Lattice QCD and Hadron Structure

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1 Lattice QCD and Hadron Structure Huey-Wen Lin University of Washington 1

2 Human Exploration Matter has many layers of structure 10 2 m 10 9 m Materials Molecules m The scientific cycle Proton 2

3 Parton Distribution Functions Form Factors Generalized Parton Distributions spin 3

4 Quantum Chromodynamics The strong interactions of quarks and gluons (SU(3) gauge) Confinement no free quarks allowed Asymptotic freedom weak interactions at large energies Hadrons Pion Proton α s α(q) s (Q) The Nobel Prize in Physics 2004 Q (GeV) 4

5 Difficulties at Low Energy Strong interactions make analytic calculation impossible Direct QCD calculation is desired Lattice QCD

6 Outline The tool = Lattice Gauge Theory Topics in Hadron Structure Parton distribution functions Trademark of JLQCD Form factors Summary and Outlook 6

7 QCD 7

8 Lattice QCD x, y, z L t a 8

9 Lattice QCD quark field gluon field x, y, z L t a 9

10 Actions Guided by Symanzik Improvement (order in a) Gauge sector: O(a 2 )-improved Fermion sector: O(a)-improved $$ $$$ $$ $ 10

11 Actions Guided by Symanzik Improvement (order in a) Gauge sector: O(a 2 )-improved Fermion sector: O(a)-improved Complicated matrix elements, mixings Spectroscopy Form factors $$ $$ $$$ $ Smaller a and m π Larger V 11

12 Actions Guided by Symanzik Improvement (order in a) Gauge sector: O(a 2 )-improved Fermion sector: O(a)-improved Needs a lot more computational resources Mixings and renormalization $$ $$ $$$ $ Complex flavor mixing! 12

13 Actions Guided by Symanzik Improvement (order in a) Gauge sector: O(a 2 )-improved Fermion sector: O(a)-improved Needs a lot more computational resources Mixings and renormalization $$ $$ $$$ $ Complex flavor mixing! 13

14 The Dark Side... Currently, not running with the physical pion mass Lighter quark simulations require $$$ Example: BMW Collaboration, Science (2008) 14

15 Lattice in the News Post-dictions of well known quantities Example: BMW Collaboration, Science 2008 Proves all the systematics are under control 15

16 From Lattice 2009 Post-dictions of well known quantities Consistent results from various actions/groups BMW n f = 2+1 Clover, PACS-CS n f = 2+1 Clover, m π 190 MeV m π 160 MeV HSC n f = 2+1 anisoclover, m π 370 MeV ETMC n f = 2 Twisted Wilson,m π 300 MeV MILC n f = 2+1 Staggered, m π 240 MeV LHPC n f = 2+1 DWF/Stag., m π 300 MeV 16

17 Omega spectroscopy From Particle Data Book Prediction Lattice HSC,

18 Probing Insights into Hadrons 18

19 Probing Insights into Hadrons BNL JLab 19

20 Parton Distribution Function Deep inelastic scattering Probing nucleon structure The symmetric, unpolarized, spin-averaged The anti-symmetric, polarized 20

21 Moments of the Structure Function No light-cone operator directly calculated on the lattice Operator product expansion Polarized Unpolarized e 1, e 2, c 1, c 2 are Wilson coefficients x n q, x n q, d n are forward nucleon matrix elements 21

22 Green Functions Three-point function with connected piece only O: V μ = q γ μ q, A μ = q γ μ γ 5 q, or others J = Two topologies: Isovector quantities O u d disconnected diagram cancelled 22

23 Nucleon Structure Functions The first moment of the quark momentum fraction x q 2+1f : u/d + s 2f : u/d 0f : Ø HWL et al., Phys. Rev. D78, (2008) and RBC/UKQCD arxiv: [hep-lat]; K. Orginos et al., Phys.Rev.D73: (2005); LHPC, arxiv: [hep-lat]; D. Dolgov et al., Phys. Rev. D66, (2002); M. Guertler et al., PoS(LAT2006)107; D. Pleiter et al., PoS(LAT2006)120; 23

