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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