Constituent Quark Number Scaling from Strange Hadron Spectra

Home , Constituent quark

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dN dN κ 2 φ , e/)srneqakand quark strange GeV/c) Ω ch ch hc sacersga of signal clear a is which and p /d /d T Λ ain ra (GeV/c) , 〉η 〉η China , =6.1 =17.5 Ξ p − P olsosa H energies LHC at collisions -Pb √ , s Ω 0.05 0.15 pp 0.2 2, = − 0.1 0.1 0 0 , † e 2]adtepre- the and [26] TeV 7 φ pp olsosat collisions p/φ . . . 3 2.5 2 1.5 1 0.5 3 2.5 2 1.5 1 0.5 and pp collisions κ dN/dp φ, ai is ratio K olsosat collisions Ω p (d) (b) P olsosat collisions -Pb ∗ nfu multiplic- four in 〈 〈 50-100% class 5-30% class in T dN dN dy ch ch p /d /d T aaof data (GeV/c) 〉η 〉η =2.9 =10.4 √ pp √ vely. s s col- are Ω = = − 2

0.2 0.2 signal of the quark number scaling property for hadronic dy) (a) pp 7TeV (INEL) (b) pp 13TeV (INEL) T pT spectra in pp collisions. Considering the production 1/2 − 0.15 f φ (2p ) 0.15 of baryon Ω (sss) and meson φ (ss¯), their momentum T 1/3 κ× distribution functions f (p ) dN/dp in QCM under f Ω (3p ) φ, Ω T T 0.1 T 0.1 equal velocity combination approximation≡ read as scaled dN/(dp 0.05 0.05

3 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 pT p (GeV/c) p (GeV/c) f (p )= κ f , (1) T T Ω T Ω s 3 2 h pT i pT pT Figure 2. The scaling property for the dN/dpT dy data of f (p )= κ f f = κ f . (2) − φ T φ s 2 s¯ 2 φ s 2 Ω and φ at midrapidity in inelastic events in pp collisions at h i √s = 7 and 13 TeV. The coefficient κφ,Ω is taken to be (2.0, 1.5), respectively. Data of Ω− and φ are taken from Refs. [26, 27]. Here, κφ and κΩ are coefficients independent of momentum. fs,s¯ (pT ) is s (s¯) quark distribution at hadronization and we assume fs (pT ) = fs¯ (pT ) in the center rapidity region at LHC energies. With above two formulas, we On the other hand, we run event generators Pythia8 get a production correlation between Ω− and φ in QCM [30, 31] and Herwig6.5 as the naive test of the prediction of string and cluster fragmentation mechanism in pp collisions at √s = 13 TeV. Fig. 3 shows results of the scaled p spectra of Ω and φ at mid-rapidity in two event gen- 1/2 1/3 1/2 T fφ (2pT )= κφ,ΩfΩ (3pT )= κφ fs (pT ) , (3) erators. Here we adopt Pythia version 8240 and Herwig version 6521 to simulate. We choose two event classes, inelastic non-diffractive events (INEL) and high multiplicity events with dN /dy 15, to check the multiplic- 1/2 1/3 ch ≥ where κφ,Ω = κφ /κΩ is independent of momentum. ity dependence of the prediction. In Pythia8 simulations, In order to check this scaling property, we take following we further check the prediction with default string frag- − operation on the dN/dpT dy data of Ω and φ at midra- mentation tune (marked as Pythia8 in Fig. 3) and that − pidity [26]: (i) divide the pT bin of Ω (φ) by 3 (2) and with rope hadronization mechanism (marked as Pythia8 (ii) take the 1/3 (1/2) power of the measured dN/dpT dy rope in Fig. 3). Panels (a)-(c) show the scaled spectra of − 1/3 for Ω (φ), and (iii) multiply (dNΩ/dpT dy) by a con- Ω and φ where coefficient κ is chosen so that two spectra stant factor κφ,Ω so that data points at small pT (pT . are coincident with each other at small pT . Panel (d) 0.5 GeV/c) are in coincidence with the scaled data of φ shows the ratio of two scaled spectra. We see that the as much as possible. We show in Fig.1 the scaled data of constituent quark number scaling property in two event Ω− and φ in different multiplicity classes in pp collisions generators with current tunes is violated by more than at √s = 7 TeV. The relative statistical uncertainties of 20% at pT & 1.5GeV/c. the scaled data are only a few percentages and are shown as rectangles with filled colors in the figure. We see that (a) 1/3 (b) f Ω (3p ) κ× in the high multiplicity classes, e.g., Fig.1(a) and (b), the T 1/2 − f φ (2p ) T scaled data of Ω are consistent very well with those of −1 −1 φ, and therefore the quark number scaling property holds 10 10 well. This verifies our argument in the recent work [21] scaled spectra and is a clear signal of quark combination hadronization in pp collisions at LHC. In the low multiplicity classes, Pythia8 Pythia8 rope − Fig. 1(c) and (d), the scaled data of Ω are somewhat 0.5 1 1.5 2 0.5 1 1.5 2 κ 1/3 (d) flatter than those of φ as p & 1 GeV/c and the quark (c) 1.4 fΩ (3p ) T T 1/2 f φ (2p ) number scaling property seems to be broken to a certain T 1.2 extent. We note that this is probably due to the thresh- −1 10 Pythia8 INEL old effects of strange quark production [21]. scaled spectra scaled spectra 1 Pythia8 rope INEL Herwig6.5 INEL ≥ − Pythia8 dNch/dy 15 In Fig. 2, we show the scaled data of Ω and φ in pp scaled-spectrum ratio ≥ Herwig6.5 0.8 Pythia8 rope dNch/dy 15 ≥ collisions at √s = 7 and 13 TeV [26–29] as a guide of Herwig6.5 dN ch/dy 15 0.5 1 1.5 2 0.5 1 1.5 2 energy dependence. We see that the quark number scal- pT (GeV/c) pT (GeV/c) ing property in inelastic events in pp collisions at √s =

