Production of \(\Lambda \)hyperons in inelastic p+p interactions at 158 \({\mathrm{GeV}}\!/\!c\)
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Abstract
Inclusive production of \(\Lambda \)hyperons was measured with the large acceptance NA61/SHINE spectrometer at the CERN SPS in inelastic p+p interactions at beam momentum of 158 \({\mathrm{GeV}}\!/\!c\). Spectra of transverse momentum and transverse mass as well as distributions of rapidity and x\(_{_F}\) are presented. The mean multiplicity was estimated to be \(0.120\,{\pm }\,0.006\;(stat.)\;{\pm }0.010\;(sys.)\). The results are compared with previous measurements and predictions of the Epos, Urqmd and Fritiof models.
Keywords
Systematic Uncertainty Chebyshev Polynomial Liquid Hydrogen Transverse Mass Primary Vertex1 Introduction
Hyperon production in proton–proton (p+p) interactions has been studied in a long series of fixed target and collider experiments. However, the resulting experimental data suffers from low statistics, incomplete beam momentum coverage, and large differences between the measurements reported by different experiments. Also popular models of proton–proton interactions mostly fail to reproduce the measurements. The data on \(\Lambda \) production and the model predictions are reviewed at the end of this paper.
At the same time rather impressive progress was made in measurements of hyperon production in nucleus–nucleus (A+A) collisions [1]. This has two reasons. Firstly, mean multiplicities of all hadrons in central heavy ion collisions are typically two to three orders of magnitude higher than the corresponding multiplicities in inelastic p+p interactions. Secondly, the hyperon yields per nucleon are enhanced by substantial factors in A+A collisions with respect to p+p interactions. This enhancement, which increases with the strangeness content of the hyperon in question, has raised considerable interest over the past decades. It has in particular been brought into connection with production of the Quark–Gluon Plasma, a ’deconfined’ state of matter at that time hypothetical [2, 3]. Nowadays, for the energies well below the LHC energy range, nucleus–nucleus collisions are investigated mainly to find the critical point of strongly interacting matter as well as to study the properties of the onset of deconfinement [4, 5]. In particular, precise measurements of inclusive hadron production properties as a function of beam momentum (13A–158A \({\mathrm{GeV}}\!/\!c\)) and size of colliding nuclei (p+p, p+Pb, Be+Be, Ar+Sc, Xe+La) are performed by NA61/SHINE [6]. Results on inelastic p+p interactions are an important part of this scan.
NA61/SHINE already published results on \(\pi ^{\pm }\), K\(^{\pm }\), proton, \(\Lambda \) and \(K^0_S\) production in p+C interactions at beam momentum of 31 \({\mathrm{GeV}}\!/\!c\) [7, 8, 9, 10], as well as \(\pi ^\) production in p+p collisions at 20–158 \({\mathrm{GeV}}\!/\!c\) [11].
This paper presents the first NA61/SHINE results on strange particle production in p+p interactions. Since all \(\Sigma ^0\) hyperons decay electromagnetically via \(\Sigma ^0\rightarrow \Lambda \gamma \), which is indistinguishable from direct \(\Lambda \) production, \(\Lambda \) in the following denotes the sum of both \(\Lambda \) directly produced in strong p+p interactions and \(\Lambda \) from decays of \(\Sigma ^0\) hyperons produced in these interactions.
2 The experimental setup
The NA61/SHINE experiment [6] uses a large acceptance hadron spectrometer located in the H2 beamline at the CERN SPS accelerator complex. The layout of the experiment is schematically shown in Fig. 1. Hereby we describe only the components relevant for the analysis. The main detector system is a set of large volume time projection chambers (TPCs). Two of them (VTPC1 and VTPC2) are placed inside superconducting magnets (VTX1 and VTX2) with a combined bending power of 9 Tm. The standard current setting for data taking at 158 \({\mathrm{GeV}}\!/\!c\) corresponds to full field, 1.5 T, in the first and reduced field, 1.1 T, in the second magnet. Two large TPCs (MTPCL and MTPCR) are positioned downstream of the magnets, symmetrically to the undeflected beam. A fifth small TPC (GAPTPC) is placed between VTPC1 and VTPC2 directly on the beam line and covers the gap between the sensitive volumes of the other TPCs. The NA61/SHINE TPC system allows a precise measurement of the particle momenta p with a resolution of \(\sigma (p)/p^2\approx (0.3  7)\!\times \!10^{4}\;\mathrm {(GeV/c)}^{1}\) at the full magnetic field used for data taking at 158 \({\mathrm{GeV}}\!/\!c\) and provides particle identification via the measurement of the specific energy loss, \(\mathrm{d}E\!/\!\mathrm{d}x\), with relative resolution of about 4.5 %.
A set of scintillation and Cherenkov counters, as well as beam position detectors (BPDs) upstream of the main detection system provide a timing reference, as well as identification and position measurements of the incoming beam particles. The 158 \({\mathrm{GeV}}\!/\!c\) secondary hadron beam was produced by 400 \({\mathrm{GeV}}\!/\!c\) primary protons impinging on a 10 cm long beryllium target. Hadrons produced at the target are transported downstream to the NA61/SHINE experiment by the H2 beamline, in which collimation and momentum selection occur. Protons from the secondary hadron beam are identified by a differential Cherenkov counter (CEDAR) [12]. Two scintillation counters, S1 and S2, together with the three veto counters V0, V1 and V1\(^p\) were used to select beam particles. Thus, beam particles were required to satisfy the coincidence \(S1\cdot S2\cdot \overline{V0}\cdot \overline{V1}\cdot \overline{V1^p}\cdot \)CEDAR in order to become accepted as a valid proton. Trajectories of individual beam particles were measured in a telescope of beam position detectors placed along the beam line (BPD1/2/3 in Fig. 1). These are multiwire proportional chambers with two orthogonal sense wire planes and cathode strip readout, allowing to determine the transverse coordinates of the individual beam particle at the target position with a resolution of about 100 \(\mu \)m. For data taking on p+p interactions a liquid hydrogen target (LHT) of 20.29 cm length (2.8 % interaction length) and 3 cm diameter was placed 88.4 cm upstream of VTPC1.
Data taking with inserted and removed liquid hydrogen (LH) in the LHT was alternated in order to calculate a databased correction for interactions with the material surrounding the liquid hydrogen. Interactions in the target are selected by requiring an anticoincidence of the selected beam protons with the signal from a small scintillation counter of 2 cm diameter (S4) placed on the beam trajectory between the two spectrometer magnets. Further details on the experimental setup, beam and the data acquisition can be found in Ref. [6].
3 Analysis technique
In the following section the analysis technique is described, starting with the event reconstruction followed by the event and V\(^0\) selections. Next the \(\Lambda \) signal extraction and the calculation of \(\Lambda \)yields are presented. Then the correction procedure and the estimation of statistical and systematic uncertainties are discussed. Finally quality tests are performed on the final results. More details can be found in Ref. [13].
