1 Introduction

The study of strange baryon resonances in proton–proton (pp) collisions contributes to the understanding of hadron production mechanisms and provides a reference for tuning QCD-inspired event generators. The strange-quark content makes these baryons a valuable tool in understanding production mechanisms, since the initial state colliding projectiles contain no strange valence quarks and therefore all strange particles are created in the collision.

In addition, a measurement of resonance production in the pp system serves as a reference for understanding resonance production in heavy-ion collisions, where resonances, due to their lifetime of a few fm/\(c\) being comparable to the lifetime of the hadronic phase, are sensitive probes of the dynamical evolution of the fireball. Previous measurements at a collision energy of \(\sqrt{s}\) \(=\) 0.2 TeV with the STAR detector at the RHIC have shown that the yields of \({\Sigma }\)(1385) in Au–Au in comparison to pp collisions indicate the presence of rescattering and regeneration in the time span between chemical and kinetic freezeout [1]. Forthcoming analysis of strange baryon resonances in Pb–Pb collisions by the ALICE collaboration will further explore those effects at higher energy and density of the colliding system. The results for the \({\Sigma }\)(1385)\(^{\pm }\) and \({\Xi (1530)^{0}}\) baryons in pp collisions will therefore serve as benchmark.

Measurements of differential (\(\mathrm{d}^2N/(\mathrm{d}y \mathrm{d}p_{\mathrm{T}})\)) and integrated (d\(N\)/d\(y\)) yields of the \({\Sigma }\)(1385)\(^{\pm }\) and \({\Xi (1530)^{0}}\) baryons are presented at mid-rapidity (\(\left| y \right| \) \(<0.5\)) in inelastic (INEL) pp collisions at \(\sqrt{s}\) \(=\) 7 TeV, collected with the ALICE detector [2] at the LHC. The differential spectra are compared to Monte Carlo (MC) event generators. The mean transverse momentum \(\langle p_{\mathrm{T}}\rangle \) is compared to those of other particles measured in pp collisions with the ALICE detector at both \(\sqrt{s}\) \(=\) 7 TeV and \(\sqrt{s}\) \(=\) 0.9 TeV, and with the STAR detector at \(\sqrt{s}\) \(=\) 0.2 TeV.

The \(\Xi \)(1530) reconstruction channel \(\Xi \pi \) is additionally analysed to investigate evidence of the \(\phi (1860)\) pentaquark, previously reported by the NA49 experiment [3]. No such signal was observed by other experiments at different energies and with different beams and reactions [414].

This article is organized as follows. Section 2 gives a brief description of the main detectors used for this analysis and the experimental conditions. Section 2.1 describes track and topological selections. Signal extraction methods are presented in Sect. 2.2, and the efficiency corrections in Sect. 2.3. The evaluation of systematic uncertainties is discussed in Sect. 2.4. In Sect. 3, the \(p_{\mathrm{T}}\) spectra and the integrated yields of the studied particle species are given and compared to model predictions. In Sect. 4 the search for the \(\phi (1860)\) pentaquark is discussed. Conclusions are presented in Sect. 5.

Table 1 Particles involved in this analysis and their PDG parameters [17]. Antiparticles are not listed for brevity. From [17], \({\Xi (1530)^{0}}\) \(\longrightarrow \Xi + \pi \) has a branching ratio of \(\sim \) 100 %, then \({\Xi (1530)^{0}}\) \(\longrightarrow \) \({\Xi }\) \(^{-}+\) \({\uppi ^+}\) has a branching ratio of \(\sim \) 66.7 % due to isospin considerations
Table 2 Track selection criteria. PV primary vertex, DCA \({_r}\) and DCA \(_{z}\) distances of closest approach in the transverse plane and in the longitudinal direction, respectively

2 Experiment and data analysis

The ALICE detector [2] is designed to study a variety of colliding systems, including pp and lead-lead (Pb–Pb) collisions, at TeV-scale energies. The sub-detectors used in this analysis are described in the following. A six-layer silicon inner tracking system (ITS) [15] and a large-volume time projection chamber (TPC) [16] enable charged particle reconstruction with excellent momentum and spatial resolution in full azimuth down to a \(p_{\mathrm{T}}\) of 100 MeV\(/c\) in the pseudorapidity range \(|\eta |<0.9\). The primary interaction vertex is determined with the TPC and ITS detectors with a resolution of 200 \(\upmu \)m for events with few tracks (\(N_\mathrm{{ch}}\simeq 3\)) and below 100 \(\upmu \)m for events with higher multiplicity (\(N_\mathrm{{ch}}\gtrsim 25\)). In addition, both detectors are able to provide particle identification (PID) via energy-loss measurements.

The data analysis is carried out using a sample of \(\sim \) 250 million minimum-bias pp collisions at \(\sqrt{s}\) \(=\) 7 TeV collected during 2010.

