Hadron Structure from Lattice QCD

Hadron Structure from Lattice QCD
Πανεπιστήμιο Κύπρου Κ. Αλεξάνδρου Hadron Structure from Lattice QCD Status and Outlook

1. γN Δ matrix element on the lattice Quenched results
Outline Lattice QCD 1. Introduction 2. Some recent results Hadron Deformation: 1. γN Δ matrix element on the lattice Quenched results Full QCD results with DWF 2. Charge density distribution Diquark dynamics Pentaquarks Status and perspectives for dynamical simulations Conclusions

Introduction Quantum ChromoDynamics is the fundamental theory of strong interactions. It is formulated in terms of quarks and gluons with only parameters the coupling constant and the masses of the quarks. Self energy diverges  LQCD + regularization renormalization define a physical theory Produces the rich and complex structures of strongly interacting matter in our Universe

Fundamental properties of QCD
g QCD QED e γ Self interactive gluons unlike photons Interaction increases with distance  Confinement Interaction decreases at large energies  asympotic freedom Coupling constant αs >> αEM

Solving QCD At large energies, where the coupling constant is small, perturbation theory is applicable  has been successful in describing high energy processes At very low energies chiral perturbation theory becomes applicable At energies ~ 1 GeV the coupling constant is of order unity  need a non-perturbative approach Present analytical techniques inadequate  Numerical evaluation of path integrals on a space-time lattice  Lattice QCD – a well suited non-perturbative method that uses directly the QCD Langragian and therefore no new parameters enter pQCD ChPT E L a t t i c e Q C D

Lattice QCD Finite lattice spacing
Lattice QCD is a discretised version of the QCD Lagrangian  finite lattice spacing a must take a0 to recover continuum physics a many ways to discretise the theory all of which give the same classical continuum Lagrangian. Physics results independent of discretization All lattice quantities are measured in units of a  dimensionless and therefore the scale is set by using one known physical observable

Finite lattice spacing a
What do we accomplish by discretising space-time? Discrete space-time defines a non-perturbative regularization scheme: Finite a provides an utraviolet cut off π/a  In principle we can apply all the methods that we know in perturbation theory using lattice regularization. In practice these calculations are more difficult than perturbative calculations in the continuum and therefore are not practical It can be solved numerically on the computer using similar methods to those used in Statistical Mechanics  Finite volume and Wick rotation into Euclidean time must take the spatial volume to infinity

Lattice QCD in short a U1 U2 U3+ U4+ UP=U1 U2 U3+ U4+ ψ(n) μ
Uμ(n)=e igaAμ(n) S=Sg+Sw gauge invariant quantities smallest possible P

Observables in LQCD Integrating out the Grassmann variables is possible since Sample with M.C. Generate a sample of N independent gauge fields U Calculate propagator D-1 for each U  Easy to simulate since Sg[U] is local: use M. C. methods from Statistical Mechanics like the Metropolis algorithm Quenched approximation: set det D=1 omit pair creation For a lattice of size 323 x 64 we have 108 gluonic variables

Precision results in the quenched approximation
The quenched light quark spectrum from CP-PACS, Aoki et al., PRD 67 (2003) Lattice spacing a  0 Chiral extrapolation Infinite volume limit

Computational aspects
Lattice spacing a small enough to have continuum physics Quark mass small to reproduce the physical pion mass Lattice size large: Fermion determinant – Full QCD L mπ-1 L (fm) mπ (MeV) most quenched calculations reaching ~300 MeV pions Computational cost ~ (mq)-4.5 ~ (mπ)-9 Currently we are starting to do full QCD by including the fermionic determinant but we are still limited to rather heavy pions  understand dependence on quark mass Physical results require terascale machines

Fermion Discretizations
Dynamical Wilson: 1. conceptually well-founded many years of experience at least so we thought until Lattice 2005 but expensive M. Lüscher Lattice 2005: Schwartz-preconditioned HMC Within errors and with mπ=500 ΜeV the data are compatible with 1-loop ChPT Fπ =80(7) MeV New algorithm: Numerical simulations with Wilson fermions much less expensive than previously thought e.g. a lattice of size 483x96 with a=0.06fm and mπ=270 ΜeV can be simulated on a PC cluster with 288 nodes

