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Shameless Baylor Advertisement
• Baylor’s Physics Department – 15 research-active faculty. Rank #18 in the percentage of faculty whose works are cited and #16 in dollars per research grant.
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• Have active experimental research programs at FermiLab (CDF) and the LHC (CMS).
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Disconnected Diagrams
and Lattice QCD
What is Lattice QCD?Taming disconnected diagrams
Collaborators: Victor Guerrero and Ron Morgan
Quantum Chromodynamics (QCD)
--- the fundamental theory of the strong interaction (quarks and gluons)
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
LQCD =1
2 Tr FμνFμν +q(γ μDμ + mq )q
Field strength tensor: Fμν =∂Aμ −∂Aν +g[Aμ ,Aν ]
Covariant derivative: Dμ =∂μ +gAμ
Lattice Quantum Chromodynamics (QCD) Variables
Example: quark-antiquark potential
Example : s
Example: hadron masses (Durr et al)
Ugly gluon lines
quark loop
creationdestruction
The HARD problem:
Krylov subspace:
Starting, residual vectors:
q is poly of degree m or less that has value 1 at 0.
},...,,,{ 01
02
00 rArAArrSpan m−
r = r0 - Ax̂
r = q(A)r0 = βiq(λi )zi∑r0 = βizi∑
Deflation basics
(Solving Ax = b)
(r0 = b - Ax0 )
Matrix: bidiagonal, diagonal is 0.1, 1, 2, 3, …1999, superdiagonal is all 1’s
GMRES polynomial of degree 10
0 100 200 300 400 500 600 700 800 900 1000-0.2
0
0.2
0.4
0.6
0.8
1
1.2
GMRES polynomial of degree 100(close up view)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-0.2
0
0.2
0.4
0.6
0.8
1
1.2
GMRES polynomial of degree 150(close up view)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Residual norm curve
0 50 100 15010-3
10-2
10-1
100
101
102Solution of the Linear Equations
Residual Norm
Matrix-vector Products
Matrix vector products
2+1 CP-PACS 163 x 32 lattice, kappa=0.13980.
NonHermitian deflation
Hermitian deflation: CG vs. D-CG (M+M)
Getting M from M+M; 203 x 32 lattice, 1st rhs: Lan-DR(200,k).
Hermitian deflation: CG vs. D-CG (5 M)
Getting M from 5 M; same 203 x 32 lattice
γ
γ
Hermitian deflation: Minres vs. D-Minres (5 M)
Using5 M to get M using D-Minres; same 203 x 32 lattice
γ
γ
• Noise matrices:
• Variance:
• Z(N) ( ) noise:
Xmn ≡1L
ηmlηnl*
l=1
L
∑ < Xmn >= δmn
Error =VN
€
N ≥ 3
V[Tr{QX}}=1L
|qmn |2m≠n∑
Subtraction basics
(Ax =ηl )
• 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
M −1 =1
1−κD; M pert
−1 =1+κD+ κD( )2 + ...
QPS ≡M −1 −M pert−1
κ
κ
QES ≡M −1 −MES−1
M ES−1 =
1λqq=1
Q
∑ eRq eL
q( )†; eL
q'( )†⋅eR
q =δqq'
30x30 pseudolattice spectrum
ES=eigenspectrum subtraction, PE=perturbative subtraction, DS=direct sum (no subtraction).
Eigenspectrum Subtraction (30x30 matrix)
Noises
500 x 500 bidiagonal matrix (eig: 0.1,1,2,3,...)
TraceError
#subtracted eigenvalues
84 Wilson lattice spectrum at _critical
Imag
Real
κ
Noi
NS=no subtraction, PE=perturbative subtraction, #ev=no. of eigen. subtracted, PEc+#ev=corrected perturbative subtraction
on #ev method.
84 Lattice Results; =0.15701
Noises
κ
• 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