(2020) Charm Quark Dynamics in Quark-Gluon Plasma with 3 + 1D

PHYSICAL REVIEW C 102, 044907 (2020)

Charm quark dynamics in quark-gluon plasma with 3 + 1D viscous hydrodynamics

Manu Kurian,1,2,* Mayank Singh ,2,† Vinod Chandra,1 Sangyong Jeon ,2 and Charles Gale2 1Indian Institute of Technology Gandhinagar, Gandhinagar-382355, Gujarat, India 2Department of Physics, McGill University, 3600 University Street, Montreal, QC, H3A 2T8, Canada

(Received 24 July 2020; revised 18 September 2020; accepted 28 September 2020; published 19 October 2020)

The drag and diffusion coefﬁcients are studied within the framework of Fokker-Planck dynamics for the case of a charm quark propagating in an expanding quark-gluon plasma. The space-time evolution of the nuclear matter created in the relativistic heavy-ion collision is modeled using MUSIC,a3+ 1D relativistic viscous hydrodynamic approach. The effect of viscous corrections to the heavy quark transport coefﬁcients is explored by considering scattering processes with thermal quarks and gluons in the medium. It is observed that the momentum diffusion of the heavy quarks is sensitive to the shear and bulk viscosity to entropy ratios. The collisional energy loss of the charm quark in the viscous quark-gluon plasma is analyzed.

DOI: 10.1103/PhysRevC.102.044907

I. INTRODUCTION at the LHC [15]. The HQs are propagating through the QGP while interacting with the constituent particles and can be The heavy-ion collision experiments pursued at the Rela- treated with Boltzmann transport. Because of their large mass tivistic Heavy Ion Collider (RHIC) at Brookhaven National as compared with the QGP temperature scale, the scattering Laboratory and at the Large Hadron Collider (LHC) at CERN of HQs is amenable to a treatment in terms of Brownian have confirmed the existence of a new state of matter: the motion [16,17]. The relativistic Boltzmann equation reduces quark-gluon plasma (QGP) [1,2]. The success of hydrody- to the Fokker-Planck equation under the constraint of soft namics in describing the space-time evolution of the QGP momentum transfer in the HQ-thermal particle interactions opened new horizons in the study of relativistic heavy-ion and has been used to describe the propagation of HQ in the collisions [3]. Early works focused on ideal hydrodynamics QGP [18–21]. The interactions of the HQs with other quarks [4], and later the dissipative effects in the QGP evolution were and gluons can be incorporated in the drag and diffusion co- incorporated and helped to explain the quantitative behav- efficients. The HQ drag force can be related to the collisional ior of experimental observables in the heavy-ion collisions energy loss in the medium in the formulation of the Fokker- [5,6]. Several studies have been done in the determination Planck equation [22]. There have been several attempts to of shear viscosity to entropy ratio η/s from the final hadron study the dynamics of HQs within the scope of Brownian data. Recently, the significance of nonzero bulk viscosity to motion and to interpret related physical observables such as entropy ratio ζ/s in the evolution of the QGP has also been nuclear suppression factor R , heavy-baryon-to-meson ra- emphasized [7]. AA tio, and elliptic flow [23–33]. However, many calculations Heavy quarks (HQs), namely, charm and bottom, serve supposed the QGP is a static and thermalized medium. In as effective probes to investigate the properties of the QGP Ref. [34], propagation of the charm quark in the equilibrating [8–10] because they are mostly created in the initial moments medium is investigated by considering a purely longitudinal of the collision via hard scattering. The thermalization time boost-invariant expansion of the system. Recently, the radia- of HQs is estimated to be on the order of 10–15 fm/c for the tive energy loss of the HQ is further studied in the longitudinal charm and 25–30 fm/c for bottom quarks created at RHIC and expansion [35]. It is therefore an interesting task to investigate the LHC [11–13]. This means that the HQs can report on the the HQ dynamics with a realistic description of the viscous QGP evolution, as the lifetime of the QGP is expected in the QGP evolution. order of 4–5 fm/c at the RHIC [14] and about 10–12 fm/c The focus of the current analysis is to investigate the HQ drag and momentum diffusion in the expanding viscous QGP, and explore the sensitivity of HQ transport coefficients and *[email protected] collisional energy loss to a nonzero viscosity to entropy ra- †[email protected] tio. This requires relativistic hydrodynamical modeling of the evolution of the medium created in the relativistic heavy-ion Published by the American Physical Society under the terms of the collision. The viscous hydrodynamic equations up to sec- Creative Commons Attribution 4.0 International license. Further ond order in flow velocity gradients are the standard input distribution of this work must maintain attribution to the author(s) to characterize the bulk medium created in the collisions and the published article’s title, journal citation, and DOI. Funded [36–38]. This investigation incorporates the viscous effects by SCOAP3. in the HQ dynamics in the QGP that enters through the

