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

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 -68+20-18 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: 1501.00705 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

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) = -0.6480.0700.0130.007 for mKK2 <1.5 GeV2 ACPlow(KKK) = -0.2260.0200.0040.007 for 1.2< mKK, low2 <2.0 GeV2, mKK, high2 <15GeV2 ACPlow() = 0.5840.0820.0270.007 for m, low2 <0.4 GeV2, m, high2 > 15 GeV2 ACPlow(K) = 0.6780.0780.0320.007 for 0.08< m, low2 <0.66 GeV2, mK2 <15 GeV2

K-+- K+K+K- ++- K+K+- 4

|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 |

B-  + -- 2014 LHCb 2013 LHCb ACPincl() = 0.0580.014 inclusive (2013 data: 0.1170.024) ACPres() = 0.1720.027 for 1.0 < m2 < 2.25 GeV2 ACPlow() = 0.5840.087 for m, low2 < 0.4 GeV2, m, high2 > 15 GeV2 6

B-  K-+- 2013 LHCb 2014 LHCb ACPincl(K) = 0.0250.009 inclusive (2013 data: 0.0320.012) ACPres(K) = 0.1210.022 for 1.0 < m2 < 2.25 GeV2 ACPlow(K) = 0.6780.085 for 0.08< m, low2 < 0.66 GeV2, mK2 < 15 GeV2

B-  K+ K-- 2014 LHCb 2013 LHCb ACPincl(KK) = -0.1230.022 inclusive (2013 data: -0.1410.044) ACPres(KK) = -0.3280.041 for 1.0 < mKK2 < 2.25 GeV2 ACPlow(KK) = -0.6480.072 for mKK2 < 1.5 GeV2

B-  K-K+K- 2013 LHCb 2014 LHCb ACPincl(KKK) = -0.0360.008 inclusive (2013 data: -0.0430.012) ACPres(KKK) = -0.2110.014 for 1.0 < mKK2 < 2.25 GeV2 ACPlow(KKK) = -0.2260.022 for 1.2< mKK, low2 < 2.0 GeV2, mKK,high2 < 15 GeV2 9

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

Bhattacharya, Gronau, Rosner [1306.2625] Xu, Li, He [1307.7186] Zhang, Guo, Yang [1303.3676] Bhattacharya, Gronau, Rosner [1306.2625] Xu, Li, He [1307.7186] Bediaga, Frederico, Lourenco [1307.8164] Gronau [1308.3448] Cheng, Chua [1308.5139] Zhang, Guo, Yang [1308.5242] Lesniak, Zenczykowski [1309.1689] Di Salvo [1309.7448] Xu, Li, He [1311.3714] Cheng, Chua [1401.5514] Ying Li [1401.5948] Bhattacharya, Gronau, Imbeault, London, Rosner [1402.2909] Wang, Hu, Li, Lu [1402.5280] Ying Li [1402.6052] He, Li, Xu [1410.0476] Krankl, Mannel, Virto [1505.04111] C. Wang, Zhang, Z. Wang, Guo [1506.00324] Nogueira, Bediaga, Cavalcante, Frederico, Lourenco [1506.08332] Bediaga, Magalhaes [1512.09284] Cheng, Chua, Zhang [1607.08313] Nogueira, Frederico, Lourenço [1609.01568] Wang, Li [1609.04614] Virto [1609.07430] 11

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

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+- 22.1+3.6-3.0 41.9+5.3-5.7 B-→K-+- 17.1+12.5-2.5 34.0+3.0-2.8 B0→K-+0 19.73.6 15.67.7 B-→+-- 34.9+9.0-6.2 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)

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

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)]

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

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

+,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

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

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

<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](3.39+0.18-0.21) GeV Cheng,Chua,Soni (’05) 21 from K+K- spectrum of K+K-KS from KSKSKS rate

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

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

B-→K+K-K- NR rates: mostly from b→s (via <KK|ss|0>) BF(10-6) 2.90.0+0.5-0.5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

B-→K-+- Power corrections from QCDF added to K*0- and 0K- 2.40.0+0.6-0.50.0 31.03.03.8 +1.7-1.6 0.650.0+0.69-0.19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