3.1 Track and main vertex reconstruction
 (i)
cluster finding in the TPC raw data, calculation of the cluster centreofgravity and total charge,
 (ii)
reconstruction of local track segments in each TPC separately,
 (iii)
matching of track segments into global tracks,
 (iv)
track fitting through the magnetic field and determination of track parameters at the first measured TPC cluster,
 (v)
determination of the interaction vertex using the beam trajectory (x and y coordinates) fitted in the BPDs and the trajectories of tracks reconstructed in the TPCs (z coordinate),
 (vi)
matching of ToF hits with the TPC tracks.
3.2 Event selection
A total of \(3.5\times 10^{6}\) events recorded with the LH inserted (denoted I) and \(0.43\times 10^6\) with the LH removed from the target (denoted R) were used for the analysis. The two configurations were realised by filling the target vessel with LH and emptying it.
 (i)
no offtime beam particle was detected 1 \(\mu \)s before and after the trigger particle,
 (ii)
the trajectory of the beam particle was measured in at least one of BPD1 or BPD2 and in the BPD3 detector and was well reconstructed (BPD3 is positioned close to and upstream of the LHT),
 (iii)
the fit of the zcoordinate of the primary interaction vertex converged and the fitted z position is found within \(\pm 40\) cm of the centre of the LHT.
3.3 V\(^0\) reconstruction and selection
 (i)
For each track, the minimum number of clusters in at least one of VTPC1 and VTPC2 was required to be 15.
 (ii)
Proton and pion candidates were selected by requiring their specific energy loss measured by the TPCs to be within 3 \(\sigma \) around the nominal Bethe–Bloch value. This cut was applied only to experimental data.
 (iii)
For the simulated data (see below) the background was totally discarded by matching, i.e. by using only those reconstructed tracks which were identified as originating from the corresponding \(\Lambda \) decay. The identification was performed by matching the clusters found in the TPCs with the clusters generated in the simulation. In case more than one reconstructed track was matched to a \(\Lambda \) decay daughter the one with the largest number of matched clusters was selected.
 (iv)
The combinatorial background concentrated in the vicinity of the primary vertex is reduced by imposing a distance cut on the difference between the z coordinate of the primary and \(\Lambda \) vertex (\(\Delta z = z_\Lambda  z_{primary}\), see Fig. 3). To maximise the fraction of rejected background while minimising the number of lost \(\Lambda \) candidates, a rapidity dependent cut was applied: \(\Delta z>10\) cm for \(y<0.25\), \(\Delta z>15\) cm for \(y\in [0.25,0.75]\), \(\Delta z>40\) cm for \(y\in [0.75,1.25]\), and \(\Delta z>60\) cm for higher rapidities.
 (v)
A further significant part of the background (e.g. pairs from photon conversions) was rejected by imposing a cut on \(\cos {\phi }\), where \(\phi \) is defined as the angle between the vectors \(y'\), and n, where \(y'\) is the vector perpendicular to the momentum of the V\(^0\)particle which lies in the plane spanned by the yaxis and the V\(^0\)momentum vector, and n is a vector normal to the decay plane (see Fig. 4). A rapidity dependent cut was used: \(\cos {\phi }<0.95\) for \(y<0.25\), \(\cos {\phi }<0.9\) for \(y\in [0.25,0.75]\), \(\cos {\phi }<0.8\) for higher rapidities.
 (vi)
The trajectories of the \(\Lambda \) candidates were calculated using the decay vertex and the momentum vectors of the decay particles. Extrapolation back to the primary vertex plane resulted in impact parameters \(b_{x}\) (in the magnetic bending plane) and \(b_{y}\) (see Fig. 2). As the resolution of impact parameters is approximately twice better in y than in x direction, an elliptic cut \(\sqrt{(b_{x}/2)^2+b_{y}^{2}}< 1\) cm was imposed in order to reduce the background from \(\Lambda \) candidates which do not originate from the primary vertex.
3.4 Signal extraction
3.5 Correction factors
 1.The contribution from interactions in the material outside of the liquid hydrogen volume of the target was subtracted:The normalisation factor B was derived by comparing the distribution of the fitted z coordinate of the interaction vertex far away from the target [9] for filled and empty target vessel:$$\begin{aligned} \frac{n^I(k,l)Bn^R(k,l)}{N^IBN^R}. \end{aligned}$$(2)where \(N^I_{far\;z}\) (\(N^R_{far\;z}\)) is the number of events in the region \(100 < z < 280 \) cm downstream of the target centre for the data sample with inserted (removed) hydrogen in the target vessel.$$\begin{aligned} B=\frac{N^I_{far\;z}}{N^R_{far\;z}}~=3.93, \end{aligned}$$(3)
 2.The loss of the \(\Lambda \) hyperons due to the \(\mathrm{d}E\!/\!\mathrm{d}x\) requirement, was corrected by a constant factorwhere \(\epsilon = 0.9973\) is the probability for the proton (pion) to lie within 3\(\sigma \) around the nominal Bethe–Bloch value.$$\begin{aligned} c_{\mathrm{d}E\!/\!\mathrm{d}x} =\frac{1}{\epsilon ^{2}} = 1.005, \end{aligned}$$(4)
 3.
A detailed Monte Carlo simulation was performed to correct for geometrical acceptance, reconstruction efficiency, losses due to the trigger bias, the branching ratio of the \(\Lambda \) decay, the feeddown from hyperon decays as well as the quality cuts applied in the analysis. The correction factors are based on \(20\times 10^6\) inelastic p+p events produced by the Epos1.99 event generator [19]. The particles in the generated events were tracked through the NA61/SHINE apparatus using the Geant3 package [20]. The TPC response was simulated by dedicated NA61/SHINE software packages which take into account all known detector effects. The simulated events were reconstructed with the same software as used for real events and the same selection cuts were applied (except the identification cut). As seen from Fig. 7 the shape and position of the \(\Lambda \) peak is well reproduced by the simulation while the width is about 10 % narrower. More details on MC validation can be found in Ref. [11].
For each (k, l) bin, the correction factor \(c_{_{MC}}(k,l)\) was calculated aswhere$$\begin{aligned} c_{_{MC}}(k,l)=\frac{n_{MC}^{gen}(k,l)}{N_{MC}^{gen}}\Bigg /\frac{n_{MC}^{acc}(k,l)}{N_{MC}^{acc}}, \end{aligned}$$(5)These factors also include the correction for feeddown from weak decays (mostly of \(\Xi ^{}\) and \(\Xi ^{0}\), see Fig. 8). The \(\Xi ^{}\) yields as function of rapidity generated by the Epos1.99 simulation agree within 10 % with the measurements reported in Ref. [21]. The values of the correction factors are presented in Fig. 9. Statistical errors of the correction factors were calculated using the following approach: The correction factor (\(c_{_{MC}}\)) consists of two parts:
\(n_{MC}^{gen}(k,l)\) is the number of \(\Lambda \) hyperons produced in a given (k, l) bin in the primary interactions, including \(\Lambda \) hyperons from the \(\Sigma ^0\) decays,

