1. Bc Hadronic Production (New Developments) Oct. 12-15
Chao-Hsi Chang (Zhao-Xi Zhang)
ITP, AS, Beijing
(in collaboration with C. Driouich, P. Eerola, J.-X. Wang and X.-G. Wu)
hep-ph/ , CPC (Computer Physics Communication) 159, 192 (2004)
hep-ph/ (appear in Eur.Phys. J. C); hep-ph/

2. Bc Hadronic Production
Introduction
Formulation for the production
(LO PQCD) Approaches & Mechanisms
Generator for Bc hadronic production (S-wave) and uncertanties
P-wave excited Bc production
Summary

3. I. Introduction Bc Meson Special Interests Double Heavy Flavored
Lifetime τ, mass mBc
Decays
Production (hadronic)
Experimental observation (CDF & D0)
Special Interests
The decay possibilities for the two heavy flavor comparable
Vcb2 mb5/Vcs2 mc5~O(1) (annihilation~fBc2Vcb2 )
To study two flavor simultaneously (Vcb, Vcs)
To be a source of precisely tagged Bs mesons, to observe
χc0, χc1, χc2 and hc etc via Bc weak decay etc.
Comparatively less mechanism in hadronic production than
the hidden flavored heavy quarkonium.

4. I. Introduction Production Uncertainties Generators
LO PQCD calculation
Masses of the heavy quarks (two energy scales mb, mc)
Parameters from potential model
Factorization energy scale
Characteristic energy scale
Generators
Efficiency

5. II. Formulation for the Production
PQCD Factorization
LO calculation

6. II.Formulation for the Production
1. Fragmentation approach
subprocess (mechanism):
(It integrates the accompany jets, we do not describe here, although it is easy to do LL, NLL etc and has disadvantages else.)
2. αs4 complete approach
fragmentation function Bc
Keep the information about the accompany b and anti-c jets !

7. II. Formulation for the Production
To match the wave functions correctly (special attention on the spin structure), we start with the Mandelstam formulation on BS solution:
Here

8. II.Formulation for the Production
Namely under NRQCD framework, the production is factorized
For color-singlet component, we need (should) to work out the precise connections between matrix element and the wave functions when lattice results are not avialable:
and ( ) relation.
Therefore we start with the Mandelstam formulation on BS solution!

9. II. Formulation for the Production
Under the non-relativistic approximation (spin structure)
S-wave:
P-wave:
We introduce the definitions:

10. II. Formulation for the Production
From BS wave functions to the instantaneous (potential model) wave functions
For S-wave, the instantaneous wave function
at origin

11. II. Formulation for the Production
For P-wave, the instantaneous wave function
at origin

12. II. Formulation for the Production
with the definitions:

13. II. Formulation for the Production
We have the expansion
For S-wave only
and
contribute
The kth term of the amplitude:

14. II. Formulation for the Production
For P-wave, the kth term of the amplitude:
By straightforward calculation we obtain the cross section:
Note: in
MS,P_=mc+mb : qc22=mc2, qb12=mb2, P2=(qb1+qc2)2=MS,P2,
we must have
either MP=MS, mcP=m cS and mbP=m bS S-wave, P-wave degenerate,
or MP ≠ MS, mcP ≠ m cS and mbP ≠ m bS S-wave, P-wave does not degenerate !
We favor MP=MS, mcP=m cS and mbP=m bS , but Russian !
(mb, mc involved)

15. III. Generator and Uncertainties for S-wave
S-wave (available):
hep-ph/ , CPC (Computer Physics Communication) 159,
p-192 (2004) and hep-ph/ (Eur. Phy. J.C)
Program: CPC-Lib and
Uncertainties: hep-ph/ (appear in Eur. Phys. J. C)
For experimental applications (M. C. simulation) efficiency is very crucial:
Particle helicity technique (Chinese Magic)
The techniques for simplifying the amplitude are applied.
The efficiency to generate the Bc events is increased greatly and consistency with PYTHA very well (compared by G.M. Chen and S.H.Zhang et all).

16. III. Generator and Uncertainties for S-wave
(ours)
(The figure is offered by G.-M. Chen and S.H. Zhang)
Note: PYTHIA is on the parton shower, but for generating Bc the efficiency is too low (two min. an event at CERN computer)

17. III. Generator and Uncertainties for S-wave
Several uncertainties (S-wave production):
Quark masses: mc, mb
(Here as pointed, we take M= mc+ mb=6.4 GeV and mc=1.5GeV)

18. III. Generator and Uncertainties for S-wave
Energy scale Q2 uncertainty (S-wave production):

19. III. Generator and Uncertainties for S-wave
Comparison between two mechanisms (S-wave production) with the subprocess:
and
LHC
LHC
LHC
LHC

20. IV. P-wave Excited Bc Production
The subprocess
pt and y distributions at
1P1
3P1
3P2
3P0
3P0
3P1
1P1
3P2

21. IV. P-wave Excited Bc Production
At LHC, the P-wave & S-wave production, Pt and y distribution
(mc=1.5 GeV, mb=4.9 GeV and M=mc+mb)
1P1
3P1
3P0
3P2
1S0
3S1
1P1
3P1
3P0
3P2
3S1
1S0
1P1

22. IV. P-wave Excited Bc Production
At TEVATRON, the P-wave & S-wave production, Pt and y distribution
(mc=1.5 GeV, mb=4.9 GeV and M=mc+mb)

23. IV. P-wave Excited Bc Production
At TEVATRON and LHC, the P-wave production, the total cross section (mc=1.5 GeV, mb=4.9 GeV and M=mc+mb)
Roughly speaking, summed cross sections for P-wave production can be so great as 60% of the ground state production.

24. IV. P-wave Excited Bc Production
Pt distribution of the P-wave production: 1. mc=1.5 GeV, mb=4.9 GeV and M=mc+mb (without S-P wave splitting) ; 2. mc=1.7 GeV, mb=5.0 GeV and M=mc+mb (considering the S-P wave splitting).
LHC
TEVATRON
From LHC and TEVATRON results, it seems that we cannot attribute the effects to the phase space difference only.

25. IV. P-wave Excited Bc Production
The summed Pt distribution and y distribution of all the P-wave states
for different factorization scale 2F and renormalization scale 2 at LHC
The upper edge of the band corresponds to 2F=4MPt2;  2=MPt2/4; and the lower edge corresponds to that of 2F=MPt2/4;  2=4MPt2. The solid line, the dotted line and the dashed line corresponds to that of 2F=2 =MPt2; 2F=  2= 4MPt2 ; 2F=  2= MPt2/4.

26. IV. P-wave Excited Bc Production
The summed Pt distribution and y distribution of all the P-wave states
for different factorization scale 2F and renormalization scale 2 at TEVATRON
The upper edge of the band corresponds to 2F=4Mt2;  2=Mt2/4; and the lower edge corresponds to that of 2F=MPt2/4;  2=4MPt2. The solid line, the dotted line and the dashed line corresponds to that of 2F=2 =MPt2; 2F=  2= 4MPt2 ; 2F=  2= MPt2/4.

27. V. Summary
Hadronic Bc production depends on two masses: 9mc2 ~ mb2 (two energy scales), so higher order calculations are more difficult. To decrease theoretical uncertainties, NLO is more complicated than hidden flavor heavy quarkonia although there are less mechanisms .
The event generator for S-wave is available now.
Summed cross sections for P-wave production can be so great as 60% of the ground state production.
P-wave production is investigated and its generator will be available soon.
A method to treat the ground and the excited state production properly (splitting) is requested.

28. Thank you !