During the data-taking period, the luminosity at the interaction point was kept in the range \(0.6{-}1.2\times 10^\mathrm{29}\) cm\(^\mathrm{{-2}}\) s\(^\mathrm{{-1}}\). Runs with a mean pile-up probability per event larger than 2.9 % are excluded from the analysis. The vertex of each collision is required to be within \(\pm \)10 cm of the detector’s centre along the beam direction. The event vertex range is selected to optimize the reconstruction efficiency of particle tracks within the ITS and TPC acceptance.

2.1 Particle selections

The resonances are reconstructed via their hadronic decay channel, shown in Table 1 together with the branching ratio (BR).

For \({\Sigma }\)(1385), all four charged species (\({\Sigma }\)(1385)\(^{+}\), \({\Sigma }\)(1385)\(^{-}\), \({\overline{\Sigma }}\)(1385)\(^{-}\) and \({\overline{\Sigma }}\)(1385)\(^{+}\)) are measured separately.

\({\Xi (1530)^{0}}\) is measured together with its antiparticle (\({\overline{\Xi } (1530)^{0}}\)) due to limited statistics. Therefore in this paper, unless otherwise specified, \({\Xi (1530)^{0}}\) \(\equiv (\) \({\Xi (1530)^{0}}\)+\({\overline{\Xi } (1530)^{0}}\) \()/2\).

Note that, for brevity, antiparticles are not listed and the selection criteria, described in the following, are discussed for particles; equivalent criteria hold for antiparticles.

Several quality criteria, summarized in Table 2, are used for track selection.

Charged pions from the strong decay of both \({\Sigma }\)(1385) and \({\Xi (1530)^{0}}\) are not distinguishable from primary particles and therefore primary track selections are used. They are requested to have a distance of closest approach (DCA) to the primary interaction vertex of less than 2 cm along the beam direction and a DCA in the transverse plane smaller than 7 \(\sigma _{\mathrm {DCA}}\)(\(p_{\mathrm{T}}\)), where \(\sigma _{\mathrm {DCA}}\)(\(p_{\mathrm{T}}\)\(=\) (0.0026 \(+\) 0.0050 GeV\(/c\) \(\times \) \(p_{\mathrm{T}}\) \(^{-1}\)) cm is the parametrization which accounts for the \(p_{\mathrm{T}}\)-dependent resolution of the DCA in the transverse plane [18]. Primary tracks are also required to have at least one hit in one of the two innermost layers of the ITS (silicon pixel detector, SPD) and at least 70 reconstructed clusters in the TPC out of the maximum 159 available, which keeps the contamination from secondary and fake tracks small, while ensuring a high efficiency and good d\(E\)/d\(x\) resolution.

Tracks close to the TPC edge or with transverse momentum \(p_{\mathrm{T}}\) \(< 0.15\) GeV\(/c\) are rejected because the resolution of track reconstruction deteriorates.

In the \({\Sigma }\)(1385) analysis, PID is implemented for \({\uppi ^\pm }\) and p from \({\Lambda }\).

Particles are identified based on a comparison of the energy deposited in the TPC drift gas and an expected value computed using a Bethe–Bloch parametrization [19]. The filter is set to 3 \(\sigma _\mathrm{TPC}\), where \(\sigma \) is the resolution estimated by averaging over reconstructed tracks. An averaged value of \(\sigma _\mathrm{TPC}\) \(=\) 6.5 % is found over all reconstructed tracks [20].

PID selection criteria are not applied in the \({\Xi }\)(1530) analysis as the combinatorial background is sufficiently removed through topological selection.

\(\Lambda \) produced in the decay of \({\Sigma }\)(1385) decays weakly into \(\pi ^{-}\)p with \( {c}\tau =\) 7.89 cm [17]. These pions and protons do not originate from the primary collision vertex, and thus they are selected using a DCA to the interaction point greater than 0.05 cm. At least 70 reconstructed clusters in the TPC are requested for these tracks. Further selection criteria to identify \({\Lambda }\) are applied on the basis of the decay topology as described in [19]. Selection criteria for \({\Lambda }\) used in the \({\Sigma }\)(1385) analysis are summarized in Table 3.

\(\Xi ^{-}\) produced in the decay of the \({\Xi (1530)^{0}}\) decays weakly into \(\Lambda \pi ^{-}\) with \( {c}\tau =\) 4.91 cm [17]. Pions are selected from tracks with a DCA to the interaction point greater than 0.05 cm. Pions and protons from \(\Lambda \) are required to have a DCA to the interaction point greater than 0.04 cm. All pions and protons are requested to have at least 70 reconstructed clusters in the TPC. Decay topologies for \(\Xi ^{-}\) and \(\Lambda \) are used as described in [19]. Selection criteria are summarized in Table 4.

