QCD Vacuum, Centre Vortices and Flux Tubes

Transcription

1 QCD Vacuum, Centre Vortices and Flux Tubes Derek Leinweber Centre for the Subatomic Structure of Matter and Department of Physics University of Adelaide QCD Vacuum, Centre Vortices and Flux Tubes p.1/50

2 Overview Visualizations of QCD vacuum structure Action and Topological Charge densities QCD Vacuum, Centre Vortices and Flux Tubes p.2/50

3 Overview Visualizations of QCD vacuum structure Action and Topological Charge densities Role of Instantons and Anti-instantons Dynamical mass generation Accurate algorithms required to reveal these configurations QCD Vacuum, Centre Vortices and Flux Tubes p.2/50

4 Overview Visualizations of QCD vacuum structure Action and Topological Charge densities Role of Instantons and Anti-instantons Dynamical mass generation Accurate algorithms required to reveal these configurations Centre Vortices What are they? Why are they interesting? What happens if they are removed from QCD? QCD Vacuum, Centre Vortices and Flux Tubes p.2/50

5 Overview Visualizations of QCD vacuum structure Action and Topological Charge densities Role of Instantons and Anti-instantons Dynamical mass generation Accurate algorithms required to reveal these configurations Centre Vortices What are they? Why are they interesting? What happens if they are removed from QCD? Potential Energy between heavy quarks Y versus shape flux-tubes in baryons Emphasize how the nature of the flux tube revolutionizes the concept of a constituent-quark. QCD Vacuum, Centre Vortices and Flux Tubes p.2/50

6 CSSM Lattice Collaboration Sundance Bilson-Thompson Francois Bissey Sharada Boinepalli Frederic Bonnet Patrick Bowman Fu-Guang Cao Paul Coddington Seungho Choe Patrick Fitzhenry John Hedditch Urs Heller Waseem Kamleh Adrian Kitson Daniel Kusterer Kurt Langfeld Ben Lasscock Frank Lee Derek Leinweber Wally Melnitchouk Maria Parappilly Tony Signal Lorenz von Smekal Mark Stanford Tony Williams James Zanotti Jianbo Zhang QCD Vacuum, Centre Vortices and Flux Tubes p.3/50

7 One-Loop F µν F. Bonnet et.al, Phys.Rev.D62:094509,2000 QCD Vacuum, Centre Vortices and Flux Tubes p.4/50

8 Two-loop Improvement (2) O (x) = µν c U (x) x 1 2 µ + c x U (x) µ ν ν µ µ Generalized to five-loop improvement in S. O. Bilson-Thompson, D. B. Leinweber and A. G. Williams, arxiv:hep-lat/ QCD Vacuum, Centre Vortices and Flux Tubes p.5/50

9 Five-Loop Improvement An O(a 4 )-improved field-strength tensor is given by the following sum of Clover contributions C m n µν for m n loops: F Imp µν = k 1 C (1 1) µν + k 2 C (2 2) µν + k 3 C (1 2) µν + k 4 C (1 3) µν + k 5 C (3 3) µν, where k 1 = 19/9 55 k 5, k 2 = 1/36 16 k 5, k 3 = 64 k 5 32/45, k 4 = 1/15 6 k 5, QCD Vacuum, Centre Vortices and Flux Tubes p.6/50

10 Five-Loop Improvement An O(a 4 )-improved field-strength tensor is given by the following sum of Clover contributions C m n µν for m n loops: F Imp µν = k 1 C (1 1) µν + k 2 C (2 2) µν + k 3 C (1 2) µν + k 4 C (1 3) µν + k 5 C (3 3) µν, where k 1 = 19/9 55 k 5, k 2 = 1/36 16 k 5, k 3 = 64 k 5 32/45, k 4 = 1/15 6 k 5, k 5 is a tunable free parameter. QCD Vacuum, Centre Vortices and Flux Tubes p.6/50

11 Five-Loop Improvement An O(a 4 )-improved field-strength tensor is given by the following sum of Clover contributions C m n µν for m n loops: F Imp µν = k 1 C (1 1) µν + k 2 C (2 2) µν + k 3 C (1 2) µν + k 4 C (1 3) µν + k 5 C (3 3) µν, where k 1 = 19/9 55 k 5, k 2 = 1/36 16 k 5, k 3 = 64 k 5 32/45, k 4 = 1/15 6 k 5, k 5 is a tunable free parameter. Governs O(a 6 ) errors. When k 5 = 1/90, k 3 = k 4 = 0 providing a 3-loop F µν. Setting k 5 = 0, provides 4-loop F µν. We consider k 5 = 1/180, as the 5-loop F µν. QCD Vacuum, Centre Vortices and Flux Tubes p.6/50