$Nucleon Structure Functions The first moment of the quark momentum fraction x q HWL et al., Phys. Rev. D78, 014505 (2008) and RBC/UKQCD arxiv:1003.3387[hep-lat]; K. Orginos et al.$

24 Nucleon Structure Functions The first moment of the quark momentum fraction x q HWL et al., Phys. Rev. D78, (2008) and RBC/UKQCD arxiv: [hep-lat]; K. Orginos et al., Phys.Rev.D73: (2005); LHPC, arxiv: [hep-lat] D. Dolgov et al., Phys. Rev. D66, (2002); M. Guertler et al., PoS(LAT2006)107; D. Pleiter et al., PoS(LAT2006)120; 24

25 Nucleon Structure Functions The first moment of the quark helicity distribution x Δq HWL et al., Phys. Rev. D78, (2008) and RBC/UKQCD arxiv: [hep-lat]; K. Orginos et al., Phys.Rev.D73: (2005); LHPC, arxiv: [hep-lat] D. Dolgov et al., Phys. Rev. D66, (2002); M. Guertler et al., PoS(LAT2006)107; D. Pleiter et al., PoS(LAT2006)120; 25

26 Nucleon Structure Functions The first moment of the quark helicity distribution x Δq HWL et al., Phys. Rev. D78, (2008) and RBC/UKQCD arxiv: [hep-lat]; K. Orginos et al., Phys.Rev.D73: (2005); LHPC, arxiv: [hep-lat] D. Dolgov et al., Phys. Rev. D66, (2002); M. Guertler et al., PoS(LAT2006)107; D. Pleiter et al., PoS(LAT2006)120; 26

27 Nucleon Structure Functions World data: the zeroth moment of the transversity 1 δq HWL et al., Phys. Rev. D78, (2008) and RBC/UKQCD arxiv: [hep-lat]; K. Orginos et al., Phys.Rev.D73: (2005); LHPC, arxiv: [hep-lat] D. Dolgov et al., Phys. Rev. D66, (2002); M. Guertler et al., PoS(LAT2006)107; D. Pleiter et al., PoS(LAT2006)120; 27

28 Nucleon Structure Functions World data: the zeroth moment of the transversity Trustworthy? 1 δq HWL et al., Phys. Rev. D78, (2008) and RBC/UKQCD arxiv: [hep-lat]; K. Orginos et al., Phys.Rev.D73: (2005); LHPC, arxiv: [hep-lat] D. Dolgov et al., Phys. Rev. D66, (2002); M. Guertler et al., PoS(LAT2006)107; D. Pleiter et al., PoS(LAT2006)120; 28

29 Nucleon Structure Functions Higher moments? Yes Isovector unpolarized moments LHPC (SCRI, SESAM): 2f, Wilson and clover QCDSF: 0f clover, multiple lattice spacings x 2 q D. Dolgov et al., Phys. Rev. D66, (2002); M. Gockeler et al. Phys. Rev. D71, (2005). x 3 q 29

30 Nucleon Structure Functions Higher moments? Yes... But at n 4: mixing with lower-dimension operators Getting around W. Detmold et. al. Phys.Rev.D73: (2006) Direct calculation K. Liu, Phys.Rev.D62: (2000) Gluon structure such as x g? x m π 650 MeV 2σ away from zero Disconnected? 5σ x u,d x 2 0 χqcd, Phys.Rev.D79: (2009) M. Deka et al. (2008),

31 Origin of Nucleon Spin Nucleon total spin is ½, but how does it add up? Quark contribution ΔΣ: spin L: orbital angular momentum m π 2 M π2 (GeV 2 ) 300 MeV LHPC, arxiv: [hep-lat]; M. Ohtani et al, PoS (Lat2007)

32 Origin of Nucleon Spin Nucleon total spin is ½, but how does it add up? Quark contribution ΔΣ: spin L: orbital angular momentum Disconnected contribution How much contribution from gluons? m π 2 M π2 (GeV 2 ) 32

