1. Hadronic 3-body B decays
Hai-Yang Cheng
Academia Sinica, Taipei
FPCP2008, Taipei, May 6, 2008

2. Introduction focus on charmless B→PPP
Many three-body B decays have been observed with rates ~10-5
useful for extracting CKM angles, CP violation
Most of quasi-2-body B decays (B→VP,SP) are extracted from Dalitz plot analysis of 3-body decays
A(B→P1P2P3)= resonant + nonresonant (NR)
NR signal is less than 10% in D decays. Is NR component also
small in B decays ?
There is no any theoretical study addressing both resonant and NR effects. Some works are based on flavor symmetry
Chua, Soni, and I (2007) have applied the factorization approach to study the dynamics of 3-body decays
Gronau, Rosner
focus on charmless B→PPP

3. KKK:  90% K:  35-40% by Belle, 5% by BaBar K0: 12-15%
Two striking features:
Large NR fractions in penguin-dominated modes
Nonresonant fraction (%)
BaBar
Belle
B-→K+K-K-
14118
74.8±3.6
B0→K+K-K0
112±15
B0→K0+-
41.95.5
B-→K-+-
4.51.5
34.02.9
B0→K-+0
12.42.9
15.67.7
<25.7
B-→+--
13.66.1
KKK:  90%
K:  35-40% by Belle,
5% by BaBar
K0: 12-15%
:  14%
NR contributions are essential in penguin-dominated B decays
One of our goals is to identify the origin of NR signals

4. David Asner (Friday): DP in D decays
NR amplitude in charm decays is usually treated as a constant over Dalitz plot. However, this is no longer true in B decays due to large energy release. Both BaBar & Belle employ the parametrization
to study the NR component in B→KKK decay, but differ in the NR analysis in B→K: BaBar adopted the LASS parametrization which is an effective range NR component + Breit-Wigner form for K0*(1430)
difficult to disentangle resonant & NR effects due to interference
Recall that NR accounts for (95±7)% of D+→K-++ in old experiments. With + included by E791, NR fraction is reduced to (8.6±0.8)%, confirmed recently by CLEO.
David Asner (Friday): DP in D decays

5. New broad scalar resonances fX(1550) & fX(1300)
A broad scalar resonance fX(1500) [or X(1550) by BaBar] has been seen in K+K+K-, K+K-KS, K+K-- at energies ~1.5 GeV. It cannot be identified with f0(1500), otherwise it will decay to +- five times more frequently than to K+K- . Its nature is not clear.
Production puzzle: The fraction of fX(1500) in K+K+K- is ~120% by BaBar and 63% by Belle, whereas it is  4% in K+K-KS by BaBar
Belle
BaBar
B+→K+K+K-
B0→K+K-K0
Likewise, fX(1300) was seen in K++- and K0+-. Its mass & width are consistent with f0(1500)

6. Three factorizable amplitudes for B0→K+K-K0
current-induced process: <B0→K0><0→K+K->
transition process: <B0 → K-K0><0→K+>
annihilation process: <B0→0><0→K+K-K0>
b→s
b→u

7. b→u
almost pure NR
Early attempt: Apply HMChPT to evaluate form factors r and 
Wise, Yan et al. Donoghue et al. (1992)
Bajc,Fajfer,Oakes,Pham; Deandrea et al. (’99) K- K0 K- B0
+,r B0 B- r K0 K0 K0 K- B0
B*0s
+,-,r B0 K-
B*0s B- r

8. NR rates for B→KKK,K,  will become too large
For example, Br(B0→K+K-K0)NR=7710-6 larger than total BR=2510-6
⇒ HMChPT is applicable only to soft mesons !
Ways of improving the use of HMChPT have been suggested before
We now propose to write NR amplitude as
Fajfer et al. Yang, HYC,…
-- HMChPT is recovered in soft meson limit, p2, p3→0
-- The parameter NR » 1/(2mB) is constrained from B-→+--

9. V=, , ,…, S=f0(980), f0(1370), f0(1500), fX(1500),…
b→s
V=, , ,…, S=f0(980), f0(1370), f0(1500), fX(1500),…
Decay constants for scalar mesons have been evaluated using QCDSR Chua,Yang,HYC

10. <K+K-|qq|0> is related to the kaon’s e.m. form factors
ch, x1, x2 fitted from kaon e.m. data
Chua,Hou,Shiau,Tsai
motivated by asymptotic constraint from QCD counting rules
Brodsky, Farrar NR
NR is constrained by KSKSKS rate and K+K- mass spectrum

11. B0→K+K-K BaBar: PRL, 99, (2007)
Final state
BRexpt (10-6)
BRtheory
K0
2.98±0.45
f0(980)K0
9.57±2.51
X0(1550)K0
0.98±0.44 NR
26.7±4.6
total
23.8±2.6
NR rates: 88% from b→s (via <KK|ss|0>)
and 3% from b→u transitions

12. B-→K+K-K- BR(10-6) 1st theory error: NR
BaBar: PRD, 74, (2006)
Belle: PRD, 71, (2005)
BR(10-6)
B-→K+K-K-
1st theory error: NR
2nd theory error: ms, NR, form factors
3rd theory error: 
The predicted NR rate agrees with Belle
The large fraction of X0(1550), 121% by BaBar and 63% by Belle, is entirely unexpected, recalling that it is only 4% in K+K-K0

13. B-→K-+- BaBar Belle Evidence for direct CP violation in B→0K:
ACP=(30± )% by Belle, PRL 96, (2006)
ACP=(44± )% by BaBar, arXiv:
9.3±
BaBar & Belle have very different results for NR fractions: ~4.5% by BaBar, ~34% by Belle
BaBar Belle
calculable for the first time
K0*(1430): LASS parametrization Relativistic Breit-Wigner
K0*(1430) resonance with an effective range NR component
NR: phase space (constant amplitude) exponential
total nonres= NR(p.s.)+NR(LASS)
arXiv:

