Transverse momentum and multiplicity dependence of Λ + c / D 0 ratio in pp collisions at √ s = 13 TeV

We apply an equal-velocity quark combination model to study the Λ + c / D 0 ratio in the range p T (cid:2) 10 GeV/c in pp collisionsat √ s = 13TeV.Wedecomposetheratiointo four parts which are related to quark numbers, light-ﬂavor quark p T spectrum, charm quark p T spectrum, momentum correlation between light and charm quarks, respectively. Their inﬂuence on Λ + c / D 0 ratio is individually studied. The curvature property of light-ﬂavor quark p T spectrum is the mainreasonofthenon-monotonic p T dependenceof Λ + c / D 0 ratioexhibitedinhighmultiplicityevents.Moreover,themul-tiplicity dependence of Λ + c / D 0 ratio as the function of p T is mainly because of the multiplicity dependence of light-ﬂavor quark p T spectrum. Using the light-ﬂavor quark p T spectrum obtained from experimental data of light-ﬂavor hadrons and charm quark p T spectrum obtained from FONLL and/or PYTHIA calculations, the p T dependence of experimental data of Λ + c / D 0 ratio in high multiplicity events and that in low multiplicity events in pp collisions at √ s = 13 TeV are reasonably understood.


I. INTRODUCTION
In recent years, LHC experiments provided rich experimental data of hadron production in pp collisions at LHC energies, from which many new features of hadron production are found.For example, in production of lightflavor hadrons, experiments found the enhancement of baryon to meson ratio (such as p/π, Λ/K 0 s , Ω/φ) at intermediate p T [1,2] and the enhancement of multi-strange hadrons in high multiplicity events [3].We also find a quark number scaling property for p T spectra of hadrons at mid-rapidity by further analyzing experimental data of ALICE collaboration [4][5][6].In production of open heavyflavor hadrons, LHC experiments found the enhancement of Λ + c /D 0 in the low p T range (p T 10 GeV/c) in pp collisions at LHC energies in comparison with measurements in e + e − and ep collisions at early years [7][8][9][10].
The production enhancement of light-flavor baryons and, in particular, heavy-flavor baryons such as Λ + c attracts lots of theoretical studies.Many new phenomenological mechanism and/or details at the hadronization of final-parton system created in pp collisions at LHC energies, either in fragmentation framework [11][12][13][14][15][16][17] or in quark (re-)combination framework, are proposed to explain these new experimental data, which greatly enrich people's understandings for the property of hadron production in high energy collisions.Recently, ALICE collaboration report their precise measurement for the multiplicity dependence of Λ + c /D 0 ratio in the low p T range [18] and the preliminary data Λ + c /D 0 ratio at small p T (p T < 1 GeV/c) [19,20].These newest data will further test the existing hadronization models [21][22][23][24].
In this paper, we apply a quark combination model [4,24,25] to study the multiplicity and p T dependence of Λ + c /D 0 ratio in pp collisions at √ s = 13 TeV.The model In this paper, we apply a quark combination model proposed in previous works [4,24,25] to study ratio Λ + c /D 0 .This model is inspired by the quark number scaling property found from experimental data for p T spectra of light-flavor hadrons at mid-rapidity in pp and p-Pb collisions at LHC energies [4,26].We have applied the model to describe the yield and p T spectra of lightflavor hadrons and those of single-charm hadrons in pp, p-Pb and AA collision at RHIC and LHC energies, and found generally good agreement with experimental data [6,[27][28][29][30].In this section, we briefly introduce the model and, in particular, the relevant physical approximations and parameters in the model which may influence the ratio Λ + c /D 0 .

A. general framework
We start from the inclusive momentum distribution of single-charm hadron in general framework of quark combination mechanism Here, f c l(p 1 , p 2 ) is the joint momentum distribution of charm (c) quark and light anti-quark ( l).R M c l (p 1 , p 2 ; p) is the combination function denoting the probability density for the given c l with momenta p 1 , p 2 combining into a meson M c l with momentum p.It is similar for the baryon formula.
