Beauty production in two-photon interactions at LEP2: kt-factorization versus data
aa r X i v : . [ h e p - ph ] N ov Beauty production in two-photon interactions at LEP2: k T -factorization versus data A.V. Lipatov, N.P. ZotovNovember 10, 2018
D.V. Skobeltsyn Institute of Nuclear Physics,M.V. Lomonosov Moscow State University,119991 Moscow, Russia
Abstract
Inclusive beauty quark production in photon-photon collisions at CERN LEP2 is con-sidered in the framework of the k T -factorization approach. Both direct and resolved photoncontributions are taken into account. The unintegrated gluon distributions in a photon areeither obtained from the full CCFM evolution equation or from the Kimber-Martin-Ryskinprescription. The predicted beauty cross section reasonably agrees with the recent experi-mental data taken by the ALEPH collaboration. We argue that theoretical and experimentalstudies of the azimuthal correlations in heavy quark production at high energies can serveas a crucial probe of the unintegrated gluon densities.PACS number(s): 12.38.-t, 13.85.-tThe problem of beauty quark production at high energies continues to be a subject ofpointed discussions and intense theoretical studies up to now [1]. First results [2] on the b -quark cross section in ep -collisions at HERA were significantly higher than the QCD pre-dictions calculated at next-to-leading order (NLO) approximation. Similar observations weremade in hadron-hadron collisions at the Fermilab Tevatron [3] and also in photon-photoninteractions at LEP2 [4]. In the latter case the theoretical NLO QCD predictions were belowthe experimental data by three standard deviations. Although the latest measurements [5]do not confirm the large excess of the first HERA data over the NLO QCD, the problem isnot solved so far. The disagrement between the experimental data at the Tevatron and NLOQCD predictions was reduced by adopting a special nonperturbative fragmentation functionof the b -quark into the B -meson [6] . A more exotic solution to this problem was proposed in [7]. k T -factorization approach [9]. The k T -factorization approach has also been used for a detaileddescription of numerous experimental data on b -quark production at HERA [10]. Howeverthe problem of the b -quark production in γγ interactions is not solved so far in the k T -factorization approach [11–14].Recently the ALEPH collaboration at LEP2 has presented the result on open beautyproduction cross section in γγ collisions [15]. This is the first published measurement in whichthe lifetime information has been used to identify the heavy flavor in two-photon physics .The cross section of the process e + e − → e + e − b ¯ b X has been found to be 5 . ± . . ) ± . . ) pb which is fully inconsistent with the previous results quoted by the L3 and OPALcollaborations [4], namely 12 . ± . . ) ± . . ) pb and 14 . ± . . ) − . . (syst . ) pb,respectively. In the present note we would like to demonstrate that the ALEPH experimentaldata can be described in the k T -factorization approach also and to propose an additionaltest to distinguish the different unintegrated gluon distribution functions, which are the mainungredient of the k T -factorization (see, for example, [16]).Theoretically, heavy quarks in γγ collisions can be produced via direct and resolvedproduction mechanisms. In the direct events, two photons couple directly to a heavy quarkpair. This contribution is governed by simple QED amplitudes (which are independent ofthe gluon density in the photon). In the resolved events, one photon (”single-resolved”) orboth photons (”double-resolved”) fluctuate into a hadronic state and a gluon or a quarkfrom of this hadronic fluctuation takes part in the hard interaction. At LEP2 conditionsthe heavy quark production via the double resolved processes is highly suppressed [17] and,therefore, it will not be taken into account in our analysis.The single-resolved contribution to the γγ → b ¯ b process is dominated by the gluoncomponent of the photon and has the following form in the k T -factorization approach: dσ − res ( γγ → b ¯ b X ) dyd p T = Z π ( xs ) (1 − α ) A γ ( x, k T , µ ) | ¯ M| ( γg ∗ → b ¯ b ) d k T dφ b π dφ π , (1)where A γ ( x, k T , µ ) is the unintegrated gluon distribution in the photon, | ¯ M| ( γg ∗ → b ¯ b ) isthe off-shell (i.e. depending