1. Hadronic Production of P-Wave Bc-excited States Heavy Quarkonium Workshop June , at BNL
Chao-Hsi Chang (Zhao-Xi Zhang)
ITP, AS, Beijing
(in collaboration with C.-F. Qiao, J.-X. Wang and X.-G. Wu)
PRD , (2004); PRD , (2005); CPC , (2006);
PRD , (2005); hep-ph/
June 29, 2006
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2. Outlines Introduction
Hadronic Production (P-wave color singlet & color octet & massive mass effects etc)
Upgraded generator for hadronic Bc production (BCVEGPY2.0,BCVEGPY2.1)
Outlook
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3. I. Introduction
To Understand the Production Mechanism of Heavy Quarkonium (S-wave & P-wave, NRQCD)
To Be References for Bc Observations (an indirect source for Bc: Pt and y distributions etc)
Characteristics of P-Wave Bc Events in Hadron Collisions (a direct observation of the P-Wave Bc states: its production & decays for spectroscopy of (c\bar{b})-system)
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4. II. P-wave Bc Hadronic Production
P-wave excited Bc production:
Color singlet
Color octet (scaling rule of NRQCD: more important for P-wave Bc production than for S-wave one)
Generator for Bc hadronic production upgraded (BCVEGPY2.0, BCVEGPY2.1)
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5. IIa. P-wave production (Color singlet)
PQCD Factorization
LO calculation
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6. 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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7. Formulation for the Production
Under the non-relativistic approximation (spin structure for color-singlet)
S-wave:
P-wave:
Introduce the definitions:
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8. Formulation for the Production
Namely under NRQCD framework, the production is factorized
For color-singlet components, we prefer to work out the precise connections between the matrix element and the wave functions (when lattice results are not available):
and ( ).
We would like to start with the Mandelstam formulation which is based on BS solutions (the color singlet components of excited states and ground state of Bc are treated at the same approximation level).
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9. Formulation for the Production
From BS wave functions to the instantaneous (potential model) wave functions
For S-wave, the instantaneous wave function
at origin
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10. Formulation for the Production
For P-wave, the instantaneous wave function
The derivative at origin
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11. Formulation for the Production
with the definitions:
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12. Formulation for the Production
We have the expansion
For S-wave only
and
contribute
The kth term of the amplitude:
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13. 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 : P2=(qb1+qc2)2, mc: qc22=mc2, mb: qb12=mb2,
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 take MP=MS, mcP=m cS and mbP=m bS in the estimates mainly.
(mb, mc involved)
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14. P-wave Bc Production (color singlet)
The subprocess
pt and y distributions at
1P1
3P1
3P2
3P0
3P0
3P1
1P1
3P2
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15. P-wave Bc Production (color singlet)
At LHC, the P-wave & S-wave production, Pt and y distribution
(Color-singlet: 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
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16. P-wave Bc Production (color singlet)
At TEVATRON, the P-wave & S-wave production, Pt and y distribution
(mc=1.5 GeV, mb=4.9 GeV and M=mc+mb)
3S1
1S0
3P2
3P2
3S1
1P1
3P1
3P1
3P0
1P1
1S0
3P0
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17. P-wave Bc Production (color-singlet)
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
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18. IIb. P-wave production (color octet)
Formulation is similar (only ‘color-flue’ is different)
Nonperturbative matrix element (for color-octet) can be estimated :
and
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19. P-wave Production (color-octet)
3S1
3S1
1S0
1S0
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20. P-wave Bc Production (color-octet)
3S1
3S1
1S0
1S0
Color-octet contributions are smaller than color-singlet ones.
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21. III. Upgraded generator for Bc hadronic production
Version BCVEGPY2.0:
The amplitudes for the hadronic production of the color-singlet components corresponding to the four P-wave states and are included;
The amplitudes for P-wave production via the two color-octet components and are included;
The integration efficiency over the momentum fractions are improved.
Version BCVEGPY2.1:
Technical improvements are involved.
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22. Upgraded generator for Bc hadronic production (versions BCVEGPY2)
BCVEGPY2 contains P-wave production additionally.
For comparison, the S-wave ( and ) hadronic production via the light quark-antiquark annihilation mechanism is also included;
For convenience, 24 data files to record the information of the generated events in one run are added;
An additional file, parameter.for, is added to set the initial values of the parameters;
Two parameters, `IOUTPDF' and `IPDFNUM', are added to determine which type of PDFs to use;
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23. Upgraded generator for Bc hadronic production (versions BCVEGPY2)
Two new parameters 'IMIX' (IMIX=0 or 1) and 'IMIXTYPE' (IMIXTYPE=1, 2 or 3) are added to meet the needs of generating the events for simulating `mixing' or `separate' event samples for various Bc and its excited states correctly;
One switch, `IVEGGRADE', is added to determine whether to use the existed importance sampling function to generate a more precise importance sampling function or not;
The color-flow decomposition for the amplitudes is rewritten by an approximate way, that is adopted in PYTHIA.
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24. Upgraded generator for Bc hadronic production (BCVEGPY2.1: the latest)
Available under LINUX system (to meet the needs for most experimental group);
With a GNU C compiler, the events in respect to the experimental environments may simulated very conveniently (better modularity and less dependency among various modules) ;
A special and convenient executable-file run as default is available: the GNU command make compiles the codes requested by precise purpose with the help of a master makefile in the main code directory.
It has been implemented in the experimental generators, such as ATHENA (ATLAS group), Gauss (LHCb group) and SIMUB (CMS group) etc.
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25. IV. Outlook (to meet Exp. Needs & better understanding)
The massive mass effects in the
production: Various schemes
Decrease the uncertainties:
NLO calculations
More application (e.g. double charm
baryon\Xicc production at SELEX etc)
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26. IVa. Massive Quark Effects and Schemes
Heavy quarks b & c production: Below the threshold ‘decoupled’; Above (close to) the threshold: effects great; Much above the threshold: ‘zero mass’ 4 or 5 flavor FFN.
General-mass variable-flavor-number GM-VFN scheme and fixed flavor number FFN scheme :
gq-fusion:
gg-fusion:
`Double counting’ due to structure functions, so one must deduct it when summing up the contributions from the two mechanisms.
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27. ‘Intrinsic’ charm contribution & double counting problem
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28. ‘Intrinsic’ charm contribution
LHC:
Tevatron:
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29. ‘Intrinsic’ charm contribution
GM-VFN intrinsic + gg-fusion
LHC:
FFN gg-fusion
Tevatron:
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30. IVb. Suppress the uncertainties
Uncertainties from quark mass,
from energy scale,
etc
(see the following slides)
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31. Uncertainties in P-wave Bc Production due to the quark masses (color-singlet)
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 only to the phase space difference.
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32. Uncertainties in P-wave Bc Production due to the factorization energy scale
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.
June 29, 2006
HQG-06

33. Uncertainties in P-wave Bc Production due to the factorization energy scale
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.
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34. In progress
Suppress the uncertainties: NLO PQCD calculations may be helpful
NLO (αs5) precise calculations suppress the uncertainties from μF.
‘Intrinsic’ charm and bottom in GM-VFN scheme in NLO.
Non-perturbative ‘intrinsic’ charm and bottom contributions in GM-VFN.
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35. Thank you !
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