In our model, we assume that the charm hadron is formed mainly by the combination of charm quark with light-flavor (anti-)quarks with equal velocity.In this approximation, the combination function is simply the product of Dirac delta functions Here κ M c l and κ B cll ′ are independent of momentum but are dependent on quarks numbers due to the unitarity constraint of hadronization.Momentum fraction reads as x i = m i /(m 1 + m 2 ) in meson formula with momentum conservation constraint x 1 + x 2 = 1 and Substituting Eqs. ( 3) and ( 4) into ( 1) and ( 2), we obtain For the integral of joint distribution of quarks, we have where N ll ′ equals to N l N l ′ as l = l ′ and N l (N l − 1) as l = l ′ .Obviously, N c l is the number of all c l pair and N cll ′ is the number of all possible cll ′ combinations.Under EVC constraint, the integrals become where we use coefficient A c l to denote the effect of equalvelocity constraint to the effective number of c l pairs and A cll ′ to denote that to effective number of cll ′ combinations.Integrating Eqs. ( 5) and ( 6) over p, we obtain the number of charm hadrons We see that A c l has the meaning of the average probability of a c l pair forming a meson, and it is similar for baryon term.
If we neglect the contribution of multi-charm hadrons, the unitarity of charm quark hadronization gives which means At charm quark hadronization, light-flavor quarks serve as the background and their property, i.e., numbers N l and momentum distributions f l (p) relating to A, can be freely changed.With this consideration, we expect with N q = l Nl and N q = l N l so that the unitarity constraint can be satisfied in an easy manner.In this philosophy, we firstly introduce a dynamic parameter R (c) B/M to denote the competition between the formation of charm baryon and the formation of charm meson as a charm quark hadronizes.Then we can write Following the spirit in our previous works [24,25], we can take the following parameterizations which can be understood as follows.In the second equality, ) denotes the number of charm quarks that will form charm mesons.N cq equals to N c N q and denotes the number of all possible cq pair.Then B/M )/N cq denotes the average probability of a cq pair forming a charm meson.Because cq pair can form the charm meson with different total angular moments, here we introduce a parameter C M c l to denote the probability of a c l pair forming a given state M c l. Obviously, unitarity requires M C M c l = 1 where summation runs over all meson states with the same c l composition.It is similar for Eq. ( 18) of the baryon.Here, N iter,ll ′ is the permutation factor of ll ′ and is take 2 for l = l ′ and 1 for l = l ′ , respectively.We expect Finally, numbers of charm hadrons are and momentum distribution of charm hadrons are where f p) are the normalized distributions under integral over p.

B. application to high-energy collisions
In this paper, we study the production of charm hadron at mid-rapidity in pp collisions at √ s = 13 TeV and compare our theoretical results with experimental data at mid-rapidity.The quark momentum distribution dN/dp at rapidity y = 0 in our model is reduced to f (p T ) ≡ dN/dp T .The joint quark momentum distributions are also reduced to f c l(p T,c , p T, l) and f cll ′ (p T,c , p T,l , p T,l ′ ).The rapidity densities of charm and light-flavor quarks are denoted as N l and N c for convenience.
In experimental measurement, momentum spectra and yield densities of hadrons are mainly reported via their averaged values in the selected event class.We can extend the above formulas to follows experimental statistics by re-defining quark momentum distributions f c l(p T,c , p T, l) and f cll ′ (p T,c , p T,l , p T,l ′ ) and quark numbers N l , N c as these in the select event class.
Finally, p T spectra of hadrons in our model are with In the following studies, we also assume several symmetry property for numbers and momentum distributions of quarks produced at mid-rapidity in pp collisions at LHC energy.We consider the iso-spin symmetry between up and down quarks, i.e., N u = N d and f u (p T ) = f d (p T ), and also the charge conjugation symmetry, i.e. , N l = Nl and f l (p T ) = fl(p T ).These symmetry assumptions will greatly simplify our theoretical expressions.

III. pT DEPENDENCE OF RATIO Λ +
c /D 0 According to Eqs. ( 5) and ( 6), p T spectra of Λ + c and D 0 are where for the baryon and x We express the joint distribution functions of (anti-)quarks as where f qi (p T ) is inclusive distribution of q i -flavor quarks and C cud (p T,c , p T,u , p T,d ) is the correlation term.By qi (p T ) being the normalized distribution, we can further write the joint distribution as With the above form of f cud (p T,c , p T,u , p T,d ) and f ūc (p T,c , p T,ū ), we can calculate p T spectra of Λ + c and D 0 by formulas in Sec.II A and then consider the event average convention in Sec.II B, and finally we get the ratio Here, we have used the symmetry property f in the mid-rapidity range at LHC energies.In the second line, we split the ratio into several parts which reflect the influence of different physical ingredients on the ratio With Eqs. ( 9), ( 10) and ( 31), we can see that which means that terms in square bracket in Eq. ( 33), as a global quantity, mainly influences the shape of the Λ + c /D 0 ratio but not its global magnitude.Therefore, R (Nq i ) in Eq. ( 33) plays the role of controlling the global magnitude of Λ + c /D 0 ratio.In this section, we discuss the property of R (Nq i ) defined in Eq. ( 34).
