Energy Loss of Heavy Quarkonia in Hot QCD Plasmas

PHYSICAL REVIEW C 103, 054907 (2021)

Energy loss of heavy quarkonia in hot QCD plasmas

Juhee Hong and Su Houng Lee Department of Physics and Institute of Physics and Applied Physics, Yonsei University, Seoul 03722, Korea

(Received 4 March 2021; accepted 3 May 2021; published 18 May 2021)

We compute the energy loss of heavy quarkonia in high-temperature QCD plasmas and investigate the energy- loss effects on quarkonium suppression. Based on the effective vertex derived from the Bethe-Salpeter amplitude for quarkonium, the collisional and radiative energy loss are determined by quarkonium-gluon elastic scattering 2 and the associated gluon-bremsstrahlung, respectively. In the energy regime E < mϒ /T the collisional energy loss is dominant over the radiative one, and the total energy loss increases with the plasma temperature and the initial energy of quarkonium. Our numerical analysis indicates that the medium-induced energy loss of the ϒ(1S) results in stronger suppression at higher momentum, although the energy-loss effects are found to be small compared with the previous estimates of quarkonium dissociation in heavy-ion collisions.

DOI: 10.1103/PhysRevC.103.054907

I. INTRODUCTION We are interested in the energy loss of a color singlet quarkonium in QCD media such as QGPs or large nuclei. The depletion of high-momentum particle production with After production at an early stage of heavy-ion collisions, respect to pp collisions signals the formation of a quark-gluon quarkonia undergo not only dissociation (and regeneration) plasma (QGP) in heavy-ion collisions. Especially, heavy but also energy loss. A quarkonium state loses its energy by quarks and quarkonia which are mostly formed from the initial elastic scattering (g + ϒ → g + ϒ) which can induce gluon fusion of partons at an early stage are important probes to radiation. Quarkonium diffusion and energy loss have been investigate the transport and thermal properties of the high- discussed with potential nonrelativistic QCD (pNRQCD) in temperature and -density matter. While quarkonia suppression the regime E T [11]. In this work, we will use a formalism can be influenced by various mechanisms including disso- b developed in our previous work [12]: for quarkonium dissociation and energy loss, the energy loss of heavy quarkonia ciation through the color-dipole interaction [13], an effective has not been computed and is not described by the heavy vertex based on the Bethe-Salpeter amplitude has been in- quark-antiquark potential. troduced to calculate the next-to-leading order cross sections Energetic particles traversing QCD plasmas suffer energy which agree with the results of pNRQCD in the regime T loss by elastic scattering or gluon-bremsstrahlung: drag and E [14]. In the current kinematic range up to q ∼ 30 GeV diffusion cause particles to lose their energies, and incoming b T [15], heavy quarkonia are not ultrarelativistic (E < m2 /T ) high-energy particles can be radiatively deprived of fractions ϒ and thus the collisional energy loss can be considerable. of their energies. There have been many studies about the We will discuss how the energy loss of heavy quarkonia energy loss of partons. The diffusion processes are dominated can be calculated using our formalism of the effective vertex, by t-channel gluon exchange with soft momentum transfer and estimate the energy-loss effects on quarkonium spectrum which requires hard-thermal-loop resummation [1,2]. In the in heavy-ion collisions. In Sec. II, we calculate the momentum presence of multiple scatterings, destructive interference oc- diffusion coefficient and the collisional energy loss of weakly curs when the formation time is large compared with the bound quarkonia at high temperature. In Sec. III, we discuss mean-free path [3–6]. The radiative energy loss is naively the radiative energy loss by gluon-bremsstrahlung associated of order g2 higher than the collisional one, but there can with quarkonium-gluon elastic scattering. Shifting the trans- be enhancement in the limit of soft and collinear emission: verse momentum spectra of the ϒ(1S) by its mean energy loss, both radiative and collisional processes of heavy quarks can we estimate the energy-loss effects on the nuclear modifica- be of the same order in the coupling constant, − dE ∼ g4T 2 dx tion factor in Sec. IV. Finally, a summary is given in Sec. V. neglecting logarithmic corrections [2,7]. The radiative energy loss is dominant over the collisional one for ultrarelativistic partons, whereas the collisional energy loss is not negligible for heavy quarks [8–10]. II. COLLISIONAL ENERGY LOSS We are interested in diffusion and energy loss of weakly bound quarkonia in a high-temperature regime where the Published by the American Physical Society under the terms of the binding energy is smaller than the temperature scale. Due Creative Commons Attribution 4.0 International license. Further to the high melting temperature Tmelt ≈ 600 MeV [16], the distribution of this work must maintain attribution to the author(s) ground state of bottomonium survives as a color singlet and and the published article’s title, journal citation, and DOI. undergoes diffusion in QGP below Tmelt.