\(n_{MC}^{acc}(k,l)\) is the number of reconstructed \(\Lambda \) hyperons in a given (k, l) bin, determined by matching the reconstructed \(\Lambda \) candidates to the simulated \(\Lambda \) hyperons based on the cluster positions,

\(N_{MC}^{gen}\) is the number of generated inelastic p+p interactions (\(19\,961\times 10^3\)),

\(N_{MC}^{acc}\) is the number of accepted p+p events (\(15\,607\times 10^3\)),

\(k=y\) or \(x_{_F}\), and \(l=p_{_T}\) or \(m_{_T}m_{_\Lambda }\).
where \(\alpha \) describes the loss of inelastic events due to the event selection, and \(\beta \) takes into account the loss of \(\Lambda \) hyperons due to the \(V^0\)cuts, efficiency, and the other aforementioned effects. The error of \(\alpha \) was calculated assuming a binomial distribution, while the part \(\beta \) involving the fitting procedure takes into account the error of the fit:$$\begin{aligned} c_{_{MC}}\left( k,l\right)&=\frac{n_{MC}^{gen}(k,l)}{N_{MC}^{gen}}\Bigg /\frac{n_{MC}^{acc}(k,l)}{N_{MC}^{acc}}\nonumber \\&=\frac{N^{acc}_{MC}}{N^{gen}_{MC}}\Bigg /\frac{n^{acc}_{MC}(k,l)}{n^{gen}_{MC}(k,l)}=\frac{\alpha }{\beta (k,l)}, \end{aligned}$$(6)$$\begin{aligned} \Delta \alpha =\sqrt{\frac{\alpha (1\alpha )}{N_{MC}^{gen}}}, \end{aligned}$$(7)where \(\Delta n^{acc}_{MC}(k,l)\) is the uncertainty of the fit, and \(\Delta n^{gen}_{MC}(k,l)=\sqrt{n^{gen}_{MC}(k,l)}\). The total statistical error of \(c_{_{MC}}\) was calculated as follows:$$\begin{aligned} \Delta \beta (k,l)=\sqrt{\left( \frac{\Delta n^{acc}_{MC}(k,l)}{n^{gen}_{MC}(k,l)}\right) ^2+ \left( \frac{n^{acc}_{MC}(k,l)\Delta n^{gen}_{MC}(k,l)}{(n^{gen}_{MC}(k,l))^2}\right) ^2}, \end{aligned}$$(8)$$\begin{aligned} \Delta c_{_{MC}}=\sqrt{\left( \frac{\Delta \beta }{\alpha }\right) ^2+\left( \frac{\beta \Delta \alpha }{\alpha ^2}\right) ^2}. \end{aligned}$$(9) 

\(n^{I/R}\) the uncorrected number of \(\Lambda \) hyperons for the hydrogen inserted/removed target configurations,

\(N^{I/R}\) the number of events for the hydrogen inserted/removed data after event cuts,

\(c_{dE/dx}\), \(c_{_{MC}}\) the correction factors described in Sec. 3.5,

B the normalisation factor (defined in Sec. 3.5),

\(k=y\) or \(x_{_F}\), and \(l=p_{_T}\) or \(m_{_T}m_{_\Lambda }\),

\(\Delta k\) and \(\Delta l\) the bin widths.
3.6 Statistical and systematic uncertainties
The statistical errors of the corrected double differential yields (see Eq. 10) take into account the statistical errors of \(c_{_{MC}}\) (see Eq. (9)) and the statistical errors on the fitted \(\Lambda \) yields in the LH inserted and removed configurations. The statistical errors on B and \(c_{dE/dx}\) were neglected.
The systematic uncertainties were estimated taking into account four sources. For each source modifications to the standard analysis procedure were applied and the deviation of the results from the standard procedure were calculated. As the effects of the modifications are partially correlated, the maximal positive and negative deviation from the standard procedure was determined for each bin and source separately. Then, the positive (negative) systematic uncertainties were calculated separately by adding in quadrature the positive (negative) contribution from each source.
 (i)The uncertainty due to the signal extraction procedure:

The standard function used for background fit, a Chebyshev polynomial of 2nd order, was changed for a Chebyshev polynomial of 3rd order and for a standard polynomial of 2nd order.

