Triple-gluon and quark-gluon vertex from lattice QCD in Landau gauge

Transcription

1 Triple-gluon and quark-gluon vertex from lattice QCD in Landau gauge André Sternbeck Friedrich-Schiller-Universität Jena, Germany Lattice 2016, Southampton (UK) Overview in collaboration with 1) Motivation 2) Triple-Gluon Balduf (HU Berlin) 3) Quark-Gluon Kızılersü & Williams (Adelaide U), Oliveira & Silva (Coimbra U), Skullerud (NUIM, Maynooth) A. Sternbeck (FSU Jena) Triple-gluon and quark-gluon vertex from lattice QCD Lattice / 18

2 Motivation Research in hadron physics Successful but not restricted to lattice QCD Other nonperturbative frameworks exist (for better or for worse) Bound-state equations / Dyson-Schwinger equations Functional Renormalization group (FRG) equation (aka Wetterich equation) A. Sternbeck (FSU Jena) Triple-gluon and quark-gluon vertex from lattice QCD Lattice / 18

3 Motivation Research in hadron physics Successful but not restricted to lattice QCD Other nonperturbative frameworks exist (for better or for worse) Bound-state equations / Dyson-Schwinger equations Functional Renormalization group (FRG) equation (aka Wetterich equation) Bound-state equations Bethe-Salpether equations: Mesonic systems (q q) Faddeev/ quark-diquark equations: Baryonic systems (qqq) no restriction to Euclidean metric (makes it simpler) for lattice QCD Euclidean metric mandatory (can calculate static quantities: masses, etc. or equilibrium properties) A. Sternbeck (FSU Jena) Triple-gluon and quark-gluon vertex from lattice QCD Lattice / 18

4 Motivation Research in hadron physics Successful but not restricted to lattice QCD Other nonperturbative frameworks exist (for better or for worse) Bound-state equations / Dyson-Schwinger equations Functional Renormalization group (FRG) equation (aka Wetterich equation) Bound-state equations Bethe-Salpether equations: Mesonic systems (q q) Faddeev/ quark-diquark equations: Baryonic systems (qqq) no restriction to Euclidean metric (makes it simpler) for lattice QCD Euclidean metric mandatory (can calculate static quantities: masses, etc. or equilibrium properties) Input: nonperturbative n-point Green s functions (in a gauge) typically taken from numerical solutions of their Dyson-Schwinger equations Note: Greens function enter in a certain gauge, but physical content obtained from BSEs (masses, decay constants) is gauge independent Main problem: truncation of system of equations required A. Sternbeck (FSU Jena) Triple-gluon and quark-gluon vertex from lattice QCD Lattice / 18

5 Motivation: Meson-BSE as an example Meson-BSE (meson = two-particle bound state) Γ(P, p) = Λ q K αγ,δβ (p, q, P) { } S(q + σp) Γ(P, q) S(q + (σ 1)P) }{{}}{{} q + q γδ Scattering Kernel K Quark Propagator S (DSE) Observables [nice review: Eichmann et al., ] Masses (Reduction to an eigenvalue equation for fixed J PC. Masses: λ(p 2 = M 2 i ) = 1) Form factors (Build electrom. current from BS-amplitude Γ(q, P 2 = M 2 ) (solution) and the full quark propagator and quark-photon vertex; and project on tensor structure)... A. Sternbeck (FSU Jena) Triple-gluon and quark-gluon vertex from lattice QCD Lattice / 18

6 Motivation: Meson-BSE as an example Meson-BSE (meson = two-particle bound state) Γ(P, p) = Λ q K αγ,δβ (p, q, P) { } S(q + σp) Γ(P, q) S(q + (σ 1)P) }{{}}{{} q + q γδ Scattering Kernel K Quark Propagator S (DSE) Gluon propagator DSE Quark-Gluon-Vertex DSE (infinite tower of equations) A. Sternbeck (FSU Jena) Triple-gluon and quark-gluon vertex from lattice QCD Lattice / 18