2469-9985/2021/103(5)/054907(5) 054907-1 Published by the American Physical Society JUHEE HONG AND SU HOUNG LEE PHYSICAL REVIEW C 103, 054907 (2021)

(a) (b)

K1 K2 K1 K2 K1 K2

= +

P1 Q1 Q2 Q1 Q2 Q1 Q2 P2

FIG. 1. Elastic scattering (g + ϒ → g + ϒ) contributing to quarkonium diffusion and energy loss.

To determine the momentum diffusion coefficient of heavy The diffusion coefficient of quarkonium is suppressed by 2 2 4 3 quarkonia, we consider quarkonium-gluon elastic scattering. T /m compared with the heavy quark diffusion κHQ ∼ g T In Figs. 1(a) and 1(b), the dipole interaction of color charge [8]. In the QGP temperature region, Eq. (5)isofthesame with gluon can be described by the following effective vertex order of magnitude as the results in Ref. [11] but is much derived from the Bethe-Salpeter amplitude [17]: smaller than the momentum broadening rate for tightly bound ∂ψ ∂ψ quarkonia in Ref. [18]. μν mϒ (p) μ0 (p) μi < 2 / V (K) =−g k · δ + k δ In the energy regime E mϒ T , the collisional energy ∂ 0 ∂ i Nc p p loss per unit length is obtained by the interaction rate of 0 0 quarkonium weighted by (k − k )/v (v is the quarkonium ν 1 + γ 1 − γ 2 1 × δ j γ j T a, (1) velocity) [1,2] 2 2 where p = (p − p )/2 is the relative momentum between 3 3 1 2 −dE = 1 d q2 d k1 heavy quark and antiquark. With two effective vertices and π 3 π 3 dx 2q10 (2 ) 2q20 (2 ) 2k10 heavy quark propagators (P), the quarkonium-gluon elastic scattering in Fig. 1(a) has the following amplitude [11]: d3k × 2 n(k )[1 + n(k )](2π )4 3 1 2 4 (2π ) 2k20 μνρσ d P M (a) = (P )V μν (K )(P )V ρσ∗(K ), el π 4 1 1 2 2 (k − k ) (2 ) × δ4 + − − |M |2 2 1 , (K1 Q1 K2 Q2 ) el (6) 2 3 v g mϒ d p ∂ψ(p) ∂ψ(p) = i δabk k δμi δρk 2N 10 20 (2π )3 ∂ pi ∂ pk θ c where cos k1k2 in Eq. (3) is rewritten in a covariant form, δν jδσ l Tr[γ jγ l ] × , (2) (K · K )m2 p2 θ = − 1 2 ϒ . 2 k10 − Eb − cos k1k2 1 (7) m (K1 · Q1)(K2 · Q1) and similarly for Fig. 1(b) except the denominator has The q integration is done by the three-dimensional δ − − − p2 , 2 2( k20 Eb m ). For k10 k20 Eb, the total matrix ele- function, and the remaining phase-space integral is over ment squared (averaged over the quarkonium polarization) is , ,θ ,θ φ θ θ k1 k2 q1k1 q1k2 , and q1;k1k2 , where q1k1 ( q1k2 )isthe φ 32 polar angle between q1 and k1 (k2) and q1;k1k2 is the |M |2 = 4 2 + 2 θ el g mϒ 1 cos k1k2 azimuthal angle between the q -k and q -k planes. We in- 81 1 1 1 2 troduce a dummy variable, dωδ(ω − k + vk cos θ ) = 2 1 1 q1k1 d3 p p2 1[8] and integrate over the polar angles using the re- × + E |∇ψ|2 , (3) π 3 b δ 2 = − 2 (2 ) m maining functions. From Q2 (Q1 L) with the mo- L2 mentum transfer L, we obtain Q1 · L = . Then k1 − k2 = where θk k is the angle between k1 and k2. 2 1 2 L2 vk1 cos θq k − vk2 cos θq k − and the energy conserva- The momentum diffusion coefficient is defined by the 1 1 1 2 2q10 L2 mean-squared momentum transfer per unit time [8], tion yields δ(k2 − ω − vk2 cos θq k − ), where the last 1 2 2q10 3 3 3 L2 1 d q d k1 d k2 term − is negligible in comparison with the other terms 3κ = 2 2q10 π 3 π 3 π 3 · = θ θ + 2q10 (2 ) 2q20 (2 ) 2k10 (2 ) 2k20 of order T . Thus, with k1 k2 k1k2(cos q1k1 cos q1k2 θ θ φ 4 4 sin q1k1 sin q1k2 cos q1;k1k2 ), the energy loss is × n(k1)[1 + n(k2 )](2π ) δ (K1 + Q1 ω ω 2 2 − K − Q )|M | (k − k ) , (4) dE 1 1−v 1−v 2 2 el 1 2 − = dω dk dk 4 3 1 2 / dx 16E(2π ) v ω ω where n(k) = 1/(ek T − 1) is a thermal distribution of 1+v 1+v gluon. Using a Coulombic bound state, |∇ψ (p)|2 = 1S × dφ n(k )[1 + n(k )] 210πa7 p2/[(a p)2 + 1]6, with a2 = 1/(mE ) which is satis- q1;k1k2 1 2 0 0 0 b fied for the Coulombic binding energy, we have − (k2 k1) 2 × |Mel| , (8) 4 5 −ω −ω 128πg T q k1 k2 20 cos θq k = v , cos θq k = v κ = . (5) 1 1 k1 1 2 k2 1215m2

