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Excited states and scattering phase shifts from lattice QCD · Stockton, CA past CMU grad students:...
Transcript of Excited states and scattering phase shifts from lattice QCD · Stockton, CA past CMU grad students:...
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Excited states and scatteringphase shifts from lattice QCD
Colin MorningstarCarnegie Mellon University
Workshop of the APS Topical Group on Hadron Physics
Baltimore, MD
April 9, 2015
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Overview
goals:comprehensive survey of QCD stationary states in finite volumehadron scattering phase shifts, decay widths, matrix elementsfocus: large 323 anisotropic lattices, mπ ∼ 240 MeV
extracting excited-stateenergiessingle-hadron andmulti-hadron operatorsthe stochastic LapH methodlevel identification issuesresults for I = 1, S = 0, T+
1u channel100× 100 correlator matrix, all needed 2-hadron operators
other channelsI = 1 P-wave ππ scattering phase shifts and width of ρfuture work
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Dramatis Personae
current grad students: former CMU postdocs:
Jake FallicaCMU
Andrew HanlonPitt
Justin FoleySoftware,NVIDIA
Jimmy JugeFaculty,
Stockton, CA
past CMU grad students:
Brendan Fahy2014
Postdoc KEKJapan
You-CyuanJhang2013
Silicon Valley
David Lenkner2013
Data ScienceAuto., PGH
Ricky Wong2011
PostdocGermany
John Bulava2009
Faculty,Dublin
Adam Lichtl2006
SpaceX, LA
thanks to NSF Teragrid/XSEDE:Athena+Kraken at NICSRanger+Stampede at TACC
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Temporal correlations from path integrals
stationary-state energies from N×N Hermitian correlation matrix
Cij(t) = 〈0|Oi(t+t0) Oj(t0) |0〉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
Cij(t) =
∫D(ψ,ψ,U) Oi(t + t0) Oj(t0) exp
(−S[ψ,ψ,U]
)∫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 matrixSG[U] is gluon action
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Integrating the quark fields
integrals over Grassmann-valued quark fields done exactlymeson-to-meson example:∫
D(ψ,ψ) ψaψb ψcψd exp(−ψKψ
)=
(K−1
ad K−1bc − K−1
ac K−1bd
)det K.
baryon-to-baryon example:∫D(ψ,ψ) ψa1ψa2ψa3 ψb1
ψb2ψb3
exp(−ψKψ
)=
(−K−1
a1b1K−1
a2b2K−1
a3b3+ K−1
a1b1K−1
a2b3K−1
a3b2+ K−1
a1b2K−1
a2b1K−1
a3b3
− K−1a1b2
K−1a2b3
K−1a3b1− K−1
a1b3K−1
a2b1K−1
a3b2+ K−1
a1b3K−1
a2b2K−1
a3b1
)det K
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Monte Carlo integration
correlators have form
Cij(t) =
∫DU det K[U] K−1[U] · · ·K−1[U] exp (−SG[U])∫
DU det K[U] exp (−SG[U])
resort to Monte Carlo method to integrate over gluon fieldsuse Markov chain to generate sequence of gauge-fieldconfigurations
U1,U2, . . . ,UN
most computationally demanding parts:including det K in updatingevaluating K−1 in numerator
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Lattice QCD
Monte Carlo method using computers requires hypercubicspace-time latticequarks reside on sites, gluons reside on links between sitesfor gluons, 8 dimensional integral on each link
path integral dimension 32NxNyNzNt
268 million for 323×256 lattice
Metropolis method with globalupdating proposal
RHMC: solve Hamilton equationswith Gaussian momentadet K estimates with integral overpseudo-fermion fields
systematic errors− discretization− finite volume
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Excited states from correlation matrices
in finite volume, energies are discrete (neglect wrap-around)
Cij(t) =∑
n
Z(n)i Z(n)∗
j e−Ent, Z(n)j = 〈0| Oj |n〉
not practical to do fits using above formdefine new correlation matrix C(t) using a single rotation
C(t) = U† C(τ0)−1/2 C(t) C(τ0)−1/2 U
columns of U are eigenvectors of C(τ0)−1/2 C(τD) C(τ0)−1/2
choose τ0 and τD large enough so C(t) diagonal for t > τD
effective energiesmeffα (t) =
1∆t
ln
(Cαα(t)
Cαα(t + ∆t)
)tend to N lowest-lying stationary state energies in a channel
