K ∗ (892 ) 0 meson production in inelastic p+p interactions at 158 GeV/ c beam momentum measured by NA61/SHINE at the CERN SPS
NA61/SHINE Collaboration, A. Aduszkiewicz, E.V. Andronov, T. Antićić, V. Babkin, M. Baszczyk, S. Bhosale, A. Blondel, M. Bogomilov, A. Brandin, A. Bravar, W. Bryliński, J. Brzychczyk, M. Buryakov, O. Busygina, A. Bzdak, H. Cherif, M. Ćirković, M. Csanad, J. Cybowska, T. Czopowicz, A. Damyanova, N. Davis, M. Deliyergiyev, M. Deveaux, A. Dmitriev, W. Dominik, P. Dorosz, J. Dumarchez, R. Engel, G.A. Feofilov, L. Fields, Z. Fodor, A. Garibov, M. Gaździcki, O. Golosov, V. Golovatyuk, M. Golubeva, K. Grebieszkow, F. Guber, A. Haesler, S.N. Igolkin, S. Ilieva, A. Ivashkin, S.R. Johnson, K. Kadija, E. Kaptur, N. Kargin, E. Kashirin, M. Kiełbowicz, V.A. Kireyeu, V. Klochkov, V.I. Kolesnikov, D. Kolev, A. Korzenev, V.N. Kovalenko, S. Kowalski, M. Koziel, A. Krasnoperov, W. Kucewicz, M. Kuich, A. Kurepin, D. Larsen, A. László, T.V. Lazareva, M. Lewicki, K. Łojek, B. Łysakowski, V.V. Lyubushkin, M. Maćkowiak-Pawłowska, Z. Majka, B. Maksiak, A.I. Malakhov, D. Manić, A. Marcinek, A.D. Marino, K. Marton, H.-J. Mathes, T. Matulewicz, V. Matveev, G.L. Melkumov, A.O. Merzlaya, B. Messerly, Ł. Mik, S. Morozov, S. Mrówczyński, Y. Nagai, M. Naskręt, V. Ozvenchuk, V. Paolone, O. Petukhov, R. Płaneta, P. Podlaski, B.A. Popov, B. Porfy, M. Posiadała-Zezula, D.S. Prokhorova, D. Pszczel, S. Puławski, J. Puzović, et al. (40 additional authors not shown)
EEUROPEAN ORGANISATION FOR NUCLEAR RESEARCH (CERN) published in Eur. Phys. J. C80, 460 (2020)DOI: 10.1140/epjc/s10052-020-7955-1 CERN-EP-2020-002May 26, 2020 K ∗ (892) meson production in inelastic p + pinteractions at 158 GeV / c beam momentummeasured by NA61 / SHINE at the CERN SPS
The NA61 / SHINE Collaboration
The measurement of K ∗ (892) resonance production via its K + π − decay mode in inelas-tic p + p collisions at beam momentum 158 GeV / c ( √ s NN = . / SHINE hadron spectrometer at the CERN Super ProtonSynchrotron. The template method was used to extract the K ∗ (892) signal and double-di ff erential transverse momentum and rapidity spectra were obtained. The full phase-spacemean multiplicity of K ∗ (892) mesons was found to be (78 . ± . ± . · − .The NA61 / SHINE results are compared with the E pos + p and nucleus-nucleus collisions. c (cid:13) / SHINE Collaboration.Reproduction of this article or parts of it is allowed as specified in the CC-BY-4.0 license. a r X i v : . [ nu c l - e x ] M a y Introduction and motivation
Strange hadron production is believed to be an important tool to study the dynamics of high-energycollisions. In collisions achieving high energy densities strangeness production was predicted to be en-hanced [1] as a result of the decrease of the mass of strangeness carriers due to partial chiral symmetryrestoration. The K ∗ (892) resonance state contains an ¯ s valence quark and is therefore sensitive to thelevel of strangeness production. Thus, the data on K ∗ (892) meson production provide a more completeunderstanding of hadron chemistry.Measurements of the production of short-lived resonances are a unique tool to understand the less knownaspects of high energy collisions, especially their time evolution. In heavy ion collisions the yields ofresonances may help to distinguish between two possible freeze-out scenarios: the sudden and the grad-ual one [2]. Namely, the ratio of K ∗ (892) to charged kaon production may allow to estimate the timeinterval between chemical (end of inelastic collisions) and kinetic (end of elastic collisions) freeze-out.The lifetime of the K ∗ (892) resonance ( ≈ / c ) is comparable to the expected duration of the rescatter-ing hadronic gas phase between the two freeze-out stages. Consequently, a certain fraction of K ∗ (892) resonances will decay inside the fireball. The momenta of their decay products are expected to be sig-nificantly modified by elastic scatterings, preventing the experimental reconstruction of the resonancevia an invariant mass analysis. In such a case a suppression of the observed K ∗ (892) yield is expected.Such an e ff ect was indeed observed in nucleus-nucleus collisions at Super Proton Synchrotron (SPS)and Relativistic Heavy Ion Collider (RHIC) energies [3, 4, 5, 6, 7, 8]. The ratio of K ∗ / K production( K ∗ stands for K ∗ (892) , K ∗ (892) or K ∗± , and K denotes K + or K − ) showed a decrease with increasingsystem size as expected due to the increasing rescattering time between chemical and kinetic freeze-out.The same e ff ect was recently reported also by the ALICE Collaboration at the Large Hadron Collider(LHC) [9, 10, 11, 12].When looking at the energy dependence of the K ∗ / K − ratio in central Pb + Pb or Au + Au collisions, a bitlarger suppression of K ∗ is observed for the 2.76 TeV LHC energy [10] when compared to the top RHIC( √ s NN =
200 GeV) energy [7], namely K ∗ / K − = . ± .
027 (0 . ± . + Pb reactions at LHC and 0 . ± .
04 for the 0–10% most central Au + Au interactions at RHIC.Those values can be compared with those for p + p interactions, which are 0 . ± .
043 at LHC [10] and0 . ± .
05 at RHIC [7]. Thus, the K ∗ / K − ratio in central Pb + Pb collisions at LHC (2.76 TeV) dropsto 59 (61)% of the value found for p + p interactions. For RHIC energies this drop is similar and equals59%.In the NA49 experiment at the CERN SPS K ∗ (892) and K ∗ (892) meson production was analyzed sep-arately and the corresponding (almost 4 π ) mean multiplicities obtained in the 23.5% most central Pb + Pbcollisions at √ s NN = . ± . . ± .
7, respectively [4]. They can be rescaled(using the mean number of wounded nucleons; factor 362 / . ± . . ± .
3, respectively. Their aver-age, divided by the (cid:104) K − (cid:105) multiplicity (51 . ± .
6) for the 5% most central Pb + Pb collisions [13] re-sults in the ratio 0 . · ( (cid:104) K ∗ (892) (cid:105) + (cid:104) K ∗ (892) (cid:105) ) / (cid:104) K − (cid:105) of 0 . ± .
04 which is similar to the value K ∗ / K − = . ± .
04 measured in the 10% most central Au + Au collisions at RHIC [7]. Finally, theratio 0 . · ( (cid:104) K ∗ (892) (cid:105) + (cid:104) K ∗ (892) (cid:105) ) / (cid:104) K − (cid:105) for p + p interactions at the same SPS energy can be estimatedas 0 . ± .
