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 antidiquark 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

/ [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, • 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

Vr) " ATO Vr> *2 (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 (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). B (z) etc. To see this more clearly ve write J

P£ -g£(0) - / f.(z)dz (10) o

and are the z where from eqs. (3) p,+p2 » I, P3+P& • 1. Pj(P2> Vfo^ integrated probabilities that a meson (baryon) is the first emitted particle from a quark or diquark, respectively. Then one can show that 6

1 plP4 + P2P3 lim M (z) - lim M.(z) » - A—*-=- (IIa) z-K) q z-0 d z A

1 P2P4 lim B (z) - lim B (z) - lim B ,(z) - lim B ,(z) = - -~ (I lb) z-0 q z-0 q z-0 d z-0 d Z A with

1 1

A - - p4 / [f|(z)+f2(z)]ln z dz - p2 / [f3(z)+f4(z)jln z dz . (12) 0 O

This means that the height of the rapidity plateau for a given particle type is the same for quark and diquark fragments. The ratio of the plateau heights for mesons and baryons is given by

M (z) M (z) p p

B (2) B (z) p P z^ q Z+0 d 2 4

It depends only on the parameters p. and not on the shape of the momentum sharing functions f.(z). Furthermore, in the scaling limit where eqs. (I) are supposed to hold, antibaryons develop a plateau with the same height as baryons. These results also hold, if flavour is taken into account (see eqs. (19), (23) and (25) to (29) below). The quantities p. are the basic parameters of the model.

III. Inclusion of Flavour

This can be achieved in a way analogous to that of ref. I. Let us call Y the probability of creating a quark-antiquark pair with flavour a Ä

and Y h the probability of creating a diquark-antidiquark pair with flavour content ab. With the normalizations (2) and (3) we have

lYa-lYab-l. (14) a ab

We call M ab(z) and Babc(z) the probabilities thac in the jet of the quark with flavour q a primary meson with quark content (ab) or a primary baryon with quark content (abc^, respectively, is produced with longitudinal momentum fraction z. md B (z) are the analogously defined qiqa q-iqa 7

functions for a diquark jet of flavour q.q2« An analogous notation for the antibaryons is used. We can then rewrite eqs. (1) as

b <"> - vvi<'-> • }*i,w heMf(i) * )% f2<„) [v fH-;,£) n ^ z e z ef (15a)

n labe} n z e z • i*.£f#

•/?V"I'rf

z e

5 ab7 5 8bC B « (z) « / ^ £.<„> I J[Y B (^)+Y B ^)] • / AI f .

These are the basic equations for our study of quark and diquark frag mentation. Here and in the following £ means the sum over all different {abc} permutations of the quarks a,b,c (notice that in the convention used the order of q. and q. has to be kept fixed). In the following it will also bejnore convenient to work with the Mellin transforms of eqs. (15).

M*b(r), ÄÄbc(r)etc. then denote the Mellin transforms of M*(z), Babc(z), etc. defined analogously to eq. (4). Then e.g. from the Hellin transform ations of eqs. (15a) and (I5d) we get

b XY.äfCr) - YaYbh,(r) • g,(r) bjt*M • g^r) I YefM^ (r) (16a) e e ef

^VfMef(r) " Vbh3(r) + g3(r) JVÄf(r) *g 4(r) Wb(r)' (l6b) Eqs. (16a) and (16b) together with eqs. (6a) and (6d) lead to

It. ifw • V* V° (,7)

U, > " Vb *«<'> (18> ef ef From eqs. (6a), (15a), (17) and (18) one then obtains

Mqb(*> * Vb fl°-Z) + VblM,(,> " f»(,"Z)1 * °9)

From eq. (14) and (15d) one furthermore gets

e H2 Hl 1 -•*• g,(r) ;l n2 z J _ (20) + g3(D lYeYf 8fi(r) • 2g4(r) ^ hff (r)} . ef e

Inserting this and eqs. (17) Mid (18) again in eq. (15d) and using eqs. (6) gives

»J*0 (r) - U& +6 )y. p h,(r) + y„yA$LM ~ h.(r)] W 7 qla V b 1 - I ,(r) 3 * b d l-i i,(r) 3 I8 V 2 63x (21) Defining a function K(Z) by

