Quark-gluon interactions in QCD

Transcription

1 Quark-gluon interactions in QCD AYSE KIZILERSU (CSSM, University of Adelaide) Australia Collaborators : Jonivar Skullerud (Maynooth, Ireland ) Andre Sternbeck (Jena, Germany) QCD DOWNUNDER! -4 July 7! Cairns-Australia! Orlando Oliveira (Coimbra, Portugal) Paulo J. Silva (Coimbra, Portugal) Anthony G. Williams (Adelaide)

2 Pictorial Non-Perturbative QCD Gluon-Propagator Quark-Propagator Ghost-Propagator S = Z d 4 x ( g i µ A a µt a ij j + i(i 6@ m) QCD ) 4 F µ F a a µ F a µ µ A A a µ + gf abc A b µa c Quark-Gluon Vertex -Gluon Vertex Ghost-Gluon Vertex 4-Gluon Vertex

3 Perturbative vs Non-Perturbative : Related to the dynamical chiral symmetry breaking, confinement, hadron spectroscopy

4 Non-Perturbative Methods: Lattice QCD calculations vs Schwinger-Dyson Calculations BOTH starts from first principles(partition Function): Z Z = DA µ D D e S QCD Z D + Gauge fixing Gauge independent physical quantities Truncation of the infinite system

5 Fermion-Anti-fermion Scattering kernel Schwinger-Dyson Equations = / / /6 / + + =

6 Danger of reintroducing the gauge dependence Non-Perturbative Methods: Lattice QCD calculations vs Schwinger-Dyson Calculations BOTH starts from first principles(partition Function): Z Z = DA µ D D Extrapolation to continuum is needed Control: Finite volume effects Lattice discretization effects e S QCD Z D + Gauge fixing Gauge independent physical quantities BOTH ARE COMPLEMENTARY Truncation of the infinite system

7 Quark Propagator in Landau Gauge = In Continuum In Discrete S F (p) = = Z(p ) i 6p + M(p ) A(p )i 6p + B(p ) S L (pa) = Z L (pa) ia 6K(p)+aM L (pa) K µ (p) a sin(p µa) Z /A M B/A Quark (Fermion) Wave-Function Renormalisation Running Mass Function Z L M L

8 Improved Wilson, non-prturbative improved (Sheikholeslami-Wohlert) clover fermions : S SW = W i a 4 g X X S c SW (x) µ F µ (x) (x) SW = S W i a 4 g X X c SW (x) µ F µ (x) (x) x µ x µ Tree-level improvement In Continuum (x)(i 6D + m) (x)+ (x)d (x)(i 6D + m) (x)+ (x)d (x)+ (x) µ F µ (x) (x) (x)+ (x) µ F µ (x) (x) Other fermion actions such as Staggered, overlap fermions are used in different studies However continuum limit must stay the same

9 SW action, an O(a) improved quark propagator: The O(a) improved quark propagator suffers from large tree-level lattice artifacts (due to broken, rotational symmetry and finite lattice spacing) which complicate the determination of the physical momentum dependence of the form factors. Lattice quark propagator: S L (pa) = Z L (pa) ia 6K(p)+aM L (pa) Need to remove lattice artifacts: momentum cuts (cylinder cut with a radius of lattice momentum unit) select momenta close to the diagonal in 4-dim momentum space, which have the smallest hypercubic artifacts. tree level correction Z(ap) = ZL (ap) Z () (ap) am(p) = M L (pa) a M ( ) (pa) Z (+) m tree-level quark wave function

10 Improved Wilson, Clover Fermion Action Rotated (Sheikholeslami-Wohlert) in Discrete Simulation Parameters: The quark propagator using state-of-the art gauge configurations with N f = flavors of O(a) improved Wilson fermion The configurations from Regensburg Collaboration(RQCD) is used for this project G. S. Bali et all, Phys Rev D9, 545 (4) [arxiv:4.76] The calculations ( gauge fixing, propagators) where performed using the HLRN (Germany) supercomputing

11 Quark wave function renormalisation in Landau Gauge All 4 ensembles on the x 64 volume, using the tree-level corrected Z(p) S F (p) = Z(p ) 6p M(p )

