1. Branching Fractions and Direct CP
Violation in Hadronic 3-body B decays
Hai-Yang Cheng
Academia Sinica, Taipei
in collaboration with Chun-Khiang Chua
ICHEP, Valencia, July 4, 2014

2. Introduction
Many charmless three-body B decays have been studied at B factories and LHCb (with Dalitz-plot analyses for many of Bu,d decay modes). Most with BFs ~ (BFs ~ 10-6 for B KK & Bs KKK)
useful for extracting CKM angles & CP violation
A(B→P1P2P3) = resonant + nonresonant (NR)
All the quasi-2-body B decays, B→VP, SP (except 00, ) are extracted from Dalitz plot analysis of 3-body decays
NR signal is less than 10% in D decays. Is this also true in B decays ?
(LHCb)

3. Direct CP asymmetries (3-body)
LHCb (’13) found evidence of inclusive CP asymmetry in
B- +--, K+K-K-, K+K--
BaBar(%)
Belle(%)
LHCb(%)
Average
+ - -
3.2
11.72.11.1
10.52.2
K+ K- K-
1.4
-4.30.90.8
-3.71.0
K- + -
2.82.02.3
4.92.62.0
3.20.80.8
3.31.0
K+ K- -
0103
-14.14.01.9
-11.94.1
Large asymmetries observed in localized regions of p.s.
ACP(KK) = 0.0700.013 for mKK2 <1.5 GeV2
ACP(KKK) = 0.0200.004 for 1.2< mKK, low2 <2.0 GeV2, mKK, high2 <15 GeV2
ACP() = 0.5840.0820.027 for m, low2 <0.4 GeV2, m, high2 > 15 GeV2
ACP(K) = 0.6780.0780.032 for 0.08< m, low2 <0.66 GeV2, mK2 <15 GeV2

4. B-  K-+- ACP(K) = 0.0320.0080.0040.007 inclusive
ACP(K) = 0.6780.0780.032 for 0.08< m, low2 <0.66 GeV2, mK2 <15 GeV2

5. Three-body B decays KKK:  70-90% K:  35-40% by Belle, 20% by BaBar
Two striking features:
1. Large NR fractions in penguin-dominated modes
Nonresonant fraction (%)
BaBar
Belle
B-→K+K-K-
6824
78±10
B0→K+K-K0
~ 130
B0→K-KSKS
~ 196
B0→K0+-
B-→K-+-
B0→K-+0
19.73.6
15.67.7
B-→+--
KKK:  70-90%
K:  35-40% by Belle,
20% by BaBar
K0: 15-20%
:  35%
NR contributions are essential in penguin-dominated B decays
One of our goals is to identify the origin of NR signals
HYC, Chua, Soni (’07)

6. 2. A new broad scalar resonance fX(1500) ?
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). So what is its nature ?
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
BaBar(’06)
Belle(’04)
B+→K+K+K-
B0→K+K-K0
New angular momentum analysis by BaBar (’12) fX(1500) is a sum of f0(1500), f0(1710) & f’2(1525) and no more production puzzle 6

7. Most of theory studies focus only on either resonant or NR effects
We take the factorization approximation as a working hypothesis to discuss resonant & NR contributions
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

8. +,r r +,-,r r NR contribution of
Early attempt: Apply heavy meson chiral perturbation theory (HMChPT) to evaluate form factors r and 
Bajc, Fajfer, Oakes, Pham; Deandrea et al. (’99)
Yan et al.; Donoghue et al.; Wise (’92) K- K0 K- B0
+,r B0 B- r K0 K0 K0 K- B0
B*0s
+,-,r B0 K-
B*0s B- r 8

9. NR rates for tree-dominated B→KK,  will become too large
For example, Br(B-→K+K--)NR = 3310-6 larger than total BF, 510-6
⇒ HMChPT is applicable only to soft mesons !
Ways of improving the use of HMChPT have been suggested before
We write tree-induced NR amplitude as
Fajfer et al; Yang, HYC,… p2 p1
-- HMChPT is recovered in soft meson limit, p1, p2→0
-- The parameter NR » 1/(2mB) is constrained from B-→+--

10. V=, , …, S=f0(980), f0(1370), f0(1500), f(1710),…
b→s
V=, , …, S=f0(980), f0(1370), f0(1500), f(1710),…
Decay constants for scalar mesons have been evaluated in various approaches Chua,Yang, HYC; L.D. Lu et al.
How about the NR contributions ? 10