7 TeV is broken to a certain extent but it is well fulﬁlled Figure 3. The scaled pT spectra of Ω and φ at midrapidity in inelastic events in pp collisions at √s = 13 TeV. This in pp collision at √s = 13 TeV in event generators Pythia8 is an indication of the quark combination hadronization and Herwig6.5. See text for detailed description. at higher collision energies. 3

In this paper, we apply a specific quark combination and/or antiquarks. We have, for Bi(q1q2q3) model proposed in our recent works [17, 21] to systematically study the production of identified hadrons in pp f (n) (p )= A f (n) (x p ) f (n) (x p ) f (n) (x p ) , Bi B Bi q1 1 B q2 2 B q3 3 B collisions at √s = 13 TeV. We mainly calculate the pT (6) distributions and yields of identified hadrons and focus and for Mi (q1q¯2) on various ratios or correlations for hadronic yields and (n) (n) pT spectra. We compare our results with the available (n) fM (pM )= AMi fq1 (x1pM ) fq¯2 (x2pM ) , (7) experimental data to systematically test the quark com- i bination hadronization in pp collisions at LHC energies. (n) where fq (p) dnq/dp is the momentum distribution Predictions are made for the further test in future. ≡ −1 3 (n) of quarks normalized to one. A = dp fq (xip) The paper is organized as follows: Sec. II briefly intro- Bi i=1 i −1 (n) (n) duces a specific model of quark (re)combination mech- and A = dpfq1 (x1p) f (x2p) are normalization Mi q¯2 R Q anism under equal velocity combination approximation. coefficients for baryon Bi and meson Mi, respectively. Sec. III and Sec. IV present our results and relevant Momentum fractionsR x are given by recalling momentum discussions in inelastic events and different multiplicity p = mγv m, classes, respectively. A summary and discussion is given ∝ at last in Sec. V. xi = mi/ mj , (8) j X

II. QUARK COMBINATION MODEL UNDER where indexes i, j = 1, 2, 3 for baryon and i, j = 1, 2 for EQUAL VELOCITY COMBINATION meson. Quark masses are taken to be constituent masses APPROXIMATION ms = 500 MeV and mu = md = 330 MeV. Multiplicities of baryon and meson are Quark (re-)combination/coalescence mechanism was NB = Nq q q Pq q q →B , (9) proposed in 1970s [32] and has many applications both i 1 2 3 1 2 3 i − + N = N P . (10) in elementary e e , pp and heavy-ion collisions [33–39]. Mi q1q¯2 q1q¯2→Mi In particular, ultra-relativistic heavy-ion collisions create Here N is the number of all possible three the deconfined hot quark matter of large volume whose q1q2q3 quark combinations relating to B formation and microscopic hadronization process can be described by i is taken to be 6N N N , 3N (N 1) N and QCM naturally [40–45]. In this section, we briefly intro- q1 q2 q3 q1 q1 q2 N (N 1) (N 2) for cases of three− different fla- duce a quark combination model proposed in previous q1 q1 q1 vors, two− identical− flavor and three identical flavor, re- works [17, 21] within QCM framework under the equal spectively. Factors 6 and 3 are numbers of permutation velocity combination approximation. We take the con- relating to different quark flavors. N = N N is the stituent quarks and antiquarks as the effective degrees of q1q¯2 q1 q¯2 number of all possible q q¯ pairs relating to M forma- freedom of the soft parton system created in collisions 1 2 i tion. just at hadronization. The combination of these con- Considering the flavor independence of strong inter- stituent quarks and antiquarks with equal velocity forms action, we assume the probability of q q q forming a the identified baryons and/or mesons. 1 2 3 baryon and the probability of q1q¯2 forming a meson are flavor independent, the combination probability can be written as A. Hadron production at given numbers of quarks and antiquarks

