# NLO Productions of \(\omega \) and \(K^0_{\mathrm{S}}\) with a global extraction of the jet transport parameter in heavy-ion collisions

## Abstract

In this work, we pave the way to calculate the productions of \(\omega \) and \(K^0_{\mathrm{S}}\) mesons with large \(p_T\) in p+p and A+A collisions both at RHIC and LHC. The fragmentation functions (FFs) of the \(\omega \) meson in vacuum at next-to-leading order (NLO) are obtained by evolving the NLO DGLAP evolution equations with rescaled \(\omega \) FFs at initial scale \(Q_0^2=1.5\) \(\hbox {GeV}^2\) from a broken SU(3) model, and the FFs of \(K^0_{\mathrm{S}}\) in vacuum are taken from AKK08 parametrization directly. Within the framework of the NLO pQCD improved parton model, we arrive at good descriptions of the experimental data on \(\omega \) and \(K^0_{\mathrm{S}}\) in p+p both at RHIC and LHC. With the higher-twist approach, to take into account jet quenching effect by medium-modified FFs, nuclear modification factors for \(\omega \) meson and \(K^0_{\mathrm{S}}\) meson both at RHIC and LHC are presented with different sets of jet transport coefficients \({\hat{q}}_0\). Then we make a global extraction of \({\hat{q}}_0\) both at RHIC and LHC by confronting our model calculations with all available data on six identified mesons: \(\pi ^0\), \(\eta \), \(\rho ^0\), \(\phi \), \(\omega \), and \(K^0_{\mathrm{S}}\). The minimum value of total \(\chi ^2/d.o.f\) for productions of these mesons gives the best value of \({\hat{q}}_0=0.5\mathrm ~GeV^2/fm\) for Au+Au collisions with \(\sqrt{s_{\mathrm{NN}}}=200\) GeV at RHIC, and \({\hat{q}}_0=1.2\mathrm ~GeV^2/fm\) for Pb+Pb collisions with \(\sqrt{s_{\mathrm{NN}}}=2.76\) TeV at LHC, respectively, with the QGP spacetime evolution given by the event-by-event viscous hydrodynamics model IEBE-VISHNU. With these global extracted values of \({\hat{q}}_0\), nuclear modification factors of \(\pi ^0\), \(\eta \), \(\rho ^0\), \(\phi \), \(\omega \), and \(K^0_{\mathrm{S}}\) in A+A collisions are presented, and predictions of yield ratios such as \(\omega /\pi ^0\) and \(K^0_{\mathrm{S}}/\pi ^0\) at the high-\(p_T\) regime in heavy-ion collisions both at RHIC and LHC are provided.

## 1 Introduction

The jet quenching effect describes energy dissipation of an energetic parton when it traverses through the hot and dense QCD medium, which is produced shortly after high energy nuclear collisions [1]. The single hadron production suppression at the high-\(p_T\) regime when compared to scaled p+p data is one of the primary perturbative probes to study the properties of this de-coupled quark and gluon QCD matter [2]. \(\pi ^0\) is the best measured final-state hadron, and its nuclear modification factor \(R_{\mathrm{AA}}\) as a function of transverse momentum \(p_T\) is interpreted as the consequence of the jet quenching effect and it did help us to constrain the strength of the jet–medium interaction and the properties of the QCD medium [3, 4, 5, 6, 7, 8, 9]. In 2013, the Jet Collaboration summarized different jet quenching theoretical frameworks and compared the jet transport parameter extracted by different energy loss models using the same hydro description of QCD medium [10]. Meanwhile, the \(R_{\mathrm{AA}}\) for different final-state identified hadrons have been measured, and their production suppressions as well as patterns of their yield ratios have been observed [4, 5, 11, 12, 13, 14, 15, 16]. It is of interest and it is a challenge to describe the cross sections of leading hadrons of different types and their yield ratios with each other in heavy-ion collisions (HIC) both at RHIC and LHC with a unified model of jet quenching, which should shed light on the flavor dependence of the parton energy loss and the intrinsic properties of the identified hadron productions in p+p and A+A reactions [17, 18, 19, 20, 21, 22, 23, 24].

