Page 1
arXiv:1004.1889v1 [hepex] 12 Apr 2010
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH
CERNPHEP/2009029
November 22, 2009
Inclusive production of charged kaons in p+p collisions
at 158 GeV/c beam momentum and a new evaluation
of the energy dependence of kaon production
up to collider energies
T. Anticic12, B. Baatar5, J. Bartke4, L. Betev6, H. Białkowska11, B. Boimska11, J. Bracinik1,a,
V. Cerny1, O. Chvala8,b, J. Dolejsi8, V. Eckardt7, H.G. Fischer6, Z. Fodor3, E. Gładysz4,
K. Kadija12, A. Karev6, V. Kolesnikov5, M. Kowalski4, M. Kreps1,c, M. Makariev10,
A. Malakhov5, M. Mateev9, G. Melkumov5, A. Rybicki4, N. Schmitz7, P. Seyboth7, T. Susa12,
P. Szymanski11, V. Trubnikov11, D. Varga2, G. Vesztergombi3, S. Wenig6,1)
(The NA49 Collaboration)
1Comenius University, Bratislava, Slovakia
2E¨ otv¨ os Lor´ and University, Budapest, Hungary
3KFKI Research Institute for Particle and Nuclear Physics, Budapest, Hungary
4H. Niewodnicza´ nski Institute of Nuclear Physics, Polish Academy of Sciences, Cracow,
Poland
5Joint Institute for Nuclear Research, Dubna, Russia
6CERN, Geneva, Switzerland
7MaxPlanckInstitut f¨ ur Physik, Munich, Germany
8Charles University, Faculty of Mathematics and Physics, Institute of Particle and Nuclear
Physics, Prague, Czech Republic
9Atomic Physics Department, Sofia University St. Kliment Ohridski, Sofia, Bulgaria
10Institute for Nuclear Research and Nuclear Energy, BAS, Sofia, Bulgaria
11Institute for Nuclear Studies, Warsaw, Poland
12Rudjer Boskovic Institute, Zagreb, Croatia
anow at School of Physics and Astronomy, University of Birmingham, Birmingham, UK
bnow at UC Riverside, Riverside, CA, USA
cnow at Institut fur Experimentelle Kernphysik, Karlsruhe, DE
to be published in EPJC
1)Corresponding author: Siegfried.Wenig@cern.ch
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Abstract
New data on the production of charged kaons in p+p interactions are presented. The data
come from a sample of 4.8 million inelastic events obtained with the NA49 detector at
the CERN SPS at 158 GeV/c beam momentum. The kaons are identified by energy loss
in a large TPC tracking system. Inclusive invariant cross sections are obtained in inter
vals from 0 to 1.7 GeV/cin transverse momentum and from 0to 0.5 in Feynman x. Using
these data as a reference, a new evaluation of the energy dependence of kaon production,
including neutral kaons, is conducted over a range from 3 GeV to p+p collider energies.
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1 Introduction
Following the detailed investigation of inclusive pion [1] and baryon [2] production in
p+p interactions,thepresent paperconcentrates on thestudyofcharged kaons. It thuscompletes
a series of publications aimed at the exploration of final state hadrons in p+p collisions by using
a new set of high precision data from the NA49 detector at the CERN SPS [3]. The data have
been obtained at a beam momentum of 158 GeV/c corresponding to a centerofmass system
(cms) energy of 17.2 GeV. This matches the highest momentum per nucleon obtainable with
lead beams at the SPS, permitting the direct comparison of elementary and nuclear reactions.
In addition, the chosen cms energy marks, concerning kaon production, the transition from
thresholddominated effects with strong sdependences to the more gentle approach to higher
energies where scaling concepts become worth investigating. On the other hand the character
istic differences between K+and K−production which are directly related to the underlying
production mechanisms, as for instance associate kaon+hyperon versus K+K pair production,
are still well developed at SPS energy. They are manifest in the strong evolution of the K+/K−
ratio as a function of the kinematic variables. One of the aims of this paper is in addition the
attempt to put the availableresults from other experimentsinto perspectivewith the present data
in order to come to a quantitative evaluation of the experimental situation.
A critical assessment of the complete sdependence of kaon production seems the more
indicated as its evolution in heavy ion interactions, especially in relation to pions, is promul
gated since about two decades as a signature of ”new” physics by the creation of a deconfined
state of matter in these interactions. As all claims of this nature have to rely completely on