N B Pq1q2q3→Bi = CBi , (11) The momentum distributions of identified baryon and Nqqq meson are denoted as N M Pq1 q¯2→Mi = CMi . (12) f (p )= N f (n) (p ) , (4) Nqq¯ Bi B Bi Bi B f (p )= N f (n) (p ) . (5) Mi M Mi Mi M Here N B/Nqqq denotes the average (or flavor blinding) probability of three quarks combining into a baryon. Here pB and pM are momenta of baryon Bi and me- N B is the average number of total baryons and Nqqq = son M , respectively. N and N are the momentum- N (N 1)(N 2) is the number of all possible three i Bi Mi q q − q − integrated multiplicities of Bi and Mi, respectively. The quark combinations with Nq = f Nf the total quark superscript (n) denotes the distribution function is nor- number. CB is the probability of selecting the correct i P malized to one. Under the equal velocity combination discrete quantum number such as spin relating to Bi as approximation, also called comoving approximation, mo- q1q2q3 is destined to form a baryon. Similarly, N M /Nqq¯ mentum distributions of baryon and meson can be simply approximately denotes the average probability of a quark obtained by the product of those of constituent quarks and an antiquark combining into a meson and CMi is the 4 branch ratio to Mi as q1q¯2 is destined to from a me- B. Quark number fluctuations and some threshold son. N M is total meson number and Nqq¯ = NqNq¯ is the effects of hadron production number of all possible quark antiquark pairs for meson formation. As quark numbers at hadronization are small, identi- In this paper, we only consider the ground state J P = fied hadron production will suffer some threshold effects. 0−, 1− mesons and J P = (1/2)+, (3/2)+ baryons in fla- For example, baryon production is forbidden for events vor SU(3) group. For mesons − with Nq < 3. For events with Ns < 3, Ω baryon pro- 1 for J P =0− mesons duction is forbidden. In pp collisions at LHC energies, 1+RV /P C = (13) the event-averaged number of strange quarks Ns . 1 Mj RV /P for J P =1− mesons, h i ( 1+RV /P in midrapidity region ( y < 0.5) for inelastic events and not-too high multiplicity| | event classes. Therefore, the where we introduce a parameter RV/P which represents yield of Ω− is no longer completely determined by the P − the relative production weight of the J = 1 vector average number of strange quarks but is strongly influ- P − mesons to the J =0 pseudo-scalar mesons of the same enced by the distribution of strange quark number. Sim- flavor composition. For baryons ilar case for Ξ which needs two strange quarks, etc. Here we use ( N , N ) to denote the distribution of 1 for J P = (1/2)+ baryons P { qi } {h qi i} 1+RD/O quark numbers around the event average, and obtain the CB = R (14) j D/O for J P = (3/2)+ baryons, averaged multiplicity of identified hadrons by ( 1+RD/O except that CΛ = CΣ0 = 1/(2 + RD/O), CΣ∗0 = Nh = ( Nq , Nq ) Nh , (15) RD/O/(2 + RD/O), C∆++ = C∆− = CΩ− = 1. The pa- h i i P { i } {h i i} i rameter R stands for the relative production weight Nq D/O {Xi } of the J P = (3/2)+ decuplet to the J P = (1/2)+ octet baryons of the same flavor content. Here, RV/P is taken where N is given by Eqs. (9) and (10) and is the func- to be 0.45 by fitting the data of K∗/K ratios in pp col- hi tion of Nqi . lisions at √s = 7 TeV and p-Pb collisions at √sNN = { } For simplicity, we assume flavor-independent quark 5.02 TeV [46] and RD/O is taken to be 0.5 by fitting the data of Ξ∗/Ξ and Σ∗/Λ [47]. The fraction of baryons number distribution relative to mesons is N /N 0.085 at vanishing net- B M ≈ quarks [18, 21, 45]. Using the unitarity constraint of ( N , N )= (N , N ) , (16) P { qi } {h qi i} P f h f i hadronization NM +3NB = Nq, NBi and NMi can be f calculated using above formulas for the given quark num- Y bers at hadronization. We summarize the main underlying dynamics of the where f runs over u, d, s flavors. Here we neglect the model. Constituent quarks and antiquarks are assumed fluctuation of net-charges and take Nf = Nf¯ in each to be effective degrees of freedom of soft parton system events. The distribution of u and d quarks is based on at hadronization. The combination of these constituent the Poisson distribution P oi Nu(d), Nu(d) . As afore- mentioned discussion, we particularlyh tune strangei quark quarks and antiquarks with equal velocity forms baryons and mesons. This is similar to constituent quark model, distribution. Because in minimum bias events and small multiplicity classes in pp collisions N . 1 and Pois- i.e., the summation of masses (and momenta in mo- h si tion) of constituent quarks properly constructs the mass son distribution P oi (Ns, Ns ) in this case has a long tail as N 3 which mayh over-weighti the events with (and momentum in motion) of hadron. Model param- s ≥ Ns 3, we distort the Poisson distribution by a suppres- eters RV/P and RD/O contain unclear non-perturbative sion≥ factor γ , that is, (N , N )= P oi (N , N ) dynamics and are obtained by fitting the relevant ex- s P s h si N s h si × perimental data and are assumed to be relatively sta- [γs Θ (Ns 3)+Θ(3 Ns)] where Θ (x) is the Heaviside step function− and −is normalization constant. γ is ble in/at different collision systems/energies. Also, the N s normalization of hadronization process is a prerequisite taken to be 0.8 in inelastic (INEL>0) events and various for quark combination. Quark number conservation is multiplicity classes. not only globally satisfied via NM + 3NB = Nq and There are other possible effects of small quark num- NM +3NB¯ = Nq¯ but also satisfied for each quark fla- bers. For example, in events with Ns = Ns¯ = 1, be- vor via h nqi,hNh = Nqi . Here h runs over all hadron cause s and s¯ are most likely created from the same one species and qi = d, u, s, d,¯ u,¯ s¯. nq ,h is the number of vacuum excitation and therefore they are not likely to P i constituent quark qi in hadron h. Therefore, this model directly constitute color singlet and therefore φ produc- is a statistical model based on constituent quark degrees tion in these events is suppressed. In addition, quark of freedom and is different from those popular parton momentum distributions are more or less dependent on recombination/coalescence models [40, 41] which adopt the quark numbers (in other words, system size) and we the Wigner wave function method under instantaneous neglect such dependence for the given multiplicity classes hadronization approximation. and its potential effects are studied in the future works. 5

