Production of single-charm hadrons by quark combination mechanism in $p$-Pb collisions at $\sqrt{s_{NN}}=5.02$ TeV
Hai-hong Li Feng-lan Shao Jun Song Rui-qin Wang
hep-ph





School of Physics and Engineering, Qufu Normal University, Shandong 273165, China
Department of Physics, Jining University, Shandong 273155, China 


shaofl@mail.sdu.edu.cn
School of Physics and Engineering, Qufu Normal University, Shandong 273165, China


songjun2011@jnxy.edu.cn
Department of Physics, Jining University, Shandong 273155, China


School of Physics and Engineering, Qufu Normal University, Shandong 273165, China


If QGP-like medium is created in p-Pb collisions at extremely high collision energies, charm quarks that move in the medium can hadronize by capturing the co-moving light quark(s) or anti-quark(s) to form the charm hadrons. Using light quark p_T spectra extracted from the experimental data of light-flavor hadrons and a charm quark p_T spectrum that is consistent with perturbative QCD calculations, the central-rapidity data of p_T spectra and the spectrum ratios for D mesons in the low p_T range (p_T≲7 GeV/c) in minimum-bias p-Pb collisions at √(s_NN)=5.02 TeV are well described by quark combination mechanism in equal-velocity combination approximation. The Λ_c^+/D^0 ratio in quark combination mechanism exhibits the typical increase-peak-decrease behavior as the function of p_T, and the shape of the ratio for p_T≳3 GeV/c is in agreement with the data of ALICE collaboration in central rapidity region -0.96<y<0.04 and the preliminary data of LHCb collaboration in forward rapidity region 1.5<y<4.0. The global production of single-charm baryons is quantified using the data and the possible enhancement (relative to light flavor baryons) is discussed. The p_T spectra of Ξ_c^0, Ω_c^0 in minimum-bias events and those of single-charm hadrons in high-multiplicity event classes are predicted, which serves as the further test of the possible change of the hadronization characteristic for low p_T charm quarks in the small system created in p-Pb collisions at LHC energies. 


13.85.Ni, 25.75.Nq, 25.75.Dw
Production of single-charm hadrons by quark combination mechanism in p-Pb collisions at √(s_NN)=5.02 TeV
    Rui-qin Wang
    December 30, 2023
========================================================================================================





§ INTRODUCTION


Quark–gluon plasma (QGP) in which quarks and gluons are deconfined is a new state of the matter of QCD <cit.>, which is significantly different from the normal nuclear matter. Ultra-relativistic heavy-ion collisions are served as the main experimental approach to study the creation and property of this new state of matter. However, recent ALICE, CMS, and ATLAS experiments find with great surprise that the production of hadrons in high-multiplicity p-Pb and pp collisions at LHC exhibits a series of remarkable similarities with that in heavy-ion collisions where QGP is created. These striking observations include long range angular correlations <cit.> and collectivity <cit.>, enhanced strangeness <cit.> and enhanced baryon to meson ratio at low transverse momentum (p_T) <cit.>, etc. In heavy ion collisions these phenomena are usually attributed to the creation of QGP. Theoretical explanations on these striking observations in small collision systems usually focus on the creation of mini-QGP or phase transition <cit.>, multiple parton interaction <cit.>, color re-connection and string overlap at hadronization <cit.>, etc. In the latest work <cit.>, we found that the experimental data of p_T spectra for ϕ, Ω^-, Ξ^*(1530), K^*(892) and other identified hadrons in the low p_T range (p_T≲8 GeV/c) in p-Pb collisions at √(s_NN)=5.02 TeV exhibit an interesting constituent quark number scaling. Such scaling behavior indicates the change of hadronization characteristic in low p_T range from the traditional string/cluster fragmentation to the quark (re-)combination mechanism in light (up, down, strange) sector, and is a possible signature for the formation of small dense parton medium in collisions. 

In this paper, we take the p_T spectra of single-charm hadrons containing only a charm or anti-charm quark as another probe for the property of the soft parton system created in p-Pb collisions at √(s_NN)=5.02 TeV, and we study the quark (re-)combination hadronization for low p_T charm quarks. This is mainly motivated by the fact that, even though the production of charm quarks is the perturbative QCD process by the initial hard collisions of partons in the incoming nucleons, the hadronization of charm quarks is dependent on the surrounding parton environment. At hadronization, if the charm quark is surrounded by the medium with relatively abundant soft partons, the charm quark can pick up a light antiquark or two light quarks co-moving with it to form a heavy flavor hadron, where the momentum characteristic is the combination p_H=p_c+p_q̅,qq. Otherwise, in the case of the lack of the co-moving neighbor partons, it will color neutralize by connecting with the faraway parton(s) and fragment into the charm hadron of momentum p_H=xp_c with x<1. This different characteristic of hadronization will be reflected by the p_T spectra of charm hadrons in the low p_T range. We study such possible change of hadronization characteristic by focusing on the p_T spectra of mesons D^±,0, D_s^+ and D^*, baryons Λ_c^+, Ξ_c^0 and Ω_c^0, and in particular the ratios among them. 

The paper is organized as follows: Sec. II will introduce a working model in quark (re-)combination mechanism for charm quark hadronization.  Sec. III presents our results and relevant discussions. Summary is given at last in Sec. IV. 



§ CHARM QUARK HADRONIZATION IN QCM


The (re-)combination of heavy quarks with surrounding light quarks and antiquarks has been suggested in early 1980s <cit.>, and has successfully explained the flavor asymmetry of D mesons at forward rapidities in hadronic collisions through the recombination of charm quarks with valence and/or sea quarks from projectile <cit.>.  The (re-)combination mechanism is also phenomenologically successful in heavy-ion collisions <cit.>, where the QGP provides a natural source of thermal light quarks and antiquarks to color neutralize heavy quarks at hadronization. As the aforementioned discussions, if the small dense quark matter is created in p-Pb collisions at LHC energies, the low-p_T charm quarks will prefer to pick up the co-moving light quarks or antiquarks to form the charm hadrons. In this section, we present a working model for the production of single-charm hadrons in the low p_T range in quark (re-)combination mechanism (QCM) in momentum space. 



 §.§ formulism in momentum space


For a charm meson M_cl̅ composed of a c and a light antiquark l̅, and a charm baryon B_cll' composed of a c and two light quarks l l', their momentum distributions in QCM, as formulated in e.g. <cit.> in general, can be obtained by 

    f_M_cl̅(p)     =∫ dp_1dp_2f_cl̅(p_1,p_2)  R_M_cl̅(p_1,p_2;p),
    
    f_B_cll'(p)     =∫ dp_1dp_2dp_3f_cll'(p_1,p_2,p_3)  R_B_cll'(p_1,p_2,p_3;p).