The range within which the raw number of \(\Lambda \) particles is summed up was changed from 3\(\Gamma \) to 2.5\(\Gamma \) and \(3.5\Gamma \).

The lower limit of the fitting range was changed from 1.08 \({\mathrm{GeV}}\!/\!c^2\) (1.076 for \(y=0.5\), 1.073 for \(y=1.0\)) to 1.083 \({\mathrm{GeV}}\!/\!c^2\) (1.079 for \(y=0.5\), 1.076 for \(y=1.0\)).

The initial value of the \(\Gamma \) parameter of the signal function was changed by \(\pm \)8 %.

the initial value for the mass parameter of the Lorentz function was changed by \(\pm \)0.3 MeV.

 (ii)The effect of the event and quality cuts were checked by performing the analysis with the following cuts changed compared to the values presented in Secs. 3.2 and 3.3.

The cut on the zposition of the interaction vertex was changed from \(\pm \)40 to \(\pm \)30 cm and \(\pm \)50 cm with respect to the centre of the target.

The window in which offtime beam particles are not allowed was increased from 1 to 1.5 \(\mu \)s.

The elliptic cut on the impact parameters was reduced by a factor of 2: \(\sqrt{b_{x}^2+(2b_{y})^{2}}< 1\) cm.

The \(\mathrm{d}E\!/\!\mathrm{d}x\) cut was modified to \(\pm \)2.8\(\sigma \) or 3.2\(\sigma \) to estimate possible systematic effects of \(\mathrm{d}E\!/\!\mathrm{d}x\) calibration.

The matching procedure used to reject background in the simulation was turned off.

The required minimal number of charge clusters in at least one of the VTPCs for both \(V^0\)decay products was decreased to 12 or increased to 18.
 The cut on \(\Delta z\), the distance between the decay and the primary interaction vertex, was changed from the standard values to the values shown in columns A and B in the following table:
Minimal \(\Delta z\) (cm) allowed
\(y_{min}\)
\(y_{max}\)
Standard
A
B
\(\)1.75
0.25
10
7.5
12.5
0.25
0.75
15
11.25
18.75
0.75
1.25
40
30
50
 The limits for the cut on \(\cos {\phi }\) were changed from the standard values to the values shown in columns A and B in the following table:
Maximal \(\cos {\phi }\) allowed
\(y_{min}\)
\(y_{max}\)
Standard
A
B
\(\)1.75
\(\)0.25
0.95
0.975
0.925
\(\)0.25
0.75
0.9
0.95
0.85
0.75
1.25
0.8
0.85
0.75

 (iii)
In order to find the systematic uncertainty of the normalisation factor B in Eq. (3) for the LH removed configuration, the limits of the region for which this parameter was calculated was varied in steps of 0.1 m. For each combination of the lower limit (ranging from 0.8 to 1.8 m from the target) and upper limit in z (from 2.8 to 3.8 m from the target) the Bfactor was calculated. The smallest and the highest value of B obtained in this way is taken as the systematic uncertainty range of B.
 (iv)
For estimation of the uncertainty due to the feeddown correction a conservative systematic uncertainty of 30 % on the \(\Xi ^{}\) and \(\Xi ^{0}\) yields predicted by Epos1.99 was assumed.
The distribution of the proper lifetime of \(\Lambda \) hyperons was obtained using an analysis procedure analogous to the one presented in Sec. 4. The data for the lifetime analysis were binned in rapidity \(k=y\) (from \(\)1.5 to +1.0, in steps of 0.5) and lifetime normalised to the mean lifetime \(t/\tau _{PDG}\) [18] (from 0.00 to 4.75, in steps of 0.25) with \(c\tau _{PDG}=7.89\) cm. The lifetime was calculated using the distance r between the V\(^0\)decay vertex and the interaction vertex of the V\(^0\)candidates (\(t=r/(\gamma \beta )\), where \(\gamma \), \(\beta \) are the Lorentz variables). Then \(d^2n/(dydt)\) was calculated and an exponential function was fitted to the lifetime distribution for each rapidity bin separately (see the example in Fig. 10a for \(y=1.0\)). The ratio of the fitted mean lifetime \(\tau \) to the corresponding PDG value \(\tau _{PDF}\) is shown in Fig. 10b as a function of rapidity. The fitted mean lifetimes are seen to agree with the PDG value for all rapidities indicating good accuracy of the correction procedure.
The expected forwardbackward symmetry of the data was also checked. The final double and singledifferential distributions used for this test were found to agree for the corresponding backward and forward rapidities within the statistical errors.
4 Results
4.1 Formalism
Doubledifferential yield \(\frac{d^2n}{dydp_{T}}\)
y  \(p_{T}\)  \(\frac{d^2n}{dydp_{T}}\)  \(\Delta _{stat}\)  \(\Delta _{sys}^{}\)  \(\Delta _{sys}^{+}\) 

