# Energy loss of heavy quarks in the isotropic collisional hot QCD medium

## Abstract

The collisional energy loss of heavy partons (charm and bottom quarks) has been determined within the framework of semi-classical transport theory implying the Bhatnagar–Gross–Krook (BGK) collisional kernel. Hot QCD medium effects have been incorporated while employing a quasi-particle description of the medium in terms of effective gluons, quarks and antiquarks with respective temperature dependent effective fugacities. The momentum dependence of the energy loss for the charm and the bottom quark has been investigated. It is observed that with the increase in momentum of the heavy quarks, the loss increases sharply for the smaller values and reaches saturation later. Furthermore, as compared to the charm quark, the bottom quark loses less energy at a particular momentum and collisional frequency. The energy loss is seen to increase with increasing collisional frequency. We also provide a comparative study of the results obtained using the BGK kernel rather than those using the relaxation time approximation (RTA) kernel and found them to be consistent with each other. The medium effects in all the situations are seen to play a quite significant role.

## 1 Introduction

The quark–gluon plasma (QGP) produced in the Relativistic Heavy-Ion Collider (RHIC) at Brookhaven National Laboratory and Large Hadron Collider (LHC) at CERN provides an opportunity to study the universe at the age of a few microseconds and the different phases of quantum chromodynamics (QCD). The observed QGP is seen to behave more like a near-perfect fluid (a tiny value of \(\eta /s\)) [1, 2, 3, 4, 5, 6, 7, 8]. There have been proposed several indirect probes for the QGP in heavy-ion collisions. Among them, collective flow, jet quenching, quarkonia suppression and the suppression of high \(p_T\) hadrons are the most reliable ones indicating the creation of a QGP. The suppression of high \(p_T\) hadrons is mainly caused by the energy loss of moving heavy partons in the QGP medium [9, 10, 11, 12, 13, 14]. This set the motivation for the present work.

Let us now discuss the current understanding of energy loss due to a fast charged particle in the interacting plasma medium. In classical electrodynamics, the energy loss of a fast-moving charged particle, passing through a plasma is of particular importance in which one can relate the stopping power to the dielectric permittivity of the medium [15]. In QCD, the analog of this problem is the energy loss of high energy partons moving in the hot QCD medium. The high energy partons/heavy quarks that are created in the initial hard scatterings, in ultra-relativistic heavy-ion collisions, pass through the hot and dense matter produced after the collision and lose their energy through various processes (collisions, radiation, *etc*). Some of the initial investigations primarily involve work by Bjorken, who studied the collisional energy loss suffered by the high energy partons due to the elastic scatterings off thermal quarks and gluons in the QCD plasma [16]. Later on, Thoma and Gyulassy developed a formalism [17] in which they obtained the collisional energy loss in terms of the longitudinal and transverse dielectric functions. In this approach, the infrared divergence is self-regulated due to the collective plasma effects. Within the finite temperature field theory approach, Braaten and Thoma [18, 19] had constructed a systematic framework of the energy loss for both soft and hard momentum transfers [20, 21, 22]. As the (momentum) anisotropy is present in all the stages of the system expansion, the authors in Refs. [23, 24, 25, 26, 27] have studied the anisotropic effects in the context of heavy quarks energy loss. Apart from that, there are several excellent articles in which authors have discussed the energy loss of heavy quarks either through radiative or collisional means [28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53]. Recently, the polarization energy loss of heavy quarks, considering a hot viscous QGP, has also been studied by Jiang et al. [54, 55]. The heavy quarks’ collisional energy loss inside the QGP medium within the framework of the transport approach, employing the finite RTA, has been studied in Ref. [56]. The same with the BGK collisional kernel has been investigated in Ref. [57]. The energy loss of moving heavy quarks has also been studied considering the ADS/CFT approach in Refs. [58, 59, 60].

