The Quark-Gluon Vertex in Landau-Gauge QCD

The Quark-Gluon Vertex in Landau-GaugeQCDA. Windisch, M. Hopfer and R. AlkoferLight Cone 20128 th − 13 th July 2012Cracow, Polish Academy of Arts and Sciences

Outline➣ Introduction➣ The Quark Gluon Vertex➣ How we do itAndreas Windisch, University of GrazThe Quark-Gluon Vertex in Landau Gauge QCD

⎡Introduction: Dyson-Schwinger Eq.⎤⎢⎣−−⎥⎦ −EquationAndreas Windisch, University of GrazThe Quark-Gluon Vertex in Landau Gauge QCD

Dyson-Schwinger ApproachI: Generating Functional∫∫Z [J] = D[φ] exp{−S[φ] +φ(x)J(x)dx}II: Connected DiagramsW [J] = log{Z [J]}IV: Effective Action: 1PI∫Γ[φ] = −W [J] +III: Legendre TransformationδW [J]δJ(x) = φ(x); δΓ[φ]δφ(x) = J(x)φ(x)J(x)dxAndreas Windisch, University of GrazThe Quark-Gluon Vertex in Landau Gauge QCD

Dyson-Schwinger ApproachV: Derivation∫D[φ] δδφ exp{−S[φ] + ∫φ(x)J(x)dx} = 0VI: DSEδΓ[φ]δφ= δSδφ [φ op]C. Fischer, J. Pawlowski, Phys.Rev. D75 & ref. therein∫δφ op = dyD(x, y)δφ(y) + φ(x)VII: DiagrammaticAndreas Windisch, University of GrazThe Quark-Gluon Vertex in Landau Gauge QCD

Dyson-Schwinger Equations• non-perturbative• infinite tower• µ − T -plane accessibleAndreas Windisch, University of GrazThe Quark-Gluon Vertex in Landau Gauge QCD

Getting DirectionsAndreas Windisch, University of GrazThe Quark-Gluon Vertex in Landau Gauge QCD

Infrared Behavior: C. Fischer, J. Pawlowski, Phys.Rev. D75 & ref. thereinKugo-Ojima Confinement ScenarioGluonGhostp 2 〈A(p)A(−p)〉 p2 →0→ 0p 2 〈C(p) ¯C(−p)〉 p2 →0→ ∞Andreas Windisch, University of GrazThe Quark-Gluon Vertex in Landau Gauge QCD

Infrared Behavior: C. Fischer, J. Pawlowski, Phys.Rev. D75 & ref. thereinKugo-Ojima Confinement ScenarioGluonGhostp 2 〈A(p)A(−p)〉 p2 →0→ 0p 2 〈C(p) ¯C(−p)〉 p2 →0→ ∞1. Linear rising quark potential?Andreas Windisch, University of GrazThe Quark-Gluon Vertex in Landau Gauge QCD

Infrared Behavior: C. Fischer, J. Pawlowski, Phys.Rev. D75 & ref. thereinKugo-Ojima Confinement ScenarioGluonGhostp 2 〈A(p)A(−p)〉 p2 →0→ 0p 2 〈C(p) ¯C(−p)〉 p2 →0→ ∞1.2.Linear rising quark potential?DχSB: Enhanced QGVAndreas Windisch, University of GrazThe Quark-Gluon Vertex in Landau Gauge QCD

Infrared Behavior: C. Fischer, J. Pawlowski, Phys.Rev. D75 & ref. thereinKugo-Ojima Confinement ScenarioGluonGhostp 2 〈A(p)A(−p)〉 p2 →0→ 0p 2 〈C(p) ¯C(−p)〉 p2 →0→ ∞1.2.3.Linear rising quark potential?DχSB: Enhanced QGVYM-running coupling: IR-FPAndreas Windisch, University of GrazThe Quark-Gluon Vertex in Landau Gauge QCD

The Quark-Gluon Vertex.Andreas Windisch, University of GrazThe Quark-Gluon Vertex in Landau Gauge QCD

Lorentz invariance:’only’ 12 comb.Basis.⎧⎪⎨⎪⎩1✚✚p 1✚✚p 2✚✚p 1 ✚✚p 2⎫⎪ ⎬⎪ ⎭⊗⎧⎨⎩γ µp µ 1p µ 2⎫⎬⎭Andreas Windisch, University of GrazThe Quark-Gluon Vertex in Landau Gauge QCD