We have achieved a better understanding of the suppression patterns of different mesons by conducting calculations and by analysis of the \(R_{\mathrm{AA}}\) and particle ratios for other identified hadron productions [25, 26, 27]. We find that understanding of the suppression pattern of the leading hadron requires one to take into account all three factors [25]: the initial hard jet spectrum, the energy loss mechanism and the parton fragmentation functions in vacuum. The flavor dependence of the energy loss will result in a decrease of the fraction of gluon fragmenting contribution in the nuclear–nuclear collision and an increase of the fraction of the quark fragmenting contribution. The productions of final-state mesons, like \(\pi ^0\), \(\rho ^0\), \(\eta \) [25, 26, 27], are dominated by the quark fragmentation contribution at the high-\(p_T\) regime in p+p collisions. The energy loss effect will only enhance the domination of the quark fragmenting contribution in A+A collisions. Therefore \(\rho ^0/\pi ^0\) and \(\eta /\pi ^0\) in p+p and A+A collisions coincide at large-\(p_T\), since they are only determined by the ratios of the FFs in vacuum. But the production of \(\phi \) meson is dominated by the gluon fragmenting contribution in p+p collision, therefore we can observe a separation of \(\phi /\pi ^0\) in p+p and A+A. It is of great interest to gain better understanding of the suppression pattern of different final-state hadrons, and furthermore to provide systematical pQCD predictions for the current experimental measurement of the identified hadrons. In this letter, we mainly investigate one of the low-mass vector mesons (\(\omega \)) and one of the pseudoscalar mesons (\(K_S^0\)), together with \(\pi ^0\), \(\eta \), \(\rho ^0\) and \(\phi \) to achieve these goals.

In this article, we investigate the production of two other mesons, \(\omega \) and \(K^0_{\mathrm{S}}\), with large \(p_T\) in A+A collisions, which has never been computed so far, to the best of our knowledge. The \(\omega \) meson is constituted of a similar valence quark of \(\pi ^0\) with larger mass 782.65 MeV and spin 1. Kaons are a group of lightest mesons, carrying strangeness components, \(K^0_{\mathrm{S}}(\frac{\mathrm{d}{\bar{\mathrm{s}}}+\mathrm{s}{\bar{\mathrm{d}}}}{2})\) is one type of kaon, consisting of \(\mathrm s\)-quark, \(\mathrm d\)-quark and their corresponding anti-quarks. The productions of these two mesons have been measured in p+p and A+A collisions both at RHIC and LHC, but a theoretical description is lacking. Within the NLO pQCD improved parton model, we calculate \(\omega \) and \(K^0_{\mathrm{S}}\) yields at the high-\(p_T\) regime in heavy-ion collisions, by employing medium-modification fragmentation functions (FFs) due to gluon radiation in the hot/dense QCD medium in the higher-twist approach of jet quenching [22, 23, 28, 29, 30], the same approach as in our calculations on the productions of \(\eta \), \(\rho ^0\) and \(\phi \) mesons in HIC [25, 26, 27].

It is noted in the previous studies that, for consistency, we implemented the Hirano hydro description [31, 32] to describe the spacetime evolution of the QGP fireball. In this work, we utilize a state-of-the-art, event-by-event (2+1)-D viscous hydrodynamics model (IEBE-VISHNU) [33] to give the spacetime evolution information of the hot and dense medium. Due to the changes of the medium description, the strength of the jet–medium interaction characterized by the jet transport parameter \({\hat{q}}_0\) should be re-extracted. Taking advantage of the systematical study of the nuclear modification factors with respect to \(p_T\) and the large amount of experimental data of \(\pi ^0\), \(\eta \), \(\rho ^0\) and \(\phi \) both at RHIC and LHC, with two more mesons \(\omega \) and \(K^0_{\mathrm{S}}\) calculated in this article, we can make a global extraction of the jet transport parameter \({\hat{q}}_0\) with all the available experimental data of these six identified mesons \(R_{\mathrm{AA}}\).