a comparison with elementary collisions, the detailed study of the behaviour of kaon produc
tion in p+p reactions from threshold up to RHIC and collider energies should be regarded as
a necessity in particular as the last global evaluation of this type dates back by more than 30
years [4]. A complete coverage of phase space, as far as a comparison of different experiments
is concerned, is made possible in this paper, as compared to pions [1] and baryons [2], by the
fact that there is no concern about feeddown corrections from weak hyperon decays, with the
exception of Ω decay which is negligible for all practical purposes.
This paper is arranged in the same fashion as the preceding publications [1,2]. A sum
mary of the phase space coverage of the available data from other experiments in Sect. 2 is
followed by a short presentation of the NA49 experiment, its acceptance coverage and the cor
responding binning scheme in Sect. 3. Section 4 gives details on the particle identification via
energy loss measurement as they are specific to the problem of kaon yield extraction. The eval
uation of the inclusive cross sections and of the necessary corrections is described in Sect. 5,
followed by the data presentation including a detailed data interpolation scheme in Sect. 6.
K+/K−, K/π and K/baryon ratios are presented in Sect. 7. A first step of data comparison with
data in the SPS/Fermilab energy range is taken in Sect. 8. Section 9 deals with the data inte
grated over transverse momentum and the total measured kaon yields. The data comparison is
extended, in a second step, over the range from√s ∼ 3 to ISR, RHIC and p+p collider energies
in Sect. 10. Section 11 concentrates on an evaluation of K0
and on a discussion of total kaon multiplicities as a function of√s. A comment on the influ
ence of resonance decay on the observed patterns of pTand s dependence is given in Sect. 12.
In Sect. 13 a global overview of charged and neutral kaon yields as they result from the study
of sdependence in this paper is presented, both for the pTintegrated invariant yields at xF= 0
and for the total kaon multiplicities. A summary of results and conclusion is given in Sect. 14.
Syields in relation to charged kaons
1
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2The experimental situation
Thispaper considersthedoubledifferentialinclusivecross sectionsofidentified charged
kaons,
d2σ
dxFdp2
T
,
(1)
as a function of the phase space variables defined as transverse momentum pT and reduced
longitudinal momentum
xF=
pL
√s/2
(2)
where pLdenotes the longitudinal momentum component in the cms.
If the phase space coverage of the existing data has been shown to be incomplete and
partially incompatible for pion and baryon production in the preceding publications [1,2], the
situation is even more unsatisfactory for charged kaons. A wide range of data covering essen
tially the complete energy range from kaon threshold via the PS and AGS up to the ISR and
RHIC energy has been considered here. One advantage concerning the data comparison for
kaons is the absence of feeddown from weak decays with the exception of Ω−decay which can
be safely neglected at least up to ISR energies. An overview of the available data sets is given
in Fig. 1 for K+and Fig. 2 for K−in the xF/pTplane.
00.20.4
x
0.60.8
[12]
[13]
[14]
d)
+
K
c)
+
K
[11]
b)
[7,8]
[9]
[10]
+
K
0
2
0.5
1
1.5
2
[5]
[6]
a)
+
K
00.20.4
x
0.60.8
g)
+
KNA49
00.20.4
x
0.60.8
f)
+
K
[2325]
[26,27]
[2830]
00.20.4
x
0.60.8
0
0.5
1
1.5
e)
+
K
[1619]
[20]
[21]
[22]
[GeV/c]
p
T
[GeV/c]
p
T
FFF
F
Figure 1: Phase space coverage of the existing K+data: a) Cosmotron/PPA [5,6], b) PS/AGS
[7–10], c)Serpukhov[11], d)SPS/Fermilab [12–14], e)ISR [16–22],f)RHIC [23–30], g)NA49
Thesubpanelsa)throughg)showsuccessivelytheenergyrangesoftheCosmotron/PPA
[5,6], PS/AGS [7–10], Serpukhov [11], SPS/Fermilab [12–14], ISR [15–22] and RHIC [23–30]
accelerators in comparison to the new data from NA49. The scarcity of data in the important
intermediate energy range around√s ∼ 10 GeV and the general lack of coverage in the low
pTand lowxFregions are clearly visible. The coverage of the NA49 data, Figs. 1g and 2g, is
2
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00.20.4
x
0.60.8
[12]
[13]
[14]
d)

