QCD. Nucleon structure from Lattice QCD at nearly physical quark masses. Gunnar Bali for RQCD

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1 Nucleon structure from Lattice QCD at nearly physical quark masses Gunnar Bali for RQCD with Sara Collins, Benjamin Gläßle, Meinulf Göckeler, Johannes Najjar, Rudolf Rödel, Andreas Schäfer, Wolfgang Söldner and André Sternbeck QCD APS Hadronic Physics Baltimore, April 10, 2015

2 Outline Importance of proton structure beyond QCD Lattice QCD set-up Mass: σ-terms Spin: The q s and g A Other couplings Momentum fraction: x u d Summary Gunnar Bali (Regensburg) Nucleon structure QCD 2 / 40

3 Protons in use e.g. at the LHC Gunnar Bali (Regensburg) Nucleon structure QCD 3 / 40

4 What is known about parton distribution functions? The u and d PDFs are well-known from experiment, e.g., at DESY. Strangeness and gluonic PDFs have much larger uncertainties. Generated using from the NNPDF2.3 data set. NNPDF: R D Ball et al, NPB 867 (13) 244 Gunnar Bali (Regensburg) Nucleon structure QCD 4 / 40

5 Nucleons as dark matter probes: XENON1T at Gran Sasso y-scale of shaded areas depends on scalar couplings m q N qq N. Gunnar Bali (Regensburg) Nucleon structure QCD 5 / 40

6 Proton structure calculations are essential to constrain beyond-the-standard-model (BSM) dark matter candidates, relating predictions to experimental limits.... important to predict cross-sections for processes on the quark-gluon level. Experiment e.g. unable to directly measure strangeness and gluon PDFs.... needed to relate QCD to low energy effective theories that are also relevant for precision experiments. Here I concentrate on How is the mass distributed among the partons? (scalar couplings) How is the spin distributed? (axial couplings) Proton-neutron transition couplings. (g S, g T, g T, g P, g P ) How is the momentum distributed? (moments of PDFs) Gunnar Bali (Regensburg) Nucleon structure QCD 6 / 40

7 Lattice QCD Na o o o o o o o o o o o a o o o o o o o o o o o o o o plaquette link site typical values: a 1 = 2 5 GeV, Na = 2 7 fm continuum limit: a 0, Na fixed infinite volume: Na O = 1 S[U,ψ, ψ] [du] [dψ][d ψ] O[U]e Z Measurement : average over a representative ensemble of gluon configurations {U i } with probability P(U i ) [dψ][d ψ]e S[U,ψ, ψ] O = 1 n n i=1 O(U i ) + O O 1 n n 0 Gunnar Bali (Regensburg) Nucleon structure QCD 7 / 40

8 Input: discretized L QCD = 1 16πα L (a) FF + q f (D/ + m f (a))q f mπ latt /mn latt m latt N = m phys N a = mπ phys /m phys N m u (a) m d (a) Output: hadron masses, matrix elements, decay constants, etc... Required: 1. L = Na : FSE suppressed with exp( Lm π ) Lm π m latt q m latt ud mq phys : chiral perturbation theory (χpt) helps for m ud but must be sufficiently small to start with (m π 200 MeV?). 3. a 0: functional form known: O(a 2 ), O(α s a) 4 lattice spacings. Gunnar Bali (Regensburg) Nucleon structure QCD 8 / 40

9 Landscape of recent lattice simulations (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 '08 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) M π [MeV] a[fm] Figures taken from C Hoelbling, arxiv: 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] Gunnar Bali (Regensburg) Nucleon structure QCD 9 / 40

10 Computational challenges Cost of simulation is proportional to number of points: (L/a) 4 condition number of linear system: 1/m 2 π L 1/2 /m π in (Omelyan) time integration within hybrid Monte Carlo 1/a 2 critical slowing down (autocorrelations) Adjusting L 1/m π this means: cost 1 a 6 m 7.5 π In addition: for baryonic observables at small m π serious signal/noise problem. State of the art: sites, corresponding to ( ) 2 (sparse) complex matrices. Tremendous progress in Hybrid Monte Carlo, solver, noise reduction. Gunnar Bali (Regensburg) Nucleon structure QCD 10 / 40