\(\times 10^{3}\)  \(\left( \frac{1}{GeV/c}\right) \)  
\(\)1.5  0.1  16.0  2.8  2.2  1.0 
0.3  35.3  4.8  5.0  3.5  
0.5  30.3  3.4  2.9  1.6  
0.7  27.1  2.7  1.9  1.7  
0.9  14.0  1.7  0.5  0.7  
1.1  6.2  1.1  0.4  0.5  
1.3  3.22  0.71  0.44  0.19  
1.5  1.36  0.45  0.18  0.22  
\(\)1.0  0.1  14.7  1.2  1.3  1.1 
0.3  28.2  1.7  1.4  1.1  
0.5  27.7  1.7  1.4  1.2  
0.7  20.9  1.3  0.9  0.8  
0.9  12.16  0.89  0.89  0.46  
1.1  6.96  0.64  0.26  0.22  
1.3  2.93  0.39  0.15  0.12  
1.5  1.80  0.30  0.12  0.13  
1.7  0.59  0.16  0.04  0.04  
1.9  0.38  0.14  0.04  0.02  
\(\)0.5  0.1  10.74  0.59  1.11  0.37 
0.3  24.31  0.95  2.34  0.85  
0.5  25.5  1.0  2.2  0.8  
0.7  20.68  1.00  1.41  0.72  
0.9  12.05  0.77  0.96  0.42  
1.1  6.61  0.55  0.28  0.23  
1.3  3.74  0.41  0.18  0.13  
1.5  1.62  0.25  0.12  0.09  
1.7  0.87  0.19  0.07  0.05  
1.9  0.42  0.14  0.06  0.06  
0.0  0.1  10.06  0.56  0.47  0.42 
0.3  22.89  0.87  1.75  0.61  
0.5  23.26  0.91  1.83  0.73  
0.7  18.58  0.89  1.44  0.75  
0.9  11.50  0.78  0.66  0.63  
1.1  5.63  0.59  0.20  0.33  
1.3  3.74  0.49  0.22  0.22  
1.5  1.45  0.27  0.09  0.08  
1.7  0.69  0.20  0.05  0.04  
1.9  0.34  0.11  0.06  0.08  
0.5  0.1  10.79  0.63  1.55  0.24 
0.3  23.89  0.95  4.07  0.66  
0.5  27.4  1.0  3.9  0.6  
0.7  18.03  0.88  1.94  0.51  
0.9  11.04  0.78  1.13  0.36  
1.1  5.87  0.57  0.27  0.15  
1.3  3.50  0.49  0.14  0.34  
1.5  1.47  0.30  0.14  0.15  
1.7  0.65  0.21  0.09  0.05  
1.9  0.45  0.14  0.46  0.01  
1.0  0.1  11.91  0.91  1.55  0.68 
0.3  26.6  1.4  3.2  1.2  
0.5  29.0  1.5  6.6  1.2  
0.7  21.2  1.4  2.1  1.1  
0.9  13.4  1.1  1.7  0.6  
11  7.65  0.80  2.26  0.11  
1.3  3.32  0.56  0.20  0.29  
1.5  1.06  0.28  0.00  0.00 
Doubledifferential yield \(\frac{d^2n}{dydm_{_T}}\)
y  \(m_{_T}\)  \(\frac{d^2n}{dydm_{_T}}\)  \(\Delta _{stat}\)  \(\Delta _{sys}^\)  \(\Delta _{sys}^+\) 

\(m_{_\Lambda }\)  \(\times 10^3\)  \(\left( \frac{1}{GeV/c^2}\right) \)  
\(\)1.5  0.05  62.0  6.2  8.8  4.5 
0.15  31.9  3.2  2.3  1.5  
0.25  18.5  2.1  0.8  1.1  
0.35  8.4  1.3  0.5  0.3  
0.45  4.51  0.92  0.30  0.72  
0.55  2.91  0.73  0.43  0.21  
0.65  1.26  0.43  0.15  0.08  
\(\)1.0  0.05  53.8  2.5  2.4  1.7 
0.15  27.3  1.6  1.5  1.1  
0.25  14.6  1.0  0.9  0.7  
0.35  8.40  0.72  0.48  0.31  
0.45  5.10  0.54  0.25  0.34  
0.55  2.50  0.36  0.19  0.08  
0.65  1.44  0.28  0.11  0.14  
0.75  1.16  0.24  0.06  0.08  
0.85  0.58  0.16  0.05  0.03  
\(\)0.5  0.05  45.9  1.4  4.3  1.5 
0.15  25.9  1.1  1.7  0.8  
0.25  14.56  0.86  1.04  0.61  
0.35  8.28  0.63  0.63  0.29  
0.45  4.54  0.45  0.21  0.23  
0.55  3.23  0.38  0.19  0.10  
0.65  1.83  0.28  0.07  0.10  
0.75  0.89  0.18  0.05  0.06  
0.85  0.79  0.19  0.09  0.06  
0.95  0.51  0.16  0.06  0.07  
0.0  0.05  43.4  1.3  3.0  1.3 
0.15  22.63  0.93  1.82  1.19  
0.25  13.38  0.79  0.66  0.48  
0.35  7.67  0.67  0.68  0.42  
0.45  4.34  0.52  0.20  0.32  
0.55  2.56  0.39  0.39  0.14  
0.65  1.89  0.34  0.19  0.35  
0.75  1.00  0.24  0.06  0.22  
0.85  0.61  0.18  0.09  0.11  
0.5  0.05  46.2  1.4  6.9  7.6 
0.15  25.7  1.0  3.5  4.2  
0.25  13.10  0.79  1.12  2.15  
0.35  7.28  0.63  0.76  1.20  
0.45  4.17  0.48  0.13  0.69  
0.55  3.05  0.43  0.17  0.60  
0.65  1.44  0.32  0.09  0.27  
0.75  0.77  0.21  0.10  0.14  
0.85  0.92  0.25  0.11  0.21  
0.95  0.32  0.13  0.03  0.06  
1.0  0.05  52.8  2.1  9.4  8.2 
0.15  26.6  1.5  3.0  4.3  
0.25  16.5  1.2  1.9  2.7  
0.35  9.04  0.94  1.19  1.97  
0.45  4.00  0.61  0.67  0.88  
0.55  2.51  0.49  0.31  0.33  
0.65  1.90  0.49  0.16  0.46  
0.75  0.63  0.22  0.08  0.13 
4.2 Spectra
Doubledifferential yields, \(\frac{d^{2}n}{x_{_F}p_{_T}}\) and \(f_n(x_{_F},p_{T})\), for \(x_{_F}<0\)
\(x_{_F}\)  \(p_{_T}\)  \(\frac{d^2n}{dx_{_F}dp_{_T}}\)  \(\Delta _{stat}\)  \(\Delta _{sys}^\)  \(\Delta _{sys}^+\)  \(f_n(x_{_F},p_{_T})\)  \(\Delta _{stat}\)  \(\Delta _{sys}^\)  \(\Delta _{sys}^+\) 

