1. 1. What data show 2. Polarization analysis & Puzzle ? 3.Results by PQCD & speculation 4. Summary Chuan-Hung Chen Physics Department, National Cheng-Kung University, Tainan, Taiwan Polarization Problem in B decays

2. What data show in B→  K * decays

3. What data show in B→  K * decays

4. What data show in B→  (  )  (  ) decays

5. Polarization analysis u b u s Tree W V ub V us Effective operators Tree u b u s g penguin b s qq t WV tb V ts g Effective interactions for penguin b s qq g Penguin C 3 (m W ) ~ -C4 (m W ) /N c ; C 5 (m W ) ~ -C 6 (m W )/N c

7. Vector mesons are composed of quark and anti-quark and spin=1  =anti-s s, K *0 =anti-s d,  + =anti-d u moving direction +  + 0 Meson helicity

8. For B → V 1 V 2 decays, V 1,2 : vector mesons Helicity basis: ss bs VAVAVAVA mm ss bs m K* ss bs mm mBmB

9. H 00 >> H  >> H ++ If one sets the hadronic effects to be the same, i.e. F 00 ~ F  ~ F ++ The resultants are Polarization fractions:

10. Estimation: 1.Set the QCD approach you like, naïve factorization, generalized factorization, QCDF, PQCD,… and so on 2. For simplicity, let’s concentrate on naïve factorization. By gluon penguin, Fierz identities

12. Notation: Helicity basis

13. By polarization basis By taking A 1 ~ A 2 ~ V and neglecting m 2 V /m 2 B  In SM, with naïve factorization, no way to solve the small longitudinal polarization.  Unless we make some fine-tuning on the form factors; however, small BR will be the problem.

14. The results of PQCD in the SM

15. Inevitably, annihilation and nonfactorizable effects should be included, in which in general the nonfactorization includes final state interactions. H.Y. Cheng, C.K. Chua, A. Soni, Phys. Rev. D71 (2005) 014030  Why annihilation is important ? By Fierz transformation and equation of motion, (V-A)(V+A)→(S-P)(S+P) Only O 6 is important

16. Tree dominant processes: C 1 >> C 4,6 u b u d Tree W V ub = A 3 R b V ud ~O(1) penguin b d qq t WV tb V td g Since tree only has (V  A)  (V  A), for those color-allowed processes, annihilation contributions are negligible The nonfactorizable effects are associated with C 2 /N c, as known C 2 << C 1, we expect that they are also small. Expectably, the results will be similar to the estimations of naïve factorization

17. The results by PQCD are summarized as follows:

18. No tree contributions in neutral B decay, purely penguin process Since there involve (V  A)  (V+A) operators, annihilation contributions are important Nonfactorizable effects are associated with C 3 (  )/N c, C 4 (  )/N c, C 5 (  )/N c, C 6 (  )/N c  =1.5 GeV 8.83 10  3,  1.67 10  2, 4.35 10  3,  2.45 10  2  =2.5 GeV 6.31 10  3,  1.27 10  2, 3.51 10  3,  1.70 10  2 and non-negligible

19. By conventional PQCD with including the transverse momentum, k T, and with the chosen condition for hard scale b 1 : the conjugate variable of k 1T, light quark momentum of B b : the conjugate variable of k T, light quark momentum of meson

20. How do annihilation and nonfactorizable effects affect ? Can we make the results be better ? H.N. Li, Phys.Lett. B622, 63 (2005)

21. By modifying the condition for hard scale to be Nonperturbative wave functions as expansion of Gegenbauer polynomials: Usually  is set to be 1 GeV

22. penguin dominant processes tree is important in B d →   K *+ By conventional PQCD with including the transverse momentum, k T, and with the chosen condition for hard scale

23. By modifying the condition for hard scale to be Since the tree of B u →  + K *0 is annihilation, it is negligible; that is, it is proper to take it as a pure penguin dominant decay Compare to B d →K *0 , LP of 65% with  =1, it seems a little bit large; Why is it different ? Are the annihilation and nonfactorizable effects important for polarization fractions ?

24. Comparisons: the decay amplitudes for B→VV V ts V * tb Factorized parts Nonfactorized parts annihilation mode f V F Le 10  3 M Le 10  4 f B F La 10  3 M La 10  5 B d → K *0   3.41 3.77 + i 5.11 0.15  i 1.35 0.26  i 9.52 B u →  0 K *+  2.50 1.88  i 3.16  0.47  i 1.51  3.71  i 6.50 The real parts of F Le in B d →K *0  and B u →  0 K *+ are opposite in sign If we artificially change the sign of B u →  0 K *+ |A 0 | will change from 78% to 68%

25. Summary : It is clear that only considering the factorized effects in the SM has no chance to solve the polarization problem of B  K * It is easy to understand the polarization fractions of B→  (  )  (  ) Is any anomaly between A || and A  ? By modifying the chosen condition of PQCD for hard scale, the BRs of B  K * are consistent with the data, and longitudinal polarizations could be around 60% By PQCD, the annihilation contributions are strongly depend on the wave functions of involving mesons; consequently, the longitudinal polarizations are different in K *  and  K *

26. mode F Te 10  3 M Te 10  4 F Ta 10  3 M Ta 10  5 B d → K *0   4.49  3.56  i 0.22  2.82 + i 5.82 0.17 + i 0.66 B u →  0 K *+  3.21  1.80  i 0.36 0.03 + i 8.77  0.06 + i 1.19