We take Here, we use the approximations N ū = N d = N u = N d and N q = N q in the mid-rapidity range at LHC energies.We neglect quark number correlations between different flavors by considering the small off-diagonal susceptibilities of quark flavors in Lattice-QCD calculations [31].
In order to obtain intuitive expression, we define a strangeness suppression factor By noticing that The branch fraction parameter C Λc and C D 0 directly influence R (Nq i ) .However, experimental data of Λ + c and D 0 usually contain the contribution of strong and electromagnetic decays of other single-charm hadrons.Considering the decay contributions of Σ ++,+,0 c and Σ * ++,+,0 c , the yield of final-state Because the measured D 0 contains the decay contribution of D * 0,+ , the yield of final-state D 0 is with decay branch ratio B D * + →D 0 = 0.677 from particle data group [32].The branch fraction parameter C D * + is about 0.6 since the ratio D * 0 /D 0 is about 0.43 [33].With Eqs. ( 44) and ( 45) we obtain R (Nq i ) for final-state According to our estimations in previous work [5,26], strangeness suppression factor λ s in low multiplicity events is about 0.3 and that in high multiplicity events is about 0.36 in pp collisions at LHC energies.This change magnitude of λ s causes little influence on R (Nq i ) through the term 2+λ s and therefore contribute little multiplicity dependence of B/M is a dynamic parameter in our model that denotes the production competition between baryon formation via combination with two light-flavor quarks and meson formation via combination with an anti-quark at a charm quark hadronization.In the environment of abundant light-flavor quarks and antiquarks, charm quark has sufficient chance to interact with surrounding light-flavor quarks and antiquarks, and the baryon to meson production competition is sufficient and therefore we expect R (c) B/M should be saturated.However, in the low multiplicity events this baryonto-meson production competition is not sufficient due to the restricted numbers of light-flavor quarks and/or anti-quarks.For example, in event multiplicity class X with mid-rapidity dN ch /dη = 2.52 in pp collisions at √ s = 13 TeV [34], the up quark number density is dN u /dy = 1.48 at mid-rapidity according to our previous study of multiplicity dependence of light-flavor hadron production [5].Events with a charm quark and up/down quark with numbers (N u , N d ) = (2, 0), (0, 2) can not form Λ + c .Events with (N u , N d ) = (2, 1) only provide one effective ud pair for Λ + c formation but can provide three light-flavor antiquarks for charm meson formation.Therefore, the formation of Λ + c in these events is suppressed relative to D mesons.The fraction of these events should not be too small due to the small values of dN u,d /dy in low multiplicity events.Therefore, we can expect the suppression of R (c) B/M and corresponding Λ + c /D 0 ratio to a certain extent in low multiplicity events in pp collisions at √ s = 13 TeV.The exact magnitude of such suppression is dependent on the property of quark number distribution event-byevent, which is little known at present.Here, we take the Poisson distribution as an example to roughly estimate the magnitude of this suppression.We assume the joint distribution of quark numbers as with mean number N u = N d .The number of ū and that of d are taken to be N ū = N u and N d = N d , respectively.That is, we consider the pair production in each event, for simplicity.For events with V0M multiplicity classes IX and X with dN ch /dη = 2.52, 4.64 [34], the u quark number is about N u = 1.5 − 2.5 according to our previous study of light-flavor hadrons in pp at √ s = 13 TeV [5].The fraction of events that can form D 0 but not Λ + c is with P nor = Nu,N d P (N u , N d ) − P (0, 0).In addition, the fraction of events that can form D 0 but not Σ 0 c or Σ * 0 c which finally decay into Λ + c is The fraction of events that can not form Σ ++ c or Σ * ++ c is also P ′′ .These estimation therefore indicates a magnitude of about 20% suppression for R (c) B/M in low multiplicity events in comparison with that in high multiplicity events where the above suppression is absent.In our previous works in study of single-charm hadrons in pp and pPb collisions at LHC energies [24,25], we use R (c) B/M = 0.425 ± 0.025.If we take this value as the possibly saturated value, we estimate that R (c) B/M in low mul-tiplicity events is about 0.34.Therefore, we expect Because m c ≈ 5m u , the momentum fraction of up/down quark x The difference between them is quite small In the studied range p T 8 GeV/c, the momentum difference ∆x u p T 0.2 GeV/c is also small.In order to simplify R (l) ∆x l (p T ), we can take Taylor expansion for where the contribution of perturbative terms are quite small, i.e., about 0.01 ∼ 0.07 as p T < 8 GeV/c.Therefore, the light-flavor part R We see that R (l) ∆x l (p T ) directly depends on the property of p T spectrum of up/down quarks.
