Charm-strange meson production in ultra-relativistic heavy-ion collisions at the CERN-LHC energies

The nuclear modification factor ${R}_{\rm AA}$ and the elliptic flow coefficient ${v}_{\rm 2}$ of charm-strange meson $D^{+}_{s}$ is systematically studied in Pb--Pb collisions at $\sqrt{s_{\rm NN}}=5.02~{\rm TeV}$ and $2.76~{\rm TeV}$. During the modeling, the coupling strength between the injected charm quark and the incident medium constituents, is extracted from the lattice QCD calculations: $2\pi TD_{s}=7$ (\textbf{Model-A}) and $2\pi TD_{s}=1.3 + (T/T_{c})^2$ (\textbf{Model-B}). We find that, comparing ${R}_{\rm AA}(D^{+}_{s})$ with ${R}_{\rm AA}(non-strange)$, the heavy-light coalescence effect is more pronounced for the former one, resulting in an enhancement behavior in the range $2\lesssim {p}_{\rm T}\lesssim5~{\rm GeV}$. The predictions of ${R}_{\rm AA}(D^{+}_{s})$ and ${R}_{\rm AA}(non-strange)$ favor Model-A to have a better description of the measured ${p}_{\rm T}$ dependence in both energies, while their ${v}_{\rm 2}$ prefer Model-B at moderate ${p}_{\rm T}$ ($2\lesssim {p}_{\rm T}\lesssim4~{\rm GeV}$). Therefore, it is necessary to consider the temperature- and/or momentum-dependence of $2\pi TD_{s}$ to describe simultaneously ${R}_{\rm AA}(D^{+}_{s})$ and ${v}_{\rm 2}(D^{+}_{s})$ in different centrality classes in Pb--Pb collisions.


I. INTRODUCTION
Ultra-relativistic heavy-ion collisions provide the unique opportunity to produce and study the properties of the strongly-interacting matter within the extrem high temperature and energy density enviorment, where a phase transition is expected from the ordinary hadron state to its deconfined constituents, namely Quark-Gluon Plasma (QGP) [1,2].Heavy quarks (HQ) such as charm and bottom are of particular interest amongst the various probes of the QGP [3][4][5].Due to the large mass, they are mainly produced at the early stage of the collisions via the hard scattering process, and subsequently interact with the QGP constituents without affecting their mass, resulting in the negligible re-generation propagating through the medium.Meanwhile, the HQ flavour is conserved during the interation with QGP constituents, therefore, the initial produced HQ will experience the full evolution of the hot and dense medium.
While traversing the QGP medium, a heavy quark will interact with the medium constituents and thus, lose part of its initial energy via both elastic (2 → 2, collisional porcesses [6]) and inelastic scatterings (2 → 2 + X, including gluon radiation [7]), naming the collisional and radiative energy loss, respectively.The energy loss effect together with the HQ hadronization mechanisms can be investigated by measuring the nuclear modification factor of the final heavy-flavor productions such as the open charmed mesons (i.e.D mesons including D 0 , D + , D * + and D + s [8,9]), where, d 2 σ AA /dp T dy is the p T and y double-differential production cross section in nucleus-nucleus collisions, scaled by the number of binary nucleon-nucleon collisions; d 2 σ pp /dp T dy is the doubledifferential result in nucleon-nucleon collisions.The deviation of R AA from unity is sensitive to the nulcear effects, e.g. the initial (anti-)shadowing and the subsequent in-medium energy loss.In addition, the elliptic flow coefficient allows to describe the anisotropy of the transverse momentum, hence, v 2 is sensitive to the EoS and initial conditions in the low p T region, and it is also able to reflect path-length-dependence of the energy loss at high p T .Many models were developed [10][11][12][13][14][15] to study the comprehensive sets of the available measurements of nonstrange charmed meson, e.g.D 0 , D + and D * + .It was realized [16][17][18][19] that the simultaneous description of their R AA and v 2 requiring further understanding of the temperature-dependence of the coupling strength (2πT D s ) between the injected (heavy) quark and the incident medium constituent.The charm-strange meson D + s (cs) production is more interesting with respect to non-strange charmed mesons, since its valence quark content consists of charm and (anti-)strange quark, which will couple the well-know strangness enhancement [20].D + s spectra will be therefore affected by both the charm conservation and the strangeness enhancement effects in heavy-ion collisions.However, few models [21] were dedicated to investigate the D + s meson spectra, as well as its R AA and v 2 until now.
Based on the previous work, we try to adress this question by taking into account the various temperaturedependence of 2πT D s which are phenomenologically ex-arXiv:1805.05807v1[hep-ph] 15 May 2018 tracted from the lattice QCD calculation, and then investigate their effects on the observables (R AA and v 2 ), in particular for the charm-strange meson D + s at the LHC energies.Meanwhile, as pointed in Ref. [22], we will explore the propagation of theoretical uncertainties in energy-loss predictions, for instance the pp baseline calcultion and the (anti-)shadowing parametrization, in this analysis.
This paper proceeds as follows: Section II is dedicated to the introduce the general steps of our hybrid model, including the initial condistion, hydrodynamics expansion of the fireball, heavy quark Brownian motion and the subsequent hadronization processes.Section III presents the results such as the production cross section, R AA and v 2 of D + s meson in pp and Pb-Pb collisions.The comparison with available measurements are performed as well.Section IV contains the summary and conclusion.