04 [4, 14]. Thus, at SPS energy the resonance to non-resonance ratio in central Pb + Pb drops In ALICE at LHC and STAR at RHIC papers. e.g. Refs. [10, 12, 7], the results for K ∗ (892) and K ∗ (892) were combinedand averaged and denoted by the symbol K ∗ ; the ratios were measured at mid-rapidity. The K ∗ / K − ratios in Pb + Pb collisions at √ s NN =
2o about 43–44% of the value for p + p interactions. This e ff ect is even stronger than the one observedat RHIC and LHC and might suggest that the lifetime (calculated in the K ∗ rest frame; see Eq. (21) inSec. 5.4) of the hadron gas system created in central nucleus-nucleus collisions at the SPS is longer thanthat at higher energies. Eventually, resonance regeneration processes start to play a role for higher ener-gies counteracting the K ∗ suppression due to rescattering. It should also be pointed out that the wholepicture assumes that the conditions at chemical freeze-out of p + p and Pb + Pb collisions are the same.More detailed calculations of the time between freeze-outs, both in the K ∗ rest frame and in collisioncenter-of-mass reference system, are given in Sec. 5.4.The results for p + p collisions provide an important base-line for heavier nucleus-nucleus systems. So farthe K ∗ / K − ratio for p + p interactions did not show large di ff erences between the top RHIC and four LHCenergies [15, 10, 16, 12]. Most of the results at lower energies are less reliable due to large uncertainties,see the compilation in Ref. [15], and new points in Refs. [4, 10, 16, 12]. This emphasizes the need toobtain high precision p + p data at energies lower than the top RHIC energy. Continuing considerations forp + p collisions, a very intriguing e ff ect was reported in the most recent ALICE analysis of the multiplicitydependence in p + p collisions [17, 18]. The K ∗ / (cid:104) K ± (cid:105) and K ∗ / K S ratios decrease when going from low-multiplicity to high-multiplicity p + p interactions at the LHC energies. This may be an indication of ahadronic phase with significant non-zero lifetime even in p + p collisions.The transverse mass spectra and yields of K ∗ (892) mesons are also important inputs for Blast-Wavemodels (determining kinetic freeze-out temperature and transverse flow velocity) and Hadron ResonanceGas models (determining chemical freeze-out temperature, baryochemical potential, strangeness under-saturation factor, system volume, etc.). Those models significantly contribute to our understanding ofthe phase diagram of strongly interacting matter. In principle, the precise determination of transverseflow velocity is attractive due to the fact, that recent LHC, RHIC and even SPS results suggest that denseand collectively behaving system may appear also in collisions of small nuclei, or even in elementaryinteractions. Finally, the study of resonances in elementary interactions contributes to the understandingof hadron production, due to the fact that products of resonance decays represent a large fraction of thefinal state particles. Resonance spectra and yields provide an important reference for tuning Monte Carlostring-hadronic models.The study of K ∗ (892) and / or K ∗ (892) production in p + p collisions at RHIC energies was performed bythe STAR [5] and PHENIX [19] experiments and at LHC energies by ALICE [15, 10, 16, 12, 20, 18].The NA49 experiment performed the measurements in inelastic p + p collisions at beam momentum of158 GeV / c (CERN SPS) [4]. Also the LEBC-EHS facility at the CERN SPS measured K ∗ (892) and K ∗ (892) production in p + p collisions at 400 GeV / c [21]. Finally, results obtained at the energies of theCERN Intersecting Storage Rings (ISR) were published in Refs. [22, 23].This paper reports measurements of K ∗ (892) resonance production via its K + π − decay mode in inelasticp + p collisions at beam momentum of 158 GeV / c ( √ s NN = . [24]. The data were recorded bythe NA61 / SHINE hadron spectrometer [25] at the CERN SPS. Unlike in the previous NA49 analysis [4]at the same beam momentum, the template method was used to extract the K ∗ (892) signal. This methodwas found to allow a more precise background subtraction than the standard procedure. Moreover, thelarge statistics NA61 / SHINE data (about 52.5M events recorded with the interaction trigger compared to2.5M p + p events analysed in NA49 [26, 27]) allowed to obtain high quality double-di ff erential transversemomentum and rapidity spectra of K ∗ (892) mesons. The paper is organized as follows. Section 2 brieflydescribes the NA61 / SHINE detector. Section 3 discusses the analysis procedures, including event and The analysis of K ∗ (892) as well as K ∗ (892) and K ∗ (892) at lower SPS energies is a subject of future NA61 / SHINE paper.
13 m ToF-LToF-R PSDToF-FMTPC-RMTPC-LVTPC-2VTPC-1 Vertex magnetsTarget GAPTPCBeam S4 S5
S2S1BPD-1 BPD-2 BPD-3V1V1V0THCCEDAR zxy p Figure 1: (Color online) The schematic layout of the NA61 / SHINE experiment at the CERN SPS (horizontal cut,not to scale). The beam and trigger detector configuration used for data taking in 2009 is shown in the inset (seeRefs. [28, 29] for detailed description). The chosen coordinate system is drawn on the lower left: its origin lies inthe middle of the VTPC-2, on the beam axis. track cuts, method of signal extraction, corrections, and evaluation of uncertainties. The final resultsare presented in Section 4 and their comparison with world data and models in Section 5. A summarySection 6 closes the paper.
The NA61 / SHINE experiment [25] uses a large acceptance hadron spectrometer located in the CERNNorth Area. The schematic layout of the NA61 / SHINE detector is shown in Fig. 1. The detailed descrip-tion of the full detector can be found in Ref. [25]. Here only the detector components, which were usedin this analysis, are described.A set of scintillation and Cherenkov counters as well as beam position detectors (BPDs) upstream ofthe spectrometer provide timing reference, identification and position measurements of incoming beamparticles. The trigger scintillator counter S4 placed downstream of the target is used to select events withcollisions in the target area by the absence of a charged particle hit.Secondary beams of positively charged hadrons at 158 GeV / c are produced from 400 GeV / c protons ex-tracted from the SPS accelerator. Particles of the secondary hadron beam are identified by two Cherenkovcounters, a CEDAR [30] (for 158 GeV / c beam CEDAR-N) and a threshold counter (THC). The CEDARcounter, using a coincidence of six out of the eight photo-multipliers placed radially along the Cherenkovring, provides positive identification of protons, while the THC, operated at pressure lower than the pro-ton threshold, is used in anti-coincidence in the trigger logic. A selection based on signals from theCherenkov counters allowed one to identify beam protons with a purity of about 99%. A consistentvalue for the purity was found by bending the beam into the TPCs with the full magnetic field and usingidentification based on its specific ionization energy loss d E / d x [31].4he main tracking devices of the spectrometer are four large volume Time Projection Chambers (TPCs).Two of them, the vertex TPCs (VTPC-1 and VTPC-2), are located in the magnetic fields of two super-conducting dipole magnets with a combined bending power of 9 Tm which corresponds to about 1.5 Tand 1.1 T fields in the upstream and downstream magnets, respectively.Two large main
TPCs (MTPC-L and MTPC-R) are positioned downstream of the magnets symmetricallyto the beam line. The fifth small TPC (GAP TPC) is placed between VTPC-1 and VTPC-2 directly on thebeam line. It closes the gap between the beam axis and the sensitive volumes of the other TPCs. The TPCsare filled with Ar and CO gas mixtures. Particle identification in the TPCs is based on measurements ofthe specific energy loss (d E / d x ) in the chamber gas.The p + p data sets, which are the topic of this paper, were recorded with the proton beam incident on aliquid hydrogen target (LHT), a 20 cm long cylinder positioned about 80 cm upstream of VTPC-1. The results for p + p interactions are based on high-statistics data runs (in years 2009, 2010, and 2011)which recorded about 56 . × collisions (52 . × events selected by the interaction trigger)of the proton beam with a 20 cm long liquid hydrogen target (LHT). The conditions during the threeruns were very similar as demonstrated in Fig. 2 (left) where the z -position (along the beam line) of thereconstructed p + p interaction vertex is shown. For the analysis the range of z -position of the main vertexwas selected to cover mostly the LHT (see Sec. 3.3) in order to maximize the number of good events andminimize the contamination by o ff -target interactions. Figure 2 (right) shows that for the 2009 productionthe ratio of the number of events in the target-removed sample to the number of events in the target-inserted sample (ratio calculated in the range − < z < −
572 cm; histograms normalized in the range − < z < −
300 cm) is on the level of 4.8%, and therefore no correction for non-target interactions wasapplied. An alternative method of analysis (see for example Ref. [28]) would be to measure and subtractthe resonance yields in the target-removed data, but both the standard method and the template -fittingmethod used in this paper cannot be applied to data sets with small statistics such as the target removeddata recorded by NA61 / SHINE. In order to estimate the systematic biases related to the contamination byo ff -target interactions the window of z -position of the main vertex was varied (see Sec. 3.10).Table 1 presents the details of data sets collected in the three separate data taking periods. The numberof events recorded with the interaction trigger, as well as the number of events selected for the analysis(see Sec. 3.3) are shown. One sees that only 44–56% of the events were used for the analysis. Thisdrop is caused mainly by BPD reconstruction ine ffi ciencies and o ff -target interactions accepted by thetrigger. The number of tracks, given in the Table 1, refers to tracks registered in accepted events only.The agreement of the fractions of accepted tracks in the three analyzed data sets confirms the similarityof the data recorded in 2009, 2010 and 2011. For the analysis of K ∗ (892) production these three datasets were combined at the level of preparing invariant mass distributions (Sec. 3.6).5 - - - - - - z (cm) en t r i e s ( a r b . un i t s ) p+p, 2009p+p, 2010p+p, 2011 700 - - - - - - - - - z (cm) en t r i e s ( a r b . un i t s ) p+p, 2009p+p, 2009 empty target Figure 2: (Color online) Left: Distributions of the z -coordinate of the reconstructed interaction vertex ( z ) for eventsrecorded with the target inserted (2009, 2010 and 2011 data). Histograms are normalized to the same integral inthe range − < z < −
