SShock Treatment: Heavy QuarkEnergy Loss in a Novel AdS / CFT Geometry
W. A. Horowitz
Department of Physics, The Ohio State University191 West Woodru ff Avenue, Columbus OH 43210, USA
Abstract
We first note the failures of traditional pQCD techniques as applied to high- p T heavy ion physicsand the suggestion of examining the double ratio of charm to bottom nuclear modification factorsto generically distinguish between these weak coupling ideas and the strong coupling ideas ofAdS / CFT. In order to gain confidence in the use of AdS / CFT (and to increase the likelihood offalsifying it and / or pQCD) we extend its application to heavy quark energy loss in both thermaland nonthermal media by calculating the string drag in a shock metric.
1. Introduction
Despite the early successes of perturbative QCD (pQCD) in describing the highly averagedhigh- p T light hadron R AA physics at RHIC (see, e.g., [1] and references therein), closer exami-nation of a number of experimental observables (e.g. high- p T v , nonphotonic electron R AA and v , see, e.g., [1], and even I AA [2]) shows that these pQCD techniques do not currently provide aquantitatively consistent picture of all the known data. Worse, pQCD does not simultaneouslydescribe any two of these quantitatively. On the other hand the large coupling techniques ofAdS / CFT are known to give a reasonable qualitative understanding of a number of RHIC phe-nomena (see, e.g., [1]). Recent work [1] suggests examining the double ratio of charm to bottom R AA as a means of falsifying either the usual pQCD approach or the AdS / CFT one (or both) tohigh- p T jet quenching.Specifically the application of AdS / CFT to heavy quark energy loss has shown particularpromise [3, 4]. Previous work considered a string hanging in the fifth dimension as the repre-sentation of a heavy quark in the 4D theory in an empty space metric [3] and in a black holemetric [4], the former to calculate the vacuum energy loss of an accelerating heavy quark andthe latter to calculate the energy loss of a heavy quark in a thermalized medium of N = / Schwarzschild results; however the shockcan describe matter with any isotropic distribution of momentum, and we therefore conclude thatwe have generalized the string drag calculation.
Preprint submitted to Nuclear Physics A December 12, 2018 a r X i v : . [ nu c l - t h ] S e p . Shock Metric, Drag Force, and Limits of the Calculation We will work in an asymptotic AdS space with metric [6] ds ≡ G µν dx µ dx ν = L z (cid:104) − dx + dx − + µ z θ ( x − ) dx − + dx ⊥ + dz (cid:105) (1a) = L z (cid:104) − (cid:16) − µ z θ ( x − ) (cid:17) dt − µ z θ ( x − ) dt dx + (cid:16) + µ z θ ( x − ) (cid:17) dx + dx ⊥ + dz (cid:105) , (1b)where we have used the x ± = ( t ± x ) / √ x = x and dropped the d Ω standard metric of the five-sphere in AdS × S . As usual L is the radius ofthe S space, and dx ⊥ = ( dx ) + ( dx ) is the transverse part of the metric.The dual of a finite mass quark in the fundamental representation in AdS / CFT is an openNambu-Goto string terminating on a D7 brane at z = z M = √ λ/ (2 π M q ) [4]; the other end of thestring ends on the stack of N c color branes at z = ∞ . The test string action is S NG = − T (cid:90) d τ d σ √− g , g = det g ab , g ab = G µν ∂ a X µ ∂ b X ν , (2)where G µν is the spacetime metric of Eq. (1b), Greek indices refer to spacetime coordinates,and Latin indices to worldsheet coordinates. X µ = X µ ( σ ) specifies the mapping from the stringworldsheet coordinates σ a to spacetime coordinates x µ . The backreaction of the fundamentalstring ( O ( N c )) is neglected as compared to the O ( N c ) contributions from the adjoint fields of N = Figure 1: (Color online) An illustration of the shock of the metric Eq. (1a) colliding with a heavy quark Q in its restframe. (a) before the collision the Nambu-Goto string, Eq. (2), hangs straight down; (b) afterwards the string is