Hadron Structure from Lattice QCD

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1 Hadron Structure from Lattice QCD André Sternbeck a,b On behalf of the RQCD collaboration (Regensburg) a Friedrich-Schiller-Universität Jena, Germany b SFB/TRR 55 Hadron Physics from Lattice QCD September 2015 International School of Nuclear Physics in Erice, Sicily A. Sternbeck (Uni Jena) Hadron Structure from Lattice QCD September / 38

2 Contents Lattice QCD / hadron physics in a nutshell Hadronic quantities from lattice QCD Lattice results of the RQCD collaboration near the physical point (N f = 2) g A, g S, g T,..., x u d, σ πn, Σ, s + s Summary Add-on: New input for functional approaches to hadron physics Lattice results for Quark-Gluon and 3-Gluon vertex in Landau-gauge A. Sternbeck (Uni Jena) Hadron Structure from Lattice QCD September / 38

3 Lattice regularization of QCD (LQCD) QCD action S QCD = d 4 x [ 1 2g 0 2 Tr F 2 [A] + f ψ f x ] ( ) /D[A] m0 f ψx f Discretization of Euclidean space-time Introduce 4-dim lattice L 3 T x = n a 4, n Z 4 Quark fields ψ dwell on sites But: naive discretization of gluon field A µ(x) not gauge-invariant at finite a ψx ψx a A. Sternbeck (Uni Jena) Hadron Structure from Lattice QCD September / 38

4 Lattice regularization of QCD (LQCD) introduced by K. Wilson ( ) Parallel transporter between x and x + ˆµ U µ(n) = Pe iag 0 n+ ˆµ n A µ(z)dz SU c(3) Using links U µ(n) for discretization gives a gauge-invariant action for any a. Example: unimproved Wilson action (a 1) gauge part {}}{ SLQCD W = β { ( 1 1 ReTr ) }}{ 3 n,µν + ψ nd f nm[u, W κ f ]ψm f n,µ<ν n,m,f fermionic part a 0 S QCD Wilson fermion matrix: D W nm [U, κ f ] = δ nm κ f ±µ δ m,n+ ˆµ(1 + γ µ)u nµ (breaks chiral symmetry) QCD Parameters: β 6/g 2 0, κ f 1/(2m f 0 8) Other (improved) lattice actions possible, requirement: S LQCD a 0 S QCD A. Sternbeck (Uni Jena) Hadron Structure from Lattice QCD September / 38

5 Lattice QCD calculation Expectation value of an observable via path integral O = 1 du nµ d Z ψ n dψ n O[U, ψ, ψ] e (Sg [U]+S f [U, ψ,ψ]) nµ No gauge-fixing needed! Integrate over fermionic fields exactly which results in a modified action S eff and O (non-local functionals of the links U) S eff [U; β, κ] = S g [U; β] + log det D[U; κ] Side remark: integration over Grassmann numbers gives det D ɛ k1k2 kn }{{} j 1 j 2 j n D 1 k 1 i D 1 i k ni n = d ψ ndψ n ψ j1 ψ i1 ψ n1 ψ in e ψdψ S S k eff 1 k n n }{{} O Ô Leaves us with Master integral for expectation value O β,κ,l = 1 du nµ Ô[U] e S eff [U,β,κ] Z nµ A. Sternbeck (Uni Jena) Hadron Structure from Lattice QCD September / 38

6 Lattice QCD calculation Monte-Carlo calculation Master integral is very-high dimensional integral O β,κ,l = 1 du nµ Ô[U] e S eff [U,β,κ] Z Number of integration variables (N c = 3, N d = 4) (N 2 c 1) N d = (N 2 c 1) N d = Calculate integrals stochastically Monte-Carlo integration: Sample links U with probability density ( important sampling ) P[U (i) ] = 1 Z e S eff [U (i) ] Estimate for expectation value: nµ e S[U] S[U] Markov chain: U (1), U (2),..., U (N) SU(3) 1 }{{} O N data N Ô[U (i) ] i=1 N O β,κ,l A. Sternbeck (Uni Jena) Hadron Structure from Lattice QCD September / 38

