1. Hadronic B decays involving tensor mesons Hai-Yang Cheng ( 鄭海揚 ) Academia Sinica Properties of tensor mesons QCD factorization Comparison with experiment April 5, 2011 in collaboration with Kwei-Chou Yang 2011 Cross Strait Meeting on Particle Physics and Cosmology

2. 22 Even-parity mesons 2 Scalar mesons (J PC = 0 ++ ) Axial-vector mesons 3P13P1 1P11P1 ( J PC =1 ++ ) ( J PC =1 +- ) Kwei-Chou Yang, Nucl. Phys. B776, 187-257 (2007).  1 GeV  1 GeV

3. 333 Tensor mesons For J P =2 + tensor mesons 3 P 2 nonet: I=0: f 2 (1270), f’ 2 (1525), I=1/2: K 2 * (1430) I=1: a 2 (1320) close to ideal mixing,  f2  5.8 o

4. 4 B  SM (M=P,V): HYC, Chua, Yang in QCD factorization (’ 06, ’ 08) C.D. Lu et al. in pQCD ( ’ 06, ’ 07, ’ 09) Delepine et al. ( ’ 08) Z. J. Xiao et al. in pQCD ( ’ 08 - ’ 10) B  AM: HYC, Yang in QCDF (’ 07) C.D. Lu et al. in pQCD ( ’ 07) B  TM: last enterprise

5. 5 To study B → TM (M=P,V) decays, we need to know mixing angles decay constants light-cone distribution amplitudes form factors for B → T transition vertex corrections, spectator interactions, annihilation for decay amplitudes HYC, Koike, Yang (’10) HYC, Yang (’10) W. Wang (’10), Yang (’10), Z.G. Wang (’10) Aliev & Shifman ( ’ 82) Braun & Kivel (’01) ISGW (’89,’95), CCH ( ’ 01)

6. 6 Decay constants Tensor meson cannot be produced from local V-A current owing to   p =0 Can be created from local current involving covariant derivatives with Previous estimates: Aliev & Shifman (’82); Aliev, Azizi, Bashiry (’10) Based on QCD sum rules we obtain (HYC, Koike, Yang, arXiv:1007.3526)

7. 77 Form factors for B → T 7 ISGW (Isgur-Scora-Grinstein-Wise) non-relativistic quark model (’89,’95) Covariant light-front quark model (Chua, Hwang, HYC, ’04) Relativistic effects in B-to-light transitions at q 2 =0 are important Large energy effective theory (LEET) (Charles et al. ’99) pQCD approach (W. Wang, arXiv:1008.5326) QCD sum rules (K.C. Yang, arXiv:1010.2144; Z.G. Wang, arXiv:1011.3200)

8. 88 Light-cone distribution amplitudes (LCDAs) twist-2:  ∥,   twist-3: g v, g a, h t, h s twist-4: g 3, h 3 8 C i 3/2 : Gegenbauer polynomial Due to even G-parity, these LCDAs are anti-symmetric under the replacement u→1-u in SU(3) limit first studied by Braun & Kivel (‘01)

9. 9 Longitudinal & transverse helicity projectors for tensor mesons: Transverse momentum derivative terms should be included before taking collinear approximation Helicity projectors for vector mesons:

10. 10 B → TM in QCDF Apply QCD factorization to B→TM ( Beneke, Buchalla, Neubert, Sachrajda) vertex & penguin spectator int. annihilation

11. Data Previous studies based on naïve or generalized factorization predict rates typically too small by 1-2 orders of magnitude compared to experiment dominated by BaBar, f 2 K modes are due to Belle

12. 12 Penguin-dominated B  TP

13. 13 Beyond naïve factorization, contributions  f T defined from local currents involving covariant derivatives can be produced from nonfactorizable contributions such as vertex, penguin and hard spectator corrections B -  K 2 *0   vanishes in naïve factorization, while its BR is measured to be ~ 5.6  10 -6  importance of nonfactorizble effects Penguin annihilation is needed in QCDF to account for rates & CP asymmetries   TP =0.83,   TP = -70 o   PT =0.75,   PT = -30 o similar to the parameters for B  PP

14. 14 Penguin-dominated B  TP

15. 15 B  K 2 * , K 2 *  ’ Interference between (b) & (c) is constructive for K 2 *  ’ and destructive for K 2 *   large rate of K 2 *  ’ than K 2 *  C.S. Kim et al. obtained Br(B  K 2 *  ’)/Br(B  K 2 *  ) ~ 45, while it is ~ 2 experimentally. This is because the matrix elements do not have correct chiral limit behavior due to anomaly and should be replaced by

16. 16 Tree-dominated B  TP

17. 17 Penguin-dominated B  TV

18. 18 Rate puzzle in B  K 2 *  decays It is naively expected that Experimentally, Br(B  K 2 *  )  Br(B  K 2 *  ). This can be accommodated by having penguin annihilation such that   (K 2 *  ) >>   (  K 2 * ). But why ? What is the dynamical origin ?

19. 19 Polarization puzzle in charmless B→VV decays Why is f T so sizable ~ 0.5 in penguin-dominated B  K * , K * , K *0  0 decays ? In transversity basis 19 A 00 >> A -- >> A ++

20. 20 constructive (destructive) interference in A - (A 0 ) ⇒ f L  0.58 NLO corrections alone can lower f L and enhance f T significantly ! Beneke,Rohere,Yang HYC,Yang Although f L is reduced to 60% level, polarization puzzle is not completely resolved as the predicted rate, BR  4.3  10 -6, is too small compared to the data, ~ 10  10 -6 for B →K *  Kagan (S-P)(S+P) (S-P)(S+P) penguin annihilation contributes to A -- & A 00 with similar amount

21. 21 Polarization puzzle in B  K 2 *  f L (K 2 *+  ) = 0.56  0.11, f L (K 2 *0  ) = 0.45  0.12, f L (K 2 *+  ) = 0.80  0.10, f L (K 2 *0  ) = 0.901 +0.059 -0.069 f L (K 2 *  ) = 0.88, 0.72, 0.48 for  A TV = -30 o, -45 o, -60 o, f L (K 2 *  )= 0.68, 0.66, 0.64 for  A VT = -30 o, -45 o, -60 o In QCDF, f L is very sensitive to the phase  A TV for B  K 2 * , but not so sensitive to  A VT for B  K 2 *  Why is f T / f L <<1 for B  K 2 *  and f T /f L  1 for B  K 2 *  ? Rates & polarization fractions can be accommodated in QCDF BaBar but no dynamical explanation is offered Why is that f T behaves differently in K 2 *  and K *  ?

22. 22 Conclusions Tensor meson cannot be created from local V-A current, but its decay constant can be defined through non-local current or local current with covariant derivative. Some decays e.g. B -  K 2 *0  - prohibited in naïve factorization receive sizable nonfactorizable corrections Predictions of QCD factorization in general agree with experiment for B  TM (M=P,V), but there remains puzzles to be resolved: rate of K 2 *  and polarization of K 2 * 