Here, we present the study of energy loss of bottom and charm quarks traversing through the isotropic collisional QGP within the effective transport theory approach. The collisions have been modeled using the BGK collisional kernel in the Boltzmann–Vlasov transport equation. Whenever a charged quark passes through the hot QCD medium, it induces the chromo-electric field that generates the Lorentz force which acts back on the moving quark. Hence, the incident quark loses its energy. The induced chromo-electric field is obtained in terms of the longitudinal and transverse parts of the medium’s dielectric permittivity, which, in turn, can be expressed in terms of the gluon selfenergy. The gluon selfenergy has been obtained using the transport theory approach in the Abelian limit. This matches with the one-loop results from the HTL effective theory [61, 62, 63]. In this limit, one needs to consider a high temperature, where the perturbation theory is relevant. To incorporate the non-ideal hot QCD medium interaction effects in the analysis, the effective fugacity quasi-particle model (EQPM) [64, 65, 66] is employed which has recently been studied in Refs. [67, 68, 69]. The results are then compared with those obtained on considering the ideal case (or leading order (LO)). A systematic comparison of the results as regards the energy loss of heavy quarks employing the BGK and RTA kernels (collisions) is presented here.

The paper is organized as follows. In Sect. 2, we shall discuss the energy loss of heavy quarks moving in the hot QCD medium. Here, we shall obtain its expression in terms of dielectric functions using the BGK kernel within a semi-classical transport theory approach. It is important to note that in our previous work [70], we have already calculated the gluon selfenergy using the BGK kernel within the same method. We shall use a few of the results directly from there. In Sect. 3, we shall discuss the various outcomes and provide a comparative analysis. Section 4 is dedicated to the summary and future possibilities of the present work.

## 2 Energy loss of a moving heavy parton

*g*is the coupling constant and \(F_{a}^{\mu \nu }\) is the chromodynamic field strength tensor. \(\tau \), \(x^{\mu }\) and \(p^{\mu }(\tau )\) are the proper time, trajectory and four momentum of the parton, respectively. The four velocity is \(u^{\mu }=\gamma (1,\mathbf{u})=\frac{p^{\mu }(\tau )}{m}\) (where

*m*is the mass of the particle). Here, we have \(N_{c}^{2}-1\) chromo-electric/magnetic fields which belong to the \(SU(N_c)\) gauge group, \(A^{\mu }_{a}\) is the four potential. The expression of the energy loss can be obtained from Wong’s equations (given in Eq. (1)), following two assumptions. First, we consider the gauge condition \(u_{\mu }A^{\mu }_{a} = 0\) which says that \(q^{a}\) is independent of \(\tau \), and second the quark’s momentum and energy evolve in time without changing the magnitude of its velocity while interacting with the chromodynamic field. Now considering the zeroth component, \(\mu =0\), in the second Wong equation (Eq. (1)) one can obtain the energy loss per unit length:

*i.e.,*\(\left\langle \mathbf{E}_{fl}^{a}\right\rangle =0\). Therefore, the main contribution to the energy loss comes from the polarization of the chromo-electric field that can be expressed as in [17, 22, 56, 57],

*etc.*). Here, we mainly focus on the polarization energy loss. Therefore, it is sufficient to consider the zeroth component.

*SU*(3).

Now to solve Eq. (18), we need to know the form of the transverse and longitudinal components of the dielectric permittivity. We shall discuss this while considering the collisional effects using the BGK kernel in the next subsection.