1 , j go(r) / <(z) zrdz - —i-p (22)

0 1 - j g3(r) the inverse Mellin transform of eq. (21) can be performed leading to

CoU) -Xa+Va)YbIf3(1-z)+JM(2)1 + VblMd-f3(,-2)-JM(*)] 12 12 (23) with

JM(z) -/^-c(n) f3(l -f) • (24) z

In a similar way one obtains 9

BqbC(r> * , I ,{VbCf2(1'l) + VbClBQ(l)-f2(,-l)1} U5)

{6 + ^ <*) " I n A I.Y f4d-t) • i(6„ .Yv • « hYjY„ J-(*) q q q b c 4 2 q a b q b a c B I 2 {abc} V 2 l 2 (26) + VbctBd(z) - V»-«> - V2)1} with

JB(z) -2 /^K(n) f4(l -f) (27) as well as •">' • <.L>V" •>' (28)

In analogy to eqs. (17) and (18) the following equalities must hold

K*T<* ' , I ,Vbc Bq(2) (30) e {abc} n I B*C« - [ »SC(,) - , J , V*> • <3') ef V ef Vf ** {abc} Vbc

By expanding equ. (21) into powers of •=• s3(r) it can be seen that the

terms Ju(z) and J_(z) in equs. (23) and (26), respectively, come from such contributions where one of the quarks in the original diquark runs as a spectator through the whole chain until the last emitted particle. Denoting the meson ab by its symbol M and the baryon abc by its symbol B, the results for the fragmentation functions, equs. (19), (23), (25) and (26) can be cast into the more convenient form

DJU) -ojjftjf,(!-«) + «^[M («)-£,(!-«)] (32a)

B DJ - oBAjf2(l-z) • oBB [Bq(z)-f2(l-z)] (32b)

ll D£ q («) - c^cJJ [f3(l-z)+JM(z)] + oMB [Md(s)-f3(l-z)-J||(s)] (32c)

iM) 1 E J Z + lB Z f ] ^2 ' V^qjV -«' * °B qiq2 B< > °/ d< >- 4<'^"V^ (32d) 10

M M where the coefficients A • 6 y, and B «K Y, are already listed e.g. q qa'b a'b * in table 1 of ref. 1. Putting y • YJ " Y» Y " 1 - 2y and y • 0 for the heavy quarks the quantities CT • -A 6 +5 )y, as well as 7 2 a a b q,q2 Y can be qlq2 {.be} V q2b c qlq2 2 {abc} V b q2b a C calculated. They are listed in tables 1 and 2 (f.i. y * 0.4 was taken in B r ref. !, y » 0.44 in ref. 8). In order to compute A • ) 6 „y.^ and q {abc} qa bC

B • £ Y Yh an assumption about the probabilities y. has to be made. {abc} For the sake of simplicity we shall put y. • y.y . The resulting coefficients A and B are given in table 3. Finally, the coefficient OL, takes into account the relative probability that a pseudoscalar or a vector meson is made from the quark pair (ab). We shall assume Op « QL. - 0.5. Similarly, a- measures the relative production rate of spin 1/2 to spin 3/2 baryons from the quarks abc. Again we shall assume a.,- * a,,- * ^.5 if spin 1/2 baryon production is possible, a.»- »0 and 013/9 " ' otherwise. Production of higher resonances is neglected. The treatment of flavour as presented here is exact and the results eqs. (32) are obtained in a closed form whereas in ref. 8 the inclusion of flavour is done only in an iterative way up to third order.

IV. Solutions of the Model Equations

From eqs. (32) one can see that also in the case of including flavour it suffices to know the solution of the equations (1). It is amazing that although eqs. (1) look rather complicated there are special cases which allow exact analytic solutions in a closed form being not only simple but also realistic and useful for practical purposes. We shall first present such an example and then discuss a way oi obtaining approximate solutions for more general cases. a) Exact Analytic Solution

The Mellin transforms of the solutions of eqs. (I) are already given in eqs. (7) and the problem would be solved if one could do the inverse Mellin transformation eq. (9) analytically. For arbitrary input functions 11

f.(n) this is in general not possible. However for the special case of a power law ansatz

d. f^n) - p.(d.+l)n X (33) we have d. + l g.(r) - p. * . (34) l ri r+d.+l l T(d.+2)r(r+l) Vr> - Pi r(r+d.+2) <35> and the problem is reduced to solving an algebraic equation for finding all zeros of A(r) in eq. (8)(besides that one at r - 0). Specializing further we put d - d, • 1, d„ - d- • 3, i.e.