draggedby the shock behind its endpoint. Varying the action, Eq. (2), yields the equations of motion: ∇ a P a µ = , P a µ = π a µ / √− g = − T G µν ∂ a X ν , where ∇ a is the covariant derivative with respect to the induced metric, g ab , andthe π a µ are the canonical momenta, π a µ = − T ∂ √− g /∂ ( ∂ a X µ ). Limiting our attention to onlythe nontrivial directions of t , z , and x , X µ ( σ ) maps into the ( t , x , z ) coordinates; choosing thestatic gauge, σ a = ( t , z ), the string embedding is described by a single function, x ( t , z ). We findthe equations of motion for x ( t , z ) by plugging it into the action and then varying with respectto t and z . Denoting ∂ t x = ˙ x and ∂ z x = x (cid:48) we find − g = L (cid:0) + x (cid:48) − ˙ x − µ z (1 − ˙ x ) (cid:1) / z and ∂ t (cid:0) ( µ z − (1 + µ z ) ˙ x ) / z √− g (cid:1) + ∂ z (cid:0) x (cid:48) / z √− g (cid:1) =
0. If we assume a static solution ansatz x ( t , z ) = ξ ( z ) in a shock that fills spacetime the equations of motion become ∂ z ( ξ (cid:48) / z √− g ) = C one may solve for ξ (cid:48) : ξ (cid:48) = ± Cz (cid:112) (1 − µ z ) / (1 − C z ). There are two cases of interest for a string hanging from small z to z = ∞ : C = C (cid:44)
0. For the latter case C is set by requiring reality of the solution. At2mall z both the numerator and denominator of the equation for ξ (cid:48) are positive; at large z both arenegative. For their ratio to always be real C = √ µ . Trivial integration yields ξ ( z ) = x ± √ µ z / z = z h ) is x ( t , z ) ≈ x + vt ± z / (3 z h ). For C = ξ = x is immediately found; the string hangs straight down. We plot the three solutions in Fig. 2.One resolves the sign ambiguity in the pair of solutions for C = √ µ , resulting from thetime-reversal symmetry of the problem, by taking the physical one, which trails behind thequark and has the positive sign; the negative sign solution has the string “trailing” in frontof the quark. Additionally plugging the straight solution back into the action Eq. (2) gives S = − T (cid:82) dt (cid:82) ∞ z M dz (cid:112) − µ z / z . The IR region of the z integration contributes an infinite imag-inary part to the action; we interpret this solution as infinitely unstable that would immediatelydecay into the physical, trailing string solution. Figure 2: The three solutions to the static equations of motion, x ( t , z ) = ξ ( z ) = x , x ± √ µ z / . The drag force on the heavy quark in the SYM theory corresponds to the momentum flowfrom the direction of heavy quark propagation down the string, d p / dt = − π x . From the canonicalmomenta, in the rest frame of the heavy quark, d p / dt = − π x = √ λµ/ π . Formally the metric,Eq. (1a), has the shock on the light cone; thinking of this as an approximation to a shock nearlyon the light cone one has a well-defined rest frame for the shock. This then is the lab frame,where we want to know to momentum loss of the heavy quark, and also allows us to relate µ toproperties of the medium.Following [7], we assume the medium is made up of N c valence gluons of the N = Λ —with associated inter-particle spacing of order 1 / Λ —thenthe 00 component of the stress-energy tensor in the rest frame of the shock is (cid:104) T (cid:48) (cid:105) ∝ N c Λ ,where primes denote quantities in the rest frame of the medium and proportionality is up to aconstant numerical factor. Transforming into lightcone coordinates, boosting into the rest frameof the heavy quark, and dropping the O (1) constant of proportionality, yields (cid:104) T −− (cid:105) = N c Λ γ = N c Λ ( p (cid:48) / M ) , where we assumed ultrarelativistic motion for the heavy quark in the medium restframe, p (cid:48) (cid:39) M γ . Comparing this with the energy momentum tensor found above we read o ff µ = π Λ ( p (cid:48) / M ) . Figure 3: (a) Shock and quark system as viewed in the heavy quark rest frame. (b) In the shock rest frame (the labframe).
