New feature of low $p_{T}$ charm quark hadronization in $pp$ collisions at $\sqrt{s}=7$ TeV

Treating the light-flavor constituent quarks and antiquarks that can well describe the data of light-flavor hadrons in $pp$ collisions at $\sqrt{s}=7$ TeV as the underlying source of chromatically neutralizing the charm quarks of low transverse momenta ($p_{T}$), we show that the experimental data of $p_{T}$ spectra of single-charm hadrons $D^{0,+}$, $D^{*+}$ $D_{s}^{+}$, $\Lambda_{c}^{+}$ and $\Xi_{c}^{0}$ at mid-rapidity in the low $p_{T}$ range ($2\lesssim p_{T}\lesssim7$ GeV/$c$) in $pp$ collisions at $\sqrt{s}=7$ TeV can be well understood by the equal-velocity combination of perturbatively-created charm quarks and those light-flavor constituent quarks and antiquarks. This suggests a possible new scenario of low $p_{T}$ charm quark hadronization, in contrast to the traditional fragmentation mechanism, in $pp$ collisions at LHC energies. This is also another support for the exhibition of the effective constituent quark degrees of freedom for the small parton system created in $pp$ collisions at LHC energies.


I. INTRODUCTION
The experimental study of the quark-gluon plasma (QGP), a new state of matter of QCD, is mainly through the heavy-ion collisions which can create the big thermal parton system with relatively long lifetime. Relative to heavy ion collisions, proton-nucleus (pA) collisions create the "intermediate" parton system and proton-proton (pp) collisions create the "small" parton system. The deconfined medium is usually assumed to be not created in pA and pp collisions, at least up to RHIC energies. In particular, the data of pp collisions, in the context of heavy-ion physics, are usually served as the baseline to study the effects and/or properties of cold and hot nuclear matter in pA and AA collisions, respectively.
Large Hadron Collider (LHC) pushes the center of mass energy per colliding nucleon up to TeV level, which brings new properties even for the small parton system created in pp collisions. Recent measurements in pp collisions at LHC energies from CMS and ALICE collaborations find several remarkable similarities with heavy-ion collisions. In high-multiplicity events of pp collisions, the phenomena such as long range angular correlations [1,2] and collectivity [3,4], strangeness enhancement [5,6], and the increased baryon to meson ratio at low transverse momentum (p T ) [7][8][9] are observed. These phenomena were already observed in heavy-ion collisions at RHIC and LHC energies and are usually regarded as the typical behaviors related to the formation of QGP. Theoretical studies of these striking observations focus on what happens on the small parton system created in pp collisions at LHC energies through different phenomenological/theoretical methods such as the mini-QGP creation or phase transition [10][11][12][13][14][15], multiple parton interaction [16], string overlap and color re-connection at hadronization [17][18][19][20], etc. * shaofl@mail.sdu.edu.cn In the recent work [21], we found that the mid-rapidity data of p T spectra of light-flavor hadrons in the low p T range (p T 6 GeV/c) in pp collisions at √ s = 7 TeV can be well understood by the equal-velocity combination of up/down and strange quarks and antiquarks with constituent masses at hadronization. This suggests that the constituent quark degrees of freedom (CQdof) play an important role in low p T hadron production in pp collisions at LHC energies, which indicates the possible existence of the underlying source with soft CQdof, a kind of new property for the small parton system created in pp collisions at LHC energies.
The hadronization of charm quarks in pp collisions is usually described by the traditional fragmentation mechanism or fragmentation function. In this paper, we study the possibility of new feature for the hadronization of low p T charm quarks in pp collisions at LHC energies. As the aforementioned discussion, the production of lightflavor hadrons in pp collisions at LHC energies can be well described by the combination of light-flavor constituent quarks and antiquarks of low p T . These constituent quarks and antiquarks also serve as an underlying source for the color neutralization of charm quarks at hadronization to form the single-charm hadrons. Specifically, the charm quark can pick up a co-moving light antiquark or two co-moving quarks to form a single-charm meson or baryon, where the momentum characteristic is the combination p H = p c + pq ,qq . This (re-)combination mechanism of charm quark hadronization will reflect in the momentum spectra of charm hadrons and, in particular, the ratio of charm baryon to charm meson. Therefore, in this paper, we study the mid-rapidity p T spectra of single-charm mesons D 0,+ , D * + , D + s and baryons Λ + c , Ξ 0 c and the ratios among them in the framework of quark (re-)combination mechanism (QCM), and compare our results with available experimental data and several theoretical predictions by fragmentation mechanism.
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.

