Effect of the chromo-electromagnetic field fluctuations on heavy quark propagation at the LHC energies

We consider the effect of the chromo-electromagnetic field fluctuations in addition to the collisional as well as the radiative energy loss suffered by heavy quarks while propagating through the hot and densed deconfined medium of quarks and gluons created in relativistic heavy ion collisions. The chromo-electromagnetic field fluctuations play an important role as it leads to an energy gain of heavy quarks of all momentum, significantly effective at the lower momentum region. With this, we have computed, for the first time, the nuclear modification factor (R_{AA}) of heavy mesons, viz., D-mesons and B-mesons and compared with the those experimental measurements in Pb-Pb collisions at \sqrt{s_{NN}} = 2.76 TeV and \sqrt{s_{NN}} = 5.02 TeV by the CMS and ALICE experiments at the LHC. Our results are found to be in very good agreement with those available data measured by CMS and ALICE experiments.


Introduction
The main goal of relativistic heavy-ion collisions at Relativistic Heavy Ion Collider (RHIC) at BNL and Large Hadron Collider (LHC) at CERN is to produce a hot and dense deconfined state of QCD matter, so called quark-gluon plasma (QGP).It is believed that this new deconfined state of matter has been formed during relativistic heavy-ion collisions at RHIC[1] and LHC [2].One of the features of this deconfined plasma created in heavy ion collisions is the suppression of high energy hadrons compare to the case of p− p collisions, called jet quenching.This jet quenching is caused due to the energy loss of initial hard partons via collisional and radiative energy loss inside the deconfined medium.It was anticipated first by Bjorken [3] as a crucial probe of this deconfined medium.
Heavy quarks are mostly produced in early stage of the heavy ion collisions from the initial fusion of partons They may also be produced in the QGP, if initial temperature of QGP is high enough than the mass of the heavy quarks.
However, no heavy quarks are produced at the latter stage and none in the hadronic matters.Hence, the total number of heavy quarks becomes frozen at the very early stage in the history of the collisions, which makes them a good probe of the QGP.These heavy quarks immediately after their production will propagate through the dense medium and will start losing energy during their path of travel.This energy loss suffered by the heavy quarks are reflected in the transverse momentum spectra and nuclear modification factor of heavy mesons.
Heavy quarks lose energy in two different fashions in the QGP: one is caused by elastic collisons with the light partons of the thermal background (QGP) and the other one is by radiating gluons, viz., bremsstrahlung process due to the deceleration of the charge particles.
The energy loss in the QGP are usually obtained by treating the medium in an average manner and the fluctuations are ignored.Since QGP is a statistical ensemble of mobile coloured charge particles, which could also be characterised by omnipresent stochastic fluctuations.This microscopic fluctuations generally couple with the external perturbations and affect the response of the medium.
The effect of electromagnetic field fluctuations during the passage of charged particles though a non-relativistic classical plasma has been calculated by several authors in the literature [29,30,31,32,33,34].On the other hand the effect of chromo-electromagnetic fluctuations in the QGP leads to an energy gain of heavy quarks of all momentum and significantly at the lower momentum ones [25].This is because the moving parton in the QGP encounters a statistical change in the energy due to the fluctuations of the chromo-electromagnetic fields as well as the velocity of the particle under the influence of this field.The effect of such fluctuation was not considered in earlier literature for studying the hadron spectra in the perspective of heavy ion collisions.
In this Letter, we investigate for the first time the effect of the chromo- The paper is organised in as follows: In sec.2 we brifely outline the basic setup containing, heavy quark production and fragmentation, models for both collisional and radiative energy loss, and energy gain due to field fluctuations, medium evolution and initial conditions etc, for the purpose.Here we consider the collisional energy loss of heavy quarks by Peigne and Pashier (PP) formalism [18] and the radiative energy loss by Abir, Jamil, Mustafa and Srivastava (AJMS) formalism [26] along with the energy gain due to the chromoelectric field fluctuations prescription by Chakraborty, Mustafa and Thoma (CMT) in Ref. [25].In sec.3 we discuss our results and a conclusion in sec.4.