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--

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

Direct CP asymmetries NR Resonant NR + Res Expt (%) (+ - -)incl 25.0+4.9-4.1 5.3+1.6-1.3 8.3+1.7-1.9 5.81.4 (K+K- -)incl -25.6+2.8 -3.2 -16.3+0.9-0.8 -10.2+2.2-2.9 -12.32.2 (K- + -)incl 9.1+2.6-2.7 6.9+2.1-1.8 7.3+2.1-2.0 2.50.9 (K- K+K-)incl -7.8+1.9-1.7 1.2+0.0-0.0 -6.0+2.0-1.5 -3.60.8 (+ - -)low 58.3+4.5-3.5 4.5+1.6-1.2 21.9+3.0-3.3 58.48.7 (K+K- -)low -25.0+3.9-6.0 -4.9+0.5-0.4 -17.5+1.8-1.7 -64.87.2 (K- + -)low 48.9+10.3-13.3 57.1+8.0-16.6 49.4+9.5-14.2 67.88.5 (K- K+K-)low -13.0+3.4-3.4 1.6+0.1-0.1 -16.8+4.5-3.9 -22.62.2 (+ - -)resc 36.7+7.0-5.9 7.0+1.8-1.5 13.4+2.1-2.4 17.22.7 (K+K- -)resc -27.7+4.3-6.5 -5.6+0.5-0.4 -20.4+2.3-2.5 -32.84.1 (K- + -)resc 31.8+6.5-8.1 1.1+0.6-0.5 4.1+0.9-1.0 12.12.2 (K- K+K-)resc -10.8+2.8-2.8 0.96+0.02-0.02 -3.8+1.6-1.1 -21.11.4 pQCD (Wang et al.) : 51.9+16.7-23.9

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

CP violation in B- 0- ACP changes sign at m(+-)  m In naïve factorization, ACP(0-) ~ 0.05. BaBar obtained ACP =0.18+0.09-0.17. 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: 0.47 < m(+-)low <0.77 GeV, cos>0, II: 0.77 < 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

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 25.0+4.9-4.1 5.3+1.6-1.3 8.3+1.7-1.9 (+ - -)low 58.48.7 58.3+4.5-3.5 4.5+1.6-1.2 21.9+3.0-3.3 (+ - -)resc 17.22.7 36.7+7.0-5.9 7.0+1.8-1.5 13.4+2.1-2.4 -16.3 -6.7 -16.8 6.0 -11.4 0.4 An important issue needs to be resolved! 31

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.

+-- 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= 20-12 GeV2 Negative at m2()= 9.5-10.5 GeV2, m2(K)=10-18 GeV2 and at m2()= 2-6 GeV2, m2(K)=20.5-21.5 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, -0.29 -0.09 -0.04 0.36 -0.44     0.11 0.41

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.

Spare slides

Strong phase     Expt (%) NR Resonant NR + Res (K+K- -)incl -12.32.2 -25.6+2.8 -3.2 -16.3+0.9-0.8 -10.2+2.2-2.9 (K- + -)incl 2.50.9 9.1+2.6-2.7 6.9+2.1-1.8 7.3+2.1-2.0 (K+K- -)low -64.87.2 -25.0+3.9-6.0 -4.9+0.5-0.4 -17.5+1.8-1.7 (K- + -)low 67.88.5 48.9+10.3-13.3 57.1+8.0-16.6 49.4+9.5-14.2 (K+K- -)resc -32.84.1 -27.7+4.3-6.5 -5.6+0.5-0.4 -20.4+2.3-2.5 (K- + -)resc 12.12.2 31.8+6.5-8.1 1.1+0.6-0.5 4.1+0.9-1.0   

Strong phase   = 0 Expt (%) NR Resonant NR + Res (K+K- -)incl -12.32.2 17.4+2.8 -3.2 -16.3+0.9-0.8 4.9+2.2-2.9 (K- + -)incl 2.50.9 -3.5+2.6-2.7 6.9+2.1-1.8 -0.8+2.1-2.0 (K+K- -)low -64.87.2 22.3+3.9-6.0 -4.9+0.5-0.4 4.6+1.8-1.7 (K- + -)low 67.88.5 -19.0+10.3-13.3 57.1+8.0-16.6 40.7+9.5-14.2 (K+K- -)resc -32.84.1 25.2+4.3-6.5 -5.6+0.5-0.4 10.1+2.3-2.5 (K- + -)resc 12.12.2 -11.5+6.5-8.1 1.1+0.6-0.5 -6.4+0.9-1.0  = 0

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

+-  K+K- rescattering seems to head in wrong direction Expt (%) NR + Res NR+RES+FSI (+ - -)incl 5.81.4 8.3+1.7-1.9 -15.6 (K+K- -)incl -12.32.2 4.9+1.1-1.0 8.1 (K- + -)incl 2.50.9 -0.8+0.9-0.6 0.7 (K+ K-K-)incl -3.60.8 -6.0+2.0-1.5 -6.1 (+ - -)low 58.48.7 21.9+3.0-3.3 -17.6 (K+K- -)low -64.87.2 4.6+0.9-1.0 13.2 (K- + -)low 67.88.5 40.7+5.9-8.9 2.3 (K+ K-K-)low -22.62.2 -16.8+4.5-3.9 -16.7 (+ - -)resc 17.22.7 13.4+2.1-2.4 10.4 (K+K- -)resc -32.84.1 10.1+1.8-1.7 20.0 (K- + -)resc 12.12.2 -6.4+1.1-0.7 -1.3 (K+ K-K-)resc -21.11.4 -3.8+1.6-1.1 -4.0 Final-state +-  K+K- rescattering seems to head in wrong direction