2-exponential fits to Cαα(t) yield energies Eα and overlaps Z(n)j
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Building blocks for single-hadron operators
building blocks: covariantly-displaced LapH-smeared quark fieldsstout links Uj(x)
Laplacian-Heaviside (LapH) smeared quark fields
ψaα(x) = Sab(x, y) ψbα(y), S = Θ(σ2
s + ∆)
3d gauge-covariant Laplacian ∆ in terms of U
displaced quark fields:
qAaαj = D(j)ψ(A)
aα , qAaαj = ψ
(A)
aα γ4 D(j)†
displacement D(j) is product of smeared links:
D(j)(x, x′) = Uj1(x) Uj2(x+d2) Uj3(x+d3) . . . Ujp(x+dp)δx′, x+dp+1
to good approximation, LapH smearing operator is
S = VsV†scolumns of matrix Vs are eigenvectors of ∆
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Extended operators for single hadrons
quark displacements build up orbital, radial structure
ΦABαβ(p, t) =
∑x eip·(x+ 1
2 (dα+dβ))δab qBbβ(x, t) qA
aα(x, t)
ΦABCαβγ(p, t) =
∑x eip·xεabc qC
cγ(x, t) qBbβ(x, t) qA
aα(x, t)
group-theory projections onto irreps of lattice symmetry group
Ml(t) = c(l)∗αβ Φ
ABαβ(t) Bl(t) = c(l)∗
αβγ ΦABCαβγ(t)
definite momentum p, irreps of little group of p
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Importance of smeared fields
effective masses of3 selected nucleonoperators shownnoise reduction ofdisplaced-operatorsfrom link smearingnρρ = 2.5, nρ = 16quark-fieldsmearingσs = 4.0, nσ = 32reducesexcited-statecontamination
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Early results on small 163 and 243 lattices
Bob Sugar in 2005: “You’ll never see more than 2 levels”I = 1, S = 0 energies on 243 lattice, mπ ∼ 390 MeV in 2010use of single-meson operators onlyshaded region shows where two-meson energies expected
+
1gA+
2gA+gE
+
1gT+
2gT+
1uA+
2uA+uE
+
1uT+
2uT
1gA
2gA
gE
1gT
2gT
1uA
2uA
uE
1uT
2uT
1b
ρ
0a
π
nu
cle
on
m/m
0
0.5
1
1.5
2
2.5
3
3.5 * E
ta
0
0.2
0.4
0.6
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Early results on small lattices
kaons on 163 lattice, mπ ∼ 390 MeV in 2008use of single-meson operators only
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Early results on small lattices
N, ∆ baryons on 163 lattice, mπ ∼ 390 MeV in 2008use of single-baryon operators only
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Early results on small lattices
Σ, Λ, Ξ baryons on 163 lattice, mπ ∼ 390 MeV in 2008use of single-baryon operators only
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Two-hadron operators
our approach: superposition of products of single-hadronoperators of definite momenta
cI3aI3bpaλa; pbλb
BIaI3aSapaΛaλaia BIbI3bSb
pbΛbλbib
fixed total momentum p = pa + pb, fixed Λa, ia,Λb, ibgroup-theory projections onto little group of p and isospin irrepsrestrict attention to certain classes of momentum directions
on axis ±x, ±y, ±zplanar diagonal ±x± y, ±x± z, ±y± zcubic diagonal ±x± y± z
crucial to know and fix all phases of single-hadron operators forall momenta
each class, choose reference direction prefeach p, select one reference rotation Rp
ref that transforms pref into p
efficient creating large numbers of two-hadron operatorsgeneralizes to three, four, . . . hadron operators
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Quark propagation
quark propagator is inverse K−1 of Dirac matrixrows/columns involve lattice site, spin, colorvery large Ntot × Ntot matrix for each flavor
Ntot = NsiteNspinNcolor
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 sourcevectors y neededmulti-hadron operators and isoscalar mesons require largenumber of source vectors y
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Quark line diagrams
temporal correlations involving our two-hadron operators needslice-to-slice quark lines (from all spatial sites on a time slice to allspatial sites on another time slice)sink-to-sink quark lines
isoscalar mesons also require sink-to-sink quark lines
solution: the stochastic LapH method!