### 2.1 Dielectric permittivity in the presence of collisions

*i*represents the plasma species (quark, anti-quark and gluon). \(\theta _{i}\in \{\theta _g,\theta _q,\theta _{\bar{q}}\}\) have the values \(\theta _{g}=\theta _{q}=1\) and \(\theta _{\bar{q}}=-1\). \(C^i_a(p,X)\) is the collisional kernel which describes the effects of collisions between hard particles in the hot QCD medium. Here, we initially focus on \(C^{i}_a(p,X)\) consiedred to be of BGK-type [73, 74, 77], which is defined by

*i.e.,*while using the BGK kernel we have

## 3 Results and discussion

*LB*and \(HTL_{pt}\), respectively, in the plots). The different collision frequencies, \(\nu \), have been chosen to investigate their impact on the energy loss and also have been compared with the collision-less case \(\nu =0.0\). Little work considering the LO case using the BGK kernel is already available in the literature [57]. Our numbers for the LO are slightly different. The reason is that in the present case the coupling constant and the Debye mass are not fixed. Instead, they are temperature dependent. Here, we are working at temperature, \(T=2~T_c\) where \(T_c=0.17 ~GeV\).

In Figs. 1 and 2, the energy loss of the charm and bottom quarks have been plotted for the collision-less case \(\nu =0\) (left), with collision frequency \(\nu =0.1~m_{D}\) (center) and \(\nu =0.3~m_{D}\) (right), respectively. We have noticed that the energy loss initially increases and then saturates with the increasing particle’s momentum, which matches with the results that are already present in the literature [57]. While considering the non-ideal EoSs (*LB* and \(HTL_{pt}\)) the energy loss (for both heavy quarks) have been found to be suppressed as compared to the ideal one (*LO*) at fixed collision frequency. An increase in the collision frequency causes a higher energy loss. In Fig. 3, we compare the energy loss of charm and bottom quarks at \(\nu =0.0 ~\text {and}~0.1~m_{D}\) for the leading order case and observed that the charm quark loses more energy at fixed momentum than the bottom quark. This supports the idea that a heavy particle loses less energy while moving in a medium than the lighter one, given the same conditions.

## 4 Summary and future aspects

The energy loss of the heavy quarks moving through the isotropic collisional hot QCD medium produced in the relativistic heavy-ion collision experiments has been investigated. The expression of the energy loss is obtained in terms of the longitudinal and transverse part of the dielectric permittivity. Employing the effective kinetic theory in the high-temperature limit (considering the Abelian part) using the BGK kernel, the gluon selfenergy and the dielectric permittivity tensor have been obtained. We found that the energy loss increases initially with the momentum and then saturates. The energy loss also is found to be higher for higher collision frequency. Moreover, the bottom quark is found to lose less energy than the charm quark for the same collision frequency and momentum and, therefore, thermalizes late as compared to the charm quark. We also performed the same analysis considering the RTA kernel and provided a comparative study. It has been observed that the expression of the dielectric permittivity was modified and a slight deviation has been found in the results. Considering the same values of momentum and collision frequency, a higher energy loss is observed in the RTA case than in the BGK case.

We intend to incorporate the momentum anisotropy in the formalism in the near future. The inclusion of viscous effects by employing the effective quasi-particle picture will also be an immediate extension to the present work. In addition, \(R_{AA}\) would be another important quantity to investigate, as it is essential to relate the theoretical estimations with the experimental observations.

## Notes

### Acknowledgements

M. Y. Jamal would like to thank Prof. Jitesh R. Bhatt and Dr. Avdhesh Kumar for fruitful discussions and valuable inputs that helped in improving the present manuscript. M. Y. Jamal further acknowledges NISER Bhubaneswar for providing a postdoctoral position. V. Chandra would like to sincerely acknowledge DST, Govt. of India, for Inspire Faculty Award -IFA13/PH-15 and Early Career Research Award (ECRA) Grant 2016. We would also like to acknowledge the people of India for their generous support for the research in the fundamental sciences in the country.

