Universal scaling of strange particle p_{\rm T} spectra in pp collisions
Liwen Yang, Yanyun Wang, Wenhui Hao, Na Liu, Xiaoling Du, Wenchao Zhang
aa r X i v : . [ h e p - ph ] F e b EPJ manuscript No. (will be inserted by the editor)
Universal scaling of strange particle p T spectra in pp collisions Liwen Yang, Yanyun Wang, Wenhui Hao, Na Liu, Xiaoling Du, and Wenchao Zhang
School of Physics and Information Technology, Shaanxi Normal University, Xi’an 710119, People’s Republic of Chinae-mail: [email protected]
Received: date / Revised version: date
Abstract.
As a complementary study to that performed on the transverse momentum ( p T ) spectra ofcharged pions, kaons and protons in proton-proton (pp) collisions at LHC energies 0.9, 2.76 and 7 TeV,we present a scaling behaviour in the p T spectra of strange particles ( K S , Λ, Ξ and φ ) at these threeenergies. This scaling behaviour is exhibited when the spectra are expressed in a suitable scaling variable z = p T /K , where the scaling parameter K is determined by the quality factor method and increases withthe center of mass energy ( √ s ). The rates at which K increases with ln √ s for these strange particles arefound to be identical within errors. In the framework of the colour string percolation model, we argue thatthese strange particles are produced through the decay of clusters that are formed by the colour stringsoverlapping. We observe that the strange mesons and baryons are produced from clusters with differentsize distributions, while the strange mesons (baryons) K S and φ (Λ and Ξ) originate from clusters with thesame size distributions. The cluster’s size distributions for strange mesons are more dispersed than thosefor strange baryons. The scaling behaviour of the p T spectra for these strange particles can be explainedby the colour string percolation model in a quantitative way. PACS.
The transverse momentum ( p T ) spectra of final state par-ticles are important observables in high energy collisions.They play an essential role in understanding the mecha-nism of particle productions. In many studies, searchingfor a scaling behaviour of the p T spectra is useful to re-veal the mechanism. In ref. [1], a scaling behaviour waspresented in the pion p T spectra in Au-Au collisions atthe Relativistic Heavy Ion Collider (RHIC). It was inde-pendent of the centrality of the collision. This scaling be-haviour was later extended to the proton and anti-proton p T spectra with different centralities in Au-Au collisionsat RHIC [2].Recently, a similar scaling behaviour was found in the p T spectra of inclusive charged hadrons as well as identi-fied charged hadrons (charged pions, kaons and protons)in proton-proton (pp) collisions at the Large Hadron Col-lider (LHC) [3,4]. This scaling behaviour was independentof the center of mass energy ( √ s ). It was exhibited whenthe spectra were expressed in a suitable scaling variable z = p T /K , where K is the scaling parameter relying on √ s . In pp collisions, the hadrons produced are predomi-nantly pions, kaons and protons. As the strange quark isheavier than the up and down quarks, the strange parti-cles such as K S , Λ, Ξ and φ only constitute a small frac-tion of final state particles. However, the investigation oftheir spectra is an important ingredient in understandingthe mechanism of particle production in high energy colli- sions. Thus, in this paper, we will focus on the p T spectraof K S , Λ, Ξ and φ produced in pp collisions at 0.9, 2.76and 7 TeV [5–10]. The p T spectra of Ω are not consideredin this work, as their spectra at 0.9 TeV are not availableso far. A scaling behaviour independent of the collisionenergy will be searched for among these strange particlespectra. If the scaling behaviour