12 Finite Volume Effects Quark Wave-function for two lattice volumes at b = 5.9, k =.6

13 Mass function All 4 ensembles on the x 64 volume, using the tree-level corrected scheme for M(p). M = 49.7 ± 5. p= 6 MeV for m π =9 MeV

14 Finite Volume Effects Mass function for two lattice volumes at β = 5.9, κ =.6

15 Summary on the Quark Propagator: M = 9 ± 6 p= 6 MeV for m p =5 MeV M = 49.7 ± 5. p= 6 MeV for m p =9 MeV lattice artifacts under control for Z(p ) but not for for M(p ) quark wave function seems to be lattice space independent at UV but not in the IR and it has very weak volume dependence. Z(p ) suppressed in IR and further suppressed as the quark mass and lattice spacing reduced but the suppresion is weakened as the volume is increased.

16 Quark Gluon Vertex

17 Determining Quark-Gluon Vertex -Determining qqg vertex perturbatively - Non-perturbatively in continuum by using SDE -Non-perturbatively in discreet space by using lattice QCD

18 -Determining qqg vertex perturbatively: J.S. Ball and T.W. Chiu, Phys.Rev. D (98) 54 (Feynman gauge, -loop) J.S. Ball and T.W. Chiu, Phys.Rev. D (98) 55 (Feynman gauge,-loop) A. Kızılersu, M. Reenders, and M.R. Pennington, Phys. Rev. D5(995) 4 (General covariant gauge, -loop) A. I. Davydychev, P. Osland and L. Saks, Phys.Rev.D6() 4 (General covariant gauge, -loop) A. Bashir, Y. Concha-Sanchez, Y. and R. Delbourgo, Phys. Rev.D76 (7), 659 (Scalar theory, -loop)

19 - Non-perturbatively in continuum by using SDE a-either by solving DSE for the vertex itself = + b-by solving coupled SDE for the propagators self consistently by imposing constraints on Green s functions c- By using Ward-Green-Takahashi identities D.Curtis and M.R. Pennington,Phys.Rev.D48 (99) 49 Bashir, A. Kızılersu, and M.R. Pennington A. Bashir and M. R. Pennington, Phys. Rev. D5, 7679 (994) D.Atkinson, J.C.R.Bloch, V.P.Gusynin, M.R.Pennington, M. Reenders,Phys. Lett.B9 (994) 7. Bashir, A. Kizilersu, and M. R. Pennington, Phys. Rev. D57, 4 (998) P. Maris and P.C. Tandy, Phys.Rev. C6 (999) 554 A.Kızılersu and M.R. Pennington, Phys. Rev. D79 (9) 5 A.Bashir R. Bermudez L. Chang and C. D. Roberts, Phys.Rev.C85 () 455 A.C. Aguilar, J.Phys.Conf.Ser. 6 (5) no., 58 E. Rojas, B. El-Bennich, J.P.B.C. De Melo and M.Ali Paracha, Few Body Syst. 56 (5)69 A.C. Aguilar, D. Binosi, D. Ibañez, J. Papavassiliou, Phys.Rev. D9 (4) 657 R. Williams, Eur.Phys.J. A5 (5) 57 M. Hopfer,. Windisch, R. Alkofer PoS ConfinementX () 7 R. Alkofer, C. S. Fischer, F. J. Llanes-Estrada, K. Schwenzer, Annals Phys. 4 (9) 6. R.Williams, C.S.Fischer, W.Heupel, Phys.Rev.D9 (6) 46 S-X Qin,C. D. Roberts, S. M. Schmidt,Phys.Lett. B7 (4) -8 Bashir, R. Bermudez, L. Chang and C.D., Phys. Rev.C85 (), 455 L.Fernandez-Rangel, A.Bashir, L-X Gutierrez-Guerrero and Y. Concha-Sanchez, Phys. Rev.D9(6)", 65 M. Vujinovic, R. Alkofer, G. Eichmann and R. Williams, Acta Phys. Polon. Supp.(4), 67 D. Binosi, Daniele, L. Chang, J. Papavassiliou S.-X. Qin, C.D. Roberts, 6, nucl-th