f,(n) - p, 2n (36a)

*2(n) - P2 4^ (36b)

3 f3(n) - P3 4n (36c)

f4(n) - PA 2n (36d)

which is suggested by counting rules 112], and we obtain from eq. (18)

r(r+2p,+4p ) A " (r+2)U) ' (37)

Inserting into eqs. (7) the inverse Hellin transform is obtainable in a straightforward way, giving

8(P|P4+P2P3) , 2pr8P|p4-16p2P3 »6p2P3 2

KZ) z \ " 2p2+4p4 7 2p2+4p4-l 2p2+4p4-3 (38a)

8p p 2 3 3 —U, * - (2p2+4p4-2) Q(.) 12

4p 4p 3 2p _Üi__i 4"' 4~ 2 3 + 2 + Z Bq(z) » 4p2 {2p2+4p^ 2 ' 2p2+4p4-l 2p2+4p4-3 2p2+4p4-4 (38b)

12p2 2p2+4p4-l

(2p2+4p4)(2p2+4p4-l)(2p244p4-3)(2p2+4p4-4)

2 4 H 1 1 1 z V P4 } (38C) Bq(z) - 8p2p4 {-^-^- - - 2p2+4p^, + (2p2+4P4)(2p2+4p4-l)

8(PlP^P2P3) x 12(P3-2p2P3-plP4) APl(p4-6p3)

M UJ + Z d " 2p2+4p4 z 2p2+4p4-l 2p2+4p4-2 (38d) 4p,d+2p ) + 2p!^p -3 z2 + (2P +4P4~4) Q(2) 2 4 2

4p9 , 3-6p„ 2p

B^(z1 ) » 2pfc,, l A{^--f---i + d ' 42p2+4p4 z 2p2+4p4-l 2p2+4p4-2

2p2+4p4-l (38e)

4p4(2p2+4p4-4) z

J (2p2+4p4)(2p2+4p4-l)(2p2+4p4~2)

? i \ ft 1 1 3 3 I t Z BdU) - »P2P4 2p2+Ap4 I " 2p2+4p4-l 2p2+4p4-2 ' 2p2+4p4-3 (38f) 2p2+4p4-l

+ 6z }

(2p2+4p4)(2p2+4p4-l)(2p2+4p4-2)(2p2+Ap4-3)' with

4pIp3 QU) - [ l 3 (2p2+4p4)(2p2+4p4-l)(2p2+4p4-2)

96p2p3 ^ 2p2+4p4-l :1* (2p2+4p4)(2p2+4p4-l)(2p2+4p4-2)(2p2+4p4-3)(2p2+4p4>4)• (39)

Note that by eqs. (3) only p. and p. (or p2 and p4) are independent para

meters. As one would expect B (z) (B,(z)) vanishes when p2 •• 0 (p4 •* 0), but M (z) and M.(z) vanish only it both p. and p- go to zero. We have plotted zM (z), zM.(z), zB (z), and zB.(z) in fig. 2. The parameters were chosen as p. - 0.92, p« • 0.5, as this corresponds also to the realistic situation when one compares with experimental data. For comparison we 13 have shown in fig. 2 also zF(z) from ref. 1 which is obtained with an

input function f.(n) » 1 - a + 3an2, with a =0.7. As one can see, the difference between zF(z) and zM (z) is less than 20 percent, apart from the range z > 0.9 although rather different functions f.(z) are used. Usually, the analysis of ueson spectra in quark jets is done with a momentum sharing function f,(z) = p (1 - a + 3az2) like in ref. 1. One can find an exact analytic solution to eqs. (1) also in this case. The resulting formulae are somewhat lengthy and, therefore, are not explicitly given here. For a - 0.7 the numerical difference to the solution (38) is less than 10 percent. In Figs. 3a-d we have plotted the fragmentation functions for uu and ud diquarks into primary pseudoscalar mesons and spin 1/2 baryons, following from eqs. (32c), (32d), (38d) and (38e). From eqs. (22), (24) and (27) one can also calculate in this case

