Constituent Quark Number Scaling from Strange Hadron Spectra in Pp

Chinese Physics C Vol. 44, No. 1 (2020) 014101

Constituent quark number scaling√ from strange hadron spectra in pp collisions at s = 13 TeV*

1 2 1;1) 2;2) Jian-Wei Zhang(张建伟) Hai-Hong Li(李海宏) Feng-Lan Shao(邵凤兰) Jun Song(宋军) 1School of Physics and Engineering, Qufu Normal University, Shandong 273165, China 2Department of Physics, Jining University, Shandong 273155, China √ − Abstract: We show that the pT spectra of Ω and ϕ at midrapidity in the inelastic events in pp collisions at s = 13 TeV exhibit a constituent quark number scaling property, which is a clear signal of quark combination mechanism at hadronization. We use a quark combination model with equal velocity combination approximation to systematically √ study the production of identified hadrons in pp collisions at s= 13 TeV. The midrapidity pT spectra for protons, Λ, Ξ−, Ω−, ϕ and K∗ in the inelastic events are simultaneously fitted by the model. The multiplicity dependence of the yields of these hadrons are also well understood. The strong pT dependence of the p/ϕ ratio is well explained by the model, which further suggests that the production of two hadrons with similar masses is determined by their quark content at hadronization. The p spectra of strange hadrons at midrapidity in different multiplicity classes in pp colli- √ T sions at s = 13 TeV are predicted for further tests of the model. The midrapidity p spectra of soft (p < 2 GeV/c) √ T T strange quarks and up/down quarks at hadronization in pp collisions at s = 13 TeV are extracted. Keywords: strange hadron production, quark combination, hadronization, quark-gluon plasma DOI: 10.1088/1674-1137/44/1/014101

1 Introduction ness [4-6]. Theoretical studies of these new phenomena mainly focus on how the features of a small partonic system are related to these observations by considering dif- Most hadrons produced in high energy collisions have ferent mechanisms, such as the color re-connection, string a relatively low (transverse) momentum perpendicular to the beam axis. The production of soft hadrons is mainly overlap and/or color rope [7-10], or by considering the driven by the soft QCD processes, and in particular the creation of mini-QGP or the phase transition [11-16]. non-perturbative hadronization. Experimental and theor- In our latest works [17-21], by studying the available etical studies of soft hadron production are important for data for the hadronic pT spectra and yields, we proposed understanding the properties of the soft parton system a new understanding of the novel features of hadron pro- created in collisions, and for testing and/or developing duction in small quark/parton systems created in pp phenomenological models. Heavy-ion physics at the SPS, and/or p-Pb collisions at the LHC energies, i.e. a change RHIC and LHC energies shows that quark-gluon plasma of the hadronization mechanism from the traditional frag- (QGP) is created in the early stage of collisions. In pp mentation to quark (re-)combination. In the quark (re-) and/or pp¯ collisions, it is usually assumed that QGP is combination mechanism (QCM), some typical features of not created, at least not up to the RHIC energies. the identified hadron production appear, such as the en- However, recent measurements at the LHC energies show hanced baryon-to-meson ratio and quark number scaling a series of new hadron production phenomena in pp colli- of hadron elliptical flow at intermediate pT . These fea- sions, such as the ridge and collectivity behavior [1-3], tures were observed in the relativistic heavy-ion colli- increased baryon-to-meson ratio, and increased strange- sions [22-24], and recently also in pp and p-Pb collisions

Received 3 June 2019, Revised 1 October 2019, Published online 19 November 2019 * Supported by National Natural Science Foundation of China (11575100), and by Shandong Province Natural Science Foundation(ZR2019YQ06, ZR2019MA053), and by A Project of Shandong Province Higher Educational Science and Technology Program (J18KA228) 1) E-mail: [email protected], Corresponding author 2) E-mail: [email protected], Corresponding author Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must main- tain attribution to the author(s) and the title of the work, journal citation and DOI. Article funded by SCOAP3 and published under licence by Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Sciences and the Institute of Modern Physics of the Chinese Academy of Sciences and IOP Pub- lishing Ltd