33 33

34 Electromagnetic Form Factors Experimentally studied through elastic scattering process Two definitions Dirac and Pauli form factors F 1, F 2 Sachs form factors G E, G M 34

35 EM Form Factors: Experiment MAMI JLab/Hall C Pun05 Gay02 JLab/CLAS JLab/Hall A : Rosenbluth data (SLAC, JLab) JLab Hall A recoil pol. data new JLab Hall C recoil pol. exp t extension up to Q GeV 2 new JLab/Hall A double-pol. exp t extension to Q GeV 2 completed 35

36 Higher-Q 2 Form Factors Higher-Q 2 data will help us to understand hadrons and challenge QCD-based models New Proposal Challenge for lattice-qcd calculations Typical Q 2 range for nucleon form factors is < 3.0 GeV 2 Higher-Q 2 calculations suffer from poor noise-to-signal ratios 36

37 Conventional Calculation Higher-Q 2 calculations suffer from poor noise-to-signal ratios 37

38 Lattice High-Q 2 Form Factors Problem: traditional approach simplify to one-state problem 38

39 Lattice High-Q 2 Form Factors Problem: traditional approach simplify to one-state problem 600 MeV pion 39

40 Lattice High-Q 2 Form Factors Problem: traditional approach Solution: confront excited states directly and allow operators to couple to both ground and excited states 40

41 Form Factors The form factors are buried in the amplitudes n = n = 0 gives us nucleon Matrix Element and solve linear equations for form factors 41

42 Form Factors The form factors are buried in the amplitudes n = n = 0 gives us nucleon ME F 2 (0) and solve linear equations for form factors 42

43 Higher-Q 2 Nucleon Form Factors N f = 0 anisotropic lattices, M π 480, 720, 1080 MeV G E p G M p G E n G M n HWL et al., Phys. Rev. D78, (2008); HWL et al. [ ] (GeV 2 ) 43

44 Higher-Q 2 Nucleon Form Factors N f = 2+1 anisotropic lattices, M π 450, 580, 875 MeV G E p G M p G E n G M n HWL et al., Phys. Rev. D78, (2008); HWL et al. [ ] (GeV 2 ) 44

45 Higher-Q 2 Nucleon Form Factors N f = 2+1 anisotropic lattices, M π 450, 580, 875 MeV F 1 u d F 2 u d HWL et al., arxiv:

46 How High is High Enough? For example, how does high-q 2 affect charge density? G. A. Miller, arxiv: Magnetization density Red band uses lattice data 2.0 GeV 2 Blue band uses lattice data 4.0 GeV 2 HWL et al., arxiv:

47 b y (fm) Transverse Charge Density Proton Neutron Magnetization density HWL et al., arxiv: b x (fm) 47

48 Transverse Charge Density Proton Neutron Magnetization density b (fm) HWL et al., arxiv:

49 b y (fm) Transverse Magnetization Density Proton Neutron Magnetization density HWL et al., arxiv: b x (fm) 49

50 Transverse Magnetization Density Proton Neutron Magnetization density b (fm) HWL et al., arxiv:

51 Nucleon Axial Form Factors N f = 2+1 anisotropic lattices, M π 450, 580, 875 MeV Polarized Transverse Dist n Preliminary 51

52 Pion Form Factors N f = 2+1 anisotropic lattices, M π 875 MeV Preliminary 52

53 Miscellany Disconnected contribution O(10 2 ) for EM form factor Small for most of the form factors but could be significant for neutron electric form factor To get larger momentum, we use O(ap) 1 Rome was not built in a day Methodology for improving a traditional lattice calculation Possible future improvement Step-scaling through multiple lattice spacings and volumes Higher momentum transfer 53

54 Exciting era using LQCD for studying hadron structure Improvement Summary Huge leaps due to increasing computational resources world-wide and improved algorithms Universality Different lattice actions/groups with independent calculations provide consistency checks: so far so good Confidence Reproducing well measured experimental values gives us confidence for predicting quantities that haven t/couldn t be measured by experiment Variety There are many different aspects of hadron structure; only presented a few examples 54

