1. Theoretical issues with S in 3-body decays
Hai-Yang Cheng
Academia Sinica, Taipei
with Chun-Khiang Chua & Amarjit Soni
CKM2006
Nagoya, December 14, 2006

2. Time-dependent CP asymmetries in 3-body decays
K+K-KL (BaBar): sin2eff=0.090.37, ACP=-0.540.24
K+K-K0 (BaBar): eff=0.3610.087, ACP=-0.0340.082
KS00 (BaBar): sin2eff= 0.71, ACP=-0.270.54

3. Sf  -f sin2eff, f: CP eigenvalue of the final state f
KSKSKS: CP-even eigenstate, f=1
KSKSKL: CP-odd eigenstate, f=-1
K+K-KS: CP-even dominated
f=2f+-1, f+= 0.890.10 (BaBar), 0.930.10 (Belle)
K+K-KL: CP-odd dominated
f= -(2f--1), f-= 0.920.09 (BaBar)
KS00: CP-even eigenstate, f=1

4. (ii) Final-state rescattering of , KK near threshold
Symmetry consideration
SU(3)F symmetry: Grossman, Ligeti, Nir, Quinn; Engelhard, Nir, Raz
isospin & U-spin symmetries: Gronau, Rosner
Dynamical approach
(i) Factorization
HYC, Chua, Soni
(ii) Final-state rescattering of , KK near threshold
Furman, Kaminski, Lesniak, Loiseau

5. B0K+K-K0 decay BaBar: Dalitz plot analysis (see talk by E. Di Marco)
Final state
Fraction(%)
K0
12.91.3
f0(980)K0
22.38.9
X0(1550)K0
4.11.8
NR (KK)
9119
c0K0
2.840.77
D+K-
3.180.89
DS+K-
1.720.65
Dominated by S-wave non-resonant (NR) decays
Recall that NR contributions are at most 10% in existing 3-body decays of D mesons
Destructive interference between resonant & NR amplitudes
It is important to take into account NR effects in any realistic model calculations
Large BR due to NR is also seen in B+→K+K+K- by both BaBar & Belle

6. NR amplitude
Cannot be treated as a constant over Dalitz plot due to large energy release. Both BaBar & Belle employ the parametrization
to study the NR background in B→KKK decay, where mij2=(pi+pj)2
For B→K, BaBar adopted the LASS parametrization: an effective range NR component + Breit-Wigner form for K0*(1430)
⇒ different results for NR fractions in B+→K++-
4.51.5% by BaBar and 34.0 % by Belle

7. Three factorizable amplitudes for B0→K+K-K0
transition process: <B0→K0><0→K+K->
current-induced process: <B0 → K-K0><0→K+>
annihilation process: <B0→0><0→K+K-K0>
b→s
b→u
The first diagram (b→s transition) prefers a small m(K+K-)
The second diagram (b→u transition) prefers a small m(K+K0) and hence large m(K+K-)
 Interference between b→u and b→s is suppressed

8. <K+K-|qq|0> is related to the kaon’s e.m. form factors
b→s
<K+K-|qq|0> is related to the kaon’s e.m. form factors
ch, x1, x2 fitted from kaon e.m. data
motivated by asymptotic constraint from QCD counting rules
i=f0(980), X(1550)  is constrained from KSKSKS rate

9. B→KKK,K,  will become too large (larger than total rate)
b→u
pure NR
If HMChPT is applied to evaluate form factors r and , NR rates for
B→KKK,K,  will become too large (larger than total rate)
⇒ HMChPT is applicable only to soft mesons ! K- K0 K- B0
+,r B0 B- r K0 K0 K0 K- B0
B*0s
+,-,r B0 K-
B*0s B- r

10. We shall follow BaBar to write
Previously we have introduced form factors to account for the off-shell effect and the deviation from chiral limit → somewhat arbitrary and too small NR rates
We shall follow BaBar to write
Chua,Soni,HYC
dominant NR term & destructive interference between NR & pole
amplitudes ⇒ a phase between them
We found  » ||exp(i300) with || determined from B0KSKSKS

11. B0→K+K-K0
Final state
Theory (%)
Fraction(%)
K0
12.1
12.91.3
f0(980)K0
28.1
22.38.9
X(1550)K0
4.3
4.11.8 NR 84
9119
NR amplitudes are half from b→s and half from b→u transitions
Br(B0→K+K-K0)=( )10-6
consistent with expt=(12.41.2)10-6

12. K+K-KS(L) & KSKSKS(L) rates
KS KS KS rate is used as an input to fix the NR amplitude denoted the  parameter
SD rates agree with data within errors
c quark not considered yet
a small room for LD contribution
1st error: ms, B→K form factors, 
2rd error: 
3rd error: cNR, 

13. CP-odd K+K-KS decay spectrum
b→s
b→u
low mKK peak: fKS + NR
middle mKK hump: interference between b→s & b→u amplitudes

14. 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 X(1550)

15. Expressions of S & C Hence, (Gronau 89)  0.42 (=0.22)
 0.42 (=0.22)
Hence, (Gronau 89)
Sources of S: u-penguin, color-suppressed tree for two-body modes
color-allowed tree for three-body modes (e.g. K+K-K0)
or LD u-penguin and color-allowed tree induced from FSI

16. CP asymmetries in K+K-KS & KSKSKS
sin2b=O(+0.1) and ACP 10% in K+K-KS due to color-allowed tree contribution
DS, ACP are small in KsKsKs: no b→u tree diagram
sin2=0.6750.026 (all charmonium), (CKM fit)

17. Uncertainties How to describe NR amplitudes dominated in B→KKK ?
relative phases between pole & NR amplitudes ?
(In principle, the NR parameters xi, ,… are complex)
Full Dalitz plot analysis will be very helpful
form factors, , ms,…
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
Chua,Soni,HYC K-
Ds+
D*0 B0 K+ D-
Ds*+ KS

18. Conclusions Need to understand non- resonant contributions to 3-body
decays
S & ACP in B0→K+K+KS are subject to color-allowed tree
pollution and can be as large as O(10%)
KSKSKS(L) appears very clean: uncertainties in S & ACP are
negligible.