Relativistic Effects in Two Photon Decay of Quarkonium

1. Relativistic Effects in Two Photon Decay of Quarkonium H.Q.Zhou Institute of High Energy Physics, CAS, Beijing100039,P.R.China with B.S.Zou

2. 1:Introduction 2:Basic formalism 3:Methods in literatures and Our work 4:Application to and our result 5:Further discussion

3. 1: Introduction The is the most simple and pure process related to QCD bound state, it is an ideal place to test various description of QCD bound state from theoretical point of view. Up to now there are two main methods to study this process: NRQCD and Salpeter equation+Mandelstam form . NRQCD can study the corrections order by order in and while Salpeter equation can study the whole relativistic effects

4. 2:Basic formalism Decay width in Mandelstam form for is momentum of meson relative momentum momentum of photons polarization of photons

5. When we discuss relativistic effects we choose leading order in pQCD for The relativistic effects are mainly contained in BS wave function

6. 3 Methods in literature and Our work Methods in most of the literatures to discuss relativistic effects are in fact equivalent to assuming the following form for BS function: We assume (in meson rest frame) In this form all the relativistic effects just come from momentum distribution This gives two new kinds of corrections which relate with energy distribution when considering the quark confinement and complicated property ofquark propagator and spinor form

7. Why there are such two new kinds of corrections ? complicated quark propagator free spinor form not free spinor form heavy quark limit quark confinement: pole structure of quark propagator heavy quark limit or (in [5]) (in [1,2,3,4]) obey a Schrodinger -type equation

8. These two new kinds of effects are not known clearly (while definitely exist) and are different from the situation in QED because of confinement. For simplicity we do the following parameterization based on heavy quark limit and some physical consideration : with is mean-square-radius of meson

9. In heavy quark limit we have In this limit the above BS wave function will give the same result with the literatures

10. 4: Application to and our result With the formalism, we analysis the following situations and give their comparison. 1: Heavy quark limit 2: Momentum distribution with spinor form 1

11. 3.Momentum distribution with spinor form 2 4.Momentum and energy distribution with spinor form 1 5.Momentum and energy distribution with spinor form 2

12. Result： Dependence of decay width on NR: non-relativistic static limit Re-1: for relativistic case with spinor assuming Re-2: for relativistic case with spinor assuming

14. Conclusion for the relativistic effects of 1: relative momentum distribution : ------ give about -50% correction comparing with static NR approximation 2: various treatments for the bound quark spinor : ------ cause about 6% uncertainty 3: relative energy distribution ------ gives little correction of -2% level.

15. 5:Further discussion We also extend the formalism directly to situation and get the following result: Result of correction: momentum distribution: 75% spinor form: 10% energy distribution: 25%

16. Finally I think the following two problems are interesting 1:What will really happen in the situation of light quarks 2: The comparison of the BS wave function form in heavy quark limit and chiral limit

17. cc THE END THANKS

18. Appendix: 1: discussion of the parameter : 1.2Gev-1.6Gev : : : 2: literatures [1]W-Y Keung, I.J. Muzinich Phys.Rev.D27:1518,1983 [2]Z.P. Li, F.E. Close, T Barnes Phys.Rev.D43:2161-2170,1991 [3]E.S. Ackleh ,T Barnes Phys.Rev.D45:232-240,1992 [4]S.N.Gupta, J M. Johnson, W. W. Repko Phys.Rev.D54:2075-2080,1996 [5]D. Ebert, R.N. Faustov, V.O. Galkin Mod.Phys.Lett.A18:601-608,2003

19. 3:decay width