Table 3 Selection criteria used in the \({\Sigma }\)(1385) analysis. PV primary vertex, \(R_{r}\) transverse radius of the decay vertex

All these criteria are optimized to obtain maximum signal significance. Values for the significance are presented in Sect. 2.2.3.

2.2 Signal extraction

2.2.1 Combinatorial background and event-mixing

Due to their very short lifetime of a few fm/\(c\), resonance decay products originate from a position that is indistinguishable from the primary vertex. Thus, the computation of invariant mass distributions for potential resonance decay candidates has significant combinatorial background that has to be subtracted to ensure reliable yield determination.

This is shown in the left panels of Figs. 1 and 2 (for \({\Sigma }\)(1385)\(^{+}\) and \({\Sigma }\)(1385)\(^{-}\), respectively) and Fig. 3 (for the \({\Xi (1530)^{0}}\)).

Figures similar to Figs. 1 and 2 are obtained for the antiparticles \({\overline{\Sigma }}\)(1385)\(^{-}\) and \({\overline{\Sigma }}\)(1385)\(^{+}\). In Fig. 2 the peak from \(\Xi ^{-}\longrightarrow \Lambda + \pi ^{-}\) is visible.

Table 4 Same as Table 3 but for the \({\Xi }\)(1530) analysis
Fig. 1
figure 1

Left panel The \({\Lambda }\) \({\uppi ^+}\) invariant mass distribution in \(\left| y \right| \) \(<\) 0.5 for the transverse momentum bin 1.2 \(<\) \(p_{\mathrm{T}}\) \(<\)1.4 GeV\(/c\)  in pp collisions at \(\sqrt{s}\) \(=\) 7 TeV. The background shape estimated using pairs from different events (event-mixing) is shown as open red squares. The mixed-event background is normalized in the range 1.56 \(<M<\) 2.0 GeV\(/c^2\), where \(M\) is the \({\Lambda }\) \({\uppi ^+}\) invariant mass. Right panel The invariant mass distribution after mixed-event background subtraction for 1.2 \(<\) \(p_{\mathrm{T}}\) \(<\)1.4 GeV\(/c\). The solid curve is the result of the combined fit (see text for details) and the dashed lines describes the residual background

Fig. 2
figure 2

Same as Fig. 1 but for \({\Sigma }\)(1385)\(^{-}\) \(\longrightarrow \) \({\Lambda }\) \(+\) \({\uppi ^-}\). Note the peak at around the \(\Xi (1321)^{-}\) mass, which is absent in Fig. 1

Fig. 3
figure 3

Left panel The \({\Xi }\) \(^{-}\) \({\uppi ^+}\) invariant mass distribution in \(\left| y \right| \) \(<\) 0.5 for the transverse momentum bin \(1.2<\) \(p_{\mathrm{T}}\) \(~<1.6\) GeV\(/c\)  in pp collisions at \(\sqrt{s}\) \(=\) 7 TeV. The background shape estimated using pairs from different events (event-mixing) is shown as open red squares. The mixed-event background is normalized in the range 1.49 \(<M<\) 1.51 GeV\(/c^2\). Right panel The invariant mass distribution after mixed-event background subtraction for \(1.2<\) \(p_{\mathrm{T}}\) \( <1.6\) GeV\(/c\). The solid curve is the result of the combined fit and the dashed line describes the residual background

The combinatorial background distributions are obtained and subtracted from the invariant mass distribution by means of a mixed-event technique, in which a reference background distribution is built with uncorrelated candidates from different events. To avoid mismatch due to different acceptances and to ensure a similar event structure, only tracks from events with similar vertex positions \(z\) (\(\Delta z <1\) cm) and track multiplicities \(n\) (\(\Delta n <10\)) are mixed. In order to reduce statistical uncertainties, each event is mixed with several other events (5 in the \({\Sigma }\)(1385) analysis and \(>20\) in the \({\Xi (1530)^{0}}\) analysis), so that the total number of entries in the mixed-event invariant mass distribution is higher than the total number of entries in the distribution from the same event. Thus the mixed-event distribution needs to be scaled before it can be used to describe the background in the same-event distribution. For \({\Sigma }\)(1385), the regions for the normalization of the mixed-event distribution are selected in the rightmost part of the invariant mass window, where the residual background is absent (see Sect. 2.2.2 for a description of the residual background). These regions are different for the different \(p_{\mathrm{T}}\) bins, ranging from 1.48 \(<M<\) 2.0 GeV\(/c^2\), for the lowest \(p_{\mathrm{T}}\) bin, to 1.95 \(<M<\) 2.0 GeV\(/c^2\), for the highest \(p_{\mathrm{T}}\) bin (\(M\) being the invariant mass of \({\Sigma }\)(1385) and 2.0 GeV\(/c^2\) being the upper extreme of the invariant mass window). The reason for this \(p_{\mathrm{T}}\)-dependent choice is due to the reach of the residual background, which is higher in invariant mass for higher \(p_{\mathrm{T}}\). Fixed regions, 1.6 \(<M<\) 1.8 GeV\(/c^2\) and 1.8 \(<M<\) 2.0 GeV\(/c^2\), have also been tried, giving a systematic uncertainty of \(\sim \) 1 %. For \({\Xi (1530)^{0}}\) a fixed region 1.49 \(<M<\) 1.51 GeV\(/c^2\), just at the left of the signal, is selected. A fixed region can be selected because for all \(p_{\mathrm{T}}\) intervals the background shape is similar and the invariant mass resolution on the reconstructed peak is the same. The uncertainty in the normalization (\(\sim \) 1 %), which is included in the quoted systematic uncertainty for signal extraction, is estimated by using another normalization region, 1.56 \(<M<\) 1.58 GeV\(/c^2\), just at the right of the signal. The open squares in the left panels of Figs. 12 and 3 correspond to the properly scaled mixed-event invariant mass distribution.