12 2, 3, 4 and 5-Loop Improved F µν QCD Vacuum, Centre Vortices and Flux Tubes p.7/50

13 Revealing the Structure of Gluon Fields Consider the Action density S(x) = 1 2 F ab µν(x) F ba µν(x). S(x) is a scalar field in four dimensions. Consider a three-dimensional slice of the lattice. QCD Vacuum, Centre Vortices and Flux Tubes p.8/50

14 Revealing the Structure of Gluon Fields Consider the Action density S(x) = 1 2 F ab µν(x) F ba µν(x). S(x) is a scalar field in four dimensions. Consider a three-dimensional slice of the lattice. To view the structure of typical vacuum gluon-field configurations Render areas of intense action density in red. Render areas of moderate action density in blue. Low action-density regions are not rendered to allow us to see into the gluon field. QCD Vacuum, Centre Vortices and Flux Tubes p.8/50

15 Exposing Long-Distance Physics Short-distance physics in QCD is well understood. Interactions are weak at short distances. Quarks behave as approximately free particles. Corrections may be estimated via perturbation theory. QCD Vacuum, Centre Vortices and Flux Tubes p.9/50

16 Exposing Long-Distance Physics Short-distance physics in QCD is well understood. Interactions are weak at short distances. Quarks behave as approximately free particles. Corrections may be estimated via perturbation theory. This physics may be removed by locally minimizing the action. Cabibbo-Marinari algorithm identifies the link which maximally reduces the local action. Process is called Cooling. QCD Vacuum, Centre Vortices and Flux Tubes p.9/50

17 Exposing Long-Distance Physics Short-distance physics in QCD is well understood. Interactions are weak at short distances. Quarks behave as approximately free particles. Corrections may be estimated via perturbation theory. This physics may be removed by locally minimizing the action. Cabibbo-Marinari algorithm identifies the link which maximally reduces the local action. Process is called Cooling. Each frame in the animation follows one sweep of Cooling. All links are updated to locally minimize the action. Highly-improved lattice operators are utilized. Both O(a 2 ) and O(a 4 ) errors are removed O(a 6 ) errors tuned to stabilize nonperturbative phenomena QCD Vacuum, Centre Vortices and Flux Tubes p.9/50

18 Revealing the Structure of Gluon Fields Consider the Topological Charge density q(x) = g2 32π 2 ɛ µνρσ F ab µν(x) F ba ρσ(x). q(x) is a measure of the winding of the gluon field lines QCD Vacuum, Centre Vortices and Flux Tubes p.10/50

19 Revealing the Structure of Gluon Fields Consider the Topological Charge density q(x) = g2 32π 2 ɛ µνρσ F ab µν(x) F ba ρσ(x). q(x) is a measure of the winding of the gluon field lines q(x) is a scalar field in four dimensions. Consider a three-dimensional slice of the lattice. QCD Vacuum, Centre Vortices and Flux Tubes p.10/50

20 Revealing the Structure of Gluon Fields Consider the Topological Charge density q(x) = g2 32π 2 ɛ µνρσ F ab µν(x) F ba ρσ(x). q(x) is a measure of the winding of the gluon field lines q(x) is a scalar field in four dimensions. Consider a three-dimensional slice of the lattice. To view the structure of typical vacuum gluon-field configurations Render areas of positive charge density in red. Render areas of negative charge density in blue. Low charge-density regions are not rendered to allow us to see into the gluon field. QCD Vacuum, Centre Vortices and Flux Tubes p.10/50

21 Instanton Facts Gluon fields having nontrivial winding represented by the topological charge Q = x q(x) = ±1. Classical solutions to the QCD equations of motion They live in four dimensions. They have small finite action, S 0 = 8π 2 /g 2 Approximately 10,000 times smaller than the action of a typical field configuration QCD Vacuum, Centre Vortices and Flux Tubes p.11/50

22 Instanton Facts Gluon fields having nontrivial winding represented by the topological charge Q = x q(x) = ±1. Classical solutions to the QCD equations of motion They live in four dimensions. In isolation, the topological charge density is spherically symmetric q(x) = ± 6 π 2 ρ 4 (x 2 + ρ 2 ) 4, where ρ is a size parameter. Similarly for the action. QCD Vacuum, Centre Vortices and Flux Tubes p.11/50