exists, then one may asktwo questions: (1) Is the dependence of the scaling pa-rameter K on √ s for K S , Λ, Ξ and φ the same as thatfor charged pions, kaons and protons? (2) Can the stringpercolation model utilized in ref. [4] be adopted to explainthe scaling behaviour of strange particles?The organization of the paper is as follows. In sect.2, the method to search for the scaling behaviour will bedescribed briefly. In sect. 3, the scaling behaviour of the K S , Λ, Ξ and φ spectra will be presented. In sect. 4, wewill discuss the scaling behaviour of the strange particlespectra in the framework of the colour string percolationmodel. Finally, the conclusion is given in sect. 5. As done in ref. [4], we will search for the scaling be-haviour of the K S p T spectra with the following steps.A scaling variable, z = p T /K , and a scaled p T spectrum,Φ( z ) = A (2 πp T ) − d N/dp T dy | p T = Kz will be defined first.Here y is the rapidity of K S , (2 πp T ) − d N/dp T dy is theinvariant yield of K S . With suitable scaling parameters Liwen Yang, et al.: Universal scaling of strange particle p T spectra in pp collisions K and A that depend on √ s , the data points of the K S p T spectra at 0.9, 2.76 and 7 TeV can be coalesced intoone curve. In ref. [4], K and A for the charged pion, kaonand proton spectra at 2.76 TeV were set to be 1. Thischoice was made due to reason that the p T coverage ofthe spectra at 2.76 TeV is much larger than the cover-age at 0.9 and 2.76 TeV. In this work, the K S spectrumat 2.76 TeV covers a p T range from 0.225 to 19 GeV/c,which is larger the ranges of the spectra at 0.9 and 7 TeV,0.1 to 9 GeV/c and 0.1 to 9 GeV/c. Therefore, to keepthe similarity and consistency with ref. [4], we prefer toset the K and A for the K S spectrum at 2.76 TeV tobe 1. K and A values at 0.9 and 7 TeV will be deter-mined by the quality factor method [11, 12]. Obviously,the scaling function Φ( z ) depends on the choice of K and A at 2.76 TeV. This arbitrariness could be eliminated ifthe spectra are presented in u = z/ h z i = p T / h p T i . Here h z i = R ∞ z Φ( z ) zdz (cid:14) R ∞ Φ( z ) zdz . The normalized scalingfunction then is Ψ( u ) = h z i Φ( h z i u ) (cid:14) R ∞ Φ( z ) zdz . WithΨ( u ), the spectra at 0.9 and 7 TeV can be parameterizedas f ( p T ) = R ∞ Φ( z ) zdz/ ( A h z i )Ψ ( p T / ( K h z i )), where K and A are the scaling parameters at these energies. Themethods to search for the scaling behaviour of the Λ, Ξand φ spectra are similar to that for the K S spectra. K S , Λ , Ξ and φ p T spectra The K S , Λ and Ξ p T spectra in pp collisions at 0.9 and 7TeV were published by the CMS collaboration [5]. Here Λand Ξ refer to (Λ+ ¯Λ) / + +Ξ − ) / K S , Λ and Ξ p T spectra at 2.76 TeV, so far there areno official data. As the K S spectrum is theoretically thesame as the charged kaon spectrum, and the charged kaonspectrum at 2.76 TeV were officially published in ref. [13],we utilize the charged kaon spectrum instead of the K S spectrum at this energy. For the Λ and Ξ spectra at 2.76TeV, we use the preliminary results of the ALICE collab-oration at this energy instead. They are publicly availablein refs. [6, 7]. The φ spectra at 0.9, 2.76 and 7 TeV werepublished by the ALICE collaboration [8–10]. Since thescaling parameters K and A at 2.76 TeV are chosen tobe 1, the scaling function Φ( z ) is exactly the K S , Λ, Ξor φ p T spectrum at this energy. As described in ref. [14],due to the reason that the temperature of the hadronizingsystem fluctuates from event to event, the p T spectrumof final state hadrons produced in high energy collisionsfollows a non-extensive statistical distribution, the Tsal-lis distribution [15]. Thus, the scaling function Φ( z ) forstrange particles can be parameterized as follows [4]Φ( z ) = C q " − (1 − q ) √ m + z − mz − q , (1)where C q , q and z are free parameters, 1 − q is a measureof the non-extensivity, m is the strange particle mass. Ineq. (1), 1 / ( q −