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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(ηiη∗j ) = δij
Z4 noise is used 1, i,−1,−isolve K[U]X(r) = η(r) for each of NR noise vectors η(r), thenobtain a Monte Carlo estimate of all elements of K−1
K−1ij ≈
1NR
NR∑r=1
X(r)i η
(r)∗j
variance reduction using noise dilutiondilution introduces projectors
P(a)P(b) = δabP(a),∑
a
P(a) = 1, P(a)† = P(a)
defineη[a] = P(a)η, X[a] = K−1η[a]
to obtain Monte Carlo estimate with drastically reduced variance
K−1ij ≈
1NR
NR∑r=1
∑a
X(r)[a]i η
(r)[a]∗j
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Stochastic LapH method
introduce ZN noise in the LapH subspace
ραk(t), t = time, α = spin, k = eigenvector number
four dilution schemes:
P(a)ij = δij a = 0 (none)
P(a)ij = δijδai a = 0, 1, . . . ,N−1 (full)
P(a)ij = δijδa,Ki/N a = 0, 1, . . . ,K−1 (interlace-K)
P(a)ij = δijδa,i mod k a = 0, 1, . . . ,K−1 (block-K)
apply dilutions totime indices (full for fixed src, interlace-16 for relative src)spin indices (full)LapH eigenvector indices (interlace-8 mesons, interlace-4 baryons)
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The effectiveness of stochastic LapH
comparing use of lattice noise vs noise in LapH subspaceND is number of solutions to Kx = y
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Quark line estimates in stochastic LapH
each of our quark lines is the product of matrices
Q = D(j)SK−1γ4SD(k)†
displaced-smeared-diluted quark source and quark sink vectors:
%[b](ρ) = D(j)VsP(b)ρ
ϕ[b](ρ) = D(j)SK−1γ4 VsP(b)ρ
estimate in stochastic LapH by (A,B flavor, u, v compound:space, time, color, spin, displacement type)
Q(AB)uv ≈ 1
NRδAB
NR∑r=1
∑b
ϕ[b]u (ρr) %[b]
v (ρr)∗
occasionally use γ5-Hermiticity to switch source and sink
Q(AB)uv ≈ 1
NRδAB
NR∑r=1
∑b
%[b]u (ρr) ϕ[b]
v (ρr)∗
defining %(ρ) = −γ5γ4%(ρ) and ϕ(ρ) = γ5γ4ϕ(ρ)
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Source-sink factorization in stochastic LapH
baryon correlator has form
Cll = c(l)ijk c(l)∗
ijkQA
iiQBjjQ
Ckk
stochastic estimate with dilution
Cll ≈1
NR
∑r
∑dAdBdC
c(l)ijk c(l)∗
ijk
(ϕ
(Ar)[dA]i %
(Ar)[dA]∗i
)×
(ϕ
(Br)[dB]j %
(Br)[dB]∗j
)(ϕ
(Cr)[dC]k %
(Cr)[dC]∗k
)define baryon source and sink
B(r)[dAdBdC]l (ϕA, ϕB, ϕC) = c(l)
ijk ϕ(Ar)[dA]i ϕ
(Br)[dB]j ϕ
(Cr)[dC]k
B(r)[dAdBdC]l (%A, %B, %C) = c(l)
ijk %(Ar)[dA]i %
(Br)[dB]j %
(Cr)[dC]k
correlator is dot product of source vector with sink vector
Cll ≈1
NR
∑r
∑dAdBdC
B(r)[dAdBdC]l (ϕA, ϕB, ϕC)B(r)[dAdBdC]
l(%A, %B, %C)∗
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Correlators and quark line diagrams
baryon correlator
Cll ≈1
NR
∑r
∑dAdBdC
B(r)[dAdBdC]l (ϕA, ϕB, ϕC)B(r)[dAdBdC]
l(%A, %B, %C)∗
express diagrammatically
meson correlator
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More complicated correlators
two-meson to two-meson correlators (non isoscalar mesons)
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Quantum numbers in toroidal box
periodic boundary conditions incubic box
not all directions equivalent⇒using JPC is wrong!!