2469-9985/2020/102(4)/044907(11) 044907-1 Published by the American Physical Society KURIAN, SINGH, CHANDRA, JEON, AND GALE PHYSICAL REVIEW C 102, 044907 (2020) momentum distribution of constituent particles in the medium HQ(p) + l(q) → HQ(p ) + l(q), where l represents quarks and through the screening mechanism. A collision integral or gluons in the medium, as that takes account of the 2 → 2 elastic HQ-thermal particle collisions in the QGP medium is considered in the analysis. 1 1 d3q d3p d3q The significance of viscous coefficients of the QGP medium A = i 0 3 0 3 0 3 0 has already been discussed in dilepton emission, photon pro- γc 2P (2π ) 2Q (2π ) 2P (2π ) 2Q duction, heavy quarkonia, anisotropic flow, and other relevant 4 4 2 × (2π ) δ (P + Q − P − Q ) |M , , | observables of heavy-ion collisions at the RHIC and the LHC HQ g q [39–45]. × ± − fg,q(Q)[1 fg,q(Q )](p p )i The rest of the article is organized as follows: In Sec. II, ≡ − , a brief description of HQ drag and momentum diffusion is (p p )i (5) presented within the framework of Fokker-Planck dynamics. Section III is devoted to the details of the relativistic hydrody- and namical modeling to calculate the evolution of the background QGP, followed by the description of viscous corrections to the 1 1 d3q d3p d3q HQ transport coefficients. The results are discussed in Sec. IV B = ij 0 3 0 3 0 3 0 and, finally, we conclude in Sec. V. 2γc 2P (2π ) 2Q (2π ) 2P (2π ) 2Q 4 4 2 × (2π ) δ (P + Q − P − Q ) |MHQ,g,q| II. HEAVY QUARK DRAG AND DIFFUSION × fg,q(Q)[1 ± fg,q(Q )](p − p ) (p − p ) In the present analysis, we adopt the formalism developed i j by Svetitsky [18] to investigate the HQ dynamics in the QGP 1 ≡ (p − p) (p − p) , (6) medium. The dynamics of HQ can be described by the rela- 2 i j tivistic Boltzmann equation as ∂ f μ HQ where γ is the statistical degeneracy of the HQ and f , p ∂μ fHQ = , (1) c g q ∂t c is the momentum distribution of the thermal particles in P = E , p Q = E , q where fHQ is HQ momentum distribution function. The term the bulk medium. Here, ( p ), ( q ) denote the ∂ fHQ energy-momenta of the HQ and thermal particles in the en- ( ∂ )c denotes the collision term that quantifies the rate t = , = , of change of f due to the interactions or scattering with trance channel and P (Ep p ), Q (Eq q ) represent the HQ energy-momenta after scattering. The HQ-thermal particles thermal quarks and gluons in the medium. The relativistic → |M | collision integral for the two-body collision takes the form 2 2 scattering matrix element, HQ,g,q , can be obtained from Feynman diagrams as described in Ref. [18]. The drag ∂ fHQ 3 force and momentum diffusion respectively measure the ther- = d k[ω(p + k, k) fHQ(p + k) ∂t c mal average of the momentum transfer and its square, due to the HQ-thermal particles scattering in the QGP medium. − ω(p, k) fHQ(p)], (2) Since Ai and Bij depend only on p, they can be decomposed where ω(p, k) is the collision rate per unit momentum phase- as follows: space of the HQ with quarks and gluons that change its momentum from p to p − k. The collision integral can A = p A(p2, T ), (7) be simplified by employing the Landau approximation [46] i i which assumes small momentum transfer in the HQ-thermal pi p j 2 pi p j 2 B = δ − B (p , T ) + B (p , T ), (8) particles scattering, p k, where k = p − p . Using the ex- ij ij p2 0 p2 1 pansion of ω(p + k, k) fHQ(p + k) up to the second order of momentum transfer, with p2 =|p|2. Here, A is the HQ drag coefficient and B ω + , + ≈ ω , ij (p k k) fHQ(p k) (p k) fHQ(p) follows longitudinal-transverse decomposition where B0 and ∂ 1 ∂2 B1 denotes the independent transverse and longitudinal diffu- + k · ω f + k k ω f , ∂ [ HQ] i j ∂ ∂ [ HQ] (3) sion coefficients. The coefficients can be defined in terms of p 2 pi p j interaction amplitude as follows: the relativistic Boltzmann equation can be simplified to Fokker-Planck dynamics, 2 A =1 − p · p /p , (9) ∂ ∂ ∂ fHQ = + , = 1 2 − · 2/ 2 , Ai(p) fHQ [Bij(p) fHQ] (4) B0 4 [ p (p p ) p ] (10) ∂t ∂ pi ∂ p j = 1 · 2/ 2 − · + 2 . B1 2 [ (p p ) p 2 p p p 1 ] (11) where Ai and Bij are the drag force and momentum diffusion of the HQs in the QGP medium. Here, i, j = 1, 2, 3 denote the spatial components of the three-vectors. The HQ drag and The integrals can be simplified by solving the kinematics momentum diffusion take the following forms for the process in the center-of-momentum frame of the colliding particles