K
c)

K
[11]
b)
[7,8]
[9]
[10]

K
0
2
0.5
1
1.5
2
a)

K
00.20.4
x
0.60.8
g)

KNA49
00.20.4
x
0.60.8
f)

K
[2325]
[26,27]
[2830]
00.20.4
x
0.60.8
0
0.5
1
1.5
e)

K
[1517]
[20]
[21]
[22]
[GeV/c]
p
T
[GeV/c]
p
T
FFF
F
Figure 2: Phase space coverage of the existing K−data: a) Cosmotron/PPA, b) PS/AGS [7–10],
c) Serpukhov [11], d) SPS/Fermilab [12–14], e) ISR [15–17,20–22], f) RHIC [23–30], g) NA49
essentially only limited by counting statistics towards high pT and by limitations concerning
particle identification towards high xF, in particular for K+, see Sect. 4 below.
The task of establishing data consistency over the wide range of energies considered
here is a particularly ardent one for kaons, as will be shown in the data comparison, see Sects. 8
and 10 below. This concerns especially any attempt at establishing total integrated yields where
the existing efforts evidently suffer from a gross underestimation of systematic errors. Their
relation to the total yields of K0
to SPS/Fermilab energies as well as their eventual comparison with strangeness production in
nuclear collisions should therefore be critically reconsidered.
Swhich are established with considerably higher reliability up
3The NA49 experiment, acceptance coverage and binning
The basic features of the NA49 detectors have been described in detail in [1–3]. The top
view shown in Fig. 3 recalls the main components.
The beam is a secondary hadron beam produced by 450 GeV/c primary protons imping
ing on a 10 cm long Be target. It is defined by a CEDAR Cerenkov counter, several scintillation
counters (S1, S2, V0) and a set of high precision proportional chambers (BPD13). The hydro
gen target is placed in front of two superconducting Magnets (VTX1 and VTX2). Four large
volumeTimeProjectionChambers (VTPC1 and VTPC2 insidethemagneticfields, MTPCLand
MTPCR downstream of the magnets) provide for charged particle tracking and identification.
A smaller Time Projection Chamber (GTPC) placed between the two magnets together with
two Multiwire Proportional Chambers (VPC1 and VPC2) in forward direction allows tracking
in the high momentum region through the gaps between the principal track detectors. A Ring
Calorimeter (RCal) closes the detector setup 18 m downstream of the target.
The phase space region accessible to kaon detection is essentially only limited by the
availablenumberof4.6 M inelasticevents.It spans arange oftransversemomentabetween 0.05
and 1.7 GeV/c for K+and K−and Feynman xFbetween 0 and 0.5 for K−. For K+a limitation
3
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Target
VTPC 1VTPC 2
MTPC R
MTPC L
GTPC
13.05 m
x
1 m
z
1.8 m
S4
VPC1
VPC2
RCal
55 m
34 m
10 m
4 m
Target
BPD1 BPD2BPD3
VTX 1VTX 2
S1
S2V0 S4
CEDAR
erenkovC
Beam and trigger definition elements
Figure 3: NA49 detector layout and real tracks of a typical mean multiplicity p+p event. The
open circles are the points registered in the TPC’s, the dotted lines are the interpolated trajec
tories between the track segments and the extrapolations to the event vertex in the LH2target.
The beam and trigger definition counters are presented in the inset
to xF≤ 0.4 is imposed by the constraints on particle identification discussed in Sect. 4 below.
These kinematical regions are subdivided into bins in the xF/pTplane which vary ac
cording to the measured particle yields, effects of finite bin widths being corrected for in the
evaluation of the inclusive cross sections (Sect. 5). The resulting binning schemes are shown in
Fig. 4 also indicating different ranges of the corresponding statistical errors.
[GeV/c]
p
T
F
x
00.10.20.30.40.50.6
< 3 %
310 %
> 10 %
b)