23 Instanton Facts Gluon fields having nontrivial winding represented by the topological charge Q = x q(x) = ±1. Classical solutions to the QCD equations of motion They live in four dimensions. In isolation, the topological charge density is spherically symmetric q(x) = ± 6 π 2 ρ 4 (x 2 + ρ 2 ) 4, where ρ is a size parameter. Similarly for the action. Revealed in lattice QCD only after extensive cooling. QCD Vacuum, Centre Vortices and Flux Tubes p.11/50

24 The Effective Mass of Quarks Short-distance physics in QCD is well understood. Interactions are weak at short distances. Quarks behave as approximately free particles. Corrections may be estimated via perturbation theory. This property of QCD is called asymptotic freedom. QCD Vacuum, Centre Vortices and Flux Tubes p.12/50

25 The Effective Mass of Quarks Short-distance physics in QCD is well understood. Interactions are weak at short distances. Quarks behave as approximately free particles. Corrections may be estimated via perturbation theory. This property of QCD is called asymptotic freedom. At long distances, a rich structure in the QCD vacuum is revealed. QCD Vacuum, Centre Vortices and Flux Tubes p.12/50

26 The Effective Mass of Quarks Short-distance physics in QCD is well understood. Interactions are weak at short distances. Quarks behave as approximately free particles. Corrections may be estimated via perturbation theory. This property of QCD is called asymptotic freedom. At long distances, a rich structure in the QCD vacuum is revealed. Quarks and gluons do not propagate as free particles. QCD Vacuum, Centre Vortices and Flux Tubes p.12/50

27 The Effective Mass of Quarks Short-distance physics in QCD is well understood. Interactions are weak at short distances. Quarks behave as approximately free particles. Corrections may be estimated via perturbation theory. This property of QCD is called asymptotic freedom. At long distances, a rich structure in the QCD vacuum is revealed. Quarks and gluons do not propagate as free particles. The quark acquires an effective mass due to its interaction with the QCD vacuum. QCD Vacuum, Centre Vortices and Flux Tubes p.12/50

28 Massless Quarks in the QCD Vacuum QCD Vacuum, Centre Vortices and Flux Tubes p.13/50

29 Propagator Spectral Representation Eigenmodes of the Dirac Operator are defined by D ψ i = λ i ψ i. QCD Vacuum, Centre Vortices and Flux Tubes p.14/50

30 Propagator Spectral Representation Eigenmodes of the Dirac Operator are defined by D ψ i = λ i ψ i. The spectral representation of the quark propagator is S = 1 D + m q = i ψ i 1 λ i + m ψ i. QCD Vacuum, Centre Vortices and Flux Tubes p.14/50

31 Propagator Spectral Representation Eigenmodes of the Dirac Operator are defined by D ψ i = λ i ψ i. The spectral representation of the quark propagator is S = 1 D + m q = i ψ i 1 λ i + m ψ i. For small quark masses (like the u or d quarks in nature) Eigenmodes having small eigenvalues, λ i, dominate the nature of how quarks propagate in the QCD vacuum. QCD Vacuum, Centre Vortices and Flux Tubes p.14/50

32 Propagator Spectral Representation Eigenmodes of the Dirac Operator are defined by D ψ i = λ i ψ i. The spectral representation of the quark propagator is S = 1 D + m q = i ψ i 1 λ i + m ψ i. For λ i small, the real scalar field P i (x) = x ψ i ψ i x = ψ(x) ψ (x), describes the probable locations of the quarks in the vacuum as they propagate. QCD Vacuum, Centre Vortices and Flux Tubes p.14/50

33 Low-lying Eigenmode Density Low-lying eigenmodes of the Dirac operator are located on the topological structures giving rise to them. Each low-lying nondegenerate mode is associated with a single topological structure. QCD Vacuum, Centre Vortices and Flux Tubes p.15/50

34 What are Centre Vortices? 1. Gauge fix gluon configurations to Maximal Centre Gauge Bring the links U µ (x) close to the centre elements of SU(3) Z = exp ( 2πi m 3 ) I, with m = 1, 0, 1 On the lattice, search for the gauge transformation Ω tr U Ω µ (x) x,µ 2 Ω max QCD Vacuum, Centre Vortices and Flux Tubes p.16/50