label stationary states of QCD in a periodic box using irreps ofcubic space group even in continuum limit
zero momentum states: little group Oh
A1a,A2ga,Ea, T1a, T2a, G1a,G2a,Ha, a = g, uon-axis momenta: little group C4v
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 −)
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Spin content of cubic box irreps
numbers of occurrences of Λ irreps in J subduced
J A1 A2 E T1 T2
0 1 0 0 0 01 0 0 0 1 02 0 0 1 0 13 0 1 0 1 14 1 0 1 1 15 0 0 1 2 16 1 1 1 1 27 0 1 1 2 2
J G1 G2 H J G1 G2 H12 1 0 0 9
2 1 0 232 0 0 1 11
2 1 1 252 0 1 1 13
2 1 2 272 1 1 1 15
2 1 1 3
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Common hadrons
irreps of commonly-known hadrons at rest
Hadron Irrep Hadron Irrep Hadron Irrep
π A−1u K A1u η, η′ A+1u
ρ T+1u ω, φ T−1u K∗ T1u
a0 A+1g f0 A+
1g h1 T−1g
b1 T+1g K1 T1g π1 T−1u
N,Σ G1g Λ,Ξ G1g ∆,Ω Hg
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Ensembles and run parameters
plan to use three Monte Carlo ensembles(323|240): 412 configs 323 × 256, mπ ≈ 240 MeV, mπL ∼ 4.4(243|240): 584 configs 243 × 128, mπ ≈ 240 MeV, mπL ∼ 3.3(243|390): 551 configs 243 × 128, mπ ≈ 390 MeV, mπL ∼ 5.7
anisotropic improved gluon action, clover quarks (stout links)QCD coupling β = 1.5 such that as ∼ 0.12 fm, at ∼ 0.035 fmstrange quark mass ms = −0.0743 nearly physical (using kaon)work in mu = md limit so SU(2) isospin exactgenerated using RHMC, configs separated by 20 trajectories
stout-link smearing in operators ξ = 0.10 and nξ = 10
LapH smearing cutoff σ2s = 0.33 such that
Nv = 112 for 243 latticesNv = 264 for 323 lattices
source times:4 widely-separated t0 values on 243
8 t0 values used on 323 lattice
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Use of XSEDE resources
use of XSEDE resources crucialMonte Carlo generation of gauge-field configurations:∼ 200 million core hoursquark propagators: ∼ 100 million core hourshadrons + correlators: ∼ 40 million core hoursstorage: ∼ 300 TB
Kraken at NICS Stampede at TACC
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Status report
correlator software last_laph completed summer 2013testing of all flavor channels for single and two-mesons completedfall 2013testing of all flavor channels for single baryon and meson-baryonscompleted summer 2014
small-a expansions of all operators completedfirst focus on the resonance-rich ρ-channel: I = 1, S = 0, T+
1u
results from 63× 63 matrix of correlators (323|240) ensemble10 single-hadron (quark-antiquark) operators“ππ” operators“ηπ” operators, “φπ” operators“KK” operators
inclusion of all possible 2-meson operators3-meson operators currently neglectedstill finalizing analysis code sigmondnext focus: the 20 bosonic channels with I = 1, S = 0
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Operator accounting
numbers of operators for I = 1, S = 0, P = (0, 0, 0) on 323 lattice
(322|240) A+1g A+
1u A+2g A+
2u E+g E+
u T+1g T+
1u T+2g T+
2uSH 9 7 13 13 9 9 14 23 15 16“ππ” 10 17 8 11 8 17 23 30 17 27“ηπ” 6 15 10 7 11 18 31 20 21 23“φπ” 6 15 9 7 12 19 37 11 23 23“KK” 0 5 3 5 3 6 9 12 5 10Total 31 59 43 43 43 69 114 96 81 99
(322|240) A−1g A−1u A−2g A−2u E−g E−u T−1g T−1u T−2g T−2uSH 10 8 11 10 12 9 21 15 19 16“ππ” 3 7 7 3 8 11 22 12 12 15“ηπ” 26 15 10 12 24 21 25 33 28 30“φπ” 26 15 10 12 27 22 26 38 30 32“KK” 11 3 4 2 11 5 12 5 12 6Total 76 48 42 39 82 68 106 103 101 99
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Operator accounting
numbers of operators for I = 1, S = 0, P = (0, 0, 0) on 243 lattice
(242|390) A+1g A+
1u A+2g A+
2u E+g E+