K
F
x
00.10.20.30.40.50.6
0
0.5
1
1.5
2
< 3 %
310 %
> 10 %
a)
+
K
Figure 4: Binning schemes in xFand pTfor a) K+and b) K−together with information on the
statistical errors
4
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4 Particle identification
The identification of kaons by their ionization energy loss in the gas of the TPC detector
system meets with specific problems if compared to pion [1] and baryon [2] selection. This
specificity has several reasons:
– Corresponding to the momentum range of the NA49 data the ionization energy loss has
to be determined in the region of the relativisticrise of the energy deposit, with the kaon
energy loss positioned in between the one for baryons and for pions.
– The relative distance in dE/dx between the different particle species is small and varies
from only 4.5 to 7% for kaons with respect to protons and from 6.5 to 14% with respect
to pions, over the xF range of the present data, with an rms width of the energy loss
distributions of typically 3%. This creates an appreciable overlap problem over most of
the phase space investigated.
– High precision in the determination of the absolute position of the mean truncated en
ergy loss per particle species and of the corresponding widths is therefore mandatory.
– The relative production yield of kaons is generally small as compared to pions, with
K/π ratios on the level of 5–30% for K+and 5–20% for K−. In addition, for K+the fast
decrease of the K+/p ratio from typically 1 at xF= 0 to less than 5% at xF= 0.4 finally
imposes a limit on the applicability of dE/dx identification towards high xFvalues.
This general situation may be visualized by looking at a couple of typical dE/dx distri
butions for different xFregions as shown in Fig. 5.
Entries
Entries
dE/dx [MIP]dE/dx [MIP]
π

Kp
= 0.4 GeV/c
= 0.05
F
x
T
p
b)
0
5000
10000
15000
20000
+
π
+
Kp
= 0.4 GeV/c
= 0.05
F
x
T
p
a)
11.52
π

Kp
= 0.4 GeV/c
= 0.25
F
x
T
p
d)
11.52
0
1000
2000
3000
4000
+
π
+
K
p
= 0.4 GeV/c
= 0.25
F
x
T
p
c)
Figure 5: dE/dx distributions for K+and K−bins at xF= 0.05, pT= 0.4 GeV/c and xF= 0.25,
pT= 0.4 GeV/c superimposed with results of the fitted distributions
As already described in [2] a considerable effort has been invested into the improved
control of the analog response of the detector. Several aspects and results of this work, in partic
ular as far as kaon identification is concerned, will be discussed in the following subsections.
5
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4.1 NonGaussian shape of the dE/dx distributions
Due to the small K/π and K/p ratios mentioned above, the precise description of the
tails of the energy loss distributions of the dominant particle species becomes important. The
extraction of kaon yields becomes indeed sensitive to small deviations in the upper tail of the
proton and in the lower tail of the pion distributions for the extreme yield ratios mentioned
above,asisalso apparentfromtheexamplesshowninFig. 5. Eventualasymmetrieswithrespect
to the generally assumed Gaussian shape of the energy loss distributions have therefore to be
carefully investigated as they will influence both the fitted central position and the extracted
yields of the kaons. A detailed study of the shape of the dE/dx distributions has therefore been
performed both experimentally and by analytical calculation.
By selecting long tracks in the NA49 TPC system which pass both through the VTPC
and the MTPC detectors one may use the energy deposit in one of the TPC’s to sharply select
a specific particle type of high yield, for instance pions or protons. The dE/dx deposit in the
other TPC will then allow a precise shape determination. An example is shown in Fig. 6 for the
selection of pions at xF= 0.02 and pT= 0.3 GeV/c in the VTPC. The corresponding distribution
of the truncated mean for 90 samples in the MTPC is presented in Fig. 6a together with a
Gaussian fit.
dE/dx [MIP]
1.21.3 1.41.5
Data/Gaussian
0
0.5
1
1.5
2
b)
= 0.02
F
x
= 0.2 GeV/c
= 0.3 GeV/c
T
p
p
T
dE/dx [MIP]
1.21.31.4 1.5
Entries
10
2
10
3
10
4
10
a)
= 0.02
F
x
= 0.3 GeV/c
T
p