## References

- 1.J. Adams et al., STAR Collaboration, Nucl. Phys. A
**757**, 102 (2005)Google Scholar - 2.Adcox K. et al., PHENIX Collaboration, Nucl. Phys. A
**757**, 184 (2005)Google Scholar - 3.B.B. Back et al., PHOBOS Collaboration, Nucl. Phys. A
**757**, 28 (2005)Google Scholar - 4.I. Arsene et al., BRAHMS Collaboration, Nucl. Phys. A
**757**, 1 (2005)Google Scholar - 5.K. Aamodt et al., The Alice Collaboration, Phys. Rev. Lett.
**105**, 252302 (2010)Google Scholar - 6.K. Aamodt et al., (The Alice Collaboration), Phys. Rev. Lett.
**105**, 252301 (2010)Google Scholar - 7.K. Aamodt et al., The Alice Collaboration, Phys. Rev. Lett.
**106**, 032301 (2011)Google Scholar - 8.U.W. Heinz, arXiv:hep-ph/0407360
- 9.M. Gyulassy, P. Levai, I. Vitev, Nucl. Phys. B
**571**, 197 (2000)ADSCrossRefGoogle Scholar - 10.B.G. Zakharov, JETP Lett.
**73**, 49 (2001)ADSCrossRefGoogle Scholar - 11.M. Djordjevic, U.W. Heinz, Phys. Rev. Lett.
**101**, 022302 (2008)ADSCrossRefGoogle Scholar - 12.R. Baier, Y.L. Dokshitzer, A.H. Mueller, D. Schiff, JHEP
**0109**, 033 (2001)ADSCrossRefGoogle Scholar - 13.S. Jeon, G.D. Moore, Phys. Rev. C
**71**, 034901 (2005)ADSCrossRefGoogle Scholar - 14.P. Roy, J.E. Alam, A.K. Dutt-Mazumder, J. Phys. G
**35**, 104047 (2008)ADSCrossRefGoogle Scholar - 15.E. Infeld, Basic Principles of Plasma Physics: a Statistical Approach. By S. Ichimaru. Benjamin Frontiers in Physics, 1973. 324 pp. \$19.50 (hardcover), 12.50 (paperback), J. Plasma Phys.
**13**(3), 571–572 (1975). https://doi.org/10.1017/S0022377800025289 ADSCrossRefGoogle Scholar - 16.James D. Bjorken, Fermilab preprint 82/59-THY, (1982), unpublishedGoogle Scholar
- 17.M.H. Thoma, M. Gyulassy, Nucl. Phys. B
**351**, 491 (1991)ADSCrossRefGoogle Scholar - 18.E. Braaten, M.H. Thoma, Phys. Rev. D
**44**, 1298 (1991)ADSCrossRefGoogle Scholar - 19.E. Braaten, M.H. Thoma, Phys. Rev. D
**44**, R2625 (1991)ADSCrossRefGoogle Scholar - 20.S. Mrowczynski, Phys. Lett. B
**269**, 383 (1991)ADSCrossRefGoogle Scholar - 21.M.H. Thoma, Phys. Lett. B
**273**, 128 (1991)ADSCrossRefGoogle Scholar - 22.Y. Koike, T. Matsui, Phys. Rev. D
**45**, 3237 (1992)ADSCrossRefGoogle Scholar - 23.P. Romatschke, M. Strickland, Phys. Rev. D
**69**, 065005 (2004)ADSCrossRefGoogle Scholar - 24.P. Romatschke, M. Strickland, Phys. Rev. D
**71**, 125008 (2005)ADSCrossRefGoogle Scholar - 25.R. Baier, Y. Mehtar-Tani, Phys. Rev. C
**78**, 064906 (2008)ADSCrossRefGoogle Scholar - 26.M.E. Carrington, K. Deja, S. Mrowczynski, Phys. Rev. C
**92**, 044914 (2015)ADSCrossRefGoogle Scholar - 27.M.E. Carrington, K. Deja, S. Mrowczynski, Phys. Rev. C
**95**, 024906 (2017)ADSCrossRefGoogle Scholar - 28.R. Baier, D. Schiff, B.G. Zakharov, Ann. Rev. Nucl. Part. Sci.