The article is organized as follows. We first present the theoretical framework of computing single hadron cross sections in p+p collision in Sect. 2, and we give the p+p baseline in investigating the in-medium modification of these productions. In Sect. 3, we discuss the inclusive hadron production in A+A collisions with the medium-modified FFs, based on the higher-twist approach of parton energy loss, and we present the nuclear modification factors \(R_{\mathrm{AA}}\) for \(\omega \) and \(K^0_{\mathrm{S}}\) both at RHIC and LHC. Section 4 shows the global extraction of the jet transport coefficient \({\hat{q}}_0\) both at RHIC and LHC by confronting our model calculations with all available data on six identified mesons: \(\pi ^0\), \(\eta \), \(\rho ^0\), \(\phi \), \(\omega \), and \(K^0_{\mathrm{S}}\). In Sect. 5, we make predictions on the identified hadron yield ratios \(\omega /\pi ^0\) and \(K^0_{\mathrm{S}}/\pi ^0\) in p+p and A+A collisions, and we compare them with experimental data if applicable. We give a brief summary in Sect. 6.

## 2 Large \(p_{\mathrm{T}}\) yield of \(\omega \) and \(K^0_{\mathrm{S}}\) meson in p+p

*a*,

*b*,

*c*,

*d*,

*e*) defined in

*V*, \(\gamma \), \(D_g\) and a few additional parameters defined for each vector meson such as the strangeness suppression factor \(\lambda \), the vector mixing angle \(\theta \), the sea suppression factor \(f_{\mathrm{sea}}^\omega \), \(f_1^u(\omega )\) and \(f_g^\omega \) have been determined in the broken SU(3) model in Refs. [39, 40], and we multiply the parameter \(a_i\) by 3 to make the best fit to the yield of the \(\omega \) meson in p+p collisions. All these parameters for fitting the initial FFs of \(\omega \) meson are listed in Table 1. To obtain a NLO parton FF for \(\omega \) at any energy scale

*Q*, we employ the numerical NLO DGLAP evolution program provided in Ref. [41] with the initial parton FFs starting scale \(Q_0^2= 1.5~\mathrm{GeV}^2\) as input.

*Q*at fixed \(z_h=0.4\) and \(z_h=0.6\) on the right. We find in the typical fraction region (\(z_h=0.4 \rightarrow 0.7\)) \(D_g^\omega > D_s^\omega \gg D_{u(d)}^\omega \), unlike the case \(D_s^\phi > D_g^\phi \gg D_{u(d)}^\phi \) in the \(\phi \) FFs [27]. Due to the fact that in parton FFs the gluon contribution exceeds the quark contribution, an overwhelming advantage of the gluon fragmenting contribution of the final-state \(\omega \) production in p+p collision is expected in the competition with the quark fragmenting contribution. In Fig. 2, we plot the NLO FFs of \(K^0_{\mathrm{S}}\) from the AKK08 parametrization in the same manner as the \(\omega \) meson in Fig. 1. We find the strange quark FF \(D_s^{K^0_{\mathrm{S}}}>D_g^{K^0_{\mathrm{S}}}\) in the typical \(z_h=0.4 \rightarrow 0.7\) region, therefore competition of the gluon and quark fragmenting contributions is expected in the \(K^0_{\mathrm{S}}\) production in p+p collisions based on the pattern discovered in the previous study of the \(\pi ^0\), \(\eta \), \(\rho ^0\) and \(\phi \) mesons.

## 3 Large \(p_{\mathrm{T}}\) yield of \(\omega \) and \(K^0_{\mathrm{S}}\) meson in A+A

*b*in A+B collisions; it can be calculated by using the Glauber model [44]. \(f_{a/A}(x_{a},\mu ^{2})\) represents the effective PDFs inside a nucleus. In our calculations, we employed EPPS16 NLO nuclear PDFs [45] to include the initial-state cold nuclear matter effects on single hadron productions [46].

*b*[3]:

The theoretical results of \(R_{\mathrm{AA}}\) at various values of \({\hat{q}}_0 =0.4 - 0.7 \mathrm ~GeV^2/fm\) for both \(\omega \) and \(K_{\mathrm{s}}\) mesons are presented in Figs. 5 and 6.

