0-%012)+*%3#$4$5€¦ · $f"./+&! @;& 8&7$%"2$...
TRANSCRIPT
![Page 1: 0-%012)+*%3#$4$5€¦ · $F"./+&! @;& 8&7$%"2$ lattice%QCD%exotic%meson%spectrum%circa%2006 0.1 0.2 0.3 0.4 0.5 0.6 1.0 1.5 2.0 2.5 quenched dynamical m π MeV others?](https://reader035.vdocuments.us/reader035/viewer/2022071002/5fbf72459e28ac23fd699ed5/html5/thumbnails/1.jpg)
Does QCD predict light hybrid mesons?
Jozef DudekOld Dominion University & Jefferson Lab
JLab Users’ Group Workshop
Robert Edwards (JLab)Balint Joo (JLab)David Richards (JLab)Christopher Thomas (JLab)Mike Peardon (Trinity Coll., Dublin)
for the Hadron Spectrum Collaboration
![Page 2: 0-%012)+*%3#$4$5€¦ · $F"./+&! @;& 8&7$%"2$ lattice%QCD%exotic%meson%spectrum%circa%2006 0.1 0.2 0.3 0.4 0.5 0.6 1.0 1.5 2.0 2.5 quenched dynamical m π MeV others?](https://reader035.vdocuments.us/reader035/viewer/2022071002/5fbf72459e28ac23fd699ed5/html5/thumbnails/2.jpg)
2
constituent quark model
π(1300
)π(
1800
)
ρ
π
ρ(14
50)
ρ(1700
)
ρ3
π 2(1670)π 2
(188
0)
b1
a0(980)
a 0(145
0)
a 1(126
0) a2
a 4(204
0)
(historical) empirical motivation
![Page 3: 0-%012)+*%3#$4$5€¦ · $F"./+&! @;& 8&7$%"2$ lattice%QCD%exotic%meson%spectrum%circa%2006 0.1 0.2 0.3 0.4 0.5 0.6 1.0 1.5 2.0 2.5 quenched dynamical m π MeV others?](https://reader035.vdocuments.us/reader035/viewer/2022071002/5fbf72459e28ac23fd699ed5/html5/thumbnails/3.jpg)
“1S”
2
constituent quark model
π(1300
)π(
1800
)
ρ
π
ρ(14
50)
ρ(1700
)
ρ3
π 2(1670)π 2
(188
0)
b1
a0(980)
a 0(145
0)
a 1(126
0) a2
a 4(204
0)
(historical) empirical motivation
![Page 4: 0-%012)+*%3#$4$5€¦ · $F"./+&! @;& 8&7$%"2$ lattice%QCD%exotic%meson%spectrum%circa%2006 0.1 0.2 0.3 0.4 0.5 0.6 1.0 1.5 2.0 2.5 quenched dynamical m π MeV others?](https://reader035.vdocuments.us/reader035/viewer/2022071002/5fbf72459e28ac23fd699ed5/html5/thumbnails/4.jpg)
1P?
“1S”
2
constituent quark model
π(1300
)π(
1800
)
ρ
π
ρ(14
50)
ρ(1700
)
ρ3
π 2(1670)π 2
(188
0)
b1
a0(980)
a 0(145
0)
a 1(126
0) a2
a 4(204
0)
(historical) empirical motivation
![Page 5: 0-%012)+*%3#$4$5€¦ · $F"./+&! @;& 8&7$%"2$ lattice%QCD%exotic%meson%spectrum%circa%2006 0.1 0.2 0.3 0.4 0.5 0.6 1.0 1.5 2.0 2.5 quenched dynamical m π MeV others?](https://reader035.vdocuments.us/reader035/viewer/2022071002/5fbf72459e28ac23fd699ed5/html5/thumbnails/5.jpg)
1P?
2S
“1S”
2
constituent quark model
π(1300
)π(
1800
)
ρ
π
ρ(14
50)
ρ(1700
)
ρ3
π 2(1670)π 2
(188
0)
b1
a0(980)
a 0(145
0)
a 1(126
0) a2
a 4(204
0)
(historical) empirical motivation
![Page 6: 0-%012)+*%3#$4$5€¦ · $F"./+&! @;& 8&7$%"2$ lattice%QCD%exotic%meson%spectrum%circa%2006 0.1 0.2 0.3 0.4 0.5 0.6 1.0 1.5 2.0 2.5 quenched dynamical m π MeV others?](https://reader035.vdocuments.us/reader035/viewer/2022071002/5fbf72459e28ac23fd699ed5/html5/thumbnails/6.jpg)
1P?
1D
2S
“1S”
2
constituent quark model
π(1300
)π(
1800
)
ρ
π
ρ(14
50)
ρ(1700
)
ρ3
π 2(1670)π 2
(188
0)
b1
a0(980)
a 0(145
0)
a 1(126
0) a2
a 4(204
0)
(historical) empirical motivation
![Page 7: 0-%012)+*%3#$4$5€¦ · $F"./+&! @;& 8&7$%"2$ lattice%QCD%exotic%meson%spectrum%circa%2006 0.1 0.2 0.3 0.4 0.5 0.6 1.0 1.5 2.0 2.5 quenched dynamical m π MeV others?](https://reader035.vdocuments.us/reader035/viewer/2022071002/5fbf72459e28ac23fd699ed5/html5/thumbnails/7.jpg)
1P?
