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Nuclear and Particle Physics Proceedings 289–290 (2017) 217–220 www.elsevier.com/locate/nppp
RAA vs. v2 of heavy and light hadrons within a linear Boltzmann transport model Shanshan Caoa,b , Long-Gang Pangc , Tan Luod , Yayun Hed,a , Guang-You Qind , Xin-Nian Wangd,a a
Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA of Physics and Astronomy, Wayne State University, 666 W. Hancock St., Detroit, MI 48201, USA c Frankfurt Institute for Advanced Studies, Ruth-Moufang-Strasse 1, 60438 Frankfurt am Main, Germany d Institute of Particle Physics and Key Laboratory of Quark and Lepton Physics (MOE), Central China Normal University, Wuhan, 430079, China b Department
Abstract We establish a linear Boltzmann transport (LBT) model coupled to hydrodynamical background to study hard parton evolution in heavy-ion collisions. Both elastic and inelastic scatterings are included in our calculations; and heavy and light flavor partons are treated on the same footing. Within this LBT model, we provide good descriptions of heavy and light hadron suppression from RHIC to the LHC energies. Possible solutions to the RAA vs. v2 puzzle are discussed. Keywords: relativistic nuclear collisions, jet energy loss, linear Boltzmann transport 1. Introduction Heavy quarks and high pT light partons are produced at the primordial stage via hard scatterings and thus serve as valuable probes of the full evolution history of the color deconfined quark-gluon plasma (QGP) matter created in relativistic heavy-ion collisions. In the past decade, sophisticated studies have been implemented in understanding the parton-medium interaction via various jet quenching observables. However, a fully consistent picture has not been achieved about the dynamical evolution and nuclear modification of partons and hadrons of different flavors in different collision systems and energies. For instance, the comparable nuclear modification factors of D mesons and light flavor hadrons at high pT [1, 2, 3] seem contradictory to the conventional expectation from the color and mass dependences of parton energy loss. In addition, a great challenge still remains in the simultaneous description of the suppression (RAA ) and elliptic flow (v2 ) for both heavy and light flavor hadrons. We establish a linear Boltzmann transport (LBT) model coupled to hydrodynamic background to describe the temporal evolution of hard partons inside QGP. Both elastic and inelastic scatterings are incorporate in our https://doi.org/10.1016/j.nuclphysbps.2017.05.048 2405-6014/© 2017 Elsevier B.V. All rights reserved.
LBT model; and the energy loss of heavy and light flavor partons are treated on the same footing. Within this newly developed approach, a simultaneous description of heavy and light flavor hadron RAA is naturally obtained for various collision systems at RHIC and the LHC. Possible solutions to the RAA vs. v2 puzzle are explored. 2. A linear Boltzmann transport model We follow our previous work [4, 5] and describe the scatterings of energetic partons inside a thermal medium using the Boltzmann equation, in which the phase space evolution of an incoming parton “1” is written as p1 · ∂ f1 (x1 , p1 ) = E1 (Cel + Cinel ),
(1)
in which Cel and Cinel are collision integrals for elastic and inelastic scatterings. For the elastic scattering process, the collision term Cel is evaluated with the leading-order matrix elements for all possible 12 → 34 processes between the given jet parton “1” and a thermal parton “2” excited from the medium background. To regulate the collinear (u, t → 0) divergence of the matrix element, S 2 (s, t, u) =
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Figure 1: (Color online) RAA of D and π from RHIC to the LHC energies.
θ(s ≥ 2μ2D )θ(−s + μ2D ≤ t ≤ −μ2D ) is imposed in which μ2D = g2 T 2 (Nc + N f /2)/3 is the Debye screening mass. In our work, light partons are assumed massless, and mc =1.27 GeV and mb =4.19 GeV are used for charm and beauty quarks respectively. With this setup, one can obtain the scattering rate of parton “1” as follows: γ2 d 3 p2 d 3 p3 d 3 p4 3 3 2E1 (2π) 2E2 (2π) 2E3 (2π)32E4 2,3,4 × f2 (p2 ) 1 ± f3 (p3 ) 1 ± f4 (p4 ) S 2 (s, t, u)
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× (2π)4 δ(4) (p1 + p2 − p3 − p4 )|M12→34 |2 .
