Full text of DOI:10.1007/s100500050394

Eur. Phys. J. A 7, 293–298 (2000)
THE EUROPEAN
PHYSICAL JOURNAL A
c
Societa Italiana di Fisica
Springer-Verlag 2000
On the Absorption of antibaryons in nuclei
M. Gonina, O. Hansen
Niels Bohr Institute for Astronomy, Physics and Geophysics, University of Copenhagen, 2100 Copenhagen, Denmark
Received: 15 July 1999 / Revised version: 10 November 1999
Communicated by P. Kienle
Abstract. Antiproton (p) and antilambda ( ) production has been measured for minimum bias in p+A
collisions and central A1+A2 collisions at the CERN-SPS by the collaborations NA35/49 and NA44. The
measurements are extrapolated from rapidity distributions to absolute minimum bias cross sections. It is
shown that the p cross sections divided by A1 A2 follow an exponential trend as a function of a characteristic
length obtained from a Glauber type absorption model, while the cross sections divided by A1 A2 are
constant. The exponential trend also holds for p production at the lower energies of the Brookhaven AGS.
A discussion of the physics interpretation of the established trends in terms of an e ective absorption cross
section is presented.
PACS. 25.75.-q Relativistic heavy-ion collisions – 25.75.Dw Particle and resonance production
1 Introduction
the global trends are established and Sect. 6 deals with
the physics interpretation of the demonstrated systemat-
ics and in particular with the problem of the use of a
Glauber type recipe which is not clearly a priori justi -
able.
The production of anti-baryons in high energy p+A and
A1+A2 collisions is of interest because mechanisms di er-
ent from those in p+p collisions may come into play and
because absorption in the hot and dense nuclear medium is
of importance (see e.g. [1–3]). This paper presents an anal-
ysis of empirical p+A→ p( ) and A1+A2 → p( ) cross
sections with the purpose of establishing global trends
as function of (A1,A2). The inspiration comes from the
successful demonstration of such global trends for the
p+A→J/ and A1+A2 →J/ cross sections in terms of
a simple Glauber type absorption model [4,5]. In the J/
case, with one exception, all the measured minimum bias
total reduced cross sections follow a single exponential
trend as function of a characteristic length, L, derived
from the absorption model; the term ”reduced cross sec-
tion” stands for the cross section divided by A1 A2. The
exception from this trend in the J/ case occurs for non-
peripheral Pb+Pb collisions.
2 Calculation of L
The absorption lengths, L used in the analysis of J/
production are derived from a Glauber type statistical ab-
sorption model, see e.g. [6,7], for p+A processes. In the
model the projectile and the created rare particle move
forward in straight line trajectories. The rare particle (X)
is created at a well de ned location z’ along the incom-
ing projectile direction inside the target nucleus and un-
dergoes absorption on its forward trajectory out through
the target nucleus with an absorption cross section . The
probability distribution of the creation location z’ through
the nucleus is assumed to be uniform.
It is demonstrated in the present paper that the p re-
duced cross sections also follow exponentials as a function
of L, while the reduced cross sections are constant with
L. It is further shown that this implies a simple depen-
dence with A1,A2.
In Sect. 2 the recipe for deriving the characteristic
length L from the absorption model is described and the
A1,A2 dependence of L is established. Section 3 presents
the empirical results as well as the somewhat intricate
procedure for extrapolating the data to a uniform body
of total minimum bias cross sections. In Sects. 4 and 5
After integration over z’ the formulae for total mini-
mum bias p+A cross sections [7] are
(p + A → X)
= exp( LpA/ ) =
A (p + p → X)
Z
1
db(1 exp( LpA(b)/ )), (1)
(A 1)
where
Z
∞
a
Also at Ecole Polytechnique, L.P.N.H.E., 91128 Palaiseau,
LpA(b)/ = ∞ dz(A 1) (b, z)
(2)
France

294
M. Gonin, O. Hansen: On the Absorption of antibaryons in nuclei
and
The L-values thus to good accuracy have a simple mass
dependence, except for the lightest system.
4
3
1/ =
=
/( r03).