3-2p3 J K(Z) - 2p3 z (40)

3_2p3

2 3 T (i\ m ftri2r, ,' 3Z 3Z Z 3Z , . ... + J ikl) JMU; »P3l3-2p3 ' 2-2p3 l-2p3 2p3 " p3(3-2p3)(2-2p3)(l-2p3)

1 z z3'2'3 J (Z) 8 p [ + ] (42) B " P3 4 5=2pJ " 2=fpJ (3-2p3)(2-2p3) -

b) Solution by Approximate Inverse Mellin Transformation

In the analysis of deep inelastic scattering several methods were developed to approximately reconstruct the structure functions of the nucleon from their moments. For arbitrary input functions f.(z) the moments of the fragmentation functions are known from eqs. (7). Thus, if an exact solution cannot be obtained one can try such a method of approximate inversion here,too. We shall do this by following a procedure proposed in ref. 13 which uses the normalized Bernstein polynomials

b(H.«U).a^Y__l^_-, ,.0......

At each of the points 14

the average over the function M (z), for example, is calculated by

(N k) *fytk> - J dz b * (z) Mq(z) which by eqs. (43) is given as

Analogous expressions hold for the other fragmentation functions. The average values of the fragmentation functions, M (z„ .) etc. are directly obtainable from their moments eq. (7). The z interval IL. . over which the functions are averaged, is given by

2 N-k+1 (46) H. k (N+2)2(N+3)

For k • 2,3,...N also a correction term to eq. (45) can be calculated which is

N-k

i i N* i i • • ni**ir* i • 5M q N'* l k!(N+2)2(N+3) £-0 *KN k l)1 q

For further details we refer to ref. 13. With the same input functions and parameters as for the exact solution we calculated the fragmentation functions according to eqs. (45) and (47) for N • 12. The results -or zM (z), zM.(z), zB (z), and zB.(z) are also plotted in fig. 2, where the length of the averaging interval, A» . , is also shown. As one can see, the method works well for M (z) and B.(z). For M (z) and B (z) which vary q d u q more strongly, deviations from the exact solution up to 40 percent occur. The corrections according to eq. (47) were less than 2 percent. It turned out that for N >_ 15 the results become numerically unstable. As we can compare with an exact solution, we have provided in this way a test of this method of approximate moment inversion. 15

V. Comparison with Experiment and Discussion

An essential amount of the particles observed in a jet is not directly produced but come from resonance decays. For a comparison with experimental data it is therefore necessary to also include the production and decay of resonances. This is most conveniently done by simulating the generation of a quark or a diquark jet by the Monte Carlo method. Our procedure is an extension of that of Field and Feynman by including particle production involving diquarks according to figs, lb-d in addition to fig. la. Without inclusion of resonance decay this is of course equivalent to the analytic solution of the integral equations (15), as discussed previously. As usual the primary mesons are pseudoscalars or vectors in the ratio 1:1. For baryon production only the I • 0 states with spin 1/2 and 3/2 according to the ratio 1:1 (except of course for Ü , A , A ) are taken into account. The two particle decays are assumed to be isotropic in their rest system and the three particle decays are supposed to gc via two-particle decays. For the created quarks and diquarks we have included transverse momentum

according to a Gaussian distribution with oq « oqq • 330 MeV. The actual

value taken for aqq has practically no influence on the longitudinal momentum distributions.

It has not been our intention to present detailed fits to all avail able experimental data. Instead we shall rather concentrate on cha cter- istic examples to discuss the main features and determine the parameters. For this purpose we shall deal here only with jets in neutrino- and muon- production because in this case one quark flavour dominates in the inter action. These processes are therefore better suited for the study of fragmentation of light quarks and diquarks. At present we shall not treat fragmentation of heavy quarks (c, b,...) as the weak decays play an important role here. Therefore we shall for L1 e moment not compare with e e data. dj

Choosing f.(n) • P.O ~ a + 3an2), a « 0.7 and f. « P;(di • On for i • 2,3,4 the parameters to be determined are p., p. and the powers d.. The parameter p. - l-p. is given by the rate of baryon production in quark jets. With the choice p~ • 0.08 we got a satisfactory description of proton production by high energy muons [14] as can be seen in fig. 4. 16