014101-1 Chinese Physics C Vol. 44, No. 1 (2020) 014101 at the LHC energies in the high multiplicity classes [3, 4, Ω− and ϕ in QCM 6, 25]. In particular, a quark number scaling property of 1/2 = κ 1/3 = κ1/2 , fϕ (2pT ) ϕ,Ω fΩ (3pT ) ϕ fs (pT ) (3) the hadron transverse momentum spectra was observed in√ = . κ = κ1/2/κ1/3 p-Pb collisions at sNN 5 02 TeV [17]. where ϕ,Ω ϕ Ω is independent of momentum. In or- Recently, the ALICE collaboration reported pT spec- der to check this scaling property, we apply the follow- − tra of the identified hadrons√ in different multiplicity ing operation on the dN/(dpT dy) data for Ω and ϕ at = − classes in pp collisions at s 7 TeV [26], and prelimin-√ midrapidity [26]: (i) we divide the pT bin for Ω (ϕ) by 3 ary data for the inelastic events in pp collisions at s = 13 (2), (ii) take the 1/3 (1/2) power of the measured − 1/3 TeV [27]. For the first time, a clear signal of the quark dN/(dpT dy) for Ω (ϕ), and (iii) multiply (dNΩ/(dpT dy)) number scaling property in the hadronic pT spectra in pp by a constant factor κϕ,Ω , so that the data points for small collisions is seen. Considering the production of baryons pT (pT ≲ 0.5 GeV/c) are in coincidence with the scaled Ω− (sss) and mesons ϕ(ss¯), the momentum distribution data for ϕ as much as possible. We show in Fig. 1 the ≡ / − ϕ functions f (pT ) dN dpT in QCM with the equal velo- scaled data for Ω √and in different multiplicity classes city combination approximation read as in pp collisions at s = 7 TeV. The relative statistical un- [ ( )] 3 pT certainties of the scaled data are only a few percent, and fΩ (p ) = κΩ f , (1) T s 3 are shown as rectangles with filled colors in the figure. ( ) ( ) [ ( )] We see that in the high multiplicity classes, e.g. Fig.1(a) 2 pT pT pT and (b), the scaled data for Ω− are consistent with those fϕ (pT ) = κϕ fs fs¯ = κϕ fs . (2) 2 2 2 for ϕ, and therefore the quark number scaling property Here, κϕ and κΩ are coefficients independent of mo- holds. This verifies our argument in the recent work [21], mentum. fs,s¯ (pT ) is the s (s¯) quark distribution at hadron- and is a clear signal of quark combination hadronization ization, and we assume fs (pT ) = fs¯ (pT ) in the center in pp collisions at the LHC energies. In the low multipli- rapidity region at the LHC energies. With the above two city classes, Fig. 1(c) and (d), the scaled data for Ω− are formulas, we get a correlation between the production of somewhat flatter than for ϕ at pT ≳ 1 GeV/c, and the

Fig. 1. (color online) The scaling property in the dN/dp dy data for Ω− and ϕ at midrapidity in different multiplicity classes in pp col- √ T − lisions at s = 7 TeV. The coefficient κϕ,Ω in the four multiplicity classes is taken as (1.76, 1.82, 1.83, 1.93). The data for Ω and ϕ are taken from Ref. [26].

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quark number scaling property seems to be broken to a tios or correlations of the hadronic yields and pT spectra. certain extent. We note that this is probably due to the We compare our results with the available experimental threshold effects of strange quark production [21]. data to test the quark combination hadronization in pp − In Fig. 2, we show the scaled data for√ Ω and ϕ in pp collisions at the LHC energies. Predictions are made for collisions at s = 7 and 13 TeV [26-29] as a guide for en- future tests. ergy dependence. We see that the quark number scaling√ The paper is organized as follows: Sec. 2 briefly in- property in the inelastic events in pp collisions at s = 7 troduces the model of quark (re)combination mechanism TeV is broken to a certain extent, but it holds well in the√ with the equal velocity combination approximation. Sec. inelastic events in pp collisions at s = 13 TeV. This is 3 and Sec. 4 present our results and relevant discussions an indication of the quark combination hadronization at for the inelastic events and different multiplicity classes. higher collision energies. A summary and discussion is given in Sec. 5. On the other hand, we performed a run of event generators Pythia8 [30, 31] and Herwig6.5 as a naive test of 2 Quark combination model with the equal the string and√ cluster fragmentation mechanisms in pp velocity combination approximation collisions at s = 13 TeV. Fig. 3 shows the results for the scaled pT spectra of Ω and ϕ at mid-rapidity given by the two event generators. Here, we adopted the Pythia ver- Quark (re-)combination/coalescence mechanism was proposed in the 1970s [32] and has many applications in sion 8240 and Herwig version 6521. We chose two event − + classes, the inelastic non-diffractive events (INEL) and high energy e e , pp and heavy-ion collisions [33-39]. In particular, ultra-relativistic heavy-ion collisions create de- the high multiplicity events with dNch/dy ⩾ 15, to check the multiplicity dependence of the predictions. In the Py- confined hot quark matter in a large volume whose mi- thia8 simulations, we further checked the predictions with croscopic hadronization process can be naturally de- the default string fragmentation tune (marked as Pythia8 scribed by QCM [40-45]. In this section, we briefly intro- in Fig. 3), and with the rope hadronization mechanism duce the quark combination model, proposed in previous (marked as Pythia8 rope in Fig. 3). Panels (a)-(c) show works [17, 21] in the framework of QCM with the equal the scaled spectra of Ω and ϕ where the coefficient κ is velocity combination approximation. We take the constituent quarks and antiquarks as the effective degrees of chosen so that the two spectra are coincident at small pT . Panel (d) shows the ratio of the two scaled spectra. We freedom of the soft parton system created at hadroniza- see that the constituent quark number scaling property tion. Combinations of constituent quarks and antiquarks given by the two event generators with the current tunes with equal velocity result in formation of identified baryons and/or mesons. is violated by more than 20% at pT ≳ 1.5 GeV/c. In this paper, we apply the quark combination model 2.1 Hadron production for a given number of quarks proposed in our recent works [17, 21] to systematically and antiquarks study the√ production of identified hadrons in pp collisions at s = 13 TeV. We calculate the pT distributions The momentum distributions of the identified bary- and yields of identified hadrons and focus on various ra- ons and mesons are denoted as