(3)
L-values can also be calculated for cross sections corre-
sponding to collisions of nite centrality. In such cases (4)
is not valid, i.e. the additivity of the p+A L-values does
not hold. Also the integrations have higher dimensions
and the simple mass dependence of (8) will not hold. In
order to preserve the simplicity of the model for total min-
imum bias cross sections, we have chosen to extrapolate
the data, rather than to calculate L-values corresponding
to the experimental centrality conditions.
0
b is the transversal position of the incoming proton rela-
tive to the center of nucleus A, expressed by the impact
parameter b =| b | and the azimuthal angle; z is the coor-
dinate along the the incoming direction and the integra-
tions are over all space and (b, z) is the nuclear density
distribution normalized to unity when integrated over all
space. A is the baryon number of the target nucleus and
X is absorbed with a cross section , which is assumed
independent of b and z. The r0 is chosen to be 1.1198 fm,
so that 0=0.17 nucleons per fm3.
3 Data extrapolations and systematic
uncertainties
For A1+A2 collisions the absorption length L satis es
[7]
The data consist mainly of multiplicities per unit rapidity
(dn/dy) near midrapidity and they are quoted in tables
1, 2 and 3 in the next two sections. In many cases part
of a dn/dy distribution has been measured, in some cases
there is only a value at midrapidity. The p+A data have
in all cases been obtained with a trigger condition that is
close to a minimum bias condition, whereas all the heavy
ion data are from central triggers. The experimental re-
sults must therefore be extrapolated in rapidity and from
a centrality condition to a minimum bias condition in or-
der to be compared to the absorption model as explained
above. The experimental cross sections are evaluated by
(A1 + A2 → X)
=
A1A2 (p + p → X)
exp( LA A / ) = exp( (LpA + LpA )/ ),
(4)
1
2
1
2
where (A1 + A2 → X)/(A1A2) is the reduced cross sec-
tion. The density distribution (b, z) is either of Saxon-
Woods shape or is a uniform spherical density distribu-
tion. The Saxon-Woods shape is
1
VA
1
√
(b, z) =
,
(5)
1 + exp(( b2 + z2 RA)/a)
(A1 + A2 → X) = (dn/dy)F1(y)F2(b)F3
,
(9)
inel
with VA denoting the volume of nucleus A, a=0.51 fm
and RA is adjusted to give an average density of 0.17
where
stands for the total geometrical cross section
inel
in fm2 parametrized as
4
3
nucleons/fm3 (or equivalently VA
=
r03A). If the uni-
form distribution is used the density is also xed at 0.17
nucleons/fm3. L
is calculated from the above for-
A1+A2
= 6.86(A1/3 + A12/3 1.32)2.
(10)
inel
1
mulae by numerical integration for a chosen value of
.
The extrapolations F1 and F2 have been calculated by
means of the event generator RQMD [8]; F3 is used only
for AGS data where decays may contribute to the p yield
(see Sect. 5). The choice of the RQMD event generator was
made because the model generally gives a good acount of
measured dn/dy-distributons and of the dependence on
centrality. Any other model with similar characteristics
could have been used.
For → 0 an expansion of the righthand part of (1)
in LpA
gives
/
to rst order and in LpA(b)/ to second order
Z
0
L( → 0) =
dbL2(b).
(6)
2(A 1)
The L-value becomes independent of
uniform spherical distribution
and for the
The F1 correction is calculated as the ratio between
the RQMD total multiplicity and the RQMD midrapid-
ity dn/dy value for particle X. The shapes of the RQMD
dn/dy distributions agree well with the observed shapes
for the rapidity intervals covered by experiment and the
correction, by construction, is independent of the absolute
RQMD multiplicity. The experimental trigger condition
was simulated by a cut on the impact parameter b, such
that the RQMD ratio between central cross section and
inelastic cross section equalled the experimentally quoted
ratio, typically near 5%, 7% or 10% for heavy ion col-
lisions. The F2 extrapolation is the ratio of the RQMD
integrated dn/dy for particle X in the minimum bias cut
and in the central cut on impact parameters.
3 A
4
1
L0HS
=
R,
(7)
A
where R is the radius of the spherical distribution R =
1
3
r0A .
Table 1 lists a selection of L-values calculated for each
A1,A2 combination both for a homogeneous spherical den-
sity distribution in the limit of no absorption and for
a Saxon-Woods distribution with a nite absorption of
=2.14 fm2. With the exception of the p+Be case, the
ratio L0HS/L
is constant to better than 1%, so one can
2.14
SW
write with this accuracy, by use of (4) and (7),
1
1
A1
1
A2
1
2.14
SW
The uncertainties given in the tables in the column for
the reduced cross sections only re ect the experimental
L
= 0.515(
A13 +
A23 ).