20 -Non-perturbatively in discreet space by using lattice gauge field theory J.I. Skullerud and A. Kızılersu, JHEP9, (). J.I. Skullerud, P.O. Bowman, A. Kızılersu, D.B. Leinweber and A.G. Williams, JHEP4()47. A. Kızılersu, J.I. Skullerud, D.B. Leinweber and A.G.Williams, Eur.Phys.J.C5(7) 87 M. Peláez, M. Tissier, N. Wschebor, Phys.Rev. D9 (5) 45 O. Oliveira, A. Kızılersü, P. Silva, J. Skullerud, A. Sternbeck, Andre and A.G. Williams, Acta Phys. Polon. Supp.(6), 6.

21 Quark Gluon Vertex in General Kinematics: P = k + p q = k p Transverse Basis 4X µ F (p, P, q) = + i= 8X i= i L µ i (p, P ) i T µ i (p, P ) T µ = i [(pq)k µ (kq)p µ ] T µ = 6P [(pq)k µ (kq)p µ ] T µ = 6qq µ q µ T 4µ = i q µ P +q µ p k T 5µ = i µ q T 6µ = (qp) µ 6qP µ i T 7µ = (qp) µ P ip µ p k T 8µ = µ p k 6pk µ + 6kp µ Slavnov-Taylor Identity Satisfying Basis L µ = µ L µ = 6PP µ L µ = ip µ L 4µ = i µ P

22 QED: Ward-Green-Takahashi Identity: q µ µ = S (k) S (p) QCD: Slavnov_Taylor Identity: q µ µ(p, q) =G P (q ) H(k, p, q)s (k) S (p)h( p, k, q) Ghost-Quark Scattering Kernel

23 Transverse Projection D µν - does not exist, so we will be looking at transverse projection T µ (p, k, q) =P T µ (q) µ = µ q µ q q µ(p, k, q) We study transverse projected vertex: a,p µ (p, q) P µ a,lat. (p, q) =S(p) V a (p, q)s(p + q) D(q ) T µ = q [ T µ + T µ + T µ + 4 T µ 4 ] + 5 T µ T µ T µ T µ 8 = q q = q = 4 = 4 + q 4

24 Quark-Gluon Vertex in Special Kinematics Ø Soft Gluon Kinematics : q µ = k µ = p µ = P µ ( a µ) ij = ig t a ij( T µ ) ( a µ) ij = ig t a ij( E µ 4 E 6pp µ i E p µ i E 4 µ p ) Tr 4 [I T µ ] = i E p µ Tr 4 [ T µ ] = E µ 4 E p p µ Tr 4 [ T µ ] = i E 4 (p µ p µ ) E E,, E 4 = E

25 Quark-Gluon Vertex in Special Kinematics Ø Soft Gluon Kinematics : q µ = k µ = p µ = P µ Tr 4 [I T µ ] (P, q = ) = = i Q (P ) iq µ(p ) E p µ Tr 4 T µ R! q= Tree-level correction : 8! (P, q = ) = < : c q 4 Q (P ) c qq (P ) 4c q K µ (p) = a sin p µa Q µ (p) = a sin pµ a C µ (p) = cos(p µ a) Q(P ) K(P ) 4 c qq (P ) Q (P ) 9 = ;

26 Quark-Gluon Vertex in Special Kinematics Ø Soft Gluon Kinematics : Tr 4 [ T µ ] q µ = k µ = p µ = P µ = E µ 4 E p p µ (P, q = ) = ( Tr 4 µ T µ R! q= Q µ (P )Q (P ) Q (P ) Tr 4 T µ R! q= ) Tree Level Expression: " (P, q = ) = 4 Q (P ) 8 # K(P ) Q(P ) 4 Q (P )! c q 4 Q (P ) + c qq (P ) 5! = Tr 4 T µ R q=,p µ =, =µ no sum over µ q=,, =µ no sum over µ Tree Level Expression: q=,p µ =, =µ no sum over µ = 4 c qq (P ) + c qq (P )