572 cm. Right: Distributions of the z -coordinate of the reconstructed interaction vertex fortarget-inserted (solid histogram) and target-removed (dash-dotted histogram) 2009 data. Histograms are normalizedto the same integral in the range − < z < −
300 cm. All event cuts were applied (see Sec. 3.3) with exception ofcut (ii) and (v). Black vertical lines indicate the cuts used for the analysis (see Sec. 3.3).2009 2010 2011 TotalNumber of events 2.87M (100%) 37.78M (100%) 11.88M (100%) 52.53M (100%)selected by interaction triggerNumber of events after cuts 1.26M (43.9%) 19.97M (52.9%) 6.62M (55.7%) 27.85M (53.0%)Number of tracks 8.62M (100%) 136.58M (100%) 45.48M (100%) 190.68M (100%)Number of tracks after cuts 4.81M (55.8%) 76.41M (55.9%) 24.91M (54.8%) 106.13M (55.7%)without d E / d x cutNumber of tracks after all cuts 2.26M (26.2%) 35.79M (26.2%) 11.74M (25.8%) 49.79M (26.1%)Table 1: Data sets used for the analysis of K ∗ (892) production. The same event and track cuts (Sec. 3.3, 3.4 and3.5) were used for all three data taking periods. The details of NA61 / SHINE calibration, track and vertex reconstruction procedures, as well as simula-tions used to correct the reconstructed data, are discussed in Refs. [28, 29, 32]. In the following sectionthe analysis technique developed for the measurement of the K ∗ (892) spectra in p + p interactions isdescribed. The procedure used for the data analysis consists of the following steps:(i) application of event and track selection criteria,(ii) selection of K + and π − candidates based on the measurement of their ionization energy loss (d E / d x )in the gas volume of the TPCs,(iii) creation of invariant mass distribution of K + π − pairs,(iv) creation of invariant mass distribution of K + π − pairs for mixed events and Monte Carlo templates,(v) extraction of K ∗ (892) signal,(vi) application of corrections (obtained from simulations) to the raw numbers of K ∗ (892) ; they includelosses of inelastic p + p interactions due to the on-line and o ff -line event selection as well as lossesof K ∗ (892) due to track and pair selection cuts and the detector geometrical acceptance.6he details of the steps are described in the following subsections. Inelastic p + p interactions were selected by the following criteria:(i) an interaction was recognized by the trigger logic (see Refs. [28, 29] for detailed description),(ii) no o ff -time beam particle was detected within ± µ s around the trigger (beam) particle,(iii) the trajectory of the beam particle was measured in at least one of BPD-1 or BPD-2 and in theBPD-3 detector and was well reconstructed,(iv) the primary interaction vertex fit converged,(v) the fit of the z -coordinate of the primary p + p interaction vertex (see Fig. 2) converged and thefitted z position was found between -590 cm and -572 cm, where the center of the LHT was at-580 cm. The range of this cut was selected to maximize the number of good events and minimizethe contamination by o ff -target interactions,(vi) events with a single, well measured positively charged track with absolute momentum close to thebeam momentum ( p > p beam − / c ) were rejected.The above event cuts select well measured inelastic p + p interactions. The background due to elasticinteractions is removed (cuts (iv) and (vi)). The contribution of o ff -target interactions is reduced (cut (v)).The losses of inelastic interactions due to the event selection procedure were corrected using simulations(see below). The number of events after these cuts is 27 . × . After the event selection criteria a set of track quality cuts were applied to individual tracks. Thesecuts were used to ensure high reconstruction e ffi ciency, proper identification of tracks and to reducethe contamination of tracks from secondary interactions, weak decays and o ff -time interactions. Theindividual tracks were selected by the following criteria:(i) the track fit including the interaction vertex converged,(ii) the total number of reconstructed points on the track should be greater than 30,(iii) the sum of the number of reconstructed points in VTPC-1 and VTPC-2 was greater than 15 or thenumber of reconstructed points in the GAP TPC was greater than 4,(iv) the distance between the track extrapolated to the interaction plane and the interaction point (impactparameter) should be smaller than 4 cm in the horizontal (bending) plane and 2 cm in the vertical(drift) plane,(v) the track momentum (in the laboratory reference system) is in the range 3 ≤ p lab ≤
158 GeV / c ,(vi) the track transverse momentum is required to be smaller than 1.5 GeV / c ,(vii) d E / d x track cuts were applied to select K + and π − candidates (see Sec. 3.5).7 - - /(GeV/c)) lab log(p d E / d x ( a r b . un i t s ) · + K + p + ep - - /(GeV/c)) lab log(p d E / d x ( a r b . un i t s ) · - K - p - ep - - /(GeV/c)) lab log(p d E / d x ( a r b . un i t s ) + K + p + ep - - /(GeV/c)) lab log(p d E / d x ( a r b . un i t s ) · - K - p - ep Figure 3: (Color online) Top: the values of d E / d x versus log( p lab / (GeV / c )) for positively (left) and negatively (right)charged particles after track cuts (i) – (vi) from Sec. 3.4. The Bethe-Bloch curves are also drawn. Bottom: selectionof K + (left) and π − (right) candidates. The number of tracks left after these cuts is about 49 . × . Charged particle identification in the NA61 / SHINE experiment is based on the measurement of theirionization energy loss (d E / d x ) in the gas of the TPCs and of the time of flight ( tof ) obtained from the ToF-Land ToF-R walls. For the region of the relativistic rise of the ionization at large momenta, the measurementof d E / d x alone allows identification. At lower momenta the d E / d x bands for di ff erent particle speciesoverlap and the identification based only on measurements of d E / d x in the TPCs (this analysis) is notenough. For this reason the track cut (v) was applied. In Fig. 3 the d E / d x values as a function of totalmomentum ( p lab ), measured in the laboratory reference system, are shown for positively and negativelycharged particles, separately. The K + and π − candidates were selected by requiring their d E / d x valuesto be within 1 . σ or 3 . σ around their nominal Bethe-Bloch values, respectively. Here σ represents thetypical standard deviation of a Gaussian fitted to the d E / d x distribution of kaons and pions. Since onlysmall variations of σ were observed for di ff erent total momentum and transverse momentum bins, fixedvalues σ = .
044 were used for K + and σ = .
052 for π − . The bands of selected K + and π − candidatesare shown in the bottom panel of Fig. 3 K ∗ (892) signal extraction The raw numbers of K ∗ (892) are usually obtained by performing fits to the invariant mass spectra withthe sum of a background and a signal function. The invariant mass is defined as: m K + π − = (cid:113) ( E K + + E π − ) − ( −−→ p K + + −−→ p π − ) , (1)8here E represents the total energy and (cid:126) p the momentum vector of daughter particles from K ∗ (892) decay.In the standard method ( mixing method ) the large combinatorial background is estimated by invariantmass spectra calculated for K + π − pairs originating from di ff erent events. Figures 4 and 5 (top, left)show combinatorial background histograms (red points) compared to the data histograms of m K + π − (bluepoints). Mixed events were normalized to the same number of pairs as in real data in the invariant massrange from 0.6 to 1.6 GeV. After subracting the normalized mixed event background the blue points inFigs. 4, 5 (bottom, left) were obtained. The K ∗ (892) signal is prominently seen, but the histogram stillshows a residual background, seen especially for low invariant mass values. This residual backgroundprobably comes from the products of other resonance decays, which are not properly accounted for bythe event-mixing, and should be subtracted. The final fit ( total fit 2 ) was performed with the function ofEq. (2) using an additional background component based on a second order polynomial: f ( m K + π − ) = d · ( m K + π − ) + e · ( m K + π − ) + f + g · BW ( m K + π − ) , (2)where d , e , f , and g are free parameters of the fit, and the Breit-Wigner ( BW ) component is described byEq. (3): BW ( m K + π − ) = A · · Γ K ∗ ( m K + π − − m K ∗ ) + Γ K ∗ , (3)where A is the normalization factor, and m K ∗ and Γ K ∗ are also fitted. The initial values of the mass ( m K ∗ )and width ( Γ K ∗ ) parameters of K ∗ (892) were taken from the Particle Data Group (PDG): m K ∗ = m = . Γ K ∗ = Γ = . polynomial background ) in Figs. 4, 5(bottom, left) show the fitted additional background component (Eq. (2) without BW ) and the brown lines( total fit 2 ) the total fit result (Eq. (2)).In order to obtain a better background description compared to the mixing method, the template method was applied. Namely, the invariant mass spectra of the data (blue data points in Figs. 4, 5 (top, right))were fitted with a function given by Eq. (4): f ( m K + π − ) = a · T MCres ( m K + π − ) + b · T DAT Amix ( m K + π − ) + c · BW ( m K + π − ) . (4)The background is described as a sum of two contributions: T MCres and T DAT Amix . T DAT Amix is the backgroundestimated based on the mixing method, which was discussed above. The T MCres template (MC stands forMonte Carlo) is the shape of background, which describes the contribution of K + π − pairs originatingfrom:(i) combination of tracks that come from decays of resonances di