35 What are Centre Vortices? 1. Gauge fix gluon configurations to Maximal Centre Gauge 2. Project the gluon field to the centre phase U µ (x) Z µ (x) where Z µ (x) = exp Implemented by 1 3 tr U µ Ω (x) = r µ (x) }{{} real ( cos ϕ µ (x) 2π ) 3 m µ(x) }{{} close to zero ( 2πi m ) µ(x), m µ (x) = 1, 0, 1 3 ( ) exp iϕ µ (x) } {{ } phase, m µ max. QCD Vacuum, Centre Vortices and Flux Tubes p.16/50

36 What are Centre Vortices? 1. Gauge fix gluon configurations to Maximal Centre Gauge 2. Project the gluon field to the centre phase U µ (x) Z µ (x) where Z µ (x) = exp 3. Vortices are identified by the centre charge ( 2πi m ) µ(x), m µ (x) = 1, 0, 1 3 z = Z µ (x) = exp ( 2πi n ) 3 If mod(n, 3) = 0 no vortex pierces the plaquette If mod(n, 3) = 1, or 1 a vortex with charge z pierces the plaquette QCD Vacuum, Centre Vortices and Flux Tubes p.16/50

37 What are Centre Vortices? 1. Gauge fix gluon configurations to Maximal Centre Gauge 2. Project the gluon field to the centre phase U µ (x) Z µ (x) where Z µ (x) = exp 3. Vortices are identified by the centre charge z = Z µ (x) = exp ( 2πi m ) µ(x), m µ (x) = 1, 0, 1 3 ( 2πi n ) 3 4. Vortices are removed by removing the centre phase U µ (x) U µ(x) = Z µ(x) U µ (x), QCD Vacuum, Centre Vortices and Flux Tubes p.16/50

38 Static Quark Potential V(r)/σ 1/ Fit to Full SU(3) potential Vortex β=4.80 No-Vortex β=4.80 No-Vortex β=4.60 Vortex β=4.60 No-Vortex β=4.38 Vortex β= r σ 1/2 QCD Vacuum, Centre Vortices and Flux Tubes p.17/50

39 Gluon Propagator QCD Vacuum, Centre Vortices and Flux Tubes p.18/50

40 Centre Vortices and Mass Generation Dynamical mass generation is associated with Dynamical Chiral Symmetry Breaking qq 0. QCD Vacuum, Centre Vortices and Flux Tubes p.19/50

41 Centre Vortices and Mass Generation Dynamical mass generation is associated with Dynamical Chiral Symmetry Breaking qq 0. Dynamical Chiral Symmetry Breaking is associated with a Finite density of zeromodes of the Dirac Operator, ρ(0), via the Casher-Banks relation qq = π ρ(0), as m q 0 QCD Vacuum, Centre Vortices and Flux Tubes p.19/50

42 Centre Vortices and Mass Generation Dynamical mass generation is associated with Dynamical Chiral Symmetry Breaking qq 0. Dynamical Chiral Symmetry Breaking is associated with a Finite density of zeromodes of the Dirac Operator, ρ(0), via the Casher-Banks relation qq = π ρ(0), as m q 0 Topologically non-trivial gauge fields (including instantons) Give rise to zeromodes QCD Vacuum, Centre Vortices and Flux Tubes p.19/50

43 Centre Vortices and Mass Generation Dynamical mass generation is associated with Dynamical Chiral Symmetry Breaking qq 0. Dynamical Chiral Symmetry Breaking is associated with a Finite density of zeromodes of the Dirac Operator, ρ(0), via the Casher-Banks relation qq = π ρ(0), as m q 0 Topologically non-trivial gauge fields (including instantons) Give rise to zeromodes Hence, a link between centre vortices and topology implies a Link between centre vortices and dynamical mass generation. QCD Vacuum, Centre Vortices and Flux Tubes p.19/50

44 Gluon Action Evolution QCD Vacuum, Centre Vortices and Flux Tubes p.20/50

45 Gluon Action Evolution QCD Vacuum, Centre Vortices and Flux Tubes p.21/50

46 Gluon Action Evolution QCD Vacuum, Centre Vortices and Flux Tubes p.22/50

47 Quark Propagator Decomposition In a covariant gauge, Lorentz invariance allows S aa (ζ; q) S(ζ; q) = Z(ζ; q 2 ) iγ q + M(q 2 ), M(q 2 ) is the Mass function Z(ζ; q 2 ) is the Renormalization function ζ is the renormalization point (3 GeV) with conditions Z(ζ; ζ 2 ) 1 M(ζ 2 ) m(ζ) For sufficiently large ζ, m(ζ) m ζ, the current quark mass. QCD Vacuum, Centre Vortices and Flux Tubes p.23/50