II. CHARM QUARK HADRONIZATION IN QCM
The (re-)combination mechanism of charm quark hadronization was proposed in early 1980s [22][23][24] and has many applications in both hadron-hadron collisions [25][26][27] and relativistic heavy ion collisions [28][29][30][31]. Because of the lack of the sufficient knowledge for the spatial information of the small parton system created in pp collisions at LHC energies, in this section, we present a working model for the (re-)combination hadronization of charm quarks in the low p T range in momentum space, which only incorporates the most basic feature of QCM, i.e., the equal-velocity combination approximation. The unclear non-perturbative dynamics such as the selection of different spin states and the formation competition between baryon and meson in the combination are treated as model parameters.

A. formulas in momentum space
The momentum distributions of the single-charm meson M cl and baryon B cll ′ in QCM, as formulated in e.g. [32,33] in general, can be obtained by f M cl (p) = dp 1 dp 2 f cl (p 1 , p 2 ) R M cl (p 1 , p 2 ; p), ( f B cll ′ (p) = dp 1 dp 2 dp 3 f cll ′ (p 1 , p 2 , p 3 ) R B cll ′ (p 1 , p 2 , p 3 ; p). ( Here, f cl (p 1 , p 2 ) is the joint momentum distribution for charm (c) quark and light anti-quark (l). R M cl (p 1 , p 2 ; p) is the combination function that is the probability density for the given cl with momenta p 1 , p 2 combining into a meson M cl with momentum p. It is similar for the baryon.
We take independent distributions for quarks of different flavors by neglecting correlations, We suppose the combination takes place mainly for quark and/or antiquark that takes a given fraction of momentum of the hadron so that the combination function is the product of Dirac delta functions where κ M cl and κ B cll ′ are constants which are independent of the momentum but dependent on other ingredients such as the quark number so that all charm quarks can be correctly exhausted (after further including multicharm hadrons). Following our works [21,34] for light-flavor hadrons in pp and p-Pb collisions at LHC energies, we adopt the co-moving approximation in combination, i.e., the charm quark combines with light quark(s) of the same velocity to form the charm hadron. Since the equal velocity implies p i = γvm i ∝ m i , the momentum fraction is where quark masses are taken to be m u = m d = 0.33 GeV, m s = 0.5 GeV, and m c = 1.5 GeV, the constituent masses in the quark model. Substituting Eqs. (3)(4) and (5-6) into Eqs. (1-2), we obtain the distributions of single-charm hadrons We rewrite the distribution functions of charm hadrons, where f qi (x i p) with the normalized charm and light quark distribution dp f R cl→M cl and R cll ′ →B cll ′ are parameterized. We use N Mc to denote the total number of all single-charm mesons. N cq = N c (Nū + Nd + Ns) is the possible number of all charm-light pairs. N Mc /N cq gives the flavoraveraged probability of a cl forming a charm meson. The where Pl ≡ Nl/Nq denotes the probability of an antiquark with the flavorl. For a given cl combination, it can form different J P states, and we use C M i,cl to denote the probability of forming the particular spin state i, and finally obtain the yield of charm meson M i,cl 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 quark flavors, and have We take R V /P = 1.5, the thermal weight value used in [33,35,36].
In baryon sector, we have where N Bc is the total number of all single-charm baryons, N iter,ll ′ P l P l ′ selects the specific light flavor ll ′ , and C B i,cll ′ selects the particular spin state. Here ability of a quark with the flavor l. N iter,ll ′ is the iteration number of ll ′ pair and is taken to be 1 for l = l ′ and 2 for l = l ′ . We consider the produc- 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 quark flavors, 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 quark flavors. We also take the the effective thermal weighs as a guideline and take R S1/T = 0.5 and R S3/S1 = 1.5, respectively. For ll ′ = uu, dd, ss, For ll ′ = ud, us, ds, We note that yields and momentum spectra of final state Λ + c , Ξ 0 c and Ω 0 c after taking decay contribution into account are not sensitive to parameters R S1/T and R S3/S1 .
Considering the single-charm mesons and baryons consume most of charm quarks produced in collisions, we have the following approximated normalization to singlecharm hadrons Here we treat the ratio R B/M = 0.425, the value in previous work [33].