The right panels show the signals for each resonance after the mixed-event combinatorial background is subtracted.

2.2.2 Residual correlated background

The mixed-event technique removes only uncorrelated background pairs in the invariant mass spectrum. The consequence is that residual correlations near the signal mass range are not subtracted by the mixed-event spectrum and correlated background pairs remain [21]. This is especially dominant for \({\Sigma }\)(1385) (see Figs. 12, right), for which the correlated residual background takes contributions from two dominant sources:

  • Type A: correlated \(\Lambda \pi \) pairs coming from the decays of other particles which have \({\Lambda }\) and \(\pi \) among the decay products.

  • Type B: correlated \(\Lambda \pi \) pairs which come from the dynamics of the collision and are not removed from the subtraction of the mixed-event background.

All these contributions are present in the MC, albeit with potentially incorrect proportions. Thus, simulations are used to determine the shapes of such contributions in invariant mass space and then these contributions are renormalized using data, as described later.

All the sources of contamination of Type A, which can potentially produce correlated \(\Lambda \pi \) pairs, are listed in Table 5. A similar scheme, not discussed for sake of brevity, is valid for the antiparticles (e.g. the \(\overline{\Xi }^{+}\) \(\longrightarrow \) \({\overline{\Lambda }}\) \({\uppi ^+}\) decay channel affects the reconstruction of \({\overline{\Sigma }}\)(1385)\(^{+}\)). Only sources A1, A5 and A6 in Table 5 give a significant contribution to the correlated residual background of Type A. This is discussed in the following.

Table 5 Potential sources of contamination in the reconstruction of \({\Sigma }\)(1385). Checkmarks show which species is potentially affected. Checkboxes further indicate whether the source gives a significant contamination (see text). A similar scheme, not shown for sake of brevity, is valid for the antiparticles

Source A1 in Table 5 is due to the primary \(\Xi ^{-}\) which decays weakly to \(\Lambda \pi ^{-}\), affecting the reconstruction of \({\Sigma }\)(1385)\(^{-}\). Since the \(\Xi ^{-}\) hyperon is metastable, it shows up in the \(\Lambda \pi ^{-}\) invariant mass spectrum as a very narrow peak at around the \(\Xi ^{-}\) mass, \(M_\mathrm{{\Xi ^{-}}}=\) 1321.71 MeV/\(c^{\mathrm 2}\) [17], just on the left tail of the \({\Sigma }\)(1385)\(^{-}\) signal. The \(\Xi ^{-}\) peak is clearly seen in Fig. 2. This contribution, which is expected to be important since the yield of \(\Xi ^{-}\) is comparable to the yield of \({\Sigma }\)(1385)\(^{-}\), is in fact suppressed, by an order of magnitude, because the filter on the DCA to the primary vertex of both \(\Lambda \) and \(\pi \) filters out most of the \({\Lambda }\) \(\pi \) pairs from \(\Xi ^{-}\). Indeed, the filter on the DCA to the primary vertex is optimized for the \({\Sigma }\)(1385) decay products, which are not distinguishable from primary particles (see Sect. 2.1), whereas \(\Lambda \) and \(\pi \) from \(\Xi ^{-}\) come from a secondary vertex, centimetres away from the primary vertex. Only a small percentage of the \(\Xi ^{-}\) yield survives the filter on the DCA. Source A1 is taken into account by adding a Gaussian function, with the mean value fixed to the \(\Xi ^{-}\) mass and the width and normalization left free, to the combined fit of the invariant mass spectrum in the reconstruction of \({\Sigma }\)(1385)\(^{-}\). The contamination from \(\Xi ^{-}\) reaches about 5–10 % of the raw \({\Sigma }\)(1385)\(^{-}\) signal and varies little with \(p_{\mathrm{T}}\).