ff erent than K ∗ (892) , for exampleone track from a ρ meson and one from a K ∗ + meson,(ii) combination of tracks where one comes from decay of a resonance and one comes from directproduction in the primary interaction.The T MCres templates were constructed by passing p + p interactions, generated with the E pos / SHINE detector sim-ulation chain and then through the same reconstruction routines as the data. The simulation keeps thehistory of particle production thus allowing to identify their identity and origin enabling the constructionthe proper templates. For the reconstructed MC events, the same event and track selection criteria, as for9 .6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 (GeV) - p + K m en t r i e s ( a r b . un i t s ) datamixed events (0.2;0.4) GeV/c, p+p @ 158 GeV/c ˛ T (0.5;1.0), p ˛ y (GeV) - p + K m en t r i e s ( a r b . un i t s ) datatotal fit 1fitted background / ndf=2.29 c (0.2;0.4) GeV/c, p+p @ 158 GeV/c ˛ T (0.5;1.0), p ˛ y (GeV) - p + K m - en t r i e s ( a r b . un i t s ) mixed events subtracteddatatotal fit 2polynomial background / ndf=1.48 c (0.2;0.4) GeV/c, p+p @ 158 GeV/c ˛ T (0.5;1.0), p ˛ y (GeV) - p + K m en t r i e s ( a r b . un i t s ) data as in Eq. (5)total fit 2polynomial background / ndf=1.35 c – =29363 * K N (0.2;0.4) GeV/c, p+p @ 158 GeV/c ˛ T (0.5;1.0), p ˛ y Figure 4: (Color online) The example of the procedure of signal extraction for K ∗ (892) in rapidity bin 0 . < y < . . < p T < . / c for p + p collisions at 158 GeV / c . Top, left: data signal (blue points), and backgroundhistogram (red points) obtained from mixed events (standard method). Top, right: data signal (blue points), andfitted background (red line) obtained from the templates. Bottom: background subtracted signal for the standardmethod (left) and template method (right) – more details in the text. Thin black vertical lines in bottom right panelcorrespond to the range of integrating fit functions while obtaining the raw number of K ∗ (892) mesons ( m ± Γ ;see the text for details). real data, were used. They also include the e ff ects of the limited acceptance of the detector. Both thetemplate and the data histograms were computed in bins of rapidity y (calculated in the center-of-massreference system) and transverse momentum p T .Finally, the signal ( BW ) is described using the Breit-Wigner distribution Eq. (3).The T MCres and T DAT Amix histograms in the fit function Eq. (4) were normalized to have the same numbers ofpairs as the real data histogram in the invariant mass range from 0.6 to 1.6 GeV. The symbols a , b and c inEq. (4) are the normalization parameters of the fit ( a + b + c = T MCres , T DAT Amix and BW to the invariant mass spectra. The mass and width of the K ∗ (892) are the parameters ofthe Breit-Wigner shape obtained within the mass window m ± Γ . The values from total fit 2 (see Fig. 4or 5 (bottom, right)) were used in the results section below.In Figs. 4, 5 (top, right), the fitted invariant mass spectra, using Eq. (4), are presented by brown curves( total fit 1 ). The red lines ( fitted background ) show the fitted function without the signal contribution( BW ). The fits (brown and red curves) were performed in the invariant mass range from 0.66 GeV to1.26 GeV. It is seen that Eq. (4) (without BW component) describes the background much better thanonly mixed events (Figs. 4, 5 (top, left)). After MC template and mixed event background subtraction (see10 .6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 (GeV) - p + K m en t r i e s ( a r b . un i t s ) datamixed events (0.2;0.4) GeV/c, p+p @ 158 GeV/c ˛ T (1.5;2.0), p ˛ y (GeV) - p + K m en t r i e s ( a r b . un i t s ) datatotal fit 1fitted background / ndf=1.45 c (0.2;0.4) GeV/c, p+p @ 158 GeV/c ˛ T (1.5;2.0), p ˛ y (GeV) - p + K m - en t r i e s ( a r b . un i t s ) mixed events subtracteddatatotal fit 2polynomial background / ndf=1.13 c (0.2;0.4) GeV/c, p+p @ 158 GeV/c ˛ T (1.5;2.0), p ˛ y (GeV) - p + K m en t r i e s ( a r b . un i t s ) data as in Eq. (5)total fit 2polynomial background / ndf=1.09 c – =14714 * K N (0.2;0.4) GeV/c, p+p @ 158 GeV/c ˛ T (1.5;2.0), p ˛ y Figure 5: (Color online) Same as Fig. 4 but for 1 . < y < . . < p T < . / c . Eq. (5)), the resulting mass distributions (blue data points) are shown in Figs. 4, 5 (bottom, right). Onesees that the remaining background (red curves) is much less significant than in the case of the standardmethod (Figs. 4, 5 (bottom, left)). In fact, a small residual background is present mostly for the y and p T bins in which the statistics is very low. To subtract it, a fit of the blue histograms was performedas the last step using the function Eq. (2). The results are shown in Figs. 4, 5 (bottom, right). The redlines ( polynomial background ) illustrate the remaining residual background (Eq. (2) without BW ) and thebrown curves ( total fit 2 ) the sum of residual background and BW signal distribution (Eq. (2)). Finally,the uncorrected number of K ∗ (892) mesons (for each separate y and p T bin) is obtained as the integralover the BW signal of total fit 2 in Figs. 4, 5 (bottom, right). The integral is calculated in the mass window m ± Γ . K ∗ (892) Figure 6 presents the uncorrected numbers of K ∗ (892) ( N K ∗ ) as obtained from the extraction proceduredescribed in Sec. 3.6. The values are shown with statistical uncertainties. For each m K + π − invariant massbin in Fig. 4 or 5 (bottom, right), the bin content N bin ( m K + π − ) was calculated as: N bin ( m K + π − ) = N ra w ( m K + π − ) − a · T MCres ( m K + π − ) − b · T DAT Amix ( m K + π − ) , (5)where N ra w ( m K + π − ) is the raw production in a given m K + π − bin, and a , b , T MCres ( m K + π − ) and T DAT Amix ( m K + π − )are described in Eq. (4). The statistical uncertainty of N bin ( m K + π − ) can be expressed as (the notation11 .5 − y ( G e V / c ) T p ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± K* N Figure 6: (Color online) Uncorrected numbers of K ∗ (892) obtained from the extraction procedure described inSec. 3.6. The values are shown with statistical uncertainties. ( m K + π − ) is omitted for simplifying the formula): ∆ N bin = (cid:113) ( ∆ N ra w ) + a ( ∆ T MCres ) + b ( ∆ T DAT Amix ) , (6)where ∆ N ra w , ∆ T MCres and ∆ T DAT Amix are the standard statistical uncertainties taken as the square root of thenumber of entries. For T MCres and T DAT Amix histograms the number of entries had to be properly normalized.Due to high statistics of data, Monte Carlo, and mixed events, the uncertainties of parameters a and b wereneglected. Finally, for each bin of ( y, p T ) in Fig. 6 the uncorrected number of K ∗ (892) , N K ∗ ( y, p T ), wascalculated as the integral over the BW signal of total fit 2 in Figs. 4, 5 (bottom, right). The integral wasobtained within the mass window m ± Γ . The statistical uncertainty of the raw number of K ∗ (892) , ∆ N K ∗ ( y, p T ), was taken as the uncertainty of the integral calculated by the ROOT [36] package usingcovariance matrix of the fitted parameters. In order to determine the number of K ∗ (892) mesons produced in inelastic p + p interactions, two correc-tions were applied to the extracted raw number of K ∗ (892) :(i) The loss of the K ∗ (892) due to the d E / d x requirement was corrected by a constant factor: c dE / dx = (cid:15) K + · (cid:15) π − = . , (7)where (cid:15) K + = .
866 and (cid:15) π − = .
997 are the probabilities (based on the cumulative Gaussian distri-bution) for K + or π − to lie within 1 . σ or 3 σ around the nominal Bethe-Bloch value.12ii) A detailed Monte Carlo simulation was performed to correct for geometrical acceptance, recon-struction e ffi ciency, losses due to the trigger bias, detector acceptance as well as the quality cutsapplied in the analysis. The width of the K ∗ (892) resonance was simulated according to the knownPDG value [37]. The correction factors are based on 227 . × inelastic p + p events producedby the E pos / SHINE apparatus using the G eant / SHINE software packages which take into account all known detector e ff ects. Thesimulated events were reconstructed with the same software as used for real events and the sameselection cuts were applied (except the identification cuts: d E / d x and total momentum p lab ).For each y and p T bin, the correction factor c MC ( y, p T ) was calculated as: c MC ( y, p T ) = n g en ( y, p T ) n sel ( y, p T ) = N g enK ∗ ( y, p T ) N g ene v ents / N selK ∗ ( y, p T ) N sele v ents , (8)where:- N g enK ∗ ( y, p T ) is the number of K ∗ (892) generated in a given (y, p T ) bin,- N selK ∗ ( y, p T ) is the number of K ∗ (892) reconstructed and selected by the cuts in a given ( y, p T )bin. The reconstructed charged particles were matched to the simulated K + and π − basedon cluster positions. Then the invariant mass was calculated for all K + π − pairs. The recon-structed number of K ∗ (892) was obtained by repeating the same steps (template method) asin raw data; they are described in Section 3.6,- N g ene v ents is the number of generated inelastic p + p interactions (227 . × ),- N sele v ents is the number of accepted p + p events (140 . × ).The uncertainty of c MC ( y, p T ) was calculated assuming that the denominator n sel ( y, p T ) is a subsetof the nominator n g en ( y, p T ) and thus has a binomial distribution. The uncertainty of c MC ( y, p T )was calculated as follows: ∆ c MC ( y, p T ) = c MC ( y, p T ) (cid:118)(cid:116) N g enK ∗ ( y, p T ) − N selK ∗ ( y, p T ) N g enK ∗ · N selK ∗ (9)The values of correction factors c MC , together with statistical uncertainties, are presented in Fig. 7 for allanalyzed ( y, p T ) bins. K ∗ (892) yields The double-di ff erential yield of K ∗ (892) per inelastic event in a bin of ( y, p T ) is calculated as follows: d nd y d p T ( y, p T ) = BR · N K ∗ ( y, p T ) N e v ents · c dE / dx · c MC ( y, p T ) ∆ y ∆ p T , (10)13 .5 − y ( G e V / c ) T p . ± . . ± . . ± . . ± . . ± .
895 0 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± .
273 0 . ± . . ± . . ± . . ± . . ± .
233 0 . ± . . ± . . ± . . ± .
923 0 . ± .
514 0 . ± . . ± . . ± .
637 0 . ± . . ± . . ± . . ± .
989 0 . ± .
593 0 . ± .
