We present a perturbative calculation of baryon pair production in two-photon collisions, γγ → BB, in which baryons are treated as quark-diquark systems. Our approach accounts for constituent mass effects in a systematic way. Taking the diquark-model parameters from foregoing studies of other electron- and photon-induced baryonic reactions, our results agree well with the most recent large momentum-transfer data for the p¯p, ΛΛ, and Σ0Σ¯0 channels.Izlažemo perturbativan račun tvorbe barionskog para, γγ → BB, u kojemu se barioni predstavljaju kao sustav kvark-dikvark. Ovaj pristup na sustavan način objašnjava učinke sastavnica (konstituenata) masa. Uzimajući parametre ranijih istraživanja drugih barionskih reakcija izazvanih elektronima i fotonima, postigli smo slaganje ishoda naših računa s najnovijim podacima za kanale p¯p, ΛΛ i Σ0 Σ¯0 na velikim prijenosima impulsa

Berger, Carola F.

Schweiger, Wolfgang

English

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ISSN1330–0016CODEN FIZBE7LETTER TO THE EDITORHARD EXCLUSIVE BARYON-ANTIBARYON PRODUCTIONIN 2-γ COLLISIONSCAROLA F. BERGERa and WOLFGANG SCHWEIGERbaI.N.F.N. Sezione di Torino, Via P. Giuria 1, I-10125 Torino, ItalyE-mail address: carola.berger@to.infn.itbInstitute of Theoretical Physics, University of Graz,A-8010 Graz, Universitätsplatz 5, AustriaE-mail address: wolfgang.schweiger@uni-graz.atReceived 5 September 2003; Accepted 26 April 2004Online 16 July 2004We present a perturbative calculation of baryon pair production in two-photoncollisions, γγ → BB̄, in which baryons are treated as quark-diquark systems.Our approach accounts for constituent mass effects in a systematic way. Tak-ing the diquark-model parameters from foregoing studies of other electron- andphoton-induced baryonic reactions, our results agree well with the most recentlarge momentum-transfer data for the pp̄, ΛΛ̄, and Σ0Σ̄0 channels.PACS numbers: 12.38.Bx, 13.60.Rj UDC 539.12.1Keywords: perturbative QCD, diquarks, hard hadronic processes, 2-γ reactionsThe present study was stimulated by recent experimental efforts of theOPAL [1], L3 [2, 3], and BELLE [4] groups to measure γγ → BB̄ cross sections atlarge energies and momentum transfers, where a theoretical description by meansof perturbative QCD is supposed to become applicable. It updates and extends pre-vious work [5 – 7] on the perturbative description of two-photon annihilation intobaryon-antibaryon pairs within a quark-diquark model of baryons. We have madethree improvements compared to these previous studies:• all octet baryon channels are considered,• calculations are performed within the full diquark model with scalar andvector diquarks taken into account,FIZIKA B 13 (2004) 1, 75–80 75berger and schweiger: hard exclusive barion-antibarion production in 2-γ . . .• constituent-mass effects are included.Our model is a modification of the usual hard-scattering mechanism [8], baryonsare described as quark-diquark systems. Diquarks are introduced to parameterizebinding effects between two quarks in a baryon. This approach allows us to modelhigher-order and non-perturbative effects which are undoubtedly present at cur-rently experimentally accessible momentum transfers. Within the hard-scatteringpicture, the hadronic amplitude M{λ}(ŝ, t̂)for γγ → BB̄ is expressed as a con-volution of a perturbatively calculable hard-scattering amplitude T̂{λ}, with non-perturbative baryon distribution amplitudes ΨB,M{λ}(ŝ, t̂)=1∫0dx1∫0dy Ψ†B (x) Ψ†B(y) T̂{λ}(x, y; ŝ, t̂). (1)Here x and y are the longitudinal momentum fractions carried by the quark andantiquark in the baryon and antibaryon, respectively. ŝ and t̂ denote masslessMandelstam variables, which we will discuss further below. The subscript {λ}denotes all possible helicity configurations of photon and baryon helicities. Thehard-scattering amplitude consists of a coherent superposition of tree graphs thatdescribe the scattering process on the constituent level. This means that the twoincoming photons are