Recent results for propagators and vertices of Yang-Mills theory

Transcription

1 Introduction Dyson-Schwinger equations Extending truncations Summary and conclusions Recent results for propagators and vertices of Yang-Mills theory Markus Q. Huber arxiv: Institute of Theoretical Physics, Giessen University Institute of Physics, University of Graz ACHT 2018 Non-Perturbative Methods in Quantum Field Theory Leibnitz, Austria September 27, 2018 Markus Q. Huber Giessen University, University of Graz September 27, /30

2 Hadronic bound states Bound state equations: E.g., meson Ingredients: Interaction kernel K Quark propagator S Markus Q. Huber Giessen University, University of Graz September 27, /30

3 Hadronic bound states Bound state equations: E.g., meson Ingredients: Interaction kernel K Quark propagator S Approaches: Phenomenological (bottom-up): Model interactions From rst principles (top-down): Piecing together the elementary pieces Markus Q. Huber Giessen University, University of Graz September 27, /30

4 The elementary pieces Markus Q. Huber Giessen University, University of Graz September 27, /30

5 The elementary pieces Gluon propagator D µν (p 2 ): 1 = Quark-gluon vertex Γ µ (p, q): = Couple to innity of equations. Gluonic part is crucial. Markus Q. Huber Giessen University, University of Graz September 27, /30

6 The elementary pieces Gluon propagator D µν (p 2 ): 1 = Quark-gluon vertex Γ µ (p, q): = Couple to innity of equations. Gluonic part is crucial. Note: Eective interaction via g 2 D µν (p)γ µ (p, q) Z 2 Z3 D (0) µν (p)γ µ G((p + q) 2 ) Markus Q. Huber Giessen University, University of Graz September 27, /30

7 Another example: QCD phase diagram Questions: Phases and transitions between them, critical point Experimental signatures Theoretical challenges: Model description Mathematical, e.g., complex action for lattice QCD Complexity, e.g., truncations of function eqs.... Markus Q. Huber Giessen University, University of Graz September 27, /30

8 Another example: QCD phase diagram Questions: Phases and transitions between them, critical point Experimental signatures Theoretical challenges: Model description Mathematical, e.g., complex action for lattice QCD Complexity, e.g., truncations of function eqs.... Markus Q. Huber Giessen University, University of Graz September 27, /30

9 Truncations Inuence of higher correlation functions? qualitative? quantitative? negligible? Markus Q. Huber Giessen University, University of Graz September 27, /30

10 Truncations Inuence of higher correlation functions? qualitative? quantitative? negligible? Hierarchy of diagrams/correlation functions? negligible diagrams? irrelevant correlation functions for specic questions? Markus Q. Huber Giessen University, University of Graz September 27, /30

11 Truncations Inuence of higher correlation functions? qualitative? quantitative? negligible? Hierarchy of diagrams/correlation functions? negligible diagrams? irrelevant correlation functions for specic questions? Model dependence Self-contained truncation? conicting requirements for models? parameter-free solution? Markus Q. Huber Giessen University, University of Graz September 27, /30

12 Truncations Inuence of higher correlation functions? qualitative? quantitative? negligible? Hierarchy of diagrams/correlation functions? negligible diagrams? irrelevant correlation functions for specic questions? Model dependence Self-contained truncation? conicting requirements for models? parameter-free solution? How to realize resummation? higher loop contributions? Markus Q. Huber Giessen University, University of Graz September 27, /30

13 Truncations Inuence of higher correlation functions? qualitative? quantitative? negligible? Hierarchy of diagrams/correlation functions? negligible diagrams? irrelevant correlation functions for specic questions? Model dependence Self-contained truncation? conicting requirements for models? parameter-free solution? How to realize resummation? higher loop contributions? Systematics and tests? comparison to other methods, self-tests? Markus Q. Huber Giessen University, University of Graz September 27, /30