u T+1g T+
1u T+2g T+
2uSH 9 7 13 13 9 9 14 23 15 16“ππ” 6 12 2 6 8 9 15 17 10 12“ηπ” 2 10 8 4 8 11 21 14 14 13“φπ” 2 10 8 4 8 11 23 3 14 13“KK” 0 4 1 4 1 4 8 10 4 6Total 19 43 32 31 34 44 81 67 57 60
(242|390) A−1g A−1u A−2g A−2u E−g E−u T−1g T−1u T−2g T−2uSH 10 8 11 10 12 9 20 15 19 16“ππ” 1 5 6 2 3 7 18 8 10 9“ηπ” 19 9 4 6 13 12 11 18 15 14“φπ” 18 9 4 6 14 12 11 19 15 15“KK” 7 2 2 2 6 4 9 4 8 4Total 55 33 27 26 48 44 69 64 67 58
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I = 1, S = 0, T+1u channel
effective energies meff(t) for levels 0 to 24energies obtained from two-exponential fits
0.0
0.1
0.2
0.3
0.4
0.5
mef
f
0.0
0.1
0.2
0.3
0.4
0.5
mef
f
0.0
0.1
0.2
0.3
0.4
0.5
mef
f
0.0
0.1
0.2
0.3
0.4
0.5
mef
f
5 10 15 20
time
0.0
0.1
0.2
0.3
0.4
0.5
mef
f
5 10 15 20
time5 10 15 20
time5 10 15 20
time5 10 15 20
time
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I = 1, S = 0, T+1u energy extraction, continued
effective energies meff(t) for levels 25 to 49energies obtained from two-exponential fits
0.1
0.2
0.3
0.4
0.5
0.6
mef
f
0.1
0.2
0.3
0.4
0.5
0.6
mef
f
0.1
0.2
0.3
0.4
0.5
0.6
mef
f
0.1
0.2
0.3
0.4
0.5
0.6
mef
f
5 10 15 20
time
0.1
0.2
0.3
0.4
0.5
0.6
mef
f
5 10 15 20
time5 10 15 20
time5 10 15 20
time5 10 15 20
time
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Level identification
level identification inferred from Z overlaps with probe operatorsanalogous to experiment: infer resonances from scattering crosssectionskeep in mind:
probe operators Oj act on vacuum, create a “probe state” |Φj〉,Z’s are overlaps of probe state with each eigenstate
|Φj〉 ≡ Oi|0〉, Z(n)j = 〈Φj|n〉
have limited control of “probe states” produced by probe operatorsideal to be ρ, single ππ, and so onuse of small−a expansions to characterize probe operatorsuse of smeared quark, gluon fieldsfield renormalizations
mixing is prevalentidentify by dominant probe state(s) whenever possible
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Level identification
overlaps for various operators
N0 10 20 30 40 50
2|Z
|
0
1
2
SS1 OA-2 Aπ SS1 -
2 Aπ
(140)π(140) π
N0 10 20 30 40 50
2|Z
|
0
1
2
3 SS1 OA2 Ac SS1 K2K A
(497)cK(497) K
N0 10 20 30 40 50
2|Z
|
0
1
2
SS0 PD-2 Aπ SS0 -
2 Aπ
(140)π(140) π
N0 10 20 30 40 50
2|Z
|
0
1
2
LSD1 OA-2 Aπ SS1 - Eη
(140)π(782) ω
N0 10 20 30 40 50
2|Z
|
0
1
2
SS0 PD2 Ac SS0 K2K A
(497)cK(497) K
N0 10 20 30 40 50
2|Z
|
0
1
2
3
4 SS1 OA-
2 Aπ SS1 - Eφ
(140)π(1020) φ
N0 10 20 30 40 50
2|Z
|
0
0.5
1
SS0 CD-2 Aπ SS0 -
2 Aπ
(140)π(140) π
N0 10 20 30 40 50
2|Z
|
0
0.5
1 SS0-1g Aπ SS0 -
1u Tη
(980)0
(782) aω
N0 10 20 30 40 50
2|Z
|
0
0.5
1
TSD0 OA-2 Aπ SS1 -
2 Aπ
(1300)π(140) π
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Identifying quark-antiquark resonances
resonances: finite-volume “precursor states”probes: optimized single-hadron operators
analyze matrix of just single-hadron operators O[SH]i (12× 12)
perform single-rotation as before to build probe operatorsO′[SH]
m =∑
i v′(m)∗i O[SH]
i
obtain Z′ factors of these probe operators
Z′(n)m = 〈0| O′[SH]
m |n〉
0 10 20 30 40 50
Level
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
|Z|2
0 10 20 30 40 50
Level
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
|Z|2
0 10 20 30 40 50
Level
0.0
0.1
0.2
0.3
0.4
0.5
0.6
|Z|2
0 10 20 30 40 50
Level
0.0
0.1
0.2
0.3
0.4
0.5
0.6
|Z|2
0 10 20 30 40 50
Level
0.0
0.1
0.2
0.3
0.4
|Z|2
0 10 20 30 40 50
Level
0.0
0.1
0.2
0.3
0.4
0.5
0.6
|Z|2
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Staircase of energy levels
stationary state energies I = 1, S = 0, T+1u channel on (323 × 256)
