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All-to-all<br />

Propagators<br />

in <strong>Lattice</strong><br />

Hadron<br />

Spectrum<br />

Calculations<br />

John Bulava<br />

All-to-all Propagators in <strong>Lattice</strong> Hadron<br />

Spectrum Calculations<br />

Background<br />

Distillation -<br />

An Exact<br />

All-to-all<br />

Method<br />

Variance-<br />

Reduced<br />

Stochastic<br />

LapH (VRSL)<br />

John Bulava<br />

DESY<br />

Zeuthen, Germany<br />

January 18 th , 2010<br />

DESY, Zeuthen<br />

Germany<br />

John Bulava<br />

All-to-all Propagators in <strong>Lattice</strong> Hadron Spectrum Calculations


Outline<br />

All-to-all<br />

Propagators<br />

in <strong>Lattice</strong><br />

Hadron<br />

Spectrum<br />

Calculations<br />

John Bulava<br />

1 Background<br />

Background<br />

Distillation -<br />

An Exact<br />

All-to-all<br />

Method<br />

Variance-<br />

Reduced<br />

Stochastic<br />

LapH (VRSL)<br />

2 Distillation - An Exact All-to-all Method<br />

3 Variance-Reduced Stochastic LapH (VRSL)<br />

John Bulava<br />

All-to-all Propagators in <strong>Lattice</strong> Hadron Spectrum Calculations


Outline<br />

All-to-all<br />

Propagators<br />

in <strong>Lattice</strong><br />

Hadron<br />

Spectrum<br />

Calculations<br />

John Bulava<br />

1 Background<br />

Background<br />

Distillation -<br />

An Exact<br />

All-to-all<br />

Method<br />

Variance-<br />

Reduced<br />

Stochastic<br />

LapH (VRSL)<br />

2 Distillation - An Exact All-to-all Method<br />

3 Variance-Reduced Stochastic LapH (VRSL)<br />

John Bulava<br />

All-to-all Propagators in <strong>Lattice</strong> Hadron Spectrum Calculations


The Hadron Spectrum Collaboration<br />

All-to-all<br />

Propagators<br />

in <strong>Lattice</strong><br />

Hadron<br />

Spectrum<br />

Calculations<br />

John Bulava<br />

Background<br />

Distillation -<br />

An Exact<br />

All-to-all<br />

Method<br />

Variance-<br />

Reduced<br />

Stochastic<br />

LapH (VRSL)<br />

Dedicated to mapping out the low-lying excited hadron<br />

spectra.<br />

Members:<br />

Jefferson Lab: H. W. Lin, S. D. Cohen, J. Dudek, R. G.<br />

Edwards, B. Joo, D. G. Richards<br />

Carnegie Mellon: J. Foley, C. Morningstar, D. Lenkner,<br />

C. H. Wong<br />

DESY, Zeuthen: JB<br />

U. of Maryland: E. Engelson, S. Wallace<br />

U. of the Pacific: K. J. Juge<br />

Tata Inst., Mumbai: N. Mathur<br />

Trinity College, Dublin: M. J. Peardon, S. M. Ryan<br />

John Bulava<br />

All-to-all Propagators in <strong>Lattice</strong> Hadron Spectrum Calculations


<strong>Lattice</strong> QCD Background<br />

All-to-all<br />

Propagators<br />

in <strong>Lattice</strong><br />

Hadron<br />

Spectrum<br />

Calculations<br />

Correlation functions between hadron operators are an<br />

ensemble average over gauge configurations.<br />

John Bulava<br />

Background<br />

Distillation -<br />

An Exact<br />

All-to-all<br />

Method<br />

Variance-<br />

Reduced<br />

Stochastic<br />

LapH (VRSL)<br />

〈0|O i<br />

[<br />

ψ, ψ, U<br />

]<br />

(t) Oj<br />

[<br />

ψ, ψ, U<br />