11 HW Hamber, E Marinari, G Parisi, C Rebbi, NPB225 (83) 475 (Appendix B) GP Lepage 89, C N (t) exp ( m N t) [ C N (t)] 2 C N (t) C N (t) exp [( m N 3 2 m π ) ] t exp ( 3m π t) Self-averaging over many source points increases statistics. Becomes increasingly important towards small m π. Gunnar Bali (Regensburg) Nucleon structure QCD 11 / 40

12 Three point functions Evaluate N qγq N (Lines: quark propagators M 1 xy, M = /D + m q ) O = q Γq N N connected disconnected q {u, d}: both quark-line connected and disconnected terms. q = s: only the disconnected term. Connected requires only 12 rows (spin colour) of M 1. Disconnected 12N 3 rows (timeslice): stochastic all-to-all methods. Disconnected cancels (m u = m d, ////// QED) from isovector combinations: proton minus neutron, i.e. p (ūγu dγd) p = p ūγd n. Gunnar Bali (Regensburg) Nucleon structure QCD 12 / 40

13 Action and ensembles N f = 2 NP improved Sheikholeslami-Wilson fermions, Wilson glue. Lm π up to 6.7, a down to 0.06 fm, m π down to 150 MeV. Two lattice spacings around m π 290 MeV, three around 425 MeV Wuppertal=Gauss smearing iterations on top of APE smearing. # β a/fm κ V m π/mev Lm π n conf t sink /a I (4) 13 II (2) 15 III (2) 15,17 IV (2) 7,9,11,13,15,17 V (2) 15 VI (2) 15 VII (2) 15 VIII (3) 9,12,15 IX (2) 17 X (2) 17 XI (2) 17 Gunnar Bali (Regensburg) Nucleon structure QCD 13 / 40

14 Ensembles II Lmπ VIII VII XI VI V I IV a 0.08 fm a 0.07 fm a 0.06 fm m π [GeV] III X II IX Lm π 6.7: Lm π > 4.1: Lm π > 3.4: Lm π 2.8: Gunnar Bali (Regensburg) Nucleon structure QCD 14 / 40

15 Decomposition of the proton (and pion) mass I m N = m q N q 1 q N q {u,d,s,...} }{{} quarks ( m N m q N q 1 q N 4 q }{{} trace anomaly 1 N (E 2 B 2 ) + qd γq 8πα L N q }{{} ) gluon interactions (Eucl. spacetime) VEV 0 qq 0 is understood to be subtracted from N qq N. Pion-nucleon σ-term: σ πn = m u N ūu N + m d N dd N = σ u + σ d. Scalar particles (Higgs, neutralino etc.) couple quark matrix elements. Gunnar Bali (Regensburg) Nucleon structure QCD 15 / 40

16 Decomposition of the proton (and pion) mass II Therefore: σ π = m ud π ūu + dd π = m ud m π m ud m π 1 2 m 3 π + }{{} 8 m π }{{} σ π gluon interactions = m π + O(m }{{ 2 π) 3. } GMOR m π }{{} trace anomaly σ π can be further decomposed into valence and sea quark contributions. Wilson fermions: singlet and non-singlet mass renormalization constants differ by r m > 1 valence > connected : r := π ūu + dd π sea π ūu + dd π = r m ( π ūu + dd π dis lat π ūu + dd π lat 1 ) + 1 Gunnar Bali (Regensburg) Nucleon structure QCD 16 / 40

17 Pion mass: σ π compared to m π /2 0.2 σπ (GeV) m π (GeV) [S Collins, D Richtmann] The theoretical expectation σ π m π /2 is confirmed. Gunnar Bali (Regensburg) Nucleon structure QCD 17 / 40

18 Pion mass: light sea quark and strange quark contribs. 0.3 r σs (GeV) m π (GeV) m π (GeV) Less than 10% of σ π (or 5% of the mass) is due to sea quarks. Strange quarks are negligible too. Nevertheless, r m = Zm singlet > 1 means at a fm about 30% of the signal originates from the disconnected contribution. So this needs to be computed even for the valence quark contribution. /Z nonsinglet m Gunnar Bali (Regensburg) Nucleon structure QCD 18 / 40

19 σ πn for the nucleon 0.3 σπn [GeV] a 0.08 fm a 0.07 fm a 0.06 fm ETMC N f = m 2 π [GeV2 ] 0.05 Lm π 6.7: Lm π > 4.1: Lm π > 3.4: [S Collins] The non-vanishing light quark masses are directly responsible for only 35 MeV of the nucleon mass but for 68 MeV of the pion mass! This may not be too surprising since m N 0 as m ud 0. Gunnar Bali (Regensburg) Nucleon structure QCD 19 / 40

