Z. Phys. A AtomicNuclei 336, 103-112 (1990)

Zeitschrift for Physik A

Atomic Nuclei 9 Springer-Verlag1990

Deuteron fragmentation d + p p + X at relativistic energies and the role of particle production processes H. Miiller Zentralinstitutffir KernforschungRossendorf,Dresden, German DemocraticRepublic Received November 13, 1989

A modified phase-space model for the description of inelastic hadronic and nuclear reactions is proposed. The probability of observing a definite final channel is assumed to be proportional to the corresponding phasespace factor modified by a transition matrix element that is based on statistics, thermodynamics, and the quark picture of hadrons. We apply the model to proton-proton reactions up to 12 GeV/c as well as deuteron fragmentation d + p ~ p + X up to 9 GeV/c and achieve good overall agreement with experimental data. The Paris deuteron wave function is used to calculate the momentum distribution of the nucleons in the deuteron. Particle production processes turn out to be decisive for the reproduction of the proton spectra from dp interactions at 9 GeWc. PACS: 25.45.-z; 13.85.Ni; 13.75.Cs; 25.40.Ep

1. Introduction The deuteron is the simplest nucleus we know and, therefore, of particular importance for an understanding of nuclear phenomena. So far, proton spectra from fragmentation of deuterons have been measured [1-5] at incidence energies from 1.25 to 7.3 GeV (corresponding to 2.5-9 GeV/c incidence momentum) mainly with the aim of investigating the deuteron wave function (DWF) at small distances. Indeed, the 0 ~ proton spectra are usually assumed to be directly proportional to the deuteron single-nucleon momentum density, provided the reaction mechanism can be described in the impulse approximation. In this case, it is assumed that during the reaction one of the nucleons of the deuteron (the participant) interacts with the target, while the other one (the spectator) proceeds undisturbed with the momentum it had before the interaction. But at energies high enough to allow particle production, there is still another point which must be considered. The centre of mass (c.m.) energy of the participant-target subprocess depends not only

on the incidence energy but also on the energy carried away by the spectator. A high-momentum spectator leaves little energy for the participant-target interaction. Therefore, particle production is impossible near the upper kinematical limit for the spectator momentum. The smaller the spectator momentum becomes, the greater the extent to which inelastic channels contribute to the spectrum. This leads to significant deviations from the proposed proportionality of the 0~ spectra to the nucleon momentum density, even in the impulse approximation. Although this problem has attracted little attention so far, we show in the present paper that a careful consideration of particle production processes is decisive for a correct description of the deuteron breakup at high energies. Our approach is based on a consideration of the kinematical and dynamical aspects of the reaction process. Kinematical constraints arising from particle production processes are implemented by calculating the phasespace factors of the final channels. The dynamics of the interaction is taken into account by using a nonrelativistic DWF to calculate the momentum distribution of the spectator and an empirical interaction matrix element to describe the nucleo~nucleon (NN) subprocess. The NN matrix element reflects statistical and thermodynamical properties of the reaction as well as the quark structure of hadrons. Since the model is implemented as a Monte-Carlo generator which samples complete events, it is not only applicable to the interpretation of experimental data but is also well suited to a simulation of new experiments. Here, we will discuss the 0 ~ proton spectra from dp collisions at different incidence energies. A common feature of these spectra is the presence of a broad shoulder in the momentum region around 0.3 GeV/c (in the deuteron rest frame), which was also observed in deuteronnucleus interactions [-1, 2, 4, 5]. Various attempts have been made to explain this peculiarity in the 9 GeV/c spectrum. In [1] the parameters of a six-quark admixture [6] to the deuteron wave function were adjusted to reproduce the data in the impulse approximation. Braun