1) determines the power law behaviour of Φ( z ) in the high p T region, while z controls the expo-nential behaviour in the low p T region. C q , q and z aredetermined by the least squares fitting of Φ( z ) to the K S ,Λ, Ξ and φ p T spectra at 2.76 TeV. The statistical andsystematic errors of the data points have been added inquadrature in the fits. Table 1 tabulates C q , q , z and theiruncertainties returned by the fits. The χ s per degrees offreedom (dof), named reduced χ s, for these fits are alsogiven in the table. Table 1. C q , q and z of Φ( z ) for the K S , Λ, Ξ and φ spectra.The uncertainties quoted are due to the quadratic sum of thestatistical and systematic errors of the data points. The lastcolumn shows the reduced χ s for the fits on the strange p T spectra at 2.76 TeV. C q q z (GeV/c) χ /dof K S (214 ± × − ± ± ± × − ± ± ± × − ± ± φ (96 ± × − ± ± As described in sect. 2, the scaling parameters K and A at 0.9 and 7 TeV will be evaluated with the qualityfactor (QF) method. Compared with the method utilizedin ref. [3], this method is more robust since it does notrely on the shape of the scaling function. To define thequality factor, a set of data points ( ρ i , τ i ) is consideredfirst. Here ρ i = p i T /K , τ i = log( A (2 πp i T ) − d N i /dp i T dy i ), ρ i are ordered, τ i are rescaled so that they are in therange between 0 and 1. Then, the QF is introduced asfollows [11, 12]QF( K, A ) = " n X i =2 ( τ i − τ i − ) ( ρ i − ρ i − ) + 1 /n − , (2)where n is the number of data points and 1 /n keeps thesum finite in the case of two points taking the same ρ value. It is obvious that a large contribution to the sum inthe QF is given if two successive data points are close in ρ and far in τ . Therefore, a set of data points are expectedto lie close to a single curve if they have a small sum (alarge QF) in eq. (2). The best set of ( K , A ) at 0.9 (7) TeVis chosen to be the one which globally maximizes the QFof the data points at 0.9 (7) and 2.76 TeV. Table 2 tabu-lates K and A for the K S , Λ, Ξ and φ spectra at 0.9, 2.76and 7 TeV. Also shown in the table is the maximum QF(QF max ). In order to determine the uncertainties of K and A at 0.9 and 7 TeV, we utilize the method mentioned inref. [11]. Let’s take the determination of the uncertainty of K ( A ) for K S at 0.9 TeV as an example. In fig. 1 we firstplot the QF as a function of K ( A ) with A ( K ) fixed to thevalue 0.24 (0.92) returned by the QF method. The peakvalue with QF > (QF max − .
01) shows a good scalingand we make a Gaussian fit to this bump. The standarddeviation of the Gaussian fit, σ K ( A ) , is taken as the un-certainty of K ( A ) for K S at 0.9 TeV. The mean value ofthe Gaussian fit, µ K ( A ) , is consistent with the value of K iwen Yang, et al.: Universal scaling of strange particle p T spectra in pp collisions 3 ( A ) returned by the QF method, thus this method to de-termine the uncertainties of scaling parameters is robust.The errors of K and A for K S at 7 TeV, Λ, Ξ and φ at0.9 and 7 TeV are determined by making Gaussian fits tothe peaks with QF > (QF max − . Table 2. K and A for the K S , Λ, Ξ and φ spectra at 0.9,2.76 and 7 TeV. The QF max is shown in the last column ofthe table. The standard deviations of the Gaussian fits to thepeaks of the QF scatter plots at 0.9 and 7 TeV are taken asthe uncertainties of K and A at these two energies. √ s (TeV) K A QF max ± ± K S ± ± ± ± ± ± ± ± ± ± ± ± φ ± ± K0.5 1 1.5 Q F µ K = 0.92 σ K = 0.01 A0 0.2 0.4 0.6QF versus AGaussian fit µ A = 0.24 σ A = 0.02 Fig. 1.