data
pion fit
Figure 6: a) Conventional Gaussian fit of the MTPC dE/dx distribution, for tracks with pion
selection using the VTPC dE/dx; b) Ratio of data and fit function
The small but very evident skewness of the truncated energy loss distribution is ex
pressed in Fig. 6b by the ratio of the experimental data to a Gaussian fit. This ratio may be
described by a cubic polynomial form with one normalization parameter Z, shown as the full
line in Fig. 6b.
(Data)/(Gaussian) ≈ 1 + Z(g3− 3g),
(3)
where g is the distance from the mean of the dE/dx distribution, normalized to the rms of the
Gaussian fit,
g =1
σ
??dE
dx
?
−
?dE
dx
??
.
(4)
6
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The parameter Z is related to the number of measured points, Np, on each track, and the
central dE/dx value by the relation
Z = z0N−β
p
?dE
dx
?γ
,
(5)
with β and γ experimentally determined to 0.5 [1] and 0.4±0.2, respectively. Together with the
relation:
σ
(dE/dx)= σ0N−β
p
?dE
dx
?α
,
(6)
assuming α = γ which is a safe assumption regarding the sizeable error in the determination of
γ, z0is obtained as
z0= 0.215 ± 0.02
z0= 0.21 ± 0.02
for the VTPC
for the MTPC.
(7)
A Monte Carlo simulation based on the Photon Absorption Ionization (PAI) model [31]
confirmed these results, demonstrating that the shape distortion is indeed a remnant of the basi
cally asymmetric Landau distribution of ionization energy loss.
4.2 Position and width of the energy loss distributions
Particle identificationproceeds, in each defined bin ofphase space, via aχ2optimization
procedure between the measured energy loss distributions and four single particle dE/dx dis
tributions of known shape but a priori unknown positions and widths for electrons, pions, kaons
and protons, respectively. Due to the generally small fraction of electrons and their position in
the density plateau of the energy loss function, and due to the known dependence of the dE/dx
resolution on the dE/dx value for each particle species [1], (Eq. 6), the problem reduces in
practice to the determination of eight quantities: three absolute positions of the energy loss of
π, K, p, one width parameter and four yield values which correspond to the particle cross sec
tions to be determined. If the fit of the predominant particle species like pions and protons in
general presents no problems, the situation is more critical for the kaons. Here it is in principle
the central kaon position and the overall rms width of the dE/dx distributions which are liable
to create systematic yield variations. In the ideal case, the detector response should reproduce
exact scaling in the p/m variable as implied by the BetheBloch function of ionization energy
loss (BB), with p the lab momentum and m the particle mass. As shown in [1–3] this scaling is
fulfilled for pions and protons in the NA49 detector on the subpercent level. The precision of
the dE/dx fitting procedure allows for a quantification of the remnant deviations δ with respect
to the BetheBloch parametrization as a function of xFand pT
δ(xF,pT) =dE
dx(xF,pT) − BB
(8)
in units of minimumionization (MIP), where dE/dx is the mean truncated energy loss [1]. This
is presented in Fig. 7 for the mean deviation of π+and protons.
The observed deviations are due to residual errors in the calibration of the detector re
sponse and in the transformation between the BetheBloch parametrizations of the different
gases used in the VTPC and MTPC detectors [3]. They stay in general below the level of
±0.005. The fitted shifts of the kaon position, as characterized by their difference to the pion
7
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[GeV/c]
T
p
0 0.51 1.5
〉
,p
+
π
δ 〈
0.005
0
0.005
F
x
0.0
0.025
0.05
0.1
0.3
0.15
0.2
0.25
0.4
Figure 7: Mean deviation ?δπ+,p?, in units of minimumionization, of π+and proton dE/dx with
respect to the BetheBloch parametrization as a function of pTfor different values of xF
position δK− δπ, are shown in Fig. 8 as a function of xFand averaged over pT, the error bars
representing the rms deviation of the averages.
F
x
0 0.20.4
〉
π
δ 