104 and Vechernin [7] came to the conclusion that rescattering of virtual pions is responsible for the appearence of the shoulder. Ignatenko and Lykasov [8] shared this point of view and, moreover, stressed that a six-quark admixture can contribute to the spectrum near the kinematical limit but not in the region of the shoulder. We will show that at 9 GeV/c the shoulder has a very natural explanation, because it is connected with the kinematical limits of the various channels contributing to the spectrum (see [9] and Sect. 3.2). At lower incidence energies (1.25 and 2.1 GeV), however, the shoulder cannot be explained and it seems that different reaction mechanisms are involved. The various processes contributing to the cross-section have extremely different angular dependences, and it is interesting to consider also the 8~ proton spectrum recently measured at 9 GeV/c [10]. Azhgirey et al. [1012] reproduced the 8~ and 0~ spectra by using a special deuteron structure function [13] and claimed that the data cannot be described by applying a nonrelativistic DWF. In contrast, we reproduce both spectra fairly well by using the nonrelativistic Paris DWF [14]. In a further analysis [15] of the 8~ spectrum, which was carried out in the framework of light-cone dynamics [16, 17], a sixquark admixture to the DWF had to be included in order to reproduce the yield of high-momentum protons. According to our calculations, this part of the spectrum simply originates from spectator contributions, which were neglected in [10-12, 15]. In the present work we want to discuss deuteron breakup in a more general context. Therefore, we also consider the inverse process, namely the interaction of protons with a deuterium target at 0.8 GeV [18], and show that the spectra of inelastically scattered protons are well reproduced, too. Since we pay special attention to the role of particle production in the deuteron fragmentation process, we fix the model parameters in a first step by reproducing experimental results from pp collisions. These parameters are then used to calculate deuteron fragmentation without any further adjustment. So the well-known concepts we employ allow us to describe pp data and are also well suited to the investigation of deuteron fragmentation. The present paper is structured as follows. In Sect. 2, the main features of the modified phase-space (MOPS) model are explained. We begin with hadronic interactions, and then we discuss the ingredients necessary to extend the approach to nuclear reactions. Section 3 contains a comparison of theoretical and experimental results on pp and dp interactions. Concluding remarks are given in Sect. 4.

NN collisions. This makes it desirable to develop a common approach to hadronic and nuclear reactions. In order to understand the mechanism of nucleus-nucleus interactions, simpler systems have to be studied first and therefore we discuss here pp and dp interactions. One has to keep in mind that the overwhelming majority of hadronic interaction events are low-momentum transfer processes, the correct theoretical description of which is still an unsolved problem. So far, a large variety of phenomenological approaches has been developed, reaching from phase-space and thermodynamical models to approaches based on the partonic substructure of hadrons. For a comprehensive list of references we refer to the review articles by Hofmann [19], Feinberg [20, 21] and Fialkowski and Kittel [22]. More recent references concerning the dual parton and LUND models can be found, for example, in the papers by Capella et al. [23] and Andersson et al. [24]. In the present work we deal with pp interactions at low incidence momenta up to 12 GeV/c, and the number of hadrons produced remains small (typically no more than five final particles). For this special situation thermodynamics and quark statistics [25] are combined with an exact calculation of the phase-space factor for the channels involved. To be more specific, we employ a two-step model, where in the first step one or both of the colliding hadrons become excited. Thus, translational energy is transformed into excitation energy, whose amount follows from thermodynamical considerations. An excited hadron is pictured as consisting of the valence quarks of the initial hadron and additional quark-antiquark pairs created during the collision. In the second stage of the reaction, it is supposed that the excited hadrons decay, whereby the available quarks recombine into hadrons according to the rules of quark statistics. We will demonstrate in Sect. 3.1 that this simple picture works well up to 12 GeV/c. (At higher energies a hadronic interaction is regarded as a sequence of parton-parton collisions, and a varying number of excited subsystems is assumed to emerge from the collision [26].) In the case of the dp reaction, we consider three types of interactions, which contribute to the inclusive proton spectra. In quasi-elastic collisions

2. The model

d+p~d*+p

2.1. Basic assumptions

encompasses all processes where both nucleons of the deuteron participate in the interaction due to, for example, multiple NN collisions, the appearance of virtual particles in intermediate states or the presence of a sixquark component in the DWF. An exact treatment of

At present, special attention is concentrated on the investigation of relativistic nucleus-nucleus reactions, which are known to proceed predominantly as a sequence of

[N]+p--*N+p

(1)

and quasi-free inelastic collisions

[N]+p~N*+p

or

N+p*

or

N*+p*

(2)

the spectator dissociates from the deuteron and the participant-target subprocess [N] +p proceeds elastically or inelastically. Cluster excitation or

d*+p*

(3)

105 such processes is rather involved and afflicted with many uncertainties. Therefore, we suppose the deuteron to behave like a single entity and treat it as a cluster in the sense of the cluster excitation model [27, 28], which was originally developed for the interpretation of particle production in the kinematically forbidden region of hadron-nucleus collisions. In the following two sections the qualitative picture discussed so far will be further specified to make it suitable for numerical calculations.