Heavy quark production and fragmentation
The heavy quarks in p − p collisions are mainly produced by fusion of gluons or light quarks [35].Their production cross section has been obtained to nextto-leading order (NLO ) accuracy with CT10 parton distribution function [36] for p-p collisions.For heavy ion collisions, the shadowing effect is taken into account by using the NLO parameters of EPS09 [37] nuclear parton distribution function.The same set of parameters as that of Nelson et.al. [38]

Medium Evolution and initial condition
As the heavy quarks lose energy during their passage through the QGP medium, it is important to figure out the path length it is traversing inside the medium.We consider a heavy quark, which is being produced at a point (r,φ) in heavy ion collisions and propagates at an angle φ with respect to r in the transverse plane.So, the path length L covered by the heavy quark inside the medium is given by [40]: where R is the radius of the colliding nuclei.The average distance travelled by the heavy quark inside the plasma is where T AA (r, b = 0) is the nuclear overlap function.We estimate L = 6.14f m for central P b − P b collisions.The effective path length of heavy quark of transverse mass m T and transverse momentum p T in the QGP of life time τ f is obtained as, We consider the medium evolution as per the isentropic cylindrical expansion as discussed in Ref. [41].The equation of state is obtained by Lattice QCD along with hadronic resonance in order to calculate temperature as a function of proper time [42].We calculate the heavy quark energy loss over QGP life time and finally averaged over the temperature evolution.The initial conditions used for the hydrodynamic medium evolution are similar to the Ref [28].We consider the initial time τ 0 = 0.

Collisional Energy Loss: Peigne and Peshier (PP) Formalism
One of the important mechanism in which heavy quarks may lose energy inside the QGP is by collisions.The calculation of collisional energy loss per unit length dE/dx has been reported by in the past by several authors [4,5,43].
The most detailed calculation of dE/dx was made by Brateen and Thoma [5] which was based on their previous QED calculation of dE/dx for muon [44].This calculation of Brateen and Thoma for dE/dx is based on an assumption that the momentum exchange in elastic collisions, q ≪ E, which is not appropriate in the domain E ≫ M 2 /T , where M is the mass of the heavy quark and T is the temperature of the medium.The improved differential energy loss expression, valid for E ≫ M 2 /T , is given by Peigne and Pashier [18] as where, µ 2 g = 4πα s T 2 (1 + n f /6) is the square of Debye screening mass, n f = 3, is the number of active quark flavours and c(n f ) ≈ 0.146n f + 0.05 and α s = 0.3, is the strong constant.

Radiative Energy Loss: Abir, Jamil, Mustafa and Srivastava (AJMS) Formalism
The most important and dominant way of energy loss from a fast partons inside the QGP is due to gluon radiation.The first attempt to estimate the radiative energy loss was made in Ref. [6].Later many authors [8,9,17,21,45,46,47] also estimated the energy loss with various ingredients and kinematical conditions.In Refs.[8,9] the soft gluon emission was estimated which was found to suppress compared to the light quarks due to the mass effect, known as dead cone effect.The radiative energy loss induced by the medium due to the dead cone effect was limited only to the forward direction.In Ref. [12] by relaxing some of the constraints imposed in Refs.[8,9], e.g., the gluon emission angle and the scaled mass of the heavy quark with its energy, a generalised dead cone was obtained which led to a very compact expression for the gluon emission probability off a heavy quark.Based on the generalised dead cone approach and the gluon emission probability, AJMS [26] computed the heavy quark radiative energy loss1 as with where and ρ QGP is the density of the QGP medium which acts as a background containing the target partons.If ρ q and ρ g are the density of quarks and gluons respectively in the medium, then the ρ QGP is given by

Energy gain by chromo-electromagnetic fields fluctuations: Chakraborty, Mustafa and Thoma (CMT) Formalism
The energy loss calculations both collisional and radiative of heavy quarks in the QGP were obtained by treating the QGP medium without considering microscopic fluctuations.However, QGP being the statistical system, it is characterised by stochastic chromo-electromagnetic field fluctuations.Since the energy loss is of topical interest for the phenomenology of heavy quark jet quenching in hot and dense medium.A quantitatively estimate of the effect of the microscopic electromagnetic fluctuations on the propagation a heavy quark was done using semiclassical approximation2 by CMT in Ref. [25].This was found to led an energy gain of the heavy quark caused due to the statistical change in the energy of the moving parton in the QGP due to the fluctuations of the chromo-electromagnetic fields as well as the velocity of the particle under the influence of this field.The leading-log (LL) contribution of the energy gain was obtained [25] as where k min = µ g = Debye mass and k max = min E, with q ∼ T is the typical momentum of the thermal partons.One can physically interpret this energy gain of a heavy quark that absorbs gluons during its propagation.as a function of its momentum, obtained using PP [18], AJMS [26] and Fluctuations [25].The energy loss of a bottom quark inside QGP medium as a function of its momentum, obtained using PP [18], AJMS [26] and Fluctuations [25].
In Fig. 1  Fig. 3 and Fig. 4 display the fractional energy loss from collisional and radiative process, and also the energy gain due to the field fluctuations for charm and bottom quarks, respectively.It is clear that the energy gain for heavy quarks is relatively more at the lower momentum region (4 − 40 MeV) than that in very higher momentum (> 40 MeV) region.The reason is that the field fluctuations and thus the energy gain become substantial in the low velocity limit.Because of this the field fluctuations, the total energy loss of a heavy quark gets reduced up to a very moderately high values of momentum beyond which its contribution gradually diminishes.The relative importance of it will be very re levant for LHC energies as we would see below.

