High Energy Physics Volume 206, Article ID 950349, 4 pages http://dx.doi.org/0.55/206/950349 Research Article R 2 Corrections to the Jet Quenching Parameter Zi-qiang Zhang, De-fu Hou, 2 Yan Wu, and Gang Chen School of Mathematics and Physics, China University of Geosciences, Wuhan 430074, China 2 Key Laboratory of Quark and Lepton Physics (MOE), Central China Normal University, Wuhan 430079, China Correspondence should be addressed to Zi-qiang Zhang; zhangzq@cug.edu.cn Received 20 November 205; Revised 25 March 206; Accepted April 206 Academic Editor: Enrico Lunghi Copyright 206 Zi-qiang Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP 3. A calculation of R 2 corrections to the jet quenching parameter from AdS/CFT correspondence is presented. It is shown that these corrections will increase or decrease the jet quenching parameter depending on the coefficients of the high curvature terms.. 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 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, perunitmeanfreepath[,2]. AdS/CFT duality can explore the strongly coupled N =4 supersymmetric Yang-Mills (SYM) plasma through the correspondence between type IIB superstring theory formulated on AdS 5 S 5 and N = 4 SYM in four dimensions [3 5]. Therefore some quantities, for instance, the jet quenching parameter,canbestudied. By using the AdS/CFT correspondences, Liu et al. [6] have calculated the jet quenching parameter for N = 4 SYM theory firstly. Interestingly the magnitude of q SYM turns out to be closer to the value extracted from RHIC data [7,8]thanpQCDresultforthetypicalvalueofthe thooft 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 2]. 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 R 2 corrections to the jet quenching parameter. The last part is devotedtoconclusionanddiscussion. 2. 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 arectangularcontourofsizel 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 parameter q defined in [] is extracted from the asymptotic expression for TL : W A [C] exp [ 4 2 ql L 2 ], () where W A [C] W F [C] 2 with W F [C] the thermal expectation value in the fundamental representation. Using AdS/CFT correspondence, we can calculate W F [C] according to W F [C] exp [ S I ], (2)
2 High Energy Physics where S I is the regulated finite on-shell string worldsheet action whose boundary corresponds to the null-like rectangular loop C. By using such a strategy, they find that [6] q SYM = π3/2 Γ (3/4) λt 3 7.53 λt 3, (3) Γ (5/4) where T is the temperature of N =4SYM plasma. 3. R 2 Corrections to q SYM We now consider the R 2 corrections to q SYM. Restricting to the gravity sector in AdS 5, the leading order higher derivative corrections can be written as [3] S= 6πG N d 5 x g[r 2Λ +l 2 (α R 2 +α 2 R μ] R μ] +α 3 R μ]ρσ R μ]ρσ )], where α i are arbitrary small coefficients and the negative cosmological constant Λ is related to the AdS space radius l by (4) The worldsheet in the bulk can be parameterized by τ and σ; then the Nambu-Goto action for the worldsheet becomes S= 2πα dτ dσ det g αβ. () 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 worldsheet, implying x 3 = const, x + = const; then the string action is given by S= 2r 2 0 L 2πα l 2 with r = σ r. The equation of motion for r(σ) reads L/2 dσ + r 2 l 2 0 fr 2, (2) r 2 =γ 2 r 2 f l 2, (3) where γ is an integration constant. From (3), it is clear that the turning point occurs at f = 0, implying r = 0 at the horizon r=r 0.Knowingthatr=r 0 at σ=0,then(3)canbe integrated as Λ= 6 l 2. (5) The black brane solution of AdS 5 space with curvaturesquared corrections is given by where with ds 2 = f(r) dt 2 + r2 l 2 dx2 + f (r) dr2, (6) f (r) = r2 l 2 ( r4 0 r 4 +α+βr8 0 ), (7) r8 where L 2 = r 0 = al2 γr 0, ldr γ fr = l2 γr 0 dt t 4 +αt 4 + β/t 4 dt a=. t 4 +αt 4 + β/t 4 Then we can find the action for the heavy quark pair (4) (5) α= 2 3 (0α +2α 2 +α 3 ), β=2α 3, where r denotes the radial coordinate of the black brane geometry, r=r 0 is the horizon, and l relates the string tension /2πα to the t Hooft coupling constant by l 2 /α = λ. The temperature of N = 4 SYM plasma with R 2 corrections is given by T = r 0 πl 2 ( + 4 α 5 4 β) = T ( + 4 α 5 β). (9) 4 In terms of the light-cone coordinate x μ =(r,x ±,x 2,x 3 ), metric (6) becomes ds 2 = [ r2 l 2 +f(r)]dx+ dx + r2 l 2 (dx2 2 +dx2 3 ) + 2 [r2 l 2 f(r)] [(dx+ ) 2 +(dx ) 2 ]+ dr2 f (r). (8) (0) S= π λl LT 2 2 2 + 4a2 π 2 T 2 L2, (6) wherewehaveusedtherelationsr 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 worldsheets; that is, S 0 = 2L 2πα dr g g rr = a λl T. (7) r 0 2 Therefore, the net on-shell action is given by S I =S S 0 λt 3 8 2a L L 2, (8) wherewehaveusedt L. Note that q is related to W A [C] according to π2 W A [C] exp [ 4 2 ql L 2 ] exp [ 2S I ]. (9)
High Energy Physics 3.05.04 β = 0.00.02 α = 0.005 q R 2 / q SYM.03 β = 0.003 q R 2 / q SYM.0 α = 0.00.02 β = 0.006.00 α = 0.00 0.004 0.006 0.008 0.00 α 0.004 0.006 0.008 0.00 β (a) (b) Figure : (a) q R 2/ q SYM versus α.fromtoptodownβ = 0.00, 0.003, 0.006.(b) q R 2/ q SYM versus β.fromtoptodownα = 0.005, 0.00, 0.00. Plugging (8) into (9), we end up with jet quenching parameter under R 2 corrections: q R 2 = π2 a λt 3 = ( + (/4) α (5/4) β) 3 π 2 λt 3. (dt/ t 4 +αt 4 + β/t 4 ) (20) Comparing (4) with (22), we find α = λ GB 2, α 2 = 2λ GB, α 3 = λ GB 2. Plugging (23) into (8), we have (23) Wenowdiscussourresult.Itisclearthatthejetquenching parameter in (3) can be derived from (20) if we neglect the effect of curvature-squared corrections by plugging α= β=0in (20). Furthermore, we compare the jet quenching parameter under R 2 corrections with its counterpart in the case of the dual gauge theory as follows: q R 2 Γ (5/4) π = q SYM Γ (3/4) ( + (/4) α (5/4) β) 3. (2) (dt/ t 4 +αt 4 + β/t 4 ) Notice that the result of (2) is related to α and β, or, in other words, it is dependent on α i. However, the exact interval of α i has not been determined; we only know that α i α /l 2 [4]. In order to choose the values of α and β properly, we employ the Gauss-Bonnet gravity [5] 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 [6]: S= 6πG N d 5 x g[r 2Λ + λ GB 2 l2 (R 2 4R μ] R μ] +R μ]ρσ R μ]ρσ )]. (22) α=λ GB, β=λ GB, (24) where λ GB canbeconstrainedbycausalityandpositivedefiniteness of the boundary energy density [] 7 36 <λ GB 9 00. (25) Moreover, there is an obstacle to the calculation of (2): we must require that the square root in the denominator be everywhere positive and the temperature in (9) be also positive. Under the above conditions, we find 7 36 <α 9 00, 0 β 9 00, α+β 0, (26) and then we can discuss the results of (2) for different values of α and β in such a range. As a concrete example, we plot q R 2/ q SYM versus α with different β and q R 2/ q SYM versus β with different α in Figure. In (a) from top to down, β = 0.00, 0.003, 0.006; q R 2/ q SYM increases at each β as α increases. While in (b) from top to
