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P and -Pb 5 fQPlk eimi rae in created is medium QGP-like If Ξ √ < y < .INTRODUCTION I. c 0 1 , s colo hsc n niern,Qf omlUniversity, Normal Qufu Engineering, and Physics of School NN Ω φ c 0 , 4 nmnmmba vnsadtoeo igecamhdosin hadrons single-charm of those and events minimum-bias in 5 = Ω pp a-ogLi, Hai-hong . 0 − h lblpouto fsnl-hr ayn squantifie is baryons single-charm of production global The . 2 , eateto hsc,Jnn nvriy hnog273155 Shandong University, Jining Physics, of Department . olsosa H xiisa exhibits LHC at collisions 02 p Ξ T p ∗ T e xii ninteresting an exhibit TeV (1530) & ag ( range p 3 T − p p e si gemn ihtepeiiaydt fAIEcollab ALICE of data preliminary the with agreement in is GeV hr ursi h ml ytmcetdin created system small the in quarks charm P olsosat collisions -Pb 0 T . ,2 1, K , 96 pcr n h pcrmrto for ratios spectrum the and spectra p p olsosat collisions T < y < ∗ T eglnShao, Feng-lan (892) pcrmta scnitn ihprubtv C calculat QCD perturbative with consistent is that spectrum . 8 p 0 e)in GeV) P olsosa xrml ihcliineege,charm energies, collision high extremely at collisions -Pb . n other and 04 n hs fLC olbrto nfradrpdt region rapidity forward in collaboration LHCb of those and √ s p NN T p 1, e t pcr xrce rmteeprmna aao light of data experimental the from extracted spectra - √ 5 = ∗ s u Song, Jun NN . 02 uefrtefraino ml es atnmdu in medium parton dense signa- small possible of a collisions. formation is the and for sector, in ture strange) mechanism down, (re-)combination (up, quark light the to mentation nlow in nte rb o h rpryo h otpro system parton soft the of in as property created quark the anti-charm for or probe charm another a only containing hadrons rtolgtqak omvn ihi ofr heavy a form the to is characteristic it combination momentum antiquark with the light where co-moving a hadron, quarks flavor up light pick two can is or quark soft quark charm abundant the charm relatively with partons, the medium if the by parton hadronization, surrounded surrounding At the hadronization on the dependent environment. nucleons, is quarks collisions incoming charm hard the of initial in the the partons is by of quarks process charm of QCD production perturbative the though even that, low for hadronization p (re-)combination quark the study rgetit h hr arno momentum of hadron and charm color parton(s) the will faraway into it the fragment with partons, connecting neighbor by co-moving neutralize the of lack the ntelow the in arnzto hrceitcb ouigo the on focusing by characteristic hadronization u eut n eeatdsusos umr sgiven is Summary presents discussions. III relevant Sec. and mech- results hadronization. (re-)combination our quark quark charm in for model anism working a duce them. among ratios the particular in and r fmesons of tra inwl erflce ythe by reflected be will tion with T nti ae,w aethe take we paper, this In h ae sognzda olw:Sc Iwl intro- will II Sec. follows: as organized is paper The e r eldsrbdb ur combination quark by described well are TeV 5 = hr urs hsi anymtvtdb h fact the by motivated mainly is This quarks. charm aira h ucinof function the as havior < x p . osbecag ftehadronization the of change possible e T he 02 2, ros sdsusd The discussed. is aryons) D ag rmtetaiinlsrn/lse frag- string/cluster traditional the from range 1 n ih ur()o niqaks to anti-quark(s) or quark(s) light ing p † Λ p eosi h low the in mesons hsdffrn hrceitco hadroniza- of characteristic different This . TeV P olsosat collisions -Pb T c + n u-i Wang Rui-qin and p D /D H ag.W td uhpsil hneof change possible such study We range. p ± P olsosa H energies. LHC at collisions -Pb hnog236,China 273165, Shandong 0 = , 0 ai nqakcombination quark in ratio , p D c sn h preliminary the using d s + + ihmlilct event high-multiplicity China , and p q,qq ¯ p p p T D tews,i h aeof case the in Otherwise, . hns in chanism T √ T p n h shape the and , ∗ T s ag ( range pcr fcamhadrons charm of spectra baryons , 1 NN pcr fsingle-charm of spectra p rto in oration T os the ions, 5 = spectra p quarks T . . 