1F
1D
2S
“1S”
2
constituent quark model
π(1300
)π(
1800
)
ρ
π
ρ(14
50)
ρ(1700
)
ρ3
π 2(1670)π 2
(188
0)
b1
a0(980)
a 0(145
0)
a 1(126
0) a2
a 4(204
0)
(historical) empirical motivation
![Page 8: 0-%012)+*%3#$4$5€¦ · $F"./+&! @;& 8&7$%"2$ lattice%QCD%exotic%meson%spectrum%circa%2006 0.1 0.2 0.3 0.4 0.5 0.6 1.0 1.5 2.0 2.5 quenched dynamical m π MeV others?](https://reader035.vdocuments.us/reader035/viewer/2022071002/5fbf72459e28ac23fd699ed5/html5/thumbnails/8.jpg)
hybrid mesons -‐ beyond the quark model
observed meson state flavor & JPC systematics suggest “constituent quarks”
![Page 9: 0-%012)+*%3#$4$5€¦ · $F"./+&! @;& 8&7$%"2$ lattice%QCD%exotic%meson%spectrum%circa%2006 0.1 0.2 0.3 0.4 0.5 0.6 1.0 1.5 2.0 2.5 quenched dynamical m π MeV others?](https://reader035.vdocuments.us/reader035/viewer/2022071002/5fbf72459e28ac23fd699ed5/html5/thumbnails/9.jpg)
hybrid mesons -‐ beyond the quark model
observed meson state flavor & JPC systematics suggest “constituent quarks”
exotic quantum numbers
![Page 10: 0-%012)+*%3#$4$5€¦ · $F"./+&! @;& 8&7$%"2$ lattice%QCD%exotic%meson%spectrum%circa%2006 0.1 0.2 0.3 0.4 0.5 0.6 1.0 1.5 2.0 2.5 quenched dynamical m π MeV others?](https://reader035.vdocuments.us/reader035/viewer/2022071002/5fbf72459e28ac23fd699ed5/html5/thumbnails/10.jpg)
hybrid mesons -‐ beyond the quark model
observed meson state flavor & JPC systematics suggest “constituent quarks”
exotic quantum numbers
but what if excited gluonic fields play a rôle -‐ a hybrid meson, ?
possibly exotic JPC & extra ‘non-‐exotic’ states
must be ‘heavier’ or ‘harder to produce’ ?
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hybrid mesons -‐ models tubes, bags & heavy glue
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hybrid mesons -‐ models tubes, bags & heavy glue
flux-‐tube model0-‐+, 1-‐+, 2-‐+, 1-‐-‐
0+-‐, 1+-‐, 2+-‐, 1++roughly degenerate
![Page 13: 0-%012)+*%3#$4$5€¦ · $F"./+&! @;& 8&7$%"2$ lattice%QCD%exotic%meson%spectrum%circa%2006 0.1 0.2 0.3 0.4 0.5 0.6 1.0 1.5 2.0 2.5 quenched dynamical m π MeV others?](https://reader035.vdocuments.us/reader035/viewer/2022071002/5fbf72459e28ac23fd699ed5/html5/thumbnails/13.jpg)
hybrid mesons -‐ models tubes, bags & heavy glue
flux-‐tube model0-‐+, 1-‐+, 2-‐+, 1-‐-‐
0+-‐, 1+-‐, 2+-‐, 1++roughly degenerate
bag model 0-‐+, 1-‐+, 2-‐+, 1-‐-‐& others heavier
![Page 14: 0-%012)+*%3#$4$5€¦ · $F"./+&! @;& 8&7$%"2$ lattice%QCD%exotic%meson%spectrum%circa%2006 0.1 0.2 0.3 0.4 0.5 0.6 1.0 1.5 2.0 2.5 quenched dynamical m π MeV others?](https://reader035.vdocuments.us/reader035/viewer/2022071002/5fbf72459e28ac23fd699ed5/html5/thumbnails/14.jpg)
hybrid mesons -‐ models tubes, bags & heavy glue
flux-‐tube model0-‐+, 1-‐+, 2-‐+, 1-‐-‐
0+-‐, 1+-‐, 2+-‐, 1++roughly degenerate
bag model 0-‐+, 1-‐+, 2-‐+, 1-‐-‐& others heavier
massive quasi-‐gluons Coulomb gauge transverse (no 3-‐body) 0++, 1++, 2++, 1+-‐
0-‐-‐, 1-‐+, ...1-‐-‐ “S-‐wave”
![Page 15: 0-%012)+*%3#$4$5€¦ · $F"./+&! @;& 8&7$%"2$ lattice%QCD%exotic%meson%spectrum%circa%2006 0.1 0.2 0.3 0.4 0.5 0.6 1.0 1.5 2.0 2.5 quenched dynamical m π MeV others?](https://reader035.vdocuments.us/reader035/viewer/2022071002/5fbf72459e28ac23fd699ed5/html5/thumbnails/15.jpg)
hybrid mesons -‐ models tubes, bags & heavy glue
flux-‐tube model0-‐+, 1-‐+, 2-‐+, 1-‐-‐
0+-‐, 1+-‐, 2+-‐, 1++roughly degenerate
bag model 0-‐+, 1-‐+, 2-‐+, 1-‐-‐& others heavier
massive quasi-‐gluons Coulomb gauge transverse (no 3-‐body) 0++, 1++, 2++, 1+-‐
0-‐-‐, 1-‐+, ...1-‐-‐ “S-‐wave”
Coulomb gauge transverse (incl. 3-‐body)
0+-‐, (2+-‐)2, ...
0-‐+, 1-‐+, 2-‐+, 1-‐-‐
1+-‐ “P-‐wave”
![Page 16: 0-%012)+*%3#$4$5€¦ · $F"./+&! @;& 8&7$%"2$ lattice%QCD%exotic%meson%spectrum%circa%2006 0.1 0.2 0.3 0.4 0.5 0.6 1.0 1.5 2.0 2.5 quenched dynamical m π MeV others?](https://reader035.vdocuments.us/reader035/viewer/2022071002/5fbf72459e28ac23fd699ed5/html5/thumbnails/16.jpg)
hybrid mesons -‐ models tubes, bags & heavy glue
flux-‐tube model0-‐+, 1-‐+, 2-‐+, 1-‐-‐
0+-‐, 1+-‐, 2+-‐, 1++roughly degenerate
without much data -‐ nothing much to constrain them ...
bag model 0-‐+, 1-‐+, 2-‐+, 1-‐-‐& others heavier
massive quasi-‐gluons Coulomb gauge transverse (no 3-‐body) 0++, 1++, 2++, 1+-‐
0-‐-‐, 1-‐+, ...1-‐-‐ “S-‐wave”
Coulomb gauge transverse (incl. 3-‐body)
0+-‐, (2+-‐)2, ...
0-‐+, 1-‐+, 2-‐+, 1-‐-‐
1+-‐ “P-‐wave”
![Page 17: 0-%012)+*%3#$4$5€¦ · $F"./+&! @;& 8&7$%"2$ lattice%QCD%exotic%meson%spectrum%circa%2006 0.1 0.2 0.3 0.4 0.5 0.6 1.0 1.5 2.0 2.5 quenched dynamical m π MeV others?](https://reader035.vdocuments.us/reader035/viewer/2022071002/5fbf72459e28ac23fd699ed5/html5/thumbnails/17.jpg)
spectrum from lattice QCD
two-‐point correlator e.g.