Sources A2, A3 and A4 give a negligible contribution. Sources A2 and A3 are due to the hadronic decay channels of \(\Xi (1530)^{-}\), with BR \(=\) 33.3 % and BR \(=\) 66.7 %, respectivelyFootnote 1, and, like A1, affect only the \({\Sigma }\)(1385)\(^{-}\) reconstruction. Source A4 is due to \(\Xi (1530)^{0}\) and potentially affects the reconstruction of both \({\Sigma }\)(1385)\(^{+}\) and \({\Sigma }\)(1385)\(^{-}\), since it involves two opposite-sign pions. The same topological considerations hold for A2 as they do for A1, since it involves a \(\Xi ^{-}\). Indeed, this \(\Xi ^{-}\) comes from the strong decay of \(\Xi (1530)^{-}\), therefore it is practically not distinguishable from the (primary) \(\Xi ^{-}\) in A1. Unlike contribution A1, a further suppression, of about an order of magnitude with respect to A1, comes from both the smaller yield of \(\Xi (1530)^{-}\) with respect to the primary \(\Xi ^{-}\), and the BR of the \(\Xi (1530)^{-}\rightarrow \Xi ^{-}\pi ^{0}\) channel. This further suppression makes contribution A2 practically negligible. Similar conclusions hold for contributions A3 and A4.

Source A5 in Table 5 is related to the second \(\Sigma (1385)\) decay channel, \(\Sigma (1385)^{\pm }\rightarrow \Sigma ^{0}\pi ^{\pm }\) (BR \(=\) \(5.8\) %Footnote 2), with \(\Sigma ^{0}\rightarrow \Lambda \gamma \) (BR \(\simeq \) 100 % [17]). \(\Lambda \) from \(\Sigma ^{0}\) is paired with \(\pi ^{\pm }\) from \(\Sigma (1385)^{\pm }\). This gives a Gaussian-like peak at around 1.306 GeV\(/c^2\), with a width of \(\sim \) 0.059 GeV\(/c^2\) (FWHM). This peak is used in the combined fit to the signal (see below) with a relative normalization with respect to the signal which accounts for the ratio (\(=\)0.067) between the BR (\(=\)5.8 %) for the \(\Sigma (1385)^{\pm }\rightarrow \Sigma ^{0}\pi ^{\pm }\) channel and the BR (\(=\)87 %) for the \(\Sigma (1385)^{\pm }\rightarrow \Lambda \pi ^{\pm }\) channel.

Source A6 in Table 5 is due to the \(\Lambda (1520)\rightarrow \Lambda \pi ^{\pm }\pi ^{\mp }\) channel (BR \(=\) \(5\) %Footnote 3). The positive (negative) pion, paired with \(\Lambda \), produces a Gaussian-like peak, which contaminates the invariant mass distribution of \({\Sigma }\)(1385)\(^{+}\) (\({\Sigma }\)(1385)\(^{-}\)). This peak is centred at \(\sim \) 1.315 GeV\(/c^2\) and has a width of \(\sim \) 0.076 GeV\(/c^2\) (FWHM). The peak is used in the combined fit to the signal. The normalization of the peak is kept free in the fit since the \(\Lambda \)(1520) yield is not measured. The contamination from \(\Lambda \)(1520) decreases with increasing \(p_{\mathrm{T}}\), ranging from about 75 % of the raw \({\Sigma }\)(1385)\(^{-}\) signal in the first \(p_{\mathrm{T}}\) interval, down to 0 for \(p_{\mathrm{T}}\) \(>\) 4 GeV\(/c\).

A third-degree polynomial is used to fit the residual background of Type B in the MC. The fit to MC data is performed in the region from 1.26 GeV\(/c^2\) (just left of the signal region) to the lower edge of the event-mixing normalization region. The fitting function is then normalized to the residual background in real data; the normalization is done in the region from 1.46 GeV\(/c^2\) (just right of the signal region) to the lower edge of the event-mixing normalization region, where other sources of contamination are absent. The lower point of the normalization region is the same for all \(p_{\mathrm{T}}\) intervals since the mean, the width and the invariant mass resolution on the reconstructed peak stay the same over all the \(p_{\mathrm{T}}\) range considered. Comparable results are obtained from using different event generators (PYTHIA 6.4, tune Perugia 0 [22], and PHOJET [23]) and other degrees for the polynomial (second and fourth). The differences of about 2 % are included in the systematic uncertainties.

The invariant mass distribution is fitted with a combined fit function: a (non-relativistic) Breit–Wigner peak plus the functions that make up the residual background (Figs. 12, right). The Breit–Wigner width \(\Gamma \) is kept fixed to the PDG value to improve the stability of the fit.

