Spectroscopy in a hybrid DWF/Asqtad
Computation, etc…
David Richards (Jefferson Laboratory/LHPC)
•Computations at light pion masses
• hybrid asqtad/DWF computations
• Hadron spectrum
• Excited resonance spectrum
• Resonance spectrum
• Group-theoretical methods
Thomas Jefferson National Accelerator Facility
Operated by the Southeastern Universities Research Association for the U.S. Department of Energy
kab/SC-PAC2001-6.19.01
ILFT – Mar 2005
LHP Collaboration
G. Fleming, Ph. Hägler, J.W. Negele, R. Edwards, K. Orginos, A.V.
Pochinski, D.B. Renner, W. Schroers
Thomas Jefferson National Accelerator Facility
Operated by the Southeastern Universities Research Association for the U.S. Department of Energy
ILFT– Mar 2005
Onward to chiral regime…
Thomas Jefferson National Accelerator Facility
Operated by the Southeastern Universities Research Association for the U.S. Department of Energy
ILFT– Mar 2005
Clusters at Jefferson Laboratory - SciDAC
128-node P4 Xeon, with Myrinet
256-node P4 Xeon, with 3d
gigE torus; 384-node “4g”
Thomas Jefferson National Accelerator Facility
Operated by the Southeastern Universities Research Association for the U.S. Department of Energy
ILFT– Mar 2005
Partially Quenched χ PT
Bär, Rupak, Shoresh, 2002, 2003
Thomas Jefferson National Accelerator Facility
Operated by the Southeastern Universities Research Association for the U.S. Department of Energy
ILFT– Mar 2005
DWF Computations
•
MILC Asqtad lattices – staggered sea quarks
•
Conventional DWF valence quarks computed on HYP-smeared lattices:
use non-perturbative HYP parameters.
PRD64, 034504
Lattices bisected in temporal direction – DWF propagators computed
using Dirichlet boundary conditions on half lattice
•
•
M5 = 1.7 – Ls = 16
•
Parameter matching – tune DWF pion to match Asqtad Goldstone pion
mass mπ
•
a ' 0.13 fm, two volumes at lightest pion mass
Thomas Jefferson National Accelerator Facility
Operated by the Southeastern Universities Research Association for the U.S. Department of Energy
ILFT– Mar 2005
DWF parameters
mπ L = 4.5
mπ L = 6.3
Alternate configs on first and second halves of lattice
Thomas Jefferson National Accelerator Facility
Operated by the Southeastern Universities Research Association for the U.S. Department of Energy
ILFT– Mar 2005
Pion mass matching..
DWF, L=20
DWF, L=28
Asqtad, L=20
(a mπ)
2
0.2
0.1
0
0
0.01
0.02
0.03
0.04
0.05
0.06
asqtad
(mq)
Thomas Jefferson National Accelerator Facility
Operated by the Southeastern Universities Research Association for the U.S. Department of Energy
ILFT– Mar 2005
Setting L5
Residual mass
• Finite L5 induces extra term in divergence of axial current
4-D PCAC from
5-D CVC
Domain walls
Ja5q
0
L5 - 1
Thomas Jefferson National Accelerator Facility
Operated by the Southeastern Universities Research Association for the U.S. Department of Energy
ILFT– Mar 2005
Residual mass….
Thomas Jefferson National Accelerator Facility
Operated by the Southeastern Universities Research Association for the U.S. Department of Energy
ILFT– Mar 2005
Pion mass….
Thomas Jefferson National Accelerator Facility
Operated by the Southeastern Universities Research Association for the U.S. Department of Energy
ILFT– Mar 2005
Nucleon mass….
Thomas Jefferson National Accelerator Facility
Operated by the Southeastern Universities Research Association for the U.S. Department of Energy
ILFT– Mar 2005
Scalar Correlator
Thomas Jefferson National Accelerator Facility
Operated by the Southeastern Universities Research Association for the U.S. Department of Energy
ILFT– Mar 2005
Meson masses…..
DWF, L=20
DWF, L=28
Asqtad, L=20
Pion
(a mπ)
2
0.2
0.1
1.5
0
0
0.01
0.02
0.03
0.04
DWF, L=20
DWF, L=28
Asqtad, L=20
0.05
0.06
asqtad
(mq)
amρ
1
Rho
0.5
0
0
0.01
0.02
0.03
0.04
asqtad
(mq)
Thomas Jefferson National Accelerator Facility
Operated by the Southeastern Universities Research Association for the U.S. Department of Energy
ILFT– Mar 2005
0.05
0.06
Baryon masses….