Fig. 2. (color online) The scaling property in the dN/dp dy data for Ω− and ϕ at midrapidity in the inelastic events in pp collisions at √ T − s = 7 and 13 TeV. The coefficient κϕ,Ω is taken as (2.0, 1.5), respectively. The data for Ω and ϕ are taken from Refs. [26, 27].

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√ Fig. 3. (color online) The scaled pT spectra of Ω and ϕ at midrapidity in pp collisions at s = 13 TeV given by the event generators Pythia8 and Herwig6.5. See the text for detailed explanation.

/∑ (n) fB (pB) = NB f (pB), (4) = , i i Bi xi mi m j (8) j (n) fM (pM) = NM f (pM). (5) i i Mi where indexes i, j = 1,2,3 are for baryons, and i, j = 1,2

Here, pB and pM are the momenta of baryons Bi and for mesons. The quark masses are taken to be the constituent masses m = 500 MeV and m = m = 330 MeV. mesons Mi, and NB and NM are the momentum-integ- s u d i i The multiplicities of baryons and mesons are rated multiplicities of Bi and Mi, respectively. The super- = → , script (n) means that the distribution function is normal- NBi Nq1q2q3 Pq1q2q3 Bi (9) ized to one. In the equal velocity combination approxima- NM = Nq q¯ Pq q¯ →M . (10) tion, also called co-moving approximation, the mo- i 1 2 1 2 i Here, Nq q q is the number of all possible combinations of mentum distributions of baryons and mesons can be 1 2 3 simply obtained as a product of the distributions for the three quarks related to the( ) Bi formation, and is taken as( )( ) 6N N N , 3N N − 1 N and N N − 1 N − 2 for constituent quarks and/or antiquarks. We have, for q1 q2 q3 q1 q1 q2 q1 q1 q1 Bi(q1q2q3) the case of three different flavors, two identical flavors (n) (n) (n) (n) and three identical flavors, respectively. The factors 6 and f (pB) = AB fq (x1 pB) fq (x2 pB) fq (x3 pB), (6) Bi i 1 2 3 3 are the number of permutations related to the different quark flavors. N = N N is the number of all possible and for Mi (q1q¯2) q1q¯2 q1 q¯2 (n) (n) (n) q1q¯2 pairs related to the Mi formation. f (pM) = AM fq (x1 pM) f (x2 pM), (7) Mi i 1 q¯2 Considering the flavor independence of the strong in- (n) where f (p) ≡ dn /dp is the momentum distribution of teraction, we assume that the probability of q1q2q3 form- q q ∫ ∏ − (n) ing a baryon and the probability of q q¯ forming a meson quarks normalized to one. A 1 = dp 3 f (x p) and 1 2 B i=1 qi i ∫ i are flavor independent. The combination probability can −1 (n) (n) A = dp fq (x1 p) f (x2 p) are the normalization coeffi- Mi 1 q¯2 then be written as cients for baryons B and mesons M , respectively. The i i N momentum fraction x is given by, recalling that the mo- = B , Pq1q2q3→Bi CBi (11) mentum p = mγv ∝ m, Nqqq

