R^2 corrections to the jet quenching parameter

A calculation of the $R^2$ corrections to the jet quenching parameter from AdS/CFT correspondence is presented. These corrections are related to curvature-squared corrections in the corresponding gravity dual. It is shown that the corrections will increase or decrease the jet quenching parameter depending on the coefficients of the high curvature terms.


I. INTRODUCTION
The experiments of ultrarelativistic nucleus-nucleus collisions at RHIC and LHC have produced a strongly-coupled quark-gluon plasma(sQGP). One of the interesting properties of the sQGP is jet quenching phenomenon: due to the interaction with the medium, high energy partons traversing the medium are strongly quenched. This phenomenon is usually characterized by the jet quenching parameter (or transport coefficient)q which describes the average transverse momentum square transferred from the traversing parton, per unit mean free path [1,2].
AdS/CFT duality can explore the strongly coupled N = 4 supersymmetric Yang-Mills(SYM) plasma through the correspondence between the type IIB superstring theory formulated on AdS 5 × S 5 and N = 4 SYM in four dimensions [3][4][5]. Therefore some quantities for instance the jet quenching parameter can be studied.
By using the AdS/CFT correspondences, H.Liu et al [6] have calculated the jet quenching parameter for N = 4 SYM theory firstly. Interestingly the magnitude ofq SY M turns out to be closer to the value extracted from RHIC data [7,8] than pQCD result for the typical value of the 't Hooft coupling, λ ≃ 6π, of QCD. This proposal has attracted lots of interest. Thus, after [6] there are many attempts to addressing jet quenching parameter [9][10][11][12]. However, in [6] the jet quenching parameter was studied only in infinite-coupling case. Therefore, it is very interesting to investigate the effect of finite-coupling corrections. The purpose of the present work is to study R 2 corrections to the jet quenching parameter.
The organization of this paper is as follows. In the next section, the result of jet quenching parameter from dual gauge theory is briefly reviewed [6]. In section 3, we consider the R 2 corrections to the jet quenching parameter. The last part is devoted to conclusion and discussion.

II.q FROM DUAL GAUGE THEORY
The eikonal approximation relates the jet quenching parameter with the expectation value of an adjoint Wilson loop W A [C] with C a rectangular contour of size L × L − , where the sides with length L − run along the light-cone and the limit L − → ∞ is taken in the end. Under the dipole approximation, which is valid for small transverse separation L, the jet-quenching parameterq defined in Ref [1] is extracted from the asymptotic expression for T L << 1 > the thermal expectation value in the fundamental representation. Using AdS/CFT correspondence, we can calculate < W F [C] > according to: where S I is the regulated finite on-shell string world-sheet action whose boundary corresponds to the null-like rectangular loop C. By using such a strategy, they find that [6] where T is the temperature of N = 4 SYM plasma.
We now consider the R 2 corrections toq SY M . Restricting to the gravity sector in AdS 5 , the leading order higher derivative corrections can be written as [13] where α i are arbitrary small coefficients and the negative cosmological constant Λ relates the AdS space radius l by The black brane solution of AdS 5 space with curvature-squared corrections is given by where with where r denotes the radial coordinate of the black brane geometry, r = r 0 is the horizon, and l relates the string tension 1 2πα ′ to the 't Hooft coupling constant by l 2 α ′ = √ λ. The temperature of N = 4 SYM plasma with the R 2 corrections is given by In terms of the light-cone coordinate x µ = (r, x ± , x 2 , x 3 ), the metric (6) becomes The world sheet in the bulk can be parameterized by τ and σ, then the Nambu-Goto action for the world-sheet becomes We set the pair of quarks at x 2 = ± L 2 and choose x − = τ, x 2 = σ. In this setup, we can ignore the effect of x − dependence of the world-sheet, implying x 3 = const, x + = const, then the string action is given by with r ′ = ∂ σ r. The equation of motion for r(σ) reads where γ is an integration constant. From Eq. (13), it is clear that the turning point occurs at f = 0, implying r ′ = 0 at the horizon r = r 0 . Knowing that r = r 0 at σ = 0, then Eq.(13) can be integrated as where Then we can find the action for the heavy quark pair where we have used the relation r 0 = πl 2 T and l 2 α ′ = √ λ. This action needs to be subtracted by the self-energy of the two quarks, given by the parallel flat string world-sheets, that is Therefore, the net on-shell action is given by where we have used T 1 L ≪ 1. Note thatq is related to < W A [C] > according to: Plugging (18) into (19), we end up with jet quenching parameter under the R 2 correctionŝ We now discuss our result. It is clear that the jet quenching parameter in (3) can be derived from (20) if we neglect the effect of curvature-squared corrections by plugging α = β = 0 in (20). Furthermore, we compare the jet quenching parameter under the R 2 corrections with its counterpart in the case of the dual gauge theory as followinĝ Notice that the result of Eq.(21) is related to α and β, or in other words, it is dependent of α i . However, the exact interval of α i has not been determined, we only know that α i ∼ α′ l 2 ≪ 1 [14]. In order to choose the values of α and β properly, we employ the Gauss-Bonnet gravity [15] which has nice properties that are absent for theories with more general ratios of the α ′ i s. The Gauss-Bonnet gravity is defined by the following action [16]  Comparing (4) with (22), we find Plugging (23) into (8), we have where λ GB can be constrained by causality and positive-definiteness of the boundary energy density [11] − 7 36 < λ GB ≤ 9 100 .
Moreover, there is an obstacle to the calculation of (21): we must require that the square root in the denominator is everywhere positive and the temperature in Eq.(9) is also positive. Under the above conditions, we find then we can discuss the results of (21) for different values of α and β in such a range. As a concrete example, we plotq R 2 /q SY M versus α with different β andq R 2 /q SY M versus β with different α in FIG.1. In the left plot from top to down, β = 0.001, 0.003, 0.006,q R 2 /q SY M increases at each β as α increases. While in the right plot from top to down α = 0.005, 0.001, −0.001, at each α,q R 2 /q SY M is a decreasing function of β.
From the figure, it is clear that R 2 corrections can affect the jet quenching parameter. With some chosen values of α and β in this paper, the jet quenching parameter can be larger than or smaller than its counterpart in infinite-coupling case.

IV. CONCLUSION AND DISCUSSION
In this paper, we have investigated the R 2 corrections to the jet quenching parameter. These corrections are related to curvature-squared corrections in the corresponding gravity dual. It is shown that the corrections will increase or decrease the jet quenching parameter depending on the coefficients of the high curvature terms. Interestingly, under R 2 corrections, the drag force is also larger than or smaller than that in the infinite-coupling case [17]. However, we should admit that we cannot predict a result for N = 4 SYM because the first higher derivative correction in weakly curved type II B backgrounds enters at order R 4 , and not R 2 .
Actually, there are some other important finite-coupling corrections to the jet quenching parameter. For example, next to leading order corrections to (3) due to world sheet fluctuations suggests [10] q ′ SY M =q SY M (1 − 1.97λ −1/2 ).
Other corrections of O(1/N c ) and higher orders in 1 √ λ are also discussed in [11,12]. However, 1/λ corrections on jet quenching parameter are as yet undetermined, we hope to report our progress in this regard in future.