Effect of thermal gluon absorption and medium fluctuations on heavy flavour nuclear modification factor at RHIC and LHC energies

Abstract

The presence of thermal gluons reduces the stimulated gluon emission off a heavy quark propagating in the quark-gluon plasma (QGP) while absorption causes reduction of the radiative energy loss. On the other hand, the chromo-electromagnetic field fluctuations present in the QGP lead to collisional energy gain of the heavy quark. The net effect of the thermal gluon absorption and field fluctuations is a reduction of the total energy loss of the heavy quark, prominent at the lower momenta. We consider both kind of the energy gains along with the usual losses, and compute the nuclear modification factor ($R_{AA}$) of heavy mesons, viz., D and B mesons. The calculations have been compared with the experimental measurements in Au–Au collisions at $\sqrt{s_{NN}} = 200$ GeV from STAR and PHENIX experiments at the RHIC and Pb–Pb collisions at $\sqrt{s_{NN}} = 2.76$ TeV and 5.02 TeV from CMS and ALICE experiments at the LHC. We find a significant effect of the total energy gain due to thermal gluon absorption and field fluctuations on heavy flavour suppression, especially at the lower transverse momenta.

Appendices

Appendix A: Radiative energy loss: generalized dead cone approach

The expression for heavy quark radiative energy loss, based on the generalised dead cone approach and the gluon emission probability [13], can be found as [47]:

$$\begin{aligned} \frac{dE}{dx}= & {} 24\alpha _{s}^{3}\rho _{QGP}\frac{1}{\mu _g}\left( 1-\beta _1\right) \nonumber \\&\times \left( \sqrt{\frac{1}{(1-\beta _1)}\log (\frac{1}{\beta _1})}-1\right) {\mathscr {F}}(\delta ), \end{aligned}$$

(A.1)

with

$$\begin{aligned} {\mathscr {F}}(\delta )= & {} 2\delta - \frac{1}{2}\log \left( \frac{1+\frac{M^2}{s}e^{2\delta }}{1+\frac{M^2}{s}e^{-2\delta }}\right) \nonumber \\&- \left( \frac{\frac{M^2}{s}\sinh {(2\delta )}}{1+2\frac{M^2}{s}\cosh {(2\delta )}+\frac{M^4}{s^2}} \right) , \end{aligned}$$

(A.2)

where

$$\begin{aligned} \delta = \frac{1}{2}\log \left[ \frac{1}{(1-\beta _1)}\log \left( \frac{1}{\beta _1}\right) \left( 1+\sqrt{1-\frac{(1-\beta _1)}{\log \frac{1}{\beta _1}}}\right) ^2\right] \end{aligned}$$

(A.3)

and $\rho _{QGP}$ is the density of the QGP medium which acts as a background containing the target partons. If $\rho _q$ and $\rho _g$ are the density of quarks and gluons respectively in the medium, then the $\rho _{QGP}$ is given by

$$\begin{aligned} \rho _{QGP}= & {} \rho _q+\frac{9}{4}\rho _g, \end{aligned}$$

(A.4)

$$\begin{aligned} \beta _1= & {} \frac{\mu _{g}^{2}}{CET}, \end{aligned}$$

(A.5)

$$\begin{aligned} C= & {} \frac{3}{2}-\frac{M^2}{4ET}\nonumber \\&+\frac{M^4}{48E^2T^2\beta _0}\log \left( \frac{M^2+6ET(1+\beta _0)}{M^2+6ET(1-\beta _0)}\right) , \end{aligned}$$

(A.6)

$$\begin{aligned} \beta _0= & {} \sqrt{1-\frac{M^2}{E^2}} \end{aligned}$$

(A.7)

It is to be noted here that later a kinematical correction of the above calculations was made in Ref. [48].

Appendix B: Collisional energy loss: Thoma Gyulassy approach

The collisional energy loss per unit length dE/dx of a heavy quark of mass m, momentum p, energy $E=\sqrt{p^2+m^2}$ and velocity $v=p/E$ has been reported by Thoma and Gyulassy [2]:

$$\begin{aligned} \frac{dE}{dx}= & {} \frac{4\pi \alpha _{s}^{2}T^{2}C_F}{3} \ln \left( \frac{k_{max}}{k_D}\right) \nonumber \\&\times \frac{1}{v^2}\left( v+\frac{v^2-1}{2}\ln \frac{1+v}{1-v}\right) \end{aligned}$$

(B.8)

where $k_{max} \approx \frac{4Tp}{E-p+4T}$ is maximal momentum transfer, with T being the temperature of the medium and $k_D = \sqrt{3}m_g$ is Debye momentum, $m_g$ being thermal gluon mass.