Deep inelastic two-particle distributions in the two-component parton model

Nuclear Physics B74 (1974) 111 - 124. Nbrth-HoUand Publishing Company

DEEP INELASTIC TWO-PARTICLE DISTRIBUTIONS IN THE TWO-COMPONENT PARTON MODEL

M. CHAICHIAN *, S. KITAKADO **, W.S. LAM and Y. ZARMI *** Department of Theoretical Physics University of Bielefeld, Germany

Received 1 November 1973 (Revised 21 January 1974)

Abstract: The two-component parton model is used for the description of inclusive two-particle distributions in the deep inelastic region. Simple predictions for the cross sections and correlation functions in electroproduction and in e+e - annihilation are presented.

1. Introduction

Two-particle inclusive spectra are useful for the investigation of hadronic properties. In this respect, the two-particle correlation functions are of interest, as they single out non-pomeron effects, when simple properties are assumed for the pomeron contribution.

The application of the parton model to deep inelastic phenomena [1 ,2] enables one to describe a large variety of experimental quantities in a simple manner. Its two-component version (namely, the quark parton model constrained by two-component duality [3] has been applied both to total cross sections [4, 5] and to single- particle distributions in the deep inelastic region [6, 7]. The model has yielded many interesting predictions, some of which, e.g. the rr+/lr - ratios in the current fragmentation region, seem to agree with experimental data.

Although it is still not clear whether this picture, which predicts a jet structure of hadron distributions in the final state in e+e - annihilation, is correct or not, its application to two-particle distributions [8] is of interest. This is especially so, as it may serve as a good test for the validity of the two-component structure of the parton model.

In this paper we study the correlation between particles emitted in opposite directions. In the framework of the parton model this case has a relatively simple des-

* On leave of absence at the Research Institute for Theoretical Physics, University of Helsinki. ** Alexander von Humboldt Fellow, on leave of absence from the Institute 6f Physics, Uni-

versity of Tokyo, Komaba. *** On leave of absence from the Weizmann Institute, Rehovot.

112 M. Chaichian et al., Two-component parton model

cription, as it is natural to assume that they are emitted independently. We briefly study a possible description of correlations among particles in the same direction.

In sect. 2 we describe the kinematics and notation for the cross sections of the processes involved. In sect. 3 we present the results of our model. Special attention is given to tests of SU(3) breaking in the emission process. Some of the results are given for Gell-Mann-Zweig (or coloured) quarks, denoted by GMZ, and for Han- Nambu quarks (HN). We conclude in sect. 4 with a few remarks.

2. Kinematics and notation

We use the notation proposed in ref. [2] with the two-component modifications of refs. [5, 7].

2.1. e+ e - annihilation

The parton model description of this process is shown in fig. 1. The virtual photon materializes into a fast q~- pair. Each of the quarks is assumed to fragment independently (on the average) with distribution functions which scale.

The total and single-particle inclusive cross sections are given by

O" t (e+e - ~ X) = 4rrofl 1 3q2 2 l~. Q2,

1 do ( e + e _ _ + h + X ) = 2 ~ ~iDh(z) , (2.1) a t dz i

1

Here q2 stands for the virtual photon mass, Qi is the charge of a parton of type i, and the summation is over all types of patrons (quarks and anti-quarks). For an observed hadron with four-momentum h, the distribution, scales and only depends on the energy fraction it carries:

z = 2 h . q/q2. (2.2)

The distribution due to the emission from a parton of type i is described by Dh(z) . For the two-particle distributions, one has to distinguish between the case of

particles moving in opposite directions (except for possible small transverse momenta), and that of particles moving in the same direction. For the first case we have the independent emission (on the average, not in individual events) by the two quarks [8]. Namely, that the cross-section scales and, after integration over transverse momenta, obtains the form

M. Chaichian et al., Two-component parton model 113

e + e -

Fig. l. Hadron production in e+e - annihilation. The photon is converted into a qq- pair and these fragment into hadrons.