212 0 . ± . MC c Figure 7: (Color online) Correction factors c MC with statistical uncertainties. where:- BR = / K ∗ (892) decay into K + π − pairs (obtained [27] from the Clebsch-Gordan coe ffi cients),- N K ∗ ( y, p T ) is the uncorrected number of K ∗ (892) , obtained by the signal extraction proceduredescribed in Sec. 3.6,- N e v ents is the number of events after cuts,- c dE / dx , c MC ( y, p T ) are the correction factors described above,- ∆ y and ∆ p T are the bin widths.The corrected double-di ff erential yields of K ∗ (892) together with their uncertainties are presented inSec. 4. The statistical uncertainties of the corrected double-di ff erential yields (see Eq. (10)) take into accountthe statistical uncertainties of c MC ( y, p T ) (see Eq. (9)) and the statistical uncertainties ∆ N K ∗ ( y, p T ) (seeSec. 3.7) of the uncorrected number of K ∗ (892) . The correction c dE / dx has no statistical uncertainty. Thefinal formula is expressed as follows: ∆ d nd y d p T ( y, p T ) = BR · (cid:115)(cid:32) c dE / dx · c MC ( y, p T ) N e v ents ∆ y ∆ p T (cid:33) ∆ N K ∗ ( y, p T ) + (cid:32) N K ∗ ( y, p T ) · c dE / dx N e v ents ∆ y ∆ p T (cid:33) ∆ c MC ( y, p T ) . (11)14he systematic uncertainties were estimated taking into account two sources. The first group of e ff ects isassociated with the signal extraction procedure and the second with event and track quality cuts.The considered sources of the systematic uncertainty and the corresponding modifications of the analysismethod were the following:(I) The uncertainty due to the signal extraction procedure:(i) the lower limit of the invariant mass fitting range (see Figs. 4, 5 (top, right)) was changedfrom 0.66 GeV to 0.69 GeV,(ii) the initial value of the Γ K ∗ parameter of the signal function was changed by ± ± a , b and c describing the contribution of the templates in the fitting function(see Eq. (4)) were changed by ± Γ K ∗ parameter of the signal function was fixed at the PDG value Γ ,(vi) the value of the m K ∗ parameter of the signal function was fixed at the PDG value m ,(vii) in the final step of the background fit (see Figs. 4, 5 (bottom, right)) the standard polynomialcurve of the 2nd order was changed into a polynomial curve of the 3rd order,(viii) the invariant mass range over which the raw number of K ∗ (892) was integrated was changedfrom m ± Γ to ± . Γ and ± . Γ ,(ix) the raw number of K ∗ (892) was calculated as the sum of points (after 2nd order polynomialsubtraction) instead of the BW signal integral.(II) The e ff ects of event and track quality cuts were checked by performing the analysis with the fol-lowing cuts changed compared to the original values:(i) the window in which o ff -time beam particles are not allowed was increased from ± µ s to ± . µ s around the trigger particle,(ii) the cut on the z -position of the interaction vertex was changed from [ − − − − − , − E / d x cuts ( ± σ for π − and ± . σ for K + ) were modified to ± . σ for π − , ± . σ for K + (narrower cut) and ± . σ for π − , ± . σ for K + (wider cut),(iv) the minimum required total number of points in all TPCs for K ∗ (892) decay products waschanged from 30 to 25 and 35,(v) the minimum required number of clusters in both VTPCs for K ∗ (892) decay products waschanged from 15 to 12 and 18,(vi) the impact parameter cuts for the tracks were turned o ff .15 T (GeV / c ) m K ∗ (MeV) Γ K ∗ (MeV)(0.0;0.2) 893 . ± . ± . . ± . ± . . ± . ± . . ± . ± . ± . ± . . ± . ± . . ± . ± . . ± . ± . . ± . ± . . ± . ± . . ± . ± . . ± . ± . . ± . ± . . ± . ± . . ± . ± . . ± . ± . Table 2: Numerical values of mass and width of K ∗ (892) mesons fitted in 0 < y < . For each of the possible sources described above the partial systematic uncertainty σ i was calculated ashalf of the di ff erence between the lowest and the highest value obtained by varying the given parameter.Then, the final systematic uncertainty was taken as: σ s y s = (cid:113)(cid:80) σ i . The contributions of uncertainties σ i to the total uncertainty are negligible for I (ii), I (iii), and I (iv). The final systematic uncertainties areshown in the figures as light red shaded bands. K ∗ (892) The values of mass and width of K ∗ (892) mesons were extracted from the fits to background subtractedinvariant mass spectra (see Sec. 3.6). They are presented in Fig. 8 in di ff erent transverse momentumbins (numerical data are listed in Table 2). The results are shown for the rapidity range 0 < y < . Γ K ∗ values are consistent with information provided by the PDG. However, oneobserves a slight increase of the m K ∗ parameter with p T with an average close to the PDG value. Thecorresponding slope is significant since a large part of the shown systematic uncertainty is due to themagnetic field uncertainty (see below). The points (with their statistical uncertainties), presented in theleft panel of Fig. 8, were fitted with a linear function resulting in the slope parameter value equal to 4.5 ± m K ∗ parameter with transverse momentum does not introduce a systematicvariation of the K ∗ (892) yield since the parameter is fitted in each ( y , p T ) bin, and the signal integrationrange ( ≈
380 MeV) is much larger than the m K ∗ change ( ≈ K S and Λ invariant mass distributions [40]. In order to check how the magnetic field calibration influences theresults, the momentum components of K ∗ (892) decay products (kaons and pions) were varied by ± ff ect K ∗ (892) width and yield significantly. However, the resulting changes ofthe mass parameter are equal or larger than uncertainties described in Sec. 3.10, and they were taken intoaccount in the calculation of the final uncertainty of the K ∗ (892) mass parameter shown in Fig. 8 (left)and Table 2.The comparison of mass and width of K ∗ (892) mesons with other experiments is shown in Sec. 5.16 (GeV/c) T p ( M e V ) K * m (GeV/c) T p ( M e V ) K * G Figure 8: (Color online) The transverse momentum dependence of mass and width of K ∗ (892) mesons fitted for0 < y < .
5. The numerical data are listed in Table 2. The horizontal lines represent PDG values m = .
55 MeVand Γ = . − y ( G e V / c ) T p -1 (GeV/c) T n/dydp d Figure 9: (Color online) Double-di ff erential K ∗ (892) spectra in inelastic p + p interaction at 158 GeV / c in bins of( y, p T ) as obtained from Eq. (10). The numerical values are given in Table 3. ff erential K ∗ (892) spectra The double-di ff erential yields of K ∗ (892) mesons in inelastic p + p interaction at 158 GeV / c in bins of( y, p T ) are presented in Fig. 9. The numerical values with statistical and systematic uncertainties arepresented in Table 3. Figure 10 shows the double-di ff erential yields of K ∗ (892) mesons as function of p T presented for sepa-rate rapidity bins. The corresponding numerical values are listed in Table 3.17 p T (GeV / c ) (-0.5;0.0) (0.0;0.5) (0.5;1.0)(0.0;0.2) 21.8 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± y p T (GeV / c ) (1.0;1.5) (1.5;2.0)(0.0;0.2) 8.97 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± Table 3: Numerical values of double-di ff erential yields d nd y dp T presented in Fig. 10, given in units of 10 − (GeV / c ) − .The first uncertainty is statistical, while the second one is systematic. In order to measure the inverse slope parameter T of transverse momentum spectra and to estimate theyield of K ∗ (892) mesons in the unmeasured high p T region, the function: f ( p T ) = A · p T exp − (cid:113) p T + m T (12)was fitted to the measurements shown in Fig. 10. The inverse slope parameters obtained from the fits arecited in the figure legends.The transverse mass ( m T ≡ (cid:113) p T + m ) spectra m T d ndm T d y were calculated based on d nd y dp T spectra accordingto: 1 m T d ndm T d y = p T d nd y d p T . (13)The results are shown in Fig. 11 and the numerical values are presented in Table 4.For the mid-rapidity region (0 < y < .