\(\times 10^3\)  \(\left( \frac{1}{GeV/c}\right) \)  \(\times 10^3\)  \(\left( \frac{1}{\left( GeV/c\right) ^2}\right) \)  
\(\)0.35  0.1  44  21  13  4  26  12  9  5 
0.3  127  34  30  12  25.2  6.8  6.4  4.7  
0.5  89  15  8  13  10.7  1.9  0.6  1.0  
0.7  78  12  6  10  6.8  1.0  0.4  0.6  
0.9  37.4  6.4  3.7  4.0  2.56  0.44  0.22  0.19  
1.1  15.4  3.4  0.8  1.0  0.88  0.19  0.06  0.04  
1.3  9.3  2.2  0.8  0.1  0.46  0.11  0.04  0.02  
1.5  4.6  1.5  0.8  0.6  0.201  0.066  0.029  0.042  
\(\)0.25  0.1  45.5  8.9  2.0  3.5  20.4  4.0  3.2  3.2 
0.3  110  14  11  10  16.5  2.2  2.6  1.1  
0.5  92  11  6  6  8.44  0.98  1.14  0.32  
0.7  78.9  7.4  2.8  2.3  5.25  0.49  0.94  0.13  
0.9  41.4  4.5  1.4  3.9  2.20  0.24  0.48  0.06  
1.1  21.8  2.8  1.0  1.0  0.97  0.13  0.17  0.04  
1.3  8.5  1.6  0.6  0.6  0.334  0.062  0.077  0.012  
1.5  4.7  1.1  0.3  0.5  0.165  0.038  0.032  0.033  
1.7  1.71  0.58  0.13  0.17  0.055  0.019  0.008  0.005  
\(\)0.15  0.1  68.7  5.8  3.7  3.0  21.7  1.8  0.9  1.5 
0.3  135.6  7.8  4.7  5.4  14.46  0.83  0.36  0.46  
0.5  132.9  7.4  5.4  4.9  8.73  0.48  0.30  0.22  
0.7  101.6  6.0  3.4  4.0  4.94  0.29  0.10  0.19  
0.9  53.4  3.9  3.0  1.9  2.11  0.15  0.09  0.05  
1.1  29.6  2.7  1.2  1.3  1.008  0.092  0.033  0.039  
1.3  11.6  1.6  0.6  0.5  0.352  0.049  0.014  0.011  
1.5  5.9  1.1  0.6  0.4  0.165  0.030  0.016  0.015  
1.7  2.64  0.73  0.20  0.11  0.069  0.019  0.005  0.002  
\(\)0.05  0.1  72.7  3.4  2.8  2.5  16.08  0.75  1.05  0.26 
0.3  161.4  5.4  6.9  4.5  12.23  0.41  0.43  0.26  
0.5  160.8  5.6  6.3  5.4  7.68  0.27  0.23  0.24  
0.7  118.5  5.2  4.4  3.5  4.33  0.19  0.11  0.07  
0.9  64.1  4.0  2.6  2.2  1.97  0.12  0.06  0.05  
1.1  33.0  2.9  1.2  1.5  0.899  0.078  0.024  0.036  
1.3  20.2  2.3  1.4  1.0  0.506  0.059  0.031  0.036  
1.5  7.2  1.2  0.5  0.4  0.169  0.029  0.016  0.008  
1.7  4.8  1.2  0.3  0.3  0.108  0.027  0.010  0.006  
1.9  0.55  0.32  0.07  0.06  0.0120  0.0070  0.0016  0.0032 
Doubledifferential yields, \(\frac{d^{2}n}{x_{_F}p_{_T}}\) and \(f_n(x_{_F},p_{T})\), for \(x_{_F}>0\)
\(x_{_F}\)  \(p_{_T}\)  \(\frac{d^2n}{dx_{_F}dp_{_T}}\)  \(\Delta _{stat}\)  \(\Delta _{sys}^\)  \(\Delta _{sys}^+\)  \(f_n(x_{_F},p_{_T})\)  \(\Delta _{stat}\)  \(\Delta _{sys}^\)  \(\Delta _{sys}^+\) 

\(\times 10^3\)  \(\left( \frac{1}{GeV/c}\right) \)  \(\times 10^3\)  \(\left( \frac{1}{\left( GeV/c\right) ^2}\right) \)  
0.05  0.1  70.3  3.5  8.9  2.7  15.56  0.77  2.49  1.11 
0.3  160.9  5.4  22.8  3.7  12.20  0.41  1.59  0.15  
0.5  166.4  5.6  20.8  4.0  7.95  0.27  0.97  0.09  
0.7  113.1  4.9  13.3  2.9  4.13  0.18  0.45  0.08  
0.9  64.4  4.2  7.4  1.7  1.97  0.13  0.16  0.04  
1.1  33.5  3.2  2.5  1.0  0.913  0.086  0.065  0.028  
1.3  17.0  2.3  1.1  0.8  0.426  0.058  0.035  0.017  
1.5  6.0  1.3  0.2  0.4  0.141  0.030  0.004  0.009  
1.7  2.56  0.82  0.33  0.14  0.058  0.018  0.007  0.003  
1.9  1.19  0.51  0.21  0.23  0.026  0.011  0.005  0.006  
0.15  0.1  63.1  4.5  8.5  3.4  19.9  1.4  4.3  1.0 
0.3  131.5  6.6  26.7  5.1  14.02  0.70  2.28  0.71  
0.5  146.8  7.0  35.4  3.5  9.64  0.46  1.79  0.35  
0.7  87.8  5.7  16.2  3.4  4.27  0.28  0.54  0.19  
0.9  61.2  4.7  6.7  1.8  2.42  0.19  0.62  0.09  
1.1  22.3  2.7  2.4  1.1  0.758  0.091  0.077  0.039  
1.3  15.1  2.5  2.2  1.3  0.460  0.077  0.039  0.043  
1.5  6.6  1.4  1.0  0.4  0.185  0.039  0.032  0.012  
1.7  1.39  0.63  0.14  0.29  0.036  0.017  0.007  0.007  
0.25  0.1  43.7  7.0  9.4  5.2  19.6  3.1  2.6  2.6 
0.3  115  10  8  8  17.3  1.5  3.4  1.2  
0.5  104.7  9.4  11.1  4.7  9.57  0.86  1.00  1.20  
0.7  91.9  8.2  13.4  5.4  6.12  0.55  1.58  0.41  
0.9  44.7  5.8  8.9  2.7  2.37  0.31  0.46  0.09  
1.1  16.8  3.3  3.5  1.5  0.75  0.15  0.17  0.05  
1.3  8.7  2.1  0.7  0.8  0.341  0.082  0.039  0.045  
1.5  2.9  1.1  0.4  0.4  0.102  0.037  0.015  0.010  
0.35  0.1  29  12  16  14  17.5  7.4  4.3  1.7 
0.3  81  17  28  23  16.1  3.4  6.4  4.6  
0.5  91  16  39  13  10.9  1.9  1.8  2.4  
0.7  61  11  25  19  5.28  0.95  0.81  1.79  
0.9  37.4  8.2  22.3  6.0  2.56  0.56  0.48  0.26  
1.1  17.3  5.1  6.3  6.0  0.99  0.29  0.29  0.13  
1.3  11.7  3.6  4.3  2.6  0.57  0.18  0.18  0.14 
\(p_{_T}\) integrated yield \(\frac{dn}{dy}\), the inverse slope parameter T and the mean transverse mass \(\langle m_{_T}\rangle  m_{\Lambda }\)
y  \(\frac{dn}{dy}\)  \(\Delta _{stat}\)  \(\Delta _{sys}^\)  \(\Delta _{sys}^+\)  T  \(\Delta _{stat}\)  \(\Delta _{sys}^\)  \(\Delta _{sys}^+\)  \(\langle m_{_T}\rangle m_{\Lambda }\)  \(\Delta _{stat}\)  \(\Delta _{sys}^\)  \(\Delta _{sys}^+\) 