on the initial gluon virtuality) matrix element squared, s is thetotal c.m. frame energy and α = q m b + p T exp( y ) / √ s . The produced beauty quark hasthe transverse momentum p T , rapidity y and azimuthal angle φ b . The initial off-shell gluonhas a fraction x of the parent photon’s longitudinal momentum, the non-zero transversemomentum k T ( k T = − k T = 0) and azimuthal angle φ . In accord with the k T -factorizationprescription [9], the off-shell gluon spin density matrix is taken in the form ǫ µ ( k ) ǫ ∗ ν ( k ) = k µT k νT k T . (2)In all other respects our calculations follow the standard Feynman rules. The analyticexpression for the | ¯ M| ( γg ∗ → b ¯ b ) is given in our previous paper [13]. Note that if weaverage Eq. (1) over the azimuthal angle φ and take the limit k T →
0, we recover thewell-known formulas corresponding to the leading-order (LO) QCD calculations. The previous measurements by L3 and OPAL collaborations [4] were based on a fiting the transversemomentum of leptons with respect to jets. e + e − collisions.In order to obtain the corresponding cross sections, the γγ cross sections need to be weightedwith the photon flux in the electron: dσ ( e + e − → e + e − b ¯ b X ) = Z f γ/e ( x ) dx Z f γ/e ( x ) dx dσ ( γγ → b ¯ b X ) , (3)where we use the Weizacker-Williams approximation for the photon distribution in the elec-tron: f γ/e ( x ) = α em π − x ) x ln Q Q + 2 m e x Q − Q !! . (4)Here α em is the fine structure constant, m e is the electron mass, Q = m e x / (1 − x ) and Q = 6 GeV [15].The unintegrated gluon distribution in the photon A γ ( x, k T , µ ) can be obtained fromthe analytical or numerical solution of the BFKL or CCFM evolution equations. In order toestimate the degree of theoretical uncertainty connected with the choice of unintegrated gluondensities, in the numerical calculations we tested two different sets, namely the CCFM [12]and KMR [18] ones. First of them was obtained in [12] from the full CCFM equationformulated for the photon, and the second one was obtained from the usual (collinear)parton densities using the Kimber-Martin-Ryskin prescription [18]. These distributionsare widely discussed in the literature (see, for example, [16]). Other essential parameterswere taken as follows: the b -quark mass m b = 4 . ± . µ = ξ q m b + h p T i , where h p T i is set to the average p T of the beautyquark and antiquark. In order to investigate the scale dependence of our results we varythe scale parameter ξ between 1 / ξ = 1. For completeness,we use the LO formula for the coupling constant α s ( µ ) with n f = 4 active quark flavoursand Λ QCD = 200 MeV, such that α s ( M Z ) = 0 . vegas [20]. Thefull C++ code is available from the authors on request . This code is identical to that usedin [13, 14].The results of our calculations are displayed in Figs. 1 — 4. Fig. 1 confronts the total crosssection σ ( e + e − → e + e − b ¯ b X ) calculated as a function of the total c.m. energy √ s with recentexperimental data [15] taken by the ALEPH collaboration. The solid and dash-dotted linescorrespond to the results obtained with the CCFM and KMR unintegrated gluon densities,respectively. The upper and lower dashed lines correspond to the CCFM gluon density with b -quark mass and scale variations as it was described above. Separately shown (as a dottedline) is the contribution from the direct production mechanism γγ → b ¯ b . It is clear that at √ s ∼
200 GeV the cross section is mostly controlled by the single-resolved contribution, i.e. γg ∗ → b ¯ b subprocess. Despite the fact that the central predictions are slightly lower thanthe measured cross section, we observe a reasonable agreement between our calculations andthe ALEPH experimental data [15] within the theoretical and experimental uncertainties.The CCFM-evolved gluon density gives slightly larger cross section compared to the KMRone, where the small- x logarithms are not taken into account [18]. A similar effect (but In the numerical calculations we have used the standard GRV (LO) parametrizations [19] of the collinearquark and gluon distributions. [email protected] µ and beauty mass m b is rather large. However, this sensitivity is ofthe same order approximately as in the massive NLO QCD calculations [21].The transverse momentum and pseudo-rapidity distributions calculated at the averagedtotal e + e − energy √ s = 196 GeV (130 < √ s <