P8 i

~ . ~

n _ r~l+ r)2

Fig. L Phase-space decomposition for inelastic 5adron-hadron interactions

dRz(s; M1, M2) and factors dR,j(Mj; c~j), which belong 2.2 Hadron-hadron interaction A description of our approach to multiparticle production in hadronic interactions is given in [29]. Here, we summarize the main points of an improved version. The first step of the interaction is assumed to lead to an excitation of the colliding hadrons

hA + h~ -+ h* + h*.

(4)

We consider the contributions to the inelastic cross-section from all possible channels 9 = (cq, c~2)defined by the numbers n~, n2 and types of particles in the final states of the two decaying subsystems h~ and h*. The associated differential cross-section da(s; ~) for c.m. energy s ~/2 and channel 9 is related to the inelastic cross-section a~, by introducing the relative probability dW(s; ~) according to

(5)

da(s; ~) = ai. dW(s; ~t) with the normalization condition

~,~dW(s; ~)= 1.

(6)

~t

to the hadronization stage of the two subsystems, result from the two-step picture of the reaction. The dynamics of the reaction is described by the matrix element 2 A i n ( ~ ) = A~e(t)

[[ Aex(Mj) A~t(aj) Aqs(~j),

which is factorized into the components Aso(t) and Aex(Mj) governing deflection and excitation of the two hadrons and into the factors Ast(aj)Aqs(~j) controlling the decay of the subsystems. In the following, the meaning of the different components is discussed in more detail. To describe the angular distribution of the excited hadrons we use the common approximation [31] Asc(t) ocexp (b t)

dW(s; ~)ocdR.(s; ~) Ain(~)

(7)

is given by the Lorentz-invariant phase-space factor

dR,(s; a)=dR,(s; rn~.... , m,) = 64

Pi

d3pl/2ei

(8)

i

times the square of the reaction matrix element Ain(~). In (8), the phase-space factor depends via s=p 2 on the four-momentum p of the initial state of the reaction as well as on the masses ml of all n=nl +n2 particles in the final state with four-momenta pi = (e~, Pi). The whole phase space is decomposed according to the recursion formula [30] 2

dR.(s; ~t)= dR2(s; M1, ME) I ] dM~ dR.~(Mj; c~j)

(9)

j=l

with the invariant masses Mj of the subsystems h*. This phase-space decomposition is schematically depicted in Fig. 1. The phase-space factor of the two excited hadrons

(11)

known from diffractive processes. The square of the transferred four-momentum is given by t and the slope parameter b depends predominantly on the size of the colliding objects. The expression

Aex (Mj) = (M/O) K~ (M/O)

The probability of populating a certain final channel

(10)

j=l

(12)

with K1 standing for the modified Hankel function is the kernel of the so-called K transformation. It is used to transform a microcanonical phase-space distribution, which depends on the total energy M j, into a canonical one, which is characterized by a temperature O (see [30]). The number of particles involved in the reactions under consideration seems to be too small to justify a canonical treatment. Therefore, we use expression (12) in a microcanonical approach to determine the amount of translational energy converted into internal excitation energy of the colliding hadrons. Since the product of the decreasing function A~x(Mj) with the increasing phasespace factor R,j(Mj, c~j) exhibits a maximum around M3ocnj O, the mean kinetic energy per particle in the rest frame of an excited hadron is proportional to the parameter O. At the same time, the fluctuations of Mj appearing for low particle numbers are exactly taken into consideration. The decay of an excited hadron is assumed to proceed statistically, i.e. the decay probability into channel c9 is proportional to the number of states per energy unit in this channel, which is given by

Z,j(M j ; aj)oCAst(C~j)~dR,j(M j ; c@

(13)