Chiral fermions New fermion formulation:
Ginsparg-Wilson relation: D-1γ5 + γ5D-1=aγ5 where D is the fermion matrix There are two equivalent approaches in constructing a D that obeys the Ginsparg-Wilson relation PRD 25 (1982) 1. Domain wall fermions: Define the Wilson fermion matrix in five dimensions Left-handed fermion right-handed fermion Kaplan, PLB 288 (1992) L5 Narayanan and Neubeger PLB 302 (1993) 2. Overlap fermions Infinite number of flavors in four dimensions In the limit the domain wall and overlap Dirac operators are the same Review: Chandrasekharan &Wiese, hep-lat/ Expensive to simulate  next stage full QCD calculations

Dynamical Kogut-Susskind (staggered) fermions
Advantages fastest to simulate Have a residual chiral symmetry giving zero pions at zero quark mass MILC collaboration simulates full QCD using improved staggered fermions. The dynamical 2+1 flavor MILC configurations are the most realistic simulations of the physical vacuum to date. Just download them! Disadvantages More fermions than in the real theory – extra flavors complicate hadron matrix elements Approximation – fourth root of the fermion determinant Hybrid calculation A practical option: use different sea and valence quarks A number of physical observables are calculated using MILC configurations and different valence quarks e.g. LHP and HPQCD collaborations to check consistency staggered DWF

Precision results in Full QCD
“Gold plated” observables (hadron with negligible widths or at least below 100 MeV decay threshold and matrix elements that involve only one hadron in the initial and final state Simulations using staggered fermions, a~0.13 and 0.09 fm C. Davies et al. PRL 92 (2004)

Recent progress Hadronic decays:
In a box of finite spatial size L the two-body energy spectrum is discrete : For two non-interacting hadrons h1 and h2 the energy is given by In the interacting case there is a a small L-dependent energy shift due to the interaction. From these energy shifts close to the resonance energy one can determine the phase shift and from those the resonance mass and width (Lüscher 1991). However one needs very accurate data. Recently the π+π+ scattering length was calculated by measuring the energy shifts. Domain wall fermions with smallest pion mass ~300 MeV were used and the result obtained in the chiral limit is in agreement with experiment (R.S. Beane et al. hep-lat/ ) C. Michael, Lattice 2005: ρ 2π using dynamical Wilson fermions Evaluate the correlation matrix of a rho meson of energy m-Δ/2 and a ππ state with energy m+ Δ/2 Use linear t-dependence to extract transition amplitude x=<ρ|ππ> Pion form factor at large Q2 within the hybrid approach G. Flemming, Lattice 2005

Ohta and Orginos hep/lat0411008 Still too high <x>exp~0.15
Nucleon structure Quenched DWF Nf=2 unquenched DWF Ohta and Orginos hep/lat Still too high <x>exp~0.15 From W. Schroers Generalized form factors (LHPC) ξ=longitudinal momentum fraction, x=1 q=delta(b_per)  slope 0  x increases slope decreases g_a isovector axial charge M. Burkardt, PRD62 (2000) In the forward limit: Ph. Hägler et al. PRL 2004

Hadron Deformation

Hadron Deformation Spin defines an orientation  hadron can be deformed with respect to its spin axis Deformation of the Δ by evaluating Q20 quadrupole operator Extract the wave function squared from current-current correlators Prolate: Q20 > 0 Sphere: Q20=0 Oblate: Q20 < 0 For spin 1/2 : Quadrupole moment not observable observable intrinsic (with respect to the body fixed frame) Make direct connection to experiment: γNΔ(1232)

γN Δ Spin-parity selection rules allow a magnetic dipole, M1, an electric quadrupole, E2, and a Coulomb quadrupole, C2, amplitude. Three transition form factors, GM1, GE2 and GC2  non-zero E2 and 2C multipoles signal deformation Experimentally the measured quantities are given usually in terms of the ratios E2/M1 or C2/M1 known as EMR (or REM) and CMR (or RSM): and

Deformed nucleon? Bates 0.0 0.2 0.4 0.6 0.8 1.0 -5 -10 -15 -20 CMR %
-5 -10 -15 -20 CMR % Q2 (GeV/c)2 MAID DMT 1 -1 -2 -3 -4 -5 EMR % Q2 (GeV/c)2 C.N. Papanicolas, “Nucleon Deformation” Eur. Phys. J. Α18, 141 (2003) OOPS collaboration Deformed Spherical

