1. Lattice QCD study of Radiative Transitions in Charmonium (with a little help from the quark model) Jo Dudek, Jefferson Lab with Robert Edwards & David Richards

2. Jo Dudek, Jefferson Lab2 Charmonium spectrum & radiative transitions most charmonium states below threshold have measured radiative transitions to a lighter charmonium state transitions are typically E1 or M1 multipoles, although transitions involving also admit M2 & E3, whose amplitude is measured through angular distributions these transitions are described reasonably well in quark potential models Eichten, Lane & Quigg PRL.89:162002,2002

3. Jo Dudek, Jefferson Lab3 Realization of QCD on a lattice Continuous space-time  4-dim Euclidean lattice Derivatives  finite differences Quarks on sites, gluons on links. Gluons U m (x) = exp(iga A m (x)) elements of group SU(3) Integrate anti-commuting fermion fields  ! det(M(U)) and M -1 (U) factors Gauge action ~ continuum:

4. Jo Dudek, Jefferson Lab4 Monte Carlo Methods Numerical (but not direct) integration Stochastic Monte Carlo method: –Generate configurations U m (i) (x) distributed with probability exp(-S G (U))det(M(U))/Z –Compute expectation values as averages: Statistical errors » 1/sqrt(N), for N configurations The det(M(U)) is expensive since matrix M is O (VxV), though sparse Full QCD: include det – improved algorithms, cost ~ O (V 5/4 ) Quenched approximation: set det(M) = 1, e.g., neglect internal quark loops.

5. Jo Dudek, Jefferson Lab5 Lattice QCD – two-point functions and masses a simple object in QCD – a meson two-point function: choose a quark bilinear of appropriate quantum numbers

6. Jo Dudek, Jefferson Lab6 Lattice QCD – two-point functions and masses e.g.

7. Jo Dudek, Jefferson Lab7 Lattice QCD – two-point functions and masses e.g. ground state “plateau” excited state contribution fermion formalism “nasties”

8. Jo Dudek, Jefferson Lab8 Smearing honesty - did not give us that clear plateau - this operator has overlap with all psuedoscalar states mocking-up the quark-antiquark wavefunction width of the Gaussian is the smearing parameter in practice we can compute two-point functions with several smearings and fit them simultaneously to a sum of exponentials “gauge-invariant Gaussian” we attempt to maximise the overlap of our operator with the ground state by “smearing it out”

9. Jo Dudek, Jefferson Lab9 Smearing & two-point fits [smeared – local] two-point

10. Jo Dudek, Jefferson Lab10 Scale setting & the spectrum this mass is in lattice units – we set the lattice spacing,, using the static quark-antiquark potential on our lattice we didn’t “tune” our charm quark mass very accurately – our whole charmonium spectrum will be too light

11. Jo Dudek, Jefferson Lab11 Spectrum appropriate time to own up to some approximations in our simulations - what have we actually been computing? and not QUENCHED nor NO DISCONNECTED

12. Jo Dudek, Jefferson Lab12 Approximations in the spectrum? do these approximations show up in the spectrum? our hyperfine splitting is too small in quark potential picture hyperfine is a short-distance effect – quenching could bite here because: * set scale with long-distance quantity (a mid-point of the potential) * neglect light quark loops which cause the coupling to run * coupling doesn’t run properly to short-distances “Fermilab” Clover

13. Jo Dudek, Jefferson Lab13 Approximations in the spectrum? what about the disconnected diagrams - in the perturbative picture disconnected diagrams might contribute to the hyperfine splitting - studies (QCD-TARO, Michael & McNeile) suggest an effect of order 10 MeV in the right direction likely to be sensitive also to finite lattice spacing effects

14. Jo Dudek, Jefferson Lab14 Three-point functions radiative transitions involve the insertion of a vector current to a quark-line simple example is the “transition” * analogue of the pion form-factor * couple the vector current only to the quark to avoid charge-conjugation

15. Jo Dudek, Jefferson Lab15 Three-point functions need to insert the current far enough away from that the excited states have died away (exponentially) form-factor is defined by the Lorentz-covariant decomposition compute these three-point functions for a range of lattice momenta - obtain by factoring out the we got from fitting the two-point functions

16. Jo Dudek, Jefferson Lab16 form-factor we fix the source field at and the sink field at - plot three-point as a function of vector current insertion time

17. Jo Dudek, Jefferson Lab17 plot as a function of form-factor

18. Jo Dudek, Jefferson Lab18 transition transitions between different states are no more difficult has a polarisation smeared-smeared decomposition in terms of one form-factor

19. Jo Dudek, Jefferson Lab19 transition radiative width: Note that this is just one Crystal Ball measurement I cooked up an alternative from

20. Jo Dudek, Jefferson Lab20 transition how do we extrapolate back to ? - take advice from the non-rel quark-model: causes problems for quark potential models this is an M1 transition which proceeds by quark spin-flip convoluting with Gaussian wavefunctions one obtains we can fit our lattice points with this form to obtain and

21. Jo Dudek, Jefferson Lab21 transition systematic problems dominate

22. Jo Dudek, Jefferson Lab22 transition we have a clue about the systematic difference: scaling of by 1.11

23. Jo Dudek, Jefferson Lab23 transition at, there are only transverse photons and this transition has only one multipole – E1 with non-zero, longitudinal photons provide access to a second: C1 multipole decomposition:

24. Jo Dudek, Jefferson Lab24 transition three-point function do the inversion to obtain the multipole form-factors several different combinations have the same value are known functions