14 Dyson-Schwinger equations 1 = = 1 = = = Markus Q. Huber Giessen University, University of Graz September 27, /30

15 Coupled systems of Dyson-Schwinger equations 1 = = 1 quark propagator + 3-point functions: [Williams, Fischer, Heupel '15] application to bound states Markus Q. Huber Giessen University, University of Graz September 27, /30

16 Coupled systems of Dyson-Schwinger equations 1 = = 1 Yang-Mills propagators, 3-point functions + 4-gluon vertex in d = 3: [MQH '16] = perm. Markus Q. Huber Giessen University, University of Graz September 27, /30

17 Coupled systems of Dyson-Schwinger equations 1 = = 1 Three- and four-point functions [MQH '17] = perm. Markus Q. Huber Giessen University, University of Graz September 27, /30

18 3PI system of equations Three-loop expansion of PI eective action [Berges '04]: 1 = = + + = Markus Q. Huber Giessen University, University of Graz September 27, /30

19 Example of a bottom-up calculation 3 2 Propagators and ghost-gluon vertex with three-gluon vertex model: One-loop truncation of gluon propagator with an optimized eective model (contains zero crossing) [MQH, von Smekal '13; lattice: Sternbeck '06]: Z p 2 4 G p p Good quantitative agreement for ghost and gluon dressings. Markus Q. Huber Giessen University, University of Graz September 27, /30

20 Example of a bottom-up calculation 3 2 Propagators and ghost-gluon vertex with three-gluon vertex model: One-loop truncation of gluon propagator with an optimized eective model (contains zero crossing) [MQH, von Smekal '13; lattice: Sternbeck '06]: Z p 2 4 G p p Good quantitative agreement for ghost and gluon dressings. QCD is only this: Can we do with only that? Markus Q. Huber Giessen University, University of Graz September 27, /30

21 UV behavior of the gluon propagator Resummed one-loop order: anomalous dimension γ = 13/22 ( 1 + α(s)11n c 12π ) γ ln p2 s Markus Q. Huber Giessen University, University of Graz September 27, /30

22 UV behavior of the gluon propagator Resummed one-loop order: anomalous dimension γ = 13/22 ( 1 + α(s)11n c 12π ) γ ln p2 s One-loop anomalous dimension Origin in resummation of higher order diagrams. However, one-loop truncation discards some terms. Markus Q. Huber Giessen University, University of Graz September 27, /30

23 UV behavior of the gluon propagator Resummed one-loop order: anomalous dimension γ = 13/22 ( 1 + α(s)11n c 12π ) γ ln p2 s One-loop anomalous dimension Origin in resummation of higher order diagrams. However, one-loop truncation discards some terms. Puts constraints on UV behavior of vertices if one wants a self-consistent solution [von Smekal, Hauck, Alkofer '97]. Markus Q. Huber Giessen University, University of Graz September 27, /30

24 Fixing the UV behavior of the gluon propagator I First possibility: Modify the UV behavior of the integrand, e.g., replace the renormalization constant by a momentum dependent function [von Smekal, Hauck, Alkofer '97] ('RG improvement'), adapt employed models. Z 1 f (p 2 ) Part of the modeling. Markus Q. Huber Giessen University, University of Graz September 27, /30

25 Fixing the UV behavior of the gluon propagator I First possibility: Modify the UV behavior of the integrand, e.g., replace the renormalization constant by a momentum dependent function [von Smekal, Hauck, Alkofer '97] ('RG improvement'), adapt employed models. Z 1 f (p 2 ) Part of the modeling. Examples: Yang-Mills propagators, e.g., [von Smekal, Hauck, Alkofer '97; Fischer, Alkofer '02; MQH, von Smekal '12, '14] quark propagator: e.g., [Maris, Tandy '97] Markus Q. Huber Giessen University, University of Graz September 27, /30

