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/0309120, CPC (Computer Physics Communication) 159, 192 (2004) hep-ph/0309121 (appear in Eur.Phys. J. C); hep-ph/0409280

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

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.

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

II. Formulation for the Production PQCD Factorization LO calculation

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 !

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

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!

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

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

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

II. Formulation for the Production with the definitions:

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

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)

III. Generator and Uncertainties for S-wave S-wave (available): hep-ph/0309120, CPC (Computer Physics Communication) 159, p-192 (2004) and hep-ph/0309121 (Eur. Phy. J.C) Program: CPC-Lib and http://www.itp.ac.cn/~zhangzx/bcvegpy1.tar.gz Uncertainties: hep-ph/0309121 (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).

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)

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)

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

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

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

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

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)

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.

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.

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.

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.

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.

Thank you !