5) the inverse slope parameter of the transverse momentumspectrum was found to be equal to T = (173 ± ±
9) MeV, where statistical uncertainty (the first one)is equal to the uncertainty of the fit parameter, and the systematic uncertainty was estimated in the waydescribed in Sec. 3.10. The NA49 experiment measured the T parameter of the p T spectrum in the rapidityrange 0 . < y < . T = (166 ± ±
10) MeV [4].18 (GeV/c) T p ) - (( G e V / c ) d y T dpn d NA61/SHINE K*(892) p+p at 158 GeV/c (-0.5;0.0) ˛ y 6) MeV – T=(175 (GeV/c) T p (0.0;0.5) ˛ y 3) MeV – T=(173 (GeV/c) T p (0.5;1.0) ˛ y 2) MeV – T=(166 (GeV/c) T p (1.0;1.5) ˛ y 2) MeV – T=(165 (GeV/c) T p (1.5;2.0) ˛ y 2) MeV – T=(136
Figure 10: (Color online) Transverse momentum spectra d nd y dp T for five bins of rapidity. The fitted function (solidline) is given by Eq. (12). The numerical values are listed in Table 3 and the fitted inverse slope parameters T foreach bin are given in the legends. (GeV) K* -m T m - - - ) - (( G e V ) d y T d m n d T m NA61/SHINE K*(892) p+p at 158 GeV/c (-0.5;0.0) ˛ y 6) MeV – T=(175 (GeV) -m T m - - - (0.0;0.5) ˛ y 3) MeV – T=(173 (GeV) -m T m - - - (0.5;1.0) ˛ y 2) MeV – T=(166 (GeV) -m T m - - - (1.0;1.5) ˛ y 2) MeV – T=(165 (GeV) -m T m - - - (1.5;2.0) ˛ y 2) MeV – T=(136
Figure 11: (Color online) Transverse mass spectra m T d ndm T d y for five bins of rapidity. The numerical values are listedin Table 4. The solid lines represent function given by Eqs. (12) and (13) with A and T parameters taken fromFig. 10. m T − m (GeV) p T (GeV / c ) (-0.5;0.0) (0.0;0.5) (0.5;1.0)0.011 (0.0;0.2) 109.0 ± ±
34 172.8 ± ±
47 112.1 ± ± ± ±
28 87.1 ± ± ± ± ± ±
13 44.42 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± y m T − m (GeV) p T (GeV / c ) (1.0;1.5) (1.5;2.0)0.011 (0.0;0.2) 89.7 ± ±
22 74.6 ± ± ± ±
13 40.0 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± Table 4: Numerical values of double-di ff erential yields m T d ndm T d y given in units of 10 − (GeV) − and presented inFig. 11; the values of m T − m specify the bin centers. The first uncertainty is statistical, while the second one issystematic. p T -integrated and extrapolated rapidity distribution The rapidity distribution dnd y was calculated by integrating and extrapolating (for the non-measured high- p T region) the d nd y dp T spectrum: dnd y = (cid:88) i d nd y d p T · d p T + A p T I p T (cid:88) i d nd y d p T · d p T , (14)where: A p T = (cid:90) + ∞ . A · p T exp − (cid:113) p T + m T d p T , I p T = (cid:90) . A · p T exp − (cid:113) p T + m T d p T . (15)The parameters T were taken from the corresponding plots in Fig. 10. The statistical uncertainties of21 - - y d y dn NA61/SHINE K*(892) p+p at 158 GeV/c
Figure 12: (Color online) The p T -integrated and extrapolated rapidity distribution. The fitted Gaussian function(solid line) is given by Eq. (17); the first point (with y <
0) was not included in the fit (see the text for details). Thenumerical data are listed in Table 5. p T -integrated and extrapolated dnd y values were calculated as follows: ∆ dnd y = (cid:118)(cid:116)(cid:32) + A p T I p T (cid:33) · (cid:88) i d p T · (cid:32) ∆ d nd y d p T (cid:33) . (16)The p T -integrated and extrapolated dnd y spectrum of K ∗ (892) mesons is plotted in Fig. 12 and the numer-ical values are listed in Table 5.A Gaussian function: f ( y ) = A · exp − y σ y (17)was fitted to the data points to measure the width σ y of the K ∗ (892) rapidity distribution. The first pointwith y < (cid:104) K ∗ (892) (cid:105) (see Sec. 4.5 for details of the procedure).The statistical uncertainty of σ y was taken from the fit and the systematic uncertainty was estimated inthe way described in Sec. 3.10. The numerical values of σ y and (cid:104) K ∗ (892) (cid:105) are listed in Table 5. K ∗ (892) The mean multiplicity of K ∗ (892) mesons was calculated as the sum of measured points in Fig. 12 (thefirst point, with y <
0, was not included in the sum) and the integral of the fitted Gaussian functionEq. (17) in the unmeasured region assuming symmetry around y = (cid:104) K ∗ (892) (cid:105) = (cid:88) i dnd y · d y + (cid:32) A y − + A y + I y (cid:33) (cid:88) i dnd y · d y, (18)22 dnd y (-0.5;0.0) (22.50 ± ± · − (0.0;0.5) (23.71 ± ± · − (0.5;1.0) (19.27 ± ± · − (1.0;1.5) (14.83 ± ± · − (1.5;2.0) (10.73 ± ± · − σ y ± ± (cid:104) K ∗ (892) (cid:105) (78.44 ± ± · − Table 5: Numerical values of the p T -integrated and extrapolated dnd y distribution presented in Fig. 12. The firstuncertainty is statistical, while the second one is systematic. Additionally, the width of the Gaussian fit to the dnd y distribution, as well as the mean multiplicity of K ∗ (892) mesons are shown (see the text for details). where: A y − = (cid:90) −∞ A · e − y σ y d y, A y + = (cid:90) + ∞ . A · e − y σ y d y, I y = (cid:90) . A · e − y σ y d y. (19)The statistical uncertainty of (cid:104) K ∗ (892) (cid:105) was obtained from the formula: ∆ (cid:104) K ∗ (892) (cid:105) = (cid:118)(cid:116)(cid:32) + A y − + A y + I y (cid:33) · (cid:88) i d y · (cid:32) ∆ dnd y (cid:33) , (20)and the systematic uncertainty was estimated in the way described in Sec. 3.10.The mean multiplicity of K ∗ (892) mesons, produced in inelastic p + p collisions at 158 GeV / c , is equal to(78.44 ± ± · − , where the first uncertainty is statistical and the second one is systematic. This section compares the NA61 / SHINE measurements in inelastic p + p interactions at 158 GeV / c withpublicly available world data as well as with predictions from microscopic and statistical models. K ∗ (892) Figure 13 shows the comparison of mass and width of K ∗ (892) mesons obtained in p + p interactionsby NA61 / SHINE, STAR (top RHIC energy), as well as in Pb + Pb and Au + Au collisions at SPS, RHICand LHC energies. For the ALICE and STAR experiments the averaged measurements of K ∗ (892) and K ∗ (892) mesons are shown. One sees that among the available results (and within the p T range coveredby the figure) the precision of the NA61 / SHINE measurements is the highest and the results are veryclose to the PDG values. For p + p collisions the STAR experiment measured lower K ∗ mass, especiallyat lower transverse momenta. 23 (GeV/c) T p ( M e V ) K * m =17.3 GeV NN sNA61/SHINE: p+p at =200 GeV NN sSTAR: p+p at =17.3 GeV NN sNA49: 0-23.5% Pb+Pb at =200 GeV NN sSTAR: 0-10% Au+Au at = 2.76 TeV NN sALICE: 0-20% Pb+Pb at PDG value (GeV/c) T p ( M e V ) K * G =17.3 GeV NN sNA61/SHINE: p+p at =200 GeV NN sSTAR: p+p at =17.3 GeV NN sNA49: 0-23.5% Pb+Pb at =200 GeV NN sSTAR: 0-10% Au+Au at = 2.76 TeV NN sALICE: 0-20% Pb+Pb at PDG value Figure 13: (Color online) The transverse momentum dependence of mass and width of K ∗ (892) (or K ∗ ) mesonsobtained by NA61 / SHINE, NA49 [4], ALICE [9] and STAR [5]. For ALICE and STAR the averaged ( K ∗ ) mea-surements of K ∗ (892) and K ∗ (892) are shown. The horizontal lines represent PDG values [35]. - - y d y dn K*(892) p+p at 158 GeV/c
NA61/SHINEEPOS 1.99
Figure 14: (Color online) Comparison of K ∗ (892) rapidity distribution from NA61 / SHINE (points) and theE pos / SHINE points (solid line) is given by Eq. (17);the first point (with y <
0) was not included in the fit (see the text for details). (cid:104) K ∗ (892) (cid:105) σ y NA61 / SHINE, p T -integrated and extrapolated dnd y (78 . ± . ± . · − . ± . ± . dnd y in wide p T bin [4] (74 . ± . ± . · − . ± . ± . pos . ± . · − - Table 6: The mean multiplicities (cid:104) K ∗ (892) (cid:105) and the widths of the rapidity distributions σ y obtained from dnd y distri-butions (see the text for details). The first uncertainty is statistical and the second systematic. pos The NA61 / SHINE measurements of the rapidity spectrum and mean multiplicity were also compared tothose predicted by the model of hadron production E pos pos K ∗ (892) production in inelastic p + p collisions at 158 GeV / c .Table 6 also shows the comparison with the NA49 result [4] for the same collision system and beammomentum. Instead of analysing in separate p T bins, as in NA61 / SHINE, the NA49 experiment used onewide p T bin (0 < p T < . / c ). The mean multiplicity of K ∗ (892) in NA49 was obtained as theintegral under the Gaussian function in the range − < y < dnd y distribution [27]. Within theuncertainties shown, the results of both experiments are consistent. (cid:104) K ∗ (892) (cid:105) at 158 A GeV / c and predictions of HGM The statistical Hadron Resonance Gas Models (HGM) are commonly used to predict particle multiplicitiesin elementary and nucleus-nucleus collisions, using as adjustable parameters the chemical freeze-out tem-perature T chem , the baryochemical potential µ B , strangeness saturation parameter γ S , etc. In the following25 K ∗ (892) (cid:105) or K ∗ (892) NA61 / SHINE, p T -integrated and extrapolated dnd y (78 . ± . ± . · − HGM, Canonical Ensemble, fit A (no φ ) [41] 74.1 · − HGM, Canonical Ensemble, fit B (with φ ) [41] 56.3 · − HGM, Grand Canonical Ensemble (with φ ) [42, 43] 80.5 · − Table 7: The mean multiplicity of K ∗ (892) mesons for 158 GeV / c inelastic p + p interactions compared to theoreticalmultiplicities obtained within Hadron Gas Models [41, 42]. the measured (cid:104) K ∗ (892) (cid:105) multplicities