Parametrization of the inclusive cross section for $\bar p$ production in p + A collisions and $\bar p$ mean multiplicity distribution

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Parametrization of the inclusive cross section for ¯p

production in p + A collisions and ¯p mean multiplicity


C.Y. Huang, M. Buenerd

To cite this version:

C.Y. Huang, M. Buenerd. Parametrization of the inclusive cross section for ¯p production in p + A



Parametrization of the Inclusive Cross Section



Production in

p+ A

Collisions and


Mean Multiplicity Distribution

Ching-Yuan Huang and Michel Buenerd

Institut des Sciences Nucleaires, CNRS/IN2P3 et Universite Joseph Fourier, 53, Avenue des Martyrs, 38026 Grenoble Cedex, France


The parametrized inclusive triple dierential cross section for antiproton production in proton-nucleus collisions is studied. An energy-dependent term is introduced in the functional form of inclusive cross section previously used. The new parametriza-tion provides a consistent agreement with both the experimental data for laboratory incident energies from 12 to 24 GeV for proton-proton and proton-nucleus collisions and also for the mean p multiplicity distribution in p-p collisions at least up to the center of mass energy of 25 GeV.

PSCS numbers: 13.85.Ni, 25.75.Dw, 13.85.Hd


1 Introduction

Antiprotons are a rare component of cosmic rays (CRs). The origin of cosmic-ray antiprotons has attracted a lot of attention since the rst observation reported by Golden et al. 1]. The data currently available on the antiproton ux have been measured, by means of magnetic spectrometers, by balloon-borne BESS, HEAT, CAPRICE and IMAX 2{5] and are expected to come out also from the space-borne experimentAMSin its test ight in June 1998. Cosmic-ray antiprotons are supposed to be produced, at least dominantly, by the interactions of galactic high-energy CRs with the interstellar medium (ISM) 6]. The measured ux however, is a superposition of antiprotons origi-nating from the galactic production with the ux produced in the atmosphere by primary CRs interacting with atmospheric nuclei. This latter production then constitutes a physical background for the study of the galactic p ux. The measured ux of antiprotons must then be corrected from the local p prouction in order to obtain the galactic antiprotons ux 7]. An important input for evaluating such a correction is the p-production cross section in proton-nucleus and nucleus-nucleus collisions. Unfortunately, the experimen-tal data are scarce in the energy range of interest. In addition, it is found that the available parametrization shows a poor agreement with the available data (see Figure 1-4).

In the past few decades, high quality secondary beams became available at pro-ton synchrotrons with increasing energies (PS, AGS, IHEP, FNAL, SPS) 8], and the studies on the multiplicity distribution of charged hadrons in high-energy hadron-hadron collisions came up to its stage. The multiplicity distri-butions of charged secondaries produced in p-p interactions were measured with high statistics at the CERN Intersecting Storage Rings (ISR), by us-ing the Split-Field Magnet (SFM) detector, by the ABCDHW Collaboration 9,10]. The experimental results indicate that, if the charged particles are pro-duced randomly and independently, their multiplicity distributions could obey a Poisson distribution 11]. The deviations from a Poissonian distribution are then a hint of the underlying production processes, and also a source of infor-mation about the nuclear medium eects.

The aim of this paper is rstly to present a more widely applicable expres-sion for p production in p + A collisions (A: nucleus). The parametrized p-production cross section is studied and then a good agreement is obtained with the existing data 12{15]. Secondly, the new parametrized cross section is applied to calculate the average multiplicitydistribution of antiprotons inpp collisions. A parametrized expression of the mean multiplicity for antiprotons in p-p collisions, based on the experimental charged multiplicity distributions 10], is then presented.


