Koichi SEO Gifu City Womens’ Col. Takayuki MATSUKI Tokyo Kasei Univ.

1. Chiral Particle/Photon emission from heavy-light mesons Koichi SEO Gifu City Womens’ Col. Takayuki MATSUKI Tokyo Kasei Univ. HNP13(07/20/2013)

2. Outline • Introduction • Spectroscopy of heavy-light system • Fundamental Formulation of one-particle decay • Relativistic Formulation of decay widths • Numerical Results for one chiral particle/one photon emission from heavy-light meson • Summary

3. §1.Introduction Mass spectrum of heavy-light mesons has been explained by our group successfully in a semi-relativistic potential model （ Matsuki & Morii , Phys. Rev. D 56, 5646(1997) Matuski et. al. , Prog. Theor. Phys. 117, 1077(2007) ; Eur. Phys. J. A31, 701(2007) ） →　Semi-Leptonic Decay Form Factor 　（ Matsuki & Seo, Prog. Theor. Phys. 118,1087(2007)　） →　πorKemittingHadronic Decay Width 　（ Matsuki & Seo, Phys. Rev. D85,014036(2012)　）

4. in units of MeV Non-relativistic method used in the atomic transition is not appropriate for B and D decays. Relativistic calculation is necessary !

5. Outline • Introduction • Spectroscopy of heavy-light system • Fundamental Formulation of one-particle decay • Relativistic Formulation of decay widths • Numerical Results for one chiral particle/one photon emission from heavy-light meson • Summary

6. §2. Spectroscopy of heavy-light system (Mass Spectraof DsJ) • Successful prediction/reproduction of Ds mass spectra using our semi-relativistic potential model • Lowering 0+ and 1+ of Ds0*(2317) and Ds1’(2460)compared with other potential models Below threshold prediction by our semi-relativisitic potential model (Prog. Theor. Phys.117(2007) 1077) prediction by conventional potential model (Godfrey & Kokski, PRD43, 1679 (1991))

7. Other Mass Spectraof Our Model Successful reproduction of the following spectra • D0*(2318) and D1’(2427) by Belle • Ds0(2860) and Ds*(2715) by BaBar & Belle (n=2; 0+ and 1- states of Ds) • B1(5720) and B2*(5745) by D0 (1+ and 2+ states of B) • Bs2*(5839) by D0 (2+ state of Bs) D Below threshold B Bs

8. Our Numerical Values/Present Exp. Status CDFdata Below BK/B*K threshold

9. Fermi-Yang Equation 0-th order wave function in 1/mQ expansion …Angular & spin wf

10. 1-st order corrections to wave function can be included as Radial wave function Input parameters

11. Outline • Introduction • Spectroscopy of heavy-light system • Fundamental Formulation of one-particle decay • Relativistic Formulation of decay widths • Numerical Results for one chiral particle/one photon emission from heavy-light meson • Summary

12. Intermultipletpionic transition § 3. Fundamental Formulation of one particle decay • Following “Excited heavy-light systems and hadronic transitions”by Di Perro and Eichten, PRD 64, 114004 (2001); Goity and Roberts PRD 60, 034001 (1999).Georgi-Manohar interaction between quarksand π, K,…（ chiral multiplets) light quark current • “Chiral multiplets of heavy-light mesons”by Bardeen, Eichten, and Hill, PRD 68, 054024 (2003)effective Lagrangian among heavy mesonandπ, K,…, γ

13. Calculation of Hadronic-Decay like Goity &Roberts (potential model) Assuming the infinitely heavy mQ, people used to use the static w.f. for the meson. In this talkCalculate decay widths by taking into account the recoileffects of mesonsDecay processes are • 1-pion decay • radiative decay

14. Transition amplitude based on the field theory (1) Wave function in the moving frame (Internal coordinate) • ξ＝0pos. of HQ (X=y) • ξ＝1pos. of LQ (X=x) (External coordinate) Wave function in the rest frame (Confining linear potential) (Color Coulomb potential)

15. Transition amplitude based on the field theory (2) Fermi-YangEquation (Eigen value problem) Transition Amplitude phase no phase factor

16. Transition amplitude based on the field theory (3) Insertand use valencequark approximation. Wave function in moving frame (equal time of two quarks) Time difference in the rest-frame Boost operator Boost matrix Velocity of the meson

17. Transition amplitude based on the field theory (4) Estimate of T. A. if a parent moves with +V in the Breit frame

18. Transition amplitude based on the field theory (5) • Comparing with the methods which have been used • Replace the pion wave function as • There are corrections to perpendicular directions which vanish if contracting with pion momentum

19. In the case of EM current Relativistic formula for the matrix elements of the EM currentin the Breit frame independent of ξ !

20. Relativistic formula for the matrix elements of the EM current in the Breit frame Breit frame: Parent meson is moving with +V. Daughter meson is moving with –V. q~k (photon momentum) “effective mass” of the light quark

21. Radiative decay widths ofheavy-light mesons in a non-relativistic potential model Bardeenetal., PRD 68, 054024 (2003) …E1and M1 Close and Swanson, PRD 72, 094004 (2005) …E1and M1 Godfrey, PRD 72, 054029 (2005)…E1only wave function in the rest frame

22. Outline • Introduction • Spectroscopy of heavy-light system • Fundamental Formulation of one-particle decay • Relativistic Formulation of decay widths • Numerical Results for one chiral particle/one photon emission from heavy-light meson • Summary

23. §4.Relativistic Formulation of Decay Width (Tensor structures of Transition Amplitude) Couples to pion

24. Tensor structures of T. A. for photon

25. Estimate of transition amplitude (1) Wave Function at Rest

26. Estimate of T. A. (2) ｋ＝-1 Valuse of parameters ｋ＝1 ｋ＝-2

27. Estimate of transition amplitude (3) （Ex.）　0+(3P0：k=+1) → 0-(1S0:k=-1)

28. Relation between the boosted wave function and the static wave function Wave function in the moving frame (equal time of two quarks) Boost operator Boost matrix (different time of two quarks) Velocity of the meson

29. 1/mQ corrections in the relation between the boosted wf&the static wf

30. Outline • Introduction • Spectroscopy of heavy-light system • Fundamental Formulation of one-particle decay • Relativistic Formulation of decay widths • Numerical Results for one chiral particle/one photon emission from heavy-light meson • Summary

31. §5Numerical resultsp/g emission from excited D meson states Xiao-Hai Lin Values in the parentheses are 0-th order results in 1/mQ expansion

32. p/g emission from excited B meson states Values in the parentheses are 0-th order results in 1/mQ expansion

33. p/gemission from excited Ds / Bs meson states 33

34. Comparison with non-relativistic calculations in units of keV Ref.1…Bardeenetal., PRD 68, 054024 (2003) Ref.2…Close and Swanson, PRD 72, 094004 (2005) Ref.3…Godfrey, PRD 72, 054029 (2005)

39. Outline • Introduction • Spectroscopy of heavy-light system • Fundamental Formulation of one-particle decay • Relativistic Formulation of decay widths • Numerical Results for one chiral particle/one photon emission from heavy-light meson • Summary

40. §6Summary • Formulas for the one pion decay/radiative decay widths were given including the 1st order corrections of wave functions(wf)& the relation of the moving wf to the static wf in 1/mQ expansion • For chargedD*or Ds*, sizable decay widths were obtained by including the 1st order corrections in 1/mQ expansion. • For DsJ, large decay widths were obtained compared with non-relativistic works.

41. §7non-relativistic limit cf）　Result of Bardeen et al.

42. non-relativistic limit (2) cf）　Result of Bardeen et al.