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N M the mass (and momentum) of a hadron. The model para- P → = C . (12) q1q¯2 Mi Mi / / Nqq¯ meters RV P and RD O contain the non-perturbative dynamics and are obtained by fitting the relevant experi- Here, N /N denotes the average (or flavor blinding) B qqq mental data. They are assumed to be relatively stable in/at probability of three quarks combining into a baryon. NB different collision systems/energies. Also, the normaliza- is the average number of total baryons and = − − tion of the hadronization process is a prerequisite for Nqqq Nq(Nq 1)(Nq 2) is the number∑ of all possible quark combination. The quark number conservation is not N = N three-quark combinations, with q f f the total only globally satisfied via N + 3N = N and C M B q quark number. Bi is the probability of selecting the cor- + = NM ∑3NB¯ Nq¯ , but is also satisfied for each quark flavor rect discrete quantum number, such as spin, in the forma- = via h nqi,hNh Nqi. Here, h runs over all hadron species, tion of Bi as q1q2q3 is destined to form a baryon. Simil- = , , , ¯, , and qi d u s d u¯ s¯. nqi,h is the number of constituent arly, N M/Nqq¯ denotes the average probability for a quark quarks qi in a hadron h. Therefore, this is a statistical and antiquark to combine into a meson, and CM is the i model based on the constituent quark degrees of freedom, branching ratio for the formation of Mi as q1q¯2 is destined and is different from the popular parton recombination/ to from a meson. N M is the total meson number, and coalescence models [40, 41] which adopt the Wigner = Nqq¯ NqNq¯ is the number of all possible quark-antiquark wave function method with the instantaneous hadroniza- pairs for meson formation. tion approximation. In this paper, we only consider the ground state JP = 0−, 1− mesons and JP = (1/2)+, (3/2)+ baryons in the 2.2 Quark number fluctuation and threshold effects in flavor SU(3) group. For mesons  hadron production   1 P = − ,  for J 0 mesons As the quark number at hadronization is small, the 1 + RV/P CM =  (13) identified hadron production will suffer some threshold j  R /  V P for JP = 1−mesons, effects. For example, baryon production is forbidden for 1 + R / − V P events with Nq < 3. For events with Ns < 3, Ω baryon where we introduce a parameter RV/P , which represents production is forbidden. In pp collisions at the LHC ener- the relative production weight of the JP = 1− vector gies, the event-averaged number of strange quarks is − mesons to the JP = 0 pseudo-scalar mesons with the ⟨Ns⟩ ≲ 1 in the midrapidity region (|y| < 0.5) for the in- same flavor composition. For baryons elastic events and not-too high multiplicity event classes.  −  Therefore, the yield of Ω is no longer completely de-  1 P = / + ,  for J (1 2) baryons termined by the average number of strange quarks but is 1 + RD/O CB =  (14) also strongly influenced by the distribution of the strange j  R /  D O P = / + , quark number. The case of Ξ, which needs two strange + for J (3 2) baryons ({ } { }) 1 RD/O P , ⟨ ⟩ quarks, is similar. We use Nqi Nqi to denote the = = / + ∗ = / + except that CΛ CΣ0 1 (2 RD/O), CΣ 0 RD/O (2 RD/O), distribution of the quark number around the event aver- ++ = − = − = C∆ C∆ CΩ 1. The parameter RD/O stands for the age, and obtain the averaged multiplicity of identified + relative production weight of the JP = (3/2) decuplet to hadrons by + ∑ the JP = (1/2) octet baryons of the same flavor content. ({ } { }) ⟨Nh ⟩ = P Nq , ⟨Nq ⟩ Nh , (15) Here, RV/P is taken as 0.45 by√ fitting the data for the i i i i ∗ {N } / = qi K K ratio in√ pp collisions at s 7 TeV and p-Pb collisions at s = 5.02 TeV [46], and R / is taken as 0.5 NN D O where{ }Nhi is given by Eqs. (9) and (10), and is a function Ξ∗/Ξ Σ∗/Λ by fitting the data for and [47]. The fraction of of Nq . / ≈ . i baryons relative to mesons is NB NM 0 085 for vanish- For simplicity, we assume the flavor-independent ing net-quarks [18, 21, 45]. Using the unitarity constraint quark number distribution ({ } { }) ∏ ( ) for hadronization, NM + 3NB = Nq, NB and NM can be cal- i i P N , ⟨N ⟩ = P N ,⟨N ⟩ , (16) culated using the above formulas for the given quark qi qi f f numbers at hadronization. f We summarize the main underlying dynamics of the where f runs over u, d, s flavors. We neglect the fluctu- = model. The constituent quarks and antiquarks are as- ation of net-charges and take N f N f¯ in all events. The sumed to be the effective degrees of freedom of the soft distribution of( u and d quarks is based on the Poisson ) dis- parton system at hadronization. Combinations of constitu- tribution Poi Nu(d),⟨Nu(d)⟩ . As discussed above, we in ent quarks and antiquarks with equal velocity result in the particular tune the strange quark distribution. Since for formation of baryons and mesons. This is similar to the the minimum bias events and small multiplicity classes in constituent quark model, i.e. the sum of the masses (and pp collisions ⟨Ns⟩ ≲ 1 , and the Poisson distribution momenta) of the constituent quarks is used to construct Poi(Ns,⟨Ns⟩) in this case has a long tail for Ns ⩾ 3 which