Left (right) panel: QF versus K ( A ) for K S at 0.9TeV, with A ( K ) fixed to 0.24 (0.92). The black solid curve isthe QF scatter plot, the red dash curve is the Gaussian fit ofthe peak with QF > . Using the scaling parameters K and A in table 2,now we can shift the K S p T spectra at 0.9 and 7 TeVto the spectrum at 2.76 TeV. They are shown in the up-per panel of fig. 2. On a log scale, most of the data pointsat different energies appear consistent with the universalcurve which is described by Φ( z ) in eq. (1) with param-eters in the second row of table 1. In order to see howwell the data points agree with the fitted curve, a ra-tio, R = (data − fitted) / data, is evaluated at 0.9, 2.76and 7 TeV. The uncertainty of R is determined to be(fitted / data) × (∆data / data), where ∆data is the totaluncertainty of the data point. The R distribution is shown in the lower panel of the figure. Except for the last threepoints in the high p T region at 0.9 TeV, all the other pointshave R values in the range between -0.3 and 0.3, which im-plies that the agreement between the data points and thefitted curve is within 30%. This agreement roughly corre-sponds to the systematic errors on R and the accuracy ofthe fits. If we take into account the systematic errors on R , then this agreement is within 22%. Φ ( z ) -10 -8 -6 -4 -2 K S0 -1 R -0.7-0.5-0.3-0.100.10.3 Fig. 2.
Upper panel: the scaling behaviour of the K S p T spec-tra presented in z at 0.9, 2.76 and 7 TeV. The solid curve isfrom Φ( z ) with parameters in the second row of table 1. Thedata points are taken from refs. [5,13]. Lower panel: the R dis-tributions. The R value for the last data point at 0.9 TeV is-1.27 and it is not shown in the lower panel. Φ ( z ) -8 -6 -4 -2 Λ -1 R -0.7-0.5-0.3-0.100.10.3 Fig. 3.
Upper panel: the scaling behaviour of the Λ p T spectrapresented in z at 0.9, 2.76 and 7 TeV. The solid curve is fromΦ( z ) with parameters in the third row of table 1. The datapoints are taken from refs. [5, 6]. Lower panel: the R distribu-tions. In the upper panels of figs. 3, 4 and 5, we present thescaling behaviour of the Λ, Ξ and φ p T spectra at 0.9, 2.76and 7 TeV. In the lower panels of these figures are the R distributions for these spectra. For the Λ spectra, exceptfor the second-to-last point at 0.9 TeV and the last pointat 7 TeV, all the other points agree with the fitted curvewithin 30%. Taking into account the systematic uncertain-ties of R , this agreement is within 11%. For the Ξ spectra, Liwen Yang, et al.: Universal scaling of strange particle p T spectra in pp collisions Φ ( z ) -8 -6 -4 -2 Ξ -1 R -0.8-0.6-0.4-0.200.20.40.6 Fig. 4.
Upper panel: the scaling behaviour of the Ξ p T spectrapresented in z at 0.9, 2.76 and 7 TeV. The solid curve is fromΦ( z ) with parameters in the fourth row of table 1. The datapoints are taken from refs. [5, 7]. Lower panel: the R distribu-tions. Φ ( z ) -10 -8 -6 -4 -2 φ R -0.3-0.100.10.3 Fig. 5.