III. RESULTS IN INELASTIC EVENTS We emphasize that, by taking advantage of the quark number scaling property, this is the first time we can We apply the above quark combination model to de- conveniently extract the momentum distributions of soft scribe the transverse production of hadrons at midrapid- quarks at hadronization from the experimental data of ity in pp collisions.The approximation of equal velocity hadronic pT spectra. The extracted quark pT spectra combination in the model is reduced to that of equal carry important information of soft parton system cre- transverse-velocity combination. Here, we only study ated in pp collisions at LHC energies. First, because of parameters bs and bu in quark distribution function in one dimensional pT distribution of hadrons by further integrating over the azimuthal angle. The p distribu- Eq. (17) are obviously smaller than one, the extracted T (n) (n) tion functions of quarks at hadronization at midrapidity fu (pT ) and fs (pT ) deviate from Boltzmann distri- are inputs of the model and are denoted as fqi (pT ) = bution in the low pT range. This indicates that thermal- (n) ¯ ization may be not reached for the small partonic system Nqi fqi (pT ) with qi = d, u, s, d, u,¯ s¯. Nqi is the num- h i h i (n) created in pp collisions at LHC. Second, we see that the ber of qi in rapidity interval y < 0.5 and fqi (pT ) (n) (n) | | ≡ ratio fs (pT ) /fu (pT ), Fig. 4 (b), increases at small dnqi /dpT is quark pT spectrum normalized to one. We assume the iso-spin symmetry between up and down quarks pT and then saturates (only slightly decreases) with pT . This property is similar to the observation in pp collisions and assume the charge conjugation symmetry between √ quark and antiquark. Finally we have only two inputs at s = 7 TeV [21] and in p-Pb collisions at √sNN = 5.02 TeV [17] and is also similar to the observation in heavy- fu(pT ) and fs(pT ) which can be fixed by fitting the data of identified hadrons. ion collisions at RHIC and LHC [48–50]. These information of constituent quarks provides important constraint of developing more sophisticated theoretical models of soft parton system created in high energy collisions. A. Quark pT distribution at hadronization

Using the scaling property in Eq. (3) and experimental B. pT spectra of identified hadrons data shown in Fig. 2, we can directly obtain the normalized pT distribution of strange quarks at hadronization, Among hadrons that are often measured by experi- which can be parameterized in the form ments, pion and kaon are most abundant particles. How- ever, because the masses of pion and kaon are signif- −ns 2 2 icantly smaller than the summed masses of their con- (n) bs pT + Ms Ms f (pT )= s (pT + as) 1+ − , stituent (anti-)quarks, the momenta of pion and kaon can s N n c p s s ! not be calculated by the simple combination of those of (17) constituent (anti-)quarks at hadronization [21]. There- where is the normalization constant and parameters Ns fore, momentum spectra of pion and kaon are not the as = 0.15 GeV/c, bs = 0.649, ns = 4.14, Ms = 0.5GeV most direct probe of the quark combination model and and cs =0.346. By fitting the data of other hadrons such their results are not shown in this paper. On the other ∗0 as proton and K , we also obtain the pT distribution of hand, proton, Λ, Ξ−, Ω−, φ and K∗0 can be well con- up/down quarks at hadronization. Taking also the pa- structed by the constituent quarks and antiquarks. These rameterization form Eq. (17), the parameters of up/down hadrons can be used to effectively test the quark combi- quarks are au = 0.15 GeV/c, bu = 0.355, nu = 3.46, nation model. M = 0.33 GeV and c = 0.358. In Fig. 4, we plot u u In Fig. 5, we show the calculation results of pT spectra (n) (n) − − ∗0 fu (pT ) and fs (pT ) as the function of pT and their of proton, Λ, Ξ , Ω , φ and K in inelastic (INEL>0) ratio in inelastic events in pp collisions at √s = 13 TeV. events in pp collisions at √s = 13 TeV using the quark spectra in Fig. 4 and quark numbers Nu = 2.8 and 1.4 h i ) T (a) (b) Ns = 0.86. Here, quark numbers are fixed by glob- 1 h i (p 1.2ratio ally fitting data of pT -integrated yield densities of these (n) q f hadrons [27]. Solid lines are QCM results which have 1 included the contribution of strong and electromagnetic

