Testing AdS/CFT Deviations from pQCD Heavy Quark Energy Loss with Pb+Pb at LHC

Heavy quark jet quenching in nuclear collisions at LHC is predicted and compared using the classical gravity AdS/CFT correspondence and Standard Model perturbative QCD. The momentum independence and inverse quark mass dependence of the drag coefficient in AdS/CFT differs substantially from the characteristic log(pT/M)/pT variation of the drag in QCD. We propose that the measurement of the momentum dependence of the double ratio of the nuclear modification factors of charm and bottom jets is a robust observable that can be used to search for strong coupling deviations from perturbative QCD predictions.

Qualitative successes of recent AdS/CFT applications [4] to nuclear collision phenomenology include the analytic account for (1) the surprisingly small (∼ 3/4) drop [16] of the entropy density in lattice QCD calculations relative to Stefan-Boltzmann, (2) the order of magnitude reduction of the viscosity to entropy ratio η/s predicted relative to pQCD needed to explain the seemingly near perfect fluid flow of the sQGP observed at RHIC, (3) the surprising large stopping power of high transverse momenta heavy quarks as inferred from the quenching and elliptic flow of leptonic decay fragments of heavy quark jets, and (4) the possible occurrence of conical "Mach" wave-like correlations of hadrons associated with jets.
While quantitative and systematic comparisons of AdS/CFT gravity dual models with nuclear collision data are still incomplete and while the conjectured double Type IIB string theory↔conformal Supersymmetric Yang-Mills (SYM) gauge theory↔non-conformal, nonsupersymmetric QCD correspondence remains under debate (see, e.g.[17]), the current successes provide strong motivation to seek more sensitive experimental tests that could help guide the development of such novel theoretical approaches.
The aim of this Letter is to propose a robust new test that could reveal possible strong coupling deviations from pQCD predictions as suggested by a specific version of the AdS/CFT correspondence.This test involves the nuclear modification of identified heavy quark jets produced in central Pb+Pb reactions at 5.5 ATeV at LHC.Similar tests can be performed at RHIC, after future detector upgrades, and will be reported elsewhere.Specifically, we propose that the double ratio of identified charm and bottom jet nuclear modification factors R Q AA (p T ) can easily distinguish between a wide class of pQCD models and a class of gravity dual models [18,19,20,21,22] of heavy quark dynamics.The current failure of pQCD models to account quantitatively for the recent RHIC data from STAR [12] and PHENIX [13] on the nonphotonic electron spectrum provides additional strong motivation to focus on heavy quark jet observables.Unlike for light quark and gluon jet observables, where pQCD predictions were found to be remarkably quantitative [11], heavy quark jet quenching, especially as inferred indirectly for bottom quarks, appears to be significantly underpredicted [23,24,25,26].However, the pQCD predictions can not yet be falsified at RHIC because of (1) remaining uncertainty in the nuclear production ratio of bottom to charm quarks and (2) current controversy over the relative magnitude of elastic versus radiative loss channels [25,26].Recent attempts to reconcile pQCD predictions with the data by adding nonperturbative hadronic final state interaction effects can be found in [27,28].
The AdS/CFT correspondence has so far been applied to QCD jet physics in three different ways.The first calculates the Wilson line correlator needed to predict the partonic radiative transport coefficient q [29,30].The second predicts the heavy quark diffusion coefficient D [31] used as an input parameter in a relativistic Langevin model with drag deduced from the Einstein relation [32].The third computes directly the heavy quark drag coefficient via a specific classical string configuration [19,20,21].We note that the first two approaches conform more closely to the original spirit of the AdS/CFT conjecture relating quantum SYM correlation functions arXiv:0706.2336v1[nucl-th] 15 Jun 2007 to the asymptotic behavior of the classical supergravity (SUGRA) correlations.Unfortunately these correspondences are hard to interpret in terms of gauge theory energy loss mechanisms since infinitely coupled SYM does not support a familiar quasiparticle basis similar to gluon and quark degrees of freedom in QCD.In contrast, the third model, while more easily interpretable, requires a stronger form of the AdS/CFT correspondence.All three approaches remain under active debate (see, e.g.[21,33,34]).
We focus in this Letter on the third proposed AdS/ CFT application that involves the most direct string theoretic inspired gravity "realization" of heavy quark dynamics [18,19,20].A heavy quark in the fundamental representation is a bent Nambu-Goto string with one end attached to a probe brane and that trails back above the horizon of a D3 black brane representing the uniform strongly coupled SYM plasma heat bath.This geometry maps the drag force problem into a modern string theoretic version of the old 1696 Brachistochrone problem that yields a remarkable, simple analytic solution for the string shape and momentum loss per unit time.