δ 〈
K
0.005
0
0.005
0.01
b)

K
F
x
0 0.2 0.4
〉
+
π
δ 
+
K
δ 〈
0.005
0
0.005
0.01
a)
+
K
Figure 8: Mean deviations in units of minimum ionization of a) K+and b) K−with respect to
the pion position ?δK± − δπ±? as a function of xF, averaged over pT
Evidently the measured positions fall well within the margin of ±0.005 in units of min
imum ionization as obtained for pions and protons. The similarity, within errors, between the
results for K+and K−indicates systematic detector response effects as the principle source of
the measured deviations.
The fitted rms widths of the dE/dx distributions, characterized by their relative devia
tion from the calculated expectation value (Eq. 6 above), are shown in Fig. 9 as a function of
xF, after averaging over pT.
The results show that the predicted widths are reproduced with an accuracy within a
few percent of the expected values, with a slight systematic upwards trend as a function of xF
closely similar for K+and K−.
8
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F
x
0 0.2 0.4
rel
σ
0.98
1
1.02
1.04
+
K
K

Figure 9: Relative rms width σrelas a function of xFfor K+and K−, averaged over pT
4.3 Estimation of systematic errors
The dependence of the fitted kaon yields on the four parameters mentioned above,
namely the positions of pions, kaons, protons, and the relative rms width of the fits, has been
studied in detail. It appears that only two of these parameters are liable to produce noticeable
systematic effects. These are the kaon position and the rms width. By enforcing a range of fixed
values of these parameters, their influence on the extraction of kaon yields may be obtained.
This is demonstrated in Fig. 10 for the dependence on kaon position and in Fig. 11 for the de
pendence on the relative rms width, the error bars in each plot indicating the rms size of the pT
dependence.
F
x
0 0.1 0.2 0.30.4
slope of the yield variation [%/0.001]
3
2
1
0
1
+
K
K

Figure 10: Slope of the yield variation given in % per assumed kaon shift of 0.001 for K+and
K−as a function of xF, averaged over pT
Several aspects of this study are noteworthy:
– As far as the influence of the kaon position uncertainty is concerned, and taking into
account the size of the measured deviations from pions and protons and their rms fluc
tuation (see Fig. 8) the related errors stay on the level of less than 1% up to xF= 0.2.
Above this value the K+yield reacts very critically on the fitted position. This is related
to the proton yield which becomes rapidly overwhelming towards high xF.
– Concerning the rms width the situation is somewhat more critical especially for K+.
Here, allowing for a systematic error of about 0.5% in the fitted relative rms, Fig. 9,
9
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F
x
0 0.1 0.20.3 0.4
slope of the yield variation [%/0.005]
10
8
6
4
2
0
+
K

K
Figure 11: Slope of the yield variation given in % per assumed change of σrelof 0.005 for K+
and K−as a function of xF, averaged over pT
the corresponding yield error reaches values of about 2% at xF= 0.2 and about 10% at
xF= 0.4. This is again measuring the influence of the large proton fraction. For K−on
the other hand, the systematic error stays below the 2% level for the whole xFregion
investigated.
The systematic errors estimated here have been included in the error estimation in Ta
ble 1.
4.4Fit stability and xFlimit for kaon yield extraction
The fitting procedure described above results in stable values for all eight parameters
involved for xFvalues below about 0.25 for K+and below 0.3 for K−. This is to be understood
in the sense that the χ2optimization procedure converges to a welldefined minimum in all
variables with reasonable values for the ratio of χ2over the degrees of freedom. For higher xF
values the fits tend to become unstable in the sense that certain variables tend to ”run away”
into unphysical configurations. In the present case of extraction of kaon yields this concerns
basically only the kaon position in the dE/dx variable and the rms width parameter of the
energy loss distributions, as the pion and proton positions are always well constrained even in
the critical regions of phase space. The problem is of course connected to the high sensitivity of
the extracted kaon yield on these two parameters in relation to the small K/π and K/p ratios as
discussed in the preceding section.
As the evolution of both the kaon position and the rms width with the phase space
variablesxFand pTshowsno indicationof any rapid variationup to thelimitsoffitting stability,
and as indeed the geometrical configuration of the tracks in the TPC detectors shows a smooth
and slow dependence on the track momenta in the regions concerned, it has been decided to
extend the xF range up to 0.4 for K+and to 0.5 for K−by imposing constraints on the two
criticalparameters.Thisisrealizedbyconstrainingthekaonpositiontofixedvalueswithrespect
to the pions, as indicated by the extrapolated lines in Fig. 8, and by also fixing the rms widths to
the values following from the smooth extrapolation indicated in Fig. 9. The expected statistical
error margins, allowing for reasonable values for the uncertainties in the quantities concerned,
see Figs. 10 and 11, have been added in quadrature to the statistical errors.
10
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4.5 Estimation of statistical errors
It has been shownin[2]that theestimationofthestatisticalerroroftheextractedparticle
yields has to take into account the dependence of the fit result on all parameters fitted via the
covariance matrix. This means that the inverse square root of the predicted numbers of each
particle species is only a first approximation to the relative statistical error. The fluctuations
of the fitted particle positions discussed above and their contributions to the error of the yield
parameters are intercorrelated with the particle ratios and with the relative distances of the
energy deposits in the dE/dx variable. The method outlined in [2] has been applied to all
extracted kaon yields and results in the statistical errors quoted in the data tables, Sect. 6 below.
The ratio Rstatbetween the full statistical error and the inverse square root of the extracted
yields is a sensitive indicator of the fluctuations inherent in the fitting method itself. It can vary
drastically over phase space according to the correlation with the particle ratios and the relative
positions with respect to the BetheBloch function. This is visible in the distributions of the
ratio Rstatdefined above and shown in Fig. 12 for K+and K−in two different regions of xF.
Entries
stat
R
1 1.52 2.5
0
5
10
< 0.2
mean = 1.39
F
x a)
+
K
0.25
≤
F
x
≤
0.2
mean = 2.11
stat
R
1 1.52 2.5
0
5
10
15
< 0.2
mean = 1.17
F
x b)