Ball-Chiu Basis, J. Ball, T. Chiu, Phys. Rev. D 22, 2542 (1980)Γ µ =4∑8∑λ i L µ i+ τ j T µ ji=1 j=1L µ 1 = γµ T µ 3 = ✚ p 3p µ 3 − p2 3 γµL µ 2 = −( ✚ p 1 +✚ p 2)(p 1 + p 2 ) µ T µ 4 = −i(p2 3 σµν (p 1 + p 2 ) ν + 2p µ 3 σ λν p λ 1 pν 2 )L µ 3 = −i(p 1 + p 2 ) µ T µ 5 = iσµν (p 3 ) νL µ 4 = −iσµν (p 1 + p 2 ) ν T µ 6 = (p2 1 − p2 2 )γµ + (p 1 + p 2 ) µ ✚ p 3T µ 1 = i(pµ 1 p 2 · p 3 − p µ 2 p 1 · p 3 ) T µ 7 = i 2 (p2 1 − p2 2 )ξµ − i(p 1 + p 2 ) µ σ λν p λ 2 pν 1T µ 2 = (pµ 1 p 2 · p 3 − p µ 2 p 1 · p 3 )(✚ p 1 +✚ p 2) T µ 8 = −γµ σ λν p λ 2 pν 1 − ✚ p 2p µ 1 + ✚ p 1p µ 2ξ µ = (✚ p 1 +✚ p2)γµ − (p 1 + p 2 ) µ ;σ µν = 1 (γµγν − γν γµ)2Andreas Windisch, University of GrazThe Quark-Gluon Vertex in Landau Gauge QCD

Quark Propagator DSEThe coupled systemQuark-Gluon Vertex DSEAndreas Windisch, University of GrazThe Quark-Gluon Vertex in Landau Gauge QCD

Truncations/ApproximationsAndreas Windisch, University of GrazThe Quark-Gluon Vertex in Landau Gauge QCD

Truncations/ApproximationsAndreas Windisch, University of GrazThe Quark-Gluon Vertex in Landau Gauge QCD

Truncations/ApproximationsAndreas Windisch, University of GrazThe Quark-Gluon Vertex in Landau Gauge QCD

Truncations/ApproximationsAndreas Windisch, University of GrazThe Quark-Gluon Vertex in Landau Gauge QCD

Truncations/ApproximationsAndreas Windisch, University of GrazThe Quark-Gluon Vertex in Landau Gauge QCD

Truncations/ApproximationsAndreas Windisch, University of GrazThe Quark-Gluon Vertex in Landau Gauge QCD

Truncations/ApproximationsAndreas Windisch, University of GrazThe Quark-Gluon Vertex in Landau Gauge QCD

Truncations/ApproximationsAndreas Windisch, University of GrazThe Quark-Gluon Vertex in Landau Gauge QCD

Truncations/ApproximationsAndreas Windisch, University of GrazThe Quark-Gluon Vertex in Landau Gauge QCD

Truncations/ApproximationsAndreas Windisch, University of GrazThe Quark-Gluon Vertex in Landau Gauge QCD

The truncated & coupled systemQuark Propagator DSEQuark-Gluon Vertex DSEAndreas Windisch, University of GrazThe Quark-Gluon Vertex in Landau Gauge QCD

The truncated & coupled systemQuark Propagator DSEQuark-Gluon Vertex DSEAndreas Windisch, University of GrazThe Quark-Gluon Vertex in Landau Gauge QCD

Results obtained with this truncated systemχSBχSB restored byIR-singular:IR-singular:1 overall1 different solution2 soft gluon: (p 2 ) − 1 2 −κ2 reduced: (p 2 ) −κIR-divergence in 1-gexchange kernel➣ linear rising➣ quark confinementReduced strength of IRdivergence➣ Coulomb type➣ non-confiningQuark confinementDχSBAndreas Windisch, University of GrazThe Quark-Gluon Vertex in Landau Gauge QCD

Results obtained with this truncated systemχSBχSB restored byIR-singular:IR-singular:1 overall1 different solution2 soft gluon: (p 2 ) − 1 2 −κ2 reduced: (p 2 ) −κIR-divergence in 1-gexchange kernel➣ linear rising➣ quark confinementReduced strength of IRdivergence➣ Coulomb type➣ non-confiningQuark confinementDχSBAndreas Windisch, University of GrazThe Quark-Gluon Vertex in Landau Gauge QCD