(8)
A1
A2

M. Gonin, O. Hansen: On the Absorption of antibaryons in nuclei
295
Table 1. Antiproton results from CERN-SPS
2.14
SW
System L0HS
L
y
dn/dy
F1
F2
/(A1A2)
Ref.
inel
p+S
p+Au
S+S
S+Ag
S+Au
Pb+Pb
p+Be
p+S
p+Pb
S+S
S+Pb
Pb+Pb
2.58
4.86
5.17
6.55
7.45
9.90
1.55
2.58
4.95
5.17
7.54
9.90
1.60
2.96
3.20
4.08
4.56
6.01
0.84
1.60
3.01
3.20
4.60
6.01
56 3.0 0.032
208 3.0 0.046
174 3.5 0.40
301 3.5 0.60
405 3.5 0.70
763 3.0 2.40
21 2.7 0.045
56 2.7 0.047
216 2.7 0.062
174 2.7 0.65
416 2.7 0.92
763 2.7 1.80
2.60 1.00 0.146 0.015
2.60 1.00 0.126 0.014
3.10 0.36 0.076 0.019
3.10 0.37 0.060 0.021
3.10 0.39 0.055 0.016
2.60 0.34 0.037 0.008
2.80 1.00 0.294 0.070
2.80 1.00 0.230 0.060
2.80 1.00 0.180 0.040
2.80 0.36 0.112 0.022
2.80 0.39 0.063 0.012
2.80 0.34 0.031 0.010
[11]
[11]
[11]
[11]
[11]
[12]
[13]
[13]
[13]
[13]
[13]
[14]
Lengths are in fm and cross sections in fm2. The absorption lengths L are de ned
in (4), and the factors F in (6). The Pb+Pb dn/dy has not been rescaled to a beam
momentum of 200 GeV/c.
uncertainties on the dn/dy measurements. The extrapo-
lations F1 and F2 both carry statistical and systematic
uncertainties. The statistical uncertainties stem from the
nite number of events calculated and are kept small com-
pared to the uncertainties of the measurements. The sys-
tematic uncertainties on F1 originate from RQMD pre-
dicted dn/dy versus y distributions that do not agree with
the true dn/dy distributions. In the four cases for mea-
surements, where part of a dn/dy distribution has been
measured, the RQMD predictions do agree quite well with
the data. Uncertainties on the order 10% are a realistic
estimate for the absolute value for F1. Note that the ab-
solute multiplicity values are not relevant for this study,
only the relative changes from one (A1,A2) combination to
another are. The correction factor F2 is due in rst order
to the geometry of the collision as can be seen from the ta-
bles and therefore small uncertainties are expected. Some
experimental checks for F2 are possible for p using the
BNL-AGS data. The E-802 collaboration has reported for
Si+Al, Si+Au [9] and Au+Au [10] dn/dy values for both
central and minimum bias triggers. From these data we
found the following values for F2: 0.45
0.14 for Si+Al,
0.42 0.10 for Si+Au and 0.46 0.10 for Au+Au. These
values can be compared with the RQMD F2 values listed
in the tables: 0.35 for Si+Al, 0.37 for Si+Au and 0.40 for
Au+Au. As can be seen the agreement is reasonable and
the fact that the measured values are systematically higher
than the RQMD values suggest that the experimental min-
imum bias data may not correspond to exactly 100% of
Fig. 1. Reduced cross sections for p measured at CERN-
SPS. The line represents the simultaneous t to the data. The
Pb+Pb data obtained at 158 GeV are rescaled to 200 GeV by
1.06
but rather to some large fraction of
because of
inel
inel
geometry (5) was used, while HS stands for a homogeneous
spherical density distribution with a sharp edge. The re-
sults are collected in Table 2. Figures 1 and 2 show the
threshold e ects in the detectors.
reduced cross sections plotted versus LA A . The L-scale
4 CERN-SPS results
2
depends on the absorption cross section, 1and the scale for
The A1+A2 → p data are collected in Table 1 together p production is the result of an iterative process. The cri-
with references to the appropriate experiments. The ab- terion for the iteration was that the cross section assumed
sorption lengths L carry a superscript that gives the ab- for L agrees with the slope of the exponential through the
sorption cross section used in the integrations (1) in units data. The results in Fig. 2 do not exhibit a slope and
of fm2. The subscript SW indicates that the Saxon-Woods the L-values were calculated for =0.

