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Exotic and excited-state meson spectroscopy and radiative transitions from lattice QCD

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

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Exotic and excited-state meson spectroscopy and radiative transitions from lattice QCD

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  1. 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

  2. Outline • Introduction • Light meson spectrum • Charmonium radiative transitions

  3. Overview Photoproduction at GlueX (JLab 12 GeV upgrade) Spectrum and photocouplings Light mesons GlueX (JLab), BESIII, PANDA Exotics (1-+, ...)?

  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 (t >> t0) Eigenvectors  optimal linear combination of operators to overlap on to a state Z(n)related to eigenvectors

  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

  7. Spin and operator construction On a lattice, 3D rotation group is broken to Octahedral Group Construct operators which only overlap on to one spin in the continuum limit ‘Subduce’ operators on to lattice irreps:

  8. Light Meson Spectroscopy • Unquenched calculation (dynamical fermions) • 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) • Mp = 833 MeV • Only connected diagrams – Isovectors (I=1) only • Show here mostly results with volume = 163 (Ls  1.96 fm) Dudek et al, arXiv:0909.0200

  9. 4-+ 4-- 1-+ Exotic First J = 4 on lattice! mπ J-- J-+

  10. 0+- 2+- 4++ More exotics J++ J+-

  11. Z values Mass / MeV Vector Hybrid?? This operator  commutator of two covariant derivatives 13D1 23S1 13S1 J--

  12. Z values – spin 2 J--

  13. What about multiparticle states? Mass / MeV Expect two-meson states above 2m Momentum constrained to discrete values on a lattice – discrete spectrum of multiparticle states Preliminary ()L=1 0--  2.4 GeV (with min mom allowed on lattice) 2m  1.7 GeV T1-- 163 203 Where are they? J--

  14. Charmonium radiative transitions BABAR, Belle, BES, CLEO-c Meson – Photon coupling Exotic 1-+ ? Dudek, Edwards & CT, PR D79 094504 (2009)

  15. Exotic 1-+ – Vector 1-- M1multipole dominates Same scale as many measured conventional charmonium transitions BUT very large for an M1 transition • Usually M1 spin flip  1/mc suppression • Spin-triplet hybrid  M1 transition without spin flip  not suppressed

  16. More charmonium results Vector (1--) hybrid candidate: Tensor – Vector transitions Identify 13P2, 13F2, 23P2 tensors from hierarchy of multipoles E1, M2, E3 Vector – Psuedoscalar Scalar – Vector Axial – Vector Quenched, only disconnected diagrams, one volume and one lattice spacing Dudek, Edwards & CT, PR D79 094504 (2009)

  17. Summary and Outlook Charmonium • Method successful – first calc. of excited state rad. trans. on lattice • Hybrid photocoupling is large: • Non-exotic vector hybrid candidate • Comparison with quark models Lighter mesons • 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 ...

  18. Extra Slides

  19. Photocouplings on the Lattice Calculate from 3-point correlators: Known from 2-point analysis These couplings are what we want. Parameterize in terms of multipoles (like form factors) m n

  20. 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 analysis Specify pf(usually pf = 0 0 0 )

  21. Charmonium radiative transitions Multipoles Electric (Ek), Magnetic (Mk) Dipole E1,QuadrupoleE2,OctupoleE3 Ji = Jfk (k > 0) No parity change: Ekfor even k, Mk for odd k Parity change: Ekfor odd k, Mk for even k Experimentally measure multipoles at Q2 = 0

  22. 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)

  23. 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

  24. 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)

  25. Charmonium radiative transitions • Caveats: • Quenched (no quark loops; no light quarks at all) • One lattice spacing and volume • Only connected diagrams Lots more results and details in paper: Dudek, Edwards & CT, PR D79 094504 (2009) Only a brief mention here... Also: Dudek et al PR D77 034501 (2008) ; Dudek & Rrapaj PR D78 094504 (2008)

  26. 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):

  27. Tensor 2++ – Vector 1-- E1, M2, E3 PDG08: 406(31) keV Quark models (13P2)  290 – 420 keV • Same hierarchy as expected: • Ratio |M2/E1| is considerably larger than experiment |E1(0)| > |M2(0)| >> |E3(0)|

  28. Tensor 2++ – Vector 1-- E1, M2, E3 Completely different hierarchy! |E3(0)| > |M2(0)| , |E1(0)|

  29. Tensor 2++ – Vector 1-- E1, M2, E3 Quark models (23P2)  50 – 80 keV Reverted to expected hierarchy: |E1(0)| > |M2(0)| >> |E3(0)|

  30. 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

  31. Vector 1-- – Pseudoscalar 0-+ Spectrum results (Dudek et al PR D77 034501 (2008) ):

  32. Vector 1-- – Pseudoscalar 0-+ Only M1 • Quark model: • spin flip ( 1/mc) gives suppression • ’ is 23S1 11S0 – further suppressed

  33. Vector 1-- – Pseudoscalar 0-+ Only M1 Quark model: 13D1 11S0 has same leading Q2 behaviour as 23S1  11S0

  34. Vector 1-- – Pseudoscalar 0-+ Only M1 Much larger than other 1-- 0-+ M1 trans Spectrum analysis suggests a vector hybrid (spin-singlet) Analogous to 1-+ hybrid to vector trans: M1 with no spin flip c.f. flux tube model 30 – 60 keV

  35. Vector 1-- – Pseudoscalar 0-+ Loops

  36. Scalar 0++ – Vector 1-- Only E1

  37. Axial 1++ – Vector 1-- E1, M2 c.f. PDG08: 320(20) keV c.f. quark models (13P1)  215 – 314 keV Expected hierarchy: |E1(0)| > |M2(0)|

  38. Axial 1++ – Vector 1-- E1, M2 c.f. quark models (23P1)  14 – 71 keV

  39. 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)

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