2 Inclusive Nuclear Cross Section

The parametrization of the inclusive cross section considered is based on the Regge phenomenology and (quark counting rules) parton model. The kinemat-ics used in the following to express the inclusive cross section forp+A collisions is based on the assumption that all processes take place in the nucleon-nucleon (NN) system. Nuclear eects are incorporated in a delicated term of the an-alytic expression of the collisions. The minor (for energies above threshold) kinematical eects due to the nuclear Fermi motion of target nucleons have been neglected. The invariant triple dierential cross section for the inclusive production of p on a nucleus A is expressed in terms of the usual kinemati-cal variables, momentum P, mass m, production angles  and , in various combinations and reference frames, as 16]:

 Ed3 d3p ! inv= 12mT d2 ddmT = 12pT d2 ddpT (1) = EP2 d2 ddP = 2PE 2 d2 dPdcos (2)

where pT is the transverse momentum mT =

q p2 T +m2 0 is the transverse mass  = 12lnE + QE ;Q is the rapidity Q = m

Tsinh is the longitudinal

mo-mentum is the Lab angle between the measured inclusive particle and the incident particle direction  is the solid angle. At high energy, the inelas-tic processes dominate hadrons collisions and, for the study on the ux of secondary antiprotons, the spectrum of produced particle number is:


d = ineltot1 dd = totinel1

Z p Tmax 0 2pT  Ed3 d3p ! invdpT (3)

where the total inelastic cross section ineltot for p + A collisions has been


Ek is the incident kinetic energy in MeV, and A is the target mass in amu

unit. 1

3 Parametrized Inclusive Cross Section

A parametrized eective proton-nucleus inclusive cross section for the process p + A! p + X has been proposed in 18] as:

 Ed3 d3p ! inv= tot inelC1A b(pT)(1 ;x) C2exp( ;C 3x)(pT) (7) (pT)=exp(;C 4p 2 T) +C5 exp(;C 6xT) (p2 T +2)4 (8) where b(pT) = 8 > < > : b0pT pT ; b0; pT > ; The (1;x)C

2 form in (7), originates from quark counting rules in hadronic interactions, while exp(;C

3x) is characterised by Regge regime 19] A is the target mass number and its exponent could be viewed as the net-eect of the multiple NN scattering in the target nucleus x = E


max is a scaling

variable, in which, E and E

max are the total energy of the inclusive particle

and its maximum possible energy in the centre of mass frame, respectively pT is the transverse momentumxT = 2ppT

s is the transverse variable. E

max = s;M 2 Xmin+m2 p 2p

s , in which, MXmin = 2mp + mA is the minimum possibleA mass of the recoiling particle in the relevant process (p + N ! p + X in this case). p

s is the invariant mass of the system. 1 In A+A collisions, one could use the form 17]:

totinel= 10r2 0A 1=3 proj+A1=3 target;] 2 (mb) (6)

where Aproj and Atarget are the mass numbers of the projectile particle and the

target respectively r0 = 1:47  = 1:12. More detailed review about the total

inelastic cross section ineltot could be found in 26].


4 Fitting Procedure and Results

The 2 minimization procedure was based on the MINUIT package 20]. The set of experimental data used was from 12{15], corresponding to incident protons with energies of 12.0 GeV, 14.6 GeV/c, 19.2 GeV/c and 24 GeV/c, while the measured transverse momentum range extended up to 1.068 GeV/c. The parameters C1


6 b0 

2, and ; given in 18] were used as starting values for the minimization procedure.

Fitting the expression (7) and (8) of the parametrized inclusive cross section to the individual data, it was noted that the second term in (pT), is of signicant

magnitude for low transverse momentumpT range, at least comparable to the

rst term. It is also investigated that, the second term in  is signicantly similar to the rst term at low pT over the entire range of p

s while in high energy systems, this term is much greater than the rst and will dominate  at high pT range. In this case, 2 cannot be treated as the modication term in  in the original KMN parametrisation. And, in independent searches over individual sets of data, it shows that good ts were obtained for energies E > 12 GeV with only the rst term. In addition, in nuclear collisions, the probability of largepT scattering must be small so that the second is required

for only low energies. Hence,in order to enforce the above points to be satised, an energy-dependent cuto function was introduced to the function  by modifying  to the form as

(pT) = exp(;C 4p 2 T) +C5 exp(;C 6xT) (p2 T +2)4 exp( ; p s) (9)

Combining all the experimental data 12{15] and using the new forms given in relations of (7) and (9), the 2 minimization has been achieved and the best t parameters obtained are given in Table 2 where they are compared to those from 18]. The corresponding ts are shown in gures 1- 4. It can be noted in Fig. 3 that the cross sections are maximum as expected around mid-rapidity (i.e., around xF=0) for both p + p and p + Al collisions. Fig. 3 also provides

evidence that p multiplicity in p + A collisions increases slightly with the target mass and also the centrality, which is consistent with the experimental measurements reported by Abbottet al. 12]. It should be also noted that, by the t shown in Fig. 1, the experimental error bars of Sugaya et al. 14] have to be set to more realistic values in the optimization procedure.