(2)
With this rate, the probability of elastic scattering for parton “1” in each time step Δt is then Pel = Γel Δt. The jet transport coefficients qˆ may be obtained from the above integral weighted by the transverse momentum broadening of parton “1”. For the inelastic process, the average number of emitted gluons from a hard parton in each time step Δt is evaluated as [6, 7, 5] dNg Ng (E, T, t, Δt) = Δt dxdk⊥2 , (3) dxdk⊥2 dt in which the radiated gluon spectrum is taken from the higher-twist energy loss formalism [8, 9, 10]: 4 dNg 2α sC A qP(x)k ˆ ⊥ 2 t − ti (4) = sin 2τ . dxdk⊥2 dt π k2 + x2 m2 4 f ⊥ x and k⊥ are the fractional energy and transverse momentum of the emitted gluon with respect to its parent parton, α s is the strong coupling constant, P(x) is the splitting function, and qˆ is the transport coefficient due to elastic scattering. Eq. (4) contains the mass dependence for heavy quark radiative energy loss. ti
denotes an “initial time” or the production time of the “parent” parton from which the gluon is radiated, and τ f = 2Ex(1 − x)/(k⊥2 + x2 m2 ) is the formation time of the radiation. A lower energy cut-off for the emitted gluon xmin = μD /E is implemented to regulate the divergence as x → 0. Multiple gluon radiation is allowed in each time step, and the number n of radiated gluons is assumed to obey a Poisson distribution with the mean as Ng . Thus, the probability for such inelastic scattering process is Pinel = 1 − e−Ng . Note for g → gg process, Ng /2 is taken as the mean to avoid double counting. Given the probabilities of elastic and inelastic scattering processes, the total probability is then Ptot = Pel + Pinel − Pel Pinel , which may be divided into two regions: pure elastic scattering with probability Pel (1 − Pinel ) and inelastic scattering with probability Pinel . Based on these probabilities, a Monte-Carlo (MC) method is applied to determine whether a given jet parton is scattered with the medium constituents at each time step, and whether the scattering is pure elastic or inelastic. With a selected channel, the energy and momentum of the outgoing partons are then sampled based on the differential spectra given by Eq. (2) and (4). To study parton evolution in heavy-ion collisions, we couple this LBT model to hydrodynamical simulation of QGP medium [11, 12, 13] that provides the local temperature and flow velocity of the QGP fireball. In each time step, we first boost each jet parton into the local rest frame in which its scattering with the medium is simulated based on our LBT model, and then we boost outgoing partons back to the global computation frame in which they propagate to the next time step. On the freeze-out hypersurface of the fireball (T c = 165 MeV), the hadronization of heavy quarks is described using a hybrid model of fragmentation plus coalescence following our earlier work [5], while for high pT light hadron production from light
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Figure 2: (Color online) Effect of the temperature dependence of the transport coefficient on (a) RAA vs. (b) v2 of D mesons.
Figure 3: (Color online) Effect of different initial conditions on (a) RAA vs. (b) v2 of pions.
partons, we use Pythia fragmentation.
heavy meson data. Here we use K p = 1 + A p e−|p| /2σ p with A p = 5 and σ p = 5 GeV following our previous study [5].
2
One crucial quantity controlling jet-medium interaction is the strong coupling constant α s . Considering the fact that the variation of the medium temperature is small compared to that of the jet energy, we use a fixed coupling α s = 0.15 for the interaction vertices connecting directly to thermal partons. But for the vertices connecting to hard partons, we take the running coupling as: α s (Q2 ) = 4π/[(11 − 2n f /3)ln(Q2 /Λ2 )], with Q2 = 2ET and Λ = 0.2 GeV. With such setup, we now present in Fig. 1 the nuclear modification factors RAA for D and π in 200 AGeV Au-Au collisions at RHIC, 2.76 ATeV and 5.02 ATeV Pb-Pb collisions at the LHC: upper panels show the RAA of D mesons, and lower panels show the RAA of π in which three curves are presented separately – upper for π from quark jet, lower for π from gluon jet and middle for the mixture. Note that since all heavy quarks are produced from initial hard scatterings, their full pT spectra of RAA can be obtained. On the other hand, reliable calculation for light hadron is only available at high pT at this moment. We also note that since our transport coefficients are obtained from LO pQCD calculation, we apply a K factor for jet transport coefficients at low momentum in order to describe the low pT
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3. Solutions towards the RAA vs. v2 puzzle While our LBT model naturally provides good descriptions of the nuclear modification factors for both heavy and light flavor hadrons in different centralities in different collision systems, it still remains a challenge to simultaneously describe their RAA and v2 . In this section, we investigate 3 directions which may affect hadron v2 while its RAA is fixed. First and foremost, we study the effect of different temperature dependence of quark/gluon transport coefficient on hadron v2 [5]. To model possible nonperturbative enhancement of the strong coupling constant near the critical temperature T c , we rescale the fixed coupling constant α s0 with a factor that peaks around T c but returns to unity at large T :
2 2 (5) α s (T ) = α s0 1 + AT e−(T −T c ) /2σT . As shown in Fig. 2(a), by adjusting α s , different temperature dependences of the transport coefficient can
S. Cao et al. / Nuclear and Particle Physics Proceedings 289–290 (2017) 217–220
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rates their elastic and inelastic scatterings inside QGP. Within this LBT framework, we have provided good descriptions of both heavy and light hadron suppression at RHIC and the LHC. Several possible solutions to the “RAA vs. v2 puzzle” have been discussed, including near T c enhancement of the transport coefficient, different bulk evolution profiles and the initial state fluctuation of the bulk matter, but more systematic investigation should be implemented in our upcoming study.