]<br />

(t0 )|0〉<br />

〈0|0〉<br />

= 〈 F ij [M −1 (U), U] 〉 U<br />

M −1 (U), is a (V × L t × N spin × N color )-dimensional<br />

matrix. Can only solve equations like<br />

M(x, y)φ(y) = η(x) → φ(x) = M −1 (x, y)η(y)<br />

‘Point-to-all’ ⇒ η(x) ∝ δ(x, x 0 )<br />

‘All-to-all’ ⇒ Use of M −1 (x, y) ∀x, y<br />

John Bulava<br />

All-to-all Propagators in <strong>Lattice</strong> Hadron Spectrum Calculations


Hadron Spectra from <strong>Lattice</strong> QCD<br />

All-to-all<br />

Propagators<br />

in <strong>Lattice</strong><br />

Hadron<br />

Spectrum<br />

Calculations<br />

John Bulava<br />

Background<br />

Distillation -<br />

An Exact<br />

All-to-all<br />

Method<br />

Variance-<br />

Reduced<br />

Stochastic<br />

LapH (VRSL)<br />

Spectral decomposition of two-point correlation functions:<br />

〈0| O(t + t 0 ) O(t 0 ) |0〉 = ∑ n<br />

| 〈0| O(t 0 ) |n〉 | 2 exp[−E n t]<br />

Difficult (or impossible) to fit sub-leading exponentials,<br />

instead form a matrix of two-point correlators:<br />

C ij (t) = 〈0| O i (t + t 0 ) O j (t 0 ) |0〉<br />

Define new operators Ω n (t) such that:<br />

〈0| Ω a (t + t 0 ) Ω b (t 0 ) |0〉 = δ ab λ a (t)<br />

Fit λ n (t) with a single exponential to obtain E n<br />

John Bulava<br />

All-to-all Propagators in <strong>Lattice</strong> Hadron Spectrum Calculations


The Need For All-to-all Propagators<br />

All-to-all<br />

Propagators<br />

in <strong>Lattice</strong><br />

Hadron<br />

Spectrum<br />

Calculations<br />

John Bulava<br />

Finite-volume stationary states are comprised of resonance<br />

states as well as scattering states.<br />

Some resonance states may have multi-particle content.<br />

Background<br />

Distillation -<br />

An Exact<br />

All-to-all<br />

Method<br />

Variance-<br />

Reduced<br />

Stochastic<br />

LapH (VRSL)<br />

〈0|B(p = 0, t)B(p = 0, t 0 )|0〉 = (1)<br />

1 ∑<br />

V 2 〈0|ϕ B (x, t)ϕ B (y, t 0 )|0〉<br />

x,y<br />

B(p, t)M(−p, t) = 1 ∑<br />

V 2 ϕ B (x, t)ϕ M (y, t)e ip·(x−y) (2)<br />

x,y<br />

Sum over y can be eliminated in Eq. 1, but not in<br />

correlation functions containing the operator from Eq. 2.<br />

Solving Mφ = η for all y is not feasible.<br />

John Bulava<br />

All-to-all Propagators in <strong>Lattice</strong> Hadron Spectrum Calculations


Simulation Details<br />

All-to-all<br />

Propagators<br />

in <strong>Lattice</strong><br />

Hadron<br />

Spectrum<br />

Calculations<br />

John Bulava<br />

Background<br />

Distillation -<br />

An Exact<br />

All-to-all<br />

Method<br />

Variance-<br />

Reduced<br />

Stochastic<br />

LapH (VRSL)<br />

Anisotropic Wilson Gauge action with spatial rectangle.<br />

(C. Morningstar, M. Peardon ‘99)<br />

Anisotropic Clover-Wilson quark action, with tadpole<br />

improvement<br />

Spatial links are stout smeared (Morningstar, Peardon<br />

’04), parameters are tuned non-perturbatively (Edwards,<br />

Lin, Joo ’08).<br />

N f = 2 + 1, a s = 0.12fm, a t = a s /3.5. Range of pion<br />

masses: ≈ 700MeV − 230MeV, and lattice sizes 12 3 × 96<br />

to 32 2 × 256.<br />

John Bulava<br />