II. METHODOLOGY
We construct a theoretical framework [23] to study the charm quark evolution in ultra-relativistic heavy-ion collisions.The general steps are outlined as follows, as well as the estimation of the theoretical uncertainties.

Initial conditions for the hydrodynamical evolution
The initial spatial distribution of heavy quark pairs is sampled according to the initial entropy density distributions.The relevent transverse profile is modeled by a Glauber-based approach [24], while the longidudinal profile is described by a data-inspired phenomenological function [23].The initial momentum distribution of heavy quark pairs is obtained via the FONLL calculations [25][26][27].Finally, the cc is generated in back-to-back before including nuclear shadowing effect [28].
The initial entropy density distributions will be taken as input of the subsequent hydrodynamical evolution, which can be described by utlizing a 3+1 dimensional relativistic viscous hydrodynamics model [29] with the start time scale τ 0 = 0.6 fm/c and the shear viscosity η/s = 1/(4π).The tuning parameters in these modules are determined by the model-to-data comparison [23].

Heavy quark diffusion
The Brownian motion of charm quark when propagating through the Quark-Gluon Plasma (QGP), is described by utilizing the Langevin Transport Equation, and it can be modified to incorporate both the collisional and radiative energy loss processes, which reads [12] with the drag force the thermal random force Note ¶ and the recoil force p Gluon indicates the momentum of the radiated gluon, which can be quantified by the pQCD Higher-Twist calculation [30].It is assumed [12] that the fluctuationdissipation relation is still validated between the drag (Eq.4) and the diffusion terms (Eq.5) in Eq. 3: where, Γ(p) and κ(p) denote the drag and the momentum diffusion coefficients, respectively, and they can be rewritten via the spatial diffusion coefficient 2πT D s [31], Note that the definition of 2πT D s is extended from zeromomentum to larger momentum region.As discussed in Ref. [23], 2πT D s can be obtained by performing a phenomenological fit analysis with the lattice QCD calculations.Two approaches are summarized as follows: 8.0 (Pb-Pb @5.02 TeV) (10) In this approach the drag coefficient behaves Γ ∝ T 2 , which is similar with the AdS/CFT or pQCD calculation [10].
where, T c denotes the critical temperature.In this approach the drag coefficient behaves a weak Tdependence, which is consistent with the results shown in Ref. [18,32].
Note ¶ Assuming a isotropic momentum-dependence of the diffusion coefficient with the post-point scheme.[33], square [34] and triangle [35].The phenomenological approaches (dashed, dotted and solid curves) are displayed as well.
Figure 1 presentes the T -dependence of 2πT D s as calculated by the lattice QCD, i.e.Banerjee (circles [33]), Kaczmarek (square [34]) and Ding (triangle [35]), as well as the results modeled via the two approaches, i.e.Model-A (dashed and dotted curves; Eq. 10) and Model-B (soild curve; Eq. 11).The corresponding results are summarized in Tab.I.It is found that most of the results obtained for the momentum diffusion coefficient κ/T 3 and HQ transport coefficient qQ /T 3 , are consistent with the other model predictions within the significant systematic uncertainties.For charm quark, the relevant thermalization time defined in zero momentum limit [31]  Based on the Model-A and Model-B approaches, the in-medium energy loss of charm quark will be larger with the latter one [23], since (1) the initial transverse momentum spectrum of charm quark is much more harder than that of medium constituent, thus, the multiple elastic scatterings among them are dominated by the drag term rather than the diffusion term; (2) a larger drag coefficient near T c with Model-B, stating a stronger interaction strentgh between the injected charm quark and the incident medium constituents, consequently, the charm quark allows to lose more its energy with Model-B approach.

Heavy quark hadronization
When the local temperature below the critical one T c = 165 GeV, the charm quark will undergo the instantaneous hadronization via a "daul" approach, including fragmentation and heavy-light coalescence mechanisms.Concerning the universal fragmentation functions, various models are adopted in this work, e.g.Lund-PYTHIA 6.4 [38], Peterson [39], Collins-Spiller [40], Braaten [41] and FONLL-style [42], which are summarized in Tab.II.Appart from the Lund-PYTHIA, the fragmentation fractions for the various hadron species are f (c [23], respectively, in the other approaches.

