1. CP Violation in Charmless 3-body B Decays
Hai-Yang Cheng (鄭海揚)
Academia Sinica, Taipei
HFCPV, SJTU
November 3, 2016

2. Direct CP asymmetries (2-body)
B-/B0
K-+
+-
K-
K*0
K*-+
K- f2(1270)
K-0

-f0(1370)
ACP(%)
-8.20.6
295
-378
195
-236
3711
-134
7222 S
13.7
5.8
4.6
3.8
3.6
3.4
3.3 B-
K*-0
ACP(%)
-5215 S
3.4
Bs0
K+-
ACP(%)
264 S
7.2
LHCb
BaBar:
In QCD factorization, direct CP violation in B M1M2 can be understood in terms of two different kinds of power corrections: penguin annihilation and soft corrections to color-suppressed tree amplitude 2 2 2 2

3. Direct CP asymmetries (3-body)
LHCb (’13) found first evidence of inclusive integrated CP asymmetry in B- +--, K+K-K-, K+K--
2013 LHCb(%)
2014 LHCb(%) 
+ - -
11.72.11.1
5.80.80.90.7
4.2
K+ K- K-
-4.30.90.8
-3.60.40.20.7
4.3
K+ K- -
-14.14.01.9
-12.31.71.20.7
5.6
K- + -
3.20.80.8
2.50.40.40.7
2.8
+-: +
K+K-: -
Large asymmetries observed in localized small invariant mass regions of p.s.
ACPlow(KK) = 0.0700.013 for mKK2 <1.5 GeV2
ACPlow(KKK) = 0.0200.004 for 1.2< mKK, low2 <2.0 GeV2, mKK, high2 <15GeV2
ACPlow() = 0.5840.0820.027 for m, low2 <0.4 GeV2, m, high2 > 15 GeV2
ACPlow(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+- 4

5. |ACPlow | >> |ACPresc | >> |ACPincl |
LHCb (’14) measured another local CP asymmetry in the rescattering regions 1.0 GeV < m,KK < 1.5 GeV
inclusive
rescattering
region
low invariant
mass region
+ - -
5.80.80.90.7
17.22.11.50.7
58.48.22.70.7
K+K-K-
-3.60.40.20.7
-21.11.10.40.7
-22.62.00.40.7
K+ K- -
-12.31.71.20.7
-32.82.82.90.7
-64.87.01.30.7
K- + -
2.50.40.40.7
12.11.21.70.7
67.87.83.20.7
Except the K+K-K- mode, local CP asymmetries in low invariant-mass region are much larger than that in rescattering region
|ACPlow | >> |ACPresc | >> |ACPincl |

6. B-  + --
2014 LHCb
2013 LHCb
ACPincl() = 0.058 inclusive (2013 data: 0.024)
ACPres() = 0.172 for 1.0 < m2 < 2.25 GeV2
ACPlow() = 0.584 for m, low2 < 0.4 GeV2, m, high2 > 15 GeV2 6

7. B-  K-+-
2013 LHCb
2014 LHCb
ACPincl(K) = 0.025 inclusive (2013 data: 0.012)
ACPres(K) = 0.121 for 1.0 < m2 < 2.25 GeV2
ACPlow(K) = 0.678 for 0.08< m, low2 < 0.66 GeV2, mK2 < 15 GeV2

8. B-  K+ K--
2014 LHCb
2013 LHCb
ACPincl(KK) =  inclusive (2013 data: 0.044)
ACPres(KK) =  for 1.0 < mKK2 < 2.25 GeV2
ACPlow(KK) =  for mKK2 < 1.5 GeV2

9. B-  K-K+K-
2013 LHCb
2014 LHCb
ACPincl(KKK) =  inclusive (2013 data: 0.012)
ACPres(KKK) =  for 1.0 < mKK2 < 2.25 GeV2
ACPlow(KKK) =  for 1.2< mKK, low2 < 2.0 GeV2, mKK,high2 < 15 GeV2 9

10. CP asymmetries in other invariant mass regions
K-+-
(a)
(b)
K+K+K-
K+K+-
(c)
(d)
++-
The study of three-body CP-violating Dalitz distributions provides a great challenge to the theorists! 10

11. Bhattacharya, Gronau, Rosner [1306.2625] Xu, Li, He [1307.7186]
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 [ ]
He, Li, Xu [ ]
Krankl, Mannel, Virto [ ]
C. Wang, Zhang, Z. Wang, Guo [ ]
Nogueira, Bediaga, Cavalcante, Frederico,
Lourenco [ ]
Bediaga, Magalhaes [ ]
Cheng, Chua, Zhang [ ]
Nogueira, Frederico, Lourenço [ ]
Wang, Li [ ]
Virto [ ] 11

