1. Exotic and excited-state meson spectroscopy and radiative transitions from lattice QCD Christopher Thomas, Jefferson Lab In collaboration with: Jo Dudek, Robert Edwards, David Richards and the Hadron Spectrum Collaboration QNP 2009, Beijing, China

2. Outline 1 Introduction Light meson spectrum Charmonium radiative transitions

3. Overview Photoproduction at GlueX (JLab 12 GeV upgrade) Light mesons GlueX (JLab), BESIII, PANDA Exotics (1 -+,...) ? 2 Spectrum and photocouplings

4. Spectroscopy on the lattice 3 Calculate energies and matrix elements (Z) from correlation functions of meson interpolating fields

5. Variational Method 4 Consider a large basis of operators  matrix of correlators C ij (t) Generalised eigenvector problem: Eigenvalues  energies Eigenvectors  optimal linear combination of operators to overlap on to a state Z (n) related to eigenvectors (t >> t 0 )

6. Spin and operator construction 5 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 6 Construct operators which only overlap on to one spin in the continuum limit On a lattice, 3D rotation group is broken to Octahedral Group IrrepA1A1 A2A2 T1T1 T2T2 E dim11332 cont. spins0,4,6,...3,6,7,...1,3,4,...2,3,4,...2,4,5,... ‘Subduce’ operators on to lattice irreps:

8. Light Meson Spectroscopy 7 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 M  = 833 MeV Unquenched calculation (dynamical fermions) Show here mostly results with volume = 16 3 (L s  1.96 fm)

9. 8 1 -+ 4 -- 4 -+ J -+ J -- Exotic First J = 4 on lattice! mπmπ

10. 9 4 ++ 0 +- 2 +- J +- J ++ More exotics

11. 10 J -- 13S113S1 23S123S1 13D113D1 Vector Hybrid?? Z values Mass / MeV This operator  commutator of two covariant derivatives

12. 11 Z values – spin 2 J --

13. What about multiparticle states? 12 Expect two-meson states above 2m  Where are they? 2m   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 -- Mass / MeV Preliminary 16 3 20 3 T 1 --

14. Charmonium radiative transitions Meson – Photon coupling BABAR, Belle, BES, CLEO-c Exotic 1 -+ ? 13 Dudek, Edwards & CT, PR D79 094504 (2009)

15. Same scale as many measured conventional charmonium transitions BUT very large for an M 1 transition Exotic 1 -+ – Vector 1 -- 14 Usually M 1  spin flip  1/m c suppression Spin-triplet hybrid  M 1 transition without spin flip  not suppressed M 1 multipole dominates

16. More charmonium results 15 Vector (1 -- ) hybrid candidate: Vector – Psuedoscalar Scalar – Vector Axial – Vector Dudek, Edwards & CT, PR D79 094504 (2009) Tensor – Vector transitions Identify 1 3 P 2, 1 3 F 2, 2 3 P 2 tensors from hierarchy of multipoles E 1, M 2, E 3 Quenched, only disconnected diagrams, one volume and one lattice spacing

17. Summary and Outlook 16 Charmonium Lighter mesons

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19. Extra Slides 18

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

21. More on 3-points 20 Source (t i ): Only (smeared) local operators (  =  5,  i, 1) Momentum selected automatically from momentum cons. Local vector current: j  (t i < t < t f ) Sink (t f ): Use ‘best’ operator, O (n), from 2- point analysis Specify p f (usually p f = 0 0 0 )

22. Charmonium radiative transitions 21 Electric (E k ), Magnetic (M k ) Dipole E 1, Quadrupole E 2, Octupole E 3 Multipoles J i = J f  k (k > 0) No parity change: E k for even k, M k for odd k Parity change: E k for odd k, M k for even k Experimentally measure multipoles at Q 2 = 0

23. Spin on the lattice 22 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: A 1, A 2, E, T 1, T 2 (+ others for half-integer) IrrepA1A1 A2A2 T1T1 T2T2 E dim11332 cont. spins0,4,6,...3,6,7,...1,3,4,...2,3,4,...2,4,5,...

24. Lattice systematics – charmonium 23 Quenched anisotropic lattice (a s /a t = 3) Clover fermion action Vector current three-point functions from sequential source technology Only connected diagrams (OZI justification?) Fixed lattice spacin g, a t -1 = 6.05 GeV  0.033 fm Fixed volume (12 3 x 48) (L s  1.2 fm) Extrapolation to Q 2 = 0

25. Lattice systematics – light mesons 24 UNQUENCHED anisotropic lattice (a s /a t = 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 spacin g, a t -1 = 5.62(4) GeV  0.035 fm First volume = 16 3 x 128 (L s  1.96 fm) (other volumes underway) First results are with M  = 833 MeV (other masses underway)

26. Charmonium radiative transitions 25 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)

27. Radiative Transition Results 26 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 (a t -1 = 6.05 GeV, from static pot.) Constant term in t dependence; fit Q 2 form (or similar):

28. Tensor 2 ++ – Vector 1 -- 27 PDG08: 406(31) keV Same hierarchy as expected: Ratio |M 2 /E 1 | is considerably larger than experiment |E 1 (0)| > |M 2 (0)| >> |E 3 (0)| Quark models (1 3 P 2 )  290 – 420 keV E 1, M 2, E 3

29. Tensor 2 ++ – Vector 1 -- 28 Completely different hierarchy! |E 3 (0)| > |M 2 (0)|, |E 1 (0)| E 1, M 2, E 3

30. Tensor 2 ++ – Vector 1 -- 29 Reverted to expected hierarchy: |E 1 (0)| > |M 2 (0)| >> |E 3 (0)| Quark models (2 3 P 2 )  50 – 80 keV E 1, M 2, E 3

31. Tensor 2 ++ – Vector 1 -- 30 Interpretation: single quark transition model In general: E 1, M 2, E 3 (k = 1,2,3) If only a single quark is involved ( 3 P 2  3 S 1 ): j = 1/2  j = 1/2 k = 1,2 only and E 3 = 0 |E 1 (0)| > |M 2 (0)| >> |E 3 (0)| If instead tensor is 3 F 2 ( 3 F 2  3 S 1 ): j = 5/2  j = 1/2 k = 2,3 only and E 1 = 0 |E 3 (0)| > |M 2 (0)| >> |E 1 (0)| Interpretation:  c2  – 1 3 P 2  ’ c2 – 1 3 F 2  ’’ c2 – 2 3 P 2 Supported by spectrum analysis

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

33. Vector 1 -- – Pseudoscalar 0 -+ 32 Only M 1 Quark model: spin flip (  1/m c ) gives suppression  ’ is 2 3 S 1  1 1 S 0 – further suppressed

34. Vector 1 -- – Pseudoscalar 0 -+ 33 Quark model: 1 3 D 1  1 1 S 0 has same leading Q 2 behaviour as 2 3 S 1  1 1 S 0 Only M 1

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

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

37. Scalar 0 ++ – Vector 1 -- 36 Only E 1

38. Axial 1 ++ – Vector 1 -- 37 c.f. PDG08: 320(20) keV c.f. quark models (1 3 P 1 )  215 – 314 keV Expected hierarchy: |E 1 (0)| > |M 2 (0)| E 1, M 2

39. Axial 1 ++ – Vector 1 -- 38 c.f. quark models (2 3 P 1 )  14 – 71 keV E 1, M 2

40. More charmonium results 39 Exotic 1 -+ : Very large for M 1 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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