55 55

56 Backup Slides 56

57 b y (fm) Proton Transverse Density We know the proton is composed of up and down quarks Magnetization density d u Charge density b x (fm) HWL et al., arxiv:

58 g P g P induced pseudoscalar coupling constant Ordinary muon capture μ + p ν μ + n Radiative muon capture μ + p γ + ν μ + n 58

59 g P g P induced pseudoscalar coupling constant g P = m μ G P (0.88 m μ2 )/2m N 59

60 Proton EM Form Factors: Exp t new MAMI/A1 data up to Q GeV 2 : Rosenbluth data (SLAC, JLab) Pun05 JLab Hall A Gay02 recoil pol. data new JLab Hall C recoil pol. exp t extension up to Q GeV 2 60

61 Neutron EM Form Factors: Exp t MAMI JLab/Hall C JLab/CLAS JLab/Hall A new MIT-Bates (BLAST) data for both p and n at low Q 2 new JLab/Hall A double-pol. exp t extension to Q GeV 2 completed 61

62 Polarized Transverse Distributions N f = 2+1 anisotropic lattices, M π 450, 580, 875 MeV 62

63 Nucleon Axial Radii Axial RMS r A u d G A r A u d M N r A u d m π 2 HWL et al., in preparation m π L 63

64 Difficulties at Low Energy Even just the vacuum of QCD is complicated Classical QCD 64

65 Lattice Inputs for LECs 65

66 Dynamical Anisotropic Lattices Baryon mass-ratio extrapolation with modified NLO HBXPT Dimensionless quantities to avoid lattice-spacingambiguities Mass ratios with finite-volume corrections m N /m Ξ m Δ /m Ω HWL et al., in preparation m Σ /m Ξ m Σ* /m Ω m Λ /m Ξ m Ξ* /m Ω l Ξ l Ω 66

67 67

68 Axial Charge Coupling Axial-vector current matrix element and axial charge coupling g A = G A u d (Q 2 =0) Well measured experimentally from neutron beta decay Finite-volume effect Phys.Rev.D68: (2003) 68

69 Axial Charge Coupling World data: statistical error-bars only 2+1f : u/d + s 2f : u/d 0f : Ø FVE HWL et al., Phys. Rev. D78, (2008) and Phys. Rev. Lett. 100: (2008); K. Orginos et al., Phys.Rev.D73: (2005); D. Dolgov et al., Phys. Rev. D66, (2002); M. Guertler et al., PoS(LAT2006)107; D. Pleiter et al., PoS(LAT2006)120; D. Renner et al., PoS(LAT2006)121 69

70 Axial Charge Coupling Finite-volume effects A+B m π2 +C f V (m π L) f V ~ e m πl Systematics: f V ~ (m π L) 3 f V ~ m π2 e m πl (m π L) 0.5 RBC/UKQCD, Phys. Rev. Lett. 100: (2008) 70

71 Axial Charge Coupling Comparison of lattice calculations HWL et al., Phys. Rev. D78, (2008) and Phys. Rev. Lett. 100: (2008); K. Orginos et al., Phys.Rev.D73: (2005); LHPC, arxiv: [hep-lat] D. Dolgov et al., Phys. Rev. D66, (2002); M. Guertler et al., PoS(LAT2006)107; D. Pleiter et al., PoS(LAT2006)120; 71

72 g πnn via 2 approaches Pion-pole domination πnn Coupling g πnn [(Q 2 +m π2 ) m N G P (Q 2 )/f π ] at Q 2 = m π 2 Goldberger-Treiman Relation g πnn m N G A (Q 2 =0) /f π 72

73 g ΞΞ and g ΣΣ Has applications such as hyperon scattering, non-leptonic decays, Cannot be determined by experiment Existing theoretical predictions: Chiral perturbation theory Large-N c M. J. Savage et al., Phys. Rev. D55, 5376 (1997); R. Flores-Mendieta et al., Phys. Rev. D58, (1998); Loose bounds on the values Lattice QCD can provide substantial improvement 73

