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Uwthph-81-28 QUARK and DIQUARK FRAGMENTATION INTO UWThPh-81-28 QUARK AND DIQUARK FRAGMENTATION INTO «SONS AND BARYONS A. Bartl Institut für Theoretische Physik Universität Wien A-1090 Wien, Austria H. Fraas Physikalisches Institut Universität Würzburg D-8700 Würzburg, Federal Republik of Germany U. Majerotto Institut für Hochenergiephysik Österreichische Akademie der Wissenschaften A-1050 Wien» Austria Abstract Quark and diquark fragmentation into mesons and baryons is treated in a cascade-type model based on six coupled integral equations. An analytic solution including flavour dependence is found. Comparison with experimental data is given. Our results indicate a probability of about 50 percent for the diquark to break up. +) Supported in part by "Fonds zur Förderung der wissenschaftlichen Forschung in Österreich", Project number 4347. 1 I. Introduction It is conmonly accepted that quarks are confined and fragment into jets of hadrons which are observed in high energy reactions. The properties of these jets are usually described in terns of so-called fragmentation functions. In the phenomenological picture which has been developed to understand this process of hadronization (see e.g. ref. 1-3) the frag­ mentation of a highly energetic quark into mesons proceeds by creation of quark-antiquark pairs in the colour field of the original quark and recursive pairing of quarks with antiquarks starting with the original quark. In quark jets also baryons are produced [4] at a rate of 5-10 percent. This can be incorporated into the phenomenological description by assuming that also diquark-antidiquark pairs are created and a quark together with a diquark forms a baryon [5]. What concerns colour the diquark behaves like an antiquark, but since it has a higher effective mass than a quark diquark-antidiquark pairs are produced less frequently. When a baryon is produced in a quark jet an antidiquark continues the chain. If the anti- diquark acts as a single unit then the next hadron produced in the chain is an antibaryon (local conservation of baryon number [6]) and strong correlations between the emitted baryon and antibaryon have to be expected. On the other hand, since the diquark is a bound object it may also break up. Then mesons are emitted before the antibaryon is formed. In this case the strong correlation between the baryon and the antibaryon is lost. Besides quark fragmentations into baryons there are further phenomena where the concept of a diquark plays an important role. In deep inelastic lepton scattering wnen a quark is knocked out from a nucleon, the diquark is the main component of the target remnant responsible for target frag­ mentation. Diquark fragmentation will occur in large p_ hadron hadron collisions, too, but there the reaction mechanism is certainly more com­ plicated. Some of the diquark fragmentation functions have already been determined phenomenologically [7], a recent calculation is given in ref. 8. The concept of the diquark also appears in the calculation of higher-twist contributions [9] and, furthermore, it might be useful in baryon spectro­ scopy [10]. 2 In a proper treatment of quark and diquark fragmentation it is necessary to simultaneously take into account emission of mesons, baryons and antibaryons from the quark as well as from the diquark. This leads to a system of six coupled integral equations for the quark and diquark fragmentation functions [5]. It is the aim of this paper to investigate this system of integral equations and to look also for other solutions besides a Monte Carlo one. In order to include resonance decays, masses etc. a Monte Carlo simulation of the emission process is certainly best suited for analyzing experimental data, but it is <*lso important to know about exact analytic solutions or other approximate ones. From a comparison with experiment one will then learn about the role of the diquark in the process of hadronization and whether it behaves as a coloured entity or not. In ref. 11 a mechanism for baryon production in quark jets based on the chromoelectric-flux-tube model has been discussed. It relies on the SU(3) nature of the colour quantum number and does not make use of the notion of the diquark. If the ideas presented there turn out to be right, then our model involving a diquark which can break up could be considered a phenomenological description of