# Nonperturbative collisional energy loss of heavy quarks in quark-gluon plasma

###### Abstract

We suggest a new mechanism for the energy loss of fast heavy quarks in quark-gluon plasma. This mechanism is based on pion production caused by the anomalous chromomagnetic quark-gluon-pion interaction induced by strong topological fluctuations of the gluon fields represented by instantons. We found that this mechanism gives a considerable contribution to the collisional energy loss of a heavy quark in quark-gluon plasma, which shows a nontrivial role of nonperturbative phenomena in strongly interacting quark-gluon plasma.

###### pacs:

The large energy loss of heavy quarks observed in high energy heavy ion collisions at Relativistic Heavy Ion Collider (RHIC) and Large Hadron Collider (LHC) is one of exciting puzzles in the physics of quark-gluon plasma (QGP). In spite of the significant progress in understanding of the possible mechanisms for the energy loss of fast heavy quarks, the complete mechanism of this effect is yet to be explored. So far, there are many suggestions on the origins of heavy quark energy loss. The first is the collisional energy loss that was considered in the pioneering work of Bjorken Bjorken82 (). (See also Refs. MT03 (); Mustafa04 (); PP08 ().) Another mechanism is the radiative energy loss due to the gluon radiation by a heavy quark induced by the interactions with QGP partons GW93 (); Zakharov96 (); BDMPS97 (); BDMS98 (); GLV00c (); AMY02 (); AJMS12 (). Other suggestions include the so-called dead cone effect that comes from the suppression of gluon emission by the quark mass at small angles DK01b (). The detailed discussions on the different mechanisms of the heavy quark energy suppression in heavy ion collisions can be found, for example, in recent publications Uphoff14 (); CQB15 (); SSS15 (); AAAB15 () and references therein.

Although the afore-mentioned effects must be crucial to understand the energy loss phenomena, these ideas are based on perturbative QCD (pQCD). This means that the system is assumed to be in the domain of pQCD for heavy quark interactions within QGP. However, it is now widely accepted that the QGP produced in relativistic heavy ion collisions is not weakly interacting pQCD-like QGP but is strongly interacting QGP (sQGP) with nonperturbative interactions between quarks and gluons Shuryak14 (). Therefore, it is natural to speculate that nonperturbative QCD effects might have a crucial role in unravelling the heavy quark energy loss problem. For example, it was suggested in Ref. Zakharov14 () that thermal monopoles in QGP may lead to a large enhancement of the parton radiative energy loss in QGP.

In this article we investigate a nonperturbative mechanism for heavy quark energy loss that is related to the anomalous chromomagnetic quark-gluon interaction induced by instantons Kochelev98 (). Instantons, being strong topological fluctuations of gluon fields in the QCD vacuum, play an important role in hadron physics and give strong influence to the properties of QGP. (For a review, see, for example, Refs. SS98b (); Diakonov02 ().) Furthermore, instantons lead to various nontrivial effective interactions such as a very specific quark-quark interaction tHooft76 (), the chromomagnetic quark-gluon interaction Kochelev98 (), and the quark-gluon-pion interaction BPW97 (); Diakonov02 (). Recently, it was demonstrated that the interaction of the last type has an important role in understaning the cross sections of the inclusive pion production in high energy proton-proton collisions KLZZ15 (). Furthermore, it may have a nontrivial role in unpolarized and polarized gluon distributions of nucleons at small Bjorken KLZZ15b (). Therefore, it is natural to expect that such nonperturbative strong interactions would have a nontrivial role in high energy heavy ion collisions as well. The purpose of the present work is thus to investigate the role of the chromomagnetic quark-gluon and quark-gluon-pion interactions in the heavy quark energy loss in QGP.

In Ref. Kochelev98 (), it was shown that instantons generate a new type of chromomagnetic quark-gluon interaction as

(1) |

where is the SU(3) Gell-Mann matrices, is the anomalous quark chromomagnetic moment, is the effective quark mass in the instanton vacuum, is the strong coupling constant, and is the gluon field strength tensor. Within the instanton model the value of is estimated as

(2) |

where is the average instanton size in QCD vacuum, is the strong coupling constant at the scale of . We refer the details to Refs. Diakonov02 (); Kochelev09 ().