26 Fixing the UV behavior of the gluon propagator I First possibility: Modify the UV behavior of the integrand, e.g., replace the renormalization constant by a momentum dependent function [von Smekal, Hauck, Alkofer '97] ('RG improvement'), adapt employed models. Z 1 f (p 2 ) Part of the modeling. Examples: Yang-Mills propagators, e.g., [von Smekal, Hauck, Alkofer '97; Fischer, Alkofer '02; MQH, von Smekal '12, '14] quark propagator: e.g., [Maris, Tandy '97] IR completion has an eect on the gluon propagator [MQH, von Smekal '14]: Markus Q. Huber Giessen University, University of Graz September 27, /30

27 Fixing the UV behavior of the gluon propagator II Second possibility: Include higher perturbative terms. Worked out analytically for φ 3 -theory [MQH '18]. Markus Q. Huber Giessen University, University of Graz September 27, /30

28 Fixing the UV behavior of the gluon propagator II Second possibility: Include higher perturbative terms. Worked out analytically for φ 3 -theory [MQH '18]. Two-loop diagrams q 1 + p q 1 q 2 p q 2 p q 1 + p q 1 q 2 q 1 q 2 Markus Q. Huber Giessen University, University of Graz September 27, /30

29 Fixing the UV behavior of the gluon propagator II Second possibility: Include higher perturbative terms. Worked out analytically for φ 3 -theory [MQH '18]. Two-loop diagrams q 1 + p q 1 q 2 p q 2 p q 1 + p q 1 q 2 q 1 q 2 Contributions also from renormalization constants in front of one-loop diagrams. Z 1 Z 1 All two-loop contributions in the gluon propagator are included. And higher contributions... Markus Q. Huber Giessen University, University of Graz September 27, /30

30 Resummed behavior Minimal requirements to obtain one-loop resummed behavior: Squint diagram (sunset has no g 4 ln 2 p 2 ) Correct anomalous dimensions of three-point functions Correct renormalization (constants) Markus Q. Huber Giessen University, University of Graz September 27, /30

31 Resummed behavior Minimal requirements to obtain one-loop resummed behavior: Squint diagram (sunset has no g 4 ln 2 p 2 ) Correct anomalous dimensions of three-point functions Correct renormalization (constants) G(p 2 ) Z(p 2 )/p 2 Z(p 2 ) p[gev] p[gev] p[gev] [propagator+ghost-gluon eqs. full, 3-gluon vertex model, bare 4-gluon vertex] G(p 2 ) Resummed behavior is recovered [MQH '17] Z(p 2 ) G UV (p 2 ) Z UV (p 2 ) p[gev] p[gev] Markus Q. Huber Giessen University, University of Graz September 27, /30

32 Extending truncations Various ways to extend truncations: Vertex tensors beyond tree-level tensor Include neglected diagrams Include neglected correlation functions Extensions also test the previous truncations! Markus Q. Huber Giessen University, University of Graz September 27, /30

33 Extending truncations Various ways to extend truncations: Vertex tensors beyond tree-level tensor Include neglected diagrams Include neglected correlation functions Extensions also test the previous truncations! In the following: Three-gluon vertex Four-point functions Coupling the equations Markus Q. Huber Giessen University, University of Graz September 27, /30

34 Three-gluon vertex DSE [Cucchieri, Maas, Mendes '08; Alkofer, MQH, Schwenzer '09; Pelaez, Tissier, Wschebor '13; Aguilar et al. '13; Blum, et al. '14; Eichmann, Alkofer, Vujinovic '14; Cyrol et al. '16; Williams, Fischer, Heupel '16; Sternbeck '16; Athenodorou et al. 16; Duarte et al. '16; Boucaud et al. '17] Markus Q. Huber Giessen University, University of Graz September 27, /30