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[34], space-time evolution of the QGP is described by the viscous hydrodynamics, it is understood that the system is not exactly 1 1 ∞ q2 1 F (p ) = dq d cos χ in thermal equilibrium. To that end, one needs to obtain the π 4γ 512 c Ep 0 Eq −1 viscous corrections to the momentum distribution function of

quarks and gluons while estimating the HQ transport coeffi- + 2 − 2 2 − 2 s mc mg,q 4smc cients in the viscous medium. × f , (E ) g q q s 1 B. Shear-viscous correction 2 × d cos θc.m. |MHQ,g,q| −1 For a given HQ-thermal particle collision process, one 2π can include the viscous corrections to the local momentum βE × dφc.m.e q fg,q(Eq )F (p ), (12) distribution of the thermal particles and thereby to the screen- 0 ing Debye mass in the medium. The first step towards the 2 2 estimation of the dissipative effects in the HQ evolution in where s = (Ep + Eq ) − (p + q) , Eq = Ep + Eq − Ep , and the QGP is to include the viscous correction to the quark p can be represented in terms of p, q, θc.m., and φc.m.. Here, and gluons distribution function. We linearize the viscous mc and mg,q are the mass of charm quark and thermal mass of correction in the HQ drag and momentum diffusion in the the gluons or quarks, respectively. μν π μν π shear-stress tensor , yielding a leading-order result in +P . The distribution function takes the following form [39]: III. HYDRODYNAMICAL MODELING AND VISCOUS , = 0 + δ , , CORRECTIONS TO HEAVY QUARK DYNAMICS fg,q(Q X ) fg,q(Q) fg,q(Q X ) (16) A. Hydrodynamical evolution of the quark-gluon plasma with μ ν j j For the purpose of this study, we consider the realistic bulk δ f , (Q, X ) = πμν Q Q S (X )S (Q, T ). (17) + g q X M evolution history of a Pb Pb collision event at 2.76 TeV. To j illustrate the viscous effects on the charm quark dynamics, we use one event with the IP-Glasma initial state [47,48]. The Equation (17) is the general form of the nonequilibrium part of hydrodynamic phase is evolved by using MUSIC,a3+ 1D the distribution function. Note that the sum over the index j is hydrodynamical approach [49]. necessary only when space- and momentum-dependent terms The shear tensor π μν and bulk-viscous pressure consti- cannot be factorized directly; see the discussions in Ref. [39]. tutes the dissipative part of the energy-momentum tensor of For the parton distribution function, the functions SX and SM the QGP, respectively take the form μν μν μν 0 ± 0 δT = π − , (13) 1 fg,q(q) 1 fg,q(q) S = , S = , (18) X 2( + P) M T 2 where μν = gμν − uμuν is the projection operator orthogo- nal to the fluid velocity uμ and gμν = diag(1, −1, −1, −1) is where and P are the energy density and pressure of the metric tensor. It is established that the dynamics of the the medium. These thermodynamical quantities are related bulk QGP is sensitive to the viscous transport (both shear through the equation of state (EoS) of the QGP. Linearizing δ and bulk viscosity) of the medium [50–52]. The stress tensor in fg,q, Ai in Eq. (5) that defines the thermal average of and bulk-viscous pressure satisfy relaxation-type equations as momentum transfer becomes, follows [53–55]: (0) + shear, Ai Ai Ai (19) τ π μν + π μν = ησ μν − δ π μν θ + φ π μπ νβ π ˙ 2 ππ 7 β in leading order, where μ − τ π σ νβ + λ σ μν , ππ β π (14) 1 1 d3q d3p d3q Ashear = μν i 0 3 0 3 0 3 0 τ ˙ + =−ζθ − δ θ + λ π σ , γc 2P (2π ) 2Q (2π ) 2P (2π ) 2Q π μν (15) μ μν × π 4δ4 + − − |M |2 with θ = ∂μu as the expansion parameter and σ = (2 ) (P Q P Q ) HQ,g,q μν ∂α β μν ≡ 1 μν + μν − 1 μν αβ u where αβ ( α β β α ) αβ de- 0 0 2 3 × δ f , Q ± f Q ± f Q δ f , Q fines the traceless, symmetric, projection operator. We use the g q( ) 1 g,q( ) g,q( ) g q( ) μν = μν αβ notation X αβX in the viscous evolution equations. × (p − p )i. (20) The values of shear and bulk viscosities are fixed to match the measured transverse momentum integrated anisotropic flow The first-order correction to the distribution function is de- coefficients and the spectra of charged particles. The shear scribed in Eq. (16). Using Eq. (17), the effect of shear viscosity over entropy density is chosen as η/s = 0.13. A viscosity on HQ drag Eq. (20) can be written as follows: temperature dependent bulk viscosity profile parametrized in shear μ ν j j A = πμν P P S (X )S¯ (P, T ), (21) Ref. [56] and used in Refs. [7,39] is used in the current analy- i X M j sis. The second-order coefficients δππ, φ7, τππ, λπ, τπ , δ, μν μν λπ , τ are related to the first-order transport coefficients, where π gμν = π uμ = 0 were employed to constrain the shear and bulk viscosities, η and ζ , respectively [54]. As the coefficient multiplying π μν . Following the same prescriptions