02 Λ 7 c + e,adwe and TeV, , p Ξ -Pb p c 0 p H and T = spec- xp Ω c 0 c , 2 at last in Sec. IV. and the three-quark combination Ncll′ = NcNll′ with N ′ = N (N ′ δ ′ ). δ ′ is Kronecker delta function. ll l l − l,l l,l The combination function contains the key informa- II. CHARM QUARK HADRONIZATION IN tion of hadronization. In sudden hadronization approx- QCM imation, it is determined by the overlap between the wave function of quarks and that of the hadron or by The (re-)combination of heavy quarks with surround- the Wigner function of the hadron [29, 31, 38]. However ing light quarks and antiquarks has been suggested in considering the non-perturbative nature of hadroniza- early 1980s [23–25], and has successfully explained the tion, beyond such approximation will be more realistic flavor asymmetry of D mesons at forward rapidities but in such case we do not know the precise form of the in hadronic collisions through recombination of charm combination function from the solid QCD phenomenol- quarks with valence and/or sea quarks from projectile ogy. Therefore, here we only take the most basic char- [26–28]. The (re-)combination mechanism is also phe- acteristic of the combination — the combination mostly nomenologically successful in heavy-ion collisions [29–36], happens for quarks and antiquarks that are neighboring where the QGP provides a natural source of thermal light in momentum space. We suppose the combination takes quarks and antiquarks to color-neutralize heavy quarks place mainly for the quark and/or antiquark which has a at hadronization. As the aforementioned discussions, if given fraction of momentum of the hadron, and we write the small dense quark matter is created in p-Pb collisions the combination function at LHC energies, the low-pT charm quarks will prefer to pick up the co-moving light quarks or antiquarks to form 2 (p ,p ; p)= κ δ(p x p), (5) the charm hadrons. In this section, we present a work- RMcl¯ 1 2 Mcl¯ i − i i=1 ing model for the production of single-charm hadrons in Y 3 the low pT range in quark (re-)combination mechanism ′ (p ,p ,p ; p)= κ ′ δ(p x p), (6) (QCM) in momentum space. RBcll 1 2 3 Bcll i − i i=1 Y
′ A. formulism in momentum space where κMcl¯ and κBcll are constants independent of p. We adopt the approximation of equal transverse velocity in combination, or called co-moving approximation for For a charm meson M ¯ composed of a c and a light cl heavy quark hadronization. Because the velocity is v = antiquark ¯l, and a charm baryon B ′ composed of a c cll p/E = p/γm, equal velocity implies p = γvm m and two light quarks ll′, their momentum distributions i i i which leads to ∝ in QCM, as formulated in e.g. [37] in general, can be obtained by xi = mi/ mj , (7) j f (p)= dp dp f ¯(p ,p ) (p ,p ; p), (1) X Mcl¯ 1 2 cl 1 2 RMcl¯ 1 2 Z where quark masses are taken to be the constituent f ′ (p)= dp dp dp f ′ (p ,p ,p ) ′ (p ,p ,p ; p). masses in the quark model. Specifically, we take mu = Bcll 1 2 3 cll 1 2 3 RBcll 1 2 3 Z md = 0.33 GeV, ms = 0.5 GeV, and mc = 1.5 GeV so (2) that the mass and