As to the power d„ it turned out that the form f»(n) " 2p»n gave better agreement with data than the counting rule ansatz f_(n) * 4p_n3. In particular with d» • 3 the rate of antiprotons at large z would have been higher than that of protons what is not seen in the data. (Notice that in fig. 4 and in ref. 14 the variable z - E./v and is therefore different n from the variable z used in our paper.) With f~ * 2p„n and p. - 0.08 also a good description of A-production by neutrinos [15] for x > 0 (see fig. 5) was obtained. The flavour dependence is essentially described by the para meter y • l-2y. A comparison with K + K production by neutrinos (fig. 5) s suggests a value y • 0.1,smaller than that originally used in ref. 1. Such s a value for y was also found experimentally in ref. 16 and used in ref. 8. Moreover, it can be seen that the x -dependence of K° + K° for x > 0 is F r reasonably well reproduced. For the sake of completeness also a comparison with the forward IT and ir neutrinoproduction data is shown in fig. 6. Due to the value p. * 0.92 this reproduces within 10 percent the prediction of the Feynman-Field model. In describing diquark fragmentation, the momentum sharing functions fo(n) " 4p~n3 anu f,(n) * 2p,n reproduce satisfactorily the observed z- behaviour. The remaining parameter is p,. With the choice of p, = 0.5 we obtain fair agreement with ir , IT , K° + K° and A-neutrinoproduction for x < 0 as can be seen in figs. 5 and 6. The rate of antibaryons produced r in quark jets is also directly proportional to p, » l-p~ (see eq. (38c)). p, * 0.5 also describes the p-production by muons (see fig. 4) quite well. We tried to determine more accurately the parameter p. which is a measure of the break-up of the diquark. On the basis of the present experimental situation we can only say that p, has to be in the range between 0.4 and 0.8. The case p- • 0 (no splitting of the diquark) seems to be excluded because then the ratesof n and ir in neutrinoproduction would practically become equal for x_ < - 0.5 contrary to the data (see fig. 6). The following quantities are sensitive to the value of p.: the ratio of ir to ir neutrinoproduction for x_ < 0, b< on production for x_ < 0, in A A particular the ratio 0 (z)/D j(z), and an.ibaryon production im muon and neutrino scattering. More accurate data on these processes are therefore necessary for a better determination of p~. 17

With f.(z) as given above we could not reproduce the proton production by (anti)neutrinos of ref. 18 for JL < - 0.6. It seems (see fig. 7) that the spectrum does not fall off as strongly as one would expect from the fragmentation functions shown in fig. 2. A possible explanation for this behaviour could be that the backward protons show the so-called leading particle effect. A further indication for this effect could also be seen in the difference between forward and backward multiplicities in neutrinoproduction [17]. We simulated such a behaviour of the proton in the follow ing simple way which is most easily implemented into the Monte Carlo procedure: if the final proton is produced as a rank one particle, its momentum dependence is not given by f.(l-z) • 2p,(l-z) but by a function

f"(z) which for simplicity we take as f"(z) - 2p46(z-0.5). The higher rank protons are distributed according to f,(n). With this simple method one gets reasonable agreement for x^, < - 0.5 (fig. 7).

In fig. 8 we show the predictions of the model for antineutrinoproduction of TT , TT , K° + K°. We compared them with the data of ref. 18. Whereas we got good agreement for the ir and u distribution, for K + K and A production our results are too high. As our model agrees with the neutrinoproduction data [15] of BEBC and one expects practically equal fragmentation functions D and D,, we have no explanation for this dis crepancy. Further data are needed to draw final conclusions. Summarizing we can say that the treatment of quark and diquark fragmentation within the system of integral equations as discussed here provides a satisfactory description of meson, baryon, antibaryon distribu tions in the current and target fragmentation regions in leptoproduction. We found an analytic solution including the flavour dependence which shows the characteristic features of the fragmentation functions and clearly exhibits the dependence on the main parameters. The experimental baryon spectra suggest that in a quark jet the first rank particle is a baryon with about 8 percent probability. Moreover, we got better agreement with

the momentum sharing function f2(n) ^ n than with the counting rule

3 behaviour f2(n) ^ n . By comparing with present data our results also indicate that the diquark has an approximately 50 percent probability to break up. So far it seems that the diquark forming a baryon in a quark jet behaves in the same way as the diquark responsible for target frag- 18 mentation. The only difference could be the leading particle effect of the target proton which» however, can be incorporated quite easily.