Here, f_cl̅(p_1,p_2) is the joint momentum distribution for c and l̅. R_M_cl̅(p_1,p_2;p) is the combination function that is the probability density for a given cl̅ with momenta p_1, p_2 combining into a meson M_cl̅ with momentum p. It is similar for baryons.

Considering the perturbative nature of charm quark production, we assume the momentum distribution of charm quarks is independent of those of light quarks. If we also take independent distributions for light quarks of different flavors, we have 

    f_cl̅(p_1,p_2)     =N_cl̅f_c^(n)(p_1)f_l̅^(n)(p_2),
    
    f_cll'(p_1,p_2,p_3)     =N_cll'f_c^(n)(p_1)f_l^(n)(p_2)f_l'^(n)(p_3).

Here we have defined the normalized momentum distribution f_c^(n)(p) with ∫ dpf_c^(n)(p)=1 and N_c is charm quark number. It is similar for f_l^(n)(p) and N_l. The number of quark-antiquark pair reads as N_cl̅=N_cN_l̅ and the three-quark combination N_cll'=N_cN_ll' with N_ll'=N_l(N_l'-δ_l,l').  δ_l,l' is Kronecker delta function. 

The combination function contains the key information of hadronization.  In sudden hadronization approximation, it is determined by the overlap between the wave function of quarks and that of the hadron or by the Wigner function of the hadron <cit.>.  However considering the non-perturbative nature of hadronization, beyond such approximation will be more realistic but in such case we do not know the precise form of the combination function from the solid QCD phenomenology. Therefore, here we only take the most basic characteristic of the combination — the combination mostly happens for quarks and antiquarks that are neighboring in momentum space. We suppose the combination takes place mainly for the quark and/or antiquark which has a given fraction of momentum of the hadron, and we write the combination function

    R_M_cl̅(p_1,p_2;p)     =κ_M_cl̅∏_i=1^2δ(p_i-x_ip),
     R_B_cll'(p_1,p_2,p_3;p)     =κ_B_cll'∏_i=1^3δ(p_i-x_ip),

where κ_M_cl̅ and κ_B_cll' are constants independent of p. We adopt the approximation of equal velocity in combination, or called co-moving approximation for heavy quark hadronization. Because the velocity is v=p/E=p/γ m, equal velocity implies p_i=γ vm_i∝ m_i which leads to 

    x_i=m_i/∑_jm_j,

where quark masses are taken to be the constituent masses in the quark model. Specifically, we take m_u=m_d=0.33 GeV, m_s=0.5 GeV, and m_c=1.5 GeV so that the mass and momentum of the hadron can be properly generated by the combination of these constituent quarks and antiquarks. We emphasize that such equal velocity approximation is shown to be quite effective in light sector in our previous work <cit.> where the data of p_T spectra for identified hadrons such as p, Λ, Ξ, Ω, ϕ, K^*, Ξ^*, and Σ^* in p-Pb collisions at √(s_NN)=5.02 TeV can be well explained by a up/down quark spectrum f_u(p_T) and a s quark spectrum f_s(p_T) at hadronization. For charm hadrons, although the charm quark carries the major part of the momentum of the hadron, light constituent quarks also influence explicitly the momentum distribution of the charm hadron, which can be clearly seen from spectrum ratios such as D_s/D, Λ_c^+/D and Ω_c^0/D, etc. 

Substituting Eqs. (<ref>-<ref>) and (<ref>-<ref>) into Eqs. (<ref>-<ref>), we have 

    f_M_cl̅(p)     =N_cl̅ κ_M_cl̅f_c^(n)(x_1p)f_l̅^(n)(x_2p),
    
    f_B_cll'(p)     =N_cll'κ_B_cll'f_c^(n)(x_1p)f_l^(n)(x_2p)f_l'^(n)(x_3p).

 By defining the normalized meson distribution

    f_M_cl̅^(n)(p)=A_M_cl̅f_c^(n)(x_1p)f_l̅^(n)(x_2p),

where A_M_cl̅^-1=∫ dp f_c^(n)(x_1p)f_l̅^(n)(x_2p),
and the normalized baryon distribution

    f_B_cll'^(n)(p)=A_B_cll' f_c^(n)(x_1p)f_l^(n)(x_2p)f_l'^(n)(x_3p),

where A_B_cll'^-1=∫ dp∏_i=1^3f_q_i^(n)(x_ip), we finally obtain the following expressions for charm hadrons 

    f_M_cl̅(p)     =N_M_cl̅ f_M_cl̅^(n)(p),
    
    f_B_cll'(p)     =N_B_cll' f_B_cll'^(n)(p_),

with yields 

    N_M_cl̅    =N_cl̅κ_M_cl̅/A_M_cl̅=N_cl̅P_cl̅→ M_cl̅,  
    
    N_B_cll'    =N_cll'κ_B_cll'/A_B_cll'=N_cll'P_cll'→ B_cll'.

We see that P_cl̅→ M_cl̅≡κ_M_cl̅/A_M_cl̅ has the proper physical meaning, i.e., the momentum-integrated combination probability for cl̅→ M_cl̅. Similarly P_cll'→ B_cll'≡κ_B_cll'/A_B_cll' denotes the momentum-integrated combination probability for cll'→ B_cll'.

The combination probabilities P_cl̅→ M_cl̅ and P_cll'→ B_cll', correspondingly κ_B_cll' and κ_M_cl̅, are parameterized.  We use N_M_c to denote the total number of all charm mesons containing one charm constituent. N_cq̅=N_c(N_u̅+N_d̅+N_s̅) is the possible number of all charm-light pairs. N_M_c/N_cq̅ is then used to estimate the average probability of a cq̅ forming a charm meson. For a specific combination cl̅, it can have different J^P states. In this paper we consider only the pseudo-scalar mesons J^P=0^-(D^+, D^0 and D_s^+) and vector mesons J^P=1^- (D^*+, D^*0 and D_s^*+) in the ground state. We introduce a parameter R_V/P to denote the relative ratio of vector meson to pseudo-scalar meson of the same flavor composition. Then the combination probability of single-charm mesons can be written as 

    P_cl̅→ M_cl̅=C_M_cl̅N_M_c/N_cq̅,

with 

    C_M_cl̅=1/1+R_V/P   for J^P=0^- mesons
    R_V/P/1+R_V/P   for J^P=1^- mesons.

By counting polarization states a naive estimation of R_V/P is 3.0. However, the mass of the hadron will influence the formation probability in the sense that the lower mass denotes the lower energy level for the bound state formation and means preferable formation.  Here, we roughly estimate such a mass effect through the effective statistical weight and take R_V/P=1.5, see Appendix <ref> for the details and relevant discussions. 