are compared with predictions of two HGM models described inRefs. [41, 42].In Ref. [41] the HGM results for K ∗ (892) multiplicities were calculated for two versions of the modelfits to particle yields. The first one, called fit B, allowed for strangeness under-saturation so the usualparametrization with γ S was applied. For p + p interactions, the fit was carried out without includingthe multiplicities of Ξ and Ω baryons. In the second fit, called A, the parameter γ S was replaced bythe mean number of strange quark pairs (cid:104) s ¯ s (cid:105) . For p + p collisions fit A was performed without the φ meson. For both fits predicted multiplicities were calculated in the Canonical Ensemble (CE) [41]. Themeasured mean multiplicity of K ∗ (892) in 158 GeV / c inelastic p + p interactions was divided by HGMpredictions based on fit A and B and compared with the value found by NA49 [4]. The results are shownin Fig. 15 for p + p interactions, as well as C + C, Si + Si, and Pb + Pb collisions measured by NA49 [4].In Ref. [41] the S-Canonical Ensemble (SCE) with exact strangeness conservation and grand-canonicaltreatment of electric charge and baryon number was used for the heavier C + C and Si + Si systems, and theGrand Canonical Ensemble (GCE) was assumed for Pb + Pb collisions. For C + C and Si + Si interactionsall available particles were used in the HGM fits, including φ meson and multi-strange baryons. ForPb + Pb data only the measured Λ (1520) yield was removed from the fitted multiplicities. Note that thecentrality of Pb + Pb collisions used in the HGM fits was 0–5% whereas the (cid:104) K ∗ (892) (cid:105) values in NA49were obtained for the 0–23.5% most central interactions. Therefore, the HGM yields had to be scaled bya factor 262 /
362 corresponding to the respective number of wounded nucleons (see Table 8).For heavier systems (including C + C and Si + Si), there is no significant di ff erence between fit A and fitB, however the deviation between the HGM predictions and experimental data increases with increasingsystem size. The p + p measurements are very close to the HGM prediction but only in case of fit A, wherethe φ meson was excluded from the fit. In the most recent paper [42], where the HGM fits were donefor the NA49 and the new NA61 / SHINE measurement in p + p interactions, it is also stressed that at SPSenergies the φ meson multiplicities in p + p collisions cannot be well fitted within the CE formulation ofthe HGM (the quality of CE fits becomes much worse when the φ meson yield is included). However,the mean multiplicity of K ∗ (892) mesons in inelastic p + p collisions at 158 GeV / c can also be comparedto the HGM prediction based on the Grand Canonical Ensemble formulation [42]. The results for theNA49 and NA61 / SHINE measurements are shown in Fig. 15 as closed cross and closed star symbols.Surprisingly, the GCE statistical model provides a good description of the K ∗ (892) yield in the smallp + p system. The numerical values of the NA61 / SHINE p + p measurement and the statistical models arepresented in the Table 7. In Fig. 15 the total uncertainty of (cid:104) K ∗ (892) (cid:105) was taken as the square root of thesum of squares of statistical and systematic uncertainties. The uncertainty of the ratio shown on verticalaxis was taken as the final uncertainty of (cid:104) K ∗ (892) (cid:105) divided by K ∗ (892) .26 W N Æ H G M / K * ( ) æ K * ( ) Æ NA61/SHINE (fit A): p+p(CE)NA61/SHINE (fit B): p+p(CE)NA49 (fit A): p+p(CE), C+C & Si+Si(SCE), Pb+Pb(GCE)NA49 (fit B): p+p(CE), C+C & Si+Si(SCE), Pb+Pb(GCE) NA61/SHINE: p+p(GCE), x-shiftedNA49: p+p(GCE), x-shifted
Figure 15: (Color online) The mean multiplicity of K ∗ (892) for p + p reactions (this analysis and NA49 measure-ment [4]), as well as results of NA49 for C + C, Si + Si and Pb + Pb [4] interactions at 158 A GeV / c divided by the HGMpredictions [41] for fit B (closed circle and closed squares) and fit A (open circle and open squares), see the textfor details. Closed star and cross symbols show p + p measurements compared to HGM predictions for the GrandCanonical Ensemble formulation [42, 43]. N W denotes the number of wounded nucleons taken from Ref. [4]. K ∗ over charged kaon ratios and time between freeze-outs The K ∗ to charged kaons ratios may allow to estimate the time interval between chemical and kineticfreeze-out in nucleus-nucleus collisions. The K ∗ mesons have identical quark (anti-quark) content as K mesons, but di ff erent mass and relative orientation of quark spins. Thus, the (cid:104) K ∗ (892) (cid:105) / (cid:104) K − (cid:105) and (cid:104) K ∗ (892) (cid:105) / (cid:104) K + (cid:105) ratios are considered as the least model dependent ratios for studying the K ∗ productionproperties as well as the freeze-out conditions.The system size dependence of the K ∗ / K ratio at SPS, RHIC and LHC energies shows a strong decreasewith increasing system size and / or multiplicity density (see Sec. 1 for a full list of references). The e ff ectseems to be stronger at the SPS than at RHIC and LHC. Figure 16 presents this dependence at the SPSfor the NA49 and NA61 / SHINE results at 158 A GeV / c . The numerical values are given in Table 8.The NA61 / SHINE (cid:104) K ∗ (892) (cid:105) / (cid:104) K + (cid:105) and (cid:104) K ∗ (892) (cid:105) / (cid:104) K − (cid:105) yield ratios for p + p interactions and the corre-sponding ratios in central Pb + Pb collisions from NA49 can be used to estimate the time interval betweenchemical and kinetic freeze-outs in Pb + Pb. Following Ref. [5]: K ∗ K | kinetic = K ∗ K | chemical · e − ∆ t τ , (21)27 W N Æ æ + / - K Æ / æ K * ( ) Æ æ + K Æ / æ K*(892) Æ NA61/SHINE: æ - K Æ / æ K*(892) Æ NA61/SHINE:
NA49 (p+p: NA61) æ + K Æ / NA49 æ K*(892) Æ NA49 (p+p: NA61) æ - K Æ / NA49 æ K*(892) Æ Figure 16: (Color online) The system size dependences of (cid:104) K ∗ (892) (cid:105) / (cid:104) K + (cid:105) and (cid:104) K ∗ (892) (cid:105) / (cid:104) K − (cid:105) yield ratios inp + p, C + C, Si + Si and Pb + Pb collisions at 158 A GeV / c . N W denotes the number of wounded nucleons taken fromRef. [4]. The numerical values are listed in Table 8. For better visibility the NA61 / SHINE points are shifted on thehorizontal axis. where:- the ratio (cid:104) K ∗ (892) (cid:105) / (cid:104) K + / − (cid:105) in inelastic p + p interactions can be treated as the one at chemicalfreeze-out,- the ratio (cid:104) K ∗ (892) (cid:105) / (cid:104) K + / − (cid:105) for central Pb + Pb (NA49) interactions can be used as the one at kineticfreeze-out,- τ is the mean K ∗ (892) lifetime of approximately 4.17 fm / c [35],- ∆ t is the time interval between chemical and kinetic freeze-outs calculated in the K ∗ (892) restframe.Assuming that the losses of K ∗ (892) before kinetic freeze-out are due to rescattering e ff ects and thatthere are no regeneration processes, the time between chemical and kinetic freeze-outs (in the resonancerest frame) can be estimated as 3.7 ± / c from the (cid:104) K ∗ (892) (cid:105) / (cid:104) K + (cid:105) ratio and 3.2 ± / c fromthe (cid:104) K ∗ (892) (cid:105) / (cid:104) K − (cid:105) ratio. These numbers correspond to 23.5% of the most central Pb + Pb interactionsbut the time is similar when using 5% of the most central events.Following Ref. [12], the above times may be expressed in the collision center-of-mass reference systemusing the multiplicative Lorentz factor: γ ≈ (cid:113) + ( (cid:104) p T (cid:105) / m c ) , (22)28 K ∗ (892) (cid:105) (cid:104) K + (cid:105) (cid:104) K − (cid:105) (cid:104) K ∗ (892) (cid:105) / (cid:104) K + (cid:105) (cid:104) K ∗ (892) (cid:105) / (cid:104) K − (cid:105) NA61 / SHINEp + p N W = ± ± ± ± ± + p N W = ± / SHINE fromNA61 / SHINE 0.317 ± ± + C (cid:104) N W (cid:105) = ± ± ± ± ± ± + Si (cid:104) N W (cid:105) = ± ± ± ± ± ± + Pb (cid:104) N W (cid:105) = ± ± ± ± ± ± + Pb (cid:104) N W (cid:105) = ± ± ± Table 8: The mean multiplicities of di ff erent particle species measured in nucleus-nucleus collisions at 158 A GeV / c by NA49 and NA61 / SHINE. The total uncertainties of (cid:104) K ∗ (892) (cid:105) , (cid:104) K + (cid:105) and (cid:104) K − (cid:105) were taken as the square rootsof the sums of squares of statistical and systematic uncertainties. For NA49 p + p data, the (cid:104) K + (cid:105) and (cid:104) K − (cid:105) resultsinclude statistical uncertainties only ( (cid:104) K + (cid:105) = ± (cid:104) K − (cid:105) = ± / SHINE (cid:104) K + (cid:105) and (cid:104) K − (cid:105) values were used inthe (cid:104) K ∗ (892) (cid:105) / (cid:104) K + (cid:105) and (cid:104) K ∗ (892) (cid:105) / (cid:104) K − (cid:105) ratios. The numbers of (cid:104) K + (cid:105) and (cid:104) K − (cid:105) and their uncertainties for the 5%most central Pb + Pb collisions were multiplied by a factor 262 /
362 in order to estimate charged kaon multiplicitiesin the 23.5% most central Pb + Pb reactions. where (cid:104) p T (cid:105) can be used as an approximation for K ∗ (892) total momentum for the measurements at mid-rapidity. The NA49 experiment published the K ∗ (892) transverse momentum spectrum for 23.5% of themost central Pb + Pb interactions in the rapidity range 0 . < y < .