momentum of the hadron can be prop- erly generated by the combination of these constituent Here, fc¯l(p1,p2) is the joint momentum distribution for c ¯ quarks and antiquarks. We emphasize that such equal and l. M ¯(p1,p2; p) is the combination function that is R cl velocity approximation is shown to be quite effective in the probability density for a given c¯l with momenta p1, light sector in our previous work [22] where the data of pT p combining into a meson M ¯ with momentum p. It is 2 cl spectra for identified hadrons such as p, Λ, Ξ, Ω, φ,K∗, similar for baryons. Ξ∗, and Σ∗ in p-Pb collisions at s = 5.02 TeV can Considering the perturbative nature of charm quark √ NN be well explained by a up/down quark spectrum f (p ) production, we assume the momentum distribution of u T and a s quark spectrum f (p ) at hadronization. For charm quarks is independent of those of light quarks. If s T charm hadrons, although the charm quark carries the ma- we also take independent distributions for light quarks of jor part of the momentum of the hadron, light constituent different flavors, we have quarks also influence explicitly the momentum distribu- (n) (n) tion of the charm hadron, which can be clearly seen from fc¯l(p1,p2)= Nc¯lfc (p1)f¯l (p2), (3) + 0 spectrum ratios such as Ds/D, Λc /D and Ωc/D, etc. (n) (n) ′ ′ (n) fcll (p1,p2,p3)= Ncll fc (p1)fl (p2)fl′ (p3). (4) Substituting Eqs. (5-6) and (3-4) into Eqs. (1-2), we have Here we have defined the normalized momentum distri- (n) (n) bution f (p) with dpf (p)=1 and N is charm (n) (n) c c c f (p)= N ¯κ f (x p)f (x p), (8) Mcl¯ cl Mcl¯ c 1 ¯l 2 quark number. It is similar for f (n) (p) and N . The R l l (n) (n) (n) fB ′ (p)= Ncll′ κB ′ f (x1p)f (x2p)f ′ (x3p). (9) number of quark-antiquark pair reads as Nc¯l = NcN¯l cll cll c l l 3
′ ′ By defining the normalized meson distribution Baryon formation probability Pcll →Bcll is obtained
similarly. We use NBc to denote the total number of (n) (n) (n) f (p)= AM ¯f (x1p) f (x2p) , (10) all charm baryons containing one charm quark. Ncqq = Mcl¯ cl c ¯l N N = N N (N 1) with N = N +N +N denotes c qq c q q − q u d s −1 (n) (n) the possible number of all cqq combinations. NB /Ncqq where A = dp fc (x1p) f (x2p), and the normal- c Mcl¯ ¯l ized baryon distribution estimates the average probability of the cqq forming a R charm baryon. For a specific cll′ combination with num- (n) (n) (n) (n) ber Ncll′ , the averaged number of the formed baryons ′ ′ fB ′ (p)= ABcll fc (x1p) fl (x2p) fl (x3p) , (11) NB cll is N ′ N ′ c . Here, N ′ is the iteration num- iter,ll cll Ncqq iter,ll ′ ′ −1 3 (n) ber of ll pair and is taken to be 1 for l = l and 2 where A ′ = dp fqi (xip), we finally obtain the Bcll i=1 for l = l′. The appearance of this factor is due to the following expressions for charm hadrons 6 R Q possible double counting in Nqq, e.g. NuNd appears (n) twice in Nqq. We consider the production of single-charm fM ¯ (p)= NM ¯ f (p) , (12) cl cl Mcl¯ + + 0 P + baryons in triplet (Λc , Ξc , Ξc ) with J = (1/2) , in sex- (n) ′ ′ + fB ′ (p)= NB ′ f ′ (p) , (13) 0 + ++ 0 + 0 P cll cll Bcll tet Σc, Σc , Σc , Ξc , Ξc , Ωc with J = (1/2) , and ∗0 ∗+ ∗++ ∗0 ∗+ ∗0 P with yields in sextet Σc , Σc , Σc , Ξc , Ξc , Ωc with J = (3/2)+, respectively, in the ground state. We intro- κM ¯ cl duce a parameter RS1/T to denote the relative ratio of NM ¯ = Nc¯l = Nc¯lPc¯l→M ¯, (14) cl A cl P + P + Mcl¯ J = (1/2) sextet baryons to J = (1/2) triplet κ ′ Bcll baryons of the same flavor composition, and a parameter ′ ′ ′ ′ ′ NBcll = Ncll = Ncll Pcll →Bcll . (15) + ′ P ABcll RS3/S1 to denote