1. Bs → PV decays and effects of next-to-leading order contributions in the perturbative QCD factorization approach
Da-cheng Yan
Ping Yang, Xin Liu, Zhen-Jun Xiao
Nanjing Normal Univercity

2. Outline
Motivation
Framework and contents
Numerical results
Summary

3. Motivation（theory）
(LO)

4. Motivation（theory）
(NLO)

5. Motivation (experiment)
First observation of the decay at LHCb
JHEP 01, 012 (2016)
Observation of and evidence for decays
New J. Phys. 16, (2014)

6. Framework and contents
Transition matrix elements:
the weak effective Hamiltonian can be written as:
In PQCD factorization approach,

7. Framework and contents
Wave functions
• B_s meson
the light-cone distribution amplitude:

8. Framework and contents
Wave fuctions
• Pseudo-scalar meson
the light-cone distribution amplitude:
Twist-2:
Twist-3:
Gegenbauer moments：

9. Framework and contents•
decay constant:
are:

10. Framework and contents
• Vector meson
the light-cone distribution amplitude:
Twist-2:
Twist-3:
Gegenbauer moments：

11. Framework and contents
Leading order contributins
factorizable emission diagrams( )
hard-spectator diagrams( )
annihilation diagrams( )
annihilation diagrams( )

12. Framework and contents
Total decay amplitudes at LO (general expression):

13. Framework and contents
Next-to-leading order contributins
(1) The NLO vertex corrections (VC).
(2) The NLO contributions from the quark-loops(QL).
(3) The NLO contributions from the chromo-magnetic
penguin (MP) operator
(4) The NLO twist-2 and twist-3 contributions to the form
factors of

14. Framework and contents
Next-to-leading order contributions
(1) Vertex corrections
redefinition of the Wilson coefficients :
where M is the meson emitted from the weak vertex.

15. Framework and contents
Vertex functions (NDR scheme)
Phys. Rev. D 72,
When M is a vector meson

16. Framework and contents
(2) Quark loops :
A kind of penguin correction ( transition):
up quark loop : ~
charm quark loop : ~
QCD penguin loop : ~

17. Framework and contents
The effective Hamiltonian ：
Phys. Rev. D 72,

18. Framework and contents
(3)Chromo magnetic penguins (MP):
A another kind of penguin correction ( transition):
The effective Hamiltonian ：

19. Framework and contents
(4) The form factors at NLO level
(Phys.Rev. D 85, )
(Phys.Rev. D 89, )
NLO twist-2 and twist-3 contributions to form factors

20. Framework and contents
Total decay amplitudes at NLO :
Branching ratios (Br) and CP asymmetries:

21. Numerical results NLO contributions to the form factor
Branching ratios

22. Numerical results
The agreements between the PQCD predictions and the three measured values are indeed improved significantly due to the inclusion of the NLO contributions.

23. Numerical results
CP-violating asymmetries

24. Summary
• The effects of the NLO contributions on the PQCD predictions for the branching ratios and CP asymmetries of the considered decay modes are channel dependent.
• From the variations of the ratios R1,2,3, one can see that the agreements between the PQCD predictions and the measured values are improved significantly due to the inclusion of the NLO contributions.
• The effects from the inclusion of the NLO contributions to the form factor
are always small: less than 10% in magnitude.

25. Thank you!