However, the Lagrangian of Eq. (1) does not respect chiral symmetry and needs to be modified by introducing the pion field for preserving chiral symmetry Diakonov02 (); BPW97 (). Then, in the single instanton approximation, the effective Lagrangian becomes

(3) |

with the pion decay constant . Using Eq. (2), it is rewritten as

(4) | |||||

where is the pion-quark coupling constant at zero temperature, . Equipped with the effective Lagrangian of the quark-gluon and quark-pion-gluon interactions, we now consider their role in the heavy quark energy loss in QGP. The relevant diagrams contributing to the nonperturbative heavy quark energy loss are presented in Fig. 1.

Our starting point is the Bjorken’s formula for the collisional energy loss Bjorken82 () with a -channel exchange, which reads

(5) |

where is the flux factor and with being the energy of the emergent parton. The parton density in QGP at temperature is given by

(6) |

where the positive sign corresponds to the quark density () with and being the number of active quark flavors in QGP, while the negative sign corresponds to the gluon density () with . In our estimation, we use by considering the light and quarks. The flux factor is , where is the angle between the momenta of two incident partons.

The cross section for the diagram of Fig. 1(a) is then calculated as

(7) |

where is the instanton form factor Kochelev09 () with being the modified Bessel function of the second kind of order 2, and we assume that, in our nonperturbative calculation, the scale in the running strong coupling constant is determined by the instanton size. Therefore, the nonperturbative contribution of this diagram to the energy loss is

(8) | |||||

which leads to

(9) |

For the diagram of Fig. 1(b), consideration of the kinematics of the process gives the relation of

(10) |

where is the invariant mass of the final system of the light quark and pion. Therefore, the collisional energy loss can be rewritten as

(11) |

By decomposing the momentum in the longitudinal and transversal components KLZZ15 (), the differential cross section becomes

(12) |

where is the fraction of the initial light quark momentum carried by the pion in the center of momentum frame, is the transverse momentum of the pion, and is the transverse momentum of the exchanged gluon, which leads to . Here the isospin factor is included for pion production. By substituting and using the relations,

(13) |

the integrals over and can be performed.

Since instantons describe the subbarrier transitions between classical QCD vacua with different topological charges, the integration over invariant mass of the light-quark–pion system is restricted by the so-called sphaleron energy defined as

(14) |

which is the height of the potential barrier between the vacua Diakonov02 (). The expression for the heavy quark energy loss induced by the nonperturbative quark-gluon-pion interaction then becomes

(15) |

This result should be compared with the pQCD collisional energy loss, which is given by PP08 ()

(16) | |||||

where and is the Debye mass.

For our numerical estimate, we neglect the possible weak temperature-dependence of the quark-pion coupling constant in the range of following Ref. EKMV93 (). This temperature range is expected to cover the temperatures achieved at the relativistic heavy ion collisions at RHIC and LHC. For the deconfinement temperature, we use MeV, and the charm quark mass is GeV. The Debye mass is a function of temperature and, following the lattice QCD calculations of Refs. PKLSW02 (); DEKKL05 (); KZ05 (), we write it as

(17) |

The integrals for and are performed with and .

The parameters of our model are determined as follows. For the running strong coupling constant, we use the widely used form given as SS97c ()

(18) |

where and MeV for . The average size of the instanton is taken to be fm, which is supported by both phenomenology and lattice calculations SS98b (). The strong coupling constant is then , which also agrees with the value of the instanton model of Ref. Diakonov02 ().