7 Hadron mass from the lattice Hadronic 2-point function C2pt( p, h t) = 1 e i x p h( x, t) h( 0, 0) VS U x Large temporal extension to select m 0 h has quantum number of hadron h π + (x) = d(x) γ 5 u(x), x = ( x, t) In practise: used smeared operators to improve overlap with ground state ( p=0) = Z 0 2 e m 0t + Z 1 2 e m 1t +... mπ (GeV) pion mass a 0.07 fm, mπ 422 MeV a 0.06 fm, mπ 426 MeV a 0.08 fm, mπ 281 MeV a 0.07 fm, Lmπ = 3.42, mπ 295 MeV a 0.07 fm, Lmπ = 4.19, mπ 289 MeV Monte-Carlo integration h π +(x)h π +(y) = 1 N N i= ( ) Tr γ 5Dxy 1 [U (i) ]γ 5 Dyx 1 [U (i) ] }{{} expensive Ô[U (i) ] tf (fm) + O(1/ N) A. Sternbeck (Uni Jena) Hadron Structure from Lattice QCD September / 38

8 Hadron mass from the lattice Hadronic 2-point function C2pt( p, h t) = 1 e i x p h( x, t) h( 0, 0) VS U x Large temporal extension to select m 0 h has quantum number of hadron h π + (x) = d(x) γ 5 u(x), x = ( x, t) In practise: used smeared operators to improve overlap with ground state ( p=0) = Z 0 2 e m 0t + Z 1 2 e m 1t +... mn (GeV) nucleon mass a 0.07 fm, mπ 422 MeV a 0.06 fm, mπ 426 MeV a 0.08 fm, mπ 281 MeV a 0.07 fm, Lmπ = 3.42, mπ 295 MeV a 0.07 fm, Lmπ = 4.19, mπ 289 MeV Monte-Carlo integration h π +(x)h π +(y) = 1 N N i=1 ( ) Tr γ 5Dxy 1 [U (i) ]γ 5 Dyx 1 [U (i) ] }{{} 0.9 expensive Ô[U (i) ] tf (fm) + O(1/ N) A. Sternbeck (Uni Jena) Hadron Structure from Lattice QCD September / 38

9 Lattice QCD accellerator A. Sternbeck (Uni Jena) Hadron Structure from Lattice QCD September / 38

10 Systematics LQCD comes with the same parameters as QCD (coupling and quark masses) plus the lattice volume O β,κ,l = 1 du xµ Ô[U] e S eff [U,β,κ] Z Parameters: β 6/g 2 0, κ f 1/(2m f 0 8), Have to tune them to physical point, e.g., am 2 am 1 (β, κ) nµ β,l large ratio(κ) κ f κ f c Mphys 2 M phys 1 Calculations close to physical point very expensive ( extrapolation) Lattice spacing: a = a(β) Monte-Carlo simulations in practise N f = 2, N f = 2+1, N f = Lattice spacings: a 0.04 fm, Volumes: L 6 fm Challenge: Control over systematic error when extrapolating to physical point A. Sternbeck (Uni Jena) Hadron Structure from Lattice QCD September / 38

11 Landscape of recent lattice simulations Lattice QCD simulations approach (start approaching) the physical point Lattice spacing vs. m π M π [MeV] ETMC '09 (2) ETMC '10 (2+1+1) MILC '10 MILC '12 QCDSF '10 (2) QCDSF-UKQCD '10 BMWc '10 BMWc'08 PACS-CS '09 RBC/UKQCD '10 JLQCD/TWQCD '09 HSC '08 BGR '10 CLS '10 (2) a[fm] Figures from Hoelbling, arxiv: A. Sternbeck (Uni Jena) Hadron Structure from Lattice QCD September / 38

12 Landscape of recent lattice simulations Lattice QCD simulations approach (start approaching) the physical point Volume vs. m π L[fm] % 0.3% 1% ETMC '09 (2) ETMC '10 (2+1+1) MILC '10 MILC '12 QCDSF '10 (2) QCDSF-UKQCD '10 BMWc '10 BMWc'08 PACS-CS '09 RBC/UKQCD '10 JLQCD/TWQCD '09 HSC '08 BGR '10 (2) CLS '10(2) M π [MeV] Figures from Hoelbling, arxiv: A. Sternbeck (Uni Jena) Hadron Structure from Lattice QCD September / 38