III. RESULTS AND DISCUSSIONS
We apply the above formulas in QCM to the onedimensional p T space and calculate the p T spectra of single-charm hadrons at mid-rapidity in pp collisions at √ s = 7 TeV. The p T distributions of quarks at hadronization are inputs of the model. We have obtained the p T spectra of light-flavor quarks in previous work [21]. The averaged quark number in the rapidity interval |y| < 0.5 is 2.5 for u quark and 0.8 for s quark, respectively.
The normalized distributions f  Fig.1(a). The charge conjugation symmetry between quark and antiquark and the iso-spin symmetry between up and down quarks are applied in calculations.  In Fig. 1(b), we show the normalized distribution of charm quarks, which is obtained from the online calculation of Fixed-Order Next-to-Leading-Logarithmic (FONLL) [37]. The points are center values and the shadow area shows the scale uncertainties, see Refs. [38,39] for details. The uncertainty due to parton distribution functions (PDFs) is not included. Because of the large theoretical uncertainty, in particular, at low p T , we only take the FONLL calculation as a guideline. The practically used p T spectrum of charm quarks is reversely extracted from the data of D * + meson [40,41] in QCM and is shown as the thick solid line in Fig. 1(b). The cross section of charm quarks in |y| < 0.5 interval is 1.2 mb. The extracted spectrum is found to be very close to the center values of FONLL calculation for p T 1.5 GeV/c and be higher than the latter to a certain extent for p T < 1.5 GeV/c but be still within the theoretical uncertainties.
In Fig. 2, we show results of differential cross sections of D mesons at mid-rapidity as the function of p T in pp collisions at √ s = 7 TeV, and compare them with experimental data [42]. We see that QCM well describes the data of D mesons for p T 7 GeV/c but under-predicts the data for larger p T . This is reasonable. In the equalvelocity combination of charm quarks and light quarks, a charm quark with p T,c 6 GeV/c will combine a light antiquark of p T,l 1.5 GeV/c. Because most of light quarks, see Fig. 1(a), are of such low p T , they provide the sufficient partners (or chance) for the hadronization of charm quarks. For a charm quark of p T,c 6 GeV/c, the combining light antiquark should have p T,l 1.5 GeV/c whose number is very small and is exponentially dropped. In this case, those light antiquarks may be not enough to provide the sufficient chance for the combination hadronization of charm quarks of p T,c 6 GeV/c, and therefore the combination may be not the dominated channel and the fragmentation will take over. In Fig. 3, we show results for the ratios of different D mesons as the function of p T in pp collisions at √ s = 7 TeV, and compare them with experimental data [42]. We see that, within experimental uncertainties, the model results are in agreement with the data. For the magnitudes of these four ratios, we can give a simple explanation from the yield (corresponding to differential cross-section) ratios of D mesons. Using Eq. (14) and taking strong and electromagnetic decay contribution into account where the data of decay branch ratios are taken from PDG [43], we have with λ s = N s /N u = 0.32 and R V /P = 1.5. Theoretical pQCD calculations with fragmentation functions were compared with experimental data of D mesons in Ref. [42]. It is shown that pQCD calculations in large p T range have small theoretical uncertainties and often well explain the data. However, pQCD calculations in small p T range have quite large theoretical uncertainties and the comparison with data is not conclusive. In contrast with those pQCD calculations with fragmentation functions, our results suggest a different mechanism for the charm quark hadronization at low p T .
The production of baryons is more sensitive to the hadronization mechanism. In Fig. 4, we show results of the p T spectrum of baryon Λ + c and the ratio to D 0 meson, and compare them with the experimental data [9]. We see that, similar to D mesons, our results of Λ + c spectrum and ratio Λ + c /D 0 are in good agreement with the data for p T 7 GeV/c. The predictions of other models or event generators [20,[44][45][46] which adopt string or cluster fragmentation mechanism for hadronization are also shown in Fig. 4(b). PYTHIA8 Monash tune, DIPSY with rope parameter, and HERWIG7 predict a small Λ + c /D 0 ratio of 0.1 and almost constant ratio at different p T . Considering the effect of string formation beyond leading color approximation, PYTHIA8 (CR Mode0) increases the ratio to a certain extent and gives the decrease tendency with p T . In Fig. 5, we show results of the differential crosssection of Ξ 0 c at mid-rapidity multiplied by the branch ratio into e + Ξ − ν e (a) and the relative ratio to D 0 (b) as the function of p T in pp collisions at √ s = 7 TeV. Because of the lack of the absolute branch ratio into e + Ξ − ν e , our result of the spectrum of Ξ 0 c is multiplied by a branch ratio 3.8%, which is within the current range of theoretical calculations (0.83%-4.2%). We see that the model result, the thick solid line in Fig. 5(a), can well describe the data of Ξ 0 c for 2 p T 7 GeV/c but significantly underestimates the first data point at p T = 1.5 GeV/c. However, the first data point, to our knowledge, is somewhat puzzlingly high if we note that the studied differential cross-section is dσ/dp T dy. The data of Λ + c in Fig. 4(a) and D 0 in Fig. 2(a) suggest that the differential crosssection tends to increase slowly with the decreasing p T for small p T 2 GeV/c and will saturate and decrease as p T → 0. The data of light-flavor hadrons for dN/dp T dy, e.g. K (892) * , show this behavior more clearly [47]. As a naive illustration, we see the first data point of Ξ 0 c at p T = 1.5 GeV/c is more than twice the exponential extrapolation from data points of larger p T , the thin dashed line, which is not the case for the data of D mesons and Λ + c . The QCM result of the ratio Ξ 0 c /D 0 is shown in Fig.  5(b). We see that the two data points within 2 p T 7 GeV/c can be well described by QCM and the first data point at p T = 1.5 GeV/c is much higher than the QCM result.  Figure 5. Differential cross-section of Ξ 0 c at mid-rapidity multiplied by the branch ratio into e + Ξ − νe (a) and the ratio to D 0 (b) as the function of pT in pp collisions at √ s = 7 TeV. Symbols are experimental data [48] and the thick solid lines are results of QCM. Results of other models or event generators in panel (b) are taken from [48].
String/cluster fragmentation usually under-predicts the production of Ξ 0 c . Here, we show predictions of several models or event generators which adopt string/cluster fragmentation at hadronization. They are taken from Ref. [48] and are shown as different kinds of thin lines in Fig. 5(b). The decay branch Ξ 0 c → e + Ξ − ν e is taken to be 4.2%, and therefore these predictions correspond to the up limits. HERWIG7 which adopts the cluster fragmentation predicts the decreasing ratio for p T 1 GeV/c but is lower than the data about an order of the magnitude. PYTHIA8 (Monash tune) and DIPSY with rope parameter which adopt sting fragmentation predict the increasing ratio as the function of p T and are significantly lower than the data. PYTHIA8 (CR mode0) which takes the color re-connection into account by considering the string formation beyond leading color approximation will increase the prediction of ratio to a large extent but the prediction is still only one third of the data.