In Fig. 1(a), we show the normalized p T spectra of up quarks f   36) and the dashed line is the approximation Eq. ( 53).We see that the approximation is very good.We further see that R (l) ∆x l (p T ) increases with p T in the range 0 < p T 3 GeV/c and decreases with p T at larger p T .This property will cause the non-monotonic p T dependence of ratio Λ + c /D 0 which is exhibited in experimental data in high multiplicity class.We emphasize that the property of f Because the difference between momentum fraction of charm quarks in D 0 and that in is not quite small, the momentum difference is ∆x c p T 1 GeV/c in the studied range p T 10 GeV/c.Taylor expansion is not generally effective to simplify . Therefore, we take the specific form of f In practice, we take the following parameterization for p T spectrum of charm quarks where b, n, M , c are parameters with positive values and N is the normalization coefficient.This function can usually well fit the p T spectrum of charm quarks obtained from theoretical method as well as these of single-charm hadrons measured by experiments.Substituting Eq. (55) into (35), we obtain which is slowly increasing with p T due to the positive exponent n and x We apply the online calculator of FONLL theoretical method [35,36] to calculate p T spectrum of charm quarks in pp collisions at √ s = 13 TeV.The result is shown in Fig. 2 ( ∆xc (p T ).We see that it monotonically increases with p T and the change magnitude is about two in the range p T < 10 GeV/c.We also run PYTHIA 8 (version 8.243) [37] to obtain the p T spectra of charm quarks in three multiplicity classes and calculate their R (c) ∆xc (p T ).Here, the low, intermediate and high multiplicity classes are defined as events within multiplicity interval [1,10], [11,19] and [20,60] with the multiplicity dN ch /dη = 4.4, 14.3 and 25.0 within the pseudo-rapidity interval |η| < 0.5, respectively.Results of R corr (p T ) = 1 and contribute trivially to Λ + c /D 0 ratio.In high energy pp collisions, charm quarks are dominantly produced by initial hard parton-parton scattering process and possibly interact with neighboring lightflavor (anti-)quarks before they hadronize.Part of these interactions is non-perturbative and is difficult to calculate from first principles.On the other hand, quark momentum correlations are also difficult to reversely obtained from experimental data which are mainly of inclusive distributions of particles at present.
In order to roughly understand the possible property of R (cl) corr (p T ) and its influence on Λ + c /D 0 ratio in pp collisions, we apply the event generator PYTHIA 8 (version 8.243) with default tunes, as an example, to study the possible property of R (cl) corr (p T ).We use PYTHIA8 to firstly calculate f cū (p T,c , p T,ū , ∆φ cū ) at |y| < 0.5 where ∆φ cū is the azimuthal angle difference between ū and c.Then we take the limit p T,ū → p T,c m u /m c and ∆φ cū → 0 ′ u p T by Eq. (30).Because the fraction of the event simultaneously consisting of u, d, and c in the mid-rapidity range |y| < 0.5 is very small, it is very hard to directly calculate f cud (x c p T , x u p T , x d p T ), in particular, in low multiplicity events.In the general multi-variable statistics abc = a b c + a C bc + b C ac + c C ab + C abc , C ab is standard two-body correlation and C abc is three-body correlation which is a higher order term compared with  two-body correlations.Here, we neglect the contribution of high-order correlation C cud and obtain and then we obtain In the second line, we have assumed that C cū is small and In Fig. 3, we show results of two-quark correlation C q1q2 as the function of p T in low multiplicity events with dN ch /dη = 4.4 and those in high multiplicity events with dN ch /dη = 25.0.We see that C ud , C cu and C cū are very small in both low and high multiplicity events.We emphasize that these statistics only serve as a qualitatively estimation because of the following reasons.Quarks and antiquarks are only small fractions of final-state parton system in PYTHIA simulations.Gluons that take large fraction of the parton system are not involved in current statistics.Non-perturbative interactions of these quarks and gluons near hadronization are not fully simulated in PYTHIA8, which may generate some momentum correlations among quarks and antiquarks of different flavors.In addition, collectivity or collective flow of partons may be formed in high multiplicity events [38][39][40][41][42], which may also generate some momentum correlations among quarks and antiquarks.These possible influence on R In this section, we combine effects of above four terms to calculate the Λ + c /D 0 ratio and compare with the latest data of ALICE collaboration [18].Here, we neglect the effect of the momentum correlation R where the property of R (Nq i ) , R  53) and (56).Coefficients A D0 and A Λc can be calculated by Eqs. ( 9) and (10) with the inclusive distributions of charm and lightflavor quarks shown in Figs. 1 and 2.
