# Flavor hierarchy of jet quenching in relativistic heavy-ion collisions

###### Abstract

Relativistic heavy-ion experiments have observed similar quenching effects for mesons as compared to charged hadrons for transverse momenta larger than 6-8 GeV, which remains a mystery since heavy quarks typically lose less energies in quark-gluon plasma than light quarks and gluons. In this work, we study the energy loss and nuclear modification for heavy and light flavor jets in high-energy heavy-ion collisions using a next-to-leading-order perturbative QCD framework combined with a linear Boltzmann transport model that describes the evolution of jet partons in quark-gluon plasma. The space-time profile of the background fireball is obtained via realistic hydrodynamics simulation. Taking into account both elastic and inelastic interactions between jet partons and medium constituents and incorporating the contributions from both quark and gluon fragmentations to light and heavy flavor hadron productions, we obtain satisfactory descriptions of the experimental data on the nuclear modification factors for charged hadrons, mesons and mesons simultaneously over a wide range of transverse momenta (8-300 GeV). Our study predicts that at transverse momenta larger than 30-40 GeV, mesons will also exhibit similar suppression effects to charged hadrons and mesons, which may be tested by future precision measurements.

Introduction – Large transverse momentum () jets are hard probes of the strongly-coupled quark-gluon plasma (QGP) created in relativistic heavy-ion collisions Wang:1991xy (); Qin:2015srf (); Blaizot:2015lma (); Majumder:2010qh (); Gyulassy:2003mc (). During their propagation through the QGP medium, jet partons tend to lose energies via elastic and inelastic interactions with the medium constituents, which is usually referred to as jet quenching. Jet quenching not only leads to the yield suppression for high hadrons Khachatryan:2016odn (); Acharya:2018qsh (); Aad:2015wga (); Burke:2013yra (); Xu:2014tda (); Chien:2015vja (); Andres:2016iys (); Cao:2017hhk (); Zigic:2018ovr () and full jets Adam:2015ewa (); Aad:2014bxa (); Khachatryan:2016jfl (); Qin:2010mn (); Young:2011qx (); Dai:2012am (); Wang:2013cia (); Blaizot:2013hx (); Mehtar-Tani:2014yea (); Cao:2017qpx (); Kang:2017frl (); He:2018xjv (), but also modifies jet-related correlations Aad:2010bu (); Chatrchyan:2012gt (); Qin:2009bk (); Chen:2016vem (); Chen:2016cof (); Chen:2017zte (); Luo:2018pto (); Zhang:2018urd (); Kang:2018wrs () and the internal structures of full jets Chatrchyan:2013kwa (); Aad:2014wha (); Chang:2016gjp (); Casalderrey-Solana:2016jvj (); Tachibana:2017syd (); KunnawalkamElayavalli:2017hxo (); Brewer:2017fqy (); Chien:2016led (); Milhano:2017nzm (), as compared to proton-proton collisions. With the increase of collision energy by more than a factor of 10 from the Relativistic Heavy-Ion Collider (RHIC) to the Large Hadron Collider (LHC), we can now produce abundant jets (and hadrons) with of hundreds of GeV, which enables us to peform more and more precise jet quenching studies for heavy-ion collisions.

Heavy (charm and bottom) quarks, due to their finite masses, are expected to lose less energies in QGP than light quarks (and also gluons due to different color factors). Thus one expects heavy flavor hadrons (e.g., and mesons) would exhibit less quenching effects than light charged hadrons. There has been tremendous effort devoted to heavy quark dynamics in relativistic heavy-ion collisions Dong:2019byy (); Rapp:2018qla (); Cao:2018ews (); Uphoff:2011ad (); He:2011qa (); Young:2011ug (); Alberico:2011zy (); Nahrgang:2013saa (); Cao:2013ita (); Cao:2015hia (); Das:2015ana (); Song:2015ykw (); Cao:2016gvr (); Cao:2017crw (); Liu:2017qah (); Li:2018izm (); Ke:2018tsh (). However, experiments have observed similar quenching effects for mesons as compared to charged hadrons at 6-8 GeV Adare:2014rly (); Adamczyk:2014uip (); ALICE:2012ab (). Such result challenges our theoretical understanding of the flavor dependence of jet-medium interaction and parton energy loss, and is usually denoted as the flavor hierarchy puzzle of jet quenching. Reference Djordjevic:2013pba () tried to solve this puzzle by suggesting that different patterns in parton fragmentation functions may play important roles in the final-state hadron suppression. But the final hadron modification pattern also strongly relies on the dependence of jet suppression. Studies in Refs. Norrbin:2000zc (); Aad:2012ma (); Huang:2013vaa (); Huang:2015mva (); Cao:2015kvb (); Kang:2016ofv () indicate that gluons could also contribute to heavy flavor jet and hadron productions. However, a satisfactory solution to tackle the flavor hierarchy puzzle via a complete calculation of both heavy and light flavor jet quenching in heavy-ion collisions is still lacking.