296
M. Gonin, O. Hansen: On the Absorption of antibaryons in nuclei
Table 2. Antilambda results from CERN-SPS
System L0SW
y
dn/dy
F1
F2
/(A1A2)
Ref.
inel
p+S
p+Au
S+S
S+Au
Pb+Pb
1.99
4.40
3.99
6.40
8.99
56 1.75-4.75 0.015
208 1.25-4.75 0.015
174 1.25-4.75 0.91
405 3.25-4.75 0.92
2.50 1.00 0.066 0.013
2.50 1.00 0.040 0.013
2.50 0.30 0.116 0.032
3.20 0.33 0.062 0.012
2.50 0.30 0.066 0.014
[11]
[11]
[11]
[11]
[15]
763
3.0
5.00
Units and de nitions are as in Table 1. Pb+Pb was not rescaled.
conditions fall below the global exponential trend. For J/
the deviation from the global exponential increases with
increasing centrality; it is unfortunately not possible to
analyse p production for Pb+Pb as a function of centrality
due to the lack of data.
As can be seen from Fig. 2, the reduced cross sections
are constant as a function of L. The value for obtained
from the t is consistent with zero.
Finally, from p+Be up to Pb+Pb collisions for p and
from p+S up to Pb+Pb collisions for , the results can
be summarized as
(A1 + A2 → p) ∝ (A1 A2) exp( 0LA +A
)
(11)
1
2
and
(A1 + A2 → ) ∝ (A1 A2)
(12)
where the result of (12) is in fact model independent.
5 BNL-AGS results
Fig. 2. Reduced cross sections for measured at CERN-SPS.
The line represents an exponential t to the data. The Pb+Pb
data obtained at 158 GeV are rescaled to 200 GeV by 1.07
The A1+A2 → p data obtained at the lower beam momen-
tum (14.6 GeV/c) are collected in Table 3 together with
references to the appropriate experiments. Relevant re-
sults concerning are not available at AGS energies. The
p data obtained at the AGS are contaminated by antipro-
tons from the decay of . An estimate [18] for the E802
experiment shows that about 50% of the antiprotons from
-decays are identi ed as antiprotons. In order to correct
The reduced p cross sections from NA44 and NA35/49
data do not agree in absolute values, but they exhibit
nearly identical slopes with L. The data from NA44 as
presented in Fig. 1 were (arbitrarily) rescaled by 0.85. The
for this contamination, the
rates were estimated from
slope value from the t of Fig. 1 is = 21.4
2 mb and
the p+p result of /p = 0.20 0.04 and the p+p→ p yield
measured at 19 GeV/c [17]. The cross section was rst
extrapolated from 19 GeV/c to the AGS heavy ion mo-
mentum of 14.6 GeV/c with the RQMD model, an extrap-
olation that agrees with the empirical p+p systematics
[19]. Next it was assumed that (12) holds at AGS energies
for production and a cross section could then be con-
structed for each A1,A2 combination and an F3 correction
could be deduced. The F3 corrections from this procedure
are shown in Table 3.
is in agreement with separate ts to the data from each
experiment.
The beam momentum per nucleon for Pb is lower than
for protons and S. The Pb beam data in Fig. 1 (but not
in the table) have been corrected upwards by 6% to cor-
respond to the S and p beam conditions. The correction
was derived from the RQMD model. No correction was
applied for the results in Fig. 2 for Pb because of the
large statistical uncertainties on the data.
The most conspicuous result of the present analysis
The F3 corrections leads to a /p ratio near one for
is that all reduced cross sections for p production fall on Au+Au minimum bias collisions in qualitative agreement
a single exponential curve ( 2/dof = 1.07). The second with the measured values for central Au+Pb reactions
qualitative result is the lack of a break in the exponential [20]. Also Cole et al. measured F3 for central Si+Au to be
for Pb+Pb→ p, in contrast with the observation for J/
0.55 0.15 [21] as compared to the minimum bias value
production [5], where the Pb+Pb result for minimum bias from Table 3 of 0.66.

