Unearthing the excited hadron resonances in
lattice QCD using NSF XSEDE resources
Colin Morningstar
Carnegie Mellon University
APS Meeting
Savannah, GA
April 6, 2014
Outline
goals
comprehensive survey of spectrum of QCD stationary states in
finite volume
hadron scattering phase shifts, decay widths, matrix elements
focus: large 323 lattices, mπ ∼ 240 MeV, all 2-hadron operators
extracting excited-state energies
single-hadron and multi-hadron operators
the stochastic LapH method
XSEDE computing resources allow computations of
unprecedented scale
level identification issues
future work
C. Morningstar
Excited States
1
Dramatis Personae
Brendan Fahy
CMU
You-Cyuan Jhang
CMU
David Lenkner
CMU
John Bulava
Trinity, Dublin
Justin Foley
NVIDIA
Jimmy Juge
U Pacific, Stockton
C. Morningstar
CMU
Ricky Wong
UC San Diego
Thanks to NSF Teragrid/XSEDE:
Athena+Kraken at NICS
Ranger+Stampede at TACC
C. Morningstar
Excited States
2
Temporal correlations from path integrals
stationary-state energies from N × N Hermitian correlation matrix
Cij (t) = h0| Oi (t+t0 ) Oj (t0 ) |0i
judiciously designed operators Oj create states of interest
Oj (t) = Oj [ψ(t), ψ(t), U(t)]
correlators from path integrals over quark ψ, ψ and gluon U fields
R
Cij (t) =
D(ψ, ψ, U) Oi (t + t0 ) Oj (t0 ) exp −S[ψ, ψ, U]
R
D(ψ, ψ, U) exp −S[ψ, ψ, U]
involves the action
S[ψ, ψ, U] = ψ K[U] ψ + SG [U]
K[U] is fermion Dirac matrix
SG [U] is gluon action
C. Morningstar
Excited States
3
Integrating the quark fields
integrals over Grassmann-valued quark fields done exactly
meson-to-meson example:
Z
D(ψ, ψ) ψa ψb ψ c ψ d exp −ψKψ
−1 −1
−1 −1
= Kad
Kbc − Kac
Kbd det K.
baryon-to-baryon example:
Z
D(ψ, ψ) ψa1 ψa2 ψa3 ψ b1 ψ b2 ψ b3 exp −ψKψ
=
−Ka−1
Ka−1
Ka−1
+ Ka−1
Ka−1
Ka−1
+ Ka−1
Ka−1
Ka−1
1 b1
2 b2
3 b3
1 b1
2 b3
3 b2
1 b2
2 b1
3 b3
− Ka−1
Ka−1
Ka−1
− Ka−1
Ka−1
Ka−1
+ Ka−1
Ka−1
Ka−1
det K
1 b2
2 b3
3 b1
1 b3
2 b1
3 b2
1 b3
2 b2
3 b1
C. Morningstar
Excited States
4
Monte Carlo integration
correlators have form
R
Cij (t) =
DU det K[U] K −1 [U] · · · K −1 [U] exp (−SG [U])
R
DU det K[U] exp (−SG [U])
resort to Monte Carlo method to integrate over gluon fields
use Markov chain to generate sequence of gauge-field
configurations
U1 , U2 , . . . , UN
most computationally demanding parts:
including det K in updating
evaluating K −1 in numerator
C. Morningstar
Excited States
5
Lattice QCD
Monte Carlo method using computers requires hypercubic
space-time lattice
quarks reside on sites, gluons reside on links between sites
for gluons, 8 dimensional integral on each link
path integral dimension 32Nx Ny Nz Nt
268 million for 323×256 lattice
Metropolis method with global
updating proposal
RHMC: solve Hamilton equations
with Gaussian momenta
det K estimates with integral over
pseudo-fermion fields
systematic errors
− discretization
− finite volume
C. Morningstar
Excited States
6
Building blocks for single-hadron operators
building blocks: covariantly-displaced LapH-smeared quark fields
e j (x)
stout links U
Laplacian-Heaviside (LapH) smeared quark fields
e
ψeaα (x) = Sab (x, y) ψbα (y),
S = Θ σs2 + ∆
e in terms of U
e
3d gauge-covariant Laplacian ∆
displaced quark fields:
(A)
qAaαj = D(j) ψeaα
,
