1. 1 Charmless Three-body Decays of B Mesons Chun-Khiang Chua Chung Yuan Christian University HEP2007, 20 July 2007, Manchester

2. 2 Introduction Many three-body B decay modes are observed with rates~10 -5. These modes usually have non-negligible (sizable) nonresonant (NR) contributions. Some modes, K S K S K S, K + K - K S, are useful for the extraction of sin2  eff Most theoretical works are based on flavor symmetry. (Gronau et al, …) We (Cheng-CKC-Soni) use a factorization approach to study these modes (FA seems to work for DKK decay) A global analysis

3. 3 Different topological decay amplitudes Tree b  u Penguin b  s, d

4. 4 Factorized into creation and transition parts Creation Transition Annihilation Penguin b  s, d Tree b  u

5. 5 b→s b→u A brief look at a typical three-body amp. The creation part is related to the kaon’s e.m. form factors, fitted from data Non-resonant part is constrained by K S K S K S rate and K + K - mass spectrum The transiton part Use HMChPT amplitude with exponential form momentum dependence (  NR fixed from  rate)

6. 6 B 0 to K + K - K 0 rate: Final state Expt (10 -6 ) Theory K0K0 2.98±0.45 2.6 +0.5 -0.4 f 0 (980)K 0 9.57±2.51 5.8 +0.1 -0.5 X 0 (1550)K 0 0.98±0.44 0.93 +0.16 -0.15 NR 26.7±4.6 18.1 +5.1 -3.9 total 23.8±2.6 19.8 +0.6 -0.6 NR rates: 88% from b→s (via ) and 3% from b→u transitions

7. 7 b→s b→u The b→s transition prefers a small m(K + K - ) Low m KK peak due mainly to  K S The b→u transition prefers a small m(K + K 0 ) and hence large m(K + K - ) ⇒ tiny interference between b→s & b→u transitions b→s b→u CP-odd K + K - K S decay spectrum

8. 8 low m KK peak: f 0 (980)K S + NR peak at m KK  1.5 GeV due to X 0 (1550) b→s b→u CP-even K + K - K S decay spectrum CP-even+CP-odd

9. 9 K + K - K S(L) & K S K S K S(L) rates K S K S K S rate is used as an input to fix the NR amplitude in 0→KK creation (denoted the  parameter) SD rates agree with data within errors  c quark not considered yet  a small room for LD contribution 1 st error: m s, B→K form factors,  2 rd error:  NR

10. 10 B - →K + K - K - decay The predicted NR rate agrees with Belle The nature of broad X 0 (1550) (or f X (1500) by Belle) is not clear. The large fraction of X 0 (1550), 121% by BaBar and 63% by Belle, is entirely unexpected, recalling that it is only 4% in K + K - K 0

11. 11 B→K  decays The predicted NR rate agrees with Belle

12. 12 B 0 →K -  +  0 decay 1st theory error:  NR 2nd theory error: ms,  NR, form factors 3rd theory error:  Calculations for K - K S K S, K S  -  0, K S  0  0, K S  +  - modes are also available

13. 13 B→KK  and  S  decays KK  rate agrees with BaBar and Belle  rate is sensitive to B →  transition (can be used to fix the  NR parameter) B 0 →  +  -  0 is predicted to have a rate (Br=26.3  10 -6 ) larger than  +  -  - as it receives  +,  - and  0 resonant contributions B - →K + K -  - B-→+--B-→+--

14. 14  sin2  is naively expected in K + K - K S due to color- allowed tree contribution, tied to NR amplitude  S, A CP are small in K S K S K S : no b→u tree diagram CP asymmetries in K + K - K S, K S K S K S sin2  =0.678  0.025 (all charmonium), 0.695 +0.018 -0.016 (CKM fit)

15. 15  sin2  eff =sin2  eff -sin2  charmonium  theory expt  sin2  (K + K - K S ) =0.050 +0.028 -0.033 -0.098 +0  18 -0.16  sin2  (K S K S K S ) =0.041 +0.027 -0.032 -0.098±0.20  sin2  (K S  0  0 ) =0.051 +0.027 -0.032 -1.398  0.71  sin2  (K S  +  - ) =0.040 +0.031 -0.032

16. 16 Conclusions It is important to understand the NR amplitudes in 3-body decays. We have identified two NR sources. We found large NR signal (35%~40%) in K  modes, in favor of Belle measurements Contribution of X 0 (1550) to K + K + K - should be clarified Intermediate vector & scalar meson contributions to 3-body decays are identified. The total rates of 3-body B decays are calculated for the first time Final-state rescattering  known to be important for penguin-induced two-body modes  some FSI effects for 3-body modes are incorporated in the propagator of resonances  LD c-penguin, u-penguin Full Dalitz plot analysis will be very helpful