Monitoring of charmed and beauty quark distributions in proton at LHC
aa r X i v : . [ h e p - ph ] J a n MONITORING OF CHARMED AND BEAUTY QUARK DISTRIBUTIONSIN PROTON AT LHC
G.I.Lykasov ∗ and V.A.Bednyakov † Joint Institute for Nuclear Research, 141980, Dubna, Russia
A short review on charmed and beauty hadron production in the lepton deep inelasticscattering off proton, in proton-proton and proton-antiproton collisions at high energies ispresented. It is shown that the existing theoretical and experimental information on charmedand beauty quark distributions in a proton is not satisfactory. A some procedure to studythese distributions at LHC energies is suggested.
PACS numbers: 14.40.Lb;14.65.Dw;25.20.LjKeywords: charmed mesons, charmed quarks, cross-section
I. INTRODUCTION
Various approaches of the perturbative QCD including the next-to-leading order calculationshave been applied to construct the distributions of quarks in a proton. The theoretical analysis ofthe lepton deep inelastic scattering (DIS) off protons and nuclei provide rather realistic informationon distribution of light quarks like u, d, s in proton. However, to find a believable distribution ofheavy quarks like c (¯ c ) and especially b (¯ b ) in proton describing the experimental data on the DIS isnon-trivial task. It is due mainly to small values of D - and B - meson yields produced in the DISat existing energies. Even at the Tevatron energies the B - meson yield is not so large. However,at LHC energies the multiplicity of D - and B - mesons produced in p − p collisions will be largersignificantly. Therefore one can try to extract a new information on the distribution of these heavymesons in proton. In this paper we suggest some proposal to study the distribution of heavy quarkslike c (¯ c ) and b (¯ b ) in proton based on the analysis of the future LHC experimental data. ∗ Also at Institute for Nuclear Research, Dubna, Russia ; Electronic address: [email protected] † Electronic address: [email protected]
II. HARD PARTON SCATTERING MODEL
Usually, the multiple hadron production in hadron-nucleon collisions at high initial energies andlarge transfers is analyzed within the hard parton scattering model (HPSM) suggested in Refs.[1, 2]and Ref[3]. For example, the inclusive spectrum of hadron produced in the hard p − p interactionis presented in the following convolution form: E h dσd p h = X q f ,g Z G p q f ,g ( x , k t ) G p q f ,g ( x , k t ) ˆ sπ dσ (ˆ s, ˆ t ) d ˆ t z D hq f ,g ( z ) δ (ˆ s + ˆ t + ˆ u ) dxd k t dz , (1)where E h , p h are the energy and three-momentum of the produced hadron h ; G p q f ,g ( x , k t ) is thedistribution of quark with flavor f or gluon in the first colliding proton depending on the Feynmanvariable x and the transverse momentum p t ; G p q f ,g ( x , k t ) is the same quark or gluon distribu-tion in the second colliding proton; ˆ s, ˆ t and ˆ u are the Mandelstam variables corresponding to thecolliding quarks; dσ (ˆ s, ˆ t ) /d ˆ t is the differential cross section of the elastic parton-parton scattering; z is the fraction of the produced hadron h [1, 2].The HPSM has been improved significantly applying the QCD parton approach implemented inthe modified minimal-subtraction renormalization and factorization scheme. It provides a rigor-ous theoretical framework for a global data analysis. In this framework there are two distinctapproaches for next-to-leading order (NLO) calculations in perturbative QCD.The first calculation scheme is the so-called massive scheme or fixed-flavor-number scheme(FFNS) developed in Refs.[4, 5] and Refs..[6, 7]. In this approach the number of active flavors inthe initial state is limited to n f = 4, e.g., u (¯ u ) , d ( ¯ d ) , s (¯ s ) and c (¯ c ) quarks being the initial partons,whereas the b (¯ b ) quark appears only in the final state. In this case the beauty quark is alwaystreated as a heavy particle, not as a parton. In this scheme the mass of heavy quarks acts as cutoffparameter for the initial- and final-state collinear singularities and sets the scale for perturbativecalculations. Actually, the FFNS with n f = 4 is limited to a rather small range of transversemomenta p t of produced D or B -mesons less than the masses of c or b quarks. In this scheme the m c,b /p t terms are fully included.The another approach is the so-called zero-mass variable-flavor-number scheme (ZM-VFNS),see Refs.