When obtaining a suitable value of \({\hat{q}}_0\) by comparing the theoretical calculations of \(R_{\mathrm{AA}}\) for \(\omega \) and \(K_{\mathrm{s}}\) with the corresponding data, there are two caveats. First, since the data of \(\omega \) and \(K^0_{\mathrm{S}}\) meson at large \(p_T\) in A+A collisions are rather limited and come with a large uncertainty, it is difficult to make a good constraint on \({\hat{q}}_0\) with these data. Second, there is the caveat that we employ the IEBE-VISHNU hydrodynamics model to describe the spacetime evolution of the fireball, which gives different information of physics quantities, such as temperature and density, from those provided by other hydro models, such as the Hirano hydro description [31, 32]. Therefore, we could not take advantage of the extracted value of \({\hat{q}}_0\) in Refs. [10, 22, 23, 25, 26, 27], where the Hirano hydro description has been utilized.

With these two cautions in mind, and realizing that with our model we are now ready to make a systematic study of six types of identified mesons such as \(\pi ^0\), \(\eta \), \(\rho ^0\), \(\phi \), \(\omega \), and \(K^0_{\mathrm{S}}\) in heavy-ion collisions, it will be of great interest to make a global extraction of the jet transport coefficient \({\hat{q}}_0\) both at RHIC and LHC by confronting our model calculations (with the IEBE-VISHNU hydro model) against all available data on six identified mesons: \(\pi ^0\), \(\eta \), \(\rho ^0\), \(\phi \), \(\omega \), and \(K^0_{\mathrm{S}}\), and then make precise calculations on the nuclear modification factors of these mesons including \(\omega \), and \(K^0_{\mathrm{S}}\), and the yield ratios of these six mesons in HIC.

## 4 Global extraction of \({\hat{q}}_0\) with \(R_{\mathrm{AA}}\) for six identified mesons

*a*. \(\sigma _i^2\) means the

*i*th systematic and statistical experimental errors. We show in the top panel of Fig. 9 the derived \(\chi ^2\) averaged by the number of the compared data points for different final-state mesons at various \({\hat{q}}_0 =0.4 \)–\( 0.7\mathrm ~GeV^2/fm\) at RHIC with \(\sqrt{s_{\mathrm{NN}}}=200\) GeV. In the bottom panel of Fig. 9 we plot the curve \(\mathrm \chi ^2/d.o.f\) as a function of \({\hat{q}}_0\) both at RHIC and LHC, where the minimum of the curve \(\mathrm \chi ^2/d.o.f\) with respect to \({\hat{q}}_0\) presents the best fit of theory with data. We then observe that at RHIC the minimum point of \(\mathrm \chi ^2/d.o.f\) of all six identified mesons gives the best value of \({\hat{q}}_0=0.5(+\,0.15/-\,0.05)\mathrm ~GeV^2/fm\). It is noted that the production of \(\pi ^0\) will carry the largest weight due to its more abundant data with relatively smaller error bars. We also derive the best value of \({\hat{q}}_0\) in the Pb+Pb collisions at LHC \(\sqrt{s_{\mathrm{NN}}}=2.76\) TeV to be \({\hat{q}}_0=1.2(+\,0.25/-\,0.15)\mathrm ~GeV^2/fm\), though the curve \(\mathrm \chi ^2/d.o.f\) at LHC is much flatter that at the RHIC. So theoretical results with \({\hat{q}}_0=1.1-1.4\mathrm ~GeV^2/fm\) should all give decent descriptions as regards the data at LHC.

It is noted that in our current model the extracted values of jet transport coefficient \({\hat{q}}_0\) both at RHIC and LHC are smaller than that by the JET Collaboration in Ref. [10] and the ones in our preceding calculations [25, 26, 27]. These differences come mainly from the different hydro models utilized in the studies. We have checked [47] that if we employ the Hirano hydro description in the calculation, the extracted values of \({\hat{q}}_0\) from our global fitting will be consistent with those in Refs. [10, 25, 26, 27], though the model in this article has the potential to give more precise extraction of the jet transport coefficients when more data of identified hadrons in A+A collisions become available in the near future.