In order to study the multiplicity dependence of the Λ + c /D 0 ratio, we take the light-flavor quark spectrum f l (p T ) in event class dN ch /dη = 25.75 (class I) and dN ch /dη = 4.64 (class IX) as examples of high multi- plicity events and low multiplicity events, respectively.For charm quark p T spectrum f c (p T ), we have two choices.One is that of FONLL calculation and we use it in both high multiplicity events and low multiplicity events.Another is that of PYTHIA8 calculation where in Sec.VI we have obtained f c (p T ) in high multiplicity events dN ch /dη = 25.0 and that in low multiplicity events dN ch /dη = 4.4.The term R (Nq i ) contains two multiplicity-dependent factors λ s and R (c) B/M .As shown by Eq. ( 46), strangeness factor λ s causes little changes on Λ + c /D 0 ratio and we can safely neglect its multiplicity dependence.Baryon-to-meson production competition factor R (c) B/M is a relatively-free parameter which cannot be fixed in our model at present.In Sec.IV, we have discussed its possible multiplicity dependence.
In Fig. 4(a), we show results of Λ + c /D 0 ratio as the function of p T in high multiplicity events and low multiplicity events at given R B/M is to let us focus on the influence of p T spectra of light-flavor quarks and charm quarks on the p T dependence of Λ + c /D 0 ratio.Here, high multiplicity corresponds to dN ch /dη ≈ 25 since we combine effect of f l (p T ) in event class dN ch /dη = 25.75 (class I) and that of f c (p T ) in events with dN ch /dη = 25 in PYTHIA8 calculations.Low multiplicity corresponds to dN ch /dη ≈ 4.5 since we combine effect of f l (p T ) in event class dN ch /dη = 4.64 (class IX) and that of f c (p T ) in events with dN ch /dη = 4.4 in PYTHIA8 calculations.The dot-dashed line with shadow band is the result when f c (p T ) calculated by FONLL and f l (p T ) in high multiplicity events are used.The dotted line with shadow band is the result when f c (p T ) calculated by FONLL and f l (p T ) in low multiplicity events are used.Comparing two results, we see an obvious multiplicity dependence of Λ + c /D 0 ratio caused by that of light-flavor quark spectrum.As p T 3 GeV/c, the Λ + c /D 0 ratio in high multiplicity events is obviously higher than that in low multiplicity events.This is because f l (p T ) in high multiplicity events is obviously flatter than that in low multiplicity events, see Fig. 1(a).As p T 3 GeV/c, the situation is reversed.This is because at the same R (c) B/M the ratio of p T -integrated yield of Λ + c to D 0 is almost the same, the suppress of Λ + c at p T 3 GeV/c in low multiplicity events is offset by the enhancement of Λ + c at low p T .We also show the result of Λ + c /D 0 ratio when the f c (p T ) calculated by PYTHIA8 in high multiplicity events is applied.The result, the solid line, is higher than that with FONLL calculated f c (p T ) in low p T range and is slightly smaller than the latter as p T 3 GeV/c.In the range 3 p T 8 GeV/c, the result of Λ + c /D 0 ratio with f c (p T ) calculated by PYTHIA8 in low multiplicity events, the dashed line, is also slightly smaller than that with FONLL calculated f c (p T ).Comparing results using FONLL calculated f c (p T ) with those using PYTHIA8 calculated f c (p T ), we see a weak influence of f c (p T ) uncertainty on Λ + c /D 0 ratio.In addition, we see that Λ + c /D 0 ratios in our model in both high and low multiplicity events always exhibit a non-monotonic p T dependence.This is mainly because of the property of R c /D 0 ratio in pp collisions at √ s = 13 TeV [18].Here, we show data of Λ + c /D 0 ratio in V0M multiplicity class with dN ch /dη = 31.5 as an example of high multiplicity events and data in V0M multiplicity class with dN ch /dη = 4.4 as an example of low multiplicity events.Because the light-flavor quark p T spectrum used in our calculation