Progress in Excited Hadron
States in Lattice QCD
Colin Morningstar
(Carnegie Mellon University)
Florida International U. Colloquium
November 20, 2009
November 20, 2009
C. Morningstar
1
The frontier awaits
experiments show many excited-state hadrons exist
z significant experimental efforts to map out QCD resonance
spectrum Æ JLab Hall B, Hall D, ELSA, etc.
z great need for ab initio calculations Æ lattice QCD
z
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2
The challenge of exploration!
z
most excited hadrons are unstable (resonances)
z
excited states more difficult to extract in Monte Carlo calculations
z
‰
correlation matrices needed
‰
operators with very good overlaps onto states of interest
must extract all states lying below a state of interest
‰
z
as pion get lighter, more and more multi-hadron states
best multi-hadron operators made from constituent hadron
operators with well-defined relative momenta
‰
z
need for all-to-all quark propagators
disconnected diagrams
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3
Hadron Spectrum Collaboration (HSC)
z
spin-off from the Lattice Hadron Physics Collaboration which was
spear-headed by Nathan Isgur and John Negele
z
current members:
‰
Justin Foley, David Lenkner, Colin Morningstar, Ricky Wong (CMU)
‰
John Bulava (DESY, Zeuthen)
‰
Eric Engelson, Steve Wallace (U. Maryland)
‰
Mike Peardon, Sinead Ryan (Trinity Coll. Dublin)
‰
Keisuke Jimmy Juge (U. of Pacific)
‰
R. Edwards, B. Joo, D. Richards, C. Thomas (Jefferson Lab.)
‰
H.W. Lin (U. Washington), J. Dudek (Old Dominion)
‰
N. Mathur (Tata Institute)
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Overview of our spectrum project
z
obtain stationary state energies of QCD in various boxes
‰ 1st milestone: quenched excited states with heavy pion Æ done
‰ 2nd milestone: Nf=2 excited states with heavy pion Æ done
‰ 3rd milestone: Nf=2+1 excited states with light pion
– multi-hadron operators needed Æ many-to-many quark propagators
– recent technology breakthrough Æ new quark smearing with
improved variance reduction
z
interpretation of finite-volume energies
‰ spectrum matching to construct effective hadron theory?
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5
Monte Carlo method
hadron operators φ = φ [ψ ,ψ , U ] ψ =quark U =gluon field
z temporal correlations from path integrals
−ψ M [U ]ψ − S [U ]
D
[
ψ
,
ψ
,
U
]
φ
(
t
)
φ
(0)
e
φ (t )φ (0) = ∫
−ψ M [U ]ψ − S [U ]
D
[
ψ
,
ψ
,
U
]
e
∫
z integrate exactly over quark Grassmann fields
z
φ (t )φ (0) =
−1
DU
det
M
[
U
]
M
[U ]
(
∫
∫ DU det M [U ] e
)e
− S [U ]
− S [U ]
resort to Monte Carlo method to integrate over gluon fields
z generate sequence of field configurations U1 , U 2 , U 3 , , U N
using Markov chain procedure
‰ use of parallel computations on supercomputers
‰ especially intensive as quark mass (pion mass) gets small
z
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6
Lattice regularization
z
z
z
z
z
hypercubic space-time lattice regulator needed for Monte Carlo
quarks reside on sites, gluons reside on links between sites
lattice excludes short wavelengths from theory (regulator)
regulator removed using standard renormalization procedures
(continuum limit)
systematic errors
‰ discretization
‰ finite volume
quarks
gluons
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Excited-state energies from Monte Carlo
z
extracting excited-state energies requires matrix of correlators
z
for a given N × N correlator matrix Cαβ (t ) = 0 Oα (t )Oβ+ (0) 0
one defines the N principal correlators λα (t , t0 ) as the eigenvalues of
C (t0 ) −1/ 2 C (t ) C (t0 ) −1/ 2
where t0 (the time defining the “metric”) is small
z
−( t −t0 ) Eα
(1 + e −tΔEα )
can show that limt →∞ λα (t , t0 ) = e
z
⎛ λα (t , t0 ) ⎞
eff
⎟⎟
⎜⎜
m
t
(
)
=
ln
N principal effective masses defined by α