054907-2 ENERGY LOSS OF HEAVY QUARKONIA IN HOT QCD PLASMAS PHYSICAL REVIEW C 103, 054907 (2021)

0.5 (a) (b) T=500MeV K1 K2 K1 K2 T=400MeV R 0.4 T=300MeV R

Q1 Q2 Q1 Q2 0.3 coll FIG. 3. Gluon-bremsstrahlung associated with quarkonium- gluon elastic scattering (Fig. 1). 0.2 -dE/dx (GeV/fm) 0.1 the following light-cone coordinates [20]: rad √ m2 K = s − √ϒ , 0, 0 , 0.0 1 s 10 20 30 40 50 ⎡ ⎤ E (GeV) √ m2 r2 = ⎣ − √ϒ , T , ⎦, R z s √ 2 rT (9) ϒ s − m√ϒ FIG. 2. The collisional and radiative energy loss of the (1S) z s s in a plasma with temperature T , as a function of its initial energy. α = . m2 √ s 0 3isused. Q = √ϒ , s, 0 , L = [l+, l−, l ], 1 s T where z is the momentum fraction of the emitted gluon relative to the maximum available, and the components of the which can be performed numerically by using the binding momentum transfer L are determined by the on-shell condi- energy computed in lattice QCD [19]. 2 = + − 2 = 2 = − 2 = 2 tions, K2 (K1 L R) 0 and Q2 (Q1 L) mϒ . The effective vertex of Eq. (1) is based on the dipole In the gauge A+ = 0, the polarization of the radiated gluon interaction of color charge with gluon which is valid when the is specified by · R = 0 and + = 0, quarkonium size is smaller than the inverse energy transfer of ⎡ ⎤ the gluon. Hence, we consider the quarkonium energy loss 2 · r at a kinematic regime where the temperature and binding = ⎣ , T T , ⎦. 0 √ 2 T (10) energy scales are smaller than 1 ≈ 1–1.5 GeV. In principle, z s − m√ϒ a0 s there can be energy loss of the virtual heavy quarks in Fig. 1 [11] and energy loss coming from the color octet state prior For soft gluon emission, the radiation processes are factorized to quarkonium formation. Assuming that the lifetime of the into elastic scattering and gluon emission whose amplitude virtual heavy quarks (in the large-Nc limit) and the color octet is the emitted gluon field multiplied by an additional gluon state is proportional to the formation time of the quarkonium propagator. In the limit rT lT , the leading contribution is as which is related to the inverse of the binding energy, the square follows: of the momentum transfer of the virtual heavy quarks and the T · rT κ 4 3 M = 2gC z M , (11) initial color octet state is roughly of order HQ ∼ g T .Inthe rad 2 el Eb Eb rT regime 1 ∼ mg2 > T , E ∼ mg4 > g2T , and the momentum r b where C is the color factor associated with Fig. 3 divided by transfer is smaller than the gT scale. Since the momentum the factor in the absence of radiation (C2 = 3). The energy of scale of the heavy quarks is at least of order T , we ignore the the emitted gluon is small compared with that of the parent energy loss by the initial color octet state as well as the virtual gluon for soft radiation, but it is still assumed to be larger than heavy quark diffusion in this work. its transverse momentum. Figure 2 shows our numerical results of the ϒ(1S) energy The radiative energy loss associated with quarkonium- loss with m = 4.8GeV,mϒ = 9.46 GeV, and α = 0.3. The s gluon elastic scattering can be determined by the emitted collisional energy loss of the ϒ(1S) increases with its energy gluon spectrum, and the plasma temperature, and it is less than about half of the bottom quark energy loss [2,7,10]. d4R M 2 dn = 2πδ(R2 ) rad g 4 (2π ) Mel dz d2r z2 = T 6g2 . (12) π 3 2 III. RADIATIVE ENERGY LOSS z (2 ) rT Quarkonia undergoing quarkonium-gluon elastic scattering The mean energy loss is the average of the probability of induce gluon radiation, and the amount of the emitted gluon emitting a gluon times the gluon energy, energy is the radiative energy loss. In terms of the elastic ⎡ ⎤ √ 2 mϒ scattering defined on the left-hand side of Fig. 1,thelowest- g2 √ m2 s − √ order contribution to the energy loss is from the processes in δE = dn r = s − √ϒ ln ⎣ s ⎦, (13) g 0 π 2 Fig. 3. Gluon emission from heavy quark lines is ignored in 4 s 2mD