11. <K+K-|qq|0> can be related to the kaon’s e.m. form factors
ch, x1, x2 fitted from kaon e.m. data
Chua,Hou,Shiau,Tsai (’03)
motivated by asymptotic constraint from QCD counting rules
Brodsky, Farrar (’75)
The fitted ch agrees with the model (~ decay constant  strong coupling)
NR  exp[i/4]( ) GeV NR
from K+K- spectrum of K+K-KS
from KSKSKS rate 11

12. Weak phase: CKM matrix elements
The decay amplitude of B0  K+K-K0 consists of two pieces:
Nonresonant: <B0 K+K-><0 K0>
<B0 K0><0 K+K->
(<B0 K0><0 K+K->)penguin
Resonant: B0 f0K0 K+K-K0 , f0 = f0(980), f0(1500), f0(1710),…
B0 VK0 K+K-K0, V = , , ,…
Weak phase: CKM matrix elements
Strong phases: (i) effective Wilson coefficients
(ii) propagator (s - m2 + im)-1
(iii) matrix element <M1M2|qq|0> for NR contribution
in the penguin sector

13. B-→K+K-K- NR rates: mostly from b→s (via <KK|ss|0>)
BF(10-6)
theory errors: (NR) , (ms, NR, form factors), ()
calculable for the first time
Large NR rate is penguin-dominated and governed by <K+K-|ss|0>NR
NR rates: mostly from b→s (via <KK|ss|0>)
and a few percentages from b→u transitions

14. We predict a larger rate of +-0 than +-- as the former receives  and 0 resonant contributions with BF of order 2010-6, while only 0 to the latter.
Belle (’13): BF(B0 K+ K- 0) = (2.170.65)10-6 is a surprise !
Recall BF(B- K+K--) = (5.00.7)10-6
At short-distance level, we obtain BF ~ 510-8
Long-distance contribution due to B0 +-0 followed by +- K+K- rescattering  BF  510-7

15. Inclusive direct CP asymmetries
Expt (%)
Theory (%)
Predictions
in 2007
+ - -
10.32.5
K+ K- K-
-3.71.0
K- + -
3.31.0
 -3.7
K+ K- -
-12.44.5
 13.1
HYC,Chua,Soni (`’07)
Relative signs between K-K+K- & -+- and between K-+- & -K+K- agree with experiment & U-spin symmetry (s  d) predictions:
Xu, Li, He; Bhattacharya, Gronau, Rosner
However, relative signs between -K+K- & -+- and between K-+- &
K-K+K- disagree with the data

16. Correlation seen by LHCb:
ACP(K-K+K-)  – ACP(K-+-), ACP(-K+K-)  – ACP(-+-)
It has been conjectured that CPT theorem & final-state rescattering of +-  K+K- may play important roles
Bediaga et al
FSI
Fit to B- K-+- 
U-spin symmetry 
U-spin symmetry which relates <K|sd|0> to <KK|ss|0> is badly broken

17. Direct CP violation in 3-body Bu,d decays
Theory (%)
Expt (%)
+ - -
10.32.5
K+ K- K-
-3.71.0
K- + -
3.31.0
K+ K- -
-12.44.5
K- K+ 0
K-K+KS
K-KSKS
45
KS+0
7534
(+ - -)region
58.48.7
(K+ K-K-)region
-22.62.2
(K- + -)region
67.88.5
(K+K- -)region
-64.87.2
Inclusive CP asymmetries
predictions
BaBar
Predicted local CP asymmetries seem to be too small except
K+K-K-

18. Regional CP asymmetries due to NR contributions
(ACPregion)NR+RES
Except K+K-K- the magnitude of local CP asymmetries is substantially reduced by nearby resonances
Wang et al
(pQCD)
Zhang, Guo, Yang advocated that local CP violation in +-- arises from interference of 0 with f0(500)
It is important to pin down the underlying mechanism responsible for large local CP violation in 3-body charged B decays

19. BFs & CP violation in 3-body Bs decays
LHCb made first observation of three charmless 3-body Bs decays
Penguin-dominated
(10-6)
(10-6)
Tree-dominated
Penguin-dominated modes K0K-+, K0K+- have largest rates, dominated by K*0(1430) resonances
Tree-dominated mode K+K-K0 is predicted to have BF ~ 1.410-6
ACP(K0K+K-)  - 2ACP(K0+-) 19

20. U-spin symmetry relations
They cannot be tested by the present available data, but can be checked by dynamical calculations.
U-spin relations are generally not well respected as U-spin symmetry is sometimes badly broken

21. Conclusions
Three-body B decays receive sizable NR contributions governed by the matrix elements of scalar densities.
Three sources of strong phases responsible for direct CP violation in 3-body B decays.
It is important to pin down the mechanism responsible for large local CP asymmetries.