⎝ λα (t + 1, t0 ) ⎠
now tend (plateau) to the N lowest-lying stationary-state energies
z
analysis:
‰
fit each principal correlator to single exponential
‰
optimize on earlier time slice, matrix fit to optimized matrix
‰
both methods as consistency check
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Operator design issues
statistical noise increases with temporal separation t
z use of very good operators is crucial or noise swamps signal
z recipe for making better operators
‰ crucial to construct operators using smeared fields
z
– link variable smearing
– quark field smearing
‰
‰
spatially extended operators
use large set of operators (variational coefficients)
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Three stage approach
( PRD72:094506,2005 )
concentrate on baryons at rest (zero momentum)
z operators classified according to the irreps of Oh
G1g , G1u , G2 g , G2u , H g , H u
z (1) basic building blocks: smeared, covariant-displaced quark fields
z
~
( D (j p )ψ~ ( x)) Aaα
p - link displaceme nt ( j = 0,±1,±2,±3)
z
(2) construct elemental operators (translationally invariant)
F
B F ( x) = φ ABC
ε abc ( Di( p )ψ ( x)) Aaα ( D (j p )ψ ( x)) Bbβ ( Dk( p )ψ ( x))Ccγ
‰ flavor structure from isospin
‰ color structure from gauge invariance
z
(3) group-theoretical projections onto irreps of Oh
dΛ
Λλ F
(Λ)
∗
+
F
Bi (t ) =
D
(
R
)
U
B
(
t
)
U
∑ λλ
R i
R
g O D R∈OhD
h
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Single-hadron operators
covariantly-displaced quark fields as building blocks
z group-theoretical projections onto irreps of lattice symmetry group
z displacements of different lengths build up radial structure
z displacements in different directions build up orbital structure
z
baryons
Δ-flux
Y-flux
mesons
z
reference: PRD72, 094506 (2005 )
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Spin identification and other remarks
z
spin identification possible by pattern matching
total numbers of operators assuming two
different displacement lengths
z
total numbers of operators is huge Æ uncharted territory
z
ultimately must face two-hadron scattering states
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12
Quark- and gauge-field smearing
smeared quark and gluon fields fields Æ dramatically reduced
coupling with short wavelength modes
z link-variable smearing (stout links PRD69, 054501 (2004))
z
‰
define C μ ( x) =
ρ μν Uν ( x)U μ ( x + νˆ )Uν ( x + μˆ )
∑
(ν μ )
+
± ≠
ρ 4k = ρ k 4 = 0
spatially isotropic
‰
exponentiate traceless Hermitian matrix
(
Ω μ = CμU μ+
‰
z
ρ jk = ρ ,
‰
iterate
U μ → U μ(1) →
)
(
)
μ̂
νˆ
x
Qμ = i Ω +μ − Ω μ − i Tr Ω +μ − Ω μ
2
2N
U ( n+1) = exp iQμ( n ) U ( n )
→Uμ
(n)
μ
~
≡Uμ
(
)
μ
initial quark-field smearing (Laplacian using smeared gauge field)
n
σs ⎞ σ
⎛
Δ ⎟ ψ ( x)
ψ (x) = ⎜1 +
4
n
⎝
σ ⎠
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Importance of smearing
states using σ s = 3.0
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0
0
1.5
20
0.5
Link Smearing Only
0
0
1.5
10
t / at
1
20
0.5
0
0
1.5
10
t / at
1
20
0.5
Both Quark and Link Smearing
0
10
t / at
20
C. Morningstar
0
0
0
1.5
10
t / at
20
10
t / at
20
10
t / at
20
1
0.5
M i (t)
0.5
10
t / at
1
M i (t)
20
1
0.5
M i (t)
10
t / at
1
0
M i (t)
M i (t)
Quark Smearing Only
M i (t)
•less noise in excited
1
Triply-Displaced-T
0.5
0
0
1.5
reduces couplings to
high-lying states
σ s = 4.0, nσ = 32
nρ ρ = 2.5, nρ = 16
1
Singly-Displaced
0.5
M i (t)
•quark-field smearing
1.5
Single-Site
•noise reduction from
link variable smearing,
especially for displaced
operators
1.5
M i (t)
selected operators
1.5
M i (t)
•Nucleon G1g channel
•effective masses of 3
0
0
1.5
1
0.5
0
10
t / at
20
0
0
14
Operator selection
operator construction leads to very large number of operators
z rules of thumb for “pruning” operator sets
‰ noise is the enemy!