044907-3 KURIAN, SINGH, CHANDRA, JEON, AND GALE PHYSICAL REVIEW C 102, 044907 (2020) as in Ref. [39], we obtain Viscous corrections to the distribution functions of quarks 2 + · 2 and gluons modify the gluon self-energy and hence the screen- j 1 P 2(u P) ¯ , = μν + μ ν ing mass in the medium. The bulk-viscous correction to the SM (P T ) 2 g 2 u u 2[(u · P) − P2] [(u · P) − P2] retarded gluon self-energy and Debye screening mass μ2 → μ ν 2 2 P P (u · P) μ ν μ + δμ is investigated in Ref. [57]. The Debye mass can + 3 − 3 (P u μ2 = = [(u · P)2 − P2] [(u · P)2 − P2] be defined by using the gluon self-energy, 00(q0 0, |q |−→0), and takes the following form 3 3 ν μ 1 1 d q d p + P u ) 3 γ 0 π 3 0 π 3 0 2 d q c 2P (2 ) 2Q (2 ) 2P μ = 4παsβ [2Nc fg(1 + fg) + 2Nf fq(1 − fq )], (2π )3 d3q × (2π )4δ4(P + Q − P − Q) (26) π 3 0 (2 ) 2Q 2 where αs = g /4π is the coupling constant, Nf is the number × − |M |2 μ ν j , (p p )i HQ,g,q Q Q SM (Q T ) of flavors and Nc denotes the number of colors. The viscous corrections to the screening mass can be defined from Eq. (26) × ± 0 ± 0 μ ν j , . 1 fg,q(Q ) fg,q(Q)Q Q SM (Q T ) as d3q (22) δμ2 = πα β δ + 0 4 s 2Nc fg 1 2 fg ¯ , (2π )3 The term SM (P T ) is a scalar that depends on the HQ mo- 0 mentum, and one can evaluate this scalar in the fluid rest + 2Nf δ fq 1 − 2 f . (27) shear q frame. The shear-viscous part of the HQ drag, Ai , follows the same decomposition as Eq. (7), and we can simplify the The first-order shear-viscous correction for the Debye screen- integral in the center-of-mass frame. Similarly, we can esti- ing mass can be obtained by substituting Eq. (17) into Eq. (27) and we have mate the shear-viscous correction to the momentum diffusion ∞ 2 1 of the HQ in the QGP medium using Eq. (16)inEq.(6). To 2 μ ν 1 1 q δμ = 4πα πμν Q Q dq d cos χ proceed further, the shear-viscous correction to the general s 3 2 ( + P) T 0 (2π ) −1 term F (p) needs to be done. The viscous correction to × 2 + 2 2 χ − 2 0 + 0 the integral in the center-of-mass frame can be defined as 2Nc mg 3q cos Eq fg (Eq ) 1 fg (Eq ) × + 0 + 2 + 2 2 χ − 2 shear μ ν 1 1 1 2 fg (Eq ) 2Nf mq 3q cos Eq F (p ) = πμν P P + 2 π 4γ ( P) T 512 c × 0 − 0 − 0 . fq (Eq ) 1 fq (Eq ) 1 2 fq (Eq ) (28) 1 × [ (p, T ) ± (p, T )], (23) The integral in Eq. (28) consists of gluonic and quark contri- 2 1 2 4Ep p χ 2 + butions. While performing the cos integral, the term (mg 2 2 2 where 1(p, T ) and 2(p, T ) take the forms 3q cos χ − E ) vanishes (within the integration limits) as q 2 + 2 = 2 2 mg,q q Eq for both quarks and gluonic terms. We con- ∞ 2 1 + 2 − 2 − 2 q s mc mg,q 4smc clude that shear-viscous correction of the screening mass of 1 = dq d cos χ 0 Eq −1 s the QGP is not directly affecting the HQ drag and diffusion in leading order. × 0 ± 0 2 + 2 2 χ − 2 fg,q(Eq ) 1 fg,q(Eq ) mg,q 3q cos Eq 1 2π 2 βE C. Bulk-viscous correction × d cos θc.m. |MHQ,g,q| dφc.m.e q −1 0 1. Distribution function × fg,q(Eq )F (p ), (24) In this section, we focus on the bulk-viscous correction to the HQ transport coefficients, considering the viscous correc- and tions through the distribution function and screening mass in 2 ∞ 2 1 + 2 − 2 − 2 the QGP medium. For the quantitative analysis, we utilize the q s mc mg,q 4smc 2 = dq d cos χ leading order bulk-viscous correction to the distribution func- 0 Eq −1 s tion obtained from the Chapman-Enskog expansion within 1 2π the relaxation-time approximation, and it takes the following × 0 θ |M |2 φ fg,q(Eq ) d cos c.m. HQ,g,q d c.m. form: −1 0 m2 0 0 g,q 3 δ f , (Q, X ) =−β f (Q) 1 ± f (Q) E − × ± 2 + 2 + χ g q g,q g,q q 1 fg,q(Eq ) fg,q(Eq ) mg,q [p pq cos E p2 q 2 1 (X ) 2 2 × c − , (29) − · − . s (p p )] Eq F (p ) (25) 3 (ζ/τR )

where Note that, here, p is a function of p, q, cos χ, and scattering θ φ ζ 1 2 angles in the center-of-mass frame, c.m. and c.m., respec- ≈ − 2 + P , 15 cs ( ) (30) tively. τR 3

