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2. Disconnected Diagrams and Lattice QCD H What is Lattice QCD? HTaming disconnected diagrams Collaborators: Victor Guerrero and Ron Morgan

3. Quantum Chromodynamics (QCD) --- the fundamental theory of the strong interaction (quarks and gluons) 2 "Actions" - gluon and quark. Does the "path integral" automatically via Monte Carlo simulation. Degrees of freedom are the points in space, colors, spin and particle/anti-particle. Lattice scale "a" set by renormalization group behavior. Quark propagators are matrix inverses of the "mass matrix". Aspects of lattice QCD

4. Lattice Quantum Chromodynamics (QCD) Variables

5. Example: quark-antiquark potential

6. Example :  s

7. Example: hadron masses (Durr et al)

9. Ugly gluon lines quark loop creation destruction H The HARD problem:

10. Krylov subspace: Starting, residual vectors: q is poly of degree m or less that has value 1 at 0. Deflation basics

11. Matrix: bidiagonal, diagonal is 0.1, 1, 2, 3, …1999, superdiagonal is all 1’s GMRES polynomial of degree 10

12. GMRES polynomial of degree 100 (close up view)

13. GMRES polynomial of degree 150 (close up view)

14. Residual norm curve Matrix vector products

15. 2+1 CP-PACS 16 3 x 32 lattice, kappa=0.13980. NonHermitian deflation

16. Hermitian deflation: CG vs. D-CG (M + M) Getting M from M + M; 20 3 x 32 lattice, 1st rhs: Lan-DR(200,k).

17. Hermitian deflation: CG vs. D-CG (  5 M) Getting M from  5 M; same 20 3 x 32 lattice

18. Hermitian deflation: Minres vs. D-Minres (  5 M) Using  5 M to get M using D-Minres; same 20 3 x 32 lattice

19. Noise matrices: Variance: Z(N) ( ) noise: Subtraction basics

20. Perturbative subtraction (PS) (Q = M -1 ): Should work best for small (large quark mass). Eigenspectrum (ES) subtraction: Should work best for large  (small quark mass). Can also combine PS and ES methods

21. 30x30 pseudolattice spectrum

22. ES=eigenspectrum subtraction, PE=perturbative subtraction, DS=direct sum (no subtraction). Eigenspectrum Subtraction (30x30 matrix) Noises

23. 500 x 500 bidiagonal matrix (eig: 0.1,1,2,3,...) Trace Error #subtracted eigenvalues

24. 8 4 Wilson lattice spectrum at _critical Imag Real Figure 1: Plot of the low eigenspectrum of the 8 4 Wilson lattice at  cr = 0.15701.

25. Noi NS=no subtraction, PE=perturbative subtraction, #ev=no. of eigen. subtracted, PEc+#ev=corrected perturbative subtraction on #ev method. 8 4 Lattice Results; =0.15701 Noises

26. Deflation is a important new method in Lattice studies, which will become more important for smaller quark masses. Effective in a Hermitian or non-Hermitian context. Eigenspectrum subtraction is helpful for disconnected diagrams at small quark masses. Extension to Fortran (larger lattices) is almost finished. Thanks to Ron Morgan and Victor Guerrero for their invaluable contributions! Summary