This is the objective of our work. We build a comprehensive jet quenching framework and study the energy loss and nuclear modification for both heavy and light flavor jets in high-energy nucleus-nucleus collisions. A next-to-leading-order (NLO) perturbative QCD framework is used to calculate the productions of high jet partons and hadrons, including both quark and gluon fragmentations to light and heavy flavor hadrons. A linear Boltzmann transport (LBT) model is utilized to describe jet evolution in the QGP medium, including both elastic and inelastic interactions between jet partons and the medium constituents. A relativistic hydrodynamics model is employed to simulate the space-time profiles of the background QGP fireball. By combining all important ingredients into our state-of-the-art jet quenching model, we obtain satisfactory descriptions of the experimental data for the nuclear modifications of charged hadrons, mesons, and mesons simultaneously over the widest range of transverse momenta ( 8-300 GeV) in literature. Our study shows that, due to the mass effect, mesons typically exhibit less suppression than light charged hadrons and mesons at not-very-high . But such mass effect diminishes as increases, and at 30-40 GeV, charged hadrons, mesons and mesons all have similar quenching effects.

Jet quenching framework – We use the NLO framework developed in Refs. Jager:2002xm (); Aversa:1988vb () to calculate jet and high- hadron productions in relativistic nuclear collisions. The differential cross section for hadron production in proton-proton collisions can be expressed as follows:

(1) | |||||

In the above equation, sums over all parton flavors, and denote parton distribution functions (PDFs) for two incoming partons, is the NLO partonic scattering cross section, and represents the parton-to-hadron fragmentation function (FF). The PDFs are taken from CTEQ parameterizations Lai:1999wy (), and the FFs are taken from Ref. Kretzer:2000yf () for charged hadrons, Ref. Kneesch:2007ey () for mesons, and Ref. Kniehl:2008zza () for mesons.

For jet and high- hadron productions in relativistic heavy-ion collisions, we need to account for two different nuclear effects. The initial-state nuclear shadowing effect is taken into account by applying EPS09 parameterizations Eskola:2009uj () for nuclear PDFs. The final-state hot medium effect is the focus of our work: high-energy jet partons experience elastic and inelastic interactions with the strongly-coupled QGP before fragmenting into high hadrons. The hot medium effect is incorporated in our study by using the LBT approach developed in Refs. Cao:2017hhk (); Cao:2016gvr (); He:2015pra ().

In the LBT model, the evolution of jet partons in the QGP medium is simulated according to the following Boltzmann equation:

(2) |

where and represent the collision integrals of elastic and inelastic processes experienced by the parton .

For elastic scatterings between jet partons and medium constituents, we take leading-order perturbative QCD matrix elements to calculate the elastic scattering rate and the elastic scattering probability for a given time step , where is the parton density in the QGP medium and is the parton scattering cross section.

For inelastic radiative process, we use higher-twist energy loss formalism, in which the medium-induced gluon radiation spectrum takes the following form Wang:2001ifa (); Zhang:2003wk (); Majumder:2009ge (),

(3) |

where is mass of the propagating parton, and are the momentum fraction and transverse momentum carried by the radiated gluon with respect to the parent parton, is the strong coupling for the splitting vertex, is the splitting function, is the transport coefficient (the transverse momentum transfer squared per mean free path) due to elastic scatterings between the propagating parton and medium constituents, is the time of the last gluon radiation, and is the gluon formation time, with being the energy of the parent parton. Here we take light partons to be massless, and for heavy quarks, we use GeV and GeV. The above gluon radiation spectrum for the propagating parton is used to calculate the inelastic scattering rate with the medium constituents, the average number of emitted gluons , and the inelastic scattering probability , in a given time step .

In the LBT model, the total scattering probability is splitted into two parts, the probability for pure elastic scatterings and the probability for inelastic scatterings with at least one gluon emission . These probabilities combined with the information about jet partons and medium profiles are used in our Monte-Carlo model to simulate the evolution of jet partons in the QGP medium. More details on the LBT model can be found in Refs. Cao:2017hhk (); Cao:2016gvr (); He:2015pra ().

Numerical results – We first show, in Fig. 1, the transverse momentum spectra for light charged hadrons and mesons in proton-proton collisions at 5.02 TeV based on the NLO perturbative QCD calculation Jager:2002xm (); Aversa:1988vb (), compared to the CMS data Khachatryan:2016odn (); Sirunyan:2017xss (). The factorization scale and the renormalization scales are all taken as the jet parton in the calculation. One can see that the NLO perturbative QCD calculation can provide a very good description of both charged hadron and meson spectra (at relatively high ). In the figure, we also show the relative contributions from quark and gluon fragmentations to charged hadron and meson productions. For charged hadrons, gluon contribution is more dominant at low , and quark contribution becomes more important at GeV. For mesons, charm quark fragmetation and gluon fragmentation contribute almost equally to the meson yield at low . Then with increasing of mesons, the gluon contribution decreases, but it still renders around 40% contribution to the meson yield at GeV. Note that the NLO perturbative QCD framework adopted here uses the zero-mass factorization scheme, thus is not valid for very small .