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τ 2 2. Bulk-viscous correction to screening mass with R as the thermal relaxation time and cs as the square of the speed of sound in the medium. It is important to note The bulk-viscous correction to the distribution function of that the effect of the running of coupling is not considered quarks and gluons in the QGP medium modiﬁes the Debye δ in the above expression of fg,q. For the general expression screening mass [57], which in turn affects the collision matrix of bulk-viscous correction to the distribution function while element for the 2 → 2 HQ–gluon-quark scattering process. considering the running coupling and its reduction to the For the t-channel HQ-quark or antiquark scattering process, non-running-coupling limit, see the discussion in Ref. [39]. the matrix element takes the following form [18]: Equation (29) can be written as 2 − 2 + 2 − 2 + 2 j j 2 2 2 mc s mc u 2mct δ f , (Q, X ) = B (X )B (Q, T ), (31) |MHQ,q| = 256Nf π α , g q X M s (t − μ2 )2 j (39) where B (X ) and B (Q, T ) for partons respectively take the X M where s, u, t are Mandelstam variables. Incorporating the forms effect of leading order bulk-viscous correction to the Debye 1 screening mass in matrix element, Eq. (39) takes the following BX (X ) = , (32) 1 − 2 + P 15 3 cs ( ) form:

2 2 2 2(1) 1 m , |M¯ , | =|M , | +|M , | , (40) , = 0 ± 0 − g q . HQ q HQ q HQ q BM (Q T ) fg,q(Q) 1 fg,q(Q) Eq (33) T Eq with Employing Eqs. (5) and (31), the effect of bulk viscosity on |M |2(1) = π 2α2δμ2 HQ,q 512Nf s HQ drag can be defined as 2 2 m2 − s + m2 − u + 2m2t Abulk = B j (X )B¯ j (P, T ), (34) × c c c . (41) i X M (t − μ2 )3 j δμ2 where Here, denotes the bulk-viscous corrections to the screening mass in the QGP medium. The bulk-viscous correction ¯ j , BM (p T ) to the screening mass can be explicitly calculated from the |M |2(1) = 1 1 d3q d3p d3q Eq. (27) by employing Eq. (31). Defining HQ,q = |M |2δμ2 0 3 0 3 0 3 0 2 and following the same prescriptions as earlier, we γc 2P (2π ) 2Q (2π ) 2P (2π ) 2Q have × π 4δ4 + − − − |M |2 (2 ) (P Q P Q )(p p )i HQ,g,q 1 1 F (p)bulk(2) = B (X ) (p, T ), (42) X π 4γ 3 × j , ± 0 ± 0 j , . 512 c Ep BM (Q T ) 1 fg,q(Q ) fg,q(Q)BM (Q T ) (35) where 2 The drag coefficient A can be described from Eq. (9)by ∞ 2 1 + 2 − 2 − 2 2αs q s mc mq 4smc following the decomposition. The bulk-viscous correction to 3 = dq d cos χ πT 0 Eq −1 s the simplified integral in the center-of-mass frame takes the 1 2π following form: β × 0 θ |M |2 φ Eq fq (Eq ) d cos c.m. 2 d c.m.e fq(Eq ) B (X ) 1 −1 0 F (p)bulk = X [ (p, T ) ± (p, T )], (36) 512π 4γ E 1 2 ∞ c p × 2 , ± 0 . F (p ) r dr 2Nc/ f BM (R T ) 1 2 fg,q(Er ) where 0 (43) 2 ∞ 2 1 s + m2 − m2 − 4sm2 q c g,q c The net bulk-viscous correction to the quark contribution to 1 = dq d cos χ 0 Eq −1 s the HQ drag and diffusion can be described from Eqs. (36) 1 2π and (43). Similarly, we incorporate the effect of bulk correc- 2 ×BM (Q, T ) d cos θc.m. |MHQ,g,q| dφc.m. tions to the screening mass into the HQ-gluon processes. In −1 0 general, these corrections due to the screening mass to the βE α ×e q fg,q(Eq )F (p ), (37) HQ transport coefficients are higher order in s.Now,from Eqs. (9)–(11), we can define the viscous corrections to the HQ and transport coefficients in the medium. 2 ∞ 2 1 + 2 − 2 − 2 q s mc mg,q 4smc 2 = dq d cos χ IV. RESULTS AND DISCUSSIONS 0 Eq −1 s A. Heavy quark transport coefficients in the 1 2π × 0 θ |M |2 φ evolving quark-gluon plasma fg,q(Eq ) d cos c.m. HQ,g,q d c.m. −1 0 We initiate the discussion with the space-time evolution × BM (Q , T )F (p ). (38) of the temperature in Pb + Pb collision at 2.76 TeV, using

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15 0.7 15 0.5 0.45 10 0.6 10 0.4 5 0.5 0.35 5

0.3 /fm) 0 0.4 2

x(fm) 0 0.25 -5 0.3 x(fm) 0.2 (GeV

-5 0 -10 0.15 0.2 B Temperature (GeV) 0.1 -15 -10 2 4 6 8 10 12 14 16 τ(fm) 0.05 -15 0 2 4 6 8 10 12 14 16 FIG. 1. Temperature evolution of QGP along the y = 0axisat τ(fm) midrapidity. White arrows denote the size and direction of velocity ﬁelds. Curved lines indicate constant-temperature contours. 15 0.6