35 Three-gluon vertex DSE [Cucchieri, Maas, Mendes '08; Alkofer, MQH, Schwenzer '09; Pelaez, Tissier, Wschebor '13; Aguilar et al. '13; Blum, et al. '14; Eichmann, Alkofer, Vujinovic '14; Cyrol et al. '16; Williams, Fischer, Heupel '16; Sternbeck '16; Athenodorou et al. 16; Duarte et al. '16; Boucaud et al. '17] Full DSE: = Markus Q. Huber Giessen University, University of Graz September 27, /30

36 Three-gluon vertex DSE [Cucchieri, Maas, Mendes '08; Alkofer, MQH, Schwenzer '09; Pelaez, Tissier, Wschebor '13; Aguilar et al. '13; Blum, et al. '14; Eichmann, Alkofer, Vujinovic '14; Cyrol et al. '16; Williams, Fischer, Heupel '16; Sternbeck '16; Athenodorou et al. 16; Duarte et al. '16; Boucaud et al. '17] Full DSE: = Perturbative one-loop truncation [Blum, MQH, Mitter von Smekal '14; Eichmann, Alkofer, Vujinovic '14; Williams, Fischer, Heupel '16]: Markus Q. Huber Giessen University, University of Graz September 27, /30

37 Three-gluon vertex DSE [Cucchieri, Maas, Mendes '08; Alkofer, MQH, Schwenzer '09; Pelaez, Tissier, Wschebor '13; Aguilar et al. '13; Blum, et al. '14; Eichmann, Alkofer, Vujinovic '14; Cyrol et al. '16; Williams, Fischer, Heupel '16; Sternbeck '16; Athenodorou et al. 16; Duarte et al. '16; Boucaud et al. '17] Full DSE: = Non-perturbative one-loop truncation [MQH '17]: Markus Q. Huber Giessen University, University of Graz September 27, /30

38 Inuence of two-ghost-two-gluon vertex Markus Q. Huber Giessen University, University of Graz September 27, /30

39 Inuence of two-ghost-two-gluon vertex Coupled system of ghost-gluon, three-gluon and four-gluon vertices with and without two-ghost-two-gluon vertex [MQH '17]: D AAA (p 2 ) 1 D k AA c _ c =0 D k AA c _ c dyn p[gev] -2-3 Markus Q. Huber Giessen University, University of Graz September 27, /30

40 Inuence of two-ghost-two-gluon vertex Coupled system of ghost-gluon, three-gluon and four-gluon vertices with and without two-ghost-two-gluon vertex [MQH '17]: D A c_ c (p 2 ) D AAA (p 2 ) D k AA c _ c =0 D k AA c _ c dyn. 1-1 AA c _ c D k =0 AA c _ c D k dyn p[gev] p[gev] Small inuence on ghost-gluon vertex (< 1.7%) Negligible inuence on three- and four-gluon vertices. Markus Q. Huber Giessen University, University of Graz September 27, /30

41 Three-gluon vertex results D AAA (p 2 ) p 2 [GeV 2 ] -2-3 one-loop -4-5 Markus Q. Huber Giessen University, University of Graz September 27, /30

42 Three-gluon vertex results D AAA (p 2 ) p 2 [GeV 2 ] -2-3 one-loop 3PI -4-5 Markus Q. Huber Giessen University, University of Graz September 27, /30

43 Three-gluon vertex results D AAA (p 2 ) D AAA (p 2 ) p 2 [GeV 2 ] p[gev] one-loop two-loop 3PI -2-3 one-loop two-loop 3PI Lattice: Cucchieri, Maas, Mendes ' Markus Q. Huber Giessen University, University of Graz September 27, /30

44 Three-gluon vertex results D AAA (p 2 ) D AAA (p 2 ) p 2 [GeV 2 ] p[gev] one-loop two-loop 3PI -2-3 one-loop two-loop 3PI Lattice: Cucchieri, Maas, Mendes ' Two-loop truncation: All diagrams except the one with a ve-point function. = One-momentum conguration approximation Markus Q. Huber Giessen University, University of Graz September 27, /30