Name
Frag.Function Parameter Lund-PYTHIA Eq. 12.11 in Ref. [ According to the heavy-light coalescence model [43], the momentum distributions of heavy-flavor mesons (Qq) are given as where, g M is the degeneracy factor; f Q ( x Q , p Q ) and f q( x q, p q) are the phase-space distributions of heavy quark and light anti-quark, respectively.The coalescence probability for Qq combination to form the heavy-flavor meson in the n th excited state, is quantified by where, are the relative coordinate and the relative momentum, respectively, in the center-of-mass frame of Qq pair.The width parameter σ M can be written as [23] (eQ+eq where, r 2 M ≈ (0.9 fm) 2 is the mean-square charge radius of D-meson; e Q and e q are the absoluate values of the charge of heavy quark and light anti-quark, respectively; the light (anti-)quark mass takes m u = m ū = m d = md = 300 MeV and m s = m s = 475 MeV.We consider the charm-strange meson species up to their first excited states (n 1), which are listed in Tab.III. Figure 2 shows the coalescence probability obtained in central (0 − 10%) Pb-Pb collisions at √ s NN = 5.02 TeV, as a function of the charm quark transverse momentum (p T ).The results for the charm quark combined with down quark (cd), up quark (cu) and strange quark (cs) are presented as dot-dashed, dashed and long dashed curves, respectively.As shown in Eq. 14 and 16, the quark mass and its charge plays the role of the weighting factor in the heavy-light coalescence model, resulting in the difference among cd, cu and cs combinations.Moreover, this difference can also be induced at a certain amount by the thermal spectrum of u/d and s quark, which is steeper for the former one, indicating a larger probability to sample the light quark with small p T .Finally, it is found that the charm quark prefers to coalesce with u and s quarks in the range p T 3 GeV.The total results (solid curve) show a decreasing behavior with increasing p T , varying from 0.7 at p T ∼ 0 to 0.2 at p T ∼ 10 GeV, hence, the HQ with low/moderate and high p T tends to hadronize via coalescence and fragmentation mechanisms, respectively.

B. Theoretical Uncertainty
In this analysis, the total theoretical uncertainty consists of three components: FONLL predictions, nuclear shadowing and fragmentation models, which are added in quadature for the final predictions.
The initial charm quark spectra are determined by the FONLL calculations [25], as well as the corresponding central values obtained by setting µ R = µ F = µ 0 ≡ p 2 T + m 2 c , where, µ R (µ F ) is the renormalization (factorization) scale; m c denotes the heavy quark mass, and its central value is m c = 1.5 GeV.The relevant uncertainties are estimated via a conservative approach [45]: The uncertainty on nuclear shadowing is estimated according to the various nPDFs sets in EPS09NLO parametrization, which are obtained by tunning the fit parameters to reproduce the availabel measurments [28].In this work, we employ the nPDFs sets up to k = 7. See Eq. 2.12 and 2.13 in Ref. [28] for details.
Based on the different fragmentation scenarios (see Tab. II), the final observables such as the production cross section are close to each other.Therefore, we take the averaged results among them as the final one, and the maximum dispersion gives the theoretical uncertainty.

A. Production cross section in pp collsions
In Fig. 3    s /D 0 and D + s /D + , the theoretical uncertainty on FONLL calculation (∼ 10% at maximum) is dominated in the range 2 p T 4 GeV/c, while the one on fragmentation models (∼ 20% at maximum) at higher p T .It is found that, within uncertainties, the measurements can be reproduced by the corresponding model predictions.
Note ¶ D + s spectrum can be obtained via dσpp/dp T = R AA • dσ AA /dp T in pp collisions at √ s = 5.02 TeV, while the corresponding R AA and dσ AA /dp T are reported in Ref. [46].