13 Landscape of recent lattice simulations Lattice QCD simulations approach (start approaching) the physical point Physical point vs. m π (2M K 2 -Mπ 2 ) 1/2 [MeV] ETMC '10 (2+1+1) MILC '10 QCDSF-UKQCD '10 BMWc'10 PACS-CS '09 JLQCD/TWQCD '09 RBC-UKQCD '10 HSC ' M π [MeV] Figures from Hoelbling, arxiv: A. Sternbeck (Uni Jena) Hadron Structure from Lattice QCD September / 38

14 Hadron masses from the lattice Lattice QCD works! 2400 H H * H s H s * B c B c * Andreas Kronfeld/Fermi Natl Accelerator Lab. (MeV) π ρ K K η η ω φ N Λ Σ Ξ Σ Ξ Ω Figure : Hadron masses from lattice QCD: experimental vs. lattice QCD results (MILC, PACS-CS, BMW, QCDSF und RBC&UKQCD). From [Kronfeld arxiv: ]. A. Sternbeck (Uni Jena) Hadron Structure from Lattice QCD September / 38

15 Physics program of the RQCD collaboration Hadron structure from lattice QCD Moments of parton distribution functions (Nucleon: arxiv: , , pion in preparation) Nucleon isovector couplings (arxiv: ), sigma terms (arxiv: , update in preparation), form factors and generalised form factors (in preparation) Moments of light-cone distribution amplitudes (Nucleon: arxiv: , Pion: arxiv: ) Charmed baryon spectroscopy (arxiv: ) Strangeness Contribution to the Proton Spin (arxiv: )... Up to now: N f = 2 calculations from m π = MeV (this talk) safer extrapolation to physical point main uncertainties: volume, lattice spacing artefacts A. Sternbeck (Uni Jena) Hadron Structure from Lattice QCD September / 38

16 RQCD-Kollaboration: N f = 2 ensembles Action Wilson gauge N f = 2 Wilson-clover fermions VI a 0.08 fm a 0.07 fm a 0.06 fm Physical regime Lattice spacings: a = fm (2 spacings m π 290 MeV 3 spacings m π 425 MeV). m π down to 150 MeV. Lm π up to 6.7, Lmπ VIII V XI I IV 3 VII mπ [GeV] Symbol Lm π 6.7 > 4.1 > III X II IX Reduction of excited-state contaminations Gauss-smearing iterations on top of APE smearing. Multi-exponential fits A. Sternbeck (Uni Jena) Hadron Structure from Lattice QCD September / 38

17 Hadronic two- and three-point functions on the lattice Hadronic matrix elements N O N are obtained via: 1 two-point correlation functions (as above) 2 three-point correlation functions C3pt(τ, O t, P, P ) = h(t, P ) Õ(τ, P) h (0, P) h and Õ denote the Fourier transform of the hadronic and the insertion operator h(t, P) = x e i P x h(t, x), Õ(τ, P) = x e i P x O( x, τ) Insertion operator: O = {γ 5, γ µ, γ νd µ,...} chosen wrt. desired matrix element, e.g. R(t, τ) = C 3pt O (τ, t) = N O N +... C 2pt (τ) 2M N A. Sternbeck (Uni Jena) Hadron Structure from Lattice QCD September / 38

18 01 01 Illustration: Measurement of a three-point function on the lattice Connected and disconnected contribution Diagrams t f t t i t f On the Lattice t t i Figure : from R. Horsley (1999) A. Sternbeck (Uni Jena) Hadron Structure from Lattice QCD September / 38

19 Nucleon axial charge g A Associated with β decay (neutron proton) Experimentally well known: g A = (23)g V (isospin symmetric limit: g V = 1) Theorie: g A = g A (0) of the nucleon axial form factor: [ p ūγ µγ 5d n = ū p(p f ) g A (q 2 )γ µ + g ] P(q 2 ) q µ γ 5u n(p i) 2M N Lattice QCD Good benchmark observable for a lattice calculation Challenge for many years (underestimate of exp. value) Conjecture: volume effect g A [QCDSF (2006)] =5.20 =5.25 =5.29 = m 2 [GeV 2 ] A. Sternbeck (Uni Jena) Hadron Structure from Lattice QCD September / 38