e (A) γ D(j)†
qAaαj = ψ
aα 4
displacement D(j) is product of smeared links:
e j1 (x) U
e j2 (x+d2 ) U
e j3 (x+d3 ) . . . U
e jp (x+dp )δx0 , x+dp+1
D(j) (x, x0 ) = U
to good approximation, LapH smearing operator is
S = Vs Vs†
e
columns of matrix Vs are eigenvectors of ∆
C. Morningstar
Excited States
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Extended operators for single hadrons
quark displacements build up orbital, radial structure
AB
Φαβ (p, t)
ABC
Φαβγ (p, t)
1
=
P
eip·(x+ 2 (dα +dβ )) δab qBbβ (x, t) qAaα (x, t)
=
P
eip·x εabc qCcγ (x, t) qBbβ (x, t) qAaα (x, t)
x
x
group-theory projections onto irreps of lattice symmetry group
(l)∗
AB
M l (t) = cαβ Φαβ (t)
(l)∗
ABC
Bl (t) = cαβγ Φαβγ (t)
definite momentum p, irreps of little group of p
C. Morningstar
Excited States
8
Two-hadron operators
our approach: superposition of products of single-hadron
operators of definite momenta
cIp3aa λI3ba ; pb λb BIpaaIΛ3aaSλa a ia BIpbbIΛ3bbSλb b ib
fixed total momentum p = pa + pb , fixed Λa , ia , Λb , ib
group-theory projections onto little group of p and isospin irreps
restrict attention to certain classes of momentum directions
on axis ±b
x, ±b
y, ±bz
planar diagonal ±b
x ±b
y, ±b
x ± bz, ±b
y ± bz
cubic diagonal ±b
x ±b
y ± bz
crucial to know and fix all phases of single-hadron operators for
all momenta
each class, choose reference direction pref
each p, select one reference rotation Rpref that transforms pref into p
efficient creating large numbers of two-hadron operators
generalizes to three, four, . . . hadron operators
C. Morningstar
Excited States
9
Quark propagation
quark propagator is inverse K −1 of Dirac matrix
rows/columns involve lattice site, spin, color
very large Ntot × Ntot matrix for each flavor
Ntot = Nsite Nspin Ncolor
for 323 × 256 lattice, Ntot ∼ 101 million
not feasible to compute (or store) all elements of K −1
solve linear systems Kx = y for source vectors y
translation invariance can drastically reduce number of source
vectors y needed
multi-hadron operators and isoscalar mesons require large
number of source vectors y
C. Morningstar
Excited States
10
Quark line diagrams
temporal correlations involving our two-hadron operators need
slice-to-slice quark lines (from all spatial sites on a time slice to all
spatial sites on another time slice)
sink-to-sink quark lines
isoscalar mesons also require sink-to-sink quark lines
solution: the stochastic LapH method!
C. Morningstar
Excited States
11
Stochastic estimation of quark propagators
do not need exact inverse of Dirac matrix K[U]
use noise vectors η satisfying E(ηi ) = 0 and E(ηi ηj∗ ) = δij
Z4 noise is used {1, i, −1, −i}
solve K[U]X (r) = η (r) for each of NR noise vectors η (r) , then
obtain a Monte Carlo estimate of all elements of K −1
NR
1 X
(r) (r)∗
Kij−1 ≈
Xi ηj
NR
r=1
variance reduction using noise dilution
dilution introduces projectors
X
P(a) P(b) = δ ab P(a) ,
P(a) = 1,
define
P(a)† = P(a)
a
η [a] = P(a) η,
X [a] = K −1 η [a]
to obtain Monte Carlo estimate with drastically reduced variance
NR X
1 X
(r)[a] (r)[a]∗
Kij−1 ≈
Xi
ηj
NR
a
r=1
C. Morningstar
Excited States
12
Stochastic LapH method
introduce ZN noise in the LapH subspace
ραk (t),