that of J = (3/2) sextet baryons to J P = (1/2)+ sextet baryons of the same flavor com- We see that P ¯ κM ¯/AM ¯ has the proper phys- cl→Mcl¯ cl cl position. We also take the effective thermal weight as ical meaning, i.e., the≡ momentum-integrated combina- a guideline and take RS1/T = 0.5 and RS3/S1 = 1.5, ¯ ′ tion probability for cl M ¯. Similarly P → ′ → cl cll Bcll ≡ respectively. ′ ′ κBcll /ABcll denotes the momentum-integrated combina- ′ Finally, the yield of a specific kind of charm baryons tion probability for cll Bcll′ . ′ → NBi,cll is The combination probabilities P ¯ and P ′→ ′ cl→Mcl¯ cll Bcll can be determined with a few parameters. We use NMc NBc to denote the total number of all charm mesons contain- ′ ′ ′ ′ NBi,cll = CBi,cll Niter,ll Ncll , (18) Ncqq ing one charm constituent. Ncq¯ = Nc (Nu¯ + Nd¯ + Ns¯) is the possible number of all charm-light pairs. NM /Ncq¯ c where C ′ is the production weight according to two is then used to estimate the average probability of a cq¯ Bi,cll parameters R and R . For ll′ = uu,dd,ss, forming a charm meson. For a specific combination c¯l, it S1/T S3/S1 can have different J P states. In this paper we consider P − + 0 1 for Σ++, Σ0, Ω0 only the pseudo-scalar mesons J = 0 (D , D and 1+RS3/S1 c c c C ′ = (19) + P − ∗+ ∗0 ∗+ Bi,cll RS3/S1 ∗++ ∗0 ∗0 Ds ) and vector mesons J = 1 (D , D and Ds ) for Σ , Σ , Ω . ( 1+RS3/S1 c c c in the ground state. We introduce a parameter RV/P to denote the relative ratio of vector meson to pseudo-scalar For ll′ = ud,us,ds, meson of the same flavor composition. Then the yield of a specific kind of charm mesons Mi,c¯l is 1 + 0 + for Λc , Ξc , Ξc 1+RS1/T (1+RS3/S1) R ′ ′ NM S1/T + 0 + c ′ for Σ , Ξ , Ξ N = C N ¯ , (16) CBi,cll = 1+R 1+R c c c Mi,cl¯ Mi,cl¯ cl S1/T ( S3/S1) Ncq¯ RS1/T RS3/S1 ∗+ ∗0 ∗+ for Σc , Ξc , Ξc . 1+RS1/T (1+RS3/S1) with (20) We note that after taking the strong and electromagnetic 1 for J P =0− mesons 1+RV /P decays into account, yields and momentum spectra of CM ¯ = R (17) i,cl V /P for J P =1− mesons. final-state Λ+, Ξ0 and Ω0 which are usually measured in ( 1+RV /P c c c LHC experiments are not sensitive to parameters RS1/T By counting polarization states a naive estimation of and RS3/S1. RV/P is 3.0. However, the mass of the hadron will in- The single-charm mesons and baryons consume most fluence the formation probability in the sense that the of charm quarks and antiquarks produced in collisions. lower mass denotes the lower energy level for the bound A rough estimation gives the relative ratios of multi- state formation and means preferable formation. Here, charm hadrons to single-charm hadrons are only about we consider the effective thermal weight used in [39, 40] N /N N /N . 0.01 and N /N N /N . Mcc¯ Mc ∼ c q Bcc Bc ∼ c q as a guideline and take RV/P =1.5 in this paper. 0.01. Therefore, we have the following approximation to 4
(n) single-charm hadrons fs¯ (pT ) is assumed at LHC energies due to the charge conjugation symmetry. The iso-spin symmetry between N + N N . (21) Mc Bc ≈ c u and d as well as the charge conjugation symmetry be- (c) (n) (n) tween u and u¯ are assumed, so fu (pT ) = f (pT ) = Here we treat the ratio R NBc /NMc as a param- d B/M ≡ (n) (n) eter of the model, which globally characterizes the rela- fu¯ (pT ) = fd¯ (pT ), which can be extracted from the tive production of single-charm baryons to single-charm spectrum of K∗ by the relation mesons. (n) (n) ∗ ∗ Some discussions on the present model in contrast fK 0 ((1 + r) pT )= Nds¯ κK 0 fs¯ (rpT ) fd (pT ) (23) with other popular (re-)combination/coalescence