Upper panel: the scaling behaviour of the φ p T spec-tra presented in z at 0.9, 2.76 and 7 TeV. The solid curve isfrom Φ( z ) with parameters in the fifth row of table 1. Thedata points are taken from refs. [8–10]. Lower panel: the R distributions. except for the points with z = 4.1, 4.3 and 6.4 GeV/c at0.9 TeV, all the other points are consistent with the fittedcurve within 20%. Taking into account the systematic er-rors of R , this consistency is within 3%. For the φ spectra,all the points are in agreement with the fitted curve within30%. With the consideration of the systematic errors of R ,this agreement is within 18%.From the above statement, we have shown that the p T spectra of K S , Λ, Ξ and φ at 0.9, 2.76 and 7 TeV exhibit ascaling behaviour independent of √ s . As described in sect.2, the scaling function Φ( z ) relies on K and A chosen at2.76 TeV. In order to get rid of this reliance, we utilize thescaling variable u = z/ h z i instead. The h z i values for the K S , Λ, Ξ and φ p T spectra are determined as 0.701 ± ± ± ± C q , q and z in table1. The corresponding normalized scaling function Ψ( u ) isΨ( u ) = C ′ q " − (1 − q ′ ) p ( m ′ ) + u − m ′ u − q ′ . (3) Here C ′ q = h z i C q / R ∞ Φ ( z ) zdz , q ′ = q , u = z / h z i and m ′ = m/ h z i . Their values are presented in table 3. Asdescribed in sect. 2, with Ψ( u ), the spectra of K S , Λ, Ξand φ at 0.9 (7) TeV can be parameterized as f ( p T ) = R ∞ Φ( z ) zdz/ ( A h z i )Ψ ( p T / ( K h z i )), where K and A arethe scaling parameters of these strange particles at 0.9 (7)TeV in table 2. In ref. [5], the CMS collaboration have pre-sented the relative production versus p T between differentstrange particle species, N (Λ) /N ( K S ) and N (Ξ) /N (Λ), at0.9 and 7 TeV. In the upper (lower) panel of fig. 6, we showthat the N (Λ) /N ( K S ) ( N (Ξ) /N (Λ)) distributions in dataat 0.9 and 7 TeV are well described by f Λ ( p T ) /f K S ( p T )( f Ξ ( p T ) /f Λ ( p T )). This agreement is a definite indicationthat the scaling behaviour exists in the p T spectra ofstrange particles at 0.9, 2.76 and 7 TeV. The p T depen-dence of the relative production can be explained as fol-lows. At low p T , f ( p T ) inclines to be an exponential dis-tribution which is controlled by the parameter z = u h z i .For N (Λ) /N ( K S ) ( N (Ξ) /N (Λ)), the z value for Λ (Ξ) islarger than that for K S (Λ), therefore both N (Λ) /N ( K S )and N (Ξ) /N (Λ) grow with p T . At high p T , f ( p T ) prefersto be a power law distribution which is dominated by1 / ( q ′ − q ′ value for Λ (Ξ) is smaller than (almost equalto) that for K S (Λ), thus N (Λ) /N ( K S ) decreases with p T while N (Ξ) /N (Λ) appears to be flat. Table 3. C ′ q , q ′ , u and m ′ of Ψ( u ) for K S , Λ, Ξ and φ . Theuncertainties quoted are due to the errors of C q , q and z intable 1. C ′ q q ′ u m ′ K S ± ± ± ± ± ± ± ± ± ± ± ± φ ± ± ± ± N ( Λ ) / N ( K S o ) . f . Λ ( p T ) f . K S ( p T ) f Λ ( p T ) f K S ( p T ) p T (GeV/c)0 2 4 6 8 10 12 N ( Ξ ) / N ( Λ ) . f . Ξ ( p T ) f . Λ ( p T ) f Ξ ( p T ) f Λ ( p T ) Fig. 6.
Upper (Lower) panel: the N (Λ) /N ( K S ) ( N (Ξ) /N (Λ))distributions at 0.9 (solid line) and 7 TeV (dashed line). Thedata points are taken from ref. [5]. The values of N (Λ) /N ( K S )for data points have been divided by 2, since here Λ refers to(Λ + ¯Λ) / p T spectra in pp collisions 5 In sect. 3, we have shown that there is indeed a scaling be-haviour in the K S , Λ, Ξ and φ p T spectra in pp collisionsat 0.9, 2.76 and 7 TeV. This scaling behaviour appearswhen the spectra are presented in terms of the scalingvariable z . Now we would like to discuss this scaling be-haviour in terms of the colour string percolation (CSP)model [16, 17].In this model, colour strings are stretched between thepartons of the projectile and target protons in pp col-lisions. These strings then will split into new ones bythe production of sea q ¯ q pairs from the vacuum. Strangeparticles such as K S , Λ, Ξ and φ are produced throughthe hadronization of these new strings. In the transverseplane, the colour strings look like discs, each of which hasan area, S = πr , r ≈ . n strings is assumed to behave as a singlestring. The colour field of the cluster Q n is the vectorialsum of the colour charge of each individual Q string, Q n = P n Q . Since the individual string colour fields areoriented arbitrarily, the average value of Q i · Q j is zeroand Q n = n Q . Q n also depends on the transverse areaof each individual string S and the transverse area of thecluster S n . Thus, Q n = p nS