For \({\Xi (1530)^{0}}\), the residual background after the mixed-event background subtraction is fitted with a first-degree polynomial. The fitting procedure is done in three stages. First, the background is fitted alone from 1.48 to 1.59 GeV\(/c^2\) while excluding the \(\Xi \)(1530)\(^0\) mass region from 1.51 to 1.56 GeV\(/c^2\). Second, a combined fit for signal and background is performed over the full range with the background polynomial fixed to the results from the first fit stage; a Voigtian function—a convolution of Breit–Wigner and Gaussian functions—is used for the signal. The Gaussian part accounts for detector resolution. Third, a fit is redone over the full range again with all parameters free but set initially to the values from the second stage.

2.2.3 Counting signal and signal characteristics

The above procedure is applied for 10 (8) \(p_{\mathrm{T}}\) bins for \({\Sigma }\)(1385) (\({\Xi (1530)^{0}}\)), from 0.7 to 6.0 (0.8 to 5.6) GeV\(/c\). For \({\Sigma }\)(1385), the fit is repeated leaving the Breit–Wigner width \(\Gamma \) free to move, and, for each \(p_{\mathrm{T}}\) interval, the difference in the yield is included in the systematic uncertainties (\(\sim \) 4 % maximum contribution). The widths of both \({\Sigma }\)(1385) and \({\Xi (1530)^{0}}\) are consistent with the PDG values for all \(p_{\mathrm{T}}\) intervals. In the \({\Sigma }\)(1385)\(^{-}\) analysis, a Gaussian function, centred at 1.321 GeV\(/c^2\) and with a starting value for the width of 2 MeV/\(c^{\mathrm 2}\), is used to help the combined fit around the \(\Xi \)(1321)\(^{-}\) peak (Fig. 2). The value of 2 MeV/\(c^{\mathrm 2}\) is obtained from the analysis of \(\Xi \)(1321)\(^{-}\) [19] and is related to the mass resolution. Since the \({\Sigma }\)(1385) mass binning of 8 MeV/\(c^{\mathrm 2}\), which is optimised for the \(\chi ^{2}\) of the combined fit, is larger than the mass resolution, only a rough description of the \(\Xi \)(1321)\(^{-}\) peak is possible. For \({\Xi (1530)^{0}}\), the standard deviation of the Gaussian component of the Voigtian peak is found to be \(\sim \) 2 MeV/\(c^{\mathrm 2}\), which is consistent with the detector resolution, as obtained from the MC simulation. At low \(p_{\mathrm{T}}\), the fitted mass values for \({\Sigma }\)(1385) are found to be slightly lower (by \(\sim \) 5 MeV/\(c^{\mathrm 2}\)) than the PDG value, which is attributed to imperfections in the corrections for energy loss in the detector material. For \({\Xi (1530)^{0}}\), the reconstructed masses are found to be in agreement with the PDG value within the statistical uncertainties.

The raw yields \(N^\mathrm{RAW}\) are obtained by integrating the Breit–Wigner function. As an alternative, \(N^\mathrm{RAW}\) is calculated by integrating the invariant mass histogram after the subtraction of the event-mixing background and subtracting the integral of the residual background (bin-counting method). The difference between the two methods of integration is lower than 2 % on average.

Significance values (defined as \(S/\sqrt{S+B}\), where \(S\) is the signal and \(B\) the background) for \({\Sigma }\)(1385)\(^+\) (\({\Xi (1530)^{0}}\)) are found to be 16.6 (16.5) in the lowest \(p_{\mathrm{T}}\) interval, and 20.9 (22.8) in the highest \(p_{\mathrm{T}}\) interval, and reached 24.2 (52.4) in the intermediate \(p_{\mathrm{T}}\) interval. Significance values comparable to those of \({\Sigma }\)(1385)\(^+\) are obtained for the other \({\Sigma }\)(1385) species.

2.3 Correction and normalization

In order to extract the baryon yields, \(N^\mathrm{RAW}\) are corrected for BR, the geometrical acceptance (\(A\)), the detector efficiency (\(\epsilon \)) and the correction factor which accounts for the GEANT3 overestimation of the \(\bar{\mathrm{p}}\) cross sections (\(\epsilon _\mathrm{GEANT3/FLUKA}\)) [24]

$$\begin{aligned} N^\mathrm{cor}(p_{\mathrm{T}})=\frac{N^\mathrm{RAW}(p_{\mathrm{T}})}{\mathrm{BR}\; (A\times \epsilon )(p_{\mathrm{T}})}\; \epsilon _\mathrm{GEANT3/FLUKA}(p_{\mathrm{T}}). \end{aligned}$$

The product of acceptance and efficiency (\(A \times \epsilon \)) is determined from MC simulations with the PYTHIA 6.4 event generator (tune Perugia 0 [22]) and a GEANT3-based simulation of the ALICE detector response [25]. The \(\epsilon _\mathrm{GEANT3/FLUKA}\) correction factor is equal to 0.99 for the protons from \({\Sigma }\)(1385)\(^{\pm }\) and \({\Xi (1530)^{0}}\) and ranges from 0.90 to 0.98, from the lowest to the highest \(p_{\mathrm{T}}\) interval, for the antiprotons from \({\overline{\Sigma }}\)(1385)\(^{\pm }\) and \({\overline{\Xi } (1530)^{0}}\). About \(200\times 10^6\) Monte-Carlo events, with the same vertex distribution as for the real events, were analysed in exactly the same way as for the data. The \(A\times \epsilon \) is determined from MC simulations as the ratio of the number of reconstructed resonances to the number of those generated in \(\left| y \right| \) \(<0.5\), differentially as a function of transverse momentum, as shown in Fig. 4.