209 GeV) are shown in Figs. 2 and 3. As arepresentative example, we have used the following cuts: p T <
20 GeV and | η | <
2. In ourcalculations we took into account for both the beauty quarks and anti-quarks. One can seeagain that the difference between the CCFM and KMR predictions is not significant, exceptat large p T (namely p T ∼
10 GeV) only. A similar observation was also made [14] in the caseof charm production at LEP2. It was shown that the shape and the absolute normalizationof D ∗ transverse momentum and pseudo-rapidity distributions practically do not depend onthe unintegrated gluon density.We would like to stress that further understanding of the process dynamics may beobtained from the angular correlation between the transverse momenta of the producedquarks. These quantities are particularly sensitive to high-order corrections. So, in the naiveLO collinear approximation of QCD, the distribution over ∆ φ = φ b − φ ¯ b must be simplya delta function δ (∆ φ − π ) since the produced quarks are back-to-back in the transverseplane. Large deviations from these values may come from higher-order QCD effects. Inthe k T -factorization approach, taking into account the non-vanishing initial gluon transversemomentum k T leads to the violation of this back-to-back kinematics even at leading order. Itis an illustration to the fact that the LO k T -factorization formalism incorporates a large partof standard (collinear) high-order corrections (see also [9, 16] for more information). Thedifferential cross section dσ/d ∆ φ calculated at √ s = 196 GeV is shown in Fig. 4. One cansee that the shape of this distribution predicted by the CCFM and KMR gluon densities arestrongly differ from each other. At large ∆ φ ∼ π both gluon densities under considerationgive similar results, whereas at low ∆ φ ∼ k T -factorization approachsupplemented with the CCFM-evolved gluon density agrees well with the numerous data onthe b -quark production at HERA and Tevatron (without any special assumption on the b -quark to B -meson fragmentation function), as it was demonstrated earlier in [10, 8]. So wecan conclude that at present there is no contradiction between the CCFM-based theoreticalpredictions and available data on the beauty production at high energies, and we believethat the k T -factorization holds a possible key to understanding the production dynamics athigh energies.We thank H. Jung for offering the CCFM code for the unintegrated gluon distributionsused in our calculations and S.P. Baranov for careful reading of the manuscript. The authorsare very grateful to P.F. Ermolov for the support and DESY Directorate for the support inthe framework of Moscow — DESY project on Monte-Carlo implementation for HERA —LHC. A.V.L. was supported in part by the grant of President of Russian Federation (MK-9820.2006.2) and the grant of Helmholtz — Russia Joint Research Group (HRJRG-002).4lso this research was supported by the FASI of Russian Federation (grant NS-8122.2006.2). References [1] G.P. Salam, Acta Phys. Pol. B , 2791 (2002); U. Karshon, I. Schienbein, P. Thompson, Proccedings of the 14th Intern. Workshop ”Deep inelastic scattering”,
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D58 , 014014(1998). 6igure 1: The beauty total cross section σ ( e + e − → e + e − b ¯ b X ) as a function of the e + e − center-of-mass energy √ s . The solid and dash-dotted lines correspond to the results obtainedwith the CCFM and KMR unintegrated gluon densities, respectively. The upper and lowerdashed lines correspond to the CCFM gluon density with variation in b -quark mass and scaleas it was described in text. Separately shown is the contribution from the direct productionmechanism (dotted line). The experimental data are from ALEPH [15].7igure 2: The differential beauty cross section dσ/dp T for the process e + e − → e + e − b ¯ b X at | η | < √ s = 196 GeV. Notation of curves is the same as in Fig. 1.8 d σ / d | η | ( pb ) | η | Figure 3: The differential beauty cross section dσ/d | η | for the process e + e − → e + e − b ¯ b X at p T <
20 GeV and √ s = 196 GeV. Notation of curves is the same as in Fig. 1.9igure 4: The differential beauty cross section dσ/ ∆ φ for the process e + e − → e + e − b ¯ b X at √ ss