![Page 18: 0-%012)+*%3#$4$5€¦ · $F"./+&! @;& 8&7$%"2$ lattice%QCD%exotic%meson%spectrum%circa%2006 0.1 0.2 0.3 0.4 0.5 0.6 1.0 1.5 2.0 2.5 quenched dynamical m π MeV others?](https://reader035.vdocuments.us/reader035/viewer/2022071002/5fbf72459e28ac23fd699ed5/html5/thumbnails/18.jpg)
spectrum from lattice QCD
two-‐point correlator e.g.
as a sum of QCD eigenstates
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spectrum from lattice QCD
two-‐point correlator e.g.
as a sum of QCD eigenstates
e.g. 1-‐-‐two-‐point correlator matrix
![Page 20: 0-%012)+*%3#$4$5€¦ · $F"./+&! @;& 8&7$%"2$ lattice%QCD%exotic%meson%spectrum%circa%2006 0.1 0.2 0.3 0.4 0.5 0.6 1.0 1.5 2.0 2.5 quenched dynamical m π MeV others?](https://reader035.vdocuments.us/reader035/viewer/2022071002/5fbf72459e28ac23fd699ed5/html5/thumbnails/20.jpg)
spectrum from lattice QCD
two-‐point correlator e.g.
as a sum of QCD eigenstates
e.g. 1-‐-‐two-‐point correlator matrix
each state comes from an orthogonal combination of
solve by using a ‘Rayleigh-‐Ritz’-‐style variational approach -‐ “diagonalize the matrix”
optimal operator :
![Page 21: 0-%012)+*%3#$4$5€¦ · $F"./+&! @;& 8&7$%"2$ lattice%QCD%exotic%meson%spectrum%circa%2006 0.1 0.2 0.3 0.4 0.5 0.6 1.0 1.5 2.0 2.5 quenched dynamical m π MeV others?](https://reader035.vdocuments.us/reader035/viewer/2022071002/5fbf72459e28ac23fd699ed5/html5/thumbnails/21.jpg)
mesonic operator basis
fermion bilinears with up to three covariant derivatives -‐ project into good JPC
actually also have to project into irreducible representations of the cubic symmetry group
# of operators
~J=0
~J=1
~J=2
~J=2
~J=3
this is by far the largest operator set ever used in a calculation like this
![Page 22: 0-%012)+*%3#$4$5€¦ · $F"./+&! @;& 8&7$%"2$ lattice%QCD%exotic%meson%spectrum%circa%2006 0.1 0.2 0.3 0.4 0.5 0.6 1.0 1.5 2.0 2.5 quenched dynamical m π MeV others?](https://reader035.vdocuments.us/reader035/viewer/2022071002/5fbf72459e28ac23fd699ed5/html5/thumbnails/22.jpg)
isovector spectrum
0.4
0.6
0.8
1.0
1.2
1.4
1.6
0.4
0.6
0.8
1.0
1.2
1.4
1.6
0.4
0.6
0.8
1.0
1.2
1.4
1.6
mπ ~ 700 MeV
![Page 23: 0-%012)+*%3#$4$5€¦ · $F"./+&! @;& 8&7$%"2$ lattice%QCD%exotic%meson%spectrum%circa%2006 0.1 0.2 0.3 0.4 0.5 0.6 1.0 1.5 2.0 2.5 quenched dynamical m π MeV others?](https://reader035.vdocuments.us/reader035/viewer/2022071002/5fbf72459e28ac23fd699ed5/html5/thumbnails/23.jpg)
isovector spectrum
0.4
0.6
0.8
1.0
1.2
1.4
1.6
0.4
0.6
0.8
1.0
1.2
1.4
1.6
0.4
0.6
0.8
1.0
1.2
1.4
1.6
mπ ~ 700 MeV
![Page 24: 0-%012)+*%3#$4$5€¦ · $F"./+&! @;& 8&7$%"2$ lattice%QCD%exotic%meson%spectrum%circa%2006 0.1 0.2 0.3 0.4 0.5 0.6 1.0 1.5 2.0 2.5 quenched dynamical m π MeV others?](https://reader035.vdocuments.us/reader035/viewer/2022071002/5fbf72459e28ac23fd699ed5/html5/thumbnails/24.jpg)
isovector spectrum
0.4
0.6
0.8
1.0
1.2
1.4
1.6
0.4
0.6
0.8
1.0
1.2
1.4
1.6
0.4
0.6
0.8
1.0
1.2
1.4
1.6
mπ ~ 700 MeV
0.1 0.2 0.3 0.4500
1000
1500
2000
2500
3000
![Page 25: 0-%012)+*%3#$4$5€¦ · $F"./+&! @;& 8&7$%"2$ lattice%QCD%exotic%meson%spectrum%circa%2006 0.1 0.2 0.3 0.4 0.5 0.6 1.0 1.5 2.0 2.5 quenched dynamical m π MeV others?](https://reader035.vdocuments.us/reader035/viewer/2022071002/5fbf72459e28ac23fd699ed5/html5/thumbnails/25.jpg)
exotic JPC -‐ before
lattice QCD exotic meson spectrum circa 2006
0.1 0.2 0.3 0.4 0.5 0.61.0
1.5
2.0
2.5
quenched
dynamical
mπ ~ 7
00 MeV
mπ ~ 5
00 MeV
mπ ~ 3
00 MeV
![Page 26: 0-%012)+*%3#$4$5€¦ · $F"./+&! @;& 8&7$%"2$ lattice%QCD%exotic%meson%spectrum%circa%2006 0.1 0.2 0.3 0.4 0.5 0.6 1.0 1.5 2.0 2.5 quenched dynamical m π MeV others?](https://reader035.vdocuments.us/reader035/viewer/2022071002/5fbf72459e28ac23fd699ed5/html5/thumbnails/26.jpg)
exotic JPC -‐ before
lattice QCD exotic meson spectrum circa 2006
0.1 0.2 0.3 0.4 0.5 0.61.0
1.5
2.0
2.5
quenched
dynamical
mπ ~ 7
00 MeV
mπ ~ 5
00 MeV
mπ ~ 3
00 MeV
others?