\(\times 10^3\)  (MeV)  \(\left( \frac{GeV}{c^2}\right) \)  
\(\)1.5  26.8  1.5  2.4  1.4  143.8  6.3  5.4  3.3  0.156  0.013  0.005  0.006 
\(\)1.0  23.30  0.65  1.02  0.73  152.8  3.8  4.2  4.1  0.1687  0.0076  0.0050  0.0046 
\(\)0.5  21.35  0.43  1.71  0.64  163.0  3.2  4.5  5.1  0.1813  0.0067  0.0050  0.0056 
0.0  19.65  0.40  1.14  0.60  160.7  3.6  4.3  5.2  0.1777  0.0076  0.0051  0.0068 
0.5  20.64  0.42  2.53  0.43  154.0  3.6  3.9  8.4  0.1697  0.0070  0.0037  0.0102 
1.0  22.98  0.62  2.96  0.65  153.9  4.1  4.0  4.6  0.1640  0.0085  0.0028  0.0084 
\(p_{_T}\) integrated yield \(\frac{dn}{x_{_F}}\) and the invariant cross section \(F(x_{_F})\)
\(x_{_F}\)  \(\frac{dn}{dx_{_F}}\)  \(\Delta _{stat}\)  \(\Delta _{sys}^\)  \(\Delta _{sys}^+\)  \(F(x_{_F})\)  \(\Delta _{stat}\)  \(\Delta _{sys}^\)  \(\Delta _{sys}^+\) 

