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Exotic and excited-state meson spectroscopy and radiative transitions from lattice QCD. Christopher Thomas, Jefferson Lab. QNP 2009, Beijing, China. In collaboration with: Jo Dudek, Robert Edwards, David Richards and the Hadron Spectrum Collaboration. Outline. Introduction - PowerPoint PPT Presentation
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Exotic and excited-state meson spectroscopy and radiative transitions from lattice QCD
Christopher Thomas, Jefferson Lab
In collaboration with: Jo Dudek, Robert Edwards, David Richardsand the Hadron Spectrum Collaboration
QNP 2009, Beijing, China
2
Outline
• Introduction
• Light meson spectrum
• Charmonium radiative transitions
3
Overview
Photoproduction at GlueX (JLab 12 GeV upgrade)
Light mesons
GlueX (JLab), BESIII, PANDA
Exotics (1-+, ...)?
Spectrum and photocouplings
4
Spectroscopy on the lattice
Calculate energies and matrix elements (Z) from correlation functions of meson interpolating fields
5
Variational Method
Consider a large basis of operators matrix of correlators Cij(t)
Generalised eigenvector problem:
Eigenvalues energies
Eigenvectors optimal linear combination of operators to overlap on to a state
Z(n) related to eigenvectors
(t >> t0)
6
Spin and operator construction
On a lattice, 3D rotation group is broken to Octahedral Group
2D Example
Eigenstates of angular momentum are
On a lattice, the allowed rotations are + /2
Can’t distinguish e.g. J = 0 and J = 4
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Spin and operator construction
Construct operators which only overlap on to one spin in the continuum limit
On a lattice, 3D rotation group is broken to Octahedral Group
Irrep A1 A2 T1 T2 E
dim 1 1 3 3 2
cont. spins 0,4,6, ... 3,6,7, ... 1,3,4, ... 2,3,4, ... 2,4,5, ...
‘Subduce’ operators on to lattice irreps:
8
Light Meson Spectroscopy
Dudek et al, arXiv:0909.0200
• Results here are with three degenerate ‘light’ quarks:
Exact SU(3) symmetry – all mesons in octet ( , K, 8) are degenerate (singlet, 1, has different mass)
• Only connected diagrams – Isovectors (I=1) only
• Mp = 833 MeV
• Unquenched calculation (dynamical fermions)
• Show here mostly results with volume = 163 (Ls 1.96 fm)
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1-+
4--
4-+
J-+ J--
Exotic
First J = 4 on lattice!
mπ
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4++
0+- 2+-
J+- J++
More exotics
11
J--
13S1
23S1
13D1
Vector Hybrid??
Z valuesMas
s / M
eV
This operator commutator of two covariant derivatives
12
Z values – spin 2
J--
13
What about multiparticle states?
Expect two-meson states above 2mp
Where are they?
2mp 1.7 GeV
( )L=1 0-- 2.4 GeV (with min mom allowed on lattice)
Momentum constrained to discrete values on a lattice – discrete spectrum of multiparticle states
J--
Mas
s / M
eV
Preli
minary
163 203
T1--
14
Charmonium radiative transitions
Meson – Photon coupling
BABAR, Belle, BES, CLEO-c
Exotic 1-
+ ?
Dudek, Edwards & CT, PR D79 094504 (2009)
15
Same scale as many measured conventional charmonium transitions BUT very large for an M1 transition
Exotic 1-+ – Vector 1--
• Usually M1 spin flip 1/mc suppression
• Spin-triplet hybrid M1 transition without spin flip not suppressed
M1 multipole dominates
16
More charmonium results
Vector (1--) hybrid candidate:
Vector – Psuedoscalar
Scalar – Vector
Axial – Vector
Dudek, Edwards & CT, PR D79 094504 (2009)
Tensor – Vector transitions
Identify 13P2, 13F2, 23P2 tensors from hierarchy of multipoles E1, M2, E3
Quenched, only disconnected diagrams, one volume and one lattice spacing
17
Summary and Outlook
• Method successful – first calc. of excited state rad. trans. on lattice• Hybrid photocoupling is large: • Non-exotic vector hybrid candidate• Comparison with quark models
• Our first results on light mesons – technology and method work• Spin identification is possible• First spin 4 state extracted and confidentially identified on lattice• Exotics (and non-exotic hybrids?)• Ongoing work: different masses and volumes,
multi-meson operators, photocouplings ...
Charmonium
Lighter mesons
18
19
Extra Slides
20
Photocouplings on the Lattice
Calculate from 3-point correlators:
These couplings are what we want.Parameterize in terms of multipoles (like form factors)
nm
Known from 2-point analysis
21
More on 3-points
Source (ti): Only (smeared) local operators ( = 5, i, 1) Momentum selected automatically from momentum cons.
