slides - Lattice Seminar - Desy
slides - Lattice Seminar - Desy
slides - Lattice Seminar - Desy
Create successful ePaper yourself
Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.
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