27 Quark-Gluon Vertex in Special Kinematics Ø Soft Gluon Kinematics : q µ = k µ = p µ = P µ Tr 4 [ T µ ] = E µ 4 E p p µ (P, q = ) = ( Q (P ) 8 Tree Level Expression: (P, q = ) = < 4 : Q (P ) Tr 4 µ T µ R +4c q +! q= c q 4 Q (P ) 4 Q µ(p )Q (P ) Q (P )! K(P ) Q(P ) Q (P ) Tr 4 T µ R K(P ) Q(P ) Q (P ) Q! (P ) c q! q= ) Q (P )! c q 4 Q (P )!)! q=, 6= q=, 6= = Tr 4 T µ R = c q q=, 6= 4 c qq (P ) c q Q! µ(p ) 8

28 Form Factors of ghost-quark scattering kernel What do we know about the Form Factors? In Continuum: Soft Gluon Kinematics : QCD i = C F C A a i + C A b i pert (-loop) - BC. i In QED: (p,,p ) A(p ) In QCD: (p,,p )=G() p [GeV] h A(p ) (p,,p )+B(p ) (p,,p )+ (p,,p ) p A(p ) (p,,p ) i

29 Soft gluon Kinematics (Asymmetric) λ with and without tree-level correction Quenched, N f = x64 Unquenched, N f = x Uncorrected Tree-leel corrected, quenched, V= 64, a=.7 fm, =6.6 Uncorrected Tree-leel corrected, =95 MeV, a=.7 fm, =5.9 without TL without TL a*p [GeV].5 Uncorrected Tree-leel corrected, V=64 64, M =9 MeV, a=.7 fm, =5.9 without TL.5 with TL a*p [GeV] Unquenched, N f = 64 x with TL a*p [GeV]

30 TL corrected, binned data 4 4 =8 MeV, a=.8 fm, =5. V=64 64, M =9 MeV, a=.7 fm, =5.9.5 =4 MeV, a=.7 fm, =5.9 =95 MeV, a=.7 fm, =5.9 =46 MeV, a=.6 fm, =5.4.5 =8 MeV, a=.8 fm, =5. =4 MeV, a=.7 fm, =5.9 =95 MeV, a=.7 fm, =5.9 =46 MeV, a=.6 fm, = (a*p) [GeV ] - - (a*p) [GeV ] 4 4 V=64 64, M =9 MeV, a=.7 fm, =5.9 V=64 64, M =9 MeV, a=.7 fm, =5.9.5 =8 MeV, a=.8 fm, =5. =4 MeV, a=.7 fm, =5.9.5 =8 MeV, a=.8 fm, =5. =4 MeV, a=.7 fm, =5.9 =95 MeV, a=.7 fm, =5.9 =46 MeV, a=.6 fm, =5.4 =95 MeV, a=.7 fm, =5.9 =46 MeV, a=.6 fm, =5.4.5 quenched, V= 64, a=.7 fm, =6.6.5 quenched, V= 64, a=.7 fm, =6.6 -loop pertubative calc (a*p) [GeV ] - - (a*p) [GeV ]

31 Soft gluon Kinematics (Asymmetric) λ Without tree-level correction 4 =95 MeV, a=.7 fm, =5.9.5 quenched, V= 64, a=.7 fm, = (a*p) [GeV ]

32 Soft gluon Kinematics (Asymmetric) Volume dependence λ With tree-level correction 4 V=64 64, M =9 MeV, a=.7 fm, =5.9.5 =95 MeV, a=.7 fm, = (a*p) [GeV ]

33 without tree-level correction λ 8 =8 MeV, a=.8 fm, =5. =4 MeV, a=.7 fm, =5.9 =95 MeV, a=.7 fm, =5.9 =46 MeV, a=.6 fm, =5.4 8 V=64 64, M =9 MeV, a=.7 fm, =5.9 =8 MeV, a=.8 fm, =5. =4 MeV, a=.7 fm, =5.9 =95 MeV, a=.7 fm, =5.9 =46 MeV, a=.6 fm, = (a*p) [GeV ] - - (a*p) [GeV ] 8 6 V=64 64, M =9 MeV, a=.7 fm, =5.9 =8 MeV, a=.8 fm, =5. =4 MeV, a=.7 fm, =5.9 =95 MeV, a=.7 fm, =5.9 =46 MeV, a=.6 fm, =5.4 quenched, V= 64, a=.7 fm, = V=64 64, M =9 MeV, a=.7 fm, =5.9 =8 MeV, a=.8 fm, =5. =4 MeV, a=.7 fm, =5.9 =95 MeV, a=.7 fm, =5.9 =46 MeV, a=.6 fm, =5.4 quenched, V= 64, a=.7 fm, =6.6 -loop pertubative calc (a*p) [GeV ] - - (a*p) [GeV ]