12. “Future Challenges in Non-Leptonic B Decays”
Bad Honnef, Germany, February 10-12, 2016 12

13. Three-body B decays KKK:  70-90% K:  35-40% by Belle, 20% by BaBar
Large nonresonant (NR) fractions in penguin-dominated B decay modes, recalling that NR signal is less than 10% in D decays
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)

14. 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 quasi-2-body decays P3 P2 P1
Two energetic (P1, P2) & one soft (P3)
(d) 14

15. Receive both resonant & NR contributions
Central 3-body region explored by QCDF or pQCD
3-body decays resemble quasi 2-body ones through the use
of 2-meson distribution amplitude in the Dalitz-plot regions
depicted by [Wang, Hu, Li, Lu (’14); Krankl, Mannel, Virto (’15)]
Regions with a soft meson emission [Suzuki (`00)]

16. Unlike charmless B M1M2 decays which can be described by QCD-inspired theories such as QCDF, pQCD, SCET, the theoretical tool for 3-body B decays is still in the stage of modeling
Early attempts:
Factorization: HYC, Yang (’02), HYC, Chua, Soni (’05,’07),
HYC, Chua (’13,’14)
pQCD: Chen, Li (’03), Wang, Hu, Li, Lu (’14)
QCDF: Krankl, Mannel, Virto (’15)
We rely on the factorization approximation to explore 3-body decays

17. Most of theory studies focus only on either resonant effects.
We discuss both resonant & NR contributions based on factorization.
Under the factorization approximation, there are 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

18. +,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 18

19. NR rates for tree-dominated B→KK,  will become too large
For example, BF(B-→K+K--)NR = 3310-6 larger than total BF, 510-6
BF(B-→+- -)NR = 7510-6 larger than total BF, 5.310-6
⇒ HMChPT is applicable only to soft mesons !
We write tree-induced NR amplitude as p2 p1
-- HMChPT is recovered in soft meson limit, p1, p2→0
-- The parameter NR  1/(2mB) is constrained from B-→+--
NR background is parametrized by BaBar and Belle as

20. For penguin-dominated modes, B- K+K-K-, K-+-, NR rates induced by b u tree transition are too small compared to b s penguin transition due to
|VubVus*| << |VcbVcs*|  |VtbVts|.
BF(B- K+K-K-)NR  1.110-6 from b u transition, while BFNRexpt 2310-6
 NR contributions to penguin-dominated modes arise from b s penguin transition

21. <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)
Fitted ch agrees with the model (~ mass  decay constant  strong coupling)
Resonant NR
NR  exp[i/4]( ) GeV
Cheng,Chua,Soni (’05) 21
from K+K- spectrum of K+K-KS
from KSKSKS rate

22. Resonant contributions
Vi = ,,,… , Si =f0(980), f0(1370), f0(1500),… for P1P2=+-,
Vi = K*(892),K*(1410),… , Si = K0*(1430),… for P1P2= K

23. 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

24. B-→K+K-K- NR rates: mostly from b→s (via <KK|ss|0>)
BF(10-6)
2.9 0.0
calculable for the first time
theory errors: (NR) , (ms, NR, form factors), ()
Power corrections from QCDF added to K-
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 tree transitions

25. B-→K-+- Power corrections from QCDF added to K*0- and 0K-
2.4 0.0
31.03.0
0.65 0.0
Power corrections from QCDF added to K*0- and 0K-
BaBar has measured K0*0(1430)- from B- KS0-0 (2015) with a
result consistent with Belle. An issue for QCDF and pQCD.
If U-spin relation with =0 is
used, it will lead to (i) too large BR’s for NR and total, (ii) CP
asymmetries with wrong signs. Fit to BR    
The K0*(1430) has the largest contributions to B K decays
BF(B- K-+-)NR  29.710-6
BF(B- K-+-)tot  68.510-6

26. Tree-dominated B-→+--, K+K--
NR B- +-- rate is used to fix the parameter NR
The predicted NR fraction is about 55% for B- K+K--

27. Direct CP violation in 3-body B decays
Correlation seen by LHCb:
ACP(K-K+K-)  – ACP(K-+-), ACP(-K+K-)  – ACP(-+-)
U-spin symmetry (s  d) predictions for the relative signs between
K-K+K- & -+- and between K-+- & -K+K- agree with experiment:
Xu, Li, He; Bhattacharya, Gronau, Rosner
However, relative signs between -K+K- & -+- and between K-+- &
K-K+K- cannot be fixed from symmetry argument alone