7 Motivation: Meson-BSE as an example Meson-BSE (meson = two-particle bound state) Γ(P, p) = Λ q K αγ,δβ (p, q, P) { } S(q + σp) Γ(P, q) S(q + (σ 1)P) }{{}}{{} q + q γδ Scattering Kernel K Quark Propagator S (DSE) Truncation, e.g. Rainbow-Ladder Simplest truncation, preserves chiral symmetry and Goldstone pion, agrees with PT Leading structure of quark-gluon vertex Γ a µ(p, k) = 14 i=1 f ip i γ µγ(k 2 )t a Effective coupling (UV known) α(k 2 ) = Z 1f g 2 Z2 2 4π Z(k2 )Γ(k 2 ) Only leading term of system. expansion Improvements are needed A. Sternbeck (FSU Jena) Triple-gluon and quark-gluon vertex from lattice QCD Lattice / 18

8 Motivation: Input from lattice QCD Greens function from lattice QCD Nonperturbative structure of n-point functions in a certain gauge (Landau gauge) are needed to improve truncations / cross-check results Lattice QCD can provide these nonperturbative + untruncated 2-point: quark, gluon (and ghost) propagators 3-point: quark anti-quark gluon, 3-gluon, (ghost-ghost-gluon) 4-point: 4-gluon vertex,... µ, a µ a k k 5-point:... q p p ν, b Already: 2-point functions are used as input to DSE studies µ a k ( ) q q p ρ, c b c ( ) A. Sternbeck (FSU Jena) Triple-gluon and quark-gluon vertex from lattice QCD Lattice / 18

9 Motivation: Input from lattice QCD Greens function from lattice QCD Nonperturbative structure of n-point functions in a certain gauge (Landau gauge) are needed to improve truncations / cross-check results Lattice QCD can provide these nonperturbative + untruncated 2-point: quark, gluon (and ghost) propagators 3-point: quark anti-quark gluon, 3-gluon, (ghost-ghost-gluon) 4-point: 4-gluon vertex,... µ, a µ a k k 5-point:... q p p ν, b Already: 2-point functions are used as input to DSE studies Most desired 3-point functions (quenched + unquenched) Quark-gluon Vertex and Triple-Gluon Vertex Improved truncations of quark-dse µ a k ( ) q q p ρ, c b c ( ) full quark DSE truncated A. Sternbeck (FSU Jena) Triple-gluon and quark-gluon vertex from lattice QCD Lattice / 18

10 Parameters of our gauge field ensembles N f = 2 and N f = 0 ensembles Allows to study β κ L 3 s Lt a [fm] mπ [GeV2 ] #config quenched vs. unquenched, quark mass dependence discretization and volume effects infrared behavior, i.e., p GeV Acknowledgements N f = 2 configurations provided by RQCD collaboration (Regensburg) Gauge-fixing and calculation of propagators at the HLRN (Germany) A. Sternbeck (FSU Jena) Triple-gluon and quark-gluon vertex from lattice QCD Lattice / 18

11 Results for Triple-gluon vertex in Landau gauge (in collaboration with MSc. P. Balduf) A. Sternbeck (FSU Jena) Triple-gluon and quark-gluon vertex from lattice QCD Lattice / 18

12 Triple-Gluon-Vertex in Landau gauge Γ µνλ(p, q) = i=1,...,14 f i (p, q) P (i) µνλ (p, q) Perturbation theory: f i known up to three-loop order Nonperturbative structure mostly unknown ν, b p µ, a (Gracey) k ρ, c (few DSE and lattice results) Most relevant ingredient for improved truncations of quark-dse DSE results Blum et al., PRD89(2014) (improved truncation) Eichmann et al., PRD89(2014) (full transverse form)... q P (1) = δ µνp λ, P (2) = δ νλ p µ,..., P (5) = δ νλ q µ,..., P (9) = 1 µ 2 qµpνq λ,... A. Sternbeck (FSU Jena) Triple-gluon and quark-gluon vertex from lattice QCD Lattice / 18

