

(i /

D − m

)ψ + ¯

ca( −∂µDab

µ

)

cb Substituting Λa = ∂µAa

µ, we see that the G.F. term behaves

gaussian: e− Λ(x)

2ξ

(23)

Ghost Propagators on the Lattice

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Gauge Fixing LCG

LCG ∂µAa

µ = Λa

(22) Recall: L = −1 4

(

F a

µν

)2 −

1 2ξ (∂µAa

µ)2 + ¯

ψ

(i /

D − m

)ψ + ¯

ca( −∂µDab

µ

)

cb Substituting Λa = ∂µAa

µ, we see that the G.F. term behaves

gaussian: e− Λ(x)

2ξ

(23) On the lattice we want ∂µAa

µ = Λa with Λa(x) ∼ N

(0, √ξ )

Ghost Propagators on the Lattice

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Gauge Fixing LCG

Recall: ELG = −

∑

µ

∫

d4x tr

[

g(x)Uµ(x)g†(x + ˆ µ)

]

δELG δωa(x) =

∑

µ

1 2a2gτ∂µAa

µ(x + aˆ

µ/2) = 0 at extremum

Ghost Propagators on the Lattice

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Gauge Fixing LCG

Recall: ELG = −

∑

µ

∫

d4x tr

[

g(x)Uµ(x)g†(x + ˆ µ)

]

δELG δωa(x) =

∑

µ

1 2a2gτ∂µAa

µ(x + aˆ

µ/2) = 0 at extremum How to modify ELG such that δELCG δωa(x) =

∑

µ

1 2a2gτ

[

∂µAa

µ(x + aˆ

µ/2) − Λa(x)

]

= 0

Ghost Propagators on the Lattice

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Gauge Fixing LCG

ELCG Minimizing ELCG = Re

∫

d4x tr

[

−

∑

µ

g(x)Uµ(x)g†(x + ˆ µ) + ig(x)Λ(x)

]

with Λa(x) ∼ N

(0, √ξ )

Gives ∂µAµ(x + aˆ µ/2) = Λ(x) at O(τ), while leaving the O

(τ 2)

term unchanged.

Ghost Propagators on the Lattice

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Gauge Fixing LCG

Investigating O( τ 2) in more detail

Mab

xy → −1

2

[

∂µDab

µ + Dab µ ∂µ]

In Landau gauge, ∂µDab

µ = Dab µ ∂µ. But in LCG,

(

∂µDab

µ

)T = Dab

µ ∂µ.

Ghost Propagators on the Lattice

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Gauge Fixing LCG

Investigating O( τ 2) in more detail

Mab

xy → −1

2

[

∂µDab

µ + Dab µ ∂µ]

In Landau gauge, ∂µDab

µ = Dab µ ∂µ. But in LCG,

(

∂µDab

µ

)T = Dab

µ ∂µ.

Mab

xy is symmetric under (x, a) ↔ (y, b).

Ghost Propagators on the Lattice

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Gauge Fixing LCG

Investigating O( τ 2) in more detail

Mab

xy → −1

2

[

∂µDab

µ + Dab µ ∂µ]

In Landau gauge, ∂µDab

µ = Dab µ ∂µ. But in LCG,

(

∂µDab

µ

)T = Dab

µ ∂µ.

Mab

xy is symmetric under (x, a) ↔ (y, b).

Mab

xy is not ∝ −∂µDab µ

Ghost Propagators on the Lattice

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

The Challenge: find gf abc∂µAc

µ

In the continuum: Dab

µ ∂µωb = ∂2ωa + gf abcAc µ∂µωb

∂µDab

µ ωb = ∂2ωa + gf abcAc µ∂µωb + gf abc(

∂µAc

µ

)

ωb We want to find the lattice equivalent of gf abc∂µAc

µ.

Ghost Propagators on the Lattice

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

The Challenge: find gf abc∂µAc

µ

In the continuum: Dab

µ ∂µωb = ∂2ωa + gf abcAc µ∂µωb

∂µDab

µ ωb = ∂2ωa + gf abcAc µ∂µωb + gf abc(

∂µAc

µ

)

ωb We want to find the lattice equivalent of gf abc∂µAc

µ.

