The isobar model and the resonance model are applied for the first analysis
of the subthreshold $K^+$-meson production in proton-carbon collisions, which
was perfomed at GSI at an emission angle of 40 degrees and a bombarding energy
of 1.2 GeV. In this study, we focus on the role of the secondary processes $\pi
N \to K^+ Y$ in normal nuclear matter density. It turns out that the present
approach can reproduce very well both the $\pi^+$- and $K^+$- meson spectra. It
is also demonstrated that the different kinds of descriptions for the $\pi N
\to K^+ \Lambda$ reactions substantially differentiate the calculated results
for the $p A \to K^+ X$ differential cross sections.Comment: Latex 6 pages with 4 figures (eps file

Faessler, Amand

Sibirtsev, A.

Tsushima, K.

English

arXiv

The isobar model and the resonance model are applied for the first analysis of the subthreshold K^+-meson production in proton-carbon collisions, which was perfomed at GSI at an emission angle of 40 degrees and a bombarding energy of 1.2 GeV. In this study, we focus on the role of the secondary processes \pi N \rightarrow K^+ Y in normal nuclear matter density. It turns out that the present approach can reproduce very well both the \pi^+- and K^+- meson spectra. It is also demonstrated that the different kinds of descriptions for the \pi N \rightarrow K^+ \Lambda reactions substantially differentiate the calculated results for the p A \rightarrow K^+ X differential cross sections

Sibirtsev, A A

Tsushima, K

Faessler, A

CERN Document Server

K+ production in p− C-collisions at a beam
energy 1.2 GeV
A. Sibirtsev, K. Tsushimayand Amand Faessler
Institut fu¨r Theoretische Physik, Universita¨t Tu¨bingen,
Auf der Morgenstelle 14, D-72076 Tu¨bingen, Germany
August 30, 1995
Abstract
The isobar model and the resonance model are applied for the rst analysis
of the subthreshold K+-meson production in proton-carbon collisions, which
was perfomed at GSI at an emission angle of 40 degrees and a bombarding en-
ergy of 1.2 GeV. In this study, we focus on the role of the secondary processes
N ! K+Y in normal nuclear matter density. It turns out that the present
approach can reproduce very well both the +- and K+- meson spectra. It is
also demonstrated that the dierent kinds of descriptions for the N ! K+
reactions substantially dierentiate the calculated results for the pA! K+X
dierential cross sections.
It has been proposed for more than one decade that K+-mesons are one of the
most promising probes for hot and dense nuclear matter formed in heavy ion colli-
sions [1]. However, in refs. [2, 3, 4, 5] a quite dierent scenario for kaon production in
heavy ion collisions was suggested that kaons may be produced mainly through sec-
ondary processes in the pure hadronic phase with nuclear matter density (0:17fm−3)
obtained in nucleus-nucleus collisions, which is far away from the highly compressed
and hot phase.
In order to clarify these arguments, the KaoS collaboration of GSI have mea-
sured recently the spectra of +- and K+- mesons in proton-nucleus collisions at
an angle of 40 degrees and for a beam energy of 1.2 GeV [6]. It was found that
the K+-dierential cross sections are substantially underestimated according to the
calculations based on only the rst step processes pN ! K+N , even together with
an inclusion of the high momentum component of the nuclear spectral function [7].
Thus, it has been claimed that the kaons might be produced by either the secondary
−N- or the multi-nucleon collisions.
One of the purposes of the present study is to analyse those data theoretically
for the rst time, with particulary focusing on the role of the secondary process
Permanent address: Laboratory of Nuclear Problems, Joint Institute of Nuclear Research,
Dubna, 141980, Russia and Institute of Theoretical and Experimental Physics, Moscow, Russia.
yAddress since October: Department of Physics and Mathematical Physics, The University of
Adelaide, Adelaide 5005 Australia.
1
N ! KY in the normal hadronic phase. We will compare our results with the
GSI data for the spectra of K+-mesons as well as those for +-mesons, which will
turn out to show the validity of the present approach.
For the calculations of the +-meson spectra at a beam energy of 1.2 GeV,
we adopt the isobar model with the assumptions that the s-wave -production
is dominant for the pN ! N reaction, and that the ’s decay isotropically in
the -isobar rest frame. at the -isobar rest system. As was demonstrated by
Cugnon et al. [8], the isobar model can reproduce the experimental data on pion
production reasonably well for beam energies below 2 GeV/u, both in proton-nucleus
and nucleus-nucleus collisions. We use the -production cross section as,
(pN ! N) =
Z
A
in(pp) +
A− Z
A
in(pn); (1)
where A and Z denote the target mass and charge, respectively. The proton-
proton and proton-neutron inelastic cross sections in(pp) and in(pn) are taken
from ref. [9].
In order to take into account the internal momentum of the target, we average
over the -dierential cross section by convoluting the spectral function (q) of the
target nucleus. Then, the dierential pion production cross section in proton-nucleus
collisions can be calculated as
E
d3
d3p
(pA! X) = Neff (p)
Z
(q) E
d3(
p
s;q)
d3p
Lo!N (
p
s; p) dq; (2)
where Ed3(
p
s;q)=d3p stands for the -dierential cross section in the center-of-
mass system of the incident proton and the target nucleus carrying the momentum
q and the corresponding invariant mass
p
s.
In eq. (2) Lo!N (
p
s; p) stands for the Lorentz transformation of the pion
momentum p from the -rest system to the laboratory system.
Here, the spectral function of the target carbon (q) necessary for the present
calculations is parametrized by the experimental data (e; e0C) and (γ; γ0C) scatter-
ings as follows [10];
(q) =
1
3
exp
 