The truncated & coupled systemQuark Propagator DSEQuark-Gluon Vertex DSEAndreas Windisch, University of GrazThe Quark-Gluon Vertex in Landau Gauge QCD

Quark Propagator DSEThe improved systemQuark-Gluon Vertex DSEAndreas Windisch, University of GrazThe Quark-Gluon Vertex in Landau Gauge QCD

The GoalQuark Propagator DSEQuark-Gluon Vertex DSEAndreas Windisch, University of GrazThe Quark-Gluon Vertex in Landau Gauge QCD

How we do it.Andreas Windisch, University of GrazThe Quark-Gluon Vertex in Landau Gauge QCD

The truncated & coupled systemQuark Propagator DSEQuark-Gluon Vertex DSEAndreas Windisch, University of GrazThe Quark-Gluon Vertex in Landau Gauge QCD

Quark Propagator DSEAlgebraic expression⎛4∑⎝ λ i (p 1 , p 2 ; p 2 − p 1 )L µ ii=1∫i✚ p 2A(p 2 2 ) + B(p2 2 ) = i ✚ p 2 + m 0 +d4 p1 B(p 2(2π) 4 γν 1 ) − i ✚ p 1A(p1 2 )p1 2A2 (p1 2) + B2 (p1 2)⎞(8∑+ τ j (p 1 , p 2 ; p 2 − p 1 )T µ ⎠jj=1δ νµ − (p )2 − p 1 ) ν (p 2 − p 1 ) µ Z ((p2 − p 1 ) 2 )(p 2 − p 1 ) 2 (p 2 − p 1 ) 2Andreas Windisch, University of GrazThe Quark-Gluon Vertex in Landau Gauge QCD

Quark-Gluon Vertex DSEAlgebraic expressionLhs⎛4∑⎝ λ i (p 1 , p 2 ; p 2 − p 1 )L µ i+i=1j=1⎞8∑τ j (p 1 , p 2 ; p 2 − p 1 )T µ ⎠jAndreas Windisch, University of GrazThe Quark-Gluon Vertex in Landau Gauge QCD

Quark-Gluon Vertex DSEAlgebraic expressionRhs(∫γ µ +⎛d 4 q(2π) 4 γ B(q 2 ) − i✚ p 1A(q 2 )4∑ρ ′ ⎝qA 2 (q 2 ) + B 2 (q 2 )i=1λ i (q, p 2 ; p 2 − q)L σ′i +8∑j=1τ j (q, p 2 ; p 2 − q)T σ′jδ ρ ′ ρ − (p )1 − q) ρ(p 1 − q) ρ ′ Z ((p1 − q) 2 [ ])((p 1 − q) 2 (p 1 − q) 2 M δ σ ′ σ − (q − p2)ρ(q − p )2) σ ′ Z ((q − p2 ) 2 )(q − p 2 ) 2 (q − p 2 ) 2⎞⎠Andreas Windisch, University of GrazThe Quark-Gluon Vertex in Landau Gauge QCD

Linear SystemLhs: ProjectionL 1 × Γ(p 1 , p 2 ; p 2 − p 1 ) → L 1 (λ i , τ j )L 2 × Γ(p 1 , p 2 ; p 2 − p 1 ) → L 2 (λ i , τ j )L 3 × Γ(p 1 , p 2 ; p 2 − p 1 ) → L 3 (λ i , τ j )L 4 × Γ(p 1 , p 2 ; p 2 − p 1 ) → L 4 (λ i , τ j )T 1 × Γ(p 1 , p 2 ; p 2 − p 1 ) → T 1 (λ i , τ j )T 2 × Γ(p 1 , p 2 ; p 2 − p 1 ) → T 2 (λ i , τ j )T 3 × Γ(p 1 , p 2 ; p 2 − p 1 ) → T 3 (λ i , τ j )T 4 × Γ(p 1 , p 2 ; p 2 − p 1 ) → T 4 (λ i , τ j )T 5 × Γ(p 1 , p 2 ; p 2 − p 1 ) → T 5 (λ i , τ j )T 6 × Γ(p 1 , p 2 ; p 2 − p 1 ) → T 6 (λ i , τ j )T 7 × Γ(p 1 , p 2 ; p 2 − p 1 ) → T 7 (λ i , τ j )T 8 × Γ(p 1 , p 2 ; p 2 − p 1 ) → T 8 (λ i , τ j )Rhs: ProjectionL 1 × Rhs → pL1L 2 × Rhs → pL2L 3 × Rhs → pL3L 4 × Rhs → pL4T 1 × Rhs → pT 1T 2 × Rhs → pT 2T 3 × Rhs → pT 3T 4 × Rhs → pT 4T 5 × Rhs → pT 5T 6 × Rhs → pT 6T 7 × Rhs → pT 7T 8 × Rhs → pT 8Andreas Windisch, University of GrazThe Quark-Gluon Vertex in Landau Gauge QCD