γN Δ matrix element γμ Δ(p’) N(p) M1 E2 C2 Sach´s form factors
At the lowest value of q2 the form factors were evaluated in both the quenched and the unquenched theory but with rather heavy sea quark masses. The main conclusion of this comparison is that quenched and unquenched results are within statistical errors. C.A., Ph. de Forcrand, Th. Lippert, H. Neff, J. W. Negele, K. Schilling. W. Schroers and A. Tsapalis, PRD 69 (2004) Quenched results are done on a lattice of spatial volume 323 at β=6.0 (~3.2 fm)3 using smaller quark masses (smallest gives mπ/mρ = 1/2) up to q2 of 1.5 GeV2 C. A., Ph. de Forcrand, H. Neff, J. W. Negele, W. Schroers and A. Tsapalis, PRL 94 (2005) MILC configurations and DWF C. A., R. Edwards, G. Koutsou, Th. Leontiou, H. Neff, J.W. Negele, W. Schroers and A. Tsapalis, Lattice 2005

Three-point functions
Matrix elements like <Δ| jμ |N>: quark line with photon needs to be evaluated  sequential propagator q =p’-p D-1(x2 ;0) all to all propagator D-1(x2;x1) D-1(x1;0) Projects momentum p’ to Δ  sum first: fixed sink technique Projects to momentum transfer q for the photon sum first: fixed current technique Since we worked on a finite lattice the momentum takes discrete values 2πk/L, k=1,…,N Use fixed sink technique  obtained all q2 values at once

γN Δ matrix element on the Lattice
Trial initial state with quantum numbers of the nucleon Trial final state with quantum numbers of the Delta x Normalize with time dependence cancels in the ratio x 1/2 Δ Rμσ ~ t2 2t1 N t1 R t1 Fit in the plateau region to extract M1, E2, C2

Technicalities The form factors are extracted from the ratio where σ is the spin index of the Δ field and the projection matrices Γ are given by We consider kinematics where the Δ is produced at rest i.e. What combination of Δ index σ , current direction μ and projection matrices Γ is optimal? Using the fixed sink technique we require an inversion for each combination of σ index and projection matrix Γi whereas we obtain the ratio Rσ for all current directions μ and momentum transfers q=p’-p without extra inversions.

Optimal choice of matrix elements Πσ
The form factors, GM1 , GE2 and GC2, can be extracted by appropriate choices of the current direction, Δ spin index and projection matrices Γj For example the magnetic dipole can be extracted from This means there are six statistically different matrix elements each requiring an inversion. e.g. If σ=3 and μ=1 then only momentum transfers in the 2-direction contribute. kinematical factor  all lattice momentum vectors in all three directions, resulting in a given Q2, contribute This combination requires only one inversion if the appropriate sum of Δ interpolating fields is used as a sink. However if we take the more symmetric combination: Smallest lattice momentum vectors: 2π/L {(1,0,0) (0,1,0) (0,0,1)} contributing to the same Q2

Linear extrapolation Lowest value of Q2
We perform a linear extrapolation in mπ2. Chiral logs are expected to be suppressed due to the finite momentum carried by the nucleon . Systematic error due to the chiral extrapolation?

Overconstrained analysis
Results are obtained in the quenched theory for β=6.0 (lattice spacing a~0.1 fm) on a lattice of 323x64 using 200 configurations at κ=0.1554, and (mπ/mρ=0.64, 0.58, 0.50) M1 Linear extrapolation in mπ2 to obtain results at the chiral limit Fits to Proton electric form factor

Results: EMR Linear extrapolation in mπ2 to obtain results at the chiral limit pQCD: ? Sato and Lee model with dressed vertices Sato and Lee model with bare vertices

Results: CMR Linear extrapolation in mπ2 to obtain results at the chiral limit pQCD: Discrepancy at low Q2 where pion cloud contributions are expected to be important

Chiral extrapolation NLO chiral extrapolation on the ratios using mπ/Μ~δ2 , Δ/Μ~δ. GM1 itself not given. V. Pascalutsa and M. Vanderhaeghen, hep-ph/

Domain wall valence quarks using MILC configurations
Fix the size of the 5th dimension: we take the same as determined by the LHP collaboration Tune the bare valence quark mass so that the resulting pion mass is equal to that obtained with valence and sea quarks the same. The spatial lattice size is 203 (2.5 fm)3 as compared to 323 (3.2 fm)3 used for our quenched calculation. Larger fluctuations We use ψγμ ψ for the current which is not conserved for finite lattice spacing  renormalization constant ZV which can be calculated Left-handed fermion right-handed fermion L5