AdS/CFT compared to pQCD Exploiting this AdS/ CFT correspondence, the drag coefficient for a massive quark moving through a strongly-coupled SYM plasma in the λ [19,20,21] as where T * is the temperature of the SYM plasma as fixed by the Hawking temperature of the dual D3 black brane.Issues related to the relaxation of the strong assumptions made in deriving, and the momentum limitation of the applicability of, Eq. (1) will be discussed later in the text.
Applying Eq. (1) to LHC requires an additional proposal that maps QCD temperatures and couplings to the SYM world and its SUGRA dual.The "obvious" first prescription [35] is to take g SY M = g s constant, T * = T QCD , and N c = 3. However it was suggested in [35] that a more physical "alternative" might be to equate energy densities, giving T * = T QCD /3 1/4 , and to fit the coupling λ = g 2 SY M N c ≈ 5.5 in order to reproduce the static quark-antiquark forces calculated via lattice QCD.
The string theoretic result for the diffusion coefficient used in the Langevin model is D/2πT * = 4/ √ λ [31].This illustrates well the problem of connecting the T * and λ of SYM to "our" QCD world.Using the "obvious" prescription with α s = .3,N c = 3, one finds D/2πT ∼ 1.2.However, D/2πT = 3 was claimed in [13,31] to fit PHENIX data somewhat better.Note that D/2πT = 3 requires an unnaturally small α s ∼ 0.05 that is very far from the assumed λ 1 't Hooft limit.We proceed by computing the nuclear modification factors, neglecting initial state shadowing or saturation ef-  1) and pQCD using the WHDG model [25] convolving elastic and inelastic parton energy loss.Possible initial gluon rapidity densities at LHC are given by dNg/dy = 1750, from a PHOBOS [6,38] extrapolation, or dNg/dy = 2900, from the KLN model of the color glass condensate (CGC) [39].The top two curves from pQCD increase with pT while the bottom two curves from AdS/CFT slowly decrease with pT .The AdS/CFT parameters here were found using the "obvious" prescription with αSY M = .05,τ0 = 1 fm/c, giving D/2πT = 3 (abbreviated to D = 3 in the figure).Similar trends were seen for the other input parameter possibilities discussed in the text.
fects.In order to correctly deconvolute such effects from the final state effects that we compute below, it will be necessary to measure nuclear modification factors in p+A as a function of (y, p T ) at LHC just as d + A was the critical control experiment [1] at RHIC [2].
Final state suppression of high-p T jets due to a fractional energy loss , p f T = (1 − )p i T , can be computed knowing the Q-flavor dependent spectral indices n where the average is over the distribution P ( ; M Q , p T , ) that depends in general on the quark mass, p T , and the path length of the jet through the sQGP.As in [25] we average over jets produced according to the binary distribution geometry and compute through a participant transverse density distribution taking into account the nuclear diffuseness.Given dN g /dy of produced gluons, the temperature is computed assuming isentropic Bjorken 1D Hubble flow.As emphasized in [25], detailed geometric path length averaging plays a crucial role in allowing consistency between π 0 , η and heavy quark quenching in pQCD.
For AdS/CFT drag, Eq. (1) gives the average fractional energy loss as ¯ = 1 − exp(−µ Q ).Energy loss is assumed to start at thermalization, τ 0 ∼ 0.6 − 1.0 fm/c, and stops when the confinement temperature, T c ∼ 160 MeV, is reached.The exponentiated T 2 dependence in µ Q leads to a significant sensitivity to the opacity of the medium, as well as to τ 0 and T c .
To understand the generic qualitative features of our numerical results it is instructive to consider the simplest case of a geometric path average over a static, finite, uniform plasma of thickness L; then where the p T dependence is carried entirely by the spectral index n Q (p T ).R AA can be interpreted for L Q ≡ 1/(n Q µ Q ) as the fraction Q /L of the Q jets that escape unstopped from the strongly coupled plasma within the AdS/CFT approximation.
FIG. 2: The double ratio of R c AA (pT ) to R b AA (pT ) predictions for LHC using Eq.(1) for AdS/CFT and WHDG [25] for pQCD with a wide range of input parameters.The generic difference between the pQCD results tending to unity contrasted to the much smaller and nearly pT -independent results from AdS/CFT can be easily distinguished at LHC.Two implementations of pQCD energy loss are used in this paper.The first is the full WHDG model convolving fluctuating elastic and inelastic loss with fluctuating path geometry [25].The second restricts WHDG to include only radiative loss in order to facilitate comparison to [30].Note that when realistic nuclear geometries with Bjorken expansion are used, the "fragility" of R AA for large q reported in [36] is absent in both implementations of WHDG.