t = time, α = spin, k = eigenvector number
four dilution schemes:
(a)
Pij
(a)
Pij
(a)
Pij
(a)
Pij
= δij
= δij δai
= δij δa,Ki/N
= δij δa,i mod k
a=0
a = 0, 1, . . . , N −1
a = 0, 1, . . . , K −1
a = 0, 1, . . . , K −1
(none)
(full)
(interlace-K)
(block-K)
apply dilutions to
time indices (full for fixed src, interlace-16 for relative src)
spin indices (full)
LapH eigenvector indices (interlace-8 mesons, interlace-4 baryons)
C. Morningstar
Excited States
13
Quark line estimates in stochastic LapH
each of our quark lines is the product of matrices
Q = D(j) SK −1 γ4 SD(k)†
displaced-smeared-diluted quark source and quark sink vectors:
%[b] (ρ)
=
D(j) Vs P(b) ρ
ϕ[b] (ρ)
=
D(j) SK −1 γ4 Vs P(b) ρ
estimate in stochastic LapH by (A, B flavor, u, v compound:
space, time, color, spin, displacement type)
N
Q(AB)
≈
uv
R X
X
1
r
[b] r ∗
ϕ[b]
δAB
u (ρ ) %v (ρ )
NR
r=1
b
occasionally use γ5 -Hermiticity to switch source and sink
N
(AB)
Quv
≈
R X
X
1
r
[b] r ∗
δAB
%[b]
u (ρ ) ϕv (ρ )
NR
r=1
b
defining %(ρ) = −γ5 γ4 %(ρ) and ϕ(ρ) = γ5 γ4 ϕ(ρ)
C. Morningstar
Excited States
14
Source-sink factorization in stochastic LapH
baryon correlator has form
(l) (l)∗
Cll = cijk cijk QAii QBjj QCkk
stochastic estimate with dilution
1 X X (l) (l)∗ (Ar)[dA ] (Ar)[dA ]∗ Cll ≈
cijk cijk ϕi
%i
NR r
dA dB dC
(Br)[dB ] (Br)[dB ]∗
(Cr)[dC ] (Cr)[dC ]∗
×
ϕj
%j
ϕk
%k
define baryon source and sink
(r)[dA dB dC ]
Bl
(ϕA , ϕB , ϕC )
(r)[dA dB dC ]
Bl
(%A , %B , %C )
(l)
(Ar)[dA ]
(l)
(Ar)[dA ] (Br)[dB ] (Cr)[dC ]
%j
%k
= cijk ϕi
(Br)[dB ]
ϕj
(Cr)[dC ]
ϕk
= cijk %i
correlator is dot product of source vector with sink vector
1 X X (r)[dA dB dC ] A B C (r)[dA dB dC ] A B C ∗
Cll ≈
Bl
(ϕ , ϕ , ϕ )Bl
(% , % , % )
NR r
dA dB dC
C. Morningstar
Excited States
15
Correlators and quark line diagrams
baryon correlator
1 X X (r)[dA dB dC ] A B C (r)[dA dB dC ] A B C ∗
Cll ≈
Bl
(ϕ , ϕ , ϕ )Bl
(% , % , % )
NR r
dA dB dC
express diagrammatically
meson correlator
C. Morningstar
Excited States
16
More complicated correlators
two-meson to two-meson correlators (non isoscalar mesons)
C. Morningstar
Excited States
17
Use of XSEDE resources
use of XSEDE resources crucial
Monte Carlo generation of gauge-field configurations:
∼ 200 million core hours
quark propagators: ∼ 100 million core hours
hadrons + correlators: ∼ 40 million core hours
storage: ∼ 300 TB
Kraken at NICS
C. Morningstar
Stampede at TACC
Excited States
18
Excited states from correlation matrices
in finite volume, energies are discrete (neglect wrap-around)
X (n) (n)∗
(n)
Cij (t) =
Zi Zj
e−En t ,
Zj = h0| Oj |ni
n
not practical to do fits using above form
e using a single rotation
define new correlation matrix C(t)
e = U † C(τ0 )−1/2 C(t) C(τ0 )−1/2 U
C(t)
columns of U are eigenvectors of C(τ0 )−1/2 C(τD ) C(τ0 )−1/2
e diagonal for t > τD
choose τ0 and τD large enough so C(t)
!
effective masses
e αα (t)
1
C
eff
e α (t) =
m
ln
e αα (t + ∆t)
∆t
C
tend to N lowest-lying stationary state energies in a channel
e αα (t) yield energies Eα and overlaps Z (n)
2-exponential fits to C
j
C. Morningstar
Excited States
19
Quantum numbers in toroidal box
periodic boundary conditions in
cubic box
not all directions equivalent ⇒
using J PC is wrong!!