is extracted from experimental data of light-flavor hadrons in V0M event class I with dN ch /dη = 25.75 and class IX with dN ch /dη = 4.64, the underlying events of our theoretical calculations are not exactly same as those for data of Λ + c /D 0 ratio but the difference should be small due to the similar multiplicity.In high multiplicity events, we see that theoreti-  cal result with FONLL calculated f c (p T ), the dot-dashed line, is consistent with the first two datum points with p T < 3 GeV/c and is higher than data at larger p T to a certain extent.Theoretical result with PYTHIA8 calculated f c (p T ) in high multiplicity events, the solid line, is more close to experimental data in comparison with that with FONLL calculated f c (p T ).Here, the baryonto-meson competition factor R B/M is taken to be 0.425, the possible saturation value in high multiplicity events.
According to discussions in Sec.IV, we expect a suppression of R (c) B/M in low multiplicity events and a rough estimation based on Poisson distribution of quark numbers gives about 20% suppression.Therefore, here we use R (c) B/M = 0.34 to calculate the Λ + c /D 0 ratio in low multiplicity events.We see a good agreement with experimental data in the available p T range.
In above comparison with experimental data, we have considered the decay contributions of other singlecharm hadrons to Λ + c /D 0 ratio by the yield term R (Nq i ) Eq. ( 46).The decay influence on the shape of Λ + c /D 0 ratio (as the function of p T ) is numerically studied and is found to be quite small and therefore the comparison with experimental data in Fig. 4(b) is little changed.
In Fig. 5, we show p T spectra of D 0 , D + s and Λ + c in inelastic pp collisions at √ s = 13 TeV and compare them with experimental data [18].Here, p T spectra of light-flavor quarks in inelastic events have been obtained in Ref. [5] and p T spectrum of charm quarks is taken from FONLL calculations shown in Fig. 2 (a) with a p Tintegrated yield density dN c /dy = 0.025.The model parameter R (c) B/M is set to be 0.38, which falls in between the value in high multiplicity events and that in low multiplicity events.The decay contributions from other singlecharm hadrons in ground state are systematically considered.We see that experimental data of D 0 , D + s and Λ + c are self-consistently explained in our model.

IX. SUMMARY
We have applied an equal-velocity quark combination model to study the Λ + c /D 0 ratio as the function of p T in pp collisions at √ s = 13 TeV.Taking advantage of the analytic feature of the model, we decomposed the Λ + c /D 0 ratio into four parts and studied their individual influence on the ratio.Finally, we combined effects of these parts to calculate Λ + c /D 0 ratio as the function of p T and compared theoretical results with experimental data of Λ + c /D 0 ratio in high and low multiplicity events in pp collisions at √ s = 13 TeV.The first part of Λ + c /D 0 decomposition is the term of light-flavor quark numbers.We summarized this part into a compact form containing a strangeness factor and a baryon-to-meson competition factor defined in our model.The multiplicity dependence of strangeness has very weak influence on the Λ + c /D 0 ratio.The global effect of light-flavor quark numbers is manifested by the baryon-to-meson competition factor.In low multiplicity events where quark numbers are relatively small, charm quark has relatively small chance to interact with two light-flavor quarks and thus the production of charm baryon should be suppressed to a certain extent.We adopted a Poisson distribution as an example to roughly estimate this suppression and found about 20% suppression of the Λ + c /D 0 ratio in low multiplicity events with mid-rapidity dN ch /dη ≈ 4.5 in comparison with that in high multiplicity events with dN ch /dη ≈ 25.