20 Chiral extrapolation of the nucleon mass preliminary t (unconstrained) 3.0 r0mn 2.5 combined t data: r 0 σ (uncorrected) data: r 0 M N (volume-corrected) data: r 0 M N physical point (r fm) (r 0 m π ) 2 Gunnar Bali (Regensburg) Nucleon structure QCD 20 / 40

21 The scalar matrix elements m q N qq N determine the coupling of the nucleon to scalar particles at zero recoil: f N m N α q f Tq + 2 m q {u,d,s} q 33 6 f α q T G. m q {c,b,t,...} q Cross section f N 2. Higgs example: α q m q /m W. f Tq m q N qq N m N are the contributions of the light quark masses to the proton mass and f TG 1 q {u,d,s} f Tq. Little about f Tq is known experimentally. Gunnar Bali (Regensburg) Nucleon structure QCD 21 / 40

22 Scalar strangeness content a 0.08 fm a 0.07 fm a 0.06 fm Engelhardt N f = ETMC N f = fts m 2 π [GeV2 ] [QCDSF: GB et al, arxiv: , RQCD: S Collins et al, in preparation]: NPI Wilson [M Engelhardt, arxiv: ]: domain wall on staggered [ETMC, C Alexandrou et al, arxiv: ]: twisted mass Gunnar Bali (Regensburg) Nucleon structure QCD 22 / 40

23 Spin of the nucleon 1 2 = 1 2 Σ + L q + J g : q, q Ji decomposition into the contributions of the (longitudinal) quark spins Σ = u + ū + d + d + s + s +, the (longitudinal) quark and antiquark orbital angular momenta L q = J q 1 2 q and the (longitudinal) gluon total angular momentum J g. Naïve non-relativistic SU(6) quark model: Σ = 1, L q = J g = s = 0. Relativistic quark models: Σ 0.6, L quarks 0.2. I will say nothing about the Jaffe and Manohar decomposition: 1 2 = 1 2 Σ + L quarks + G + L g ( J g G + L g, J q 1 ) 2 q + L q. Gunnar Bali (Regensburg) Nucleon structure QCD 23 / 40

24 The total quark angular momenta J q = 1 2 q + L q can be extracted from generalized form factors at q 2 = 0: J q + J q = 1 2 [ A q 20 (0) + Bq 20 (0)], where A q 20 (q2 ) and B q 20 (q2 ) are obtained from matrix elements of local quark bilinears of the form N, s, p + q qγ {µ 1 D µ 2} q N, s, p. Then L q = J q 1 2 q, J g = 1 2 J q. q, q Gunnar Bali (Regensburg) Nucleon structure QCD 24 / 40

25 Individual quark spin contributions (q {u, d, s}) Axial charges: ( q + q) s µ = 1 m N N, s qγ µ γ 5 q N, s = F q A (0) = Ãq 10 (0) a 3 = s µ 1 m N N, s ψγ µ γ 5 λ 3 ψ N, s = u d = g A a 8 = s µ 3 m N N, s ψγ µ γ 5 λ 8 ψ N, s = u + ū + d + d 2 s 2 s a 0 (Q 2 ) = s µ 1 m N N, s ψγ µ γ 5 1ψ N, s = u + ū + d + d + s + s = Σ(Q 2 ). ψ = (u, d, s) t, λ j are Gell-Mann flavour matrices. a 3 = g A known from neutron β decay, assuming isospin symmetry. a 8 usually estimated from hyperon β decay, assuming SU(3) F symmetry. Gunnar Bali (Regensburg) Nucleon structure QCD 25 / 40

26 Extraction of the q s from experiment DIS gives spin structure functions of proton and neutron g p,n 1 (x, Q 2 ). First moment related to a i s via OPE (leading twist): Γ p,n 1 (Q2 ) = 1 0 dx g p,n 1 (x, Q 2 ) = 1 36 [(a 8 ± 3a 3 )C NS + 4a 0 C S ] Use models to extrapolate g 1 from experimental x min to x = 0! C S/NS = C S/NS (α s (Q 2 )). Combinations of a i give q s, e.g., ( s + s)(q 2 ) = 1 3 [a 0(Q 2 ) a 8 ] SIDIS allows for direct measurements of the q(x) but requires fragmentation functions. [COMPASS, arxiv ] Naive Extrap. combined with DSSV ( s + s)(5 GeV 2 ) 0.02 ± 0.02 ± ± 0.02 ± 0.02 DSSV: [de Florian et al, arxiv: ] Gunnar Bali (Regensburg) Nucleon structure QCD 26 / 40

