photon spectrum in the soft-energy region

1. photon spectrum in the soft-energy region Pedro D. Ruiz-Femenía Nucl.Phys.B (2007), doi:10.1016/j.nuclphysb.2007.07.013 Talk at the International Workshop on Heavy Quarkonium 17-20 October 2007, DESY, Hamburg

2. Outline • Motivation • The oPs decay spectrum for • The soft-radiation loop momentum region • spectrum for • Summary • NRQED calculation • QED spectrum in the soft-energy region ________________________________________________________ International Workshop on Heavy Quarkonium, DESY, Hamburg, Oct 17th

3. Motivation • orthopositronium (o-Ps) decay spectrum for • NRQED, includes binding effects • satisfies Low´s theorem for [ Manohar, RF ’03 ] • valid for the whole region ? new NR mode? is the NREFT calculation missing something? ________________________________________________________ International Workshop on Heavy Quarkonium, DESY, Hamburg, Oct 17th

4. NRQED computation of the oPs spectrum • there is a long-distance part in the decay when one of the photons in the final state is not hard short-distance part hard photons long-distance part Coulomb Hamiltonian ________________________________________________________ International Workshop on Heavy Quarkonium, DESY, Hamburg, Oct 17th

5. Dipole approximation dipole approx. Atomic transitions in Ps Multipole expansion valid for o-Ps  3 decay with one small-energy photon [ Voloshin ’04 ] ________________________________________________________ International Workshop on Heavy Quarkonium, DESY, Hamburg, Oct 17th

6. oPs calculation without multipole expanding the e.m. field dipole approx. ________________________________________________________ International Workshop on Heavy Quarkonium, DESY, Hamburg, Oct 17th

7. EFT oPs spectrum: results ________________________________________________________ International Workshop on Heavy Quarkonium, DESY, Hamburg, Oct 17th

8. EFT oPs spectrum: ________________________________________________________ International Workshop on Heavy Quarkonium, DESY, Hamburg, Oct 17th

9. from the NRQED computation in the region QED computation • exact phase-space distribution at one-loop is known [ Adkins ’05 ] ________________________________________________________ International Workshop on Heavy Quarkonium, DESY, Hamburg, Oct 17th

10. Threshold expansion of QED 1-loop diagrams Beneke, Smirnov ‘98 • asymptotic expansion of loop integrals involving heavy massive particles close to threshold • separation of regions achieved by expanding loop momenta according to the v-scaling above (each region contributes to a single power of v) ________________________________________________________ International Workshop on Heavy Quarkonium, DESY, Hamburg, Oct 17th

11. Ladder graph ________________________________________________________ International Workshop on Heavy Quarkonium, DESY, Hamburg, Oct 17th

12. 1-loop QED result for • agrees with NLO term of NRQED calculation ________________________________________________________ International Workshop on Heavy Quarkonium, DESY, Hamburg, Oct 17th

13. Relevance for Heavy Quarkonium • photon spectrum can be written as a sum of a direct contrib. and a fragmentation contrib. ________________________________________________________ International Workshop on Heavy Quarkonium, DESY, Hamburg, Oct 17th

14. 1-loop QCD correction for ________________________________________________________ International Workshop on Heavy Quarkonium, DESY, Hamburg, Oct 17th

15. Results for the direct Upsilon decay spectrum Krämer ‘99 Catani, Hautmann ’95 Maltoni, Petrelli ‘99 ________________________________________________________ International Workshop on Heavy Quarkonium, DESY, Hamburg, Oct 17th

16. Summary • NRQED calculation of the photon spectrum valid for all • Threshold expansion of 1-loop diagrams • Application to the soft-energy direct photon spectrum in Upsilon decays is straightforward ________________________________________________________ International Workshop on Heavy Quarkonium, DESY, Hamburg, Oct 17th