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may over-weight the events with Ns ⩾ 3, we distort the rons. Poisson distribution by the suppression factor γs, i.e. we 3.1 Quark pT distribution at hadronization take P(Ns,⟨Ns⟩) = NPoi(Ns,⟨Ns⟩) × [γs Θ(Ns − 3) + Θ(3− Ns)] , where Θ(x) is the Heaviside step function and N is Using the scaling property in Eq. (3) and the experi- the normalization constant. γs is taken as 0.8 for the in- mental data shown in Fig. 2, we can directly obtain the elastic (INEL > 0) events and the various multiplicity normalized pT distribution of strange quarks at hadroniz- classes. ation, which can be parametrized in the form  √ − There are other possible effects of the small number ns  2 2   p + Ms − Ms  of quarks. For example, in the events with Ns = Ns¯ = 1, as (n)  T  = N + bs  +  , fs (pT ) s (pT as) 1  (17) s and s¯ are most likely created from the same vacuum ex-  nscs  citation and therefore are not likely to directly constitute a color singlet, the ϕ production is suppressed. In addition, where Ns is the normalization constant, and the paramet- as the distributions of quark momenta are dependent on ers are as = 0.15 GeV/c, bs = 0.649, ns = 4.14, Ms = 0.5 GeV the number of quarks (i.e. on the system size), we neg- and cs = 0.346. By fitting the data for the other hadrons ∗0 lect such dependence in the given multiplicity classes. such as protons and K , we also obtain the pT distribu- Their potential effects will be studied in a future work. tion of up/down quarks at hadronization. Taking the parametrization in Eq. (17), the parameters of up/down quarks are au = 0.15 GeV/c, bu = 0.355, nu = 3.46, Mu = 0.33 GeV 3 Results for the inelastic events (n) (n) and cu = 0.358. In Fig. 4, we plot fu (pT ) and fs (pT ) as a function of pT√ , and their ratio in the inelastic events in We use the above quark combination model to de- pp collisions at s = 13 TeV. scribe the transverse production of hadrons at midrapid- We emphasize that by taking advantage of the quark ity in pp collisions.The approximation of equal velocity number scaling property, we can conveniently extract the combination in the model is reduced to the equal trans- momentum distributions of soft quarks at hadronization verse-velocity combination. Here, we only study one di- from the experimental data for the hadronic pT spectra. mensional p distribution of hadrons by further integrat- T The extracted quark pT spectra carry important informa- ing over the azimuthal angle. The pT distribution function about the soft parton system created in pp collisions tions of quarks at hadronization and at midrapidity are in- at the LHC energies. First of all, since the parameters bs put for the model and are denoted as and bu in the quark distribution function in Eq. (17) are f (p ) = ⟨N ⟩ f (n) (p ) with q = d,u, s,d¯,u¯, s¯. ⟨N ⟩ is the (n) qi T qi qi T i qi obviously smaller than one, the extracted fu (pT ) and | | < . (n) number of qi in the rapidity interval y 0 5 , and fs (pT ) deviate from the Boltzmann distribution in the (n) ≡ / fqi (pT ) dnqi dpT is the quark pT spectrum normalized low pT range. This indicates that thermalization may be to one. We assume the iso-spin symmetry between up and not reached in the small partonic system created in pp down quarks, and also assume the charge conjugation collisions at the LHC energies. Secondly, we see that the (n) (n) symmetry between a quark and antiquark. Finally, we ratio fs (pT )/ fu (pT ), Fig. 4 (b), increases for small pT have only two input functions fu(pT ) and fs(pT ) , which and then saturates (or only slightly decreases) with pT . can be determined by fitting the data for identified had- This property is similar to that observed in pp collisions

Fig. 4. (color online) The p spectra of u and s quarks at midrapidity, and their ratio in the inelastic (INEL>0) events in pp collisions √ T at s = 13 TeV.

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√ √ ∗0 at s = 7 TeV [21] and in p-Pb collisions at sNN = 5.02 results for K are slightly below the data. If we multiply TeV [17], and in heavy-ion collisions at RHIC and the the K∗0 spectrum by a constant factor, we see that the LHC [48-50]. This information about the constituent shape is in good agreement with the data. quarks provides an important constraint for developing Besides the scaling property between the pT spectra more sophisticated theoretical models of soft parton sys- of Ω− and ϕ shown in the Introduction, the ratio (p + p¯)/ϕ tem created in high energy collisions. as a function of pT can also give an intuitive picture of the microscopic mechanism of hadron production. Pro- 3.2 pT spectra of identified hadrons tons and ϕ have similar masses but totally different quark Among the hadrons that are often measured by exper- content. In√ the central (0%-10% centrality) Pb-Pb colli- iments, pions and kaons are the most abundant particles. sions at sNN = 2.76 TeV, the data for the (p + p¯)/ϕ ratio However, because the pion and kaon masses are signific- [51], black squares in Fig. 6, are almost flat with respect antly smaller than the sum of the masses of their constitu- to pT . The flat ratio is related to the similar masses of ent (anti-)quarks, the pion and kaon momenta can not be protons and ϕ , and is usually attributed to the strong radi- calculated by a simple combination of the constituent al flow and statistical hadronization in the chemical/ (anti-)quark momenta at hadronization [21]. Therefore, thermal equilibrium in relativistic heavy-ion collisions. the momentum spectra of pions and kaons are not the However, the data for√ (p + p¯)/ϕ in the inelastic events in most direct probes of the quark combination model, and pp collisions at s = 13 TeV [52], solid circles in Fig. 6, these results are not shown here. On the other hand, pro- show a rapid decrease with increasing pT . This is an in- tons, Λ, Ξ−, Ω−, ϕ and K∗0 can be constructed from the dication of the out-of-thermal equilibrium in pp colli- constituent quarks and antiquarks. These hadrons can be sions. In QCM, the pT distributions of identified hadrons used to effectively test the quark combination model. are determined by the pT spectra of (anti-)quarks at had- In Fig. 5, we show the calculated results for the pT ronization. The (p + p¯)/ϕ ratio in QCM reflects the ratio − − ∗ spectra of protons, Λ, Ξ , Ω , ϕ and K√ 0 in the inelastic or the correlation between the third power of the u quark (INEL > 0) events in pp collisions at s = 13 TeV using spectrum and the square of the s quark spectrum. With the quark spectra in Fig. 4 and quark numbers ⟨Nu⟩ = 2.8 the quark spectra in Fig. 4 , which self-consistently de- and ⟨Ns⟩ = 0.86. The quark numbers are fixed by globally scribe the data for the hadronic pT spectra in Fig. 5, the fitting the data for the pT -integrated yield densities of calculated p/ϕ ratio in QCM, solid line in Fig. 6, shows a these hadrons [27]. The solid lines are the QCM results decreasing behavior with pT and is in good agreement which include the contribution of the strong and electro- with the data for pp collisions [52]. magnetic decay of the resonances. The symbols are the Hyperons Λ, Ξ− and Ω− contain one, two and three s preliminary data for the hadronic pT spectra at midrapid- constituent quarks, respectively. Therefore, the ratios ity measured by the ALICE collaboration [27]. We see Ξ−/Λ and Ω−/Ξ− reflect the difference in momentum dis- that the data can in general be fitted well by QCM. Our tributions of u(d) quark and s quark at hadronization. Fig. 7