label stationary states of QCD in a periodic box using irreps of
cubic space group even in continuum limit
zero momentum states: little group Oh
A1a , A2ga , Ea , T1a , T2a , G1a , G2a , Ha ,
on-axis momenta: little group C4v
a = g, u
A1 , A2 , B1 , B2 , E, G1 , G2
planar-diagonal momenta: little group C2v
A1 , A2 , B1 , B2 , G1 , G2
cubic-diagonal momenta: little group C3v
A1 , A2 , E, F1 , F2 , G
include G parity in some meson sectors (superscript + or −)
C. Morningstar
Excited States
20
Importance of smeared fields
effective masses of
3 selected nucleon
operators shown
noise reduction of
displaced-operators
from link smearing
nρ ρ = 2.5, nρ = 16
quark-field
smearing
σs = 4.0, nσ = 32
reduces
excited-state
contamination
C. Morningstar
Excited States
21
The effectiveness of stochastic LapH
comparing use of lattice noise vs noise in LapH subspace
ND is number of solutions to Kx = y
C. Morningstar
Excited States
22
Ensembles and run parameters
plan to use three Monte Carlo ensembles
(323 |240): 412 configs 323 × 256,
(243 |240): 584 configs 243 × 128,
(243 |390): 551 configs 243 × 128,
mπ ≈ 240 MeV, mπ L ∼ 4.4
mπ ≈ 240 MeV, mπ L ∼ 3.3
mπ ≈ 390 MeV, mπ L ∼ 5.7
anisotropic improved gluon action, clover quarks (stout links)
QCD coupling β = 1.5 such that as ∼ 0.12 fm, at ∼ 0.035 fm
strange quark mass ms = −0.0743 nearly physical (using kaon)
work in mu = md limit so SU(2) isospin exact
generated using RHMC, configs separated by 20 trajectories
stout-link smearing in operators ξ = 0.10 and nξ = 10
LapH smearing cutoff σs2 = 0.33 such that
Nv = 112 for 243 lattices
Nv = 264 for 323 lattices
source times:
4 widely-separated t0 values on 243
8 t0 values used on 323 lattice
C. Morningstar
Excited States
23
First results
correlator software last_laph completed summer 2013
testing of all flavor channels for single and two-mesons completed
+
first focus on the resonance-rich ρ-channel: I = 1, S = 0, T1u
experiment: ρ(770), ρ(1450), ρ(1570), ρ3 (1690), ρ(1700)
first results: 56 × 56 matrix of correlators (243 |390) ensemble
12 single-hadron (quark-antiquark) operators
17 “ππ” operators
14 “ηπ” operators, 3 “φπ” operators
10 “KK” operators
inclusion of all possible 2-meson operators
3-meson operators currently neglected
still finalizing analysis code
C. Morningstar
Excited States
24
+
I = 1, S = 0, T1u
channel
at m
e eff (t) for levels 0 to 15
effective masses m
dashed lines show energies from single exponential fits
0.2
0.3
Level 0
at mFIT = 0.15331(70)
χ2/DOF = 0.98
0.18
0.35
Level 1
at mFIT = 0.21362(97)
χ2/DOF = 0.69
0.25
0.4
Level 2
at mFIT = 0.2496(10)
χ2/DOF = 0.93
0.3
Level 3
at mFIT = 0.2616(24)
χ2/DOF = 1.47
0.35
0.3
0.16
0.25
0.2
0.14
5
10
15
20
5
t
10
15