Acknowledgements

A.B. would like to thank Prof. J. Prentki and the CERN Theory Division, Prof. S.D. Drell and the SLAC Theory Group, and Prof. D.R. Yennie and the Newman Laboratory of Nuclear Studies at Cornell University for financial support and warm hospitality. Parts of this work were done while A.B. was visitor at these laboratories. Discussions with Professors S.J. Brodsky, M. Jacob, G.P. Lepage and C. Peterson are gratefully acknowledged. We also thank Dr. J. Mittendorfer for providing his Monte Carlo program. 19

Table It Flavour coefficients for mesons in diquark jets

B q,q2 uu ud dd

* i P Y y Y 0 Yz - - 1 •> » » p o j Y y y* 0 O I 1 1 9 *»p jY yY jy y'

K+, K** I-2Y |(I-2Y) 0 Y(1-2Y)

K°,K*° 0 |(1-2Y) I-2Y Y(1"2Y)

K°, K*° 0 0 0 Y(I-2Y)

K~, K*~ 0 0 0 Y(«-2Y) n, a ^ Y sin29 ~ Y sin2e j y sin26 Y2»in29 + (I-2Y)2COS20 n',* | Y cos2e | Y C0826 I Y cog26 Y2cos2e • (I-2Y)2sin29

Here 6 is the mixing angle of isoscalar mesons. Table II; Flavour coefficients for baryons in diquark jets

BB BB

q^2 q,q2 uu ud dd uu ud dd

2 2 + 2 3 Y Y 0 2Y ^Y T Y 3Y P. * 2 • O 2 3 2 2 3 n, A 0 Y Y Y 2Y 3Y

A, 1°, 1 I** 0 •J<1-2Y) 0 Y(l-2Y) Y(1-2Y) Y(l-2Y) 3Y2(1-2Y)

+ E , E** 1-2Y 0 0 2Y(I-2Y) Y(1-2Y) 0 3Y2(1"2Y)

E", E*~ 0 0 1-2Y 0 Y(1-2Y) 2Y(1"2Y) 3Y2(1-2Y)

_o _*o 2 2 2 — t — 0 0 0 (1-2Y) y(I-2Y) 0 3Y<1-2Y)

— » — 0 0 0 0 j(l-2Y)2 (1-2Y)2 3Y(I"2Y)2

+ 2 2 3 A + 0 0 Y T 0 Y Y ±Y2

2 2 3 A" 0 0 Y 0 Y J-Y2 Y Y a" 0 0 0 0 0 0 0-2Y)3 21

Table III: Flavour coefficients for baryons in quark jets, (Notice that a slightly different convention is used in ref. 6)

AB BB q u d s

P, A 2Y2 Y2 0 3Y3 n, A Y2 2Y2 0 3Y3

2 2 A. 1°. I E*° Y(1~2Y) Y(I-2Y) Y 3Y (I-2Y)

+ E , I** 2Y(I-2Y) 0 Y2 3Y2(1"2Y)

Z , Z 0 2Y(I-2Y) Y2 3Y2(1-2Y) j.0 „»0 2 - » — (1-2Y)2 0 2Y(1-2Y) 3V(1-2Y)