10 0.5 the viscous hydrodynamical model MUSIC. For this study, we /fm) have used one event from the 0%–5% centrality class. We 5 0.4 2 use the lattice-QCD-based EoS from the HotQCD collaboration [58,59]. Viscous effects, more speciﬁcally, terms up 0 0.3 ) (GeV x(fm) to the second-order gradient expansion, are incorporated in 0 -5 0.2 the hydrodynamical evolution. We choose y = 0, midrapidity, -B 1

to illustrate the result of our calculations. Figure 1 is the (B -10 0.1 temperature-evolution profile of our system where vectors denote the size and direction of velocity fields. As expected, the -15 0 2 4 6 8 10 12 14 16 flow is larger towards the edge of the system and grows with τ time. For the quantitative estimation of HQ drag and diffusion (fm) coefficients, we consider the mass of the charm quark to be − = . FIG. 3. Diffusion coefficients B0 (top) and B1 B0 (bottom) of mc 1 5 GeV with the effective number of degrees of free- a charm quark with momentum p = 5 GeV at different space-time = . α = . dom Nf 2 5 and the coupling constant s 0 3. The drag points. Curved lines are constant value contours of plotted quantities. coefficient of the HQ with a given momentum p = 5GeV,at any space-time (τ,x) is shown in Fig. 2. This is the drag co- μ = 2 + 2 1/2, , , efficient if a charm quark with p ((mc p ) p 0 0) and hence the motion of HQ becomes more random in the = with p 5 GeV is present at that space-time point. We ob- medium. This observation is qualitatively consistent with the serve that the drag coefficient drops as the QGP expands result of Ref. [34]. The diffusion coefficients of the HQ with in space-time. This implies that the QGP offers less resis- momentum p = 5 GeV in the expanding medium is plotted in tance to the HQ motion at low-temperature regimes. The Fig. 3 as a function of space and time. Similar to the drag HQ experience more random forces in the early stage of the coefficient, the momentum diffusion of the HQ goes down evolution of the QGP as compared with its equilibrated stage in the low-temperature regime. Furthermore, we observe that the diffusion is larger when the charm quark is moving in the same direction as the background fluid, whereas the drag is 15 0.14 larger for the charm quark moving opposite to fluid. Clearly,

10 0.12 the details of the dynamics will play an important role.

0.1 5 ) B. Effect of shear and bulk-viscous corrections

0.08 -1 0 to drag and diffusion

x(fm) 0.06 We have incorporated the shear and bulk-viscous cor- A (fm -5 0.04 rections through the momentum distribution function and screening mass of the QGP. The HQ drag and momentum -10 0.02 diffusion coefﬁcients are sensitive to the nonzero η/s, i.e., variation up to 10% for η/s = 0.13. The shear-viscous effects -15 0 2 4 6 8 10 12 14 16 to the HQ drag and diffusion coefﬁcients are studied by esti- τ(fm) mating

FIG. 2. Drag coefﬁcient of a charm quark with momentum shear shear shear shear = A B B − B p 5 GeV at different space-time points. Curved lines indicate , 0 , and 1 0 , constant-A contours. A(0) (0) (0) − (0) B0 B1 B0

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15 0.02 15 0.02 0 0.015 10 10 −0.02 0.01 −0.04 5 5

0.005 (0) −0.06 (0) /A 0 0 0 −0.08 /A x(fm) x(fm)

−0.1 bulk -0.005 shear A

-5 A −5 −0.12 -0.01 −0.14 -10 −10 -0.015 −0.16 -15 -0.02 −15 −0.18 2 4 6 8 10 12 14 16 2 4 6 8 10 12 14 16 τ(fm) τ(fm) 15 15 0.04 10 0 10 0.02

5 (0) -0.1

0 5 (0) 0 /B 0 0 0 -0.2 /B x(fm) shear x(fm) bulk 0

-5 0

-0.02 B -5 -0.3 B -10 -0.04 -10 -0.4 -15 2 4 6 8 10 12 14 16 -15 τ(fm) 2 4 6 8 10 12 14 16

) τ 15 0.1 (fm) (0) 0 15 0.05 10 0.05 ) −B (0) (0) 0 10 0 0 5 1 -B (0) −0.05 )/(B 5 -0.05 0 1

x(fm) −0.1 shear 0 -0.1 )/(B −5 0 x(fm)

−0.15 bulk −B -5 -0.15 0

−10 −0.2 -B shear 1

-10 -0.2 bulk

−15 −0.25 1 2 4 6 8 10 12 14 16 (B τ(fm) -15 -0.25 (B 2 4 6 8 10 12 14 16 τ FIG. 4. Ratio of shear correction to equilibrium value for drag (fm) coefﬁcient A (top), B (middle) and B − B (bottom). 0 1 0 FIG. 5. Ratio of bulk correction to equilibrium value for drag coefﬁcient A (top), B0 (middle) and B1 − B0 (bottom).