13 Triple-Gluon-Vertex in Landau gauge Γ µνλ(p, q) = i=1,...,14 f i (p, q) P (i) µνλ (p, q) Perturbation theory: f i known up to three-loop order Nonperturbative structure mostly unknown ν, b p µ, a (Gracey) k ρ, c (few DSE and lattice results) Most relevant ingredient for improved truncations of quark-dse Lattice results (deviation from tree-level) Cucchieri et al., [PRD77(2008)094510], quenched SU(2) Yang-Mills zero-crossing at small momenta (2d, 3d) Athenodorou et al. [ ], quenched SU(3) data (4d) zero-crossing for p 2 < 0.03 GeV 2 for symmetric momentum setup Duarte et al. [ ], quenched SU(3) data (4d) zero-crossing for p GeV 2 for p = q q P (1) = δ µνp λ, P (2) = δ νλ p µ,..., P (5) = δ νλ q µ,..., P (9) = 1 µ 2 qµpνq λ,... A. Sternbeck (FSU Jena) Triple-gluon and quark-gluon vertex from lattice QCD Lattice / 18

14 Lattice calculation: nonperturbative deviation from tree-level Gauge-fix all gauge field ensembles to Landau gauge U xµ U g xµ = g xu xµg x+µ with bwd µ A a xµ[u g ] = 0 Gluon field: A a µ(p) = x eipx A a µ(x) with A a µ(x) := 2 Im Tr T a U xµ Triple-gluon Green s function (implicit color sum) G µνρ(p, q, p q) = A µ(p)a ν(q)a ρ( p q) U Gluon propagator D µν(p) = A µ(p)a ν( p) U Momenta: all pairs of nearly diagonal p = q Average data for equal a p = a q and nearby momenta Projection on lattice tree-level form G 1(p, q) = Γ(0) µνρ Γ (0) µνρ G µνρ(p, q, p q) D µλ (p)d νσ(q)d ρω(p q)γ (0) λσω A. Sternbeck (FSU Jena) Triple-gluon and quark-gluon vertex from lattice QCD Lattice / 18

15 Effect of binning and tree-level improvement β = 5.29, κ = , [a = 0.07 fm, mπ 290 MeV] 1 β = 6.16, quenched, (a = 0.07 fm) 1 G1 G p = 0 p = π/128 p = π/64 p = π/32 p = π/ a p = a q -1 lattice tree-level corrected no corrections a p = a q Achieve much reduced statistical noise per configuration through binning Lattice tree-level improvement relevant only for larger ap µ (expected) Deviations moderate if large momenta are not of interest A. Sternbeck (FSU Jena) Triple-gluon and quark-gluon vertex from lattice QCD Lattice / 18

16 1) Angular dependence 2 β = 5.20, κ = , [a = 0.08 fm, mπ 280 MeV] G1 = Γ τ1/τ φ = p = q Consider p = q and φ = (p, q) Visible but small angular dependence in all lattice data small p : φ = 180 data increase less strong than φ = 60 data large p : φ = 180 data falls less strong than φ = 60 data A. Sternbeck (FSU Jena) Triple-gluon and quark-gluon vertex from lattice QCD Lattice / 18

17 2) Quenched versus unquenched (N f = 2) G1 G1 (renormalized at p = 0.6 GeV) (renormalized at p = 2.0 GeV) a = 0.22 fm, β = 5.60, quenched a = 0.17 fm, β = 5.70, quenched a = 0.07 fm, β = 6.16, quenched a = 0.07 fm, β = 5.29, κ = a = 0.06 fm, β = 5.40, κ = a = 0.07 fm, β = 6.16, quenched 64 4 : a = 0.07 fm, β = 5.29, κ = a = 0.07 fm, β = 5.29, κ = a = 0.06 fm, β = 5.40, κ = p = q p = q Where to renormalize? Different slope at small momenta (N f = 0 vs. N f = 2) Consistent with findings from Williams et al., PRD93(2016) Unquenching effect cleary visible but small depending on ren. point A. Sternbeck (FSU Jena) Triple-gluon and quark-gluon vertex from lattice QCD Lattice / 18