B. Nuclear modification factor and Elliptic flow
The upper panel of Fig. 5 shows the average R AA of non-strange charmed meson (D 0 , D + and D * + ) at midrapidity (|y| < 0.5), with Model-A approach (Eq.10) in central (0 − 10%) Pb-Pb collisions at √ s NN = 5.02 TeV, which is contributed by the various hadronizaton mechanisms.It is found that the fragmentation component (long dashed curve) is dominated at p T 7 − 8 GeV/c, while the coalescence (dotted curve) is significant at 1 p T 5 GeV/c, and furthermore, the first excited states contribution (dot-dashed curve) is more pronounced in this region.The central prediction (solid curve) can describe the measurement in the range p T > 5 GeV/c.R AA of charm-strange meson (D + s ) is presented in the bottom panel of Fig. 5. Similar behavior is observed when comparing with the non-strange charmed mesons, however, the coalescence effect is more pronounced for D + s meson.It is further checked that R AA (average) and R AA (D + s ) calculated by considering alone the fragmentation mechanism, are close to each other.All the conclusions drawn above are the same as the ones found in semi-central (30 − 50%) Pb-Pb collisions at √ s NN = 5.02 TeV, as well as in Pb-Pb collisions at √ s NN = 2.76 TeV.Therefore, the future measurments of R AA (D + s ) with higher presision are more powerful to constrain the heavy-light coalescence effect at moderate p T (p T = 2 ∼ 5 GeV/c).s production with respect to the average one.As mentioned in Fig. 5, the enhancement behavior is mainly induced by the coalescence mechanism during the charm quark hadronization.Note that, for R AA (average), the uncertainty on FONLL calculations (∼ 80% at maximum) are dominated at 1 < p T < 3 GeV/c, while the one on nuclear shadowing ( 10%) and fragmentation functions (∼ 10 − 15%) are significant at higher p T .Similar behavior can be found for R AA (D + s ) at low p T , but the uncertianty on fragmentation functions (∼ 20 − 40%) are dominated at p T > 3 GeV/c.For comparison, the available measurements for the average non-strange D-meson (solid) and D + s meson (empty) are displayed as well.Within the experimental and theoretical uncertainties, the model calculations can reproduce the data for both the average and D + s meson.Similar results are found in central (0 − 10%) Pb-Pb collisions at √ s NN = 2.76 TeV.
See the bottom panel of Fig. 6 for details.
To compare the predictions based on Model-A (Eq.10) and Model-B approaches (Eq.11), the upper panel of Fig. 7 presents R AA (average) and R AA (D + s ) calcu-

IV. SUMMARY AND CONCLUDINGS
In this analysis, we aim to investigate the nuclear modification of D + s meson spectra together with its elliptic flow in untra-relativistic heavy-ion collisions.We utlize the theoretical framework built in our previous work to achieve this goal, and entend it to include further the theoretical uncertainty on initial charm quark spectra, nuclear shadowing and fragmentation model.and semi-central (30 − 50%) Pb-Pb collisions both at √ s NN = 5.02 TeV and 2.76 TeV, which is mainly induced by the heavy-light coalescence mechanism.Hence, the future measurments of R AA (D + s ) with higher presision are more powerful to constrain the heavy-light coalescence effect at moderate p T (p T = 2 ∼ 5 GeV/c).For the model-to-data comparisons, the predictions of R AA (D + s ) and R AA (non − strange) favor Model-A to reproduce well the measured p T dependence in both colliding energies, while their v 2 prefer Model-B at moderate p T (2 p T 4 GeV), suggesting a temperature and/or momentum dependent 2πT D s is needed to describe simultaneously the D-meson R AA and v 2 data.
Summary of the different approaches for 2πT Ds as a function of temperature (see Fig.1), as well as the relevant results obtained for κ/T 3 and qQ/T3 .The other model predictions are shown for comparison.

FIG. 2 .
FIG. 2. Comparison of the coalescence probability contributed by different combinations in central (0 − 10%) Pb-Pb collisions at √ sNN = 5.02 TeV: cd (dot-dashed curve), cu (dashed curve) and cs (long dashed curve).Both the ground states and the first excited states are considered.The combined resluts (solid curve) are presented as well.

5 FIG. 4 .
FIG.4.Ratios of D-meson production cross sections as a function of pT. the measurments for D + s /D 0 (solid) and D + s /D + (empty) are shown as boxes, while the relevant model predictions displayed as the bands.Experimental data taken from Ref.[47].

Figure 4
Figure 4  presents the ratios of the charm-strange meson D + s with respect to the non-strange charmed meson such as D 0 and D + , in pp collisions at √ s = 7 TeV.See the legend for details.For both D + s /D 0 and D + s /D + , the theoretical uncertainty on FONLL calculation (∼ 10% at maximum) is dominated in the range 2 p T 4 GeV/c, while the one on fragmentation models (∼ 20% at maximum) at higher p T .It is found that, within uncertainties, the measurements can be reproduced by the corresponding model predictions.
The coupling strength for charm quark 2πT D s is obtained by fitting the lattice QCD calculations: 2πT D s = const.(Model-A, i.e. no temperature dependence) and 2πT D s = 1.3 + (T /T c ) 2 (Model-B, i.e. weak temperature dependence).It is found that D + s spectra measured at mid-rapidity (|y| < 0.5) can be well described by the relevant model predictions in pp collisions both at √ s = 7 TeV and 5.02 TeV, as well as the derived particle ratios D + s /D 0 and D + s /D + .The nuclear modification factor R AA (D + s ) is systematically larger than R AA (non − strange) at intermediate p T (2 p T 5 GeV) in central (0 − 10%)

TABLE III
[44]arm-strange meson species taken into account in this analysis.Results adopted from Ref.[44].