−
q2
22
!
; (3)
where q = j~qj and the slope parameter  =82 MeV=c. It is assumed that the
distribution of the vector q is isotropic.
The factor Neff (p) appearing in eq. (2) is accounting for the mass number
A dependence of the  production and the nal stage of the pion absorption. It
should be mentioned that the energy dependence of the eective collision number
Neff (p) must be taken into account correctly, because there is a strong dependence
of the N cross sections (N) on the produced pion momentum p [9], which can
be understood clearly from Fig. 1a). Based on the approach used in refs. [11, 12],
this Neff (p) can be calculated as
Neff (p) =
Z +1
0
bdb
Z +1
−1
(b; z)dz
Z 2
0
d
exp

−tot(pN)
Z z
−1
(b; )d − tot(N; p)
Z +1
0
(r[])d

;(4)
2
where (r[]) is the one-particle density distribution which is taken as a harmonic
oscillator [13] and normalized to the target mass number A. r[] appearing in eq.(4)
is dened as
r[] = r0(b; 0; z) +  e^; (5)
where b and z stand for the impact parameter and the z-component of the coordinate
along the beam-axis, respectively. e^ is a unit vector in coordinate space dened by
(sincos; sinsin; cos) with  being the detection angle. In order to obtain the
collision number Neff for the other processes, one may just replace the tot(N) in
eq.(4) to the corresponding cross sections.
The eective collision number (4) calculated for the carbon target is shown in Fig.
1b) as a function of total cross section . Note that in the calculations performed in
refs. [3, 5], the factor Neff was obtained by the Glauber approach, and parametrized
by
Neff = A
0:740:01: (6)
Because the total cross section for the K+N-interaction is very small (KN) ’
12mb, one can treat it pertubatively and can get the eective collision number Neff
for (KN) from the curve plotted in Fig. 1b). Similar argument holds also for
K+-meson production through the secondary N-collisions, because the reaction
threshold corresponds to the pion momenta is above 1 GeV/c and (N) is also
small.
However, for the calculations of the +-meson spectra in proton-nucelus and
nucleus-nucleus collisions one should take into acount the pion momentum p depen-
dence of Neff , since the spectra reflect directly the contributions of pion absorption.
At rst we will discuss the results for +-meson production. The solid line in
Fig. 2a) shows the calculated spectrum for +-mesons in proton-carbon collisions
obtained by using eq. (2). The experimental results are represented by the circles.
The dashed line shows the results obtained by the phase-space calculations for the
N ! NN reactions and averaged over by convoluting the spectral function (eq.
(3)) and normalized to Neff . Here the + production cross section is given by
VerWest and Arndt [14] as follows;
(pN ! NN+) =
Z
A
(pp! pn+) +
A− Z
A
(pn! nn+): (7)
One can see that the isobar model ts the +-meson spectrum quite satisfactory.
Because the high momentum tail of the pion spectrum is reproduced pretty well and
it implies that sucient amount of high momentum pions exist, thus one can expect
that K+-meson production through the N ! K+Y reactions is quite possible.
Next, we will discuss about K+-meson poduction. The dierential kaon produc-
tion cross section in proton-nucleus collisions is given by
EK
d3
d3pK
(pA! K+X) =
Z
(p)
tot(pN)
"
E
d3
d3p
#
(q) EK
d3(
p
s)
d3pK
dp
E
dq; (8)
where the dierential pion production cross section Ed3=d3p is taken the same
form as that is given by eq. (2), and EKd
3=d3pK stands for the dierential kaon