First results in full QCD
Use MILC dynamical configurations and domain wall fermions 125 confs 75 confs 38 confs (283 lattice) ( ) masses: nucleon = 0.789(14) , Delta =0.971(27), rho = 0.637(13), pion=0.373(3)in lattice units ( ) masses nucleon =0.895(13), Delta =1.051(17), rho = 0.582(19), pion=0.306(3) in lattice units EMR and CMR too noisy so far to draw any conclusions The lightest pion mass that we would like to consider is ~250 MeV. This would require large statistics but it will be close enough to the chiral limit to begin to probe effects due to the pion cloud

Two- and three- density correlators
They provide information on the spatial distribution of quarks in a hadron Δ x γ0 j0 (x) = : u(x) γ0 u(x) : γ0 Δ x γ0 non-relativistic limit  |wave function|2 Deformation can be studied rho is elongated along its spin axis  prolate Δ+ is flatten along its spin axis  oblate sphere Deformation in full QCD: a feasibility study mπ=760 MeV C.A., Ph. de Forcrand and A. Tsapalis Can we improve this calculation? smaller quark mass, larger lattice, chiral fermions, momentum projection

Momentum projection Requires the all to all propagator
We use stochastic techniques to calculate it. We found that the best method is to use the stochastic propagator for one segment and sum over the spatial coordinates of the sink by computing the sequential propagator. t=Τ/2 t=0 j0(x) j0(x+y) z j0 (x) = : u(x) γ0 u(x) : G(z;x+y) quark propagator G(x+y;0) ρ meson distribution is broader C. A., P. Dimopoulos, G. Koutsou and H. Neff, Lattice 2005

Rho meson asymmetry quenched
Opens up the way of evaluating other physically interesting quantities like polarizabilites. unquenched

Diquark Dynamics

Diquark dynamics Originally proposed by Jaffe in 1977: Attraction between two quarks can produce diquarks: qq in 3 flavor, 3 color and spin singlet  behave like a bosonic antiquark in color and flavor D :scalar diquark s and  D q A diquark and an anti-diquark mutually attract making a meson of diquarks tetraquarks A nonet with JPC=0++  if diquarks dominate no exotics in q2q2 pentaquarks Exotic baryons? Soliton model Diakonov, Petrov and Polyakov in 1997 predicted narrow Θ+(1530) in antidecuplet

Can we study diquarks on the Lattice?
Define color antitriplet diquarks in the presence of an infinitely heavy spectator: Static quark propagator Baryon with an infinitely heavy quark Flavor symmetric  spin one Flavor antisymmetric  spin zero t t=0 light quark propagator G(x;0) R. Jaffe hep-ph/ JP color flavor diquark structure 0+ 1+ 6 qTCγ5q, qTCγ5γ0q qTCγiq, qTCσ0i q Models suggest that scalar diquark is lighter than the vector attraction: MA MS>MA MS In the quark model, one gluon exchange gives rise to color spin interacion: MS -MA ~ 2/3 (MΔ-MN)= 200 MeV and

Mass difference between ``bad`` and ``good`` diquarks
C.A., Ph. De Forcrand and B. Lucini Lattice 2005 First results using 200 quenched configurations at β=5.8 (a~0.15 fm) β=6.0 (a~0.10 fm) fix mπ~800 MeV (κ= at β=5.8 and κ=0.153 at β=6.0) heavier mass mπ ~1 GeV to see decrease in mass (κ=0.153 at β=5.8) β=6.0 κ=0.153 ΔM (GeV) β κ ΔΜ (MeV) 5.8 6.0 0.153 0.1575 70 (12) 109 (13) 143 (10) K. Orginos Lattice 2005: unquenched results with lighter light quarks

Diquark distribution Two correlators : provide information on the spatial distribution of quarks inside the heavy-light baryon j0(x) j0(y) quark propagator G(x;0) j0 (x) = : u(x) γ0 u(x) : u d θ Study the distribution of d-quark around u-quark. If there is attraction the distribution will peak at θ=0

Diquark distribution ``Good´´ diquark peaks at θ=0

Linear confining potential
A tube of chromoelectric flux forms between a quark and an antiquark. The potential between the quarks is linear and therefore the force between them constant. linear potential G. Bali, K. Schilling, C. Schlichter, 1995