Baryon formation probability P_cll'→ B_cll' is obtained similarly. We use N_B_c to denote the total number of all charm baryons containing one charm quark. N_cqq=N_cN_qq=N_cN_q(N_q-1) with N_q=N_u+N_d+N_s denotes the possible number of all cqq combinations. N_B_c/N_cqq estimates the average probability of the cqq forming a charm baryon. For a specific cll' combination with number N_cll', the averaged number of the formed baryons is N_iter,ll'N_cll'N_B_c/N_cqq. Here, N_iter,ll' is the iteration number of ll' pair and is taken to be 1 for l=l' and 2 for l≠ l'. The appearance of this factor is due to the possible double counting in N_qq, e.g. N_uN_d appears twice in N_qq. We consider the production of single-charm baryons in triplet (Λ_c^+,Ξ_c^+,Ξ_c^0) with J^P=(1/2)^+, in sextet (Σ_c^0, Σ_c^+, Σ_c^++, Ξ_c^'0, Ξ_c^'+, Ω_c^0) with J^P=(1/2)^+, and in sextet (Σ_c^*0, Σ_c^*+, Σ_c^*++, Ξ_c^*0, Ξ_c^*+, Ω_c^*0) with J^P=(3/2)^+, respectively, in the ground state.  We introduce a parameter R_S1/T to denote the relative ratio of J^P=(1/2)^+ sextet baryons to J^P=(1/2)^+ triplet baryons of the same flavor composition, and a parameter R_S3/S1 to denote that of J^P=(3/2)^+ sextet baryons to J^P=(1/2)^+ sextet baryons of the same flavor composition.  We also take the effective statistical weight as a guideline and take R_S1/T=0.5 and R_S3/S1=1.4, see Appendix <ref> for the details and discussions.

 
Finally, formation probability P_cll'→ B_cll' is written as 

    P_cll'→ B_cll'=C_B_cll'N_iter,ll'N_B_c/N_cqq,

 where C_B_cll' is the spin-related production weight according to two parameters R_S1/T and R_S3/S1. For ll'=uu,dd,ss, 

    C_B_cll'=1/1+R_S3/S1   for Σ_c^++,Σ_c^0, Ω_c^0
    R_S3/S1/1+R_S3/S1   for Σ_c^*++,Σ_c^*0, Ω_c^*0.

For ll'=ud,us,ds, 

    C_B_cll'=1/1+R_S1/T(1+R_S3/S1)   for Λ_c^+, Ξ_c^0, Ξ_c^+
    R_S1/T/1+R_S1/T(1+R_S3/S1)   for Σ_c^+, Ξ_c^'0, Ξ_c^'+
    R_S1/TR_S3/S1/1+R_S1/T(1+R_S3/S1)   for Σ_c^*+, Ξ_c^*0, Ξ_c^*+.

 We note that after taking the strong and electromagnetic decays into account, yields and momentum spectra of final-state Λ_c^+, Ξ_c^0 and Ω_c^0 which are usually measured in LHC experiments are not sensitive to parameters R_S1/T and R_S3/S1. 

The single-charm mesons and baryons consume most of charm quarks and antiquarks produced in collisions. A rough estimation gives the relative ratios of multi-charm hadrons to single-charm hadrons are only about N_M_cc̅/N_M_c∼ N_c/N_q≲0.01 and N_Bcc/N_Bc∼ N_c/N_q≲0.01.  Therefore, we have the following approximation to single-charm hadrons

    N_M_c+N_B_c≈ N_c.

Here we treat the ratio R_B/M^(c)≡ N_B_c/N_M_c as a parameter of the model, which globally characterizes the relative production of single-charm baryons to single-charm mesons. Then we have N_M_c = N_c/(1+R_B/M^(c)) and N_B_c=R_B/M^(c) N_M_c, and substituting them into Eqs. (<ref>) and (<ref>)  we can calculate the yields and momentum distributions of single-charm hadrons through Eqs. (<ref>) and (<ref>) with a few parameters. 

Some discussions on the present model in contrast with other popular (re-)combination/coalescence models applied in relativistic heavy-ion collisions <cit.> are necessary. In essence, our model is a statistical hadronization method based on the constituent quark degrees of freedom, in which unclear non-perturbative dynamics such as the selection of different spin states and the formation competition between baryon and meson in quark combination are treated as model parameters. In addition, it is still unclear at present for the geometrical or spatial structure of the soft parton system in p-Pb collisions at LHC, and therefore we do not consider the spatial distributions of quarks at hadronization in the present working model. These points are main difference from those (re-)combination/coalescence models in terms of Wigner function method applied in relativistic heavy-ion collisions <cit.>.

Finally, we clarify parameters and/or inputs of the model in study of the single-charm hadrons. Firstly, quark momentum distributions at hadronization are inputs of the model and will be fixed and discussed in the next subsection. Secondly, the values of spin selection parameters R_V/P, R_S1/T, R_S3/S1 are taken from the effective statistical weights. The baryon/meson production competition parameter R^(c)_B/M is a relatively open parameter and will be discussed in Sec. <ref>.
On the other hand, in the study of the possible creation of deconfined quark matter, results of QCM are usually compared with those of (string) fragmentation mechanism. Because these parameters also exist somehow in string fragmentation, the key phenomenological difference between two classes of hadronization mechanism, in our opinion, lies in the kinetic characteristic in momentum space, which will, for example, obviously exhibit in the ratio of baryons to mesons as the function of p_T. 



 §.§ p_T spectra of constituent quarks at hadronization


In this paper we study the production of single-charm hadrons at specific rapidities and we apply the formulas in the previous section to the one-dimensional p_T space. The p_T distributions of quarks and antiquarks at hadronization are inputs of the model. The p_T distributions of light constituent quarks in the low p_T range are unavailable from the first-principle QCD calculations. However, we can extract them from the data of p_T spectra of identified hadrons in QCM in the equal-velocity combination approximation. For example, as formulated in Refs. <cit.> which is similar to Eq. (<ref>), the p_T spectrum of ϕ is related to that of s quarks


    f_ϕ(2p_T)=N_ss̅κ_ϕ[f_s^(n)(p_T)]^2,

where κ_ϕ is a constant independent of momentum. We can extract f_s^(n)(p_T) using the data of ϕ. f_s^(n)(p_T)=f_s̅^(n)(p_T) is assumed at LHC energies due to the charge conjugation symmetry.  The iso-spin symmetry between u and d as well as the charge conjugation symmetry between u and u̅ are assumed, so f_u^(n)(p_T)=f_d^(n)(p_T)=f_u̅^(n)(p_T)=f_d̅^(n)(p_T), which can be extracted from the spectrum of K^* by the relation

    f_K^*0((1+r)p_T)=N_ds̅ κ_K^*0 f_s̅^(n)(rp_T)f_d^(n)(p_T)

with the extracted f_s^(n)(p_T) and r=m_s/m_u if the data of K^* are available. Otherwise, it can be extracted from the data of proton after subtracting the decay influence. The number of s quarks and that of u or d quarks are fitted from the data of hadronic yields in QCM. 