![Page 27: 0-%012)+*%3#$4$5€¦ · $F"./+&! @;& 8&7$%"2$ lattice%QCD%exotic%meson%spectrum%circa%2006 0.1 0.2 0.3 0.4 0.5 0.6 1.0 1.5 2.0 2.5 quenched dynamical m π MeV others?](https://reader035.vdocuments.us/reader035/viewer/2022071002/5fbf72459e28ac23fd699ed5/html5/thumbnails/27.jpg)
0.1 0.2 0.3 0.4 0.5 0.61.0
1.5
2.0
2.5
quenched
dynamical
dynamical
exotic JPC -‐ after
2+-‐, 1-‐+, 0-‐-‐
![Page 28: 0-%012)+*%3#$4$5€¦ · $F"./+&! @;& 8&7$%"2$ lattice%QCD%exotic%meson%spectrum%circa%2006 0.1 0.2 0.3 0.4 0.5 0.6 1.0 1.5 2.0 2.5 quenched dynamical m π MeV others?](https://reader035.vdocuments.us/reader035/viewer/2022071002/5fbf72459e28ac23fd699ed5/html5/thumbnails/28.jpg)
exotic JPC -‐ after
2+-‐, 1-‐+, 0-‐-‐
0.1 0.2 0.3 0.4 0.5 0.61.0
1.5
2.0
2.5
dynamical
![Page 29: 0-%012)+*%3#$4$5€¦ · $F"./+&! @;& 8&7$%"2$ lattice%QCD%exotic%meson%spectrum%circa%2006 0.1 0.2 0.3 0.4 0.5 0.6 1.0 1.5 2.0 2.5 quenched dynamical m π MeV others?](https://reader035.vdocuments.us/reader035/viewer/2022071002/5fbf72459e28ac23fd699ed5/html5/thumbnails/29.jpg)
understanding & interpreting ?
0.4
0.6
0.8
1.0
1.2
1.4
1.6
0.4
0.6
0.8
1.0
1.2
1.4
1.6
0.4
0.6
0.8
1.0
1.2
1.4
1.6
systematics of a pair
mπ ~ 700 MeV
1S
2S
3S?
1D
2D?1G?
1P
2P
1F
![Page 30: 0-%012)+*%3#$4$5€¦ · $F"./+&! @;& 8&7$%"2$ lattice%QCD%exotic%meson%spectrum%circa%2006 0.1 0.2 0.3 0.4 0.5 0.6 1.0 1.5 2.0 2.5 quenched dynamical m π MeV others?](https://reader035.vdocuments.us/reader035/viewer/2022071002/5fbf72459e28ac23fd699ed5/html5/thumbnails/30.jpg)
understanding & interpreting ?
0.4
0.6
0.8
1.0
1.2
1.4
1.6
0.4
0.6
0.8
1.0
1.2
1.4
1.6
0.4
0.6
0.8
1.0
1.2
1.4
1.6
systematics of a pair
mπ ~ 700 MeV
1S
2S
3S?
1D
2D?1G?
1P
2P
1F?
![Page 31: 0-%012)+*%3#$4$5€¦ · $F"./+&! @;& 8&7$%"2$ lattice%QCD%exotic%meson%spectrum%circa%2006 0.1 0.2 0.3 0.4 0.5 0.6 1.0 1.5 2.0 2.5 quenched dynamical m π MeV others?](https://reader035.vdocuments.us/reader035/viewer/2022071002/5fbf72459e28ac23fd699ed5/html5/thumbnails/31.jpg)
understanding & interpreting ? try (model-‐dependent) analysis of matrix elements
1-‐-‐
![Page 32: 0-%012)+*%3#$4$5€¦ · $F"./+&! @;& 8&7$%"2$ lattice%QCD%exotic%meson%spectrum%circa%2006 0.1 0.2 0.3 0.4 0.5 0.6 1.0 1.5 2.0 2.5 quenched dynamical m π MeV others?](https://reader035.vdocuments.us/reader035/viewer/2022071002/5fbf72459e28ac23fd699ed5/html5/thumbnails/32.jpg)
understanding & interpreting ? try (model-‐dependent) analysis of matrix elements
1-‐-‐
hybrid?
(spin-singlet)
![Page 33: 0-%012)+*%3#$4$5€¦ · $F"./+&! @;& 8&7$%"2$ lattice%QCD%exotic%meson%spectrum%circa%2006 0.1 0.2 0.3 0.4 0.5 0.6 1.0 1.5 2.0 2.5 quenched dynamical m π MeV others?](https://reader035.vdocuments.us/reader035/viewer/2022071002/5fbf72459e28ac23fd699ed5/html5/thumbnails/33.jpg)
understanding & interpreting ? try (model-‐dependent) analysis of matrix elements
hybrid?
ground state is dominantly
0.4
0.6
0.8
1.0
1.2
1.4
1.6
0.0 0.1 0.2 0.3 0.4
500
1000
1500
2000
2500
3000
1-‐-‐
![Page 34: 0-%012)+*%3#$4$5€¦ · $F"./+&! @;& 8&7$%"2$ lattice%QCD%exotic%meson%spectrum%circa%2006 0.1 0.2 0.3 0.4 0.5 0.6 1.0 1.5 2.0 2.5 quenched dynamical m π MeV others?](https://reader035.vdocuments.us/reader035/viewer/2022071002/5fbf72459e28ac23fd699ed5/html5/thumbnails/34.jpg)
understanding & interpreting ? try (model-‐dependent) analysis of matrix elements
hybrid?
ground state is dominantly
0.4
0.6
0.8
1.0
1.2
1.4
1.6
0.0 0.1 0.2 0.3 0.4
500
1000
1500
2000
2500
3000
1st excited state is dominantly
0.0 0.1 0.2 0.3 0.4
500
1000
1500
2000
2500
3000
1-‐-‐
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understanding & interpreting ? try (model-‐dependent) analysis of matrix elements
hybrid?
ground state is dominantly
0.4
0.6
0.8
1.0
1.2
1.4
1.6
0.0 0.1 0.2 0.3 0.4
500
1000
1500
2000
2500
3000
1st excited state is dominantly
0.0 0.1 0.2 0.3 0.4
500
1000
1500
2000
2500
3000
2nd excited state is dominantly
with some
0.0 0.1 0.2 0.3 0.4
500
1000
1500
2000
2500
3000
1-‐-‐
![Page 36: 0-%012)+*%3#$4$5€¦ · $F"./+&! @;& 8&7$%"2$ lattice%QCD%exotic%meson%spectrum%circa%2006 0.1 0.2 0.3 0.4 0.5 0.6 1.0 1.5 2.0 2.5 quenched dynamical m π MeV others?](https://reader035.vdocuments.us/reader035/viewer/2022071002/5fbf72459e28ac23fd699ed5/html5/thumbnails/36.jpg)
understanding & interpreting ? try (model-‐dependent) analysis of matrix elements
hybrid?
