Color Confinement and the Faddeev-Popov Ghosts in Coulomb

Introduction Coulomb gauge QCD Numerical simulations Summary Summary

Color conﬁnement and the Faddeev-Popov ghosts in Coulomb gauge QCD

Yoshiyuki Nakagawa1 collaboration with A. Nakamura2, T. Saito3, H. Toki1

1Research Center for Nuclear Physics, Osaka University, Ibaraki, Osaka 567-0047, Japan

2Research Institute for Information Science and Education, Hiroshima University, Higashi-Hiroshima 739-8521, Japan

3Integrated Information Center, Kochi University, Akebono-cho, Kochi 780-8520, Japan

Lattice 2007 @Regensburg, 8/2/2007 Introduction Coulomb gauge QCD Numerical simulations Summary Summary

Outline

I Introduction

I Coulomb gauge QCD

I Numerical simulations

I ghost propagator (conﬁnement/deconﬁnement phases) I instantaneous time-time gluon propagator I equal-time gluon propagator

I Summary Introduction Coulomb gauge QCD Numerical simulations Summary Summary — Introduction —

Color conﬁnement

I Absence of isolated quarks and gluons in nature

I A physical state is a color singlet state; the energy of a color non-singlet state diverges.

Several scenarios for the mechanism of color conﬁnement

I dual superconducting scenario (color monopoles) — maximal Abelian gauge

I center vortex model (center vortices) — maximal center gauge

I Gribov-Zwanziger conﬁnement scenario (F-P ghosts) — Coulomb gauge, covariant gauge

I Kugo-Ojima conﬁnement scenario (F-P ghosts) — covariant gauge (Landau gauge, etc.) Introduction Coulomb gauge QCD Numerical simulations Summary Summary — Coulomb gauge QCD —

Coulomb gauge Hamiltonian

1 Z H = d3x (E tr)2 + B2 2 i i 1 Z Z + d3y d3zρa(~y, t)Vab(~y,~z; Atr)ρb(~z, t) 2 kernel of the instantaneous interaction Z ab tr 3 −1 ac 2 −1 cb V (~y,~z; A ) = d x(M [A])~y,~x (−∇~x )(M [A])~x,~z

Color-Coulomb potential

2 a b D ab tr E Vc (~x − ~y) = g T~ · T~ V (~x,~y; A )

Time-time gluon propagator

D44(x − y) = Vc (~x − ~y)δ(x4 − y4) + P(x − y) Introduction Coulomb gauge QCD Numerical simulations Summary Summary — Coulomb gauge QCD —

Gribov-Zwanziger conﬁnement scenario

I Limit to the fundamental modular region

I Dominance of the conﬁgurations near the Gribov horizon

I enhancement of the low-lying FP eigenvalue density

I Strong infrared divergence of the ghost propagator

I Conﬁning color-Coulomb potential Introduction Coulomb gauge QCD Numerical simulations Summary Summary — Conﬁning instantaneous interaction —

Lattice simulation color-Coulomb potential 3 Singlet V(R,T), 18 x32, β=6 Linearly rising potential at 2 I large quark separations

1.5 I 2-3 times larger string tension than that of the

1 static potential V(R,T) I Zwanziger’s inequality 0.5 color-Coulomb potential V (R) ≥ V (R) static potential (from Wilson loop) c 0 0 5 10 R from: A. Nakamura and T. Saito, PTP115, 189 (2006) Introduction Coulomb gauge QCD Numerical simulations Summary Summary — Enhancement of the F-P low-lying modes —

Lattice simulation F-P operator

0.08 ab ab 4 M = −∂i Di 24 , β=6.00 4 0.07 20 , β=6.00 4 F-P eigenvalue density 16 , β=6.00 4 0.06 12 , β=6.00 N(λ, λ + ∆λ) 0.5 ρ(λ) = 2 (Nc − 1)V3∆λ

ρ(λ)/λ 0.05 In an Abelian gauge theory 0.04 √ λ ρ(λ) = 0.03 2 0 0.5 1 1.5 2 4π 2 λ [GeV ] from: Y. Nakagawa, A. Nakamura, T. Saito and H. Toki, PRD75 (2007) 014508 Introduction Coulomb gauge QCD Numerical simulations Summary Summary — Coulomb gauge QCD —