[8, 9] and Ref.[10] and references therein. It is the conventional parton model approach,the zero-mass parton approximation is applied also to the b quark, although its mass is certainlymuch larger than the asymptotic scale parameter Λ QCD . In this approach the b (¯ b ) quark is treatedas incoming parton originating from colliding protons or proton-antiproton. This approach canbe used in the region of large transverse momenta of produced charmed or beauty mesons, e.g.,at p t ≥ m c,b . Within this scheme the terms of order m c,b /p t can be neglected. Recently theexperimental inclusive p t - spectra of B -mesons in p − ¯ p collisions obtained by the CDF Collaboration[12, 13] at the Tevatron energy √ s =1.96 TeV in the rapidity region − ≤ y ≤ p t ≥ GeV /c ) using the non-perturbative structure functions. In another kinematic region, e.g., at 2 . GeV /c ≤ p t ≤ GeV /c )the FFNS model mentioned above allowed to describe the CDF data without using fragmentationfunctions of b quarks to B -mesons. Both these schemes have some uncertainties related to therenormalization parameters.Looking the CDF data on the p t -spectra of B -mesons produced in p − ¯ p collision at the Tevatronenergies published in Ref.[12] and Ref.[13] one can find a difference between them especially at highvalues of p t . Therefore it would be very useful to have more precise data at different LHC energies.The experimental inclusive p t -spectra of D -mesons obtained by the CDF Collaboration at thesame Tevatron energy and rapidity region [14] at p t ≥ c quarks.To calculate the inclusive spectrum of hadrons, for example heavy mesons exploring eq.(1)we have to know the distributions of quarks and gluons and their fragmentation functions (FF).Usually, they are calculated within the QCD using the experimental information from the DIS ofleptons off protons and in the e + − e − annihilation. The information on the gluon distributionand its fragmentation function can be extracted from the experimental data on the jet production[16] and their theoretical analysis [17, 18]. Unfortunately the theoretical QCD calculation of thejet production has different sources of uncertainty. As is shown in Ref.[16] the main contribu-tion comes from uncertainty on parton distribution functions (PDFs) and is computed within themethod suggested in Ref.[19]. At low transverse momenta of jets p jett the uncertainty is small andapproximately independent on the jet rapidity y jet . The uncertainty increases as p jett and | y jet | increase. It can become about 130 percents [16]. To analyze the jet- and heavy quark-production atlow p t and large y or large values of Feynman variable x F one can apply the another nonperturbingQCD model, the so-called Quark-Gluon String Model. (QGSM). III. THE QUARK-GLUON STRING MODEL
The QGSM is based on the 1 /N expansion in QCD suggested in Refs.[20, 21] instead of α s expansion that has the infrared divergence problem at Q →
0, here N is the number of flavorsor colors. The relation of the topological expansion over 1 /N of the hadron-hadron scatteringamplitude to its t -channel one over Regge poles has been suggested in Refs.[22, 23]. This approachhas been applied to analyze soft hadron processes at high energies, see for example Ref.[24].It has been shown [24] that the main contribution to the inclusive spectrum of hadron producedin p − p collisions at high energies comes from the so-called cylinder graphs corresponding toone-Pomeron and multi-Pomeron exchanges which are presented in Fig.1. qqqqqq qqqqqq . . .. . . Fig. 1: The cylinder graph corresponding to the one-Pomeron exchange (left diagram) and the multi-cylindergraph corresponding to the multi-Pomeron exchanges (right diagram).