**50**, 37 (2000)ADSCrossRefGoogle Scholar - 29.P. Jacobs, X.N. Wang, Prog. Part. Nucl. Phys.
**54**, 443 (2005)ADSCrossRefGoogle Scholar - 30.N. Armesto et al., Phys. Rev. C
**86**, 064904 (2012)ADSCrossRefGoogle Scholar - 31.A. Majumder, M. Van Leeuwen, Prog. Part. Nucl. Phys.
**66**, 41 (2011)ADSCrossRefGoogle Scholar - 32.M.G. Mustafa, D. Pal, D.K. Srivastava, M. Thoma, Phys. Lett. B
**428**, 234 (1998)ADSCrossRefGoogle Scholar - 33.Y.L. Dokshitzer, D.E. Kharzeev, Phys. Lett. B
**519**, 199 (2001)ADSCrossRefGoogle Scholar - 34.M. Djordjevic, M. Gyulassy, Nucl. Phys. A
**733**, 265 (2004)ADSCrossRefGoogle Scholar - 35.S. Wicks et al., Nucl. Phys. A
**783**, 493 (2007)ADSCrossRefGoogle Scholar - 36.S. Wicks et al., Nucl. Phys. A
**784**, 426 (2007)ADSCrossRefGoogle Scholar - 37.R. Abir et al., Phys. Rev. D
**85**, 054012 (2012)ADSCrossRefGoogle Scholar - 38.R. Abir et al., Phys. Lett. B
**715**, 183 (2012)ADSCrossRefGoogle Scholar - 39.G.Y. Qin, J. Ruppert, C. Gale, S. Jeon, G.D. Moore, M.G. Mustafa, Phys. Rev. Lett.
**100**, 072301 (2008)ADSCrossRefGoogle Scholar - 40.S. Cao, G.Y. Qin, S.A. Bass, Phys. Rev. C
**88**, 044907 (2013)ADSCrossRefGoogle Scholar - 41.M.G. Mustafa, M.H. Thoma, Acta Phys. Hung. A
**22**, 93 (2005)CrossRefGoogle Scholar - 42.M.G. Mustafa, M.H. Thoma, Phys. Rev. C
**72**, 014905 (2005)ADSCrossRefGoogle Scholar - 43.A.K. Dutt-Mazumder, J.E. Alam, P. Roy, B. Sinha, Phys. Rev. D
**71**, 094016 (2005)ADSCrossRefGoogle Scholar - 44.A. Meistrenko, A. Peshier, J. Uphoff, C. Greiner, Nucl. Phys. A
**901**, 51 (2013)ADSCrossRefGoogle Scholar - 45.K.M. Burke et al., [JET Collaboration], Phys. Rev. C
**90**, 014909 (2014)Google Scholar - 46.S. Peigne, A. Peshier, Phys. Rev. D
**77**, 014015 (2008)ADSCrossRefGoogle Scholar - 47.S. Peigne, A. Peshier, Phys. Rev. D
**77**, 114017 (2008)ADSCrossRefGoogle Scholar - 48.R.B. Neufeld, I. Vitev, H. Xing, Phys. Rev. D
**89**(2014), 096003 (2014)ADSCrossRefGoogle Scholar - 49.P. Chakraborty, M.G. Mustafa, M.H. Thoma, Phys. Rev. C
**75**, 064908 (2007)ADSCrossRefGoogle Scholar - 50.A. Adil, M. Gyulassy, W.A. Horowitz, S. Wicks, Phys. Rev. C
**75**, 044906 (2007)ADSCrossRefGoogle Scholar - 51.S. Peigne, P.B. Gossiaux, T. Gousset, JHEP
**0604**, 011 (2006)ADSCrossRefGoogle Scholar - 52.K. Dusling, I. Zahed, Nucl. Phys. A
**833**, 