14. Difficulties for extracting NR component by BaBar:
Substantial mixing of NR & K0*(1430) due to LASS shape
Part of LASS is really NR and should be added to phase-space NR piece
Total NR=NR(LASS) + NR(p.s.)
This leads to a better agreement with Belle, NR fraction is enhanced from 4.5% to 17.5%
No perfect agreement due to different models for NR K mass shape

15. Why is NR rate large in K++- ?
SU(3) symmetry ⇒
⇒ similar NR rates are expected in K++- and in KKK.
Why is NR fraction ~ 40% in K-+- but ~ 90% in K+K-K- ?
resonant poles in KKK: , f0(980),…
resonances in K: K*, K*0(1430), , f0(980),…
⇒ K has a total rate larger than KKK by a factor  2

16. BaBar: arXiv:
Belle: PLB, 599, 148 (2004)
B0→K-+0
BaBar: LASS + nonres
Belle: performed with simplified technique for DP; interference between quasi-two-body amplitudes was not taken into account
Just as DP analysis of B-→K-+-, it is necessary to include NR(LASS) to get total nonres for BaBar.

17. Tree dominated B→KK, 

18. B-→K+K-- dominated by b→u tree and b→d penguin
Decay rate is small and consistent with the limits set by BaBar & Belle.
Recently, BaBar [PRL 99, (2007)] obtained
Br(B+→K+K-+)=(5.0±0.5±0.5)10-6
broad peak at ~1.5 GeV in KK mass
no peak at ~ 1 GeV due to 

19. B→
dominated by intermediate  mesons
Since <|qq|0> is suppressed by penguin Wilson coefficients, NR
amplitude arises mainly from B→ transition⇒ NR is suppressed
⇒ can be used to fix the NR parameter
+--
B→+-0 is predicted to have a rate (Br=26.3£10-6) larger than +-- as it receives +, - and 0 resonant contributions

20. Quasi-two-body B decays
We compute B→P1P2P3 and then apply narrow width approximation
(B→ RP3; R￫P1P2)=(B→RP3) Br(R→P1P2) R: V,S
and
to determine the rates of quasi-two-body B decays: B→VP,SP

21. VP modes Br(-++-+)=24.0±2.5
QCDF predictions are from Beneke and Neubert
Unless specified, expt’l BRs are extracted from 3-body Dalitz plot analysis

22. SP modes
QCDF predictions are from Chua, Yang, HYC. Assumption of Br(f0(980)→+-)=0.50 has been made
f0(980)K rates are well accommodated, K*0(1430) rates are too small by a factor of 2~3 compared to the data due to destructive interference between a4 & a6 terms charming penguin ? Lesniak et al [arXiv: ]
penguin annihilation ?

23. b→sqq tCPV measurements
Sf= ± sin2eff
from b→ccs
2-body: HYC,Chua,Soni;Beneke
3-body: CCS
Also pQCD, SCET
Naïve b→s penguin average: 0.68±0.04, 0.56±0.05 (if f0K0 excluded), 0.0.1, 2.2, 2.6 deviation from b→ccs average

24. CP asymmetries in K+K-KS & KSKSKS
See C.K. Chua talk
sin2b=O(+0.1) is naively expected in K+K-KS due to color-allowed tree contribution, tied to NR amplitude
DS, ACP are small in KsKsKs: no b→u tree diagram
sin2=0.6800.025 (all charmonium), (CKM fit)

25. sin2theory is always positive and less than O(0.1)
sin2eff=sin2eff-sin2charmonium
Chua,Soni,HYC, PR,D76, (2007)
theory expt
sin2(K+K-KS) = ±0.11
sin2(KSKSKS) = ±0.20
sin2(KS00) = 0.41
sin2(KS+-) =
sin2theory is always positive
and less than O(0.1)

26. Conclusions It is important to understand the NR amplitudes in 3-body
decays. We have identified two NR sources:
We found large NR signal in K modes.
Total NR issue should be clarified
Contribution of fX(1500) 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.
m.e. of scalar density <KK|ss|0>, <K|qs|0>, BR  2010-6
tree transition, BR  210-6

27. Back-up slides

28. Different topological decay amplitudes
HYC, Yang (02’)
Tree
bu
Penguin
bs, d
K, KKK: b → s penguin
, KK: b → u tree & b → d penguin

29. Factorizable contributions
Creation
Tree
bu
Transition
Annihilation
Penguin
bs, d

30. Three-Body Branching Ratios (10-6)
3-Body Mode
BaBar
Belle
K+π+π-
54.4±1.1±4.6
48.8±1.1±3.6
K0π+π-
43.0±2.3±2.3
47.5±2.4±3.7
K+π-π0
34.9±2.1±3.9
36.6±4.1±0.8
K+K+K-
33.5±0.9±1.6
30.6±1.2±2.3
K0K+K-
23.8±2.0±1.6
28.3±3.3±4.0
++-
16.2±1.2±0.9
KSKSK+
10.7±1.2±1.0
13.4±1.9±1.5
KSKSKS
6.9±0.8±0.6
±0.8
K+K-π+
5.0±0.5±0.5
<13

31. CP-odd K+K-KS decay spectrum
b→s
b→u
b→s
b→u
The b→s transition prefers a small m(K+K-)
Low mKK peak due mainly to KS
The b→u transition prefers a small m(K+K0) and hence large m(K+K-) ⇒ tiny interference between b→s & b→u transitions

32. CP-even K+K-KS decay spectrum
CP-even+CP-odd
b→s
b→u
low mKK peak: f0(980)KS + NR
peak at mKK  1.5 GeV due to X0(1550)