054907-3 JUHEE HONG AND SU HOUNG LEE PHYSICAL REVIEW C 103, 054907 (2021) where r0 is the energy of the emitted gluon. In Eq. (12), 0.08 the maximum transverse momentum of the gluon is given by T = 525-550 MeV √ 2 0 = − m√ϒ / ∼ zk1 z( s s ) 2, and the Debye screening mass mD gT has been chosen for the minimum: our results and the 0.06 following discussion are not very sensitive to variation in 2 the limits. For an estimate, we use s mϒ + 2Ek1 with a

E/E 0.04 mean thermal energy k1 ∼ 3T . The average radiative energy loss per unit length is estimated by −dE/dx ≈ δE/λ with the wavelength λ = 1/(σel ρ), where σel is the cross section ρ = 0.02 of quarkonium-gluon elastic scattering in Fig. 1 and d3k = 16ζ (3)T 3 16 (2π )3 n(k) π 2 is the gluon density. ϒ We present the (1S) radiative energy loss in Fig. 2, com- 0.00 paring with the collisional energy loss. The ϒ(1S) radiatively 10 20 30 40 50 loses more energy in a hotter medium, but the effect is much E (GeV) smaller than the collisional energy loss, which grows more E ϒ rapidly at high energy. In comparison with the radiative energy FIG. 4. The fractional energy loss E of the (1S)foraBjorken loss of a bottom quark [21], the energy loss of the ϒ(1S)is expansion with the initial temperature T0 = 525–550 MeV, the approximately an order of magnitude smaller at least. higher temperature being the upper boundary.

IV. NUCLEAR MODIFICATION FACTOR In Ref. [23] we have computed the nuclear modification ϒ In the previous two sections, we have computed the col- factor of the √(1S) by dissociation and regeneration in PbPb = . , . lisional and radiative energy loss of the ϒ(1S). Since the collisions at sNN 2 76 5 02 TeV and found that the nu- energy loss depends on the medium temperature, it changes merical results depend on initial conditions with significant with time as the plasma expands and cools down in heavy-ion uncertainties at an early stage when quarkonia formation is collisions. In this section, we use the medium-induced energy in progress. The quarkonium energy loss can also affect the loss to calculate the total energy loss during evolution, and initial spectrum at the initial time QGP is formed (as well then investigate the energy-loss effects on nuclear modifica- as during a Bjorken expansion): the energy-loss effects at the tion factors in the central rapidity region. early stage might be important. Using the same initial con- Quarkonium production has not been fully understood and ditions as Ref. [23] without the energy loss at the beginning, ϒ the theoretical prediction involves large uncertainties even Fig. 5 shows the (1S) RAA factor determined by its energy in the absence of a nuclear medium. For these reasons, loss during time evolution. Because the collisional energy ϒ we exploit quarkonium momentum spectra measured in pp loss increases with momentum as seen in Fig. 2,the (1S) collisions and then estimate the energy loss effects in AA is more suppressed at larger momentum. For higher initial collisions. The transverse momentum spectrum of the ϒ(1S) temperature, we obtain stronger suppression. in pp collisions can be parametrized as The hot-medium effects can be obtained by including both quarkonium energy loss and dissociation-regeneration. dσ pp ∝ 1 In an effective-field theory framework, dissociation and 2 / 2 + α d qT [(qT ) 1] = . α = . ( 6 05 GeV and 2 44) from the data in Ref. [15]. 1.0 Using the momentum spectrum as an initial unquenched one, the energy-loss effects can be realized approximately by a shift of the momentum spectrum by the ϒ(1S) energy loss. For 0.8 an expanding plasma undergoing a Bjorken expansion [22], / = t0 1 3 0.6 T (t ) T0( t ) , the total energy loss during the evolution is =− t f v dE ∼ . / ∼ / AA E dt , where t0 0 3fm c and t f 7fm c at R t0 dx the phase transition (as assumed in Ref. [23]). If the energy 0.4 E loss is small ( E 1, see Fig. 4), the effects on normalized transverse momentum spectra can be approximated as [24] 0.2 Y(1S) energy loss dσAA(E ) dσpp(E + E ) = . (14) 0.0 2 2 d qT d qT 0 10 20 30 Then we estimate the nuclear modification factor by the qT (GeV) energy-loss effects, FIG. 5. The energy-loss effects on the ϒ(1S) RAA factor in σ d AA (E ) an expanding plasma with the initial temperature T = 525–550 2 0 √ d qT = RAA(qT ) = . (15) MeV (as assumed in Ref. [23]) for heavy-ion collisions at sNN dσpp (E ) 2 2.76, 5.02 TeV. d qT