attached either to the quark or diquark line and quark anddiquark are connected via one-gluon exchange. Form factors at the gauge-bosondiquark vertices ensure that the correct asymptotic (fixed angle) scaling behaviourof differential cross sections is recovered for large enough momentum transfers. Theparameterization of the diquark form factors and of the quark-diquark distributionamplitudes for the present study has been adopted from previous work [9] in whichthese parameters have been fitted to electromagnetic nucleon form factors. Theanalytic form of the quark-diquark distribution amplitudes is flavour dependentsuch that a heavy strange quark carries on average more of the baryon momentumthan a light up or down quark.Before presenting our results we want to comment on a more technical point,the treatment of constituent masses. In the standard hard-scattering approach, themasses and transverse momenta are neglected when calculating the hard-scatteringamplitude. We keep the collinear approximation, but take into account hadronmasses. In this approximation, each constituent four-momentum is proportional tothe hadron four-momentum. Likewise, the constituent mass is also proportional tothe hadron mass where the proportionality constant is the fraction of the hadronmomentum which is carried by the constituent. A variable constituent mass maylook somewhat peculiar, but one has to keep in mind that the momentum frac-tions are weighted by the distribution amplitudes so that, e.g., a quark in a protonhas, on the average, about 1/3 of the proton mass. Perturbative amplitudes arethen calculated with these variable constituent masses and expressed in terms ofthe hadronic Mandelstam variables s, t and u. As a final step massless Mandel-stam variables ŝ, t̂ and û are introduced, and the amplitudes are expanded in76 FIZIKA B 13 (2004) 1, 75–80berger and schweiger: hard exclusive barion-antibarion production in 2-γ . . .terms of (mB/√ŝ), where mB is the baryon mass. Only leading-order and next-to-leading order terms are kept in this expansion. The leading-order terms provide thehadron-helicity conserving amplitudes, the next-to-leading order terms contributeto helicity amplitudes in which the hadronic helicity is flipped by one unit. Thistreatment of mass effects has the advantage that no additional mass parameters forthe constituents are introduced, and it preserves gauge invariance with respect tophoton and gluon. Furthermore, it provides also the correct crossing relations onthe hadronic level between the s- and t-channel processes, γB → γB and γγ → BB̄,respectively. A more detailed account of the above and other technical questions,including analytical expressions for the dominant amplitudes and the treatment oftime-like diquark form factors, can be found in Ref. [10].Let us now present the results of our calculations. The energy dependence of theintegrated cross sections (| cos(θ)| ≤ 0.6) for the p, Λ, and Σ0 channels is shown inFig. 1. θ denotes the centre-of-mass scattering angle. The data are taken from theCLEO [11, 12], VENUS [13], OPAL [1] and L3 [2, 3] collaborations. Preliminarydata from BELLE, which agree nicely with the previous data, have been reportedmost recently at the Photon2003 conference in Frascati [4]. The solid lines, whichcorrespond to the full diquark-model calculation with mass effects included, areseen to lie well within the range of the data. The dashed line in the left plot hasbeen obtained by omitting the mass correction terms, that is, with the two hadronichelicity-conserving amplitudes only. It can be seen that mass effects are still siz-able in the energy and momentum-transfer range of a few GeV. The contributionsfrom the (mass dependent) hadronic helicity flip amplitudes are of the same order of2 3 4 52 3 4 50.00010.0010.010.11102 3 4 5W[GeV]σ[nbarn]γ γ → p p γ γ → Λ Λ γ γ → Σ0 Σ0-- -Fig. 1. Integrated γγ → pp̄, ΛΛ̄, and Σ0Σ̄0 cross sections (| cos(θ)| ≤ 0.6) vs. W =√s. Solid (dashed) line: predictions