18 3) Zero crossing considered here for N f = 0 1 Angular dependence 2 Quenched vs. unquenched 3 Zero-Crossing Cucchieri, Maas, Mendes (2008) 3-dim SU(2) YM theory 2 p = π/128 1 G β = 5.60, φ = 60 φ = 90 φ = 120 β = 5.70, φ = 60 φ = 90 φ = p = q [GeV] Is zero-crossing feature of 2- and 3-dim YM theory? Does it happens at much lower momentum for 4-dim? A. Sternbeck (FSU Jena) Triple-gluon and quark-gluon vertex from lattice QCD Lattice / 18

19 3) Zero crossing considered here for N f = 0 1 Angular dependence 2 Quenched vs. unquenched 3 Zero-Crossing Cucchieri, Maas, Mendes (2008) 3-dim SU(2) YM theory 2 p = π/128 1 G β = 5.60, φ = 180 β = 5.70, φ = p = q [GeV] Is zero-crossing feature of 2- and 3-dim YM theory? Does it happens at much lower momentum for 4-dim? A. Sternbeck (FSU Jena) Triple-gluon and quark-gluon vertex from lattice QCD Lattice / 18

20 Bose-symmetric transverse tensor structure Eichmann et al., PRD89(2014) Tensor structure of triple-gluon vertex Γ µνλ (p, q) = 14 i=1 f i (p, q)p (i) µνλ (p, q) Tensor structure of transversely projected triple-gluon vertex Γ T µνρ(p, q) = 4 i=1 F i (S 0, S 1, S 2) τ µνρ i (p1, p2, p3) with the Lorentz invariants S 0 S 0(p 1, p 2, p 3) = 1 6 ( ) p1 2 + p2 2 + p3 2 [p 1 = p, p 2 = q, p 3 = (p + q)] S 1 S 1(p 1, p 2, p 3) = a 2 + s 2 [0, 1] with a 3 p2 2 p 2 1 6S 0 S 2 S 2(p 1, p 2, p 3) = s(3a 2 s 2 ) [ 1, 1] and s p2 1 + p 2 2 2p 2 3 6S 0 A. Sternbeck (FSU Jena) Triple-gluon and quark-gluon vertex from lattice QCD Lattice / 18

21 First lattice results for transverse tensor structure F 1 F F1(p 2, q 2 ) (unrenormalized) F2(p 2, q 2 ) 0.0 (unrenormalized) -0.5 β = 5.60, quenched β = 5.70, quenched β = 6.16, quenched β = 5.29, κ = β = 5.40, κ = β = 5.60, quenched β = 5.70, quenched β = 6.16, quenched β = 5.29, κ = β = 5.40, κ = p = q p = q DSE study of triple-gluon vertex [Eichmann et al. (2014)] Leading form factor is F 1 F i=2,3,4 close to zero momenta Lattice results confirm that behavior A. Sternbeck (FSU Jena) Triple-gluon and quark-gluon vertex from lattice QCD Lattice / 18

22 First lattice results for transverse tensor structure F 1 F F1(p 2, q 2 ) (unrenormalized) F3(p 2, q 2 ) -0.1 (unrenormalized) -0.5 β = 5.60, quenched β = 5.70, quenched β = 6.16, quenched β = 5.29, κ = β = 5.40, κ = β = 5.60, quenched β = 5.70, quenched β = 6.16, quenched β = 5.29, κ = β = 5.40, κ = p = q p = q DSE study of triple-gluon vertex [Eichmann et al. (2014)] Leading form factor is F 1 F i=2,3,4 close to zero momenta Lattice results confirm that behavior A. Sternbeck (FSU Jena) Triple-gluon and quark-gluon vertex from lattice QCD Lattice / 18

23 Results for Quark-gluon vertex in Landau gauge (in collaboration with Kızılersü, Oliveira, Silva, Skullerud, Williams) A. Sternbeck (FSU Jena) Triple-gluon and quark-gluon vertex from lattice QCD Lattice / 18