mod- els applied in relativistic heavy-ion collisions [29, 38, 41] (n) with the extracted fs (pT ) and r = ms/mu if the data are necessary. In essence, our model is a statistical of K∗ are available. Otherwise, it can be extracted from hadronization method based on the constituent quark de- the data of proton after subtracting the decay influence. grees of freedom, in which unclear non-perturbative dy- The number of s quarks and that of u or d quarks are namics such as the selection of different spin states and fitted from the data of hadronic yields in QCM. the formation competition between baryon and meson in We have obtained these information of light quarks quark combination are treated as model parameters. In at hadronization in different multiplicity classes in p-Pb addition, it is still unclear at present for the geometri- collisions at √sNN = 5.02 TeV in previous work [22], cal or spatial structure of the soft parton system in p-Pb which is shown in Fig. 1 as the input of the present collisions at LHC, and therefore we do not consider the work. The pT spectra of light quarks in minimum-bias spatial distributions of quarks at hadronization in the events are also shown. present working model. These points are main differ-
-1 10 ence from those (re-)combination/coalescence models in (a) (b) terms of Wigner function method applied in relativistic u quark s quark heavy-ion collisions [29, 38, 41]. 10
On the other hand, in the study the possible creation dy) (GeV/c) of deconfined quark matter, results of QCM are usually T 1 compared with those of (string) fragmentation mecha- 0-5% nism. Because parameters that control the production dN/(dp 5-10% 1 10-20% weight for different spin states in hadron production such 20-40% as RV/P and RS3/S1 also exist somehow in string frag- Minimum-bias events mentation, the key phenomenological difference between 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 p (GeV/c) p (GeV/c) two classes of hadronization mechanism, in our opin- T T ion, lies in the kinetic characteristic in momentum space, which will, for example, obviously exhibit in the ratio of Figure 1. The pT spectra of light quarks at mid-rapidities in p- baryons to mesons as the function of pT . Pb collisions at √sNN = 5.02 TeV. The pT -integrated number densities dN/dy of u and s quarks are (24.6,8.8), (19.7,7.0), (16.0,5.6), (12.2,4.2), and (8.6,2.9) in event classes 0-5%, 5- 10%, 10-20%, 20-40%, and minimum-bias events, respectively. B. pT spectra of constituent quarks at hadronization
The pT spectrum of charm quarks is calculable in In this paper we study the production of single-charm perturbative QCD. Here, we take the calculation of hadrons at specific rapidities and we apply the formu- the Fixed-Order Next-to-Leading-Logarithmic (FONLL) las in the previous section to the one-dimensional pT [43, 44] in pp collisions at √s = 5.02 TeV as the guide- space. The pT distributions of quarks and antiquarks line. In Fig. 2, we show the normalized pT distribution at hadronization are inputs of the model. The pT distri- of charm quarks, which is obtained from the online cal- butions of light constituent quarks in the low pT range culation of FNOLL[45]. The points with the line are are unavailable from the first-principle QCD calculations. center values of FONLL and the shadow area shows the However, we can extract them from the data of pT spec- scale uncertainties, see Refs. [43, 44] for details. The un- tra of identified hadrons in QCM in the equal-velocity certainty of parton distribution functions (PDFs) is not combination