\(\times 10^3\)  \(\times 10^3\) (mb)  
\(\)0.35  81.3  9.0  9.3  6.5  313  34  36  70 
\(\)0.25  81.1  4.4  3.9  4.7  239  13  10  13 
\(\)0.15  108.6  2.9  3.9  3.5  231.6  6.1  5.9  4.9 
\(\)0.05  128.7  2.3  4.7  3.7  204.1  3.7  5.2  4.5 
0.05  127.2  2.3  14.4  3.0  200.9  3.7  22.3  2.2 
0.15  107.3  2.7  18.5  3.0  228.9  5.8  39.2  6.7 
0.25  86.0  3.8  9.8  4.0  253  11  29  12 
0.35  67.0  6.1  25.8  11.6  258  23  99  45 
The mean multiplicity of \(\Lambda \) hyperons (\(\langle \Lambda \rangle \)) was determined from the \(x_{_F}\) distribution. As the models applicable in the SPS energies range show large discrepancies in the region not measured by NA61/SHINE (see Fig. 20), the \(\Lambda \) yield in the unmeasured \(x_{F}\) region (\(x_{F} >\)0.4) was approximated by the straight line shown in Fig. 20. The line is defined assuming symmetry of the distribution. It crosses the points \(A_\pm =\left( \pm 0.35,\frac{1}{2}\left( \frac{dn}{dx_{_F}}(0.35)+\frac{dn}{dx_{_F}}(0.35)\right) \right) \) and \(B_\pm =(\pm 1,0)\). For the estimation of statistical part of the extrapolation error, the value of the point A was increased/decreased by \(\frac{1}{2}\left( \Delta \frac{dn}{dx_{_F}}(0.35)+\Delta \frac{dn}{dx_{_F}}(0.35)\right) \). The extrapolation amounts to 34.3 % of the total \(\Lambda \) yield and results in \(\langle \Lambda \rangle =0.120\pm 0.006\;(stat.)\) of the mean \(\Lambda \) multiplicity.
For the Epos model, not used for this extrapolation, the yield outside of NA61/SHINE acceptance to the total yield amounts to 38.0 %.
The systematic uncertainty of the mean multiplicity was calculated following the procedure described in Sec 3.6. An additional source of systematic uncertainty arises from the extrapolation of the \(\Lambda \) multiplicity to full phasespace. This was estimated by an alternative procedure based on a parametrisation of published rapidity distributions. In an iterative procedure a symmetric polynomial of 4\(^{th}\) order [22] was fitted to the \((1/\langle n\rangle )(dn/dz)\) distributions obtained by five bubblechamber experiments [23, 24, 25, 26, 27] and the NA61/SHINE data, where z stands for \(y/y_{beam}\). First, the fit included only the five bubblechamber datasets. Next, the NA61/SHINE spectrum was normalised to the fit result obtained in the first step and added as the 6\(^{th}\) set for the fit. Finally, the procedure was iterated using those six datasets until the normalisation factor converged. The ratio of the integral of the fitted function \(\frac{1}{\langle \Lambda \rangle }\frac{dn}{dz}\left( z\right) =0.394+1.99z^22.66z^4\) (see Fig. 15) for the full range of rapidity to the integral in the range outside of the NA61/SHINE acceptance was used as the extrapolation factor for the NA61/SHINE results. This ratio amounted to \(1.92\pm 0.12\), i.e. 48 % of the total production is outside of the acceptance for this procedure resulting in a mean multiplicity of \(\langle \Lambda \rangle =0.129\pm 0.008\). The difference between this result and the linear extrapolation of the \(x_F\) distribution is added in quadrature to the (positive) systematic error.
5 Comparison with world data and model predictions
Though the statistical error and the systematic uncertainty of the NA61/SHINE measurement is much smaller than for the other experiments, and the results are consistent with all the datasets used for the comparison, the general tendency obtained by fitting a symmetric polynomial of 4\(^{th}\) order does not describe well the NA61/SHINE data. On the other hand, the result of Brick et al. for which the beam momentum (147 \({\mathrm{GeV}}\!/\!c\)) differs the least from the NA61/SHINE momentum, shows the best agreement.
The mean multiplicity of \(\Lambda \) for 158 \({\mathrm{GeV}}\!/\!c\) inelastic p+p interactions is compared in Fig. 16 with the world data [28] as well as with predictions of the Epos1.99 model in its validity range. A steep rise in the threshold region is followed by a more gentle increase at higher energies that is well reproduced by the Epos1.99 model.
The dependence of the invariant spectrum on \(x_{_F}\) for NA61/SHINE and published results from bubble chamber experiments [23, 25, 26, 29, 30, 31] at nearby beam momenta is shown in Fig. 17. The NA61/SHINE results are consistent with the experiments performed at proton beams of lower energy, although the diplike structure visible at central \(x_{_F}\) in the data from the experiments operating at higher energies is not observed.
Figure 18 shows a comparison of rapidity spectra divided by the mean number of wounded nucleons \(1/\langle N_W\rangle \) in inelastic p+p interactions (this paper) and central C+C, Si+Si and Pb+Pb collisions (NA49 [32, 33]) at 158 \(A\,{\mathrm{GeV}}\!/\!c\). The yield of \(\Lambda \) hyperons per wounded nucleon increases with increasing \(\langle N_W\rangle \) as a consequence of strangeness enhancement in nucleus–nucleus collisions.
Figure 19 displays \(m_T\) spectra at midrapidity for inelastic p+p interactions (this paper) and central nucleus–nucleus collisions (NA49 [32, 33]) at 158 \(A\,{\mathrm{GeV}}\!/\!c\). The inverse slope parameter of the spectrum increases with increasing nuclear size due to increasing transverse flow.
A comparison with calculations from the models Epos1.99 [19], Urqmd3.4 [34, 35], and Fritiof7.02 [36] embedded in Hsd2.0 [37] is presented in Fig. 20.
The best agreement is found for the Epos1.99 model.
6 Summary
Inclusive production of \(\Lambda \)hyperons was measured with the large acceptance NA61/SHINE spectrometer at the CERN SPS in inelastic p+p interactions at beam momentum of 158 \({\mathrm{GeV}}\!/\!c\). Spectra of transverse momentum (up to 2 \({\mathrm{GeV}}\!/\!c\)) and transverse mass as well as distributions of rapidity (from \(\)1.75 to 1.25) and x\(_{_F}\) (from \(\)0.4 to 0.4) are presented. The mean multiplicity was found to be \(0.120\pm 0.006\;(stat.)\;\pm 0.010\;(sys.)\). The new results are in reasonable agreement with measurements from bubblechamber experiments at nearby beam momenta, but have much smaller uncertainties.
Predictions of the Epos, Urqmd and Fritiof models were compared to the new NA61/SHINE measurements reported in this paper. While Epos describes the data quite well, significant discrepancies are observed with the latter two models.
The results expand our knowledge of elementary protonproton interactions, allowing for a more precise description of strangeness production. They are expected to be used not only as an important input in the research of strongly interacting matter, but also as an input for tuning MCgenerators, including those used for cosmicray shower and neutrino beams simulations.
Notes
Acknowledgments
This work was supported by the Hungarian Scientific Research Fund (Grants OTKA 68506 and 71989), the János Bolyai Research Scholarship of the Hungarian Academy of Sciences, the Polish Ministry of Science and Higher Education (Grants 667/NCERN/2010/0, NN 202 48 4339 and NN 202 23 1837), the Polish National Science Centre (Grants 2011/03/ N/ ST2/03691, 2012/04/ M/ ST2/00816 and 2013/11/ N/ ST2/03879), the Foundation for Polish Science—MPD program, cofinanced by the European Union within the European Regional Development Fund, the Federal Agency of Education of the Ministry of Education and Science of the Russian Federation (SPbSU research Grant 11.38.193.2014), the Russian Academy of Science and the Russian Foundation for Basic Research (Grants 080200018, 090200664 and 120291503CERN), the Ministry of Education, Culture, Sports, Science and Technology, Japan, GrantinAid for Scientific Research (Grants 18071005, 19034011, 19740162, 20740160 and 20039012), the German Research Foundation (Grant GA 1480/22), the EUfunded Marie Curie Outgoing Fellowship, Grant PIOFGA2013624803, the Bulgarian Nuclear Regulatory Agency and the Joint Institute for Nuclear Research, Dubna (bilateral contract No. 4418115/17), Ministry of Education and Science of the Republic of Serbia (Grant OI171002), Swiss Nationalfonds Foundation (Grant 200020117913/1) and ETH Research Grant TH01 073. Finally, it is a pleasure to thank the European Organisation for Nuclear Research for strong support and hospitality and, in particular, the operating crews of the CERN SPS accelerator and beam lines who made the measurements possible.
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