1. Direct CP violation in 3-body B decays
Hai-Yang Cheng
Academia Sinica
in collaboration with Chun-Khiang Chua
XS2014, Hefei
May 06, 2014

2.  Direct CP asymmetries (2-body) AK  ACP(K-0) – ACP(K-+)
Bu/Bd
K-+
+-
K-
K*0
K*-+
K- f2(1270)
K-0

ACP(%)
-8.20.6
295
-378
195
-236
3711
-134 S
13.7
5.8
4.6
3.8
3.6
3.4
3.3
Bu/Bd
-
K-
0K*-
K-0
+
+K-
00
-+
K*0
ACP(%)
-145
104
3113
4.02.1
-209
2011
4324
116
4525 S
2.8
2.5
2.4
1.9
1.8  Bs
K+-
ACP(%)
264 S
7.2 B-
K-
ACP(%)
2.22.3
No CP asymmetry observed by LHCb in B- K-
AK
12.22.2
5.5
AK  ACP(K-0) – ACP(K-+)
K puzzle: AK is naively expected to vanish 2 2 2

3. Direct CP asymmetries (3-body)
LHCb 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. K-+-
K+K+K-
++-
K+K+-

5. Wen-Fei Wang’s talk on May 8th
Cheng, Chua, Soni [ ]
Zhang, Guo, Yang [ ]
Bhattacharya, Gronau, Rosner [ ]
Xu, Li, He [ ]
Bediaga, Frederico, Lourenco [ ]
Gronau [ ]
Cheng, Chua [ ]
Zhang, Guo, Yang [ ]
Lesniak, Zenczykowski [ ]
Di Salvo [ ]
Xu, Li, He [ ]
Cheng, Chua [ ]
Ying Li [ ]
Bhattacharya, Gronau, Imbeault, London, Rosner [ ]
Wang, Hu, Li, Lu [ ]
Ying Li [ ]
Wen-Fei Wang’s talk on May 8th

6. Many three-body B decays have been observed with BFs ~10-5
(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. Many argued that 3-body B decays are also dominated by resonant contributions
(LHCb) 6 6

7. Three-body B decays KKK:  70-90% K:  35-40% by Belle, 20% by BaBar
A striking feature: 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 three-body B decays
One of our goals is to identify the origin of NR signals 7
HYC, Chua, Soni (’07)

8. P2 b P1 P3 P2 P1 P3 P3 P2 P1 P3 P2 P1 All three mesons energetic (a)
All three mesons energetic, but two of them nearly parallel P1 P3
(b) P3
All three energetic & two of them nearly parallel. The spectator
quark is kicked by a hard gluon to become hard P2 P1
(c)
(b) & (c) mimic 2-body decays P3 P2 P1
Two energetic (P1, P2) & one soft (P3)
(d)

9. 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

10. +,r r +,-,r r NR contribution of
Early attempt: Apply HMChPT to evaluate form factors r and 
(CLY)2; Wise; Burdman, Donoghue
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 10

11. 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 propose to write NR amplitude as
Fajfer et al; Yang, HYC,…
-- HMChPT is recovered in soft meson limit, p1, p2→0
-- The parameter NR » 1/(2mB) is constrained from B-→+--

12. Resonant contribution of-
Resonant contribution of + B- -
V=, , …, S=f0(980), f0(1370), f0(1500), f(1710),… 12 12

13. How about the NR contributions ?
b→s
Decay constants for scalar mesons have been evaluated in various approaches Chua,Yang, HYC; C.D. Lu et al
How about the NR contributions ? 13

14. <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 x strong coupling) NR
NR  exp[i/4]( ) GeV
from K+K- spectrum of K+K-KS
from KSKSKS rate 14

15. 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

16. 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

17. 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): Br(B0 K+ K- 0) = (2.170.65)10-6 is a surprise !
Recall that Br(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  0.510-6

18. Inclusive direct CP asymmetries
Expt (%)
Theory(%)
2007
+ - -
11.72.4
K+ K- K-
-4.31.2
K- + -
3.21.1
 -3.7
K+ K- -
-14.14.4
 13.1
U-spin symmetry (s  d) 
Xu, Li, He; Bhattacharya, Gronau, Rosner
Relative signs between K-K+K- & -+- and between K-+- & -K+K- agree with experiment & U-spin symmetry predictions
However, relative signs between -K+K- & -+- and between K-+- &
K-K+K- disagree with the data

19. Naïve U-spin symmetry relations
Xu, Li, He (I, ’13)
However, momentum dependence of decay amplitudes should be taken into account
Xu, Li, He (II, ’13)
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

20. Fit to B- K-+- 
U-spin symmetry 
U-spin symmetry which relates <K|sd|0> to <KK|ss|0> is badly broken

21. Direct CP violation in 3-body Bu,d decays
Theory (%)
Expt (%)
+ - -
11.72.4
(+ - -)region
58.48.7
K+ K- K-
-4.31.2
(K+ K-K-)region
-22.62.2
K- + -
3.21.1
(K- + -)region
67.88.5
K+ K- -
-14.14.4
(K+K- -)region
-64.87.2
K- K+ 0
K-K+KS
K-KSKS
45
predictions

22. K-+-
K+K+K-
++-
K+K+- 22

23. 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
Zhang, Guo, Yang advocated that local CP violation in +-- arises from interference of 0 with f0(500)

24. 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+-)

25. 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

26. Conclusions
CP asymmetries are the ideal places to discriminate between different models.
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.