approximation. For example, as formulated included. We see that the theoretical uncertainty is large in Refs. [22, 42] which is similar to Eq. (8), the pT for the spectrum of charm quarks at low pT . If we directly spectrum of φ is related to that of s quarks use this spectrum, our results of charm hadrons also have large uncertainties and the comparison with the data will 2 be less conclusive. Therefore we only take the calculation (n) fφ (2pT )= Nss¯κφ fs (pT ) , (22) of FONLL as an important guideline. The practical pT h i spectrum of charm quarks used in this paper is extracted where κφ is a constant independent of momentum. We by fitting the data of D∗+ mesons [46, 47] in QCM and (n) (n) can extract fs (pT ) using the data of φ. fs (pT ) = is shown as the thick solid line in Fig. 2. We see that 5 the extracted spectrum is very close to the center points -1 (a) (b) 0 + of FONLL calculation for pT & 1.5 GeV and excesses the 104 D 104 D latter to a certain extent for lower pT . b(GeV/c) 3 3
µ 10 10
dy) 2 2 -1 (n) 10T 10 f c (p ) FONLL data 1 T /(dp
σ 10 QCM 10 used in QCM d 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14 dy) (GeV/c) -1 T (c) (d) *+ + − 10 1 104 D 104 Ds dn/(dp
b(GeV/c) 3 3
µ 10 10 dy) 10T 2 102 10 − 2 0 1 2 3 4 5 6 7 8 /(dp σ 10 10 p (GeV) d T 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14 p(GeV/c) p (GeV/c) T T Figure 2. The normalized pT spectrum of charm quarks in rapidity region 0.96 lisions [48]. RV/P is taken to be the thermal-weight III. COMPARISON WITH DATA AND (c) value 1.5. R is 0.26, which is the extrapolation from DISCUSSIONS B/M the production of light flavor hadrons and will be dis- cussed in the next subsection. The extracted dσc/dy In this section, we use the above working model to 158 mb is consistent with the experimental estimation≈ describe the available data of the pT spectra of D mesons +13 + 151 14(stat)−26(syst) mb [47]. In addition, the re- and Λc baryon in central and forward rapidity regions in ± sulting dσD0 /dy = 76.0 mb is also consistent with the minimum-bias p-Pb collisions at √s = 5.02 TeV. We +7.1 NN experimental estimation 79.0 7.3 (stat) (syst) mb study to what extent these data are described by QCM, −13.4 [47]. ± and discuss what information on the hadronization of low In Fig. 3, we see that results of QCM are in good pT charm quarks can be extracted from these data. We 0 agreement with the data for pT . 7 GeV but are smaller give the prediction of other single-charm baryons Ξc and 0 + 0 0 than the data for high momenta pT & 8 GeV. This is rea- Ωc as well as the pT spectra of D mesons, Λc , Ξc and Ωc baryons in different multiplicity classes in p-Pb collisions sonable. Supposing the existence of the parton medium with ample (dozens of) quarks and antiquarks with soft at √sNN =5.02 TeV for the further test of the model. momenta pTl . 2 GeV, during moving in the medium the perturbatively-created charm quark with momentum A. D meson spectra and spectrum ratios pT,c . 5 GeV has many potential co-moving light quarks or antiquarks and it can pick up one of them to form the In Fig. 3, we show the differential cross sections of D single-charm hadron. For the hadron formation at high momentum p & 8 GeV, if it hadronizes still by combi- mesons as the function of p in central rapidity region T T & 0.96 1 1 + 0 (a) *+ 0 (b) D /D D /D Table I. The pT -integrated cross sections dσ/dy and pT of ratio 0.8 0.8 h i D mesons in minimum-bias p-Pb collisions at √sNN = 5.02 0.6 0.6 TeV. 0 + + ∗+ 0.4 0.4 D D Ds D 0.2 data 0.2 dσ (mb) 76.0 31.2 17.7 32.0 QCM dy 00 2 4 6 8 10 12 14 00 2 