106 where the term nj

Ast(~j)---g(o:j) (V/(27"ch)3) ns-' H (2ai + 1) 2m~ i=1

(14)

contains the spin degeneracy factors (2ai+ 1) and the volume V in which the final hadrons are produced. The quantity g(ej) is the degeneracy factor for identical particles and prevents multiple counting of states. The flavour content of the various decay channels results from the underlying quark statistics. This leads to additional weight factors Aqs(~s), which are important for obtaining the correct relative abundances of the various types of particles produced. We assume that an excited hadron can be viewed as a system consisting of the valence quarks of the incoming hadron and an arbitrary number of quark-antiquark (q~) pairs. As usual, the production of strange q c~pairs is suppressed relative to that of up and down quarks by sampling the flavours of the quarks according to the ratio u: d: s = 1 : 1 : 2 with 2 < 1. The final hadrons are then built up by randomly selecting sequences of quarks. A sequence qg/ (or ~q) is converted into a meson and sequences qqq or 040 into baryons or antibaryons. For this end, the meson nonets with angular momenta zero and one (1So, 3St, 1pz, 3Po' sp0 and the baryons with masses up to about 1.7 GeV are taken into account [32]. A probability distribution Prob (mi)ocexp ( - mJO) suppresses the production of heavier hadrons relative to that of lighter ones in accordance with experimental findings. To calculate the number of states, see (13), masses of resonances are sampled according to Breit-Wigner distributions with parameters taken from [32]. Since we are interested in final states which contain only stable hadrons, the subsequent decay of resonances is simulated using the probabilities for the respective channels compiled in [32]. The final expression for the relative probability of the channel 9 associated with the decay of the subsystems h* and h2* 2

dW(s; ~)ocdR2(s; Mr, Mz) A~o(t) Iq dM~ A~x(Ms) j=l

9Aqs(eS) A~t(c~S)dR,s(M s ; c~j)

(15)

contains four parameters: the suppression factor of strange quarks 2, the slope parameter b introduced in (11), the "temperature" parameter O appearing in (12), and the radius R of the volume V=4zcR3/3 in (14). Since we use 2=0.1~).2 in accordance with Wroblewski [33-1 and a typical value b = 4 GeV-2 as measured in diffractive processes [34], only O and R remain as adjustable parameters. In thermodynamical approaches temperatures of the order of the pion mass are usually employed [21]. We use the correct invariant phase space instead of the noninvariant one as in pure thermodynamical models. This has the immediate effect that the temperatures fitted are considerably larger than the pion mass (see e.g. [35]). At low incidence energies this trend is further increased by taking overall energy conservation into account [29]. The parameter O is especially sensitive to the transverse

momenta Px of final particles, and the Px dependence of the cross-section can be well described (see Fig. 4) using values O ~ 320 MeV. The parameter R, in turn, is particularly sensitive to the multiplicity of final particles. By reproducing the multiplicities measured in the energy region under consideration, we obtain values R = 1.1-1.2 fm, which agree with the general expectation that R should be comparable to the radius of the initial hadron.

2.3. Deuteron-proton interaction The differential cross-section is calculated as a sum of incoherent contributions from reactions (1), (2), and (3)

da(s) = %1 d Wel(s)+ o-inE d Win(s; 0t)+ a d ~ d WcI(s; [J). (16) Here, the relative probabilities dW~l(s), dWin(s; ~t) and dW~l(s; 1ff)are normalized in analogy to (6), whereas the sum of the corresponding integrated cross-sections ael, ain, and a~l is equal to the inelastic d+p cross-section [36] at the energy under consideration. The ratio ffel/ain is taken from pp data, and aol is adopted to reproduce the 8~ spectrum at 9 GeV/c (Sect. 3.2). In the case of quasi-elastic scattering we set

dWel(S)~dR3 (s; rnx, m2, ms)]~12 Ael

(17)

and for quasi-free inelastic interactions we have

dWi.(s; ~t)ozdR,(s; ~t)11~[2Ain(00.

(18)

The phase-space decomposition for these two processes is shown in Fig. 2. There, the spectator (particle number 1 with four-momentum Pl) is assumed to dissociate from the incident deuteron with a momentum distribution described by the term l012 = qo

(q2)12

(19)

in (17) and (18), where q = (qo, q) is the four-momentum of the spectator in the deuteron rest frame and r the Paris DWF [14]. A given spectator momentum q is then transformed into Pl, the momentum in the reference frame chosen for the calculation. The four-momentum of the participant follows from pp = P a - P l with the deuteron four-momentum Pd" We neglect in the following

2 S

M1

:

S* 3

Pt

~

~

n

Pt

a}

12 n= nl+ n 2"1

b}

Fig. 2a, b. Phase-space decompositionfor (a) quasi-elastic and (b)

quasi-free inelasticdp interactions

107 that the participant is off the mass shell (ppZ=t=m 2 with m• being the nucleon mass) and describe the participanttarget subprocess as a N N interaction at c.m. energy s' 1/2 with s' given by S' = (Pt + Pp)2 = (Pt + Pd -- pl)2.