ground state is dominantly
0.4
0.6
0.8
1.0
1.2
1.4
1.6
0.0 0.1 0.2 0.3 0.4
500
1000
1500
2000
2500
3000
1st excited state is dominantly
0.0 0.1 0.2 0.3 0.4
500
1000
1500
2000
2500
3000
2nd excited state is dominantly
with some
0.0 0.1 0.2 0.3 0.4
500
1000
1500
2000
2500
3000
3rd excited state is dominantly
hybrid?
with some
0.0 0.1 0.2 0.3 0.4
500
1000
1500
2000
2500
3000
1-‐-‐
![Page 37: 0-%012)+*%3#$4$5€¦ · $F"./+&! @;& 8&7$%"2$ lattice%QCD%exotic%meson%spectrum%circa%2006 0.1 0.2 0.3 0.4 0.5 0.6 1.0 1.5 2.0 2.5 quenched dynamical m π MeV others?](https://reader035.vdocuments.us/reader035/viewer/2022071002/5fbf72459e28ac23fd699ed5/html5/thumbnails/37.jpg)
the lightest hybrid supermultiplet ?
0.4
0.6
0.8
1.0
1.2
1.4
1.6
0.4
0.6
0.8
1.0
1.2
1.4
1.6
0.4
0.6
0.8
1.0
1.2
1.4
1.6
mπ ~ 700 MeV
1S
2S
3S?
1D
2D?1G?
1P
2P
1F
0.4
0.6
0.8
1.0
1.2
1.4
1.6
0.4
0.6
0.8
1.0
1.2
1.4
1.6
0.4
0.6
0.8
1.0
1.2
1.4
1.6
0-‐+
1-‐-‐2-‐+ 1-‐+
0.4
0.6
0.8
1.0
1.2
1.4
1.6
0.4
0.6
0.8
1.0
1.2
1.4
1.6
0.4
0.6
0.8
1.0
1.2
1.4
1.6
![Page 38: 0-%012)+*%3#$4$5€¦ · $F"./+&! @;& 8&7$%"2$ lattice%QCD%exotic%meson%spectrum%circa%2006 0.1 0.2 0.3 0.4 0.5 0.6 1.0 1.5 2.0 2.5 quenched dynamical m π MeV others?](https://reader035.vdocuments.us/reader035/viewer/2022071002/5fbf72459e28ac23fd699ed5/html5/thumbnails/38.jpg)
Does QCD predict light hybrid mesons?
YES
![Page 39: 0-%012)+*%3#$4$5€¦ · $F"./+&! @;& 8&7$%"2$ lattice%QCD%exotic%meson%spectrum%circa%2006 0.1 0.2 0.3 0.4 0.5 0.6 1.0 1.5 2.0 2.5 quenched dynamical m π MeV others?](https://reader035.vdocuments.us/reader035/viewer/2022071002/5fbf72459e28ac23fd699ed5/html5/thumbnails/39.jpg)
Does QCD predict light hybrid mesons?
YES
* Light hybrid mesons may be observed only for heavy quark masses, risk of large hadronic widths has not been determined. The Hadron Spectrum Collaboration offers no guarantee that they can be produced in photoproduction.
*
![Page 40: 0-%012)+*%3#$4$5€¦ · $F"./+&! @;& 8&7$%"2$ lattice%QCD%exotic%meson%spectrum%circa%2006 0.1 0.2 0.3 0.4 0.5 0.6 1.0 1.5 2.0 2.5 quenched dynamical m π MeV others?](https://reader035.vdocuments.us/reader035/viewer/2022071002/5fbf72459e28ac23fd699ed5/html5/thumbnails/40.jpg)
models in light of what we’ve seen
0.4
0.6
0.8
1.0
1.2
1.4
1.6
0.4
0.6
0.8
1.0
1.2
1.4
1.6
0.4
0.6
0.8
1.0
1.2
1.4
1.6
flux-‐tube
1-‐+ 0+-‐ 2+-‐
bag
1-‐+ 0+-‐ 2+-‐
quasi-‐gluon1-‐-‐ “S-‐wave”
1-‐+ 0+-‐ 2+-‐ 0-‐-‐
quasi-‐gluon1+-‐ “P-‐wave”
1-‐+ 0+-‐ 2+-‐
exotic state degeneracy pattern
✘? ✔?✔? ✘✘
0.4
0.6
0.8
1.0
1.2
1.4
1.6
0.4
0.6
0.8
1.0
1.2
1.4
1.6
0.4
0.6
0.8
1.0
1.2
1.4
1.6
0-‐+
1-‐-‐2-‐+ 1-‐+
0.4
0.6
0.8
1.0
1.2
1.4
1.6
0.4
0.6
0.8
1.0
1.2
1.4
1.6
0.4
0.6
0.8
1.0
1.2
1.4
1.6
have the ‘right’ complete lightest hybrid supermultiplet
![Page 41: 0-%012)+*%3#$4$5€¦ · $F"./+&! @;& 8&7$%"2$ lattice%QCD%exotic%meson%spectrum%circa%2006 0.1 0.2 0.3 0.4 0.5 0.6 1.0 1.5 2.0 2.5 quenched dynamical m π MeV others?](https://reader035.vdocuments.us/reader035/viewer/2022071002/5fbf72459e28ac23fd699ed5/html5/thumbnails/41.jpg)
* Light hybrid mesons may be observed only for heavy quark masses, risk of large hadronic widths has not been determined. The Hadron Spectrum Collaboration offers no guarantee that they can be produced in photoproduction.
addressing the small print
calculate at lighter quark masses increased computational cost
just a matter of time ...
![Page 42: 0-%012)+*%3#$4$5€¦ · $F"./+&! @;& 8&7$%"2$ lattice%QCD%exotic%meson%spectrum%circa%2006 0.1 0.2 0.3 0.4 0.5 0.6 1.0 1.5 2.0 2.5 quenched dynamical m π MeV others?](https://reader035.vdocuments.us/reader035/viewer/2022071002/5fbf72459e28ac23fd699ed5/html5/thumbnails/42.jpg)
* Light hybrid mesons may be observed only for heavy quark masses, risk of large hadronic widths has not been determined. The Hadron Spectrum Collaboration offers no guarantee that they can be produced in photoproduction.
addressing the small print
observing states as decaying resonances
scattering of composite objects in non-‐perturbative field theory !