48 Quark Renormalization Function at m 0 q = 78 MeV QCD Vacuum, Centre Vortices and Flux Tubes p.24/50

49 Quark Mass Function at m 0 q = 116 MeV QCD Vacuum, Centre Vortices and Flux Tubes p.25/50

50 Quark Mass Function at m 0 q = 116 MeV QCD Vacuum, Centre Vortices and Flux Tubes p.26/50

51 Quark Mass Function at m 0 q = 78 MeV QCD Vacuum, Centre Vortices and Flux Tubes p.27/50

52 Quark Mass Function at m 0 q = 58 MeV QCD Vacuum, Centre Vortices and Flux Tubes p.28/50

53 Infrared Mass Function QCD Vacuum, Centre Vortices and Flux Tubes p.29/50

54 Infrared Mass Function QCD Vacuum, Centre Vortices and Flux Tubes p.30/50

55 LCG Mass Function with m 0 q = 29 MeV QCD Vacuum, Centre Vortices and Flux Tubes p.31/50

56 Potential Energy between Heavy Quarks QCD Vacuum, Centre Vortices and Flux Tubes p.32/50

57 Gluon Field Distribution in Mesons How does the Vacuum respond to the presence of Quarks? C( y, d) = Q( x) S( y + x) Q ( x + d) x Q( x) Q ( x + d) x S( x) x Q( x) denotes a quark at position x. Q ( x + d) denotes an antiquark at position x + d. S( y + x) denotes the action density at position y from the quark at x.... x denotes average over x and gluon field configurations. If there is no correlation between the action density and the locations of the quark and antiquark, C( y, d) = 1. QCD Vacuum, Centre Vortices and Flux Tubes p.33/50

58 Baryonic Wilson Loop U 3 ε a b c τ U 2 PSfrag replacements ε abc U 1 QCD Vacuum, Centre Vortices and Flux Tubes p.34/50

59 Quark Coordinates (x, y) Coordinates (lattice units) Distance (fm) # Q 1 Q 2 Q 3 F r s d qq 1 (1, 0) ( 1, 1) ( 1, 1) ( 0.42, 0) (2, 0) ( 1, 2) ( 1, 2) (0.15, 0) (3, 0) ( 1, 2) ( 1, 2) (0.15, 0) (3, 0) ( 2, 3) ( 2, 3) ( 0.27, 0) (4, 0) ( 3, 4) ( 3, 4) ( 0.69, 0) (5, 0) ( 4, 5) ( 4, 5) ( 1.11, 0) (7, 0) ( 4, 6) ( 4, 6) ( 0.54, 0) (8, 0) ( 4, 7) ( 4, 7) (0.04, 0) QCD Vacuum, Centre Vortices and Flux Tubes p.35/50

60 T- and Y-Shape Source Paths g replacements Q 2 y PSfrag replacements Q 2 y O x Q 1 O x Q 1 Q 3 Q 3 y x QCD Vacuum, Centre Vortices and Flux Tubes p.36/50

61 Gluon Field Distribution in Baryons How does the Vacuum respond to the presence of Quarks? W3Q ( r 1, r 2, r 3 ; τ) S( y, τ/2) C( y; r 1, r 2, r 3 ; τ) = W3Q ( r 1, r 2, r 3 ; τ) S( y, τ/2) If there is no correlation between the action density and the locations of the quarks, C( y, r 1, r 2, r 3 ; τ) = 1. QCD Vacuum, Centre Vortices and Flux Tubes p.37/50

62 Gluon Field Distribution in Baryons How does the Vacuum respond to the presence of Quarks? W3Q ( r 1, r 2, r 3 ; τ) S( y, τ/2) C( y; r 1, r 2, r 3 ; τ) = W3Q ( r 1, r 2, r 3 ; τ) S( y, τ/2) If there is no correlation between the action density and the locations of the quarks, C( y, r 1, r 2, r 3 ; τ) = 1. Similar results are observed for E 2 and B 2 separately. QCD Vacuum, Centre Vortices and Flux Tubes p.37/50