Experiment

(20)

The angular distribution of the elastic N N subprocess is parametrized [37] according to Ael oc exp [b (s'). t'],

Table 1. Moments of the inelastic multiplicity distribution of charged particles. The experimental results of [38] are compared to model calculations with the parameter values O = 320 MeV, R=I.1 fro. 2=0.2, b=4 GeV -2

(21)

where t' and b(s') are the square of the f o u r - m o m e n t u m transferred to the recoil target p r o t o n and the energydependent slope parameter, respectively. To describe the inelastic N N subprocess we use the matrix element Ai,(~) discussed in Sect. 2.2. The relative probabilities of the various channels /~ in the case of an excitation of the whole deuteron, dWcl(s; ~) in (16), are calculated in analogy to the corresponding quantities d W(s; ~) [-see (7)] for hadronic interactions. The n u m b e r of valence quarks is now six, and the corresponding decay channels are denoted by p. Since the parameters b and R are related to the size of the colliding objects, they are scaled with the baryon number B = 2 of the deuteron by the substitutions b ---r B 2/3 b and R ~ B 1/3 R. F o r the temperature parameter we use the value O = 140 MeV adopted earlier [27, 28] to reproduce spectra of particles emitted into the kinematically forbidden region. This O value is lower than in the case of a nucleon and reflects the fact that a nucleon group is not such a strongly bound system as a single nucleon. Moreover, a cluster can decay into its constituents, the nucleons, while the decay of a single excited nucleon is always accompanied by the production of new particles.

3. Comparison with experimental data and discussion

3.1. Particle production in proton-proton collisions In this section we demonstrate the ability of the MOPS model to reproduce the overall features of particle production processes in p r o t o n - p r o t o n interactions. The most comprehensive data set in the energy region of interest was published by the Bonn-Hamburg-Mfinchen collaboration, and we compare our results to the pp data at 12 GeV/c. An important characteristic of particle production processes is the multiplicity distribution of charged particles. It can be seen from Table 1 that the theoretical m o m e n t s of this distribution compare well with the experimental values. Table 2 contains the crosssections for the production of the various particle types, and again we find satisfactory agreement between calculated and measured values. The good overall reproduction of the relative abundances of the various particle types can be considered as a confirmation of the stochastic nature of multiparticle production discussed in Sect. 2.2. Figure 3 shows the dependence of the crosssection on Feynman's variable XF, and Fig. 4 the dependence on the transverse m o m e n t u m PT- The yield of S -

(n~h) D2

f2
3.43 -+0.03 2.05 _+0.03 --1.38 +0.04 0.715-+0.015 0.51 -+0.01 -0.20 +0.02

Calculation 3.44 2.03 --1.40 0.716 0.52 -0.20

Table 2. Integrated single-particle inclusive cross-sections. The experimental results of 1,38] (Tz-, ~o ~+, p, ~), 1-39] (K~ A, 2; +, Z-), 1-40] (K* +, K* ), 1,,413 (Z* +, Z*-), and [42] (pO) as well as values derived from a parametrization of the energy dependence of average multiplicities [43] are compared to model calculations Cross-section values [mb]

~z~0 ~z+ KK~ K+ po K* K *§ p A

ES+ S*Z* +

Experiment

Parametrization

Calculation

21.1 _+0.4 35.2 __+2.4 42.7 +0.7

20.6

20.8 46.5 38.4 O.4 0.8 1.6 3.7

36.6 0.2

0.600 • 0.012 1.3 1.80 -+0.25 + O.02 0.04 -- 0.03 0.27 -+0.03 37.5 • 0.6 1.154• 0.003 + 0.001 - 0.002 0.148 -+0.006 0.655 _ 0.035 0.07 -+0.02 0.20 • 0.03

0.04 0.5 40.4 1.4 0.001 0.1 0.5 0.02 0.25

particles is underestimated, while for the other particles the data are well reproduced at values xv < 0.6 and PT < 1 GeV/c. At higher values of xv and PT there are slight discrepancies. The invariant cross-section of pions is depicted in Figs. 5 and 6 as a function of the rapidity for different values of the transverse m o m e n t u m . While the cross-section of positive pions is underestimated in the central region for transverse m o m e n t a above 0.6 GeV/c, the negative pion data are nearly perfectly reproduced. In Fig. 7 we show the spectra of protons inelastically scattered by a hydrogen target at angles between 5 ~ and 30 ~. The kinetic energy of the b o m b a r d i n g protons is 0.8 GeV, and only pion production is possible. At such a low energy the phase-space factor dominates the calculated spectra, whereas the parameter values are of minor importance. Again the model predictions agree quite well with the data.