anisotropic lattice
Levels0
1
2
3
4
m/m
K
single-hadron dominated
two-hadron dominated
significant mixing
T1up
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Summary and comparison with experiment
right: energies of qq-dominant states as ratios over mK for(323|240) ensemble (resonance precursor states)left: experiment
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Issues
address presence of 3 and 4 meson statesin other channels, address scalar particles in spectrum
scalar probe states need vacuum subtractionshopefully can neglect due to OZI suppression
infinite-volume resonance parameters from finite-volumeenergies
Luscher method too cumbersome, restrictive in applicabilityneed for new hadron effective field theory techniques
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Bosonic I = 1, S = 0, A−1u channel
finite-volume stationary-state energies: “staircase” plot323 × 256 lattice for mπ ∼ 240 MeVuse of single- and two-meson operators onlyblue: levels of max ovelaps with SH optimized operators
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Bosonic I = 1, S = 0, E+u channel
finite-volume stationary-state energies: “staircase” plot323 × 256 lattice for mπ ∼ 240 MeVuse of single- and two-meson operators onlyblue: levels of max ovelaps with SH optimized operators
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Bosonic I = 1, S = 0, T−1g channel
finite-volume stationary-state energies: “staircase” plot323 × 256 lattice for mπ ∼ 240 MeVuse of single- and two-meson operators onlyblue: levels of max ovelaps with SH optimized operators
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Bosonic I = 1, S = 0, T−1u channel
finite-volume stationary-state energies: “staircase” plot323 × 256 lattice for mπ ∼ 240 MeVuse of single- and two-meson operators onlyblue: levels of max ovelaps with SH optimized operators
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Bosonic I = 12 , S = 1, T1u channel
kaon channel: effective energies meff(t) for levels 0 to 8results for 323 × 256 lattice for mπ ∼ 240 MeVtwo-exponential fits
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Bosonic I = 12 , S = 1, T1u channel
effective energies meff(t) for levels 9 to 17results for 323 × 256 lattice for mπ ∼ 240 MeVtwo-exponential fits
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Bosonic I = 12 , S = 1, T1u channel
effective energies meff(t) for levels 18 to 23dashed lines show energies from single exponential fits
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Bosonic I = 12 , S = 1, T1u channel
finite-volume stationary-state energies: “staircase” plot323 × 256 lattice for mπ ∼ 240 MeVuse of single- and two-meson operators onlyblue: levels of max ovelaps with SH optimized operators
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Scattering phase shifts from finite-volume energies
correlator of two-particle operator σ in finite volume
Bethe-Salpeter kernel
C∞(P) has branch cuts where two-particle thresholds beginmomentum quantization in finite volume: cuts→ series of polesCL poles: two-particle energy spectrum of finite volume theory
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Phase shift from finite-volume energies (con’t)
finite-volume momentum sum is infinite-volume integral pluscorrection F
define the following quantities: A, A′, invariant scatteringamplitude iM
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Phase shifts from finite-volume energies (con’t)
subtracted correlator Csub(P) = CL(P)− C∞(P) given by
sum geometric series
Csub(P) = A F(1− iMF)−1 A′.