IV. SUMMARY AND DISCUSSION
We have shown the experimental data of p T spectra of single-charm hadrons D 0,+ , D * + D + s , Λ + c and Ξ 0 c at mid-rapidity in the low p T range (2 p T 7 GeV/c) in pp collisions at √ s = 7 TeV can be well understood by the equal-velocity combination of perturbatively-created charm quarks and the light-flavor constituent quarks and antiquarks. We emphasize the following aspects to address the physical importance of our results. (1) The property, i.e., the p T distributions, of light-flavor constituent quarks and antiquarks at hadronization are obtained from the data of p T spectra of light-flavor hadrons in work [21] where it is found that equal-velocity combination of light-flavor quarks can reasonably describe the data of light flavor hadrons in the low p T range. The existence of the underlying source of light-flavor quarks is a new property of small parton system, maybe related to the creation of the deconfined parton system in pp collisions at LHC energies. (2) The good performance for the combination of charm quarks and those constituent quarks and antiquarks in comparison with the data suggests a new scenario of the low p T charm quark hadronization in the presence of the underlying light quark source in pp collisions at LHC energies, in contrast to the usually adopted fragmentation mechanism.
(3) Most of light quarks combine into light-flavor hadrons that reproduces the data of light-flavor hadrons. A small fraction of light quarks combine with charm quarks which also well explains the data of single-charm hadrons in low p T range. This provides a possible universal picture for the production of low p T hadrons in pp collisions at LHC energies.
Several discussions on the limitation of our model and results are necessary. The present work only focuses on the characteristic of hadron production in (transverse) momentum space. It is still unclear that what kind of the spatial property for the (light-flavor dominated) small parton system leads to the effective combination of charm quarks and those light-flavor quarks. In particular, in the light-flavor sector we adopt the concept of the constituent quarks. What kind of the spatial property for the small parton system is responsible for the exhibition of the light-flavor constituent quarks degrees of freedom? Is it related to the possible de-confinement in the small system of pp collisions at LHC energies? These interesting and important questions are deserved to study in future.

ACKNOWLEDGMENTS
This work is supported by the National Natural Science Foundation of China under Grant Nos. 11575100.