20
t
0.2
0.25
0.2
5
10
15
20
5
t
10
15
20
t
at m
0.5
Level 4
at mFIT = 0.2877(77)
χ2/DOF = 1.10
0.4
Level 5
at mFIT = 0.2962(48)
χ2/DOF = 1.24
0.4
Level 6
at mFIT = 0.3032(10)
χ2/DOF = 1.70
0.4
0.35
Level 7
at mFIT = 0.3037(76)
χ2/DOF = 1.75
0.6
0.4
0.3
0.3
0.2
5
10
15
20
t
0.2
0.3
0.25
5
10
15
20
0.2
5
t
10
15
20
5
t
10
15
20 t
at m
0.7
Level 8
at mFIT = 0.3092(33)
χ2/DOF = 1.27
0.4
Level 9
at mFIT = 0.321(18)
χ2/DOF = 1.77
0.6
0.5
0.35
0.6
Level 10
at mFIT = 0.323(15)
χ2/DOF = 2.51
0.6
Level 11
at mFIT = 0.3261(77)
χ2/DOF = 1.28
0.4
0.4
0.4
0.3
0.3
at m
0.25
5
0.6
10
15
t
0.2
15
20 t
Level 13
at mFIT = 0.3363(60)
χ2/DOF = 2.02
0.5
0.4
10
10
15
20 t
Level 14
at mFIT = 0.3381(26)
χ2/DOF = 1.35
0.5
0.4
5
10
15
20 t
Level 15
at mFIT = 0.342(96)
χ2/DOF = 0.94
0.6
0.5
0.4
0.3
0.2
5
0.6
0.6
Level 12
at mFIT = 0.3301(47)
χ2/DOF = 2.10
0.2
0.2
5
0.4
0.3
0.3
0.2
5
10
C. Morningstar
15
20
t
5
10
15
t
0.2
5
Excited States
10
15
t
0.2
5
10
15
t
25
+
I = 1, S = 0, T1u
energy extraction, continued
at m
e eff (t) for levels 16 to 31
effective masses m
dashed lines show energies from single exponential fits
0.6
0.6
Level 16
at mFIT = 0.3469(33)
χ2/DOF = 1.93
0.5
0.6
Level 17
at mFIT = 0.3489(51)
χ2/DOF = 0.49
0.5
0.4
Level 18
at mFIT = 0.3492(40)
χ2/DOF = 0.91
0.5
0.4
0.4
0.3
0.3
Level 19
at mFIT = 0.3505(67)
χ2/DOF = 0.54
0.6
0.4
0.3
at m
0.2
5
10
15
20
10
15
20
Level 20
at mFIT = 0.3532(30)
χ2/DOF = 1.43
0.4
Level 21
at mFIT = 0.3537(54)
χ2/DOF = 1.22
0.5
0.4
0.35
5
10
15
20
5
t
20 t
10
15
20
t
0.4
0.4
0.3
0.3
0.5
0.6
0.5
15
20
0.6
10
15
20 t
0.7
Level 28
at mFIT = 0.377(46)
χ2/DOF = 1.03
0.5
20
0.2
0.6
0.5
20
t
10
15
20 t
0.7
Level 26
at mFIT = 0.3684(24)
χ2/DOF = 1.59
Level 27
at mFIT = 0.3724(33)
χ2/DOF = 1.26
0.6
0.5
0.4
0.3
5
10
15
t
0.7
Level 29
at mFIT = 0.379(12)
χ2/DOF = 1.91
15
Level 23
at mFIT = 0.3598(90)
χ2/DOF = 1.77
5
t
0.3
5
t
15
0.4
0.3
10
10
0.5
0.4
10
0.2
5
0.6
0.3
5
0.2
5
0.5
0.7
Level 25
at mFIT = 0.3681(46)
χ2/DOF = 2.08
0.2
0.6
Level 22
at mFIT = 0.3577(65)
χ2/DOF = 0.53
0.5
0.7
Level 24
at mFIT = 0.3635(74)
χ2/DOF = 1.45
0.4
at m
15
0.2
0.6
0.2
10
0.3
0.3
0.25
5
t
0.6
0.45
at m
5
t
0.5
0.2
5
10
15
20 t
0.8
Level 30
at mFIT = 0.3958(31)
χ2/DOF = 1.01
0.6
0.5
Level 31
at mFIT = 0.3981(89)
χ2/DOF = 0.79
0.6
0.4
0.4
0.3
0.2
0.4
0.3
5
10
C. Morningstar
15
20 t
0.2
0.4
0.3
5
10
15
t
0.2
5
Excited States
10
15
t
0.2
5
10
15
t
26
+
I = 1, S = 0, T1u
energy extraction, continued
at m
e eff (t) for levels 32 to 47
effective masses m
dashed lines show energies from single exponential fits
0.8
Level 32
at mFIT = 0.4(98)
χ2/DOF = 0.72
0.6
Level 33
at mFIT = 0.4026(69)
χ2/DOF = 0.83
0.6
0.5
0.4
Level 34
at mFIT = 0.4091(96)
χ2/DOF = 0.98
0.6
Level 35