20 Nucleon axial charge g A Associated with β decay (neutron proton) Experimentally well known: g A = (23)g V (isospin symmetric limit: g V = 1) Theorie: g A = g A (0) of the nucleon axial form factor: [ p ūγ µγ 5d n = ū p(p f ) g A (q 2 )γ µ + g ] P(q 2 ) q µ γ 5u n(p i) 2M N Lattice QCD New data from RQCD (arxiv: ) Clear volume effect for fixed lattice spacing No significant discret. errors for fixed volume ga [RQCD: S Collins, R Rödl] a 0.08 fm a 0.07 fm a 0.06 fm Expt m 2 π [GeV2 ] Symbol Lm π 6.7 > 4.1 > A. Sternbeck (Uni Jena) Hadron Structure from Lattice QCD September / 38

21 Nucleon axial charge g A : Finite volume effects arxiv: Prediction from χpt: Similar finite volume effects for g A and F π Follow QCDSF [ ] and plot/extrapolate ratio Extrapolation to phys. point Fit: g A /F π = 13.88(29) GeV 1 ga/fπ [GeV 1 ] a 0.08 fm a 0.07 fm a 0.06 fm Exp Exp: g A /F π = (34) GeV With F π(exp) = MeV we obtain m 2 π [GeV2 ] g A = 1.280(27)(35) Exp: g A = (35) A. Sternbeck (Uni Jena) Hadron Structure from Lattice QCD September / 38

22 Comparison of recent lattice results for g A arxiv: ga QCDSF Nf = 2 Mainz Nf = 2 ETMC Nf = 2 LHPC Nf = RBC Nf = ETMC Nf = PNDME Nf = RQCD Nf = 2 RQCD Nf = 2, ga/fπ Expt m 2 π [GeV2 ] World lattice data approaches physical regime (m π 135 MeV) Reproduction of experimental value for g A if excited-states and volume effects are under control A. Sternbeck (Uni Jena) Hadron Structure from Lattice QCD September / 38

23 Nucleon (isovector) couplings arxiv: Calculated also all other couplings Besides g T, all couplings are directly obtained in the forward limit q 2 = 0 g T from the form factor data g T (q 2 0) p ūd n = g S (q 2 ) ū p(p f )u n(p i ), p ūγ 5 d n = g P (q 2 )ū p(p f )γ 5 u n(p i ), [ p ūγ µd n = ū p(p f ) g V (q 2 )γ µ + g T (q 2 ] ) iσ µνq ν u n(p i ), 2M N [ p ūγ µγ 5 d n = ū p(p f ) g A (q 2 )γ µ + g P(q 2 ] ) q µ γ 5 u n(p i ), 2M N p ūσ µνd n = g T (q 2 )ū p(p f )σ µνu n(p i ) Induced tensor charge g T known another benchmark observable Other charges more or less unknown, might be relevant for new physics processes or dark matter searches g T and g S accessible at present only through lattice simulation. A. Sternbeck (Uni Jena) Hadron Structure from Lattice QCD September / 38

24 Lattice results for g S and g T arxiv: ETMC Nf = 2 RBC Nf = LHPC Nf = PNDME Nf = ETMC Nf = RQCD Nf = g S MS (2 GeV) g T MS (2 GeV) LHPC Nf = PNDME Nf = ETMC Nf = RQCD Nf = m 2 π [GeV2 ] m 2 π [GeV2 ] We find gs MS (2 GeV) = 1.02(18)(30) and gt MS (2 GeV) = 1.005(17)(29) when approaching phys. point In agreement with other determinations Lattice spacing effects not excluded (a = fm) A. Sternbeck (Uni Jena) Hadron Structure from Lattice QCD September / 38

25 Nucleon (isovector) couplings arxiv: : Final results extrapolated to physical point Table : First errors contain statistics and systematics. The second errors are estimates of lattice spacing effects. Our result Experiment g A 1.280(44)(46) (23) gs MS (2 GeV) 1.02(18)(30) gt MS (2 GeV) 1.005(17)(29) g T 3.00(08)(31) (5) gp 8.40(40)(159) 8.06(55) Couplings by definition valence quark quantities no significant effects from including strange (or charm) sea quarks expected Chiral extrapolation to physical point under control (estimate at m π 150 MeV allows us to estimate uncertainties) Deviation of g T not yet clear A. Sternbeck (Uni Jena) Hadron Structure from Lattice QCD September / 38