[

ta, tb] = if abctc

Ghost Propagators on the Lattice

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

The Challenge: find gf abc∂µAc

µ

In the continuum: Dab

µ ∂µωb = ∂2ωa + gf abcAc µ∂µωb

∂µDab

µ ωb = ∂2ωa + gf abcAc µ∂µωb + gf abc(

∂µAc

µ

)

ωb We want to find the lattice equivalent of gf abc∂µAc

µ.

[

ta, tb] = if abctc Ac

µ(x + aˆ

µ/2) = 2 Re tr [tcUµ(x)/(iag)]

Ghost Propagators on the Lattice

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

The Challenge: find gf abc∂µAc

µ

In the continuum: Dab

µ ∂µωb = ∂2ωa + gf abcAc µ∂µωb

∂µDab

µ ωb = ∂2ωa + gf abcAc µ∂µωb + gf abc(

∂µAc

µ

)

ωb We want to find the lattice equivalent of gf abc∂µAc

µ.

[

ta, tb] = if abctc Ac

µ(x + aˆ

µ/2) = 2 Re tr [tcUµ(x)/(iag)] gf abc∂µAc

µ →

[∂µAµ]ab

xy := 2 Re tr

[[

ta, tb] (Uµ(x) − Uµ(x − aˆ µ))

]

δxy (24)

Ghost Propagators on the Lattice

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Lattice 101

Continuum: Dab

µ ∂µ = 1

2

[

∂µDab

µ + Dab µ ∂µ]

− 1 2gf abc∂µAc

µ

∂µDab

µ = 1

2

[

∂µDab

µ + Dab µ ∂µ]

+ 1 2gf abc∂µAc

µ

Lattice [Dµ∂µ]ab

xy = −Mab xy + [∂µAµ]ab xy

[∂µDµ]ab

xy = −Mab xy − [∂µAµ]ab xy

Now also on the lattice, (∂µDab

µ )T = Dab µ ∂µ

Ghost Propagators on the Lattice

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Lattice 101

Very preliminary data

For each thermalized configuration {Uµ(x)}, 20 {Λ(x)} have been averaged.

(a) SU(2), V = 244, β = 2.4469 and ξ = 0.24469 (b) SU(3), V = 244, β = 6.0 and ξ = 0.1

Both are in agreement with Landau within error bars

Ghost Propagators on the Lattice

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Conclusions & Outlook

Ghost Propagators on the Lattice

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Conclusions & Outlook

Conclusions: We have found a lattice definition of −∂µDab

µ .

Preliminary results are in agreement with Landau results within error-bars. Outlook Simulations for more V , a and ξ should be performed. ξ = 1 is of particular interest, as is small a to scan the IR in more detail. Is the ghost propagator in LCG a real quantity? Do we still suffer from zero-modes? How to define a Gribov region in LCG?

Ghost Propagators on the Lattice

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

References

Thank you for your attention! Cucchieri, Attilio et al. (2018). “Faddeev-Popov Matrix in Linear Covariant Gauge: First Results”. In: arXiv: 1809.08224 [hep-lat]. Pereira, Antonio Duarte (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

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Ghost propagator

How to get to the ghost propagator?

⟨

¯ ca(x)cb(y)

⟩

=

(

[∂µDµ]ab

xy

)−1

(25)

Ghost Propagators on the Lattice

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Ghost propagator

How to get to the ghost propagator?

⟨

¯ ca(x)cb(y)

⟩

=

(

[∂µDµ]ab

xy

)−1

(25) Direct inversion is computationally far to expensive. Attach to a source of color b0 at position y0: [∂µDµ]ab

xyϕb(y) = ψaa0(x)

”[∂µDµ]ϕ = ψ” (26)

Ghost Propagators on the Lattice

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Ghost propagator

How to get to the ghost propagator?