To rewrite d p / dt in terms of the momentum and time in the medium rest frame, d p (cid:48) / dt (cid:48) ,3ote that d p / dt is the 3-vector component of the force 4-vector in the quark rest frame: f x ≡ d p / d τ = d p / dt . π t =
0, and hence f t =
0, and the 4-force boosted into the shock rest frame is f (cid:48) x = − γ f x = − γ d p / dt , where the negative sign comes from boosting into a frame moving in theopposite direction; see Fig. 3. From the definition of the 4-force we also know that in this frame f (cid:48) x ≡ d p (cid:48) / d τ = γ d p (cid:48) / dt (cid:48) . Hence we find that d p / dt = − d p (cid:48) / dt (cid:48) . Using these we find our mainresult, d p (cid:48) dt (cid:48) = − √ λ Λ M q p (cid:48) . (3)If we take the typical medium particle momentum Λ = √ π T then our result exactly reproducesthat of the black hole metric, d p (cid:48) / dt (cid:48) = − π √ λ T p (cid:48) / (2 M q ) [4].One may readily derive a “speed limit” of applicability of this formalism in the case of a staticheavy quark solution. The momentum lost to the medium must be balanced by a momentuminput from a Born-Infeld derived electric field on the D7 brane, which has a natural cuto ff atthe energy scale of heavy quark pair production [4]. For the case of a heavy quark allowed toslow down of its own accord one may examine the speed limit of a point particle traveling at z M [8]. This gives the same limit as the one from the BI action, but it is not entirely clear that itimplies a restriction on the string whose action remains real. Nevertheless one may derive it inthe shock metric; while the shock does not support a black hole horizon the local speed of lighthas a nontrivial z dependence ( µ z − / ( µ z + ≤ v ≤
1. Setting v = µ z M ≤ z M = √ λ/ (2 π M q ) and µ = π Λ γ , along with Λ = √ π T ,we obtain γ ≤ (4 π M q ) / ( λ Λ ) = (4 M q ) / ( λ T ) . We note that this limit is identical to that for theBH metric [4, 8].
3. Conclusions
Advances in experimental heavy ion physics holds out the promise of testing both the usualpQCD and the novel AdS / CFT formalisms. A rigorous understanding of the theoretical error andregimes of applicability associated with the predictions of these theories is required in order tofalsify one or both experimentally. In this work we generalized AdS / CFT heavy quark energyloss to media of any isotropic distribution of momentum. We found that, just as in pQCD, theform of energy loss is independent of the thermal or nonthermal nature of the medium.
Acknowledgments
This work was supported by the O ffi ce of Nuclear Physics, of the O ffi ce of Science, of theU.S. Department of Energy under Grant No. DE-FG02-05ER41377. References [1] W. A. Horowitz and M. Gyulassy, Phys. Lett. B , 320 (2008) [arXiv:0706.2336 [nucl-th]].[2] H. Zhang, et al., Phys. Rev. Lett. , 212301 (2007); J. L. Nagle, arXiv:0907.2707 [nucl-ex].[3] A. Mikhailov, arXiv:hep-th / , 126005 (2006); C. P. Herzog, et al., JHEP , 013 (2006); J. Casalderrey-Solanaand D. Teaney, JHEP , 039 (2007).[5] W. A. Horowitz and Y. V. Kovchegov, arXiv:0904.2536 [hep-th].[6] R. A. Janik and R. B. Peschanski, Phys. Rev. D , 045013 (2006) [arXiv:hep-th / , 074 (2008) [arXiv:0806.1484 [hep-th]].[8] S. S. Gubser, Nucl. Phys. B , 175 (2008) [arXiv:hep-th /0612143].