We have obtained these information of light quarks at hadronization in different multiplicity classes in p-Pb collisions at √(s_NN)=5.02 TeV in previous work <cit.>, which is shown in Fig.  <ref> as the input of the present work. The p_T spectra of light quarks in minimum-bias events are also shown.



The p_T spectrum of charm quarks is calculable in perturbative QCD. Here, we take the calculation of the Fixed-Order Next-to-Leading-Logarithmic (FONLL) <cit.> in pp collisions at √(s)=5.02 TeV as the guideline. In Fig. <ref>, we show the normalized p_T distribution of charm quarks in the rapidity interval -0.96<y<0.04, which is obtained from the online calculation of FONLL[FONLL Heavy Quark Production, http://www.lpthe.jussieu.fr/~cacciari/fonll/fonllform.html].  The points with the line are center values of FONLL and the shadow area shows the scale uncertainties, see Refs. <cit.> for details. The uncertainty of parton distribution functions (PDFs) is not included. We see that the theoretical uncertainty is large for the spectrum of charm quarks at low p_T. If we directly use this spectrum, our results of charm hadrons also have large uncertainties and the comparison with the data will be less conclusive. Therefore we only take the calculation of FONLL as an important guideline. The practical p_T spectrum of charm quarks used in this paper is extracted by fitting the data of D^*+ mesons in rapidity region -0.96<y<0.04 <cit.> in minimum-bias p-Pb collisions at √(s_NN)=5.02 TeV in QCM and is shown as the thick solid line in Fig. <ref>. We see that the extracted spectrum is very close to the center points of FONLL calculation for p_T≳1.5 GeV/c and excesses the latter to a certain extent for lower p_T. 






§ COMPARISON WITH DATA AND DISCUSSIONS


In this section, we use the above working model to describe the available data of the p_T spectra of D mesons and Λ_c^+ baryon in central and forward rapidity regions in minimum-bias p-Pb collisions at √(s_NN)=5.02 TeV. We study to what extent these data are described by QCM, and discuss what information on the hadronization of low p_T charm quarks can be extracted from these data. We give the prediction of other single-charm baryons Ξ_c^0 and Ω_c^0 as well as the p_T spectra of D mesons, Λ_c^+, Ξ_c^0 and Ω_c^0 baryons in different multiplicity classes in p-Pb collisions at √(s_NN)=5.02 TeV for the further test of the model. 



 §.§ D meson spectra and spectrum ratios




In Fig. <ref>, we show the differential cross sections of D mesons as the function of p_T in central rapidity region -0.96<y<0.04 in minimum-bias p-Pb collision at √(s_NN)=5.02 TeV. Symbols are experimental data <cit.> and lines are results of QCM. The data of D^*+ are used to determine the p_T spectrum of charm quarks at hadronization. The normalized charm quark spectrum f_c^(n)(p_T) was shown in Fig. <ref>. The p_T-integrated cross section of charm quarks dσ_c/dy can be extracted from the D^*+ cross section by

    dσ_c/dy    =(2+λ_s)(1+R_B/M^(c))(1+1/R_V/P)dσ_D^*+/dy.

Here, λ_s≡dσ_s/dy/dσ_u/dy=dN_s/dy/dN_u/dy is the strangeness suppression factor and is 0.34±0.01 in p-Pb collisions <cit.>. R_V/P is taken to be the thermal-weight value 1.5. 
The fitted dσ_D^*+/dy is about 32 mb. Parameter R_B/M^(c) is taken to be 0.425 according to the study of Λ_c^+ in the next subsection. The resulting dσ_c/dy is 178 mb. We find that it is higher than the experimental extraction 151±14(stat)_-26^+13(syst) mb using the data of D^0 <cit.> and the fraction of charm quarks hadronizing into D^0 mesons f_c→ D^0=0.542±0.024 <cit.>. This overestimation is because the fraction f_c→ D^0=0.422 in our model at R_B/M^(c)=0.425 is smaller than the experimentally used fraction. The calculated dσ_D^0/dy=75.0 mb in our model is consistent with the experimental data 79.0±7.3(stat)_-13.4^+7.1(syst) mb <cit.>. 




In Fig. <ref>, we see that results of QCM are in good agreement with the data for p_T≲7 GeV/c but are smaller than the data for high momenta p_T≳8 GeV/c. This is reasonable. Supposing the existence of the parton medium with ample (dozens of) quarks and antiquarks with soft momenta p_T_l≲2 GeV/c, during  moving in the medium the perturbatively-created charm quark with momentum p_T,c≲5 GeV/c has many potential co-moving light antiquarks and it can pick up one of them to form the single-charm meson. For the hadron formation at high momentum p_T≳8 GeV/c, if it hadronizes still by combination, a charm quark with p_T,c≳6 GeV/c should find out or occur the co-moving light antiquark with p_T,l≳2 GeV/c which number drops rapidly. In this case the combination is not the dominated channel and the fragmentation will take over. 


In Fig. <ref>, we show results for the ratios of differential cross sections for D mesons as the function of p_T. Symbols are experimental data <cit.> and lines are results of QCM. The sawtooth behavior in our results is due to the finite statistics. Our results of D^+/D^0, D^*+/D^0 and D_s^+/D^0 are in good agreement with the data. The comparison for D_s^+/D^+ is not conclusive. 

The p_T-integrated cross sections dσ/dy and the averaged transverse momenta ⟨ p_T⟩ of D mesons are shown in Table <ref>. Several ratios for the cross sections of charm hadrons are interesting. Using Eq. (<ref>) and taking the strong and electromagnetic decay contribution into account, we obtain the ratios of cross sections for D mesons

    D^+/D^0=1+0.323R_V/P/1+1.677R_V/P≈0.42,



    D^*+/D^0    =R_V/P/1+1.677R_V/P≈0.43,


    D_s^+/D^0=1+R_V/P/1+1.677R_V/Pλ_s≈0.24,



    D_s^+/D^0+D^+=1/2λ_s≈0.17,

where we have used λ_s=0.34±0.01 in p-Pb collisions <cit.> and R_V/P=1.5. We see that results of D^+/D^0, D^*+/D^0 and D_s^+/D^0 are consistent with the observation in low p_T range in Fig. <ref>. In addition, ratio D_s^+/(D^0+D^+) is only dependent on the λ_s in QCM and therefore is a potentially interesting measurement. 