⟨

¯ ca(x)cb(y)

⟩

=

(

[∂µDµ]ab

xy

)−1

(25) Direct inversion is computationally far to expensive. Attach to a source of color b0 at position y0: [∂µDµ]ab

xyϕb(y) = ψaa0(x)

”[∂µDµ]ϕ = ψ” (26) Additionally, we can work orthogonal to zero modes by inverting [∂µDµ][∂µDµ]ϕ = [∂µDµ]ψ (27)

Ghost Propagators on the Lattice

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Gauge fixing - Continuum limit

Baker-Campbell-Hausdorff

g†(x + aˆ µ)g(x) = e−iτtaωa(x+aˆ

µ)eiτtaωa(x)

= eiτta[−ωa(x+aˆ

µ)+ωa(x)]+ 1

2 τ 2ωa(x+aˆ

µ)ωb(x)[ta,tb]+O(τ 3)

= e−iaτta∂µωa(x)+ i

2 τ 2ωa(x+aˆ

µ)ωb(x)f abctc+O(τ 3)

Ghost Propagators on the Lattice

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Gauge fixing - Continuum limit

Baker-Campbell-Hausdorff

g†(x + aˆ µ)g(x) = e−iτtaωa(x+aˆ

µ)eiτtaωa(x)

= eiτta[−ωa(x+aˆ

µ)+ωa(x)]+ 1

2 τ 2ωa(x+aˆ

µ)ωb(x)[ta,tb]+O(τ 3)

= e−iaτta∂µωa(x)+ i

2 τ 2ωa(x+aˆ

µ)ωb(x)f abctc+O(τ 3)

Uµ(x) = eiagtaAa

µ(x+aˆ

µ/2)

Ghost Propagators on the Lattice

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Gauge fixing - Continuum limit

Baker-Campbell-Hausdorff

g†(x + aˆ µ)g(x) = e−iτtaωa(x+aˆ

µ)eiτtaωa(x)

= eiτta[−ωa(x+aˆ

µ)+ωa(x)]+ 1

2 τ 2ωa(x+aˆ

µ)ωb(x)[ta,tb]+O(τ 3)

= e−iaτta∂µωa(x)+ i

2 τ 2ωa(x+aˆ

µ)ωb(x)f abctc+O(τ 3)

Uµ(x) = eiagtaAa

µ(x+aˆ

µ/2)

U′

µ(x) = g†(x + aˆ

µ)g(x)Uµ(x) = eitaθa

Ghost Propagators on the Lattice

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Gauge fixing - Continuum limit

Baker-Campbell-Hausdorff

g†(x + aˆ µ)g(x) = e−iτtaωa(x+aˆ

µ)eiτtaωa(x)

= eiτta[−ωa(x+aˆ

µ)+ωa(x)]+ 1

2 τ 2ωa(x+aˆ

µ)ωb(x)[ta,tb]+O(τ 3)

= e−iaτta∂µωa(x)+ i

2 τ 2ωa(x+aˆ

µ)ωb(x)f abctc+O(τ 3)

Uµ(x) = eiagtaAa

µ(x+aˆ

µ/2)

U′

µ(x) = g†(x + aˆ

µ)g(x)Uµ(x) = eitaθa iθ = log

[

g†(x + aˆ µ)g(x)Uµ(x)

]

= −iaτta∂µωa + i 2τ 2ωa(x + aˆ µ)ωbf abctc + iagtaAa

µ)

+ i a2 2 τg∂µωaAb

µf abctc + i 1

4a2τ 2gωa(x + aˆ µ)ωbf abcAd

µf cdete + O

(

τ 3)

Ghost Propagators on the Lattice

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Gauge fixing - Continuum limit

Baker-Campbell-Hausdorff

Because of the real trace we skip odd powers of θ, resulting in only O

(a2) corrections.

Re tr

(

g(x)Uµ(x)g†(x + ˆ µ)

)

= N − 1 2 Re tr

[

θ2]

Ghost Propagators on the Lattice