n /S Q . As the multiplicityof strange particles produced from the cluster is propor-tional to its colour charge, µ n = p nS n /S µ , where µ isthe multiplicity of strange particles produced by a singlestring. Since the transverse momentum is conserved beforeand after the overlapping, µ n h p i n = nµ h p i , where h p i n is the mean p of strange particles produced by thecluster, h p i is the mean p of strange particles producedby a single string. Therefore, h p i n = p nS /S n h p i ,where nS /S n is the degree of string overlap. For the casewhere strings just get in touch with each other, S n = nS , nS /S n = 1 and h p i n = h p i , which means that the n strings fragment into strange hadrons independently.For the case in which strings maximally overlap with eachother, S n = S , nS /S n = n and h p i n = √ n h p i , whichmeans that the mean p is maximally enhanced due to thepercolation. The p T spectra of strange particles producedin pp collisions can be written as a superposition of the p T distribution produced by each cluster, g ( x, p T ), weightedwith the cluster’s size distribution W ( x ), d N πp T dp T dy = C Z ∞ W ( x ) g ( x, p T ) dx, (4)where C is a normalization parameter which characterizesthe total number of clusters formed for strange particlesbefore hadronization. W ( x ) is supposed to be a gammadistribution, W ( x ) = γΓ ( κ ) ( γx ) κ − exp( − γx ) , (5)where x is proportional to 1 / h p i n , κ and γ are free pa-rameters. κ is related to the dispersion of the size distribu-tion, 1 /κ = ( h x i − h x i ) / h x i . It depends on the density of the strings, η = ( r /R ) N s , where R is the effectiveradius of the interaction region, N s is the average num-ber of strings of the cluster. γ is related to the mean x , h x i = κ/γ .In order to see whether the CSP model can describethe scaling behaviour of strange particle p T spectra, weattempt to fit eq. (4) to the combination of the scaleddata points at 0.9, 2.76 and 7 TeV with the least squaresmethod. Here the cluster’s fragmentation function in theCSP fit is chosen as the Schwinger formula [18] g ( x, p T ) = exp( − p x ) . (6) C , γ and κ returned by the fits are listed in table 4. Fromthe table, we see that the dispersion of the cluster’s sizedistribution (1/ κ ) for strange mesons ( K S and φ ) is largerthan that of strange baryons (Λ and Ξ), while the disper-sion of the cluster’s size distribution for K S (Λ) is al-most equal to that of φ (Ξ) when considering the errors.This implies that the strange mesons and baryons are pro-duced from clusters with different size distributions, whilethe strange mesons (baryons) K S and φ (Λ and Ξ) orig-inate from clusters with the same size distributions. Thecluster’s size distributions for strange mesons are moredispersed than those for strange baryons. The differencebetween the cluster’s size distributions of strange mesonsand baryons could be explained as follows. As described inref. [16], since additional quarks required to form a baryonare provided by the quarks of the overlapping strings thatform the cluster, the baryons probe a higher string densitythan mesons for the same energy of collisions. When η isabove the critical string density at which the string per-colation appears, κ increases with η [19]. Therefore the κ values for strange baryons are larger than those for strangemesons. The fit results for K S , Λ, Ξ and φ are presentedin the upper panels of figs. 7, 8, 9 and 10 respectively. The R distributions are shown in the lower panels of these fig-ures. For the K S spectra, except for the last two pointsat 0.9 TeV and the last three points at 2.76 TeV, all theother points agree with the CSP fit within 30%. For the Λspectra, except for the points with z = 6.4 and 8.2 GeV/cat 0.9 TeV, all the other data points are consistent withthe CSP fit within 30%. For the Ξ spectra, except for thepoints at z = 4.1, 4.3 and 6.4 GeV/c at 0.9 TeV, all theother data points agree with the CSP fit within 20%. Forthe φ spectra, except for the last point at 2.76 TeV, all theother points are consistent with the CSP fit within 30%. Table 4. C , γ and κ returned by the CSP fits on the combi-nation of the scaled K S , Λ, Ξ and φ spectra at 0.9, 2.76 and7 TeV. The uncertainties quoted originate from the statisticaland systematic errors of the data points added in quadrature.The last column shows the reduced χ s for the fits. C γ κ χ /dof K S (166 ± × − ± ± ± × − ± ± ± × − ± ± φ (86 ± × − ± ± p T spectra in pp collisions Φ ( z ) -10 -8 -6 -4 -2 K S0 -1 R -0.7-0.5-0.3-0.100.10.3 Fig. 7.