45 Three-gluon vertex results D AAA (p 2 ) D AAA (p 2 ) p 2 [GeV 2 ] p[gev] one-loop two-loop 3PI -2-3 one-loop two-loop 3PI Lattice: Cucchieri, Maas, Mendes ' Dierence between two-loop DSE and 3PI smaller than lattice error. Zero crossing in agreement with other approaches, e.g., [Pelaez et al. '13; Aguilar et al. '13; Cyrol et al. '15; Athenodorou et al. '16; Duarte et al. '16; Sternbeck et al. '17] g(µ 2 ) Γ(q 2,µ 2 ) β= (Wilson old) β=4.2, L=32 (tlsym) β=3.8, L=48 (tlsym) β=5.6, L=48 (Wilson new) β=3.9, L=64 (tlsym) β=5.8, L=48 (Wilson new) β=5.6,l=52 (Wilson new) q [GeV] [Athenodorou et al. '16, '18] Markus Q. Huber Giessen University, University of Graz September 27, /30

46 The two-ghost-two-gluon vertex Non-primitively divergent correlation function No guide from tree-level tensor. Use full basis. Markus Q. Huber Giessen University, University of Graz September 27, /30

47 The two-ghost-two-gluon vertex Non-primitively divergent correlation function No guide from tree-level tensor. Use full basis. Lorentz basis transverse wrt gluon legs 5 tensors τ i µν(p, q; r, s), (anti-)symmetric under exchange of gluon legs. Color basis: 8 tensors (results show that only 5 required). Markus Q. Huber Giessen University, University of Graz September 27, /30

48 The two-ghost-two-gluon vertex Non-primitively divergent correlation function No guide from tree-level tensor. Use full basis. Lorentz basis transverse wrt gluon legs 5 tensors τ i µν(p, q; r, s), (anti-)symmetric under exchange of gluon legs. Color basis: 8 tensors (results show that only 5 required). Two-ghost-two-gluon vertex Γ AA cc,abcd µν (p, q; r, s) = g 4 with 40 ρ k,abcd µν Dk(i,j) AA cc k=1 (p, q; r, s) ρ k,abcd µν = σ abcd i τ j µν, k = k(i, j) = 5(i 1) + j Markus Q. Huber Giessen University, University of Graz September 27, /30

49 The two-ghost-two-gluon vertex DSE 2 DSEs, choose the one with the ghost leg attached to the bare vertex Truncation discards only one diagram. i1 i3 i2 i4 = + i3 i4 i1 i2 + i4 i2 i3 i1 + i3 i2 i4 i1 + i4 i2 i1 i3 + i3 i2 i1 i4 + i3 i4 i1 i2 + i3 i4 i2 i1 - i4 i3 i2 i1 i2 i3 i2 i4 i1 i3 i3 i i1 i4 i3 i1 i2 i4 i4 i1 i2 i3 + i4 i1 Markus Q. Huber Giessen University, University of Graz September 27, /30

50 The two-ghost-two-gluon vertex DSE 2 DSEs, choose the one with the ghost leg attached to the bare vertex Truncation discards only one diagram. i1 i3 i2 i4 i4 i3 = + i2 i1 + i3 i2 i4 i1 + i4 i2 i1 i3 + i3 i2 i1 i4 + i3 i4 i1 i2 + i3 i4 i2 i1 + i4 i3 i2 i1 + i2 i1 i3 i4 i2 i4 i1 i3 i3 i2 i2 i i3 i1 i2 i4 i4 i1 i4 i1 Markus Q. Huber Giessen University, University of Graz September 27, /30

51 Results for the two-ghost-two-gluon vertex Kinematic approximation: one-momentum conguration Dimensionless dressing functions D k. 2 1 D1 D6 D11 D16 D D2 D7 D12 D17 D22 Each plot one Lorentz tensor p[gev] p[gev] 2 D3 D8 2 D4 D9 2 D5 D10 D13 D14 D15 1 D18 1 D19 1 D20 D23 D24 D p[gev] p[gev] p[gev] Two classes of dressings: 13 very small, 12 not small No nonzero solution for {σ 6, σ 7, σ 8 } found. [MQH '17] Markus Q. Huber Giessen University, University of Graz September 27, /30