In addition, in order to investigate the continuity properties of the func-tional forms used together with the incident beam energy, the target, etc, the parametrised p production cross section had been checked by plotting as a function of its variables over the whole variable domains. The smoothness pro-vides the evidence that this reparametrisation is valid under the constraints


of the available experiments taken into account 12{15].



Multiplicity Distribution

The p mean multiplicity is calculated ny integrating the parametrised p pro-duction cross section over the whole domaines of its variables. Figure 5 shows the mean multiplicity distribution of antiprotons produced calculated forp+p collision 9] by the new parametrisation in this work. In Figure 5, it has to be noted that the original data (black circles) were measured by removing the single-diractive parts in total cross section so that the corrected ones are given by estimating 10  15% in contribution to its multiplicity 8]. The cor-rected values from the experimental data are also shown in the gure. Hence, after such a correction, the calculated p mean multiplicity distribution has shown a good agreement with the experimental data up top

s = 25 GeV. For p

s  30 GeV, the parametrisation underestimates the p mean multiplicity. The dierence between the calculations and the measurements becomes larger at larger energies but still remains close to the experimental uncertainty of the available experiments.

It has to be noted that, whenp

s is reaching 1 TeV, the calculated pmean mul-tiplicity will turn out to decrease. Such a decrease is not physically expected. This has provided the upper energy limitof the validity of this parametrisation which is beyond the range of interest of this work.

The multiplicity of particles created in high-energy hadron collisions follows a distribution with a long tail, qualitatively similar to the distribution of ion-isation energy loss (Landau distribution). The produced hadron multiplicity is generally described in terms of the Koba-Nielsen-Olesen (KNO) variables 21], i.e., z = N

<N>, the ratio between the multiplicity and the average

mul-tiplicity of the particles produced at the centre of mass energy p

s. However, it is found that the KNO scaling is violated in high energy collisions 11,22]. Therefore, another parametrised hadron-nucleus multiplicity distribution in terms of lnp


The constants in Eqs. (10) and (11) are determined by the tting procedure. Note that a threshold value for the multiplicity was used in the parametrisa-tions (10) and (11) 9,10,24]. Also note that the leading particle contribution has been included in the factor A 23]. In these results, the mean multiplic-ity of each type of charged particle appears in a similar shape to that of the all charged particles. Therefore, the p mean multiplicity distribution in p + p collisions could be described by using the same functional forms shown in Eqs. (10) and (11). The constants for the p mean multiplicity distribution in form of (11) was reported 24] while the constants for the p mean multiplicity distribution in form of (10) was determined in this work, by the best tting. The constants for these two parametrisations are shown in Table 3.

Figure 6 shows the p mean multiplicity distributions in p + p collisions as a function of the centre of mass energy in the whole phase space from four dierent origins: the experiment 10] together with its corrected values, the values calculated by the revised parametrisation of the p production in this work, and the parametrisations from Eqs (10) 9,10,23] and (11) 24]. In this gure, both these two parametrisations of the p mean multiplicitieshave shown a better agreement with the experimental data than the results calculated by this work for the centre of mass energy p

s > 25 GeV. Nevertheless, it has to be indicated that, according the functions used in the parametrisations (10) and (11), these two parametrisations have the following lower limits of the validity: p

s = 6:4 GeV, i.e., the incident nucleon energy E ' 20:89 GeV for the parametrisation (10) p

s = 5:37 GeV, i.e., the incident nucleon energyE '14:43 GeV for the parametrisation (11). However,the energy range below these limits is still important to the p production in nuclear interactions for the present work. On the other hand, as it will be discussed later, for the interest of study on the cosmic p origins, the energy range E > 200 GeV is of less importance. In addition, the previous parametrisations of the mean p multiplicity 9{11,23,24] were obtained by data tting while the mean 

p multiplicity in this work was calculated by integrating the p production cross section. Therefore, the mean p multiplicity obtained in this work is more physical.

