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

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= - 0.840.71, ACP=-0.270.54

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

(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

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

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2.2+2.1-1.8% by Belle

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

<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

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

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

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)=(10.83+8.68+0.30-4.06-0.32)10-6 consistent with expt=(12.41.2)10-6

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, 

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

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)

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

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), 0.695+0.018-0.016 (CKM fit)

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

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.