at mFIT = 0.41(13)
χ2/DOF = 1.03
0.6
0.4
0.4
0.4
0.2
at m
0.2
5
10
15
t
0.7
0.3
5
10
t
0.2
5
10
15
10
15
t
0.7
Level 36
at mFIT = 0.4116(65)
χ2/DOF = 0.79
0.6
0.5
Level 37
at mFIT = 0.441(10)
χ2/DOF = 0.10
0.6
0.5
0.4
0.4
0.3
0.3
0.2
0.2
Level 38
at mFIT = 0.455(13)
χ2/DOF = 2.21
0.6
0.5
5
10
t
0.8
5
10
t
0.8
Level 40
at mFIT = 0.465(13)
χ2/DOF = 1.59
0.6
Level 39
at mFIT = 0.458(24)
χ2/DOF = 0.35
0.6
0.4
0.4
at m
5
t
0.3
5
10
t
0.2
5
10
t
0.8
Level 41
at mFIT = 0.501(13)
χ2/DOF = 0.54
0.6
Level 42
at mFIT = 0.5152(76)
χ2/DOF = 1.34
0.6
Level 43
at mFIT = 0.525(11)
χ2/DOF = 1.89
0.8
0.6
0.4
0.4
0.4
0.4
at m
5
10
t
0.2
5
10
5
t
10
2
t
4
6
8
10 t
1.2
Level 44
at mFIT = 0.559(11)
χ2/DOF = 0.81
1
0.8
1
0.8
0.6
0.6
0.4
0.4
0.2
2
4
6
C. Morningstar
8
10 t
Level 45
at mFIT = 0.567(11)
χ2/DOF = 0.90
0.2
2
4
6
8
Level 46
at mFIT = 0.638(12)
χ2/DOF = 2.07
1
0.8
t
1
0.8
0.6
0.6
0.4
0.4
2
Excited States
4
6
Level 47
at mFIT = 0.6599(87)
χ2/DOF = 3.01
t8
2
4
6
t8
27
Level identification
level identification inferred from Z overlaps with probe operators
analogous to experiment: infer resonances from scattering cross
sections
keep in mind:
probe operators Oj act on vacuum, create a “probe state” |Φj i,
Z’s are overlaps of probe state with each eigenstate
(n)
|Φj i ≡ Oi |0i,
Zj = hΦj |ni
have limited control of “probe states” produced by probe operators
ideal to be ρ, single ππ, and so on
use of small−a expansions to characterize probe operators
use of smeared quark, gluon fields
field renormalizations
mixing is prevalent
identify by dominant probe state(s) whenever possible
C. Morningstar
Excited States
28
Level identification
-
π A 2 SS1 π A 2 SS1 OA
3
c
K A2 SS1 K A 2 SS1 OA
π(140) π(140)
2
|Z|2
-
|Z|2
|Z|2
overlaps for various operators
-
-
π A 2 SS0 π A 2 SS0 PD
π(140) π(140)
c
K(497) K (497)
2
2
1
1
1
20
-
30
40
0
50
N
-
η E SS1 π A 2 LSD1 OA
10
20
30
40
0
50
N
c
K A2 SS0 K A 2 SS0 PD
2
ω(782) π(140)
2
0
|Z|2
10
|Z|2
0
|Z|2
0
0
10
4
20
40
50
N
-
φ(1020) π(140)
c
K(497) K (497)
30
-
φ E SS1 π A 2 SS1 OA
3
2
1
1
1
20
30
-
40
0
50
N
0
10
-
π A 2 SS0 π A 2 SS0 CD
20
-
40
1u
-
10
C. Morningstar
20
30
40
50
N
0
10
20
-
30
40
50
N
-
π A 2 SS1 π A 2 TSD0 OA
π(140) π(1300)
0.5
0.5
0
0
1
ω(782) a0(980)
0.5
0
0
50
N
η T SS0 π A 1g SS0
1
π(140) π(140)
1
30
|Z|2
10
|Z|2
0
|Z|2
0
0
10
20
30
Excited States
40
50
N
0
0
10
20
30
40
50
N
29
Small−a expansion of probes
illustrate problem with real scalar field ϕ(x)
consider three operators Φj for j = 1, 2, 3 defined by
1
ϕ(x + bj) − ϕ(x − bj) .
Φj (x) =
2a
this is forward-backward finite difference approx to derivative
carry T1 irrep of the octahedral point group O
classical small-a expansion:
1
Φj (x) ≈ ∂j ϕ(x) + a2 ∂j3 ϕ(x) + O(a4 ).