4 6 8 10 12 14 pT (GeV/c) 2.20 2.28 2.34 2.32 1 1 h i + 0 (c) + + (d) Ds/D Ds/D ratio 0.8 0.8 0.6 0.6 are extracted from the data of light-flavor hadrons using quark combination mechanism; (2) the used pT spectrum 0.4 0.4 of charm quarks is consistent with perturbative-QCD cal- 0.2 0.2 culations. Therefore, our results suggest a possibly uni- versal picture for the production of light and heavy flavor 00 2 4 6 8 10 12 14 00 2 4 6 8 10 12 14 hadrons in low p range at the hadronization of the small p(GeV/c) p (GeV/c) T T T parton system in p-Pb collisions at LHC energies. Figure 4. Ratios of differential cross sections for D mesons as the function of pT in central rapidity region 0.96 pp 7 TeV |y|<0.5 p-Pb 5.02 TeV mentation to describe the hadronization. We see that 0 PYTHIA8(Monash) (a) (b) PYTHIA8(CR Mode0) 1.5 + c DIPSY(ropes) nCTEQ15 + 0 HERWIG7 Λ EPS09NLO color re-connection give the Λc /D ratio of about 0.1 p-Pb 5.02 TeV which is significantly smaller than the data and give a -0.96 After the above discussions, we conclude that two fea- 3 3 + 0 dy) 10 10 tures of the Λc /D ratio, i.e., the magnitude of the ratio T in 2 . pT . 5 GeV range and the pT dependence of the /(dp + 2 2 Λc 1.5 σ 10 10 ratio (i.e., ratio shape), are key test for the theoretical + 0 d Λ models. In QCM, the baryon/meson ratio usually ex- c -0.96 p p −3.02 2 Ω/φ [51–55]. p 1− T T +1 N + T ( 19.1 )( 8.15 ) + 0 Λc − The Λ /D spectrum ratio in Eq. (32) is also influ- 1.15 √pT e 4.38 . c ≈ ND0 enced by the yield ratio N + /N 0 which describes the Λc D (32) + 0 global production of Λc relative to D . We illustrate 8 this point by the ratio of the pT -integrated cross section tion of single-charm baryons to single-charm mesons as + 0 of Λc to that of D . Using Eqs. (16) and (18) and tak- the charm quark hadronizes. There are two striking prop- ing the strong and electromagnetic decays into account (c) erties for the extracted RB/M . First, we see a large differ- we have (c) ence between RB/M in central rapidity region and that in Λ+ 4 1 c = 2+λs R(c) =1.215R(c) , (33) forward rapidity region. This is somewhat puzzling. For D0 1+1.677RV /P B/M B/M light-flavor hadrons, the ratios of baryons to mesons such 1+RV /P 0 as p/π and Λ/Ks yield ratios at LHC energies are rela- where λs =0.34 and RV/P =1.5 are taken. We see that tively stable at different system sizes and in different ra- (c) pidity regions [8, 48, 51, 52, 63, 64]. Models or event gen- the ratio is directly influenced by the parameter RB/M which quantifies the production competition of single- erators based on string or cluster fragmentation also usu- charm baryons to mesons at hadronization. Because of ally predict the stable baryon/meson yield ratios as the (c) function of rapidity [19, 21, 57]. In addition, the prelimi- the difficulty of non-perturbative QCD, RB/M can not + 0 nary data of LHCb collaboration [61] show Λc /D ratio be determined by first-principle calculations and here we is stable at different forward and backward rapidities. treat it as the parameter of the model. Second, the extracted R(c) is larger than that extrap- The bands in Fig. 5 (c) and (e) are our results of B/M + 0 + olated from light-flavor hadrons. If we suppose that the Λc /D ratio and Λc spectrum as the function of pT us- ing the light quark spectra in Fig. 1 and charm quark baryon-meson production competition in the formation (c) of single-charm hadrons is the same as that in the for- spectrum in Fig. 2 with R = 0.425 0.25. We see B/M mation of light-flavor hadrons, we have the relative ratio ±+ that the shape of the ratio and that of Λc spectrum in (c) (l) (l) RB/M 3RB/M =0.26. Here, RB/M NB(lll)/NM(ll¯) is 3 . pT . 