27 No continuum limit, m π 290 MeV add 20 % systematic error. Lattice 0.8 DSSV x>0.001 DSSV u+ u d+ d s+ s Σ = a 0 [QCDSF: GB et al, ] Result in the MS scheme at µ 2 = 7.4 GeV 2 : Σ = u + d + s = 0.45(4)(9) s = 0.020(10)(4) Gunnar Bali (Regensburg) Nucleon structure QCD 27 / 40

28 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.] Gunnar Bali (Regensburg) Nucleon structure QCD 28 / 40

29 J q + J q = 1 2 (Aq 20(0) + B q 20(0)) From Lattice 14 review [M Constaninou, ] Contributions to nucleon spin m 2 π (GeV 2 ) : disconnected contribution included. J u J d [LHPC: S Syritsyn et al, (N f = 2 + 1); QCDSF/UKQCD: A Sternbeck et al, (N f = 2); ETMC: C Alexandrou et al, , unpublished (N f = 2); ETMC: C Alexandrou et al, (N f = ).] Gunnar Bali (Regensburg) Nucleon structure QCD 29 / 40

30 g A = u d ga a 0.08 fm a 0.07 fm a 0.06 fm Expt Lm π 6.7: Lm π > 4.1: Lm π > 3.4: Lm π 2.8: [S Collins, R Rödl] m 2 π [GeV2 ] Comparing similar volumes: no significant discretization effects. m π 425 MeV: g A increases by 5% with Lm π m π 290 MeV: g A up by 6% with Lm π , then constant. m π 150 MeV: No difference between Lm π 2.8 and Lm π 3.5. Gunnar Bali (Regensburg) 3 Nucleon structure QCD 30 / 40

31 Finite volume effects predicted by χpt similar for g A and F π = follow QCDSF: R Horsley et al, arxiv: and plot ratio a 0.08 fm a 0.07 fm a 0.06 fm Exp ga/fπ [GeV 1 ] m 2 π [GeV2 ] Extrapolation to physical point: g A /F π = 13.88(29) GeV 1 Expt: g A /F π = (34) GeV 1. Using F π (expt) = MeV we obtain g A = 1.280(27)(35) Expt: g A = (35). Gunnar Bali (Regensburg) Nucleon structure QCD 31 / 40

32 g A : summary of recent lattice results 1.4 ga QCDSF N f = 2 Mainz N f = 2 ETMC N f = 2 LHPC N f = RBC N f = ETMC N f = PNDME N f = RQCD N f = 2 RQCD N f = 2, g A/F π Expt m 2 π [GeV2 ] QCDSF: Mainz: ETMC 2: LHPC: RBC/UKQCD: ETMC 2+1+1: PNDME: RQCD: Gunnar Bali (Regensburg) Nucleon structure QCD 32 / 40

33 Isovector scalar charge g S MS (2 GeV) a 0.08 fm a 0.07 fm a 0.06 fm g S MS (2 GeV) m 2 π [GeV2 ] LHPC Nf = PNDME Nf = ETMC Nf = RQCD Nf = m 2 π [GeV2 ] LHPC: , PNDME: , ETMC: , RQCD: Gunnar Bali (Regensburg) Nucleon structure QCD 33 / 40

34 Isovector tensor charge a 0.08 fm a 0.07 fm a 0.06 fm ETMC Nf = 2 RBC Nf = LHPC Nf = PNDME Nf = ETMC Nf = RQCD Nf = 2 g T MS (2 GeV) g T MS (2 GeV) m 2 π [GeV2 ] m 2 π [GeV2 ] ETMC 2: , RBC: , LHPC: , PNDME: , ETMC 2+1+1: , RQCD: General remark: we vary a 2 only by a factor 1.8 we cannot exclude lattice spacing effects of up to /( ) g 1.7 g. Gunnar Bali (Regensburg) Nucleon structure QCD 34 / 40