108 10 2

T

I

"~

100 ;,

1

p + p - " - a +X 12 GeV/c -~

~

P

~-~ 9

102

I

oo

I

I

I

!

L

>~

I

I

/

p +p-,--~-+x

j o--o-o~

V

A(xS)

'0

u5

Z§ 10-2 ~

~

-

~I_

10-41

_

A

0

0.4

I

I

/

10-2

o.8

I

0

ID

ropidity

XF

Fig. 3. Inclusive cross-section versus XF for p, A, ~+, 2-, and K ~ The data [38] (open and full circles) are compared to model calculations (histograms)

p+p--,-a+X

9

G v/c

/

t

i

I

I

I

p+p---p+X Ep = 0.8 GeV

-

o

102

I

5~ 11~ t0-1}

E

%0

~1o~

3.0

Fig. 6. Invariant cross-section versus rapidity for ~z- at different values of PT. Data from [38]

104 ~. 102F ~

L-.

20

@ 10~

o

!

10-4

0

0.5

1.0

1.5 PT(GeV/c)

Fig. 4. Inclusive cross-section versus PT for p, A, N+, ~-, and K ~ Data from [38] 10211

I

I

I

I

j o-o-o-o_o ~ I ~'~L~

12GeV/c

0.8

I

I I 1.0

i

0.8

9 I

I

1.2 Ptob(GeV/c}

A t incidence energies between the two values considered, experimental data are rather scarce. In this energy region we reproduce average multiplicities of charged particles [44] with parameter values 0 = 3 2 0 MeV, 2 = 0 . 1 , b = 4 G e V -2, and R varying between 1.1 and 1.2 fro. By neglecting this small energy dependence we will describe the N N subprocess in quasi-free inelastic d e u t e r o n - p r o t o n collisions using R = 1.2 fro, independent of the N N c.m. energy.

L_]

t ]0-21 I 0

I

p+p---~+X o

~ ~ o o o

I

0.4

Fig. 7. Inclusive proton spectra versus laboratory momentum. Data from [18]

I

I

g ~

0

I '~ 2.0

t I

3.2. Inclusive deuteron fragmentation 3.0

ropidity Fig. 5. Invariant cross-section versus rapidity for 7c+ at different values of PT. Data from [38]

In two recent notes [9, 45] we have analyzed the 0 ~ [3] and 8 ~ [10] p r o t o n spectra m e a s u r e d at 9 GeV/c. Here, we discuss these spectra in m o r e detail and begin

