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Slide1
Baryon Resonances from Lattice QCD
Robert Edwards
Jefferson LabN* @ high Q2, 2011
TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAAAAAAAA
Collaborators:
J.
Dudek
, B.
Joo
, D
. Richards, S. Wallace
Auspices of the
Hadron
Spectrum CollaborationSlide2
Lattice QCD
Goal:
resolve highly excited statesNf = 2 + 1 (u,d + s)
Anisotropic lattices:(as)-1 ~ 1.6 GeV
, (a
t
)
-1
~ 5.6 GeV
0810.3588, 0909.0200, 1004.4930Slide3
Spectrum from variational method
Matrix of
correlatorsDiagonalize: eigenvalues
! spectrum eigenvectors ! wave function overlapsBenefit: orthogonality for near degenerate states
Two-point correlator
3
Each state optimal combination of
©
iSlide4
Operator construction
Baryons : permutations of 3 objects
4
Color antisymmetric ! Require Space [Flavor Spin
] symmetric
Permutation group
S
3
: 3 representations
Symmetric
: 1-dimensional
e.g.,
uud+udu+duu
Antisymmetric
: 1-dimensional
e.g.,
uud-udu+duu
-…
Mixed
: 2-dimensional
e.g.,
udu
-
duu
& 2duu - udu - uud
Classify operators by these permutation symmetries: Leads to rich structure
1104.5152Slide5
Orbital angular momentum via derivatives
1104.5152
Couple derivatives onto single-site spinors:Enough D’s – build any J,M
5Use all possible
operators
up to 2 derivatives (transforms like 2 units orbital angular momentum)
Only using
symmetries
of continuum QCDSlide6
Baryon operator basis
3-quark operators with up to two covariant derivatives – projected into definite
isospin and continuum JP
6Spatial symmetry classification:Nucleons:
N
2S+1
L
¼
JPJP
#ops
Spatial symmetries
J=1/2
-
24
N
2
P
M
½
-
N
4
P
M
½
-
J=3/2-28N 2PM
3/2-N 4PM
3/2
-
J=5/2
-
16
N
4
P
M
5/2
-
J=1/2
+
24
N 2SS ½+N 2SM ½+ N 4DM ½+N 2PA ½+ J=3/2+28N 2DS3/2+ N 2DM3/2+ N 2PA 3/2+N 4SM3/2+ N 4DM3/2+J=5/2+16N 2DS5/2+ N 2DM5/2+N 4DM5/2+J=7/2+4N 4DM7/2+
By far the largest operator basis ever used for such calculations
Symmetry crucial for spectroscopySlide7
Spin identified Nucleon & Delta spectrum
m
¼ ~ 520MeV7
arXiv:1104.5152Statistical errors < 2%Slide8
Spin identified Nucleon & Delta spectrum
8
arXiv:1104.5152
45
3
1
2
3
2
1
2
2
1
1
1
SU(6)
xO
(3) counting
No parity doubling
m
¼
~
520MeVSlide9
Spin identified Nucleon & Delta spectrum
9
arXiv:1104.5152
[70,1-]P-wave
[
70,1
-
]
P-wave
m
¼
~
520MeV
[
56,0
+
]
S-wave
[
56,0
+
]
S-wave
Discern structure: wave-function overlaps Slide10
N=2 J+ Nucleon & Delta spectrum
10
Significant mixing in J
+2SS 2
S
M
4
SM 2DS 2DM 4D
M
2
P
A
13 levels/ops
2
S
M
4
S
S
2
D
M
4DS8 levels/ops
Discern structure: wave-function overlaps Slide11
Roper??
11
Near degeneracy in
½+ consistent with SU(6)O(3) but heavily mixed
Discrepancies??
Operator basis – spatial structure
What else?
Multi-particle operatorsSlide12
Spectrum of finite volume field theory
Missing states:
“continuum” of multi-particle scattering statesInfinite volume
: continuous spectrum
2m
π
2m
π
Finite volume
:
discrete spectrum
2m
π
Deviation
from
(discrete) free
energies depends
upon interaction - contains
information
about scattering
phase
shift
ΔE(L) ↔ δ(E) :
Lüscher
method
12Slide13
Finite volume scattering
E.g
. just a single elastic resonance
e.g. Lüscher
method
scattering
in a periodic cubic box (length
L
)
finite
volume energy levels
E(L)
!
δ(E)
13
At some
L ,
have discrete excited energies Slide14
I=1 ¼¼ : the “
½”
Feng, Jansen, Renner, 1011.5288Extract δ1(E) at discrete E
g½¼¼m¼2 (GeV2
)
Extracted coupling:
stable in
pion
mass
Stability a generic feature of couplings?? Slide15
Form Factors
What is a form-factor off of a resonance?
What is a resonance? Spectrum first!Extension of scattering techniques:
Finite volume matrix element modified
E
Requires excited level transition FF’s: some experience
Charmonium
E&M transition FF’s
(1004.4930)
Nucleon 1
st
attempt: “Roper”->N
(0803.3020)
Range: few
GeV
2
Limitation: spatial lattice spacing
Kinematic
factor
Phase shiftSlide16
(Very) Large Q2
Cutoff effects: lattice spacing (
as)-1 ~ 1.6
GeV 16
Standard requirements:
Appeal to renormalization group
: Finite-Size
scaling
D. Renner
“Unfold” ratio only at low
Q
2
/
s
2N
Use short-distance quantity: compute perturbatively and/or parameterize
For
Q
2
= 100
GeV
2
and N=3,
Q
2
/
s
2N
~ 1.5
GeV
2
Initial applications: factorization in
pion
-FFSlide17
Hadronic Decays
17
m
¼ ~ 400 MeV
Some candidates: determine phase shift
Somewhat elastic
¢
!
[N
¼
]
P
S
11
!
[N
¼
]
SSlide18
Prospects
Strong effort in excited state spectroscopyNew operator & correlator constructions
! high lying statesResults for baryon excited state spectrum:No “freezing” of degrees of freedom nor parity doublingBroadly consistent with non-relativistic quark modelAdd multi-particles ! baryon spectrum becomes denserShort-term plans: resonance determination!Lighter quark massesExtract couplings in multi-channel systemsForm-factors:
Use previous resonance parameters: initially, Q2 ~ few GeV2Decrease lattice spacing: (as)-1 ~ 1.6 GeV
!
3.2
GeV
, then Q2 ~ 10 GeV2Finite-size scaling:
Q
2
!
100
GeV
2
???Slide19
Backup slides
The end
19Slide20
Baryon Spectrum
“Missing resonance problem”What are collective modes?
What is the structure of the states?20PDG uncertainty on B-W mass
Nucleon spectrumSlide21
Phase Shifts demonstration: I=2 ¼¼
¼¼
isospin=2
Extract δ0(E) at discrete ENo discernible
pion
mass dependence
1011.6352
(
PRD)Slide22
Phase Shifts: demonstration
¼¼
isospin=2δ
2(E)Slide23
Nucleon J
-
23
Overlaps Little mixing in each J
-
Nearly “pure” [S= 1/2 & 3/2]
1-Slide24
N & ¢ spectrum: lower
pion mass
24
m¼ ~ 400 MeV
Still bands of states with same counting
More mixing in nucleon N=2 J
+Slide25
Operators are not states
Full basis of operators: many operators can create same state
States may have subset of allowed symmetriesTwo-point
correlator