K
0.25
≤
F
x
≤
0.2
mean = 1.49
Figure 12: Rstat= σstat/(1/√N) for the bins xF< 0.2 (solid line) and 0.2 ≤ xF≥ 0.25 (dashed
line); a) K+and b) K−
Rstatis in general bigger for K+than for K−due to the large p/p ratio. In both cases the
forward bins in xFshow a strong increase in Rstatwhich indicates the approach to the limit of
stability of the fit procedure in particular for K+. In the higher xFbins, xF= 0.3 and xF= 0.4
the constraints imposed on some fit parameters, Sect. 4.4, limit of course also the range of the
possible statistical fluctuations. Here, the problem has to be tracked by the evaluation of the
corresponding systematic errors.
5Evaluation of invariant cross sections and corrections
The experimental evaluation of the invariant cross section
f(xF,pT) = E(xF,pT) ·d3σ
dp3(xF,pT)
(9)
follows the methods described in [1]. This includes the absolutenormalization via the measured
trigger cross section of 28.23 mb and the number of events originating from the liquid hydrogen
target. The trigger is defined by a system of scintillation counters and proportional chambers on
the incoming beam plus a downstream scintillator vetoing noninteracting beam particles.
11
Page 14
5.1 Empty target correction
Due to the small empty/full target ratio of 9% and the larger fraction of zero prong
events in the empty target sample, the empty target contribution may be treated as a small
correction as argued in [1]. This correction is, within the statistical errors, equal for K+and K−
and independent on pTand xF. It is compatible with the one given for pions [1] and protons [2]
and is presented in Fig. 13 as a function of xF.
F
x
0 0.20.4
factor for ET correction
0.98
1
1.02
1.04
1.06
1.08
Figure 13: Empty target correction for K+and K−as a function of xF, averaged over pT
5.2 Trigger bias correction
This correction is necessitated by the interaction trigger which uses a small scintillator
placed between the two magnets (S4 in Fig. 3) in anticoincidence with the beam signal. This
triggervetoeseventswithfastforward particlesandtherebynecessitatesatriggerbiascorrection
whichcan in principledependbothon particletypeand onthekinematicvariables.Asdescribed
in detail in [1] the correction is quantified experimentally by increasing the diameter of the S4
veto counter offline and extrapolating the observed change in cross sections to diameter zero.
For the case of kaons, the correction turns out to be within errors independent on pTand similar
for K+and K−. Its xFdependence is shown in Fig. 14.
trig. bias corr. [%]
F
x
0 0.20.4
b)

K
→
p+p
F
x
0 0.2 0.4
0
2
4
6
8
a)
+
K
→
p+p
Figure 14: Triggerbias correction as a function of xFfor a) K+and b) K−. The lines correspond
to the parametrization of the correction
12
Page 15
5.3 Reinteraction in the target
This correction has been evaluated [1] using the PYTHIA event generator. It is pTinde
pendent within the available event statistics. The xFdependence is shown in Fig. 15.
F
x
0 0.2 0.4
target reinteraction corr. [%]
2
1
0
+
K
K

Figure 15: Target reinteraction correction as a function of xF
5.4Absorption in the detector material
The correction for kaons interacting in the detector material downstream of the target is
determined using the GEANT simulation of the NA49 detector, taking account of the K+and
K−inelastic cross sections in the mostly light nuclei (Air, Plastic foils, Ceramic rods). Due to
the nonhomogeneous material distribution the correction shows some structure both in pTand
xFas presented in Fig. 16.
F
x
0 0.20.4
det. abs. corr [%]
0
1
2
3

, K
+
K
= 0.1 GeV/c
T
= 1.1 GeV/c
T
p
p
Figure 16: Correction due to the absorption of produced kaons in the downstream detector
material as a function of xFfor two pTvalues. The lines are shown to guide the eye
5.5 Kaon weak decays
Due to their decay length of about 30 m at the lowest lab momentum studied here, the
weak decay of kaons necessitates corrections of up to 7% for for kaons produced in the lowest
13
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