Gribov-Zwanziger conﬁnement scenario

I Limit to the fundamental modular region

I Dominance of the conﬁgurations near the Gribov horizon

I enhancement of the low-lying FP eigenvalue density

I Strong infrared divergence of the ghost propagator

I Conﬁning color-Coulomb potential Introduction Coulomb gauge QCD Numerical simulations Summary Summary — Coulomb gauge QCD —

Analysis of the ghost DSE (Fischer and Zwanziger 2005)

I Infrared power law ansatz c c |~p| G(|~p|) = 1 , Dtr(|~p|) = 2 (~p 2)1+γgh (~p 2)1+γgl

I Inrared exponents satisfy

2γgh + γgl = 0

I The equal-time would-be physical gluon propagator vanishes at vanishing momentum for γgh > 0.25. Introduction Coulomb gauge QCD Numerical simulations Summary Summary — Propagators in Coulomb gauge —

Ghost propagator, equal-time and instantaneous gluon propagagator

−1 G(~x − ~y) = (M [A])~x,~y

D44(~x − ~y) = hA4(~x, t)A4(~y, t)i = D44(~x − ~y) ∂ ∂ D (~x − ~y) = hA (~x, t)A (~y, t)i = δ − i j Dtr (~x − ~y) ij i j ij ∂2 dressing functions Z (~p 2) Z (|~p|) G(~p 2) = G , D (|~p|) = 44 ~p 2 44 ~p 2

2 Renormalization: a GB (a|~p|) = Z3(a, µ)GR (|~p|; µ)

1 1 tr GR (|~p|; µ) = , D44(|~p|; µ) = , D (|~p|; µ) = 1 ~p2=µ2 µ2 ~p2=µ2 µ2 ~p2=µ2 Introduction Coulomb gauge QCD Numerical simulations Summary Summary — Simulation parameters —

Gauge conﬁgurations

I SU(3) quenched simulations

I Wilson plaquette action, Iwasaki improved action

I Volumes (statistics):

124(120), 164(80), 184(80), 244(40), 324(20), 243 × 6(40)

I Lattice couplings (Wilson): β = 5.80, 5.90, 6.00, 6.10

I Lattice couplings (Iwasaki): β = 2.20, 2.40, 2.50, 2.60 Gauge ﬁxing

I Wilson-Mandula method (iterative method) 2 −14 I Tolerance of the gauge ﬁxing: (∂i Ai ) < 10

I Temporal gauge ﬁxing is needed for D44 Introduction Coulomb gauge QCD Numerical simulations Summary Summary — Ghost propagator —

Ghost dressing function ZG (|~p|; µ) Renormalization: µ = 3 [GeV]

2.5

4 ZG (|~p| = µ) = 1 12 , β=6.00 4 18 , β=6.00 4 Infrared ﬁtting 2 24 , β=5.80 4 24 , β=5.90 c 4 ZG (|~p|) = (|p|) 24 , β=6.00 2 γgh G (~p )

Z 4 1.5 24 , β=6.10 c = 1.42(1), γgh = 0.29(2) 2 1 χ /d.o.f . ∼ 0.244

1 10 p [GeV]

I Finite volume eﬀect for small lattice volume

I The ghost dressing function diverges in the IR limit. Introduction Coulomb gauge QCD Numerical simulations Summary Summary — Ghost propagator —

Ghost dressing function ZG (|~p|; µ) Renormalization: µ = 3 [GeV]

2.5

4 ZG (|~p| = µ) = 1 24 , β=6.00 (T=0) 3 β 24 x6, =6.00 (T~1.3Tc) 2 (|p|) G Z 1.5

1 10 p [GeV]

I IR divergent behavior even in the deconﬁnement phase

I Consistent with the conﬁning color-Coulomb potential in the deconﬁnement phase Introduction Coulomb gauge QCD Numerical simulations Summary Summary — instantaneous time-time gluon propagator —

2 2 Cylinder and cone cut Cylinder cut : ki − (ki ni ) ≤ 1 12 12 4 4 β=5.80 24 , β=5.80 24 , 10 4 10 4 β 24 , β=5.90 24 , =5.90 4 4 β=6.00 24 , β=6.00 24 , 8 4 8 4 β 24 , β=6.10 24 , =6.10 (p)

(p) 6 6 44 44 Z Z

4 4

2 2

0 0 1 10 1 10 p [GeV] p [GeV]

I Scaling is not observed.