‰ prune first using intrinsic noise (diagonal correlators)
‰ prune next within operator types (single-site, singly-displaced,
etc.) based on condition number of
‰ prune across all operators based on
condition number
z best to keep a variety of different types of operators, as long as
Cij (t )
condition numbers maintained
ˆ
Cij (t ) =
,
t =1
Cii (t )C jj (t )
z typically use 16 operators to get 8 lowest lying levels
z
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Nucleon G1g effective masses
200 quenched configs, 123×48 anisotropic Wilson lattice, as~0.1 fm,
as/at~3, mπ~700 MeV
z nucleon G1g channel
z green=fixed coefficients, red=principal
z
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Nucleon Hu effective masses
200 quenched configs, 123×48 anisotropic Wilson lattice, as~0.1 fm,
as/at~3, mπ~700 MeV
z nucleon Hu channel
z green=fixed coefficients, red=principal
z
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Nucleons
Nf=2 on 243×64 anisotropic clover lattice, as~0.11 fm, as/at~3
z Left: mπ=578 MeV Right: mπ=416 MeV PRD 79, 034505 (2009)
z
z
multi-hadron thresholds above show need for multi-hadron
operators to go to lower pion masses!!
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18
Spatial summations
z
baryon at rest is operator of form
B( p = 0, t ) =
z
1
∑ϕB ( x, t )
V x
baryon correlator has a double spatial sum
0 B ( p = 0, t ) B ( p = 0, 0) 0 =
1
0 ϕ B ( x , t ) ϕ B ( y, 0) 0
2 ∑
V x, y
computing all elements of propagators exactly not feasible
z translational invariance can limit summation over source site to a
single site for local operators
z
0 B ( p = 0, t ) B( p = 0, 0) 0 =
November 20, 2009
1
∑ 0 ϕ B ( x, t ) ϕB (0, 0) 0
V x
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19
Slice-to-slice quark propagators
z
good baryon-meson operator of total zero momentum has form
B ( p, t ) M ( − p, t ) =
1
ip i ( x − y )
(
,
)
(
,
)
ϕ
x
t
ϕ
y
t
e
∑ B
M
V 2 x, y
cannot limit source to single site for multi-hadron operators
z disconnected diagrams (scalar mesons) will also need many-to-many
quark propagators
z quark propagator elements from all spatial sites to all spatial sites are
needed!
z
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20
Laplacian Heaviside quark-field smearing
new quark-field smearing method PRD80, 054506 (2009)
z judicious choice of quark-field smearing makes exact computations
with all-to-all quark propagators possible (on small volumes)
z to date, quark field smeared using covariant Laplacian
z
n
σs ⎞ σ
⎛
Δ ⎟ ψ ( x)
ψ (x) = ⎜1 +
⎝ 4nσ ⎠
z
express in term of eigenvectors/eigenvalues of Laplacian
σ
ψ (x) = ⎛⎜1 + s Δ ⎞⎟
⎝ 4nσ ⎠
=
∑
k
z
nσ
⎛ σ s λk ⎞
⎜ 1 + 4n ⎟
⎝
σ ⎠
∑ ϕk
ϕk ψ ( x)
k
nσ
ϕk ϕk ψ ( x)
truncate sum and set weights to unity Æ Laplacian Heaviside
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Getting to know the Laplacian
spectrum of the covariant Laplacian
z left: dependence on lattice size; right: dependence on link smearing
z
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22
Choosing the smearing cut-off
z
Laplacian Heaviside (Laph) quark smearing
(
)
ψ (x) = Θ σ s2 + Δ ψ ( x)
≈
N max
∑
ϕk ϕk ψ ( x)
k =1
z
choose smearing cut-off based on minimizing excited-state
contamination, keep noise small
‰ behavior of nucleon t=1 effective masses
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23
Tests of Laplacian Heaviside smearing
z
comparison of ρ-meson effective masses using same number of
gauge-field configurations
typically need about 32 modes on 163 lattice
z about 128 modes on 243 lattice
z
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Nucleon operator pruning
z
Nf=2+1 on 163×128 lattice, mπ= 380 MeV (100 configs, 32 eigvecs)
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Delta operator pruning
z
Nf=2+1 on 163×128 lattice, mπ= 380 MeV (481 configs, 32 eigvecs)
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26
Sigma operator pruning
z
Nf=2+1 on 163×128 lattice, mπ= 380 MeV (100 configs, 32 eigvecs)
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Isovector G-parity odd mesons
z
Nf=2+1 on 163×128 lattice, mπ= 380 MeV (100 configs, 32 eigvecs)
a mesons
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π mesons