172 (2010)ADSCrossRefGoogle Scholar - 53.S. Cho, I. Zahed, Phys. Rev. C
**82**, 064904 (2010)ADSCrossRefGoogle Scholar - 54.B.F. Jiang, D. Hou, J.R. Li, J. Phys. G
**42**, 085107 (2015)ADSCrossRefGoogle Scholar - 55.B.F. Jiang, D.F. Hou, J.R. Li, Nucl. Phys. A
**953**, 176 (2016)ADSCrossRefGoogle Scholar - 56.M. Elias, J. Peralta-Ramos, E. Calzetta, Phys. Rev. D
**90**, 014038 (2014)ADSCrossRefGoogle Scholar - 57.C. Han, D.F. Hou, B.F. Jiang, J.R. Li, Eur. Phys. J. A
**53**, 205 (2017)ADSCrossRefGoogle Scholar - 58.K.B. Fadafan, JHEP
**0812**, 051 (2008)CrossRefGoogle Scholar - 59.K. Bitaghsir Fadafan, Eur. Phys. J. C
**68**, 505 (2010)CrossRefGoogle Scholar - 60.K.B. Fadafan, H. Soltanpanahi, JHEP
**1210**, 085 (2012)ADSCrossRefGoogle Scholar - 61.M. Yousuf, S. Mitra, V. Chandra, Phys. Rev. D
**95**, 094022 (2017)ADSCrossRefGoogle Scholar - 62.J.P. Blaizot, E. Iancu, Nucl. Phys. B
**417**, 608 (1994)ADSCrossRefGoogle Scholar - 63.P.F. Kelly, Q. Liu, C. Lucchesi, C. Manuel, Phys. Rev. D
**50**, 4209 (1994) ADSCrossRefGoogle Scholar - 64.V. Chandra, R. Kumar, V. Ravishankar, Phys. Rev. C
**76**, 054909 (2007); [Erratum: Phys. Rev. C**76**, 069904 (2007)]Google Scholar - 65.V. Chandra, A. Ranjan, V. Ravishankar, Eur. Phys. J. A
**40**, 109–117 (2009)ADSCrossRefGoogle Scholar - 66.V. Chandra, V. Ravishankar, Phys. Rev. D
**84**, 074013 (2011)ADSCrossRefGoogle Scholar - 67.M. Kurian, S.K. Das, V. Chandra, arXiv:1907.09556 [nucl-th]
- 68.M.Y. Jamal, I. Nilima, V. Chandra, V.K. Agotiya, Phys. Rev. D
**97**(9), 094033 (2018)ADSCrossRefGoogle Scholar - 69.V.K. Agotiya, V. Chandra, M.Y. Jamal, I. Nilima, Phys. Rev. D
**94**(9), 094006 (2016)ADSCrossRefGoogle Scholar - 70.A. Kumar, M.Y. Jamal, V. Chandra, J.R. Bhatt, Phys. Rev. D
**97**(3), 034007 (2018)ADSCrossRefGoogle Scholar - 71.S.K. Wong, Nuovo Cim. A
**65**, 689 (1970)ADSCrossRefGoogle Scholar - 72.M. Laine, Y. Schroder, JHEP
**0503**, 067 (2005)ADSCrossRefGoogle Scholar - 73.Bing-feng Jiang, De-fu Hou, Jia-rong Li, Phys. Rev. D
**94**, 074026 (2016)ADSCrossRefGoogle Scholar - 74.Bjoern Schenke, Michael Strickland, Carsten Greiner, Markus H. Thoma, Phys. Rev. D
**73**, 125004 (2006)ADSCrossRefGoogle Scholar - 75.S. Mrowczynski, Phys. Lett. B
**314**, 118 (1993)ADSCrossRefGoogle Scholar - 76.P. Romatschke, M. Strickland, Phys. Rev. D
**68**, 036004 (2003)ADSCrossRefGoogle Scholar - 77.P.L. Bhatnagar, E.P. Gross, M. Krook, Phys. Rev.
**94**, 511 (1954)ADSCrossRefGoogle Scholar

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