production cross sections for the N ! K+Y reactions in the center-of-mass of the
pion and the nucleon with the corresponding invariant mass
p
s of the system. Here,
3
we may take into account only the -hyperon production reaction channel, which is
reasonable for the small values of
p
s.
In eq. (8) the factor (p) stands for the probability of the -meson interaction
with nucleons inside the target, and it is given by [11]
(p) = 1−Neff (p)=Neff ((N) = 0): (9)
It was found in ref. [15] that the angular distribution of K+-mesons induced
by the N ! K+ reactions is not isotropical at all even at energies close to
the threshold. This does not have a large influence on the calculated results for the
total kaon production cross sections [3, 5], however, this anisotropic eects should be
taken into account adequately when the calculations for the dierential K+-meson
spectrum are performed.
The angular spectra of K+-mesons induced by the reaction 0p ! K+ in the
center-of-mass system are given in Fig. 3. The results represented by the circles,
squares, triangles and stars are obtained by adopting the resonance model [16, 17],
while the solid-, dashed-, dotted- and dashed-dotted- lines represent the results
obtained by the use of the parametrization of refs. [15, 18] for the range of
p
s from
1.62 to 1.7 GeV . The explicite parametrization is given by,
d
dΩ
 1 + (
p
s)cosK ; (10)
with (
p
s) = 10:9
p
s − 17:6 and  = 1, corresponding to the ranges threshold
<
p
s < 1.7 GeV and
p
s > 1.7 GeV , respectively.
It is worth noting that the resonance model shows a strong anisotropy for K+-
meson production, whereas the function represented by eq. (10) is quite flat as a
function of cos with  being the angle between the kaon and the pion momenta.
Now we discuss the results for K+-mesons. The results are given in Fig. 2b)
for the kaon spectra in proton-carbon collisions at a kaon angle of 40 degrees and
a proton bombarding energy of 1.2 GeV by applying eq. (8). The solid line shows
the results obtained by adopting the resonance model, while the dashed line shows
the results obtained with eq. (10). The experimental data are given by the circles.
The calculations preformed by adopting the resonance model can reproduce the
experimental data quite satisfactory. This fact indicates that the importance of the
precise description for the N ! K reactions. Furthermore, this leads support to
the assumption in the present calculations, that K+-mesons may be produced in a
pure hadronic phase under the normal conditions.
To summarize, we have performed the rst theoretical analysis for K+-meson
production in proton-carbon collisions which is measured at GSI [6]. In the calcula-
tions, the isobar model and the resonance model were adopted with normal nuclear
matter conditions. By taking into account the experimental spectral function for
the carbon target, we could reproduce the experimental spectra quite well for both
the +- and K+- mesons. It was demonstrated that the secondary N-collision
processes give dominant contributions for K+-meson production in proton-nucleus
collisions below the reaction threshold in free space as was discussed by Cassing
et al. [19]. The present results support the scenario that the K+-mesons may be
produced mainly in the pure hadronic phase under the normal conditions through
the secondary processes [2, 3, 4, 5] N ! KY . Furthermore, it turned out that the
4
angular distribution of the K+-meson spectrum in proton-nucleus collisions is quite
sensitive to the N ! K dierential cross sections, and can serve to dierentiate