All-to-all Propagators in <strong>Lattice</strong> Hadron Spectrum Calculations


Outline<br />

All-to-all<br />

Propagators<br />

in <strong>Lattice</strong><br />

Hadron<br />

Spectrum<br />

Calculations<br />

John Bulava<br />

1 Background<br />

Background<br />

Distillation -<br />

An Exact<br />

All-to-all<br />

Method<br />

Variance-<br />

Reduced<br />

Stochastic<br />

LapH (VRSL)<br />

2 Distillation - An Exact All-to-all Method<br />

3 Variance-Reduced Stochastic LapH (VRSL)<br />

John Bulava<br />

All-to-all Propagators in <strong>Lattice</strong> Hadron Spectrum Calculations


LapH (Laplacian Heaviside) Smearing<br />

All-to-all<br />

Propagators<br />

in <strong>Lattice</strong><br />

Hadron<br />

Spectrum<br />

Calculations<br />

John Bulava<br />

Background<br />

Distillation -<br />

An Exact<br />

All-to-all<br />

Method<br />

Variance-<br />

Reduced<br />

Stochastic<br />

LapH (VRSL)<br />

Quark smearing damps out unwanted excited states by<br />

applying S = exp[ σ2<br />

4 ∆].<br />

N∑<br />

S ab [U; t](x|y) = e σ2<br />

4 λi[U;t] v a (i) [U; t](x) v ∗(i) [U; t](y)<br />

i=1<br />

Truncate at a finite number (n max


Volume Dependence<br />

All-to-all<br />

Propagators<br />

in <strong>Lattice</strong><br />

Hadron<br />

Spectrum<br />

Calculations<br />

John Bulava<br />

Background<br />

Distillation -<br />

An Exact<br />

All-to-all<br />

Method<br />

Variance-<br />

Reduced<br />

Stochastic<br />

LapH (VRSL)<br />

-∆ eigenvalue<br />

0.6<br />

0.5<br />

0.4<br />

0.3<br />

0.2<br />

12 3 lattice<br />

16 3 lattice<br />

0.1<br />

0 10 20 30 40 50<br />

Eigenvalue number<br />

N f = 2 + 1 ensembles:<br />

12 3 × 96 and 16 3 × 128<br />

with a s = 0.12fm,<br />

a t = a s /3.5,<br />

m π ≈ 700MeV<br />

Examine the number of<br />

eigenvalues in [0.3, 0.4]:<br />

n 12 /n 16 = (12/16) 3<br />

For fixed λ max , n max ∝ V<br />

John Bulava<br />

All-to-all Propagators in <strong>Lattice</strong> Hadron Spectrum Calculations


LapH Conclusions<br />

All-to-all<br />

Propagators<br />

in <strong>Lattice</strong><br />

Hadron<br />

Spectrum<br />

Calculations<br />

John Bulava<br />

Exact (smeared) all-to-all propagator for a finite number<br />

of inversions!<br />

Background<br />

Distillation -<br />

An Exact<br />

All-to-all<br />

Method<br />

Variance-<br />

Reduced<br />

Stochastic<br />

LapH (VRSL)<br />

n max (and thus required number of inversions) grows<br />

linearly with the volume.<br />

Cannot completely separate initial and final degrees of<br />

freedom.<br />

Still useful for smaller volumes ( L s < 2 − 2.5fm). Results!<br />

John Bulava<br />

All-to-all Propagators in <strong>Lattice</strong> Hadron Spectrum Calculations


It’s All About the Operators<br />

All-to-all<br />

Propagators<br />

in <strong>Lattice</strong><br />

Hadron<br />

Spectrum<br />

Calculations<br />

John Bulava<br />

Background<br />

Distillation -<br />

An Exact<br />

All-to-all<br />

Method<br />

Variance-<br />

Reduced<br />

Stochastic<br />

LapH (VRSL)<br />

Spatially extended operators have overlap with orbital and<br />

radial excitations.<br />