1 do A

o t dZldZ 2 h 1 h 2

- - ( e + e - ~ h 1 +h 2 + X ) = 2 ~ ~iDi (Zl)D T (z2). i

(2.3)

Here A denotes anti-parallel. For the case of parallel particles, it was suggested by Cisneros [9] that the frag-

mentation of a quark into a number of particles also scales. In our case (P stands for parallel):

°tl dZl d z ~ d ° P (e+e _ ~ h 1 + h 2 + X) = 2 ~ i ~t ~:' Dh'l'h2t (Zl' z2)" (2.4)

D hI'h2 (z 1 , z2) is a new unknown function. In the spirit of suggestions in ref. [7], we propose to decompose it as follows:

hl,h 2 h I h2/h 1 . h 2 hl/h 2 D i (Zl ,Z2)=Di ( z I ) F i (Zl ,Z2)+Di (z2) F / (z2, zl). (2.5)

Here F h2/hl (Zl, z2) is the conditional distribution of h 2 (with momentum fraction z2) after h 1 (with fraction z 1) has been emitted. However, the "memory" carried by the distribution of the second particle (concerning the identity and momentum of the first one) does not enable us to obtain definite and interesting predictions.

2.2. Electroproduction

The parton model description of the process in the deep inelastic region is shown in fig. 2. In the photon-parton Breit frame, the parton which absorbs the photon reverses its~direction of motion. It is assumed [2] that now each part of the system (left- and right-moving partons) fragments independently and in a scaling manner.

For spin-~ pattons the only independent structure functions are given by [2]:


e_ p : ( P , o , o , p )

[---~ ~' q : ( o , o , o , - 2 x P )

~ (xP, o,o, xP) 4~,..ftY'JV",d~

,, b

(xP, o , o , - x P ) ( ( I - x ) P , o , o , ( I - x ) P )

4 - ~ 2 - 2 _ _ - - . - . • • 4 1 - - - - . . . . • - - , ; : :

Fig. 2. Two-step description of electroproduction in the parton model; (a) before current-quark interaction, (b) after the interaction has occurred.

et 1 F 1 (x) = ~- . ~ Q2 d~ (x), total cross section,

l

qt, h(x, z)= t

single particle: current fragmentation,

R etl,h (x, r)= ~I ~ . Q2 d~(x)Eht ._i(x,r), single particle: target fragmentation.

Here

(2.6)

x = -q2/2q. p, z = 2h. q/q2, r (1-x) = h. q/q. p. (2.7)

Assuming that left and right emissions (see fig. 2) are independent, we easily write the two-particle cross section for one current fragment and one target fragment:

M et (x,z,r) = 1 ~ O2di h L h R l'hihR 2 i ~i t (x)Di (z)Et-i(x,r)" (2.8)

In eqs. (2.6) and (2.8) d~ (x) is the distribution function for a quark of type i in the target, DhL(z) is the fragmentation function describing the emission of a left t h moving hadron h L from the left moving parton i, and EtR i (x,r) describes the emission of a right moving hadron by the remaining system (target minus parton of type/). For details see ref. [2].)

M. Chaichian et al., Two-component parton model 115

2.3. Two-component parton model

In the two-component parton model the distribution of partons inside the target is decomposed [4, 5] into a sum of sea and valence terms:

i i (x) + i (x). (2.9) d t (x) = o t s t

The sea of partons is an SU(2) singlet. Thus, in a proton target,

s u =sff=s d =sd---s(x), ¢ =sS--s'(x) (2.10) P P P P P •

The distribution functions for hadrons emitted by the fragmenting quarks are similarly decomposed [7]:

(z) +

d~(x) E p _ (x,r) = o ti (x) (gh_i (x,r) + s t h i (x, r)) u 1)

"~h + s~(x) ( vt_h i (x,r) + St_ i (x,r)). (2.11) o s

Explicit expressions for the case of nucleon targets (in the quark parton model) are given in ref. [7].

The empirical observation [10] that the parton sea is limited to small x values (x < ~ 0.1) has led to the conjecture [7] that a similar statement should hold for the sea contributions in the hadronic distributions. Namely, we assume the existence of some typical values x0, z0, r 0 (all ~ 0.1 - "~ 0.2, say), such that

si(x) ,~ 1, x > x 0 , empirical observation ; vi(x) s (z) vh( z)- ~ 1, z > z 0 , current fragmentation ]

"~tt-hi(x'r) I conjecture.