I Diverges more stronger at large lattice couplings Introduction Coulomb gauge QCD Numerical simulations Summary Summary — instantaneous time-time gluon propagator —

Cylinder cut Cylinder and cone cut

12 12

4 4 16 , β=2.20 16 , β=2.20 10 4 10 4 16 , β=2.40 16 , β=2.40 4 4 16 , β=2.50 16 , β=2.50 8 4 8 4 16 , β=2.60 16 , β=2.60

(p) 6 (p) 6 44 44 Z Z

4 4

2 2

0 0 1 10 1 10 p [GeV] p [GeV]

I The Iwasaki improved action does not cure the non-scaling. Introduction Coulomb gauge QCD Numerical simulations Summary Summary — Equal-time gluon propagator —

with the cylinder cut with the cylinder and the cone cut

15 15

4 4 18 , β=5.80 18 , β=5.80 4 4 24 , β=5.80 24 , β=5.80 4 4 32 , β=5.80 32 , β=5.80

] 10 ] 10 -1 -1 (p) [GeV (p) [GeV tr tr

D 5 D 5

0 0 0 1 2 3 4 5 0 1 2 3 4 5 p [GeV] p [GeV]

I The cone cut reduces the ﬁnite volume eﬀect.

I We do not see a turnover of the equal-time gluon propagator. Introduction Coulomb gauge QCD Numerical simulations Summary Summary — Equal-time gluon propagator —

Wilson plaquette action Iwasaki improved action

10 10

4 4 9 24 , β=5.80 9 16 , β=2.40 4 4 8 24 , β=5.90 8 16 , β=2.50 4 4 7 24 , β=6.00 7 16 , β=2.60 ] ]

-1 6 -1 6

5 5 [GeV [GeV tr tr

D 4 D 4

3 3

2 2

1 1

0 0 1 10 1 10 p [GeV] p [GeV]

I Suppressed at large lattice couplings in the IR region

I Non-scaling both with the Wilson and the Iwasaki action Introduction Coulomb gauge QCD Numerical simulations Summary Summary — Summary—

We discussed the conﬁnement mechanism in Coulomb gauge and calculated the ghost and the gluon propagators.

I The ghost propagator diverges stronger than the free ghost propagator in the infrared limit both in the conﬁnement and the deconﬁnement phases.

I The gluon propagator is suppressed in the infrared region.

I Both the instantaneous part of the time-time gluon propagator and the equal-time gluon propagator do not scale. Introduction Coulomb gauge QCD Numerical simulations Summary Summary

Back up slides SECTION 0 - ARE YOU ENJOYING PHYSICS? 07/25/2007 02:38 PM SECTION 0 - ARE YOU ENJOYING PHYSICS? 07/25/2007 02:38 PM

home/GFIX00/P2 18^4, 6.00 201~280 ARE YOU ENJOYING PHYSICS? home/GFIX00/P3 18^4, 5.90 201~280 home/GFIX00/P4 18^4, 5.80 201~280

home/GFIX03/P2 8^4, 6.00 201~320 CONTENTS 検索 SECTION 0 SECTION 0 home/GFIX02/P2 12^4, 6.00 201~320 2007-07-13 (金) 20:41:46 (12d) (Top Page) Go! home/GFIX01/P1 18^4, 6.10 201~280 SECTION 1 サイト内 home/GFIX01/P2 18^4, 6.00 201~280 (LQCD) 未成熟な人間の特徴は、理想のために高貴な死を選ぼうとす Web SECTION 2 る点にある。 home/GFIX01/P3 18^4, 5.90 201~280 (Work) これに反して成熟した人間の特徴は、理想のために卑小な生編集操作 home/GFIX01/P4 18^4, 5.80 201~280 SECTION 3 を選ぼうとする点にある。編集 (Interest) Wilhelm Stekel home/GFIX04/P1 24^4, 6.10 201~240 添付 SECTION 4 差分 home/GFIX04/P2 24^4, 6.00 201~240 (English) † バックアップ SECTION 5 更新履歴 home/GFIX04/P3 24^4, 5.90 201~240