6
first term ∂j ϕ(x) is spin J = 1
second term contains both J = 1 and J = 3
radiative corrections modify relative weights (calculate in lattice
perturbation theory, but difficult)
C. Morningstar
Excited States
30
Small−a expansion of probes
link variables in terms of continuum gluon field
( Z
)
x+µ̂
Uµ (x) = P exp ig
dη · A(η) ,
x
classical small−a expansion of displaced quark field:
Uj (x)Uk (x + bj)ψα (x + bj + b
k) = exp(aDj ) exp(aDk ) ψα (x).
where Dj = ∂j + igAj is covariant derivative
must take smearing of fields into account
radiative corrections of expansion coefficients (hopefully small
due to smearing)
work in progress for our probe operators
C. Morningstar
Excited States
31
J PG of continuum probe operators
isovector meson continuum probe operators
Mµj1 j2 ··· = χd Γµ Dj1 Dj2 · · · ψ u ,
χ = ψγ4
where Γ0 = 1 and Γk = γk (analogous table inserting γ4 , γ5 , γ4 γ5 )
J PG
0++
1−+
1−−
0+−
1+−
2+−
0++
1+−
2++
C. Morningstar
OG
h irrep
A+
1g
+
T1u
−
T1u
−
A1g
−
T1g
Eg−
−
T2g
A+
1g
−
T1g
Eg+
+
T2g
Basis operator
M0
M1
M01
M11 + M22 + M33
M23 − M32
M11 − M22
M23 + M32
M011 + M022 + M033
M023 − M032
M011 − M022
M023 + M032
Excited States
32
J PG of continuum probe operators (continued)
isovector meson continuum probe operators
Mµj1 j2 ··· = χd Γµ Dj1 Dj2 · · · ψ u ,
χ = ψγ4
where Γ0 = 1 and Γk = γk (analogous table inserting γ4 , γ5 , γ4 γ5 )
J PG
0−−
1−+
1−+
1−−
2−−
2−+
3−+
OG
h irrep
A−
1u
+
T1u
+
T1u
−
T1u
Eu−
−
T2u
Eu+
+
T2u
+
A2u
+
T1u
+
T2u
C. Morningstar
Basis operator
M123 + M231 + M312 − M321 − M213 − M132
M111 + M122 + M133
2M111 + M221 + M331 + M212 + M313
M221 + M331 − M212 − M313
M123 + M213 − M231 − M132
M221 − M331 + M313 − M212
M123 + M213 − 2M321 − 2M312 + M231 + M132
M221 − M331 − 2M122 + 2M133 − M313 + M212
M123 + M231 + M312 + M213 + M321 + M132
2M111 − M221 − M331 − M212 − M313 − M122 − M133
M331 − M212 + M313 − M122 + M133 − M221
Excited States
33
Identifying resonances
resonances: finite-volume “precursor states”
probes: optimized single-hadron operators
[SH]
analyze matrix of just single-hadron operators Oi (12 × 12)
perform single-rotation as before to build probe operators
P 0(m)∗ [SH]
0[SH]
Om = i vi
Oi
obtain Z 0 factors of these probe operators
2
2
SH-Optimized 1
|Z|
SH-Optimized 0
2
|Z|
2
|Z|
|Z|
2
Zm0(n) = h0| O0[SH]
|ni
m
SH-Optimized 2
0.6
1
SH-Optimized 3
2
1.5
0.4
1
0.5
1
0.2
0.5
20
30
40
50
N
0
10
20
30
40
1.5
0
10
20
|Z|
|Z|
|Z|
SH-Optimized 4
0
50
N
2
10
2
0
0
30
40
50
N
0
0
10
20
30
40
2
0
SH-Optimized 5
4
SH-Optimized 6
10
1
2
5
0.5
0
0
10
20
30
40
C. Morningstar
50
N
0
0
10
20
30
40
50
N
0
0
10
Excited States
20
30
40
50
N
34
50
N
List of tentative level identifications
Level
0
1
2
3
4
5
6
7
8
9
11
12
13
14
C. Morningstar
Dominant Probe
ρ(770)
π(140)π(140) (OA)
K(497)K c (497) (OA)
π(140)π(140) (PD)
ω(782)π(140 (OA)
K(497)K c (497) (PD)
φ(1020)π(140) (OA)
π(140)π(140) (CD) (12)
ρ(1450)
η(547)ρ(770) (OA)
π(140)a1 (1260) (AR)
π(140)π(140) (CD) (7)
ρ(1570)
K ∗ (892)K c (497) (OA)
Level
16
17
18
20
21
22
30
31
36
40
41
46
48
Excited States
Dominant Probe
ω(782)π(140) (PD)
φ(1020)π(140) (PD)
φ(1020)π(140) (PD)
K(497)K c (497) (CD)
K(497)K c (497) (CD)
ρ(770)ρ(770) (OA)
η(547)ρ(770) (PD)
ρ(1690)
h1 (1170)π(140) (OA)
ρ(1700)
η(547)b1 (1235) (AR)
ρ(?)