63 30-sweep T-shape Source QCD Vacuum, Centre Vortices and Flux Tubes p.38/50

64 30-sweep Y-shape Source QCD Vacuum, Centre Vortices and Flux Tubes p.39/50

65 Effective Potential at τ = Y T (original node) T (node moved 1 LU) T (node moved 1 then 2 LU) a V 3Q Length of the piece of string in fm QCD Vacuum, Centre Vortices and Flux Tubes p.40/50

66 Flux Tube Cross Section C(y) x=0 x=1 x=2 x=3 x=4 x=5 x=6 x=7 x=8 x=9 30 steps y QCD Vacuum, Centre Vortices and Flux Tubes p.41/50

67 Cross Section Fit C(y) 0.96 Data points Fit y QCD Vacuum, Centre Vortices and Flux Tubes p.42/50

68 Baryonic Ground State Properties Flux-tube radius is 0.38(3) fm. Vacuum-field action suppressed by 7.2(6)%. QCD Vacuum, Centre Vortices and Flux Tubes p.43/50

69 Baryonic Ground State Properties Flux-tube radius is 0.38(3) fm. Vacuum-field action suppressed by 7.2(6)%. Flux-tube node is 25% larger at 0.47(2) fm. Vacuum-field action suppression is 15(3)% larger at 8.1(7)%. QCD Vacuum, Centre Vortices and Flux Tubes p.43/50

70 The Heart of the Atom QCD Vacuum, Centre Vortices and Flux Tubes p.44/50

71 The Structure of the Nucleon QCD Vacuum, Centre Vortices and Flux Tubes p.45/50

72 Information on the Web The software used to create the animations is Advanced Visual System s AVS/Express Many of these animations are available on the web. QCD Vacuum, Centre Vortices and Flux Tubes p.46/50

73 QCD Vacuum, Centre Vortices and Flux Tubes p.47/50

74 Flux Tubes in SU(2) Gauge Theory G.S. Bali, K. Schilling and C. Schlichter (Wuppertal U.) Phys. Rev. D51 (1995) 5165 QCD Vacuum, Centre Vortices and Flux Tubes p.48/50

75 Enhancement or Expulsion? It is common to express the correlation as C S ( y, d) = Q( x) S( y + x) Q ( x + d) x Q( x) Q ( x + d) x S( x) x QCD Vacuum, Centre Vortices and Flux Tubes p.49/50

76 Enhancement or Expulsion? It is common to express the correlation as C S ( y, d) = Q( x) S( y + x) Q ( x + d) x Q( x) Q ( x + d) x S( x) x In Euclidean space S( x) = 1 2 tr ( B 2 Eucl + E 2 Eucl ) = 1 2 tr ( B 2 E 2) > 0 QCD Vacuum, Centre Vortices and Flux Tubes p.49/50

77 Enhancement or Expulsion? It is common to express the correlation as C S ( y, d) = Q( x) S( y + x) Q ( x + d) x Q( x) Q ( x + d) x S( x) x In Euclidean space S( x) = 1 2 tr ( B 2 Eucl + E 2 Eucl ) = 1 2 tr ( B 2 E 2) > 0 But the Minkowski action is S M ( x) = 1 2 tr ( E 2 B 2) < 0 QCD Vacuum, Centre Vortices and Flux Tubes p.49/50

78 Enhancement or Expulsion? It is common to express the correlation as C S ( y, d) = Q( x) S( y + x) Q ( x + d) x Q( x) Q ( x + d) x S( x) x In Euclidean space S( x) = 1 2 tr ( B 2 Eucl + E 2 Eucl ) = 1 2 tr ( B 2 E 2) > 0 But the Minkowski action is S M ( x) = 1 2 tr ( E 2 B 2) < 0 The sign of the correlation may be selected freely. QCD Vacuum, Centre Vortices and Flux Tubes p.49/50

79 Gluon Field Distribution in Mesons How does the Vacuum respond to the presence of Quarks? C( y, d) = Q( x) S( y + x) Q ( x + d) x Q( x) Q ( x + d) x S( x) x Q( x) denotes a quark at position x. Q ( x + d) denotes an antiquark at position x + d. S( y + x) denotes the action density at position y from the quark at x.... x denotes average over x and gluon field configurations. If there is no correlation between the action density and the locations of the quark and antiquark, C( y, d) = 1. QCD Vacuum, Centre Vortices and Flux Tubes p.50/50