— » — 0 (I-2Y)2 2Y(1-2Y) 3Y(1-2Y)2

++ A Y2 0 0 Y3

A~ 0 Y2 0 Y3

3 Ö~ 0 0

References

[1] R.D. Field, R.P. Feynaan, Nucl. Phys. BI36, I (1978) [2] B. Andersson, G. Gustafson, C. Peterson, Nucl. Phys. BI35, 273 (1978); Z. Physik C U J05 (1979) [3] U. Sukhatme, Z. Physik C2, 321 (1979) [4] R. Brandelik et al., Phys. Lett. 94B, 444 (1980); G.S. Abrams et al., Phys. Rev. Lett. 44, 10 (1980) [5] E.H. Ilgenfritz, J. Kripfganz, A. Schiller, Acta Phys. Polonica B9, 881 (1978) [6] A. Bartl, H. Fraas, U. Majerotto, Z. Physik C £, 335 (1980) [7] M. Fontannaz, B. Pire, D. Schiff, Phys. Lett. 77B, 315 (1978); B. Andersson, G. Gustafson, C. Peterson, Phys. Lett. 69B, 221 (1977) [8] U.P. Sukhatme, K.E. Lassila, R. Orava, Fermilab preprint FERMILAB- Pub-81/20-TOY, January 1981 [9] E.L. Berger, S.J. Brodsky, Phys. Rev. Lett. 42_, 940 (1979) [10] J. Bediaga, E. Predazzi, A.F.S. Santoro, CERN preprint CERN-TH-3104, June 1981 [11] A. Casher, H. Neuberger, S. Nussinov, Phys. Rev. D20, 179 (1979) [12] S.J. Brodsky, R. Blancenbecler, Phys. Rev. Dip, 2973 (1974); S.J. Brodsky, J.F. Gunion, Phys. Rev. Pi7, 848 (1978) [13] F.J. Yndurain, Phys. Lett. 74B, 68 (1978) [14] J.J. Aubert et al., Phys. Lett. 103B, 388 (1981) [15] H. Grassier et al., CERN preprint, CERN/EP 81-56, June 1981 [16] V.V. Ammosov et al., Phys. Lett. 93B, 210 (1980) [17] N. Schmitz, talk given at the International Symposium on Lepton and Photon Interactions at High Energies, Bonn, August 1981; and talk given at the International Conference on Neutrino Physics and Astrophysics, Hawaii, July 1981 [18] M. Derrick et al., Phys. Rev. D24, 1071(1981) 23 \

Figure Captions

Fig. 1 Elementary processes for emission of primary particles: (a) meson from quark, (b) baryon from quark, (c) meson from diquark, (d) baryon from diquark.

Fig. 2 Full lines: Exact solution for the fragmentation functions M (z),

B (z), Md(z) and Bd(z) of eqs. (38) with f.(n) given in eq. (36). Dashed line: For comparison fragmentation function of ref. I, -»»--©-fragmentation functions obtained by the method of approximate inverse Meliin transformation, the length of the averaged z-interval as indicated.

Fig. 3 Diquark fragmentation functions following from eqs. (32) and (38) with the flavour coefficients as given in tables 1 and 2 (y "0.1, ctp • o„ • o. .« • a,/« " 0»^)» without resonance contributions. (a) uu •+ mesons (b) uu -• baryons (c) ud •+ mesons (d) ud * baryons.

Fig. 4 Comparison of our model with p and p data of the European Muon

Collaboration [14]. Theoretical curves obtained with f« - 2p2n,

f, • 2p,n; P2 " 0.08, p, »0.5. Here z • E./v, and jet energy

corresponding to - 13 GeV, - 15 GeV .

Fig, 5 Comparison with A and K° + K° neutrinoproduction of BEBC (ref. 15). Theoretical curves obtained with a quark or diquark jet energy of 3 GeV, corresponding to - 6 GeV.

Fig. 6 n and n neutrinoproduction of BEBC [17] compared to our model pre dictions. Full line: n , dashed line: ir~. Quark or diquark jet

energy taken to be 3 GeV. f. - p.(l - a + 3an2), a - 0.7, j>. - 0.92,

3 f. - Ap-n , p3 • 0.5. For x_ > 0 the result corresponds within 10 percent to the Feynman-Field model [1].

Fig. 7 Comparison with proton (anti) neutrinoproduction data of ref. 18 in the target fragmentation region. Full line: vp, dashed line: vp. Model calculation including the leading particle effect, with di quark jet energy of 3 GeV. 24

8 Model predictions for antineutrinoproduction of i , i , K +K» and A for quark and diquark fragmentation with quark or diquark jet energy of 3 Gev. Data for * and « production from ref. 18. — • + Full line: w (for x_ > 0), and i~ (for x < 0). Dashed line: « Dotted line: K° • K°. Dashed-dotted line: A. -fy- negative aesons, data from ref. 18, -^- positive aesons, data from ref. 18. _JIL

zMq(z)

zMd(z)

OJ-

00/-

0O01 0 ./ .2 .3 A .5 .6 .7 .a .9 I Fig 3a \

-* p»n,/i,r,-•r-rO

0.1 •

0.01 -