054907-4 ENERGY LOSS OF HEAVY QUARKONIA IN HOT QCD PLASMAS PHYSICAL REVIEW C 103, 054907 (2021) recombination have been systematically studied in interaction rate of quarkonium-gluon elastic scattering and the Refs. [25–28]. Noting that the enhancement by regeneration energy transfer. The radiative energy loss is realized through tends to grow with momentum in Ref. [23], the energy gluon-bremsstrahlung associated with the elastic scattering loss as in Fig. 5 can weaken the enhancement influence and is estimated by a convolution of the probability of emit- at high momentum to be more consistent with the data. ting a gluon and its energy. Our numerical analysis indicates As the regeneration effects are more pronounced with that the collisional energy loss is dominant over the radiative 2 softer initial distributions of quarkonia and heavy quarks, one in the energy regime E < mϒ /T and increases with the the energy-loss effects become more significant when quarkonium energy and the plasma temperature. In compari- the ϒ(1S) spectrum is softer. Although the energy loss son with the bottom quark energy loss, the energy loss of the makes a little contribution compared with the almost ϒ(1S) is smaller by a factor of two at least. momentum-independent suppression RAA ≈ 0.4 measured The energy loss of heavy quarkonia affects their transverse at the LHC [15,29], the ϒ(1S) energy loss can affect the momentum spectra and is reflected on the nuclear modifi- high-momentum spectra up to ≈15% at qT ≈ 30 GeV, cation factors, resulting in quarkonia suppression. Adopting and thus the diffusion and energy loss of heavy quarkonia, a simple shift of the momentum spectra by the energy loss together with dissociation and regeneration, need to be taken for a Bjorken expansion, we have estimated the energy-loss into account to analyze the experimental data. effects on the ϒ(1S) RAA factor. The energy loss is found to For feed-down, we can apply the calculation of quarko- provide a small effect on the ϒ(1S) suppression compared nium energy loss to the excited states of bottomonia with quarkonium dissociation, but it is necessary to con- 2 as well. Using a Coulombic bound state, |∇ψ2S (p)| = sider the hot-medium effects and to analyze the experimental 7π 7 2 2 − 1 2/ 2 + 1 8 2 a0 p [(a0 p) 2 ] [(a0 p) 4 ] , with a quarter of the data in heavy-ion collisions. Furthermore, the energy loss of binding energy of the ground state, the energy loss of the heavy quarkonia might be partially responsible for sequential ϒ(2S) is more than six times as large as the ϒ(1S) energy suppression even in small systems. As the medium-induced loss. Larger energy loss as well as thermal width of excited energy loss is sensitive to many factors including path length states might lead to sequential suppression of bottomonia. and geometrical effects in nuclear collisions, the mean energy loss is not sufficient and more systematic studies need to be V. SUMMARY undertaken. We have presented an estimate of quarkonium energy ACKNOWLEDGMENTS loss in hot QCD plasmas using an effective vertex between quarkonium and gluon. Based on our formalism derived from We would like to thank Yongsun Kim for useful dis- the Bethe-Salpeter amplitude, we have investigated the lead- cussions. This work is supported by the National Research ing effects on the transverse momentum spectra of ϒ(1S). Foundation of Korea (NRF) grant funded by the Korea gov- The collisional energy loss is obtained by convoluting the ernment (MSIT) (No. 2018R1C1B6008119).

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