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Kaons
z
Nf=2+1 on 163×128 lattice, mπ= 380 MeV (100 configs, 32 eigvecs)
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Stochastic estimation of quark propagators
new Laph quark smearing method allows exact computation of all-toall quark propagators
z but number of Laplacian eigenvectors needed becomes prohibitively
large on large lattices
‰ 128 modes needed on 243 lattice
z computational method is rather cumbersome, too
z need to resort to stochastic estimation
z
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30
Stochastic estimation
quark propagator is just inverse of Dirac matrix M
z noise vectors η satisfying E(ηi)=0 and E(ηiηj*)=δi j are useful for
stochastic estimates of inverse of a matrix M
z Z4 noise is used {1, i, −1, −i}
z define X(η)=M-1η then
z
⎛
⎞
E ( X iη ∗j ) = E ⎜ ∑ M ik−1ηkη ∗j ⎟ = ∑ M ik−1 E (η kη ∗j ) = ∑ M ik−1δ kj = M ij−1
k
⎝ k
⎠ k
z
if can solve M X(r)= η(r) for each of NR noise vectors η(r) then we have
a Monte Carlo estimate of all elements of M-1:
M
−1
ij
1
≈
NR
NR
( r ) ( r )∗
X
∑ i ηj
r =1
variances in above estimates usually unacceptably large
z introduce variance reduction using source dilution
z
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31
Source dilution for single matrix inverse
z
dilution introduces a complete set of projections:
∑P
P ( a ) P (b ) = δ ab P ( a ) ,
z
(a)
= 1,
P ( a )† = P ( a )
a
observe that
M ij−1 = M ik−1δ kj = ∑ M ik−1 Pkj( a ) =∑ M ik−1 Pkk( a′ )δ k ′j′ Pj(′ja )
a
a
= ∑ M ik−1 Pkk( a′ ) E (ηk ′η ∗j′ ) Pj(′ja ) = ∑ M ik−1 E ( Pkk( a′ )ηk ′η ∗j′ Pj(′ja ) )
a
z
define η
[a]
k
=P η ,
(a)
kk ′ k ′
η
[ a ]∗
j
a
=η P ,
∗
j′
(a)
j ′j
X k[ a ] = M kj−1η [ja ]
M ij−1 = ∑ E ( X i[ a ]η [ja ]∗ )
so that
a
z Monte Carlo estimate is now
1 NR
−1
M ij ≈
X i( r )[ a ]η (j r )[ a ]∗
∑∑
N R r =1 a
z
∗
[ a ] [ a ]∗
η
η
η
η
∑ a i j has same expected value as i j , but reduced variance
(statistical zeros Æ exact)
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32
Earlier schemes
Introduce ZN noise in color, spin, space, time
ηaα ( x , t )
z Time dilution (particularly effective)
Pa(αB;)bβ ( x , t; y, t ′ ) = δ abδαβ δ ( x , y ) δ Btδ Bt ′ ,
B = 0,1,… , Nt − 1
z
z
z
Spin dilution
Pa(αB;)bβ ( x , t ; y, t ′ ) = δ abδ Bα δ Bβ δ ( x , y ) δ tt ′ ,
Color dilution
Pa(αB;)bβ ( x , t ; y, t ′ ) = δ Baδ Bbδαβ δ ( x , y ) δ tt ′ ,
z
B = 0,1, 2,3
B = 0,1, 2
Spatial dilutions?
‰ even-odd
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Dilution tests (old method)
z
100 quenched configs, 123×48 anisotropic Wilson lattice
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New stochastic Laph method
z
Introduce ZN noise in Laph subspace
ρα k (t ) t = time, α = spin, k = eigenvector number
z
Time dilution (particularly effective)
Pα(kB;)β l ( t ; t ′ ) = δ klδαβ δ Btδ Bt ′ ,
B = 0,1,… , Nt − 1
Spin dilution
Pα(kB;)β l ( t ; t ′ ) = δ klδ Bα δ Bβ δ tt ′ ,
B = 0,1, 2,3
z
z
Laplacian eigenvector dilution
Pα(kB;)β l ( t ; t ′ ) = δ Bk δ Blδαβ δ tt ′ ,
‰ define
‰ group projectors together
B = 0,1, 2, N eig − 1
– by blocking
– as interlaced
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Old stochastic versus new stochastic
new method (open symbols) has dramatically decreased variance
z test using a triply-displaced-T nucleon operator
z
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Old stochastic versus new stochastic (zoom in)
z
zoom in of triply-displaced-T nucleon plot on last slide
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Old stochastic versus new stochastic
z
comparison using single-site π operator
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Old stochastic versus new stochastic
z
zoom in of π plot on previous slide
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Mild volume dependence
163 lattice versus 203 lattice, both old and new stochastic methods
z test using triply-displaced-T nucleon operator
z
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40
Mild volume dependence
z
zoom in of plot on previous slide
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41
Source-sink factorization
z
baryon correlator has form
( l )∗ A B C
Cll = cijk( l ) cijk
Qii Q jj Qkk
