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1. What data show 2. Polarization analysis & Puzzle ? Results by PQCD & speculation 4. Summary

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

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1. What data show 2. Polarization analysis & Puzzle ? Results by PQCD & speculation 4. Summary

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

  2. What data show in B→K* decays

  3. What data show in B→K* decays

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

  5. Effective interactions for Vtb W Vts Tree u u s b penguin W t b s g Vub Vus q q Effective operators g u u Tree b s Penguin s b penguin C3 (mW) ~ -C4 (mW) /Nc ; C5 (mW) ~ -C6 (mW )/Nc g q q Polarization analysis

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

  7. s s b s VA VA s s m b s mB s s m b s mK* For B → V1 V2 decays, V1,2 : vector mesons Helicity basis:

  8. Polarization fractions: The resultants are If one sets the hadronic effects to be the same, i.e. F00 ~ F ~ F++ H00 >> H >> H++

  9. Fierz identities Estimation: • 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,

  10. Notation: Helicity basis

  11. By polarization basis By taking A1 ~ A2 ~ V and neglecting m2V/m2B • 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.

  12. The results of PQCD in the SM

  13. 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 O6 is important

  14. Vtb W Vtd d b penguin Tree u u t g W b d q q Vud ~O(1) Vub= A3Rb • Tree dominant processes: C1 >> C4,6 • Since tree only has (VA)(VA), for those color-allowed processes, annihilation contributions are negligible • The nonfactorizable effects are associated with C2/Nc, as known C2 << C1, we expect that they are also small. • Expectably, the results will be similar to the estimations of naïve factorization

  15. The results by PQCD are summarized as follows:

  16. 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 C3()/Nc, C4 ()/Nc , C5()/Nc , C6()/Nc =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

  17. By conventional PQCD with including the transverse momentum, kT,and with the chosen condition for hard scale b1 : the conjugate variable of k1T, light quark momentum of B b : the conjugate variable of kT , light quark momentum of meson

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

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

  20. penguin dominant processes • tree is important in Bd→  K*+ • By conventional PQCD with including the transverse momentum, kT,and with the chosen condition for hard scale

  21. By modifying the condition for hard scale to be • Since the tree of Bu→ + K*0 is annihilation, it is negligible; that is, it is proper to take it as a pure penguin dominant decay • Compare to Bd→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 ?

  22. annihilation Factorized parts Nonfactorized parts VtsV*tb • Comparisons: the decay amplitudes for B→VV • The real parts of FLe in Bd→K*0  and Bu→0K*+ are opposite in sign • If we artificially change the sign of Bu→0K*+ |A0| will change from 78% to 68%

  23. 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→()() • 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* • Is any anomaly between A|| and A ?

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