The second part of Λ + c /D 0 decomposition is the term containing the normalized p T spectrum of up/down quarks.Considering the small difference between momentum fraction (x u ) of up/down quark in Λ + c and that  (x ′ u ) in D 0 , we adopted the Taylor expansion method and reduced this part to the inclusive distribution of up/down quarks f )p T with a good numerical accuracy.This suggests that the shape of up quark p T spectrum directly transmit to the p T dependence of the Λ + c /D 0 ratio.In particular, p T spectrum of up/down quarks in high multiplicity events behaves a thermal like distribution in the low p T range, which will lead to an obviously non-monotonic p T dependence of Λ + c /D 0 ratio in the low and intermediate p T range.
The third part of Λ + c /D 0 decomposition is the term containing the normalized p T spectrum of charm quarks.We adopted FONLL method and PYTHIA8 to calculate charm quark p T spectrum, respectively.This part increases with p T and therefore weaken the effect of the second part.We found that results of PYTHIA8 are roughly close to that of FONLL calculations but have a weak multiplicity dependence.The fourth part of Λ + c /D 0 decomposition is the term containing momentum correlation between up/down quarks and charm quarks.PYTHIA8 simulation on two-quark correlations show a negligible momentum correlation between up/down quarks and charm quarks.Therefore, we neglect the effect of this part in calculation of Λ + c /D 0 ratio in this paper.Finally, we combined these parts and calculate Λ + c /D 0 ratio as the function of p T in pp collisions at √ s = 13 TeV.In calculations, we chosen the p T spectrum of up/down quarks in multiplicity class IX with dN ch /dη = 4.64 as an example of low multiplicity events and that in multiplicity class I with dN ch /dη = 25.75 as an example of high multiplicity events.p T spectrum of charm quarks from PYTHIA8 calculation with dN ch /dη = 4.4 and that with dN ch /dη = 25 are taken in order to be consistent with the selected p T spectrum of light-flavor quarks at similar dN ch /dη .Finally, we obtained Λ + c /D 0 ratio as the function of p T in high multiplicity events with dN ch /dη ≈ 25 and that in low multiplicity events with dN ch /dη ≈ 4.5.We compared our theoretical results with experimental data of Λ + c /D 0 ratio in multiplicity class with dN ch /dη = 31.5 and those with dN ch /dη = 4.4.In high multiplicity events with a baryon-to-meson competition factor R (c) B/M = 0.425, we found that theoretical result with PYTHIA8 calculated charm p T spectrum is in better agreement with experiment data than that with FONLL calculated charm p T spectrum.In low multiplicity events with a suppressed R (c) B/M = 0.34, theoretical result with FONLL calculated charm p T spectrum and that with PYTHIA8 calculated charm p T spectrum are both in good agreement with experimental data.

R (c)
B/M is a dynamical parameter of our model which denotes the competition between the formation of charm baryon and that of charm meson at a charm quark hadronization and can not be predicted by the model of present version.Based on the present work and our previous studies [6,24,25,28], we found that R (c) B/M is about 0.385 − 0.425 in inelastic and relatively-high multiplicity events pp and pPb collisions and is suppressed in small multiplicity events but it may also increase to a certain extent in AA collisions [30].This possibly multiplicity-dependent property of R (c) B/M is an interesting property of charm quark hadronization, which is not seen in the hadronization of light-flavor quarks where we found an relatively stable baryon-to-meson production competition [43].Its explanation is beyond the model of current version and should introduce further dynamic considerations.For example, by considering the possi-ble production of some cll ′ resonances which is predicted by quark model but is not observed yet in experiments [23,44], the hadronization channel of charm quark to charm baryons should increase which may prompt the possible increase of R (c) B/M in high multiplicity events where those resonances will have more practical chance to produce.Also, by considering the competition of quark combination mechanism and quark fragmentation mechanism and, in particular, the multiplicity dependence of such competition [22,45], the effective R (c) B/M can also be changed in our model.These possible dynamics at charm quark hadronization are deserved to study in the future to improve our model.