Local vector current: j (ti < t < tf)
Sink (tf): Use ‘best’ operator, O(n), from 2- point analysisSpecify pf (usually pf = 0 0 0 )
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Charmonium radiative transitions
Electric (Ek), Magnetic (Mk)
Dipole E1, Quadrupole E2, Octupole E3
Multipoles
Ji = Jf k (k > 0)
No parity change: Ek for even k, Mk for odd kParity change: Ek for odd k, Mk for even k
Experimentally measure multipoles at Q2 = 0
23
Spin on the lattice
Rotation group:
• infinite number of irreducible representations (irreps)• J = 0, 1, 2, 3, 4, ...
Lattice:
• broken to octahedral group (group of rotations of a cube)• finite number of irreps: A1, A2, E, T1, T2 (+ others for half-integer)
Irrep A1 A2 T1 T2 E
dim 1 1 3 3 2
cont. spins 0,4,6, ... 3,6,7, ... 1,3,4, ... 2,3,4, ... 2,4,5, ...
24
Lattice systematics – charmonium
• Quenched anisotropic lattice (as/at = 3)
• Clover fermion action
• Vector current three-point functions from sequential source technology
• Only connected diagrams (OZI justification?)
• Fixed lattice spacing, at-1 = 6.05 GeV 0.033 fm
• Fixed volume (123 x 48) (Ls 1.2 fm)
• Extrapolation to Q2 = 0
25
Lattice systematics – light mesons
• UNQUENCHED anisotropic lattice (as/at = 3.5)
• Two light clover quarks and one strange quark• in first results strange and light degenerate (other masses
underway)
• Only connected diagrams – isovector states
• Fixed lattice spacing, at-1 = 5.62(4) GeV 0.035 fm
• First volume = 163 x 128 (Ls 1.96 fm) (other volumes underway)
• First results are with Mp = 833 MeV (other masses underway)
26
Charmonium radiative transitions
Lots more results and details in paper: Dudek, Edwards & CT, PR D79 094504 (2009)
Only a brief mention here...
• Caveats:• Quenched (no quark loops; no light quarks at all) • One lattice spacing and volume• Only connected diagrams
Also: Dudek et al PR D77 034501 (2008) ; Dudek & Rrapaj PR D78 094504 (2008)
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Radiative Transition Results
• Photon only couples to quark and not antiquark
• Don’t explicitly include the quark electric charge
• Actually compute
• Plot in terms of temporal lattice spacing (at-1 = 6.05 GeV, from static pot.)
• Constant term in t dependence; fit Q2 form (or similar):
•
28
Tensor 2++ – Vector 1--
PDG08: 406(31) keV
• Same hierarchy as expected:
• Ratio |M2/E1| is considerably larger than experiment
|E1(0)| > |M2(0)| >> |E3(0)|
Quark models (13P2) 290 – 420 keV
E1, M2, E3
29
Tensor 2++ – Vector 1--
Completely different hierarchy! |E3(0)| > |M2(0)| , |E1(0)|
E1, M2, E3
30
Tensor 2++ – Vector 1--
Reverted to expected hierarchy: |E1(0)| > |M2(0)| >> |E3(0)|
Quark models (23P2) 50 – 80 keV
E1, M2, E3
31
Tensor 2++ – Vector 1--
Interpretation: single quark transition model
In general: E1 , M2 , E3 (k = 1,2,3)
If only a single quark is involved (3P2 3S1):j = 1/2 j = 1/2 k = 1,2 only and E3 = 0|E1(0)| > |M2(0)| >> |E3(0)|
If instead tensor is 3F2 (3F2 3S1):j = 5/2 j = 1/2 k = 2,3 only and E1 = 0|E3(0)| > |M2(0)| >> |E1(0)|
Interpretation:
cc2 – 13P2
c’c2 – 13F2
c’’c2 – 23P2
Supported by spectrum analysis
32
Vector 1-- – Pseudoscalar 0-+
Spectrum results (Dudek et al PR D77 034501 (2008) ):
33
Vector 1-- – Pseudoscalar 0-+Only M1
Quark model: • spin flip ( 1/mc) gives suppression• ’ is 23S1 11S0 – further suppressed
34
Vector 1-- – Pseudoscalar 0-+
Quark model: 13D1 11S0 has same leading Q2 behaviour as 23S1 11S0
Only M1
35
Vector 1-- – Pseudoscalar 0-+
Much larger than other1-- 0-+ M1 trans
Analogous to 1-+ hybrid to vector trans:M1 with no spin flip
Spectrum analysis suggests a vector hybrid (spin-singlet)
c.f. flux tube model 30 – 60 keV
Only M1
36
Vector 1-- – Pseudoscalar 0-+
Loops
37
Scalar 0++ – Vector 1--Only E1
38
Axial 1++ – Vector 1--
c.f. PDG08: 320(20) keVc.f. quark models (13P1) 215 – 314 keVExpected hierarchy: |E1(0)| > |M2(0)|
E1, M2
39
Axial 1++ – Vector 1--
c.f. quark models (23P1) 14 – 71 keV
E1, M2
40
More charmonium results
Exotic 1-+:
Very large for M1 transition (typical 2 keV)
Vector (1--) hybrid candidate:
Vector – Psuedoscalar
Scalar – Vector
Axial – Vector
Tensor – Vector
Dudek, Edwards & CT, PR D79 094504 (2009)
41