109 with some improvements of the model. In [45] the 8 ~ proton spectrum was underestimated in the momentum region between 5 and 6 GeV/c. The inclusion of reaction (3), cluster excitation, with o-~ = 3 mb results in a better description of this momentum region (see Fig. 8), while the same process does not essentially contribute to the 0 ~ spectrum, as can be seen from Fig. 9. With o-~ adopted, the values 6~, = 22 mb and a~, = 43 mb are derived from pp and dp data as explained in Sect. 2.3. The cross-sections for quasi-free p r o t o n - p r o t o n and n e u t r o n - p r o t o n interactions are assumed to be equal c#pp_,~,p_ 1/2 o-~j" ~f,~ = crf~ = 1/2 o%). Note that only the protons contribute to the measured yield and that the corresponding factors 1/2 were incorporated into the integrated cross sections used in [9, 45, 46]. Another point concerns the values of the parameters O = 240 MeV and R = 1.0 fm used earlier [9, 45] in contrast to O = 320 MeV and R = 1.2 fm employed in the present paper. The d~fference is caused by the probability distribution P r o b ( m ~ ) o c e x p ( - m i / O ) , see Sect. 2.2, which leads to a stronger suppression of heavier resonances among the primarily produced particles and results in a better description of pp data. In both variants the parameters were adopted to reproduce the multiplicity of charged particles from pp interactions at half the deuteron momentum. In spite of the different parameter values, the results are very similar, as can be seen by comparing Figs. 8 and 9 with the corresponding figures in [45] and [9]. Thus, the peculiarities of particle production seem to be irrelevant to the description of inclusive proton spectra from deuteron fragmentation as long as such a characteristic quantity as the average particle multiplicity is reproduced, because the sum over all channels wipes out finer details. In Fig. 8 the contributions of the various processes to the 8~ spectrum are shown separately. At proton momenta be]ow 4 GeV/c the participants from quasi-free inelastic interactions dominate the spectrum. The peak around 4.3 GeV/c is due to participants arising from quasi-elastic scattering (1). Then the region follows where duster excitation (3) provides the main contribution, and at momenta above 7 GeV/c the spectators from quasifree inelastic and quasi-elastic scattering become decisive. Here, the spectator momenta in the deuteron rest frame reach values up to 1 GeV/c, and it is interesting that our nonrelativistic treatment yields reasonable results. More detailed measurements are necessary to decide whether the good agreement of the calculation with the data in this region is accidental, due to the empirical parametrization of the short-range behaviour of the Paris potential (cf. [47]), or whether it can be considered as a confirmation of the Paris potential at short distances. As we have shown in [45], the whole picture changes drastically when we apply a relativistic D W F [6]. The spectator contributions then become negligible and the spectrum is underestimated above 6 GeV/c. Dolidze and Lykasov [15] employ a six-quark admixture to the D W F in their relativistic approach in order to reproduce the high-momentum part of the spectrum. Azhgirey et al. [10-12] also take only the participant contributions into account and adjust the parameters ~Uel

--

~el

I

I

~

)

J

on

~> 1~

d +p -,--p +X ~

Pd = 9.0 fideV lc

~

v

i0 ~2 - - ~

--

10 - 42O

4.0

5.0

- 8.0

Pl.~b(GeV/c)

Fig. 8. Invariant cross-section versus laboratory momentum. The partial spectra are labelled by numbers referring to quasi-elastic (t), quasi-free inelastic (2), and cluster excitation (3) processes, while the letters a and b denote contributions from spectators and participants, respectively. The curves are calculated with a~=22 mb, ~r~ =43 mb, and a~= 3 rob. Curve 4 represents the sum of all partial spectra. Data from [10]

2.95 I

3.65

4,55

I

~

lo 4

5 65 ~

Ptab (GeV/c) 6,95 I

--

d + p---'-p +X Pd = 9.1 GeVlc 0~

4

10z ---Z 4

2b 10-2

I - O.4

j

I

\ __~

0.4 proton momentum (GeV/c) 0

Fig. 9. Invariant cross-section versus proton momentum in the deuteron rest frame. Curves are labelled as in Fig. 8. Data from [3] of a deuteron structure function [13] to give agreement with the data. Thus, there are contradictory explanations of the origin of high-momentum protons and we have proposed an exclusive experiment [46], whose realization would cast some light upon this problem. A comparison of the 0 ~ (Fig. 9) with the 8 ~ spectrum reveals considerable differences in the angular dependences of the various partial spectra. The process of cluster excitation and the participant contribution from quasi-free inelastic interactions vary weakly with the angle, while the yield of participants from quasi-elastic scattering increases by about one order of magnitude on changing the angle from 8 ~ to 0 ~ In spite of this considerable