6 Conclusions

In this paper, a modied parametrized inclusive cross section for p + A col-lisions has been proposed with a new energy-dependent term to account for the p production cross section in the 10-20 Gev transition region of incident energy. The mean number distribution of antiprotons calculated by this new parametrized cross section is in a good agreement with the measured data with p

s from threshold up to 100 GeV. A parametrized p multiplicity distribution calculated by this parametrization is also given.


This paper explored the reliability of the parametrized inclusive expression (7) with (9) over a wide range of incident energies, from around 10 GeV, up to 100-200 GeV, matching the spectrum of cosmic rays particles.

The determined parameters will then be applied to calculate the ux of the secondary antiprotons. In addition they have been used to evaluate the p production in the galactic disk 25{27]. Their results are in a good agreement with the antiprotons ux measured byBESS 2].

The results obtained in this paper will be subsequently applied to the evalu-ation of the atmospheric production of secondary antiprotons.



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Fig. 1. Antiproton cross sections measured at 5:10in function of antiproton

produc-tion momentum for p+C and p+Al collisions at an incident energy of 12 GeV compared with the experimental data 14] and the KMN parametrisation 18].

Fig. 2. Antiproton invariant spectra in function of antiproton production transverse mass dierencedmT =mT;m

0forp+Alcollision at an incident momentum of 14.6

GeV/c compared with the experimental data 12] and the KMN parametrisation


Fig. 3. Antiproton spectradn=din function of antiproton production rapidityfor

p+pand p+Al collisions at an incident momentum of 19.2 GeV/c compared with the experimental data 13] and the KMN parametrisation 18].

Fig. 4. Antiproton Lorentz invariant density in function of antiproton production momentum forp+Al and p+Becollisions at an incident momentum of 24 GeV/c compared with the experimental data 15]. The measurement angles are 17, 27, 37, 47, 57, 67, 87, 107 and 127 mrad from top to bottom. The density is plotted and multiplied by a power of 10;1, i.e., 100 for 17 mrad, 10;1 for 27 mrad, 10;2 for 37

mrad etc.


Exp. Type Ein=Pin Ppmax (GeV/c) pTpmax (GeV/c)

Allaby et al. 1970 13] p+p 19.2 GeV/c 14.5 0.91 Allaby et al. 1970 13] p+Al 19.2 GeV/c 14.5 0.91 Eichten et al. 1972 15] p+Be 24.0 GeV/c 18.0 1.068 Eichten et al. 1972 15] p+Al 24.0 GeV/c 18.0 1.068 Abbott et al. 1993 12] p+Al 14.6 GeV/c 0.78 Sugaya et al. 1998 14] p+C 12.0 GeV 2.5 0.22 Sugaya et al. 1998 14] p+Al 12.0 GeV 2.5 0.22 Table 1

List of experiments and data used.

parameters Kalinovski(1989) this paper (2001)

C1 0.08 0.042257 C2 8.6 5.9260 C3 2.30 0.9612 C4 4.20 2.1875 C5 2.0 84.344 C6 10.5 10.5 2 1.1 0.092743 b0 0.12 0.12 ; 5.0 5.0 2.2429 2 per point 7.30 0.544 Table 2

Values of the parameters and 2 by Kalinovski18] and this work. Note that the

two values ofC5 cannot be compared directly. See text.


Table 3

Constants for the parametrised pmean multiplicity distributions in p+p collisions described in forms of Eq. (10) 9,10,23] and Eq. (11) 24].


Fig. 5. Antiproton mean multiplicity distribution in the whole phase space calculated by the revised parametrisation in p+pcollision. The results are compared with the experimental data 10].

Fig. 6. Same data as in Figure 5 compared with the values calculated in this work and the parametrisations of Eqs. (10) and (11). See the text for the discussions.




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