Operators must transform irreducibly under lattice<br />

symmetries.<br />

Generate a large set of operators from simple elemental<br />

building blocks C. Morningstar, et al. (2005)<br />

✉<br />

✎☞<br />

✉ ❡ ✉<br />

✉ ✉✉ ✉<br />

✍✌<br />

♠✉ ✉ ✉ ❤✉ ✉ ❤✉ ✉ ✉<br />

singlesitdisplacedisplaced-displaced-L<br />

singly-<br />

doubly-<br />

doubly-<br />

triplydisplaced-T<br />

Select O(10) with good low-lying state overlap which form<br />

a well-conditioned correlator matrix.<br />

John Bulava<br />

All-to-all Propagators in <strong>Lattice</strong> Hadron Spectrum Calculations


Results: Baryon and Meson Spectra<br />

All-to-all<br />

Propagators<br />

in <strong>Lattice</strong><br />

Hadron<br />

Spectrum<br />

Calculations<br />

John Bulava<br />

Background<br />

Distillation -<br />

An Exact<br />

All-to-all<br />

Method<br />

Variance-<br />

Reduced<br />

Stochastic<br />

LapH (VRSL)<br />

100 configurations from N f = 2 + 1, 16 3 × 128 ensemble,<br />

with L s ≈ 2fm, a s ≈ 0.12fm, a t = a s /3.5, and<br />

m π ≈ 380MeV. Also, n max = 32.<br />

Operators transform under lattice irreps: ‘g’→ +ve parity,<br />

‘u’ → -ve parity. J = 1/2 → G 1 , J = 3/2 → H ,<br />

J = 5/2 → G 2 and H.<br />

About 12 (single-hadron) operators chosen in each<br />

channel.<br />

John Bulava<br />

All-to-all Propagators in <strong>Lattice</strong> Hadron Spectrum Calculations


Nucleon (JB) and Delta (E. Engelson)<br />

Spectra<br />

All-to-all<br />

Propagators<br />

in <strong>Lattice</strong><br />

Hadron<br />

Spectrum<br />

Calculations<br />

John Bulava<br />

Background<br />

Distillation -<br />

An Exact<br />

All-to-all<br />

Method<br />

Variance-<br />

Reduced<br />

Stochastic<br />

LapH (VRSL)<br />

M(GeV)<br />

4<br />

3.6<br />

3.2<br />

2.8<br />

2.4<br />

2<br />

1.6<br />

1.2<br />

0.8<br />

0.4<br />

0<br />

G 1g<br />

H g<br />

G 2g<br />

G 1u<br />

H u<br />

G 2u<br />

John Bulava<br />

All-to-all Propagators in <strong>Lattice</strong> Hadron Spectrum Calculations


Σ (D. Lenkner) and Ξ (C. H. Wong)<br />

Baryon Spectra<br />

All-to-all<br />

Propagators<br />

in <strong>Lattice</strong><br />

Hadron<br />

Spectrum<br />

Calculations<br />

John Bulava<br />

Background<br />

3.5<br />

3.0<br />

2.5<br />

3<br />

Simulated Sigma Spectrum, V=16<br />

3.5<br />

3.0<br />

2.5<br />

3<br />

Simulated Cascade Spectrum, V=16<br />

Distillation -<br />

An Exact<br />

All-to-all<br />

Method<br />

Variance-<br />

Reduced<br />

Stochastic<br />

LapH (VRSL)<br />

GeV<br />

2.0<br />

1.5<br />

1.0<br />

0.5<br />

0.0<br />

G1g<br />

Hg<br />

G2g<br />

G1u<br />

Hu<br />

G2u<br />

GeV<br />

2.0<br />

1.5<br />

1.0<br />

0.5<br />

0.0<br />

G1g<br />

Hg<br />

G2g<br />

G1u<br />

Hu<br />

G2u<br />

John Bulava<br />

All-to-all Propagators in <strong>Lattice</strong> Hadron Spectrum Calculations


N f = 2 + 1 Results: I G = 1 − and I G = 1 +<br />

Meson Spectrum (C. H. Wong)<br />

All-to-all<br />

Propagators<br />

in <strong>Lattice</strong><br />

Hadron<br />

Spectrum<br />

Calculations<br />

John Bulava<br />