7 GeV range are consistent with the data in ≈ ≡ central rapidity region ( 0.96 -1 0 + 0 + 0 Ξ /D Ωc/Ds ratios exhibit the typical increase-peak-decrease (a) Ξ (b) c s c 0 + 3 0 Ω /D × 3 behavior as the function of pT . In addition, we note that 10 Ω c s Ratio 0 + c the magnitude of Ξc/Ds at peak position pT 3 GeV is 0.4 ≈ + 0 b(GeV/c) 2 about 0.45 which is slightly smaller than that of Λc /D µ 10 0 + at similar peak position. Ωc /Ds ratio has the much lower 0 + dy) magnitude than Ξ /D ratio because of the strangeness T c s 10 0 + 0.2 suppression. The peak position of Ωc/Ds ratio is about /(dp pT 4 GeV which is about 1 GeV larger than those of σ + ≈ 0 0 + d 1 Λc /D and Ξc/Ds ratios. This is because the pT spec- (c) ± trum of strange quarks is flatter than that of up/down RB/M=0.425 0.025 − 10 1 0 quarks [22]. 0 2 4 6 8 10 12 0 2 4 6 8 10 12 p(GeV/c) p (GeV/c) The above results are of minimum-bias events in p- T T Pb collisions which have relatively small charged-particle 0 0 multiplicity dNch/dη = 17.6 at midrapidity [69]. In Figure 6. The prediction of pT spectra of Ξc and Ωc and fact, as indicated by the observation of collectivity and their ratios to D+ in central rapidity region 0.96 5 5 5 5 -1 10 10 10 10 0 (a) + (b) *+ (c) + (d) D D D Ds 10 4 10 4 10 4 10 4 b(GeV/c) 10 3 10 3 10 3 10 3 µ dy) 2 2 2 2 T 10 10 10 10 /(dp 10 10 10 10 σ d 0 2 4 6 8 10 12 0 2 4 6 8 10 12 0 2 4 6 8 10 12 0 2 4 6 8 10 12 -1 0 0 p (GeV/c) Λ (e) Ξ (f) Ω (g) T 10 4 c 10 4 c 10 4 c Multiplicity class 10 3 10 3 10 3 b(GeV/c) 0-5% µ 10 2 10 2 10 2 5-10% (× 2 -1) dy) T 10 10 10 10-20% ( × 2 -2) /(dp 1 1 1 σ × -3 d 20-40% ( 2 ) 10 −1 10 −1 10 −1 0 2 4 6 8 10 12 0 2 4 6 8 10 12 0 2 4 6 8 10 12 p(GeV/c) p (GeV/c) p (GeV/c) T T T Figure 7. The prediction of pT spectra of single-charm hadrons in central rapidity region 0.96 1 1 1 1 + 0 *+ 0 + 0 + + (a) D /D (b) D /D (c) Ds/D (d) Ds/D Ratio 0.8 0.8 0.8 0.8 0.6 0.6 0.6 0.6 0.4 0.4 0.4 0.4 0.2 0.2 0.2 0.2 00 2 4 6 8 10 12 00 2 4 6 8 10 12 00 2 4 6 8 10 12 00 2 4 6 8 10 12 0 0 + 0 + p (GeV/c) (e) Λ /D (f) Ξ /D 0.1 (g) Ω /D T c c s c s Ratio 0.6 0.4 Multiplicity class 0-5% 0.4 0.05 5-10% 0.2 0.2 10-20% 20-40% 00 2 4 6 8 10 12 00 2 4 6 8 10 12 00 2 4 6 8 10 12 p(GeV/c) p (GeV/c) p (GeV/c) T T T Figure 8. The prediction of ratios among different single-charm hadrons as function of pT in different multiplicity classes in (c) p-Pb collisions at √sNN = 5.02 TeV. The lines are results with RB/M = 0.425. + 0 + + ratios Ds /D and Ds /D also change little. The multi- quark combination mechanism. Considering the possi- + 0 0 0 0 0 plicity dependence of ratios Λc /D , Ξc/D and Ωc /D ble creation of the dense parton medium in p-Pb col- is more obvious than that of meson ratios but the depen- lisions at such extremely-high collision energies, the dence is still weak. Such increased dependence is because charm quark hadronizes by capturing the co-moving light the formation of single-charm baryon needs a more light quark or antiquark in dense medium to form the single- quark than that of single-charm meson and therefore suf- charm hadron. We introduced a working model in the fers more influence of the change of light quark spectra. framework of quark combination mechanism to formulate the yields and momentum distributions of single-charm hadrons formed by the combination of charm quarks and light-flavor (anti-)quarks in equal-velocity combination IV. SUMMARY 0,+ + approximation. 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