1. Singa and kappa analyses at BESII Ning Wu Institute of High Energy Physics, CAS Beijing, China January 25-26, 2007

2. Introduction  The existence of  and  had been suggested from various viewpints both theoretically and phenomenologically. Early analysis of I=0  S-wave phase shift make the conclusions against the existence of the σ particle. As a results, it had ever disappeared from PDG for about 20 years. But most re- analysis of ππ/πK scattering data support the existence of the  and  particles.  In early studies, most evidences of the existence of  and  come from ππ/πK scattering. If  and  particles exist, they should also be seen in the production processes. Searching and studying  and  in production processes is also important for us to know their properties and structures.

3. Introduction(2)  We first found evidence of the existence of σand κparticles in 7.8M BESI J/ ψ data. After BESII obtain much larger J/ ψ data sample, we moved our analysis to be based on BESII data.  Based on BES J/ ψ decay data, a low mass enhancement in ππ spectrum in J/ ψ  ωππ and a low mass enhancement in Kπ spectrum in J/ ψ  K * (892) Kπ are found. In order to prove that they are σand κparticles, we need not only to measure their pole positions, but also to determine their spin-parity. So, PWA analyses are needed to study σand κparticles in these two channels.

4. Introduction(3)  PWA analysis is widely used in BES physics analysis in past a few years. It is used to determine mass, width and branching ratio of a resonance, and to determine its spin- parity.  In this talk, we discuss PWA analyses of J/ ψ  ωππ and J/ ψ  K * (892)K .  Main contents of the talk are 1.Helicity Formalism 2.Maximum Likelihood Method 3.PWA analysis on J/ ψ  ωππ 4.PWA analysis on ψ′  ππJ/ ψ 5.PWA analysis on J/ ψ  K * (892)K  6.Summary

5. Helicity Formalism  For a two-body decay process a  b + c spinJ s b s c momentump a p b p c helicitym λ b λ c parity  a  b  c  Its S-matrix element is Reltive momentum of two final state particles in center of mass syetem

6. Helicity Formalism(2)  The decay amplitude is D-function Helicity coupling amplitude All angular information of the decay vertex are contained in the D-function, and helicity coupling amplitude is independent of all angular variables.

7. Helicity Formalism(3)  In J/ ψ hadronic or radiative decay processes, parity conservation is hold. The heliclity coupling amplitude has the following symmetry  If two final state particle b and c are identical particles, the wave function of final state system should be symmetric or antisymmetric, and

8. Helicity Formalism(4)  In experimental physics analysis, most decays we encountered are sequential decays, and resonant states appear as intermediate states.  The decay amplitude for this sequential decay is a b c d e Decay amplitude for a  b+c Decay amplitude for b  d+e Breit-Wigner function of the resonance b

9. Maximum Likelihood Method  In experimental physics analysis, after we obtain a data sample, we first need to know how many resonances appear and what is the decay mechanism. Then we need to calculate the differential cross-section =( 1, 2,…) helicities of final state particles  =(  1,  2, … )helicities of intermediate resonances mhelicity of the mother particle dΦelement of phase space BGnon-interference backgrounds icomponents considered

10. Maximum Likelihood Method(2)  Normalized probability density function which is used to describe the whole decay process is σtotal cross section W(Φ)effects of detection efficiency xquantities which are measured by experiments αunknown parameters which need to be determined in the PWA fit.

11. Maximum Likelihood Method(3)  Total cross-section is defined by N MC the total number of Monte Carlo events ( … ) j the quantity is calculated from the j-th Monte Carlo events  It is required that these Monte Carlo events are obtained through real detection simulation and have passed all cut conditions which are used to to obtain the data sample of the process.

12. Maximum Likelihood Method(4)  The maximum likelihood function is given by the adjoint probability for all the data  Define  In the data analysis, the goal is to find the set of unknown parameters α by minimizing S. Mass and width of a resonance are determined by mass and width scan. Spin-parity of a resonance is determined by comparing fit quality with different solution of different spin-parity.