## 5 Particle ratios of \(\omega /\pi ^0\) and \(K^0_{s}/\pi ^0\) in A+A

With the global extracted value of \({\hat{q}}_0\) discussed in Sect. 4, we are able to further investigate the particle ratio of \(\omega \) and \(K^0_{\mathrm{S}}\) both at RHIC and LHC. We first calculate \(\omega /\pi ^0\) ratio as a function of \(p_T\) and show the results in p+p and Au+Au at RHIC in the left panel of Fig. 10, where the PHENIX experimental data on the \(\omega /\pi ^0\) ratio in p+p are also illustrated. An enhancement of the ratio in A+A relative to that in p+p is found in the small-\(p_T\) region, whereas we have a small suppression in the high-\(p_T\) regime. We also predict the \(\omega /\pi ^0\) ratio as a function of \(p_T\) in p+p and Pb+Pb collisions with \(\sqrt{s_{NN}}=2.76\) TeV at LHC. In Fig. 10 we do not see the overlapping of the curves \(\omega /\pi ^0\) in p+p and A+A at the high-\(p_T\) regime, as we have observed for the ratios \(\eta /\pi ^0\) [25] and \(\rho ^0/\pi ^0\) [26] in p+p and A+A.

We also compute the \(K^0_{\mathrm{S}}/\pi ^0\) ratio as a function of \(p_T\) both at RHIC and LHC in Fig. 12. We find the curves in A+A and in p+p are approaching to each other with \(p_T\) increasing, and an obvious coincidence of these two curves is seen at LHC. We show the gluon and quark contribution fractions to the \(K^0_{\mathrm{S}}\) meson (and the \(\pi ^0\) meson) as functions of \(p_T\) in Fig. 13. We find in p+p collisions that the productions of both \(K^0_{\mathrm{S}}\) and \(\pi ^0\) at very large transverse momenta are dominated by quark fragmentation. In A+A collisions, the gluon contribution shall be further suppressed because the gluon generally loses more energy. Thus, both in p+p and A+A collisions, the ratio \(K^0_{\mathrm{S}}/\pi ^0\) should be largely determined by the ratio of quark FFs for \(K^0_{\mathrm{S}}\) \((D^{K^0_{\mathrm{S}}}_q(z_h, Q^2))\) to quark FFs for \(\pi ^0\) \((D^{\pi ^0}_q(z_h,Q^2))\) at very high \(p_T\), where these FFs vary slowly with the momentum fraction \(z_h\). This is very similar to the case of \(\eta _0/\pi ^0\) at high \(p_T\) [25]. Even though in A+A collisions the jet quenching effect can shift \(z_h\) of quark FFs, if the quark FFs have a rather weak dependence on \(z_h\) and \(p_T\), we can see that at the very high-\(p_T\) regime the curves for \(K^0_{\mathrm{S}}/\pi ^0\) in A+A and p+p come close to each other, and they even coincide at LHC.

## 6 Summary

In summary, we obtain the NLO FFs of the \(\omega \) meson in vacuum by evolving the rescaled \(\omega \) FFs from a broken SU(3) model at a starting scale \(Q^2_0=1.5~ \mathrm{GeV}^2\), and we directly employ NLO \(K^0_{\mathrm{S}}\) FFs in vacuum from the AKK08 parameterizations; the numerical simulation of productions of both \(\omega \) and \(K^0_{\mathrm{S}}\) matches well with the experimental data in p+p reactions. With the IEBE-VISHNU hydro profile of the QCD medium, we calculate the nuclear modification factors of the \(\omega \) and \(K^0_{\mathrm{S}}\) mesons as well as \(\pi ^0\), \(\eta \), \(\phi \), \(\rho ^0\) in A+A collisions both at RHIC and LHC, including the jet quenching effect in a higher-twist approach. The global extraction of the jet transport parameter \({\hat{q}}_0\) is made, with comparison of the theoretical calculation and the experimental data of all six identified mesons: \(\pi ^0\), \(\rho ^0\), \(\eta \), \(\phi \), \(\omega \), \(K^0_{\mathrm{S}}\) . Furthermore, we predict the yield ratios of \(\omega /\pi ^0\) both at RHIC and LHC, and a fairly good agreement of the theoretical results and experimental data is found at RHIC. Theoretical predictions of \(K^0_{\mathrm{S}}/\pi ^0\) ratios as functions of \(p_T\) at RHIC and LHC are also presented.

## Notes

### Acknowledgements

The research is supported by the NSFC of China with Project nos. 11435004 and 11805167, and partly supported by the Fundamental Research Funds for the Central Universities, China University of Geosciences (Wuhan) (no. 162301182691).

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