We note that the experimental data of p_T spectra and spectrum ratios of D mesons in p-Pb collisions at √(s_NN)=5.02 TeV can be explained by perturbative-QCD calculations with fragmentation functions <cit.> within the theoretical uncertainties.  In contrast, our results indicate another kind of the understanding for these data. We emphasize two points in our method (1) the mid-rapidity p_T spectra of light-flavor quarks in the low p_T range are extracted from the data of light-flavor hadrons using quark combination mechanism; (2) the used p_T spectrum of charm quarks is consistent with perturbative-QCD calculations.  Therefore, our results suggest a possibly universal new picture for the production of light and heavy flavor hadrons in low p_T range at the hadronization of the small parton system in p-Pb collisions at LHC energies. 



 §.§ Λ_c^+/D^0 ratio as the function of p_T


The production of baryons is sensitive to the hadronization mechanism.  In particular, the ratio of baryon to meson as the function of p_T is usually served as an effective probe of the hadron production mechanism.  The enhancement of the baryon/meson ratios at “intermediate” p_T (2 p_T5 GeV/c) is a characteristic behavior of QCM.  In experiments, the enhancement of the ratios for light-flavor hadrons such as p/π, Λ/K_s^0 and Ω/ϕ has been observed many times in relativistic heavy-ion collisions <cit.>, and it is also recently observed in p-Pb collisions at LHC energies <cit.>. The data of the Λ_c^+/D^0 ratio in central rapidity region -0.96<y<0.04 in p-Pb collisions at √(s_NN)=5.02 TeV are recently reported by ALICE collaboration,  which  show an enhancement <cit.>.  In addition, LHCb collaboration also presents the preliminary data of Λ_c^+/D^0 ratio at forward and backward rapidities, which show the similar enhancement and further indicate the nontrivial p_T dependence of the ratio. In this subsection, we use QCM to understand these recent data for the Λ_c^+/D^0 ratio. 

Firstly, we briefly discuss the experimental data of the Λ_c^+/D^0 ratio at LHC in central and forward rapidity regions and some existing theoretical predictions. The data of Λ_c^+/D^0 ratio in central rapidity region -0.96<y<0.04 in p-Pb collisions at √(s_NN)=5.02 TeV <cit.>, solid squares, are shown in Fig. <ref>(a). Here, the data of pp collisions at midrapidity (|y|<0.5) at √(s)=7 TeV, open circles, are also shown and are consistent with the data of p-Pb collisions, which implies the similarity of charm quark hadronization in pp and p-Pb collisions at LHC energies <cit.>. These data show the ratio Λ_c^+/D^0 at LHC energies reaches about 0.5 in p_T range (2≲ p_T≲5 GeV/c) and seems to decrease with the p_T as p_T≳2 GeV/c. The predictions from popular event generators PYTHIA8 <cit.>, DIPSY <cit.>, and HERWIG <cit.> are also shown in Fig. <ref>(a) with different kinds of lines. These event generators adopt the string or cluster fragmentation to describe the hadronization. We see that predictions of DIPSY, HERWIG and PYTHIA8 without color re-connection give the Λ_c^+/D^0 ratio of about 0.1 which is significantly smaller than the data and give a slightly increasing tendency with p_T. Considering the color re-connection effects in PYTHIA8 <cit.> can increase the ratio up to about 0.3 and give the decreasing tendency with p_T. In Fig. <ref>(b), the preliminary data of Λ_c^+/D^0 ratio in forward rapidity region 1.5<y<4.0 in p-Pb collisions at √(s_NN)=5.02 TeV are shown. The data in forward rapidity region show the Λ_c^+/D^0 ratio in p_T range (2≲ p_T≲5 GeV/c) is about 0.36 which is smaller than that in central rapidity region. The data show the Λ_c^+/D^0 ratio decreases with p_T as p_T≳3 GeV/c but the first data point at 2.5 GeV/c is relatively smaller than the peak and therefore may indicate the possible decrease of the ratio at smaller p_T. NLO theoretical predictions <cit.> with parton distribution functions EPS09LO, EPS09NLO and nCTEQ15 are shown in Fig.<ref>(b) as different kinds of lines, which have the proper magnitude for the ratio in p_T range (2≲ p_T≲5 GeV/c) but give the slightly increasing tendency with p_T. 



After the above discussions, we conclude that two features of the Λ_c^+/D^0 ratio, i.e., the magnitude of the ratio in 2≲ p_T≲5 GeV/c range and the p_T dependence of the ratio (i.e., ratio shape), are key test for the theoretical models. In QCM, the baryon/meson ratio usually exhibits an enhancement at “intermediate” p_T and the p_T dependence of the ratio is non-monotonic, i.e., the ratio increases at small p_T and peaks at p_T∼2-3 GeV/c and then decreases at larger p_T.  For the p_T dependence of the Λ_c^+/D^0 ratio, we can see this typical behavior from the ratio of directly produced Λ_c^+ to D^0. The extracted p_T spectra of light quarks and charm quarks at hadronization in minimum-bias events shown in Figs. <ref> and <ref> can be parameterized by the following form [We adopt this kind of parameterization instead of the usual form based on Lévy-Tsallis parameterization in order to better separate the term increasing the baryon/meson ratio from the decrease one. ]

    f_q^(n)(p_T)=𝒜_q√(p_T)exp[P_4,q(p_T)],

where P_4,q(p_T) is the polynomial of degree four and is taken to be 

    P_4,u(p_T)=-2.098p_T-0.1508p_T^2+0.1807p_T^3-0.02684p_T^4

 for u quarks in p_T range [0,3] GeV/c and 

    P_4,c(p_T)=-0.02778p_T-0.1919p_T^2+0.02197p_T^3-0.0008231p_T^4

 for c quarks in p_T range [0,10] GeV/c, respectively. 𝒜_q is the normalization. Following Eqs. (<ref>) and (<ref>), the ratio of directly produced Λ_c^+ to directly produced D^0 is

    f_Λ_c^+(p_T)/f_D^0(p_T)    =N_cudκ_Λ_c^+f_c^(n)(m_c p_T/m_c+2m_u)[f_u^(n)(m_u p_T/m_c+2m_u)]^2/N_cu̅κ_D^0f_c^(n)(m_c p_T/m_c+m_u)f_u^(n)(m_u p_T/m_c+m_u)
       ≈1.15N_Λ_c^+/N_D^0√(p_T) e^-p_T(1-p_T/19.1)((p_T-3.02/8.15)^2+1)/4.38.