Static potential for tetraquarks and pentaquarks
Main conclusion: When the distances are such that diquark formation is favored the static potentials become proportional to the minimal length flux tube joining the quarks signaling formation of a genuine multiquark state C. Α. and G. Koutsou, PRD 71 (2005)

Pentaquarks

The Storyteller, like a cat slipping in and out of the shadows
The Storyteller, like a cat slipping in and out of the shadows. Slipping in and out of reality? Θ+

First evidence for a pentaquark
SPring-8 : γ 12C  Κ+ Κ- n CLAS at Jlab: γD K+ K- pn High statistics confirmed the peak

Summary of experimental results
Negative results Positive results Experiment Reaction CDF p pPX ALEPH Hadronic Z decays L3 γγΘΘ HERA-Β pA  PX Belle KN  PX BaBar e+ e- Y Bes e+ e- J/ψ HyperCP (K+,π+,p)CuPX SELEX (p,Σ,π)p  PX FOCUS γp  PX E690 pp  PX DELPHI COMPASS μ+(6Li D) PX ZEUS ep  PX SPHINX pC ΘK0C PHENIX AuAuPX Experiment Reaction Mass (MeV) Width (MeV) LEPS γ C12K- K+ n 1540(10) <25 DIANA K+ Xe KS0 pXe’ 1539(2) <9 CLAS γ d  K- K+ np γ p  K- K+ nπ+ 1542(5) <21 SAPHIR γ pKS0 K+ n 1540(6) COSY ppΣ+ KS0 p 1530(5) <18 SVD pA KS0 pX 1526(3) <24 ITEP νAKS0pX 1533(5) <20 HERMES e+ d KS0pX 1528(3) 13(9) ZEUS e pKs0 p X 1522(3) 8(4) A. Dzierba et al., hep-ex/ P=pentaquark state (Θs,Ξ,Θc)

Properties World average
Parity: Chiral soliton model predicts positive Naïve quark model negative Isospin: probably I=0 as predicted by soliton model Width: Why so narrow? Other multiplets (Ξ ?) M. Karliner, Durham, 2004

Does lattice QCD support a Θ+?
Objective for lattice calculations: to determine whether quenched QCD supports a five quark resonance state and if it does to predict its parity.

Initial state |φtrial> with the quantum numbers of Θ+ at time t=0
Pentaquark mass Initial state |φtrial> with the quantum numbers of Θ+ at time t=0 s* u d Θ at a later time t>0 s* u d Time evolution If one state dominates then t meff Correlator: C(t) ~ w exp(-m t) Effective mass: mΘ-mKN~100 MeV Fit in plateau region C(t) ~ w0exp(-mKN t)+w1 exp(-mΘ t) +…

Interpolating fields for pentaquarks
What is a good |φtrial> for Θ+? All lattice groups have used one or some combinations of the following isoscalar interpolating fields: Jaffe and Wilczek PRL 91 (2003) Motivated by the diquark structure: L=0 L=1 u d s Motivated by KN strucutre: N K Modified NK Both local and smeared quark fields were considered : Results should be independent of the interpolating field if it has reasonable overlap with our state

Scattering states The two lowest KN scattering states with non-zero momentum n=1 n=2 E (GeV) Spacing between scattering states~1/ Ls2 Θ+ Contributes only in negative parity channel S-wave KN Correlator: If mixing is small w0~L-3  suppressed for large L Dominates if w1>>w0 and (mΘ-mKN) t <  t<10 GeV-1 assuming energy gap~100MeV or t/a<20

Study the two-pion system in I=2
Correlation matrix total momentum=0 2mπ 2mρ Ε12π C.A. and A. Tsapalis, Lattice 2005 Besides we also project to a state in which each of the two particles carries opposite momentum by evaluating the correlation matrix Besides we also project to a state in which each of the two particles carries opposite momentum by evaluating the correlation matrix Besides we also project to a state in which each of the two particles carries opposite momentum by evaluating the correlation matrix Besides we also project to a state in which each of the two particles carries opposite momentum by evaluating the correlation matrix Besides we also project to a state in which each of the two particles carries opposite momentum by evaluating the correlation matrix Besides we also project to a state in which each of the two particles carries opposite momentum by evaluating the correlation matrix E1ππ E1ππ

Project on good relative momentum p: π(p)π(-p)
Check taking p=0 on small lattice (163x32)