Upper panel: the scaling behaviour of the K S p T spectra presented in z at 0.9, 2.76 and 7 TeV. The solid curveis the CSP fit in eq. (4) with parameters in the second rowof table 4. The data points are taken from refs. [5, 13]. Lowerpanel: the R distributions. The R value for the last data pointat 0.9 TeV is -1.60 and it is not shown in the lower panel. Φ ( z ) -8 -6 -4 -2 Λ -1 R -0.9-0.7-0.5-0.3-0.100.10.30.5 Fig. 8.
Upper panel: the scaling behaviour of the Λ p T spectrapresented in z at 0.9, 2.76 and 7 TeV. The solid curve is theCSP fit in eq. (4) with parameters in the third row of table 4.The data points are taken from refs. [5, 6]. Lower panel: the R distributions. From the above statement, we see that the CSP modelcan successfully describe the scaling behaviour of the strangeparticle p T spectra at 0.9, 2.76 and 7 TeV. The reason is asfollows. W ( x ) in eq. (5) and g ( x, p T ) in eq. (6) are invari-ant under the transformation x → x ′ = λx , γ → γ ′ = γ/λ and p T → p ′ T = p T / √ λ . Here λ = h S n /nS i / , wherethe average is taken over all the clusters decaying intostrange particles [17]. As a result, the strange particle p T spectra in eq. (4) are also invariant. This invariance isexactly the scaling behaviour we are looking for. Compar-ing the p ′ T transformation in the CSP model p ′ T → p ′ T √ λ with the one utilized to search for the scaling behaviour p T → p T /K , we deduce that the scaling parameter K isproportional to h nS /S n i / . As the degree of string over-lap nS /S n nonlinearly grows with √ s [16,19], the scalingparameter K should also increase with √ s in a nonlineartrend. That’s indeed what we observed in table 2. There- Φ ( z ) -8 -6 -4 -2 Ξ -1 R -0.8-0.6-0.4-0.200.20.40.6 Fig. 9.
Upper panel: the scaling behaviour of the Ξ p T spectrapresented in z at 0.9, 2.76 and 7 TeV. The solid curve is theCSP fit in eq. (4) with parameters in the fourth row of table4. The data points are taken from refs. [5, 7]. Lower panel: the R distributions. Φ ( z ) -10 -8 -6 -4 -2 φ R -0.3-0.100.10.3 Fig. 10.
Upper panel: the scaling behaviour of the φ p T spectrapresented in z at 0.9, 2.76 and 7 TeV. The solid curve is theCSP fit in eq. (4) with parameters in the fifth row of table 4.The data points are taken from refs. [8–10]. Lower panel: the R distributions. fore the CSP model can qualitatively explain the scalingbehaviour for the K S , Λ, Ξ and φ p T spectra separately.In order to determine the nonlinear trend with which K increases with √ s , we fit the K values at 0.9, 2.76and 7 TeV for K S , Λ, Ξ and φ in table 2 with a function K = α ln( √ s )+ β , where √ s is in TeV, α and β are free pa-rameters and α characterizes the rate at which K changeswith ln √ s . In sect. 2, the scaling parameter K at 2.76 TeVis set to be 1 and it is not assigned to an uncertainty. Here,in order to do the fit, we take its uncertainty as the rela-tive error of h p T i at this energy. The α values returned bythe fits for K S , Λ, Ξ and φ are 0.109 ± ± ± ± h z i are the same for K S (Λ, Ξor φ ) at 0.9, 2.76 and 7 TeV. As K = h p T i / h z i , the ratiobetween the values of K should be equal to the ratio be-tween the values of h p T i . h p T i is evaluated in terms of the iwen Yang, et al.: Universal scaling of strange particle p T spectra in pp collisions 7 CSP model as [4] h p T i = R ∞ R ∞ W ( x ) g ( x, p T ) p dxdp T R ∞ R ∞ W ( x ) g ( x, p T ) p T dxdp T . (7)Plugging W ( x ) in eq. (5) and g ( x, p T ) in eq. (6) into eq.(7), we get h p T i = √ γπ ( κ − Γ ( κ − )2 Γ ( κ ) , (8)which depends on γ and κ . In order to determine the val-ues of γ and κ at 0.9, 2.76 and 7 TeV, we fit the strangeparticle spectra at these three energies to eq. (4) with theleast squares method. They are tabulated in table 5. Withthese γ and κ values, we can calculate the ratios betweenthe values of h p T i at 0.9 (7) and 2.76 TeV for K S , Λ,Ξ and φ . They are 0 . ± .