M. Gonin, O. Hansen: On the Absorption of antibaryons in nuclei
297
Table 3. Antiproton results from BNL-AGS
2.48
SW
System
L0HS
L
y
dn/dy
F1
F2
F3
/(A1A2) Ref.
inel
p+Be
p+Al
1.55
2.42
3.30
4.86
4.87
7.31
9.72
1.55
2.42
3.30
4.94
0.83
1.44
2.06
2.92
2.93
4.41
5.84
0.83
1.44
2.06
2.96
21 1.4 0.00038 2.0 1.00 0.86
49 1.4 0.00047 2.0 1.00 0.85
93 1.4 0.00049 2.0 1.00 0.82
208 1.4 0.00049 2.0 1.00 0.76
15.2 4.8
14.5 4.5
11.6 4.5
7.9 2.5
11.6 2.5
4.9 1.0
1.2 0.3
60.4 6.0
43.8 5.2
36.6 4.1
18.1 1.9
[16]
[16]
[16]
[16]
[9]
p+Cu
p+Au
Si+Al
Si+Au
Au+Au
p+Be
p+Al
153 1.4 0.0100
390 1.4 0.0142
727 1.4 0.0150
2.0 0.35 0.82
2.0 0.37 0.66
2.0 0.40 0.53
[9]
[10]
[17]
[17]
[17]
[17]
21 2.5 0.00074 3.5 1.00 1.00
49 2.5 0.00069 3.5 1.00 1.00
93 2.5 0.00072 3.5 1.00 1.00
214 2.5 0.00050 3.5 1.00 1.00
p+Cu
p+Pb
Units and de nitions are as in Table 1, except that all reduced cross sections have been
multiplied by 104. Note that the data from [15] are at 19.2 GeV/c and the Au+Au is at
11.2 GeV/c.
6 Discussion
The preceding analysis demonstrates that the reduced
cross sections for p production in p+A and A1+A2 colli-
sions follow exponentials as function of the characteristic
length LA A calculated from a Glauber type absorption
1
2
model, (4). As mentioned in the introduction and elabo-
rated on below, it is not a priori justi able to use the ab-
sorption model. The discussion presented here, deals with
this question and further with what physics, if any, one
can learn from the established systematics.
The ts to the exponentials may just re ect a general
mass dependence, as given by (8), (11) and (12), i.e. the
ts are not connected to the physics of the absorption
model and may in this connection be taken as fortuitous.
The data presented here can in fact be tted to other
functional mass dependences, e.g. the SPS p follow to a
similar accuracy the power law expression
(A1 + A2 → p) ∝ (A1 A2)
with =0.78 0.02.
(13)
The connection between the power law dependence
and the Glauber-type absorption description has been dis-
cussed in some detail by Capella et al. [6], and it may be
concluded from their work that the power of 0.78 found
here makes a Glauber approach very doubtful. The empir-
ical systematics, however, are very compelling and a brief
examination of the physics conditions that must necessar-
ily be ful lled if a Glauber absorption model should be
applicable is therefore presented.
Fig. 3. Reduced cross sections for p measured at BNL-AGS.
The Au+Au data obtained at 11.2 GeV are rescaled to 14.6
GeV by 2.4 while the data obtained at 19.2 GeV (Allaby et
al.) are rescaled to 14..6 GeV by 1/2.7
All AGS data are presented in Fig. 3 along with p+A
The simplest condition of the absorption model (see
data obtained at 19.2 GeV/c at CERN. Data in the gure the beginning of Sect. 2) is that particle X is a rarely pro-
at beam momenta di erent from 14.6 GeV/c have been duced particle, because the model does not allow multiple
extrapolated to this beam momentum by means of the production. The very small p+p production cross sections
RQMD model. As can be seen from the gure, the re- for p and [19] shows that this condition is well ful lled
duced cross sections for p fall on a single exponential curve at AGS and SPS energies.