24 Quark-Gluon vertex in Landau gauge Quark-Gluon Green s function G ψψa µ = Γ ψψa λ (p, q) S(p) D µλ (q) S(p + q) Up to now: only lattice data for quenched QCD Ball-Chiu parametrization with Γ ψψa µ (p, q) = Γ ST µ (p, q) + Γ T µ (p, q) Γ ST µ (p, q) = λ i (p 2, q 2 )L iµ (p, q) i=1...4 satisfies Slavnov-Taylor identities Γ T µ (p, q) = τ i (p 2, q 2 )T iµ (p, q) is transverse (q µγ T µ = 0). i=1...8 L 1µ (p, q) = γ µ, L 2µ (p, q) = γ µ(2p µ + q µ),..., T 1µ (p, q) = i[p µq 2 q µ(p q)],... A. Sternbeck (FSU Jena) Triple-gluon and quark-gluon vertex from lattice QCD Lattice / 18

25 Quark-Gluon vertex in Landau gauge Quark-Gluon Green s function G ψψa µ = Γ ψψa λ (p, q) S(p) D µλ (q) S(p + q) Up to now: only lattice data for quenched QCD Lattice calculation Calculate (on same gauge-fixed ensembles) 1 quark and gluon propagators: Sαβ ab (p) and Dab µν 2 Quark-Antiquark-Gluon Greens functions: G ψψa µ (p, q) = A a µ(q)sαβ bc (q)) Amputate: gluon and quark legs and project out tensor structure Γ ψψa G ψψa µ (p, q) λ (p, q) = D(q)S(p)S(p + q) = i=1...4 λ i (p 2, q 2 )L iµ + i=1...8 τ i (p 2, q 2 )T iµ Averaging over physically equivalent momenta, bin over nearby momenta A. Sternbeck (FSU Jena) Triple-gluon and quark-gluon vertex from lattice QCD Lattice / 18

26 Lattice results for a few form factors Soft-gluon kinematic: q = 0 λ i=1,2,3 (p, 0) 0 τ i (p, 0) = 0 Discretization effects Significant for large p 2 even with tree-level corrections λ λ1 2 1 binsize: 0.05 (mπ, a) = (280 MeV, 0.08 fm) (422 MeV, 0.07 fm) (290 MeV, 0.07 fm) (430 MeV, 0.06 fm) (quenched, 0.07 fm) Interesting Unquenching pronounced for λ 1 and λ 3 For p > 2 GeV: λ i constant (ignoring discretization effects) May explain partial success of rainbow-ladder truncation p 2 [GeV 2 ] A. Sternbeck (FSU Jena) Triple-gluon and quark-gluon vertex from lattice QCD Lattice / 18

27 Lattice results for a few form factors Soft-gluon kinematic: q = 0 λ i=1,2,3 (p, 0) 0 τ i (p, 0) = 0 Discretization effects Significant for large p 2 even with tree-level corrections λ λ1 2 1 binsize: 0.05 tl-corrected (mπ, a) = (280 MeV, 0.08 fm) (422 MeV, 0.07 fm) (290 MeV, 0.07 fm) (430 MeV, 0.06 fm) (quenched, 0.07 fm) Interesting Unquenching pronounced for λ 1 and λ 3 For p > 2 GeV: λ i constant (ignoring discretization effects) May explain partial success of rainbow-ladder truncation p 2 [GeV 2 ] A. Sternbeck (FSU Jena) Triple-gluon and quark-gluon vertex from lattice QCD Lattice / 18

28 Lattice results for a few form factors Soft-gluon kinematic: q = 0 λ i=1,2,3 (p, 0) 0 τ i (p, 0) = 0 Discretization effects Significant for large p 2 even with tree-level corrections λ λ binsize: 0.05 no TL-corrections (mπ, a) = (280 MeV, 0.08 fm) (422 MeV, 0.07 fm) (290 MeV, 0.07 fm) (430 MeV, 0.06 fm) (quenched, 0.07 fm) Interesting Unquenching pronounced for λ 1 and λ 3 For p > 2 GeV: λ i constant (ignoring discretization effects) May explain partial success of rainbow-ladder truncation p 2 [GeV 2 ] A. Sternbeck (FSU Jena) Triple-gluon and quark-gluon vertex from lattice QCD Lattice / 18