03, 0 . ± .
04, 0 . ± . . ± .
38 (1 . ± .
03, 1 . ± .
04, 1 . ± .
04 and1 . ± . γ and κ at 0.9 (7) and 2.76 TeV. Comparing these ratioswith the scaling parameters K at 0.9 and 7 TeV in table2, we find they are indeed consistent within uncertainties.Therefore, the CSP model can also explain the scaling be-haviour of the K S , Λ, Ξ and φ p T spectra in a quantitativeway.Finally, we would like to see whether the energy depen-dence of the scaling parameter K for the strange particles K S , Λ, Ξ and φ is the same as that for charged pions,kaons and protons. We fit K = α ln( √ s ) + β to the K values at 0.9, 2.76 and 7 TeV for charged pions, kaonsand protons in ref. [4]. The values of α for charged pions,kaons and protons are 0 . ± . . ± .
015 and0 . ± . α value for charged pions is smallerthan those for the strange particles while the α values forcharged kaons and protons are comparable to those forthe strange particles. Table 5. C , γ and κ returned by the CSP fits on the K S ,Λ, Ξ and φ spectra at 0.9, 2.76 and 7 TeV. The uncertaintiesquoted originate from the statistical and systematic errors ofthe data points added in quadrature. The last column showsthe reduced χ s for the fits. √ s (TeV) C γ κ χ /dof0.9 (64 ± × − ± ± K S ± × − ± ± ± × − ± ± ± × − ± ± ± × − ± ± ± × − ± ± ± × − ± ± ± × − ± ± ± × − ± ± ± × − ± ± φ ± × − ± ± ± × − ± ± In this paper, we have presented the scaling behaviourof the K S , Λ, Ξ p T and φ p T spectra at 0.9, 2.76 and7 TeV. This scaling behaviour appears when the spectraare shown in terms of the scaling variable z = p T /K . Thescaling parameter K is determined by the quality factormethod and it increases with energy. The rates at which K increases with ln √ s for these strange particles are foundto be identical within errors. In the framework of the CSPmodel, the strange particles are produced through the de-cay of clusters that are formed by the strings overlapping.We find that the strange mesons and baryons are pro-duced from clusters with different size distributions, whilethe strange mesons (baryons) K S and φ (Λ and Ξ) orig-inate from clusters with the same size distributions. Thecluster’s size distributions for strange mesons are moredispersed than those for strange baryons. The scaling be-haviour of the p T spectra for these strange particles canbe explained by the colour string percolation model quan-titatively. Acknowledgements
Liwen Yang, Yanyun Wang, Na Liu, Xiaoling Du andWenchao Zhang were supported by the Fundamental Re-search Funds for the Central Universities of China un-der Grant No. GK201502006, by the Scientific ResearchFoundation for the Returned Overseas Chinese Scholars,State Education Ministry, by Natural Science Basic Re-search Plan in Shaanxi Province of China under Grant No.2017JM1040, and by the National Natural Science Foun-dation of China under Grant Nos. 11447024 and 11505108.Wenhui Hao was supported by the National Student’sPlatform for Innovation and Entrepreneurship TrainingProgram under Grant No. 201710718043.
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