shear shear as shown in Fig. 4.ThetermsB1 and B0 respectively define the first-order shear-viscous correction to the longitu- significance of bulk viscosity is consistent with the recent dinal and transverse diffusion coefficients of the HQ, whereas study [7] that highlights the large effect of the temperature (0) (0) dependent ζ/s in the hadronic observables in the heavy-ion B1 and B0 denote the corresponding equilibrium values. The inclusion of shear viscosity quantitatively affects the HQ collisions. transport coefficient and the effect is more pronounced for The spatial diffusion coefficient Ds is defined in the limit the momentum diffusion of HQs in the QGP medium. We p → 0 from the fluctuation-dissipation theorem with the form T Ds = . The temperature behavior of Ds in the vis- observe that the shear-viscous effects to the drag and diffusion mcA(p→0,T ) coefficients are negligible in the very later stage of the QGP cous QGP is depicted in Fig. 6. In the current analysis, evolution. The bulk-viscous correction to the HQ transport we are only focusing on the elastic 2 → 2 scattering (per- coefficients is depicted in Fig. 5. The correction is largest turbative interactions) of HQs with thermal particles in the when ζ/s is large. The inclusion of bulk-viscous pressure medium via s, t, u channels, and the interferences terms. Our considerably modifies the HQ drag and diffusion coefficients, results are consistent with those of leading-order pQCD es- up to 30%, in the QGP medium. This observation of the timates in Ref. [60]. However, the Ds consists of two parts,

044907-7 KURIAN, SINGH, CHANDRA, JEON, AND GALE PHYSICAL REVIEW C 102, 044907 (2020)

100 15 0.7 Π/P = 0 Π /P = -0.1 10 0.6 Π/P = -0.3 +10 +5 Π /P = -0.5 5 +2 LO pQCD 0.5

s QPM 0 0.4

lattice QCD x(fm)

T)D 10

π −5 −2 0.3 (2 −10 −5 40 −10 Temperature (GeV) 0.2

36 −15 0 2 4 6 8 10 12 14 16 32 τ(fm) 150 200 250 300 350 400 1 1.8 150 200 250 300 350 400 -10 Temperature (MeV) 1.6 -5 1.4 -2 FIG. 6. Spatial diffusion coefﬁcient at the limit p → 0 as a func- +2 tion of temperature and comparison of the results with perturbative 1.2 +5 +10 QCD (pQCD) [60], lattice [61], and QPM [62] estimations. 1

0.8

0.6 the soft component and the pQCD part, in which soft com- 0.4 Momentum lost (GeV) ponent accounts for the nonperturbative effects [33]. The 0.2 nonperturbative effects to the Ds can be estimated from lat- 0 tice QCD [61] and quasiparticle model (QPM) [62] results. 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Local Temperature (GeV) The QPM incorporates the nonperturbative dynamics with a 30 temperature-dependent background field, bag constant, and −10 with the temperature-dependent quasiparticle mass. Note that 25 −5 nonperturbative effects, along with the radiation of color −2 20 +2 charges, need to be considered in the estimation of HQ ob- +5 servables such as nuclear suppression factor, flow coefficients, 15 +10 etc. This can be done by solving the Fokker-Plank equation stochastically by employing Langevin simulations, and we 10 intend to explore this aspect in the near future. The current 5 focus lies in the study of viscous effects to Ds considering 0 the perturbative interactions, and we observe that the viscous Percent of momentum lost effects are more prominent in the temperature regime near to −5 0 2 4 6 8 10 12 14 16 the transition temperature. τ(fm)

FIG. 7. Different trajectories for different initial charm quark C. Heavy quark energy loss in the expanding viscous medium momentum (top). The charm quark momentum loss as a function of the local temperature along the trajectory (middle). The charm HQs execute Brownian motion in the QGP medium and quark fractional momentum loss with proper time in the viscous may lose energy by elastic collisions with quarks and gluons. medium for these trajectories (bottom). The color of trajectories in The drag force which accounts for the resistance to the HQ the top plot corresponds to the color of momentum curves in the motion, leads to its energy loss in the QGP medium. The middle and bottom plots. The numbers in the labels correspond differential collisional energy loss of the HQ in the QGP is to the initial charm quark momentum in the x direction in GeV. related to the drag coefﬁcient as Momentum in the y and longitudinal direction are taken to be zero. dE − = A(p2, T )p, (44) dL where dL is the length traveled by the HQ in the medium the results of Ref. [63] with hard and soft collision process in the direction of x axis within the time interval dτ.The is done in Ref. [34] and the observation conﬁrms that the energy lost by the HQ goes into the medium and could be result from Eq. (44) is consistent with that of Ref. [63]. To accounted for by introducing a source term in the hydro- quantify the energy loss, we choose different initial momenta dynamic energy-momentum conservation equation. We have for the charm quark while propagating in the viscous QGP neglected such a source term in our analysis. A comparative medium. Different trajectories of motion of the charm quark study of the energy loss of the HQ from the drag force with for different initial momentum is depicted in Fig. 7 (top). The

044907-8 CHARM QUARK DYNAMICS IN QUARK-GLUON PLASMA … PHYSICAL REVIEW C 102, 044907 (2020)

5.2 The viscous corrections have small effects on the momentum 5 Equilibrium evolution of HQ in the medium. Equilibrium + Shear 4.8 Equilibrium + Shear + Bulk 4.6 V. CONCLUSION AND OUTLOOK 4.4 In this article, we have studied the HQ dynamics in the 4.2 expanding QGP medium using a realistic 3 + 1D hydrody- MUSIC 4 namical modeling— . The model describes the QGP expansion by considering the second-order evolution equa- 3.8

Momentum (GeV) tions for shear tensor and bulk-viscous pressure, along with the realistic initial conditions and lattice EoS. We have de- ((Eq. + Sh.) - Eq.)X10 scribed the HQ transport within the Fokker-Planck dynamics. 0.02 (Eq. + Sh. + Bu.) - Eq. We observe that the HQ drag and momentum diffusion coef- 0.01 ﬁcients drop in the later stage of the evolution of the medium. 0 We have conducted a systematic analysis in the shear and