(
2/dof = 0.86) and the tted value for the slope corre-
In order for the concept of absorption to be realistic,
sponds to = 24.8
3
5mb, where the external error the particle X must be produced within the nuclear vol-
corresponds to a systematic uncertainty of 50% on the ume and afterwards traverse part of it, in other words, the
F3-values.
formation time should be small compared to the nuclear

298
M. Gonin, O. Hansen: On the Absorption of antibaryons in nuclei
linear dimension. For hard processes the formation time global mass dependence demonstrated certainly provides
may be estimated by
0.2/ E in fm/c and the energy a rather stringent test for the various microscopic models
in GeV, which for production of a p, p pair in the nuclear in common use, e.g. RQMD, VENUS [25] and ARC [26].
rest system is 1fm/c as compared to 0.6 fm/c for J/ .
The hard processes then ful l the condition on formation
The authors acknowledge the help of Aa. Winther and J. Bon-
time, but at the AGS and SPS energies these processes
dorf with the evaluation of the absorption lengths LA
.
+A
2
1
are dominated by rst collisions and are hence not evenly
distributed through the nuclear volume, but peaked at the
contact surfaces (see e.g [1,2,22]).
There may be other mechanisms than elementary hard
collisions that are important for the production of p in
heavy ion collisions, namely excitation of heavy resonances
and the formation of di-strings or colour ropes (see e.g. [1,
2,22] and references cited there). In fact such ”enhance-
ment” mechanisms dominate the production cross sections
in the RQMD model. The formation times after the cre-
ation of the intermediate excitation are short, 1-1.5 fm/c
in the nuclear system, so the antibaryon is formed well
within the nuclear volume and undergoes absorption on
its way out. The various mechanisms may well conspire to
produce a nearly evenly distributed set of formation loca-
tions as required for the validity of the absorption model.
The absorption model of (4) relates the production
cross section to the elementary N+N (nucleon+nucleon)
processes, in which the ”enhancement” mechanisms in-
voked above, do not come into play. The absorption model,
deals with elementary collisions between nucleons in heavy
ion collisions, but not with the ”enhancement” processes
used by e.g. the RQMD model. Even if the absorption
model is not valid for the processes studied here, the va-
lidity of (4) for the (A1,A2) dependence may come about
if the ”enhancement” mechanisms are proportional (or
roughly proportional) to A1 A2. In that case (4) contains
the right ingredients, but the cross sections are not related
to the elementary N+N cross sections. The good ts to the
power law pointed out above supports the idea that extra
mechanisms may be nearly proportional to A1 A2.
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19. A. Wroblewski, in Proceedings of the Int. Symp. on Multi-
particle Dynamics, Kayserberg 1977; A. Wroblewski, Acta
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20. M.J. Bennett et al., the E878 Collaboration, Phys. Rev. C
56, 1521 (1997); T.A. Armstrong et al., the E864 Collab-
oration, nucl-ex/9709005 v2 (1997)
If this proposition is accepted, the observed exponen-
tial slopes represent e ective absorptions and not straight-
forward absorptions. The p absorption cross sections of 21
to 25 mb are much smaller than the free p + p annihila-
tion cross sections (see e.g. [23,24]). This may then re ect
a combination of ”enhanced” production and absorption,
and should not necessarily be interpreted as a quenching
of the elementary absorption mechanisms. The small ef-
fective absorption found in the case then does not repre-
sent the absence of elementary absorption processes, such
as
+
→ K +N, but rather a combination of absorption
and enhanced production, that here happens to combine
to an almost vanishing e ective absorption.
21. B.A. Cole et al., the E802 Collaboration, Nucl. Phys. A
590, 196c (1995)
In summary, the reduced cross sections for antipro-
ton and antilambda production in proton-nucleus and
nucleus-nucleus minimum bias collisions follow the mass
dependence predicted by a simple Glauber type absorp-
tion model. The model is strictly speaking not applicable,
and the agreement between data and model may be fortu-
itous or may come about because the contributing produc-
tion mechanisms are nearly proportional to A1 A2. The
22. A. Jahns et al., Zeitschrift f. Physik, A 341, 243 (1992);
Phys. Rev. Lett., 68, 2895(1992); Phys. Lett., B 308, 1993
(1993)
23. V. Flaminio et al., CERN-HERA 8-01, 1984 (unpublished)
24. S. Gavin et al., Phys. Lett. B 234, 175 (1990)
25. K. Werner, Z. Phys. C 42, 85 (1989)
26. S. Kahana et al., Phys. Rev. C 47, R1356 (1993), and
references therein