28. Direct CP asymmetries NR Resonant NR + Res Expt (%) (+ - -)incl
5.81.4
(K+K- -)incl
-12.32.2
(K- + -)incl
2.50.9
(K- K+K-)incl
-3.60.8
(+ - -)low
58.48.7
(K+K- -)low
-64.87.2
(K- + -)low
67.88.5
(K- K+K-)low
-22.62.2
(+ - -)resc
17.22.7
(K+K- -)resc
-32.84.1
(K- + -)resc
12.12.2
(K- K+K-)resc
-21.11.4
pQCD (Wang et al.) :

29. Some issues on CP asymmetries
CP violation in B- 0-
CP asymmetries in large invariant mass regions
The origin of the strong phase  for B K, KK
Final-state rescattering

30. CP violation in B- 0- ACP changes sign at m(+-)  m
In naïve factorization, ACP(0-) ~ BaBar obtained ACP = However, QCDF, pQCD, SCET & diagrammatic approach all predict a negative & sizable CP violation for B- 0-, ACP  -0.20
LHCb has measured CP asymmetries in regions dominated by vector resonances
I: < m(+-)low <0.77 GeV, cos>0,
II: < m(+-)low <0.92 GeV, cos>0,
III: 0.47 < m(+-)low <0.77 GeV, cos<0,
IV: 0.77 < m(+-)low <0.92 GeV, cos<0
ACP changes sign at
m(+-)  m
Summing over regions I-IV yields CP asymmetry consistent with zero with slightly positive central value

31. An important issue needs to be resolved!
If we follow QCDF to add power corrections to render ACP(0-) of order -0.20, then CP violation in +-- induced by , f0 resonances will become negative
Expt (%) NR
Resonant
NR + Res
(+ - -)incl
5.81.4
(+ - -)low
58.48.7
(+ - -)resc
17.22.7
An important issue needs to be resolved! 31

32. CP asymmetries at large invariant mass regions
K-+-
(a)
(b)
K+K+K-
K+K+-
(c)
(d)
++-
The study of three-body CP-violating Dalitz distributions provides a great challenge to the theorists.

33. +-- K-+- K+K-- K+K-K- 0.47, -0.29 -0.09 -0.04 0.36 -0.44 0.11
ACP is large & positive at m2(+-)low= 5-10 GeV2 and m2(+-)high= 9-12
GeV2, negative at m2(+-)low= 3-8 GeV2 and m2(+-)high= GeV2
Negative at m2()= GeV2, m2(K)=10-18 GeV2 and
at m2()= 2-6 GeV2, m2(K)= GeV2
Large & negative at m2(KK) =16-25 GeV2, m2(K) = 5-10 GeV2,
positive at m2(KK)= 5-9 GeV2, m2(K) =4-13 GeV2
Positive at (i) m2(KK)low = 3-5 GeV2, m2(KK)high =18-22 GeV2,
(ii) m2(KK)low = 8-9 GeV2, m2(KK)high =18-19 GeV2
0.47,
-0.09
-0.04
0.36
-0.44
0.11
0.41

34. 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.
In general, NR contributions alone yield large CP-violating effects.
It is important to pin down the mechanism responsible for regional CP asymmetries.

35. Spare slides

36. Strong phase     Expt (%) NR Resonant NR + Res (K+K- -)incl
-12.32.2
(K- + -)incl
2.50.9
(K+K- -)low
-64.87.2
(K- + -)low
67.88.5
(K+K- -)resc
-32.84.1
(K- + -)resc
12.12.2
  

37. Strong phase   = 0 Expt (%) NR Resonant NR + Res (K+K- -)incl
-12.32.2
(K- + -)incl
2.50.9
(K+K- -)low
-64.87.2
(K- + -)low
67.88.5
(K+K- -)resc
-32.84.1
(K- + -)resc
12.12.2
 = 0

38. Final-state rescattering
It has been conjectured that CPT theorem & final-state rescattering of +-  K+K- may play important roles to explain the CP correlation observed by LHCb. Consider +- & K+K- rescattering and neglect possible interactions with 3rd meson
Bediaga et al
Suzuki, Wolfenstein
: inelasticity, assuming KK =
For numerical calculations, we follow the parameterization
of Pelaez and Yndurain

39. +-  K+K- rescattering seems to head in wrong direction
Expt (%)
NR + Res
NR+RES+FSI
(+ - -)incl
5.81.4
-15.6
(K+K- -)incl
-12.32.2
8.1
(K- + -)incl
2.50.9
0.7
(K+ K-K-)incl
-3.60.8
-6.1
(+ - -)low
58.48.7
-17.6
(K+K- -)low
-64.87.2
13.2
(K- + -)low
67.88.5
2.3
(K+ K-K-)low
-22.62.2
-16.7
(+ - -)resc
17.22.7
10.4
(K+K- -)resc
-32.84.1
20.0
(K- + -)resc
12.12.2
-1.3
(K+ K-K-)resc
-21.11.4
-4.0
Final-state
+-  K+K- rescattering
seems to head in wrong direction