Difference (GeV) 0 2 4 6 8 10 12 14 16 τ(fm) bulk-viscous corrections to the HQ transport coefficients. The viscous corrections are incorporated through the quark FIG. 8. Charm quark momentum evolution in the QGP medium and gluon phase-space distribution functions and through the (top panel). Different colors correspond to different evolution runs HQ-thermal particle scattering matrix element via screening with and without the inclusion of viscous corrections. The initial mo- mass in the analysis. The coefficients of drag and momentum mentum is taken as p = 5 GeV. Equilibrium and equilibrium + shear diffusion of the HQ in the viscous QGP are estimated and curves almost overlap as the effect of shear correction on energy compared with the HQ coefficients obtained in a fully ther- loss is negligible. Difference in charm quark momentum between malized medium. evolution runs with and without viscous corrections (bottom panel). Results showed that the effects of shear and bulk-viscous dynamics to the drag and diffusion are non-negligible and the variation ranges from to 0%–30% for different temperature regimes. These viscous corrections are essential to maintain effects of viscous coefficients will reflect on HQ transport consistency in the theoretical description of HQ dynamics in the medium and also on the space-time evolution of the in the QGP medium which is away from the equilibrium. system. Note that the shear and bulk viscous effects on the Furthermore, we have computed the collisional energy loss HQ momentum loss can be quantified in terms of the drag co- of the charm quark in the expanding medium at the LHC. efficient A A(0) + Ashear + Abulk, whereas the viscous effects The HQ drag force accounts for the energy loss due to the in the space-time evolution of the medium are incorporated charm quark collisions with thermal particles. The energy loss in the temperature evolution of the QGP. The energy loss of the HQ is reflected in the evolution of HQ momentum in of the charm quark can be demonstrated by analyzing its the viscous QGP medium. We observe that the energy loss is momentum evolution in the QGP medium. The energy loss sensitive to the initial charm quark momentum. In addition, of the HQ depends on the temperature dependence of each we have investigated the effects of shear and bulk viscosities trajectory and initial momentum of the HQ in the medium. to the charm quark momentum evolution. The viscous effects The temperature dependence of the charm quark momentum are seen to have weaker dependence on the momentum evo- loss in the medium is plotted in Fig. 7 (middle panel). The tra- lution of the charm quark in the QGP, especially in the initial jectories traverse different high- and low-temperature regions stages of the collision. A similar analysis will hold for bottom of the QGP, which results in the unusual temperature profiles quarks, and the effects will be less pronounced because of in the middle panel. We observe that the charm quark loses their larger mass. The current analysis is important for the its momentum as the QGP cools down, and the momentum understanding of dilepton signals stemming from the decay loss depends on the value of initial momentum in the medium. of open charm and bottom mesons. In particular, the energy This can be understood from Eq. (44) because the drag coef- loss of the charm (bottom) quark causes a reduction in the ficient depends on the HQ momentum and temperature of the number of high-invariant-mass dileptons from the decay of medium [18,34]. The percentage of charm quark momentum open charm (bottom) mesons. loss for different trajectories (with different initial momenta) The analysis presented in the article is the first step to- is demonstrated in Fig. 7 (bottom). It is observed that the wards the investigation of the phenomenological implications charm quark loses up to 10%–30% of the initial momentum of the HQ propagation in the viscous expanding medium with while propagating through the QGP for the duration with time 3 + 1D relativistic hydrodynamics. The viscous corrections interval up to τmax = 14 fm due to the collisions with thermal to HQ transport coefficients determined in this work could particles in the medium. The charm quark with initial mo- affect the experimental signals such as nuclear suppression mentum p = 2 GeV loses up to 30% of its momentum while factor, elliptic flow, etc. The hydrodynamic description of propagating in the viscous QGP whereas the charm quark with the pT spectra and flow of heavy baryons could be modified p = 5 GeV has 20% of momentum loss in the QGP evolution. by incorporating the realistic temperature dependence. We The viscous effects to the momentum evolution of the charm intend to work on these interesting aspects in the near future. quark, with initial momentum p = 5 GeV, is plotted in Fig. 8. Investigating the radiative energy loss that is almost the same

044907-9 KURIAN, SINGH, CHANDRA, JEON, AND GALE PHYSICAL REVIEW C 102, 044907 (2020) order of collisional energy loss at high energy scales of HQ acknowledges the hospitality of McGill University, and the (6–10 GeV), and the effects of electromagnetic ﬁelds on HQ Indian Institute of Technology, Gandhinagar (IIT GN) for transport while including the nonequilibrium corrections, are the Overseas Research Experience Fellowship to visit McGill other interesting directions to follow. University. This work was funded in part by the Natural Sci- ences and Engineering Research Council of Canada, by SERB ACKNOWLEDGMENTS for the Early Career Research Award (ECRA/2016), and by We are thankful to Sigtryggur Hauksson, Scott McDonald, the DST, Govt. of India for INSPIRE-Faculty Fellowship and Shuzhe Shi for useful discussions and suggestions. M.K. (IFA-13/PH-55).

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