52 Results for the four-ghost vertex Kinematic approximation: one-momentum conguration Four-ghost vertex Γ c ccc,abcd (p, q, r, s) = g 4 8 σ k,abcd E k=1 c ccc k (p, q, r, s). Markus Q. Huber Giessen University, University of Graz September 27, /30

53 Results for the four-ghost vertex Kinematic approximation: one-momentum conguration Four-ghost vertex Γ c ccc,abcd (p, q, r, s) = g 4 8 σ k,abcd E k=1 c ccc k (p, q, r, s) E1 E2 p[gev] All dressings very small. [MQH '17] E3 E4 E5 Markus Q. Huber Giessen University, University of Graz September 27, /30

54 3PI system of primitively divergent correlation functions 1 = = + + = Markus Q. Huber Giessen University, University of Graz September 27, /30

55 Results for fully coupled 3PI system [lattice: Sternbeck '06] Z(p 2 ) preliminary p[gev] Markus Q. Huber Giessen University, University of Graz September 27, /30

56 Results for fully coupled 3PI system [lattice: Sternbeck '06] Z(p 2 ) preliminary p[gev] Z(p 2 )/p preliminary p[gev] Markus Q. Huber Giessen University, University of Graz September 27, /30

57 Results for fully coupled 3PI system [lattice: Sternbeck '06] Z(p 2 ) preliminary p[gev] Details of renormalization crucial! Other details also important. Very small angle dependence of three-gluon vertex. Slight bending down of gluon propagator in IR. Z(p 2 )/p C AAA (p 2 ) preliminary preliminary p[gev] p[gev] [lattice: Cucchieri, Maas, Mendes '08] Markus Q. Huber Giessen University, University of Graz September 27, /30

58 Open checks Eects of larger tensor bases, in particular of the three-gluon vertex Renormalization What tests can be done? Markus Q. Huber Giessen University, University of Graz September 27, /30

59 Couplings Couplings can be dened from every vertex, e.g., [Allés et al. '96; Alkofer et al., '05; Eichmann et al. '14]: α ghg (p 2 ) = α(µ 2 ) ( D A cc (p 2 ) ) 2 G 2 (p 2 )Z(p 2 ), α 3g (p 2 ) = α(µ 2 ) ( C AAA (p 2 ) ) 2 Z 3 (p 2 ), α 4g (p 2 ) = α(µ 2 )F AAAA (p 2 )Z 2 (p 2 ). They must agree perturbatively (STIs). This agreement is important also in coupled systems of functional equations and constitutes a highly non-trivial check of a truncation [Mitter, Pawlowski, Strodtho '14]. Markus Q. Huber Giessen University, University of Graz September 27, /30

60 Couplings Couplings can be dened from every vertex, e.g., [Allés et al. '96; Alkofer et al., '05; Eichmann et al. '14]: α ghg (p 2 ) = α(µ 2 ) ( D A cc (p 2 ) ) 2 G 2 (p 2 )Z(p 2 ), α 3g (p 2 ) = α(µ 2 ) ( C AAA (p 2 ) ) 2 Z 3 (p 2 ), α 4g (p 2 ) = α(µ 2 )F AAAA (p 2 )Z 2 (p 2 ). 10 They must agree perturbatively (STIs). This agreement is important also in coupled systems of functional equations and constitutes a highly non-trivial check of a truncation [Mitter, Pawlowski, Strodtho '14] preliminary αghg α3 g α4 g p[gev] Ghost-gluon vs. other couplings: Further checks required. Markus Q. Huber Giessen University, University of Graz September 27, /30