110 increase, the quasi-elastic participants can be neglected for the reproduction of the 0 ~ spectrum, because the spectator contributions increase much more strongly, namely by no less than five orders of magnitude, and they become decisive for the explanation of the 0 ~ spectrum. A more detailed measurement of the angular dependence would surely help to make our approach more profound, especially with respect to the cluster excitation process, because the relative importance of this process should increase at larger angles due to its weak angular dependence. The 0 ~ spectrum is dominated by participants from quasi-free inelastic collisions at laboratory momenta below 3.5 GeV/c and by spectators at higher momenta. The latter fact is the reason for the supposed proportionality of the 0 ~ spectrum to the deuteron single-nucleon momentum distribution mentioned in the introduction. Such a proportionality is approximately fulfilled as long as the momentum dependence of the phase-space factors in (17) and (18) is negligible compared to that of the momentum distribution 10l 2. This is the case for spectator momenta far enough from the kinematical limit, while near the limit the phase-space factors sharply decrease and become zero in the limit. According to (20), the maximal spectator momentum is related to the minimal c.m. energy s' 1/2 of the N N subprocess, which is given by the sum of the rest masses in the final state of the N N interaction. Thus, the onset of "phase-space dominance" occurs for each channel at different spectator momenta, depending on the masses of the particles produced, and the spectator reaches the largest momentum in the quasi-elastic channel. So, the calculated crosssection is a complicated superposition of functions, whose momentum dependences differ from channel to channel. In Fig. 9 the cross-sections arising from the quasi-elastic channel and from the sum of all channels with particle production are shown separately. U p to about 0.2 GeV/c the two curves mainly reflect the momentum dependence of ]OJ2. Then the steeper decrease of the inelastic contributions begins, and the different behaviour of the two curves reproduces the shoulder in the spectrum around 0.3 GeV/c. In this connection the behaviour of the cluster excitation process (3) is of special interest. This is our global description of phenomena such as six-quark admixture to the D W F or rescattering of virtual particles, which were discussed by several authors [1, 6-8] as the origin of the shoulder. In our present calculation cluster excitation is unimportant, because it contributes less than 15% to the cross-section in the region of the shoulder. But the data available are insufficient to draw definite conclusions concerning the role of this process, and an exclusive measurement of the channel dp---,ppn would help us to clarify the origin of the shoulder. The 0 ~ spectra measured at kinetic energies 2.1 and 1.25 GeV are shown in Figs. 10 and 11. We observe a picture similar to that at 9 GeV/c, except that the yield of spectators from quasi-free inelastic collisions is considerably smaller. The spectra are well reproduced up to momenta of about 0.2 GeV/c, while the region around 0.3 GeV/c is underestimated. It seems that the shoulders

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So far, we have dealt with protons originating from the fragmentation of the incident deuteron, while the decay products of the target protons, due to their small energy, have not played any role in the momentum regions considered. One way of investigating these reaction products too consists in looking at the inverse reaction p + d ~ p + X , where the high-momentum parts of the spectra at small angles reflect the behaviour of the scattered projectiles. As can be seen in Fig. 12, the spectra of protons inelastically scattered by a deuterium target are well reproduced. F o r the 5~ spectrum the partial spectra are depicted, and we observe the typical peak from quasi-elastic interactions and at lower momenta a broad enhancement arising from quasi-free inelastic interactions, while the contributions from cluster excitation are negligible.

4. Summary

We have developed the modified phase-space model to describe inelastic hadronic and nuclear reactions and have applied it to pp and dp interactions. Hadronic collisions are assumed to proceed via the excitation of the two incident particles. The probability of their subsequent decay into the various final channels is determined by the corresponding statistical weights. This simple picture allows us to describe pp data for a variety of experimental studies up to incidence momenta of about 12 GeV/c. In a quasi-free dp interaction the spectator is assumed to dissociate from the deuteron with its momentum distribution governed by the DWF. The remaining

participant-target collision is treated as a hadronic interaction at the corresponding c.m. energy. Interactions beyond the scope of the impulse approximation with both nucleons of the deuteron involved are described by proposing that the deuteron behaves like a single entity. The application of the MOPS model to the 0 ~ and 8~ proton spectra from dp interactions at 9 GeV/c yields excellent results, and what matters is that the channels with particle production turn out to be decisive for the reproduction of the spectra. Except in the region near the kinematical limit, the 0 ~ spectra measured at 1.25 and 2.1 GeV are also satisfactorily described. Thus, the overall features of inclusive proton spectra from dp reactions can be reproduced by using a conventional nonrelativistic wave function and taking particle production processes into account. In our opinion, the consideration of particle production in dp interactions at high energies is a necessary step before information about such important and interesting questions as, e.g., the existence of a six-quark admixture to the DWF, the use of a structure function instead of a wave function, or the difficult problem of a relativistically exact description of the deuteron structure can be derived from experimental results. There is no doubt that particle production plays an important role, while the answers to the other questions mentioned above are of more speculative character at present. The analysis of experiments like, e.g., T2o analysing power [5, 48] and proposed exclusive [46] measurements is then the next step in disentangling the interplay between the description of deuteron structure and reaction mechanism. The author would like to thank H.W. Barz, J. M6sner, H. Schulz, and R. Wfinsch for valuable comments.

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H. Miiller Zentralinstitut ffir Kernforschung DDR-8051 Rossendorf ii. Dresden German Democratic Republic