26 Moments of Parton Distributions (PDFs) Nucleon parton distribution function Important for phenomenology Known for large range of x = [0, 1] (long. mom. fraction) Good knowledge for u and d For other quarks not as good Armstrong et al. (2012) Lattice QCD allows only access to moments of PDFs Lowest moment of nucleon PDF x (q) = 1 0 dx x [q(x) + q(x)] Ever popular moment: x (u d) no disconn. contributions needed discrepancy: lattice vs. exp. A. Sternbeck (Uni Jena) Hadron Structure from Lattice QCD September / 38

27 Reduction of systematic error for x MS u d Fitting of leading excited states Access via 2- and 3-point functions (operator O with derivative) For moderate τ and τ t f also contributions from excited states C 2pt (t f ) = Z 0 2 e m 0 t f + Z 1 2 e m 1 t f +... C 3pt (t f, t) = Z 0 2 N 0 O N 0 e m 0 t f + Z 1 2 N 1 O N 1 e m 1 t f + Z 1 Z 0 N 1 O N 0 e m 0 t e m 1 (t f t) + Z 0 Z 1 N 0 O N 1 e m 1 t e m 0 (t f t) +... IV, a 0.07 fm, m π 295 MeV, Lm π = 3.42 VIII, a 0.07 fm, m π 150 MeV, Lm π = x MS u d(2 GeV) x MS u d(2 GeV) t a tf 2a t a tf 2a A. Sternbeck (Uni Jena) Hadron Structure from Lattice QCD September / 38

28 Lowest moment of nucleon PDF x u d Status x MS u d(2 GeV) a 0.08 fm a 0.07 fm a 0.06 fm QCDSF(2011) ETMC(2011) CT10 ( 14) NNPDF ( 12) ABM ( 11) MSTW ( 09) m 2 π [GeV 2 ] x MS u d(2 GeV) RQCD Nf = RBC/UKQCD ( 10), ( 13) Nf = LHPC ( 12) Nf = ETMC ( 13) Nf = ETMC ( 13) Nf = m 2 π [GeV 2 ] Deviation to phenom. value reduced (better treatment of excited-states) Discrepancy remains though Further issues? A. Sternbeck (Uni Jena) Hadron Structure from Lattice QCD September / 38

29 Physics program of the RQCD collaboration Hadron structure from lattice QCD Moments of parton distribution functions (Nucleon: arxiv: , , pion in preparation) Nucleon isovector couplings (arxiv: ), sigma terms (arxiv: , update in preparation), form factors and generalised form factors (in preparation) J q Moments of light-cone distribution amplitudes (Nucleon: arxiv: , Pion: arxiv: ) Charmed baryon spectroscopy (arxiv: ) Strangeness Contribution to the Proton Spin (arxiv: )... Up to now: N f = 2 calculations from m π = MeV (this talk) safer extrapolation to physical point main uncertainties: volume, lattice spacing artefacts A. Sternbeck (Uni Jena) Hadron Structure from Lattice QCD September / 38

30 Pion-nucleon sigma term Decomposition of nucleon mass M N = N T 44 N N N = 4 σ q + N d 3 x 3 N N q }{{} small ( 1 8πα (E2 B 2 ) + ) q qγ Dq N } {{ } dominant T energy density component of energy-momentum tensor Nucleon mass is largely generated by gluon field energy ( 900 MeV) (M N = 938 MeV, m u 2 MeV, m d 4 MeV) σ-terms Small fraction from sea and valence quarks parametrizes individual quark contribution f q = σ q/m N to nucleon mass Pion-nucleon sigma term: σ πn = σ u + σ d σ q = m q N q q N N N VEV 0 qq 0 is understood to be subtracted from N qq N Phenomenology does not give a clear answer on (45 65 MeV) Lattice allows for direct calculation (involves disconnected contributions) A. Sternbeck (Uni Jena) Hadron Structure from Lattice QCD September / 38

31 Pion-nucleon sigma term from the lattice SESAM (1998) Leinweber et al. (2003) Procura et al. (2004) Procura et al. (2006) JLQCD (2008) ETMC (2009) Thomas & Young (2009) PACS-CS (2009) BMW (2011) Thomas & Young (2012) QCDSF (2012) QCDSF (2012) Nf = 2 Nf = σ phys [MeV] Most calculations via Feynman- Hellmann theorem σ πn = mπ 2 M N (m π) mπ 2 New determinations are done directly and include contributions from disconnected quark loops (expensive and noisy) Central value changed over the years: MeV A. Sternbeck (Uni Jena) Hadron Structure from Lattice QCD September / 38