(n)u (p T ) in the low p T range (i.e., p T,u 2 GeV/c) in three multiplicity classes in pp collisions at √ s = 13 TeV, which are obtained in previous work[5] by fitting experimental data for p T spectra of proton at mid-rapidity using our EVC model.We see that f

u
(p T ) is different in three multiplicity classes.In Fig. 1(b), we show the property of R (l) ∆x l (p T ) in high multiplicity class I with dN ch /dη = 25.75 at mid-rapidity.Here, the V0M multiplicity classes are defined in Ref.[34].The solid line denotes the result of direct calculation of R (l) ∆x l (p T ) by the definition Eq. (

u
(p T ) actually determine the p T dependence of the ratio Λ + c /D 0 to a large extent.In Fig. 1(c), we show the property of R (l) ∆x l (p T ) in low multiplicity class IX with dN ch /dη = 4.64 at mid-rapidity.We see that R (l) ∆x l (p T ) only has a small shoulder structure in the range p T 1 GeV/c and then decrease with p T .This is the direct consequence of f (n) u (p T ) in the low multiplicity class which exhibits less thermal behavior in the low p T range, see the dot-dashed line in Fig. 1(a).Comparing Fig. 1 (b) and (c), we emphasize that the multiplicity dependence of light-flavor quark spectrum f (n) u (p T ) will lead to the significant multiplicity dependence of Λ + c /D 0 ratio as a function of p T .VI. THE PROPERTY OF R (c) ∆xc (pT )

Figure 1 .
Figure 1.The normalized pT spectra of up quarks in three multiplicity classes in pp collisions at √ s = 13 TeV (a), the calculated R (l) ∆x l (pT ) in multiplicity class I with dN ch /dη = 25.75 at mid-rapidity (b) and that in multiplicity class IX with dN ch /dη =4.64 (c).The solid lines in (b) and (c) are directly calculated from the definition by Eq. (36) and the dashed lines are the approximations by Eq. (53).
a), where the shadow area denotes the scale uncertainties in calculations.The solid line is the fit to center values of FONLL calculations by Eq. (55) with parameter values b = 1.3, M = 6.4 GeV/c, n = 4.16, and c = 0.34 GeV/c.Fig. 2(b) show the calculated R (c) (c) ∆xc (p T ) are shown in Fig.2(b).We see that R (c) ∆xc (p T ) calculated by PYTHIA8 has a certain multiplicity dependence and its p T dependence is similar with that of FONLL calculations.
VII. THE POSSIBLE PROPERTY OF R(cl) corr (pT ) The term R (cl) corr (p T ) defined in Eq. (37) denotes effects of momentum correlations between different quark flavors on the p T dependence of Λ + c /D 0 ratio.It contains two correlation functions C cud (x c p T , x u p T , x d p T ) and C cū (x ′ c p T , x ′ u p T ).In the case of vanishing momentum correlations C cud = 0 and C c l = 0, R

Figure 2 .
Figure 2. Panel (a): the differential cross-section of charm quarks at mid-rapidity in pp collisions at √ s = 13 TeV calculated by FONLL method.The shadow area denotes scale uncertainties.The solid line shows the fit of central points of theoretical calculations with Eq. (55).Panel (b): R (c) ∆xc (pT ) for the result of FONLL and these of PYTHIA8 in three multiplicity classes.
corr (p T ) are left for future study.In this paper, we take R (cl) corr (p T ) ≈ 1 for the moment in the following studies.VIII.RESULT OF Λ + c /D 0 RATIO (cl) corr (p T ) between charm quarks and light-flavor (anti-)quarks by taking R (cl) corr (p T ) ≈ 1.This is equivalent to the independence approximation for the joint momentum distribution f c l p T,c , p T, l = f c (p T,c ) fl p T, l and l (p T ) are already shown in Eqs.(46), (

Figure 3 .
Figure 3. Two quark correlations Cq 1 q 2 as the function of pT in low and high multiplicity events calculated by PYTHIA8.
l (p T ) in Eq. (53), which is determined by the property of p T spectrum of light-flavor quark as shown in Fig. 1(b) and (c).The term R (c) ∆xc (p T ), as shown in Fig. 2(b), strengthen the increase of the Λ + c /D 0 ratio in the low p T range (p T 2 − 3 GeV/c) but offsets the decrease of the R (l) ∆x l (p T ) as p T 3 GeV/c to a certain extent and leads to the relatively weak decrease of the Λ + c /D 0 ratio as p T 3 GeV/c.In Fig. 4(b), we test above theoretical results by experimental data of Λ +

Figure 4 .
Figure 4. Λ + c /D 0 ratio as the function of pT in low and high multiplicity events in pp collisions at √ s = 13 TeV (a) and the comparison with experimental data [18] (b).See text for the detailed discussions.

Figure 5 .
Figure 5. pT spectra of D 0 , D + s (a) and Λ + c (b) in inelastic pp collisions at √ s = 13 TeV.Lines are model results and symbols are experimental data [18].