Fig. 4
figure 4

The product of acceptance, efficiency and branching ratio of \({\Sigma }\)(1385)\(^+\) and \({\Xi }\)(1530)\(^0\), obtained with PYTHIA 6.4 [22] and GEANT3 [25], as function of \(p_{\mathrm{T}}\) in \(\left| y \right| \) \(<\)0.5. Only statistical uncertainties are reported. The dashed- and the dash-dotted lines indicate the overall branching ratio for the two reconstruction channels

The drop in efficiency at low \(p_{\mathrm{T}}\) is due to the loss of slow pions involved in the decay chain. As a cross-check, the efficiency \(\times \) acceptance has also been assessed with PHOJET [23] as event generator. The relative difference of the resulting \(A \times \epsilon \), averaged over the various \(p_{\mathrm{T}}\) intervals, is below 1 %.

Finally, corrections for the trigger inefficiency (\(\epsilon _\mathrm{trigger}\)) and the loss of candidates outside of the \(z\)-vertex range (\(\epsilon _\mathrm{vert}\)) are applied via

$$\begin{aligned} \frac{1}{N_\mathrm{INEL}}\frac{\mathrm {d}^2N}{\mathrm {d}y\mathrm {d}p_{\mathrm{T}}} = \frac{N^\mathrm{cor}(p_{\mathrm{T}})}{\Delta y \Delta p_{\mathrm{T}}} \; \frac{\epsilon _\mathrm{trigger} }{\epsilon _\mathrm{vert} }\; \frac{1}{N_ \mathrm{MB}}, \end{aligned}$$

where \(N^{\mathrm {cor}}\) and \(N_ {\mathrm {MB}}\) are the number of reconstructed \({\Sigma }\)(1385) or \({\Xi }\)(1530) and the total number of minimum bias triggers, respectively. \(\Delta \) \(y\) and \(\Delta \) \(p_{\mathrm{T}}\) are the rapidity window width and the \(p_{\mathrm{T}}\) bin width, respectively. The trigger selection efficiency for inelastic collisions \(\epsilon _{\mathrm {trigger}}\) is equal to \(0.852 ^{+0.062} _{-0.030}\) [26]. The loss of resonances due to the trigger selection, estimated by MC simulations, is negligible, less than 0.2 %. The \(\epsilon _\mathrm{vert}\) correction factor accounts for resonance losses (\(\sim \) 7 %) due to the requirement to have a primary vertex \(z\) position in the range \(\pm \)10 cm.

2.4 Systematic uncertainties of \(p_{\mathrm{T}}\) spectra

Two types of systematic uncertainties in the particle spectra are considered: \(p_{\mathrm{T}}\)-dependent systematic uncertainties, which are due to the selection efficiency and signal extraction at a given \(p_{\mathrm{T}}\), and \(p_{\mathrm{T}}\)-independent uncertainties due to the normalization to inelastic collisions and other corrections.

The minimum and maximum values of the major contributions to the point-to-point systematic uncertainties are listed in Table 6.

Table 6 Summary of the systematic uncertainties in the \({\Sigma }\)(1385) and \(\Xi (1530)\) differential yield, \(\mathrm{d}^2N/(\mathrm{d}y \mathrm{d}p_{\mathrm{T}})\)

The uncertainties introduced by tracking, topology selection and PID are obtained by varying the selection criteria for the decay products. To this purpose, the selection criteria listed in Tables 23 and 4 are changed by a certain amount which varies the raw yield in real data by \(\pm \)10 %. The maximum difference between the default yield and the alternate value obtained by varying the selection, is taken as systematic uncertainty. The uncertainties introduced by the signal extraction come from several sources: normalization of the event-mixing background, fitting function and range of the residual background, signal fitting and integration. For \({\Sigma }\)(1385), the contamination from the \(\Lambda \)(1520) introduced the largest contribution (\(\sim \) 8 %). All the sources are combined by summing in quadrature the uncertainties for each \(p_{\mathrm{T}}\).

Among the \(p_{\mathrm{T}}\)-independent uncertainties, the INEL normalization leads to a +7.3 % and \(-\)3.5 % uncertainty [26], the determination of the material thickness traversed by the particles (material budget) introduces a 4 % uncertainty and the use of FLUKA [27, 28] to correct the antiproton absorption cross section in GEANT3 leads to a further 2 % uncertainty [24]. For \({\Sigma }\)(1385), a further 1.5 % comes from the uncertainty in the branching ratio. A summary of the \(p_{\mathrm{T}}\)-independent uncertainties is presented in Table 6.