![Page 43: 0-%012)+*%3#$4$5€¦ · $F"./+&! @;& 8&7$%"2$ lattice%QCD%exotic%meson%spectrum%circa%2006 0.1 0.2 0.3 0.4 0.5 0.6 1.0 1.5 2.0 2.5 quenched dynamical m π MeV others?](https://reader035.vdocuments.us/reader035/viewer/2022071002/5fbf72459e28ac23fd699ed5/html5/thumbnails/43.jpg)
* Light hybrid mesons may be observed only for heavy quark masses, risk of large hadronic widths has not been determined. The Hadron Spectrum Collaboration offers no guarantee that they can be produced in photoproduction.
addressing the small print
observing states as decaying resonances
scattering of composite objects in non-‐perturbative field theory !
-70
-60
-50
-40
-30
-20
-10
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
HooglandLosty
CohenDurusoyex
pt.
-15
-10
-5
0
5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
HooglandLosty
CohenDurusoyex
pt.
Isospin=2 ππ scattering
![Page 44: 0-%012)+*%3#$4$5€¦ · $F"./+&! @;& 8&7$%"2$ lattice%QCD%exotic%meson%spectrum%circa%2006 0.1 0.2 0.3 0.4 0.5 0.6 1.0 1.5 2.0 2.5 quenched dynamical m π MeV others?](https://reader035.vdocuments.us/reader035/viewer/2022071002/5fbf72459e28ac23fd699ed5/html5/thumbnails/44.jpg)
* Light hybrid mesons may be observed only for heavy quark masses, risk of large hadronic widths has not been determined. The Hadron Spectrum Collaboration offers no guarantee that they can be produced in photoproduction.
addressing the small print
observing states as decaying resonances
scattering of composite objects in non-‐perturbative field theory !
-70
-60
-50
-40
-30
-20
-10
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
HooglandLosty
CohenDurusoyex
pt.
-15
-10
-5
0
5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
HooglandLosty
CohenDurusoyex
pt.
Isospin=2 ππ scattering
0
0.2
0.4
0.6
0.8
1
400 600 800 1000 1200 1400
0
0.2
0.4
0.6
0.8
1
400 600 800 1000 1200 1400
0
0.2
0.4
0.6
0.8
1
400 600 800 1000 1200 1400
0
0.2
0.4
0.6
0.8
1
400 600 800 1000 1200 1400
mπ=481MeV
mπ=421MeV
mπ=330MeV
mπ=290MeV
Isospin=1 ππ scattering
Feng, Jansen, Renner
![Page 45: 0-%012)+*%3#$4$5€¦ · $F"./+&! @;& 8&7$%"2$ lattice%QCD%exotic%meson%spectrum%circa%2006 0.1 0.2 0.3 0.4 0.5 0.6 1.0 1.5 2.0 2.5 quenched dynamical m π MeV others?](https://reader035.vdocuments.us/reader035/viewer/2022071002/5fbf72459e28ac23fd699ed5/html5/thumbnails/45.jpg)
* Light hybrid mesons may be observed only for heavy quark masses, risk of large hadronic widths has not been determined. The Hadron Spectrum Collaboration offers no guarantee that they can be produced in photoproduction.
addressing the small print
determining coupling to photons is relatively straighforward
t-‐channel meson exchange
p N
γM’
M
production modeling
![Page 46: 0-%012)+*%3#$4$5€¦ · $F"./+&! @;& 8&7$%"2$ lattice%QCD%exotic%meson%spectrum%circa%2006 0.1 0.2 0.3 0.4 0.5 0.6 1.0 1.5 2.0 2.5 quenched dynamical m π MeV others?](https://reader035.vdocuments.us/reader035/viewer/2022071002/5fbf72459e28ac23fd699ed5/html5/thumbnails/46.jpg)
* Light hybrid mesons may be observed only for heavy quark masses, risk of large hadronic widths has not been determined. The Hadron Spectrum Collaboration offers no guarantee that they can be produced in photoproduction.
addressing the small print
determining coupling to photons is relatively straighforward
t-‐channel meson exchange
p N
γM’
M
production modelingγ
M’
M
photocoupling / transition form-‐factor
lattice calculation of this
![Page 47: 0-%012)+*%3#$4$5€¦ · $F"./+&! @;& 8&7$%"2$ lattice%QCD%exotic%meson%spectrum%circa%2006 0.1 0.2 0.3 0.4 0.5 0.6 1.0 1.5 2.0 2.5 quenched dynamical m π MeV others?](https://reader035.vdocuments.us/reader035/viewer/2022071002/5fbf72459e28ac23fd699ed5/html5/thumbnails/47.jpg)
* Light hybrid mesons may be observed only for heavy quark masses, risk of large hadronic widths has not been determined. The Hadron Spectrum Collaboration offers no guarantee that they can be produced in photoproduction.
addressing the small print
determining coupling to photons is relatively straighforward
t-‐channel meson exchange
p N
γM’
M
production modelingγ
M’
M
photocoupling / transition form-‐factor
lattice calculation of this0 1 2 3 4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Q2 / GeV
V(Q
2 )
CLEO ‘08
p = 000
p = 100
e.g. in charmonium
0 1 2 3 4-0.1
0.1
0.2
0.3
0.4
0.5
0.6
0
CLEO ‘08PDG ‘08
p = 000
p = 100
Q2 / GeV
V(Q
2 )
![Page 48: 0-%012)+*%3#$4$5€¦ · $F"./+&! @;& 8&7$%"2$ lattice%QCD%exotic%meson%spectrum%circa%2006 0.1 0.2 0.3 0.4 0.5 0.6 1.0 1.5 2.0 2.5 quenched dynamical m π MeV others?](https://reader035.vdocuments.us/reader035/viewer/2022071002/5fbf72459e28ac23fd699ed5/html5/thumbnails/48.jpg)
“Isoscalar meson spectroscopy from lattice QCD” -‐ arXiv:1102.4299 [hep-‐lat] (2011) (PRD in press)“The phase-‐shift of isospin-‐2 ππ scattering from lattice QCD” -‐ PRD.83.071504 (2011)“Toward the excited meson spectrum of dynamical QCD” -‐ PRD.82.034508 (2010)“Highly excited and exotic meson spectrum from dynamical lattice QCD” -‐ PRL.103.262001 (2009)“A novel quark-‐field creation operator construction for hadronic physics in lattice QCD” -‐ PRD.80.054506 (2009)
Robert Edwards (JLab)Balint Joo (JLab)David Richards (JLab)Christopher Thomas (JLab)Mike Peardon (Trinity Coll., Dublin)
for the Hadron Spectrum Collaboration
Does QCD predict light hybrid mesons?