,~ ~ 1, r > r 0 , target fragmentation (2.12) (rth_i (x,r)

The sea part in the hadron distribution is related to the logarithmically increasing multiplicity. Finite multiplicities are directly related to the absence of sea contributions (S h, s h ) .

Let us conclude with the following remark: I f the D-functions satisfy SU(3) symmetry relations (namely, the hadrons are classified in SU(3) multiplets), then for the 0 - meson octet there are only three independent functions [2]: rvr+ rvr+ r~rr

~ U ' ~ d ~ S " The two-component description reduces the number to two: W r, S ~r.


3. Results

3.1. e+ e - ann ih i la t ion

For the reasons mentioned in sect. 2, we concentrate on the case of anti-parallel particles. The situation becomes particularly simple when one studies the emission of two fast particles (Zl, z 2 > Zo). Then, only the valence part (V/h (z)) of the D- functions in eq. (2.3) contributes (see eq. (2.12)).

In particular, for the 0 - mesons one finds for

1 do (e+e _ + h 2 + X ) a t dz I dz 2 ~ h 1

the following expressions (see also Walsh and Zerwas [8]):

7r+.- . ~ [3] vlr(gl)vTr(z2) '

~%o : ~ [1~1 V"(z 1) V"(z2),

Tr+K 0 : ~ [{] V~r(Zl ) V K (z2), (3.1)

7r+K- : 1 [¼] V~r(Zl ) V K ( z2 ) '

K+K 0 : ~ [~] vK(zl ) v'K(z2 ),

notation: GMZ [HN],

K + K - : -~ [¼] VK(z 1) VK(z2) + 1~ [~] V 'I( (z 1) V'K(z2),

zr+zr +, 7r+K +, zr+K -if, K+K +, K+K 0 all zero for Z1, Z 2 > Z 0 .

The numerical coefficients in eq. (3.1) are for GMZ [HN] quarks. V K (V 'K) is the valence component in the emission of K-mesons from non-strange (strange) quarks.

The vanishing of some of the cross sections, is a trivial consequence of the production mechanism. When one of the particles (say hl) is emitted by a quark of type i, the other (h2) is emitted by the anti-quark T. The valence component of the emission (only it contributes to fast particle production) is non-zero only when quark i is a valence constituent of h l , and i- is a valence quark of h 2.

The factorization of the cross sections is a direct consequence of the assumption of independent emission.

A simple consequence of eq. (3.1) is, that the two-pion cross sections (for fast mesons) are proportional to products of single pion cross sections (the proportionality coefficients are determined by eqs. (2.3) and (3.1)). For example, for GMZ [HN] quarks,

M. Chaichian et al., Two.component parton model 117

Ot doA(Tr +, ~r-) = ~ [-}] ~-tl da (.+) a-tl d o ( . - ) . (3.2)

A similar statement holds for cross sections involving K-mesons if the emission process is SU(3)-invariant (then V n = V K = v'K). For example, for GMZ [HN] quarks

1 do(K +)1 d o ( K - ) . 1__ do A (K+K _) = 6 [~_] ~-t b~t o t

(3.3)

Another simple test of SU(3) invariance is the numerical proportionality among all the cross sections appearing in eq. (3.1).

doA(rr +, zr-) = daA(K +, K- ) = ~ [3] do A (zr +, K-) , (3.4)

where the numerical coefficient is for GMZ [HN] quarks. Another useful relation is +

1_._ (daA(rr +, 7r-) -- do A 0r +, rr+)) = ~ [4 3-] Wr(zl ) Wr(z2 ), (3.5) ° t

which is true for all z-values (independent of the z 1 > z 0 condition). Thus, the valence component of the D-functions can be cleanly isolated by use of eq. (3.5). This cannot be done at the single particle distribution level (as da (rt +) = do(n 0) = do(n-) for all z). Similar relations hold for the rrK and KK cross sections.

Let us now turn to the two-meson correlation functions. They are defined as

1 1 do(h!) 1 do(h2) 1 daA(hl,h2) cA(h 1,h 2 ) = ~ - o t dz 1 at dz 2 a t dz 1 dz 2

h 2 h 1 h 2 = iOpl (z,) jDj (z2)- i (Zl)O r (zg. i ] i

The factor of i in eq. (3.6) is a consequence of the limitation to anti-parallel particles (only half of the whole solid angle integration taken into account).