Unlike the AdS/CFT dynamics, pQCD predicts [23,24,25] that the average energy loss fraction in a static uniform plasma is approximately ¯ ≈ κL 2 q log(p T /M Q )/p T , with κ a proportionality constant and q = µ 2 D /λ g .The most important feature in pQCD relative to AdS/CFT is that ¯ pQCD → 0 asymptotically at high-p T while ¯ AdS remains constant.n Q (p T ) is a slowly increasing function of momentum; thus R pQCD AA increases with p T whereas R AdS AA decreases.This generic difference can be observed in Fig. 1, which shows representative predictions from the full numerical calculations of charm and bottom R AA (p T ) at LHC.Double Ratio of charm to bottom R Q AA A disadvantage of the R Q AA (p T ) observable alone is that its normalization and slow p T dependence can be fit with different model assumptions compensated by using very different medium parameters.In particular, high value extrapolations of the q parameter proposed in [26] could simulate the flat p T independent prediction from AdS/CFT.
We propose to use the double ratio of charm to bottom R AA to amplify the observable difference between the mass and p T dependencies of the AdS/CFT drag and pQCD-inspired energy loss models.One can see in Fig. 2 that not only are most overall normalization differences canceled, but also that the curves remarkably bunch to either AdS/CFT-like or pQCD-like generic results regardless of the input parameters used.
The numerical value of R cb shown in Fig. 2 for AdS/CFT can be roughly understood analytically from Eq. ( 2) as, where in this approximation all λ, T * , L, and n c (p T ) ≈ n b (p T ) dependences drop out.The pQCD trend in Fig. 2 can be understood qualitatively from the expected behavior of ¯ pQCD noted above giving (with where p cb = κn(p T )L 2 log(M b /M c )q sets the relevant momentum scale.Thus R cb → 1 more slowly for higher opacity.One can see this behavior reflected in the full numerical results shown in Fig. 2 for moderate suppression, but that the extreme opacity q = 100 case deviates from Eq. ( 4).The maximum momentum for which string theoretic predictions for R cb can be trusted is not well understood.Eq. (1) was derived assuming a constant heavy quark velocity.Supposing this is maintained by the presence of an electromagnetic field, the Born-Infeld action gives a "speed limit" of γ c = M 2 /λ(T * ) 2 [37].The work of [19] relaxed the assumptions of infinite quark mass and constant velocity; nevertheless Eq. (1) well approximates the full results.Requiring a time-like endpoint on the probe brane for a constant velocity string representing a finite mass quark leads to [21] a parametrically similar cutoff, There is no known limit yet for the dynamic velocity case.To get a sense of the p T scale where the AdS/CFT approximation may break down, we plot the momentum cutoffs from Eq. ( 5) for the given SYM input parameters corresponding to T * (τ 0 ) and T * c .These are depicted by "O" and "|" in the figures, respectively.
Conclusions Possible strong coupling deviations from pQCD in nuclear collisions were studied based on a recent AdS/CFT model of charm and bottom energy loss.The predicted nuclear modification factors, R Q AA , were found to decrease as a function of p T , as compared to increasing as predicted from pQCD.The distinction between these dynamical models is amplified by studying the p T dependence of the double ratio R c AA (p T )/R b AA (p T ), which clearly illustrates a quark mass independence of pQCD energy loss at asymptotically high momenta in contrast to the momentum independence and inverse mass dependence prediction of the AdS/CFT drag coefficient.A clear signal of novel nonperturbative physics would be the observation of a large deviation of the double ratio from unity at p T M b soon to be accessible at LHC and RHIC.

FIG. 1 :
FIG. 1: (Color Online) R c AA (pT ) and R b AA (pT ) predicted for central Pb+Pb at LHC comparing AdS/CFT Eq. (1) and pQCD using the WHDG model[25] convolving elastic and inelastic parton energy loss.Possible initial gluon rapidity densities at LHC are given by dNg/dy = 1750, from a PHOBOS[6,38] extrapolation, or dNg/dy = 2900, from the KLN model of the color glass condensate (CGC)[39].The top two curves from pQCD increase with pT while the bottom two curves from AdS/CFT slowly decrease with pT .The AdS/CFT parameters here were found using the "obvious" prescription with αSY M = .05,τ0 = 1 fm/c, giving D/2πT = 3 (abbreviated to D = 3 in the figure).Similar trends were seen for the other input parameter possibilities discussed in the text.