Background<br />

Distillation -<br />

An Exact<br />

All-to-all<br />

Method<br />

Variance-<br />

Reduced<br />

Stochastic<br />

LapH (VRSL)<br />

John Bulava<br />

All-to-all Propagators in <strong>Lattice</strong> Hadron Spectrum Calculations


Outline<br />

All-to-all<br />

Propagators<br />

in <strong>Lattice</strong><br />

Hadron<br />

Spectrum<br />

Calculations<br />

John Bulava<br />

1 Background<br />

Background<br />

Distillation -<br />

An Exact<br />

All-to-all<br />

Method<br />

Variance-<br />

Reduced<br />

Stochastic<br />

LapH (VRSL)<br />

2 Distillation - An Exact All-to-all Method<br />

3 Variance-Reduced Stochastic LapH (VRSL)<br />

John Bulava<br />

All-to-all Propagators in <strong>Lattice</strong> Hadron Spectrum Calculations


Stochastic All-to-all<br />

All-to-all<br />

Propagators<br />

in <strong>Lattice</strong><br />

Hadron<br />

Spectrum<br />

Calculations<br />

John Bulava<br />

Background<br />

Distillation -<br />

An Exact<br />

All-to-all<br />

Method<br />

Variance-<br />

Reduced<br />

Stochastic<br />

LapH (VRSL)<br />

Generate N r independent stochastic sources<br />

η (r)<br />

cα(x, t) ∈ Z 4<br />

and solve Mφ (r) = η (r) for φ (r) .<br />

Can estimate the quark propagator:<br />

M −1 ≈ 1 ∑N r<br />

φ (r) η (r)∗<br />

N r<br />

r=1<br />

Variance Reduction: Dilute each noise source<br />

η (r)[d] = P [d] η (r) , with ∑ d=1 P[d] = 1 and<br />

P [d] P [d′] = δ dd ′P [d] .<br />

John Bulava<br />

All-to-all Propagators in <strong>Lattice</strong> Hadron Spectrum Calculations


Noise in the Subspace Only<br />

All-to-all<br />

Propagators<br />

in <strong>Lattice</strong><br />

Hadron<br />

Spectrum<br />

Calculations<br />

John Bulava<br />

Background<br />

Distillation -<br />

An Exact<br />

All-to-all<br />

Method<br />

Variance-<br />

Reduced<br />

Stochastic<br />

LapH (VRSL)<br />

A new way to introduce noise:<br />

η cα(x, (r) ∑N ev<br />

t) =<br />

n=1<br />

Dilute in the subspace only<br />

η cα (r)[d] ∑N ev<br />

(x, t) =<br />

n=1<br />

αn[t] v c<br />

(n) [U; t](x)<br />

a (r)<br />

a (r)[d]<br />

αn<br />

[t] v (n) [U; t](x)<br />

a αn<br />

(r)[d] [t] = P [d]<br />

(αn|α ′ n ′ ) (t|t′ ) a (r)<br />

α ′ n<br />

[t ′ ] ′<br />

c<br />

Is VRSL more efficient than conventional dilution What<br />

about the volume dependence<br />

John Bulava<br />

All-to-all Propagators in <strong>Lattice</strong> Hadron Spectrum Calculations


VRSL vs. Conventional Dilution<br />

All-to-all<br />

Propagators<br />

in <strong>Lattice</strong><br />

Hadron<br />

Spectrum<br />

Calculations<br />

John Bulava<br />

Background<br />

Distillation -<br />

An Exact<br />

All-to-all<br />

Method<br />

Variance-<br />

Reduced<br />

Stochastic<br />

LapH (VRSL)<br />

)/σ g<br />

σ(N<br />

inv<br />

900<br />

800<br />

700<br />

600<br />

500<br />

400<br />

300<br />

200<br />

100<br />

Time LAPH<br />

Time+Every Other<br />

Time+2 Groups<br />

Time+Spin<br />

Time+Every 4th<br />

Time+ 4 Groups<br />

Time+Every 8th<br />

Time+8 Groups<br />

Time+Every 12th<br />

Time+Every 16th<br />