We see that the p_T dependence of the ratio is controlled by two terms. The first term is √(p_T) which increases the ratio with p_T.  The second term is the exponential term which monotonically decreases with p_T. The competition between two terms leads to that the ratio increases at small p_T and then reaches the peak at p_T≈3 GeV/c and decreases for larger p_T. We emphasize that this non-monotonic p_T dependence of the Λ_c^+/D^0 ratio essentially comes from the non-monotonic shape of quark distribution dn/dp_Tdy, see Figs. <ref> and <ref>. The p_T spectrum of baryon is the product of p_T spectra of three quarks, which is the third power of quark's non-monotonic shape with p_T. The p_T spectrum of meson is the product of that of quark and that of antiquark, which is the square of quark's non-monotonic shape with p_T. Therefore, the Λ_c^+/D^0 ratio has the non-monotonic p_T dependence and it is a general property of baryon/meson ratio in QCM.  In addition, taking the decay contribution into account only slightly changes the shape of the ratio. 

In heavy-ion collisions, the light quark spectra at low p_T exhibit the thermal feature, i.e. f_l^(n)(p_T)=dn/dp_T∝ p_Texp[-√(p_T^2+m^2)/T].  Therefore, the first term that controls the increase of the baryon/meson ratio is p_T instead of √(p_T) and the ratio can reach higher peak value at “intermediate” p_T in heavy-ion collisions, which had been observed for light-flavor hadrons such as p/π, Λ/K_s^0 and Ω/ϕ <cit.>. 

The Λ_c^+/D^0 spectrum ratio in Eq. (<ref>) is also influenced by the yield ratio N_Λ_c^+/N_D^0 which describes the global production of Λ_c^+ relative to D^0. We illustrate this point by the ratio of the p_T-integrated cross section of Λ_c^+ to that of D^0. Using Eqs. (<ref>) and (<ref>) and taking the strong and electromagnetic decays into account we have

    Λ_c^+/D^0=41/2+λ_s/1+1.677R_V/P/1+R_V/PR_B/M^(c)=1.215R_B/M^(c),

where λ_s=0.34 and R_V/P=1.5 are taken. We see that the ratio is directly influenced by the parameter R_B/M^(c) which quantifies the production competition of single-charm baryons to mesons at hadronization. Because of the difficulty of non-perturbative QCD, R_B/M^(c) can not be determined by first-principle calculations and here we treat it as the parameter of the model.

The bands in Fig. <ref> (c) and (e) are our results of Λ_c^+/D^0 ratio and Λ_c^+ spectrum as the function of p_T using light quark spectra in Fig. <ref> and the charm quark spectrum in Fig. <ref> with R_B/M^(c)=0.425±0.025.  We see that the shape of the ratio and that of Λ_c^+ spectrum in 3≲ p_T≲7 GeV/c range are consistent with the data in central rapidity region -0.96<y<0.04. We also use a u quark spectrum f_u(p_T) and a charm quark spectrum f_c(p_T) to fit the data of p_T spectra of Λ_c^+ and D^0 at forward rapidities 1.5<y<4.0 and results with R_B/M^(c)=0.29 are shown in Fig.  <ref> (f). Here we note that the extracted f^(n)_c(p_T) at the forward rapidity is also very close to the center values of FONLL calculation  <cit.>. The result of Λ_c^+/D^0 ratio in forward rapidity region 1.5<y<4 is shown in Fig.  <ref> (d). We see that the data of p_T spectra of Λ_c^+ and D^0 and their ratio in 3≲ p_T≲7 GeV/c range can be well fitted by QCM. 

In addition, we find that, except the difference in global magnitude due to R_B/M^(c), the shape of Λ_c^+/D^0 ratio with respect to p_T in forward rapidity region 1.5<y<4.0 is very close to that in central rapidity region -0.96<y<0.04 in our model. This is because of the following reasons which can be seen from Eq. (<ref>). Since the charm quark carries most of momenta of the Λ_c^+ and D^0 in QCM, the change of charm quark spectrum at different rapidities is largely canceled in Λ_c^+/D^0 ratio. In addition, the momenta of light quarks that take part in the formation of Λ_c^+ and D^0 are usually p_T≲2 GeV/c. The change of light quark spectra for p_T≲2 GeV/c at different rapidities is not significant and is only weakly passed to the ratio because the light (anti-)quarks carry small fraction of momenta of the Λ_c^+ and D^0.  This is different from that in light-flavor hadrons where the baryon/meson ratios have nontrivial dependence on the change of quark spectra because there each quark carries similar fraction of the momenta of the formed hadrons due to the similar constituent quark masses. 

An important result in the above fitting is that we extract R_B/M^(c)=0.425±0.025 in central rapidity region and R_B/M^(c)=0.29 in forward rapidity region. The parameter R_B/M^(c) characterizes the relative production of single-charm baryons to single-charm mesons as the charm quark hadronizes. There are two striking properties for the extracted R_B/M^(c). First, we see a large difference between R_B/M^(c) in central rapidity region and that in forward rapidity region. This is somewhat puzzling.  For light-flavor hadrons, the ratios of baryons to mesons such as p/π and Λ/K_s^0 yield ratios at LHC energies are relatively stable at different system sizes and in different rapidity regions <cit.>.  Models or event generators based on string or cluster fragmentation also usually predict the stable baryon/meson yield ratios with the rapidity at LHC <cit.>.  In addition, the preliminary data of LHCb collaboration <cit.> show Λ_c^+/D^0 ratio is stable at different forward and backward rapidities. Second, the extracted R_B/M^(c) is larger than that extrapolated from light-flavor hadrons. If we suppose that the baryon/meson production competition in the formation of single-charm hadrons is the same as that in the formation of light-flavor hadrons, we have the relative ratio R_B/M^(c)≈3R_B/M^(l)=0.26, see Appendix <ref> for the derivation of this relation.  Here, R_B/M^(l)≡ N_B(lll)/N_M(ll̅) is the ratio of light-flavor baryons to mesons and is about 0.086 by fitting the data of light-flavor hadrons in our previous work <cit.>.  This extrapolated value of R_B/M^(c)≈0.26 is close to (10% lower than) the extracted R_B/M^(c) in forward rapidity region but is significantly (60%) lower than that in central rapidity region. On the other hand, the Λ_c^+/D^0 ratio as R_B/M^(c)≈0.26 is about 0.32, see Eq. (<ref>), which is close to the thermal estimation Λ_c^+/D^0≃0.25-0.3 by statistical hadronization model <cit.>. In addition, we note that the parameter R_V/P=1.5 in meson production which can well describe the data of D meson ratios in Fig. <ref> is also from the thermal weight. 