As an example, we give the lr+Tr - correlation function;

(3.6)

cA( + , - ) = _ ~ [~] VlrVlr _ ~ [ 3] (V~r(STr_S,~r)+(Su_S,.) VTr)

5 3 - ig [~] (S'-S") (S'-S'"), (3.7)

where in each term the first function in the product depends on z 1 and the second one on z 2. S 'r is the sea component in the emission of pions from non-strange quarks,

+ Valid also when SU(3) is broken.


C"(r:+, R - }

Z~

oi-_,---I---

Z2

Ii ~:, 01~!-- + . . . .

I-', +

o/- 1 - 11

s~_- s'~

Z;l-i . . . . . . l + i

0 i . . . . . I -_--- + ]

'z~

'z,

Fig. 3. Possible sign combinations for correlations between anti-parallel pions in e+e - annihilation. Sign changes are expected to occur for z 1, z 2 ~ 0.1 - 0.2. I fS lr = S 'Tr (SU(3)) all correlations vanish for z 1 or z 2 --* 0.

and S '~r for strange quarks. These two functions may be different if the pion sea is an SU(2) singlet only. In the SU(3) limit S 7r = S '~r.

For fast pions the correlation is negative (only the V ~r V ~r term remains) as expected, since i f a quark can emit a ~r +, the anti-quark will prefer to emit a 7r-.

SU(3) breaking at the level of emission of mesons from quarks is expected to work in the direction S 7r > S '~r (more pions from non-strange quarks than f r o m strange ones). This implies that C A (rr +, r r - ) is negative for all values of z 1 , z 2 . The unlikely case of S 7r < S '~r manifests itself in the existence of two regions (z 1 < z 0, z 2 > z 0 ;z 0 "" 0.1 - 0.2) in which cA(rr +, 7r-) should be positive. CA(n +, rr +) can be analyzed in a similar manner. The various possibilities for the signs of c A ( n +, rr -+) are shown in fig. 3. If SU(3) is not broken (S ~r = S 'Tr) all correlations vanish when either z 1 or z 2 ~ 0.

The values of z 1 or z 2 at which the correlation functions change sign (in some of the possible cases) are of the order o f z 0 (~" 0.1 - "~ 0.2). The signs of the correlations depend on the dynamical picture o f independent emission and the two-component description of the D i. These gross features are not sensitive to the type of quarks (Gell-Mann-Zweig or Han-Nambu).

Since average quantities will be measured before the details of inclusive distributions are obtained, let us finish with a few remarks about average joint multiplicities. Denote the average mult ipl ici ty of a hadron h in e+e - annihilation by (nh), and the

joint multiplici ty of h I , h 2 (when coming out in opposite direction) by <nhl • r/h2> A .

M. Chaichian et al., Two-component patton model 119

From eq. (3.6) one has:

fdZldZ C A (hi, h2 ) = 1 (nhl). (nh2) _ (nhl . nh2)A. (3.8)

In the parton model one expects the average multiplicity to grow like In q2. This is due to the dz/z behaviour of the sea functions at small z, which implies (see Gro- nau et al. [2]):

f Dhi (z) ~ ~ In q2. (3.9)

As a result, both terms on the right-hand-side of eq. (3.8) grow like (In q2)2. A study of ch(rr +, rr-) [eq. (3.7)] and similarly cA(n +, lr +) reveals that for large q2:

(i) If S ~r 4=S'7r: 21- (n r)2 - (nr + • n r~.) A < 0 ,

) 2 _ ( n . n _ ) A <0" 21- (n n +

(ii) I fS ~r = S '~r (SU(3) not broken): The terms proportional to In q2 and On q2)2 in ½ (nrt) 2 and (n~r+ nTr_+) A are the same. Moreover, ½ (n~r)2 - (n~r+ n r+) A becomes positive.

