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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

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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

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I. Introduction Bc Meson 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 mb

5/Vcs2 mc

5~O(1) (annihilation~fBc2Vcb

2 ) 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.

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I. Introduction Production Uncertainties 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

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II. Formulation for the Production

PQCD Factorization LO calculation

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cBbbbgg c

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

bcBgg c

II.Formulation for the Production

Bc

Keep the information about the accompany b and anti-c jets !

fragmentation function

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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

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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

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II. Formulation for the ProductionUnder the non-relativistic approximation (spin structure)

S-wave:

P-wave:

We introduce the definitions:

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From BS wave functions to the instantaneous (potential model) wave functions .

For S-wave, the instantaneous wave function

II. Formulation for the Production

at origin

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For P-wave, the instantaneous wave function

II. Formulation for the Production

at origin

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II. Formulation for the Production

with the definitions:

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II. Formulation for the Production

We have the expansion

For S-wave only

and contribute

The kth term of the amplitude:

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For P-wave, the kth term of the amplitude:

II. Formulation for the Production

By straightforward calculation we obtain the cross section:

Note: in MS,P_=mc+mb : qc2

2=mc2, qb1

2=mb2, P2=(qb1+qc2)2=MS,P

2, 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)

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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.gzUncertainties: 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).

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III. Generator and Uncertainties for S-wave

Note: PYTHIA is on the parton shower, but for generating Bc the efficiency is too low (two min. an event at CERN computer)

(ours)

(The figure is offered by G.-M. Chen and S.H. Zhang)

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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)

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III. Generator and Uncertainties for S-wave

Energy scale Q2 uncertainty (S-wave production):

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III. Generator and Uncertainties for S-wave

Comparison between two mechanisms (S-wave production) with the subprocess: and

cbBgg c cbBqq c

LHC

LHC

LHC

LHC

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IV. P-wave Excited Bc ProductionThe subprocess pt and y distributions at

1P1

3P1 3P23P0

3P0

3P1

1P1

3P2

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IV. P-wave Excited Bc Production

1P1

At LHC, the P-wave & S-wave production, Pt and y distribution (mc=1.5 GeV, mb=4.9 GeV and M=mc+mb)

1P13P1

3P0

3P2

3S1

1S01P1

3P1

3P0

3P21S0

3S1

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

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IV. P-wave Excited Bc ProductionAt TEVATRON and LHC, the P-wave production, the total cross sect

ion (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.

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IV. P-wave Excited Bc ProductionPt 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).

From LHC and TEVATRON results, it seems that we cannot attribute the effects to the phase space difference only.

LHC TEVATRON

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IV. P-wave Excited Bc ProductionThe summed Pt distribution and y distribution of all the P-wave states for different factorization scale 2

F and renormalization scale 2 at LHC

The upper edge of the band corresponds to 2F=4MPt

2; 2=MPt2/4; and the

lower edge corresponds to that of 2F=MPt

2/4; 2=4MPt2. The solid line, the

dotted line and the dashed line corresponds to that of 2F=2 =MPt

2; 2F= 2=

4MPt2 ; 2

F= 2= MPt2/4.

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IV. P-wave Excited Bc ProductionThe summed Pt distribution and y distribution of all the P-wave states for different factorization scale 2

F and renormalization scale 2 at TEVATRON

The upper edge of the band corresponds to 2F=4Mt

2; 2=Mt2/4; and the

lower edge corresponds to that of 2F=MPt

2/4; 2=4MPt2. The solid line, the

dotted line and the dashed line corresponds to that of 2F=2 =MPt

2; 2F= 2=

4MPt2 ; 2

F= 2= MPt2/4.

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V. Summary Hadronic Bc production depends on two masses: 9mc

2 ~ 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 avail

able soon. A method to treat the ground and the excited state production p

roperly (splitting) is requested.

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Thank you !