Linear SystemLhs: ProjectionL 1 × Γ(p 1 , p 2 ; p 2 − p 1 ) → L 1 (λ i , τ j )L 2 × Γ(p 1 , p 2 ; p 2 − p 1 ) → L 2 (λ i , τ j )L 3 × Γ(p 1 , p 2 ; p 2 − p 1 ) → L 3 (λ i , τ j )L 4 × Γ(p 1 , p 2 ; p 2 − p 1 ) → L 4 (λ i , τ j )T 1 × Γ(p 1 , p 2 ; p 2 − p 1 ) → T 1 (λ i , τ j )T 2 × Γ(p 1 , p 2 ; p 2 − p 1 ) → T 2 (λ i , τ j )T 3 × Γ(p 1 , p 2 ; p 2 − p 1 ) → T 3 (λ i , τ j )T 4 × Γ(p 1 , p 2 ; p 2 − p 1 ) → T 4 (λ i , τ j )T 5 × Γ(p 1 , p 2 ; p 2 − p 1 ) → T 5 (λ i , τ j )T 6 × Γ(p 1 , p 2 ; p 2 − p 1 ) → T 6 (λ i , τ j )T 7 × Γ(p 1 , p 2 ; p 2 − p 1 ) → T 7 (λ i , τ j )T 8 × Γ(p 1 , p 2 ; p 2 − p 1 ) → T 8 (λ i , τ j )Equate============Rhs: ProjectionL 1 × Rhs → pL1L 2 × Rhs → pL2L 3 × Rhs → pL3L 4 × Rhs → pL4T 1 × Rhs → pT 1T 2 × Rhs → pT 2T 3 × Rhs → pT 3T 4 × Rhs → pT 4T 5 × Rhs → pT 5T 6 × Rhs → pT 6T 7 × Rhs → pT 7T 8 × Rhs → pT 8Solve system once (in advance) and get λ i , τ jAndreas Windisch, University of GrazThe Quark-Gluon Vertex in Landau Gauge QCD

Renormalization➣ Z 2➣ Z m➣ Z 1f➣ MOM-schemeAndreas Windisch, University of GrazThe Quark-Gluon Vertex in Landau Gauge QCD

NumericsUsing GPUs➣ M. Hopfer et al.,arXiv:1206.1779➣ AW et al.,arXiv:1205.0752Massive parallelismCode Development➣ CUDA-C/FORTRAN➣ OpenACC➣ GPU- and CPU ClusterAndreas Windisch, University of GrazThe Quark-Gluon Vertex in Landau Gauge QCD

NumericsUsing GPUs➣ M. Hopfer et al.,arXiv:1206.1779➣ AW et al.,arXiv:1205.0752Massive parallelismCode Development➣ CUDA-C/FORTRAN➣ OpenACC➣ GPU- and CPU ClusterAndreas Windisch, University of GrazThe Quark-Gluon Vertex in Landau Gauge QCD

SummarySummary1Quark-confinement, DχSB2Systematically improve on thetruncation3Solve the full vertex-DSEQuark Propagator DSEQuark-Gluon Vertex DSEAndreas Windisch, University of GrazThe Quark-Gluon Vertex in Landau Gauge QCD

OutlookOutlook1Include YM-System2Extend towards µ − T3Scalar vertex models (Talk: M.Hopfer)4Replace scalar vertices with quarksAndreas Windisch, University of GrazThe Quark-Gluon Vertex in Landau Gauge QCD

Thank You ForYour Attention!Andreas Windisch, University of GrazThe Quark-Gluon Vertex in Landau Gauge QCD