Time<br />

Time+Space_eo<br />

Time+Color<br />

Time+Spin<br />

Time+Color+Space_eo<br />

Time+Spin+Space_eo<br />

Time+Spin+Color<br />

Time+Spin+Color+Space_eo<br />

Max Dil. Limit<br />

0<br />

0.00 0.01 0.02 0.03 0.04 0.05 0.06<br />

N<br />

−1/2<br />

inv<br />

John Bulava<br />

All-to-all Propagators in <strong>Lattice</strong> Hadron Spectrum Calculations


VRSL Volume Dependence<br />

All-to-all<br />

Propagators<br />

in <strong>Lattice</strong><br />

Hadron<br />

Spectrum<br />

Calculations<br />

John Bulava<br />

Background<br />

Distillation -<br />

An Exact<br />

All-to-all<br />

Method<br />

Variance-<br />

Reduced<br />

Stochastic<br />

LapH (VRSL)<br />

)/σ g<br />

σ(N<br />

inv<br />

30<br />

25<br />

20<br />

15<br />

10<br />

5<br />

Time+Spin 16^3<br />

Time+Every 4th 16^3<br />

Time+4 Groups 16^3<br />

Time+Every 8th 16^3<br />

Time+8 Groups 16^3<br />

Time+Every 12th 16^3<br />

Time+Every 16th 16^3<br />

Time+Spin 20^3<br />

Time+Every 4th 20^3<br />

Time+4 Groups 20^3<br />

Time+Every 8th 20^3<br />

Time+8 Groups 20^3<br />

Time+Every 12th 20^3<br />

Time+Every 16th 20^3<br />

Max Dil. Limit<br />

0<br />

0.000 0.005 0.010 0.015 0.020 0.025 0.030<br />

N<br />

−1/2<br />

inv<br />

John Bulava<br />

All-to-all Propagators in <strong>Lattice</strong> Hadron Spectrum Calculations


Same-Time (‘Disconnected’) Diagrams<br />

All-to-all<br />

Propagators<br />

in <strong>Lattice</strong><br />

Hadron<br />

Spectrum<br />

Calculations<br />

John Bulava<br />

Background<br />

Distillation -<br />

An Exact<br />

All-to-all<br />

Method<br />

Variance-<br />

Reduced<br />

Stochastic<br />

LapH (VRSL)<br />

Required for flavor-singlet quantities and single<br />

hadron/multi-hadron correlators.<br />

For ‘Connected’ correlators, only need η(x 0 , t 0 ) for a single<br />

t 0 but here we need many t 0 ’s.<br />

Exact distillation result on all t 0 ’s for a small lattice: 100<br />

cfgs., N f = 2 + 1, 12 3 × 96, m π ≈ 700MeV, n max = 12,<br />

N inv = 4608.<br />

Try different time, spin, and eigenvector dilution schemes.<br />

(C. H. Wong, M. Peardon)<br />

John Bulava<br />

All-to-all Propagators in <strong>Lattice</strong> Hadron Spectrum Calculations


Disc. Term from η ′ Meson Correlator -<br />

192 Inversions<br />

All-to-all<br />

Propagators<br />

in <strong>Lattice</strong><br />

Hadron<br />

Spectrum<br />

Calculations<br />

John Bulava<br />

1<br />

Eta’ , V=12 3 x96, in the Chosen Dilution Scheme<br />

No. of Interlace Projectors in (time, vector, spin) in the labels<br />

96,12,4(Dist)<br />

12,4,4<br />

Background<br />

Distillation -<br />

An Exact<br />

All-to-all<br />

Method<br />

Variance-<br />

Reduced<br />

Stochastic<br />

LapH (VRSL)<br />

C(τ)<br />

0.1<br />

0.01<br />

0 5 10 15 20 25<br />

τ<br />

John Bulava<br />

All-to-all Propagators in <strong>Lattice</strong> Hadron Spectrum Calculations


Disc. Term from σ Meson Correlator -<br />

192 Inversions<br />

All-to-all<br />

Propagators<br />

in <strong>Lattice</strong><br />

Hadron<br />

Spectrum<br />

Calculations<br />

