Κ. Αλεξάνδρου

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

2. 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

3. Self energy diverges  LQCD + regularization renormalization define a physical theory 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. Produces the rich and complex structures of strongly interacting matter in our Universe

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

5. L a t t i c e Q C D ChPT pQCD E 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

6. a must take a0 to recover continuum physics Lattice QCD Finite lattice spacing Lattice QCD is a discretised version of the QCD Lagrangian  finite lattice spacing 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

7. 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

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

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

10. 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

11. mπ-1 L L (fm) mπ(MeV) 1.6 560 2.5 360 4.5 200 6.4 140 most quenched calculations reaching ~300 MeV pions 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 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

12. at least so we thought until Lattice 2005 M. Lüscher Lattice 2005: Schwartz-preconditioned HMC 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 • Within errors and with mπ=500 ΜeV the data are compatible with 1-loop ChPT Fπ =80(7) MeV Fermion Discretizations • Dynamical Wilson: 1. conceptually well-founded 2. many years of experience but expensive

13. L5 Kaplan, PLB 288 (1992) right-handed fermion Left-handed fermion 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 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/0405024 Expensive to simulate  next stage full QCD calculations

14. 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 staggered DWF A number of physical observables are calculated using MILC configurations and different valence quarks e.g. LHP and HPQCD collaborations to check consistency 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

15. 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)

16. 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/0506013) 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 thehybrid approach G. Flemming, Lattice 2005

17. Quenched DWF Still too high <x>exp~0.15 Nf=2 unquenched DWF Ohta and Orginos hep/lat0411008 Nucleon structure From W. Schroers Generalized form factors (LHPC) M. Burkardt, PRD62 (2000) In the forward limit: Ph. Hägler et al. PRL 2004

18. Hadron Deformation

19. 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 Oblate: Q20< 0 Prolate: Sphere: Q20 > 0 Q20=0 • For spin 1/2 : Quadrupole moment not observable intrinsic (with respect to the body fixed frame) observable Make direct connection to experiment: γNΔ(1232) Hadron Deformation

20. 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,GE2and 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

21. 0 1 0 -1 -2 MAID -5 EMR % Bates -10 CMR % -3 -4 -5 -15 DMT -20 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Q2 (GeV/c)2 Q2 (GeV/c)2 OOPS collaboration Deformed Spherical C.N. Papanicolas, “Nucleon Deformation” Eur. Phys. J. Α18, 141 (2003) Deformed nucleon?

22. Δ(p’) N(p) C2 E2 M1 γN Δ matrix element γμ Sach´s form factors • At the lowest value of q2 the form factors were evaluated in boththe quenched and the unquenched theorybut 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

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

24. t1 t2 0 Rμσ ~ 1/2 2t1 t2 0 N 0 N Δ Δ 2t1 0 Δ Δ x x time dependence cancels in the ratio 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 Normalize with R t1 Fit in the plateau region to extract M1, E2, C2

25. 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 ofσ indexand projection matrix Γi whereas we obtain the ratio Rσ for all current directions μ and momentum transfersq=p’-p without extra inversions. 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

26. This means there are six statistically different matrix elements each requiring an inversion. kinematical factor e.g. If σ=3and μ=1 then only momentum transfers in the 2-direction contribute. However if we take the more symmetric combination:  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. Smallest lattice momentum vectors: 2π/L {(1,0,0) (0,1,0) (0,0,1)} contributing to the same Q2 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

27. Lowest value of Q2 Linear extrapolation 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?

28. Proton electric form factor Fits to 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, 0.1558and 0.1562 (mπ/mρ=0.64, 0.58, 0.50) M1 Linear extrapolation in mπ2to obtain results at the chiral limit

29. Sato and Lee model with bare vertices Sato and Lee model with dressed vertices Results: EMR Linear extrapolation in mπ2to obtain results at the chiral limit pQCD: ?

30. Results: CMR Linear extrapolation in mπ2to obtain results at the chiral limit pQCD: Discrepancy at low Q2 where pion cloud contributions are expected to be important

31. 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/0508060

32. We use ψγμ ψ for the current which is not conserved for finite lattice spacing  renormalization constant ZV which can be calculated L5 right-handed fermion Left-handed fermion 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

33. 125 confs 75 confs 38 confs (283 lattice) 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 First results in full QCD Use MILC dynamical configurations and domain wall fermions

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

35. j0(x) z j0 (x) = : u(x) γ0 u(x) : t=0 t=Τ/2 j0(x+y) 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. 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

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

37. Diquark Dynamics

38. Originally proposed by Jaffe in 1977: Attraction between two quarks can produce diquarks: qqin 3 flavor, 3 color and spin singlet  behave like a bosonic antiquark in color and flavorD :scalar diquark s and D 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 Exotic baryons? pentaquarks q q Diquark dynamics Soliton model Diakonov, Petrov and Polyakov in 1997 predicted narrow Θ+(1530) in antidecuplet

39. Can we study diquarks on the Lattice? Static quark propagator Baryon with an infinitely heavy quark Flavor symmetric  spin one Define color antitriplet diquarks in the presence of an infinitely heavy spectator: Flavor antisymmetric  spin zero t t=0 light quark propagator G(x;0) R. Jaffe hep-ph/0409065 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

40. 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=0.1575at β=5.8 and κ=0.153at β=6.0) • heavier mass mπ ~1 GeV to see decrease in mass(κ=0.153at β=5.8) β=6.0 κ=0.153 ΔM (GeV) K. Orginos Lattice 2005: unquenched results with lighter light quarks

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

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

43. 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

44. q q q q q q q q q 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. andG. Koutsou, PRD 71 (2005)

45. Pentaquarks

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

47. First evidence for a pentaquark SPring-8 : γ12C Κ+ Κ- n CLAS at Jlab: γD K+ K- pn High statistics confirmed the peak

48. Summary of experimental results Negative results Positive results A. Dzierba et al., hep-ex/0412077 P=pentaquark state (Θs,Ξ,Θc)

49. 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

50. 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.