35 Isovector electromagnetic formfactors p ūγ µ d n = ū p (p f ) [ g V (q 2 )γ µ + g T (q 2 ] ) iσ µν q ν u n (p i ) 2m N Dirac FF: g V (q 2 ) = F p 1 (q2 ) F1 n(q2 ) q2 0 1 Pauli FF: g T (q 2 ) = F p 2 (q2 ) F2 n(q2 ) q2 0 κ p κ n (5) [ g V (Q 2 ) = 1 r Q2 + O(Q 4 ), g T (Q 2 ) = g T (0) Proton radius: rp 2 r g T (0) 2mN 2. Dipole fit to determine the induced tensor charge g T = g T (0): g T (Q 2 ) = g T (0) ( 1 + r 2 2 Q 2 /12 ) 2. 1 r Q2 + O(Q 4 ) Gunnar Bali (Regensburg) Nucleon structure QCD 35 / 40 ]

36 Extrapolation of the Pauli formfactor at m π = 290 MeV Difference between magnetic moment anomalies g T (0) = κ p κ n. 4.0 Lm π 3.4 Lm π 4.2 Lm π 6.7 Lm π 3.4 Lm π 4.2 Lm π g T /g V g T Q 2 [GeV 2 ] Q 2 [GeV 2 ] Extrapolation error decreases with smaller Q 2 min = π2 /L 2. Therefore, invisible FSE for Lm π > 3.4 at m π = 290 MeV (L > 2.3 fm) do not necessarily imply they are irrelevant within the smaller statistical errors at m π = 150 MeV (L > 4.5 fm). Gunnar Bali (Regensburg) Nucleon structure QCD 36 / 40

37 Induced isovector tensor charge Extrapolating in the usual way... however, FSE are unquantifiable at the lightest mass point and O(a) improvement is not yet included a 0.08 fm a 0.07 fm a 0.06 fm Expt gt 3 gt m 2 π [GeV2 ] QCDSF Nf = 2 Mainz Nf = 2 ETMC Nf = 2 LHPC Nf = RBC Nf = ETMC Nf = PNDME Nf = RQCD Nf = 2 Expt m 2 π [GeV] QCDSF: , Mainz: , ETMC 2: , LHPC: , RBC: , ETMC 2+1+1: , PNDME: , RQCD: Gunnar Bali (Regensburg) Nucleon structure QCD 37 / 40

38 Isovector quark momentum fraction: x MS u d(2 GeV) [RQCD: GB et al, arxiv: ] N f = 2 x MS u d(2 GeV) a 0.08 fm a 0.07 fm a 0.06 fm CT10 ( 14) NNPDF ( 12) ABM ( 11) MSTW ( 09) m 2 π [GeV 2 ] Lm π 6.7: Lm π > 4.1: Lm π > 3.4: Lm π 2.8: Mild dependence on V, m π. Renormalised non-perturbatively. O(a) leading errors, a varied from 0.08 to 0.06 fm. Improvement on earlier calculations which suffered from excited state contamination x MS u d (2 GeV) Near physical point but more work needs to be done lattice spacing dependence? Gunnar Bali (Regensburg) Nucleon structure QCD 38 / 40

39 x MS u d(2 GeV): summary of recent lattice results 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 ] RQCD: GB et al, ; RBC/UKQCD: Y Aoki et al, arxiv: ; LHPC: J Green et al, arxiv: ; ETMC 2+1+1: C Alexandrou et al, arxiv: ; ETMC 2: C Alexandrou et al, arxiv: PDFs from S Alekhin et al, ; CT10: J Gao et al, ; NNPDF: R Ball et al ; A Martin et al [ETMC, arxiv: ]: disconnected contributions small predictions for x MS (2 GeV) soon. Mixing between quarks and glue! q Gunnar Bali (Regensburg) Nucleon structure QCD 39 / 40

40 Summary Lattice can contribute to many quantities that are hard to constrain by experiment, e.g., σ πn, f Ts, g S, g T. Lattice calculations are important to determine the spin content of the nucleon: q, Σ, J q, x q,.... In the past disconnected quark line diagrams were often omited and differences quoted: g A, x u d,..., but no s, Σ, J q, x q etc. Improved methods now allow for the calculation of these contributions. g A seems to approach the physical value, once Lm π > 4. x u d comes out 20% bigger than expected. lattice spacing effects? Renormalization? Precision physics requires an extrapolation a 0. For quite a few quantities however errors of 20% are acceptable. High Mellin moments almost impossible to compute recent interest also in quasi parton distribution functions. Gunnar Bali (Regensburg) Nucleon structure QCD 40 / 40