34 =95 MeV, a=.7 fm, =5.9 quenched, V= 64, a=.7 fm, = (a*p) [GeV ]

35 Volume dependence λ Without tree-level correction Soft gluon Kinematics (Asymmetric) 8 V=64 64, M =9 MeV, a=.7 fm, =5.9 =95 MeV, a=.7 fm, = (a*p) [GeV ]

36 λ Without tree-level correction =8 MeV, a=.8 fm, =5. -8 =4 MeV, a=.7 fm, =5.9 V= 64, M -9 =95 MeV, a=.7 fm, =5.9 =46 MeV, a=.6 fm, = (a*p) [GeV ] V=64 64, M =9 MeV, a=.7 fm, =5.9 =8 MeV, a=.8 fm, =5. -8 =4 MeV, a=.7 fm, =5.9 V= 64, M -9 =95 MeV, a=.7 fm, =5.9 =46 MeV, a=.6 fm, = (a*p) [GeV ] V=64 64, M =9 MeV, a=.7 fm, =5.9-7 =8 MeV, a=.8 fm, =5. =4 MeV, a=.7 fm, =5.9-8 =95 MeV, a=.7 fm, =5.9-9 =46 MeV, a=.6 fm, =5.4 quenched, V= 64, a=.7 fm, = (a*p) [GeV ] (a*p) [GeV ] V=64 64, M =9 MeV, a=.7 fm, =5.9 =8 MeV, a=.8 fm, =5. =4 MeV, a=.7 fm, =5.9 =95 MeV, a=.7 fm, =5.9 =46 MeV, a=.6 fm, =5.4 quenched, V= 64, a=.7 fm, =6.6 -loop pertubative calc.

37 Soft gluon Kinematics (Asymmetric) Volume dependence λ Without tree-level correction V=64 64, M =9 MeV, a=.7 fm, =5.9 =95 MeV, a=.7 fm, = (a*p) [GeV ]

38 λ With tree-level correction Unquenched versus Quenched =95 MeV, a=.7 fm, =5.9 quenched, V= 64, a=.7 fm, = (a*p) [GeV ]

39 Hirarchy of Form Factors Soft gluon Kinematics (Asymmetric) V= 6 Unquenched, Nf = 4 V= 64, M =4 MeV, a=.7 fm, =5.9 64, M =4 MeV, a=.7 fm, =5.9-4p -p i Unquenched, Nf = ( aµ )ij = p [GeV] -4.5 V= ig taij (.5 p [GeV] E.5 µ 4 E 6 ppµ i E pµ V= 64, M =8 MeV, a=.7 fm, =6.6 i E 4 µ p ) 64, M =8 MeV, a=.7 fm, = Quenched, Nf = 7-4p -p 6 5 i 4 - Quenched, Nf = p [GeV] p [GeV].5

40 Conclusion: Quark-Gluon Vertex in Special Kinematics Ø Soft Gluon Kinematics :,,, 4 = Ø Hard Reflection Kinematics :, 5 q µ = k µ = p µ = P µ P µ = k µ = p µ = q µ Ø Orthogonal Kinematics : q P = k = p,,, 4, 5, 7 Ø General Kinematics :,,, 4, 5, 6, 7, 8

41 Summary and Outlook u high statistics quark propagator and quark-gluon vertex computation closer to the physical point u l IR enhanced u l IR strongly enhanced u l IR suppressed u Finite volume effects for Z(p) are under control but for mass M(p) needs to be better understood u qualitative agreement with quark-gluon models u explore further kinematics + form factors of the vertex u need to have a better understanding of the lattice artefacts