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

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shows our prediction for the ratios Ξ−/Λ and Ω−/Ξ− as a √function of pT in the inelastic events in pp collisions at s = 13 TeV. We see that the two ratios initally increase with pT , and then tend to saturate at intermediate pT ∼ 6 GeV/c, which is due to the difference between the pT spectra of strange quarks and of up/down quarks in the (n) (n) range pT ≲ 2GeV/c, see the ratio fs (pT )/ fu (pT ) shown in Fig. 4(b).

4 Results for different multiplicity classes

Using the preliminary data for the pT spectra of pro- ∗0 Fig. 6. (color online) The ratio (p + p¯)/ϕ as a function of p tons, K and ϕ in different multiplicity classes [52, 53], √ T in the inelastic events in pp collisions at s = 13 TeV. The we can determine the corresponding pT spectra of con- solid line is the result of QCM and the symbols are the ex- stituent quarks at hadronization and predict the pT spec- perimental data [51, 52]. tra of the other identified hadrons. Fig. 8 shows the extracted quark pT spectra (using the parametrization in Eq. (17)) at midrapidity in different multiplicity classes. Since the parameter bq in the quark spectrum in high multiplicity classes tends to one, the distribution function in Eq. (17) asymptotically tends to the Boltzmann distribution in the low pT range, and therefore we see a thermal behavior of the quark spectrum. This is related to the increasing multiple parton interactions in these event classes. In small multiplicity classes, the parameter bq is relatively small and the quark spectrum deviates from the thermal behavior. 4.1 Hadronic yields and yield ratios

In Fig. 9, we show the pT -integrated yields of identi- 1) fied hadrons (including kaons) in different multiplicity Fig. 7. (color online) Prediction of the ratios Ξ−/Λ and classes and compare them with the preliminary data in√ pp Ω−/Ξ− as a function of p at midrapidity in the inelastic √T collisions at s = 13 TeV [52, 54]. In general, the results = events in pp collisions at s 13 TeV. of QCM, solid lines, are in good agreement with the data

√ Fig. 8. (color online) The p spectra of u and s quarks at midrapidity in different multiplicity classes in pp collisions at s = 13 TeV. T

1) Because the strangeness is conserved during the combination, the number of kaon can be effectively predicted by the current quark combination model.

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√ Fig. 9. (color online) The yield densities of identified hadrons at midrapidity in different multiplicity classes in pp collisions at s = 13 TeV. The solid lines are the results of QCM and the symbols are the preliminary data of the ALICE collaboration [52, 54].

(with the maximum deviation of about 10%). The yield ratios of different hadrons can considerably cancel the dependence on the model parameters and/or model input. Therefore, they are a more direct test of the basic physics of the model when confronted with the experimental data. In Fig. 10, we show the yield ratios of hyperons Ω−, Ξ− and Λ to pions divided by the number of √inclusive INEL>0 events. The data for pp collisions at = √s 7 [5] and 13 TeV [52], and for p-Pb collisions at sNN = 5.02 TeV [55, 56], are presented in order to get a clear tendency with respect to the multiplicity of charged particles at midrapidity. The solid lines are the numerical results of QCM, which are found to be in agreement with the data. We emphasize that the strangeness-related hierarchy is closely related to the strange quark content of these hyperons at hadronization, which can be easily understood using the analytical relation in QCM. Taking the yield formulas Eqs. (9) and (11) and considering the