According to the QGSM, between quark q (diquark qq ) and diquark qq (quark q ) for collidingprotons (Fig.1, left graph) the colorless strings are formed, then after their brake q ¯ q pairs arecreated which fragmentate to the hadron h . The right diagram in Fig.1 corresponds to the creationof two colorless quark-diquark and diquark-quark strings and 2( n −
1) chains between see quark(antiquark) and antiquark(quark). The inclusive spectrum ρ h of hadron h produced in p − p collisioncorresponding to the one-Pomeron graph (Fig.1, left graph) has the following form [25, 26]: ρ h ( x, p t ) = σ F hq ( x + , p t ) F hqq ( x − , p t ) /F hqq (0 , p t ) + F hqq ( x + , p t ) F hq ( x − , p t ) /F hq (0 , p t ) , (2)where σ is the cross section of the 2-chains production, corresponding to the s -channel discon-tinuity of the cylinder (one-Pomeron) graph, usually it is calculated within the quasi-eikonal ap-proximation [27]; F hq ( qq ) ( x ± , p t ) = X flavors Z x ± dx Z d p t d p t f q ( qq ) ( x , p t ) G hq ( qq ) ( x ± x , p t ) δ (2) ( p t + p t − p t ) , (3)Here f q ( qq ) ( x , p t ) is the quark or diquark distribution function depending on the Feynman variable x and the transverse momentum of quark or diquark, z − G hq ( qq ) ( z, p t ) = D hq ( qq ) ( z, p t ) is thefragmentation function of quark or diquark to the hadron h ; x ± = ( p x t + x ± x ) and x t =2 m ht / √ s, m ht = q m h + p t , s is the square of the initial energy in the p − p c.m.s. Actually, theinteraction function F hq ( qq ) ( x + , p t ) corresponds to the fragmentation of the upper quark (diquark)to the hadron h , whereas F hqq ( q ) ( x − , p t ) corresponds to the fragmentation of the down diquark(quark) to h , see Fig.1 (left diagram). The expression for the inclusive spectrum of the hadron h produced in p − p collision corresponding to the right graph of Fig.2 has more complicated form,see the details in Refs.[25, 26].All the quark distributions and fragmentation functions are related to the intercepts and slopesof Regge trajectories. For example,the distribution of c (¯ c ) quarks in a proton obtained within theQGSM and the Regge theory has the following form [25]: f ( n ) c (¯ c ) = C ( n ) see δ c (¯ c ) x − α ψ (0) (1 − x ) α R (0) − α N (0)+( α R (0) − α ψ (0)+ n − , (4)where n is the number of Pomeron exchanges, α R (0) , α N (0) are the intercepts of the Reggeon andnucleon Regge trajectories, α ψ (0) is the intercept of the ψ -Regge trajectory, δ c (¯ c ) is a probabilityfraction of c (¯ c ) pairs in a quark see of the proton.The intercepts α R (0) and α N (0) are known very well from the experimental data on the softhadron processes and the corresponding Regge trajectories have linear behavior as a function ofthe transfer t . As for the Regge trajectory α ψ ( t ), the information on its t -dependence is ratherpoor. As a function of t , it can be linear or nonlinear.The b (¯ b ) quark distributions and the fragmentation functions of b (¯ b ) to B -mesons can be alsoobtained within the Regge theory and the QGSM. They are related to the conventional Reggetrajectories of light mesons and the Regge trajectory of the Υ-meson consisting of b ¯ b pair. Theinformation on the t -dependence of the Υ-Regge trajectory is also uncertain.The modified version of the QGSM suggested in Refs.[25, 26] including the transverse momen-tum dependence of the interaction functions F hq,qq ( x ± , p t ) allowed to describe the experimental dataon inclusive spectra of D -mesons produced in p − p collisions at the ISR energies and at moderatevalues of transverse momenta until p t ≃ − GeV /c ). However, the results are very sensitive tothe value of the intercept α ψ (0). Note, that the QGSM in contrast to the perturbative QCD hasnot uncertainty related to the mass parameter. IV. PROPOSAL
Concluding a short review on the HPMS and the QGSM we would like to propose to do acomplex theoretical analysis both the jet production in p − p and p − ¯ p collisions within theperturbative QCD and the semi-hard production of D - and B -mesons in these reactions within themodified version of the QGSM. The semi-hard hadron processes mean the inclusive production ofheavy mesons at not large transverse momenta p t and not small values of x F . We are going toextract the gluon distribution in a proton and its fragmentation functions to heavy mesons fromthe QCD analysis of the jet production in p − p and p − ¯ p inelastic processes. Constructing a newmodified version of the QGSM based on Refs.[25, 26] we intent to include also the gluon-gluon andgluon-quark interactions using the obtained gluon distributions in proton and its FF. Then we willapply the suggested approach to describe all the existing experimental data on D - and B -mesonsproduced in p − p and p − ¯ p collisions at high energies and perform corresponding predictions atLHC energies. From this analysis we propose to extract a new information on the distribution ofcharmed and beauty quarks in proton. Acknowledgments
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