1.5
DWF, L=20
DWF, L=28
Asqtad, L=20
Nucleon
mN
1
1.5
0.5
DWF, L=20
DWF, L=28
Asqtad, L=20
1
0
0
0.01
0.02
0.03
0.04
0.05
0.06
a m∆
asqtad
(mq)
0.5
Delta
0
0
0.01
0.02
0.03
0.04
asqtad
(mq)
Thomas Jefferson National Accelerator Facility
Operated by the Southeastern Universities Research Association for the U.S. Department of Energy
ILFT– Mar 2005
0.05
0.06
Spectrum summary…
1.4
1.2
m∆
mN
aM
1
0.8
mρ
0.6
0.4
0.2
0
0
0.1
0.2
2
(amπ)
Thomas Jefferson National Accelerator Facility
Operated by the Southeastern Universities Research Association for the U.S. Department of Energy
ILFT– Mar 2005
Resonance spectrum
S. Basak, R. Edwards, R. Fiebig, G. Fleming, U. Heller,
C. Morningstar, DGR, I. Sato, S. Wallace
•
Higher resonances provide information about dynamics and confinement
•
Aim to compute masses of lowest few states of given quantum number, and
learn about their structure…
12
2
*−+
0
3
r0mG
8
2
*++
6
0
++
2
0
3−−
2−−
1
2
*−+
++
Morningstar and Peardon
PRD60, 034509
Quenched glueball calculations
provide road-map
• Method more demanding for
particles containing quarks
−−
+−
10
4
+−
3
−+
3
+−
1
−+
2
4
2
++
0
1
++
−+
PC
+−
−−
Thomas Jefferson National Accelerator Facility
Operated by the Southeastern Universities Research Association for the U.S. Department of Energy
ILFT– Mar 2005
mG (GeV)
Glueballs
+−
0
Variational Methods….
Measure many interpolating operators, and measure N × N matrix
Eigenvalues of matrix give excited states
Eigenvalues are real, and lowest N states decouple:
u’s are orthogonal with metric C(t0) – information about eigenstates
Thomas Jefferson National Accelerator Facility
Operated by the Southeastern Universities Research Association for the U.S. Department of Energy
ILFT– Mar 2005
Example: baryon fields and operators
•
States are classified according to
• Flavor structure, F
• Parity
• Total Spin, J, and Helicity, or third component of spin, Jz.
•
In lattice computation, need to construct operators BiΛ λ F(t, x) transforming
irreducibly under lattice symmetries Λ, λ and form correlation matrix
•
Begin by considering flavor and orbital radial structure
Thomas Jefferson National Accelerator Facility
Operated by the Southeastern Universities Research Association for the U.S. Department of Energy
ILFT– Mar 2005
Flavor Structure
Assume Isospin is good symmetry – classify states according tom say, SU(2)
flavour
•
Similarly for Ι = 3/2 ∆
Thomas Jefferson National Accelerator Facility
Operated by the Southeastern Universities Research Association for the U.S. Department of Energy
ILFT– Mar 2005
Orbital and Radial Structure
Build up radial structure using displacement operators D
Provide our elemental operators having correct flavour and colour structure
Thomas Jefferson National Accelerator Facility
Operated by the Southeastern Universities Research Association for the U.S. Department of Energy
ILFT– Mar 2005
Cubic Group of Lattice
•
Rotations restricted to isotropic cubic lattice form Octohedral group O
• O has 24 elements
• O has five conjugacy classes, hence five single-valued representations,
A1(1), A2(1), E(2), T1(3), T2(3)
• Under elements R of O, operator lying in Λ irreducible representation
transforms as (λ is row)
• Spatial inversions Is ! Oh, with 48 elements; parities labelled g or u
Thomas Jefferson National Accelerator Facility
Operated by the Southeastern Universities Research Association for the U.S. Department of Energy
ILFT– Mar 2005
Double Octohedral Group
•
Has 48 elements
•
Contains irreducible representations of O, together with 3 spinor irreps G1,
G2, H: R.C.Johnson, PLB114, 147 (82)
• Note that states with J > 5/2 lie in
representations with lower spins.
• Masses of different components of state
in different representations only agree in
continuum limit
Final task is to decompose the elemental operators BiF(t,x) into irreducible
representations of OhD.
Thomas Jefferson National Accelerator Facility
Operated by the Southeastern Universities Research Association for the U.S. Department of Energy
ILFT– Mar 2005
N* Resonance…
mH
mG2
M5/2
1720
a
1535
1670
1675
5/2+
5/2-
1520
938
1/2+
1/2-
3/2+
3/2-
Thomas Jefferson National Accelerator Facility
Operated by the Southeastern Universities Research Association for the U.S. Department of Energy
ILFT– Mar 2005
Projection formula
• Central formula
•
Λ is representation of OhD, λ is row of representation
• g is order of OhD, dΛ is dimension of Λ irreducible representation
• Dλ λ(Λ)*(R) is representation in Λ of transformation R.