obtained with the standard parameterizationof the diquark model (cf. Ref. [9]) with (without) mass corrections. The shadedline (band) is the result of the handbag approach [15]. The data are taken from theCLEO [11, 12] (), VENUS [13] (N), OPAL [1] (•), and L3 [2, 3] (◦) collaborations,respectively.FIZIKA B 13 (2004) 1, 75–80 77berger and schweiger: hard exclusive barion-antibarion production in 2-γ . . .magnitude as those from the hadronic helicity conserving ones. At this point, itshould be mentioned that the predictions for γγ → BB̄ obtained within the purequark hard-scattering approach are less satisfactory [14] and lie at least one orderof magnitude below the data.There exists an alternative description of hard baryon-antibaryon production intwo-photon collisions, the generalized parton picture, which also achieves a goodagreement with the experimental data [15]. Within this approach, annihilation isassumed to be dominated by the handbag contribution. This contribution is for-mally power-suppressed compared to the leading twist one, calculated in the purequark hard-scattering approach. Nevertheless, as mentioned at the beginning, theregime where leading twist fully dominates seems out of reach for current exper-iments. The authors of Ref. [15] factorize the handbag amplitude for γγ → BB̄into a hard γγ → qq̄ amplitude and into form factors encoding the soft physics ofthe transition qq̄ → BB̄. The integrated cross section may then be written as acombination of these form factors,σ(γγ → BB̄)|| cos(θ)|≤0.6 = 181 nbarn GeV21s×{|RBA(s) + RBP(s)|2 +s4m2B|RBP(s)|2 + 0.134|RBV(s)|2}. (2)Here, RBV, RBA and RBP are vector, axial and pseudoscalar form factors, respectively,related to the generalized parton distribution functions for a baryon B. The numeri-cal prefactors are the result of integrating the hard amplitude over the angular range| cos(θ)| ≤ 0.6. The authors of Ref. [15] argue that RBV is negligible. They considerthe rest as an effective form factor(RBeff)2= |RBA(s) + RBP(s)|2 + s/(4m2B)|RBP(s)|2and fix it by means of the integrated γγ → pp̄ cross-section data. Predictions forthe other hyperon channels are then obtained by means of isospin and U -spin re-lations. The down/up ratio ρ = F d,pi /Fu,pi , i =V,A,P of the various proton formfactors remains as a parameter, where Rpi =∑q e2qFq,pi , with eq the charge of quarkq. The shaded bands in Fig. 1 correspond to 0.25 ≤ ρ ≤ 0.75. With this parame-terization, the handbag approach provides results comparable with those from thediquark model, as shown in the figure.Freund et al. [16] have investigated the time reversed-process, pp̄ → γγ, withinthe generalized parton picture. They have tried to model the time-like double distri-butions for pp̄ → qq̄ directly and obtain a value of 0.25×10−9 fm2 for the integratedcross section (| cos(θ)| ≤ 0.7) at s = 10 GeV2. The corresponding prediction fromthe diquark model is much larger, namely 0.14 × 10−7 fm2 [10]. This, however, isthe order of magnitude one would naively expect from the γγ → pp̄ data and agreesalso with the findings of Diehl and collaborators [15]. These considerations couldbe interesting in the light of plans for a pp̄ storage ring at GSI in Darmstadt [17].The diquark-model provides, of course, also predictions for octet baryon chan-nels different from p, Λ, or Σ0 (see Ref. [10]). Those agree roughly with estimatesbased on SU(3) flavour-symmetry relations. The ΛΣ̄0 cross section is, for example,78 FIZIKA B 13 (2004) 1, 75–80berger and schweiger: hard exclusive barion-antibarion production in 2-γ . . .small in agreement with the upper limits quoted by the L3 collaboration. Σ+Σ̄−,Σ−Σ̄+ and Ξ−Ξ̄+ cross sections are, on the other hand, of the same order of mag-nitude as the ΛΛ̄ cross section, so that there is a chance for their measurement.There are indeed efforts from the BELLE collaboration to determine Σ+Σ̄− andΞ0Ξ̄0 cross sections [4]. A comparison of different octet baryon channels couldbe useful to gain information on the