Hadronic Production of P-Wave Bc-excited States Heavy Quarkonium Workshop 2006 June. 27-30, 2006 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 70 114019, (2004); PRD 71 074012, (2005); CPC 174 241, (2006); PRD 72 114009, (2005); hep-ph/0604238. June 29, 2006 HQG-06

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 June 29, 2006 HQG-06

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) June 29, 2006 HQG-06

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) June 29, 2006 HQG-06

IIa. P-wave production (Color singlet) PQCD Factorization LO calculation June 29, 2006 HQG-06

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 June 29, 2006 HQG-06

Formulation for the Production Under the non-relativistic approximation (spin structure for color-singlet) S-wave: P-wave: Introduce the definitions: June 29, 2006 HQG-06

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). June 29, 2006 HQG-06

Formulation for the Production From BS wave functions to the instantaneous (potential model) wave functions . For S-wave, the instantaneous wave function at origin June 29, 2006 HQG-06

Formulation for the Production For P-wave, the instantaneous wave function The derivative at origin June 29, 2006 HQG-06

Formulation for the Production with the definitions: June 29, 2006 HQG-06

Formulation for the Production We have the expansion For S-wave only and contribute The kth term of the amplitude: June 29, 2006 HQG-06

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) June 29, 2006 HQG-06

P-wave Bc Production (color singlet) The subprocess pt and y distributions at 1P1 3P1 3P2 3P0 3P0 3P1 1P1 3P2 June 29, 2006 HQG-06

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 June 29, 2006 HQG-06

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 June 29, 2006 HQG-06

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 June 29, 2006 HQG-06

IIb. P-wave production (color octet) Formulation is similar (only ‘color-flue’ is different) Nonperturbative matrix element (for color-octet) can be estimated : and June 29, 2006 HQG-06

P-wave Production (color-octet) 3S1 3S1 1S0 1S0 June 29, 2006 HQG-06

P-wave Bc Production (color-octet) 3S1 3S1 1S0 1S0 Color-octet contributions are smaller than color-singlet ones. June 29, 2006 HQG-06

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. June 29, 2006 HQG-06

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; June 29, 2006 HQG-06

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. June 29, 2006 HQG-06

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. June 29, 2006 HQG-06

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) June 29, 2006 HQG-06

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. June 29, 2006 HQG-06

‘Intrinsic’ charm contribution & double counting problem June 29, 2006 HQG-06

‘Intrinsic’ charm contribution LHC: Tevatron: June 29, 2006 HQG-06

‘Intrinsic’ charm contribution GM-VFN intrinsic + gg-fusion LHC: FFN gg-fusion Tevatron: June 29, 2006 HQG-06

IVb. Suppress the uncertainties Uncertainties from quark mass, from energy scale, etc (see the following slides) June 29, 2006 HQG-06

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. June 29, 2006 HQG-06

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

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. June 29, 2006 HQG-06

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. June 29, 2006 HQG-06

Thank you ! June 29, 2006 HQG-06