74 g ΞΞ and g ΣΣ Pion mass: MeV HWL and K. Orginos, Phys.Rev.D79:034507,2009 First lattice calculation of these quantities; mixed-action full-qcd Combine with g A for study of SU(3) symmetry breaking SU(3) simultaneous fits among three coupling constants D, F, and other low-energy constants 74

75 Axial Charge Coupling Comparison of lattice calculations SU(3)-constrained fit gives Lin et al. smaller extrapolated statistical error than LHPC Lighter m π, finer a, multiple V essential for precise calculation HWL et al., Phys. Rev. D78, (2008) and Phys. Rev. Lett. 100: (2008); HWL et al., Phys.Rev.D79:034507,2009 K. Orginos et al., Phys.Rev.D73: (2005); LHPC, arxiv: [hep-lat] D. Dolgov et al., Phys. Rev. D66, (2002); M. Guertler et al., PoS(LAT2006)107; D. Pleiter et al., PoS(LAT2006)120; 75

76 πyy Coupling Goldberger-Treiman Relation g πnn m N G A (Q 2 =0) / f π 76

77 πy 1 Y 2 Coupling Goldberger-Treiman Relation g πnn m N G A (Q 2 =0) / f K What s the right relation for this????? 77

78 g ΞΣ and g ΣN Matrix element of the hyperon β-decay process with SU(3) breaking 78

79 g ΞΣ and g ΣN Matrix element of the hyperon β-decay process with 79

80 Hyperon-Decay Experiments Experiments: CERN WA2, Fermilab E715, BNL AGS, Fermilab KTeV, CERN NA48 Summary N. Cabibbo et al In experiment, only measure linear combination of g 1 /f 1 and g 2 /f 1 usually assume g 2 0 Systematic from ignoring g 2? Better g 1 /f 1 from lattice calculations? 80

81 Lattice Studies Two quenched calculations, different channels Pion mass > 700 MeV g 1 (0) / f 1 (0) = 0.287(52) stat. Exp t: 0.340(17) g 2 (0) / f 1 (0) = 0.63(26) stat. Guadagnoli et al., Nucl.Phys.B761:63-91 (2007) Pion mass MeV g 1 (0) / f 1 (0) = 1.248(29) stat. Exp t: 1.31( ) g 2 (0) / g 1 (0) = 0.68(18) stat. Sasaki et al., [hep-ph] No systematic error estimate from quenching effects! 81

82 First Dynamical Study N f = 2+1 mixed action, M π MeV Report on Σ n results for now Naïve chiral extrapolation We got g 1 (0) / f 1 (0) = 0.348(37) stat g 2 (0) / f 1 (0) = +0.29(20) stat 82

83 Topics in Nuclear Physics What happens when we put hadrons together? 83

84 Summary Nuclear Physics is not essential for our existence but also for universe we live in Many new experiments are either under construction or in plan Lattice QCD is a hard probe for the theory (QCD) Helpful in the cases when experimental limitations or challenges Isolated neutrons or pion source (in medium) Unstable particles due to weak decay (hyperons) Predictions of new phenomena to guide experiment There is no free lunch The difficulties = The opportunities And some of them have been concurred, as demonstrated in this talk Filter for New Physics 84

85 Summary Nuclear Physics is essential for our existence but also for universe we live in Investment in experiment continues (12-GeV, FRIB, RHIC or EIC) Lattice QCD is an excellent probe of the theory (QCD) Helpful in the cases when experiment is limited Isolated neutrons or pion source (in medium) Unstable particles due to weak decay (hyperons) Predictions of new phenomena to guide experiment There is no free lunch The difficulties = opportunities And some of them have been conquered, as demonstrated in this talk Filter for New Physics 85

Hadron Structure from Lattice QCD

Hadron Structure from Lattice QCD Huey-Wen Lin University of Washington 1 Outline Lattice QCD Overview Nucleon Structure PDF, form factors, GPDs Hyperons Axial coupling constants, charge radii... Summary