Jozef DudekOld Dominion University & Jefferson Lab
![Page 49: 0-%012)+*%3#$4$5€¦ · $F"./+&! @;& 8&7$%"2$ lattice%QCD%exotic%meson%spectrum%circa%2006 0.1 0.2 0.3 0.4 0.5 0.6 1.0 1.5 2.0 2.5 quenched dynamical m π MeV others?](https://reader035.vdocuments.us/reader035/viewer/2022071002/5fbf72459e28ac23fd699ed5/html5/thumbnails/49.jpg)
isoscalar mesons
isovector correlator :
![Page 50: 0-%012)+*%3#$4$5€¦ · $F"./+&! @;& 8&7$%"2$ lattice%QCD%exotic%meson%spectrum%circa%2006 0.1 0.2 0.3 0.4 0.5 0.6 1.0 1.5 2.0 2.5 quenched dynamical m π MeV others?](https://reader035.vdocuments.us/reader035/viewer/2022071002/5fbf72459e28ac23fd699ed5/html5/thumbnails/50.jpg)
isoscalar mesons
isovector correlator :
isoscalar (with just light quarks) correlator :
-‐2
![Page 51: 0-%012)+*%3#$4$5€¦ · $F"./+&! @;& 8&7$%"2$ lattice%QCD%exotic%meson%spectrum%circa%2006 0.1 0.2 0.3 0.4 0.5 0.6 1.0 1.5 2.0 2.5 quenched dynamical m π MeV others?](https://reader035.vdocuments.us/reader035/viewer/2022071002/5fbf72459e28ac23fd699ed5/html5/thumbnails/51.jpg)
isoscalar mesons
isovector correlator :
isoscalar (with just light quarks) correlator :
-‐2 noisy
need inversion from all t
GPUs ideal for the ‘grunt’ work
![Page 52: 0-%012)+*%3#$4$5€¦ · $F"./+&! @;& 8&7$%"2$ lattice%QCD%exotic%meson%spectrum%circa%2006 0.1 0.2 0.3 0.4 0.5 0.6 1.0 1.5 2.0 2.5 quenched dynamical m π MeV others?](https://reader035.vdocuments.us/reader035/viewer/2022071002/5fbf72459e28ac23fd699ed5/html5/thumbnails/52.jpg)
isoscalar mesons
isoscalar (with light & strange) :
-‐2
-‐
⎧|
⎩
⎫|
⎭||
||
diagonalising gives the mixing
-‐√2
-‐√2
![Page 53: 0-%012)+*%3#$4$5€¦ · $F"./+&! @;& 8&7$%"2$ lattice%QCD%exotic%meson%spectrum%circa%2006 0.1 0.2 0.3 0.4 0.5 0.6 1.0 1.5 2.0 2.5 quenched dynamical m π MeV others?](https://reader035.vdocuments.us/reader035/viewer/2022071002/5fbf72459e28ac23fd699ed5/html5/thumbnails/53.jpg)
isoscalar spectrum
ETMC (2009) [2-‐flavour, extrap. to phys. quark mass]ρ/ω splitting of 27(10) MeV
very few results due to the difficulty of calculation
C. Michael et al (UKQCD, 2001) [heavy quarks, 2-‐flavour theory]found f1/a1 , b1/h1 , ρ/ω splittings consistent with zero
![Page 54: 0-%012)+*%3#$4$5€¦ · $F"./+&! @;& 8&7$%"2$ lattice%QCD%exotic%meson%spectrum%circa%2006 0.1 0.2 0.3 0.4 0.5 0.6 1.0 1.5 2.0 2.5 quenched dynamical m π MeV others?](https://reader035.vdocuments.us/reader035/viewer/2022071002/5fbf72459e28ac23fd699ed5/html5/thumbnails/54.jpg)
isoscalar spectrum
0.5
1.0
1.5
2.0
2.5
exotics
isoscalar
isovector
YM glueball
negative parity positive parity
M / GeV
η
η’
mπ ~ 420 MeV
RBC/UKQCD (2010)
ETMC (2009) [2-‐flavour, extrap. to phys. quark mass]ρ/ω splitting of 27(10) MeV
very few results due to the difficulty of calculation
C. Michael et al (UKQCD, 2001) [heavy quarks, 2-‐flavour theory]found f1/a1 , b1/h1 , ρ/ω splittings consistent with zero
![Page 55: 0-%012)+*%3#$4$5€¦ · $F"./+&! @;& 8&7$%"2$ lattice%QCD%exotic%meson%spectrum%circa%2006 0.1 0.2 0.3 0.4 0.5 0.6 1.0 1.5 2.0 2.5 quenched dynamical m π MeV others?](https://reader035.vdocuments.us/reader035/viewer/2022071002/5fbf72459e28ac23fd699ed5/html5/thumbnails/55.jpg)
isoscalar spectrum
0.5
1.0
1.5
2.0
2.5
exotics
isoscalar
isovector
YM glueball
negative parity positive parity
![Page 56: 0-%012)+*%3#$4$5€¦ · $F"./+&! @;& 8&7$%"2$ lattice%QCD%exotic%meson%spectrum%circa%2006 0.1 0.2 0.3 0.4 0.5 0.6 1.0 1.5 2.0 2.5 quenched dynamical m π MeV others?](https://reader035.vdocuments.us/reader035/viewer/2022071002/5fbf72459e28ac23fd699ed5/html5/thumbnails/56.jpg)
isoscalar spectrum
0.5
1.0
1.5
2.0
2.5
exotics
isoscalar
isovector
YM glueball
negative parity positive parity
![Page 57: 0-%012)+*%3#$4$5€¦ · $F"./+&! @;& 8&7$%"2$ lattice%QCD%exotic%meson%spectrum%circa%2006 0.1 0.2 0.3 0.4 0.5 0.6 1.0 1.5 2.0 2.5 quenched dynamical m π MeV others?](https://reader035.vdocuments.us/reader035/viewer/2022071002/5fbf72459e28ac23fd699ed5/html5/thumbnails/57.jpg)
isoscalar spectrum
0.5
1.0
1.5
2.0
2.5
exotics
isoscalar
isovector
YM glueball
negative parity positive parity
![Page 58: 0-%012)+*%3#$4$5€¦ · $F"./+&! @;& 8&7$%"2$ lattice%QCD%exotic%meson%spectrum%circa%2006 0.1 0.2 0.3 0.4 0.5 0.6 1.0 1.5 2.0 2.5 quenched dynamical m π MeV others?](https://reader035.vdocuments.us/reader035/viewer/2022071002/5fbf72459e28ac23fd699ed5/html5/thumbnails/58.jpg)
0.0 0.1 0.2 0.3 0.4
500
1000
1500
2000
2500
3000
0.0 0.1 0.2 0.3 0.4
500
1000
1500
2000
2500
3000
0.0 0.1 0.2 0.3 0.4
500
1000
1500
2000
2500
3000
1D
1F1-‐+
M / GeV
level ordering ?