0.8 decay of resonances. Symbols are preliminary data of (n) − 1 f (p ) hadronic pT spectra at midrapidity measured ALICE col- 10 u (n) (n) (n) T 0.6 f s (p f)/ u (p ) laboration [27]. We see that the data can be well ﬁtted fs (p ) T T T ∗0 0 0.5 1 1.5 2 2.5 3 0.4 0 0.5 1 1.5 2 2.5 3 in general by QCM. Our result of K is slightly smaller p (GeV/c) p (GeV/c) than the data. If we multiply the result of K∗0 spectrum T T by a constant factor and we will see the shape is in good

Figure 4. The pT spectra of u and s quarks at midrapidity agreement with the data. and the ratio between them in inelastic (INEL>0) events in Besides the scaling property between the pT spectra of pp collisions at √s = 13 TeV. Ω− and φ shown in the introduction (Sec. I), (p +¯p) /φ ratio as the function of pT can also give an intuitive 6

] -1 − 1 10 − − 1 2 10 10

− 2 − 2 10 10 − 3

dy)(GeV/c) 10 T − 3 − 3 10 10 − 4 10 dN/(dp + − 4 - − 4 Ξ Ξ 10 p + p 10 ( Λ + Λ)/2 + − 5 10 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7

− 3 -1 − 1 10 10 − 2 10 − 2 10 − 4 − 3 dy)(GeV/c) 10

T 10 − 3 + 10 - Ω + Ω − 4 dN/(dp *0 *0 − 5 10 φ K + K − 4 10 10 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 p (GeV/c) p (GeV/c) p (GeV/c) T T T

Figure 5. The pT spectra of identiﬁed hadrons at midrapidity in inelastic (INEL>0) events in pp collisions at √s = 13 TeV. Solid lines are results of QCM and symbols are preliminary data of ALICE collaboration [27].

picture of microscopic mechanism of hadron production. φ Proton and φ have similar masses but totally different )/p pp 13TeV INEL quark contents. In the central (0-10% centrality) Pb-Pb 10 (p+ QCM, pp 13TeV collisions at √sNN = 2.76 TeV, the data of (p +p ¯) /φ ratio [51], black squares in Fig. 6, are almost flat with PbPb 2.76TeV 0-10% respect to pT . This flat ratio is usually related to similar masses of proton and φ and is usually attributed to the strong radial flow and statistical hadronization un- 5 der chemical/thermal equilibrium in relativistic heavy- ion collisions. However, data of (p +¯p) /φ in inelastic events in pp collisions at √s = 13 TeV [52], solid circles in Fig. 6, show a rapid decrease with the increasing 0 1 2 3 4 5 6 7 8 pT . This is an indication of out of thermal equilibrium p (GeV/c) T in pp collisions. In QCM, pT distributions of identified hadrons are determined by pT spectra of (anti-)quarks at hadronization. (p +¯p) /φ ratio in QCM reflects the ratio Figure 6. Ratio (p +p ¯) /φ as the function of pT in inelastic events in pp collisions at √s = 13 TeV. Solid line is the result or correlation between the third power of u quark spec- of QCM and symbols are experimental data [51, 52]. trum and the square of s quark spectrum. With quark spectra in Fig. 4 which self-consistently describe data of hadronic pT spectra in Fig. 5, the calculated p/φ ratio in QCM, the solid line in Fig. 6, shows a decreasing behavior with pT and is in good agreement with the data of pp collisions [52]. Hyperons Λ, Ξ− and Ω− contain one, two and three s constituent quarks, respectively. Therefore, ratios Ξ−/Λ and Ω−/Ξ− can reflect the difference in momentum distribution between u(d) quark and s quark at hadronization. Fig. 7 shows our prediction of ratios Ξ−/Λ and − − Ω /Ξ as the function of pT in inelastic events in pp collisions at √s = 13 TeV. We see that two ratios increase tween pT spectrum of strange quark and that of up/down (n) (n) with pT and then tend to be saturate at intermediate quark in the range pT . 2GeV/c, see fs (pT )/fu (pT ) p 6 GeV/c, which is directly due to the difference be- ratio shown Fig. 4(b). T ∼ 7