ρ(?)
35
Summary and comparison with experiment
3
3
2.5
2.5
2
ρ(1700)
1.5
ρ(1570)
1
ρ3(1690)
2
ρ(1450)
1.5
ρ(770)
0.5
1
0.5
0
0
Lattice
C. Morningstar
m / mN
m / mN
left: energies of qq-dominant states as ratios over mN for
(243 |390) ensemble (resonance precursor states)
right: experiment
Experiment
Excited States
36
Issues
address presence of 3 and 4 meson states
in other channels, address scalar particles in spectrum
scalar probe states need vacuum subtractions
hopefully can neglect due to OZI suppression
infinite-volume resonance parameters from finite-volume
energies
Luscher method too cumbersome, restrictive in applicability
need for new hadron effective field theory techniques
C. Morningstar
Excited States
37
Bosonic I = 12 , S = 1, T1u channel
now starting to get results for 323 × 256 lattice with pion mass
mπ ∼ 240 MeV
example: kaon channel: I = 12 , S = 1, T1u
experiment: K ∗ (892), K ∗ (1410), K ∗ (1680), K3∗ (1780)
results: 103 × 103 matrix of correlators (323 |240) ensemble
25 single-hadron (quark-antiquark) operators
35 “Kπ” operators
21 “Kη” operators, 22 “Kφ” operators
C. Morningstar
Excited States
38
Bosonic I = 12 , S = 1, T1u channel
e eff (t) for levels 0 to 8
effective masses m
dashed lines show energies from two exponential fits
C. Morningstar
Excited States
39
Bosonic I = 12 , S = 1, T1u channel
e eff (t) for levels 9 to 17
effective masses m
dashed lines show energies from two exponential fits
C. Morningstar
Excited States
40
Bosonic I = 12 , S = 1, T1u channel
e eff (t) for levels 18 to 23
effective masses m
dashed lines show energies from single exponential fits
C. Morningstar
Excited States
41
Bosonic I = 12 , S = 1, T1u channel
spectrum of single- and two-meson energies: “staircase” plot
C. Morningstar
Excited States
42
Bosonic I = 1, S = 0, A−
1u channel
spectrum of single- and two-meson energies: “staircase” plot
C. Morningstar
Excited States
43
Bosonic I = 1, S = 0, A−
2g channel
spectrum of single- and two-meson energies: “staircase” plot
C. Morningstar
Excited States
44
Bosonic I = 1, S = 0, A+
2g channel
spectrum of single- and two-meson energies: “staircase” plot
C. Morningstar
Excited States
45
Bosonic I = 1, S = 0, A−
2u channel
spectrum of single- and two-meson energies: “staircase” plot
C. Morningstar
Excited States
46
Bosonic I = 1, S = 0, A+
2u channel
spectrum of single- and two-meson energies: “staircase” plot
C. Morningstar
Excited States
47
References
S. Basak et al., Group-theoretical construction of extended
baryon operators in lattice QCD, Phys. Rev. D 72, 094506 (2005).
S. Basak et al., Lattice QCD determination of patterns of excited
baryon states, Phys. Rev. D 76, 074504 (2007).
C. Morningstar et al., Improved stochastic estimation of quark
propagation with Laplacian Heaviside smearing in lattice QCD,
Phys. Rev. D 83, 114505 (2011).
C. Morningstar et al., Extended hadron and two-hadron operators
of definite momentum for spectrum calculations in lattice QCD,
Phys. Rev. D 88, 014511 (2013).
C. Morningstar
Excited States
48
Conclusion and future work
goal: comprehensive survey of energy spectrum of QCD
stationary states in a finite volume
stochastic LapH method works very well
allows evaluation of all needed quark-line diagrams
source-sink factorization facilitates large number of operators
last_laph software completed for evaluating correlators
+
showed first results in ρ-channel: I = 1, S = 0, T1u
using 56 × 56
matrix of correlators
now starting to get results in large number of channels in
323 × 256 lattice with mπ ∼ 240 MeV
can evaluate and analyze correlator matrices of unprecedented
size 100 × 100 due to XSEDE resources
much work still to do, especially with level identification
study various scattering phase shifts also planned
infinite-volume resonance parameters from finite-volume
energies −→ need new effective field theory techniques
C. Morningstar
Excited States
49