Here we have used the notation (n +) - (n~r). As a result, a study of the sign and ~l~e q2 dependence of

½ (nhl) (nh2) -- (nhinh2) A

Ahl,h 2 -- ½ (nhl) (nh2) (3.10)

can tell us something about the SU(3) properties of the sea component in the D- functions. To summarize, for large q2:

C + / t -

( C + > 0 , C < 0 ) : S ~ = S ''~, A + _+(q2) ~ (ln q2)2 7r 7r-

~" ,/~" C 2 c 1 + ( c 1 < 0 ) : s ~ 4:s ' ,~ (3.11)

In q2

The constants appearing in eq. (3.11) depend on the quark charges and on the details of the unknown functions W r, S ~r, S 'rr. A lower bound for e 1 is obtained by putting S '~r = 0. This gives -~- [-½] < c 1 < 0.

One still has, of course, the unlikely possibility o fS ~r = S 'Tr = 0 (no plateau in the distribution of hadrons in e+e - annihilations). In this case, for large q2, (nh), (nhl • nh2) a and Ahlh2 are all independent o fq 2.

3.2. Deep inelastic electroproduction

In this process, the correlations between parallel particles (both in the current or target fragmentation region) do not bear any simple predictions. Let us, there-


fore, concentrate on anti-parallel particles (one current and one target fragment; see eq. (2.7)) and limit ourselves to fast pions (z,r > ~ 0.1 - 0.2). Eq. (2.7) then obtains the form (for GMZ quarks):

... + ~,r + V"(z) + + (40U(x) V~_uv (x,r) + 5 s(x) Vp. 3 (x,r) 18 : ~'L rrR'

MePLTr R I , T r (4vU(x) "~ n (x,r) + 5 s(x) V ~r (x,r)) Vn(z) • + 7rR , =, P- o p;s ~ - - "/rL (3.12)

,+ ~ + V~(z) (od(x) Vp_% (x,r) + 5 s(x) V" (x,r)) : ~-~ + P~ 18 rrR'

5 s(x) ? ~- (x,r) V~(z) p;s 18 : lrL 7rR"

Here L stands for a left moving (current fragment) particle, and R for a right moving (target fragment) one. Similar expressions can be written for neutron targets. In the last line of eq. (3.12) the obviously expected result implies that the cross section for two fast anti-parallel 7r-'s vanishes when s(x) is negligible (x > "" 0.1 - 0.2).

In ref. [7] it was argued that one should expect

Vgs(x,r) > Vp;s ~x,0.

The reason was that the parton sea is limited to very small x (< "0 .1 , say). Thus after the emission of a slow-sea quark, the momentum content of the right moving system (see fig. 2) is unaffected, and the two u-valence quarks will emit more rr+'s than the d-valence quark will emit fast ~r-'s.

It was moreover argued in ref. [7], that for x -) 1 V "+ ~ V-~u • In this case the fast u-valence quark has been ejected (od/o u) ~-(~V'as 60 v 1)Oand the remaining u- and d-valence quarks are in an antisymmetric state (see Feynman [2]). For co -~ 0% ou "~ 2od seems to be a good approximation [5]. These qualitative assumptions enable us to obtain a general idea of the co-dependence (co = I / x ) of the ratios of various two-pion cross sections. Some ratios are shown in fig. 4. There, R 0 > 1 is determined by

Vp;"~ lVp" (co -, oo).

Eq. (3.12) trivially implies for proton (and by analogy, for neutron and deuteron) cross sections the following inequalities:

daeP(TrL ,+ 7r R+ ), daeP(ir~, rt R ), daeP(TrL , rt~) > daeP(frL , 7rR) ,

daen(TrL, 7tR) ' daen (lrL ' lrR) ' en + en + + + de (rrL, rr R ) > do (n'L, ~'R) ,

do ed (.~, .~) > doe~(~, .~), ed + do (TrL, zr~) ) doed(rrL , 7rR).

M. Chaichian et aL, Two-component parton model 121

Ro

l -I

Ro

T b) 4

I ~ 3 - ~ --)" ~

R~ ~ 0

c)

1 - - - - )r

Fig. 4. Cross features of the to-dependence of ratios of two fast pions cross sections as predicted in the two-component GMZ quark patton model. Targets: (a) proton, (b) neutron, (c) deuteron

• + + + 4 - - - - - 4 - 4 - - - - - "4"

Ratios: (1) ~r L ZrR/ZrLZr ~, (2) ~rL~r R, (3) ¢r L zrRbr ~, n~, (4) ¢rLZrR/rr L ~r R.