poles of Csub(P) are poles of CL(P) from det(1− iMF) = 0
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Phase shifts from finite-volume energies (con’t)
work in spatial L3 volume with periodic b.c.total momentum P = (2π/L)d, where d vector of integersmasses m1 and m2 of particle 1 and 2calculate lab-frame energy E of two-particle interacting state inlattice QCDboost to center-of-mass frame by defining:
Ecm =√
E2 − P2, γ =E
Ecm,
q2cm =
14
E2cm −
12
(m21 + m2
2) +(m2
1 − m22)2
4E2cm
,
u2 =L2q2
cm
(2π)2 , s =
(1 +
(m21 − m2
2)
E2cm
)d
E related to S matrix (and phase shifts) by
det[1 + F(s,γ,u)(S− 1)] = 0,
where F matrix defined next slideC. Morningstar Excited States 52
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Phase shifts from finite-volume energies (con’t)
F matrix in JLS basis states given by
F(s,γ,u)J′mJ′L′S′a′; JmJLSa =
ρa
2δa′aδS′S
δJ′JδmJ′mJδL′L
+W(s,γ,u)L′mL′ ; LmL
〈J′mJ′ |L′mL′ , SmS〉〈LmL, SmS|JmJ〉,
total angular mom J, J′, orbital mom L,L′, intrinsic spin S, S′
a, a′ channel labelsρa = 1 distinguishable particles, ρa = 1
2 identical
W(s,γ,u)L′mL′ ; LmL
=2i
πγul+1Zlm(s, γ, u2)
∫d2Ω Y∗L′mL′
(Ω)Y∗lm(Ω)YLmL (Ω)
Rummukainen-Gottlieb-Lüscher (RGL) shifted zeta functions Zlm
defined next slideF(s,γ,u) diagonal in channel space, mixes different J, J′
recall S diagonal in angular momentum, but off-diagonal inchannel space
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RGL shifted zeta functions
compute Zlm using
Zlm(s, γ, u2) =∑n∈Z3
Ylm(z)(z2 − u2)
e−Λ(z2−u2)
+δl0γπeΛu2(
2uD(u√
Λ)− Λ−1/2)
+ilγ
Λl+1/2
∫ 1
0dt(π
t
)l+3/2eΛtu2 ∑
n∈Z3n 6=0
eπin·sYlm(w) e−π2w2/(tΛ)
where
z = n− γ−1[ 12 + (γ − 1)s−2n · s
]s,
w = n− (1− γ)s−2s · ns, Ylm(x) = |x|l Ylm(x)
D(x) = e−x2∫ x
0dt et2
(Dawson function)
choose Λ ≈ 1 for convergence of the summationintegral done Gauss-Legendre quadrature, Dawson with Rybicki
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P-wave I = 1 ππ scattering
for P-wave phase shift δ1(Ecm) for ππ I = 1 scatteringdefine
wlm =Zlm(s, γ, u2)
γπ3/2ul+1
d Λ cot δ1
(0,0,0) T+1u Re w0,0
(0,0,1) A+1 Re w0,0 + 2√
5Re w2,0
E+ Re w0,0 − 1√5Re w2,0
(0,1,1) A+1 Re w0,0 + 1
2√
5Re w2,0 −
√65 Im w2,1 −
√3
10 Re w2,2,
B+1 Re w0,0 − 1√
5Re w2,0 +
√65 Re w2,2,
B+2 Re w0,0 + 1
2√
5Re w2,0 +
√65 Imw2,1 −
√3
10 Re w2,2
(1,1,1) A+1 Re w0,0 + 2
√65 Im w2,2
E+ Re w0,0 −√
65 Im w2,2
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Finite-volume ππ I = 1 energies
ππ-state energies for various d2
dashed lines are non-interacting energies, shaded region aboveinelastic thresholds
A +1
E+0.10
0.12
0.14
0.16
0.18
0.20
atE
π[0]−π[1]
π[1]−π[2] π[1]−π[2]