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Figure 4: (Color online) Effect of initial state fluctuation on pion v2 .
provide similar RAA of D mesons. However, as shown in Fig. 2(b), while their RAA is fixed, the stronger α s is around T c , the larger v2 one obtains. The physical picture is that if the interaction around T c is stronger, larger part of heavy quark energy loss will be shifted towards the freeze-out hypersurface of the QGP where the anisotropic flow of the bulk matter is stronger, and therefore heavy quarks pick up a larger v2 from the medium. This is consistent with the findings presented in Refs. [14, 15]. Next we investigate how different bulk evolutions affect the v2 of hard probes in Fig. 3. Two different initial conditions – MC-Glauber and “pseudo” IP-Glasma mimicked by the Trento [16] model are compared. One observes that while these two bulk profiles provide the same RAA of high pT pion, they lead to non-negligible difference in its v2 between 10-20 GeV. In our earlier work, we showed that KLN initialization would lead to even larger v2 due to its larger eccentricity [17]. Last but not least, it was recently pointed out that the initial state fluctuation of the bulk evolution is also important in understanding the v2 of hard probes [18]: the v2 of hard probes that experimentalists measure is actually the average v2 theorists calculate without considsoft 2 ering the bulk fluctuation times [1 + (δvsoft 2 /v2 ) ], in soft soft which δv2 /v2 quantifies the bulk fluctuation. However, as shown in Fig. 4, only 10% larger high pT pion v2 is observed within our LBT model after the inclusion of the bulk fluctuation. This is consistent with the 10% soft value of the soft fluctuation δvsoft 2 /v2 extracted from our CLVisc hydro [12] with the Trento initialization. 4. Summary To conclude, we have established a linear Boltzmann transport model that treats heavy and light parton evolution on the same footing and simultaneously incorpo-
Acknowlegements This work is funded by the U.S. Department of Energy under Contract No. DE-AC02-05CH11231 and DE-SC0013460, by the Natural Science Foundation of China (NSFC) under Grant No. 11221504 and No. 11375072, by the Chinese Ministry of Science and Technology under Grant No. 2014DFG02050 and by the Major State Basic Research Development Program in China (No. 2014CB845404). References [1] PHENIX Collaboration, A. Adare et al., Phys. Rev. C84, 044905 (2011), arXiv:1005.1627. [2] STAR Collaboration, L. Adamczyk et al., Phys. Rev. Lett. 113, 142301 (2014), arXiv:1404.6185. [3] ALICE Collaboration, B. Abelev et al., JHEP 1209, 112 (2012), arXiv:1203.2160. [4] X.-N. Wang and Y. Zhu, Phys. Rev. Lett. 111, 062301 (2013), arXiv:1302.5874. [5] S. Cao, T. Luo, G.-Y. Qin, and X.-N. Wang, Phys. Rev. C94, 014909 (2016), arXiv:1605.06447. [6] S. Cao, G.-Y. Qin, and S. A. Bass, Phys. Rev. C88, 044907 (2013), arXiv:1308.0617. [7] S. Cao, G.-Y. Qin, and S. A. Bass, Phys. Rev. C92, 024907 (2015), arXiv:1505.01413. [8] X.-F. Guo and X.-N. Wang, Phys. Rev. Lett. 85, 3591 (2000), arXiv:hep-ph/0005044. [9] A. Majumder, Phys. Rev. D85, 014023 (2012), arXiv:0912.2987. [10] B.-W. Zhang, E. Wang, and X.-N. Wang, Phys. Rev. Lett. 93, 072301 (2004), arXiv:nucl-th/0309040. [11] Z. Qiu, C. Shen, and U. Heinz, Phys. Lett. B707, 151 (2012), arXiv:1110.3033. [12] L. Pang, Q. Wang, and X.-N. Wang, Phys. Rev. C86, 024911 (2012), arXiv:1205.5019. [13] L.-G. Pang, Y. Hatta, X.-N. Wang, and B.-W. Xiao, Phys. Rev. D91, 074027 (2015), arXiv:1411.7767. [14] S. K. Das, F. Scardina, S. Plumari, and V. Greco, Phys. Lett. B747, 260 (2015), arXiv:1502.03757. [15] J. Xu, J. Liao, and M. Gyulassy, Chin. Phys. Lett. 32, 9 (2015), arXiv:1411.3673. [16] J. S. Moreland, J. E. Bernhard, and S. A. Bass, Phys. Rev. C92, 011901 (2015), arXiv:1412.4708. [17] S. Cao, G.-Y. Qin, and S. A. Bass, J. Phys. G40, 085103 (2013), arXiv:1205.2396. [18] J. Noronha-Hostler, B. Betz, J. Noronha, and M. Gyulassy, Phys. Rev. Lett. 116, 252301 (2016), arXiv:1602.03788.