13. Study of  Particle at BES Clear signals of σ particle are found in two channels at BES: 1)J/ψ→ωππ 2)Ψ ′ →ππJ/ψ

14.  Pole in J/ψ→ωπ + π - 1)This channel was ever studied by MARKIII, DM2 and BES. 2)In the early studies, the low mass enhancement does not obtain enough attention. 3)Since 2000, BES had performed careful study on the structure of the low mass enhancement, and measured parameters of its pole position. 4)Data sample: BESI 7.8M J/ψ events BESII 58M J/ψ events

15. Study of σ Based on 58M BESII J/ψ Events π 0 and ω Signal BESII

16. Invariant mass spectrum and Daliz plot BESI BESII

17. Possible Origin of the low mass enhancement 1.Backgrounds 2.Phase space effect 3.Threshold Effects 4.Resonance

18. Two different kinds of backgrounds (1)Contain ωparticle in the decay sequence: J/ ψ  ωX (2)Do not contain ωparticle in the decay sequence ω side-band does not contain the low mass enhancement, so it does not come from the second kind of backgrounds. Background Study BESII After side-band subtraction

19. Monte Carlo simulation of some J/ψ decay channels. All these backgrounds can not produced the low mass enhancement, so it can not also come from the first kind of backgrounds. Background Study

20. Generate 50M J/ψ  anything Monte Carlo events. The generator is based on Lund-Charm model. It contain almost all known J/ψ decay channels. It contains the backgrounds of inclusive J/ψ decays. Backgound Study

21. Not a phase space effect. Phase Space Effect BESII Enents are not unifromly scattered in the whole phase space, the shape of the low mass enhancement is also different from that of phase space.

22. A clear peak is seen in the phase space and efficiency corrected spectrum. Threshold effect should decrease monotonically at the threshold. Threshold Effect BESII

23. Summary on σ origin Not from backgrounds Not a phase space effect Not threshold effect It should be a resonance

24. PWA Analysis Two Independent PWA analysis are performed: 1)Using the method of relativistic helicity coupling amplitude analysis to analyze the spectrum of lower mass region 2)Using the Zemach formalism to analyze the spectrum of the whole mass region. Results obtained from two independent analysis are basically consistent.

25. To avoid complicity in the higher mass region, and concentrate our study on the low mass enhancement, PWA analysis is performed only on the 0 — 1.5 GeV mass region. PWA analysis ： 0-1.5 GeV BESII

26. Components The following components are considered: σ f 2 (1270) f 0 (980) b 1 (1235) Background

27. PWA Analysis The dominant backgrounds are phase space backgrounds and ρ3π backgrounds 。 Three different methods are used to fit BG.(free fit, directly side-band subtraction, fix BG to different level) Large uncertainties comes from the fit on backgrounds, which is the main sources of uncertainties.

28. 0 ++ 2 ++ 4 ++ Angular distributions of the low mass enhancement Spin-Parity

29. Compare the fit quality Spin-Parity

30. Parametrizations There does not exist a mature method to parametrize a wide resonance near threshold, so different parametrizations are tried in the fit. Constant width With contains ρ(s) Zheng’s parametrization

31. Three different parametrizations are used in this fit. Mass and width are obtained through the fit, pole positions are calculated theoretically. Mass, width and pole positions Eq.(9)BW of constant width Eq.(13)BW of width contains ρ Eq.(14)Zheng’s parametrization

32. A 0 ++ resonance is used to fit σ particle. Fit on the angular distributions of the lower mass region

33. Contribution from σ particle

34. Fit on angular distributions

35. Fit on Dalitz Plot

36. Method II Another independent PWA analysis is performed in this channel. It analyze the whole mass region and the following processes are added into the fit.

37. Method II (continued) The σ particle is also needed in the fit of the low mass enhancement. The dominant contribution of the low mass enhancement also comes from the σ particle.