25. Jo Dudek, Jefferson Lab25 transition

26. Jo Dudek, Jefferson Lab26 transition

27. Jo Dudek, Jefferson Lab27 transition good plateaux were suggested by the two-point functions

28. Jo Dudek, Jefferson Lab28 transition

29. Jo Dudek, Jefferson Lab29 transition we’ll again constrain our extrapolation using a quark model form convoluting with Gaussian wavefunctions one obtains this is an E1 transition which proceeds by the electric-dipole moment where the photon 3-momentum at virtuality is given by in the rest frame of a decaying

30. Jo Dudek, Jefferson Lab30 transition

31. Jo Dudek, Jefferson Lab31 transition also obtain the physically unobtainable* C1 multipole *unless someone can measure quark model form

32. Jo Dudek, Jefferson Lab32 transition two physical multipoles contribute: E1, M2 also one longitudinal multipole: C1 multipole decomposition:

33. Jo Dudek, Jefferson Lab33 transition experimental studies of angular dependence give the M2/E1 ratio

34. Jo Dudek, Jefferson Lab34 transition we might have some trouble with the three-point functions - go back to the at rest two-point function plateau is borderline – our smearing isn’t ideal for this state non-zero momentum states are even worse

35. Jo Dudek, Jefferson Lab35 transition we anticipate borderline plateaux:

36. Jo Dudek, Jefferson Lab36 transition we anticipate borderline plateaux:

37. Jo Dudek, Jefferson Lab37 transition we anticipate borderline plateaux:

38. Jo Dudek, Jefferson Lab38 transition

39. Jo Dudek, Jefferson Lab39 transition quark model structure: has an additional spin-flip beyond the E1 multipole - cost of this is

40. Jo Dudek, Jefferson Lab40 transition

41. Jo Dudek, Jefferson Lab41 transition

42. Jo Dudek, Jefferson Lab42 how well do we do? wavefunction extent: quark model charmonium wavefunctions (from Coulomb + Linear potential) have typically *using lattice simul. masses lat* PDG CLEO transition widths & multipole ratios: lat PDG Grotch et al

43. Jo Dudek, Jefferson Lab43 how well do we do? pretty successful, although still systematics to content with no obvious sign of our approximations: quenching & connected - these transitions are long-distance effects quark model extrapolation forms are very powerful – need to verify them with lattice data points at smaller once current simulation run is complete we’ll make a prediction : excited state transitions are rather tricky – limits the ease with which we can get

44. Jo Dudek, Jefferson Lab44 nearer to our current simulations have points very near to : very small, negative no plateaux

45. Jo Dudek, Jefferson Lab45 future running right now – isotropic lattice (equal spatial & temporal lattice size) - might cure some of our systematics larger volume – gets us closer to but with a simulation-time cost ultimate aim of this project is to do JLab physics e.g. hybrid meson GlueX:

46. Jo Dudek, Jefferson Lab46 extras

47. Jo Dudek, Jefferson Lab47 What were those wiggles? back to the wiggles at small times in the two-point functions they come about because of our choice of formalism for fermions on the lattice - we used Domain Wall Fermions lattice theorist on learning that we’re using anisotropic DWF for charmonium 3pt functions

48. Jo Dudek, Jefferson Lab48 Domain Wall Fermions usually thought of as being good for chiral quarks charm quarks at our lattice spacings are feasible advantage over Wilson is automatic O(a) improvement more computer time, less human time than tuning a Wilson Clover action

49. Jo Dudek, Jefferson Lab49 Domain Wall Wiggles ghosts – don’t have a quadratic dispersion relation

50. Jo Dudek, Jefferson Lab50 Anisotropic Lattices Lattice QCD is a cutoff theory, with. So to model charmonium we will need, i.e. a fine lattice spacing. we also require a lattice volume large enough to hold the state, so appear to need a lot of sites,, and hence a lot of computing time there’s a handy trick to get us out of this - the only large scale is, which is conjugate to time - typical charmonium momenta are only need a fine spacing in the time direction – space directions can be coarse – “Anisotropic Lattice” fine spacing in the time direction allowed us to see all the structure in the two-point functions

51. Jo Dudek, Jefferson Lab51 Anisotropic Lattices the gluon (Yang-Mills) piece of the action gains a parameter, the quark-gluon piece of the action features both and a second parameter,, sometimes called the “bare speed-of-light” (ratio of spat. to temp. derivatives) this is chosen to get the desired anisotropy is tuned to ensure physical particles have the correct dispersion relation i.e. (up to lattice artifacts)

52. Jo Dudek, Jefferson Lab52 Dispersion relation tests we extract the spectrum of states with non-zero momentum using two-point functions because our lattice is of finite volume we don’t have continuous momenta – rather a discrete set

53. Jo Dudek, Jefferson Lab53 Dispersion relation tests

54. Jo Dudek, Jefferson Lab54 Dispersion relation tests display the dispersion relation via the quantity perfect tuning would be we’ve not tuned perfectly – a hazard of using anisotropy!

55. Jo Dudek, Jefferson Lab55 Lattice QCD – two-point functions and masses e.g.

56. Jo Dudek, Jefferson Lab56 ZV for the vector current we use, which isn’t conserved on a lattice. Hence it gets renormalised (multiplicatively) obtain it from the “pion” form-factor at zero Qsq NB we see an 11% discrepancy between the value at pf=000 and at pf=100 same 11% doesn’t seen to be present using the rho form-factor?