The following discussions are helpful to understand the large value for the extracted R_B/M^(c). First is the naive estimation from the stochastic color combination. When the charm quark moves in the medium, supposing the colors of neighboring light quarks and/or antiquarks are stochastic, then the probability of the charm quark with specific color (e.g. R) occurring the light anti-quark with right anti-color (R̅) to form the color singlet is 1/3, and the probability of this charm quark occurring two light quarks with right colors (corresponding to R̅) to form the color singlet is 1/9. Then we have R_B/M^(c)=1/9/1/3=1/3 by the stochastic color combination, which is close to the extraction in the forward rapidity region. Second is the possible effect of the correlated color combination. As the charm quark hadronizes, the surrounding medium (those light quarks and antiquarks) is also in the vicinity of the hadronization. The colors of two light quarks neighboring in phase space will tend to be 3̅ states that have attractive force.  Then in the single-charm baryon formation the probability of the charm quark occurring two light quarks with right colors can be greater than 1/9 (the stochastic one) and R_B/M^(c)>1/3. Such possible enhancement has been studied by considering the existence of diquark in QGP near hadronization in Refs. <cit.>, which can raise the yield ratio Λ_c^+/D^0>1.3 (corresponding to parameter R_B/M^(c)≳1 in our model). In addition, the wave functions of charm hadrons will also influence the combination probability, for example, in several quark coalescence models through hadronic Wigner function <cit.>, which suggests the significantly enhanced ratio Λ_c^+/D^0>0.8 in relativistic heavy-ion collisions <cit.> (corresponding to parameter R_B/M^(c)>0.7 in our model). 

We finally give a short summary of this subsection. The Λ_c^+/D^0 ratio in our model exhibits the typical behavior of baryon/meson ratios as the function of p_T in quark combination mechanism. The shape of the calculated Λ_c^+/D^0 ratio is consistent with the experimental data in central and forward rapidity regions in p-Pb collisions at √(s_NN)=5.02 TeV <cit.>.  We suggest the measurement of the more precise data points, especially those for p_T≲3 GeV/c, in order to better test the quark combination characteristic for charm quark hadronization and quantify the enhancement of Λ_c^+ baryon in low p_T range in p-Pb collisions at LHC energies. 



 §.§ predictions for other hadrons and multiplicity classes




The production of other single-charm baryons such as Ξ_c^0 and Ω_c^0 will also exhibit the similar enhancement in QCM. In Fig. <ref>, we predict the p_T spectra of Ξ_c^0 and Ω_c^0 and their ratios to D_s^+ in central rapidity region -0.96<y<0.04 in minimum-bias p-Pb collisions at √(s_NN)=5.02 TeV. The parameter R_B/M^(c)=0.425±0.025 is taken according to the discussion of Λ_c^+ production in previous subsection. We see that both Ξ_c^0/D_s^+ and Ω_c^0/D_s^+ ratios exhibit the typical increase-peak-decrease behavior as the function of p_T. In addition, we note that the magnitude of Ξ_c^0/D_s^+ at peak position p_T≈3 GeV/c is about 0.45 which is slightly smaller than that of Λ_c^+/D^0 at similar peak position. Ω_c^0/D_s^+ ratio has the much lower magnitude than Ξ_c^0/D_s^+ ratio because of the strangeness suppression. The peak position of Ω_c^0/D_s^+ ratio is about p_T≈4 GeV/c which is about 1 GeV/c larger than those of Λ_c^+/D^0 and Ξ_c^0/D_s^+ ratios.  This is because the p_T spectrum of strange quarks is flatter than that of up/down quarks <cit.>. 

The above results are of minimum-bias events in p-Pb collisions which have relatively small charged-particle multiplicity dN_ch/dη=17.6 at midrapidity <cit.>. In fact, as indicated by the observation of collectivity and strangeness enhancement <cit.>, QGP-like medium is most probably created in high-multiplicity events of p-Pb collisions at LHC, and QCM should be preferably applied in these events. Results of QCM for light-flavor hadrons have shown this point <cit.>. Therefore, we make predictions for p_T spectra of single-charm hadrons in high-multiplicity events for the future test. The selected multiplicity classes are I(0-5%), II(5-10%), III(10-20%) and IV(20-40%), which are four highest multiplicity classes in measurements of ALICE collaboration. The cross sections of charm quarks dσ_c/dy in the four multiplicity classes are fixed in QCM by fitting the experimental data of the multiplicity dependence of D meson cross sections <cit.>. With R_B/M^(c)=0.425±0.025, they are taken to be (594±35, 448±26, 360±21, 254±15) mb in four classes, respectively.  We neglect the possible effects of energy loss for charm quarks traveling in the medium before hadronization and use the extracted spectrum of charm quarks in the minimum-bias events. 

In Fig. <ref>, we show p_T spectra of single-charm hadrons in central rapidity region in different multiplicity classes in p-Pb collisions at √(s_NN)=5.02 TeV. In Table <ref>, we show the p_T-integrated cross sections and ⟨ p_T⟩ of single-charm hadrons for R_B/M^(c)=0.425. We see that the ⟨ p_T⟩ of charm hadrons slightly increase with multiplicity and the increase of baryons is slightly larger than that of mesons. This is due to the increased ⟨ p_T⟩ of light quarks in high-multiplicity events. 







In Fig. <ref>, we show the ratios among different single-charm hadrons as the function of p_T in different multiplicity classes.  We see that ratios D^+/D^0 and D^*+/D^0 are hardly changed in different multiplicity classes. We first note that their p_T-integrated ratios Eqs. (<ref>) and (<ref>) are independent of the multiplicity. Even though the p_T spectrum of up/down quarks is weakly changed in different multiplicity classes, as shown in Fig. <ref>, this change will influence the inclusive spectra of D^+, D^*+ and D^0 but is significantly canceled in their ratios. As indicated by Eq. (<ref>), spectrum ratios D_s^+/D^0 and D_s^+/D^+ are explicitly dependent on the strangeness suppression factor λ_s. Because the change of λ_s is only about 0.02 in the studied four multiplicity classes <cit.>, the spectrum ratios D_s^+/D^0 and D_s^+/D^+ also change little. The multiplicity dependence of ratios Λ_c^+/D^0, Ξ_c^0/D^+_s and Ω_c^0/D^+_s is more obvious than that of meson ratios. Such increasing dependence is because the formation of single-charm baryon needs one  more light quark than that of single-charm meson and therefore the baryon/meson ratios suffer more influence of the change of light quark spectra. 



§ SUMMARY


We have studied the production of single-charm hadrons in p-Pb collisions at √(s_NN)=5.02 TeV in quark combination mechanism.  Considering the possible creation of the dense parton medium in p-Pb collisions at such extremely-high collision energies, the charm quark hadronizes by capturing the co-moving light quark or antiquark in dense medium to form the single-charm hadron. We introduced a working model in the 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 approximation. The data of p_T spectra of D^0,+, D_s^+ and D^*+ mesons and spectrum ratios in central rapidity region in low p_T range p_T≲7 GeV/c in minimum-bias events are well described by the quark combination mechanism. We emphasized two important foundations in calculations: (1) the p_T spectra of light-flavor quarks in the low p_T range are extracted from the data of light-flavor hadrons using quark combination mechanism also in equal-velocity combination approximation; (2) the used p_T spectrum of charm quarks is consistent with perturbative-QCD calculations. 