(Memo) SECTION 9にてスペイン滞在記始めました。 home/GFIX04/P4 24^4, 5.80 201~240 SECTION 6 (For 気になる本 Mac) ↑ 魔の系譜 (文庫) 201~240(q † home/GFIX05/P1 32^4, 6.10 SECTION 7 気になるリンク谷川健一 (著) ) (Miscellanea) ビジネス英語雑記帳 201~240(q 気になるMac home/GFIX05/P2 32^4, 6.00 SECTION 8 ) シャア専用iPod (Links) google docs 201~240(q SECTION 9 home/GFIX05/P3 32^4, 5.90 ガンダムの名言でタイピングの練習 ) (Travels) << 2007.7 >> Lectures by Feynman 201~240(q home/GFIX05/P4 32^4, 5.80 [SECTION 0] ) 日月火水木金土エンタメ進化系2.0 ドゥードゥル・テレビ home/GFIX14/P1 24^3x6, 6.10 201~240 1 2 3 4 5 6 7 シャア専用Macbook 8 9 10 11 12 13 14 home/GFIX14/P2 24^3x6, 6.00 201~240 15 16 17 18 19 20 21 QCDLAB project home/GFIX14/P3 24^3x6, 5.90 201~240 22 23 24 25 26 27 28 29 30 31 Mac OS X メンテナンス home/GFIX14/P4 24^3x6, 5.82 201~240 Introduction Coulomb gauge QCD Numerical simulations Summary Summary ドキュメント ↑ home/GFIX10/P1 16^4, 2.60 201~240 † メニューバーの編仕事メモ home/GFIX10/P2 16^4, 2.50 201~240 集 ↑ home/GFIX10/P3 16^4, 2.40 201~240 ライトバーの編集 Lattice cutoff † pukiwiki Home home/GFIX10/P4 16^4, 2.20 201~240 Page Wilson cutoff spacing Iwasaki ヘルプ -- coupling [GeV] [fm] coupling 18^4 lattice require 1 hour for 120000 iterations. PukiWikiで編集す 0.9259 0.216 2.20 24^4 lattice require 2 hour for 60000 iterations. るには？ ↑ テキスト整形の 5.80 1.333 0.150 2.40 Lowest 1000 eigenvalues of the FP operator † ルール(詳細版) 5.82 1.389 0.144 プラグインマニュ home/FPGEigen 8^4,6.00 201~320(0.3GB)(q) 5.90 1.621 0.124 2.50 アル home/FPGEigen 12^4,6.00 201~320(0.6GB)(q) 6.00 1.931 0.104 2.60 home/FPGEigen 18^4,6.00 201~280(2.0GB)(q) 6.10 2.271 0.088

↑ home/FPGEigen 24^4,6.00 201~240(5.8GB)(q)

↑ Gauge Updata † Ghost propagator † home/IM10 16^4, 2.6,2.5,2.4,2.2 201~300 home/FPGProp 8^4,6.00 201~320 home/IM11 24^4, 2.6,2.5,2.4,2.2 201~300

↑ home/FPGProp 12^4,6.00 201~320

201~280(q Gauge fixing † home/FPGProp 18^4,6.10 ) home/GFIX00/P1 18^4, 6.10 201~280 http://www.rcnp.osaka-u.ac.jp/~nkgw/index.php Page 1 of 3 http://www.rcnp.osaka-u.ac.jp/~nkgw/index.php Page 2 of 3 Introduction Coulomb gauge QCD Numerical simulations Summary Summary

— ReTrUµ —

Wilson plaquette action Iwasaki improved action 4 4 18 with the Wilson plaquette action 16 with the Iwasaki improved action 1 1

0.9 0.9

0.8 0.8

0.7 0.7

0.6 0.6 µ µ 0.5 0.5

ReTrU 0.4 ReTrU 0.4

0.3 0.3

0.2 0.2 µ=1 µ=1 0.1 µ=4 0.1 µ=4

0 0 5.8 5.9 6 6.1 2.2 2.3 2.4 2.5 2.6 β β Introduction Coulomb gauge QCD Numerical simulations Summary Summary — Equal-time gluon propagator —

I For a massive particle

Z dp0 Deq(|~p|) = p2 + m2 1 ∼ p~p 2 + m2

where Z p p dp0D (p0, |~p|) = δ − i j Dtr(|~p|) ij ij |~p|2 Introduction Coulomb gauge QCD Numerical simulations Summary Summary — Equal-time gluon propagator —

Wilson plaquette action Iwasaki improved action

100 100

4 4 24 , β=5.80 16 , β=2.40 4 4 24 , β=5.90 16 , β=2.50 4 4 24 , β=6.00 16 , β=2.60 10 10 ] ] -1 -1 [GeV [GeV tr tr D D 1 1

0.1 0.1 1 10 1 10 p [GeV] p [GeV]