In Fig. 2, we show the nuclear modification factor as a function of for charged hadrons in central 0-10% Pb+Pb collisions at 5.02A TeV at the LHC, compared to the CMS 0-10% and ALICE 0-5% data Khachatryan:2016odn (); Acharya:2018qsh (). In this study, the space-time evolution of the QGP fireball in 5.02A TeV Pb+Pb collisions is obtained via a (3+1)-dimensional viscous hydrodynamics model CLVisc Pang:2012he (); Pang:2018zzo () in which fm, and MeV are employed to describe the soft hadron spectra. In the LBT model, the strong coupling for the interaction vertex connecting to thermal partons is taken as . For the vertices connecting to jet partons, we take the running coupling as: , with GeV. Note that the default version of the LBT model only considers leading-order elastic scattering processes, thus the distribution for the exchanged transverse momentum between jet partons and medium constituents typically has a hard power-law tail. To account for the possible contributions from multiple soft scatterings whose transverse momentum distribution is typically a Gaussian, we impose an effective momentum cutoff for transverse momentum exchange between jet and medium (). Such setup reduces the energy loss of jet partons and also weakens the energy dependence of parton energy loss. More detailed study on the interplay between single hard and multiple soft scatterings and their influences on final-state observables will be explored in the future effort. In the figure, we also show ’s for charged hadrons produced from light quarks and gluons, respectively. One can see that due to the color effect, quark-initiated hadrons exhibit less quenching effects than gluon-initiated hadrons. After combining both quark and gluon fragmentations to charged hadrons, our model gives a nice description of charged hadron over a wide range of transverse momenta (- GeV).

Figure 3 shows the nuclear modification factor as a function of for mesons in central 0-10% Pb+Pb collisions at 5.02A TeV at the LHC, compared to the CMS and ALICE data Sirunyan:2017xss (); Acharya:2018hre (). In the figure, we also show ’s for mesons produced from charm quarks and gluons, respectively. Similar to charged hadrons, we can see that mesons produced from charm quark fragmentation have less quenching than mesons from gluon fragmentation. Again, after combining both charm quark and gluon contributions to meson production, we obtain successful description of meson data from CMS for 8-100 GeV.

In our study, both elastic scattering and inelastic radiative processes are included in the LBT simulation. The relative contributions from collisional and radiative energy loss components to the nuclear modifications of mesons are shown in Fig. 4 for central 0-10% Pb+Pb collisions at 5.02A TeV at the LHC. One can see that while radiative energy loss provides more dominant contributions to the nuclear modification factor in the range explored here, collisional energy loss also gives sizable contributions to at not-very-high regime and such contribution diminishes with increasing .

The above results clearly show that our calculation can simultaneously describe both meson and light charged hadron ’s for central 0-10% Pb+Pb collisions at 5.02A TeV at the LHC. It is very interesting to see whether our model can describe meson suppression as well since beauty quarks have much larger mass. In Fig. 5, we show the nuclear modification factor as a function of for mesons together with ’s for charged hadrons and mesons, for 0-80% Pb+Pb collisions at 5.02A TeV at the LHC. Also shown are the CMS minimum bias data Khachatryan:2016odn (); Sirunyan:2017xss (); Sirunyan:2017oug () for comparison. It is worth noting that we compute for 0-80% as follows: , where is the probability of finding jet events in a given centrality bin. If one uses an average medium profile via averaging the hydrodynamics profiles or initial conditions over different centralities, much less jet quenching effects would be obtained for the minimum bias calculation. From the figure, one can see that our model can simultaneously describe the nuclear modifications of charged hadrons, mesons and mesons. Below 30-40 GeV, mesons exhibit less quenching than charged hadrons and mesons, while above 30-40 GeV, our model predicts similar quenching for mesons to charged hadrons and mesons. Future high luminosity LHC experiments should be able to test our result.

Summary – In this work, we have built a comprehensive jet quenching framework to study the energy loss and nuclear modification for heavy and light flavor jets in high-energy heavy-ion collisions. Our state-of-the-art jet quenching model combines a NLO perturbative QCD framework to calculate the productions of high transverse momentum jet partons and hadrons, a linear Boltzmann transport model to simulate the evolution of heavy and light flavor jets in the QGP, and a realistic hydrodynamic model to describe the space-time evolution of the QGP fireball. It includes both quark and gluon contributions to light and heavy flavor hadron productions and incorporates both elastic and inelastic interactions between jet partons and the medium constituents. With all important ingredients implemented in our jet quenching model, we obtain satisfactory descriptions of the experimental data for the nuclear modification factors of charged hadrons, mesons and mesons over the widest range of transverse momenta ( 8-300 GeV) in literature. Our work provides a natural solution to the flavor hierarchy puzzle of jet quenching in relativistic heavy-ion collisions. Based on our jet quenching model, we predict that at transverse momenta 30-40 GeV, mesons will also exhibit similar suppression effects to charged hadrons and mesons, which can be tested by future high luminosity precision measurements.

Acknowledgments – We thank Z. Kang for helpful discussions. This work is supported in part by Natural Science Foundation of China (NSFC) under grant Nos. 11775095, 11890711 and 11375072. S. C. is supported by U.S. Department of Energy under Contract No. DE-SC0013460. H. X. is supported by NSFC under grant No. 11435004.

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