78 [4]. The (cid:104) p T (cid:105) can be obtainedfrom the fitted exponential function in the range 0 < p T < / c . The average transverse momentum of K ∗ (892) mesons was found to be 0.908 GeV / c that results in γ ≈ .
42. Finally, the Lorentz boosted timeinterval between chemical and kinetic freeze-outs can be estimated as 5.3 fm / c for the (cid:104) K ∗ (892) (cid:105) / (cid:104) K + (cid:105) ratio or 4.6 fm / c for the (cid:104) K ∗ (892) (cid:105) / (cid:104) K − (cid:105) ratio.Similar calculations can be performed for the published RHIC ( √ s NN =
200 GeV) and LHC ( √ s NN = K ∗ / K − ratio was found to be 0 . ± .
04 for the10% most central Au + Au collisions, and 0 . ± .
05 for p + p interactions [7]. Thus, the time betweenfreeze-outs (calculated in the K ∗ rest frame) is equal to 2.2 ± / c . The average transverse mo-mentum of K ∗ mesons in Au + Au collisions at mid-rapidity ( | y | < .
5) was found to be 1.09 GeV / c [7],which corresponds to γ ≈ / c . This value is smaller than the onesobtained at SPS.In the ALICE experiment at LHC, the K ∗ / K − ratio was found to be 0 . ± .
027 for the 5% most centralPb + Pb collisions, and 0 . ± .
043 for p + p interactions [10]. Following Eq. (21), ∆ t can be evaluatedas 2.2 ± / c . The (cid:104) p T (cid:105) of K ∗ mesons in Pb + Pb collisions at mid-rapidity ( | y | < .
5) was found tobe 1.310 GeV / c [10], which corresponds to γ ≈ / c .29he above numbers may imply that, in central heavy ion collisions, the lifetime of the hadronic periodof the fireball after chemical freeze-out is longer at SPS than at RHIC or even at LHC energies. Oneshould, however, remember that such a conclusion is valid only under the assumption that there areno regeneration processes of K ∗ mesons before kinetic freeze-out. As the K ∗ (892) regeneration mayhappen at all energies, the obtained time interval values should be considered as lower limits of the timebetween chemical and kinetic freeze-outs. In this paper the NA61 / SHINE measurement of K ∗ (892) meson production via its K + π − decay mode ininelastic p + p collisions at beam momentum 158 GeV / c ( √ s NN = . template method was used to extract raw K ∗ (892) signals. In this method the background is described as a sumof two contributions: background due to uncorrelated pairs modeled by event mixing and backgroundof correlated pairs modeled by E pos K ∗ (892) production the template method was found toprovide a better background description than the standard one which relies on mixed events only. Themass and width of the K ∗ (892) were extracted from the fits to background subtracted invariant massspectra. Their values, for di ff erent transverse momentum bins, are close to the PDG results, however, aslight increase of the K ∗ (892) mass with transverse momentum can be observed.With the large statistics of NA61 / SHINE data (52.53M events selected by the interaction trigger) it waspossible to obtain double-di ff erential transverse momentum and rapidity spectra of K ∗ (892) mesons. Thefull phase-space mean multiplicity of K ∗ (892) mesons, obtained from the p T -integrated and extrapolatedrapidity distribution, was found to be (78 . ± . ± . · − , where the first uncertainty is statisticaland the second one is systematic. The result agrees with the previous NA49 measurement for the samesystem and energy.The NA61 / SHINE result was compared with predictions of statistical Hadron Resonance Gas models inCanonical and Grand Canonical formulations. Surprisingly, the GCE model provides a good descriptionof the NA61 / SHINE measurement of the K ∗ (892) multiplicity in p + p collisions. The CE model alsoagrees provided that the φ meson is excluded from the fits.Finally, based on the previous results of NA49 from central Pb + Pb collisions and the new measurementsof NA61 / SHINE on p + p interactions, an attempt was made to estimate the time between chemical andkinetic freeze-outs in central Pb + Pb reactions at 158 A GeV / c . This time was found to be larger than atRHIC, suggesting that either the system life-time between freeze-outs is indeed higher at SPS or the K ∗ (892) regeneration e ff ects start to play a significant role at higher collision energies. Acknowledgements
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A. Aduszkiewicz , E.V. Andronov , T. Anti´ci´c , V. Babkin , M. Baszczyk , S. Bhosale ,A. Blondel , M. Bogomilov , A. Brandin , A. Bravar , W. Bryli´nski , J. Brzychczyk ,M. Buryakov , O. Busygina , A. Bzdak , H. Cherif , M. ´Cirkovi´c , M. Csanad , J. Cybowska ,T. Czopowicz , , A. Damyanova , N. Davis , M. Deliyergiyev , M. Deveaux , A. Dmitriev ,W. Dominik , P. Dorosz , J. Dumarchez , R. Engel , G.A. Feofilov , L. Fields , Z. Fodor , ,A. Garibov , M. Ga´zdzicki , , O. Golosov , V. Golovatyuk , M. Golubeva , K. Grebieszkow ,F. Guber , A. Haesler , S.N. Igolkin , S. Ilieva , A. Ivashkin , S.R. Johnson , K. Kadija ,E. Kaptur , N. Kargin , E. Kashirin , M. Kiełbowicz , V.A. Kireyeu , V. Klochkov ,V.I. Kolesnikov , D. Kolev , A. Korzenev , V.N. Kovalenko , S. Kowalski , M. Koziel ,A. Krasnoperov , W. Kucewicz , M. Kuich , A. Kurepin , D. Larsen , A. László , T.V. Lazareva ,M. Lewicki , K. Łojek , B. Łysakowski , V.V. Lyubushkin , M. Ma´ckowiak-Pawłowska ,Z. Majka , B. Maksiak , A.I. Malakhov , D. Mani´c , A. Marcinek , A.D. Marino , K. Marton ,H.-J. Mathes , T. Matulewicz , V. Matveev , G.L. Melkumov , A.O. Merzlaya , B. Messerly ,Ł. Mik , S. Morozov , , S. Mrówczy´nski , Y. Nagai , M. Naskr ˛et , V. Ozvenchuk , V. Paolone ,O. Petukhov , R. Płaneta , P. Podlaski , B.A. Popov , , B. Porfy , M. Posiadała-Zezula ,D.S. Prokhorova , D. Pszczel , S. Puławski , J. Puzovi´c , M. Ravonel , R. Renfordt , E. Richter-W ˛as , D. Röhrich , E. Rondio , M. Roth , B.T. Rumberger , M. Rumyantsev , A. Rustamov , ,M. Rybczynski , A. Rybicki , A. Sadovsky , K. Schmidt , I. Selyuzhenkov , A.Yu. Seryakov ,P. Seyboth , M. Słodkowski , P. Staszel , G. Stefanek , J. Stepaniak , M. Strikhanov ,H. Ströbele , T. Šuša , A. Taranenko , A. Tefelska , D. Tefelski , V. Tereshchenko , A. Toia ,R. Tsenov , L. Turko , R. Ulrich , M. Unger , F.F. Valiev , D. Veberiˇc , V.V. Vechernin ,A. Wickremasinghe , , Z. Włodarczyk , O. Wyszy´nski , E.D. Zimmerman , and R. Zwaska National Nuclear Research Center, Baku, Azerbaijan Faculty of Physics, University of Sofia, Sofia, Bulgaria Ru ¯der Boškovi´c Institute, Zagreb, Croatia LPNHE, University of Paris VI and VII, Paris, France Karlsruhe Institute of Technology, Karlsruhe, Germany University of Frankfurt, Frankfurt, Germany Wigner Research Centre for Physics of the Hungarian Academy of Sciences, Budapest, Hungary University of Bergen, Bergen, Norway Jan Kochanowski University in Kielce, Poland Institute of Nuclear Physics, Polish Academy of Sciences, Cracow, Poland National Centre for Nuclear Research, Warsaw, Poland Jagiellonian University, Cracow, Poland AGH - University of Science and Technology, Cracow, Poland University of Silesia, Katowice, Poland University of Warsaw, Warsaw, Poland University of Wrocław, Wrocław, Poland Warsaw University of Technology, Warsaw, Poland Institute for Nuclear Research, Moscow, Russia Joint Institute for Nuclear Research, Dubna, Russia National Research Nuclear University (Moscow Engineering Physics Institute), Moscow, Russia St. Petersburg State University, St. Petersburg, Russia University of Belgrade, Belgrade, Serbia 35 University of Geneva, Geneva, Switzerland Fermilab, Batavia, USA University of Colorado, Boulder, USA26