Lattice QCD study of Radiative Transitions in Charmonium

Transcription

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 2

3 Simulation details anisotropic Wilson gauge action domain wall fermions for charm quark propagators quenched approximation & no disconnected diagrams e.g. two-point Jo Dudek, Jefferson Lab 3

4 Smearing & two-point fits smeared local psuedoscalar two-point simultaneous multi-exponential fit to multiple correlators Jo Dudek, Jefferson Lab 4

5 Smeared-local two-point functions Require early-time plateau in two-points for good three-points smeared local Jo Dudek, Jefferson Lab 5

6 Scale setting & the spectrum Sommer scale and 1P-1S produce compatible lattice spacings we didn t tune our charm quark mass -parameter very accurately our whole charmonium spectrum is too light small residual differences from discretisation, quenching & lack of disconnected Jo Dudek, Jefferson Lab 6

7 Three-point functions radiative transitions involve the insertion of a vector current to a quark-line computation of this correlator is by the sequential-source method: red line is one propagator new calc n for each new used local vector current (hence not-conserved) gets renormalised multiplicatively Jo Dudek, Jefferson Lab 7

8 Fit method for three-point functions are extracted from the two-point function fits is what we intend to extract alternative ratio method is less simple to implement when we have multiple form-factors (multipoles) Jo Dudek, Jefferson Lab 8

9 form-factor & setting true has no electromagnetic form-factor by charge conjugation invariance - at quark level by coupling to - non-zero if just couple to systematic problem extract from the plateaux Jo Dudek, Jefferson Lab 9

10 form-factor plot as a function of Jo Dudek, Jefferson Lab 10

11 transition transitions between different states are no more difficult decomposition in terms of one form-factor has a polarisation smeared-smeared Jo Dudek, Jefferson Lab 11

12 transition radiative width: Note that this is just one Crystal Ball measurement I cooked up an alternative from Jo Dudek, Jefferson Lab 12

13 transition how do we extrapolate back to? - take advice from the non-rel quark-model: 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 Jo Dudek, Jefferson Lab 13

14 transition we have a clue about the systematic difference: scaling of by 1.11 Jo Dudek, Jefferson Lab 14

15 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: Jo Dudek, Jefferson Lab 15

16 transition three-point function several different combinations have the same value are known functions do the inversion to obtain the multipole form-factors Jo Dudek, Jefferson Lab 16

17 transition Jo Dudek, Jefferson Lab 17

18 transition Jo Dudek, Jefferson Lab 18

19 transition we ll again constrain our extrapolation using a quark model form this is an E1 transition which proceeds by the electric-dipole moment convoluting with Gaussian wavefunctions one obtains where the photon 3-momentum at virtuality is given by in the rest frame of a decaying Jo Dudek, Jefferson Lab 19

20 transition Jo Dudek, Jefferson Lab 20

21 transition also obtain the physically unobtainable C1 multipole Jo Dudek, Jefferson Lab 21

22 Quark model extrapolation error shrinks as!? property of the extrapolation form: at the error on is irrelevant; only error on matters great benefit of this form but only if it s right! Jo Dudek, Jefferson Lab 22

23 transition two physical multipoles contribute: E1, M2 also one longitudinal multipole: C1 multipole decomposition: Jo Dudek, Jefferson Lab 23

24 transition Jo Dudek, Jefferson Lab 24

25 nearer to our simulations do have points very near to : very small, negative no plateaux! Jo Dudek, Jefferson Lab 25

26 how well do we do? wavefunction extent: quark model charmonium wavefunctions (from Coulomb + Linear potential) have typically transition widths & multipole ratios: lat* PDG CLEO lat PDG Grotch et al *using lattice simul. masses Jo Dudek, Jefferson Lab 26

27 extras Jo Dudek, Jefferson Lab 27

28 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 Jo Dudek, Jefferson Lab 28

29 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 Jo Dudek, Jefferson Lab 29

30 transition we anticipate borderline plateaux: Jo Dudek, Jefferson Lab 30

31 transition we anticipate borderline plateaux: Jo Dudek, Jefferson Lab 31

32 transition we anticipate borderline plateaux: Jo Dudek, Jefferson Lab 32

33 Anisotropic Lattices the gluon (Yang-Mills) piece of the action gains a parameter, this is chosen to get the desired anisotropy 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) is tuned to ensure physical particles have the correct dispersion relation i.e. (up to lattice artifacts) Jo Dudek, Jefferson Lab 33

34 Jo Dudek, Jefferson Lab 34

35 Dispersion relation tests display the dispersion relation via the quantity perfect tuning would be we ve not tuned perfectly a hazard of using anisotropy! Jo Dudek, Jefferson Lab 35