Jo Dudek, Jefferson Lab

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. 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 Jo Dudek, Jefferson Lab

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

4. Monte Carlo Methods • Numerical (but not direct) integration • Stochastic Monte Carlo method: • Generate configurations Um(i)(x)distributed with probability exp(-SG(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 isO(VxV), though sparse • Full QCD: include det – improved algorithms, cost ~ O(V5/4) • Quenched approximation: set det(M) = 1, e.g., neglect internal quark loops. Jo Dudek, Jefferson Lab

5. 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 Jo Dudek, Jefferson Lab

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

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

8. Smearing we attempt to maximise the overlap of our operator with the ground state by “smearing it out” “gauge-invariant Gaussian” 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 honesty - did not give us that clear plateau - this operator has overlap with all psuedoscalar states Jo Dudek, Jefferson Lab

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

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

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

12. Approximations in the spectrum? 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 do these approximations show up in the spectrum? our hyperfine splitting is too small Jo Dudek, Jefferson Lab

13. Approximations in the spectrum? 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 what about the disconnected diagrams - in the perturbative picture Jo Dudek, Jefferson Lab

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

15. 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 Jo Dudek, Jefferson Lab

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

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

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

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

20. transition 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 causes problems for quark potential models how do we extrapolate back to ? - take advice from the non-rel quark-model: Jo Dudek, Jefferson Lab

21. transition systematic problems dominate Jo Dudek, Jefferson Lab

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

23. transition multipole decomposition: 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 Jo Dudek, Jefferson Lab

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

25. transition Jo Dudek, Jefferson Lab

26. transition Jo Dudek, Jefferson Lab

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

28. transition Jo Dudek, Jefferson Lab

29. transition 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 we’ll again constrain our extrapolation using a quark model form convoluting with Gaussian wavefunctions one obtains Jo Dudek, Jefferson Lab

30. transition Jo Dudek, Jefferson Lab

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

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

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

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

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

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

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

38. transition Jo Dudek, Jefferson Lab

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

40. transition Jo Dudek, Jefferson Lab

41. transition Jo Dudek, Jefferson Lab

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

43. 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 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 quark model extrapolation forms are very powerful – need to verify them with lattice data points at smaller Jo Dudek, Jefferson Lab

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

45. future ultimate aim of this project is to do JLab physics e.g. GlueX: hybrid meson 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 Jo Dudek, Jefferson Lab

46. extras Jo Dudek, Jefferson Lab

47. 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 Jo Dudek, Jefferson Lab

48. 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 Jo Dudek, Jefferson Lab

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

50. 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 Jo Dudek, Jefferson Lab