strong and electromagnetic decays, we have Fig. 10. (color online) The yield ratios of hyperons to pions λ3 > s divided by the number of inclusive INEL 0 events. The NΩ ≈ N , (18) √ 3 B data for pp collisions at s = 7 [5] and 13 TeV [52], and p- (2 + λs) √ Pb collisions at sNN = 5.02 TeV [55, 56], are presented. λ2 3 s The solid lines are the numerical results of QCM, and the NΞ ≈ N , (19) 3 B (2 + λs) dotted lines are the analytic approximation in QCM. ( ) + . 2 0 88RD/O RD/O 6λs λ = ⟨ ⟩/⟨ ⟩ NΛ ≈ + 0.94 N adopt the strangeness suppression factor s Ns Nu . + + + λ 3 B 2 RD/O 1 RD/O (2 s) Due to the contribution of complex decays, the pion yield 7.73λ ≈ s N , (20) has a complex expression [57], and here we write + λ 3 B (2 s) Nπ = aπ⟨Nq⟩ with the coefficient aπ almost constant. The where we neglect the effects of small quark numbers, and double ratios in Fig. 10 then have simple approximate ex-

014101-9 Chinese Physics C Vol. 44, No. 1 (2020) 014101 pressions ( ) / pp 3 / ′3 NΩ NΩ λ λ ≈ s ( s ) , (21) π π + λ 3 ′ 3 N N INEL>0 (2 s) 2 + λs ( ) / pp 2 / ′2 NΞ NΞ λ λ ≈ s ( s ) , (22) π π + λ 3 ′ 3 N N INEL>0 (2 s) 2 + λs ( ) / pp / ′ NΛ NΛ λ λ ≈ s ( s ) , (23) π π + λ 3 ′ 3 N N INEL>0 (2 s) 2 + λs λ′ where s is the strangeness suppression factor in the IN- EL >0 events in pp collisions. Here, we see a clear hierarchy structure among the three double ratios in terms of λs. The dotted lines in Fig. 10 are the results of the above analytic formulas[ with a naively( tuned strangeness)] sup- λ = λ′ + . ⟨ / η⟩ / . pression s s 1 0 165log dNch d |η|<0.5 6 0 with λ′ = . s 0 31. They fit well the experimental data for the ⟨ / η⟩ ≳ double ratios for dNch d |η|<0.5 10. In the small multi- Fig. 11. (color online) The yield ratio Ξ/ϕ as a function of plicity classes ⟨dN /dη⟩|η|< . ≲ 6 , the small quark num- ch 0 5 ⟨dN /dη⟩|η|< . . The preliminary data for pp collisions at √ ch 0 5 ber effects are not negligible so that the analytic approx- s = 13 TeV, solid squares, and for p-Pb collisions at imations give larger values than the experimental data. √ sNN = 5.02 TeV, solid circles, are taken from Refs. [52, Our numerical results include the small quark number ef- 58]. The solid line is the result of QCM. fects and are found to be closer to the data. The yield ratio Ξ/ϕ is also influenced by the small quark number effects. If we neglect them, we have λ2 RV/P s Nϕ ≈ N (24) + 2 M 1 RV/P (2 + λs) and using Eq. (19), we get the ratio

NΞ + NΞ¯ 1 + RV/P 3 1.7 = 2 RB/M ≈ , (25) Nϕ RV/P 2 + λs 2 + λs which slightly decreases with increasing λs , and therefore slightly decreases with an increase of multiplicity ⟨dNch/dη⟩ because λs increases with ⟨dNch/dη⟩. This is in contradiction with the experimental data. However, taking into account the small quark number effects in QCM, we get the correct behavior of the ratio Ξ/ϕ, as shown by the solid line in Fig. 11. The production of Ξ− needs not only two s quarks but also a d quark, which is different from ϕ which needs only an s and s¯. Therefore, in small

multiplicity events or for small quark numbers, the form- Fig. 12. (color online) The yield ratio ϕ/p as a function of ation of Ξ− is suppressed to a certain extent (or occasion- √ ⟨dN /dη⟩|η|< . . The data for pp collisions at s = 7 and 13 ϕ ch 0 5 ally forbidden) in comparison with . We see that the cal- TeV, solid circles and squares, and the data for p-Pb colli- culated ratio Ξ/ϕ using QCM increases with system mul- √ sions at sNN = 5.02 TeV, solid circles, are taken from Refs. ⟨ / η⟩ tiplicity dNch d , and is consistent with the √ experiment- [52, 58]. The solid line is the numerical result of QCM. The = al data for√ pp collisions at s 13 TeV and for p-Pb col- short dashed lines are the analytical approximation. lisions at sNN = 5.02 TeV [52, 58]. 1 1 Protons and ϕ have similar masses but different quark N ≈ N . (26) p 3 B content. The yield ratio ϕ/p can test the flavor-domin- 4 (2 + λs) ated features of hadron production in QCM. Neglecting Using Eq. (24), we get the yield ratio the small quark number effects, the proton yield, after Nϕ R / ≈ 1 V P λ2 + λ N M ≈ . λ2 + λ , taking into account the decay of ∆ resonances, has a s (2 s) 0 91 s (2 s) (27) N 4 1 + R / simple expression p V P NB