Results and Discussions
In Fig. 5 and Fig. 6 we have displayed the nuclear modification factor, R AA , for D 0 -meson in (0 − 10)% and (0 − 100)% centrality, respectively, in P b − P b collisions, considering both collisional and radiative energy loss along with the energy gain due to the field fluctuations and compared with ALICE [48] and CMS data [49].We observe that only the radiative energy loss (AJMS) or the with CMS data [51].The radiative energy loss itself produces a small suppression but when the collisional one is added generates more suppression than the measured CMS data.When the energy gain due to field fluctuation is taken into account in addition to both radiative and collisional losses, the suppression is found to be very closer to the measured data within their uncertainties.

Conclusion
The energy loss encountered by an energetic parton in a QGP medium reveals the dynamical properties of that medium in view of jet quenching of high energy partons.This is usually reflected in the transverse momentum spectra electromagnetic field fluctuations leading to energy gain of heavy quarks in addition to both the collisional and the radiative energy loss on the nuclear modification factor for D and B mesons and compared with the measurements of both ALICE and CMS experiments in P b − P b collisions at √ s N N = 2.76 TeV and CMS experiment at √ s N N = 5.02 TeV.We found that the chromoelectromagnetic field fluctuations play an important role on the propagation of the heavy quark jets in a QGP vis-a-vis the nuclear modification factor of heavy flavoured hadrons.It is interesting to note that only collisional or radiative or both energy loss can not explain the data satisfactorily.If the energy gain due to fluctuations is included along with the collisional and radiative energy loss, then the data can be explained very satisfactorily from low to moderately high value of transverse momentum.

Figure 1 :
Figure 1: The energy loss of a charm quark inside QGP medium Figure 2: The energy loss of a bot-

Figure 3 :Figure 4 :
Figure 3: Fractional energy loss of charm quark inside QGP due to fluctuations, collisions (PP) and radiations (AJMS) with its momentum.The path length considered is L = 5 fm.

Figure 5 :Figure 6 :Fig. 7 Figure 7 :Figure 8 :
Figure 5: Nuclear modification factor R AA of D 0 -meson with collisional (PP) and radiative (AJMS) energy loss along with the effect of fluctuations as a function of transverse momentum p T for (0 − 10)% centrality at P b − P b collisions at √ s NN = 2.76 TeV.The data for D 0 -meson are taken from the measurement of AL-ICE [48] and CMS experiment [49].
and nuclear modification factor of mesons which are measured in heavy ion experiments.For the phenomenology of the heavy quarks jet quenching the field fluctuations in the QGP medium were not considered in the literature before.In this article, for the first time, we have considered the propagation of high energy heavy quarks by including the energy gain due to field fluctuations along with the energy loss caused by the collisions and gluon radiations inside the QGP medium.The nuclear modification factors R AA for D-mesons and B-mesons in P b − P b collisions at √ s N N = 2.76 TeV and √ s N N = 5.02 TeV are calculated by including the both energy losses and the field fluctuations effect.We note that the radiative energy loss alone can describe the D-mesons suppressions at higher transverse momentum.Nevertheless, the nuclear modification factors for both D and B mesons are found to agree quite well with those data in the entire p T range measured by CMS and ALICE experiments at LHC energies, if the energy gain due the field fluctuations are taken into account in addition to the collisional and radiative loss in the medium.The effect of field fluctuations in hot and dense QGP medium is found to play an important role in the propagation of heavy quarks also in describing the experimental data for heavy quarks quenching.