amount of SU(3)-symmetry breaking in thebaryon distribution amplitudes.From the integrated cross sections shown in Fig. 1 it may seem that the pertur-bative QCD approach works already at relatively low energies (W =√s ≈ 2.5 GeV)and momentum transfers. Recent differential cross section data, however, revealthat the perturbative predictions for W <∼ 3 GeV have to be taken with some cau-tion. Angular distributions for γγ → pp̄ for different values of the photon centre-of-mass energy W are shown in Fig. 2. The data show an enhancement in thecross sections around θ = 90◦ which decreases with increasing energy. Such anenhancement can neither be explained within the diquark model nor within thehandbag approach. It is a clear signal for the dominance of low partial waves andis most likely caused by non-perturbative production mechanisms like, e.g., pp̄ pro-duction via intermediate-state resonances. As one would expect, the perturbativedescription of the differential cross section becomes better with increasing W . AtW ≈ 3 GeV, the theoretically predicted angular distribution is already close to themeasured one. There are also attempts by the BELLE collaboration to determinethe angular distributions for the ΛΛ̄ and the Σ0Σ̄0 channels [4]. It will be inter-esting to see whether these exhibit a similar behaviour as the pp̄ differential crosssections.051015200 0.2 0.4 0.6 0 0.2 0.4 0.6 0 0.2 0.4 0.6dσ/d|cos(θ)| [nbarn]|cos(θ)|W = 2.35 GeV W = 2.75 GeVdσ/d|cos(θ)|×10W = 3.5 GeVdσ/dcos(θ)×100Fig. 2. Differential cross sections dσ(γγ → pp̄)/d| cos(θ)|) for different values ofW =√s. Solid line and symbols for the data as in Fig. 1.Finally, we want to mention that there are efforts from the L3 collaborationto determine the γγ → ∆++∆̄−− cross section. This is a particularly interestingFIZIKA B 13 (2004) 1, 75–80 79berger and schweiger: hard exclusive barion-antibarion production in 2-γ . . .process for the diquark-model description, because it involves only vector diquarksand would thus allow to constrain the model parameters for the vector diquarks.Corresponding theoretical investigations are presently in progress [18].References[1] G. Abbiendi et al., Eur. Phys. J. C 28 (2003) 45.[2] P. Achard et al., Phys. Lett. B 536 (2002) 24.[3] P. Achard et al., Phys. Lett. B 571 (2003) 11.[4] Ch.-Ch. Kuo et al., in Proc. Photon 2003, Frascati, Italy, eds. F. Anulli et al., Nucl.Phys. Proc. Suppl. 126 (2004).[5] M. Anselmino et al., Int. J. Mod. Phys. A 4 (1989) 5213.[6] P. Kroll et al., Int. J. Mod. Phys. A 6 (1991) 4107.[7] P. Kroll et al., Phys. Lett. B 316 (1993) 546.[8] See, for example, S. J. Brodsky and G. P. Lepage, in Perturbative Quantum Chromo-dynamics, ed. by A. H. Mueller, World Scientific, Singapore (1989).[9] R. Jakob et al., Z. Phys. A 347 (1993) 109.[10] C. F. Berger and W. Schweiger, Eur. Phys. J.C 28 (2003) 249.[11] M. Artuso et al., Phys. Rev. D 50 (1994) 5484.[12] S. Anderson et al., Phys. Rev. D 56 (1997) 2485.[13] H. Hamasaki et al., Phys. Lett. B 407 (1997) 185.[14] G. R. Farrar et al., Nucl. Phys. B 311 (1989) 585.[15] M. Diehl et al., Eur. Phys. J. C 26 (2003) 567.[16] A. Freund et al., Phys. Rev. Lett. 90 (2003) 092001.[17] H. H. Gutbrod et al., An International Accelerator Facility for Beams of Ions andAntiprotons, Conceptual Design Report, GSI Darmstadt (2001).[18] C. F. Berger and W. Schweiger, in preparation.TVRDA EKSKLUZIVNA TVORBA BARIONA-ANTIBARIONAU SUDARIMA 2-γIzlažemo perturbativan račun tvorbe barionskog para, γγ → BB̄, u kojemu sebarioni predstavljaju kao sustav kvark-dikvark. Ovaj pristup na sustavan načinobjašnjava učinke sastavnica (konstituenata) masa. Uzimajući parametre ranijihistraživanja drugih barionskih reakcija izazvanih elektronima i fotonima, postiglismo slaganje ishoda naših računa s najnovijim podacima za kanale pp̄, ΛΛ̄ i Σ0Σ̄0na velikim prijenosima impulsa.80 FIZIKA B 13 (2004) 1, 75–80

Tvrda ekskluzivna tvorba bariona-antibariona u sudarima 2-γ

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