between the two models. We therefore urge the experimental group at GSI to mea-
sure such an angular distribution of the kaons. For this purpose we give in gure 4
the angular distribution of the kaons integrated from the kaon momentum pK = 0:5
to 0:7 Gev/c for the reaction 12C(p;K+) at 1.2 GeV proton bombarding energy.
Acknowledgement The authors express their sincere thanks to E. Grosse, P. Sen-
ger and M. Debowski for allowing us to use the experimental data prior to their
publication.
References
[1] S. Schnetzer et al. Phys. Rev. Lett. 49 (1982) 989; S. Nagamiya, Phys. Rev.
Lett. 49 (1982) 1383; D. Miskowiec et al. (KaoS collaboration), Phys. Rev. Lett.
72 (1994) 3767.
[2] J. Kapusta and A. Mekjian, Phys. Rev. D33 (1986) 1304.
[3] W. Cassing et al., Phys. Rep. 188 (1990) 363.
[4] J. Cleymans and H. Satz, Z. Phys. C57 (1993) 135.
[5] A. Sibirtsev and M. Bu¨scher, Z. Phys. A347 (1994) 191.
[6] P. Senger, private communication.
[7] M. Debowski, Ph.D. Thesis.
[8] J. Cugnon, D. Kinet and J. Vandermeulen, Nucl. Phys. A379 (1982) 553.
[9] J. Cugnon and M.-C. Lemaire, Nucl. Phys. A489 (1988) 781.
[10] S. Frullani and J. Mougey, Adnv. Nucl. Phys. 14 (1984) 1.
[11] B. Margolis, Phys. Lett. B26 (1968) 524.
[12] E. Vercellin et al., Nuovo Cim. A106 (1993) 861.
[13] R.J. Glauber and J. Mathiae, Nucl. Phys. B21 (1970) 135.
[14] B.J. VerWest and R.A. Arndt, Phys. Rev. C25 (1982) 1979.
[15] J. Cugnon and R.M. Lombard, Nucl. Phys. A422 (1984) 635.
[16] K. Tsushima, S. W. Huang and Amand Faessler, Phys. Lett. B337 (1994) 245.
[17] K. Tsushima, S. W. Huang and Amand Faessler, J. Phys. G21 (1995) 33.
[18] A. Sibirtsev, Sov. J. Nucl. Phys. 55 (1992) 145.
[19] W. Cassing et al., Phys. Lett. B238 (1990) 25; Cassing Phys. Scr. 48 (1993)
194.
5
Figure captions
Figure 1: Total +p cross section as a function of pion momentum p a), and
the calculated eective collision number for 12C as a function of total cross section
 b).
Figure 2: The spectra of +- and K+- mesons in proton-carbon collisions at an
angle of 40 degrees and a beam energy of 1.2 GeV. The circles in Figs. stand for
the experimental data. The solid- and the dashed- lines in Fig. a) show the results
obtained by the isobar model and the phase space distributions, respectively. The
solid- and the dashed- lines in Fig. b) represent the results obtained by adopting
the resonance model [16] and eq. (10), respectively.
Figure 3: The angular spectra of K+-mesons for the 0p ! K+ reaction. The
dierent curves are plotted as a function of the total center of mass energy
p
s.
They represent the following theoretical results: The circles and the solid line give
the results for
p
s =1.62 GeV, the squares and the dashed line for
p
s =1.64 GeV,
the triangles and the dotted line for
p
s =1.66 GeV, and the stars and the dashed-
dotted line for
p
s =1.70 GeV, respectively. Here, the circles, squares, triangles and
stars show the results obtained by applying the resonance model, while the solid-,
dashed-, dotted- and dashed-dotted- lines show the results obtained with eq. (10).
Figure 4: The angular distribution of kaons with the experimental data for the
12C(p;K+) reaction at 1.2 GeV proton bombarding energy. The results shown in
gure are obtained by integrating the kaon momentum from pK = 0:5 to 0:7 GeV/c.
The solid- and the dashed- lines show the results obtained by adopting the resonance
model [16] and eq. (10), respectively.
6


K^+ production in p-C-collisions at a beam energy 1.2 GeV

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