z
stochastic estimates with dilution
Cll ≈
z
1
NR
∑∑
(
r d A d B dC
( l ) ( l )∗
cijk
cijk (ϕi( Ar )[ d A ]η i( Ar )[ d A ]∗ )
)(
× ϕ (j Br )[ d B ]η (j Br )[ d B ]∗ ϕ k(Cr )[ dC ]η k( Cr )[ dC ]∗
define
)
( l ) ( Ar )[ d A ] ( Br )[ d B ] ( Cr )[ dC ]
Γl( r )[ d Ad B dC ] = cijk
ϕi
ϕj
ϕk
( l ) ( Ar )[ d A ] ( Br )[ d B ] ( Cr )[ dC ]
Ωl( r )[ d Ad B dC ] = cijk
ηi
ηj
ηk
z
correlator becomes dot product of source vector with sink vector
1
( r )[ d A d B dC ]
( r )[ d A d B dC ]∗
Cll ≈
Γ
Ω
∑∑ l
l
N R r d A d B dC
z
store ABC permutations to handle Wick orderings
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Moving π and a mesons
z
first step towards including multi-hadron operators:
‰ moving single hadrons
‰ results below have one unit of on-axis momentum
‰ projections onto space group irreps (see J. Foley talk)
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43
Configuration generation
z
z
z
z
z
significant time on USQCD (DOE) and NSF computing resources
anisotropic clover fermion action (with stout links) and anisotropic
improved gauge action
‰ tunings of couplings, aspect ratio, lattice spacing done
anisotropic Wilson configurations generated during clover tuning
current goal:
‰ three lattice spacings: a = 0.125 fm, 0.10 fm, 0.08 fm
‰ three volumes: V = (3.2 fm)4, (4.0 fm)4, (5.0 fm)4
‰ 2+1 flavors, mπ ~ 350 MeV, 220 MeV, 180 MeV
USQCD Chroma software suite
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Resonances in a box: an example
Consider simple 1D quantum mechanics example
z Hamiltonian
2
z
H = 1 p 2 + V ( x)
2
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V ( x ) = ( x 4 − 3) e − x
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/2
45
1D example spectrum
z
Spectrum has two bound states, two resonances for E<4
transmission coefficient
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Scattering phase shifts
z
define even- and odd-parity phase shifts δ±
‰ phase between transmitted and incident wave
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Spectrum in box (periodic b.c.)
spectrum is discrete in box (momentum quantized)
z narrow resonance is avoided level crossing, broad resonance?
z
Dotted curves are V=0 spectrum
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Unstable particles (resonances)
our computations done in a periodic box
‰ momenta quantized
‰ discrete energy spectrum of stationary
states Æ single hadron, 2 hadron, …
z how to extract resonance info from box info?
z approach 1: crude scan
‰ if goal is exploration only Æ “ferret” out resonances
‰ spectrum in a few volumes
‰ placement, pattern of multi-particle states known
‰ resonances Æ level distortion near energy with little volume
dependence
‰ short-cut tricks of McNeile/Michael, Phys Lett B556, 177 (2003)
z
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49
Unstable particles (resonances)
z
approach 2: phase-shift method
‰ if goal is high precision Æ work much harder!
‰ relate finite-box energy of multi-particle
model to infinite-volume phase shifts
‰ evaluate energy spectrum in several volumes to compute phase
shifts using formula from previous step
‰ deduce resonance parameters from phase shifts
‰ early references
– B. DeWitt, PR 103, 1565 (1956) (sphere)
– M. Luscher, NPB364, 237 (1991) (ρ-ππ in cube)
approach 3: histogram method
‰ recent work for pion-nucleon system:
‰ V. Bernard et al, arXiv:0806.4495 [hep-lat]
z new approach: construct effective theory of hadrons?
z
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Summary
z
goal: to wring out hadron spectrum from QCD Lagrangian using
Monte Carlo methods on a space-time lattice
‰ baryons, mesons (and glueballs, hybrids, tetraquarks, …)
z
discussed extraction of excited states in Monte Carlo calculations
z
‰
correlation matrices needed
‰
operators with very good overlaps onto states of interest
must extract all states lying below a state of interest
‰
z
as pion get lighter, more and more multi-hadron states
multi-hadron operators Æ relative momenta
‰
need for slice-to-slice quark propagators
z
new stochastic Laph method Æ end game in sight?
z
interpretation of finite-box energies
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