One of the important parameter of our model is the effective quark mass in instanton vacuum. The Lagrangians of Eqs. (1) and (4) are obtained in the effective single instanton approximation and, in this approximation, multi-instanton effects are included in the effective quark mass in the zero-mode-like propagator in the instanton field. Then a careful analysis on various correlation functions within the instanton model leads to MeV FS01 (). We will use this value in the present work. But, since our results are rather dependent of the quark mass, we also discuss the results with a higher mass, MeV that was estimated in the earlier work of Ref. Shuryak83 ().

Shown in Figs. 2 and 3 are the numerical results of the present calculation. In Fig. 2, the charm quark energy loss from the perturbative and nonperturbative parts are shown as functions of energy and temperature. We find that the contribution from the quark-gluon interaction without pion emission [Fig. 1(a)] is very small as shown by the dotted lines in Fig. 2 and, therefore, can be safety neglected. However, the nonperturbative contribution with pion emission [Fig. 1(b)] to the collisional heavy quark energy loss is found to be similar or even larger than that of the pQCD contribution. To quantify the difference, in Fig. 3, we plot the ratio defined as

(19) |

Figure 3 shows that in the considered region of energy and temperature. We also found that the energy dependence of the ratio is very weak in the energy range between 10 GeV and 30 GeV, while it has sensitive dependence on temperature. This is because the energy dependence is determined by the cross sections that are almost saturated in the considered energy region, while the temperature dependence is governed by the parton density distribution.

As stated before, the radiative energy loss is expected to be one of the major sources of heavy
quark energy loss.
However, there are large uncertainties in the estimation of the radiative energy
loss GW93 (); Zakharov96 (); BDMPS97 (); BDMS98 (); GLV00c (); AMY02 (); AJMS12 ().
In Refs. UFXG12 (); MUGP12 (), it was claimed that a phenomenological factor
for the ratio of the total energy loss to the pQCD collisional energy loss is needed to explain
the measured data.
This means that the contributions from the mechanisms other than pQCD contribution
should be larger than the pQCD contribution by a factor of .
Therefore, we conclude that a smaller radiative energy loss, namely, about 1.3 times the pQCD collisional
contribution, would be enough to resolve the heavy quark energy loss puzzle due to the
nonperturbative contribution considered in the present work.^{1}^{1}1We found that the ratio is rather sensitive to the effective quark mass .
If we use MeV, it becomes as large as .
Therefore, the ambiguity in brings in another uncertainty originating from the
nonperturbative calculation.

We finally make a qualitative comment on the nonperturbative energy loss of light quarks. In this case, the virtuality of the fast quark is not large and, therefore, additional diagrams with pion radiation from the fast light quark can give nontrivial contributions to the total energy loss. Examples of such diagrams are shown in Fig. 4. In this case, it is evident that the direct pion production from fast quarks will give enhancement of total pion production in heavy ion collisions which, therefore, will increase the nuclear modification factor . However, on the other hand, the direct pion emission should lead to the energy loss of the fast quark similar to the heavy quark case, and this will decrease . Therefore, we have two contributions of opposite roles and the empirical will be determined by the competition between them. This requires more careful and complex analyses and will be discussed elsewhere.

To summarize, we suggest a new nonperturbative mechanism for heavy quark energy loss in QGP. It was shown that the nonperturbative chromomagnetic quark-gluon-pion interaction in QGP may give a nontrivial contribution to the heavy quark collisional energy loss. Therefore, this mechanism will be important to understand the mechanisms of the heavy quark energy loss combined with other mechanisms. Our finding again shows the important role of nonperturbative phenomena in the understanding of the dynamics of QGP observed in high energy heavy ion collisions.

###### Acknowledgements.

This work was partially supported by the National Natural Science Foundation of China under Grant Nos. 11035006 and 11175215 and by the Chinese Academy of Sciences visiting professorship for senior international scientists under Grant No. 2013T2J0011. Support from the MSIP of the Korean Government through the Brain Pool Program No. 121S-1-3-0318 is also gratefully acknowledged. H.-J.L. and Y.O. were supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education under Grant Nos. NRF-2013R1A1A2009695 and NRF-2013R1A1A2A10007294, respectively.## References

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