The Λ_c^+/D^0 ratio in quark combination mechanism exhibits a typical increase-peak-decrease behavior as the function of p_T. The shape of the ratio for p_T≳3 GeV/c is in agreement with the data of ALICE collaboration in the central rapidity region and those of LHCb collaboration in the forward rapidity region. The enhanced production of single-charm baryons is parameterized by R_B/M^(c) in the model, which is found to be quite high at central rapidities by fitting ALICE data, i.e., R_B/M^(c)≈0.43. It is significantly larger than that from the empirical extrapolation from light flavor hadrons (R_B/M^(c)≈0.26). The R_B/M^(c) extracted from the preliminary data of Λ_c^+/D^0 ratio in forward rapidity region reported by LHCb collaboration is about 0.29, which is close to the extrapolation from light flavor hadrons.  We made discussions on the possible reasons of the enhancement of R_B/M^(c) existing in literature. 

We made predictions on the p_T spectra of Ξ_c^0 and Ω_c^0 baryons and their ratios to D_s^+ in central rapidity region to make further test of such significant enhancement of charm baryons.  In particular, the Ξ_c^0/D_s^+ ratio shows the similar enhancement with Λ_c^+/D^0. In addition, we predicted the production of D mesons, Λ_c^+, Ξ_c^0 and Ω_c^0 baryons in high multiplicity classes in p-Pb collisions at √(s_NN)=5.02 TeV for the future test. 


The authors thank Qun Wang for helpful discussions. This work is supported by the National Natural Science Foundation of China under Grant Nos.  11575100, 11505104.


apsrev4-1




§  ESTIMATION OF R_V/P, R_S3/S1 AND R_S1/T BY EFFECTIVE STATISTICAL WEIGHTS  


In the combination of a given cl̅ or a cll, the formed meson or baryon can have different spin states.  We use an effective statistical model to roughly estimate the relative production weight of different spin states. We recall the fact that the data of multiplicities of light-flavor hadrons and open charm hadrons in elementary e^+e^-, pp and pp̅ reactions were shown to be well described by the statistical model <cit.>, which indicates the statistical description of hadron multiplicity captures the key characteristics of the non-perturbative hadronization <cit.>. 

    In the Boltzmann limit of Bose and Fermi statistics, the hadron multiplicity is the phase space integral of Boltzmann distribution, 

    N    = (2J+1) V/(2π)^3∫ d^3p  e^-√(p^2+m^2)/T 
       = (2J+1) V T/2π^2 m^2  K_2(m/T),

where m is mass and J is spin. V is the proper volume of the system and T is temperature. K_2 is the modified Bessel function of order 2. 

Using Eq. (<ref>), we get the ratio of vector meson D^* (m_D^*=2.01 GeV) to pseudo-scalar meson D (m_D=1.87 GeV)

    D^*/D = 3m_D^*^2 K_2(m_D^*/T)/m_D^2 K_2(m_D/T)≈ 1.45,

and that of D_s^* (m_D_s^*=2.11 GeV) to D_s (m_D_s=1.97 GeV), 

    D_s^*/D_s≈ 1.45

at temperature T=170 MeV <cit.>. We take the above statistical weights as a guideline and take R_V/P=1.5 in our calculation. 

For baryons, we have 

    Σ_c/Λ_c^+≈ 0.41,  Ξ'_c/Ξ_c≈ 0.57,

with m_Σ_c=2.46 GeV, m_Λ_c^+=2.29 GeV, m_Ξ'_c=2.58 GeV, and m_Ξ_c=2.47 GeV.  We take the average of above two statistical weights as a guideline and take R_S1/T=0.5 in our calculation.

For sextet baryons of spins 1/2 and 3/2, we have 

    Σ_c^*/Σ_c≈ 1.41, Ξ_c^*/Ξ'_c≈ 1.38, Ω_c^*/Ω_c≈ 1.37,

with m_Σ_c^*=2.52 GeV,  m_Ξ_c^*=2.65 GeV, m_Ω_c^*=2.77 GeV, m_Ωc=2.70 GeV. We take the average of above  statistical weights as a guideline and take R_S3/S1= 1.4 in our calculation.




§ R_B/M^(C) EXTRAPOLATED FROM LIGHT-FLAVOR HADRONS 


For simplicity, we use l to denote light-flavor quarks and c charm quarks, and N_l and N_c their numbers at hadronization.  N_q=N_l+N_c is the total number of quarks. The numbers of the formed light-flavor baryons (B_lll), single-charm baryons (B_cll), multiple-charm baryons (B_ccl and B_ccc) are N_B_lll, N_B_cll, N_B_ccl and N_B_ccc.
The total number of baryons is 

    N_B=N_B_lll+N_B_cll+N_B_ccl+N_B_ccc.

If we assume the flavor-blind baryon formation probability, we have

    N_B_f_1f_2f_3    =N_f_1f_2f_3 P_f_1f_2f_3→ B, 
    
    P_f_1f_2f_3→ B    =N_iter,f_1f_2f_3N_B/N_qqq,

where indexes f_1,f_2,f_3=l,c. The coefficient N_iter,f_1f_2f_3 is (1, 3, 3, 1) for cases (lll, cll, ccl, ccc), respectively, so that N_lll+3N_cll+3N_ccl+N_ccc=N_qqq and the normalization Eq. (<ref>) is satisfied. Then we get 

    P_cll→ B_cll=3 P_lll→ B_lll.


Similarly, the numbers of the formed light-flavor mesons (M_ll̅), single-charm mesons M_cl̅ and M_c̅l, and charmonium M_cc̅ are N_M_ll̅, N_M_cl̅, N_M_c̅l, and N_M_cc̅. The total meson number is N_M=N_M_ll̅+N_M_cl̅+N_M_c̅l+N_M_cc̅.
In flavor-blind approximation, we have 

    N_M_f_1f̅_2    =N_f_1f̅_2 P_f_1f̅_2→ M, 
    
    P_f_1f̅_2→ M    =N_M/N_qq̅,

where indexes f_1,f_2=l,c. Then we get 

    P_cl̅→ M_cl̅=P_ll̅→ M_ll̅.

Using Eqs. (<ref>), (<ref>), (<ref>) and (<ref>), the ratio of single-charm baryons to single-charm mesons is 

    R_B/M^(c)    =N_B_cll/N_M_cl̅=3N_cll/N_cl̅P_lll→ B_lll/P_ll̅→ M_ll̅
       =3N_cll/N_cl̅N_ll̅/N_lllN_B_lll/N_M_ll̅
        =3N_l/N_l-2N_B_lll/N_M_ll̅
       ≈3N_B_lll/N_M_ll̅=3R_B/M^(l)