38. Method II (continued) Pole positions of the σ particle obtained by this method is consistent with above. The combined results are (541±39) –i (252±42) MeV. Constant With ρ

39. σ particle in Ψ ′ →ππJ/ψ The ππ mass spectrum can be fit phenomenonlogically. It can also be fit by σ particle destructively interfere with a broad scalar structure, i.e. |BW(σ)+IPS| 2.

40. σ particle in Ψ’→ππJ/ψ Strong destructive interference, so that the amplitude at threshold is almost zero. Three different BW parametrizations are also tried in the fit. The shape of the BW given by different parametrizations are almost the same.

41. σ particle in Ψ’→ππJ/ψ Different parametrizations are tried in the PWA fit. Results on pole positions given by these parametrizations are quite consistent.

42. Summary on σ pole positions BESI J/ψ data BESII J/ψ data ωππ system BESII J/ψ data 5π system BESII ψ ′ data

43. Study of κ Particle at BES Clear signal of κ particle is found in the Kπ invariant mass spectrum in the decay channel J/ψ→K * (892) 0 Kπ. It is seen at both BESI and BESII data.

44. κ particle in J/ψ→K * (892) 0 K + π - BESII data For our BESII data, the statistics are much larger.

45. Recoil mass spectrum against K * (892) 0 κ signal is clear in the invariant mass spectrum.

46. Dalitz Plot κ signal is clear in the Dalitz plot.

47. Charge conjugate channel The spectrum is almost the same as that of the charge conjugate channel.

48. Charge conjugate channel 共轭道

49. 1.Backgrounds 2.Phase space effects 3.Threshold effects 4.Resonance Possible origin

50. Background Study K * (892) 0 side-band

51. Background Study Monte Carlo simulation

52. Background Study 50M inclusive Monte Carlo Use the J/ψ  anything Monte Carlo to study the backgrounds from inclusive J/ψ decay.

53. Background Study 50M inclusive Monte Carlo Data MC

54. Background Study 50M inclusive Monte Carlo MC Data All these results show that the low mass enhancement does not come from the backgrounds of inclusive J/ψ decay.

55. Phase space effect Not a phase space effect

56. Summary on κ origin Not from backgrounds 。 Not a phase space effect 。 Threshold effect? (limit ststistics) Should be a resonance

57. Decay sequence In the recoil mass spectrum against κ, only K * (892) 0 can be clearly see, so it is produced through J/ψ→ K * (892) 0 κ 。

58. Decay Mechanism In the PWA analysis, the following four decay processes are considered.

59. Two Independent PWA Analysis Method A is based on covariant helicity amplitude analysis. Method B is based VMW method. Two analyses are based on the same data sample. Two PWA analysis are performed independently.

60. Spin-parity It is a 0 + resonance.

61. Statistical significance

62. Pole position Poles given by different parametrizations are close to each other.

63. Comparison Δφ=φ(s)-φ(s t ) Width contains ρ Constant width Zheng’s parametrization Though mass and width parameters of different parametrizations are different, the the shape given by them are almost the same.

64. Global fit

66. Total 0 ++ contribution

67. Fit on Dalitz plot PWA fit Data

68. Method B Method B is based on VMW method. The least χ 2 method is used in the fit. The fit is independently performed. Its theoretical formula of the decay amplutude, fit method and components added into the fit are different from those of method A. Final PWA results are consistent with those of Method A.

69. VMW Method Analysis formula is based on the Lagrangian of strong interactions.

70. VMW Method Statistical significance In the Mthod B, the κ particle is used to fit the low mass enhancement. Its statistical signigicance is also high.

71. VMW Method Components are different in two analysis.

72. Pole Position Combined:

73. VMW Method

75. Summary 1)In BES J/ψ decay data, σ and κ particles are clearly seen in both invariant mass spectra and Dalitz plots. 2)The spin-parity of the low mass enhancements are determined to be 0 ++. They are considered to be the σ and κ particles respectively. 3) σ and κ particles are highly needed in the fit of the corresponding spectrum. 4)Different parametrizations are used to fit them. And the final pole positions given by these parametrizations are quite close to each other.

76. Thanks !