Similarly, a few equalities can be derived:

dtreP(rt L, 7rR ) = doen(Tr~, 7r~),

doeP(TrE ,+ 7r~) _ doen(TrL, + 7rR) = 4 (doen(Tr L, 7rR ) -- doeP(Tr L , / r~)) , (3.14/

e l l +

doeP ( . ~ , 7 r R ) - doeP(TrL, 7 rR)=4 ~doen(TrL , 7r~) -- do (TrL' 7rR))"

122 M. ~rhaichian et aL, Two-component patton model

When s(x) is negligible in eq. (3.12) (x > ~ 0.1 - 0.2) one has:

- _ ed + + =4doed0rL, daeP(Tr~, 7r~) = 4 dr7 en (rrL, rtR) - do (~'L' 7rR) rtR)'

doepor~, 7rR) = 4 daenorL, 7r~), (3.15)

daen + _ =4doeP0rL, + (7i" L , 71" R ) 7rR),

daeP0rL ' ~'R ) = daen (~'L' + 7rR) + - - 0.

The factor of 4 in eqs. (3.14) and (3.1 5) is for GMZ quarks. For HN quarks it is replaced by 2.

Eq. (3.12) also implies that the ratio of any two cross sections (for proton, neutron or deuteron targets) is independent o f z (scaling variable for left moving pion). This, again, is a direct consequence of independent emission and the limitation to fast pions in the two-component model.

4. Remarks and conclusions

The two-component parton model was constructed for the description of single hadron inclusive distributions in deep inelastic reactions. In this paper we have used it in order to derive a few simple and general features of two-particle distributions in deep inelastic e+e - annihilation and electroproduction. The basic assumption of course, is the independent emission of particles moving in opposite direction. However, simplicity and predictive power are obtained due to the conjecture, that the fragmentation into fast hadrons proceeds only via the valence component of

h . h ~h ~h • h the distribution functions: V~ m D i and Vp_iv (Vp;s) m Ep_ i. We have derived a few tests for the SU(3) properties of the D-functions. In parti-

cular, the q2 dependence of the joint left-right multiplicity is of interest. Let us mention that, if one assumes SU(3) symmetry for the D-functions, inde-

pent emission implies that in e+e - annihilation (independently of the two-component constraints) one has

CA(h,h ') = 0. (4.1) h'

Here h is any hadron, and h' runs over a whole SU(3) multiplet (for mesons) and its conjugate (for baryons.) This implies that the Predazzi-Veneziano sum rule [11] for left-right correlation functions is trivially satisfied in the SU(3) limit. This sum rule states that the first moment of the left-right correlations, when summed over all particles, should vanish.

For some of the quantities discussed here, we presented results for both GMZ (Gell-Mann-Zweig, with or without colour) and HN quarks. It is not clear yet, which

M. Chaiehian et al., Two-component parton model 123

type of quarks one should choose. Moreover, one has the possibility of a transition from the predictions of a GMZ model to a HN model [12]. We have, therefore, tried to concentrate on qualitative predictions which do not depend strongly on the quark charges.

Concerning the experimental situation, as yet there are no results on two hadrons detected. The observed non-constancy of

e -e + ~ hadrons R =

e-e + _~/a-/2 +

could be explained in different ways, like for instance, the threshold effects or the two-state set-up of Gell-Mann-Zweig and Han-Nambu cases (cf. Bjorken's talk at the Bonn Conference [12]) and/or a later set-up of scaling in the case of the time-like photon.

The non-scaling observed for z < 0.5 (while scaling for z > 0.5) could be explained, for instance, again as a result of a later set-up of scaling in the case of the time-like photon. Some specific models of this type are indeed being constructed.

These points, however, are not within the scope of the present paper.

Helpful discussions with F. Bopp are acknowledged. M. Chaichian would like to thank the Research Institute for Theoretical Physics, University of Helsinki, for hospitality. S. Kitakado thanks the Alexander yon Humboldt-Stiftung for financial support. Y. Zarmi thanks H. Satz for hospitality in the Department of Theoretical Physics, University of Bielefeld.

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124 M. Chaichian et al., Two-component patton model

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Deep inelastic two-particle distributions in the two-component parton model