3 Results

The corrected baryon yields per \(p_{\mathrm{T}}\) interval per unit rapidity (1/\(N_\mathrm{INEL}\) \(\times \) \(\mathrm{d}^2N/(\mathrm{d}y \mathrm{d}p_{\mathrm{T}})\)) are shown in Fig. 5. They cover the ranges 0.7 \(<\) \(p_{\mathrm{T}}\) \(<\) 6.0 GeV\(/c\) for \({\Sigma }\)(1385) and 0.8 \(<\) \(p_{\mathrm{T}}\) \(<\) 5.6 GeV\(/c\)  for \({\Xi (1530)^{0}}\).

Fig. 5
figure 5

Inelastic baryon yields, \(\mathrm{d}^2N/(\mathrm{d}y \mathrm{d}p_{\mathrm{T}})\), per \(p_{\mathrm{T}}\) interval per unit rapidity for \({\Sigma }\)(1385) and \({\Xi (1530)^{0}}\). Statistical and systematic uncertainties are summed in quadrature, excluding the \(p_{\mathrm{T}}\)-independent uncertainties, which affect only the overall normalization of the spectra and are not considered in the fit. Spectra are fitted with a Lévy–Tsallis function. The ratio data/fit is shown in the lower panel. For the sake of visibility, only \({\Sigma }\)(1385)\(^{+}\) is shown in the lower panel, but similar ratios have been obtained for the other three \({\Sigma }\)(1385) species. For the ratio, the integral of the fitting function in each corresponding \(p_{\mathrm{T}}\) interval is considered. Spectra points are represented at the centre of the \(p_{\mathrm{T}}\) interval

Table 7 Parameters extracted from the Lévy–Tsallis (LT) fits (Eq. 3) to the transverse momentum spectra. The values of d\(N\)/d\(y\) are calculated using the spectra in the measured range and the extrapolation of the fitted Lévy–Tsallis function outside the measured range. Systematic uncertainties quoted here are the ones derived from Lévy–Tsallis fit only (see text)

The vertical error bars in Fig. 5 represent the sum in quadrature of the statistical and systematic uncertainties, excluding the \(p_{\mathrm{T}}\)-independent uncertainties, which affect only the normalization.

All spectra are fitted with a Lévy–Tsallis function [29], which is used for most of the identified particle spectra in pp collisions [19, 20, 3032],

$$\begin{aligned}&\frac{1}{N_\mathrm{INEL}}\frac{\mathrm {d}^{2}N}{\mathrm {d}y\mathrm {d}p_{\mathrm{T}}} \nonumber \\&\quad = \frac{(n-1)(n-2)}{nC [nC + m_{0}(n-2)]} \frac{\mathrm {d}N}{\mathrm {d}y} \; {p_{\mathrm{T}}} \left( 1+ \frac{m_{\mathrm {T}}-m_{0}}{nC}\right) ^{-n} , \end{aligned}$$

where \(m_\mathrm{T}=\sqrt{m_{0}^2+p_{\mathrm{T}}^2}\) and \(m_{0}\) denotes the PDG particle mass. This function, quantified by the inverse slope parameter \(C\) and the exponent parameter \(n\), describes both the exponential shape of the spectrum at low \(p_{\mathrm{T}}\) and the power law distribution at large \(p_{\mathrm{T}}\). The parameter d\(N\)/d\(y\) represents the particle yield per unit rapidity per INEL event. d\(N\)/d\(y\), \(C\) and \(n\) are the free parameters considered for this function. Table 7 presents the parameter outcome of the Lévy–Tsallis fit, together with the mean transverse momentum, \(\langle p_{\mathrm{T}}\rangle \), and the reduced \(\chi ^{2}\).

The values of d\(N\)/d\(y\) in Table 7 are obtained by adding the integral of the experimental spectrum in the measured range and the extrapolations with the fitted Lévy–Tsallis function to both \(p_{\mathrm{T}}\) \(=0\) and high \(p_{\mathrm{T}}\). The contribution of the low-\(p_{\mathrm{T}}\) extrapolation to the total d\(N\)/d\(y\) is \(\sim \) 30 % for both \({\Sigma }\)(1385) and \({\Xi (1530)^{0}}\). The contribution of the high-\(p_{\mathrm{T}}\) extrapolation is negligible.

For each species considered here, such a composite d\(N\)/d\(y\) differs very little (\(<\) 1 %) from the value of d\(N\)/d\(y\) as the first free parameter returned by the fit, i.e. from the integration of the fit function from 0 to infinity.

In order to obtain the systematic uncertainty on the parameters of the Lévy–Tsallis fit (d\(N\)/d\(y\),