baryon production although the diquark would then be a less fundamental object. In section II we describe our basic system of equations for the fragmentation functions and draw some general conclusions. In section III we show how to include flavour. An exact analytic solution and an appro­ ximate one for the fragmentation functions are presented in section IV. In section V we compare with experimental data using a Monte Carlo pro­ cedure and discuss our results. II. The Model Equations The elementary processes for emitting a single (first rank) meson or baryon from a quark or diquark are shown in Fig. I. We shall denote by f,(n) the probability for emission of a meson from a quark jet (Fig. la) regardless of flavour when the meson carries away the fraction l-n of the longitudinal momentum of the quark and leaves longitudinal momentum fraction n to the rest of the chain. Similarly f9(n)f f.(n) and f,(n) are 3 the probability functions for emission of a baryon fro» a quark» a meson from a diquark and a baryon from a diquark, respectively (Fig. Ib-d). In particular f«(n) + 0 means that a diquark can split and the two quarks become part of different hadrons. If we then call M (z), B (z) and B (z) 1 1 «I the probabilities of finding in a quark jet a primary meson, baryon or antibaryon, respectively, independent of rank and with momentum fraction z, and analogously M.(z), B.(z) and B.(z) the probabilities to find a primary meson, baryon or antibaryon, respectively, in a diquark jet, the following set of equations can be written down [5] 1 . 1 . M(z) - f,(l-z) • / SO. f („) !•(£) + / 511 f (n) M.(f) (la) q i zn qn'nz dn I . l . B (z) « f.(l-z) • / ÄL f (n) B (£) + / S3, f (n) i tly (|b) q z ' n i qn'nz dn Z Z V«> - / f f ,(n) Bq(i) • / 41 f2(n) Bd(i) (le) z z Md(z) - f3(l-z) • / f £ (n) Md(i) • J *L f4(n) M (i) (id) z z Bd(z) - f4(.-z) + / & f3(n) Bd(i) • / Ä £4(n) B (f) (le) z z B>> - / T f3<"> Vf> + / T f4^> Bq^> ' (,£) z z Here we have made use of charge conjugation invariance, i.e. B«(z) - B.(z), B (z) • B-(z), etc. The right hand side of eq.(la) follows because the emitted meson can be of rank one, or it can be of higher rank, the first emitted particle being a meson or being a baryon, giving rise to the first, second and third term, respectively. In an analogous way also eqs. (lb) to (If) can be understood. Here transverse momentum of the emitted particles is not taken into account, as we discuss longitudinal momentun distributions only. Equation (la) without the last term is just the equation of Feymnan and Field [I]. Eq. (lb) without the last term has been used in ref. 6. Equations similar to (Id) and (le) without the last term were used in ref. 8 as a model for diquark fragmentation. Conservation of longitudinal momentum gives the normalization conditions: I / s[M (z) T• B (•) • B* (z)]dz - 1 v„ " W » o^zjiaz - l (2a) o q q / s[Md(i) + Bd(z) • Bd(Z)]dz - 1 (2b) and these lead to I / [f,(z) + f2U)]dz - I (3a) o I / [f3(z) • f4(z)]dz « 1 . (3b) o One way to obtain a solution of eqs.(l) is by performing a Mellin trans­ formation. Defining moments as I r ftq(r) - / Mq(z) z dz (4) and in an analogous way for B (r), B (r), M.(r), B.(r) and B.(r) as well as I g.(r) « / f.(z) zrdz (5a) o 1 h.(r) - / f (!-z) zrdz , £ (5b) o eqs. (1) are transformed into \W - h,<r> • g,(r) i*q(r) • g2(r) fyr) (6aj r fy > - h2(r) + g,(r) &q(r) • g2(r) §d(r) (6b) yr)-g,(r) Bq(r) • g2(r) td(r) (6c) ftQ(r) - h3(r) • g3(r) &d(r) + g4(r) ftq(r) (6d) (r) r (r) (r + r) r *d " V > * «3 *d > *4< V > (6«) r) V - g3(r) Bd(r) • g4(r) frq(r) . (6f) 5 The solution of eqs. (6) is easily obtained as Sfq(r) • jgj {h,(r)[l - g3(r)] • g2(r) h3(r)} (7a) Bq(r) • ^y h2(r) [I - g3<r)] (7b) Vr) " ATO Vr> *2<r> (7c> ftd(r) -^y (h3(r) [1 - g,(r)] + hj(r) gA(r)} (7d) *d(r) "Mr7h4(r) [l -*iiT>] (7e) *d(r) * -ÄfrJ h2(r) *4<r> (7f) with A(r) - [I - g,(r)][1 - g3(r)] - g2(r) g4(r) . (8) The inverse of the Mellin transformation of eqs. (7) . c+i» . Mq(2) ' 2iT /. dr « "q(r) (9) "» c-i» n etc. are in general not obtainable in closed form. In section IV we shall discuss a special case where this is possible. But some general remarks can already be made here. From eq. (8) one can see that A(r) vanishes when r •*• 0. Consequently, the functions fll (r), $ (r) etc. in eqs. (7) have poles for r •+ 0. These poles then completely control the z •+ 0 behaviour of the functions M (z).
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