Scaling of spectral weights
Consider the ratio Fit the correlators on different volumes to extract the spectral weights and check or scaling. Fix upper fit range and look at the ratio of spectral weights WL1/WL2 and vary the lower fit range ti Mathur et al. PRD 70 (2004) Lowest eigenstate Use largest possible range Need accurate results same results even if 2x2 matrix with Jπ and Jρ Upper fit range 56 Upper fit range 26 E1ππ

Pentaquark Perform a similar analysis as in the two-pion system using Jdiqaurk and JKN Negative parity mK+mN mK+mN

Spectral weights for pentaquark
Different from two pion system  can not exclude a resonance Ratio WL1/WL2 ~1 for ti/a up to 26 which is the range available on the small lattices Negative parity

Identifying the Θ+ on the Lattice
There is agreement among lattice groups on the raw data but the interpretation differs depending on the criterion used Negative parity Alexandrou & Tsapalis (2.9 fm) Lasscock et al. (2.6 fm) Mathur et al. (2.4 fm) Ishii et al. (2.15 fm) Mathur et al. (3.2 fm) Csikor et al. (1.9 fm) Sasaki (2.2 fm) From Lassock et al. hep-lat/

How do we identify a resonance?
Method used: Identify the two lowest states and check for volume dependence of their mass Extract the spectral weights and check their scaling with the spatial volume Change from periodic to antiperiodic boundary condition in the spatial directions and check if the mass in the negative parity channel changes Check whether the binding increases with the quark mass

Summary of lattice computations
Group Method of analysis/criterion Conclusion Alexandrou and Tsapalis Correlation matrix, Scaling of weights Can not exclude a resonance state. Mass difference seen in positive channel of right order but mass too large Chiu et al. Correlation matrix Evidence for resonance in the positive parity channel Csikor et al. Correlation matrix, scaling of energies First paper supported a pentaquark , second paper with different interpolating fields produces a negative result Holland and Juge Negative result Ishii et al. Hybrid boundary conditions Negative result in the negative parity channel Lasscosk et al. Binding energy Mathur et al. Scaling of weights Sasaki Double plateau Evidence for a resonance state in the negative parity channel. Takahashi et al. Correlation matrix, scaling of weights J. Negele, Lattice 2005 Evidence for a resonance state?

Dynamical calculations-status and perspectives

Dynamical calculations-status and perspectives
A. Kennedy, Lattice 2004 No continuum limit Two lattice spacings or fewer Autocorrelations ignored Depend on the physics anyhow Naïve volume scaling V for R algorithm V5/4 for HMC Dynamical quarks Two or three flavours

Three choices: Commercial supercomputers : general purpose, rather expensive PC Clusters: general purpose, cheaper than commercial but scalability a problem custom-design machines (APEnext, QCDOC): optimized for QCD Commercial supercomputers SGI Altrix (LRZ Munich) 16.4 TFlop/s peak 34.6 TFlop/s peak BlueGene/L very good scalability 11.2 TFlop/s peak at Jülich 5.6 TFlop/s peak at Edinburgh, BU, MIT T. Wettig, Lattice 2005 PC clusters

Custom-design machines
APEnext QCDOC Planned Installations ( 1 rack=512 nodes =0.66 TFlop/s peak) Planned Installations ( 1 rack=1024 nodes =0.83 TFlop/s peak) UKQCD nodes DOE nodes RIKEN-BNL nodes Columbia nodes Regensburg nodes INFN nodes (12 racks) Bielefeld nodes (6 racks) DESY nodes (3 racks) Orsay nodes (1 rack) Cost: $0.6 per peak MFlop/s Cost: $0.45 per peak MFlop/s Tens of Teraflop/s will be available for lattice QCD

Conclusions Lattice QCD is entering an era where it can make significant contributions in the interpretation of current experimental results. A valuable method for understanding hadronic phenomena γN Δ : GM1, GE2 and GC2 calculated in quenched approximation in the range of Q2 where recent measurements are taken. Results in full QCD are within reach. Non-zero values  deformation of the nucleon/Δ Diquark dynamics Studies of exotics and two-body decays Computer technology and new algorithms will deliver 10´s of Teraflop/s in the next five years Provide dynamical gauge configurations in the chiral regime Enable the accurate evaluation of more involved matrix elements

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Hadron Structure from Lattice QCD

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