d2 =1
A +1 B +
1 B +2
0.10
0.12
0.14
0.16
0.18
0.20
0.22
atE
π[0]−π[2]
π[1]−π[3]
π[1]−π[5]
π[1]−π[3]π[2]−π[2]
π[1]−π[1]
π[2]−π[2]
d2 =2
A +1
E+0.10
0.12
0.14
0.16
0.18
0.20
0.22
atE
π[0]−π[3]
π[1]−π[2]
π[1]−π[6]
π[1]−π[2]
d2 =3
A +1
E+0.10
0.12
0.14
0.16
0.18
0.20
0.22
atE
π[0]−π[4]
π[1]−π[5]
π[2]−π[2]
π[1]−π[5]π[3]−π[3]
d2 =4
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Pion dispersion relation
boost to cm frame requires aspect ratio on anisotropic latticeaspect ratio ξ from pion dispersion
(atE)2 = (atm)2 +1ξ2
(2πas
L
)2
d2
slope below equals (π/(16ξ))2, where ξ = as/at
0 1 2 3 4 5 6 7 8d2
0.000
0.005
0.010
0.015
0.020
0.025
0.030
E2
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I = 1 ππ scattering phase shift and width of the ρ
preliminary results 323×256, mπ≈240 MeVadditional collaborator: Ben Hoerz (Dublin)
0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22
atEcm
−50
0
50
100
150
200
250δ 1/°
d2 =0
d2 =1
d2 =2
d2 =3
d2 =4
T +1u
A +1
B +1
B +2
E+
Breit-Wigner g=5.04±0.48 mr =0.1284±0.0010 χ2 /dof=2.1474Breit-Wigner g=5.04±0.48 mr =0.1284±0.0010 χ2 /dof=2.1474
fit tan(δ1) =Γ/2
mr − E+ A and Γ =
g2
48πm2r
(m2r − 4m2
π)3/2
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References
S. Basak et al., Group-theoretical construction of extendedbaryon operators in lattice QCD, Phys. Rev. D 72, 094506 (2005).
S. Basak et al., Lattice QCD determination of patterns of excitedbaryon states, Phys. Rev. D 76, 074504 (2007).
C. Morningstar et al., Improved stochastic estimation of quarkpropagation with Laplacian Heaviside smearing in lattice QCD,Phys. Rev. D 83, 114505 (2011).
C. Morningstar et al., Extended hadron and two-hadron operatorsof definite momentum for spectrum calculations in lattice QCD,Phys. Rev. D 88, 014511 (2013).
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Conclusion
goal: comprehensive survey of energy spectrum of QCDstationary states in a finite volumestochastic LapH method works very well
allows evaluation of all needed quark-line diagramssource-sink factorization facilitates large number of operatorslast_laph software completed for evaluating correlators
analysis software sigmond urgently being developedanalysis of 20 channels I = 1, S = 0 for (243|390) and (323|240)ensembles nearing completioncan evaluate and analyze correlator matrices of unprecedentedsize 100× 100 due to XSEDE resourcesstudy various scattering phase shifts also plannedinfinite-volume resonance parameters from finite-volumeenergies −→ need new effective field theory techniques
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