61 Renormalization with a hard UV cuto The breaking of gauge covariance by the UV regularization leads to spurious (quadratic) divergences. Markus Q. Huber Giessen University, University of Graz September 27, /30

62 Renormalization with a hard UV cuto The breaking of gauge covariance by the UV regularization leads to spurious (quadratic) divergences. Due to the anomalous running, this behaves as (at one-loop resummed level) Λ 2 QCD p 2 ( 1) 2δ Γ(1 + 2δ, ln(λ 2 /Λ 2 QCD)) Note: Appears already perturbatively! Many ways to deal with them, e.g., Brown-Pennington projector, modications of integrands/vertices, tting, seagull identities,.... Markus Q. Huber Giessen University, University of Graz September 27, /30

63 Examples for renormalization with a hard UV cuto Here: Second renormalization condition Breaking of gauge covariance mass counter term [Collins '84] Renormalization condition: D(0) = c [Meyers, Swanson '14] Markus Q. Huber Giessen University, University of Graz September 27, /30

64 Examples for renormalization with a hard UV cuto Here: Second renormalization condition Breaking of gauge covariance mass counter term [Collins '84] Renormalization condition: D(0) = c [Meyers, Swanson '14] Extreme example: One-loop truncation with bare vertices in three dimensions [MQH '16] Better example: Full system with one-momentum conguration approximation. Z(p 2 )/p preliminary p[gev] Markus Q. Huber Giessen University, University of Graz September 27, /30

65 Results for fully coupled 3PI system revisited Vary the renormalization condition D(0): [lattice: Sternbeck '06] Z(p 2 ) preliminary p[gev] Markus Q. Huber Giessen University, University of Graz September 27, /30

66 Results for fully coupled 3PI system revisited Vary the renormalization condition D(0): [lattice: Sternbeck '06] Z(p 2 ) preliminary p[gev] Z(p 2 )/p preliminary p[gev] Two solutions on top of each other. D(0) is not a parameter of the system. Markus Q. Huber Giessen University, University of Graz September 27, /30

67 Four-gluon vertex Four-point functions have 6 kinematic variables. Organize via S 4 permutation group [Eichmann, Fischer, Heupel '15] and restrict to singlet and doublet. Three variables. Markus Q. Huber Giessen University, University of Graz September 27, /30

68 Four-gluon vertex Four-point functions have 6 kinematic variables. Organize via S 4 permutation group [Eichmann, Fischer, Heupel '15] and restrict to singlet and doublet. Three variables. F AAAA (p 2 ) p[gev] 10 preliminary Deviations from leading perturbative behavior small, but larger angle dependence than three-gluon vertex. Markus Q. Huber Giessen University, University of Graz September 27, /30

69 Summary and conclusions Towards a systematic understanding of truncations of functional equations to establish them as a rst principles method. Markus Q. Huber Giessen University, University of Graz September 27, /30

70 Summary and conclusions Towards a systematic understanding of truncations of functional equations to establish them as a rst principles method. Hierarchy of correlation functions exists. Negligible diagrams identied. Self-tests of results are useful. Markus Q. Huber Giessen University, University of Graz September 27, /30

71 Summary and conclusions Towards a systematic understanding of truncations of functional equations to establish them as a rst principles method. Hierarchy of correlation functions exists. Negligible diagrams identied. Self-tests of results are useful. Outlook and possibilities: Non-classical tensors in gluonic vertices Add quarks Finite temperature Bound states Finite density Markus Q. Huber Giessen University, University of Graz September 27, /30

72 Summary and conclusions Towards a systematic understanding of truncations of functional equations to establish them as a rst principles method. Hierarchy of correlation functions exists. Negligible diagrams identied. Self-tests of results are useful. Outlook and possibilities: Non-classical tensors in gluonic vertices Add quarks Finite temperature Bound states Finite density Thank you for your attention! Markus Q. Huber Giessen University, University of Graz September 27, /30