John Bulava<br />

Background<br />

1<br />

0.1<br />

Sigma, V=12 3 x96, for the Chosen Dilution Scheme<br />

No. of interlace projectors in (time, vector, spin) in the labels<br />

96,12,4(Dist)<br />

12,4,4<br />

Distillation -<br />

An Exact<br />

All-to-all<br />

Method<br />

Variance-<br />

Reduced<br />

Stochastic<br />

LapH (VRSL)<br />

C(τ)<br />

0.01<br />

0.001<br />

0.0001<br />

0 5 10 15 20 25<br />

τ<br />

John Bulava<br />

All-to-all Propagators in <strong>Lattice</strong> Hadron Spectrum Calculations


Single Term from Scalar Meson Decay -<br />

192 Inversions<br />

All-to-all<br />

Propagators<br />

in <strong>Lattice</strong><br />

Hadron<br />

Spectrum<br />

Calculations<br />

John Bulava<br />

1<br />

Scalar Decay,V=12 3 x96, for the Chosen Dilution Scheme<br />

No. of Interlace Projectors in [t,v,s][t,v,s] for the labels(1 st [] is t 0 -t 1 ,2 nd [] is t 0 -t 0 )<br />

[96,12,4][96,12,4](dist)<br />

[96,6,4][12,4,4]<br />

Background<br />

Distillation -<br />

An Exact<br />

All-to-all<br />

Method<br />

Variance-<br />

Reduced<br />

Stochastic<br />

LapH (VRSL)<br />

Re(-C(τ))<br />

0.1<br />

0.01<br />

0.001<br />

0 5 10 15 20 25<br />

τ<br />

John Bulava<br />

All-to-all Propagators in <strong>Lattice</strong> Hadron Spectrum Calculations


VRSL Conclusions<br />

All-to-all<br />

Propagators<br />

in <strong>Lattice</strong><br />

Hadron<br />

Spectrum<br />

Calculations<br />

John Bulava<br />

Background<br />

Distillation -<br />

An Exact<br />

All-to-all<br />

Method<br />

Variance-<br />

Reduced<br />

Stochastic<br />

LapH (VRSL)<br />

VRSL is more efficient than conventional dilution. For a<br />

fixed number of inversions, VRSL gives a considerably<br />

lower error.<br />

The smeared subspace is low-dimensional, so the<br />

gauge-noise (σ g ) limit can be obtained with a MUCH<br />

smaller number of projectors.<br />

Unlike exact method, volume dependence seems mild.<br />

A moderate level of dilution projectors works for<br />

disconnected terms.<br />

John Bulava<br />

All-to-all Propagators in <strong>Lattice</strong> Hadron Spectrum Calculations


Future Plans<br />

All-to-all<br />

Propagators<br />

in <strong>Lattice</strong><br />

Hadron<br />

Spectrum<br />

Calculations<br />

John Bulava<br />

Background<br />

Distillation -<br />

An Exact<br />

All-to-all<br />

Method<br />

Variance-<br />

Reduced<br />

Stochastic<br />

LapH (VRSL)<br />

Employ the VRSL method to study multi-hadron<br />

(specifically nucleon-pion) operators.<br />

J. Foley: Group theory for moving hadrons<br />

J. Juge, C. H. Wong, J. Bulava: First Multi-pion results<br />

Repeat the spectrum calculations for a variety of larger<br />

volumes (up to 32 3 × 256, L s ≈ 3.8fm) with lighter quarks<br />

(pion mass down to ≈ 230MeV) including multi-hadron<br />

operators.<br />

Differentiate resonances from multi-hadron states and<br />

identify resonance quantum numbers.<br />

John Bulava<br />

All-to-all Propagators in <strong>Lattice</strong> Hadron Spectrum Calculations

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