A. Hadronic yields and yield ratios ratio 0.2 In Fig. 9, we show the pT -integrated yields of identified hadrons (including kaon) [54] in different multiplicity 0.15 classes and compare them with the preliminary data in pp collisions at √s = 13 TeV [52, 55]. In general, results of QCM, solid lines, are in good agreement with the data 0.1 (with maximum deviation about 10%). Yield ratios of different hadrons can significantly can- Ξ - Λ 0.05 / cel the dependence of model parameters and/or model Ω- Ξ - inputs. Therefore, they are more direct test of the ba- / sic physics of the model in confronting with the exper- 00 1 2 3 4 5 6 7 imental data. In Fig. 10, we show the yield ratios of p (GeV/c) − − T hyperons Ω , Ξ and Λ to pions divided by the values in the inclusive INEL>0 events. Data of pp collisions at √s = 7 [5] and 13 TeV [52] and those of p-Pb col- Figure 7. Prediction of ratios Ξ−/Λ and Ω−/Ξ− as the func- lisions at √sNN = 5.02 TeV [56, 57] are all presented tion of pT at midrapidity in inelastic events in pp collisions at √s = 13 TeV. in order to get a visible tendency with respect to multiplicity of charged particles at midrapidity. Solid lines are numerical results of QCM, which are found to be in agreement with the data. We emphasize that such strangeness-related hierarchy behavior is closely related to the strange quark content of these hyperons during IV. RESULTS IN DIFFERENT MULTIPLICITY their production at hadronization, which can be under- CLASSES stood easily via an analytical relation in QCM. Taking yield formulas Eqs. (9), (11) and considering the strong Using the preliminary data of pT spectra of proton, and electromagnetic decays, we have ∗0 K and φ in different multiplicity classes [52, 53], we can 3 λs determine the corresponding pT spectra of constituent N N , (18) Ω ≈ 3 B quarks at hadronization and predict pT spectra of other (2 + λs) 2 identified hadrons. Fig. 8 shows the extracted quark 3λs NΞ N B, (19) pT spectra (using the parameterized form Eq. (17)) ≈ (2 + λ )3 at midrapidity in different multiplicity classes. Because s 2+0.88RD/O RD/O 6λs the parameter bq of quark spectrum in high multiplicity N +0.94 N Λ ≈ 2+ R 1+ R 3 B classes tends to one, the distribution function in Eq. (17) D/O D/O (2 + λs) will asymptotically tend to Boltzmann distribution in low (20) pT range and therefore we see a thermal behavior for 7.73λs quark spectrum. This is related to the increasing mul- N B, ≈ (2 + λ )3 tiple parton interactions in these event classes. In small s multiplicity classes, parameter bq of quark spectrum is where we neglect the effects of small quark numbers relatively small and the quark spectrum deviates from and adopt the strangeness suppression factor λs = thermal behavior. Ns / Nu . Because of complex decay contributions, hpioni yieldh i has a complex expression [58] and here we can ) ) write Nπ = aπ Nq with coefficient aπ being almost a T 10 T (a) (b) constant. Thenh thei double ratios in Fig. 10 have simple (p (p u s f f 1 approximate expressions ′ pp 3 3 1 NΩ NΩ λs λs I / / , (21) N N ≈ 3 ′ 3 II − 1 π π INEL>0 (2 + λs) (2 + λs) 10 ′ III VII pp 2 2 − 1 N N λ λ 10 IV VIII Ξ / Ξ s / s , (22) V IX 3 ′ 3 Nπ Nπ INEL>0 ≈ (2 + λs) (2 + λ ) VI X s − 2 ′ 10 pp 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 NΛ NΛ λs λs p (GeV/c) p (GeV/c) / / ′ , (23) T T N N ≈ 3 3 π π INEL>0 (2 + λs) (2 + λs) ′ Figure 8. The pT spectra of u and s quarks at midrapidity in where λs is the strangeness suppression factor in INEL>0 different multiplicity classes in pp collisions at √s = 13 TeV. events in pp collisions. Here, we see a clear hierarchy structure among three double ratios in terms of λs. 8

(a) (b) (c)

/dy 1 h + - K + K −1 dN 1 10

*0 − *0 φ 10 1 (K +K )/2 p + p

−2 −1 0 5 10 15 20 25 30 10 0 5 10 15 20 25 30 10 0 5 10 15 20 25 30

1 − (d)10 1 (e)−2 (f)

/dy 10 h dN

0 0 + (Λ + Λ ) (Ξ +Ξ ) −3 - − −2 10 (Ω + Ω ) 10 1 10

0 5 10 15 20 25 30 0 5 10 15 20 25 30 0 5 10 15 20 25 30 〈dN /d 〉η (|η|<0.5) 〈dN /d 〉η (|η|<0.5) 〈dN /d 〉η (|η|<0.5) ch ch ch

Figure 9. Yield densities of identiﬁed hadrons at midrapidity in diﬀerent multiplicity classes in pp collisions at √s = 13 TeV. Solid lines are results of QCM and symbols are preliminary data of ALICE collaboration [52, 55].

The dotted lines in Fig. 10 are results of above ana- ΩΞΛ lytic formulas with a naively tuned strangeness suppres- pp 7TeV

′ pp,7TeV INEL>0 sion λs = λ 1+0.165 log dNch/dη |η|<0.5/6.0 with )

s π 2 ′ h i pp 13TeV λ = 0.31. They can well ﬁt the experimental data of s pPb 5.02TeV

)/(h/ these double ratios as dNch/dη |η|<0.5 & 10. In small h i π multiplicity classes dNch/dη |η|<0.5 . 6 small quark