• UR is the operator that implements R.
•
Group theory performed in Maple – requires explicit Dirac basis.
Thomas Jefferson National Accelerator Facility
Operated by the Southeastern Universities Research Association for the U.S. Department of Energy
ILFT– Mar 2005
Simple example: local nucleon operators…
•
Elemental operators:
•
Constraints:
•
20 linearly-independent operators
•
3 embeddings of G1g, G1u , 1 embedding each of Hg, Hu .
•
Three linearly independent nucleon operators, and a single N*3/2 operator
Thomas Jefferson National Accelerator Facility
Operated by the Southeastern Universities Research Association for the U.S. Department of Energy
ILFT– Mar 2005
Operator table….
Thomas Jefferson National Accelerator Facility
Operated by the Southeastern Universities Research Association for the U.S. Department of Energy
ILFT– Mar 2005
Orthogonality of States
Thomas Jefferson National Accelerator Facility
Operated by the Southeastern Universities Research Association for the U.S. Department of Energy
ILFT– Mar 2005
Non-local sources…
• Compute all possible operators for quarks from local source and from
displaced source of length p in direction j
• Enable us to construct operators in G2g and G2u, not possible with only
local sources
• 27 embeddings of G1
• 8 embeddings of G2
• 33 embeddings of H
Thomas Jefferson National Accelerator Facility
Operated by the Southeastern Universities Research Association for the U.S. Department of Energy
ILFT– Mar 2005
Example….
123 × 48 lattice, anisotropic Wilson with ξ = 3.0
at-1 ' 6 GeV
mπ ' 1100 MeV
175 Cfgs.
Thomas Jefferson National Accelerator Facility
Operated by the Southeastern Universities Research Association for the U.S. Department of Energy
ILFT– Mar 2005
Higher spin states…
Mπ ' 500 MeV
0.5
Effective Mass
163 × 64 lattice, anisotropic
Wilson with ξ = 3.0
G1g
G2g
Hg
0.4
0.3
0.2
0.1
5
10
Time
15
Thomas Jefferson National Accelerator Facility
Operated by the Southeastern Universities Research Association for the U.S. Department of Energy
ILFT– Mar 2005
20
Principle eigenvalues…
• Beginning studies on 163 ×
64 lattices and 243 × 64
lattices using improved
anisotropic actions
• Aim to identify basis of
interpolating operators for
baryons, hybrid mesons, and
pentaquarks/molecular states.
• Use basis as guidance in
subsequent full QCD
computations
Thomas Jefferson National Accelerator Facility
Operated by the Southeastern Universities Research Association for the U.S. Department of Energy
ILFT– Mar 2005
Conclusions and Perspective
•
Use of hybrid Asqtad/DWF calculation providing entry into pion-cloud
regime for range of physical quantities – spectrum (including pentaquarks),
nucleon and meson form factors and transition form factors, moments of
PDF’s and GPD’s…
•
Determined minimal value of Ls for which physical hadron masses appear
insensitive to residual mass.
•
Computations being extended both to smaller values of lattice spacing
(a=0.9, 0.6 fm), and to lighter mπ = 250 MeV – reveal effects of pion cloud.
•
First exploratory studies using fully consistent chiral action for valence and
sea quarks.
•
Crucial to develop operators and variational methods for excited resonance
spectrum.
Thomas Jefferson National Accelerator Facility
Operated by the Southeastern Universities Research Association for the U.S. Department of Energy
ILFT– Mar 2005
Implementation…
•
On each configuration, form generalised baryon correlator
•
Antisymmetrise over α,γ - I = 1/2.
•
Symmetrise over α and β – Wick contractions
•
Form correlators according to table.
•
Illustration: DWF valence propagators, on MILC Asqtad lattices: mπ ' 600
MeV
Thomas Jefferson National Accelerator Facility
Operated by the Southeastern Universities Research Association for the U.S. Department of Energy
ILFT– Mar 2005
Effective masses
G1g and G1u
Hg and Hu
Thomas Jefferson National Accelerator Facility
Operated by the Southeastern Universities Research Association for the U.S. Department of Energy
ILFT– Mar 2005
Orthogonality of States
Thomas Jefferson National Accelerator Facility
Operated by the Southeastern Universities Research Association for the U.S. Department of Energy
ILFT– Mar 2005