4 High Energy Physics down α = 0.005, 0.00, 0.00, at each α; q R 2/ q SYM 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. 4. Conclusion and Discussion In this paper, we have investigated 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 [7]. However, we should admit that we cannot predict a result for N =4SYM because the first higher derivativecorrectioninweaklycurvedtypeiibbackgrounds 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, the subleading order corrections to the jet quenching parameter (3) due to worldsheet fluctuations can be found in [0]: q SYM = q SYM (.97λ /2 ). (27) Other corrections of O(/N c ) and higher orders in / λ are also discussed in [, 2]. However, /λ corrections on jet quenching parameter are as yet undetermined; we hope to reportourprogressinthisregardinfuture. Competing Interests The authors declare that they have no competing interests. Acknowledgments The authors would like to thank Professor Hai-cang Ren for useful discussions. This research is partly supported by the Ministry of Science and Technology of China (MSTC) under the 973 Project no. 205CB856904(4). Zi-qiang Zhang is supported by the NSFC under Grant no. 547204. Gang Chen is supported by the NSFC under Grant no. 47549. De-fu Hou is partly supported by NSFC under Grant nos. 375070 and 22504. [3] J. M. Maldacena, The large N limit of superconformal field theories and supergravity, Theoretical and Mathematical Physics,vol.2,pp.23 252,998,International Journal of Theoretical Physics, vol. 38, no. 4, pp. 3 33, 999. [4] S. S. Gubser, I. R. Klebanov, and A. M. Polyakov, Gauge theory correlators from non-critical string theory, Physics Letters B, vol. 428, no. -2, pp. 05 4, 998. [5] O. Aharony, S. S. Gubser, J. Maldacena, H. Ooguri, and Y. Oz, Large N field theories, string theory and gravity, Physics Reports,vol.323,no.3-4,pp.83 386,2000. [6] H. Liu, K. Rajagopal, and U. A. Wiedemann, Calculating the jet quenching parameter from AdS/CFT, Physical Review Letters, vol.97,articleid8230,2006. [7] K.J.Eskola,H.Honkanen,C.A.Salgado,andU.A.Wiedemann, The fragility of high-pt hadron spectra as a hard probe, Nuclear Physics A, vol. 747, no. 2 4, pp. 5 529, 2005. [8] A. Dainese, C. Loizides, and G. Paic, Leading-particle suppression in high energy nucleus-nucleus collisions, European Physical Journal C,vol.38,no.4,pp.46 474,2005. [9] K. B. Fadafan, Charge effect and finite t Hooft coupling correction on drag force and jet quenching parameter, The European Physical Journal C,vol.68,no.3,pp.505 5,200. [0] Z.-Q. Zhang, D.-F. Hou, and H.-C. Ren, The finite thooft coupling correction on the jet quenching parameter in a N =4 super Yang-Mills plasma, High Energy Physics, vol. 203,article32,203. [] A. Ficnar, S. S. Gubser, and M. Gyulassy, Shooting string holography of jet quenching at RHIC and LHC, Physics Letters B,vol. 738, pp. 464 47, 204. [2] N. Armesto, J. D. Edelstein, and J. Mas, Jet quenching at finite t Hooft coupling and chemical potential from AdS/CFT, Journal of High Energy Physics, no. 9, p. 39, 2006. [3] S. M. Carroll, Spacetime and Geometry: An Introduction to General Relativity, Addison-Wesley, San Francisco, Calif, USA, 2004. [4]M.Brigante,H.Liu,R.C.Myers,S.Shenker,andS.Yaida, Viscosity bound violation in higher derivative gravity, Physical Review D,vol.77,no.2,ArticleID26006,2008. [5] R.-G. Cai, Gauss-Bonnet black holes in AdS spaces, Physical Review D,vol.65,ArticleID08404,2002. [6] B. Zwiebach, Curvature squared terms and string theories, Physics Letters B,vol.56,no.5-6,pp.35 37,985. [7] K. B. Fadafan, R 2 curvature-squared corrections on drag force, High Energy Physics, vol. 2008, no. 2, article 05, 2008. References [] R. Baier, Y. L. Dokshitzer, A. H. Mueller, S. Peigne, and D. Schiff, Radiative energy loss and p - broadening of high energy partons in nuclei, Nuclear Physics B,vol.484,no.-2,pp.265 282, 997. [2] M. Gyulassy, I. Vitev, X.-N. Wang, and B.-W. Zhang, Jet quenching and radiative energy loss in dense nuclear matter, in Quark-Gluon Plasma, R.C.HwaandX.-N.Wang,Eds.,p.23, World Scientific, Singapore, 2004.
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