(h/ 1.5 number effects are noth negligiblei and the analytic approximation is larger than the experimental data to a certain extent. Our numerical results have included small quark number effects and are found to be closer to the data. 1 Yield ratio Ξ/φ is also influenced by the small quark number effects. If we neglect small quark number effects, QCM we have analytic approx 0.5 R λ2 N V/P s N (24) φ ≈ 1+ R 2 M 10 V/P (2 + λs) 〈dN /d 〉η (|η|<0.5) ch and using Eq. (19) we get the ratio Figure 10. Yield ratios to pions divided by the values in the NΞ + NΞ¯ 1+ RV/P 3 1.7 inclusive INEL>0 events. Data of pp collisions at √s = 7 =2 RB/M , (25) Nφ RV/P 2+ λs ≈ 2+ λs [5] and 13 TeV [52] and p-Pb collisions at √sNN = 5.02 TeV [56, 57] are presented. Solid lines are numerical results of which slightly decreases with the increase of λs and there- QCM and dotted lines are analytic approximation in QCM. fore will slightly decrease with the increase of multiplicity dNch/dη because λs increases with dNch/dη . This is inh contradictioni with the experimentalh data.i However, considering small quark number effects in QCM will pre- events of small multiplicity or small quark numbers, the dict the correct behavior of the ratio Ξ/φ, see the solid formation of Ξ− will be suppressed to a certain extent (or line in Fig. 11. The formation of Ξ− needs not only two forbidden occasionally) due to the need of one more light s quarks but also a d quark, which is different from the quark, in comparing with the formation of φ. We see that formation of φ needing only a s and a s¯. Therefore, in the calculated ratio Ξ/φ using QCM increases with sys- 9 tem multiplicity dN /dη and the increased magnitude ch /p 0.4 of the ratio is consistenth withi the experimental data of φ pp collisions at √s = 13 TeV and those of p-Pb collisions 0.35 at √sNN = 5.02 TeV [52, 59].

φ 0.3 )/

+ λ s=0.36 Ξ 0.7 +

- 0.25 Ξ

( λ =0.32 pp 7TeV 0.6 s 0.2 pp 13TeV pPb 5.02TeV 0.5 0.15 QCM analytic approx 0.4 0.1 10 pp 13TeV 〈dN /d 〉η (|η|<0.5) ch 0.3 pPb 5.02TeV

Figure 12. Yield ratio φ/p as a function of dNch/dη 0 5. QCM h i|η|< . 0.2 Data in pp collisions at √s = 7 and 13 TeV, solid triangles 10 and squares, and data in p-Pb collisions at √sNN = 5.02 〈dN /d 〉η (|η|<0.5) TeV, solid circles, are taken from Refs. [52, 59]. The solid ch line is the numerical results of QCM. The short dashed lines are estimation under analytical approximation. Figure 11. Yield ratio Ξ/φ as a function of dNch/dη |η|<0.5. The preliminary data in pp collisions at √sh= 13 TeV,i solid squares, and in p-Pb collisions at √sNN = 5.02 TeV, solid circles, are taken from Refs. [52, 59]. The solid line is the result of QCM.

B. pT spectra of identified hadrons Proton and φ have similar masses but different quark contents. Yield ratio φ/p can further test the flavor- In Fig. 13, we show the fit of data of pT spectra of dominated feature of hadron production in QCM. Ne- proton, K∗∗ and φ [52, 53] using QCM and the predic- glecting small quark number effects, proton yield after tion of other identified hadrons in different multiplicity taking into account decay contribution of ∆ resonances classes in pp collisions at √s = 13 TeV. Note that classes has a simple expression IV and V are combined for the K∗0 data and the same for our results. Beside directly comparing the predic- 1 1 Np 3 N B. (26) tion of single hadron spectrum with the future data, we ≈ 4 (2 + λs) emphasize that QCM can be more effectively tested by some spectrum ratios and/or scaling. The first is to test Using Eq. (24), we get yield ratio whether the constituent quark number scaling property − between the pT spectrum data of Ω and φ holds in dif- Nφ 1 RV/P 2 N M λs(2 + λs) ferent multiplicity classes. The second is to study the N ≈ 4 1+ R − − p V/P N B ratio Ω /φ as the function of pT . Ω /φ ratio in QCM is 2 0.91λs(2 + λs), (27) solely determined by the strange quark pT spectrum at ≈ hadronization and the ratio usually exhibits a nontriv- which shows a significant dependence on the strangeness ial pT dependence, as shown in Fig. 14(a), which is a suppression factor λ . We get N /N 0.22 with typical behavior of baryon-to-meson ratio in QCM and s φ p ≈ λs = 0.32 in low multiplicity classes and Nφ/Np 0.28 is absent or unapparent in the traditional fragmentation ≈ − with λs = 0.36 in high multiplicity classes. The short picture. We also see that the ratio Ω /φ in higher multi- dashed lines in Fig. 12 are above two values under ana- plicity classes can reach higher peak values and the peak lytical approximation. They are slightly lower than the position of Ω−/φ in high multiplicity classes is also en- experimental data [52, 59], symbols in the figure. Small larged in comparing with that in low multiplicity classes. quark number effects will increase the ratio to a certain The third is to study p/φ ratio as the function of pT to extent in terms of the suppression of proton yield. We clarify the pT dependence of the ratio is flavor originated further show the numerical results of our model including or mass originated? Results of QCM are shown in Fig. small quark number effects, the solid line, and we see a 14(b) which decrease with pT and show relatively weak good agreement with the data. multiplicity dependence. 10