29 Lattice results for a few form factors Soft-gluon kinematic: q = 0 λ i=1,2,3 (p, 0) 0 τ i (p, 0) = 0 Discretization effects Significant for large p 2 even with tree-level corrections λ λ binsize: 0.05 no TL-corrections (mπ, a) = (280 MeV, 0.08 fm) (422 MeV, 0.07 fm) (290 MeV, 0.07 fm) (430 MeV, 0.06 fm) (quenched, 0.07 fm) p 2 [GeV 2 ] Interesting Unquenching pronounced for λ 1 and λ 3 For p > 2 GeV: λ i constant (ignoring discretization effects) May explain partial success of rainbow-ladder truncation Work in progress: Other kinematics τ i (p, q) and 64 4 lattices. A. Sternbeck (FSU Jena) Triple-gluon and quark-gluon vertex from lattice QCD Lattice / 18

30 Summary New lattice data for triple-gluon and quark-gluon vertex in Landau gauge Quenched/unquenched (N f = 2) data Different quark masses, lattice spacings and volumes Until now, almost nothing available from the lattice (Recent papers by Athenodorou et al. [ ] and Duarte et al. [ ]) Important for cross-checks / input to continuum functional approaches Data still preliminary, but suggest Triple-gluon: Lattice results qualitatively agree with recent DSE results Stronger momentum dependence only at small momenta Leading form factor of triple-gluon vertex is F 1, others F i>1 0 Clear unquenching effect, mild quark-mass dependence Zero crossing of G 1 no clear answer (down to p GeV 2!) A. Sternbeck (FSU Jena) Triple-gluon and quark-gluon vertex from lattice QCD Lattice / 18

31 Summary New lattice data for triple-gluon and quark-gluon vertex in Landau gauge Quenched/unquenched (N f = 2) data Different quark masses, lattice spacings and volumes Until now, almost nothing available from the lattice (Recent papers by Athenodorou et al. [ ] and Duarte et al. [ ]) Important for cross-checks / input to continuum functional approaches Data still preliminary, but suggest Triple-gluon: Lattice results qualitatively agree with recent DSE results Stronger momentum dependence only at small momenta Leading form factor of triple-gluon vertex is F 1, others F i>1 0 Clear unquenching effect, mild quark-mass dependence Zero crossing of G 1 no clear answer (down to p GeV 2!) Quark-gluon: Significant momentum dependence only below p < 2 GeV 2 For p > 3 GeV large discretization effects Tree-Level corrections have to be implemented for all λ i and τ i (sufficient?) Analysis of 64 4 data will gives us access to lower p (work in progress) A. Sternbeck (FSU Jena) Triple-gluon and quark-gluon vertex from lattice QCD Lattice / 18

32 Summary New lattice data for triple-gluon and quark-gluon vertex in Landau gauge Quenched/unquenched (N f = 2) data Different quark masses, lattice spacings and volumes Until now, almost nothing available from the lattice (Recent papers by Athenodorou et al. [ ] and Duarte et al. [ ]) Important for cross-checks / input to continuum functional approaches Data still preliminary, but suggest Triple-gluon: Lattice results qualitatively agree with recent DSE results Stronger momentum dependence only at small momenta Leading form factor of triple-gluon vertex is F 1, others F i>1 0 Clear unquenching effect, mild quark-mass dependence Zero crossing of G 1 no clear answer (down to p GeV 2!) Quark-gluon: Significant momentum dependence only below p < 2 GeV 2 For p > 3 GeV large discretization effects Tree-Level corrections have to be implemented for all λ i and τ i (sufficient?) Analysis of 64 4 data will gives us access to lower p (work in progress) Thank you for coming Friday evening! A. Sternbeck (FSU Jena) Triple-gluon and quark-gluon vertex from lattice QCD Lattice / 18