014101-10 Chinese Physics C Vol. 44, No. 1 (2020) 014101 which shows a significant dependence on the strangeness predictions for the other identified hadrons√ in different suppression factor λs. We get Nϕ/Np ≈ 0.22 with λs = 0.32 multiplicity classes for pp collisions at s = 13 TeV. ∗0 in the low multiplicity classes, and Nϕ/Np ≈ 0.28 with Note that the classes IV and V are combined for the K λs = 0.36 in the high multiplicity classes. The short data and for our results. Beside comparing the predic- dashed lines in Fig. 12 are for the above two values in the tions of single hadron spectra with the data, we emphas- analytical approximation. They are slightly lower than the ize that QCM can be more effectively tested by spectrum experimental data [52, 58], symbols in the figure. The ratios and/or scaling. The first test is whether the con- small quark number effects increase the ratio to a certain stituent quark number scaling holds in the pT spectra for extent as they suppress the proton yield. We show the nu- Ω− and ϕ in different multiplicity classes. The second is − − merical results of our model including the small quark to study the ratio Ω /ϕ as a function of pT . The Ω /ϕ ra- number effects, solid line, and see a good agreement with tio in QCM is solely determined by the strange quark pT the data. spectrum at hadronization, and this ratio usually exhibits a nontrivial p dependence, as shown in Fig. 14(a), which 4.2 p spectra of identified hadrons T T is a typical behavior of the baryon-to-meson ratio in In Fig. 13, we show the fit of the data for the pT spec- QCM and is absent or unapparent in the traditional frag- − tra of protons, K∗0 and ϕ [52, 53] using QCM, and the mentation picture. We also see that the ratio Ω /ϕ in

√ Fig. 13. (color online) The pT spectra of identified hadrons at midrapidity in different multiplicity classes in pp collisions at s = 13 TeV. The solid lines are the results of QCM and the symbols are the preliminary data of the ALICE collaboration [52, 53].

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√ Fig. 14. (color online) Prediction of the ratios Ω−/ϕ and p/ϕas a function of p at midrapidity in pp collisions at s = 13 TeV. T √ higher multiplicity classes can reach higher peak values. s = 13 TeV shows a clear decrease with increasing pT , The peak position of Ω−/ϕ in high multiplicity classes is which indicates that the statistical hadronization model is also enlarged compared with the low multiplicity classes. not the reason for this behavior. We demonstrated that the The third test is to study the ratio p/ϕ as a function of pT data are naturally explained by the quark combination Λ/π Ξ−/π Ω−/π so as to clarify if its pT dependence is flavor dominated model. (3) The yield ratios , and divided or mass dominated. The results of QCM are shown in by the number of inelastic events as a function of system ⟨ / η⟩ Fig. 14(b) , which decrease with pT and show a relatively multiplicity dNch d at midrapidity show a strangeness- weak multiplicity dependence. related hierarchy structure. We demonstrated that the hierarchy structure is closely related to the strange quark 5 Summary and discussion content of these hyperons as the result of the combination of strange quarks and up/down quarks during their production. Taking advantage of the available experimental data Using the quark number scaling property, the pT for the hadronic pT spectra and yields at midrapidity, we spectrum of strange quarks was extracted from the data have systematically studied the production of soft had- √ for Ω and ϕ. The pT spectra of up/down quarks was ex- rons in pp collisions at s = 13 TeV in the framework of tracted from the data for the other hadrons containing the quark combination mechanism at hadronization. We up/down constituent quarks. The extracted quark mo- applied the quark combination model which assumes that mentum distribution functions are important results the constituent quarks and antiquarks are the effective de- which describe the properties of the strongly interacting grees of freedom of the parton system at hadronization, partonic system at hadronization in the language of con- and used the equal velocity combination approximation in stituent quarks. hadron formation. We applied the model to systematic- To confirm the new features of hadronization dynam- ally calculate the pT spectra and yields of soft strange ics in high energy pp collisions, we should carefully hadrons in the inelastic events (INEL>0) and in different study all related experimental data. We have made pre- multiplicity classes. dictions of the pT spectra and spectrum ratios of strange√ We found several interesting results which are sensit- hadrons in pp collisions at s = 13 TeV to further test our ive to the hadronization mechanism. (1) The pT spectra of model with the experimental data that will be available in Ω− ϕ and √ in the inelastic events (INEL>0) in pp colli- the future. On the other hand, compared to our previous√ sions at s = 13 TeV exhibit a constituent quark number study of pp collisions at s = 7 TeV [21] which gave a scaling property. The high multiplicity classes in√ pp col- first indication, the current study provides a stronger sig- lisions at s = 7 TeV also show this scaling property. nal of quark combination hadronization in high energy pp This is the first time that such a scaling property is ob- collisions. We still need further systematical studies of served in high energy pp collisions, and is an obvious ex- pp collisions at the other LHC energies to test the univer- perimental signal of the quark combination mechanism at sality of the new hadronization features, and to study hadronization in high energy pp collisions. (2) The p/ϕ their relation with the possible creation of mini-QGP in ratio in the inelastic events (INEL > 0) in pp collisions at small collision systems.

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