102 2 -1 (a) (b)10 (c)2 (d) 102 10 p + p 10 10

1 1 1 1

−1 dy)(GeV/c) −1

T 10 − 10 10 2 −2 − 2 − 10 10 10 2 9 − dN/(dp 4 I(× 2 ) −3 10 −3 × 8 10 10 II( 2 ) −4 × 7 10 III( 2 ) − 6 X 4 −4 −6 × 10 10 10 IV( 2 ) IX(× 2) 5 2 V(× 2 ) × − − *0 VIII( 2 ) 5 5 *0 − 4 3 φ 6 VI(× 2 ) VII(× 2 ) 10 10 (K +K )/2 10 (Λ + Λ )/2 −8 10 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 10

-1 (e)1 (f) (g) (h) 10 10

− 10 1 − − 2 − 10 1 10 10 1 dy)(GeV/c) T −3 − 10 −3 4 −3 10 10 10 dN/(dp

−5 − 10 −5 6 −5 10 10 10

0 + *- *0 Ξ0 Ξ Ω- Ω Σ*+ Σ Ξ*0 Ξ ( + )/2 ( + )/2 ( + )/2 −7 ( + )/2 − −8 − 10 10 7 10 10 7 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 p(GeV/c) p (GeV/c) p (GeV/c) p (GeV/c) T T T T

Figure 13. The pT spectra of identiﬁed hadrons at midrapidity in diﬀerent multiplicity classes in pp collisions at √s = 13 TeV. Solid lines are results of QCM and symbols are preliminary data of ALICE collaboration [52, 53].

0.1 I II III φ V. SUMMARY AND DISCUSSION

φ (b) IV V VI 8 / p/ - VII VIII IX Ω X INEL 6 Taking advantage of available experimental data of hadronic pT spectra and yields at midrapidity, we have 0.05 (a) 4 systematically studied the production of soft hadrons in pp collisions at √s = 13 TeV within a framework of quark 2 combination mechanism for hadronization. We applied a quark combination model which assumes the constituent 00 1 2 3 4 5 6 00 1 2 3 4 5 6 quarks and antiquarks being the effective degrees of free- p (GeV/c) p (GeV/c) T T dom for the parton system at hadronization and takes equal velocity combination approximation in hadron for- Figure 14. Prediction of ratios Ω−/φ and p/φ as the function mation. We used the model to systematically calcu- of pT at midrapidity in pp collisions at √s = 13 TeV. late the pT spectra and yields of soft strange hadrons in inelastic events (INEL>0) and in different multiplicity classes. We found several interesting results which are sensitive to the hadronization mechanism. (1) Data of pT spectra of Ω− and φ in inelastic events (INEL>0) in pp collisions 11 at √s = 13 TeV exhibit a constituent quark number scal- quarks. ing property. Data in high multiplicity classes in pp col- To confirm such a new feature of hadronization dy- lisions at √s=7 TeV also show this scaling property. It is namics in high energy pp collisions, we should carefully the first time that such a scaling property is observed in study all the related experimental data. We have made high energy pp collisions. This is an obvious experimental plenty of predictions on pT spectra and spectrum ratios signal of the quark combination mechanism at hadroniza- of strange hadrons in pp collisions at √s = 13 TeV to tion in high energy pp collisions. (2) Data of p/φ ratio further test our model by the future experimental data. in inelastic events (INEL>0) in pp collisions at √s = On the other hand, compared to our previous study in 13 TeV show an obvious decrease with the increasing pT , pp collisions at √s = 7 TeV [21] which gives the first which indicates the statistical hadronization model is not indication, the current study provides a stronger sugges- responsible for this observation. We demonstrated that tion of quark combination hadronization in high energy data are naturally explained by the quark combination pp collisions. We still need further systematical studies − − model. (3) Data of yield ratios Λ/π, Ξ /π and Ω /π di- in pp collisions at other available LHC energies so that vided by the values in inelastic events as the function we can test the universality of this new hadronization of system multiplicity dN /dη at midrapidity show h ch i feature and study its relation with the possible creation a strangeness-related hierarchy structure. We demon- of mini-QGP in small collision systems. strated that the hierarchy structure is closely related to the strange quark content of these hyperons during their production via the combination of strange quarks and up/down quarks. ACKNOWLEDGMENTS By the quark number scaling property, the pT spectrum of strange quarks was directly extracted from data of Ω and φ. The pT spectrum of up/down quarks was ex- This work is supported by the National Natu- tracted from data of other hadrons containing up/down ral Science Foundation of China (11575100), and constituent quarks. These extracted quark momentum by Shandong Province Natural Science Founda- distribution functions are important results which de- tion(ZR2019YQ06,ZR2019MA053), and by A Project scribe the property of strongly-interacting partonic sys- of Shandong Province Higher Educational Science and tem at hadronization in the language of constituent Technology Program (J18KA228).

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