32 Pion-nucleon sigma term from the lattice New data from RQCD, to be published soon 0.3 σπn [GeV] a 0.08 fm 0.05 a 0.07 fm a 0.06 fm ETMC Nf = m 2 π [GeV2 ] Lm π 6.7: Lm π > 4.1: Lm π > 3.4: [S Collins] Linear Fit with and without constraint σ πn (0) = 0 gives σ πn 35 MeV The non-vanishing light quark masses responsible for only 35 MeV of M N Not unexpected: M N 0 as m ud 0 Note: quark contribution to pion mass m π/2! A. Sternbeck (Uni Jena) Hadron Structure from Lattice QCD September / 38

33 Decomposition of pion mass and the Pion sigma term m π 1 2 mπ + 3 }{{} 8 mπ + 1 }{{} 8 mπ }{{} σ π gluon int. trace anomaly with σ π = m ud π ūu + dd π = m ud m π m ud 0.2 σπ (GeV) [S Collins, D Richtmann] m π (GeV) The theoretical expectation σ π m π/2 is confirmed. A. Sternbeck (Uni Jena) Hadron Structure from Lattice QCD September / 38

34 Side remark: Chiral extrapolation of the nucleon mass Lattice QCD simulations produce bare numbers Need to calculate physical quantity (mass, decay constant,... ) Need to extrapolate to physical point (systematic uncertainty) Extrapolation of M N data desired but yet too uncertain Sigma-term values (slope) constrain the chiral fit (basically only offset is left free) r0mn 3.0 preliminary t (unconstrained) 2.5 combined t data: r0σ (uncorrected) data: r0mn (volume-corrected) data: r0mn physical point (r fm) (r0mπ) 2 σ πn values make chiral extrapolation of M N data much less ambiguous. A. Sternbeck (Uni Jena) Hadron Structure from Lattice QCD September / 38

35 Summary Lattice can contribute to many quantities that are hard to constrain by experiment: σ πn, g S, g T, q, Σ, J q, x q, Benchmark observables help to improve lattice methods Hadron masses (2-point functions) come out nicely g A (3-point functions) seems to approach the physical value, once Lm π > 4 x u d comes out 20% bigger than expected Up to now: N f = 2 calculations from m π = MeV with lattice spacings: a = fm (this talk) Ultimately precision physics requires an extrapolation a 0. Upcoming: N f = calculations targetting physical point and continuum limit (part of the CLS effort). Disconnected quark line diagrams will more and more be included in the past often omitted and differences quoted: g A, x u d,..., but no s, Σ, J q, x q etc. See, e.g., quark spin contribution to proton spin Σ A. Sternbeck (Uni Jena) Hadron Structure from Lattice QCD September / 38

36 Quark spin contributions to proton spin Bali et al., PRL108 (2012) Proton Spin 1 2 = 1 Σ + Lq + G 2 Results for MS scheme at µ 2 = 7.4 GeV Lattice DSSV x>0.001 DSSV u+ u d+ d s+ s Σ = a 0 Σ = u + ū + d + d + s + s = 0.45(4)(9) s + s = 0.020(10)(4) Warning: no continuum limit, m π 290 MeV add 20 % systematic error. A. Sternbeck (Uni Jena) Hadron Structure from Lattice QCD September / 38

37 Quark spin contributions to proton spin Comparison of recent lattice calculations s + s QCDSF 11 N f = 2 Engelhardt 12 N f = ETMC 13 N f = χ-qcd 14 N f = m 2 π GeV 2 Consistency between different determinations: small s + s. ETMC result shows statistical accuracy that is possible. Systematics! [QCDSF: GB et al, ; M Engelhardt, ; ETMC: A Abdel-Rehim et al, ; χqcd: Y Yang et al, unpublished.] A. Sternbeck (Uni Jena) Hadron Structure from Lattice QCD September / 38

38 Leaving the traditional way lattice QCD is used for hadron physics Lattice QCD gauge-invariant nonperturbative approach to QCD Systematically improvable (larger and finer lattices, tune bare parameters towards physical point) Allows access to many interesting quantities without a need to fix a gauge A. Sternbeck (Uni Jena) Hadron Structure from Lattice QCD September / 38