mπ ~ 4
00 MeV
mπ ~ 7
00 MeV
![Page 59: 0-%012)+*%3#$4$5€¦ · $F"./+&! @;& 8&7$%"2$ lattice%QCD%exotic%meson%spectrum%circa%2006 0.1 0.2 0.3 0.4 0.5 0.6 1.0 1.5 2.0 2.5 quenched dynamical m π MeV others?](https://reader035.vdocuments.us/reader035/viewer/2022071002/5fbf72459e28ac23fd699ed5/html5/thumbnails/59.jpg)
real* resonances
but we can’t be satisfied with this ...
APS April meeting 2011
*complex
![Page 60: 0-%012)+*%3#$4$5€¦ · $F"./+&! @;& 8&7$%"2$ lattice%QCD%exotic%meson%spectrum%circa%2006 0.1 0.2 0.3 0.4 0.5 0.6 1.0 1.5 2.0 2.5 quenched dynamical m π MeV others?](https://reader035.vdocuments.us/reader035/viewer/2022071002/5fbf72459e28ac23fd699ed5/html5/thumbnails/60.jpg)
real* resonances
but we can’t be satisfied with this ...
APS April meeting 2011
*complex
the spectrum should not be this simple
enhancements in the meson-‐meson scattering continuum
excited states should be resonances
![Page 61: 0-%012)+*%3#$4$5€¦ · $F"./+&! @;& 8&7$%"2$ lattice%QCD%exotic%meson%spectrum%circa%2006 0.1 0.2 0.3 0.4 0.5 0.6 1.0 1.5 2.0 2.5 quenched dynamical m π MeV others?](https://reader035.vdocuments.us/reader035/viewer/2022071002/5fbf72459e28ac23fd699ed5/html5/thumbnails/61.jpg)
real* resonances
but we can’t be satisfied with this ...
in finite volume only discrete meson-‐meson states
but we aren’t seeing them !
APS April meeting 2011
*complex
the spectrum should not be this simple
enhancements in the meson-‐meson scattering continuum
excited states should be resonances
![Page 62: 0-%012)+*%3#$4$5€¦ · $F"./+&! @;& 8&7$%"2$ lattice%QCD%exotic%meson%spectrum%circa%2006 0.1 0.2 0.3 0.4 0.5 0.6 1.0 1.5 2.0 2.5 quenched dynamical m π MeV others?](https://reader035.vdocuments.us/reader035/viewer/2022071002/5fbf72459e28ac23fd699ed5/html5/thumbnails/62.jpg)
real* resonances
but we can’t be satisfied with this ...
in finite volume only discrete meson-‐meson states
but we aren’t seeing them !
APS April meeting 2011
*complex
the spectrum should not be this simple
enhancements in the meson-‐meson scattering continuum
excited states should be resonances
0.4
0.6
0.8
1.0
1.2
1.4
1.6
increasingvolume
AB
![Page 63: 0-%012)+*%3#$4$5€¦ · $F"./+&! @;& 8&7$%"2$ lattice%QCD%exotic%meson%spectrum%circa%2006 0.1 0.2 0.3 0.4 0.5 0.6 1.0 1.5 2.0 2.5 quenched dynamical m π MeV others?](https://reader035.vdocuments.us/reader035/viewer/2022071002/5fbf72459e28ac23fd699ed5/html5/thumbnails/63.jpg)
real* resonances
but we can’t be satisfied with this ...
in finite volume only discrete meson-‐meson states
but we aren’t seeing them !
APS April meeting 2011
*complex
the spectrum should not be this simple
enhancements in the meson-‐meson scattering continuum
excited states should be resonances
0.4
0.6
0.8
1.0
1.2
1.4
1.6
increasingvolume
AB
our operators closely resemble single hadrons ...
and not meson-‐meson pairs ~
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28
parity doubling & chiral symmetry restoration ?
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0.0 0.1 0.2 0.3 0.4
500
1000
1500
2000
2500
3000
0.0 0.1 0.2 0.3 0.4
500
1000
1500
2000
2500
3000
0.0 0.1 0.2 0.3 0.4
500
1000
1500
2000
2500
3000
1-‐-‐
0-‐+
2-‐+
overlaps with decreasing quark mass
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cubic complications ...
integer spin not a good quantum number
restricted rotational symmetry of a cube
APS April meeting 2011
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cubic complications ...
integer spin not a good quantum number
restricted rotational symmetry of a cube
APS April meeting 2011
E
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cubic complications ...
integer spin not a good quantum number
restricted rotational symmetry of a cube
APS April meeting 2011
⎭⎬⎫E
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cubic complications ...
integer spin not a good quantum number
restricted rotational symmetry of a cube
APS April meeting 2011
⎭⎬⎫ ⎫ ⎬ | ⎭|
E
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cubic complications ...
‘solved’ by careful operator construction
APS April meeting 2011
construct operators of definite J in the continuum
“subduce” into the cubic group irreps
if the rotational symmetry is “restored”and then
operators respect cubic symmetry, but are ‘preconditioned’ to be J-‐diagonal
... but does it work in practice ?
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cubic complications ...
APS April meeting 2011
0.6
0.8
1.0
1.2
1.4
1.6
(26)
clear dominance of a single spin for each state
J=1
J=3
J=4