39 Leaving the traditional way lattice QCD is used for hadron physics Lattice QCD gauge-invariant nonperturbative approach to QCD Systematically improvable (larger and finer lattices, tune bare parameters towards physical point) Allows access to many interesting quantities without a need to fix a gauge Lattice QCD allows more... Apply a gauge condition access to nonperturbative and untruncated elementary n-point functions of QCD 2-point functions: quark, gluon and ghost propagators Many lattice studies in the past helped to settle momentum dependence 3-point functions: quark-gluon, 3-gluon, 4-gluon, ghost-gluon vertex Essential information for functional approaches to hadron physics (e.g. QCD Dyson-Schwinger and bound-state equations) perturbative, semi-perturbative and non-perturbative regimes e.g: validity of Rainbow-Ladder truncations etc. A. Sternbeck (Uni Jena) Hadron Structure from Lattice QCD September / 38

40 New lattice results for Quark-Gluon-Vertex in Landau gauge in collaboration with Kizilersu, Oliveira, Silva, Skullerud Quark-Gluon vertex G ψψa µ = Γ ψψa λ (p, q) S(p) D µλ (q) S(p + q) Non-perturbative structure mostly unknown, but crucial for many functional approaches to hadron physics (Dyson-Schwinger, Bethe-Salpether equations etc.) Up to now: only lattice data for quenched QCD Ball-Chiu-Parametrisation with Γ ψψa µ (p, q) = Γ ST µ (p, q) + ΓT µ (p, q) 4 Γ ST µ (p, q) = λ i (p 2, q 2 )L iµ (p, q) i=1 8 Γ T µ (p, q) = i=1 satisfies Slavnov-Taylor identies τ i (p 2, q 2 )T iµ (p, q) is transverse (q µγ T µ = 0). The tensors L µ and T µ can be found in the literature: 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 (Uni Jena) Hadron Structure from Lattice QCD September / 38

41 New lattice results for Quark-Gluon-Vertex in Landau gauge (Be careful, results are preliminary yet!) Example: q = 0 (soft gluon) τi (p, 0) = 0 10 (mπ, a) = (280 MeV, 0.08 fm) (422 MeV, 0.07 fm) (290 MeV, 0.07 fm) (430 MeV, 0.06 fm) 9 8 (mπ, a) = (280 MeV, 0.08 fm) (422 MeV, 0.07 fm) (290 MeV, 0.07 fm) (430 MeV, 0.06 fm) λ λ (without TL-corrections) [GeV2 ] p2 0 1 (with TL-corrections) p λ (without TL-corrections) [GeV2 ] -6 Discretization effect not yet under control (hypercubic artefacts for p2 > 3 due to lattice) (mπ, a) = (280 MeV, 0.08 fm) (422 MeV, 0.07 fm) (290 MeV, 0.07 fm) (430 MeV, 0.06 fm) Deviation from tree-level for p < 1 GeV 1 p [GeV2 ] For p > 1 GeV: Vertex tree-level (might explain why Rainbow ladder truncation so successful) A. Sternbeck (Uni Jena) Hadron Structure from Lattice QCD September / 38

42 Lattice results for 3-Gluon vertex together with BSc. Balduf (preliminary) a [fm] m π [GeV 2 ] A B C F D E 0.07 p = q, θ = angle(p, q) Statistical noise grows with p = q Small quark-mass dependence (slight upward shift) for the available sets, but a clear quenching effect (vertical shift) No zero-crossing for p > 0.3 GeV (p > 0.8 GeV quenched) A. Sternbeck (Uni Jena) Hadron Structure from Lattice QCD September / 38

43 Lattice results for 3-Gluon vertex together with BSc. Balduf (preliminary) a [fm] m π [GeV 2 ] A B C F D E 0.07 p = q, θ = angle(p, q) Statistical noise grows with p = q Small quark-mass dependence (slight upward shift) for the available sets, but a clear quenching effect (vertical shift) No zero-crossing for p > 0.3 GeV (p > 0.8 GeV quenched) A. Sternbeck (Uni Jena) Hadron Structure from Lattice QCD September / 38

44 Thank you for your attention! A. Sternbeck (Uni Jena) Hadron Structure from Lattice QCD September / 38